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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 1st June 1999.

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\centerline{\Largebf THIRD SUPPLEMENT TO AN ESSAY ON THE}

\vskip12pt

\centerline{\Largebf THEORY OF SYSTEMS OF RAYS}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (Transactions of the Royal Irish Academy, vol.~17,
part~1 (1837), pp. 1--144.)}

\vskip36pt

\vfill

\centerline{\largerm Edited by David R. Wilkins}

\vskip 12pt

\centerline{\largerm 2001}

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\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

The {\it Third Supplement to an Essay on the Theory of Systems of
Rays} by William Rowan Hamilton was originally published in
volume~17 of the {\it Transactions of the Royal Irish Academy}.
It is included in {\it The Mathematical Papers of Sir William
Rowan Hamilton, Volume I: Geometrical Optics}, edited for the
Royal Irish Academy by A.~W. Conway and J.~L. Synge, and published
by Cambridge University Press in 1931.

There are a small number of obvious typographical errors in the
original publication which are corrected in this edition
(following the edition edited by Conway and Synge).  These
include the following:---

\smallskip

\item{---}
in the article numbered 3, following equation (R), the line
$$ \left( {\delta V \over \delta x} \right),\quad
   \left( {\delta V \over \delta y} \right),\quad
   \left( {\delta V \over \delta y} \right),$$
in the original publication has been corrected to read
$$ \left( {\delta V \over \delta x} \right),\quad
   \left( {\delta V \over \delta y} \right),\quad
   \left( {\delta V \over \delta z} \right),$$

\smallskip

\item{---}
in article~10, equations (H${}^6$), the final term
$\displaystyle
\left( \delta \beta - v'
   \, \delta' {\delta \Omega \over \delta \upsilon} \right)$
in the original publication has been corrected to read
$\displaystyle
\left( \delta \gamma - v'
   \, \delta' {\delta \Omega \over \delta \upsilon} \right)$;

\smallskip

\item{---}
in article~23, equation (L${}^{16}$),
$\displaystyle {\delta^2 W \over \delta \sigma \, \delta \tau}$
has been replaced on the second line of the equation by
$\displaystyle {\delta^2 W \over \delta \tau \, \delta \upsilon}$;

\smallskip

\item{---}
in article~29, equations (W${}^{19}$), $\tau$ has been replaced
by $\tau^2$.

\smallskip

The table of contents in the original publication differred
in places from the headings of the sections in the main body of
the paper.  This edition follows the edition edited by Conway
and Synge in revising the table of contents so as to agree with
the section headings in the body of the paper.  Accordingly the
following changes have been made to the table of contents:---

\smallskip

\item{---}
in the description of article~16, the words `guiding
surface' have been replaced by `guiding paraboloid';

\smallskip

\item{---}
in the description of article~17, the word `corresponding' has
been replaced by `respective';

\smallskip

\item{---}
the description of article~25 in the original publication is
`Principal rays and principal foci for straight or curved
systems.  General method of determining the arrangement and
aberrations of the rays, near a principal focus, or other point
of vergency.'

\smallskip

A small number of changes have been made to the spelling and
punctuation of the original publication.

\nobreak\bigskip

\leftline{\hskip.5\hsize David R. Wilkins}

\vskip3pt

\leftline{\hskip.5\hsize Dublin, June 1999}

\vskip0pt

\leftline{\hskip.5\hsize Edition corrected October 2001}

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{\largeit\noindent
Third Supplement to an Essay on the Theory of Systems of Rays.
\hskip 0pt plus10pt minus0pt
By {\largerm WILLIAM R. HAMILTON}, A. B., M. R. I. A.,
M. R.\ Ast.\ Soc.\ London, M. G.\ Soc.\ Dublin,
Hon.\ M.\ Soc.\ Arts for Scotland,
Hon.\ M.\ Portsmouth Lit.\ and Phil.\ Soc.,
Member of the British Association for the Advancement of Science,
Fellow of the American Academy of Arts and Sciences,
Andrews' Professor of Astronomy in the University of Dublin,
and Royal Astronomer of Ireland.}

\bigbreak

\centerline{Read January 23, 1832 and October 22, 1832.}

\bigbreak

\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~17, part~1 (1837), pp. 1--144.]}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

\centerline{\largerm INTRODUCTION.}

\bigskip

The present Supplement contains a system of general methods for
the solution of Optical Problems, together with some general
results, deduced from the fundamental formula and view of Optics,
which have been proposed in my former memoirs.  The copious
analytical headings, prefixed to the several numbers, and
collected in the Table of Contents, will sufficiently explain the
plan of the present communication; and it is only necessary to
say a few words here, respecting some of the principal results.

Of these the theory of external and internal conical refraction,
deduced by my general methods from the principles of
{\sc Fresnel}, will probably be thought the least undeserving of
attention.  It is right, therefore, to state that this theory had
been deduced, and was communicated to a general meeting of the
Royal Irish Academy, not at the earlier, but at the later of the
two dates prefixed to the present Supplement.  After making this
communication to the Academy, in October, 1832, I requested
Professor {\sc Lloyd} to examine the question experimentally, and
to try whether he could perceive any such phenomena in biaxal
crystals, as my theory of conical refraction had led me to
expect.  The experiments of Professor {\sc Lloyd}, confirming my
theoretical expectations, have been published by him in the
numbers of the London and Edinburgh Philosophical Magazine, for
the months of February and March, 1833; and they will be found
with fuller details in the present Volume of the Irish
Transactions.

I am informed that {\sc James Mac Cullagh}, Esq., F.T.C.D., who
published in the last preceding Volume of these Transactions a
series of elegant Geometrical Illustrations of {\sc Fresnel's}
theory, has, since he heard of the experiments of Professor
{\sc Lloyd}, employed his own geometrical methods to confirm my
results respecting the existence of those conoidal cusps and
circles of contact on {\sc Fresnel's} wave, from which I had been
led to the expectation of conical refraction.  And on my lately
mentioning to him that I had connected these cusps and circles on
{\sc Fresnel's} wave, with circles and cusps of the same kind on
a certain other surface discovered by {\sc M.~Cauchy}, by a
general theory of reciprocal surfaces, which I stated last year
at a general meeting of the Royal Irish Academy, Mr.~{\sc Mac
Cullagh} said that he had arrived independently at similar
results, and put into my hands a paper on the subject, which I
have not yet been able to examine, but which will (I hope) be
soon presented to the Academy, and published in their
Transactions.

I ought also to mention, that on my writing in last November to
Professor {\sc Airy}, and communicating to him my results
respecting the cusps and circles of {\sc Fresnel's} wave, and my
expectation of conical refraction which had not then been
verified, Professor {\sc Airy} replied that he had long been
aware of the existence of the conoidal cusps, which indeed it is
surprising that {\sc Fresnel} did not perceive.  Professor
{\sc Airy}, however, had not perceived the existence of the
circles of contact, nor had he drawn from either cusps or circles
any theory of conical refraction.

The latter theory was deduced, by my general methods, from the
hypothesis of transversal vibrations in a luminous ether, which
hypothesis seems to have been first proposed by Dr.~{\sc Young},
but to have been independently framed and far more perfectly
developed by {\sc Fresnel}; and from {\sc Fresnel's} other
principle, of the existence of three rectangular axes of
elasticity within a biaxal crystallised medium.  The
verification, therefore, of this theory of conical refraction, by
the experiments of Professor {\sc Lloyd}, must be considered as
affording a new and important probability in favour of
{\sc Fresnel's} views: that is, a new encouragement to reason
from those views, in combining and predicting appearances.

The length to which the present Supplement has already extended,
obliges me to reserve, for a future communication, many other
results deduced by my general methods from the principle of the
characteristic function: and especially a general theory of the
focal lengths and aberrations of optical instruments of
revolution.

\line{\hfil WILLIAM R. HAMILTON.}

\nobreak\bigskip

{\sc Observatory}, {\it June}, 1833.

\vfill\eject

\centerline{\largerm CONTENTS OF THE THIRD SUPPLEMENT.}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

1.
Fundamental formula of Mathematical Optics.  Design of the present
Supplement.

2.
Fundamental problem of Mathematical Optics, and solution by the
fundamental formula.  Partial differential equations, respecting the
characteristic function~$V$, and common to all optical combinations.
Deduction of the medium-functions $\Omega$, $v$, from this
characteristic function~$V$.  Remarks on the new symbols
$\sigma$,~$\tau$,~$\upsilon$.

3.
Connexion of the characteristic function~$V$, with the formation and
integration of the general equations of a curved ray, ordinary or
extraordinary.

4.
Transformations of the fundamental formula.  New view of the
auxiliary function~$W$; new auxiliary function~$T$.  Deductions of
the characteristic and auxiliary functions $V$, $W$, $T$, each from
each.  General theorem of maxima and minima, which includes all the
details of such deductions.  Remarks on the respective advantages of
the characteristic and auxiliary functions.

5.
General transformations, by the auxiliary functions $W$, $T$, of the
partial differential equations in $V$.  Other partial differential
equations in $V$, for extreme uniform media.  Integration of these
equations, by the functions $W$, $T$.

6.
General deductions and transformations of the differential and
integral equations of a curved or straight ray, ordinary of
extraordinary, by the auxiliary functions $W$, $T$.

7.
General remarks on the connexions between the partial differential
coefficients of the second order of the functions $V$, $W$, $T$.
General method of investigating those connexions.  Deductions of the
coefficients of $V$ from those of $W$, when the final medium is
uniform.

8.
Deduction of the coefficients of $W$ from those of $V$.  Homogeneous
transformations.

9.
Deductions of the coefficients of $T$ from those of $W$, and
reciprocally.

10.
General remarks and cautions, with respect to the foregoing
deductions. Case of a single uniform medium.  Connexions between the
coefficients of the functions $v$, $\Omega$, $\upsilon$, for any
single medium.

11.
General formula for reflection or refraction, ordinary or
extraordinary.  Changes of $V$, $W$, $T$.  The difference
$\Delta V$ is $= 0$; $\Delta W = \Delta T =$ a homogeneous function of
the first dimension of the differences $\Delta \sigma$, $\Delta \tau$,
$\Delta \upsilon$, depending on the shape and position of the
reflecting or refracting surface.  Theorem of maxima and minima, for
the elimination of the incident variables.  Combinations of reflectors
or refractors.  Compound and component combinations.

12.
Changes of the coefficients of the second order, of $V$, $W$, $T$,
produced by reflection or refraction.

13.
Changes produced by transformation of co-ordinates.  Nearly all the
foregoing results may be extended to oblique co-ordinates.  The
fundamental formula may be presented so as to extend even to polar or
to any other marks of position, and new auxiliary functions may then
be found, analogous to, and including, the functions $W$, $T$:
together with new and general differential and integral equations for
curved and polygon rays, ordinary and extraordinary.

14.
General geometrical relations of infinitely near rays.  Classification
of twenty-four independent coefficients, which enter into the
algebraical expressions of these general relations.  Division of the
general discussion into four principal problems.

15.
Discussion of the four problems.  Elements of arrangement of near
luminous paths.  Axis and constant of chromatic dispersion.  Axis of
curvature of ray.  Guiding paraboloid, and constant of deviation.
Guiding planes, and conjugate guiding axes.

16.
Application of the elements of arrangement.  Connexion of the two
final vergencies, and planes of vergency, and guiding lines, with the
two principal curvatures and planes of curvature of the guiding
paraboloid, and with the constant of deviation.  The planes of
curvature are the planes of extreme projection of the final ray-lines.

17.
Second application of the elements.  Arrangement of the near final-ray
lines from an oblique plane.  Generalisation of the theory of the
guiding paraboloid and constant of deviation.  General theory of
deflexures of surfaces.  Circles and axes of deflexure.  Rectangular
planes and axes of extreme deflexure.  Deflected lines, passing
through these axes, and having the centres of deflexure for their
respective foci by projection.  Conjugate planes of deflexure, and
indicating cylinder of deflexion.

18.
Construction of the new auxiliary paraboloid, (or of an osculating
hyperboloid,) and of the new constant of deviation, for ray-lines from
an oblique plane, by the former elements of arrangement.

19.
Condition of intersection of two near final ray-lines.  Conical locus
of the near final points in a variable medium which satisfy this
condition.  Investigations of {\sc Malus}.  Illustration of the condition
of intersection, by the theory of the auxiliary paraboloid, for ray-lines
from an oblique plane.

20.
Other geometrical illustrations of the condition of intersection, and
of the elements of arrangement.  Composition of partial deviations.
Rotation round the axis of curvature of a final ray.

21.
Relations between the elements of arrangement, depending only on the
extreme points, directions, and colour of a given luminous path, and
on the extreme media.  In a final uniform medium, ordinary or
extraordinary, the two planes of vergency are conjugate planes of
deflexure of any surface of a certain class determined by the final
medium; and also of a certain analogous surface determined by the
whole combination.  Relations between the visible magnitudes and
distortions of any two small objects viewed from each other through
any optical combination.  Interchangeable eye-axes and object-axes of
distortion.  Planes of no distortion.

22.
Calculation of the elements of arrangement, for arbitrary axes of
co-ordinates.

23.
The general linear expressions for the arrangement of near rays, fail
at a point of vergency.  Determination of these points, and of their
loci, the caustic surfaces, in a straight or curved system, by the
methods of the present Supplement.

24.
Connexion of the conditions of initial and final intersection of two
near paths of light, polygon or curved, with the maxima or minima of
the time or action-function $V + V_\prime = \sum \int v\,ds$.
Separating planes, transition planes, and transition points, suggested
by these maxima and minima.  The separating planes divide the near
points of less from those of greater action, and they contain the
directions of osculation or intersection of the surfaces for which $V$
and $V_\prime$ are constant; the transition planes touch the caustic
pencils, and the transition points are on the caustic curves. Extreme
osculating waves, or action surfaces.  Law of osculation.  Analogous
theorems for sudden reflection or refraction.

25.
Formul{\ae} for the principal foci and principal rays of a
straight or curved system, ordinary or extraordinary.  General method
of investigating the arrangement and aberrations of the rays, near a
principal focus, or other point of vergency.

26.
Combination of the foregoing view of optics with the undulatory
theory of light.  The quantities $\sigma$,~$\tau$,~$\upsilon$, or
$${\delta V \over \delta x},
  {\delta V \over \delta y},
  {\delta V \over \delta z},$$
that is, the partial differential coefficients of the first order of
the characteristic function~$V$, taken with respect to the final
co-ordinates, are, in the undulatory theory of light, the components of
normal slowness of propagation of a wave.  The fundamental formula
(A) may easily be explained and proved by the principles of
the same theory.

27.
Theory of {\sc Fresnel}.  New formulae, founded on that theory,
for the velocities and polarisations of a plane wave or wave-element.
New method of deducing the equation of {\sc Fresnel's} curved wave,
propagated from a point in a uniform medium with three unequal
elasticities.  Lines of single ray-velocity, and of single
normal-velocity, discovered by {\sc Fresnel}.

28.
New properties of {\sc Fresnel's} wave.  This wave has four conoidal
cusps, at the ends of the lines of single ray-velocity; it has also
four circles of contact, of which each is contained on a touching
plane of single normal-velocity.  The lines of single ray-velocity
may therefore be called cusp-rays; and the lines of single
normal-velocity may be called normals of circular contact.

29.
New consequences of {\sc Fresnel's} principles.  It follows from those
principles, that crystals of sufficient biaxal energy ought to exhibit
two kinds of conical refraction, an external and an internal: a
cusp-ray giving an external cone of rays, and a normal of circular
contact being connected with an internal cone.

30.
Theory of conical polarisation.  Lines of vibration.  These lines, on
{\sc Fresnel's} wave, are the intersections of two series of
concentric and co-axal ellipsoids.

31.
In any uniform medium, the curved wave propagated from a point is
connected with a certain other surface, which may be called the
surface of components, by relations discovered by {\sc M.~Cauchy}, and
by some new relations connected with a general theorem of reciprocity.
This new theorem of reciprocity gives a new construction for the wave
in any undulatory theory of light: and it connects the cusps and
circles of contact on {\sc Fresnel's} wave, with circles and cusps of
the same kind upon the surface of components.

\vfill\eject

\centerline{\largerm THIRD SUPPLEMENT.}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

{\sectiontitle
Fundamental Formula of Mathematical Optics.  Design of the present
Supplement.\par}

\nobreak\bigskip

1.
When light is considered as propagated, according to that
known general law which is called the law of least action, or of
swiftest propagation, along any curved or polygon ray, ordinary or
extraordinary, describing each element of that ray
$ds = \surd(dx^2 + dy^2 + dz^2)$ with a molecular velocity or
undulatory slowness~$v$, which is supposed to depend, in the most
general case, on the nature of the medium, the position and direction
of the element, and the colour of the light, having only a finite
number of values when these are given, and being therefore a function
of the three rectangular co-ordinates, or marks of position,
$x$,~$y$,~$z$, the three differential ratios or cosines of direction,
$$\alpha = {dx \over ds},\quad
  \beta  = {dy \over ds},\quad
  \gamma = {dx \over ds},$$
and a chromatic index or measure of colour, $\chi$, the form of which
function~$v$ depends on and characterises the medium; then if we
denote as follows the variation of this function,
$$\delta v = {\delta v \over \delta x} \delta x
           + {\delta v \over \delta y} \delta y
           + {\delta v \over \delta z} \delta z
           + {\delta v \over \delta \alpha} \delta \alpha
           + {\delta v \over \delta \beta } \delta \beta
           + {\delta v \over \delta \gamma} \delta \gamma
           + {\delta v \over \delta \chi  } \delta \chi,$$
and if, by the help of the relation
$\alpha^2 + \beta^2 + \gamma^2 = 1$, we determine
$${\delta v \over \delta \alpha},\quad
  {\delta v \over \delta \beta },\quad
  {\delta v \over \delta \gamma},$$
so as to satisfy the condition
$$      \alpha {\delta v \over \delta \alpha}
      + \beta  {\delta v \over \delta \beta }
      + \gamma {\delta v \over \delta \gamma}
   = v,$$
namely, by making $v$ homogeneous of the first dimension with respect
to $\alpha$,~$\beta$,~$\gamma$; it has been shown, in my First
Supplement, that the variation of the definite integral
$V = \int v \,ds$, considered as a function, which I have called the
{\it Characteristic Function\/} of the final and initial co-ordinates,
that is, the {\it variation of the action, or the time, expended by
light of any one colour, in going from one variable point to another},
is
$$\delta V
   = ( \delta \int v\,ds \mathrel{=)}
        {\delta v  \over \delta \alpha } \delta x
      - {\delta v' \over \delta \alpha'} \delta x'
      + {\delta v  \over \delta \beta  } \delta y
      - {\delta v' \over \delta \beta' } \delta y'
      + {\delta v  \over \delta \gamma } \delta z
      - {\delta v' \over \delta \gamma'} \delta z':
   \eqno {\rm (A)}$$
the accented being the initial quantities.  This general equation,
(A), which I have called the {\it Equation of the
Characteristic Function}, involves very various and extensive
consequences, and appears to me to include the whole of mathematical
optics.  I propose, in the present Supplement, to offer some
additional remarks and methods, connected with the characteristic
function~$V$, and the fundamental formula (A); and in
particular to point out a new view of the auxiliary function~$W$,
introduced in my former memoirs, and a new auxiliary function~$T$,
which may be employed with advantage in many optical researches: I
shall also give some other general transformations and applications of
the fundamental formula, and shall speak of the connection of my view
of optics with the undulatory theory of light.

\bigbreak

{\sectiontitle
Fundamental Problem of Mathematical Optics, and Solution by the
Fundamental Formula.  Partial Differential Equations, respecting the
Characteristic Function~$V$, and common to all optical combinations.
Deduction of the Medium-Functions $\Omega$, $v$, from this
Characteristic Function~$V$.  Remarks on the new symbols
$\sigma$,~$\tau$,~$\upsilon$.\par}

\nobreak\bigskip

2.
It may be considered as a {\it fundamental problem\/} in
Mathematical Optics, to which all others are reducible, {\it to
determine, for any proposed combination of media, the law of
dependence of the two extreme directions of a curved or polygon ray,
ordinary or extraordinary, on the positions of the two extreme points
which are visually connected by that ray, and on the colour of the
light\/}: that is, in our present notation, to determine the law of
dependence of the extreme {\it direction-cosines\/}
$\alpha$~$\beta$~$\gamma$ $\alpha'$~$\beta'$~$\gamma'$,
on the extreme co-ordinates
$x$~$y$~$z$ $x'$~$y'$~$z'$, and on the chromatic index~$\chi$.
This fundamental problem is resolved by our fundamental formula
(A); or by the six following equations into which
(A) resolves itself, and which express the law of dependence
required:
$$\left.
\eqalign{
     {\delta V \over \delta x}
      &= {\delta v  \over \delta \alpha};\cr
   - {\delta V \over \delta x'}
      &= {\delta v' \over \delta \alpha'};\cr}\quad
\eqalign{
     {\delta V \over \delta y}
      &= {\delta v  \over \delta \beta };\cr
   - {\delta V \over \delta y'}
      &= {\delta v' \over \delta \beta '};\cr}\quad
\eqalign{
     {\delta V \over \delta z}
      &= {\delta v  \over \delta \gamma};\cr
   - {\delta V \over \delta z'}
      &= {\delta v' \over \delta \gamma'}.\cr} \right\}
   \eqno {\rm (B)}$$
These equations appear to require, for their application to any
proposed combination, not only the knowledge of the form of the
{\it Characteristic Function\/}~$V$, that is, the law of dependence of
the action or time on the extreme positions and on the colour, but
also on the knowledge of the forms of the functions $v$, $v'$, that
is, the optical properties of the final and initial media; but these
final and initial {\it medium-functions}~$v$, $v'$, may themselves be
deduced from the one characteristic function~$V$, by reasonings of the
following kind.

Whatever be the nature of the final medium, that is, whatever be the
law of dependence of $v$ on the position, direction, colour, we have
supposed, in deducing the general formula (A), that the
expression of this dependence has been so prepared as to make the
medium-function~$v$ homogeneous of the first dimension relatively to
the direction-cosines $\alpha$,~$\beta$,~$\gamma$; the partial
differential coefficients
$${\delta v \over \delta \alpha},\quad
  {\delta v \over \delta \beta },\quad
  {\delta v \over \delta \gamma},$$
of this homogeneous function, are themselves homogeneous, but of the
dimension zero; that is, they are functions of the two ratios
$${\alpha \over \gamma},\quad {\beta \over \gamma},$$
involving also, in general, the co-ordinates $x$~$y$~$z$, and the
chromatic index~$\chi$: if then we conceive the two ratios
$${\alpha \over \gamma},\quad {\beta \over \gamma},$$
to be eliminated between the three first of equations (B), and
if, in like manner, we conceive
$${\alpha' \over \gamma'},\quad {\beta' \over \gamma'},$$
to be eliminated between the last three equations (B), we see
that such eliminations would give two partial differential equations
of the first order, between the characteristic function~$V$ and the
co-ordinates and colour, of the form
$$\left. \eqalign{
0 &= \Omega \left(
         {\delta V \over \delta x},
         {\delta V \over \delta y},
         {\delta V \over \delta z},
         x,y,z,\chi \right),\cr
0 &= \Omega' \left(
       - {\delta V \over \delta x'},
       - {\delta V \over \delta y'},
       - {\delta V \over \delta z'},
         x', y' ,z' ,\chi \right),\cr} \right\}
   \eqno {\rm (C)}$$
which both conduct to the following general equation, of the second
order and third degree, common to all optical combinations,
$$\eqalign{&\mathrel{\phantom{=}}
      {\delta^2 V \over \delta x \, \delta x'}\,
      {\delta^2 V \over \delta y \, \delta y'}\,
      {\delta^2 V \over \delta z \, \delta z'}
    + {\delta^2 V \over \delta x \, \delta y'}\,
      {\delta^2 V \over \delta y \, \delta z'}\,
      {\delta^2 V \over \delta z \, \delta x'}
    + {\delta^2 V \over \delta x \, \delta z'}\,
      {\delta^2 V \over \delta y \, \delta x'}\,
      {\delta^2 V \over \delta z \, \delta y'} \cr
   &= {\delta^2 V \over \delta z \, \delta x'}\,
      {\delta^2 V \over \delta y \, \delta y'}\,
      {\delta^2 V \over \delta x \, \delta z'}
    + {\delta^2 V \over \delta z \, \delta y'}\,
      {\delta^2 V \over \delta y \, \delta z'}\,
      {\delta^2 V \over \delta x \, \delta x'}
    + {\delta^2 V \over \delta z \, \delta z'}\,
      {\delta^2 V \over \delta y \, \delta x'}\,
      {\delta^2 V \over \delta x \, \delta y'}.\cr}
   \eqno {\rm (D)}$$
If now we put, for abridgment,
$$\left.
\eqalign{ {\delta V \over \delta x}  &= \sigma,   \cr
        - {\delta V \over \delta x'} &= \sigma',  \cr}\quad
\eqalign{ {\delta V \over \delta y}  &= \tau,     \cr
        - {\delta V \over \delta y'} &= \tau',    \cr}\quad
\eqalign{ {\delta V \over \delta z}  &= \upsilon, \cr
        - {\delta V \over \delta z'} &= \upsilon',\cr} \right\}
   \eqno {\rm (E)}$$
and if between the first three of these equations (E) we
eliminate two of the three initial co-ordinates $x'$~$y'$~$z'$, it is
easy to perceive, by (C) or (D), that in every optical
combination the third co-ordinate will disappear; and similarly that
between the three last equations (E) we can eliminate all the
three final co-ordinates, by eliminating any two of them; and that
these eliminations will conduct to the relations (C) under the
form
$$\left. \eqalign{
0 &= \Omega(\sigma, \tau, \upsilon, x, y, z, \chi),\cr
0 &= \Omega'(\sigma', \tau', \upsilon', x', y', z', \chi),\cr}
   \right\}
   \eqno {\rm (F)}$$
which can thus be obtained, by differentiation and elimination, from
the characteristic function~$V$ alone: and which, as we are about to
see, determine the forms of $v$, $v'$, that is, the properties of the
extreme media.  Comparing the differentials of the relations
(F), with the following, that is, with the conditions of
homogeneity of $v$, $v'$, prepared by the definitions (E)
and by the relations (B),
$$\left. \eqalign{
v  &=   \alpha  {\delta v  \over \delta \alpha}
      + \beta   {\delta v  \over \delta \beta}
      + \gamma  {\delta v  \over \delta \gamma}
    = \alpha \sigma + \beta \tau + \gamma \upsilon,\cr
v' &=   \alpha' {\delta v' \over \delta \alpha'}
      + \beta'  {\delta v' \over \delta \beta'}
      + \gamma' {\delta v' \over \delta \gamma'}
    = \alpha' \sigma' + \beta' \tau' + \gamma' \upsilon',\cr}
   \right\}
   \eqno {\rm (G)}$$
and with their differentials, that is with
$$\left. \eqalign{
        \alpha  \,\delta \sigma
      + \beta   \,\delta \tau
      + \gamma  \,\delta \upsilon
   &=   {\delta v  \over \delta x}  \delta x
      + {\delta v  \over \delta y}  \delta y
      + {\delta v  \over \delta z}  \delta z
      + {\delta v  \over \delta \chi} \delta \chi,\cr
        \alpha' \,\delta \sigma'
      + \beta'  \,\delta \tau'
      + \gamma' \,\delta \upsilon'
   &=   {\delta v' \over \delta x'} \delta x'
      + {\delta v' \over \delta y'} \delta y'
      + {\delta v' \over \delta z'} \delta z'
      + {\delta v' \over \delta \chi} \delta \chi,\cr}
   \right\}
   \eqno {\rm (H)}$$
we find
$$\left.
\eqalign{
{\alpha  \over v}
   &= {\delta \Omega  \over \delta \sigma   },\cr
{\alpha' \over v'}
   &= {\delta \Omega' \over \delta \sigma'  },\cr}\quad
\eqalign{
{\beta   \over v}
   &= {\delta \Omega  \over \delta \tau     },\cr
{\beta'  \over v'}
   &= {\delta \Omega' \over \delta \tau'    },\cr}\quad
\eqalign{
{\gamma  \over v}
   &= {\delta \Omega  \over \delta \upsilon },\cr
{\gamma' \over v'}
   &= {\delta \Omega' \over \delta \upsilon'},\cr}
   \right\}
   \eqno {\rm (I)}$$
and also
$$\left.
\eqalign{ - {1 \over v }{\delta v  \over \delta x }
   &= {\delta \Omega  \over \delta x },\cr
          - {1 \over v'}{\delta v' \over \delta x'}
   &= {\delta \Omega' \over \delta x'},\cr}\quad
\eqalign{ - {1 \over v }{\delta v  \over \delta y }
   &= {\delta \Omega  \over \delta y },\cr
          - {1 \over v'}{\delta v' \over \delta y'}
   &= {\delta \Omega' \over \delta y'},\cr}\quad
\eqalign{ - {1 \over v }{\delta v  \over \delta z }
   &= {\delta \Omega  \over \delta z },\cr
          - {1 \over v'}{\delta v' \over \delta z'}
   &= {\delta \Omega' \over \delta z'},\cr}\quad
\eqalign{ - {1 \over v }{\delta v  \over \delta \chi}
   &= {\delta \Omega  \over \delta \chi},\cr
          - {1 \over v'}{\delta v' \over \delta \chi}
   &= {\delta \Omega' \over \delta \chi},\cr}
   \right\}
   \eqno {\rm (K)}$$
if we so prepare the expressions of the relations (F) as to have
$$\left. \eqalign{
      \sigma    {\delta \Omega  \over \delta \sigma   }
    + \tau      {\delta \Omega  \over \delta \tau     }
    + \upsilon  {\delta \Omega  \over \delta \upsilon }
   &= 1,\cr
      \sigma'   {\delta \Omega' \over \delta \sigma'  }
    + \tau'     {\delta \Omega' \over \delta \tau'    }
    + \upsilon' {\delta \Omega' \over \delta \upsilon'}
   &= 1;\cr} \right\}
   \eqno {\rm (L)}$$
which can be done by putting those relations under the form
$$\left. \eqalign{
0 &= (\sigma^2  + \tau^2  + \upsilon^2 )^{1 \over 2} \omega  - 1
      = \Omega,\cr
0 &= (\sigma'^2 + \tau'^2 + \upsilon'^2)^{1 \over 2} \omega' - 1
      = \Omega';\cr} \right\}
   \eqno {\rm (M)}$$
in which $\omega$, $\omega'$, that is
$(\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$, and
$(\sigma'^2 + \tau'^2 + \upsilon'^2)^{-{1 \over 2}}$,
are to be expressed as functions respectively of
$\sigma (\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$,
$\tau (\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$,
$\upsilon (\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$,
$x$,~$y$,~$z$,~$\chi$, and of
$\sigma' (\sigma'^2 + \tau'^2 + \upsilon'^2)^{-{1 \over 2}}$,
$\tau' (\sigma'^2 + \tau'^2 + \upsilon'^2)^{-{1 \over 2}}$,
$\upsilon' (\sigma'^2 + \tau'^2 + \upsilon'^2)^{-{1 \over 2}}$,
$x'$,~$y'$,~$z'$,~$\chi$.  After this preparation the partial
differential coefficients
$${\delta \Omega \over \delta \sigma  },\quad
  {\delta \Omega \over \delta \tau    },\quad
  {\delta \Omega \over \delta \upsilon},$$
are homogeneous of dimension zero relatively to
$\sigma$,~$\tau$,~$\upsilon$; and in like manner
$${\delta \Omega' \over \delta \sigma'  },\quad
  {\delta \Omega' \over \delta \tau'    },\quad
  {\delta \Omega' \over \delta \upsilon'},$$
are homogeneous of dimension zero relatively to
$\sigma'$,~$\tau'$,~$\upsilon'$; if, therefore, between the three
first equations (I), we eliminate any two of the three final
quantities $\sigma$,~$\tau$,~$\upsilon$, the third will disappear;
and similarly all the three initial quantities
$\sigma'$,~$\tau'$,~$\upsilon'$, can be eliminated together, between
the last three of the equations (I): and by these eliminations
we shall be conducted to two relations of the form
$$\left. \eqalign{
0 &= \Psi({\alpha \over v}, {\beta  \over v}, {\gamma \over v},
          x, y, z, \chi),\cr
0 &= \Psi'({\alpha' \over v'}, {\beta'  \over v'}, {\gamma' \over v'},
          x', y', z', \chi),\cr}
   \right\}
   \eqno {\rm (N)}$$
which determine the forms of the final and initial medium-functions
$v$, $v'$; so that these forms can be deduced from the form of the
characteristic function~$V$.  We can therefore reduce to the study of
this one function~$V$, that general problem of mathematical optics
which has been already mentioned.

The partial differential coefficients of the characteristic
function~$V$, taken with respect to the co-ordinates $x$,~$y$,~$z$, are
of continual occurrence in the optical methods of my present and
former memoirs; I have therefore thought it useful to denote them in
this Supplement by separate symbols $\sigma$,~$\tau$,~$\upsilon$, and
I shall show in a future number their meanings in the undulatory
theory: namely, that they denote, in it, the components of normal
slowness of propagation of a wave.

\bigbreak

{\sectiontitle
Connection of the Characteristic Function~$V$, with the Formation and
Integration of the General Equations of a Curved Ray, Ordinary or
Extraordinary.\par}

\nobreak\bigskip

3.
It may be considered as a particular case of the foregoing
general problem, to determine general forms for the differential
equations of a curved ray, ordinary or extraordinary; that is, to
connect the general changes of direction with those of position, in
the passage of light through a variable medium.  The following forms,
$$d {\delta v \over \delta \alpha}
  = {\delta v \over \delta x} ds,\quad
  d {\delta v \over \delta \beta}
  = {\delta v \over \delta y} ds,\quad
  d {\delta v \over \delta \gamma}
  = {\delta v \over \delta z} ds,
   \eqno {\rm (O)}$$
(which are of the second order, because $\alpha$,~$\beta$,~$\gamma$,
$\alpha'$,~$\beta'$,~$\gamma'$, are defined by the equations
$$\left.
  \eqalign{\alpha  &= {dx  \over ds },\cr
           \alpha' &= {dx' \over ds'},\cr}\quad
  \eqalign{\beta   &= {dy  \over ds },\cr
           \beta'  &= {dy' \over ds'},\cr}\quad
  \eqalign{\gamma  &= {dz  \over ds },\cr
           \gamma' &= {dz' \over ds'},\cr} \right\}
   \eqno {\rm (P)}$$
{\it the symbol~$d$ referring, throughout the present Supplement, to
motion along a ray, while $\delta$ refers to arbitrary infinitesimal
changes of position, direction, and colour}, and $ds'$ being the
initial element of the ray,) were deduced, in the First Supplement, by
the Calculus of Variations, from the law of least action.  The same
forms (O), which are equivalent to but two distinct equations,
may be deduced from the fundamental formula (A), by the
properties of the characteristic function~$V$.  For, if we
differentiate the first equation (C), (which involves the
coefficients of this function~$V$, and was deduced from the formula
(A),) with reference to each of the three co-ordinates
$x$,~$y$,~$z$, considered as three independent variables, and with
reference to the index of colour~$\chi$, we find, by the foregoing
number,
$$\left. \eqalign{
\alpha {\delta^2 V \over \delta x^2}
      + \beta  {\delta^2 V \over \delta x \,\delta y}
      + \gamma {\delta^2 V \over \delta x \,\delta z}
   &= {\delta v \over \delta x},\cr
\alpha {\delta^2 V \over \delta x \,\delta y}
      + \beta  {\delta^2 V \over \delta y^2}
      + \gamma {\delta^2 V \over \delta y \,\delta z}
   &= {\delta v \over \delta y},\cr
\alpha {\delta^2 V \over \delta x \,\delta z}
      + \beta  {\delta^2 V \over \delta y \,\delta z}
      + \gamma {\delta^2 V \over \delta z^2}
   &= {\delta v \over \delta z},\cr
\alpha {\delta^2 V \over \delta x \,\delta \chi}
      + \beta  {\delta^2 V \over \delta y \,\delta \chi}
      + \gamma {\delta^2 V \over \delta z \,\delta \chi}
   &= {\delta v \over \delta \chi},\cr} \right\}
   \eqno {\rm (Q)}$$
and the three first of the equations (Q), by the general
relations (B), which were themselves deduced from
(A), and by the meanings (P) of
$\alpha$,~$\beta$,~$\gamma$, may easily be transformed to (O).
The differential equations (O) may also be regarded as the
limits of the following,
$$\sigma   - \sigma'   = \left( {\delta V \over \delta x} \right),
      \quad
  \tau     - \tau'     = \left( {\delta V \over \delta y} \right),
      \quad
  \upsilon - \upsilon' = \left( {\delta V \over \delta z} \right),
   \eqno {\rm (R)}$$
in which
$$\left( {\delta V \over \delta x} \right),\quad
  \left( {\delta V \over \delta y} \right),\quad
  \left( {\delta V \over \delta z} \right),$$
are obtained by differentiating $V$ considered as a function of the
seven variables $x$,~$y$,~$z$, $\Delta x$, $\Delta y$, $\Delta z$,
$\chi$, if $\Delta x = x - x'$, $\Delta y = y - y'$,
$\Delta z = z - z'$; the variation of $V$, when so considered, being
by (A), and by the definitions (E),
$$\eqalign{\delta V
   &=   (\sigma   - \sigma'  ) \delta x
      + (\tau     - \tau'    ) \delta y
      + (\upsilon - \upsilon') \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \left( {\delta V \over \delta \Delta x} \right)
               \delta \Delta x
      + \left( {\delta V \over \delta \Delta y} \right)
               \delta \Delta y
      + \left( {\delta V \over \delta \Delta z} \right)
               \delta \Delta z
      + {\delta V \over \delta \chi} \delta \chi,\cr}
   \eqno {\rm (S)}$$
in which
$$\left( {\delta V \over \delta \Delta x} \right) = \sigma',\quad
  \left( {\delta V \over \delta \Delta y} \right) = \tau',\quad
  \left( {\delta V \over \delta \Delta z} \right) = \upsilon'.
   \eqno {\rm (T)}$$

If we differentiate the first equation (C) relatively to
$x'$,~$y'$,~$z'$, we find, by the foregoing number,
$$\left. \eqalign{
0  &=   \alpha {\delta^2 V \over \delta x \,\delta x'}
      + \beta  {\delta^2 V \over \delta y \,\delta x'}
      + \gamma {\delta^2 V \over \delta z \,\delta x'},\cr
0  &=   \alpha {\delta^2 V \over \delta x \,\delta y'}
      + \beta  {\delta^2 V \over \delta y \,\delta y'}
      + \gamma {\delta^2 V \over \delta z \,\delta y'},\cr
0  &=   \alpha {\delta^2 V \over \delta x \,\delta z'}
      + \beta  {\delta^2 V \over \delta y \,\delta z'}
      + \gamma {\delta^2 V \over \delta z \,\delta z'},\cr} \right\}
   \eqno {\rm (U)}$$
of which, in virtue of (D), any two include the third, and
which may be put by (P) under the form
$$0 = d {\delta V \over \delta x'};\quad
  0 = d {\delta V \over \delta y'};\quad
  0 = d {\delta V \over \delta z'};
   \eqno {\rm (V)}$$
and these differential equations (V) of the first order, in
which the initial co-ordinates and the colour are constant, belong to
the ray, and may be regarded as integrals of (O).  They have,
themselves, for integrals
$${\delta V \over \delta x'} = {\rm const.},\quad
  {\delta V \over \delta y'} = {\rm const.},\quad
  {\delta V \over \delta z'} = {\rm const.},
   \eqno {\rm (W)}$$
the constants being, by (B), the values of the initial
quantities
$$- {\delta v' \over \delta \alpha'},\quad
  - {\delta v' \over \delta \beta'} ,\quad
  - {\delta v' \over \delta \gamma'}.$$

In like manner, by differentiating the last equation (C), we
find the following equations, which are analogous to (Q) and
(U),
$$\left. \eqalign{
\alpha' {\delta^2 V \over \delta x'^2}
      + \beta'  {\delta^2 V \over \delta x' \,\delta y'}
      + \gamma' {\delta^2 V \over \delta x' \,\delta z'}
   &= - {\delta v' \over \delta x'},\cr
\alpha' {\delta^2 V \over \delta x' \,\delta y'}
      + \beta'  {\delta^2 V \over \delta y'^2}
      + \gamma' {\delta^2 V \over \delta y' \,\delta z'}
   &= - {\delta v' \over \delta y'},\cr
\alpha' {\delta^2 V \over \delta x' \,\delta z'}
      + \beta'  {\delta^2 V \over \delta y' \,\delta z'}
      + \gamma' {\delta^2 V \over \delta z'^2}
   &= - {\delta v' \over \delta z'},\cr
\alpha' {\delta^2 V \over \delta x' \,\delta \chi}
      + \beta'  {\delta^2 V \over \delta y' \,\delta \chi}
      + \gamma' {\delta^2 V \over \delta z' \,\delta \chi}
   &= - {\delta v' \over \delta \chi},\cr} \right\}
   \eqno {\rm (X)}$$
and
$$\left. \eqalign{
0  &=   \alpha' {\delta^2 V \over \delta x \,\delta x'}
      + \beta'  {\delta^2 V \over \delta x \,\delta y'}
      + \gamma' {\delta^2 V \over \delta x \,\delta z'},\cr
0  &=   \alpha' {\delta^2 V \over \delta y \,\delta x'}
      + \beta'  {\delta^2 V \over \delta y \,\delta y'}
      + \gamma' {\delta^2 V \over \delta y \,\delta z'},\cr
0  &=   \alpha' {\delta^2 V \over \delta z \,\delta x'}
      + \beta'  {\delta^2 V \over \delta z \,\delta y'}
      + \gamma' {\delta^2 V \over \delta z \,\delta z'},\cr} \right\}
   \eqno {\rm (Y)}$$
The second members of the three first equations (X) vanish
when the initial medium is uniform, and those of the three first
equations (Q) when the final medium is so; and in this latter
case, of a final uniform medium, the final portion of the ray is
straight, and in its whole extent we have not only the equations
(W) but also the following,
$${\delta V \over \delta x} = {\rm const.},\quad
  {\delta V \over \delta y} = {\rm const.},\quad
  {\delta V \over \delta z} = {\rm const.},
   \eqno {\rm (Z)}$$
the constants being by (B) those functions of the final
direction-cosines and of the colour which we have denoted by
$${\delta v \over \delta \alpha},\quad
  {\delta v \over \delta \beta} ,\quad
  {\delta v \over \delta \gamma},$$
and which are here independent of the co-ordinates.  In general, if we
consider the final co-ordinates and the colour as constant, the
relations (Z) between the initial co-ordinates are forms for
the equations of a ray.  And though we have hitherto considered
rectangular co-ordinates only, yet we shall show in a future number
that there are analogous results for oblique and even for polar
co-ordinates.

\bigbreak

{\sectiontitle
Transformations of the Fundamental Formula.  New View of the
Auxiliary Function $W$; New Auxiliary Function $T$.  Deductions
of the Characteristic and Auxiliary Functions, $V$, $W$, $T$,
each from each.  General Theorem of Maxima and Minima, which
includes the details of such deductions.  Remarks on the
respective advantages of the Characteristic and Auxiliary
Functions.\par}

\nobreak\bigskip

4.
The fundamental equation (A) may be put under the form
$$\delta V
  =   \sigma   \, \delta x - \sigma'   \, \delta x'
    + \tau     \, \delta y - \tau'     \, \delta y'
    + \upsilon \, \delta z - \upsilon' \, \delta z'
    + {\delta V \over \delta \chi} \, \delta \chi,
   \eqno {\rm (A')}$$
employing the definitions (E), and introducing the variation of
colour; it admits also of the two following general
transformations,
$$\delta W
  =   x \, \delta \sigma
    + y \, \delta \tau
    + z \, \delta \upsilon
    + \sigma'   \, \delta x'
    + \tau'     \, \delta y'
    + \upsilon' \, \delta z'
    - {\delta V \over \delta \chi} \, \delta \chi,
   \eqno {\rm (B')}$$
and
$$\delta T
  =   x \, \delta \sigma   - x' \, \delta \sigma'
    + y \, \delta \tau     - y' \, \delta \tau'
    + z \, \delta \upsilon - z' \, \delta \upsilon'
    - {\delta V \over \delta \chi} \, \delta \chi,
   \eqno {\rm (C')}$$
in which
$$W = - V + x \sigma + y \tau + z \upsilon,
   \eqno {\rm (D')}$$
and
$$T = W - x' \sigma' - y' \tau' - z' \upsilon'.
   \eqno {\rm (E')}$$
In the two foregoing Supplements, the quantity $W$ was
introduced, and was considered as a function of the final
direction-cosines $\alpha$,~$\beta$,~$\gamma$, the final medium
being regarded as uniform, and the luminous origin and colour as
given; we shall now take another and a more general view of this
auxiliary function $W$, and shall consider it as depending, by
(B${}'$), for all optical combinations, on the seven quantities
$\sigma$~$\tau$~$\upsilon$ $x'$~$y'$~$z'$~$\chi$.
In like manner, we shall consider the new auxiliary function $T$
as depending, by the new transformation (C${}'$), on the seven
quantities
$\sigma$~$\tau$~$\upsilon$
$\sigma'$~$\tau'$~$\upsilon'$~$\chi$.
The forms of these auxiliary functions $W$, $T$, are connected
with each other, and with the characteristic function $V$, by
relations of which the knowledge is important, in the theory of
optical systems.  Let us therefore consider how the form of each
of the three functions $V$, $W$, $T$, can be deduced from the
form of either of the other two.

These deductions may all be affected by suitable applications of
the three forms (A${}'$) (B${}'$) (C${}'$), of our fundamental
equation (A), together with the definitions (D${}'$) (E${}'$), as
we shall soon see more in detail, by means of the following
remarks.

When the form of the characteristic function $V$ is known, and it
is required to deduce the form of the auxiliary function $W$, we
are to eliminate the three final co-ordinates $x$,~$y$,~$z$,
between the equation (D${}'$) and the three first of the
equations (E); and similarly when it is required to deduce the
form of $T$ from that of $V$, we are to eliminate the six final
and initial co-ordinates
$x$~$y$~$z$ $x'$~$y'$~$z'$
between the six equations (E), (which are all included in the
formula (A${}'$),) and the following,
$$T = - V
      + x \sigma - x' \sigma'
      + y \tau - y' \tau'
      + z \upsilon - z' \upsilon':
   \eqno {\rm (F')}$$
and if it be required to deduce the form of $T$ from that of $W$,
we are to eliminate the three initial co-ordinates
$x'$~$y'$~$z'$, between the equation (E${}'$) and the three
following general equations,
$$\sigma'   = {\delta W \over \delta x'},\quad
  \tau'     = {\delta W \over \delta y'},\quad
  \upsilon' = {\delta W \over \delta z'}.
   \eqno {\rm (G')}$$
But when it is required to deduce reciprocally $V$ from $T$ or
from $W$, or $W$ from $T$, we must distinguish between the cases
of variable and of uniform media; because we must then use the
equations into which (B${}'$) and (C${}'$) resolve themselves,
and this resolution, when the extreme media are not both
variable, requires the consideration of the connexion that then
exists between the quantities
$\sigma$~$\tau$~$\upsilon$
$\sigma'$~$\tau'$~$\upsilon'$~$\chi$:
which circumstance also, of a connexion between these variable
quantities, leave a partial indeterminateness in the forms of $T$
and $W$ as deduced from $V$, and in the form of $T$ as deduced
from $W$, for the case of uniform media.

When the final medium is variable, then
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$, may in general vary
independently, and the equation (B${}'$) gives
$${\delta W \over \delta \sigma}   = x,\quad
  {\delta W \over \delta \tau}     = y,\quad
  {\delta W \over \delta \upsilon} = z,\quad
  {\delta W \over \delta \chi}
      = - {\delta V \over \delta \chi};
   \eqno {\rm (H')}$$
and, in this case, $V$ can in general be deduced from $W$ by
eliminating $\sigma$,~$\tau$,~$\upsilon$, between the equation
(D${}'$), and the three first equations (H${}'$).  But if the
final medium be uniform, then
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$, are connected by the first
of the relations (F), from which, in this case, the final
co-ordinates disappear; and instead of the four equations (H${}'$)
we have the three following
$$  {\displaystyle {\delta W \over \delta \sigma}   - x
         \over \displaystyle {\delta \Omega \over \delta \sigma}}
  = {\displaystyle {\delta W \over \delta \tau}     - y
         \over \displaystyle {\delta \Omega \over \delta \tau}}
  = {\displaystyle {\delta W \over \delta \upsilon} - z
         \over \displaystyle {\delta \Omega \over \delta \upsilon}}
  = {\displaystyle {\delta W \over \delta \chi}
            +  {\delta V \over \delta \chi}
         \over \displaystyle {\delta \Omega \over \delta \chi}};
   \eqno {\rm (I')}$$
by means of the two first of which, combined with the relation
already mentioned, namely,
$$0 = \Omega(\sigma, \tau, \upsilon, \chi),
   \eqno {\rm (K')}$$
which depends on, and characterises, the nature of the final
uniform medium, we can eliminate $\sigma$,~$\tau$,~$\upsilon$,
from equation (D${}'$), and so deduce $V$ from $W$.

In like manner, if both the extreme media be variable, then the
seven quantities
$\sigma$~$\tau$~$\upsilon$
$\sigma'$~$\tau'$~$\upsilon'$~$\chi$
may in general vary independently, and the equation (C${}'$)
resolves itself into the seven following,
$${\delta T \over \delta \sigma}    = x,\quad
  {\delta T \over \delta \tau}      = y,\quad
  {\delta T \over \delta \upsilon}  = z,\quad
  {\delta T \over \delta \chi}      = - {\delta V \over \delta \chi},\quad
  {\delta T \over \delta \sigma'}   = - x',\quad
  {\delta T \over \delta \tau'}     = - y',\quad
  {\delta T \over \delta \upsilon'} = - z',
   \eqno {\rm (L')}$$
by the three first and three last of which we can eliminate
$\sigma$~$\tau$~$\upsilon$ $\sigma'$~$\tau'$~$\upsilon'$
from (F${}'$), and so deduce $V$ from $T$.  And in the same case,
or even in the case when only the initial medium is variable, the
three last of the equations (L${}'$) are true, and suffice to
eliminate $\sigma'$,~$\tau'$,~$\upsilon'$, from (E${}'$), and so
to deduce $W$ from $T$.

But if the final medium be uniform, the initial being still
variable, then $\sigma$,~$\tau$,~$\upsilon$,~$\chi$, are
connected by the relation (K${}'$), while
$\sigma'$~$\tau'$~$\upsilon'$ remain independent; and instead
of the four first equations (L${}'$) we have the three following,
$$  {\displaystyle {\delta T \over \delta \sigma}   - x
         \over \displaystyle {\delta \Omega \over \delta \sigma}}
  = {\displaystyle {\delta T \over \delta \tau}     - y
         \over \displaystyle {\delta \Omega \over \delta \tau}}
  = {\displaystyle {\delta T \over \delta \upsilon} - z
         \over \displaystyle {\delta \Omega \over \delta \upsilon}}
  = {\displaystyle {\delta T \over \delta \chi}
            +  {\delta V \over \delta \chi}
         \over \displaystyle {\delta \Omega \over \delta \chi}};
   \eqno {\rm (M')}$$
by the two first of which, combined with the relation (K${}'$),
and with the three last equations (L${}'$), we can eliminate
$\sigma$,~$\tau$,~$\upsilon$, $\sigma'$,~$\tau'$,~$\upsilon'$,
from (F${}'$), and so deduce $V$ from $T$.

If both the extreme media be uniform, we have then not only the
relation (K${}'$) for the final medium, but also an analogous
relation
$$0 = \Omega' (\sigma', \tau', \upsilon', \chi)
   \eqno {\rm (N')}$$
for the initial; and instead of the seven equations (L${}'$), we
have the two first of the equations (M${}'$), and the two
following,
$$  {\displaystyle {\delta T \over \delta \sigma'}   + x'
         \over \displaystyle {\delta \Omega \over \delta \sigma'}}
  = {\displaystyle {\delta T \over \delta \tau'}     + y'
         \over \displaystyle {\delta \Omega \over \delta \tau'}}
  = {\displaystyle {\delta T \over \delta \upsilon'} + z'
         \over \displaystyle {\delta \Omega \over \delta \upsilon'}},
   \eqno {\rm (O')}$$
together with this equation,
$${\delta T \over \delta \chi}
      + {\delta V \over \delta \chi}
  = \lambda {\delta \Omega \over \delta \chi}
      + \lambda' {\delta \Omega' \over \delta \chi},
   \eqno {\rm (P')}$$
in which $\lambda$ is the common value of the three first equated
quantities in (M${}'$), and $\lambda'$ is the common value of the
three equated quantities in (O${}'$).  And in this case, by means
of the two equations (O${}'$), and the two that remain of
(M${}'$), combined with the two relations (K${}'$) (N${}'$), we
can eliminate
$\sigma$,~$\tau$,~$\upsilon$, $\sigma'$,~$\tau'$,~$\upsilon'$,
from (F${}'$), and so deduce $V$ from $T$: while, in the same
case, or even if the initial medium alone be uniform, we are to
deduce $W$ from $T$, by eliminating
$\sigma'$,~$\tau'$,~$\upsilon'$, between the equations
(E${}'$) (N${}'$) (O${}'$).

When all the media of the combination are not only uniform, but
bounded by plane surfaces, which happens in investigations
respecting prisms, ordinary or extraordinary, then of the seven
quantities
$\sigma$,~$\tau$,~$\upsilon$,
$\sigma'$,~$\tau'$,~$\upsilon'$,~$\chi$,
only three are independent; two other relations existing besides
(K${}'$) and (N${}'$), which may be thus denoted,
$$\left. \eqalign{
0 &= \Omega''  (\sigma, \tau, \upsilon,
                \sigma', \tau', \upsilon', \chi),\cr
0 &= \Omega''' (\sigma, \tau, \upsilon,
                \sigma', \tau', \upsilon', \chi);\cr}
   \right\}
   \eqno {\rm (Q')}$$
because, in this case, the initial direction, and the colour,
determine the final direction.  In this case, we may still treat
the variations of
$\sigma$,~$\tau$,~$\upsilon$,
$\sigma'$,~$\tau'$,~$\upsilon'$,~$\chi$,
as independent, in $\delta T$, by introducing the variations
of the four conditions (K${}'$) (N${}'$) (Q${}'$), multiplied by
factors
$\lambda$, $\lambda'$, $\lambda''$, $\lambda'''$, that is by
putting
$$\delta T
  =   x \, \delta \sigma   - x' \, \delta \sigma'
    + y \, \delta \tau     - y' \, \delta \tau'
    + z \, \delta \upsilon - z' \, \delta \upsilon'
    - {\delta V \over \delta \chi} \, \delta \chi
    + \lambda    \, \delta \Omega
    + \lambda'   \, \delta \Omega'
    + \lambda''  \, \delta \Omega''
    + \lambda''' \, \delta \Omega''':
   \eqno {\rm (R')}$$
an equation which decomposes itself into the seven following,
$$\left. \eqalign{
{\delta T \over \delta \sigma}   - x
   &=    \lambda    {\delta \Omega    \over \delta \sigma}
       + \lambda''  {\delta \Omega''  \over \delta \sigma}
       + \lambda''' {\delta \Omega''' \over \delta \sigma},\cr
{\delta T \over \delta \tau}     - y
   &=    \lambda    {\delta \Omega    \over \delta \tau}
       + \lambda''  {\delta \Omega''  \over \delta \tau}
       + \lambda''' {\delta \Omega''' \over \delta \tau},\cr
{\delta T \over \delta \upsilon} - z
   &=    \lambda    {\delta \Omega    \over \delta \upsilon}
       + \lambda''  {\delta \Omega''  \over \delta \upsilon}
       + \lambda''' {\delta \Omega''' \over \delta \upsilon},\cr
{\delta T \over \delta \sigma'}   + x'
   &=    \lambda'   {\delta \Omega'   \over \delta \sigma'}
       + \lambda''  {\delta \Omega''  \over \delta \sigma'}
       + \lambda''' {\delta \Omega''' \over \delta \sigma'},\cr
{\delta T \over \delta \tau'}     + y'
   &=    \lambda'   {\delta \Omega'   \over \delta \tau'}
       + \lambda''  {\delta \Omega''  \over \delta \tau'}
       + \lambda''' {\delta \Omega''' \over \delta \tau'},\cr
{\delta T \over \delta \upsilon'} + z'
   &=    \lambda'   {\delta \Omega'   \over \delta \upsilon'}
       + \lambda''  {\delta \Omega''  \over \delta \upsilon'}
       + \lambda''' {\delta \Omega''' \over \delta \upsilon'},\cr
{\delta T \over \delta \chi} + {\delta V \over \delta \chi}
   &=    \lambda    {\delta \Omega    \over \delta \chi}
       + \lambda'   {\delta \Omega'   \over \delta \chi}
       + \lambda''  {\delta \Omega''  \over \delta \chi}
       + \lambda''' {\delta \Omega''' \over \delta \chi},\cr}
   \right\}
   \eqno {\rm (S')}$$
between the six first of which, and the five equations marked
(F${}'$) (K${}'$) (N${}'$) (Q${}'$), we can eliminate the ten
quantities
$\sigma$,~$\tau$,~$\upsilon$, $\sigma'$,~$\tau'$,~$\upsilon'$,
$\lambda$, $\lambda'$, $\lambda''$, $\lambda'''$, and thus deduce
the relation between
$V$, $x$,~$y$,~$z$, $x'$,~$y'$,~$z'$, $\chi$,
from that between
$T$, $\sigma$,~$\tau$,~$\upsilon$,
$\sigma'$,~$\tau'$,~$\upsilon'$, $\chi$.
It is easy to extend this method to other cases, in which there
exists a mutual dependence, expressed by any number of equations,
between the seven quantities
$\sigma$,~$\tau$,~$\upsilon$,
$\sigma'$,~$\tau'$,~$\upsilon'$,~$\chi$.

And all the foregoing details respecting the mutual deductions of
the function $V$, $W$, $T$, may be summed up in this one rule or
theorem: that each of these three functions may be deduced from
either of the other two, by using one of the three equations
(D${}'$) (E${}'$) (F${}'$) and by making the sought function a
maximum or minimum with respect to the variables that are to be
eliminated.  For example we may deduce $T$ from $V$, by making
the expression (F${}'$) a maximum or minimum with respect to the
initial and final co-ordinates.

An optical combination is more perfectly characterised by the
original function~$V$, than by either of the two connected
auxiliary functions $W$, $T$; because $V$ enables us to determine
the properties of the extreme media, which $W$ and $T$ do not;
but there is an advantage in using these latter functions when
the extreme media are uniform and known, because the known
relations which in this case exist, of the forms (K${}'$) and
(N${}'$), (together with the other relations (Q${}'$) which arise
when the combination is prismatic,) leave fewer independent
variables in the auxiliary than in the original function.  At the
same time, as has been already remarked, and will be afterwards
more fully shown, the existence of relations between the
variables produces a partial indeterminateness in the forms of
the auxiliary functions, from which the characteristic function
$V$ is free, but which is rather advantageous than the contrary,
because it permits us to introduce suppositions and
transformations, that contribute to elegance or simplicity.

\bigbreak

{\sectiontitle
General Transformations, by the Auxiliary Functions $W$, $T$, of
the Partial Differential Equations in $V$.  Other Partial
Differential Equations in $V$, for Extreme Uniform Media.
Integration of these Equations, by the Functions $W$, $T$.\par}

\nobreak\bigskip

5.
Another advantage of the auxiliary functions $W$, $T$, is that
they serve to transform, and in the case of extreme uniform media
to integrate, the partial differential equations (C), which the
characteristic function $V$ must satisfy.  In fact, if the final
medium be variable, the first of the two partial differential
equations (C) may be put by the foregoing number under the two
following forms,
$$\left. \eqalign{
0 &= \Omega \left( \sigma, \tau, \upsilon,
            {\delta W \over \delta \sigma},
            {\delta W \over \delta \tau},
            {\delta W \over \delta \upsilon},
            \chi \right),\cr
0 &= \Omega \left( \sigma, \tau, \upsilon,
            {\delta T \over \delta \sigma},
            {\delta T \over \delta \tau},
            {\delta T \over \delta \upsilon},
            \chi \right);\cr}
   \right\}
   \eqno {\rm (T')}$$
and if the initial medium be variable, the second of the two
partial differential equations (C) may be put under these two
forms,
$$\left. \eqalign{
0 &= \Omega' \left(
            {\delta W \over \delta x'},
            {\delta W \over \delta y'},
            {\delta W \over \delta z'},
            x', y', z', \chi \right),\cr
0 &= \Omega' \left( \sigma', \tau', \upsilon',
            - {\delta T \over \delta \sigma'},
            - {\delta T \over \delta \tau'},
            - {\delta T \over \delta \upsilon'},
            \chi \right);\cr}
   \right\}
   \eqno {\rm (U')}$$
of which indeed the first is general.  But if the final medium be
uniform, then $W$ remains an arbitrary function of the four
variables
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$,
which are in this case connected with each other by the relation
(K${}'$); and the two equations (D${}'$) (K${}'$), together with
the two first of those marked (I${}'$), compose a system, which
is a form for the integral of the partial differential equation
$$0 = \Omega \left(
            {\delta V \over \delta x},
            {\delta V \over \delta y},
            {\delta V \over \delta z},
            \chi \right),
   \eqno {\rm (V')}$$
to which the first equation (C) in this case reduces itself.  In
like manner, if both the extreme media be uniform, in which case
the second equation (C) reduces itself to the form
$$0 = \Omega' \left(
            - {\delta V \over \delta x'},
            - {\delta V \over \delta y'},
            - {\delta V \over \delta z'},
            \chi \right),
   \eqno {\rm (W')}$$
the system of partial differential equations (V${}'$) (W${}'$)
has for integral the system composed of the equations
(F${}'$) (K${}'$) (N${}'$) (O${}'$), and the two first equations
(M${}'$), in which $T$ is considered an arbitrary function of
$\sigma$,~$\tau$,~$\upsilon$,
$\sigma'$,~$\tau'$,~$\upsilon'$,~$\chi$.
It will be found that these integrals are extensively useful, in
the study of optical combinations.

The two partial differential equations, (V${}'$) (W${}'$), of the
first order, are themselves integrals of the two following, of
the second order,
$$\eqalignno{
         {\delta^2 V \over \delta x^2}
         {\delta^2 V \over \delta y^2}
         {\delta^2 V \over \delta z^2}
     + 2 {\delta^2 V \over \delta x \, \delta y}
         {\delta^2 V \over \delta y \, \delta z}
         {\delta^2 V \over \delta z \, \delta x}
      \hskip -72pt \cr
   &=    {\delta^2 V \over \delta x^2} \left(
         {\delta^2 V \over \delta y \, \delta z}
         \right)^2
       + {\delta^2 V \over \delta y^2} \left(
         {\delta^2 V \over \delta z \, \delta x}
         \right)^2
       + {\delta^2 V \over \delta z^2} \left(
         {\delta^2 V \over \delta x \, \delta y}
         \right)^2,
   &{\rm (X')}\cr}$$
and
$$\eqalignno{
         {\delta^2 V \over \delta x'^2}
         {\delta^2 V \over \delta y'^2}
         {\delta^2 V \over \delta z'^2}
     + 2 {\delta^2 V \over \delta x' \, \delta y'}
         {\delta^2 V \over \delta y' \, \delta z'}
         {\delta^2 V \over \delta z' \, \delta x'}
      \hskip -72pt \cr
   &=    {\delta^2 V \over \delta x'^2} \left(
         {\delta^2 V \over \delta y' \,\delta z'}
         \right)^2
       + {\delta^2 V \over \delta y'^2} \left(
         {\delta^2 V \over \delta z' \,\delta x'}
         \right)^2
       + {\delta^2 V \over \delta z'^2} \left(
         {\delta^2 V \over \delta x' \,\delta y'}
         \right)^2,
   &{\rm (Y')}\cr}$$
which are obtained by elimination from (Q) and (X), after making
$$\left. \multieqalign{
{\delta v \over \delta x} &= 0, &
{\delta v \over \delta y} &= 0, &
{\delta v \over \delta z} &= 0; \cr
{\delta v' \over \delta x'} &= 0, &
{\delta v' \over \delta y'} &= 0, &
{\delta v' \over \delta z'} &= 0.\cr}
   \right\}
   \eqno {\rm (Z')}$$
The system of the three first of these six equations (Z${}'$), or
the partial differential equation of the second order (X${}'$),
or its integral of the first order (V${}'$), expresses that the
final medium is uniform; and the uniformity of the initial medium
is, in like manner, expressed by the three last equations
(Z${}'$), or by the partial differential equation (Y${}'$), or by
its integral of the first order (W${}'$).  The integral systems
of equations, also, which we have already assigned, express
properties peculiar to optical combinations that have one or
both of the extreme media uniform.

The first equation (U${}'$) has for transformation the second
equation (U${}'$), when the initial medium is variable; and it
has for integral, when the initial medium is uniform, the system
(E${}'$) (N${}'$) (O${}'$), by which, in that case, $W$ is
deduced from the arbitrary function $T$: while, in the same case,
of an initial uniform medium, the first equation (U${}'$) becomes
of the form
$$0 = \Omega' \left(
            {\delta W \over \delta x'},
            {\delta W \over \delta y'},
            {\delta W \over \delta z'},
            \chi \right),
   \eqno {\rm (A^2)}$$
and is an integral of the following equation of the second order,
analogous to (Y${}'$),
$$\eqalignno{
         {\delta^2 W \over \delta x'^2}
         {\delta^2 W \over \delta y'^2}
         {\delta^2 W \over \delta z'^2}
     + 2 {\delta^2 W \over \delta x' \, \delta y'}
         {\delta^2 W \over \delta y' \, \delta z'}
         {\delta^2 W \over \delta z' \, \delta x'}
      \hskip -108pt \cr
   &=    {\delta^2 W \over \delta x'^2} \left(
         {\delta^2 W \over \delta y' \,\delta z'}
         \right)^2
       + {\delta^2 W \over \delta y'^2} \left(
         {\delta^2 W \over \delta z' \,\delta x'}
         \right)^2
       + {\delta^2 W \over \delta z'^2} \left(
         {\delta^2 W \over \delta x' \,\delta y'}
         \right)^2.
   &{\rm (B^2)}\cr}$$

When the final medium is variable, the function $W$ satisfies the
following partial differential equation, analogous to the general
equation (D),
$$\eqalignno{
         {\delta^2 W \over \delta \sigma   \, \delta x'}
         {\delta^2 W \over \delta \tau     \, \delta y'}
         {\delta^2 W \over \delta \upsilon \, \delta z'}
       + {\delta^2 W \over \delta \sigma   \, \delta y'}
         {\delta^2 W \over \delta \tau     \, \delta z'}
         {\delta^2 W \over \delta \upsilon \, \delta x'}
       + {\delta^2 W \over \delta \sigma   \, \delta z'}
         {\delta^2 W \over \delta \tau     \, \delta x'}
         {\delta^2 W \over \delta \upsilon \, \delta y'}
      \hskip -216pt \cr
   &=    {\delta^2 W \over \delta \upsilon \, \delta x'}
         {\delta^2 W \over \delta \tau     \, \delta y'}
         {\delta^2 W \over \delta \sigma   \, \delta z'}
       + {\delta^2 W \over \delta \upsilon \, \delta y'}
         {\delta^2 W \over \delta \tau     \, \delta z'}
         {\delta^2 W \over \delta \sigma   \, \delta x'}
       + {\delta^2 W \over \delta \upsilon \, \delta z'}
         {\delta^2 W \over \delta \tau     \, \delta x'}
         {\delta^2 W \over \delta \sigma   \, \delta y'};
   &{\rm (C^2)}\cr}$$
and when both the extreme media are variable, the function $T$
satisfies the following analogous equation,
$$\eqalignno{
         {\delta^2 T \over \delta \sigma   \, \delta \sigma'}
         {\delta^2 T \over \delta \tau     \, \delta \tau'}
         {\delta^2 T \over \delta \upsilon \, \delta \upsilon'}
       + {\delta^2 T \over \delta \sigma   \, \delta \tau'}
         {\delta^2 T \over \delta \tau     \, \delta \upsilon'}
         {\delta^2 T \over \delta \upsilon \, \delta \sigma'}
       + {\delta^2 T \over \delta \sigma   \, \delta \upsilon'}
         {\delta^2 T \over \delta \tau     \, \delta \sigma'}
         {\delta^2 T \over \delta \upsilon \, \delta \tau'}
      \hskip -252pt \cr
   &=    {\delta^2 T \over \delta \upsilon \, \delta \sigma'}
         {\delta^2 T \over \delta \tau     \, \delta \tau'}
         {\delta^2 T \over \delta \sigma   \, \delta \upsilon'}
       + {\delta^2 T \over \delta \upsilon \, \delta \tau'}
         {\delta^2 T \over \delta \tau     \, \delta \upsilon'}
         {\delta^2 T \over \delta \sigma   \, \delta \sigma'}
       + {\delta^2 T \over \delta \upsilon \, \delta \upsilon'}
         {\delta^2 T \over \delta \tau     \, \delta \sigma'}
         {\delta^2 T \over \delta \sigma   \, \delta \tau'}.
   &{\rm (D^2)}\cr}$$

\bigbreak

{\sectiontitle
General Deductions and Transformations of the Differential and
Integral Equations of a Curved or Straight Ray, Ordinary or
Extraordinary, by the Auxiliary Functions $W$, $T$.\par}

\nobreak\bigskip

6.
The auxiliary functions $W$, $T$, give new equations for the
initial and final portions of a curved or polygon ray.  Thus the
function $W$ gives generally the following equations, between the
final quantities $\sigma$,~$\tau$,~$\upsilon$, analogous to the
equations (W),
$${\delta W \over \delta x'} = \hbox{const.},\quad
  {\delta W \over \delta y'} = \hbox{const.},\quad
  {\delta W \over \delta z'} = \hbox{const.},
   \eqno {\rm (E^2)}$$
in which $x'$~$y'$~$z'$ are the co-ordinates of some fixed
point on the initial portion, and the constants are, by (G${}'$),
the corresponding values of the initial quantities
$\sigma'$,~$\tau'$,~$\upsilon'$.
The equations (E${}^2$) have for differentials the following,
$$\left. \eqalign{
0  &=    {\delta^2 W \over \delta \sigma   \, \delta x'} \, d \sigma
       + {\delta^2 W \over \delta \tau     \, \delta x'} \, d \tau   
       + {\delta^2 W \over \delta \upsilon \, \delta x'} \, d \upsilon,\cr
0  &=    {\delta^2 W \over \delta \sigma   \, \delta y'} \, d \sigma
       + {\delta^2 W \over \delta \tau     \, \delta y'} \, d \tau   
       + {\delta^2 W \over \delta \upsilon \, \delta y'} \, d \upsilon,\cr
0  &=    {\delta^2 W \over \delta \sigma   \, \delta z'} \, d \sigma
       + {\delta^2 W \over \delta \tau     \, \delta z'} \, d \tau   
       + {\delta^2 W \over \delta \upsilon \, \delta z'} \, d \upsilon;\cr}
   \right\}
   \eqno {\rm (F^2)}$$
$d$ still referring to motion along a ray: and if we combine these
with the following,
$$\left. \eqalign{
0  &=    {\delta v \over \delta x}
            {\delta^2 W \over \delta \sigma   \, \delta x'}
       + {\delta v \over \delta y}
            {\delta^2 W \over \delta \tau     \, \delta x'}
       + {\delta v \over \delta z}
            {\delta^2 W \over \delta \upsilon \, \delta x'},\cr
0  &=    {\delta v \over \delta x}
            {\delta^2 W \over \delta \sigma   \, \delta y'}
       + {\delta v \over \delta y}
            {\delta^2 W \over \delta \tau     \, \delta y'}
       + {\delta v \over \delta z}
            {\delta^2 W \over \delta \upsilon \, \delta y'},\cr
0  &=    {\delta v \over \delta x}
            {\delta^2 W \over \delta \sigma   \, \delta z'}
       + {\delta v \over \delta y}
            {\delta^2 W \over \delta \tau     \, \delta z'}
       + {\delta v \over \delta z}
            {\delta^2 W \over \delta \upsilon \, \delta z'},\cr}
   \right\}
   \eqno {\rm (G^2)}$$
which are obtained by differentiating the first equation (T${}'$)
relatively to the initial co-ordinates $x'$~$y'$~$z'$, and by
attending to the relations (K), we see that for a curved ray the
differentials
$d \sigma$, $d \tau$, $d \upsilon$,
are proportional to
$${\delta v \over \delta x},\quad
  {\delta v \over \delta y},\quad
  {\delta v \over \delta z};$$
and from this proportionality, combined with the relation
$$\alpha \, \delta \sigma
      + \beta  \, \delta \tau
      + \gamma \, \delta \upsilon
   =  \left(
         \alpha {\delta v \over \delta x}
       + \beta  {\delta v \over \delta y}
       + \gamma {\delta v \over \delta z}
      \right) \, ds,
   \eqno {\rm (H^2)}$$
which results from (H) and (P), we can easily infer the equations
(O): these differential equations (O) for the final portion of a
curved ray, which can be extended to the initial portion by
merely accenting the symbols, may therefore be deduced from the
consideration of the auxiliary function $W$.  The equations (O)
for a curved ray, may also be deduced from the function $W$, by
combining the differentials $d$ of the three first equations
(H${}'$), with the partial differentials of the first equation
(T${}'$), taken with respect to $\sigma$,~$\tau$,~$\upsilon$.

The same auxiliary function $W$ gives for the final straight
portion of a polygon ray, the two first equations (I${}'$), which
may be thus written,
$${1 \over \alpha} \left(
      x - {\delta W \over \delta \sigma} \right)
    = {1 \over \beta} \left(
      y - {\delta W \over \delta \tau} \right)
    = {1 \over \gamma} \left(
      z - {\delta W \over \delta \upsilon} \right):
   \eqno {\rm (I^2)}$$
these equations may also be put under the form
$$\left. \eqalign{
      x {\delta \sigma   \over \delta \theta}
    + y {\delta \tau     \over \delta \theta}
    + z {\delta \upsilon \over \delta \theta}
   &= {\delta W \over \delta \theta},\cr
      x {\delta \sigma   \over \delta \phi}
    + y {\delta \tau     \over \delta \phi}
    + z {\delta \upsilon \over \delta \phi}
   &= {\delta W \over \delta \phi},\cr}
   \right\}
   \eqno {\rm (K^2)}$$
if in virtue of (K${}'$), we consider
$\sigma$,~$\tau$,~$\upsilon$,
as functions, each, of $\chi$, and of two other independent
variables denoted by $\theta$,~$\phi$, and consider $W$ as a
function of the six independent variables
$\theta$,~$\phi$, $\chi$, $x'$,~$y'$,~$z'$.
We may choose $\sigma$,~$\tau$, for the independent variables
$\theta$,~$\phi$, considering $\upsilon$ as, by (K${}'$), a
function of $\sigma$,~$\tau$, $\chi$, such that by (H),
$${\delta \upsilon \over \delta \sigma}
   = - {\alpha \over \gamma},\quad
  {\delta \upsilon \over \delta \tau}
   = - {\beta  \over \gamma},\quad
  {\delta \upsilon \over \delta \chi}
   = {1 \over \gamma} {\delta v \over \delta \chi},
   \eqno {\rm (L^2)}$$
and considering $W$ as a function of the six independent
variables
$\sigma$,~$\tau$, $\chi$, $x'$,~$y'$,~$z'$;
and then the equations (I${}^2$) or (K${}^2$), for the final
straight portion of a polygon ray, ordinary or extraordinary,
will take these simpler forms, which we shall have frequent
occasion to employ,
$$x - {\alpha \over \gamma} z
    = {\delta W \over \delta \sigma};\quad
  y - {\beta  \over \gamma} z
    = {\delta W \over \delta \tau}.
   \eqno {\rm (M^2)}$$

The other auxiliary function, $T$, gives the following equations
between $\sigma$,~$\tau$,~$\upsilon$, for the final portion,
straight or curved, when the initial medium is variable,
$${\delta T \over \delta \sigma'}   = \hbox{const.},\quad
  {\delta T \over \delta \tau'}     = \hbox{const.},\quad
  {\delta T \over \delta \upsilon'} = \hbox{const.},
   \eqno {\rm (N^2)}$$
in which $\sigma'$,~$\tau'$,~$\upsilon'$, belong to some point on
the initial portion, and in which the constants are, by (L${}'$),
the negatives of the co-ordinates of that point; it gives, in
like manner, for the initial portion, when the final medium is
variable, the following equations between
$\sigma'$,~$\tau'$,~$\upsilon'$,
$${\delta T \over \delta \sigma}   = \hbox{const.},\quad
  {\delta T \over \delta \tau}     = \hbox{const.},\quad
  {\delta T \over \delta \upsilon} = \hbox{const.},
   \eqno {\rm (O^2)}$$
$\sigma$,~$\tau$,~$\upsilon$, belonging to some point upon the
final portion, and the constants being the co-ordinates of that
point: and from these equations we might deduce the differential
equations (O), by processes analogous to those already mentioned.
When both the extreme media are uniform, and therefore both the
extreme portions straight, we have, for those straight portions,
the following equations, deduced from (M${}'$) (O${}'$) (I),
$$\left. \eqalign{
      {1 \over \alpha} \left(
      x - {\delta T \over \delta \sigma} \right)
   &= {1 \over \beta} \left(
      y - {\delta T \over \delta \tau} \right)
    = {1 \over \gamma} \left(
      z - {\delta T \over \delta \upsilon} \right),\cr
      {1 \over \alpha'} \left(
      x' + {\delta T \over \delta \sigma'} \right)
   &= {1 \over \beta'} \left(
      y' + {\delta T \over \delta \tau'} \right)
    = {1 \over \gamma'} \left(
      z' + {\delta T \over \delta \upsilon'} \right);\cr}
   \right\}
   \eqno {\rm (P^2)}$$
which may be thus transformed,
$$\left. \eqalign{
0   &=   x  {\delta \sigma    \over \delta \theta}
       + y  {\delta \tau      \over \delta \theta}
       + z  {\delta \upsilon  \over \delta \theta}
       -    {\delta T         \over \delta \theta},\cr
0   &=   x  {\delta \sigma    \over \delta \phi}
       + y  {\delta \tau      \over \delta \phi}
       + z  {\delta \upsilon  \over \delta \phi}
       -    {\delta T         \over \delta \phi},\cr
0   &=   x' {\delta \sigma'   \over \delta \theta'}
       + y' {\delta \tau'     \over \delta \theta'}
       + z' {\delta \upsilon' \over \delta \theta'}
       +    {\delta T         \over \delta \theta'},\cr
0   &=   x' {\delta \sigma'   \over \delta \phi'}
       + y' {\delta \tau'     \over \delta \phi'}
       + z' {\delta \upsilon' \over \delta \phi'}
       +    {\delta T         \over \delta \phi'},\cr}
   \right\}
   \eqno {\rm (Q^2)}$$
if, as before, by virtue of (K${}'$), we consider
$\sigma$,~$\tau$,~$\upsilon$,
as functions, each, of $\chi$ and of two other independent
variables $\theta$,~$\phi$, considering similarly
$\sigma'$,~$\tau'$,~$\upsilon'$,
as functions, each, by (N${}'$), of three independent variables
$\theta'$,~$\phi'$,~$\chi$;
and $T$ as a function of the five independent variables
$\theta$,~$\phi$, $\theta'$,~$\phi'$,~$\chi$.
If we choose the independent variables $\theta$,~$\phi$, so as to
coincide with $\sigma$,~$\tau$, and if in like manner we take
$\sigma'$,~$\tau'$, for the independent variables
$\theta'$,~$\phi'$, making, by (H),
$${\delta \upsilon' \over \delta \sigma'}
   = - {\alpha' \over \gamma'},\quad
  {\delta \upsilon' \over \delta \tau'}
   = - {\beta'  \over \gamma'},\quad
  {\delta \upsilon' \over \delta \chi}
   = {1 \over \gamma'} {\delta v' \over \delta \chi},
   \eqno {\rm (R^2)}$$
and considering $T$ as a function of the five independent
variables
$\sigma$,~$\tau$, $\sigma'$,~$\tau'$,~$\chi$,
we have the following transformed equations for the extreme
straight portions of a polygon ray, ordinary or extraordinary,
$$\left. \multieqalign{
0  &= x  - {\alpha \over \gamma} z
         - {\delta T \over \delta \sigma}; &
0  &= y  - {\beta  \over \gamma} z
         - {\delta T \over \delta \tau};\cr
0  &= x' - {\alpha' \over \gamma'} z'
         + {\delta T \over \delta \sigma'}; &
0  &= y' - {\beta'  \over \gamma'} z'
         + {\delta T \over \delta \tau'}:\cr}
   \right\}
   \eqno {\rm (S^2)}$$
which are analogous to the equations (M${}^2$) and, like them,
will often be found useful.

It may be remarked here, that from the differential equations (O)
of a curved ray, ordinary or extraordinary, to which, in the
present and former numbers, we have been conducted by so many
processes, the following may be deduced,
$$\left. \eqalign{
0  &= {d\sigma   \over dV}
      + {\delta \Omega \over \delta x};\quad
0   = {d\tau     \over dV}
      + {\delta \Omega \over \delta y};\quad
0   = {d\upsilon \over dV}
      + {\delta \Omega \over \delta z}:\cr
dW &= dT = \left(
               x {\delta v \over \delta x}
             + y {\delta v \over \delta y}
             + z {\delta v \over \delta z}
            \right) \, ds
    = x \, d\sigma + y \, d\tau + z \, d\upsilon.\cr}
   \right\}
   \eqno {\rm (T^2)}$$

We may also remark, that when the final medium is uniform, and
when therefore the quantities
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$, are connected by a relation
(K${}'$), the quantity
$$W (\sigma^2 + \tau^2 + \upsilon^2)^{-{n \over 2}}$$
may, in general, by means of this relation, be expressed as a
function of
$${\sigma \over \upsilon}, {\tau \over \upsilon},
  x', y', z', \chi,$$
and that
$\displaystyle T (\sigma^2 + \tau^2 + \upsilon^2)^{-{n \over 2}}$
may, in like manner, be expressed as a function of
$${\sigma \over \upsilon}, {\tau \over \upsilon},
  \sigma', \tau', \upsilon', \chi;$$
and that therefore $W$, $T$, may both be made homogeneous
functions, of any assumed dimension~$n$, relatively to
$\sigma$,~$\tau$,~$\upsilon$,
so as to satisfy the following conditions
$$\left. \eqalign{
         \sigma   {\delta W \over \delta \sigma}
       + \tau     {\delta W \over \delta \tau}
       + \upsilon {\delta W \over \delta \upsilon}
   &= nW,\cr
         \sigma   {\delta T \over \delta \sigma}
       + \tau     {\delta T \over \delta \tau}
       + \upsilon {\delta T \over \delta \upsilon}
   &= nT.\cr}
   \right\}
   \eqno {\rm (U^2)}$$
With this preparation, the two first equations (I${}'$), and the
two first equations (M${}'$), which belong to the straight final
portion of the ray, may be transformed by (L) to the following,
$$\left. \eqalign{
x   - {\delta \Omega \over \delta \sigma}
      (\sigma x + \tau y + \upsilon z)
   &= {\delta W \over \delta \sigma}
      - n W {\delta \Omega \over \delta \sigma}
    = {\delta T \over \delta \sigma}
      - n T {\delta \Omega \over \delta \sigma},\cr
y   - {\delta \Omega \over \delta \tau}
      (\sigma x + \tau y + \upsilon z)
   &= {\delta W \over \delta \tau}
      - n W {\delta \Omega \over \delta \tau}
    = {\delta T \over \delta \tau}
      - n T {\delta \Omega \over \delta \tau},\cr
z   - {\delta \Omega \over \delta \upsilon}
      (\sigma x + \tau y + \upsilon z)
   &= {\delta W \over \delta \upsilon}
      - n W {\delta \Omega \over \delta \upsilon}
    = {\delta T \over \delta \upsilon}
      - n T {\delta \Omega \over \delta \upsilon}.\cr}
   \right\}
   \eqno {\rm (V^2)}$$
If then we make $n = 1$, that is if we make $W$ homogeneous of
the first dimension relatively to $\sigma$,~$\tau$,~$\upsilon$,
and if we attend to the relation (D${}'$), we see that the
equations of this straight final portion may be thus written,
$$x =    {\delta W \over \delta \sigma}
       + V {\delta \Omega \over \delta \sigma},\quad
  y =    {\delta W \over \delta \tau}
       + V {\delta \Omega \over \delta \tau},\quad
  z =    {\delta W \over \delta \upsilon}
       + V {\delta \Omega \over \delta \upsilon},
   \eqno {\rm (W^2)}$$
of which any two include the third, and which we shall often
hereafter employ, on account of their symmetry.

In like manner, when the initial medium is uniform, and therefore
the initial portion straight, the equations (O${}'$) of this
straight portion may be put under the form,
$$\left. \eqalign{
x'  - {\delta \Omega' \over \delta \sigma'}
      (\sigma' x' + \tau' y' + \upsilon' z')
   &= - {\delta T \over \delta \sigma'}
      + n' T {\delta \Omega \over \delta \sigma'},\cr
y'  - {\delta \Omega' \over \delta \tau'}
      (\sigma' x' + \tau' y' + \upsilon' z')
   &= - {\delta T \over \delta \tau'}
      + n' T {\delta \Omega \over \delta \tau'},\cr
z'  - {\delta \Omega' \over \delta \upsilon'}
      (\sigma' x' + \tau' y' + \upsilon' z')
   &= - {\delta T \over \delta \upsilon'}
      + n' T {\delta \Omega \over \delta \upsilon'},\cr}
   \right\}
   \eqno {\rm (X^2)}$$
by making $T$ homogeneous of dimension $n'$ relatively to
$\sigma'$,~$\tau'$,~$\upsilon'$, so as to have
$$\sigma'   {\delta T \over \delta \sigma'}
       + \tau'     {\delta T \over \delta \tau'}
       + \upsilon' {\delta T \over \delta \upsilon'}
   = n' T.
   \eqno {\rm (Y^2)}$$

If both the extreme media be uniform, and if we make $n = 0$,
$n' = 0$, that is, if we express $W$ as a function of
$${\sigma \over \upsilon}, {\tau   \over \upsilon},
  x', y', z', \chi,$$
and $T$ as a function of
$${\sigma \over \upsilon}, {\tau \over \upsilon},
  {\sigma' \over \upsilon'}, {\tau' \over \upsilon'},
  \chi,$$
we find the following form for the equations of the extreme
straight portions of a polygon ray, ordinary or extraordinary,
less simple than (S${}^2$), but more symmetric,
$$\left. \matrix{
\eqalign{
x   - {\delta \Omega \over \delta \sigma}
      (\sigma x + \tau y + \upsilon z)
   &= {\delta W \over \delta \sigma}
    = {\delta T \over \delta \sigma},\cr
y   - {\delta \Omega \over \delta \tau}
      (\sigma x + \tau y + \upsilon z)
   &= {\delta W \over \delta \tau}
    = {\delta T \over \delta \tau},\cr
z   - {\delta \Omega \over \delta \upsilon}
      (\sigma x + \tau y + \upsilon z)
   &= {\delta W \over \delta \upsilon}
    = {\delta T \over \delta \upsilon},\cr} \cr
   \noalign{\vskip 3pt} \eqalign{
x'  - {\delta \Omega' \over \delta \sigma'}
      (\sigma' x' + \tau' y' + \upsilon' z')
   &= - {\delta T \over \delta \sigma'},\cr
y'  - {\delta \Omega' \over \delta \tau'}
      (\sigma' x' + \tau' y' + \upsilon' z')
   &= - {\delta T \over \delta \tau'},\cr
z'  - {\delta \Omega' \over \delta \upsilon'}
      (\sigma' x' + \tau' y' + \upsilon' z')
   &= - {\delta T \over \delta \upsilon'}.\cr} \cr}
   \right\}
   \eqno {\rm (Z^2)}$$

The case of prismatic combinations may be treated as in the
fourth number.

\bigbreak

{\sectiontitle
General Remarks on the Connexions between the Partial
Differential Coefficients of the Second Order of the Functions
$V$, $W$, $T$.  General Method of investigating those Connexions.
Deductions of the Coefficients of $V$ from those of $W$, when the
Final Medium is uniform.\par}

\nobreak\bigskip

7.
It is easy to see, from the manner in which the equations of
a ray involve the partial differential coefficients of the first
order, of the functions $V$, $W$, $T$, that the partial
differential coefficients of the second order, of the same three
functions, must present themselves in investigations respecting
the geometrical relations between infinitely near rays of a
system; and that therefore it must be useful to know the general
connexions between these coefficients of the second order.
Connexions of this kind, between the coefficients of the second
order of the characteristic function~$V$, taken with respect to
the final co-ordinates, and those of the auxiliary function~$W$,
considered as belonging to a final system of straight rays of a
given colour, which issued originally from a given luminous
point, were investigated in the First Supplement; but these
connexions will now be considered in a more general manner, and
will be extended to the new auxiliary function~$T$, which was not
introduced before: the new investigations will differ also from
the former, by making $W$ depend on the quantities
$\sigma$,~$\tau$,~$\upsilon$, rather than on
$\alpha$,~$\beta$,~$\gamma$.

The general problem of investigating these connexions may be
decomposed into many particular problems, according to the way in
which we pair the functions, and according as we suppose the
extreme media to be uniform or variable; but all these particular
problems may be resolved by attending to the following general
principle, that the connexions between the partial differential
coefficients of the three functions, whether of the second or of
higher orders, are to be obtained by differentiating and
comparing the equations which connect the three functions
themselves: that is, by differentiating and comparing the three
forms (A${}'$) (B${}'$) ($C{}'$) of the fundamental equation (A),
and the equations into which these forms
(A${}'$) (B${}'$) (C${}'$) resolve themselves.

Thus to deduce the twenty-eight partial differential coefficients
of the second order, of the characteristic function~$V$, taken
with respect to the extreme co-ordinates and the colour, from the
coefficients of the same order of the auxiliary function $W$, or
$T$, we are to differentiate the equations into which (B${}'$) or
($C{}'$) resolves itself, together with the relations between the
variables on which $W$ or $T$ depends, if any such relations
exist; and then by elimination to deduce the variations of the
first order of the seven coefficients of the variation (A${}'$)
as linear functions of the seven variations of the first order of
the extreme co-ordinates and the colour: these seven linear
functions will have forty-nine coefficients, of which, however,
only twenty-eight will be distinct, and these will be the
coefficients sought.

More particularly, if the final medium be variable, and if it be
required to deduce the coefficients of the second order of $V$
from those of $W$, we first obtain expressions for
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$, as linear
functions of
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$,
from the differentials of the three first equations (H${}'$),
deduced from (B${}'$), expressions which will necessarily satisfy
the first condition (H); we then substitute these expressions for
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$, in the
differentials of the three equations (G${}'$), deduced from
(B${}'$), so as to get analogous expressions for
$\delta \sigma'$, $\delta \tau'$, $\delta \upsilon'$,
which must satisfy the second condition (H); and substituting the
same expressions for
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$,
in the differential of the last equation (H${}'$), also deduced
from (B${}'$), we get an expression of the same kind for
$\displaystyle \delta {\delta V \over \delta \chi}$:
after which, we have only to compare the expressions so obtained,
with the following, that is, with the differentials of the
equations into which the formula (A${}'$) resolves itself,

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
\delta \sigma
   &=   {\delta^2 V \over \delta x^2}               \, \delta x
      + {\delta^2 V \over \delta x \, \delta y}     \, \delta y
      + {\delta^2 V \over \delta x \, \delta z}     \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta x \, \delta x'}    \, \delta x'
      + {\delta^2 V \over \delta x \, \delta y'}    \, \delta y'
      + {\delta^2 V \over \delta x \, \delta z'}    \, \delta z'
      + {\delta^2 V \over \delta x \, \delta \chi}  \, \delta \chi,\cr
\delta \tau
   &=   {\delta^2 V \over \delta x \, \delta y}     \, \delta x
      + {\delta^2 V \over \delta y^2}               \, \delta y
      + {\delta^2 V \over \delta y \, \delta z}     \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta y \, \delta x'}    \, \delta x'
      + {\delta^2 V \over \delta y \, \delta y'}    \, \delta y'
      + {\delta^2 V \over \delta y \, \delta z'}    \, \delta z'
      + {\delta^2 V \over \delta y \, \delta \chi}  \, \delta \chi,\cr
\delta \upsilon
   &=   {\delta^2 V \over \delta x \, \delta z}     \, \delta x
      + {\delta^2 V \over \delta y \, \delta z}     \, \delta y
      + {\delta^2 V \over \delta z^2}               \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta z \, \delta x'}    \, \delta x'
      + {\delta^2 V \over \delta z \, \delta y'}    \, \delta y'
      + {\delta^2 V \over \delta z \, \delta z'}    \, \delta z'
      + {\delta^2 V \over \delta z \, \delta \chi}  \, \delta \chi,\cr
- \delta \sigma'
   &=   {\delta^2 V \over \delta x \, \delta x'}    \, \delta x
      + {\delta^2 V \over \delta y \, \delta x'}    \, \delta y
      + {\delta^2 V \over \delta z \, \delta x'}    \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta x'^2}              \, \delta x'
      + {\delta^2 V \over \delta x' \, \delta y'}   \, \delta y'
      + {\delta^2 V \over \delta x' \, \delta z'}   \, \delta z'
      + {\delta^2 V \over \delta x' \, \delta \chi} \, \delta \chi,\cr
- \delta \tau'
   &=   {\delta^2 V \over \delta x \, \delta y'}    \, \delta x
      + {\delta^2 V \over \delta y \, \delta y'}    \, \delta y
      + {\delta^2 V \over \delta z \, \delta y'}    \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta x' \, \delta y'}   \, \delta x'
      + {\delta^2 V \over \delta y'^2}              \, \delta y'
      + {\delta^2 V \over \delta y' \, \delta z'}   \, \delta z'
      + {\delta^2 V \over \delta y' \, \delta \chi} \, \delta \chi,\cr
- \delta \upsilon'
   &=   {\delta^2 V \over \delta x \, \delta z'}    \, \delta x
      + {\delta^2 V \over \delta y \, \delta z'}    \, \delta y
      + {\delta^2 V \over \delta z \, \delta z'}    \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta x' \, \delta z'}   \, \delta x'
      + {\delta^2 V \over \delta y' \, \delta z'}   \, \delta y'
      + {\delta^2 V \over \delta z'^2}              \, \delta z'
      + {\delta^2 V \over \delta z' \, \delta \chi} \, \delta \chi,\cr
\delta {\delta V \over \delta \chi}
   &=   {\delta^2 V \over \delta x \, \delta \chi}  \, \delta x
      + {\delta^2 V \over \delta y \, \delta \chi}  \, \delta y
      + {\delta^2 V \over \delta z \, \delta \chi}  \, \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta x' \, \delta \chi} \, \delta x'
      + {\delta^2 V \over \delta y' \, \delta \chi} \, \delta y'
      + {\delta^2 V \over \delta z' \, \delta \chi} \, \delta z'
      + {\delta^2 V \over \delta \chi^2}            \, \delta \chi.\cr}
   \right\}
   \eqno {\rm (A^3)}$$

But if the final medium be uniform, then
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$, are not independent, but
related by (K${}'$); and the formula (B${}'$) resolves itself,
not into the seven equations (G${}'$) and (H${}'$) but into the
six equations (G${}'$) and (I${}'$), the differentials of which
are to be combined with the differential of the relation
(K${}'$), so as to give the expressions for
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$,
$\delta \sigma'$, $\delta \tau'$, $\delta \upsilon'$,
$\displaystyle \delta {\delta V \over \delta \chi}$,
which are to be compared with (A${}^3$) as before.  And in this
case, of a final uniform medium, we may employ, instead of the
two first equations (I${}'$), any of the transformations of those
equations in the foregoing number; or we may employ the following
transformations of (I${}'$),
$$x + z {\delta \upsilon \over \delta \sigma}
    = {\delta W \over \delta \sigma};\quad
  y + z {\delta \upsilon \over \delta \tau}
    = {\delta W \over \delta \tau};\quad
  z {\delta \upsilon \over \delta \chi}
    = {\delta V \over \delta \chi}
       + {\delta W \over \delta \chi}:
   \eqno {\rm (B^3)}$$
in which, $W$ is considered as a function of the six independent
variables $\sigma$,~$\tau$,~$\chi$, $x'$,~$y'$,~$z'$,
obtained by substituting for $\upsilon$ its value as a function
of $\sigma$,~$\tau$,~$\chi$; the form of which function
$\upsilon$ depends on and characterises the properties of the
final medium, and is deduced from the relation (K${}'$).  It may
be useful here to go through the process last indicated, both to
explain its nature more fully, and to have its results ready for
future researches.

Differentiating therefore the first two equations (B${}^3$), we
obtain
$$\left. \eqalign{
\delta x
      + {\delta \upsilon \over \delta \sigma} \, \delta z
      - \delta' {\delta W \over \delta \sigma}
      + z {\delta^2 \upsilon \over \delta \sigma \, \delta \chi} \, \delta \chi
   &=   \left(
         {\delta^2 W \over \delta \sigma^2}
         - z {\delta^2 \upsilon \over \delta \sigma^2}
         \right) \, \delta \sigma
      + \left(
         {\delta^2 W \over \delta \sigma \, \delta \tau}
         - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
         \right) \, \delta \tau,\cr
\delta y
      + {\delta \upsilon \over \delta \tau} \, \delta z
      - \delta' {\delta W \over \delta \tau}
      + z {\delta^2 \upsilon \over \delta \tau \, \delta \chi} \, \delta \chi
   &=   \left(
         {\delta^2 W \over \delta \sigma \, \delta \tau}
         - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
         \right) \, \delta \sigma
      + \left(
         {\delta^2 W \over \delta \tau^2}
         - z {\delta^2 \upsilon \over \delta \tau^2}
         \right) \, \delta \tau,\cr}
   \right\}
   \eqno {\rm (C^3)}$$
in which we have put for abridgment
$$\left. \eqalign{
\delta' {\delta W \over \delta \sigma}
   &=   {\delta^2 W \over \delta \sigma \, \delta x'}   \, \delta x'
      + {\delta^2 W \over \delta \sigma \, \delta y'}   \, \delta y'
      + {\delta^2 W \over \delta \sigma \, \delta z'}   \, \delta z'
      + {\delta^2 W \over \delta \sigma \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta W \over \delta \tau}
   &=   {\delta^2 W \over \delta \tau   \, \delta x'}   \, \delta x'
      + {\delta^2 W \over \delta \tau   \, \delta y'}   \, \delta y'
      + {\delta^2 W \over \delta \tau   \, \delta z'}   \, \delta z'
      + {\delta^2 W \over \delta \tau   \, \delta \chi} \, \delta \chi,\cr}
   \right\}
   \eqno {\rm (D^3)}$$
$\delta'$ referring only to the variations of the initial
co-ordinates and of the colour: and if we put
$$w''
   =     \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right)
         \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right)
       - \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)^2,
   \eqno {\rm (E^3)}$$
the equations (C${}^3$) give, by elimination,
$$\left. \eqalign{
w'' \, \delta \sigma
   &=    \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right)
         \left(
            \delta x
            + {\delta \upsilon \over \delta \sigma} \, \delta z
            - \delta' {\delta W \over \delta \sigma}
            + z {\delta^2 \upsilon
               \over \delta \sigma \, \delta \chi} \, \delta \chi
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
         \left(
            \delta y
            + {\delta \upsilon \over \delta \tau} \, \delta z
            - \delta' {\delta W \over \delta \tau}
            + z {\delta^2 \upsilon
               \over \delta \tau \, \delta \chi} \, \delta \chi
            \right),\cr
w'' \, \delta \tau
   &=    \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right)
         \left(
            \delta y
            + {\delta \upsilon \over \delta \tau} \, \delta z
            - \delta' {\delta W \over \delta \tau}
            + z {\delta^2 \upsilon
               \over \delta \tau \, \delta \chi} \, \delta \chi
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
         \left(
            \delta x
            + {\delta \upsilon \over \delta \sigma} \, \delta z
            - \delta' {\delta W \over \delta \sigma}
            + z {\delta^2 \upsilon
               \over \delta \sigma \, \delta \chi} \, \delta \chi
            \right);\cr}
   \right\}
   \eqno {\rm (F^3)}$$
and hence by (A${}^3$) we can deduce already, without any farther
differentiation,
$$\left. \multieqalign{
{\delta^2 V \over \delta x^2}
   &= \mathbin{\phantom{-}} {1 \over w''} \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right); &
{\delta^2 V \over \delta x \, \delta z}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x^2}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta x \, \delta y};\cr
{\delta^2 V \over \delta x \, \delta y}
   &= - {1 \over w''} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right); &
{\delta^2 V \over \delta y \, \delta z}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta y}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta y^2};\cr
{\delta^2 V \over \delta y^2}
   &= \mathbin{\phantom{-}} {1 \over w''} \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right); &
{\delta^2 V \over \delta z^2}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta z}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta z};\cr}
   \right\}
   \eqno {\rm (G^3)}$$
observing, in deducing the sixth of these equations (G${}^3$),
that by the definitions (E), and by the dependence of $\upsilon$
on $\sigma$,~$\tau$,~$\chi$, we have
$$\delta {\delta V \over \delta z}
  = (\delta \upsilon = ) \,
         {\delta \upsilon \over \delta \sigma}
            \, \delta {\delta V \over \delta x}
       + {\delta \upsilon \over \delta \tau}
            \, \delta {\delta V \over \delta y}
       + {\delta \upsilon \over \delta \chi}
            \, \delta \chi.
   \eqno {\rm (H^3)}$$
The equations (A${}^3$) (F${}^3$) (H${}^3$) give also
$$\left. \eqalign{
{\delta^2 V \over \delta x \, \delta x'}
   &=   {1 \over w''} {\delta^2 W \over \delta \tau \, \delta x'} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
      - {1 \over w''} {\delta^2 W \over \delta \sigma \, \delta x'} \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right);\cr
{\delta^2 V \over \delta x \, \delta y'}
   &=   {1 \over w''} {\delta^2 W \over \delta \tau \, \delta y'} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
      - {1 \over w''} {\delta^2 W \over \delta \sigma \, \delta y'} \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right);\cr
{\delta^2 V \over \delta x \, \delta z'}
   &=   {1 \over w''} {\delta^2 W \over \delta \tau \, \delta z'} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
      - {1 \over w''} {\delta^2 W \over \delta \sigma \, \delta z'} \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right);\cr
{\delta^2 V \over \delta y \, \delta x'}
   &=   {1 \over w''} {\delta^2 W \over \delta \sigma \, \delta x'} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
      - {1 \over w''} {\delta^2 W \over \delta \tau \, \delta x'} \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right);\cr
{\delta^2 V \over \delta y \, \delta y'}
   &=   {1 \over w''} {\delta^2 W \over \delta \sigma \, \delta y'} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
      - {1 \over w''} {\delta^2 W \over \delta \tau \, \delta y'} \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right);\cr
{\delta^2 V \over \delta y \, \delta z'}
   &=   {1 \over w''} {\delta^2 W \over \delta \sigma \, \delta z'} \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right)
      - {1 \over w''} {\delta^2 W \over \delta \tau \, \delta z'} \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right);\cr
{\delta^2 V \over \delta z \, \delta x'}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta x'}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta x'};\cr
{\delta^2 V \over \delta z \, \delta y'}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta y'}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta y'};\cr
{\delta^2 V \over \delta z \, \delta z'}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta z'}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta z'};\cr}
   \right\}
   \eqno {\rm (I^3)}$$
and
$$\left. \eqalign{
{\delta^2 V \over \delta x \, \delta \chi}
   &=   {1 \over w''}  \left(
            {\delta^2 W \over \delta \tau \, \delta \chi}
            - z {\delta^2 \upsilon \over \delta \tau \, \delta \chi}
            \right) \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      - {1 \over w''} \left(
            {\delta^2 W \over \delta \sigma \, \delta \chi}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \chi}
            \right) \left(
            {\delta^2 W \over \delta \tau^2}
            - z {\delta^2 \upsilon \over \delta \tau^2}
            \right);\cr
{\delta^2 V \over \delta y \, \delta \chi}
   &=   {1 \over w''}  \left(
            {\delta^2 W \over \delta \sigma \, \delta \chi}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \chi}
            \right) \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
            - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      - {1 \over w''} \left(
            {\delta^2 W \over \delta \tau \, \delta \chi}
            - z {\delta^2 \upsilon \over \delta \tau \, \delta \chi}
            \right) \left(
            {\delta^2 W \over \delta \sigma^2}
            - z {\delta^2 \upsilon \over \delta \sigma^2}
            \right);\cr
{\delta^2 V \over \delta z \, \delta \chi}
   &=   {\delta \upsilon \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta \chi}
      + {\delta \upsilon \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta \chi}
      + {\delta \upsilon \over \delta \chi}.\cr}
   \right\}
   \eqno {\rm (K^3)}$$
We have therefore found expressions (G${}^3$) (I${}^3$)
(K${}^3$), for eighteen out of the twenty-eight partial
differential coefficients of $V$ of the second order; and with
respect to nine of the remaining ten, namely all except
$\displaystyle {\delta^2 V \over \delta \chi^2}$,
we may obtain expressions for these by differentiating the three
equations (G${}'$), and comparing the differentials with
(A${}^3$); for thus we find,
$$\left. \eqalign{
{\delta^2 V \over \delta x'^2}
   &= - {\delta^2 W \over \delta x'^2}
      - {\delta^2 W \over \delta \sigma \, \delta x'}
            {\delta^2 V \over \delta x \, \delta x'}
      - {\delta^2 W \over \delta \tau \, \delta x'}
            {\delta^2 V \over \delta y \, \delta x'};\cr
{\delta^2 V \over \delta y'^2}
   &= - {\delta^2 W \over \delta y'^2}
      - {\delta^2 W \over \delta \sigma \, \delta y'}
            {\delta^2 V \over \delta x \, \delta y'}
      - {\delta^2 W \over \delta \tau \, \delta y'}
            {\delta^2 V \over \delta y \, \delta y'};\cr
{\delta^2 V \over \delta z'^2}
   &= - {\delta^2 W \over \delta z'^2}
      - {\delta^2 W \over \delta \sigma \, \delta z'}
            {\delta^2 V \over \delta x \, \delta z'}
      - {\delta^2 W \over \delta \tau \, \delta z'}
            {\delta^2 V \over \delta y \, \delta z'};\cr
{\delta^2 V \over \delta x' \, \delta y'}
   &= - {\delta^2 W \over \delta x' \, \delta y'}
      - {\delta^2 W \over \delta \sigma \, \delta x'}
            {\delta^2 V \over \delta x \, \delta y'}
      - {\delta^2 W \over \delta \tau \, \delta x'}
            {\delta^2 V \over \delta y \, \delta y'};\cr
{\delta^2 V \over \delta y' \, \delta z'}
   &= - {\delta^2 W \over \delta y' \, \delta z'}
      - {\delta^2 W \over \delta \sigma \, \delta y'}
            {\delta^2 V \over \delta x \, \delta z'}
      - {\delta^2 W \over \delta \tau \, \delta y'}
            {\delta^2 V \over \delta y \, \delta z'};\cr
{\delta^2 V \over \delta z' \, \delta x'}
   &= - {\delta^2 W \over \delta z' \, \delta x'}
      - {\delta^2 W \over \delta \sigma \, \delta z'}
            {\delta^2 V \over \delta x \, \delta x'}
      - {\delta^2 W \over \delta \tau \, \delta z'}
            {\delta^2 V \over \delta y \, \delta x'};\cr}
   \right\}
   \eqno {\rm (L^3)}$$
and
$$\left. \eqalign{
{\delta^2 V \over \delta x' \, \delta \chi}
   &= - {\delta^2 W \over \delta x' \delta \chi}
      - {\delta^2 W \over \delta \sigma \, \delta x'}
            {\delta^2 V \over \delta x \, \delta \chi}
      - {\delta^2 W \over \delta \tau \, \delta x'}
            {\delta^2 V \over \delta y \, \delta \chi};\cr
{\delta^2 V \over \delta y' \, \delta \chi}
   &= - {\delta^2 W \over \delta y' \delta \chi}
      - {\delta^2 W \over \delta \sigma \, \delta y'}
            {\delta^2 V \over \delta x \, \delta \chi}
      - {\delta^2 W \over \delta \tau \, \delta y'}
            {\delta^2 V \over \delta y \, \delta \chi};\cr
{\delta^2 V \over \delta z' \, \delta \chi}
   &= - {\delta^2 W \over \delta z' \delta \chi}
      - {\delta^2 W \over \delta \sigma \, \delta z'}
            {\delta^2 V \over \delta x \, \delta \chi}
      - {\delta^2 W \over \delta \tau \, \delta z'}
            {\delta^2 V \over \delta y \, \delta \chi}:\cr}
   \right\}
   \eqno {\rm (M^3)}$$
the equations (G${}'$) give also
$$\left. \eqalign{
{\delta^2 V \over \delta x' \, \delta y'}
   &= - {\delta^2 W \over \delta x' \, \delta y'}
      - {\delta^2 W \over \delta \sigma \, \delta y'}
            {\delta^2 V \over \delta x \, \delta x'}
      - {\delta^2 W \over \delta \tau \, \delta y'}
            {\delta^2 V \over \delta y \, \delta x'};\cr
{\delta^2 V \over \delta y' \, \delta z'}
   &= - {\delta^2 W \over \delta y' \, \delta z'}
      - {\delta^2 W \over \delta \sigma \, \delta z'}
            {\delta^2 V \over \delta x \, \delta y'}
      - {\delta^2 W \over \delta \tau \, \delta z'}
            {\delta^2 V \over \delta y \, \delta y'};\cr
{\delta^2 V \over \delta z' \, \delta x'}
   &= - {\delta^2 W \over \delta z' \, \delta x'}
      - {\delta^2 W \over \delta \sigma \, \delta x'}
            {\delta^2 V \over \delta x \, \delta z'}
      - {\delta^2 W \over \delta \tau \, \delta x'}
            {\delta^2 V \over \delta y \, \delta z'};\cr}
   \right\}
   \eqno {\rm (N^3)}$$
but these three expressions (N${}^3$) agree with the
corresponding expressions (L${}^3$), because, by (I${}^3$),
$$\left. \eqalign{
      {\delta^2 W \over \delta \sigma \, \delta x'}
            {\delta^2 V \over \delta x \, \delta y'}
      + {\delta^2 W \over \delta \tau \, \delta x'}
            {\delta^2 V \over \delta y \, \delta y'}
   &= {\delta^2 W \over \delta \sigma \, \delta y'}
            {\delta^2 V \over \delta x \, \delta x'}
      + {\delta^2 W \over \delta \tau \, \delta y'}
            {\delta^2 V \over \delta y \, \delta x'};\cr
      {\delta^2 W \over \delta \sigma \, \delta y'}
            {\delta^2 V \over \delta x \, \delta z'}
      + {\delta^2 W \over \delta \tau \, \delta y'}
            {\delta^2 V \over \delta y \, \delta z'}
   &= {\delta^2 W \over \delta \sigma \, \delta z'}
            {\delta^2 V \over \delta x \, \delta y'}
      + {\delta^2 W \over \delta \tau \, \delta z'}
            {\delta^2 V \over \delta y \, \delta y'};\cr
      {\delta^2 W \over \delta \sigma \, \delta z'}
            {\delta^2 V \over \delta x \, \delta x'}
      + {\delta^2 W \over \delta \tau \, \delta z'}
            {\delta^2 V \over \delta y \, \delta x'}
   &= {\delta^2 W \over \delta \sigma \, \delta x'}
            {\delta^2 V \over \delta x \, \delta z'}
      + {\delta^2 W \over \delta \tau \, \delta x'}
            {\delta^2 V \over \delta y \, \delta z'}.\cr}
   \right\}
   \eqno {\rm (O^3)}$$
Finally, with respect to the twenty-eighth coefficient
$\displaystyle {\delta^2 V \over \delta \chi^2}$,
this may be obtained by differentiating the third equation
(B${}^3$), which gives
$${\delta^2 V \over \delta \chi^2}
   = z {\delta^2 \upsilon \over \delta \chi^2}
         - {\delta^2 W \over \delta \chi^2}
         + \left(
               z {\delta^2 \upsilon \over \delta \sigma \, \delta \chi}
               - {\delta^2 W \over \delta \sigma \, \delta \chi}
               \right) {\delta^2 V \over \delta x \, \delta \chi}
         + \left(
               z {\delta^2 \upsilon \over \delta \tau \, \delta \chi}
               - {\delta^2 W \over \delta \tau \, \delta \chi}
               \right) {\delta^2 V \over \delta y \, \delta \chi}.
   \eqno {\rm (P^3)}$$
And if we would generalise the twenty-eight expressions
(G${}^3$) (I${}^3$) (K${}^3$) (L${}^3$) (M${}^3$) (P${}^3$), so
as to render them independent of the particular supposition, that
$W$ has been made, by a previous elimination of $\upsilon$, a
function involving only the six independent variables
$\sigma$,~$\tau$,~$\chi$, $x'$,~$y'$,~$z'$,
we may do so by suitably generalising fifteen out of the
twenty-one coefficients of $W$, of the second order, which result
from the foregoing suppositions; that is by leaving unchanged the
six that are formed by differentiating only with respect to
$x'$,~$y'$,~$z'$, but changing
$\displaystyle {\delta^2 W \over \delta \sigma^2}$,
\&c., to the following more general expressions
$\displaystyle \left[ {\delta^2 W \over \delta \sigma^2} \right]$,
\&c.;
$$\left. \eqalign{
\left[ {\delta^2 W \over \delta \sigma^2} \right]
   &= {\delta^2 W \over \delta \sigma^2}
      + 2 {\delta^2 W \over \delta \sigma \, \delta \upsilon}
            {\delta \upsilon \over \delta \sigma}
      + {\delta^2 W \over \delta \upsilon^2}
            \left( {\delta \upsilon \over \delta \sigma} \right)^2
      + {\delta W \over \delta \upsilon}
            {\delta^2 \upsilon \over \delta \sigma^2};\cr
\left[ {\delta^2 W \over \delta \tau^2} \right]
   &= {\delta^2 W \over \delta \tau^2}
      + 2 {\delta^2 W \over \delta \tau \, \delta \upsilon}
            {\delta \upsilon \over \delta \tau}
      + {\delta^2 W \over \delta \upsilon^2}
            \left( {\delta \upsilon \over \delta \tau} \right)^2
      + {\delta W \over \delta \upsilon}
            {\delta^2 \upsilon \over \delta \tau^2};\cr
\left[ {\delta^2 W \over \delta \chi^2} \right]
   &= {\delta^2 W \over \delta \chi^2}
      + 2 {\delta^2 W \over \delta \chi \, \delta \upsilon}
            {\delta \upsilon \over \delta \chi}
      + {\delta^2 W \over \delta \upsilon^2}
            \left( {\delta \upsilon \over \delta \chi} \right)^2
      + {\delta W \over \delta \upsilon}
            {\delta^2 \upsilon \over \delta \chi^2};\cr
\left[ {\delta^2 W \over \delta \sigma \, \delta \tau} \right]
   &= {\delta^2 W \over \delta \sigma \, \delta \tau}
      +   {\delta^2 W \over \delta \tau \, \delta \upsilon}
            {\delta \upsilon \over \delta \sigma}
      +   {\delta^2 W \over \delta \sigma \, \delta \upsilon}
            {\delta \upsilon \over \delta \tau}
      + {\delta^2 W \over \delta \upsilon^2}
            {\delta \upsilon \over \delta \sigma}
            {\delta \upsilon \over \delta \tau}
      + {\delta W \over \delta \upsilon}
            {\delta^2 \upsilon \over \delta \sigma \, \delta \tau};\cr
\left[ {\delta^2 W \over \delta \sigma \, \delta \chi} \right]
   &= {\delta^2 W \over \delta \sigma \, \delta \chi}
      +   {\delta^2 W \over \delta \chi \, \delta \upsilon}
            {\delta \upsilon \over \delta \sigma}
      +   {\delta^2 W \over \delta \sigma \, \delta \upsilon}
            {\delta \upsilon \over \delta \chi}
      + {\delta^2 W \over \delta \upsilon^2}
            {\delta \upsilon \over \delta \sigma}
            {\delta \upsilon \over \delta \chi}
      + {\delta W \over \delta \upsilon}
            {\delta^2 \upsilon \over \delta \sigma \, \delta \chi};\cr
\left[ {\delta^2 W \over \delta \tau \, \delta \chi} \right]
   &= {\delta^2 W \over \delta \tau \, \delta \chi}
      +   {\delta^2 W \over \delta \chi \, \delta \upsilon}
            {\delta \upsilon \over \delta \tau}
      +   {\delta^2 W \over \delta \tau \, \delta \upsilon}
            {\delta \upsilon \over \delta \chi}
      + {\delta^2 W \over \delta \upsilon^2}
            {\delta \upsilon \over \delta \tau}
            {\delta \upsilon \over \delta \chi}
      + {\delta W \over \delta \upsilon}
            {\delta^2 \upsilon \over \delta \tau \, \delta \chi};\cr
\left[ {\delta^2 W \over \delta \sigma \, \delta x'} \right]
   &= {\delta^2 W \over \delta \sigma \, \delta x'}
      +   {\delta^2 W \over \delta \upsilon \, \delta x'}
            {\delta \upsilon \over \delta \sigma};\quad
\left[ {\delta^2 W \over \delta \tau \, \delta x'} \right]
    = {\delta^2 W \over \delta \tau \, \delta x'}
      +   {\delta^2 W \over \delta \upsilon \, \delta x'}
            {\delta \upsilon \over \delta \tau};\cr
\left[ {\delta^2 W \over \delta \sigma \, \delta y'} \right]
   &= {\delta^2 W \over \delta \sigma \, \delta y'}
      +   {\delta^2 W \over \delta \upsilon \, \delta y'}
            {\delta \upsilon \over \delta \sigma};\quad
\left[ {\delta^2 W \over \delta \tau \, \delta y'} \right]
    = {\delta^2 W \over \delta \tau \, \delta y'}
      +   {\delta^2 W \over \delta \upsilon \, \delta y'}
            {\delta \upsilon \over \delta \tau};\cr
\left[ {\delta^2 W \over \delta \sigma \, \delta z'} \right]
   &= {\delta^2 W \over \delta \sigma \, \delta z'}
      +   {\delta^2 W \over \delta \upsilon \, \delta z'}
            {\delta \upsilon \over \delta \sigma};\quad
\left[ {\delta^2 W \over \delta \tau \, \delta z'} \right]
    = {\delta^2 W \over \delta \tau \, \delta z'}
      +   {\delta^2 W \over \delta \upsilon \, \delta z'}
            {\delta \upsilon \over \delta \tau};\cr
\left[ {\delta^2 W \over \delta \chi \, \delta x'} \right]
   &= {\delta^2 W \over \delta \chi \, \delta x'}
      +   {\delta^2 W \over \delta \upsilon \, \delta x'}
            {\delta \upsilon \over \delta \chi};\cr
\left[ {\delta^2 W \over \delta \chi \, \delta y'} \right]
   &= {\delta^2 W \over \delta \chi \, \delta y'}
      +   {\delta^2 W \over \delta \upsilon \, \delta y'}
            {\delta \upsilon \over \delta \chi};\cr
\left[ {\delta^2 W \over \delta \chi \, \delta z'} \right]
   &= {\delta^2 W \over \delta \chi \, \delta z'}
      +   {\delta^2 W \over \delta \upsilon \, \delta z'}
            {\delta \upsilon \over \delta \chi}:\cr}
   \right\}
   \eqno {\rm (Q^3)}$$
obtained by differentiating the three corresponding expressions
of the first order,
$$\left[ {\delta W \over \delta \sigma} \right]
  = {\delta W \over \delta \sigma}
      + {\delta W \over \delta \upsilon}
         {\delta \upsilon \over \delta \sigma};\quad
  \left[ {\delta W \over \delta \tau} \right]
  = {\delta W \over \delta \tau}
      + {\delta W \over \delta \upsilon}
         {\delta \upsilon \over \delta \tau};\quad
  \left[ {\delta W \over \delta \chi} \right]
  = {\delta W \over \delta \chi}
      + {\delta W \over \delta \upsilon}
         {\delta \upsilon \over \delta \chi},
   \eqno {\rm (R^3)}$$
which are to be substituted in (B${}^3$), in place of
$${\delta W \over \delta \sigma},\quad
  {\delta W \over \delta \tau},\quad
  {\delta W \over \delta \chi}.$$

\bigbreak

{\sectiontitle
Deduction of the Coefficients of $W$ from those of $V$.
Homogeneous Transformations.\par}

\nobreak\bigskip

8.
Reciprocally, if it be required to deduce the partial
differential coefficients of $W$, of the second order, from those
of $V$, in the case of a final variable medium, we have only to
compare the expressions for
$$\delta x,\quad
  \delta y,\quad
  \delta z,\quad
  \delta \sigma',\quad
  \delta \tau',\quad
  \delta \upsilon',\quad
  - \delta {\delta V \over \delta \chi},$$
as linear functions of
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$, deduced
from the equations (A${}^3$), with those that are obtained by
differentiating the seven equations (G${}'$) (H${}'$), into which
(B${}'$) resolves itself: that is, with the developed expressions
for the variations of
$${\delta W \over \delta \sigma},\quad
  {\delta W \over \delta \tau},\quad
  {\delta W \over \delta \upsilon},\quad
  {\delta W \over \delta x'},\quad
  {\delta W \over \delta y'},\quad
  {\delta W \over \delta z'},\quad
  {\delta W \over \delta \chi}.$$

But if the final medium be uniform, then (B${}'$) no longer
furnishes the seven equations (G${}'$) (H${}'$), nor can
$\delta x$, $\delta y$, $\delta z$, themselves, but only
certain combinations of them, be deduced from (A${}^3$); and the
auxiliary function $W$ is no longer completely determined in
form, by the mere knowledge of the form of the characteristic
function $V$, with which it is connected; because, in this case,
the seven variables on which $W$ depends, are not independent of
each other, four of them being connected by the relations
(K${}'$), by means of which relation the dependence of $W$ on the
seven may be changed in an infinite variety of ways, while the
dependence of $V$ on its seven variables, and the properties of
the optical combination, remain unaltered.  Accordingly this
indeterminateness of $W$, as deduced from $V$, in the case of a
final uniform medium, produces an indeterminateness, in the same
case, in the partial differential coefficients of $W$; and
whereas $W$, considered as a function of seven variables, has
thirty-five partial differential coefficients of the first and
second orders, we have only twenty-seven relations between these
thirty-five coefficients, unless we make some particular
supposition respecting the form of $W$; such as the supposition,
already mentioned, that one of the related variables, for example
$\upsilon$, has been removed by a previous elimination, which
gives the eight conditions,
$$\left. \eqalign{
 &{\delta W \over \delta \upsilon} = 0,\quad
  {\delta^2 W \over \delta \sigma \, \delta \upsilon} = 0,\quad
  {\delta^2 W \over \delta \tau \, \delta \upsilon} = 0,\quad
  {\delta^2 W \over \delta \upsilon^2} = 0,\quad
  {\delta^2 W \over \delta \upsilon \, \delta \chi} = 0,\cr
 &{\delta^2 W \over \delta \upsilon \, \delta x'} = 0,\quad
  {\delta^2 W \over \delta \upsilon \, \delta y'} = 0,\quad
  {\delta^2 W \over \delta \upsilon \, \delta z'} = 0.\cr}
   \right\}
   \eqno {\rm (S^3)}$$
This last supposition removes the indeterminateness of $W$
itself, and therefore of its partial differential coefficients;
of which, for the two first orders, eight vanish by (S${}^3$),
and the remaining twenty-seven are determined, (when the
variables and coefficients of $V$ are known,) by the six
equations (G${}'$), (B${}^3$), the three left-hand equations
(G${}^3$), the six first (I${}^3$), the two first (K${}^3$), and
the ten (L${}^3$) (M${}^3$) (P${}^3$); in resolving which
equations it is useful to observe, that by (E${}^3$) and
(G${}^3$),
$${1 \over w''}
    = {\delta^2 V \over \delta x^2} {\delta^2 V \over \delta y^2}
       - \left( {\delta^2 V \over \delta x \, \delta y} \right)^2.
   \eqno {\rm (T^3)}$$
And the twenty-seven expressions thus found for the coefficients
of $W$ of the two first orders, on the supposition of a previous
elimination of one of the seven related variables, may be
generalised, by (Q${}^3$) and (R${}^3$), into the twenty-seven
relations already mentioned as existing between the thirty-five
coefficients on any other supposition; which supposition, if it
be sufficient to determine the form of $W$, will give the eight
remaining conditions analogous to the conditions (S${}^3$), that
are necessary to determine the coefficients sought.

If, for example, we determine $W$ by supposing it made
homogeneous of the first dimension with respect to
$\sigma$,~$\tau$,~$\upsilon$, we shall have the eight following
conditions,
$$\sigma {\delta W \over \delta \sigma}
      + \tau {\delta W \over \delta \tau}
      + \upsilon {\delta W \over \delta \upsilon}
   = W,
   \eqno {\rm (U^3)}$$
and
$$\left. \eqalign{
         \sigma   {\delta^2 W \over \delta \sigma^2}
       + \tau     {\delta^2 W \over \delta \sigma \, \delta \tau}
       + \upsilon {\delta^2 W \over \delta \sigma \, \delta \upsilon}
   &= 0,\cr
         \sigma   {\delta^2 W \over \delta \sigma \, \delta \tau}
       + \tau     {\delta^2 W \over \delta \tau^2}
       + \upsilon {\delta^2 W \over \delta \tau \, \delta \upsilon}
   &= 0,\cr
         \sigma   {\delta^2 W \over \delta \sigma \, \delta \upsilon}
       + \tau     {\delta^2 W \over \delta \tau \, \delta \upsilon}
       + \upsilon {\delta^2 W \over \delta \upsilon^2}
   &= 0,\cr
         \sigma   {\delta^2 W \over \delta \sigma   \, \delta x'}
       + \tau     {\delta^2 W \over \delta \tau     \, \delta x'}
       + \upsilon {\delta^2 W \over \delta \upsilon \, \delta x'}
   &= {\delta W \over \delta x'},\cr
         \sigma   {\delta^2 W \over \delta \sigma   \, \delta y'}
       + \tau     {\delta^2 W \over \delta \tau     \, \delta y'}
       + \upsilon {\delta^2 W \over \delta \upsilon \, \delta y'}
   &= {\delta W \over \delta y'},\cr
         \sigma   {\delta^2 W \over \delta \sigma   \, \delta z'}
       + \tau     {\delta^2 W \over \delta \tau     \, \delta z'}
       + \upsilon {\delta^2 W \over \delta \upsilon \, \delta z'}
   &= {\delta W \over \delta z'},\cr
         \sigma   {\delta^2 W \over \delta \sigma   \, \delta \chi}
       + \tau     {\delta^2 W \over \delta \tau     \, \delta \chi}
       + \upsilon {\delta^2 W \over \delta \upsilon \, \delta \chi}
   &= {\delta W \over \delta \chi},\cr}
   \right\}
   \eqno {\rm (V^3)}$$
to be combined with the twenty-seven which are independent of the
form of $W$, and are deduced by the general method already
mentioned.  But this supposition of homogeneity appears to
deserve a separate investigation, on account of the symmetry of
the processes and results to which it leads.

Let us therefore resume the equations
$$x = {\delta W \over \delta \sigma}
         + V {\delta \Omega \over \delta \sigma},\quad
  y = {\delta W \over \delta \tau}
         + V {\delta \Omega \over \delta \tau},\quad
  z = {\delta W \over \delta \upsilon}
         + V {\delta \Omega \over \delta \upsilon},
   \eqno {\rm (W^2)}$$
which were deduced in the sixth number from the homogeneous form
that we now assign to $W$, and which are to be combined with the
following
$$0 = {\delta W \over \delta \chi}
       + {\delta V \over \delta \chi}
       + V {\delta \Omega \over \delta \chi},
   \eqno {\rm (W^3)}$$
and with the general equations of the fourth number,
$$\sigma'   = {\delta W \over \delta x'},\quad
  \tau'     = {\delta W \over \delta y'},\quad
  \upsilon' = {\delta W \over \delta z'}:
   \eqno {\rm (G')}$$
and let us eliminate
$$\delta x,\quad
  \delta y,\quad
  \delta z,\quad
  \delta \sigma',\quad
  \delta \tau',\quad
  \delta \upsilon',\quad
  \delta {\delta V \over \delta \chi},$$
by (A${}^3$), from the differentials of these seven equations,
(W${}^2$) (G${}'$) (W${}^3$), that is from the seven following,
$$\left. \eqalign{
\delta {\delta W \over \delta \sigma}
      + V \, \delta {\delta \Omega \over \delta \sigma}
   &= \delta x
      - {\delta \Omega \over \delta \sigma}   \, \delta V,\cr
\delta {\delta W \over \delta \tau}
      + V \, \delta {\delta \Omega \over \delta \tau}
   &= \delta y
      - {\delta \Omega \over \delta \tau}     \, \delta V,\cr
\delta {\delta W \over \delta \upsilon}
      + V \, \delta {\delta \Omega \over \delta \upsilon}
   &= \delta z
      - {\delta \Omega \over \delta \upsilon} \, \delta V,\cr
\delta {\delta W \over \delta x'}
    = \delta \sigma',\quad
\delta {\delta W \over \delta y'}
   &= \delta \tau',\quad
\delta {\delta W \over \delta z'}
    = \delta \upsilon',\cr
\delta {\delta W \over \delta \chi}
      + V \delta {\delta \Omega \over \delta \chi}
   &= - \delta {\delta V \over \delta \chi}
      - {\delta \Omega \over \delta \chi} \, \delta V.\cr}
   \right\}
   \eqno {\rm (X^3)}$$
This elimination gives

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
\lambda^{(1)} \, \delta \Omega
   &= - \delta \sigma
      + \delta' {\delta V \over \delta x}
      + {\delta^2 V \over \delta x^2} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right)
      + {\delta^2 V \over \delta x \, \delta y} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta x \, \delta z} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr
\lambda^{(2)} \, \delta \Omega
   &= - \delta \tau
      + \delta' {\delta V \over \delta y}
      + {\delta^2 V \over \delta x \, \delta y} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right)
      + {\delta^2 V \over \delta y^2} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta y \, \delta z} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr
\lambda^{(3)} \, \delta \Omega
   &= - \delta \upsilon
      + \delta' {\delta V \over \delta z}
      + {\delta^2 V \over \delta x \, \delta z} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right)
      + {\delta^2 V \over \delta y \, \delta z} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta z^2} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr
\lambda^{(4)} \, \delta \Omega
   &=   \delta  {\delta W \over \delta x'}
      + \delta' {\delta V \over \delta x'}
      + {\delta^2 V \over \delta x \, \delta x'} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right)
      + {\delta^2 V \over \delta y \, \delta x'} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta z \, \delta x'} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr
\lambda^{(5)} \, \delta \Omega
   &=   \delta  {\delta W \over \delta y'}
      + \delta' {\delta V \over \delta y'}
      + {\delta^2 V \over \delta x \, \delta y'} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right)
      + {\delta^2 V \over \delta y \, \delta y'} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta z \, \delta y'} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr
\lambda^{(6)} \, \delta \Omega
   &=   \delta  {\delta W \over \delta z'}
      + \delta' {\delta V \over \delta z'}
      + {\delta^2 V \over \delta x \, \delta z'} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right)
      + {\delta^2 V \over \delta y \, \delta z'} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta z \, \delta z'} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr
\lambda^{(7)} \, \delta \Omega
   &=   \delta  {\delta W \over \delta \chi}
      + V \, \delta {\delta \Omega \over \delta \chi}
      + \delta' {\delta V \over \delta \chi}
      + {\delta^2 V \over \delta x \, \delta \chi} \left(
            \delta {\delta W \over \delta \sigma}
            + V \, \delta {\delta \Omega \over \delta \sigma}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 V \over \delta y \, \delta \chi} \left(
            \delta {\delta W \over \delta \tau}
            + V \, \delta {\delta \Omega \over \delta \tau}
            \right)
      + {\delta^2 V \over \delta z \, \delta \chi} \left(
            \delta {\delta W \over \delta \upsilon}
            + V \, \delta {\delta \Omega \over \delta \upsilon}
            \right);\cr}
   \right\}
   \eqno {\rm (Y^3)}$$
if we put for abridgment

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
\delta' {\delta V \over \delta x}
   &=   {\delta^2 V \over \delta x \, \delta x'} \, \delta x'
      + {\delta^2 V \over \delta x \, \delta y'} \, \delta y'
      + {\delta^2 V \over \delta x \, \delta z'} \, \delta z'
      + {\delta^2 V \over \delta x \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta V \over \delta y}
   &=   {\delta^2 V \over \delta y \, \delta x'} \, \delta x'
      + {\delta^2 V \over \delta y \, \delta y'} \, \delta y'
      + {\delta^2 V \over \delta y \, \delta z'} \, \delta z'
      + {\delta^2 V \over \delta y \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta V \over \delta z}
   &=   {\delta^2 V \over \delta z \, \delta x'} \, \delta x'
      + {\delta^2 V \over \delta z \, \delta y'} \, \delta y'
      + {\delta^2 V \over \delta z \, \delta z'} \, \delta z'
      + {\delta^2 V \over \delta z \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta V \over \delta x'}
   &=   {\delta^2 V \over \delta x'^2} \, \delta x'
      + {\delta^2 V \over \delta x' \, \delta y'} \, \delta y'
      + {\delta^2 V \over \delta x' \, \delta z'} \, \delta z'
      + {\delta^2 V \over \delta x' \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta V \over \delta y'}
   &=   {\delta^2 V \over \delta x' \, \delta y'} \, \delta x'
      + {\delta^2 V \over \delta y'^2} \, \delta y'
      + {\delta^2 V \over \delta y' \, \delta z'} \, \delta z'
      + {\delta^2 V \over \delta y' \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta V \over \delta z'}
   &=   {\delta^2 V \over \delta x' \, \delta z'} \, \delta x'
      + {\delta^2 V \over \delta y' \, \delta z'} \, \delta y'
      + {\delta^2 V \over \delta z'^2} \, \delta z'
      + {\delta^2 V \over \delta z' \, \delta \chi} \, \delta \chi,\cr
\delta' {\delta V \over \delta \chi}
   &=   {\delta^2 V \over \delta x' \, \delta \chi} \, \delta x'
      + {\delta^2 V \over \delta y' \, \delta \chi} \, \delta y'
      + {\delta^2 V \over \delta z' \, \delta \chi} \, \delta z'
      + {\delta^2 V \over \delta \chi^2} \, \delta \chi,\cr}
   \right\}
   \eqno {\rm (Z^3)}$$
using $\delta'$ as in the notation (D${}^3$); and if we observe
that the partial differential equation of the fifth number,
$$0 = \Omega \left(
         {\delta V \over \delta x},
         {\delta V \over \delta y},
         {\delta V \over \delta z},
         \chi
         \right),
   \eqno {\rm (V')}$$
gives
$$\left. \eqalign{
0  &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x^2}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta x \, \delta y}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta x \, \delta z},\cr
0  &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta y}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta y^2}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta y \, \delta z},\cr
0  &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta z}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta z}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta z^2},\cr
0  &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta x'}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta x'}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta z \, \delta x'},\cr
0  &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta y'}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta y'}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta z \, \delta y'},\cr
0  &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta z'}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta z'}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta z \, \delta z'},\cr
- {\delta \Omega \over \delta \chi}
   &=    {\delta \Omega \over \delta \sigma}
            {\delta^2 V \over \delta x \, \delta \chi}
       + {\delta \Omega \over \delta \tau}
            {\delta^2 V \over \delta y \, \delta \chi}
       + {\delta \Omega \over \delta \upsilon}
            {\delta^2 V \over \delta z \, \delta \chi}.\cr}
   \right\}
   \eqno {\rm (A^4)}$$
We have introduced, in the equations (Y${}^3$), the terms
$\lambda^{(1)} \, \delta \Omega, \ldots \,
 \lambda^{(7)} \, \delta \Omega$,
that we may treat as independent the variations
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$, $\delta \chi$,
which are connected by the condition $\delta \Omega = 0$.

To determine the multipliers
$\lambda^{(1)}, \ldots \, \lambda^{(7)}$,
we are to observe that in deducing the foregoing equations, the
relation $\Omega = 0$ between the four variables
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$, has been supposed to have
been so expressed, by the method mentioned in the second number,
that the function $\Omega$ when increased by unity becomes
homogeneous of the first dimension with respect to
$\sigma$,~$\tau$,~$\upsilon$; in such a manner that we have
identically, for all values of the four variables
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$,
$$      \sigma   {\delta \Omega \over \delta \sigma}
      + \tau     {\delta \Omega \over \delta \tau}
      + \upsilon {\delta \Omega \over \delta \upsilon}
   = \Omega + 1,
   \eqno {\rm (B^4)}$$
and therefore,
$$\left. \eqalign{
        \sigma   {\delta^2 \Omega \over \delta \sigma^2}
      + \tau     {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
      + \upsilon {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}
   &= 0,\cr
        \sigma   {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
      + \tau     {\delta^2 \Omega \over \delta \tau^2}
      + \upsilon {\delta^2 \Omega \over \delta \tau   \, \delta \upsilon}
   &= 0,\cr
        \sigma   {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}
      + \tau     {\delta^2 \Omega \over \delta \tau   \, \delta \upsilon}
      + \upsilon {\delta^2 \Omega \over \delta \upsilon^2}
   &= 0,\cr
        \sigma   {\delta^2 \Omega \over \delta \sigma   \, \delta \chi}
      + \tau     {\delta^2 \Omega \over \delta \tau     \, \delta \chi}
      + \upsilon {\delta^2 \Omega \over \delta \upsilon \, \delta \chi}
   &= {\delta \Omega \over \delta \chi}.\cr}
   \right\}
   \eqno {\rm (C^4)}$$
Hence, and from the conditions (V${}^3$), relative to the
homogeneity of the function $W$, it is easy to infer that the
multipliers have the following values;
$$\left. \eqalign{
\lambda^{(1)} &= - \sigma;\quad
\lambda^{(2)}  = - \tau;\quad
\lambda^{(3)}  = - \upsilon;\cr
\lambda^{(4)} &= \sigma';\quad
\lambda^{(5)}  = \tau';\quad
\lambda^{(6)}  = \upsilon';\quad
\lambda^{(7)}  = - {\delta V \over \delta \chi}:\cr}
   \right\}
   \eqno {\rm (D^4)}$$
attending to (G${}'$) and (W${}^3$).  If we substitute these
values of the multipliers, in the seven equations (Y${}^3$), we
may decompose each of those equations into seven others, by
treating the seven variations
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$,
as independent; and thus obtain forty-nine equations of the first
degree, of which however only twenty-eight are distinct, for the
determination of the twenty-eight partial differential
coefficients of the second order, of $W$ considered as a function
of $\sigma$,~$\tau$,~$\upsilon$, $x'$,~$y'$,~$z'$,~$\chi$,
which relatively to $\sigma$,~$\tau$,~$\upsilon$, is homogeneous
of the first dimension: the corresponding coefficients of the
first order being determined by the seven equations (G${}'$)
(W${}^2$) (W${}^3$).

Instead of calculating in this manner the coefficients of $W$ of
the second order, by eliminating between the equations into which
the system (Y${}^3$) may be decomposed, it is simpler to
eliminate between the equations (Y${}^3$) themselves, and thus to
obtain expressions for the variations
$$\delta {\delta W \over \delta \sigma},\ldots \,
  \delta {\delta W \over \delta \chi}$$
of the coefficients of the first order, from which expressions
the coefficients of the second order will then immediately
result.  Eliminating, therefore, between the three first
equations (Y${}^3$), in order to get expressions for the three
variations
$$\delta {\delta W \over \delta \sigma},\quad
  \delta {\delta W \over \delta \tau},\quad
  \delta {\delta W \over \delta \upsilon},$$
we find, after some symmetric reductions,
$$\left. \eqalign{
\delta {\delta W \over \delta \sigma}
    = - V \, \delta {\delta \Omega \over \delta \sigma}
      + {1 \over v^2 V''} \left(
               \tau     {\delta^2 V \over \delta x \, \delta z}
             - \upsilon {\delta^2 V \over \delta x \, \delta y}
            \right) \left\{
               \upsilon \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
             - \tau     \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
            \right\} \phantom{;} \cr
      - {\delta \Omega \over \delta \sigma} \, \delta' V
      + {1 \over v^2 V''} \left(
               \tau     {\delta^2 V \over \delta y \, \delta z}
             - \upsilon {\delta^2 V \over \delta y^2}
            \right) \left\{
               \sigma   \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
             - \upsilon \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
            \right\} \phantom{;} \cr
      + {1 \over v^2 V''} \left(
               \tau     {\delta^2 V \over \delta z^2}
             - \upsilon {\delta^2 V \over \delta y \, \delta z}
            \right) \left\{
               \tau     \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
             - \sigma   \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
            \right\};\cr
\delta {\delta W \over \delta \tau}
    = - V \, \delta {\delta \Omega \over \delta \tau}
      + {1 \over v^2 V''} \left(
               \upsilon {\delta^2 V \over \delta x^2}
             - \sigma   {\delta^2 V \over \delta x \, \delta z}
            \right) \left\{
               \upsilon \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
             - \tau     \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
            \right\} \phantom{;} \cr
      - {\delta \Omega \over \delta \tau} \, \delta' V
      + {1 \over v^2 V''} \left(
               \upsilon {\delta^2 V \over \delta x \, \delta y}
             - \sigma   {\delta^2 V \over \delta y \, \delta z}
            \right) \left\{
               \sigma   \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
             - \upsilon \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
            \right\} \phantom{;} \cr
      + {1 \over v^2 V''} \left(
               \upsilon {\delta^2 V \over \delta x \, \delta z}
             - \sigma   {\delta^2 V \over \delta z^2}
            \right) \left\{
               \tau     \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
             - \sigma   \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
            \right\};\cr
\delta {\delta W \over \delta \upsilon}
    = - V \, \delta {\delta \Omega \over \delta \upsilon}
      + {1 \over v^2 V''} \left(
               \sigma   {\delta^2 V \over \delta x \, \delta y}
             - \tau     {\delta^2 V \over \delta x^2}
            \right) \left\{
               \upsilon \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
             - \tau     \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
            \right\} \phantom{;} \cr
      - {\delta \Omega \over \delta \upsilon} \, \delta' V
      + {1 \over v^2 V''} \left(
               \sigma   {\delta^2 V \over \delta y^2}
             - \tau     {\delta^2 V \over \delta x \, \delta y}
            \right) \left\{
               \sigma   \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
             - \upsilon \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
            \right\} \phantom{;} \cr
      + {1 \over v^2 V''} \left(
               \sigma   {\delta^2 V \over \delta y \, \delta z}
             - \tau     {\delta^2 V \over \delta x \, \delta z}
            \right) \left\{
               \tau     \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
             - \sigma   \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
            \right\};\cr}
   \right\}
   \eqno {\rm (E^4)}$$
in which,
$$\left. \matrix{
\displaystyle
V''  =  {\delta^2 V \over \delta x^2} {\delta^2 V \over \delta y^2}
      - \left( {\delta^2 V \over \delta x \, \delta y} \right)^2
      + {\delta^2 V \over \delta y^2} {\delta^2 V \over \delta z^2}
      - \left( {\delta^2 V \over \delta y \, \delta z} \right)^2
      + {\delta^2 V \over \delta z^2} {\delta^2 V \over \delta x^2}
      - \left( {\delta^2 V \over \delta z \, \delta x} \right)^2,\cr
   \noalign{\vskip 3pt}
\displaystyle
\hbox{and} \quad {1 \over v^2}
    =   \left( {\delta \Omega \over \delta \sigma} \right)^2
      + \left( {\delta \Omega \over \delta \tau} \right)^2
      + \left( {\delta \Omega \over \delta \upsilon} \right)^2,\cr}
   \right\}
   \eqno {\rm (F^4)}$$
$v$ having the same meaning as before: $\delta'$ also referring,
as before, to the variations of $x'$~$y'$~$z'$~$\chi$, alone,
and $V''$ having the same meaning as in the First Supplement.  In
effecting this elimination, we have attended to the forms of the
functions $W$, $\Omega$ which give
$$    \sigma   \left(
         \delta {\delta W \over \delta \sigma}
         + V \, \delta {\delta \Omega \over \delta \sigma}
         \right)
    + \tau     \left(
         \delta {\delta W \over \delta \tau}
         + V \, \delta {\delta \Omega \over \delta \tau}
         \right)
    + \upsilon \left(
         \delta {\delta W \over \delta \upsilon}
         + V \, \delta {\delta \Omega \over \delta \upsilon}
         \right)
  = - \delta' V;
   \eqno {\rm (G^4)}$$
we have also employed the equations (A${}^4$), which give, by
(F${}^4$),
$$\left. \multieqalign{
{\delta^2 V \over \delta y^2} {\delta^2 V \over \delta z^2}
      - \left( {\delta^2 V \over \delta y \, \delta z} \right)^2
   &= V'' v^2 \left( {\delta \Omega \over \delta \sigma} \right)^2; &
{\delta^2 V \over \delta x \, \delta y}
         {\delta^2 V \over \delta z \, \delta x}
    - {\delta^2 V \over \delta x^2}
         {\delta^2 V \over \delta y \, \delta z}
   &= V'' v^2 {\delta \Omega \over \delta \tau}
              {\delta \Omega \over \delta \upsilon};\cr
{\delta^2 V \over \delta z^2} {\delta^2 V \over \delta x^2}
      - \left( {\delta^2 V \over \delta z \, \delta x} \right)^2
   &= V'' v^2 \left( {\delta \Omega \over \delta \tau} \right)^2; &
{\delta^2 V \over \delta y \, \delta z}
         {\delta^2 V \over \delta x \, \delta y}
    - {\delta^2 V \over \delta y^2}
         {\delta^2 V \over \delta z \, \delta x}
   &= V'' v^2 {\delta \Omega \over \delta \upsilon}
              {\delta \Omega \over \delta \sigma};\cr
{\delta^2 V \over \delta x^2} {\delta^2 V \over \delta y^2}
      - \left( {\delta^2 V \over \delta x \, \delta y} \right)^2
   &= V'' v^2 \left( {\delta \Omega \over \delta \upsilon} \right)^2; &
{\delta^2 V \over \delta z \, \delta x}
         {\delta^2 V \over \delta y \, \delta z}
    - {\delta^2 V \over \delta z^2}
         {\delta^2 V \over \delta x \, \delta y}
   &= V'' v^2 {\delta \Omega \over \delta \sigma}
              {\delta \Omega \over \delta \tau}.\cr}
   \right\}
   \eqno {\rm (H^4)}$$
Having thus obtained expressions (E${}^4$) for the three variations
$$\delta {\delta W \over \delta \sigma},\quad
  \delta {\delta W \over \delta \tau},\quad
  \delta {\delta W \over \delta \upsilon},$$
it only remains to substitute these expressions in the four last
equations (Y${}^3$), and so to deduce, without any new
elimination, the four other variations
$$\delta {\delta W \over \delta x'},\quad
  \delta {\delta W \over \delta y'},\quad
  \delta {\delta W \over \delta z'},\quad
  \delta {\delta W \over \delta \chi};$$
after which, we shall have immediately the twenty-eight
coefficients of $W$, of the second order.  The six coefficients,
for example, of this order, which are formed by differentiating
$W$ with respect to $\sigma$,~$\tau$,~$\upsilon$, are expressed
by the six following equations, deduced from (E${}^4$):
$$\left. \eqalign{
{\delta^2 W \over \delta \sigma^2}
   &= - V \, {\delta^2 \Omega \over \delta \sigma^2}
      + {1 \over v^2 V''} \left(
               \tau^2            {\delta^2 V \over \delta z^2}
             + \upsilon^2        {\delta^2 V \over \delta y^2}
             - 2 \tau \upsilon   {\delta^2 V \over \delta y \, \delta z}
            \right);\cr
{\delta^2 W \over \delta \tau^2}
   &= - V \, {\delta^2 \Omega \over \delta \tau^2}
      + {1 \over v^2 V''} \left(
               \upsilon^2        {\delta^2 V \over \delta x^2}
             + \sigma^2          {\delta^2 V \over \delta z^2}
             - 2 \upsilon \sigma {\delta^2 V \over \delta z \, \delta x}
            \right);\cr
{\delta^2 W \over \delta \upsilon^2}
   &= - V \, {\delta^2 \Omega \over \delta \upsilon^2}
      + {1 \over v^2 V''} \left(
               \sigma^2          {\delta^2 V \over \delta y^2}
             + \tau^2            {\delta^2 V \over \delta x^2}
             - 2 \sigma \tau     {\delta^2 V \over \delta x \, \delta y}
            \right);\cr
{\delta^2 W \over \delta \sigma \, \delta \tau}
   &= - V \, {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
      + {1 \over v^2 V''} \left(
             - \upsilon^2        {\delta^2 V \over \delta x \, \delta y}
             + \tau \upsilon     {\delta^2 V \over \delta z \, \delta x}
             + \upsilon \sigma   {\delta^2 V \over \delta y \, \delta z}
             - \sigma \tau       {\delta^2 V \over \delta z^2}
            \right);\cr
{\delta^2 W \over \delta \tau \, \delta \upsilon}
   &= - V \, {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
      + {1 \over v^2 V''} \left(
             - \sigma^2          {\delta^2 V \over \delta y \, \delta z}
             + \upsilon \sigma   {\delta^2 V \over \delta x \, \delta y}
             + \sigma \tau       {\delta^2 V \over \delta z \, \delta x}
             - \tau \upsilon     {\delta^2 V \over \delta x^2}
            \right);\cr
{\delta^2 W \over \delta \upsilon \, \delta \sigma}
   &= - V \, {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
      + {1 \over v^2 V''} \left(
             - \tau^2            {\delta^2 V \over \delta z \, \delta x}
             + \sigma \tau       {\delta^2 V \over \delta y \, \delta z}
             + \tau \upsilon     {\delta^2 V \over \delta x \, \delta y}
             - \upsilon \sigma   {\delta^2 V \over \delta y^2}
            \right):\cr}
   \right\}
   \eqno {\rm (I^4)}$$
which may be shown to agree with the less simple equations of the
same kind in the First Supplement, and may be thus summed up,
$$\eqalignno{
v^2 V'' ( \delta''^2 W + V \, \delta''^2 \Omega)
  &=   {\delta^2 V \over \delta x^2}
         (\tau \, \delta \upsilon - \upsilon \, \delta \tau)^2
     + 2 {\delta^2 V \over \delta y \, \delta z}
         (\upsilon \, \delta \sigma - \sigma \, \delta \upsilon)
         (\sigma \, \delta \tau - \tau \, \delta \sigma) \cr
   &\mathrel{\phantom{=}} \mathord{}
     + {\delta^2 V \over \delta y^2}
         (\upsilon \, \delta \sigma - \sigma \, \delta \upsilon)^2
     + 2 {\delta^2 V \over \delta z \, \delta x}
         (\sigma \, \delta \tau - \tau \, \delta \sigma)
         (\tau \, \delta \upsilon - \upsilon \, \delta \tau) \cr
   &\mathrel{\phantom{=}} \mathord{}
     + {\delta^2 V \over \delta z^2}
         (\sigma \, \delta \tau - \tau \, \delta \sigma)^2
     + 2 {\delta^2 V \over \delta x \, \delta y}
         (\tau \, \delta \upsilon - \upsilon \, \delta \tau)
         (\upsilon \, \delta \sigma - \sigma \, \delta \upsilon),
   &{\rm (K^4)}\cr}$$
the mark of variation $\delta''$ referring only to the variables
$\sigma$,~$\tau$,~$\upsilon$, as $\delta'$ referred only to
$x'$,~$y'$,~$z'$,~$\chi$.

And the whole system of the twenty-eight expressions for the
twenty-eight coefficients of $W$, of the second order, may be
summed up in this one formula:

\vfill\eject  % Page break necessary with current page size

$$\eqalignno{
v^2 V'' (\delta^2 W + V \, \delta^2 \Omega
            + 2 \, \delta' V \, \delta \Omega
            + \delta'^2 V)
      \hskip -156pt \cr
   &=   {\delta^2 V \over \delta x^2} \left\{
               \tau     \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
             - \upsilon \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
            \right\}^2 \cr
     &+ {\delta^2 V \over \delta y^2} \left\{
               \upsilon \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
             - \sigma   \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
            \right\}^2 \cr
     &+ {\delta^2 V \over \delta z^2} \left\{
               \sigma   \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
             - \tau     \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
            \right\}^2 \cr
     &+ 2 {\delta^2 V \over \delta x \, \delta y} \left\{
               \tau     \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
             - \upsilon \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
            \right\} \left\{
               \upsilon \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
             - \sigma   \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
            \right\} \cr
     &+ 2 {\delta^2 V \over \delta y \, \delta z} \left\{
               \upsilon \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
             - \sigma   \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
            \right\} \left\{
               \sigma   \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
             - \tau     \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
            \right\} \cr
     &+ 2 {\delta^2 V \over \delta z \, \delta x} \left\{
               \sigma   \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
             - \tau     \left(
                  \delta \sigma   - \delta' {\delta V \over \delta x}
                  \right)
            \right\} \left\{
               \tau     \left(
                  \delta \upsilon - \delta' {\delta V \over \delta z}
                  \right)
             - \upsilon \left(
                  \delta \tau     - \delta' {\delta V \over \delta y}
                  \right)
            \right\};\cr
   & &{\rm (L^4)}\cr}$$
in which the symbols $\delta^2$, $\delta'^2$ are easily
understood by what precedes, and in which the seven variations
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$,
may be treated as independent of each other.

The formula (K${}^4$) has an inverse, deduced from (X${}^3$),
namely
$$\eqalignno{
{\delta'''^2 V \over v^2 V''}
   &= \left( {\delta^2 W \over \delta \sigma^2}
                  + V {\delta^2 \Omega \over \delta \sigma^2}
               \right) \left(
                  {\delta \Omega \over \delta \tau} \, \delta z
                  - {\delta \Omega \over \delta \upsilon} \, \delta y
               \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left( {\delta^2 W \over \delta \tau^2}
                  + V {\delta^2 \Omega \over \delta \tau^2}
               \right) \left(
                  {\delta \Omega \over \delta \upsilon} \, \delta x
                  - {\delta \Omega \over \delta \sigma} \, \delta z
               \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left( {\delta^2 W \over \delta \upsilon^2}
                  + V {\delta^2 \Omega \over \delta \upsilon^2}
               \right) \left(
                  {\delta \Omega \over \delta \sigma} \, \delta y
                  - {\delta \Omega \over \delta \tau} \, \delta x
               \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 \left( {\delta^2 W \over \delta \sigma \, \delta \tau}
                  + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
               \right) \left(
                  {\delta \Omega \over \delta \tau} \, \delta z
                  - {\delta \Omega \over \delta \upsilon} \, \delta y
               \right) \left(
                  {\delta \Omega \over \delta \upsilon} \, \delta x
                  - {\delta \Omega \over \delta \sigma} \, \delta z
               \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 \left( {\delta^2 W \over \delta \tau \, \delta \upsilon}
                  + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
               \right) \left(
                  {\delta \Omega \over \delta \upsilon} \, \delta x
                  - {\delta \Omega \over \delta \sigma} \, \delta z
               \right) \left(
                  {\delta \Omega \over \delta \sigma} \, \delta y
                  - {\delta \Omega \over \delta \tau} \, \delta x
               \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 \left( {\delta^2 W \over \delta \upsilon \, \delta \sigma}
                  + V {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
               \right) \left(
                  {\delta \Omega \over \delta \sigma} \, \delta y
                  - {\delta \Omega \over \delta \tau} \, \delta x
               \right) \left(
                  {\delta \Omega \over \delta \tau} \, \delta z
                  - {\delta \Omega \over \delta \upsilon} \, \delta y
               \right),
   &{\rm (M^4)}\cr}$$
in which $\delta'''$ refers to $x$,~$y$,~$z$, and in which $V''$
may be deduced from $W$ by the relation
$$\eqalignno{
{\sigma^2 + \tau^2 + \upsilon^2 \over V'' v^2}
   &=    \left( {\delta^2 W \over \delta \sigma^2}
                  + V {\delta^2 \Omega \over \delta \sigma^2}
               \right)
         \left( {\delta^2 W \over \delta \tau^2}
                  + V {\delta^2 \Omega \over \delta \tau^2}
               \right)
       - \left( {\delta^2 W \over \delta \sigma \, \delta \tau}
                  + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
               \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 W \over \delta \tau^2}
                  + V {\delta^2 \Omega \over \delta \tau^2}
               \right)
         \left( {\delta^2 W \over \delta \upsilon^2}
                  + V {\delta^2 \Omega \over \delta \upsilon^2}
               \right)
       - \left( {\delta^2 W \over \delta \tau \, \delta \upsilon}
                  + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
               \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 W \over \delta \upsilon^2}
                  + V {\delta^2 \Omega \over \delta \upsilon^2}
               \right)
         \left( {\delta^2 W \over \delta \sigma^2}
                  + V {\delta^2 \Omega \over \delta \sigma^2}
               \right)
       - \left( {\delta^2 W \over \delta \upsilon \, \delta \sigma}
                  + V {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
               \right)^2:
   &{\rm (N^4)}\cr}$$
and the more extensive formula (L${}^4$) has an inverse also, namely,
$$\eqalignno{
   & {1 \over V'' v^2} (\delta^2 V + V \, \delta'^2 \Omega
            + 2 \, \delta V \, \delta' \Omega
            + \delta'^2 W) \cr
   &=    \left( {\delta^2 W \over \delta \sigma^2}
                  + V {\delta^2 \Omega \over \delta \sigma^2}
               \right)
         \left\{
            {\delta \Omega \over \delta \tau} \left(
               \delta z
                - \delta' {\delta W \over \delta \upsilon}
                - V \, \delta' {\delta \Omega \over \delta \upsilon}
            \right)
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta y
                - \delta' {\delta W \over \delta \tau}
                - V \, \delta' {\delta \Omega \over \delta \tau}
            \right)
         \right\}^2 \cr
      &+ \left( {\delta^2 W \over \delta \tau^2}
                  + V {\delta^2 \Omega \over \delta \tau^2}
               \right)
         \left\{
            {\delta \Omega \over \delta \upsilon} \left(
               \delta x
                - \delta' {\delta W \over \delta \sigma}
                - V \, \delta' {\delta \Omega \over \delta \sigma}
            \right)
          - {\delta \Omega \over \delta \sigma} \left(
               \delta z
                - \delta' {\delta W \over \delta \upsilon}
                - V \, \delta' {\delta \Omega \over \delta \upsilon}
            \right)
         \right\}^2 \cr
      &+ \left( {\delta^2 W \over \delta \upsilon^2}
                  + V {\delta^2 \Omega \over \delta \upsilon^2}
               \right)
         \left\{
            {\delta \Omega \over \delta \sigma} \left(
               \delta y
                - \delta' {\delta W \over \delta \tau}
                - V \, \delta' {\delta \Omega \over \delta \tau}
            \right)
          - {\delta \Omega \over \delta \tau} \left(
               \delta x
                - \delta' {\delta W \over \delta \sigma}
                - V \, \delta' {\delta \Omega \over \delta \sigma}
            \right)
         \right\}^2 \cr
      &+ 2 \left( {\delta^2 W \over \delta \sigma \, \delta \tau}
                  + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
               \right)
   \hskip -0.3em
         \left\{
   \hskip -0.2em
         \matrix{
            \displaystyle
            \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \tau} \left(
               \delta z
                - \delta' {\delta W \over \delta \upsilon}
                - V \, \delta' {\delta \Omega \over \delta \upsilon}
            \right) \cr
            \displaystyle
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta y
                - \delta' {\delta W \over \delta \tau}
                - V \, \delta' {\delta \Omega \over \delta \tau}
            \right) \cr}
   \hskip -0.2em
         \right\}
   \hskip -0.3em
         \left\{
   \hskip -0.2em
         \matrix{
            \displaystyle
            \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \upsilon} \left(
               \delta x
                - \delta' {\delta W \over \delta \sigma}
                - V \, \delta' {\delta \Omega \over \delta \sigma}
            \right) \cr
            \displaystyle
          - {\delta \Omega \over \delta \sigma} \left(
               \delta z
                - \delta' {\delta W \over \delta \upsilon}
                - V \, \delta' {\delta \Omega \over \delta \upsilon}
            \right) \cr}
   \hskip -0.2em
         \right\} \cr
      &+ 2 \left( {\delta^2 W \over \delta \tau \, \delta \upsilon}
                  + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
               \right)
   \hskip -0.3em
         \left\{
   \hskip -0.2em
         \matrix{
            \displaystyle
            \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \upsilon} \left(
               \delta x
                - \delta' {\delta W \over \delta \sigma}
                - V \, \delta' {\delta \Omega \over \delta \sigma}
            \right) \cr
            \displaystyle
          - {\delta \Omega \over \delta \sigma} \left(
               \delta z
                - \delta' {\delta W \over \delta \upsilon}
                - V \, \delta' {\delta \Omega \over \delta \upsilon}
            \right) \cr}
   \hskip -0.2em
         \right\}
   \hskip -0.3em
         \left\{
   \hskip -0.2em
         \matrix{
            \displaystyle
            \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \sigma} \left(
               \delta y
                - \delta' {\delta W \over \delta \tau}
                - V \, \delta' {\delta \Omega \over \delta \tau}
            \right) \cr
            \displaystyle
          - {\delta \Omega \over \delta \tau} \left(
               \delta x
                - \delta' {\delta W \over \delta \sigma}
                - V \, \delta' {\delta \Omega \over \delta \sigma}
            \right) \cr}
   \hskip -0.2em
         \right\} \cr
      &+ 2 \left( {\delta^2 W \over \delta \upsilon \, \delta \sigma}
                  + V {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
               \right)
   \hskip -0.3em
         \left\{
   \hskip -0.2em
         \matrix{
            \displaystyle
            \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \sigma} \left(
               \delta y
                - \delta' {\delta W \over \delta \tau}
                - V \, \delta' {\delta \Omega \over \delta \tau}
            \right) \cr
            \displaystyle
          - {\delta \Omega \over \delta \tau} \left(
               \delta x
                - \delta' {\delta W \over \delta \sigma}
                - V \, \delta' {\delta \Omega \over \delta \sigma}
            \right) \cr}
   \hskip -0.2em
         \right\}
   \hskip -0.3em
         \left\{
   \hskip -0.2em
         \matrix{
            \displaystyle
            \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \tau} \left(
               \delta z
                - \delta' {\delta W \over \delta \upsilon}
                - V \, \delta' {\delta \Omega \over \delta \upsilon}
            \right) \cr
            \displaystyle
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta y
                - \delta' {\delta W \over \delta \tau}
                - V \, \delta' {\delta \Omega \over \delta \tau}
            \right) \cr}
   \hskip -0.2em
         \right\}
   \hskip -0.3em
         ,\cr
   & &{\rm (O^4)}\cr}$$
$\delta'$ retaining its recent meaning, so that, as $\Omega$ does
not contain $x'$,~$y'$,~$z'$, we have, in the last formula,
$$\left. \multieqalign{
\delta' \Omega
   &= {\delta \Omega \over \delta \chi} \, \delta \chi, &
\delta'^2 \Omega
   &= {\delta^2 \Omega \over \delta \chi^2} \, \delta \chi^2,\cr
\delta' {\delta \Omega \over \delta \sigma}
   &= {\delta^2 \Omega \over \delta \sigma   \, \delta \chi}
         \, \delta \chi, &
\delta' {\delta \Omega \over \delta \tau}
   &= {\delta^2 \Omega \over \delta \tau     \, \delta \chi}
         \, \delta \chi, &
\delta' {\delta \Omega \over \delta \upsilon}
   &= {\delta^2 \Omega \over \delta \upsilon \, \delta \chi}
         \, \delta \chi.\cr}
   \right\}
   \eqno {\rm (P^4)}$$

If we do not choose to suppose $W$ homogeneous of the first
dimension with respect to $\sigma$,~$\tau$,~$\upsilon$, and if we
put for abridgment
$$    \sigma   {\delta W \over \delta \sigma}
    + \tau     {\delta W \over \delta \tau}
    + \upsilon {\delta W \over \delta \upsilon}
    - W
   = w_1,
   \eqno {\rm (Q^4)}$$
and denote by $\delta W_1$ and $\delta^2 W_1$, the expressions
already found on this particular supposition, for the variations
of $W$, of the two first orders, so that, for the first order, by
(G${}'$) (W${}^2$) (W${}^3$),
$$\delta W_1
  =   x \, \delta \sigma
    + y \, \delta \tau
    + z \, \delta \upsilon
    + \sigma'   \, \delta x'
    + \tau'     \, \delta y'
    + \upsilon' \, \delta z'
    - {\delta V \over \delta \chi} \, \delta \chi
    - V \, \delta \Omega,
   \eqno {\rm (R^4)}$$
and, for the second order, $\delta^2 W_1 =$ the value of
$\delta^2 W$ assigned by the formula (L${}^4$); we may generalise
these particular values $\delta W_1$, $\delta^2 W_1$, by the
following relations,
$$\left. \eqalign{
\delta W_1
   &= \delta W - w_1 \, \delta \Omega,\cr
\delta^2 W_1
   &= \delta^2 W - w_1 \, \delta^2 \Omega
         - 2 \, \delta w_1 \, \delta \Omega \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left(
            \sigma   {\delta w_1 \over \delta \sigma}
          + \tau     {\delta w_1 \over \delta \tau}
          + \upsilon {\delta w_1 \over \delta \upsilon}
         \right) \, \delta \Omega^2,\cr}
   \right\}
   \eqno {\rm (S^4)}$$
in which $\delta W$, $\delta^2 W$, are general expressions,
independent of the condition of homogeneity $w_1 = 0$, and of
every other particular supposition respecting the form of $W$.
It is, however, here understood that the final medium is uniform,
and that in forming the variations of the function~$W$, the
quantities
$\sigma$,~$\tau$,~$\upsilon$, $\chi$, $x'$,~$y'$,~$z'$,
on which it depends, are treated as if they were seven
independent variables.

And if we would deduce expressions $\delta W_n$, $\delta^2 W_n$,
for the variations of $W$, of the two first orders, on the
supposition that $W$ is made, before differentiation, homogeneous
of any dimension~$n$, with respect to
$\sigma$,~$\tau$,~$\upsilon$,
we may put
$$    \sigma   {\delta W \over \delta \sigma}
    + \tau     {\delta W \over \delta \tau}
    + \upsilon {\delta W \over \delta \upsilon}
    - n W
   = w_n,
   \eqno {\rm (T^4)}$$
and we shall have the following relations
$$\left. \eqalign{
\delta W_n
   &= \delta W - w_n \, \delta \Omega,\cr
\delta^2 W_n
   &= \delta^2 W - w_n \, \delta^2 \Omega
         - 2 \, \delta w_n \, \delta \Omega \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left(
            \sigma   {\delta w_n \over \delta \sigma}
          + \tau     {\delta w_n \over \delta \tau}
          + \upsilon {\delta w_n \over \delta \upsilon}
          + w_n - n w_n
         \right) \, \delta \Omega^2,\cr}
   \right\}
   \eqno {\rm (U^4)}$$
which include the relations (S${}^4$).  The general analysis of
these homogeneous transformations is interesting, but we cannot
dwell upon it here.

\bigbreak

{\sectiontitle
Deductions of the Coefficients of $T$ from those of $W$, and
reciprocally.\par}

\nobreak\bigskip

9.
The general principles of investigation, respecting the
connexions between the partial differential coefficients of the
second order, of the characteristic and auxiliary functions,
having been sufficiently explained by the remarks made at the
beginning of the seventh number, and by the details into which we
have since entered; we shall confine ourselves, in the remaining
research of such connexions, for the new auxiliary function $T$,
to the case of extreme uniform media.  And having already treated
of the mutual connexions between the coefficients of the two
functions $V$ and $W$, it will be sufficient now to connect the
coefficients of either of these two, for example, the
coefficients of $W$, with those of $T$, of the first and second
orders: since the connexions between the coefficients of all
three functions will thus be sufficiently known.  We shall also
suppose that $W$ has been made, before differentiation,
homogeneous of the first dimension with respect to
$\sigma$,~$\tau$,~$\upsilon$, that our results may be the more
easily combined with the symmetric expressions already deduced
from this supposition, expressions which can be generalised in
the manner that has been explained: and similarly we shall
suppose that $T$ is made homogeneous of the first dimension with
respect to $\sigma$,~$\tau$,~$\upsilon$, and also with respect to
$\sigma'$,~$\tau'$,~$\upsilon'$.  Let us then seek to express the
partial differential coefficients of the two first orders, of
$T$, by means of those of $W$, both functions being thus
symmetrically prepared.

In this inquiry, we have, as before, the conditions of homogeneity
(U${}^3$) (V${}^3$), relative to the function $W$, and analogous
conditions relative to $T$, namely, for the first order,
$$\left. \eqalign{
        \sigma    {\delta T \over \delta \sigma}
      + \tau      {\delta T \over \delta \tau}
      + \upsilon  {\delta T \over \delta \upsilon}
   &= T,\cr
        \sigma'   {\delta T \over \delta \sigma'}
      + \tau'     {\delta T \over \delta \tau'}
      + \upsilon' {\delta T \over \delta \upsilon'}
   &= T;\cr}
   \right\}
   \eqno {\rm (V^4)}$$
and, for the second order,
$$\left. \multieqalign{
0 &=   \sigma    {\delta^2 T \over \delta \sigma^2}
     + \tau      {\delta^2 T \over \delta \sigma \, \delta \tau}
     + \upsilon  {\delta^2 T \over \delta \sigma \, \delta \upsilon}; &
0 &=   \sigma'   {\delta^2 T \over \delta \sigma'^2}
     + \tau'     {\delta^2 T \over \delta \sigma' \, \delta \tau'}
     + \upsilon' {\delta^2 T \over \delta \sigma' \, \delta \upsilon'}; \cr
0 &=   \sigma    {\delta^2 T \over \delta \sigma \, \delta \tau}
     + \tau      {\delta^2 T \over \delta \tau^2}
     + \upsilon  {\delta^2 T \over \delta \tau \, \delta \upsilon}; &
0 &=   \sigma'   {\delta^2 T \over \delta \sigma' \, \delta \tau'}
     + \tau'     {\delta^2 T \over \delta \tau'^2}
     + \upsilon' {\delta^2 T \over \delta \tau' \, \delta \upsilon'}; \cr
0 &=   \sigma    {\delta^2 T \over \delta \sigma \, \delta \upsilon}
     + \tau      {\delta^2 T \over \delta \tau \, \delta \upsilon}
     + \upsilon  {\delta^2 T \over \delta \upsilon^2}; &
0 &=   \sigma'   {\delta^2 T \over \delta \sigma' \, \delta \upsilon'}
     + \tau'     {\delta^2 T \over \delta \tau' \, \delta \upsilon'}
     + \upsilon' {\delta^2 T \over \delta \upsilon'^2}; \cr
{\delta T \over \delta \sigma'}
  &=   \sigma    {\delta^2 T \over \delta \sigma \, \delta \sigma'}
     + \tau      {\delta^2 T \over \delta \tau \, \delta \sigma'}
     + \upsilon  {\delta^2 T \over \delta \upsilon \, \delta \sigma'}; &
{\delta T \over \delta \sigma}
  &=   \sigma'   {\delta^2 T \over \delta \sigma \, \delta \sigma'}
     + \tau'     {\delta^2 T \over \delta \sigma \, \delta \tau'}
     + \upsilon' {\delta^2 T \over \delta \sigma \, \delta \upsilon'}; \cr
{\delta T \over \delta \tau'}
  &=   \sigma    {\delta^2 T \over \delta \sigma \, \delta \tau'}
     + \tau      {\delta^2 T \over \delta \tau \, \delta \tau'}
     + \upsilon  {\delta^2 T \over \delta \upsilon \, \delta \tau'}; &
{\delta T \over \delta \tau}
  &=   \sigma'   {\delta^2 T \over \delta \tau \, \delta \sigma'}
     + \tau'     {\delta^2 T \over \delta \tau \, \delta \tau'}
     + \upsilon' {\delta^2 T \over \delta \tau \, \delta \upsilon'}; \cr
{\delta T \over \delta \upsilon'}
  &=   \sigma    {\delta^2 T \over \delta \sigma \, \delta \upsilon'}
     + \tau      {\delta^2 T \over \delta \tau \, \delta \upsilon'}
     + \upsilon  {\delta^2 T \over \delta \upsilon \, \delta \upsilon'}; &
{\delta T \over \delta \upsilon}
  &=   \sigma'   {\delta^2 T \over \delta \upsilon \, \delta \sigma'}
     + \tau'     {\delta^2 T \over \delta \upsilon \, \delta \tau'}
     + \upsilon' {\delta^2 T \over \delta \upsilon \, \delta \upsilon'}; \cr
{\delta T \over \delta \chi}
  &=   \sigma    {\delta^2 T \over \delta \sigma \, \delta \chi}
     + \tau      {\delta^2 T \over \delta \tau \, \delta \chi}
     + \upsilon  {\delta^2 T \over \delta \upsilon \, \delta \chi}; &
{\delta T \over \delta \chi}
  &=   \sigma'   {\delta^2 T \over \delta \sigma' \, \delta \chi}
     + \tau'     {\delta^2 T \over \delta \tau' \, \delta \chi}
     + \upsilon' {\delta^2 T \over \delta \upsilon' \, \delta \chi}; \cr}
   \right\}
   \eqno {\rm (W^4)}$$
together with the conditions relative to $\Omega$, $\Omega'$,
namely (B${}^4$), (C${}^4$), and the following,
$$\left. \eqalign{
      \sigma'   {\delta \Omega' \over \delta \sigma'}
    + \tau'     {\delta \Omega' \over \delta \tau'}
    + \upsilon' {\delta \Omega' \over \delta \upsilon'}
   &= \Omega' + 1  = 1,\cr
      \sigma'   {\delta^2 \Omega' \over \delta \sigma'^2}
    + \tau'     {\delta^2 \Omega' \over \delta \sigma' \, \delta \tau'}
    + \upsilon' {\delta^2 \Omega' \over \delta \sigma' \, \delta \upsilon'}
   &= 0,\cr
      \sigma'   {\delta^2 \Omega' \over \delta \sigma' \, \delta \tau'}
    + \tau'     {\delta^2 \Omega' \over \delta \tau'^2}
    + \upsilon' {\delta^2 \Omega' \over \delta \tau' \, \delta \upsilon'}
   &= 0,\cr
      \sigma'   {\delta^2 \Omega' \over \delta \sigma' \, \delta \upsilon'}
    + \tau'     {\delta^2 \Omega' \over \delta \tau' \, \delta \upsilon'}
    + \upsilon' {\delta^2 \Omega' \over \delta \upsilon'^2}
   &= 0,\cr
      \sigma'   {\delta^2 \Omega' \over \delta \sigma' \, \delta \chi}
    + \tau'     {\delta^2 \Omega' \over \delta \tau' \, \delta \chi}
    + \upsilon' {\delta^2 \Omega' \over \delta \upsilon' \, \delta \chi}
   &= {\delta \Omega' \over \delta \chi};\cr}
   \right\}
   \eqno {\rm (X^4)}$$
we have also the general equations
$${\delta W \over \delta x'} = \sigma',\quad
  {\delta W \over \delta y'} = \tau',\quad
  {\delta W \over \delta z'} = \upsilon',
   \eqno {\rm (G')}$$
by combining which with the foregoing conditions and with the
partial differential equation (A${}^2$), we find the following,
analogous to (A${}^4$),
$$\left. \eqalign{
0  &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta x'^2}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta x' \, \delta y'}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta x' \, \delta z'},\cr
0  &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta x' \, \delta y'}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta y'^2}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta y' \, \delta z'},\cr
0  &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta x' \, \delta z'}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta y' \, \delta z'}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta z'^2},\cr
{\delta \Omega \over \delta \sigma}
   &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta \sigma \, \delta x'}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta \sigma \, \delta y'}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta \sigma \, \delta z'},\cr
{\delta \Omega \over \delta \tau}
   &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta \tau \, \delta x'}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta \tau \, \delta y'}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta \tau \, \delta z'},\cr
{\delta \Omega \over \delta \upsilon}
   &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta \upsilon \, \delta x'}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta \upsilon \, \delta y'}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta \upsilon \, \delta z'},\cr
{\delta \Omega \over \delta \chi}
       - {\delta \Omega' \over \delta \chi}
   &=    {\delta \Omega' \over \delta \sigma'}
         {\delta^2 W \over \delta \chi \, \delta x'}
       + {\delta \Omega' \over \delta \tau'}
         {\delta^2 W \over \delta \chi \, \delta y'}
       + {\delta \Omega' \over \delta \upsilon'}
         {\delta^2 W \over \delta \chi \, \delta z'};\cr}
   \right\}
   \eqno {\rm (Y^4)}$$
and if we combine the conditions of homogeneity of the two
functions $W$, $T$, with the fundamental relation (E${}'$)
between these two functions, and with the properties of $\Omega$,
$\Omega'$, and attend to (G${}'$), we find the following
expressions for the partial differential coefficients of $T$, of
the first order,
$$\left. \multieqalign{
{\delta T \over \delta \sigma}
   &= {\delta W \over \delta \sigma}
         + (T - W) {\delta \Omega \over \delta \sigma}; &
{\delta T \over \delta \sigma'}
   &= - x' + W {\delta \Omega' \over \delta \sigma'};\cr
{\delta T \over \delta \tau}
   &= {\delta W \over \delta \tau}
         + (T - W) {\delta \Omega \over \delta \tau}; &
{\delta T \over \delta \tau'}
   &= - y' + W {\delta \Omega' \over \delta \tau'};\cr
{\delta T \over \delta \upsilon}
   &= {\delta W \over \delta \upsilon}
         + (T - W) {\delta \Omega \over \delta \upsilon}; &
{\delta T \over \delta \upsilon'}
   &= - z' + W {\delta \Omega' \over \delta \upsilon'};\cr
{\delta T \over \delta \chi}
   &= {\delta W \over \delta \chi}
         + (T - W) {\delta \Omega \over \delta \chi}
         + W       {\delta \Omega' \over \delta \chi}. &
  & \cr}
   \right\}
   \eqno {\rm (Z^4)}$$
Differentiating the expressions (Z${}^4$), and eliminating
$\delta x'$, $\delta y'$, $\delta z'$, by means of the
differentials of the general equations (G${}'$), we obtain, by
(Y${}^4$), the following system, analogous to the system
(Y${}^3$);

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
\lambda_1 \, \delta \Omega + \lambda_1' \, \delta \Omega'
   &= {\delta^2 W \over \delta x'^2}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta x' \, \delta y'}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta x' \, \delta z'}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right)
    - \delta_\prime {\delta W \over \delta x'}
      + \delta \sigma';\cr
\lambda_2 \, \delta \Omega + \lambda_2' \, \delta \Omega'
   &= {\delta^2 W \over \delta x' \, \delta y'}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta y'^2}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta y' \, \delta z'}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right)
    - \delta_\prime {\delta W \over \delta y'}
      + \delta \tau';\cr
\lambda_3 \, \delta \Omega + \lambda_3' \, \delta \Omega'
   &= {\delta^2 W \over \delta x' \, \delta z'}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta y' \, \delta z'}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta z'^2}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right)
    - \delta_\prime {\delta W \over \delta z'}
      + \delta \upsilon';\cr
\lambda_4 \, \delta \Omega + \lambda_4' \, \delta \Omega'
   &= {\delta^2 W \over \delta \sigma \, \delta x'}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta \sigma \, \delta y'}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta \sigma \, \delta z'}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \delta_\prime {\delta W \over \delta \sigma}
      + \delta \left( {\delta T \over \delta \sigma}
                      - T {\delta \Omega \over \delta \sigma}
                      \right)
      + W \, \delta {\delta \Omega \over \delta \sigma};\cr
\lambda_5 \, \delta \Omega + \lambda_5' \, \delta \Omega'
   &= {\delta^2 W \over \delta \tau \, \delta x'}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta \tau \, \delta y'}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta \tau \, \delta z'}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \delta_\prime {\delta W \over \delta \tau}
      + \delta \left( {\delta T \over \delta \tau}
                      - T {\delta \Omega \over \delta \tau}
                      \right)
      + W \, \delta {\delta \Omega \over \delta \tau};\cr
\lambda_6 \, \delta \Omega + \lambda_6' \, \delta \Omega'
   &= {\delta^2 W \over \delta \upsilon \, \delta x'}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta \upsilon \, \delta y'}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta \upsilon \, \delta z'}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \delta_\prime {\delta W \over \delta \upsilon}
      + \delta \left( {\delta T \over \delta \upsilon}
                      - T {\delta \Omega \over \delta \upsilon}
                      \right)
      + W \, \delta {\delta \Omega \over \delta \upsilon};\cr
\lambda_7 \, \delta \Omega + \lambda_7' \, \delta \Omega'
   &= {\delta^2 W \over \delta \chi \, \delta x'}
      \left( \delta {\delta T \over \delta \sigma'}
             - W \, \delta {\delta \Omega' \over \delta \sigma'}
             \right)
    + {\delta^2 W \over \delta \chi \, \delta y'}
      \left( \delta {\delta T \over \delta \tau'}
             - W \, \delta {\delta \Omega' \over \delta \tau'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 W \over \delta \chi \, \delta z'}
      \left( \delta {\delta T \over \delta \upsilon'}
             - W \, \delta {\delta \Omega' \over \delta \upsilon'}
             \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \delta_\prime {\delta W \over \delta \chi}
      + \delta \left( {\delta T \over \delta \chi}
                      - T {\delta \Omega \over \delta \chi}
                      \right)
      + W \, \delta \left( {\delta \Omega \over \delta \chi}
                           - {\delta \Omega' \over \delta \chi}
                           \right):\cr}
   \right\}
   \eqno {\rm (A^5)}$$
in which $\delta_\prime$ refers only to the four variations
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$, $\delta \chi$,
and in which we may treat the seven variations
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$, $\delta \chi$,
$\delta \sigma'$, $\delta \tau'$, $\delta \upsilon'$,
as independent, if we assign to the fourteen multipliers
$\lambda_1, \ldots \, \lambda_7'$, the following values;
$$\left. \eqalign{
\lambda_1
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta x'^2}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta x' \, \delta y'}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta x' \, \delta z'}
       - {\delta W \over \delta x'};\cr
\lambda_2
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta x' \, \delta y'}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta y'^2}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta y' \, \delta z'}
       - {\delta W \over \delta y'};\cr
\lambda_3
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta x' \, \delta z'}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta y' \, \delta z'}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta z'^2}
       - {\delta W \over \delta z'};\cr
\lambda_4
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta \sigma \, \delta x'}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta \sigma \, \delta y'}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta \sigma \, \delta z'}
       - T {\delta \Omega \over \delta \sigma};\cr
\lambda_5
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta \tau \, \delta x'}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta \tau \, \delta y'}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta \tau \, \delta z'}
       - T {\delta \Omega \over \delta \tau};\cr
\lambda_6
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta \upsilon \, \delta x'}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta \upsilon \, \delta y'}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta \upsilon \, \delta z'}
       - T {\delta \Omega \over \delta \upsilon};\cr
\lambda_7
   &=    {\delta T \over \delta \sigma'}
            {\delta^2 W \over \delta \chi \, \delta x'}
       + {\delta T \over \delta \tau'}
            {\delta^2 W \over \delta \chi \, \delta y'}
       + {\delta T \over \delta \upsilon'}
            {\delta^2 W \over \delta \chi \, \delta z'}
       - T {\delta \Omega \over \delta \chi}
       + W {\delta \Omega' \over \delta \chi};\cr
\lambda_1' &= \sigma';\quad
\lambda_2'  = \tau';\quad
\lambda_3'  = \upsilon';\cr
\lambda_4' &= {\delta W \over \delta \sigma}
               - W {\delta \Omega \over \delta \sigma};\quad
\lambda_5'  = {\delta W \over \delta \tau}
               - W {\delta \Omega \over \delta \tau};\cr
\lambda_6' &= {\delta W \over \delta \upsilon}
               - W {\delta \Omega \over \delta \upsilon};\quad
\lambda_7'  = {\delta W \over \delta \chi}
               - W {\delta \Omega \over \delta \chi}:\cr}
   \right\}
   \eqno {\rm (B^5)}$$
the values of $\lambda_1, \ldots \, \lambda_7$, may also be
thus expressed,
$$\left. \multieqalign{
\lambda_1 &= - {\delta w' \over \delta x'}, &
\lambda_4 &= - {\delta w' \over \delta \sigma}
             + (W - T) {\delta \Omega \over \delta \sigma}, \cr
\lambda_2 &= - {\delta w' \over \delta y'}, &
\lambda_5 &= - {\delta w' \over \delta \tau}
             + (W - T) {\delta \Omega \over \delta \tau}, \cr
\lambda_3 &= - {\delta w' \over \delta z'}, &
\lambda_6 &= - {\delta w' \over \delta \upsilon}
             + (W - T) {\delta \Omega \over \delta \upsilon}, \cr
   &  &
\lambda_7 &= - {\delta w' \over \delta \chi}
             + (W - T) {\delta \Omega \over \delta \chi},\cr}
   \right\}
   \eqno {\rm (C^5)}$$
if we put for abridgment
$$w' =   x' {\delta W \over \delta x'}
       + y' {\delta W \over \delta y'}
       + z' {\delta W \over \delta z'},
   \eqno {\rm (D^5)}$$
and consider $w'$, like $W$, as a function of
$\sigma$,~$\tau$,~$\upsilon$, $\chi$, $x'$,~$y'$,~$z'$, which,
relatively to $\sigma$,~$\tau$,~$\upsilon$, is homogeneous of the
first dimension.  The four last equations (A${}^5$) give, by
addition, after multiplying them respectively by
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$, $\delta \chi$,
$$\left. \eqalign{
\delta^2 T
   &= (T - W) \, \delta^2 \Omega + W \, \delta^2 \Omega'
      + (W - T) \, \delta \Omega^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
      + (\delta T - \delta_\prime w') \, \delta \Omega
      + (\delta_\prime W - W \, \delta \Omega) \, \delta \Omega'
      + \delta_\prime^2 W \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \left( \delta_\prime {\delta W \over \delta x'}
               - \delta \sigma'
               \right)
        \left( \delta {\delta T \over \delta \sigma'}
               - W \, \delta {\delta \Omega' \over \delta \sigma'}
               \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \left( \delta_\prime {\delta W \over \delta y'}
               - \delta \tau'
               \right)
        \left( \delta {\delta T \over \delta \tau'}
               - W \, \delta {\delta \Omega' \over \delta \tau'}
               \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \left( \delta_\prime {\delta W \over \delta z'}
               - \delta \upsilon'
               \right)
        \left( \delta {\delta T \over \delta \upsilon'}
               - W \, \delta {\delta \Omega' \over \delta \upsilon'}
               \right),\cr}
   \right\}
   \eqno {\rm (E^5)}$$
$\delta_\prime$ still referring only to the variations of
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$; and the three first equations
(A${}^5$) give, by elimination,

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
\delta {\delta T \over \delta \sigma'}
      - W \, \delta {\delta \Omega' \over \delta \sigma'}
         \hskip -66pt \cr
   &= {\delta \Omega' \over \delta \sigma'}
            (\delta_\prime W - W \, \delta \Omega)
      + {\delta T \over \delta \sigma'} \, \delta \Omega \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \upsilon'   {\delta^2 W \over \delta x' \, \delta y'}
                - \tau'     {\delta^2 W \over \delta x' \, \delta z'}
                \right)
        \left\{ \tau'       \left( \delta_\prime {\delta W \over \delta z'}
                                   - \delta \upsilon'
                                   \right)
                - \upsilon' \left( \delta_\prime {\delta W \over \delta y'}
                                   - \delta \tau'
                                   \right)
                \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \upsilon'   {\delta^2 W \over \delta y'^2}
                - \tau'     {\delta^2 W \over \delta y' \, \delta z'}
                \right)
        \left\{ \upsilon'   \left( \delta_\prime {\delta W \over \delta x'}
                                   - \delta \sigma'
                                   \right)
                - \sigma'   \left( \delta_\prime {\delta W \over \delta z'}
                                   - \delta \upsilon'
                                   \right)
                \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \upsilon'   {\delta^2 W \over \delta y' \, \delta z'}
                - \tau'     {\delta^2 W \over \delta z'^2}
                \right)
        \left\{ \sigma'     \left( \delta_\prime {\delta W \over \delta y'}
                                   - \delta \tau'
                                   \right)
                - \tau'     \left( \delta_\prime {\delta W \over \delta x'}
                                   - \delta \sigma'
                                   \right)
                \right\};\cr
\delta {\delta T \over \delta \tau'}
      - W \, \delta {\delta \Omega' \over \delta \tau'}
         \hskip -66pt \cr
   &= {\delta \Omega' \over \delta \tau'}
            (\delta_\prime W - W \, \delta \Omega)
      + {\delta T \over \delta \tau'} \, \delta \Omega \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \sigma'     {\delta^2 W \over \delta x' \, \delta z'}
                - \upsilon' {\delta^2 W \over \delta x'^2}
                \right)
        \left\{ \tau'       \left( \delta_\prime {\delta W \over \delta z'}
                                   - \delta \upsilon'
                                   \right)
                - \upsilon' \left( \delta_\prime {\delta W \over \delta y'}
                                   - \delta \tau'
                                   \right)
                \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \sigma'     {\delta^2 W \over \delta y' \, \delta z'}
                - \upsilon' {\delta^2 W \over \delta x' \, \delta y'}
                \right)
        \left\{ \upsilon'   \left( \delta_\prime {\delta W \over \delta x'}
                                   - \delta \sigma'
                                   \right)
                - \sigma'   \left( \delta_\prime {\delta W \over \delta z'}
                                   - \delta \upsilon'
                                   \right)
                \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \sigma'     {\delta^2 W \over \delta z'^2}
                - \upsilon' {\delta^2 W \over \delta x' \, \delta z'}
                \right)
        \left\{ \sigma'     \left( \delta_\prime {\delta W \over \delta y'}
                                   - \delta \tau'
                                   \right)
                - \tau'     \left( \delta_\prime {\delta W \over \delta x'}
                                   - \delta \sigma'
                                   \right)
                \right\};\cr
\delta {\delta T \over \delta \upsilon'}
      - W \, \delta {\delta \Omega' \over \delta \upsilon'}
         \hskip -66pt \cr
   &= {\delta \Omega' \over \delta \upsilon'}
            (\delta_\prime W - W \, \delta \Omega)
      + {\delta T \over \delta \upsilon'} \, \delta \Omega \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \tau'       {\delta^2 W \over \delta x'^2}
                - \sigma'   {\delta^2 W \over \delta x' \, \delta y'}
                \right)
        \left\{ \tau'       \left( \delta_\prime {\delta W \over \delta z'}
                                   - \delta \upsilon'
                                   \right)
                - \upsilon' \left( \delta_\prime {\delta W \over \delta y'}
                                   - \delta \tau'
                                   \right)
                \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \tau'       {\delta^2 W \over \delta x' \, \delta y'}
                - \sigma'   {\delta^2 W \over \delta y'^2}
                \right)
        \left\{ \upsilon'   \left( \delta_\prime {\delta W \over \delta x'}
                                   - \delta \sigma'
                                   \right)
                - \sigma'   \left( \delta_\prime {\delta W \over \delta z'}
                                   - \delta \upsilon'
                                   \right)
                \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {1 \over v'^2 W'''} \left(
                \tau'       {\delta^2 W \over \delta x' \, \delta z'}
                - \sigma'   {\delta^2 W \over \delta y' \, \delta z'}
                \right)
        \left\{ \sigma'     \left( \delta_\prime {\delta W \over \delta y'}
                                   - \delta \tau'
                                   \right)
                - \tau'     \left( \delta_\prime {\delta W \over \delta x'}
                                   - \delta \sigma'
                                   \right)
                \right\};\cr}
   \right\}
   \eqno {\rm (F^5)}$$
in which
$$W'''
   =    {\delta^2 W \over \delta x'^2}
           {\delta^2 W \over \delta y'^2}
      - \left( {\delta^2 W \over \delta x' \, \delta y'} \right)^2
      + {\delta^2 W \over \delta y'^2}
           {\delta^2 W \over \delta z'^2}
      - \left( {\delta^2 W \over \delta y' \, \delta z'} \right)^2
      + {\delta^2 W \over \delta z'^2}
           {\delta^2 W \over \delta x'^2}
      - \left( {\delta^2 W \over \delta z' \, \delta x'} \right)^2,
   \eqno {\rm (G^5)}$$
and
$${1 \over v'^2}
   =   \left( {\delta \Omega' \over \delta \sigma'} \right)^2
     + \left( {\delta \Omega' \over \delta \tau'} \right)^2
     + \left( {\delta \Omega' \over \delta \upsilon'} \right)^2,
   \eqno {\rm (H^5)}$$
$v'$ having the same meaning as in the second number.  In
effecting the last elimination, we have attended to the relations
(Y${}^4$), which give
$$\left. \eqalign{
{\delta^2 W \over \delta y'^2} {\delta^2 W \over \delta z'^2}
      - \left( {\delta^2 W \over \delta y' \, \delta z'} \right)^2
   &= W''' v'^2 \left( {\delta \Omega' \over \delta \sigma'} \right)^2;\cr
{\delta^2 W \over \delta z'^2} {\delta^2 W \over \delta x'^2}
      - \left( {\delta^2 W \over \delta z' \, \delta x'} \right)^2
   &= W''' v'^2 \left( {\delta \Omega' \over \delta \tau'} \right)^2;\cr
{\delta^2 W \over \delta x'^2} {\delta^2 W \over \delta y'^2}
      - \left( {\delta^2 W \over \delta x' \, \delta y'} \right)^2
   &= W''' v'^2 \left( {\delta \Omega' \over \delta \upsilon'} \right)^2;\cr
{\delta^2 W \over \delta x' \, \delta y'}
         {\delta^2 W \over \delta z' \, \delta x'}
    - {\delta^2 W \over \delta x'^2}
         {\delta^2 W \over \delta y' \, \delta z'}
   &= W''' v'^2
         {\delta \Omega' \over \delta \tau'}
         {\delta \Omega' \over \delta \upsilon'};\cr
{\delta^2 W \over \delta y' \, \delta z'}
         {\delta^2 W \over \delta x' \, \delta y'}
    - {\delta^2 W \over \delta y'^2}
         {\delta^2 W \over \delta z' \, \delta x'}
   &= W''' v'^2
         {\delta \Omega' \over \delta \upsilon'}
         {\delta \Omega' \over \delta \sigma'};\cr
{\delta^2 W \over \delta z' \, \delta x'}
         {\delta^2 W \over \delta y' \, \delta z'}
    - {\delta^2 W \over \delta z'^2}
         {\delta^2 W \over \delta x' \, \delta y'}
   &= W''' v'^2
         {\delta \Omega' \over \delta \sigma'}
         {\delta \Omega' \over \delta \tau'}.\cr}
   \right\}
   \eqno {\rm (I^5)}$$
And combining (E${}^5$) (F${}^5$) we obtain the following
formula for $\delta^2 T$, analogous to the formula (L${}^4$),
which completes the solution of our present problem, because it
it equivalent to twenty-eight expressions for the twenty-eight
partial differential coefficients of $T$, of the second order,
deduced from the coefficients of $W$;
$$\eqalignno{0
  &= v'^2 W''' \{ \delta^2 T + (W - T) \, \delta^2 \Omega
      - W \, \delta^2 \Omega'
      - 2 \, \delta_\prime W \mathbin{.} \delta \Omega'
      - \delta_\prime^2 W \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 2 (   x' \, \delta \sigma'
            + y' \, \delta \tau'
            + z' \, \delta \upsilon') \, \delta \Omega \} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 W \over \delta x'^2} \left\{
            \tau'     \left( \delta \upsilon'
                             - \delta_\prime {\delta W \over \delta z'}
                             \right)
          - \upsilon' \left( \delta \tau'
                             - \delta_\prime {\delta W \over \delta y'}
                             \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 W \over \delta y'^2} \left\{
            \upsilon' \left( \delta \sigma'
                             - \delta_\prime {\delta W \over \delta x'}
                             \right)
          - \sigma'   \left( \delta \upsilon'
                             - \delta_\prime {\delta W \over \delta z'}
                             \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\delta^2 W \over \delta z'^2} \left\{
            \sigma'   \left( \delta \tau'
                             - \delta_\prime {\delta W \over \delta y'}
                             \right)
          - \tau'     \left( \delta \sigma'
                             - \delta_\prime {\delta W \over \delta x'}
                             \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 2 {\delta^2 W \over \delta x' \, \delta y'} \left\{
            \tau'     \left( \delta \upsilon'
                             - \delta_\prime {\delta W \over \delta z'}
                             \right)
          - \upsilon' \left( \delta \tau'
                             - \delta_\prime {\delta W \over \delta y'}
                             \right)
         \right\} \cr
   &\mathrel{\phantom{=}} \qquad \times \left\{
            \upsilon' \left( \delta \sigma'
                             - \delta_\prime {\delta W \over \delta x'}
                             \right)
          - \sigma'   \left( \delta \upsilon'
                             - \delta_\prime {\delta W \over \delta z'}
                             \right)
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 2 {\delta^2 W \over \delta y' \, \delta z'} \left\{
            \upsilon' \left( \delta \sigma'
                             - \delta_\prime {\delta W \over \delta x'}
                             \right)
          - \sigma'   \left( \delta \upsilon'
                             - \delta_\prime {\delta W \over \delta z'}
                             \right)
         \right\} \cr
   &\mathrel{\phantom{=}} \qquad \times \left\{
            \sigma'   \left( \delta \tau'
                             - \delta_\prime {\delta W \over \delta y'}
                             \right)
          - \tau'     \left( \delta \sigma'
                             - \delta_\prime {\delta W \over \delta x'}
                             \right)
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 2 {\delta^2 W \over \delta z' \, \delta x'} \left\{
            \sigma'   \left( \delta \tau'
                             - \delta_\prime {\delta W \over \delta y'}
                             \right)
          - \tau'     \left( \delta \sigma'
                             - \delta_\prime {\delta W \over \delta x'}
                             \right)
         \right\} \cr
   &\mathrel{\phantom{=}} \qquad \times \left\{
            \tau'     \left( \delta \upsilon'
                             - \delta_\prime {\delta W \over \delta z'}
                             \right)
          - \upsilon' \left( \delta \tau'
                             - \delta_\prime {\delta W \over \delta y'}
                             \right)
         \right\}.
   &{\rm (K^5)}\cr}$$
And if we denote by $\delta^2 T_{1,1}$ the value of the second
differential $\delta^2 T$ assigned by the formula (K${}^5$), and
determined on the supposition that $T$ has been made, before
differentiation, homogeneous of the first dimension with respect
to $\sigma$,~$\tau$,~$\upsilon$, and also with respect to
$\sigma'$,~$\tau'$,~$\upsilon'$, and denote by $\delta T_{1,1}$
the corresponding value of $\delta T$, determined by the
coefficients (Z${}^4$), we may generalise thse values by means of
the following relations, analogous to (S${}^4$);
$$\left. \eqalign{
\delta T_{1,1}
   &= \delta T
      - \delta \Omega  \mathbin{.} \nabla_1  T
      - \delta \Omega' \mathbin{.} \nabla_1' T;\cr
\delta^2 T_{1,1}
   &= \delta^2 T
      - \delta^2 \Omega  \mathbin{.} \nabla_1  T
      - \delta^2 \Omega' \mathbin{.} \nabla_1' T
      - 2 \, \delta \Omega  \mathbin{.} \delta \nabla_1  T
      - 2 \, \delta \Omega' \mathbin{.} \delta \nabla_1' T \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \delta \Omega^2  \mathbin{.} \nabla_1  (\nabla_1  + 1) T
      + 2 \, \delta \Omega \mathbin{.} \delta \Omega'
            \mathbin{.} \nabla_1  \nabla_1' T
      + \delta \Omega'^2 \mathbin{.} \nabla_1' (\nabla_1' + 1) T:\cr}
   \right\}
   \eqno {\rm (L^5)}$$
$\nabla_1$, $\nabla_1'$, being here characteristics of operation,
defined by the following symbolic equations,
$$\left. \eqalign{
\nabla_1
   &=   \sigma    {\delta \over \delta \sigma}
      + \tau      {\delta \over \delta \tau}
      + \upsilon  {\delta \over \delta \upsilon}
      - 1;\cr
\nabla_1'
   &=   \sigma'   {\delta \over \delta \sigma'}
      + \tau'     {\delta \over \delta \tau'}
      + \upsilon' {\delta \over \delta \upsilon'}
      - 1.\cr}
   \right\}
   \eqno {\rm (M^5)}$$

More generally, if we denote by $T_{n,n'}$ the function deduced
from $T$ by the homogeneous preparation mentioned in the sixth
number, which coincides with $T$ when the variables
$\sigma$~$\tau$~$\upsilon$ $\sigma'$~$\tau'$~$\upsilon'$~$\chi$
are connected by the relations $\Omega = 0$, $\Omega' = 0$, and
which is, for arbitrary values of those variables, homogeneous of
the dimension $n$ with respect to $\sigma$,~$\tau$,~$\upsilon$,
and of the dimension $n'$ with respect to
$\sigma'$,~$\tau'$,~$\upsilon'$, we have the following
expressions, analogous to (U${}^4$),
$$\left. \eqalign{
\delta T_{n,n'}
   &= \delta T
      - \delta \Omega  \mathbin{.} \nabla_n  T
      - \delta \Omega' \mathbin{.} \nabla_{n'}' T;\cr
\delta^2 T_{n,n'}
   &= \delta^2 T
      - \delta^2 \Omega  \mathbin{.} \nabla_n  T
      - \delta^2 \Omega' \mathbin{.} \nabla_{n'}' T
      - 2 \, \delta \Omega  \mathbin{.} \delta \nabla_n  T
      - 2 \, \delta \Omega' \mathbin{.} \delta \nabla_{n'}' T \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \delta \Omega^2  \mathbin{.} \nabla_n  (\nabla_n  + 1) T
      + 2 \, \delta \Omega \mathbin{.} \delta \Omega'
            \mathbin{.} \nabla_n  \nabla_{n'}' T
      + \delta \Omega'^2 \mathbin{.} \nabla_{n'}' (\nabla_{n'}' + 1) T:\cr}
   \right\}
   \eqno {\rm (N^5)}$$
defining the characteristics $\nabla_n$, $\nabla_{n'}'$, as
follows,
$$\left. \eqalign{
\nabla_n
   &=   \sigma    {\delta \over \delta \sigma}
      + \tau      {\delta \over \delta \tau}
      + \upsilon  {\delta \over \delta \upsilon}
      - n;\cr
\nabla_{n'}'
   &=   \sigma'   {\delta \over \delta \sigma'}
      + \tau'     {\delta \over \delta \tau'}
      + \upsilon' {\delta \over \delta \upsilon'}
      - n'.\cr}
   \right\}
   \eqno {\rm (O^5)}$$

Reciprocally to deduce the coefficients of $W$, of the second
order, from those of $T$, on the same suppositions of
homogeneity, and with the same dimensions $n = 1$, $n' = 1$, we
are to eliminate
$\delta \sigma'$, $\delta \tau'$, $\delta \upsilon'$,
between the differentials of (G${}'$) and (Z${}^4$), and we find
the following system,

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
\lambda_1'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \sigma'^2}
                - W {\delta^2 \Omega' \over \delta \sigma'^2}
                \right)
            \delta {\delta W \over \delta x'}
       + \left( {\delta^2 T \over \delta \sigma' \, \delta \tau'}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \tau'}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \sigma' \, \delta \upsilon'}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \upsilon'}
                \right)
            \delta {\delta W \over \delta z'}
       + \delta_\prime {\delta T \over \delta \sigma'}
       - W \, \delta_\prime {\delta \Omega' \over \delta \sigma'}
       + \delta x'
       - {\delta \Omega' \over \delta \sigma'} \delta W;\cr
\lambda_2'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \sigma' \, \delta \tau'}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \tau'}
                \right)
            \delta {\delta W \over \delta x'}
       + \left( {\delta^2 T \over \delta \tau'^2}
                - W {\delta^2 \Omega' \over \delta \tau'^2}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \tau' \, \delta \upsilon'}
                - W {\delta^2 \Omega' \over \delta \tau' \, \delta \upsilon'}
                \right)
            \delta {\delta W \over \delta z'}
       + \delta_\prime {\delta T \over \delta \tau'}
       - W \, \delta_\prime {\delta \Omega' \over \delta \tau'}
       + \delta y'
       - {\delta \Omega' \over \delta \tau'} \delta W;\cr
\lambda_3'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \sigma' \, \delta \upsilon'}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \upsilon'}
                \right)
            \delta {\delta W \over \delta x'}
       + \left( {\delta^2 T \over \delta \tau' \, \delta \upsilon'}
                - W {\delta^2 \Omega' \over \delta \tau' \, \delta \upsilon'}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \upsilon'^2}
                - W {\delta^2 \Omega' \over \delta \upsilon'^2}
                \right)
            \delta {\delta W \over \delta z'}
       + \delta_\prime {\delta T \over \delta \upsilon'}
       - W \, \delta_\prime {\delta \Omega' \over \delta \upsilon'}
       + \delta z'
       - {\delta \Omega' \over \delta \upsilon'} \delta W;\cr
\lambda_4'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \sigma \, \delta \sigma'}
                - {\delta T \over \delta \sigma'}
                     {\delta \Omega \over \delta \sigma}
                \right)
            \delta {\delta W \over \delta x'}
       + \left( {\delta^2 T \over \delta \sigma \, \delta \tau'}
                - {\delta T \over \delta \tau'}
                     {\delta \Omega \over \delta \sigma}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \sigma \, \delta \upsilon'}
                - {\delta T \over \delta \upsilon'}
                     {\delta \Omega \over \delta \sigma}
                \right)
            \delta {\delta W \over \delta z'}
       - \delta {\delta W \over \delta \sigma}
       + \delta_\prime {\delta T \over \delta \sigma} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (W - T) \, \delta {\delta \Omega \over \delta \sigma}
       + {\delta \Omega \over \delta \sigma} (\delta W - \delta_\prime T);\cr
\lambda_5'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \tau \, \delta \sigma'}
                - {\delta T \over \delta \sigma'}
                     {\delta \Omega \over \delta \tau}
                \right)
            \delta {\delta W \over \delta x'}
       + \left( {\delta^2 T \over \delta \tau \, \delta \tau'}
                - {\delta T \over \delta \tau'}
                     {\delta \Omega \over \delta \tau}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \tau \, \delta \upsilon'}
                - {\delta T \over \delta \upsilon'}
                     {\delta \Omega \over \delta \tau}
                \right)
            \delta {\delta W \over \delta z'}
       - \delta {\delta W \over \delta \tau}
       + \delta_\prime {\delta T \over \delta \tau} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (W - T) \, \delta {\delta \Omega \over \delta \tau}
       + {\delta \Omega \over \delta \tau} (\delta W - \delta_\prime T);\cr
\lambda_6'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \upsilon \, \delta \sigma'}
                - {\delta T \over \delta \sigma'}
                     {\delta \Omega \over \delta \upsilon}
                \right)
            \delta {\delta W \over \delta x'}
       + \left( {\delta^2 T \over \delta \upsilon \, \delta \tau'}
                - {\delta T \over \delta \tau'}
                     {\delta \Omega \over \delta \upsilon}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \upsilon \, \delta \upsilon'}
                - {\delta T \over \delta \upsilon'}
                     {\delta \Omega \over \delta \upsilon}
                \right)
            \delta {\delta W \over \delta z'}
       - \delta {\delta W \over \delta \upsilon}
       + \delta_\prime {\delta T \over \delta \upsilon} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (W - T) \, \delta {\delta \Omega \over \delta \upsilon}
       + {\delta \Omega \over \delta \upsilon} (\delta W - \delta_\prime T);\cr
\lambda_7'' \, \delta \Omega
   &=    \left( {\delta^2 T \over \delta \sigma' \, \delta \chi}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \chi}
                - {\delta T \over \delta \sigma'}
                     {\delta \Omega \over \delta \chi}
                \right)
            \delta {\delta W \over \delta x'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \tau' \, \delta \chi}
                - W {\delta^2 \Omega' \over \delta \tau' \, \delta \chi}
                - {\delta T \over \delta \tau'}
                     {\delta \Omega \over \delta \chi}
                \right)
            \delta {\delta W \over \delta y'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \upsilon' \, \delta \chi}
                - W {\delta^2 \Omega' \over \delta \upsilon' \, \delta \chi}
                - {\delta T \over \delta \upsilon'}
                     {\delta \Omega \over \delta \chi}
                \right)
            \delta {\delta W \over \delta z'} \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \delta {\delta W \over \delta \chi}
       + \delta_\prime {\delta T \over \delta \chi}
       + (W - T) \, \delta {\delta \Omega \over \delta \chi}
       + {\delta \Omega \over \delta \chi} (\delta W - \delta_\prime T)
       - W \, \delta_\prime {\delta \Omega' \over \delta \chi}
       - {\delta \Omega' \over \delta \chi} \delta W;\cr}
   \right\}
   \eqno {\rm (P^5)}$$
$\delta_\prime$ still referring only to variations of
$\sigma$,~$\tau$,~$\upsilon$,~$\chi$,
and the values of the multipliers being,
$$\left. \multieqalign{
\lambda_1'' &= - x'; &
\lambda_4'' &= {\delta W \over \delta \sigma}
             - T {\delta \Omega \over \delta \sigma}; \cr
\lambda_2'' &= - y'; &
\lambda_5'' &= {\delta W \over \delta \tau}
             - T {\delta \Omega \over \delta \tau}; \cr
\lambda_3'' &= - z'; &
\lambda_6'' &= {\delta W \over \delta \upsilon}
             - T {\delta \Omega \over \delta \upsilon}; \cr
   &  &
\lambda_7'' &= {\delta W \over \delta \chi}
             - T {\delta \Omega \over \delta \chi}. \cr}
   \right\}
   \eqno {\rm (Q^5)}$$
Hence may be deduced, by reasonings analogous to those already
employed, the following formula for $\delta^2 W$, which is
equivalent to twenty-eight separate expressions for the partial
differential coefficients of $W$, of the second order, considered
as deduced from the coefficients of $T$, on the foregoing
suppositions of homogeneity:
$$\eqalign{
0 &= {1 \over v'^2 W'''} \{
         \delta^2 W - \delta_\prime^2 T + (T - W) \, \delta^2 \Omega
       + W \, \delta_\prime^2 \Omega'
       + 2 \, \delta W \, (\delta_\prime \Omega' - \delta \Omega)
       + 2 \, \delta_\prime W \mathbin{.} \delta \Omega \} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \sigma'^2}
                - W \, {\delta^2 \Omega' \over \delta \sigma'^2}
                \right) D^2
       + 2 \left( {\delta^2 T \over \delta \tau' \, \delta \upsilon'}
                - W {\delta^2 \Omega' \over \delta \tau' \, \delta \upsilon'}
                \right) D' D'' \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \tau'^2}
                - W \, {\delta^2 \Omega' \over \delta \tau'^2}
                \right) D'^2
       + 2 \left( {\delta^2 T \over \delta \upsilon' \, \delta \sigma'}
                - W {\delta^2 \Omega' \over \delta \upsilon' \, \delta \sigma'}
                \right) D'' D \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \upsilon'^2}
                - W \, {\delta^2 \Omega' \over \delta \upsilon'^2}
                \right) D''^2
       + 2 \left( {\delta^2 T \over \delta \sigma' \, \delta \tau'}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \tau'}
                \right) D D';\cr}
   \eqno {\rm (R^5)}$$
in which we have put for abridgment,
$$\left. \eqalign{
D   &=   {\delta \Omega' \over \delta \tau'} \left(
            \delta z' + z' \, \delta \Omega
            + \delta_\prime {\delta T \over \delta \upsilon'}
            - W \, \delta_\prime {\delta \Omega' \over \delta \upsilon'}
            \right)
       - {\delta \Omega' \over \delta \upsilon'} \left(
            \delta y' + y' \, \delta \Omega
            + \delta_\prime {\delta T \over \delta \tau'}
            - W \, \delta_\prime {\delta \Omega' \over \delta \tau'}
            \right),\cr
D'  &=   {\delta \Omega' \over \delta \upsilon'} \left(
            \delta x' + x' \, \delta \Omega
            + \delta_\prime {\delta T \over \delta \sigma'}
            - W \, \delta_\prime {\delta \Omega' \over \delta \sigma'}
            \right)
       - {\delta \Omega' \over \delta \sigma'} \left(
            \delta z' + z' \, \delta \Omega
            + \delta_\prime {\delta T \over \delta \upsilon'}
            - W \, \delta_\prime {\delta \Omega' \over \delta \upsilon'}
            \right),\cr
D'' &=   {\delta \Omega' \over \delta \sigma'} \left(
            \delta y' + y' \, \delta \Omega
            + \delta_\prime {\delta T \over \delta \tau'}
            - W \, \delta_\prime {\delta \Omega' \over \delta \tau'}
            \right)
       - {\delta \Omega' \over \delta \tau'} \left(
            \delta x' + x' \, \delta \Omega
            + \delta_\prime {\delta T \over \delta \sigma'}
            - W \, \delta_\prime {\delta \Omega' \over \delta \sigma'}
            \right),\cr}
   \right\}
   \eqno {\rm (S^5)}$$
and in which $W'''$ can be deduced from $T$, by the relation
$$\eqalignno{
{\sigma'^2 + \tau'^2 + \upsilon'^2 \over v'^2 W'''}
   &=    \left( {\delta^2 T \over \delta \sigma'^2}
                - W \, {\delta^2 \Omega' \over \delta \sigma'^2}
                \right)
         \left( {\delta^2 T \over \delta \tau'^2}
                - W \, {\delta^2 \Omega' \over \delta \tau'^2}
                \right)
       - \left( {\delta^2 T \over \delta \sigma' \, \delta \tau'}
                - W {\delta^2 \Omega' \over \delta \sigma' \, \delta \tau'}
                \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \tau'^2}
                - W \, {\delta^2 \Omega' \over \delta \tau'^2}
                \right)
         \left( {\delta^2 T \over \delta \upsilon'^2}
                - W \, {\delta^2 \Omega' \over \delta \upsilon'^2}
                \right)
       - \left( {\delta^2 T \over \delta \tau' \, \delta \upsilon'}
                - W {\delta^2 \Omega' \over \delta \tau' \, \delta \upsilon'}
                \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left( {\delta^2 T \over \delta \upsilon'^2}
                - W \, {\delta^2 \Omega' \over \delta \upsilon'^2}
                \right)
         \left( {\delta^2 T \over \delta \sigma'^2}
                - W \, {\delta^2 \Omega' \over \delta \sigma'^2}
                \right)
       - \left( {\delta^2 T \over \delta \upsilon' \, \delta \sigma'}
                - W {\delta^2 \Omega' \over \delta \upsilon' \, \delta \sigma'}
                \right)^2.\cr
   & &{\rm (T^5)}\cr}$$

\bigbreak

{\sectiontitle
General Remarks and Cautions, with respect to the foregoing deductions.
Case of a Single Uniform Medium.  Connexions between the Coefficients
of the Functions $v$, $\Omega$, $\upsilon$, for any Single Medium.\par}

\nobreak\bigskip

10.
We are then able, by combining the formul{\ae} of the three
preceding numbers, to deduce the partial differential
coefficients of the two first orders, of any one of the three
functions $V$, $W$, $T$, from those of either of the other two,
when the extreme media are uniform and known: since we have
expressed the coefficients of $V$ by those of $W$, and the
coefficients of $W$ by those of $T$, and reciprocally, for this
case of uniform media.  And if the extreme media be not uniform,
but variable, that is, if they be atmospheres, ordinary or
extraordinary, we can still connect the partial differential
coefficients of the three functions, by the general method
mentioned at the beginning of the seventh number: which method
extends to orders higher than the second, without much additional
difficulty of elimination, but with the results of greater
complexity, and of less interesting application.

This general method consists, as has been said, in
differentiating and comparing the equations into which the
general expressions (A${}'$) (B${}'$) (C${}'$) for the variations
of the three functions resolve themselves: and {\it in making
this preliminary resolution of the general expressions\/}
(A${}'$) (B${}'$) (C${}'$), {\it it is necessary to attend with
care to the relations between the variables\/}
$\sigma$,~$\tau$,~$\upsilon$,
$\sigma'$,~$\tau'$,~$\upsilon'$,~$\chi$,
or between
$\sigma$,~$\tau$,~$\upsilon$, $x'$,~$y'$,~$z'$,~$\chi$,
when any such relations exist.  The investigations into which
we have entered in the three last numbers, for the case of
extreme uniform media, {\it suppose that the variables are
connected only by the relations\/} $\Omega = 0$, $\Omega' = 0$,
which arise from and express the optical properties of these
media; and {\it other but analogous processes must be deduced
from the general method, when any additional relations\/}
$\Omega'' = 0$, $\Omega''' = 0,\ldots$ between the variables of
the question, arise from the particular nature of a combination
which we wish to study.  In the very simple case, for instance,
of a single uniform medium, we have the three relations
$$\sigma' = \sigma,\quad
  \tau' = \tau,\quad
  \upsilon' = \upsilon,
   \eqno {\rm (U^5)}$$
which are to be combined with the relation $\Omega = 0$; and with
this combination of relations, the general expression (C${}'$)
for the variation of $T$ will no longer admit of being resolved
in the same way as when more of the quantities on which $T$
depends could vary independently of each other.

In the case last mentioned,of a {\it single uniform medium}, the
characteristic function~$V$ involves the co-ordinates
$x$,~$y$,~$z$, $x'$,~$y'$,~$z'$,
only by involving their differences
$x - x'$, $y - y'$, $z - z'$,
and is, with respect to these differences, homogeneous of the
first dimension, being determined by an equation of the form
$$0 = \Psi \left(
         {x - x' \over V},
         {y - y' \over V},
         {z - z' \over V},
         \chi
         \right),
   \eqno {\rm (V^5)}$$
which results from the equation (N) for the medium function~$v$,
by first suppressing in that equation the co-ordinates on account
of the supposed uniformity, and then making
$${\alpha \over v} = {x - x' \over V},\quad
  {\beta  \over v} = {y - y' \over V},\quad
  {\gamma \over v} = {z - z' \over V}.
   \eqno {\rm (W^5)}$$
The relation (V${}^5$) may also be deduced from the relation
$\Omega = 0$, by eliminating the ratios of
$\sigma$,~$\tau$,~$\upsilon$,
between the three following equations,
$${x - x' \over V} = {\delta \Omega \over \delta \sigma},\quad
  {y - y' \over V} = {\delta \Omega \over \delta \tau},\quad
  {z - z' \over V} = {\delta \Omega \over \delta \upsilon}.
   \eqno {\rm (X^5)}$$
We have also, in this case of a single uniform medium,
$$V = \sigma (x - x') + \tau (y - y') + \upsilon (z - z'),
   \eqno {\rm (Y^5)}$$
and therefore, by (D${}'$) (E${}'$) (U${}^5$),
$$\left. \eqalign{
W &= \sigma x' + \tau y' + \upsilon z',\cr
T &= 0:\cr}
   \right\}
   \eqno {\rm (Z^5)}$$
the last of which results may be verified by observing that the
general expression for the auxiliary function~$T$ may be put
under the form
$$T =   x  {\delta V \over \delta x}
      + y  {\delta V \over \delta y}
      + z  {\delta V \over \delta z}
      + x' {\delta V \over \delta x'}
      + y' {\delta V \over \delta y'}
      + z' {\delta V \over \delta z'}
      - V,
   \eqno {\rm (A^6)}$$
so that $T$ vanishes when $V$ is homogeneous of the first
dimension with respect to the six extreme co-ordinates.  The
formul{\ae} of the last number, for the partial differential
coefficients of $T$, all fail in this case of a single uniform
medium, for the reason already assigned; but we may consider
all these coefficients of $T$ as vanishing, like $T$ itself: we
may however give any other values to these coefficients which when
combined with the relations between the variables will make the
variations of $T$ vanish.  The coefficients of $W$ may be obtained
by differentiating the expression (Z${}^5$), which is of the
homogeneous form that we have already found convenient to adopt;
they are, for the first two orders, included in the two following
formul{\ae},
$$\left. \eqalign{
\delta W
   &=   x' \, \delta \sigma
      + y' \, \delta \tau
      + z' \, \delta \upsilon
      + \sigma   \, \delta x'
      + \tau     \, \delta y'
      + \upsilon \, \delta z',\cr
\delta^2 W
   &=   2 \, \delta \sigma   \, \delta x'
      + 2 \, \delta \tau     \, \delta y'
      + 2 \, \delta \upsilon \, \delta z',\cr}
   \right\}
   \eqno {\rm (B^6)}$$
and they vanish for orders higher than the second.  And the
coefficients of $V$, of the two first orders, may be deduced from
those of $W$ by the formul{\ae} of the eighth number, which are
not vitiated by the existence of the relations (U${}^5$), because
those relations do not affect the variables that enter into the
composition of $V$ and $W$.  The variation of $V$, of the first
order, is
$$\delta V
   =   \sigma   (\delta x - \delta x')
     + \tau     (\delta y - \delta y')
     + \upsilon (\delta z - \delta z')
     - V {\delta \Omega \over \delta \chi} \, \delta \chi;
   \eqno {\rm (C^6)}$$
and that of the second order is given by the following equation,
deduced from (O${}^4$), (N${}^4$), (B${}^6$),

\vfill\eject  % Page break necessary with current page size

$$\eqalignno{
V \left\{
         {\delta^2 \Omega \over \delta \sigma^2}
            {\delta^2 \Omega \over \delta \tau^2}
       - \left(
            {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
            \right)^2
       + {\delta^2 \Omega \over \delta \tau^2}
            {\delta^2 \Omega \over \delta \upsilon^2}
       - \left(
            {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
            \right)^2
       + {\delta^2 \Omega \over \delta \upsilon^2}
            {\delta^2 \Omega \over \delta \sigma^2}
       - \left(
            {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
            \right)^2
      \right\}
      \hskip -324pt \cr
   &\mathrel{\phantom{=}}
      \quad \times
   \left(
      {\delta^2 V + V \, \delta'^2 \Omega
         + 2 \, \delta V \, \delta' \Omega
         \over \sigma^2 + \tau^2 + \upsilon^2}
         \right) \cr
   &= {\delta \Omega^2 \over \delta \sigma^2} \left\{
            {\delta \Omega \over \delta \tau} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right)
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta \Omega^2 \over \delta \tau^2} \left\{
            {\delta \Omega \over \delta \upsilon} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right)
          - {\delta \Omega \over \delta \sigma} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta \Omega^2 \over \delta \upsilon^2} \left\{
            {\delta \Omega \over \delta \sigma} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right)
          - {\delta \Omega \over \delta \tau} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta \Omega^2 \over \delta \sigma \, \delta \tau} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \tau} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr}
         \right\} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \upsilon} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \sigma} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr}
      \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta \Omega^2 \over \delta \tau \, \delta \upsilon} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \upsilon} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \sigma} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr}
         \right\} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \sigma} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \tau} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr}
      \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta \Omega^2 \over \delta \upsilon \, \delta \sigma} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \sigma} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \tau} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr}
         \right\} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \tau} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr}
      \right\};
   &{\rm (D^6)}\cr}$$
in which the symbol $\delta'$ has the same meaning as before, so
that as $x'$~$y'$~$z'$ do not enter into the composition of
the function $\Omega$, $\delta'$ refers here to the variation of
colour only.  This equation (D${}^6$) may be put under the
following simpler form,

\vfill\eject  % Page break necessary with current page size

$$\eqalignno{
{V \over v} (\delta^2 V + V \, \delta'^2 \Omega
         + 2 \, \delta V \, \delta' \Omega)
         \hskip -72pt \cr
   &= {\delta^2 v \over \delta \alpha^2} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 v \over \delta \beta^2} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 v \over \delta \gamma^2} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta^2 v \over \delta \beta \, \delta \gamma} \left(
               \delta y - \delta y' - V \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta^2 v \over \delta \gamma \, \delta \alpha} \left(
               \delta z - \delta z' - V \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \left(
               \delta x - \delta x' - V \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right),
   &{\rm (E^6)}\cr}$$
if we attend to the equations already established, in the second
number,
$$\multieqalign{
{\alpha \over v} &= {\delta \Omega \over \delta \sigma}, &
{\beta  \over v} &= {\delta \Omega \over \delta \tau}, &
{\gamma \over v} &= {\delta \Omega \over \delta \upsilon}, &
- {1 \over v} {\delta v \over \delta \chi}
   &= {\delta \Omega \over \delta \chi},\cr
\sigma   &= {\delta v \over \delta \alpha}, &
\tau     &= {\delta v \over \delta \beta}, &
\upsilon &= {\delta v \over \delta \gamma}, \cr}$$
and to the relations which result from these, by differentiation
and elimination.  For thus we obtain
$$\left. \eqalign{
\delta {\alpha \over v}
      - \delta' {\delta \Omega \over \delta \sigma}
   &=   {\delta^2 \Omega \over \delta \sigma^2}
         \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}
         \, \delta {\delta v \over \delta \gamma},\cr
\delta {\beta  \over v}
      - \delta' {\delta \Omega \over \delta \tau}
   &=   {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \tau^2}
         \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
         \, \delta {\delta v \over \delta \gamma},\cr
\delta {\gamma \over v}
      - \delta' {\delta \Omega \over \delta \upsilon}
   &=   {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}
         \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
         \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \upsilon^2}
         \, \delta {\delta v \over \delta \gamma},\cr
- \delta \left( {1 \over v} {\delta v \over \delta \chi} \right)
      - \delta' {\delta \Omega \over \delta \chi}
   &=   {\delta^2 \Omega \over \delta \sigma   \, \delta \chi}
         \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \tau     \, \delta \chi}
         \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \upsilon \, \delta \chi}
         \, \delta {\delta v \over \delta \gamma},\cr}
   \right\}
   \eqno {\rm (F^6)}$$
in which $v$ is considered as a homogeneous function of the first
dimension of $\alpha$,~$\beta$,~$\gamma$, involving also the
colour $\chi$; and in which, although the three variations
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
are connected by the relation
$\alpha \, \delta \alpha + \beta \, \delta \beta + \gamma \, \delta \gamma
   = 0$,
yet we may treat these variations as independent; because, if we
introduced indeterminate multipliers of
$\alpha \, \delta \alpha + \beta \, \delta \beta + \gamma \, \delta \gamma$,
in (F${}^6$), to allow for the relation, we should find that
these multipliers vanish, on account of the homogeneity of $v$.
And if we put for abridgment
$$\omega''
    =   {\delta^2 \Omega \over \delta \sigma^2}
         {\delta^2 \Omega \over \delta \tau^2}
      - \left(
         {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \right)^2
      + {\delta^2 \Omega \over \delta \tau^2}
         {\delta^2 \Omega \over \delta \upsilon^2}
      - \left(
         {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
         \right)^2
      + {\delta^2 \Omega \over \delta \upsilon^2}
         {\delta^2 \Omega \over \delta \sigma^2}
      - \left(
         {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
         \right)^2,
   \eqno {\rm (G^6)}$$
the equations (F${}^6$) give the following formula for
$\delta^2 v$, that is, for the second variation of $v$, taken as
if $\alpha$~$\beta$~$\gamma$~$\chi$, were four independent
variables,
$$\eqalignno{
{v \omega'' \over \sigma^2 + \tau^2 + \upsilon^2}
   (\delta^2 v + v \, \delta'^2 \Omega
      + 2 \, \delta v \, \delta' \Omega)
   \hskip -144pt \cr
   &= {\delta^2 \Omega \over \delta \sigma^2} \left\{
            {\delta \Omega \over \delta \upsilon} \left(
               \delta \beta  - v \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right)
          - {\delta \Omega \over \delta \tau} \left(
               \delta \gamma - v \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 \Omega \over \delta \tau^2} \left\{
            {\delta \Omega \over \delta \sigma} \left(
               \delta \gamma - v \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right)
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta \alpha - v \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {\delta^2 \Omega \over \delta \upsilon^2} \left\{
            {\delta \Omega \over \delta \tau} \left(
               \delta \alpha - v \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right)
          - {\delta \Omega \over \delta \sigma} \left(
               \delta \beta  - v \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right)
         \right\}^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta^2 \Omega \over \delta \sigma \, \delta \tau} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \upsilon} \left(
               \delta \beta  - v \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \tau} \left(
               \delta \gamma - v \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr}
         \right\} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \sigma} \left(
               \delta \gamma - v \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta \alpha - v \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta^2 \Omega \over \delta \tau \, \delta \upsilon} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \sigma} \left(
               \delta \gamma - v \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \upsilon} \left(
               \delta \alpha - v \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr}
         \right\} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \tau} \left(
         \displaystyle
               \delta \alpha - v \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \sigma} \left(
               \delta \beta  - v \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + 2 {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \tau} \left(
               \delta \alpha - v \, \delta'
                  {\delta \Omega \over \delta \sigma}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \sigma} \left(
               \delta \beta  - v \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr}
         \right\} \left\{
      \matrix{
         \displaystyle
         \mathbin{\phantom{-}}
            {\delta \Omega \over \delta \upsilon} \left(
               \delta \beta  - v \, \delta'
                  {\delta \Omega \over \delta \tau}
                  \right) \cr
         \displaystyle
          - {\delta \Omega \over \delta \tau} \left(
               \delta \gamma - v \, \delta'
                  {\delta \Omega \over \delta \upsilon}
                  \right) \cr}
         \right\};
   &{\rm (H^6)}\cr}$$
which justifies the passage from (D${}^6$) to (E${}^6$), and
expresses the law of dependence of the partial differential
coefficients of the second order of the function $v$ on those of
$\Omega$, for the case of a uniform medium.

If the medium be not uniform, and if we would still express the
law of this dependence, we have only to change $\delta'$, in the
four equations (F${}^6$), to a new
characteristic~$\delta_{\prime\prime}$ referring to the
variations of $x$~$y$~$z$~$\chi$, and to combine the four thus
altered with the three following,
$$\left. \eqalign{
- \delta \left( {1 \over v} {\delta v \over \delta x} \right)
      - \delta_{\prime\prime} {\delta \Omega \over \delta x}
   &=   {\delta^2 \Omega \over \delta \sigma   \, \delta x}
            \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \tau     \, \delta x}
            \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \upsilon \, \delta x}
            \, \delta {\delta v \over \delta \gamma},\cr
- \delta \left( {1 \over v} {\delta v \over \delta y} \right)
      - \delta_{\prime\prime} {\delta \Omega \over \delta y}
   &=   {\delta^2 \Omega \over \delta \sigma   \, \delta y}
            \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \tau     \, \delta y}
            \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \upsilon \, \delta y}
            \, \delta {\delta v \over \delta \gamma},\cr
- \delta \left( {1 \over v} {\delta v \over \delta z} \right)
      - \delta_{\prime\prime} {\delta \Omega \over \delta z}
   &=   {\delta^2 \Omega \over \delta \sigma   \, \delta z}
            \, \delta {\delta v \over \delta \alpha}
      + {\delta^2 \Omega \over \delta \tau     \, \delta z}
            \, \delta {\delta v \over \delta \beta}
      + {\delta^2 \Omega \over \delta \upsilon \, \delta z}
            \, \delta {\delta v \over \delta \gamma},\cr}
   \right\}
   \eqno {\rm (I^6)}$$
in which $\delta_{\prime\prime}$ is the same new characteristic,
and which are deduced from the equations already established for
variable media,
$$- {1 \over v} {\delta v \over \delta x}
   = {\delta \Omega \over \delta x},\quad
  - {1 \over v} {\delta v \over \delta y}
   = {\delta \Omega \over \delta y},\quad
  - {1 \over v} {\delta v \over \delta z}
   = {\delta \Omega \over \delta z}:$$
and we are conducted to a formula for $\delta^2 v$, which no
otherwise differs from (H${}^6$) than by having
$\delta_{\prime\prime}$ instead of $\delta'$ throughout.

And if, reciprocally, we would express the law of dependence of
the coefficients of $\Omega$ of the second order, on those of
$v$, we may do so by the following general formula,
$$\eqalignno{
v'' v^2 ( v \, \delta^2 \Omega + \delta_{\prime\prime}^2 v
      + 2 \, \delta_{\prime\prime} v \, \delta \Omega)
   \hskip -108pt \cr
   &= {\delta^2 v \over \delta \alpha^2} \left\{
            \upsilon \left(
               \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta}
                  \right)
          - \tau     \left(
               \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma}
                  \right)
         \right\}^2 \cr
   &+ {\delta^2 v \over \delta \beta^2} \left\{
            \sigma   \left(
               \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma}
                  \right)
          - \upsilon \left(
               \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha}
                  \right)
         \right\}^2 \cr
   &+ {\delta^2 v \over \delta \gamma^2} \left\{
            \tau     \left(
               \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha}
                  \right)
          - \sigma   \left(
               \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta}
                  \right)
         \right\}^2 \cr
   &+ 2 {\delta^2 v \over \delta \alpha \, \delta \beta} \left\{
            \upsilon \left(
               \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta}
                  \right)
          - \tau     \left(
               \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma}
                  \right)
         \right\} \left\{
            \sigma   \left(
               \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma}
                  \right)
          - \upsilon \left(
               \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha}
                  \right)
         \right\} \cr
   &+ 2 {\delta^2 v \over \delta \beta \, \delta \gamma} \left\{
            \sigma   \left(
               \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma}
                  \right)
          - \upsilon \left(
               \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha}
                  \right)
         \right\} \left\{
            \tau     \left(
         \displaystyle
               \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha}
                  \right)
          - \sigma   \left(
               \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta}
                  \right)
         \right\} \cr
   &+ 2 {\delta^2 v \over \delta \gamma \, \delta \alpha} \left\{
            \tau     \left(
               \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha}
                  \right)
          - \sigma   \left(
               \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta}
                  \right)
         \right\} \left\{
            \upsilon \left(
               \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta}
                  \right)
          - \tau     \left(
               \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma}
                  \right)
         \right\};\cr
   & &{\rm (K^6)}\cr}$$
in which $\delta_{\prime\prime}$ refers still to the variations
$x$,~$y$,~$z$,~$\chi$, and in which $v''$ has the same meaning as
in the First Supplement, namely
$$v''
    =   {\delta^2 v \over \delta \alpha^2}
         {\delta^2 v \over \delta \beta^2}
      - \left(
         {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right)^2
      + {\delta^2 v \over \delta \beta^2}
         {\delta^2 v \over \delta \gamma^2}
      - \left(
         {\delta^2 v \over \delta \beta \, \delta \gamma}
         \right)^2
      + {\delta^2 v \over \delta \gamma^2}
         {\delta^2 v \over \delta \alpha^2}
      - \left(
         {\delta^2 v \over \delta \gamma \, \delta \alpha}
         \right)^2:
   \eqno {\rm (L^6)}$$
this quantity $v''$ is also connected with the $\omega''$ of
(G${}^6$) (H${}^6$), by the relation
$$v'' \omega'' = {\sigma^2 + \tau^2 + \upsilon^2 \over v^4}.
   \eqno {\rm (M^6)}$$
The formula (K${}^6$) is equivalent to twenty-eight separate
expressions for the partial differential coefficients of
$\Omega$, of the second order, which extend to variable as well as
to uniform media: the formula gives, for example, the six
following general expressions, which enable us to introduce the
coefficients of the function $v$, of the second order, instead of
those of $\Omega$, if it be thought desirable so to do, in many
of the general equations of the present memoir, as the
expressions contained in (H${}^6$) would enable us to introduce
$\Omega$ instead of $v$, in many of those of the First
Supplement:

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
{\delta^2 \Omega \over \delta \sigma^2}
   &= {1 \over v'' v^3} \left(
            \tau^2     {\delta^2 v \over \delta \gamma^2}
          + \upsilon^2 {\delta^2 v \over \delta \beta^2}
          - 2 \tau \upsilon {\delta^2 v \over \delta \beta \, \delta \gamma}
         \right);\cr
{\delta^2 \Omega \over \delta \tau^2}
   &= {1 \over v'' v^3} \left(
            \upsilon^2 {\delta^2 v \over \delta \alpha^2}
          + \sigma^2   {\delta^2 v \over \delta \gamma^2}
          - 2 \upsilon \sigma {\delta^2 v \over \delta \gamma \, \delta \alpha}
         \right);\cr
{\delta^2 \Omega \over \delta \upsilon^2}
   &= {1 \over v'' v^3} \left(
            \sigma^2   {\delta^2 v \over \delta \beta^2}
          + \tau^2     {\delta^2 v \over \delta \alpha^2}
          - 2 \sigma \tau {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right);\cr
{\delta^2 \Omega \over \delta \sigma \, \delta \tau}
   &= {1 \over v'' v^3} \left(
          - \upsilon^2      {\delta^2 v \over \delta \alpha \, \delta \beta}
          + \tau \upsilon   {\delta^2 v \over \delta \gamma \, \delta \alpha}
          + \upsilon \sigma {\delta^2 v \over \delta \beta \, \delta \gamma}
          - \sigma \tau     {\delta^2 v \over \delta \gamma^2}
         \right);\cr
{\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
   &= {1 \over v'' v^3} \left(
          - \sigma^2        {\delta^2 v \over \delta \beta \, \delta \gamma}
          + \upsilon \sigma {\delta^2 v \over \delta \alpha \, \delta \beta}
          + \sigma \tau     {\delta^2 v \over \delta \gamma \, \delta \alpha}
          - \tau \upsilon   {\delta^2 v \over \delta \alpha^2}
         \right);\cr
{\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
   &= {1 \over v'' v^3} \left(
          - \tau^2          {\delta^2 v \over \delta \gamma \, \delta \alpha}
          + \sigma \tau     {\delta^2 v \over \delta \beta \, \delta \gamma}
          + \tau \upsilon   {\delta^2 v \over \delta \alpha \, \delta \beta}
          - \upsilon \sigma {\delta^2 v \over \delta \beta^2}
         \right).\cr}
   \right\}
   \eqno {\rm (N^6)}$$

To make more complete this theory of the coefficients of the
function $\Omega$, which determines the nature of the final
uniform or variable medium by the manner of its dependence on the
seven variables
$\sigma$~$\tau$~$\upsilon$ $x$~$y$~$z$~$\chi$,
and is supposed to have been so prepared that $\Omega + 1$ is
homogeneous of the first dimension relatively to
$\sigma$~$\tau$~$\upsilon$, let us investigate the connexion of
these coefficients of $\Omega$ with those of the simpler though
less symmetric function $\upsilon$, considered as depending on
the six other variables
$\sigma$~$\tau$ $x$~$y$~$z$~$\chi$
by the relation $\Omega = 0$.  For this purpose we are to combine
the differentials of that relation with the conditions of
homogeneity (B${}^4$) (C${}^4$), and with the following other
conditions of the same kind, which are only useful in variable
media,
$$\left. \eqalign{
      \sigma   {\delta^2 \Omega \over \delta \sigma   \, \delta x}
    + \tau     {\delta^2 \Omega \over \delta \tau     \, \delta x}
    + \upsilon {\delta^2 \Omega \over \delta \upsilon \, \delta x}
   &= {\delta \Omega \over \delta x},\cr
      \sigma   {\delta^2 \Omega \over \delta \sigma   \, \delta y}
    + \tau     {\delta^2 \Omega \over \delta \tau     \, \delta y}
    + \upsilon {\delta^2 \Omega \over \delta \upsilon \, \delta y}
   &= {\delta \Omega \over \delta y},\cr
      \sigma   {\delta^2 \Omega \over \delta \sigma   \, \delta z}
    + \tau     {\delta^2 \Omega \over \delta \tau     \, \delta z}
    + \upsilon {\delta^2 \Omega \over \delta \upsilon \, \delta z}
   &= {\delta \Omega \over \delta z}.\cr}
   \right\}
   \eqno {\rm (O^6)}$$
In this manner we find, for the first order,
$$\delta \Omega = \lambda \left(
      \delta \upsilon
       - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
       - {\delta \upsilon \over \delta \tau}   \, \delta \tau
       - {\delta \upsilon \over \delta x}      \, x
       - {\delta \upsilon \over \delta y}      \, y
       - {\delta \upsilon \over \delta z}      \, z
       - {\delta \upsilon \over \delta \chi}   \, \delta \chi
      \right);
   \eqno {\rm (P^6)}$$
that is
$$\left. \multieqalign{
{\delta \Omega \over \delta \sigma}
   &= - \lambda {\delta \upsilon \over \delta \sigma}; &
{\delta \Omega \over \delta \tau}
   &= - \lambda {\delta \upsilon \over \delta \tau}; &
{\delta \Omega \over \delta \upsilon}
   &= \lambda; \cr
{\delta \Omega \over \delta x}
   &= - \lambda {\delta \upsilon \over \delta x}; &
{\delta \Omega \over \delta y}
   &= - \lambda {\delta \upsilon \over \delta y}; &
{\delta \Omega \over \delta z}
   &= - \lambda {\delta \upsilon \over \delta z}; &
{\delta \Omega \over \delta \chi}
   &= - \lambda {\delta \upsilon \over \delta \chi}; \cr}
   \right\}
   \eqno {\rm (Q^6)}$$
$\lambda$ being a multiplier introduced for the purpose of
treating the variations of
$\sigma$~$\tau$~$\upsilon$ $x$~$y$~$z$~$\chi$
as independent; and to determine the value of this multiplier we
have, by the condition of homogeneity (B${}^4$),
$$\lambda \left(
   \upsilon
      - \sigma {\delta \upsilon \over \delta \sigma}
      - \tau   {\delta \upsilon \over \delta \tau}
      \right)
   = \Omega + 1 = 1:
   \eqno {\rm (R^6)}$$
the coefficients of $\Omega$ of the first order are therefore
known, and we have for example,
$${\delta \Omega \over \delta \upsilon}
   = \lambda
   = {1 \over \displaystyle \upsilon
      - \sigma {\delta \upsilon \over \delta \sigma}
      - \tau   {\delta \upsilon \over \delta \tau}}.
   \eqno {\rm (S^6)}$$
Again, for the second order,
$$\left. \eqalign{
\delta {\delta \Omega \over \delta \sigma}
   &= - \delta \mathbin{.} \lambda {\delta \upsilon \over \delta \sigma}
         + \lambda_1 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr
\delta {\delta \Omega \over \delta \tau}
   &= - \delta \mathbin{.} \lambda {\delta \upsilon \over \delta \tau}
         + \lambda_2 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr
\delta {\delta \Omega \over \delta \upsilon}
   &= \phantom{-} \delta \lambda
            \hskip 20pt
         + \lambda_3 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr
\delta {\delta \Omega \over \delta x}
   &= - \delta \mathbin{.} \lambda {\delta \upsilon \over \delta x}
         + \lambda_4 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr
\delta {\delta \Omega \over \delta y}
   &= - \delta \mathbin{.} \lambda {\delta \upsilon \over \delta y}
         + \lambda_5 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr
\delta {\delta \Omega \over \delta z}
   &= - \delta \mathbin{.} \lambda {\delta \upsilon \over \delta z}
         + \lambda_6 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr
\delta {\delta \Omega \over \delta \chi}
   &= - \delta \mathbin{.} \lambda {\delta \upsilon \over \delta \chi}
         + \lambda_7 \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right);\cr}
   \right\}
   \eqno {\rm (T^6)}$$
in which, by (C${}^4$) (O${}^6$) (Q${}^6$), the multipliers
$\lambda_1,\ldots\, \lambda_7$, have the following values,
$$\left. \eqalign{
\lambda_1
   &= \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \mathbin{.} \lambda {\delta \upsilon \over \delta \sigma};\cr
\lambda_2
   &= \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \mathbin{.} \lambda {\delta \upsilon \over \delta \tau};\cr
\lambda_3
   &= - \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \lambda;\cr
\lambda_4
   &= \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \mathbin{.} \lambda {\delta \upsilon \over \delta x}
      - \lambda^2 {\delta \upsilon \over \delta x};\cr
\lambda_5
   &= \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \mathbin{.} \lambda {\delta \upsilon \over \delta y}
      - \lambda^2 {\delta \upsilon \over \delta y};\cr
\lambda_6
   &= \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \mathbin{.} \lambda {\delta \upsilon \over \delta z}
      - \lambda^2 {\delta \upsilon \over \delta z};\cr
\lambda_7
   &= \lambda \left(
            \sigma {\delta \over \delta \sigma}
          + \tau {\delta \over \delta \tau}
         \right) \mathbin{.} \lambda {\delta \upsilon \over \delta \chi}
      - \lambda^2 {\delta \upsilon \over \delta \chi};\cr}
   \right\}
   \eqno {\rm (U^6)}$$
$\lambda$, like $\upsilon$, being here treated as a function of
$\sigma$,~$\tau$, $x$,~$y$,~$z$,~$\chi$: and if we put, as usual,
$$\delta^2 \Omega
   =    \delta \sigma   \, \delta {\delta \Omega \over \delta \sigma}
     +  \delta \tau     \, \delta {\delta \Omega \over \delta \tau}
     +  \delta \upsilon \, \delta {\delta \Omega \over \delta \upsilon}
     +  \delta x \, \delta {\delta \Omega \over \delta x}
     +  \delta y \, \delta {\delta \Omega \over \delta y}
     +  \delta z \, \delta {\delta \Omega \over \delta z}
     +  \delta \chi     \, \delta {\delta \Omega \over \delta \chi},
   \eqno {\rm (V^6)}$$
and similarly
$$\delta^2 \upsilon
   =    \delta \sigma   \, \delta {\delta \upsilon \over \delta \sigma}
     +  \delta \tau     \, \delta {\delta \upsilon \over \delta \tau}
     +  \delta x \, \delta {\delta \upsilon \over \delta x}
     +  \delta y \, \delta {\delta \upsilon \over \delta y}
     +  \delta z \, \delta {\delta \upsilon \over \delta z}
     +  \delta \chi     \, \delta {\delta \upsilon \over \delta \chi},
   \eqno {\rm (W^6)}$$
we find
$$\eqalignno{
\delta^2 \Omega
   &= - \lambda \, \delta^2 \upsilon \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 2 \, \delta \lambda \mathbin{.} \left(
         \delta \upsilon
          - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
          - {\delta \upsilon \over \delta \tau}   \, \delta \tau
          - {\delta \upsilon \over \delta x}      \, \delta x
          - {\delta \upsilon \over \delta y}      \, \delta y
          - {\delta \upsilon \over \delta z}      \, \delta z
          - {\delta \upsilon \over \delta \chi}   \, \delta \chi
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \lambda^3 \left(
            \sigma^2 {\delta^2 \upsilon \over \delta \sigma^2}
          + 2 \sigma \tau
               {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
          + \tau^2 {\delta^2 \upsilon \over \delta \tau^2}
         \right) \left(
            \delta \upsilon
             - {\delta \upsilon \over \delta \sigma} \, \delta \sigma
             - \hbox{\&c.}
            \right)^2,
   &{\rm (X^6)}\cr}$$
in which
$$\delta \lambda
   = \lambda^2 \left(
            \sigma \, \delta {\delta \upsilon \over \delta \sigma}
          + \tau   \, \delta {\delta \upsilon \over \delta \tau}
          - {\delta \upsilon \over \delta x} \, \delta x
          - {\delta \upsilon \over \delta y} \, \delta y
          - {\delta \upsilon \over \delta z} \, \delta z
          - {\delta \upsilon \over \delta \chi} \, \delta \chi
         \right),
   \eqno {\rm (Y^6)}$$
and which is equivalent to twenty-eight expressions for the
partial differential coefficients of $\Omega$ of the second
order: it gives, for example,
$${\delta^2 \Omega \over \delta \upsilon^2}
   = {\displaystyle
            \sigma^2 {\delta^2 \upsilon \over \delta \sigma^2}
          + 2 \sigma \tau
               {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
          + \tau^2 {\delta^2 \upsilon \over \delta \tau^2}
      \over \displaystyle \left(
            \sigma {\delta \upsilon \over \delta \sigma}
          + \tau {\delta \upsilon \over \delta \tau}
          - \upsilon
         \right)^3}.
   \eqno {\rm (Z^6)}$$

And since the forms of the connected functions $\Omega$,
$\upsilon$, $v$, of which each expresses the optical properties
of the final medium, may be deduced, by the method of the second
number, from the form of the characteristic function~$V$, it is
evident that their partial differential coefficients also, of all
orders, are not only related to each other, but may be deduced
from the coefficients of that one characteristic function.

\bigbreak

{\sectiontitle
General Formula for Reflection or Refraction, Ordinary or
Extraordinary.  Changes of $V$, $W$, $T$.  The Difference
$\Delta V$ is $= 0$; $\Delta W = \Delta T = $ a Homogeneous
Function of the First Dimension of the Differences
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$,
depending on the Shape and Position of the Reflecting or
Refracting Surface.  Theorem of Maxima and Minima, for the
Elimination of the Incident Variables.  Combinations of
Reflectors or Refractors.  Compound and Component
Combinations.\par}

\nobreak\bigskip

11.
Let us now endeavour to improve our theory of the
characteristic and related functions, by applying the methods of
the present memoir to improve the determination given in the
First Supplement, of the sudden changes produced in these
functions and in their coefficients, by reflexion or refraction,
ordinary or extraordinary.

The general formula of such changes, which easily results from
the nature of the characteristic function $V$, is
$$0 = \Delta V = V_2 - V_1;
   \eqno {\rm (A^7)}$$
$V_1$, $V_2$, being the two successive forms of the function $V$,
before and after the reflexion or refraction; and the final
co-ordinates $x$,~$y$,~$z$, in these forms, being connected by the
equation
$$0 = u(x,y,z)
   \eqno {\rm (B^7)}$$
of the reflecting or refracting surface.  The formula (A${}^7$)
may be differentiated any number of times with reference to the
final and initial co-ordinates and the colour, attending to the
relation (B${}^7$); and such differentiation, combined with the
properties of the final uniform or variable media, conducts to
the general laws of reflexion and refraction, and to all the
conditions necessary for determining the changes of the
coefficients of $V$, and therefore also of the connected
coefficients of $W$ and $T$, as well as to the laws of change of
the functions $V$, $W$, $T$, themselves.

Thus, for the first order, we have the general formula
$$\delta V_2 - \delta V_1 = \delta \Delta V
   = \lambda \, \delta u,
   \eqno {\rm (C^7)}$$
which, on account of the multiplier $\lambda$, and the
definitions $(E)$, resolves itself into the seven following,
$$\left. \matrix{\displaystyle
\Delta \sigma   = \lambda {\delta u \over \delta x};\quad
\Delta \tau     = \lambda {\delta u \over \delta y};\quad
\Delta \upsilon = \lambda {\delta u \over \delta z};\cr
   \noalign{\vskip 3pt} \displaystyle
\Delta \sigma'   = 0;\quad
\Delta \tau'     = 0;\quad
\Delta \upsilon' = 0;\quad
\Delta {\delta V \over \delta \chi} = 0:\cr}
   \right\}
   \eqno {\rm (D^7)}$$
the symbol $\Delta$ referring, as in (A${}^7$), to the finite
changes produced at the surface (B${}^7$), so that
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$,
denote the differences
$\sigma_2 - \sigma_1$, $\tau_2 - \tau_1$, $\upsilon_2 - \upsilon_1$,
between the new and the old values of
$\sigma$,~$\tau$,~$\upsilon$,
that is, of the partial differential coefficients of the first
order, of the characteristic function $V$, taken with respect to
the final co-ordinates.  The three first of the equations
(D${}^7$) contain the general laws of the sudden reflexion or
refraction of a straight or curved ray, ordinary or extraordinary;
because, when combined with the equation of the form (F),
$$0 = \Omega_2( \sigma_2, \tau_2, \upsilon_2, x, y, z, \chi ),
   \eqno {\rm (E^7)}$$
which expresses the nature of the final medium, they suffice, in
general, when that final medium is known, to determine, or at
least to restrict to a finite variety, the new values
$\sigma_2$,~$\tau_2$,~$\upsilon_2$, of the quantities
$\sigma$,~$\tau$,~$\upsilon$, on which the direction of the
reflected or refracted ray depends, if we know the old values
$\sigma_1$~$\tau_1$~$\upsilon_1$, which depend on the direction
of the incident ray and on the properties of the medium
containing it, and if we know also
$\chi$,~$x$,~$y$,~$z$, and the ratios of
$\displaystyle {\delta u \over \delta x}$,
$\displaystyle {\delta u \over \delta y}$,
$\displaystyle {\delta u \over \delta z}$,
that is, the colour, the point of incidence, and the normal to
the reflecting or refracting surface at that point.  A remarkable
case of indeterminateness, however, or rather two such cases,
will appear, when we come to treat, in a future number, of
external and internal conical refraction.

With respect to the new form $V_2$ of the characteristic
function $V$, it is to be determined by the two following
conditions; first, by the condition of satisfying, at the surface
(B${}^7$), the equation in finite differences (A${}^7$), that is,
by the condition of becoming equal to the value of the old form
$V_1$, when the final co-ordinates $x$,~$y$,~$z$, are connected by
the relation $u = 0$; and secondly by the condition of
satisfying, when the final co-ordinates are considered as
arbitrary, the partial differential equation of the form (C),
$$0 = \Omega_2 \left(
            {\delta V_2 \over \delta x},
            {\delta V_2 \over \delta y},
            {\delta V_2 \over \delta z},
            x, y, z, \chi \right),
   \eqno {\rm (F^7)}$$
if the final medium be variable, or the simpler partial
differential equation of the form (V${}'$), if that final medium
be uniform.  And as it has been already shown that the partial
differential equations relative to the characteristic function $V$,
may be transformed, and in the case of uniform media integrated,
by the help of the auxiliary functions $W$, $T$, it is useful to
consider here the changes of those auxiliary functions, which are
also otherwise interesting.

It easily follows from the definitions of $W$, $T$, that the
increments of these two functions, acquired in reflexion or
refraction, are equal to each other, and may be thus expressed,
$$\Delta W = \Delta T
   =   x \, \Delta \sigma
     + y \, \Delta \tau
     + z \, \Delta \upsilon.
   \eqno {\rm (G^7)}$$
And because the differences
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$,
are, by the general equations of reflexion or refraction
(D${}^7$), proportional to
$\displaystyle {\delta u \over \delta x}$,
$\displaystyle {\delta u \over \delta y}$,
$\displaystyle {\delta u \over \delta z}$,
we may consider these differences as equal to the projections, on
the rectangular axes of the co-ordinates $x$,~$y$,~$z$, of a
straight line
$= \surd ( \Delta \sigma^2 + \Delta \tau^2 + \Delta \upsilon^2)$,
perpendicular to the reflecting or refracting surface at the
point of incidence, and making with the axes of co-ordinates
angles of which the cosines may be called $n_x$,~$n_y$,~$n_z$; in
such a manner that we shall have
$$\left. \eqalign{
\Delta \sigma
   &= n_x \surd ( \Delta \sigma^2 + \Delta \tau^2 + \Delta \upsilon^2);\cr
\Delta \tau
   &= n_y \surd ( \Delta \sigma^2 + \Delta \tau^2 + \Delta \upsilon^2);\cr
\Delta \upsilon
   &= n_z \surd ( \Delta \sigma^2 + \Delta \tau^2 + \Delta \upsilon^2);\cr
\Delta W = \Delta T
   &= (x n_x + y n_y + z n_z)
          \surd ( \Delta \sigma^2 + \Delta \tau^2 + \Delta \upsilon^2).\cr}
   \right\}
   \eqno {\rm (H^7)}$$
Now the quantity $x n_x + y n_y + z n_z$ is equal, abstracting
from sign, to the perpendicular let fall from the origin of
co-ordinates on the plane which touches the reflecting or
refracting surface at the point of incidence; it is therefore
constant if that surface be plane, and in general it may be
considered as a function of the ratios of
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$,
because when those ratios are given we know the direction of the
normal, and therefore, if the surface be curved and given, we
know the point of incidence, or at least can in general restrict
that point to a finite number of positions: we have therefore in
general
$$\Delta W = \Delta T
   = f( \Delta \sigma, \Delta \tau, \Delta \upsilon ),
   \eqno {\rm (I^7)}$$
the function $f$ being homogeneous of the first dimension, and
depending for its form on the shape and position of the
reflecting or refracting surface, from the equation (B${}^7$) of
which surface it is to be deduced, by eliminating
$x$~$y$~$z$~$\lambda$ between the equations
(B${}^7$) (G${}^7$) and the three first of those marked
(D${}^7$).  We have also
$$\left. \matrix{\displaystyle
{f \over \Delta \upsilon}
   = \phi \left(
            {\Delta \sigma \over \Delta \upsilon},
            {\Delta \tau   \over \Delta \upsilon}
            \right);\quad
{\Delta \sigma \over \Delta \upsilon}
   = - {\delta z \over \delta x};\quad
{\Delta \tau   \over \Delta \upsilon}
   = - {\delta z \over \delta y};\cr
   \noalign{\vskip 3pt}\displaystyle
z - x {\delta z \over \delta x} - y {\delta z \over \delta y}
   = \phi \left(
            - {\delta z \over \delta x},
            - {\delta z \over \delta y}
            \right);\cr}
   \right\}
   \eqno {\rm (K^7)}$$
the form therefore of the homogeneous function~$f$ may easily be
deduced from the equation of the surface (B${}^7$), by so
preparing that equation as to express
$\displaystyle z - x {\delta z \over \delta x} - y {\delta z \over \delta y}$
as a function $\phi$ of
$\displaystyle - {\delta z \over \delta x}$,
$\displaystyle - {\delta z \over \delta y}$,
which function $\phi$ reduces itself to a constant when the
surface is plane: and we have a simple expression for the
variation of the homogeneous function~$f$, namely
$$\delta f
   =   x \, \delta \Delta \sigma
     + y \, \delta \Delta \tau
     + z \, \delta \Delta \upsilon,
   \eqno {\rm (L^7)}$$
which, when the reflecting or refracting surface is curved,
resolves itself into the following remarkable expressions for the
co-ordinates of the point of incidence,
$$x = {\delta f \over \delta \Delta \sigma},\quad
  y = {\delta f \over \delta \Delta \tau},\quad
  z = {\delta f \over \delta \Delta \upsilon};
   \eqno {\rm (M^7)}$$
so that these co-ordinates, which, for a curved surface, we knew
before to be functions of the ratios
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$,
are now seen to be, for such a surface, the partial differential
coefficients of the homogeneous function~$f$.  When the surface
(B${}^7$) is plane, the differences
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$,
are no longer independent, since their ratios are then given; and
although the expression (L${}^7$) for $\delta f$ still holds, it
no longer resolves itself into the three equations (M${}^7$).

Having thus studied some of the chief properties of the common
increment~$f$, which the functions $W$, $T$, receive, in the act
of reflexion or refraction, we are prepared to investigate the
new forms $W_2$, $T_2$ of these functions $W$, $T$, considered as
depending on the new quantities
$\sigma_2$,~$\tau_2$,~$\upsilon_2$,
instead of the old
$\sigma_1$,~$\tau_1$,~$\upsilon_1$.
For this purpose we have first the equations
$$\left. \eqalign{
W_2 &= W_1 + f( \sigma_2 - \sigma_1, \tau_2 - \tau_1,
         \upsilon_2 - \upsilon_1),\cr
T_2 &= T_1 + f( \sigma_2 - \sigma_1, \tau_2 - \tau_1,
         \upsilon_2 - \upsilon_1),\cr}
   \right\}
   \eqno {\rm (N^7)}$$
by which $W_2$, $T_2$, at the reflecting or refracting surface,
are expressed as explicit functions of
$\sigma_1$~$\tau_1$~$\upsilon_1$ $\sigma_2$~$\tau_2$~$\upsilon_2$;
the expression of $W_2$ involving also
$x'$~$y'$~$z'$~$\chi$,
and the expression of $T_2$ involving
$\sigma'$~$\tau'$~$\upsilon'$~$\chi$:
and to eliminate from these expressions the incident quantities
$\sigma_1$~$\tau_1$~$\upsilon_1$
we have, if the surface be curved, the following equations, in
which the symbol
$\delta_{\sigma_1, \tau_1, \upsilon_1}$
refers to the variation of those incident quantities,
$$\left. \matrix{\displaystyle
\delta_{\sigma_1, \tau_1, \upsilon_1} \mathbin{.} f
   =  - x \, \delta \sigma_1
      - y \, \delta \tau_1
      - z \, \delta \upsilon_1
   =  - \delta_{\sigma_1, \tau_1, \upsilon_1} \mathbin{.} W_1
   =  - \delta_{\sigma_1, \tau_1, \upsilon_1} \mathbin{.} T_1;\cr
   \noalign{\vskip 3pt}\displaystyle
   \hbox{and $\raise1ex\hbox{.}\,.\raise1ex\hbox{.}$}\quad
\delta_{\sigma_1, \tau_1, \upsilon_1} \mathbin{.} W_2 = 0;\quad
\delta_{\sigma_1, \tau_1, \upsilon_1} \mathbin{.} T_2 = 0;\cr}
   \right\}
   \eqno {\rm (O^7)}$$
we are therefore to disengage the incident quantities from the
expressions for $W_2$, $T_2$, by making each of those expressions
a maximum or minimum with respect to those quantities, attending
to the relation $\Omega_1 = 0$, between them; the phrase
{\it maximum or minimum\/} being employed with the usual
latitude.  For the case of a plane surface this method of
elimination fails, the form of $f$ becoming indeterminate, on
account of the constant ratios which then exist, by (K${}^7$)
or (D${}^7$), between
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$;
but these very ratios, combined with the relation $\Omega_1 = 0$,
between the quantities
$\sigma_1$~$\tau_1$~$\upsilon_1$,
enable us in this case to eliminate those quantities from $W_2$,
$T_2$.  And when we have thus determined the new forms $W_2$,
$T_2$, of the functions $W$, $T$, for the points of the reflecting
or refracting surface, we may extend these forms to the other
points of the final medium, if that medium be uniform, because
then the final rays are straight, and for any one such ray the
quantities
$\sigma_2$~$\tau_2$~$\upsilon_2$ $W_2$~$T_2$,
are constant; but if the final medium be variable, then the final
rays are curved, and the general forms of $W_2$, $T_2$, for
arbitrary points of the medium, are to be determined by
combinations of partial differential equations and equations in
finite differences, analogous to the combinations of such
equations in $V_2$, and easily deduced from the principles
already laid down.

It is easy to extend the foregoing remarks to any combination of
reflexions or refractions, and to show, for example, that in the
case of any combination of uniform media, producing any system of
polygon rays, ordinary or extraordinary, the auxiliary
function $T$ is equal to the following expression,
$$T = {\textstyle\sum} f ( \Delta \sigma, \Delta \tau, \Delta \upsilon),
   \eqno {\rm (P^7)}$$
that is, to the sum of all the homogeneous functions $f$ of the
differences of the quantities
$\sigma$,~$\tau$,~$\upsilon$,
obtained by considering the successive reflecting or refracting
surfaces: from which expression the intermediate quantities of
the form
$\sigma$,~$\tau$,~$\upsilon$,
are to be eliminated by making the expression a maximum or minimum
with respect to those intermediate quantities, attending to the
relations between them which result from the properties of the
media, and using, for plane surfaces, the other method of
elimination, founded on the ratios of
$\Delta \sigma$, $\Delta \tau$, $\Delta \upsilon$.
And when the function $T$ is known, we can deduce from it, by the
methods of the fourth number, the other auxiliary function $W$,
and the characteristic function $V$.

In general, for all optical combinations, whether with uniform or
with variable media, we have, by the definitions of the functions
$V$, $W$, $T$, and by the results of former numbers, the
following expressions,
$$\left. \eqalign{
V &= \int_0^s v \, ds;\quad
T  = \int_0^s  \left(
                  x {\delta v \over \delta x}
                + y {\delta v \over \delta y}
                + z {\delta v \over \delta z}
               \right) \, ds;\cr
W &= x' \sigma' + y' \tau' + z' \upsilon'
         + \int_0^s \left(
                  x {\delta v \over \delta x}
                + y {\delta v \over \delta y}
                + z {\delta v \over \delta z}
               \right) \, ds:\cr}
   \right\}
   \eqno {\rm (Q^7)}$$
$ds$ being, as before, the element of the curved or polygon ray;
and hence it follows that if we consider any total combination,
of $m + n - 1$ media, whether uniform or variable, as resulting
from two partial combinations, of $m$ and of $n$ media
respectively, combined so that the last medium of the one partial
combination ($m$) is the first of the other partial combination
($n$), and so that the final rays of the one partial combination
are the initial rays of the other, then the functions $V$, $T$,
(but not in general $W$,) for the total combination, are the sums
of the corresponding functions for the partial combinations: it
follows also from the general expressions for the variations of
these functions, that the intermediate variables, belonging to
the last medium of the first partial combination, or to the first
medium of the second, are to be eliminated from the sum, by the
condition of making that sum a maximum or minimum with respect to
them.  Analogous remarks apply to compound combinations, composed
of more than two component combinations. These properties of the
functions $V$, $T$, for total or resultant combinations, will be
useful in the theory of double and triple object-glasses, and
other compound optical instruments.

\vfill\eject  % Page break necessary with current page size

{\sectiontitle
Changes of the Coefficients of the Second Order, of $V$, $W$,
$T$, produced by Reflexion or Refraction.\par}

\nobreak\bigskip

12.
With respect to the changes produced by reflexion or
refraction in the coefficients of the second order, of the
characteristic function $V$, and therefore also of the connected
functions $W$, $T$, they may be deduced from the following
formula, analogous to (C${}^7$),
$$\delta^2 \Delta V = \delta^2 \mathbin{.} \lambda u
   = \lambda \, \delta^2 u + 2 \, \delta \lambda \, \delta u;
   \eqno {\rm (R^7)}$$
$u$, $\lambda$, having the same meanings as in
(B${}^7$) (C${}^7$); and the multiplier $\lambda$, which was
introduced also in the First Supplement, and was there regarded
as a function of the final co-ordinates $x$,~$y$,~$z$, being now
considered as involving also the initial co-ordinates
$x'$,~$y'$,~$z'$, and the chromatic index $\chi$.  The seven
variations
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$,
may be treated as independent in (R${}^7$), if we assign a proper
value to $\delta \lambda$, as a linear function of these seven
variations; so that we may deduce from (R${}^7$) the seven
following equations,
$$\left. \eqalign{
\Delta \delta {\delta V \over \delta x}
   &= \lambda \, \delta {\delta u \over \delta x}
      + {\delta \lambda \over \delta x} \, \delta u
      + {\delta u \over \delta x} \, \delta \lambda;\cr
\Delta \delta {\delta V \over \delta y}
   &= \lambda \, \delta {\delta u \over \delta y}
      + {\delta \lambda \over \delta y} \, \delta u
      + {\delta u \over \delta y} \, \delta \lambda;\cr
\Delta \delta {\delta V \over \delta z}
   &= \lambda \, \delta {\delta u \over \delta z}
      + {\delta \lambda \over \delta z} \, \delta u
      + {\delta u \over \delta z} \, \delta \lambda;\cr
\Delta \delta {\delta V \over \delta x'}
   &=   {\delta \lambda \over \delta x'} \, \delta u;\quad
\Delta \delta {\delta V \over \delta y'}
    =   {\delta \lambda \over \delta y'} \, \delta u;\quad
\Delta \delta {\delta V \over \delta z'}
    =   {\delta \lambda \over \delta z'} \, \delta u;\cr
\Delta \delta {\delta V \over \delta \chi}
   &=   {\delta \lambda \over \delta \chi} \, \delta u:\cr}
   \right\}
   \eqno {\rm (S^7)}$$
of which each may again be decomposed into seven others.  But of
the forty-nine expressions thus obtained for the changes of the
twenty-eight coefficients of $V$ of the second order, only
twenty-eight expressions are distinct; and these involve seven
multipliers as yet unknown, namely, the seven partial
differential coefficients of $\lambda$: however we can determine
these seven multipliers, and the twenty-eight coefficients of
$V_2$ of the second order, by introducing the seven additional
equations obtained by differentiating the partial differential
equation (F${}^7$), with respect to
$x$~$y$~$z$ $x'$~$y'$~$z'$~$\chi$.

The differential of the equation (F${}^7$) is
$$0 =    {\delta \Omega_2 \over \delta \sigma_2} \,
            \delta {\delta V_2 \over \delta x}
       + {\delta \Omega_2 \over \delta \tau_2} \,
            \delta {\delta V_2 \over \delta y}
       + {\delta \Omega_2 \over \delta \upsilon_2} \,
            \delta {\delta V_2 \over \delta z}
       + {\delta \Omega_2 \over \delta x} \, \delta x
       + {\delta \Omega_2 \over \delta y} \, \delta y
       + {\delta \Omega_2 \over \delta z} \, \delta z
       + {\delta \Omega_2 \over \delta \chi} \, \delta \chi;
   \eqno {\rm (T^7)}$$
and this, when combined with the three first equations (S${}^7$),
conducts to the following formula,
$$\eqalignno{
0 &=     {\delta \Omega_2 \over \delta \sigma_2} \,
            \delta {\delta V_1 \over \delta x}
       + {\delta \Omega_2 \over \delta \tau_2} \,
            \delta {\delta V_1 \over \delta y}
       + {\delta \Omega_2 \over \delta \upsilon_2} \,
            \delta {\delta V_1 \over \delta z}
       + {\delta \Omega_2 \over \delta x} \, \delta x
       + {\delta \Omega_2 \over \delta y} \, \delta y
       + {\delta \Omega_2 \over \delta z} \, \delta z
       + {\delta \Omega_2 \over \delta \chi} \, \delta \chi \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \lambda \left(
         {\delta \Omega_2 \over \delta \sigma_2} \,
            \delta {\delta u \over \delta x}
       + {\delta \Omega_2 \over \delta \tau_2} \,
            \delta {\delta u \over \delta y}
       + {\delta \Omega_2 \over \delta \upsilon_2} \,
            \delta {\delta u \over \delta z}
      \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \delta u \mathbin{.} \left(
         {\delta \Omega_2 \over \delta \sigma_2} \,
            {\delta \lambda \over \delta x}
       + {\delta \Omega_2 \over \delta \tau_2} \,
            {\delta \lambda \over \delta y}
       + {\delta \Omega_2 \over \delta \upsilon_2} \,
            {\delta \lambda \over \delta z}
      \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \delta \lambda \mathbin{.} \left(
         {\delta \Omega_2 \over \delta \sigma_2} \,
            {\delta u \over \delta x}
       + {\delta \Omega_2 \over \delta \tau_2} \,
            {\delta u \over \delta y}
       + {\delta \Omega_2 \over \delta \upsilon_2} \,
            {\delta u \over \delta z}
      \right);
   &{\rm (U^7)}\cr}$$
which resolves itself into seven separate equations, sufficient
to determine the seven multipliers
$${\delta \lambda \over \delta x},\quad
  {\delta \lambda \over \delta y},\quad
  {\delta \lambda \over \delta z},\quad
  {\delta \lambda \over \delta x'},\quad
  {\delta \lambda \over \delta y'},\quad
  {\delta \lambda \over \delta z'},\quad
  {\delta \lambda \over \delta \chi}.$$
Three of these seven equations into which (U${}^7$) resolves
itself, give, by a proper combination, a value for the trinomial
$$       {\delta \Omega_2 \over \delta \sigma_2} \,
            {\delta \lambda \over \delta x}
       + {\delta \Omega_2 \over \delta \tau_2} \,
            {\delta \lambda \over \delta y}
       + {\delta \Omega_2 \over \delta \upsilon_2} \,
            {\delta \lambda \over \delta z},$$
which enables us to eliminate that trinomial from (U${}^7$) and
so to deduce a value for $\delta \lambda$, which being combined
with (R${}^7$) gives,
$$\eqalignno{
   &\mathrel{\phantom{=}} \mathbin{\phantom{+}}
   \left\{ \eqalign{
     &\mathbin{\phantom{+}}
         \left( {\delta \Omega_2 \over \delta \sigma_2} \right)^2
         \left(   {\delta^2 V_1 \over \delta x^2}
                  + \lambda {\delta^2 u \over \delta x^2}
                  \right)
      + 2 {\delta \Omega_2 \over \delta \tau_2}
          {\delta \Omega_2 \over \delta \upsilon_2}
         \left(   {\delta^2 V_1 \over \delta y \, \delta z}
                  + \lambda {\delta^2 u \over \delta y \, \delta z}
                  \right) \cr
     &+  \left( {\delta \Omega_2 \over \delta \tau_2} \right)^2
         \left(   {\delta^2 V_1 \over \delta y^2}
                  + \lambda {\delta^2 u \over \delta y^2}
                  \right)
      + 2 {\delta \Omega_2 \over \delta \upsilon_2}
          {\delta \Omega_2 \over \delta \sigma_2}
         \left(   {\delta^2 V_1 \over \delta z \, \delta x}
                  + \lambda {\delta^2 u \over \delta z \, \delta x}
                  \right) \cr
     &+  \left( {\delta \Omega_2 \over \delta \upsilon_2} \right)^2
         \left(   {\delta^2 V_1 \over \delta z^2}
                  + \lambda {\delta^2 u \over \delta z^2}
                  \right)
      + 2 {\delta \Omega_2 \over \delta \sigma_2}
          {\delta \Omega_2 \over \delta \tau_2}
         \left(   {\delta^2 V_1 \over \delta x \, \delta y}
                  + \lambda {\delta^2 u \over \delta x \, \delta y}
                  \right) \cr
     &+   {\delta \Omega_2 \over \delta \sigma_2}
          {\delta \Omega_2 \over \delta x}
      +   {\delta \Omega_2 \over \delta \tau_2}
          {\delta \Omega_2 \over \delta y}
      +   {\delta \Omega_2 \over \delta \upsilon_2}
          {\delta \Omega_2 \over \delta z} \cr}
   \right\} \mathbin{.} \delta u^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
   - 2 \left(  {\delta \Omega_2 \over \delta \sigma_2}
                  {\delta u \over \delta x}
             + {\delta \Omega_2 \over \delta \tau_2}
                  {\delta u \over \delta y}
             + {\delta \Omega_2 \over \delta \upsilon_2}
                  {\delta u \over \delta z}
            \right) \, \delta u \cr
   &\mathrel{\phantom{=}}
   \times \left\{ \eqalign{
      &  {\delta \Omega_2 \over \delta \sigma_2} \left(
            \delta {\delta V_1 \over \delta x}
            + \lambda \, \delta {\delta u \over \delta x}
            \right)
       + {\delta \Omega_2 \over \delta \tau_2} \left(
            \delta {\delta V_1 \over \delta y}
            + \lambda \, \delta {\delta u \over \delta y}
            \right)
       + {\delta \Omega_2 \over \delta \upsilon_2} \left(
            \delta {\delta V_1 \over \delta z}
            + \lambda \, \delta {\delta u \over \delta z}
            \right) \cr
      &+ {\delta \Omega_2 \over \delta x} \, \delta x
       + {\delta \Omega_2 \over \delta y} \, \delta y
       + {\delta \Omega_2 \over \delta z} \, \delta z
       + {\delta \Omega_2 \over \delta \chi} \, \delta \chi \cr}
      \right\} \cr
   &= (\delta^2 V_2 - \delta^2 V_1 - \lambda \, \delta^2 u)
         \left(   {\delta \Omega_2 \over \delta \sigma_2}
                     {\delta u \over \delta x}
                + {\delta \Omega_2 \over \delta \tau_2}
                     {\delta u \over \delta y}
                + {\delta \Omega_2 \over \delta \upsilon_2}
                     {\delta u \over \delta z}
               \right)^2:
   &{\rm (V^7)}\cr}$$
a formula that is equivalent to twenty-eight separate expressions
for the twenty-eight coefficients of $V_2$, of the second order.
This formula supposes the rays to be reflected or refracted into
a variable medium; but it can be adapted to the simpler
supposition of reflexion or refraction into an uniform medium, by
merely making the quantities
$\displaystyle {\delta \Omega_2 \over \delta x}$,
$\displaystyle {\delta \Omega_2 \over \delta y}$,
$\displaystyle {\delta \Omega_2 \over \delta z}$,
vanish.  Whether the last medium be variable or uniform, the
formula (V${}^7$) gives,
$$\delta'^2 V_2 = \delta'^2 V_1;
   \eqno {\rm (W^7)}$$
$\delta'$ referring, as in former numbers of this Supplement, to
the variations $x'$,~$y'$,~$z'$,~$\chi$, alone, that is, to the
variations of the initial co-ordinates and of the colour; and the
final co-ordinates $x$~$y$~$z$ being those of any point on the
reflecting or refracting surface.  Thus the ten differential
coefficients, of the second order, of the characteristic
function $V$, like the four of the first order, taken with
respect to the initial co-ordinates and the colour, undergo no
sudden change by reflexion or refraction; but the differential
coefficients of both orders, which involve the final co-ordinates,
take suddenly new values which we have shown how to determine:
and from these new coefficients of $V$, we can deduce those of
$W$ and $T$, by the methods of the foregoing numbers.  The
coefficients thus found, of $W_2$ and $T_2$, remain unchanged
through the whole extent of the last reflected or refracted
portion of the ray, when this last portion is straight, the final
medium being uniform; but the coefficients of $V_2$, of the
second order, change gradually in passing from one point to
another, even of this straight portion, according to laws
deducible from their connexion, already explained, with the
constant coefficients of $W_2$.

The coefficients of $W_2$ and $T_2$ of the second and higher
orders may also be calculated, whether the last medium be uniform
or variable, by differentiating the expressions (N${}^7$), and
eliminating the variations of
$\sigma_1$~$\tau_1$~$\upsilon_1$
by the help of the conditions already mentioned, of maximum or
minimum.

Another method of calculating the changes produced in the partial
differential coefficients of $V$ of the second order, by
reflexion or refraction, ordinary or extraordinary, into a medium
uniform or variable, is to develope the second differential of the
general formula (A${}^7$), considering $\Delta V$ as a function
of the seven variables
$x$,~$y$,~$z$, $x'$,~$y'$,~$z'$,~$\chi$,
and considering $x$,~$y$,~$z$, as themselves functions of two
independent variables; for example, considering $z$ as a function
of $x$, $y$, of which the form is determined by the equation of
the reflecting or refracting surface.  In this manner we obtain,
besides the formula (W${}^7$), which is equivalent to ten
equations, the eleven following;
$$\left. \eqalign{
0  &= {\delta^2 \Delta V \over \delta x^2}
      + 2 {\delta^2 \Delta V \over \delta x \, \delta z}
           {\delta z \over \delta x}
      + {\delta^2 \Delta V \over \delta z^2}
           \left( {\delta z \over \delta x} \right)^2
      + {\delta \Delta V \over \delta z}
           {\delta^2 z \over \delta x^2};\cr
0  &= {\delta^2 \Delta V \over \delta y^2}
      + 2 {\delta^2 \Delta V \over \delta y \, \delta z}
           {\delta z \over \delta y}
      + {\delta^2 \Delta V \over \delta z^2}
           \left( {\delta z \over \delta y} \right)^2
      + {\delta \Delta V \over \delta z}
           {\delta^2 z \over \delta y^2};\cr
0  &= {\delta^2 \Delta V \over \delta x \, \delta y}
      + {\delta^2 \Delta V \over \delta x \, \delta z}
           {\delta z \over \delta y}
      + {\delta^2 \Delta V \over \delta y \, \delta z}
           {\delta z \over \delta x}
      + {\delta^2 \Delta V \over \delta z^2}
           {\delta z \over \delta x}
           {\delta z \over \delta y}
      + {\delta \Delta V \over \delta z}
           {\delta^2 z \over \delta x \, \delta y};\cr
0  &= {\delta^2 \Delta V \over \delta x \, \delta x'}
      + {\delta^2 \Delta V \over \delta z \, \delta x'}
           {\delta z \over \delta x};\quad
0   = {\delta^2 \Delta V \over \delta y \, \delta x'}
      + {\delta^2 \Delta V \over \delta z \, \delta x'}
           {\delta z \over \delta y};\cr
0  &= {\delta^2 \Delta V \over \delta x \, \delta y'}
      + {\delta^2 \Delta V \over \delta z \, \delta y'}
           {\delta z \over \delta x};\quad
0   = {\delta^2 \Delta V \over \delta y \, \delta y'}
      + {\delta^2 \Delta V \over \delta z \, \delta y'}
           {\delta z \over \delta y};\cr
0  &= {\delta^2 \Delta V \over \delta x \, \delta z'}
      + {\delta^2 \Delta V \over \delta z \, \delta z'}
           {\delta z \over \delta x};\quad
0   = {\delta^2 \Delta V \over \delta y \, \delta z'}
      + {\delta^2 \Delta V \over \delta z \, \delta z'}
           {\delta z \over \delta y};\cr
0  &= {\delta^2 \Delta V \over \delta x \, \delta \chi}
      + {\delta^2 \Delta V \over \delta z \, \delta \chi}
           {\delta z \over \delta x};\quad
0   = {\delta^2 \Delta V \over \delta y \, \delta \chi}
      + {\delta^2 \Delta V \over \delta z \, \delta \chi}
           {\delta z \over \delta y};\cr}
   \right\}
   \eqno {\rm (X^7)}$$
which may be put under the form
$$\left. \eqalign{
0 &= \Delta \left\{
      {\delta^2 V \over \delta x^2}
      + 2 {\delta^2 V \over \delta x \, \delta z}
           {\delta z \over \delta x}
      + {\delta^2 V \over \delta z^2}
           \left( {\delta z \over \delta x} \right)^2
      + {\delta V \over \delta z}
           {\delta^2 z \over \delta x^2};
      \right\};\cr
   &\mathrel{\phantom{=}} \hbox{\&c.};\cr}
   \right\}
   \eqno {\rm (Y^7)}$$
and are deduced by differentiation from the analogous equations
of the first order
$$0 = \Delta \left(
            {\delta V \over \delta x}
            + {\delta V \over \delta z} {\delta z \over \delta x}
            \right);\quad
  0 = \Delta \left(
            {\delta V \over \delta y}
            + {\delta V \over \delta z} {\delta z \over \delta y}
            \right).
   \eqno {\rm (Z^7)}$$
And the eleven equations thus deduced, when combined with the ten
given by (W${}^7$), and with the seven into which (T${}^7$)
resolves itself, suffice, in general, to determine the
twenty-eight coefficients of $V_2$ of the second order.

\bigbreak

{\sectiontitle
Changes produced by Transformation of Co-ordinates.  Nearly all
the foregoing Results may be extended to Oblique Co-ordinates.
The Fundamental Formula may be presented so as to extend even to
Polar or to any other marks of position, and new Auxiliary
Functions may then be found, analogous to, and including, the
Functions $W$, $T$: together with New and General Differential
and Integral Equations for Curved and Polygon Rays, Ordinary or
Extraordinary.\par}

\nobreak\bigskip

13.
In all the foregoing investigations, it has been supposed
that the final and initial co-ordinates,
$x$,~$y$,~$z$, $x'$,~$y'$,~$z'$, were referred to one common set
of rectangular axes.  But since it may be often convenient to
change the mode of marking the final and initial positions, let
us now express the old rectangular co-ordinates as linear
functions of new and more general co-ordinates
$x_\prime$, $y_\prime$, $z_\prime$, and
$x_\prime'$, $y_\prime'$, $z_\prime'$,
which may or may not be rectangular, and may or may not be
referred to one common set of final or initial axes.  For this
purpose we may employ the following formul{\ae},
$$\left. \eqalign{
x &= x_0 + x_{x_\prime} x_\prime
         + x_{y_\prime} y_\prime
         + x_{z_\prime} z_\prime;\cr
y &= y_0 + y_{x_\prime} x_\prime
         + y_{y_\prime} y_\prime
         + y_{z_\prime} z_\prime;\cr
z &= z_0 + z_{x_\prime} x_\prime
         + z_{y_\prime} y_\prime
         + z_{z_\prime} z_\prime;\cr
x' &= x_0' + x'_{x_\prime'} x_\prime'
           + x'_{y_\prime'} y_\prime'
           + x'_{z_\prime'} z_\prime';\cr
y' &= y_0' + y'_{x_\prime'} x_\prime'
           + y'_{y_\prime'} y_\prime'
           + y'_{z_\prime'} z_\prime';\cr
z' &= z_0' + z'_{x_\prime'} x_\prime'
           + z'_{y_\prime'} y_\prime'
           + z'_{z_\prime'} z_\prime';\cr}
   \right\}
   \eqno {\rm (A^8)}$$
in which each of the eighteen coefficients of the form
$x_{x_\prime}$ is the cosine of the angle between the directions
of the two corresponding semiaxes, so that these coefficients are
connected by the six following relations, on account of the
rectangularity of the old co-ordinates,
$$\left. \multieqalign{
x_{x_\prime}{}^2 + y_{x_\prime}{}^2 + z_{x_\prime}{}^2 &= 1; &
x'_{x_\prime'}{}^2 + y'_{x_\prime'}{}^2 + z'_{x_\prime'}{}^2 &= 1;\cr
x_{y_\prime}{}^2 + y_{y_\prime}{}^2 + z_{y_\prime}{}^2 &= 1; &
x'_{y_\prime'}{}^2 + y'_{y_\prime'}{}^2 + z'_{y_\prime'}{}^2 &= 1;\cr
x_{z_\prime}{}^2 + y_{z_\prime}{}^2 + z_{z_\prime}{}^2 &= 1; &
x'_{z_\prime'}{}^2 + y'_{z_\prime'}{}^2 + z'_{z_\prime'}{}^2 &= 1.\cr}
   \right\}
   \eqno {\rm (B^8)}$$
Let us also establish, according to the analogy of our former
notation, the following definitions similar to (P),
$$\left. \multieqalign{
\alpha_\prime &= {dx_\prime \over ds}, &
\beta_\prime  &= {dy_\prime \over ds}, &
\gamma_\prime &= {dz_\prime \over ds},\cr
\alpha_\prime' &= {dx_\prime' \over ds'}, &
\beta_\prime'  &= {dy_\prime' \over ds'}, &
\gamma_\prime' &= {dz_\prime' \over ds'},\cr}
   \right\}
   \eqno {\rm (C^8)}$$
and the following, similar to (E),
$$\left. \multieqalign{
\sigma_\prime   &= {\delta V \over \delta x_\prime}, &
\tau_\prime     &= {\delta V \over \delta y_\prime}, &
\upsilon_\prime &= {\delta V \over \delta z_\prime},\cr
\sigma_\prime'   &= - {\delta V \over \delta x_\prime'}, &
\tau_\prime'     &= - {\delta V \over \delta y_\prime'}, &
\upsilon_\prime' &= - {\delta V \over \delta z_\prime'},\cr}
   \right\}
   \eqno {\rm (D^8)}$$
we shall then have
$$\left. \eqalign{
\alpha &=   \alpha_\prime x_{x_\prime}
          + \beta_\prime  x_{y_\prime}
          + \gamma_\prime x_{z_\prime};\cr
\beta  &=   \alpha_\prime y_{x_\prime}
          + \beta_\prime  y_{y_\prime}
          + \gamma_\prime y_{z_\prime};\cr
\gamma &=   \alpha_\prime z_{x_\prime}
          + \beta_\prime  z_{y_\prime}
          + \gamma_\prime z_{z_\prime};\cr
\alpha' &=   \alpha_\prime' x'_{x_\prime'}
           + \beta_\prime'  x'_{y_\prime'}
           + \gamma_\prime' x'_{z_\prime'};\cr
\beta'  &=   \alpha_\prime' y'_{x_\prime'}
           + \beta_\prime'  y'_{y_\prime'}
           + \gamma_\prime' y'_{z_\prime'};\cr
\gamma' &=   \alpha_\prime' z'_{x_\prime'}
           + \beta_\prime'  z'_{y_\prime'}
           + \gamma_\prime' z'_{z_\prime'}:\cr}
   \right\}
   \eqno {\rm (E^8)}$$
and
$$\left. \multieqalign{
\sigma_\prime    &=   \sigma    x_{x_\prime}
                    + \tau      y_{x_\prime}
                    + \upsilon  z_{x_\prime}; &
\sigma_\prime'   &=   \sigma'   x'_{x_\prime'}
                    + \tau'     y'_{x_\prime'}
                    + \upsilon' z'_{x_\prime'};\cr
\tau_\prime      &=   \sigma    x_{y_\prime}
                    + \tau      y_{y_\prime}
                    + \upsilon  z_{y_\prime}; &
\tau_\prime'     &=   \sigma'   x'_{y_\prime'}
                    + \tau'     y'_{y_\prime'}
                    + \upsilon' z'_{y_\prime'};\cr
\upsilon_\prime  &=   \sigma    x_{z_\prime}
                    + \tau      y_{z_\prime}
                    + \upsilon  z_{z_\prime}; &
\upsilon_\prime' &=   \sigma'   x'_{z_\prime'}
                    + \tau'     y'_{z_\prime'}
                    + \upsilon' z'_{z_\prime'}.\cr}
   \right\}
   \eqno {\rm (F^8)}$$
And if, by substituting in the former homogeneous
medium-functions, $v$,~$v'$, the expressions (E${}^8$) for
$\alpha$,~$\beta$,~$\gamma$, $\alpha'$,~$\beta'$,~$\gamma'$,
we obtain $v$ under a new form, as a homogeneous function of
$\alpha_\prime$,~$\beta_\prime$,~$\gamma_\prime$,
of the first dimension, and $v'$ as a homogeneous function of the
same dimension of
$\alpha_\prime'$,~$\beta_\prime'$,~$\gamma_\prime'$,
and then differentiate these new forms of $v$,~$v'$, with
reference to their new variables, we find, by (E${}^8$), the
following relations between the new and the old coefficients,
$$\left. \eqalign{
{\delta v \over \delta \alpha_\prime}
   &=   {\delta v \over \delta \alpha} x_{x_\prime}
      + {\delta v \over \delta \beta}  y_{x_\prime}
      + {\delta v \over \delta \gamma} z_{x_\prime};\cr
{\delta v \over \delta \beta_\prime}
   &=   {\delta v \over \delta \alpha} x_{y_\prime}
      + {\delta v \over \delta \beta}  y_{y_\prime}
      + {\delta v \over \delta \gamma} z_{y_\prime};\cr
{\delta v \over \delta \gamma_\prime}
   &=   {\delta v \over \delta \alpha} x_{z_\prime}
      + {\delta v \over \delta \beta}  y_{z_\prime}
      + {\delta v \over \delta \gamma} z_{z_\prime};\cr
{\delta v' \over \delta \alpha_\prime'}
   &=   {\delta v' \over \delta \alpha'} x'_{x'_\prime}
      + {\delta v' \over \delta \beta'}  y'_{x'_\prime}
      + {\delta v' \over \delta \gamma'} z'_{x'_\prime};\cr
{\delta v' \over \delta \beta_\prime'}
   &=   {\delta v' \over \delta \alpha'} x'_{y'_\prime}
      + {\delta v' \over \delta \beta'}  y'_{y'_\prime}
      + {\delta v' \over \delta \gamma'} z'_{y'_\prime};\cr
{\delta v' \over \delta \gamma_\prime'}
   &=   {\delta v' \over \delta \alpha'} x'_{z'_\prime}
      + {\delta v' \over \delta \beta'}  y'_{z'_\prime}
      + {\delta v' \over \delta \gamma'} z'_{z'_\prime};\cr}
   \right\}
   \eqno {\rm (G^8)}$$
from which relations, combined with (D${}^8$) (F${}^8$), and with
the equations (B) (E), of the second number, we obtain the
following generalisations of the equations (B),
$$\left. \multieqalign{
{\delta V \over \delta x_\prime}
   &= {\delta v \over \delta \alpha_\prime}; &
{\delta V \over \delta y_\prime}
   &= {\delta v \over \delta \beta_\prime}; &
{\delta V \over \delta z_\prime}
   &= {\delta v \over \delta \gamma_\prime};\cr
- {\delta V \over \delta x_\prime'}
   &= {\delta v' \over \delta \alpha_\prime'}; &
- {\delta V \over \delta y_\prime'}
   &= {\delta v' \over \delta \beta_\prime'}; &
- {\delta V \over \delta z_\prime'}
   &= {\delta v' \over \delta \gamma_\prime'}:\cr}
   \right\}
   \eqno {\rm (H^8)}$$
and therefore the following important generalisation of the
fundamental formula (A),
$$\delta V
   =   {\delta v  \over \delta \alpha_\prime}  \, \delta x_\prime
     - {\delta v' \over \delta \alpha_\prime'} \, \delta x_\prime'
     + {\delta v  \over \delta \beta_\prime}   \, \delta y_\prime
     - {\delta v' \over \delta \beta_\prime'}  \, \delta y_\prime'
     + {\delta v  \over \delta \gamma_\prime}  \, \delta z_\prime
     - {\delta v' \over \delta \gamma_\prime'} \, \delta z_\prime',
   \eqno {\rm (I^8)}$$
which is thus shown to extend to oblique co-ordinates, and not
even to require that the initial should coincide with the final
axes.

We may adopt nearly all the foregoing reasonings and results, of
the present Supplement, to this more general view.  We have, for
example, partial differential equations of the first order in
$V$, analogous to the equations (C), and of the form
$$\left. \eqalign{
0 &= \Omega_\prime \left(
         {\delta V \over \delta x_\prime},
         {\delta V \over \delta y_\prime},
         {\delta V \over \delta z_\prime},
         x_\prime, y_\prime, z_\prime, \chi \right),\cr
0 &= \Omega_\prime' \left(
       - {\delta V \over \delta x_\prime'},
       - {\delta V \over \delta y_\prime'},
       - {\delta V \over \delta z_\prime'},
         x_\prime', y_\prime', z_\prime', \chi \right),\cr}
   \right\}
   \eqno {\rm (K^8)}$$
which conduct to a partial differential equation of the second
order, analogous to (D): and if we put the equations (K${}^8$)
under the form
$$\left. \eqalign{
0 &= \Omega_\prime \left(
         \sigma_\prime, \tau_\prime, \upsilon_\prime,
         x_\prime, y_\prime, z_\prime, \chi \right),\cr
0 &= \Omega_\prime' \left(
         \sigma_\prime', \tau_\prime', \upsilon_\prime',
         x_\prime', y_\prime', z_\prime', \chi \right),\cr}
   \right\}
   \eqno {\rm (L^8)}$$
and suppose them so prepared, by the method indicated in the
second number, that the function $\Omega_\prime + 1$ shall be
homogeneous of the first dimension with respect to
$\sigma_\prime$, $\tau_\prime$, $\upsilon_\prime$,
and that $\Omega_\prime' + 1$ shall be homogeneous of the same
dimension with respect to
$\sigma_\prime'$, $\tau_\prime'$, $\upsilon_\prime'$,
we shall have
$$\left. \multieqalign{
{\alpha_\prime \over v}
   &= {\delta \Omega_\prime \over \delta \sigma_\prime}, &
{\beta_\prime \over v}
   &= {\delta \Omega_\prime \over \delta \tau_\prime}, &
{\gamma_\prime \over v}
   &= {\delta \Omega_\prime \over \delta \upsilon_\prime},\cr
{\alpha_\prime' \over v'}
   &= {\delta \Omega_\prime' \over \delta \sigma_\prime'}, &
{\beta_\prime' \over v'}
   &= {\delta \Omega_\prime' \over \delta \tau_\prime'}, &
{\gamma_\prime' \over v'}
   &= {\delta \Omega_\prime' \over \delta \upsilon_\prime'},\cr}
   \right\}
   \eqno {\rm (M^8)}$$
with many other relations, analogous to those of the second
number.  The differential equations of a curved ray, ordinary
or extraordinary, in the third number, may be generalised as
follows,
$${d \over ds} {\delta v \over \delta \alpha_\prime}
   = {\delta v \over \delta x_\prime};\quad
  {d \over ds} {\delta v \over \delta \beta_\prime}
   = {\delta v \over \delta y_\prime};\quad
  {d \over ds} {\delta v \over \delta \gamma_\prime}
   = {\delta v \over \delta z_\prime};
   \eqno {\rm (N^8)}$$
and their integrals may be extended to oblique co-ordinates, under
the form,
$${\delta V \over \delta x_\prime'} = \hbox{const.};\quad
  {\delta V \over \delta y_\prime'} = \hbox{const.};\quad
  {\delta V \over \delta z_\prime'} = \hbox{const.}:
   \eqno {\rm (O^8)}$$
while, if the final portion of the ray be straight, we have also,
for that final portion,
$${\delta V \over \delta x_\prime} = \hbox{const.};\quad
  {\delta V \over \delta y_\prime} = \hbox{const.};\quad
  {\delta V \over \delta z_\prime} = \hbox{const.}
   \eqno {\rm (P^8)}$$
The formula (A${}^7$) of reflexion or refraction, ordinary or
extraordinary, namely
$$\Delta V = 0,$$
extends to oblique co-ordinates; and if we introduce new
auxiliary functions $W_\prime$, $T_\prime$, analogous to $W$,
$T$, and defined by the new equations
$$\left. \eqalign{
W_\prime &= - V
            + x_\prime \sigma_\prime
            + y_\prime \tau_\prime
            + z_\prime \upsilon_\prime,\cr
T_\prime &= W_\prime
            - x_\prime' \sigma_\prime'
            - y_\prime' \tau_\prime'
            - z_\prime' \upsilon_\prime',\cr}
   \right\}
   \eqno {\rm (Q^8)}$$
analogous to the definitions (D${}'$) (E${}'$), and attend to the
meanings and properties of the symbols
$\sigma_\prime$~$\tau_\prime$~$\upsilon_\prime$
$\sigma_\prime'$~$\tau_\prime'$~$\upsilon_\prime'$,
we shall obtain the following expressions for the variations of
$V$, $W_\prime$, $T_\prime$,
$$\left. \eqalign{
\delta V
   &=    \sigma_\prime    \, \delta x_\prime
       - \sigma_\prime'   \, \delta x_\prime'
       + \tau_\prime      \, \delta y_\prime
       - \tau_\prime'     \, \delta y_\prime'
       + \upsilon_\prime  \, \delta z_\prime
       - \upsilon_\prime' \, \delta z_\prime'
       + {\delta V \over \delta \chi} \, \delta \chi;\cr
\delta W_\prime
   &=    x_\prime         \, \delta \sigma_\prime
       + \sigma_\prime'   \, \delta x_\prime'
       + y_\prime         \, \delta \tau_\prime
       + \tau_\prime'     \, \delta y_\prime'
       + z_\prime         \, \delta \upsilon_\prime
       + \upsilon_\prime' \, \delta z_\prime'
       - {\delta V \over \delta \chi} \, \delta \chi;\cr
\delta T_\prime
   &=    x_\prime         \, \delta \sigma_\prime
       - x_\prime'        \, \delta \sigma_\prime'
       + y_\prime         \, \delta \tau_\prime
       - y_\prime'        \, \delta \tau_\prime'
       + z_\prime         \, \delta \upsilon_\prime
       - z_\prime'        \, \delta \upsilon_\prime'
       - {\delta V \over \delta \chi} \, \delta \chi;\cr}
   \right\}
   \eqno {\rm (R^8)}$$
which resemble the expressions (A${}'$) (B${}'$) (C${}'$), and
lead to analogous results.  Thus, the partial differential
coefficients of the new auxiliary functions $W_\prime$,
$T_\prime$, may be deduced, by methods similar to those already
employed, from the new coefficients of the characteristic
function~$V$, which may themselves be deduced from the old
coefficients of that function, by the following general formula,
$$\left. \eqalign{
      \left( {\delta \over \delta x_\prime}  \right)^i
      \left( {\delta \over \delta y_\prime}  \right)^{i'}
      \left( {\delta \over \delta z_\prime}  \right)^{i''}
      \left( {\delta \over \delta x_\prime'} \right)^k
      \left( {\delta \over \delta y_\prime'} \right)^{k'}
      \left( {\delta \over \delta z_\prime'} \right)^{k''}
      \left( {\delta \over \delta \chi}      \right)^l
      V
         \hskip - 216pt\cr
   &= \left(
            x_{x'} {\delta \over \delta x}
          + y_{x'} {\delta \over \delta y}
          + z_{x'} {\delta \over \delta z}
      \right)^i
      \left(
            x'_{x_\prime'} {\delta \over \delta x'}
          + y'_{x_\prime'} {\delta \over \delta y'}
          + z'_{x_\prime'} {\delta \over \delta z'}
      \right)^k \cr
   &\mathrel{\phantom{=}}
      \left(
            x_{y'} {\delta \over \delta x}
          + y_{y'} {\delta \over \delta y}
          + z_{y'} {\delta \over \delta z}
      \right)^{i'}
      \left(
            x'_{y_\prime'} {\delta \over \delta x'}
          + y'_{y_\prime'} {\delta \over \delta y'}
          + z'_{y_\prime'} {\delta \over \delta z'}
      \right)^{k'} \cr
   &\mathrel{\phantom{=}}
      \left(
            x_{z'} {\delta \over \delta x}
          + y_{z'} {\delta \over \delta y}
          + z_{z'} {\delta \over \delta z}
      \right)^{i''}
      \left(
            x'_{z_\prime'} {\delta \over \delta x'}
          + y'_{z_\prime'} {\delta \over \delta y'}
          + z'_{z_\prime'} {\delta \over \delta z'}
      \right)^{k''}
      {\delta^l V \over \delta \chi^l}:\cr}
   \right\}
   \eqno {\rm (S^8)}$$
and the equations of a straight final ray may be put under the
forms,
$$\left. \eqalign{
{1 \over \alpha_\prime} \left(
            x_\prime
          - {\delta W_\prime \over \delta \sigma_\prime}
         \right)
  &= {1 \over \beta_\prime} \left(
            y_\prime
          - {\delta W_\prime \over \delta \tau_\prime}
         \right)
   = {1 \over \gamma_\prime} \left(
            z_\prime
          - {\delta W_\prime \over \delta \upsilon_\prime}
         \right),\cr
{1 \over \alpha_\prime} \left(
            x_\prime
          - {\delta T_\prime \over \delta \sigma_\prime}
         \right)
  &= {1 \over \beta_\prime} \left(
            y_\prime
          - {\delta T_\prime \over \delta \tau_\prime}
         \right)
   = {1 \over \gamma_\prime} \left(
            z_\prime
          - {\delta T_\prime \over \delta \upsilon_\prime}
         \right),\cr}
   \right\}
   \eqno {\rm (T^8)}$$
while those of a straight initial ray may be put under these
other forms,
$${1 \over \alpha_\prime'} \left(
            x_\prime'
          + {\delta T_\prime \over \delta \sigma_\prime'}
         \right)
   = {1 \over \beta_\prime'} \left(
            y_\prime'
          + {\delta T_\prime \over \delta \tau_\prime'}
         \right)
   = {1 \over \gamma_\prime'} \left(
            z_\prime'
          + {\delta T_\prime \over \delta \upsilon_\prime'}
         \right);
   \eqno {\rm (U^8)}$$
these new equations (T${}^8$) (U${}^8$) being analogous to
(I${}^2$) and (P${}^2$).  It is evident that the arbitrary
constants introduced by these transformations of co-ordinates must
often assist to simplify the solution of optical problems.  In
the comparison, for example, of a given polygon ray, ordinary or
extraordinary, of any given system, with other near rays of the
same system, it will often be found convenient to choose the
final portion of the given polygon ray for the axis of
$z_\prime$, and the initial portion for the axis of $z_\prime'$,
a choice which will make
$\alpha_\prime$~$\beta_\prime$ $\alpha_\prime'$~$\beta_\prime'$
and many of the new partial differential coefficients vanish,
without producing, by this simplification, any real loss of
generality.

We may even carry these transformations farther, and introduce
polar co-ordinates, or any other marks of initial and final
position, and still obtain results having much analogy to the
foregoing.  For if we suppose that the final co-ordinates
$x$,~$y$,~$z$ are functions of any three quantities
$\rho$,~$\theta$,~$\phi$, and that in like manner the initial
co-ordinates $x'$,~$y'$,~$z'$ are functions of any other three
quantities $\rho'$,~$\theta'$,~$\phi'$, so that
$$\left. \multieqalign{
\delta x
   &=    {\delta x \over \delta \rho}   \, \delta \rho
       + {\delta x \over \delta \theta} \, \delta \theta
       + {\delta x \over \delta \phi}   \, \delta \phi, &
dx
   &=    {\delta x \over \delta \rho}   \, d\rho
       + {\delta x \over \delta \theta} \, d\theta
       + {\delta x \over \delta \phi}   \, d\phi,\cr
\delta y
   &=    {\delta y \over \delta \rho}   \, \delta \rho
       + {\delta y \over \delta \theta} \, \delta \theta
       + {\delta y \over \delta \phi}   \, \delta \phi, &
dy
   &=    {\delta y \over \delta \rho}   \, d\rho
       + {\delta y \over \delta \theta} \, d\theta
       + {\delta y \over \delta \phi}   \, d\phi,\cr
\delta z
   &=    {\delta z \over \delta \rho}   \, \delta \rho
       + {\delta z \over \delta \theta} \, \delta \theta
       + {\delta z \over \delta \phi}   \, \delta \phi, &
dz
   &=    {\delta z \over \delta \rho}   \, d\rho
       + {\delta z \over \delta \theta} \, d\theta
       + {\delta z \over \delta \phi}   \, d\phi,\cr
\delta x'
   &=    {\delta x' \over \delta \rho'}   \, \delta \rho'
       + {\delta x' \over \delta \theta'} \, \delta \theta'
       + {\delta x' \over \delta \phi'}   \, \delta \phi', &
dx'
   &=    {\delta x' \over \delta \rho'}   \, d\rho'
       + {\delta x' \over \delta \theta'} \, d\theta'
       + {\delta x' \over \delta \phi'}   \, d\phi',\cr
\delta y'
   &=    {\delta y' \over \delta \rho'}   \, \delta \rho'
       + {\delta y' \over \delta \theta'} \, \delta \theta'
       + {\delta y' \over \delta \phi'}   \, \delta \phi', &
dy'
   &=    {\delta y' \over \delta \rho'}   \, d\rho'
       + {\delta y' \over \delta \theta'} \, d\theta'
       + {\delta y' \over \delta \phi'}   \, d\phi',\cr
\delta z'
   &=    {\delta z' \over \delta \rho'}   \, \delta \rho'
       + {\delta z' \over \delta \theta'} \, \delta \theta'
       + {\delta z' \over \delta \phi'}   \, \delta \phi', &
dz'
   &=    {\delta z' \over \delta \rho'}   \, d\rho'
       + {\delta z' \over \delta \theta'} \, d\theta'
       + {\delta z' \over \delta \phi'}   \, d\phi',\cr}
   \right\}
   \eqno {\rm (V^8)}$$
we may consider $V$ as a function of
$\rho$~$\theta$~$\phi$ $\rho'$~$\theta'$~$\phi'$~$\chi$,
obtained by substituting for
$x$~$y$~$z$ $x'$~$y'$~$z'$
their values; and if we substitute also the values of
$dx$, $dy$, $dz$, in the differential $dV$, or $v \, ds$, which
was before a homogeneous function of the first dimension of
$dx$, $dy$, $dz$, such that by our fundamental formula
$$\left. \eqalign{
{\delta dV \over \delta dx}
   &= {\delta \mathbin{.} v \, ds \over \delta dx}
    = {\delta v \over \delta \alpha}
    = {\delta V \over \delta x},\cr
{\delta dV \over \delta dy}
   &= {\delta \mathbin{.} v \, ds \over \delta dy}
    = {\delta v \over \delta \beta}
    = {\delta V \over \delta y},\cr
{\delta dV \over \delta dz}
   &= {\delta \mathbin{.} v \, ds \over \delta dz}
    = {\delta v \over \delta \gamma}
    = {\delta V \over \delta z},\cr}
   \right\}
   \eqno {\rm (W^8)}$$
we may consider this differential $dv = v \, ds$ as becoming now
a homogeneous function of $d\rho$, $d\theta$, $d\phi$, of the
first dimension, such that

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
{\delta \mathbin{.} v \, ds \over \delta d\rho}
   &= {\delta dV \over \delta d\rho}
    =    {\delta V \over \delta x}{\delta x \over \delta \rho}
       + {\delta V \over \delta y}{\delta y \over \delta \rho}
       + {\delta V \over \delta z}{\delta z \over \delta \rho}
    = {\delta V \over \delta \rho},\cr
{\delta \mathbin{.} v \, ds \over \delta d\theta}
   &= {\delta dV \over \delta d\theta}
    =    {\delta V \over \delta x}{\delta x \over \delta \theta}
       + {\delta V \over \delta y}{\delta y \over \delta \theta}
       + {\delta V \over \delta z}{\delta z \over \delta \theta}
    = {\delta V \over \delta \theta},\cr
{\delta \mathbin{.} v \, ds \over \delta d\phi}
   &= {\delta dV \over \delta d\phi}
    =    {\delta V \over \delta x}{\delta x \over \delta \phi}
       + {\delta V \over \delta y}{\delta y \over \delta \phi}
       + {\delta V \over \delta z}{\delta z \over \delta \phi}
    = {\delta V \over \delta \phi},\cr}
   \right\}
   \eqno {\rm (X^8)}$$
the symbol $d$ referring still to motion along a ray.  In like
manner we may consider the initial differential element of $V$,
namely $v' \, ds'$, as a homogeneous function of the first
dimension of
$d\rho'$, $d\theta'$, $d\phi'$,
and then we shall find that the partial differential coefficients
of the first order of this function, are equal respectively to
$$- {\delta V \over \delta \rho'},\quad
  - {\delta V \over \delta \theta'},\quad
  - {\delta V \over \delta \phi'};$$
we may therefore generalise the fundamental formula (A) as
follows
$$\eqalignno{
\delta V
   &=    {\delta \mathbin{.} v \, ds \over \delta d\rho}
            \, \delta \rho
       + {\delta \mathbin{.} v \, ds \over \delta d\theta}
            \, \delta \theta
       + {\delta \mathbin{.} v \, ds \over \delta d\phi}
            \, \delta \phi \cr
   &\mathrel{\phantom{=}} \mathord{}
       - {\delta \mathbin{.} v' \, ds' \over \delta d\rho'}
            \, \delta \rho'
       - {\delta \mathbin{.} v' \, ds' \over \delta d\theta'}
            \, \delta \theta'
       - {\delta \mathbin{.} v' \, ds' \over \delta d\phi'}
            \, \delta \phi'
       + {\delta V \over \delta \chi}
            \, \delta \chi.
   &{\rm (Y^8)}\cr}$$
And the auxiliary functions $W$, $T$, correspond to the following
more general functions,
$$- V + \rho   {\delta V \over \delta \rho}
      + \theta {\delta V \over \delta \theta}
      + \phi   {\delta V \over \delta \phi},
   \quad \hbox{and} \quad
  - V + \rho   {\delta V \over \delta \rho}
      + \theta {\delta V \over \delta \theta}
      + \phi   {\delta V \over \delta \phi}
      + \rho'   {\delta V \over \delta \rho'}
      + \theta' {\delta V \over \delta \theta'}
      + \phi'   {\delta V \over \delta \phi'};$$
of which the first may be regarded as a function of
$${\delta V \over \delta \rho},\quad
  {\delta V \over \delta \theta},\quad
  {\delta V \over \delta \phi},\quad
  \rho',\quad \theta',\quad \phi',\quad \chi,$$
and the second as a function of
$${\delta V \over \delta \rho},\quad
  {\delta V \over \delta \theta},\quad
  {\delta V \over \delta \phi},\quad
  - {\delta V \over \delta \rho'},\quad
  - {\delta V \over \delta \theta'},\quad
  - {\delta V \over \delta \phi'},\quad
  \chi.$$
It is easy also to establish the following general differential
equations of a curved ray, ordinary or extraordinary, and the
following general integrals analogous to and including those
already assigned for rectangular and oblique co-ordinates,
$$\left. \multieqalign{
d {\delta dV \over \delta d\rho}
   &= {\delta dV \over \delta \rho}; &
d {\delta dV \over \delta d\theta}
   &= {\delta dV \over \delta \theta}; &
d {\delta dV \over \delta d\phi}
   &= {\delta dV \over \delta \phi}:\cr
{\delta V \over \delta \rho'}   &= \hbox{const.}; &
{\delta V \over \delta \theta'} &= \hbox{const.}; &
{\delta V \over \delta \phi'}   &= \hbox{const.}.\cr}
   \right\}
   \eqno {\rm (Z^8)}$$

\bigbreak

{\sectiontitle
General geometrical Relations of infinitely near Rays.
Classification of twenty-four independent Coefficients, which
enter into the algebraical Expressions of these general
Relations.  Division of the general Discussion into four
principal Problems.\par}

\nobreak\bigskip

14.
It is an important general problem of mathematical optics,
included in that fundamental problem which was stated in the
second number, to investigate {\it the general relations of
infinitely near rays}, or paths of light; and especially to
examine {\it how the extreme directions change, for any
infinitely small changes of the extreme points, and of the
colour\/}: that is, in the notation of this Supplement, to
examine the general dependence of the variations
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$, on
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$.
This important case of our fundamental problem is easily resolved
by the application of our general methods, and by the partial
differential coefficients, of the two first orders, of the
characteristic and related functions: it may also be resolved by
the partial differentials of the three first orders, of the
characteristic function $V$ alone.  For from these we can in
general deduce six linear expressions for
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$,
in terms of
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$,
involving forty-two coefficients, of which however only
twenty-four are independent, because they are connected by
fourteen relations included in the formul{\ae}
$\alpha \, \delta \alpha
   + \beta \, \delta \beta
   + \gamma \, \delta \gamma = 0$,
$\alpha' \, \delta \alpha'
   + \beta' \, \delta \beta'
   + \gamma' \, \delta \gamma' = 0$,
and by four more included in the conditions that the final
direction does not change when the initial point takes any new
position on the given luminous path, nor the initial direction
when the final point is removed to any new point on that given
path.

Thus, if we employ the characteristic function $V$, and the final
and initial medium-functions $v$, $v'$, we have, by (B), the
following general relations:
$$\left. \multieqalign{
\delta {\delta V \over \delta x}
   &= \delta {\delta v \over \delta \alpha}; &
\delta {\delta V \over \delta y}
   &= \delta {\delta v \over \delta \beta}; &
\delta {\delta V \over \delta z}
   &= \delta {\delta v \over \delta \gamma};\cr
- \delta {\delta V \over \delta x'}
   &= \delta {\delta v' \over \delta \alpha'}; &
- \delta {\delta V \over \delta y'}
   &= \delta {\delta v' \over \delta \beta'}; &
- \delta {\delta V \over \delta z'}
   &= \delta {\delta v' \over \delta \gamma'}:\cr}
   \right\}
   \eqno {\rm (A^9)}$$
in which, by the last number, we are at liberty to assign
different origins and different and oblique directions to the
axes of the final and initial co-ordinates, if we assign new and
corresponding values to the marks of final and initial direction,
$\alpha$,~$\beta$,~$\gamma$, $\alpha'$,~$\beta'$,~$\gamma'$, so
as to have still the equations (P),
$$\alpha = {dx \over ds},\quad
  \beta  = {dy \over ds},\quad
  \gamma = {dz \over ds},\quad
  \alpha' = {dx' \over ds'},\quad
  \beta'  = {dy' \over ds'},\quad
  \gamma' = {dz' \over ds'},$$
$ds$ being still the final, and $ds'$ the initial element of the
curved or polygon path.  We may suppose, for example, that both
sets of co-ordinates are rectangular, but that the origins of the
final and initial co-ordinates are respectively the final and
initial points of a given ordinary or extraordinary path, and
that the positive semiaxes of $z$,~$z'$, coincide with the final
and initial directions, so as to give
$$\left. \eqalign{
   &x = 0,\quad y = 0,\quad z = 0,\quad
      \alpha = 0,\quad \beta = 0,\quad \gamma = 1,\quad
      \delta \gamma = 0;\cr
   &x' = 0,\quad y' = 0,\quad z' = 0,\quad
      \alpha' = 0,\quad \beta' = 0,\quad \gamma' = 1,\quad
      \delta \gamma' = 0;\cr}
   \right\}
   \eqno {\rm (B^9)}$$
and then the six equations (A${}^9$), of which only four are
distinct, reduce themselves to the four following,

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
    \mathbin{\phantom{+}}
         {\delta^2 v \over \delta \alpha^2}
            \, \delta \alpha
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
            \, \delta \beta
   &=    {\delta^2 V \over \delta x \, \delta x'} \, \delta x'
       + {\delta^2 V \over \delta x \, \delta y'} \, \delta y'
       + \left( {\delta^2 V \over \delta x \, \delta \chi}
             - {\delta^2 v \over \delta \alpha \, \delta \chi}
            \right) \, \delta \chi \cr
   &\hskip -84pt
       + \left( {\delta^2 V \over \delta x^2}
             - {\delta^2 v \over \delta \alpha \, \delta x}
            \right) \, \delta x
       + \left( {\delta^2 V \over \delta x \, \delta y}
             - {\delta^2 v \over \delta \alpha \, \delta y}
            \right) \, \delta y
       + \left( {\delta v \over \delta x}
             - {\delta^2 v \over \delta \alpha \, \delta z}
            \right) \, \delta z;\cr
    \mathbin{\phantom{+}}
         {\delta^2 v \over \delta \alpha \, \delta \beta}
            \, \delta \alpha
       + {\delta^2 v \over \delta \beta^2}
            \, \delta \beta
   &=    {\delta^2 V \over \delta y \, \delta x'} \, \delta x'
       + {\delta^2 V \over \delta y \, \delta y'} \, \delta y'
       + \left( {\delta^2 V \over \delta y \, \delta \chi}
             - {\delta^2 v \over \delta \beta  \, \delta \chi}
            \right) \, \delta \chi \cr
   &\hskip -84pt
       + \left( {\delta^2 V \over \delta x \, \delta y}
             - {\delta^2 v \over \delta \beta  \, \delta x}
            \right) \, \delta x
       + \left( {\delta^2 V \over \delta y^2}
             - {\delta^2 v \over \delta \beta  \, \delta y}
            \right) \, \delta y
       + \left( {\delta v \over \delta y}
             - {\delta^2 v \over \delta \beta \, \delta z}
            \right) \, \delta z;\cr
       - {\delta^2 v' \over \delta \alpha'^2}
            \, \delta \alpha'
       - {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            \, \delta \beta'
   &=    {\delta^2 V \over \delta x \, \delta x'} \, \delta x
       + {\delta^2 V \over \delta y \, \delta x'} \, \delta y
       + \left( {\delta^2 V \over \delta x' \, \delta \chi}
             + {\delta^2 v' \over \delta \alpha' \, \delta \chi}
            \right) \, \delta \chi \cr
   &\hskip -84pt
       + \left( {\delta^2 V \over \delta x'^2}
             + {\delta^2 v' \over \delta \alpha' \, \delta x'}
            \right) \, \delta x'
       + \left( {\delta^2 V \over \delta x' \, \delta y'}
             + {\delta^2 v' \over \delta \alpha' \, \delta y'}
            \right) \, \delta y'
       - \left( {\delta v' \over \delta x'}
             - {\delta^2 v' \over \delta \alpha' \, \delta z'}
            \right) \, \delta z';\cr
       - {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            \, \delta \alpha'
       - {\delta^2 v' \over \delta \beta'^2}
            \, \delta \beta'
   &=    {\delta^2 V \over \delta x \, \delta y'} \, \delta x
       + {\delta^2 V \over \delta y \, \delta y'} \, \delta y
       + \left( {\delta^2 V \over \delta y' \, \delta \chi}
             + {\delta^2 v' \over \delta \beta'  \, \delta \chi}
            \right) \, \delta \chi \cr
   &\hskip -84pt
       + \left( {\delta^2 V \over \delta x' \, \delta y'}
             + {\delta^2 v' \over \delta \beta'  \, \delta x'}
            \right) \, \delta x'
       + \left( {\delta^2 V \over \delta y'^2}
             + {\delta^2 v' \over \delta \beta'  \, \delta y'}
            \right) \, \delta y'
       - \left( {\delta v' \over \delta y'}
             - {\delta^2 v' \over \delta \beta' \, \delta z'}
            \right) \, \delta z':\cr}
   \right\}
   \eqno {\rm (C^9)}$$
they give therefore, by easy eliminations, expressions for
$\delta \alpha$, $\delta \beta$, $\delta \alpha'$, $\delta \beta'$,
of the form
$$\left. \eqalign{
\delta \alpha
   &=    {\delta \alpha  \over \delta x}    \, \delta x
       + {\delta \alpha  \over \delta y}    \, \delta y
       + {\delta \alpha  \over \delta z}    \, \delta z
       + {\delta \alpha  \over \delta x'}   \, \delta x'
       + {\delta \alpha  \over \delta y'}   \, \delta y'
       + {\delta \alpha  \over \delta \chi} \, \delta \chi,\cr
\delta \beta
   &=    {\delta \beta   \over \delta x}    \, \delta x
       + {\delta \beta   \over \delta y}    \, \delta y
       + {\delta \beta   \over \delta z}    \, \delta z
       + {\delta \beta   \over \delta x'}   \, \delta x'
       + {\delta \beta   \over \delta y'}   \, \delta y'
       + {\delta \beta   \over \delta \chi} \, \delta \chi,\cr
\delta \alpha'
   &=    {\delta \alpha' \over \delta x'}   \, \delta x'
       + {\delta \alpha' \over \delta y'}   \, \delta y'
       + {\delta \alpha' \over \delta z'}   \, \delta z'
       + {\delta \alpha' \over \delta x}    \, \delta x
       + {\delta \alpha' \over \delta y}    \, \delta y
       + {\delta \alpha' \over \delta \chi} \, \delta \chi,\cr
\delta \beta'
   &=    {\delta \beta'  \over \delta x'}   \, \delta x'
       + {\delta \beta'  \over \delta y'}   \, \delta y'
       + {\delta \beta'  \over \delta z'}   \, \delta z'
       + {\delta \beta'  \over \delta x}    \, \delta x
       + {\delta \beta'  \over \delta y}    \, \delta y
       + {\delta \beta'  \over \delta \chi} \, \delta \chi,\cr}
   \right\}
   \eqno {\rm (D^9)}$$
which involve twenty-four coefficients, and enable us to
determine the general geometrical relations between the final and
initial tangents to the near luminous paths.

If the extreme media be ordinary, that is, if the functions $v$,
$v'$, be independent of the directions of the rays, we have
$$v  = \mu  \surd (\alpha^2  + \beta^2  + \gamma^2),\quad
  v' = \mu' \surd (\alpha'^2 + \beta'^2 + \gamma'^2),
   \eqno {\rm (E^9)}$$
$\mu$, $\mu'$ being functions of the colour $\chi$, of which
$\mu$ involves also the final co-ordinates, and $\mu'$ the initial
co-ordinates, when the exteme media are atmospheres: and then the
equations (C${}^9$) reduce themselves at once to the following
expressions of the form (D${}^9$),
$$\left. \eqalign{
\delta \alpha
   &= {1 \over \mu} \left(
            {\delta^2 V \over \delta x^2}               \, \delta x
          + {\delta^2 V \over \delta x \, \delta y}     \, \delta y
          + {\delta \mu \over \delta x}                 \, \delta z
          + {\delta^2 V \over \delta x \, \delta x'}    \, \delta x'
          + {\delta^2 V \over \delta x \, \delta y'}    \, \delta y'
          + {\delta^2 V \over \delta x \, \delta \chi}  \, \delta \chi
         \right),\cr
\delta \beta
   &= {1 \over \mu} \left(
            {\delta^2 V \over \delta x \, \delta y}     \, \delta x
          + {\delta^2 V \over \delta y^2}               \, \delta y
          + {\delta \mu \over \delta y}                 \, \delta z
          + {\delta^2 V \over \delta y \, \delta x'}    \, \delta x'
          + {\delta^2 V \over \delta y \, \delta y'}    \, \delta y'
          + {\delta^2 V \over \delta y \, \delta \chi}  \, \delta \chi
         \right),\cr
\delta \alpha'
   &= - {1 \over \mu'} \left(
            {\delta^2 V \over \delta x'^2}              \, \delta x'
          + {\delta^2 V \over \delta x' \, \delta y'}   \, \delta y'
          - {\delta \mu' \over \delta x'}               \, \delta z'
          + {\delta^2 V \over \delta x \, \delta x'}    \, \delta x
          + {\delta^2 V \over \delta y \, \delta x'}    \, \delta y
          + {\delta^2 V \over \delta x' \, \delta \chi} \, \delta \chi
         \right),\cr
\delta \beta'
   &= - {1 \over \mu'} \left(
            {\delta^2 V \over \delta x' \, \delta y'}   \, \delta x'
          + {\delta^2 V \over \delta y'^2}              \, \delta y'
          - {\delta \mu' \over \delta y'}               \, \delta z'
          + {\delta^2 V \over \delta x \, \delta y'}    \, \delta x
          + {\delta^2 V \over \delta y \, \delta y'}    \, \delta y
          + {\delta^2 V \over \delta y' \, \delta \chi} \, \delta \chi
         \right).\cr}
   \right\}
   \eqno {\rm (F^9)}$$
In general we see that the twenty-four coefficients of the
expressions (D${}^9$) can easily be deduced, by (C${}^9$), from
the partial differentials of the two first orders of the
characteristic function $V$, and of the extreme medium-functions
$v$, $v'$: we have for example
$$\left. \eqalign{
{\delta \alpha \over \delta x}
   &=    {1 \over v''}
         {\delta^2 v \over \delta \beta^2} \left(
            {\delta^2 V \over \delta x^2}
          - {\delta^2 v \over \delta \alpha \, \delta x}
         \right)
       - {1 \over v''}
         {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \beta \, \delta x}
         \right),\cr
{\delta \alpha \over \delta y}
   &=    {1 \over v''}
         {\delta^2 v \over \delta \beta^2} \left(
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha \, \delta y}
         \right)
       - {1 \over v''}
         {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
            {\delta^2 V \over \delta y^2}
          - {\delta^2 v \over \delta \beta \, \delta y}
         \right),\cr
{\delta \beta \over \delta x}
   &=    {1 \over v''}
         {\delta^2 v \over \delta \alpha^2} \left(
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \beta \, \delta x}
         \right)
       - {1 \over v''}
         {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
            {\delta^2 V \over \delta x^2}
          - {\delta^2 v \over \delta \alpha \, \delta x}
         \right),\cr
{\delta \beta \over \delta y}
   &=    {1 \over v''}
         {\delta^2 v \over \delta \alpha^2} \left(
            {\delta^2 V \over \delta y^2}
          - {\delta^2 v \over \delta \beta \, \delta y}
         \right)
       - {1 \over v''}
         {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha \, \delta y}
         \right),\cr}
   \right\}
   \eqno {\rm (G^9)}$$
$v''$ having the same meaning as in the tenth number.  The same
twenty-four coefficients of (D${}^9$) may also be deduced (as we
have said) from the partial differentials of the two first orders
of the other related and auxiliary functions: or even from the
partial differentials of the three first orders of the
characteristic function $V$ alone.  Let us therefore suppose that
these twenty-four coefficeints of the expressions (D${}^9$) are
known, and let us consider their geometrical meanings and uses:
that is, their connexions with questions respecting the
infinitely small variations of the extreme directions or tangents
of a luminous path, arising from variations of the extreme points
and of the colour.

In discussing these connexions, it is evidently permitted, by the
linear form of the differential expressions (D${}^9$), to
consider separately and successively the influence of the seven
variations
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$,
of the extreme co-ordinates and the colour, or the influence of
any groupes of these seven variations, on the four variations
$\delta \alpha$, $\delta \beta$, $\delta \alpha'$, $\delta \beta'$,
of the extreme small cosines of direction.  Thus, if it be
required to compare the extreme directions of a given path of
ordinary or extraordinary light of the colour $\chi$, from a
given initial point $A$ to a given final point $B$, which path we
shall denote as follows,
$$(A,B)_\chi,
   \eqno {\rm (H^9)}$$
with the extreme directions of an infinitely near path of
infinitely near colour $\chi + \delta \chi$ from an infinitely
near initial point $A'$ to an infinitely near final point $B'$,
which path we shall in like manner denote thus
$$(A',B')_{\chi + \delta \chi},
   \eqno {\rm (I^9)}$$
we may do so by comparing separately the extreme directions of
the given path $(A,B)_\chi$ with those of the three following
other infinitely near paths;
$$\hbox{1st.}\enspace (A,B)_{\chi + \delta \chi};\quad
  \hbox{2d.}\enspace  (A,B')_\chi;\quad
  \hbox{3d.}\enspace  (A',B)_\chi:
   \eqno {\rm (K^9)}$$
which are obtained by changing, successively and separately, the
colour $\chi$, the final point~$B$, and the initial point~$A$.
We are therefore led, by this consideration, to examine
separately and successively the meanings and uses of the three
following groupes, out of the twenty-four coefficients of
(D${}^9$):
$$\left. \vcenter{\halign{&\hfil #\quad&$\displaystyle #$\hfil\cr
1st groupe&
   {\delta \alpha  \over \delta \chi},\quad
   {\delta \beta   \over \delta \chi},\quad
   {\delta \alpha' \over \delta \chi},\quad
   {\delta \beta'  \over \delta \chi};\cr
   \noalign{\vskip 3pt}
2d groupe&
   {\delta \alpha  \over \delta x},\quad
   {\delta \alpha  \over \delta y},\quad
   {\delta \alpha  \over \delta z},\quad
   {\delta \beta   \over \delta x},\quad
   {\delta \beta   \over \delta y},\quad
   {\delta \beta   \over \delta z},\quad
   {\delta \alpha' \over \delta x},\quad
   {\delta \alpha' \over \delta y},\quad
   {\delta \beta'  \over \delta x},\quad
   {\delta \beta'  \over \delta y};\cr
   \noalign{\vskip 3pt}
3d groupe&
   {\delta \alpha  \over \delta x'},\quad
   {\delta \alpha  \over \delta y'},\quad
   {\delta \beta   \over \delta x'},\quad
   {\delta \beta   \over \delta y'},\quad
   {\delta \alpha' \over \delta x'},\quad
   {\delta \alpha' \over \delta y'},\quad
   {\delta \alpha' \over \delta z'},\quad
   {\delta \beta'  \over \delta x'},\quad
   {\delta \beta'  \over \delta y'},\quad
   {\delta \beta'  \over \delta z'}.\cr}}
   \right\}
   \eqno {\rm (L^9)}$$
But we may simplify and improve the plan of our investigation, by
means of the following considerations.

Of the three comparisons, of the given path (H${}^9$) with the
three near paths (K${}^9$), the third is evidently of the same
kind with the second, and need not be treated as distinct;
because, of the two extreme points of a luminous path, it is
indifferent which we consider as initial and which as final.  We
may therefore omit the third comparison (K${}^9$), and confine
ourselves to the first and second, that is, we may omit the
consideration of the third groupe (L${}^9$), in forming a theory
of the general relations of infinitely near rays.  For a similar
reason we may omit the consideration of the two last coefficients
of the first groupe (L${}^9$), and so may reduce the study of the
whole twenty-four to the study of half that number.

On the other hand, the second comparison (K${}^9$) may
conveniently be decomposed into two: for instead of the arbitrary
infinitesimal line $\overline{B B'}$, connecting the given final
point $B$ with the near point $B'$, we may conveniently consider
the two projections of this line, on the final element or tangent
of the given luminous path, and on the plane perpendicular to
this element: that is, we may put
$$\overline{B B'}^2
   = \overline{B B'_d}^2 + \overline{B B'_\delta}^2,
   \eqno {\rm (M^9)}$$
$\overline{B B'_d}$ being the projection on the element, and
$\overline{B B'_\delta}$ the projection on the perpendicular
plane, and we may consider separately the two near points
$B_d$, $B_\delta$, upon this element and plane, and the two
corresponding paths,
$$(A,B_d)_\chi,\quad
  (A,B_\delta)_\chi,
   \eqno {\rm (N^9)}$$
instead of considering the more general near point $B'$, and the
near path $(A,B')_\chi$.  In this manner we are led to consider
separately, as one subordinate class or set, suggested by the
path $(A,B_d)_\chi$, the system of the two coefficients
$\displaystyle {\delta \alpha \over \delta z}$, $\displaystyle
{\delta \beta  \over \delta z}$; distinguishing these from the
eight other coefficients of the second groupe (L${}^9$), which
correspond to the other near path $(A,B_\delta)_\chi$; and these
eight may again be conveniently divided into two distinct
classes, according as we consider the changes of final or initial
direction.

We are then led to arrange the twelve retained coefficients of
the expressions (D${}^9$), in {\it four new sets\/} or classes,
suggesting {\it four separate problems\/}:
$$\left. \vcenter{\halign{&#\hfil\quad&$\displaystyle #$\hfil\cr
First set&
{\delta \alpha \over \delta \chi},\quad
{\delta \beta  \over \delta \chi};\quad &
Second,&
{\delta \alpha \over \delta z},\quad
{\delta \beta  \over \delta z};\cr
   \noalign{\vskip 3pt}
Third,&
{\delta \alpha \over \delta x},\quad
{\delta \alpha \over \delta y},\quad
{\delta \beta  \over \delta x},\quad
{\delta \beta  \over \delta y};\quad &
Fourth,&
{\delta \alpha' \over \delta x},\quad
{\delta \alpha' \over \delta y},\quad
{\delta \beta'  \over \delta x},\quad
{\delta \beta'  \over \delta y}.\quad \cr}}
   \right\}
   \eqno {\rm (O^9)}$$
In each of these four problems, the initial point is considered
as given, and may be supposed to be a luminous origin, common to
all the infinitely near paths of which we compare the extreme
directions.  In the first problem, the final point also is given,
but the colour $\chi$ is variable; and we study the final
chromatic dispersion of the different near paths of heterogeneous
light, connecting the given final point with the given luminous
origin: whereas, in the three remaining problems, the light is
considered as homogeneous, but the luminous path varies by the
variation of its final point.  In the second problem, the new
final point $B_d$ is on the original path, or on that path
prolonged; and we examine whether and in what manner the final
direction varies, on account of the final curvature of that
original path.  In the third problem, the new final point
$B_\delta$ is on an infinitely small line
$$\delta l = \overline{B B_\delta},
   \eqno {\rm (P^9)}$$
which is drawn from the given final point of the original path,
perpendicular to the given final element of that path, namely to
the element
$$ds = \overline{B B_d};
   \eqno {\rm (Q^9)}$$
and we inquire into the mutual arrangement and relations of the
final system of right lines which coincide with and mark the
final directions of the near luminous paths, at the several near
points $B_\delta$ where they meet the given final plane
perpendicular to the given element $ds$.  In the fourth problem,
we consider the initial system of right lines, which mark, at the
luminous origin, the initial directions of the same near paths of
homogeneous light; and we compare these initial directions with
the positions of the points $B_\delta$.  Let us now consider
separately these four principal problems, respecting the
geometrical relations of infinitely near rays.

\bigbreak

{\sectiontitle
Discussion of the Four Problems.  Elements of Arangement of near
Luminous Paths.  Axis and Constant of Chromatic Dispersion.  Axis
of Curvature of Ray.  Guiding Paraboloid, and Constant of
Deviation.  Guiding Planes, and Conjugate Guiding Axes.\par}

\nobreak\bigskip

15.
The {\it first\/} of these four problems, namely that in
which it is required to determine the final chromatic dispersion,
by means of the two coefficients
$\displaystyle {\delta \alpha \over \delta \chi}$,
$\displaystyle {\delta \beta  \over \delta \chi}$,
is very easily resolved: since we have the following equations
for the magnitude and plane of this dispersion,
$$\left. \eqalign{
   &\hbox{\it Final angle of chromatic dispersion}
         = \xi \, \delta \chi;\quad
      \xi = \sqrt{ \left( {\delta \alpha \over \delta \chi} \right)^2
                 + \left( {\delta \beta  \over \delta \chi} \right)^2}:\cr
   &\hbox{\it Final plane of dispersion}\ldots\ldots\ldots\ldots\enspace
         y {\delta \alpha \over \delta \chi}
       = x {\delta \beta  \over \delta \chi}.\cr}
   \right\}
   \eqno {\rm (R^9)}$$
We may geometrically construct the effect of this dispersion, by
making the given final line of direction of the original luminous
path revolve through the small angle $\xi \, \delta \chi$, in
which $\xi$ may be called the {\it constant of final chromatic
dispersion}, round the following line which may be called
{\it the axis of final chromatic dispersion},
$$x {\delta \alpha \over \delta \chi}
       + y {\delta \beta  \over \delta \chi}
   = 0,\quad z = 0.
   \eqno {\rm (S^9)}$$

The {\it second\/} problem, which relates to the final curvature
of the given luminous path, is resolved by the analogous
equations,
$$\left. \eqalign{
   &\hbox{\it Final curvature of ray}
          = \sqrt{ \left( {\delta \alpha \over \delta z} \right)^2
                 + \left( {\delta \beta  \over \delta z} \right)^2};\cr
   &\hbox{\it Plane of curvature}\ldots\ldots\enspace
         y {\delta \alpha \over \delta z}
       = x {\delta \beta  \over \delta z};\cr}
   \right\}
   \eqno {\rm (T^9)}$$
we have also the following equations for the axis of curvature,
that is, for the axis of the circle of curvature, or of the final
osculating circle to the given luminous path,
$$x {\delta \alpha \over \delta z}
       + y {\delta \beta  \over \delta z}
   = 1,\quad z = 0:
   \eqno {\rm (U^9)}$$
and in all these equations of curvature we may, consistently with
the notation of the present Supplement, express the coefficients
$\displaystyle {\delta \alpha \over \delta z}$,
$\displaystyle {\delta \beta  \over \delta z}$
by the symbols
$\displaystyle {d \alpha \over dz}$,
$\displaystyle {d \beta  \over dz}$,
because they relate to motion along a given luminous path.  It is
evident that these coefficients vanish, when the final portion of
this path is straight.  But when this final portion is curved, we
may geometrically construct the effect of this curvature on the
final direction, by making the final element $ds$ revolve through
an infinitely small angle round the final axis of curvature.

The two remaining problems are more complicated, because each
involves two independent variations $\delta x$, $\delta y$,
namely the two rectangular co-ordinates of the near point
$B_\delta$ on the final plane of $xy$, which point is
considered as the final point of a near luminous path.  The
equations of the right line, which is the final portion or final
tangent of this near path, are
$$\left. \eqalign{
x &= \delta x
      + z \left(   {\delta \alpha \over \delta x} \, \delta x
                 + {\delta \alpha \over \delta y} \, \delta y
                 \right),\cr
y &= \delta y
      + z \left(   {\delta \beta  \over \delta x} \, \delta x
                 + {\delta \beta  \over \delta y} \, \delta y
                 \right);\cr}
   \right\}
   \eqno {\rm (V^9)}$$
and the equations of the right line, which is the initial portion
or the initial tangent of the same near path, are
$$\left. \eqalign{
x' &=  z' \left(   {\delta \alpha' \over \delta x} \, \delta x
                 + {\delta \alpha' \over \delta y} \, \delta y
                 \right),\cr
y' &=  z' \left(   {\delta \beta'  \over \delta x} \, \delta x
                 + {\delta \beta'  \over \delta y} \, \delta y
                 \right).\cr}
   \right\}
   \eqno {\rm (W^9)}$$
Our {\it third\/} problem is to investigate the geometrical
relations of the system of right lines (V${}^9$), which we shall
call {\it final ray-lines}, with each other, and with the
co-ordinates $\delta x$, $\delta y$; and our {\it fourth\/}
problem is to investigate the connexion of the same co-ordinates
or variations with the right lines of the system (W${}^9$), which
may be called {\it initial ray-lines}.

The {\it third\/} problem may be considered as resolved, if we
can assign any surface to which the final ray-lines (V${}^9$) are
normals, or with which they are determinately connected by any
other known geometrical relation.  Let us therefore examine
whether the ray-lines of the system (V${}^9$) are normals to any
common surface, which passes through the given final point of the
original luminous path.  If so, this surface may be considered,
in our present order of approximation, as perpendicular to the
final rays themselves.  Now, in general,  when rays of a given
colour diverge from a given luminous point, and undergo any
number of ordinary or extraordinary and gradual or sudden
reflexions or refractions, they are, or are not, perpendicular
in their final state to a common surface, according as the
following differential equation
$$\alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z
   = 0
   \eqno {\rm (X^9)}$$
is or is not integrable; and if there be any one surface
perpendicular to all the final rays, there is also a series of
such surfaces, represented by the integral of this equation.
Hence, in the present question, the normal surface sought is
such, if it exists at all, as to satisfy the conditions
$\delta z = 0$, and
$$\delta^2 z + \delta \alpha \, \delta x + \delta \beta \, \delta y
   = 0;
   \eqno {\rm (Y^9)}$$
that is, if it exist, it must touch the given final plane of
$xy$, and must have contact of the second order with the
following paraboloid, which may therefore in our present order of
approximation be employed instead of it,
$$2z  + {\delta \alpha \over \delta x} x^2
      + \left(   {\delta \alpha \over \delta y}
               + {\delta \beta  \over \delta x} \right) xy
      + {\delta \beta \over \delta y} y^2
   = 0.
   \eqno {\rm (Z^9)}$$
The normals to this paraboloid, near its summit, that is, near
the final point of the given luminous path, or the origin of the
final co-ordinates, have for their approximate equations,
$$\left. \eqalign{
x &= \delta x
      + z \left(   {\delta \alpha \over \delta x} \, \delta x
                 + {\delta \alpha \over \delta y} \, \delta y
                 \right)
      + zn \, \delta y,\cr
y &= \delta y
      + z \left(   {\delta \beta  \over \delta x} \, \delta x
                 + {\delta \beta  \over \delta y} \, \delta y
                 \right)
      - zn \, \delta x,\cr}
   \right\}
   \eqno {\rm (A^{10})}$$
if we put for abridgment
$$n = {\textstyle {1\over 2}} \left(
                 {\delta \beta  \over \delta x}
               - {\delta \alpha \over \delta y} \right):
   \eqno {\rm (B^{10})}$$
they coincide therefore with the ray-lines (V${}^9$) when the
following condition is satisfied,
$${\delta \beta  \over \delta x}
   = {\delta \alpha \over \delta y},
   \eqno {\rm (C^{10})}$$
which is in fact the condition of integrability of the
differential equation (X${}^9$), because we have made
$\alpha$~$\beta$ vanish by our choice of the axis of $z$.  The
condition (C${}^{10}$) is satisfied, by (F${}^9$), when the final
medium is ordinary; and in fact the final rays whether straight
or curved are then perpendicular to the series of surfaces
represented by the equation
$$V = \hbox{const.}:
   \eqno {\rm (D^{10})}$$
which is, for ordinary rays, the integral of the equation
(X${}^9$), and gives, as an approximate equation of the normal
surface at the origin, the following,
$$0 = \delta V + {\textstyle {1\over 2}} \delta^2 V,
   \quad \hbox{or} \quad
  0 = \mu z
      + {\textstyle {1\over 2}} {\delta^2 V \over \delta x^2} x^2
      + {\delta^2 V \over \delta x \, \delta y} xy
      + {\textstyle {1\over 2}} {\delta^2 V \over \delta y^2} y^2;
   \eqno {\rm (E^{10})}$$
agreeing, by (F${}^9$), with the equation of the paraboloid
(Z${}^9$).  In general, the condition (C${}^{10}$) for the
existence of a normal surface, may be put, by (G${}^9$), under
the form
$$\eqalign{
   &\mathrel{\phantom{=}}
      {\delta^2 v \over \delta \alpha^2} \left(
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \beta \, \delta x}
         \right)
    - {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
            {\delta^2 V \over \delta x^2}
          - {\delta^2 v \over \delta \alpha \, \delta x}
         \right) \cr
   &= {\delta^2 v \over \delta \beta^2} \left(
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha \, \delta y}
         \right)
    - {\delta^2 v \over \delta \alpha \, \delta \beta} \left(
            {\delta^2 V \over \delta y^2}
          - {\delta^2 v \over \delta \beta \, \delta y}
         \right):\cr}
   \eqno {\rm (F^{10})}$$
and it is not satisfied by extraordinary rays, except in
particular cases.  We may however always consider the paraboloid
(Z${}^9$) as an auxiliary surface, with which the final ray-lines
of the proposed system (V${}^9$) are connected by a remarkable and
simple relation. For if we take the rectangular planes of
curvature of this paraboloid for the co-ordinate planes of $xz$,
$yz$, and denote the two curvatures corresponding by
$r$, $t$, so as to have the following form for the equation of
the paraboloid
$$z = {\textstyle {1 \over 2}} r x^2 + {\textstyle {1 \over 2}} t y^2,
   \eqno {\rm (G^{10})}$$
we shall satisfy the condition
$${\delta \alpha \over \delta y}
      + {\delta \beta  \over \delta x}
   = 0,
   \eqno {\rm (H^{10})}$$
and may employ the following expressions for the four
coefficients of our problem,
$${\delta \alpha \over \delta x}= - r,\quad
  {\delta \alpha \over \delta y}= - n,\quad
  {\delta \beta  \over \delta x}= + n,\quad
  {\delta \beta  \over \delta y}= - t:\quad
   \eqno {\rm (I^{10})}$$
the ray-lines of our system (V${}^9$) may therefore be thus
represented
$$\left. \eqalign{
x &= \delta x - z ( r \, \delta x + n \, \delta y),\cr
y &= \delta y - z ( t \, \delta y - n \, \delta x),\cr}
   \right\}
   \eqno {\rm (K^{10})}$$
while the normals to the paraboloid are represented by these
equations
$$x = \delta x - z r \, \delta x,\quad
  y = \delta y - z t \, \delta y;
   \eqno {\rm (L^{10})}$$
from which it follows that the angle $\delta \nu$ between a
ray-line (K${}^{10}$) and the corresponding normal (L${}^{10}$)
may be thus expressed
$$\delta \nu = n \, \delta l,
   \quad \hbox{in which} \quad
  \delta l = \sqrt{\delta x^2 + \delta y^2},
   \eqno {\rm (M^{10})}$$
$\delta l$ being the same small line $\overline{B B_\delta}$ as
before; and that the plane of this angle $\delta \nu$, or in
other words, the plane containing the ray-line and the normal,
has for equation
$$x \, \delta x + y \, \delta y
   = \delta l^2 - z (r \, \delta x^2 + t \, \delta y^2):
   \eqno {\rm (N^{10})}$$
this plane therefore contains also the right line having for
equations
$$x \, \delta x + y \, \delta y = 0,\quad
  z = {\delta l^2 \over r \, \delta x^2 + t \, \delta y^2},
   \eqno {\rm (O^{10})}$$
that is, the axis of the osculating circle of curvature of the
normal or diametral section of the paraboloid, of which the line
$\delta l$ is an element; and {\it the normal may be brought to
coincide with the ray-line by being made to revolve around the
element $\delta l$, through an angle $\delta v$ proportional to
$\delta l$, and equal to that element multiplied by the constant
$n$\/}: the direction of the rotation depending on the sign of
the constant.  On account of this simple law of deviation of the
final ray-lines from the normals of the paraboloid, we shall call
this paraboloid the {\it guiding surface\/}: and the
constant~$n$, we shall call the {\it constant of deviation}.  And
we may consider this theory, of the guiding paraboloid and the
constant of deviation, as containing an adequate solution of our
third general problem, in the discussion of the geometrical
relations of infinitely near rays: since this theory shows
adequately the general arrangement of the final system of
ray-lines (V${}^9$), and the geometrical meanings of the third
set of coefficients (O${}^9$), namely,
$${\delta \alpha \over \delta x},\quad
  {\delta \alpha \over \delta y},\quad
  {\delta \beta  \over \delta x},\quad
  {\delta \beta  \over \delta y}.$$

The geometrical construction suggested by this theory may be
still farther simplified by observing that the infinitely near
normals to the guiding surface all pass through two rectangular
lines, namely, the axes of the two principal circles of curvature
of the surface; it is therefore sufficient to draw through any
proposed point $B_\delta$ two planes containing respectively
these two given axes of curvature, and then to make the line of
intersection of these two planes revolve round the proposed small
line $\delta l$ or $\overline{B B_\delta}$, through the same
small angle $n \, \delta l$ as before, in order to obtain the
sought final ray-line for the proposed final point.

Finally, to compare, as required in the {\it fourth\/} problem,
the initial system of ray-lines (W${}^9$) with the corresponding
final points $B_\delta$ on the given final plane, we may denote
these initial ray-lines by the equations
$$x' = z' \, \delta \theta' \mathbin{.} \cos \phi',\quad
  y' = z' \, \delta \theta' \mathbin{.} \sin \phi',
   \eqno {\rm (P^{10})}$$
if we put
$$\delta \alpha' = \delta \theta' \mathbin{.} \cos \phi',\quad
  \delta \beta'  = \delta \theta' \mathbin{.} \sin \phi':
   \eqno {\rm (Q^{10})}$$
and if in like manner we put
$$\delta x = \delta l \mathbin{.} \cos \phi,\quad
  \delta y = \delta l \mathbin{.} \sin \phi,
   \eqno {\rm (R^{10})}$$
we shall have the following relations, between
$\phi$, $\phi'$, $\delta l$, $\delta \theta'$,
and the fourth set of partial differential coefficients
(O${}^9$),
$$\left. \eqalign{
\delta \theta' \mathbin{.} \cos \phi' = \left(
            {\delta \alpha' \over \delta x} \cos \phi
          + {\delta \alpha' \over \delta y} \sin \phi
         \right) \, \delta l,\cr
\delta \theta' \mathbin{.} \sin \phi' = \left(
            {\delta \beta'  \over \delta x} \cos \phi
          + {\delta \beta'  \over \delta y} \sin \phi
         \right) \, \delta l.\cr}
   \right\}
   \eqno {\rm (S^{10})}$$
These relations give
$$\tan \phi'
  = {\displaystyle
            {\delta \beta'  \over \delta x}
          + {\delta \beta'  \over \delta y} \tan \phi
         \over \displaystyle
            {\delta \alpha' \over \delta x}
          + {\delta \alpha' \over \delta y} \tan \phi};
   \eqno {\rm (T^{10})}$$
they enable us therefore to determine, for any given value of
$\phi$, that is, for any proposed direction of the small final
line $\delta l$, or $\overline{B B_\delta}$, the corresponding
value of $\phi'$, that is, the direction of the initial plane of
ray-lines, having for equation
$$y' = x' \tan \phi'.
   \eqno {\rm (U^{10})}$$
Thus the final line $\delta l$ and initial plane~$\phi'$
revolve together, but not in general with equal rapidity; and
arbitary rectangular directions of the one do not in general give
rectangular directions of the other, because the conditions
$$\left. \matrix{\displaystyle
\tan \phi_1'
  = {\displaystyle
            {\delta \beta'  \over \delta x}
          + {\delta \beta'  \over \delta y} \tan \phi_1
         \over \displaystyle
            {\delta \alpha' \over \delta x}
          + {\delta \alpha' \over \delta y} \tan \phi_1},\quad
\tan \phi_2'
  = {\displaystyle
            {\delta \beta'  \over \delta x}
          + {\delta \beta'  \over \delta y} \tan \phi_2
         \over \displaystyle
            {\delta \alpha' \over \delta x}
          + {\delta \alpha' \over \delta y} \tan \phi_2},\cr
   \noalign{\vskip 3pt} \displaystyle
\phi_2  = \phi_1  + {\pi \over 2},\quad
\phi_2' = \phi_1' + {\pi \over 2},\cr}
   \right\}
   \eqno {\rm (V^{10})}$$
(in which $\pi$ is the semicircumference to the radius unity,)
give the following formula for the angle $\phi_1$,
$$2 \left(  {\delta \beta'  \over \delta x}
               {\delta \beta'  \over \delta y}
          + {\delta \alpha' \over \delta x}
               {\delta \alpha' \over \delta y}
         \right) \cotan 2 \phi_1
  =    \left( {\delta \beta'  \over \delta x} \right)^2
     - \left( {\delta \beta'  \over \delta y} \right)^2
     + \left( {\delta \alpha' \over \delta x} \right)^2
     - \left( {\delta \alpha' \over \delta y} \right)^2,
   \eqno {\rm (W^{10})}$$
which is not in general satisfied by arbitrary values of that
angle.  There are however in general two rectangular final
directions determined by this formula, which correspond to two
rectangular initial planes; and if we take these rectangular
directions and planes respectively for the directions of
$x$, $y$, and for the planes of $x' z'$, $y' z'$, we shall
have
$${\delta \alpha' \over \delta y} = 0,\quad
  {\delta \beta'  \over \delta x} = 0.\quad
   \eqno {\rm (X^{10})}$$
We may also in general satisfy, at the same time, by a proper
choice of the semiaxes of co-ordinates, the following other
conditions,
$${\delta \beta'  \over \delta y} > 0,\quad
  {\delta \alpha' \over \delta x} > {\delta \beta'  \over \delta y}.
   \eqno {\rm (Y^{10})}$$
By this choice of co-ordinates, the relations (S${}^{10}$) are
simplified, and become
$$\left. \eqalign{
\delta \theta' \mathbin{.} \cos \phi'
   = {\delta \alpha' \over \delta x}
         \mathbin{.} \delta l \mathbin{.} \cos \phi;\cr
\delta \theta' \mathbin{.} \sin \phi'
   = {\delta \beta'  \over \delta y}
         \mathbin{.} \delta l \mathbin{.} \sin \phi:\cr}
   \right\}
   \eqno {\rm (Z^{10})}$$
while the equations (W${}^9$) of the initial ray-lines reduce
themselves to the following,
$$x' = z' {\delta \alpha' \over \delta x} \, \delta x;\quad
  y' = z' {\delta \beta'  \over \delta y} \, \delta y.
   \eqno {\rm (A^{11})}$$
If, then, these initial ray-lines form a circular cone having for
equation
$$x'^2 + y'^2 = z'^2 \, \delta \theta'^2,
   \eqno {\rm (B^{11})}$$
the corresponding locus of the final point $B_\delta$, on the
final plane of $xy$, will not in general be a circle, but an
ellipse, having for its equation
$$    \left( {\delta \alpha' \over \delta x} \right)^2 \, \delta x^2
    + \left( {\delta \beta'  \over \delta y} \right)^2 \, \delta y^2
   = \delta \theta'^2,
   \eqno {\rm (C^{11})}$$
of which, by (Y${}^{10}$), the axis of $x$ coincides with the
least and the axis of $y$ with the greatest axis; and
reciprocally if the final locus be a circle having for equation
$$\delta x^2 + \delta y^2 = \delta l^2,
   \eqno {\rm (D^{11})}$$
the initial cone of ray-lines will have for equation
$$    x'^2 \left( {\delta \alpha' \over \delta x} \right)^{-2}
    + y'^2 \left( {\delta \beta'  \over \delta y} \right)^{-2}
  = z'^2 \, \delta l^2,
   \eqno {\rm (E^{11})}$$
so that its perpendicular sections are ellipses, having their
greater axes in the plane of $x' z'$, and their lesser axes in
the plane of $y' z'$.  It is evident that a circle equal to
the final circle (D${}^{11}$) may be obtained from the elliptic
cone (E${}^{11}$), by cutting that elliptic cone by any one of
the four following planes,
$$z' =   \pm \left( {\delta \alpha' \over \delta x} \right)^{-1}
         \pm y' \sqrt{
             \left( {\delta \alpha' \over \delta x} \right)^2
             \left( {\delta \beta'  \over \delta y} \right)^{-2}
               - 1};
   \eqno {\rm (F^{11})}$$
and in like manner the four elliptic sections of the circular
cone (B${}^{11}$), made by the same four planes, are all equal
and similar to the final ellipse (C${}^{11}$).  In general it is
easy to prove by the equations of the initial ray-lines
(A${}^{11}$), that whatever final locus we take for the point
$B_\delta$, represented by the equation
$$\delta y = f(\delta x),
   \eqno {\rm (G^{11})}$$
the corresponding initial cone
$${y' \over z'} \left( {\delta \beta'  \over \delta y} \right)^{-1}
   = f   \left( {x' \over z'}
                \left( {\delta \alpha' \over \delta x} \right)^{-1}
         \right)
   \eqno {\rm (H^{11})}$$
will have four sections equal and similar to this final locus,
namely, the sections by the four planes (F${}^{11}$).  We may
therefore consider these as {\it four guiding planes\/} for the
initial ray, since {\it each contains for any proposed final
curve or locus\/} (G${}^{11}$) {\it of the final point
$B_\delta$, an equal and similar guiding curve or locus, which is
a section of the sought initial cone, and by which therefore that
cone may be determined}.  If, then, we know these four {\it
guiding planes}, or any one of them, and the corresponding system
of final and initial rectangular directions, or {\it conjugate
guiding axes}, of which two are determined by a guiding plane, we
shall be able to construct the initial ray-line or ray-cone
corresponding to any final position or locus of the point
$B_\delta$.  The fourth and last general problem of those
proposed above, may therefore be considered as resolved, by this
theory of the guiding planes and guiding axes.

We see then that in order to compare completely the extreme
directions of any two near luminous paths
$$(A,B)_\chi,\quad (A',B')_{\chi + \delta \chi},$$
in which $A$ is the initial and $B$ the final point of a given
path, and $A'$, $B'$, are any other initial and final points
infinitely near to these, the following geometrical {\it elements
of arrangement}, or some data equivalent to them, are necessary
and sufficient to be known.

First.  The final axis, and the initial axis, of chromatic
dispersion; and the corresponding final and initial constants
$\xi$, $\xi'$, with their proper signs, to indicate the
directions, as well as the quantities of dispersion.

Second.  The final axis, and the initial axis, of curvature of
the given path.

Third.  The final pair, and the initial pair, of axes of
curvature of the guiding paraboloids, at the ends of this given
path; and the final and initial constants of deviation $n$, $n'$.

Fourth.  A guiding plane for the initial ray-lines, and a guiding
plane for the final ray-lines; together with the final system and
the initial system of rectangular directions, or conjugate
guiding axes, connected with these guiding planes.

When these different elements of arrangement of the extreme
ray-lines are known, we can deduce from them the dependence of
$\delta \alpha$, $\delta \beta$, $\delta \alpha'$, $\delta \beta'$,
and more generally of
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$,
on
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$;
and reciprocally when this latter dependence has been deduced
from the partial differential coefficients of the characteristic
or related functions, we can deduce it from the geometrical
elements above mentioned.

\bigbreak

{\sectiontitle
Application of the Elements of Arrangement.  Connexion of the two
final Vergencies, and Planes of Vergency, and Guiding Lines, with
the two principal Curvatures and Planes of Curvature of the
Guiding Paraboloid, and with the Constant of Deviation.  The
Planes of Curvature are the Planes of Extreme Projection of the
final Ray-Lines.\par}

\nobreak\bigskip

16.
To give now an example of the application of these
geometrical elements of arrangement, let us employ them to
determine the {\it conditions of intersection of two near final
ray-lines}, corresponding to a given colour and to a given
luminous origin; and let us suppose, for simplicity, that one of
these two straight ray-lines being the final portion or final
tangent of a given luminous path $(A,B)_\chi$, the other
corresponds (as in the third of the foregoing problems) to a
final point $B_\delta$ on the given final plane perpendicular to
this given path at $B$.  Then if the constant~$n$ of deviation
vanishes, so that the final ray-lines are normals to the guiding
paraboloid, the condition of intersection requires evidently that
the near point $B_\delta$ should be in one of the two principal
diametral planes, that is, on one of the two rectangular tangents
to the lines of curvature on this surface; and the corresponding
point of intersection must be one of the two centres of
curvature.  But when $n$ does not vanish, the deviation of the
ray-lines obliges us to alter this result.  The intersection of
the near ray-line with the given ray-line will not now take place
for the directions of the lines of curvature; but for those other
directions, if any, for which the angular deviation
$n \, \delta l$ of the ray-line from the normal is equal and
contrary to the angular deviation of the normal from the
corresponding plane of normal section, that is, from the
corresponding diametral plane of the guiding paraboloid.  This
latter deviation, abstracting from sign, is, by the general
properties of normals, equal to the semidifference of curvatures
multiplied by the element of the normal section $\delta l$, and
by the sine of twice the inclination of this element to either of
the lines of curvature; it cannot therefore destroy the deviation
$n \, \delta l$ of the ray-line from the normal, unless the
semidifference of the two principal curvatures of the paraboloid
is greater, or at least not less, abstracting from sign, than the
constant of deviation~$n$; this then is a necessary condition for
the possibility of the intersection sought.  But when the
semidifference of curvatures is greater (abstracting from sign)
than $n$, then there are two distinct directions $P_1$, $P_2$, of
the normal or diametral plane of section, symmetrically placed
with respect to the two principal planes of curvature, and such
that if the element of section $\delta l$ be contained in either
of these two planes $P_1$, $P_2$, (but not if the element $\delta
l$ be in any other normal plane,) the corresponding ray-line from
the extremity of that element will be contained in the same
normal plane $P_1$ or $P_2$, and will intersect the given
ray-line as required; and the point of intersection of these two
near ray-lines will be the centre of curvature of the
corresponding normal section.  We may therefore call the
curvatures of these two diametral sections the
{\it two vergencies\/} of the final ray-lines; and the two
corresponding planes $P_1$, $P_2$, we may call the
{\it two planes of vergency}.

The same conclusions may be deduced algebraically from the
equations (K${}^{10}$), which give the following conditions of
intersection of a near ray-line with the given ray-line or axis
of $z$,
$$0 = (z^{-1} - r) \, \delta x - n \, \delta y;\quad
  0 = (z^{-1} - t) \, \delta y + n \, \delta x;
   \eqno {\rm (I^{11})}$$
$z$ being the sought ordinate of intersection, and therefore
$z^{-1}$ the vergency: for thus we find by elimination the
following quadratic to determine the ratio of
$\delta x$, $\delta y$, that is, the direction of $\delta l$,
$$(t - r) \, \delta x \, \delta y
   = n ( \delta y^2 + \delta x^2),
   \eqno {\rm (K^{11})}$$
which may be put under the form
$$\sin 2 \phi = {2n \over t - r},
   \eqno {\rm (L^{11})}$$
the angle $\phi$ being, as in (R${}^{10}$), the inclination of
$\delta l$ to the axis of $x$, that is, to one of the tangents of
the lines of curvature, while $r$, $t$, are the two curvatures
themselves, of the guiding paraboloid; there are therefore two
real directions of $\delta l$, or one, or none, corresponding to
the intersection supposed, according as we have
$$\left( {t - r \over 2} \right)^2
   >, \hbox{ or } =, \hbox{ or } < n^2;
   \eqno {\rm (M^{11})}$$
so that we are thus conducted anew to the same conditions of
reality, and to the same symmetric directions of the two planes
of vergency, which we obtained before by a reasoning of a more
geometrical kind.  The same conditions may also be obtained by
considering the quadratic for the vergency itself, namely,
$$(z^{-1} - r)(z^{-1} - t) + n^2 =0,
   \eqno {\rm (N^{11})}$$
which results from the equations (I${}^{11}$) and shows that the
sum and product of the two vergencies may be thus expressed, by
means of the curvatures $r$, $t$, and the constant of deviation
$n$,
$$z_1^{-1} + z_2^{-1} = r + t;\quad
  z_1^{-1} z_2^{-1} = rt + n^2.
   \eqno {\rm (O^{11})}$$
The equations (I${}^{11}$) give also, by elimination of $n$,
$$z^{-1} = r \cos \phi^2 + t \sin \phi^2;
   \eqno {\rm (P^{11})}$$
we see, therefore, as before, that the two vergencies, when real,
of the final ray-lines, are the curvatures of the two
corresponding sections of the guiding paraboloid.  In general the
centre of curvature of any section of this surface, made by a
normal plane drawn through the given final ray-line, is the
common {\it focus by projection\/} of all the near ray-lines from
the points of that section; that is, the projections of these
near ray-lines on this plane, all pass through this centre of
curvature.  {\it The two rectangular planes of curvature, or
principal diametral planes, of the guiding paraboloid, may
therefore be called the planes of extreme projection\/}; under
which view they were considered in the First Supplement, for the
case of an uniform medium, and were proposed as {\it a pair of
natural co-ordinate planes\/} passing through any given straight
ray.  The two planes of vergency, for the case of straight final
rays, were also considered in that First Supplement, in connexion
with the two developable pencils or ray-surfaces which pass
through a given straight ray, and of which the two tangent planes
contain rays infinitely near, and therefore coincide with the two
planes of vergency.

When the planes of vergency are real and distinct, then, whether
the final rays are straight or curved, there exist {\it two
guiding lines\/} perpendicular to the given final ray-line, which
are both intersected by all the near final ray-lines from the
points $B_\delta$ on the given final plane of $xy$, and which
therefore suffice to determine the geometrical arrangement and
relations of that system of final ray-lines.  To prove the
existence and determine the positions of these two guiding lines,
let us examine what conditions are necessary and sufficient, in
order that a right line having for equations
$$y = x \, \tan \Phi,\quad z = Z,
   \eqno {\rm (Q^{11})}$$
should be intersected by all the near final ray-lines of the
system (K${}^{10}$).  These conditions are
$$Z^{-1} = r + n \, \cotan \Phi = t - n \, \tan \Phi;
   \eqno {\rm (R^{11})}$$
they give
$$\sin 2 \Phi = {2n \over t - r},
   \eqno {\rm (S^{11})}$$
and
$$(Z^{-1} - r)(Z^{-1} - t) + n^2 = 0:
   \eqno {\rm (T^{11})}$$
when therefore
$$(t - r)^2 > 4 n^2,
   \eqno {\rm (U^{11})}$$
that is, when there are two real vergencies there are also two
real guiding lines of the kind explained above; and these two
guiding lines are contained in the two planes of vergency, and
cross the final ray-line in the two corresponding points in which
it is crossed by other ray-lines of the same system: the
intersection of each guiding line with the given final ray-line
being the point of convergence or divergence of the near
ray-lines contained in that plane of vergency which contains the
other guiding line.  When the constant of deviation $n$ vanishes,
these guiding lines are necessarily real, and are the axes of the
two principal circles of curvature of the guiding paraboloid.
And when the final rays are straight, then, whether $n$ vanishes
or not, {\it the two guiding lines\/} (if real) {\it are tangents
to the two caustic surfaces\/}; that is, to {\it the two surfaces
which are touched by the final rays, and are the loci of the two
points of vergency}.  If the guiding lines are imaginary then the
points of vergency are so too, and the final rays are not all
tangents to any common surface.  We shall have occasion to resume
hereafter the theory of the caustic and developable surfaces.

If it happen that
$$t - r = \pm 2n,
   \eqno {\rm (V^{11})}$$
without $t - r$ and $n$ separately vanishing, then the two planes
of vergency close up into one plane, bisecting one pair of the
right angles formed by the two principal planes of curvature of
the guiding paraboloid; the two vergencies reduce themselves to
a single vergency, corresponding to this single plane, and equal
to the semisum of the two curvatures of the same surface: and the
two guiding lines reduce themselves to a single guiding-line,
passing through the corresponding point of convergence or
divergence, and having still the property of being intersected by
all the near final ray-lines, although this property is not now
sufficient to determine this system of ray-lines.

But if the two members of (V${}^{11}$) vanish separately, that is,
if the difference of the curvatures and the constant of deviation
are separately equal to zero, then the guiding paraboloid is a
surface of revolution, having its summit at the given final
point~$B$, and all the near final ray-lines are normals to this
paraboloid of revolution, and (with the same order of
approximation) to the osculating sphere at its summit, and they
all pass through the centre of this sphere.  Reciprocally, if
there be any one point $0, 0, Z$, through which all the final
ray-lines pass, the equations (K${}^{10}$) give
$$n = 0,\quad t = r = Z^{-1}:
   \eqno {\rm (W^{11})}$$
and the more general equations (V${}^9$), in which the
rectangular axes of $x$ and $y$ are arbitrary, give
$${\delta \alpha \over \delta x}
    = {\delta \beta \over \delta y} = - Z^{-1};\quad
  {\delta \alpha \over \delta y} = 0;\quad
  {\delta \beta \over \delta x} = 0;
   \eqno {\rm (X^{11})}$$
that is, by (G${}^9$), or (C${}^9$),
$$\left. \eqalign{
{\delta^2 V \over \delta x^2}
      + Z^{-1} {\delta^2 v \over \delta \alpha^2}
   &= {\delta^2 v \over \delta \alpha \, \delta x};\cr
{\delta^2 V \over \delta x \, \delta y}
      + Z^{-1} {\delta^2 v \over \delta \alpha \, \delta \beta}
   &= {\delta^2 v \over \delta \alpha \, \delta y}
    = {\delta^2 v \over \delta \beta  \, \delta x};\cr
{\delta^2 V \over \delta y^2}
      + Z^{-1} {\delta^2 v \over \delta \beta^2}
   &= {\delta^2 v \over \delta \beta  \, \delta y}.\cr}
   \right\}
   \eqno {\rm (Y^{11})}$$

When the final rays are straight, and satisfy these last
conditions (Y${}^{11}$), which then reduce themselves to the
following,
$${\delta^2 V \over \delta x^2}
      + Z^{-1} {\delta^2 v \over \delta \alpha^2}
   = 0,\quad
  {\delta^2 V \over \delta x \, \delta y}
      + Z^{-1} {\delta^2 v \over \delta \alpha \, \delta \beta}
   = 0,\quad
  {\delta^2 V \over \delta y^2}
      + Z^{-1} {\delta^2 v \over \delta \beta^2}
   = 0,
   \eqno {\rm (Z^{11})}$$
the given final ray becomes one of those which we have called
{\it principal rays\/} in former memoirs, and the point of
convergence or divergence $0, 0, Z$, is what we have called a
{\it principal focus}.

\bigbreak

{\sectiontitle
Second Application of the Elements.  Arrangements of the Near
Final Ray-lines from an Oblique Plane.  Generalisation of the
Theory of the Guiding Paraboloid and Constant of Deviation.
General Theory of Deflexures of Surfaces.  Circles and Axes of
Deflexure.  Rectangular Planes and Axes of Extreme Deflexure.
Deflected Lines passing through these Axes, and having the
Centres of Deflexure for their respective Foci by Projection.
Conjugate Planes of Deflexure, and Indicating Cylinder of
Deflexion.\par}

\nobreak\bigskip

17.
The foregoing theorems respecting the mutual relations of
the final ray-lines, suppose that the near final point $B_\delta$
is on the given plane which is perpendicular to the given luminous
path $(A,B)_\chi$ at its given final point~$B$: but analogous
theorems can be found for the more general case where the near
final point $B'$ is not in this given perpendicular plane, by
combining the solutions of the second and third of the four
problems lately discussed; that is, by considering jointly the
second and third sets of coefficients (O${}^9$), and therefore by
employing the following equations for a final ray-line,
$$\left. \eqalign{
x &= \delta x + z \left(
            {\delta \alpha \over \delta x} \, \delta x
          + {\delta \alpha \over \delta y} \, \delta y
          + {\delta \alpha \over \delta z} \, \delta z
         \right),\cr
y &= \delta y + z \left(
            {\delta \beta  \over \delta x} \, \delta x
          + {\delta \beta  \over \delta y} \, \delta y
          + {\delta \beta  \over \delta z} \, \delta z
         \right).\cr}
   \right\}
   \eqno {\rm (A^{12})}$$
If, in these equations, we establish no relation between
$\delta x$, $\delta y$, $\delta z$,
then the system of these final ray-lines (A${}^{12}$) is what has
been called (in my Theory of Systems of Rays) {\it a System of
the Third Class}, because the equations of a ray-line in this
system involve {\it three\/} arbitrary elements of position,
namely the co-ordinates
$\delta x$, $\delta y$, $\delta z$,
of the near point $B'$; but to study more conveniently the
properties of this total system of the third class, we may
decompose it into partial {\it systems of the second class}, that
is, systems with only two arbitrary elements of position, by
assuming some relation, with an arbitrary parameter, between the
three co-ordinates
$\delta x$, $\delta y$, $\delta z$,
or, in other words, by assuming some arbitrary and variable
surface, as a locus for the near point $B'$.  For example we may
assume, as this locus, an oblique plane passing through the given
point $B$, and having for equation
$$\delta z = p \, \delta x + q \, \delta y,
   \eqno {\rm (B^{12})}$$
in which one of the two parameters $p$, $q$, is arbitrary, and
the other depends on it by some assumed law; and then, for every
such assumed plane locus (B${}^{12}$), we shall have to consider
a partial system of the second class, deduced from and included
in the total system of the third class (A${}^{12}$); namely a
system in which the ray lines are as follows,
$$\left. \eqalign{
x &= \delta x
      + z \left( {\delta \alpha \over \delta x}
          + p {\delta \alpha \over \delta z} \right) \, \delta x
      + z \left( {\delta \alpha \over \delta y}
          + q {\delta \alpha \over \delta z} \right) \, \delta y;\cr
y &= \delta y
      + z \left( {\delta \beta  \over \delta x}
          + p {\delta \beta  \over \delta z} \right) \, \delta x
      + z \left( {\delta \beta  \over \delta y}
          + q {\delta \beta  \over \delta z} \right) \, \delta y.\cr}
   \right\}
   \eqno {\rm (C^{12})}$$
Let us therefore consider the geometrical arrangement and
properties of this system of final ray-lines (C${}^{12}$),
corresponding to the oblique plane locus (B${}^{12}$) of the
final point $B'$.

The system (C${}^{12}$), of ray-lines from the arbitrary oblique
plane (B${}^{12}$), includes, as a particular case, the system of
ray-lines from the plane of no obliquity: that is, the system
(V${}^9$), considered in a former number.  And as the ray-lines
of that particular system (V${}^9$) were found to have a
remarkable connection with the guiding paraboloid (Z${}^9$),
which touched the given perpendicular plane locus of the near
final point $B_\delta$, and which satisfied the differential
condition of the second order (Y${}^9$): so, the ray-lines of the
more general system (C${}^{12}$) may be shown to be connected in
an analogous manner with the following more general paraboloid,
which satisfies the same differential condition (Y${}^9$), and
touches the more general oblique plane locus (B${}^{12}$) at the
given final point $B$,
$$z = p x + q y
      + {\textstyle {1 \over 2}} r x^2
      + s xy
      + {\textstyle {1 \over 2}} t y^2;
   \eqno {\rm (D^{12})}$$
in which $p$, $q$, retain their recent meanings, and the
coefficients $r$, $s$, $t$ have the following values,
$$\left. \eqalign{
r  &= - \left( {\delta \alpha \over \delta x}
               + p {\delta \alpha \over \delta z} \right);\quad
t   = - \left( {\delta \beta  \over \delta y}
               + q {\delta \beta  \over \delta z} \right);\cr
s  &= - {\textstyle {1 \over 2}} \left(
               {\delta \beta  \over \delta x}
             + {\delta \alpha \over \delta y}
             + p {\delta \beta  \over \delta z}
             + q {\delta \alpha \over \delta z}
            \right).\cr}
   \right\}
   \eqno {\rm (E^{12})}$$
But in order to develope this more general connexion, between the
ray-lines (C${}^{12}$) and the paraboloid (D${}^{12}$), it will
be useful previously to establish some general theorems
respecting the deflexure of curved surfaces, which include some
of the known theorems respecting their curvatures and planes of
curvature.


Let us then consider the paraboloid (D${}^{12}$), or any other
curved surface which has, at the origin of co-ordinates, a
complete contact of the second order therewith, and which is
therefore approximately represented by the same equation: that
is, (on account of the arbitrary position of the origin, and
arbitrary values of the coefficients $p$, $q$, $r$, $s$, $t$,)
any surface of continuous curvature, near any assumed point on
this surface.  The tangent plane at this arbitrary point or
origin has for equation
$$z = p x + q y;
   \eqno {\rm (F^{12})}$$
and the {\it deflexion\/} from this tangent plane, measured in
the direction of the arbitrary axis of $z$, which we shall call
the {\it axis of deflexion}, or in any direction infinitely near
to this, is, for any point $B'$ infinitely near to the point of
contact $B$,
$$\hbox{\it Deflexion} = {\textstyle {1 \over 2}} \delta^2 z
   =    {\textstyle {1 \over 2}} r \, \delta x^2
      + s \, \delta x \, \delta y
      + {\textstyle {1 \over 2}} t \, \delta y^2.
   \eqno {\rm (G^{12})}$$
This deflexion depends therefore on the perpendicular distance
$\delta l$ of the near point $B'$ from the axis of deflexion, and
on the direction of the plane containing this point and axis; in
such a manner that if we put, as in (R${}^{10}$),
$$\delta x = \delta l \mathbin{.} \cos \phi,\quad
  \delta y = \delta l \mathbin{.} \sin \phi,$$
and give the name of {\it deflexure\/} (after the analogy of the
known name {\it curvature\/}) to the quotient
$\displaystyle {\delta^2 z \over \delta l^2}$,
that is, to the double deflexion divided by the square of the
perpendicular distance from the axis of deflexion, we shall have
the following law of dependence of this {\it deflexure}, which we
shall denote by $f$, on the angle $\phi$,
$$\hbox{\it Deflexure} = f = {\delta^2 z \over \delta l^2}
   =   r \, \cos \phi^2
     + 2s \, \cos \phi \, \sin \phi
     + t \, \sin \phi^2.
   \eqno {\rm (H^{12})}$$
There are, therefore, {\it two rectangular planes of extreme
deflexure}, corresponding to angles $\phi_1$, $\phi_2$,
determined by the following formula,
$$\tan 2 \phi = {2s \over r - t};
   \eqno {\rm (I^{12})}$$
and if we take these for the co-ordinate planes of
$xz$, $yz$,
and denote the {\it two extreme deflexures\/} corresponding to
$f_1$, $f_2$, we have
$$r = f_1,\quad s = 0,\quad t = f_2,
   \eqno {\rm (K^{12})}$$
and the general formula for the deflexure becomes
$$f = f_1 \, \cos \phi^2 + f_2 \, \sin \phi^2:
   \eqno {\rm (L^{12})}$$
which is analogous to, and includes, the known formula for the
curvature of a normal section.  And as it is usual to consider a
system of circles of curvature, for any given point of a curved
surface, namely, the osculating circles of the normal sections of
that surface, so we may now more generally consider a system of
{\it circles of deflexure\/}: namely, in each plane of
deflexure~$\phi$, a circle passing through the given point of the
surface; and having its centre on the given axis of deflexion,
and its curvature equal to the deflexure~$f$; so that the radius
of this circle, or the ordinate of its centre, which we may call
the {\it radius of deflexure}, is
$\displaystyle {1 \over f}$,
and so that the equations of the circle of deflexure are,
$$y = x \, \tan \phi,\quad x^2 + y^2 + z^2 = {2z \over f}.
   \eqno {\rm (M^{12})}$$
We may also give the name of {\it axis of deflexure}, to the axis
of this circle, that is, to the right line having for equations
$$y = - x \, \cotan \phi,\quad z = {1 \over f}:
   \eqno {\rm (N^{12})}$$
and we easily see that there are {\it two principal circles of
deflexure}, analogous to the two principal circles of curvature,
namely, the two circles having for equations
$$\left. \vcenter{\halign{#\hfil&&\quad $\displaystyle #$\hfil\cr
First&  y = 0, & x^2 + z^2 = {2z \over f_1}; \cr
   \noalign{\vskip 3pt}
Second& x = 0, & y^2 + z^2 = {2z \over f_2}; \cr}}
   \right\}
   \eqno {\rm (O^{12})}$$
and {\it two principal rectangular axes of deflexure}, namely,
$$\hbox{First}\enspace  x = 0,\quad z = {1 \over f_1};\quad
  \hbox{Second}\enspace y = 0,\quad z = {1 \over f_2}.
   \eqno {\rm (P^{12})}$$
These principal axes of deflexure are analogous to the principal
axes of curvature, that is, to the axes of the two principal
osculating circles of the normal sections, in the less general
theory of normals.  And as, in that theory, the near normals all
pass through the two principal axes of curvature, so we may now
consider a more general system of right lines, which we shall
call the {\it deflected lines,} all near the arbitrary axis of
deflexion, and all passing through the two corresponding
principal axes of deflexure, and therefore having for equations,
$$x = \delta x - z f_1 \, \delta x,\quad
  y = \delta y - z f_2 \, \delta y,
   \eqno {\rm (Q^{12})}$$
when the co-ordinates are chosen as before.  These deflected lines
are normals, in the present order of approximation, to the
locus of the circles of deflexure (M${}^{12}$), that is, to the
surface of the fourth degree
$$x^2 + y^2 + z^2 = {2z(x^2 + y^2) \over f_1 x^2 + f_2 y^2};
   \eqno {\rm (R^{12})}$$
and they might be defined by this condition, or by the condition
that they are normals, in the same order of approximation, to the
following paraboloid,
$$z = {\textstyle {1 \over 2}} (f_1 x^2 + f_2 y^2),
   \eqno {\rm (S^{12})}$$
which osculates to the locus (R${}^{12}$), and has the property
that its ordinates measure the deflexions (G${}^{12}$) of the
given surface.

A deflected line of the system (Q${}^{12}$) is in the
corresponding plane of deflexure
$$y \, \delta x = x \, \delta y,
   \eqno {\rm (T^{12})}$$
if that plane coincide with either of those two principal
rectangular planes of deflexure, which we have taken for
co-ordinate planes; but otherwise the deflected line makes with
the plane of deflexure an infinitesimal angle $\delta \psi$,
expressed as follows,
$$\delta \psi
   = {\textstyle {1 \over 2}} (f_1 - f_2)
         \, \delta l \mathbin{.} \sin 2 \phi:
   \eqno {\rm (U^{12})}$$
this angle, therefore, is equal to the semidifference of the
extreme deflexures multiplied by the infinitesimal perpendicular
distance from the axis of deflexion, and by the sine of twice the
inclination $\phi$ of this perpendicular (or of the plane of
deflexure containing it) to one of the two rectangular planes of
extreme deflexure.  In this general case, the deflected line
(Q${}^{12}$) does not intersect the given axis of deflexion,
which we have made the axis of $z$; but the deflected line
(Q${}^{12}$) always intersects its own axis of deflexure
(N${}^{12}$), in a point of which the co-ordinates may be thus
expressed
$$x = - {\delta \psi \over f} \mathbin{.} \sin \phi,\quad
  y =   {\delta \psi \over f} \mathbin{.} \cos \phi,\quad
  z = {1 \over f},
   \eqno {\rm (V^{12})}$$
the symbols $f$, $\phi$, and $\delta \psi$, retaining their
recent meanings.  It is easy to see that if a near deflected line
be projected on the corresponding plane of deflexure, the
projection will cross the axis of deflexion in the centre of the
circle of deflexure; and therefore that this centre of deflexure
may be considered as a {\it focus by projection}, and that {\it
the planes of extreme deflexure are planes of extreme
projection}.

The foregoing results respecting the deflexures and deflected
lines of a curved surface, near any given point upon that
surface, and for any given axis of deflexion, may easily be
expressed by general formul{\ae} extending to an arbitrary
origin and arbitrary axes of co-ordinates.  If, for simplicity, we
still suppose the co-ordinates rectangular, and still take the
given point upon the surface for origin, and the given axis of
deflexion for axis of $z$, but leave the rectangular co-ordinate
planes of $xz$ and $yz$ arbitrary, so that the
coefficient~$s$ in the equation of the surface shall not in
general vanish, then the equations of a deflected line become
$$x = \delta x - z (r \, \delta x + s \, \delta y),\quad
  y = \delta y - z (s \, \delta x + t \, \delta y);
   \eqno {\rm (W^{12})}$$
since the equation of the paraboloid (S${}^{12}$), to which they
are nearly normals, and of which the ordinates measure the
deflexions (G${}^{12}$) of the given surface, becomes
$$z =   {\textstyle {1 \over 2}} r \, \delta x^2
      + s \, \delta x \, \delta y
      + {\textstyle {1 \over 2}} t \, \delta y^2.
   \eqno {\rm (X^{12})}$$
The deflexure for any plane~$\phi$ is expressed by the general
formula (H${}^{12}$); and in like manner the general formul{\ae}
(M${}^{12}$) (N${}^{12}$) determine still the circle and axis of
deflexure.  The two principal planes of deflexure $\phi_1$,
$\phi_2$, are still determined by the formula (I${}^{12}$), while
the corresponding extreme deflexures $f_1$, $f_2$, are the roots
of the following quadratic
$$f^2 - f (r + t) + rt - s^2 = 0:
   \eqno {\rm (Y^{12})}$$
and the angular deviation $\delta \psi$ of a deflected line from
the corresponding plane of deflexure, is thus expressed,
$$\delta \psi
   = {\textstyle {1 \over 2}} (f_1 - f_2)
         \mathbin{.} \sin (2 \phi - 2 \phi_1)
         \mathbin{.} \delta l
   = \left( {r - t \over 2} \mathbin{.} \sin 2 \phi
         - s \mathbin{.} \cos 2\phi \right) \, \delta l.
   \eqno {\rm (Z^{12})}$$

Before we proceed to apply these general remarks on the
deflexures of surfaces to the optical question proposed in the
present number, that is, to the study of the connexion of the
ray-lines (C${}^{12}$) with the paraboloid (D${}^{12}$), we may
remark that the theory which {\sc M.~Dupin} has given, in his
excellent {\it D\'{e}veloppements de G\'{e}om\'{e}trie}, of the
{\it indicating curves\/} and {\it conjugate tangents\/}
of a surface, may be extended from curvatures to deflexures.  For
if we consider the deflexion
(${1 \over 2} \delta^2 z = {1 \over 2} f \, \delta l^2$)
in the given arbitrary direction of $z$ as equal to any given
infinitesimal quantity of the second order, that is, if we cut
the given surface by a plane
$$z - px - qy = {\textstyle {1 \over 2}} \delta^2 z
   = \hbox{\it deflexion} = \hbox{const.},
   \eqno {\rm (A^{13})}$$
parallel and infinitely near to the given tangent plane
(F${}^{12}$), we obtain in general a plane curve of section which
may be considered as of the second degree, namely, the
{\it indicating curve\/} considered by {\sc M.~Dupin}, of which
the axes by their directions and values indicate the shape of the
given surface near the given point, by indicating its curvatures
and planes of curvature.  This indicating curve is on the
following {\it cylinder of the second degree}, which has for its
indefinite axis the axis of deflexion, and which we shall call
the {\it indicating cylinder of deflexion},
$$r x^2 + 2 s x y + t y^2 = \delta^2 z = \hbox{const.};
   \eqno {\rm (B^{13})}$$
and it is easy to see that the two principal planes of deflexure,
$\phi_1$, $\phi_2$, are the principal diametral planes of this
indicating cylinder, and that the two principal deflexures
$f_1$, $f_2$, positive or negative, are equal respectively to the
given double deflexion $\delta^2 z$ divided by the squares of the
real or imaginary principal semidiameters or seimiaxes of the
cylinder, perpendicular to its indefinite axis.  In general, the
positive or negative deflexure~$f$, corresponding to any plane of
deflexure~$\phi$, is equal to the given double deflexion
$\delta^2 z$ divided by the square of the real or imaginary
semidiameter of the cylinder, contained in this plane of
deflexure, and perpendicular to the axis of deflexion, that is,
to the indefinite axis of the cylinder.  Hence it follows, that
if we consider any two conjugate diametral planes $\phi$,
$\phi_\prime$, which we shall call {\it conjugate planes of
deflexure}, and which are connected by the relation
$$0 = r + s ( \tan \phi + \tan\phi_\prime)
      + t \mathbin{.} \tan \phi \, \tan \phi_\prime,
   \eqno {\rm (C^{13})}$$
{\it the sum of the two corresponding conjugate radii of
deflexure
$\displaystyle {1 \over f} + {1 \over f_\prime}$,
is constant, and equal to the sum of the two extreme or principal
radii\/}: that is, we have
$${1 \over f} + {1 \over f_\prime}
   = {1 \over f_1} + {1 \over f_2},
   \eqno {\rm (D^{13})}$$
a relation which might also have been deduced from the general
expression for the deflexure, without its being necessary to
employ the indicating cylinder.  We may remark that any two
conjugate planes of deflexure, connected by the relation
(C${}^{13}$), intersect the tangent plane of the surface in two
conjugate tangents of the kind considered by {\sc M.~Dupin}.

Let us now resume the system of ray-lines (C${}^{12}$), of which
the equations may be put by (E${}^{12}$) under the form
$$\left. \eqalign{
x  &= \delta x - z (r \, \delta x + s \, \delta y)
         - z n \, \delta y,\cr
y  &= \delta y - z (s \, \delta x + t \, \delta y)
         + z n \, \delta x,\cr}
   \right\}
   \eqno {\rm (E^{13})}$$
if we make
$$n = {\textstyle {1 \over 2}} \left(
                  {\delta \beta  \over \delta x}
                - {\delta \alpha \over \delta y}
              + p {\delta \beta  \over \delta z}
              - q {\delta \alpha \over \delta z}
            \right):
   \eqno {\rm (F^{13})}$$
and let us compare these ray-lines with the deflected lines from
the auxiliary paraboloid (D${}^{12}$), which have for equations
$$x = \delta x - z ( r \, \delta x + s \, \delta y),\quad
  y = \delta y - z ( s \, \delta x + t \, \delta y).
   \eqno {\rm (W^{12})}$$
We easily see, by this comparison, that the infinitesimal angle
of deviation $\delta \nu$ of a ray-line (E${}^{13}$) from the
corresponding deflected line (W${}^{12}$), is still determined by
the same formula (M${}^{10}$)
$$\delta \nu = n \, \delta l,$$
as in the simpler theory of the guiding paraboloid explained in
the fifteenth number; that is, this angular deviation
$\delta \nu$ is still equal to the perpendicular distance
$\delta l$ of the near final point from the given final ray-line,
multiplied by a constant of deviation $n$.  The plane of this
angle $\delta \nu$, that is, the plane containing the ray-line
(E${}^{13}$) and the deflected line (W${}^{12}$), has for
equation
$$x \, \delta x + y \, \delta y
   = \delta l^2 - z ( r \, \delta x^2
          + 2s \, \delta x \, \delta y + t \, \delta y^2 ),
   \eqno {\rm (G^{13})}$$
and therefore contains the right line having for equations
$$x \, \delta x + y \, \delta y = 0,\quad
   z = {\delta x^2 + \delta y^2
            \over r \, \delta x^2 + 2s \, \delta x \, \delta y
                  + t \delta y^2},
   \eqno {\rm (H^{13})}$$
that is, the axis of deflexure (N${}^{12}$): results which are
analogous to those of the fifteenth number, expressed by the
equations (N${}^{10}$) (O${}^{10}$).  And we may construct the
final ray-line (E${}^{13}$) by a process of rotation analogous to
that already employed, namely, by making the deflected line
(W${}^{12}$), which passes through the two rectangular axes of
deflexure of the auxiliary paraboloid (D${}^{12}$), revolve round
the perpendicular $\delta l$, through the infinitesimal angle
$\delta \nu$, proportional to that perpendicular.  The theory,
therefore, of the guiding paraboloid and constant of deviation,
which was given in the fifteenth number, for the ray-lines from
the near points $B_\delta$ on the final perpendicular plane,
extends with little modification to the ray-lines from the points
$B'$ on any final oblique plane locus passing through the given
final point: namely, by employing a more general auxiliary
paraboloid, and by considering deflexures and deflected lines,
instead of curvatures and normals.  And we may transfer to this
more general auxiliary paraboloid, and to its connected constant
of deviation, the reasonings of the sixteenth number, respecting
the system of final ray-lines; for example, the reasoning
respecting the foci by projection, and those respecting the
condition of intersection of such ray-lines.  And since for any
given values of $p$, $q$, that is, for any given position of the
oblique plane (B${}^{12}$), we can construct the new auxiliary
paraboloid (D${}^{12}$), and its new constant of deviation
(F${}^{13}$), by the coefficients
$${\delta \alpha \over \delta x},\quad
  {\delta \beta  \over \delta x},\quad
  {\delta \alpha \over \delta y},\quad
  {\delta \beta  \over \delta y},\quad
  {\delta \alpha \over \delta z},\quad
  {\delta \beta  \over \delta z},$$
that is, by means of the former guiding paraboloid (Z${}^9$) and
the former constant of deviation (B${}^{10}$), and by the
magnitude and plane of curvature (T${}^9$) of the final ray, we
may be considered as having reduced the theory of the geometrical
arrangement and relations of the system of final ray-lines
(C${}^{12}$), from an oblique plane (B${}^{12}$), to the theory
of the {\it elements of arrangement}, which was given in the
fifteenth number.

\bigbreak

{\sectiontitle
Construction of the New Auxiliary Paraboloid, (or of an
Osculating Hyperboloid,) and of the New Constant of Deviation,
for Ray-lines from an Oblique Plane, by the former Elements of
Arrangement.\par}

\nobreak\bigskip

18.
To construct the new auxiliary paraboloid (D${}^{12}$) by
the former elements of arrangement, we may observe that this new
paraboloid not only touches the given oblique plane (B${}^{12}$)
at the given final point $B$ of the original luminous path, but
osculates in all directions at that given point to a certain
hyperboloid, represented by the following equation,
$$z = px + qy
      + {\textstyle {1 \over 2}} r_0 x^2
      + s_0 x y
      + {\textstyle {1 \over 2}} t_0 y^2
      - {\textstyle {1 \over 2}} z \left(
               x {\delta \alpha \over \delta z}
             + y {\delta \beta  \over \delta z}
            \right);
   \eqno {\rm (I^{13})}$$
in which $r_0$~$s_0$~$t_0$ are the particular values
$$r_0 = - {\delta \alpha \over \delta x},\quad
  s_0 = - {\textstyle {1 \over 2}} \left(
                  {\delta \alpha \over \delta y}
                + {\delta \beta  \over \delta x}
               \right),\quad
  t_0 = - {\delta \beta  \over \delta y},
   \eqno {\rm (K^{13})}$$
of the coefficients $r$~$s$~$t$, deduced from the general
expressions (E${}^{12}$) by making
$$p = 0,\quad q = 0,
   \eqno {\rm (L^{13})}$$
that is, by passing to the case of no obliquity; so that the
equation (Z${}^9$) of the guiding paraboloid may be put under the
form
$$z =   {\textstyle {1 \over 2}} r_0 x^2
      + s_0 x y
      + {\textstyle {1 \over 2}} t_0 y^2,
   \eqno {\rm (M^{13})}$$
which includes the form (G${}^{10}$).  Reciprocally, the sought
paraboloid (D${}^{12}$) is the only parabol\-oid which has its
indefinite axis parallel to the given final ray-line, and
osculates in all directions at the given final point to the
hyperboloid (I${}^{13}$): it is therefore sufficient to construct
this osculating hyperboloid, in order to deduce the sought
paraboloid (D${}^{12}$).  We might even employ the hyperboloid as
a new guiding surface for the ray-lines from the oblique plane,
instead of employing the paraboloid, since these two osculating
surfaces have the same deflexures and deflected lines, near their
given point of osculation.

Now to construct the osculating hyperboloid (I${}^{13}$), by the
oblique plane (B${}^{12}$) or (F${}^{12}$), and by the former
elements of arrangement, that is, by the guiding paraboloid
(M${}^{13}$), and by the coefficients
$\displaystyle {\delta \alpha \over \delta z}$,
$\displaystyle {\delta \beta  \over \delta z}$,
which determine the magnitude and plane of curvature of the final
ray, we may compare the sought hyperboloid (I${}^{13}$) with the
following new paraboloid
$$z = px + qy
      + {\textstyle {1 \over 2}} r_0 x^2
      + s_0 x y
      + {\textstyle {1 \over 2}} t_0 y^2,
   \eqno {\rm (N^{13})}$$
which may be called the {\it guiding paraboloid removed}, since
it is equal and similar to the guiding paraboloid (M${}^{13}$),
and may be obtained by transporting that guiding paraboloid
without rotation to a new position such that it touches the given
oblique plane at the given point.  The intersection of the
hyperboloid (I${}^{13}$) and paraboloid (N${}^{13}$) consists in
general of an ellipse or hyperbola in the given plane
$$z = 0,
   \eqno {\rm (O^{13})}$$
perpendicular to the given final ray, and of a parabola in the
plane
$$x {\delta \alpha \over \delta z}
      + y {\delta \beta  \over \delta z}
   = 0,
   \eqno {\rm (P^{13})}$$
which contains the given final ray-line or ray-tangent, and is
perpendicular to the final plane of curvature of the ray.  If
then, we make this final plane of curvature the plane of
$xz$, so that its equation shall be
$$y = 0,
   \eqno {\rm (Q^{13})}$$
and so that, by (T${}^9$),
$${\delta \beta \over \delta z} = 0,
   \eqno {\rm (R^{13})}$$
we shall have the following equations for the two curves of
intersection; first, for the ellipse or hyperbola,
$$z = 0,\quad
  px + qy
      + {\textstyle {1 \over 2}} r_0 x^2
      + s_0 x y
      + {\textstyle {1 \over 2}} t_0 y^2
    = 0;
   \eqno {\rm (S^{13})}$$
and secondly, for the parabola,
$$x = 0,\quad z = q y + {\textstyle {1 \over 2}} t_0 y^2:
   \eqno {\rm (T^{13})}$$
and these two curves may be considered as known, since they are
the intersections of two known planes with the known guiding
paraboloid removed to a known position.  To examine now how far a
surface of the second degree is restricted by the condition of
containing these two known curves, and what other conditions are
necessary, in order to oblige this surface to be the hyperboloid
sought, let us employ the following general form for the equation
of a surface of the second degree,
$$A x^2 + B y^2 + C z^2 + D xy + E yz + F zx + G x + H y + I z + K
   = 0,
   \eqno {\rm (U^{13})}$$
and let us seek the relations which restrict the coefficients of
this equation when the surface is obliged to contain the two
known curves.  The condition of containing the parabola
(T${}^{13}$) gives
$$K = 0,\quad H = - Iq,\quad E = 0,\quad C = 0,\quad
  B = - {\textstyle {1 \over 2}} I t_0;
   \eqno {\rm (V^{13})}$$
so that, by this condition alone, the general equation
(U${}^{13}$) is reduced to the following form,
$$z = q y
      + {\textstyle {1 \over 2}} t_0 y^2
      - {x \over I} (G + Fz + Dy + Ax).
   \eqno {\rm (W^{13})}$$
In order that this less general surface of the second degree
(W${}^{13}$), should contain the ellipse or hyperbola
(S${}^{13}$), it is necessary and sufficient that we should have
the relations,
$$G = - I p,\quad
  D = - I s_0,\quad
  A = - {\textstyle {1 \over 2}} I r_0:
   \eqno {\rm (X^{13})}$$
the general equation, therefore, of those surfaces of the second
degree which contain at once the two known curves
(S${}^{13}$) (T${}^{13}$), involves only one arbitrary
coefficient, and may be put under the form
$$z = px + qy
      + {\textstyle {1 \over 2}} r_0 x^2
      + s_0 x y
      + {\textstyle {1 \over 2}} t_0 y^2
      + \lambda xz.
   \eqno {\rm (Y^{13})}$$
This general equation, with the arbitrary coefficient $\lambda$,
belongs to the guiding paraboloid removed, that is, to the
surface (N${}^{13}$), when we suppose
$$\lambda = 0;
   \eqno {\rm (Z^{13})}$$
and the same general equation belongs by (R${}^{13}$) to the
sought hyperboloid (I${}^{13}$), when
$$\lambda
   = - {\textstyle {1 \over 2}} {\delta \alpha \over \delta z}.
   \eqno {\rm (A^{14})}$$
To put this last condition under a geometrical form, let us, as
we have already considered the intersections of the hyperboloid
with the two rectangular co-ordinate planes of $xy$ and
$yz$, consider now its intersection with the third co-ordinate
plane of $xz$, that is, with the plane of curvature
(Q${}^{13}$) of the given final ray.  This intersection is the
following hyperbola,
$$y = 0,\quad
  z = px + {\textstyle {1 \over 2}} r_0 x^2
      - {\textstyle {1 \over 2}} {\delta \alpha \over \delta z} xz,
   \eqno {\rm (B^{14})}$$
and the corresponding intersection for the surface (Y${}^{13}$)
is
$$y = 0,\quad
  z = px + {\textstyle {1 \over 2}} r_0 x^2
      + \lambda x z;
   \eqno {\rm (C^{14})}$$
the condition (A${}^{14}$) is therefore equivalent to an
expression of the coincidence of these two intersections; and if
we oblige the surface of the second degree (U${}^{13}$) to
contain the three curves (S${}^{13}$) (T${}^{13}$) (B${}^{14}$),
in the three rectangular co-ordinate planes, we shall thereby
oblige it to become the sought hyperboloid (I${}^{13}$).  It is
not necessary, however, though it is sufficient, to assign the
hyperbola (B${}^{14}$), as a third curve upon this hyperboloid.
For, in general, if we know  the intersections of a surface of
the second degree with two known planes, there remains only one
unknown quantity in the equation of that surface, and the
intersection with a third known plane is more than sufficient to
determine it.  Thus, in the present question, if the intersection
(C${}^{14}$) be distinct from the following parabola
$$y = 0,\quad
  z = px + {\textstyle {1 \over 2}} r_0 x^2,
   \eqno {\rm (D^{14})}$$
that is, if the surface (Y${}^{13}$), containing the two known
curves (S${}^{13})$ (T${}^{13}$), be distinct from the known
guiding paraboloid removed, which also contains the same two
curves, the intersection (C${}^{14}$) with the plane of curvature
of the ray is in general a hyperbola, which touches the known
parabola (D${}^{14}$) at the known origin of co-ordinates, and
meets this parabola again in another known point on the axis of
$x$, that is, on the radius of curvature of the known final ray,
namely, in the point
$$x = - {2p \over r_0},\quad y = 0,\quad z = 0;
   \eqno {\rm (E^{14})}$$
the hyperbola (C${}^{14}$) has also one asymptote parallel to the
known final ray-line or axis of $z$, namely, the asymptote having
for equations
$$x = {1 \over \lambda},\quad y = 0,
   \eqno {\rm (F^{14})}$$
and it will be entirely determined, if, in addition to the
foregoing properties, we know also a line parallel to its other
asymptote, namely, to that which has for equations,
$$x = - 2 \left( {\lambda \over r_0} \right) z
      - {1 \over \lambda} - {2p \over r_0},\quad
  y = 0:
   \eqno {\rm (G^{14})}$$
it will therefore be obliged to coincide with the hyperbola
(B${}^{14}$), if only we oblige its second asymptote (G${}^{14}$)
to be parallel to the following known right line,
$$x = {z \over r_0}{\delta \alpha \over \delta z},\quad
  y = 0,
   \eqno {\rm (H^{14})}$$
in which the coefficient
$${1 \over r_0} {\delta \alpha \over \delta z}
   = {\hbox{\it curvature of the final ray}
      \over \hbox{\it deflexure of the guiding paraboloid}},
   \eqno {\rm (I^{14})}$$
the plane of the deflexure $r_0$ being the plane of curvature of
the ray.  We see, then, that this last condition, respecting the
direction of the second asymptote (G${}^{14}$) of the hyperbolic
section (C${}^{14}$), is sufficient, when combined with the
conditions of containing the two known curves
(S${}^{13}$) (T${}^{13}$), to determine completely the sought
hyperboloid (I${}^{13}$).  Even the conditions of containing the
two curves (S${}^{13}$) (T${}^{13}$) are not perfectly distinct
and independent; nor would their coexistence be possible, in the
determination of a surface of the second degree, if the two
points in which the parabola (T${}^{13}$) is intersected by the
axis of $y$, that is, by the intersection-line of the planes of
the two curves, namely, the origin and the point
$$x = 0,\quad y = - {2q \over t_0},\quad z = 0,
   \eqno {\rm (K^{14})}$$
were not also contained on the ellipse or hyperbola (S${}^{13}$).
But we may confine ourselves to the last chosen conditions, of
having these two known curves as the intersections of the
hyperboloid with two known planes, and of having known directions
for the asymptotes of its hyperbolic curve of intersection with a
third known plane, as adequate and sufficiently simple conditions
for the construction of the sought hyperboloid, and thereby of
the auxiliary paraboloid (D${}^{12}$), to which that hyperboloid
osculates.  And with respect to the new constant of deviation
$n$, connected with this auxiliary paraboloid, we may put its
general value (F${}^{13}$) under the form
$$n = n_0
      + {\textstyle {1 \over 2}} p {\delta \beta  \over \delta z}
      - {\textstyle {1 \over 2}} q {\delta \alpha \over \delta z},
   \eqno {\rm (L^{14})}$$
$n_0$ being the particular value
$$n_0 = {\textstyle {1 \over 2}} \left(
               {\delta \beta  \over \delta x}
             - {\delta \alpha \over \delta y}
            \right)
   \eqno {\rm (M^{14})}$$
for the plane of no obliquity, that is, the value (B${}^{10}$)
connected with the guiding paraboloid (Z${}^9$) in the theory of
the elements of arrangement which was given in a former number:
we may therefore construct the new constant $n$ as the ordinate
$z$ of a plane
$$z = p x + q y + n_0,
   \eqno {\rm (N^{14})}$$
which is parallel to the given oblique plane (B${}^{12}$), and
contains the point
$$x = 0,\quad y = 0,\quad z = n_0,
   \eqno {\rm (O^{14})}$$
so that it intersects the axis of $z$ at a distance from the
origin $=$ the old constant of deviation $n_0$.  The other
co-ordinates $x$,~$y$, to which the ordinate $z = n$ corresponds,
are
$$x =   {\textstyle {1 \over 2}} {\delta \beta  \over \delta z},\quad
  y = - {\textstyle {1 \over 2}} {\delta \alpha \over \delta z},
   \eqno {\rm (P^{14})}$$
so that the corresponding line $\sqrt{x^2 + y^2}$ is equal to
half the curvature of the ray, and is perpendicular to the radius
of that curvature.

The details of the present number have been given, in order to
illustrate the subject, by combining it more closely with
geometrical conceptions; but the new auxiliary paraboloid, and
the new constant of deviation, might have been considered as
sufficiently defined by their former algebraical expressions.

\bigbreak

{\sectiontitle
Condition of Intersection of Two Near Final Ray-lines.  Conical
Locus of the Near Final Points, in a variable medium, which satisfy
this condition.  Investigations of {\sc Malus}.  Illustration of the
Condition of Intersection, by the Theory of the Auxiliary
Paraboloid, for Ray-lines from an Oblique Plane.\par}

\nobreak\bigskip

19.
Returning now to the system of final ray-lines (C${}^{12}$)
from an oblique plane (B${}^{12}$), let us consider the condition
necessary in order that one of these near ray-lines (C${}^{12}$)
may intersect the given final ray-line or axis of $z$.  This
condition may be at once obtained by making $x$ and $y$ vanish in
the equations (C${}^{12}$), and then eliminating $z$; it may
therefore be thus expressed,
$$\delta x \mathbin{.} \left\{
         \left( {\delta \beta  \over \delta x}
            + p {\delta \beta  \over \delta z} \right) \, \delta x
       + \left( {\delta \beta  \over \delta y}
            + q {\delta \beta  \over \delta z} \right) \, \delta y
      \right\}
   = \delta y \mathbin{.} \left\{
         \left( {\delta \alpha \over \delta x}
            + p {\delta \alpha \over \delta z} \right) \, \delta x
       + \left( {\delta \alpha \over \delta y}
            + q {\delta \alpha \over \delta z} \right) \, \delta y
      \right\},
   \eqno {\rm (Q^{14})}$$
or more concisely thus, on account of the equation of the oblique
plane (B${}^{12}$),
$$\delta x \mathbin{.} \left(
               {\delta \beta  \over \delta x} \, \delta x
             + {\delta \beta  \over \delta y} \, \delta y
             + {\delta \beta  \over \delta z} \, \delta z
            \right)
   = \delta y \mathbin{.} \left(
               {\delta \alpha \over \delta x} \, \delta x
             + {\delta \alpha \over \delta y} \, \delta y
             + {\delta \alpha \over \delta z} \, \delta z
            \right),
   \eqno {\rm (R^{14})}$$
that is,
$$\delta x \, \delta \beta = \delta y \, \delta \alpha;
   \eqno {\rm (S^{14})}$$
{\it it is therefore necessary and sufficient, for the
intersection sought, that the near final point $B'$ should be on a
certain conical locus of the second degree}, determined by the
equation (R${}^{14}$), between the co-ordinates
$\delta x$, $\delta y$, $\delta z$.
A conical locus of this kind appears to have been first
discovered by {\sc Malus}.  That excellent mathematician and observer
had occasion, in his {\it Trait\'{e} D'Optique}, to make some
remarks on the general properties of a system of right-lines in
space, represented by equations of the form
$${x - x' \over m} = {y - y' \over n} = {z - z' \over o},$$
in which $m$,~$n$,~$o$, are any given functions of the
co-ordinates $x'$,~$y'$,~$z'$, of a point through which the line
is supposed to pass, and by which it is supposed to be
determined; and he remarked that the condition of intersection of a
line thus determined, with the corresponding near line from a
point infinitely near, was expressed by an equation of the second
degree between the differentials of the co-ordinates
$x'$,~$y'$,~$z'$, which might be considered as the equation of a
conical locus of the second degree for the infinitely near point.
The theory of systems of rays which was given by {\sc Malus}, differs
much, in form and in extent, from that proposed in the present
Supplement; especially because, in the former theory, the
coefficients which mark the direction of a ray were left as
independent and unconnected functions, whereas, in the latter,
they are shown to be connected with each other, and to be
deducible by uniform methods from one characteristic function.
But the mere consideration of the existence of some functional
laws, whether connected or arbitrary, of dependence of the
coefficients $m$~$n$~$o$ on the co-ordinates $x'$~$y'$~$z'$,
or of $\alpha$~$\beta$~$\gamma$ on $x$~$y$~$z$, conducts
easily, as we have seen, to a conical locus of the kind
(R${}^{14}$).  This result may however be illustrated by the
theory which we have given of the geometrical relations of the
near final ray-lines from an oblique plane with the deflected
lines of a certain auxiliary paraboloid, and with a certain law
and constant of deviation.

For, according to the theory of these relations, the ray-line
from a near final point $B'$ on a given oblique plane drawn
through the given point $B$, will or will not intersect the given
final ray-line from $B$, according as its deviation $\delta \nu$
from its own deflected line does or does not compensate for the
deviation $\delta \psi$ of that deflected line from the
corresponding plane of deflexure, by these two deviations being
equal in magnitude but opposite in direction; the condition of
intersection may therefore be thus expressed,
$$\delta \nu + \delta \psi = 0;
   \eqno {\rm (T^{14})}$$
or, by the values of the deviations $\delta \nu$, $\delta \psi$,
established in the seventeenth number,
$$n = {t - r \over 2} \mathbin{.} \sin 2 \phi
                  + s \mathbin{.} \cos 2 \phi,
   \eqno {\rm (U^{14})}$$
that is,
$$n ( \delta x^2 + \delta y^2)
   = (t - r) \, \delta x \, \delta y
      + s (\delta x^2 - \delta y^2):
   \eqno {\rm (V^{14})}$$
and the condition of intersection thus obtained, by the
consideration of two equal and opposite deviations, is, on
account of the meanings (E${}^{12}$) (F${}^{13}$) of $n$, $r$,
$s$, $t$, equivalent to (Q${}^{14}$), and therefore to the
equation (R${}^{14}$) of the cone of the second degree.  In this
manner, then, as well as by the former less geometrical process,
we might perceive that the two planes of vergency for the
ray-lines from an oblique plane, (determined by (U${}^{14}$) or
(V${}^{14}$), and analogous to the two less general planes of
vergency considered in the sixteenth number,) intersect the
oblique plane in the same two lines in which that plane
intersects a certain cone of the second degree, through the
centre of which cone it passes; and that the planes of vergency
are imaginary when the oblique plane does not intersect this
cone.  We may remark that the intersection of the oblique plane
with the cone, or of a near final ray-line from the oblique plane
with the given final ray-line, is impossible, when the constant
of deviation corresponding to the oblique plane is greater
(abstracting from its sign) than the semidifference of the
extreme deflexures of the auxiliary paraboloid: for then the
compensation of the two deviations $\delta \nu$, $\delta \psi$,
is impossible, the near ray-line always deviating more from the
corresponding deflected line of the auxiliary paraboloid, than
this deflected line from the corresponding plane of deflexure.
And when the compensation and therefore the intersection becomes
possible, by the constant of deviation being less than the
semidifference of the two extreme deflexures, then the two real
planes of vergency of the near final ray-lines from the oblique
plane are symmetrically situated with respect to the two
rectangular planes of extreme deflexure: which latter planes may
also, for a reason already alluded to, be called the planes of
extreme projection of the final ray-lines.

\bigbreak

{\sectiontitle
Other Geometrical Illustrations of the Condition of Intersection,
and of the Elements of Arrangement.  Composition of Partial
Deviations.  Rotation round the Axis of Curvature of a Final
Ray.\par}

\nobreak\bigskip

20.
The condition of intersection of two near final ray-lines
may also be illustrated, and might have obtained, by other
geometrical considerations, on which we shall dwell a little,
because they will help to illustrate and improve the theory of
the elements of arrangement.

It was remarked, in the fourteenth number, that the general
comparison of a given luminous path $(A,B)_\chi$ with a near path
$(A',B')_{\chi + \delta \chi}$ might be decomposed into several
particular comparisons, such as the comparisons with the less
general near paths $(A,B_d)_\chi$, $(A,B_\delta)_\chi$, and
others, on account of the linear form of the expressions
(D${}^9$) for the variations
$\delta \alpha$, $\delta \beta$, $\delta \alpha'$, $\delta \beta'$,
of the extreme small cosines of direction, which form permits us
to consider separately and successively the influence of the
variations of the extreme co-ordinates and colour, or the
influence of any groupes of these variations.  Accordingly, by an
{\it Analysis\/} founded on this remark, we decomposed the
general discussion of the geometrical relations of infinitely
near rays into four less general problems, which were treated of,
in the fifteenth number.  The applications, in the sixteenth
number, to questions respecting the mutual intersections of the
final ray-lines from the final perpendicular plane, may be
considered as only illustrations and corollaries of the third of
those four problems: but the questions since discussed,
respecting the ray-lines from an oblique plane, require a
combination of the solutions of the second and third of the four
problems, and furnish, therefore, an example of the
{\it Synthesis\/} of those elements of arrangement of near rays,
to which the former {\it Analysis\/} had conducted.  This
synthesis, however, has in the foregoing numbers been itself
{\it algebraically\/} performed, (namely, by the algebraical
addition of certain partial variations,) although many of the
results were enunciated geometrically, and combined with
geometrical conceptions: but a {\it geometrical\/} idea and
method, of the Synthesis of the Elements of Arrangement, may be
obtained by considering, in a general manner, the geometrical
composition of partial deviations.

To understand more fully the occasion of such composition, let us
remember that our theory of the Elements of Arrangement enables
us to pass from the extreme directions of a given luminous path
$(A,B)_\chi$, to the four following sets of near extreme
directions, by the solution of the four problems considered in
the fifteenth number.

First.  The extreme directions of the near path
$(A,B)_{\chi + \delta \chi}$, which has the same extreme points
$A$, $B$, but differs by chromatic dispersion.

Second.  The final direction of $(A,B_d)_\chi$, that is, of the
original path prolonged at the end, and the initial direction of
$(A_d,B)_\chi$, that is, of the same path prolonged at the
beginning; these near extreme directions being in general affected
by curvature.

Third.  The final direction of $(A,B_\delta)_\chi$, and the
initial direction of $(A_\delta,B)_\chi$; the small lines
$\overline{A A_\delta}$, $\overline{B B_\delta}$, being
perpendicular to the given path at its extremities.

Fourth.  The initial direction of $(A,B_\delta)_\chi$, and the
final direction of $(A_\delta,B)_\chi$.

We saw also that the initial direction of $(A,B_d)_\chi$ and the
final direction of $(A_d,B)_\chi$ do not differ from the
corresponding extreme directions of the original luminous path.

If then we would apply this theory to determine the final
direction of an arbitrary near path
$(A',B')_{\chi + \delta \chi}$, we would have to consider and
compound, algebraically or geometrically, the following partial
deviations from the given final direction of the given path
$(A,B)_\chi$: first, the chromatic deviation of the final
direction of the near path $(A,B)_{\chi + \delta \chi}$ from that
given final direction; second, the deviation of curvature of the
final direction of $(A,B_d)_\chi$; third, the final deviation of
the path $(A, B_\delta)_\chi$, to be determined by the theory of
the final guiding paraboloid; and fourth, the deviation of the
final direction of $(A_\delta, B)_\chi$, to be found by the
theory of the guiding planes and conjugate guiding axes.  A
similar composition of four partial deviations is required for
the determination of the initial direction of the same arbitrary
near path $(A',B')_{\chi + \delta \chi}$.

Now to compound in a geometric manner the four preceding partial
deviations of the final ray-line, we may proceed as follows.  We
may construct each partial deviation, by drawing the deviated
final ray-line corresponding, or a line parallel thereto, through
the given final point $B$; the line thus drawn will differ little
in direction from the given final ray-line or axis of $z$, and if
we take its length equal to unity, then its small projection on
the given final plane of $xy$, to which it is nearly
perpendicular, will measure the magnitude and will indicate the
direction of the deviation: and if we compound all these
projections according to the usual geometrical rule of
composition of forces, the result will be the projection of the
equal line which represents in direction the resultant or total
deviation.  And similarly we may compound the four partial
deviations of a near initial ray-line.

The geometrical synthesis of the partial deviations may also be
performed in other ways.  For example, we may consider each
partial deviation as arising from a partial or component
rotation, and we may compound these several rotations by the
geometrical methods proper for such composition.

In particular, we may compound the final deviation of curvature
with any of the other partial deviations, by making the deviated
ray-line, obtained without considering the final curvature of the
ray, revolve through an infinitely small angle round the axis of
final curvature, that is, round the axis of the final osculating
circle of the given final ray.  By this rotation, the projection
$B_\delta$ of a near final point $B'$ on the final perpendicular
plane, will be brought into the position $B'$; and, by the same
rotation, the near final ray-line, which had been obtained by
abstracting from the final curvature, and by considering
$B_\delta$ as the final point, will be brought, at the same time,
into the position of the sought ray-line, which corresponds to a
final point at $B'$.

Applying now these general principles to the particular question
respecting the condition of intersection of two near final
ray-lines, from two near final points $B$, $B'$, (the colour
$\chi$ and the initial point $A$ being considered as common and
given,) we see that if the projection $B_\delta$ of $B'$ be
given, the small projecting perpendicular
$\overline{B' B_\delta}$ or $\delta z$ and therefore also the
near point $B'$ itself may in general be determined so as to
satisfy the condition of intersection: for the final ray-line
from $B_\delta$ may in general be brought to intersect the given
final ray-line, by revolving through an infinitesimal angle round
the axis of curvature of the given final ray.  We see also that
the angular quantity of rotation and therefore the length
$\delta z = \overline{B_\delta B'}$ depends on the position of
the projection $B_\delta$, that is, on the co-ordinates
$\delta x$, $\delta y$; and therefore that there must be some
determined surface as the locus of the near final point $B'$,
when the final ray-line from that point is supposed to intersect
the given final ray-line.

To investigate the form of this locus, by the help of the
foregoing geometrical conceptions, we may observe that the only
point, on the near ray-line from $B_\delta$, which is brought by
the supposed rotation to meet the given final ray-line, is the
point contained in the final plane of curvature of the given
final ray; and that if we call this point, where the ray-line
from $B_\delta$ intersects the given plane of curvature, the
point $P$, the angle of rotation required is the angle between
the line $\overline{BP}$ and the given final ray-line; because
the same infinitesimal rotation which brings the near ray-line
from $B_\delta$, that is, the line $\overline{B_\delta P}$, into
a new position in which it intersects the given final ray-line,
brings also the line $\overline{BP}$ into the position of the
given final ray-line itself.  Translating now these geometrical
results into algebraical language, and taking the given final
plane of curvature for the plane of $xz$, so as to satisfy
the condition (R${}^{13}$), we find the following co-ordinates of
the point $P$ of intersection of this plane of curvature with the
ray-line (V${}^9$) from $B_\delta$,

\vfill\eject  % Page break necessary with current page size

$$x = \delta x
      - {\displaystyle
         \delta y \mathbin{.} \left(
               {\delta \alpha \over \delta x} \, \delta x
             + {\delta \alpha \over \delta y} \, \delta y
            \right)
         \over \displaystyle
               {\delta \beta  \over \delta x} \, \delta x
             + {\delta \beta  \over \delta y} \, \delta y};\quad
  y = 0;\quad
  z =   {- \delta y
         \over \displaystyle
               {\delta \beta  \over \delta x} \, \delta x
             + {\delta \beta  \over \delta y} \, \delta y}:
   \eqno {\rm (W^{14})}$$
so that the angle between the line $\overline{BP}$ which connects
this point with the origin of co-ordinates, and the given final
ray-line or axis of $z$, is
$$\delta \theta = \left( -{x \over z} = \right) \,
      {1 \over \delta y} \mathbin{.} \delta x \mathbin{.} \left(
               {\delta \beta  \over \delta x} \, \delta x
             + {\delta \beta  \over \delta y} \, \delta y
            \right)
       - \left(
               {\delta \alpha \over \delta x} \, \delta x
             + {\delta \alpha \over \delta y} \, \delta y
            \right);
   \eqno {\rm (X^{14})}$$
and this being equal to the infinitesimal angle of rotation, that
is, to the small line $\delta z$ or $\overline{B_\delta B'}$
multiplied by
$\displaystyle {\delta \alpha \over \delta z}$
or by the final curvature of the given ray taken with its proper
sign, we have the following equation for the locus of the near
point $B'$, when the condition of intersection is to be
satisfied,
$${\delta \alpha \over \delta z} \, \delta z
   = {1 \over \delta y} \, \delta x \left(
               {\delta \beta  \over \delta x} \, \delta x
             + {\delta \beta  \over \delta y} \, \delta y
            \right)
       - \left(
               {\delta \alpha \over \delta x} \, \delta x
             + {\delta \alpha \over \delta y} \, \delta y
            \right):
   \eqno {\rm (Y^{14})}$$
which is, accordingly, the equation of the former conical locus
(R${}^{14}$), only simplified by the condition (R${}^{13}$),
arising from a choice of co-ordinates.  Without making that
choice, we might easily have deduced in a similar manner the
equation (R${}^{14}$), under the form
$$\delta z = {\displaystyle
         \delta x \left(
               {\delta \beta  \over \delta x} \, \delta x
             + {\delta \beta  \over \delta y} \, \delta y
            \right)
       - \delta y \left(
               {\delta \alpha \over \delta x} \, \delta x
             + {\delta \alpha \over \delta y} \, \delta y
            \right)
      \over \displaystyle
               {\delta \alpha \over \delta z} \, \delta y
             - {\delta \beta  \over \delta z} \, \delta x},
   \eqno {\rm (Z^{14})}$$
in which each member is an expression for the infinitesimal angle
of rotation divided by the curvature of the ray.

Another way of applying the foregoing geometrical principles to
investigate the condition of intersection of two near final
ray-lines, is to consider the infinitesimal angle by which the
ray-line from $B_\delta$ deviates from the plane containing the
given final ray-line and the near point $B_\delta$.  This angular
deviation is expressed by the numerator of the fraction
(Z${}^{14}$), divided by $\delta l$, that is, divided by the
small line $\overline{B B_\delta}$; and the denominator of the
same fraction (Z${}^{14}$), divided also by $\delta l$, is equal
to the final curvature of the ray multiplied by the sine of the
inclination of the line $\delta l$ to the radius of this final
curvature: and hence it is easy to see, by geometrical
considerations, that the fraction in the second member of
(Z${}^{14}$) is equal to the infinitesimal angle of rotation
required for destroying the last mentioned deviation, divided
by the curvature of the ray, and therefore equal to the ordinate
$\delta z$ of the sought locus of the near point $B'$, as
expressed by the first member.  We might therefore easily have
obtained, by calculations founded on this other geometrical view,
the same condition of intersection as before, and the same
conical locus.

\bigbreak

{\sectiontitle
Relations between the Elements of Arrangement, depending only on
the Extreme Points, Directions, and Colour, of a Given Luminous
Path, and on the Extreme Media.  In a Final Uniform Medium,
Ordinary or Extraordinary, the two Planes of Vergencey are
Conjugate Planes of Deflexure of any Surface of a certain class,
determined by the Final Medium; and also of a certain Analogous
Surface determined by the whole combination.  Relations between
the Visible Magnitudes and Distortions of any two small objects
viewed from each other through any Optical Combination.
Interchangable Eye-axes and Object-axes of Distortion.  Planes of
No Distortion.\par}

\nobreak\bigskip

21.
It was shown in the fourteenth number, and the result has
since been developed in detail, that the general geometrical
relations between the extreme directions of infinitely near rays
are determined by the coefficients of the linear variations
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$,
of the six marks of extreme direction, considered as functions of
the six extreme co-ordinates and of the colour; and that, between
the forty-two general coefficients of these six linear
variations, there exists eighteen general relations, leaving only
twenty-four coefficients arbitrary, if we suppose for simplicity
that the final and initial co-ordinates are referred to
rectangular axes.  But besides these eighteen general relations
which are common to all optical combinations, there arise certain
other relations between the coefficients, when the extreme media
are considered as given, and when the extreme points, directions,
and colour, of any one luminous path, are also supposed to be
known.  For, if we then employ the general equations (A${}^9$),
we may consider the extreme medium functions $v$, $v'$, and their
partial differentials, as known, and may deduce general
expressions for the coefficients before mentioned of the linear
variations of the extreme cosines of direction, involving only,
as unknown quantities, twenty-seven partial differentials of the
second order of the characteristic function $V$, namely, all of
this order, which are not relative to the variation of colour
only; but these twenty-seven are connected by the fourteen
general relations (Q) (U) (X) (Y), deduced in the third number,
of which however only thirteen are distinct, because the two
systems (U) (Y) conduct both to one common equation (D); there
remain, therefore, as independent quantities, only fourteen of
the partial differentials of $V$, in the general expressions of
those twenty-four coefficients of the linear variations of the
extreme direction-cosines, which had before been considered as
independent, when the extreme medium-functions $v$, $v'$ were
supposed unknown and arbitrary: and if we eliminate the fourteen
independent differentials of $V$ between the expressions of these
twenty-four coefficients, we shall obtain {\it ten general
relations, between the elements of arrangement of infinitely near
rays, involving only the extreme points, directions, and colour,
of the given luminous path, and the properties of the extreme
media.}

The simplest manner of obtaining these ten general  relations
is to eliminate the fourteen differentials of $V$ which enter
into the twenty-four expressions, deducible from (C${}^9$), from
the twenty-four coefficients (D${}^9$).  The ten relations thus
obtained, may be arranged in three different groupes: the first
groupe containing the two following
$$\left. \eqalign{
         {\delta^2 v \over \delta \alpha^2}
            {\delta \alpha \over \delta z}
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta \beta  \over \delta z}
       + {\delta^2 v \over \delta \alpha \, \delta z}
   &= {\delta v \over \delta x},\cr
         {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta \alpha \over \delta z}
       + {\delta^2 v \over \delta \beta^2}
            {\delta \beta  \over \delta z}
       + {\delta^2 v \over \delta \beta  \, \delta z}
   &= {\delta v \over \delta y},\cr}
   \right\}
   \eqno {\rm (A^{15})}$$
and two others similar to these, but with accented or initial
symbols; the second groupe containing the final relation
$$       {\delta^2 v \over \delta \alpha^2}
            {\delta \alpha \over \delta y}
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta \beta  \over \delta y}
       + {\delta^2 v \over \delta \alpha \, \delta y}
   =     {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta \alpha \over \delta x}
       + {\delta^2 v \over \delta \beta^2}
            {\delta \beta  \over \delta x}
       + {\delta^2 v \over \delta \beta  \, \delta x},
   \eqno {\rm (B^{15})}$$
and a similar initial relation; and the third groupe comprising
the four following,
$$\left. \eqalign{
         {\delta^2 v  \over \delta \alpha^2}
            {\delta \alpha \over \delta x'}
       + {\delta^2 v  \over \delta \alpha \, \delta \beta}
            {\delta \beta  \over \delta x'}
       + {\delta^2 v' \over \delta \alpha'^2}
            {\delta \alpha' \over \delta x}
       + {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            {\delta \beta'  \over \delta x}
   &= 0,\cr
         {\delta^2 v  \over \delta \alpha^2}
            {\delta \alpha \over \delta y'}
       + {\delta^2 v  \over \delta \alpha \, \delta \beta}
            {\delta \beta  \over \delta y'}
       + {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            {\delta \alpha' \over \delta x}
       + {\delta^2 v' \over \delta \beta'^2}
            {\delta \beta'  \over \delta x}
   &= 0,\cr
         {\delta^2 v  \over \delta \alpha \, \delta \beta}
            {\delta \alpha \over \delta x'}
       + {\delta^2 v  \over \delta \beta^2}
            {\delta \beta  \over \delta x'}
       + {\delta^2 v' \over \delta \alpha'^2}
            {\delta \alpha' \over \delta y}
       + {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            {\delta \beta'  \over \delta y}
   &= 0,\cr
         {\delta^2 v  \over \delta \alpha \, \delta \beta}
            {\delta \alpha \over \delta y'}
       + {\delta^2 v  \over \delta \beta^2}
            {\delta \beta  \over \delta y'}
       + {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            {\delta \alpha' \over \delta y}
       + {\delta^2 v' \over \delta \beta'^2}
            {\delta \beta'  \over \delta y}
   &= 0.\cr}
   \right\}
   \eqno {\rm (C^{15})}$$

The two first relations of the first groupe, namely, the
equations (A${}^{15}$), are equivalent to the two first
differential equations (O) of a curved ray, and express that the
magnitude and plane of final curvature of a luminous path, in a
final variable medium, are determined, in general, by the
properties of that medium, the colour of the light, the position
of the final point, and the direction of the final tangent.  And
the two other relations of the same groupe express, in like
manner, a dependence of the initial magnitude and plane of
curvature of a luminous path, on the initial medium, colour,
point, and tangent.

The equation (B${}^{15}$), belonging to the second groupe, is a
relation between the four coefficients
$\displaystyle {\delta \alpha \over \delta x}$,
$\displaystyle {\delta \alpha \over \delta y}$,
$\displaystyle {\delta \beta  \over \delta x}$,
$\displaystyle {\delta \beta  \over \delta y}$,
and therefore a relation between the guiding paraboloid and
constant of deviation for the final ray-lines, depending on the
final medium, colour, point and tangent.  And similarly the other
equation of the second groupe expresses an analogous relation for
the initial medium.

In the extensive case of a final uniform medium, the equation
(B${}^{15}$) reduces itself to the following,
$$0 =    {\delta^2 v \over \delta \alpha^2}
            {\delta \alpha \over \delta y}
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
            \left( {\delta \beta  \over \delta y}
                   - {\delta \alpha \over \delta x} \right)
       - {\delta^2 v \over \delta \beta^2}
            {\delta \beta  \over \delta x};
   \eqno {\rm (D^{15})}$$
and, in the same case, the general conical locus of the second
degree (R${}^{14}$), connected with the condition of intersection
of the final ray-lines, reduces itself to two real or imaginary
planes of vergency, represented by the quadratic
$$0 =    {\delta \alpha \over \delta y} \, \delta y^2
          + \left( {\delta \alpha \over \delta x}
                   - {\delta \beta  \over \delta y} \right)
               \, \delta x \, \delta y
       - {\delta \beta  \over \delta x} \, \delta x^2,
   \eqno {\rm (E^{15})}$$
and coinciding with the two planes of vergency considered in the
sixteenth number: attending therefore to (C${}^{13}$), the
relation (D${}^{15}$) may be geometrically enunciated by saying,
that {\it in a final uniform medium the two planes of vergency
are conjugate planes of deflexure of any surface of a certain
class determined by the nature of the medium}, namely, that class
for which, at the origin of co-ordinates,
$${\delta^2 z \over \delta x^2}
    = \lambda {\delta^2 v \over \delta \alpha^2},\quad
  {\delta^2 z \over \delta x \, \delta y}
    = \lambda {\delta^2 v \over \delta \alpha \, \delta \beta},\quad
  {\delta^2 z \over \delta y^2}
    = \lambda {\delta^2 v \over \delta \beta^2},
   \eqno {\rm (F^{15})}$$
and therefore nearly, for points near to this origin,
$$z = px + qy + {\lambda \over 2} \left(
            x^2  {\delta^2 v \over \delta \alpha^2}
          + 2 xy {\delta^2 v \over \delta \alpha \, \delta \beta}
          + y^2  {\delta^2 v \over \delta \beta^2},
         \right)
   \eqno {\rm (G^{15})}$$
the given final ray or axis of $z$ being taken as the axis of
deflexion, and the constants $p$, $q$, $\lambda$, being
arbitrary.  This relation may be still farther simplified, by
choosing the arbitrary constants as follows,
$$p = - {1 \over v} {\delta v \over \delta \alpha},\quad
  q = - {1 \over v} {\delta v \over \delta \beta},\quad
  \lambda = {1 \over vZ},
   \eqno {\rm (H^{15})}$$
$Z$ being any constant ordinate; for then, (by the theory of the
characteristic function $V$ for a single uniform medium, which
was given in the tenth number,) the surface (G${}^{15}$) acquires
a simple optical property, and becomes, in the final uniform
medium, the approximate locus of the points $x$,~$y$,~$z$, for
which
$$V_\prime = \int v \, ds = v \rho = \hbox{const.},
   \eqno {\rm (I^{15})}$$
the integral
$V_\prime = \int v \, ds$
being taken here, in the positive direction, along the variable
line $\rho$, from the fixed point $0, 0, Z$, to the variable point
$x, y, z$, or from the latter to the former, according as $Z$ is
negative or positive.  And though the equation (G${}^{15}$) is
only an approximate representation of the medium-surface
(I${}^{15}$), which was called in the First Supplement a {\it
spheroid of constant action}, and which is in the undulatory
theory a curved {\it wave\/} propagated from or to a point in the
final medium, yet since the equation (G${}^{15}$) gives a correct
development of the ordinate $z$ of this surface as far as terms
of the second dimension inclusive, when the constants are
determined by (H${}^{15}$), the conclusion respecting the
deflexures applies rigorously to the surface (I${}^{15}$); and
{\it the two planes of vergency\/} (E${}^{15}$), {\it in a final
uniform medium, are conjugate planes of deflexure of the spheroid
or wave\/} (I${}^{15}$).  We shall soon resume this result, and
endeavour to illustrate and extend it.  In the mean time we may
remark that the same planes of vergency (E${}^{15}$) are also
conjugate planes of deflexure of a certain analogous surface,
determined by the whole combination, and not merely by the final
uniform medium, namely, the surface (D${}^{10}$), for which
$$\int v \, ds \, (= V) \, = \hbox{const.},
   \eqno {\rm (K^{15})}$$
the integral being here extended to the whole luminous path, and
being therefore equal to the characteristic function $V$ of the
whole optical combination; an additional property of the planes
of vergency, which is proved by the following relation, analogous
to (D${}^{15}$), and deducible from (C${}^9$) or (G${}^9$),
$$0 =    {\delta^2 V \over \delta x^2}
            {\delta \alpha \over \delta y}
       + {\delta^2 V \over \delta x \, \delta y}
            \left( {\delta \beta  \over \delta y}
                   - {\delta \alpha \over \delta x} \right)
       - {\delta^2 V \over \delta y^2}
            {\delta \beta  \over \delta x};
   \eqno {\rm (L^{15})}$$

Finally, with respect to the four remaining equations, of the
third groupe (C${}^{15}$), it is evident that they express
certain general relations depending on the extreme media, between
the coefficients which determine the guiding planes and conjugate
guiding axes, for the final and inital ray-lines.  In the
extensive case of extreme ordinary media, they reduce themselves
to the four following, which may also be deduced from (F${}^9$),
$$\left. \multieqalign{
         \mu  {\delta \alpha  \over \delta x'}
       + \mu' {\delta \alpha' \over \delta x}
   &= 0, &
         \mu  {\delta \alpha  \over \delta y'}
       + \mu' {\delta \beta'  \over \delta x}
   &= 0,\cr
         \mu  {\delta \beta   \over \delta x'}
       + \mu' {\delta \alpha' \over \delta y}
   &= 0, &
         \mu  {\delta \beta   \over \delta y'}
       + \mu' {\delta \beta'  \over \delta y}
   &= 0,\cr}
   \right\}
   \eqno {\rm (M^{15})}$$
$\mu$, $\mu'$ being the {\it indices\/} of the media; and they
conduct to some simple conclusions, respecting the general
relations between the visible magnitudes and distortions of a
small plane object, placed alternately at each end of any given
luminous path, and viewed from the other end, through any
ordinary or extraordinary combination: at least so far as we
suppose these distortions and magnitudes to be measured by the
shape and size of the initial and final ray-cones.  For then the
conjugate guiding axes, initial and final, perpendicular to the
given path at its extremities, and determined in the fifteenth
number, may be called the {\it eye-axes and object-axes of
distortion}, for a small object placed in the final perpendicular
plane, and viewed from the initial point; and if we take these
for the axes of initial and final co-ordinates, so as to have, by
(X${}^{10}$) (Y${}^{10}$),
$${\delta \alpha' \over \delta y} = 0,\quad
  {\delta \beta'  \over \delta x} = 0,\quad
  {\delta \beta'  \over \delta y} > 0,\quad
  {\delta \alpha' \over \delta x} > {\delta \beta'  \over \delta y},$$
we shall then have also, by (M${}^{15}$), (the extreme media
being supposed ordinary, and their indices $\mu$, $\mu'$
positive,)
$${\delta \alpha  \over \delta y'} = 0,\quad
  {\delta \beta   \over \delta x'} = 0,\quad
  - {\delta \beta   \over \delta y'} > 0,\quad
  - {\delta \alpha  \over \delta x'} > - {\delta \beta   \over \delta y'};
   \eqno {\rm (N^{15})}$$
that is, in this case, {\it the guiding axes for the initial
ray-lines are also the guiding axes of the same kind for the final
ray-lines measured backward\/}; which is already a remarkable
relation, and may be enunciated by saying that {\it the eye-axes
and object-axes of distortion are interchangeable, when the
extreme media are ordinary\/}: that is, for such extreme
media, {\it the eye-axes of distortion become object-axes, and
the object-axes become eye-axes, when the object is removed from
the final to the initial perpendicular plane, and is viewed from
the final instead of the initial point}.  And while the equations
of the fifteenth number,
$$x' = z' {\delta \alpha' \over \delta x} \, \delta x,\quad
  y' = z' {\delta \beta'  \over \delta y} \, \delta y,
   \eqno {\rm (A^{11})}$$
represent the initial visual ray-line corresponding to a final
visible point $B'$ which has for co-ordinates
$\delta x$, $\delta y$, $\delta z$, the following other
equations,
$$x = - z \mathbin{.} {\mu' \over \mu}
            {\delta \alpha' \over \delta x} \, \delta x',\quad
  y = - z \mathbin{.} {\mu' \over \mu}
            {\delta \beta'  \over \delta y} \, \delta y',
   \eqno {\rm (O^{15})}$$
will represent by (M${}^{15}$) the final visual ray-line
corresponding to an initial visible point $A'$ which has for
co-ordinates $\delta x'$, $\delta y'$, $\delta z'$; the initial
visual ray-cone corresponding to any small object
$$\delta y = f( \delta x )
   \eqno {\rm (G^{11})}$$
in the final perpendicular plane is therefore represented by the
equation
$${y' \over z'} \left( {\delta \beta'  \over \delta y} \right)^{-1}
   = f   \left( {x' \over z'}
                \left( {\delta \alpha' \over \delta x} \right)^{-1}
         \right),
   \eqno {\rm (H^{11})}$$
and the final visual ray-cone corresponding to any small object
$$\delta y' = f'( \delta x' )
   \eqno {\rm (P^{15})}$$
in the initial perpendicular plane is represented by the
following analogous equation
$$- {y \over z} {\mu \over \mu'}
         \left( {\delta \beta'  \over \delta y} \right)^{-1}
   = f'  \left( - {x \over z} {\mu \over \mu'}
                \left( {\delta \alpha' \over \delta x} \right)^{-1}
         \right);
   \eqno {\rm (Q^{15})}$$
if therefore these two small objects (G${}^{11}$) (P${}^{15}$),
at the ends of a given luminous path, be equal and similar and
similarly placed with respect to the conjugate axes of
distortion, that is, if the final and initial functions
$f$, $f'$ be the same, and if we cut the two ray-cones
(H${}^{11}$) (Q${}^{15}$) respectively by perpendicular planes
having for equations
$$z' = \mu' R,\quad z = - \mu R,
   \eqno {\rm (R^{15})}$$
in which $R$ is any constant length, while $\mu$, $\mu'$ are the
same constant indices as before of the extreme ordinary media,
the two perpendicular sections thus obtained will be equal and
similar to each other; and if, besides, we put, by (Y${}^{10}$),
$${\delta \beta' \over \delta y}
   = {\delta \alpha' \over \delta x} \, \cos G,
   \eqno {\rm (S^{15})}$$
($G$ being by (F${}^{11}$) the inclination of an initial guiding
plane to the plane perpendicular to the given initial ray-line,)
and determine also the arbitrary quantity $R$ as follows,
$$R = {1 \over \mu'}
            \left( {\delta \alpha' \over \delta x} \right)^{-1}
    = - {1 \over \mu}
            \left( {\delta \alpha  \over \delta x'} \right)^{-1},
   \eqno {\rm (T^{15})}$$
the perpendicular sections of the initial and final ray-cones may
then be represented as follows,
$$y' = \cos G \mathbin{.} f(x'),\quad
  z' = \left( {\delta \alpha' \over \delta x} \right)^{-1},
   \eqno {\rm (U^{15})}$$
and
$$y  = \cos G \mathbin{.} f(x),\quad
  z  = \left( {\delta \alpha  \over \delta x'} \right)^{-1}:
   \eqno {\rm (V^{15})}$$
{\it the visible distortions\/} therefore, depending on the
inclination $G$, {\it are the same for any two small equal
objects, thus perpendicularly and similarly placed at the ends of
any given luminous path, and viewed from each other along that
path, through any optical combination}.

The distortion here considered will in general change, if the
object at either end of the given luminous path be made to
revolve in the perpendicular plane at that end, so as to change
its position with respect to the axes of distortion.  For
example, if the object be a small right-angled triangle in the
final perpendicular plane, having the summit of the right angle
at the given final point $B$ of the path, we know, by the theory
given in the fifteenth number, that the right angle will appear
right to an eye placed at the initial point $A$, when the
rectangular directions  of its sides $\phi_1'$, $\phi_2'$,
coincide with those of the final guiding axes, or object-axes of
distortion; but that otherwise the right angle
$\phi_2' - \phi_1'$ will appear acute or obtuse, its apparent
magnitude $\phi_2 - \phi_1$ being determined by the formula
$$- \tan \left( \phi_2 - \phi_1 - {\pi \over 2} \right)
   = {\displaystyle
            \left( {\delta \alpha' \over \delta x} \right)^2
          - \left( {\delta \beta'  \over \delta y} \right)^2
         \over \displaystyle
            2 {\delta \alpha' \over \delta x}
              {\delta \beta'  \over \delta y}}
      \mathbin{.} \sin 2 \phi_1',
   \eqno {\rm (W^{15})}$$
which may, by (S${}^{15}$), be reduced to the following,
$$- \tan \left( \phi_2 - \phi_1 - {\pi \over 2} \right)
   = {\textstyle {1 \over 2}} \sin G
         \mathbin{.} \tan G
         \mathbin{.} \sin 2 \phi_1'.
   \eqno {\rm (X^{15})}$$
The law of change of the distortion, corresponding to a
rotation in the final perpendicular plane, may also be deduced
from the theory of the guiding planes, explained in the fifteenth
number.

The distortion will also change if the small plane object be
removed into an oblique instead of a perpendicular plane.  In
this case we may still employ the equations
(A${}^{11}$) (O${}^{15}$) for the initial and final ray-lines,
and may still represent the initial and final ray-cones by the
equations (H${}^{11}$) (Q${}^{15}$); but we are now to consider
the equations (G${}^{11}$) (P${}^{15}$), for the final and
initial objects, as representing the projections of those objects
on the extreme perpendicular planes; or rather the {\it
projecting cylinders}, which contain the objects, and which
determine their visible magnitudes and distortions, by
determining the connected ray-cones.  For example, the equation
(C${}^{11}$) may be considered as representing a final elliptic
cylinder, of which any section near the final point $B$ of the
given luminous path will correspond to an initial circular
ray-cone (B${}^{11}$), and will therefore appear a circle to an
eye placed at the initial point $A$; while on the other hand we
may regard the equation (D${}^{11}$) as representing a final
circular cylinder, such that any section of this cylinder, near
the final point $B$, will give an initial elliptic ray-cone
(E${}^{11}$), and will appear an ellipse at $A$.  And as the
elliptic ray-cone (E${}^{11}$) conducted, by its circular
sections, to the guiding planes (F${}^{11}$) for the initial
ray-lines, so, for small plane final objects, the planes
$$z = \pm x \, \tan G,
   \eqno {\rm (Y^{15})}$$
namely, by (S${}^{15}$), {\it the planes of circular section of
the elliptic cylinder\/} (C${}^{11}$), {\it are planes of no
distortion\/}; in such a manner that not only, by what has been
said, the circular sections themselves in these two planes appear
each circular, but every other small final object in either of
the same two planes appears with its proper shape to an eye
placed at the initial point $A$ of the given luminous path; the
angular magnitude of the final object thus placed, being the same
as if it were viewed perpendicularly by straight rays, without
any refracting or reflecting surface or medium interposed, from a
final distance
$\displaystyle
   = \left( {\delta \beta'  \over \delta y} \right)^{-1}$.
In like manner, the planes
$$z' = \pm x' \, \tan G,
   \eqno {\rm (Z^{15})}$$
which are the planes of circular section of an analogous initial
elliptic cylinder, are {\it initial planes of no distortion}, of
the same kind as the final planes (Y${}^{15}$); since any small
initial object, placed in either of these two initial planes
(Z${}^{15}$), and viewed from the final point $B$ of the given
luminous path, will appear with its proper shape, and with the
same angular magnitude as if it were viewed directly from an
initial distance
$\displaystyle
   = - \left( {\delta \beta \over \delta y'} \right)^{-1}
   = {\mu \over \mu'} \left( {\delta \beta' \over \delta y} \right)^{-1}$.

This theory of the {\it planes of no distortion\/} gives a simple
determination of the visible shape and size of any small object
placed in any manner near either end of a given luminous path;
since we have only to project the object on one of the two planes
of no distortion at that end, by lines parallel to the
corresponding extreme direction of the path, and then to suppose
this projection viewed directly from a final or initial distance
determined as above.  We might, for example, deduce from this
theory the property of the guiding planes, the circular and
elliptic appearances (B${}^{11}$) (E${}^{11}$) of the ellipse and
circle (C${}^{11}$) (D${}^{11}$), and the acute or obtuse
appearance (X${}^{15}$) of a right angle in the final
perpendicular plane, when the directions of the sides of this
angle are different from those of the object-axes of distortion.
And the relations (M${}^{15}$) for extreme ordinary media may be
expressed by the following theorems: first, that {\it the
angle\/} $(2G)$ {\it between the final pair of planes of no
distortion\/} (Y${}^{15}$), {\it is equal to that between the
initial pair\/} (Z${}^{15}$); second, {\it the visible angular
magnitudes of any small and equal linear objects in final and
initial planes of no distortion, are proportional to the indices
of the final and initial media}, when the objects are viewed along
a given luminous path, from the initial and final points; and
third, {\it the two intersection-lines of the two pairs of planes
of no distortion coincide each with the visible direction of the
other}, when viewed along the path.
 
\bigbreak

{\sectiontitle
Calculation of the Elements of Arrangement, for Arbitrary Axes of
Co-ordinates.\par}

\nobreak\bigskip

22.
In the foregoing formul{\ae} for the elements of arrangement
of near rays, we have chosen for simplicity the final and initial
points of a given luminous path, as the respective origins of two
sets of rectangular co-ordinates, final and initial, and we have
made the final and initial ray-lines, or tangents to the given
path, the axes of $z$ and $z'$; a choice of co-ordinates which had
the convenience of reducing to zero eighteen of the forty-two
general coefficients in the expressions of
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$,
as linear functions of
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$.
The twenty-four remaining coefficients (D${}^9$) may however be
easily deduced, by the methods already established, and by the
partial differential coefficients of the characteristic and
related functions, from other systems of final and initial
co-ordinates, for example, from any other rectangular sets of
final and initial axes.

In effecting this deduction, it will be useful to distinguish by
lower accents the particular co-ordinates and cosines of
direction, which enter into the expressions (D${}^9$), and are
referred to particular axes of the kind already described; and
then we may connect these particular co-ordinates and cosines with
the more general analogous quantities
$x$~$y$~$z$ $x'$~$y'$~$z'$
$\alpha$~$\beta$~$\gamma$ $\alpha'$~$\beta'$~$\gamma'$,
by the formul{\ae} of transformation given in the thirteenth
number, which may easily be shown to extend to the case of two
distinct rectangular sets of given or unaccented co-ordinates.  In
this manner the axes of $z_\prime$ and $z_\prime'$, considered in
the thirteenth number, become the final and initial ray-lines,
and we have, by (A${}^8$),
$$\left. \eqalign{
\delta x &=    x_{x_\prime}   \, \delta x_\prime
             + x_{y_\prime}   \, \delta y_\prime
             + \alpha         \, \delta z_\prime,\cr
\delta y &=    y_{x_\prime}   \, \delta x_\prime
             + y_{y_\prime}   \, \delta y_\prime
             + \beta          \, \delta z_\prime,\cr
\delta z &=    z_{x_\prime}   \, \delta x_\prime
             + z_{y_\prime}   \, \delta y_\prime
             + \gamma         \, \delta z_\prime,\cr
\delta x' &=   x'_{x_\prime'} \, \delta x_\prime'
             + x'_{y_\prime'} \, \delta y_\prime'
             + \alpha'        \, \delta z_\prime',\cr
\delta y' &=   y'_{x_\prime'} \, \delta x_\prime'
             + y'_{y_\prime'} \, \delta y_\prime'
             + \beta'         \, \delta z_\prime',\cr
\delta z' &=   z'_{x_\prime'} \, \delta x_\prime'
             + z'_{y_\prime'} \, \delta y_\prime'
             + \gamma'        \, \delta z_\prime',\cr}
   \right\}
   \eqno {\rm (A^{16})}$$
because
$$\left. \multieqalign{
x_{z_\prime} &= \alpha, &
y_{z_\prime} &= \beta, &
z_{z_\prime} &= \gamma,\cr
x'_{z_\prime'} &= \alpha', &
y'_{z_\prime'} &= \beta', &
z'_{z_\prime'} &= \gamma';\cr}
   \right\}
   \eqno {\rm (B^{16})}$$
we have also
$$\left. \multieqalign{
\alpha_\prime  &= 0, &
\beta_\prime   &= 0, &
\gamma_\prime  &= 1, &
\delta \gamma_\prime  &= 0,\cr
\alpha_\prime' &= 0, &
\beta_\prime'  &= 0, &
\gamma_\prime' &= 1, &
\delta \gamma_\prime' &= 0,\cr}
   \right\}
   \eqno {\rm (C^{16})}$$
and therefore, by (E${}^8$),
$$\left. \multieqalign{
\delta \alpha  &=   x_{x_\prime}   \, \delta \alpha_\prime
                  + x_{y_\prime}   \, \delta \beta_\prime; &
\delta \alpha' &=   x'_{x_\prime'} \, \delta \alpha_\prime'
                  + x'_{y_\prime'} \, \delta \beta_\prime';\cr
\delta \beta   &=   y_{x_\prime}   \, \delta \alpha_\prime
                  + y_{y_\prime}   \, \delta \beta_\prime; &
\delta \beta'  &=   y'_{x_\prime'} \, \delta \alpha_\prime'
                  + y'_{y_\prime'} \, \delta \beta_\prime';\cr
\delta \gamma  &=   z_{x_\prime}   \, \delta \alpha_\prime
                  + z_{y_\prime}   \, \delta \beta_\prime; &
\delta \gamma' &=   z'_{x_\prime'} \, \delta \alpha_\prime'
                  + z'_{y_\prime'} \, \delta \beta_\prime';\cr}
   \right\}
   \eqno {\rm (D^{16})}$$
and substituting these values (A${}^{16}$) (D${}^{16}$) for the
twelve variations
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$,
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$,
in the general linear relations (A${}^9$) between these twelve
variations and the variation of colour $\delta \chi$, or in any
other linear relations of the same kind, deduced from the
characteristic and related functions, and referred to arbitrary
rectangular co-ordinates, we shall easily discover the particular
dependence, of the form (D${}^9$), of
$\delta \alpha_\prime$, $\delta \beta_\prime$, on
$\delta x_\prime$, $\delta y_\prime$, $\delta z_\prime$,
$\delta x_\prime'$, $\delta y_\prime'$,
$\delta \chi$, and of
$\delta \alpha_\prime'$, $\delta \beta_\prime'$, on
$\delta x_\prime$, $\delta y_\prime$,
$\delta x_\prime'$, $\delta y_\prime'$, $\delta z_\prime'$,
$\delta \chi$.

We seem by this transformation, to introduce twelve arbitrary
cosines or coefficients, namely,
$$x_{x_\prime},\quad
  y_{x_\prime},\quad
  z_{x_\prime},\quad
  x_{y_\prime},\quad
  y_{y_\prime},\quad
  z_{y_\prime},\quad
  x'_{x_\prime'},\quad
  y'_{x_\prime'},\quad
  z'_{x_\prime'},\quad
  x'_{y_\prime'},\quad
  y'_{y_\prime'},\quad
  z'_{y_\prime'};$$
but these twelve coefficients are connected by ten relations,
arising from the rectangularity of each of the four sets of
co-ordinates, and from the given directions of the semiaxes of
$z_\prime$ and $z_\prime'$; so that there remain only two
arbitrary quantities, corresponding to the arbitrary planes of
$x_\prime z_\prime$, $x_\prime' z_\prime'$, of which planes
we often, lately, disposed at pleasure, so as to make them
coincide with certain given planes of curvature, or otherwise to
simplify the recent geometrical discussions.  Thus, although we
may assign to the semiaxis of $x_\prime$ any position in the
given final plane perpendicular to the luminous path, and
therefore may assign to its cosines of direction
$x_{x_\prime}$,~$y_{x_\prime}$,~$z_{x_\prime}$, any values
consistent with the first equation (B${}^8$), namely
$$x_{x_\prime}^2 + y_{x_\prime}^2 + z_{x_\prime}^2 = 1,$$
and with the following
$$\alpha x_{x_\prime} + \beta y_{x_\prime} + \gamma z_{x_\prime}
   = 0,
   \eqno {\rm (E^{16})}$$
yet when the axis of $x_\prime$ has been so assumed, the
perpendicular axis of $y_\prime$ in the final perpendicular plane
is determined, and we have
$$\left. \eqalign{
x_{y_\prime} &= \pm (\beta  z_{x_\prime} - \gamma y_{x_\prime}),\cr
y_{y_\prime} &= \pm (\gamma x_{x_\prime} - \alpha z_{x_\prime}),\cr
z_{y_\prime} &= \pm (\alpha y_{x_\prime} - \beta x_{x_\prime}),\cr}
   \right\}
   \eqno {\rm (F^{16})}$$
the upper or lower signs being here obliged to accompany each
other: and similarly for the initial axes of $x_\prime'$ and
$y_\prime'$.

The characteristic and related functions give immediately, by their
partial differentials of the first order, the dependence of the
quantities which we have denoted by
$\sigma$,~$\tau$,~$\upsilon$, $\sigma'$,~$\tau'$,~$\upsilon'$,
rather than that of
$\alpha$,~$\beta$,~$\gamma$, $\alpha'$,~$\beta'$,~$\gamma'$,
on the extreme co-ordinates and the colour; and therefore the same
functions give immediately, by their partial differentials of the
second order, the variations of
$\delta \sigma$, $\delta \tau$, $\delta \upsilon$,
$\delta \sigma'$, $\delta \tau'$, $\delta \upsilon'$,
rather than
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
$\delta \alpha'$, $\delta \beta'$, $\delta \gamma'$,
in terms of
$\delta x$, $\delta y$, $\delta z$,
$\delta x'$, $\delta y'$, $\delta z'$, $\delta \chi$.
But we can easily deduce the variations of
$\alpha$~$\beta$~$\gamma$ $\alpha'$~$\beta'$~$\gamma'$
from those of
$\sigma$~$\tau$~$\upsilon$ $\sigma'$~$\tau'$~$\upsilon'$
and of
$x$~$y$~$z$ $x'$~$y'$~$z'$~$\chi$,
by differentiating the relations
$$\multieqalign{
\sigma    &= {\delta v  \over \delta \alpha}, &
\tau      &= {\delta v  \over \delta \beta}, &
\upsilon  &= {\delta v  \over \delta \gamma},\cr
\sigma'   &= {\delta v' \over \delta \alpha'}, &
\tau'     &= {\delta v' \over \delta \beta'}, &
\upsilon' &= {\delta v' \over \delta \gamma'},\cr}$$
which have often been employed already in the present Supplement; for
thus we obtain

\vfill\eject  % Page break necessary with current page size

$$\left. \eqalign{
         {\delta^2 v \over \delta \alpha^2}
            \, \delta \alpha
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
            \, \delta \beta
       + {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \, \delta \gamma
   &= \delta \sigma   - \delta_{\prime\prime}
            {\delta v \over \delta \alpha},\cr
         {\delta^2 v \over \delta \alpha \, \delta \beta}
            \, \delta \alpha
       + {\delta^2 v \over \delta \beta^2}
            \, \delta \beta
       + {\delta^2 v \over \delta \beta \, \delta \gamma}
            \, \delta \gamma
   &= \delta \tau     - \delta_{\prime\prime}
            {\delta v \over \delta \beta},\cr
         {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \, \delta \alpha
       + {\delta^2 v \over \delta \beta \, \delta \gamma}
            \, \delta \beta
       + {\delta^2 v \over \delta \gamma^2}
            \, \delta \gamma
   &= \delta \upsilon - \delta_{\prime\prime}
            {\delta v \over \delta \gamma},\cr
         {\delta^2 v' \over \delta \alpha'^2}
            \, \delta \alpha'
       + {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            \, \delta \beta'
       + {\delta^2 v' \over \delta \alpha' \, \delta \gamma'}
            \, \delta \gamma'
   &= \delta \sigma'   - \delta'
            {\delta v' \over \delta \alpha'},\cr
         {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
            \, \delta \alpha'
       + {\delta^2 v' \over \delta \beta'^2}
            \, \delta \beta'
       + {\delta^2 v' \over \delta \beta' \, \delta \gamma'}
            \, \delta \gamma'
   &= \delta \tau'     - \delta'
            {\delta v' \over \delta \beta'},\cr
         {\delta^2 v' \over \delta \alpha' \, \delta \gamma'}
            \, \delta \alpha'
       + {\delta^2 v' \over \delta \beta' \, \delta \gamma'}
            \, \delta \beta'
       + {\delta^2 v' \over \delta \gamma'^2}
            \, \delta \gamma'
   &= \delta \upsilon' - \delta'
            {\delta v' \over \delta \gamma'},\cr}
   \right\}
   \eqno {\rm (G^{16})}$$
$\delta_{\prime\prime}$ referring, as in former numbers, to the
variations of $x$,~$y$,~$z$,~$\chi$, and $\delta'$ to those of
$x'$,~$y'$,~$z'$,~$\chi$: and hence we have, by some symmetric
eliminations,
$$\left. \eqalign{
v'' \, \delta \alpha
      \hskip -12pt \cr
   &= \left(   {\delta^2 v \over \delta \beta^2}
             + {\delta^2 v \over \delta \gamma^2} \right)
         \left( \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha} \right)
      - {\delta^2 v \over \delta \alpha \, \delta \beta}
         \left( \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta} \right)
      - {\delta^2 v \over \delta \gamma \, \delta \alpha}
         \left( \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma} \right),\cr
v'' \, \delta \beta
      \hskip -12pt \cr
   &= \left(   {\delta^2 v \over \delta \gamma^2}
             + {\delta^2 v \over \delta \alpha^2} \right)
         \left( \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta} \right)
      - {\delta^2 v \over \delta \beta \, \delta \gamma}
         \left( \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma} \right)
      - {\delta^2 v \over \delta \alpha \, \delta \beta}
         \left( \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha} \right),\cr
v'' \, \delta \gamma
      \hskip -12pt \cr
   &= \left(   {\delta^2 v \over \delta \alpha^2}
             + {\delta^2 v \over \delta \beta^2} \right)
         \left( \delta \upsilon - \delta_{\prime\prime}
                  {\delta v \over \delta \gamma} \right)
      - {\delta^2 v \over \delta \gamma \, \delta \alpha}
         \left( \delta \sigma   - \delta_{\prime\prime}
                  {\delta v \over \delta \alpha} \right)
      - {\delta^2 v \over \delta \beta \, \delta \gamma}
         \left( \delta \tau     - \delta_{\prime\prime}
                  {\delta v \over \delta \beta} \right),\cr
v''' \, \delta \alpha'
      \hskip -12pt \cr
   &= \left(   {\delta^2 v' \over \delta \beta'^2}
             + {\delta^2 v' \over \delta \gamma'^2} \right)
         \left( \delta \sigma'   - \delta'
                  {\delta v' \over \delta \alpha'} \right)
      - {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
         \left( \delta \tau'     - \delta'
                  {\delta v' \over \delta \beta'} \right)
      - {\delta^2 v' \over \delta \gamma' \, \delta \alpha'}
         \left( \delta \upsilon' - \delta'
                  {\delta v' \over \delta \gamma'} \right),\cr
v''' \, \delta \beta'
      \hskip -12pt \cr
   &= \left(   {\delta^2 v' \over \delta \gamma'^2}
             + {\delta^2 v' \over \delta \alpha'^2} \right)
         \left( \delta \tau'     - \delta'
                  {\delta v' \over \delta \beta'} \right)
      - {\delta^2 v' \over \delta \beta' \, \delta \gamma'}
         \left( \delta \upsilon' - \delta'
                  {\delta v' \over \delta \gamma'} \right)
      - {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
         \left( \delta \sigma'   - \delta'
                  {\delta v' \over \delta \alpha'} \right),\cr
v''' \, \delta \gamma'
      \hskip -12pt \cr
   &= \left(   {\delta^2 v' \over \delta \alpha'^2}
             + {\delta^2 v' \over \delta \beta'^2} \right)
         \left( \delta \upsilon' - \delta'
                  {\delta v' \over \delta \gamma'} \right)
      - {\delta^2 v' \over \delta \gamma' \, \delta \alpha'}
         \left( \delta \sigma'   - \delta'
                  {\delta v' \over \delta \alpha'} \right)
      - {\delta^2 v' \over \delta \beta' \, \delta \gamma'}
         \left( \delta \tau'     - \delta'
                  {\delta v' \over \delta \beta'} \right),\cr}
   \right\}
   \eqno {\rm (H^{16})}$$
$v''$ having the meaning (L${}^6$), and $v'''$ the analogous meaning
$$v'''
    =   {\delta^2 v' \over \delta \alpha'^2}
         {\delta^2 v' \over \delta \beta'^2}
      - \left(
         {\delta^2 v' \over \delta \alpha' \, \delta \beta'}
         \right)^2
      + {\delta^2 v' \over \delta \beta'^2}
         {\delta^2 v' \over \delta \gamma'^2}
      - \left(
         {\delta^2 v' \over \delta \beta' \, \delta \gamma'}
         \right)^2
      + {\delta^2 v' \over \delta \gamma'^2}
         {\delta^2 v' \over \delta \alpha'^2}
      - \left(
         {\delta^2 v' \over \delta \gamma' \, \delta \alpha'}
         \right)^2.
   \eqno {\rm (I^{16})}$$
We might also deduce the variations of
$\alpha$~$\beta$~$\gamma$ $\alpha'$~$\beta'$~$\gamma'$
from those of
$\sigma$~$\tau$~$\upsilon$ $\sigma'$~$\tau'$~$\upsilon'$
$x$~$y$~$z$ $x'$~$y'$~$z'$~$\chi$,
by differentiating the equations (I) of the second number, and by
applying the functions $\Omega$, $\Omega'$, instead of $v$, $v'$.

\bigbreak

{\sectiontitle
The general Linear Expressions for the Arrangement of Near Rays,
fail at a Point of Vergency.  Determination of these Points, and
of their Loci, the Caustic Surfaces, in a Straight or Curved
System, by the Methods of the present Supplement.\par}

\nobreak\bigskip

23.
We have hitherto supposed that the infinitesimal or limiting
expressions of the variations of the extreme cosines of direction
of a luminous path, are linear functions of the variations of the
extreme co-ordinates and colour.  But although this supposition is
in general true, it admits of an important and extensive
exception; for the linear form becomes inapplicable when the
given luminous path $(A,B)_\chi$, with which other near paths are
to be compared, is intersected in its initial and final points
$A$, $B$, by another path infinitely near, and having the same
colour~$\chi$: since then the extreme directions may undergo
certain infinitesimal variations, while the extreme positions
$A$, $B$, and the colour~$\chi$, remain unaltered.  It is
therefore an important general problem of mathematical optics, to
determine, for any proposed optical combination, {\it the
relations between the extreme co-ordinates and the colour of a
luminous path which is intersected in its extreme points by
another infinitely near path of the same colour.}  This general
problem, of which the solution includes the general theory of the
caustic surfaces touched by the straight or curved rays of any
proposed optical system, may easily be resolved by the methods of
the present Supplement.

In applying these methods to the present question, we are to
differentiate the general equations which connect the extreme
directions with the extreme positions and colour, by the partial
differential coefficients of the first order of the
characteristic and related functions, and then to suppress the
variations of
$x$~$y$~$z$ $x'$~$y'$~$z'$ $\chi$.
And of the partial differential coefficients of the second order,
introduced by such differentiation, it is easy to see by
(A${}^9$) that those of the characteristic function $V$, or at
least some of them, are infinite in the present research: it is
therefore advantageous here to employ one of the auxiliary
functions $W$, $T$, combined if necessary with the functions
$v$,~$v'$, or $\Omega$,~$\Omega'$, which express by their form the
properties of the extreme media.

Thus, when the final medium is uniform, and therefore the final
rays straight, we may conveniently employ the following
equations, which involve the coefficients of the functions $W$,
$\Omega$, and were established in the sixth number,
$$x = {\delta W \over \delta \sigma}
             + V {\delta \Omega \over \delta \sigma},\quad
  y = {\delta W \over \delta \tau}
             + V {\delta \Omega \over \delta \tau},\quad
  z = {\delta W \over \delta \upsilon}
             + V {\delta \Omega \over \delta \upsilon}.
   \eqno {\rm (W^2)}$$
Differentiating these equations with respect to
$\sigma$~$\tau$~$\upsilon$ as the only variables, and
suppressing the variation of the first order of $V$, as well as
those of
$x$~$y$~$z$ $x'$~$y'$~$z'$~$\chi$,
we obtain
$$\left. \eqalign{
0  &=    \left(
            {\delta^2 W \over \delta \sigma^2}
             + V {\delta^2 \Omega \over \delta \sigma^2}
         \right) \, \delta \sigma
       + \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
             + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \right) \, \delta \tau
       + \left(
            {\delta^2 W \over \delta \sigma \, \delta \upsilon}
             + V {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}
         \right) \, \delta \upsilon,\cr
0  &=    \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
             + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \right) \, \delta \sigma
       + \left(
            {\delta^2 W \over \delta \tau^2}
             + V {\delta^2 \Omega \over \delta \tau^2}
         \right) \, \delta \tau
       + \left(
            {\delta^2 W \over \delta \tau \, \delta \upsilon}
             + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
         \right) \, \delta \upsilon,\cr
0  &=    \left(
            {\delta^2 W \over \delta \sigma \, \delta \upsilon}
             + V {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}
         \right) \, \delta \sigma
       + \left(
            {\delta^2 W \over \delta \tau \, \delta \upsilon}
             + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
         \right) \, \delta \tau
       + \left(
            {\delta^2 W \over \delta \upsilon^2}
             + V {\delta^2 \Omega \over \delta \upsilon^2}
         \right) \, \delta \upsilon,\cr}
   \right\}
   \eqno {\rm (K^{16})}$$
and hence, by a symmetric elimination, and by the forms of $W$, $\Omega$,
$$\eqalignno{
0  &= \mathbin{\phantom{+}}
         \left(
            {\delta^2 W \over \delta \sigma^2}
             + V {\delta^2 \Omega \over \delta \sigma^2}
         \right) \left(
            {\delta^2 W \over \delta \tau^2}
             + V {\delta^2 \Omega \over \delta \tau^2}
         \right)
       - \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
             + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 W \over \delta \tau^2}
             + V {\delta^2 \Omega \over \delta \tau^2}
         \right) \left(
            {\delta^2 W \over \delta \upsilon^2}
             + V {\delta^2 \Omega \over \delta \upsilon^2}
	 \right)
       - \left(
            {\delta^2 W \over \delta \tau \, \delta \upsilon}
             + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
         \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 W \over \delta \upsilon^2}
             + V {\delta^2 \Omega \over \delta \upsilon^2}
         \right) \left(
            {\delta^2 W \over \delta \sigma^2}
             + V {\delta^2 \Omega \over \delta \sigma^2}
	 \right)
       - \left(
            {\delta^2 W \over \delta \upsilon \, \delta \sigma}
             + V {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
         \right)^2:
   &{\rm (L^{16})}\cr}$$
which is a form of the condition required, for the final and
initial intersections of two near luminous paths, of any common
colour, the final medium being uniform.  The condition
(L${}^{16}$) is quadratic with respect to $V$, and determines,
for any final system of straight rays, corresponding to any given
luminous or initial point $A$, and to any given colour $\chi$,
two real or imaginary points of vergency $B_1$, $B_2$, on any one
straight final ray, that is, two points in which this ray is
intersected by infinitely near rays of the same final system;
and the joint equation in $x$~$y$~$z$, (involving also
$x'$~$y'$~$z'$~$\chi$ as parameters,) of the {\it two caustic
surfaces\/} which are touched by all the final rays and are the
loci of the points of vergency, may be obtained by eliminating
$\sigma$~$\tau$~$\upsilon$ between the equations (W${}^2$)
and the quadratic (L${}^{16}$): which quadratic, by the
homogeneity of the functions $W$ and $\Omega + 1$, may be put
under the following simpler form,
$$       \left(
            {\delta^2 W \over \delta \sigma^2}
             + V {\delta^2 \Omega \over \delta \sigma^2}
         \right) \left(
            {\delta^2 W \over \delta \tau^2}
             + V {\delta^2 \Omega \over \delta \tau^2}
         \right)
       - \left(
            {\delta^2 W \over \delta \sigma \, \delta \tau}
             + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
         \right)^2
   = 0,
   \eqno {\rm (M^{16})}$$
and admits of several other transformations.  When $V$ has either
of the two values determined by this quadratic, that is, when the
final point $B$ of the luminous path has any position $B_1$ or
$B_2$ on either of the two caustic surfaces, then the equations
deduced from (W${}^2$) by differentiating with respect to
$x$~$y$~$z$ as well as $\sigma$~$\tau$~$\upsilon$, namely,
$$\left. \eqalign{
\delta x - {\delta \Omega \over \delta \sigma}
               (   \sigma   \, \delta x
                 + \tau     \, \delta y
                 + \upsilon \, \delta z )
   &= \delta {\delta W \over \delta \sigma}
         + V \, \delta {\delta \Omega \over \delta \sigma},\cr
\delta y - {\delta \Omega \over \delta \tau}
               (   \sigma   \, \delta x
                 + \tau     \, \delta y
                 + \upsilon \, \delta z )
   &= \delta {\delta W \over \delta \tau}
         + V \, \delta {\delta \Omega \over \delta \tau},\cr
\delta z - {\delta \Omega \over \delta \upsilon}
               (   \sigma   \, \delta x
                 + \tau     \, \delta y
                 + \upsilon \, \delta z )
   &= \delta {\delta W \over \delta \upsilon}
         + V \, \delta {\delta \Omega \over \delta \upsilon},\cr}
   \right\}
   \eqno {\rm (N^{16})}$$
conduct to a linear relation between
$\delta x$, $\delta y$, $\delta z$,
which may be put under several forms, for example under the
following,
$$\eqalign{
      {1 \over \lambda} \left\{
      \delta x - {\delta \Omega \over \delta \sigma}
               (   \sigma   \, \delta x
                 + \tau     \, \delta y
                 + \upsilon \, \delta z )
      \right\}
   &= {1 \over \lambda'} \left\{
      \delta y - {\delta \Omega \over \delta \tau}
               (   \sigma   \, \delta x
                 + \tau     \, \delta y
                 + \upsilon \, \delta z )
      \right\} \cr
   &= {1 \over \lambda''} \left\{
      \delta z - {\delta \Omega \over \delta \upsilon}
               (   \sigma   \, \delta x
                 + \tau     \, \delta y
                 + \upsilon \, \delta z )
      \right\},\cr}
   \eqno {\rm (O^{16})}$$
in which we may assign to $\lambda$~$\lambda'$~$\lambda''$ any
of the following systems of values,
$$\left. \multieqalign{
   &\hbox{First} &
\lambda
   &= {\delta^2 W \over \delta \sigma^2}
          + V {\delta^2 \Omega \over \delta \sigma^2}, &
\lambda'
   &= {\delta^2 W \over \delta \sigma \, \delta \tau}
          + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau} &
\lambda''
   &= {\delta^2 W \over \delta \sigma \, \delta \upsilon}
          + V {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon};\cr
   &\hbox{Second} &
\lambda
   &= {\delta^2 W \over \delta \sigma \, \delta \tau}
          + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}, &
\lambda'
   &= {\delta^2 W \over \delta \tau^2}
          + V {\delta^2 \Omega \over \delta \tau^2} &
\lambda''
   &= {\delta^2 W \over \delta \tau \, \delta \upsilon}
          + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon};\cr
   &\hbox{Third} &
\lambda
   &= {\delta^2 W \over \delta \sigma \, \delta \upsilon}
          + V {\delta^2 \Omega \over \delta \sigma \, \delta \upsilon}, &
\lambda'
   &= {\delta^2 W \over \delta \tau \, \delta \upsilon}
          + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}, &
\lambda''
   &= {\delta^2 W \over \delta \upsilon^2}
          + V {\delta^2 \Omega \over \delta \upsilon^2}:\cr}
   \right\}
   \eqno {\rm (P^{16})}$$
and it is easy to see that the linear relation thus deduced,
between $\delta x$, $\delta y$, $\delta z$, is the differential
equation, or equation of the tangent plane, of the caustic
surface at the point of vergency $x$~$y$~$z$.  The same
linear equation represents also the plane of vergency, or the
tangent plane to the developable pencil of straight rays,
corresponding to the other or conjugate point of vergency on the
given final ray.

When the final medium is variable, the three first equations
(H${}'$), namely,
$$x = {\delta W \over \delta \sigma},\quad
  y = {\delta W \over \delta \tau},\quad
  z = {\delta W \over \delta \upsilon},$$
are to be differentiated with respect to
$\sigma$,~$\tau$,~$\upsilon$; and thus we obtain
$$\left. \eqalign{
         {\delta^2 W \over \delta \sigma^2} \, \delta \sigma
       + {\delta^2 W \over \delta \sigma \, \delta \tau} \, \delta \tau
       + {\delta^2 W \over \delta \sigma \, \delta \upsilon} \, \delta \upsilon
   &= 0,\cr
         {\delta^2 W \over \delta \sigma \, \delta \tau} \, \delta \sigma
       + {\delta^2 W \over \delta \tau^2} \, \delta \tau
       + {\delta^2 W \over \delta \tau \, \delta \upsilon} \, \delta \upsilon
   &= 0,\cr
         {\delta^2 W \over \delta \sigma \, \delta \upsilon} \, \delta \sigma
       + {\delta^2 W \over \delta \tau \, \delta \upsilon} \, \delta \tau
       + {\delta^2 W \over \delta \upsilon^2} \, \delta \upsilon
   &= 0,\cr}
   \right\}
   \eqno {\rm (Q^{16})}$$
and consequently, by elimination,
$${\delta^2 W \over \delta \sigma^2}
         {\delta^2 W \over \delta \tau^2}
         {\delta^2 W \over \delta \upsilon^2}
     + 2 {\delta^2 W \over \delta \sigma \, \delta \tau}
         {\delta^2 W \over \delta \tau \, \delta \upsilon}
         {\delta^2 W \over \delta \upsilon \, \delta \sigma}
   =     {\delta^2 W \over \delta \sigma^2}
         \left( {\delta^2 W \over \delta \tau \, \delta \upsilon} \right)^2
       + {\delta^2 W \over \delta \tau^2}
         \left( {\delta^2 W \over \delta \upsilon \, \delta \sigma} \right)^2
       + {\delta^2 W \over \delta \upsilon^2}
         \left( {\delta^2 W \over \delta \sigma \, \delta \tau} \right)^2:
   \eqno {\rm (R^{16})}$$
this equation, therefore, (which may be put under other forms,)
takes the place, when the final medium is variable, of the
quadratic (L${}^{16}$) for a final uniform medium; and if we
eliminate from it $\sigma$~$\tau$~$\upsilon$ by (H${}'$), it
will give, for any proposed initial point and colour, the
equation of the {\it single or multiple caustic surface, touched
by the curved rays\/} of the corresponding final system.

The auxiliary function $T$ may also be employed for the case of
curved rays, but it is chiefly useful when both the extreme media
are uniform.  In that case the extreme portions of a luminous
path are straight, and we may employ for these extreme straight
portions the equations (S${}^2$) under the form
$$x = {\delta S \over \delta \sigma},\quad
  y = {\delta S \over \delta \tau},\quad
  x' = - {\delta S \over \delta \sigma'},\quad
  y' = - {\delta S \over \delta \tau'},
   \eqno {\rm (S^{16})}$$
in which we have put, for abridgement,
$$S = T - z \upsilon + z' \upsilon',
   \eqno {\rm (T^{16})}$$
and in which we consider $\upsilon$ as a function of
$\sigma$,~$\tau$,~$\chi$; $\upsilon'$ as a function of
$\sigma'$,~$\tau'$,~$\chi$; $T$ as a function of
$\sigma$,~$\tau$, $\sigma'$,~$\tau'$,~$\chi$;
and $S$ as a function of
$z$,~$z'$, $\sigma$,~$\tau$, $\sigma'$,~$\tau'$,~$\chi$.
Differentiating these equations (S${}^{16}$) with respect to
$\sigma$,~$\tau$, $\sigma'$,~$\tau'$, we find that if the extreme
straight portions, ordinary or extraordinary, of two infinitely
near paths of light of the same colour, intersect in an initial
point $x'$~$y'$~$z'$, and in a final point $x$~$y$~$z$, the
final and initial variations
$\delta \sigma$, $\delta \tau$, $\delta \sigma'$, $\delta \tau'$,
and the final and initial ordinates of intersection $z$, $z'$,
must satisfy the four following conditions,
$$\left. \eqalign{
0  &=    {\delta^2 S \over \delta \sigma^2}
               \, \delta \sigma
       + {\delta^2 S \over \delta \sigma \, \delta \tau}
               \, \delta \tau
       + {\delta^2 S \over \delta \sigma \, \delta \sigma'}
               \, \delta \sigma'
       + {\delta^2 S \over \delta \sigma \, \delta \tau'}
               \, \delta \tau',\cr
0  &=    {\delta^2 S \over \delta \sigma \, \delta \tau}
               \, \delta \sigma
       + {\delta^2 S \over \delta \tau^2}
               \, \delta \tau
       + {\delta^2 S \over \delta \tau \, \delta \sigma'}
               \, \delta \sigma'
       + {\delta^2 S \over \delta \tau \, \delta \tau'}
               \, \delta \tau',\cr
0  &=    {\delta^2 S \over \delta \sigma \, \delta \sigma'}
               \, \delta \sigma
       + {\delta^2 S \over \delta \tau \, \delta \sigma'}
               \, \delta \tau
       + {\delta^2 S \over \delta \sigma'^2}
               \, \delta \sigma'
       + {\delta^2 S \over \delta \sigma' \, \delta \tau'}
               \, \delta \tau',\cr
0  &=    {\delta^2 S \over \delta \sigma \, \delta \tau'}
               \, \delta \sigma
       + {\delta^2 S \over \delta \tau \, \delta \tau'}
               \, \delta \tau
       + {\delta^2 S \over \delta \sigma' \, \delta \tau'}
               \, \delta \sigma'
       + {\delta^2 S \over \delta \tau'^2}
               \, \delta \tau';\cr}
   \right\}
   \eqno {\rm (U^{16})}$$
which give, by eliminating between the two first,
$$\left. \eqalign{
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
         \right) \, \delta \sigma'
   &=    \left(
            {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
         \right) \, \delta \sigma \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
         \right) \, \delta \tau;\cr
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
         \right) \, \delta \tau'
   &=    \left(
            {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma   \, \delta \sigma'}
         \right) \, \delta \sigma \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
          - {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
         \right) \, \delta \tau;\cr}
   \right\}
   \eqno {\rm (V^{16})}$$
and therefore, by substituting these values of
$\delta \sigma'$, $\delta \tau'$, in the two last,

\vfill\eject  % Page break necessary with current page size

$$\eqalignno{
0 &= \delta \sigma \biggl\{
         {\delta^2 S \over \delta \sigma'^2}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
         \right)
       + {\delta^2 S \over \delta \sigma' \, \delta \tau'}
         \left(
            {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma   \, \delta \sigma'}
         \right) \cr
   &\mathrel{\phantom{=}} \hskip 144pt
       + {\delta^2 S \over \delta \sigma \, \delta \sigma'}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
         \right) \biggr\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \delta \tau \biggl\{
         {\delta^2 S \over \delta \sigma'^2}
         \left(
            {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
         \right)
       + {\delta^2 S \over \delta \sigma' \, \delta \tau'}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
          - {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
         \right) \cr
   &\mathrel{\phantom{=}} \hskip 144pt
       + {\delta^2 S \over \delta \tau \, \delta \sigma'}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
         \right) \biggr\};\cr
0 &= \delta \sigma \biggl\{
         {\delta^2 S \over \delta \sigma' \, \delta \tau'}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
         \right)
       + {\delta^2 S \over \delta \tau'^2}
         \left(
            {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma   \, \delta \sigma'}
         \right) \cr
   &\mathrel{\phantom{=}} \hskip 144pt
       + {\delta^2 S \over \delta \sigma \, \delta \tau'}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
         \right) \biggr\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \delta \tau \biggl\{
         {\delta^2 S \over \delta \sigma' \, \delta \tau'}
         \left(
            {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
         \right)
       + {\delta^2 S \over \delta \tau'^2}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
          - {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
         \right) \cr
   &\mathrel{\phantom{=}} \hskip 144pt
       + {\delta^2 S \over \delta \tau \, \delta \tau'}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau   \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau   \, \delta \sigma'}
         \right) \biggr\};\cr
   & &{\rm (W^{16})}\cr}$$
so that by a new elimination we obtain, between the final and
initial ordinates $z$, $z'$, the following equation, which, by
the form of $S$, is quadratic with respect to each ordinate
separately, and involves the product of their squares:
$$\eqalignno{
0  &= \left\{
            {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau^2}
          - \left( {\delta^2 S \over \delta \sigma \, \delta \tau} \right)^2
      \right\}
      \left\{
            {\delta^2 S \over \delta \sigma'^2}
            {\delta^2 S \over \delta \tau'^2}
          - \left( {\delta^2 S \over \delta \sigma' \, \delta \tau'} \right)^2
      \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau \, \delta \sigma'}
      \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       - {\delta^2 S \over \delta \sigma'^2} \left\{
            {\delta^2 S \over \delta \sigma^2}
            \left( {\delta^2 S \over \delta \tau \, \delta \tau'} \right)^2
          - 2 {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau \, \delta \tau'}
          + {\delta^2 S \over \delta \tau^2}
            \left( {\delta^2 S \over \delta \sigma \, \delta \tau'} \right)^2
      \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 2 {\delta^2 S \over \delta \sigma' \, \delta \tau'}
         \biggl\{
            {\delta^2 S \over \delta \sigma^2}
            {\delta^2 S \over \delta \tau \, \delta \sigma'}
            {\delta^2 S \over \delta \tau \, \delta \tau'}
          - {\delta^2 S \over \delta \sigma \, \delta \tau}
            \left(
               {\delta^2 S \over \delta \sigma \, \delta \sigma'}
               {\delta^2 S \over \delta \tau \, \delta \tau'}
             + {\delta^2 S \over \delta \sigma \, \delta \tau'}
               {\delta^2 S \over \delta \tau \, \delta \sigma'}
            \right) \cr
   &\mathrel{\phantom{=}} \hskip 72pt
          + {\delta^2 S \over \delta \tau^2}
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \sigma \, \delta \tau'}
         \biggr\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       - {\delta^2 S \over \delta \tau'^2} \left\{
            {\delta^2 S \over \delta \sigma^2}
            \left( {\delta^2 S \over \delta \tau \, \delta \sigma'} \right)^2
          - 2 {\delta^2 S \over \delta \sigma \, \delta \tau}
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau \, \delta \sigma'}
          + {\delta^2 S \over \delta \tau^2}
            \left( {\delta^2 S \over \delta \sigma \, \delta \sigma'} \right)^2
      \right\}.
   &{\rm (X^{16})}\cr}$$
When the point of intersection of the infinitely near initial
rays removes to an infinite distance, this equation reduces
itself to the following,

\vfill\eject  % Page break necessary with current page size

$$\eqalignno{
0  &=    {\delta^2 S \over \delta \sigma^2}
         {\delta^2 S \over \delta \tau^2}
       - \left( {\delta^2 S \over \delta \sigma \, \delta \tau} \right)^2 \cr
   &= \left(
         {\delta^2 T \over \delta \sigma^2}
         - z {\delta^2 \upsilon \over \delta \sigma^2}
      \right)
      \left(
         {\delta^2 T \over \delta \tau^2}
         - z {\delta^2 \upsilon \over \delta \tau^2}
      \right)
    - \left(
         {\delta^2 T \over \delta \sigma \, \delta \tau}
         - z {\delta^2 \upsilon \over \delta \sigma \, \delta \tau}
      \right)^2:
   &{\rm (Y^{16})}\cr}$$
and when in like manner the two infinitely near final rays become
parallel, it gives the following quadratic to determine the two
corresponding positions of the point of initial intersection,
$$\eqalignno{
0  &=    {\delta^2 S \over \delta \sigma'^2}
         {\delta^2 S \over \delta \tau'^2}
       - \left( {\delta^2 S \over \delta \sigma' \, \delta \tau'} \right)^2 \cr
   &= \left(
         {\delta^2 T \over \delta \sigma'^2}
         + z' {\delta^2 \upsilon \over \delta \sigma'^2}
      \right)
      \left(
         {\delta^2 T \over \delta \tau'^2}
         + z' {\delta^2 \upsilon \over \delta \tau'^2}
      \right)
    - \left(
         {\delta^2 T \over \delta \sigma' \, \delta \tau'}
         + z' {\delta^2 \upsilon \over \delta \sigma' \, \delta \tau'}
      \right)^2.
   &{\rm (Z^{16})}\cr}$$

The caustic surfaces of straight systems, ordinary or
extraordinary, were determined in the First Supplement: but it
seemed useful to resume the subject in a more general manner
here, and to treat it by the new methods of the present memoir.

\bigbreak

{\sectiontitle
Connexion of the Conditions of Initial and Final Intersection of
two Near Paths of Light, Polygon or Curved, with the Maxima or
Minima of the Time or Action-Function
$V + V_\prime = \sum \int v \, ds$.
Separating Planes, Transition Planes, and Transition Points,
suggested by these Maxima and Minima.  The Separating Planes
divide the Near Points of less from those of greater Action, and
they contain the Directions of Osculation or Intersection of the
Surfaces for which $V$ and $V_\prime$ are constant; the
Transition Planes touch the Caustic Pencils, and the Transition
Points are on the Caustic Curves.  Extreme Osculating Waves, or
Action-Surfaces: Law of Osculation.  Analogous Theorems for
Sudden Reflexion or Refraction.\par}

\nobreak\bigskip

24.
The conditions of initial and final intersection of two near
luminous paths, have a remarkable connexion with the maxima and
minima of the integral in the law of least action, that is, with
those of the characteristic function $V$, or rather with those of
the sum of two such integrals or functions, which may be
investigated in the following manner.

Let $A$, $B$, $C$, be three successive points, at finite
intervals, on one common luminous path.  Let the rectangular
co-ordinates of these three points be $x'$,~$y'$,~$z'$ for $A$;
$x$,~$y$,~$z$ for $B$; and $x_\prime$, $y_\prime$, $z_\prime$ for
$C$.  Let $V(A,B)$ denote the integral $\int v \, ds$ taken from
the first point~$A$ to the second point~$B$; let $V(B,C)$ denote
the same integral, taken from the second point~$B$ to the third
point~$C$; and similarly, let $V(A,C)$ be the integral from $A$
to $C$, which is evidently equal to the sum of the two former,
$$V(A,C) = V(A,B) + V(B,C),
   \eqno {\rm (A^{17})}$$
so that, if we put for abridgment
$$V(A,B) = V,\quad V(B,C) = V_\prime,
   \eqno {\rm (B^{17})}$$
we shall have, by the continuity of the integral,
$$V(A,C) = V + V_\prime.
   \eqno {\rm (C^{17})}$$
If we do not suppose that the intermediate point $B$ is a point
of sudden reflexion or refraction, the final direction of the
part $(A,B)$ will coincide with the initial direction of the part
$(B,C)$, and the final direction-cosines
$\alpha$~$\beta$~$\gamma$ of the one part will be equal to the
initial direction-cosines of the other; considering $V$
therefore, as usual, as a function of
$x$~$y$~$z$ $x'$~$y'$~$z'$~$\chi$,
and $V_\prime$ as a function of
$x_\prime$~$y_\prime$~$z_\prime$ $x$~$y$~$z$~$\chi$,
we have, by our fundamental formula (A),
$${\delta V \over \delta x}
   = {\delta v \over \delta \alpha}
   = - {\delta V_\prime \over \delta x},\quad
  {\delta V \over \delta y}
   = {\delta v \over \delta \beta}
   = - {\delta V_\prime \over \delta y},\quad
  {\delta V \over \delta z}
   = {\delta v \over \delta \gamma}
   = - {\delta V_\prime \over \delta z};
   \eqno {\rm (D^{17})}$$
that is, we have
$$\delta V + \delta V_\prime = 0,
   \eqno {\rm (E^{17})}$$
for any infinitesimal variations of the co-ordinates $x$~$y$~$z$,
and therefore, to the accuracy of the first order,
$$V(A,B') + V(B',C) = V(A,B) + V(B,C) = V(A,C),
   \eqno {\rm (F^{17})}$$
$B'$ being any new intermediate point infinitely near to $B$, and
the path $(B',C)$ being not in general a continuation of the path
$(A,B')$.  If therefore we regard the extreme points $A$, $C$, as
fixed, but consider the intermediate point $B$ as variable and as
not necessarily situated on the path $(A,C)$, the function
$V + V_\prime$, or $\sum \int v \, ds$, composed of the two
partial and now not necessarily continuous integrals
(B${}^{17}$), will acquire what may be called a {\it stationary
value\/} when the paths $(A,B)$ $(B,C)$ become continuous, that
is, when the intermediate point $B$ takes any position on the
path $(A,C)$ from one given extreme point to the other: since
then the change of this function will be infinitely small of the
the second order, for any infinitely small alteration
$\overline{B B'}$, of the first order, in the position of the
point $B$.  The stationary value thus determined, namely
$V(A,C)$, might be called, by that customary latitude of
expression which leads to the received phrase of {\it least
action}, a {\it maximum\/} or {\it minimum\/} of the function
$V + V_\prime$: but in order that this value should really be
greater than all the neighbouring values, or less than all, a new
condition is necessary.  To find this new condition, we may
observe that the relations
$$\left. \eqalign{
         \alpha {\delta^2 V \over \delta x^2}
       + \beta  {\delta^2 V \over \delta x \, \delta y}
       + \gamma {\delta^2 V \over \delta x \, \delta z}
   &= {\delta v \over \delta x}
    = - \left(
         \alpha {\delta^2 V_\prime \over \delta x^2}
       + \beta  {\delta^2 V_\prime \over \delta x \, \delta y}
       + \gamma {\delta^2 V_\prime \over \delta x \, \delta z}
      \right),\cr
         \alpha {\delta^2 V \over \delta x \, \delta y}
       + \beta  {\delta^2 V \over \delta y^2}
       + \gamma {\delta^2 V \over \delta y \, \delta z}
   &= {\delta v \over \delta y}
    = - \left(
         \alpha {\delta^2 V_\prime \over \delta x \, \delta y}
       + \beta  {\delta^2 V_\prime \over \delta y^2}
       + \gamma {\delta^2 V_\prime \over \delta y \, \delta z}
      \right),\cr
         \alpha {\delta^2 V \over \delta x \, \delta z}
       + \beta  {\delta^2 V \over \delta y \, \delta z}
       + \gamma {\delta^2 V \over \delta z^2}
   &= {\delta v \over \delta y}
    = - \left(
         \alpha {\delta^2 V_\prime \over \delta x \, \delta z}
       + \beta  {\delta^2 V_\prime \over \delta y \, \delta z}
       + \gamma {\delta^2 V_\prime \over \delta z^2}
      \right),\cr}
   \right\}
   \eqno {\rm (G^{17})}$$
which result from the third number, give
$$\eqalignno{
\delta^2 V + \delta^2 V_\prime
   &=    \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V_\prime \over \delta x^2}
         \right)
         \left( \delta x - {\alpha \over \gamma} \, \delta z \right)^2
       + \left(
            {\delta^2 V \over \delta y^2}
          + {\delta^2 V_\prime \over \delta y^2}
         \right)
         \left( \delta y - {\beta  \over \gamma} \, \delta z \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
     + 2 \left(
            {\delta^2 V \over \delta x \, \delta y}
          + {\delta^2 V_\prime \over \delta x \, \delta y}
         \right)
         \left( \delta x - {\alpha \over \gamma} \, \delta z \right)
         \left( \delta y - {\beta  \over \gamma} \, \delta z \right);
   &{\rm (H^{17})}\cr}$$
the {\it condition of existence of a maximum or minimum, properly
so called, of the function\/} $V + V_\prime$, is therefore,
$$Q > 0,
   \quad \hbox{if} \quad
  Q =    \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V_\prime \over \delta x^2}
         \right)
         \left(
            {\delta^2 V \over \delta y^2}
          + {\delta^2 V_\prime \over \delta y^2}
         \right)
       - \left(
            {\delta^2 V \over \delta x \, \delta y}
          + {\delta^2 V_\prime \over \delta x \, \delta y}
         \right)^2.
   \eqno {\rm (I^{17})}$$
When we have on the contrary
$$Q < 0,
   \eqno {\rm (K^{17})}$$
the variation of the second order
$\delta^2 V + \delta^2 V_\prime$
admits of changing sign, in passing from one set of values of
$\delta x$, $\delta y$, $\delta z$
to another, that is, in passing from one near point $B'$ to
another; and since, to the accuracy of the second order,
$$V(A,B') + V(B',C) - V(A,C)
   = {\textstyle {1 \over 2}} (\delta^2 V + \delta^2 V_\prime),
   \eqno {\rm (L^{17})}$$
we shall have the one or the other of the two following opposite
inequalities
$$V(A,B') + V(B',C) > \hbox{ or } < V(A,C),
   \eqno {\rm (M^{17})}$$
according as the near point $B'$ is in one or the other pair of
opposite diedrate angles formed by {\it two separating planes\/}
$P'$~$P''$ determined by the following equation
$$\delta^2 V + \delta^2 V_\prime = 0,
   \eqno {\rm (N^{17})}$$
which is, by (H${}^{17}$), quadratic with respect to the ratio
$${\displaystyle
      \delta y - {\beta  \over \gamma} \, \delta z
   \over \displaystyle
      \delta x - {\alpha \over \gamma} \, \delta z}.$$
{\it These two separating planes\/} $P'$~$P''$ {\it contain each
the ray-line or element of the path $(A,B,C)$ at $B$; and they
divide the near points of less from those of greater action, or
those of shorter from those of longer time, when the continuous
integral $V + V' = V(A,C)$ is not greater than all, or less than
all, the adjacent values of the sum\/} $\sum \int v \, ds$.  The
directions of these planes depend on the positions of the points
$A$, $B$, $C$; so that if we consider $A$ and $B$ as fixed, but
suppose $C$ to move along the prolongation $(B,C)$ of the path
$(A,B)$, the separating planes $P'$,~$P''$, will in general
revolve about the ray-line at $B$.  The will even become
imaginary, when by this motion of $C$ the quantity~$Q$ becomes
$>$ instead of $< 0$, so as to satisfy the condition of existence
of a maximum or minimum of the function $V + V_\prime$; and in
this transition from the real to the imaginary state the two
separating planes $P'$~$P''$ will close up into one real {\it
transition-plane\/} $P$, determined by either of the two
following equations,
$$\left. \eqalign{
0  &=    \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V_\prime \over \delta x^2}
         \right)
         \left( \delta x - {\alpha \over \gamma} \, \delta z \right)
       + \left(
            {\delta^2 V \over \delta x \, \delta y}
          + {\delta^2 V_\prime \over \delta x \, \delta y}
         \right)
         \left( \delta y - {\beta  \over \gamma} \, \delta z \right),\cr
0  &=    \left(
            {\delta^2 V \over \delta x \, \delta y}
          + {\delta^2 V_\prime \over \delta x \, \delta y}
         \right)
         \left( \delta x - {\alpha \over \gamma} \, \delta z \right)
       + \left(
            {\delta^2 V \over \delta y^2}
          + {\delta^2 V_\prime \over \delta y^2}
         \right)
         \left( \delta y - {\beta  \over \gamma} \, \delta z \right);\cr}
   \right\}
   \eqno {\rm (O^{17})}$$
while the corresponding position of the point $C$, which we may
call by analogy a {\it transition-point}, will satisfy the
condition
$$Q = 0,
   \quad \hbox{that is,} \quad
         \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V_\prime \over \delta x^2}
         \right)
         \left(
            {\delta^2 V \over \delta y^2}
          + {\delta^2 V_\prime \over \delta y^2}
         \right)
   =     \left(
            {\delta^2 V \over \delta x \, \delta y}
          + {\delta^2 V_\prime \over \delta x \, \delta y}
         \right)^2.
   \eqno {\rm (P^{17})}$$

We are now prepared to perceive a remarkable connexion between
the transition-planes and transition-points to which we have been
thus conducted by the consideration of the maxima and minima of
the function $V + V_\prime$, and the condition of final and
initial intersection of two near luminous paths.  For these
conditions of intersection may be obtained by supposing that not
only the point $B$, having the co-ordinates $x$~$y$~$z$, is on a
given path $(A,C)$, so as to satisfy the equations (D${}^{17}$),
but that also an infinitely near point $B'$, having for
co-ordinates
$x + \delta x$, $y + \delta y$, $z + \delta z$,
is on another path of the same colour connecting the same extreme
points $A$ and $C$, so as to give the differential equations
$$\delta {\delta V \over \delta x}
     = - \delta {\delta V_\prime \over \delta x},\quad
  \delta {\delta V \over \delta y}
     = - \delta {\delta V_\prime \over \delta y},\quad
  \delta {\delta V \over \delta z}
     = - \delta {\delta V_\prime \over \delta z}:
   \eqno {\rm (Q^{17})}$$
and since these last equations may be reduced, by the relations
(G${}^{17}$), to the forms (O${}^{17}$), we see that when the
conditions of initial and final intersection of a given path
$(A,B,C)$ with a near path $(A,B',C)$ are satisfied, and when we
consider the initial point $A$ as fixed, the near intermediate
point $B'$ must be in a transition-plane~$P$ of the form
(O${}^{17}$), and the final point of intersection $C$ must be a
transition-point of the form (P${}^{17}$).  Continuing therefore
to regard the initial point $A$ as the fixed origin of a system
of luminous paths, polygon or curved, of any common colour, which
undergo any number of refractions or reflexions, ordinary or
extraordinary, and gradual or sudden, it is easy to see that we
may consider these paths as touching a certain set of
{\it caustic curves}, in the final state of the system, and
therefore grouped into certain sets of consecutively intersecting
paths, and as having for their loci certain corresponding sets of
ray-surfaces, which may be called {\it caustic pencils\/}: and
that {\it these caustic pencils are touched by the
transition-planes\/} (O${}^{17}$), while the {\it
transition-points\/} (P${}^{17}$) {\it are on the caustic
curves}, and therefore on their loci the caustic surfaces.  The
transition-points are also evidently the points of consecutive
intersection, or of vergency, of the luminous paths from $A$, in
the final state of the system.  And it is manifest, from the
foregoing remarks, that these final points of intersection are
also transition-points in the following other sense, that when
the point~$C$, in moving along the prolongation of the path
$(A,B)$, arrives at any one of these points of intersection, the
condition of existence of maximum or minimum of the function
$V + V_\prime$ begins or ceases to be satisfied.

The separating planes $P'$~$P''$ have, when real, another
remarkable property, that of containing the directions of mutual
osculation, at the point $B$, of the two action-surfaces or waves
determined by the equations
$$V = \hbox{const.},\quad
  V_\prime = \hbox{const.};
   \eqno {\rm (R^{17})}$$
for these equations may be put approximately under the following
forms, (when we choose the point $B$ for origin and the final
direction of the path $(A,B)$ for the positive semiaxis of $z$,
so as to have $\alpha = 0$, $\beta = 0$, $\gamma = 1$,)
$$\left. \eqalign{
z  &= p x + q y
            + {\textstyle {1 \over 2}} r x^2
            + s xy
            + {\textstyle {1 \over 2}} t y^2,\cr
z_\prime &= p_\prime x + q_\prime y
            + {\textstyle {1 \over 2}} r_\prime x^2
            + s_\prime xy
            + {\textstyle {1 \over 2}} t_\prime y^2,\cr}
   \right\}
   \eqno {\rm (S^{17})}$$
in which the coefficients have the following relations
$$\left. \eqalign{
p_\prime &= p,\quad q_\prime = q,\cr
r_\prime - r &= {1 \over v}
         \left(
            {\delta^2 V_\prime \over \delta x^2}
          + {\delta^2 V \over \delta x^2}
         \right),\cr
s_\prime - s &= {1 \over v}
         \left(
            {\delta^2 V_\prime \over \delta x \, \delta y}
          + {\delta^2 V \over \delta x \, \delta y}
         \right),\cr
t_\prime - t &= {1 \over v}
         \left(
            {\delta^2 V_\prime \over \delta y^2}
          + {\delta^2 V \over \delta y^2}
         \right),\cr}
   \right\}
   \eqno {\rm (T^{17})}$$
and therefore the planes
$$0 =    (r_\prime - r) x^2
       + 2 (s_\prime - s) xy
       + (t_\prime - t) y^2,
   \eqno {\rm (U^{17})}$$
which pass through the given ray-line at the point $B$, and
contain the directions of osculation of the second order of the
two touching surfaces (R${}^{17}$) or (S${}^{17}$), are the
separating planes (N${}^{17}$).  We might also characterise these
separating planes, or planes of osculation, as containing the
directions of mutual intersection of the same two touching
surfaces for which $V$ and $V_\prime$ are constant; or as the
planes in which the deflexures of these two surfaces are equal,
the ray-line at $B$ being made the axis of deflexion.

The comparison of the same two waves or action-surfaces
(R${}^{17}$) gives a new property of the planes and points of
transition; for the equations which determine a plane and point
of this kind may be put under the form
$$(r - r_\prime) x + (s - s_\prime) y = 0,\quad
  (s - s_\prime) x + (t - t_\prime) y = 0,
   \quad \hbox{or,} \quad
  \delta p_\prime = \delta p,\quad
  \delta q_\prime = \delta q:
   \eqno {\rm (V^{17})}$$
they express, therefore, that when $C$ is a transition-point, the
two surfaces (R${}^{17}$) touch one another not only at the
point~$B$, but in the whole extent of an infinitely small arc
contained in the transition-plane.

The point~$C$ may be called the {\it focus\/} of the second wave
or action-surface $V_\prime$, since all the corresponding paths
of light $(B',C)$ are supposed to meet in it; and in like manner
the point $A$ may be called the focus of the first surface $V$ of
the same kind, since all the paths $(A,B')$ are supposed to
diverge from $A$.  The focus~$A$ and the point of osculation~$B$
remaining fixed, we may change the focus~$C$, and thereby the
directions of osculation; but there are, in general, certain
{\it extreme or limiting positions for the osculating focus~$C$,
corresponding to extreme osculating waves or action-surfaces\/}
$V_\prime$, and it is easy to show that {\it these extreme
osculating foci coincide with the transition-points or points of
vergency\/}: and that {\it the transition-planes or
tangent-planes of the caustic pencils contain the directions of
such extreme or limiting osculation}.

These theorems of intersection and osculation include several
less general theorems of the same kind, assigned in former
memoirs.  It is easy also to see that they extend to the
case when the order of the points $A$~$B$~$C$ on a luminous
path is different, so that $B$ is not intermediate between
$A$ and $C$, and so that the paths $(A,B)$ $(A,B')$, which go
from $A$ to the points $B$ and $B'$, coincide at those points
with the paths $(C,B)$ $(C,B')$, and not with the opposite paths
$(B,C)$ $(B',C)$, that is, tend {\it from\/} the point $C$, not
{\it to\/} it; observing only that we must then employ the
{\it difference\/} instead of the {\it sum\/} of the two
integrals $\int v \, ds$, or of the two functions $V$ and
$V_\prime$.

When the point~$C$ is on a given straight ray in a given uniform
medium, we can easily prove, by the theory of the partial
differential coefficients of the second order of the
characteristic and related functions which was explained in
former numbers, that the equation (P${}^{17}$) becomes quadratic
with respect to $z_\prime$ or $V_\prime$, and assigns, in
general, two real or imaginary positions $C_1$,~$C_2$, for the
transition-point, or point of vergency; and that the equations
(O${}^{17}$) assign two corresponding real or imaginary
transition-planes $P_1$~$P_2$, or tangent planes of caustic
pencils.  And when, besides, the points $B$, $C$, are both in one
common or uniform medium, so that the paths $(B,C)$ $(B',C)$ are
straight, then each of the caustic pencils, or ray-surfaces,
composed of such straight paths consecutively intersecting each
other and touching one caustic curve, becomes a {\it developable
pencil}, and its tangent plane becomes a {\it plane of
vergency}, of the kind considered in the sixteenth number.  The
relations also between the two planes of vergency in a final
uniform medium, which were pointed out in the twenty-first
number, may easily be deduced from the present more general view
and from the recent theorems of osculation; for thus we are led to
consider a series of waves or action-surfaces $V_\prime$, similar
and similarly placed, and determined in shape but not in size or
focus by the uniform medium, and then to seek the extreme or
limiting surfaces of this set which osculate to the given
surface~$V$ at the given point~$B$; and since it can be shown
that {\it in general among any series of surfaces, similar and
similarly placed, but having arbitrary magnitudes, and osculating
to a given surface at a given point, there are two extreme
osculating surfaces}, real or imaginary, and that {\it the
tangents which mark the two corresponding directions of
calculation are conjugate tangents\/} (of the kind discovered by
{\sc M.~Dupin}) {\it on each surface of the osculating series,
and also on the given surface}, it follows as before that the
conjugate planes of vergency in a final uniform medium are
conjugate planes of deflexure of each medium-surface $V_\prime$,
and also of the surface $V$ determined by the whole combination.
When the final medium is ordinary as well as uniform, then the
osculating surfaces $V_\prime$ are spheres, and the directions of
extreme osculation are the rectangular directions of the lines of
curvature on the surface $V$, which is now perpendicular to the
rays; in this case, therefore, and more generally when a given
ray in a final uniform medium corresponds to an {\it umbilical
point\/} or point of spheric curvature on the medium-surface
$V_\prime$, the planes of vergency cut that surface, and the
surface $V$ to which it osculates, in two rectangular directions,
because two conjugate tangents at an umbilical point are always
perpendicular to each other: and, in like manner, the planes of
vergency being conjugate planes of deflexure will (by the
seventeenth number) be themselves rectangular, if the final ray
whether ordinary or extraordinary be such that taking it for the
axis of deflexion of the medium-surface $V_\prime$ the indicating
cylinder of deflexion is circular.

The foregoing principles give us also the {\it law of
osculation\/} of the variable medium-surface $V_\prime$ between
its extreme positions, in a final uniform medium, namely, that
{\it the distances of the variable osculating focus from the two
points of vergency, are proportional to the squares of the sines
of the inclinations of the variable plane of osculation to the
two planes of vergency, multiplied respectively by certain
constant factors}.  A formula expressing this law was deduced in
the First Supplement; but the constant and in general unequal
factors, (in the formula $\zeta$ and $1$,) for the squares of the
sines of the inclinations, were inadvertently omitted in the
enunciation.  Our present methods would enable us to investigate
without difficulty the law for the more complicated case, when
the osculating focus $C$ being still in a uniform medium, the
point of osculation $B$ is in another uniform medium, or even in
an atmosphere ordinary or extraordinary.

We might extend the reasonings of the present number to the case
of sudden reflexion or refraction, ordinary or extraordinary, and
obtain analogous results, which would include, in like manner, the
results of former memoirs.  In this case we should find a certain
analogous condition for the existence of a maximum or minimum of
the function $\sum \int v \, ds$; and when this condition is not
satisfied, we should have to consider {\it two pairs of
separating planes}, which cross the tangent plane of the
reflecting or refracting surface in one common pair of
{\it separating lines\/}: the two pairs of planes passing together
from the real to the imaginary state, and in this passage closing
up into {\it two transition-planes, which touch the caustic
pencil before and after the sudden reflexion or refraction}, and
intersect in one common  {\it transition-line, on the tangent
plane of the reflector or refractor}, connected with a {\it
transition-point upon the caustic curve of the pencil}, and with
certain {\it extreme osculating waves or action-surfaces and
focal reflectors or refractors}, of a kind easily discovered from
the analogy of the foregoing results.

\bigbreak

{\sectiontitle
Formul{\ae} for the Principal Foci and Principal Rays of a
Straight or Curved System, Ordinary or Extraordinary.  General
method of investigating the Arrangement and  Aberrations of the
Rays, near a Principal Focus, or other point of vergency.\par}

\nobreak\bigskip

25.
Among the various points of consecutive intersection of the
rays of an optical system, there are in general certain eminent
points of vergency, in which certain particular luminous paths
are intersected each by all the infinitely near paths of the
system.  These eminent points and paths have been pointed out in
my former memoirs, and have been called {\it principal foci}, and
{\it principal rays}.  They may be determined for straight final
systems, by the characteristic function $V$, and by any three of
the six following equations,
$$\left. \multieqalign{
{\delta^2 V \over \delta x^2}
      + {1 \over R} {\delta^2 v \over \delta \alpha^2}
   &= 0, &
{\delta^2 V \over \delta x \, \delta y}
      + {1 \over R} {\delta^2 v \over \delta \alpha \, \delta \beta}
   &= 0,\cr
{\delta^2 V \over \delta y^2}
      + {1 \over R} {\delta^2 v \over \delta \beta^2}
   &= 0, &
{\delta^2 V \over \delta y \, \delta z}
      + {1 \over R} {\delta^2 v \over \delta \beta \, \delta \gamma}
   &= 0,\cr
{\delta^2 V \over \delta z^2}
      + {1 \over R} {\delta^2 v \over \delta \gamma^2}
   &= 0, &
{\delta^2 V \over \delta z \, \delta x}
      + {1 \over R} {\delta^2 v \over \delta \gamma \, \delta \alpha}
   &= 0,\cr}
   \right\}
   \eqno {\rm (W^{17})}$$
$x$,~$y$,~$z$ being the co-ordinates of any point on a principal
ray, and $x + \alpha R$, $y + \beta R$, $z + \gamma R$ being the
co-ordinates of the principal focus; they may also be deduced from
the auxiliary function $W$, when made homogeneous of the first
dimension with respect to $\sigma$,~$\tau$,~$\upsilon$, by the
equations
$$\left. \multieqalign{
{\delta^2 W \over \delta \sigma^2}
      + V {\delta^2 \Omega \over \delta \sigma^2}
   &= 0, &
{\delta^2 W \over \delta \sigma \, \delta \tau}
      + V {\delta^2 \Omega \over \delta \sigma \, \delta \tau}
   &= 0,\cr
{\delta^2 W \over \delta \tau^2}
      + V {\delta^2 \Omega \over \delta \tau^2}
   &= 0, &
{\delta^2 W \over \delta \tau \, \delta \upsilon}
      + V {\delta^2 \Omega \over \delta \tau \, \delta \upsilon}
   &= 0,\cr
{\delta^2 W \over \delta \upsilon^2}
      + V {\delta^2 \Omega \over \delta \upsilon^2}
   &= 0, &
{\delta^2 W \over \delta \upsilon \, \delta \sigma}
      + V {\delta^2 \Omega \over \delta \upsilon \, \delta \sigma}
   &= 0,\cr}
   \right\}
   \eqno {\rm (X^{17})}$$
of which only three are distinct, and in which $V$ corresponds to
the focus: or from the function~$T$, when expressed as depending
on $\sigma$,~$\tau$, $\sigma'$,~$\tau'$,~$\chi$, by the
following,
$$\eqalignno{
   &\left( {\delta^2 S \over \delta \sigma'^2} \right)^{-1}
      \left\{
         {\delta^2 S \over \delta \sigma^2}
         \left( {\delta^2 S \over \delta \tau \, \delta \sigma'} \right)^2
       - 2 {\delta^2 S \over \delta \sigma \, \delta \tau}
         {\delta^2 S \over \delta \sigma \, \delta \sigma'}
         {\delta^2 S \over \delta \tau \, \delta \sigma'}
       + {\delta^2 S \over \delta \tau^2}
         \left( {\delta^2 S \over \delta \sigma \, \delta \sigma'} \right)^2
      \right\} \cr
   &= \left( {\delta^2 S \over \delta \sigma' \, \delta \tau'} \right)^{-1}
      \biggl\{
         {\delta^2 S \over \delta \sigma^2}
         {\delta^2 S \over \delta \tau \, \delta \sigma'}
         {\delta^2 S \over \delta \tau \, \delta \tau'}
       - {\delta^2 S \over \delta \sigma \, \delta \tau}
         \left(
            {\delta^2 S \over \delta \sigma \, \delta \sigma'}
            {\delta^2 S \over \delta \tau \, \delta \tau'}
          + {\delta^2 S \over \delta \sigma \, \delta \tau'}
            {\delta^2 S \over \delta \tau \, \delta \sigma'}
         \right) \cr
   &\mathrel{\phantom{=}} \hskip 72pt
       + {\delta^2 S \over \delta \tau^2}
         {\delta^2 S \over \delta \sigma \, \delta \sigma'}
         {\delta^2 S \over \delta \sigma \, \delta \tau'}
      \biggr\} \cr
   &= \left( {\delta^2 S \over \delta \tau'^2} \right)^{-1}
      \left\{
         {\delta^2 S \over \delta \sigma^2}
         \left( {\delta^2 S \over \delta \tau \, \delta \tau'} \right)^2
       - 2 {\delta^2 S \over \delta \sigma \, \delta \tau}
         {\delta^2 S \over \delta \sigma \, \delta \tau'}
         {\delta^2 S \over \delta \tau \, \delta \tau'}
       + {\delta^2 S \over \delta \tau^2}
         \left( {\delta^2 S \over \delta \sigma \, \delta \tau'} \right)^2
      \right\} \cr
   &=    {\delta^2 S \over \delta \sigma^2}
         {\delta^2 S \over \delta \tau^2}
       - \left( {\delta^2 S \over \delta \sigma \, \delta \tau} \right)^2,
   &{\rm (Y^{17})}\cr}$$
in which, as before,
$S = T - z \upsilon + z' \upsilon'$.

When the final medium is variable, we may employ the following equations,
$$\left. \eqalign{
      {\displaystyle
         \, {\delta^2 W \over \delta \sigma^2} \,
      \over \displaystyle
         \, \left( {\delta \Omega \over \delta \sigma} \right)^2 \,}
   &= {\displaystyle
         \, {\delta^2 W \over \delta \tau^2} \,
      \over \displaystyle
         \, \left( {\delta \Omega \over \delta \tau} \right)^2 \,}
    = {\displaystyle
         \, {\delta^2 W \over \delta \upsilon^2} \,
      \over \displaystyle
         \, \left( {\delta \Omega \over \delta \upsilon} \right)^2 \,}
    = {\displaystyle
         \, {\delta^2 W \over \delta \sigma \, \delta \tau} \,
      \over \displaystyle
         \, {\delta \Omega \over \delta \sigma}
            {\delta \Omega \over \delta \tau} \,}
    = {\displaystyle
         \, {\delta^2 W \over \delta \tau \, \delta \upsilon} \,
      \over \displaystyle
         \, {\delta \Omega \over \delta \tau}
            {\delta \Omega \over \delta \upsilon} \,}
    = {\displaystyle
         \, {\delta^2 W \over \delta \upsilon \, \delta \sigma} \,
      \over \displaystyle
         \, {\delta \Omega \over \delta \upsilon}
            {\delta \Omega \over \delta \sigma} \,} \cr
   &= - \left(
            {\delta \Omega \over \delta \sigma}
            {\delta \Omega \over \delta x}
          + {\delta \Omega \over \delta \tau}
            {\delta \Omega \over \delta y}
          + {\delta \Omega \over \delta \upsilon}
            {\delta \Omega \over \delta z}
      \right)^{-1},\cr
\hbox{or,} \quad
      {1 \over \alpha^2}
               {\delta^2 W \over \delta \sigma^2}
   &= {1 \over \beta^2}
               {\delta^2 W \over \delta \tau^2}
    = {1 \over \gamma^2}
               {\delta^2 W \over \delta \upsilon^2}
    = {1 \over \alpha \beta}
               {\delta^2 W \over \delta \sigma \, \delta \tau}
    = {1 \over \beta \gamma}
               {\delta^2 W \over \delta \tau \, \delta \upsilon}
    = {1 \over \gamma \alpha}
               {\delta^2 W \over \delta \upsilon \, \delta \sigma} \cr
   &= \left(
            \alpha {\delta v \over \delta x}
          + \beta  {\delta v \over \delta y}
          + \gamma {\delta v \over \delta z}
      \right)^{-1},\cr}
   \right\}
   \eqno {\rm (Z^{17})}$$
of which only three are distinct, but which are sufficient to
determine the {\it principal foci and principal rays of a curved
system, ordinary or extraordinary}, by the auxiliary function
$W$, considered as depending on
$\sigma$,~$\tau$,~$\upsilon$, $x'$,~$y'$,~$z'$,~$\chi$,
in confirmity to the new view of that function, proposed in the
present Supplement.  The new function $T$ might also be employed
for the same purpose, but with somewhat less facility.

It was remarked, in a former number, that at a point of vergency
the general linear expressions for the relations of near rays
fail; but the more complex expressions by which these linear
forms must be replaced at a principal focus or other point of
vergency, and generally when it is proposed to determine the
aberrational corrections of the first approximate or limiting
relations, can always be obtained without difficulty by
developing to the required order of accuracy the general and
rigorous equations which we have given for a luminous path.  An
example of such deduction will occur, when we come to consider the
theory of {\it instruments of revolution}, which on account of
its extent and importance must be reserved for a future occasion.

\bigbreak

{\sectiontitle
Combination of the foregoing View of Optics with the Undulatory Theory
of Light.  The quantities $\sigma$,~$\tau$,~$\upsilon$, or
$\displaystyle {\delta V \over \delta x}$,
$\displaystyle {\delta V \over \delta y}$,
$\displaystyle {\delta V \over \delta z}$,
that is, the Partial Differential Coefficients of the First Order of
the Characteristic Function~$V$, taken with respect to the Final
Co-ordinates, are, in the Undulatory Theory of Light, the Components of
Normal Slowness of Propagation of a Wave.  The Fundamental Formula
{\largerm (A)} may easily be explained and proved by the principles of
the same theory.\par}

\nobreak\bigskip

26.
It remains, for the execution of the design announced at
the beginning of this Supplement, to illustrate the mathematical view
of optics proposed in this and in former memoirs, by connecting it
more closely with the undulatory theory of light.

For this purpose we shall begin by examining the undulatory meanings
of the symbols $\sigma$,~$\tau$,~$\upsilon$, of which, in the present
Supplement, we have made so frequent a use, and which we have defined
by the equations (E),
$$\sigma  = {\delta V \over \delta x},\quad
 \tau     = {\delta V \over \delta y},\quad
 \upsilon = {\delta V \over \delta z},$$
$V$ being the undulatory time of propagation of light of some given
colour, from some origin $x'$,~$y'$,~$z'$, to a point $x$,~$y$,~$z$,
through any combination of media.  It is evident that these quantities
$\sigma$,~$\tau$,~$\upsilon$ are proportional to the direction-cosines
of the normal to the wave for which the time~$V$ is constant, and
which has for its differential equation
$$0 = \delta V = \sigma \,\delta x + \tau \,\delta y
     + \upsilon \,\delta z;
   \eqno {\rm (A^{18})}$$
and if, as in the second number, we denote
$(\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$ by $\omega$, these
direction-cosines themselves will be $\sigma \omega$, $\tau \omega$,
$\upsilon \omega$; and $\omega$ will be the {\it normal velocity},
because the infinitesimal time $\delta V$, during which the wave
propagates itself in the direction of its own normal through the
infinitesimal line $\delta l$, from the point $x$,~$y$,~$z$, to the
point
$x + \sigma   \omega \mathbin{.} \delta l$,
$y + \tau     \omega \mathbin{.} \delta l$,
$z + \upsilon \omega \mathbin{.} \delta l$, is
$$\delta V
   = \sigma   \mathbin{.} \sigma   \omega \mathbin{.} \delta l
   + \tau     \mathbin{.} \tau     \omega \mathbin{.} \delta l
   + \upsilon \mathbin{.} \upsilon \omega \mathbin{.} \delta l
   = {1 \over \omega} \,\delta l:
   \eqno {\rm (B^{18})}$$
we may therefore call the quantities $\sigma$,~$\tau$,~$\upsilon$,
{\it the components of normal slowness}, because they are equal to the
reciprocal of the normal velocity, that is, to {\it the normal
slowness, multiplied respectively by the direction-cosines of the
normal}, that is, by the cosines of the angles which it makes with the
rectangular axes of co-ordinates.

Such then may be said to be the optical meaning of our quantities
$\sigma$,~$\tau$,~$\upsilon$, in the theory of the propagation of light
by waves.  And we might easily deduce from this meaning, and from the
first principles of the undulatory theory, the general expression
(A) for the variation of the characteristic function~$V$,
which has been proposed in the present and former memoirs, as
fundamental in mathematical optics.  For it is an immediate
consequence of the dynamical ideas of the undulatory theory of light,
that for a plane wave of a given direction and colour, in a given
uniform medium, the normal velocity of propagation is determined, or
at least restricted to a finite variety of values; so that this normal
velocity may be considered as a function of its cosines of direction,
involving also the colour, and depending for its form on the nature of
the uniform medium, and on the positions of the axes of co-ordinates,
to which the angles of direction are referred: and if the medium be
variable instead of uniform, and the wave curved instead of plane, we
must suppose that the normal velocity~$\omega$ is still a function of
its direction-cosines
$\sigma   (\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$,
$\tau     (\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$,
$\upsilon (\sigma^2 + \tau^2 + \upsilon^2)^{-{1 \over 2}}$,
and of the colour $\chi$, involving also, in this more general case,
the co-ordinates $x$,~$y$,~$z$.  In this manner we are conducted, by
the principles of the undulatory theory, to a relation between
$\sigma$,~$\tau$,~$\upsilon$, $x$,~$y$,~$z$,~$\chi$, of the kind
already often employed in the present Supplement, namely,
$$0 = \Omega
   = (\sigma^2 + \tau^2 + \upsilon^2)^{1 \over 2} \omega - 1,
   \eqno {\rm (M)}$$
$\Omega + 1$ being a homogeneous function of
$\sigma$,~$\tau$,~$\upsilon$, of the first dimension, which
satisfies therefore the condition
$$\sigma      {\delta \Omega \over \delta \sigma}
   + \tau     {\delta \Omega \over \delta \tau}
   + \upsilon {\delta \Omega \over \delta \upsilon}
   = \Omega + 1,$$
and which involves also in general the co-ordinates $x$,~$y$,~$z$, and
the colour $\chi$, and depends for its form on the optical properties
of the medium in which the point $x$~$y$~$z$ is placed.  {\it To
connect now}, for any given point and colour, the {\it velocity and
direction of the ray with the direction of the normal of the wave}, we
may suppose, at first, that the medium is uniform, and that the wave
is plane.  The two positions of this plane wave, at the time~$V$, and
at the time $V + \Delta V$, may be denoted by the equations
$$\left.
\vcenter{\halign{#\hfil\enspace&&$#$\hfil\hskip2pt\cr
   First  &\sigma x
          &\mathord{} + \tau y
          &\mathord{} + \upsilon z
      &= V + W,\cr
   Second &\sigma \Delta x
          &\mathord{} + \tau \Delta y
          &\mathord{} + \upsilon \Delta z
      &= \Delta V,\cr}} \right\}
   \eqno {\rm (C^{18})}$$
in which $\sigma$,~$\tau$,~$\upsilon$, $W$, are constants; and by the
principles of the same undulatory theory, if the point $x + \Delta x$,
$y + \Delta y$, $z + \Delta z$, on the second plane wave,
corresponding to the time $V + \Delta V$, be upon the ray that passes
through the point $x$~$y$~$z$ of the first plane wave, it will be
also on all the other infinitely near plane waves which correspond to
the same time $V + \Delta V$, these other waves having passed through
the point $x$~$y$~$z$ at the time $V$, and having made infinitely
small angles with the first plane wave; we are therefore to find the
co-ordinates $x + \Delta x$, $y + \Delta y$, $z + \Delta z$, of the
second point upon the ray, by seeking the intersection of the second
wave (C${}^{18}$) with all those other waves which are obtained
from it by assigning to $\sigma$,~$\tau$,~$\upsilon$, any infinitely
small variations consistent with the relation
$$0 = \delta \Omega
    = {\delta \Omega \over \delta \sigma} \delta \sigma
       + {\delta \Omega \over \delta \tau} \delta \tau
       + {\delta \Omega \over \delta \upsilon} \delta \upsilon;$$
and thus we find
$${\alpha \over v} = {\Delta x \over \Delta V}
                   = {\delta \Omega \over \delta \sigma},\quad
  {\beta  \over v} = {\Delta y \over \Delta V}
                   = {\delta \Omega \over \delta \tau},\quad
  {\gamma \over v} = {\Delta z \over \Delta V}
                   = {\delta \Omega \over \delta \upsilon},
   \eqno {\rm (D^{18})}$$
as in the second number of this Supplement, and therefore
$$\eqalign{
v        &= \alpha \sigma + \beta \tau + \gamma \upsilon,\cr
0        &= \alpha \,\delta \sigma + \beta \,\delta \tau
               + \gamma \,\delta \upsilon,\cr
\delta v &= \sigma \,\delta \alpha + \tau \,\delta \beta
               + \upsilon \,\delta \gamma,\cr}$$
and finally
$${\delta v \over \delta \alpha} = \sigma,\quad
  {\delta v \over \delta \beta}  = \tau,\quad
  {\delta v \over \delta \gamma} = \upsilon,
   \eqno {\rm (E^{18})}$$
if we denote by $v$ the reciprocal of the undulatory velocity with
which the light is propagated along the ray, and by
$\alpha$,~$\beta$,~$\gamma$, the cosines of the angles which the
ray makes with the axes of co-ordinates.  We see, therefore, by the
foregoing reasoning, which it is easy to extend to the case of
curved waves and of variable media, that {\it the components
$\sigma$,~$\tau$,~$\upsilon$, of normal slowness of a wave, or the
partial differential coefficients of the first order of the
time-function~$V$, are equal to the partial differential coefficients
of the first order,
$\displaystyle{\delta v \over \delta \alpha}$,
$\displaystyle{\delta v \over \delta \beta}$,
$\displaystyle{\delta v \over \delta \gamma}$,
of the undulatory slowness $V$ of propagation along the ray, when this
latter slowness is expressed as a homogeneous function of the first
dimension of the direction-cosines $\alpha$~$\beta$~$\gamma$ of the
ray\/}: which is the general theorem of mathematical optics, expressed
by our fundamental formula (A).

That general theorem does not appear to have been perceived by other
writers; nor do they seem to have distinctly thought of the components
of normal slowness, nor of the function of which these components are
partial differential coefficients, that is, the time~$V$ of
propagation of light from one variable point to another, through any
combination of uniform or variable media, considered as depending on
the final and initial co-ordinates and on the colour: much less do
those who have hitherto written upon light, appear to have thought of
{\it the time-function~$V$ as a} {\sc characteristic function},
{\it to the study of which may be reduced all the problems of
mathematical optics}.  But the problem of connecting by general
equations the direction and velocity of a ray with the direction and
with the law of normal velocity of a wave, has been elegantly resolved
by {\sc M.~Cauchy}, in the 50th {\it Livraison\/} of the {\it Exercices de
Math\'{e}matiques\/}: and the formul{\ae} which have been there deduced
by considering the normal velocity as a homogeneous function of the
first dimension of its three cosines of direction, may easily be shown
to agree with the equations (D${}^{18}$).

\bigbreak

{\sectiontitle
Theory of {\sc Fresnel}.  New Formul{\ae}, founded on that
theory, for the Velocities and Polarisations of a Plane Wave or
Wave-Element.  New method of deducing the Equation of
{\sc Fresnel's} Curved Wave, propagated from a Point in a
Uniform Medium with Three Unequal Elasticities.  Lines of Single
Ray-Velocity, and of Single Normal-Velocity, discovered by
{\sc Fresnel}.\par}

\nobreak\bigskip

27.
Let us now consider more particularly the undulatory theory
of {\sc Fresnel}.

In that theory, the small displacements of the vibrating etherial
points are confined to the surface of the wave, the ether being
supposed to be sensibly incompressible, and so to resist and prevent
any sensible normal vibration: and the tangential forces, which
regulate the tangential or transversal vibrations, result in general
from the elasticity of the ether, combined with this normal
resistance.  It is also supposed that the etherial medium has in
general three principal unequal elasticities, corresponding to
displacements in the directions of three rectangular {\it axes of
elasticity}; in such a manner that if we take these for the axes of
co-ordinates, any small component displacements $\delta x$, $\delta y$,
$\delta z$ parallel to these three axes will produce elastic forces
$-a^2\,\delta x$, $-b^2 \delta y$, $-c^2 \,\delta z$ parallel to the
same axes, and equal to the displacements taken with contrary signs
and multiplied by certain constant positive factors $a^2$, $b^2$,
$c^2$; and any small resultant displacement, $\delta l$, in any other
direction, having $\delta x$, $\delta y$, $\delta z$ for its
components or projections, will produce a corresponding elastic force
$-E\,\delta l$, of which the components are $-a^2\,\delta x$,
$-b^2\,\delta y$, $-c^2\,\delta z$, and which has not in general the
same direction as the displacement $\delta l$, nor a direction exactly
opposite to that.  Light, polarised in any plane, $P$, is supposed to
correspond to vibrations perpendicular to that plane, and propagated
without change of direction; and in order that a vibration should thus
preserve its direction unchanged, while the plane wave or wave-element
to which it belongs is propagated through the uniform medium with a
normal velocity $\omega$, it is necessary and sufficient that the
elastic force $-E\,\delta l$, when combined with a normal resistance
arising from the incompressibility of the ether, should produce a
tangential force $- \omega^2 \,\delta l$, in the direction opposite to
the displacement $\delta l$, and equal to this displacement taken with
a contrary sign, and multiplied by the square of the normal velocity
of propagation, so that its components are $-\omega^2 \,\delta x$,
$-\omega^2 \,\delta y$, $-\omega^2 \,\delta z$: that is, we must have
the equations
$${1 \over \sigma}      (\omega^2 - a^2) \delta x
   = {1 \over \tau}     (\omega^2 - b^2) \delta y
   = {1 \over \upsilon} (\omega^2 - c^2) \delta z,
   \eqno {\rm (F^{18})}$$
in which $\sigma$,~$\tau$,~$\upsilon$, are, as before, the components
of normal slowness, so that the equation of the wave-element
containing the transversal vibration is
$$\sigma \, \delta x + \tau \, \delta y + \upsilon \, \delta z
   = 0.
   \eqno {\rm (A^{18})}$$
{\it These equations\/ {\rm (A${}^{18}$) (F${}^{18}$)} suffice
in general to determine, on {\sc Fresnel's} principles, the velocities
of propagation and the planes of polarisation for any given wave-element
in any known crystallised medium.}

Thus, eliminating the components of displacement $\delta x$,
$\delta y$, $\delta z$, between the equations (A${}^{18}$)
(F${}^{18}$), we find the following {\it law of the normal
velocity\/}~$\omega$, considered as depending on the normal direction,
that is, on the ratios of $\sigma$,~$\tau$,~$\upsilon$,
$${\sigma^2 \over \omega^2 - a^2}
  + {\tau^2 \over \omega^2 - b^2}
  + {\upsilon^2 \over \omega^2 - c^2}
  = 0.
   \eqno {\rm (G^{18})}$$
To deduce hence the {\it direction and velocity of a ray, for any
given normal direction and normal velocity}, compatible with the
foregoing law, that is, for any given values of the components of
normal slowness $\sigma$,~$\tau$,~$\upsilon$, compatible with the
relation (G${}^{18}$), we are to make, by (M),
$$\omega^2 = {(\Omega + 1)^2 \over \sigma^2 + \tau^2 + \upsilon^2},
   \eqno {\rm (H^{18})}$$
and we then find, by (I), or by (D${}^{18}$), the
following expressions for the components of the velocity of the ray,
$$\left. \eqalign{
{\alpha \over v}
   &= {\delta \Omega \over \delta \sigma}
      = {\sigma \omega^2 \over \Omega + 1}\,{\lambda^2 - a^2 \over
           \omega^2 - a^2},\cr
{\beta \over v}
   &= {\delta \Omega \over \delta \tau}
      = {\tau \omega^2 \over \Omega + 1}\,{\lambda^2 - b^2 \over
           \omega^2 - b^2},\cr
{\gamma \over v}
   &= {\delta \Omega \over \delta \upsilon}
      = {\upsilon \omega^2 \over \Omega + 1}\,{\lambda^2 - c^2 \over
           \omega^2 - c^2},\cr} \right\}
   \eqno {\rm (I^{18})}$$
if we put for abridgment
$$\lambda^2 = {\displaystyle
                 \left({a^2 \sigma   \over \omega^2 - a^2} \right)^2
              +  \left({b^2 \tau     \over \omega^2 - b^2} \right)^2
              +  \left({c^2 \upsilon \over \omega^2 - c^2} \right)^2
   \over \displaystyle
                 \left({a   \sigma   \over \omega^2 - a^2} \right)^2
              +  \left({b   \tau     \over \omega^2 - b^2} \right)^2
              +  \left({c   \upsilon \over \omega^2 - c^2} \right)^2}.
   \eqno {\rm (K^{18})}$$
And to deduce the {\it law of the velocity $\displaystyle {1 \over v}$
of the ray, considered as depending on its own direction}, that is, on
the cosines $\alpha$~$\beta$~$\gamma$ of its inclinations to the
semiaxes $a$~$b$~$c$ of elasticity, we are to eliminate (according
to the general method of the second number) the ratios of
$\sigma$~$\tau$~$\upsilon$ between the three expressions (I${}^{18}$),
and so deduce the relations between the three components of velocity
$\displaystyle {\alpha \over v}$, $\displaystyle {\beta \over v}$,
$\displaystyle {\gamma \over v}$; now the equations (I${}^{18}$)
given evidently, by (K${}^{18}$),
$${a^2 \alpha^2 \over \lambda^2 - a^2}
   + {b^2 \beta^2 \over \lambda^2 - b^2}
   + {c^2 \gamma^2 \over \lambda^2 - c^2}
   = 0;
   \eqno {\rm (L^{18})}$$
they give also, when we attend to (G${}^{18}$),
$$\biggl( {\alpha \over v} \biggr)^2 + \biggl( {\beta \over v} \biggr)^2
     + \biggl( {\gamma \over v} \biggr)^2
   = \lambda^2:
   \eqno {\rm (M^{18})}$$
$\lambda$ therefore is the velocity of the ray, or the radius vector
of the curved {\it unit-wave}, propagated in all directions from the
origin of co-ordinates during the unit of time; and the {\it equation
of the wave\/} in rectangular co-ordinates $x$~$y$~$z$, parallel to
the axes of elasticity, is
$${a^2 x^2 \over x^2 + y^2 + z^2 - a^2}
      + {b^2 y^2 \over x^2 + y^2 + z^2 - b^2}
      + {b^2 z^2 \over x^2 + y^2 + z^2 - c^2}
   = 0,
   \eqno {\rm (N^{18})}$$
or, when freed from fractions,
$$(x^2 + y^2 + z^2)(a^2 x^2 + b^2 y^2 + c^2 z^2) + a^2 b^2 c^2
   = a^2 (b^2 + c^2) x^2 + b^2 (c^2 + a^2) y^2 + c^2 (a^2 + b^2) z^2.
   \eqno {\rm (O^{18})}$$
This method of determining the equation of {\sc Fresnel's} {\it Wave},
will perhaps be thought simpler than that which was employed by the
illustrious discover, and than others which have since been proposed.

Reciprocally to determine by our general methods the normal direction
and velocity, or the components of normal slowness
$\sigma$,~$\tau$,~$\upsilon$, for any proposed direction and velocity
of a ray compatible with this form of the wave, that is, for any values of
$\alpha$~$\beta$~$\gamma$~$\lambda$ compatible with the relation
(L${}^{18}$), we are to substitute for the ray-velocity~$\lambda$
in that relation its value (M${}^{18}$), and we find, by
(E${}^{18}$),
$$\left. \eqalign{
   \sigma   &= {\delta v \over \delta \alpha}
                = {\alpha \over v} \mathbin{.} {1 - a^2 \nu^2 \over
                   \lambda^2 - a^2},\cr
   \tau     &= {\delta v \over \delta \beta}
                = {\beta \over v} \mathbin{.} {1 - b^2 \nu^2 \over
                   \lambda^2 - b^2},\cr
   \upsilon &= {\delta v \over \delta \gamma}
                = {\gamma \over v} \mathbin{.} {1 - c^2 \nu^2 \over
                   \lambda^2 - c^2},\cr} \right\}
   \eqno {\rm (P^{18})}$$
if we put for abridgment
$$\nu^2 = {\displaystyle
                 \left({  \alpha \over \lambda^2 - a^2} \right)^2
              +  \left({  \beta  \over \lambda^2 - b^2} \right)^2
              +  \left({  \gamma \over \lambda^2 - c^2} \right)^2
   \over \displaystyle
                 \left({a \alpha \over \lambda^2 - a^2} \right)^2
              +  \left({b \beta  \over \lambda^2 - b^2} \right)^2
              +  \left({c \gamma \over \lambda^2 - c^2} \right)^2}.
   \eqno {\rm (Q^{18})}$$
It is easy to see that the value of $\nu$ thus determined is the
normal slowness, or reciprocal of $\omega$, because the expressions
(P${}^{18}$) give, by (L${}^{18}$),
$$\sigma^2 + \tau^2 + \upsilon^2 = \nu^2;
   \eqno {\rm (R^{18})}$$
and since the same expressions given also evidently, by
(Q${}^{18}$),
$$      {\sigma^2   \over 1 - a^2 \nu^2}
      + {\tau^2     \over 1 - b^2 \nu^2}
      + {\upsilon^2 \over 1 - c^2 \nu^2}
   = 0,
   \eqno {\rm (S^{18})}$$
we easily deduce the law (G${}^{18}$) of dependence of the normal
velocity on the normal direction, from the form of {\sc Fresnel's} wave,
as we had deduced the latter from the former.

The equations (L${}^{18}$) (M${}^{18}$) which gave us the
equation of the wave in rectangular co-ordinates, give also the
following polar equation for the reciprocal of its radius-vector, that
is, for the slowness $v$ of the ray,
$$\eqalign{0
   &= v^4 - v^2 \{ \alpha^2 (b^{-2} + c^{-2}) 
                 + \beta^2  (c^{-2} + a^{-2}) 
                 + \gamma^2 (a^{-2} + b^{-2}) \} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + (\alpha^2 + \beta^2 + \gamma^2) (
      \alpha^2 b^{-2} c^{-2} + \beta^2 c^{-2} a^{-2}
         + \gamma^2 a^{-2} b^{-2}),}
   \eqno {\rm (T^{18})}$$
and therefore the following double expression for the square of this
slowness,
$$\eqalign{v^2
   &= {\textstyle {1 \over 2}} (c^{-2} + a^{-2})
            (\alpha^2 + \beta^2 + \gamma^2) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\textstyle {1 \over 2}} (c^{-2} - a^{-2})
      \{ A' A'' \pm \sqrt{\alpha^2 + \beta^2 + \gamma^2 - A'^2}
                    \sqrt{\alpha^2 + \beta^2 + \gamma^2 - A''^2} \},}
   \eqno {\rm (U^{18})}$$
if we put for abridgment
$$\left. \eqalign{
A'  &=  \alpha \sqrt{ \displaystyle
                      {b^{-2} - a^{-2} \over c^{-2} - a^{-2}} }
      + \gamma \sqrt{ \displaystyle
                      {c^{-2} - b^{-2} \over c^{-2} - a^{-2}} }, \cr
A'' &=  \alpha \sqrt{ \displaystyle
                      {b^{-2} - a^{-2} \over c^{-2} - a^{-2}} }
      - \gamma \sqrt{ \displaystyle
                      {c^{-2} - b^{-2} \over c^{-2} - a^{-2}} }: \cr}
  \right\}
   \eqno {\rm (V^{18})}$$
supposing therefore $a^2 > b^2 > c^2$, the {\it polar equation of the
wave\/} may be put under the form
$$\rho^{-2} = {\textstyle {1 \over 2}} (c^{-2} + a^{-2})
            + {\textstyle {1 \over 2}} (c^{-2} - a^{-2})
              \cos ((\rho \rho') \pm (\rho \rho'')),
   \eqno {\rm (W^{18})}$$
$\rho$ being the radius-vector or velocity and $(\rho \rho')$
$(\rho \rho'')$ being the angles which this radius $\rho$ makes with
two constant radii $\rho'$,~$\rho''$, determined by the following
cosines of their inclinations to the semiaxes of $x$~$y$~$z$, or of
$a$~$b$~$c$,
$$\rho_a' = \rho_a''
     = \sqrt{ \displaystyle
              {b^{-2} - a^{-2} \over c^{-2} - a^{-2}} },\quad
  \rho_b' = \rho_b'' = 0,\quad
  \rho_c' = - \rho_c''
     = \sqrt{ \displaystyle
              {c^{-2} - b^{-2} \over c^{-2} - a^{-2}} }.
   \eqno {\rm (X^{18})}$$
The expression (W${}^{18}$), for the reciprocal of the square of
the velocity of a ray, has been assigned by {\sc Fresnel}, who has
also remarked that it gives always two unequal velocities unless the
direction $\rho$ of the ray coincide with some one of the four
directions $\pm \rho'$,~$\pm \rho''$, which are opposite two by two, and
situated in the plane~$ac$ of the extreme axes of elasticity.
{\sc Fresnel} has shown in like manner that any given normal
direction corresponds to two unequal normal velocities, except four
particular directions, which we may call $\pm \omega'$,
$\pm \omega''$, and which are determined by the following cosines
of direction,
$$\omega_a' = - \omega_a''
     = \sqrt{ \displaystyle
              {a^2 - b^2 \over a^2 - c^2} },\quad
  \omega_b' = \omega_b'' = 0,\quad
  \omega_c' = \omega_c''
     = \sqrt{ \displaystyle
              {b^2 - c^2 \over a^2 - c^2} }:
   \eqno {\rm (Y^{18})}$$
and in fact it is easy to establish the following expression for the
double value of the square of the normal velocity, analogous to the
expression (W${}^{18}$),
$$\omega^2  = {\textstyle {1 \over 2}} (a^2 + c^2)
            + {\textstyle {1 \over 2}} (a^2 - c^2)
              \cos ((\omega \omega') \pm (\omega \omega'')),
   \eqno {\rm (Z^{18})}$$
which cannot reduce itself to a single value, unless the sine of
$(\omega \omega')$ or of $(\omega \omega'')$ vanishes.  {\sc Fresnel}
has given the name of {\it optic axes} sometimes to the one and
sometimes to the other of the two sets of directions
(X${}^{18}$) (Y${}^{18}$); but to prevent the confusion which
might arise from this double use of a term, we shall, for the present,
call the set $\pm \rho'$,~$\pm \rho''$, by the longer but more
expressive name of the directions or {\it lines of single
ray-velocity\/}: and similarly we shall call the set $\pm \omega'$,
$\pm \omega''$, the directions or {\it lines of single normal
velocity}.

\bigbreak

{\sectiontitle
New Properties of {\sc Fresnel's} Wave.  This Wave has Four
Conoidal Cusps, at the Ends of the Lines of Single Ray-Velocity;
it has also Four Circles of Contact, of which each is contained on
a Touching Plane of Single Normal-Velocity.  The Lines of Single
Ray-Velocity may therefore be called Cusp-Rays; and the Lines of
Single Normal-Velocity may be called Normals of Circular Contact.\par}

\nobreak\bigskip

28.
The reasonings of the foregoing number suppose that the axes
of co-ordinates coincide with the axes of elasticity; but it is easy to
extend the result thus obtained, to any other axes of co-ordinates, by
the formul{\ae} of transformation which were given in the thirteenth
number.  We shall content ourselves at present with considering two
remarkable transformations of this kind, suggested by the two
foregoing sets of lines of single velocity, which conduct to some new
properties of {\sc Fresnel's} wave, and to some new consequences of his
theory.

The polar equation (W${}^{18}$) of the wave may be put under the
form
$$1 = {\textstyle {1 \over 2}} (c^{-2} + a^{-2}) \rho^2
      + {\textstyle {1 \over 2}} (c^{-2} - a^{-2}) \{
         r' r'' \pm \sqrt{\rho^2 - r'^2} \sqrt{\rho^2 - r''^2} \},
   \eqno {\rm (A^{19})}$$
if we put for abridgment
$$r'  = A'  \rho = x \rho_a'  + z \rho_c',\quad
  r'' = A'' \rho = x \rho_a'' + z \rho_c'',
   \eqno {\rm (B^{19})}$$
so that $r'$,~$r''$, are the projections of the radius-vector~$\rho$
on the directions $\rho'$,~$\rho''$, of single ray-velocity; and if we
take new rectangular co-ordinates, $x_\prime$~$y_\prime$~$z_\prime$,
such that the plane of $x_\prime z_\prime$ is still the plane
$ac$ of the extreme axes of elasticity, but that the positive semiaxis
of $z_\prime$ coincides with the line~$\rho'$, we may employ the
following formul{\ae} of transformation
$$x = x_\prime \rho_c' + z_\prime \rho_a',\quad
  y = y_\prime,\quad
  z = - x_\prime \rho_a' + z_\prime \rho_c',
   \eqno {\rm (C^{19})}$$
which give
$$\rho^2 = x_\prime^2 + y_\prime^2 + z_\prime^2,\quad
  r' = z_\prime,\quad
  r'' = x_\prime \sin (\rho' \rho'')
         + z_\prime \cos (\rho' \rho''),
   \eqno {\rm (D^{19})}$$
and change the equation (A${}^{19}$) of the wave to the form
$$\eqalign{1
   &= b^{-2} z_\prime^2
      + {\textstyle {1 \over 2}} z_\prime x_\prime (c^{-2} - a^{-2})
         \sin (\rho' \rho'') + {\textstyle {1 \over 2}}
         (c^{-2} + a^{-2})(x_\prime^2 + y_\prime^2) \cr
   &\mathrel{\phantom{=}} \mathord{}
      \pm {\textstyle {1 \over 2}} (c^{-2} - a^{-2})
         \sqrt{x_\prime^2 + y_\prime^2}
         \sqrt{ ( z_\prime \sin (\rho' \rho'')
                 - x_\prime \cos (\rho' \rho'') )^2
                 + y_\prime^2}.\cr}
   \eqno {\rm (E^{19})}$$
This equation enables us easily to examine the shape of the wave near
the end of the radius~$\rho'$, that is, near the point having for its
new co-ordinates
$$x_\prime = 0,\quad y_\prime = 0,\quad z_\prime = b;
   \eqno {\rm (F^{19})}$$
for it takes, near that point, the following approximate form,
$$z_\prime
   = b - {\textstyle {1 \over 2}} b^2
      \sqrt{c^{-2} - b^{-2}} \, \sqrt{b^{-2} - a^{-2}} \,
      (x_\prime \pm \sqrt{x_\prime^2 + y_\prime^2}),
   \eqno {\rm (G^{19})}$$
which shows that at the point (F${}^{19}$) {\it the wave has a
conoidal cusp}, and is touched not by one determined tangent plane but
by {\it a tangent cone of the second degree}, represented rigorously by
the equation (G${}^{19}$).  {\sc Fresnel} does not appear to have
been aware of the existence of this tangent cone to his wave; he seems to
have thought that at the end of a radius $\rho'$ of single ray-velocity
the wave was touched only by two right lines, contained in the plane
of $ac$, namely, by the tangents to a certain circle and ellipse, the
intersections of the wave with that plane: but it is evident from the
foregoing transformation that every other section of the wave, made by
a plane containing the radius-vector $\rho'$, is touched, at the end
of that radius, by two tangent lines, contained on the cone
(G${}^{19}$).  It is evident also that there are {\it four such
conoidal cusps}, at the ends of the four lines of single ray-velocity,
$\pm \rho'$, $\pm \rho''$.  They are determined by the following
co-ordinates, when referred to the axes of elasticity,
$$x = \pm c \sqrt{ {a^2 - b^2 \over a^2 - c^2} },\quad
  y = 0,\quad
  z = \pm a \sqrt{ {b^2 - c^2 \over a^2 - c^2} };
   \eqno {\rm (H^{19})}$$
and they are the four intersections of {\sc Fresnel's} circle and
ellipse, in the plane of $ac$, which have for their equations in
that plane
$$x^2 + z^2 = b^2,\quad a^2 x^2 + c^2 z^2 = a^2 c^2.
   \eqno {\rm (I^{19})}$$

Again, if we employ the following new formul{\ae} of transformation,
$$x = x_{\prime\prime} \omega_c' + z_{\prime\prime} \omega_a',\quad
  y = y_{\prime\prime},\quad
  z = - x_{\prime\prime} \omega_a' + z_{\prime\prime} \omega_c',
   \eqno {\rm (K^{19})}$$
so as to pass to a new system of rectangular co-ordinates such that the
plane of $x_{\prime\prime} z_{\prime\prime}$ coincides with the plane of
$ac$, and the positive semiaxis of $z_{\prime\prime}$ with the line
$\omega'$ of single normal velocity, we find a new transformed
equation of the wave, which may be thus written,
$$(x_{\prime\prime}^2 + y_{\prime\prime}^2
      + x_{\prime\prime} z_{\prime\prime} b^{-2}
         \sqrt{a^2 - b^2} \sqrt{b^2 - c^2})^2
   = Q(1 - z_{\prime\prime}^2 b^{-2}),
   \eqno {\rm (L^{19})}$$
if we put for abridgment
$$Q = (a^2 + c^2) \rho^2 + (a^2 - c^2) r' r''
      - a^2 c^2 (1 + z_{\prime\prime}^2 b^{-2});
   \eqno {\rm (M^{19})}$$
and hence it is easy to prove that {\it the plane}
$$z_{\prime\prime} = b,
   \eqno {\rm (N^{19})}$$
{\it which is perpendicular to the line $\omega'$ at its extremity,
touches the wave in the whole extent of a circle\/}; the equation of
this circle of contact being, in its own plane,
$$x_{\prime\prime}^2 + y_{\prime\prime}^2
      + x_{\prime\prime} b^{-1} \sqrt{a^2 - b^2} \sqrt{b^2 - c^2}
   = 0.
   \eqno {\rm (O^{19})}$$
It is evident that there are {\it four such circles of plane contact
at the ends of the four lines} $\pm \omega'$, $\pm \omega''$, {\it of
single normal-velocity}.  They are all equal to each other, and the
common magnitude of their diameters is
$b^{-1} \sqrt{a^2 - b^2} \sqrt{b^2 - c^2}$.
The same conclusions may be drawn from {\sc Fresnel's} equation of the
wave in co-ordinates $x$~$y$~$z$, referred to the axes of elasticity:
the equations of the {\it four planes of circular contact\/} being, in
these co-ordinates,
$$z \sqrt{b^2 - c^2} \pm x \sqrt{a^2 - b^2}
   = \pm b \sqrt{a^2 - c^2}.
   \eqno {\rm (P^{19})}$$
{\sc Fresnel} however does not himself appear to have suspected the
existence of these circles of contact, nor do they seem to have been
since perceived by any other person.  We shall find that the circles
and cusps, pointed out in the present number, conduct to some
remarkable theoretical conclusions respecting the laws of refraction
in biaxal crystals.

\bigbreak

{\sectiontitle
New Consequences of {\sc Fresnel's} Principles.  It follows from
those Principles, that Crystals of sufficient Biaxal Energy ought to
exhibit two kinds of Conical Refraction, an External and an Internal: a
Cusp-Ray giving an External Cone of Rays, and a Normal of Circular
Contact being connected with an Internal Cone.}

\nobreak\bigskip

29.
The general formul{\ae} for reflexion or refraction, ordinary
or extraordinary, which we have deduced from the nature of the
characteristic function~$V$, become simply
$$\Delta \sigma = 0,\quad \Delta \tau = 0,
   \eqno {\rm (Q^{19})}$$
when we take for the plane of $xy$ the tangent plane to the reflecting
or refracting surface; they show therefore that {\it the components of
normal slowness parallel to this tangent plane are not changed}, which
is a new and general form for the laws of reflexion and refraction.
It is easy to combine this general theorem with {\sc Fresnel's} law of
velocity, and so to deduce new consequences from that law with respect
to biaxal crystals.

For this deduction, our theorem may be expressed as follows,
$$0 = \Delta \left(   a_t {\delta v \over \delta \alpha}
                    + b_t {\delta v \over \delta \beta}
                    + c_t {\delta v \over \delta \gamma}
                    \right),
   \eqno {\rm (R^{19})}$$
in which $v$ is the undulatory slowness of a ray considered as a
homogeneous function of the first dimension of the cosines
$\alpha$~$\beta$~$\gamma$ of its inclinations to any three rectangular
semiaxes $a$~$b$~$c$, while $\Delta$ refers to the changes produced
by reflexion or refraction, the unaltered trinomial to which it is
prefixed being the component of normal slowness in the direction of
any line~$t$ on the tangent plane of the reflecting or refracting
surface, and $a_t$~$b_t$~$c_t$ being the cosines of the inclinations
of this line to the semiaxes $a$~$b$~$c$: and in order to combine
this theorem with the principles of {\sc Fresnel}, we have only to suppose
that the rectangular semiaxes $a$~$b$~$c$ in each medium are the
semiaxes of elasticity of that medium, and that the form of the
function~$v$ is determined as in the twenty-seventh number.

Thus, to calculate the refraction of light on entering from a vacuum
into a biaxal crystal $a$~$b$~$c$ bounded by a plane face~$F$, we
may denote by $\alpha_0$~$\beta_0$~$\gamma_0$ the cosines of the
inclinations of the internal or incident ray to two rectangular lines
$s$,~$t$ upon the face~$F$, and to the inward normal, and we shall have
the two equations following,
$$\left.  \eqalign{
\alpha_0 &=   a_s {\delta v \over \delta \alpha}
            + b_s {\delta v \over \delta \beta}
            + c_s {\delta v \over \delta \gamma}
            \enspace (= \sigma a_s + \tau b_s + \upsilon c_s),\cr
\beta_0  &=   a_t {\delta v \over \delta \alpha}
            + b_t {\delta v \over \delta \beta}
            + c_t {\delta v \over \delta \gamma}
            \enspace (= \sigma a_t + \tau b_t + \upsilon c_t),\cr}
   \right\}
   \eqno {\rm (S^{19})}$$
which contain the required connexions between
$\alpha_0$~$\beta_0$~$\gamma_0$ and $\alpha$~$\beta$~$\gamma$,
that is, between the external and internal directions.  In this
manner we find in general two incident rays for one refracted, and
two refracted for one incident; because a given system of values
of $\alpha$~$\beta$~$\gamma$, that is, a given direction of the
internal ray, corresponds in general to two systems of values of
the internal components of normal slowness
$\sigma$~$\tau$~$\upsilon$, and therefore to two systems of values
$\alpha_0$~$\beta_0$~$\gamma_0$, that is, to two external
directions; while, reciprocally, a given system of two linear
relations between $\sigma$,~$\tau$,~$\upsilon$, deduced by
${\rm (S^{19})}$ from a given external direction, corresponds in
general to two directions of the internal ray.  But there are two
remarkable exceptions, connected with the two sets of lines of
single velocity, and with the conoidal cusps and circles of contact
on {\sc Fresnel's} wave.

For we have seen that at a conoidal cusp the tangent plane to the wave
is indeterminate; it is evident therefore that a {\it cusp-ray\/} must
correspond to an infinite variety of systems of components of
normal slowness $\sigma$,~$\tau$,~$\upsilon$, within the biaxal
crystal, and therefore also to an infinite variety of systems of
direction-cosines $\alpha_0$~$\beta_0$~$\gamma_0$ of the external
ray; so that {\it this one internal cusp-ray must correspond to an
external cone of rays, according to a new theoretical law of light\/}
which may be called {\sc External Conical Refraction}.

And again, at a circle of contact, the wave has one common tangent
plane for all the points of that circle, and therefore the infinite
variety of internal rays which correspond to these different points
have all one common wave-normal, which may be called a {\it normal of
circular contact}, and all these internal rays have one common system
of components of normal slowness $\sigma$~$\tau$~$\upsilon$ within
the crystal, and consequently correspond to one common external ray:
so that {\it this one external ray is connected with an internal cone
of rays, according to another new theoretical law of light}, which may
be called {\sc Internal Conical Refraction}.

To develope, somewhat more fully, these two new consequences from
{\sc Fresnel's} principles, let us begin by considering {\it external
conical refraction\/}: and let us seek the equation of the external
cone of rays, corresponding to the internal cusp-ray $\rho'$.  The
approximate equation ${\rm (G^{19})}$ of the wave, near the end of
this cusp-ray, in the transformed co-ordinates
$x_\prime$~$y_\prime$~$z_\prime$, gives the following approximate
expression for the undulatory slowness~$v$ of a near ray, considered
as a homogeneous function of the first dimension of the cosines
$\alpha_\prime$~$\beta_\prime$~$\gamma_\prime$ of its inclinations
to the positive semiaxes of these co-ordinates
$x_\prime$~$y_\prime$~$z_\prime$,
$$v = b^{-1} \gamma_\prime + r_\prime (\alpha_\prime
         \pm \sqrt{\alpha_\prime^2 + \beta_\prime^2}),
   \eqno {\rm (T^{19})}$$
in which
$$r_\prime = {\textstyle {1 \over 2}} b
                \sqrt{c^{-2} - b^{-2}} \sqrt{b^{-2} - a^{-2}};
   \eqno {\rm (U^{19})}$$
it gives therefore by our general method, the following components of
normal slowness parallel to the same semiaxes of
$x_\prime$~$y_\prime$~$z_\prime$,
$$\left. \eqalign{
\sigma \rho_c' - \upsilon \rho_a'
   &= \sigma_\prime
      = {\delta v \over \delta \alpha_\prime}
      = r_\prime \pm {r_\prime \alpha_\prime \over
         \sqrt{\alpha_\prime^2 + \beta_\prime^2}}, \cr
\tau
   &= \tau_\prime
      = {\delta v \over \delta \beta_\prime}
      = \phantom{r_\prime} \pm {r_\prime \beta_\prime \over
         \sqrt{\alpha_\prime^2 + \beta_\prime^2}}, \cr
\sigma \rho_a' + \upsilon \rho_c'
   &= \upsilon_\prime
      = {\delta v \over \delta \gamma_\prime}
      = b^{-1},\cr} \right\}
   \eqno {\rm (V^{19})}$$
the expressions for $\sigma_\prime$~$\tau_\prime$ becoming
indefinitely more accurate as $\alpha_\prime$~$\beta_\prime$
diminish, that is, as the near internal ray approaches to the cusp-ray
$\rho'$, and the expression for $\upsilon_\prime$ being rigorous:
{\it the relations between the components of normal slowness
$\sigma$~$\tau$~$\upsilon$ of the cusp-ray\/}~$\rho'$ are therefore
$$(\sigma \rho_c' - \upsilon \rho_a')^2 + \tau^2
   = 2r_\prime (\sigma \rho_c' - \upsilon \rho_a'),\quad
  \sigma \rho_a' + \upsilon \rho_c' = b^{-1},
   \eqno {\rm (W^{19})}$$
and {\it the equation (in $\alpha_0$ $\beta_0$) of the external cone
of rays corresponding to the one internal cusp-ray $\rho'$ is to be
found by eliminating these three internal components
$\sigma$~$\tau$~$\upsilon$ between the two relations\/} (W${}^{19}$)
{\it and the two equations of refraction\/}~(S${}^{19}$).

For example, if the internal cusp-ray $\rho'$ coincide with the inward
normal to the refracting face~$F$ of the crystal, we may take, for the
semiaxes $s$,~$t$ upon that face, the projection of $a$, and the
semiaxis $b$ of elasticity; and then the equations of refraction
(S${}^{19}$) becoming
$$\alpha_0 = \sigma \rho_c' - \upsilon \rho_a',\quad
   \beta_0 = \tau,
   \eqno {\rm (X^{19})}$$
we have, by (W${}^{19}$), the following polar equation of the
external cone of rays,
$$\alpha_0^2 + \beta_0^2 = 2 r_\prime \alpha_0;
   \eqno {\rm (Y^{19})}$$
or, in rectangular co-ordinates, an equation of the fourth degree,
$$(x_0^2 + y_0^2)^2 = 4 r_\prime^2 x_0^2 ( x_0^2 + y_0^2 + z_0^2).
   \eqno {\rm (Z^{19})}$$
This cone is nearly circular in all the known biaxal crystals, because
the coefficient $r_\prime$ is small, by (U${}^{19}$), when the
biaxal energy is weak, that is, when the semiaxes of elasticity
$a$~$b$~$c$ are nearly equal to each other: and rigorously the external
cone (Z${}^{19}$) meets the concentric sphere of radius unity in a
curve contained on a circular cylinder of radius $= r_\prime$, one
side of this cylinder coinciding with a ray of the cone.

With respect to the {\it internal conical refraction}, the {\it
equation of the internal cone of rays corresponding to the internal
wave-normal\/}~$\omega'$, or {\it normal of circular contact}, is
always, by (N${}^{19}$) (O${}^{19}$),
$$x_{\prime\prime}^2 + y_{\prime\prime}^2
       + 2 r_{\prime\prime} x_{\prime\prime} z_{\prime\prime}
     = 0, \hbox{ if }
  r_{\prime\prime} = {\textstyle {1 \over 2}} b^{-2}
         \sqrt{a^2 - b^2} \sqrt{b^2 - c^2},
   \eqno {\rm (A^{20})}$$
when referred to the rectangular co-ordinates
$x_{\prime\prime}$~$y_{\prime\prime}$~$z_{\prime\prime}$ by the
transformation (K${}^{19}$); and in the simpler rectangular
co-ordinates $x$~$y$~$z$ which are parallel to the axes of elasticity
the equation of this cone is
$$(x \omega_c' - z \omega_a')^2 + y^2 + 2r_{\prime\prime}
      (x \omega_c' - z \omega_a')(x \omega_a' + z \omega_c')
   = 0,
   \eqno {\rm (B^{20})}$$
in which we may change the co-ordinates $x$~$y$~$z$ to the
direction-cosines $\alpha$~$\beta$~$\gamma$ of an internal ray of
the cone: while the one external ray corresponding is determined by
the following direction-cosines
$$\alpha_0 = b^{-1} \omega_s',\quad
  \beta_0  = b^{-1} \omega_t';
   \eqno {\rm (C^{20})}$$
or by the ordinary law of proportional sines, since the internal
wave-normal of circular contact~$\omega'$, which is one ray of the
internal cone, is connected with the external ray by this ordinary
law, if we take as the refracting index of the crystal the reciprocal
$b^{-1}$ of the mean semiaxis of elasticity.  It is evident hence that
if the internal cone emerge at a new plane face, it will {\it emerge
a cylinder}, whether the two faces be parallel or inclined, that is,
whether the crystal be a plate or a prism.

\bigbreak

{\sectiontitle
Theory of Conical Polarisation.  Lines of Vibration.  These Lines, on
{\sc Fresnel's} Wave, are the Intersections of Two Series of
Concentric and Co-axal Ellipsoids.\par}

\nobreak\bigskip

30.
A given direction of a wave-normal in a biaxal crystal
corresponds in general to two directions of vibration, and therefore
to two planes of polarisation, determined by the equations
(F${}^{18}$), namely one for each of the two values $\omega_1^2$,
$\omega_2^2$ of the square of the normal velocity deduced by
(G${}^{18}$) from the given system of ratios of
$\sigma$,~$\tau$,~$\upsilon$; and these two directions of vibration,
or the two planes of polarisation, that is, the two normal planes of
the wave perpendicular to these vibrations, are perpendicular to each
other, since we can easily deduce from (G${}^{18}$) the following
relation between $\omega_1^2$,~$\omega_2^2$,
$${\sigma^2       \over (\omega_1^2 - a^2)(\omega_2^2 - a^2)}
    + {\tau^2     \over (\omega_1^2 - b^2)(\omega_2^2 - b^2)}
    + {\upsilon^2 \over (\omega_1^2 - c^2)(\omega_2^2 - c^2)}
  = 0:
   \eqno {\rm (D^{20})}$$
which general rectangularity of the two vibrations on any one plane
wave has been otherwise established by {\sc Fresnel}, and is an
important result of his theory.  But besides this general {\it double
polarisation\/} connected with the general {\it double refraction\/}
in biaxal crystals, we may consider two other kinds which may be
called {\it conical polarisation}, connected with the two kinds of
{\it conical refraction}, which were pointed out in the foregoing
number.

To examine the law of the conical polarisation connected with the
internal conical refraction, and therefore with the planes of circular
contact, we may employ the co-ordinates
$x_{\prime\prime}$~$y_{\prime\prime}$~$z_{\prime\prime}$ defined by
(K${}^{19}$), and thus transform the general equations of polarisation
(A${}^{18}$) (F${}^{18}$) into the following equally general,
$$\left. \eqalign{
{\omega_c' \,\delta x_{\prime\prime}
         + \omega_a' \,\delta z_{\prime\prime}
      \over \omega_c' \sigma_{\prime\prime}
         + \omega_a' \upsilon_{\prime\prime}}
      (\omega^2 - a^2)
   = {\delta y_{\prime\prime} \over \tau_{\prime\prime}}
      (\omega^2 - b^2)
  &= {-\omega_a' \,\delta x_{\prime\prime}
         + \omega_c' \,\delta z_{\prime\prime}
      \over -\omega_a' \sigma_{\prime\prime}
         + \omega_c' \upsilon_{\prime\prime}}
      (\omega^2 - c^2),\cr
\sigma_{\prime\prime}         \,\delta x_{\prime\prime}
    + \tau_{\prime\prime}     \,\delta y_{\prime\prime}
    + \upsilon_{\prime\prime} \,\delta z_{\prime\prime}
   &= 0;\cr}
   \right\}
   \eqno {\rm (E^{20})}$$
which give, for the projection of a vibration on the plane
$x_{\prime\prime} y_{\prime\prime}$ of single normal velocity, the
rigorous formula
$${\delta y_{\prime\prime} \over \delta x_{\prime\prime}}
   = {(\omega^2 - a^2)(\omega^2 - c^2) \over \omega^2 - b^2}
      \,{\tau_{\prime\prime} \over \upsilon_{\prime\prime}
            \sqrt{a^2 - b^2} \sqrt{b^2 - c^2}
         + \sigma_{\prime\prime} (\omega^2 + b^2 - a^2 - c^2)},
   \eqno {\rm (F^{20})}$$
and for any plane wave slightly inclined to this plane of
$x_{\prime\prime} y_{\prime\prime}$ the following approximate relation
between the components of normal slowness,
$$\upsilon_{\prime\prime}
   = b^{-1} + r_{\prime\prime} (\sigma_{\prime\prime}
      \pm \sqrt{\sigma_{\prime\prime}^2 + \tau_{\prime\prime}^2}),
   \eqno {\rm (G^{20})}$$
retaining the meaning (A${}^{20}$) of $r_{\prime\prime}$; and if
we attend to the general connexions, established in this Supplement,
between the direction-cosines of a ray and the components of normal
slowness of a wave, we easily deduce from (G${}^{20}$), by
differentiation, the following other relations,
$${\alpha_{\prime\prime} \over \gamma_{\prime\prime}}
   = - {\delta \upsilon_{\prime\prime}
         \over \delta \sigma_{\prime\prime}}
   = - r_{\prime\prime} \left(
      1 \pm {\sigma_{\prime\prime} \over
         \sqrt{\sigma_{\prime\prime}^2 + \tau_{\prime\prime}^2}}
      \right),\quad
{\beta_{\prime\prime} \over \gamma_{\prime\prime}}
   = - {\delta \upsilon_{\prime\prime}
         \over \delta \tau_{\prime\prime}}
   = {\mp r_{\prime\prime} \tau_{\prime\prime} \over
         \sqrt{\sigma_{\prime\prime}^2 + \tau_{\prime\prime}^2}};
   \eqno {\rm (H^{20})}$$
and finally for the vibrations of a near wave
$${\delta y_{\prime\prime} \over \delta x_{\prime\prime}}
   = {\tau_{\prime\prime} \over \sigma_{\prime\prime}
      \pm \sqrt{\sigma_{\prime\prime}^2 + \tau_{\prime\prime}^2}}
   = {\beta_{\prime\prime} \over \alpha_{\prime\prime}}.
   \eqno {\rm (I^{20})}$$
This formula contains the theory of the conical polarisation connected
with internal conical refraction.  It shows that {\it the vibrations
at the circle of contact on {\sc Fresnel's} wave are in the chords of
that circle drawn from the extremity of the normal~$\omega'$ of single
velocity\/}; and therefore that {\it the corresponding planes of
polarisation all pass through another parallel normal at the opposite
point of the circle}.  The plane of polarisation, therefore, in
passing from one position to another, {\it revolves only half as
rapidly\/} as the revolving radius, so that the angle between any two
planes of polarisation is only {\it half\/} the angle between the two
corresponding radii of this circle on {\sc Fresnel's} wave.  And if we
suppose that the direction of the external incident ray coincides with
the wave-normal~$\omega'$, and therefore also with the normal to the
refracting face of the crystal, then the small internal components of
normal slowness $\sigma_{\prime\prime}$~$\tau_{\prime\prime}$,
parallel to this refracting face, are equal (by our general theorem of
refraction) to the small external direction-cosines
$\alpha_0$~$\beta_0$ of the inclinations of a near incident ray
to the semiaxes of $x_{\prime\prime}$ and $y_{\prime\prime}$; from
which it follows, by (I${}^{20}$) that {\it the plane of external
incidence containing this near incident ray revolves twice as
rapidly as the corresponding plane of refraction}.

For the other kind of conical polarisation, connected with the
external conical refraction, and therefore with the conoidal cusps on
{\sc Fresnel's} wave, we find by a similar process,
$${\delta y_\prime \over \delta x_\prime}
   = {\tau_\prime \over \sigma_\prime}
   = {\beta_\prime \over \alpha_\prime
      \pm \sqrt{\alpha_\prime^2 + \beta_\prime^2}},
   \eqno {\rm (K^{20})}$$
and
$$\delta z_\prime = -2 b r_\prime \,\delta x_\prime,
   \eqno {\rm (L^{20})}$$
$r_\prime$ having the meaning (U${}^{19}$).  The formula
(K${}^{20}$) shows that the normal plane to the wave, containing
any vibration near the cusp, contains either the cusp-ray itself, or a
line parallel to this ray; so that the direction of any near vibration
coincides with or is parallel to the projection of the cusp-ray on the
corresponding tangent plane of the wave, or of the cone which touches
it at the cusp: and the formula (L${}^{20}$) shows that all these
near vibrations are parallel to one common plane, which is easily seen
to be perpendicular to the plane of $ac$, and to contain the tangent
at the cusp to the elliptic section (I${}^{19}$) of the wave, made
by this latter plane; so that {\it all the planes of polarisation near
the cusp, contain, or are parallel to, the normal of this elliptic
section}.  And the direction of any near vibration on the wave, or on
its tangent cone, may be obtained by cutting the corresponding tangent
plane of this wave or cone by a plane perpendicular to this elliptic
normal.

If the cusp-ray be incident perpendicularly on a refracting face of
the crystal, then the internal components
$\sigma_\prime$~$\tau_\prime$ are equal to the direction-cosines
$\alpha_0$~$\beta_0$ of the corresponding ray of the emerging
external cone; and therefore, by (K${}^{20}$), the plane of
refraction of this external ray contains the internal vibration,
and therefore also, by {\sc Fresnel's} principles, the external
vibration corresponding: so that, {\it in the external conical
polarisation, produced by the perpendicular internal incidence of
a cusp-ray, the plane of polarisation of an external ray is
perpendicular to its plane of refraction; and therefore revolves
about half as rapidly as the plane containing this emergent ray
and passing through the approximate axis of the nearly circular
emergent cone, when the biaxal energy is small}.  We see also, by
(K${}^{20}$), that the plane containing the cusp-ray and
containing or parallel to a near internal ray, revolves with double
the rapidity of the plane containing the cusp-ray and parallel to
the near wave-normal; and therefore, in the case of perpendicular
incidence of the cusp-ray, the plane of incidence of a near internal
ray revolves with double the rapidity of the plane of external
refraction, which, as we have seen, contains here the external
vibrations.

In general, the equations of polarisation (F${}^{18}$), which we
have deduced from {\sc Fresnel's} principles, conduct, by (I${}^{18}$)
(L${}^{18}$), to the following simple formula
$$a^2 \alpha \,\delta x + b^2 \beta \,\delta y + c^2 \gamma \,\delta z
  = 0,
   \eqno {\rm (M^{20})}$$
$\delta x$, $\delta y$, $\delta z$ being still the components of
displacement parallel to the semiaxes $a$,~$b$,~$c$, and
$\alpha$,~$\beta$,~$\gamma$ being still the cosines of the inclinations
of the ray to the same semiaxes of elasticity: and this formula
(M${}^{20}$), when combined with the equation of transversal
vibrations,
$$\delta V = 0,\quad \hbox{ or,}\quad
\sigma \,\delta x + \tau \,\delta y + \upsilon \,\delta z
   = 0,
   \eqno {\rm (A^{18})}$$
determines easily the direction of vibration for any given direction
and velocity of a ray, that is, for any point of {\sc Fresnel's} curved
wave propagated from a luminous origin within a biaxal crystal.  And we
easily see that {\it on any wave in a biaxal crystal}, whether
propagated from within or from without, the differential equation
(M${}^{20}$) determines {\it a series of lines of vibration},
having the property that at any point of such a line the vibration is
in the direction of the line itself.  To find these lines on
{\sc Fresnel's} wave (O${}^{18}$), we may change
$\alpha$~$\beta$~$\gamma$ to $x$~$y$~$z$ in the differential
equation (M${}^{20}$), and we then find, by integration,
$$a^2 x^2 + b^2 y^2 + c^2 z^2 = \varepsilon^4,
   \eqno {\rm (N^{20})}$$
$\varepsilon$ being an arbitrary constant; and since this integral, when
combined with the equation (O${}^{18}$) of the wave itself, gives
$$(a^4 + \varepsilon^4) x^2 + (b^4 + \varepsilon^4) y^2
      + (c^4 + \varepsilon^4) z^2
   = (a^2 + b^2 + c^2) \varepsilon^4 - a^2 b^2 c^2,
   \eqno {\rm (O^{20})}$$
we see that {\it the lines of vibration on {\sc Fresnel's} wave,
propagated from a point in a biaxal crystal, are the intersections
of two series} (N${}^{20}$) (O${}^{20}$) {\it of concentric
and co-axal ellipsoids}.

By this general integration, extending to the whole wave, or by
integrating the approximate equations for vibrations near the conoidal
cusps and circles of contact, obtained from (K${}^{20}$)
(I${}^{20}$) by changing the direction-cosines of a ray to the
proportional co-ordinates of the wave, we find that near a cusp the
lines of vibration coincide nearly with small parabolic arcs on the
tangent cone of the wave, in planes perpendicular to the elliptic
normal already mentioned; and that in crossing a circle of contact the
course of each line of vibration is directed towards that point of the
circle which is the end of the corresponding wave-normal of single
velocity, that is, towards the foot of the perpendicular let fall from
the centre of the wave on the plane of circular contact.

\bigbreak

{\sectiontitle
In any Uniform Medium, the Curved Wave propagated from a point is
connected with a certain other surface, which may be called the
surface of components, by relations discovered by {\sc M.~Cauchy},
and by some new relations connected with a General Theorem of Reciprocity.
This new Theorem of Reciprocity gives a new construction for the Wave,
in any Undulatory Theory of Light: and it connects the Cusps and
Circles of Contact on {\sc Fresnel's} Wave, with Circles and Cusps
of the same kind on the Surface of Components.\par}

\nobreak\bigskip

31.
The theory of the wave propagated from a point in any
uniform medium may be much illustrated by comparing this wave with a
certain other surface which appears to have been first discovered by
{\sc M.~Cauchy}, who has pointed out some of its properties in the
{\it Livraison\/} already referred to.  In that {\it Livraison},
{\sc M.~Cauchy} has treated of the propagation of plane waves in a
system of mutually attracting or repelling particles; and has been
conducted to a relation between the normal velocity of propagation,
which he calls~$s$, and the cosines of its inclinations to the positive
semiaxes of $x$,~$y$,~$z$, which cosines he denotes by $a$,~$b$,~$c$.
The relation thus found being expressed by equating to zero a certain
homogeneous function (of the sixth dimension) of $s$,~$a$,~$b$,~$c$,
it has suggested to {\sc M.~Cauchy} the consideration of $s$ as a
homogeneous function of the first dimension of the cosines
$a$,~$b$,~$c$, whereas we have preferred to treat the normal velocity
(denoted in this Supplement by $\omega$) as a homogeneous function
of its cosines of direction of the dimension zero; a difference in
method which makes no real difference in the results, because the
relation existing between the cosines (namely, that the sum of
their squares is unity,) permits us to transform in an infinite
variety of ways any equation into which they enter.  {\sc M.~Cauchy}
deduces from his view of the relation between the normal velocity
and cosines of normal direction, the following equations between
the time~$t$ and the co-ordinates $x$~$y$~$z$ of a ray from the
origin of co-ordinates,
$${x \over t} = {ds \over da},\quad
  {y \over t} = {ds \over db},\quad
  {z \over t} = {ds \over dc},$$
which were alluded to in the twenty-sixth number of the present
Supplement, as substantially equivalent to our equations
(D${}^{18}$).  He deduces also an equation of the form
$$F \left( {a \over s}, {b \over s}, {c \over s} \right)
   = 0,$$
which he constructs by a surface having
$\displaystyle {a \over s}$, $\displaystyle {b \over s}$,
$\displaystyle {c \over s}$, for its co-ordinates.  Our methods suggest
immediately the same surface, as the construction of the same equation
under the form
$$\Omega(\sigma, \tau, \upsilon) = 0,$$
which has been so frequently employed in this Supplement; and from the
optical meanings that we have pointed out for the co-ordinates
$\sigma$,~$\tau$,~$\upsilon$, of this surface $\Omega = 0$, we shall
call it the surface of components of normal slowness, or simply
{\it the surface of components}.  {\sc M.~Cauchy} shows that this surface
is connected with the curved wave propagated from the origin of
co-ordinates in the unit of time, (which we have called the
{\it unit-wave\/} and may denote by the equation
$$V = 1,)$$
by two remarkable relations, which can easily be deduced from our
formul{\ae}, and may be thus enunciated: first, {\it the sum of the
products of their corresponding co-ordinates}, or, in other words,
{\it the product of any two corresponding radii multiplied by the
cosine of the included angle, is unity\/}; and secondly, {\it the wave
is the enveloppe of the planes which cut perpendicularly the radii of
the surface of components at distances from the centre equal to the
reciprocals of those radii}.

To these two relations, discovered by {\sc M.~Cauchy}, we may add a
third, not less remarkable, which he does not seem to have perceived:
namely, that {\it the surface of components is the enveloppe of the planes
which cut perpendicularly the radii of the wave at distances from its
centre equal to the reciprocals of those radii}, that is, equal to the
slowness of the rays.  For it is a general theorem of reciprocity
between surfaces, which can easily be deduced from the evident
coexistence of the three equations
$$\left. \eqalign{
x x'          + y y'          + z z'          &= 1,\cr
x \,\delta x' + y \,\delta y' + z \,\delta z' &= 0,\cr
x' \,\delta x + y' \,\delta y + z' \,\delta z &= 0,\cr} \right\}
   \eqno {\rm (P^{20})}$$
that {\it if one surface~$B$ be deduced from another~$A$ by drawing
radii vectores to the latter from an arbitrary origin~$O$, and
altering the lengths of these radii to their reciprocals without
changing their directions, and seeking the enveloppe~$B$ of the planes
perpendicular at the extremities to these altered radii of $A$, then
reciprocally, the surface~$A$ may be deduced from $B$ by a repetition
of the same construction}, employing the same origin~$O$, and the same
arbitrary unit of length.  For example, if the surface~$A$ be formed
by the revolution of an ellipse about its major axis, and if we place
the arbitrary origin~$O$ at one focus of this ellipsoid~$A$, and take
the arbitrary unit equal to the semiaxis minor, the enveloped
surface~$B$ will be a sphere, having its diameter equal to the axis
major of the ellipsoid, and its centre on that axis major, the
interval between the centres of the two surfaces being bisected by the
origin~$O$; and if from this excentric origin we draw radii to the
sphere~$B$, and change these unequal radii to their reciprocals, and
draw perpendicular planes at the extremities of these new radii, the
enveloppe of the planes so drawn will be the ellipsoid~$A$.  Another
particular case of this general theory of {\it reciprocal surfaces},
namely the case of two concentric and co-axal ellipsoids, referred to
their centre as origin, and having the semiaxes of one equal to the
reciprocals of those of the other, has been perceived by
Mr.~{\sc Mac Cullagh}, and elegantly proved by him, in the Second Part
of the Sixteenth Volume of the {\it Transactions of the Royal Irish
Academy}.

This general theorem of reciprocity, when applied to the unit-wave and
surface of components, gives {\it a new construction for the unit-wave
in any uniform medium, and for any law of velocity\/}: namely, that
{\it the wave is the locus of the points obtained by letting fall
perpendiculars from the centre on the tangent planes of the surface of
components, and then altering the lengths of these perpendiculars to
their reciprocals, without altering their directions}.

It follows also from this general theory of {\it reciprocal surfaces},
that {\it a conoidal cusp on any surface~$A$ corresponds in general to
a curve of plane contact on the reciprocal surface~$B$}, and
reciprocally; and, accordingly the cusps and circles on {\sc Fresnel's}
wave are connected with circles and cusps on the corresponding surface
of components, which latter surface is indeed deducible from the former
by merely changing the semiaxes of elasticity $a$~$b$~$c$ to their
reciprocals.  And it was in fact by this general theorem that I was
led to discover the four circles of contact on {\sc Fresnel's} wave, by
concluding that this wave must touch four planes in curves instead of
points of contact, as soon as I had perceived the existence of four
conoidal cusps on the surface of components, by obtaining (in some
investigations respecting the aberrations of biaxal lenses) the
formula (G${}^{20}$), which is the approximate equation of such a
cusp.  I easily found also that there were {\it only four\/} such
cusps on each of the two reciprocal surfaces, and therefore concluded
that there were {\it only four\/} curves of plane contact on each.  I
may mention that though I have taken care to attribute to {\sc M.~Cauchy}
the discovery of the surface of components, yet before I met the
{\it Exercices de Math\'{e}matiques}, I was familiar, in my own
investigations, with the existence and with the foregoing properties
of this surface: it is indeed immediately suggested by the first
principles of my view of optics, since it constructs the fundamental
partial differential equation
$$\Omega \left(
   {\delta V \over \delta x},
   {\delta V \over \delta y},
   {\delta V \over \delta z} \right) = 0$$
which my characteristic function~$V$ must satisfy in a final uniform
medium.

The surface of components possesses many other interesting properties,
for example the following, that in a final uniform medium any two
conjugate planes of vergency (E${}^{15}$) are perpendicular to two
conjugate tangents on it: which is analogous to the less simple
relations considered in the twenty-first number.  But the length to
which this Supplement has extended, confines me here to remarking,
that the general equations of reflexion or refraction,
$$\Delta \sigma = 0,\quad \Delta \tau = 0,
   \eqno {\rm (Q^{19})}$$
may be thus enunciated; {\it the corresponding points\/}
($\sigma$,~$\tau$,~$\upsilon$, and $\sigma + \Delta \sigma$,
$\tau + \Delta \tau$, $\upsilon + \Delta \upsilon$) {\it upon
the surface or surfaces of components\/} ($0 = \Omega$,
$0 = \Omega + \Delta \Omega$,) {\it before and after any reflexion
or refraction, ordinary or extraordinary, are situated on one common
perpendicular to the plane which touches the reflecting or refracting
surface at the point of reflexion or refraction\/}; a new geometrical
relation, which gives a new and general construction to determine
a reflected or refracted ray, simpler in many cases than the
construction proposed by {\sc Huyghens}.

\bye

