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

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\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.~15
(1828), pp.~69--174.)}

\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 Theory of Systems of Rays} by William Rowan Hamilton was
originally published in volume~15 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.

Although the table of contents describes the contents of {\it Part
Second\/} and {\it Part Third\/}, neither of these were published
in Hamilton's lifetime.  {\it Part Second\/} was 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.  (It is not included in this
edition.)  The editors of this volume were unable to locate a
manuscript for {\it Part Third}.

The printed date `Dec.~13, 1824' in fact refers to the date
on which the paper {\it On Caustics\/} was read before the
Royal Irish Academy.  {\it On Caustics: Part First\/} was not
accepted for publication by the Academy, and was published
for the first time in volume~1 of the {\it The Mathematical
Papers of Sir William Rowan Hamilton}.  The {\it Theory of
Systems of Rays\/} was in fact read before the Royal Irish
Academy on April 23rd, 1827.  The history of the papers
{\it On Caustics\/} and the {\it Theory of Systems of Rays}
is described on page~462 of the first volume of
{\it The Mathematical Papers of Sir William Rowan Hamilton}.

This edition corrects various errata noted by Hamilton, and
listed at the end of the original publication, and at the
end of the {\it First Supplement\/} in volume~16, part~2.

A small number of changes have been made to the spelling and
punctuation of the original publication.  Also the following
discrepancies in the original publication have been corrected:

\smallskip

\item{---}
in the Table of Contents, the heading for Part Second is `PART
SECOND: ON SYSTEMS OF REFRACTED RAYS', though elsewhere it is
give as `PART SECOND: ON ORDINARY SYSTEMS OF REFRACTED RAYS';

\smallskip

\item{---}
in the final sentence of the article numbered~58, the first occurrence of $\pi$
is denoted by $\varpi$, though the second occurrence in that sentence
is denoted by $\pi$;

\smallskip

\item{---}
in article~65, the formula preceding (I${}'''$) as been corrected
(following Conway and Synge) by the replacement of $dx$ by $dy$
following $d^2 q''$ and $d^2 q$;

\smallskip

\item{---}
in article~69, parentheses have been inserted into the numerators
of the expression for the quantity $\mu^{(t)}$ preceding
(X${}''''$) (following Conway and Synge) in order to resolve the
ambiguities which would otherwise be present in this expression;

\smallskip

\item{---}
in article~75, equation (Z${}^{(8)}$), $\pi$ is denoted by $\varpi$;

\smallskip

\item{---} in article~69, from equation (X${}^{(6)}$) onwards, the character
${\cal C}$ has been used in this edition to represent a character,
which resembles a script~C in the original publication,  and which
denotes a quantity defined in equation (Y${}^{(6)}$).

\bigbreak\bigskip

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

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

\vskip0pt

\leftline{\hskip.5\hsize Corrected October 2001}

\vfill\eject

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\null\vskip36pt

{\largeit\noindent
Theory of Systems of Rays.
\hskip 0pt plus10pt minus0pt
By {\largerm W. R. Hamilton}, Professor of Astronomy in the
University of Dublin.}

\bigbreak

\centerline{Read Dec.~13, 1824.\footnote*{Since this paper
was first read before the Academy, various delays have occurred,
which postponed the printing until the present time.  I have
availed myself of these delays, to add some developments and
applications of my Theory, which would, I thought, be useful.}}

\bigbreak

\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~15, (1828), pp. 69--174.]}

\bigbreak

\centerline{INTRODUCTION.}

\nobreak\bigskip

Those who have hitherto written upon the properties of Systems of
Rays, have confined themselves for the most part to the
considerations of those particular systems, which are produced by
ordinary reflexion and refraction at plane surfaces and at
surfaces of revolution.  Malus, indeed, in his {\it Trait\'{e}
D'Optique}, has considered the subject in a more general manner,
and has made some valuable remarks upon systems of rays, disposed
in any manner in space, or issuing from any given surface
according to any given law; but besides that those remarks are
far from exhausting the subject, Malus appears to me to have
committed some important errors, in the application of his theory
to the systems produced by combinations of mirrors and lenses.
And with the exception of this author, I am not aware that any
one has hitherto sought to investigate, in all their generality,
the properties of optical systems; much less to establish
principles respecting systems of rays in general, which shall be
applicable not only to the theory of light, but also to that of
sound and of heat.  To establish such principles, and to
investigate such properties is the aim of the following essay.  I
hope that mathematicians will find its results and reasonings
interesting, and that they will pardon any defects which they may
perceive in the execution of so abstract and extensive a design.

\nobreak\bigskip

{\it Observatory},

{\it June\/} 1827.

\vfill\eject

\centerline{\largerm CONTENTS.}

\nobreak\bigskip

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

\nobreak\bigskip

\centerline{\vbox{\halign{#\hfil\cr
{\sc Part First}: On Ordinary Systems of Reflected Rays.\cr
{\sc Part Second}: On Ordinary Systems of Refracted Rays.\cr
{\sc Part Third}: On Extraordinary Systems, and Systems of
   Rays in general.\cr}}}

\nobreak\bigskip

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

\bigbreak

\centerline{PART FIRST.  ON ORDINARY SYSTEMS OF REFLECTED RAYS.}

\nobreak\bigskip

\centerline{I.  {\it Analytic expressions of the Law of Ordinary Reflexion.}}

\nobreak\bigskip

The sum of the cosines of the angles which an incident and a
reflected ray, measured from the mirror, make with any assumed
line, is equal to the cosine of the angle which the normal to the
mirror makes with the same line, multiplied by twice the cosine
of incidence; this theorem determines immediately the angles
which a reflected ray makes with three rectangular axes, when we
know the corresponding angles for the incident ray, and the
tangent plane to the mirror.
\dotfill \hbox{Arts.~1, 2.}

Principle of Least Action; the sum of the distances of the point
of incidence, from any two assumed points, situated on the two
rays, is equal to the corresponding sum, for any point,
indefinitely near, upon the mirror, (the distances being counted
negative when the assumed points are on the rays produced):
consequences respecting ellipsoid, hyperboloid, paraboloid, and
plane mirrors.
\dotfill \hbox{3, 4, 5.}

\bigbreak

\centerline{II.  {\it Theory of Focal Mirrors.}}

\nobreak\bigskip

A {\it Focal Mirror\/} is one which would reflect to a given
point the rays of a given system; differential equation of such
mirrors.
\dotfill \hbox{6.}

In order that this equation be integrable, the incident rays must
be perpendiculars to a surface.
\dotfill \hbox{7, 8.}

When this condition is satisfied, the integral expresses, that the
whole bent path traversed by the light, in going from the
perpendicular surface to the Focal Mirror, and from this to the
Focus, is of a constant length, the same for all the rays.
\dotfill \hbox{9.}

The Focal Mirror is the enveloppe of a certain series of
ellipsoids.
\dotfill \hbox{10.}

\bigbreak

\centerline{III.  {\it Surfaces of Constant Action.}}

\nobreak\bigskip

When rays issuing from a luminous point, or from a
perpendicular surface, have been any number of times reflected,
they are cut perpendicularly by a series of surfaces, possessing
this property, that the whole polygon path traversed by the
light, in arriving at any one of them, is of a constant length,
the same for all the rays.
\dotfill \hbox{11, 12.}

Reasons for calling these surfaces, {\it Surfaces of Constant
action}.
\dotfill \hbox{13.}

Distinction of these surfaces into {\it positive} and {\it negative}.
\dotfill \hbox{14.}

Each surface of constant action is the enveloppe of a certain
series of spheres; if it be itself a sphere, the final rays all
pass through the centre of that sphere; it is always possible to
choose the final mirror, so as to satisfy this condition.
\dotfill \hbox{15.}

\bigbreak

\centerline{IV.  {\it Classification of Systems of Rays.}}

\nobreak\bigskip

{\it Elements of Position\/} of a ray; a system in which there is
but one such element, is a {\it system of the first class\/}; a
system with two elements of position, is a {\it system of the
second class\/}; the principal systems of optics belong to these
two classes.
\dotfill \hbox{16, 17.}

A system is {\it rectangular\/} when the rays are perpendiculars
to a surface.
\dotfill \hbox{18.}

In such a system, the cosines of the angles that a ray makes with
the axes of coordinates, are equal to the partial differentials
of a certain {\it characteristic function}.
\dotfill \hbox{19, 20.}

\bigbreak

\centerline{V.  {\it On the pencils of a Reflected System.}}

\nobreak\bigskip

The rays that are reflected from any assumed curve upon the
mirror, compose a partial system of the first class, and have a
{\it pencil\/} for their locus.
\dotfill \hbox{21, 22.}

An infinite number of these pencils may be composed by the rays
of a given reflected system; functional equation of these
pencils.
\dotfill \hbox{23.}

The arbitrary function in this equation, may be determined by the
condition, of passing through a given curve, or enveloping a
given surface; application of these principles to problems of
painting and perspective.
\dotfill \hbox{24.}

We may also eliminate the arbitrary function, and thus obtain a
partial differential equation of the first order, representing
all the pencils of the system.
\dotfill \hbox{25.}

\bigbreak

\centerline{VI.  {\it On the developable pencils, the two foci of
a ray, and the caustic curves and surfaces.}}

\nobreak\bigskip

Each ray of a reflected system has two developable pencils
passing through it, and therefore touches two caustic curves, in
two corresponding foci, which are contained upon two caustic
surfaces.
\dotfill \hbox{26.}

Equations which determine these several circumstances.
\dotfill \hbox{27, 28, 29.}

Examples.
\dotfill \hbox{30.}

Remarks upon the equations of the caustic curves and surfaces.
\dotfill \hbox{31.}

\bigbreak

\centerline{VII.  {\it Lines of reflection on a mirror.}}

\nobreak\bigskip

The curves in which the developable pencils meet the mirror, are
called the lines of reflexion; differential equations of these
lines; example.
\dotfill \hbox{32, 33, 34.}

Formul{\ae} which determine at once the foci, and the lines of
reflexion; example.
\dotfill \hbox{35.}

\bigbreak

\centerline{VIII.  {\it On Osculating Focal Mirrors.}}

\nobreak\bigskip

Object of this section.
\dotfill \hbox{36.}

When parallel rays fall on a curved mirror, the directions of the
two lines of reflexion are the directions of osculation of the
greatest and least osculating paraboloids; and the two foci of
the reflected ray are the foci of those paraboloids.
\dotfill \hbox{37.}

In general the directions of the lines of reflexion are the
directions of osculation of the greatest and least osculating
focal mirrors; and the two foci of the ray are the foci of those
two mirrors.
\dotfill \hbox{38.}

The variation of the osculating focal length, between its extreme
limits, follows an analogous law, to the variation of the radius
of an osculating sphere.
\dotfill \hbox{39.}

If on the plane passing through a given reflected ray and through
a given direction of osculation, we project the ray reflected
from a consecutive point on that direction, the projection will
cross the given ray in the osculating focus corresponding.
\dotfill \hbox{40.}

\bigbreak

\centerline{IX.  {\it On thin and undevelopable pencils.}}

\nobreak\bigskip

Functional equation of {\it thin pencils}.
\dotfill \hbox{41.}

When we look at a luminous point by any combination of mirrors,
every perpendicular section of the {\it bounding pencil of
vision\/} is an ellipse, except two which are circular; namely,
the section at the eye, and the section whose distance from the
eye is an harmonic mean between the distances of the two foci;
when the eye is beyond the foci the radius of this harmonic
circular section is less than the semiaxis of any of the elliptic
sections.
\dotfill \hbox{42.}

Whatever be the shape of a thin pencil, provided it be closed,
the area of a perpendicular section varies as the product of the
focal distances.
\dotfill \hbox{43.}

The tangent plane to an undevelopable pencil does not touch the
pencil in the whole extent of a ray; it is inclined to a certain
{\it limiting plane}, at an angle whose tangent is equal to a
constant coefficient divided by the distance of the point of
contact from a certain fixed point upon the ray; properties of
the fixed point, the constant coefficient, and the limiting
plane.
\dotfill \hbox{44, 45.}

\bigbreak

\centerline{X.  {\it On the axes of a Reflected System.}}

\nobreak\bigskip

The intersection of the two caustic surfaces of a reflected
system, reduces itself in general to a finite number of isolated
points, at which the density of light is greatest; these points
may be called the {\it principal foci}, and the corresponding
rays the {\it axes of the system}; determination of these points
and rays by means of the characteristic function.
\dotfill \hbox{46.}

Each axis is intersected in its own focus by all the rays
indefinitely near; this focus belongs to an osculating focal
mirror, which has contact of the second order with the given
mirror, at a point which may be called the {\it vertex}.
\dotfill \hbox{47.}

The principal focus is also the centre of a series of spheres,
which have contact of the second order with the surfaces of
constant action.
\dotfill \hbox{48.}

Examples.
\dotfill \hbox{49, 50.}

\bigbreak

\centerline{XI.  {\it Images formed by Mirrors.}}

\nobreak\bigskip

The image of a luminous point, formed by any given combination of
mirrors, is the principal focus of the last reflected system; the
image of a curve or surface is the locus of the images of its
points.
\dotfill \hbox{51.}

Example; the image of a planet's disk, formed by a single mirror,
is in general an ellipse; its projection on a plane perpendicular
to the reflected rays is a circle, the radius of which is equal
to the focal length of the mirror multiplied by the semidiameter
of the planet.
\dotfill \hbox{52, 53.}

General theorem respecting the images of small objects, formed by
any combination of mirrors.
\dotfill \hbox{54.}

There are, in general, one or more ways of placing a given mirror
so as to produce an undistorted image of a planet; the points
which are to be used as vertices for this purpose, are determined
by two relations between the partial differentials, third order,
of the mirror.
\break.\dotfill \hbox{55.}

\bigbreak

\centerline{XII.  {\it Aberrations.}}

\nobreak\bigskip

General series for calculating the lateral aberrations by means
of the characteristic function; the longitudinal aberrations do
not exist for reflected systems in general; but there are certain
analogous quantities, calculated in the third part of this essay.
\dotfill \hbox{56, 57.}

First application; aberrations measured on a plane which does not
pass through either focus; the rays which make with the given ray
angles not exceeding some given small angle, are diffused over
the area of an ellipse.
\dotfill \hbox{58.}

Second application; aberrations measured on a plane passing
through one focus; the rays which were before diffused over the
area of an ellipse, are now diffused over a mixt-lined space,
bounded partly by a curve shaped like a figure of eight, and
partly by an arc of a common parabola, which envelopes the other
curve; quadrature of this mixt-lined space, and calculation of
the coefficients of the result, by means of the curvatures of the
caustic surface.
\break.\dotfill \hbox{59, 60, 61.}

Third application; aberrations measured on a plane passing
through a principal focus; the rays which make with the given ray
a given small angle, cut the plane of aberration in an ellipse;
if the focus be inside this ellipse, the intermediate rays are
diffused over the area of that curve; otherwise they are diffused
over a mixt-lined space, bounded partly by an arc of the ellipse,
and partly by two {\it limiting lines}, namely the tangents drawn
from the focus.
\dotfill \hbox{62.}

The distinction between these two cases depends on the nature of
the roots of a certain quadratic equation; when the focus is
inside the ellipse, the caustic surfaces do not intersect the
plane of aberration; but when the focus is outside, then the
caustic surfaces intersect that plane in the limiting lines
before mentioned.
\dotfill \hbox{63.}

This distinction depends also on the roots of a certain cubic
equation; when the focus is inside there are three directions of
{\it focal inflexion\/} on the mirror, and of {\it spheric\/}
inflexion on the surfaces of constant action, but when the focus
is outside, there is but one such direction; the aberrations of
the second order vanish, when there is contact of the third order
between the mirror and the focal surface, or between the surfaces
of constant action and their osculating spheres.
\dotfill \hbox{64, 65.}

\bigbreak

\centerline{XIII.  {\it Density.}}

\nobreak\bigskip

Method by which Malus computed the density for points not upon
the caustic surfaces; other method founded on the principles of
this essay; along a given ray, the density varies inversely as
the product of the focal distances; near a caustic surface it
varies inversely as the square root of the perpendicular distance
from that surface.
\dotfill \hbox{66, 67, 68.}

Law of the density at the caustic surfaces; this density is
greatest at the principal foci, and at a bright edge, the locus
of the points upon the caustic curves, at which their radius of
curvature vanishes.
\dotfill \hbox{69, 70.}

Density near a principal focus; ellipses or hyperbolas, upon the
plane of aberration, at which this density is constant; the axes
of these curves may be considered as natural axes of coordinates;
the density at the principal focus itself, is expressed by an
elliptic integral, the value of which depends on the excentricity
of the ellipses or hyperbolas, at which the density is constant.
\dotfill \hbox{71, 72, 73, 74, 75, 76.}

\bigbreak

\centerline{PART SECOND.  ON ORDINARY SYSTEMS OF REFRACTED RAYS.}

\nobreak\bigskip

\centerline{XIV.  {\it Analytic expressions of the law of ordinary
refraction.}}

\nobreak\bigskip

Fundamental formula of dioptrics; principle of least action;
cartesian surfaces.
\dotfill \hbox{77, 78, 79.}

\bigbreak

\centerline{XV.  {\it On focal refractors, and on the surfaces of
constant action.}}

\nobreak\bigskip

Differential equation of focal refractors; this equation is
integrable, when the incident rays are perpendicular to a
surface; form of the integral; the focal refractor is the
enveloppe of a certain series of cartesian surfaces.
\dotfill \hbox{80.}

When homogeneous rays have been any number of times reflected and
refracted, they are cut perpendicularly by the surfaces of
constant action.
\dotfill \hbox{81.}

\bigbreak

\centerline{XVI.  {\it Characteristic Function.}}

\nobreak\bigskip

The systems produced by ordinary reflexion and refraction being
all rectangular, the properties of every such system may be
deduced from the form of one characteristic function, whose
partial differentials of the first order, are proportional to the
cosines of the angles that the ray makes with the axes.
\dotfill \hbox{82.}

\bigbreak

\centerline{XVII.  {\it Principal properties of a refracted system.}}

\nobreak\bigskip

The results contained in the preceding part, respecting the
pencils of a reflected system, the lines of reflexion, the
caustic curves and surfaces, the osculating focal surfaces, the
axes of the system, the principal foci, the images, aberrations,
and density, may all be applied, with suitable modifications to
refracted systems also.
\dotfill \hbox{83, 84, 85, 86.}

\bigbreak

\centerline{XVIII.  {\it On the determination of reflecting and
refracting surfaces, by their}}
\nobreak\vskip 3pt
\centerline{{\it lines of reflexion and refraction.}}

\nobreak\bigskip

Analogy to questions in the application of analysis to geometry.
\dotfill \hbox{87.}

Remarks on a question of this kind, which has been treated by
Malus; solution of the same question on the principles of this
essay.
\dotfill \hbox{88, 89.}

Questions of this kind conduct in general to partial differential
equations of the second order; another example, which conducts to
a case of the equation of vibrating chords.
\dotfill \hbox{90.}

The partial differential equation, which expresses the condition
for the lines of reflexion or refraction coinciding with one
another, resolves itself into two distinct equations; the
surfaces represented by the integral, are the focal reflectors or
refractors.
\dotfill \hbox{91.}

\bigbreak

\centerline{XIX.  {\it On the determination of reflecting and
refracting surfaces, by means of their}}
\centerline{{\it caustic surfaces.}}

\nobreak\bigskip

Object of this section.
\dotfill \hbox{92.}

Remarks on the analogous questions treated of by Monge.
\dotfill \hbox{93.}

Method of reducing to those questions, the problems of the
present section.
\dotfill \hbox{94.}

Another method of treating the same problems, which conducts to
partial differential equations of an order higher by unity;
example, in the case where it is required to find a mirror, which
shall have one set of its foci upon a given sphere, the incident
rays being parallel; the complete integral, with two arbitrary
functions, represents here the enveloppe of a series of
paraboloids, which have their foci upon the given sphere; there
is also a singular primitive, of the first order, representing
the mirrors which have the sphere for one of their caustic
surfaces.
\dotfill \hbox{95, 96.}

Generalization of the preceding results.
\dotfill \hbox{97.}

Remarks on other questions of the same kind, which conduct to
equations in ordinary differentials.
\dotfill \hbox{98.}

The partial differential equation which expresses the condition
for the two caustic surfaces coinciding, resolves itself into two
other equations which are peculiar to the focal reflectors or
refractors: however, these two equations determine, on some
particular reflecting and refracting surfaces, a {\it line of
focal curvature}, analogous to the line of spheric curvature,
which exists upon some curved surfaces.
\dotfill \hbox{99.}

\bigbreak

\centerline{XX.  {\it On the caustics of a given reflecting or
refracting curve.}}

\nobreak\bigskip

General theorem respecting the arc of a caustic curve; means of
finding by this theorem, the curves corresponding to a given
caustic.
\dotfill \hbox{100.}

As an infinite number of curves correspond to the same given
caustic, so also an infinite number of caustics correspond to the
same given curve; these caustics have for their locus a
{\it surface of circular profile}, and are the shortest lines
betweeen two points upon it; this surface envelopes all the
caustic surfaces, corresponding to the reflectors or refractors,
upon which the given curve is a line of reflection or refraction.
\dotfill \hbox{101, 102.}

\bigbreak

\centerline{XXI.  {\it On the conditions of Achromatism.}}

\nobreak\bigskip

The coordinates of the image of a luminous point, formed by any
given combination of lenses and mirrors, are functions of the
colour of the rays; hence may be deduced series for the chromatic
aberrations, and conditions for achromatism, perfect or
approximate.
\dotfill \hbox{103, 104.}

\bigbreak

\centerline{XXII.  {\it On systems of atmospheric rays.}}

\nobreak\bigskip

Equations of an atmospheric ray.
\dotfill \hbox{105.}

These rays are perpendicular to the surfaces of constant action;
and the properties of the system may all be deduced from the form
of one characteristic function.
\dotfill \hbox{106.}

\bigbreak

\centerline{PART THIRD.  ON EXTRAORDINARY SYSTEMS, AND SYSTEMS OF}

\nobreak\vskip 3pt

\centerline{RAYS IN GENERAL.}

\nobreak\bigskip

\centerline{XXIII.  {\it On plane systems of rays.}}

\nobreak\bigskip

Object of this part.
\dotfill \hbox{107.}

Equation of a ray in a plane system, formul{\ae} for the focal
coordinates, and for the arc and curvature of the caustic curve.
\dotfill \hbox{108.}

General series and approximate formul{\ae} for the aberrations,
lateral and longitudinal; principal foci, and axes of a plane
system.
\dotfill \hbox{109, 110, 111.}

Properties of the rectangular trajectories.
\dotfill \hbox{112.}

Least linear space, into which can be collected a given parcel of
rays near an axis of a plane system.
\dotfill \hbox{113.}

On finding the system by means of the caustic.
\dotfill \hbox{114.}

On plane emanating systems; general theorem respecting the focal
lengths of plane reflecting and refracting curves, ordinary and
extraordinary.
\dotfill \hbox{115.}

Plane curves having a given caustic; focal curves.
\dotfill \hbox{116.}

\bigbreak

\centerline{XXIV.  {\it On developable systems.}}

\nobreak\bigskip

A developable system is a system of the first class, in which the
rays have a developable pencil for their locus; equations of a
ray; condition of developability.
\dotfill \hbox{117.}

Formul{\ae} for the caustic curve.
\dotfill \hbox{118.}

Aberrations; formula for the radius of curvature of a curve in
space; principal foci and axes of a developable system.
\dotfill \hbox{119, 120, 121.}

Remarks upon some properties of developable pencils, considered
as curve surfaces.
\break.\dotfill \hbox{122, 123, 124.}

\bigbreak

\centerline{XXV.  {\it On undevelopable systems.}}

\nobreak\bigskip

Generalisation of the results of the IXth section, respecting the
tangent plane, the limiting plane, the virtual focus, and the
coefficient of undevelopability.
\dotfill \hbox{125.}

{\it Virtual caustic}, and axes of an undevelopable pencil.
\dotfill \hbox{126.}

{\it Directrix\/} of the pencil; every undevelopable surface
composed of right lines may be generated by one of the indefinite
sides of a rectangle of variable breadth, whose other indefinite
side constantly touches the directrix, while its plane constantly
osculates to that curve.
\break.\dotfill \hbox{127, 128.}

Pencils having a given directrix; {\it isoplatal surfaces}.
\dotfill \hbox{129, 130.}

The surfaces of centres of curvature of an undevelopable pencil,
are the enveloppe of a series of hyperboloids; formul{\ae} for
the two radii of curvature; these two radii are turned in
opposite directions; {\it lines of equal and opposite curvature}.
\dotfill \hbox{131.}

When the ray is an axis of the undevelopable pencil, the locus of
centres of curvature is a common hyperbola; {\it point of
evanescent curvature\/} at which the normal to the pencil is an
asymptote to the hyperbola of centres; point in which the ray
touches the directrix; these two points are equally distant from
the focus.
\dotfill \hbox{132.}

On every surface whose curvatures are opposite there exist two
series of lines, which may be called the {\it lines of
inflexion\/}; properties of these lines; on the surfaces of least
area, the lines of inflexion cut at right angles.
\dotfill \hbox{133.}

Aberrations in an undevelopable system; virtual developments of
the pencil.
\dotfill \hbox{134, 135.}

\bigbreak

\centerline{XXVI.  {\it On systems of the second class.}}

\nobreak\bigskip

General formul{\ae} for these systems; condition of
rectangularity; equations of the surfaces which cut the rays
perpendicularly when this condition is satisfied.
\dotfill \hbox{136.}

Pencils of the system; caustic surfaces; when the system is
rectangular, the intersection of these surfaces reduces itself to
a finite number of points.
\dotfill \hbox{137, 138.}

Virtual foci of a ray; virtual caustic surfaces; diametral
surface; principal virtual foci.
\break.\dotfill \hbox{139.}

Law of the variation of the virtual focus; the planes of extreme
virtual foci cut one another at right angles; generalization of
the results of the IXth section respecting the properties of thin
pencils.
\dotfill \hbox{140.}

On emanating systems.
\dotfill \hbox{141.}

Foci by projection; the planes corresponding to the extreme foci
by projection, coincide with the planes of extreme virtual foci;
they furnish a pair of natural coordinates, which are of
extensive use in optics.
\dotfill \hbox{142.}

Osculating focal surfaces; the greatest and least have their foci
upon the caustic surfaces, and osculate in the directions in
which the developable pencils intersect the surface from which
the rays proceed.
\dotfill \hbox{143.}

Applications of this theory.
\dotfill \hbox{144, 145, 146, 147.}

Caustics of a given curve; conditions of integrability, which are
necessary for the existence of focal surfaces.
\dotfill \hbox{148.}

\bigbreak

\centerline{XXVII.  {\it On systems of the third class.}}

\nobreak\bigskip

Object of this section; limiting surface enveloped by all the
cones and other pencils of the system; condition for this surface
being touched by all the rays; remarks on the inverstigations of
Malus, respecting systems of this kind.
\dotfill \hbox{149, 150, 151.}

\bigbreak

\centerline{XXVIII.  {\it On extraordinary systems produced by
single-axed crystals.}}

\nobreak\bigskip

Object of this section; analytic expression of the law of
Huygens; principle of least action; characteristic function of an
extraordinary system.
\dotfill \hbox{152, 153.}

The surfaces of constant action are touched by spheroids, having
their centres on the extraordinary rays, and the rays may be
considered as proceeding from these surfaces, according to a
simple law; when the extraordinary rays converge to one focus,
the surfaces of constant action become a series of concentric
spheroids, and it is always possible to assign such a form to the
surface of the crystal as to satisfy this condition: hence it
follows, by XXVI.\ that the extraordinary rays are in general
tangents to two caustic surfaces, which contain the foci of the
greatest and least osculating crystals; the directions of
osculation are the directions of the {\it limes of extraordinary
refraction}, analogous to those lines of ordinary reflexion and
refraction which were before considered: the caustic surfaces of
the extraordinary system contain also the centres of the greatest
and least spheroids which osculate to the surfaces of constant
action; the intersection of the caustic surfaces reduces itself
in general to a finite number of {\it principal foci}, analogous
to the principal foci of ordinary systems, considered in the two
first parts; each principal focus is the centre of a spheroid
which has contact of the second order with a surface of constant
action; it is also the focus of a focal crystal which has contact
of the second order with the surface of the given crystal; the
partial differential equations which represent crystals
corresponding to a given caustic surface, are to be integrated
after the manner of the second part.
\dotfill \hbox{154, 155.}

Condition for the rectangularity of an extraordinary system,
expressed by a partial differential equation of the second order;
integration of this equation; the integral expresses that the
normals to a surface of constant action are tangents to a
cylindric surface, whose generating line is parallel to the axis
of the crystal.
\dotfill \hbox{156.}

The preceding results may be extended to the extraordinary
systems produced by reflexion at the interior surface of the
crystal; when the extraordinary rays recover their ordinary
velocity they become again perpendicular to the surfaces of
constant action: this theorem enables us to apply the results of
the two first parts, to systems produced by combinations of
crystals, mirrors, and lenses.
\dotfill \hbox{157.}

\bigbreak

\centerline{XXIX.  {\it On other extraordinary systems.}}

\nobreak\bigskip

Law of extraordinary refraction in crystals with two axes;
characteristic function of a system produced by such a crystal;
spheroids of Brewster; surfaces of constant action; the results
of the preceding sections may be extended to these systems.
\dotfill \hbox{158, 159.}

Remarks on systems produced by crystallized mediums of
continually varying nature.
\break.\dotfill \hbox{160.}

\bigbreak

\centerline{XXX.  {\it Law of least action.}}

\nobreak\bigskip

General expressions of this law; development of these expressions
by means of the calculus of variations.  In every optical system,
the {\it action\/} may be considered as a {\sc characteristic
function}, from the form of which function may be deduced all the
other properties of the system.  This function (when we know the
luminous point, and the reflecting or refracting media) depends
only on the coordinates and on the colour; its partial
differentials, of the first order, taken with respect to the
coordinates, are in ordinary systems of the form
$${\delta i \over \delta x} = v \mathbin{.} \alpha.\quad
  {\delta i \over \delta y} = v \mathbin{.} \beta.\quad
  {\delta i \over \delta z} = v \mathbin{.} \gamma,$$
and in extraordinary systems of the form
$${\delta i \over \delta x} = {\delta v \over \delta \alpha},\quad
  {\delta i \over \delta y} = {\delta v \over \delta \beta},\quad
  {\delta i \over \delta z} = {\delta v \over \delta \gamma},$$
$\alpha$, $\beta$, $\gamma$, being the cosines of the angles
which the ray makes with the axes of $x$, $y$, $z$, and $v$ being
the velocity, estimated on the material hypothesis, and
considered, in extraordinary systems, as a homogeneous function,
first dimension, of the cosines $\alpha$, $\beta$, $\gamma$.
Reason for calling this principle the {\sc Principle of Constant
Action}; analogous principle respecting the motion of a system of
bodies.  General consequences of this principle: generalization
of the properties of optical systems, considered in former
sections of this essay; the phenomena of coloured systems depend
on the partial differentials of the characteristic function,
taken with respect to the colour.
\dotfill \hbox{161 to the end.}

\bigbreak

\centerline{CONCLUSION}

\nobreak\bigskip

Review of the chief results of this essay, and of the manner in
which they may be useful; and remarks on the researches of former
writers, respecting the properties of systems of rays.

\vfill\eject

\centerline{\largerm PART FIRST.}

\nobreak\bigskip

\centerline{\largerm ON ORDINARY SYSTEMS OF REFLECTED RAYS.}

\bigbreak

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

\nobreak\bigskip

\centerline{SECTION I.}

\nobreak\bigskip

\centerline{%
\it Analytic expressions of the law of ordinary reflexion.}

\nobreak\bigskip

[1.]
When a ray of light is reflected at a mirror, we know by
experience, that the normal to the mirror, at the point of
incidence, bisects the angle between the incident and the
reflected rays.  If therefore two forces, each equal to unity,
were to act at the point of incidence, in the directions of the
two rays, their resultant would act in the direction of the
normal, and would be equal to twice the cosine of the angle of
incidence.  If then we denote by ($\rho l)$ ($\rho' l)$ ($n l)$
the angles which the two rays and the normal make respectively
with any assumed line ($l$), and by ($I$) the angle of incidence,
we shall have the following formula,
$$\cos \rho l + \cos \rho' l
   = 2 \cos I \mathbin{.} \cos nl
   \eqno {\rm (A)}$$
which is the analytic representation of the known law of
reflexion, and includes the whole theory of catoptrics.

\bigbreak

[2.]
It follows from (A) that if we denote by
$\rho x$, $\rho y$, $\rho z$, $\rho' x$, $\rho' y$, $\rho' z$,
$nx$, $ny$, $nz$,
the angles which the two rays and the normal make respectively
with three rectangular axes, we shall have the three following
equations,
$$\left. \eqalign{
\cos \rho x + \cos \rho' x
   &= 2 \cos I \mathbin{.} \cos nx \cr
\cos \rho y + \cos \rho' y
   &= 2 \cos I \mathbin{.} \cos ny \cr
\cos \rho z + \cos \rho' z
   &= 2 \cos I \mathbin{.} \cos nz \cr}
   \right\}
   \eqno {\rm (B)}$$
which determine the direction of the reflected ray, when we know
that of the incident ray, and the tangent plane to the mirror.

\bigbreak

[3.]
Let $(x, y, z)$ be the coordinates of the point of incidence;
$x + \delta x$, $y + \delta y$, $z + \delta z$, those of a point
infinitely near; if this point be upon the mirror we shall have
$$    \cos nx \mathbin{.} \delta x
    + \cos ny \mathbin{.} \delta y
    + \cos nz \mathbin{.} \delta z
   = 0,$$
and therefore, by (B),
$$\eqalignno{
0  &=   \cos \rho  x \mathbin{.} \delta x
      + \cos \rho  y \mathbin{.} \delta y
      + \cos \rho  z \mathbin{.} \delta z \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \cos \rho' x \mathbin{.} \delta x
      + \cos \rho' y \mathbin{.} \delta y
      + \cos \rho' z \mathbin{.} \delta z.
   &{\rm (C)}\cr}$$
Now if we assume any point $X$~$Y$~$Z$ on the incident ray, at
a distance~$\rho$ from the mirror, and another point
$X'$~$Y'$~$Z'$ on the reflected ray at a distance~$\rho'$ from
the mirror, the distances of those assumed points from the point
$x + \delta x$, $y + \delta y$, $z + \delta z$, will be
$$\eqalign{
\rho  + \delta \rho
   &= \rho  + {d\rho  \over dx} \mathbin{.} \delta x
            + {d\rho  \over dy} \mathbin{.} \delta y
            + {d\rho  \over dz} \mathbin{.} \delta z,\cr
\rho' + \delta \rho'
   &= \rho  + {d\rho' \over dx} \mathbin{.} \delta x
            + {d\rho' \over dy} \mathbin{.} \delta y
            + {d\rho' \over dz} \mathbin{.} \delta z;\cr}$$
and because
$$\eqalign{
\rho^2  &= (X  - x)^2 + (Y  - y)^2 + (Z  - z)^2,\cr
\rho'^2 &= (X' - x)^2 + (Y' - y)^2 + (Z' - z)^2,\cr}$$
we shall have
$$\multieqalign{
{d \rho  \over dx} &= - {X  - x \over \rho}, &
{d \rho  \over dy} &= - {Y  - y \over \rho}, &
{d \rho  \over dz} &= - {Z  - z \over \rho}, \cr
{d \rho' \over dx} &= - {X' - x \over \rho'}, &
{d \rho' \over dy} &= - {Y' - y \over \rho'}, &
{d \rho' \over dz} &= - {Z' - z \over \rho'}, \cr}$$
that is
$$\multieqalign{
{d\rho  \over dx} &= - \cos \rho  x, &
{d\rho  \over dy} &= - \cos \rho  y, &
{d\rho  \over dz} &= - \cos \rho  z, \cr
{d\rho' \over dx} &= - \cos \rho' x, &
{d\rho' \over dy} &= - \cos \rho' y, &
{d\rho' \over dz} &= - \cos \rho' z; \cr}$$
and finally, by (C)
$$\delta \rho + \delta \rho' = 0.
   \eqno {\rm (D)}$$

This equation (D) is called the {\it Principle of least Action},
because it expresses that if the coordinates of the point of
incidence were to receive any infinitely small variations
consistent with the nature of the mirror, the bent path
$(\rho + \rho')$ would have its variation nothing; and if light
be a material substance, moving with a velocity unaltered by
reflection, this bent path $\rho + \rho'$ measures what in
mechanics is called the {\it Action}, from the one assumed point
to the other.  Laplace has deduced the formula (D),
together with analogous formul{\ae} for ordinary and
extraordinary refraction, by supposing light to consist of
particles of matter, moving with certain determined velocities,
and subject only to forces which are insensible at sensible
distances.  The manner in which I have deduced it, is independent
of any hypothesis about the nature or the velocity of light; but
I shall continue to call it, from analogy, the principle of least
action.

\bigbreak

[4.]
The formula (D) expresses, that if we assume any two points, one
on each ray, the sum of the distances of these two assumed points
from the point of incidence, is equal to the sum of their
distances from any infinitely near point upon the mirror.  If
therefore we construct an ellipsoid of revolution, having its two
foci at the two assumed points, and its axis equal to the bent
path traversed by the light in going from the one point to the
other, this ellipsoid will touch the mirror at the point of
incidence.  Hence it may be inferred, that every normal to an
ellipsoid of revolution bisects the angle between the lines drawn
to the two foci; and therefore that rays issuing from one focus
of an ellipsoid mirror, would be reflected accurately to the
other.

\bigbreak

[5.]
These theorems about the ellipsoid have long been known; to
deduce the known theorems corresponding, about the hyperboloid
and plane, I observe that from the manner in which the formula
(D) has been obtained, we must change the signs of the distances
$\rho$,~$\rho'$, if the assumed points
$X$,~$Y$,~$Z$, $X'$,~$Y'$,~$Z'$,
to which they are measured, be not upon the rays themselves, but
on the rays produced.  If therefore, we assume one point
$X$,~$Y$,~$Z$, upon the incident ray, and the other point
$X'$,~$Y'$,~$Z'$, on the reflected ray produced behind the
mirror, the equation (D) expresses that the difference of the
distances of these two points from the point of incidence, is the
same as the difference of their distances from any infinitely
near point upon the mirror; so that if we construct a
hyperboloid, having its axis equal to this difference, and having
its foci at the two assumed points, this hyperboloid will touch
the mirror.  The normal to a hyperboloid bisects therefore the
angle between the line drawn to one focus, and the produced part
of the line drawn to the other focus; from which it follows, that
rays issuing from one focus of a hyperboloid mirror, would after
reflection diverge from the other focus.  A plane is a
hyperboloid whose axis is nothing, and a paraboloid is an
ellipsoid whose axis is infinite; if, therefore, rays issued from
the focus of a paraboloid mirror, they would be reflected
parallel to its axis; and if rays issuing from a luminous point
any where situated fall upon a plane mirror, they diverge after
reflection from a point situated at an equal distance behind the
mirror.  These are the only mirrors giving accurate convergence
or divergence, which have hitherto been considered by
mathematicians: in the next section I shall treat the subject in
a more general manner, and examine what must be the nature of a
mirror, in order that it may reflect to a given point the rays of
a given system.

\bigbreak

\centerline{%
II.  \it Theory of focal mirrors.}

\nobreak\bigskip

[6.]
The question, to find a mirror which shall reflect to a given
focus the rays of a given system, is analytically expressed by
the following differential equation,
$$    (\alpha + \alpha') \, dx
    + (\beta  + \beta' ) \, dy
    + (\gamma + \gamma') \, dz
  = 0,
   \eqno {\rm (E)}$$
$x$,~$y$,~$z$, being the coordinates of the mirror, and
$\alpha$,~$\beta$,~$\gamma$, $\alpha'$,~$\beta'$,~$\gamma'$,
representing for abridgment the cosines of the angles which the
incident and reflected rays make with the axes of coordinates.
In this equation, which follows immediately from (C), or from
(B), $\alpha$,~$\beta$,~$\gamma$, are to be considered as given
functions of $x$,~$y$,~$z$, depending on the nature of the incident
system, and $\alpha'$,~$\beta'$,~$\gamma'$, as other given
functions of $x$,~$y$,~$z$, depending on the position of the
focus; and when these functions are of such a nature as to render
integrable the equation (E), the integral will represent an
infinite number of different mirrors, each of which will
possess the property of reflecting to the given focus, the rays
of the given system, and which for that reason I shall call
{\it focal mirrors}.

\bigbreak

[7.]
To find under what circumstances the equation (E) is integrable,
I observe that the part
$$\alpha' \, dx + \beta' \, dy + \gamma' \, dz$$
is always an exact differential; for if we represent by
$X'$,~$Y'$,~$Z'$ the coordinates of the given focus, and by
$\rho'$ the distance of that focus from the point of incidence,
we shall have the equations
$$X' - x = \alpha' \rho',\quad
  Y' - y = \beta'  \rho',\quad
  Z' - z = \gamma' \rho',$$
and therefore
$$\alpha' \, dx + \beta' \, dy + \gamma' \, dz = - d \rho'$$
because
$$\alpha'^2 + \beta'^2 + \gamma'^2 = 1,\quad
  \alpha' \, d\alpha' + \beta' \, d\beta' + \gamma' \, d\gamma'
   = 0.$$

If therefore the equation (E) be integrable, that is, if it can
be satisfied by any unknown relation between $x$,~$y$,~$z$, it is
necessary that in establishing this unknown relation between those
three variables, the part
($\alpha \mathbin{.} dx + \beta \mathbin{.} dy + \gamma \mathbin{.} dz$)
should also be an exact differential of a function of the two
variables which remain independent; the condition of this
circumstance is here
$$    (\alpha + \alpha') \left(
            {d\beta  \over dz} - {d\gamma \over dy} \right)
    + (\beta  + \beta' ) \left(
            {d\gamma \over dx} - {d\alpha \over dz} \right)
    + (\gamma + \gamma') \left(
            {d\alpha \over dy} - {d\beta  \over dx} \right)
  = 0,
   \eqno {\rm (F)}$$
and I am going to shew, from the relations which exist between
the functions $\alpha$, $\beta$, $\gamma$, that this condition
cannot be satisfied, unless we have separately
$${d\beta  \over dz} - {d\gamma \over dy} = 0,\quad
  {d\gamma \over dx} - {d\alpha \over dz} = 0,\quad
  {d\alpha \over dy} - {d\beta  \over dx} = 0,
   \eqno {\rm (G)}$$
that is, unless the formula
($\alpha \mathbin{.} dx + \beta \mathbin{.} dy + \gamma \mathbin{.} dz$)
be an exact differential of a function of $x$,~$y$,~$z$,
considered as three independent variables.

\bigbreak

[8.]
For this purpose I observe, that since the functions $\alpha$,
$\beta$, $\gamma$, are the cosines of the angles which the
incident ray passing through the point $(x, y, z)$ makes with the
axes, they will not vary when the coordinates $x$,~$y$,~$z$,
receive any variations $\delta x$, $\delta y$, $\delta z$
proportional to those cosines $\alpha$,~$\beta$,~$\gamma$;
because then the point
$x + \delta x$,~$y + \delta y$,~$z + \delta z$,
will belong to the same incident ray as the point $x$,~$y$,~$z$.
This remark gives us the following equations,
$$\eqalign{
     \alpha \mathbin{.} {d\alpha \over dx}
   + \beta  \mathbin{.} {d\alpha \over dy}
   + \gamma \mathbin{.} {d\alpha \over dz}
   &= 0,\cr
     \alpha \mathbin{.} {d\beta  \over dx}
   + \beta  \mathbin{.} {d\beta  \over dy}
   + \gamma \mathbin{.} {d\beta  \over dz}
   &= 0,\cr
     \alpha \mathbin{.} {d\gamma \over dx}
   + \beta  \mathbin{.} {d\gamma \over dy}
   + \gamma \mathbin{.} {d\gamma \over dz}
   &= 0,\cr}$$
and combining these with the relations
$$\eqalign{
     \alpha \mathbin{.} {d\alpha \over dx}
   + \beta  \mathbin{.} {d\beta  \over dx}
   + \gamma \mathbin{.} {d\gamma \over dx}
   &= 0,\cr
     \alpha \mathbin{.} {d\alpha \over dy}
   + \beta  \mathbin{.} {d\beta  \over dy}
   + \gamma \mathbin{.} {d\gamma \over dy}
   &= 0,\cr
     \alpha \mathbin{.} {d\alpha \over dz}
   + \beta  \mathbin{.} {d\beta  \over dz}
   + \gamma \mathbin{.} {d\gamma \over dz}
   &= 0,\cr}$$
which result from the known formula
$$\alpha^2 + \beta^2 + \gamma^2 = 1,$$
we find that the three quantities
$${d\beta  \over dz} - {d\gamma \over dy},\quad
  {d\gamma \over dx} - {d\alpha \over dz},\quad
  {d\alpha \over dy} - {d\beta  \over dx},$$
are proportional to $(\alpha, \beta, \gamma)$, and therefore that
the condition (F) resolves itself into the three equations (G).

\bigbreak

[9.]
These conditions (G) admit of a simple geometrical enunciation;
they express that the rays of the incident system are cut
perpendicularly by a series of surfaces, having for equation
$$\int ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
   = \hbox{const.}
   \eqno {\rm (H)}$$

Let $X$, $Y$, $Z$, be the point in which an incident ray is
crossed by any given surface of this series (H), and let $\rho$
be its distance from the point of incidence $(x, y, z)$: we shall
have
$$X - x = \alpha \rho,\quad
  Y - y = \beta  \rho,\quad
  Z - z = \gamma \rho,$$
and therefore,
$$\alpha \mathbin{.} dx + \beta  \mathbin{.} dy + \gamma \mathbin{.} dz
   = - d \rho,$$
because
$$\alpha \mathbin{.} dX + \beta  \mathbin{.} dY + \gamma \mathbin{.} dZ
   = 0.$$
We may therefore put the differential equation of the mirror (E)
under the form
$$d\rho + d\rho' = 0,$$
of which the integral
$$\rho + \rho' = \hbox{const.}
   \eqno {\rm (I)}$$
expresses that the whole bent path traversed by the light in
going from the perpendicular surface (H) to the mirror, and from
the mirror to the focus, is of a constant length, the same for
all the rays.  In this interpretation of the integral (I) we have
supposed the distances, $\rho$,~$\rho'$, positive; that is, we
have supposed them measured upon the rays themselves; if on the
contrary, they are measured on the rays produced behind the
mirror, they are then to be considered as negative.

\bigbreak

[10.]
Then, in general, if it be required to find a mirror which shall
reflect to a given focus the rays of a given system, we must try
whether the rays of that system are cut perpendicularly by any
series of surfaces; for unless this condition be satisfied, the
problem is impossible.  When we have found a surface cutting the
incident rays perpendicularly, we have only to take upon each of
the rays a point such that the sum or difference of its
distances, from the perpendicular surface and from the given
focus, may be equal to any constant quantity; the locus of the
points thus determined will be a focal mirror, possessing the
property required.  Or, which comes to the same thing, we may
make an ellipsoid or hyperboloid of revolution, having a constant
axis, but a variable excentricity, move in such a manner that one
focus may traverse in all directions the surface that cuts the
incident rays perpendicularly, while the other focus remains
fixed at the point through which all the reflected rays are to
pass; the surface that envelopes the ellipsoid or hyperboloid, in
all its different positions, will be the mirror required, and
each ellipsoid or hyperboloid thus moving will in general have
two such enveloppes.  And to determine whether the reflected rays
converge to the given focus, or diverge from it, it is only
necessary to determine the sign of the distance~$\rho'$, which is
positive in the first case, and negative in the second.

\bigbreak

\centerline{%
III.  \it Surfaces of constant action.}

\nobreak\bigskip

[11.]
We have seen, in the preceding section, that if it be possible to
find a mirror, which shall reflect to a given focus the rays of a
given system, those rays must be perpendicular to a series of
surfaces; and that the whole bent path traversed by the light,
from any one of these perpendicular surfaces to the mirror, and
from the mirror to the focus, is a constant quantity, the same
for all the rays.  Hence it follows, reciprocally, that when rays
issuing from a luminous point have been reflected at a mirror,
the rays of the reflected system are cut perpendicularly by a
series of surfaces; and that these surfaces may be determined, by
taking upon every reflected ray a point such that the whole bent
path from the luminous point to it, may be equal to any constant
quantity.  I am going to shew, in general, that when rays issuing
from a luminous point, or from a perpendicular surface, have been
any number of times reflected, by any combination of mirrors, the
rays of the final system are cut perpendicularly by a series of
surfaces, possessing this remarkable property, that the whole
polygon path traversed by the light, in arriving at any one of
them, is of a constant length, the same for all the rays.

\bigbreak

[12.]
To prove this theorem I observe, that if upon every ray of the
final system we take a point, such that the whole polygon path to
it, from the luminous point or perpendicular surface, may be
equal to any constant quantity, the locus of the points thus
determined will satisfy the differential equation
$$    {d\rho \over dx} \mathbin{.} dX
    + {d\rho \over dy} \mathbin{.} dY
    + {d\rho \over dz} \mathbin{.} dZ
   = 0,
   \eqno {\rm (K)}$$
$X$, $Y$, $Z$, being the coordinates of the point, and $\rho$ the
last side of the polygon; because by hypothesis the variation of
the whole path is nothing, and also that part which arises from
the variation of the first point or origin of the polygon, and by
the principle of least action, the part arising from the
variation of the several points of incidence, is nothing;
therefore the variation arising from the last point of the
polygon must be nothing also, which is the condition expressed by
the equation (K), and which requires either that this last point
should be a fixed focus through which all the rays of the final
system pass, or else that its locus should be a surface cutting
these rays perpendicularly.

\bigbreak

[13.]
We see then that when rays issuing from a luminous point, or from
a perpendicular surface, have been any number of times reflected,
the rays of the final system are cut perpendicularly by that
series of surfaces, for which
$${\textstyle\sum} (\rho) = \hbox{const.},
   \eqno {\rm (L)}$$
$\sum (\rho)$ representing the sum of the several paths or sides
of the polygon.  When we come to consider the systems produced by
ordinary refraction, we shall see that the rays of such a system
are cut perpendicularly by a series of surfaces having for
equation
$${\textstyle\sum} \mathbin{.} (m\rho) = \hbox{const.},$$
$\sum \mathbin{.} (m\rho)$ representing the sum of the several
paths, multiplied each by the refractive power of the medium in
which it lies.  In the systems also, produced by atmospheric and
by extraordinary refraction, there are analogous surfaces
possessing remarkable properties, which render it desirable that
we should agree upon a name by which we may denote them.  Since
then in mechanics the sum obtained by adding the several elements
of the path of a particle, multiplied each by the velocity with
which it is described, is called the {\it Action\/} of the
particle; and since if light be a material substance its velocity
in uncrystallized mediums is proportional to the refractive
power, and is not altered by reflection: I shall call the
surfaces (L) the {\it surfaces of constant action\/}; intending
only to express a remarkable analogy, and not assuming any
hypothesis about the nature or velocity of light.

\bigbreak

[14.]
We have hitherto supposed all the sides of the polygon positive,
that is, we have supposed them all to be actually traversed by
the light.  This is necessarily the case for all the sides
between the first and last; but if the point to which the
last side of the polygon is measured were a focus from which the
final rays diverge, or it it were on a perpendicular surface
situated behind the last mirror, this last side would then be
negative; and in like manner, if the first point, or origin of
the polygon, were a focus to which the first incident rays
converged, or if it were on a perpendicular surface behind the
first mirror, we should have to consider the first side as
negative.  With these modifications the equation (L) represents
all the surfaces that cut the rays perpendicularly; and to mark
the analytic distinction between those which cut the rays
themselves, and those which only cut the rays produced, we may
call the former {\it positive}, and the latter {\it negative\/}:
the positive surfaces of constant action lying at the front of
the mirror, and the negative ones lying at the back of it.

\bigbreak

[15.]
It follows from the preceding theorems, that if with each point
of the last mirror for centre, and with a radius equal to any
constant quantity, increased or diminished by the sum of the
sides of the polygon path, which the light has traversed in
arriving at that point, we construct a sphere, the enveloppe of
these spheres will be a surface cutting the final rays
perpendicularly.  These spheres will also have another enveloppe
perpendicular to the incident rays.  It follows also, that when
rays, either issuing from a luminous point, or perpendicular to a
given surface, have been reflected by any combination of mirrors,
it is always possible to find a focal mirror which shall reflect
the final rays, so as to make them all pass through any given
point; namely, by choosing it so, that the sum of the sides of
the whole polygon path measured to that given focus, and taken
with their proper signs, may be equal to any constant quantity.

\bigbreak

\centerline{%
IV.  \it Classification of Systems of Rays.}

\nobreak\bigskip

[16.]
Before proceeding any farther in our investigations about
reflected systems of rays, it will be useful to make some remarks
upon systems of rays in general, and to fix upon a classification
of such systems which may serve to direct our researches.  By a
{\it Ray}, in this Essay, is meant a line along which light is
propagated; and by a {\it System of Rays\/} is meant an infinite
number of such lines, connected by any analytic law, or any
common property.  Thus, for example, the rays which proceed from
a luminous point in a medium of uniform density, compose one
system of rays; the same rays, after being reflected or
refracted, compose another system.  And when we represent a ray
analytically by two equations between its three coordinates, the
coefficients of those equations will be connected by one or more
relations depending on the nature of the system, so that they may
be considered as functions of one or more arbitrary quantities.
These arbitrary quantities, which enter into the equations of the
ray, may be called its {\it Elements of Position}, because they
serve to particularise its situation in the system to which it
belongs.  And the {\it number\/} of these arbitrary quantities,
or elements of position, is what I shall take for the basis of my
classification of systems of rays; calling a system with one
element of position a {\it system of the First Class\/}: a system
with two elements of position, a {\it system of the Second Class},
and so on.

\bigbreak

[17.]
Thus, if we are considering a system of rays emanating in all
directions from a luminous point $(a,b,c)$, the equations of a
ray are of the form
$$\eqalign{
x - a &= \mu (z - c) \cr
y - b &= \nu (z - c),\cr}$$
which involve only {\it two\/} arbitrary quantities, or elements
of position, namely $\mu$, $\nu$, the tangents of the angles
which the two projections of the ray, on the vertical planes of
coordinates, make with the axis of $(z)$; a system of this kind
is therefore a system of the second class.  If among the rays
thus emanating in all directions from a luminous point
$(a,b,c)$, we consider those only which are contained on a given
plane passing through that point, and having for equation
$$z - c = A (x - a) + B (y - b),$$
then the two quantities $\mu$, $\nu$, are connected by the
relation
$$1 = A \mu + B \nu,$$
so that one only remains arbitrary, and the system is of the
first class.  In general if we consider only those rays which
belong to a given cone, having the luminous point for centre, and
for equation
$${y - b \over z - c} = \phi \left( {x - a \over z - c} \right),$$
$\phi$ denoting any given function, the two quantities $\mu$,
$\nu$, will be connected by the given relation
$$\nu = \phi(\mu),$$
and the system will be of the first class.  If now we suppose a
system of rays thus emanating from a luminous point, to be any
number of times modified by reflection or refraction, it is
evident that the class of the system will not be altered; that
is, there will be the same number of arbitrary constants, or
elements of position, in the final system as in the original
system: provided that we do not take into account the dispersion
of the differently coloured rays.  But if we do take this
dispersion into account, it will introduce in refracted systems a
new element of position depending on the colour of the ray, and
thus will raise the system to a class higher by unity.

\bigbreak

[18.]
From the preceding remarks, it is evident that optics, considered
mathematically, relates for the most part, to the properties of
systems of rays, of the first and second class.  In the third
part of this essay I shall consider the properties of those two
classes in the most general point of view; but at present I shall
confine myself to such as are more immediately connected with
catoptrics.  And I shall begin by making some remarks upon the
general properties of those systems, in which the rays are cut
perpendicularly by a series of surfaces; a system of this kind I
shall call a {\it Rectangular System}.  The properties of such
systems are of great importance in optics; for, by what I have
already proved, they include all systems of rays which after
issuing from a luminous point, or from a perpendicular surface,
have been any number of times reflected, by any combination of
mirrors; we shall see also, in the next part, that they include
also the systems produced by ordinary refraction.

\bigbreak

[19.]
In any system of the second class, a ray may in general be
determined by the condition of passing through an assigned point
of space, for this condition furnishes two equations between the
coefficients of the ray, which are in general sufficient to
determine the two arbitrary elements of position.  We may
therefore consider the cosines $(\alpha, \beta, \gamma)$ of the
angles which the ray makes with the axes as functions of the
coordinates $(x, y, z)$ of any point upon the ray; because, if
the latter be given, the former will be determined.  And if the
system be rectangular, that is, if the rays be cut
perpendicularly by any series of surfaces, it may be proved by
the reasonings in Section~II. of this essay, that these functions
must be of such a nature as to render the formula
$$(\alpha \mathbin{.} dx + \beta \mathbin{.} dy + \gamma \mathbin{.} dz)$$
an exact differential, independently of any relation between
$(x, y, z)$; that is, the cosines $(\alpha, \beta, \gamma)$ of
the angles which the ray passing through any assigned point
$(x, y, z)$ makes with the axes, must be equal to the partial
differential coefficients
$${dV \over dx},\quad {dV \over dy},\quad {dV \over dz},$$
of a function of $(x, y, z)$ considered as three independent
variables.

\bigbreak

[20.]
The properties of any one rectangular system, as distinguished
from another, may all be deduced by analytic reasoning from the
form of the function ($V$); and it is the method of making this
deduction, together with the calculation of the form of the
{\it characteristic function\/} ($V$) for each particular system,
that appears to me to be the most complete and simple definition
that can be given, of the {\it Application of analysis to
optics\/}; so far as regards the systems produced by ordinary
reflection and refraction, which, as I shall shew, are all
rectangular.  And although the systems produced by extraordinary
refraction, do not in general enjoy this property; yet I shall
shew that with respect to them, there exists an analogous
characteristic function, from which all the circumstances of the
system may be deduced: by which means optics acquires, as it
seems to me, an uniformity and simplicity in which it has been
hitherto deficient.

\bigbreak

\centerline{%
V.  \it On the pencils of a Reflected System.}

\nobreak\bigskip

[21.]
When a rectangular system of rays, that is a system the rays of
which are cut perpendicularly by a series of surfaces, is
reflected at a mirror, we have seen that the reflected system is
also rectangular; the rays being cut perpendicularly by the
surfaces of constant action, (III.); and that therefore the
cosines of the angles which a reflected ray makes with the axes,
are equal to the partial differential coefficients of a certain
function~($V$) which I have called the {\it characteristic\/} of
the system, because all the properties of the system may be
deduced from it.  It is this deduction which we now proceed to;
and before we occupy ourselves with the entire reflected system,
we are going to investigate some of the properties of the various
partial systems that can be formed, by establishing any assumed
relation between the rays, that is by considering only those
which are reflected from any assumed curve upon the mirror.

\bigbreak

[22.]
A partial system of this kind, is a {\it system of the first
class\/}; that is, the position of a ray in such a system depends
only on one arbitrary element; for example, on any one coordinate
of the assumed curve upon the mirror.  And if we eliminate this
one element, between the two equations of the ray, we shall
obtain the equation of a surface, which is the locus of the rays
of the partial system that we are considering.  The form of this
surface depends on the arbitrary curve upon the mirror, from which
the rays of the partial system proceed; so that according to the
infinite variety of curves which we can trace upon the given
mirror, we shall have an infinite number of surfaces composed by
rays of a given reflected system.  And since these surfaces
possess many important properties, which render it expedient that
we should denote them by a name, I shall call them
{\it pencils\/}: defining a pencil to be the locus of the rays of
a system of the first class, that is, of a system with but one
arbitrary constant.

\bigbreak

[23.]
Although, as we have seen, an infinite number of pencils may be
formed by the rays of a given reflected system, yet there are
certain properties common to them all, which render them
susceptible of being included in one common analytic expression.
For, if we denote by ($V$) the characteristic function [20.]\ of
the given reflected system, so that
$$\alpha = {dV \over dx},\quad
  \beta  = {dV \over dy},\quad
  \gamma = {dV \over dz},
   \eqno {\rm (M)}$$
$(\alpha, \beta, \gamma)$ being the cosines of the angles which
the reflected ray passing through any assigned point of space
$(x, y, z)$ makes with the axes of coordinates; we shall have,
for all the points of any one ray, the three equations
$${dV \over dx} = \hbox{const.},\quad
  {dV \over dy} = \hbox{const.},\quad
  {dV \over dz} = \hbox{const.},$$
which are equivalent to but two distinct relations, because
$$    \left( {dV \over dx} \right)^2
    + \left( {dV \over dy} \right)^2
    + \left( {dV \over dz} \right)^2
  = \alpha^2 + \beta^2 + \gamma^2 = 1.$$

If then we consider the rays of any of the partial systems,
produced by establishing an arbitrary relation between the rays
of the entire reflected system; the locus of these rays, that is,
the pencil of this partial system, will have for its equation
$${dV \over dy} = f \left( {dV \over dx} \right),
   \eqno {\rm (N)}$$
$f$ representing an arbitrary function, the form of which depends
upon the nature of the partial system.

\bigbreak

[24.]
The form of this function ($f$) may be determined, if we know any
curve through which the rays of the pencil pass, or any surface
which they envelope.  For first, the latter of these two
questions may be reduced to the former, by determining upon the
enveloped surface the locus of the points of contact; this is
done by means of the formula
$$    {dV \over dx} \mathbin{.} {du \over dx}
    + {dV \over dy} \mathbin{.} {du \over dy}
    + {dV \over dz} \mathbin{.} {du \over dz}
  = 0,$$
which expresses that the rays of the unknown pencil are tangents
to the given enveloped surface $u = 0$.  And when we know a curve
$u = 0$, $v = 0$, through which all the rays of the pencil pass,
we have only to eliminate $(x, y, z)$ between the two equations
of this curve, and the two following,
$$\alpha = {dV \over dx},\quad\beta  = {dV \over dy};$$
we shall thus obtain the relation between $(\alpha, \beta)$ which
characterises the rays that pass through the given curve: and
substituting, in this relation, the values of $(\alpha, \beta)$
considered as functions of $(x, y, z)$ we shall have the equation
of the pencil.

In this manner we can determine the shadow of any opaque body,
produced by the rays of a given reflected system, if we know the
equation of the body, and that of the skreen upon which the
shadow is thrown; we can also determine the boundary of light and
darkness upon the body, which is the curve of contact with the
enveloping pencil; and if we consider visual instead of luminous
rays, we can determine, on similar principles, the perspective of
reflected light, that is, the apparent form and magnitude of a
body seen by any combination of mirrors; at least so far as that
form and magnitude depend on the shape and size of the visual
cone.

\bigbreak

[25.]
Besides the general analytic expression
$${dV \over dy} = f \left( {dV \over dx} \right),
   \eqno {\rm (N)}$$
which represents all the pencils of the system, by means of the
arbitrary function ($f$), we can also find another analytic
expression for those pencils, by eliminating that arbitrary
function, and introducing instead of it the partial differential
coefficients of the pencil of the first order.  In this manner we
find, by differentiating (N) for ($x$) and ($y$) successively,
and eliminating the differential coefficient of the arbitrary
function ($f$),
$$\eqalign{
{d^2 V \over dx^2} \mathbin{.} {d^2 V \over dy^2}
      - \left( {d^2 V \over dx \mathbin{.} dy} \right)^2
   &= \mathbin{\phantom{+}}
         \left\{
            {d^2 V \over dx \mathbin{.} dy}
            \mathbin{.}
            {d^2 V \over dy \mathbin{.} dz}
          - {d^2 V \over dy^2}
            \mathbin{.}
            {d^2 V \over dx \mathbin{.} dz}
         \right\}
         {dz \over dx} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left\{
            {d^2 V \over dx \mathbin{.} dy}
            \mathbin{.}
            {d^2 V \over dx \mathbin{.} dz}
          - {d^2 V \over dx^2}
            \mathbin{.}
            {d^2 V \over dy \mathbin{.} dz}
         \right\}
         {dz \over dy};\cr}$$
and since the general relation
$\alpha^2 + \beta^2 + \gamma^2 = 1$, that is
$$    \left( {dV \over dx} \right)^2
    + \left( {dV \over dy} \right)^2
    + \left( {dV \over dz} \right)^2
   = 1,$$
gives by differentiation
$$\eqalign{
         \alpha . {d^2 V \over dx^2}
       + \beta  . {d^2 V \over dx \mathbin{.} dy}
       + \gamma . {d^2 V \over dx \mathbin{.} dz}
   &= 0,\cr
         \alpha . {d^2 V \over dx \mathbin{.} dy}
       + \beta  . {d^2 V \over dy^2}
       + \gamma . {d^2 V \over dy \mathbin{.} dz}
   &= 0,\cr}$$
and therefore
$$\eqalign{
         {d^2 V \over dx \mathbin{.} dy}
         \mathbin{.}
         {d^2 V \over dy \mathbin{.} dz}
       - {d^2 V \over dy^2}
         \mathbin{.}
         {d^2 V \over dx \mathbin{.} dz}
   &= {\alpha \over \gamma} \mathbin{.}
         \left\{
            {d^2 V \over dx^2} \mathbin{.} {d^2 V \over dy^2}
          - \left( {d^2 V \over dx \mathbin{.} dy} \right)^2
         \right\},\cr
         {d^2 V \over dx \mathbin{.} dy}
         \mathbin{.}
         {d^2 V \over dx \mathbin{.} dz}
       - {d^2 V \over dx^2}
         \mathbin{.}
         {d^2 V \over dy \mathbin{.} dz}
   &= {\beta  \over \gamma} \mathbin{.}
         \left\{
            {d^2 V \over dx^2} \mathbin{.} {d^2 V \over dy^2}
          - \left( {d^2 V \over dx \mathbin{.} dy} \right)^2
         \right\},\cr}$$
the partial differential equation of the pencils becomes finally
$$    {\alpha \over \gamma} \mathbin{.} {dz \over dx}
    + {\beta  \over \gamma} \mathbin{.} {dz \over dy}
   = 1,
   \eqno {\rm (O)}$$
which expresses that the tangent plane to the pencil contains the
ray passing through the point of contact.

\bigbreak

\centerline{%
VI.  \it On the developable pencils, the two foci of a ray, and the
caustic curves and surfaces.}

\nobreak\bigskip

[26.]
Among all the pencils of a given rectangular system, there is
only a certain series developable; namely, those which pass
through the lines of curvature on the surfaces that cut the rays
perpendicularly.  It follows from the known properties of normals
to surfaces, that each ray has two of these developable pencils
passing through it, and is therefore a common tangent to
{\it two caustic curves}, the {\it ar\^{e}tes de rebroussement\/}
of these pencils; the points in which it touches those two
caustic curves may be called the {\it two foci\/} of the ray; and
the locus of these foci forms {\it two caustic surfaces}, touched
by all the rays.

\bigbreak

[27.]
To determine analytically these several properties of the system,
let us represent by $(a, b, c)$ the coordinates of the point in
which a ray crosses a given perpendicular surface; these
coordinates will be determined, if the ray be given, so that they
may be considered as functions of $(\alpha, \beta)$; we may
therefore put their differentials under the form
$$\eqalign{
da &=    {da \over d\alpha} \mathbin{.} d\alpha
       + {da \over d\beta}  \mathbin{.} d\beta,\cr
db &=    {db \over d\alpha} \mathbin{.} d\alpha
       + {db \over d\beta}  \mathbin{.} d\beta,\cr
dc &=    {dc \over d\alpha} \mathbin{.} d\alpha
       + {dc \over d\beta}  \mathbin{.} d\beta.\cr}$$
We have also
$\alpha \, da + \beta \, db + \gamma \, dc = 0$, which gives
$$\eqalign{
{dc \over d\alpha}
   &=  - \left(
            {\alpha \over \gamma} \mathbin{.} {da \over d\alpha}
          + {\beta  \over \gamma} \mathbin{.} {db \over d\alpha}
         \right),\cr
{dc \over d\beta}
   &=  - \left(
            {\alpha \over \gamma} \mathbin{.} {da \over d\beta}
          + {\beta  \over \gamma} \mathbin{.} {db \over d\beta}
         \right);\cr}$$
with respect to the coefficients
$\displaystyle {da \over d\alpha}$,
$\displaystyle {da \over d\beta}$,
$\displaystyle {db \over d\alpha}$,
$\displaystyle {db \over d\beta}$,
these are to be determined by differentiating the two following
equations,
$${dV \over da} = \alpha,\quad {dV \over db} = \beta,$$
which give
$$\eqalign{
      {d^2 V \over da^2} \mathbin{.} da
    + {d^2 V \over da \mathbin{.} db} \mathbin{.} db
    + {d^2 V \over da \mathbin{.} dc} \mathbin{.} dc
   &= d\alpha,\cr
      {d^2 V \over da \mathbin{.} db} \mathbin{.} da
    + {d^2 V \over db^2} \mathbin{.} db
    + {d^2 V \over db \mathbin{.} dc} \mathbin{.} dc
   &= d\beta,\cr}$$
and therefore
$$\eqalign{
M \mathbin{.} da
   &=    \left(
            \gamma \mathbin{.} {d^2 V \over db^2}
          - \beta  \mathbin{.} {d^2 V \over db \mathbin{.} dc}
         \right) \mathbin{.} d\alpha
       + \left(
            \beta  \mathbin{.} {d^2 V \over da \mathbin{.} dc}
          - \gamma \mathbin{.} {d^2 V \over da \mathbin{.} db}
         \right) \mathbin{.} d\beta,\cr
M \mathbin{.} db
   &=    \left(
            \alpha \mathbin{.} {d^2 V \over db \mathbin{.} dc}
          - \gamma \mathbin{.} {d^2 V \over da \mathbin{.} db}
         \right) \mathbin{.} d\alpha
       + \left(
            \gamma \mathbin{.} {d^2 V \over da^2}
          - \alpha \mathbin{.} {d^2 V \over da \mathbin{.} dc}
         \right) \mathbin{.} d\beta,\cr}$$
if we put for abridgment
$$\eqalign{
M  &= \mathbin{\phantom{+}}
         \alpha \mathbin{.}
         \left\{
            {d^2 V \over da \mathbin{.} db}
            \mathbin{.}
            {d^2 V \over db \mathbin{.} dc}
          - {d^2 V \over da \mathbin{.} dc}
            \mathbin{.}
            {d^2 V \over db^2}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \beta  \mathbin{.}
         \left\{
            {d^2 V \over da \mathbin{.} db}
            \mathbin{.}
            {d^2 V \over da \mathbin{.} dc}
          - {d^2 V \over db \mathbin{.} dc}
            \mathbin{.}
            {d^2 V \over da^2}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \gamma \mathbin{.}
         \left\{
            {d^2 V \over da^2}
            \mathbin{.}
            {d^2 V \over db^2}
          - \left( {d^2 V \over da \mathbin{.} db} \right)^2
         \right\}.\cr}$$

This being laid down, let $(x, y, z)$ be any other point upon the
ray, at a distance ($\rho$) from the given perpendicular surface;
we shall have
$$x = a + \alpha \rho,\quad
  y = b + \beta  \rho,\quad
  z = c + \gamma \rho,$$
$$d\rho = \alpha \, dx + \beta \, dy + \gamma \, dz,$$
$$dx - \alpha \, d\rho = da + \rho \, d\alpha,\quad
  dy - \beta  \, d\rho = db + \rho \, d\beta:$$
and if the coordinates $(x, y, z)$ belong to a caustic curve, the
first members of these two last equation will vanish, so that on
this hypothesis,
$$da + \rho \mathbin{.} d\alpha = 0,\quad
  db + \rho \mathbin{.} d\beta = 0;$$
eliminating ($\rho$) we find, for the developable pencils,
$$da \mathbin{.} d\beta - db \mathbin{.} d\alpha = 0,
   \eqno {\rm (P)}$$
and eliminating
$\displaystyle {d\beta \over d\alpha}$
we find, for the caustic surfaces,
$$    \left( \rho + {da \over d\alpha} \right)
      \left( \rho + {db \over d\beta}  \right)
    - {da \over d\beta} \mathbin{.} {db \over d\alpha}
   = 0;
   \eqno {\rm (Q)}$$
substituting for
$\displaystyle {da \over d\alpha}$,
$\displaystyle {da \over d\beta}$,
$\displaystyle {db \over d\alpha}$,
$\displaystyle {db \over d\beta}$,
their values, and observing that by the general relation
$$    \left( {dV \over da} \right)^2
    + \left( {dV \over db} \right)^2
    + \left( {dV \over dc} \right)^2
  = 1,$$
we have
$$\eqalign{
      \alpha \mathbin{.} {d^2 V \over da^2}
    + \beta  \mathbin{.} {d^2 V \over da \mathbin{.} db}
    + \gamma \mathbin{.} {d^2 V \over da \mathbin{.} dc}
   &= 0,\cr
      \alpha \mathbin{.} {d^2 V \over da \mathbin{.} db}
    + \beta  \mathbin{.} {d^2 V \over db^2}
    + \gamma \mathbin{.} {d^2 V \over db \mathbin{.} dc}
   &= 0,\cr
      \alpha \mathbin{.} {d^2 V \over da \mathbin{.} dc}
    + \beta  \mathbin{.} {d^2 V \over db \mathbin{.} dc}
    + \gamma \mathbin{.} {d^2 V \over dc^2}
   &= 0,\cr}$$
$$\gamma \mathbin{.} M
  =   {d^2 V \over da^2}
      \mathbin{.}
      {d^2 V \over db^2}
    - \left( {d^2 V \over da \mathbin{.} db} \right)^2,$$
we find this other form for the equation of the caustic surfaces,
$${1 \over \gamma^2}
      \left\{
         {d^2 V \over da^2}
         \mathbin{.}
         {d^2 V \over db^2}
       - \left( {d^2 V \over da \mathbin{.} db} \right)^2
      \right\} \mathbin{.} \rho^2
    + \left\{
         {d^2 V \over da^2}
       + {d^2 V \over db^2}
       + {d^2 V \over dc^2}
      \right\} \mathbin{.} \rho
    + 1
  = 0.
   \eqno {\rm (R)}$$

\bigbreak

[28.]
The manner in which these formul{\ae} are to be employed is
evident.  We are to integrate (P) considered as a differential
equation, of the first order and second degree, between
$\alpha$,~$\beta$, or between the corresponding functions of
$x$,~$y$,~$z$,
$$\alpha = {dV \over dx},\quad\beta = {dV \over dy};$$
the integral will be of the form
$${dV \over dy} = f \left( {dV \over dx}, C \right),$$
$C$ being an arbitrary constant; the condition of passing through
a given ray will determine the two values of this constant,
corresponding to the two developable pencils: and the equations
of the caustic curves, considered as the {\it ar\^{e}tes de
rebroussement\/} of those pencils, will follow by the known
methods from the equations of the pencils themselves.  The
points in which a given ray touches these caustic curves, that is
the two foci of the ray, are determined, without any integration,
by means of (Q) or (R); and thus we can determine, by elimination
alone, the equations of the two caustic surfaces, the locus of
these points or foci.

\bigbreak

[29.]
In the preceding reasonings, we have supposed given the form of
the characteristic function $V$, whose partial differential
coefficients of the first order, are equal to the cosines of the
angles that the reflected ray makes with the axes; let us now see
how the partial differential coefficients of the second order of
this function, which enter into the formul{\ae} that we have
found for the developable pencils and for the caustic surfaces,
depend on the curvature of the mirror, and on the characteristic
function of the incident system.  Let ($V'$) represent this
latter function, so that we shall have
$$\alpha' = {dV' \over dx},\quad
  \beta'  = {dV' \over dy},\quad
  \gamma' = {dV' \over dz},$$
$\alpha'$, $\beta'$, $\gamma'$ being the cosines of the angles
that the incident ray, measured from the mirror, makes with the
axes of coordinates; and let $p$, $q$, $r$, $s$, $t$, be the
partial differential coefficients of the mirror, of the first and
second orders, so that
$$dz = p \, dx + q \, dy,\quad
  dp = r \, dx + s \, dy,\quad
  dq = s \, dx + t \, dy,$$
$x$,~$y$,~$z$ being the coordinates of the mirror.  Then, by the
first section of this essay we shall have the two equations
$$\alpha + \alpha' + p(\gamma + \gamma') = 0,\quad
  \beta  + \beta'  + q(\gamma + \gamma') = 0,$$
which give by differentiation,
$$\eqalign{
0 &= (\gamma + \gamma') \mathbin{.} r
      + {d^2 V \over dx^2} + {d^2 V' \over dx^2}
      + 2p \mathbin{.}
         \left(
            {d^2 V  \over dx \mathbin{.} dz}
          + {d^2 V' \over dx \mathbin{.} dz}
         \right)
      + p^2 \mathbin{.}
         \left(
            {d^2 V  \over dz^2}
          + {d^2 V' \over dz^2}
         \right),\cr
0 &= (\gamma + \gamma') \mathbin{.} t
      + {d^2 V \over dy^2} + {d^2 V' \over dy^2}
      + 2q \mathbin{.}
         \left(
            {d^2 V  \over dy \mathbin{.} dz}
          + {d^2 V' \over dy \mathbin{.} dz}
         \right)
      + q^2 \mathbin{.}
         \left(
            {d^2 V  \over dz^2}
          + {d^2 V' \over dz^2}
         \right),\cr
0 &= (\gamma + \gamma') \mathbin{.} s
      + {d^2 (V + V') \over dx \mathbin{.} dy}
      + p \mathbin{.} {d^2 (V + V')  \over dy \mathbin{.} dz}
      + q \mathbin{.} {d^2 (V + V')  \over dx \mathbin{.} dz}
      + pq \mathbin{.} {d^2 (V + V')  \over dz^2}.\cr}$$
Combining these three equations with the three which result from
differentiating the equation
$$    \left( {dV \over dx} \right)^2
    + \left( {dV \over dy} \right)^2
    + \left( {dV \over dz} \right)^2
   = 1$$
we shall have the partial differential coefficients, second
order, of $V$, when we know those of $V'$ and of $z$, that is,
when we know the incident system and the mirror: it will then
remain to substitute them in the formul{\ae} of the preceding
paragraph, in order to find the developable pencils, and the
caustic surfaces, in which we may change the partial differential
coefficients of $V$, taken with respect to $(a, b, c)$, to the
corresponding coefficients with respect to $(x, y, z)$.

\bigbreak

[30.]
Suppose, to give an example of the application of the preceding
reasonings, that the incident rays are parallel, and that we take
for the axes of ($x$) and ($y$), the tangents to the lines of
curvature on the mirror at the point of incidence, so that the
normal at that point shall be vertical; the partial differential
coefficients of the second order of ($V'$) will vanish, and we
shall have
$$x = 0,\quad y = 0,\quad z = 0,\quad
  p = 0,\quad q = 0,\quad s = 0,$$
$$\alpha + \alpha' = 0,\quad
  \beta  + \beta'  = 0,\quad
  \gamma = \gamma' = \cos I,$$
$I$ being the angle of incidence; the formul{\ae} for the partial
differential coefficients of the second order of ($V$) become
$${d^2 V \over dx^2} = - 2 \gamma r,\quad
  {d^2 V \over dx \mathbin{.} dy} = 0,\quad
  {d^2 V \over dy^2} = - 2 \gamma t,$$
$${d^2 V \over dx \mathbin{.} dz} = 2 \alpha r,\quad
  {d^2 V \over dy \mathbin{.} dz} = 2 \beta  t,\quad
  {d^2 V \over dz^2}
    = - {2  (\alpha^2 r + \beta^2 t ) \over \gamma},$$
and the formula (R) for the two foci, which may be thus written
$${\rho^2 \over \gamma^2}
      \left\{
         {d^2 V \over dx^2}
         \mathbin{.}
         {d^2 V \over dy^2}
       - \left( {d^2 V \over dx \mathbin{.} dy} \right)^2
      \right\}
    + \rho \mathbin{.}
      \left\{
         {d^2 V \over dx^2}
       + {d^2 V \over dy^2}
       + {d^2 V \over dz^2}
      \right\}
    + 1
  = 0,
   \eqno {\rm (R)}$$
becomes
$$4rt \mathbin{.} \rho^2
      - {2 \rho \over \gamma} \mathbin{.}
      \{ (\alpha^2 + \gamma^2) r + (\beta^2 + \gamma^2) t \}
      + 1
  = 0.
   \eqno {\rm (S)}$$
If the incident rays be perpendicular to the mirror, at the given
point of incidence, then
$$\gamma = 1,\quad\alpha = 0,\quad\beta = 0,$$
and the two roots of (S) are
$$\rho = {1 \over 2} \mathbin{.} {1 \over r},\quad
  \rho = {1 \over 2} \mathbin{.} {1 \over t},$$
that is the two focal distances are the halves of the two radii
of curvature of the mirror.

If without being perpendicular to the mirror, the incident ray is
contained in the plane of $(xz)$, that is in the plane of the
greatest or the least osculating circle to the mirror, we shall
have
$\beta = 0$, $\alpha^2 + \gamma^2 = 1$, and the two roots of (S)
will be
$$\rho = {1 \over 2} \mathbin{.} \gamma \mathbin{.} {1 \over r},\quad
  \rho = {1 \over 2} \mathbin{.} {1 \over \gamma} \mathbin{.} {1 \over t};$$
the first root is quarter of the chord of curvature, that is,
quarter of the portion of the reflected ray intercepted within
the osculating circle before mentioned; and the other root is
equal to the distance of the point, where the reflected ray meets
a parallel to the incident rays, passing through the centre of
the other osculating circle.  In general, it will appear, when we
come to treat of osculating focal mirrors, that the two foci
determined by the formula (S), are the foci of the greatest and
least paraboloids of revolution which, having their axis parallel
to the incident rays, osculate to the mirror at the point of
incidence.

\bigbreak

[31.]
I shall conclude this section by remarking, that the equation of
the caustic surfaces is a singular primitive of the partial
differential equation (O), which we found in the preceding
section to represent all the pencils of the system, and that the
equations,
$${dx \over \alpha} = {dy \over \beta} = {dz \over \gamma},$$
of which the complete integral represents all the rays, are also
satisfied, as a singular solution, by the equations of the
caustic curves: from which it may be proved, that the portion of
any ray, or the arc of any caustic curve, intercepted between any
two given points, is equal to the increment that the
characteristic function ($V$) receives in passing from the one
point to the other.

\bigbreak

\centerline{%
VII.  \it Lines of Reflection on a mirror.}

\nobreak\bigskip

[32.]
We have seen that the rays of a reflected system are in general
tangents to two series of caustic curves, and compose two
corresponding series of developable pencils; the intersections of
these pencils with the mirror, form two series of remarkable
curves upon that surface, which were first discovered by Malus,
and which were called by him the {\it Lines of Reflexion}.  We
propose, in the present section to investigate the differential
equation of these curves, and some of their principal properties;
and at the same time to make some additional remarks, on the
manner of calculating the foci, and the caustic surfaces.

\bigbreak

[33.]
To find the differential equation of the curves of reflexion, we
may employ the formula of the preceding section,
$$\eqalignno{
   &\mathrel{\phantom{=}}
      d\beta \mathbin{.}
      \left\{
         \left(
            \beta  \mathbin{.} {d^2 V \over dx \mathbin{.} dz}
          - \gamma \mathbin{.} {d^2 V \over dx \mathbin{.} dy}
         \right) \, d\beta
       + \left(
            \gamma \mathbin{.} {d^2 V \over dy^2}
          - \beta  \mathbin{.} {d^2 V \over dy \mathbin{.} dz}
         \right) \, d\alpha
      \right\} \cr
   &= d\alpha \mathbin{.}
      \left\{
         \left(
            \gamma \mathbin{.} {d^2 V \over dx^2}
          - \alpha \mathbin{.} {d^2 V \over dx \mathbin{.} dz}
         \right) \, d\beta
       + \left(
            \alpha \mathbin{.} {d^2 V \over dy \mathbin{.} dz}
          - \gamma \mathbin{.} {d^2 V \over dx \mathbin{.} dy}
         \right) \, d\alpha
      \right\},
   &{\rm (P)}\cr}$$
considering $(\alpha, \beta, \gamma)$ as given functions of the
coordinates of the point of incidence, such that
$$\eqalign{
d\alpha
   &=    {d^2 V \over dx^2}              \mathbin{.} dx
       + {d^2 V \over dx \mathbin{.} dy} \mathbin{.} dy
       + {d^2 V \over dx \mathbin{.} dz} \mathbin{.} dz,\cr
d\beta
   &=    {d^2 V \over dx \mathbin{.} dy} \mathbin{.} dx
       + {d^2 V \over dy^2}              \mathbin{.} dy
       + {d^2 V \over dy \mathbin{.} dz} \mathbin{.} dz,\cr}$$
and deducing the partial differential coefficients of the
characteristic function $V$, either immediately from the form of
that function itself, if it be given, or from the equation of the
mirror and from the nature of the incident system, according to
the method already explained.  But in this latter case, that is,
when we are only given the incident system and the mirror, it
will be simpler to treat the question immediately, by reasonings
analogous to those by which the formula (P) was deduced.

Let, therefore, $X$,~$Y$,~$Z$, represent the coordinates of a
point upon a caustic curve, at a distance ($\rho$) from the
mirror; we shall have
$$X = x + \alpha \rho,\quad
  Y = y + \beta  \rho,\quad
  Z = z + \gamma \rho,$$
$$d\rho =   \alpha \mathbin{.} d(X - x)
          + \beta  \mathbin{.} d(Y - y)
          + \gamma \mathbin{.} d(Z - z),$$
$$\eqalign{
dX &= \alpha \mathbin{.}
         ( \alpha \, dX + \beta  \, dY + \gamma \, dZ ),\cr
dY &= \beta  \mathbin{.}
         ( \alpha \, dX + \beta  \, dY + \gamma \, dZ ),\cr
dZ &= \gamma \mathbin{.}
         ( \alpha \, dX + \beta  \, dY + \gamma \, dZ ),\cr}$$
$$\eqalign{
dx - \alpha \mathbin{.}
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      + \rho \, d\alpha
   &= 0,\cr
dy - \beta  \mathbin{.}
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      + \rho \, d\beta
   &= 0,\cr
dz - \gamma \mathbin{.}
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      + \rho \, d\gamma
   &= 0;\cr}$$
eliminating $d\alpha$, $d\beta$, $d\gamma$, by these equations,
from those which are obtained by differentiating the formul{\ae}
already found,
$$\alpha + \alpha' + p (\gamma + \gamma') = 0,\quad
  \beta  + \beta'  + q (\gamma + \gamma') = 0,$$
we get the two following equations,
$$\left. \eqalign{
0  &= \rho \mathbin{.}
         \{ (\gamma + \gamma') \, dp + d\alpha'
            + p \mathbin{.} d\gamma' \}
      + (\alpha + \gamma p)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      - (dx + p \, dz),\cr
0  &= \rho \mathbin{.}
         \{ (\gamma + \gamma') \, dq + d\beta'
            + q \mathbin{.} d\gamma' \}
      + (\beta  + \gamma q)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      - (dy + q \, dz),\cr}
   \right\}
   \eqno {\rm (T)}$$
which give by elimination of $\rho$, the following general
equation for the lines of reflexion,
$$   {(\gamma + \gamma') \, dq + d\beta'  + q \mathbin{.} d\gamma'
      \over
      (\gamma + \gamma') \, dp + d\alpha' + p \mathbin{.} d\gamma'}
   = {(\beta  + \gamma q)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      - (dy + q \, dz)
      \over
      (\alpha + \gamma p)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz )
      - (dx + p \, dz)}.
   \eqno {\rm (U)}$$

\bigbreak

[34.]
Suppose, to give an example, that the incident rays are parallel,
and that the axes of coordinates are chosen as in [30.], the normal
at some given point of incidence for the axis of ($z$), and the
tangents to the lines of curvature for the axes of ($x$) and
($y$); our general formula (U) will then become
$${t \mathbin{.} dy \over r \mathbin{.} dx}
   = {\beta  \mathbin{.} (\alpha \, dx + \beta  \, dy) - dy
      \over
      \alpha \mathbin{.} (\alpha \, dx + \beta  \, dy) - dx};$$
that is
$$\alpha \beta \mathbin{.}
      (t \mathbin{.} dy^2 - r \mathbin{.} dx^2)
   - \{ (\beta^2 + \gamma^2) t - (\alpha^2 + \gamma^2) r \}
         \, dx \mathbin{.} dy
   = 0.
   \eqno {\rm (V)}$$

We shall see, in the next section, that the two directions
determined by this formula, are the directions of osculation of
the greatest and least paraboloids, which, having their axes
parallel to the incident rays, osculate to the mirror at the
point of incidence; in the mean time we may remark, that if the
plane of incidence coincides with either the plane of the
greatest or the least osculating circle to the mirror, or if the
point of incidence be a point of spheric curvature, one of the
two directions of the lines of reflexion is contained in the
plane of incidence, while the other is perpendicular to that
plane; and it is easy to prove, by means of the formula (V), that
these are the only cases in which the lines of reflexion are
perpendicular to one another, the incident rays being parallel.

\bigbreak

[35.]
The formul{\ae} (T) determine not only, as we have seen, the
lines of reflexion, but also the two focal distances, and
therefore the caustic surfaces.  For as, by elimination of
($\rho$), they conduct to the differential equation of the lines
of reflexion, so by elimination of the differentials they conduct
to a quadratic equation in ($\rho$), which is equivalent to the
formula (R), and which determines the two focal distances.  As an
example of this, let us take the following general problem, to
find the caustic surfaces and lines of reflexion of a mirror,
when the incident rays diverge from a given luminous point
$X'$,~$Y'$,~$Z'$.  We have here
$$X' = x + \alpha' \rho',\quad
  Y' = y + \beta'  \rho',\quad
  Z' = z + \gamma' \rho',$$
$\rho'$ being the distance of the luminous point from the mirror;
$$d\rho' = - (\alpha' \, dx + \beta'  \, dy + \gamma' \, dz),$$
$$- \rho' \mathbin{.} d\alpha' = dx + \alpha' \mathbin{.} d\rho',\quad
  - \rho' \mathbin{.} d\beta'  = dy + \beta'  \mathbin{.} d\rho',\quad
  - \rho' \mathbin{.} d\gamma' = dz + \gamma' \mathbin{.} d\rho',$$
and because
$$\alpha' + \gamma' p = - (\alpha + \gamma p),\quad
  \beta'  + \gamma' q = - (\beta  + \gamma q),$$
$$\alpha' \, dx + \beta' \, dy + \gamma' \, dz
   = - ( \alpha \, dx + \beta \, dy + \gamma \, dz ),$$
the equations (T) become
$$\eqalign{
(\gamma + \gamma') \mathbin{.} dp
   &= \left( {1 \over \rho} + {1 \over \rho'} \right)
      \{ dx + p \, dz
          - (\alpha + \gamma p)
            (\alpha \, dx + \beta \, dy + \gamma \, dz) \},\cr
(\gamma + \gamma') \mathbin{.} dq
   &= \left( {1 \over \rho} + {1 \over \rho'} \right)
      \{ dy + q \, dz
          - (\beta  + \gamma q)
            (\alpha \, dx + \beta \, dy + \gamma \, dz) \}:\cr}$$
eliminating $\rho$, we find, for the lines of reflexion,
$$\eqalignno{
   &\mathrel{\phantom{=}}
      dq \mathbin{.}
      \{ dx + p \, dz
          - (\alpha + \gamma p)
            (\alpha \, dx + \beta \, dy + \gamma \, dz) \} \cr
   &= dp \mathbin{.}
      \{ dy + q \, dz
          - (\beta  + \gamma q)
            (\alpha \, dx + \beta \, dy + \gamma \, dz) \},
   &{\rm (W)}\cr}$$
and eliminating the differentials, we find, for the focal
distances,
$$\eqalignno{
   &\mathrel{\phantom{=}}
      \left\{
         (1 + p^2 - (\alpha + \gamma p)^2 )
         \left( {1 \over \rho} + {1 \over \rho'} \right)
       - (\gamma + \gamma') . r
      \right\} \cr
   &\times
      \left\{
         (1 + q^2 - (\beta  + \gamma q)^2 )
         \left( {1 \over \rho} + {1 \over \rho'} \right)
       - (\gamma + \gamma') . t
      \right\} \cr
   &=
      \left\{
         (pq - (\alpha + \gamma p) (\beta  + \gamma q) )
         \left( {1 \over \rho} + {1 \over \rho'} \right)
       - (\gamma + \gamma') . s
      \right\}^2.
   &{\rm (X)}\cr}$$

We may remark, that since ($\rho'$) has disappeared from the
equation ($W$) of the lines of reflexion, the direction of those
lines at any given point upon the mirror depends only on the
direction of the incident ray, and not on the distance of the
luminous point; we see also, from the form of the equation (X),
that the harmonic mean between that distance ($\rho'$) and either
of the two focal distances ($\rho$), does not depend on
($\rho'$): so that if the luminous point were to move along the
incident ray, the two foci of the reflected ray would indeed
change position, but the line joining each to the luminous point,
would constantly pass through the same fixed point upon the
normal.

\bigbreak

\centerline{%
VIII.  \it On osculating focal mirrors.}

\nobreak\bigskip

[36.]
It has long been known that a paraboloid of revolution possesses
the property of reflecting to its focus, rays which are incident
parallel to its axis; and that an ellipsoid in like manner will
reflect to one of its two foci, rays that diverge from the other:
but I do not know that any one has hitherto applied these
properties of accurately reflecting mirrors, to the investigation
of the caustic surfaces, and lines of reflexion of mirrors in
general.  There exists however a remarkable connexion between
them, analogous to the connexion between the properties of
spheres and of normals; and it is this connexion, not only for
paraboloids and ellipsoids, but also for that general class of
focal mirrors, pointed out in Section~II.\ of this Essay, that we
are now going to consider.

\bigbreak

[37.]
To begin with the simplest case, I observe that the general
equation of a paraboloid of revolution may be put under the form
$$\rho = P  + \alpha' \mathbin{.} (x - X)
            + \beta'  \mathbin{.} (y - Y)
            + \gamma' \mathbin{.} (z - Z),$$
($P$) being the semiparameter, ($\rho$) the distance from the
focus $(X, Y, Z)$, and $\alpha'$,~$\beta'$,~$\gamma'$, the
cosines of the angles which the axis of the paraboloid, measured
from the vertex, makes with the axes of coordinates: and that the
partial differential coefficients of ($z$), of the first and
second orders, which we shall denote by
$(p', q', r', s', t')$, are determined by the following
equations,
$$\eqalign{
x - X + p' \mathbin{.} (z - Z)
   &= \rho \mathbin{.} (\alpha' + \gamma' p'),\cr
y - Y + q' \mathbin{.} (z - Z)
   &= \rho \mathbin{.} (\beta'  + \gamma' q'),\cr
1 + p'^2 + r' \mathbin{.} (z - Z)
   &= \rho \gamma' r' + (\alpha' + \gamma' p')^2,\cr
1 + q'^2 + t' \mathbin{.} (z - Z)
   &= \rho \gamma' t' + (\beta'  + \gamma' q')^2,\cr
p' q'    + s' \mathbin{.} (z - Z)
   &= \rho \gamma' s' + (\alpha' + \gamma' p') (\beta'  + \gamma' q').\cr}$$

This being laid down, if we suppose that the three constants
$(\alpha', \beta', \gamma')$ determined by the condition that the
axis of the paraboloid shall be parallel to a given system of
incident rays, we may propose to determine the other four
constants ($X$,~$Y$,~$Z$,~$P$) by the condition of osculating to
a given mirror, at a given point, in a given direction.  The
condition of passing through the given point, will serve to
determine, or rather eliminate ($P$), and the condition of
contact produces the two equations
$$p' = p,\quad q' = q,$$
which express that the focus of the paraboloid is somewhere on
the reflected ray, and which are therefore equivalent to the
three following,
$$X - x = \alpha \rho,\quad
  Y - y = \beta  \rho,\quad
  Z - z = \gamma \rho,$$
$(\alpha, \beta, \gamma)$ being the cosines of the angles which
the reflected ray makes with the axes.  To determine the
remaining constant ($\rho$), by the condition that the paraboloid
shall osculate to the mirror in a given direction, we are to
employ the formula
$$(r' - r) \, dx^2 + 2 (s' - s) dx \mathbin{.} dy
      + (t' - t) \, dy^2 = 0,$$
$r$,~$s$,~$t$, being the given partial differential coefficients,
second order, of the mirror, and $r'$,~$s'$,~$t'$, the
corresponding coefficients of the paraboloid, which involve the
unknown distance ($\rho$), being determined by the equations,
$$\eqalign{
\rho \mathbin{.} (\gamma + \gamma') \mathbin{.} r'
   &= 1 + p^2 - (\alpha' + \gamma' p)^2,\cr
\rho \mathbin{.} (\gamma + \gamma') \mathbin{.} s'
   &= pq      - (\alpha' + \gamma' p) (\beta'  + \gamma' q),\cr
\rho \mathbin{.} (\gamma + \gamma') \mathbin{.} t'
   &= 1 + q^2 - (\beta'  + \gamma' q)^2.\cr}$$
To simplify our calculations, let us, as in [30.], take the normal
to the mirror for the axis of ($z$), and the tangents to the
lines of curvature for the axes of ($x$) and ($y$); we shall then
have
$$p = 0,\quad q = 0,\quad s = 0,\quad
  \alpha + \alpha' = 0,\quad
  \beta  + \beta'  = 0,\quad
  \gamma = \gamma',$$
$$2 \gamma \rho . r' = \beta^2  + \gamma^2,\quad
  2 \gamma \rho . s' = - \alpha \beta,\quad
  2 \gamma \rho . t' = \alpha^2 + \gamma^2,$$
and the condition of osculation becomes
$$2 \gamma \rho \mathbin{.} (r + t \tau^2)
   = \beta^2 + \gamma^2 - 2 \alpha \beta \tau
      + (\alpha^2 + \gamma^2) \tau^2,
   \eqno {\rm (Y)}$$
if we put $dy = \tau \mathbin{.} dx$.  This formula (Y)
determines the osculating paraboloid for any given value of
($\tau$), that is, for any given direction of osculation;
differentiating it with respect to ($\tau$), in order to find the
greatest and least osculating paraboloids, we get
$$\eqalign{
2 \gamma \rho t \mathbin{.} \tau
   &= - \alpha \beta + (\alpha^2 + \gamma^2) \tau,\cr
2 \gamma \rho r
   &= \beta^2 + \gamma^2 - \alpha \beta \tau,\cr}$$
equations which give, by elimination,
$$\eqalign{
\{ 2 \gamma \rho t - (\alpha^2 + \gamma^2) \}
         \{ 2 \gamma \rho r - (\beta^2 + \gamma^2) \}
      - \alpha^2 \beta^2
   &= 0,\cr
\alpha \beta (t \mathbin{.} \tau^2 - r)
      + \{ (\alpha^2 + \gamma^2) r - (\beta^2 + \gamma^2) t \} \tau
   &= 0:\cr}$$
and since these coincide with the formul{\ae} (S) (V) of the two
preceding sections, it follows, that when parallel rays are
incident upon a mirror, the two foci of any given reflected ray,
that is, the two points in which it touches the caustic surfaces,
are the foci of the greatest and least paraboloids, which having
their axis parallel to the incident rays, osculate to the mirror
at the given point of incidence; and that the directions of the
two lines of reflexion passing through that point, are the
directions of osculation corresponding.

\bigbreak

[38.]
In general when the incident system is rectangular, which is
always the case in nature, it follows from the principles already
established that we can find an infinite number of focal mirrors,
possessing the property of reflecting the rays to any given point
$(X, Y, Z)$, and having for their differential equation,
$$d\rho = \alpha' \, dx + \beta' \, dy + \gamma' \, dz = dV',$$
$V'$ being the characteristic function of the incident system,
and $\rho$ the distance from the point of incidence $(x, y, z)$
to the point $(X, Y, Z)$, the focus of the focal mirror.  The
condition of touching the given mirror at a given point,
furnishes two equations of the form
$$p' = p,\quad q' = q,$$
which expresses that the focus $(X, Y, Z)$ is somewhere on the
given reflected ray; and the condition of osculating in a given
direction furnishes the equation
$$(r' - r) \mathbin{.} dx^2
      + 2 (s' - s) \mathbin{.} dx \mathbin{.} dy
      + (t' - t) \mathbin{.} dy^2
   = 0,$$
$(r, s, t)$ being given, but $(r', s', t')$ depending on the
unknown focal distance ($\rho$); and if we wish to make this
distance a maximum or minimum, we are to satisfy the two
conditions
$$(r' - r) \mathbin{.} dx + (s' - s) \mathbin{.} dy = 0,\quad
  (s' - s) \mathbin{.} dx + (t' - t) \mathbin{.} dy = 0,$$
which may thus be written
$$dp' = dp,\quad dq' = dq,$$
$p'$, $q'$, being the partial differentials, first order, of the
focal mirror, and $p$, $q$, those of the given mirror.  Now the
general equation of focal mirrors,
$d\rho = dV' = \alpha' \, dx + \beta' \, dy + \gamma' \, dz$,
gives
$$\eqalign{
x - X + p' \mathbin{.} (z - Z)
   &= \rho \mathbin{.} (\alpha' + \gamma' p'),\cr
y - Y + q' \mathbin{.} (z - Z)
   &= \rho \mathbin{.} (\beta'  + \gamma' q'),\cr}$$
and therefore
$$\eqalign{
dx + p' \, dz - (\alpha' + \gamma' p') \, d \rho
   &= \rho \mathbin{.} (d\alpha' + p' \, d\gamma')
      + (Z - z + \gamma' \rho) \, dp',\cr
dy + q' \, dz - (\beta'  + \gamma' q') \, d \rho
   &= \rho \mathbin{.} (d\beta'  + q' \, d\gamma')
      + (Z - z + \gamma' \rho) \, dq';\cr}$$
if then we put $p' = p$, $q' = q$, in order to express that the
focal mirror touches the given mirror, we shall have, to
determine $dp'$, $dq'$, two equations which may be thus written,
$$\left. \eqalign{
\rho \mathbin{.}
         \{ (\gamma + \gamma') \, dp' + d\alpha' + p \mathbin{.} d\gamma' \}
   = dx + p \, dz
      - (\alpha + \gamma p)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz ),\cr
\rho \mathbin{.}
         \{ (\gamma + \gamma') \, dq' + d\beta'  + q \mathbin{.} d\gamma' \}
   = dy + q \, dz
      - (\beta  + \gamma q)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz ),\cr}
   \right\}
   \eqno {\rm (Z)}$$
and if in these equations (Z) we change $(dp', dq')$ to
$(dp, dq)$ in order to find the greatest and least osculating
focal mirrors, they become the formul{\ae} (T) of the preceding
section.  Hence it follows, that in general, the foci of the
greatest and least osculating mirrors, are the points in which
the reflected ray touches the two caustic surfaces; and that the
directions of the lines of reflexion, are the directions of
osculation corresponding.

\bigbreak

[39.]
The equations (Z) determine not only the maximum and minimum
values of the osculating focal distance $(\rho)$, but also the law
by which that distance varies for intermediate directions of
osculation.  To find this law, we are to employ the formula,
$$(r' - r) \mathbin{.} dx^2
      + 2 (s' - s) \mathbin{.} dx \mathbin{.} dy
      + (t' - t) \mathbin{.} dy^2 = 0,$$
that is
$$dp' \mathbin{.} dx + dq' \mathbin{.} dy
   = dp \mathbin{.} dx + dq \mathbin{.} dy.$$
Adding therefore the two equations (Z), multiplied respectively
by $(dx, dy)$, then changing $(dp' \, dx + dq' \, dy)$ to
$(dp \, dx + dq \, dy)$, and reducing; we find the following
general expression for the osculating focal distance
$$\rho
   = {dx^2 + dy^2 + dz^2 - d\rho^2
      \over
      (\gamma + \gamma') \mathbin{.}
         (dp \mathbin{.} dx + dq \mathbin{.} dy)
      + d\alpha' \mathbin{.} dx
      + d\beta'  \mathbin{.} dy
      + d\gamma' \mathbin{.} dz}.
   \eqno {\rm (A')}$$
To simplify this formula, let us take the given reflected ray for
the axis of ($z$); the numerator then reduces itself to
($dx^2 + dy^2$), and the denominator may be put under the form
$$\epsilon \, dx^2 + \zeta \mathbin{.} dx \, dy + \eta \, dy^2,$$
the coefficients $\epsilon$, $\zeta$, $\eta$, being independent
of $\rho$, and of the differentials; if then we put
$$dy = dx \mathbin{.} \tan \psi,$$
so that ($\psi$) shall be the angle which the plane, passing
through the ray and through the direction of osculation, makes
with the plane of ($xz$), we shall have
$${1 \over \rho}
   =   \epsilon \mathbin{.} \cos^2 \psi
     + \zeta    \mathbin{.} \sin \psi \mathbin{.} \cos \psi
     + \eta     \mathbin{.} \sin^2 \psi.
   \eqno {\rm (B')}$$
This formula may be still further simplified, by taking for the
planes of $(x, z)$, $(y, z)$, the tangent planes to the
developable pencils, which, by what we have proved, correspond to
the maximum and minimum of ($\rho$).  To find these planes we are
to put
$\displaystyle {d\rho \over d\psi} = 0$,
which gives,
$$\tan 2 \psi = {\zeta \over \epsilon - \eta};$$
if then we take them for the planes of $(x, z)$, $(y, z)$ we shall
have
$$\zeta = 0,\quad
  \epsilon = {1 \over \rho_1},\quad
  \eta = {1 \over \rho_2},$$
and the formula for the osculating focal distance becomes
$${1 \over \rho}
   =   {1 \over \rho_1} \mathbin{.} \cos^2 \psi
     + {1 \over \rho_2} \mathbin{.} \sin^2 \psi.
   \eqno {\rm (C')}$$
$\rho_1$, $\rho_2$, being the extreme values of $\rho$, namely
the distances of the two points in which the ray touches the two
caustic surfaces.  The analogy of this formula (C${}'$) to the
known formula for the radius of an osculating sphere, is evident;
and it is important to observe, that although the reciprocal of
($\rho$) is included between two given limits, the quantity
($\rho$) itself is not always included between the corresponding
limits, but is on the contrary excluded from between them, when
those limits are of opposite algebraic signs, that is, when the
two foci of the ray are at opposite sides of the mirror: so that,
in this case, there is some impropriety in the term
{\it greatest\/} osculating focal distance, since there are some
directions of osculation for which that distance is infinite,
namely, the two directions determined by the condition
$${1 \over \rho} = 0,\quad
  \tan^2 \psi = - {\rho_2 \over \rho_1}.$$

I shall however continue to employ it, both on account of the
analytic theorem which it expresses, and also on account of its
analogy to the received phrase of {\it greatest osculating
sphere}, to which the same objection may be made, when the two
concavities of the surface are turned in opposite directions.

\bigbreak

[40.]
I shall conclude this section, by pointing out another remarkable
property of the osculating focal mirrors; which is, that if upon
the plane, passing through a given direction of osculation, we
project the ray reflected from the consecutive point on that
direction, the projection will cross the given ray in the
osculating focus corresponding.  To prove this theorem, I
observe, that when the given ray is taken for axis of ($z$), the
point where it meets the mirror for origin, and the tangent
planes of the developable pencils for the planes of
$(x, z)$, $(y, z)$, the partial differentials second order of the
characteristic function ($V$) become, at the origin,
$${d^2 V \over dx^2} = - {1 \over \rho_1},\quad
  {d^2 V \over dx \mathbin{.} dy} = 0,\quad
  {d^2 V \over dy^2} = - {1 \over \rho_2},\quad
  {d^2 V \over dx \mathbin{.} dz} = 0,\quad
  {d^2 V \over dy \mathbin{.} dz} = 0,\quad
  {d^2 V \over dz^2} = 0,$$
and therefore the cosines of the angles which an infinitely near
ray makes with the axes of ($x$) and ($y$), are
$$d\alpha = - {dx \over \rho_1},\quad
  d\beta  = - {dy \over \rho_2}.$$
Hence it follows that the equations of this infinitely near ray
are of the form
$$\rho_1 x' + (z' - \rho_1) \, dx = 0,\quad
  \rho_2 y' + (z' - \rho_2) \, dy = 0;$$
and if we project this ray on the plane
$$x' \, dy - y' \, dx = 0,$$
which passes through the given ray and through the consecutive
point on the mirror, the projecting plane will have for equation
$$    {\rho_1 \mathbin{.} x' + (z' - \rho_1) \mathbin{.} dx
         \over (h - \rho_1) \mathbin{.} dx}
    - {\rho_2 \mathbin{.} y' + (z' - \rho_2) \mathbin{.} dy
         \over (h - \rho_2) \mathbin{.} dy}
   = 0,$$
($h$) being the height of the point where the projection crosses
the given ray, which is to be determined by the condition that
the latter plane shall be perpendicular to the former, that is,
by the equation,
$$    {\rho_1 \mathbin{.} dy \over (h - \rho_1) \, dx}
    + {\rho_2 \mathbin{.} dx \over (h - \rho_2) \, dy}
   = 0;$$
which, when we put $dy = dx \mathbin{.} \tan \psi$, becomes
$$h = {\rho_1 \mathbin{.} \rho_2
      \over
         \rho_2 \mathbin{.} \cos^2 \psi
       + \rho_1 \mathbin{.} \sin^2 \psi},
   \eqno {\rm (D')}$$
a formula that evidently coincides with the one that we found
before, for the height of the osculating focus.

\bigbreak

\centerline{%
IX.  \it On thin and undevelopable pencils.}

\nobreak\bigskip

[41.]
Having examined some of the most important properties of the
developable pencils of a reflected system, we propose in this
section to make some remarks upon pencils not developable; and we
shall begin by considering {\it thin pencils}, that is, pencils
composed of rays that are very near to a given ray; because in
all the most useful applications of optical theory, it is not an
entire reflected or refracted system that is employed, but only a
small parcel of the rays belonging to that system.

To simplify our calculations, let us take the given ray for the
axis of ($z$), and let us choose the coordinate planes as in the
preceding paragraph; the cosines of the angles which a near ray
makes with the axes of ($x$) and ($y$), will be, nearly,
$$\alpha = - {x \over \rho_1},\quad\beta = - {y \over \rho_2},$$
$x$,~$y$, being coordinates of the point in which it meets the
mirror; and the equations of this near ray will be, nearly,
$$x' = x + \alpha z',\quad y' = y + \beta z',$$
that is
$$x' = \alpha \mathbin{.} (z' - \rho_1),\quad
  y' = \beta  \mathbin{.} (z' - \rho_2),
   \eqno {\rm (E')}$$
$x'$,~$y'$,~$z'$, being the coordinates of the near ray.  And if
we eliminate $\alpha$, $\beta$, by these equations, from the
general equation (N)
$$\beta = f(\alpha),$$
which represents all the pencils of the system, we find for the
general equation of thin pencils,
$${y' \over z' - \rho_2} = f \left( {x' \over z' - \rho_1} \right).
   \eqno {\rm (F')}$$

\bigbreak

[42.]
These equations (E${}'$), (F${}'$) include nearly the whole
theory of thin pencils.  As a first application of them, let us
suppose that we are looking at a luminous point, by means of any
combination of mirrors; the rays that enter the eye will not in
general diverge from any one focus, and therefore will not be
bounded by a cone, but by a pencil of another shape, which I
shall call the {\it Bounding Pencil of Vision}, and the
properties of which I am now going to investigate.

Suppose for this purpose, that the optic axis coincides with that
given ray of the reflected system which we have taken for the
axis of ($z$), and let ($\delta$) represent the distance of the
eye from the mirror; the circumference of the pupil will have for
equations
$$z = \delta,\quad x^2 + y^2 = e^2,$$
$(e)$ being the radius of the pupil; the rays of the bounding
pencil of vision pass through this circumference, and therefore
satisfy the condition
$$\alpha^2 \mathbin{.} (\delta - \rho_1)^2
      + \beta^2 \mathbin{.} (\delta - \rho_2)^2
   = e^2;$$
and eliminating $\alpha$,~$\beta$, from this, by means of
(E${}'$), we find the following equation for the bounding pencil
of vision,
$$       \left( {\delta - \rho_1 \over z' - \rho_1} \right)^2
            \mathbin{.} x'^2
       + \left( {\delta - \rho_2 \over z' - \rho_2} \right)^2
            \mathbin{.} y'^2
   = e^2.
   \eqno {\rm (G')}$$

It is evident, from this equation, that every section of the
pencil by a plane perpendicular to the optic axis, that is, to
the given ray, is a little ellipse, having its centre on that
ray, and its semiaxes situated in the tangent planes to the two
developable pencils, that is in the planes of $(x, z)$, $(y, z)$.
Denoting these semiaxes by ($a$), ($b$), we have
$$a = \pm e \mathbin{.} {z' - \rho_1 \over \delta - \rho_1},\quad
  b = \pm e \mathbin{.} {z' - \rho_2 \over \delta - \rho_2};
   \eqno {\rm (H')}$$
these semiaxes become equal, that is, the little elliptic section
becomes circular, first when
$$z' = \delta,\quad a = b = e,$$
that is, at the eye itself, and secondly when
$$z' = \delta
      - {2 \mathbin{.} (\delta - \rho_1)  (\delta - \rho_2)
        \over (\delta - \rho_1) + (\delta - \rho_2)},\quad
  a = b = \pm {e \mathbin{.} (\rho_1 - \rho_2)
        \over 2 \delta - (\rho_1 + \rho_2)},$$
that is, at a distance from the eye equal to the harmonic mean
between the distances of the eye from the two foci of that
reflected ray, which coincides with the optic axis.  It may also
be proved, that when the eye is beyond the two foci, the radius
of this harmonic section, (which is to the radius of the pupil as
the semi-interval between the two foci is to the distance of the
eye from the middle point between them,) is less than the
semiaxis major of any of the elliptic sections, that is, than the
extreme aberration of the visual rays at any other distance from
the eye; so that, in this case, we may consider the centre of the
harmonic section as the {\it visible image\/} of the luminous
point, seen by the given combination of mirrors; observing
however that the {\it apparent distance\/} of the luminous point
will depend on other circumstances of brightness, distinctness
and magnitude, as it does in the case of direct vision with the
naked eye.

\bigbreak

[43.]
One of the principal properties of thin pencils, is that the area
of a perpendicular section of such a pencil is always
proportional to the product of its distances from the two foci of
the given ray.  We may verify this theorem, in the case of the
bounding pencil of vision, by means of the formul{\ae} (H${}'$)
for the semiaxes of the little elliptic section; in general if we
represent by $\Sigma$ the area of the section of any given thin
pencil, corresponding to any given value of ($z'$), we shall have
by (E${}'$)
$$2 \mathbin{.} \Sigma
   = \int (y' \, dx' - x' \, dy')
   = (z' - \rho_1) (z' - \rho_2) \mathbin{.}
         \int (\beta \, d\alpha - \alpha \, d\beta),
   \eqno {\rm (I')}$$
and the definite integral
$\int (\beta \, d\alpha - \alpha \, d\beta)$,
depending only on the relation between $\alpha$, $\beta$, is
constant when the pencil is given.  It follows from this theorem,
that along a given ray the density of the reflected light varies
inversely as the product of the distances from the two foci, and
is infinite at the caustic surfaces.

\bigbreak

[44.]
The same equations (E${}'$), from which we have deduced the
theory of thin pencils, serve also to investigate the properties
of other undevelopable surfaces, composed by the rays of the
system.  The most remarkable difference between an undevelopable
and a developable pencil, consists in this, that the tangent
plane to the latter always touches it in the whole extent of a
ray; whereas in the former, when the point of contact moves along
a given ray, the tangent plane changes position, and turns round
that ray, like a hinge.  To find the law of this rotation let the
coordinate planes be chosen as before, the given ray for axis of
($z$), the point where it meets the mirror for origin, and the
tangent planes to the two developable pencils for the planes of
$(xz)$, $(yz)$; then by (E${}'$), the equations of an
infinitely near ray will be
$$x' = (z' - \rho_1) \mathbin{.} d\alpha,\quad
  y' = (z' - \rho_2) \mathbin{.} d\beta,
   \eqno {\rm (K')}$$
and if it belong to a given undevelopable pencil having for
equation $\beta = f(\alpha)$, we shall have
$$d\beta = f' \mathbin{.} d\alpha,$$
$f'$ being a given quantity; the tangent plane to this pencil, at
any given distance ($z'$) from the mirror, being obliged to
contain the given ray, and to pass through a point on the
consecutive, has for equation
$${y \over x} = {z' - \rho_2 \over z' - \rho_1} \mathbin{.} f';
   \eqno {\rm (L')}$$
when $z'$ increases, that is, when the point of contact recedes
indefinitely from the mirror, this tangent plane approaches to
the limiting position
$${y \over x} = f',
   \eqno {\rm (M')}$$
which is evidently parallel to the consecutive ray (K${}'$); and
the angle ($P$) which it makes with this limiting position, is
given by the formula
$$\tan P
   = {(\rho_1 - \rho_2) \mathbin{.} f'
      \over z' \mathbin{.} (1 + f'^2)
         - (\rho_1 + \rho_2 \mathbin{.} f'^2)},$$
that is, if we put $f' = \tan L$,
$$\tan P
   = {(\rho_1 - \rho_2) \mathbin{.} \sin L
            \mathbin{.} \cos L
      \over z' - ( \rho_1 \mathbin{.} \cos^2 L + \rho_2
            \mathbin{.} \sin^2 L )},$$
or, finally,
$$\tan P = {u \over \delta},
   \eqno {\rm (N')}$$
($u$) being a constant coefficient,
$$u = (\rho_1 - \rho_2) \mathbin{.} \sin L
            \mathbin{.} \cos L,
   \eqno {\rm (O')}$$
and ($\delta$) being the distance of the point of contact from a
certain fixed point upon the ray, whose distance from the mirror
is
$$z' =     \rho_1 \mathbin{.} \cos^2 L
         + \rho_2 \mathbin{.} \sin^2 L.
   \eqno {\rm (P')}$$

\bigbreak

[45.]
The quantity ($u$), which thus enters as a constant coefficient
into the law of rotation of the tangent plane of an undevelopable
pencil, I shall call the {\it coefficient of undevelopability}.
In the third part of this essay, I shall treat more fully of its
properties, and of those of the fixed point determined by the
formula (P${}'$); in the mean time, I shall observe, that if we
cut the consecutive ray (K${}'$) by any plane perpendicular to
the given ray, at a distance $\delta$ from this fixed point
(P${}'$), the interval between the two rays, corresponding to
this distance ($\delta$), is
$$\Delta = \surd (u^2 + \delta^2) \mathbin{.} d\theta,
   \eqno {\rm (Q')}$$
($d\theta$) being the angle between the rays; from which it
follows, that the fixed point (P${}'$) may be called the
{\it virtual focus\/} of the given ray, in the given
undevelopable pencil, because it is the nearest point to an
infinitely near ray of that pencil; and that the coefficient of
undevelopability ($u$), is equal to the least distance between
the given ray and the consecutive ray, divided by the angle
between them.  We may also observe, that although a given ray has
in general an infinite number of undevelopable pencils passing
through it, and therefore an infinite number of virtual foci
corresponding, yet these virtual foci are all included between
the two points where the ray touches the two caustic surfaces,
because the expression (P${}'$)
$$z' =     \rho_1 \mathbin{.} \cos^2 L
         + \rho_2 \mathbin{.} \sin^2 L,$$
is always included between the limits $\rho_1$ and $\rho_2$.  And
whenever, in this essay, the term {\it foci of a ray\/} shall
occur, the two points of contact with the caustic surfaces are to
be understood except when the contrary is expressed.

\bigbreak

\centerline{%
X.  \it On the axes of a reflected system.}

\nobreak\bigskip

[46.]
We have seen that the density of light in a reflected system is
greatest at the caustic surfaces; from which it is natural to
infer, that this density is greatest of all at the intersection
of these surfaces: a remark which has already been made by
Malus, and which will be still farther confirmed, when we
come to consider the aberrations.  It is important therefore to
investigate the nature and position of the intersection of the
caustic surfaces.  I am going to shew that this intersection is
not in general a curve, but reduces itself to a finite number of
isolated points, the foci of a finite number of rays, which are
intersected in those points by all the rays infinitely near them.
For this purpose I resume the formula (Q) found in Section~VI.
$$    \left( \rho + {da \over d\alpha} \right)
      \left( \rho + {db \over d\beta}  \right)
    - {da \over d\beta} \mathbin{.} {db \over d\alpha}
   = 0,
   \eqno {\rm (Q)}$$
which determines the two foci of a given ray, and in which the
coefficients
$\displaystyle {da \over d\alpha}$,
$\displaystyle {da \over d\beta}$,
$\displaystyle {db \over d\alpha}$,
$\displaystyle {db \over d\beta}$,
are connected by the following relation, deduced from the same
section,
$$\alpha \beta \mathbin{.}
         \left( {da \over d\alpha} - {db \over d\beta} \right)
       - (\alpha^2 + \gamma^2) \mathbin{.} {da \over d\beta}
       + (\beta^2  + \gamma^2) \mathbin{.} {db \over d\alpha}
   = 0.
   \eqno {\rm (R')}$$
The condition of equal roots in (Q), is
$$\left( {da \over d\alpha} - {db \over d\beta} \right)^2
   + 4 \mathbin{.} {da \over d\beta} \mathbin{.} {db \over d\alpha}
   = 0;$$
this then is the equation which determines the relation between
$\alpha$,~$\beta$, that belongs to the rays passing through the
intersection of the caustic surfaces; and it is easy to prove, by
means of the formula (R${}'$), that it resolves itself into the
three following, which however, in consequence of the same
formula, are equivalent to but two distinct equations:
$${da \over d\beta} = 0,\quad {db \over d\alpha} = 0,\quad
  {da \over d\alpha} - {db \over d\beta} = 0.
   \eqno {\rm (S')}$$
The rays determined by these equations, I shall call the
{\it axes of the reflected system}, and their foci, for which
$$\rho = - {da \over d\alpha} = - {db \over d\beta},
   \eqno {\rm (T')}$$
I shall call the {\it principal foci}.

\bigbreak

[47.]
We have seen, that a given ray has, in general, an infinite
number of virtual foci, corresponding to the undevelopable
pencils, and determined by the formula (P${}'$),
$$z' =     \rho_1 \mathbin{.} \cos^2 L
         + \rho_2 \mathbin{.} \sin^2 L,$$
and an infinite number of osculating foci, corresponding to the
osculating focal mirrors, and determined by the formula (C${}'$)
$${1 \over \rho}
   =   {1 \over \rho_1} \mathbin{.} \cos^2 \psi
     + {1 \over \rho_2} \mathbin{.} \sin^2 \psi.$$
But when $\rho_2 = \rho_1$, that is, when the ray is an axis of
the system, the the variable angles disappear from these
formul{\ae}, and all the virtual and all the osculating foci
close up into one single point, namely, the principal focus
corresponding to that axis.  Hence, and from the coefficient of
undevelopability vanishing, it follows, that each axis of the
system is intersected, at its own focus, by all the rays
infinitely near; and that this focus, is the focus of a focal
mirror, which has, with the given mirror, complete contact of the
second order.  A point of contact of this kind, that is, a point
where the given mirror is met by an axis of the reflected system,
I shall call a {\it vertex\/} of the mirror.

\bigbreak

[48.]
Another remarkable property of the principal foci, is that they
are the centres of spheres, which have complete contact of the
second order with the surfaces that cut the rays perpendicularly;
which may be proved by means of the following formul{\ae},
deduced from (S${}'$) and (T${}'$), combined with the formul{\ae}
of [27.],
$$\left. \multieqalign{
{d^2 V \over dx^2} &= {\alpha^2 - 1 \over \rho}, &
{d^2 V \over dy^2} &= {\beta^2  - 1 \over \rho}, &
{d^2 V \over dz^2} &= {\gamma^2 - 1 \over \rho},\cr
{d^2 V \over dx \mathbin{.} dy} &= {\alpha \beta  \over \rho}, &
{d^2 V \over dx \mathbin{.} dz} &= {\alpha \gamma \over \rho}, &
{d^2 V \over dy \mathbin{.} dz} &= {\beta  \gamma \over \rho},\cr}
   \right\}
   \eqno {\rm (U')}$$
And if we substitute these expressions (U${}'$) in the
formul{\ae} of [29.], we find the following equations,
$$\left. \eqalign{
(\gamma + \gamma') \mathbin{.} r + {d^2 V' \over dx^2}
      + 2p \mathbin{.} {d^2 V' \over dx \mathbin{.} dz}
      + p^2 \mathbin{.} {d^2 V' \over dz^2}
   &= {1 + p^2 - (\alpha + \gamma p)^2 \over \rho},\cr
(\gamma + \gamma') \mathbin{.} t + {d^2 V' \over dy^2}
      + 2q \mathbin{.} {d^2 V' \over dy \mathbin{.} dz}
      + q^2 \mathbin{.} {d^2 V' \over dz^2}
   &= {1 + q^2 - (\beta  + \gamma q)^2 \over \rho},\cr
(\gamma + \gamma') \mathbin{.} s + {d^2 V' \over dx \mathbin{.} dy}
      +  p \mathbin{.} {d^2 V' \over dy \mathbin{.} dz}
      +  q \mathbin{.} {d^2 V' \over dx \mathbin{.} dz}
      + pq \mathbin{.} {d^2 V' \over dz^2}
   &= {pq - (\alpha + \gamma p)(\beta  + \gamma q) \over \rho},\cr}
   \right\}
   \eqno {\rm (V')}$$
which determine the vertices, the axes, and the principal foci,
when we know the equation of the mirror, and the characteristic
function of the incident system.  These formul{\ae} (V${}'$) may
also be deduced from the equations (Z) of Section~VIII.\ by means
of the theorem that we have already established, respecting the
complete contact of the second order, which exists, at a vertex,
between the given mirror and the osculating focal surface
corresponding: and they may be reduced to the two following; that
is, to the equations (T) of Section~VII.
$$\left. \eqalign{
\rho \mathbin{.}
         \{ (\gamma + \gamma') \mathbin{.} dp + d\alpha'
            + p \mathbin{.} d\gamma' \}
   = dx + p \mathbin{.} dz
      - (\alpha + \gamma p)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz ),\cr
\rho \mathbin{.}
         \{ (\gamma + \gamma') \mathbin{.} dq + d\beta'
            + q \mathbin{.} d\gamma' \}
   = dy + q \mathbin{.} dz
      - (\beta  + \gamma q)
         ( \alpha \, dx + \beta  \, dy + \gamma \, dz ),\cr}
   \right\}
   \eqno {\rm (T)}$$
by observing, that these equations, which in general determine
the lines of reflexion on the mirror, are, at a vertex, satisfied
independently of the ratio between the differentials $(dx, dy)$,
provided that we assign to ($\rho$) its proper value, namely the
distance of the principal focus.

\bigbreak

[49.]
As an application of the preceding theory, let us suppose that
the incident rays diverge from a luminous point $(X', Y', Z')$,
and let us seek the vertices, the axes, and the principal foci of
the reflected system.  In this question, the equations (T)
become, by [35.],
$$\eqalign{
(\gamma + \gamma') \mathbin{.} dp
   &= \left( {1 \over \rho} + {1 \over \rho'} \right)
      \{ dx + p \, dz
          - (\alpha + \gamma p)
            (\alpha \, dx + \beta \, dy + \gamma \, dz) \},\cr
(\gamma + \gamma') \mathbin{.} dq
   &= \left( {1 \over \rho} + {1 \over \rho'} \right)
      \{ dy + q \, dz
          - (\beta  + \gamma q)
            (\alpha \, dx + \beta \, dy + \gamma \, dz) \},\cr}$$
$\rho'$ being the distance of the luminous point from the mirror;
and since these equations are to be satisfied independently of the
ratio between $(dx, dy)$, they resolve themselves into the three
following,
$$\left. \eqalign{
(\gamma + \gamma') \mathbin{.} r
   &= \left( {1 \over \rho} + {1 \over \rho'} \right) \mathbin{.}
         \{ 1 + p^2 - (\alpha + \gamma p)^2 \},\cr
(\gamma + \gamma') \mathbin{.} t
   &= \left( {1 \over \rho} + {1 \over \rho'} \right) \mathbin{.}
         \{ 1 + q^2 - (\beta  + \gamma q)^2 \},\cr
(\gamma + \gamma') \mathbin{.} s
   &= \left( {1 \over \rho} + {1 \over \rho'} \right) \mathbin{.}
         \{ pq - (\alpha + \gamma p)(\beta  + \gamma q) \},\cr}
   \right\}
   \eqno {\rm (W')}$$
which contain the solution of the problem.

To shew the geometrical meaning of these equations (W${}'$), let
us take the vertex for origin, the normal at that point for the
axis of ($z$), and the tangents to the lines of curvature for the
axes of ($x$) and ($y$); we shall then have
$$p = 0,\quad q = 0,\quad s = 0,\quad
  r = {1 \over R},\quad t = {1 \over R'},\quad
  \gamma = \gamma' = \cos I,$$
$I$ being the angle of incidence, and $R$, $R'$, being the two
radii of curvature of the mirror; and the formul{\ae} (W${}'$)
become
$$h \mathbin{.} \cos I
      = R (1 - \alpha^2) = R' (1 - \beta^2),\quad
   \alpha \beta = 0,
   \eqno {\rm (X')}$$
($h$) being the harmonic mean between the conjugate focal
distances, so that
$${2 \over h} = {1 \over \rho} + {1 \over \rho'}.$$
The equation $\alpha \beta = 0$, shews that the plane of
incidence must coincide either with the plane of the greatest or
the least osculating circle to the mirror; and if we put
$\beta = 0$, that is, if we choose the plane of incidence for the
plane of $(x, z)$ we shall have $R' = R (1 - \alpha^2)$,
$R > R'$, so that it is with the plane of the greatest osulating
circle that the plane of incidence coincides.  We have also
$1 - \alpha^2 = \cos^2 I$,
$$\left. \eqalign{
h &= R \mathbin{.} \cos I = R' \mathbin{.} \mathop{\rm sec.} I,\cr
h^2 &= R R',\quad R' = R \mathbin{.} \cos^2 I,\cr}
   \right\}
   \eqno {\rm (Y')}$$
from which it follows, that the harmonic mean between the
conjugate focal distances, is equal to the geometric mean between
the radii of curvature of the mirror; and that the square of the
cosine of the angle of incidence is equal to the ratio of those
two radii of curvature.  It follows also, that the line joining
the luminous point to its conjugate focus, (that is, the axis of
the osculating ellipsoid) passes through the centre of the least
osculating circle to the mirror; and since it is also contained
in the plane of the greatest osculating circle, it is tangent to
one surface of centres of curvature of the mirror.

\bigbreak

[50.]
As another application, let us take the case of parallel rays
reflected by a combination of two given mirrors.  Let
$\alpha$,~$\beta$,~$\gamma$, $\alpha'$,~$\beta'$,~$\gamma'$,
be still the cosines of the angles which the last reflected and
last incident ray, measured from the last mirror, make with the
axes of coordinates;
$\alpha''$,~$\beta''$,~$\gamma''$,
the given cosines of the angles which the first incident
ray, measured towards the first mirror, makes with the same axes;
$x$, $y$, $z$, $p$, $q$, $r$, $s$, $t$,
the coordinates and partial differential coefficients of the last
mirror, and
$x'$, $y'$, $z'$, $p'$, $q'$, $r'$, $s'$, $t'$,
the corresponding quantities of the first.  We have then,
$$\alpha' + \alpha'' + p' \mathbin{.} (\gamma' + \gamma'')
   = 0,\quad
  \beta'  + \beta''  + q' \mathbin{.} (\gamma' + \gamma'')
   = 0;$$
and therefore
$$\eqalign{
d\alpha' + p' \mathbin{.} d\gamma'
         + (\gamma' + \gamma'') \mathbin{.} dp'
   &= 0,\cr
d\beta'  + q' \mathbin{.} d\gamma'
         + (\gamma' + \gamma'') \mathbin{.} dq'
   &= 0;\cr}$$
we have also
$$x' = x + \alpha' \rho',\quad
  y' = y + \beta'  \rho',\quad
  z' = z + \gamma' \rho',$$
$$dx' =dx + d \mathbin{.} (\alpha' \rho'),\quad
  dy' =dy + d \mathbin{.} (\beta'  \rho'),\quad
  dz' =dz + d \mathbin{.} (\gamma' \rho'),$$
$$d\rho'
   =   \alpha' \mathbin{.} (dx' - dx)
     + \beta'  \mathbin{.} (dy' - dy)
     + \gamma' \mathbin{.} (dz' - dz),$$
$\rho'$ being the path traversed by the light, in going from the
one mirror to the other; by means of these equations we can find,
for the quantities
($d\alpha' + p \, d\gamma'$),
($d\beta'  + q \, d\gamma'$),
which enter into (T), expressions which may be shewn to be of the
form
$$\eqalign{
d\alpha' + p \, d \gamma' &= A \, dx + B \, dy,\cr
d\beta'  + q \, d \gamma' &= B \, dx + C \, dy,\cr}$$
$A$, $B$, $C$, involving
$p$, $q$, $p'$, $q'$, $r'$, $s'$, $t'$, $\rho'$:
and to determine the vertices, the axes, and the principal foci,
of the last reflected system, we shall have the following
equations,
$$\left. \eqalign{
\rho \mathbin{.} \{ (\gamma + \gamma') \mathbin{.} r + A \}
   &= 1 + p^2 - (\alpha + \gamma p)^2,\cr
\rho \mathbin{.} \{ (\gamma + \gamma') \mathbin{.} s + B \}
   &= pq - (\alpha + \gamma p)(\beta  + \gamma q),\cr
\rho \mathbin{.} \{ (\gamma + \gamma') \mathbin{.} t + C \}
   &= 1 + q^2 - (\beta  + \gamma q)^2.\cr}
   \right\}
   \eqno {\rm (Z')}$$

\bigbreak

\centerline{%
XI.  \it On the images formed by mirrors.}

\nobreak\bigskip

[51.]
It appears from the preceding section, that when rays issuing
from a luminous point have been reflected at a given mirror, the
two caustic surfaces touched by the reflected rays intersect one
another in a finite number of isolated points, at which the
density of reflected light is greatest, and of which each is the
conjugate focus of an ellipsoid of revolution, that has its other
focus at the given luminous point, and that has contact of the
second order with the given mirror.  It is evident that these
points of maximum of density are the images of the given luminous
point, formed by the given mirror; and that in like manner, the
image or images of a given point, formed by a given combination of
mirrors, are the corresponding points of maximum density, to which the
intersection of the last pair of caustic surfaces reduces itself, and
which are the foci of focal mirrors that have contact of the second
order with the last given mirror.  And on similar principles are we to
determine the image of a curve or of a surface, formed by any given
mirror, or combination of mirrors; namely, by considering the image of
the curve or surface as the locus of the images of its points.

\bigbreak

[52.]
Let us apply these principles to the investigation of the image
of a planet formed by a curved mirror.  The image of the planet's
centre is the focus of a paraboloid of revolution, which has its
axis pointed to that centre, and which has complete contact of
the second order with the mirror.  To find this image, together
with the corresponding point of contact, or vertex on the mirror,
we have the equations
$$\alpha + \alpha' + p \mathbin{.} (\gamma + \gamma') = 0,\quad
  \beta  + \beta'  + q \mathbin{.} (\gamma + \gamma') = 0,
   \eqno {\rm (A'')}$$
$$\left. \eqalign{
(\gamma + \gamma') \mathbin{.} \rho \, dp
   &= dx + p \, dz - (\alpha + \gamma p)
            (\alpha \, dx + \beta \, dy + \gamma \, dz),\cr
(\gamma + \gamma') \mathbin{.} \rho \, dq
   &= dy + q \, dz - (\beta  + \gamma q)
            (\alpha \, dx + \beta \, dy + \gamma \, dz),\cr}
   \right\}
   \eqno {\rm (B'')}$$
$\alpha$, $\beta$, $\gamma$, $\alpha'$, $\beta'$, $\gamma'$,
being, as before, the cosines of the angles which the reflected
and incident rays make with the axes of coordinates, and ($\rho$)
being the focal distance; the formul{\ae} (B${}''$) are satisfied
by every infinitely near point upon the mirror, and therefore are
equivalent to three distinct equations, which contain the
conditions for the contact of the second order between the
paraboloid and the mirror.  Differentiating the equations
(A${}''$), in order to pass from the centre to the disk of the
planet, and eliminating $(dp, dq)$ by means of (B${}''$), we find
$$\eqalign{
0  &= \rho \mathbin{.} (d\alpha' + p \, d\gamma')
      + dx + \rho \, d\alpha
      + p \mathbin{.} (dz + \rho \, d\gamma)
      - (\alpha + \gamma p) (\alpha \, dx + \beta \, dy + \gamma \, dz),\cr
0  &= \rho \mathbin{.} (d\beta'  + q \, d\gamma')
      + dy + \rho \, d\beta
      + q \mathbin{.} (dz + \rho \, d\gamma)
      - (\beta  + \gamma q) (\alpha \, dx + \beta \, dy + \gamma \, dz),\cr}$$
that is,
$$\left. \eqalign{
0  &= \rho \mathbin{.} (d\alpha' + p \, d\gamma')
      + da + p \, dc
      - (\alpha + \gamma p) (\alpha \, da + \beta \, db + \gamma \, dc),\cr
0  &= \rho \mathbin{.} (d\beta'  + q \, d\gamma')
      + db + q \, dc
      - (\beta  + \gamma q) (\alpha \, da + \beta \, db + \gamma \, dc),\cr}
   \right\}
   \eqno {\rm (C'')}$$
if we put $(a,b,c)$ to represent the coordinates of the image, so that
$$a - x = \alpha \rho,\quad
  b - y = \beta  \rho,\quad
  c - z = \gamma \rho.$$
Differentiating also the three distinct equations which are
included in (B${}''$), and eliminating, we shall get a result of
the form
$$\rho \mathbin{.} d\gamma' = A \, da + B \, db + C \, dc,
   \eqno {\rm (D'')}$$
$A$, $B$, $C$, involving the partial differentials of the mirror,
as high as the third order.  These equations (C${}''$),
(D${}''$), combined with the identical relation
$\alpha' \, d\alpha' + \beta' \, d\beta' + \gamma' \, d\gamma' = 0$,
and with the following formula,
$$d\alpha'^2 + d\beta'^2 + d\gamma'^2 = \sigma^2,
   \eqno {\rm (E'')}$$
in which $\sigma$ is the semidiameter of the planet, contain the
solution of the question; for they determine the image of any
given point on the disk; and if we eliminate
$d\alpha'$, $d\beta'$, $d\gamma'$, between them, we shall find
the two relations between $da$, $db$, $dc$, which belong to the
locus of those images, that is, to the image of the disk itself.

\bigbreak

[53.]
To simplify this elimination, let us take the central reflected
ray for the axis of ($z$), that is, let us put
$\alpha = 0$, $\beta = 0$, $\gamma = 1$.  We shall then have by
(A${}''$),
$$p = {- \alpha' \over 1 + \gamma'},\quad
  q = {- \beta' \over 1 + \gamma'},$$
and the formul{\ae} (C${}''$) will become
$$\rho \, d\alpha' + da = {\alpha' \rho \, d\gamma' \over 1 + \gamma'},\quad
  \rho \, d\beta'  + db = {\beta'  \rho \, d\gamma' \over 1 + \gamma'},$$
which give, by the identical relations
$$\alpha'^2 + \beta'^2 + \gamma'^2 = 1,\quad
      \alpha' \mathbin{.} d\alpha'
    + \beta'  \mathbin{.} d\beta'
    + \gamma' \mathbin{.} d\gamma'
  = 0,$$
$$\left. \eqalign{
\rho \, d\gamma'
   &= \alpha' \, da + \beta' \, db,\cr
\rho \, d\alpha'
   &= - da
      + {\alpha' \mathbin{.} (\alpha' \, da + \beta' \, db)
            \over 1 + \gamma'},\cr
\rho \, d\beta'
   &= - db
      + {\beta'  \mathbin{.} (\alpha' \, da + \beta' \, db)
            \over 1 + \gamma'}.\cr}
   \right\}
   \eqno {\rm (F'')}$$

Eliminating $d\alpha'$, $d\beta'$, $d\gamma'$, by these
formul{\ae}, from the equations (D${}''$) and (E${}''$), we find,
for the equations of the image,
$$\left. \vcenter{\halign{#\hfil\quad&\hfil $\displaystyle #$\hfil\cr
1st.&
      (A - \alpha') \mathbin{.} da
    + (B - \beta')  \mathbin{.} db
    +  C            \mathbin{.} dc
   = 0,\cr
2d.&
da^2 +db^2 = \rho^2 \mathbin{.} \sigma^2;\cr}}
   \right\}
   \eqno {\rm (G'')}$$
the image is therefore, in general, an ellipse, the plane of
which depends on the quantities $A$,~$B$,~$C$, which enter into
the 1st of its two equations, and therefore on the partial
differentials of the mirror, as high as the third order; but the
2d. of its two equations (G${}''$), is independent of those
partial differentials, and contains this remarkable theorem, that
the projection of the image of the disk, on a plane perpendicular
to the reflected rays, is a circle, whose radius is equal to the
focal distance ($\rho$), multiplied by ($\sigma$) the sine of the
semidiameter of the planet.

\bigbreak

[54.]
The theorem that has been just demonstrated, respecting the
projection of a planet's image, is only a particular case of the
following theorem, respecting reflected images in general, which
easily follows from the principles of the preceding section,
respecting the axes of a reflected system.  This theorem is, that
if we want to find the image of any small object, formed by any
given combination of mirrors, and have found the image of any
given point upon the object, together with the corresponding
vertex upon the last mirror of the given combination; the rays
which come to this given vertex, from the several points of the
object, pass after reflection through the corresponding points of
the image.

\bigbreak

[55.]
It follows from this theorem, that in order to form, by a single
mirror, an undistorted image of any small plane object, whose
plane is perpendicular to the incident rays, it is necessary and
sufficient that the plane of the image be perpendicular to the
reflected rays.  This condition furnishes two relations between
the partial differential coefficients, third order, of the
mirror, which will in general determine the manner in which the
object and mirror are to be placed with respect to one another,
in order to produce an undistorted image.  Thus, if it were
required to find, how we ought to turn a given mirror, in order
to produce a circular image of a planet; we should have the
following condition,
$$d\rho = \alpha' \, dx + \beta' \, dy + \gamma' \, dz,
   \eqno {\rm (H'')}$$
which expresses that the reflected rays are perpendicular to the
plane of the image; $\alpha'$,~$\beta'$,~$\gamma'$, being the
cosines of the angles which the incident ray makes with the axes
of coordinates; and $\rho$ being the focal length of the mirror,
which by [49.]\ is equal to half the geometric mean between the
radii of curvature; so that it is a given function of the partial
differentials, first and second orders, of the mirror,
$$\rho = {\textstyle {1 \over 2}} \mathbin{.}
         {1 + p^2 + q^2 \over \sqrt{(rt - s^2)}};
   \eqno {\rm (I'')}$$
the cosines $(\alpha', \beta', \gamma')$ may also be considered
as given functions of $(p, q, r, s, t)$, because, by [49.]\ the
incident ray at the vertex is contained in the plane of the
greatest osculating circle to the mirror, and the square of the
cosine of angle of incidence is equal to the ratio of the radii
of curvature.  The two equations therefore, into which (H${}''$)
resolves itself, by putting separately $dy = 0$, $dx = 0$, will
furnish two relations between the partial differentials of the
mirror, up to the third order; these are the two relations which
express the condition for the image of the planet being circular:
they are identically satisfied in the case of a spheric mirror,
for then the first member of (H${}''$) vanishes, on account of
the focal length being constant, and the second member on account
of the incident ray coinciding with the normal; and accordingly,
whatever point of a spheric mirror we choose for vertex, it will
form a circular image of a planet; but when the mirror is not
spheric, these two relations will in general determine a finite
number of points upon it, proper to be used as vertices, in order
to form an undistorted image.  And when we shall have found these
points, which I shall call the {\it Vertices of Circular Image},
it will then remain to direct towards the planet, one of the two
lines which at any such vertex are contained in the plane of the
greatest osculating circle to the mirror, and which make with the
normal, at either side, angles, the square of whose cosine is
equal to the ratio of the radii of curvature.

\bigbreak

\centerline{%
XII.  \it Aberrations.}

\nobreak\bigskip

[56.]
After the preceding investigations respecting the two foci of a
reflected ray, or points of intersections with rays infinitely
near; and respecting the axes of a reflected system, each of which
is intersected, in one and the same point, by all the rays that
are infinitely near it; we come now to consider the
{\it Aberrations} of rays at a small but finite distance:
quantities which have long been calculated for certain simple
cases, but which have not, I believe, been hitherto investigated
for reflected systems in general.

\bigbreak

[57.]
When rays fall on a mirror of revolution, from a luminous point
in its axis, the reflected rays all intersect that axis, and the
distances of those intersections from the focus, are called the
longitudinal aberrations.  But in general, the rays of a
reflected system do not all intersect any one ray of that system;
and therefore the {\it longitudinal\/} aberrations do not in
general exist, in the same manner as they do for those
particular cases, which have been hitherto considered.  However I
shall shew, in a subsequent part of this essay, that there are
certain other quantities which in a manner take their place, and
follow analogous laws: but at present I shall confine myself to
the {\it lateral\/} aberrations measured on a plane perpendicular
to a given ray, of which the theory is simpler, as well as more
important.

Let therefore $(x', y', z')$ represent the coordinates of the
point in which the plane of aberration is crossed by any
particular ray; these coordinates may be considered as functions
of any two quantities which determine the position of that ray;
for example, of the cosines of the angles which the ray makes
with the axes of ($x$) and ($y$).  They may therefore be
developed in series of the form

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

$$\left. \eqalign{
x' &= X  + {dX \over d\alpha} \mathbin{.} \alpha_\prime
         + {dX \over d\beta}  \mathbin{.} \beta_\prime
         + {\textstyle {1 \over 2}}
            \left\{
               {d^2 X \over d\alpha^2}
                  \mathbin{.} \alpha_\prime^2
             + 2 {d^2 X \over d\alpha \mathbin{.} d\beta}
                  \mathbin{.} \alpha_\prime \beta_\prime
             + {d^2 X \over d\beta^2}
                  \mathbin{.} \beta_\prime^2
            \right\}
         + \hbox{\&c.}\cr
y' &= Y  + {dY \over d\alpha} \mathbin{.} \alpha_\prime
         + {dY \over d\beta}  \mathbin{.} \beta_\prime
         + {\textstyle {1 \over 2}}
            \left\{
               {d^2 Y \over d\alpha^2}
                  \mathbin{.} \alpha_\prime^2
             + 2 {d^2 Y \over d\alpha \mathbin{.} d\beta}
                  \mathbin{.} \alpha_\prime \beta_\prime
             + {d^2 Y \over d\beta^2}
                  \mathbin{.} \beta_\prime^2
            \right\}
         + \hbox{\&c.}\cr
z' &= Z  + {dZ \over d\alpha} \mathbin{.} \alpha_\prime
         + {dZ \over d\beta}  \mathbin{.} \beta_\prime
         + {\textstyle {1 \over 2}}
            \left\{
               {d^2 Z \over d\alpha^2}
                  \mathbin{.} \alpha_\prime^2
             + 2 {d^2 Z \over d\alpha \mathbin{.} d\beta}
                  \mathbin{.} \alpha_\prime \beta_\prime
             + {d^2 Z \over d\beta^2}
                  \mathbin{.} \beta_\prime^2
            \right\}
         + \hbox{\&c.}\cr}
   \right\}
   \eqno {\rm (K'')}$$
$X$, $Y$, $Z$, being their values for the given ray, that is the
coordinates of the point from which aberration is measured,
and $(\alpha_\prime, \beta_\prime)$ being the small but finite
increments which the cosines $(\alpha, \beta)$ receive, in
passing from the given ray to the near ray.  These equations
(K${}''$) contain the whole theory of lateral aberration; but in
order to apply them, we must shew how to calculate the partial
differential coefficients of $(X, Y, Z)$, considered as functions
of $(\alpha, \beta)$.  For this purpose I observe, that $\alpha$,
$\beta$, being themselves the partial differential coefficients
of the characteristic of the system, (Section~V.), may be
considered as functions of the coordinates $a$, $b$, of the
projection of the point in which the ray crosses any given
perpendicular surface;
$$\alpha = {dV \over da},\quad \beta = {dV \over db},$$
$$\multieqalign{
{d\alpha \over da}
   &= {d^2 V \over da^2}
      + {d^2 V \over da \mathbin{.} dc} \mathbin{.} {dc \over da}, &
{d\alpha \over db}
   &= {d^2 V \over da \mathbin{.} db}
      + {d^2 V \over da \mathbin{.} dc} \mathbin{.} {dc \over db},\cr
{d\beta  \over da}
   &= {d^2 V \over da \mathbin{.} db}
      + {d^2 V \over db \mathbin{.} dc} \mathbin{.} {dc \over da}, &
{d\beta  \over db}
   &= {d^2 V \over db^2}
      + {d^2 V \over db \mathbin{.} dc} \mathbin{.} {dc \over db};\cr}$$
($c$) being the other coordinate of the perpendicular surface,
connected with $a$, $b$, by the relation
$$V = \hbox{const.},$$
which gives
$$      {dV \over da} \mathbin{.} da
      + {dV \over db} \mathbin{.} db
      + {dV \over dc} \mathbin{.} dc
   = 0,$$
that is
$$\alpha \, da + \beta \, db + \gamma \, dc = 0:$$
and if we represent by ($\rho$) the portion of the ray,
intercepted between this perpendicular surface and the plane of
aberration, we shall have
$$X = a + \alpha \rho,\quad
  Y = b + \beta  \rho,\quad
  Z = c + \gamma \rho.$$
By means of these formul{\ae}, combined with the equations
$$\eqalign{
      \alpha \mathbin{.} dX
    + \beta  \mathbin{.} dY
    + \gamma \mathbin{.} dZ
   &= 0,\cr
      \alpha \mathbin{.} d^2 X
    + \beta  \mathbin{.} d^2 Y
    + \gamma \mathbin{.} d^2 Z
   &= 0,\cr
      \alpha \mathbin{.} d^n X
    + \beta  \mathbin{.} d^n Y
    + \gamma \mathbin{.} d^n Z
   &= 0,\cr}$$
we can calculate the partial differential coefficients of the five
quantities $X$, $Y$, $Z$, $\alpha$, $\beta$, considered as
functions of $(a, b)$; and if we wish to deduce hence, their
partial differential coefficients relatively to one another, we
can do so by means of the following formul{\ae},
$$\eqalign{
dX &=    {dX \over d\alpha}   \mathbin{.} d\alpha
       + {dX \over d\beta}    \mathbin{.} d\beta,\cr
d^2 X
   &=    {dX \over d\alpha}   \mathbin{.} d^2 \alpha
       + {dX \over d\beta}    \mathbin{.} d^2 \beta
       + {dX \over d\alpha^2} \mathbin{.} d\alpha^2
       + 2 {dX \over d\alpha \, d\beta}
                              \mathbin{.} d\alpha \mathbin{.} d\beta
       + {dX \over d\beta^2}  \mathbin{.} d\beta^2, \hbox{ \&c.}\cr}$$
together with the corresponding formul{\ae} for $Y$ and $Z$.

\bigbreak

[58.]
As a first application of the preceding theory, let us suppose
the distance between the two rays so small, that we may neglect
the squares and products of $(\alpha_\prime, \beta_\prime)$; let
us also suppose, that the perpendicular surface of which
$(a, b, c)$ are coordinates, crosses the given ray at the point
where that ray meets the mirror, and let us take that point for
origin, the given ray for the axis of ($z$), and the tangent
planes to the two developable pencils passing through it for the
planes of $(x, z)$, $(y, z)$: we shall then have, [40.],
$$\alpha = 0,\quad
  \beta  = 0,\quad
  \gamma = 1,\quad
  a = 0,\quad
  b = 0,\quad
  c = 0,\quad
  X = 0,\quad
  Y = 0,\quad
  Z = \rho,$$
$$dc = 0,\quad
  d\rho = 0,\quad
  dZ = 0,\quad
  dX = da + \rho \, d\alpha,\quad
  dY = db + \rho \, d\beta,$$
$$\eqalign{
d\alpha
   &=    {d^2 V \over dx^2}              \mathbin{.} da
       + {d^2 V \over dx \mathbin{.} dy} \mathbin{.} db
       + {d^2 V \over dx \mathbin{.} dz} \mathbin{.} dc
    =  - {da \over \rho_1},\cr
d\beta
   &=    {d^2 V \over dx \mathbin{.} dy} \mathbin{.} da
       + {d^2 V \over dy^2}              \mathbin{.} db
       + {d^2 V \over dy \mathbin{.} dz} \mathbin{.} dc
    =  - {db \over \rho_2},\cr}$$
$${dX \over d\alpha} = \rho - \rho_1,\quad
  {dX \over d\beta}  = 0,\quad
  {dY \over d\alpha} = 0,\quad
  {dY \over d\beta}  = \rho - \rho_2,$$
$\rho_1$, $\rho_2$, being the focal lengths of the mirror; and
substituting these values for the partial differential
coefficients of $X$, $Y$, in the general expressions (K${}''$)
for the lateral aberrations, we find
$$x' = (\rho - \rho_1) \mathbin{.} \alpha_\prime,\quad
  y' = (\rho - \rho_2) \mathbin{.} \beta_\prime,
   \eqno {\rm (L'')}$$
$\rho$ being the distance from the perpendicular surfaces at the
mirror to the plane on which the aberration is measured.  These
formul{\ae} (L${}''$) are only the equations (E${}'$) of the IXth
section, under another form; and it follows from the principles
of that section, that the whole lateral aberration may be thus
expressed,
$$\surd (x'^2 + y'^2)
   = \theta \mathbin{.} \surd (u^2 + \delta^2),$$
$\theta = \surd (\alpha_\prime^2 + \beta_\prime^2)$
being the angle which the near ray makes with the given ray;
($u$) a constant coefficient, depending on the position of the
near ray, and determined by the equation
$$u = (\rho_1 - \rho_2)
         \mathbin{.} \sin L \mathbin{.}\cos L$$
($L$ being the angle which the plane of $(x, z)$ makes with a
plane drawn through the given ray parallel to the near ray, so
that
$\beta_\prime = \alpha_\prime \mathbin{.} \tan L$): and
$$\delta = \rho - (  \rho_1 \mathbin{.} \cos^2 L
                   + \rho_2 \mathbin{.} \sin^2 L)$$
being the distance of the point where the aberration is measured
from the point at which that aberration is least.  It follows
also, that if we consider any small parcel of the near rays, the
area on the plane of aberration over which these rays are
diffused, is equal to the product of the distances of that plane
from the two foci of the given ray, multiplied by a constant
quantity depending on the nature of the parcel.  If, for
instance, we consider only those rays which make with the given
ray angles not exceeding some small given angle ($\theta$), these
rays are diffused over the area of an ellipse, having for
equation
$$    {x'^2 \over (\rho - \rho_1)^2}
    + {y'^2 \over (\rho - \rho_2)^2}
  = \theta^2,
   \eqno {\rm (M'')}$$
and this area is equal to the product of the focal distances
($\rho - \rho_1$), ($\rho - \rho_2$), multiplied by
$\pi \mathbin{.} \theta^2$, $\pi$ being the semicircumference of
a circle whose radius is equal to unity.

\bigbreak

[59.]
As a second application, let us take the case where the plane of
aberration passes through one of the two foci of the given ray,
for example, through the second, so that $\rho = \rho_2$.  In
this case the formul{\ae} (L${}''$) become
$$x' = (\rho_2 - \rho_1) \mathbin{.} \alpha_\prime,\quad
  y' = 0,$$
so that if we continue to neglect terms of the second order, the
points in which the near rays cross the plane of aberration, are
all contained on the axis of ($x'$), that is, on the tangent to
the caustic surface.  But if we take into account the aberrations
of the second order, that is, if we do not neglect the squares
and products of $\alpha_\prime$, $\beta_\prime$, which enter into
the general expression (K${}''$) for $y'$, then the rays will
cross the plane of aberration at a small but finite distance from
the axis of ($x'$); that is, $y'$ will have a small but finite
value, which we now propose to investigate.  For this purpose,
that is, to calculate the coefficients in the expression
$$y' = {\textstyle {1 \over 2}}
            \left\{
               {d^2 Y \over d\alpha^2}
                  \mathbin{.} \alpha_\prime^2
             + 2 {d^2 Y \over d\alpha \mathbin{.} d\beta}
                  \mathbin{.} \alpha_\prime \beta_\prime
             + {d^2 Y \over d\beta^2}
                  \mathbin{.} \beta_\prime^2
            \right\},
   \eqno {\rm (N'')}$$
I observe that the general formula, [57.],
$$d^2 Y =
         {dY \over d\alpha}   \mathbin{.} d^2 \alpha
       + {dY \over d\beta}    \mathbin{.} d^2 \beta
       + {d^2 Y \over d\alpha^2} \mathbin{.} d\alpha^2
       + 2 {d^2 Y \over d\alpha \, d\beta}
                              \mathbin{.} d\alpha \mathbin{.} d\beta
       + {d^2 Y \over d\beta^2}  \mathbin{.} d\beta^2,$$
(in which $\alpha$, $\beta$, $Y$, are considered as functions of
the independent variables $(a, b)$, and which is equivalent to
three distinct equations) gives, in the present case,
$${d^2 Y \over d\alpha^2}
   = \rho_1^2 \mathbin{.} {d^2 Y \over da^2},\quad
  {d^2 Y \over d\alpha \mathbin{.} d\beta}
   = \rho_1 \rho_2 \mathbin{.} {d^2 Y \over da \mathbin{.} db},\quad
  {d^2 Y \over d\beta^2}
   = \rho_2^2 \mathbin{.} {d^2 Y \over db^2},$$
because
$${dY \over d\alpha} = 0,\quad
  {dY \over d\beta} = 0,\quad
  d\alpha = - {da \over \rho_1},\quad
  d\beta  = - {db \over \rho_2}.$$
Again, the equation $Y = b + \beta \rho$ gives
$d^2 Y = \rho_2 \mathbin{.} d^2 \beta$, when we put
$$d^2 b = 0,\quad \beta = 0,\quad d\rho = 0,\quad \rho =\rho_2;$$
we have therefore
$${d^2 Y \over da^2}
   = \rho_2 \mathbin{.} {d^2 \beta \over da^2},\quad
  {d^2 Y \over da \mathbin{.} db}
   = \rho_2 \mathbin{.} {d^2 \beta \over da \mathbin{.} db},\quad
  {d^2 Y \over db^2}
   = \rho_2 \mathbin{.} {d^2 \beta \over db^2},$$
and the question is reduced to calculating these partial
differential coefficients of ($\beta$).  Now, the equation
$$d\beta
   =     {d^2 V \over dx \mathbin{.} dy} \mathbin{.} da
       + {d^2 V \over dy^2}              \mathbin{.} db
       + {d^2 V \over dy \mathbin{.} dz} \mathbin{.} dc,$$
gives, (when we put $d^2 a = 0$, $d^2 b = 0$, $dc = 0$,
$\displaystyle {d^2 V \over dy \mathbin{.} dz}  = 0$,)
$$d^2 \beta
   =     {d^3 V \over dx^2 \mathbin{.} dy} \mathbin{.} da^2
     + 2 {d^3 V \over dx \mathbin{.} dy^2} \mathbin{.} da \mathbin{.} db
     +   {d^3 V \over dy^3}                \mathbin{.} db^2,$$
and therefore
$${d^2 \beta \over da^2}
   =  {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 \beta \over da \mathbin{.} db}
   =  {d^3 V \over dx \mathbin{.} dy^2},\quad
  {d^2 \beta \over db^2}
   =  {d^3 V \over dy^3},$$
($V$) being the characteristic of the system; so that finally,
the coefficients in the formula (N${}''$) have for expressions
$${d^2 Y \over d\alpha^2}
   =  \rho_1^2 \mathbin{.} \rho_2   \mathbin{.}
      {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 Y \over d\alpha \mathbin{.} d\beta}
   =  \rho_1   \mathbin{.} \rho_2^2 \mathbin{.}
      {d^3 V \over dx \mathbin{.} dy^2},\quad
  {d^2 Y \over d\beta^2}
   =  \rho_2^3 \mathbin{.}
      {d^3 V \over dy^3}.
   \eqno {\rm (O'')}$$

They may therefore be calculated, either immediately from the
characteristic function ($V$), if the form of that function be
given; or from the equation of the mirror, and the characteristic
of the incident system, according to the method of Section~VI.

\bigbreak

[60.]
The formula (N${}''$), which for conciseness may be written thus
$$y' = {\textstyle {1 \over 2}}
         (A \alpha_\prime^2 + 2 B \alpha_\prime \beta_\prime
            + C \beta_\prime^2).$$
combined with the equation $x' = i \mathbin{.} \alpha_\prime$, in
which ($i$) denotes the interval ($\rho_2 - \rho_1$) between the
two foci of the ray, enables us to find the curve in which any
thin pencil $\beta_\prime = f(\alpha_\prime)$, is cut by a
perpendicular plane passing through the focus of the given ray; a
question for which the formul{\ae} of Section~IX.\ are not
sufficient; since, by those formul{\ae}, the curve would reduce
itself to a right line, namely the tangent to the caustic
surface.  Suppose, for example, that all the rays of the thin
pencil make with the given ray some given small angle ($\theta$),
in which case we have seen that an ordinary section of the pencil
is a little ellipse (M${}''$); we then have to eliminate
$\alpha_\prime$, $\beta_\prime$, between the three equations
$$x' = i \alpha_\prime,\quad
  y' = {\textstyle {1 \over 2}}
         (A \alpha_\prime^2 + 2 B \alpha_\prime \beta_\prime
            + C \beta_\prime^2),\quad
  \alpha_\prime^2 + \beta_\prime^2 = \theta^2,$$
and we find for the equation of the section
$$2 i^2 y'
   = A x'^2 \pm 2 B x' \mathbin{.}
         \surd (i^2 \theta^2 - x'^2)
       + C (i^2 \theta^2 - x'^2);
   \eqno {\rm (P'')}$$
which evidently represents a curve shaped like an hour-glass, or
figure of eight, having its node on the axis of ($y'$), that is,
on the normal to the caustic surface, at a distance
$= {1 \over 2} C \mathbin{.} \theta^2$
from the focus, and bounded by the two tangents
$x' = \pm i \theta$.
The area of this curve is the double of the definite integral
$\displaystyle {2B \over i^2}
      \int \surd (i^2 \theta^2 - x'^2) \mathbin{.} x' \, dx'$,
taken from $x' = 0$ to $x' = i\theta$; it is therefore
$$\Sigma
   = {4 \over 3} \mathbin{.} B \mathbin{.} i
      \mathbin{.} \theta^3.
   \eqno {\rm (Q'')}$$
But we must not suppose that this area, like the area of the
elliptic section (M${}''$), is the entire space over which all
the intermediate rays, that is, all the rays making with the
given ray angles less than ($\theta$), are diffused upon the
plane of aberration; for it is clear that these intermediate rays
intersect the plane of aberration partly inside the curve
(P${}''$), and partly outside it; since the focus itself, that is,
the point $x' = 0$, $y' = 0$, is outside that curve.  We must
therefore, in order to find the whole space occupied by the
intermediate rays, investigate the enveloppe of all the curves
similar to (P$''$), which can be formed by assigning different
values to ($\theta$), and then add to the area ($\Sigma$) of the
curve (P${}''$) itself, the area of the additional space included
between it and its enveloppe.  Differentiating therefore, the
equation (P${}''$) for ($\theta$) as the only variable, we find
$$Bx' \pm C \mathbin{.} \surd (i^2 \theta^2 - x'^2) = 0,\quad
  x'^2 = {C^2 i^2 \theta^2 \over B^2 + C^2},\quad
  y' = {{1 \over 2} C \mathbin{.} \theta^2 \mathbin{.} (AC - B^2)
         \over B^2 + C^2},$$
so that the enveloppe sought is a common parabola, having for
equation,
$$2C \mathbin{.} i^2 \mathbin{.} y'
   = (AC - B^2) \mathbin{.} x'^2,
   \eqno {\rm (R'')}$$
and the additional space ($\Sigma'$), included between it and the
curve which it envelopes, being equal to the double of the
definite integral
$${1 \over 2 i^2 C} \mathbin{.}
      \int \{ Bx' - C \mathbin{.} \surd (i^2 \theta^2 - x'^2) \}^2
      \mathbin{.} dx',$$
taken from $x' = 0$, to
$\displaystyle x'
   = {i \mathbin{.} C \mathbin{.} \theta \over \surd (B^2 + C^2)}$,
has for expression
$$\Sigma'
   = {2 \over 3} i \mathbin{.} \theta^3 \mathbin{.}
         \{ -B + \surd (B^2 + C^2) \};$$
so that the whole space over which the intermediate rays are
diffused, has for expression
$$\Sigma + \Sigma'
   = {2 \over 3} i \mathbin{.} \theta^3 \mathbin{.}
         \{ B + \surd (B^2 + C^2) \}.
   \eqno {\rm (S'')}$$

In these calculations $A$, $B$, $C$, $i$, have been supposed
positive: but the formula (S${}''$) holds also when all or any of
them are negative, provided that we then substitute their numeric
for their algebraic values.

\bigbreak

[61.]
To find the geometric meanings of the coefficients $A$, $B$, $C$,
which enter into the preceding expressions for the aberrations
measured from a focus, let us investigate the curvatures of the
caustic surface.  The two focal lengths of a ray, measured from
the given perpendicular surface, are determined by the formula
(Q) of Section~VI.
$$    \left( \rho + {da \over d\alpha} \right)
      \left( \rho + {db \over d\beta}  \right)
    - {da \over d\beta} \mathbin{.} {db \over d\alpha}
   = 0,$$
which when we make
$\displaystyle {da \over d\alpha} = - \rho_1$,
$\displaystyle {da \over d\beta}  = 0$,
$\displaystyle {db \over d\alpha} = 0$,
$\displaystyle {db \over d\beta}  = - \rho_2$,
$\rho = \rho_2$, gives by differentiation,
$\displaystyle d\rho
   = - d \mathbin{.} \left( {db \over d\beta} \right)$.
We have also, by the same section,
$$    \left\{
         {d^2 V \over dx^2} \mathbin{.} {d^2 V \over dy^2}
       - \left( {d^2 V \over dx \mathbin{.} dy} \right)^2
      \right\}
      \mathbin{.} {db \over d\beta}
   =  \gamma \mathbin{.}
      \left(
         \gamma \mathbin{.} {d^2 V \over dx^2}
       - \alpha \mathbin{.} {d^2 V \over dx \mathbin{.} dz}
      \right),$$
which when we put
$$\alpha = 0,\quad \gamma = 1,\quad d\gamma = 0,\quad
  {d^2 V \over dx \mathbin{.} dy} = 0,\quad
  {d^2 V \over dx \mathbin{.} dz} = 0,$$
$${d^2 V \over dx^2} = - {1 \over \rho_1},\quad
  {d^2 V \over dy^2} = - {1 \over \rho_2},\quad
  {d b \over d\beta} = - \rho_2,$$
gives by differentiation
$$d \mathbin{.} \left( {db \over d\beta} \right)
   = - \rho_2^2 \mathbin{.} d \mathbin{.}
         \left( {d^2 V \over dy^2} \right)
   = B \mathbin{.} d\alpha + C \mathbin{.} d\beta,$$
and therefore $d\rho = - (B \, d\alpha + C \, d\beta)$.  If then
we denote by $x_\prime$,~$y_\prime$,~$z_\prime$, the coordinates
of the caustic surface, considered as functions of $a$ and $b$,
we have
$$x_\prime = a + \alpha \rho,\quad
  y_\prime = b + \beta  \rho,\quad
  z_\prime = c + \gamma \rho,$$
$$dx_\prime = da + \rho \, d \alpha = i \, d\alpha,\quad
  dy_\prime = 0,\quad
  dz_\prime = d\rho = - (B \, d\alpha + C \, d\beta),$$
$$d^2 y_\prime
   = \rho_2 \mathbin{.} d^2 \beta + 2 d\beta \mathbin{.} d\rho
   = A \mathbin{.} d\alpha^2 - C \mathbin{.} d\beta^2,$$
so that the focus of a near ray has for coordinates
$$x_\prime = i \alpha_\prime,\quad
  y_\prime = {\textstyle {1 \over 2}}
               (A \alpha_\prime^2 - C \beta_\prime^2),\quad
  z_\prime = \rho_2 - (B \alpha_\prime + C \beta_\prime);$$
eliminating $\alpha_\prime$, $\beta_\prime$, we find for the
approximate equation of the caustic surface,
$$2 i^2 C \mathbin{.} y_\prime
      + \{ i \mathbin{.} (z_\prime - \rho_2) + B x_\prime \}^2
      - AC \mathbin{.} x_\prime^2
   = 0,$$
which shews, that the radius of curvature $R$ of a normal section
of this surface, is given by the following equation,
$${i^2 C \over R}
   = i^2 \mathbin{.} \cos^2 \omega
      + 2iB \mathbin{.} \sin \omega
         \mathbin{.} \cos \omega
      + (B^2 - AC) \mathbin{.} \sin^2 \omega,$$
($\omega$) being the angle which the plane of the section makes
with the plane of $(yz)$.  Making $\omega = 0$, we get
$R = C$; and the maximum and minimum of $R$ are given by the
equation
$$A \mathbin{.} R^2 + (B^2 - AC + i^2) \mathbin{.} R
     - C \mathbin{.} i^2 = 0;$$
from which it follows that $C$ is the radius of curvature of the
caustic curve, and that if we denote by ($\omega$) the angle at
which this curve crosses either line of curvature on the caustic
surface, we shall have
$$C = {R' R'' \over R' \cos^2 \omega + R'' \sin^2 \omega},\quad
  A = {- i^2 \over R' \cos^2 \omega + R'' \sin^2 \omega},$$
$$B = {i \mathbin{.} (R' - R'') \mathbin{.}
         \sin \omega \mathbin{.} \cos \omega
         \over R' \cos^2 \omega + R'' \sin^2 \omega}
   \eqno {\rm (T'')}$$
$R'$, $R''$ being the two radii of curvature of the caustic
surface.  It appears from these formul{\ae} (T${}''$), that when
the ray touches either line of curvature upon the caustic
surface, (which is always the case when the reflected system
consists of rays, which after issuing from a luminous point, have
been reflected by any combination of mirrors of revolution, that
have a common axis passing through the luminous point), or when
the focus is a point of spheric curvature on its own caustic
surface, then $B$ vanishes, and the area (Q${}''$) of the little
hour-glass curve is equal to nothing.  In fact, in this case,
that curve changes shape, and becomes confounded with a little
parabolic arc, which has for equation
$2i^2 y' = A x'^2 + C (i^2 \theta^2 - x'^2)$,
and which is comprised between the limits $x' = \pm i\theta$;
this parabolic arc is crossed at its extremities by the parabola
(R${}''$), of which the equation becomes $2i^2 y' = A x'^2$: and
the whole space included between these two parabolas, that is,
the whole space over which the near rays are diffused, has for
expression,
$$\Sigma' = {2 \over 3} i \mathbin{.} C \mathbin{.} \theta^3.
   \eqno {\rm (U'')}$$

\bigbreak

[62.]
As a third application, let us consider the case of aberrations
from a principal focus.  In this case we have $i = 0$, and the
expressions for $\Sigma$, $\Sigma'$, vanish; we must therefore
have recourse to new calculations, and introduce terms of the
second order, in the expression of $x'$, as well as in that of
$y'$.  We find
$$\eqalign{
x' &= {\textstyle {1 \over 2}} \rho^3 \mathbin{.}
         \left(
              {d^3 V \over dx^3} \mathbin{.} \alpha_\prime^2
          + 2 {d^3 V \over dx^2 \mathbin{.} dy}
               \mathbin{.} \alpha_\prime \beta_\prime
          +   {d^3 V \over dx \mathbin{.} dy^2} \mathbin{.} \beta_\prime^2
         \right),\cr
y' &= {\textstyle {1 \over 2}} \rho^3 \mathbin{.}
         \left(
              {d^3 V \over dx^2 \mathbin{.} dy} \mathbin{.} \alpha_\prime^2
          + 2 {d^3 V \over dx \mathbin{.} dy^2}
               \mathbin{.} \alpha_\prime \beta_\prime
          +   {d^3 V \over dy^3} \mathbin{.} \beta_\prime^2
         \right),\cr}$$
expressions which may be thus written
$$\left. \eqalign{
x' &= (A \alpha_\prime^2 + 2 B \alpha_\prime \beta_\prime
         + C \beta_\prime^2),\cr
y' &= (B \alpha_\prime^2 + 2 C \alpha_\prime \beta_\prime
         + D \beta_\prime^2),\cr}
   \right\}
   \eqno {\rm (V'')}$$
$A$, $B$, $C$, having different meanings here, from what they had
in the preceding paragraphs.  And if we eliminate
$\alpha_\prime$, $\beta_\prime$, between those equations, by
means of the relation
$$\alpha_\prime^2 + \beta_\prime^2 = \theta^2,$$
which expresses that the near rays make with the given ray an
angle $= \theta$; we find, for the curve of aberration, that is,
for the locus of the points in which those rays cross the
perpendicular plane drawn through the principal focus, the
following equation,
$$\eqalign{
    4 \{ (B^2 - AC) \theta^2 - By' + Cx' \}
      \{ (C^2 - BD) \theta^2 + By' - Cx' \} \hskip -216pt \cr
   &= \{ (A - C) y' + (D - B) x'
         + (BC - AD) \mathbin{.} \theta^2 \}^2,\cr}$$
which may be thus written
$$A'' y'^2 + 2 B'' x' y' + C'' x'^2
      - (D'' y' + E'' x') \theta^2
      + F'' \mathbin{.} \theta^4 = 0,
   \eqno {\rm (W'')}$$
if we put for abridgment
$$A'' = (A - C)^2 + 4 B^2,\quad
  B'' = (A - C)(D - B) - 4 BC,\quad
  C'' = (D - B)^2 + 4 C^2,$$
$$D'' = (A + C) \mathbin{.} B'' + (D + B) \mathbin{.} A'',\quad
  E'' = (A + C) \mathbin{.} C'' + (D + B) \mathbin{.} B'',$$
$$F'' = (AD - BC)^2 - 4 (B^2 - AC) (C^2 - BD).$$
These values give
$$A'' C'' - B''^2 = 4 \{ C (A - C) + B (D - B) \}^2,$$
so that the curve (W${}''$) is an ellipse; the centre of this
little ellipse has for coordinates
$$a'' = {\textstyle {1 \over 2}} (A + C) \mathbin{.} \theta^2,\quad
  b'' = {\textstyle {1 \over 2}} (D + B) \mathbin{.} \theta^2,$$
and its area is
$$\Sigma
   = \pm {\textstyle {1 \over 2}} \pi
         \{ C \mathbin{.} (C - A) + B \mathbin{.} (B - D) \}
         \mathbin{.} \theta^4.
   \eqno {\rm (X'')}$$

If now we consider those intermediate rays, which make with the
given ray some given small angle ($\theta'$), less than
($\theta$), the points in which these rays cut the plane of
aberration will form another similar ellipse, having for equation
$$A'' y'^2 + 2 B'' x' y' + C'' x'^2
      - (D'' y' + E'' x') \theta'^2
      + F'' \mathbin{.} \theta'^4 = 0,$$
and if (F${}''$) be negative, this ellipse is entirely inside the
other, and all the rays that make with the given ray angles not
exceeding ($\theta$) are diffused over the elliptic area
(X${}''$).  But if  (F${}''$) be positive, that is, if the focus
be outside the little ellipse of aberration (W${}''$), then the
intermediate rays are not all diffused over the area of the
ellipse, but cut the plane of aberration partly inside that area
and partly outside it.  To find therefore, in this case, the
whole space over which these near rays are diffused, we must seek
the enveloppe of all the little ellipses similar to (W${}''$), and
then add to the area of that curve (W${}''$) itself, the area of
the space included between it and its enveloppe.  This enveloppe
has for equation
$$(D'' y' + E'' x')^2
   = 4 F'' \mathbin{.} (A'' y'^2 + 2 B'' x' y' + C'' x'^2);
   \eqno {\rm (Y'')}$$
when (F${}''$) is negative it has no existence, and when
(F${}''$) is positive it consists of two right lines passing
through the focus, which are common tangents to all the little
ellipses, and which may be called the {\it Limiting Lines of
Aberration\/}; the space included between them and the ellipse
(W${}''$), has for expression
$$\Sigma'
   = {\Sigma \over \pi} \mathbin{.} (\tan \psi - \psi),
   \eqno {\rm (Z'')}$$
$\Sigma$ being the area of the ellipse, and ($\psi$) an angle
whose cosine, multiplied by the focal distance of the centre of
that ellipse, is equal to the semidiameter whose prolongation
passes through the focus; we have therefore
$$\tan \psi = {\sqrt{F''} \over C (C - A) + B (B - D)},
   \eqno {\rm (A''')}$$
and the entire space over which the intermediate rays are
diffused is
$$\Sigma + \Sigma'
   = {\textstyle {1 \over 2}}
      [ \sqrt{F''} + (\pi - \psi) \{ C (C - A) + B (B - D) \} ]
      \mathbin{.} \theta^4.
   \eqno {\rm (B''')}$$

\bigbreak

[63.]
We have just seen, that in investigating the aberrations from a
principal focus, it is necessary to distinguish two cases,
essentially different from one another.  In the one case, all the
rays that make with the given ray angles not exceeding some given
small angle ($\theta$), are diffused over the area of a little
ellipse; in the other case they are diffused over a mixtilinear
space, bounded partly by an elliptic arc, and partly by two right
lines, which touch that elliptic arc, and which pass through the
principal focus.  The analytic distinction between these two
cases depends on the sign of a certain quantity $F''$, which is
negative in the first case, and positive in the second.  It is
therefore interesting to examine, for any proposed system,
whether this quantity be positive or negative.  I am going to
shew that this depends on the reality of the roots of a certain
cubic equation, which determines the directions of spheric
inflexion on the surfaces that cut the rays perpendicularly; I
shall shew also that the sign of the same quantity, is the
criterion of the reality of the roots of a certain quadratic
equation, which determines the directions in which the plane of
aberration is cut by the two caustic surfaces.

First then, with respect to the caustic surfaces, it may be
proved, by reasonings similar to those of [61.], that the two foci
of a near ray have for coordinates
$$x_\prime = x' + \rho_\prime \alpha_\prime,\quad
  y_\prime = y' + \rho_\prime \beta_\prime, \quad
  z_\prime = \rho + \rho_\prime,$$
$x'$, $y'$, being the coordinates of the point in which the near
ray crosses the plane of aberration, determined by the formul{\ae}
(V${}''$), and ($\rho_\prime$) having a double value determined by
the following quadratic equation
$$    ({\textstyle {1 \over 2}} \rho_\prime
         + A \alpha_\prime + B \beta_\prime)
      ({\textstyle {1 \over 2}} \rho_\prime
         + C \alpha_\prime + D \beta_\prime)
      - (B \alpha_\prime + C \beta_\prime)^2
   = 0,$$
in which $A$, $B$, $C$, $D$, have the same meanings as in the
preceding paragraph.  The intersection therefore of the caustic
surfaces with the plane of aberration, is to be found by putting
$\rho_\prime = 0$, which gives
$z' = \rho$, $x_\prime = x'$, $y_\prime = y'$,
$$( A \alpha_\prime + B \beta_\prime)( C \alpha_\prime + D \beta_\prime)
      - (B \alpha_\prime + C \beta_\prime)^2
   = 0;
   \eqno {\rm (C''')}$$
the condition for the roots being real, in this quadratic
(C${}'''$), is
$$(AD - BC)^2 - 4 (B^2 - AC)(C^2 - BD) > 0,
   \eqno {\rm (D''')}$$
that is, $F'' > 0$, so that unless this condition be satisfied,
the caustic surfaces do not intersect the plane of aberration;
and when this condition is satisfied, the intersection consists
of two right lines, which are determined by the equation
$$(A y' - B x') (C y' - D x')
   = (B y' - C x')^2,
   \eqno {\rm (E''')}$$
and which may easily be shewn to be the same with those limiting
lines which we have already considered.

\bigbreak

[64.]
Secondly, respecting the surfaces that cut the rays
perpendicularly, and which are given by the differential equation
$$\alpha \, da + \beta \, db + \gamma \, dc = 0;$$
we have seen in a former section that the principal foci are the
centres of spheres that have contact of the second order with
these perpendicular surfaces; and if we wish to find the
directions in which they are cut by those osculating spheres, we
must express that the sum of the terms of the third order in the
development of the ordinate of the sphere, is equal to the
corresponding sum, in the development of the perpendicular
surface.  This condition, when the ray is taken for the axis of
($z$), gives $d^3 c = 0$, that is
$d^2 \alpha \mathbin{.} da + d^2 \beta \mathbin{.} db = 0$,
which produces the following cubic equation, (see [59.])
$$0 = {d^3 V \over dx^3} \mathbin{.} da^3
      + 3 \mathbin{.} {d^3 V \over dx^2 \mathbin{.} dy}
            \mathbin{.} da^2 \mathbin{.} db
      + 3 \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2}
            \mathbin{.} da \mathbin{.} db^2
      + {d^3 V \over dy^3} \mathbin{.} db^3.
   \eqno {\rm (F''')}$$

This equation determines the directions of {\it spheric
inflexion\/} upon the perpendicular surface, that is, the
directions in which it is cut by its osculating sphere; and the
condition for there being three such directions, that is, for the
three roots of this cubic equation being real, is
$$\eqalignno{
   &\mathrel{\phantom{<}}
      \left\{
         {d^3 V \over dx^3} \mathbin{.} {d^3 V \over dy^3}
       - {d^3 V \over dx^2 \mathbin{.} dy}
            \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2}
      \right\}^2 \cr
   &\mathrel{\phantom{<}}
    - 4 \mathbin{.}
      \left\{
         \left( {d^3 V \over dx^2 \mathbin{.} dy} \right)^2
       - {d^3 V \over dx^3} \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2}
      \right\}
      \left\{
         \left( {d^3 V \over dx \mathbin{.} dy^2} \right)^2
       - {d^3 V \over dx^2 \mathbin{.} dy} \mathbin{.} {d^3 V \over dy^3}
      \right\} \cr
   &< 0,
   &{\rm (G''')}\cr}$$
that is $F'' < 0$.  When, therefore, the principal focus is
inside the little ellipses of aberration, there are three
directions of spheric inflexion on the surfaces that cut the rays
perpendicularly; and when it is outside those little ellipses,
there is but one such direction.  It appears also, from the
formula (F${}'''$), that the aberrations of the second order do
not vanish, unless the surfaces that cut the rays perpendicularly
have contact of the third order with the osculating spheres that
have their centre at the principal focus; this condition is
expressed by four equations which are not in general satisfied:
and for this reason I shall dispense with considering the
aberrations of the third order, because they only present
themselves in some particular cases; for example, in spheric
mirrors, the theory of which has perhaps been sufficiently
studied by others.

\bigbreak

[65.]
I shall conclude this section by shewing that the conditions for
contact of the third order between the perpendicular surface and
its osculating sphere, which, as we have just seen, are the
conditions for the aberrations of the second order vanishing, are
also the conditions for contact of the third order, between the
mirror and the osculating focal surface (Section~VIII.); and that
the sign of that quantity ($F''$) which distinguishes between the
two different kinds of aberration from a principal focus, and
which, as we have seen, depends on the number of directions in
which the perpendicular surface is cut by the osculating sphere,
depends also on the number of directions in which the mirror is
cut by its osculating focal surface.

To prove these theorems, I observe that if we denote by
$(p'', q'')$ the partial differential coefficients, first order,
of the focal surface, that is, of the surface which would reflect
accurately the rays of the given incident system to the focus
$(X, Y, Z)$, the condition that determines the directions, in
which this surface cuts the given mirror, with which (by
Section~X.) it has complete contact of the second order, is
$$d^2 p'' \mathbin{.} dx + d^2 q'' \mathbin{.} dy
   = d^2 p \mathbin{.} dx + d^2 q \mathbin{.} dy,
   \eqno {\rm (H''')}$$
and that this same equation, when it is to be satisfied
independently of the ratio between $dx$, $dy$ resolves itself
into four distinct equations, which are the conditions for
contact of the third order, between the given mirror and its
osculating focal surface.  Now, if we represent by
$\alpha''$,~$\beta''$,~$\gamma''$,
the cosines of the angles which the reflected ray would make with
the axes, if it came from the focal surface, and not from the
given mirror, we shall have (Section~II.)
$$\alpha'' + \alpha' + (\gamma'' + \gamma') p'' = 0,\quad
  \beta''  + \beta'  + (\gamma'' + \gamma') q'' = 0,$$
and therefore
$$\eqalign{
d\alpha'' + d\alpha' + (d\gamma'' + d\gamma') \mathbin{.} p''
      + (\gamma'' + \gamma') \mathbin{.} dp''
   &= 0,\cr
d\beta''  + d\beta'  + (d\gamma'' + d\gamma') \mathbin{.} q''
      + (\gamma'' + \gamma') \mathbin{.} dq''
   &= 0,\cr}$$
$$\eqalign{
d^2 \alpha'' + d^2 \alpha'
      + (d^2 \gamma'' + d^2 \gamma') \mathbin{.} p''
      + 2 (d\gamma'' + d\gamma') \mathbin{.} dp''
      + (\gamma'' + \gamma') \mathbin{.} d^2 p''
   &= 0,\cr
d^2 \beta''  + d^2 \beta'
      + (d^2 \gamma'' + d^2 \gamma') \mathbin{.} q''
      + 2 (d\gamma'' + d\gamma') \mathbin{.} dq''
      + (\gamma'' + \gamma') \mathbin{.} d^2 q''
   &= 0,\cr}$$
$\alpha'$, $\beta'$, $\gamma'$ being the cosines of the angles
which the incident ray makes with the axes; in the same manner,
we have for the given mirror,
$$\alpha + \alpha' + (\gamma + \gamma') \mathbin{.} p = 0,\quad
  \beta  + \beta'  + (\gamma + \gamma') \mathbin{.} q = 0,$$
$$\eqalign{
d\alpha + d\alpha' + (d\gamma + d\gamma') \mathbin{.} p
      + (\gamma + \gamma') \mathbin{.} dp
   &= 0,\cr
d\beta  + d\beta'  + (d\gamma + d\gamma') \mathbin{.} q
      + (\gamma + \gamma') \mathbin{.} dq
   &= 0,\cr}$$
$$\eqalign{
d^2 \alpha + d^2 \alpha'
      + (d^2 \gamma + d^2 \gamma') \mathbin{.} p
      + 2 (d\gamma + d\gamma') \mathbin{.} dp
      + (\gamma + \gamma') \mathbin{.} d^2 p
   &= 0,\cr
d^2 \beta  + d^2 \beta'
      + (d^2 \gamma + d^2 \gamma') \mathbin{.} q
      + 2 (d\gamma + d\gamma') \mathbin{.} dq
      + (\gamma + \gamma') \mathbin{.} d^2 q
   &= 0,\cr}$$
and because of the contact of the second order, which exists
between the two surfaces, we have
$$p'' = p,\quad
  q'' = q,\quad
  \alpha'' = \alpha,\quad
  \beta''  = \beta, \quad
  \gamma'' = \gamma,$$
$$dp'' = dp,\quad
  dq'' = dq,\quad
  d\alpha'' = d\alpha,\quad
  d\beta''  = d\beta, \quad
  d\gamma'' = d\gamma,$$
$$\eqalign{
   &(\gamma + \gamma')
         (d^2 p'' \mathbin{.} dx + d^2 q'' \mathbin{.} dy)
      + 2 (d\gamma + d\gamma')
         (dp \mathbin{.} dx + dq \mathbin{.} dy) \cr
   &\quad
      + d^2 (\alpha'' + \alpha') \mathbin{.} dx
      + d^2 (\beta''  + \beta' ) \mathbin{.} dy
      + d^2 (\gamma'' + \gamma') \mathbin{.} dz
   = 0,\cr
   &(\gamma + \gamma')
         (d^2 p   \mathbin{.} dx + d^2 q   \mathbin{.} dy)
      + 2 (d\gamma + d\gamma')
         (dp \mathbin{.} dx + dq \mathbin{.} dy) \cr
   &\quad
      + d^2 (\alpha   + \alpha') \mathbin{.} dx
      + d^2 (\beta    + \beta' ) \mathbin{.} dy
      + d^2 (\gamma   + \gamma') \mathbin{.} dz
   = 0;\cr}$$
the condition (H${}'''$) may therefore be thus written,
$$    d^2 \alpha'' \mathbin{.} dx
    + d^2 \beta''  \mathbin{.} dy
    + d^2 \gamma'' \mathbin{.} dz
  =   d^2 \alpha   \mathbin{.} dx
    + d^2 \beta    \mathbin{.} dy
    + d^2 \gamma   \mathbin{.} dz:
   \eqno {\rm (I''')}$$
besides, when the given ray, or axis of the system, is made the
axis of ($z$), and when we take for the two independent variables
the two quantities $(a, b)$, that is, the coordinates of the
projection of the point in which the ray crosses the
perpendicular surface, [57.], we have, from [59.], and from the
nature of the functions $\alpha''$,~$\beta''$,~$\gamma''$,
$$d^2 \alpha'' = 0,\quad
  d^2 \beta''  = 0,\quad
  d^2 \gamma'' = d^2 \gamma,\quad
  da = dx,\quad
  db = dy,$$
$$\eqalign{
d^2 \alpha
   &=      {d^3 V \over dx^3} \mathbin{.} dx^2
       + 2 \mathbin{.}
           {d^3 V \over dx^2 \mathbin{.} dy} \mathbin{.} dx \mathbin{.} dy
       +   {d^3 V \over dx \mathbin{.} dy^2} \mathbin{.} dy^2,\cr
d^2 \beta
   &=      {d^3 V \over dx^2 \mathbin{.} dy} \mathbin{.} dx^2
       + 2 \mathbin{.}
           {d^3 V \over dx \mathbin{.} dy^2} \mathbin{.} dx \mathbin{.} dy
       +   {d^3 V \over dy^3} \mathbin{.} dy^2,\cr}$$
so that (I${}'''$) becomes
$$         {d^3 V \over dx^3} \mathbin{.} dx^3
       + 3 \mathbin{.}
           {d^3 V \over dx^2 \mathbin{.} dy} \mathbin{.} dx^2 \mathbin{.} dy
       + 3 \mathbin{.}
           {d^3 V \over dx \mathbin{.} dy^2} \mathbin{.} dx \mathbin{.} dy^2
       +   {d^3 V \over dy^3} \mathbin{.} dy^3
   = 0;
   \eqno {\rm (K''')}$$
this then is the cubic equation which determines on the given
mirror, the directions of {\it focal inflection}, that is, the
directions in which it is cut by the osculating focal mirror; and
comparing this with the cubic equation (F${}'''$) which
determines the directions of spheric inflexion on the
perpendicular surfaces, we see that the planes which pass through
these directions of spheric inflexion, and through the axes of
the system, pass also through the directions of focal inflexion
on the mirror; so that the number of the latter directions is the
same as the number of the former.  If then there be but one
direction of focal inflexion on the mirror, that is, if the cubic
equation (K${}'''$) have two of its roots imaginary, the
principal focus is outside the little ellipses of aberration, and
the caustic surfaces cross the plane of aberration, in those two
limiting lines, or tangents to the little ellipses, which we have
considered in [62.]; but if there be three directions of focal
inflexion, that is, if the three roots of (K${}'''$) be real,
then the limiting lines of aberration become imaginary, and the
principal focus is inside the little ellipses.  And if the
equation (K${}'''$) be identically satisfied, that is, if the
mirror have contact of the third order with its osculating focal
surface, then the little ellipses themselves disappear, and the
aberrations of the second order vanish.

\bigbreak

\centerline{%
XIII.  \it Density.}

\nobreak\bigskip

[66.]
Malus, who first discovered that the rays of a reflected
system are in general tangents to two caustic surfaces, has given
in his {\it Trait\'{e} D'Optique}, (published among the
{\it M\'{e}moirs des Savans \'{E}trangers\/}) the following method
for investigating the density of the reflected light at any given
point of the system.  He considers two infinitely near pairs of
developable surfaces formed by the rays; and as he believed
himself to have demonstrated that the two surfaces of such a pair
are not in general perpendicular to one another, when the rays
have been more than once reflected, he concludes that the
perpendicular section of the parcel of rays comprised between the
four developable surfaces, will be in general shaped as an
oblique angled parallelogram, whose area is equal to the product
of the two focal distances of the section, multiplied by the sine
of the angle formed by the two developable surfaces of each pair.
He then compares this area with the area over which the same rays
would be diffused, if they had proceeded without interruption to
an equal distance from the luminous point; and he takes the
reciprocal ratio of these areas for the measure of the density of
the reflected light, compared with that of the direct light.  The
calculations required in this method are of considerable
intricacy, and the most remarkable result to which they lead, is
that along a given ray the density varies inversely as the
product of the focal distances, being infinite at the caustic
surfaces, and greatest at their intersection.  The same result
follows from the theorem which I have pointed out in
[43.]\ respecting small parcels of rays bounded by any thin pencil,
of whatever shape; and that theorem furnishes a general method
for investigating the density of the reflected light, at points
not upon the caustic surfaces, which appears to me simpler than
that of Malus, and which for that reason I am going here to
explain.

Suppose then that rays issuing from a luminous point have been
any number of times reflected by any combination of mirrors; let
us put $\Delta$ to represent the density of the direct light at
the distance unity from the luminous point, and let us put
($s$) to represent the space over which any given small parcel of
that light, bounded by any thin cone, is perpendicularly diffused
at that distance.  Then, if we represent by ($\rho$) the first
side of the polygon, that is, the portion of any given incident
ray comprised between the luminous point and the first mirror,
the perpendicular section of the incident parcel, at that
distance from the luminous point, will have its area
$\Sigma = \rho^2 \mathbin{.} s$;
and the space over which the parcel is diffused upon the mirror,
has for expression
$\displaystyle {\rho^2 \mathbin{.} s \over \cos I}$,
$I$ being the angle of incidence.  Immediately after reflexion,
the parcel will again have its perpendicular section
$\rho^2 \mathbin{.} s = \Sigma$; and if we represent by
$F_1'$,~$F_2'$, the two focal lengths of the first mirror, that
is, the distances from the point of incidence to the two points
where the first reflected ray touches the first pair of caustic
surfaces, we shall have by [43.]\ the following expression for the
perpendicular section of the reflected parcel, at any distance
($\rho'$) from the first mirror:
$$\Sigma'
   =  {\Sigma \mathbin{.} (\rho' - F_1') (\rho' - F_2')
         \over F_1' \mathbin{.} F_2'},
   \hbox{ in which }
   \Sigma = \rho^2 \mathbin{.} s.
   \eqno {\rm (L''')}$$
Now let $\rho'$ be the second side of the polygon, that is, the
path run over by the light in going from the first mirror to the
second, and let $(F_1'', F_2'')$ be the two focal lengths of the
second mirror; we shall have, in a similar manner, for the area
of the perpendicular section of the parcel, after the second
reflexion, at a distance~$\rho''$ from the second mirror,
$$\Sigma''
   =  {\Sigma' \mathbin{.} (\rho'' - F_1'') (\rho'' - F_2'')
         \over F_1'' \mathbin{.} F_2''};
   \eqno {\rm (M''')}$$
and so on, for any number of reflexions.  Having thus got the
space over which the reflected rays are perpendicularly diffused,
the density is obtained by this formula
$$\Delta^{(n)} = {s \mathbin{.} \Delta \over \Sigma^{(n)}}.
   \eqno {\rm (N''')}$$

For instance, if the rays have been but once reflected, then the
density is
$$\Delta'
   =  {s \mathbin{.} \Delta \over \Sigma'}
   =  {\Delta \over \rho^2} \mathbin{.}
      {F_1' \mathbin{.} F_2' \over (\rho' - F_1') (\rho' - F_2')},
   \eqno {\rm (O''')}$$
a formula which agrees with that of Malus; after two reflections,
the density is
$$\Delta''
   =  {s \mathbin{.} \Delta \over \Sigma''}
   =  {\Delta' \mathbin{.} F_1'' \mathbin{.} F_2''
         \over (\rho'' - F_1'') (\rho'' - F_2'')},
   \eqno {\rm (P''')}$$
$\Delta'$ being the density immediately before the second
reflexion: a formula which is different from that of Malus, and
which appears by me to be simpler.

\bigbreak

[67.]
The two preceding methods, namely, that of Malus, and that of the
preceding paragraph, fail when the density is to be measured at
the caustic surfaces; for they only shew that the density at those
surfaces is infinitely greater than at other points of the
system, without shewing by what law the density varies in passing
from one point of a caustic surface to another.  This question,
which has not been treated by Malus, appears to me too important
to be passed over, although the discussion of it is more
difficult than the investigation of the density at ordinary
points of the system.

Let us then, as a first approximation, resume the formul{\ae} of
[60.]
$$x' = i \alpha_\prime,\quad
  y' = {\textstyle {1 \over 2}}
         (A \alpha_\prime^2
            + 2 B \alpha_\prime \beta_\prime
            + C \beta_\prime^2),$$
$x'$, $y'$ being the coordinates of the point in which a near ray
crosses the plane of aberration, that is, a plane perpendicular
to the given ray, passing through the focus of that ray;
$\alpha_\prime$,~$\beta_\prime$, small but finite quantities,
namely, the cosines of the angles which the near ray makes with
the axes of $(x')$ and $(y')$, the former of which axes is a
tangent and the other a normal to the caustic surface;
$A$,~$B$,~$C$, coefficients depending on the curvatures of that
surface, and on the interval ($i$) between the two foci of the
ray.  To find by these equations the space over which the rays,
that pass through any given small area on the plane of
aberration, are diffused upon another perpendicular plane, which
crosses the given reflected ray at the point where that ray meets
the mirror, we are to employ these other formul{\ae} (see [58.])
$$a= - \rho_1 \mathbin{.} \alpha_\prime,\quad
  b= - \rho_2 \mathbin{.} \beta_\prime;$$
$a$,~$b$, being the coordinates of the point in which a near ray
crosses this latter plane, and $\rho_1$,~$\rho_2$, the distances
of that point from the two caustic surfaces, that is, the two
focal lengths of the mirror.  In this manner we find, that to any
given point $(x', y')$ on the plane of aberration, correspond two
other points on the other perpendicular plane, determined by the
equations
$$a = - {\rho_1 \mathbin{.} x' \over i},\quad
  b = {\rho_2 \over Ci} \mathbin{.} ( B x'
        \mp \surd \{ 2 C i^2 y' + (B^2 - AC) x'^2 \});
   \eqno {\rm (Q''')}$$
understanding however that these two points become imaginary,
when the quantity under the radical sign is negative, that is,
when the point $(x', y')$ is at the wrong side of the enveloping
parabola (R${}'')$,~[60.]; which parabola, within the small extent
in which it is taken, may be considered as confounded with the
normal section of the caustic surface made by the plane of
aberration.  Now, if we consider any infinitely little rectangle
upon this latter plane, having for the coordinates of its four
corners
$$\hbox{1st.}\enspace x',\, y',\quad
  \hbox{2d.}\enspace  x' + dx',\, y',\quad
  \hbox{3d.}\enspace  x',\, y' + dy',\quad
  \hbox{4th.}\enspace x' + dx',\, y' + dy',$$
the rays which pass inside this little rectangle are diffused
over two little parallelograms on the other perpendicular plane;
the four corners of the one having for coordinates,
$$\hbox{1st.}\enspace a,\, b,\quad
  \hbox{2d.}\enspace  a + da,\,
         b + {db \over dx'} \mathbin{.} dx',$$
$$\hbox{3d.}\enspace  a,\,
         b + {db \over dy'} \mathbin{.} dy',\quad
  \hbox{4th.}\enspace a + da,\,
         b + {db \over dx'} \mathbin{.} dx'
           + {db \over dy'} \mathbin{.} dy',$$
and the four corners of the other having for coordinates,
$$\hbox{1st.}\enspace a,\, b',\quad
  \hbox{2d.}\enspace  a + da,\,
         b' + {db' \over dx'} \mathbin{.} dx',$$
$$\hbox{3d.}\enspace  a,\,
         b' + {db' \over dy'} \mathbin{.} dy',\quad
  \hbox{4th.}\enspace a + da,\,
         b' + {db' \over dx'} \mathbin{.} dx'
           + {db' \over dy'} \mathbin{.} dy',$$
$b$, $b'$ being the two values of ($b$) given by the
formul{\ae}~(Q${}'''$).  The areas of these two parallelograms
are each equal to
$\displaystyle \left(
      da \mathbin{.} {db \over dy'} \mathbin{.} dy' \right)$,
that is to
$${\rho_1 \mathbin{.} \rho_2 \mathbin{.} dx' \mathbin{.} dy'
      \over \surd \{ 2C i^2 y' + (B^2 - AC) x'^2 \}};$$
and the area of the little rectangle on the plane of aberration
is $dx' \mathbin{.} dy'$; if then we denote by $\Delta^{(\mu)}$
the density at the mirror, we shall have for the density at the
point $x'$,~$y'$, on the plane of aberration, the following
approximate expression
$$\Delta^{(\alpha)}
   =  {2 \mathbin{.} \rho_1 \mathbin{.} \rho_2 \mathbin{.}
         \Delta^{(\mu)}
      \over \surd \{ 2C i^2 y' + (B^2 - AC) x'^2 \}};
   \eqno {\rm (R''')}$$
an expression which shews that at the caustic surface the density
is infinitely greater than at the mirror; and that near the
caustic surface the density is not uniform, but varies nearly
inversely as the square root of the perpendicular distance from
that surface; so that we may consider this density as constant in
any one of the little parabolic bands comprised between two
infinitely near parallels to the enveloping curve (R${}''$)~[60.].

\bigbreak

[68.]
To treat this question, respecting the variation of density upon
the plane of aberration, in a more accurate manner, let us take
into account the remaining terms of the developments of $x'$ und
$y'$, as given by the general theory, which we have explained at
the beginning of the preceding section.  For although we were at
liberty to neglect these terms, when we were only in quest of
approximate formul{\ae} to represent the manner in which certain
of the near rays are diffused over the plane of aberration; yet,
when we are returning from this latter plane to the perpendicular
plane at the mirror, it is not safe to neglect any term on
account of its smallness, until we have examined whether, in thus
returning, its influence may not be magnified in such a manner as
to become sensible.

Let us then resume the general series (K${}''$)~[57.]
$$\eqalign{
x' &= X + {dX \over d\alpha} \mathbin{.} \alpha_\prime
        + {dX \over d\beta}  \mathbin{.} \beta_\prime
        + {\textstyle {1 \over 2}} \mathbin{.}
         \left\{
            {d^2 X \over d\alpha^2} \mathbin{.} \alpha_\prime^2
          + 2 {d^2 X \over d\alpha \mathbin{.} d\beta}
               \mathbin{.} \alpha_\prime \beta_\prime
          + {d^2 X \over d\beta^2}  \mathbin{.} \beta_\prime^2
         \right\}
        + \hbox{\&c.},\cr
y' &= Y + {dY \over d\alpha} \mathbin{.} \alpha_\prime
        + {dY \over d\beta}  \mathbin{.} \beta_\prime
        + {\textstyle {1 \over 2}} \mathbin{.}
         \left\{
            {d^2 Y \over d\alpha^2} \mathbin{.} \alpha_\prime^2
          + 2 {d^2 Y \over d\alpha \mathbin{.} d\beta}
               \mathbin{.} \alpha_\prime \beta_\prime
          + {d^2 Y \over d\beta^2}  \mathbin{.} \beta_\prime^2
         \right\}
        + \hbox{\&c.},\cr}$$
in which we have at present
$$X = 0,\quad Y = 0,\quad
  {dX \over d\alpha} = i,\quad
  {dX \over d\beta}  = 0,\quad
  {dY \over d\alpha} = 0,\quad
  {dY \over d\beta}  = 0,$$
$${d^2 X \over d\alpha^2}
   = \rho_1^3 \mathbin{.} {d^3 V \over dx^3},\quad
  {d^2 X \over d\alpha \mathbin{.} d\beta}
   = \rho_1^2 \mathbin{.} \rho_2 \mathbin{.}
      {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 X \over d\beta^2}
   = \rho_1 \mathbin{.} \rho_2^2 \mathbin{.}
      {d^3 V \over dx \mathbin{.} dy^2},$$
$${d^2 Y \over d\alpha^2}
   = \rho_1^2 \mathbin{.} \rho_2 \mathbin{.}
      {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 Y \over d\alpha \mathbin{.} d\beta}
   = \rho_1 \mathbin{.} \rho_2^2 \mathbin{.}
      {d^3 V \over dx \mathbin{.} dy^2},\quad
  {d^2 Y \over d\beta^2}
   = \rho_2^3 \mathbin{.} {d^3 V \over dy^3},$$
$V$ being the characteristic function; so that
$${d^2 X \over d\alpha \mathbin{.} d\beta}
  = {d^2 Y \over d\alpha^2},
  \quad\hbox{and}\quad
  {d^2 X \over d\beta^2}
  = {d^2 Y \over d\alpha \mathbin{.} d\beta}.$$
We have in like manner,
$$\eqalign{
\alpha_\prime
   &=    {d\alpha \over da} \mathbin{.} a
       + {d\alpha \over db} \mathbin{.} b
       + {\textstyle {1 \over 2}} \mathbin{.}
         \left\{
            {d^2 \alpha \over da^2} \mathbin{.} a^2
          + 2 {d^2 \alpha \over da \mathbin{.} db}
               \mathbin{.} a b
          + {d^2 \alpha \over db^2} \mathbin{.} b^2
         \right\}
       + \hbox{\&c.},\cr
\beta_\prime
   &=    {d\beta  \over da} \mathbin{.} a
       + {d\beta  \over db} \mathbin{.} b
       + {\textstyle {1 \over 2}} \mathbin{.}
         \left\{
            {d^2 \beta  \over da^2} \mathbin{.} a^2
          + 2 {d^2 \beta  \over da \mathbin{.} db}
               \mathbin{.} a b
          + {d^2 \beta  \over db^2} \mathbin{.} b^2
         \right\}
       + \hbox{\&c.},\cr}$$
in which, at present,
$${d\alpha \over da} = - {1 \over \rho_1},\quad
  {d\alpha \over db} = 0,\quad
  {d\beta  \over da} = 0,\quad
  {d\beta  \over db} = - {1 \over \rho_2},$$
$${d^2 \alpha \over da^2}
   = {d^3 V \over dx^3},\quad
  {d^2 \alpha \over da \mathbin{.} db}
   = {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 \alpha \over db^2}
   = {d^3 V \over dx \mathbin{.} dy^2};$$
$${d^2 \beta  \over da^2}
   = {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 \beta  \over da \mathbin{.} db}
   = {d^3 V \over dx \mathbin{.} dy^2},\quad
  {d^2 \beta  \over db^2}
   = {d^3 V \over dy^3}.$$
And if we substitute these expressions for
$\alpha_\prime$,~$\beta_\prime$,
in the two series for $x'$ and $y'$, we shall get two other
series of the form
$$\left. \eqalign{
x' &=    {dx' \over da} \mathbin{.} a
       + {dx' \over db} \mathbin{.} b
       + {\textstyle {1 \over 2}} \mathbin{.}
         \left\{
            {d^2 x' \over da^2} \mathbin{.} a^2
          + 2 {d^2 x' \over da \mathbin{.} db}
               \mathbin{.} a b
          + {d^2 x' \over db^2} \mathbin{.} b^2
         \right\}
       + \hbox{\&c.},\cr
y' &=    {dy' \over da} \mathbin{.} a
       + {dy' \over db} \mathbin{.} b
       + {\textstyle {1 \over 2}} \mathbin{.}
         \left\{
            {d^2 y' \over da^2} \mathbin{.} a^2
          + 2 {d^2 y' \over da \mathbin{.} db}
               \mathbin{.} a b
          + {d^2 y' \over db^2} \mathbin{.} b^2
         \right\}
       + \hbox{\&c.},\cr}
   \right\}
   \eqno {\rm (S''')}$$
in which, at present,
$${dx' \over da} = - {i \over \rho_1},\quad
  {dx' \over db} = 0,\quad
  {dy' \over da} = 0,\quad
  {dy' \over db} = 0,$$
$${d^2 x' \over da^2}
   = \rho_2 \mathbin{.} {d^3 V \over dx^3},\quad
  {d^2 x' \over da \mathbin{.} db}
   = \rho_2 \mathbin{.} {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 x' \over db^2}
   = \rho_2 \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2},$$
$${d^2 y' \over da^2}
   = \rho_2 \mathbin{.} {d^3 V \over dx^2 \mathbin{.} dy},\quad
  {d^2 y' \over da \mathbin{.} db}
   = \rho_2 \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2},\quad
  {d^2 y' \over db^2}
   = \rho_2 \mathbin{.} {d^3 V \over dy^3},$$
and in which the other coefficients can also be calculated by
means of the characteristic function.

This being laid down, let us put
$x' = r \mathbin{.} \cos v$,
$y' = r \mathbin{.} \sin v$,
and let us develope ($a$) and ($b$) according to the powers of
($r$).  The developments will be of the form
$$\left. \eqalign{
a &=     r^m     \mathbin{.} u
       + r^{m'}  \mathbin{.} u'
       + r^{m''} \mathbin{.} u''
       + \ldots,\cr
b &=     r^n     \mathbin{.} w
       + r^{n'}  \mathbin{.} w'
       + r^{n''} \mathbin{.} w''
       + \ldots,\cr}
   \right\}
   \eqno {\rm (T''')}$$
$m, m', m'' \, \ldots$ $n, n', n'' \, \ldots$ being positive and
increasing exponents, which may or may not be fractional, and
$u, u', u'' \, \ldots$ $w, w', w'' \, \ldots$ being functions of
the angle ($v$): which functions, as well as the exponents of the
terms that multiply them, we have now to determine. Substituting
therefore the values (T${}'''$) in the series (S${}'''$) we find
the following equations:
$$\eqalign{
\hbox{1st.}\ldots\ldots \,
0  &= - r \mathbin{.} \cos v
      + {dx' \over da}
            \mathbin{.} (r^m \mathbin{.} u + \ldots)
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 x' \over da^2}
            \mathbin{.} (r^m \mathbin{.} u + \ldots)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
      +  {d^2 x' \over da \mathbin{.} db}
            \mathbin{.} (r^m \mathbin{.} u + \ldots)
                        (r^n \mathbin{.} w + \ldots)
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 x' \over db^2}
            \mathbin{.} (r^n \mathbin{.} w + \ldots)^2
      + \hbox{\&c.},\cr
\hbox{2d.}\ldots\ldots \,
0  &= - r \mathbin{.} \sin v
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 y' \over da^2}
            \mathbin{.} (r^m \mathbin{.} u + \ldots)^2
      +  {d^2 y' \over da \mathbin{.} db}
            \mathbin{.} (r^m \mathbin{.} u + \ldots)
                        (r^n \mathbin{.} w + \ldots) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 y' \over db^2}
            \mathbin{.} (r^n \mathbin{.} w + \ldots)^2
      + \hbox{\&c.} \cr}$$
In order that these two equations should be identically
satisfied, we must have, in the first place, for the exponents of
the lowest powers of $r$ in the developments (T${}'''$)
$$m = 1,\quad n = {\textstyle {1 \over 2}};$$
and for the corresponding coefficients, $(u, w)$,
$$\eqalign{
\hbox{(1)}\ldots \,
0  &= - \cos v + {dx' \over da} \mathbin{.} u
      + {\textstyle {1 \over 2}} \mathbin{.}
        {d^2 x' \over db^2} \mathbin{.} w^2,\cr
\hbox{(2)}\ldots \,
0  &= - \sin v
      + {\textstyle {1 \over 2}} \mathbin{.}
        {d^2 y' \over db^2} \mathbin{.} w^2,\cr}$$
that is, in the notation of the preceding paragraph,
$$\left. \eqalign{
2 \rho_1 \rho_2 \mathbin{.} \cos v
      + 2 i \rho_2 \mathbin{.} u
   &= B \mathbin{.} w^2,\cr
2 \rho_2^2 \mathbin{.} \sin v
   &= C \mathbin{.} w^2.\cr}
   \right\}
   \eqno {\rm (U''')}$$
In a similar manner we find for the next greater exponents
$m' = {3 \over 2}$, $n' = 1$;
and for the corresponding coefficients $u'$, $w'$,
$$\eqalign{
\hbox{(1)${}'$}\ldots \,
0  &=   {dx' \over da} \mathbin{.} u'
      + {d^2 x' \over da \mathbin{.} db} \mathbin{.} u w
      + {d^2 x' \over db^2} \mathbin{.} w w'
      + {\textstyle {1 \over 6}} \mathbin{.}
        {d^3 x' \over db^3} \mathbin{.} w^3,\cr
\hbox{(2)${}'$}\ldots \,
0  &=   {d^2 y' \over da \mathbin{.} db} \mathbin{.} u
      + {d^2 y' \over db^2} \mathbin{.} w'
      + {\textstyle {1 \over 6}} \mathbin{.}
        {d^3 y' \over db^3} \mathbin{.} w^2.\cr}$$
And so proceeding, we can find as many of the exponents and
coefficients of the developments (T${}'''$), as may be necessary;
the exponents forming the two following series,
$$m = 1,\quad
  m' = {\textstyle {3 \over 2}},\quad
  m'' = 2,
      \quad\ldots\quad
  m^{(t)} = {t + 2 \over 2},$$
$$n = {\textstyle {1 \over 2}},\quad
  n' = 1,\quad
  n'' = {\textstyle {3 \over 2}}
      \quad\ldots\quad
  n^{(t)} = {t + 1 \over 2};$$
and the coefficients being successively determined by equations
of the following form,
$$\left. \eqalign{
\hbox{(1)${}^{(t)}$}\ldots \,
0  &=   {dx' \over da} \mathbin{.} u^{(t)}
      + {d^2 x' \over db^2} \mathbin{.} w \mathbin{.} w^{(t)}
      + k_1^{(t)},\cr
\hbox{(2)${}^{(t)}$}\ldots \,
0  &=   {d^2 y' \over db^2} \mathbin{.} w \mathbin{.} w^{(t)}
      + k_2^{(t)},\cr}
   \right\}
   \eqno {\rm (V''')}$$
$k_1^{(t)} \mathbin{.} r^{{t + 2 \over 2}}$,
$k_2^{(t)} \mathbin{.} r^{{t + 2 \over 2}}$,
representing for abridgment the sums of the known terms of the
dimension
$\displaystyle \left( {t + 2 \over 2} \right)$
in the expansions of $x'$ and $y'$, according to the powers of
$r$, obtained by substituting in (S${}'''$) the assumed
developments (T${}'''$) in place of $a$ and $b$.  The quantities
$k_1^{(t)}$, $k_2^{(t)}$, are therefore rational functions of the
preceding coefficients
$u, u', \ldots \, u^{(t-1)}$, $w, w', \ldots \, w^{(t-1)}$,
and therefore finally of $u$ and $w$; and these functions do, or
do not, change sign along with $w$, according as $t$ is an odd or
an even number.  Hence it follows, that the developments
(T${}'''$), which represent the coordinates of the points, where
the near rays passing through any assigned point upon the plane
of aberration are intersected by the perpendicular plane at the
mirror, are of the form
$$\eqalign{
a  &= r \mathbin{.}
         (u + r \mathbin{.} u'' + r^2 \mathbin{.} u'''' + \ldots)
      \pm r^{{3 \over 2}} \mathbin{.}
         (u' + r \mathbin{.} u''' + \ldots),\cr
b  &= \pm r^{{1 \over 2}} \mathbin{.}
         (w + r \mathbin{.} w'' + r^2 \mathbin{.} w'''' + \ldots)
      + r \mathbin{.}
         (w' + r \mathbin{.} w''' + \ldots),\cr}$$
the coefficients of the fractional powers being real or
imaginary, according as $w$ is real or imaginary, that is, by
(U${}'''$), according as
$\displaystyle \left( {\sin v \over C} \right)$
is positive or negative; or finally, by [61.], according as the
assigned point $(r, v)$ upon the plane of aberration, is, or is
not, situated at that side of the tangent plane of the caustic
surface, towards which is turned the convexity of the caustic
curve.  However, when the polar angle ($v$) approaches to ($0$)
or ($180^\circ$), that is when the right line joining the focus
of the given ray to the assigned point upon the plane of
aberration, tends to become a tangent to the caustic surface, the
numeric value of ($\sin v$), and therefore of ($w$),
diminishes indefinitely; and consequently the coefficients which
contain negative powers of that quantity, increase without limit,
so that the series (T${}'''$) become at length illusory.  In this
case, therefore, it becomes necessary to have recourse to new
developments, which will be indicated in the succeeding
paragraph.  But abstracting for the present from this case,
which, in examining the variation of the density of the reflected
light upon the plane of aberration, may usually be avoided by a
proper choice of the focus from which the aberrations are to be
measured: it may easily be shewn, by reasonings similar to those
of the preceding paragraph, that if we consider any infinitely
small polar rectangle upon the plane of aberration, having its
base $= r \mathbin{.} dv$, and its altitude $= dr$; the rays
which pass inside this little rectangle, are, at the mirror,
diffused nearly perpendicularly over two little parallelograms,
whose areas are
$$\left. \eqalign{
\Sigma_1^{(\mu)}
   &= \left(
         {da  \over dv} \mathbin{.} {db  \over dr}
       - {da  \over dr} \mathbin{.} {db  \over dv}
      \right) \mathbin{.} dr \mathbin{.} dv,\cr
\Sigma_2^{(\mu)}
   &= \left(
         {da' \over dr} \mathbin{.} {db' \over dv}
       - {da' \over dv} \mathbin{.} {db' \over dr}
      \right) \mathbin{.} dr \mathbin{.} dv,\cr}
   \right\}
   \eqno {\rm (W''')}$$
$a$, $b$, $a'$, $b'$, being the coordinates of the two points,
determined by the series (T${}'''$).  Substituting for these
coordinates their values, we find that the two areas (W${}'''$)
are the two values of the following expression,
$$\eqalign{
\Sigma^{(\mu)}
   &= {\textstyle {1 \over 2}} \mathbin{.} r^{1 \over 2}
         \mathbin{.} dr \mathbin{.} dv \mathbin{.}
         \left\{
            {du \over dv}
          + r \mathbin{.} {du'' \over dv}
          + r^2 \mathbin{.} {du'''' \over dv}
          + \ldots
          \pm r^{1 \over 2} \mathbin{.}
            \left(
               {du' \over dv}
             + r \mathbin{.} {du''' \over dv}
             + \ldots
            \right)
         \right\} \cr
   &\qquad \times
         \{ w
          + 3 \mathbin{.} r \mathbin{.} w''
          + 5 \mathbin{.} r^2 \mathbin{.} w''''
          + \ldots
          \pm 2 \mathbin r^{1 \over 2} \mathbin{.}
            (  w'
             + 2 \mathbin{.} r \mathbin{.} w'''
             + \ldots )
         \} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - {\textstyle {1 \over 2}} \mathbin{.} r^{1 \over 2}
         \mathbin{.} dr \mathbin{.} dv \mathbin{.}
         \left\{
            {dw \over dv}
          + r \mathbin{.} {dw'' \over dv}
          + r^2 \mathbin{.} {dw'''' \over dv}
          + \ldots
          \pm r^{1 \over 2} \mathbin{.}
            \left(
               {dw' \over dv}
             + r \mathbin{.} {dw''' \over dv}
             + \ldots
            \right)
         \right\} \cr
   &\qquad \times
         \{ 2 \mathbin{.}
            (  u
             + 2 \mathbin{.} r \mathbin{.} u''
             + 3 \mathbin{.} r^2 \mathbin{.} u''''
             + \ldots )
          \pm \mathbin r^{1 \over 2} \mathbin{.}
            (  3 \mathbin{.} u'
             + 5 \mathbin{.} r \mathbin{.} u'''
             + \ldots )
         \} \cr
   &= {\textstyle {1 \over 2}} \mathbin{.} r^{1 \over 2}
         \mathbin{.} dr \mathbin{.} dv \mathbin{.}
         \biggl\{
            \left(
               {du \over dv}
             + r \mathbin{.} {du'' \over dv}
             + \ldots
            \right)
            ( w + 3 \mathbin{.} r \mathbin{.} w'' + \ldots ) \cr
   &\qquad - 2 \mathbin{.}
            \left(
               {dw \over dv}
             + r \mathbin{.} {dw'' \over dv}
             + \ldots
            \right)
            ( u + 2 \mathbin{.} r \mathbin{.} u'' + \ldots )
         \biggr\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\textstyle {1 \over 2}} \mathbin{.} r^{3 \over 2}
         \mathbin{.} dr \mathbin{.} dv \mathbin{.}
         \biggl\{
            2 \mathbin{.}
            \left(
               {du' \over dv}
             + r \mathbin{.} {du''' \over dv}
             + \ldots
            \right)
            ( w' + 2 \mathbin{.} r \mathbin{.} w''' + \ldots ) \cr
   &\qquad -
            \left(
               {dw' \over dv}
             + r \mathbin{.} {dw''' \over dv}
             + \ldots
            \right)
            ( 3 \mathbin{.} u' + 5 \mathbin{.} r \mathbin{.} u''' + \ldots )
         \biggr\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      \pm {\textstyle {1 \over 2}} \mathbin{.} r
         \mathbin{.} dr \mathbin{.} dv \mathbin{.}
         \biggl\{
            \left(
               {du' \over dv}
             + r \mathbin{.} {du''' \over dv}
             + \ldots
            \right)
            ( w + 3 \mathbin{.} r \mathbin{.} w'' + \ldots ) \cr
   &\qquad -
            \left(
               {dw \over dv}
             + r \mathbin{.} {dw'' \over dv}
             + \ldots
            \right)
            ( 3 \mathbin{.} u' + 5 \mathbin{.} r \mathbin{.} u''' + \ldots )
         \biggr\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      \pm r \mathbin{.} dr \mathbin{.} dv \mathbin{.}
         \biggl\{
            \left(
               {du \over dv}
             + r \mathbin{.} {du'' \over dv}
             + \ldots
            \right)
            ( w' + 2 \mathbin{.} r \mathbin{.} w''' + \ldots ) \cr
   &\qquad -
            \left(
               {dw' \over dv}
             + r \mathbin{.} {dw''' \over dv}
             + \ldots
            \right)
            ( u + 2 \mathbin{.} r \mathbin{.} u'' + \ldots )
         \biggr\},\cr}$$
which is of the form
$$\eqalignno{
\Sigma^{(\mu)}
   &= {\textstyle {1 \over 2}} \mathbin{.} r^{1 \over 2} \mathbin{.}
      (  U^{(0)}
       + U^{(1)} \mathbin{.} r
       + U^{(2)} \mathbin{.} r^2 + \ldots )
      \mathbin{.} dr \mathbin{.} dv \cr
   &\mathrel{\phantom{=}} \mathord{}
      \pm {\textstyle {1 \over 2}} \mathbin{.} r \mathbin{.}
      (  U_\prime^{(0)}
       + U_\prime^{(1)} \mathbin{.} r
       + U_\prime^{(2)} \mathbin{.} r^2 + \ldots )
      \mathbin{.} dr \mathbin{.} dv,
   &{\rm (X''')}\cr}$$
the coefficients
$U^{(0)}, U^{(1)},\ldots$
$U_\prime^{(0)}, U_\prime^{(1)},\ldots$
being functions of the polar angles ($v$).  The densities of the
reflected light, at these little parallelograms (W${}'''$), have
for developments
$$\left. \eqalign{
\Delta_1
   &=    \Delta^{(\mu)}
       + {d \mathbin{.} \Delta^{(\mu)} \over da} \mathbin{.} a
       + {d \mathbin{.} \Delta^{(\mu)} \over db} \mathbin{.} b
       + \ldots,\cr
\Delta_2
   &=    \Delta^{(\mu)}
       + {d \mathbin{.} \Delta^{(\mu)} \over da} \mathbin{.} a'
       + {d \mathbin{.} \Delta^{(\mu)} \over db} \mathbin{.} b'
       + \ldots,\cr}
   \right\}
   \eqno {\rm (Y''')}$$
$\Delta^{(\mu)}$ being, as in the preceding paragraph, the
density at the point $a = 0$, $b = 0$, that is, at the point
where the given ray meets the mirror: and substituting in these
developments (Y${}'''$), the values of $a$, $b$, $a'$, $b'$,
given by the series (T${}'''$), we find that the two densities
$\Delta_1$, $\Delta_2$, are the two values of the following
expression:
$$\eqalign{
\Delta = \Delta^{(\mu)}
   &+ {d \mathbin{.} \Delta^{(\mu)} \over da}
      \{ r \mathbin{.}
            (u + r \mathbin{.} u'' + \ldots )
         \pm r^{3 \over 2} \mathbin{.}
            (u' + r \mathbin{.} u''' + \ldots ) \} \cr
   &+ {d \mathbin{.} \Delta^{(\mu)} \over db}
      \{ \pm r^{1 \over 2} \mathbin{.}
            (w + r \mathbin{.} w'' + \ldots )
         + r \mathbin{.}
            (w' + r \mathbin{.} w''' + \ldots ) \} \cr
   &+ \hbox{\&c.},\cr}$$
which is of the form
$$\eqalignno{\Delta
   &= \Delta^{(\mu)}
      + \Delta_1^{(\mu)} \mathbin{.} r
      + \Delta_2^{(\mu)} \mathbin{.} r^2
      + \ldots \cr
   &\mathrel{\phantom{=}} \mathord{}
      \pm r^{1 \over 2} \mathbin{.}
         (  \Delta_{\left( {1 \over 2} \right)}^{(\mu)}
          + \Delta_{\left( {3 \over 2} \right)}^{(\mu)}
               \mathbin{.} r + \ldots),
   &{\rm (Z''')}\cr}$$
the coefficients being functions of the polar angle ($v$).

Similarly, if we denote by $\gamma_1$, $\gamma_2$, the cosines of
the angles which the two near rays, passing through the two
points $(a, b)$, $(a', b')$, make with the axis of ($z$), that
is, with the given ray, these cosines will have developments of
the form
$$\eqalignno{
\gamma
   &= 1 + {\textstyle {1 \over 2}}
         \left\{
            {d^2 \gamma \over da^2} \mathbin{.} a^2
          + 2 {d^2 \gamma \over da \mathbin{.} db} \mathbin{.} ab
          + {d^2 \gamma \over db^2} \mathbin{.} b^2
         \right\}
      + \ldots \cr
   &= 1
      + \Gamma^{(1)} \mathbin{.} r
      + \Gamma^{(2)} \mathbin{.} r^2
      \, \ldots \cr
   &\mathrel{\phantom{=}} \mathord{}
      \pm r^{1 \over 2} \mathbin{.}
         (  \Gamma_\prime^{(1)} \mathbin{.} r
          + \Gamma_\prime^{(2)} \mathbin{.} r^2
          + \ldots ),
   &{\rm (A'''')}\cr}$$
the coefficients being also functions of the polar angle ($v$);
and the whole number of the near reflected rays, which pass
within the little rectangle
($r \mathbin{.} dr \mathbin{.} dv$) upon the plane of aberration,
being equal to the sum of the two values of the product
$\gamma \mathbin{.} \Delta \mathbin{.} \Sigma^{(\mu)}$,
will be expressed by a development of the form
$$Q^{(\alpha)}
   = r^{1 \over 2} \mathbin{.} dr \mathbin{.} dv \mathbin{.}
      (  Q^{(0)}
       + Q^{(1)} \mathbin{.} r
       + Q^{(2)} \mathbin{.} r^2
       + \ldots ),
   \eqno {\rm (B'''')}$$
where $Q^{(0)} = \Delta^{(\mu)} \mathbin{.} U^{(0)}$, and the
other coefficients $Q^{(1)}, Q^{(2)},\ldots$ are other functions
of the polar angle ($v$), which may be determined by the
formul{\ae} (X${}'''$), (Z${}'''$), (A${}''''$).  Confining
ourselves to the first term of this development, and dividing by
$r \mathbin{.} dr \mathbin{.} dv$, that is by the area of the
little polar rectangle upon the plane of aberration, we find the
following approximate expression for the density of the reflected
light at the point $(r, v)$ upon this latter plane,
$$\Delta^{(\alpha)}
   = {Q^{(\alpha)} \over r \mathbin{.} dr \mathbin{.} dv}
   = Q^{(0)} \mathbin{.} r^{- {1 \over 2}};
   \eqno {\rm (C'''')}$$
which nearly agrees with the formula (R${}'''$) of the preceding
paragraph, because, as we have seen,
$$Q^{(0)} = \Delta^{(\mu)} \mathbin{.} U^{(0)}
   = \Delta^{(\mu)} \mathbin{.}
      \left(
         w \mathbin{.} {du \over dv}
       - 2 \mathbin{.} u \mathbin{.} {dw \over dv}
      \right)
   = {\Delta^{(\mu)} \mathbin{.} \rho_1 \mathbin{.} \rho_2
      \over i \mathbin{.} \surd
         ( {1 \over 2} C \mathbin{.} \sin v)},$$
and therefore
$$Q^{(0)} \mathbin{.} r^{- {1 \over 2}}
   = {\Delta^{(\mu)} \mathbin{.} \rho_1 \mathbin{.} \rho_2
      \over i \mathbin{.} \sqrt{{1 \over 2} C \mathbin{.} y'}}.$$

More accurately, the density $\Delta^{(\mu)}$ being equal to the
sum of the two quotients obtained by dividing the quantity of
light corresponding to each of the little parallelograms
(W${}'''$), by the space over which that quantity is
perpendicularly diffused at the point $(r, v)$, has for
expression
$$\eqalignno{
\Delta^{(\alpha)}
   &=    {\gamma_1 \mathbin{.} \Delta_1 \mathbin{.} \Sigma_1^{(\mu)}
            \over \gamma_1 \mathbin{.} r \mathbin{.} dr \mathbin{.} dv}
       + {\gamma_2 \mathbin{.} \Delta_2 \mathbin{.} \Sigma_2^{(\mu)}
            \over \gamma_2 \mathbin{.} r \mathbin{.} dr \mathbin{.} dv}
    = {  \Delta_1 \mathbin{.} \Sigma_1^{(\mu)}
       + \Delta_2 \mathbin{.} \Sigma_2^{(\mu)}
            \over r \mathbin{.} dr \mathbin{.} dv} \cr
   &= r^{-{1 \over 2}} \mathbin{.}
         ( U^{(0)} + U^{(1)} \mathbin{.} r + \ldots )
         (  \Delta^{(\mu)}
          + \Delta_1^{(\mu)}
               \mathbin{.} r + \ldots) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + r^{1 \over 2} \mathbin{.}
         ( U_\prime^{(0)} + U_\prime^{(1)} \mathbin{.} r + \ldots )
         (  \Delta_{\left( {1 \over 2} \right)}^{(\mu)}
          + \Delta_{\left( {3 \over 2} \right)}^{(\mu)}
               \mathbin{.} r + \ldots).
   &{\rm (D'''')}\cr}$$
The first term of this development being the same as the
approximate expression (C${}''''$), and therefore agreeing
nearly with the formula (R${}'''$) of [67.], we see, by this
method, as well as by the less accurate one of the 67th
paragraph, that the density upon the plane of aberration varies
nearly inversely as the square root of the perpendicular distance
from the caustic surface: a conclusion which might also be
deduced from the general theorem [43.], that along a given ray the
density varies inversely as the product of the distances from its
two foci.  But the present method has the advantage of enabling
us to take into account as many of the remaining terms of the
density as may be necessary, by means of the formula (D${}''''$);
it gives also, by integration of the formul{\ae} (B${}''''$) and
(X${}'''$), the whole number of the near reflected rays which
pass within any small assigned space
$\displaystyle \int \!\!\! \int r \mathbin{.} dr \mathbin{.} dv$,
upon the plane of aberration, and the whole corresponding space
on the perpendicular plane at the mirror; since this latter space
is expressed by the sum of the following integrals:
$$S^{(\mu)}
  = \int \!\!\! \int (\Sigma_1^{(\mu)} + \Sigma_2^{(\mu)})
  = \int \!\!\! \int U^{(0)} \mathbin{.} r^{1 \over 2}
            \mathbin{.} dr \mathbin{.} dv
    + \int \!\!\! \int U^{(1)} \mathbin{.} r^{3 \over 2}
            \mathbin{.} dr \mathbin{.} dv
    + \hbox{\&c.},
   \eqno {\rm (E'''')}$$
and the corresponding quantity of light is expressed by this
other sum,
$$Q^{(s)}
  = \int \!\!\! \int Q^{(\alpha)}
  = \int \!\!\! \int Q^{(0)} \mathbin{.} r^{1 \over 2}
            \mathbin{.} dr \mathbin{.} dv
    + \int \!\!\! \int Q^{(1)} \mathbin{.} r^{3 \over 2}
            \mathbin{.} dr \mathbin{.} dv
    + \hbox{\&c.},
   \eqno {\rm (F'''')}$$
the integrals in these developments being taken within the same
limits as the given integral
$\displaystyle \int \!\!\! \int r \mathbin{.} dr \mathbin{.} dv$,
which represents the assigned space upon the plane of aberration,
and the extreme values of ($v$) being supposed such as not to
render the series (T${}'''$) illusory.  These series (T${}'''$)
serve also to correct the approximate expression of the preceding
paragraph, for the first term of ($a$); which first term was
there taken as
$$a = - {\rho_1 \mathbin{.} x' \over i},
   \eqno {\rm (Q''') \quad [67.]}$$
whereas by employing the remaining terms in the development of
$x'$ and $y'$, we have now found it to be
$$a = u \mathbin{.} r
   = \left(  {B \rho_2 \mathbin{.} \sin v
            - C \rho_1 \mathbin{.} \cos v
            \over C \mathbin{.} i}
            \right) r
   = {B \rho_2 \mathbin{.} y' - C \rho_1 \mathbin{.} x'
            \over C \mathbin{.} i},$$
a value which differs from the preceding, by the addition of
$\displaystyle \left( {B \mathbin{.} \rho_2 \mathbin{} y'
            \over C \mathbin{.} i} \right)$.
And if, by means of this corrected value, and by using as many of
the remaining terms of (T${}'''$), as the question may render
necessary, we eliminate ($r$) and ($v$) from the polar equation
of any given curve upon the plane of aberration; for example,
from that of the boundary of the space
$\displaystyle \int \!\!\! \int r \mathbin{.} dr \mathbin{.} dv$,
for which we have already determined the corresponding quantity
of light, and the area over which that quantity is diffused on
the perpendicular plane at the mirror; we shall find the
approximate equation of the boundary of this latter area, and
thus resolve a new and extensive class of questions respecting
thin pencils, for which the formulae of Section IX.\ and those of
the 60th paragraph would be either inadequate or inconvenient.

As an example of application of the reasonings of the present
paragraph, let us conceive a small circular sector, upon the
plane of aberration, having its centre at the focus of the given
ray, and having its radius ($r$) so small, that we may confine
ourselves, in each development, to the lowest powers of that
radius.  Let ($\psi$) denote the semiangle of this sector, and
let ($v''$) be the polar angle which the bisecting radius makes
with the axis of ($x'$); then $v'' - \psi$, $v'' + \psi$, will be
the extreme values of the polar coordinate ($v$), while the
corresponding limits of the radius vector will be ($0$) and
($r$).  Denoting by ($S^{(c)}$) the whole space occupied on the
perpendicular plane at the mirror, by the rays which pass within
the given little circular sector, and by ($Q^{(c)}$) the number
of these near reflected rays; the formul{\ae} (E${}''''$)
(F${}''''$) give, for these quantities
$$\eqalign{
S^{(c)}
  &= \int \!\!\! \int U^{(0)} \mathbin{.} r^{1 \over 2}
            \mathbin{.} dr \mathbin{.} dv
   = {\textstyle {2 \over 3}} \mathbin{.} r^{3 \over 2}
      \mathbin{.} \int \!\!\! \int U^{(0)} \mathbin{.} dv,\cr
Q^{(c)}
  &= \int \!\!\! \int Q^{(0)} \mathbin{.} r^{1 \over 2}
            \mathbin{.} dr \mathbin{.} dv
   = {\textstyle {2 \over 3}} \mathbin{.} r^{3 \over 2}
      \mathbin{.} \int \!\!\! \int Q^{(0)} \mathbin{.} dv,\cr}$$
or, substituting for $U^{(0)}$, $Q^{(0)}$ their values,

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

$$\left. \eqalign{
S^{(c)}
   & = {2 \mathbin{.} \rho_1 \mathbin{.} \rho_2 \mathbin{.} r^{3 \over 2}
      \over 3 \mathbin{.} i \mathbin{.} \sqrt{ {1 \over 2} C } }
      \mathbin{.} \int {dv \over \sqrt{ \sin v }},\cr
Q^{(c)}
   & = {2 \mathbin{.} \rho_1 \mathbin{.} \rho_2
         \mathbin{.} \Delta^{(\mu)} \mathbin{.} r^{3 \over 2}
      \over 3 \mathbin{.} i \mathbin{.} \sqrt{ {1 \over 2} C } }
      \mathbin{.} \int {dv \over \sqrt{ \sin v }},\cr}
   \right\}
   \eqno {\rm (G'''')}$$
the integral in each expression being taken from
$v = v'' - \psi$, to $v = v'' + \psi$; so that we have the
relation
$$Q^{(c)} = \Delta^{(\mu)} \mathbin{.} S^{(c)}.
   \eqno {\rm (H'''')}$$

If the semiangle of the sector be so small that we may neglect
its cube and higher powers, the definite integral
$\displaystyle U^{(0)} \mathbin{.} dv$,
being the difference of the developments
$$\eqalign{
U^{(0)} \mathbin{.} \psi
      + {d U^{(0)} \over dv''}
         \mathbin{.} {\psi^2 \over 2}
      + {d^2 U^{(0)} \over dv''^2}
         \mathbin{.} {\psi^3 \over 2 \mathbin{.} 3}
      + \ldots,\cr
- U^{(0)} \mathbin{.} \psi
      + {d U^{(0)} \over dv''}
         \mathbin{.} {\psi^2 \over 2}
      - {d^2 U^{(0)} \over dv''^2}
         \mathbin{.} {\psi^3 \over 2 \mathbin{.} 3}
      + \ldots,\cr}$$
is nearly equal to $2 \mathbin{.} U^{(0)} \mathbin{.} \psi$; and
the quantities $S^{(c)}$, $Q^{(c)}$, may be thus expressed,
$$\left. \eqalign{
S^{(c)}
   &= {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2
         \over 3 \mathbin{.} i \mathbin{.} \sqrt{ {1 \over 2} C }}
      \mathbin{.}
      {r^{3 \over 2} \psi \over \sqrt{ \sin v'' }}
    = {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2 \mathbin{.} s''
         \over 3 \mathbin{.} i \mathbin{.}
            \sqrt{ {1 \over 2} C \mathbin{.} y''}},\cr
Q^{(c)}
   &= {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2 \mathbin{.} \Delta^{(\mu)}
         \over 3 \mathbin{.} i \mathbin{.} \sqrt{ {1 \over 2} C }}
      \mathbin{.}
      {r^{3 \over 2} \psi \over \sqrt{ \sin v'' }}
    = {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2
         \mathbin{.} \Delta^{(\mu)} \mathbin{.} s''
         \over 3 \mathbin{.} i \mathbin{.}
            \sqrt{ {1 \over 2} C \mathbin{.} y''}},\cr}
   \right\}
   \eqno {\rm (I'''')}$$
$s''$ being the area of the little circular sector, and $y''$
being the projection of its bisecting radius upon the normal to
the caustic surface; so that if the sector were to receive a
rotation in its own plane round its own centre, that is, in the
plane of aberration round the focus of the given ray, the area at
the mirror ($S^{(c)}$) and the quantity of light ($Q^{(c)}$)
would vary nearly inversely as the square root of the cosine of
the angle, which the bisecting radius of the sector made with the
normal to the caustic surface.  If, on the contrary, without
supposing the angles ($v''$) or ($\psi$)  to vary, we alter the
length of the radius, or transport the centre of the sector to
any other point on either of the two caustic surfaces, so as to
produce another sector, similar and similarly situated; it follows
from (G${}''''$) that the quantities $S^{(c)}$ and $Q^{(c)}$ will
vary as the following expressions,
$$\rho_1 \mathbin{.} \rho_2 \mathbin{.} r^{3 \over 2}
         \mathbin{.} i^{-1} \mathbin{.} C^{-{1 \over 2}},\quad
  \rho_1 \mathbin{.} \rho_2 \mathbin{.} \Delta^{(\mu)}
         \mathbin{.} r^{3 \over 2}
         \mathbin{.} i^{-1} \mathbin{.} C^{-{1 \over 2}};$$
so that if the centre of the sector be fixed, they vary as the
sesquiplicate power of its radius; and if the radius be given,
but not the centre, then they vary, the one as the product of the
two focal lengths of the mirror, divided by the difference of
those two focal lengths and by the square root of the radius of
curvature of the caustic curve; and the other, as this latter
quotient, multiplied by the density of the reflected light at the
corresponding point of the mirror.  These latter theorems, being
founded on the formul{\ae} (G${}''''$), do not require that we
should neglect any of the powers of $\psi$, that is of the
semiangle of the circular sector; they may even be extended, by
means of the equations (E${}''''$) (F${}''''$), to the case of
similar and similarly situated sectors, bounded by lines of any
other form.  If, for instance, we suppose any small isosceles
triangle, having its height $= h$, and its base
$= 2 \mathbin{.} h \mathbin{.} \tan \theta$,
to move in such a manner that its summit is always situated on
one of the two caustic surfaces, while the ray passing through
that point is perpendicular to its plane, and the bisector of its
vertical angle is perpendicular to the caustic surface; and if we
put $Q^{(i)}$, $S^{(i)}$, to denote, respectively, the number of
the near reflected rays that pass inside this little triangle,
and the space over which those rays are diffused, on the
perpendicular plane at the mirror; we shall have the approximate
equations,
$$\eqalign{
Q^{(i)}
   &= \int \!\!\! \int Q^{(0)} \mathbin{.} r^{1 \over 2}
            \mathbin{.} dr \mathbin{.} dv
    = {\rho_1 \mathbin{.} \rho_2 \mathbin{.} \Delta^{(\mu)}
         \over i \mathbin{.} \sqrt{ {1 \over 2} C }}
      \mathbin{.} \int \!\!\! \int
      {r^{1 \over 2} \mathbin{.} dr \mathbin{.} dv
         \over \sqrt{ \sin v }},\cr
S^{(i)}
   &= \int \!\!\! \int U^{(0)} \mathbin{.} r^{1 \over 2}
            \mathbin{.} dr \mathbin{.} dv
    = {\rho_1 \mathbin{.} \rho_2
         \over i \mathbin{.} \sqrt{ {1 \over 2} C }}
      \mathbin{.} \int \!\!\! \int
      {r^{1 \over 2} \mathbin{.} dr \mathbin{.} dv
         \over \sqrt{ \sin v }},\cr}$$
in which the integrals are to be taken from $r = 0$ to
$\displaystyle r = {h \over \sin v}$,
and from
$v = {1 \over 2} \mathbin{.} \pi - \theta$, to
$v = {1 \over 2} \mathbin{.} \pi + \theta$.
Performing these integrations, we find
$$\left. \eqalign{
Q^{(i)}
   &= {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2 \mathbin{.} \Delta^{(\mu)}
            \mathbin{.} h^{3 \over 2}
         \over 3 \mathbin{.} i \mathbin{.} \sqrt{ {1 \over 2} C }}
      \mathbin{.} \tan \theta
    = {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2 \mathbin{.} \Delta^{(\mu)}
            \mathbin{.} s^{(i)}
         \over 3 \mathbin{.} i
         \mathbin{.} \sqrt{ {1 \over 2} C \mathbin{.} h }},\cr
S^{(i)}
   &= {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2
            \mathbin{.} h^{3 \over 2}
         \over 3 \mathbin{.} i \mathbin{.} \sqrt{ {1 \over 2} C }}
      \mathbin{.} \tan \theta
    = {4 \mathbin{.} \rho_1 \mathbin{.} \rho_2
            \mathbin{.} s^{(i)}
         \over 3 \mathbin{.} i
         \mathbin{.} \sqrt{ {1 \over 2} C \mathbin{.} h }},\cr}
   \right\}
   \eqno {\rm (K'''')}$$
$s^{(i)}$ being the area of the little isosceles triangle;
expressions analogous to those which we found before, for the
case of the circular sector, and leading to similar results.

Returning to the case of the sector, we have yet to determine the
boundary of the space ($S^{(c)}$) on the perpendicular plane at
the mirror.  For this purpose, we are to eliminate ($r$) and
($v$) by means of the following expressions, (T${}'''$),
$$\eqalign{
a  &= u \mathbin{.} r
    = (B \rho_2 \mathbin{.} \sin v
      - C \rho_1 \mathbin{.} \cos v)
      \mathbin{.} i^{-1} \mathbin{.} C^{-1} \mathbin{.} r,\cr
b  &= \pm w \mathbin{.} r^{1 \over 2}
    = \pm \rho_2 \mathbin{.} \sqrt{2} \mathbin{.} C^{-{1 \over 2}}
      \mathbin{.} \sqrt{r \mathbin{.} \sin v},\cr}$$
from the polar equations of the boundaries of the sector, namely
$$\vcenter{\halign{#\hfil&\quad $\displaystyle #$\hfil \cr
Ist.
   &v = v'' - \psi = v_1,\cr
   \noalign{\vskip 3pt}
IId.
   &v = v'' + \psi = v_2,\cr
   \noalign{\vskip 3pt}
IIId.
   &r = r,\cr}}$$
of which the two first represent the bounding radii, and the
third the circular arc.  Putting for abridgment,
$$\rho_2 \mathbin{.} \surd 2  \mathbin{.} C^{-{1 \over 2}}
   = \surd \epsilon,\quad
  B \rho_2 \mathbin{.} i^{-1} \mathbin{.} C^{-1}
   = \epsilon \mathbin{.} P^{-1},\quad
  C \rho_1
   = B \rho_2 \mathbin{.} \tan v',$$
conditions which give
$\epsilon = 2 \rho_2^2 \mathbin{.} C^{-1}$,
$P = 2 i \mathbin{.} \rho_2 \mathbin{.} B^{-1}$;
and supposing, for simplicity, that $\surd \epsilon$ is real,
and that
$\displaystyle v' < {\pi \over 2}$,
that is, supposing $C$ and $\tan v'$ positive, a condition
which may always be satisfied by a proper direction of the
positive portions of the axes of $y'$ and $x'$; our expressions
for $a$, $b$, become
$$a = \epsilon \mathbin{.} P^{-1} \mathbin{.} r
      \mathbin{.} \mathop{\rm sec.} v'
      \mathbin{.} \sin (v - v'),\quad
  b = \pm \surd ( \epsilon \mathbin{.} r \mathbin{.} \sin v ),
   \eqno {\rm (L'''')}$$
and we find the following equations for the boundary of the space
$S^{(c)}$,
$$\left. \vcenter{\halign{#\hfil&\quad $\displaystyle #$\hfil \cr
1st.
   &P \mathbin{.} a
      = b^2 \mathbin{.} \mathop{\rm sec.} v' \mathbin{.} \cosec v_1
         \mathbin{.} \sin (v_1 - v'),\cr
   \noalign{\vskip 3pt}
2d.
   &P \mathbin{.} a
      = b^2 \mathbin{.} \mathop{\rm sec.} v' \mathbin{.} \cosec v_2
         \mathbin{.} \sin (v_2 - v'),\cr
   \noalign{\vskip 3pt}
3d.
   &P \mathbin{.} a
      = b^2 \mp \tan v'
         \mathbin{.} \surd (\epsilon^2 \mathbin{.} r^2 - b^4) .\cr}}
   \right\}
   \eqno {\rm (M'''')}$$

The two first of these equations represent parabolic arcs, having
their common vertex at the origin of ($a$) and ($b$), that is at
the point where the given ray meets the mirror, and having their
common axis coincident with the axis of ($x$) or of ($a$), and
therefore parallel to the tangent of the curve in which the
caustic surface is cut by the plane of aberration.  It is, then,
in the points of these little parabolic arcs, that the rays which
pass through the bounding radii of the little circular sector,
are intersected by the perpendicular plane at the mirror; and
from the manner in which their parameters depend on the
inclination of those bounding radii to the tangent of the caustic
surface, it is evident that any intermediate radius of the sector
has an intermediate parabola corresponding.  The ends of these
little parabolic arcs are contained on two equal and opposite
portions of a curve of the fourth degree represented by the third
of the three equations (M${}''''$), it is then in these two
opposite portions of this third curve, that the rays which pass
through the bounding arc of the sector are crossed by the
perpendicular plane at the mirror.  With respect to the form of
this third curve, considered in its whole extent, it is easy to
see that it is in general shaped like a {\it heart}, being
bisected, first by the axis of ($a$), which we may call the
{\it diameter\/} of the curve, and secondly by a parabola
$$b^2 = P \mathbin{.} a,
   \eqno {\rm (N'''')}$$
which we may call its {\it diametral parabola}, and bounded by
the four following tangents,
$$\hbox{1st.}\enspace
  b = + \sqrt{ \epsilon \mathbin{.} r };\quad
  \hbox{2d.} \enspace
  b = - \sqrt{ \epsilon \mathbin{.} r };$$
$$\hbox{3d.} \enspace
  a = - \epsilon \mathbin{.} r \mathbin{.} P^{-1}
         \mathbin{.} \tan v';\quad
  \hbox{4th.}\enspace
  a = \epsilon \mathbin{.} r \mathbin{.} P^{-1}
         \mathbin{.} \mathop{\rm sec.} v';
   \eqno {\rm (O'''')}$$
of which the two first are parallel to the diameter, and the two
last perpendicular thereto.  We may remark that the diametral
parabola, (N${}''''$), corresponds to the rays that pass through
the axis of $y'$, that is, through the normal to the caustic
surface; and that the two points where it meets the curve, are
the points of contact corresponding to the two first of the four
tangents (O${}''''$).  The point of contact corresponding to the
third of these tangents, is situated at what may be called the
negative end of the diameter; and the fourth touches the curve in
two distinct points, whose common distance from the diameter is
$b = \pm \surd (\epsilon \mathbin{.} r \mathbin{.} \cos v')$,
and which may be called the two {\it summits\/} of the heart.
The curve has also another tangent parallel to the axis of (b),
which touches it at the point
$$a = + \epsilon \mathbin{.} r \mathbin{.} P^{-1}
      \mathbin{.} \tan v',\quad
  b = 0,
   \eqno {\rm (P'''')}$$
that is, at the positive end of the diameter; and which crosses
the curve in two other points, equally distant from the diameter,
and having for coordinates,
$$a = \epsilon \mathbin{.} r \mathbin{.} P^{-1}
      \mathbin{.} \tan v',\quad
  b = \pm \surd (\epsilon \mathbin{.} r \mathbin{.} \sin 2 v'),$$
And the whole area of this heartlike curve is equal to the
following definite integral,
$$\Pi = 4 P^{-1} \mathbin{.} \tan v'
      \mathbin{.} \int \surd (\epsilon^2 \mathbin{.} r^2 - b^4)
      \mathbin{.} db,
   \eqno {\rm (Q'''')}$$
the integral being taken from $b = 0$, to
$b = \sqrt{\epsilon \mathbin{.} r}$.
In the next paragraph we shall return to this definite integral,
and shew its optical value.

\bigbreak

[69.]
But the preceding calculations only shew how the density varies
{\it near\/} the caustic surface; to find the law of the
variation {\it at\/} that surface, we must reason in a different
manner.  For if the infinitely small rectangle on the plane of
aberration, which we have considered in the preceding paragraph,
have one of its corners on the caustic surface, we can no longer
consider the density as uniform, even in the infinitely small
extent of that rectangle.  But if we consider the rays that pass
within a given infinitely small distance ($dr$) from a given
point upon the caustic surface, for example, from the focus of
the given ray, we can find the space over which these rays are
diffused upon the perpendicular plane at the mirror; and this
space, multiplied by the density at the mirror, may be taken for
the measure of the density at the given focus, not as compared
with the density at the mirror, but with the density at other
points upon the caustic surface.

To calculate this measure, let us consider the following more
general question, to find the whole number ($Q^{(r)}$) of the near
reflected rays which pass within any small but finite distance
($r$) from the focus of the given ray, and the space ($S^{(r)}$)
over which these rays are diffused, on the perpendicular plane at
the mirror.  This question evidently comes to supposing the
little circular sector ($r^2 \mathbin{.} \psi$) of the preceding
paragraph completed into an entire circle, and consequently may
be solved by integrating the formul{\ae} (E${}''''$) (F${}''''$)
of that paragraph, within the double limits afforded by the
equation of the circle on the one hand, and by that of the
section of the caustic surface on the other; since it is easy to
see that only a part of the little circular area
($\pi \mathbin{.} r^2$) is illumined, namely, the part which lies
at that side of the caustic surface, towards which is turned the
convexity of the caustic curve.

But as the formul{\ae} of the preceding paragraph were founded on
the developments (T${}'''$) which, as we have before remarked,
become illusory when the polar radius ($r$) approaches to become
a tangent to the caustic surface, (a position of that radius which
we are not now at liberty to neglect,) it becomes necessary to
investigate other developments, and to transform the double
integrals (E${}''''$) (F${}''''$) of [68.]\ into others better
suited for the question that we are now upon.  And to effect this
the more clearly, it seems convenient to consider separately the
four following problems: 1st, to find general expressions for the
coefficients $u^{(t)}$, $w^{(t)}$, which enter into the
developments (T${}'''$), and to examine what negative powers they
contain of the sine of the polar angle ($v$); 2d., to eliminate
these negative powers, and so transform the two series (T${}'''$)
into others which shall contain none but ascending powers of any
variable quantity; 3d., to effect corresponding transformations
on the integral formul{\ae} (E${}''''$) (F${}''''$) of the
preceding paragraph; and 4th, to perform the double integrations
within the limits of the question.

In this manner we shall obtain developments proceeding according
to the ascending powers of the little circular radius ($r$), to
represent the optical quantities which we have denoted by
$Q^{(r)}$, $S^{(r)}$; it will then remain to suppose ($r$)
infinitely small, and the resulting expressions $Q^{(dr)}$,
$S^{(dr)}$, which must evidently satisfy the relation
$$Q^{(dr)} = \Delta^{(\mu)} \mathbin{.} S^{(dr)},
   \eqno {\rm (R'''')}$$
$\Delta^{(\mu)}$ being the density at the mirror, will each serve
to measure the density at the caustic surface in the sense that
we have already explained.

\bigbreak

(I.)
First then, with respect to the coefficients $u^{(t)}$,
$w^{(t)}$, of the series (T${}'''$),
$$\eqalign{
a  &= u r + u' r^{3 \over 2} + \ldots \,
         u^{(t)} \mathbin{.} r^{t + 2 \over 2} + \ldots,\cr
b  &= w r^{1 \over 2} + w' r + \ldots \,
         w^{(t)} \mathbin{.} r^{t + 1 \over 2} + \ldots;\cr}$$
it is evident that if we differentiate these series with respect
to $\surd r $, we shall have, supposing $\surd r$ to vanish after
the differentiations,
$${da \over d \surd r} = 0,\quad
  {d^2 a \over d \surd r^2} = 2 \mathbin{.} u,\quad
  {d^3 a \over d \surd r^3} = 2 \mathbin{.} 3 \mathbin{.} u',\ldots,$$
$${db \over d \surd r} = w,\quad
  {d^2 b \over d \surd r^2} = 2 \mathbin{.} w',\quad
  {d^3 b \over d \surd r^3} = 2 \mathbin{.} 3 \mathbin{.} w'',\ldots,$$
and in general
$${d^s a \over d \surd r^s} = [s]^s \mathbin{.} u^{(s-2)},\quad
  {d^s b \over d \surd r^s} = [s]^s \mathbin{.} w^{(s-1)}:
   \eqno {\rm (S'''')}$$
$[s]^s$ expressing, according to the notation of Vandermonde, the
factorial quantity
$1 \mathbin{.} 2 \mathbin{.} 3 \mathbin{.} \ldots
      \mathbin{.} (s - 1) \mathbin{.} s$.
If then we differentiate, with respect to $\surd r$, the
equations
$$r \mathbin{.} \cos v = x',\quad
  r \mathbin{.} \sin v = y',$$
considering $x'$, $y'$, as functions of $a$ and $b$, and these as
functions of $\surd r$; the resulting equations,
$${d^s \mathbin{.} r \mathbin{.} \cos v \over d \surd r^s}
   = {d^s \mathbin{.} x' \over d \surd r^s},\quad
  {d^s \mathbin{.} r \mathbin{.} \sin v \over d \surd r^s}
   = {d^s \mathbin{.} y' \over d \surd r^s},
   \eqno {\rm (T'''')}$$
wll serve, with the help of the formul{\ae} (S${}''''$), to
determine successively the coefficients $u'$,~$w'$,
$u''$,~$w'',\ldots$ $u^{(t)}$,~$w^{(t)}$, as functions of those
which precede them; observing that the partial differentials
$\displaystyle {dx' \over da}, {dx' \over db}, {d^2 x' \over da^2},\ldots$
$\displaystyle {dy' \over da}, {dy' \over db}, {d^2 y' \over da^2},\ldots$
are the coefficients of the series (S${}'''$) of [68.], and are to
be deduced from the characteristic function of the system in the
manner there described.  To develope these equations (T${}''''$),
we have, for the first members,
$$\multieqalign{
{d \mathbin{.} r \mathbin{.} \cos v \over d \surd r}
   &= 2 \surd r \mathbin{.} \cos v, &
{d \mathbin{.} r \mathbin{.} \sin v \over d \surd r}
   &= 2 \surd r \mathbin{.} \sin v; \cr
{d^2 \mathbin{.} r \mathbin{.} \cos v \over d \surd r^2}
   &= 2 \mathbin{.} \cos v, &
{d^2 \mathbin{.} r \mathbin{.} \sin v \over d \surd r^2}
   &= 2 \mathbin{.} \sin v; \cr
{d^3 \mathbin{.} r \mathbin{.} \cos v \over d \surd r^3}
   &= 0, &
{d^3 \mathbin{.} r \mathbin{.} \sin v \over d \surd r^3}
   &= 0: \cr}$$
and in general, when $s > 2$,
$${d^s \mathbin{.} r \mathbin{.} \cos v \over d \surd r^s}
   = 0,\quad
  {d^s \mathbin{.} r \mathbin{.} \sin v \over d \surd r^s}
   = 0:$$
and for the second members,
$$\left. \eqalign{
{d^s \mathbin{.} x' \over d \surd r^s}
   &= [s]^s \mathbin{.} \sum \mathbin{.}
         {d^{\Sigma \alpha + \Sigma \beta} x'
            \over da^{\Sigma \alpha} \mathbin{} db^{\Sigma \beta}}
            \mathbin{.} \lambda^{(s)},\cr
{d^s \mathbin{.} y' \over d \surd r^s}
   &= [s]^s \mathbin{.} \sum \mathbin{.}
         {d^{\Sigma \alpha + \Sigma \beta} y'
            \over da^{\Sigma \alpha} \mathbin{} db^{\Sigma \beta}}
            \mathbin{.} \lambda^{(s)},\cr}
   \right\}
   \eqno {\rm (U'''')}$$
if we put for abridgment,
$$\eqalignno{
\lambda^{(s)}
   &= \left( {da    \over d \surd r}   \right)^{\alpha_1}
         \mathbin{.}
      \left( {d^2 a \over d \surd r^2} \right)^{\alpha_2}
         \ldots
      \left( {d^s a \over d \surd r^s} \right)^{\alpha_s}
         \times
      \left( {db    \over d \surd r}   \right)^{\beta_1}
         \mathbin{.}
      \left( {d^2 b \over d \surd r^2} \right)^{\beta_2}
         \ldots
      \left( {d^s b \over d \surd r^s} \right)^{\beta_s} \cr
   &\mathrel{\phantom{=}} \mathord{}
      \times ([0]^{-1})^{\alpha_1}
         \mathbin{.} ([0]^{-2})^{\alpha_2}
         \ldots ([0]^{-s})^{\alpha_s}
      \times ([0]^{-1})^{\beta_1}
         \mathbin{.} ([0]^{-2})^{\beta_2}
         \ldots ([0]^{-s})^{\beta_s} \cr
   &\mathrel{\phantom{=}} \mathord{}
      \times [0]^{-\alpha_1}
         \mathbin{.} [0]^{-\alpha_2}
         \ldots [0]^{-\alpha_s}
      \times [0]^{-\beta_1}
         \mathbin{.} [0]^{-\beta_2}
         \ldots [0]^{-\beta_s};
   &{\rm (V'''')}\cr}$$
$\alpha_1, \alpha_2,\ldots \, \alpha_s$,
$\beta_1, \beta_2,\ldots \, \beta_s$,
being any positive integers which satisfy the following relation,
$$\eqalignno{s
   &= \alpha_1 + 2 \alpha_2 + 3 \alpha_3 + \ldots \, s \mathbin{.} \alpha_s \cr
   &+ \beta_1 + 2 \beta_2 + 3 \beta_3 + \ldots \, s \mathbin{.} \beta_s;
   &{\rm (W'''')}\cr}$$
and $\sum$ being the symbol of a sum, so that
$$\sum \alpha = \alpha_1 + \alpha_2 + \ldots \, \alpha_s,\quad
  \sum \beta  = \beta_1 + \beta_2 + \ldots \, \beta_s.$$

Developing in this manner the equations (T${}''''$), and
observing by our present choice of the coordinate planes, we
have, [68.],
$${dx' \over db} = 0;\quad
  {dy' \over da} = 0;\quad
  {dy' \over db} = 0;\quad
  {da \over d \surd r} = 0:$$
we arrive again at the same equations which in the preceding
paragraph we deduced by substituting in (S${}'''$), for the
components $a$, $b$, of aberration at the mirror, their assumed
developments (T${}'''$), and by then comparing the corresponding
powers of $r$, that is, of the aberration at the caustic surface.
Thus, if we make $s = 1$, the equations (T${}''''$) become
identical; if $s = 2$, they become
$$\eqalign{
2 \cos v
   &=    {dx' \over da} \mathbin{.} {d^2 a \over d \surd r^2}
       + {dx' \over db} \mathbin{.} {d^2 b \over d \surd r^2} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {d^2 x' \over da^2}
            \mathbin{.} \left( {da \over d \surd r} \right)^2
       + 2 \mathbin{.} {d^2 x' \over da \mathbin{.} db}
            \mathbin{.} {da \over d \surd r}
            \mathbin{.} {db \over d \surd r}
       + {d^2 x' \over db^2}
            \mathbin{.} \left( {db \over d \surd r} \right)^2,\cr
2 \sin v
   &=    {dy' \over da} \mathbin{.} {d^2 a \over d \surd r^2}
       + {dy' \over db} \mathbin{.} {d^2 b \over d \surd r^2} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {d^2 y' \over da^2}
            \mathbin{.} \left( {da \over d \surd r} \right)^2
       + 2 \mathbin{.} {d^2 y' \over da \mathbin{.} db}
            \mathbin{.} {da \over d \surd r}
            \mathbin{.} {db \over d \surd r}
       + {d^2 y' \over db^2}
            \mathbin{.} \left( {db \over d \surd r} \right)^2,\cr}$$
that is,
$$\eqalign{
2 \cos v
   &= {dx' \over da} \mathbin{.} 2u + {d^2 x' \over db^2} w^2,\cr
2 \sin v
   &= \phantom{{dy' \over da} \mathbin{.} 2u +} \, \,
      {d^2 y' \over db^2} w^2,\cr}$$
as in the formul{\ae} (1), (2), or (U${}'''$) of [68.]; if $s = 3$,
they become after reductions,
$$\eqalign{
0  &= {dx' \over da} \mathbin{.} {d^3 a \over d \surd r^3}
      + 3 \mathbin{.}
         \left(
            {d^2 x' \over da \mathbin{.} db}
               \mathbin{.} {d^2 a \over d \surd r^2}
          + {d^2 x' \over db^2}
               \mathbin{.} {d^2 b \over d \surd r^2}
         \right)
         \mathbin{.} {db \over d \surd r}
      + {d^3 x' \over db^3}
         \mathbin{.} \left( {db \over d \surd r} \right)^3,\cr
0  &= \phantom{{dy' \over da} \mathbin{.} {d^3 a \over d \surd r^3} +} \,\,
        3 \mathbin{.}
         \left(
            {d^2 y' \over da \mathbin{.} db}
               \mathbin{.} {d^2 a \over d \surd r^2}
          + {d^2 y' \over db^2}
               \mathbin{.} {d^2 b \over d \surd r^2}
         \right)
         \mathbin{.} {db \over d \surd r}
      + {d^3 y' \over db^3}
         \mathbin{.} \left( {db \over d \surd r} \right)^3,\cr}$$
that is,
$$\eqalign{
0  &= 6 \mathbin{.} {dx' \over da} \mathbin{.} u'
      + 3 \mathbin{.}
         \left(
            {d^2 x' \over da \mathbin{.} db} \mathbin{.} 2u
          + {d^2 x' \over db^2} \mathbin{.} 2w'
         \right)
         \mathbin{.} w
      + {d^3 x' \over db^3} \mathbin{.} w^3,\cr
0  &= \phantom{6 \mathbin{.} {dy' \over da} \mathbin{.} u' +} \,\,
        3 \mathbin{.}
         \left(
            {d^2 y' \over da \mathbin{.} db} \mathbin{.} 2u
          + {d^2 y' \over db^2} \mathbin{.} 2w'
         \right)
         \mathbin{.} w
      + {d^3 y' \over db^3} \mathbin{.} w^3,\cr}$$
as in the formul{\ae} (1)${}'$, (2)${}'$ of the same paragraph;
$s = 4$ gives

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

$$\eqalign{
0  &= {dx' \over da} \mathbin{.} {d^4 a \over d \surd r^4}
      + 4 \mathbin{.}
         \left(
            {d^2 x' \over da \mathbin{.} db}
               \mathbin{.} {d^3 a \over d \surd r^3}
          + {d^2 x' \over db^2}
               \mathbin{.} {d^3 b \over d \surd r^3}
         \right)
         \mathbin{.} {db \over d \surd r} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 3 \mathbin{.}
         \left(
            {d^2 x' \over da^2}
               \mathbin{.} \left( {d^2 a \over d \surd r^2} \right)^2
          + 2 \mathbin{.} {d^2 x' \over da \mathbin{.} db}
               \mathbin{.} {d^2 a \over d \surd r^2}
               \mathbin{.} {d^2 b \over d \surd r^2}
          + {d^2 x' \over db^2}
               \mathbin{.} \left( {d^2 b \over d \surd r^2} \right)^2
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 6 \mathbin{.}
         \left(
            {d^3 x' \over da \mathbin{.} db^2}
               \mathbin{.} {d^2 a \over d \surd r^2}
          + {d^3 x' \over db^3}
               \mathbin{.} {d^2 b \over d \surd r^2}
         \right)
         \mathbin{.} \left( {db \over d \surd r} \right)^2
      + {d^4 x' \over db^4}
         \mathbin{.} \left( {db \over d \surd r} \right)^4,\cr
0  &= 4 \mathbin{.}
         \left(
            {d^2 y' \over da \mathbin{.} db}
               \mathbin{.} {d^3 a \over d \surd r^3}
          + {d^2 y' \over db^2}
               \mathbin{.} {d^3 b \over d \surd r^3}
         \right)
         \mathbin{.} {db \over d \surd r} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 3 \mathbin{.}
         \left(
            {d^2 y' \over da^2}
               \mathbin{.} \left( {d^2 a \over d \surd r^2} \right)^2
          + 2 \mathbin{.} {d^2 y' \over da \mathbin{.} db}
               \mathbin{.} {d^2 a \over d \surd r^2}
               \mathbin{.} {d^2 b \over d \surd r^2}
          + {d^2 y' \over db^2}
               \mathbin{.} \left( {d^2 b \over d \surd r^2} \right)^2
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 6 \mathbin{.}
         \left(
            {d^3 y' \over da \mathbin{.} db^2}
               \mathbin{.} {d^2 a \over d \surd r^2}
          + {d^3 y' \over db^3}
               \mathbin{.} {d^2 b \over d \surd r^2}
         \right)
         \mathbin{.} \left( {db \over d \surd r} \right)^2
      + {d^4 y' \over db^4}
         \mathbin{.} \left( {db \over d \surd r} \right)^4,\cr}$$
that is,
$$\eqalign{
(1)''\ldots \quad 0
   &= 24 \mathbin{.} {dx' \over da} \mathbin{.} u''
      + 24 \mathbin{.}
         \left(
            {d^2 x' \over da \mathbin{.} db} u'
          + {d^2 x' \over db^2} \mathbin{.} w''
         \right)
         \mathbin{.} w \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 12 \mathbin{.}
         \left(
            {d^2 x' \over da^2} \mathbin{.} u^2
          + 2 \mathbin{.} {d^2 x' \over da \mathbin{.} db}
               \mathbin{.} u \mathbin{.} w'
          + {d^2 x' \over db^2} \mathbin{.} w'^2
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 12 \mathbin{.}
         \left(
            {d^3 x' \over da \mathbin{.} db^2} \mathbin{.} u
          + {d^3 x' \over db^3} \mathbin{.} w'
         \right)
         \mathbin{.} w^2
      + {d^4 x' \over db^4} \mathbin{.} w^4,\cr
(2)''\ldots \quad 0
   &= 24 \mathbin{.}
         \left(
            {d^2 y' \over da \mathbin{.} db} u'
          + {d^2 y' \over db^2} \mathbin{.} w''
         \right)
         \mathbin{.} w \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 12 \mathbin{.}
         \left(
            {d^2 y' \over da^2} \mathbin{.} u^2
          + 2 \mathbin{.} {d^2 y' \over da \mathbin{.} db}
               \mathbin{.} u \mathbin{.} w'
          + {d^2 y' \over db^2} \mathbin{.} w'^2
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 12 \mathbin{.}
         \left(
            {d^3 y' \over da \mathbin{.} db^2} \mathbin{.} u
          + {d^3 y' \over db^3} \mathbin{.} w'
         \right)
         \mathbin{.} w^2
      + {d^4 y' \over db^4} \mathbin{.} w^4;\cr}$$
and generally, if we make the differential index $s = t + 2$, and
divide by the factorial quantity $[t + 2]^{t + 2}$, we shall
reproduce the equations (V${}'''$) [68.]\ under the form
$$\eqalign{
(1)^{(t)}\ldots \quad 0
   &= {dx' \over da} \mathbin{.} u^{(t)}
      + {d^2 x' \over db^2} \mathbin{.} w \mathbin{.} w^{(t)}
      + \sum \mathbin{.} {d^{\Sigma \alpha + \Sigma \beta} x'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
         \mathbin{.} \mu^{(t)},\cr
(2)^{(t)}\ldots \quad 0
   &= \phantom{{dy' \over da} \mathbin{.} u^{(t)} +} \,\,
      {d^2 y' \over db^2} \mathbin{.} w \mathbin{.} w^{(t)}
      + \sum \mathbin{.} {d^{\Sigma \alpha + \Sigma \beta} y'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
         \mathbin{.} \mu^{(t)},\cr}$$
where
$$\mu^{(t)}
   =  {(u)^{\alpha_2} \over [\alpha_2]^{\alpha_2}}
      \mathbin{.}
      {(u')^{\alpha_3} \over [\alpha_3]^{\alpha_3}}
      \mathbin{\ldots}
      {(u^{(t-1)})^{\alpha_{t+1}} \over [\alpha_{t+1}]^{\alpha_{t+1}}}
      \times
      {(w)^{\beta_1} \over [\beta_1]^{\beta_1}}
      \mathbin{.}
      {(w')^{\beta_2} \over [\beta_2]^{\beta_2}}
      \mathbin{\ldots}
      {(w^{(t-1)})^{\beta_t} \over [\beta_t]^{\beta_t}},$$
$$\sum \alpha = \alpha_2 + \alpha_3 + \ldots \, \alpha_{t+1},\quad
  \sum \beta = \beta_1 + \beta_2 + \ldots \, \beta_t,$$
and $\alpha_2, \alpha_3 \, \ldots \, \beta_1, \beta_2 \, \ldots$
are any positive integers which satisfy the following relation:
$$t + 2 = 2 \alpha_2 + 3 \alpha_3 + \ldots \, (t + 1) \alpha_{t + 1}
      + \beta_1 + 2 \beta_2 + \ldots \, t \beta_t.
   \eqno {\rm (X'''')}$$
It is easy to see, that if by these equations we calculate
successively the coefficients $u^{(t)}$, $w^{(t)}$, as functions
of $u$ and $w$, and if we eliminate $u$ by the assumed relation
$$u= z \mathbin{.} w^2,
   \eqno {\rm (Y'''')}$$
$z$ being a new variable; the resulting expressions will be of
the form
$$\left.
\eqalign{
u' &= u_1 \mathbin{.} w^3,\cr
w' &= w_1 \mathbin{.} w^2,\cr} \quad
\eqalign{
u'' &= u_2 \mathbin{.} w^4, \, \ldots\cr
w'' &= w_2 \mathbin{.} w^3, \, \ldots\cr} \quad
\eqalign{
u^{(t)} &= u_t \mathbin{.} w^{t+2},\cr
w^{(t)} &= w_t \mathbin{.} w^{t+1},\cr}
   \right\}
   \eqno {\rm (Z'''')}$$
$u_t$, $w_t$, being rational and integer functions of $z$, not
exceeding the $t^{\rm th}$~dimension; so that we may put
$$\left. \eqalign{
u_t &= u_{t,0} + u_{t,1} \mathbin{.} z + u_{t,2} \mathbin{.} z^2
         + \ldots \, u_{t,t'} \mathbin{.} z^{t'}
         + \ldots \, u_{t,t} \mathbin{.} z^{t},\cr
w_t &= w_{t,0} + w_{t,1} \mathbin{.} z + w_{t,2} \mathbin{.} z^2
         + \ldots \, w_{t,t'} \mathbin{.} z^{t'}
         + \ldots \, w_{t,t} \mathbin{.} z^{t},\cr}
   \right\}
   \eqno {\rm (A^{(5)})}$$
$u_{t,0}, u_{t,1},\ldots \, w_{t,0}, w_{t,1},\ldots$ being
constant quantities, not containing the polar angle~$v$, and
depending only on the position of the given ray, and on the
nature of the reflected system.  In order therefore to complete
our determination of the polar functions $u^{(t)}$, $w^{(t)}$, it
is sufficient to calculate general expressions for the constants
$u_{t,t'}$, $w_{t,t'}$, considered as functions of the indices
$t$, $t'$, and of the partial differentials
$\displaystyle {dx' \over da}$,
$\displaystyle {dx' \over db}$,
$\displaystyle {d^2 x' \over da^2},\ldots$
$\displaystyle {dy' \over da} \, \ldots$;
since these differentials may, as we have before remarked, be
deduced from the differentials of the characteristic function of
the system.

To calculate these constants, the method which first presents
itself, is to substitute in the equations $(1)^{(t)}$,
$(2)^{(t)}$, in place of $u'$, $u''$, $w'$, $w'',\ldots$ their
values (Z${}''''$), (A${}^{(5)}$), and to compare the
corresponding powers of $z$.  Thus if we confine ourselves to the
constants $u_{t,t}$, $w_{t,t}$, which multiply the highest powers
of $z$, as the most important in our present investigations,
because when $w$ diminishes $z$ increases without limit; we are
to retain only those values of $\mu^{(t)}$ which give terms
multiplied by $z^t$, and it is easy to see that these terms are
distinguished by the relation
$$2 + \alpha_2 = 2 \sum \alpha + \sum \beta;
   \eqno {\rm (B^{(5)})}$$
putting them, then, under the form
$\mu_{t,t} \mathbin{.} w^{t+2} \mathbin{.} z^t$, we have when
$t = 1$,
$$\sum \mathbin{.}
   {d^{\Sigma \alpha + \Sigma \beta} x'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
      \mathbin{.} \mu_{1,1}
   = {d^2 x' \over da \mathbin{.} db};\quad
  \sum \mathbin{.}
   {d^{\Sigma \alpha + \Sigma \beta} y'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
      \mathbin{.} \mu_{1,1}
   = {d^2 y' \over da \mathbin{.} db};$$
when $t = 2$,
$$\eqalign{
\sum \mathbin{.}
   {d^{\Sigma \alpha + \Sigma \beta} x'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
      \mathbin{.} \mu_{2,2}
   &= {\textstyle {1 \over 2}} {d^2 x' \over da^2}
      + {d^2 x' \over da \mathbin{.} db} \mathbin{.} w_{1,1}
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 x' \over db^2}
            \mathbin{.} w_{1,1}^2,\cr
\sum \mathbin{.}
   {d^{\Sigma \alpha + \Sigma \beta} y'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
      \mathbin{.} \mu_{2,2}
   &= {\textstyle {1 \over 2}} {d^2 y' \over da^2}
      + {d^2 y' \over da \mathbin{.} db} \mathbin{.} w_{1,1}
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 y' \over db^2}
            \mathbin{.} w_{1,1}^2;\cr}$$
and, when $t > 2$,
$$\eqalign{
\sum \mathbin{.}
   {d^{\Sigma \alpha + \Sigma \beta} x'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
      \mathbin{.} \mu_{t,t}
   &= {d^2 x' \over da \mathbin{.} db} \mathbin{.} w_{t-1,t-1}
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 x' \over db^2}
            \mathbin{.} \sum \mathbin{.} w_{s,s} \mathbin{.} w_{t-s,t-s},\cr
\sum \mathbin{.}
   {d^{\Sigma \alpha + \Sigma \beta} y'
         \over da^{\Sigma \alpha} db^{\Sigma \beta}}
      \mathbin{.} \mu_{t,t}
   &= {d^2 y' \over da \mathbin{.} db} \mathbin{.} w_{t-1,t-1}
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 y' \over db^2}
            \mathbin{.} \sum \mathbin{.} w_{s,s} \mathbin{.} w_{t-s,t-s},\cr}$$
the sums in the second members being taken from $s = 1$, to
$s = t - 1$: and since the equations (1)${}^{(t)}$,
(2)${}^{(t)}$, give, by comparison of the highest powers of $z$,
$$\eqalign{
0  &= {dx' \over da} \mathbin{.} u_{t,t}
      + {d^2 x' \over db^2} \mathbin{.} w_{t,t}
      + {d^{\Sigma \alpha + \Sigma \beta} x'
            \over da^{\Sigma \alpha} db^{\Sigma \beta}}
         \mathbin{.} \mu_{t,t},\cr
0  &= \phantom{{dy' \over da} \mathbin{.} u_{t,t} +} \,\,
      {d^2 y' \over db^2} \mathbin{.} w_{t,t}
      + {d^{\Sigma \alpha + \Sigma \beta} y'
            \over da^{\Sigma \alpha} db^{\Sigma \beta}}
         \mathbin{.} \mu_{t,t},\cr}$$
we have successively
$$\eqalign{
(1)_{(1,1)}\ldots \quad 0
   &= {dx' \over da} \mathbin{.} u_{1,1}
      + {d^2 x' \over db^2} \mathbin{.} w_{1,1}
      + {d^2 x' \over da \mathbin{.} db},\cr
(2)_{(1,1)}\ldots \quad 0
   &= \phantom{{dy' \over da} \mathbin{.} u_{1,1} +} \,\,
        {d^2 y' \over db^2} \mathbin{.} w_{1,1}
      + {d^2 y' \over da \mathbin{.} db},\cr
(1)_{(2,2)}\ldots \quad 0
   &= {dx' \over da} \mathbin{.} u_{2,2}
      + {d^2 x' \over db^2} \mathbin{.} w_{2,2}
      + {\textstyle {1 \over 2}} {d^2 x' \over da^2}
      + {d^2 x' \over da \mathbin{.} db} \mathbin{.} w_{1,1}
      + {\textstyle {1 \over 2}} {d^2 x' \over db^2}
         \mathbin{.} w_{1,1}^2,\cr
(2)_{(2,2)}\ldots \quad 0
   &= \phantom{{dy' \over da} \mathbin{.} u_{2,2} +} \,\,
        {d^2 y' \over db^2} \mathbin{.} w_{2,2}
      + {\textstyle {1 \over 2}} {d^2 y' \over da^2}
      + {d^2 y' \over da \mathbin{.} db} \mathbin{.} w_{1,1}
      + {\textstyle {1 \over 2}} {d^2 y' \over db^2}
         \mathbin{.} w_{1,1}^2,\cr
(1)_{(3,3)}\ldots \quad 0
   &= {dx' \over da} \mathbin{.} u_{3,3}
      + {d^2 x' \over db^2} \mathbin{.} w_{3,3}
      + {d^2 x' \over da \mathbin{.} db} \mathbin{.} w_{2,2}
      + {d^2 x' \over db^2} \mathbin{.} w_{1,1} \mathbin{.} w_{2,2},\cr
(2)_{(3,3)}\ldots \quad 0
   &= \phantom{{dy' \over da} \mathbin{.} u_{3,3} +} \,\,
        {d^2 y' \over db^2} \mathbin{.} w_{3,3}
      + {d^2 y' \over da \mathbin{.} db} \mathbin{.} w_{2,2}
      + {d^2 y' \over db^2} \mathbin{.} w_{1,1} \mathbin{.} w_{2,2},\cr}$$
the two last of which equations reduce themselves, by means of
the two first, to the following form:
$$u_{3,3} = u_{1,1} \mathbin{.} w_{2,2},\quad w_{3,3} = 0;$$
a similar reduction gives in general, when $t > 3$,
$$\eqalign{
(1)_{(t,t)}\ldots \quad u_{t,t}
   &= u_{1,1}\mathbin{.} w_{t-1, t-1},\cr
(2)_{(t,t)}\ldots \quad \hskip1em 0
   &= w_{t,t} + {\textstyle {1 \over 2}}
            \mathbin{.} \sum \mathbin{.} w_{s,s}
            \mathbin{.} w_{t-s, t-s},\cr}$$
the sum being taken from $s = 2$, to $s = t - 2$; so that the
four first constants $u_{1,1}$, $w_{1,1}$, $u_{2,2}$, $w_{2,2}$,
being determined by the four equations (1)${}_{1,1}$,
(2)${}_{1,1}$, (1)${}_{2,2}$, (2)${}_{2,2}$, all the succeeding
constants of the same kind, $u_{3,3}$, $u_{4,4},\ldots$
$w_{3,3}$, $w_{4,4},\ldots$ are given by the following general
expressions, which may be deduced from the formul{\ae}
(1)${}_{t,t}$, (2)${}_{t,t}$, either by successive elimination,
or by the calculus of finite differences;
$$\left. \multieqalign{
w_{2\tau, 2\tau}
   &= 2^\tau \mathbin{.} [{\textstyle {1 \over 2}}]^\tau
         \mathbin{.} [0]^{-\tau}
         \mathbin{.} (w_{2,2})^\tau; &
w_{2\tau + 1, 2\tau + 1} &= 0;\cr
u_{2\tau + 1, 2\tau + 1}
   &= u_{1,1} \mathbin{.} w_{2\tau, 2\tau}; &
u_{2\tau + 2, 2\tau + 2}
   &= 0;\cr}
   \right\}
   \eqno {\rm (C^{(5)})}$$
$\tau$ being any integer number $> 0$, and
$[{\textstyle {1 \over 2}}]^\tau$, $[0]^{-\tau}$, being known
factorial symbols.  In a similar manner we might calculate
general expressions for the other constants of the form
$u_{t, t'}$, $w_{t, t'}$; but it seems preferable to employ the
following method, founded on the properties of partial
differentials, and on the development of functions into series.

To make use of these properties, I observe that if we put
$$r \mathbin{.} w^2 = \theta^2,\quad
  a = \zeta \mathbin{.} \theta^2,\quad
  b = \eta  \mathbin{.} \theta,
   \eqno {\rm (D^{(5)})}$$
and substitute for $u^{(t)}$, $w^{(t)}$ their expressions
(Z${}''''$), (A${}^{(5)}$), the series (T${}'''$) will take the
form
$$\left. \eqalign{
\zeta
   &= z + (u_{1,0} + u_{1,1} \mathbin{.} z) \mathbin{.} \theta
        + (u_{2,0} + u_{2,1} \mathbin{.} z + u_{2,2} \mathbin{.} z^2)
            \mathbin{.} \theta^2
        + \ldots
        + u_{t, t'} \mathbin{.} z^{t'} \mathbin{.} \theta^t
        + \hbox{\&c.},\cr
\eta
   &= 1 + (w_{1,0} + w_{1,1} \mathbin{.} z) \mathbin{.} \theta
        + (w_{2,0} + w_{2,1} \mathbin{.} z + w_{2,2} \mathbin{.} z^2)
            \mathbin{.} \theta^2
        + \ldots
        + w_{t, t'} \mathbin{.} z^{t'} \mathbin{.} \theta^t
        + \hbox{\&c.},\cr}
   \right\}
   \eqno {\rm (E^{(5)})}$$
equations which give by differentiation
$$\eqalign{
{d^{t+t'} \zeta \over d \theta^t \mathbin{.} dz^{t'}}
   &= [t]^t \mathbin{.} [t']^{t'} \mathbin{.} u_{t,t'}
      + [t + 1]^t \mathbin{.} [t']^{t'}
         \mathbin{.} u_{t+1, t'} \mathbin{.} \theta
      + [t]^t \mathbin{.} [t' + 1]^{t'}
         \mathbin{.} u_{t, t'+1} \mathbin{.} z
      + \hbox{\&c.},\cr
{d^{t+t'} \eta  \over d \theta^t \mathbin{.} dz^{t'}}
   &= [t]^t \mathbin{.} [t']^{t'} \mathbin{.} w_{t,t'}
      + [t + 1]^t \mathbin{.} [t']^{t'}
         \mathbin{.} w_{t+1, t'} \mathbin{.} \theta
      + [t]^t \mathbin{.} [t' + 1]^{t'}
         \mathbin{.} w_{t, t'+1} \mathbin{.} z
      + \hbox{\&c.},\cr}$$
and therefore, when $\theta = 0$, $z = 0$,
$$u_{t,t'}
   = [0]^{-t} \mathbin{.} [0]^{-t'}
      \mathbin{.} {d^{t+t'} \zeta \over d\theta^t \mathbin{.} dz^{t'}};\quad
  w_{t,t'}
   = [0]^{-t} \mathbin{.} [0]^{-t'}
      \mathbin{.} {d^{t+t'} \eta  \over d\theta^t \mathbin{.} dz^{t'}};
   \eqno {\rm (F^{(5)})}$$
in order therefore to obtain general expressions for the
constants $u_{t,t'}$, $w_{t,t'}$ it is sufficient to calculate
expressions for these partial differentials of $\zeta$, $\eta$.
Now, if from the two equations (S${}'''$), [68.], we subtract the
two others
$$x' = {dx' \over da} \mathbin{.} ur
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 x' \over db^2}
         \mathbin{.} w^2 r,\quad
  y' =  {\textstyle {1 \over 2}} \mathbin{.} {d^2 y' \over db^2}
         \mathbin{.} w^2 r,$$
which result from the formul{\ae} (U${}'''$) of the same
paragraph; if we then eliminate $b^2 - w^2 r$, and put for
abridgment,
$${\textstyle {1 \over 2}} \mathbin{.} {d^2 y' \over da^2}
         \mathbin{.} a^2
      + {d^2 y' \over da \mathbin{.} db}
         \mathbin{.} ab
      + {\textstyle {1 \over 6}} \mathbin{.} {d^3 y' \over da^3}
         \mathbin{.} a^3
      + \hbox{\&c.}
   = - {d^2 y' \over db^2} \mathbin{.} \Phi(a,b),$$
$${d^2 x' \over db^2} \mathbin{.} \Phi(a,b)
      + {\textstyle {1 \over 2}} \mathbin{.} {d^2 x' \over da^2}
         \mathbin{.} a^2
      + {d^2 x' \over da \mathbin{.} db}
         \mathbin{.} ab
      + {\textstyle {1 \over 6}} \mathbin{.} {d^3 x' \over da^3}
         \mathbin{.} a^3
      + \hbox{\&c.}
   = - {dx' \over da} \mathbin{.} F(a,b);$$
we shall have the following two equations,
$$a = ur + F,\quad b^2 = w^2 r + 2 \Phi;
   \eqno {\rm (G^{(5)})}$$
which, when we put
$$u = z w^2,\quad
  w^2 r = \theta^2,\quad
  a = \zeta \theta^2,\quad
  b = \eta \theta,$$
become
$$\zeta  = z + f(\zeta, \eta, \theta),\quad
  \eta^2 = 1 + 2 \phi(\zeta, \eta, \theta);
   \eqno {\rm (H^{(5)})}$$
$f$, $\phi$, being functions such that
$$F = \theta^2 \mathbin{.} f,\quad
  \Phi = \theta^2 \mathbin{.} \phi,
   \eqno {\rm (I^{(5)})}$$
and therefore
$$\left. \eqalign{
0 &= {d^2 y' \over db^2} \mathbin{.} \phi
      + \sum \mathbin{.} [0]^{-m} \mathbin{.} [0]^{-m'}
         \mathbin{.} {d^{m+m'} y' \over da^m \mathbin{.} db^{m'}}
         \mathbin{.} \zeta^m \mathbin{.} \eta^{m'}
         \mathbin{.} \theta^{2m + m' - 2},\cr
- {dx' \over da} \mathbin{.} f
  &= {d^2 x' \over db^2} \mathbin{.} \phi
      + \sum \mathbin{.} [0]^{-m} \mathbin{.} [0]^{-m'}
         \mathbin{.} {d^{m+m'} x' \over da^m \mathbin{.} db^{m'}}
         \mathbin{.} \zeta^m \mathbin{.} \eta^{m'}
         \mathbin{.} \theta^{2m + m' - 2},\cr}
   \right\}
   \eqno {\rm (K^{(5)})}$$
$m$, $m'$, being any positive integers which satisfy the
following relation
$$2m + m' > 2.
   \eqno {\rm (L^{(5)})}$$
If then we eliminate $\zeta$, $\eta$, between the equations
(H${}^{(5)}$), so as to find expressions for those variables as
functions of $z$ and $\theta$; it will remain to differentiate
those expressions, $(t)$ times for $\theta$, and $(t')$ times for
$z$, and to put after the differentiations $\theta = 0$, $z = 0$;
since the partial differentials thus obtained, multiplied by the
factorial quantity $[0]^{-t} \mathbin{.} [0]^{-t'}$, will give,
by (F${}^{(5)}$), the general expressions that we are in search
of, for the constants $u_{t,t'}$, $w_{t,t'}$.

To perform this elimination, we may employ the theorems which
Laplace has given, in the second book of the
{\it M\'{e}canique C\'{e}leste}, for the development of functions
into series.  Laplace has there shewn that if we have any number
($r$) of equations of the form
$$x = \phi(t + \alpha z),\quad
  x' = \psi(t' + \alpha' z'),\quad
  x'' = \Pi(t'' + \alpha'' z''),\,\hbox{\&c.},$$
in which $z$, $z'$ ,$z''$, \&c. are functions of $x$, $x'$,
$x''$, \&c. and if we develope any other function~$u$ of the same
variables, according to the powers and products of $\alpha$,
$\alpha'$, $\alpha''$, \&c.\ in a series of which the general
term is represented by
$q_{n, n', n'',\ldots} \alpha^n
      \mathbin{.} \alpha'^{n'}
      \mathbin{.} \alpha''^{n''}
      \mathbin{.} \hbox{\&c.}$;
we shall have, to determine the coefficient
$q_{n, n', n'',\ldots}$,
a formula which may thus be written,
$$q_{n, n', n'',\ldots}
   = [0]^{-n} \mathbin{.} [0]^{-n'} \mathbin{.} [0]^{-n''} \, \ldots \,
      {d^{n + n' + n'' + \ldots - r}
         \over dt^{n-1}
            \mathbin{.} dt'^{n' - 1}
            \mathbin{.} dt''^{n'' - 1} \ldots}
         \left(
            {d^r u_\prime \over d\alpha
            \mathbin{.} d\alpha'
            \mathbin{.} d\alpha''
            \ldots}
         \right),$$
$u_\prime$ being a function formed by changing in $u$ the
original variables $x$, $x'$, $x'',\ldots$ into other variables
determined by the following equations
$$x = \phi(t + \alpha z^n),\quad
  x' = \psi(t' + \alpha' z'^{n'}),\quad
  x'' = \Pi(t'' + \alpha'' z''^{n''}),\, \hbox{\&c.},$$
the functions $\phi$, $\psi$, $\Pi$, $z$, $z'$, $z''$,
\&c.\ retaining the same forms as before, and
$\alpha$, $\alpha'$, $\alpha'',\ldots$ being supposed to vanish
after the differentiations.  Laplace has also shewn, that
when there are but two variables $x$, $x'$, the partial
differential
$\displaystyle
   \left(
      {d^r u_\prime \over d\alpha
      \mathbin{.} d\alpha'
      \mathbin{.} d\alpha''
      \ldots}
   \right)$,
determined in this manner, reduces itself to
$$\left( {ddu_\prime \over d\alpha \mathbin{.} d\alpha'} \right)
   =  Z^n \mathbin{.} Z'^{n'} \mathbin{.}
         \left( {dd{\rm u} \over dt \mathbin{.} dt'} \right)
      + Z'^{n'} \mathbin{.}
         \left({d \mathbin{.} Z^n \over dt'} \right)
         \mathbin{.}
         \left( {d{\rm u} \over dt} \right)
      + Z^n \mathbin{.}
         \left({d \mathbin{.} Z'^{n'} \over dt} \right)
         \mathbin{.}
         \left( {d{\rm u} \over dt'} \right),$$
in which $Z$, $Z'$, ${\rm u}$ represent the values that $z$,
$z'$, $u$, take, when we suppose $\alpha = 0$, $\alpha' = 0$.  If
then the original equations are of the form
$$x = t + \alpha z,\quad x' = \surd (t' + \alpha' z'),$$
and if we change the function $u$ to $x$ and $x'$ successively,
we find the following developments for those two variables,
according to the powers and products of $\alpha$, $\alpha'$;
$$\eqalign{
x  &= \sum [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} \alpha^n \alpha'^{n'}
         \mathbin{.} {d^{n + n' - 2}
            \over dt^{n-1} \mathbin{.} dt'^{n'-1}}
         \left(
            {Z'^{n'} \over 2 \surd t'}
            \mathbin{.}
            {d \mathbin{.} Z^n \over d \surd t'}
         \right),\cr
x' &= \sum [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} \alpha^n \alpha'^{n'}
         \mathbin{.} {d^{n + n' - 2}
            \over dt^{n-1} \mathbin{.} dt'^{n'-1}}
         \left(
            {Z^{n} \over 2 \surd t'}
            \mathbin{.}
            {d \mathbin{.} Z^n \over dt}
         \right),\cr}$$
in which $Z$, $Z'$ are formed from $z$, $z'$, by changing $x$ to
$t$, and $x'$ to $\surd t'$.  Applying these results to the
equations (H${}^{(5)}$), which are of the form
$$\zeta = z + \alpha f,\quad \eta = \surd (z' + \alpha' \phi);$$
we find the following expressions for $\zeta$, $\eta$, as
functions of $z$ and $\theta$,
$$\left. \eqalign{
\zeta
   &= \sum \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'} \mathbin{.}
         {d^{n + n' - 2} \mathbin{.} \zeta^{(n, n')}
            \over dz^{n - 1} \mathbin{.} dz'^{n'-1}},\cr
\eta
   &= \sum \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'} \mathbin{.}
         {d^{n + n' - 2} \mathbin{.} \eta^{(n, n')}
            \over dz^{n - 1} \mathbin{.} dz'^{n'-1}},\cr}
   \right\}
   \eqno {\rm (M^{(5)})}$$
in which
$$\zeta^{(n,n')}
   =  {\phi^{n'} \over 2 \surd z'}
      \mathbin{.}
      {d \mathbin{.} f^n \over d \surd z'},\quad
  \eta^{(n,n')}
   =  {f^n \over 2 \surd z'}
      \mathbin{.}
      {d \mathbin{.} \phi^{n'} \over dz},
   \eqno {\rm (N^{(5)})}$$
$f$, $\phi$, being deduced from the formul{\ae} (K${}^{(5)}$) by
changing $\zeta$ to $z$, and $\eta$ to $\surd z'$, and in which
we may make after the differentiations $z' = 1$.  And
differentiating these developments (M${}^{(5)}$) in the manner
already prescribed, we find, finally, the following general
expressions for the constants $u_{t,t'}$, $w_{t,t'}$;
$$\left. \eqalign{
u_{t,t'}
   &= [0]^{-t} \mathbin{.} [0]^{-t'}
      \mathbin{.} \sum \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'} \mathbin{.}
         {d^{t + t' + n + n' - 2} \mathbin{.} \zeta^{(n, n')}
            \over d\theta^t \mathbin{.} dz^{t' + n - 1}
               \mathbin{.} dz'^{n'-1}},\cr
w_{t,t'}
   &= [0]^{-t} \mathbin{.} [0]^{-t'}
      \mathbin{.} \sum \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'} \mathbin{.}
         {d^{t + t' + n + n' - 2} \mathbin{.} \eta^{(n, n')}
            \over d\theta^t \mathbin{.} dz^{t' + n - 1}
               \mathbin{.} dz'^{n'-1}},\cr}
   \right\}
   \eqno {\rm (O^{(5)})}$$
$n$, $n'$, being any positive integers, and $(\theta, z, z')$
being changed after differentiations to $(0,0,1)$.  It may be
useful to observe, that by the formulae (N${}^{(5)}$), and by the
nature of the developments, we are to make
$$\left. \multieqalign{
{d^{-1} \mathbin{.} \zeta^{(0,n')} \over dz^{-1}}
   &= 0; &
{d^{-1} \mathbin{.} \eta^{(n,0)}   \over dz'^{-1}}
   &= 0; \cr
{d^{-1} \mathbin{.} \zeta^{(n,0)}  \over dz'^{-1}}
   &= f^n; &
{d^{-1} \mathbin{.} \eta^{(0,n')}  \over dz^{-1}}
   &= {\phi^{n'} \over 2 \surd z'}; \cr
{d^{-2} \mathbin{.} \zeta^{(0,0)}
         \over dz^{-1} \mathbin{.} dz'^{-1}}
   &= z; &
{d^{-2} \mathbin{.} \eta^{(0,0)}
         \over dz^{-1} \mathbin{.} dz'^{-1}}
   &= \surd z'. \cr}
   \right\}
   \eqno {\rm (P^{(5)})}$$
These expressions (O${}^{(5)}$), may be put under other forms,
some of which are more convenient for calculation.  If, for
abridgment, we write them thus
$$u_{t,t'} = \sum \mathbin{.} u_{t,t'}^{(n,n')},\quad
  w_{t,t'} = \sum \mathbin{.} w_{t,t'}^{(n,n')},
   \eqno {\rm (Q^{(5)})}$$
$u_{t,t'}^{(n,n')}$, $w_{t,t'}^{(n,n')}$, denoting the terms of
$u_{t,t'}$, $w_{t,t'}$, which correspond to any given values of
the integers $n$, $n'$; we have, by (O${}^{(5)}$),
$$\left. \eqalign{
u_{t,t'}^{(n,n')}
   &= [0]^{-t} \mathbin{.} [0]^{-t'}
      \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'}
         {d^{t + t' + n + n' - 2} \mathbin{.} \zeta^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}};\cr
w_{t,t'}^{(n,n')}
   &= [0]^{-t} \mathbin{.} [0]^{-t'}
      \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'}
         {d^{t + t' + n + n' - 2} \mathbin{.} \eta^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}};\cr}
   \right\}
   \eqno {\rm (R^{(5)})}$$
and if in (N${}^{(5)}$) we change $f$, $\phi$, to their values
(I${}^{(5)}$),
$$\eqalign{
f  &= \theta^{-2} \mathbin{.} F(a,b)
    = \theta^{-2} \mathbin{.} F( \theta^2 z, \theta \surd z'),\cr
\phi
   &= \theta^{-2} \mathbin{.} \Phi(a,b)
    = \theta^{-2} \mathbin{.} \Phi( \theta^2 z, \theta \surd z'),\cr}$$
we find the following developments,
$$\zeta^{(n,n')} = \sum \mathbin{.} \zeta_{m,m'}^{(n,n')};\quad
  \eta^{(n,n')}  = \sum \mathbin{.} \eta_{m,m'}^{(n,n')};
   \eqno {\rm (S^{(5)})}$$
in which
$$\left. \eqalign{
\zeta_{m,m'}^{(n,n')}
   &= {\textstyle {1 \over 2}}
      \mathbin{.} [0]^{-m} \mathbin{.} [0]^{-m'}
      \mathbin{.} \theta^{2m + m' + 1 - 2(n + n')}
      \mathbin{.} z^m \mathbin{.} z'{}^{m' - 1 \over 2}
      {\displaystyle d^{m + m'}
            \left(
               \Phi^{n'} \mathbin{.} {d \mathbin{.} F^n \over db}
            \right)
         \over da^m \mathbin{.} db^{m'}},\cr
\eta_{m,m'}^{(n,n')}
   &= {\textstyle {1 \over 2}}
      \mathbin{.} [0]^{-m} \mathbin{.} [0]^{-m'}
      \mathbin{.} \theta^{2m + m' + 1 - 2(n + n')}
      \mathbin{.} z^m \mathbin{.} z'{}^{m' - 1 \over 2}
      {\displaystyle d^{m + m'}
            \left(
               F^n \mathbin{.} {d \mathbin{.} \Phi^{n'} \over da}
            \right)
         \over da^m \mathbin{.} db^{m'}},\cr}
   \right\}
   \eqno {\rm (T^{(5)})}$$
$a$, $b$, being supposed to vanish after the differentiations,
and $m$, $m'$, being any integer numbers: but the only values of
these integers which do not make the partial differentials
$${d^{t + t' + n + n' - 2} \zeta_{m,m'}^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}},\quad
  {d^{t + t' + n + n' - 2} \eta_{m,m'}^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}},$$
vanish when $\theta = 0$, $z = 0$, are those for which, in
$\zeta_{m,m'}^{(n,n')}$,
$$t = 2m + m' + 1 - 2(n + n'),\quad t' + n - 1 = m,$$
and, in $\eta_{m,m'}^{(n,n')}$,
$$t = 2m + m' + 2 - 2(n + n'),\quad t' + n - 1 = m;$$
we have, therefore, when $\theta = 0$, $z = 0$,
$$\left. \eqalign{
{d^{t + t' + n + n' - 2} \zeta^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}}
   =  {d^{t + t' + n + n' - 2} \zeta_{t' + n - 1, t + 1 + 2n' - 2t'}^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}},\cr
  {d^{t + t' + n + n' - 2} \eta^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}}
   =  {d^{t + t' + n + n' - 2} \eta_{t' + n - 1, t + 2n' - 2t'}^{(n, n')}
            \over d\theta^t \, dz^{t' + n - 1} \, dz'^{n'-1}},\cr}
   \right\}
   \eqno {\rm (U^{(5)})}$$
in which the second members may be calculated by (T${}^{(5)}$).
In this manner, the expressions (R${}^{(5)}$) become, after
reductions,
$$\left. \eqalign{
u_{t,t'}^{(n,n')}
   &= [0]^{-t'} \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'} \mathbin{.} [0]^{-(t + 2 + 2n' - 2t')}
         \mathbin{.} \left[ {t + 2 \over 2} + n' - t' \right]^{n'} \cr
   &\mathrel{\phantom{=}} \mathord{}
         \times {\displaystyle d^{t - t' + n + 2n'}
            \left(
               \Phi^{n'} \mathbin{.}
               {d \mathbin{.} F^n \over db}
            \right)
            \over d^{t' + n - 1} \, db^{t + 1 + 2n' - 2t'}},\cr
w_{t,t'}^{(n,n')}
   &= [0]^{-t'} \mathbin{.} [0]^{-n} \mathbin{.} [0]^{-n'}
         \mathbin{.} 2^{n'} \mathbin{.} [0]^{-(t + 1 + 2n' - 2t')}
         \mathbin{.} \left[ {t + 1 \over 2} + n' - t' \right]^{n'} \cr
   &\mathrel{\phantom{=}} \mathord{}
         \times {\displaystyle d^{t - t' + n + 2n' - 1}
            \left(
               F^n \mathbin{.}
               {d \mathbin{.} \Phi^{n'} \over da}
            \right)
            \over d^{t' + n - 1} \, db^{t + 2n' - 2t'}};\cr}
   \right\}
   \eqno {\rm (V^{(5)})}$$
and substituting these expressions in the developments
(Q${}^{(5)}$), we get new developments for the constants
$u_{t,t'}$, $w_{t,t'}$, in which we have only to differentiate
for the two variables $(a,b)$ instead of the three
$(\theta, z, z')$.

Again, if we observe that by the nature of the functions $F$,
$\Phi$, we have when $a = 0$, $b = 0$,
$$\left. \multieqalign{
F &= 0 &
{dF \over da} &= 0, &
{dF \over db} &= 0, &
{d^2 F \over db^2} &= 0, \cr
\Phi &= 0 &
{d\Phi \over da} &= 0, &
{d\Phi \over db} &= 0, &
{d^2 \Phi \over db^2} &= 0; \cr}
   \right\}
   \eqno {\rm (W^{(5)})}$$
we shall easily deduce relations between the integer numbers $t$,
$t'$, $n$, $n'$, which reduce the summation of these developments
to the addition of a finite number of terms.  For we may prove,
either by these equations (W${}^{(5)}$), or by the condition
(L${}^{(5)}$) which contains them, that the partial differentials
$${d^{m + m'} \over da^m \, db^{m'}}
      \left(
         \Phi^{n'} \mathbin{.} {d \mathbin{.} F^n \over db}
      \right),\quad
   {d^{m + m'} \over da^m \, db^{m'}}
      \left(
         F^n \mathbin{.} {d \mathbin{.} \Phi^{n'} \over da}
      \right),$$
which enter into the formul{\ae} (T${}^{(5)}$), vanish, unless
in the first
$$m + m' + 2 > 2(n + n');\quad 2m + m' + 2 > 3(n + n');$$
and, in the second,
$$m + m' + 2 > 2(n + n');\quad 2m + m' + 3 > 3(n + n').$$
Hence, by (V${}^{(5)}$), the partial differentials which enter
into the expression for $u_{t,t'}^{(n,n')}$, $w_{t,t'}^{(n,n')}$,
vanish unless in the first
$$n < t - t' + 2;\quad n + n' < t + 1;
   \eqno {\rm (X^{(5)})}$$
and, in the second
$$n < t - t' + 1;\quad n + n' < t + 1.
   \eqno {\rm (Y^{(5)})}$$
Thus, in calculating the constants $u_{t,t'}$, $w_{t,t'}$, by the
formul{\ae} (Q${}^{(5)}$),
$$u_{t,t'} = \sum u_{t,t'}^{(n,n')},\quad
  w_{t,t'} = \sum w_{t,t'}^{(n,n')},$$
we may reject all values of $n$, $n'$, which are too great to
satisfy these relations (X${}^{(5)}$), (Y${}^{(5)}$); we may
also, by (V${}^{(5)}$), reject not only all negative values of
the same integers $n$, $n'$, but all for which the factorial
index $t + 2 + 2n' - 2t'$ is negative in $u_{t,t'}^{(n,n')}$, or
$t + 1 + 2n' - 2t'$ in $w_{t,t'}^{(n,n')}$; and by (R${}^{(5)}$),
(P${}^{(5)}$) we may reject the value $n = 0$ in the former, and
$n' = 0$ in the latter.  Finally, we may remark, that since a
factorial vanishes, when its base is less than its index, if both
be positive integers, the expression (V${}^{(5)}$) for
$u_{t,t'}^{(n,n')}$ vanishes if $t$ be even, and
$\displaystyle t' > {t + 2 \over 2}$;
and similarly the expression for $w_{t,t'}^{(n,n')}$ vanishes if
$t$ be odd, and
$\displaystyle t' > {t + 1 \over 2}$:
from which it follows, that if the developments (T${}'''$) of the
preceding paragraph, be put by (Y${}''''$) (D${}^{(5)}$),
(E${}^{(5)}$), under the form
$$\left. \eqalign{
a  &= \sum \mathbin{.} u_{t,t'} \mathbin{.} z^{t'} \theta^{t + 2}
    = \sum \mathbin{.} u_{t,t'} \mathbin{.} u^{t'}
         \mathbin{.} w^{t + 2 - 2t'} \surd r^{t + 2},\cr
b  &= \sum \mathbin{.} w_{t,t'} \mathbin{.} z^{t'} \theta^{t + 1}
    = \sum \mathbin{.} w_{t,t'} \mathbin{.} u^{t'}
         \mathbin{.} w^{t + 1 - 2t'} \surd r^{t + 1},\cr}
   \right\}
   \eqno {\rm (Z^{(5)})}$$
the negative even powers of ($\theta$) or of ($w$) will all
disappear.

Let us verify these general results, respecting the constants
$u_{t,t'}$, $w_{t,t'}$, by applying them to the particular case
$t' = t$, which as we have before remarked, is the most important
in our present investigations, and which we have already resolved
by an entirely different method.  In this case, when $t = 1$, we
find by our present method,
$$u_{1,1} = u_{1,1}^{(1,0)} = {d^2 F \over da \mathbin{.} db};\quad
  w_{1,1} = w_{1,1}^{(0,1)} = {d^2 \Phi \over da \mathbin{.} db};$$
when $t = 2$,
$$\eqalign{
u_{2,2}
   &= u_{2,2}^{(1,0)} + u_{2,2}^{(1,1)}
    = {\textstyle {1 \over 2}}
         {\displaystyle d^{2 - 1}
               \left( \Phi^0 {dF \over db} \right)
            \over da^2 \, db^{-1}}
      + {\textstyle {1 \over 2}}
         {\displaystyle d^{2 + 1}
               \left( \Phi^1 {dF \over db} \right)
            \over da^2 \, db^1}
    = {\textstyle {1 \over 2}} {d^2 F \over da^2}
      + {d^2 F \over da \mathbin{.} db} {d^2 \Phi \over da \mathbin{.} db},\cr
w_{2,2}
   &= w_{2,2}^{(0,1)} + w_{2,2}^{(0,2)}
    = {\textstyle {1 \over 2}} {d^2 \Phi \over da^2}
      + {\textstyle {1 \over 8}} {d^4 \mathbin{.} \Phi^2 \over da^2 \, db^2}
    = {\textstyle {1 \over 2}} {d^2 \Phi \over da^2}
      + {\textstyle {1 \over 2}}
         \left( {d^2 \mathbin{.} \Phi
               \over da  \mathbin{.} db} \right)^2;\cr}$$
and when $t > 2$, if we put
$$u_{t,t} = u_{2\tau + 1, 2\tau + 1},\quad
  w_{t,t} = w_{2\tau, 2\tau},$$
(since $u_{t,t}$ vanishes if $t$ be an even number $> 2$, and
$w_{t,t}$ if $t$ be an odd number $> 2$), we have the following
formul{\ae},
$$u_{2\tau + 1, 2\tau + 1}
   = \sum \mathbin{.} u_{2\tau + 1, 2\tau + 1}^{(1,n')};\quad
  w_{2\tau, 2\tau}
   = \sum \mathbin{.} w_{2\tau, 2\tau}^{(0,n')};
   \eqno {\rm (A^{(6)})}$$
the sum being taken in each from $n' = \tau$ to $n' = 2\tau$.  We
have also, by (V${}^{(5)}$),
$$\left. \eqalign{
u_{2\tau + 1, 2\tau + 1}^{(1,n')}
   &= [0]^{-(2\tau + 1)} \mathbin{.} [0]^{-n'}
         \mathbin{.} [0]^{-(2n' - 2\tau + 1)} \mathbin{.} 2^{n'}
         \mathbin{.} [n' - \tau + {\textstyle {1 \over 2}}]^{n'}
         \mathbin{.} {\displaystyle
               d^{2n' + 1} \mathbin{.} \Phi^{n'} {dF \over db}
            \over da^{2\tau + 1} db^{2n' - 2\tau}},\cr
w_{2\tau, 2\tau}^{(0,n')}
   &= [0]^{-2\tau} \mathbin{.} [0]^{-n'}
         \mathbin{.} [0]^{-(2n' - 2\tau + 1)} \mathbin{.} 2^{n'}
         \mathbin{.} [n' - \tau + {\textstyle {1 \over 2}}]^{n'}
         \mathbin{.} {d^{2n'} \mathbin{.} \Phi^{n'}
            \over da^{2\tau} db^{2n' - 2\tau}},\cr}
   \right\}
   \eqno {\rm (B^{(6)})}$$
in which, by (W${}^{(5)}$),
$$\left. \eqalign{
{\displaystyle d^{2n' + 1} \mathbin{.} \Phi^{n'} {dF \over db}
            \over da^{2\tau + 1} \mathbin{.} db^{2n' - 2\tau}}
   &= (2\tau + 1) \mathbin{.} {d^2 F \over da \mathbin{.} db}
         \mathbin{.} {d^{2n'} \mathbin{.} \Phi^{n'}
            \over da^{2\tau} \mathbin{.} db^{2n' - 2\tau}},\cr
{d^{2n'} \mathbin{.} \Phi^{n'}
            \over da^{2\tau} \mathbin{.} db^{2n' - 2\tau}}
   &= [2\tau]^{2\tau} [n']^{2n' - 2\tau} \mathbin{.} 2^{n' - 2\tau}
         \left( {d^2 \Phi \over da^2} \right)^{2\tau - n'}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{2n' - 2\tau},\cr}
   \right\}
   \eqno {\rm (C^{(6)})}$$
and, by the properties of factorials,
$$\left. \eqalign{
[0]^{-(2n' - 2\tau + 1)} [n' - \tau + {\textstyle {1 \over 2}}]^{n'}
         \mathbin{.} 2^{2n' - 2\tau}
   &= [\textstyle {1 \over 2}]^\tau \mathbin{.} [0]^{-(n' - \tau)},\cr
[0]^{-n'} \mathbin{.} [n']^{2n' - 2\tau}
   &= [0]^{-\tau} \mathbin{.} [\tau]^{n' - \tau};\cr}
   \right\}
   \eqno {\rm (D^{(6)})}$$
thus the formul{\ae} (A${}^{(6)}$) reduce themselves to
$$\left. \eqalign{
u_{2\tau + 1, 2\tau + 1}
   &= {d^2 F \over da \mathbin{.} db}
         \mathbin{.} [0]^{-\tau} \! \mathbin{.} [{\textstyle {1 \over 2}}]^\tau
         \mathbin{.} \sum \mathbin{.} [0]^{-(n' - \tau)} [\tau]^{n' - \tau}
         \left( {d^2 \Phi \over da^2} \right)^{\tau - (n' - \tau)}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{2(n' - \tau)},\cr
w_{2\tau, 2\tau}
   &= \phantom{{d^2 F \over da \mathbin{.} db} \mathbin{.}} \,\,
         [0]^{-\tau} \! \mathbin{.} [{\textstyle {1 \over 2}}]^\tau
         \mathbin{.} \sum \mathbin{.} [0]^{-(n' - \tau)} [\tau]^{n' - \tau}
         \left( {d^2 \Phi \over da^2} \right)^{\tau - (n' - \tau)}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{2(n' - \tau)},\cr}
   \right\}
   \eqno {\rm (E^{(6)})}$$
or, finally, effecting the summations,
$$\left. \eqalign{
u_{2\tau + 1, 2\tau + 1}
   &= {d^2 F \over da \mathbin{.} db}
         \mathbin{.} [0]^{-\tau} \mathbin{.} [{\textstyle {1 \over 2}}]^\tau
         \left(
            {d^2 \Phi \over da^2}
            + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)^\tau,\cr
w_{2\tau, 2\tau}
   &= \phantom{{d^2 F \over da \mathbin{.} db} \mathbin{.}} \,\,
         [0]^{-\tau} \mathbin{.} [{\textstyle {1 \over 2}}]^\tau
         \left(
            {d^2 \Phi \over da^2}
            + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)^\tau:\cr}
   \right\}
   \eqno {\rm (F^{(6)})}$$
and if we eliminate
$\displaystyle {d^2 \Phi \over da^2}$,
$\displaystyle {d^2 \Phi \over da \mathbin{.} db}$,
$\displaystyle {d^2 F \over da^2}$,
$\displaystyle {d^2 F \over da \mathbin{.} db}$,
by the following relations
$$\left. \eqalign{
0  &= {d^2 y' \over db^2} \mathbin{.}
         {d^{m + m'} \Phi \over da^m \mathbin{.} db^{m'}}
       + {d^{m + m'} y'   \over da^m \mathbin{.} db^{m'}},\cr
-{dx' \over da} \mathbin{.}
         {d^{m + m'} F    \over da^m \mathbin{.} db^{m'}}
   &= {d^2 x' \over db^2} \mathbin{.}
         {d^{m + m'} \Phi \over da^m \mathbin{.} db^{m'}}
       + {d^{m + m'} x'   \over da^m \mathbin{.} db^{m'}},\cr}
   \right\}
   \eqno {\rm (G^{(6)})}$$
in which $m$, $m'$, are any positive integers satisfying the
conditions (L${}^{(5)}$), we arrive again at the same expressions
for the constants of the form $u_{t,t}$, $w_{t,t}$, as those
given by the equations (1)${}_{1,1}$, (2)${}_{1,1}$,
(1)${}_{2,2}$, (2)${}_{2,2}$, (C${}^{(5)}$), which we obtained
before by reasonings of so different a nature.  It results from
these equations, that if in the developments (Z${}^{(5)}$), for
the components of aberration at the mirror, we confine ourselves
to the terms of the form
$$u_{2\tau + 1, 2\tau + 1}
         \mathbin{.} u^{2\tau + 1} \mathbin{.} w^{1 - 2\tau}
         \mathbin{.} \surd r^{2\tau + 3},\quad
  w_{2\tau, 2\tau}
         \mathbin{.} u^{2\tau} \mathbin{.} w^{1 - 2\tau}
         \mathbin{.} \surd r^{2\tau + 1},$$
which correspond to the greatest negative powers of $w$, or of
the sine of the polar angle~$v$; the sums of these terms, taken
from $\tau = 0$, to $\tau = \infty$, may be calculated by the
binomial theorem, and are thus expressed:
$$\left. \eqalign{
\sum\nolimits_0^\infty \mathbin{.} w_{2\tau, 2\tau}
         u^{2\tau} w^{1 - 2\tau} \surd r^{2\tau + 1}
   &= w \surd r \mathbin{.} \sum\nolimits_0^\infty
         \mathbin{.} [0]^{-\tau}
         \mathbin{.} [{\textstyle {1 \over 2}}]^\tau
         (2 w_{2,2} u^2 w^{-2} r)^\tau \cr
   &= \surd (w^2 r + 2 w_{2,2} u^2 r^2);\cr
\sum\nolimits_0^\infty \mathbin{.} u_{2\tau + 1, 2\tau + 1}
         u^{2\tau + 1} w^{1 - 2\tau} \surd r^{2\tau + 3}
   &= u_{1,1} \mathbin{.} ur \mathbin{.}
         \surd (w^2 r + 2 w_{2,2} u^2 r^2).\cr}
   \right\}
   \eqno {\rm (H^{(6)})}$$
We shall return to this remarkable result, and examine its
optical meaning.

As another application of our general formul{\ae} for the
constants $u_{t,t'}$, $w_{t,t'}$, let us take the terms of the
form
$$w_{2\tau + 2, 2\tau + 1}
         \mathbin{.} u^{2\tau + 1} \mathbin{.} w^{1 - 2\tau}
         \mathbin{.} \surd r^{2\tau + 3},$$
which correspond to the next greatest negative powers of $w$, or
of the sine of the polar angle~$v$, in the development
(Z${}^{(5)}$) for $b$, that is, for the aberration at the mirror
measured in a direction perpendicular to the tangent plane of the
caustic surface, and considered as depending on the polar
coordinates $r$ and $v$, which determine the magnitude and
direction of the aberration on the perpendicular plane at the
focus.  We have, by what precedes,
$$\eqalignno{
w_{2\tau + 2, 2\tau + 1}
   &= \sum \mathbin{.} w_{2\tau + 2, 2\tau + 1}^{(0,n')}
      + \sum \mathbin{.} w_{2\tau + 2, 2\tau + 1}^{(1,n')} \cr
   &= [0]^{-(2\tau + 1)} \! \mathbin{.} \sum \mathbin{.}
         [0]^{-n'} 2^{n'} [0]^{-(2n' - 2\tau + 1)}
         [n' - \tau + {\textstyle {1 \over 2}}]^{n'}
         {\displaystyle
               d^{2n' + 1} \left( \Phi^{n'} + F \mathbin{.}
               {d \mathbin{.} \Phi^{n'} \over da} \right)
            \over da^{2\tau + 1} \mathbin{.} db^{2n' - 2\tau}};\cr
   & &{\rm (I^{(6)})}\cr}$$
and the summation here indicated, with reference to the variable
integer $n'$, may be performed by partial differential and
factorial transformations, similar to those which we have already
employed in finding the sums of the expressions (B${}^{(6)}$).
Thus, we may eliminate the variable $n'$ from the partial
differential index, by putting, in virtue of (W${}^{(5)}$),
$$\eqalignno{
[0]^{-(2\tau + 1)} [0]^{-n'}
         {\displaystyle d^{2n' + 1}
               \left(
                  \Phi^{n'}
                + F \mathbin{.} {d \mathbin{.} \Phi^{n'} \over da}
               \right)
            \over da^{2\tau + 1} \mathbin{.} db^{2n' - 2\tau}}
    = [2n' - 2\tau]^{2n' - 2\tau}
         \hskip -288pt \cr
   &\times \sum \mathbin{.} [0]^{-s} \!\! \mathbin{.} [0]^{-(3 - s)}
         [0]^{-s'} \!\! \mathbin{.} [0]^{-(n' - 1 - s')}
         \left(
            {\textstyle {1 \over 2}} \mathbin{.} {d^2 \Phi \over da^2}
         \right)^{n' - 1 - s'}
         \!\!
         \left(
            {d^2 \Phi \over da \mathbin{.} db}
         \right)^{s'}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \mathbin{.} db^s},\cr
   & & {\rm (K^{(6)})}\cr}$$
the integers $s$, $s'$, being new variables connected by the
relation
$$s + s' = 2n' - 2\tau,
   \eqno {\rm (L^{(6)})}$$
which gives, by the properties of factorials,
$$[2n' - 2\tau]^{2n' - 2\tau} [0]^{-s'}
   = [2n' - 2\tau]^s
   = 2^s [n' - \tau]^s + 2^{s - 2} [s]^2 [n' - \tau]^{s - 1},
   \eqno {\rm (M^{(6)})}$$
observing that by (K${}^{(6)}$), $s$ is included between the
limits $0$ and $3$; and, by the same properties,
$$\left. \eqalign{
[0]^{-(n' - 1 - s')}
   &= [0]^{-(\tau - \tau')} [\tau - \tau']^{n' - \tau - s + 1 - \tau'},\cr
[0]^{-(2n' - 2\tau + 1)} 2^{n'} [n' - \tau + {\textstyle {1 \over 2}}]^{n'}
   &= [{\textstyle {1 \over 2}}]^{\tau'}
         [{\textstyle {1 \over 2}} - \tau']^{\tau - \tau'}
         2^{2\tau - n'} [0]^{(n' - \tau)},\cr}
   \right\}
   \eqno {\rm (N^{(6)})}$$
$\tau'$ being an arbitrary integer; so that if we put
$$\left. \eqalign{
W  &= [0]^{-(n' - \tau - s)} [\tau - \tau']^{n' - \tau - s + 1 - \tau'}
      \left( {d^2 \Phi \over da^2} \right)^{2\tau - n' + s - 1}
      \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{2n' - 2\tau - s},\cr
W' &= [0]^{-(n' - \tau - s + 1)} [\tau - \tau']^{n' - \tau - s + 1 - \tau'}
      \left( {d^2 \Phi \over da^2} \right)^{2\tau - n' + s - 1}
      \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{2n' - 2\tau - s},\cr}
   \right\}
   \eqno {\rm (O^{(6)})}$$
the expression (I${}^{(6)}$) resolves itself into the two following parts:
$$\eqalignno{
w_{2\tau + 2, 2\tau + 1}
   &= 2 [{\textstyle {1 \over 2}}]^{\tau'}
         [{\textstyle {1 \over 2}} - \tau']^{\tau - \tau'}
         [0]^{-(\tau - \tau')}
         \sum\nolimits^{(s)}
         [0]^{-s} [0]^{-(3 - s)}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \, db^s}
         \mathbin{.} \sum\nolimits^{(n')} W \cr
   &\hskip -36pt
    + 2^{-1} [{\textstyle {1 \over 2}}]^{\tau'}
         [{\textstyle {1 \over 2}} - \tau']^{\tau - \tau'}
         [0]^{-(\tau - \tau')}
         \sum\nolimits^{(s)}
         [0]^{-(s - 2)} [0]^{-(3 - s)}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \, db^s}
         \mathbin{.} \sum\nolimits^{(n')} W',\cr
   & &{\rm (P^{(6)})}\cr}$$
in which $\sum^{(n')}$, $\sum^{(s)}$, denote summations with
reference to the two independent variables $n'$ and $s$, and
which can be calculated separately, by making in the first
$\tau' = 1$, and in the second $\tau' = 0$: for this gives, by
the binomial theorem,
$$\left. \eqalign{
\sum\nolimits^{(n')} W
   &= \left( {d^2 \Phi \over da \mathbin{.} db} \right)^s
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)^{\tau - 1},\cr
\sum\nolimits^{(n')} W'
   &= \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{s - 2}
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)^\tau,\cr}
   \right\}
   \eqno {\rm (Q^{(6)})}$$
and reduces the expression (P${}^{(6)}$) to the following form,
$$\eqalignno{
w_{2\tau + 2, 2\tau + 1}
   &= [-{\textstyle {1 \over 2}}]^{\tau - 1} [0]^{-(\tau - 1)}
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)^{\tau - 1} \cr
   &\mathrel{\phantom{= +}}
         \times \sum\nolimits^{(s)} [0]^{-s} [0]^{-(3 - s)}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \mathbin{.} db^s}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^s \cr
   &\mathrel{\phantom{=}} \mathord{}
    + [{\textstyle {1 \over 2}}]^\tau [0]^{-\tau}
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)^\tau \cr
   &\mathrel{\phantom{= +}}
         \times \sum\nolimits^{(s)}
         {[0]^{-(s - 2)} [0]^{-(3 - s)} \over 2}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \mathbin{.} db^s}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{s - 2},
   &{\rm (R^{(6)})}\cr}$$
in which the first sum contains only four terms, and the second
only two, however great may be the value of ($\tau$).  And if we
multiply this expression (R${}^{(6)}$) by
$u^{2\tau + 1} \mathbin{.} w^{1 - 2\tau} \surd r^{2\tau + 3}$,
and sum with reference to ($\tau$), from $\tau = 0$ to
$\tau = \infty$, we find
$$\eqalignno{
\sum\nolimits_0^\infty \mathbin{.} w_{2\tau + 2, 2 \tau + 1}
         u^{2\tau + 1} w^{1 - 2\tau} \surd r^{2\tau + 3}
   \hskip -144pt \cr
   &= u^3 r^3 \{ w^2 r + 2 w_{2,2} \mathbin{.} u^2 r^2 \}^{-{1 \over 2}}
         \sum\nolimits^{(s)} [0]^{-s} [0]^{-(3 - s)}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \mathbin{.} db^s}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^s \cr
   &\mathrel{\phantom{=}} \mathord{}
    + u r \{ w^2 r + 2 w_{2,2} \mathbin{.} u^2 r^2 \}^{1 \over 2}
         \sum\nolimits^{(s)} {[0]^{-(s-2)} [0]^{-(3 - s)} \over 2}
         {\displaystyle d^3 \!\! \mathbin{.}
               \left(
                  \Phi
                + F \mathbin{.} {d \mathbin{.} \Phi \over da}
               \right)
            \over da^{3 - s} \mathbin{.} db^s}
         \left( {d^2 \Phi \over da \mathbin{.} db} \right)^{s - 2},\cr
   & &{\rm (S^{(6)})}\cr}$$
if we observe that
$\displaystyle w_{2,2}
   = {\textstyle {1 \over 2}}
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)$.
We might easily extend the principles of these summations, but it
is better to make use of the results to which we have already
arrived, for the solution of our second problem.

\bigbreak

(II.)
We proposed, first, to find general expressions for the polar
functions $u^{(t)}$, $w^{(t)}$, which enter as coefficients into
the developments (T${}'''$), and to examine what negative powers
they contain of the sine of the polar angle~$v$; and secondly, to
eliminate these negative powers, and so to transform the series
(T${}'''$) into others which shall contain none but positive
powers of any variable quantity.  The I${}^{\rm st.}$ of these
problems has been completely resolved by the discussions in which
we have just been engaged.  We have seen that the functions
$u^{(t)}$, $w^{(t)}$, are of the form
$$u^{(t)}
   = \sum\nolimits_{(t')}{}_0^t \mathbin{.} u_{t,t'}
         \mathbin{.} u^{t'} \mathbin{.} w^{t + 2 - 2t'};\quad
  w^{(t)}
   = \sum\nolimits_{(t')}{}_0^t \mathbin{.} w_{t,t'}
         \mathbin{.} u^{t'} \mathbin{.} w^{t + 1 - 2t'};
   \eqno {\rm (T^{(6)})}$$
$\sum\nolimits_{(t')}{}_0^t$ denoting a summation with reference
to $t'$ from $t' = 0$ to $t' = t$; $u_{t,t'}$, $w_{t,t'}$
constants, which we have given general formul{\ae} to determine;
and $u$, $w$, functions, which in the notation of
(L${}''''$)~[68.], have for expressions
$$u = {\epsilon \mathbin{.} \sin (v - v')
         \over P \mathbin{.} \cos v'},\quad
  w = \pm \surd (\epsilon \mathbin{.} \sin v),
   \eqno {\rm (U^{(6)})}$$
$v$ being the polar angle, and $\epsilon$, $P$, $v'$ constants
which enter into the equations of the curves (M${}''''$).
Substituting these values in the series (T${}'''$) which may be
thus written,
$$\left. \eqalign{
a  &= \sum\nolimits_{(\tau)}{}_0^\infty
         \mathbin{.} u^{(2\tau)} r^{\tau + 1}
      \pm \surd r \mathbin{.} \sum\nolimits_{(\tau)}{}_0^\infty
         \mathbin{.} u^{(2\tau + 1)} \mathbin{.} r^{\tau + 1},\cr
b  &= \sum\nolimits_{(\tau)}{}_0^\infty
         \mathbin{.} w^{(2\tau + 1)} r^{\tau + 1}
      \pm \surd r \mathbin{.} \sum\nolimits_{(\tau)}{}_0^\infty
         \mathbin{.} w^{(2\tau)} \mathbin{.} r^{\tau},\cr}
   \right\}
   \eqno {\rm (V^{(6)})}$$
and observing that as the negative powers of $w$ are all odd,
those of ($\sin v$) are all fractional, we find the
following transformed developments:
$$\left. \eqalign{
a   &= \sum\nolimits_{(\tau)}{}_0^\infty
         (\epsilon r \sin v)^{\tau + 1}
         \mathbin{.} \sum\nolimits_{(t')}{}_0^{\tau + 1}
         \mathbin{.} u_{2\tau, t'} \mathbin{.} P^{-t'}
         \left(
            {\sin (v - v')
               \over \cos v' \mathbin{.} \sin v}
         \right)^{t'} \cr
   &\mathrel{\phantom{=}} \mathord{}
    \pm \sum\nolimits_{(\tau)}{}_0^\infty
         (\epsilon r \sin v)^{\tau + {3 \over 2}}
         \mathbin{.} \sum\nolimits_{(t')}{}_0^{2\tau + 1}
         \mathbin{.} u_{2\tau + 1, t'} \mathbin{.} P^{-t'}
         \left(
            {\sin (v - v')
               \over \cos v' \mathbin{.} \sin v}
         \right)^{t'},\cr
b   &= \sum\nolimits_{(\tau)}{}_0^\infty
         (\epsilon r \sin v)^{\tau + 1}
         \mathbin{.} \sum\nolimits_{(t')}{}_0^{\tau + 1}
         \mathbin{.} w_{2\tau + 1, t'} \mathbin{.} P^{-t'}
         \left(
            {\sin (v - v')
               \over \cos v' \mathbin{.} \sin v}
         \right)^{t'} \cr
   &\mathrel{\phantom{=}} \mathord{}
    \pm \sum\nolimits_{(\tau)}{}_0^\infty
         (\epsilon r \sin v)^{\tau + {1 \over 2}}
         \mathbin{.} \sum\nolimits_{(t')}{}_0^{2\tau}
         \mathbin{.} w_{2\tau, t'} \mathbin{.} P^{-t'}
         \left(
            {\sin (v - v')
               \over \cos v' \mathbin{.} \sin v}
         \right)^{t'},\cr}
   \right\}
   \eqno {\rm (W^{(6)})}$$
which have the advantage of exhibiting to the eye, the manner
wherein the rectangular components $a$, $b$, of aberration at the
mirror, depend on the polar components $r$, $v$, of aberration at
the caustic surface.  To eliminate from these developments
(W${}^{(6)}$) the negative powers of ($\sin v$), without
introducing those of any other variable, or the positive powers
of any quantity which (like the $z$ of the preceding problem)
becomes infinite when the polar radius~$r$ assumes a particular
direction; let us resume the summations, expressed by the
equations (H${}^{(6)}$).  It results from those equations, or
from the formul{\ae} (C${}^{(5)}$) (F${}^{(6)}$), on which they
were founded, that if, in order to begin with the greatest
negative powers of ($\sin v$), we reject at first all but
the greatest values of $t'$ in the developments (W${}^{(6)}$),
namely $t' = 2\tau + 1$ in $a$, and $t' = 2\tau$ in $b$, and
denote by $a_1$, $b_1$, the sums of the terms that remain, we
shall have
$$a_1 = {d^2 F \over da \mathbin{.} db} \mathbin{.}
         {r \mathbin{.} \sin (v - v')
            \over P \mathbin{.} \cos v'}
         \mathbin{.} \epsilon^{3 \over 2} \scriptC^{1 \over 2},\quad
  b_1 = \epsilon^{1 \over 2} \scriptC^{1 \over 2};
   \eqno {\rm (X^{(6)})}$$
in which
$$\scriptC = r \mathbin{.} \sin v +
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)
         \left(
            {\surd \epsilon \mathbin{.} r \mathbin{.} \sin (v - v')
               \over P \mathbin{.} \cos v'}
         \right)^2,
   \eqno {\rm (Y^{(6)})}$$
$F$, $\Phi$ having the same meanings as in the foregoing problem.
To find the optical meaning of the binomial function
($\scriptC$), let us consider the points upon the plane of
aberration for which that function vanishes.  It is evident that
at these points ($\sin v$) is small; if then we change
$r \mathbin{.} \sin (v - v')$ to
$- r \mathbin{.} \sin v'$,
the condition $\scriptC = 0$ becomes by (Y${}^{(6)}$)
$$0 = \sin v +
         \left(
            {d^2 \Phi \over da^2}
          + \left( {d^2 \Phi \over da \mathbin{.} db} \right)^2
         \right)
         \mathbin{.}
         (\surd \epsilon \mathbin{.} P^{-1} \mathbin{.} \tan v')^2
         \mathbin{.} r,$$
that is, in the notation of paragraph [61.],
$$2 i^2 C \mathbin{.} \sin v = (AC - B^2) r,
   \eqno {\rm (Z^{(6)})}$$
which, by the same paragraph, is the equation of the osculating
circle to the section of the caustic surface; from which it
follows, that in this approximation, the function ($\scriptC$)
is, for any other point upon the plane of aberration, the distance
of that point from the osculating circle just mentioned, measured
in a direction parallel to the normal of the caustic surface.
More accurately, if we put
$$r \mathbin{.} \sin v
   = y'' \mathbin{.} \sin v',\quad
  r \mathbin{.} \sin (v - v')
   = x'' \mathbin{.} \sin v',
   \eqno {\rm (A^{(7)})}$$
$x''$, $y''$, will be the oblique coordinates of the point $r$,
$v$, referred to two axes in the plane of aberration, of which
one touches the caustic surface at the focus of the given ray,
while the other is inclined to this tangent at an angle $= v'$;
and the equation (Y${}^{(6)}$) will become
$$\scriptC = y'' \mathbin{.} \sin v'
      - \left( {AC - B^2 \over 2 i^2 C} \right)
         \mathbin{.} x''^2,
   \eqno {\rm (B^{(7)})}$$
which shews that ($\scriptC$) vanishes for the points of a
parabola, which has its diameter parallel to the axis of $y''$,
and has contact of the second order with the section of the
caustic surface; and that for any other point upon the plane of
aberration, ($\scriptC$) is equal to the distance from this
parabola measured in a direction parallel to its own diameter,
and then projected upon the normal.  If, therefore, in the
developments (W${}^{(6)}$), we change
$r \mathbin{.} \sin v$,
$r \mathbin{.} \sin (v - v')$,
to their values
$y'' \mathbin{.} \sin v'$,
$x'' \mathbin{.} \sin v'$
(A${}^{(7)}$); if we then eliminate
$y'' \mathbin{.} \sin v'$,
by changing it, in virtue of (B${}^{(7)}$), to the binomial
$$\scriptC + \left( {AC - B^2 \over 2 i^2 C} \right)
         \mathbin{.} x''^2,$$
and develope every fractional power of this binomial according to
the ascending powers of $x''$, and the descending powers of
$\scriptC$, we see that the new developments will contain no
negative powers of this latter variable, except those which arise
from the terms that we rejected in effecting the summations
(X${}^{(6)}$): and I am going to shew, that if in place of the
parabola $\scriptC = 0$, which has contact of the second order
with the section of the caustic surface, we take that section
itself, whose equation referred to the coordinates $(x'', y'')$
is of the form $\scriptC' = 0$, in which
$$\scriptC'
   = \scriptC
      - \sum\nolimits_{(\nu)}{}_0^\infty
         \mathbin{.} [0]^{-(\nu + 3)}
         \left( {d^{\nu + 3} \scriptC \over dx''^{\nu + 3}} \right)
         \mathbin{.} x''^{\nu + 3}
   = (y'' - y_0'') \sin v',
   \eqno {\rm (C^{(7)})}$$
$y_0''$ being the ordinate of the section, and $\scriptC'$ the
distance from that curve, measured in a direction parallel to the
axis of $y''$, and then projected on the normal; it is sufficient
to change the fractional powers of
$y'' \mathbin{.} \sin v'$
to those of
$\scriptC' + y_0'' \mathbin{.} \sin v'$,
in order to obtain development for $a$ and $b$, which shall
satisfy the condition of the question, containing no negative
powers of any variable quantity, but only positive and integer
powers and products of $x''$ and of $\surd \scriptC'$.

To demonstrate this theorem, let us resume the equations
(G${}^{(5)}$), putting them by (U${}^{(6)}$), (A${}^{(7)}$) under
the form
$$\epsilon P^{-1} \tan v' \mathbin{.} x''
   = a - F(a, b);\quad
  \epsilon \mathbin{.} \sin v' \mathbin{.} y''
   = b^2 - 2 \Phi(a, b).
   \eqno {\rm (D^{(7)})}$$
Conceive a parallel to the axis of $y''$, drawn through the point
$x''$, $y''$, upon the plane of aberration; this parallel will
meet the section of the caustic surface in a point having for
coordinates $x''$, $y_0''$, and the ray which has that point for
focus will cross the perpendicular plane at the mirror in another
point whose coordinates may be called $a_0$, $b_0$; to determine
these coefficients we have by (D${}^{(7)}$),
$$\epsilon P^{-1} \tan v' \mathbin{.} x''
   = a_0 - F_0;\quad
  \epsilon \mathbin{.} \sin v' \mathbin{.} y_0''
   = b_0^2 - 2 \Phi_0,
   \eqno {\rm (E^{(7)})}$$
$F_0$, $\Phi_0$, representing for abridgment the functions
$F(a_0, b_0)$, $\Phi(a_0, b_0)$; we have also, by the nature of
$y_0''$,
$${dx'' \over da_0} \mathbin{.} {dy_0'' \over db_0}
   = {dx'' \over db_0} \mathbin{.} {dy_0'' \over da_0},
   \quad \hbox{that is,}\quad
  \left( 1 - {dF_0 \over da_0} \right)
  \left( b_0 - {d\Phi_0 \over db_0} \right)
   = {dF_0 \over db_0} \mathbin{.} {d\Phi_0 \over da_0},
   \eqno {\rm (F^{(7)})}$$
from which it follows that the locus of the point $a_0$, $b_0$,
on the perpendicular plane at the mirror, has for tangent the
right line
$$b_0 = a_0 \mathbin{.} {d^2 \Phi \over da \mathbin{.} db}
   = a_0 \tan (v' + {\textstyle {1 \over 2}} \pi),
   \eqno {\rm (G^{(7)})}$$
and that we can develope $a_0$, $b_0$ in series of the form
$$\left. \eqalign{
a_0 &= \epsilon P^{-1} \tan v'' \mathbin{.} x''
      + {d^2 a_0 \over dx''^2} \mathbin{.} {x''^2 \over 2}
      + {d^3 a_0 \over dx''^3} \mathbin{.} {x''^3 \over 2 \mathbin{.} 3}
      + \hbox{\&c.},\cr
b_0 &= - \epsilon \mathbin{.} P^{-1} \mathbin{.} x''
      + {d^2 b_0 \over dx''^2} \mathbin{.} {x''^2 \over 2}
      + {d^3 b_0 \over dx''^3} \mathbin{.} {x''^3 \over 2 \mathbin{.} 3}
      + \hbox{\&c.} \cr}
         \right),
   \eqno {\rm (H^{(7)})}$$
This being laid down, let us subtract (E${}^{(7)}$) from
(D${}^{(7)}$); we find
$$\left. \eqalign{
a - a_0
   &= F - F_0 \cr
   &=    {dF_0 \over da_0} (a - a_0)
       + {dF_0 \over db_0} (b - b_0)
       + \sum \mathbin{.}
         {d^{m + m'} F_0 \over da_0^m \mathbin{.} db_0^{m'}}
         \mathbin{.} {(a - a_0)^m \over [m]^m}
         \mathbin{.} {(b - b_0)^{m'} \over [m']^{m'}},\cr
{\textstyle {1 \over 2}} (b^2 - b_0^2 - \epsilon \scriptC')
   &= \Phi - \Phi_0 \cr
   &=    {d\Phi_0 \over da_0} (a - a_0)
       + {d\Phi_0 \over db_0} (b - b_0)
       + \sum \mathbin{.}
         {d^{m + m'} \Phi_0 \over da_0^m \mathbin{.} db_0^{m'}}
         \mathbin{.} {(a - a_0)^m \over [m]^m}
         \mathbin{.} {(b - b_0)^{m'} \over [m']^{m'}},\cr}
   \right\}
   \eqno {\rm (I^{(7)})}$$
and therefore, by (F${}^{(7)}$),
$$\eqalignno{
(b - b_0)^2
   &= \epsilon \scriptC'
      + 2 \mathbin{.} \sum \mathbin{.}
         {d^{m + m'} \Phi_0 \over da_0^m \mathbin{.} db_0^{m'}}
         \mathbin{.} {(a - a_0)^m \over [m]^m}
         \mathbin{.} {(b - b_0)^{m'} \over [m']^{m'}} \cr
   &\mathrel{\phantom{=}} \mathord{}
      +  {\displaystyle 2 \mathbin{.} {d\Phi_0 \over da_0}
            \over \displaystyle 1 - {dF_0 \over da_0}}
         \mathbin{.} \sum \mathbin{.}
         {d^{m + m'} F_0 \over da_0^m \mathbin{.} db_0^{m'}}
         \mathbin{.} {(a - a_0)^m \over [m]^m}
         \mathbin{.} {(b - b_0)^{m'} \over [m']^{m'}},
   &{\rm (K^{(7)})}\cr}$$
in which $m + m' > 1$,
$$\left. \eqalign{
{d^{m + m'} F_0    \over da_0^m \mathbin{.} db_0^{m'}}
   &= \sum \mathbin{.}
         {d^{m + m' + m'' + m'''} F
            \over da^{m + m''} \mathbin{.} db^{m' + m'''}}
         \mathbin{.} {a_0^{m''} \over [m'']^{m''}}
         \mathbin{.} {b_0^{m'''} \over [m''']^{m'''}},\cr
{d^{m + m'} \Phi_0 \over da_0^m \mathbin{.} db_0^{m'}}
   &= \sum \mathbin{.}
         {d^{m + m' + m'' + m'''} \Phi
            \over da^{m + m''} \mathbin{.} db^{m' + m'''}}
         \mathbin{.} {a_0^{m''} \over [m'']^{m''}}
         \mathbin{.} {b_0^{m'''} \over [m''']^{m'''}},\cr}
   \right\}
   \eqno {\rm (L^{(7)})}$$
$a$, $b$, being supposed to vanish after the differentiations in
these second members: and it is easy to see that by means of
these equations we can develope $a - a_0$, $b - b_0$, and
therefore also $a$, $b$, according to the positive integer powers
and products of $x''$, $\surd \scriptC'$.  With respect to the
coefficients of these developments, they may be calculated by
differentiating the equations that we have just established; they
may also be deduced from the coefficients of the series
(T${}'''$), by relations which will be elsewhere indicated.

In the mean time let us remark, that instead of measuring the
distance from the caustic section in a direction parallel to the
axis of $y''$, we may measure it parallel to any other line upon
the plane of aberration.  If for instance, to simplify our
remaining calculations, we resume the rectangular coordinates
$x'$, $y'$, of which the former is a tangent, and the latter a
normal to the section; if from the point $(x', y')$ we draw a
line parallel to this normal, and denote by $x_0'$, $y_0'$, the
coordinates of the point where this line meets the section, and
by $\delta$ the intercepted portion, so that
$$x' - x_0' = 0,\quad y' - y_0' = \delta;
   \eqno {\rm (M^{(7)})}$$
if also we call $a_0$, $b_0$, the coordinates of the point in
which the ray that passes through $(x_0', y_0')$ is crossed by
the perpendicular plane at the mirror; we shall have the
equations
$$\left. \eqalign{
x' &= {dx' \over da} \mathbin{.} a
      + \sum \mathbin{.}
         {d^{m + m'} x' \over da^m \mathbin{.} db^{m'}}
         \mathbin{.} {a^m \over [m]^m}
         \mathbin{.} {b^{m'} \over [m']^{m'}},\cr
y' &= \phantom{{dy' \over da} \mathbin{.} a +} \,\,
        \sum \mathbin{.}
         {d^{m + m'} y' \over da^m \mathbin{.} db^{m'}}
         \mathbin{.} {a^m \over [m]^m}
         \mathbin{.} {b^{m'} \over [m']^{m'}},\cr}
   \right\}
   \eqno {\rm (S''') \quad [68.]}$$
$$\left. \eqalign{
x_0' &= {dx' \over da} \mathbin{.} a_0
      + \sum \mathbin{.}
         {d^{m + m'} x' \over da^m \mathbin{.} db^{m'}}
         \mathbin{.} {a_0^m \over [m]^m}
         \mathbin{.} {b_0^{m'} \over [m']^{m'}},\cr
y_0' &= \phantom{{dy' \over da} \mathbin{.} a_0 +} \,\,
        \sum \mathbin{.}
         {d^{m + m'} y' \over da^m \mathbin{.} db^{m'}}
         \mathbin{.} {a_0^m \over [m]^m}
         \mathbin{.} {b_0^{m'} \over [m']^{m'}},\cr}
   \right\}
   \eqno {\rm (N^{(7)})}$$
$$\left. \eqalign{
0  &=    {dx_0' \over da_0} \mathbin{.} (a - a_0)
       + {dx_0' \over db_0} \mathbin{.} (b - b_0)
       + \sum \mathbin{.}
         {d^{m + m'} x_0' \over da_0^m \mathbin{.} db_0^{m'}}
         \mathbin{.} {(a - a_0)^m \over [m]^m}
         \mathbin{.} {(b - b_0)^{m'} \over [m']^{m'}},\cr
\delta
   &=    {dy_0' \over da_0} \mathbin{.} (a - a_0)
       + {dy_0' \over db_0} \mathbin{.} (b - b_0)
       + \sum \mathbin{.}
         {d^{m + m'} y_0' \over da_0^m \mathbin{.} db_0^{m'}}
         \mathbin{.} {(a - a_0)^m \over [m]^m}
         \mathbin{.} {(b - b_0)^{m'} \over [m']^{m'}},\cr}
   \right\}
   \eqno {\rm (O^{(7)})}$$
$${dx_0' \over da_0} \mathbin{.} {dy_0' \over db_0}
   =  {dx_0' \over db_0} \mathbin{.} {dy_0' \over da_0}
   \eqno {\rm (P^{(7)})}$$
in which $m + m' > 1$; and by these equations we can change the
developments (T${}'''$) [68.]\ into series of the form
$$\left. \eqalign{
a  &= {da \over dx'} \mathbin{.} x'
      \pm {da \over d \surd \delta} \mathbin{.} \surd \delta
      +   {d^2 a \over dx'^2} \mathbin{.} {x'^2 \over 2}
      \pm {d^2 a \over dx' \mathbin{.} d \surd \delta}
            \mathbin{.} x' \surd \delta
    + \hbox{\&c.},\cr
b  &= {db \over dx'} \mathbin{.} x'
      \pm {db \over d \surd \delta} \mathbin{.} \surd \delta
      +   {d^2 b \over dx'^2} \mathbin{.} {x'^2 \over 2}
      \pm {d^2 b \over dx' \mathbin{.} d \surd \delta}
            \mathbin{.} x' \surd \delta
    + \hbox{\&c.},\cr}
   \right\}
   \eqno {\rm (Q^{(7)})}$$
which contain no negative powers of any variable quantity, and
which we are going to apply to the solution of the succeeding
problems.

\bigbreak

(III.)
We must be more brief in the discussion of these remaining
problems, namely to transform the integral expressions of the
preceding paragraph, and to effect the double integrations within
the limits of the present.  Applying to the series (Q${}^{(7)}$)
the geometrical and optical reasonings of [68.], we find for the
quantities $\Delta^{(\alpha)}$, $S^{(\mu)}$, $Q^{(s)}$, which
were there represented by the developments (D${}''''$),
(E${}''''$), (F${}''''$), the following transformed expressions:
$$\Delta^{(\alpha)} = {D \over \surd \delta};\quad
   S^{(\mu)} = \int \!\!\! \int
         {S \mathbin{.} dx' \mathbin{.} dy' \over \surd \delta};\quad
   Q^{(s)} = \int \!\!\! \int
         {Q \mathbin{.} dx' \mathbin{.} dy' \over \surd \delta};
   \eqno {\rm (R^{(7)})}$$
in which $\delta$ is, as in (M${}^{(7)}$), the distance of any
assigned point $x'$, $y'$ upon the plane of aberration from the
section of the caustic surface, measured in a direction parallel
to the normal of that curve; and $D$, $S$, $Q$, are rational and
integer functions of $x'$ and $\delta$, or of $x'$ and $y'$,
which when those variables vanish, that is, at the focus of the
given ray, reduce themselves to the following values:
$$D = {\Delta^{(\mu)} \mathbin{.} \rho_1 \mathbin{.} \rho_2
         \over i \sqrt{ {1 \over 2} C }};\quad
  S = {\rho_1 \mathbin{.} \rho_2
         \over i \sqrt{ {1 \over 2} C }};\quad
  Q = {\Delta^{(\mu)} \mathbin{.} \rho_1 \mathbin{.} \rho_2
         \over i \sqrt{ {1 \over 2} C }};
   \eqno {\rm (S^{(7)})}$$
$\Delta^{(\mu)}$, $\rho_1$, $\rho_2$, $i$, $c$, having the same
meanings as in [68.].  If then we integrate the two last of these
expressions (R${}^{(7)}$) within the double limits afforded by
the equations
$$\delta = 0,\quad x'^2 + y'^2 = r^2,
   \eqno {\rm (T^{(7)})}$$
of which the former represents the caustic section, and the
latter the circular circumference, we shall have the required
expressions for the quantities that we denoted by $S^{(r)}$,
$Q^{(r)}$, at the beginning of the present paragraph.

\bigbreak

(IV.)
To effect these double integrations, let us put the functions
$S$, $Q$, under the form
$$S = \sum \mathbin{.} S_{m,m'}
         \mathbin{.} x'^m \mathbin{.} \delta^{m'},\quad
  Q = \sum \mathbin{.} Q_{m,m'}
         \mathbin{.} x'^m \mathbin{.} \delta^{m'},
   \eqno {\rm (U^{(7)})}$$
and let us change the differential product
$dx' \mathbin{.} dy'$ to $dx' \mathbin{.} d\delta$, which is
permitted, because in forming this product $y'$ varies
independently of $x'$.  In this manner the expressions
(R${}^{(7)}$) become
$$\int 2 \, d\surd \delta \, \int S \, dx',\quad
  \int 2 \, d\surd \delta \, \int Q \, dx',$$
in which
$$\left. \eqalign{
\int S \, dx'
   &= \sum \mathbin{.} S_{m,m'} \mathbin{.}
         {x_2'^{m+1} - x_1'^{m+1} \over m + 1}
         \mathbin{.} \delta^{m'},\cr
\int Q \, dx'
   &= \sum \mathbin{.} Q_{m,m'} \mathbin{.}
         {x_2'^{m+1} - x_1'^{m+1} \over m + 1}
         \mathbin{.} \delta^{m'},\cr}
   \right\}
   \eqno {\rm (V^{(7)})}$$
$x_1'$, $x_2'$, being the extreme values of $x'$, corresponding
to any given value of $\delta$, that is, the absciss{\ae} of the
points where the little circumference is crossed by any given
parallel to the section of the caustic surface. To determine
these values, we have the equation
$$x'^2 + (y_0' + \delta)^2 = r^2,
   \eqno {\rm (W^{(7)})}$$
$y_0'$ being the ordinate of the section, and $r$ the radius of
the circle: and putting this equation under the form
$$x'^2 + y_0'^2 = \delta'^2 - 2 y_0' \delta,
   \quad \hbox{in which}\quad
   \delta' = \pm \sqrt{r^2 - \delta^2},
   \eqno {\rm (X^{(7)})}$$
we can, by Laplace's theorem, develope $x'^{m+1}$ according
to the positive integer powers of $\delta$, $\delta'$, the term
of least dimension being $\delta'^{m+1}$; from which it follows
that the integrals (V${}^{(7)}$) may be put under the form
$$\left. \eqalign{
\int S \, dx'
   &= 2 \delta' S'
    = 2 \sum \mathbin{.} S'_{n,n'}
         \mathbin{.} \delta^n \delta'^{2n' + 1},\cr
\int Q \, dx'
   &= 2 \delta' Q'
    = 2 \sum \mathbin{.} Q'_{n,n'}
         \mathbin{.} \delta^n \delta'^{2n' + 1},\cr}
   \right\}
   \eqno {\rm (Y^{(7)})}$$
in which
$$S'_{0,0} = S_{0,0},\quad
  Q'_{0,0} = Q_{0,0},
   \eqno {\rm (Z^{(7)})}$$
$S_{0,0}$, $Q_{0,0}$, being the values of $S$, $Q$, assigned by
the formul{\ae} (S${}^{(7)}$).  Multiplying (Y${}^{(7)}$) by
$2d \surd \delta$, integrating with reference to $\delta$ from
$\delta = 0$ to $\delta = r$, and putting for abridgment
$$I_{n,n'} = \int_0^1
         \mathbin{.} z^n (1 - z^2)^{n' + {1 \over 2}}
         d \surd z,
   \eqno {\rm (A^{(8)})}$$
we find finally
$$\left. \eqalign{
S^{(r)} &= 4 r^{3 \over 2} \mathbin{.} \sum \mathbin{.}
            S'_{n,n'} \mathbin{.} I_{n,n'} \mathbin{.} r^{n + 2n'},\cr
Q^{(r)} &= 4 r^{3 \over 2} \mathbin{.} \sum \mathbin{.}
            Q'_{n,n'} \mathbin{.} I_{n,n'} \mathbin{.} r^{n + 2n'},\cr}
   \right\}
   \eqno {\rm (B^{(8)})}$$
and therefore when we suppose $r$ infinitely small,
$$\left. \eqalign{
S^{(dr)} &= 4 S'_{0,0} \mathbin{.} I_{0,0} \mathbin{.} (dr)^{3 \over 2}
    = {4 \rho_1 \mathbin{.} \rho_2 \mathbin{.} (dr)^{3 \over 2}
         \over i \sqrt{ {1 \over 2} C }}
         \int_0^1 \mathbin{.} \sqrt{1 - z^2} \mathbin{.} d \surd z,\cr
Q^{(dr)} &= 4 Q'_{0,0} \mathbin{.} I_{0,0} \mathbin{.} (dr)^{3 \over 2}
    = {4 \Delta^{(\mu)} \rho_1 \mathbin{.} \rho_2 \mathbin{.} (dr)^{3 \over 2}
         \over i \sqrt{ {1 \over 2} C }}
         \int_0^1 \mathbin{.} \sqrt{1 - z^2} \mathbin{.} d \surd z,\cr}
   \right\}
   \eqno {\rm (C^{(8)})}$$
values which satisfy the relation (R${}''''$) at the beginning of
the present paragraph; and which shew, by the principles there
laid down, that the density at the caustic surface is
proportional to the following expression:
$$\Delta^{(\chi)}
   = {\Delta^{(\mu)} \mathbin{.} \rho_1 \mathbin{.} \rho_2
      \over i \surd C};
   \eqno {\rm (D^{(8)})}$$
that is, in passing from one point to another upon such a
surface, or from a point upon one caustic surface to a point upon
the other, the density of the reflected light varies directly as
the density at the mirror multiplied by the product of the two
focal distances, and inversely as the difference of these
distances multiplied by the square root of the radius of
curvature of the caustic curve.  We see also that the definite
integral (Q${}''''$), which represents the area of the
heartshaped curve that we considered in [68.], is equal to the
first term of the development (B${}^{(8)}$) for $S^{(r)}$,
$$\Pi = 4 \epsilon^{3 \over 2} r^{3 \over 2} P^{-1}
         \mathbin{.} \tan v' \mathbin{.}
         \int_0^1 \mathbin{.} \surd (1 - z^2)
         \mathbin{.} d \surd z
   = 4 S'_{0,0} \mathbin{.} I_{0,0} \mathbin{.} r^{3 \over 2};
   \eqno {\rm (E^{(8)})}$$
on which account we may call that curve a {\it pycnoid}, because
if $r$ be given, its area is proportional to the density at the
caustic surface divided by the density at the mirror.

\bigbreak

[70.]
The expression that we have just found for the density at a
caustic surface, becomes infinite in two cases, which require to
be considered separately; namely, first when $i = 0$, that is, at
the intersection of the two caustic surfaces, which, as I have
shewn, reduces itself to a finite number of isolated points, the
principal foci of the system; and secondly, when $C = 0$, that
is, when the radius of curvature of the caustic curve vanishes.
A point at which this latter circumstance takes place, is in
general a cusp upon the caustic curve; and the locus of these
points forms in general a curve consisting of two branches, each
of which is a sharp edge on one of the two caustic surfaces.
These cusps are also connected by remarkable relations, with the
pencils to which the caustic curves belong; on which account we
shall reserve the investigation of the density at such a cusp,
until we come to treat more fully of the developable pencils of
the system.

\bigbreak

[71.]
Let us then consider the points where the interval ($i$)
vanishes, that is, let us investigate an expression for the
density at a principal focus.  In this case we have by the
XIIth.\ section, the following approximate formul{\ae}:
$$\left. \multieqalign{
x &= A \alpha^2 + 2 B \alpha \beta + C \beta^2, &
a &= - \rho \alpha,\cr
y &= B \alpha^2 + 2 C \alpha \beta + D \beta^2, &
b &= - \rho \beta,\cr}
   \right\}
   \eqno {\rm (F^{(8)})}$$
$(x, y)$ being the coordinates of the point in which the near ray
intersects the plane of aberration; $(a, b)$ the coordinates of
the point in which it intersects the perpendicular plane at the
mirror; ($\rho$) the focal length or interval between these two
planes; $(\alpha, \beta)$ the cosines of the angles which the
near ray makes with the axes of ($x$) and ($y$), the given ray
being the axis of ($z$); and $(A, B, C, D)$ coefficients
calculated in [62.], which have not the same meanings here, as in
the four preceding paragraphs.  These formul{\ae} give, by
elimination of $\alpha$, the following biquadratic equation,
$$F'' \mathbin{.} \beta^4
      - 2 \beta^2 \mathbin{.}
         \{ 2 (B^2 - AC) (By - Cx)
            + (AD  - BC) (Ay - Bx) \}
      + (Ay - Bx)^2
   = 0,
   \eqno {\rm (G^{(8)})}$$
in which
$F'' = (AD - BC)^2 - 4 (B^2 - AC) (C^2 - BD)$;
when $F''$ is negative, that is, when the principal focus is
inside the little ellipses of aberration, [62.], this biquadratic
(G${}^{(8)}$) has two of its roots real, and the other two
imaginary; but when $F''$ is positive, that is, when the
principal focus is outside those ellipses, then the roots are
either all real, or all imaginary; so that in the first case, any
given point $(x, y)$, near the focus, will have two rays passing
through it; whereas, in the second case, it will either have four
such rays or none.  As these two cases are thus essentially
distinct, it will be convenient to consider them separately; let
us therefore begin by investigating the density in the case where
the principal focus is inside the little ellipses of aberration.

\bigbreak

\centerline{Ist. {\sc Case}.  $F'' < 0$.}

\nobreak\bigskip

[72.]
In this case, if we consider any rectangle upon the plane of
aberration, having for its four corners,
$$\hbox{1st.}\enspace x,\enspace y;\quad
  \hbox{2d.} \enspace x + dx,\enspace y;\quad
  \hbox{3d.} \enspace x,\enspace y + dy;\quad
  \hbox{4th.}\enspace x + dx,\enspace y + dy;$$
the rays that pass inside this little rectangle are diffused over
two little parallelograms on the perpendicular plane at the
mirror, the corners of the one being
$$\hbox{1st.}\enspace a,\enspace b;\quad
  \hbox{2d.} \enspace a + {da \over dx} \mathbin{.} dx,\enspace
                       b + {db \over dx} \mathbin{.} dx,$$
$$\hbox{3d.} \enspace a + {da \over dy} \mathbin{.} dy,\enspace
                       b + {db \over dy} \mathbin{.} dy,\quad
  \hbox{4th.}\enspace a + {da \over dx} \mathbin{.} dx
                         + {da \over dy} \mathbin{.} dy,\enspace
                       b + {db \over dx} \mathbin{.} dx
                         + {db \over dy} \mathbin{.} dy,$$
and those of the other being composed in a similar manner of
$a'$, $b'$; $a$, $b$, $a'$, $b'$ being the two points in which
the two rays that pass through the point $(x,y)$ are crossed by
the perpendicular plane at the mirror.  The areas of these little
parallelograms, have for expressions
$$ \left(
      {da \over dy} \mathbin{.} {db \over dx}
    - {da \over dx} \mathbin{.} {db \over dy}
   \right)
   \mathbin{.} dx \mathbin{.} dy,\quad
   \left(
      {da' \over dy} \mathbin{.} {db' \over dx}
    - {da' \over dx} \mathbin{.} {db' \over dy}
   \right)
   \mathbin{.} dx \mathbin{.} dy,$$
and they are equal to one another, because $b' = -b$, $a' = -a$;
also the area of the little retangle on the plane of aberration
is
$dx \mathbin{.} dy$;
if then we denote by $\Delta^{(\mu)}$ the density at the mirror,
the density at the point $(x,y)$ will be nearly
$$\Delta^{(\alpha)}
   = 2 \Delta^{(\mu)}
      \left(
         {da \over dy} \mathbin{.} {db \over dx}
       - {da \over dx} \mathbin{.} {db \over dy}
      \right)
   \eqno {\rm (H^{(8)})}$$
and it remains to calculate the coefficient in the second member.
For this purpose, I observe that in general, when any four
quantities $a$, $b$, $x$, $y$, are connected by two relations, so
that $a$, $b$, are functions of $x$, $y$, and reciprocally, their
partial differentials are connected by the following relation,
$$    \left(
         {da \over dy} \mathbin{.} {db \over dx}
       - {da \over dx} \mathbin{.} {db \over dy}
      \right)
      \left(
         {dy \over da} \mathbin{.} {dx \over db}
       - {dy \over db} \mathbin{.} {dx \over da}
      \right)
   = 1;
   \eqno {\rm (I^{(8)})}$$
it is sufficient therefore to calculate
$\displaystyle
      {dy \over da} \mathbin{.} {dx \over db}
    - {dy \over db} \mathbin{.} {dx \over da}$.
Now, the equations (F${}^{(8)}$) give
$$\eqalign{
{\textstyle {1 \over 2}} \rho^2 \mathbin{.} dx
   &= (Aa + Bb) \mathbin{.} da + (Ba + Cb) \mathbin{.} db,\cr
{\textstyle {1 \over 2}} \rho^2 \mathbin{.} dy
   &= (Ba + Cb) \mathbin{.} da + (Ca + Db) \mathbin{.} db,\cr}$$
$$\eqalign{
{\textstyle {1 \over 4}} \mathbin{.} \rho^4 \mathbin{.}
         \left(
            {dy \over da} \mathbin{.} {dx \over db}
          - {dy \over db} \mathbin{.} {dx \over da}
         \right)
   &= (Ba + Cb)^2 - (Aa + Bb) (Ca + Db) \cr
   &= \rho^2 \mathbin{.}
         \{ (B \alpha + C \beta)^2
            - (A \alpha + B \beta) (C \alpha + D \beta) \};\cr}$$
and if we put
$(B \alpha + C \beta)^2
            - (A \alpha + B \beta) (C \alpha + D \beta)
   = M$,
we have by the same equations
$$\left. \eqalign{
M \mathbin{.} \alpha
   &= (B \alpha + C \beta) \mathbin{.} y
      - (C \alpha + D \beta) \mathbin{.} x,\cr
M \mathbin{.} \beta
   &= (B \alpha + C \beta) \mathbin{.} x
      - (A \alpha + B \beta) \mathbin{.} y;\cr}
   \right\}
   \eqno {\rm (K^{(8)})}$$
we have therefore
$$\Delta^{(\alpha)} = {\Delta^{(\mu)} \mathbin{.} \rho^2 \over 2 M},\quad
  M = \surd \{ (By - Cx)^2 + (Ay - Bx) (Dx - Cy) \}.
   \eqno {\rm (L^{(8)})}$$

It results from this expression, that when the principal focus is
inside the little ellipses of aberration considered in [62.], there
exists another remarkable series of ellipses upon the plane of
aberration, determined by the equation
$$M = \hbox{const.}
   \eqno {\rm (M^{(8)})}$$
and possessing this characteristic property that along every such
ellipse the density of the reflected light is constant. The
ellipses of this series (M${}^{(8)}$) are all concentric and
similar, having their common centre at the principal focus, and
having their axes situated on two remarkable lines, which are
perpendicular to each other and to the given ray, and form with
that ray {\it three natural axes\/} of coordinates passing
through the principal focus.

\bigbreak

[73.]
Suppose then that we have taken for our axes of coordinates, the
three natural axes just mentioned, the ray from which the
aberrations are to be measured being still the axis of $z$, we
shall have the relation
$$AD - BC = 0,
   \eqno {\rm (N^{(8)})}$$
and the expression for the density at a point $(r, v)$ upon the
plane of aberration will become
$$\Delta^{(\alpha)}
   + {{\textstyle {1 \over 2}} \Delta^{(\mu)}
         \mathbin{.} \rho^2 \mathbin{.} r^{-1}
      \over \surd \{
            (B^2 - AC) \mathbin{.} \sin^2 v
          + (C^2 - BD) \mathbin{.} \cos^2 v \}},
   \eqno {\rm (O^{(8)})}$$
in which $B^2 - AC$, $C^2 - BD$, will both be positive, and
proportional to the squares of the semiaxes of the ellipses of
uniform density.  Multiplying this expression by
$r \mathbin{.} dr \mathbin{.} dv$,
and integrating from $r = 0$ to $r = r$, and from $v = 0$ to
$v = 2\pi$, we find for the whole number of the near reflected
rays that pass within a small given distance ($r$) from the
focus, the following approximate formula:
$$\int \!\!\! \int \Delta^{(\alpha)} \mathbin{.} r \, dr \, dv
    = 2 \Delta^{(\mu)} \mathbin{.} \rho^2 \mathbin{.} r
      \int_0^{\pi \over 2} {dv
         \over \surd \{
               (B^2 - AC) \mathbin{.} \sin^2 v
             + (C^2 - BD) \mathbin{.} \cos^2 v \}},
   \eqno {\rm (P^{(8)})}$$
a transcendental of known form, which can be calculated either by
elliptic arcs or by series.  And if we denote this transcendental
by $T$, and reason as in [69.], we find the following expression
for the density at the focus itself, as compared with the density
at another point of the same kind,
$$\Delta^{(\phi)} = \Delta^{(\mu)} \mathbin{.} \rho^2 \mathbin{.} T.
   \eqno {\rm (Q^{(8)})}$$

\bigbreak

\centerline{IId. {\sc Case}.  $F'' > 0$.}

\nobreak\bigskip

[74.]
Let us now consider the case where $F'' > 0$, that is, where the
principal focus is outside the little ellipses, [62.].  In this case
the points in which the near reflected rays intersect the plane
of aberration, are all comprised within the angle formed by the
two limiting lines (Y${}''$), [62.], namely, the common tangents to
those ellipses of aberration; and if we take the bisector of this
angle for the axis of $x$, the relation (N${}^{(8)}$) will
reappear, and the equation of the limiting lines will become
$${y^2 \over x^2} = {C^2 - BD \over AC - B^2}
   = {C \over A} = {D \over B}.
   \eqno {\rm (R^{(8)})}$$
Moreover, the rays that pass inside any little rectangle
$dx \mathbin{.} dy$,
within the angle formed by these limiting lines, are at the
mirror diffused nearly perpendicularly over four little
parallelograms, which are equal to one another, and have their
sum
$\displaystyle
   = {\rho^2 \mathbin{.} dx \mathbin{.} dy \over M}$,
$M$ being the same function as in [72.], we have, therefore, for
the density at the point $x$, $y$, the following approximate
expression,
$$\Delta_\prime^{(\alpha)} = {\Delta^{(\mu)} \mathbin{.} \rho^2 \over M},
   \eqno {\rm (S^{(8)})}$$
and the {\it lines of uniform density\/} are still given by the
equation
$$M = \hbox{const.},$$
which now represents a series of concentric and similar
hyperbolas, having the principal focus for their common centre,
and the limiting lines of aberration for their common asymptotes.
And if we multiply this expression for the density
$\Delta_\prime^{(\alpha)}$ by
$r \, dr \, dv$,
and integrate from $r = 0$ to $r = r$, and from $v = - v'$, to
$v = + v'$, $v'$ being the semiangle between the limiting lines,
and consequently
$$\tan v'
   = \sqrt{ \vphantom{C \over A} } {C \over A}
   = \sqrt{ \vphantom{D \over B} } {D \over B};
   \eqno {\rm (T^{(8)})}$$
we find for the whole number of the near rays that pass within a
given distance~$r$ from the focus
$$\int \!\!\! \int \Delta_\prime^{(\alpha)}
         \mathbin{.} r \, dr \, dv
   = 2 \Delta^{(\mu)} \mathbin{.} \rho^2 \mathbin{.} rT_\prime,
   \eqno {\rm (U^{(8)})}$$
and, therefore, for the density at the focus itself as compared
with that at another point of the same kind,
$$\Delta^{(\phi)} = \Delta^{(\mu)} \mathbin{.} \rho^2 T_\prime,
   \eqno {\rm (V^{(8)})}$$
$T_\prime$ denoting the transcendental
$$T_\prime = \int_0^{v'} {dv
         \over \surd \{
               (C^2 - BD) \mathbin{.} \cos^2 v
             - (AC - B^2) \mathbin{.} \sin^2 v \}}.
   \eqno {\rm (W^{(8)})}$$

\bigbreak

[75.]
The preceding expressions may be put under other forms, some of
which are simpler.  Thus, if we still suppose the axes of
coordinates to coincide with the natural axes determined by the
equation (N${}^{(8)}$), so that the axis of the reflected system
may be the axis of $z$, and the common transverse axis of the
lines of uniform density the axis of $x$; if also we denote by
$\displaystyle {\Delta' \over x}$
the density of a point upon this latter axis, and by
$\displaystyle {\Delta'' \over y}$
or
$\displaystyle {\Delta'' \surd -1 \over y}$
the density of a point upon the axis of $y$; we shall have
(O${}^{(8)}$) (S${}^{(8)}$) the following approximate expressions
for the density at any other point upon the plane of aberration,
$$    \left(
         {x^2 \over \Delta'^2} \pm {y^2 \over \Delta''^2}
      \right)^{-{1 \over 2}}
   = \Delta' r^{-1} (1 - e^2 \mathbin{.} \sin^2 v)^{-{1 \over 2}},
   \eqno {\rm (X^{(8)})}$$
$r$, $v$ being the polar coordinates of the point, and $e$ the
excentricity of the ellipses or hyperbolas at which the density
is constant; and the formul{\ae} (Q${}^{(8)}$) (V${}^{(8)}$) for
the density at the principal focus become
$$\left. \eqalign{
\Delta^{(\phi)}
   = \Delta' \mathbin{.}
   \int_0^{\pi \over 2} {dv
         \over (1 - e^2 \mathbin{.} \sin^2 v)^{1 \over 2}},\cr
\Delta_\prime^{(\phi)}
   = \Delta' \mathbin{.}
   \int_0^{\pi \over 2} {dv
         \over (1 - e^2 \mathbin{.} \sin^2 v)^{1 \over 2}},\cr}
   \right\}
   \eqno {\rm (Y^{(8)})}$$
$e$ being less than unity in the first and greater in the second.
With respect to the value of this excentricity, it is equal to the
cosecant of the imaginary or real angle $v'$ determined by the
formula (T${}^{(8)}$); it is also connected with the position of
the ellipses of aberration, [62.], by this remarkable relation,
that the segments into which the principal focus divides that
diameter of such an ellipse upon which it is situated, are
proportional to the squares of the semiaxes of the lines of
uniform density; in such a manner, that when the principal focus
is situated at the centre of the ellipses of aberration, the
excentricity $e$ vanishes, and the lines of uniform density
become a series of concentric circles; and when on the contrary,
the principal focus is on the circumference of the ellipses of
aberration, then $e$ becomes equal to unity, and the lines of
uniform density become a set of rectilinear parallels to the axis
of $y$, which axis in this case coincides with the common tangent
to the little ellipses of aberration, drawn through the principal
focus.  When the latter circumstance happens, the two expressions
(Y${}^{(8)}$) for the density at this principal focus, coincide
with one another, and become
$$\Delta' \mathbin{.} \int_0^{\pi \over 2} {dv \over \cos v}
   = \infty;
   \eqno {\rm (Z^{(8)})}$$
in this case therefore, we should be obliged to have recourse to
new calculations, and to introduce the consideration of
aberrations of the third order.  We may remark that the quantity
$F''$, the sign of which distinguishes between the two chief
cases of aberration from a principal focus, becomes $= 0$, in the
case which we have just been considering; and since, by
Section~XII., the sign of this quantity $F''$ determines also the
nature of the roots of the cubic equation
$${d^3 V \over dx^3} \mathbin{.} dx^3
      + 3 \mathbin{.} {d^3 V \over dx^2 \mathbin{.} dy}
         \mathbin{.} dx^2 \mathbin{.} dy
      + 3 \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2}
         \mathbin{.} dx \mathbin{.} dy^2
      + {d^3 V \over dy^3} \mathbin{.} dy^3
   = 0,$$
which by the same section assigns the directions of spheric
inflexion upon the surfaces of constant action, and of focal
inflexion on the osculating focal mirror; it follows that, in the
present case this cubic equation has two of its roots equal, and
therefore that two of the directions of focal or of spheric
inflexion coincide.  With respect to the value of these equal
roots, we have from our present choice of the coordinate planes
the equations $A = 0$, $B = 0$, and therefore by [62.],
$\displaystyle {d^3 V \over dx^3} = 0$,
$\displaystyle {d^3 V \over dx^2 \mathbin{.} dy} = 0$:
thus the cubic equation becomes
$$3 \mathbin{.} {d^3 V \over dx \mathbin{.} dy^2}
         \mathbin{.} dx \mathbin{.} dy^2
      + {d^3 V \over dy^3} \mathbin{.} dy^3
   = 0,$$
from which it follows that the two directions of inflexion which
coincide with one another are contained in the plane of $xz$,
that is, in a plane passing through the axis of the reflected
system, and cutting perpendicularly the lines of uniform density.

\bigbreak

[76.]
Many other remarks remain to be made, in order to illustrate and
complete the theory of the present section; but as we shall have
occasion, in treating of refracted systems, to resume this theory
under a more general point of view, we shall only here add, that
the function which we have called the {\it density}, may differ
sensibly in many instances from the observed intensity of light;
because in calculating this function, we have abstracted from all
physical causes not included in that fundamental law of
catoptrics, which is expressed by our original equation,
$$\cos \rho l + \cos \rho' l
   = 2 \cos I \mathbin{.} \cos nl,
   \eqno {\rm (A) \quad [1.]:}$$
or in the resulting formula
$$\alpha \, dx + \beta \, dy + \gamma \, dz = dV,$$
$\alpha$, $\beta$, $\gamma$ being the cosines of the angles which
the ray passing through the point $x$~$y$~$z$ makes with the
axes of coordinates, and $V$ the characteristic function.  To
distinguish therefore that mathematical affection of the system
to which the preceding calculations relate, from the physical
intensity of which it is an element, we may give to it a separate
name, and call it the {\it Geometrical Density}.

\vfill\eject

\centerline{CONCLUSION TO THE FIRST PART.}

\nobreak\bigskip

The preceding pages contain the execution of the first part of
our plan; being an attempt to establish general principles
respecting the systems of rays produced by the ordinary reflexion
of light, at any mirror or combination of mirrors, shaped and
placed in any manner whatsoever; and to shew that the
mathematical properties of such a system may all be deduced by
analytic methods from the form of {\sc one characteristic
function}:  as, in the application of analysis to geometry, the
properties of a plane curve, or of a curve surface, may all be
deduced by uniform methods from the form of the function which
characterises its equation.  It remains to extend these
principles to other optical systems; to shew that in every such
system, whether the rays be straight or curved, whether ordinary
or extraordinary, there exists a Characteristic Function
analogous to that which we have already pointed out for the case
of the systems produced by the ordinary reflexion of light; to
simplify and generalise the methods that we have given, for
calculating from the form of this function all the other
properties of the system; to integrate various equations which
present themselves in the determination of mirrors, lenses, and
crystals satisfying assigned conditions; to establish some more
general principles in the theory of {\sc Systems of Rays}, and to
terminate with a brief review of our own results, and of the
discoveries of former writers.  But we have trespassed too long
at present on the indulgence of mathematicians, and of the
Academy, and must defer to another occasion the completion of
this extensive design.

\line{\hfil W. R. HAMILTON.}

\nobreak\bigskip

{\it Observatory},

{\it April\/} 1828.

\bye
