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% Trinity College, 2000.

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\centerline{\Largebf ON QUATERNIONS, OR ON A NEW SYSTEM OF}

\vskip12pt

\centerline{\Largebf IMAGINARIES IN ALGEBRA}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (Philosophical Magazine, (1844--1850))}

\vskip36pt

\vfill

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

\vskip 12pt

\centerline{\largerm 2000}

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\null\vskip36pt

\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

The paper {\it On Quaternions; or on a new System of Imaginaries
in Algebra}, by Sir William Rowan Hamilton,
appeared in 18 instalments in volumes xxv--xxxvi of
{\it The London, Edinburgh and Dublin Philosophical Magazine
and Journal of Science} (3rd Series), for the years 1844--1850.
Each instalment (including the last) ended with the words
`{\it To be continued}'.

The articles of this paper appeared as follows:

\bigskip

\hskip\parindent\hbox{\vbox{\halign{#\hfil &&\quad #\hfil\cr
   articles 1--5   &July      1844 &vol.~xxv    (1844), pp.\ 10--13,\cr
   articles 6--11  &October   1844 &vol.~xxv    (1844), pp.\ 241--246,\cr
   articles 12--17 &March     1845 &vol.~xxvi   (1845), pp.\ 220--224,\cr
   articles 18--21 &July      1846 &vol.~xxix   (1846), pp.\ 26--31,\cr
   articles 22--27 &August    1846 &vol.~xxix   (1846), pp.\ 113--122,\cr
   article  28     &October   1846 &vol.~xxix   (1846), pp.\ 326--328,\cr
   articles 29--32 &June      1847 &vol.~xxx    (1847), pp.\ 458--461,\cr
   articles 33--36 &September 1847 &vol.~xxxi   (1847), pp.\ 214--219,\cr
   articles 37--50 &October   1847 &vol.~xxxi   (1847), pp.\ 278--283,\cr
   articles 51--55 &Supplementary
                              1847 &vol.~xxxi   (1847), pp.\ 511--519,\cr
   articles 56--61 &May       1848 &vol.~xxxii  (1848), pp.\ 367--374,\cr
   articles 62--64 &July      1848 &vol.~xxxiii (1848), pp.\ 58--60,\cr
   articles 65--67 &April     1849 &vol.~xxxiv  (1849), pp.\ 295--297,\cr
   articles 68--70 &May       1849 &vol.~xxxiv  (1849), pp.\ 340--343,\cr
   articles 71--81 &June      1849 &vol.~xxxiv  (1849), pp.\ 425--439,\cr
   articles 82--85 &August    1849 &vol.~xxxv   (1849), pp.\ 133--137,\cr
   articles 86--87 &September 1849 &vol.~xxxv   (1849), pp.\ 200--204,\cr
   articles 88--90 &April     1850 &vol.~xxxvi  (1850), pp.\ 305--306.\cr}}}

\bigbreak

(Articles 51--55 appeared in the supplementary number of the
{\it Philosophical Magazine} which appeared at the end of 1847.)

\bigbreak

Various errata noted by Hamilton have been corrected.  In addition,
the following corrections have been made:---

\smallbreak
 
\item{}
in the footnote to article~19, the title of the paper
by Cayley in the {\it Philosophical Magazine}
in February 1845 was wrongly given as ``On Certain
Results respecting Quaternions'', but has been corrected
to ``On Certain Results related to Quaternions'' (see
the {\it Philosophical Magazine\/} (3rd series) vol.~{\sc xxvi}
(1845), pp.\ 141--145);

\smallskip

\item{}
a sentence has been added to the footnote to article~70,
giving the date of the meeting of the Royal Irish
Academy at which the relevant communication in fact
took place (see the {\it Proceedings of the Royal Irish
Academy}, vol.~{\sc iv} (1850), pp.~324--325).

\bigskip

In this edition, `small capitals' (${\sc a}$, ${\sc b}$, ${\sc c}$,
etc.) have been used throughout to denote points of space.
(This is the practice in most of the original text in the
{\it Philosophical Magazine}, but some of the early articles of
this paper used normal-size roman capitals to denote points of
space.)

The spelling `co-ordinates' has been used throughout.  (The
hyphen was present in most instances of this word in the
original text, but was absent in articles 12, 15 and 16.)

\bigbreak

The paper {\it On Quaternions; or on a new System of Imaginaries
in Algebra}, is included in 
{\it The Mathematical Papers of Sir William Rowan Hamilton},
vol.~iii (Algebra), edited for the Royal Irish Academy
by H. Halberstam and R. E. Ingram (Cambridge University Press,
Cambridge, 1967).

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

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\line{\hfil Dublin, March 2000}

\vfill\eject

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\noindent
{\largeit On Quaternions; or on a new System of Imaginaries
in Algebra\/}\footnote*{A communication, substantially the same
with that here published, was made by the present writer to the
Royal Irish Academy, at the first meeting of that body after the
last summer recess, in November 1843.}.
{\largeit By\/} {\largerm Sir}
{\largesc William Rowan Hamilton}, {\largeit LL.D., P.R.I.A.,
F.R.A.S., Hon.\ M. R. Soc.\ Ed.\ and Dub., Hon.\ or Corr.\ M. of
the Royal or Imperial Academies of St.~Petersburgh, Berlin,
Turin, and Paris, Member of the American Academy of Arts and
Sciences, and of other Scientific Societies at Home and Abroad,
Andrews' Prof.\ of Astronomy in the University of Dublin, and
Royal Astronomer of Ireland.}

\bigbreak

\vskip 12pt

\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and}}

\vskip 3pt

\centerline{{\it Journal of Science}, 1844--1850.]}

\bigskip

1.
Let an expression of the form
$${\rm Q} = w + ix + jy + kz$$
be called a {\it quaternion}, when $w$,~$x$,~$y$,~$z$, which we
shall call the four {\it constituents\/} of the
quaternion~${\rm Q}$, denote any real quantities, positive or
negative or null, but $i$,~$j$,~$k$ are symbols of three
imaginary quantities, which we shall call {\it imaginary units},
and shall suppose to be unconnected by any linear relation with
each other; in such a manner that if there be another expression
of the same form,
$${\rm Q}' = w' + ix' + jy' + kz',$$
the supposition of an equality between these two quaternions,
$${\rm Q} = {\rm Q}',$$
shall be understood to involve four separate equations between
their respective constituents, namely, the four following,
$$w = w',\quad x = x',\quad y = y',\quad z = z'.$$
It will then be natural to define that the {\it addition\/} or
{\it subtraction\/} of quaternions is effected by the formula
$${\rm Q} \pm {\rm Q}'
   =  w \pm w' + i (x \pm x') + j  (y \pm y') + k (z \pm z');$$
or, in words, by the rule, that {\it the sums or differences of
the constituents of any two quaternions, are the constituents of
the sum or difference of those two quaternions themselves}.  It
will also be natural to define that the product
${\rm Q} {\rm Q}'$, of the multiplication of ${\rm Q}$ as a
multiplier into ${\rm Q}'$ as a multiplicand, is capable of being
thus expressed:
$$\eqalign{
{\rm Q} {\rm Q}'
   &= \mathbin{\phantom{\mathord{} + \mathord{}}}
      ww' + iwx' + jwy' + kwz' \cr
   &\mathrel{\phantom{=}} \mathord{}
    + ixw' + i^2 xx' + ijxy' + ikxz' \cr
   &\mathrel{\phantom{=}} \mathord{}
    + jyw' + jiyx' + j^2 yy' + jkyz' \cr
   &\mathrel{\phantom{=}} \mathord{}
    + kzw' + kizx' + kjzy' + k^2 zz';\cr}$$
but before we can reduce this product to an expression of the
quaternion form, such as
$${\rm Q} {\rm Q}' = {\rm Q}'' = w'' + ix'' + jy'' + kz'',$$
it is necessary to fix on quaternion-expressions (or on real
values) for the nine squares or products,
$$i^2,\quad ij,\quad  ik,\quad
  ji,\quad  j^2,\quad jk,\quad
  ki,\quad  kj,\quad  k^2.$$

\bigbreak

2.
Considerations, which it might occupy too much space to give an
account of on the present occasion, have led the writer to adopt
the following system of values or expressions for these nine
squares or products:
$$i^2 = j^2 = k^2 = -1;
   \eqno ({\rm A}.)$$
$$ij = k,\quad jk = i,\quad ki = j;
   \eqno ({\rm B}.)$$
$$ji = - k,\quad kj = - i,\quad ik = - j;
   \eqno ({\rm C}.)$$
though it must, at first sight, seem strange and almost
unallowable, to define that the product of two imaginary factors
in one order differs (in sign) from the product of the same
factors in the opposite order ($ji = - ij$).  It will, however,
it is hoped, be allowed, that in entering on the discussion of a
new system of imaginaries, it may be found necessary or
convenient to surrender {\it some\/} of the expectations
suggested by the previous study of products of real quantities,
or even of expressions of the form $x + iy$, in which
$i^2 = - 1$.  And whether the choice of the system of
definitional equations, (A.), (B.), (C.), has been a judicious,
or at least a happy one, will probably be judged by the event,
that is, by trying whether those equations conduct to results of
sufficient consistency and elegance.

\bigbreak

3.
With the assumed relations (A.), (B.), (C.), we have the four
following expressions for the four constituents of the product of
two quaternions, as functions of the constituents of the
multiplier and multiplicand:
$$\left. \eqalign{
w'' &= w w' - x x' - y y' - z z', \cr
x'' &= w x' + x w' + y z' - z y', \cr
y'' &= w y' + y w' + z x' - x z', \cr
z'' &= w z' + z w' + x y' - y x'. \cr}
   \right\}
   \eqno ({\rm D}.)$$
These equations give
$$w''^2 + x''^2 + y''^2 + z''^2
   =  (w^2 + x^2 + y^2 + z^2) (w'^2 + x'^2 + y'^2 + z'^2);$$
and therefore
$$\mu'' = \mu \mu',
   \eqno ({\rm E}.)$$
if we introduce a system of expressions for the constituents, of
the forms
$$\left. \eqalign{
w  &= \mu \cos \theta, \cr
x  &= \mu \sin \theta \cos \phi, \cr
y  &= \mu \sin \theta \sin \phi \cos \psi, \cr
z  &= \mu \sin \theta \sin \phi \sin \psi, \cr}
   \right\}
   \eqno ({\rm F}.)$$
and suppose each $\mu$ to be positive.  Calling, therefore, $\mu$
the {\it modulus\/} of the quaternion~${\rm Q}$, we have this
theorem: that {\it the modulus of the product~${\rm Q}''$ of any
two quaternions ${\rm Q}$ and ${\rm Q}'$, is equal to the product
of their moduli}.

\bigbreak

4.
The equations (D.) give also
$$\eqalign{
w' w'' + x' x'' + y' y'' + z' z''
  &=  w  (w'^2 + x'^2 + y'^2 + z'^2),\cr
w  w'' + x  x'' + y  y'' + z  z''
  &=  w' (w^2 + x^2 + y^2 + z^2);\cr}$$
combining, therefore, these results with the first of those
equations (D.), and with the trigonometrical expressions (F.), and
the relation (E.) between the moduli, we obtain the three
following relations between the angular co-ordinates
$\theta$~$\phi$~$\psi$,
$\theta'$~$\phi'$~$\psi'$,
$\theta''$~$\phi''$~$\psi''$
of the two factors and the product:
$$\left. \eqalign{
\cos \theta''
   &=    \cos \theta   \cos \theta'
       - \sin \theta   \sin \theta'
         (  \cos \phi   \cos \phi'
          + \sin \phi   \sin \phi'  \cos( \psi   - \psi'  )),\cr
\cos \theta
   &=    \cos \theta'  \cos \theta''
       + \sin \theta'  \sin \theta''
         (  \cos \phi'  \cos \phi''
          + \sin \phi'  \sin \phi'' \cos( \psi'  - \psi'' )),\cr
\cos \theta'
   &=    \cos \theta'' \cos \theta
       + \sin \theta'' \sin \theta
         (  \cos \phi'' \cos \phi
          + \sin \phi'' \sin \phi   \cos( \psi'' - \psi   )).\cr}
   \right\}
   \eqno ({\rm G}.)$$
These equations (G.) admit of a simple geometrical construction.
Let $x \, y \, z$ be considered as the three rectangular
co-ordinates of a point in space, of which the radius vector is
$= \mu \sin \theta$, the longitude $= \psi$, and the co-latitude
$= \phi$; and let these three latter quantities be called also
the {\it radius vector}, the {\it longitude\/} and the
{\it co-latitude of the quaternion\/}~${\rm Q}$; while $\theta$
shall be called the {\it amplitude\/} of that quaternion.  Let
${\sc r}$ be the point where the radius vector, prolonged if
necessary, intersects the spheric surface described about the
origin of co-ordinates with a radius $=$~unity, so that $\phi$ is
the co-latitude and $\psi$ is the longitude of ${\sc r}$; and let
this point~${\sc r}$ be called the {\it representative point\/}
of the quaternion~${\rm Q}$.  Let ${\sc r}'$ and ${\sc r}''$ be,
in like manner, the representative points of ${\rm Q}'$ and
${\rm Q}''$; then the equations (G.) express that {\it in the
spherical triangle\/} ${\sc r} \, {\sc r}' \, {\sc r}''$,
{\it formed by the representative points of the two factors and
the product\/} (in any multiplication of two quaternions),
{\it the angles are respectively equal to the amplitudes of those
two factors, and the supplement of the amplitude of the
product\/} (to two right angles); in such a manner that we have
the three equations:
$${\sc r} = \theta,\quad
  {\sc r}' = \theta',\quad
  {\sc r}'' = \pi - \theta''.
   \eqno ({\rm H}.)$$

\bigbreak

5.
The system of the four very simple and easily remembered
equations (E.) and (H.), may be considered as equivalent to the
system of the four more complex equations (D.), and as containing
within themselves a sufficient expression of the rules of
multiplication of quaternions; with this exception, that they
leave undetermined the {\it hemisphere\/} to which the
point~${\sc r}''$ belongs, or the {\it side\/} of the
arc~${\sc r} \, {\sc r}'$ on which that
{\it product-point\/}~${\sc r}''$ falls, after the
{\it factor-points\/} ${\sc r}$ and ${\sc r}'$, and the
amplitudes $\theta$ and $\theta'$ have been assigned.  In fact,
the equations (E.) and (H.) have been obtained, not immediately
from the equations (D.), but from certain combinations of the
last-mentioned equations, which combinations would have been
unchanged, if the signs of the three functions,
$$y z' - z y',\quad
  z x' - x z',\quad
  x y' - y x',$$
had all been changed together.  This latter change would
correspond to an alteration in the assumed conditions (B.) and
(C.), such as would have consisted in assuming $ij = -k$,
$ji = +k$, \&c., that is, in taking the cyclical order
$k$~$j$~$i$ (instead of $i$~$j$~$k$), as that in which the
product of any two imaginary units (considered as multiplier and
multiplicand) is equal to the imaginary unit following, taken
positively.  With this remark, it is not difficult to perceive
that {\it the product-point~${\sc r}''$ is always to be taken to
the right, or always to the left of the
multiplicand-point~${\sc r}'$, with respect to the
multiplier-point\/}~${\sc r}$, according as the semiaxis of $+ z$
is to the right or left of the semiaxis of $+ y$, with respect to
the semiaxis of $+ x$; or, in other words, according as the
positive direction of rotation in longitude is to the right or to
the left.  This {\it rule of rotation}, combined with the
{\it law of the moduli\/} and with the {\it theorem of the
spherical triangle}, completes the transformed system of
conditions, connecting the product of any two quaternions with
the factors, and with their order.

\bigbreak

6.
It follows immediately from the principles already explained,
that if ${\sc r} \, {\sc r}' \, {\sc r}''$ be any spherical
triangle, and if $\alpha \, \beta \, \gamma$ be the rectangular
co-ordinates of ${\sc r}$, $\alpha' \, \beta' \, \gamma'$ of
${\sc r}'$, and $\alpha'' \, \beta'' \, \gamma''$ of ${\sc r}''$,
the centre~${\sc o}$  of the sphere being origin and the radius
unity, and the positive semiaxis of $z$ being so chosen as to lie
to the right or left of the positive semiaxis of $y$, with
respect to the positive semiaxis of $x$, according as the radius
${\sc o} \, {\sc r}''$ lies to the right or left of
${\sc o} \, {\sc r}'$ with respect to ${\sc o} \, {\sc r}$, then the
following imaginary or symbolic {\it formula of multiplication\/}
of quaternions will hold good: 
$$\eqalignno{
\{ \cos {\sc r} + (i \alpha + j \beta + k \gamma) \sin {\sc r} \}
      \{ \cos {\sc r}' + (i \alpha' + j \beta' + k \gamma')
         \sin {\sc r}' \}
   \hskip -15em \cr
   &= -  \cos {\sc r}'' + (i \alpha'' + j \beta'' + k \gamma'')
         \sin {\sc r}'';
   &({\rm I}.)\cr}$$
the squares and products of the three imaginary units,
$i$,~$j$,~$k$, being determined by the nine equations of
definition, assigned in a former article, namely,
$$i^2 = j^2 = k^2 = -1;
   \eqno ({\rm A}.)$$
$$ij = k,\quad jk = i,\quad ki = j;
   \eqno ({\rm B}.)$$
$$ji = - k,\quad kj = - i,\quad ik = - j.
   \eqno ({\rm C}.)$$
Developing and decomposing the imaginary formula (I.) by these
conditions, it resolves itself into the four following real
equations of spherical trigonometry:
$$\left. \eqalign{
- \cos {\sc r}''
   &= \cos {\sc r} \cos {\sc r}'
       - (\alpha \alpha' + \beta \beta' + \gamma \gamma')
         \sin {\sc r} \sin {\sc r}',\cr
\alpha'' \sin {\sc r}''
   &= \alpha \sin {\sc r} \cos {\sc r}'
       + \alpha' \sin {\sc r}' \cos {\sc r}
       + (\beta  \gamma' - \gamma \beta' ) \sin {\sc r} \sin {\sc r}',\cr
\beta''  \sin {\sc r}''
   &= \beta  \sin {\sc r} \cos {\sc r}'
       + \beta'  \sin {\sc r}' \cos {\sc r}
       + (\gamma \alpha' - \alpha \gamma') \sin {\sc r} \sin {\sc r}',\cr
\gamma'' \sin {\sc r}''
   &= \gamma \sin {\sc r} \cos {\sc r}'
       + \gamma' \sin {\sc r}' \cos {\sc r}
       + (\alpha \beta'  - \beta  \alpha') \sin {\sc r} \sin {\sc r}';\cr}
   \right\}
   \eqno ({\rm K}.)$$
of which indeed the first answers to the well-known relation
(already employed in this paper), connecting a side with the
three angles of a spherical triangle.  The three other
equations~(K.) correspond to and contain a theorem (perhaps new),
which may be enunciated thus: that if three forces be applied at
the centre of the sphere, one equal to
$\sin {\sc r} \cos {\sc r}'$
and directed to the point~${\sc r}$, another equal to
$\sin {\sc r}' \cos {\sc r}$
and directed to ${\sc r}'$, and the third equal to
$\sin {\sc r} \sin {\sc r}' \sin {\sc r} \, {\sc r}'$
and directed to that pole of the arc~${\sc r} \, {\sc r}'$ which
lies towards the same side of this arc as the point~${\sc r}''$,
the resultant of these three forces will be directed to
${\sc r}''$, and will be equal to $\sin {\sc r}''$.  It is not
difficult to prove this theorem otherwise, but it may be regarded
as interesting to see that the four real equations (K.) are all
included so simply in the single imaginary formula~(I.), and can
so easily be deduced from that formula by the rules of the
{\it multiplication of quaternions\/}; in the same manner as the
fundamental theorems of plane trigonometry, for the cosine and
sine of the sum of any two arcs, are included in the well-known
formula for the multiplication of {\it couples}, that is,
expressions of the form $x + iy$, or more particularly
$\cos \theta + i \sin \theta$, in which $i^2 = -1$.  A new sort
of algorithm, or {\it calculus for spherical trigonometry}, would
seem to be thus given or indicated.

And if we suppose the spherical triangle
${\sc r} \, {\sc r}' \, {\sc r}''$
to become indefinitely small, by each of its corners tending to
coincide with the point of which the co-ordinates are
$1$,~$0$,~$0$, then each co-ordinate~$\alpha$ will tend to become
$= 1$, while each $\beta$ and $\gamma$ will ultimately vanish,
and the sum of the three angles will approach indefinitely to the
value~$\pi$; the formula~(I.) will therefore have for its limit
the following,
$$(\cos {\sc r} + i \sin {\sc r}) (\cos {\sc r}' + i \sin {\sc r}')
   =  \cos ({\sc r} + {\sc r}') + i \sin ({\sc r} + {\sc r}'),$$
which has so many important applications in the usual theory of
imaginaries.

\bigbreak

7.
In that theory there are only two different square roots of
negative unity, and they differ only in their signs.  In the
theory of quaternions, in order that the square of
$w + ix + jy + kz$ should be equal to $-1$, it is necessary and
sufficient that we should have
$$w = 0,\quad x^2 + y^2 + z^2 = 1;$$
for, in general, by the expressions (D.) of this paper for the
constituents of a product, or by the definitions (A.), (B.),
(C.), we have
$$(w + ix + jy + kz)^2
   =  w^2 - x^2 - y^2 - z^2 + 2w (ix + jy + kz).$$

There are, therefore, in this theory, {\it infinitely many
different square roots of negative one}, which have all one
common modulus $= 1$, and one common amplitude
$\displaystyle = {\pi \over 2}$,
being all of the form
$$\sqrt{-1}
   =  i \cos \phi + j \sin \phi \cos \psi + k \sin \phi \sin \psi,
   \eqno ({\rm L}.)$$
but which admit of all varieties of {\it directional
co-ordinates}, that is to say, co-latitude and longitude, since
$\phi$ and $\psi$ are arbitrary; and we may call them all
{\it imaginary units}, as well as the three original imaginaries,
$i$,~$j$,~$k$, from which they are derived.  To distinguish one
such root or unit from another, we may denote the second member
of the formula~(L.) by $i_{\phi,\psi}$, or more concisely by
$i_{\sc r}$, if ${\sc r}$ denote (as before) that particular
point of the spheric surface (described about the origin as
centre with a radius equal to unity) which has its co-latitude
$= \phi$, and its longitude $= \psi$.  We shall then have
$$i_{\sc r} = i \alpha + j \beta + k \gamma,\quad
  i_{\sc r}^2 = - 1,
   \eqno ({\rm L}'.)$$
in which
$$\alpha = \cos \phi,\quad
  \beta =  \sin \phi \cos \psi,\quad
  \gamma = \sin \phi \sin \psi,$$
$\alpha$,~$\beta$,~$\gamma$ being still the rectangular
co-ordinates of ${\sc r}$, referred to the centre as their
origin.  The formula~(I.) will thus become, for any spherical
triangle,
$$(\cos {\sc r} + i_{\sc r} \sin {\sc r})
      (\cos {\sc r}' + i_{{\sc r}'} \sin {\sc r}')
   = - \cos {\sc r}'' + i_{{\sc r}''} \sin {\sc r}''.
   \eqno ({\rm I}'.)$$

\bigbreak

8.
To {\it separate the real and imaginary parts\/} of this last
formula, it is only necessary to effect a similar separation for
the product of the two imaginary units which enter into the first
member.  By changing the angles ${\sc r}$ and ${\sc r}'$ to right
angles, without changing the points ${\sc r}$ and ${\sc r}'$ upon
the sphere, the imaginary units $i_{\sc r}$ and $i_{{\sc r}'}$
undergo no change, but the angle~${\sc r}''$ becomes equal to the
arc ${\sc r} \, {\sc r}'$, and the point ${\sc r}''$ comes to
coincide with the {\it positive pole\/} of that arc, that is,
with the pole to which the least rotation from ${\sc r}'$ round
${\sc r}$ is positive.  Denoting then this pole by ${\sc p}''$,
we have the equation
$$i_{\sc r} i_{{\sc r}'}
   =  - \cos {\sc r} \, {\sc r}'
      + i_{{\sc p}''} \sin {\sc r} \, {\sc r}',
   \eqno ({\rm M}.)$$
which is included in the formula (I${}'$.), and reciprocally
serves to transform it; for it shows that while the comparison of
the real parts reproduces the known equation
$$\cos {\sc r} \cos {\sc r}'
       - \sin {\sc r} \sin {\sc r}' \cos {\sc r} \, {\sc r}'
   =  - \cos {\sc r}'',
   \eqno ({\rm K}'.)$$
the comparison of the imaginary parts conducts to the following
symbolic expression for the theorem of the 6th article:
$$i_{\sc r} \sin {\sc r} \cos {\sc r}'
       + i_{{\sc r}'} \sin {\sc r}' \cos {\sc r}
       + i_{{\sc p}''} \sin {\sc r} \sin {\sc r}'
            \sin {\sc r} \, {\sc r}' 
   =  i_{{\sc r}''} \sin {\sc r}''.
   \eqno ({\rm K}''.)$$

As a verification we may remark, that when the triangle (and with
it the arc ${\sc r} \, {\sc r}'$) tends to vanish, the two last
equations tend to concur in giving the property of the plane
triangle,
$${\sc r} + {\sc r}' + {\sc r}'' = \pi.$$

\bigbreak

9.
The expression (M.) for the {\it product of any two imaginary
units}, which admits of many applications, may be immediately
deduced from the fundamental definitions (A.), (B.), (C.),
respecting the squares and products of the three original
imaginary units, $i$,~$j$,~$k$, by putting it under the form
$$\eqalignno{
(i \alpha + j \beta + k \gamma)
      (i \alpha' + j \beta' + k \gamma')
   \hskip -15em \cr
   &=  - (\alpha \alpha' + \beta \beta' + \gamma \gamma')
       + i (\beta  \gamma' - \gamma \beta'  )
       + j (\gamma \alpha' - \alpha \gamma' )
       + k (\alpha \beta'  - \beta  \alpha' );
   &({\rm M}'.)\cr}$$
and it is evident, either from this last form or from
considerations of rotation such as those already explained, that
if the {\it order\/} of any two pure imaginary factors be
changed, the real part of the product remains unaltered, but the
imaginary part changes sign, in such a manner that the
equation~(M.) may be combined with this other analogous equation,
$$i_{{\sc r}'} i_{\sc r}
   =   - \cos {\sc r} \, {\sc r}'
       - i_{{\sc p}''} \sin {\sc r} \, {\sc r}'.
   \eqno ({\rm N}.)$$
In fact, we may consider $- i_{{\sc p}''}$ as
$= i_{{\sc p}_\prime''}$, if ${\sc p}_\prime''$ be the point
diametrically opposite to ${\sc p}''$, and consequently the
positive pole of the reversed arc ${\sc r}' {\sc r}$ (in the
sense already determined), though it is the {\it negative pole\/}
of the arc~${\sc r} \, {\sc r}'$ taken in its former direction.

And since in general the product of any two quaternions, which
differ only in the signs of their imaginary parts, is real and
equal to the square of the modulus, or in symbols,
$$\mu (\cos \theta + i_{\sc r} \sin \theta)
      \times \mu (\cos \theta - i_{\sc r} \sin \theta)
   = \mu^2,
   \eqno ({\rm O}.)$$
we see that the product of the two different products, (M.) and
(N.), obtained by multiplying any two imaginary units together in
different orders, is real and equal to unity, in such a manner
that we may write
$$i_{\sc r} i_{{\sc r}'} \mathbin{.} i_{{\sc r}'} i_{\sc r} = 1;
   \eqno ({\rm P}.)$$
and the two quaternions, represented by the two products
$i_{\sc r} i_{{\sc r}'}$ and $i_{{\sc r}'} i_{\sc r}$, may be said to
be {\it reciprocals\/} of each other.  For example, it follows
immediately from the fundamental definitions (A.), (B.), (C.),
that
$$ij \mathbin{.} ji = k \times - k = - k^2 = 1;$$
the products $ij$ and $ji$ are therefore reciprocals, in the
sense just now explained.  By supposing the two imaginary
factors, $i_{\sc r}$ and $i_{{\sc r}'}$, to be mutually
{\it rectangular}, that is, the arc~${\sc r} \, {\sc r}' =$ a
quadrant, the two products (M.) and (N.) become
$\pm i_{{\sc p}''}$; and thus, or by a process more direct, we
might show that if two imaginary units be mutually
{\it opposite\/} (one being the negative of the other), they are
also mutually {\it reciprocal}.

\bigbreak

10.
The equation~(P.), which we shall find to be of use in the
{\it division of quaternions}, may be proved in a more purely
algebraical way, or at least in one more abstracted from
considerations of directions in space, as follows.  It will be
found that, in virtue of the definitions (A.), (B.), (C.), every
equation of the form
$$\iota \mathbin{.} \kappa \lambda = \iota \kappa \mathbin{.} \lambda$$
is true, if the three factors $\iota$,~$\kappa$,~$\lambda$,
whether equal or unequal among themselves, be equal, each, to one
or other of the three imaginary units $i$,~$j$,~$k$; thus, for
example,
$$i \mathbin{.} jk \, = ( \, i \mathbin{.} i = -1
   = k \mathbin{.} k = \, ) \, ij \mathbin{.} k,$$
$$j \mathbin{.} ji = \, ( \, j \mathbin{.} -k = -i
   = -1 \mathbin{.} i = \, ) \, jj \mathbin{.} i.$$
Hence, whatever three quaternions may be denoted by
${\rm Q}$,~${\rm Q}'$,~${\rm Q}''$, we have the equation
$${\rm Q} \mathbin{.} {\rm Q}' {\rm Q}''
   =  {\rm Q} {\rm Q}' \mathbin{.} {\rm Q}'';
   \eqno ({\rm Q}.)$$
and in like manner, for any four quaternions,
$${\rm Q} \mathbin{.} {\rm Q}' {\rm Q}'' {\rm Q}'''
   =  {\rm Q} {\rm Q}' \mathbin{.} {\rm Q}'' {\rm Q}'''
   =  {\rm Q} {\rm Q}' {\rm Q}'' \mathbin{.} {\rm Q}''',
   \eqno ({\rm Q}'.)$$
and so on for any number of factors; the notation
${\rm Q} {\rm Q}' {\rm Q}''$
being employed, in the formula (Q${}'$.), to denote that one
determined quaternion which, in virtue of the theorem (Q.), is
obtained, whether we first multiply ${\rm Q}''$ as a multiplicand
by ${\rm Q}'$ as a multiplier, and then multiply the product
${\rm Q}' {\rm Q}''$ as a multiplicand by ${\rm Q}$ as a
multiplier; or multiply first ${\rm Q}'$ by ${\rm Q}$, and then
${\rm Q}''$ by ${\rm Q} {\rm Q}'$.  With the help of this
principle we may easily prove the equation (P.), by observing that
$$i_{\sc r} i_{{\sc r}'} \mathbin{.} i_{{\sc r}'} i_{\sc r}
   =  i_{\sc r} \mathbin{.} i_{{\sc r}'}^2 \mathbin{.} i_{\sc r}
   =  - i_{\sc r}^2 = +1.$$

\bigbreak

11.
The theorem expressed by the formul{\ae} (Q.), (Q${}'$), \&c., is of
great importance in the {\it calculus of quaternions}, as tending
(so far as it goes) to assimilate this system of calculations to
that employed in ordinary algebra.  In ordinary multiplication we
may distribute any factor into any number of parts, real or
imaginary, and collect the partial products; and the same process
is allowed in operating on quaternions:
{\it quaternion-multiplication\/} possesses therefore the
{\it distributive\/} character of multiplication commonly so
called, or in symbols,
$${\rm Q} ({\rm Q}' + {\rm Q}'')
   =  {\rm Q} {\rm Q}' + {\rm Q} {\rm Q}'',\quad
 ({\rm Q} + {\rm Q}') {\rm Q}''
   =  {\rm Q} {\rm Q}'' + {\rm Q}' {\rm Q}'',
   \quad\hbox{\&c.}$$
But in ordinary algebra we have also
$${\rm Q} {\rm Q}' = {\rm Q}' {\rm Q};$$
which equality of products of factors, taken in opposite orders,
does {\it not\/} in general hold good for quaternions
($ji = - ij$); the {\it commutative\/} character of ordinary
multiplication is therefore in general {\it lost\/} in passing to
the new operation, and ${\rm Q} {\rm Q}' - {\rm Q}' {\rm Q}$,
instead of being a symbol of zero, comes to represent a pure
imaginary, but not (in general) an evanescent quantity.  On the
other hand, for quaternions as for ordinary factors, we
{\it may\/} in general {\it associate the factors among
themselves, by groups, in any manner which does not alter their
order\/}; for example,
$${\rm Q} \mathbin{.} {\rm Q}' {\rm Q}'' \mathbin{.}
         {\rm Q}''' {\rm Q}^{\rm IV}
   =  {\rm Q} {\rm Q}' \mathbin{.} {\rm Q}'' {\rm Q}'''
         {\rm Q}^{\rm IV};$$
this, therefore, which may be called the {\it associative\/}
character of the operation, is (like the distributive character)
{\it common\/} to the multiplication of quaternions, and to that
of ordinary quantities, real or imaginary.

\bigbreak

12.
A quaternion,~${\rm Q}$, divided by its modulus,~$\mu$, may in
general (by what has been shown) be put under the form,
$$\mu^{-1} {\rm Q} = \cos \theta + i_{\sc r} \sin \theta;$$
in which $\theta$ is a real quantity, namely the amplitude of the
quaternion; and $i_{\sc r}$ is an imaginary unit, or square root
of a negative one, namely that particular root, or unit, which is
distinguished from all others by its two directional co-ordinates,
and is constructed by a straight line drawn from the origin of
co-ordinates to the representative point~${\sc r}$; this
point~${\sc r}$ being on the spheric surface which is described
about the origin as centre, with a radius equal to unity.
Comparing this expression for $\mu^{-1} {\rm Q}$ with the formula
(M.) for the product of any two imaginary units, we see that if
with the point~${\sc r}$ as a positive pole, we describe on the
same spheric surface an arc ${\sc p}' {\sc p}''$ of a great
circle, and take this arc $= \pi - \theta =$ the supplement of
the amplitude of ${\rm Q}$; and then consider the points
${\sc p}'$ and ${\sc p}''$ as the representative points of two
new imaginary units $i_{{\sc p}'}$ and $i_{{\sc p}''}$, we shall
have the following {\it general transformation for any given
quaternion},
$${\rm Q} = \mu i_{{\sc p}'} i_{{\sc p}''};$$
the arc~${\sc p}' {\sc p}''$ being given in length and in
direction, except that it may turn round in its own plane (or on
the great circle to which it belongs), and may be increased or
diminished by any whole number of circumferences, without
altering the value of ${\rm Q}$.

\bigbreak

13.
Consider now the {\it product of several successive quaternion
factors\/} ${\rm Q}_1, {\rm Q}_2,\ldots$ under the condition that
their amplitudes $\theta_1, \theta_2,\ldots$ shall be
respectively equal to the angles of the spherical polygon which
is formed by their representative points
${\sc r}_1, {\sc r}_2,\ldots$
taken in their order.  To fix more precisely what is to be
understood in speaking here of these angles, suppose that
${\sc r}_m$ is the representative point of the $m$th quaternion
factor, or the $m$th corner of the polygon, the next preceding
corner being ${\sc r}_{m-1}$, and the next following being
${\sc r}_{m+1}$; and let the angle, or (more fully) the
{\it internal angle}, of the polygon, at the point~${\sc r}_m$,
be denoted by the same symbol~${\sc r}_m$, and be defined to be
the least angle of rotation through which the
arc~${\sc r}_m {\sc r}_{m+1}$ must revolve in the positive
direction round the point~${\sc r}_m$, in order to come into the
direction of the arc~${\sc r}_m {\sc r}_{m-1}$.  Then, the
rotation $2\pi - {\sc r}_m$ would bring ${\sc r}_m {\sc r}_{m-1}$
to coincide in direction with ${\sc r}_m {\sc r}_{m+1}$; and
therefore the rotation $\pi - {\sc r}_m$, performed in the same
sense or in the opposite, according as it is positive or
negative, would bring the {\it prolongation\/} of the preceding
arc~${\sc r}_{m-1} \, {\sc r}_m$ to coincide in direction with
the following arc~${\sc r}_m {\sc r}_{m+1}$; on which account we
shall call this angle $\pi - {\sc r}_m$, taken with its proper
sign, the {\it external angle\/} of the polygon at the
point~${\sc r}_m$.  The same rotation $\pi - {\sc r}_m$ would
bring the positive pole, which we shall call ${\sc p}_{m+1}$, of
the preceding side~${\sc r}_{m-1} \, {\sc r}_m$ of the polygon,
to coincide with the positive pole ${\sc p}_{m+2}$ of the
following side~${\sc r}_m {\sc r}_{m+1}$ thereof, by turning
round the corner~${\sc r}_m$ as a pole, in an arc of a great
circle, and in a positive or negative direction of rotation
according as the external angle $\pi - {\sc r}_m$ of the polygon
is itself positive or negative; consequently, by the last
article, we shall have the formula
$$\mu_m^{-1} {\rm Q}_m = \cos {\sc r}_m + i_{{\sc r}_m} \sin {\sc r}_m
   = i_{{\sc p}_{m+1}} i_{{\sc p}_{m+2}}.$$
Multiplying together in their order the $n$ formul{\ae} of this sort
for the $n$ corners of the polygon, and attending to the
{\it associative\/} character of quaternion multiplication, which
gives, as an extension of the formula~(P.), the following,
$$i_{{\sc p}_1} i_{{\sc p}_2} \mathbin{.} i_{{\sc p}_2} i_{{\sc p}_3}
      \mathbin{.} \ldots \, i_{{\sc p}_n} i_{{\sc p}_1}
   = (-1)^n,
   \eqno ({\rm P}'.)$$
we see that under the supposed conditions as to the amplitudes we
have this expression for the product of the $n$ quaternion
factors,
$${\rm Q}_1 {\rm Q}_2 {\rm Q}_3 \, \ldots \, {\rm Q}_n
   =  (-1)^n \mu_1 \mu_2 \mu_3 \, \ldots \, \mu_n;$$
from which it follows, that for {\it any spherical polygon\/}
${\sc r}_1 {\sc r}_2 \, \ldots \, {\sc r}_n$,
(even with salient and re-entrant angles), this general equation
holds good:
$$    (\cos {\sc r}_1 + i_{{\sc r}_1} \sin {\sc r}_1)
      (\cos {\sc r}_2 + i_{{\sc r}_2} \sin {\sc r}_2)
      \, \ldots \,
      (\cos {\sc r}_n + i_{{\sc r}_n} \sin {\sc r}_n)
   =  (-1)^n.
   \eqno ({\rm R}.)$$

\bigbreak

14.
For the case of a spherical triangle
${\sc r} \, {\sc r}' \, {\sc r}''$, this relation becomes
$$    (\cos {\sc r} + i_{\sc r} \sin {\sc r})
      (\cos {\sc r}' + i_{{\sc r}'} \sin {\sc r}')
      (\cos {\sc r}'' + i_{{\sc r}''} \sin {\sc r}'')
   =  -1;
   \eqno ({\rm I}''.)$$
and reproduces the formula (I${}'$.), when we multiply each member,
as multiplier, into $\cos {\sc r}'' - i_{{\sc r}''} \sin {\sc r}''$
as multiplicand.  The restriction, mentioned in a former article,
on the direction of the positive semiaxis of one co-ordinate after
those of the two other co-ordinates had been chosen, was designed
merely to enable us to consider the three angles of the triangle
as being each positive and less than two right angles, according
to the usage commonly adopted by writers on spherical
trigonometry.  It would not have been difficult to deduce
reciprocally the theorem (R.) for any spherical polygon, from the
less general relation (I${}'$.) or (I${}''$.) for the case of a
spherical triangle, by assuming any point~${\sc p}$ upon the
spherical surface as the common vertex of $n$ triangles which
have the sides of the polygon for their $n$ bases, and by
employing the associative character of multiplication, together
with the principle that codirectional quaternions, when their
moduli are supposed each equal to unity, are multiplied by adding
their amplitudes.  This last principle gives also, as a
verification of the formula (R.), for the case of an infinitely
small, or in other words, a {\it plane polygon}, the known
equations,
$$\cos \Sigma {\sc r} = (-1)^n,\quad \sin \Sigma {\sc r} = 0.$$

\bigbreak

15.
The associative character of multiplication, or the formula (Q.),
shows that if we assume any three quaternions
${\rm Q}$,~${\rm Q}'$,~${\rm Q}''$, and derive two others
${\rm Q}_\prime$, ${\rm Q}_{\prime\prime}$ from them, by the
equations
$${\rm Q} {\rm Q}' = {\rm Q}_\prime,\quad
  {\rm Q}' {\rm Q}'' = {\rm Q}_{\prime\prime},$$
we shall have also the equations
$${\rm Q}_\prime {\rm Q}'' = {\rm Q} {\rm Q}_{\prime\prime}
   = {\rm Q}''',$$
${\rm Q}'''$ being a third derived quaternion, namely the ternary
product ${\rm Q} {\rm Q}' {\rm Q}''$.  Let
${\sc r} \, {\sc r}' \, {\sc r}''$
${\sc r}_\prime \, {\sc r}_{\prime\prime} \, {\sc r}'''$
be the six representative points of these six quaternions, on the
same spheric surface as before; then, by the general construction
of a product assigned in a former article\footnote*{In the Number
of this Magazine for July 1844, S.~3. vol.~{\sc xxv}.},
we shall have the following expressions for the six amplitudes of
the same six quaternions:
$$\multieqalign{
\theta
   &= {\sc r}' {\sc r} {\sc r}_\prime
    = {\sc r}_{\prime\prime} \, {\sc r} \, {\sc r}'''; &
\theta_\prime
   &= {\sc r}'' {\sc r}_\prime {\sc r}'''
    = \pi - {\sc r} \, {\sc r}_\prime {\sc r}'; \cr
\theta'
   &= {\sc r}_\prime {\sc r}' {\sc r}
    = {\sc r}'' {\sc r}' {\sc r}_{\prime\prime}; &
\theta_{\prime\prime}
   &= {\sc r}''' {\sc r}_{\prime\prime} \, {\sc r}
         =  \pi - {\sc r}' {\sc r}_{\prime\prime} \, {\sc r}''; \cr
\theta''
   &= {\sc r}_{\prime\prime} \, {\sc r}'' {\sc r}'
    = {\sc r}''' {\sc r}'' {\sc r}_\prime; &
\theta'''
   &= \pi - {\sc r}_\prime {\sc r}''' {\sc r}''
         =  \pi - {\sc r} \, {\sc r}''' {\sc r}_{\prime\prime}; \cr}$$
${\sc r}' \, {\sc r} \, {\sc r}_\prime$ being the spherical angle at
${\sc r}$, measured from ${\sc r} \, {\sc r}'$ to
${\sc r} \, {\sc r}_\prime$, and similarly in other cases.  But
these equations between the spherical angles of the figure are
precisely those which are requisite in order that the two points
${\sc r}_\prime$ and ${\sc r}_{\prime\prime}$ should be the
{\it two foci of a spherical conic inscribed in the spherical
quadrilateral\/} ${\sc r} \, {\sc r}' \, {\sc r}'' \, {\sc r}'''$,
or touched by the four great circles of which the arcs
${\sc r} \, {\sc r}'$, ${\sc r}' \, {\sc r}''$,
${\sc r}'' {\sc r}'''$, ${\sc r}''' {\sc r}$
are parts; this geometrical relation between the six
representative points
${\sc r} \, {\sc r}' \, {\sc r}'' \, {\sc r}_\prime \,
      {\sc r}_{\prime\prime} \, {\sc r}'''$
of the six quaternions
${\rm Q}$, ${\rm Q}'$, ${\rm Q}''$, ${\rm Q} {\rm Q}'$,
${\rm Q}' {\rm Q}''$, ${\rm Q} {\rm Q}' {\rm Q}''$,
which may conveniently be thus denoted,
$${\sc r}_\prime {\sc r}_{\prime\prime}
         (.\,.)
         {\sc r} {\sc r}' {\sc r}'' {\sc r}''',
   \eqno ({\rm Q}''.)$$
is therefore a consequence, and may be considered as an
interpretation, of the very simple  algebraical theorem for three
quaternion factors,
$${\rm Q} {\rm Q}' \mathbin{.} {\rm Q}''
   = {\rm Q} \mathbin{.} {\rm Q}' {\rm Q}''.
   \eqno ({\rm Q}.)$$
It follows at the same time, from the theory of spherical conics,
that the two straight lines, or {\it radii vectores}, which are
drawn from the origin of co-ordinates to the points
${\sc r}_\prime$,~${\sc r}_{\prime\prime}$,
and {\it which construct the imaginary parts of the two binary
quaternion products
${\rm Q} {\rm Q}'$, ${\rm Q}' {\rm Q}''$,
are the two focal lines of a cone of the second degree, inscribed
in the pyramid which has for its four edges the four radii which
construct the imaginary parts of the three quaternion factors
${\rm Q}$,~${\rm Q}'$,~${\rm Q}''$,
and of their continued (or ternary) product
${\rm Q} {\rm Q}' {\rm Q}''$.}

\bigbreak

16.
We had also, by the same associative character of multiplication,
analogous formul{\ae} for any four independent factors,
$${\rm Q} \mathbin{.} {\rm Q}' {\rm Q}'' {\rm Q}'''
   = {\rm Q} {\rm Q}' \mathbin{.} {\rm Q}'' {\rm Q}'''
   = \hbox{\&c.};
   \eqno ({\rm Q}'.)$$
if then we denote this continued product by ${\rm Q}^{\rm IV}$,
and make
$${\rm Q} {\rm Q}' = {\rm Q}_\prime,\quad
  {\rm Q}' {\rm Q}'' = {\rm Q}_\prime',\quad
  {\rm Q}'' {\rm Q}''' = {\rm Q}_\prime'',$$
$${\rm Q} {\rm Q}' {\rm Q}'' = {\rm Q}_\prime''',\quad
  {\rm Q}' {\rm Q}'' {\rm Q}''' = {\rm Q}_\prime^{\rm IV},$$
and observe that whenever ${\sc e}$ and ${\sc f}$ are foci of a
spherical conic inscribed in a spherical quadrilateral
${\sc a} {\sc b} {\sc c} {\sc d}$,
so that, in the notation recently proposed,
$${\sc e} {\sc f} (.\,.) {\sc a} {\sc b} {\sc c} {\sc d},$$
then also we may write
$${\sc f} {\sc e} (.\,.) {\sc a} {\sc b} {\sc c} {\sc d},
   \quad\hbox{and}\quad
  {\sc e} {\sc f} (.\,.) {\sc b} {\sc c} {\sc d} {\sc a},$$
we shall find, without difficulty, by the help of the formula
(Q${}''$.), the five following geometrical relations, in which
each ${\sc r}$ is the representative point of the corresponding
quaternion~${\rm Q}$:
$$\left. \eqalign{
   &{\sc r}_\prime {\sc r}_\prime' (.\,.)
         {\sc r} {\sc r}' {\sc r}'' {\sc r}_\prime''';\cr
   &{\sc r}_\prime' {\sc r}_\prime'' (.\,.)
         {\sc r}' {\sc r}'' {\sc r}''' {\sc r}_\prime^{\rm IV};\cr
   &{\sc r}_\prime'' {\sc r}_\prime''' (.\,.)
         {\sc r}'' {\sc r}''' {\sc r}^{\rm IV} {\sc r}_\prime;\cr
   &{\sc r}_\prime''' {\sc r}_\prime^{\rm IV} (.\,.)
         {\sc r}''' {\sc r}^{\rm IV} {\sc r} {\sc r}_\prime';\cr
   &{\sc r}_\prime^{\rm IV} {\sc r}_\prime (.\,.)
         {\sc r}^{\rm IV} {\sc r} {\sc r}' {\sc r}_\prime''.\cr}
   \right\}
   \eqno ({\rm Q}'''.)$$

These five formul{\ae} establish a remarkable {\it connexion between
one spherical pentagon and another\/} (when constructed according
to the foregoing rules), through the medium of {\it five
spherical conics\/}; of which five curves each touches two sides
of one pentagon and has its foci at two corners of the other.  If
we suppose for simplicity that each of the ten moduli is $= 1$,
the dependence of six quaternions by multiplication on four (as
their three binary, two ternary, and one quaternary product, all
taken without altering the order of succession of the factors)
will give eighteen distinct equations between the ten amplitudes
and the twenty polar co-ordinates of the ten quaternions here
considered; it is therefore in general permitted to assume at
pleasure twelve of these co-ordinates, or to choose six of the
ten points upon the sphere.  Not only, therefore, may we in
general take {\it one of the two pentagons arbitrarily}, but
also, at the same time, may assume {\it one corner of the other
pentagon\/} (subject of course to exceptional cases); and, after
a suitable choice of the ten amplitudes, the five relations
(Q${}'''$.), between the two pentagons and the five conics, will
still hold good.

\bigbreak

17.
A very particular (or rather limiting) yet not inelegant case of
this theorem is furnished by the consideration of the plane and
regular pentagon of elementary geometry, as compared with that
other and interior pentagon which is determined by the
intersections of its five diagonals.  Denoting by
${\sc r}_\prime$ that corner of the interior pentagon which is
nearest to the side ${\sc r} \, {\sc r}'$ of the exterior one;
by ${\sc r}_\prime'$ that corner which is nearest to
${\sc r}' {\sc r}''$, and so on to ${\sc r}_\prime^{\rm IV}$; the
relations (Q${}'''$) are satisfied, the symbol $(.\,.)$ now
denoting that the two points written before it are foci of an
ordinary (or plane) ellipse, inscribed in the plane quadrilateral
whose corners are the four points written after it.  We may add,
that (in this particular case) two points of contact for each of
the five quadrilaterals are corners of the interior pentagon; and
that the axis major of each of the five inscribed ellipses is
equal to a side of the exterior figure.

\bigbreak

18.
The separation of the real and imaginary parts of a quaternion is
an operation of such frequent occurrence, and may be regarded as
being so fundamental in this theory, that it is convenient to
introduce symbols which shall denote concisely the two separate
results of this operation.  The algebraically {\it real\/} part
may receive, according to the question in which it occurs, all
values contained on the one {\it scale\/} of progression of
number from negative to positive infinity; we shall call it
therefore the {\it scalar part}, or simply the {\it scalar\/} of
the quaternion, and shall form its symbol by prefixing, to the
symbol of the quaternion, the characteristic ${\rm Scal.}$, or
simply ${\rm S}$., where no confusion seems likely to arise from
using this last abbreviation.  On the other hand, the
algebraically {\it imaginary\/} part, being geometrically
constructed by a straight line, or radius vector, which has, in
general, for each determined quaternion, a determined length and
determined direction in space, may be called the {\it vector
part}, or simply the {\it vector\/} of the quaternion; and may be
denoted by prefixing the characteristic ${\rm Vect.}$, or
${\rm V}$.  We may therefore say that {\it a quaternion is in
general the sum of its own scalar and vector parts}, and may
write
$${\rm Q}
   =  \mathop{\rm Scal.} {\rm Q} + \mathop{\rm Vect.} {\rm Q}
   =  {\rm S} \mathbin{.} {\rm Q} + {\rm V} \mathbin{.} {\rm Q},$$
or simply
$${\rm Q} = {\rm S} {\rm Q} + {\rm V} {\rm Q}.$$
By detaching the characteristics of operation from the signs of
the operands, we may establish, for this notation, the general
formul{\ae}:
$$1 = {\rm S} + {\rm V};\quad
  1 - {\rm S} = {\rm V};\quad
  1 - {\rm V} = {\rm S};$$
$${\rm S} \mathbin{.} {\rm S} = {\rm S};\quad
  {\rm S} \mathbin{.} {\rm V} = 0;\quad
  {\rm V} \mathbin{.} {\rm S} = 0;\quad
  {\rm V} \mathbin{.} {\rm V} = {\rm V};$$
and may write
$$({\rm S} + {\rm V})^n = 1,$$
if $n$ be any positive whole number.  The scalar or vector of a
sum or difference of quaternions is the sum or difference of the
scalars or vectors of those quaternions, which we may express by
writing the formul{\ae}:
$${\rm S} \Sigma = \Sigma {\rm S};\quad
  {\rm S} \Delta = \Delta {\rm S};\quad
  {\rm V} \Sigma = \Sigma {\rm V};\quad
  {\rm V} \Delta = \Delta {\rm V}.$$

\bigbreak

19.
Another general decomposition of a quaternion, into factors
instead of summands, may be obtained in the following
way:---Since the square of a scalar is always positive, while the
square of a vector is always negative, the algebraical excess of
the former over the latter square is always a positive number; if
then we make
$$({\rm T} {\rm Q})^2
   = ({\rm S} {\rm Q})^2 - ({\rm V} {\rm Q})^2,$$
and if we suppose ${\rm T} {\rm Q}$ to be always a real and
positive or absolute number, which we may call the {\it tensor\/}
of the quaternion~${\rm Q}$, we shall not thereby diminish the
generality of that quaternion.  The {\it tensor\/} is what was
called in former articles the {\it modulus\/};\footnote*{The
writer believes that what originally led him to use the terms
``modulus'' and ``amplitude,'' was a recollection of M.~Cauchy's
nomenclature respecting the usual imaginaries of algebra.  It was
the use made by his friend John~T. Graves, Esq., of the word
``constituents,'' in connexion with the ordinary imaginary
expressions of the form $x + \sqrt{-1} \, y$, which led Sir
William Hamilton to employ the same term in connexion with his
own imaginaries.  And if he had not come to prefer to the word
``modulus,'' in this theory, the name ``tensor,'' which suggested
the characteristic~${\rm T}$, he would have borrowed the
symbol~${\rm M}$, with the same signification, from the valuable
paper by Mr.~Cayley, ``On Certain Results relating to
Quaternions,'' which appeared in the Number of this Magazine for
February~1845.  It will be proposed by the present writer, in a
future article, to call the {\it logarithmic modulus\/} the
``mensor'' of a quaternion, and to denote it by the foregoing
characteristic~${\rm M}$; so as to have
$${\rm M} {\rm Q} = \mathop{\rm log.} {\rm T} {\rm Q},\quad
  {\rm M} \mathbin{.} {\rm Q} {\rm Q}'
   =  {\rm M} {\rm Q} + {\rm M} {\rm Q}'.$$}
but there seem to be some conveniences in not obliging ourselves
to retain here a term which has been used in several other
senses by writers on other subjects; and the word tensor has (it
is conceived) some reasons in its favour, which will afterwards
more fully appear.  Meantime we may observe, as some
justification of the use of this word, or at least as some
assistance to the memory, that it enables us to say that {\it the
tensor of a pure imaginary}, or vector, is the number expressing
the {\it length\/} or linear {\it extension of the straight
line\/} by which that algebraical imaginary is geometrically
constructed.  If such an imaginary be divided by its own tensor,
the quotient is an imaginary or vector {\it unit}, which marks
the {\it direction\/} of the constructing line, or the region of
space towards which that line is {\it turned\/}; hence, and for
other reasons, we propose to call this quotient the
{\it versor\/} of the pure imaginary: and generally to say that
{\it a quaternion is the product of its own tensor and versor
factors}, or to write
$${\rm Q} = {\rm T} {\rm Q} \mathbin{.} {\rm U} {\rm Q},$$
using ${\rm U}$ for the characteristic of versor, as ${\rm T}$
for that of tensor.  This is the other general decomposition of
a quaternion, referred to at the beginning of the present
article; and in the same notation we have
$${\rm T} \mathbin{.} {\rm T} {\rm Q} = {\rm T} {\rm Q};\quad
  {\rm T} \mathbin{.} {\rm U} {\rm Q} = 1;\quad
  {\rm U} \mathbin{.} {\rm T} {\rm Q} = 1;\quad
  {\rm U} \mathbin{.} {\rm U} {\rm Q} = {\rm U} {\rm Q};$$
so that the tensor of a versor, or the versor of a tensor, is
unity, as it was seen that the scalar of a vector, or the vector
of a scalar, is zero.

The tensor of a positive scalar is equal to that scalar itself,
but the tensor of a negative scalar is equal to the positive
opposite thereof.  The versor of a positive or negative scalar is
equal to positive or negative unity; and in general, by what has
been shown in the 12th article, the versor of a quaternion is the
product of two imaginary units.  The tensor and versor of a
vector have been considered in the present article.  A tensor
cannot become equal to a versor, except by each becoming equal to
positive unity; as a scalar and a vector cannot be equal to each
other, unless each reduces itself to zero.

\bigbreak

20.
If we call two quaternions {\it conjugate\/} when they have the
same scalar part, but have opposite vector parts, then because,
by the last article,
$$({\rm T} {\rm Q})^2
   =  ({\rm S} {\rm Q} + {\rm V} {\rm Q})
      ({\rm S} {\rm Q} - {\rm V} {\rm Q}),$$
we may say that the {\it product of two conjugate quaternions},
${\rm S} {\rm Q} + {\rm V} {\rm Q}$ and
${\rm S} {\rm Q} - {\rm V} {\rm Q}$,
is equal to the {\it square of their common tensor},
${\rm T} {\rm Q}$; from which it follows that {\it conjugate
versors are the  reciprocals of each other}, one quaternion being
called the {\it reciprocal\/} of another when their product is
positive unity. If ${\rm Q}$ and ${\rm Q}'$ be any two
quaternions, the two products of their vectors, taken in opposite
orders, namely
${\rm V} {\rm Q}  \mathbin{.} {\rm V} {\rm Q}'$ and
${\rm V} {\rm Q}' \mathbin{.} {\rm V} {\rm Q}$,
are conjugate quaternions, by the definition given above, and by
the principles of the 9th article; and the conjugate of the sum
of any number of quaternions is equal to the sum of their
conjugates; therefore the products
$$    ({\rm S} {\rm Q} + {\rm V} {\rm Q})
      ({\rm S} {\rm Q}' + {\rm V} {\rm Q}')
   \quad\hbox{and}\quad
      ({\rm S} {\rm Q}' - {\rm V} {\rm Q}')
      ({\rm S} {\rm Q}  - {\rm V} {\rm Q})$$
are conjugate; therefore
${\rm T} \mathbin{.} {\rm Q} {\rm Q}'$,
which is the tensor of the first, is equal to the square root of
their product, that is, of
$$    ({\rm S} {\rm Q} + {\rm V} {\rm Q})
      ({\rm T} {\rm Q}')^2
      ({\rm S} {\rm Q} - {\rm V} {\rm Q}),
   \quad\hbox{or of}\quad
      ({\rm T} {\rm Q})^2 ({\rm T} {\rm Q}')^2;$$
we have therefore the formula
$${\rm T} \mathbin{.} {\rm Q} {\rm Q}'
   =  {\rm T} {\rm Q} \mathbin{.} {\rm T} {\rm Q}',$$
which gives also
$${\rm U} \mathbin{.} {\rm Q} {\rm Q}'
   =  {\rm U} {\rm Q} \mathbin{.} {\rm U} {\rm Q}';$$
that is to say, the {\it tensor of the product\/} of any two
quaternions is equal to the {\it product of the tensors}, and in
like manner the {\it versor of the product\/} is equal to the
{\it product of the versors}.  Both these results may easily be
extended to any number of factors, and by using $\Pi$ as the
characteristic of a product, we may write, generally,
$${\rm T} \Pi {\rm Q} = \Pi {\rm T} {\rm Q};\quad
  {\rm U} \Pi {\rm Q} = \Pi {\rm U} {\rm Q}.$$

It was indeed shown, so early as in the 3rd article, that the
modulus of a product is equal to the product of the moduli; but
the process by which an equivalent result has been here deduced
does not essentially depend upon that earlier demonstration: it has
also the advantage of showing that {\it the continued product of
any number of quaternion factors is conjugate to the continued
product of the respective conjugates of those factors, taken in
the opposite order\/}; so that we may write
$$({\rm S} - {\rm V}) \mathbin{.}
         {\rm Q} {\rm Q}' {\rm Q}'' \, \ldots
   =  \ldots \,
      ({\rm S} {\rm Q}'' - {\rm V} {\rm Q}'')
      ({\rm S} {\rm Q}' - {\rm V} {\rm Q}')
      ({\rm S} {\rm Q} - {\rm V} {\rm Q}),$$
a formula which, when combined with this other,
$$({\rm S} + {\rm V}) \mathbin{.}
         {\rm Q} {\rm Q}' {\rm Q}'' \, \ldots
   =  ({\rm S} {\rm Q} + {\rm V} {\rm Q})
      ({\rm S} {\rm Q}' + {\rm V} {\rm Q}')
      ({\rm S} {\rm Q}'' + {\rm V} {\rm Q}'') \, \ldots ,$$
enables us easily to develope
${\rm S} \Pi {\rm Q}$ and ${\rm V} \Pi {\rm Q}$,
that is, the scalar and vector of any product of
quaternions, in terms of the scalars and vectors of the several
factors of that product.  For example, if we agree to use, in
these calculations, the small Greek letters $\alpha$,~$\beta$,
\&c., with or without accents, as symbols of vectors (with the
exception of $\pi$, and with a few other exceptions, which
shall be either expressly mentioned as they occur, or clearly
indicated by the context), we may form the following table:---
$$\multieqalign{
2 {\rm S} \mathbin{.} \alpha
   &= \alpha - \alpha = 0; &
2 {\rm V} \mathbin{.} \alpha
   &= \alpha + \alpha = 2 \alpha; \cr
2 {\rm S} \mathbin{.} \alpha \alpha'
   &= \alpha \alpha' + \alpha' \alpha; &
2 {\rm V} \mathbin{.} \alpha \alpha'
   &= \alpha \alpha' - \alpha' \alpha; \cr
2 {\rm S} \mathbin{.} \alpha \alpha' \alpha''
   &= \alpha \alpha' \alpha'' - \alpha'' \alpha' \alpha; &
2 {\rm V} \mathbin{.} \alpha \alpha' \alpha''
   &= \alpha \alpha' \alpha'' + \alpha'' \alpha' \alpha; \cr
   & \hbox{\&c.} &
   & \hbox{\&c.} \cr}$$
of which the law is evident.

\bigbreak

21.
The fundamental rules of multiplication in this calculus give, in
the recent notation, for the scalar and vector parts of the
product of any two vectors, the expressions,
$$\eqalign{
{\rm S} \mathbin{.} \alpha \alpha'
   &= - (x x' + y y' + z z');\cr
{\rm V} \mathbin{.} \alpha \alpha'
   &= i (y z' - z y') + j (z x' - x z') + k (x y' - y z');\cr}$$
if we make
$$\alpha = ix + jy + kz,\quad
  \alpha' = ix' + jy' + kz',$$
$x$,~$y$,~$z$ and $x'$,~$y'$,~$z'$ being real and rectangular
co-ordinates, while $i$,~$j$,~$k$ are the original imaginary units
of this theory.  The {\it geometrical meanings\/} of the symbols
${\rm S} \mathbin{.} \alpha \alpha'$,
${\rm V} \mathbin{.} \alpha \alpha'$,
are therefore fully known. The former of these two symbols will
be found to have an intimate connexion with the theory of
{\it reciprocal polars\/}; as may be expected, if it be observed
that the equation
$${\rm S} \mathbin{.} \alpha \alpha' = - a^2$$
expresses that {\it with reference to the sphere of which the
equation is\/}
$$\alpha^2 = - a^2,$$
that is, with reference to the sphere of which the centre is at
the origin of vectors, and of which the radius has its length
denoted by $a$, {\it the vector~$\alpha'$ terminates in the polar
plane of the point which is the termination of the
vector~$\alpha$}.  The latter of the same two symbols, namely
${\rm V} \mathbin{.} \alpha \alpha'$,
denotes, or may be constructed by a straight line, which is in
direction perpendicular to both the lines denoted by $\alpha$ and
$\alpha'$, being also such that the rotation round it from
$\alpha$ to $\alpha'$ is positive; and bearing, in length, to
the unit of length, the same ratio which the area of the
{\it parallelogram under the two factor lines\/} bears to the
unit of area.  The {\it volume of the parallelepipedon\/} under
any three coinitial lines, or the {\it sextuple volume of the
tetrahedron\/} of which those lines are conterminous edges, may
easily be shown, on the same principles, to be equal to the
{\it scalar of the product of the three vectors\/}
corresponding; this scalar
${\rm S} \mathbin{.} \alpha \alpha' \alpha''$,
which is equal to
${\rm S} ( {\rm V} \mathbin{.} \alpha \alpha' \mathbin{.} \alpha'')$,
being positive or negative according as $\alpha''$ makes an
obtuse or an acute angle with
${\rm V} \mathbin{.} \alpha \alpha'$,
that is, according as the rotation round $\alpha''$ from
$\alpha'$ towards $\alpha$ is positive or negative.  To express
that two proposed lines $\alpha$,~$\alpha'$ are rectangular, we
may write the following {\it equation of perpendicularity},
$${\rm S} \mathbin{.} \alpha \alpha' = 0;
   \quad\hbox{or}\quad
  \alpha \alpha' + \alpha' \alpha = 0.$$
To express that two lines are similar or opposite in direction,
we may write the following {\it equation of coaxality}, or of
parallelism,
$${\rm V} \mathbin{.} \alpha \alpha' = 0;
   \quad\hbox{or}\quad
  \alpha \alpha' - \alpha' \alpha = 0.$$
And to express that three lines are in or parallel to one common
plane, we may write the {\it equation of coplanarity},
$${\rm S} \mathbin{.} \alpha \alpha' \alpha'' = 0;
   \quad\hbox{or}\quad
  \alpha \alpha' \alpha'' - \alpha'' \alpha' \alpha = 0;$$
either because the volume of the parallelepipedon under the three
lines then vanishes, or because one of the three vectors is then
perpendicular to the vector part of the product of the other two.

\bigbreak

22.
The {\it geometrical considerations\/} of the foregoing article
may often suggest {\it algebraical transformations\/} of
functions of the new imaginaries which enter into the present
theory.  Thus, if we meet the function
$$\alpha S \mathbin{.} \alpha' \alpha''
   - \alpha' S \mathbin{.} \alpha'' \alpha,
   \eqno (1.)$$
we may see, in the first place, that in the recent notation this
function is algebraically a pure imaginary, or {\it vector form},
which may be constructed geometrically in this theory by a
straight line having length and direction in space; because the
three symbols $\alpha$,~$\alpha'$,~$\alpha''$ are supposed to be
themselves such vector forms, or to admit of being constructed by
three such lines; while
${\rm S} \mathbin{.} \alpha' \alpha''$ and
${\rm S} \mathbin{.} \alpha'' \alpha$
are, in the same notation, two {\it scalar forms}, and denote
some two real numbers, positive negative, or zero.  We may
therefore equate the proposed function~(1.) to a new small Greek
letter, accented or unaccented, for example to $\alpha'''$,
writing
$$\alpha'''
   =  \alpha \, S \mathbin{.} \alpha' \alpha''
   - \alpha' \, S \mathbin{.} \alpha'' \alpha.
   \eqno (2.)$$
Multiplying this equation by $\alpha''$, and taking the scalar
parts of the two members of the product, that is, operating on it
by the characteristic ${\rm S} \mathbin{.} \alpha''$; and
observing that, by the properties of scalars,
$$\eqalign{
      {\rm S} \mathbin{.} \alpha'' \alpha \,
      {\rm S} \mathbin{.} \alpha' \alpha''
   &= {\rm S} \mathbin{.} \alpha'' \alpha
      \mathbin{.} {\rm S} \mathbin{.} \alpha' \alpha'' \cr
   =  {\rm S} \mathbin{.} \alpha'' \alpha'
      \mathbin{.} {\rm S} \mathbin{.} \alpha'' \alpha
   &= {\rm S} \mathbin{.} \alpha'' \alpha' \,
      {\rm S} \mathbin{.} \alpha'' \alpha,\cr}$$
in which the notation
${\rm S} \mathbin{.} \alpha'' \alpha \,
      {\rm S} \mathbin{.} \alpha' \alpha''$
is an abridgment for
${\rm S} ( \alpha'' \alpha \,
      {\rm S} \mathbin{.} \alpha' \alpha'' )$,
and the notation
${\rm S} \mathbin{.} \alpha'' \alpha
      \mathbin{.} {\rm S} \mathbin{.} \alpha' \alpha''$
is abridged from
$( {\rm S} \mathbin{.} \alpha'' \alpha )
      \mathbin{.} ( {\rm S} \mathbin{.} \alpha' \alpha'' )$,
while
${\rm S} \mathbin{.} \alpha' \alpha''$
is a symbol equivalent to
${\rm S} ( \alpha' \alpha'' )$,
and also, by article~20, to
${\rm S} ( \alpha'' \alpha' )$,
or to
${\rm S} \mathbin{.} \alpha'' \alpha'$,
although $\alpha' \alpha''$ and $\alpha'' \alpha'$ are not
themselves equivalent symbols; we are conducted to the equation
$${\rm S} \mathbin{.} \alpha'' \alpha''' = 0,
   \eqno (3.)$$
which shows, by comparison with the general {\it equation of
perpendicularity\/} assigned in the last article, that {\it the
new vector~$\alpha'''$ is perpendicular to the given
vector\/}~$\alpha''$, or that these two vector forms represent two
rectangular straight lines in space.  Again, because the squares
of vectors are scalars (being real, though negative numbers), we
have
$$\eqalign{
      \alpha (\alpha \, {\rm S} \mathbin{.} \alpha' \alpha'')
         \mathbin{.} \alpha'
   =  \alpha^2 \alpha' \, {\rm S} \mathbin{.} \alpha' \alpha''
   =  \alpha' (\alpha \, {\rm S} \mathbin{.} \alpha' \alpha'')
         \mathbin{.} \alpha,\cr
      \alpha' (\alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha)
         \mathbin{.} \alpha
   =  \alpha'^2 \alpha \, {\rm S} \mathbin{.} \alpha'' \alpha
   =  \alpha (\alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha)
         \mathbin{.} \alpha';\cr}$$
therefore the equation (2.) gives also
$$\alpha \alpha''' \alpha' = \alpha' \alpha''' \alpha;
   \eqno (4.)$$
a result which, when compared with the general {\it equation of
coplanarity\/} assigned in the same preceding article, shows
that {\it the new vector~$\alpha'''$ is coplanar with the two
other given vectors},~$\alpha$ and $\alpha'$; it is therefore
perpendicular to the vector of their product,
${\rm V} \mathbin{.} \alpha \alpha'$,
which is perpendicular to both those given vectors.  We have
therefore two known vectors, namely
${\rm V} \mathbin{.} \alpha \alpha'$
and $\alpha''$, to both of which the sought vector~$\alpha'''$ is
perpendicular; it is therefore parallel to, or coaxal with, the
vector of the product of the known vectors last mentioned, or is
equal to this vector of their product, multiplied by some scalar
coefficient~$x$; so that we may write the transformed expression,
$$\alpha'''
   =  x \, {\rm V} ( {\rm V} \mathbin{.}
         \alpha \alpha' \mathbin{.} \alpha'' ).
   \eqno (5.)$$
And because the function~$\alpha'''$ is, by the equation (2.),
homogeneous of the dimension unity with respect to each
separately of the three vectors
$\alpha$,~$\alpha'$,~$\alpha''$,
while the function
${\rm V} ( {\rm V} \mathbin{.} \alpha \alpha' \mathbin{.} \alpha'' )$
is likewise homogeneous of the same dimension with respect to
each of those three vectors, we see that the scalar
coefficient~$x$ must be either an entirely constant number, or
else a homogeneous function of the dimension zero, with respect
to each of the same three vectors; we may therefore assign to
these vectors any arbitrary lengths which may most facilitate the
determination of this scalar coefficient~$x$.  Again, the two
expressions (2.) and (5.) both vanish if $\alpha''$ be
perpendicular to the plane of $\alpha$ and $\alpha'$; in order
therefore to determine~$x$, we are permitted to suppose that
$\alpha$,~$\alpha'$,~$\alpha''$ are three coplanar vectors: and,
by what was just now remarked, we may suppose their lengths to be
each equal to the assumed unit of length.  In this manner we are
led to seek the value of $x$ in the equation
$$x \, {\rm V} \mathbin{.} \alpha \alpha' \mathbin{.} \alpha''
   =  \alpha S \mathbin{.} \alpha' \alpha''
   - \alpha' S \mathbin{.} \alpha'' \alpha,
   \eqno (6.)$$
under the conditions
$${\rm S} \mathbin{.} \alpha \alpha' \alpha'' = 0,
   \eqno (7.)$$
and
$$\alpha^2 = \alpha'^2 = \alpha''^2 = -1;
   \eqno (8.)$$
so that $\alpha$, $\alpha'$, $\alpha''$ are here {\it three
coplanar and imaginary units}.  Multiplying each member of the
equation (6.), as a multiplier, {\it into\/} $- \alpha''$ as a
multiplicand, and taking the vector parts of the two products;
observing also that
$${\rm V} \mathbin{.} \alpha' \alpha''
   =  - {\rm V} \mathbin{.} \alpha'' \alpha',
   \quad\hbox{and}\quad
  - {\rm V} \mathbin{.} \alpha \alpha''
   =  {\rm V} \mathbin{.} \alpha'' \alpha;$$
we obtain this other equation,
$$x \, {\rm V} \mathbin{.} \alpha \alpha'
   =     {\rm V} \mathbin{.} \alpha'' \alpha
         \mathbin{.} {\rm S} \mathbin{.} \alpha' \alpha''
       - {\rm V} \mathbin{.} \alpha'' \alpha'
         \mathbin{.} {\rm S} \mathbin{.} \alpha'' \alpha;
   \eqno (9.)$$
in which the three vectors
${\rm V} \mathbin{.} \alpha'' \alpha$,
${\rm V} \mathbin{.} \alpha'' \alpha'$,
${\rm V} \mathbin{.} \alpha \alpha'$
are coaxal, being each perpendicular to the common plane of the
three vectors $\alpha$,~$\alpha'$,~$\alpha''$; they bear therefore
scalar ratios to each other, and are proportional (by the last
article) to the areas of the parallelograms under the three pairs
of unit-vectors, $\alpha''$ and~$\alpha$,
$\alpha''$ and~$\alpha'$, and $\alpha$ and~$\alpha'$,
respectively; that is, to the sines of the angles $a$, $a'$, and
$a' - a$, if $a$ be the rotation from $\alpha''$ to $\alpha$, and
$a'$ the rotation from $\alpha''$ to $\alpha'$, in the common
plane of these three vectors.  At the same time we have (by the
principles of the same article) the expressions:
$$- {\rm S} \mathbin{.} \alpha'' \alpha = \cos a;\quad
  {\rm S} \mathbin{.} \alpha'' \alpha' = - \cos a';$$
so that the equation (9.) reduces itself to the following very
simple form,
$$x \sin (a' - a) = \sin a' \cos a - \sin a \cos a',
   \eqno (10.)$$
and gives immediately
$$x = 1.
   \eqno (11.)$$
Such then is the value of the coefficient~$x$ in the transformed
expression~(5.); and by comparing this expression with the
proposed form~(1.), we find that we may write, for {\it any three
vectors}, $\alpha$,~$\alpha'$,~$\alpha''$, not necessarily
subject to any conditions such as those of being equal in length
and coplanar in direction (since those conditions were not used
in {\it discovering the form\/}~(5.), but only in {\it determining
the value\/}~(11.),) the following {\it general transformation\/}:
$$\alpha \, {\rm S} \mathbin{.} \alpha' \alpha''
   - \alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha
   =  {\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
         \mathbin{.} \alpha'');
   \eqno (12.)$$
which will be found to have extensive applications.

\bigbreak

23.
But although it is possible thus to employ geometrical
considerations, in order to {\it suggest\/} and even to
{\it demonstrate\/} the validity of many general transformations,
yet it is always desirable to know how to obtain the same
{\it symbolic results}, from the {\it laws of combination of the
symbols\/}: nor ought the {\it calculus of quaternions\/} to be
regarded as complete, till all such {\it equivalences of form\/}
can be deduced from such symbolic laws, by the fewest and
simplest principles.  In the example of the foregoing article,
the symbolic transformation may be effected in the following way.

When a scalar form is multiplied by a vector form, or a vector by
a scalar, the product is a vector form; and the sum or difference
of two such vector forms is itself a vector form; therefore the
expression (1.) of the last article is a vector form, and may be
equated as such to a small Greek letter; or in other words, the
equation~(2.) is allowed.  But every vector form is equal to its
own vector part, or undergoes no change of signification when it
is operated on by the characteristic~${\rm V}$; we have therefore
this other expression, after interchanging, as is allowed, the
places of the two vector factors $\alpha'$~$\alpha''$ of a binary
product under the characteristic~${\rm S}$,
$$\alpha'''
   =  {\rm V} ( \alpha \, {\rm S} \mathbin{.} \alpha'' \alpha'
       - \alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha ).
   \eqno (1.)'$$
Substituting here for the characteristic~${\rm S}$, that which
is, by article~18, symbolically equivalent thereto, namely the
characteristic $1 - {\rm V}$, and observing that
$$0 = {\rm V} (\alpha \alpha'' \alpha' - \alpha' \alpha'' \alpha),
   \eqno (2.)'$$
because, by article~20,
$\alpha \alpha'' \alpha' - \alpha' \alpha'' \alpha$
is a scalar form, we obtain this other expression,
$$\alpha'''
   =  {\rm V} ( \alpha' {\rm V} \mathbin{.} \alpha'' \alpha
       - \alpha \, {\rm V} \mathbin{.} \alpha'' \alpha' ).
   \eqno (3.)'$$
The expression (1.)${}'$ may be written under the form
$$\alpha'''
   =  {\rm V} ( \alpha \, {\rm S} \mathbin{.} \alpha' \alpha''
       - \alpha' \, {\rm S} \mathbin{.} \alpha \alpha'' );
   \eqno (4.)'$$
and (3.)${}'$ under the form
$$\alpha'''
   =  {\rm V} ( \alpha \, {\rm V} \mathbin{.} \alpha' \alpha''
       - \alpha' \, {\rm V} \mathbin{.} \alpha \alpha'' ),
   \eqno (5.)'$$
obtained by interchanging the places of two vector-factors in
each of two binary products under the sign~${\rm V}$, and by
then changing the signs of those two products; taking then the
semisum of these two forms (4.)${}'$, (5.)${}'$, and using the
symbolic relation of article~18, ${\rm S} + {\rm V} = 1$, we find
$$\eqalignno{
\alpha'''
   &= {\textstyle {1 \over 2}}
         {\rm V} ( \alpha \alpha' \alpha'' - \alpha' \alpha \alpha'' ) \cr
   &= {\rm V} \left( {\textstyle {1 \over 2}}
         ( \alpha \alpha' - \alpha' \alpha )
         \mathbin{.} \alpha'' \right);
   &(6.)'\cr}$$
in which, by article~20,
${1 \over 2} ( \alpha \alpha' - \alpha' \alpha )
   =  {\rm V} \mathbin{.} \alpha \alpha'$;
we have therefore finally
$$\alpha'''
   =  {\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
         \mathbin{.} \alpha'' ): 
   \eqno (7.)'$$
that is, we are conducted by this purely symbolical process, from
laws of combination previously established, to the transformed
expression~(12.) of the last article.

\bigbreak

24.
A relation of the form~(4.), art.~22, that is an {\it equation
between the two ternary products of three vectors taken in two
different and opposite orders}, or an evanescence of the scalar
part of such a ternary product, may (and in fact does) present
itself in several researches; and although we know, by
art.~21, the {\it geometrical interpretation\/} of such a
symbolic relation between three vector forms, namely that it is
the condition of their representing {\it three coplanar lines},
which interpretation may suggest a transformation of one of them,
as a {\it linear function with scalar coefficients}, of the two
other vectors, because any one straight line in any given plane
may be treated as the diagonal of a parallelogram of which two
adjacent sides have any two given directions in the same given
plane; yet it is desirable, for the reason mentioned at the
beginning of the last article, to know how to obtain the same
general transformation of the same symbolic relation, without
having recourse to geometrical considerations.

Suppose then that any research has conducted to the relation,
$$\alpha \alpha' \alpha'' - \alpha'' \alpha' \alpha = 0,
   \eqno (1.)$$
which is not in this theory an identity, and which it is required
to transform.  [We propose for convenience to commence from time
to time a new numbering of the {\it formul{\ae}}, but shall take
care to avoid all danger of confusion of reference, by naming,
where it may be necessary, the {\it article\/} to which a formula
belongs; and when no such reference to an article is made, the
formula is to be understood to belong to the {\it current
series\/} of formul{\ae}, connected with the existing
investigation.]  By article~20, we may write the recent
relation~(1.) under the form,
$${\rm S} \mathbin{.} \alpha \alpha' \alpha'' = 0;
   \eqno (2.)$$
and because generally, for any three vectors, we have the
formula~(12.) of art.~22, if we make, in that formula,
$\alpha'' = {\rm V} \mathbin{.} \beta \beta'$,
and observe that
${\rm S} ( {\rm V} \mathbin{.} \beta \beta' \mathbin{.} \alpha )
   =  {\rm S} ( \alpha \, {\rm V} \mathbin{.} \beta \beta' )
   =  {\rm S} \mathbin{.} \alpha \beta \beta'$,
we find, for {\it any four vectors\/}
$\alpha$~$\alpha'$~$\beta$~$\beta'$,
the equation:
$${\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
         \mathbin{.} {\rm V} \mathbin{.} \beta \beta' )
   =  \alpha \, {\rm S} \mathbin{.} \alpha' \beta \beta'
    - \alpha' \, {\rm S} \mathbin{.} \alpha  \beta \beta';
   \eqno (3.)$$
making then, in this last equation, $\beta = \alpha'$,
$\beta' = \alpha''$, we find, for {\it any three vectors},
$\alpha$~$\alpha'$~$\alpha''$, the formula:
$${\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
         \mathbin{.} {\rm V} \mathbin{.} \alpha' \alpha'' )
   =  - \alpha' \, {\rm S} \mathbin{.} \alpha \alpha' \alpha''.
   \eqno (4.)$$

If then the scalar of the product $\alpha \alpha' \alpha''$ be
equal to zero, that is, if the condition (2.) or (1.) of the
present article be satisfied, the product of the two vectors
${\rm V} \mathbin{.} \alpha \alpha'$ and
${\rm V} \mathbin{.} \alpha' \alpha''$
is a scalar, and therefore the latter of these two vectors, or
the opposite vector
${\rm V} \mathbin{.} \alpha'' \alpha'$,
is in general equal to the former vector
${\rm V} \mathbin{.} \alpha \alpha'$,
multiplied by some scalar coefficient~$b$; we may therefore
write, under this condition (1.), the equation
$${\rm V} \mathbin{.} \alpha'' \alpha'
   = b \, {\rm V} \mathbin{.} \alpha \alpha',
   \eqno (5.)$$
that is,
$${\rm V} \mathbin{.} (\alpha'' - b \alpha) \alpha' = 0,
   \eqno (6.)$$
so that the one vector factor $\alpha'' - b \alpha$ of this last
product must be equal to the other vector factor~$\alpha'$
multiplied by some new scalar~$b'$; and we may write the formula,
$$\alpha'' = b \alpha + b' \alpha',
   \eqno (7.)$$
as a transformation of (1.) or of (2.).  We may also write, more
symmetrically, the equation
$$a \alpha + a' \alpha' + a'' \alpha'' = 0,
   \eqno (8.)$$
introducing {\it three\/} scalar coefficients $a$,~$a'$,~$a''$,
which have however only {\it two arbitrary ratios}, as a symbolic
transformation of the proposed equation
$\alpha \alpha' \alpha'' - \alpha'' \alpha' \alpha = 0$.
And it is remarkable that while we have thus {\it lowered by two
units the dimension of that proposed equation}, considered as
involving three variable vectors,
$\alpha$,~$\alpha'$,~$\alpha''$,
we have at the same time {\it introduced\/} (what may be regarded
as) {\it two arbitrary constants}, namely the two ratios of
$a$,~$a'$,~$a''$.  A converse process would have served to
{\it eliminate two arbitrary constants}, such as these two
ratios, or the two scalar coefficients $b$ and $b'$, from a
linear equation of the form (8.) or (7.), between three variable
vectors, at the same time {\it elevating the dimension of the
equation by two units}, in the passage to the form (2.) or (1.).
And the analogy of these two converse transformations to
{\it integrations and differentiations of equations\/} will
appear still more complete, if we attend to the intermediate
stage~(5.) of either transformation, which is of an
{\it intermediate degree}, or dimension, and involves {\it one
arbitrary constant\/}~$b$; that is to say, {\it one more\/} than
the equation of the highest dimension~(1.), and {\it one fewer\/}
than the equation of lowest dimension~(7.).

\bigbreak

25.
As the equation
${\rm S} \mathbin{.} \alpha \alpha' \alpha'' = 0$
has been seen to express that the three vectors
$\alpha$~$\alpha'$~$\alpha''$
represent coplanar lines, or that any one of these three lines,
for example the line represented by the vector~$\alpha$, is in
the plane determined by the other two, when they diverge from a
common origin; so, if we make for abridgment
$$\left. \eqalign{
\beta
   &= {\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
         \mathbin{.} {\rm V} \mathbin{.}
         \alpha''' \alpha^{\rm IV} ),\cr
  \beta'
   &= {\rm V} ( {\rm V} \mathbin{.} \alpha' \alpha''
         \mathbin{.} {\rm V} \mathbin{.}
         \alpha^{\rm IV} \alpha^{\rm V} ),\cr
  \beta''
   &= {\rm V} ( {\rm V} \mathbin{.} \alpha'' \alpha'''
         \mathbin{.} {\rm V} \mathbin{.}
         \alpha^{\rm V} \alpha ),\cr}
   \right\}
   \eqno (1.)$$
the equation
$${\rm S} \mathbin{.} \beta \beta' \beta'' = 0
   \eqno (2.)$$
may easily be shown to express that {\it the six vectors
$\alpha$~$\alpha'$~$\alpha''$~$\alpha'''$~$\alpha^{\rm IV}$~$\alpha^{\rm V}$
are homoconic}, or represent {\it six edges of one cone of the
second degree}, if they be supposed to be all drawn from one
common origin of vectors.  For if we regard the five vectors
$\alpha'$~$\alpha''$~$\alpha'''$~$\alpha^{\rm IV}$~$\alpha^{\rm V}$
as given, and the remaining vector~$\alpha$ as variable, then
first the equation (2.) will give for the locus of this variable
vector~$\alpha$, some cone of the second degree; because, by the
definitions~(1.) of $\beta$,~$\beta'$,~$\beta''$, if we change
$\alpha$ to $a \alpha$, $a$ being any scalar, each of the two
vectors $\beta$ and $\beta''$ will also be multiplied by $a$,
while $\beta'$ will not be altered: and therefore the function
${\rm S} \mathbin{.} \beta \beta' \beta''$
will be multiplied by $a^2$, that is by the square of the
scalar~$a$, by which the vector~$\alpha$ is multiplied.  In the
next place, this conical locus of $\alpha$ will contain the given
vector~$\alpha'$; because if we suppose $\alpha = \alpha'$, we
have $\beta = 0$, and the equation~(2.) is satisfied: and in like
manner the locus of $\alpha$ contains the vector
$\alpha^{\rm V}$, because the supposition
$\alpha = \alpha^{\rm V}$ gives $\beta'' = 0$.  In the third
place, the cone contains $\alpha''$ and $\alpha^{\rm IV}$; for if
we suppose $\alpha = \alpha''$, then, by the principle contained
in the formula~(4.) of the last article, we have
$$\beta''
   =  - {\rm V} ( {\rm V} \mathbin{.}
         \alpha^{\rm V} \alpha'' \mathbin{.} {\rm V}
         \mathbin{.} \alpha'' \alpha''' )
   =  \alpha'' \, {\rm S} \mathbin{.}
         \alpha^{\rm V} \alpha'' \alpha''';$$
and by the same principle, under the same condition,
$$\eqalign{
{\rm V} \mathbin{.} \beta \beta'
   &= {\rm V} ( {\rm V} ( {\rm V} \mathbin{.}
         \alpha''' \alpha^{\rm IV} \mathbin{.} {\rm V}
         \mathbin{.} \alpha' \alpha'' ) \mathbin{.}
         {\rm V} ( {\rm V} \mathbin{.}
         \alpha' \alpha'' \mathbin{.} {\rm V}
         \mathbin{.} \alpha^{\rm IV} \alpha^{\rm V} ) ) \cr
   &= - {\rm V} \mathbin{.} \alpha' \alpha'' \mathbin{.}
         {\rm S} ( {\rm V} \mathbin{.}
         \alpha''' \alpha^{\rm IV} \mathbin{.} {\rm V}
         \mathbin{.} \alpha' \alpha'' \mathbin{.} {\rm V}
         \mathbin{.} \alpha^{\rm IV} \alpha^{\rm V} ); \cr}$$
but
${\rm S} ( {\rm V} \mathbin{.}
         \alpha' \alpha'' \mathbin{.} \alpha'' )
   =  {\rm S} \mathbin{.} \alpha' \alpha'' \alpha''
   =   0$;
therefore
${\rm S} \mathbin{.} \beta \beta' \beta''
   =  {\rm S} ( {\rm V} \mathbin{.} \beta \beta'
         \mathbin{.} \beta'' ) = 0$;
and in like manner this last condition is satisfied, if
$\alpha = \alpha^{\rm IV}$,
because $\beta$ and ${\rm V} \mathbin{.} \beta' \beta''$
then differ only by scalar coefficients from
$\alpha^{\rm IV}$ and
${\rm V} \mathbin{.} \alpha^{\rm IV} \alpha^{\rm V}$,
respectively, so that the scalar of their product is zero.
Finally, the conical locus of $\alpha$ contains also the
remaining vector~$\alpha'''$, because if we suppose
$\alpha = \alpha'''$, we have
$$\beta
   =  \alpha''' \, {\rm S} \mathbin{.}
         \alpha' \alpha''' \alpha^{\rm IV},\quad
  \beta''
   =  \alpha''' \, {\rm S} \mathbin{.}
         \alpha'' \alpha''' \alpha^{\rm V},$$
and therefore in this case
${\rm S} \mathbin{.} \beta \beta' \beta'' = 0$
because the scalar of the product of $\alpha'''$ and
$\beta' \alpha'''$ is zero.  The locus of $\alpha$ is therefore a
cone of the second degree, containing the five vectors
$\alpha'$,~$\alpha''$,~$\alpha'''$,~$\alpha^{\rm IV}$,~$\alpha^{\rm V}$;
and in exactly the same manner it may be shown without difficulty
that {\it whichever of the six vectors
$\alpha \, \ldots \, \alpha^{\rm V}$
may be regarded as the variable vector, its locus assigned by the
equation\/}~(2.), of the present article, {\it is a cone of the
second degree, containing the five other vectors}.  We may
therefore say that this equation,
$${\rm S} \mathbin{.} \beta \beta' \beta'' = 0,$$
when the symbols $\beta$,~$\beta'$,~$\beta''$ have the meanings
assigned by the definitions~(1.), or (substituting for those
symbols their values) we may say that the following equation
$${\rm S} \{
      {\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
         \mathbin{.} {\rm V} \mathbin{.}
         \alpha''' \alpha^{\rm IV} ) \mathbin{.}
      {\rm V} ( {\rm V} \mathbin{.} \alpha' \alpha''
         \mathbin{.} {\rm V} \mathbin{.}
         \alpha^{\rm IV} \alpha^{\rm V} ) \mathbin{.}
      {\rm V} ( {\rm V} \mathbin{.} \alpha'' \alpha'''
         \mathbin{.} {\rm V} \mathbin{.}
         \alpha^{\rm V} \alpha ) \}
   =  0,
   \eqno (3.)$$
is the {\it equation of homoconicism}, or of {\it uniconality},
expressing that, when it is satisfied, one common cone of the
second degree passes through all the six vectors
$\alpha$~$\alpha'$~$\alpha''$~$\alpha'''$~$\alpha^{\rm IV}$~$\alpha^{\rm V}$,
and enabling us to deduce from it all the properties of this
common cone.

\bigbreak

26.
The considerations employed in the foregoing article might leave
a doubt whether {\it no other\/} cone of the same degree could
pass through the same six vectors; to remove which doubt, by a
method consistent with the spirit of the present theory, we may
introduce the following considerations respecting conical
surfaces in general.

Whatever four vectors may be denoted by
$\alpha$,~$\alpha'$,~$\beta$,~$\beta'$, we have
$${\rm V} ( {\rm V} \mathbin{.} \alpha \alpha' \mathbin{.}
         {\rm V} \mathbin{.} \beta \beta' )
       + {\rm V} ( {\rm V} \mathbin{.} \beta \beta' \mathbin{.}
         {\rm V} \mathbin{.} \alpha \alpha' )
   =  0;
   \eqno (1.)$$
substituting then for the first of these two opposite vector
functions the expression~(3.) of art.~24, and for the second
the expression formed from this by interchanging each $\alpha$
with the corresponding $\beta$, we find, for any four vectors,
$$       \alpha  \, {\rm S} \mathbin{.} \alpha' \beta  \beta'
       - \alpha' \, {\rm S} \mathbin{.} \alpha  \beta  \beta'
       + \beta   \, {\rm S} \mathbin{.} \beta'  \alpha \alpha'
       - \beta'  \, {\rm S} \mathbin{.} \beta   \alpha \alpha'
   =  0.
   \eqno (2.)$$
Again, it follows easily from principles and results already
stated, that the scalar of the product of three vectors changes
sign when any two of those three factors change places among
themselves, so that
$$       {\rm S} \mathbin{.} \alpha \beta \gamma
   =   - {\rm S} \alpha \gamma \beta
   =     {\rm S} \gamma \alpha \beta
   =   - {\rm S} \gamma \beta  \alpha
   =     {\rm S} \beta  \gamma \alpha
   =   - {\rm S} \beta  \alpha \gamma.
   \eqno (3.)$$
Assuming therefore any three vectors,
$\iota$,~$\kappa$,~$\lambda$,
of which the scalar of the product does not vanish, we may express
any fourth vector~$\alpha$ in terms of these three vectors, and
of the scalars of the three products
$\alpha \kappa \lambda$,
$\iota  \alpha \lambda$,
$\iota  \kappa \alpha$,
by the formula:
$$\alpha \, {\rm S} \mathbin{.} \iota \kappa \lambda
   =  \iota   \, {\rm S} \mathbin{.} \alpha \kappa \lambda
    + \kappa  \, {\rm S} \mathbin{.} \iota  \alpha \lambda
    + \lambda \, {\rm S} \mathbin{.} \iota  \kappa \alpha.
   \eqno (4.)$$
Let $\alpha$ be supposed to be a vector function of one scalar
variable~$t$, which supposition may be expressed by writing the
equation
$$\alpha = \phi(t);
   \eqno (5.)$$
and make for abridgment
$${{\rm S} \mathbin{.} \alpha \kappa \lambda
         \over {\rm S} \mathbin{.} \iota  \kappa \lambda}
   =  f_1(t);\quad
  {{\rm S} \mathbin{.} \iota  \alpha \lambda
         \over {\rm S} \mathbin{.} \iota  \kappa \lambda}
   =  f_2(t);\quad
  {{\rm S} \mathbin{.} \iota  \kappa \alpha
         \over {\rm S} \mathbin{.} \iota  \kappa \lambda}
   =  f_3(t);
   \eqno (6.)$$
the forms of these three scalar functions $f_1$~$f_2$~$f_3$
depending on the form of the vector function~$\phi$, and on the
three assumed vectors $\iota$~$\kappa$~$\lambda$, and being
connected with these and with each other by the relation
$$\phi(t) = \iota f_1(t) + \kappa f_2(t) + \lambda f_3(t).
   \eqno (7.)$$
Conceive $t$ to be eliminated between the expressions for the
ratios of the three scalar functions $f_1$~$f_2$~$f_3$, and an
equation of the form
$$F(f_1(t), f_2(t), f_3(t)) = 0
   \eqno (8.)$$
to be thus obtained, in which the function~$F$ is scalar (or
real), and homogeneous; it will then be evident that while the
equation~(5.) may be regarded as the {\it equation of a curve in
space\/} (equivalent to a system of three real equations between
the three co-ordinates of a curve of double curvature and an
auxiliary variable~$t$, which latter variable may represent the
{\it time}, in a motion along this curve), the {\it equation of
the cone\/} which passes through this arbitary curve, and has its
vertex at the origin of vectors, is
$$F (
      {\rm S} \mathbin{.} \alpha \kappa \lambda, \,
      {\rm S} \mathbin{.} \iota  \alpha \lambda, \,
      {\rm S} \mathbin{.} \iota  \kappa \alpha )
   = 0.
   \eqno (9.)$$
Such being a form in this theory for the equation of an
{\it arbitrary conical surface}, we may write, in particular, as
a {\it definition of the cone of the $n${\rm th} degree}, the
equation:
$$\Sigma ( A_{p,q,r}
         ({\rm S} \mathbin{.} \alpha \kappa \lambda)^p
         \mathbin{.}
         ({\rm S} \mathbin{.} \iota  \alpha \lambda)^q
         \mathbin{.}
         ({\rm S} \mathbin{.} \iota  \kappa \alpha )^r )
   = 0;
   \eqno (10.)$$
$p$,~$q$,~$r$ being any three whole numbers, positive or null, of
which the sum is $n$; $A_{p,q,r}$ being a scalar function of
these three numbers; and the summation indicated by $\Sigma$
extending to all their systems of values consistent with the
last-mentioned conditions, which may be written thus:
$$\left. \eqalign{
\sin p \pi = \sin q \pi = \sin r \pi = 0;\cr
  p \geq 0,\quad q \geq 0,\quad r \geq 0;\cr
  p + q + r = n.\cr}
   \right\}
   \eqno (11.)$$

When $n = 2$, these conditions can be satisfied only by
{\it six\/} systems of values of $p$,~$q$,~$r$; therefore, in
this case, there enter only six coefficients~$A$ into the
equation~(10.); consequently {\it five\/} scalar ratios of these
six coefficients are sufficient to particularize a cone of the
second degree; and these can in general be found, by ordinary
elimination between five equations of the first degree, when five
particular vectors are given, such as
$\alpha'$,~$\alpha''$,~$\alpha'''$,~$\alpha^{\rm IV}$,~$\alpha^{\rm V}$,
through which the cone is to pass, or which its surface must
contain upon it.  Hence, as indeed is known from other
considerations, it is in general a determined problem to find the
particular cone of the second degree which contains on its
surface five given straight lines: and the general solution of
this problem is contained in the equation of homoconicism,
assigned in the preceding article.  The proof there given that
the six vectors
$\alpha \, \ldots \, \alpha^{\rm V}$
are homoconic, when they satisfy that equation, does not involve
any property of conic sections, nor even any property of the
circle: on the contrary, that equation having once been
established, by the proof just now referred to, might be used as
the basis of a complete theory of conic sections, and of cones of
the second degree.

\bigbreak

27.
To justify this assertion, without at present attempting to
effect the actual development of such a theory, it may be
sufficient to deduce from the equation  of homoconicism assigned
in article~25, that great and fertile property of the circle, or
of the cone with circular base, which was discovered by the
genius of Pascal.  And this deduction is easy; for the three
auxiliary vectors $\beta$,~$\beta'$,~$\beta''$, introduced in the
equations~(1.) of the 25th article, are evidently, by the
principles stated in other recent articles of this paper, the
respective lines of intersection of three pairs of planes, as
follows:---The planes of $\alpha \alpha'$ and
$\alpha''' \alpha^{\rm IV}$ intersect in $\beta$, those of
$\alpha' \alpha''$ and $\alpha^{\rm IV} \alpha^{\rm V}$ in
$\beta'$; and those of $\alpha'' \alpha'''$ and
$\alpha^{\rm V} \alpha$ in $\beta''$; and in the form~(2.),
article~25, of the equation  of homoconicism, expresses that
these three lines, $\beta$~$\beta'$~$\beta''$, are coplanar.
{\it If then a hexahedral angle be inscribed in a cone of the
second degree, and if each of the six plane faces be prolonged\/}
(if necessary) {\it so as to meet its opposite in a straight
line, the three lines of meeting of opposite faces}, thus
obtained, {\it will be situated in one common plane\/}: which is
a form of the theorem of Pascal.

\bigbreak

28.
The known and purely {\it graphic\/} property of the cone of the
second degree which constitutes the theorem of Pascal, and which
expresses the coplanarity of the three lines of meeting of
opposite plane faces of an inscribed hexahedral angle, may be
transformed into another known but purely {\it metric\/} property
of the same cone of the second degree, which is a form of the
theorem of M.~Chasles, respecting the constancy of an anharmonic
ratio.  This transformation may be effected without difficulty,
on the plan of the present paper; for if we multiply into
${\rm V} \mathbin{.} \gamma \gamma'$ both members of the
equation~(3.) of the 24th article, and then operate by the
characteristic~${\rm S}$., attending to the general properties of
scalars of products, we find, for {\it any six vectors\/}
$\alpha$~$\alpha'$~$\beta$~$\beta'$~$\gamma$~$\gamma'$, the
formula
$${\rm S} (
      {\rm V} \mathbin{.} \alpha  \alpha' \mathbin{.}
      {\rm V} \mathbin{.} \beta   \beta'  \mathbin{.}
      {\rm V} \mathbin{.} \gamma  \gamma' )
   =  {\rm S} \mathbin{.} \alpha  \gamma \gamma' \mathbin{.}
      {\rm S} \mathbin{.} \alpha' \beta  \beta'
    - {\rm S} \mathbin{.} \alpha' \gamma \gamma' \mathbin{.}
      {\rm S} \mathbin{.} \alpha  \beta  \beta';
   \eqno (1.)$$
which gives, for any {\it five\/} vectors
$\alpha$~$\alpha'$~$\alpha''$~$\gamma$~$\gamma'$, this other:
$${\rm S} (
      {\rm V} \mathbin{.} \alpha  \alpha'  \mathbin{.}
      {\rm V} \mathbin{.} \alpha' \alpha'' \mathbin{.}
      {\rm V} \mathbin{.} \gamma  \gamma' )
   =  {\rm S} \mathbin{.} \alpha  \alpha' \alpha'' \mathbin{.}
      {\rm S} \mathbin{.} \gamma \alpha' \gamma'.
   \eqno (2.)$$
If, then, we take six arbitrary vectors
$\alpha$~$\alpha'$~$\alpha''$~$\alpha'''$~$\alpha^{\rm IV}$~$\alpha^{\rm V}$,
and deduce nine other vectors from them by the expressions
$$\left. \multieqalign{
\alpha_0 &= {\rm V} \mathbin{.} \alpha \alpha', &
\alpha_1 &= {\rm V} \mathbin{.} \alpha' \alpha'', &
\alpha_2 &= {\rm V} \mathbin{.} \alpha'' \alpha''', \cr
\alpha_3 &= {\rm V} \mathbin{.} \alpha''' \alpha^{\rm IV}, &
\alpha_4 &= {\rm V} \mathbin{.} \alpha^{\rm IV} \alpha^{\rm V}, &
\alpha_5 &= {\rm V} \mathbin{.} \alpha^{\rm V} \alpha, \cr
\beta    &= {\rm V} \mathbin{.} \alpha_0 \alpha_3, &
\beta'   &= {\rm V} \mathbin{.} \alpha_1 \alpha_4, &
\beta''  &= {\rm V} \mathbin{.} \alpha_2 \alpha_5; \cr}
   \right\}
   \eqno (3.)$$
we shall have, {\it generally},
$$\left. \eqalign{
{\rm S} \mathbin{.} \beta \beta' \beta''
   &= {\rm S} \mathbin{.} \alpha_0 \alpha_2 \alpha_5 \mathbin{.}
      {\rm S} \mathbin{.} \alpha_3 \alpha_1 \alpha_4
    - {\rm S} \mathbin{.} \alpha_3 \alpha_2 \alpha_5 \mathbin{.}
      {\rm S} \mathbin{.} \alpha_0 \alpha_1 \alpha_4 \cr
   &= {\rm S} \mathbin{.} \alpha_0 \alpha_1 \alpha_4 \mathbin{.}
      {\rm S} \mathbin{.} \alpha_2 \alpha_3 \alpha_5
    - {\rm S} \mathbin{.} \alpha_3 \alpha_4 \alpha_1 \mathbin{.}
      {\rm S} \mathbin{.} \alpha_5 \alpha_0 \alpha_2 \cr
   &= {\rm S} \mathbin{.} \alpha \alpha' \alpha''
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha^{\rm IV} \alpha' \alpha^{\rm V}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha'' \alpha''' \alpha^{\rm IV}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha^{\rm V} \alpha''' \alpha \cr
   &\mathrel{\phantom{=}} \mathord{}
    - {\rm S} \mathbin{.} \alpha''' \alpha^{\rm IV} \alpha^{\rm V}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha' \alpha^{\rm IV} \alpha''
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha^{\rm V} \alpha \alpha'
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha'' \alpha \alpha''' \cr
   &= {\rm S} \mathbin{.} \alpha \alpha' \alpha''
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha'' \alpha''' \alpha^{\rm IV}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha \alpha''' \alpha^{\rm V}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha^{\rm V} \alpha' \alpha^{\rm IV} \cr
   &\mathrel{\phantom{=}} \mathord{}
    - {\rm S} \mathbin{.} \alpha \alpha''' \alpha''
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha'' \alpha' \alpha^{\rm IV}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha \alpha' \alpha^{\rm V}
         \mathbin{.}
      {\rm S} \mathbin{.} \alpha^{\rm V} \alpha''' \alpha^{\rm IV}.\cr}
   \right\}
   \eqno (4.)$$
Thus if, in particular, the six vectors
$\alpha \, \ldots \, \alpha^{\rm V}$
are such as to satisfy the condition
$${\rm S} \mathbin{.} \beta \beta' \beta'' = 0,
   \eqno (5.)$$
they will satisfy also this other condition, or this other form
of the same condition:
$$    {{\rm S} \mathbin{.} \alpha \alpha' \alpha''
      \over {\rm S} \mathbin{.} \alpha \alpha''' \alpha''}
      \mathbin{.}
      {{\rm S} \mathbin{.} \alpha'' \alpha''' \alpha^{\rm IV}
      \over {\rm S} \mathbin{.} \alpha'' \alpha' \alpha^{\rm IV}}
   =  {{\rm S} \mathbin{.} \alpha \alpha' \alpha^{\rm V}
      \over {\rm S} \mathbin{.} \alpha \alpha''' \alpha^{\rm V}}
      \mathbin{.}
      {{\rm S} \mathbin{.} \alpha^{\rm V} \alpha''' \alpha^{\rm IV}
      \over {\rm S} \mathbin{.} \alpha^{\rm V} \alpha' \alpha^{\rm IV}};
   \eqno (6.)$$
and reciprocally the former of these two conditions will be
satisfied if the latter be so.

These two equations (5.) and (6.), express, therefore, each in
its own way, the existence of one and the same geometrical
relation between the six vectors
$\alpha$~$\alpha'$~$\alpha''$~$\alpha'''$~$\alpha^{\rm IV}$~$\alpha^{\rm V}$:
and a slight study of the {\it forms\/} of these equations
suffices to render evident that they both agree in expressing
that these six vectors are {\it homoconic}, in the sense of the
25th article; or in other words, that the six vectors are sides
(or edges) of one common cone of the second degree.  Indeed the
equation (5.) of the present article, in virtue of the
definitions~(3.), coincides with the equation~(2.) of the article
just cited, the symbols $\beta$,~$\beta'$,~$\beta''$ retaining
in the one the significations which they had received in the
other.  The recent transformations show, therefore, that the
{\it equation of homoconicism}, assigned in article~25, may be
put under the form~(6.) of the present article, which is
different, and in {\it some\/} respects simpler.  The former
expresses a {\it graphic\/} property, or relation between
{\it directions}, namely that the three lines
$\beta$,~$\beta'$,~$\beta''$, which are the respective
intersections of the three pairs of planes
$(\alpha \alpha', \alpha''' \alpha^{\rm IV})$,
$(\alpha' \alpha'', \alpha^{\rm IV} \alpha^{\rm V})$,
$(\alpha'' \alpha''', \alpha^{\rm V} \alpha)$,
are all situated in one common plane, if the six homoconic
vectors be supposed to diverge from one common origin; the latter
expresses the {\it metric\/} property, or relation between
{\it magnitudes}, that the ratio compounded of the two ratios of
the two pyramids
$(\alpha \alpha' \alpha'')$
$(\alpha'' \alpha''' \alpha^{\rm IV})$
to the two other pyramids
$(\alpha \alpha''' \alpha'')$
$(\alpha'' \alpha' \alpha^{\rm IV})$,
or that the product of the volumes of the first pair of pyramids
divided by the product of the volumes of the second pair, does
not vary, when the vector~$\alpha''$, which is the common edge
of these four pyramids, is changed to the new but homoconic
vector~$\alpha^{\rm V}$, as their new common edge, the four
remaining homoconic and coinitial edges
$\alpha$~$\alpha'$~$\alpha'''$~$\alpha^{\rm IV}$ of the pyramids
being supposed to undergo no alteration.  The one is the
expression of the property of the {\it mystic hexagram\/} of
Pascal; the other is an expression of the constancy of the
{\it anharmonic ratio\/} of Chasles.\footnote*{Although the
foregoing process of calculation, and generally the method of
treating geometrical problems by quaternions, which has been
extended by the writer to questions of dynamics and thermology,
appears to him to be new, yet it is impossible for him, in
mentioning here the name of Chasles, to abstain from
acknowledging the deep intellectual obligations under which he
feels himself to be, for the information, and still more for the
impulse given to his mind by the perusal of that very interesting
and excellent History of Geometrical Science, which is so widely
known by its own modest title of {\it Aper\c{c}u Historique\/}
(Brussels, 1837).  He has also endeavoured to profit by a study
of the Memoirs by M.~Chasles, on Spherical Conics and Cones of
the Second Degree, which have been translated, with Notes and an
Appendix, by the Rev.\ Charles Graves (Dublin 1841); and desires
to take this opportunity of adding, that he conceives himself to
have derived assistance, as well as encouragement, in his
geometrical researches generally, from the frequent and familiar
intercourse which he has enjoyed with the last-mentioned
gentleman.}
The calculus of Quaternions (or the method of scalars and
vectors) enables us, as we have seen, to pass, by a very short
and simple symbolical transition, from either to the other of
these two great and known properties of the cone of the second
degree.

\bigbreak

29.
If we denote by $\alpha$ and $\beta$ two constant vectors, and by
$\rho$ a variable vector, all drawn from one common origin; if
also we denote by $u$ and $v$ two variable scalars, depending on
the foregoing vectors $\alpha$,~$\beta$,~$\rho$ by the relations
$$\left. \eqalign{
u  &= 2 {\rm S} \mathbin{.} \alpha \rho
    = \alpha \rho + \rho \alpha;\cr
v^2 &= - 4 ( {\rm V} \mathbin{.} \beta \rho )^2
    = - (\beta \rho - \rho \beta )^2;\cr}
   \right\}
   \eqno (1.)$$
we may then represent the central surfaces of the second degree
by equations of great simplicity, as follows:---

An ellipsoid, with three unequal axes, may be represented by the
equation
$$u^2 + v^2 = 1.
   \eqno (2.)$$
One of its circumscribing cylinders of revolution has for
equation
$$v^2 = 1;
   \eqno (3.)$$
the plane of the ellipse of contact is represented by
$$u = 0;
   \eqno (4.)$$
and the system of the two tangent planes of the ellipsoid,
parallel to the plane of this ellipse, by
$$u^2 = 1.
   \eqno (5.)$$
A hyperboloid of one sheet, touching the same cylinder in the
same ellipse, is denoted by the equation
$$u^2 - v^2 = -1;
   \eqno (6.)$$
its asymptotic cone by
$$u^2 - v^2 = 0;
   \eqno (7.)$$
and a hyperboloid of two sheets, with the same asymptotic
cone~(7.), and with the two tangent planes~(5.), is represented
by this other equation,
$$u^2 - v^2 = 1.
   \eqno (8.)$$
By changing $\rho$ to $\rho - \gamma$, where $\gamma$ is a third
arbitrary but constant vector, we introduce an arbitrary origin
of vectors, or an arbitrary position of the centre of the
surface, as referred to such an origin.  And the general problem
of determining that individual surface of the second degree
(supposed to have a centre, until the calculation shall show in
any particular question that it has none), which shall pass
through {\it nine given points}, may thus be regarded as
equivalent to the problem of finding {\it three constant
vectors}, $\alpha$,~$\beta$~$\gamma$, which shall, for nine given
values of the variable vector~$\rho$, satisfy one equation of the
form
$$\{ \alpha (\rho - \gamma) + (\rho - \gamma) \alpha \}^2
   \pm \{ \beta (\rho - \gamma) - (\rho - \gamma) \beta \}^2
   =  \pm 1;
   \eqno (9.)$$
with suitable selections of the two ambiguous signs, depending
on, and in their turn determining, the particular species of the
surface.

\bigbreak

30.
The equation of the ellipsoid with three unequal axes, referred
to its centre as the origin of vectors, may thus be presented
under the following form (which was exhibited to the Royal Irish
Academy in December 1845):
$$(\alpha \rho + \rho \alpha)^2 - (\beta \rho - \rho \beta)^2
   =  1;
   \eqno (1.)$$
and which decomposes itself into two factors, as follows:
$$    ( \alpha \rho + \rho \alpha + \beta \rho - \rho \beta )
      ( \alpha \rho + \rho \alpha - \beta \rho + \rho \beta )
   = 1.
   \eqno (2.)$$
These two factors are not only separately linear with respect to
the variable vector~$\rho$, but are also (by art.~20,
Phil.\ Mag.\ for July 1846) {\it conjugate quaternions\/}; they
have therefore a common {\it tensor}, which must be equal to
unity, so that we may write the equation of the ellipsoid under
this other form,
$${\rm T} ( \alpha \rho + \rho \alpha + \beta \rho - \rho \beta )
   =  1;
   \eqno (3.)$$
if we use, as in the 19th article, Phil.\ Mag., July 1846, the
characteristic~${\rm T}$ to denote the operation of taking the
tensor of a quaternion.  Let $\sigma$ be an auxiliary vector,
connected with the vector~$\rho$ of the ellipsoid by the equation
$$\sigma = \rho (\alpha - \beta) \rho^{-1};
   \eqno (4.)$$
we shall then have, by (3.), and by the general law for the tensor
of a product,
$${\rm T} (\alpha + \beta + \sigma) \mathbin{.} {\rm T} \rho
   =  1;
   \eqno (5.)$$
but also
$$(\alpha - \beta + \sigma) \rho
   =  (\alpha - \beta) \rho + \rho (\alpha - \beta),
   \eqno (6.)$$
where the second member is scalar; therefore, using the
characteristic~${\rm U}$ to denote the operation of taking the
{\it versor\/} of a quaternion, as in the same art.~19, we
have the equation
$${\rm U} (\alpha - \beta + \sigma) \mathbin{.} {\rm U} \rho
   = \mp 1;
   \eqno (7.)$$
and the dependence of the variable vector~$\rho$ of the ellipsoid
on the auxiliary vector~$\sigma$ is expressed by the formula
$$\rho = \pm {{\rm U} (\alpha - \beta + \sigma)
            \over {\rm T} (\alpha + \beta + \sigma)}.
   \eqno (8.)$$
Besides, the length of this auxiliary vector~$\sigma$ is
constant, and equal to that of $\alpha - \beta$, because the
equation~(4.) gives
$${\rm T} \sigma = {\rm T} (\alpha - \beta);
   \eqno (9.)$$
we may therefore regard $\alpha - \beta$ as the vector of the
centre~${\sc c}$ of a certain auxiliary sphere, of which the
surface passes through the centre~${\sc a}$ of the ellipsoid; and
may regard the vector $\alpha - \beta + \sigma$ as a variable and
auxiliary {\it guide-chord\/} ${\sc a} {\sc d}$ of the same
{\it guide-sphere}, which chord determines the (exactly similar
or exactly opposite) direction of the variable radius vector
${\sc a} {\sc e}$ (or $\rho$) of the ellipsoid.  At the same
time, the constant vector $-2\beta$, drawn from the same constant
origin as before, namely the centre~${\sc a}$ of the ellipsoid,
will determine the position of a certain fixed point~${\sc b}$,
having this remarkable property, that its {\it distance\/} from
the extremity~${\sc d}$ of the variable guide-chord
from~${\sc a}$, will represent the {\it reciprocal of the length
of the radius vector\/}~$\rho$, or {\it the proximity\/}
$({\sc a} {\sc e})^{-1}$ of the point~${\sc e}$ on the surface of
the ellipsoid to the centre (the use of this word ``proximity''
being borrowed from Sir John Herschel).  Supposing then, for
simplicity, that the fixed point~${\sc b}$ is external to the
fixed sphere, which does not essentially diminish the generality
of the question; and taking, for the unit of length, the length
of a tangent to that sphere from that point; we may regard
${\sc a} {\sc e}$ and ${\sc b} {\sc d}'$ as two equally long
lines, or may write the equation
$$\overline{{\sc a} {\sc e}} = \overline{{\sc b} {\sc d}'},
   \eqno (10.)$$
if ${\sc d}'$ be the other point of intersection of the straight
line ${\sc b} {\sc d}$ with the sphere.

\bigbreak

31.
Hence follows this very simple
{\it construction\/}\footnote*{This construction has already been
printed in the Proceedings of the Royal Irish Academy for
1846; but it is conceived that its being reprinted here may be
acceptable to some of the readers of the London, Edinburgh and
Dublin Philosophical Magazine; in which periodical (namely in the
Number for July~1844) the first {\it printed\/} publication of
the fundamental equations of the theory of quaternions
($i^2 = j^2 = k^2 = -1$, $ij = k$, $jk = i$, $ki = j$, $ji = -k$,
$kj = -i$, $ik = -j$) took place, although those equations had
been communicated to the Royal Irish Academy in November 1843,
and had been exhibited at a meeting of the Council during the
preceding month.}
{\it for an ellipsoid\/} (with three unequal axes), by means of a
sphere and an external point, to which the author was led by the
foregoing process, but which may also be deduced from principles
more generally known.  From a fixed point~${\sc a}$ on the
surface of a sphere, draw a variable chord~${\sc a} {\sc d}$; let
${\sc d}'$ be the second point of intersection of the spheric
surface with the secant~${\sc b} {\sc d}$, drawn to the variable
extremity~${\sc d}$ of this chord~${\sc a} {\sc d}$ from a fixed
external point~${\sc b}$; take the radius
vector~${\sc a} {\sc e}$ equal in length to ${\sc b} {\sc d}'$,
and in direction either coincident with, or opposite to, the
chord~${\sc a} {\sc d}$; {\it the locus of the point~${\sc e}$,
thus constructed, will be an ellipsoid}, which will pass through
the point~${\sc b}$.

\bigbreak

32.
We may also say that {\it if of a quadrilateral\/}
(${\sc a} {\sc b} {\sc e} {\sc d}'$),
{\it of which one side\/}~(${\sc a} {\sc b}$) {\it is given in
length and in position, the two diagonals\/}
(${\sc a} {\sc e}$, ${\sc b} {\sc d}'$) {\it be equal to each
other in length, and intersect\/} (in~${\sc d}$) {\it on the
surface of a given sphere\/} (with centre~${\sc c}$), {\it of
which a chord\/} (${\sc a} {\sc d}'$) {\it is a side of the
quadrilateral adjacent to the given side\/} (${\sc a} {\sc b}$),
{\it then the other side\/} (${\sc b} {\sc e}$), {\it adjacent to
the same given side, is a chord of a given ellipsoid}.  The form,
position, and magnitude of an {\it ellipsoid\/} (with three
unequal axes), may thus be made to depend on the form, position,
and magnitude of a {\it generating
triangle\/}~${\sc a} {\sc b} {\sc c}$.  Two sides of this
triangle, namely ${\sc b} {\sc c}$ and ${\sc c} {\sc a}$, are
perpendicular to the {\it two planes of circular section\/}; and
the third side~${\sc a} {\sc b}$ is perpendicular to {\it one of
the two planes of circular projection\/} of the ellipsoid,
because it is the axis of revolution of one of the two
circumscribed circular cylinders.  This {\it triple reference to
circles\/} is perhaps the cause of the extreme facility with
which it will be found that many fundamental properties of the
ellipsoid may be deduced from this mode of generation.  As an
example of such deduction, it may be mentioned that the known
proportionality of the difference of the squares of the
reciprocals of the semiaxes of a diametral section to the product
of the sines of the inclinations of its plane to the two planes
of circular section, presents itself under the form of a
proportionality of the same difference of squares to the
rectangle under the projections of the two sides
${\sc b} {\sc c}$ and ${\sc c} {\sc a}$ of the generating
triangle on the plane of the elliptic section.

\bigbreak

33.
For the sake of those mathematical readers who are familiar with
the method of co-ordinates, and not with the method of
quaternions, the writer will here offer an investigation, by the
former method, of that general property of the ellipsoid to which
he was conducted by the latter method, and of which an account
was given in a recent Number of this Magazine (for June 1847).

Let $x$~$y$~$z$ denote, as usual, the three rectangular
co-ordinates of a point, and let us introduce two real functions
of these three co-ordinates, and of six arbitrary but real
constants, $l$~$m$~$n$ $l'$~$m'$~$n'$, which functions shall be
denoted by $u$ and $v$, and shall be determined by the two
following relations:
$$u (l l' + m m' + n n')
   =  l' x + m' y + n' z;$$
$$v^2 (l l' + m m' + n n')^2
   =  (ly - mx)^2 + (mz - ny)^2 + (nx - lz)^2;$$
then the equation
$$u^2 + v^2 = 1
   \eqno (1.)$$
will denote (as received principles suffice to show) that the
curved surface which is the locus of the point $x$~$y$~$z$ is an
ellipsoid, having its centre at the origin of co-ordinates; and
conversely this equation $u^2 + v^2 = 1$ may represent any such
ellipsoid, by a suitable choice of the six real constants
$l$~$m$~$n$ $l'$~$m'$~$n'$.  At the same time the equation
$$u^2 = 1$$
will represent a system of two parallel planes, which touch the
ellipsoid at the extremities of the diameter denoted by the
equation
$$v = 0;$$
and this diameter will be the axis of revolution of a certain
circumscribed cylinder, namely of the cylinder denoted by the
equation
$$v^2 = 1;$$
the equation of the plane of the ellipse of contact, along which
this circular cylinder envelopes the ellipsoid, being, in the
same notation,
$$u = 0;$$
all which may be inferred from ordinary principles, and agrees
with what was remarked in the 29th article of this paper.

\bigbreak

34.
This being premised, let us next introduce three new constants,
$p$,~$q$,~$r$, depending on the six former constants by the three
relations
$$2p = l + l',\quad 2q = m + m',\quad 2r = n + n'.$$
We shall then have
$$l' x + m' y + n' z = 2 (px + qy + rz) - (lx + my + nz);$$
and the equation~(1.) of the ellipsoid will become
$$\eqalign{
(ll' + mm' + nn')^2
   &= (l^2 + m^2 + n^2) (x^2 + y^2 + z^2)
       - 4 (lx + my + nz) (px + qy + rz) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 4 (px + qy + rz)^2 \cr
   &= (x^2 + y^2 + z^2)
         \{ (l - x')^2 + (m - y')^2 + (n - z')^2 \},\cr}$$
if we introduce three new variables, $x'$,~$y'$,~$z'$, depending
on the three old variables $x$,~$y$,~$z$, or rather on their
ratios, and on the three new constants $p$,~$q$,~$r$, by the
conditions,
$${x' \over x} = {y' \over y} = {z' \over z}
   = {2 (px + qy + rz) \over x^2 + y^2 + z^2}.$$

These three last equations give, by elimination of the two
ratios of $x$,~$y$,~$z$, the relation
$$x'^2 + y'^2 + z'^2 = 2 (p x' + q y' + r z');$$
the new variables $x'$,~$y'$,~$z'$, are therefore co-ordinates of
a new point, which has for its locus a certain spheric surface,
passing through the centre of the ellipsoid; and the same new
point is evidently contained on the radius vector drawn from that
centre of the ellipsoid to the point $x \, y \, z$, or on that
radius vector prolonged.  We see, also, that the length of this
radius vector of the ellipsoid, or the distance of the point
$x \, y \, z$ from the origin of the co-ordinates, is inversely
proportional to the distance of the new point $x' \, y' \, z'$ of
the spheric surface from the point $l \, m \, n$, which latter is
a certain fixed point upon the surface of the ellipsoid.  This
result gives already an easy and elementary mode of generating
the latter surface, which may however be reduced to a still
greater degree of simplicity by continuing the analysis as
follows.

\bigbreak

35.
Let the straight line which connects the two points
$x' \, y' \, z'$ and $l \, m \, n$ be prolonged, if necessary, so
as to cut the same spheric surface again in another point
$x'' \, y'' \, z''$; we shall then have the equation
$$x''^2 + y''^2 + z''^2 = 2 (p x'' + q y'' + r z''),$$
from which the new co-ordinates $x''$,~$y''$,~$z''$ may be
eliminated by substituting the expressions
$$x'' = l + t (x' - l),\quad
  y'' = m + t (y' - m),\quad
  z'' = n + t (z' - n);$$
and the root that is equal to unity is then to be rejected, in
the resulting quadratic for $t$.  Taking therefore for~$t$ the
product of the roots of that quadratic, we find
$$t = { l^2 + m^2 + n^2 - 2 (lp + mq + nr)
         \over (x' - l)^2 + (y' - m)^2 + (z' - n)^2 };$$
therefore also, by the last article,
$$t = { x^2 + y^2 + z^2
         \over l^2 + m^2 + n^2 - 2 (lp + mq + nr) };$$
consequently
$$t^2 = {x^2 + y^2 + z^2
         \over (x' - l)^2 + (y' - m)^2 + (z' - n)^2 };$$
and finally,
$$(x'' - l)^2 + (y'' - m)^2 + (z'' - n)^2
   =  x^2 + y^2 + z^2.
   \eqno (2.)$$

Denoting by ${\sc a}$, ${\sc b}$, ${\sc c}$, the three fixed
points of which the co-ordinates are respectively $(0, 0, 0)$,
$(l, m, n)$, $(p, q, r)$; and by ${\sc d}$, ${\sc d}'$,
${\sc e}$, the three variable points of which the co-ordinates are
$(x', y', z')$, $(x'', y'', z'')$, $(x, y, z)$;
${\sc a} \, {\sc b} \, {\sc e} \, {\sc d}'$
may be regarded as a plane quadrilateral, of which the diagonals
${\sc a} {\sc e}$ and ${\sc b} {\sc d}'$ intersect each other in
a point~${\sc d}$ on a fixed spheric surface, which has its
centre at ${\sc c}$, and passes through ${\sc a}$ and ${\sc d}'$;
so that one side ${\sc d}' {\sc a}$ of the quadrilateral,
adjacent to the fixed side ${\sc a} {\sc b}$, is a chord of this
fixed sphere.  And the equation (2.) expresses that the
{\it other side~${\sc b} {\sc e}$ of the same plane
quadrilateral, adjacent to the same fixed side~${\sc a} {\sc b}$,
is a chord of a fixed ellipsoid, if the two diagonals
${\sc a} {\sc e}$, ${\sc b} {\sc d}'$ of the quadrilateral be
equally long\/}; so that a general and characteristic property of
the ellipsoid, sufficient for the construction of that surface,
and for the investigation of all its properties, is included in
the remarkably simple and eminently geometrical formula
$$\overline{{\sc a} {\sc e}}
   =  \overline{{\sc b} {\sc d}'};
   \eqno (3.)$$
the locus of the point~${\sc e}$ being an ellipsoid, which passes
through~${\sc b}$, and has its centre at~${\sc a}$, when this
condition is satisfied.

This formula~(3.), which has already been printed in this
Magazine as the equation~(10.) of article~30 of this paper, may
therefore be deduced, as above, from generally admitted
principles, by the Cartesian method of co-ordinates; although it
had not been known to geometers, so far as the present writer has
hitherto been able to ascertain, until he was led to it, in the
summer of 1846\footnote*{See the Proceedings of the Royal Irish
Academy.},
by an entirely different method; namely by applying his calculus
of quaternions to the discussion of one of those new forms for
the equations of central surfaces of the second order, which he
had communicated to the Royal Irish Academy in December 1845.

\bigbreak

36.
As an example (already alluded to in the 32nd article of this
paper) of the {\it geometrical\/} employment of the formula~(3.),
or of the equality which it expresses as existing between the
lengths of the two diagonals of a certain plane quadrilateral
connected with that new construction of the ellipsoid to which
the writer was thus led by quaternions, let us now propose to
investigate geometrically, by the help of that equality of
diagonals, the difference of the squares of the reciprocals of
the greatest and least semi-diameters of any plane and diametral
section of an ellipsoid (with three unequal axes).  Conceive then
that the ellipsoid, and the auxiliary sphere employed in the
above-mentioned construction, are both cut by a plane
${\sc a} {\sc b}' {\sc c}'$, on which ${\sc b}'$ and ${\sc c}'$
are the orthogonal projections of the fixed points ${\sc b}$ and
${\sc c}$; the auxiliary point~${\sc d}$ may thus be conceived to
move on the circumference of a circle, which passes through
${\sc a}$, and has its centre at ${\sc c}'$; and since
${\sc a} {\sc e}$, being equal in length to ${\sc b} {\sc d}'$
(because these are the two equal diagonals of the quadrilateral
in the construction), must vary inversely as ${\sc b} {\sc d}$
(by an elementary property of the sphere), we are to seek the
difference of the squares of the extreme values of
${\sc b} {\sc d}$, or of ${\sc b}' {\sc d}$, because the square
of the perpendicular ${\sc b} {\sc b}'$ is constant for the
section.  But the longest and shortest straight lines,
${\sc b}' {\sc d}_1$, ${\sc b}' {\sc d}_2$, which can thus be
drawn to the auxiliary circle round ${\sc c}'$, from the fixed
point~${\sc b}'$ in its plane, are those drawn to the extremities
of that diameter ${\sc d}_1 {\sc c}' {\sc d}_2$ of this circle
which passes through or tends towards ${\sc b}'$; so that the
four points
${\sc d}_1 \, {\sc c}' \, {\sc d}_2 \, {\sc b}'$
are on one straight line, and the difference of the squares of
${\sc b}' {\sc d}_1$, ${\sc b}' {\sc d}_2$ is equal to four times
the rectangle under ${\sc b}' {\sc c}'$ and ${\sc c}' {\sc d}_1$,
or under ${\sc b}' {\sc c}'$ and ${\sc c}' {\sc a}$.  We see
therefore that the shortest and longest semi-diameters
${\sc a} {\sc e}_1$, ${\sc a} {\sc e}_2$ of the diametral section
of the ellipsoid, are perpendicular to each other, because (by
the construction above-mentioned) they coincide in their
directions respectively with the two supplementary chords
${\sc a} {\sc d}_1$, ${\sc a} {\sc d}_2$ of the section of the
auxiliary sphere, and an angle in a semicircle is a right angle;
and at the same time we see also that the difference of the
squares of the reciprocals of these two rectangular semiaxes of a
diametral section of the ellipsoid varies, in passing from one
such section to another, proportionally to the rectangle under
the projections, ${\sc b}' {\sc c}'$ and ${\sc c}' {\sc a}$, of
the two fixed lines ${\sc b} {\sc c}$ and ${\sc c} {\sc a}$, on
the plane of the variable section.  The difference of the squares
of these reciprocals of the semi-axes of a section therefore
varies (as indeed it is well-known to do) proportionally to the
product of the sines of the inclinations of the plane of the
section to two fixed diametral planes, which cut the ellipsoid in
circles; and we see that the normals to these two latter or
cyclic planes have precisely the directions of the sides
${\sc b} {\sc c}$, ${\sc c} {\sc a}$ of the {\it generating
triangle\/} ${\sc a} {\sc b} {\sc c}$, which has for its corners
the three fixed points employed in the foregoing construction: so
that the auxiliary and {\it diacentric sphere}, employed in the
same construction, touches one of those two cyclic planes at the
centre~${\sc a}$ of the ellipsoid.  If we take, as we are allowed
to do, the point~${\sc b}$ external to this sphere, then the
distance~${\sc b} {\sc c}$ of this external point~${\sc b}$ from
the centre~${\sc c}$ of the sphere is (by the construction) the
semisum of the greatest and least semiaxes of the ellipsoid,
while the radius ${\sc c} {\sc a}$ of the sphere is the
semidifference of the same two semiaxes: and (by the same
construction) these greatest and least semiaxes of the ellipsoid,
or their prolongations, intersect the surface of the same
diacentric sphere in points which are respectively situated on
the finite straight line~${\sc b} {\sc c}$ itself, and on the
prolongation of that line.  The remaining side~${\sc a} {\sc b}$
of the same fixed or generating triangle
${\sc a} {\sc b} {\sc c}$ is a semidiameter of the ellipsoid,
drawn in the direction of the axis of one of the two
circumscribed cylinders of revolution; a property which was
mentioned in the 32nd article, and which may be seen to hold
good, not only from the recent analysis conducted by the
Cartesian method, but also and more simply from the geometrical
consideration that the constant rectangle under the two straight
lines ${\sc b} {\sc d}$ and ${\sc a} {\sc e}$, in the
construction, exceeds the double area of the triangle
${\sc a} {\sc b} {\sc e}$, and therefore exceeds the rectangle
under the fixed line ${\sc a} {\sc b}$ and the perpendicular let
fall thereon from the variable point~${\sc e}$ of the ellipsoid,
except at the limit where the angle~${\sc a} {\sc d} {\sc b}$ is
right; which last condition determines a circular locus for
${\sc d}$, and an elliptic locus for ${\sc e}$, namely that
ellipse of contact along which a cylinder of revolution round
${\sc a} {\sc b}$ envelops the ellipsoid, and which here presents
itself as a section of the cylinder by a plane.  The radius of
this cylinder is equal to the line ${\sc b} {\sc g}$, if
${\sc g}$ be the point of intersection, distinct from ${\sc a}$,
of the side~${\sc a} {\sc b}$ of the generating triangle with the
surface of the diacentric sphere; which line ${\sc b} {\sc g}$ is
also easily shown, on similar geometrical principles, as a
consequence of the same construction, to be equal to the common
radius of the two circular sections, or to the mean semiaxis of
the ellipsoid, which is perpendicular to the greatest and the
least.  Hence also the side ${\sc a} {\sc b}$ of the generating
triangle is, in length, a fourth proportional to the three
semiaxes, that is to the mean, the least, and the greatest, or to
the mean, the greatest, and the least, of the three principal and
rectangular semidiameters of the ellipsoid.

\bigbreak

37.
Resuming now the quaternion form of the equation of the
ellipsoid,
$$(\alpha \rho + \rho \alpha)^2 - (\beta \rho - \rho \beta)^2
   =  1,
   \eqno (1.)$$
and making
$$\alpha + \beta = {\iota  \over \iota^2 - \kappa^2},\quad
  \alpha - \beta = {\kappa \over \iota^2 - \kappa^2},
   \eqno (2.)$$
and
$${\iota \rho + \rho \kappa \over \iota^2 - \kappa^2}
   =  {\rm Q},\quad
  {\rho \iota + \kappa \rho \over \iota^2 - \kappa^2}
   =  {\rm Q}',
   \eqno (3.)$$
the two linear factors of the first member of the equation~(1.)
become the two conjugate quaternions ${\rm Q}$ and ${\rm Q}'$, so
that the equation itself becomes
$${\rm Q} {\rm Q}' = 1.
   \eqno (4.)$$
But by articles 19 and 20 (Phil.\ Mag.\ for July 1846), the
product of any two conjugate quaternions is equal to the square
of their common tensor; this common tensor of the two quaternions
${\rm Q}$ and ${\rm Q}'$ is therefore equal to unity.  Using,
therefore, as in those articles, the letter~${\rm T}$ as the
characteristic of the operation of {\it taking the tensor\/} of a
quaternion, the equation of the ellipsoid reduces itself to the
form
$${\rm T} {\rm Q} = 1;
   \eqno (5.)$$
or, substituting for ${\rm Q}$ its expression~(3.),
$${\rm T} \left(
      {\iota \rho + \rho \kappa \over \iota^2 - \kappa^2}
   \right) = 1;
   \eqno (6.)$$
which latter form might also have been obtained, by the
substitutions~(2.), from the equation (3.) of the 30th article
(Phil.\ Mag., June 1847), namely from the
following:\footnote*{See equation~(35.) of the Abstract in the
Proceedings of the Royal Irish Academy for July 1846.  The
equation of the ellipsoid marked (1.) in article~37 of the
present paper, was communicated to the Academy in December~1845,
and is numbered (21.) in the Proceedings of that date.}
$${\rm T} (\alpha \rho + \rho \alpha + \beta \rho - \rho \beta)
   =  1.
   \eqno (7.)$$

\bigbreak

38.
In the geometrical construction or generation of the ellipsoid,
which was assigned in the preceding articles of this paper (see
the Numbers of the Philosophical Magazine for June and September
1847), the significations of some of the recent symbols are the
following.  The two constant vectors $\iota$ and $\kappa$ may be
regarded as denoting, respectively, (in lengths and in
directions,) the two sides of the generating triangle
${\sc a} {\sc b} {\sc c}$, which are drawn from the
centre~${\sc c}$ of the auxiliary and diacentric sphere, to the
fixed superficial point~${\sc b}$ of the ellipsoid, and to the
centre~${\sc a}$ of the same ellipsoid; the third side of the
triangle, or the vector from ${\sc a}$ to ${\sc b}$, being
therefore denoted (in length and in direction) by
$\iota - \kappa$: while $\rho$ is the radius vector of the
ellipsoid, drawn from the centre~${\sc a}$ to a variable
point~${\sc e}$ of the surface; so that the constant vector
$\iota - \kappa$ is, by the construction, a particular value of
this variable vector~$\rho$.  The vector from ${\sc a}$ to
${\sc c}$, being the opposite of that from ${\sc c}$ to
${\sc a}$, is denoted by $-\kappa$; and if ${\sc d}$ be still
the same auxiliary point on the surface of the auxiliary sphere,
which was denoted by the same letter in the account already
printed of the construction, then the vector from ${\sc c}$ to
${\sc d}$, which may be regarded as being (in a sense to be
hereafter more fully considered) the {\it reflexion\/} of
$-\kappa$ with respect to $\rho$, is $= - \rho \kappa \rho^{-1}$;
and consequently the vector from ${\sc d}$ to ${\sc b}$ is
$= \iota + \rho \kappa \rho^{-1}$.  The lengths of the two
straight lines ${\sc b} {\sc d}$, and ${\sc a} {\sc e}$, are
therefore respectively denoted by the two tensors
${\rm T} (\iota + \rho \kappa \rho^{-1})$ and $T \rho$; and the
rectangle under those two lines is represented by the product of
these two tensors, that is by the tensor of the product, or by
${\rm T} ( \iota \rho + \rho \kappa)$.  But by the fundamental
equality of the lengths of the diagonals, ${\sc a} {\sc e}$,
${\sc b} {\sc d}'$, of the plane quadrilateral
${\sc a} {\sc b} {\sc e} {\sc d}'$ in the construction, this
rectangle under ${\sc b} {\sc d}$ and ${\sc a} {\sc e}$ is equal
to the constant rectangle under ${\sc b} {\sc d}$ and
${\sc b} {\sc d}'$, that is under the whole secant and its
external part, or to the square on the tangent from ${\sc b}$, if
the point~${\sc b}$ be supposed external to the auxiliary sphere,
which has its centre at ${\sc c}$, and passes through ${\sc d}$,
${\sc d}'$ and ${\sc a}$.  Thus
${\rm T} (\iota \rho + \rho \kappa)$
is equal to $({\rm T} \iota)^2 - ({\rm T} \kappa)^2$, or to
$\kappa^2 - \iota^2$, which difference is here a positive scalar,
because it is supposed that ${\sc c} {\sc b}$ is longer than
${\sc c} {\sc a}$, or that
$${\rm T} \iota > {\rm T} \kappa;
   \eqno (8.)$$
and the quaternion equation (6.) of the ellipsoid reproduces
itself, as a result of the geometrical construction, under the
slightly simplified form\footnote*{See the Proceedings of the
Royal Irish Academy for July 1846, equation (44.).}
$${\rm T} (\iota \rho + \rho \kappa)
   =  \kappa^2 - \iota^2.
   \eqno (9.)$$
And to verify that this equation relative to $\rho$ is satisfied
(as we have seen that it ought to be) by the particular value
$$\rho = \iota - \kappa,
   \eqno (10.)$$
which corresponds to the particular position~${\sc b}$ of the
variable point~${\sc e}$ on the surface of the ellipsoid, we have
only to observe that, identically,
$$\iota (\iota - \kappa) + (\iota - \kappa) \kappa
   =  \iota^2 - \iota \kappa + \iota \kappa - \kappa^2
   =  \iota^2 - \kappa^2
   =  - (\kappa^2 - \iota^2);$$
and that (by article 19) the tensor of a negative scalar is equal
to the positive opposite thereof.

\bigbreak

39.
The foregoing article contains a sufficiently simple process for
the {\it retranslation\/} of the geometrical
construction\footnote*{The brevity and novelty of this rule for
constructing that important surface may perhaps justify the
reprinting it here.  It was as follows: From a fixed
point~${\sc a}$ on the surface of a sphere, draw a variable chord
${\sc a} {\sc d}$; let ${\sc d}'$ be the second point of
intersection of the spheric surface with the
secant~${\sc b} {\sc d}$, drawn to the variable
extremity~${\sc d}$ of the chord~${\sc a} {\sc d}$ from a fixed
external point~${\sc b}$; take the radius vector
${\sc a} {\sc e}$ equal in length to ${\sc b} {\sc d}'$, and in
direction either coincident with, or opposite to, the
chord~${\sc a} {\sc d}$; the locus of the point~${\sc e}$, thus
constructed, will be an ellipsoid, which will pass through the
point~${\sc b}$ (and will have its centre at~${\sc a}$).  See
Proceedings of the Royal Irish Academy for July 1846.}
of the ellipsoid described in article~31, into the language of
the calculus of quaternions, from the construction itself had
been originally derived, in the manner stated in the 30th article
of this paper.  Yet it may not seem obvious to readers unfamiliar
with this calculus, why the expression $- \rho \kappa \rho^{-1}$
was taken, in that foregoing article~38, as one denoting, in
length and in direction, that radius of the auxiliary sphere
which was drawn from ${\sc c}$ to ${\sc d}$; not in what sense,
and for what reason, this expression $- \rho \kappa \rho^{-1}$
has been said to represent the reflexion of the vector $-\kappa$
with respect to $\rho$.  As a perfectly clear answer to each of
these questions, or a distinct justification of each of the
assumptions or assertions thus referred to, may not only be
useful in connexion with the present mode of considering the
ellipsoid, but also may throw light on other applications of
quaternions to the treatment of geometrical and physical
problems, we shall not think it an irrelevant digression to enter
here into some details respecting this expression
$- \rho \kappa \rho^{-1}$, and respecting the ways in which it
might present itself in calculations such as the foregoing.  Let
us therefore now denote by $\sigma$ the vector, whatever it may
be, from ${\sc c}$ to ${\sc d}$ in the construction (${\sc c}$
being still the centre of the sphere); and let us propose to find
an expression for this sought vector~$\sigma$, as a function of
$\rho$ and of $\kappa$, by the principles of the calculus of
quaternions.

\bigbreak

40.
For this purpose we have first the equation between tensors,
$${\rm T} \sigma = {\rm T} \kappa;
   \eqno (11.)$$
which expresses that the two vectors $\sigma$ and $\kappa$ are
equally long, as being both radii of one common auxiliary sphere,
namely those drawn from the centre~${\sc c}$ to the points
${\sc d}$ and ${\sc a}$.  And secondly, we have the equation
$${\rm V} \mathbin{.} (\sigma - \kappa) \rho = 0,
   \eqno (12.)$$
where ${\rm V}$ is the characteristic of the operation of
{\it taking the vector\/} of a quaternion; which equation
expresses immediately that the product of the two vectors
$\sigma - \kappa$ and $\rho$ is scalar, and therefore that
these two vector-factors are either exactly similar or exactly
opposite in direction; since otherwise their product would be a
quaternion, having always a vector part, although the scalar part
of this quaternion-product $(\sigma - \kappa) \rho$ might vanish,
namely by the factors becoming perpendicular to each other.  Such
being the immediate and general signification of equation (12.),
the justification of our establishing it in the present question
is derived from the consideration that the radius vector~$\rho$,
drawn from the centre~${\sc a}$ to the surface~${\sc e}$ of the
ellipsoid, has, by the construction, a direction either exactly
similar or exactly opposite to the direction of that
{\it guide-chord\/} of the auxiliary sphere which is drawn from
${\sc a}$ to ${\sc d}$, that is, from the end of the radius
denoted by $\kappa$ to the end of the radius denoted by $\sigma$.
For, that the chord so drawn is properly denoted, in length and
in direction, by the symbol $\sigma - \kappa$, follows from
principles respecting {\it addition and subtraction of directed
lines}, which are indeed {\it essential}, but are {\it not
peculiar}, to the geometrical applications of quaternions; had
occurred, in various ways, to several independent inquirers,
before quaternions (as {\it products of quotients of directed
lines in space\/}) were thought of; and are now extensively
received.

\bigbreak

41.
The two equations (11.) and (12.) are evidently both satisfied
when we suppose $\sigma = \kappa$; but because the
point~${\sc d}$ is in general different from ${\sc a}$, we must
endeavour to find another value of the vector~$\sigma$, distinct
from $\kappa$, which shall satisfy the same two equations.  Such
a value, or expression, for this sought vector~$\sigma$ may be
found at once, so far as the equation (12.) is concerned, by
observing that, in virtue of this latter equation,
$\sigma - \kappa$ must bear some scalar ratio to $\rho$, or must
be equal to this vector~$\rho$ multiplied by some scalar
coefficient~$x$, so that we may write
$$\sigma = \kappa + x \rho;
   \eqno (13.)$$
and then, on substituting this expression for $\sigma$ in the
former equation~(11.), we find that $x$ must satisfy the
condition
$${\rm T} (\kappa + x \rho) = {\rm T} \kappa,
   \eqno (14.)$$
in which this sought coefficient~$x$ is supposed to be some
scalar different from zero, that is, in other words, some
positive or negative number.  Squaring both members of this last
condition, and observing that by article~19 the square of the
tensor of a vector is equal to the negative of the square of that
vector, we find the new equation
$$- (\kappa + x \rho)^2 = - \kappa^2.
   \eqno (15.)$$
But also, generally, if $\kappa$ and $\rho$ be vectors and $x$ a
scalar,
$$(\kappa + x \rho)^2
   =  \kappa^2 + x (\kappa \rho + \rho \kappa) + x^2 \rho^2;$$
adding therefore $\kappa^2$ to both members of (15.), dividing by
$-x$, and then eliminating $x$ by (13.), which is done by merely
changing $\kappa \rho + x \rho^2$ to $\sigma \rho$, we find the
equation
$$\sigma \rho + \rho \kappa = 0;
   \eqno (16.)$$
and finally
$$\sigma = - \rho \kappa \rho^{-1}:
   \eqno (17.)$$
so that the expression already assigned for the vector from
${\sc c}$ to ${\sc d}$, presents itself as the result of this
analysis.  And in fact the tensor of this expression (17.) is
equal to ${\rm T} \kappa$, by the general rule for the tensor of
a product, or because
$(- \rho \kappa \rho^{-1})^2
   =  \rho \kappa \rho^{-1} \rho \kappa \rho^{-1}
   =  \rho \kappa^2 \rho^{-1}
   =  \kappa^2$,
since $\kappa^2$ is a (negative) scalar; while the product
$(\sigma - \kappa) \rho$, being
$= - (\kappa \rho + \rho \kappa)$,
is equal, by article~20, to an expression of scalar form.

\bigbreak

42.
Conversely if, in any investigation conducted on the present
principles, we meet with the expression
$- \rho \kappa \rho^{-1}$, we may perceive in the way just now
mentioned, that it denotes a vector of which the square is equal
to that of $\kappa$; and that, if $\kappa$ be subtracted from it,
the remainder gives a scalar product when it is multiplied into
$\rho$; so that, if we denote this expression by $\sigma$, or
establish the equation~(17.), the equations (11.) and (12.) will
then be satisfied, and the vector~$\sigma$ will have the same
length as $\kappa$, while the directions of $\sigma - \kappa$ and
$\rho$ will be either exactly similar or exactly opposite to each
other.  We may therefore be thus led to regard, subject to this
condition (17.) or (16.), the two vector-symbols $\sigma$ and
$\kappa$ as denoting, in length and in direction, two radii of
one common sphere, such that the chord-line $\sigma - \kappa$
connecting their extremities has the direction of the
line~$\rho$, or of that line reversed.  Hence also, by the
elemetary property of a plane isosceles triangle, we may see
that, under the same condition, the inclination of $\sigma$ to
$\rho$ is equal to the inclination of $\kappa$ to $- \rho$, or of
$- \kappa$ to $\rho$; in such a manner that the bisector of the
external vertical angle of the isosceles triangle, or the
bisector of the angle at the centre of the sphere between the two
radii $\sigma$ and $-\kappa$, is a new radius parallel to $\rho$,
because it is parallel to the base of the triangle
(${\sc a} {\sc c} {\sc d}$), or to the chord (${\sc a} {\sc d}$)
just now mentioned.  And by conceiving a diameter of the sphere
parallel to this chord, or to $\rho$, and supposing $-\kappa$ to
denote that reversed radius which coincides in situation with the
radius~$\kappa$, but is drawn from the surface to the centre
(that is, in the recent construction, from ${\sc a}$ to
${\sc c}$), while $\sigma$ is still drawn from centre to surface
(from ${\sc c}$ to ${\sc d}$), we may be led to regard $\sigma$,
or $-\rho \kappa \rho^{-1}$, as the {\it reflexion\/} of
$-\kappa$ with respect to the diameter parallel to $\rho$, or
simply with respect to $\rho$ itself, as was remarked in the 38th
article; since the vector-symbols $\rho$,~$\sigma$, \&c.\ are
supposed, in these calculations, to indicate indeed the
{\it lengths and directions, but not the situations}, of the
straight lines which they are employed to denote.

\bigbreak

43.
The same geometrical interpretation of the symbol
$- \rho \kappa \rho^{-1}$ may be obtained in several other ways,
among which we shall specify the following.  Whatever the lengths
and directions of the two straight lines denoted by $\rho$ and
$\kappa$ may be, we may always conceive that the latter line,
regarded as a vector, is or may be decomposed, by two different
projections, into two partial or component vectors, $\kappa'$ and
$\kappa''$, of which one is parallel and the other is
perpendicular to $\rho$; so that they satisfy respectively the
equations of parallelism and perpendicularity (see article~21),
and that we have consequently,
$$\kappa = \kappa' + \kappa'';\quad
  {\rm V} \mathbin{.} \kappa' \rho = 0;\quad
  {\rm S} \mathbin{.} \kappa'' \rho = 0;
   \eqno (18.)$$
where ${\rm S}$ is the characteristic of the operation of
{\it taking the scalar\/} of a quaternion.  The equation of
parallelism gives $\rho \kappa' = \kappa' \rho$, and the equation
of perpendicularity gives $\rho \kappa'' = - \kappa'' \rho$; hence
the proposed expression $- \rho \kappa \rho^{-1}$ resolves itself
into the two parts,
$$\left. \eqalign{
- \rho \kappa' \rho^{-1}
   &= - \kappa' \rho \rho^{-1} = - \kappa';\cr
- \rho \kappa'' \rho^{-1}
   &= + \kappa'' \rho \rho^{-1} = + \kappa'';\cr}
   \right\}
   \eqno (19.)$$
so that we have, upon the whole,
$$- \rho \kappa \rho^{-1}
   =  - \rho ( \kappa' + \kappa'' ) \rho^{-1}
   =  - \kappa' + \kappa''.
   \eqno (20.)$$
The part $-\kappa'$ of this last expression, which is parallel to
$\rho$, is the same as the corresponding part of $-\kappa$; but
the part $+\kappa''$, perpendicular to $\rho$, is the same with
the corresponding part of $+\kappa$, or is opposite to the
corresponding part of $-\kappa$; we may therefore be led by this
process also to regard the expression~(17.) as denoting the
reflexion of the vector $-\kappa$, with respect to the
vector~$\rho$, regarded as a reflecting line; and we see that the
direction of $\rho$, or that of $-\rho$, is exactly intermediate
between the two directions of $-\kappa$ and
$- \rho \kappa \rho^{-1}$, or between those of $\kappa$ and of
$\rho \kappa \rho^{-1}$.

\bigbreak

44.
The equation~(9.) of the ellipsoid, in article~38, or the
equation~(4.) in article~37, may be more fully written thus:
$$(\iota \rho + \rho \kappa) (\rho \iota + \kappa \rho)
   =  (\kappa^2 - \iota^2)^2.
   \eqno (21.)$$
And to express that we propose to cut this surface by any
diametral plane, we may write the equation
$$\varpi \rho + \rho \varpi = 0,
   \eqno (22.)$$
where $\varpi$ denotes a vector to which that cutting plane is
perpendicular: thus, if in particular, we change $\varpi$ to
$\kappa$, we find, for the corresponding plane through the
centre, the equation
$$\kappa \rho + \rho \kappa = 0,
   \eqno (23.)$$
which, when combined with (21.), gives
$$(\kappa^2 - \iota^2)^2
   =  (\iota - \kappa) \rho \mathbin{.} \rho (\iota - \kappa)
   =  (\iota - \kappa) \rho^2 (\iota - \kappa)
   =  (\iota - \kappa)^2 \rho^2,$$
that is,
$$\rho^2
   =  \left( {\kappa^2 - \iota^2 \over \iota - \kappa} \right)^2;
   \eqno (24.)$$
but this is the equation of a sphere concentric with the
ellipsoid; therefore the diametral plane~(23.) cuts the ellipsoid
in a {\it circle}, or the plane itself is a {\it cyclic plane}.
We see also that the vector~$\kappa$, as being perpendicular to
this plane~(23.), is one of the {\it cyclic normals}, or normals
to planes of circular section; which agrees with the
construction, since we saw, in article~36, that the auxiliary or
diacentric sphere, with centre~${\sc c}$, touches one cyclic
plane at the centre~${\sc a}$ of the ellipsoid.  The same
construction shows that the other cyclic plane ought to be
perpendicular to the vector~$\iota$; and accordingly the equation
$$\iota \rho + \rho \iota = 0
   \eqno (25.)$$
represents this second cyclic plane; for, when combined with the
equation~(21.) of the ellipsoid, it gives
$$(\kappa^2 - \iota^2)^2
   =  \rho (\kappa - \iota) \mathbin{.} (\kappa - \iota) \rho
   =  \rho (\kappa - \iota)^2 \rho
   =  (\kappa - \iota)^2 \rho^2,$$
and therefore conducts to the same equation~(24.) of a concentric
sphere as before; which sphere~(24.) is thus seen to contain the
intersection of the ellipsoid~(21.) with the plane~(25.), as well
as that with the plane~(23.).  If we use the form~(9.), we have
only to observe that whether we change $\rho \kappa$ to
$- \kappa \rho$, or $\iota \rho$ to $- \rho \iota$, we are
conducted in each case to the following expression for the length
of the radius vector of the ellipsoid, which agrees with the
equation (24.):
$${\rm T} \rho
   =  {\kappa^2 - \iota^2 \over {\rm T} (\iota - \kappa)}.
   \eqno (26.)$$
And because $\kappa^2 - \iota^2$ denotes the square upon the
tangent drawn to the auxiliary sphere from the external
point~${\sc b}$, while ${\rm T} (\iota - \kappa)$ denotes the
length of the side ${\sc b} {\sc a}$ of the generating triangle,
we see by this easy calculation with quaternions, as well as by
the more purely geometrical reasoning which was alluded to, and
partly stated, in the 36th article, that the common radius of the
two diametral and circular sections of the ellipsoid is equal to
the straight line which was there called ${\sc b} {\sc g}$, and
which had the direction of ${\sc b} {\sc a}$, while terminating,
like it, on the surface of the auxiliary sphere; so that the two
last lines ${\sc b} {\sc a}$, and ${\sc b} {\sc g}$, were
connected with that sphere and with each other, in this or in the
opposite order, as the whole secant and the external part.  In
fact, as the point~${\sc d}$, in the construction approaches, in
any direction, on the surface of the auxiliary sphere, to
${\sc a}$, the point~${\sc d}'$ approaches to ${\sc g}$; and
${\sc b} {\sc d}'$, and therefore also ${\sc a} {\sc e}$, tends
to become equal in length to ${\sc b} {\sc g}$; while the
direction of ${\sc a} {\sc e}$, being the same with that of
${\sc a} {\sc d}$, or opposite thereto, tends to become
tangential to the sphere, or perpendicular to ${\sc a} {\sc c}$:
the line~${\sc b} {\sc g}$ is therefore equal to the radius of
that diametral and circular section of the ellipsoid which is
made by the plane that touches the auxiliary sphere at ${\sc a}$.
And again, if we conceive the point~${\sc d}'$ to revolve on the
surface of the sphere from ${\sc g}$ to ${\sc g}$ again, in a
plane perpendicular to ${\sc b} {\sc c}$, then the lines
${\sc a} {\sc d}$ and ${\sc a} {\sc e}$ will revolve together in
another plane parallel to that last mentioned, and perpendicular
likewise to ${\sc b} {\sc c}$; while the length of
${\sc a} {\sc e}$ will be still equal to the same constant
line~${\sc b} {\sc g}$ as before: which line is therefore found
to be equal to the common radius of both the diametral and
circular sections of the ellipsoid, whether as determined by the
geometrical construction which the calculus of quaternions
suggested, or immediately by that calculus itself.

\bigbreak

45.
We may write the equation~(21.) of the ellipsoid as follows:
$$f(\rho) = 1,
   \eqno (27.)$$
if we introduce a scalar function~$f$ of the variable
vector~$\rho$, defined as follows:
$$(\kappa^2 - \iota^2)^2 f(\rho)
   =  (\iota \rho + \rho \kappa) (\rho \iota + \kappa \rho)
   =  \iota \rho^2 \iota + \iota \rho \kappa \rho
       + \rho \kappa \rho \iota + \rho \kappa^2 \rho;$$
or thus, in virtue of article~20,
$$(\kappa^2 - \iota^2)^2 f(\rho)
   =  (\iota^2 + \kappa^2) \rho^2
       + 2 {\rm S} \mathbin{.} \iota \rho \kappa \rho.
   \eqno (28.)$$

Let $\rho + \tau$ denote another vector from the centre to the
surface of the same ellipsoid; we shall have, in like manner,
$$f(\rho + \tau) = 1,
   \eqno (29.)$$
where
$$f(\rho + \tau)
   = f(\rho) + 2 {\rm S} \mathbin{.} \nu \tau + f(\tau),
   \eqno (30.)$$
if we introduce a new vector symbol~$\nu$, defined by the
equation
$$(\kappa^2 - \iota^2)^2 \nu
   =  (\iota^2 + \kappa^2) \rho
       + \iota \rho \kappa + \kappa \rho \iota;
   \eqno (31.)$$
because generally, for any two vectors $\rho$ and $\tau$,
$$(\rho + \tau)^2
   =  \rho^2 + 2 {\rm S} \mathbin{.} \rho \tau + \tau^2,
   \eqno (32.)$$
and, for any four vectors, $\iota$, $\kappa$, $\rho$, $\tau$,
$$    {\rm S} \mathbin{.} \iota \tau \kappa \rho
    = {\rm S} \mathbin{.} \tau \kappa \rho \iota
    = {\rm S} \mathbin{.} \kappa \rho \iota \tau
    = {\rm S} \mathbin{.} \rho \iota \tau \kappa;
   \eqno (33.)$$
which last principle, respecting certain transpositions of vector
symbols, as factors of a product under the sign~${\rm S}$., shows,
when combined with the equations (27.), (28.), and (31.), that we
have also this simple relation:
$${\rm S} \mathbin{.} \nu \rho = 1.
   \eqno (34.)$$

Subtracting (27.) from (29.), attending to (30.), changing $\tau$
to ${\rm T} \tau \mathbin{.} {\rm U} \tau$, where ${\rm U}$ is,
as in article~19, the characteristic of the operation of
{\it taking the versor\/} of a quaternion (or of a vector), and
dividing by ${\rm T} \tau$, we find:
$$0 = { f(\rho + \tau) - f(\rho) \over {\rm T} \tau }
   =  2 {\rm S} \mathbin{.} \nu \, {\rm U} \tau
       + {\rm T} \tau \mathbin{.} f( {\rm U} \tau ).
   \eqno (35.)$$

This is a rigorous equation, connecting the {\it length\/} or the
{\it tensor\/}~${\rm T} \tau$, of any chord~$\tau$ of the
ellipsoid, drawn from the extremity of the semidiameter~$\rho$,
with the {\it direction\/} of that chord~$\tau$, or with the
{\it versor\/}~${\rm U} \tau$; it is therefore only a new form of
the equation of the ellipsoid itself, with the origin of vectors
removed from the centre to a point upon the surface.  If we now
conceive the chord~$\tau$ to diminish in length, the term
${\rm T} \tau \mathbin{.} f({\rm U} \tau)$
of the right-hand member of this equation~(35.) tends to become
$= 0$, on account of the factor~${\rm T} \tau$; and therefore the
other term $2 {\rm S} \mathbin{.} \nu {\rm U} \tau$ of the same
member must tend to the same limit zero.  In this way we arrive
easily at an equation expressing the {\it ultimate law of the
directions of the evanescent chords\/} of the ellipsoid, at the
extremity of any given or assumed semidiameter~$\rho$; which
equation is
$0 = 2 {\rm S} \mathbin{.} \nu {\rm U} \tau$,
or simply,
$$0 = {\rm S} \mathbin{.} \nu \tau,
   \eqno (36.)$$
if $\tau$ be a tangential vector.  The vector~$\nu$ is therefore
perpendicular to all such tangents, or infinitesimal chords of
the ellipsoid, at the extremity of the semidiameter~$\rho$; and
consequently it has the direction of the {\it normal\/} to that
surface, at the extremity of that semidiameter.  The {\it tangent
plane\/} to the same surface at the same point is represented by
the equation~(34.), if we treat, therein, the normal vector~$\nu$
as constant, and if we regard the symbol~$\rho$ as denoting, in
the same equation~(34.), a variable vector, drawn from the centre
of the ellipsoid to any point upon that tangent plane.  This
equation~(34.) of the tangent plane may be written as follows:
$${\rm S} \mathbin{.} \nu (\rho - \nu^{-1}) = 0;
   \eqno (37.)$$
and under this form it shows easily that the symbol~$\nu^{-1}$
represents, in length and in direction, the perpendicular let
fall from the origin of the vectors~$\rho$, that is from the
centre of the ellipsoid, upon the plane which is thus represented
by the equation (34.) or (37.); so that the vector~$\nu$ itself,
as determined by the equation~(31.) may be called the {\it vector
of proximity\/}\footnote*{This name, ``vector of proximity,'' was
suggested to the writer by a phraseology of Sir John Herschel's;
and the equation (31.), of article~45, which determines this
vector for the ellipsoid, was one of a few equations which were
designed to have been exhibited to the British Association at its
meeting in 1846: but were accidentally forwarded at the last
moment to Collingwood, instead of Southampton, and did not come
to the hands of the eminent philosopher just mentioned, until it
was too late for him to do more than return the paper, with some
of those encouraging expressions by which he delights to cheer,
as opportunities present themselves, all persons whom he
conceives to be labouring usefully for science.}
{\it of the tangent plane\/} of the ellipsoid, or of an element
of that surface, to the centre, at the end of that
semidiameter~$\rho$ from which $\nu$ is deduced by that equation.

\bigbreak

46.
Conceive now that at the extremity of an infinitesimal chord
${\rm d} \rho$ or $\tau$, we draw another normal to the
ellipsoid; the expression for any arbitrary point on the former
normal, that is the symbol for the vector of this point, drawn
from the centre of the ellipsoid, or from the origin of the
vectors~$\rho$, is of the form $\rho + n \nu$, where $n$ is an
arbitrary scalar; and in like manner the corresponding expression
for an arbitrary point on the latter and infinitely near normal,
or for its vector from the same centre of the ellipsoid, is
$\rho + {\rm d} \rho + (n + {\rm d} n) (\nu + {\rm d} \nu)$,
where ${\rm d} n$ is an arbitrary but infinitesimal scalar, and
${\rm d} \nu$ is the differential of the vector of
proximity~$\nu$, which may be found as a function of the
differential~${\rm d} \rho$ by differentiating the
equation~(31.), which connects the two vectors~$\nu$ and $\rho$
themselves.  In this manner we find, from (31.),
$$(\kappa^2 - \iota^2)^2 \, {\rm d} \nu
   =  (\iota^2 + \kappa^2) {\rm d} \rho
       + \iota  \, {\rm d} \rho \, \kappa
       + \kappa \, {\rm d} \rho \, \iota;
   \eqno (38.)$$
and the condition required for the intersection of the two near
normals, or for the existence of a point common to both, is
expressed by the formula
$$\rho + {\rm d} \rho + (n + {\rm d} n) (\nu + {\rm d} \nu)
   =  \rho + n \nu;
   \eqno (39.)$$
which may be more concisely written as follows:
$${\rm d} \rho + {\rm d} \mathbin{.} n \nu = 0;
   \eqno (40.)$$
or thus:
$${\rm d} \rho + n \, {\rm d} \nu + {\rm d} n \, \nu = 0.
   \eqno (41.)$$
We can eliminate the two scalar coefficients, $n$ and
${\rm d} n$, from this last equation, according to the rules of
the calculus of quaternions, by the method exemplified in the
24th article of this paper (Phil.\ Mag., August 1846), or by
operating with the characteristic
${\rm S} \mathbin{.} \nu \, {\rm d} \nu$,
because generally
$${\rm S} \mathbin{.} \nu \mu^2 = 0,\quad
  {\rm S} \mathbin{.} \nu \mu \nu = 0,$$
whatever vectors $\mu$ and $\nu$ may be; so that here,
$${\rm S} \mathbin{.} \nu \, {\rm d} \nu \, n \, {\rm d} \nu
   = 0,\quad
  {\rm S} \mathbin{.} \nu \, {\rm d} \nu \, {\rm d} n \, \nu
   = 0.$$
In this manner we find from (41.) the following very simple
formula:
$${\rm S} \mathbin{.} \nu \, {\rm d} \nu \, {\rm d} \rho = 0;
   \eqno (42.)$$
which is easily seen, on the same principles, to hold good, as
the {\it quaternion form of the differential equation of the
lines of curvature on a curved surface generally}, if $\nu$ be
still the {\it vector of proximity of the superficial element\/}
of the curved surface to the origin of the vectors~$\rho$, which
vector~$\nu$ is determined by the general condition
$${\rm S} \mathbin{.} \nu \, {\rm d} \rho = 0,
   \eqno (43.)$$
combined with the equation already written,
$${\rm S} \mathbin{.} \nu \rho = 1
   \qquad\hbox{(34.);}$$
or simply if $\nu$ be a {\it normal vector}, satisfying the
condition (43.) alone.  Substituting, therefore, in the case of
the ellipsoid, the expression for ${\rm d} \nu$ given by (38.),
and observing that
${\rm S} \mathbin{.} \nu \, {\rm d} \rho^2 = 0$,
we find that we may write the equation of the lines of curvature
for this particular surface as follows:
$${\rm S} \mathbin{.} \nu
         (  \iota \, {\rm d} \rho \, \kappa
          + \kappa \, {\rm d} \rho \, \iota )
         \, {\rm d} \rho
    =   0;
   \eqno (44.)$$
which equation, when treated by the rules of the present
calculus, admits of being in many ways symbolically transformed,
and may also, with little difficulty, be translated into
geometrical enunciations.

\bigbreak

47.
Thus if we observe that, by article~20,
$\iota \tau \kappa - \kappa \tau \iota$
is a {\it scalar form}, whatever three vectors may be denoted by
$\iota$, $\kappa$, $\tau$; and if we attend to the equation
(43.), which expresses that the normal~$\nu$ is perpendicular to
the linear element, or infinitesimal chord, ${\rm d} \rho$; we
shall perceive that, for {\it every\/} direction of that element,
the following equation holds good:
$${\rm S} \mathbin{.} \nu
         (  \iota \, {\rm d} \rho \, \kappa
          - \kappa \, {\rm d} \rho \, \iota )
         \, {\rm d} \rho
   =  0.
   \eqno (45.)$$
We have therefore, from (44.), for those {\it particular\/}
directions which belong to the lines of curvature, this
simplified equation;
$${\rm S} \mathbin{.} \nu \iota \, {\rm d} \rho
         \, \kappa \, {\rm d} \rho
   =  0;
   \eqno (46.)$$
which may be still a little abridged, by writing instead of
${\rm d} \rho$ the symbol~$\tau$ of a tangential vector, already
used in (36.); for thus we obtain the formula:
$${\rm S} \mathbin{.} \nu \iota \tau \kappa \tau = 0.
   \eqno (47.)$$
We might also have observed that by the same article~20 (Phil.\
Mag., July 1846),
$\iota \tau \kappa + \kappa \tau \iota$
and therefore
$\iota \, {\rm d} \rho \, \kappa
          + \kappa \, {\rm d} \rho \, \iota$
is a {\it vector form}, and that by article~26 (Phil.\ Mag.,
August 1846), three vector-factors under the
characteristic~${\rm S}$ may be in any manner transposed, with
only a change (at most) in the positive or negative sign of the
resulting scalar; from which it would have followed, by a process
exactly similar to the foregoing, that the equation (44.) of the
lines of curvature on an ellipsoid may be thus written,
$${\rm S} \mathbin{.} \nu \, {\rm d} \rho \, \iota
      \, {\rm d} \rho \, \kappa = 0;
   \eqno (48.)$$
or, substituting for the linear element ${\rm d} \rho$ the
tangential vector~$\tau$,
$${\rm S} \mathbin{.} \nu \tau \iota \tau \kappa = 0;
   \eqno (49.)$$
or finally, by the principles of the same 20th article,
$$\nu \tau \iota \tau \kappa
       - \kappa \tau \iota \tau \nu
   = 0.
   \eqno (50.)$$

\bigbreak

48.
Under this last form, it was one of a few equations selected in
September 1846, for the purpose of being exhibited to the
Mathematical Section of the British Association at Southampton;
although it happened\footnote*{See the note to article~45.}
that the paper containing those equations did not reach its
destination in time to be so exhibited.  The equations here
marked (49.) and (50.) were however published before the close of
the year in which that meeting was held, as part of the abstract
of a communication which had been made to the Royal Irish Academy
in the summer of that year.  (See the Proceedings of the Academy
for July 1846, equations (46.) and (47.).)  From the somewhat
discursive character of the present series of communications on
Quaternions, and from the desire which the author feels to render
them, to some extent, complete within themselves, or at least
intelligible to those mathematical readers of the
Philosophical Magazine who may be disposed to favour him
with their attention, to the degree which the novelty of the
conceptions and method may require, without its being
{\it necessary\/} for such readers to refer to other publications
of his own, he is induced, and believes himself to be authorized,
to copy here a few other equations from that short and hitherto
unpublished Southampton paper, and to annex to them another
formula which may be found in the Proceedings, already
cited, of the Royal Irish Academy: together with a more extensive
formula, which he believes to be new.

\bigbreak

49.
Besides the equation of the ellipsoid,
$$(\iota \rho + \rho \kappa) (\rho \iota + \kappa \rho)
   =  (\kappa^2 - \iota^2)^2
   \qquad\hbox{(21.), art.~44;}$$
with the expression derived from it, for the vector of proximity
of that surface to its centre,
$$(\kappa^2 - \iota^2)^2 \nu
   =  (\iota^2 + \kappa^2) \rho
       + \iota \rho \kappa + \kappa \rho \iota
   \qquad\hbox{(31.), art.~45;}$$
the equation for the lines of curvature on the ellipsoid,
$$\nu \tau \iota \tau \kappa
       - \kappa \tau \iota \tau \nu
   = 0
   \qquad\hbox{(50.), art.~47;}$$
and the equation
$$\nu \tau + \tau \nu = 0,
   \eqno (51.)$$
which is a form of the relation
${\rm S} \mathbin{.} \nu \tau = 0$,
that is of the equation (36.), article~45, of the present series
of communications; the author gave, in the paper which has been
above referred to, the following symbolic transformation, for the
well-known characteristic of operation,
$$    \left( {{\rm d} \over {\rm d} x} \right)^2
    + \left( {{\rm d} \over {\rm d} y} \right)^2
    + \left( {{\rm d} \over {\rm d} z} \right)^2,$$
which seems to him to open a wide and new field of analytical
research, connected with many important and difficult departments
of the mathematical study of nature.

A {\sc quaternion}, {\it symbolically considered}, being
(according to the views originally proposed by the author in
1843) an algebraical quadrinomial of the form
$w + ix + jy + kz$, where $w \, x \, y \, z$ are any four real
numbers (positive or negative or zero), while $i \, j \, k$ are
three co-ordinate imaginary units, subject to the fundamental
laws of combination (see Phil.\ Mag.\ for July 1844):
$$\left. \eqalign{
   & i^2 = j^2 = k^2 = -1;\cr
   & ij = k;\quad jk = i;\quad ki = j;\cr
   & ji = -k;\quad kj = -i;\quad ik = -j;\cr}
   \right\}
   \eqno ({\rm a}.)$$
it results at once from these definitions, or laws of symbolic
combination, (a.), that if we introduce a new characteristic of
operation,~$\vecd$, defined with relation to these three
symbols $i \, j \, k$, and to the known operation of partial
differentiation, performed with respect to three independent but
real variables $x \, y \, z$, as follows:
$$\vecd
   =  {i {\rm d} \over {\rm d} x}
    + {j {\rm d} \over {\rm d} y}
    + {k {\rm d} \over {\rm d} z};
   \eqno ({\rm b}.)$$
{\it this new characteristic~$\vecd$ will have the negative of
its symbolic square expressed by the following formula:}
$$- \vecd^2
   =  \left( {{\rm d} \over {\rm d} x} \right)^2
    + \left( {{\rm d} \over {\rm d} y} \right)^2
    + \left( {{\rm d} \over {\rm d} z} \right)^2;
   \eqno ({\rm c}.)$$
of which it is clear that the applications to analytical physics
must be extensive in a high degree.  In the paper\footnote*{In
that paper itself, the characteristic was written~$\nabla$; but
this more common sign has been so often used with other meanings,
that it seems desirable to abstain from appropriating it to the
new signification here proposed.}
designed for Southampton it was remarked, as an illustration,
that this result enables us to put the known thermological
equation,
$$    {{\rm d}^2 v \over {\rm d} x^2}
    + {{\rm d}^2 v \over {\rm d} y^2}
    + {{\rm d}^2 v \over {\rm d} z^2}
    + a {{\rm d} v \over {\rm d} t}
   =  0,$$
under the new and more symbolic form,
$$\left( \vecd^2 - {a {\rm d} \over {\rm d} t} \right) v = 0;
   \eqno ({\rm d}.)$$
while $\vecd v$ denotes, in quantity and in direction, the
{\it flux\/} of heat, at the time~$t$ and at the point
$x \, y \, z$.

\bigbreak

50.
In the Proceedings of the Royal Irish Academy for July
1846, it will be found to have been noticed that the same new
characteristic~$\vecd$ gives also this other general
transformation, perhaps not less remarkable, nor having less
extensive consequences, and which presents itself under the form
of a quaternion:
$$\vecd( it + ju + kv)
   =   - \left(
            {{\rm d} t \over {\rm d} x}
          + {{\rm d} u \over {\rm d} y}
          + {{\rm d} v \over {\rm d} z}
         \right)
       + i
         \left(
            {{\rm d} v \over {\rm d} y}
          - {{\rm d} u \over {\rm d} z}
         \right)
       + j
         \left(
            {{\rm d} t \over {\rm d} z}
          - {{\rm d} v \over {\rm d} x}
         \right)
       + k
         \left(
            {{\rm d} u \over {\rm d} x}
          - {{\rm d} t \over {\rm d} y}
         \right).
   \eqno ({\rm e}.)$$
In fact the equations (a.) give generally (see art.~21 of the
present series),
$$(ix + jy + kz) (it + ju + kv)
   = - (xt + yu + zv) + i (yv - zu) + j (zt - xv) + k (xu - yt),
   \eqno ({\rm f}.)$$
if $x \, y \, z \, t \, u \, v$ denote any six real numbers; and
the calculations by which this is proved, show, still more
generally, that the same transformation must hold good, if each
of the three symbols $i$,~$j$,~$k$, subject still to the
equations~(a.), be commutative in arrangement, as a symbolic
factor, with each of the three other symbols $x$,~$y$,~$z$; even
though the latter symbols, like the former, should not be
commutative in that way among themselves; and even if they should
denote symbolical instead of numerical multipliers, possessing
still the distributive character.  We may therefore change the
three symbols $x$,~$y$,~$z$, respectively, to the three
characteristics of partial differentiation,
$\displaystyle {{\rm d} \over {\rm d} x}$,
$\displaystyle {{\rm d} \over {\rm d} y}$,
$\displaystyle {{\rm d} \over {\rm d} z}$;
and thus the formula~(e.) is seen to be included in the
formula~(f.).  And if we then, in like manner, change the three
symbols $t$,~$u$,~$v$, regarded as factors, to
$\displaystyle {{\rm d} \over {\rm d} x'}$,
$\displaystyle {{\rm d} \over {\rm d} y'}$,
$\displaystyle {{\rm d} \over {\rm d} z'}$,
that is, to the characteristics of three partial differentiations
performed with respect to three new and independent variables
$x'$,~$y'$,~$z'$, we shall thereby change
$\displaystyle {{\rm d} t \over {\rm d} x}$ to
$\displaystyle {{\rm d} \over {\rm d} x} {{\rm d} \over {\rm d} x'}$,
and so obtain the formula:
$$\left. \eqalign{
      \left(
         i {{\rm d} \over {\rm d} x}
       + j {{\rm d} \over {\rm d} y}
       + k {{\rm d} \over {\rm d} z}
      \right)
      \left(
         i {{\rm d} \over {\rm d} x'}
       + j {{\rm d} \over {\rm d} y'}
       + k {{\rm d} \over {\rm d} z'}
      \right)
   =  -
      \left(
         {{\rm d} \over {\rm d} x}
         {{\rm d} \over {\rm d} x'}
       + {{\rm d} \over {\rm d} y}
         {{\rm d} \over {\rm d} y'}
       + {{\rm d} \over {\rm d} z}
         {{\rm d} \over {\rm d} z'}
      \right) \cr
    + i
      \left(
         {{\rm d} \over {\rm d} y}
         {{\rm d} \over {\rm d} z'}
       - {{\rm d} \over {\rm d} z}
         {{\rm d} \over {\rm d} y'}
      \right)
    + j
      \left(
         {{\rm d} \over {\rm d} z}
         {{\rm d} \over {\rm d} x'}
       - {{\rm d} \over {\rm d} x}
         {{\rm d} \over {\rm d} z'}
      \right)
    + k
      \left(
         {{\rm d} \over {\rm d} x}
         {{\rm d} \over {\rm d} y'}
       - {{\rm d} \over {\rm d} y}
         {{\rm d} \over {\rm d} x'}
      \right); \cr}
   \right\}
   \eqno ({\rm g}.)$$
which includes the formula~(c.), and is now for the first time
published.

This formula~(g.) is, however, seen to be a very easy and
immediate consequence from the author's fundamental equations of
1843, or from the relations~(a.) of the foregoing article, which
admit of being concisely summed up in the following continued
equation:
$$i^2 = j^2 = k^2 = ijk = -1.
   \eqno ({\rm h}.)$$
The geometrical interpretation of the equation
${\rm S} \mathbin{.} \nu \tau \iota \tau \kappa = 0$
of the lines of curvature on the ellipsoid, with some other
applications of quaternions to that important surface, must be
reserved for future articles of the present series, of which some
will probably appear in an early number of this Magazine.

\bigbreak

51.
It has been shown\footnote*{See the Philosophical Magazine for
October 1847; or Proceedings of the Royal Irish Academy for July
1846.}
that if the two symbols $\iota$,~$\kappa$ denote certain constant
vectors, perpendicular to the two cyclic planes of an ellipsoid,
and if $\nu$,~$\tau$ denote two other and variable vectors, of
which the former is normal to the ellipsoid at any proposed point
upon its surface, while the latter is tangential to a line of
curvature at that point, then the {\it directions\/} of these
four vectors $\iota$,~$\kappa$,~$\nu$,~$\tau$ are so related to
each other as to satisfy the condition
$${\rm S} \mathbin{.} \nu \tau \iota \tau \kappa = 0
   \qquad\hbox{(49.), article~47;}$$
${\rm S}$ being the characteristic of the operation of taking the
scalar part of a quaternion.  And because the two latter of these
four directions, namely the directions of the normal and
tangential vectors $\nu$ and $\tau$, are always perpendicular to
each other, this additional equation has been seen to hold good:
$${\rm S} \mathbin{.} \nu \tau = 0
   \qquad\hbox{(36.), article~45.}$$
Retaining the same significations of the symbols, and carrying
forward for convenience the recent numbering of the formul{\ae}, it
is now proposed to point out some of the modes of combining,
transforming, and interpreting the system of these two equations,
consistently with the principles and rules of the Calculus of
Quaternions, from which the equations themselves have been
derived.

\bigbreak

52.
Whatever two vectors may be denoted by $\iota$ and $\tau$, the
ternary product $\tau \iota \tau$ is always a {\it vector form},
because (by article~20) its scalar part is zero; and on the other
hand the square~$\tau^2$ is a pure scalar: therefore we may
always write
$$\tau \iota \tau = \mu \tau^2,\quad
  \tau \iota = \mu \tau,
   \eqno (52.)$$
where $\mu$ is a new vector, expressible in terms of $\iota$ and
$\tau$ as follows:
$$\mu = \tau \iota \tau^{-1};
   \eqno (53.)$$
so that it is, in general, by the principles of articles 40, 41,
42, 43, the {\it reflexion\/} of the vector~$\iota$ with respect
to the vector~$\tau$, and that thus the direction of $\tau$ is
exactly intermediate between the directions of $\iota$ and $\mu$.
In the present question, this new vector~$\mu$, defined by the
equation~(53.) may therefore represent the reflexion of the first
cyclic normal~$\iota$, with respect to any reflecting line which
is parallel to the vector~$\tau$, which latter vector is
tangential to one of the curves of curvature on the ellipsoid.
Substituting for $\tau \iota \tau$ its value~(52.) in the lately
cited equation~(49.), and suppressing the scalar factor~$\tau^2$,
we find this new equation:
$${\rm S} \mathbin{.} \nu \mu \kappa = 0;
   \eqno (54.)$$
which, in virtue of the general {\it equation of coplanarity\/}
assigned in the 21st article (Phil.\ Mag.\ for July 1846),
expresses that the reflected vector~$\mu$, the normal
vector~$\nu$, and the second cyclic normal~$\kappa$, are parallel
to one common plane.  This result gives already a
characteristical geometric property of the lines of curvature on
an ellipsoid, from which the directions of those curved lines, or
of their tangents~($\tau$), can generally be assigned, at any
given point upon the surface, when the direction of the normal
($\nu$) at that point, and those of the two cyclic normals
($\iota$ and $\kappa$), are known.  For it shows that if a
straight line~$\mu$ be found, in any plane parallel to the given
lines~$\nu$ and $\kappa$, such that the bisector~$\tau$ of the
angle between this line~$\mu$ and a line parallel to the other
given line~$\iota$ shall be perpendicular to the given
line~$\nu$, then this bisecting line~$\tau$ will have the sought
direction of a tangent to a line of curvature.  But it is
possible to deduce a geometrical determination, or construction,
more simple and direct than this, by carrying the calculation a
little further.

\bigbreak

53.
The equation (52.) gives
$$(\mu + \iota) \tau = \tau \iota + \iota \tau
   =  {\rm V}^{-1} 0,
   \eqno (55.)$$
this last symbol ${\rm V}^{-1} 0$ denoting generally any
quaternion of which the vector part vanishes; that is any pure
scalar, or in other words any real number, whether positive or
negative or null.  Hence $\mu + \iota$ and $\tau$ denote, in the
present question, two coincident or parallel vectors, of which
the directions are either exactly similar or else exactly
opposite to each other; since if they were inclined at any actual
angle, whether acute or obtuse, their product would be a
quaternion, of which the vector part would not be equal to zero.
Accordingly the expression~(53.) gives this equation between
tensors,
$${\rm T} \mu = {\rm T} \iota;
   \eqno (56.)$$
so that the symbols $\mu$ and $\iota$ denote here two equally
long straight lines; and therefore one diagonal of the
equilateral parallelogram (or rhombus) which is constructed with
those lines for two adjacent sides bisects the angle between
them.  But by the last article, this bisector has the direction
of $\tau$ (or of $-\tau$); and by one of those fundamental
principles of the geometrical interpretation of symbols, which
are {\it common\/} to the calculus of quaternions and to several
earlier and some later systems, the symbol $\mu + \iota$ denotes
generally the intermediate diagonal of a parallelogram
constructed with the lines denoted by $\mu$ and $\iota$ for two
adjacent sides: we might therefore in this way also have seen
that the vector $\mu + \iota$ has, in the present question, the
direction of $\pm \tau$.  This vector $\mu + \iota$ is therefore
perpendicular to $\nu$, and we have the equation
$$0 = {\rm S} \mathbin{.} \nu (\mu + \iota),
   \quad\hbox{or}\quad
  {\rm S} \mathbin{.} \nu \mu = - {\rm S} \mathbin{.} \nu \iota.
   \eqno (57.)$$
But by (56.), and by the general rule for the tensor of a product
(see art.~20), we have also
$${\rm T} \mathbin{.} \nu \mu
   =  {\rm T} \mathbin{.} \nu \iota;
   \eqno (58.)$$
and in general (by art.~19), the square of the tensor of a
quaternion is equal to the square of the scalar part, minus the
square of the vector part of that quaternion; or in symbols
(Phil.\ Mag., July 1846),
$$({\rm T} {\rm Q})^2
  =  ({\rm S} {\rm Q})^2 - ({\rm V} {\rm Q})^2.$$
Hence the two quaternions $\nu \mu$ and $\nu \iota$, since they
have equal tensors and opposite scalar parts, must have the
squares of their vector parts equal, and those vector parts
themselves must have their tensors equal to each other; that is,
we may write
$$( {\rm V} \mathbin{.} \nu \mu )^2
   =  ( {\rm V} \mathbin{.} \nu \iota )^2,\quad
  {\rm T} {\rm V} \mathbin{.} \nu \mu
   =  {\rm T} {\rm V} \mathbin{.} \nu \iota:
   \eqno (59.)$$
and may regard these two vector parts of these two quaternions,
or of the products $\nu \mu$ and $\nu \iota$, as denoting two
equally long straight lines.  Consequently the vector
$\pm \nu \tau$, which has the direction of the line represented
by the pure vector product $\nu (\mu + \iota)$, or by the sum
${\rm V} \mathbin{.} \nu \mu + {\rm V} \mathbin{.} \nu \iota$
of two equally long vectors, has at the same time a direction of
the sum of the two corresponding {\it versors\/} of those
vectors, or that of the sum of their {\it vector-units\/}; so
that we may write the equation
$$t \nu \tau
   =  {\rm U} {\rm V} \mathbin{.} \nu \mu
    + {\rm U} {\rm V} \mathbin{.} \nu \iota,
   \eqno (60.)$$
where ${\rm U}$ is (as in art.~19) the characteristic of the
operation of taking the versor of a quaternion, or of a vector;
and $t$ is a scalar coefficient.  Again, the equation
$0 = {\rm S} \mathbin{.} \nu \mu \kappa$, (54.), which expresses
that the three vectors $\nu$, $\mu$, $\kappa$ are coplanar, shows
also that the two vectors
${\rm V} \mathbin{.} \nu \mu$ and
${\rm V} \mathbin{.} \nu \kappa$
are parallel to each other, as being both perpendicular to that
common plane to which $\nu$, $\mu$ and $\kappa$ are parallel;
hence we have the following equation between two versors of
vectors, or between two vector-units,
$${\rm U} {\rm V} \mathbin{.} \nu \mu
   =  \pm {\rm U} {\rm V} \mathbin{.} \nu \kappa;
   \eqno (61.)$$
and therefore instead of the formula (60.) we may write
$$t \tau
   =  \nu^{-1} \, {\rm U} {\rm V} \mathbin{.} \nu \iota
      \pm \nu^{-1} \, {\rm U} {\rm V} \mathbin{.} \nu \kappa.
   \eqno (62.)$$
In this expression for a vector touching a line of curvature, or
parallel to such a tangent, the two terms connected by the
sign~$\pm$ are easily seen to denote (on the principles of the
present calculus) two equally long vectors, in the directions
respectively of the projections of the two cyclic normals $\iota$
and $\kappa$ on a plane perpendicular to $\nu$; that is, on the
tangent plane to the ellipsoid at the proposed point, or on any
plane parallel thereto.  If then we draw two straight lines
through the point of contact, bisecting the acute and obtuse
angles which will in general be formed at that point by the
projections on the tangent plane of two indefinite lines drawn
through the same point in the directions of the two cyclic
normals, or in directions perpendicular to the two planes of
circular section of the surface, {\it the two rectangular
bisectors of angles, so obtained, will be the tangents to the two
lines of curvature\/}: which very simple construction agrees
perfectly with known geometrical results, as will be more clearly
seen, when it is slightly transformed as follows.

\bigbreak

54.
If we multiply either of the two tangential vectors~$\tau$ by the
normal vector~$\nu$, the product of these two rectangular vectors
will be, by one of the fundamental and
{\it peculiar\/}\footnote*{See the author's Letter of October~17,
1843, to John T. Graves, Esq., printed in the Supplementary
Number of the Philosophical Magazine for December 1844:
in which Letter, the three fundamental symbols, $i$,~$j$,~$k$
were what it has been since proposed to name
{\it direction-units}.}
principles of the calculus of quaternions, a third vector
rectangular to both; we shall therefore only pass by this
multiplication, so far as {\it directions\/} are concerned, from
one to the other of the tangents of the two lines of curvature:
consequently we may omit the factor~$\nu^{-1}$ in the second
member of (62.), at least if we change (for greater facility of
comparison of the results among themselves) the ambiguous
sign~$\pm$ to its opposite.  We may also suppress the scalar
coefficient~$t$, if we only wish to form an expression for a
line~$\tau$ which shall have the required {\it direction\/} of a
tangent, without obliging the {\it length\/} of this line~$\tau$
to take any previously chosen value.  The formula for the system
of the two tangents to the two lines of curvature thus takes the
simplified form:
$$\tau
   =  {\rm U} {\rm V} \mathbin{.} \nu \iota
      \mp {\rm U} {\rm V} \mathbin{.} \nu \kappa;
   \eqno (63.)$$
in which the two terms connected by the sign~$\mp$ are two
vector-units, in the respective directions of the traces of the
two cyclic planes upon the tangent plane.  The tangents to the
two lines of curvature at any point of the surface of an
ellipsoid (and the same result holds good also for other surfaces
of the second order), are therefore parallel to the two
rectangular straight lines which bisect the angles between those
traces; or they are themselves the bisectors of the angles made
at the point of contact by the traces of planes parallel to the
two cyclic planes.  The discovery of this remarkable geometrical
theorem appears to be due to M.~Chasles.  It is only brought
forward here for the sake of the {\it process\/} by which it has
been above deduced (and by which the writer was in fact led to
perceive the theorem before he was aware that it was already
known), through an application of the method of quaternions, and
as a corollary from the geometrical construction of the ellipsoid
itself to which that method conducted him.\footnote*{See the
Numbers of the Philosophical Magazine for June,
September, and October 1847; or the Proceedings of the Royal
Irish Academy for July 1846.}
For that new geometrical {\it construction\/} has been shown (in
a recent Number of this Magazine) to admit of being
easily {\it retranslated\/} into that quaternion form of the
{\it equation\/}\footnote\dag{Another very simple construction,
derived from the same quaternion equation, and serving to
generate, by a moving sphere, a system of {\it two\/} reciprocal
ellipsoids, will be given in an early Number of this Magazine.}
of the ellipsoid, namely
$${\rm T} (\iota \rho + \rho \kappa)
   = \kappa^2 - \iota^2,
   \qquad\hbox{equation~(9.), art.~(38.),}$$
as an {\it interpretation\/} of which equation it had been
assigned by the present writer; and then a {\it general\/} method
for investigating by quaternions the directions of the lines of
curvature on {\it any\/} curved surface whatever, conducts, as
has been shown (in articles~46 and 47), to the equation of those
lines for the ellipsoid,
$${\rm S} \mathbin{.} \nu \tau \iota \tau \kappa = 0
   \qquad\hbox{(49.);}$$
from which, when combined with the general equation
${\rm S} \mathbin{.} \nu \tau = 0$,
the formula (63.) has been deduced and geometrically interpreted
as above.

\bigbreak

55.
Another mode of investigating generally the directions of those
tangential vectors~$\tau$ which satisfy the system of the two
conditions in art.~51, may be derived from observing that
those conditions fail to distinguish one such tangential vector
from another in each of the two cases where the variable
normal~$\nu$ coincides in direction with either of the two fixed
cyclic normals, $\iota$ and $\kappa$; that is, at the four
{\it umbilical points\/} of the ellipsoid, as might have been
expected from the known properties of that surface.  In fact if
we suppose
$$\nu = m \iota,\quad
  {\rm S} \mathbin{.} \iota \tau = 0,
   \eqno (64.)$$
where $m$ is a scalar coefficient, that is if we attend to either
of those two opposite {\it umbilics\/} at which $\nu$ has the
direction of $\iota$, we find the value
$$\nu \tau \iota \tau \kappa = m (\iota \tau)^2 \kappa,
   \eqno (65.)$$
which is here a vector-form, because by (64.) the
product~$\iota \tau$ denotes in this case a {\it pure vector}, so
that {\it its square (like that of every other vector in this
theory) will be a negative scalar}, by one of the fundamental and
{\it peculiar\/}\footnote*{See the author's letter of
October 17, 1843, already cited in a note to article~54.}
principles of the present calculus; the scalar part of the
product $\nu \tau \iota \tau \kappa$ therefore vanishes, or the
condition~(49.) is satisfied by the suppositions~(64.).  Again,
if we suppose
$$\nu = m' \kappa,
   \eqno (66.)$$
$m'$ being another scalar coefficient, that is, if we consider
either of those two other opposite umbilics at which $\nu$ has
the direction of $\kappa$, we are conducted to this other
expression,
$$\nu \tau \iota \tau \kappa = m' \kappa \tau \iota \tau \kappa;
   \eqno (67.)$$
which also is a vector-form, by the principles of the 20th
article.  In this manner we may be led to see that if in general
we decompose, by orthogonal projections, each of the two cyclic
normals, $\iota$ and $\kappa$, into two partial or component
vectors, $\iota'$,~$\iota''$, and $\kappa'$,~$\kappa''$, of which
$\iota'$ and $\kappa'$ shall be tangential to the surface, or
perpendicular to the variable normal~$\nu$, but $\iota''$ and
$\kappa''$ parallel to that normal, in such a manner as to
satisfy the two sets of equations,
$$\left. \multieqalign{
\iota &= \iota' + \iota''; &
{\rm S} \mathbin{.} \iota'   \nu &= 0; &
{\rm V} \mathbin{.} \iota''  \nu &= 0; \cr
\kappa &= \kappa' + \kappa''; &
{\rm S} \mathbin{.} \kappa'  \nu &= 0; &
{\rm V} \mathbin{.} \kappa'' \nu &= 0; \cr}
   \right\}
   \eqno (68.)$$
then, on substituting these values for $\iota$ and $\kappa$ in
the condition~(49.) or in the equation
$0 = {\rm S} \mathbin{.} \nu \tau \iota \tau \kappa$,
the terms involving $\iota''$ and $\kappa''$ will vanish of
themselves, and the equation to be satisfied will become
$$0 = {\rm S} \mathbin{.} \nu \tau \iota' \tau \kappa';
   \eqno (69.)$$
which is thus far a simplified form of the equation (49.), that
three of the four directions to be compared (namely those of
$\iota'$,~$\kappa'$, and $\tau$) are now parallel to one common
plane, namely to the plane which touches the ellipsoid at the
proposed point, and to which the fourth direction (that of $\nu$)
is perpendicular.  Decomposing the two quaternion products,
$\tau \iota'$ and $\tau \kappa'$, into their respective scalar
and vector parts, by the general formul{\ae},
$$\left. \eqalign{
\tau \iota'
   &= {\rm S} \mathbin{.} \tau \iota'
       + {\rm V} \mathbin{.} \tau \iota';\cr
\tau \kappa'
   &= {\rm S} \mathbin{.} \tau \kappa'
       + {\rm V} \mathbin{.} \tau \kappa';\cr}
   \right\}
   \eqno (70.)$$
and observing that the vectors ${\rm V} \mathbin{.} \tau \iota'$
and ${\rm V} \mathbin{.} \tau \kappa'$ both represent lines
parallel to $\nu$, because $\nu$ is perpendicular to the common
plane of $\tau$, $\iota'$, $\kappa'$; so that the three following
binary products
${\rm V} \mathbin{.} \tau \iota' \mathbin{.}
      {\rm V} \mathbin{.} \tau \kappa'$,
$\nu \, {\rm V} \mathbin{.} \tau \iota'$,
$\nu \, {\rm V} \mathbin{.} \tau \kappa'$,
are in the present question scalars; we find that we may write
$${\rm S} \mathbin{.} \nu \tau \iota' \tau \kappa'
   =  \nu \, {\rm S} \mathbin{.} \tau \iota' \mathbin{.}
         {\rm V} \mathbin{.} \tau \kappa'
    + \nu \, {\rm V} \mathbin{.} \tau \iota' \mathbin{.}
         {\rm S} \mathbin{.} \tau \kappa'.
   \eqno (71.)$$
Hence the equation (69.) or (49.) reduces itself, after being
multiplied by $\nu^{-1}$, to the form
$$    {\rm S} \mathbin{.} \tau \iota' \mathbin{.}
         {\rm V} \mathbin{.} \tau \kappa'
    + {\rm V} \mathbin{.} \tau \iota' \mathbin{.}
         {\rm S} \mathbin{.} \tau \kappa'
   =  0;
   \eqno (72.)$$
which gives, in general, by the rules of the present calculus,
$$    {{\rm V} \mathbin{.} \iota' \tau
         \over {\rm S} \mathbin{.} \iota' \tau}
   =  {{\rm V} \mathbin{.} \tau \kappa'
         \over {\rm S} \mathbin{.} \tau \kappa'};
   \eqno (73.)$$
and by another transformation,
$$    {{\rm V} \mathbin{.} \iota' \tau^{-1}
         \over {\rm S} \mathbin{.} \iota' \tau^{-1}}
   =  - {{\rm V} \mathbin{.} \kappa' \tau^{-1}
         \over {\rm S} \mathbin{.} \kappa' \tau^{-1}};
   \eqno (74.)$$
which may perhaps be not inconveniently written also thus:
$${{\rm V} \over {\rm S}} \mathbin{.} {\iota' \over \tau}
   =   - {{\rm V} \over {\rm S}} \mathbin{.} {\kappa' \over \tau};
   \eqno (75.)$$
in using which abridged notation, we must be careful to remember,
respecting the characteristic
$\displaystyle {{\rm V} \over {\rm S}}$,
of which the effect is to form or to denote the {\it quotient of
the vector part divided by the scalar part\/} of any quaternion
expression to which it is prefixed, that {\it this new
characteristic of operation is not\/} (like ${\rm S}$ and
${\rm V}$ themselves) {\it distributive relatively to the
operand}.  The vector denoted by the first member of (74.) or
(75.) is a line perpendicular to the plane of $\iota'$ and
$\tau$, that is to the tangent plane of the ellipsoid; and its
length is the trigonometric tangent of the angle of rotation in
that plane from the direction of the line~$\tau$ to that of the
line~$\iota'$; while a similar interpretation applies to the
second member of either of the same two equations, the sign~$-$
in that second member signifying here that the two equally long
angular motions, or rotations, from $\tau$ to $\iota'$, and from
$\tau$ to $\kappa'$ are performed in opposite directions.  Thus
the vector~$\tau$, which touches a line of curvature, coincides
in direction with the bisector of the angle in the tangent plane
between the projections, $\iota'$ and $\kappa'$, of the cyclic
normals thereupon; or with that other line, at right angles to
this last bisector, which bisects in like manner the other and
supplementary angle in the same tangent plane, between the
directions of $\iota'$ and $- \kappa'$: since $\kappa'$ may be
changed to $- \kappa'$, without altering essentially any one of
the four last equations between $\tau$,~$\iota'$,~$\kappa'$.
Those two rectangular and known directions of the tangents to the
lines of curvature at any point of an ellipsoid, which were
obtained by the process of article~53, are therefore obtained
also by the process of the present article; which conducts, by
the help of the geometrical reasoning above indicated, to the
following expression for the system of those two tangents~$\tau$,
as the symbolical solution (in the language of the present
calculus) of any one of the four last equations
(72.)~$\ldots$~(75.):
$$\tau = t' ( {\rm U} \iota' \pm {\rm U} \kappa');
   \eqno (76.)$$
where $t'$ is a scalar coefficient.

The agreement of this symbolical result with that marked (62.)
may be made evident observing that the equations~(68.) give
$$\iota'  = \nu^{-1} \, {\rm V} \mathbin{.} \nu \iota;\quad
  \kappa' = \nu^{-1} \, {\rm V} \mathbin{.} \nu \kappa;
   \eqno (77.)$$
so that if we establish, as we may, the relation
$$t t' = ({\rm T} \nu)^{-1},
   \eqno (78.)$$
between the arbitrary scalar coefficients $t$ and $t'$, which
enter into the formul{\ae} (62.) and (76.), those formul{\ae} will
coincide with each other.  And to show, without introducing
geometrical considerations, that (for example) the form~(73.) of
the recent condition relatively to $\tau$ is symbolically
satisfied by the expression~(76.), we may remark that this
expression, when operated upon according to the {\it general
rules\/} of this calculus, gives
$$\left. \multieqalign{
{\rm T} \kappa' \mathbin{.} {\rm V} \mathbin{.} \iota' \tau
   &= \pm t' \, {\rm V} \mathbin{.} \iota' \kappa'; &
{\rm T} \kappa' \mathbin{.} {\rm S} \mathbin{.} \iota' \tau
   &= t' ( - {\rm T} \mathbin{.} \iota' \kappa'
         \pm {\rm S} \mathbin{.} \iota' \kappa' );\cr
{\rm T} \iota'  \mathbin{.} {\rm V} \mathbin{.} \tau \kappa'
   &= t' \, {\rm V} \mathbin{.} \iota' \kappa'; &
{\rm T} \iota'  \mathbin{.} {\rm S} \mathbin{.} \tau \kappa'
   &= t' ( {\rm S} \mathbin{.} \iota' \kappa'
         \mp {\rm T} \mathbin{.} \iota' \kappa' );\cr}
   \right\}
   \eqno (79.)$$
and that therefore the two members of (73.) do in fact receive,
in virtue of (76.) one common symbolical value, namely one or
other of the two which are included in the ambiguous form
$${{\rm V} \mathbin{.} \iota' \kappa'
      \over {\rm S} \mathbin{.} \iota' \kappa'
         \mp {\rm T} \mathbin{.} \iota' \kappa'};
   \eqno (80.)$$
respecting which form it may not be useless to remark that the
product of its two values is unity.

\bigbreak

56.
If we denote by $b$ the length of the common radius of the two
diametral and circular sections, or the mean semiaxis of the
ellipsoid, which is also the radius of that concentric sphere of
which the equation~(24.) was assigned in art.~44, we shall
have, by the formula~(26.) of that article, the following
expression for this radius, or semiaxis:
$$b = {\kappa^2 - \iota^2 \over {\rm T} (\iota - \kappa)}.
   \eqno (81.)$$
And hence, on account of the general formula,
$$\iota \rho + \rho \kappa
   =  (\iota - \kappa)
         \left(
            \rho
          + {\kappa \rho + \rho \kappa \over \iota - \kappa}
         \right),
   \eqno (82.)$$
which holds good for {\it any\/} three vectors,
$\iota$,~$\kappa$,~$\rho$, the quaternion equation of the
ellipsoid may be changed from a form already assigned, namely
$${\rm T} (\iota \rho + \rho \kappa)
   =  \kappa^2 - \iota^2,
   \qquad\hbox{(9.), art.~38,}$$
to the following equivalent form:
$${\rm T}
      \left(
         \rho
       + {\kappa \rho + \rho \kappa \over \iota - \kappa}
      \right)
   = b.
   \eqno (83.)$$
If then we introduce a new vector-symbol~$\lambda$, denoting a
line of variable length, but one drawn in the fixed direction of
$\iota - \kappa$, or in the exactly opposite direction of
$\kappa - \iota$, and determined by the condition
$$\lambda (\kappa - \iota) = \kappa \rho + \rho \kappa,
   \eqno (84.)$$
we shall have also
$${\rm T} (\rho - \lambda) = b;
   \eqno (85.)$$
and thus the equation~(83.) of the ellipsoid may be regarded as
the result of the elimination of the auxiliary
vector-symbol~$\lambda$ between the two last equations (84.) and
(85.).  But if we suppose that this symbol~$\lambda$ receives any
{\it given\/} and constant value, of the form
$$\lambda = h (\iota - \kappa),
   \eqno (86.)$$
where $h$ is a scalar coefficient, which we here suppose to be
constant and given, and if we still conceive the symbol~$\rho$ to
denote a variable vector, drawn from the centre of the ellipsoid
as an origin, the equation~(84.) will then express that this
vector~$\rho$ terminates in a point which is contained on a
{\it given plane\/} parallel to that one of the two cyclic planes
of the ellipsoid which has for its equation
$$\kappa \rho + \rho \kappa = 0,
   \qquad\hbox{(23.), art.~44;}$$
while the equation~(85.) will express that the same vector~$\rho$
terminates also on a given spheric surface, of which the vector
of the centre (drawn from the same centre of the ellipsoid) is
$\lambda$, and of which the radius is $= b$.  The {\it system of
the two equations}, (84.) and (85.), expresses therefore that,
for any given value of the auxiliary vector~$\lambda$, or for any
given value of the scalar coefficient~$h$ in the formula~(86.),
the termination of the vector~$\rho$ is contained on the
circumference of a {\it given circle}, which is the mutual
intersection of the plane~(84.) and of the sphere~(85.).  And the
equation~(83.) of the ellipsoid, as being derived, or at least
derivable, by elimination of $\lambda$, from that system of
equations (84.) and (85.), is thus seen to express the known
theorem, that the surface of an ellipsoid may be regarded as the
locus of a certain {\it system of circular circumferences}, of
which the planes are parallel to a fixed plane of diametral and
circular section.

\bigbreak

57.
{\it One set\/} of the known circular sections of the ellipsoid,
in planes parallel to {\it one\/} of the two cyclic planes, may
therefore be assigned in this manner, as the result of a very
simple calculation; and the {\it other set\/} of such known
circular sections, parallel to the {\it other\/} cyclic plane,
may be symbolically determined, with equal facility, as the
result of an entirely similar process of calculation with
quaternions.  For if, instead of (82.), we employ this other
general formula, which likewise holds good for any three vectors,
$$\iota \rho + \rho \kappa
   =  \left(
         \rho
       + {\iota \rho + \rho \iota \over \kappa - \iota}
      \right)
      (\kappa - \iota),
   \eqno (87.)$$
we shall thereby transform the lately cited equation~(9.) of the
ellipsoid into this other form,
$${\rm T}
      \left(
         \rho
       + {\iota \rho + \rho \iota \over \kappa - \iota}
      \right)
   =  b;
   \eqno (88.)$$
which is analogous to the form (83.), and from which similar
inferences may be drawn.  Thus, we may treat this equation~(88.)
as the result of elimination of a new auxiliary vector
symbol~$\mu$ between the two equations,
$$\mu (\iota - \kappa) = \iota \rho + \rho \iota;
   \eqno (89.)$$
$${\rm T} (\rho - \mu) = b;
   \eqno (90.)$$
of which the former, namely the equation~(89.), is, relatively to
$\rho$, the equation of a {\it new plane}, parallel to that other
cyclic plane of the ellipsoid for which we have seen that
$$\iota \rho + \rho \iota = 0,
   \qquad\hbox{(25.), art.~44;}$$
while the latter equation, namely (90.), is that of a {\it new
sphere}, with the same radius~$b$ as before, but with $\mu$ for
the vector of its centre: which sphere~(90.), determines, by its
intersection with the plane~(89.), a {\it new circle\/} as the
locus of the termination of $\rho$, when $\mu$ receives any given
value of the form
$$\mu = h' (\kappa - \iota),
   \eqno (91.)$$
where $h'$ is a new scalar coefficient.  The ellipsoid~(9.) is
therefore the locus of all the circles of this second system
also, answering to the equations (89.), (90.), as it was seen to
be the locus of all those of the first system, represented by the
equations (84.), (85.); which agrees with the known properties of
the surface.

\bigbreak

58.
For any three vectors, $\iota$,~$\kappa$,~$\rho$, we have
(because $\rho^2$, $\kappa^2$, and $\kappa \rho + \rho \kappa$
are scalars) the general transformations,
$$\left. \eqalign{
(\iota \rho + \rho \iota) (\kappa \rho + \rho \kappa)
   &= \iota (\kappa \rho + \rho \kappa) \rho
       + \rho (\kappa \rho + \rho \kappa) \iota \cr
   &= (\iota \kappa + \kappa \iota) \rho^2
      + \iota \rho \kappa \rho
      + \rho \kappa \rho \iota \cr
   &= - (\iota - \kappa)^2 \rho^2
      + (\iota \rho + \rho \kappa) (\rho \iota + \kappa \rho);\cr}
   \right\}
   \eqno (92.)$$
and therefore, with the recent significations of the symbols $b$,
$\lambda$, $\mu$, expressed by the formul{\ae} (81.), (84.), (89.),
the equation of the ellipsoid assigned in a foregoing article,
namely
$$(\iota \rho + \rho \kappa) (\rho \iota + \kappa \rho)
   =  (\kappa^2 - \iota^2)^2,
   \qquad\hbox{(21.), art.~44,}$$
takes easily this shorter form,
$$\rho^2 + b^2 = \lambda \mu.
   \eqno (93.)$$
If now we cut this surface by the system of two planes, parallel
respectively to the two cyclic planes (23.) and (25.), and
included in the joint equation
$$    \{ \lambda - h  (\iota - \kappa) \}
      \{ \mu     - h' (\kappa - \iota) \}
   =  0,
   \eqno (94.)$$
which is derived by multiplication from the equations (86.) and
(91.), we are conducted to this other equation,
$$\rho^2 + b^2
   =  h  (\iota \rho + \rho \iota)
    + h' (\kappa \rho + \rho \kappa)
    + h h' (\iota - \kappa)^2;
   \eqno (95.)$$
which may be put under the form
$$- b^2
   =  (\rho - h \iota - h' \kappa)^2
       - (h + h') (h \iota^2 + h' \kappa^2);
   \eqno (96.)$$
or under this other form,
$${\rm T} (\rho - \xi) = r,
   \eqno (97.)$$
if we write, for abridgment,
$$\xi = h \iota + h' \kappa,
   \eqno (98.)$$
and
$$r = \surd \{ b^2 - (h + h') (h \iota^2 + h' \kappa^2) \}.
   \eqno (99.)$$

Any two circular sections of the ellipsoid, parallel to two
different cyclic planes, or belonging to two {\it different\/}
systems, are therefore contained upon one {\it common\/}
sphere~(97.), of which the radius~$r$, and the vector of the
centre~$\xi$, are assigned by these last formul{\ae}: which again
agrees with the known properties of surfaces of the second order.
And the equation of the {\it mean sphere\/} which contains the
two {\it diametral\/} and circular sections, is seen to reduce
itself, in this system of algebraical geometry, to the very
simple form\footnote*{Compare article~21, in the Phil.\ Mag.\ for
July 1846.}
$$\rho^2 + b^2 = 0.
   \eqno (100.)$$

\bigbreak

59.
The expressions (86.), (91.), (98.), for $\lambda$, $\mu$, $\xi$,
give
$$    {\xi - \lambda \over \kappa}
   =  {\xi - \mu \over \iota}
   =  {\lambda - \mu \over \iota - \kappa}
   =  h + h';
   \eqno (101.)$$
if then we regard $\lambda$, $\mu$, $\xi$ as the vectors of the
three corners ${\sc l}$,~${\sc m}$,~${\sc n}$ of a plane
triangle, and observe that $0$, $\iota - \kappa$, and $- \kappa$
were seen to be the vectors of the three corners
${\sc a}$,~${\sc b}$,~${\sc c}$, of the {\it generating
triangle\/} described in our construction of the ellipsoid, we
see that the new triangle ${\sc l} {\sc m} {\sc n}$ is similar to
that generating triangle ${\sc a} {\sc b} {\sc c}$, and similarly
situated in one common plane therewith, namely in the plane of
the greatest and least axes of the ellipsoid; the sides
${\sc l} {\sc m}$, ${\sc m} {\sc n}$, ${\sc n} {\sc l}$ of the
one triangle being parallel and proportional to the sides
${\sc a} {\sc b}$, ${\sc b} {\sc c}$, ${\sc c} {\sc a}$ of the
other, while the points ${\sc l}$ and ${\sc m}$ are situated on
the same indefinite straight line as ${\sc a}$,~${\sc b}$; that
is, on the axis of that circumscribed cylinder of revolution
which has been considered in former articles.  The vectors of the
points ${\sc d}$,~${\sc e}$, in the same construction of the
ellipsoid, (if drawn from its centre as their origin,) having
been seen to be respectively $\sigma - \kappa$ and $\rho$,
(compare article~40,) the equation
$$\sigma \rho + \rho \kappa = 0,
   \qquad\hbox{(16.), art.~41,}$$
combined with (84.) and (86.), gives for their product the
expressions:
$$(\sigma - \kappa) \rho = \lambda (\iota - \kappa)
   =  h (\iota - \kappa)^2;
   \eqno (102.)$$
and in general if two pairs of co-initial vectors, as here
$\sigma - \kappa$,~$\rho$ and $\lambda$,~$\iota - \kappa$, give,
when respectively multiplied, one common scalar product, they
terminate on four concircular points: the four points
${\sc d}$,~${\sc e}$,~${\sc l}$,~${\sc b}$ are therefore
contained on the circumference of one common circle: and
consequently the point~${\sc l}$ may be found by an elementary
construction, derived from this simple calculation with
quaternions, namely as the second point of intersection of the
circle ${\sc b} {\sc d} {\sc e}$ with the straight line
${\sc a} {\sc b}$ (which is situated in the plane of the circle).
Again, the equations (85.) and (90.) give
$${\rm T} (\rho - \lambda) = {\rm T} (\rho - \mu);
   \eqno (103.)$$
therefore the point~${\sc e}$ of the ellipsoid is the vertex of
an isosceles triangle, constructed on ${\sc l} {\sc m}$ as base;
and the point~${\sc m}$ may thus be found as the intersection of
the same straight line~${\sc a} {\sc b}$ or ${\sc a} {\sc l}$,
with a circle described round the point~${\sc e}$ as centre, and
having its radius $= \overline{{\sc e} {\sc l}} = b =$ the mean
semiaxis of the ellipsoid.  When the two points ${\sc l}$ and
${\sc m}$ have thus been found, the third point~${\sc n}$ can
then be deduced from them, in an equally simple geometrical
manner, by drawing parallels ${\sc l} {\sc n}$, ${\sc m} {\sc n}$
to the sides ${\sc a} {\sc c}$, ${\sc b} {\sc c}$ of the
generating triangle ${\sc a} {\sc b} {\sc c}$, from which the
ellipsoid itself has been constructed; these sides
${\sc l} {\sc n}$, ${\sc m} {\sc n}$, of the new and variable
triangle ${\sc l} {\sc m} {\sc n}$, will thus be parallel to the
two cyclic normals of the ellipsoid; and the foregoing analysis
shows that they will be portions of the axes of the two circles,
which are contained upon the surface of that ellipsoid, and pass
through the point~${\sc e}$ on that surface: while the
point~${\sc n}$, of intersection of those two axes, is the centre
of that common sphere~(97.), which contains both those two
circular sections.  It is evident that this common sphere must
{\it touch\/} the ellipsoid at ${\sc e}$, since it is itself
touched at that point by the two distinct tangents to the two
circular sections of the surface; and hence we might infer that
the semidiameter ${\sc n} {\sc e}$ or $\xi - \rho$ of the sphere,
of which the length~$r$ has been assigned in the formula~(99.),
and which is terminated at the point~${\sc n}$ by the plane of
the generating triangle, must coincide in direction with the
{\it normal\/}~$\nu$ to the ellipsoid: of which latter normal the
direction may thus be found by a simple geometrical construction,
and an expression for it be obtained without the employment of
differentials.  But to show that this geometrical result agrees
with the symbolical expression already found for~$\nu$, by means
of differentials and quaternions, we have only to substitute, on
the one hand, in the expression~(98.) for $\xi$, the following
values for $h$ and $h'$, derived from (84.), (86.), and from
(89.), (91.):
$$h  = {\kappa \rho + \rho \kappa
         \over - (\iota - \kappa)^2};\quad
  h' = {\iota  \rho + \rho \iota
         \over - (\iota - \kappa)^2};
   \eqno (104.)$$
and to observe, on the other hand, that the equation~(31.), which
has served to determine the normal vector of proximity~$\nu$, may
be thus written:
$$(\kappa^2 - \iota^2)^2 \nu
   =  (\iota - \kappa)^2 \rho
       + \iota  (\kappa \rho + \rho \kappa)
       + \kappa (\iota  \rho + \rho \iota);
   \eqno (105.)$$
for thus we are conducted, by means of (81.), to the formula:
$$\xi - \rho = b^2 \nu;
   \eqno (106.)$$
which expresses the agreement of the recent construction with the
results that had been previously obtained.

\bigbreak

60.
If we introduce two new constant vectors $\iota'$ and $\kappa'$,
connected with the two former constant vectors $\iota$,~$\kappa$,
by the equations
$$\iota \kappa' = \iota' \kappa
   =  {\rm T} \mathbin{.} \iota \kappa,
   \eqno (107.)$$
which give
$$\iota'^2 = \iota^2,\quad
  \kappa'^2 = \kappa^2,\quad
  \iota' \kappa' = \kappa \iota,
   \eqno (108.)$$
then one of the lately cited forms of the equation of the
ellipsoid, namely the equation
$${\rm T} (\iota \rho + \rho \kappa) = \kappa^2 - \iota^2
   \qquad\hbox{(9.), art.~38,}$$
takes easily, by the rules of this calculus, the new but
analogous form:
$${\rm T} (\iota' \rho + \rho \kappa')
   =  \kappa'^2 - \iota'^2.
   \eqno (109.)$$
The perfect similarity of these two forms, (9.) and (109.),
renders it evident that all the conclusions which have been
deduced from the one form can, with suitable and easy
modifications, be deduced from the other also.  Thus if we still
regard the centre~${\sc a}$ as the origin of vectors, and treat
$\iota' - \kappa'$ and $- \kappa'$ as the vectors of two new
fixed points ${\sc b}'$ and ${\sc c}'$, we may consider
${\sc a} {\sc b}' {\sc c}'$ as a {\it new generating triangle},
and may derive from it the {\it same ellipsoid\/} as before, by a
geometrical process of generation or construction, which is
similar in all respects to the process already assigned.  (See
the Numbers of the Philosophical Magazine for June,
September, and October, 1847; or the Proceedings of the
Royal Irish Academy for July,~1846.)  Hence the two new sides
${\sc b}' {\sc c}'$ and ${\sc c}' {\sc a}$, which indeed are
parallel by (107.) to the two old sides ${\sc a} {\sc c}$ and
${\sc c} {\sc b}$, or to $\kappa$ and $\iota$, must have the
directions of the two cyclic normals; and the third new side,
${\sc a} {\sc b}'$, or $\iota' - \kappa'$, must be the axis of a
{\it second cylinder of revolution}, circumscribed round the same
ellipsoid.  If we determine on this new axis two new points,
${\sc l}'$ and ${\sc m}'$, as the extremities of two new vectors
$\lambda'$ and $\mu'$, analogous to the recently considered
vectors $\lambda$ and $\mu$, and assigned by equations similar to
(84.) and (89.), namely
$$\lambda' (\kappa' - \iota')
   =  \kappa' \rho + \rho \kappa',\quad
  \mu' (\iota' - \kappa')
   =  \iota' \rho + \rho \iota',
   \eqno (110.)$$
we shall have results analogous to (85.) and (90.), namely
$${\rm T} (\rho - \lambda') = b;\quad
  {\rm T} (\rho - \mu') = b;
   \eqno (111.)$$
with others similar to (101.), namely
$$    {\xi - \lambda' \over \kappa'}
   =  {\xi - \mu' \over \iota'}
   =  {\lambda' - \mu' \over \iota' - \kappa'};
   \eqno (112.)$$
the common value of these three quotients being a new scalar, but
$\xi$ being still the same vector as before, namely that vector
which terminates in the point~${\sc n}$, where the normal to the
surface at ${\sc e}$ meets the common plane of the new and old
generating triangles, or the plane of the greatest and least axes
of the ellipsoid.  It is easy hence to infer that the new
variable triangle ${\sc l}' {\sc m}' {\sc n}$ is similar to the
new generating traingle ${\sc a} {\sc b}' {\sc c}'$, and
similarly situated in the same fixed plane therewith; and that
the sides ${\sc l}' {\sc n}$, ${\sc m}' {\sc n}$, having
respectively the same directions as ${\sc a} {\sc c}'$,
${\sc b}' {\sc c}'$ have likewise the same directions as
${\sc b} {\sc c}$, ${\sc a} {\sc c}$, and therefore also as
${\sc m} {\sc n}$, ${\sc l} {\sc n}$, or else directions opposite
to these; in such a manner that the two straight lines,
${\sc l}' {\sc m}$, ${\sc m}' {\sc l}$, must cross each other in
the point~${\sc n}$.  But these two lines may be regarded as the
diagonals of a certain quadrilateral inscribed in a circle,
namely the plane quadrilateral
${\sc l}' {\sc m}' {\sc m} {\sc l}$;
of which the four corners are, by (85.), (90.), and (111.), at
one common and constant distance $= b$, from the variable
point~${\sc e}$ of the ellipsoid.  If then we assume it as known
that the vector $b^2 \nu$, which is in direction opposite and is
in length reciprocal to the perpendicular let fall from the
centre~${\sc a}$ on the tangent plane at ${\sc e}$, must
terminate in a point~${\sc f}$ on the surface of {\it another
ellipsoid, reciprocal\/} (in a well-known sense) to that former
ellipsoid which contains the point~${\sc e}$ itself, or the
termination of the vector~$\rho$; we may combine the recent
results, so as to obtain the following geometrical
construction,\footnote*{This is the construction referred to in a
note to article~54.  It was communicated by the author to the
Royal Irish Academy, at the meeting of November~30, 1847.  See
the Proceedings of that date.}
which serves {\it to generate a system of two reciprocal
ellipsoids, by means of a moving sphere}.

\bigbreak

61.
Let then a sphere of constant magnitude, with centre~${\sc e}$,
move so that it always intersects two fixed and mutually
intersecting straight lines, ${\sc a} {\sc b}$,
${\sc a} {\sc b}'$, in four points
${\sc l}$,~${\sc m}$,~${\sc l}'$,~${\sc m}'$, of which ${\sc l}$
and ${\sc m}$ are on ${\sc a} {\sc b}$, while ${\sc l}'$ and
${\sc m}'$ are on ${\sc a} {\sc b}'$; and let one
diagonal~${\sc l} {\sc m}'$, of the inscribed quadrilateral
${\sc l} {\sc m} {\sc m}' {\sc l}'$, be constantly parallel to a
third fixed line~${\sc a} {\sc c}$, which will oblige the other
diagonal~${\sc m} {\sc l}'$ of the same quadrilateral to move
parallel to a fourth fixed line ${\sc a} {\sc c}'$.  Let
${\sc n}$ be the point in which the diagonals intersect, and draw
${\sc a} {\sc f}$ equal and parallel to ${\sc e} {\sc n}$; so
that ${\sc a} {\sc e} {\sc n} {\sc f}$ is a parallelogram: then
{\it the locus of the centre~${\sc e}$ of the moving sphere is
one ellipsoid, and the locus of the opposite corner~${\sc f}$ of
the parallelogram is another ellipsoid reciprocal thereto}.
These two ellipsoids have a common centre~${\sc a}$, and a common
mean axis, which is equal to the diameter of the moving sphere,
and is a mean proportional between the greatest axis of either
ellipsoid and the least axis of the other, of which two
last-mentioned axes the directions coincide.  Two sides,
${\sc a} {\sc e}$, ${\sc a} {\sc f}$, of the parallelogram
${\sc a} {\sc e} {\sc n} {\sc f}$, are thus two semidiameters
which may be regarded as mutually {\it reciprocal}, one of the
one ellipsoid, and the other of the other: but because they fall
at {\it opposite\/} sides of the {\it principal plane\/}
(containing the four fixed lines and the greatest and least axes
of the ellipsoids), it may be proper to call them, more fully,
{\it opposite reciprocal semidiameters\/}; and to call the points
${\sc e}$ and ${\sc f}$, in which they terminate, {\it opposite
reciprocal points}.  The two other sides ${\sc e} {\sc n}$,
${\sc f} {\sc n}$, of the same variable parallelogram, are the
{\it normals\/} to the two ellipsoids, meeting each other in the
point~${\sc n}$, upon the same principal plane.  In that plane,
the two former fixed lines, ${\sc a} {\sc b}$,
${\sc a} {\sc b}'$, are the {\it axes of two cylinders of
revolution}, circumscribed about the first ellipsoid; and the two
latter fixed lines, ${\sc a} {\sc c}$, ${\sc a} {\sc c}'$, are
the {\it two cyclic normals\/} of the same first ellipsoid: while
the diagonals, ${\sc l} {\sc m}'$, ${\sc m} {\sc l}'$, of the
inscribed quadrilateral in the construction, are the {\it axes of
the two circles\/} on the surface of that first ellipsoid, which
circles pass through the point~${\sc e}$, that is through the
centre of the moving sphere; and the intersection~${\sc n}$ of
those two diagonals is the centre of another sphere, which cuts
the first ellipsoid in the system of those two circles: all which
is easily adapted, by suitable interchanges, to the other or
reciprocal ellipsoid, and flows with facility from the quaternion
equations above given.

\bigbreak

62.
The equations (85.), (90.), and (111.), of articles 56, 57, and
60, give
$${\rm T} (\rho - \lambda)
   =  {\rm T} (\rho - \mu') = b;
   \eqno (113.)$$
and
$${\rm T} (\rho - \mu) = {\rm T} (\rho - \lambda') = b;
   \eqno (114.)$$
whence, by the meanings of the signs employed, the two following
mutually connected constructions may be derived, for
{\it geometrically generating an ellipsoid from a rhombus of
constant perimeter}, or for geometrically describing an arbitrary
curve on the surface of such an ellipsoid by the motion of a
corner of such a rhombus, which the writer supposes to be new.

1st Generation.
Let a rhombus
${\sc l} {\sc e} {\sc m}' {\sc e}'$,
of which each side preserves constantly a fixed length $= b$, but
of which the angles vary, move so that the two opposite corners
${\sc l}$,~${\sc m}'$ traverse two fixed and mutually
intersecting straight lines ${\sc a} {\sc b}$,
${\sc a} {\sc b}'$, (the point~${\sc l}$ moving along the
line~${\sc a} {\sc b}$, and the point~${\sc m}'$ along
${\sc a} {\sc b}'$,) while the diagonal ${\sc l} {\sc m}'$,
connecting these two opposite corners of the rhombus, remains
constantly parallel to a third fixed right line ${\sc a} {\sc c}$
(in the plane of the two former right lines); then,
{\it according to whatever arbitrary law the plane of the rhombus
may turn}, during its motion, {\it its two remaining corners
${\sc e}$,~${\sc e}'$ will describe curves upon the surface of a
fixed ellipsoid\/}; which surface is thus the {\it locus of all
the pairs of curves\/} that can be described by this first mode
of generation.

2nd Generation.
Let now {\it another rhombus},
${\sc l}' {\sc e}'' {\sc m} {\sc e}'''$,
with the {\it same constant perimeter\/} $= 4b$, move so that its
opposite corners ${\sc l}'$,~${\sc m}$ traverse the {\it same two
fixed lines\/} ${\sc a} {\sc b}$, ${\sc a} {\sc b}'$, as before,
but in such a manner that the diagonal ${\sc l}' {\sc m}$,
connecting these two corners, remains parallel (not to the third
fixed line~${\sc a} {\sc c}$, but) to a {\it fourth fixed
line\/}~${\sc a} {\sc c}'$; then, whatever may be the arbitrary
law according to which the plane of this new rhombus turns,
provided that the angles ${\sc b} {\sc a} {\sc b}'$,
${\sc c} {\sc a} {\sc c}'$, between the first and second, and
between the third and fourth fixed lines, have one {\it common
bisector}, the {\it two remaining corners
${\sc e}''$,~${\sc e}'''$ of this second rhombus will describe
curves upon the surface of the same fixed ellipsoid}, as that
determined by the former generation: which surface is thus the
{\it locus of all the new pairs of curves}, described in this
second mode, as it was just now seen to be the locus of all the
old pairs of curves, obtained in the first mode of description.

\bigbreak

63.
The ellipsoid (with three unequal axes), thus generated, is
therefore the {\it common locus of the four curves}, described by
the four points ${\sc e}$~${\sc e}'$~${\sc e}''$~${\sc e}'''$; of
which four curves, the first and third may be made to coincide
with {\it any arbitrary curves on that ellipsoid\/}; but the
second and fourth become determined, when the first and third
have been chosen.  And in this new {\it system of two connected
constructions for generating an ellipsoid}, as well as in that
other construction\footnote*{See Phil. Mag. for May 1848; or
Proceedings of Royal Irish Academy for November 1847.}
which was given in article~61 for a {\it system of two reciprocal
ellipsoids}, the two former fixed lines,
${\sc a} {\sc b}$, ${\sc a} {\sc b}'$, are the {\it axes of two
cylinders of revolution}, circumscribed about the ellipsoid which
is the locus of the point~${\sc e}$; while the two latter fixed
lines, ${\sc a} {\sc c}$, ${\sc a} {\sc c}'$, are the {\it two
cyclic normals\/} (or the normals to the two planes of circular
section) of that ellipsoid.  The common (internal and external)
bisectors, at the centre~${\sc a}$, of the angles
${\sc b} {\sc a} {\sc b}'$, ${\sc c} {\sc a} {\sc c}'$, made by
the first and second, and by the third and fourth fixed lines,
coincide in direction with the {\it greatest and least axes\/} of
the ellipsoid; and the constant length~$b$, of the side of either
rhombus, is the length of the {\it mean semiaxis}.  The diagonal
${\sc l} {\sc m}'$ of the first rhombus is the {\it axis of a
first circle\/} on the ellipsoid, of which circle a diameter
coincides with the second diagonal ${\sc e} {\sc e}'$ of the same
rhombus; and, in like manner, the diagonal ${\sc l}' {\sc m}$ of
the second rhombus is the {\it axis of a second circle\/} on the
same ellipsoid, belonging to the second (or {\it subcontrary\/})
system of circular sections of that surface: while the other
diagonal ${\sc e}'' {\sc e}'''$, of the same second rhombus, is a
diameter of the same second circle.  In the quaternion analysis
employed, the first of these two circular sections of the
ellipsoid corresponds to the equations (113.); and the second
circular section is represented by the equations (114.), of the
foregoing article.

\bigbreak

64.
We may also present the interpretation of those quaternion
equations, or the recent double construction of the ellipsoid, in
the following other way, which also appears to be new; although
the writer is aware that there would be no difficulty in proving
its correctness, or in deducing it anew, either by the method of
co-ordinates, or in a more purely geometrical mode.  {\it Conceive
two equal spheres to slide within two cylinders\/} (of
revolution, whose axes intersect each other, and of which each
touches its own sphere along a great circle of contact), {\it in
such a manner that the right line joining the centres of the
spheres shall be parallel to a fixed right line\/}; then {\it the
locus of the varying circle in which the two spheres intersect
each other will be an ellipsoid, inscribed at once in both the
cylinders}, so as to touch one cylinder along one ellipse of
contact, and the other cylinder along another such ellipse.  And
the {\it same\/} ellipsoid may be generated as the locus of
{\it another\/} varying circle, which shall be the intersection
of {\it two other equal spheres sliding within the same two
cylinders of revolution}, but with a connecting line of centres
which now moves parallel to {\it another fixed right line\/};
provided that the angle between these two fixed lines, and the
angle between the axes of the two cylinders, have both one common
pair of (internal and external) bisectors, which will then
coincide in direction with the greatest and least axes of the
ellipsoid, while the diameter of each of the {\it four sliding
spheres\/} is equal to the mean axis.  In fact, we have only to
conceive (with the recent significations of the letters), that
four spheres, with the same common radius $= b$, are described
about the points ${\sc l}$,~${\sc m}'$, and
${\sc l}'$,~${\sc m}$, as centres; for then the first pair of
spheres will cross each other in that circular section of the
ellipsoid which has ${\sc e} {\sc e}'$ for a diameter; and the
second pair of spheres will cross in the circle of which the
diameter is ${\sc e}'' {\sc e}'''$; after which the other
conclusions above stated will follow, from principles already
laid down.

\bigbreak

65.
If we make
$$\rho - \lambda = \lambda_\prime;\quad
  \rho - \mu = \mu_\prime;\quad
  \rho - \lambda' = \lambda_\prime';\quad
  \rho - \mu' = \mu_\prime';
   \eqno (115.)$$
and in like manner, (see (106.),)
$$\rho - \xi = - b^2 \nu = \xi_\prime;
   \eqno (116.)$$
and if we regard these five new vectors,
$\lambda_\prime$,~$\mu_\prime$,~$\lambda_\prime'$,~$\mu_\prime'$,
and $\xi_\prime$, as lines which, being drawn from the
centre~${\sc a}$, terminate respectively in five new points,
${\sc l}_\prime$,~${\sc m}_\prime$,~${\sc l}_\prime'$,~${\sc m}_\prime'$,
and ${\sc h}$; while the vector~$\rho$, drawn from the same
centre~${\sc a}$, still terminates in the point~${\sc e}$, upon
the surface of the ellipsoid; then the equations (113.), (114.),
of art.~62, will give:
$${\rm T} \lambda_\prime = {\rm T}\mu_\prime
   =  {\rm T} \lambda_\prime' = {\rm T} \mu_\prime'
   =  b;
   \eqno (117.)$$
while the equations (101.) will enable us to write
$$    {\lambda_\prime - \xi_\prime \over \kappa}
   =  {\mu_\prime - \xi_\prime \over \iota}
   =  {\mu_\prime - \lambda_\prime \over \iota - \kappa}
   =  {\rm V}^{-1} 0;
   \eqno (118.)$$
and in like manner, (see (112.),)
$$    {\lambda_\prime' - \xi_\prime \over \kappa'}
   =  {\mu_\prime' - \xi_\prime \over \iota'}
   =  {\mu_\prime' - \lambda_\prime' \over \iota' - \kappa'}
   =  {\rm V}^{-1} 0;
   \eqno (119.)$$
this symbol ${\rm V}^{-1} 0$ denoting (as already explained) a
{\it scalar}.  We shall have also, by (84.), (89.),
$$    {\rho - \lambda_\prime \over \iota - \kappa}
   =  {\lambda \over \iota - \kappa}
   =  {\rm V}^{-1} 0;\quad
      {\rho - \mu_\prime \over \kappa - \iota}
   =  {\mu \over \kappa - \iota}
   =  {\rm V}^{-1} 0;
   \eqno (120.)$$
the scalars denoted by the symbol ${\rm V}^{-1} 0$ being not
generally obliged to be equal to each other, and being, in these
last equations (120.), respectively equal, by (86.), (91.), to
those which have been denoted by $h$ and $h'$.  In like manner,
by (110.),
$$    {\rho - \lambda_\prime' \over \iota' - \kappa'}
   =  {\lambda' \over \iota' - \kappa'}
   =  {\rm V}^{-1} 0;\quad
      {\rho - \mu_\prime' \over \kappa' - \iota'}
   =  {\mu' \over \kappa' - \iota'}
   =  {\rm V}^{-1} 0.
   \eqno (121.)$$
And because, by (107.), $\iota'$ has a scalar ratio to $\kappa$,
and $\kappa'$ has a scalar ratio to $\iota$, we may infer, from
(118.), (119.), the existence of the two following other scalar
ratios:
$${\mu_\prime' - \xi_\prime \over \lambda_\prime - \xi_\prime}
   =  {\rm V}^{-1} 0;\quad
  {\lambda_\prime' - \xi_\prime \over \mu_\prime - \xi_\prime}
   =  {\rm V}^{-1} 0.
   \eqno (122.)$$
Finally we may observe that, by (120.), (121.), there exist
scalar ratios between certain others also of the foregoing
vector-differences, and especially the following:
$${\rho - \lambda_\prime \over \rho - \mu_\prime}
   =  {\rm V}^{-1} 0;\quad
  {\rho - \lambda_\prime' \over \rho - \mu_\prime'}
   =  {\rm V}^{-1} 0.
   \eqno (123.)$$

\bigbreak

66.
Proceeding now to consider the geometrical signification of the
equations in the last article, we see first, from the equations
(117.), that the four new points,
${\sc l}_\prime$,~${\sc m}_\prime$,~${\sc l}_\prime'$~${\sc m}_\prime'$,
are all situated upon the surface of that {\it mean sphere},
which is described on the mean axis of the ellipsoid as a
diameter; because the equation of that mean sphere has been
already seen to be\footnote*{This {\it form\/} of the equation of
the {\it sphere\/} was published in the Philosophical Magazine
for July 1846; and it is an immediate and a very easy consequence
of that fundamental formula of the whole theory of Quaternions,
namely
$$i^2 = j^2 = k^2 = ijk = -1,$$
which was communicated under a slightly more developed form, to
the Royal Irish Academy, on the 13th of November 1843.  (See
Phil.\ Mag.\ for July 1844.)

It may perhaps be thought not unworthy of curious notice
hereafter, that {\it after\/} the publication of this form of the
equation of the {\it sphere}, there should have been found in
England, and in 1846, a person with any mathematical character to
lose, who could profess publicly his inability to distinguish the
method of {\it quaternions\/} from that of {\it couples\/}; and
who could thus confound the system of the present writer with
those of Argand and of Fran\c{c}ais, of Mourey and of Warren.}
$$\rho^2 + b^2 = 0
   \qquad\hbox{equation~(100.), article~58;}$$
which may also be thus written, by the principles and notations
of the calculus of quaternions:
$${\rm T} \rho = b.
   \eqno (124.)$$
From the relations (122.) it follows that the two chords
${\sc l}_\prime {\sc m}_\prime'$ and
${\sc l}_\prime' {\sc m}_\prime$,
of this mean sphere, both pass through the point~${\sc h}$, of
which the vector~$\xi_\prime$ is assigned by the formula~(116.);
for the first equation~(122.) shows that the three vectors
$\lambda_\prime$,~$\mu_\prime'$,~$\xi_\prime$,
which are all drawn from one common point, namely the
centre~${\sc a}$ of the ellipsoid, all terminate on one straight
line; since otherwise the quotient of their differences,
$\mu_\prime' - \xi_\prime$ and $\lambda_\prime - \xi_\prime$,
would be a {\it quaternion},\footnote\dag{A Quaternion,
{\it geometrically\/} considered, is the {\it product, or the
quotient, of any two directed lines in space}.}
of which the vector part would not be equal to zero: and in like
manner, the second equation~(122.) expresses that the three lines
$\lambda_\prime'$,~$\mu_\prime$,~$\xi_\prime$, all terminate on
another straight line.  The four-sided figure
${\sc l}_\prime \, {\sc m}_\prime \, {\sc l}_\prime' \, {\sc m}_\prime'$
is therefore a {\it plane quadrilateral, inscribed\/} (generally)
{\it in a small circle of the mean sphere}, and having the
point~${\sc h}$ for the intersection of its second and fourth
sides, ${\sc m}_\prime {\sc l}_\prime'$ and
${\sc m}_\prime' {\sc l}_\prime$, or of those two sides
prolonged.  And these two sides, having respectively the
directions of ${\sc h} {\sc m}_\prime$ and
${\sc h} {\sc l}_\prime$, or of the vector-differences
$\mu_\prime - \xi_\prime$ and $\lambda_\prime - \xi_\prime$, are
respectively parallel, by (118.), to the two fixed vectors,
$\iota$ and $\kappa$; or (by what was shown in former articles),
to the two cyclic normals, ${\sc a} {\sc c}'$ and
${\sc a} {\sc c}$, of the original ellipsoid.  The plane of the
quadrilateral inscribed in the mean sphere is therefore
constantly parallel to the {\it principal plane\/}
${\sc c} {\sc a} {\sc c}'$ of that ellipsoid, namely to the plane
of the greatest and least axes, which contains those two cyclic
normals.  The first and third sides,
${\sc l}_\prime {\sc m}_\prime$ and
${\sc l}_\prime' {\sc m}_\prime'$,
of the same inscribed quadrilateral, being in the directions of
$\mu_\prime - \lambda_\prime$ and
$\mu_\prime' - \lambda_\prime'$, are parallel, by (118.), (119.),
to two other constant vectors, namely $\iota - \kappa$ and
$\iota' - \kappa'$, or to the axes ${\sc a} {\sc b}$, ${\sc a}
{\sc b}'$, of the two cyclinders of revolution which can be
circumscribed about the same ellipsoid.  And the point of
intersection of this other pair of opposite sides of the same
inscribed quadrilateral is, by (123.), the extremity of the
vector~$\rho$, or the point~${\sc e}$ on the surface of the
original ellipsoid; while the point~${\sc h}$, which has been
already seen to be the intersection of the former pair of
opposite sides of the quadrilateral, since it has, by (116.), its
vector $\xi_\prime = - b^2 \nu$, is the {\it reciprocal point},
on the surface of that {\it other\/} and {\it reciprocal
ellipsoid}, which was considered in article~61; namely the point
which is, on that reciprocal ellipsoid, diametrically
{\it opposite\/} to the point which was named ${\sc f}$ in that
article, and had its vector $= b^2 \nu$.

\bigbreak

67.
Conversely it is easy to see, that the foregoing analysis by
quaternions conducts to the following mode of
{\it constructing},\footnote*{This construction, of two
reciprocal ellipsoids from one sphere, was communicated to the
Royal Irish Academy in June 1848; together with an extension of
it to a mode of generating two reciprocal cones of the second
degree from one rectangular cone of revolution; and also to a
construction of two reciprocal hyperboloids, whether of one
sheet, or of two sheets, from one equilateral hyperboloid of
revolution, of one or of two sheets.}
{\it or generating, geometrically}, and by a {\it graphic\/}
rather than by a {\it metric\/} process, {\it a system of two
reciprocal ellipsoids, derived from one fixed sphere\/}; and of
determining, also {\it graphically}, for each point on either
ellipsoid, the {\it reciprocal point\/} on the other.

Inscribe in the fixed sphere a plane quadrilateral
(${\sc l}_\prime {\sc m}_\prime {\sc l}_\prime' {\sc m}_\prime'$),
of which the four sides
(${\sc l}_\prime {\sc m}_\prime$,
${\sc m}_\prime {\sc l}_\prime'$,
${\sc l}_\prime' {\sc m}_\prime'$,
${\sc m}_\prime' {\sc l}_\prime$)
shall be respectively parallel to four fixed right lines
(${\sc a} {\sc b}$, ${\sc a} {\sc c}'$,
${\sc a} {\sc b}'$, ${\sc a} {\sc c}$),
diverging from the centre~(${\sc a}$) of the sphere; and prolong
(if necessary) the first and third sides of this inscribed
quadrilateral, till they meet in a point~${\sc e}$; and the
second and fourth sides of the same quadrilateral, till they
intersect in another point~${\sc h}$.  Then {\it these two
points, of intersection\/} ${\sc e}$ and ${\sc h}$, {\it thus
found from two pairs of opposite sides of this inscribed
quadrilateral, will be two reciprocal points on two reciprocal
ellipsoids\/}; which ellipsoids will have a common mean axis,
namely that diameter of the fixed sphere which is perpendicular
to the plane of the four fixed lines: and those lines,
${\sc a} {\sc b}$, ${\sc a} {\sc c}'$,
${\sc a} {\sc b}'$, ${\sc a} {\sc c}$,
will be related to the two ellipsoids which are thus the loci of
the two points ${\sc e}$ and ${\sc h}$, according to the laws
enunciated in article~61, in connexion with a different
construction of a system of two reciprocal ellipsoids (derived
there from one common {\it moving sphere\/}); which former
construction {\it also\/} was obtained by the aid of the calculus
of quaternions.  Thus the lines ${\sc a} {\sc c}$,
${\sc a} {\sc c}'$ will be the two cyclic normals of the
ellipsoid which is the locus of ${\sc e}$, but will be the axes
of circumscribed cylinders of revolution, for that reciprocal
ellipsoid which is the locus of ${\sc h}$; and conversely, the
lines ${\sc a} {\sc b}$, ${\sc a} {\sc b}'$ will be the axes of
the two cylinders of revolution circumscribed about the
ellipsoid~(${\sc e}$), but will be the cyclic normals, or the
perpendiculars to the cyclic planes, for the reciprocal
ellipsoid~(${\sc h}$).

\bigbreak

68.
The equation of the ellipsoid (see Philosophical Magazine for
October 1847, or Proceedings of the Royal Irish Academy for July
1846),
$${\rm T} (\iota \rho + \rho \kappa) = \kappa^2 - \iota^2,
   \qquad\hbox{eq.~(9.), art.~38,}$$
which has so often presented itself in these researches, may be
anew transformed as follows.  Writing it thus,
$${\rm T} {(\iota \rho + \rho \kappa) (\iota - \kappa)
         \over \kappa^2 - \iota^2}
   =  {\rm T} (\iota - \kappa),
   \eqno (125.)$$
which we are allowed to do, because the tensor of a product is
equal to the product of the tensors, we may observe that while
the denominator of the fraction in the first member is a pure
scalar, the numerator is a pure vector; for the identity,
$$\iota \rho + \rho \kappa
   =  {\rm S} \mathbin{.} (\iota + \kappa) \rho
    + {\rm V} \mathbin{.} (\iota - \kappa) \rho,
   \eqno (126.)$$
gives
$${\rm S} \mathbin{.} (\iota \rho + \rho \kappa) (\iota - \kappa)
   =  0:
   \eqno (127.)$$
the fraction itself is therefore a pure vector, and the
sign~${\rm T}$, of the operation of taking the tensor of a
quaternion, may be changed to the sign ${\rm T} {\rm V}$, of the
generally distinct but in this case equivalent operation, of
taking the tensor of the vector part.  But, under the
sign~${\rm V}$, we may reverse the order of any {\it odd\/}
number of vector factors (see article~20 in the Philosophical
Magazine for July 1846); and thus may change, in the numerator of
the fraction in (125.), the partial product
$\iota \rho (\iota - \kappa)$ to $(\iota - \kappa) \rho \iota$.
Again, it is always allowed to {\it divide\/} (though {\it not},
generally, in this calculus, to {\it multiply\/}) {\it both\/}
the numerator and denominator of a quaternion fraction,
{\it by\/} any {\it common\/} quaternion, or by any common
vector; that is, to multiply both numerator and denominator
{\it into the reciprocal\/} of such common quaternion or vector:
namely by writing the symbol of this new factor to the
{\it right\/} (but not generally to the left) of both the symbols
of numerator and denominator, above and below the fractional bar.
{\it Dividing\/} therefore thus above and below
{\it by\/}~$\iota$, or {\it multiplying into\/}~$\iota^{-1}$,
after that permitted transposition of factors which was just now
specified, and after the change of ${\rm T}$ to
${\rm T} {\rm V}$, we find that the equation~(125.) of the
ellipsoid assumes the following form:
$${\rm T} {\rm V}
      {(\iota - \kappa) \rho + \rho (\kappa - \kappa^2 \iota^{-1})
         \over (\iota - \kappa)   + (\kappa - \kappa^2 \iota^{-1})}
   =  {\rm T} (\iota - \kappa);
   \eqno (128.)$$
the new denominator first presenting itself under the form
$\kappa^2 \iota^{-1} - \iota$, but being changed for greater
symmetry to that written in (128.), which it is allowed to do,
because, under the sign~${\rm T}$, or under the
sign~${\rm T} {\rm V}$, we may multiply by negative unity.

\bigbreak

69.
In the last equation of the ellipsoid, since
$$\kappa - \kappa^2 \iota^{-1}
   =  \kappa (\iota - \kappa) \iota^{-1},$$
we have
$${\rm T} (\kappa - \kappa^2 \iota^{-1})
   =  {\rm T} \kappa \, {\rm T} (\iota - \kappa) \, {\rm T} \iota^{-1};
   \eqno (129.)$$
and under the characteristic~${\rm U}$, of the operation of
taking the versor of a quaternion, we may multiply by any
positive scalar, such as $- \kappa^2$ is, because $\kappa^2$ and
$\kappa^{-2}$ are negative\footnote*{By this, which is one of the
earliest and most fundamental principles of the whole quaternion
theory (see the author's letter to John T. Graves, Esq., of
October~17th, 1843, printed in the Supplementary Number of the
Philosophical Magazine for December 1844), namely by the
principle that {\it the square of\/} {\sc every vector} (or
directed straight line in tridimensional space) is to be regarded
as a {\sc negative number}, this theory is not merely
{\it distinguished from}, but sharply {\sc contrasted} {\it with,
every other system\/} of algebraic geometry of which the writer
has hitherto acquired any knowledge, or received any intimation.
In saying this, he hopes that he will not be supposed to desire to
depreciate the labours of any other past or present inquirer into
the properties of that important and precious Symbol in Geometry,
$\sqrt{-1}$.  And he gladly takes occasion to repeat the
expression of his sense of the assistance which he received, in
the progress of his own speculations, from the study of
Mr.~Warren's work, before he was able to examine any of those
earlier essays referred to in Dr.~Peacock's Report: however
{\it distinct}, and even {\it contrasted}, on several
{\it fundamental\/} points, may be (as was above observed) the
methods of the {\sc Calculus of Quaternions} from those of what
Professor De Morgan has happily named {\sc Double Algebra}.}
scalars; whereas to multiply by a negative scalar, under the same
sign~${\rm U}$, is equivalent to multiplying the versor itself by
$-1$: hence,
$${\rm U} (\kappa - \kappa^2 \iota^{-1})
   =  - {\rm U} (\kappa^2 \iota^{-1} - \kappa)
   =  - {\rm U} (\kappa^{-1} - \iota^{-1}).
   \eqno (130.)$$
If then we introduce two new fixed vectors, $\eta$ and $\theta$,
defined by the equations,
$$\eta = {\rm T} \iota \, {\rm U} (\iota - \kappa);\quad
  \theta = {\rm T} \kappa \, {\rm U} (\kappa^{-1} - \iota^{-1});
   \eqno (131.)$$
and if we remember that any quaternion is equal to the product of
its own tensor and versor (Phil.\ Mag.\ for July 1846); we
shall obtain the transformations,
$$\iota - \kappa
   =  \eta \, {\rm T} {\iota - \kappa \over \iota};\quad
  \kappa - \kappa^2 \iota^{-1}
   =  - \theta \, {\rm T} {\iota - \kappa \over \iota};
   \eqno (132.)$$
which will change the equation of the ellipsoid (128.) to the
following:
$${\rm T} {\rm V} {\eta \rho - \rho \theta \over \eta - \theta}
   =  {\rm T} (\iota - \kappa).
   \eqno (133.)$$

\bigbreak

70.
To complete the elimination of the two old fixed vectors,
$\iota$,~$\kappa$, and the introduction, in their stead, of the
two new fixed vectors, $\eta$,~$\theta$, we may observe that the
two equations (132.) give, by addition,
$$\iota- \kappa^2 \iota^{-1}
   =  (\eta - \theta) {\rm T} {\iota - \kappa \over \iota};
   \eqno (134.)$$
taking then the tensors of both members, dividing by
$\displaystyle {\rm T} {\iota - \kappa \over \iota}$,
and attending to the expression~(81.) in article~56,
(Phil.\ Mag.\ for May 1848,) for the mean semiaxis~$b$ of the
ellipsoid, we find this new expression for that semiaxis:
$${\rm T} (\eta - \theta)
   =  {\kappa^2 - \iota^2 \over {\rm T} (\iota - \kappa)}
   =  b.
   \eqno (135.)$$
But also, by (131.), or by (132.),
$${\rm T} \eta = {\rm T} \iota;\quad
  {\rm T} \theta = {\rm T} \kappa;
   \eqno (136.)$$
and therefore,
$$\theta^2 - \eta^2 = \kappa^2 - \iota^2.
   \eqno (137.)$$
Hence, by (135.), we obtain the expression,
$${\rm T} (\iota - \kappa)
   =  {\theta^2 - \eta^2 \over {\rm T} (\eta - \theta)};
   \eqno (138.)$$
which may be substituted for the second member of the equation
(133.), so as to complete the required elimination of $\iota$ and
$\kappa$.  And if we then multiply on both sides by
${\rm T} (\eta - \theta)$, we obtain this new form\footnote*{This
form was communicated to the Royal Irish Academy, at the stated
meeting of that body on March~16th, 1849, in a note addressed by
the present writer to the Rev.\ Charles Graves.
[The Proceedings of the Royal Irish Academy show that this
communication was in fact made at the meeting on
April~9th, 1849.---{\sc ed}.]
It was remarked, in that note, that the {\it directions\/} of the
two fixed vectors, $\eta$,~$\theta$, are those of the two
{\it asymptotes\/} to the focal hyperbola; while their
{\it lengths\/} are such that the two extreme {\it semiaxes\/} of
the ellipsoid may be expressed as follows:
$$a = {\rm T} \eta + {\rm T} \theta;\quad
  c = {\rm T} \eta - {\rm T} \theta;$$
the {\it mean\/} semiaxis being, at the same time, expressed (as
in the text of the present paper) by the formula
$$b = {\rm T} (\eta - \theta).$$
It was observed, further, that $\eta - \theta$ has the direction
of {\it one cyclic normal\/} of the ellipsoid, and that
$\eta^{-1} - \theta^{-1}$ has the direction of the {\it other\/}
cyclic normal; that $\eta + \theta$ is the vector of {\it one
umbilic}, and that $\eta^{-1} + \theta^{-1}$ has the direction
of {\it another\/} umbilicar vector, or umbilicar semidiameter of
the ellipsoid; that the {\it focal ellipse\/} is represented
by the system of the two equations
$${\rm S} \mathbin{.} \rho \, {\rm U} \eta
   =  {\rm S} \mathbin{.} \rho \, {\rm U} \theta,$$
and
$${\rm T} {\rm V} \mathbin{.} \rho \, {\rm U} \eta
   =  2 {\rm S} \sqrt{ \eta \theta },$$
of which the first represents its {\it plane}, while
the second, which (it was remarked) might also be thus written,
$${\rm T} {\rm V} \mathbin{.} \rho \, {\rm U} \theta
   =  2 {\rm S} \sqrt{ \eta \theta },$$
represents a {\it cylinder of revolution\/} (or, under the latter
form, a {\it second\/} cylinder of the same kind), whereon the
focal ellipse is situated; and that the {\it focal hyperbola\/}
is adequately expressed or represented by the {\it single\/}
equation,
$${\rm V} \mathbin{.} \eta \rho \mathbin{.} \rho \theta
   =  ({\rm V} \mathbin{.} \eta \theta)^2.$$
To which it may be added, that by changing the two fixed vectors
$\eta$ and $\theta$ to others of the forms $t^{-1} \eta$ and
$t \theta$, we pass to a {\it confocal\/} surface.}
of the equation of the ellipsoid:
$${\rm T} {\rm V}
      {\eta \rho - \rho \theta \over {\rm U} (\eta - \theta)}
   =  \theta^2 - \eta^2;
   \eqno (139.)$$
which will be found to include several interesting geometrical
significations.

\bigbreak

71.
Before entering on any discussion of this new form of the
equation of the ellipsoid, namely the form
$${\rm T} {\rm V}
      {\eta \rho - \rho \theta \over {\rm U} (\eta - \theta)}
   =  \theta^2 - \eta^2,
   \qquad\hbox{eq.~(139.), art.~70,}$$
it may be useful to point out another manner of arriving at the
same equation of the ellipsoid, by a different process of
calculation, from that construction or generation of the surface,
as the locus of the circle which is the mutual intersection of a
pair of equal spheres, sliding within two fixed cylinders of
revolution whose axes intersect each other; while the right line,
connecting the centres of the two sliding spheres, moves parallel
to itself, or remains constantly parallel to a fixed right line
in the plane of the fixed axes of the cylinders: which mode of
generating the ellipsoid was published in the Philosophical
Magazine for July 1848 (having also been communicated to the
Royal Irish Academy in the preceding May), as a deduction from
the Calculus of Quaternions.  And whereas the fixed right line,
through the centre of the ellipsoid, to which the line connecting
the centres of the two sliding spheres is parallel, may have
either of two positions, since it may coincide with either of the
two cyclic normals, we shall here suppose it to have the
direction of the cyclic normal~$\iota$, or shall consider the
second pair of sliding spheres mentioned in article~64, of which
the quaternion equations are, by article~62 (Phil.\ Mag.\ for
July 1848),
$${\rm T} (\rho - \mu) = {\rm T} (\rho - \lambda') = b.
   \qquad\hbox{(114.).}$$

\bigbreak

72.
Here (see Phil.\ Mag.\ for May 1848), we have for $\mu$ the
value,
$$\mu = h' (\kappa - \iota),
   \qquad\hbox{eq.~(91.), art.~57;}$$
and
$$\lambda' ( \kappa' - \iota' ) = \kappa' \rho + \rho \kappa',
   \qquad\hbox{eq.~(110.), art.~60;}$$
also
$$\iota \kappa' = \iota' \kappa
   =  {\rm T} \mathbin{.} \iota \kappa,
   \qquad\hbox{eq.~(107.), same article;}$$
whence we derive for $\lambda'$ the expression,
$$\lambda'
   =  {\iota^{-1} \rho + \rho \iota^{-1}
         \over \iota^{-1} - \kappa^{-1}}
   =  {\iota \rho + \rho \iota
         \over \iota - \iota^2 \kappa^{-1}}.
   \eqno (140.)$$
But
$$(\iota - \iota^2 \kappa^{-1})^{-1}
   =  \{ \iota (\kappa - \iota) \kappa^{-1} \}^{-1}
   =  \kappa (\kappa - \iota)^{-1} \iota^{-1};
   \eqno (141.)$$
and by (104.),
$$\iota \rho + \rho \iota = - h' (\kappa - \iota)^2;
   \eqno (142.)$$
therefore
$$\lambda'
   =  - h' \kappa (\kappa - \iota) \iota^{-1}
   =  h' (\kappa - \kappa^2 \iota^{-1}).
   \eqno (143.)$$
If then we make, for abridgment,
$$g = - h' \, {\rm T} {\iota - \kappa \over \iota},
   \eqno (144.)$$
and employ the two new fixed vectors $\eta$ and $\theta$, defined
by the equations (see Phil.\ Mag.\ for May 1849),
$$\eta = {\rm T} \iota \, {\rm U} (\iota - \kappa),\quad
  \theta = {\rm T} \kappa \, {\rm U} (\kappa^{-1} - \iota^{-1}),
   \qquad\hbox{(131.)}$$
which have been found to give
$$\iota - \kappa
   =  \eta \, {\rm T} {\iota - \kappa \over \iota},\quad
  \kappa - \kappa^2 \iota^{-1}
   =  - \theta \, {\rm T} {\iota - \kappa \over \iota},
   \qquad\hbox{(132.)}$$
we shall have the values,
$$\mu = g \eta;\quad
  \lambda' = g \theta;
   \eqno (145.)$$
and the lately cited equations (114.) of the two sliding spheres
will become,
$${\rm T} (\rho - g \eta) = b;\quad
  {\rm T} (\rho - g \theta) = b;
   \eqno (146.)$$
between which it remains to eliminate the scalar coefficient~$g$,
in order to find the equation of the ellipsoid, regarded as the
locus of the circle in which the two spheres intersect each
other.

\bigbreak

73.
Squaring the equations (146.), we find (by the general rules of
this Calculus) for the two sliding spheres the two following more
developed equations:
$$\left. \eqalign{
0  &= b^2 + \rho^2 - 2 g {\rm S} \mathbin{.} \eta \rho
         + g^2 \eta^2;\cr
0  &= b^2 + \rho^2 - 2 g {\rm S} \mathbin{.} \theta \rho
         + g^2 \theta^2.\cr}
   \right\}
   \eqno (147.)$$
Taking then the difference, and dividing by $g$, we find the
equation
$$g ( \theta^2 - \eta^2)
   =  2 {\rm S} \mathbin{.} (\theta - \eta) \rho;
   \eqno (148.)$$
which, relatively to $\rho$, is linear, and may be considered as
the equation of the plane of the varying circle of intersection
of the two sliding spheres; any one position of that plane being
distinguished from any other by the value of the coefficient~$g$.
Eliminating therefore that coefficient~$g$, by substituting in
(146.) its value as given by (148.), we find that the equation of
the ellipsoid, regarded as the locus of the varying circle, may
be presented under either of the two following new forms:
$${\rm T}
         \left(
            \rho
          - {2 \eta \, {\rm S} \mathbin{.} (\theta - \eta) \rho
               \over \theta^2 - \eta^2}
         \right)
   =  b;
   \eqno (149.)$$
$${\rm T}
         \left(
            \rho
          - {2 \theta \, {\rm S} \mathbin{.} (\eta - \theta) \rho
               \over \eta^2 - \theta^2}
         \right)
   =  b;
   \eqno (150.)$$
respecting which two forms it deserves to be noticed, that either
may be obtained from the other, by interchanging $\eta$ and
$\theta$.  And we may verify that these two last equations of the
ellipsoid are consistent with each other, by observing that the
seimisum of the two vectors under the sign~${\rm T}$ is
perpendicular to their semidifference (as it ought to be, in
order to allow of those two vectors themselves having any common
length, such as $b$); or that the condition of rectangularity,
$$\rho - {(\theta + \eta) \, {\rm S} \mathbin{.} (\theta - \eta) \rho
               \over \theta^2 - \eta^2}
  \perp \theta - \eta,
   \eqno (151.)$$
is satisfied: which may be proved by showing (see
Phil.\ Mag.\ for July 1846) that the scalar of the product of
these two last vectors vanishes, as in fact it does, since the
identity
$$(\theta - \eta) (\theta + \eta)
   =  \theta^2 + \theta \eta - \eta \theta - \eta^2,$$
resolves itself into the two following formul{\ae}:
$$\left. \eqalign{
{\rm S} \mathbin{.} (\theta - \eta) (\theta + \eta)
   &= \theta^2 - \eta^2;\cr
{\rm V} \mathbin{.} (\theta - \eta) (\theta + \eta)
   &= \theta \eta - \eta \theta;\cr}
   \right\}
   \eqno (152.)$$
of which the first is sufficient for our purpose.  We may also
verify the recent equations (149.), (150.) of the ellipsoid, by
observing that they concur in giving the mean semiaxis~$b$ as the
length~${\rm T} \rho$ of the radius of that diametral and
circular section, which is made by the cyclic plane having for
equation
$${\rm S} \mathbin{.} (\theta - \eta) \rho = 0;
   \eqno (153.)$$
this plane being found by the consideration that $\eta - \theta$
has the direction of the cyclic normal~$\iota$, or by making the
coefficient $g = 0$, in the formula (148.).

\bigbreak

74.
The equation (149.) of the ellipsoid may be successively
transformed as follows:
$$\eqalignno{
b (\theta^2 - \eta^2)
   &= {\rm T} \{ (\theta^2 - \eta^2) \rho
          - 2 \eta \, {\rm S} \mathbin{.}
            (\theta - \eta) \rho \} \cr
   &= {\rm T} \{ (\theta^2 - \eta^2) \rho
          - \eta (\theta - \eta) \rho
          - \eta \rho (\theta - \eta) \} \cr
   &= {\rm T} \{ \theta^2 \rho
          - \eta (\theta \rho + \rho \theta)
          + \eta \rho \eta \} \cr
   &= {\rm T} {\rm V} \{ (\theta - \eta) \theta \rho
          - \eta \rho (\theta - \eta) \} \cr
   &= {\rm T} {\rm V} \mathbin{.}
            (\rho \theta - \eta \rho) (\theta - \eta) \cr
   &= {\rm T} {\rm V} \mathbin{.}
            (\eta \rho - \rho \theta) (\eta - \theta);
   &(154.)\cr}$$
and by a similar series of transformations, performed on the
equation (150.), we find also (remembering that
$\theta^2 - \eta^2$, being equal to $\kappa^2 - \iota^2$, is
positive),
$$b (\theta^2 - \eta^2)
   =  {\rm T} {\rm V} \mathbin{.}
            (\rho \eta - \theta \rho) (\eta - \theta).
   \eqno (155.)$$
The same result (155.) may also be obtained by interchanging
$\eta$ and $\theta$ in either of the two last transformed
expressions (154.), for the positive product
$b (\theta^2 - \eta^2)$; and we may otherwise establish the
agreement of these recent results, by observing that, in general,
if ${\rm Q}$ and ${\rm Q}'$ be any two {\it conjugate
quaternions\/} (see Phil.\ Mag.\ for July 1846), such as here
$\eta \rho - \rho \theta$ and $\rho \eta - \theta \rho$, and if
$\alpha$ be any vector, then
$${\rm T} {\rm V} \mathbin{.} {\rm Q} \alpha
   =  {\rm T} {\rm V} \mathbin{.} {\rm Q}' \alpha;
   \eqno (156.)$$
for
$$\left. \eqalign{
{\rm V} \mathbin{.} {\rm Q} \alpha
   &= \alpha \, {\rm S} {\rm Q}
       - {\rm V} \mathbin{.} \alpha \, {\rm V} {\rm Q},\cr
{\rm V} \mathbin{.} {\rm Q}' \alpha
   &= \alpha \, {\rm S} {\rm Q}
       + {\rm V} \mathbin{.} \alpha \, {\rm V} {\rm Q};\cr}
   \right\}
   \eqno (157.)$$
and because
$$0 = {\rm S} \mathbin{.} \alpha {\rm V} \mathbin{.}
         \alpha \, {\rm V} {\rm Q},
   \eqno (158.)$$
the common value of the two members of the formula (156.) is
$${\rm T} {\rm V} \mathbin{.} {\rm Q} \alpha
   =  \surd \{ ( {\rm T} {\rm V} \mathbin{.}
            \alpha \, {\rm V} {\rm Q} )^2
          + ( {\rm T} \alpha \mathbin{.}
            {\rm S} {\rm Q} )^2 \}.
   \eqno (159.)$$
If then we substitute for $b$ its value,
$$b = {\rm T} (\eta - \theta),
   \qquad\hbox{eq.~(135.), art.~70,}$$
and divide both sides by this value of $b$, we see, from (154.),
(155.), that the equation of the ellipsoid may be put under
either of these two other forms:
$${\rm T} {\rm V} \mathbin{.}
         (\eta \rho - \rho \theta) \, {\rm U} (\eta - \theta)
   =  \theta^2 - \eta^2,
   \eqno (160.)$$
$${\rm T} {\rm V} \mathbin{.}
         (\rho \eta - \theta \rho) \, {\rm U} (\eta - \theta)
   =  \theta^2 - \eta^2.
   \eqno (161.)$$
But the versor of {\it every\/} vector is, in this calculus, a
square root of negative unity; we have therefore in particular,
$$( {\rm U} (\eta - \theta) )^2 = -1;
   \eqno (162.)$$
and under the sign~${\rm T} {\rm V}$, as under the
sign~${\rm T}$, it is allowed to divide by $-1$, without
affecting the value of the tensor: it is therefore permitted to
write the equation (160.) under the form
$${\rm T} {\rm V} \mathbin{.}
         {\eta \rho - \rho \theta \over {\rm U} (\eta - \theta)}
   =  \theta^2 - \eta^2,
   \qquad\hbox{(139.)}$$
which form is thus demonstrated anew.

\bigbreak

75.
A few connected transformations may conveniently be noticed here.
Since, for any quaternion~${\rm Q}$,
$$({\rm T} {\rm V} {\rm Q})^2
   =  - ({\rm V} {\rm Q})^2
   =  ({\rm T} {\rm Q})^2 - ({\rm S} {\rm Q})^2,
   \eqno (163.)$$
while the tensor of a product is the product of the tensors, and
the tensor of a versor is unity; and since
$${\rm S} \mathbin{.} (\rho \eta - \theta \rho) (\eta - \theta)
   =  {\rm S} (\rho \eta^2 - \rho \eta \theta
          - \theta \rho \eta + \theta \rho \theta)
   =  - 2 {\rm S} \mathbin{.} \eta \theta \rho,
   \eqno (164.)$$
because
$$0 = {\rm S} \mathbin{.} \rho \eta^2
   =  {\rm S} \mathbin{.} \theta \rho \theta,
   \quad\hbox{and}\quad
  {\rm S} \mathbin{.} \rho \eta \theta
   =  {\rm S} \mathbin{.} \theta \rho \eta
   =  {\rm S} \mathbin{.} \eta \theta \rho;
   \eqno (165.)$$
we have therefore, {\it generally},
$$\left. \eqalign{
{\rm T} \mathbin{.} (\rho \eta - \theta \rho)
      \, {\rm U} (\eta - \theta)
   &= {\rm T} (\rho \eta - \theta \rho);\cr
{\rm S} \mathbin{.} (\rho \eta - \theta \rho)
      \, {\rm U} (\eta - \theta)
   &= -2 {\rm T} (\eta - \theta)^{-1}
      \, {\rm S} \mathbin{.} \eta \theta \rho;\cr}
   \right\}
   \eqno (166.)$$
and there results the equation,
$${\rm T} {\rm V} \mathbin{.}
      (\rho \eta - \theta \rho)
      \, {\rm U} (\eta - \theta)
   =  \surd \{ {\rm T} (\rho \eta - \theta \rho)^2
          - 4 {\rm T} (\eta - \theta)^{-2}
            ({\rm S} \mathbin{.} \eta \theta \rho)^2 \},
   \eqno (167.)$$
as a general formula of transformation, valid for {\it any three
vectors}, $\eta$,~$\theta$,~$\rho$.  We may also, by the general
rules of the present calculus, write the last result as follows,
$${\rm T} {\rm V} \mathbin{.}
      (\rho \eta - \theta \rho)
      \, {\rm U} (\eta - \theta)
   =  \surd \{
            (\rho \eta - \theta \rho) (\eta \rho - \rho \theta)
          + (\eta - \theta)^{-2}
            (\eta \theta \rho - \rho \theta \eta)^2 \};
   \eqno (168.)$$
the signs ${\rm S}$ and ${\rm T}$ thus disappearing from the
expression of the radical.  For the ellipsoid, this radical,
being thus equal to the left-hand member of the formula (167.),
or to that of (168.), must, by (161.), receive the constant value
$\theta^2 - \eta^2$; so that, by squaring on both sides, we find
as a new form of the equation (161.) of the ellipsoid, the
following:
$$(\theta^2 - \eta^2)^2
   =  (\rho \eta - \theta \rho) (\eta \rho - \rho \theta)
       + (\eta - \theta)^{-2}
         (\eta \theta \rho - \rho \theta \eta)^2.
   \eqno (169.)$$
Or, by a partial reintroduction of the signs ${\rm S}$ and
${\rm T}$, we find this somewhat shorter form:
$${\rm T} (\rho \eta - \theta \rho)^2
       + 4 (\eta - \theta)^{-2}
         ({\rm S} \mathbin{.} \eta \theta \rho)^2
   =  (\theta^2 - \eta^2)^2.
   \eqno (170.)$$
And instead of the square of the tensor of the quaternion
$\rho \eta - \theta \rho$, we may write any one of several
general expressions for that square, which will easily suggest
themselves to those who have studied the transformations (already
printed in this Magazine), of the earlier and in some
respects simpler equation of the ellipsoid, proposed by the
present writer, namely the equation
$${\rm T} (\iota \rho + \rho \kappa)
   =  \kappa^2 - \iota^2,
   \qquad\hbox{eq.~(9.), art.~38.}$$
For instance, we may employ any of the following general
equalities, which all flow with little difficulty from the
principles of the present calculus:
$$\eqalignno{
{\rm T} (\rho \eta - \theta \rho)^2
   &= {\rm T} (\eta \rho - \rho \theta)^2 \cr
   &= (\rho \eta - \theta \rho) (\eta \rho - \rho \theta)
    = (\eta \rho - \rho \theta) (\rho \eta - \theta \rho) \cr
   &= (\eta^2 + \theta^2) \rho^2
       - \rho \eta \rho \theta
       - \theta \rho \eta \rho \cr
   &= (\eta^2 + \theta^2) \rho^2
       - \eta \rho \theta \rho
       - \rho \theta \rho \eta \cr
   &= (\eta + \theta)^2 \rho^2
       - (\eta \rho + \rho \eta) (\theta \rho + \rho \theta) \cr
   &= (\eta^2 + \theta^2) \rho^2
       - 2 {\rm S} \mathbin{.} \eta \rho \theta \rho \cr
   &= (\eta + \theta)^2 \rho^2
       - 4  {\rm S} \mathbin{.} \eta \rho \mathbin{.}
            {\rm S} \mathbin{.} \theta \rho \cr
   &= (\eta - \theta)^2 \rho^2
       + 4  {\rm S} ( {\rm V} \mathbin{.} \eta \rho \mathbin{.}
            {\rm V} \mathbin{.} \rho \theta );
   &(171.)\cr}$$
and which all hold good, independently of any relation between
the three vectors $\eta$,~$\theta$,~$\rho$.

\bigbreak

76.
As bearing on the last of these transformations it seems not
useless to remark, that a general formula published in the
Philosophical Magazine of August 1846, for any three
vectors $\alpha$,~$\alpha'$,~$\alpha''$, namely the formula
$$\alpha \, {\rm S} \mathbin{.} \alpha' \alpha''
       - \alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha
   =  {\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
            \mathbin{.} \alpha'' ),
   \qquad\hbox{eq.~(12.) of art.~22,}$$
which is found to be extensively useful, and indeed of constant
recurrence in the applications of the calculus of quaternions,
may be proved symbolically in the following way, which is shorter
than that employed in the 23rd article:
$$\eqalignno{
{\rm V} ( {\rm V} \mathbin{.} \alpha \alpha'
            \mathbin{.} \alpha'' )
   &= {\textstyle {1 \over 2}}
         (  {\rm V} \mathbin{.} \alpha \alpha'
            \mathbin{.} \alpha''
          - \alpha'' \, {\rm V} \mathbin{.} \alpha \alpha' )
    = {\textstyle {1 \over 2}}
         (  \alpha \alpha' \mathbin{.} \alpha''
          - \alpha'' \mathbin{.} \alpha \alpha' ) \cr
   &= {\textstyle {1 \over 2}} \alpha
         ( \alpha' \alpha'' + \alpha'' \alpha')
       - {\textstyle {1 \over 2}}
         ( \alpha \alpha'' + \alpha'' \alpha) \alpha'
    = \alpha \, {\rm S} \mathbin{.} \alpha' \alpha''
       - \alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha.
   &(172.)\cr}$$
The formula may be also written thus:
$${\rm V} \mathbin{.} \alpha''
         \, {\rm V} \mathbin{.} \alpha' \alpha
   =  \alpha \, {\rm S} \mathbin{.} \alpha' \alpha''
    - \alpha' \, {\rm S} \mathbin{.} \alpha \alpha'';
   \eqno (173.)$$
whence easily flows this other general and useful transformation,
for the vector part of the product of any three vectors,
$\alpha$,~$\alpha'$,~$\alpha''$:
$${\rm V} \mathbin{.} \alpha'' \alpha' \alpha
   =  \alpha \, {\rm S} \mathbin{.} \alpha' \alpha''
    - \alpha' \, {\rm S} \mathbin{.} \alpha'' \alpha
    + \alpha'' \, {\rm S} \mathbin{.} \alpha \alpha'.
   \eqno (174.)$$
Operating on this by ${\rm S} \mathbin{.} \alpha'''$, we find,
for the scalar part of the product of any {\it four vectors}, the
expression:
$${\rm S} \mathbin{.} \alpha''' \alpha'' \alpha' \alpha
   =     {\rm S} \mathbin{.} \alpha''' \alpha
         \mathbin{.} {\rm S} \mathbin{.} \alpha' \alpha''
       - {\rm S} \mathbin{.} \alpha''' \alpha'
         \mathbin{.} {\rm S} \mathbin{.} \alpha'' \alpha
       + {\rm S} \mathbin{.} \alpha''' \alpha''
         \mathbin{.} {\rm S} \mathbin{.} \alpha \alpha'.
   \eqno (175.)$$
But a quaternion, such as is $\alpha' \alpha$, or
$\alpha''' \alpha''$, is always equal to the sum of its own
scalar and vector parts; and the product of a scalar and a vector
is a vector, while the scalar of a vector is zero; therefore
$$\alpha' \alpha
   =  {\rm S} \mathbin{.} \alpha' \alpha
    + {\rm V} \mathbin{.} \alpha' \alpha,\quad
  \alpha''' \alpha''
   =  {\rm S} \mathbin{.} \alpha''' \alpha''
    + {\rm V} \mathbin{.} \alpha''' \alpha'',
   \eqno (176.)$$
and
$${\rm S} \mathbin{.} \alpha''' \alpha'' \alpha' \alpha
   =     {\rm S} \mathbin{.} \alpha''' \alpha''
         \mathbin{.} {\rm S} \mathbin{.} \alpha' \alpha
      +  {\rm S} ( {\rm V} \mathbin{.} \alpha''' \alpha''
         \mathbin{.} {\rm V} \mathbin{.} \alpha' \alpha ).
   \eqno (177.)$$
Comparing then (175.) and (177.), and observing that
$${\rm S} \mathbin{.} \alpha \alpha'
   =  + {\rm S} \mathbin{.} \alpha' \alpha,\quad
  {\rm V} \mathbin{.} \alpha \alpha'
   =  - {\rm V} \mathbin{.} \alpha' \alpha,
   \eqno (178.)$$
we obtain the following general expression for the scalar part of
the product of the vectors of any two binary products of vectors:
$${\rm S} ( {\rm V} \mathbin{.} \alpha''' \alpha'' \mathbin{.}
         {\rm V} \mathbin{.} \alpha' \alpha )
   =  {\rm S} \mathbin{.} \alpha''' \alpha \mathbin{.}
      {\rm S} \mathbin{.} \alpha' \alpha''
    - {\rm S} \mathbin{.} \alpha''' \alpha' \mathbin{.}
      {\rm S} \mathbin{.} \alpha'' \alpha;
   \eqno (179.)$$
while the vector part of the same product of vectors is easily
found, by similar processes, to admit of being expressed in
either of the two following ways (compare equation~(3.) of
article~24):
$$\eqalignno{
{\rm V} ( {\rm V} \mathbin{.} \alpha''' \alpha'' \mathbin{.}
         {\rm V} \mathbin{.} \alpha' \alpha )
   &= \alpha''' \, {\rm S} \mathbin{.}
         \alpha'' \alpha' \alpha
    - \alpha'' \, {\rm S} \mathbin{.}
         \alpha''' \alpha' \alpha \cr
   &= \alpha \, {\rm S} \mathbin{.}
         \alpha''' \alpha'' \alpha'
    -  \alpha' \, {\rm S} \mathbin{.}
         \alpha''' \alpha'' \alpha;
   &(180.)\cr}$$
of which the combination conducts to the following general
expression for any fourth vector~$\alpha'''$, or $\rho$, in terms
of any three given vectors $\alpha$,~$\alpha'$,~$\alpha''$, which
are not parallel to any one common plane (compare equation~(4.)
of article~26):
$$\rho \, {\rm S} \mathbin{.} \alpha'' \alpha' \alpha
   =  \alpha \, {\rm S} \mathbin{.} \alpha'' \alpha' \rho
    + \alpha' {\rm S} \mathbin{.} \alpha'' \rho \alpha
    + \alpha'' {\rm S} \mathbin{.} \rho \alpha' \alpha.
   \eqno (181.)$$
If we further suppose that
$$\alpha'' = {\rm V} \mathbin{.} \alpha' \alpha,
   \eqno (182.)$$
we shall have
$${\rm S} \mathbin{.} \alpha'' \alpha' \alpha
   =  ( {\rm V} \mathbin{.} \alpha' \alpha )^2
   =  \alpha''^2;
   \eqno (183.)$$
and after dividing by $\alpha''^2$, the recent equation (181.)
will become
$$\rho = \alpha \, {\rm S} {\alpha' \rho \over \alpha''}
       + \alpha' \, {\rm S} {\rho \alpha \over \alpha''}
       + {{\rm S} \mathbin{.} \alpha'' \rho \over \alpha''};
   \eqno (184.)$$
whereby an arbitrary vector~$\rho$ may be expressed, in terms of
any two given vectors $\alpha$,~$\alpha'$, which are not parallel
to any common line, and of a third vector~$\alpha''$,
perpendicular to both of them.  And if, on the other hand, we
change $\alpha$,~$\alpha'$,~$\alpha''$,~$\alpha'''$ to
$\theta$,~$\rho$,~$\rho$,~$\eta$, in the general formula~(179.),
we find that generally, for any three vectors
$\eta$,~$\theta$,~$\rho$, the following identity holds good:
$${\rm S} ( {\rm V} \mathbin{.} \eta \rho \mathbin{.}
         {\rm V} \mathbin{.} \rho \theta )
   =  \rho^2 \, {\rm S} \mathbin{.} \eta \theta
    - {\rm S} \mathbin{.} \eta \rho \mathbin{.}
         {\rm S} \mathbin{.} \rho \theta;
   \eqno (185.)$$
which serves to connect the two last of the expressions (171.),
and enables us to transform either into the other.

\bigbreak

77.
To show the geometrical meaning of the equation~(185.), let us
divide it on both sides by
${\rm T} \mathbin{.} \rho^2 \eta \theta$;
it then becomes, after transposing,
$$- {\rm S} {\rm U} \mathbin{.} \eta \theta
   =  {\rm S} {\rm U} \mathbin{.} \eta \rho \mathbin{.}
         {\rm S} {\rm U} \mathbin{.} \rho \theta
    + {\rm S} ( {\rm V} {\rm U} \mathbin{.} \eta \rho \mathbin{.}
         {\rm V} {\rm U} \mathbin{.} \rho \theta ).
   \eqno (186.)$$
Here, by the general principles of the geometrical interpretation
of the symbols employed in this calculus (see the remarks in the
Philosophical Magazine for July 1846), the symbol
${\rm S} {\rm U} \mathbin{.} \eta \theta$
is an expression for the cosine of the supplement of the angle
between the two arbitrary vectors~$\eta$ and $\theta$; and
therefore the symbol
$- {\rm S} {\rm U} \mathbin{.} \eta \theta$
is an expression for the cosine of that angle itself.  In like
manner,
$- {\rm S} {\rm U} \mathbin{.} \eta \rho$ and
$- {\rm S} {\rm U} \mathbin{.} \rho \theta$
are expressions for the cosines of the respective inclinations of
those two vectors $\eta$ and $\theta$ to a third arbitrary
vector~$\rho$; and at the same time
${\rm V} {\rm U} \mathbin{.} \eta \rho$ and
${\rm V} {\rm U} \mathbin{.} \rho \theta$
are vectors, of which the lengths represent the sines of the same
two inclinations last mentioned, while they are directed towards
the poles of the two positive rotations corresponding; namely the
rotations from $\eta$ to $\rho$, and from $\rho$ to $\theta$,
respectively.  The vectors
${\rm V} {\rm U} \mathbin{.} \eta \rho$ and
${\rm V} {\rm U} \mathbin{.} \rho \theta$
are therefore inclined to each other at an angle which is the
supplement of the dihedral or spherical angle, subtended at the
unit-vector~${\rm U} \rho$, or at its extremity on the
unit-sphere, by the two other unit-vectors ${\rm U} \eta$ and
${\rm U} \theta$, or by the arc between their extremities: so
that the scalar part of their product, in the formula~(186.),
represents the cosine of this spherical angle itself (and not of
its supplement), multiplied into the product of the sines of the
two sides or arcs upon the sphere, between which that angle is
included.  If then we denote the three sides of the spherical
triangle, formed by the extremities of the three unit-vectors
${\rm U} \eta$, ${\rm U} \theta$, ${\rm U} \rho$, by the symbols,
${\mathop{\eta \theta}\limits^\frown}$,
${\mathop{\eta \rho}\limits^\frown}$,
${\mathop{\rho \theta}\limits^\frown}$,
and the spherical angle opposite to the first of them by the
symbol $\widehat{\eta \rho \theta}$, the equation~(186.) will
take the form
$$\cos {\mathop{\eta \theta}\limits^\frown}
   =  \cos {\mathop{\eta \rho}\limits^\frown}
      \cos {\mathop{\rho \theta}\limits^\frown}
    + \sin {\mathop{\eta \rho}\limits^\frown}
      \sin {\mathop{\rho \theta}\limits^\frown}
      \cos \widehat{\eta \rho \theta};
   \eqno (187.)$$
which obviously coincides with the well-known and fundamental
formula of spherical trigon\-ometry, and is brought forward here
merely as a verification of the consistency of the results of
this calculus, and as an example of their geometrical
interpretability.

A more interesting example of the same kind is furnished by the
general formula~(179.) for {\it four\/} vectors, which, when
divided by the tensor of their product, becomes
$${\rm S} ( {\rm V} {\rm U} \mathbin{.}
         \alpha''' \alpha'' \mathbin{.}
         {\rm V} {\rm U} \mathbin{.}
         \alpha' \alpha )
   =  {\rm S} {\rm U} \mathbin{.}
         \alpha''' \alpha \mathbin{.}
         {\rm S} {\rm U} \mathbin{.}
         \alpha' \alpha''
    - {\rm S} {\rm U} \mathbin{.}
         \alpha''' \alpha' \mathbin{.}
         {\rm S} {\rm U} \mathbin{.}
         \alpha'' \alpha;
   \eqno (188.)$$
and signifies, when interpreted on the same principles, that
$$    \sin {\mathop{\alpha \alpha'}\limits^\frown} \mathbin{.}
      \sin {\mathop{\alpha'' \alpha'''}\limits^\frown} \mathbin{.}
      \cos ( {\mathop{\alpha \alpha'}\limits^\frown}
         \widehat{\hskip 1em}
         {\mathop{\alpha'' \alpha'''}\limits^\frown} )
   =  \cos {\mathop{\alpha \alpha''}\limits^\frown} \mathbin{.}
      \cos {\mathop{\alpha' \alpha'''}\limits^\frown}
    - \cos {\mathop{\alpha \alpha'''}\limits^\frown} \mathbin{.}
      \cos {\mathop{\alpha' \alpha''}\limits^\frown};
   \eqno (189.)$$
where the spherical angle between the two arcs from $\alpha$ to
$\alpha'$ and from $\alpha''$ to $\alpha'''$ may be replaced by
the interval between the poles of the two positive rotations
corresponding.  The same result may be otherwise stated as
follows: If ${\sc l}$,~${\sc l}'$,~${\sc l}''$,~${\sc l}'''$,
denote any four points upon the surface of an unit-sphere, and
$A$ the angle which the arcs ${\sc l} {\sc l}$,
${\sc l}'' {\sc l}'''$ form where they meet each other, (the
arcs which include this angle being measured in the directions of
the progressions from ${\sc l}$ to ${\sc l}'$, and from
${\sc l}''$ to ${\sc l}'''$ respectively,) then the following
equation will hold good:
$$    \cos {\sc l} {\sc l}'' \mathbin{.} \cos {\sc l}' {\sc l}'''
    - \cos {\sc l} {\sc l}''' \mathbin{.} \cos {\sc l}' {\sc l}''
   =  \sin {\sc l} {\sc l}' \mathbin{.} \sin {\sc l}'' {\sc l}'''
         \mathbin{.} \cos A.
   \eqno (190.)$$
Accordingly this last equation has been incidentally given, as an
auxiliary theorem or lemma, at the commencement of those profound
and beautiful researches, entitled {\it Disquisitiones Generales
circa Superficies Curvas}, which were published by Gauss at
G\"{o}ttingen in 1828.  That great mathematician and philosopher
was content to prove the last written equation by the usual
formul{\ae} of spherical and plane trigonometry; but, however simple
and elegant may be the demonstration thereby afforded, it appears
to the present writer that something is gained by our being able
to present the result (190.) or (189.), under the form (188.) or
(179.), as an identity in the quaternion calculus.  In general,
all the results of plane and spherical trigonometry take the form
of {\it identities\/} in this calculus; and their expressions,
when so obtained, are associated with a reference to
{\it vectors}, which is usually suggestive of {\it graphic\/} as
well as {\it metric\/} relations.

\bigbreak

78.
Since
$$\rho \eta - \theta \rho
   =  {\rm S} \mathbin{.} \rho (\eta - \theta)
    + {\rm V} \mathbin{.} \rho (\eta + \theta),
   \eqno (191.)$$
the quaternion $\rho \eta - \theta \rho$ gives a pure vector as a
product, or as a quotient, if it be multiplied or divided by the
vector $\eta + \theta$ (compare article~68); we may therefore
write
$$\rho \eta - \theta \rho = \lambda_1 (\eta + \theta),
   \eqno (192.)$$
$\lambda_1$ being a new vector-symbol, of which the value may be
thus expressed:
$$\lambda_1
   =  \rho - 2 (\eta + \theta)^{-1} {\rm S}
         \mathbin{.} \theta \rho.
   \eqno (193.)$$
The equation (192.) will then give,
$$\left. \eqalign{
{\rm T} (\rho \eta - \theta \rho)
   &= {\rm T} \lambda_1 \mathbin{.} {\rm T} (\eta + \theta);\cr
{\rm T} (\rho \eta - \theta \rho)^2
   &= \lambda_1^2 (\eta + \theta)^2.\cr}
   \right\}
   \eqno (194.)$$
We have also the identity,
$$(\theta^2 - \eta^2)^2
   =  (\eta - \theta)^2 (\eta + \theta)^2
    + (\eta \theta - \theta \eta)^2;
   \eqno (195.)$$
which may be shown to be such, by observing that
$$\eqalignno{
(\eta - \theta)^2 (\eta + \theta)^2
   &= (\eta^2 + \theta^2 - 2 {\rm S} \mathbin{.} \eta \theta)
      (\eta^2 + \theta^2 + 2 {\rm S} \mathbin{.} \eta \theta) \cr
   &= (\eta^2 + \theta^2)^2
       - 4 ( {\rm S} \mathbin{.} \eta \theta )^2
    = (\eta^2 - \theta^2)^2
       + 4 ( {\rm T} \mathbin{.} \eta \theta )^2
       - 4 ( {\rm S} \mathbin{.} \eta \theta )^2 \cr
   &= (\eta^2 - \theta^2)^2
       - 4 ( {\rm V} \mathbin{.} \eta \theta )^2
    = (\theta^2 - \eta^2)^2 - (\eta \theta - \theta \eta)^2;
   &(196.)\cr}$$
or by remarking that (see equations (152.), (163.)),
$$\left. \eqalign{
\eta^2 - \theta^2
   =  {\rm S} \mathbin{.} (\eta - \theta) (\eta + \theta),\quad
\eta \theta - \theta \eta
   =  {\rm V} \mathbin{.} (\eta - \theta) (\eta + \theta),\cr
\hbox{and}\quad
(\eta - \theta)^2 (\eta + \theta)^2
   =  ( {\rm T} \mathbin{.} (\eta - \theta) (\eta + \theta))^2;\cr}
   \right\}
   \eqno (197.)$$
or in several other ways.  Introducing then a new
vector~$\epsilon$, such that
$$\eta \theta - \theta \eta
   =  \epsilon \, {\rm T} (\eta + \theta),
   \quad\hbox{or,}\quad
  \epsilon = 2 {\rm V} \mathbin{.} \eta \theta \mathbin{.}
         {\rm T} (\eta + \theta)^{-1};
   \eqno (198.)$$
and that therefore
$$(\eta \theta - \theta \eta)^2
   =  - \epsilon^2 (\eta + \theta)^2,
   \eqno (199.)$$
and
$$2 {\rm S} \mathbin{.} \eta \theta \rho
   =  {\rm S} \mathbin{.} \epsilon \rho \mathbin{.}
      {\rm T} (\eta + \theta),\quad
  4 ( {\rm S} \mathbin{.} \eta \theta \rho )^2
   =  -  ( {\rm S} \mathbin{.} \epsilon \rho )^2
         (\eta + \theta)^2;
   \eqno (200.)$$
while, by (135.),
$${\rm T} (\eta - \theta) = b,\quad
  (\eta - \theta)^2 = - b^2;
   \eqno (201.)$$
we find that the equation~(170.) of the ellipsoid, after being
divided by $(\eta + \theta)^2$, assumes the following form:
$$\lambda_1^2  + b^{-2} ( {\rm S} \mathbin{.} \epsilon \rho )^2
       + b^2 + \epsilon^2
   =  0.
   \eqno (202.)$$
But also, by (193.), (198.),
$${\rm S} \mathbin{.} \epsilon \lambda_1
   = {\rm S} \mathbin{.} \epsilon \rho;
   \eqno (203.)$$
the equation (202.) may therefore be also written thus:
$$0 = (\lambda_1 - \epsilon)^2
       + (b + b^{-1} \, {\rm S} \mathbin{.} \epsilon \rho)^2;
   \eqno (204.)$$
and the scalar $b + b^{-1} {\rm S} \mathbin{.} \epsilon \rho$ is
positive, even at an extremity of the mean axis of the ellipsoid,
because, by (195.), (199.), (201.), we have
$$(\theta^2 - \eta^2)^2
   =  - (b^2 + \epsilon^2) (\eta + \theta)^2
   =  (b^2 - {\rm T} \epsilon^2) \, {\rm T} (\eta + \theta)^2,
   \eqno (205.)$$
and therefore
$${\rm T} \epsilon < b.
   \eqno (206.)$$

We have then this new form of the equation of the ellipsoid,
deduced by transposition and extraction of square roots
(according to the rules of the present calculus), from the form
(204.):
$${\rm T} (\lambda_1 - \epsilon)
   =  b + b^{-1} \, {\rm S} \mathbin{.} \epsilon \rho.
   \eqno (207.)$$
By a process exactly similar to the foregoing, we find also the
form
$${\rm T} (\lambda_1 + \epsilon)
   =  b - b^{-1} \, {\rm S} \mathbin{.} \epsilon \rho.
   \eqno (208.)$$
which differs from the equation last found, only by a change of
sign of the auxiliary and constant vector~$\epsilon$: and hence,
by addition of the two last equations, we find still another
form, namely,
$$    {\rm T} (\lambda_1 - \epsilon)
    + {\rm T} (\lambda_1 + \epsilon)
   =  2b;
   \eqno (209.)$$
or substituting for $\lambda_1$, $\epsilon$, and $b$ their values
in terms of $\eta$,~$\theta$, and $\rho$, and multiplying into
${\rm T} (\eta + \theta)$,
$$    {\rm T}
      \left(
         {\rho \eta - \theta \rho \over {\rm U} (\eta + \theta)}
       - 2 {\rm V} \mathbin{.} \eta \theta
      \right)
    + {\rm T}
      \left(
         {\rho \eta - \theta \rho \over {\rm U} (\eta + \theta)}
       + 2 {\rm V} \mathbin{.} \eta \theta
      \right)
   =  2 {\rm T} \mathbin{.} (\eta - \theta) (\eta + \theta).
   \eqno (210.)$$

\bigbreak

79.
The locus of the termination of the auxiliary and variable
vector~$\lambda_1$, which is {\it derived\/} from the
vector~$\rho$ of the original ellipsoid by the {\it linear\/}
formula (193.), is expressed or represented by the
equation~(209.); it is therefore evidently a certain {\it new\/}
ellipsoid, namely an {\it ellipsoid of revolution}, which has the
mean axis~$2b$ of the old or given ellipsoid for its major axis,
or for its axis of revolution, while the vectors of its two
{\it foci\/} are denoted by the symbols $+ \epsilon$ and
$- \epsilon$.  If $a$ denote the greatest, and $c$ the least
semiaxis, of the original ellipsoid, while $b$ still denotes its
mean semiaxis, then, by what has been shown in former articles,
we have the values,
$${\rm T} \eta = {\rm T} \iota
   =  {\textstyle {1 \over 2}} (a + c);\quad
  {\rm T} \theta = {\rm T} \kappa
   =  {\textstyle {1 \over 2}} (a - c);
   \eqno (211.)$$
and consequently (compare the note to art.~70),
$$a = {\rm T} \eta + {\rm T} \theta;\quad
  c = {\rm T} \eta - {\rm T} \theta;
   \eqno (212.)$$
therefore
$$ac = {\rm T} \eta^2 - {\rm T} \theta^2 = \theta^2 - \eta^2;
   \eqno (213.)$$
also
$$\eqalignno{
{\rm T} (\eta + \theta)^2 + b^2
   &=  - (\eta + \theta)^2 - (\eta - \theta)^2
    =  - 2 \eta^2 - 2 \theta^2 \cr
   &= 2 {\rm T} \eta^2 + 2 {\rm T} \theta^2
    =    ({\rm T} \eta + {\rm T} \theta)^2
       + ({\rm T} \eta - {\rm T} \theta)^2,
   &(214.)\cr}$$
and
$${\rm T} (\eta + \theta)^2 = a^2 - b^2 + c^2;
   \eqno (215.)$$
whence, by (205.),
$${\rm T} \epsilon^2
   =  b^2 - {a^2 c^2 \over a^2 - b^2 + c^2}
   =  {(a^2 - b^2) (b^2 - c^2) \over a^2 - b^2 + c^2}.
   \eqno (216.)$$

Such, then, is the expression for the square of the distance of
either focus of the new or derived ellipsoid of revolution, which
has $\lambda_1$ for its varying vector, from the common centre of
the new and old ellipsoids, which centre is also the common
origin of the vectors~$\lambda_1$ and $\rho$: while these two
foci of the new ellipsoid are situated upon the mean axis of the
old one.  There exist also other remarkable relations, between
the original ellipsoid with three unequal semiaxes $a$,~$b$,~$c$,
and the new ellipsoid of revolution, of which some will be
brought into view, by pursuing the quaternion analysis in a way
which we shall proceed to point out.

\bigbreak

80.
The geometrical construction already mentioned (in articles 64,
71, \&c.), of the original ellipsoid as the locus of the circle
in which two sliding spheres intersect, shows easily (see
art.~72) that the scalar coefficient~$g$, in the equations
(146.) of that pair of sliding spheres, becomes equal to the
number~2, at one of those limiting positions of the pair, for
which, after cutting, they {\it touch}, before they cease to meet
each other.  In fact, if we thus make
$$g = 2,
   \eqno (217.)$$
the values (145.) of the vectors of the centres will give, for
the interval between those two centres of the two sliding
spheres, the expression
$${\rm T} (\mu - \lambda')
   =  g \, {\rm T} (\eta - \theta)
   =  2b;
   \eqno (218.)$$
this interval will therefore be in this case equal to the
diameter of either sliding sphere, because it will be equal to
the mean axis of the ellipsoid: and the two spheres will touch
one another.  Had we assumed a value for $g$, less by a very
little than the number~$2$, the two spheres would have cut each
other in a very small circle, of which the circumference would
have been (by the construction) entirely contained upon the
surface of the ellipsoid; and the plane of this little circle
would have been parallel and very near to that other plane, which
was the common tangent plane of the two spheres, and also of the
ellipsoid, when $g$ received the value~$2$ itself.  It is clear,
then, that this value~$2$ of $g$ corresponds to an {\it umbilicar
point\/} on the ellipsoid; and that the equation
$${\rm S} \mathbin{.} (\theta - \eta) \rho
   =  \theta^2 - \eta^2,
   \eqno (219.)$$
which is obtained from the more general equation~(148.) of the
plane of a circle on the ellipsoid, by changing $g$ to $2$,
represents an {\it umbilicar tangent plane}, at which the normal
has the direction of the vector $\eta - \theta$.  Accordingly it
has been seen that this last vector has the direction of the
cyclic normal~$\iota$: in fact, the expressions (131.), for
$\eta$ and $\theta$ in terms of $\iota$ and $\kappa$, give
conversely these other expressions for the latter vectors in
terms of the former,
$$\iota  = {\rm T} \eta   \, {\rm U} (\eta - \theta);\quad
  \kappa = {\rm T} \theta \, {\rm U} (\theta^{-1} - \eta^{-1}):
   \eqno (220.)$$
whence (it may here be noted) follow the two parallelisms,
$${\rm U} \iota - {\rm U} \kappa
   =  {\rm U} (\eta - \theta) + {\rm U} (\eta^{-1} - \theta^{-1})
   \parallel  {\rm U} \eta + {\rm U} \theta;
   \eqno (221.)$$
$${\rm U} \iota + {\rm U} \kappa
   =  {\rm U} (\eta - \theta) - {\rm U} (\eta^{-1} - \theta^{-1})
   \parallel  {\rm U} \eta - {\rm U} \theta;
   \eqno (222.)$$
the members of (221.) having each the direction of the greatest
axis of the ellipsoid, and the members of (222.) having each the
direction of the least axis; as may easily be proved, for the
first members of these formul{\ae}, by the construction with the
{\it diacentric sphere}, which was communicated by the writer to
the Royal Irish Academy in 1846, and was published in the present
Magazine in the course of the following year.  The
equation~(219.) may be verified by observing that it gives, for
the length of the perpendicular let fall from the centre of the
ellipsoid on an umbilicar tangent plane, the expression
$$p = (\theta^2 - \eta^2) {\rm T} (\eta - \theta)^{-1}
   =  a c b^{-1};
   \eqno (223.)$$
agreeing with known results.  And the vector~$\omega$ of the
umbilicar point itself must be the semisum of the vectors of the
centres of the two equal and sliding spheres, in that limiting
position of the pair in which (as above) they touch each other;
this {\it umbilicar vector\/}~$\omega$ is therefore expressed as
follows:
$$\omega = \eta + \theta;
   \eqno (224.)$$
because this is the semisum of $\mu$ and $\lambda'$ in (145.), or
of $g \eta$ and $g \theta$ when $g = 2$.  (Compare the note to
article~70.)  As a verification, we may observe that this
expression (224.) gives, by (215.), the following known value for
the length of an umbilicar semidiameter of the ellipsoid,
$$u = {\rm T} \omega = {\rm T} (\eta + \theta)
   =  \surd (a^2 - b^2 + c^2).
   \eqno (225.)$$
By similar reasonings it may be shown that the expression
$$\omega'
   =  {\rm T} \eta \, {\rm U} \theta
       + {\rm T} \theta \, {\rm U} \eta,
   \eqno (226.)$$
which may also be thus written, (see same note to art.~70,)
$$\omega'
   =  - {\rm T} \mathbin{.} \eta \theta \mathbin{.}
            (\eta^{-1} + \theta^{-1}),
   \eqno (227.)$$
represents {\it another\/} umbilicar vector; in fact, we have, by
(224.) and (226.),
$${\rm T} \omega' = {\rm T} \omega,
   \eqno (228.)$$
and
$$\left. \eqalign{
\omega + \omega'
   &= ({\rm T} \eta + {\rm T} \theta)
      ({\rm U} \eta + {\rm U} \theta),\cr
\omega - \omega'
   &= ({\rm T} \eta - {\rm T} \theta)
      ({\rm U} \eta - {\rm U} \theta);\cr}
   \right\}
   \eqno (229.)$$
so that the vectors $\omega$~$\omega'$ are equally long, and the
angle between them is bisected by
${\rm U} \eta + {\rm U} \theta$,
or (see (221.)) by the axis major of the ellipsoid; while the
supplementary angle between $\omega$ and $- \omega'$ is bisected
by
${\rm U} \eta - {\rm U} \theta$,
or (as is shown by (222.)) by the axis minor.  It is evident that
$- \omega$ and $- \omega'$ are also umbilicar vectors; and it is
clear, from what has been shown in former articles, that the
vectors $\eta$ and $\theta$ have the directions of the axes of
the two cylinders of revolution, which can be circumscribed about
that given or original ellipsoid, to which all the remarks of the
present article relate.

\bigbreak

81.
These remarks being premised, let us now resume the consideration
of the variable vector~$\lambda_1$, of art.~78, which has been
seen to terminate on the surface of a certain derived ellipsoid
of revolution.  Writing, under a slightly altered form, the
expression (193.) for that vector~$\lambda_1$, and combining with
it three other analogous expressions, for three other vectors,
$\lambda_2$,~$\lambda_3$,~$\lambda_4$, as follows,
$$\lambda_1
   =  {\rho \eta - \theta \rho \over \eta + \theta};\quad
  \lambda_2
   =  {\rho \theta - \eta \rho \over \eta + \theta};\quad
  \lambda_3
   =  {\rho \theta^{-1} - \eta^{-1} \rho \over \eta^{-1} + \theta^{-1}};\quad
  \lambda_4
   =  {\rho \eta^{-1} - \theta^{-1} \rho \over \eta^{-1} + \theta^{-1}};
   \eqno (230.)$$
it is easy to prove that
$$    {\rm T} \lambda_1
   =  {\rm T} \lambda_2
   =  {\rm T} \lambda_3
   =  {\rm T} \lambda_4;
   \eqno (231.)$$
and that
$$    {\rm S} \mathbin{.} \eta \theta \lambda_1
   =  {\rm S} \mathbin{.} \eta \theta \lambda_2
   =  {\rm S} \mathbin{.} \eta \theta \lambda_3
   =  {\rm S} \mathbin{.} \eta \theta \lambda_4
   =  {\rm S} \mathbin{.} \eta \theta \rho;
   \eqno (232.)$$
whence it follows that the four vectors,
$\lambda_1$,~$\lambda_2$,~$\lambda_3$,~$\lambda_4$,
being supposed to be all drawn from the centre~${\sc a}$ of the
original ellipsoid, terminate in four points,
${\sc l}_1$,~${\sc l}_2$,~${\sc l}_3$,~${\sc l}_4$,
which are the corners of a quadrilateral inscribed in a circle of
the derived ellipsoid of revolution; the plane of this circle
being parallel to the plane of the greatest and least axes of the
original ellipsoid, and passing through the point~${\sc e}$ of
that ellipsoid, which is the termination of the vector~$\rho$.
We shall have also the equations,
$${\lambda_2 - \rho \over \lambda_1 - \rho}
   =  {{\rm S} \mathbin{.} \eta \rho
         \over {\rm S} \mathbin{.} \theta \rho}
   =  {\rm V}^{-1} 0;\quad
  {\lambda_3 - \rho \over \lambda_4 - \rho}
   =  {{\rm S} \mathbin{.} \eta^{-1} \rho
         \over {\rm S} \mathbin{.} \theta^{-1} \rho}
   =  {\rm V}^{-1} 0;
   \eqno (233.)$$
which show that the two opposite sides ${\sc l}_1 {\sc l}_2$,
${\sc l}_3 {\sc l}_4$, of this inscribed quadrilateral, being
prolonged, if necessary, intersect in the lately-mentioned
point~${\sc e}$ of the original ellipsoid.  And because the
expressions (230.) give also
$${\rm V} {\lambda_2 - \lambda_1 \over \eta + \theta} = 0,\quad
  {\rm V} {\lambda_4 - \lambda_3 \over \eta^{-1} + \theta^{-1}} = 0,
   \eqno (234.)$$
these opposite sides ${\sc l}_1 {\sc l}_2$,
${\sc l}_3 {\sc l}_4$, of the plane quadrilateral thus inscribed
in a circle of the derived ellipsoid of revolution, are parallel
respectively to the vectors $\eta + \theta$,
$\eta^{-1} + \theta^{-1}$, or to the two umbilicar vectors
$\omega$,~$\omega'$, of the original ellipsoid, with the semiaxes
$a \, b \, c$.  At the same time, the equations
$${\rm V} {\lambda_3 - \lambda_2 \over \eta} = 0,\quad
  {\rm V} {\lambda_1 - \lambda_4 \over \theta} = 0,
   \eqno (235.)$$
hold good, and show that the two other mutually opposite sides of
the same inscribed quadrilateral, namely the sides
${\sc l}_2 {\sc l}_3$, ${\sc l}_4 {\sc l}_1$, are respectively
parallel to the two vectors $\eta$,~$\theta$, or to the axes of
the two cylinders of revolution which can be circumscribed about
the same original ellipsoid.  Hence it is easy to infer the
following theorem, which the author supposes to be new:---{\it If
on the mean axis~$2b$ of a given ellipsoid, $abc$, as the major
axis, and with two foci, ${\sc f}_1$,~${\sc f}_2$, of which the
common distance from the centre~${\sc a}$ is\/}
$$\overline{{\sc a} {\sc f}_1}
   =  \overline{{\sc a} {\sc f}_2}
   =   e
   =  {\surd (a^2 - b^2) \surd (b^2 - c^2)
         \over \surd (a^2 - b^2 + c^2)},
   \eqno (236.)$$
{\it we construct an ellipsoid of revolution; and if, in any
circular section of this new ellipsoid, we inscribe a
quadrilateral,
${\sc l}_1 {\sc l}_2 {\sc l}_3 {\sc l}_4$,
of which the two opposite sides
${\sc l}_1 {\sc l}_2$,~${\sc l}_3 {\sc l}_4$,
are respectively parallel to the two umbilicar diameters of the
given ellipsoid; while the two other and mutually opposite sides,
${\sc l}_2 {\sc l}_3$,~${\sc l}_4 {\sc l}_1$,
of the same inscribed quadrilateral, are respectively parallel to
the axes of the two cylinders of revolution which can be
circumscribed about the same given ellipsoid; then the point of
intersection~${\sc e}$ of the first pair of opposite sides\/}
(namely of those parallel to the umbilicar diameters), {\it will
be a point upon that given ellipsoid}.  It seems to the present
writer that, in consequence of this remarkable relation between
these two ellipsoids, the two foci ${\sc f}_1$,~${\sc f}_2$ of
the above described ellipsoid of revolution, which have been seen
to be situated upon the mean axis of the original ellipsoid, of
which the three unequal semiaxes are denoted by $a$,~$b$,~$c$,
may not inconveniently be called the {\sc two medial foci} of
that original ellipsoid: but he gladly submits the question of
the propriety of such a designation, to the judgement of other
and better geometers.  Meanwhile it may be noticed that the two
ellipsoids intersect each other in a system of two ellipses, of
which the planes are perpendicular to the axes of the two
cylinders of revolution above mentioned; and that those four
common tangent planes of the two ellipsoids, which are parallel
to their common axis, that is to the mean axis of the original
ellipsoid~$a \, b \, c$, are parallel also to its umbilicar
diameters.

\bigbreak

82.
This seems to be a proper place for inserting some notices of
investigations and results, respecting the inscription of
rectilinear (but not generally plane) polygons, in spheres, and
other surfaces of the second degree.

Let $\rho$ and $\sigma$ be any two unit-vectors, or directed
radii of an unit-sphere; so that, according to a fundamental
principle of the present Calculus, we may write
$$\rho^2 = \sigma^2 = - 1.
   \eqno (237.)$$
We shall then have also,
$$0 = \sigma^2 - \rho^2
   =  \sigma (\sigma - \rho) + (\sigma - \rho) \rho,
   \eqno (238.)$$
and consequently
$$\sigma = - (\sigma - \rho) \rho (\sigma - \rho)^{-1}
   =  - \lambda \rho \lambda^{-1},
   \eqno (239.)$$
if $\lambda$ be the directed chord $\sigma - \rho$ itself, or any
portion or prolongation thereof, or any vector parallel thereto.
If then $\rho, \rho_1, \rho_2,\ldots \, \rho_n$, be any series or
succession of unit-vectors, while
$\lambda_1, \lambda_2,\ldots \, \lambda_n$
are any vectors respectively coincident with, or parallel to, the
successive and rectilinear chords of the unit-sphere, connecting
the successive points where the vectors
$\rho \, \ldots \, \rho_n$ terminate; and if we introduce the
quaternions,
$$q_1 = \lambda_1;\quad
  q_2 = \lambda_2 \lambda_1;\quad
  q_3 = \lambda_3 \lambda_2 \lambda_1;
  \quad\hbox{\&c.},
   \eqno (240.)$$
we shall have the expressions,
$$\rho_1 = - q_1 \rho q_1^{-1};\quad
  \rho_2 = + q_2 \rho q_2^{-1};\quad
  \rho_3 = - q_3 \rho q_3^{-1};
  \quad\hbox{\&c.}
   \eqno (241.)$$
Hence if we write the equation
$$\rho_n = \rho,
   \eqno (242.)$$
to express the conception of a {\it closed\/} polygon of $n$
sides, inscribed in the sphere, we shall have the general
formula,
$$\rho q_n = (-1)^n q_n \rho;
   \eqno (243.)$$
which is immediately seen to decompose itself into the two
following principal cases, according as the number~$n$ of the
sides is even or odd:
$$\rho q_{2m} = + q_{2m} \rho;
   \eqno (244.)$$
$$\rho q_{2m+1} = - q_{2m+1} \rho.
   \eqno (245.)$$
The equation (244.) admits also of being written thus, by the
general rules of quaternions,
$$0 = {\rm V} \mathbin{.} \rho \, {\rm V} q_{2m};
   \eqno (246.)$$
and the equation (245.) resolves itself, by the same general
rules, into the two equations following:
$$0 = {\rm S} q_{2m+1};\quad
  0 = {\rm S} \mathbin{.} q_{2m+1} \rho.
   \eqno (247.)$$
We shall now proceed to consider some of the consequences which
follow from the formul{\ae} thus obtained.

\bigbreak

83.
An immediate consequence of the equations (247.), or rather a
translation of those equations into words, is the following
quaternion theorem:---{\it If any rectilinear polygon, with any
odd number of sides, be inscribed in a sphere, the continued
product of those sides is a vector, tangential to the sphere at
the first corner of the polygon}.  It is understood that, in
forming this continued product of sides, their {\it directions\/}
and {\it order\/} are attended to: the first side being
multiplied as a vector by the second, so as to form a certain
quaternion product; and this product being afterwards multiplied,
in succession, by the third side, then by the fourth, the fifth,
\&c., so as to form a series of quaternions, of which the
{\it last\/} will (by the theorem) have its {\it scalar\/} part
equal to zero; while the {\it vector\/} part, or the product
itself, will be constructed by a right line with a certain
definite direction, which will (by the same theorem) be that of a
certain rectilinear tangent to the sphere, at the point or corner
where the first side of the inscribed polygon begins.  [The
{\it tensor\/} of the resulting vector, or the {\it length\/} of
the product line, will of course represent, at the same time, by
the general law of tensors, the product of the lengths of the
factor lines, with the usual reference to some assumed unit of
length.] And conversely, whenever it happens that an odd number
of successive right lines in space, being multiplied together
successively by the rules of the present Calculus, give a
{\it line\/} as their continued product, that is to say, when the
scalar of the quaternion obtained by this multiplication
vanishes, then those right lines may be inferred to have the
{\it directions\/} of the successive sides of a polygon inscribed
in a sphere.

\bigbreak

84.
Already, even as applied to the case of an inscribed gauche
{\it pentagon}, the theorem of the last article expresses a
{\it characteristic\/} property of the {\it sphere}, which may be
regarded as being of a {\it graphic\/} rather than of a
{\it metric\/} character; inasmuch as it concerns immediately
{\it directions\/} rather than {\it magnitudes}, although there
is no difficulty in deducing from it metric relations also: as
will at once appear by considering the formula which expresses
it, namely the following,
$$0 = {\rm S} \mathbin{.} (\rho - \rho_4) (\rho_4 - \rho_3)
         (\rho_3 - \rho_2) (\rho_2 - \rho_1) (\rho_1 - \rho).
   \eqno (248.)$$
(See the Proceedings of the Royal Irish Academy for July
1846, where this quaternion theorem for the case of the inscribed
pentagon was given.)  For the theorem assigns, and in a simple
manner expresses, to those who accept the language of this
Calculus, a relation between the {\it five\/} successive
directions of the sides of a gauche pentagon inscribed in a
sphere, which appears to the present writer to be
{\it analogous\/} to (although necessarily more complex than) the
angular relation established in the third book of Euclid's
Elements, between the {\it four\/} directions of the sides of a
plane quadrilateral inscribed in a circle.  Indeed, it will be
found to be easy to deduce the property of the plane inscribed
quadrilateral, from the theorem respecting the inscribed gauche
pentagon.  For, by conceiving the fifth side ${\sc p}_4 {\sc p}$
of the pentagon ${\sc p} \, \ldots \, {\sc p}_4$ to tend to
vanish, and therefore to become tangential at the first
corner~${\sc p}$, it is seen that the vector part of the
quaternion which is the continued product of the four first sides
must tend, at the same time, to become normal to the sphere at
${\sc p}$; in order that, when multiplied into an arbitrary
tangential vector there, it may give a vector as the product.
Hence the vector part of the product of the four successive sides
of an inscribed gauche quadrilateral
${\sc p} {\sc p}_1 {\sc p}_2 {\sc p}_3$,
is constructed by a right line which is normal to the sphere at
the first corner; and more generally, either by the same
geometrical reasoning applied to the theorem of art.~83, or by
considering the signification of the formula (246.), we may
deduce this other theorem, that {\it the vector of the continued
product of the successive sides of an inscribed gauche polygon
${\sc p} \, \ldots \, {\sc p}_{2m-1}$, of any even number of
sides, is normal to the sphere at the first corner\/}~${\sc p}$.
Suppose now the inscribed quadrilateral, or more generally the
polygon of $2m$ sides, to flatten into a {\it plane\/} figure; it
will thus come to be inscribed in a {\it circle}, and
consequently in infinitely many spheres {\it at once\/}; and the
only way to escape a resulting indeterminateness in the value for
the vector of the product, is by that vector vanishing: which
accordingly it may be otherwise proved to do, although the
present mode of proof will appear sufficient to those who
examine its principles with care.  And thus we shall find
ourselves conducted to the well-known graphic property of the
quadrilateral inscribed in the circle, and more generally to a
corresponding theorem respecting inscribed hexagons, octagons,
\&c., under the form of the following proposition in quaternions,
which expresses a characteristic property of the
{\it circle\/}:---{\it The vector part of the product of the
successive sides of any polygon, with any even number of sides,
inscribed in a circle, vanishes\/}; or, in other words, the
product thus obtained, instead of being a {\it complete\/}
quaternion, reduces itself simply to a positive or negative
{\it number}.  On the other hand, it is easy to see, from what
precedes, that {\it the product of the successive sides of a
triangle, pentagon, or other polygon of any odd number of sides,
inscribed in a circle, is a vector, which touches the circle at
the first corner of the polygon}, or is parallel to such a
tangent.

\bigbreak

85.
Although the {\it precise law\/} of the relation between the
directions of the sides of an inscribed gauche pentagon,
heptagon, \&c., expressed by the first formul{\ae} (247.), is
{\it peculiar\/} to the {\it sphere\/}; yet it is easy to
abstract from that relation a {\it part}, which shall hold good,
as a law of a {\it more general\/} character, for {\it other\/}
surfaces of the second order.  For we may easily infer, from that
formula, especially when combined with the other equations of
art.~82, that {\it if the first $2m$ sides of an inscribed
polygon of $2m + 1$ sides,
${\sc p}' {\sc p}_1' {\sc p}_2' \, \ldots \, {\sc p}_{2m}'$,
be respectively parallel to the successive sides of another
polygon of $2m$ sides,
${\sc p} {\sc p}_1 \, \ldots \, {\sc p}_{2m-1}$,
inscribed in the same surface, then the last side,
${\sc p}_{2m}' {\sc p}'$, of the former polygon, will be parallel
to the plane which touches the surface at the first
corner~${\sc p}$ of the latter polygon\/}: and under {\it this\/}
form of enunciation, it is obvious that the theorem must admit of
being extended, by deformation, to ellipsoids, and other surfaces
of the second degree.  We may then enunciate also this other
theorem, respecting the inscription of rectilinear polygons in
such surfaces (which theorem was communicated to the Royal Irish
Academy in March 1849):---{\it If, after inscribing, in a surface
of the second degree, any gauche polygon of $2m$ sides,
${\sc p} {\sc p}_1 \, \ldots \, {\sc p}_{2m-1}$,
we then inscribe in the same surface another gauche polygon, of
$4m + 1$ sides,
${\sc p}' {\sc p}_1' \, \ldots \, {\sc p}_{4m}'$,
under the following $4m$ conditions of parallelism:}
$${\sc p}'   {\sc p}_1' \parallel {\sc p}   {\sc p}_1;\quad
  {\sc p}_1' {\sc p}_2' \parallel {\sc p}_1 {\sc p}_2;\quad
      \ldots\quad
  {\sc p}_{2m-1}' {\sc p}_{2m}' \parallel {\sc p}_{2m-1} {\sc p};
   \eqno (249.)$$
and
$${\sc p}_{2m}' {\sc p}_{2m+1}' \parallel {\sc p} {\sc p}_1;\quad
  {\sc p}_{2m+1}' {\sc p}_{2m+2}' \parallel {\sc p}_1 {\sc p}_2;\quad
      \ldots\quad
  {\sc p}_{4m-1}' {\sc p}_{4m}' \parallel {\sc p}_{2m-1} {\sc p};
   \eqno (250.)$$
(the first corner~${\sc p}'$ of the second polygon being
{\it assumed\/} at pleasure on the surface, and the other corners
${\sc p}_1'$, \&c., of that polygon, being successively
{\it derived\/} from this one, by drawing two series of parallels
as here directed;) {\it then the diagonal plane
${\sc p}' {\sc p}_{2m}' {\sc p}_{4m}'$,
which contains the first, middle, and last corners of the polygon
with $4m + 1$ sides, will be parallel to the plane which touches
the surface at the first corner~${\sc p}$ of the polygon with
$2m$ sides}.  In fact, the two rectilinear diagonals,
${\sc p}' {\sc p}_{2m}'$ and ${\sc p}_{2m}' {\sc p}_{4m}'$, will,
by a former theorem of the present article, be parallel to that
tangent plane.  For example, if the first, second, third and
fourth sides, of a gauche {\it quadrilateral\/} inscribed in a
surface of the second order, be parallel to the first, second,
third, and fourth, and {\it also\/} to the fifth, sixth, seventh,
and eighth sides respectively, of a gauche {\it enneagon\/}
inscribed in the same surface; than that diagonal plane of the
enneagon which contains the first, fifth and ninth corners
thereof, will be parallel to the plane which touches the surface
at the first corner of the quadrilateral.

\bigbreak

86.
The same sort of quaternion analysis, proceeding from the
formul{\ae} in art.~82, and from others analogous to them, has
conducted the author to many other geometrical theorems,
respecting the inscription of gauche polygons in surfaces of the
second degree.  An outline of some of these was given to the
Royal Irish Academy in June 1849; and some of them may be
mentioned here.  To avoid, at first, imaginary\footnote*{While
acknowledging, as the author is bound to do, the great courtesy
towards himself that has been shown by several recent and able
writers, on subjects having some general connexion or resemblance
with those on which he has been engaged, he hopes that he may be
allowed to say,---yet rather as requesting a favour than as
claiming a right,---that he will be happy if the inventor of the
{\it Pluquaternions\/} shall consent to his adopting or rather
retaining a {\it word\/}, namely ``biquaternion,'' which the
Rev.\ Mr.~Kirkman has indeed {\it employed}, with reference to the
{\it octaves\/} of Mr.~J.~T. Graves and Mr.~Cayley, but does not
appear to {\it want}, for any of his {\it own\/} purposes:
whereas Sir W.~Rowan Hamilton has for years been accustomed to
use this word {\sc biquaternion},---though perhaps hitherto
without printed publication,---and indeed could not, without
sensible inconvenience, have dispensed with it, to denote an
expression {\it entirely distinct from those octaves}, namely one
of the form
$${\rm Q}+ \sqrt{-1} {\rm Q}';$$
where $\sqrt{-1}$ is the old and {\it ordinary imaginary of
algebra\/} (and is therefore quite {\it distinct\/} from
$i$,~$j$,~$k$), while ${\rm Q}$ and ${\rm Q}'$ are abridged
symbols for {\it two different quaternions\/} of the kind
$w + ix + jy + kz$, introduced into analysis in 1843.
{\it Biquaternions\/} of {\it this\/} sort have repeatedly
{\it forced\/} themselves on the attention of Sir W.~R.~H., in
questions respecting {\it geometrical impossibility, ideal
intersections, imaginary deformations}, and the like.}
deformations, in passing from an original sphere, the surface in
which the polygons are inscribed shall be supposed, for the
present, to be an {\it ellipsoid}.  Results of the same
{\it general\/} character, but with {\it some\/} important
modifications (connected with the {\it ordinary\/} square root of
negative unity,) hold good for the inscription of such polygons
in {\it other\/} surfaces of the same order, as the writer may
afterwards point out.  He is aware, indeed, that the
corresponding class of questions, respecting the inscription of
{\it plane\/} polygons in {\it conics}, has attained sufficient
celebrity; and feels that his own acquaintance with what has been
already done in that department of geometrical science is
inferior to the knowledge of its history possessed by several of
his contemporaries, for instance, by Professor Davies.  He knows
also that some of the published methods for inscribing in a
circle, or plane conic, a polygon whose sides shall pass through
the same number of given points, can be adapted to the case of a
polygon formed by {\it arcs\/} of great circles on the surface of
a sphere, and inscribed in a {\it spherical\/} conic; and he has,
by quaternions, been conducted to some such methods himself, for
the solution of the latter problem.  But he acknowledges that he
shall feel some little surprise, though perhaps not entitled to
do so, if it shall turn out that the results of which he proceeds
to give an outline, respecting the {\it inscription of
rectilinear but gauche polygons in an ellipsoid}, have been
wholly (or even partially) anticipated.  They have certainly
been, in his own case, results of the application of the
quaternion calculus: but whatever geometrical truth has been
attained by any one {\it general mathematical method\/} (such as
the Quaternions claim to be), may also be found, or at least
{\it proved}, by any {\it other\/} method equally general.  And
those who shall take the pains of {\it proving\/} for themselves,
by the Cartesian Coordinates, or by some less algebraical and
more purely geometrical method, the following theorems, (if not
already known), which have thus been {\it found\/} by the
Quaternions, will doubtless be led to perceive {\it many\/} new
truths, connected with them, which have escaped the present
writer; although he too has arrived at other connected results,
which he must suppress in the following notice.

\bigbreak

87.
I.
An {\it ellipsoid\/} $({\sc e})$ being given, and also a system
of any {\it even\/} number of points of space,
${\sc a}_1, {\sc a}_2,\ldots \, {\sc a}_{2m}$, of which points it
is here supposed that none are situated on the surface of the
ellipsoid; it is, in general, possible to {\it inscribe\/} in
this ellipsoid, {\it two}, and {\it only two}, distinct and
{\it real polygons of $2m$ sides},
${\sc b} {\sc b}_1 \, \ldots \, {\sc b}_{2m-1}$
and
${\sc b}' {\sc b}_1' \, \ldots \, {\sc b}_{2m-1}'$,
such that the sides of each of these two polygons
$({\sc b})$~(${\sc b}')$ shall pass, respectively and
successively, through the $2m$ given points; or in other words,
so that
${\sc b} {\sc a}_1 {\sc b}_1,
   {\sc b}_1 {\sc a}_2 {\sc b}_2,\ldots \,
   {\sc b}_{2m-1} {\sc a}_{2m} {\sc b}$,
and also
${\sc b}' {\sc a}_1 {\sc b}_1',
   {\sc b}_1' {\sc a}_2 {\sc b}_2',\ldots \,
   {\sc b}_{2m-1}' {\sc a}_{2m} {\sc b}'$,
shall be straight lines; while
${\sc b}, {\sc b}_1,\ldots \, {\sc b}_{2m-1}$, and also
${\sc b}', {\sc b}_1',\ldots \, {\sc b}_{2m-1}'$,
shall be points upon the surface of the ellipsoid.

\medbreak

[It should be noted that there are also, in general, what may, by
the use of a known phraseology, be called {\it two other}, but
{\it geometrically imaginary}, modes of inscribing a polygon,
under the same conditions, in an {\it ellipsoid\/}: which modes
{\it may become real, by imaginary deformation}, in passing to
{\it another\/} surface of the second order.]

\medbreak

II.
If we now take any {\it other\/} and {\it variable
point\/}~${\sc p}$ on the ellipsoid $({\sc e})$ {\it instead\/}
of ${\sc b}$ or ${\sc b}'$, and make {\it it\/} the {\it first
corner\/} of an inscribed polygon of $2m + 1$ sides, of which the
{\it first $2m$ sides\/} shall pass, respectively and
successively, through the $2m$ given points $({\sc a})$; in such
a manner that
${\sc p} {\sc a}_1 {\sc p}_1,
   {\sc p}_1 {\sc a}_2 {\sc p}_2,\ldots \,
   {\sc p}_{2m-1} {\sc a}_{2m} {\sc p}_{2m}$,
shall be straight lines, while
${\sc p}, {\sc p}_1, {\sc p}_2,\ldots \, {\sc p}_{2m}$
shall all be points on the surface of the ellipsoid: then the
{\it last side}, or {\it closing chord}, ${\sc p}_{2m} {\sc p}$,
of this new and {\it variable polygon\/} $({\sc p})$, thus
inscribed in the ellipsoid $({\sc e})$, shall {\it touch}, in all
its positions, a certain {\it other ellipsoid\/} $({\sc e}')$.

\medbreak

III.
This {\it new\/} ellipsoid $({\sc e}')$ is itself
{\it inscribed\/} in the given ellipsoid $({\sc e})$, having
{\it double contact\/} therewith, but being elsewhere interior
thereto.

\medbreak

IV.
The {\it two points of contact\/} of these two ellipsoids are the
points~${\sc b}$ and ${\sc b}'$; that is, they are the {\it first
corners of the two inscribed polygons of $2m$ sides}, $({\sc b})$
and $({\sc b}')$, which were considered in I.

\medbreak

[So far, the results are evidently analogous to known theorems,
respecting polygons in conics; what follows is more peculiar to
space.]

\medbreak

V.
If the two ellipsoids, $({\sc e})$ and $({\sc e}')$, be cut by
any plane parallel to either of their two common tangent planes,
the sections will be {\it two similar and similarly situated
ellipses}.

\medbreak

[For example, if the {\it original\/} ellipsoid reduce itself to
a {\it sphere}, then the two points of contact, ${\sc b}$ and
${\sc b}'$, become two of the four {\it umbilics on the inscribed
ellipsoid}.]

\medbreak

VI.
The closing chords ${\sc p} {\sc p}_{2m}$ are also tangents to a
certain series or {\it system of curves\/} $({\sc c}')$, not
generally plane, on the surface of the inscribed ellipsoid
$({\sc e}')$; and therefore may be arranged into a {\it system of
developable surfaces}, $({\sc d}')$, of which these curves
$({\sc c}')$ are the {\it ar\^{e}tes de rebroussement}.

\medbreak

VII.
The same closing chords may also be arranged into a {\it second
system of developable surfaces}, $({\sc d}'')$, which
{\it envelope the inscribed ellipsoid\/} $({\sc e}')$ and have
{\it their ar\^{e}tes de rebroussement\/} $({\sc c}'')$ all
situated on a certain {\it other surface\/}~$({\sc e}'')$, which
is, in its turn, enveloped by the {\it first\/} set of
developable surfaces $({\sc d}')$; so that {\it the closing
chords ${\sc p} {\sc p}_{2m}$ are all tangents to a second set of
curves, $({\sc c}'')$, and to a second surface,~$({\sc e}'')$.}

\medbreak

VIII.
This second surface~$({\sc e}'')$ is a {\it hyperboloid of two
sheets}, having {\it double contact\/} with the given
ellipsoid~$({\sc e})$, and {\it also\/} with the inscribed
ellipsoid $({\sc e}')$, at the points ${\sc b}$ and ${\sc b}'$;
one sheet having external contact with each ellipsoid at one of
those two points, and the other at the other.

\medbreak

IX.
If either sheet of this hyperboloid~$({\sc e}'')$ be cut by a
plane parallel to either of the two common tangent planes,
{\it the elliptic section of the sheet is similar to a parallel
section of either ellipsoid, and is similarly situated
therewith}.

\medbreak

[For example, the points of contact ${\sc b}$ and ${\sc b}'$ are
two of the {\it umbilics of the hyperboloid\/} $({\sc e}'')$,
when the given surface $({\sc e})$ is a {\it sphere}.]

\medbreak

X.
The {\it centres\/} of the three surfaces,
$({\sc e})$ $({\sc e}')$ $({\sc e}'')$, are situated {\it on one
straight line}.

\medbreak

XI.
The two systems of developable surfaces, cut the original
ellipsoid,~$({\sc e})$, in {\it two new series of curves},
$({\sc f}')$, $({\sc f}'')$, not generally plane, which
everywhere so cross each other on $({\sc e})$, that at any one
such point of crossing, ${\sc p}$, {\it the tangents to the two
curves $({\sc f}')$~$({\sc f}'')$ are parallel to two conjugate
semidiameters\/} of the surface~$({\sc e})$ on which the curves
are contained.

\medbreak

[For example, if the original surface $({\sc e})$ be a
{\it sphere}, then these two sets of curves
$({\sc f}')$~$({\sc f}'')$ cross each other everywhere {\it at
right angles}, upon that spheric surface.]

\medbreak

XII.
{\it Each closing chord ${\sc p} {\sc p}_m$ is cut harmonically},
at the two points, ${\sc c}'$,~${\sc c}''$, where it touches the
inscribed ellipsoid~$({\sc e}')$, and the exscribed
hyperboloid~$({\sc e}'')$; or {\it where it touches the curves\/}
$({\sc c}')$ and $({\sc c}'')$.

\medbreak

XIII.
The closing chords, or {\it the positions of the last side of the
variable polygon $({\sc p})$, are not, in general, all cut
perpendicularly by any one common surface\/} (notwithstanding the
analogy of their arrangement, or distribution in space, in many
respects, to that of the normals to a surface).  In fact, the two
systems of developable surfaces, $({\sc d}')$ and $({\sc d}'')$,
are {\it not\/} generally rectangular to each other, in the
arrangement {\it here\/} considered, though they {\it are\/} so
for any system of normals.

\medbreak

XIV.
{\it Through any given point of space},~${\sc a}_{2m+1}$, which
is at once {\it exterior\/} to the inscribed
ellipsoid~$({\sc e}')$, and to {\it both\/} sheets of the
exscribed hyperboloid~$({\sc e}'')$, it is in general possible to
draw {\it two}, and {\it only two}, distinct and {\it real
straight lines}, ${\sc p}' {\sc p}_{2m}'$ and
${\sc p}'' {\sc p}_{2m}''$, of which {\it each shall touch at
once a curve\/}~$({\sc c}')$ on $({\sc e}')$, {\it and a
curve\/}~$({\sc c}'')$ on $({\sc e}'')$, and of which {\it each
shall coincide with one of the positions of the closing
chord},~${\sc p} {\sc p}_{2m}$; in such a manner as to be
{\it the last side of a rectilinear polygon of $2m + 1$ sides},
${\sc p}' {\sc p}_1' {\sc p}_2' \, \ldots \, {\sc p}_{2m}'$, or
${\sc p}'' {\sc p}_1'' {\sc p}_2'' \, \ldots \, {\sc p}_{2m}''$,
{\it inscribed in the given ellipsoid~$({\sc e})$, under the
condition that its sides shall pass, respectively and
successively, through the $2m + 1$ given points},
${\sc a}_1 {\sc a}_2 \, \ldots \, {\sc a}_{2m+1}$.
But if the last of these points were given {\it on either of the
two enveloped surfaces}, $({\sc e}')$, $({\sc e}'')$, the problem
of such inscription would in general admit of {\it only one
distinct solution}, obtained by drawing through the given point
the tangent to the particular curve~$({\sc c}')$ or
$({\sc c}'')$, on which that point was situated.  And if the last
given point~${\sc a}_{2m+1}$ were situated {\it within\/} the
inscribed ellipsoid~$({\sc e}')$, or {\it within either sheet\/}
of the exscribed hyperboloid~$({\sc e}'')$, the problem of the
inscription of the polygon of $2m + 1$ sides would then become
{\it geometrically impossible\/}: though it might still be said
to admit, in that case, of {\it two imaginary modes of solution}.

\bigbreak

88.
The writer desires to put on record, in this place, the following
enunciations of one or two other theorems, out of many to which
the quaternion analysis has conducted him, respecting the
inscription of gauche polygons in surfaces of the second order;
without {\it yet\/} entering on any fuller account of that
analysis itself, than what is given or suggested in some of the
preceding articles.  See the Numbers of the Philosophical
Magazine for August and September 1849.  And in the first place
he will here transcribe the memorandum of a communication,
hitherto unprinted, which was sent to him, in the month last
mentioned, to the Mathematical and Physical Section of the
Meeting of the British Association at Birmingham.

\bigbreak

89.
\hskip 0pt plus3pt minus0pt
Conceive that any rectilinear (but generally gauche) polygon of
$n$ sides,\break
${\sc b} {\sc b}_1 {\sc b}_2 \, \ldots \, {\sc b}_{n-1}$,
has been inscribed in any surface of the second order; and that
$n$ fixed points,
${\sc a}_1, {\sc a}_2,\ldots \, {\sc a}_n$, {\it not\/} on that
surface, have been assumed on its $n$ successive sides, namely
${\sc a}_1$ on ${\sc b} {\sc b}_1$, ${\sc a}_2$ on
${\sc b}_1 {\sc b}_2$, \&c.  Take then at pleasure any
point~${\sc p}$ upon the same surface, and draw the chords
${\sc p} {\sc a}_1 {\sc p}_1,
   {\sc p}_1 {\sc a}_2 {\sc p}_2,\ldots
   {\sc p}_{n-1} {\sc a}_n {\sc p}_n$,
passing respectively through the $n$ fixed points~$({\sc a})$.
Again, begin with ${\sc p}_n$, and draw, through the same $n$
points~$({\sc a})$, $n$ other successive chords,
${\sc p}_n {\sc a}_1 {\sc p}_{n+1},
   {\sc p}_{n+1} {\sc a}_2 {\sc p}_{n+2},\ldots
   {\sc p}_{2n-1} {\sc a}_n {\sc p}_{2n}$.
Again begin with ${\sc p}_{2n}$, and draw in like manner the $n$
chords,
${\sc p}_{2n} {\sc a}_1 {\sc p}_{2n+1},
   {\sc p}_{2n+1} {\sc a}_2 {\sc p}_{2n+2},\ldots \,
   {\sc p}_{3n-1} {\sc a}_n {\sc p}_{3n}$.
Then one or other of the two following Theorems will hold good,
according as the number~$n$ is {\it odd\/} or {\it even}.

\medbreak

{\it Theorem\/}~I.
If $n$ be {\it odd}, and if we draw {\it two tangent planes\/} to
the surface at the points ${\sc p}_n$,~${\sc p}_{2n}$, meeting
the two new chords, ${\sc p} {\sc p}_{2n}$,
${\sc p}_n {\sc p}_{3n}$, respectively, in two new points,
${\sc r}$,~${\sc r}'$; then {\it the three points
${\sc b} {\sc r} {\sc r}'$
shall be situated on one straight line}.

\medbreak

{\it Theorem\/}~II.
If $n$ be {\it even}, and if we describe {\it two pairs of plane
conics on the surface}, each conic being determined by the
condition of passing through {\it three points\/} thereon, as
follows: the first pair of conics passing through
${\sc b} {\sc p} {\sc p}_{2n}$,
and
${\sc p}_n {\sc p}_{2n} {\sc p}_{3n}$;
and the second pair through
${\sc b} {\sc p}_n {\sc p}_{3n}$,
and
${\sc p} {\sc p}_n {\sc p}_{2n}$;
it will then be possible to trace, {\it on the same surface, two
other plane conics}, of which {\it the first shall touch the two
conics of the first pair, at the two points\/}~${\sc b}$ and
${\sc p}_n$; while {\it the second new conic shall touch the two
conics of the second pair, at the two points\/}~${\sc b}$ and
${\sc p}_{2n}$.

\bigbreak

90.
With respect to the {\it first\/} of the two theorems thus
communicated, it may be noticed now, that it gives an easy mode
of resolving the following {\it Problem}, analogous to a
celebrated problem in plane conics:---To find the {\it two\/}
(real or imaginary) polygons,
${\sc b} {\sc b}_1 {\sc b}_2 \, \ldots \, {\sc b}_{n-1}$
and
${\sc b}' {\sc b}_1' {\sc b}_2' \, \ldots \, {\sc b}_{n-1}'$,
with any given {\it odd\/} number~$n$ of sides, which can be
inscribed in a given {\it surface\/} of the second order, so that
their $n$ successive sides, namely
${\sc b} {\sc b}_1, {\sc b}_1 {\sc b}_2,\ldots$
for one polygon, and 
${\sc b}' {\sc b}_1', {\sc b}_1' {\sc b}_2',\ldots$
for the other polygon thus inscribed, shall pass respectively
through $n$ given points
${\sc a}_1, {\sc a}_2,\ldots \, {\sc a}_n$,
which are not themselves situated upon the surface.  For we have
only to assume at pleasure {\it any\/} point~${\sc p}$ upon that
surface, and to deduce thence the {\it two\/} non-superficial
points lately called ${\sc r}$ and ${\sc r}'$, by the
construction assigned in the theorem; since by then joining the
two points thus found, {\it the joining line ${\sc r} {\sc r}'$
will cut the given surface of the second order in the two\/}
(real or imaginary) {\it points}, ${\sc b}$,~${\sc b}'$, which
are adapted to be, respectively, {\it the first corners of the
two polygons required}.---That there {\it are\/} (in general)
{\it two\/} such (real or imaginary) polygons, {\it when the
number of sides is odd}, had been previously inferred by the
writer, from the quaternion analysis which he employed.  Indeed,
it may have been perceived to be, through geometrical
deformation, a consequence of what was stated in \S~XIV.\ of
article~87 of this series of papers on Quaternions, for the
particular case of the ellipsoid, in the Philosophical Magazine
for September 1849.   See also the account, in the
Proceedings of the Royal Irish Academy, of the author's
communication to that body, at the meeting of June 25th, 1849; in
which account, indeed, will be found (among many others) both the
theorems of the preceding article~89; the second of those
theorems being however there enunciated under a {\it metric},
rather than under a {\it graphic\/} form.

\bigbreak

\centerline{[{\it To be continued.}]}

\bye
