% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 1st June 1999.

\magnification=\magstep1
\vsize=227 true mm \hsize=170 true mm
   \voffset=-0.4 true mm \hoffset=-5.4 true mm

\def\folio{\ifnum\pageno>0 \number\pageno \else
   \ifnum\pageno<0 \romannumeral-\pageno \else\fi\fi}

\def\multieqalign#1{\null\,\vcenter{\openup1\jot \mathsurround=0pt
   \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
   &&\quad \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
    \crcr#1\crcr}}\,}

\def\sum{\mathop{\Sigma}\nolimits}

\def\cos{\mathop{\rm cos.}\nolimits}

\font\Largebf=cmbx10  scaled \magstep2
\font\largerm=cmr12
\font\largeit=cmti12
\font\sc=cmcsc10

\pageno=0

\null\vskip72pt

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

\vskip12pt

\centerline{\Largebf THEORY OF SYSTEMS OF RAYS}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (Transactions of the Royal Irish Academy, vol.~16,
part~2 (1831), pp. 93--125.)}

\vskip36pt

\vfill

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

\vskip 12pt

\centerline{\largerm 2001}

\vskip36pt\eject

\begingroup
\hsize=.8\hsize
\advance \hoffset by 0.1\hsize
\emergencystretch=15pt

\pageno=-1

\null\vskip36pt

\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

The {\it Second Supplement to an Essay on the Theory of Systems
of Rays} by William Rowan Hamilton was originally published in
volume~16, part~2 of the {\it Transactions of the Royal Irish
Academy}.  It is included in {\it The Mathematical Papers of Sir
William Rowan Hamilton, Volume I: Geometrical Optics}, edited for
the Royal Irish Academy by A.~W. Conway and J.~L. Synge, and
published by Cambridge University Press in 1931.

\bigbreak\bigskip

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

\vskip3pt

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

\vskip 0pt

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

\vfill\eject
\endgroup

\pageno=1

\null\vskip36pt

{\largeit\noindent
Second Supplement to an Essay on the Theory of Systems of Rays.
\hskip 0pt plus10pt minus0pt
By {\largerm WILLIAM R. HAMILTON}, Royal Astronomer of Ireland,
\&c.\ \&c.}

\bigbreak

\centerline{Read October 25, 1830.}

\bigbreak

\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~16, part~2 (1831), pp. 93--125.]}

\nobreak\bigskip

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

\nobreak\bigskip

\centerline{\largerm INTRODUCTION.}

\nobreak\bigskip

The present Supplement contains the integration of some partial
differential equations, to which I have been conducted by the
view of mathematical optics, proposed in my former memoirs.
According to that view, the geometrical properties of an optical
system of rays may be deduced by analytic methods, from the form
of one characteristic function; of which the partial differential
coefficients of the first order, taken with respect to the three
rectangular coordinates of any proposed point of the system, are,
in the case of ordinary light, equal to the index of refraction
of the medium, multiplied by the cosines of the angles which the
ray passing through the point makes with the axes of coordinates:
and as these cosines are connected by the known relation that the
sum of their squares is unity, there results a corresponding
connexion between the partial differential coefficients to which
they are proportional.  This connexion is expressed by an
equation which it is interesting to study and to integrate,
because it contains a general property of ordinary systems of
rays, and because its integral is a general form for the
characteristic function of such a system.  The integral which I
have given in the present memoir, is deduced from equations
assigned in my former Supplement; an elimination which had been
before supposed, being now effected, by the theorems which
Laplace has established in the second Book of the {\it
M\'{e}canique C\'{e}leste}, for the development of functions
into series.  The development thus obtained, proceeds according
to the ascending powers of the perpendicular distances of a
variable point from the tangent planes of the two rectangular
developable pencils which pass through an assumed ray of the
system, and according to the descending powers of the distances
of the projection of the variable point upon the assumed ray,
from the points in which that ray touches the two caustic
surfaces.  In the case of rays contained in one plane, or
symmetric about one axis, the partial differential equation takes
simpler forms, of which I have assigned the integrals, and have
given an example of their optical use, by briefly shewing their
connexion with the longitudinal aberrations of curvature.  I
hope, in a future memoir, to point out other methods of
integrating the general equation for the characteristic function
of ordinary systems of rays, and other applications of the
resulting expressions, to the solution of optical problems.

\nobreak\bigskip

\line{\hfil WILLIAM R. HAMILTON.}

\nobreak\bigskip

{\sc Observatory}, {\it October\/} 1830.

\vfill\eject

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

\nobreak\bigskip

\begingroup
\everypar{\parindent=0pt \hangindent=20pt \hangafter=1}

Introduction.

Statement and Integration of the Partial Differential Equation,
which determines the Characteristic Function of Ordinary Systems
of Rays, produced by any Number of successive Reflexions or
Refractions,
\dotfill 1,~2

Transformation and Development of the Integral,
\dotfill 3

Verifications of the foregoing Developments,
\dotfill 4

Case of a Plane System,
\dotfill 5

Case of a System of Revolution,
\dotfill 6

Verification of the Approximate Integral for Systems of
Revolution,
\dotfill 7

Other Method of obtaining the Approximate Integral,
\dotfill 8

Connexion of the Longitudinal Aberration, in a System of
Revolution, with the Development of the Characteristic
Function~$V$,
\dotfill 9

Changes of a System of Revolution, produced by Ordinary
Refraction,
\dotfill 10

Example; Spheric Refraction; Mr.~Herschel's Formula for the
Aberration of a thin Lens,
.\dotfill 11

\endgroup

\vfill\eject

\centerline{\largerm SECOND SUPPLEMENT.}

\nobreak\bigskip

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

\nobreak\bigskip

\centerline{\it
Statement and Integration of the Partial Differential Equation,
which determines}
\centerline{\it the Characteristic Function of Ordinary Systems
of Rays, produced by any Number}
\centerline{\it of successive Reflexions or Refractions.}

\nobreak\bigskip

1.
Suppose that rays of a given colour diverge from a given luminous
origin, and undergo any number of successive changes of
direction, according to the known laws of ordinary reflexion and
refraction, at surfaces having any given shapes and positions,
and enclosing media of any given refractive indices.  Let
$\alpha$,~$\beta$,~$\gamma$, be the cosines of the angles which
the direction of a final ray makes with three rectangular axes,
and let $x$,~$y$,~$z$, be the three rectangular coordinates,
referred to the same axes, of a point upon this final ray; then
$\alpha$,~$\beta$,~$\gamma$, will in general be functions of
$x$,~$y$,~$z$, such that if $\mu$ denote the refractive index of
the final medium, for rays of the given colour, the expression
$$\mu (\alpha \, dx + \beta \, dy + \gamma \, dz)$$
is equal to the differential of a certain function~$V$, of
which I have shewn the existence and the meaning in former
memoirs, and which I have called the {\it characteristic
function\/} of the final system.  The design of the present
Supplement, is to point out some new properties and uses of this
function, resulting from the partial differential equation
$$    \left( {dV \over dx} \right)^2
    + \left( {dV \over dy} \right)^2
    + \left( {dV \over dz} \right)^2
   = \mu^2,
   \eqno {\rm (A)}$$
which we obtain by eliminating the three cosines
$\alpha$,~$\beta$,~$\gamma$, between the three equations
$${dV \over dx} = \mu \alpha,\quad
  {dV \over dy} = \mu \beta,\quad
  {dV \over dz} = \mu \gamma,
   \eqno {\rm (B)}$$
by the help of the known relation
$$\alpha^2 + \beta^2 + \gamma^2 = 1.$$

\bigbreak

2.
The equation (A) is a particular case of a more general
differential equation, for all optical systems of rays, ordinary
or extraordinary, obtained by eliminating the same three cosines,
$\alpha$,~$\beta$,~$\gamma$, by the same known relation between
the three following equations, assigned in my former memoirs,
$${dV \over dx} = {\delta v \over \delta \alpha},\quad
  {dV \over dy} = {\delta v \over \delta \beta},\quad
  {dV \over dz} = {\delta v \over \delta \gamma};$$
in which $V$ is the characteristic function of the system, and
$v$ is a homogeneous function of $\alpha$,~$\beta$,~$\gamma$, of
the first dimension, representing the velocity of the light,
estimated on the hypothesis of emission, and differentiated as if
$\alpha$,~$\beta$,~$\gamma$, were three independent variables.
And the integral of (A), is a particular case of a more general
integral, extending to all optical systems of straight rays, and
consisting of the following combination of equations, assigned in
my former Supplement:
$$\eqalign{
W + V
   &=    x {\delta v \over \delta \alpha}
       + y {\delta v \over \delta \beta}
       + z {\delta v \over \delta \gamma},\cr
{\delta W \over \delta \alpha}
   &=    x {\delta^2 v \over \delta \alpha^2}
       + y {\delta^2 v \over \delta \alpha \, \delta \beta}
       + z {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
{\delta W \over \delta \beta}
   &=    x {\delta^2 v \over \delta \alpha \, \delta \beta}
       + y {\delta^2 v \over \delta \beta^2}
       + z {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
{\delta W \over \delta \gamma}
   &=    x {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + y {\delta^2 v \over \delta \beta  \, \delta \gamma}
       + z {\delta^2 v \over \delta \gamma^2};\cr}$$
between which the three quantities $\alpha$,~$\beta$,~$\gamma$,
are to be eliminated; $W$ being an arbitrary but homogeneous
function of these three quantities, of the dimension zero; and
the partial differential coefficients in which the sign~$\delta$
occurs, being formed by differentiating the homogeneous functions
$W$, $v$, as if $\alpha$,~$\beta$,~$\gamma$, were three
independent variables.  In applying these general results to
ordinary systems of rays, we are to put
$$v = \mu (\alpha^2 + \beta^2 + \gamma^2)^{1 \over 2};$$
$${\delta v \over \delta \alpha} = {\mu^2 \alpha \over v},\quad
  {\delta v \over \delta \beta}  = {\mu^2 \beta  \over v},\quad
  {\delta v \over \delta \gamma} = {\mu^2 \gamma \over v};$$
$${\delta^2 v \over \delta \alpha^2}
   = {\mu^2 \over v^3} (v^2 - \mu^2 \alpha^2),\quad
  {\delta^2 v \over \delta \beta^2}
   = {\mu^2 \over v^3} (v^2 - \mu^2 \beta^2),\quad
  {\delta^2 v \over \delta \gamma^2}
   = {\mu^2 \over v^3} (v^2 - \mu^2 \gamma^2);$$
$${\delta^2 v \over \delta \alpha \, \delta \beta}
   = - {\mu^4 \alpha \beta \over v^3},\quad
  {\delta^2 v \over \delta \beta \, \delta \gamma}
   = - {\mu^4 \beta  \gamma \over v^3},\quad
  {\delta^2 v \over \delta \gamma \, \delta \alpha}
   = - {\mu^4 \gamma \alpha \over v^3}:$$
or, (making after the differentiations
$\alpha^2 + \beta^2 + \gamma^2 = 1$,)
$$v = \mu,\quad
  {\delta v \over \delta \alpha} = \mu \alpha,\quad
  {\delta v \over \delta \beta}  = \mu \beta,\quad
  {\delta v \over \delta \gamma} = \mu \gamma,$$
$${\delta^2 v \over \delta \alpha^2} = \mu (1 - \alpha^2),\quad
  {\delta^2 v \over \delta \beta^2}  = \mu (1 - \beta^2),\quad
  {\delta^2 v \over \delta \gamma^2} = \mu (1 - \gamma^2),$$
$${\delta^2 v \over \delta \alpha \, \delta \beta}
   = - \mu \alpha \beta,\quad
  {\delta^2 v \over \delta \beta  \, \delta \gamma}
   = - \mu \beta  \gamma,\quad
  {\delta^2 v \over \delta \gamma \, \delta \alpha}
   = - \mu \gamma \alpha;$$
and therefore,
$$\left. \eqalign{
W + V
   &= \mu (\alpha x + \beta y + \gamma z),\cr
{\delta W \over \delta \alpha}
   &= \mu x - \mu \alpha (\alpha x + \beta y + \gamma z),\cr
{\delta W \over \delta \beta}
   &= \mu y - \mu \beta  (\alpha x + \beta y + \gamma z),\cr
{\delta W \over \delta \gamma}
   &= \mu z - \mu \gamma (\alpha x + \beta y + \gamma z).\cr}
   \right\}
   \eqno {\rm (C)}$$

This system of equations (C) is one form for the integral of the
partial differential equation (A); the quantities
$\alpha$,~$\beta$,~$\gamma$, being supposed to be eliminated, and
$W$ being an arbitrary function of these quantities, of the kind
already mentioned.

\bigbreak

\centerline{\it
Transformation and Development of the Integral.}

\nobreak\bigskip

3.
The system of equations (C) may be transformed into the
following:
$$\mu x = {dU \over d\alpha},\quad
  \mu y = {dU \over d\beta},\quad
  V + U = \alpha {dU \over d\alpha} + \beta {dU \over d\beta};
   \eqno {\rm (D)}$$
in which $U$ is a function of the three independent variables
$\alpha$,~$\beta$,~$z$, obtained from the function~$W$ by putting
$$U = W - \mu \gamma z,
   \eqno {\rm (E)}$$
and by considering $\gamma$ as a function of $\alpha$,~$\beta$.
Let us now proceed to eliminate $\alpha$,~$\beta$, between the
three equations (D), by the theorems which Laplace has given in
the second Book of the {\it M\'{e}canique C\'{e}leste}, for the
development of functions into series.

This elimination may be simplified by a proper choice of the
coordinates.  The rays of an ordinary system being perpendicular
to the surfaces which have for equation
$$V = \hbox{const.},$$
compose in general two series of rectangular developable pencils,
and are tangents to two caustic surfaces.  Let us therefore
denote by $x_\prime$,~$y_\prime$,~$z_\prime$, three rectangular
coordinates so chosen that the axis of $z_\prime$ coincides with
some given ray, and that the planes of $x_\prime z_\prime$ and
$y_\prime z_\prime$ are the tangent planes of the two developable
pencils to which that ray belongs; and let
$\alpha$~$\beta$~$\gamma$ denote, for any proposed ray of the
system, the cosines of the angles which the ray makes with the
axes of $x_\prime$~$y_\prime$~$z_\prime$.  The equations (A) (B)
(C) (D) (E) will apply to the coordinates thus chosen, by simply
changing $x$~$y$~$z$ to $x_\prime$~$y_\prime$~$z_\prime$; and by
changing $\gamma$ to its value
$$\gamma = \sqrt{1 - \alpha^2 - \beta^2}
   = 1 - {\alpha^2 + \beta^2 \over 2}
       - \gamma^{(4)} - \gamma^{(6)} - \hbox{\&c.},$$
in which
$$\gamma^{(2i + 4)}
    = {1 \mathbin{.} 3 \mathbin{.} 5 \, \ldots \, (2i + 1)
      \over 2 \mathbin{.} 4 \mathbin{.} 6 \, \ldots \, (2i + 2)}
      {(\alpha^2 + \beta^2)^{i + 2} \over 2i + 4},$$
the function $W$ will in general admit of being thus developed,
$$W = \mu W^{(0)}
      + {\mu \over 2} (A \alpha^2 + B \beta^2)
      + \mu W^{(3)} + \mu W^{(4)} + \hbox{\&c.},
   \eqno {\rm (F)}$$
$W^{(0)}$, $A$, $B$, being constants, and $W^{(3)}$, $W^{(4)}$,
$W^{(i)}$, being rational homogeneous functions of the two small
variables $\alpha$,~$\beta$, of the dimensions $3$, $4$, $i$,
respectively.  The constants $A$, $B$, are here the distances
upon the ray, from the point in which it touches the two caustic
surfaces, to the origin of the coordinates
$x_\prime$~$y_\prime$~$z_\prime$; and the terms proportional to
$\alpha$, $\beta$, $\alpha \beta$, disappear from the development
of $W$, by the choice which we have made of these coordinates,
and by the principles of the former Supplement.  In this manner
the function~$U$ becomes
$$U = \mu W^{(0)} - \mu z_\prime
      + {\mu \over 2}
         \{ (z_\prime + A) \alpha^2 + (z_\prime + B) \beta^2 \}
      + \mu U^{(3)} + \mu U^{(4)} + \hbox{\&c.},
   \eqno {\rm (G)}$$
in which
$$U^{(2i + 3)} = W^{(2i + 3)};\quad
  U^{(2i + 4)} = W^{(2i + 4)} + z_\prime \gamma^{(2i + 4)};$$
and the two first of the equations (D) become
$$\alpha = \alpha_\prime + (z_\prime + A)^{-1} {d\phi \over d\alpha};\quad
  \beta  = \beta_\prime  + (z_\prime + B)^{-1} {d\phi \over d\beta},
   \eqno {\rm (H)}$$
if we put for abridgment
$$\alpha_\prime = {x_\prime \over z_\prime + A},\quad
  \beta_\prime  = {y_\prime \over z_\prime + B},\quad
  \phi = - (U^{(3)} + U^{(4)} + \hbox{\&c.}).
   \eqno {\rm (I)}$$

On account of the smallness of
$\displaystyle {d\phi \over d\alpha}$,
$\displaystyle {d\phi \over d\beta}$,
the quantities $\alpha_\prime$,~$\beta_\prime$, are approximate
values of $\alpha$,~$\beta$; and to develope $\alpha$,~$\beta$,
themselves, or any function of them, $F(\alpha, \beta)$, in a
series of ascending powers of these approximate values, we have,
by the theorems of Laplace before referred to,
$$\eqalignno{
F(\alpha, \beta)
   &= F_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle {d^n \over d\alpha_\prime^n}
               \left(
                  {dF_\prime \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
               \right)
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
          + {\displaystyle {d^n \over d\beta_\prime^n}
               \left(
                  {dF_\prime \over d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n+1}
               \right)
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}}
         \right\} \cr
     &+ \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left\{
                  \vcenter{\halign{\hfil $\displaystyle #$\cr
                  {d^2 F_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1} \cr
   \noalign{\vskip 3pt}
                + {dF_\prime \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1}
                  {d \over d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1} \cr
   \noalign{\vskip 3pt}
                + {dF_\prime \over d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
                  {d \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1} \cr}}
               \right\}
            \over [n + 1]^{n+1} [n' + 1]^{n' + 1}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}},\cr
   & &{\rm (K)}\cr}$$
the functions $F_\prime$, $\phi_\prime$, being formed from $F$,
$\phi$, by changing $\alpha$,~$\beta$, to
$\alpha_\prime$,~$\beta_\prime$, and $[n + 1]^{n + 1}$,
$[n' + 1]^{n' + 1}$, being known factorial symbols; we have
therefore,

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

$$\left. \eqalign{
\alpha
   &= \alpha_\prime
       + \sum\nolimits_{(n)}{}_0^\infty \,
         {\displaystyle {d^n \over d\alpha_\prime^n}
               \mathbin{.}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left(
                  {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^n
                  \!\!
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1}
               \right)
            \over [n]^n [n' + 1]^{n' + 1}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}};\cr
\beta
   &= \beta_\prime
       + \sum\nolimits_{(n)}{}_0^\infty \,
         {\displaystyle {d^n \over d\beta_\prime^n}
               \mathbin{.}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left(
                  {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n + 1}
                  \!\!
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'}
               \right)
            \over [n + 1]^{n+1} [n']^{n'}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}}.\cr}
   \right\}
   \eqno {\rm (L)}$$
Now, if we differentiate $V$ as a function of the three
independent variables
$\alpha_\prime$,~$\beta_\prime$,~$z_\prime$, we have by (B) and
(I),
$${dV \over d\alpha_\prime}
   = \mu \alpha (z_\prime + A),\quad
  {dV \over d\beta_\prime}
   = \mu \beta (z_\prime + B),\quad
  {dV \over dz_\prime}
   = \mu (\alpha \alpha_\prime + \beta \beta_\prime + \gamma);
   \eqno {\rm (M)}$$
we have also $V = \mu z_\prime - \mu W^{(0)}$,
when $\alpha_\prime$,~$\beta_\prime$, vanish; and therefore,
$$V = \mu z_\prime - \mu W^{(0)}
      + \mu \int \{ (z_\prime + A) \alpha \, d\alpha_\prime
            + (z_\prime + B) \beta \, d\beta_\prime \},
   \eqno {\rm (N)}$$
$z_\prime$ being considered as constant in the integration, and
the integral being so determined as to vanish with
$\alpha_\prime$,~$\beta_\prime$.  Substituting in this expression
(N), the developments of $\alpha$,~$\beta$, and performing the
integration, we find the following development for
$\displaystyle {V \over \mu}$,
$$\eqalignno{
{V \over \mu}
   &= z_\prime - W^{(0)} + {\textstyle {1 \over 2}}
         \{ (z_\prime + A) \alpha_\prime^2
          + (z_\prime + B) \beta_\prime^2 \} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \phi_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle {d^n \over d\alpha_\prime^n}
                  \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+2}
            \over [n + 2]^{n+2} (z_\prime + A)^{n+1}}
          + {\displaystyle {d^n \over d\beta_\prime^n}
                  \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n+2}
            \over [n + 2]^{n+2} (z_\prime + B)^{n+1}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left(
                  {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n + 1}
                  \!
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n' + 1}
               \right)
            \over [n + 1]^{n+1} [n' + 1]^{n'+1}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}};
   &{\rm (O)}\cr}$$
which is another form for the integral of the partial
differential equation (A), obtained from the elimination (D).
And if we wish to introduce any other rectangular coordinates
$x$,~$y$,~$z$, into the expression of this integral (O), instead
of $x_\prime$,~$y_\prime$,~$z_\prime$, we may do so by the known
methods, by putting
$$\left. \eqalign{
x_\prime
   &=    (x - x_{\prime\prime}) \cos x x_\prime
       + (y - y_{\prime\prime}) \cos y x_\prime
       + (z - z_{\prime\prime}) \cos z x_\prime,\cr
y_\prime
   &=    (x - x_{\prime\prime}) \cos x y_\prime
       + (y - y_{\prime\prime}) \cos y y_\prime
       + (z - z_{\prime\prime}) \cos z y_\prime,\cr
z_\prime
   &=    (x - x_{\prime\prime}) \cos x z_\prime
       + (y - y_{\prime\prime}) \cos y z_\prime
       + (z - z_{\prime\prime}) \cos z z_\prime,\cr}
   \right\}
   \eqno {\rm (P)}$$
$x_{\prime\prime}$,~$y_{\prime\prime}$,~$z_{\prime\prime}$,
being the values of $x$,~$y$,~$z$, that belong to the point upon
the ray which had been taken for origin.

\bigbreak

\centerline{\it
Verifications of the foregoing Developments.}

\nobreak\bigskip

4.
We may verify the form (O) which we have thus found for the
integral of (A), by the folowing condition, resulting from (M),
$${d \over dz_\prime} \mathbin{.} {V \over \mu}
      - {\alpha_\prime \over z_\prime + A}
         {d \over d\alpha_\prime} \mathbin{.} {V \over \mu}
      - {\beta_\prime \over z_\prime + B}
         {d \over d\beta_\prime} \mathbin{.} {V \over \mu}
   = \sqrt{ 1 - \alpha^2 - \beta^2 },
   \eqno {\rm (Q)}$$
of which each member is an expression for the cosine~$\gamma$
of the small angle which a near ray makes with the ray that we
have taken for the axis of $z_\prime$.  The condition (Q) may be
put under the form
$${d \over dz_\prime} \mathbin{.} {V \over \mu}
   - (\alpha \alpha_\prime + \beta \beta_\prime)
   = \sqrt{ 1 - \alpha^2 - \beta^2 }
   \eqno {\rm (R)}$$
in which, by (O),
$$\eqalignno{
{d \over dz_\prime} \mathbin{.} {V \over \mu}
   &= 1 + {\alpha_\prime^2 + \beta_\prime^2 \over 2}
         + {d\phi_\prime \over dz_\prime} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \left(
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               {d^2 \phi_\prime \over d\alpha_\prime \, dz_\prime}
            \right)
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
          + {\displaystyle {d^n \over d\beta_\prime^n}
            \left(
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n + 1}
               {d^2 \phi_\prime \over d\beta_\prime \, dz_\prime}
            \right)
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle {d^n \over d\alpha_\prime^n}
               \mathbin{.}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 2}
            \over [n]^n (n + 2) (z_\prime + A)^{n+2}}
          + {\displaystyle {d^n \over d\beta_\prime^n}
               \mathbin{.}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n + 2}
            \over [n]^n (n + 2) (z_\prime + B)^{n+2}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         {\displaystyle
         {d^{n+n'+1} \over d\alpha_\prime^n \, d\beta_\prime^{n'} \, dz_\prime}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n' + 1}
         \right)
            \over [n + 1]^{n+1} [n' + 1]^{n'+1}
               (z_\prime + A)^{n+1} (z_\prime + B)^{n'+1}} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         \left(
            {z_\prime + A \over n + 1}
          + {z_\prime + B \over n' + 1}
         \right)
         {\displaystyle
         {d^{n+n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n' + 1}
         \right)
            \over [n]^n [n']^{n'}
               (z_\prime + A)^{n+2} (z_\prime + B)^{n'+2}};\cr
   & &{\rm (S)}\cr}$$
and, by (L),
$$\eqalignno{
\alpha \alpha_\prime + \beta \beta_\prime
   &= \alpha_\prime^2 + \beta_\prime^2
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle
            \alpha_\prime \mathbin{.}
            {d^n \over d\alpha_\prime^n}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
          + {\displaystyle
            \beta_\prime \mathbin{.}
            {d^n \over d\beta_\prime^n}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n + 1}
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \alpha_\prime
         \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         {\displaystyle
         {d^{n+n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^n
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n' + 1}
         \right)
            \over [n]^n [n' + 1]^{n'+1}
               (z_\prime + A)^{n+1} (z_\prime + B)^{n'+1}} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \beta_\prime
         \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         {\displaystyle
         {d^{n+n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n'}
         \right)
            \over [n + 1]^{n+1} [n']^{n'}
               (z_\prime + A)^{n+1} (z_\prime + B)^{n'+1}};
   &{\rm (T)}\cr}$$
while the development of
$$\sqrt{ 1 - \alpha^2 - \beta^2 }$$
may be deduced from the general formula (K) by changing
$$F(\alpha, \beta)
   \quad\hbox{to}\quad
   \gamma = \sqrt{ 1 - \alpha^2 - \beta^2 },$$
$$F_\prime
   \quad\hbox{to}\quad
   \gamma_\prime = \sqrt{ 1 - \alpha_\prime^2 - \beta_\prime^2 }.$$

To compare these several developments, and to examine whether they
satisfy the condition (R), we are to observe, that from the
nature of the function $\phi$, we have by the foregoing number,
$$\left. \vcenter{\halign{\hfil$\displaystyle #$\hfil\cr
{d \phi_\prime \over dz_\prime}
   = - (\gamma_\prime^{(4)} + \gamma_\prime^{(6)} + \hbox{\&c.})
   = \gamma_1 - 1 + {\alpha_\prime^2 + \beta_\prime^2 \over 2};\cr
   \noalign{\vskip 3pt}
{d^2 \phi_\prime \over d\alpha_\prime \, dz_\prime}
   = \alpha_\prime + {d\gamma_\prime \over d\alpha_\prime};\quad
{d^2 \phi_\prime \over d\beta_\prime \, dz_\prime}
   = \beta_\prime + {d\gamma_\prime \over d\beta_\prime};\quad
{d^3 \phi_\prime \over d\alpha_\prime \, d\beta_\prime \, dz_\prime}
   = {d^2 \gamma_\prime \over d\alpha_\prime \, d\beta_\prime};\cr}}
   \right\}
   \eqno {\rm (U)}$$
and therefore
$$\eqalign{
{d \over dz_\prime}
      \left(
         {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
         \left(
            {d\phi_\prime \over d\alpha_\prime}
         \right)^{n + 1}
         \left(
            {d\phi_\prime \over d\beta_\prime}
         \right)^{n' + 1}
      \right)
   &= {d^2 \gamma_\prime \over d\alpha_\prime \, d\beta_\prime}
         \left(
            {d\phi_\prime \over d\alpha_\prime}
         \right)^{n + 1}
         \left(
            {d\phi_\prime \over d\beta_\prime}
         \right)^{n' + 1} \cr
   &\mathrel{\phantom{=}} \mathord{}
      +  \left(
            \alpha_\prime + {d\gamma_\prime \over d\alpha_\prime}
         \right)
         \left(
            {d\phi_\prime \over d\beta_\prime}
         \right)^{n' + 1}
         {d \over d\beta_\prime} \mathbin{.}
         \left(
            {d\phi_\prime \over d\alpha_\prime}
         \right)^{n + 1} \cr
   &\mathrel{\phantom{=}} \mathord{}
      +  \left(
            \beta_\prime + {d\gamma_\prime \over d\beta_\prime}
         \right)
         \left(
            {d\phi_\prime \over d\alpha_\prime}
         \right)^{n + 1}
         {d \over d\alpha_\prime} \mathbin{.}
         \left(
            {d\phi_\prime \over d\beta_\prime}
         \right)^{n' + 1};\cr}$$
by which means the difference of the developments (S) and (T)
becomes
$$\eqalignno{
{d \over dz_\prime} \mathbin{.} {V \over \mu}
      - (\alpha \alpha_\prime + \beta \beta_\prime)
   &= \gamma_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle
            {d^n \over d\alpha_\prime^n}
            \left(
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               {d\gamma_\prime \over d\alpha_\prime}
            \right)
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
          + {\displaystyle
            {d^n \over d\beta_\prime^n}
            \left(
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n + 1}
               {d\gamma_\prime \over d\beta_\prime}
            \right)
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left\{
                  \vcenter{\halign{\hfil $\displaystyle #$\cr
                  {d^2 \gamma_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1} \cr
   \noalign{\vskip 3pt}
                + {d\gamma_\prime \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1}
                  {d \over d\beta_\prime} \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1} \cr
   \noalign{\vskip 3pt}
                + {d\gamma_\prime \over d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
                  {d \over d\alpha_\prime} \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1} \cr}}
               \right\}
            \over [n + 1]^{n+1} [n' + 1]^{n' + 1}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}},\cr
   & & {\rm (V)}\cr}$$
and the series in this second member being exactly that which
would result in the development of
$$\gamma = \sqrt{ 1 - \alpha^2 - \beta^2 },$$
from the formula (K), we see that the condition (Q) or (R) is
satisfied, and the sought verification is obtained.

Another verification of the foregoing developments may be
obtained by applying the general expression in series (K), for
any function~$F$ of the cosines $\alpha$,~$\beta$, to the case
where this function is
$\displaystyle = {d\phi \over d\alpha}$.
We find, first
$$\eqalignno{
{d\phi \over d\alpha}
   &= {d\phi_\prime \over d\alpha_\prime}
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle
            {d^n \over d\alpha_\prime^n}
            \left(
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               {d^2 \phi_\prime \over d\alpha_\prime^2}
            \right)
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
          + {\displaystyle
            {d^n \over d\beta_\prime^n}
            \left(
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n + 1}
               {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
            \right)
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left\{
                  \vcenter{\halign{\hfil $\displaystyle #$\cr
                  {d^3 \phi_\prime \over d\alpha_\prime^2 \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1} \cr
   \noalign{\vskip 3pt}
                + {d^2 \phi_\prime \over d\alpha_\prime^2}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1}
                  {d \over d\beta_\prime} \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1} \cr
   \noalign{\vskip 3pt}
                + {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
                  {d \over d\alpha_\prime} \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n'+1} \cr}}
               \right\}
            \over [n + 1]^{n+1} [n' + 1]^{n' + 1}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}},\cr
   & & {\rm (W)}\cr}$$
which may be put under the form
$${d\phi \over d\alpha}
   = \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle
            {d^n \over d\alpha_\prime^n}
            \mathbin{.}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
            \over [n + 1]^{n+1} (z_\prime + A)^n}
      + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         {\displaystyle
         {d^{n+n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^n
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n' + 1}
         \right)
            \over [n]^n [n' + 1]^{n'+1}
               (z_\prime + A)^n (z_\prime + B)^{n'+1}},
   \eqno {\rm (X)}$$
that is, by (L),
$${d\phi \over d\alpha} = (z_\prime + A) (\alpha - \alpha_\prime),
   \eqno {\rm (Y)}$$
which agrees with the conditions (H).  A similar verification may
be obtained by the same conditions (H), by considering the
development of
$\displaystyle {d\phi \over d\beta}$.

Finally, we may observe that the condition
$${V \over \mu}
   = \alpha x_\prime + \beta y_\prime + \gamma z_\prime - {W \over \mu}
   = \alpha x_\prime + \beta y_\prime - {U \over \mu}
   \eqno {\rm (Z)}$$
becomes, by (G) and (I),
$${V \over \mu}
   = z_\prime - W^{(0)}
      + (z_\prime + A)
         \left(
            \alpha \alpha_\prime - {\alpha^2 \over 2}
         \right)
      + (z_\prime + B)
         \left(
            \beta \beta_\prime - {\beta^2 \over 2}
         \right)
      + \phi;
   \eqno {\rm (A')}$$
in which, by (K) and (L),
$$\eqalign{
\alpha \alpha_\prime - {\alpha^2 \over 2}
   &= {\alpha_\prime^2 \over 2}
      - \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle
            {n + 1 \over n + 2}
            {d^n \over d\alpha_\prime^n}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 2}
            \over [n + 1]^{n+1} (z_\prime + A)^{n + 2}} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         {\displaystyle
         {d^{n+n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n' + 1}
         \right)
            \over [n]^n [n' + 1]^{n'+1}
               (z_\prime + A)^{n+2} (z_\prime + B)^{n'+1}},\cr
\beta \beta_\prime - {\beta^2 \over 2}
   &= {\beta_\prime^2 \over 2}
      - \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle
            {n + 1 \over n + 2}
            {d^n \over d\beta_\prime^n}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n + 2}
            \over [n + 1]^{n+1} (z_\prime + B)^{n + 2}} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
         {\displaystyle
         {d^{n+n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
         \left(
            {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
               \left(
                  {d\phi_\prime \over d\alpha_\prime}
               \right)^{n + 1}
               \left(
                  {d\phi_\prime \over d\beta_\prime}
               \right)^{n' + 1}
         \right)
            \over [n + 1]^{n+1} [n']^{n'}
               (z_\prime + A)^{n+1} (z_\prime + B)^{n'+2}},\cr}$$
$$\eqalignno{
\phi
   &= \phi_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
         \left\{
            {\displaystyle {d^n \over d\alpha_\prime^n}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+2}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
          + {\displaystyle {d^n \over d\beta_\prime^n}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n+2}
            \over [n + 1]^{n+1} (z_\prime + B)^{n+1}}
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty \,
            {\displaystyle
            (n + n' + 3)
               {d^{n + n'} \over d\alpha_\prime^n \, d\beta_\prime^{n'}}
               \left(
                  {d^2 \phi_\prime \over d\alpha_\prime \, d\beta_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n + 1}
                  \left(
                     {d\phi_\prime \over d\beta_\prime}
                  \right)^{n' + 1}
               \right)
            \over [n + 1]^{n+1} [n' + 1]^{n'+1}
               (z_\prime + A)^{n+1}
               (z_\prime + B)^{n'+1}};
   &{\rm (B')}\cr}$$
so that we are conducted by this other method to the same
expression (O) for the characteristic function of an ordinary
optical system, as that which we before obtained by performing the
integrations (N).  In all these expressions the sign
$\sum\nolimits_{(n,n')}{}_{0,}^{\infty,} {}_0^\infty$
denotes a summation with reference to the variable integers $n$,
$n'$, from zero to infinity.

\bigbreak

\centerline{\it
Case of a Plane System.}

\nobreak\bigskip

5.
A similar analysis may be applied to integrate the partial
differential equation
$$\left( {dV \over dx} \right)^2 + \left( {dV \over dz} \right)^2
   = \mu^2,
   \eqno {\rm (C')}$$
to which the equation (A) of this Supplement reduces itself, when
we consider a system of rays of ordinary light, contained in the
plane of $xz$.  In this case, if we put
$$\left. \eqalign{
x_\prime
   &=    (x - x_{\prime\prime}) \cos x x_\prime
       + (z - z_{\prime\prime}) \cos z x_\prime,\cr
z_\prime
   &=    (x - x_{\prime\prime}) \cos x z_\prime
       + (z - z_{\prime\prime}) \cos z z_\prime,\cr}
   \right\}
   \eqno {\rm (D')}$$
we may suppose $x_\prime$~$z_\prime$ to be new rectangular
coordinates, in the same plane as $x$~$z$, and such that the
axis of $z_\prime$ coincides with the direction of some given ray
of the system: and we may denote by $\alpha$,~$\gamma$, the
cosines of the angles which any near ray makes with these new
axes, so that
$$\gamma = \sqrt{ 1 - \alpha^2 }.$$
We shall then have for one form of the integral of the partial
differential equation (C${}'$), the following combination of
equations:
$$\mu x_\prime = {dU \over d\alpha},\quad
  V + U = \alpha {dU \over d\alpha},
   \eqno {\rm (E')}$$
between which $\alpha$ is conceived to be eliminated, and in
which
$$\left. \eqalign{
U   = W - \mu \gamma z_\prime
   &= \mu W^{(0)} - \mu z_\prime
      + {\mu (z_\prime + A) \alpha^2 \over 2} - \mu \phi;\cr
- \phi
   &= \sum\nolimits_{(i)}{}_0^\infty \,
         ( W^{(i+3)} + z_\prime \gamma^{(2i+4)} );\cr
W^{(i+3)}
   &= \alpha^{i+3} \mathbin{.} w_{i+3};\quad
\gamma^{(2i+4)}
    = {1 \mathbin{.} 3 \mathbin{.} 5 \, \ldots \, (2i + 1)
         \over 2 \mathbin{.} 4 \mathbin{.} 6 \, \ldots \, (2i + 2)}
      \mathbin{.} {\alpha^{2i+4} \over 2i + 4};\cr}
   \right\}
   \eqno {\rm (F')}$$
$W^{(0)}$, $w_{i+3}$, being constant coefficients in the
development of the function~$W$, according to the powers of
$\alpha$, and $A$ being another constant in that development,
namely, the distance upon the given ray, from the point where it
touches the caustic curve of the plane system, to the origin of
$x_\prime$ and $z_\prime$.  The first of the two equations
(E${}'$) becomes
$$\alpha = \alpha_\prime
      + {1 \over z_\prime + A} {d\phi \over d\alpha},$$
when we put
$$\alpha_\prime = {x_\prime \over z_\prime + A};$$
and gives therefore, by the well-known theorem of Lagrange, for
functions of a single variable,
$$F(\alpha) = F_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \mathbin{.}
            {dF_\prime \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}},
   \eqno {\rm (G')}$$
$F(\alpha)$ denoting any function of $\alpha$, which admits of
being developed according to positive integer powers of
$\alpha_\prime$, and $F_\prime$, $\phi_\prime$, being formed from
$F$, $\phi$, by changing $\alpha$ to $\alpha_\prime$.  The
cosines $\alpha$,~$\gamma$, may therefore be thus developed,
$$\left. \eqalign{
\alpha
   &= \alpha_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}} \cr
\gamma
   &= \gamma_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \mathbin{.}
            {d\gamma_\prime \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}} \cr}
   \right\}
   \eqno {\rm (H')}$$
if we put
$$\gamma_\prime = \sqrt{ 1 - \alpha_\prime^2 }.$$
And since $V$ may be thus expressed,
$$V = \mu z_\prime - \mu W^{(0)}
      + \mu (z_\prime + A) \int_0^{\alpha_\prime} \alpha \, d\alpha_\prime,
   \eqno {\rm (I')}$$
because
$$dV = \mu (\alpha \, dx_\prime + \gamma \, dz_\prime)
   = \mu \alpha (z_\prime + A) \, d\alpha_\prime
      + \mu (\alpha \alpha_\prime + \gamma) \, dz_\prime,$$
and because $V$ becomes $\mu z_\prime - \mu W^{(0)}$ when
$\alpha_\prime = 0$, we find, finally,
$${V \over \mu}
   = z_\prime - W^{(0)} + {(z_\prime + A) \alpha_\prime^2 \over 2}
      + \phi_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+2}
            \over [n + 2]^{n+2} (z_\prime + A)^{n+1}}.
   \eqno {\rm (K')}$$

This form (K${}'$) for the integral of the partial differential
equation (C${}'$), may be verified by observing that it satisfies
the condition
$${1 \over \mu} {dV \over dz_\prime}
   = \alpha \alpha_\prime + \gamma,
   \eqno {\rm (L')}$$
$V$ being differentiated for $z_\prime$ and $\alpha_\prime$ as
two independent variables; because
$${d\phi_\prime \over dz_\prime}
   = - \sum\nolimits_{(i)}{}_0^\infty \gamma_\prime^{(2i+4)}
   = \gamma_\prime - 1 - {\alpha_\prime^2 \over 2},\quad
  {d^2 \phi_\prime \over d\alpha_\prime \, dz_\prime}
   = {d\gamma_\prime \over d\alpha_\prime} + \alpha_\prime,$$
$${d^{n+1} \over d\alpha_\prime^n \, dz_\prime} \mathbin{.}
      \left( {d\phi_\prime \over d\alpha_\prime} \right)^{n+2}
   = (n + 2) {d^n \over d\alpha_\prime^n}
      \left\{
         \left( {d\phi_\prime \over d\alpha_\prime} \right)^{n+1}
         \left(
            {d\gamma_\prime \over d\alpha_\prime} + \alpha_\prime
         \right)
      \right\},$$
$${d^{n+1} \over d\alpha_\prime^{n+1}} \mathbin{.}
      \alpha_\prime
      \left( {d\phi_\prime \over d\alpha_\prime} \right)^{n+2}
    = \alpha_\prime {d^{n+1} \over d\alpha_\prime^{n+1}} \mathbin{.}
         \left( {d\phi_\prime \over d\alpha_\prime} \right)^{n+2}
      + (n + 1) {d^n \over d\alpha_\prime^n} \mathbin{.}
         \left( {d\phi_\prime \over d\alpha_\prime} \right)^{n+2},
   \eqno {\rm (M')}$$
and therefore, differentiating (K') as if $\alpha_\prime$ were
constant,
$${1 \over \mu} {dV \over dz_\prime}
   = \gamma_\prime + \alpha_\prime^2
      + \alpha_\prime
        \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \mathbin{.}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}}
      + \sum\nolimits_{(n)}{}_0^\infty \,
            {\displaystyle {d^n \over d\alpha_\prime^n}
            \mathbin{.}
            {d\gamma_\prime \over d\alpha_\prime}
                  \left(
                     {d\phi_\prime \over d\alpha_\prime}
                  \right)^{n+1}
            \over [n + 1]^{n+1} (z_\prime + A)^{n+1}},
   \eqno {\rm (N')}$$
that is, by (H${}'$)
$${1 \over \mu} {dV \over dz_\prime} = \alpha \alpha_\prime + \gamma.$$

\bigbreak

\centerline{\it
Case of a System of Revolution.}

\nobreak\bigskip

6.
Another particular case of the partial differential equation (A)
deserves to be considered specially; namely the case of systems
of revolution, symmetric about some single ray.  In this case, if
we take for the axis of $z$, the ray which is the axis of the
system, $V$ will be a function of $z$ and of $x^2 + y^2$; and if
we put
$$x^2 + y^2 = \eta,
   \eqno {\rm (O')}$$
we may in general suppose $V$ developed according to positive
integer powers of $\eta$, in a series satisfying the condition,
$$\left( {dV \over dz} \right)^2 + 4 \eta \left( {dV \over d\eta} \right)^2
   = \mu^2.
   \eqno {\rm (P')}$$

To integrate this partial differential equation (P${}'$), which
is a particular case of (A), we may employ the corresponding case
of the general system of equations (D), (E), putting for
abridgment
$$\alpha^2 + \beta^2 = \epsilon,
   \eqno {\rm (Q')}$$
and considering the quantities $W$, $U$, as functions of
$\epsilon$, which we shall suppose capable of being developed
according to positive integer powers of that variable.  In this
manner we shall obtain
$${dU \over d\alpha} = 2 \alpha {dU \over d\epsilon},\quad
  {dU \over d\beta}  = 2 \beta  {dU \over d\epsilon},
   \eqno {\rm (R')}$$
and therefore by (D),
$$\mu^2 \eta = 4 \epsilon \left( {dU \over d\epsilon} \right)^2,\quad
  V + U = 2\epsilon {dU \over d\epsilon}.
   \eqno {\rm (S')}$$

We have also by (E),
$$U = W - \mu z \sqrt{1 - \epsilon},\quad
  {dU \over d\epsilon} = {dW \over d\epsilon} + {\mu z \over 2\gamma},
   \eqno {\rm (T')}$$
in which
$$\gamma = \sqrt{1 - \epsilon};$$
and we may put
$$\left. \eqalign{
W  &= \mu W^{(0)} + {\mu A \epsilon \over 2}
      + \mu \sum\nolimits_{(i)}{}_0^\infty \,
            \epsilon^{i + 2} w_{2i + 4};\cr
{dW \over d \epsilon}
   &= {\mu A \over 2}
      + \mu \sum\nolimits_{(i)}{}_0^\infty \,
            (i + 2) \epsilon^{i + 1} w_{2i + 4};\cr
{dU \over d\epsilon}
   &= {\mu (z + A) \over 2} - \mu \psi;\quad
- \psi = \sum\nolimits_{(i)}{}_0^\infty U_{i+1} \epsilon^{i+1};\cr
U_{i+1}
   &= (i + 2) w_{2i + 4}
      + {1 \mathbin{.} 3 \mathbin{.} 5 \, \ldots \, (2i + 1)
         \over 2 \mathbin{.} 4 \mathbin{.} 6 \, \ldots \, (2i + 2)}
      {z \over 2};\cr}
   \right\}
   \eqno {\rm (U')}$$
in which $W^{(0)}$, $w_{2i+4}$, $A$, are constants of the same
kind as before, $A$ denoting the distance of the origin from the
focus of central rays.  Hence, if we put for abridgment,
$$\epsilon_\prime = {\eta \over (z + A)^2}
   = {x^2 + y^2 \over (z + A)^2},
   \eqno {\rm (V')}$$
$\epsilon_\prime$ is an approximate value of $\epsilon$, and we
have the following relation between $\epsilon$ and
$\epsilon_\prime$,
$$\epsilon
   = \epsilon_\prime
      + {4  \epsilon \psi \over z + A}
      - {4  \epsilon \psi^2 \over (z + A)^2},
   \eqno {\rm (W')}$$
which gives, by the theorems before referred to,
$$f(\epsilon) = f_\prime
      + \sum\nolimits_{(n)}{}_0^\infty \,
         {\displaystyle
         4^{n+1}
         {d^n \over d\epsilon_\prime^n}
         \left\{
            {df_\prime \over d\epsilon_\prime}
            \left(
               \epsilon_\prime \psi_\prime
             - {\epsilon_\prime \psi_\prime^2 \over z + A}
            \right)^{n+1}
         \right\}
         \over [n + 1]^{n+1} (z + A)^{n+1}},
   \eqno {\rm (X')}$$
$f(\epsilon)$ being a function of $\epsilon$ and $f_\prime$,
$\psi_\prime$, being formed from $f$, $\psi$, by changing
$\epsilon$ to $\epsilon_\prime$.  We have also, by (S${}'$)
(T${}'$) (U${}'$),
$$\eqalignno{
{V \over \mu}
   &= z - W^{(0)} + {1 \over \mu} \int_0^\epsilon
         \left(
            {dU \over d\epsilon}
          + 2 \epsilon {d^2 U \over d\epsilon^2}
         \right)
         \, d\epsilon \cr
   &= z - W^{(0)} + {(z + A) \epsilon \over 2} - 2 \epsilon \psi
         + \int_0^\epsilon \psi \, d\epsilon,
   &{\rm (Y')}\cr}$$
and therefore, by (X${}'$),
$$\eqalignno{
{V \over \mu}
   &= z - W^{(0)} + {(z + A) \epsilon_\prime \over 2}
         - 2 \epsilon_\prime \psi_\prime
         + \int_0^{\epsilon_\prime} \psi_\prime \, d\epsilon_\prime \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \sum\nolimits_{(n)}{}_0^\infty \,
         {4^{n+1} \over [n + 1]^{n+1} (z + A)^{n+1}}
         {d^n \over d\epsilon_\prime^n}
            \left\{
               \left(
                  {z + A \over 2}
                - 2 \epsilon_\prime {d\psi_\prime \over d\epsilon_\prime}
                - \psi_\prime
               \right)
               \left(
                  \epsilon_\prime \psi_\prime
                - {\epsilon_\prime \psi_\prime^2 \over z + A}
               \right)^{n+1}
            \right\};\cr
   & &{\rm (Z')}\cr}$$
in which,
$$\psi_\prime
   = - \sum\nolimits_{(i)}{}_0^\infty \,
         U_{i+1} \epsilon_\prime^{i+1};\quad
  {d\psi_\prime \over d\epsilon_\prime}
   = - \sum\nolimits_{(i)}{}_0^\infty \,
         (i + 1) U_{i+1} \epsilon_\prime^i;$$
$$\int_0^{\epsilon_\prime} \psi_\prime \, d\epsilon_\prime
   = - \sum\nolimits_{(i)}{}_0^\infty \,
         U_{i+1} \mathbin{.} {\epsilon_\prime^{i+2} \over i + 2};\quad
  \epsilon_\prime = {\eta \over (z + A)^2}.$$

The development (Z${}'$) is one form of the integral of the
partial differential equation (P${}'$); another form of the same
integral may be obtained from the expression (K${}'$) for the
characteristic function of a plane system, by changing
$\alpha_\prime$ to
$\displaystyle {\sqrt{\eta} \over z + A}$,
and $z_\prime$ to $z$, and supposing $w_{2i+3} = 0$, and is,
$${V \over \mu} 
   = z - W^{(0)} + {\eta \over 2 (z + A)} + \phi_\prime
      + 4 \sum\nolimits_{(n)}{}_0^\infty \,
            {2^n (z + A)^{n+1} \over [n + 2]^{n+2}}
            {d^n \over (d \surd \eta)^n}
            \left(
               \eta^{1 + {n \over 2}}
               \left( {d\phi_\prime \over d\eta} \right)^{n+2}
            \right),
   \eqno {\rm (A'')}$$
in which
$$\phi_\prime
   = - \sum\nolimits_{(i)}{}_0^\infty \,
         {\eta^{i+2} \over (z + A)^{2i+4}}
         \left(
            w_{2i+4}
          + {1 \mathbin{.} 3 \mathbin{.} 5 \, \ldots \, (2i + 1)
            \over 2 \mathbin{.} 4 \mathbin{.} 6 \, \ldots \, (2i + 2)}
            {z \over 2i + 4}
         \right).
   \eqno {\rm (B'')}$$

Each of these forms gives, when we neglect $\eta^4$, the
following approximate expression for the characteristic
function~$V$ of a system of ordinary rays, symmetric about the
axis of $z$,
$${V \over \mu} 
   = z - W^{(0)} + {\eta \over 2 (z + A)}
      - {(z + 8 w_4) \eta^2 \over 8 (z + A)^4}
      - {(z + 16 w_6) \eta^3 \over 16 (z + A)^6}
      + {(z + 8 w_4)^2 \eta^3 \over 8 (z + A)^7};
   \eqno {\rm (C'')}$$
in which $\eta = x^2 + y^2$, and $W^{(0)}$, $A$, $w_4$, $w_6$,
are constants in the development of the connected function~$W$,
such that when we neglect the eighth power of the sine of the
angle contained between a near ray, and the axis of revolution of
the system, we have
$${W \over \mu}
   = W^{(0)} + {A (\alpha^2 + \beta^2) \over 2}
      + w_4 (\alpha^2 + \beta^2)^2
      + w_6 (\alpha^2 + \beta^2)^3,
   \eqno {\rm (D'')}$$
$\alpha$,~$\beta$, being, as before, the cosines of the angles
that the near ray makes with the axes of $x$ and $y$, to which it
is nearly perpendicular.

\bigbreak

\centerline{\it
Verification of the Approximate Integral for Systems of
Revolution.}

\nobreak\bigskip

7.
The approximate expression (C${}''$) for the characteristic
function of an optical system of revolution, admits of extensive
applications: it is therefore useful to consider other methods,
by which it may be obtained or verified.  An immediate
verification may be derived from the partial differential
equation (P${}'$) of which (C${}''$) ought to be an approximate
integral; namely, by computing from (C${}''$) the approximate
expressions of
$${1 \over \mu^2} \left( {dV \over dz} \right)^2
   \quad\hbox{and}\quad
  {4\eta \over \mu^2} \left( {dV \over d\eta} \right)^2,$$
and trying whether their sum is unity, when $\eta^4$ is
neglected.  Putting for this purpose the expression (C${}''$)
under the form
$$\eqalignno{
{V \over \mu} 
   &= z - W^{(0)}
      + {\eta \over 2 (z + A)}
      - {\eta^2 \over 8 (z + A)^3}
      + {\eta^3 \over 16 (z + A)^5} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {(A - 8 w_4) \eta^2 \over 8 (z + A)^4}
      + {(32 w_4 - 16 w_6 - 3A) \eta^3 \over 16 (z + A)^6}
      + {(A - 8 w_4)^2 \eta^3 \over 8 (z + A)^7},
   &{\rm (E'')}\cr}$$
we find, by differentiation,
$$\left. \eqalign{
{1 \over \mu} {dV \over dz} 
   &= 1
      - {\eta \over 2 (z + A)^2}
      + {3\eta^2 \over 8 (z + A)^4}
      - {5\eta^3 \over 16 (z + A)^6}
      - {(A - 8 w_4) \eta^2 \over 2 (z + A)^5} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - {3 (32 w_4 - 16 w_6 - 3A) \eta^3 \over 8 (z + A)^7}
      - {7 (A - 8 w_4)^2 \eta^3 \over 8 (z + A)^8};\cr
{2 \over \mu} {dV \over d\eta} 
   &=   {1 \over z + A}
      - {\eta \over 2 (z + A)^3}
      + {3 \eta^2 \over 8 (z + A)^5}
      + {(A - 8 w_4) \eta \over 2 (z + A)^4} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {3 (32 w_4 - 16 w_6 - 3A) \eta^2 \over 8 (z + A)^6}
      + {3 (A - 8 w_4)^2 \eta^2 \over 4 (z + A)^7};\cr}
   \right\}
   \eqno {\rm (F'')}$$
and therefore, neglecting $\eta^4$,
$$\left. \eqalign{
{1 \over \mu^2} \left( {dV \over dz} \right)^2
   &= 1
      - {\eta \over (z + A)^2}
      + {\eta^2 \over (z + A)^4}
      - {\eta^3 \over (z + A)^6}
      - {(A - 8 w_4) \eta^2 \over (z + A)^5} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {(11 A - 112 w_4 + 48 w_6)\eta^3 \over 4 (z + A)^7}
      - {7 (A - 8 w_4)^2 \eta^3 \over 4 (z + A)^8};\cr
{4\eta \over \mu^2} \left( {dV \over d\eta} \right)^2
   &=   {\eta \over (z + A)^2}
      - {\eta^2 \over (z + A)^4}
      + {\eta^3 \over (z + A)^6}
      + {(A - 8 w_4) \eta^2 \over (z + A)^5} \cr
   &\mathrel{\phantom{=}} \mathord{}
      - {(11 A - 112 w_4 + 48 w_6)\eta^3 \over 4 (z + A)^7}
      + {7 (A - 8 w_4)^2 \eta^3 \over 4 (z + A)^8};\cr}
   \right\}
   \eqno {\rm (G'')}$$
expressions of which the sum is unity, as it ought to be.  We may
remark that the former of these two expressions represents the
square of the cosine, and the latter the square of the sine, of
the angle which a near ray makes with the axis of revolution of
the system.

\bigbreak

\centerline{\it
Other Method of obtaining the Approximate Integral.}

\nobreak\bigskip

8.
Again, the approximate integral (C${}''$) of the partial
differential equation (P${}'$), may be obtained in the following
manner.  Since $V$ is supposed capable of being developed
according to positive integer powers of $\eta$, let us assume
$${V \over \mu}
   = V^{(0)} + V^{(1)} \eta + V^{(2)} \eta^2 + V^{(3)} \eta^3,
   \eqno {\rm (H'')}$$
neglecting $\eta^4$, and considering $V^{(0)}$, $V^{(1)}$,
$V^{(2)}$, $V^{(3)}$, as functions of $z$, of which the forms are
to be determined.  To determine these forms, we have, when
$\eta = 0$,
$$\left. \multieqalign{
V                     &=   \mu V^{(0)} &
{dV \over d\eta}      &=   \mu V^{(1)}; &
{d^2 V \over d\eta^2} &= 2 \mu V^{(2)}; &
{d^3 V \over d\eta^3} &= 6 \mu V^{(3)}; \cr
{dV \over dz}               &=   \mu {dV^{(0)} \over dz} &
{d^2 V \over d\eta \, dz}   &=   \mu {dV^{(1)} \over dz}; &
{d^3 V \over d\eta^2 \, dz} &= 2 \mu {dV^{(2)} \over dz}; &
{d^4 V \over d\eta^3 \, dz} &= 6 \mu {dV^{(3)} \over dz}. \cr}
   \right\}
   \eqno {\rm (I'')}$$

The equation (P${}'$) shews that
$\displaystyle {dV \over dz} = \pm \mu$,
when $\eta = 0$; and
$\displaystyle {dV \over dz}$
is positive, if we suppose the motion of the light directed from
the negative towards the positive part of the axis of $z$; we
have therefore, by (I${}''$),
$${dV^{(0)} \over dz} = 1.
   \eqno {\rm (K'')}$$

The equation (P${}'$) gives also, by differentiating it with
respect to $\eta$,
$$\left. \eqalign{
0  &=   {dV \over dz} {d^2 V \over d\eta \, dz}
      + 4\eta {dV \over d\eta} {d^2 V \over d\eta^2}
      + 2 \left( {dV \over d\eta} \right)^2;\cr
0  &=   {dV \over dz} {d^3 V \over d\eta^2 \, dz}
      + \left( {d^2 V \over d\eta \, dz} \right)^2
      + 4\eta {dV \over d\eta} {d^3 V \over d\eta^3}
      + 4\eta \left( {d^2 V \over d\eta^2} \right)^2
      + 8 {dV \over d\eta} {d^2 V \over d\eta^2};\cr
0  &=   {dV \over dz} {d^4 V \over d\eta^3 \, dz}
      + 3 {d^2 V \over d\eta \, dz} {d^3 V \over d\eta^2 \, dz}
      + 4\eta {dV \over d\eta} {d^4 V \over d\eta^4}
      + 12\eta {d^2 V \over d\eta^2} {d^3 V \over d\eta^3}
      + 12 {dV \over d\eta} {d^3 V \over d\eta^3}
      + 12 \left( {d^2 V \over d\eta^2} \right)^2;\cr}
   \right\}
   \eqno {\rm (L'')}$$
and, making $\eta = 0$, we find by (I${}''$) the following
equations in ordinary differentials, from which $V^{(1)}$,
$V^{(2)}$, $V^{(3)}$, are to be deduced:
$$\left. \eqalign{
0  &= {dV^{(1)} \over dz} + 2 V^{(1)2};\cr
0  &= {dV^{(2)} \over dz} + 8 V^{(1)} V^{(2)}
      + {\textstyle {1 \over 2}}
         \left( {dV^{(1)} \over dz} \right)^2;\cr
0  &= {dV^{(3)} \over dz} + 12 V^{(1)} V^{(3)}
      + {dV^{(1)} \over dz} {dV^{(2)} \over dz} + 8 V^{(2)2}.\cr}
   \right\}
   \eqno {\rm (M'')}$$

These three differential equations, when divided respectively by
$V^{(1)2}$, $V^{(1)4}$, $V^{(1)6}$, can easily be integrated, and
give, when combined with the integral of (K${}''$),
$$\left. \eqalign{
v_0   &= V^{(0)} - z;\cr
v_1   &= {1 \over V^{(1)}} - 2z;\cr
v_2   &= {V^{(2)} \over V^{(1)4}} + 2z;\cr
v_3   &= {V^{(3)} \over V^{(1)6}} - 4z
         - {4(v_1 + v_2)^2 \over 2z + v_1};\cr}
   \right\}
   \eqno {\rm (N'')}$$
$v_0$, $v_1$, $v_2$, $v_3$, being the four arbitrary constants
introduced by the four integrations.  The functions $V^{(0)}$,
$V^{(1)}$, $V^{(2)}$, $V^{(3)}$, are therefore of the form
$$\left. \eqalign{
   &V^{(0)} = z + v_0;\quad
    V^{(1)} = {1 \over 2z + v_1};\quad
    V^{(2)} = {v_2 - 2z \over (2z + v_1)^4};\cr
   &V^{(3)} = {4z + v_3 \over (2z + v_1)^6}
         + {4(v_1 + v_2)^2 \over (2z + v_1)^7};\cr}
   \right\}
   \eqno {\rm (O'')}$$
and these forms for the coefficients of the development
(H${}''$), agree perfectly with the development (C${}''$) or
(E${}''$), when we establish the following relations between the
constants:
$$v_0 = - W^{(0)};\quad
  v_1 = 2A;\quad
  v_2 = -16 w_4;\quad
  v_3 = 8 (16 w_4 - 8 w_6 - A):
   \eqno {\rm (P'')}$$
we see, therefore, that the present method of integration
confirms the former results.

\bigbreak

\centerline{\it
Connexion of the Longitudinal Aberration, in a System of
Revolution, with the}
\centerline{\it Development of the Characteristic Function~$V$.}

\nobreak\bigskip

9.
To give now an example of the optical use of the development
which has been thus obtained, let us consider its connexion with
the aberrations of the near rays, from the principal or central
focus.  We have already remarked that the constant~$A$ denotes
the distance of the origin of coordinates, upon the central ray,
beyond this principal focus, in such a manner that the focal
ordinate is $= -A$.  For the ordinate~$Z$, of intersection of any
near ray with the central ray, we have by the fourth of the
equations (C), of the present Supplement,
$$Z = {1 \over \mu (\alpha^2 + \beta^2)} {\delta W \over \delta \gamma},
   \eqno {\rm (Q'')}$$
if we form the coefficient of
$\displaystyle {\delta W \over \delta \gamma}$
by putting $W$ under the form of a homogeneous function of
$\alpha$,~$\beta$,~$\gamma$, of the dimension zero, with the help
of the relation
$\alpha^2 +\beta^2 + \gamma^2 = 1$,
and then by differentiating this function, as if $\alpha$,
$\beta$, were constant, and $\gamma$ the only variable.
Employing therefore for
$\displaystyle {W \over \mu}$
the development
$${W \over \mu} = W^{(0)}
      + {A (\alpha^2 + \beta^2) \over 2(\alpha^2 + \beta^2 + \gamma^2)}
      + \sum\nolimits_{(i)}{}_0^\infty \,
            {w_{2i+4} (\alpha^2 + \beta^2)^{i+2}
               \over (\alpha^2 + \beta^2 + \gamma^2)^{i+2}},
   \eqno {\rm (R'')}$$
which is of the homogeneous form required, and, after
differentiating for $\gamma$, making
$\alpha^2 +\beta^2 + \gamma^2 = 1$,
we find for the ordinate~$Z$,
$$Z = -A + A(1 - \gamma)
      - \gamma \sum\nolimits_{(i)}{}_0^\infty \,
            (2i + 4) w_{2i+4} (\alpha^2 + \beta^2)^{i+1},
   \eqno {\rm (S'')}$$
a series of which the term $-A$ being the ordinate of the central
focus, the remainder is the longitudinal aberration: $\gamma$ is
the cosine of the angle which the near ray makes with the central
ray, and $\alpha^2 + \beta^2$ is the square of the sine of that
angle.  If therefore we denote the aberration $Z + A$ by
$\Lambda$, we may develope $\Lambda$ in a series of the form
$$\Lambda
   = L (\alpha^2 + \beta^2) + L_1 (\alpha^2 + \beta^2)^2 + \hbox{\&c.},
   \eqno {\rm (T'')}$$
in which
$$L = {\textstyle {1 \over 2}} A - 4 w_4,\quad
  L_1 = {\textstyle {1 \over 8}} A + 2 w_4 - 6 w_6.
   \eqno {\rm (U'')}$$

And if by these relations (U${}''$), we eliminate $w_4$, $w_6$
from the approximate expression (E${}''$), we find the following
formula:
$$\eqalignno{
V  &= \mu (z - W^{(0)})
      + {\mu \eta \over 2 (z + A)}
      + {\mu \eta^2 \over 4 (z + A)^3}
         \left( {L \over z + A} - {\textstyle {1 \over 2}} \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\mu \eta^3 \over 12 (z + A)^5}
         \left\{
            {2L_1 - 5L \over z + A}
          + {6 L^2 \over (z + A)^2}
          + {\textstyle {3 \over 4}}
         \right\},
   &{\rm (V'')}\cr}$$
which shews the connexion in a system of revolution between the
development of the longitudinal aberration $\Lambda$, and that of
the characteristic function~$V$.

\bigbreak

\centerline{\it
Changes of a System of Revolution, produced by Ordinary Refraction.}

\nobreak\bigskip

10.
Suppose now that the rays of this system of revolution fall upon
a refracting surface of revolution, having for axis the axis of
the system, and having for equation
$$z = z_0 + z_1 \eta+ z_2 \eta^2 + z_3 \eta^3 + \hbox{\&c.},
   \eqno {\rm (W'')}$$
in which $\eta$ is still $= x^2 + y^2 =$ the square of the
perpendicular distance of a point $x$~$y$~$z$, from the axis;
and let $\mu'$ be the refracting index of the new medium into
which the rays pass after refraction.  It is evident that in this
new medium, the rays will compose a new system of revolution,
symmetric about the same axis as before; and we may in general
suppose the characteristic function $V'$ of this new system,
which is analogous to $V$ of the old, developed in a series
similar to (V${}''$),
$$\eqalignno{
V' &= \mu' (z - W'^{(0)})
      + {\mu' \eta \over 2 (z + A')}
      + {\mu' \eta^2 \over 4 (z + A')^3}
         \left( {L' \over z + A'} - {\textstyle {1 \over 2}} \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + {\mu' \eta^3 \over 12 (z + A')^5}
         \left\{
            {2L'_1 - 5L' \over z + A'}
          + {6 L'^2 \over (z + A')^2}
          + {\textstyle {3 \over 4}}
         \right\}:
   &{\rm (X'')}\cr}$$
the constants $A'$~$L'$~$L'_1$ being similar to $A$~$L$~$L_1$,
in such a manner that the ordinate $Z'$ of intersection of
the axis with a near ray, is
$$Z' = - A' + L' (\alpha'^2 + \beta'^2)
         + L_1' (\alpha'^2 + \beta'^2)^2,
   \eqno {\rm (Y'')}$$
if $\alpha'^2 + \beta'^2$ denote the square of the sine of the
angle which the near ray makes with the axis, and if we neglect
the sixth power of this sine.  To connect the new and old
constants in the development of the characteristic function, we
have, by the nature of this function, and by the principles of my
former memoirs, the condition
$$0 = \Delta V = V' - V;
   \eqno {\rm (Z'')}$$
which is to be satisfied for all the points of the refracting
surface, and which may therefore be differentiated, considering
$\Delta V$ as a function of $z$,~$\eta$, namely the difference of
the developments (V${}''$) (X${}''$), and considering $z$ as
itself a function of $\eta$, assigned by the equation of the
refracting surface (W${}''$).  In this manner we find,
transposing the symbols $\Delta$,~$d$,
$$\left. \eqalign{
0  &= \Delta {dV \over d\eta}
      + {dz \over d\eta} \, \Delta {dV \over dz};\cr
0  &= \Delta {d^2 V \over d\eta^2}
      + 2 {dz \over d\eta} \, \Delta {d^2 V \over dz \, d\eta}
      + \left( {dz \over d\eta} \right)^2 \, \Delta {d^2 V \over dz^2}
      + {d^2 z \over d\eta^2} \, \Delta {dV \over dz};\cr
0   &= \Delta {d^3 V \over d\eta^3}
      + 3 {dz \over d\eta} \, \Delta {d^3 V \over dz \, d\eta^2}
      + 3 \left( {dz \over d\eta} \right)^2
            \, \Delta {d^3 V \over dz^2 \, d\eta}
      + \left( {dz \over d\eta} \right)^3
            \, \Delta {d^3 V \over dz^3} \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 3 {d^2 z \over d\eta^2}
            \, \Delta {d^2 V \over dz \, d\eta}
      + 3 {dz \over d\eta} {d^2 z \over d\eta^2}
            \, \Delta {d^2 V \over dz^2}
      + {d^3 z \over d\eta^3}
            \, \Delta {dV \over dz};\cr}
   \right\}
   \eqno {\rm (A''')}$$
and, making after the differentiations $\eta = 0$, we have
$$\left. \vcenter{\halign{\hfil$\displaystyle #$\hfil\cr
z = z_0;\quad
{dz \over d\eta} = z_1;\quad
{d^2 z \over d\eta^2} = 2 z_2;\quad
{d^3 z \over d\eta^3} = 6 z_3;\cr
   \noalign{\vskip 3pt}
\Delta {dV \over dz} = \Delta \mu;\quad
\Delta {d^2 V \over dz^2} = 0;\quad
\Delta {d^3 V \over dz^3} = 0;\cr
   \noalign{\vskip 3pt}
\Delta {dV \over d\eta}
   = {\textstyle {1 \over 2}} \Delta \mathbin{.} {\mu \over z + A};\quad
\Delta {d^2 V \over dz \, d\eta}
   = {\textstyle {1 \over 2}} \Delta \mathbin{.} {-\mu \over (z + A)^2};\quad
\Delta {d^3 V \over dz^2 \, d\eta}
   = \Delta \mathbin{.} {\mu \over (z + A)^3};\cr
   \noalign{\vskip 3pt}
\Delta {d^2 V \over d\eta^2}
   = \Delta
      \left\{
         {\mu L \over 2(z + A)^4}
       - {\mu \over 4(z + A)^3}
      \right\};\quad
\Delta {d^3 V \over dz \, d\eta^2}
   = \Delta
      \left\{
         {-2 \mu L \over (z + A)^5}
       + {3\mu \over 4(z + A)^4}
      \right\};\cr
   \noalign{\vskip 3pt}
\Delta {d^3 V \over d\eta^3}
   = \Delta \mathbin{.} {\mu \over (z + A)^5}
      \left(
         {2L_1 - 5L \over 2(z + A)}
       + {3 L^2 \over (z + A)^2}
       + {\textstyle {3 \over 8}}
      \right).\cr}}
   \right\}
   \eqno {\rm (B''')}$$
We have therefore, I${}^{\rm st}$, for the change of $-A$, the
ordinate of the central focus,
$$0 = \Delta {\mu \over z + A} + 2 z_1 \, \Delta \mu:
   \eqno {\rm (C''')}$$
II${}^{\rm nd}$, for the change of $L$, the first or principal
coefficient of aberration,
$$0 = \Delta
         \left\{
            {2\mu L \over (z + A)^4}
          - {\mu \over (z + A)^3}
         \right\}
      - 4 z_1 \, \Delta \mathbin{.} {\mu \over (z + A)^2}
      + 8 z_2 \, \Delta \mu:
   \eqno {\rm (D''')}$$
III${}^{\rm rd}$, for the change of $L_1$, the coefficient of the
fourth power of the sine of the angular aberration, in the
expression of the longitudinal,
$$\eqalignno{
0  &= \Delta \mathbin{.} {\mu \over (z + A)^5}
         \left(
            {2L_1 - 5L \over 2(z + A)}
          + {3 L^2 \over (z + A)^2}
          + {\textstyle {3 \over 8}}
         \right)
      + 3 z_1 \, \Delta \mathbin{.} {\mu \over (z + A)^4}
         \left(
            {-2L \over z + A} + {\textstyle {3 \over 4}}
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
      + 3 z_1^2 \, \Delta \mathbin{.} {\mu \over (z + A)^3}
      - 3 z_2 \, \Delta \mathbin{.} {\mu \over (z + A)^2}
      + 6 z_3 \, \Delta \mu;
   &{\rm (E''')}\cr}$$
$z$ being the ordinate of the point of central incidence.  With
respect to the present meaning of the sign~$\Delta$, we may
remark, that the first of the three equations (C${}'''$)
(D${}'''$) (E${}'''$) is equivalent to the following:
$${\mu \over z + A} + 2 z_1 \mu
   = {\mu' \over z + A'} + 2 z_1 \mu';
   \eqno {\rm (F''')}$$
and the two others are to be similarly interpreted.

\bigbreak

\centerline{\it
Example; Spheric Refraction; Mr.~Herschel's Formula for the
Aberration of a thin Lens.}

\nobreak\bigskip

11.
These general equations for refracting surfaces of revolution may
be adapted to the case of a refracting spheric surface, by making
$$z_1 = {1 \over 2r};\quad
  z_2 = {1 \over 8r^3};\quad
  z_3 = {1 \over 16r^5};
   \eqno {\rm (G''')}$$
the two first, for example, becoming
$$0 = \Delta \left( {\mu \over z + A} + {\mu \over r} \right);
   \eqno {\rm (H''')}$$
$$0 = \Delta
         \left(
            {2 \mu L \over (z + A)^4}
          - {\mu \over (z + A)^3}
          - {2\mu \over r (z + A)^2}
          + {\mu \over r^3}
         \right);
   \eqno {\rm (I''')}$$
which contain under a convenient form, the known theorems for the
change of a central focus, and of the principal coefficient of
aberration, by refraction of a spheric surface; $r$ being the
radius of this surface, and being considered as positive or
negative, according as the convexity or concavity is turned
towards the incident rays.

If, for instance, we consider an infinitely thin lens in vacuo,
having $\mu$ for its refractive index, and having $r$,~$r'$, for
the radii of its two spheric surfaces, (positive when those
surfaces are convex towards the incident rays,) we may take the
point of central incidence for origin, and the equation
(H${}'''$) will become,
$$0 = {\mu \over A'} - {1 \over A} + {\mu - 1 \over r};\quad
  0 = {1 \over A''} - {\mu \over A'} + {1 - \mu \over r'},
   \eqno {\rm (K''')}$$
$-A$, $-A'$, $-A''$, being the ordinates of the central focus in
the three successive states of the system; and similarly,
(I${}'''$) will give
$$\left. \eqalign{
0  &= {2\mu L' \over A'^4} - {2L \over A^4}
      - \left( {\mu \over A'^3} - {1 \over A^3} \right)
      - \left( {2\mu \over r A'^2} - {2 \over r A^2} \right)
      + {\mu - 1 \over r^3};\cr
0  &= {2 L'' \over A''^4} - {2\mu L' \over A'^4}
      - \left( {1 \over A''^3} - {\mu \over A'^3} \right)
      - \left( {2 \over r' A''^2} - {2\mu \over r' A'^2} \right)
      + {1 - \mu \over r'^3};\cr}
   \right\}
   \eqno {\rm (L''')}$$
$L$, $L'$, $L''$, being the three successive values of the
principal coefficient of aberration.  Adding the two equations
(L${}'''$), the intermediate coefficient $L'$ disappears, and we
find,
$$0 = {2 L'' \over A''^4} - {2L \over A^4}
      - \left( {1 \over A''^3} - {1 \over A^3} \right)
      + {2\mu \over A'^2} \left( {1 \over r'} - {1 \over r} \right)
      - \left( {2 \over r' A''^2} - {2 \over r A^2} \right)
      + (\mu - 1) \left( {1 \over r^3} - {1 \over r'^3} \right),
   \eqno {\rm (M''')}$$
in which, by (K${}'''$),
$${1 \over A'}
   = {1 \over \mu A} - {\mu - 1 \over \mu r};\quad
  {1 \over A''}
   = {1 \over A} + (\mu - 1) \left( {1 \over r'} - {1 \over r} \right);
   \eqno {\rm (N''')}$$
and therefore,
$${L'' \over A''^4} - {L \over A^4}
   =  \left( {\mu - 1 \over 2} \right)
      \left( {1 \over r'} - {1 \over r} \right)
      \left( M^{(0)} + {M^{(1)} \over A} + {M^{(2)} \over A^2} \right),
   \eqno {\rm (O''')}$$
if we put for abridgment
$$\left. \eqalign{
M^{(0)}
   &= {\mu^2 \over r'^2} + {1 + 2\mu - 2\mu^2 \over r' r}
      + {\displaystyle {2 \over \mu} - 2\mu + \mu^2 \over r^2};\cr
M^{(1)}
   &= {1 + 3\mu \over r'}
      + {\displaystyle {4 \over \mu} + 3 - 3\mu \over r};\cr
M^{(2)}
   &= {2 \over \mu} + 3.\cr}
   \right\}
   \eqno {\rm (P''')}$$

It is easy to see that the formula (O${}'''$) agrees with the
expression for the spherical aberration of an infinitely thin
lens, which {\sc Mr.~Herschel} has deduced by reasonings of a
different kind, in his memoir ``On Aberrations of Compound Lenses
and Object-Glasses,'' published in the second part of the
{\it Philosophical Transactions\/} for the year 1821; and in his
excellent ``Treatise on Light,'' published in the
{\it Encyclop{\ae}dia Metropolitana}.

The elegance of this formula of Mr.~Herschel, and the important
consequences which he has obtained from it, have induced us to
shew how the same expression may be derived from the development
of the characteristic function of an ordinary system of
revolution, assigned in the present Supplement.  The same form of
development, and those other forms which we have assigned in the
same Supplement, for systems not of revolution, contain the
solution of other optical problems, of which we hope to treat
hereafter.

\bye