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% Trinity College, 1st June 1999.

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

\vskip12pt

\centerline{\Largebf THEORY OF SYSTEMS OF RAYS}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Transactions of the Royal Irish Academy, vol.~16,
part~1 (1830), pp. 1--61.)}

\vskip36pt

\vfill

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

\vskip 12pt

\centerline{\largerm 2001}

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\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

The {\it Supplement to an Essay on the Theory of Systems of Rays}
by William Rowan Hamilton was originally published in volume~16,
part~1 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.

This edition corrects various errata noted by Hamilton, and
listed at the end of the original publication of both the
{\it Supplement\/} and the {\it Second Supplement\/} to the
essay on the {\it Theory of Systems of Rays}.

In the original publication, the word {\it ar\^{e}tes\/} was
spelt as {\it aretes} in article~8, and as {\it ar\'{e}tes}
in article~9.

\bigbreak\bigskip

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

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

\vskip0pt

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

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{\largeit\noindent
Supplement to an Essay on the Theory of Systems of Rays.
\hskip 0pt plus1em minus0pt
By {\largerm WILLIAM R. HAMILTON, A. B., M. R. I. A.,}
M. Ast.\ Soc.\ Lond.,
Hon.\ M.\ Soc.\ Arts for Scotland,
Andrews' Professor of Astronomy in the University of Dublin,
and Royal Astronomer of Ireland.}

\bigbreak

\centerline{Read April 26, 1830.}

\bigbreak

\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~16, part~1 (1830), pp. 1--61.]}

\nobreak\bigskip

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

\nobreak\bigskip

\centerline{\largerm INTRODUCTION.}

\nobreak\bigskip

The present Supplement contains some developments of a view of
Mathematical Optics, which was proposed by me in the foregoing
volume of the Transactions of this Academy.  According to that
view, the geometrical properties of an optical system of rays,
whether straight or curved, whether ordinary or extraordinary,
may be deduced by analytic methods, from one fundamental formula,
and one characteristic function: the formula being an expression
for the variation which the definite integral, called {\it
action}, receives, when the coordinates of its limits vary; and
the characteristic function being this integral itself,
considered as depending on those coordinates.  Although this view
was stated, and the formula announced, in the Table of Contents
prefixed to my preceding Memoir, yet the demonstration was not
given in the part already published, except for the Systems
produced by the ordinary reflection of light; it has therefore
been thought advisable to give in the present Supplement, the
general demonstration of the formula, and some of its general
consequences.  The demonstration is founded on the principles of
the calculus of variations, and on the known optical principle of
least action.  The result deduced from these principles, is, that
the coefficients of the variations of the final coordinates, in
the variation of the integral called action, are equal to the
coefficients of the variations of the cosines of the angles which
the element of the ray makes with the axes of coordinates, in the
variation of a certain homogeneous function of those cosines:
this homogeneous function, which is of the first dimension, being
equal to the multiplier of the element of the ray under the
integral sign, and therefore to the velocity of that element,
estimated on the hypothesis of emission.  It was proposed, in my
former Memoir, to call this result the {\it principle of constant
action\/}: partly to mark its connexion with the known law of
{\it least action}, and partly because it gives immediately the
differential equation of that important class of surfaces, which,
on the hypothesis of undulation are called {\it waves}, and
which, on the hypothesis of molecular emission may be named
{\it surfaces of constant action}.  But in the present
Supplement, it is proposed to designate the fundamental formula
by the less hypothetical name of the {\it Equation of the
Characteristic Function\/}: because, whatever may be the nature
of light, the definite integral in this equation is, as we have
before observed, a function of the coordinates of its limits, on
the analytic form of which function the properties of the system
depend.  In the applications of this formula, to systems of
straight rays, ordinary or extraordinary, it is advantageous to
introduce the consideration of a characteristic function of
another kind, depending on the direction rather than on the
coordinates of the ray, but connected with the former function,
and with the geometrical properties of the system, by relations
which form the chief subject of the present Memoir.  The theory
of these relations, from the generality of its nature, will,
perhaps, be interesting to Mathematicians: I am aware that it
admits of being much farther extended, and that much remains to
be done, in order to render it practically useful.

\line{\hfil WILLIAM R. HAMILTON.}

\nobreak\bigskip

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

\vfill\eject

\centerline{\largerm CONTENTS OF THE SUPPLEMENT.}

\nobreak\bigskip

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

Fundamental formula of optical systems ordinary and
extraordinary, or equation of the characteristic function;
$$V = \int v \, ds;\quad
  \delta V
   =    {\delta v \over \delta \alpha} \, \delta x
      + {\delta v \over \delta \beta}  \, \delta y
      + {\delta v \over \delta \gamma} \, \delta z;
   \eqno \hbox{numbers 1,2,3.}$$

Other characteristic function, for systems of straight rays;
$$W + V
   =    x {\delta v \over \delta \alpha}
      + y {\delta v \over \delta \beta}
      + z {\delta v \over \delta \gamma};\quad
  \delta W
   =    x \, \delta {\delta v \over \delta \alpha}
      + y \, \delta {\delta v \over \delta \beta}
      + z \, \delta {\delta v \over \delta \gamma};
   \eqno \hbox{4.}$$

Connexions between the partial differential coefficients of the
two characteristic functions, and changes produced by reflexion
or refraction, ordinary or extraordinary;
$$\delta \Delta V = \lambda \, \delta u;\quad
  \Delta W = \lambda
         \left(
            x {\delta u \over dx}
          + y {\delta u \over dy}
          + z {\delta u \over dz}
         \right);
   \eqno \hbox{5, 6, 7.}$$

Connexions of the characteristic functions with the developable
pencils, and with the caustic curves and surfaces;
$$\rho \, \delta {\delta V \over \delta x}
   = \delta' {\delta v \over \delta \alpha},\quad
  \rho \, \delta {\delta V \over \delta y}
   = \delta' {\delta v \over \delta \beta},\quad
  \rho \, \delta {\delta V \over \delta z}
   = \delta' {\delta v \over \delta \gamma};$$
$$\delta {\delta W \over \delta \alpha}
   = \delta {\delta W' \over \delta \alpha},\quad
  \delta {\delta W \over \delta \beta}
   = \delta {\delta W' \over \delta \beta},\quad
  \delta {\delta W \over \delta \gamma}
   = \delta {\delta W' \over \delta \gamma};
   \eqno \hbox{8, 9, 10.}$$

On osculating focal systems; the foci of the extreme systems are
contained upon the caustic surfaces, and their planes of
osculation touch the developable pencils; the foci of the other
osculating systems are inside or outside the interval of the
extreme foci, according to the sign of a certain function $v''$,
which does not change by transformation of coordinates; the
distances of any one of these other foci from the two caustic
surfaces are proportional to the squares of the sines of the
angles which the corresponding plane of osculation makes with the
tangent planes to the two developable pencils;
$$\delta^2 W = \delta^2 W';\quad
  {R - R_1 \over R_2 - R}
   = \zeta
      \left(
         {\sin (\phi - \phi_1) \over \sin (\phi_2 - \phi)}
      \right)^2;\quad
  {\zeta\over v''} > 0;
   \eqno \hbox{11, 12, 13.}$$

Principal foci, and principal rays: the principal foci belong to
those osculating systems, for which the osculation is most
complete; they are the only points in which the caustic surfaces
intersect, when the function $v''$ is positive; this condition is
satisfied for ordinary systems, and for the systems produced by
crystals with one axis;
$${\delta^2 W \over \delta \alpha^2}
   = {\delta^2 W' \over \delta \alpha^2},\quad
  {\delta^2 W \over \delta \beta^2}
   = {\delta^2 W' \over \delta \beta^2},\quad
  {\delta^2 W \over \delta \gamma^2}
   = {\delta^2 W' \over \delta \gamma^2};
   \eqno \hbox{14.}$$

On osculating spheroids, and surfaces of constant action; the
extreme spheroids have their centres upon the caustic surfaces,
and their planes of osculation touch the developable pencils; the
centre of the osculating spheroid coincides with the focus of the
osculating focal reflector or refractor, when the point of
contact and the plane of osculation are the same; the principal
foci are centres of spheroids which have complete contact of the
second order with the surfaces of constant action; they are also
foci of osculating focal reflectors or refractors which have
contact of the same kind with the last given reflector or
refractor;
$$\delta^2 V = \delta^2 V';\quad
  {\rho_1^{-1} - \rho^{-1} \over \rho^{-1} - \rho_2^{-1}}
   = \zeta
      \left(
         {\sin (\psi - \psi_1) \over \sin (\psi_2 - \psi)}
      \right)^2;
   \eqno \hbox{15, 16.}$$

On foci by projection and virtual foci; the planes of extreme
projection coincide with the planes of extreme virtual foci;
these planes are perpendicular to each other, and furnish in
general a pair of natural coordinates; the tangent planes to the
developable pencils are symmetrically situated with respect to
these natural coordinate planes, and coincide with them when the
system is rectangular; this latter condition is satisfied in
ordinary systems, and the osculating foci of such systems
coincide with the foci by projection;
$$p = - {\delta x'^2 + \delta y'^2 + \delta z'^2
         \over \delta \alpha \, \delta x'
             + \delta \beta  \, \delta y'
             + \delta \gamma \, \delta z'};\quad
  {1 \over p}
   =    {(\cos \Pi)^2 \over p_1}
      + {(\sin \Pi)^2 \over p_2};$$
$$r = - {\delta \alpha \, \delta x_\prime
       + \delta \beta  \, \delta y_\prime
       + \delta \gamma \, \delta z_\prime
         \over \delta \alpha^2 + \delta \beta^2 + \delta \gamma^2};\quad
  r = r_1 (\cos \omega)^2 + r_2 (\sin \omega)^2;
   \eqno \hbox{17, 18.}$$

Theory of lateral aberrations; aberration from a principal focus;
elliptic or hyperbolic lines of uniform condensation; natural
axes of the system;
$$\Delta x
   =    [\delta x]
      + {1 \over 2} [\delta^2 x]
      + {1 \over 2 \mathbin{.} 3} [\delta^3 x]
      + \hbox{\&c.};$$
$$M'' =
      \left\{
         {\delta^3 W \over \delta \alpha^3}
         {\delta^3 W \over \delta \alpha \, \delta \beta^2}
       - \left(
            {\delta^3 W \over \delta \alpha^2 \, \delta \beta}
         \right)^2
      \right\} a^2
    + \left\{
         {\delta^3 W \over \delta \alpha^2 \, \delta \beta}
         {\delta^3 W \over \delta \beta^3}
       - \left(
            {\delta^3 W \over \delta \alpha \, \delta \beta^2}
         \right)^2
      \right\} b^2;$$
$$    {\delta^3 W \over \delta \alpha^3}
      {\delta^3 W \over \delta \beta^3}
    = {\delta^3 W \over \delta \alpha^2 \, \delta \beta}
      {\delta^3 W \over \delta \alpha \, \delta \beta^2};
   \eqno \hbox{19, 20.}$$

\endgroup

\vfill\eject

\centerline{\largerm SUPPLEMENT, \&c. \&c.}

\nobreak\bigskip

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

\nobreak\bigskip

\centerline{\it Fundamental Formula of Optical Systems, or Equation of the
Characteristic Function.}

\nobreak\bigskip

1.
The fundamental formula that we shall employ in our
investigations respecting the geometrical properties of optical
systems of rays, straight or curved, ordinary or extraordinary,
which, after issuing from any luminous origin, have been any
number of times reflected and refracted by any combination of
media, according to any laws compatible with the known condition
of least action, is the following:
$$\delta \int v \, ds
   =    {\delta v \over \delta \alpha} \, \delta x
      + {\delta v \over \delta \beta}  \, \delta y
      + {\delta v \over \delta \gamma} \, \delta z.
   \eqno {\rm (A)}$$
In this equation, $x$,~$y$,~$z$, are the coordinates of any point
of the system, referred to three rectangular axes; $\alpha$,
$\beta$, $\gamma$, are the cosines of the angles which the
tangent to the ray at that point, or the direction of the element
$ds$, makes with the axes of coordinates; $v$ is the quantity
which in the hypothesis of molecular emission represents the
velocity of this element, and is supposed to be in general a
function of the six quantities
$x$,~$y$,~$z$, $\alpha$,~$\beta$,~$\gamma$,
depending on the nature of the medium, and involving also the
colour of the light; the partial differential coefficients,
$${\delta v \over \delta \alpha},\quad
  {\delta v \over \delta \beta},\quad
  {\delta v \over \delta \gamma},$$
are obtained by putting $v$ under the form of a homogeneous
function of $\alpha$,~$\beta$,~$\gamma$, of the first dimension,
with the help of the relation
$\alpha^2 + \beta^2 + \gamma^2 = 1$, and by then differentiating
this homogeneous function, as if $\alpha$,~$\beta$,~$\gamma$,
were three independent variables; finally, the definite integral
$\int v \, ds$ is taken from the luminous origin to the point
$x$,~$y$,~$z$, and the variation
$\delta \int v \, ds$
is obtained by supposing the coordinates of this last point to
receive any infinitely small changes, the colour remaining the
same.

\bigbreak

2.
To deduce the equation (A) from the known condition of least
action, let us observe that by the calculus of variations,
$$\delta \int v \, ds
   =  \int (\delta v \mathbin{.} ds + v \mathbin{.} \delta ds);$$
in which, by what we have laid down respecting the form of $v$,
$$\delta v
   =    {\delta v \over \delta x} \, \delta x
      + {\delta v \over \delta y} \, \delta y
      + {\delta v \over \delta z} \, \delta z
      + {\delta v \over \delta \alpha} \, \delta \alpha
      + {\delta v \over \delta \beta}  \, \delta \beta
      + {\delta v \over \delta \gamma} \, \delta \gamma,$$
$$v =   \alpha {\delta v \over \delta \alpha}
      + \beta  {\delta v \over \delta \beta}
      + \gamma {\delta v \over \delta \gamma};$$
while, by the nature of $\alpha$,~$\beta$,~$\gamma$,
$$\eqalign{
\delta \alpha \mathbin{.} ds + \alpha \mathbin{.} \delta ds
   &= \delta \mathbin{.} \alpha \, ds
    = \delta \mathbin{.} dx = d \mathbin{.} \delta x,\cr
\delta \beta  \mathbin{.} ds + \beta  \mathbin{.} \delta ds
   &= \delta \mathbin{.} \beta  \, ds
    = \delta \mathbin{.} dy = d \mathbin{.} \delta y,\cr
\delta \gamma \mathbin{.} ds + \gamma \mathbin{.} \delta ds
   &= \delta \mathbin{.} \gamma \, ds
    = \delta \mathbin{.} dz = d \mathbin{.} \delta z;\cr}$$
we have therefore,
$$\eqalign{
\delta  \int v \, ds
   &= \int
         \left(
            {\delta v \over \delta x} \, \delta x
          + {\delta v \over \delta y} \, \delta y
          + {\delta v \over \delta z} \, \delta z
         \right) \, ds
      + \int
         \left(
            {\delta v \over \delta \alpha} \, d \delta x
          + {\delta v \over \delta \beta}  \, d \delta y
          + {\delta v \over \delta \gamma} \, d \delta z
         \right) \cr
   &=   {\delta v  \over \delta \alpha}  \, \delta x
      - {\delta v' \over \delta \alpha'} \, \delta x'
      + {\delta v  \over \delta \beta}   \, \delta y
      - {\delta v' \over \delta \beta'}  \, \delta y'
      + {\delta v  \over \delta \gamma}  \, \delta z
      - {\delta v' \over \delta \gamma'} \, \delta z' \cr
   &\mathrel{\phantom{=}} \mathord{}
      + \int \delta x \,
         \left(
            {\delta v \over \delta x} \, ds
          - d {\delta v \over \delta \alpha}
         \right)
      + \int \delta y \,
         \left(
            {\delta v \over \delta y} \, ds
          - d {\delta v \over \delta \beta}
         \right)
      + \int \delta z \,
         \left(
            {\delta v \over \delta z} \, ds
          - d {\delta v \over \delta \gamma}
         \right),\cr}$$
the accented quantities belonging to the first limit of integral,
and disappearing when that limit is fixed.  The condition of
least action requires that the quantities which remain under the
integral sign, as coefficients of
$\delta x$, $\delta y$, $\delta z$,
should also vanish, and furnishes thereby the following general
differential equations of a ray,
$${\delta v \over \delta x} \, ds
   = d {\delta v \over \delta \alpha},\quad
  {\delta v \over \delta y} \, ds
   = d {\delta v \over \delta \beta},\quad
  {\delta v \over \delta z} \, ds
   = d {\delta v \over \delta \gamma},
   \eqno {\rm (B)}$$
of which any two include the third.  And rejecting the evanescent
quantities in the expression for
$\delta \int v \, ds$,
we find the formula (A), which it was required to demonstrate.

\bigbreak

3.
The fundamental formula thus obtained, resolves itself into the
three following equations:
$${\delta \int v \, ds \over \delta x}
   = {\delta v \over \delta \alpha},\quad
  {\delta \int v \, ds \over \delta y}
   = {\delta v \over \delta \beta},\quad
  {\delta \int v \, ds \over \delta z}
   = {\delta v \over \delta \gamma},$$
which we shall thus write:
$${\delta V \over \delta x}
   = {\delta v \over \delta \alpha},\quad
  {\delta V \over \delta y}
   = {\delta v \over \delta \beta},\quad
  {\delta V \over \delta z}
   = {\delta v \over \delta \gamma},
   \eqno {\rm (C)}$$
representing, for abridgment, the definite integral
$\displaystyle \int v \, ds$
by $V$, and considering the integral as a function of
$x$,~$y$,~$z$, of which the form depends upon the nature of the
system, the medium, and the light, and of which the partial
differential coefficients of the first order are denoted by
$${\delta V \over \delta x},\quad
  {\delta V \over \delta y},\quad
  {\delta V \over \delta z}.$$
When the form of $V$ is given, we can obtain these coefficients
by differentiation; and if we know also the form of $v$, which
depends only on the nature of the medium and of the light, we can
by the equations (C) determine $\alpha$,~$\beta$,~$\gamma$, as
functions of $x$,~$y$,~$z$; that is, we can find the direction of
the ray or rays passing through any proposed point of the system.
The geometrical properties of one system as distinguished from
another, for any given medium and any given kind of light, may
therefore be deduced by analytic reasonings from the form of the
function~$V$; on which account we shall call this function~$V$,
the {\it characteristic function of the system\/}; and the
fundamental formula (A), that expresses its variation, namely:
$$\delta V
   =    {\delta v \over \delta \alpha} \, \delta x
      + {\delta v \over \delta \beta}  \, \delta y
      + {\delta v \over \delta \gamma} \, \delta z,$$
we shall call the {\it equation of the characteristic function}.

\bigbreak

\centerline{\it
Other Characteristic Function for Systems of Straight Rays.}

\nobreak\bigskip

4.
In the remaining reasonings of the present Supplement, we shall
confine ourselves to the consideration of homogeneous systems of
straight rays not parallel; and in investigating the properties
of such systems, it will be useful to employ another function,
connected with the function~$V$ by many remarkable relations.
This new function, which we shall call $W$, is determined by the
condition:
$$W + V
   =    x {\delta v \over \delta \alpha}
      + y {\delta v \over \delta \beta}
      + z {\delta v \over \delta \gamma},
   \eqno {\rm (D)}$$
which gives, on account of (A), or (C),
$$\delta W
   =    x \, \delta {\delta v \over \delta \alpha}
      + y \, \delta {\delta v \over \delta \beta}
      + z \, \delta {\delta v \over \delta \gamma}.
   \eqno {\rm (E)}$$

It results from this differential equation (E) (in which we
employ the sign of variation~$\delta$ to mark the connexion with
the definite integral $\int v \, ds$, a remark which applies to
the whole of the present Supplement,) that if the variations of
$x$,~$y$,~$z$, be such as to leave $\alpha$,~$\beta$,~$\gamma$,
and consequently
$${\delta v \over \delta \alpha},\quad
  {\delta v \over \delta \beta},\quad
  {\delta v \over \delta \gamma}$$
unchanged, that is, if we pass from any one point of the system
to any other point situated upon the same ray, the function~$W$
will not vary.  We may, therefore, consider $W$ as a function of
$\alpha$,~$\beta$,~$\gamma$, of which the form can be determined
from that of $V$, by eliminating $x$,~$y$,~$z$, between the
equations (C) and (D), when the nature of the medium and of the
light is known.  Reciprocally, if the connexion between $W$,
$\alpha$,~$\beta$,~$\gamma$, be given, that which exists between
$V$, $x$,~$y$,~$z$, can be found.  For if we suppose that for the
sake of symmetry, $W$ has been put under the form of a
homogeneous function of the dimension~$i$, by the help of the
relation
$\alpha^2 + \beta^2 + \gamma^2 = 1$,
and then differentiated as if $\alpha$,~$\beta$,~$\gamma$, were
three independent variables, we shall have, by (E), and by the
nature of homogeneous functions,
$$\eqalign{
{\delta W \over \delta \alpha}
   = i W \alpha
      + x {\delta^2 v \over \delta \alpha^2}
      + y {\delta^2 v \over \delta \alpha \, \delta \beta}
      + z {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
{\delta W \over \delta \beta}
   = i W \beta
      + x {\delta^2 v \over \delta \alpha \, \delta \beta}
      + y {\delta^2 v \over \delta \beta^2}
      + z {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
{\delta W \over \delta \gamma}
   = i W \gamma
      + x {\delta^2 v \over \delta \alpha \, \delta \gamma}
      + y {\delta^2 v \over \delta \beta  \, \delta \gamma}
      + z {\delta^2 v \over \delta \gamma^2},\cr}$$
in which we shall for simplicity suppose the dimension $i = 0$;
and eliminating $\alpha$,~$\beta$,~$\gamma$, by means of these
equations, from that marked (D), we shall deduce the relation
between $V$,~$x$,~$y$,~$z$, from the relation between
$W$,~$\alpha$,~$\beta$,~$\gamma$.  We may therefore consider $W$
as itself a characteristic function, which distinguishes any one
homogeneous system of straight rays not parallel, from any other
such system, composed of light of the same kind, and contained in
the same medium.  It is evident that on some occasions it must be
advantageous to attend to the function~$W$ instead of $V$,
because $V$ changes in passing from one point to another of the
same ray, whereas $W$ is constant, when the ray and the system
are given.  On the other hand, in any sudden change of the system
by reflection or refraction, the function~$W$ receives a sudden
alteration, while the change of $V$ is gradual; it is therefore
convenient to employ $V$ instead of $W$, in investigating the
effects of such changes.  Accordingly, in the remainder of this
memoir, we shall consider both these functions, and examine the
relations between them: and shall begin by investigating the
connexions between their partial differential coefficients.

\bigbreak

\centerline{\it
Connexions between the Partial Differential Coefficients of the two}
\centerline{\it Characteristic Functions.}

\nobreak\bigskip

5.
The connexions between these coefficients, are to be obtained by
differentiating the preceding expressions for
$${\delta V \over \delta x},\quad
  {\delta V \over \delta y},\quad
  {\delta V \over \delta z},\quad
  {\delta W \over \delta \alpha},\quad
  {\delta W \over \delta \beta},\quad
  {\delta W \over \delta \gamma},$$
and by attending to the homogeneous forms which we have assigned
to $v$ and $W$.  The dimension of $W$ being supposed $= 0$, we
have by the nature of homogeneous functions,
$$\left. \eqalign{
        \alpha {\delta W \over \delta \alpha}
      + \beta  {\delta W \over \delta \beta}
      + \gamma {\delta W \over \delta \gamma}
   &= 0;\cr
               {\delta W \over \delta \alpha}
      + \alpha {\delta^2 W \over \delta \alpha^2}
      + \beta  {\delta^2 W \over \delta \alpha \, \delta \beta}
      + \gamma {\delta^2 W \over \delta \alpha \, \delta \gamma}
   &= 0;\cr
               {\delta W \over \delta \beta}
      + \alpha {\delta^2 W \over \delta \alpha \, \delta \beta}
      + \beta  {\delta^2 W \over \delta \beta^2}
      + \gamma {\delta^2 W \over \delta \beta  \, \delta \gamma}
   &= 0;\cr
               {\delta W \over \delta \gamma}
      + \alpha {\delta^2 W \over \delta \alpha \, \delta \gamma}
      + \beta  {\delta^2 W \over \delta \beta  \, \delta \gamma}
      + \gamma {\delta^2 W \over \delta \gamma^2}
   &= 0;\cr
             2 {\delta^2 W \over \delta \alpha^2}
      + \alpha {\delta^3 W \over \delta \alpha^3}
      + \beta  {\delta^3 W \over \delta \alpha^2 \, \delta \beta}
      + \gamma {\delta^3 W \over \delta \alpha^2 \, \delta \gamma}
   &= 0;\cr
\hbox{\&c.}\qquad\qquad\cr}
   \right\}
   \eqno {\rm (F)}$$

We have also, by the homogeneous nature of $v$, which we have put
under the form of a function of the first dimension, the
following relations:

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

$$\left. \eqalign{
        \alpha {\delta v \over \delta \alpha}
      + \beta  {\delta v \over \delta \beta}
      + \gamma {\delta v \over \delta \gamma}
   &= v;\cr
        \alpha {\delta^2 v \over \delta \alpha^2}
      + \beta  {\delta^2 v \over \delta \alpha \, \delta \beta}
      + \gamma {\delta^2 v \over \delta \alpha \, \delta \gamma}
   &= 0;\cr
        \alpha {\delta^2 v \over \delta \alpha \, \delta \beta}
      + \beta  {\delta^2 v \over \delta \beta^2}
      + \gamma {\delta^2 v \over \delta \beta  \, \delta \gamma}
   &= 0;\cr
        \alpha {\delta^2 v \over \delta \alpha \, \delta \gamma}
      + \beta  {\delta^2 v \over \delta \beta  \, \delta \gamma}
      + \gamma {\delta^2 v \over \delta \gamma^2}
   &= 0;\cr
               {\delta^2 v \over \delta \alpha^2}
      + \alpha {\delta^3 v \over \delta \alpha^3}
      + \beta  {\delta^3 v \over \delta \alpha^2 \, \delta \beta}
      + \gamma {\delta^3 v \over \delta \alpha^2 \, \delta \gamma}
   &= 0;\cr
\hbox{\&c.}\qquad\qquad\cr}
   \right\}
   \eqno {\rm (G)}$$

These relations give
$$      \alpha \, \delta {\delta v \over \delta \alpha}
      + \beta  \, \delta {\delta v \over \delta \beta}
      + \gamma \, \delta {\delta v \over \delta \gamma}
   = 0,$$
and therefore, by (C),
$$      \alpha \, \delta {\delta V \over \delta x}
      + \beta  \, \delta {\delta V \over \delta y}
      + \gamma \, \delta {\delta V \over \delta z}
   = 0,$$
a condition which resolves itself into the three following,
$$\left. \eqalign{
        \alpha {\delta^2 V \over \delta x^2}
      + \beta  {\delta^2 V \over \delta x \, \delta y}
      + \gamma {\delta^2 V \over \delta x \, \delta z}
   &= 0;\cr
        \alpha {\delta^2 V \over \delta x \, \delta y}
      + \beta  {\delta^2 V \over \delta y^2}
      + \gamma {\delta^2 V \over \delta y  \, \delta z}
   &= 0;\cr
        \alpha {\delta^2 V \over \delta x \, \delta z}
      + \beta  {\delta^2 V \over \delta y  \, \delta z}
      + \gamma {\delta^2 V \over \delta z^2}
   &= 0;\cr}
   \right\}
   \eqno {\rm (H)}$$
and combining these three equations (H) with those which are
obtained by differentiating (C), we find,
$$\left. \eqalign{
V'' \, \delta x'
   &=    \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
          + {\delta^2 V \over \delta z^2}
         \right)
         \, \delta {\delta v \over \delta \alpha}
       - \left(
            {\delta^2 V \over \delta x^2}
               \, \delta {\delta v \over \delta \alpha}
          + {\delta^2 V \over \delta x \, \delta y}
               \, \delta {\delta v \over \delta \beta}
          + {\delta^2 V \over \delta x \, \delta z}
               \, \delta {\delta v \over \delta \gamma}
         \right),\cr
V'' \, \delta y'
   &=    \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
          + {\delta^2 V \over \delta z^2}
         \right)
         \, \delta {\delta v \over \delta \beta}
       - \left(
            {\delta^2 V \over \delta x \, \delta y}
               \, \delta {\delta v \over \delta \alpha}
          + {\delta^2 V \over \delta y^2}
               \, \delta {\delta v \over \delta \beta}
          + {\delta^2 V \over \delta y \, \delta z}
               \, \delta {\delta v \over \delta \gamma}
         \right),\cr
V'' \, \delta z'
   &=    \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
          + {\delta^2 V \over \delta z^2}
         \right)
         \, \delta {\delta v \over \delta \gamma}
       - \left(
            {\delta^2 V \over \delta x \, \delta z}
               \, \delta {\delta v \over \delta \alpha}
          + {\delta^2 V \over \delta y \, \delta z}
               \, \delta {\delta v \over \delta \beta}
          + {\delta^2 V \over \delta z^2}
               \, \delta {\delta v \over \delta \gamma}
         \right),\cr}
   \right\}
   \eqno {\rm (I)}$$
in which
$$V'' =  {\delta^2 V \over \delta x^2} {\delta^2 V \over \delta y^2}
       - \left( {\delta^2 V \over \delta x \, \delta y} \right)^2
       + {\delta^2 V \over \delta y^2} {\delta^2 V \over \delta z^2}
       - \left( {\delta^2 V \over \delta y \, \delta z} \right)^2
       + {\delta^2 V \over \delta z^2} {\delta^2 V \over \delta x^2}
       - \left( {\delta^2 V \over \delta z \, \delta x} \right)^2,$$
and
$$\eqalign{
\delta x'
   &= \delta x - \alpha (
         \alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z ),\cr
\delta y'
   &= \delta y - \beta  (
         \alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z ),\cr
\delta z'
   &= \delta z - \gamma (
         \alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z );\cr}$$
so that
$$\alpha \, \delta x' + \beta \, \delta y' + \gamma \, \delta z'
   = 0.$$

Now, if we differentiate the expressions,
$$\left. \eqalign{
{\delta W \over \delta \alpha}
   &=   x {\delta^2 v \over \delta \alpha^2}
      + y {\delta^2 v \over \delta \alpha \, \delta \beta}
      + z {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
{\delta W \over \delta \beta}
   &=   x {\delta^2 v \over \delta \alpha \, \delta \beta}
      + y {\delta^2 v \over \delta \beta^2}
      + z {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
{\delta W \over \delta \gamma}
   &=   x {\delta^2 v \over \delta \alpha \, \delta \gamma}
      + y {\delta^2 v \over \delta \beta  \, \delta \gamma}
      + z {\delta^2 v \over \delta \gamma^2},\cr}
   \right\}
   \eqno {\rm (K)}$$
which result from the foregoing number, and put for abridgment,
$$\eqalign{
         \delta \alpha \mathbin{.}
            \delta {\delta W \over \delta \alpha}
       + \delta \beta  \mathbin{.}
            \delta {\delta W \over \delta \beta}
       + \delta \gamma \mathbin{.}
            \delta {\delta W \over \delta \gamma}
   &= \delta^2 W,\cr
         \delta \alpha \mathbin{.}
            \delta {\delta^2 v \over \delta \alpha^2}
       + \delta \beta  \mathbin{.}
            \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + \delta \gamma \mathbin{.}
            \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
   &= \delta^2 {\delta v \over \delta \alpha},\cr
         \delta \alpha \mathbin{.}
            \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + \delta \beta  \mathbin{.}
            \delta {\delta^2 v \over \delta \beta^2}
       + \delta \gamma \mathbin{.}
            \delta {\delta^2 v \over \delta \beta \, \delta \gamma}
   &= \delta^2 {\delta v \over \delta \beta},\cr
         \delta \alpha \mathbin{.}
            \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + \delta \beta  \mathbin{.}
            \delta {\delta^2 v \over \delta \beta \, \delta \gamma}
       + \delta \gamma \mathbin{.}
            \delta {\delta^2 v \over \delta \gamma^2}
   &= \delta^2 {\delta v \over \delta \gamma},\cr}$$
$$\eqalign{
         \delta x \mathbin{.}
            {\delta^2 v \over \delta \alpha^2}
       + \delta y \mathbin{.}
            {\delta^2 v \over \delta \alpha \, \delta \beta}
       + \delta z \mathbin{.}
            {\delta^2 v \over \delta \alpha \, \delta \gamma}
   &= \delta' {\delta v \over \delta \alpha},\cr
         \delta x \mathbin{.}
            {\delta^2 v \over \delta \alpha \, \delta \beta}
       + \delta y \mathbin{.}
            {\delta^2 v \over \delta \beta^2}
       + \delta z \mathbin{.}
            {\delta^2 v \over \delta \beta \, \delta \gamma}
   &= \delta' {\delta v \over \delta \beta},\cr
         \delta x \mathbin{.}
            {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + \delta y \mathbin{.}
            {\delta^2 v \over \delta \beta \, \delta \gamma}
       + \delta z \mathbin{.}
            {\delta^2 v \over \delta \gamma^2}
   &= \delta' {\delta v \over \delta \gamma},\cr}$$
we find
$$\eqalign{
\delta^2 W
       - \left(
            x \, \delta^2 {\delta v \over \delta \alpha}
          + y \, \delta^2 {\delta v \over \delta \beta}
          + z \, \delta^2 {\delta v \over \delta \gamma}
         \right)
   &=       \delta \alpha \, \delta' {\delta v \over \delta \alpha}
          + \delta \beta  \, \delta' {\delta v \over \delta \beta}
          + \delta \gamma \, \delta' {\delta v \over \delta \gamma},\cr
   &=       \delta x'     \, \delta  {\delta v \over \delta \alpha}
          + \delta y'     \, \delta  {\delta v \over \delta \beta}
          + \delta z'     \, \delta  {\delta v \over \delta \gamma},\cr}$$
and therefore, by (I),
$$\eqalignno{
V'' \, \delta^2 W
   &= V''
         \left(
            x \, \delta^2 {\delta v \over \delta \alpha}
          + y \, \delta^2 {\delta v \over \delta \beta}
          + z \, \delta^2 {\delta v \over \delta \gamma}
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
          + {\delta^2 V \over \delta z^2}
         \right)
         \left\{
            \left( \delta {\delta v \over \delta \alpha} \right)^2
          + \left( \delta {\delta v \over \delta \beta}  \right)^2
          + \left( \delta {\delta v \over \delta \gamma} \right)^2
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \biggl\{
            {\delta^2 V \over \delta x^2}
               \left( \delta {\delta v \over \delta \alpha} \right)^2
          + {\delta^2 V \over \delta y^2}
               \left( \delta {\delta v \over \delta \beta}  \right)^2
          + {\delta^2 V \over \delta z^2}
               \left( \delta {\delta v \over \delta \gamma} \right)^2
          +  2 {\delta^2 V \over \delta x \, \delta y}
               \left( \delta {\delta v \over \delta \alpha} \right)
               \left( \delta {\delta v \over \delta \beta}  \right) \cr
   &\mathrel{\phantom{=}}\qquad \mathord{}
          +  2 {\delta^2 V \over \delta y \, \delta z}
               \left( \delta {\delta v \over \delta \beta}  \right)
               \left( \delta {\delta v \over \delta \gamma} \right)
          +  2 {\delta^2 V \over \delta z \, \delta x}
               \left( \delta {\delta v \over \delta \gamma} \right)
               \left( \delta {\delta v \over \delta \alpha} \right)
         \biggr\},
   &{\rm (L)}\cr}$$
in which, without violating the conditions (F), the variations
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
may be considered as independent, and which is consequently
equivalent to six expressions for the six partial differential
coefficients of $W$, of the second order.

These six expressions may be put under the following form:
$$\left. \eqalign{
{\delta^2 W \over \delta \alpha^2}
   &=    x {\delta^3 v \over \delta \alpha^3}
       + y {\delta^3 v \over \delta \alpha^2 \, \delta \beta}
       + z {\delta^3 v \over \delta \alpha^2 \, \delta \gamma}
       + {S \over V''}   {\delta^2 v \over \delta \alpha^2}
       - {v'' \over V''} {\delta^2 V \over \delta x^2},\cr
{\delta^2 W \over \delta \beta^2}
   &=    x {\delta^3 v \over \delta \alpha \, \delta \beta^2}
       + y {\delta^3 v \over \delta \beta^3}
       + z {\delta^3 v \over \delta \beta^2 \, \delta \gamma}
       + {S \over V''}   {\delta^2 v \over \delta \beta^2}
       - {v'' \over V''} {\delta^2 V \over \delta y^2},\cr
{\delta^2 W \over \delta \gamma^2}
   &=    x {\delta^3 v \over \delta \alpha \, \delta \gamma^2}
       + y {\delta^3 v \over \delta \beta  \, \delta \gamma^2}
       + z {\delta^3 v \over \delta \gamma^3}
       + {S \over V''}   {\delta^2 v \over \delta \gamma^2}
       - {v'' \over V''} {\delta^2 V \over \delta z^2},\cr
{\delta^2 W \over \delta \alpha \, \delta \beta}
   &=    x {\delta^3 v \over \delta \alpha^2 \, \delta \beta}
       + y {\delta^3 v \over \delta \alpha   \, \delta \beta^2}
       + z {\delta^3 v \over \delta \alpha \, \delta \beta \, \delta \gamma}
       + {S \over V''}   {\delta^2 v \over \delta \alpha \, \delta \beta}
       - {v'' \over V''} {\delta^2 V \over \delta x \, \delta y},\cr
{\delta^2 W \over \delta \beta  \, \delta \gamma}
   &=    x {\delta^3 v \over \delta \alpha \, \delta \beta \, \delta \gamma}
       + y {\delta^3 v \over \delta \beta^2 \, \delta \gamma}
       + z {\delta^3 v \over \delta \beta \, \delta \gamma^2}
       + {S \over V''}   {\delta^2 v \over \delta \beta  \, \delta \gamma}
       - {v'' \over V''} {\delta^2 V \over \delta y \, \delta z},\cr
{\delta^2 W \over \delta \gamma \, \delta \alpha}
   &=    x {\delta^3 v \over \delta \alpha^2 \, \delta \gamma}
       + y {\delta^3 v \over \delta \alpha \, \delta \beta \, \delta \gamma}
       + z {\delta^3 v \over \delta \alpha \, \delta \gamma^2}
       + {S \over V''}   {\delta^2 v \over \delta \gamma \, \delta \alpha}
       - {v'' \over V''} {\delta^2 V \over \delta z \, \delta x};\cr}
   \right\}
   \eqno {\rm (M)}$$
in which
$$v'' =  {\delta^2 v \over \delta \alpha^2} {\delta^2 v \over \delta \beta^2}
       - \left( {\delta^2 v \over \delta \alpha \, \delta \beta} \right)^2
       + {\delta^2 v \over \delta \beta^2} {\delta^2 v \over \delta \gamma^2}
       - \left( {\delta^2 v \over \delta \beta \, \delta \gamma} \right)^2
       + {\delta^2 v \over \delta \gamma^2} {\delta^2 v \over \delta \alpha^2}
       - \left( {\delta^2 v \over \delta \gamma \, \delta \alpha} \right)^2,$$
and
$$\eqalign{
S  &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
          + {\delta^2 V \over \delta z^2}
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \left(
            {\delta^2 v \over \delta \alpha^2} {\delta^2 V \over \delta x^2}
          + {\delta^2 v \over \delta \beta^2} {\delta^2 V \over \delta y^2}
          + {\delta^2 v \over \delta \gamma^2} {\delta^2 V \over \delta z^2}
          + 2 {\delta^2 v \over \delta \alpha \, \delta \beta}
               {\delta^2 V \over \delta x \, \delta y}
          + 2 {\delta^2 v \over \delta \beta  \, \delta \gamma}
               {\delta^2 V \over \delta y \, \delta z}
          + 2 {\delta^2 v \over \delta \gamma \, \delta \alpha}
               {\delta^2 V \over \delta z \, \delta x}
         \right).\cr}$$

These expressions enable us to deduce the partial differential
coefficients of $W$, of the second order, from the corresponding
differentials of $V$; they may also be employed to deduce the
latter from the former.  For if we put
$$\eqalign{
{\delta^2 W \over \delta \alpha^2}
       - \left(
            x {\delta^3 v \over \delta \alpha^3}
          + y {\delta^3 v \over \delta \alpha^2 \, \delta \beta}
          + z {\delta^3 v \over \delta \alpha^2 \, \delta \gamma}
         \right)
   &= M,\cr
{\delta^2 W \over \delta \beta^2}
       - \left(
            x {\delta^3 v \over \delta \alpha \, \delta \beta^2}
          + y {\delta^3 v \over \delta \beta^3}
          + z {\delta^3 v \over \delta \beta^2  \, \delta \gamma}
         \right)
   &= N,\cr
{\delta^2 W \over \delta \gamma^2}
       - \left(
            x {\delta^3 v \over \delta \alpha \, \delta \gamma^2}
          + y {\delta^3 v \over \delta \beta  \, \delta \gamma^2}
          + z {\delta^3 v \over \delta \gamma^3}
         \right)
   &= P,\cr
{\delta^2 W \over \delta \alpha \, \delta \beta}
       - \left(
            x {\delta^3 v \over \delta \alpha^2 \, \delta \beta}
          + y {\delta^3 v \over \delta \alpha \, \delta \beta^2}
          + z {\delta^3 v \over \delta \alpha \, \delta \beta \, \delta \gamma}
         \right)
   &= M',\cr
{\delta^2 W \over \delta \beta \, \delta \gamma}
       - \left(
            x {\delta^3 v \over \delta \alpha \, \delta \beta \, \delta \gamma}
          + y {\delta^3 v \over \delta \beta^2 \, \delta \gamma}
          + z {\delta^3 v \over \delta \beta \, \delta \gamma^2}
         \right)
   &= N',\cr
{\delta^2 W \over \delta \gamma \, \delta \alpha}
       - \left(
            x {\delta^3 v \over \delta \alpha^2 \, \delta \gamma}
          + y {\delta^3 v \over \delta \alpha \, \delta \beta \, \delta \gamma}
          + z {\delta^3 v \over \delta \alpha \, \delta \gamma^2}
         \right)
   &= P',\cr}$$
$$MN - M'^2 + NP - N'^2 + PM - P'^2 = W'',$$
$$\eqalign{
(M + N + P)
         \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
       - \biggl(
            M {\delta^2 v \over \delta \alpha^2}
          + N {\delta^2 v \over \delta \beta^2}
          + P {\delta^2 v \over \delta \gamma^2} \cr
          + 2M' {\delta^2 v \over \delta \alpha \, \delta \beta}
          + 2N' {\delta^2 v \over \delta \beta  \, \delta \gamma}
          + 2P' {\delta^2 v \over \delta \gamma \, \delta \alpha}
         \biggr)
   &= S',\cr}$$
we find, by the equations (M),
$$V'' W'' = v''^2,\quad V'' S' = v'' S:
   \eqno {\rm (N)}$$
and therefore
$$\left. \multieqalign{
{d^2 V \over \delta x^2}
   &= {S' \over W''} {\delta^2 v \over \delta \alpha^2}
      - {v'' M \over W''}, &
{d^2 V \over \delta x \, \delta y}
   &= {S' \over W''} {\delta^2 v \over \delta \alpha \, \delta \beta}
      - {v'' M' \over W''}, \cr
{d^2 V \over \delta y^2}
   &= {S' \over W''} {\delta^2 v \over \delta \beta^2}
      - {v'' N \over W''}, &
{d^2 V \over \delta y \, \delta z}
   &= {S' \over W''} {\delta^2 v \over \delta \beta  \, \delta \gamma}
      - {v'' N' \over W''}, \cr
{d^2 V \over \delta z^2}
   &= {S' \over W''} {\delta^2 v \over \delta \gamma^2}
      - {v'' P \over W''}, &
{d^2 V \over \delta z \, \delta x}
   &= {S' \over W''} {\delta^2 v \over \delta \gamma \, \delta \alpha}
      - {v'' P' \over W''}. \cr}
   \right\}
   \eqno {\rm (O)}$$

The coefficients of the form
$\displaystyle {\delta^2 V \over \delta x^2}$,
may also be deduced from those of the form
$\displaystyle {\delta^2 W \over \delta \alpha^2}$,
in the following manner.  Differentiating the equations (K) we obtain
$$\left. \eqalign{
M  \, \delta \alpha + M' \, \delta \beta + P' \, \delta \gamma
   &= \delta' {\delta v \over \delta \alpha},\cr
M' \, \delta \alpha + N  \, \delta \beta + N' \, \delta \gamma
   &= \delta' {\delta v \over \delta \beta},\cr
P' \, \delta \alpha + N' \, \delta \beta + P  \, \delta \gamma
   &= \delta' {\delta v \over \delta \gamma};\cr}
   \right\}
   \eqno {\rm (P)}$$
we have also
$$\eqalign{
0 &= \alpha M  + \beta M' + \gamma P',\cr
0 &= \alpha M' + \beta N  + \gamma N',\cr
0 &= \alpha P' + \beta N' + \gamma P; \cr}$$
and therefore
$$\left. \eqalign{
W'' \, \delta \alpha
   &= (M + N + P) \delta' {\delta v \over \delta \alpha}
       - \left(
            M  \, \delta' {\delta v \over \delta \alpha}
          + M' \, \delta' {\delta v \over \delta \beta}
          + P' \, \delta' {\delta v \over \delta \gamma}
         \right),\cr
W'' \, \delta \beta
   &= (M + N + P) \delta' {\delta v \over \delta \beta}
       - \left(
            M' \, \delta' {\delta v \over \delta \alpha}
          + N  \, \delta' {\delta v \over \delta \beta}
          + N' \, \delta' {\delta v \over \delta \gamma}
         \right),\cr
W'' \, \delta \gamma
   &= (M + N + P) \delta' {\delta v \over \delta \gamma}
       - \left(
            P' \, \delta' {\delta v \over \delta \alpha}
          + N' \, \delta' {\delta v \over \delta \beta}
          + P  \, \delta' {\delta v \over \delta \gamma}
         \right).\cr}
   \right\}
   \eqno {\rm (Q)}$$

Now, if we put
$$\delta^2 V
   =    \delta x \, \delta {\delta V \over \delta x}
      + \delta y \, \delta {\delta V \over \delta y}
      + \delta z \, \delta {\delta V \over \delta z},$$
we shall have
$$\delta^2 V
   =    \delta \alpha \, \delta' {\delta v \over \delta \alpha}
      + \delta \beta  \, \delta' {\delta v \over \delta \beta}
      + \delta \gamma \, \delta' {\delta v \over \delta \gamma},$$
and therefore by (Q),
$$\eqalignno{
W'' \, \delta^2 V
   &= (M + N + P)
         \left\{
            \left( \delta' {\delta v \over \delta \alpha} \right)^2
          + \left( \delta' {\delta v \over \delta \beta}  \right)^2
          + \left( \delta' {\delta v \over \delta \gamma} \right)^2
         \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \biggl\{
            M  \left( \delta' {\delta v \over \delta \alpha} \right)^2
          + N  \left( \delta' {\delta v \over \delta \beta}  \right)^2
          + P  \left( \delta' {\delta v \over \delta \gamma} \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
          + 2M'   \left( \delta' {\delta v \over \delta \alpha} \right)
                  \left( \delta' {\delta v \over \delta \beta}  \right)
          + 2N'   \left( \delta' {\delta v \over \delta \beta}  \right)
                  \left( \delta' {\delta v \over \delta \gamma} \right)
          + 2P'   \left( \delta' {\delta v \over \delta \gamma} \right)
                  \left( \delta' {\delta v \over \delta \alpha} \right)
         \biggr\};
   &{\rm (R)}\cr}$$
an equation in which $\delta x$, $\delta y$, $\delta z$, are
independent, so that it is equivalent to six separate
expressions, for the six partial differential coefficients of
$V$, of the second order: and these expressions may easily be
shewn to coincide with those marked (O).  And on similar
principles we can investigate the relations between the partial
differential coefficients of the functions $V$ and $W$, for the
third and higher orders.

\bigbreak

\centerline{\it
Changes produced by Reflexions or Refractions, Ordinary or
Extraordinary.}

\nobreak\bigskip

6.
Let us now consider the sudden changes in these partial
differential coefficients of the characteristic functions of the
system, produced by reflexion or refraction, ordinary or
extraordinary.  The general formula for such changes, is, from
the nature of the integral~$V$,
$$\Delta V \, (= V_2 - V_1) = 0,
   \eqno {\rm (S)}$$
$\Delta$ being here the symbol of a finite difference, and $V_1$,
$V_2$, being two successive forms of the function~$V$, before and
after reflexion or refraction.  The condition (S) may be
considered as a form of the equation of the reflecting or
refracting surface; and if $u = 0$ be any other form for the
equation of this surface, we may, by introducing a
multiplier~$\lambda$, differentiate the following formula:
$$\Delta V \, (= V_2 - V_1) = \lambda u,
   \eqno {\rm (T)}$$
as if the coordinates $x$, $y$, $z$, were three independent
variables.  Differentiating in this manner the equation (T), and
making, after the differentiations, $u = 0$, we find
$$\left. \eqalign{
\Delta {\delta V \over \delta x}
   &= {\delta V_2 \over \delta x} - {\delta V_1 \over \delta x}
    = \lambda {\delta u \over \delta x},\cr
\Delta {\delta V \over \delta y}
   &= {\delta V_2 \over \delta y} - {\delta V_1 \over \delta y}
    = \lambda {\delta u \over \delta y},\cr
\Delta {\delta V \over \delta z}
   &= {\delta V_2 \over \delta z} - {\delta V_1 \over \delta z}
    = \lambda {\delta u \over \delta z};\cr}
   \right\}
   \eqno {\rm (U)}$$
$$\left. \eqalign{
\Delta {\delta^2 V \over \delta x^2}
   &= {\delta^2 V_2 \over \delta x^2} - {\delta^2 V_1 \over \delta x^2}
    = \lambda {\delta^2 u \over \delta x^2}
      + 2 {d\lambda \over \delta x} {\delta u \over \delta x},\cr
\Delta {\delta^2 V \over \delta y^2}
   &= {\delta^2 V_2 \over \delta y^2} - {\delta^2 V_1 \over \delta y^2}
    = \lambda {\delta^2 u \over \delta y^2}
      + 2 {d\lambda \over \delta y} {\delta u \over \delta y},\cr
\Delta {\delta^2 V \over \delta z^2}
   &= {\delta^2 V_2 \over \delta z^2} - {\delta^2 V_1 \over \delta z^2}
    = \lambda {\delta^2 u \over \delta z^2}
      + 2 {d\lambda \over \delta z} {\delta u \over \delta z},\cr
\Delta {\delta^2 V \over \delta x \, \delta y}
   &= {\delta^2 V_2 \over \delta x \, \delta y}
      - {\delta^2 V_1 \over \delta x \, \delta y}
    = \lambda {\delta^2 u \over \delta x \, \delta y}
      + {d\lambda \over \delta x} {\delta u \over \delta y}
      + {d\lambda \over \delta y} {\delta u \over \delta x},\cr
\Delta {\delta^2 V \over \delta y \, \delta z}
   &= {\delta^2 V_2 \over \delta y \, \delta z}
      - {\delta^2 V_1 \over \delta y \, \delta z}
    = \lambda {\delta^2 u \over \delta y \, \delta z}
      + {d\lambda \over \delta y} {\delta u \over \delta z}
      + {d\lambda \over \delta z} {\delta u \over \delta y},\cr
\Delta {\delta^2 V \over \delta z \, \delta x}
   &= {\delta^2 V_2 \over \delta z \, \delta x}
      - {\delta^2 V_1 \over \delta z \, \delta x}
    = \lambda {\delta^2 u \over \delta z \, \delta x}
      + {d\lambda \over \delta z} {\delta u \over \delta x}
      + {d\lambda \over \delta x} {\delta u \over \delta z}.\cr}
   \right\}
   \eqno {\rm (V)}$$

The equations marked (U), contain the laws of reflexion and
refraction, ordinary and extraordinary; since, when put by means
of (C) under the form
$$\left. \eqalign{
\Delta {\delta v \over \delta \alpha}
   &= {\delta v_2 \over \delta \alpha_2} - {\delta v_1 \over \delta \alpha_1}
    = \lambda {\delta u \over \delta x},\cr
\Delta {\delta v \over \delta \beta}
   &= {\delta v_2 \over \delta \beta_2}  - {\delta v_1 \over \delta \beta_1}
    = \lambda {\delta u \over \delta y},\cr
\Delta {\delta v \over \delta \gamma}
   &= {\delta v_2 \over \delta \gamma_2} - {\delta v_1 \over \delta \gamma_1}
    = \lambda {\delta u \over \delta z},\cr}
   \right\}
   \eqno {\rm (W)}$$
and combined with the relation
$\alpha_2^2 + \beta_2^2 + \gamma_2^2 = 1$,
they suffice to determine, for any given forms of the functions
$v_1$,~$v_2$, and for any given directions of the incident ray and
of the tangent plane to the reflecting or refracting surface, the
cosines $\alpha_2$, $\beta_2$, $\gamma_2$, of the angles which
the reflected or refracted ray makes with the axes of
coordinates, and the value of the multiplier~$\lambda$; observing
that the ratio
$${\displaystyle
         \alpha_2 \left( {\delta u \over \delta x} \right)
       + \beta_2  \left( {\delta u \over \delta y} \right)
       + \gamma_2 \left( {\delta u \over \delta z} \right)
   \over \displaystyle
         \alpha_1 \left( {\delta u \over \delta x} \right)
       + \beta_1  \left( {\delta u \over \delta y} \right)
       + \gamma_1 \left( {\delta u \over \delta z} \right)},$$
is positive in the case of refraction, and negative in that of
reflexion.  The equations (V), when combined with the relations
(H), determine the six partial differential coefficients of $V_2$
of the second order, together with the three quantities
$${\delta \lambda \over \delta x},\quad
  {\delta \lambda \over \delta y},\quad
  {\delta \lambda \over \delta z};$$
since they give, for these three latter quantities, the
conditions
$$\left. \eqalign{
0  &=    \alpha_2
            \left(
               {\delta^2 V_1 \over \delta x^2}
             + \lambda {\delta^2 u \over \delta x^2}
            \right)
       + \beta_2
            \left(
               {\delta^2 V_1 \over \delta x \, \delta y}
             + \lambda {\delta^2 u \over \delta x \, \delta y}
            \right)
       + \gamma_2
            \left(
               {\delta^2 V_1 \over \delta x \, \delta z}
             + \lambda {\delta^2 u \over \delta x \, \delta z}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\delta \lambda \over \delta x}
            \left(
               \alpha_2 {\delta u \over \delta x}
             + \beta_2  {\delta u \over \delta y}
             + \gamma_2 {\delta u \over \delta z}
            \right)
       + {\delta u \over \delta x}
            \left(
               \alpha_2 {\delta \lambda \over \delta x}
             + \beta_2  {\delta \lambda \over \delta y}
             + \gamma_2 {\delta \lambda \over \delta z}
            \right);\cr
0  &=    \alpha_2
            \left(
               {\delta^2 V_1 \over \delta x \, \delta y}
             + \lambda {\delta^2 u \over \delta x \, \delta y}
            \right)
       + \beta_2
            \left(
               {\delta^2 V_1 \over \delta y^2}
             + \lambda {\delta^2 u \over \delta y^2}
            \right)
       + \gamma_2
            \left(
               {\delta^2 V_1 \over \delta y \, \delta z}
             + \lambda {\delta^2 u \over \delta y \, \delta z}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\delta \lambda \over \delta y}
            \left(
               \alpha_2 {\delta u \over \delta x}
             + \beta_2  {\delta u \over \delta y}
             + \gamma_2 {\delta u \over \delta z}
            \right)
       + {\delta u \over \delta y}
            \left(
               \alpha_2 {\delta \lambda \over \delta x}
             + \beta_2  {\delta \lambda \over \delta y}
             + \gamma_2 {\delta \lambda \over \delta z}
            \right);\cr
0  &=    \alpha_2
            \left(
               {\delta^2 V_1 \over \delta x \, \delta z}
             + \lambda {\delta^2 u \over \delta x \, \delta z}
            \right)
       + \beta_2
            \left(
               {\delta^2 V_1 \over \delta y \, \delta z}
             + \lambda {\delta^2 u \over \delta y \, \delta z}
            \right)
       + \gamma_2
            \left(
               {\delta^2 V_1 \over \delta z^2}
             + \lambda {\delta^2 u \over \delta z^2}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {\delta \lambda \over \delta z}
            \left(
               \alpha_2 {\delta u \over \delta x}
             + \beta_2  {\delta u \over \delta y}
             + \gamma_2 {\delta u \over \delta z}
            \right)
       + {\delta u \over \delta z}
            \left(
               \alpha_2 {\delta \lambda \over \delta x}
             + \beta_2  {\delta \lambda \over \delta y}
             + \gamma_2 {\delta \lambda \over \delta z}
            \right):\cr}
   \right\}
   \eqno {\rm (X)}$$
in which the trinomial
$$ \left(
      \alpha_2 {\delta \lambda \over \delta x}
    + \beta_2  {\delta \lambda \over \delta y}
    + \gamma_2 {\delta \lambda \over \delta z}
   \right)$$
can be determined by the following relation:
$$\eqalign{
0  &=    \alpha_2^2
            \left(
               {\delta^2 V_1 \over \delta x^2}
             + \lambda {\delta^2 u \over \delta x^2}
            \right)
       + \beta_2^2
            \left(
               {\delta^2 V_1 \over \delta y^2}
             + \lambda {\delta^2 u \over \delta y^2}
            \right)
       + \gamma_2^2
            \left(
               {\delta^2 V_1 \over \delta z^2}
             + \lambda {\delta^2 u \over \delta z^2}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 2 \alpha_2 \beta_2
            \left(
               {\delta^2 V_1 \over \delta x \, \delta y}
             + \lambda {\delta^2 u \over \delta x \, \delta y}
            \right)
       + 2 \beta_2 \gamma_2
            \left(
               {\delta^2 V_1 \over \delta y \, \delta z}
             + \lambda {\delta^2 u \over \delta y \, \delta z}
            \right)
       + 2 \gamma_2 \alpha_2
            \left(
               {\delta^2 V_1 \over \delta z \, \delta x}
             + \lambda {\delta^2 u \over \delta z \, \delta x}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
        + 2 \left(
               \alpha_2 {\delta \lambda \over \delta x}
             + \beta_2  {\delta \lambda \over \delta y}
             + \gamma_2 {\delta \lambda \over \delta z}
            \right)
            \left(
               \alpha_2 {\delta u \over \delta x}
             + \beta_2  {\delta u \over \delta y}
             + \gamma_2 {\delta u \over \delta z}
            \right).\cr}$$

In a similar manner we can calculate the new values which are
given, by reflexion or refraction, to the partial differential
coefficients of $V$, of the third and higher orders; and can
thence deduce the corresponding changes in the coefficients of
the function~$W$, by means of the relations which we have already
pointed out, between these two characteristic functions;
observing, that while the value of $V$ itself is not altered in
the act of reflexion or refraction, but only its form and its
differentials, the value of $W$ receives a sudden increment,
which has for expression,
$$\eqalignno{
\Delta W
   &=    x \, \Delta {\delta v \over \delta \alpha}
       + y \, \Delta {\delta v \over \delta \beta}
       + z \, \Delta {\delta v \over \delta \gamma} \cr
   &= \lambda
         \left(
            x {\delta u \over \delta x}
          + y {\delta u \over \delta y}
          + z {\delta u \over \delta z}
         \right).
   &{\rm (Y)}\cr}$$

\bigbreak

7.
By the help of the foregoing formul{\ae}, we can compute the
partial differential coefficients of any given order, of the
characteristic functions $V$ and $W$, for any homogeneous system
of straight rays, produced by any finite number of successive
reflexions and refractions ordinary or extraordinary, when we
know the nature of the light and of the mediums, and know also
the coordinates of the luminous origin and the equations of the
reflecting or refracting surfaces.  To shew this more fully, let
us observe, that in a system of straight rays diverging from a
luminous point, and not yet reflected or refracted, we may put
$$x - X = \alpha \rho,\quad
  y - Y = \beta  \rho,\quad
  z - Z = \gamma \rho,$$
$\rho$ being the distance from the luminous origin $X$,~$Y$,~$Z$,
to any other point $x$,~$y$,~$z$; and that we have the equations,
$$\left. \eqalign{
V  &= v \rho
    =    (x - X) {\delta v \over \delta \alpha}
       + (y - Y) {\delta v \over \delta \beta}
       + (z - Z) {\delta v \over \delta \gamma},\cr
W  &=    X {\delta v \over \delta \alpha}
       + Y {\delta v \over \delta \beta}
       + Z {\delta v \over \delta \gamma},\cr}
   \right\}
   \eqno {\rm (Z)}$$
from which we can deduce the partial differentials of the
functions $V$ and $W$, in this first state of the system; those
of the second order, for example, being given by the following
expressions:
$$\eqalign{
\rho \, \delta^2 V
   &=    \delta x \, \delta' {\delta v \over \delta \alpha}
       + \delta y \, \delta' {\delta v \over \delta \beta}
       + \delta z \, \delta' {\delta v \over \delta \gamma},\cr
\delta^2 W
   &=    X \, \delta^2 {\delta v \over \delta \alpha}
       + Y \, \delta^2 {\delta v \over \delta \beta}
       + Z \, \delta^2 {\delta v \over \delta \gamma},\cr}$$
in which the symbols
$$\delta' {\delta v \over \delta \alpha},\quad
  \delta^2 {\delta v \over \delta \alpha},$$
have the same meanings as before.  Knowing, in this manner, the
differential coefficients of $V$, before the first reflexion or
refraction, we can, by the method of the preceding number,
calculate the corresponding coefficients of $V$, and thence of
$W$, immediately after that change; the coefficients of $W$, thus
deduced, will remain the same, in passing from the point of first
reflexion or refraction to the second point at which the
direction of the ray is altered, and, by the methods of the fifth
number, we can deduce from these coefficients of $W$ the
corresponding coefficients of $V$, immediately before that second
change; and so proceeding, we can calculate the alterations in
the partial differentials of the two characteristic functions,
produced by any finite number of successive reflexions or
refractions.

\bigbreak

\centerline{\it
Connexions of the two Characteristic Functions with the
Developable}
\centerline{\it Pencils and the Caustic Curves and Surfaces.}

\nobreak\bigskip

8.
Let us now suppose these partial differentials known, and let us
examine their connexion with the geometrical properties of the
system.  One of the most remarkable of these geometrical
properties is, that the rays are in general tangents to {\it two
series of caustic curves}, which are contained upon {\it two
caustic surfaces}, and form the {\it ar\^{e}tes de
rebroussement\/} of two series of {\it developable pencils\/};
that is, two series of developable surfaces, composed by rays of
the system: a property which was first discovered by {\sc Malus},
and to which I also had arrived in my own researches, before I
was aware of the existence of his.  To investigate the connexion
of these curves and surfaces with the characteristic functions
$V$ and $W$, let us consider the conditions which must be
satisfied, in order that a curve having for coordinates
$x''$,~$y''$~$z''$, should be touched by an infinite number of
rays of the system.  Let $x$,~$y$,~$z$, be the coordinates of any
point on such a ray, and $\rho$ its distance from the point of
contact $x''$~$y''$~$z''$, in such a manner that we may put
$$x = x'' + \alpha \rho,\quad
  y = y'' + \beta  \rho,\quad
  z = z'' + \gamma \rho,$$
and therefore
$$\delta \rho
   =    \alpha (\delta x - \delta x'')
      + \beta  (\delta y - \delta y'')
      + \gamma (\delta z - \delta z''):$$
we shall then have
$$\left. \eqalign{
\delta x'
   &= \delta x
      - \alpha (
            \alpha \, \delta x
          + \beta  \, \delta y
          + \gamma \, \delta z )
    = \rho \, \delta \alpha,\cr
\delta y'
   &= \delta y
      - \beta  (
            \alpha \, \delta x
          + \beta  \, \delta y
          + \gamma \, \delta z )
    = \rho \, \delta \beta,\cr
\delta z'
   &= \delta z
      - \gamma (
            \alpha \, \delta x
          + \beta  \, \delta y
          + \gamma \, \delta z )
    = \rho \, \delta \gamma,\cr}
   \right\}
   \eqno {\rm (A')}$$
assigning to $\delta x'$ $\delta y'$ $\delta z'$ the same
meanings as in the fifth number, and observing that by the nature
of $x''$~$y''$~$z''$, the variations
$\delta x''$~$\delta y''$~$\delta z''$
are proportional to $\alpha$, $\beta$, $\gamma$, so that
$$\eqalign{
\delta x''
   &= \alpha (
            \alpha \, \delta x''
          + \beta \,  \delta y''
          + \gamma \, \delta z'' ),\cr
\delta y''
   &= \beta (
            \alpha \, \delta x''
          + \beta  \, \delta y''
          + \gamma \, \delta z'' ),\cr
\delta z''
   &= \gamma (
            \alpha \, \delta x''
          + \beta  \, \delta y''
          + \gamma \, \delta z'' ).\cr}$$
The formul{\ae} (A${}'$) give
$$\left. \eqalign{
\delta' {\delta v \over \delta \alpha}
   &= \rho \, \delta {\delta v \over \delta \alpha}
    = \rho \, \delta {\delta V \over \delta x},\cr
\delta' {\delta v \over \delta \beta}
   &= \rho \, \delta {\delta v \over \delta \beta}
    = \rho \, \delta {\delta V \over \delta y},\cr
\delta' {\delta v \over \delta \gamma}
   &= \rho \, \delta {\delta v \over \delta \gamma}
    = \rho \, \delta {\delta V \over \delta z},\cr}
   \right\}
   \eqno {\rm (B')}$$
$\displaystyle \delta' {\delta v \over \delta \alpha}$
having the same meaning as in the fifth number: and these
equations (B${}'$) contain the whole theory of the developable
pencils and of the caustic curves and surfaces.  Putting them
under the form,
$$\left. \eqalign{
0  &=    \left(
            \rho {\delta^2 V \over \delta x^2}
          - {\delta^2 v \over \delta \alpha^2}
         \right)
         \, \delta x
       + \left(
            \rho {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right)
         \, \delta y
       + \left(
            \rho {\delta^2 V \over \delta x \, \delta z}
          - {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \right)
         \, \delta z,\cr
0  &=    \left(
            \rho {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right)
         \, \delta x
       + \left(
            \rho {\delta^2 V \over \delta y^2}
          - {\delta^2 v \over \delta \beta^2}
         \right)
         \, \delta y
       + \left(
            \rho {\delta^2 V \over \delta y \, \delta z}
          - {\delta^2 v \over \delta \beta  \, \delta \gamma}
         \right)
         \, \delta z,\cr
0  &=    \left(
            \rho {\delta^2 V \over \delta x \, \delta z}
          - {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \right)
         \, \delta x
       + \left(
            \rho {\delta^2 V \over \delta y \, \delta z}
          - {\delta^2 v \over \delta \beta  \, \delta \gamma}
         \right)
         \, \delta y
       + \left(
            \rho {\delta^2 V \over \delta z^2}
          - {\delta^2 v \over \delta \gamma^2}
         \right)
         \, \delta z,\cr}
   \right\}
   \eqno {\rm (C')}$$
we find by eliminating the differentials, and attending to the
relations (G), (H), the following quadratic equation
$$0 = \rho^2 V'' - \rho S + v'',
   \eqno {\rm (D')}$$
which may also be thus transformed,
$$0 = \rho^2 v'' - \rho S' + W'':
   \eqno {\rm (E')}$$
the symbols $v''$, $V''$, $W''$, $S$, $S'$, having the same
meanings as in the fifth number.  The form (D${}'$) serves to
connect the distance~$\rho$ with the function~$V$, and the form
(E${}'$) with $W$.  By either of these forms, we obtain in
general two values of $\rho$, and therefore two points
$x''$~$y''$~$z''$, which are the only points wherein the ray can
touch a caustic curve: and the locus of the points thus obtained,
composes the {\it two caustic surfaces}.  The joint equation of
these surfaces, in $x''$~$y''$~$z''$, may be found by
eliminating $\alpha$,~$\beta$,~$\gamma$, between the four
following equations:
$$\left. \eqalign{
x'' &= x_\prime + \alpha ( \alpha x'' + \beta y'' + \gamma z'' ),\cr
y'' &= y_\prime + \beta  ( \alpha x'' + \beta y'' + \gamma z'' ),\cr
z'' &= z_\prime + \gamma ( \alpha x'' + \beta y'' + \gamma z'' ),\cr
0   &=   ( \alpha x'' + \beta y'' + \gamma z'' )^2 v''
       + ( \alpha x'' + \beta y'' + \gamma z'' ) S_\prime'
       + W_\prime'';\cr}
   \right\}
   \eqno {\rm (F')}$$
in which $S_\prime'$, $W_\prime''$, are formed from $S'$, $W''$,
by changing $x$,~$y$,~$z$, to $x_\prime$,~$y_\prime$,~$z_\prime$,
these latter symbols being abridged expressions for the following
quantities,
$$\eqalign{
x - \alpha ( \alpha x + \beta y + \gamma z ) &= x_\prime,\cr
y - \beta  ( \alpha x + \beta y + \gamma z ) &= y_\prime,\cr
z - \gamma ( \alpha x + \beta y + \gamma z ) &= z_\prime,\cr}$$
and being considered as functions of $\alpha$,~$\beta$,~$\gamma$,
determined by the conditions
$$\left. \eqalign{
0  &= \alpha x_\prime + \beta y_\prime + \gamma z_\prime,\cr
{\delta W \over \delta \alpha}
   &=    x_\prime {\delta^2 v \over \delta \alpha^2}
       + y_\prime {\delta^2 v \over \delta \alpha \, \delta \beta}
       + z_\prime {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
{\delta W \over \delta \beta}
   &=    x_\prime {\delta^2 v \over \delta \alpha \, \delta \beta}
       + y_\prime {\delta^2 v \over \delta \beta^2}
       + z_\prime {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
{\delta W \over \delta \gamma}
   &=    x_\prime {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + y_\prime {\delta^2 v \over \delta \beta  \, \delta \gamma}
       + z_\prime {\delta^2 v \over \delta \gamma^2};\cr}
   \right\}
   \eqno {\rm (G')}$$
which give, after elimination,
$$\left. \eqalign{
v'' x_\prime
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         {\delta W \over \delta \alpha}
       - \left(
            {\delta^2 v \over \delta \alpha^2}
               {\delta W \over \delta \alpha}
          + {\delta^2 v \over \delta \alpha \, \delta \beta}
               {\delta W \over \delta \beta}
          + {\delta^2 v \over \delta \alpha \, \delta \gamma}
               {\delta W \over \delta \gamma}
         \right),\cr
v'' y_\prime
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         {\delta W \over \delta \beta}
       - \left(
            {\delta^2 v \over \delta \alpha \, \delta \beta}
               {\delta W \over \delta \alpha}
          + {\delta^2 v \over \delta \beta^2}
               {\delta W \over \delta \beta}
          + {\delta^2 v \over \delta \beta  \, \delta \gamma}
               {\delta W \over \delta \gamma}
         \right),\cr
v'' z_\prime
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         {\delta W \over \delta \gamma}
       - \left(
            {\delta^2 v \over \delta \alpha \, \delta \gamma}
               {\delta W \over \delta \alpha}
          + {\delta^2 v \over \delta \beta  \, \delta \gamma}
               {\delta W \over \delta \beta}
          + {\delta^2 v \over \delta \gamma^2}
               {\delta W \over \delta \gamma}
         \right).\cr}
   \right\}
   \eqno {\rm (H')}$$

The equation of the caustic surfaces in $x$,~$y$,~$z$, may also
be deduced from the characteristic function~$W$, by eliminating
$\alpha$,~$\beta$,~$\gamma$, between the equations (K) and the
following
$$W'' = 0:
   \eqno {\rm (I')}$$
or from the function~$V$, by simply putting
$${1 \over V''} = 0.
   \eqno {\rm (K')}$$

\bigbreak

9.
The formul{\ae} of the preceding number determine by
differentiations and eliminations alone, the equation of the two
caustic surfaces; but when it is required to determine also the
two series of caustic curves contained on these two surfaces, or
the two series of developable pencils composed by the tangents to
these curves, we must then have recourse to integration.  The
differential equation in $x$,~$y$,~$z$, which determines the
developable pencils, may be found by eliminating $\rho$ between
the formul{\ae} marked (B${}'$), and may be put under any one of
the three following forms:
$$\left. \eqalign{
\delta' {\delta v \over \delta \alpha}
         \mathbin{.} \delta {\delta V \over \delta y}
   =  \delta' {\delta v \over \delta \beta}
         \mathbin{.} \delta {\delta V \over \delta x},\cr
\delta' {\delta v \over \delta \beta}
         \mathbin{.} \delta {\delta V \over \delta z}
   =  \delta' {\delta v \over \delta \gamma}
         \mathbin{.} \delta {\delta V \over \delta y},\cr
\delta' {\delta v \over \delta \gamma}
         \mathbin{.} \delta {\delta V \over \delta x}
   =  \delta' {\delta v \over \delta \alpha}
         \mathbin{.} \delta {\delta V \over \delta z},\cr}
   \right\}
   \eqno {\rm (L')}$$
in which $\alpha$,~$\beta$,~$\gamma$, are considered as given
function of $x$,~$y$,~$z$, deduced from the equations (C).  The
developable pencils having been thus determined, by integrating
the equations (L${}'$), the caustic curves will be known, because
they are the {\it ar\^{e}tes de rebroussement\/} of those
pencils; the caustic curves may also be found by the condition of
being contained at once on the developable pencils and on the
caustic surfaces; or, finally, we may find the differential
equations of those curves in $x''$,~$y''$,~$z''$, without
reference to the developable pencils, by combining with the
formul{\ae} (F${}'$) the differential relation between
$\alpha$,~$\beta$,~$\gamma$, which results from the equations
(B${}'$) and admits of being put under any one of the three
following forms:
$$\left. \eqalign{
\delta' {\delta v \over \delta \alpha}
         \mathbin{.} \delta {\delta v \over \delta \beta}
   =  \delta' {\delta v \over \delta \beta}
         \mathbin{.} \delta {\delta v \over \delta \alpha},\cr
\delta' {\delta v \over \delta \beta}
         \mathbin{.} \delta {\delta v \over \delta \gamma}
   =  \delta' {\delta v \over \delta \gamma}
         \mathbin{.} \delta {\delta v \over \delta \beta},\cr
\delta' {\delta v \over \delta \gamma}
         \mathbin{.} \delta {\delta v \over \delta \alpha}
   =  \delta' {\delta v \over \delta \alpha}
         \mathbin{.} \delta {\delta v \over \delta \gamma};\cr}
   \right\}
   \eqno {\rm (M')}$$
$$\delta' {\delta v \over \delta \alpha},\quad
  \delta' {\delta v \over \delta \beta},\quad
  \delta' {\delta v \over \delta \gamma},$$
being changed to their expressions (P), or rather to the
equivalent expressions,
$$\left. \eqalign{
\delta' {\delta v \over \delta \alpha}
   &=    M_\prime  \, \delta \alpha
       + M_\prime' \, \delta \beta
       + P_\prime' \, \delta \gamma
       + (\alpha x + \beta y + \gamma z)
            \, \delta {\delta v \over \delta \alpha},\cr
\delta' {\delta v \over \delta \beta}
   &=    M_\prime' \, \delta \alpha
       + N_\prime  \, \delta \beta
       + N_\prime' \, \delta \gamma
       + (\alpha x + \beta y + \gamma z)
            \, \delta {\delta v \over \delta \beta},\cr
\delta' {\delta v \over \delta \gamma}
   &=    P_\prime' \, \delta \alpha
       + N_\prime' \, \delta \beta
       + P_\prime  \, \delta \gamma
       + (\alpha x + \beta y + \gamma z)
            \, \delta {\delta v \over \delta \gamma},\cr}
   \right\}
   \eqno {\rm (N')}$$
from which $\alpha x + \beta y + \gamma z$ will disappear, when
substituted in the equations (M${}'$), and in which
$$\left. \eqalign{
         M_\prime  \, \delta \alpha
       + M_\prime' \, \delta \beta
       + P_\prime' \, \delta \gamma
   &= \delta {\delta W \over \delta \alpha}
       - \left(
            x_\prime
            \, \delta {\delta^2 v \over \delta \alpha^2}
          + y_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
          + z_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \right),\cr
         M_\prime' \, \delta \alpha
       + N_\prime  \, \delta \beta
       + N_\prime' \, \delta \gamma
   &= \delta {\delta W \over \delta \beta}
       - \left(
            x_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
          + y_\prime
            \, \delta {\delta^2 v \over \delta \beta^2}
          + z_\prime
            \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma}
         \right),\cr
         P_\prime' \, \delta \alpha
       + N_\prime' \, \delta \beta
       + P_\prime  \, \delta \gamma
   &= \delta {\delta W \over \delta \gamma}
       - \left(
            x_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
          + y_\prime
            \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma}
          + z_\prime
            \, \delta {\delta^2 v \over \delta \gamma^2}
         \right).\cr}
   \right\}
   \eqno {\rm (O')}$$

\bigbreak

10.
A remarkable transformation of the equations (B${}'$), which
determine, as we have seen, the developable pencils, and the
caustic curves and surfaces, may be obtained in the following
manner.  We have by (P),
$$\delta' {\delta v \over \delta \alpha}
    = \delta {\delta W \over \delta \alpha}
       - \left(
            x \, \delta {\delta^2 v \over \delta \alpha^2}
          + y \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
          + z \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \right),$$
which gives
$$\delta' {\delta v \over \delta \alpha}
      - \rho \, \delta {\delta v \over \delta \alpha}
    = \delta {\delta W \over \delta \alpha}
       - \left(
            x'' \, \delta {\delta^2 v \over \delta \alpha^2}
          + y'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
          + z'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \right),$$
when we substitute for $x$,~$y$,~$z$, their expressions
$x'' + \alpha \rho$,
$y'' + \beta  \rho$,
$z'' + \gamma \rho$,
and attend to the relations (G).  And by similar substitutions in
the expressions for
$$\delta' {\delta v \over \delta \beta},
      \quad\hbox{and}\quad
  \delta' {\delta v \over \delta \gamma},$$
the equations (B${}'$) become,
$$\left. \eqalign{
\delta {\delta W \over \delta \alpha}
   &=    x'' \, \delta {\delta^2 v \over \delta \alpha^2}
       + y'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + z'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
\delta {\delta W \over \delta \beta}
   &=    x'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + y'' \, \delta {\delta^2 v \over \delta \beta^2}
       + z'' \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
\delta {\delta W \over \delta \gamma}
   &=    x'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + y'' \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma}
       + z'' \, \delta {\delta^2 v \over \delta \gamma^2}.\cr}
   \right\}
   \eqno {\rm (P')}$$

Now, if we conceive another system of rays, composed of the same
kind of light, and contained in the same medium, but all
converging to or diverging from the one point,
$x''$,~$y''$,~$z''$, and represent by $W'$, the characteristic
function, which in this new system, corresponds to $W$ in the
old, we shall have
$$\left. \eqalign{
{\delta W' \over \delta \alpha}
   &=    x''           {\delta^2 v \over \delta \alpha^2}
       + y''           {\delta^2 v \over \delta \alpha \, \delta \beta}
       + z''           {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
{\delta W' \over \delta \beta}
   &=    x''           {\delta^2 v \over \delta \alpha \, \delta \beta}
       + y''           {\delta^2 v \over \delta \beta^2}
       + z''           {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
{\delta W' \over \delta \gamma}
   &=    x''           {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + y''           {\delta^2 v \over \delta \beta  \, \delta \gamma}
       + z''           {\delta^2 v \over \delta \gamma^2},\cr
\delta {\delta W' \over \delta \alpha}
   &=    x'' \, \delta {\delta^2 v \over \delta \alpha^2}
       + y'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + z'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
\delta {\delta W' \over \delta \beta}
   &=    x'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + y'' \, \delta {\delta^2 v \over \delta \beta^2}
       + z'' \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
\delta {\delta W' \over \delta \gamma}
   &=    x'' \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + y'' \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma}
       + z'' \, \delta {\delta^2 v \over \delta \gamma^2};\cr}
   \right\}
   \eqno {\rm (Q')}$$
the equations which determine the developable pencils, and the
caustic curves and surfaces, may therefore be thus written:
$$    \delta {\delta W  \over \delta \alpha}
   =  \delta {\delta W' \over \delta \alpha};\quad
      \delta {\delta W  \over \delta \beta}
   =  \delta {\delta W' \over \delta \beta};\quad
      \delta {\delta W  \over \delta \gamma}
   =  \delta {\delta W' \over \delta \gamma}.
   \eqno {\rm (R')}$$

\bigbreak

\centerline{\it
On Osculating Focal Systems.}

\nobreak\bigskip

11.
The equations which we have thus obtained, as transformations of
the formul{\ae} (B${}'$), are not only remarkable in an analytic
view, but contain an interesting geometrical property of the
caustic surfaces.  To explain this property, it is necessary to
introduce the consideration of osculating systems of rays.  Let
us therefore conceive a system, placed in the same medium, and
composed of the same kind of light, as that given system of rays
which has $W$ for its characteristic function, but converging to
or diverging from some one point $X$,~$Y$,~$Z$; and let us denote
by $W'$, the corresponding characteristic function of this new
system, which becomes equal to the $W'$ of the preceding number,
when the point $X$,~$Y$,~$Z$, coincides with the point
$x''$,~$y''$,~$z''$; then the general expression for this
function $W'$ is
$$W' =   X \, {\delta v \over \delta \alpha}
       + Y \, {\delta v \over \delta \beta}
       + Z \, {\delta v \over \delta \gamma}
       + C,
   \eqno {\rm (S')}$$
$C$ being an arbitrary constant; and the system which thus has
$W'$ for its characteristic function, we shall call a {\it focal
system}.  The four arbitrary quantities, $X$,~$Y$,~$Z$,~$C$,
which enter into the general expression (S${}'$) for $W'$, may be
determined by the condition that for some given ray of the given
system, that is, for some given values of
$\alpha$,~$\beta$,~$\gamma$,
certain of the first terms of the development of $W'$, according
to the positive powers of the variations of
$\alpha$,~$\beta$,~$\gamma$,
may be equal to the corresponding terms in the development of the
given function~$W$; and when the form of $W'$ has been
particularized by this condition, we shall call the corresponding
system of rays an {\it osculating focal system}.  Now, if we
suppose
$\alpha$,~$\beta$,~$\gamma$,
to be changed into
$\alpha + \delta \alpha$,
$\beta + \delta \beta$,
$\gamma + \delta \gamma$,
we may express the altered values of $W$ and $W'$ by means of the
following developments:
$$W  + \delta W  + {1 \over 2} \delta^2 W
      + {1 \over 2 \mathbin{.} 3} \delta^3 W  + \hbox{\&c.},$$
$$W' + \delta W' + {1 \over 2} \delta^2 W'
      + {1 \over 2 \mathbin{.} 3} \delta^3 W' + \hbox{\&c.},$$
in which
$$\eqalign{
\delta W
   &=    {\delta W \over \delta \alpha} \, \delta \alpha
       + {\delta W \over \delta \beta}  \, \delta \beta
       + {\delta W \over \delta \gamma} \, \delta \gamma,\cr
\delta^2 W
   &=    {\delta^2 W \over \delta \alpha^2} \, \delta \alpha^2
       + {\delta^2 W \over \delta \beta^2}  \, \delta \beta^2
       + {\delta^2 W \over \delta \gamma^2} \, \delta \gamma^2
       + 2 {\delta^2 W \over \delta \alpha \, \delta \beta}
            \, \delta \alpha \, \delta \beta
       + 2 {\delta^2 W \over \delta \beta  \, \delta \gamma}
            \, \delta \beta  \, \delta \gamma
       + 2 {\delta^2 W \over \delta \gamma \, \delta \alpha}
            \, \delta \gamma \, \delta \alpha,\cr
   &\hbox{\&c.}\cr}$$

The equations
$$W' = W,\quad \delta W' = \delta W,
   \eqno {\rm (T')}$$
will be satisfied independently of the ratios of the variations
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
if we take the point $X$, $Y$, $Z$, any where upon the given ray,
and suppose,
$$C = W - \left(
            X \, {\delta v \over \delta \alpha}
          + Y \, {\delta v \over \delta \beta}
          + Z \, {\delta v \over \delta \gamma}
         \right).$$
There remains therefore one arbitrary constant of the focal
system to be determined, and this is to be done by equating the
next terms of the developments, that is by putting
$$\delta^2 W' = \delta^2 W,
   \eqno {\rm (U')}$$
and assigning some limiting ratios to the variations
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
consistent with the differential equation
$$\alpha \, \delta \alpha + \beta \, \delta \beta + \gamma \, \delta \gamma
   = 0,$$
which results from
$\alpha^2 + \beta^2 + \gamma^2 = 1$.
And, from the nature of the functions $W$, $W'$, the equation
(U${}'$) may be put under the following form:
$$\eqalignno{
0 &=     \left(
            {\delta^2 W' \over \delta \alpha^2}
          - {\delta^2 W  \over \delta \alpha^2}
         \right)
         \left(
            \delta \alpha - {\alpha \over \gamma} \, \delta \gamma
         \right)^2
     + 2 \left(
            {\delta^2 W' \over \delta \alpha \, \delta \beta}
          - {\delta^2 W  \over \delta \alpha \, \delta \beta}
         \right)
         \left(
            \delta \alpha - {\alpha \over \gamma} \, \delta \gamma
         \right)
         \left(
            \delta \beta - {\beta \over \gamma} \, \delta \gamma
         \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 W' \over \delta \beta^2}
          - {\delta^2 W  \over \delta \beta^2}
         \right)
         \left(
            \delta \beta - {\beta \over \gamma} \, \delta \gamma
         \right)^2;
   &{\rm (V')}\cr}$$
which shews that there are in general an infinite number of
osculating focal systems corresponding to any given ray, that is,
an infinite number of different values for the arbitrary
parameter which enters into the expressions of
$${\delta^2 W' \over \delta \alpha^2},\quad
  {\delta^2 W' \over \delta \alpha \, \delta \beta},\quad
  {\delta^2 W' \over \delta \beta^2},$$
according to the infinite variety of values that we may assign to
the ratio
$${\gamma \, \delta \beta - \beta \, \delta \gamma
      \over \gamma \, \delta \alpha - \alpha \, \delta \gamma};$$
but that the values of this arbitrary parameter, which do not
change for an infinitely small alteration in the ratio on which
they depend, are determined by the following equations:
$$\left. \eqalign{
0  &=    \left(
            {\delta^2 W' \over \delta \alpha^2}
          - {\delta^2 W  \over \delta \alpha^2}
         \right)
         \left(
            \delta \alpha - {\alpha \over \gamma} \, \delta \gamma
         \right)
       + \left(
            {\delta^2 W' \over \delta \alpha \, \delta \beta}
          - {\delta^2 W  \over \delta \alpha \, \delta \beta}
         \right)
         \left(
            \delta \beta - {\beta \over \gamma} \, \delta \gamma
         \right),\cr
0  &=    \left(
            {\delta^2 W' \over \delta \alpha \, \delta \beta}
          - {\delta^2 W  \over \delta \alpha \, \delta \beta}
         \right)
         \left(
            \delta \alpha - {\alpha \over \gamma} \, \delta \gamma
         \right)
       + \left(
            {\delta^2 W' \over \delta \beta^2}
          - {\delta^2 W  \over \delta \beta^2}
         \right)
         \left(
            \delta \beta - {\beta \over \gamma} \, \delta \gamma
         \right);\cr}
   \right\}
   \eqno {\rm (W')}$$
which give, by elimination,
$$    \left(
         {\delta^2 W' \over \delta \alpha^2}
       - {\delta^2 W  \over \delta \alpha^2}
      \right)
      \left(
         {\delta^2 W' \over \delta \beta^2}
       - {\delta^2 W  \over \delta \beta^2}
      \right)
   =  \left(
         {\delta^2 W' \over \delta \alpha \, \delta \beta}
       - {\delta^2 W  \over \delta \alpha \, \delta \beta}
      \right)^2.
   \eqno {\rm (X')}$$

   The systems that correspond to these extreme values of the
arbitrary parameter, we shall call the {\it extreme osculating
focal systems\/}; and since, by the nature of the functions $W$,
$W'$, the equations (W${}'$) are equivalent to the formul{\ae}
(R${}'$), the foci of these extreme osculating systems are
contained upon the caustic surfaces: and the ratios of
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
in these extreme systems, are the same as in the developable
pencils.

\bigbreak

12.
Let us consider the law of the variation of the focus of the
osculating system, between its limiting positions.  This law is
analytically expressed by the formula (U${}'$); in which we may
geometrically interpret
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
by considering these infinitely small variations of
$\alpha$,~$\beta$,~$\gamma$, as arising in the passage from the
given ray to an infinitely near ray of the system.  The plane
which passes through the given ray, and is parallel to the
infinitely near ray, may be called the {\it plane of
osculation\/}; since, if it be known, we shall know the ratios of
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
and can determine, by the formula (U${}'$), the position of the
focus of the osculating system.  To simplify this determination,
let us put
$$X = x_\prime + \alpha R,\quad
  Y = y_\prime + \beta  R,\quad
  Z = z_\prime + \gamma R,
   \eqno {\rm (Y')}$$
$X$, $Y$, $Z$, being the coordinates of the focus, and
$x_\prime$,~$y_\prime$,~$z_\prime$, having the same meanings as
in the eighth number; the formula (U${}'$) then becomes, by the
nature of $W'$, and by the relations (G),
$$R \, \delta^2 v + \delta^2 W
   =     x_\prime \, \delta^2 {\delta v \over \delta \alpha}
       + y_\prime \, \delta^2 {\delta v \over \delta \beta}
       + z_\prime \, \delta^2 {\delta v \over \delta \gamma},
   \eqno {\rm (Z')}$$
$\delta^2 v$ denoting
$$    \delta \alpha \, \delta {\delta v \over \delta \alpha}
    + \delta \beta  \, \delta {\delta v \over \delta \beta}
    + \delta \gamma \, \delta {\delta v \over \delta \gamma}.$$

The second member of this equation (Z${}'$) vanishes when the ray
passes through the origin; and if we suppose the ray to coincide
with the axis of $z$, we shall have also $\delta \gamma = 0$, and
the equation will become,
$$0 =    \left(
            R {\delta^2 v \over \delta \alpha^2}
          +   {\delta^2 W \over \delta \alpha^2}
         \right)
         \, \delta \alpha^2
     + 2 \left(
            R {\delta^2 v \over \delta \alpha \, \delta \beta}
          +   {\delta^2 W \over \delta \alpha \, \delta \beta}
         \right)
         \, \delta \alpha \, \delta \beta
       + \left(
            R {\delta^2 v \over \delta \beta^2}
          +   {\delta^2 W \over \delta \beta^2}
         \right)
         \, \delta \beta^2,
   \eqno {\rm (A'')}$$
which expresses the dependence of the parameter~$R$, on the ratio
of $\delta \beta$ to $\delta \alpha$; $R$ being now the distance
from the origin, upon the ray, to the focus of the osculating
system; and the ratio
$\displaystyle {\delta \beta \over \delta \alpha}$
being the tangent of the angle~$\phi$, comprised between the
plane of $xz$ and the plane having for equation,
$${y \over x}
   = {\delta \beta \over \delta \alpha}
   = \tan \phi,
   \eqno {\rm (B'')}$$
that is the plane of osculation.  This plane is tangent to one of
the developable pencils, when the distance~$R$ attains either of
its extreme values, corresponding to the two points where the ray
touches the caustic surfaces, and is determined by the equation,
$$       \left(
            R {\delta^2 v \over \delta \alpha^2}
          +   {\delta^2 W \over \delta \alpha^2}
         \right)
         \left(
            R {\delta^2 v \over \delta \beta^2}
          +   {\delta^2 W \over \delta \beta^2}
         \right)
       = \left(
            R {\delta^2 v \over \delta \alpha \, \delta \beta}
          +   {\delta^2 W \over \delta \alpha \, \delta \beta}
         \right)^2,
   \eqno {\rm (C'')}$$
which results by elimination from the two following:
$$\left. \eqalign{
0  &=    \left(
            R {\delta^2 v \over \delta \alpha^2}
          +   {\delta^2 W \over \delta \alpha^2}
         \right)
       + \left(
            R {\delta^2 v \over \delta \alpha \, \delta \beta}
          +   {\delta^2 W \over \delta \alpha \, \delta \beta}
         \right)
         \tan \phi,\cr
0  &=    \left(
            R {\delta^2 v \over \delta \alpha \, \delta \beta}
          +   {\delta^2 W \over \delta \alpha \, \delta \beta}
         \right)
       + \left(
            R {\delta^2 v \over \delta \beta^2}
          +   {\delta^2 W \over \delta \beta^2}
         \right)
         \tan \phi.\cr}
   \right\}
   \eqno {\rm (D'')}$$

Let $R_1$, $R_2$, be the two values of $R$, determined by the
formula (C${}''$), and $\phi_1$, $\phi_2$, the two corresponding
values of the angle~$\phi$, which may be deduced from the
following equation:
$$\eqalignno{
   &\mathrel{\phantom{=}}
         \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \alpha \, \delta \beta}
            \tan \phi
         \right)
         \!\!
         \left(
            {\delta^2 W \over \delta \alpha \, \delta \beta}
          + {\delta^2 W \over \delta \beta^2}
            \tan \phi
         \right) \cr
   &=    \left(
            {\delta^2 v \over \delta \alpha \, \delta \beta}
          + {\delta^2 v \over \delta \beta^2}
            \tan \phi
         \right)
         \!\!
         \left(
            {\delta^2 W \over \delta \alpha^2}
          + {\delta^2 W \over \delta \alpha \, \delta \beta}
            \tan \phi
         \right);
   &{\rm (E'')}\cr}$$
then the general relation (A${}''$) between $R$ and $\phi$, may
be put under the following form:
$${R - R_1 \over R_2 - R}
   = \zeta
      \left(
         {\sin (\phi - \phi_1) \over \sin (\phi_2 - \phi)}
      \right)^2,
   \eqno {\rm (F'')}$$
$\zeta$ being a coefficient which is independent of $R$ and
$\phi$, and is positive or negative according as the quantity
$${\delta^2 v \over \delta \alpha^2} {\delta^2 v \over \delta \beta^2}
    - \left( {\delta^2 v \over \delta \alpha \, \delta \beta} \right)^2$$
is positive or negative.  This latter quantity is the same with
that which we have before denoted by $v''$, because the remaining
parts of the general expressions for $v''$, namely
$$v'' =  {\delta^2 v \over \delta \alpha^2}
         {\delta^2 v \over \delta \beta^2}
       - \left( {\delta^2 v \over \delta \alpha \, \delta \beta} \right)^2
       + {\delta^2 v \over \delta \beta^2}
         {\delta^2 v \over \delta \gamma^2}
       - \left( {\delta^2 v \over \delta \beta \, \delta \gamma} \right)^2
       + {\delta^2 v \over \delta \gamma^2}
         {\delta^2 v \over \delta \alpha^2}
       - \left( {\delta^2 v \over \delta \gamma \, \delta \alpha} \right)^2,$$
vanish when $\alpha = 0$, $\beta = 0$.  If therefore $v''$ be
positive, and if we denote by $R_2$ the greater of the two values
$R_1$, $R_2$, that is the one nearer to positive infinity, we
shall have by (F${}''$), for all other values of $R$,
$$R > R_1,\quad R < R_2,\quad (v'' > 0);
   \eqno {\rm (G'')}$$
so that in this case the foci of the osculating systems are all
ranged upon that finite portion of the ray which lies between the
caustic surfaces.  If, on the contrary, $v''$ is negative, then
the two differences $R - R_1$ and $R - R_2$ are both positive or
both negative, so that
$${R - R_1 \over R - R_2} > 0,\quad (v'' < 0);
   \eqno {\rm (H'')}$$
in this case, therefore, the foci of the osculating systems are
all contained upon the remainder of the ray, that is upon the two
indefinite portions which lie outside the former interval.  And
in each case, the distances of the focus of any osculating system
from the two points in which the ray touches the two caustic
surfaces, are proportional to the squares of the sines of the
angles which the plane of osculation makes with the two tangent
planes to the developable pencils.  In the foregoing
investigations we have supposed that $W$, and its analogous
function $W'$, which we consider for symmetry as homogeneous, are
put under the form of functions of the dimension zero; a
supposition which permits us to adopt the expressions (K) for the
partial differentials
$${\delta W \over \delta \alpha},\quad
  {\delta W \over \delta \beta},\quad
  {\delta W \over \delta \gamma},$$
instead of the less simple and more general expressions given in
the fourth number: but if we had assigned any other value to the
dimension~$i$, in those more general expressions, we should have
deduced the same results respecting the law of osculation.

\bigbreak

13.
The function $v''$, the sign of which distinguishes between the
two preceding cases of osculation, has this remarkable property,
that it is independent of the direction of the coordinate axes;
in such a manner that if $\alpha$,~$\beta$,~$\gamma$, be, as
before, the cosines of the angles which the ray makes with three
given rectangular axes, and if we denote by
$\alpha'$,~$\beta'$,~$\gamma'$, the new values which these
cosines acquire when we refer the ray to three new rectangular
axes, we shall have
$$\eqalignno{
   &  {\delta^2 v \over \delta \alpha^2}
      {\delta^2 v \over \delta \beta^2}
    - \left( {\delta^2 v \over \delta \alpha \, \delta \beta} \right)^2
    + {\delta^2 v \over \delta \beta^2}
      {\delta^2 v \over \delta \gamma^2}
    - \left( {\delta^2 v \over \delta \beta \, \delta \gamma} \right)^2
    + {\delta^2 v \over \delta \gamma^2}
      {\delta^2 v \over \delta \alpha^2}
    - \left( {\delta^2 v \over \delta \gamma \, \delta \alpha} \right)^2\cr
   &\qquad =
      {\delta^2 v \over \delta \alpha'^2}
      {\delta^2 v \over \delta \beta'^2}
    - \left( {\delta^2 v \over \delta \alpha' \, \delta \beta'} \right)^2
    + {\delta^2 v \over \delta \beta'^2}
      {\delta^2 v \over \delta \gamma'^2}
    - \left( {\delta^2 v \over \delta \beta' \, \delta \gamma'} \right)^2
    + {\delta^2 v \over \delta \gamma'^2}
      {\delta^2 v \over \delta \alpha'^2}
    - \left( {\delta^2 v \over \delta \gamma' \, \delta \alpha'} \right)^2:\cr
   & &{\rm (I'')}\cr}$$
$v$ being, in the first member, a homogeneous function of
$\alpha$,~$\beta$,~$\gamma$, and, in the second member of
$\alpha'$,~$\beta'$,~$\gamma'$, of the first dimension.  To
demonstrate this theorem, let us observe that by the known
formul{\ae} for the transformation of coordinates, we may put
$$\left. \multieqalign{
\alpha  &= \alpha' A   + \beta' B   + \gamma' C, &
\alpha' &= \alpha  A   + \beta  A'  + \gamma  A'',\cr
\beta   &= \alpha' A'  + \beta' B'  + \gamma' C', &
\beta'  &= \alpha  B   + \beta  B'  + \gamma  B'',\cr
\gamma  &= \alpha' A'' + \beta' B'' + \gamma' C'', &
\gamma' &= \alpha  C   + \beta  C'  + \gamma  C'';\cr}
   \right\}
   \eqno {\rm (K'')}$$
$A$, $B$, $C$, $A'$, $B'$, $C'$, $A''$, $B''$, $C''$, being
constant quantities of which only three are arbitrary, and which
satisfy the following conditions:
$$\left. \multieqalign{
A^2   + B^2   + C^2   &= 1, &
A^2   + A'^2  + A''^2 &= 1, \cr
A'^2  + B'^2  + C'^2  &= 1, &
B^2   + B'^2  + B''^2 &= 1, \cr
A''^2 + B''^2 + C''^2 &= 1, &
C^2   + C'^2  + C''^2 &= 1, \cr
A A'    + B B'    + C C'    &= 0, &
A B     + A' B'   + A'' B'' &= 0, \cr
A' A''  + B' B''  + C' C''  &= 0, &
B C     + B' C'   + B'' C'' &= 0, \cr
A'' A   + B'' B   + C'' C   &= 0, &
C A     + C' A'   + C'' A'' &= 0. \cr}
   \right\}
   \eqno {\rm (L'')}$$

This being laid down, we have, by (K${}''$), and by the nature of
partial differentials,
$$\eqalign{
{\delta v \over \delta \alpha'}
   &=    A   {\delta v \over \delta \alpha}
       + A'  {\delta v \over \delta \beta}
       + A'' {\delta v \over \delta \gamma},\cr
{\delta v \over \delta \beta'}
   &=    B   {\delta v \over \delta \alpha}
       + B'  {\delta v \over \delta \beta}
       + B'' {\delta v \over \delta \gamma},\cr
{\delta v \over \delta \gamma'}
   &=    C   {\delta v \over \delta \alpha}
       + C'  {\delta v \over \delta \beta}
       + C'' {\delta v \over \delta \gamma}.\cr}$$
and, continuing the differentiations,
$$\eqalign{
{\delta^2 v \over \delta \alpha'^2}
   &=    A^2   {\delta^2 v \over \delta \alpha^2}
       + A'^2  {\delta^2 v \over \delta \beta^2}
       + A''^2 {\delta^2 v \over \delta \gamma^2}
       + 2 A A'   {\delta^2 v \over \delta \alpha \, \delta \beta}
       + 2 A' A'' {\delta^2 v \over \delta \beta \, \delta \gamma}
       + 2 A'' A  {\delta^2 v \over \delta \gamma \, \delta \alpha},\cr
{\delta^2 v \over \delta \beta'^2}
   &=    B^2   {\delta^2 v \over \delta \alpha^2}
       + B'^2  {\delta^2 v \over \delta \beta^2}
       + B''^2 {\delta^2 v \over \delta \gamma^2}
       + 2 B B'   {\delta^2 v \over \delta \alpha \, \delta \beta}
       + 2 B' B'' {\delta^2 v \over \delta \beta \, \delta \gamma}
       + 2 B'' B  {\delta^2 v \over \delta \gamma \, \delta \alpha},\cr
{\delta^2 v \over \delta \gamma'^2}
   &=    C^2   {\delta^2 v \over \delta \alpha^2}
       + C'^2  {\delta^2 v \over \delta \beta^2}
       + C''^2 {\delta^2 v \over \delta \gamma^2}
       + 2 C C'   {\delta^2 v \over \delta \alpha \, \delta \beta}
       + 2 C' C'' {\delta^2 v \over \delta \beta \, \delta \gamma}
       + 2 C'' C  {\delta^2 v \over \delta \gamma \, \delta \alpha},\cr}$$
$$\eqalign{
{\delta^2 v \over \delta \alpha' \, \delta \beta'}
   &=    A
            \left(
               B   {\delta^2 v \over \delta \alpha^2}
             + B'  {\delta^2 v \over \delta \alpha \, \delta \beta}
             + B'' {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \right)
       + A'
            \left(
               B   {\delta^2 v \over \delta \alpha \, \delta \beta}
             + B'  {\delta^2 v \over \delta \beta^2}
             + B'' {\delta^2 v \over \delta \beta  \, \delta \gamma}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + A''
            \left(
               B   {\delta^2 v \over \delta \alpha \, \delta \gamma}
             + B'  {\delta^2 v \over \delta \beta  \, \delta \gamma}
             + B'' {\delta^2 v \over \delta \gamma^2}
            \right),\cr
{\delta^2 v \over \delta \beta' \, \delta \gamma'}
   &=    B
            \left(
               C   {\delta^2 v \over \delta \alpha^2}
             + C'  {\delta^2 v \over \delta \alpha \, \delta \beta}
             + C'' {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \right)
       + B'
            \left(
               C   {\delta^2 v \over \delta \alpha \, \delta \beta}
             + C'  {\delta^2 v \over \delta \beta^2}
             + C'' {\delta^2 v \over \delta \beta  \, \delta \gamma}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + B''
            \left(
               C   {\delta^2 v \over \delta \alpha \, \delta \gamma}
             + C'  {\delta^2 v \over \delta \beta  \, \delta \gamma}
             + C'' {\delta^2 v \over \delta \gamma^2}
            \right),\cr
{\delta^2 v \over \delta \gamma' \, \delta \alpha'}
   &=    C
            \left(
               A   {\delta^2 v \over \delta \alpha^2}
             + A'  {\delta^2 v \over \delta \alpha \, \delta \beta}
             + A'' {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \right)
       + C'
            \left(
               A   {\delta^2 v \over \delta \alpha \, \delta \beta}
             + A'  {\delta^2 v \over \delta \beta^2}
             + A'' {\delta^2 v \over \delta \beta  \, \delta \gamma}
            \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + C''
            \left(
               A   {\delta^2 v \over \delta \alpha \, \delta \gamma}
             + A'  {\delta^2 v \over \delta \beta  \, \delta \gamma}
             + A'' {\delta^2 v \over \delta \gamma^2}
            \right);\cr}$$
and substituting these values for
$${\delta^2 v \over \delta \alpha'^2},\quad
  {\delta^2 v \over \delta \beta'^2},\quad
  {\delta^2 v \over \delta \gamma'^2},\quad
  {\delta^2 v \over \delta \alpha' \, \delta \beta'},\quad
  {\delta^2 v \over \delta \beta'  \, \delta \gamma'},\quad
  {\delta^2 v \over \delta \gamma'  \, \delta \alpha'};$$
in the second member of (I${}''$), and reducing by the relations
(K${}''$), (L${}''$), and (G), we obtain the function in the
first member.  The function $v''$, which composes the first
member of (I${}''$), may therefore be obtained by assigning to
the axes of coordinates, any arbitrary but rectangular
directions, which may most facilitate the calculation.  For
example, when we are considering an extraordinary system of rays
in a one-axed crystal, we may take the axis of the crystal for
the axis of $z$, and then the function~$v$ will take the form
$$v = \sqrt{ m^2 \gamma^2 + n^2 (\alpha^2 + \beta^2) },
   \eqno {\rm (M'')}$$
the quantities $m$, $n$, being independent of
$\alpha$,~$\beta$,~$\gamma$;
and we find by differentiation,
$$\left. \multieqalign{
v {\delta v \over \delta \alpha} &= n^2 \alpha, &
v {\delta v \over \delta \beta} &= n^2 \beta, &
v {\delta v \over \delta \gamma} &= m^2 \gamma, \cr
{v^3 \over n^2} {\delta^2 v \over \delta \alpha^2}
   &= m^2 \gamma^2 + n^2 \beta^2, &
{v^3 \over n^2} {\delta^2 v \over \delta \beta^2}
   &= m^2 \gamma^2 + n^2 \alpha^2, &
{v^3 \over n^2} {\delta^2 v \over \delta \gamma^2}
   &= m^2 (\alpha^2 + \beta^2), \cr
{v^3 \over n^2} {\delta^2 v \over \delta \alpha \, \delta \beta}
   &= - n^2 \alpha \beta, &
{v^3 \over n^2} {\delta^2 v \over \delta \beta \, \delta \gamma}
   &= - m^2 \beta \gamma, &
{v^3 \over n^2} {\delta^2 v \over \delta \gamma \, \delta \alpha}
   &= - m^2 \gamma \alpha, \cr}
   \right\}
   \eqno {\rm (N'')}$$
values which may be verified by the relations (G), and which give
$$v'' = {m^2 n^4 (\alpha^2 + \beta^2 + \gamma^2) \over v^4}
      = {m^2 n^4 \over v^4}:
   \eqno {\rm (O'')}$$
we may therefore conclude that whatever be the directions of the
rectangular axes of coordinates in an extraordinary system of
this kind, the function~$v''$ is essentially positive, and is
equal to the square of the constant~$m$, multiplied by the fourth
power of the constant~$n$, and divided by the fourth power of
$v$; $v$ being the velocity of the extraordinary rays of some
given colour, estimated on the hypothesis of molecular emission,
and the constants $m$, $n$, being the values which $v$ assumes
when the ray becomes respectively parallel and perpendicular to
the optical axis of the crystal.  Hence it follows, that in
extraordinary systems of this kind, the foci of the osculating
systems, considered in the preceding number, are all comprised
between the two points in which the given ray touches the two
caustic surfaces.  It is evident that this result extends to the
case of ordinary systems of rays, to which the expressions
(M${}''$), (N${}''$), for $v$, and for its partial differentials,
may be adapted by making $n = m$, a change which gives, by
(O${}''$), $v'' = m^2$.

\bigbreak

\centerline{\it
Principal Foci and Principal Rays.}

\nobreak\bigskip

14.
Another important property of the function~$v''$, is that when,
by the nature of the light and of the medium, this function is
essentially greater than zero, (which we have seen to be the case
for all ordinary systems of rays, and for the extraordinary
systems produced by one-axed crystals,) the intersection of the
two caustic surfaces reduces itself in general to a finite number
of isolated points.  To prove this theorem, let us resume the
formul{\ae} of the twelfth number, and let us suppose that the ray
which coincides with the axis of $z$, passes through a point of
intersection of the caustic surfaces, so that the two roots of
the quadratic (C${}''$) are equal; then the two values of
$\tan \phi$, deduced from the quadratic (E${}''$), will be
equal also; and if we put this quadratic under the form
$$E (\tan \phi)^2 - E' \tan\phi + E'' = 0,
   \eqno {\rm (P'')}$$
in which
$$\eqalign{
E     &=    {\delta^2 v \over \delta \alpha \, \delta \beta}
               {\delta^2 W \over \delta \beta^2}
          - {\delta^2 v \over \delta \beta^2}
               {\delta^2 W \over \delta \alpha \, \delta \beta},\cr
E'    &=    {\delta^2 v \over \delta \beta^2}
               {\delta^2 W \over \delta \alpha^2}
          - {\delta^2 v \over \delta \alpha^2}
               {\delta^2 W \over \delta \beta^2},\cr
E''   &=    {\delta^2 v \over \delta \alpha^2}
               {\delta^2 W \over \delta \alpha \, \delta \beta}
          - {\delta^2 v \over \delta \alpha \, \delta \beta}
               {\delta^2 W \over \delta \alpha^2},\cr}$$
we must have
$$E'^2 - 4 E E'' = 0.
   \eqno {\rm (Q'')}$$

Now the coefficients $E$, $E'$, $E''$, are connected by the
following relation:
$$    E   {\delta^2 v \over \delta \alpha^2}
    + E'  {\delta^2 v \over \delta \alpha \, \delta \beta}
    + E'' {\delta^2 v \over \delta \beta^2}
   = 0;
   \eqno {\rm (R'')}$$
and it results from this relation, that if
$${\delta^2 v \over \delta \alpha^2} {\delta^2 v \over \delta \beta^2}
      - \left( {\delta^2 v \over \delta \alpha \, \delta \beta} \right)^2
   > 0,$$
the condition (Q${}''$) cannot be satisfied without supposing
separately
$$E = 0,\quad E' = 0,\quad E'' = 0.
   \eqno {\rm (S'')}$$

We may therefore put
$${\delta^2 W \over \delta \alpha^2}
   = \mu {\delta^2 v \over \delta \alpha^2},\quad
  {\delta^2 W \over \delta \alpha \, \delta \beta}
   = \mu {\delta^2 v \over \delta \alpha \, \delta \beta},\quad
  {\delta^2 W \over \delta \beta^2}
   = \mu {\delta^2 v \over \delta \beta^2},$$
$\mu$ being a quantity which can be determined by substituting
these values in the quadratic (C${}''$); for this substitution
gives,
$$v'' (R + \mu)^2 = 0,\quad \mu = - R,$$
$R$ being the common value of the two equal roots.  Hence it
follows, that when $R$ is made equal to this value in the
equation (A${}''$) for the focus of an osculating system, that
is, when we place this focus at the intersection of the caustic
surfaces, the coefficients of
$\delta \alpha^2$, $2 \, \delta \alpha \, \delta \beta$, $\delta \beta^2$,
namely,
$$R {\delta^2 v \over \delta \alpha^2}
    + {\delta^2 W \over \delta \alpha^2},\quad
  R {\delta^2 v \over \delta \alpha \, \delta \beta}
    + {\delta^2 W \over \delta \alpha \, \delta \beta},\quad
  R {\delta^2 v \over \delta \beta^2}
    + {\delta^2 W \over \delta \beta^2},$$
become separately $= 0$; and it is easy to prove that in like
manner the coefficients of
$$    \left(
         \delta \alpha - {\alpha \over \gamma} \, \delta \gamma
      \right)^2,\quad
    2 \left(
         \delta \alpha - {\alpha \over \gamma} \, \delta \gamma
      \right)
      \left(
         \delta \beta - {\beta \over \gamma} \, \delta \gamma
      \right),\quad
      \left(
         \delta \beta - {\beta \over \gamma} \, \delta \gamma
      \right)^2,$$
must separately vanish, in the more general equation (V${}'$) of
the eleventh number; we have therefore generally, for the
intersection of the caustic surfaces, when the function $v''$ is
essentially $> 0$, the following equations:
$$\left. \multieqalign{
{\delta^2 W' \over \delta \alpha^2}
   &=  {\delta^2 W  \over \delta \alpha^2}, &
{\delta^2 W' \over \delta \alpha \, \delta \beta}
   &=  {\delta^2 W  \over \delta \alpha \, \delta \beta}, &
{\delta^2 W' \over \delta \beta^2}
   &=  {\delta^2 W  \over \delta \beta^2}, \cr
{\delta^2 W' \over \delta \alpha \, \delta \gamma}
   &=  {\delta^2 W  \over \delta \alpha \, \delta \gamma}, &
{\delta^2 W' \over \delta \beta  \, \delta \gamma}
   &=  {\delta^2 W  \over \delta \beta  \, \delta \gamma}, &
{\delta^2 W' \over \delta \gamma^2}
   &=  {\delta^2 W  \over \delta \gamma^2}, \cr}
   \right\}
   \eqno {\rm (T'')}$$
of which the three latter result from the three former.  These
six equations, which are all expressed by the one formula
(U${}'$) or (Z${}'$), provided that we consider
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
as independent, will give in general a finite number of real or
imaginary values for $\alpha$,~$\beta$,~$\gamma$, $R$, and thus
will determine a finite number of isolated points, as the
intersection of the caustic surfaces.  We shall call these points
the {\it Principal Foci\/}; and the rays to which they belong, we
shall call the {\it Principal Rays\/} of the system.  In general,
whether $v''$ be greater than or less than zero, we may employ
the equations (T${}''$) to determine a finite number of isolated
points and rays, to which we shall give the same denominations.
It results from the equations by which these points and rays are
determined, that if the focus of an osculating system be placed
at a principal focus of a given system, the osculation of the
second order will be most complete, since it will be independent
of the direction of the plane of osculation (B${}''$); the three
first terms of the two developments in the eleventh number,
namely,
$$W  + \delta W  + {\textstyle {1 \over 2}} \delta^2 W ,\quad
  W' + \delta W' + {\textstyle {1 \over 2}} \delta^2 W',$$
becoming equal, independently of the ratios of
$\delta \alpha$, $\delta \beta$, $\delta \gamma$.
The principal foci of an optical system possess many other
remarkable properties, some of which we shall indicate in the
course of the present supplement.

\bigbreak

\centerline{\it
On Osculating Spheroids and Surfaces of Constant Action.}

\nobreak\bigskip

15.
To develope one of the properties of the principal foci and
principal rays of an optical system, we must introduce the
consideration of osculating spheroids, and surfaces of constant
action.  The characteristic function~$V$, the mode of dependence
of which upon the coordinates $x$,~$y$,~$z$, distinguishes any
one system of rays from any other, having the same kind of light
and contained in the same medium, is equal, as we have seen, to
the definite integral
$\int v \, ds$,
that is to the {\it action\/} of the light, taken from the
luminous origin of the system to the point $x$,~$y$,~$z$; the
word {\it action\/} being used in the same sense as in that known
law, which is called the law of least action.  We may therefore
give the name of {\it surfaces of constant action}, to that
series of surfaces for each of which the characteristic
function~$V$ is equal to some constant quantity, and which have
for their differential equation,
$$\delta V = 0
   =   {\delta v \over \delta \alpha} \, \delta x
     + {\delta v \over \delta \beta}  \, \delta y
     + {\delta v \over \delta \gamma} \, \delta z.
   \eqno {\rm (U'')}$$

In like manner, if we denote by $V'$ the analogous characteristic
function of one of those focal systems considered in the eleventh
number, which have their light of the same kind and in the same
medium, but converging towards or diverging from one focus; the
general expression of this function $V'$ will be
$V' = v \rho + \hbox{const.}$, $\rho$ being the distance from the
focus; and the differential equation
$$\delta \mathbin{.} v \rho = 0 = \delta V',
   \eqno {\rm (V'')}$$
will represent a series of surfaces, which are analogous to the
surfaces (U${}''$).  In the case of ordinary light, these
surfaces (V${}''$) are spheres, and they may called in general,
{\it spheroids of constant action\/}; the focus of the focal
system being called the {\it centre\/} of the spheroid.  The
general equation of such a spheroid contains four arbitrary
constants, of which three are the coordinates of the centre; and
if we determine these four constants, by the condition that for
some given values of $x$,~$y$,~$z$, that is for some given point
of a given system, certain first terms of the development
$$V' = \delta V' + {\textstyle {1 \over 2}} \delta^2 V' + \hbox{\&c.}$$
may be equal to the corresponding terms of the development
$$V  = \delta V  + {\textstyle {1 \over 2}} \delta^2 V  + \hbox{\&c.}$$
the spheroid thus determined will be an {\it osculating
spheroid}, to the surface of constant action which passes through
the given point of the system.  The conditions
$$V' = V,\quad \delta V' = \delta V,
   \eqno {\rm (W'')}$$
may be satisfied independently of the ratios of
$\delta x$, $\delta y$, $\delta z$,
by taking the centre of the spheroid any where upon the given
ray, that is, by establishing between the three coordinates of
this centre the two equations of the ray, and by assigning a
proper value to the other arbitrary constant; there still remains
therefore, after satisfying the conditions (W${}''$), an arbitrary
parameter depending on the position of the centre, which we may
determine by the equation,
$$\delta^2 V' = \delta^2 V,
   \eqno {\rm (X'')}$$
assigning any arbitrary ratios to the three variations
$\delta x$, $\delta y$, $\delta z$, or rather any value to the
one ratio
$${\gamma \, \delta x - \alpha \, \delta z
      \over \gamma \, \delta y - \beta \, \delta z};$$
because, by the relations (H),
$$\delta^2 V
   =  {\delta^2 V \over \delta x^2}
      \left(
         \delta x - {\alpha \over \gamma} \, \delta z
      \right)^2
    + 2 {\delta^2 V \over \delta x \, \delta y}
      \left(
         \delta x - {\alpha \over \gamma} \, \delta z
      \right)
      \left(
         \delta y - {\beta \over \gamma} \, \delta z
      \right)
    + {\delta^2 V \over \delta y^2}
      \left(
         \delta y - {\beta \over \gamma} \, \delta z
      \right)^2,$$
so that the condition (X${}''$) may be thus written:
$$\eqalignno{
0  &= \left(
         {\delta^2 V' \over \delta x^2}
       - {\delta^2 V  \over \delta x^2}
      \right)
      \left(
         \delta x - {\alpha \over \gamma} \, \delta z
      \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
  + 2 \left(
         {\delta^2 V' \over \delta x \, \delta y}
       - {\delta^2 V  \over \delta x \, \delta y}
      \right)
      \left(
         \delta x - {\alpha \over \gamma} \, \delta z
      \right)
      \left(
         \delta y - {\beta \over \gamma} \, \delta z
      \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left(
         {\delta^2 V' \over \delta y^2}
       - {\delta^2 V  \over \delta y^2}
      \right)
      \left(
         \delta y - {\beta \over \gamma} \, \delta z
      \right)^2,
   &{\rm (Y'')}\cr}$$
or, by a further transformation,
$$\eqalignno{
0  &= \left(
         {1 \over \rho} \mathbin{.}
         {\delta^2 v \over \delta \alpha^2}
       - {\delta^2 V  \over \delta x^2}
      \right)
      \left(
         \delta x - {\alpha \over \gamma} \, \delta z
      \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
  + 2 \left(
         {1 \over \rho} \mathbin{.}
         {\delta^2 v \over \delta \alpha \, \delta \beta}
       - {\delta^2 V  \over \delta x \, \delta y}
      \right)
      \left(
         \delta x - {\alpha \over \gamma} \, \delta z
      \right)
      \left(
         \delta y - {\beta \over \gamma} \, \delta z
      \right) \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \left(
         {1 \over \rho} \mathbin{.}
         {\delta^2 v \over \delta \beta^2}
       - {\delta^2 V  \over \delta y^2}
      \right)
      \left(
         \delta y - {\beta \over \gamma} \, \delta z
      \right)^2,
   &{\rm (Z'')}\cr}$$
$\rho$ being here the distance of the point $x$~$y$~$z$ upon
the ray, beyond the centre of the spheroid.  This equation
(Z${}''$) contains the law of osculation of the spheroid, since
it expresses the dependence of the distance~$\rho$ on the
direction of the plane passing through the ray and through the
consecutive point
$x + \delta x$, $y + \delta y$, $z + \delta z$.
We shall call this plane the plane of osculation of the spheroid;
and we see, by comparing (Z${}''$) with (C${}'$), that the
extreme values of $\rho$ correspond to those directions of the
plane of osculation in which it touches the developable pencils;
while the corresponding extreme positions of the centre of the
osculating spheroid, are contained upon the caustic surfaces.
And when the ray is one of those principal rays determined in the
preceding number, it is easy to prove that the equation (Z${}''$)
is satisfied independently of the ratios of the differentials, if
we assign to $\rho$ the value which belongs to the principal
focus; the principal foci are therefore the centres of spheroids,
which have complete contact of the second order with the surfaces
of constant action.  The equations which express this property of
the principal foci are
$$\left. \multieqalign{
{1 \over \rho} {\delta^2 v \over \delta \alpha^2}
   &= {\delta^2 V \over \delta x^2}, &
{1 \over \rho} {\delta^2 v \over \delta \alpha \, \delta \beta}
   &= {\delta^2 V \over \delta x \, \delta y}, &
{1 \over \rho} {\delta^2 v \over \delta \beta^2}
   &= {\delta^2 V \over \delta y^2}, \cr
{1 \over \rho} {\delta^2 v \over \delta \alpha \, \delta \gamma}
   &= {\delta^2 V \over \delta x \, \delta z}, &
{1 \over \rho} {\delta^2 v \over \delta \beta  \, \delta \gamma}
   &= {\delta^2 V \over \delta y \, \delta z}, &
{1 \over \rho} {\delta^2 v \over \delta \gamma^2}
   &= {\delta^2 V \over \delta z^2}, \cr}
   \right\}
   \eqno {\rm (A''')}$$
of which any three include the rest; they may also be thus
written,
$$\left. \multieqalign{
{\delta^2 V' \over \delta x^2}
   &= {\delta^2 V \over \delta x^2}, &
{\delta^2 V' \over \delta y^2}
   &= {\delta^2 V \over \delta y^2}, &
{\delta^2 V' \over \delta z^2}
   &= {\delta^2 V \over \delta z^2}, \cr
{\delta^2 V' \over \delta x \, \delta y}
   &= {\delta^2 V \over \delta x \, \delta y}, &
{\delta^2 V' \over \delta y \, \delta z}
   &= {\delta^2 V \over \delta y \, \delta z}, &
{\delta^2 V' \over \delta z \, \delta x}
   &= {\delta^2 V \over \delta z \, \delta x}; \cr}
   \right\}
   \eqno {\rm (B''')}$$
and may be summed up in the one equation (X${}''$), by
considering $\delta x$, $\delta y$, $\delta z$, as independent.
With respect to those rays which are not the principal rays of the
system, and for which the equation (X${}''$) can only be
satisfied by assigning some particular value to the ratio
$${\gamma \, \delta x - \alpha \, \delta z
      \over \gamma \, \delta y - \beta \, \delta z},$$
that is, some particular position to the plane of osculation of
the spheroid, we find, by reasonings similar to those of the
twelth number, the following law of osculation:
$${\displaystyle {1 \over \rho_1} - {1 \over \rho}
      \over \displaystyle {1 \over \rho} - {1 \over \rho_2}}
   = \zeta
      \left(
         {\sin (\psi - \psi_1) \over \sin (\psi_2 - \psi)}
      \right)^2:
   \eqno {\rm (C''')}$$
$\rho_1$, $\rho_2$, being the extreme values of $\rho$;
$\psi_1$,~$\psi_2$, the corresponding values of the angle~$\psi$,
comprised between the plane of osculation and any fixed plane
that passes through the ray; and the coefficient~$\zeta$ being
independent of $\rho$ and $\psi$, and having the same meaning as
before.  The formula (C${}'''$) may be written in the following
manner:
$${\rho - \rho_1 \over \rho_2 - \rho}
   = {\zeta \rho_1 \over \rho_2} \mathbin{.}
      \left(
         {\sin (\psi - \psi_1) \over \sin (\psi_2 - \psi)}
      \right)^2:
   \eqno {\rm (D''')}$$
in this kind of osculation, therefore, as in the former, the
distances of the variable focus or centre from the points where
the ray touches the two caustic surfaces, are proportional to the
squares of the sines of the angles which the plane of osculation
makes with the tangent planes to the developable pencils.

\bigbreak

\centerline{\it
On Osculating Focal Reflectors or Refractors.}

\nobreak\bigskip

16.
Besides the two preceding kinds of osculation, it is interesting
to consider a third kind, which exists between the last
reflecting or refracting surface, and certain other surfaces,
which would have reflected or refracted to or from one focus the
rays of the last incident system, and which we shall therefore
call {\it focal reflectors or refractors}.  Let $V_1$,~$V_2$,
denote, as in the sixth number, any two successive forms of the
characteristic function~$V$, of which we shall suppose that $V_2$
belongs to the system in its given state, and $V_1$ to the same
system before its last reflexion or refraction; then, by the
number cited, the equation $V_1 - V_2 = 0$, will be a form for
the equation of the reflector or refractor, at which the state of
the system was last changed, and which we shall consider as
known.  Let $V_2'$ be the form which $V_2$ would have, if the rays
of the final system all converged or diverged from one focus,
this form being such as was assigned in the fifteenth number, and
depending only on the nature of the light and of the final
medium, but involving four arbitrary constants, of which three
are the coordinates of the focus; then it is easy to prove that
the equation with four arbitrary constants, of the focal surface,
which would have reflected or refracted to or from one focus the
rays of the last incident system, is
$$V_1 - V_2' = 0.
   \eqno {\rm (E''')}$$

We may determine the four arbitrary constants of $V_2'$ in this
equation, by the condition that the focal reflector or refractor
shall touch the given reflector or refractor at a given point,
and osculate in a given direction.  The condition of contact, of
the first order, is expressed by the equations
$$V_2 = V_2',\quad \delta V_2 = \delta V_2',
   \eqno {\rm (F''')}$$
and may be satisfied by establishing between the three
coordinates of the focus the two equations of the ray, and by
assigning a proper value to the remaining arbitrary constant; and
the position of the focus upon the given ray, may be determined
by the condition of osculation in the given direction, which is
expressed by the equation
$$\delta^2 V_2 = \delta^2 V_2',
   \eqno {\rm (G''')}$$
assigning the given ratios to the variations
$\delta x$, $\delta y$, $\delta z$.
This equation (G${}'''$) being the same with that marked
(X${}''$) in the foregoing number, we can deduce from it the same
consequences; the osculation therefore between the focal surface
(E${}'''$) and the given reflector or refractor, follows the same
law as the osculation between the spheroid of constant action
(V${}''$) and the given surface (U${}''$) for which the
function~$V$ is constant; in such a manner that the focus of the
focal reflector or refractor coincides with the centre of the
spheroid, if the point of contact, and the plane of osculation be
the same.  The distances therefore of the focus of the focal
reflector or refractor from the points in which the ray touches
the two caustic surfaces, are proportional to the squares of the
sines of the angles which the plane of osculation makes with the
tangent planes to the two developable pencils.  And when the ray
is one of those principal rays, assigned in the fourteenth
number, (the focus of the focal surface being at the principal
focus corresponding,) then the contact of the second order is
most complete, and the two reflectors or refractors osculate to
each other in all directions.

\bigbreak

\centerline{\it
On Foci by Projection, and Virtual Foci.}

\nobreak\bigskip

17.
Another kind of focus, of which the law is similar, though not
the same, may be deduced in the following manner.  If we conceive
a plane passing through a given ray of a given optical system,
and through a point infinitely near to this given ray; the ray
which passes through the near point may be projected on the
plane, and the intersection of its projection with the given ray
may be called a {\it focus by projection}.  Suppose, to simplify
the first calculations, that the given ray is the axis of $z$,
and that the infinitely near point is contained in the plane of
$xy$; its coordinates in this plane being denoted by
$\delta x$, $\delta y$,
and the cosines of the angles which the near ray makes with the
axes of $x$ and $y$, being $\delta \alpha$, $\delta \beta$: then,
if we denote the general coordinates of this near ray by
$x_{\prime\prime}$~$y_{\prime\prime}$~$z_{\prime\prime}$,
its equations may be thus written,
$$x_{\prime\prime} = \delta x + z_{\prime\prime} \, \delta \alpha,\quad
  y_{\prime\prime} = \delta y + z_{\prime\prime} \, \delta \beta,
   \eqno {\rm (H''')}$$
and the connexions between
$\delta x$, $\delta y$, $\delta \alpha$, $\delta \beta$,
will be expressed by the two following conditions:
$$\left. \eqalign{
         {\delta^2 V \over \delta x^2} \, \delta x
       + {\delta^2 V \over \delta x \, \delta y} \, \delta y
   &=    {\delta^2 v \over \delta \alpha^2} \, \delta \alpha
       + {\delta^2 v \over \delta \alpha \, \delta \beta} \, \delta \beta,\cr
         {\delta^2 V \over \delta x \, \delta y} \, \delta x
       + {\delta^2 V \over \delta y^2} \, \delta y
   &=    {\delta^2 v \over \delta \alpha \, \delta \beta} \, \delta \alpha
       + {\delta^2 v \over \delta \beta^2} \, \delta \beta,\cr}
   \right\}
   \eqno {\rm (I''')}$$
which are obtained by differentiating (C) and making
$\delta z = 0$, $\delta \gamma = 0$.
The equation of the plane on which the near ray (H${}'''$) is to
be projected, may be put under the form
$${y_{\prime\prime} \over x_{\prime\prime}}
   = {\delta y \over \delta x};
   \eqno {\rm (K''')}$$
and if $p$ be the vertical ordinate of the focus by projection,
the equation of the projecting plane is
$${y_{\prime\prime} - \delta y - z_{\prime\prime} \, \delta \beta
      \over
   x_{\prime\prime} - \delta x - z_{\prime\prime} \, \delta \alpha}
   = {\delta y + p \mathbin{.} \delta \beta
      \over \delta x + p \mathbin{.} \delta \alpha},
   \eqno {\rm (L''')}$$
$p$ being determined by the condition that the two planes
(K${}'''$) (L${}'''$) shall be perpendicular to each other, which
gives
$${1 \over p}
   = - {\delta x \, \delta \alpha + \delta y \, \delta \beta
         \over \delta x^2 + \delta y^2}.
   \eqno {\rm (M''')}$$

In general, whatever arbitrary position we assign to the
rectangular axes, if we represent by
$x + \alpha p$, $y + \beta p$, $z + \gamma p$,
the coordinates of the focus by projection, those of the given
point being $x$,~$y$,~$z$, and those of the near point
$x + \delta x$, $y + \delta y$, $z + \delta z$,
we shall find by a similar process,
$$-{1 \over p}
   = {   \delta \alpha \, \delta x'
       + \delta \beta  \, \delta y'
       + \delta \gamma \, \delta z'
         \over \delta x'^2 + \delta y'^2 + \delta z'^2 }
   = {   \delta \alpha \, \delta x
       + \delta \beta  \, \delta y
       + \delta \gamma \, \delta z
         \over \delta x^2 + \delta y^2 + \delta z^2
         - (\alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z)^2 },
   \eqno {\rm (N''')}$$
$\delta x'$, $\delta y'$, $\delta z'$, having the same meanings
as in the fifth number.  And since the equations (C) give, by
differentiation and elimination,
$$\left. \eqalign{
v'' \, \delta \alpha
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         \, \delta {\delta V \over \delta x}
       - \left(
            {\delta^2 v \over \delta \alpha^2}
            \, \delta {\delta V \over \delta x}
          + {\delta^2 v \over \delta \alpha \, \delta \beta}
            \, \delta {\delta V \over \delta y}
          + {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \, \delta {\delta V \over \delta z}
         \right),\cr
v'' \, \delta \beta
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         \, \delta {\delta V \over \delta y}
       - \left(
            {\delta^2 v \over \delta \alpha \, \delta \beta}
            \, \delta {\delta V \over \delta x}
          + {\delta^2 v \over \delta \beta^2}
            \, \delta {\delta V \over \delta y}
          + {\delta^2 v \over \delta \beta  \, \delta \gamma}
            \, \delta {\delta V \over \delta z}
         \right),\cr
v'' \, \delta \gamma
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         \, \delta {\delta V \over \delta z}
       - \left(
            {\delta^2 v \over \delta \alpha \, \delta \gamma}
            \, \delta {\delta V \over \delta x}
          + {\delta^2 v \over \delta \beta  \, \delta \gamma}
            \, \delta {\delta V \over \delta y}
          + {\delta^2 v \over \delta \gamma^2}
            \, \delta {\delta V \over \delta z}
         \right),\cr}
   \right\}
   \eqno {\rm (O''')}$$
and therefore
$$\eqalignno{
      v'' ( \delta \alpha \, \delta x
          + \delta \beta  \, \delta y
          + \delta \gamma \, \delta z )
   &=    \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         \, \delta^2 V \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \left(
            \delta' {\delta v \over \delta \alpha}
            \, \delta {\delta V \over \delta x}
          + \delta' {\delta v \over \delta \beta}
            \, \delta {\delta V \over \delta y}
          + \delta' {\delta v \over \delta \gamma}
            \, \delta {\delta V \over \delta z}
         \right),
   &{\rm (P''')}\cr}$$
we find, finally,
$$\eqalignno{
{v'' \over p} (\delta x'^2 + \delta y'^2 + \delta z'^2)
   &=       \delta {\delta V \over \delta x}
            \, \delta' {\delta v \over \delta \alpha}
          + \delta {\delta V \over \delta y}
            \, \delta' {\delta v \over \delta \beta}
          + \delta {\delta V \over \delta z}
            \, \delta' {\delta v \over \delta \gamma} \cr
   &\mathrel{\phantom{=}} \mathord{}
       - \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
          + {\delta^2 v \over \delta \gamma^2}
         \right)
         \, \delta^2 V:
   &{\rm (Q''')}\cr}$$
$v''$ being the same function as before.  It results from this
equation (Q${}'''$) or from (M${}'''$) and (I${}'''$) that when
the given ray is taken for the axis of $z$ we shall have
$$\eqalignno{
{v'' \over p}
   &=    \left(
            {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \beta^2}
            {\delta^2 V \over \delta x^2}
         \right)
         (\cos \Pi)^2
       + \left(
            {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha^2}
            {\delta^2 V \over \delta y^2}
         \right)
         (\sin \Pi)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left\{
            {\delta^2 v \over \delta \alpha \, \delta \beta}
            \left(
               {\delta^2 V \over \delta x^2}
             + {\delta^2 V \over \delta y^2}
            \right)
          - {\delta^2 V \over \delta x \, \delta y}
            \left(
               {\delta^2 v \over \delta \alpha^2}
             + {\delta^2 v \over \delta \beta^2}
            \right)
         \right\}
         \sin \Pi \, \cos \Pi,
   &{\rm (R''')}\cr}$$
if we put $\delta y = \delta x \, \tan \Pi$, so that $\Pi$
denotes the angle which the plane of projection makes with the
plane of $xz$.  Differentiating (R${}'''$) for $\Pi$ only, we
find that the values of this angle which correspond to the
extreme positions of the focus by projection are determined by
the condition
$$    \left(
         {\delta^2 v \over \delta \alpha^2}
         {\delta^2 V \over \delta y^2}
       - {\delta^2 v \over \delta \beta^2}
         {\delta^2 V \over \delta x^2}
      \right)
      \tan 2\Pi
   =     {\delta^2 v \over \delta \alpha \, \delta \beta}
         \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
         \right)
       - {\delta^2 V \over \delta x \, \delta y}
         \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
         \right):
   \eqno {\rm (S''')}$$
the {\it planes of extreme projection}, that is, the planes which
correspond to the extreme values of $p$, are therefore
perpendicular to each other; and if we suppose them taken for the
planes of $xz$, $yz$, and denote by $p_1$, $p_2$, the
corresponding values of $p$, we shall have
$$\left. \eqalign{
0  &=    {\delta^2 v \over \delta \alpha \, \delta \beta}
         \left(
            {\delta^2 V \over \delta x^2}
          + {\delta^2 V \over \delta y^2}
         \right)
       - {\delta^2 V \over \delta x \, \delta y}
         \left(
            {\delta^2 v \over \delta \alpha^2}
          + {\delta^2 v \over \delta \beta^2}
         \right),\cr
{v'' \over p_1}
   &=    {\delta^2 v \over \delta \alpha \, \delta \beta}
         {\delta^2 V \over \delta x \, \delta y}
       - {\delta^2 v \over \delta \beta^2}
         {\delta^2 V \over \delta x^2},\quad
{v'' \over p_2}
    =    {\delta^2 v \over \delta \alpha \, \delta \beta}
         {\delta^2 V \over \delta x \, \delta y}
       - {\delta^2 v \over \delta \alpha^2}
         {\delta^2 V \over \delta y^2},\cr}
   \right\}
   \eqno {\rm (T''')}$$
and finally the dependence of $p$ upon $\Pi$, that is, the law of
the focus by projection will be expressed by the following
formula:
$${1 \over p}
   =     {1 \over p_1} (\cos \Pi)^2
       + {1 \over p_2} (\sin \Pi)^2.
   \eqno {\rm (U''')}$$

When the given ray is one of those principal rays determined in
the foregoing numbers, the angle~$\Pi$ disappears from this
formula, and all the foci by projection coincide in the principal
focus, the condition (S${}'''$) being at the same time
identically satisfied, and failing to determine the planes of
extreme projection: but in general these planes can be determined
by that condition, and have a remarkable connexion with the
tangent planes to the developable pencils, which can be deduced
from the equation (L${}'$) of the ninth number,
$$\delta' {\delta v \over \delta \alpha}
      \, \delta {\delta V \over \delta y}
    = \delta' {\delta v \over \delta \beta}
      \, \delta {\delta V \over \delta x}.$$

For, when we suppose $\delta z = 0$,
$\delta y = \delta x \, \tan \Pi$,
we find from this equation (L${}'$) the following quadratic
equation to determine the two values of $\tan \Pi$
corresponding to the tangent planes of the two developable
pencils:
$$\eqalignno{
0  &=       {\delta^2 v \over \delta \alpha^2}
            {\delta^2 V \over \delta x \, \delta y}
          - {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta^2 V \over \delta x^2}
       + \left(
            {\delta^2 v \over \delta \alpha \, \delta \beta}
            {\delta^2 V \over \delta y^2}
          - {\delta^2 v \over \delta \beta^2}
            {\delta^2 V \over \delta x \, \delta y}
         \right)
         (\tan \Pi)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \left(
            {\delta^2 v \over \delta \alpha^2}
            {\delta^2 V \over \delta y^2}
          - {\delta^2 v \over \delta \beta^2}
            {\delta^2 V \over \delta x^2}
         \right)
         \tan \Pi:
   &{\rm (V''')}\cr}$$
and if the first condition (T${}'''$) be satisfied, that is, if
the planes of extreme projection be taken for the planes of $xz$,
$yz$, the product of the two values of $\tan \Pi$
determined by this quadratic will be unity; the tangent planes to
the developable pencils are therefore symmetrically situated with
respect to the planes of extreme projection, the bisectors of the
angles formed by the one pair of planes bisecting also the angles
of the other pair.  The tangent planes to the developable pencils
are not always perpendicular to each other, and therefore are not
always fit to be taken for rectangular coordinate planes, however
remarkable they may be in other respects; but the planes of
extreme projection, determined in the present number, possess
this important property, and may be considered as furnishing for
any given straight ray of an optical system, ordinary or
extraordinary, (except the principal rays,) two {\it natural
coordinate planes}, which contain the given ray, and are
perpendicular to each other.  And whenever the developable
pencils are also perpendicular to each other, the tangent planes
to these pencils will coincide with the planes of extreme
projection, and the extreme foci by projection will be contained
upon the caustic surfaces.  This perpendicularity of the
developable pencils requires that there should exist a series of
surfaces perpendicular to the rays of the system, and having for
their differential equation
$$\alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z
   = 0;
   \eqno {\rm (W''')}$$
and reciprocally when this equation is integrable, the
perpendicularity here spoken of, exists, and we shall say that
the system is {\it rectangular}.  This condition is satisfied in
the case of ordinary systems, because, for such systems, the
differential equation (U${}''$) of the surfaces of constant
action becomes
$$\delta V
   = m ( \alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z )
   = 0,$$
and consequently coincides with the equation (W${}'''$), $m$
having the same meaning as in the thirteenth number; the rays of
an ordinary system are therefore perpendicular to the surfaces
for which the function~$V$ is constant, and their planes of
extreme projection are touched by the developable pencils.  We
may also remark that for such systems $\zeta = 1$, and the
osculating foci coincide with the foci by projection.

\bigbreak

18.
There is yet another kind of foci which we shall call
{\it Virtual Foci}, and which it may be interesting to consider,
because they conduct to the same pair of natural coordinate
planes as those which we have deduced in the foregoing number,
and because they furnish new applications of the characteristic
functions of the system.  By a {\it virtual focus\/} of a given
ray, we shall understand a point in which it is nearest to an
infinitely near ray of the system.  To explain this more fully,
let us observe, that if we establish any arbitrary relation
between $\alpha$,~$\beta$,~$\gamma$, distinct from the relation
$\alpha^2 + \beta^2 + \gamma^2 = 1$,
we shall obtain some corresponding relation between
$${\delta V \over \delta x},\quad
  {\delta V \over \delta y},\quad
  {\delta V \over \delta z},$$
by eliminating $\alpha$,~$\beta$,~$\gamma$, between the
equations~(C); the result of this elimination, which we may
represent by
$$F   \left(
         {\delta V \over \delta x},
         {\delta V \over \delta y},
         {\delta V \over \delta z}
      \right)
   = 0,$$
$F$ denoting an arbitrary function, will be the equation of a
{\it pencil}, that is of a surface of right lines, composed by
rays of the system: and unless this surface be one of the
developable pencils determined in the ninth number, the rays of
which it is composed will not intersect consecutively, so that
there will be only a {\it virtual intersection}, or nearest
approach, even between two infinitely near rays.  To find the
coordinates of this virtual intersection, we are to seek the
minimum of
$\delta x^2 + \delta y^2 + \delta z^2$, or of
$\delta x'^2 + \delta y'^2 + \delta z'^2$,
corresponding to given values of
$\alpha$,~$\beta$,~$\gamma$,
$\delta \alpha$, $\delta \beta$, $\delta \gamma$.
Now if we put
$r = \alpha x + \beta y + \gamma z$,
we shall have
$$\left. \multieqalign{
x &= x_\prime + \alpha r, &
y &= y_\prime + \beta  r, &
z &= z_\prime + \gamma r, \cr
\delta x &= \delta x_\prime + \delta \mathbin{.} \alpha r, &
\delta y &= \delta y_\prime + \delta \mathbin{.} \beta  r, &
\delta z &= \delta z_\prime + \delta \mathbin{.} \gamma r, \cr}
   \right\}
   \eqno {\rm (X''')}$$
and therefore
$$\left. \eqalign{
\delta x'
   &= r \, \delta \alpha + \delta x_\prime
       - \alpha (   \alpha \, \delta x_\prime
                  + \beta  \, \delta y_\prime
                  + \gamma \, \delta z_\prime ),\cr
\delta y'
   &= r \, \delta \beta  + \delta y_\prime
       - \beta  (   \alpha \, \delta x_\prime
                  + \beta  \, \delta y_\prime
                  + \gamma \, \delta z_\prime ),\cr
\delta z'
   &= r \, \delta \gamma + \delta z_\prime
       - \gamma (   \alpha \, \delta x_\prime
                  + \beta  \, \delta y_\prime
                  + \gamma \, \delta z_\prime ),\cr}
   \right\}
   \eqno {\rm (Y''')}$$
$x_\prime$, $y_\prime$, $z_\prime$, and
$\delta x'$, $\delta y'$, $\delta z'$,
having the same meanings as before; and the condition of minimum
gives
$$r = - {   \delta \alpha \, \delta x_\prime
          + \delta \beta  \, \delta y_\prime
          + \delta \gamma \, \delta z_\prime
            \over \delta \alpha^2 + \delta \beta^2 + \delta \gamma^2 },
   \eqno {\rm (Z''')}$$
which may also be thus written
$$0 =    \delta \alpha \, \delta x'
       + \delta \beta  \, \delta y'
       + \delta \gamma \, \delta z'
    =    \delta \alpha \, \delta x
       + \delta \beta  \, \delta y
       + \delta \gamma \, \delta z:
   \eqno {\rm (A^4)}$$
or, by the foregoing number,
$$    \left(
         {\delta^2 v \over \delta \alpha^2}
       + {\delta^2 v \over \delta \beta^2}
       + {\delta^2 v \over \delta \gamma^2}
      \right)
      \delta^2 V
   =  \delta' {\delta v \over \delta \alpha}
         \, \delta {\delta V \over \delta x}
    + \delta' {\delta v \over \delta \beta}
         \, \delta {\delta V \over \delta y}
    + \delta' {\delta v \over \delta \gamma}
         \, \delta {\delta V \over \delta z}.
   \eqno {\rm (B^4)}$$

Another transformation of this condition, which shall involve the
function~$W$ instead of $V$, may be obtained in the following
manner.  Let $W_\prime$ be the form which the characteristic
function~$W$ would have, for a system of rays of the same light
and in the same medium, but all converging towards or diverging
from the one point
$x_\prime$,~$y_\prime$,~$z_\prime$;
so that, by the theory already given,
$$\left. \eqalign{
\delta {\delta W_\prime \over \delta \alpha}
   &=    x_\prime
            \, \delta {\delta^2 v \over \delta \alpha^2}
       + y_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + z_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma},\cr
\delta {\delta W_\prime \over \delta \beta}
   &=    x_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \beta}
       + y_\prime
            \, \delta {\delta^2 v \over \delta \beta^2}
       + z_\prime
            \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma},\cr
\delta {\delta W_\prime \over \delta \gamma}
   &=    x_\prime
            \, \delta {\delta^2 v \over \delta \alpha \, \delta \gamma}
       + y_\prime
            \, \delta {\delta^2 v \over \delta \beta  \, \delta \gamma}
       + z_\prime
            \, \delta {\delta^2 v \over \delta \gamma^2}:\cr}
   \right\}
   \eqno {\rm (C^4)}$$
then, by differentiating the equations (G${}'$), and attending to
the formul{\ae} (Y${}'''$), we find
$$\left. \eqalign{
\delta {\delta (W - W_\prime) \over \delta \alpha}
   &=    {\delta^2 v \over \delta \alpha^2}
            (\delta x' - r \, \delta \alpha)
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
            (\delta y' - r \, \delta \beta)
       + {\delta^2 v \over \delta \alpha \, \delta \gamma}
            (\delta z' - r \, \delta \gamma),\cr
\delta {\delta (W - W_\prime) \over \delta \beta}
   &=    {\delta^2 v \over \delta \alpha \, \delta \beta}
            (\delta x' - r \, \delta \alpha)
       + {\delta^2 v \over \delta \beta^2}
            (\delta y' - r \, \delta \beta)
       + {\delta^2 v \over \delta \beta  \, \delta \gamma}
            (\delta z' - r \, \delta \gamma),\cr
\delta {\delta (W - W_\prime) \over \delta \gamma}
   &=    {\delta^2 v \over \delta \alpha \, \delta \gamma}
            (\delta x' - r \, \delta \alpha)
       + {\delta^2 v \over \delta \beta  \, \delta \gamma}
            (\delta y' - r \, \delta \beta)
       + {\delta^2 v \over \delta \gamma^2}
            (\delta z' - r \, \delta \gamma),\cr
0  &=    \alpha (\delta x' - r \, \delta \alpha)
       + \beta  (\delta y' - r \, \delta \beta)
       + \gamma (\delta z' - r \, \delta \gamma);\cr}
   \right\}
   \eqno {\rm (D^4)}$$
and therefore
$$\left. \eqalign{
v'' (\delta x' - r \, \delta \alpha)
   &= \left(
         {\delta^2 v \over \delta \alpha^2}
       + {\delta^2 v \over \delta \beta^2}
       + {\delta^2 v \over \delta \gamma^2}
      \right)
      \, \delta {\delta (W - W_\prime) \over \delta \alpha} \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \left\{
         {\delta^2 v \over \delta \alpha^2}
         \, \delta {\delta (W - W_\prime) \over \delta \alpha}
       + {\delta^2 v \over \delta \alpha \, \delta \beta}
         \, \delta {\delta (W - W_\prime) \over \delta \beta}
       + {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \, \delta {\delta (W - W_\prime) \over \delta \gamma}
      \right\},\cr
v'' (\delta x' - r \, \delta \beta)
   &= \left(
         {\delta^2 v \over \delta \alpha^2}
       + {\delta^2 v \over \delta \beta^2}
       + {\delta^2 v \over \delta \gamma^2}
      \right)
      \, \delta {\delta (W - W_\prime) \over \delta \beta} \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \left\{
         {\delta^2 v \over \delta \alpha \, \delta \beta}
         \, \delta {\delta (W - W_\prime) \over \delta \alpha}
       + {\delta^2 v \over \delta \beta^2}
         \, \delta {\delta (W - W_\prime) \over \delta \beta}
       + {\delta^2 v \over \delta \beta  \, \delta \gamma}
         \, \delta {\delta (W - W_\prime) \over \delta \gamma}
      \right\},\cr
v'' (\delta x' - r \, \delta \gamma)
   &= \left(
         {\delta^2 v \over \delta \alpha^2}
       + {\delta^2 v \over \delta \beta^2}
       + {\delta^2 v \over \delta \gamma^2}
      \right)
      \, \delta {\delta (W - W_\prime) \over \delta \gamma} \cr
   &\mathrel{\phantom{=}} \mathord{}
    - \left\{
         {\delta^2 v \over \delta \alpha \, \delta \gamma}
         \, \delta {\delta (W - W_\prime) \over \delta \alpha}
       + {\delta^2 v \over \delta \beta  \, \delta \gamma}
         \, \delta {\delta (W - W_\prime) \over \delta \beta}
       + {\delta^2 v \over \delta \gamma^2}
         \, \delta {\delta (W - W_\prime) \over \delta \gamma}
      \right\}.\cr}
   \right\}
   \eqno {\rm (E^4)}$$
By these equations the conditions (A${}^4$) may be transformed
into the following:
$$\left. \eqalign{
   &v'' r (\delta \alpha^2 + \delta \beta^2 + \delta \gamma^2)
    + \left(
         {\delta^2 v \over \delta \alpha^2}
       + {\delta^2 v \over \delta \beta^2}
       + {\delta^2 v \over \delta \gamma^2}
      \right)
      \, \delta^2 (W - W_\prime) \cr
   &\qquad
   =  \delta {\delta v \over \delta \alpha}
         \, \delta {\delta (W - W_\prime) \over \delta \alpha}
    + \delta {\delta v \over \delta \beta}
         \, \delta {\delta (W - W_\prime) \over \delta \beta}
    + \delta {\delta v \over \delta \gamma}
         \, \delta {\delta (W - W_\prime) \over \delta \gamma}.\cr}
   \right\}
   \eqno {\rm (F^4)}$$

To find the geometrical law expressed by this last formula, let
us take the given ray for the axis of $z$, and let us choose the
planes of $xz$, $yz$, in such a manner that the bisectors of
their angles shall bisect also the angles formed by the
developable pencils; we shall then have, by the fourteenth
number, $E = E''$, that is,
$${\delta^2 v \over \delta \alpha \, \delta \beta}
      \left(
         {\delta^2 W \over \delta \alpha^2}
       + {\delta^2 W \over \delta \beta^2}
      \right)
  =   {\delta^2 W \over \delta \alpha \, \delta \beta}
      \left(
         {\delta^2 v \over \delta \alpha^2}
       + {\delta^2 v \over \delta \beta^2}
      \right),
   \eqno {\rm (G^4)}$$
and the formula (F${}^4$) will become
$$v'' r
  =   \left(
         {\delta^2 v \over \delta \alpha \, \delta \beta}
         {\delta^2 W \over \delta \alpha \, \delta \beta}
       - {\delta^2 v \over \delta \beta^2}
         {\delta^2 W \over \delta \alpha^2}
      \right)
      {\delta \alpha^2 \over \delta \alpha^2 + \delta \beta^2}
    + \left(
         {\delta^2 v \over \delta \alpha \, \delta \beta}
         {\delta^2 W \over \delta \alpha \, \delta \beta}
       - {\delta^2 v \over \delta \alpha^2}
         {\delta^2 W \over \delta \beta^2}
      \right)
      {\delta \beta^2 \over \delta \alpha^2 + \delta \beta^2};
   \eqno {\rm (H^4)}$$
or finally
$$r = r_1 (\cos \omega)^2 + r_2 (\sin \omega)^2,
   \eqno {\rm (I^4)}$$
when we put
$$v'' r_1
  =      {\delta^2 v \over \delta \alpha \, \delta \beta}
         {\delta^2 W \over \delta \alpha \, \delta \beta}
       - {\delta^2 v \over \delta \beta^2}
         {\delta^2 W \over \delta \alpha^2},\quad
  v'' r_2
  =      {\delta^2 v \over \delta \alpha \, \delta \beta}
         {\delta^2 W \over \delta \alpha \, \delta \beta}
       - {\delta^2 v \over \delta \alpha^2}
         {\delta^2 W \over \delta \beta^2},\quad
  \delta \beta = \delta \alpha \, \tan \omega:
   \eqno {\rm (K^4)}$$
$\omega$ being the angle which the plane passing through the
given ray and parallel to the near ray makes with the plane of
$xz$; and $r_1$, $r_2$ being the extreme values of $r$.

The equation (I${}^4$) expresses in a simple manner the law of
the virtual focus.  It shews that the extreme positions of that
focus correspond to the same pair of {\it natural coordinate
planes}, passing through the given ray, which we considered in
the preceding number, and which we may therefore call the
{\it planes of extreme virtual foci}, as well as the {\it planes
of extreme projection}.  Indeed, when the given ray is one of the
principal rays of the system, assigned in the fourteenth number,
then all the virtual foci, as well as all the other foci hitherto
considered, coincide in the principal focus: and the planes of
extreme virtual foci become, in this case, indeterminate.
However, we shall shew that their place is then supplied by
another remarkable pair of planes, which pass through the principal
ray, and complete the system of natural coordinates: but for this
purpose it is necessary to enter briefly on the theory of
aberration from a principal focus, which we shall do in the
following number.

\bigbreak

\centerline{\it
Aberrations from a Principal Focus.}

\nobreak\bigskip

19.
If we conceive a plane cutting a given ray perpendicularly at a
given point, this plane will be nearly perpendicular to the near
rays, and will cut those rays in points near to the given point:
the distances of these near points from the given point, are the
{\it lateral aberrations\/} of the near rays, and the cutting
plane may be called the {\it plane of aberration}.  Let
$x$,~$y$,~$z$, be the coordinates of the given point, and
$x + \Delta x$, $y + \Delta y$, $z + \Delta z$,
the coordinates of the point in which a near ray is cut by the
plane of aberration, $\Delta$ being here the mark of a finite
difference; we shall have the condition
$$0 = \alpha \, \Delta x + \beta \, \Delta y + \gamma \, \Delta z,
   \eqno {\rm (L^4)}$$
$\alpha$,~$\beta$,~$\gamma$, being the cosines of the angles
which the given ray makes with the axes of $x$,~$y$,~$z$: and if
we determine the successive differentials of $x$,~$y$,~$z$, with
reference to $\alpha$,~$\beta$,~$\gamma$, by differentiating the
equations (C) or (K) as if $\alpha$,~$\beta$,~$\gamma$, were
three independent variables, and by putting
$$\left. \eqalign{
0  &= \alpha \, \delta x + \beta \, \delta y + \gamma \, \delta z,\cr
0  &= \alpha \, \delta^2 x + \beta \, \delta^2 y + \gamma \, \delta^2 z,\cr
0  &= \alpha \, \delta^3 x + \beta \, \delta^3 y + \gamma \, \delta^3 z,\cr
   &\hbox{\&c.}\cr}
   \right\}
   \eqno {\rm (M^4)}$$
we shall have
$$\left. \eqalign{
\Delta x &= [\delta x]
         + {1 \over 2} [\delta^2 x]
         + {1 \over 2 \mathbin{.} 3} [\delta^3 x]
         + \hbox{\&c.},\cr
\Delta y &= [\delta y]
         + {1 \over 2} [\delta^2 y]
         + {1 \over 2 \mathbin{.} 3} [\delta^3 y]
         + \hbox{\&c.},\cr
\Delta z &= [\delta z]
         + {1 \over 2} [\delta^2 z]
         + {1 \over 2 \mathbin{.} 3} [\delta^3 z]
         + \hbox{\&c.},\cr}
   \right\}
   \eqno {\rm (N^4)}$$
the expressions $[\delta x]$, $[\delta^2 x]$, \&c., being formed
from $\delta x$, $\delta^2 x$, \&c., by changing the
differentials
$\delta \alpha$, $\delta \beta$, $\delta \gamma$,
to the finite differences
$\Delta \alpha$, $\Delta \beta$, $\Delta \gamma$:
and finally, the lateral aberration of the near ray will have for
expression
$$\sqrt{ (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 }.$$

Let us apply this general theory to the case when the ray from
which the aberrations are measured, is a principal ray of the
system: and in order to simplify the calculations, let us take
this ray for the axis of $z$, and the principal focus for origin.
Then, if we neglect the squares and products of
$\Delta \alpha$, $\Delta \beta$,
we find by the preceding theory,
$$\Delta x = \rho \, \Delta \alpha,\quad
  \Delta y = \rho \, \Delta \beta,\quad
  \Delta z = 0,
   \eqno {\rm (O^4)}$$
$\rho$ being the distance from the principal focus to the plane
of aberration; if, therefore, we suppose this distance~$\rho$ to
be unity, and represent by $a$,~$b$, the corresponding values of
$\Delta \alpha$,~$\Delta \beta$, we shall have,
$$\Delta \alpha = a,\quad \Delta \beta = b;
   \eqno {\rm (P^4)}$$
and if we take the principal focus for origin, the coordinates of
the point in which the near ray intersects the plane of
aberration will be $a$,~$b$,~$1$.  If now we conceive another
plane of aberration, perpendicular to the principal ray and
passing through the principal focus, we shall have, for this new
plane, $\rho = 0$, and the expressions (O${}^4$) for the
components of aberration vanish: in this case, therefore, it is
necessary to carry the approximation farther, and take account of
terms of the second dimension, in the variations of
$\alpha$,~$\beta$,~$\gamma$.  For this purpose we may
differentiate twice successively the equations (K), as if
$\alpha$,~$\beta$,~$\gamma$, were independent, making after the
differentiations,
$x$,~$y$,~$z$, $\delta x$,~$\delta y$,~$\delta z$, $\delta^2 z$,
each $ = 0$, and changing
$\delta \alpha$,~$\delta \beta$,~$\delta \gamma$,
$\delta^2 x$,~$\delta^2 y$,
to
$\Delta \alpha$,~$\Delta \beta$,~$\Delta \gamma$,
$2 \Delta x$,~$2 \Delta y$.  In this manner we find
$$\left. \eqalign{
{1 \over 2} \left[ \delta^2 {\delta W \over \delta \alpha} \right]
   &=    {\delta^2 v \over \delta \alpha^2} \, \Delta x
       + {\delta^2 v \over \delta \alpha \, \delta \beta} \, \Delta y,\cr
{1 \over 2} \left[ \delta^2 {\delta W \over \delta \beta} \right]
   &=    {\delta^2 v \over \delta \alpha \, \delta \beta} \, \Delta x
       + {\delta^2 v \over \delta \beta^2} \, \Delta y,\cr}
   \right\}
   \eqno {\rm (Q^4)}$$
in which we may put
$$\left. \eqalign{
\left[ \delta^2 {\delta W \over \delta \alpha} \right]
   &=    {\delta^3 W \over \delta \alpha^3} a^2
       + {\delta^3 W \over \delta \alpha^2 \, \delta \beta} ab
       + {\delta^3 W \over \delta \alpha \, \delta \beta^2} b^2,\cr
\left[ \delta^2 {\delta W \over \delta \beta} \right]
   &=    {\delta^3 W \over \delta \alpha^2 \, \delta \beta} a^2
       + {\delta^3 W \over \delta \alpha \, \delta \beta^2} ab
       + {\delta^3 W \over \delta \beta^3} b^2,\cr}
   \right\}
   \eqno {\rm (R^4)}$$
changing $\Delta \alpha$, $\Delta \beta$, to their expressions
(P${}^4$), and observing that the general relation
$$    (\alpha + \Delta \alpha)^2
    + (\beta + \Delta \beta)^2
    + (\gamma + \Delta \gamma)^2
   = \alpha^2 + \beta^2 + \gamma^2 = 1,$$
gives here
$$0 = 2 \, \Delta \gamma
    + (\Delta \alpha)^2
    + (\Delta \beta)^2
    + (\Delta \gamma)^2,$$
so that the terms
$\Delta \alpha \, \Delta \gamma$,
$\Delta \beta  \, \Delta \gamma$,
$\Delta \gamma^2$,
in the developments of
$$\left[ \delta^2 {\delta W \over \delta \alpha} \right],\quad
  \left[ \delta^2 {\delta W \over \delta \beta} \right],$$
may be neglected, as being of the third dimension.  And if, for
further abridgment, we put $x$,~$y$, instead of
$\Delta x$,~$\Delta y$, in the equations (Q${}^4$) to denote the
coordinates of the intersection of the near ray with the plane of
$xy$, that is, with the plane of aberration passing through the
principal focus, and denote the partial differential coefficients
$${\delta^3 W \over \delta \alpha^3},\quad
  {\delta^3 W \over \delta \alpha^2 \, \delta \beta},\quad
  {\delta^3 W \over \delta \alpha \, \delta \beta^2},\quad
  {\delta^3 W \over \delta \beta^3},$$
by $A$, $B$, $C$, $D$, we shall have
$$\left. \eqalign{
      x {\delta^2 v \over \delta \alpha^2}
    + y {\delta^2 v \over \delta \alpha \, \delta \beta}
   &= {\textstyle {1 \over 2}}
         (A a^2 + 2B ab + Cb^2),\cr
      x {\delta^2 v \over \delta \alpha \, \delta \beta}
    + y {\delta^2 v \over \delta \beta^2}
   &= {\textstyle {1 \over 2}}
         (B a^2 + 2C ab + Db^2),\cr}
   \right\}
   \eqno {\rm (S^4)}$$
and by elimination,
$$\left. \eqalign{
2 v'' x
   &=    \left(
            A {\delta^2 v \over \delta \beta^2}
          - B {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right) a^2
     + 2 \left(
            B {\delta^2 v \over \delta \beta^2}
          - C {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right) ab
       + \left(
            C {\delta^2 v \over \delta \beta^2}
          - D {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right) b^2,\cr
2 v'' y
   &=    \left(
            B {\delta^2 v \over \delta \alpha^2}
          - A {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right) a^2
     + 2 \left(
            C {\delta^2 v \over \delta \alpha^2}
          - B {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right) ab
       + \left(
            D {\delta^2 v \over \delta \alpha^2}
          - C {\delta^2 v \over \delta \alpha \, \delta \beta}
         \right) b^2,\cr}
   \right\}
   \eqno {\rm (T^4)}$$
$v''$ having the same meaning as before.

\bigbreak

\centerline{\it
Natural Axes of a System.}

\nobreak\bigskip

20.
The equations (S${}^4$), or (T${}^4$), express the connexion
between the coordinates $x$,~$y$, of the intersection of a near
ray with the plane of aberration passing through the principal
focus, and the coordinates $a$, $b$, of the intersection of the
same near ray with another plane of aberration, parallel to the
former, and at a distance from it equal to unity: they serve
therefore to resolve the questions that have reference to this
connexion.  The most interesting questions of this kind, are
those which relate to the comparative condensation of the near
rays, in crossing the two planes of aberration.  Let us therefore
consider an infinitely small rectangle
$\delta a \mathbin{.} \delta b$
on the plane of $a$,~$b$, having for coordinates of its four
corners,
$$\hbox{Ist.}\enspace a,\enspace b;\quad
  \hbox{IInd.}\enspace a + \delta a,\enspace b;\quad
  \hbox{IIId.}\enspace a,\enspace b + \delta b;\quad
  \hbox{IVth.}\enspace a + \delta a,\enspace b + \delta b:$$
the rays which pass inside this little rectangle, will, at the
plane of $xy$, be diffused over a little parallelogram, of which
the coordinates of the corners are
$$\hbox{Ist.}\enspace
      x,\enspace y,\quad
  \hbox{IInd.}\enspace
      x + {\delta x \over \delta a} \, \delta a,\enspace
      y + {\delta y \over \delta a} \, \delta a;\quad
  \hbox{IIId.}\enspace
      x + {\delta x \over \delta b} \, \delta b,\enspace
      y + {\delta y \over \delta b} \, \delta b;$$
$$\hbox{IVth.}\enspace
      x + {\delta x \over \delta a} \, \delta a
        + {\delta x \over \delta b} \, \delta b,\enspace
      y + {\delta y \over \delta a} \, \delta a
        + {\delta y \over \delta b} \, \delta b:$$
the partial differential coefficients
$${\delta x \over \delta a},\quad
  {\delta x \over \delta b},\quad
  {\delta y \over \delta a},\quad
  {\delta y \over \delta b},$$
being obtained by differentiating the equations (S${}^4$), or
(T${}^4$).  The area of the parallelogram on the plane of $xy$ is
$$\pm \left(
         {\delta x \over \delta a}
         {\delta y \over \delta b}
       - {\delta x \over \delta b}
         {\delta y \over \delta a}
      \right)
      \, \delta a \, \delta b;$$
its ratio to the rectangle $\delta a \, \delta b$, is therefore
expressed by
$$\pm \left(
         {\delta x \over \delta a}
         {\delta y \over \delta b}
       - {\delta x \over \delta b}
         {\delta y \over \delta a}
      \right):$$
and by the equations (T${}^4$), or (S${}^4$),
$$    \left(
         {\delta x \over \delta a}
         {\delta y \over \delta b}
       - {\delta x \over \delta b}
         {\delta y \over \delta a}
      \right)
   =  {M'' \over v''},
   \eqno {\rm (U^4)}$$
if we put
$$M'' = (A a + B b) (C a + D b) - (B a + C b)^2.
   \eqno {\rm (V^4)}$$

The smaller the quantity $M''$ is, the more will the rays which
pass through the little rectangle $\delta a \, \delta b$, be
condensed at the principal focus; so that the curves upon the
plane of $a$,~$b$, which have for equation
$$M'' = \hbox{const.},
   \eqno {\rm (W^4)}$$
may be called {\it lines of uniform condensation\/}; and we see,
by (V${}^4$), that these curves will be ellipses or hyperbolas,
according as $N''$ is positive or negative, if we put for
abridgment,
$$4 (B^2 - AC) (C^2 - BD) - (AD - BC)^2 = N''.
   \eqno {\rm (X^4)}$$
These elliptic or hyperbolic curves are all concentric and
similar, and their axes are all contained on the same pair of
indefinite right lines, which are perpendicular to each other and
to the given ray; and the planes which pass through the ray, and
through these axes of the line (W${}^4$), will coincide with the
planes of $xz$, $yz$, if the following condition be satisfied:
$$AD - BC = 0,
   \eqno {\rm (Y^4)}$$
that is
$${\delta^3 W \over \delta \alpha^3}
      {\delta^3 W \over \delta \beta^3}
   =  {\delta^3 W \over \delta \alpha^2 \, \delta \beta}
      {\delta^3 W \over \delta \alpha \, \delta \beta^2}.
   \eqno {\rm (Z^4)}$$

This condition is independent of the magnitude of the unit of
distance, by which we have supposed the two planes of aberration
to be separated: there are therefore an infinite number of
systems of ellipses or hyperbolas, similar to the system
(W${}^4$), and all having their axes contained in the same pair
of rectangular planes, which pass through the principal ray: and
it is natural to take these planes for the planes of $xz$, $yz$,
the plane of $xy$ being still the same plane of aberration as
before.  And thus, the intersections of these three rectangular
planes, may be considered as furnishing, in general, {\it three
natural axes of an optical system}, which are perpendicular each
to each, and cross in the principal focus.  These natural axes
possess many other properties, of which we hope to treat
hereafter; but in the foregoing remarks we have only aimed to
shew, by some selected instances, the possibility of deducing the
geometrical properties of optical systems of rays, from the
fundamental formula (A),
$$\delta \int v \, ds
   =    {\delta v \over \delta \alpha} \, \delta x
      + {\delta v \over \delta \beta}  \, \delta y
      + {\delta v \over \delta \gamma} \, \delta z,$$
with the assistance of the characteristic function~$V$, and of
the connected function~$W$: and believing that this object has
been accomplished, we shall conclude the present Supplement.

\bye


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