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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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\title{On a general Method of expressing the Paths of Light, and of
the Planets, by the Coefficients of a Characteristic Function}
\author{By William R. Hamilton, Royal Astronomer of Ireland}
\date{[Dublin University Review and Quarterly Magazine,\\
Vol.~I, 1833, pp. 795--826.]}
\maketitle

The law of seeing in straight lines was known from the infancy of
optics, being in a manner forced upon men's notice by the most
familiar and constant experience.  It could not fail to be observed
that when a man looked at any object, he had it in his power to
interrupt his vision of that object, and hide it at pleasure from his
view, by interposing his hand between his eyes and it; and that then,
by withdrawing his hand, he could see the object as before: and thus
the notion of straight lines or rays of communication, between a
visible object and a seeing eye, must very easily and early have
arisen.  This notion of straight lines of vision, was of of course
confirmed by the obvious remark that objects can usually be seen on
looking through a straight but not through a bent tube; and the most
familiar facts of perspective supplied, we may suppose, new
confirmations and new uses of the principle.  A globe, for example,
from whatever point it may be viewed, appears to have a circular
outline; while a plate, or a round table, seems oval when viewed
obliquely: and these facts may have been explained, and reduced to
mathematical reasoning, by shewing that the straight rays or lines of
vision, which touch any one given globe and pass through any one given
point, are arranged in a hollow cone of a perfectly circular shape;
but that the straight rays, which connect an eye with the round edge
of a plate or table, compose, when they are oblique, an elliptical or
oval cone.  The same principle, of seeing in straight lines, must have
been continually employed from the earliest times in the explanation
of other familiar appearances, and in interpreting the testimony of
sight respecting the places of visible bodies.  It was, for example,
an essential element in ancient as in modern astronomy.

The shapes and sizes of shadows, again, could not fail to suggest the
notion of straight illuminating rays: although opinions, now rejected,
respecting the nature of light and vision, led some of the ancients to
distinguish the lines of luminous from those of visual communication,
and to regard the latter as a kind of feelers by which the eye became
aware of the presence of visible objects.  It appears, however, that
many persons held, even in the infancy of Optics, the modern view of
the subject, and attributed vision, as well as illumination, to an
influence proceeding from the visible or luminous body.  But what
finally established this view, and along with it the belief of a
finite velocity of progress of the luminous influence, was the
discovery made by Roemer, of the gradual propagation of light from
objects to the eye, in the instance of the satellites of Jupiter; of
which we have good reason to believe, from astronomical observation,
that the eclipses are never seen by us, till more than half an hour
after they have happened; the interval, besides, being found to be so
much the greater, as Jupiter is more distant from the Earth.  Galileo
had indeed proposed terrestrial experiments to measure the velocity of
light, which he believed to be finite; and Des Cartes, who held that
the communication of light was instantaneous, had perceived that
astronomical consequences ought to follow, if the propagation of light
were gradual: but experiments such as Galileo proposed, were not, and
could not, be made on a scale sufficient for the purpose; and the
state of astronomical observation in the time of Des Cartes did not
permit him to verify the consequences which he perceived, and seemed
rather to justify the use that he made of their non-verification, as
an argument against the opinion with which he had shown them to be
logically connected.  But when astronomers had actually observed
appearances, which seemed and still seem explicable only by this
opinion of the gradual propagation of light from objects to the eye,
the opinion itself became required, and was adopted, in the legitimate
process of induction.

By such steps, then, it has become an established theorem, fundamental
in optical science, that the communication, whether between an
illuminating body and a body illuminated, or between an object seen
and a beholding eye, is effected by the gradual but very rapid passage
of some thing, or influence, or state, called light, from the luminous
or visible body, along mathematical or physical lines, usually called
\emph{rays}, and found to be, under the most common circumstances,
exactly or nearly straight.

Again, it was very early perceived that in appearances connected with
mirrors, flat or curved, the luminous or visual communication is
effected in bent lines.  When we look into a flat mirror, and seem to
see an object, such as a candle, behind it, we should err if we were
to extend to this new case the rules of our more familiar experience.
We should not now come to touch the candle by continuing the straight
line from the eye to a hand or other obstacle, so placed between the
eye and the mirror as to hide the candle; this line continued would
meet the mirror in a certain place from which it would be necessary to
draw a new and different straight line, if we wished to reach the real
or tangible candle: and the whole bent line, made up of these two
straight parts, is found to be now the line of visual communication,
and is to be regarded now as the linear path of the light.  An opaque
obstacle, placed any where on either part of this bent line, is found
to hide the reflected candle from the eye; but an obstacle, placed any
where else, produces no such interruption.  And the law was very early
discovered, that for every such bent line of luminous or visual
communication, the angle between any two successive straight parts is
bisected by the normal, or perpendicular, to the mirror at the point
of bending.

Another early and important observation, was that of the broken or
refracted lines of communication, between an object in water and an
eye in air, and generally between a point in one ordinary medium and a
point in another.  A valuable series of experiments on such refraction
was made and recorded by Ptolemy; but it was not till long afterwards
that the law was discovered by Snellius.  He found that if two
lengths, in a certain ratio or proportion determined by the natures of
the two media, be measured, from the point of breaking, or of bending,
on the refracted ray and on the incident ray prolonged, these lengths
have one common projection on the refracting surface, or on its
tangent plane.  This law of ordinary refraction has since been
improved by Newton's discovery of the different refrangibility of the
differently coloured rays; and has been applied to explain and to
calculate the apparent elevation of the stars, produced by the
atmosphere of the earth.

The phenomena presented by the passage of light through crystals were
not observed until more lately.  Bartolinus seems to have been the
first to notice the double refraction of Iceland spar; and Huygens
first discovered the laws of this refraction.  The more complicated
double refraction produced by biaxial crystals was not observed until
the present century; and the discovery of conical refraction in such
crystals is still more recent, the experiments of Professor Lloyd on
arragonite (undertaken at my request) having been only made last year.

For the explanation of the laws of the linear propagation of light,
two principal theories have been proposed, which still divide the
suffrages of scientific men.

The theory of Newton is well known.  He compared the propagation of
light to the motion of projectiles; and as, according to that First
Law of Motion, of which he had himself established the truth by so
extensive and beautiful an induction, an ordinary projectile continues
in rectilinear and uniform progress, except so far as its course is
retarded or disturbed by the influence of some foreign body; so, he
thought, do luminous and visible objects shoot off little luminous or
light-making projectiles, which then, until they are accelerated or
retarded, or deflected one way or another, by the attractions or
repulsions of some refracting or reflecting medium, continue to move
uniformly in straight lines, either because they are not acted on at
all by foreign bodies, or because the foreign actions are nearly equal
on all sides, and thus destroy or neutralise each other.  This theory
was very generally received by mathematicians during the last century,
and still has numerous supporters.

Another theory however, proposed about the same time by another great
philosopher, has appeared to derive some strong confirmations from
modern inductive discoveries.  This other is the theory of Huygens,
who compared the gradual propagation of light, not to the motion of a
projectile, but to the spreading of sound through air, or of waves
through water.  It was, according to him, no \emph{thing}, in the
ordinary sense, no \emph{body}, which moved from the sun to the earth,
or from a visible object to the eye; but a \emph{state}, a
\emph{motion}, a \emph{disturbance}, was first in one place, and
afterwards in another.  As, when we hear a cannon which has been fired
at a distance, no bullet, no particle even of air, makes its way from
the cannon to our ears; but only the aerial motion spreads, the air near
the cannon is disturbed first, then that which is a little farther,
and last of all the air that touches us.  Or like the waves that
spread and grow upon some peaceful lake, when a pebble has stirred its
surface; the floating water-lilies rise and fall, but scarcely quit
their place, while the enlarging wave passes on and moves them in
succession.  So that great ocean of ether which bathes the farthest
stars, is ever newly stirred, by waves that spread and grow, from
every source of light, till they move and agitate the whole with their
minute vibrations: yet like sounds through air, or waves on water,
these multitudinous disturbances make no confusion, but freely mix and
cross, while each retains its identity, and keeps the impress of its
proper origin.  Such is the view of Light which Huygens adopted, and
which justly bears his name; because, whatever kindred thoughts
occurred to others before, he first shewed clearly how this view
conducted to the laws of optics, by combining it with that essential
principle of the undulatory theory which was first discovered by
himself, the principle of accumulated disturbance.

According to this principle, the minute vibrations of the elastic
luminous ether cannot perceptibly affect our eyes, cannot produce any
sensible light, unless they combine and concur in a great and, as it
were, infinite multitude; and on the other hand, such combination is
possible, because particular or secondary waves are supposed in this
theory to spread from every vibrating particle, as from a separate
centre, with a rapidity of propagation determined by the nature of the
medium.  And hence it comes, thought Huygens, that light in any one
uniform medium diffuses itself only in straight lines, so as only to
reach those parts of space to which a straight path lies open from its
origin; because an opaque obstacle, obstructing such straight
progress, though it does not hinder the spreading of weak particular
waves into the space behind it, yet prevents their accumulation within
that space into one grand general wave, of strength enough to generate
light.  This want of accumulation of separate vibrations behind an
obstacle, was elegantly proved by Huygens: the mutual destruction of
such vibrations by interference, is an important addition to the
theory, which has been made by Young and by Fresnel.  Analogous
explanations have been offered for the laws of reflexion and
refraction.

Whether we adopt the Newtonian or the Huygenian, or any other physical
theory, for the explanation of the laws that regulate the lines of
luminous or visual communication, we may regard these laws themselves,
and the properties and relations of these linear paths of light, as an
important separate study, and as constituting a separate science,
called often \emph{mathematical optics}.  This science of the laws and
relations of luminous rays, is, however, itself a branch of another
more general science, which may perhaps be called the \emph{Theory of
Systems of Rays}.  I have published, in the XVth and XVIth volumes of
the Transactions of the Royal Irish Academy, a series of
investigations in that theory; and have attempted to introduce a new
principle and method for the study of optical systems.  Another
supplementary memoir, which has been lately printed for the same
Transactions, will appear in the XVIIth volume; but having been
requested to resume the subject here, and to offer briefly some new
illustrations of my view, I shall make some preliminary remarks on the
state of deductive optics, and on the importance of a general method.

The science of optics, like every other physical science, has two
different directions of progress, which have been called the ascending
and the descending scale, the inductive and the deductive method, the
way of analysis and of synthesis.  In every physical science, we must
ascend from facts to laws, by the way of induction and analysis; and
must descend from laws to consequences, by the deductive and synthetic
way.  We must gather and groupe appearances, until the scientific
imagination discerns their hidden law, and unity arises from variety:
and then from unity must re-deduce variety, and force the discovered
law to utter its revelations of the future.

It was with such convictions that Newton, when approaching to the
close of his optical labours, and looking back on his own work,
remarked, in the spirit of Bacon, that ``As in Mathematics, so in
Natural Philosophy, the investigation of difficult things by the
method of Analysis ought ever to precede the method of Composition.
This analysis consists in making experiments and observations, and in
drawing general conclusions from them by induction, and admitting of
no objections against the conclusions but such as are drawn from
experiments or other certain truths.''  ``And although the arguing
from experiments and observations by induction be no demonstration of
general conclusions; yet it is the best way of arguing which the
nature of things admits of, and may be looked upon as so much the
stronger, by how much the induction is more general.  And if no
exception occur from phenomena, the conclusion may be pronounced
generally.  But if at any time afterwards, any exception shall occur
from experiments, it may then begin to be pronounced with such
exceptions as occur.  By this way of analysis, we may proceed from
compounds to ingredients, and from motions to the forces producing
them; and, in general, from effects to their causes, and from
particular causes to more general ones, till the argument end in the
most general.  This is the method of analysis: and the synthesis
consists in assuming the causes discovered, and established as
principles, and by them explaining the phenomena proceeding from them,
and proving the explanations.''  ``And if Natural Philosophy in all
its parts, by pursuing this method, shall at length be perfected, the
bounds of Moral Philosophy will also be enlarged.  For, so far as we
can know by Natural Philosophy, what is the First Cause, what power
He has over us, and what benefits we receive from Him, so far our
duty towards Him, as well as that towards one another, will appear
to us by the light of nature.''

In the science of optics, which has engaged the attention of almost
every mathematician for the last two thousand years, many great
discoveries have been attained by both these ways.  It is, however,
remarkable that, while the laws of this science admit of being stated
in at least as purely mathematical a form as any other physical
results, their mathematical consequences have been far less fully
traced than the consequences of many other laws; and that while modern
experiments have added so much to the \emph{inductive} progress of
optics, the \emph{deductive} has profited so little in proportion from
the power of the modern algebra.

It was known to Euclid and to Ptolemy, that the communication between
visible objects and a beholding eye is usually effected in straight
lines; and that when the line of communication is bent, by reflexion,
at any point of a plane or of a spheric mirror, the angle of bending
at this point, between the two straight parts of the bent line, is
bisected by the normal to the mirror.  It was known also that this law
extends to successive reflexions.  Optical induction was therefore
sufficiently advanced two thousand years ago, to have enabled a
mathematician to understand, and, so far as depended on the knowledge
of physical laws, to resolve the following problem: to determine the
arrangement of the final straight rays, or lines of vision, along
which a shifting eye should look, in order to see a given luminous
point, reflected by a combination of two given spherical mirrors.
Yet, of two capital deductions respecting this arrangement, without
which its theory must be regarded as very far from perfect---namely,
that the final rays are in general \emph{tangents to a pair}, and that
they are \emph{perpendicular to a series} of surfaces---the one is a
theorem new and little known, and the other is still under dispute.
For Malus, who first discovered that the rays of an ordinary reflected
or refracted system are in general tangents to a pair of caustic
surfaces, was led, by the complexity of his calculations, to deny the
general existence (discovered by Huygens) of surfaces perpendicular to
such rays; and the objection of Malus has been lately revived by an
eminent analyst of Italy, in a valuable memoir on caustics, which was
published last year in the correspondence of the observatory of
Brussels.

To multiply such instances of the existing imperfection of
mathematical or deductive optics would be an unpleasant task, and
might appear an attempt to depreciate the merit of living
mathematicians.  It is better to ascend to the source of the
imperfection, the want of a general method, a presiding idea, to guide
and assist the deduction.  For although the deductive, as opposed to
the inductive process, may be called itself a \emph{method}, yet so
wide and varied is its range, that it needs the guidance of some one
central principle, to give it continuity and power.

Those who have meditated on the beauty and utility, in theoretical
mechanics, of the general method of Lagrange---who have felt the power
and dignity of that central dynamical theorem which he deduced, in the
\emph{M\'{e}chanique Analytique}, from a combination of the principle
of virtual velocities with the principle of D'Alembert---and who have
appreciated the simplicity and harmony which he introduced into the
research of the planetary perturbations, by the idea of the variation
of parameters, and the differentials of the disturbing function, must
feel that mathematical optics can only \emph{then} attain a coordinate
rank with mathematical mechanics, or with dynamical astronomy, in
beauty, power, and harmony, when it shall possess an appropriate
method, and become the unfolding of a central idea.

This fundamental want forced itself long ago on my attention; and I
have long been in possession of a method, by which it seems to me to
be removed.  But in thinking so, I am conscious of the danger of a
bias.  It may happen to me, as to others, that a meditation which has
long been dwelt on shall assume an unreal importance; and that a
method which has for a long time been practised shall acquire an only
seeming facility.  It must remain for others to judge how far my
attempts have been successful, and how far they require to be
completed, or set aside, in the future progress of the science.

Meanwhile it appears that if a general method in deductive optics can
be attained at all, it must flow from some law or principle, itself of
the highest generality, and among the highest results of induction.
What, then, may we consider as the highest and most general axiom, (in
the Baconian sense,) to which optical induction has attained,
respecting the rules and conditions of the lines of visual and
luminous communication?  The answer, I think, must be, the principle or
law, called usually the Law of Least Action; suggested by questionable
views, but established on the widest induction, and embracing every
known combination of media, and every straight, or bent, or curved
line, ordinary or extraordinary, along which light (whatever light may
be) extends its influence successively in space and time: namely, that
this linear path of light, from one point to another, is always found
to be such, that if it be compared with the other infinitely various
lines by which in thought and in geometry the same two points might be
connected, a certain integral or sum, called often \emph{Action}, and
depending by fixed rules on the length, and shape, and position of the
path, and on the media which are traversed by it, is less than all the
similar integrals for the other neighbouring lines, or, at least,
possesses, with respect to them, a certain \emph{stationary} property.
From this Law, then, which may, perhaps, be named the \textsc{Law of
Stationary Action}, it seems that we may most fitly and with best hope
set out, in the synthetic or deductive process, and in search of a
mathematical method.

Accordingly, from this known law of least or stationary action, I
deduced (long since) another connected and coextensive principle,
which may be called, by analogy, the \textsc{Law of Varying Action}, and
which seems to offer naturally a method such as we are seeking: the
one law being as it were the last step in the ascending scale of
induction, respecting linear paths of light, while the other law may
usefully be made the first in the descending and deductive way.  And
my chief purpose, in the present paper, is to offer a few
illustrations and consequences of these two coordinate laws.

The former of these two laws was discovered in the following manner.
The elementary principle of straight rays shewed that light, under the
most simple and usual circumstances, employs the direct, and,
therefore, the shortest course to pass from one point to another.
Again, it was a very early discovery, (attributed by Laplace to
Ptolemy,) that in the case of a plane mirror, the bent line formed by
the incident and reflected rays is shorter than any other bent line,
having the same extremities, and having its point of bending on the
mirror.  These facts were thought by some to be instances and results
of the simplicity and economy of nature; and Fermat, whose researches
on maxima and minima are claimed by the continental mathematicians as
the germ of the differential calculus, sought anxiously to trace some
similar economy in the more complex case of refraction.  He believed
that by a metaphysical or cosmological necessity, arising from the
simplicity of the universe, light always takes the course which it can
traverse in the shortest time.  To reconcile this metaphysical opinion
with the law of refraction, discovered experimentally by Snellius,
Fermat was led to suppose that the two lengths, or \emph{indices},
which Snellius had measured on the incident ray prolonged and on the
refracted ray, and had observed to have one common projection on a
refracting plane, are inversely proportional to the two successive
velocities of the light before and after refraction, and therefore
that the velocity of light is diminished on entering those denser
media in which it is observed to approach the perpendicular: for
Fermat believed that the time of propagation of light along a line
bent by refraction was represented by the sum of the two products, of
the incident portion multiplied by the index of the first medium, and
of the refracted portion multiplied by the index of the second medium;
because he found, by his mathematical method, that this sum was less,
in the case of a plane refractor, than if light went by any other than
its actual path from one given point to another; and because he
perceived that the supposition of a velocity inversely as the index,
reconciled his mathematical discovery of the minimum of the foregoing
sum with his cosmological principle of least time.  Des Cartes
attacked Fermat's opinions respecting light, but Leibnitz zealously
defended them; and Huygens was led, by reasonings of a very different
kind, to adopt Fermat's conclusions of a velocity inversely as the
index, and of a \emph{minimum time} of propagation of light, in
passing from one given point to another through an ordinary refracting
plane.  Newton, however, by his theory of emission and attraction, was
led to conclude that the velocity of light was \emph{directly}, not
\emph{inversely}, as the index, and that it was \emph{increased}
instead of being \emph{diminished} on entering a denser medium; a
result incompatible with the theorem of shortest time in refraction.
The theorem of shortest time was accordingly abandoned by many, and
among the rest by Maupertuis, who, however, proposed in its stead, as
a new cosmological principle, that celebrated \emph{law of least
action} which has since acquired so high a rank in mathematical
physics, by the improvements of Euler and Lagrange.  Maupertuis gave
the name of \emph{action} to the product of space and velocity, or
rather to the sum of all such products for the various elements of any
motion; conceiving that the more space has been traversed and the less
time it has been traversed in, the more action may be considered to
have been expended: and by combining this idea of action with Newton's
estimate of the velocity of light, as increased by a denser medium,
and as proportional to the refracting index, and with Fermat's
mathematical theorem of the minimum sum of the products of paths and
indices in ordinary refraction at a plane, he concluded that the
course chosen by light corresponded always to the \emph{least possible
action}, though not always to the least possible time.  He proposed
this view as reconciling physical and metaphysical principles, which
the results of Newton had seemed to put in opposition to each other;
and he soon proceeded to extend his law of least action to the
phenomena of the shock of bodies.  Euler, attached to Maupertuis, and
pleased with these novel results, employed his own great mathematical
powers to prove that the law of least action extends to all the curves
described by points under the influence of central forces; or, to
speak more precisely, that if any such curve be compared with any
other curve between the same extremities, which differs from it
indefinitely little in shape and in position, and may be imagined to
be described by a neighbouring point with the same law of velocity,
and if we give the name of \emph{action} to the integral of the
product of the velocity and an element of a curve, the difference of
the two neighbouring values of this action will be indefinitely less
than the greatest linear distance (itself indefinitely small) between
the two near curves; a theorem which I think may be advantageously
expressed by saying that the action is \emph{stationary}.  Lagrange
extended this theorem of Euler to the motion of a system of points or
bodies which act in any manner on each other; the action being in this
case the sum of the masses by the foregoing integrals.  Laplace has
also extended the use of the principle in optics, by applying it
to the refraction of crystals; and has pointed out an analogous
principle in mechanics, for all imaginable connexions between force
and velocity.  But although the law of least action has thus attained
a rank among the highest theorems of physics, yet its pretensions
to a cosmological necessity, on the ground of economy in the
universe, are now generally rejected.  And the rejection appears
just, for this, among other reasons, that the quantity pretended
to be economised is in fact often lavishly expended.  In optics,
for example, though the sum of the incident and reflected portions
of the path of light, in a single ordinary reflexion at a plane,
is always the shortest of any, yet in reflexion at a curved mirror
this economy is often violated.  If an eye be placed in the interior
but not at the centre of a reflecting hollow sphere, it may see
itself reflected in two opposite points, of which one indeed is
the nearest to it, but the other on the contrary is the furthest;
so that of the two different paths of light, corresponding to these
two opposite points, the one indeed is the shortest, but the other
is the longest of any.  In mathematical language, the integral
called action, instead of being always a minimum, is often a maximum;
and often it is neither the one nor the other: though it has always
a certain \emph{stationary} property, of a kind which has been
already alluded to, and which will soon be more fully explained.
We cannot, therefore, suppose the economy of this quantity to have
been designed in the divine idea of the universe: though a simplicity
of some high kind may be believed to be included in that idea.
And though we may retain the name of \emph{action} to denote the
stationary integral to which it has become appropriated---which we
may do without adopting either the metaphysical or (in optics) the
physical opinions that first suggested the name---yet we ought not
(I think) to retain the epithet \emph{least}:  but rather to adopt
the alteration proposed above, and to speak, in mechanics and in
optics, of the \emph{Law of Stationary Action}.

To illustrate this great law, and that other general law, of
\emph{varying action}, which I have deduced from it, we may
conveniently consider first the simple case of rectilinear paths of
light.  For the rectilinear course, which is evidently the
\emph{shortest} of any, is also distinguished from all others by a
certain \emph{stationary property}, and \emph{law of variation},
which, being included in the general laws of stationary and varying
action, may serve as preparatory examples.

The length $V$ of any given line, straight or curved, may evidently be
denoted by the following integral:
\begin{equation}
V = \int dV = \int \sqrt{dx^2 + dy^2 + dz^2}.
\end{equation}
If now we pass from this to another neighbouring line, having the same
extremities, and suppose that the several points of the latter line
are connected with those of the former, by equations between their
co-ordinates, of the form
\begin{equation}
x_\varepsilon = x + \varepsilon \xi,\quad
y_\varepsilon = y + \varepsilon \eta,\quad
z_\varepsilon = z + \varepsilon \zeta,
\end{equation}
$\varepsilon$ being any small constant, and $\xi$, $\eta$, $\zeta$, being
any arbitrary functions of $x$, $y$, $z$, which vanish for the extreme
values of those variables, that is, for the extreme points of the
given line, and do not become infinite for any of the intermediate
points, nor for the value $\varepsilon = 0$, though they may in general
involve the arbitrary constant $\varepsilon$; the length $V_\varepsilon$ of
the new line may be represented by the new integral,
\begin{eqnarray}
V_\varepsilon
   &=& \int \sqrt{ dx_\varepsilon^2 + dy_\varepsilon^2
                 + dz_\varepsilon^2} \nonumber\\
   &=& \int \sqrt{ (dx + \varepsilon d\xi)^2
                 + (dy + \varepsilon d\eta)^2
                 + (dz + \varepsilon d\zeta)^2},
\end{eqnarray}
taken between the same extreme values of $x$, $y$, $z$, as the former;
and this new length $V_\varepsilon$ may be considered as a function of
$\varepsilon$, which tends to the old length $V$, when $\varepsilon$ tends
to $0$, the quotient
\[ \frac{1}{\varepsilon} (V_\varepsilon - V) \]
tending in general at the same time to a finite limit, which may be
thus expressed,
\begin{eqnarray}
\mathrm{lim.} \frac{1}{\varepsilon} (V_\varepsilon - V)
   &=& \int \frac{dx\,d\xi + dy\,d\eta
       + dz\,d\zeta}{\sqrt{dx^2 + dy^2 + dz^2}} \nonumber\\
\label{eqn-4}
   &=&-\int \left(
         \xi\,d\frac{dx}{dV}
       + \eta\,d\frac{dy}{dV}
       + \zeta\,d\frac{dz}{dV} \right);
\end{eqnarray}
the last of these forms being obtained from the preceding by
integrating by parts, and by employing the condition already
mentioned, that the functions $\xi$, $\eta$, $\zeta$, vanish at the
extremities of the integral.  When the original line is such that the
limit (\ref{eqn-4}) vanishes, independently of the forms of the
functions $\xi$, $\eta$, $\zeta$, and therefore that the difference of
the lengths $V_\varepsilon - V$ bears ultimately an evanescent ratio to
the small quantity $\varepsilon$, (which quantity determines the
difference between the second line and the first, and bears itself a
finite ratio to the greatest distance between these two lines,) we may
say that the original line has a \emph{stationary length}, $V$, as
compared with all the lines between the same extremities, which
differ from it infinitely little in shape and in position.  And since
it easily follows, from the last form of the limit (\ref{eqn-4}), that
this limit cannot vanish independently of the forms of $\xi$, $\eta$,
$\zeta$, unless
\begin{equation}
d\frac{dx}{dV} = 0,\quad
d\frac{dy}{dV} = 0,\quad
d\frac{dz}{dV} = 0,
\end{equation}
that is, unless the ratios
\[ \frac{dx}{dV},\quad \frac{dy}{dV},\quad \frac{dz}{dV}, \]
are constant throughout the original line, but that the limit vanishes
when this condition is satisfied, we see that the property of
\emph{stationary length} belongs (in free space) to straight lines and
to such only.  The foregoing proof of this property of the straight
line may, perhaps, be useful to those who are not familiar with the
Calculus of Variations.

To illustrate, by examples, this stationary property of the length of
a straight line, let us consider such a line as the common chord of a
series of circular arcs, and compare its length with theirs, and
theirs with one another.  The length of the straight line being called
$V$, let $\frac{1}{2} \varepsilon V$ be the height or sagitta of the
circular arch upon this chord; so that
$\frac{1}{2} (\varepsilon + \varepsilon^{-1}) V$ shall be the diameter
of the circle, and $\varepsilon$ the trigonometric tangent of the quarter
of an arc having the same number of degrees, to a radius equal to unity:
we shall then have the following expression for the length~$V_\varepsilon$
of the circular arch upon the given chord $V$,
\begin{equation}
V_\varepsilon = V(\varepsilon + \varepsilon^{-1})
         \mathrm{tan.}^{-1} \varepsilon.
\end{equation}
This expression may be put under the form,
\begin{equation}
\frac{V_\varepsilon}{V}
   = 1 + 4 \left( \int_0^\varepsilon d\varepsilon \right)^2 \varepsilon^{-3}
         \int _0^\varepsilon
         \frac{\varepsilon^2 \,d\varepsilon}{(1 + \varepsilon^2)^2},
\end{equation}
which shows not only that the ratio of the circular arch to its chord
is always $> 1$, but also, that since
\begin{equation}
\frac{dV_\varepsilon}{d\varepsilon}
   = 4V \int_0^\varepsilon \left( \varepsilon^{-3}
      \int_0^\varepsilon
      \frac{\varepsilon^2\,d\varepsilon}{(1 + \varepsilon^2)^2}
      \right) d\varepsilon,
\end{equation}
the arch $V_\varepsilon$ increases continually with its height at an
increasing rate; its differential coefficient being positive and
increasing, when $\varepsilon$ is positive and increases, but vanishing
with $\varepsilon$, and showing, therefore, that in this series of
circular arcs and chord the property of stationary length belongs to
the straight line only.

Again, we may imagine a series of semi-ellipses upon a given common
axis~$V$, the other axis conjugate to this being a variable quantity
$\varepsilon V$.  The length of such a semi-elliptic arch is
\begin{equation}
V_\varepsilon = V \int_0^{\frac{\pi}{2}} (\mathrm{cos.} \phi^2
                + \varepsilon^2 \mathrm{sin.} \phi^2)^{\frac{1}{2}}
                \, d\phi,
\end{equation}
an expression which may be thus transformed,
\begin{equation}
\frac{V_\varepsilon}{V} = 1 +
      \left( \int_0^\varepsilon d\varepsilon \right)^2
      \int_0^{\frac{\pi}{2}}
      \frac{\mathrm{sin.} \phi^2 \mathrm{cos.} \phi^2
         \,d\phi}{(\mathrm{cos.} \phi^2
         + \varepsilon^2 \mathrm{sin.} \phi^2)^{\frac{3}{2}}};
\end{equation}
thus the ratio of the elliptic arch $V_\varepsilon$ to its given base or
axis $V$ is not only greater than unity, and continually increases
with the height, but increases at an increasing rate, which vanishes
for an evanescent height; so that in this series of semi-elliptic arcs
and axis, the latter alone has the property of stationary length.

In more familiar words, if we construct on a base of a given length,
suppose one hundred feet, a series of circular or of semi-elliptic
arches, having that base for chord or for axis, the lengths of those
arches will not only increase with their heights, but every additional
foot or inch of height will augment the length more than the foregoing
foot or inch had done; and the lower or flatter any two such arches
are made, the less will be the difference of their lengths as compared
with the difference of their heights, till the one difference becomes
less than any fraction that can be named of the other.  For example,
if we construct, on the supposed base of one hundred feet, two
circular arches, the first fifty feet high, the second fifty-one feet
high, of which the first will thus be a semicircle, and the second
greater than a semicircle, the difference of lengths of these two
arches will be a little more than double the difference of their
heights, that is, it will be about two feet; but if on the same base
we construct one circular arch with only one foot of height, and
another with only two feet, the difference of lengths of these two low
arches will not be quite an inch, though the difference of their
heights remains a foot as before; and if we imagine the two circular
arches, on the same base or common chord of one hundred feet, to have
their heights reduced to one and two inches respectively, the
difference of their lengths will thereby be reduced to less than the
hundred-and-fiftieth part of an inch.

We see then that a straight ray, or rectilinear path of light, from
one given point to another, has a \emph{stationary length}, as
compared with all the lines which differ little from it in shape and
in position, and which are drawn between the same extremities.  If,
however, we suppose the extremities of the neighbouring line to
differ from those of the ray, we shall then obtain in general a
\emph{varying} instead of stationary length.  To investigate the law
of this variation, which is the simplest case of the second general
law above proposed to be illustrated, we may resume the foregoing
comparison of the lengths $V$, $V_\varepsilon$, of any two neighbouring
lines; supposing now that these two lines have different extremities,
or in other words, that the functions $\xi$, $\eta$, $\zeta$, do not
vanish at the limits of the integral.  The integration by parts gives
now, along with the last expression (\ref{eqn-4}) for the limit of
\[ \frac{1}{\varepsilon} (V_\varepsilon - V), \]
the following additional terms,
\[ \xi \frac{dx}{dV}
   + \eta \frac{dy}{dV}
   + \zeta \frac{dz}{dV}
   + \xi' \frac{d'x'}{d'V}
   + \eta' \frac{d'y'}{d'V}
   + \zeta' \frac{d'z'}{d'V}, \]
which belong to the extremities of the given line, the accented being
the initial quantities, and $d'$ referring to the infinitesimal
changes produced by a motion of the initial point along the initial
element of the line, so that $d'V$ is this initial element taken
negatively,
\begin{equation}
d'V = - \sqrt{d'x'^2 + d'y'^2 + d'z'^2};
\end{equation}
when, therefore, the last integral (\ref{eqn-4}) vanishes, by the
original line being straight, and when we compare this line with
another infinitely near, the \emph{law of varying length} is expressed
by the following equation:
\begin{eqnarray}
\mathrm{lim.} \frac{1}{\varepsilon} (V_\varepsilon - V)
   &=& \xi \frac{dx}{dV}
       + \eta \frac{dy}{dV}
       + \zeta \frac{dz}{dV}
       + \xi' \frac{d'x'}{d'V}
       + \eta' \frac{d'y'}{d'V}
       + \zeta' \frac{d'z'}{d'V} \nonumber\\
   &=& (\xi - \xi') \frac{dx}{dV} + (\eta - \eta') \frac{dy}{dV}
        + (\zeta - \zeta') \frac{dz}{dV};
\end{eqnarray}
it may also be thus expressed,
\begin{equation}
\delta V = \frac{dx}{dV} ( \delta x - \delta x')
           + \frac{dy}{dV} ( \delta y - \delta y')
           + \frac{dz}{dV} ( \delta z - \delta z'),
\end{equation}
and shows that the length $V + \delta V$ of any other line which
differs infinitely little from the straight ray in shape and in
position, may be considered as equal to its own projection on the ray.

It must be observed that in certain singular cases, the distance
between two lines may be made less, throughout, than any quantity
assigned, without causing thereby their lengths to tend to equality.
For example, a given straight line may be subdivided into a great
number of small parts, equal or unequal, and on each part a semicircle
may be constructed; and then the waving line composed of the small but
numerous semicircumferences will every where be little distant from
the given straight line, and may be made as little distant as we
please, to any degree short of perfect coincidence; while yet the
length of the undulating line will not tend to become equal to the
length of the straight line, but will bear to that length a constant
ratio greater than unity, namely the ratio of $\pi$ to $2$.  But it is
evident that such cases as these are excluded from the foregoing
reasoning, which supposes an approach of the one line to the other in
shape, as well as a diminution of the linear distance between them.

From the \emph{law of varying length} of a straight ray we may easily
perceive (what is also otherwise evident) that the straight rays
diverging from a given point $x'$ $y'$ $z'$, or converging to a
given point $x$ $y$ $z$, are cut perpendicularly by a series of
concentric spheres, having for their common equation,
\begin{equation}
\label{eqn-14}
V = \mbox{const.};
\end{equation}
and more generally, that if a set of straight rays be perpendicular to
any one surface, they are also perpendicular to a series of surfaces,
determined by the equation (\ref{eqn-14}), that is, by the condition
that the intercepted portion of a ray between any two given surfaces
of the series shall have a constant length.  Analogous consequences
will be found to follow in general from the law of varying action.

It may be useful to dwell a little longer on the case of rectilinear
paths, and on the consequences of the mathematical conception of
luminous or visual communication as a motion from point to point along
a mathematical straight line or ray, before we pass to the properties
of other less simple paths.

It is an obvious consequence of this conception, that from any one
point ($A$), considered as initial, we may imagine light, if
unobstructed, as proceeding to any other point ($B$), considered as
final, along one determined ray, or linear path; of which the shape,
being straight, is the same whatever point its ends may be; but of
which the length and the position depend on the places of those ends,
and admit of infinite variety, corresponding to the infinite variety
that can be imagined of pairs of points to be connected.  So that if
we express by one set of numbers the places of the initial and of the
final points, and by another set the length and position of the ray,
the latter set of numbers must, in mathematical language, be
\emph{functions} of the former; must admit of being deduced from them
by some fixed mathematical rules.  To make this deduction is an easy
but a fundamental problem, which may be resolved in the following
manner.

Let each of the two points $A$, $B$, be referred to one common set
of three rectangular semiaxes $OX$, $OY$, $OZ$, diverging from any
assumed origin~$O$; let the positive or negative co-ordinates of
the final point $B$, \emph{to} which the light comes, be denoted
by $x$, $y$, $z$, and let the corresponding co-ordinates of the
initial point $A$, \emph{from} which the light sets out, be denoted
similarly by $x'$, $y'$ , $z'$; let $V$ be the length of the straight
ray, or line $AB$, and let $\alpha$, $\beta$, $\gamma$, be the
positive or negative cosines of the acute or obtuse angles which
the direction of this ray makes with the positive semiaxes of
co-ordinates: the problem is then to determine the laws of the
functional dependence of the positive number $V$, and of the three
positive or negative numbers $\alpha$, $\beta$, $\gamma$, on the
six positive or negative numbers $x$, $y$, $z$, $x'$, $y'$, $z'$;
and this problem is resolved by the following evident formul\ae;
\begin{equation}
\label{eqn-15}
V = \sqrt{(x - x')^2 + (y - y')^2 + (z - z')^2};
\end{equation}
\begin{equation}
\label{eqn-16}
\alpha = \frac{x - x'}{V},\quad
\beta  = \frac{y - y'}{V},\quad
\gamma = \frac{z - z'}{V}.
\end{equation}
It is a simple but important corollary to this solution, that the laws
of the three cosines of direction $\alpha$, $\beta$, $\gamma$,
expressed by the equations (\ref{eqn-16}), are connected with the law
of the length~$V$, expressed by the formula (\ref{eqn-15}), in a
manner which may be stated thus;
\begin{equation}
\label{eqn-17}
\alpha = \frac{\delta V}{\delta x},\quad
\beta  = \frac{\delta V}{\delta y},\quad
\gamma = \frac{\delta V}{\delta z}:
\end{equation}
$\delta$ being here a characteristic of partial differentiation.  We
find, in like manner,
\begin{equation}
\label{eqn-18}
\alpha = - \frac{\delta V}{\delta x'},\quad
\beta  = - \frac{\delta V}{\delta y'},\quad
\gamma = - \frac{\delta V}{\delta z'},
\end{equation}
differentiating the function $V$ with respect to the initial
co-ordinates.  And since the three cosines of direction $\alpha$,
$\beta$, $\gamma$, are evidently connected by the relation
\begin{equation}
\alpha^2 + \beta^2 + \gamma^2 = 1,
\end{equation}
we see that the function~$V$ satisfies simultaneously the two
following partial differential equations of the first order and second
degree,
\begin{equation}
\label{eqn-20}
\left. \begin{array}{lllll}
\displaystyle \left( \frac{\delta V}{\delta x} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta y} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta z} \right)^2
   &=& 1,\\[12pt]
\displaystyle \left( \frac{\delta V}{\delta x'} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta y'} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta z'} \right)^2
   &=& 1.
\end{array} \right\}
\end{equation}

The equations (\ref{eqn-17}), (\ref{eqn-18}), (\ref{eqn-20}), will
soon be greatly extended; but it seemed well to notice them here,
because they contain the germ of my general method for the
investigation of the paths of light and of the planets, by the partial
differential coefficients of one \emph{characteristic function}.  For
the equations (\ref{eqn-17}) and (\ref{eqn-18}), which involve the
coefficients of the first order of the function~$V$, that is, in the
present case, of the length, may be considered as equations of the
straight ray which passes with a given direction through a given
initial or a given final point: and I have found analogous equations
for all other paths of light, and even for the planetary orbits under
the influence of their mutual attractions.

The equations (\ref{eqn-16}) when put under the form
\begin{equation}
x - x' = \alpha V,\quad
y - y' = \beta  V,\quad
z - z' = \gamma V,
\end{equation}
give evidently by differentiation
\begin{equation}
\label{eqn-22}
dx = \alpha \, dV,\quad
dy = \beta  \, dV,\quad
dz = \gamma \, dV,
\end{equation}
and therefore
\begin{equation}
\label{eqn-23}
dV^2 = dx^2 + dy^2 + dz^2,
\end{equation}
the symbol~$d$ referring here to an infinitesimal change of the final
point~$B$, by a motion along the ray prolonged at its extremity; in
such a manner that the equations (\ref{eqn-22}) may be regarded as
differential equations of that ray.  They give the expressions
\begin{equation}
\alpha = \frac{dx}{dV},\quad
\beta  = \frac{dy}{dV},\quad
\gamma = \frac{dz}{dV},
\end{equation}
which may, by (\ref{eqn-23}), be put under the form
\begin{equation}
\label{eqn-25}
\alpha = \frac{\delta dV}{\delta dx},\quad
\beta  = \frac{\delta dV}{\delta dy},\quad
\gamma = \frac{\delta dV}{\delta dz},
\end{equation}
$\delta$ implying still a partial differentiation, and $dV$ being
treated here as a function of $dx$, $dy$, $dz$.  And comparing the
expressions (\ref{eqn-25}) and (\ref{eqn-17}), we obtain the following
results, which we shall soon find to be very general, and to extend
with analogous meanings to all linear paths of light,
\begin{equation}
\label{eqn-26}
\frac{\delta V}{\delta x} = \frac{\delta dV}{\delta dx},\quad
\frac{\delta V}{\delta y} = \frac{\delta dV}{\delta dy},\quad
\frac{\delta V}{\delta z} = \frac{\delta dV}{\delta dz}.
\end{equation}

It must not be supposed that these equations are identical; for the
quantities in the first members are the partial differential
coefficients of one function, $V$, while those in the second members
are the coefficients of another function $dV$.

In like manner, if we employ (as before) the characteristic $d'$ to
denote the infinitesimal changes arising from a change of the initial
point $A$, by a motion along the initial element of the ray, we have
the differential equations
\begin{equation}
d'x' = - \alpha \, d'V,\quad
d'y' = - \beta  \, d'V,\quad
d'z' = - \gamma \, d'V,
\end{equation}
$d'V$ being as before the initial element taken negatively, so that
\begin{equation}
d'V^2 = d'x'^2 + d'y'^2 + d'z'^2;
\end{equation}
we have therefore
\begin{equation}
\alpha = - \frac{\delta d'V}{\delta d'x'},\quad
\beta  = - \frac{\delta d'V}{\delta d'y'},\quad
\gamma = - \frac{\delta d'V}{\delta d'z'},
\end{equation}
and consequently, by (\ref{eqn-18}),
\begin{equation}
\label{eqn-30}
\frac{\delta V}{\delta x'} = \frac{\delta d'V}{\delta d'x'},\quad
\frac{\delta V}{\delta y'} = \frac{\delta d'V}{\delta d'y'},\quad
\frac{\delta V}{\delta z'} = \frac{\delta d'V}{\delta d'z'}.
\end{equation}
The same remarks apply to these last results, as to the equations
(\ref{eqn-26}).

The general law of stationary action, in optics, may now be thus
stated.

The optical quantity called \emph{action}, for any luminous path
having $i$ points of sudden bending by reflexion or refraction, and
having therefore $i+1$ separate branches, is the sum of $i+1$ separate
integrals,
\begin{eqnarray}
\mbox{\textit{Action}} = V = \sum \int dV^{(r)}
   &=& V^{(1)} + V^{(2)} + V^{(3)} \nonumber\\
   & &+ \cdots + V^{(r)} + \cdots + V^{(i+1)},
\end{eqnarray}
of which each is determined by an equation of the form
\begin{equation}
V^{(r)} = \int dV^{(r)}
   = \int v^{(r)} \sqrt{ dx^{(r)2} + dy^{(y)2} + dz^{(r)2}},
\end{equation}
the coefficient $v^{(r)}$ of the element of the path, in the $r$th
medium, depending, in the most general case, on the optical properties
of that medium, and on the position, direction, and colour of the
element, according to rules discovered by experience, and such, for
example, that if the $r$th medium be ordinary, $v^{(r)}$ is the index
of that medium; so that $dV^{(r)}$ is always a homogeneous function of
the first dimension of the differentials $dx^{(r)}$, $dy^{(r)}$,
$dz^{(r)}$, which may also involve the undifferentiated co-ordinates
$x^{(r)}$ $y^{(r)}$ $z^{(r)}$ themselves, and has in general a
variation of the form
\begin{eqnarray}
\label{eqn-33}
\delta dV^{(r)}
   &=& \sigma^{(r)} \, \delta d x^{(r)}
       + \tau^{(r)} \, \delta d y^{(r)}
       + \upsilon^{(r)} \, \delta d z^{(r)} \nonumber\\
   & &\quad + \left(
            \frac{\delta v^{(r)}}{\delta x^{(r)}} \delta x^{(r)}
          + \frac{\delta v^{(r)}}{\delta y^{(r)}} \delta y^{(r)}
          + \frac{\delta v^{(r)}}{\delta z^{(r)}} \delta z^{(r)}
        \right) ds^{(r)},
\end{eqnarray}
if we put for abridgment
\begin{equation}
\sigma^{(r)}   = \frac{\delta dV^{(r)}}{\delta dx^{(r)}},\quad
\tau^{(r)}     = \frac{\delta dV^{(r)}}{\delta dy^{(r)}},\quad
\upsilon^{(r)} = \frac{\delta dV^{(r)}}{\delta dz^{(r)}},
\end{equation}
and
\begin{equation}
ds^{(r)} = \sqrt{dx^{(r)2} + dy^{(r)2} + dz^{(r)2}}:
\end{equation}
we have also, by the homogeneity of $dV^{(r)}$,
\begin{equation}
\label{eqn-36}
dV^{(r)} = \sigma^{(r)} \, dx^{(r)}
           + \tau^{(r)} \, dy^{(r)} + \upsilon^{(r)} \, dz^{(r)}.
\end{equation}

If we now change the co-ordinates $x^{(r)}$ $y^{(r)}$ $z^{(r)}$ of
the luminous path to any near connected co-ordinates
\begin{equation}
\label{eqn-37}
x_\varepsilon^{(r)} = x^{(r)} + \varepsilon \xi^{(r)},\quad
y_\varepsilon^{(r)} = y^{(r)} + \varepsilon \eta^{(r)},\quad
z_\varepsilon^{(r)} = z^{(r)} + \varepsilon \zeta^{(r)},
\end{equation}
$\varepsilon$ being any small constant, and $\xi^{(r)}$ $\eta^{(r)}$
$\zeta^{(r)}$ any functions of $\varepsilon$ and of the co-ordinates
$x^{(r)}$ $y^{(r)}$ $z^{(r)}$, which do not become infinite for
$\varepsilon = 0$, nor for any point on the $r$th portion of the path,
and which satisfy at the meeting of two such portions the equation of
the corresponding reflecting or refracting surface, and vanish at the
ends of the whole path; we shall pass hereby to a near line having the
same extremities as the luminous path, and having its points of
bending on the same reflecting or refracting surfaces; and the
\emph{law of stationary action} is, that if we compare the integral or
sum, $V = \sum \int dV^{(r)}$, for the luminous path, with the
corresponding integral $V_{\varepsilon}$ for this near line, the
difference of these two integrals or actions bears an indefinitely
small ratio to the quantity $\varepsilon$, (which makes the one line
differ from the other,) when this quantity $\varepsilon$ becomes itself
indefinitely small: so that we have the limiting equation,
\begin{equation}
\mathrm{lim.} \frac{1}{\varepsilon} (V_\varepsilon - V) = 0,
\end{equation}
that is
\begin{equation}
\mathrm{lim.} \sum \frac{1}{\varepsilon} (V_\varepsilon^{(r)} - V^{(r)})
  = \sum \int \mathrm{lim.} \frac{1}{\varepsilon}
         (dV_\varepsilon^{(r)} - dV^{(r)})
  = 0,
\end{equation}
or finally
\begin{equation}
\label{eqn-40}
\sum \int \left(
      \frac{\delta d V_\varepsilon^{(r)}}{\delta \varepsilon} \right)
   = 0.
\end{equation}

To develop this last equation, we have, by (\ref{eqn-33}) and
(\ref{eqn-37}),
\begin{eqnarray}
\left( \frac{\delta dV_\varepsilon^{(r)}}{\delta \varepsilon} \right)
   &=& \sigma^{(r)}   \, d \xi^{(r)}
     + \tau^{(r)}     \, d \eta^{(r)}
     + \upsilon^{(r)} \, d \zeta^{(r)} \nonumber\\
   & &\quad + \left(
            \xi^{(r)}   \frac{\delta v^{(r)}}{\delta x^{(r)}}
          + \eta^{(r)}  \frac{\delta v^{(r)}}{\delta y^{(r)}}
          + \zeta^{(r)} \frac{\delta v^{(r)}}{\delta z^{(r)}}
        \right) ds^{(r)};
\end{eqnarray}
and therefore, integrating by parts, and accenting the symbols which
belong to the beginning of the $r$th portion of the path,
\begin{eqnarray}
\int \left( \frac{\delta dV_\varepsilon^{(r)}}{\delta \varepsilon} \right)
   &=& \sigma^{(r)} \xi^{(r)}
       - \sigma'^{(r)} \xi'^{(r)}
       + \tau^{(r)} \eta^{(r)}
       - \tau'^{(r)} \eta'^{(r)}
       + \upsilon^{(r)} \zeta^{(r)}
       - \upsilon'^{(r)} \zeta'^{(r)} \nonumber\\
   & &\quad +
      \int \xi^{(r)}   \left(
         \frac{\delta v^{(r)}}{\delta x^{(r)}} ds^{(r)}
         - d \sigma^{(r)} \right) \nonumber\\
   & &\quad +
      \int \eta^{(r)}  \left(
         \frac{\delta v^{(r)}}{\delta y^{(r)}} ds^{(r)}
         - d \tau^{(r)} \right) \nonumber\\
   & &\quad +
      \int \zeta^{(r)}  \left(
         \frac{\delta v^{(r)}}{\delta z^{(r)}} ds^{(r)}
         - d \upsilon^{(r)} \right).
\end{eqnarray}

And since the extreme values, and values for the points of juncture,
of the otherwise arbitrary functions $\xi$ $\eta$ $\zeta$, are
subject to the following conditions:
\begin{equation}
\label{eqn-43}
\xi'^{(1)} = 0,\quad
\eta'^{(1)} = 0,\quad
\zeta'^{(1)} = 0,\quad
\xi^{(i+1)} = 0,\quad
\eta^{(i+1)} = 0,\quad
\zeta^{(i+1)} = 0,
\end{equation}
and
\begin{equation}
\xi^{(r)} = \xi'^{(r+1)},\quad
\eta^{(r)} = \eta'^{(r+1)},\quad
\zeta^{(r)} = \zeta'^{(r+1)},
\end{equation}
$r$ varying from $1$ to $i$; and finally, for every value of $r$
within the same range, to the condition
\begin{equation}
n_x^{(r)} \xi^{(r)} + n_y^{(r)} \eta^{(r)} + n_z^{(r)} \zeta^{(r)}
   = 0,
\end{equation}
$n^{(r)}$ being either seminormal to the $r$th reflecting or
refracting surface at the $r$th point of incidence, and $n_x^{(r)}$
$n_y^{(r)}$ $n_z^{(r)}$ being the cosines of the angles which
$n^{(r)}$ makes with the three rectangular positive semiaxes of
co-ordinates $x$ $y$ $z$; the law of stationary action (\ref{eqn-40})
resolves itself into the following equations:
\begin{equation}
\label{eqn-46}
d\sigma^{(r)}     = \frac{\delta v^{(r)}}{\delta x^{(r)}} ds^{(r)};\quad
d\tau^{(r)}       = \frac{\delta v^{(r)}}{\delta y^{(r)}} ds^{(r)};\quad
d\upsilon^{(r)}   = \frac{\delta v^{(r)}}{\delta z^{(r)}} ds^{(r)};
\end{equation}
and
\[ \sigma'^{(r+1)} - \sigma^{(r)}     = \lambda^{(r)} n_x^{(r)};\quad
\tau'^{(r+1)} - \tau^{(r)}         = \lambda^{(r)} n_y^{(r)};\]
\begin{equation}
\label{eqn-47}
\upsilon'^{(r+1)} - \upsilon^{(r)} = \lambda^{(r)} n_z^{(r)};
\end{equation}
in which $\lambda^{(r)}$ is an indeterminate multiplier.  The three
equations (\ref{eqn-46}), which may by the condition (\ref{eqn-36})
be shown to be consistent with each other, express the gradual
changes, if any, of a ray, between its points of sudden bending; and
the equations (\ref{eqn-47}) contain the rules of ordinary and
extraordinary reflexion and refraction.  All these results of that
known law, which I have called the law of stationary action, are fully
confirmed by experience, when suitable forms are assigned to the
functions denoted by $v^{(r)}$.

For example, in the case of an uniform medium, ordinary or
extraordinary, the function $v^{(r)}$ is to be considered as
independent of the undifferentiated co-ordinates $x^{(r)}$ $y^{(r)}$
$z^{(r)}$, and the differential equations (\ref{eqn-46}) of the $r$th
portion of the luminous path become simply
\begin{equation}
d\sigma^{(r)}   = 0,\quad
d\tau^{(r)}     = 0,\quad
d\upsilon^{(r)} = 0,
\end{equation}
and give by integration
\begin{equation}
\sigma^{(r)}   = \mbox{const.},\quad
\tau^{(r)}     = \mbox{const.},\quad
\upsilon^{(r)} = \mbox{const.};
\end{equation}
they express, therefore, the known fact of the rectilinear propagation
of light in a uniform medium, because in such a medium $\sigma^{(r)}$
$\tau^{(r)}$ $\upsilon^{(r)}$ depend only on the colour and
direction, but not on the co-ordinates of the path, and are functions
of $\alpha^{(r)}$ $\beta^{(r)}$ $\gamma^{(r)}$ not including
$x^{(r)}$ $y^{(r)}$ $z^{(r)}$, if we put for abridgment
\begin{equation}
\alpha^{(r)} = \frac{dx^{(r)}}{ds^{(r)}},\quad
\beta^{(r)}  = \frac{dy^{(r)}}{ds^{(r)}},\quad
\gamma^{(r)} = \frac{dz^{(r)}}{ds^{(r)}},
\end{equation}
so that $\alpha^{(r)}$, $\beta^{(r)}$, $\gamma^{(r)}$, represent the
cosines of the inclination (in this case constant) of any element of
the $r$th portion of the path to the positive semiaxes of co-ordinates.
The formul\ae\ (\ref{eqn-46}) give also the known differential
equations for a ray in the earth's atmosphere.

With respect to the rules of reflexion or refraction of light,
expressed by the equations (\ref{eqn-47}), they may in general be thus
summed up;
\begin{equation}
\label{eqn-51}
t_x^{(r)} \Delta \sigma^{(r)} + t_y^{(r)} \Delta \tau^{(r)}
      + t_z^{(r)} \Delta \upsilon^{(r)}
   = 0:
\end{equation}
in which $\Delta$ refers to the sudden changes produced by reflexion
or refraction, and $t_x^{(r)}$ $t_y^{(r)}$ $t_z^{(r)}$ are the
cosines of the inclinations to the positive semiaxes of co-ordinates,
of any arbitrary line $t^{(r)}$, which touches the $r$th reflecting or
refracting surface, at the $r$th point of incidence, so that
\begin{equation}
t_x^{(r)} n_x^{(r)} + t_y^{(r)} n_y^{(r)} + t_z^{(r)} n_z^{(r)}
   = 0.
\end{equation}
In the case of ordinary media, for example, we have
\begin{equation}
\sigma^{(r)} = v^{(r)} \alpha^{(r)},\quad
\tau^{(r)} = v^{(r)} \beta^{(r)},\quad
\upsilon^{(r)} = v^{(r)} \gamma^{(r)};
\end{equation}
and the equation (\ref{eqn-51}) may be put under the form
\begin{equation}
\Delta. v^{(r)} v_t^{(r)} = 0,
\end{equation}
in which
\begin{equation}
v_t^{(r)} = \alpha^{(r)} t_x^{(r)}
          + \beta^{(r)}  t_y^{(r)}
          + \gamma^{(r)} t_z^{(r)},
\end{equation}
so that the unchanged quantity $v^{(r)} v_t^{(r)}$ is the projection
of the index $v^{(r)}$ on the arbitrary tangent $t^{(r)}$, each index
being measured from the point of incidence in the direction of the
corresponding ray: which agrees with the law of Snellius.  In general,
if we put
\begin{equation}
\nu^{(r)} = \sqrt{\sigma^{(r)2} + \tau^{(r)2} + \upsilon^{(r)2}},
\end{equation}
and
\begin{equation}
\sigma^{(r)}   = \nu^{(r)} \nu_x^{(r)},\quad
\tau^{(r)}     = \nu^{(r)} \nu_y^{(r)},\quad
\upsilon^{(r)} = \nu^{(r)} \nu_z^{(r)},
\end{equation}
we may consider $\sigma^{(r)}$ $\tau^{(r)}$ $\upsilon^{(r)}$ as the
projections, on the axes of co-ordinates, of a certain straight line
$\nu^{(r)}$, of which the length and direction depend (according to
rules expressed by the foregoing equations) on the form of the
function $v^{(r)}$ or $dV^{(r)}$, and on the direction and colour of
the element of the luminous path, before or after incidence; and if we
put
\begin{equation}
\nu_t^{(r)} = \nu_x^{(r)} t_x^{(r)}
            + \nu_y^{(r)}  t_y^{(r)}
            + \nu_z^{(r)} t_z^{(r)},
\end{equation}
the equation (\ref{eqn-51}) will take the form
\begin{equation}
\Delta.\nu^{(r)} \nu_t^{(r)} = 0,
\end{equation}
which expresses that the projection of this straight line $\nu^{(r)}$
on any arbitrary tangent $t^{(r)}$ to the reflecting or refracting
surface, at the point of incidence, is not changed by reflection or
refraction, ordinary or extraordinary: which is a convenient general
form for all the known rules of sudden change of direction of a path
of light.  In the undulatory theory, I have found that the line
$\nu^{(r)}$ is the reciprocal of the normal velocity of propagation of
the wave; and its projections may therefore be called \emph{components
of normal slowness}: so that the foregoing property of unchanged
projection of the line $\nu^{(r)}$, may be expressed, in the language
of this theory, by saying that the component of normal slowness in the
direction of any line which touches any ordinary or extraordinary
reflecting or refracting surface at any point of incidence is not
changed by reflection or refraction.   It was, however, by a different
method that I originally deduced this general enunciation of the rules
of optical reflexion and refraction, namely, by employing my principle
of the characteristic function, and that other general law, of which
it is now time to speak.

This other general law, the \emph{law of varying action}, results
from the known law above explained, by considering the extreme points
of a luminous path as variable: that is, by not supposing the six
extreme functions (\ref{eqn-43}) to vanish.  Denoting, for abridgment,
the three final functions of this set by $\xi$ $\eta$ $\zeta$, and
the three initial functions by $\xi'$ $\eta'$ $\zeta'$, and writing
similarly $v$, $dV$, \&c., instead of the final quantities
$v^{(i+1)}$, $dV^{(i+1)}$, \&c. and $v'$, $dV'$, \&c., instead of the
initial quantities $v^{(1)}$, $dV^{(1)}$, \&c., we find this new
equation,
\begin{eqnarray}
\mathrm{lim.} \frac{1}{\varepsilon} (V_\varepsilon - V)
   &=& \sum \int \left(
          \frac{\delta dV_\varepsilon^{(r)}}{\delta \varepsilon} \right)
          \nonumber\\
   &=& \sigma \xi - \sigma' \xi' + \tau \eta - \tau' \eta'
       + \upsilon \zeta - \upsilon' \zeta',
\end{eqnarray}
which is a form of my general result.  It may also be put conveniently
under this other form,
\begin{equation}
\delta V = \sigma   \,\delta x - \sigma'   \,\delta x'
         + \tau     \,\delta y - \tau'     \,\delta y'
         + \upsilon \,\delta z - \upsilon' \,\delta z';
\end{equation}
in which
\begin{equation}
\left. \begin{array}{lllll}
   \sigma   &=& \displaystyle \frac{\delta dV}{\delta dx}
            &=& \displaystyle \frac{\delta . v\,ds}{\delta dx},\\[12pt]
   \tau     &=& \displaystyle \frac{\delta dV}{\delta dy}
            &=& \displaystyle \frac{\delta . v\,ds}{\delta dy},\\[12pt]
   \upsilon &=& \displaystyle \frac{\delta dV}{\delta dz}
            &=& \displaystyle \frac{\delta . v\,ds}{\delta dz},
   \end{array} \right\}
\end{equation}
and
\begin{equation}
\left. \begin{array}{lllll}
   - \sigma'   &=& \displaystyle
                   - \left( \frac{\delta .  v\,ds}{\delta dx} \right)'
               &=& \displaystyle \frac{\delta d'V}{\delta d'x'},\\[12pt]
   - \tau'     &=& \displaystyle
                   - \left( \frac{\delta .  v\,ds}{\delta dy} \right)'
               &=& \displaystyle \frac{\delta d'V}{\delta d'y'},\\[12pt]
   - \upsilon' &=& \displaystyle
                   - \left( \frac{\delta .  v\,ds}{\delta dz} \right)'
               &=& \displaystyle \frac{\delta d'V}{\delta d'z'},
   \end{array} \right\}
\end{equation}
the symbols
\[ \left( \frac{\delta . v\,ds}{\delta dx} \right)', \mbox{ \&c.}, \]
representing the initial quantities which correspond to
\[ \frac{\delta . v\,ds}{\delta dx}, \mbox{ \&c.}; \]
and $d'V$ being, according to the same analogy of notation, the
infinitesimal change of the whole integral~$V$, arising from the
infinitesimal changes $d'x'$, $d'y'$, $d'z'$, of the initial
co-ordinates, that is, from a motion of the initial point $x'$ $y'$
$z'$ along the initial element of the luminous path; so that $d'V$ is
the initial element of the integral taken negatively,
\begin{equation}
d'V = - v' \sqrt{ d'x'^2 + d'y'^2 + d'z'^2}.
\end{equation}
If then we consider the integral or \emph{action}~$V$ as a function
(which I have called the \emph{characteristic function}) of the six
extreme co-ordinates, and if we differentiate this function with
respect to these co-ordinates, we see that its six partial differential
coefficients of the first order may be represented generally by the
equations (\ref{eqn-26}) and (\ref{eqn-30}), which were already proved
to be true for the simple case of rectilinear paths of light.  And as,
in that simple case, those equations, being then equivalent to the
formulae (\ref{eqn-17}) and (\ref{eqn-18}), were seen to determine the
course of the straight ray, which passed with a given direction
through a given initial or a given final point; so, generally, when we
know the initial co-ordinates, direction, and colour of a luminous
path, and the optical properties of the initial medium, we can
determine, or at least restrict (in general) to a finite variety, the
values of the initial coefficients
\[ \frac{\delta d'V}{\delta d'x'},\quad
   \frac{\delta d'V}{\delta d'y'},\quad
   \frac{\delta d'V}{\delta d'z'},\]
which form the second members of the equations (\ref{eqn-30}); and
therefore we may regard as known the first members of the same
equations, namely the partial differential coefficients
\[ \frac{\delta V}{\delta x'},\quad
   \frac{\delta V}{\delta y'},\quad
   \frac{\delta V}{\delta z'},\]
of the characteristic function~$V$, taken with respect to the known
initial co-ordinates: so that if the form of the function~$V$ be known,
we have between the final co-ordinates $x$, $y$, $z$, considered as
variable, the three following equations of the path, or at least of
its final branch,
\begin{equation}
\frac{\delta V}{\delta x'} = \mbox{const.},\quad
\frac{\delta V}{\delta y'} = \mbox{const.},\quad
\frac{\delta V}{\delta z'} = \mbox{const.}
\end{equation}
These three equations are compatible with each other, and are
equivalent only to two distinct relations between the variable
co-ordinates $x$ $y$ $z$, because in general $V$ must satisfy a
partial differential equation of the form
\begin{equation}
\label{eqn-66}
0 = \Omega'( \sigma', \tau', \upsilon', x', y', z' ),
\end{equation}
in which, by what has been shown,
\begin{equation}
\sigma'   = - \frac{\delta V}{\delta x'},\quad
\tau'     = - \frac{\delta V}{\delta y'},\quad
\upsilon' = - \frac{\delta V}{\delta z'},
\end{equation}
and which is therefore analogous to the second formula (\ref{eqn-20}):
this equation (\ref{eqn-66}) being obtained by eliminating the ratios
of $d'x'$, $d'y'$, $d'z'$, between the general formul\ae\
(\ref{eqn-30}).  In like manner the formul\ae\ (\ref{eqn-26}) give
generally a partial differential equation of the form
\begin{equation}
0 = \Omega \left(
      \frac{\delta V}{\delta x},
      \frac{\delta V}{\delta y},
      \frac{\delta V}{\delta z},
      x, y, z \right),
\end{equation}
analogous to the first of those marked (\ref{eqn-20}), and the three
following compatible equations between the variable initial
co-ordinates $x'$, $y'$, $z'$, of a path of light which is obliged to
pass with a given direction through a given final point,
\begin{equation}
\frac{\delta V}{\delta x} = \mbox{const.},\quad
\frac{\delta V}{\delta y} = \mbox{const.},\quad
\frac{\delta V}{\delta z} = \mbox{const.}
\end{equation}
But for the integration and use of these partial differential
equations, the limits of the present communication oblige me to refer
to the volumes, already mentioned, of the Transactions of the
Royal Irish Academy.

I may, however, mention here, that my employment of the characteristic
function~$V$, in all questions of reflexion and refraction, is founded
on an equation in finite differences, which, by the integral nature of
this function~$V$, is evidently satisfied, namely,
\begin{equation}
\label{eqn-70}
\Delta V = 0 = \lambda u;
\end{equation}
$\Delta$ referring, as before, to the sudden changes produced at any
reflecting or refracting surface having for its equation
\begin{equation}
\label{eqn-71}
u = 0;
\end{equation}
and $\lambda$ being an indeterminate multiplier, employed for the
purpose of being able to treat the co-ordinates of incidence as three
independent variables.  For example, the formulae (\ref{eqn-47}), for
a sudden change of direction, result immediately from (\ref{eqn-70}),
under the form
\begin{equation}
\Delta \frac{\delta V}{\delta x}
   = \lambda \frac{\delta u}{\delta x},\qquad
\Delta \frac{\delta V}{\delta y}
   = \lambda \frac{\delta u}{\delta y},\qquad
\Delta \frac{\delta V}{\delta z}
   = \lambda \frac{\delta u}{\delta z},
\end{equation}
by differentiating with respect to the co-ordinates of incidence, as
three independent variables, and then reducing by the equation
(\ref{eqn-71}) of the ordinary or extraordinary reflecting or
refracting surface.  These results respecting the change of direction
of a luminous path may be put under the form
\begin{equation}
\Delta \frac{\displaystyle
   \hspace{6pt} \frac{\delta V}{\delta x} \hspace{6pt}}{\displaystyle
   \frac{\delta u}{\delta x}}
   = \Delta \frac{\displaystyle
   \hspace{6pt} \frac{\delta V}{\delta y} \hspace{6pt}}{\displaystyle
   \frac{\delta u}{\delta y}}
   = \Delta \frac{\displaystyle
   \hspace{6pt} \frac{\delta V}{\delta z} \hspace{6pt}}{\displaystyle
   \frac{\delta u}{\delta z}};
\end{equation}
or under the following,
\begin{equation}
\left. \begin{array}{lll}
\displaystyle \Delta \left(
   \frac{\delta u}{\delta x} \frac{\delta V}{\delta y}
   - \frac{\delta u}{\delta y} \frac{\delta V}{\delta x}
   \right) = 0,\\[12pt]
\displaystyle \Delta \left(
   \frac{\delta u}{\delta y} \frac{\delta V}{\delta z}
   - \frac{\delta u}{\delta z} \frac{\delta V}{\delta y}
   \right) = 0,\\[12pt]
\displaystyle \Delta \left(
   \frac{\delta u}{\delta z} \frac{\delta V}{\delta x}
   - \frac{\delta u}{\delta x} \frac{\delta V}{\delta z}
   \right) = 0:
\end{array} \right\}
\end{equation}
and in general, all theorems respecting the changes produced by
reflexion or refraction in the properties of an optical system, may be
expressed, by the help of the formula (\ref{eqn-70}), as
\emph{permanences} of certain other properties.  The remarkable
permanence, already stated, of the components of normal slowness of
propagation of a luminous wave, was suggested to me by observing that
my function $V$ is (in the undulatory theory) the time of propagation
of light from the initial to the final point, and therefore that the
waves (in the same theory) are represented by the general equation
\begin{equation}
\label{eqn-75}
V = \mbox{const.},
\end{equation}
and the components of normal slowness by the partial differential
coefficients of $V$ of the first order.  The properties of the
function~$V$, on which my whole optical method depends, supplied me
also, long since, with a simple proof of the contested theorem of
Huygens already mentioned, namely, that the rays of any ordinary
homogeneous system, which after issuing originally from any luminous
point, or being (in an initial and ordinary state) perpendicular to
any common surface, have undergone any number of reflexions or
refractions ordinary or extraordinary, before arriving at their final
state, are in that state perpendicular to a series of surfaces,
namely, to the series (\ref{eqn-75}), which are waves in the theory of
Huygens: because, by the properties of my function, the differential
equation of that series is
\begin{equation}
\alpha \,\delta x + \beta \,\delta y + \gamma \,\delta z = 0,
\end{equation}
$\alpha$, $\beta$ and $\gamma$ being the cosines which determine the
final direction of a ray.  It was also by combining the properties of
the same characteristic function~$V$ with the physical principles of
Fresnel, that I was first led, (from perceiving an indeterminateness
in two particular cases in the relations between the coefficients
\[ \frac{\delta dV}{\delta dx},\quad
   \frac{\delta dV}{\delta dy},\quad
   \frac{\delta dV}{\delta dz}, \]
and the ratios of $dx$, $dy$, $dz$,) to form that theoretical
expectation of two kinds of \emph{conical refraction} which I
communicated in last October (1832) to the Royal Irish Academy and to
Professor Lloyd, and which the latter has since verified
experimentally.  Mr. MacCullagh has lately informed me that the same
two indeterminate cases in Fresnel's theory had occurred to him from
geometrical considerations, some years ago, and that he had intended
to try to what geometrical and physical consequences they would lead.

The method of the characteristic function has conducted me to many
other consequences, besides those which I have already published in
the Transactions of the Royal Irish Academy: and I think that
it will hereafter acquire, in the hands of other mathematicians, a
rank in deductive optics, of the same kind as that which the method of
co-ordinates has attained in algebraical geometry.  For as, by the
last-mentioned method, Des Cartes reduced the study of a plane curve,
or of a curved surface, to the study of that one function which
expresses the law of the ordinate, and made it possible thereby to
discover general formul\ae\ for the tangents, curvatures, and all
other geometrical properties of the curve or surface, and to regard
them as included all in that one law, that central algebraical
relation: so I believe that mathematicians will find it possible to
deduce all properties of optical systems from the study of that one
central relation which connects, for each particular system, the
optical function~$V$ with the extreme co-ordinates and the colour, and
which has its partial differential coefficients connected with the
extreme directions of a ray, by the law of varying action, or by the
formul\ae\ (\ref{eqn-26}) and (\ref{eqn-30}).

It only now remains, in order to conclude the present remarks, that I
should briefly explain the allusions already made to my view of an
analogous function and method in the research of the planetary and
cometary orbits under the influence of their mutual perturbations.
The view itself occurred to me many years ago, and I gave a short
notice or announcement of it in the XVth volume (page 80) of the
Transactions of the Royal Irish Academy; but I have only lately
resumed the idea, and have not hitherto published any definite
statement on the subject.

To begin with a simple instance, let us attend first to the case of a
comet, considered as sensibly devoid of mass, and as moving in an
undisturbed parabola about the sun, which latter body we shall regard
as fixed at the origin of co-ordinates, and as having an attracting
mass equal to unity.  Let $r$ be the comet's radius vector at any
moment~$t$ considered as final, and $r'$ the radius vector of the same
comet at any other moment $t'$ considered as initial; let also $r''$ be
the chord joining the ends of $r$ and $r'$, and let us put for
abridgment
\begin{equation}
\label{eqn-77}
V = 2 \sqrt{r + r' + r''} \mp 2 \sqrt{r + r' - r''};
\end{equation}
then I find, that the final and initial components of velocity of the
comet, parallel to any three rectangular semiaxes of co-ordinates, may
be expressed as follows by the coefficients of the function~$V$,
\begin{equation}
\label{eqn-78}
\left. \begin{array}{llllll}
    \displaystyle \frac{dx}{dt}  
         &\displaystyle = \phantom{-} \frac{\delta V}{\delta x},\quad
   &\displaystyle \frac{dy}{dt}
         &\displaystyle = \phantom{-} \frac{\delta V}{\delta y},\quad
   &\displaystyle \frac{dz}{dt}
         &\displaystyle = \phantom{-} \frac{\delta V}{\delta z};\\[12pt]
    \displaystyle \frac{dx'}{dt'}
         &\displaystyle = -           \frac{\delta V}{\delta x'},\quad
   &\displaystyle \frac{dy'}{dt'}
         &\displaystyle = -           \frac{\delta V}{\delta y'},\quad
   &\displaystyle \frac{dz'}{dt'}
         &\displaystyle = -           \frac{\delta V}{\delta z'};
   \end{array} \right\}
\end{equation}
and that this function~$V$ satisfies the two following partial
differential equations,
\begin{equation}
\label{eqn-79}
\left. \begin{array}{lllll}
\displaystyle \left( \frac{\delta V}{\delta x} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta y} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta z} \right)^2
   &=& \displaystyle \frac{2}{r},\\[12pt]
\displaystyle \left( \frac{\delta V}{\delta x'} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta y'} \right)^2
   &+\displaystyle \left( \frac{\delta V}{\delta z'} \right)^2
   &=& \displaystyle \frac{2}{r'},
\end{array} \right\}
\end{equation}
which reconcile the expressions (\ref{eqn-78}) with the known law of a
comet's velocity.  I find also that all the other properties of a
comet's parabolic motion agree with and are included in the
formul\ae\ (\ref{eqn-78}), when the form (\ref{eqn-77}) is assigned to
the function~$V$.  They give, for example, by an easy combination, the
theorem discovered by Euler for the dependence of the time $(t - t')$
on the parabolic chord ($r''$) and on the sum ($r + r'$) of the radii
drawn to its extremities.

More generally, in any system of points or bodies which attract or
repel one another according to any function of the distance, for
example, in the solar system, I have found that the final and initial
components of momentum may be expressed in a similar manner, by the
partial differential coefficients of the first order of some one
central or \emph{characteristic function}~$V$ of the final and initial
co-ordinates; so that we have generally, by a suitable choice of $V$,
\begin{equation}
\label{eqn-80}
\left. \begin{array}{lll}
    \displaystyle m_1 \frac{dx_1}{dt}
      =  \frac{\delta V}{\delta x_1};\quad
   &\displaystyle m_1 \frac{dy_1}{dt}
      =  \frac{\delta V}{\delta y_1};\quad
   &\displaystyle m_1 \frac{dz_1}{dt}
      =  \frac{\delta V}{\delta z_1};\\[12pt]
    \displaystyle m_2 \frac{dx_2}{dt}
      =  \frac{\delta V}{\delta x_2};\quad
   &\mbox{\&c.};
\end{array} \right\}
\end{equation}
and
\begin{equation}
\label{eqn-81}
\left. \begin{array}{lll}
    \displaystyle m_1 \frac{dx_1'}{dt'}
      = -\frac{\delta V}{\delta x_1'};\quad
   &\displaystyle m_1 \frac{dy_1'}{dt'}
      = -\frac{\delta V}{\delta y_1'};\quad
   &\displaystyle m_1 \frac{dz_1'}{dt'}
      = -\frac{\delta V}{\delta z_1'};\\[12pt]
    \displaystyle m_2 \frac{dx_2'}{dt'}
      = -\frac{\delta V}{\delta x_2'};\quad
   &\mbox{\&c.}:
\end{array} \right\}
\end{equation}
$m_1$, $m_2$, \&c., being the masses of the system, and the
function~$V$ being obliged to satisfy two partial differential
equations of the first order and second degree, which are analogous to
(\ref{eqn-79}), and may be thus denoted
\begin{equation}
\label{eqn-82}
\left. \begin{array}{lll}
\displaystyle \sum. \frac{1}{m} \left\{
        \left( \frac{\delta V}{\delta x} \right)^2
      + \left( \frac{\delta V}{\delta y} \right)^2
      + \left( \frac{\delta V}{\delta z} \right)^2
     \right\} &=& 2F,\\[12pt]
\displaystyle \sum. \frac{1}{m} \left\{
        \left( \frac{\delta V}{\delta x'} \right)^2
      + \left( \frac{\delta V}{\delta y'} \right)^2
      + \left( \frac{\delta V}{\delta z'} \right)^2
     \right\} &=& 2F';
\end{array} \right\}
\end{equation}
the function~$F$ involving the final co-ordinates, and the
function~$F'$ involving similarly the initial co-ordinates, and the
common form of these two functions depending on the law of attraction
or repulsion.  In the solar system
\begin{equation}
\left. \begin{array}{lll}
F  &=& \displaystyle \sum
      \frac{m_i m_k}{\sqrt{(x_i - x_k)^2 + (y_i - y_k)^2
         + (z_i - z_k)^2}} + H,\\[18pt]
F' &=& \displaystyle \sum
      \frac{m_i m_k}{\sqrt{(x_i' - x_k')^2 + (y_i' - y_k')^2
         + (z_i' - z_k')^2}} + H,
\end{array} \right\}
\end{equation}
$H$ being a certain constant; and in general the partial differential
equations (\ref{eqn-82}) contain the law of living forces, which the
other known general laws or integrals of the equations of motion are
expressed by other general and simple properties of the same
characteristic function~$V$: the coefficients of which function, when
combined with the relations (\ref{eqn-80}) and (\ref{eqn-81}), are
sufficient to determine all circumstances of the motion of a system.
By this view the research of the most complicated orbits, in lunar,
planetary, and sidereal astronomy, is reduced to the study of the
properties of a single function~$V$; which is analogous to my optical
function, and represents the \emph{action} of the system from one
position to another.  If we knew, for example, the form of this one
function~$V$ for a system of three bodies attracting according to
Newton's law, (suppose the system of Sun, Earth, and Moon, or of the
Sun, Jupiter and Saturn,) we should need no further integration in
order to determine the separate paths and the successive
configurations of these three bodies; the eight relations, independent
of the time, between their nine variable co-ordinates, would be given
at once by differentiating the one function~$V$, and employing the
nine initial equations of the form (\ref{eqn-81}), which in
consequence of the second equation (\ref{eqn-82}) are only equivalent
to eight distinct relations, the positions and velocities being given
for some one initial epoch; and the variable time~$t$ of arriving at
any one of the subsequent states of the system would be given by a
single integration of any combination of these relations with the
equations (\ref{eqn-80}).  The development of this view, including its
extension to other analogous questions, appears to me to open in
mechanics and astronomy an entirely new field of research.  I shall
only add, that the view was suggested by a general law of varying
action in dynamics, which I had deduced from the known dynamical law
of least or stationary action, by a process analogous to that general
reasoning in optics which I have already endeavoured to illustrate.

\begin{quote}
\textit{Observatory of Trinity College, Dublin,\\
September, 1833.}
\end{quote}

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