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to common

where (X, X), (X, Y), etc., denote constants, viz., the values of Application d'o do

optics, the differential coefficients

da dxdy'
etc., when each of the or kinetics

of a single

particle. six co-ordinates x, y, z, x', y', z vanishes; and R denotes the remainder after the terms of the second degree. According to Cauchy's principles regarding the convergence of Taylor's theorem, we have a rigorous expression for - A in the same form, without R, if the coefficients (X, X), etc., denote the values of the differential coefficients with some variable values intermediate between 0 and the actual values of x, y, etc., substituted for these elements. Hence, provided the values of the differential coefficients are infinitely nearly the same for any infinitely small values of the co-ordinates as for the vanishing values, R becomes infinitely smaller than the terms preceding it, when x, y, etc., are each infinitely small. Hence when each of the variables x, y, z, x', y', is infinitely small, we may omit R in the expression (1) for -A. Now, as in the proposition to be proved, let us suppose z and z' each to be rigorously zero : and we have

=(X, X) x + (X, Y)y + (X, X') x' +(X, Y') y';

(Y, Y) y +(X, Y) x + (Y, X') oc' + (Y, Y') y'.
These expressions, if in them we make x = 0, and y=0, be-
come the component velocities parallel to 0X, OY, of a particle
passing through O having been projected from P'. Hence, if

, n, denote its co-ordinates, an infinitely small time, after
it passes through 0, we have $ = a, and
f={{X, X') ở + (X, Y sỹ, n={(Y, X') 2 + (X, Y sỹ...(2).
Here & and n are the rectangular co-ordinates of the point & in
which, in the second case, the supposed target is struck. And
by hypothesis
x" + y^2 = po.......

If we eliminate ac', y between these three equations, we have
clearly an ellipse; and the former two express the relation of the
“corresponding” points. Corresponding equations with x and
y for ac and y'; with &', n for $, n; and with - (X, X'),
-(Y, X'), -(X, Y'), - (Y, 1"), in place of (-X, I'), (X, Y'),


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Application to common optics, or kinetics of a single particle. Application to common optics.

(Y, Y''), (Y, Y'), express the first case. Hence the proposition, as is most easily seen by choosing OX and O' I so that (X, Y)

and (Y, X') may each be zero. 335. The most obvious optical application of this remarkable result is, that in the use of any optical apparatus whatever, if the eye and the object be interchanged without altering the position of the instrument, the magnifying power is unaltered. This is easily understood when, as in an ordinary telescope, microscope, or opera-glass (Galilean telescope), the instrument is symmetrical about an axis, and is curiously contradictory of the common idea that a telescope “diminishes " when looked through the wrong way, which no doubt is true if the telescope is simply reversed about the middle of its length, eye and object remaining fixed. But if the telescope be removed from the eye till its eye-piece is close to the object, the part of the object seen will be seen enlarged to the same extent as when viewed with the telescope held in the usual manner. This is easily verified by looking from a distance of a few yards, in through the object-glass of an opera-glass, at the eye of another person holding it to his eye in the usual way.

The more general application may be illustrated thus :- Let the points, O, O' (the centres of the two circles described in the preceding enunciation), be the optic centres of the eyes of two persons looking at one another through any set of lenses, prisms, or transparent media arranged in any way between them. If their pupils are of equal sizes in reality, they will be seen as similar ellipses of equal apparent dimensions by the two observers. Here the imagined particles of light, projected from the circumference of the pupil of either eye, are substituted for the projectiles from the circumference of either circle, and the retina of the other eye takes the place of the target receiv

ing them, in the general kinetic statement. Application

336. If instead of one free particle we have a conservative free mutu-" system of any number of mutually influencing free particles, the fluencing same statement may be applied with reference to the initial particles.

position of one of the particles and the final position of another, or with reference to the initial positions or to the final positions

ally in

of two of the particles. It serves to show how the influence of application an infinitely small change in one of those positions, on the di- free muturection of the other particle passing through the other position, fluencing is related to the influence on the direction of the former particle passing through the former position produced by an infinitely small change in the latter position. A corresponding statement, and to gein terms of generalized co-ordinates, may of course be adapted system. to a system of rigid bodies or particles connected in any way. All such statements are included in the following very general proposition :

The rate of increase of any one component momentum, corresponding to any one of the co-ordinates, per unit of increase of any other co-ordinate, is equal to the rate of increase of the component momentum corresponding to the latter per unit increase or diminution of the former co-ordinate, according as the two coordinates chosen belong to one configuration of the system, or one of them belongs to the initial configuration and the other to the final.

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Let 4 and


be two out of the whole number of co-ordinates constituting the argument of the Hamiltonian characteristic function A ; and g, n the corresponding momentums, . We have [330 (18)]

dA dA


the upper or lower sign being used according as it is a final or
an initial co-ordinate that is concerned. Hence


dn dydy dx


and therefore

dx dy
if both co-ordinates belong to one configuration, or


if one belongs to the initial configuration, and the other to the
final, which is the second proposition. The geometrical inter-
pretation of this statement for the case of a free particle, and two
co-ordinates both belonging to one position, its final position, for


Application to system of free mutu. ally in fluencing particles, and to generalized system.

instance, gives merely the proposition of $ 332 above, for the
case of particles projected from one point, with equal velocities
in all directions; or, in other words, the case of the arbitrary
surface of that enunciation, being reduced to a point. To com-
plete the set of variational equations derived from $ 330 we have
dt dn

which expresses another remarkable property of con-
servative motion.

Slig'itly 337. By the help of Lagrange's form of the equations of disturbed equilibrium. motion, $ 318, we may now, as a preliminary to the considera

tion of stability of motion, investigate the motion of a system infinitely little disturbed from a position of equilibrium, and left free to move, the velocities of its parts being initially infinitely small. The resulting equations give the values of the independent co-ordinates at any future time, provided the displacements continue infinitely small; and the mathematical expressions for their values must of course show the nature of the equilibrium, giving at the same time an interesting example of the coeristence of small motions, $ 89.

The method consists simply in finding what the equations of motion, and their integrals, become for co-ordinates which differ infinitely little from values corresponding to a configuration of equilibriumand for an infinitely small initial kinetic energy. The solution of these differential equations is always easy, as they are linear and have constant coefficients. If the solution indicates that these differences remain infinitely small, the position is one of stable equilibrium; if it shows that one or more of them may increase indefinitely, the result of an infinitely small displacement from or infinitely small velocity through the position of equilibrium may be a finite departure from it—and thus the equilibrium is unstable.

Since there is a position of equilibrium, the kinematic relations must be invariable. As before,

T= } {(4,4) 42 + (, ) $+ 2 (4,0) 40+ etc....}... (1), which cannot be negative for any values of the co-ordinates. Now, though the values of the coefficients in this expression are not generally constant, they are to be taken as constant in the approximate investigation, since their variations, depending on




It dj

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the infinitely small variations of y, $, etc., can only give rise to Slightly terms of the third or higher orders of small quantities. Hence equilibrium. Lagrange's equations become simply d /dT

didt Y,

= 0, etc..................(2), dt

\do and the first member of each of these equations is a linear function of it, $, etc., with constant coefficients.

Now, since we may take what origin we please for the generalized co-ordinates, it will be convenient to assume that y, 0, 0, etc., are measured from the position of equilibrium considered ; and that their values are therefore always infinitely small.

Hence, infinitely small quantities of higher orders being neglected, and the forces being supposed to be independent of the velocities, we shall have linear expressions for V, R, etc., in terms of 4, 6, etc., which we may write as follows :

V = ax + b + co +
Q=a'y + b'o + c'A +


etc. Equations (2) consequently become linear differential equations of the second order, with constant coefficients; as many in number as there are variables y, $, etc., to be determined.

The regular processes explained in elementary treatises on differential equations, lead of course, independently of any particular relation between the coefficients, to a general form of solution ($ 343 below). But this form has very remarkable characteristics in the case of a conservative system; which we therefore examine particularly in the first place. In this case we have




dy' where V is, in our approximation, a homogeneous quadratic function of 4, 0, if we take the origin, or configuration of equilibrium, as the configuration from which (s 273) the potential energy is reckoned. Now, it is obvious*, from the theory

ous tralisformation of two

• For in the first place any such assumption as

Simultaney=A4, + BP, +. ø= A'y, + B'°,+...

quadratic etc., etc.

to sums of gives equations for y, ¢, etc., in terms of y, $, etc., with the same coefficients, squares. A, B, etc., if these are independent of t. Hence (the co-ordinates being i in


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