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Properties of surfaces of equal action.

finitely small difference of action from (x, y, z) to any other point (x + dx, y + dy, ≈ + d), we have

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Hence, by (1),

whence, by (2),

which is the second proposition.


Let the second point be at an infinitely small distance, e, from the first, in the direction of the normal to the surface of equal action; that is to say, let

Sx = eλ, dy = ep, dz=ev.

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333. Irrespectively of methods for finding the "characterof varying istic function" in kinetic problems, the fact that any case of


motion whatever can be represented by means of a single function in the manner explained in § 331, is most remarkable, and, when geometrically interpreted, leads to highly important and interesting properties of motion, which have valuable applications in various branches of Natural Philosophy. One of the many applications of the general principle made by Hamilton* led to a general theory of optical instruments, comprehending the whole in one expression.

Some of its most direct applications; to the motions of planets, comets, etc., considered as free points, and to the celebrated problem of perturbations, known as the Problem of Three Bodies, are worked out in considerable detail by Hamilton (Phil. Trans., 1834-35), and in various memoirs by Jacobi, Liouville, Bour, Donkin, Cayley, Boole, etc. The now abandoned, but still interesting, corpuscular theory of light furnishes a good and exceedingly simple illustration. In this theory light is supposed to consist of material particles not mutually influencing one another, but subject to molecular forces from the particles of bodies-not sensible at sensible distances, and therefore not causing any deviation from uniform rectilinear motion in a homogeneous medium, except within an indefinitely small dis

* On the Theory of Systems of Rays. Trans. R. I. A., 1824, 1830, 1832.

of varying

tance from its boundary. The laws of reflection and of single Examples refraction follow correctly from this hypothesis, which therefore action. suffices for what is called geometrical optics.

to common

We hope to return to this subject, with sufficient detail, Application in treating of Optics. At present we limit ourselves to state optics, a theorem comprehending the known rule for measuring the magnifying power of a telescope or microscope (by comparing the diameter of the object-glass with the diameter of pencil of parallel rays emerging from the eye-piece, when a point of light is placed at a great distance in front of the object-glass), as a particular case.

of a single

334. Let any number of attracting or repelling masses, or or kinetics perfectly smooth elastic objects, be fixed in space. Let two particle. stations, O and O', be chosen. Let a shot be fired with a stated velocity, V, from O, in such a direction as to pass through O'. There may clearly be more than one natural path by which this may be done; but, generally speaking, when one such path is chosen, no other, not considerably diverging from it, can be found; and any infinitely small deviation in the line of fire from O, will cause the bullet to pass infinitely near to, but not through, O'. Now let a circle, with infinitely small radius r, be described round O as centre, in a plane perpendicular to the line of fire from this point, and let—all with infinitely nearly the same velocity, but fulfilling the condition that the sum of the potential and kinetic energies is the same as that of the shot from 0-bullets be fired from all points of this circle, all directed infinitely nearly parallel to the line of fire from O, but each precisely so as to pass through O'. Let a target be held at an infinitely small distance, a', beyond O', in a plane perpendicular to the line of the shot reaching it from O. The bullets fired from the circumference of the circle round O, will, after passing through O', strike this target in the circumference of an exceedingly small ellipse, each with a velocity (corresponding of course to its position, under the law of energy) differing infinitely little from V', the common velocity with which they pass through O. Let now a circle, equal to the former, be described round O', in the plane perpendicular to the central path through O', and let bullets be fired from points in its circumference, each

Application with the proper velocity, and in such a direction infinitely nearly parallel to the central path as to make it pass through


or kinetics

of a single O. These bullets, if a target is held to receive them perpen


dicularly at a distance a = a'

, beyond 0, will strike it along

V V the circumference of an ellipse equal to the former and placedin a "corresponding" position; and the points struck by the individual bullets will correspond; according to the following law of "correspondence":-Let Pand P' be points of the first and second circles, and Q and Q' the points on the first and second targets which bullets from them strike; then if P' be in a plane containing the central path through O' and the position which Q would take if its ellipse were made circular by a pure strain (§ 183); Q and Q' are similarly situated on the two ellipses.

For, let XOY be a plane perpendicular to the central path through 0; and XO'Y' the corresponding plane through O'. Let A be the "action" from 0 to O', and the action from a point P(x, y, z), in the neighbourhood of O, specified with reference to the former axes of co-ordinates, to a point P' (x', y', '), in the neighbourhood of O', specified with reference to the latter.

The function - A vanishes, of course, when x = 0, y = 0, z=0, x'=0, y' = 0, z'=0. Also, for the same values of the

co-ordinates, its differential coefficients


dф dф
dx' dy'


must vanish, and

must be respectively equal to

do dz' V and V', since, for any values whatever of the co-ordinates, do do

and are the component velocities parallel to the two lines dx dy

OX, OY, of the particle passing through P, when it comes from
P', and


do da


are the components parallel to OX', OY,


of the velocity through P' directed so as to reach P. Hence by Taylor's (or Maclaurin's) theorem we have

$ − A = − V'z' + Vz

do dz

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+ } {(X, X) x2 + (Y, Y) y2 + ... + (X', X') x22 + ..

+ 2 (Y, Z) yz + ... + 2 (Y', Z') y'' +....

+ 2 (X, X') xx2 + 2 (Y, Y') yy' + 2 (Z, Z') ≈≈′

+ 2 (X, Y') xy' + 2 (X, Z') xz + ... + 2 (Z, Y') ≈y'} + R ...(1),

where (X, X), (X, Y), etc., denote constants, viz., the values of Application

ď ď

to common optics,

the differential coefficients


etc., when each of the or kinetics of a single particle.

dx dxdy' 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', z' 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

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These expressions, if in them we make x=0, and y = 0, become the component velocities parallel to OX, OY, of a particle passing through O having been projected from P'. Hence, if


, 7, denote its co-ordinates, an infinitely small time, it passes through 0, we have (= a, and

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§ = {(X, X') x' + (X, Y') y'} 7, n={(Y, X') x' + (Y, Y′) y'} † ..... (2).

Here έ and ʼn are the rectangular co-ordinates of the point Q' in which, in the second case, the supposed target is struck. And by hypothesis


x2 + y'3 = r3........


If we eliminate x', 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 and y'; with ', n' for έ, n; and with (X, X'),

− (Y, X'), − (X, Y'), − (Y, Y'), in place of (X, X'), (X, Y'),

to common
or kinetics

of a single


to common optics.

(Y, X'), (Y, Y'), express the first case.

Hence the proposition,

as is most easily seen by choosing OX and O'X' 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 receiving them, in the general kinetic statement.


to of

336. If instead of one free particle we have a conservative free mutu system of any number of mutually influencing free particles, the

ally influencing particles.

same statement may be applied with reference to the initial position of one of the particles and the final position of another, or with reference to the initial positions or to the final positions

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