viating infinitely little from the course OPO'. Then, as OPO is a natural course,

which proves that the chief, or quadratic, term in the other expression (3) for the same, vanishes. Hence one at least of the coefficients A,, A,,... must vanish, and if one only, A., = 0 for instance, we must have

These equations express the condition that P, lies on a natural course from 0 to O'.

(g) Conversely if one or more of the coefficients A ̧, 1⁄4 ̧, etc., vanishes, if for instance A,, = 0, S must be a kinetic focus. For if we take P, so that

(h) Thus we have proved that at a kinetic focus conjugate to O the action from O is not a minimum of the first order*, and that the last configuration, up to which the action from O is a minimum of the first order, is a kinetic focus conjugate to 0.

(i) It remains to be proved that the action from O ceases to be a minimum when the first kinetic focus conjugate to O is passed. Let, as above (§ 360), O...P...O'...P' be a natural course extending beyond O', the first kinetic focus conjugate to 0. Let P and P' be so near one another that there is no focus conjugate to either, between them; and let O...P,...O' be a natural course from 0 to O' deviating infinitely little from 0...P...O'. By what we have just proved (e), the action 00' along 0...P,...O' differs only by R, an infinitely small quantity of the third order, from the action 00' along O...P...O', and therefore

Ac. (O...P...O...P') = Ac. (O...P,...O') + O'P' + R

=OP,+P,O' + O'P' + R.

* A maximum or minimum "of the first order" of any function of one or more variables, is one in which the differential of the first degree vanishes, but not that of the second degree.

But, by a proper application of (e) we see that

PO' + O'P' = PP' + Q

Natural
Course

proved not
a course of
minimum

where Q denotes an infinitely small quantity of the second order, action, which is essentially positive. Hence

Ac (0...P...O'...P') = OP, + P,P' + Q + R,

and therefore, as R is infinitely small in comparison with Q,

Ac (0...P...O'...P')> OP ̧+ P ̧P'.

Hence the broken course 0...P, P...P' has less action than the natural course 0...P...O'...P', and therefore, as the two are infinitely near one another, the latter is not a minimum.

beyond a kinetic focus.

which in

focus con

either ex

362. As it has been proved that the action from any con- A course figuration ceases to be a minimum at the first conjugate kinetic cludes no focus, we see immediately that if O' be the first kinetic focus jugate to conjugate to O, reached after passing O, no two configurations tremity includes on this course from 0 to 0' can be kinetic foci to one another, no pair of conjugate For, the action from O just ceasing to be a minimum when O' foci. is reached, the action between any two intermediate configurations of the same course is necessarily a minimum.

kinetic foci

363. When there are i degrees of freedom to move there How many are in general, on any natural course from any particular con- in any case. figuration, O, at least i-1 kinetic foci conjugate to 0. Thus, for example, on the course of a ray of light emanating from a luminous point 0, and passing through the centre of a convex lens held obliquely to its path, there are two kinetic foci conjugate to O, as defined above, being the points in which the line of the central ray is cut by the so-called " focal lines"* of a pencil of rays diverging from O and made convergent after passing through the lens. But some or all of these kinetic foci may be on the course previous to 0; as for instance in the case of a common projectile when its course passes obliquely downwards through 0. Or some or all may be lost; as when, in the optical illustration just referred to, the lens is only strong enough to produce convergence in one of the principal planes, or too weak to produce convergence in either. Thus

* In our second volume we hope to give all necessary elementary explanations on this subject.

How many

kinetic foci in any case.

Theorem of maximum

action.

also in the case of the undisturbed rectilineal motion of a point, or in the motion of a point uninfluenced by force, on an anticlastic surface (§ 355), there are no real kinetic foci. In the motion of a projectile (not confined to one vertical plane) there can only be one kinetic focus on each path, conjugate to one given point; though there are three degrees of freedom. Again, there may be any number more than i-1, of foci in one course, all conjugate to one configuration, as for instance on the course of a particle uninfluenced by force, moving round the surface of an anchor-ring, along either the outer great circle, or along a sinuous geodetic such as we have considered in § 361, in which clearly there are an infinite number of foci each conjugate to any one point of the path, at equal successive distances from one another.

Referring to the notation of § 361 (ƒ), let S′ be gradually moved on until first one of the coefficients, A., for instance, vanishes; then another, A.,, etc.; and so on. We have seen that each of these positions of S is a kinetic focus: and thus by the successive vanishing of the i-1 coefficients we have i−1 foci. If none of the coefficients can ever vanish, there are no kinetic foci. If one or more of them, after vanishing, comes to a minimum, and again vanishes, as S is moved on, there may be any number more than i-1 of foci each conjugate to the same configuration, O.

*

364. If 1 distinct courses from a configuration O, each differing infinitely little from a certain natural course

-19

cut it in configurations O1, 0, 0,,...O11, and if, besides these, there are not on it any other kinetic foci conjugate to O, between O and Q, and no focus at all, conjugate to E, between E and Q, the action in this natural course from 0 to Q is the maximum for all courses 0...P, P... Q; P, being a configuration infinitely nearly agreeing with some configuration between E and 0, of the standard course 0...E...0....0...O... Q, and O...P, P...Q

* Two courses are not called distinct if they differ from one another only in the absolute magnitude, not in the proportions of the components, of the deviations by which they differ from the standard course.

denoting the natural courses between 0 and P,, and P, and Q, Theorem of which deviate infinitely little from this standard course.

27

In § 361 (i), let O' be any one, O,, of the foci 0,, O2, ... O¡-19 and let P, be called P, in this case. The demonstration there given shows that

Hence there are i-1 different broken courses

0... P1, P, ... Q; 0... P2, P, ... Q; etc.,

2

in each of which the action is less than in the standard course
from 0 to Q. But whatever be the deviation of P,, it may
clearly be compounded of deviations P to P,, P to P,, P to P,,

P to P,, corresponding to these i-1 cases respectively;
and it is easily seen from the analysis that

1

OP,+PQ-OQ = (OP1 + P ̧Q − OQ) + (OP ̧+P ̧Q−OQ) +
Hence OP,+PQ0Q, which was to be proved.

...

maximum action.

tions to two

freedom.

365. Considering now, for simplicity, only cases in which Applicathere are but two degrees (§§ 195, 204) of freedom to move, degrees of we see that after any infinitely small conservative disturbance of a system in passing through a certain configuration, the system will first again pass through a configuration of the undisturbed course, at the first configuration of the latter at which the action in the undisturbed motion ceases to be a minimum. For instance, in the case of a particle, confined to a surface, and subject to any conservative system of force, an infinitely small conservative disturbance of its motion through any point, O, produces a disturbed path, which cuts the undisturbed path at the first point, O', at which the action in the undisturbed path from O ceases to be a minimum. Or, if projectiles, under the influence of gravity alone, be thrown from one point, O, in all directions with equal velocities, in one vertical plane, their paths, as is easily proved, intersect one another consecutively in a parabola, of which the focus is 0, and the vertex the point reached by the particle projected directly upwards. The actual course of each particle from O is the course of least possible action to any point, P, reached before the enveloping parabola, but is not a course of minimum action to any point, Q, in its path after the envelope is passed.

Applica tions to two

freedom.

366. Or again, if a particle slides round along the greatest degrees of circle of the smooth inner surface of a hollow anchor-ring, the "action," or simply the length of path, from point to point, will be least possible for lengths (§ 351) less than T√ab. Thus, if a string be tied round outside on the greatest circle of a perfectly smooth anchor-ring, it will slip off unless held in position by staples, or checks of some kind, at distances of not less than Vab from one another in succession round the circle. With reference to this example, see also § 361, above.

Hamilton's

second form.

Liouville's kinetic

theorem.

π

Or, of a particle sliding down an inclined cylindrical groove, the action from any point will be the least possible along the straight path to any other point reached in a time less than that of the vibration one way of a simple pendulum of length equal to the radius of the groove, and influenced by a force equal g cos i, instead of g the whole force of gravity. But the action will not be a minimum from any point, along the straight path, to any other point reached in a longer time than this. The case in which the groove is horizontal (i=0) and the particle is projected along it, is particularly simple and instructive, and may be worked out in detail with great ease, without assuming any of the general theorems regarding action.

367. In the preceding account of the Hamiltonian principle, and of developments and applications which it has received, we have adhered to the system (S$ 328, 330) in which the initial and final co-ordinates and the constant sum of potential and kinetic energies are the elements of which the action is supposed to be a function. Another system was also given by Hamilton, according to which the action is expressed in terms of the initial and final co-ordinates and the time prescribed for the motion; and a set of expressions quite analogous to those with which we have worked, are established. For practical applications this method is generally less convenient than the other; and the analytical relations between the two are so obvious that we need not devote any space to them here.

368. We conclude by calling attention to a very novel analytical investigation of the motion of a conservative system, by Liouville (Comptes Rendus, June 16, 1856), which leads im