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anchor-ring considered above, the action, or length of path, is clearly a minimum for any one point to the point reached at any subsequent time. The action is not merely a minimum, but is the least possible, from any point of the circular path to any other, through less than half a circumference of the circle.) On the other hand, although the path from any point in the greatest circle of the ring to any other at a distance from it along the circle, less than a Vab, is clearly least possible if along the circumference; the path of absolutely least length is not along the circumference between two points at a greater circular distance than a Vab from one another, nor is the path along the circumference between them a minimum at all in this latter case. On any surface whatever which is everywhere anticlastic; or along a geodetic of any surface which passes altogether through an anticlastic region, the motion is thoroughly unstable. For if it were stable from any point 0, we should have the given undisturbed path, and the disturbed path from 0 cutting it at some point (- two different geodetic lines joining two points; which is impossible on any anticlastic surface, inasmuch as the sum of the exterior angles of any closed figure of geodetic lines exceeds four right angles when the integral curvature of the enclosed area is negative, which is the case for every portion of surface thoroughly anticlastic. But, on the other hand, it is easily proved that if we have an endless rigid band of curved surface everywhere synclastic, with a geodetic line running through its middle, the motion of a particle projected along this line will be stable throughout, and an infinitely slight disturbance will give a disturbed path cutting the given undisturbed path again and again for ever at successive distances differing according to the different specific curvatures of the intermediate portions of the surface.
310. If, from any one configuration, two courses differing infinitely little from one another, have again a configuration in common, this second configuration will be called a kinetic focus relatively to the first: or (because of the reversibility of the motion) these two configurations will be called conjugate kinetic foci. Optic foci, if for a moment we adopt the corpuscular theory of light, are included as a particular case of kinetic foci in general. But it is not difficult to prove that there must be finite intervals of space and time between two conjugate foci in every motion of every kind of system, only provided the kinetic energy does not vanish.
311. Now it is obvious that, provided only a sufficiently short course is considered, the action, in any natural motion of a system, is less than for any other course between its terminal configurations. It will be proved presently ($ 318) that the first configuration up to which the action, reckoned from a given initial configuration, ceases to be a minimum, is the first kinetic focus; and conversely, that when the first kinetic focus is passed, the action, reckoned from the initial configuration, ceases to be a minimum; and therefore of course can never again be a minimum, because a course of shorter action,
deviating infinitely little from it, can be found for a part, without altering the remainder of the whole, natural course.
312. In such statements as this it will frequently be convenient to indicate particular configurations of the system by single letters, as O, P, Q, R; and any particular course, in which it moves through configurations thus indicated, will be called the course O...P...Q...R. The action in any natural course will be denoted simply by the terminal letters, taken in the order of the inotion. Thus OR will denote the action from 0 to R; and therefore OR=-RO. When there are more real natural courses from 0 to R than one, the analytical expression for OR will have more than one real value; and it may be necessary to specify for which of these courses the action is reckoned. Thus we may have
OR for 0...E...R
OR for 0... E"... R,
313. In terms of this notation the preceding statement ($ 311) may be expressed thus :-If, for a conservative system, moving on a certain course 0...,P...O...P', the first kinetic focus conjugate to O be 0, the action OP, in this course, will be less than the action along any other course deviating infinitely little from it: but, on the other hand, OP is greater than the actions in some courses from O to pdeviating infinitely little from the specified natural course 0...P...O...P.
314. It must not be supposed that the action along OP is necessarily the least possible from 0 to P. There are, in fact, cases in which the action ceases to be least of all possible, before kinetic focus is reached. Thus if OEAPOE' A' be a sinuous geodetic line cutting the outer circle of an anchor-ring, or the equator of an oblate spheroid, in successive points 0, A, A', it is easily seen that 0, the first kinetic focus conjugate to 0, must lie somewhat beyond A. But the length OEAP, although a minimum (a stable position for a stretched string), is not the shortest distance on the surface from Oto P, as this must obviously be a line lying entirely on one side
of the great circle. From 0 to any point, Q, short of A, the distance along the geodetic OEQA is clearly the least possible: but if Q be near enough to A (that is to say, between A and the point in which the envelop of the geodetics drawn from 0, cuts OĒA), there will also be two other geodetics from 0 to Q. The length of one of these will be a minimum, and that of the other not a minimum. If Q is moved forward to A, the former becomes OE, A, equal and similar to OEA, but on the other side of the great circle : and the latter becomes the great circle from 0 to A. If now l be moved on to P, beyond A, the minimum geodetic OEAP ceases to be the less of the two minima, and the geodetic OFP lying altogether on the other side of the great circle becomes the least possible line from O to P. But until P is advanced beyond the point 0, in which it is cut by another geodetic from O lying infinitely nearly along it, the length OEAP remains a minimum according to the general proposition of 311.
315. As it has been proved that the action from any configuration ceases to be a minimum at the first conjugate kinetic focus, we see immediately that if O be the first kinetic focus conjugate to O, reached after passing 0, no two configurations on this course from 0 to 0 can be kinetic foci to one another. For, the action from O just ceasing to be a minimum when O' is reached, the action between any two intermediate configurations of the same course is necessarily a minimum.
316. When there are i degrees of freedom to move there are in general, on any natural course from any particular configuration, 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 0, as defined above, being the points in which the line of the central ray is cut by the so-called ' focal lines '1 of a pencil of rays diverging from 0 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 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 ($ 309), there are no real kinetic foci. In the motion of a projectile (not confined to one vertical plane) there can be only one kinetic focus on each path, conjugate to one given point; though there are three degrees of freedom. Again,
1 In our second volume we hope to give all necessary elementary explanations on this subject.
more than 3-1 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 § 311, 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.
317. If i – 1 distinct courses from a configuration O, each differing infinitely little from a certain natural course 0..E..0,..02.. 0;- . . Q, cut it in configurations (,, 0,, 0g,... 0-1, and if, besides these, there are not on it any other kinetic foci conjugate to 0, between 0 and l, and no focus at all, conjugate to E, between
E and l, the action in this natural course from 0 to l is the maximum for all courses 0...P,, P,...Q; P, being a configuration infinitely nearly agreeing with some configuration between E and , of the standard course 0.. E..0,..0....0;- . . l, and 0...P,,P,... Q denoting the natural courses between 0 and P,, and P, and Q, which deviate infinitely little from this standard course.
318. Considering now, for simplicity, only cases in which there are but two degrees (§ 165) of freedom to move, 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, 0, 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, 0, 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 envelop is passed.
319. Or again, if a particle slides round along the greatest 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 ($ 305) less than a Vab. 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
1 Two courses are here called not 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.
distances of not less than this amount, a Vab, from one another in succession round the circle. With reference to this example, see also
§ 314, above.
Or, if a particle slides down an inclined hollow cylinder, 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 cylinder, and influenced by a force equal to g cos i, instead of 9
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.