556 Sir W. Thomson on Periodic Motion 



24. Now let P' P designate a particular path * from position 

 (y]r', 0', x'j - • which for brevity we shall call P', to position 

 (ty, (f>, %, . . .) which we shall call P. Let P' P he a part of 

 a known periodic path, from which P P is infinitely little dis- 

 tant. But first, whether P' P is periodic or not, provided 

 it is infinitely near to P ; P, and provided P' and P are 

 infinitely near to P', and P, respectively, we have, by Taylor's 

 theorem, and by (1), 



= A( ^r, 00, 0%, • • •; of, 00', 0%', • • • ) 



+ ov(^-o+) + of(*- *)+ • • .-o^'-of')-o?W-o^)- ■ • 



y (2; 



+ \(w>~^-^+--- } 



o\ dyjrdcp 



25. Let us now simplify by choosing our coordinates so 

 that the values of 0, %, &c, are each zero for every position 

 of the path P' P ; and let i/r, for any position of this path, 

 be the action along it reckoned from zero at P'. These 

 assumptions, expressed in symbols, are as follows : — 



dA A dA A dA •, dA A dA A ^ 



d0 f? X dyfr r dfi d%! \ ^ (3) 



for all values of yjr and f, if = 0, %=0, . . .; 0' = O, %' = . . . ) 



26. Taking now 



^ = 0, f = of, 0^ = 0, oX = 0,...o+' = 0, 0^ = 0, o%' = 0-.., • . . (4) ; 

 we have 

 A(„t, o*, oX, ... , of, o<j>', oX', ■■■) =Mot, 0, 0, . . . 0, 0, . . .) . . (5) 



* For any given value of E, the total energy (§ 3 above), the problem 

 of finding a path from any position P' to any position P is determinate. 

 Its solution is, for each coordinate of the system, a determinate function of 

 the coordinates which define P and P' and of t, the time reckoned from 

 the instant of passing through P'. The solution is single for the case of 

 a particle moving under the influence of no force ; every path being an 

 infinite straight line. For a single particle moving under the influence of 

 a uniform force in parallel lines (as gravity in small-scale terrestrial bal- 

 listics) the solution is duplex or imaginary. For every constrainedly finite 

 system the solution is infinitely multiple ; as is virtually well known by 

 every billiard player for the case of a Boscovichian atom flying about 

 within an enclosing surface, and by every tennis player for the parabolas 

 with which he is concerned, and their reflexions from walls or pavement. 



