1891.] On Instability of Periodic Motion. 195 



value is given for every possible configuration of the system. Let 

 v, f, rj, ", . . . . , and v ', f ', ?/, *, . . . . , be the generalised component 

 momentums of the system as it passes through the configurations 

 (T/T, 0, . . ) and (y/, 0', . . . . ) respectively. If by any means we 

 have fully solved the problem of the motion of the system under the 

 given forcive* (of which Y is the potential energy), we know A for 

 every given set of values of yr, 0, . . . . , y/, 0' } . . . . , that is to say, it 

 is a known function of (y^, . . . . , y/, 0', . . . . ). Then, by Hamil- 

 ton's principle [Thomson and Tait's ' Natural Philosophy,' 330 (18)], 

 we have 



dA. K dA. dA -. dA 



V = ^~ ~' ? 



_ _ 



*~ ~~d$" ~ d x " 



2. Now let P'P designate a particular pathf from position 

 (Y^j 0'i .x'> ) which for brevity we shall call P', to position 

 (Y^ 0> X ) which we shall call P. Let P'oP be a part of a known 

 periodic path, from which P'P is evidently little distant. But first, 

 whether P'oP is periodic or not, provided it is evidently near to 

 P'P, and provided P' and P are infinitely near to P', and P, respec- 

 tively, we have, by Taylor's theorem, and by (1), 



,0, %,....,^', 0, X,-...) 



= A(oYr, o0, 0%, ---- , 0&, 00', OX'> ---- ) 



(0-o0) + .. -V(Y / -oYO -oF(0'-o0')- .... 



..(2). 



* This is a term introduced by my brother, Professor James Thomson, to denote 

 a force-system. 



f For any given value of E, the sum. of potential and kinetic energies, 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 ballistics) 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. 



