Periodic Motion of a Finite Conservative System. 555 

 when x is positive, and 



= JJa + 2Ua2 ( - ) l e mix cos miz 

 when x is negative,] of the form 



such as is considered by Stokes ; but the real coefficients p 

 are discontinuous, changing sign toitli x, so that e pz vanishes 

 both for x= + go , and for x = — go ." 



I have here corrected a slight misprint in the exponential, 

 which would, however, cause no difficulty. I have also added 

 for the sake of clearness within square brackets the expansion 

 to which reference only was made,, and it may be noted that 

 though I explain that (5) may be so expanded, I make no 

 suggestion that such an expansion should or could conveniently 

 be used. 



LXV. On Instability of Periodic Motion, being a continuation 

 of Article on Periodic Motion of a Finite Conservative 

 System*. By Sir William Thomson f. 



23. "If ET ^r, cf>, %, §, he generalized coordinates of a system; 

 J-^ and let A (yjr, $, . . . ty' ', <£',...) be the action in 

 a path (§2 above) from the configuration (yj/, ty , . . .) to the 

 configuration (-v/r, <p, . . .) with kinetic energy (E — V) with 

 any given constant value for E, the total energy ; V being 

 the potential energy (§ 3 above), of which the value is given 

 for every possible configuration of the system. Let v, %, n, f . . . 

 and v' , £', v 1 ', J' ... be the generalized component momentums 

 of the system as it passes through the configurations (yjr, (/>,...) 

 and (^r r , </>'. . . .) respectively. If by any means we have 

 fully solved the problem of the motion of the system under 

 the given forcive* (of which V is the potential energy), we 

 know A for every given set of values of y[r, <f>, . . . yjr', cf>', . . . ; 

 that is to say, it is a know T n function of {yjr, <f> . . ., ty , 0', . . .). 

 Then, by Hamilton's principle [Thomson and Tait's ' Natural 

 Philosophy,' § 330 (18)], we have 



1 

 I 



> • (1) 

 I 



J 



* Phil. Mag. Oct. 1891. 



t Communicated by the Author, having been communicated to the 

 Koyal Society on the 26th of November, 1891. 



X This is a term introduced by my brother, Prof. James Thomson, to 

 denote a force-system. 



V = 



dA 



i = 



dA 



w v = 



dA dA 



d % > *-<!*> 



v' = 



dA 



dyfr n 



e- 



dA , 



7 dp v = 



dA y , dA 

 Htf * ~ d$ n 



