538 Mr. G. W. Walker 07i Unsipmnetrkal 



The complete solution in the usual notation for elliptic^ 

 functions is 



yLt sin (X,^ + e) -j- 1 



sin/i:?= 



/x-f sin [\tA-e) 



where e is an arbitrary constant. 

 fM is a root of the equation 



mp \ m^'ir J mp 



\' 



■-2 ^-.2 



nip m p> 



and the modulus of the elliptic functions is given by 



7 2 _ ^>^T ^^V^ 



r At- + 2a/>6 — — ^-^ 



In general the motion will consist of a linear part along 

 with a periodic motion. As a first approximation the periodic 

 part will be simply harmonic, but the period will differ from 

 the period of the incident waves. 



4K 



The period will be——, where K is the quarter-period of 



A, 



the elliptic functions. 



If we take the larger root of the equation in /x, and pro- 

 ceed as far as squares of Xq, we find that 



\ m^p>^ r ) 



m^p"^ c 



Hence the periodic time is 



27r r. 1 r4-3«2^'^Xo^ 



-^. 1 + 





Kc {^ 4 e 



We note that this is independent of the sign of e. but as 

 the negative particles have a much smaller mass than the 

 positive particles, the effect of the former will be of much 

 greater importance than the effect of the latter. 



