230 Mr. Nicholson on Electrical Vibrations 



Let X = b 2 sinh 2 f, and it becomes 



^==(0-m*smtf£)L, .... (18) 



which is known as the equation of the elliptic cylinder. 

 Let E(f), F(f) be two principal solutions. 

 I£ /n= —b 2 sin 2 rj, we find similarly 



r/ 2 M 



^ 2= -(^ + ^sin^)M. . . . (19) 



And if solutions be E^), F^rj), then the magnetic force is 

 given by 



c = {V(e)+A¥[£)\{E 1 ( v )+BF 1 ( v )}cos{kVt + e), . (20) 



the summation being for all possible values of 0. 

 While the surface condition gives, by (14), 



&]- 



(fJ -°; (2i) 



therefore if f i, f 2 denote the elliptic surfaces, the free periods 

 of the system arise from the values of h given by 



g«0 YlIM , 99 i 



F'(?o = *'(&) K } 



Owing to the impossibility of finding even a definite integral 

 to represent the functions E, F, whose properties may be 

 found in Forsyth (' Theory of Differential Equations ') , this 

 solution is of little interest. We now proceed to examine 

 certain approximations. 



If two very nearly spherical boundaries be considered, L 

 must return to its old value as a point defined by 77 describes 

 a path on a particular boundary. 



We will therefore regard (—0) as the square of a positive 

 integer, and write it n 2 . In general, 



e=-n 2 + a (kb) 2 + a 1 (kby.:., 



where a , a l5 &c. decrease to zero when n is great. 

 Then the equation for L is 



^ + ( ?t 2 + ^ 2sinll 2j )L = (23) 



It will be noticed that the periods form a doubly infinite 

 system, for the values of n are singly infinite in number, and 

 each value of n gives rise to a singly infinite number of 

 periods. We proceed to the case where n is very great. 



