232 Mr. Nicholson on Elliptical Vibrations 



^ * = d«(jU) -. (30) 



where c=\/~ ]_. 



Hence if A = b 2 sinh-' f , a set of free periods between the 

 surfaces (AiA 2 ) is given by 



to«{E(ft **)_e(&, "YuJ r- si : h2 iL, T, (3D 



provided ^ be very great. 



We may here recall the fact that b 2 is the difference of the 

 squared semiaxes of any confocal, and X is the minor semi- 

 axis. Treating the equation for M in a similar manner, 



M _ C cosn n $i "+ D sinh ?i^>x ,o 9 \ 



~ T^ 7 " -•■•■• 



where . C* 



<*>i=l v/1 — e 9 sin 2 i7rfi7, (33) 



Jo 



which is also readilv expressible as an elliptic function 

 E (,,»). 



The magnetic force is then 



p A 



° = - /T7j- -, cos (rc<£ 4- e) cosh (?i^ + e 2 ) cos (AV* + e 2 ). (34) 

 v 9 9i 



And the electric force jnay be obtained by the proper 

 operations. 



For vibrations inside a single elliptic cylinder, we must have 



L=B5™£* (35) 



in order that the forces may be everywhere finite. 



The period equation for the cylinder defined by f, for large 

 values of n, is accordingly 



, 7 f smhE(f) I 



±\_ _W 1=0, .... (36) 



g, <*? [(l + e 2 sinh 2 £)*J 



where the function E has a modulus e = — • 



n 



a ■ 



