between Con focal Elliptic Cylinders. 235 



The components of electrical force are 

 Z=0, 



— iAY.e lkVi / il\+ ( 



= ~^/x^ll ' U/ C ° S ( /cv//>2 + ^ + 6 { sinh (*vW^ + e,) 



\/b 2 + /x 



cosh(H / i 8 >X. + € a )} (46) 



-iAV. 



+ — 2 ^ + A co * (/V^+A + e 2 ) | cosh (h/F+ m. -t 6 2 ) (47) 



where e x is known from the condition that Y = at boundaries, 

 and k is one of the roots of (41). 



If 6 is the inclination of the resultant electric force to the 

 normal drawn to the confocal ellipse through that point, 



tan0 



= * = k) — Vb—>' m 



{ tanh ( V^ + M - kb) - ^~ 



And therefore the differential equation to the rays is 



x ^ b 2 + A • tan (AV6- -i- X +- e x ) + —^ — [* V b* + p tanh (& >/ 6 2 4- n — kb)— ~^ — , 



• • (49) 

 since they are propagated perpendicularly to the electric 

 force in the normal section, and 



tan 0= — -r-. 



p 2 </A 



These rays are closed curves round the axis of 2, in the 

 plane of each section. 



When the quantity n becomes appreciably great in the 

 differential equation, if the asymptotic expansion be found, it 

 appears that the ensuing waves cease to be short. The above 

 theory therefore applies to very short waves in general, pro- 

 vided they are possible to the enclosed space. This possi- 

 bility is determined by equation (41), and exists if the root #, 

 to which they correspond, is large. The approximation 



