between Confocal Elliptic Cylinders. 233 



Detailed Investigation of very Short Waves in the Space. 



In the case of short waves, h is great. 



We first take the case when n is an integer too small to be 

 comparable with k. The dependence of on k ceases when 

 k is very great, by analogy with the corresponding circular 

 problem. Moreover, the result is periodic in 97, of period 2ir. 



Then the equation for L is 



^+L.Wsinh 2 £=0; . . . (37) 



and by the previous method, the asymptotic expansion for 

 large values of k is found to be 



L =\/co^hf { A cos (kb cosh f) +B sin (kb cosh £) }. ■ (38) 



Similarly, 



M = \/cosec 7) {0 cos (kb cos y) + D sin (kb cos rj)\. (39) 

 Now X = & 2 sinh 2 £; 



_ A cos (k\/W +\) + B si n (k\/tf + \) ^ 



The period equation therefore becomes 



a/6 2 + \ 1 _ \/W\ 2 



tan % (^&S + V- v/^^+XO = 2ai 2 ^ 



1+ x /b 2 + \ 1 .b 2 + \ 2 

 U 2 Xt\ 2 



- ] ^2 V b 2 + ^ - XiyZ/r' + ^2 \ 



2k\{K 2 



since & is large . 



Hence ^ is a short possible wave-length, if k is a large 

 root of 



tan k{ (V+X^-QJi + X^] = j^^ + X^-Xx^ + XQjj^ (41) 



For a very elongated section, b is large, and we obtain 



The larger & may be, the more accurate is the approxi- 

 mation to the free period. 



