betiveen Confocal Elliptic Cylinders. 231 



Lei — =c, when n is large, 



• •• ^ 4-^ 2 (l + e 2 sinh 2 ?)L = 0. . . (24) 



The approximate value of the solution in this case may be 

 obtained by a method due to Webb (Proc. Roy. Soc. lxxiv. 

 1904, p. 315). 



Put L=^.^, 



where (yjr, </>) are functions of f . 



Substituting in the equation, and equating coefficients of 

 n, n 2 to zero, since the terms are of different orders, 



•'• -(jf) 2 +a+^»h 2 i)=o, 



and 



d$ dr d^ z 



rs 



•'• <£=± v/l + e a sinli 2 0.df, • • (25) 



'o 



j _ A cos n<f> -f B sin n(j> 



\/~cj>> 



(27) 



where (A, B) are functions of (77, t) only. 



The free periods are given by — = 0, at (j> — <j>^ </> 2 , which 

 leads to ^ 



tan n (fa — <j> 2 ) 





',f„ ' ' 



To the present order of approximation 



tann(^-^)= ^-^ 2 . . . • (28) 



This can be expressed in terms of elliptic functions, for 



kb 

 n 



-J> 



</> = v/l+e 2 siiih 2 0, 6 = 

 Hence if E (0, k) denote the incomplete elliptic function, 

 ]^T^nfisti?Fd0 (29) 



