Because of the extreme values of the parameters encountered in these 

 computations, it was necessary to make extensive use of the following 

 expansions to supplement available tables of elliptic functions: 



for k" « 1 



k^ 

 snlk, v] = sin 2; — - cos v{v — sin v cos v) + 



k^ 

 cn[ky v] = cos ?; + — sin v{v — sin !; cos v) + 



for k' ^ (1 - kY' « 1 



Klk] 



-'4-^H-')t-^ 



sn[k, v] = tanh v -\- — sech ?;(sinh v cosh v — v) 



+ •••• 



cn[k, v] = sech v — — tanh v sech z;(sinh v cosh v — v) 



+ .... 

 Also useful were; 

 for (f> « 1 



for <j>» 1 



fnl±^ = tn 2 1-* 



K^^^ 



l-<^ !-</> 2 2 



In the determination of B from (43) the problem arises of solving 

 the transcendental equation 



k'K[(l - k'')] = a « 1 



where a is a given positive quantity. This equation must be solved by 

 iteration or plotting and a reasonably good estimate of the root saves a 

 great deal of labor. Making use of the approximation vahd for /b' <<C 1 



K[{1 - k^] ^ IvAlk' 



