58 Prof. Jones on the Calculation of the Coefficient oj 



Substituting the above value of J V d<^' in equation (I.), we 



ro Aacos0 



, ( ^w Aa cos <i 



The sum of the last two of these integrals and the second half 

 of the first is zero ; and hence 



^^^p^-eAocos^g^ 



J-e « _____ 



r Tv/r r^^-®Aacos(^, yl-(H)+ ^a'-^ + £-^@%^^ 



or M = I ^ loff d(b\ . 



J-e « ^ « ^ 



If B is an integral multiple of •ot [i. e. if @ = ?zot), we may 

 express M in a series of powers of | —^ K the coefficients 



2V'Aa 



of which are functions of the quantity —. , and the com- 



A "T a 



plete elliptic integrals to that modulus. 



For in this case 



^^r-Aocos^g^ 



Jo 

 The general term in this expression for M is 



^ ^ 2 . 4 . 6 ... 2m 2^/1+1 Jq u^"'+^' 



Now 



I t^cos <pd(f) _ C'^ 



•^0 ^0 



COS ^ d(f) 

 (A2 + a^-2Aacos(/))"2— 



2 p cos 2^ f7^ 



2m + r 



/■ A I ,,\2m+l 1 2m- 



^A + aj Jq (l_c2sin^^)"2 

 where c = 



2^Aa 



A + a 



(A + a)2'«+i '"' ^ 

 ^ ^ ^2 cos20de 



where P = I zm+i' 



