1884.] K', E', J', G' in powers of the modulus. 195 



§ 14. Substituting for K, E, &c. and for K % , E, 2 , &c. their 

 approximate values given in § 12, we have 



k'= (i + m log | -w> 



E' = 



P 2 log | + 1 -IF, 



/'= - (1-P 2 ) log l+l-yf^ 4 , 



G'= -pMog|+l-i^ 



0" = - Hi + ^ 2 ) log |+l -W, 



V'= TU' 4 l0g| + l-iF, 



Tr=- 1(1 -IV) log | + 1 -IF. 



Writing the terms in order of magnitude and retaining them as 

 far as those involving F, we obtain the formulas : 



K' = - log k + 2 log 2 - |F log & + |F (2 log 2 - 1), 



E'= 1 $2? log & + IF (4 log 2 - 1), 



F = log k + 1 - 2 log 2 - £F log & + |F log 2, 



G'= 1 +!Flog£-iF(41og2 + l), 



Z7 = | log & + 1 - log 2 + |F log k - iF (2 log 2 + 1), 



T = 1 - iF, 



1F= | log & + 1 - log 2 - §F log & + iF (6 log 2 - 1). 



§ 15. Neglecting terms of the order F log&, 



