1884.] K' , E' , J' , G' in powers of the modulus. 193 



factors in 2?7 X is such that they cannot be continued backwards 

 one term. 



The quantities U 2 , V 2 , W 2 are such that 



w 2 =h(E 2 +i 2 ). 



§ 10. Denoting by K 1} E t , I 1} &c. and K 2 , E 2 , I 2 .., the series 

 to which they are equated in the preceding section, the expansions 

 of K ', E' , F, &c. in powers of & 2 are given by the equations : 



K'= Klog^-K 2 , 



I = -E l \og^ + E 2> 

 G' = -G x \o^ + G 2 , 

 U'—V t log±+V 9l 



r—u^ogl+u,, 



W' = -W t ]ogz + W % . 



§ 11. The values of K', E', I', &c. in terms of K, E, I, &c. 

 and the series K 2 , E 2 , I 2> &c. are given by the equations : 



K = — log T - A 2 , 



IT ° K 2 



E ' = - — log t ■ + / 2 , 



7T 



& 



V = - — loo- - + if 

 2W 4 



Tr = -^io g |+ir 2 . 



