186 Mr Glaisher, On the developments of [Nov. 24, 



Formula for K', E' , &c. in terms of k, § 3. 



§ 3. The formulas to which the present paper principally 

 relates are those giving the expansions of K', E', &c. in ascending 

 powers of k 2 . 



I have found that these developments may be written in the 

 following form : 



1 2 .3 A 4 2 2 1 , 1\ 74 



+ 2 2 T4r g X^l + 2-3 + 4J^ 



1 2 .3 2 .5 A 4 2 2 2.2 1 , IN , s 

 + 2 2 T4 2 T6r^-T + 2-3- t -4-5 + 6J A; 



+ &c. 



J'=log|-l 



1 A 4 1 2\ 72 



1 2 .3 A 4 2 2 1 2\ /4 

 -2 2 T4 2 ( l0g ^-I + 2-3- f 4j^ 



1 2 .3 2 .5 f A ,4_2 2_2 2_1 

 ~2 2 .4 2 .6 2 V g & 12 3 + 4 5 + 



— &c. 



