1884.] K\ E', J', G' in powers of the modulus. 205 



Legendre's formulce for K' and E, §§ 27, 28. 



§ 27. The series for K' and E' in terms of h are due to 

 Legendre. Changing only his b, F\ E 1 into k, K', E', the follow- 

 ing is the form in which they were given by him * : 



4 

 K'= log^ 



1 2 .3 2 /. 4 , 2 



+ 55 , 5 « log T — 1 - 7T- , 



2 a . 4 2 V A 3.4 



l".3"-:5 ,, , e / 1 4 _ 2 

 + 2^7^H l0g ^" "374 



+ &c. 



£' = 1 



1 7 2 A 4 1 



+ 2^( l0 ^--r72 



1 2 .3 2 5 76 /, 4 * „ 1 \ 

 + WA>-6 k { l08 k- A "5T6J 



where A' = 1, 



2 2 



3.4 '5.6' 

 &c. &c. 



§ 28. Legendre remarks that the coefficients in the series for 

 E' agree with those which he had given under another form in 

 the Memoir es de I'Academie for 1786. 



* Traite des Fonctions Elliptiques, Vol. i. pp. 67, 68. 



