278 Mr. Nicholson on Legendre's Functions and the 



integrands of other integrals, and the integration is now with 

 respect to their amplitude, and not their modulus as in the 

 above treatment. E(X) now will denote 



\ */l-k 2 sin 2 \d\ 9 and F(\)= 1 



Jo Jo 



* d\ 



Vl-k 2 sm'\ 



6 B 



In all cases, the modulus is k= cos- for E, F, and sin 7- =k' 



for E', F. 2 2 



If z is between and w, 



it 1 



— (tt — 2z)= cos z + ^ cos oz + ; 



o o 



(2rc + l) 2_ 8j e 



(7r — (j))d<j) 



/«*»£- 



' cos 2 ^ — cos 2 - 



But • V cos 'i 



V cos 2 



:0s 2 5 



=K(ti) p -w b >'< 9 > 



Jo /— 7* -0-^0^ + 1^'^ ^o(2n+l)^ 56 > 



V C0S V~ COS 2 



Put * , 6 , 



tan| = tan- . cos A; 



J: 



<ty 



===== = 2F(M : 



/ 9 9^ 



-w cos 2 £ —cos- - 



f / y >-[^<»]:-€g- Fw " 



A/ cos J |- — cos J - 



-f 



F'(\)sinXrfX 6> 



* tan 2 ! 



*'o I + tan 2 2 cos 2 A, 



since . 



^, -2smX.tan^ 



dX = j? 



1 + tan 2 . cos 2 \ 



