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



and by the change of k into p , K' is changed into k'(K' — iK). 

 Thus, by the double transformation, 



k' — ik 

 k' is changed into -r, — ^ = e~ 2i9 , 

 k + ik 



and K' „ „ %(k' + ik)(K'-iK) = ±e ie (K'-iK). 



Replacing k by k' in the first formula in § 2, we have 



9 IT' 12 12 Q2 -12 C2 e2 



w - 1 + 2 2 /c + 2 2 .4 2 + 2 2 .4 2 .6 2 + " 

 whence, changing k' into e~ 2ie , 



e ie (K'-iK) „ , l 2 ,.„ 1 2 .3 2 ... . 



that is, 



2 2 2 2 .4 



«-* + 55 e~™ +~ 2 e-w» +&c. 



giving 



-12 1 2 02 



cos ^ + ^ cos 50 + 92 * 2 cos 90 + &c. 



/ J2 J 2 3 2 \ 



— * f sin -f ^2 sin 50 + 02-T2 sm 90 + & c - ) 



i£' l 2 l 2 3 2 



= COS + jrs COS 50 + ~ a ' , a cos 90 + &c, 



ir 2 2 2 2 . 4 2 



K . „ . I 1 . ., 1 2 .3 2 

 22sm 50 + 2»-|s 



and - = sin + ^2 SU1 50 + ~ a * . 2 sin 90 + &c. 



We may obtain corresponding formulae for G and W by a 

 similar method, as follows. 



1 — k 



§ 24. The change of k' into - j corresponds to the change 



J. "T" A? 



of q into 2~, and the change of k into -^ corresponds to the change 



of q into — q. The double transformation therefore corresponds 

 to the change of q into iqK 



