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



jr 12 12 02 12 02 g2 



— = sin + ^ 2 sin 50 + ^—^ sin 90 + g, ' ^ 2 ' 6 - 2 sin 130 + &c, 



iT' l 2 l 2 3 2 l 2 3 2 5 2 



— = cos + ^ cos 50 + 2^2 cos 90 + ^ ' ^ ' Q2 cos 130 + &c, 



2W 1 l 2 3 l 2 3 2 5 



— = - sin 30 + ^4 sin ^ e + 2 2 ' 4 2 ' 6 sin 11<? + &C *' 



2JT 1 _. 1 2 .3 „ a l 2 -3 2 .5 11fl „ 



- g cos 30 - ^-j cos 70 - ^^^ cos 110 - &c, 



46? 1 l 2 l 2 3 2 



— = sin - ^ sin 30 - ^-^ sin 70 - ^ ^ 2 62 sm 110 - &c, 



4(7' l 2 l 2 l 2 3 2 



= COS + sa cos 30 + 2^ cos 70 + 2 2 4 2 6 2 cos 11^ + &e - 



The series for K and W assume the form x x when = 0. 

 When = ^77-, they are infinite in value, as they should be. 

 Except in these critical cases, and the corresponding cases for 

 K' and W, the series are convergent for all values of 0. 



Similar series for E and I, § 26. 



§ 2G. Since 



E = W+\K, 



and I=W-hK, 



K 2W 

 we mav at once deduce from the series for — and in the 



J 7T 7T 



2E 2Z 



last section, the following series for — and — : 



. 7T 7T 



9 77 1 I 2 l 2 3 



— = sin + ^ sin 30 + 22 sin 50 + 22-^ sin 70 



l 2 3 2 l 2 3 2 5 l 2 3 2 5 2 



+ 2^42 sin 90 + 28 ' 4 ,' 6 sin 110 + ^'^'^ sin 130 + &c, 



or 1 1 2 1 2 3 



— = - sin + ^ sin 30 - ^ sin 50 + ^x sin 70 



l 2 3 2 l 2 3 2 5 l 2 3 2 5 2 



- 2*74* sin 96> + 2^74^76 Sin Ud " 2 T 74 2 T6 2 sin 18 ^ + &C * 



