CHAPTER XXII 

 186. 



o 



fsin n0 



i* r~ ~~i 



(a) I sin 7i0 dQ = -- cos w0 = (cos 2mr 1) = 

 Jo nL J Q n 



= -- cos 710 = -- (cos nu cos ( nn)} = 

 nL J. w n ( 



sin n0 d0 = -- cos n0 = (cos TITC 1) 

 Jo L J o w 



= when TI is even 



2 



= - when 7i is odd 

 n 



(b) 



I cos w0 d0 = - sin nO =0 



Jo wL J 



since sin nn = if n is a positive or negative integer. 

 (c) Isin w0 sin mQ dQ = -I (cos ( 



- cos 



and it follows at once from (b) that this integral vanishes between 

 the limits and 2rc ; TC and 71 ; and and TC. 



(d) I cos n0 cos w0 dQ = -I (cos (n + m)0 + cos (n - w)0} d0 



and it follows at once from (b) that this integral vanishes between 

 the limits and 2rc ; TC and TC ; and and TC. 



(e) si 

 Jo 



sin ?i0 cos m0 



= - { sin (n + 7w)0 + sin (n w)0} 



376 



