Harmonics of the Second Type. 13". 



method. We have 



C Q„ (cos 6) {cos md - cos (wi + 2)0} dO 



Jo 



= 2 \ Q n (cos 0) sin sin (m + 1) dd 



Jo 



r2sin(m + l)0 . 2 ^Q»~T 

 L ?i(n + l) ¥J 



+ 2[m + l) _\ ' sin cos ( fn + l)0#rf*' 



= r 2sin(m+l)^ sin2g ^Q„ 

 L n(n-\- 1) c^/u, 



2(m + l) . „ , ■ ^ ^ l 7 *" 

 H — 7 — -^rv sin cos ( m + 1 )6 . (X 

 n(n + l) v ^J 



- ; ,iV l { cos cos (771 + 1)0 

 »(w + l) Jo 



- (m + 1) sin sin (m + 1)0} Q„ dO. 

 The integrated terms vanish. If 



(cos 0) cos mOdO . . . (32) 



f*TT 



Jo 



(m being an integer), the equation is readily reduced to 



u m+2 _ (n — m) (n + m-f-1) 

 u m (n — m — l)(n+m + 2) 



In particular, if m = n, u m+2 = 0, so that 



Q ?i (cos (9) cos (n + 2)0 .d6 = 



(33), 



j: 



'0 



for all integer values of n. This involves the further 

 consequence that 



i 



Q n (cos 0) cos (n + 2r) .dQ = 



if ?i and r* are both positive integers. It is, in fact, odd in 

 cos0. Again, if m=n— 1, u m = 0, so that 



I ^Q* (cos 0) cos (*-l) 0.^0 = 0, . . (34), 



