18 Zonal Harmonics of the Second Type. 



This, by successive reduction of the order of Q, is equiva- 

 lent to 



2 .4. ... 2n 2n + 2 . 2n + 4 . ...4n 



1.3.... 2n—l' 2n + 3 . ... 4^ + 1 



f 7 " Q (cos 0) cos (2n + 1) dO. 



Jo 



By the other reduction formula, we find 



I *" Q (cos 0) cos (2n + 1)0 dO 



1.3....2n-l 2. 4.. ..277, f 

 = 2,4...,2n ' 3.5....2n+l j Qo(cos0)cos0 



and therefore 



f ,r Q2»(cos 0) cos (27i + l) 6 d6 



A- 4 '. ; w\ Qo (cos ^ cos ° dd - 



The last integral is 



f *■ 1 (V/2 



1 - cos log cot 2 - d0 = 2 \ cos2(j)\ogGot(j)d(j> = 7r, 

 Jo * J Jo 



and thus 



f * Q a » (cos 0) cos (2ti + 1) dd = ir . 2± n j^j^ . (44) 



Moreover, 



I cos (2 



1.3. ... 2r-l 472 + 2 . 4n + 4 . ... An+2i 



\ * Qsm (cos 0) cos (2n + l + 2r)0d0 

 Jo ... ^ 



2 .4 .... 2r 4/1 + 3 . ... 47i + 2r + l 



X i 7r Q 2 nCOs(272+l)6>^, 



whence 



. . . (45) 



