68 



C. V. L. Cliarlier 



(150) / cos '+j+'B sin ''B dB . 



It 



~ 2 



But 



cos '+j+'B = ^0 + cos 5 + ^2 cos 2B + yJj cos 3B + ... 

 where either all coefficients Ai having even indices (i) or all coefficients having odd 

 indices vanish. 



Further we have, k being odd, 



sin '<B = B^ sin B ^ B^ sin ?>B+ ... 



and hence 



cos sin ''B = C\ sin £ + 6\ sin 2B + C'3 sin 35 + . . . , 



so that the integral (150) has the value zero. 



Taking into consideration that F{ü) is a symmetrical function of Z7, Fand W 

 it follows that the integral (150) also vanishes if i or j is an odd number. 



Hence we have 



The not-vanishing coefficients are: 



"200 ' '^020' ^002 

 ^400' ^040' ^004' "022 ' ^202 1 ^220 



We here consider only terms of the second order. 



47. Discussion of the integral T, 



We have 



(151) T=^\düÜ^I^, 

 ^ lo ' 



where 



n 



(0^ dh 



(152) I=K' 



K \, DW (D^ — d.y 



CO" - A 4- (Ï' or Ul-hrt'W A + l^/'oi 



and 



(153) c« = l/2ß. 



SJ" 



(154) <ï> = _L_e 



