Approximate Spheres and Cylinders. 181 



Again, by direct integration of (4) with retention of C 3 , 



<£ I =-H logK6)+Jg(C 1 F 1 + C 2 F 2 + C 3 F3+ . . .) 



(H 1 « F 1 + 2H 2 « 2 F 2 + 3fWF 3 + . . .) 

 4-iH„rg(C 1 F 1 + C 2 F s +C 3 F s + . . .f 



+J||(C 1 F 1 +C 2 F 2 +C 3 F 3 + . . .) 2 (H 2 u 2 F 2 + 3H 3 VF 3 



+ . . . +ip(p-l)H p u PF P ). 



In the last integral we may substitute the first approxi- 

 mate value of H P from (8). Thus in extension of (11) 



</> 1 /Q = 21og(^)-i{3C 1 2 + 5C 2 2 + 7C 3 2 + . . .} 

 + C 1 I 1 +2C 2 T 2 + 3C 3 I 3 -f ... 



f 27r dQ 

 J * ( C i F i + ° 2 F 2 + C 3 F 3 + . . .) 2 {C 2 F 2 + 3C 3 F 3 + . . . 



+ L p{p _ 1)Cp¥p} , . . (18) 



The additional integrals required in (18) are of the same 

 form (17) as those needed for l n . 



As regards the integral (17), it may be written 



>27T 



dQ cos nd cos p6 cos q6. 



'o 



Now four times the latter integral is equal to the sum of 

 integrals of cosines of (n—p — q)0, (n—p + q)0, (n + p — q)0, 

 and (n+p + q)0, of which the last vanishes in all cases. 

 We infer that (17) vanishes unless one of the three 

 quantities n, p, q is equal to the sum of the other two. In 

 the excepted cases 



(17) = |* (19) 



If p and q are equal, (17) vanishes unless n = 2p ; also 

 whenever n, p, q are all odd. 



We may consider especially the case in which only 

 C P occurs, so that 



u = u (l + C p cos p6) (20) 



In (16) 



I, 



i 





so that In vanishes unless n = 2p. But I 2p disappears in 

 (18), presenting itself only in association with (u 2 v> which 



