Approximate Spheres and Cylinders. 185 



These definite integrals have been evaluated by Ferrers * 

 and Adams f. In Adams' notation n+p + q=2s, and 



, 2 A(s-n) .A(s-p).A(s-q) ( 



^°)-2sTl Mi) ' * ' {61) 



where 



_ 1.3.5.. .(2n-l) 

 "~ 1.2.3 ... n 



(38) 



In order that the integral may be finite, no one of the 

 quantities n, p, q must be greater than the sum o£ the other 

 two, and n-\-p + q must be an even integer. The condition 

 in order that the integral may be finite is less severe than 

 we found before in the two dimensional problem, and this, 

 in general, entails a greater complication. 



But the case of a single term in 8u, say C! P P P (//,), remains 

 simple. In (36) J n occurs only when multiplied by C«, so 

 that only J p appears, and 



J v =(p + l)C p *$P/dfi (39) 



Thus (36) becomes 



0^1-J^lc^ fr+iX^ q-.Jfofr, 



'. . . (40) 

 When p is odd, the integral vanishes, and we fall back 

 upon the former result ; when p is even, by (37), (38), 



For example, if p = 2, 



and 



j> 



*dfi = 



2 



\A(\p)y 



'6p+. 



L A(|^) 



p = 2, 









0i/Q«o 



! 0> = 

 = 1- 



4 



12 



_ 3P 2 i _ (* 3 

 5^2 +3 5 ^2. 



(41) 



(42) 



Again, if two terms with coefficients C p , G q occur in 8u, we 

 have to deal only with J P , J q . The integrals to be evaluated 

 are limited to 



jP P 3 ^, jP P 2 P^/*, JP P P, 2 ^ JP ? 3 ^. 



If p be odd, the first and third of these vanish, and if q 



* ' Spherical Harmonics,' London, 1877, p. 156. 

 t Pioc. Roy. Soc. vol. xxvii. p. 63 (1878). 



