Prof. J. J. Sylvester on Spherical Harmonics. 299 

 I becomes the surface-integral of 1 _a;.. , ;2 > an d i s equal to 



Therefore 



A+8+a+...-jgj, 



or 



S.A= 



47T 



where 



2yu, + r 



°* 2(2^-1) ^ 2.4.(2/*-l)(2^-3) 



^-l)( / ,~2)(^-3)( / ,-4)(^-5) 

 2.4.6.(2^-l)(2/*-3)(2/*-5) '*" 

 This series admits of summation. And I find 

 ft _ 1 q _ 2 q _ ' 2 q_ 8 q_ 8 c _ 16 



*h-h b 2 _g, ^-g, ^~ B5 ; ^- 63 , ^-sTfZil 

 a 16 _ 128 H _ 120 



3.11. 13' 8 ~ 3 . 11 . 13 . 15' 9_ 5 .11 .13'. 17' 



256 



"" 11 . 13 . IV . iy 



in general 



2. 4. 6... (2m) 



11.13.17.19"" 

 and in general 



£>2m = 



(2,?i + l)(2/?i + 3)(2m + 5) . . . (4m- 1) 

 and 



Q _ 2m +1 q 



fe2w2+1 ""4^+I fe2m; 



that is to say, S /x is the reciprocal of the coefficient of h" in 



(i-»r». ' 



Hence the values of A, B, C . . . in 1^ are completely deter- 

 mined, and I M , and consequently the value of the complete 

 integral of 



is known for all values of a, b, e ; a, ft, 7 — and this by a me- 

 thod which is applicable step by step to any number of vari- 



