208 Prof. J. J. Sylvester on /Spherical Harmonics. 



say V? operating on the whole of I gives the result zero, the 

 same must obviously be true for each part 1^. 

 Now V« p is obviously equal to 



and 



V0 q = 2q(2q + l)p / 6 q - 1 ; 

 for 



= X(2qp^+Aq(q-l)p.p^ 



Also 



S7$Pp^—p^^AsPsPS/p q —2pqX(^s^p\sP-\p ( i- 1 = 4,pqsP.p9- i . 



Therefore 



\/ s p6*= (p 2 —2^) sP ~ 2 P q P /q+l + (k )( l + V + %q) sP ' P q ~ l - P /9 > 

 or 



S/s> J - 2 W^=(fM-2j)(^-2j-l)sP- 2 (pp / y- 2 Jp f 



+ 2j(2f,-2j+l)sP(pp>y-V-i.p>. 



Hence, equating to zero the coefficients of the different combi- 

 nations of p, p' ', cr, we easily obtain by writing for^' succes- 

 sively 0, 1, 2, 3, . . . , 



AM>-l)A + 2(2yu-l)B = 0, 



Oa-2)0&-3)B + 4(2/*--3)C=O, 



Ou-4)0*-5)C + 6(2/a-5)D = 0, 



^~ 2(2/*- 1) A ' 



2.4(2a6-1)(2^-3) ' 



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

 2.4.6(2^-l)(2^-3)(2/x-5) ' 



To find the value of A, I observe that when k = 0, Z = 0, 

 Jc f =0 } Z r = 0, and hf=zh, I becomes 



(A + B + 0+...)P\ 



But in that case, taking the radius of the sphere equal to unity, 



