592 
Proceedings of the Royal Society 
= - 22 :^ 
2i+r 
In this there is no term in which the powers of li and h' are 
different, hence we have 
— 1 
J Qi Qy clfJy — 0 
°+l 
(7). 
in all cases unless j = i. In this special case we have 
—1 
/ Qi dfx 
+1 
2 
2i +1 
( 8 .) 
Hence the left hand member of (6) vanishes unless / = i, and in 
that case we have 
y"'a-„■/($)'* 
+1 
k’ + s 
2t + l 
2 - — s 
(9). 
We might have proved (7) from (6) by exchanging i and/, and 
showing that unless i - /, we cannot have 
| i + s | j 4- s 
1 — s 
J “ s 
5. The equation (3), which is satisfied by Qy, is a mere particular 
case of the general equation of surface harmonics— 
*( l + l) Si + i-ja 3 df ' dp 
1 cZ 2 Sy ^ d 
(<>--■)© 
= 0 ( 10 ). 
which maybe obtained by putting Y$ = Sy in the ordinary equa¬ 
tion of Laplace— 
■ d \rYj) 1 cZ 2 Yy d 
dr 2 1 — /x 2 c/£> 2 cZyu. 
after differentiating the first term. That 
fact, 
differentiation gives, in 
= i(i + 1) V<. 
