592 
Proceedings of the Royal Society 
= -2 2 0 
* (hhy 
2i + V 
In this there is no term in which the powers of h and h' are 
different, hence we have 
—l 
J Q i Qj djL ■ 
in all cases unless j = i. In this special case we have 
—l 
/ Q? d/x 
J +i 
2 
2* + l 
(7). 
( 8 .) 
Hence the left hand member of (6) vanishes unless j = i, and in 
that case we have 
• to . 
+1 ' 
We might have proved (7) from (6) by exchanging i and/, and 
showing that unless i — /, we cannot have 
V \J + S _ 1 / + s 
1 i - s~ \j - s * 
5. The equation (3), which is satisfied by Q*-, is a mere particular 
case of the general equation of surface harmonics — 
*•(>•+ 1) Si + + |-(W) §)=0 (10). 
which maybe obtained by putting V < = Si in the ordinary equa- 
tion of Laplace — 
r d%^i) + 1 dfVj + d Aj _ JA _ Q 
dr 2 1 — (a? d<p 2 d/x\ d/xj * 
after differentiating the first term. That differentiation gives, in 
fact, 
