INVOLVING THE PEODUCT OF TWO LAPLACE’S COEFFICIENTS. 
589 
Hence if H denote the intensity of the radiation which comes in direction (jJ, <p', the 
general value of S„ is 
c 4?H - 1 1-1. 3.5. ..(2w— 3) 
1 + 2.4.6... (2» + 2) 
xjj HQ 2 „(cos $ cos ^+sin 0 sin O' cos(cp—(p'))d[jj'd<p', 
the integration going all round the sphere. 
Now (4w+l)JjHQ 2n ^'^<p' is the same as 4 tH 2 „, where H 2n is the harmonic element 
of H of order 2 n ; so that 
y, 1.1.3. ..(2ra — 3) 27rI4 2n 
& 2n — { J-J 2.4.6...(2n + 2) h | 2 nk’ 
a 
S 0 =^H 0 , 
S, 
2 k\ 
4 ( A+ T y 
H, 
S 4 
h 4 . 
It is remarkable that the odd terms in H (except HJ are altogether without influence. 
The reason is simply that they do not affect the total heat falling on any point of the 
surface. 
For this is expressed by 
n dpd<p. 
the point considered being taken as pole of which involves no loss of generality. 
Now (Thomson and Tait, p. 149) 
s=n 
H„ = 2 (A, cos s<p + B s sin sp)0 s „, 
s=0 
where 0® is a function of p not containing <p. 
When the integration with respect to <p is effected, all the terms will vanish except 
that whose coefficient is A 0 . For this purpose, therefore, we may take 
H„=A o 0" or A 0 Q„, 
and we know that p Q/fyt, vanishes if n be odd and different from unity*. 
* The proof given is sufficient for the object in view ; but it may be well to notice that the essential thing is 
that the two surface harmonics which are multiplied together are either both odd or both even. A harmonic 
