588 THE HON. J. W. STRUTT ON THE VALUES OF A DEFINITE INTEGRAL 
The mean temperature S 0 is seen to be independent of the conductivity and of the 
size of the sphere. 
The case where the heat which falls on the sphere proceeds from a single radiant- 
point is not only important in itself, but may be made the foundation of the general 
solution in virtue of the principle of superposition. Taking the axis in the direction of 
the radiant-point, we have 
F(E)=P 
7T 
over the positive hemisphere, that is, from 0=0 to $ = 
while over the negative hemisphere F(E) = 0. 
It is required to expand F in a series of spherical harmonics. 
Let F then/* is a function of (m, which is equal to ^ over the positive hemi- 
sphere and to — \fjj over the negative. The problem therefore reduces itself to the ex- 
pression of over the positive hemisphere in a series of functions Q of even order . 
The same series will then give — over the negative hemisphere. 
Assume 
— -^-o “l - -A-2Q2 T A4Q4 T 
Multiplying by Q 2n , and integrating with respect to p from ^=0 to g- = l, 
iC Q1Q2 iA\J> (Q2 n) 
Jo Jo 
all the other terms on the right vanishing. 
N °w f (Q 2n ?dp= 
J Q 1 Q 2 »^i !A = coefficient of e 2n in the expansion of 
1 (l+e 2 )f-l 
1 . 1 .3.5 ... . (2n — 3) 
2 . 4 . 6 . . . . ( 2 » + 2 ) ’ 
or 
A 2 „= — ( — l) 1 
4n+l 1.1.3. 
2 2.4.67 
(2w— 3) 
(2i» + 2)’ 
Accordingly 
F(E)=i+iQ 1 +Aa 
AQ4+ 
-(■ 
.4/1+1 1 . 1 .3 .5 . . . . (2m— 3) 
' 2 '2.4.6.... (2ji + 2) 
Q 2 «+- 
When n is great the coefficient of Q 2 „ approximates to 
This completes the solution for a sphere exposed to the radiation from an infinitely 
distant source of heat situated over the point ^ = 1. 
If its coordinates are <p', it is only necessary to replace p in Q 2 „ by 
cos Q cos & + sin 6 sin d cos (<p — <p'), 
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