INVOLVING THE PEODUCT OF TWO LAPLACE’S COEFFICIENTS. 
587 
I ( 1 + Y se 1 ) ( 1 ' 
These are the only terms in the expansion of — log — 
Y 1 *f" < 
m/Y 1 
in which 
+ e‘- 
the index of e is one higher than that of e 1 . 
Having regard, now, to the symmetry in e and e\ we see that generally 
| Qo ;i Q 2H -i^ — I + 1 
«./ o <^o 
As an application of some of the results of this investigation I will take the following 
physical problem. A spherical ball of uniform material is exposed to the radiation 
from infinitely distant surrounding objects. It is required to find the stationary con- 
dition. For the sake of simplicity, the surface of the sphere will be supposed to be per- 
fectly black, that is, to absorb all the radiant heat that falls upon it, and Newton’s law 
of cooling will be employed, at least provisionally. 
If V denote the temperature, it is to be determined by the equations 
(j?+!+£) V =°- (A) 
*^=F(E)-AV, (B) 
where F(E) is a function of the position of the point E on the surface, and denotes the 
heat received per unit area at that point, Tc is the conductivity, and h the coefficient of 
radiation. Equation (A) is to be satisfied throughout the interior and (B) over the 
surface of the sphere. 
If Y be expanded in Laplace’s series, 
V=S. + S,j+S,j + ...; ^L = i(S, + 2S s +3S J+ ...); 
and if 
f=f 0 +f,+f 2 +.... 
be the expansion of F in a similar series of surface harmonics, we obtain, on substituting 
in (B) and equating to zero the terms of any order, 
s,= -A 
a 
s.= 
F 
J- n 
h + 
nk 
a 
4 L 
(C) 
MDCCCLXX. 
