OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
147 
r+i 
and if we denote | 3- m ,l 3- m n dv by (n, n, to, X), or shortly by (n, n), we obtain the 
following:— 
(n, n) = a 0 a' 0 j m m ++ a s a' 2 j m m+4! + . . . 
(n, n) =(a 0 )% m +(a 1 )% m+i + . . . 
It may be easily shown that 
(n, n') = 0, n not — n .(G3) 
for the differential equations satisfied by 3- m n , 3-,/ are 
cPS* d& n w? 
(l-^-2^-r^y=XWy- W 
1 civ 1 dv l—v~ 
fj Q.»' 0)7^ 
■ • (64) 
and therefore 
(fe-fe,) f 
+i 
3- H S-*'dv= 
+i 
= 0 . 
-l 
From this result we infer that any function may be expanded in a series of ^-func¬ 
tions ; and, first of all, we may invert the series which expresses the ^-functions of 
the a-group. This is manifest algebraically ; for if we solve the equations for 0° . . . 3 r 
in terms of P„“. . . P„/ M+2 g we obtain for any one of the P’s the following:— 
where 
¥S=f ( £ 0 +f 1 * 1 +fJ>+ 
f s (s, s) — I P m n 3- s dv= (— 1 ) r a,j m r . 
■ -l 
> 
(65) 
Now we may assume that any function of v (at least any series of powers of v com¬ 
mencing with v m ) may be expressed in one or other of the three forms SAP 
2BP;/ +2i ' +1 , S(AP„y +2 ' -j-BP„/" +2r+1 ), and therefore in ^-functions of the a-group, 6-group, 
d 2 3 
or a combination of these. In particular v z 3-, v l 3, — . . . will give rise to ^-functions of 
dS- 
the same type as 3-, and v3-, - ... to functions of the opposite type. In general, if 
F(f)=2C i 5 i , (i, t)Ci=£ 
( 66 ) 
The expansion of v~3 may be obtained without difficulty ; for if we combine the first 
of equations (64) with 
u 2 
