180 
DR. T. J. I’A. BROMWICH ON THE 
( 2 - 2 ) 
Ca = 
a^v 
a?-' 
mK a^v 
= 
r do a?-’ p dcpKc dt j ’ 
Cy = - 
1 a^T 1 a /'Kau\ 
a^ dr rdO\c dt j 
P — r sin 0, 
where U, Y are any two solutions of the equation 
(2-3) 
pK a^u 
C-' a^^ 
TO 
dr'- 
+ 
sin 0 do ' do j sin^ 0 d(j>^ 
A solution of (2’3) which is sufficiently general for the applications in view may be 
found by assuming tliat U and V can be expressed as sums of terms of the type 
F(r, ^)x Y(0, 0). 
It is easy to see that then (2'3) leads to the equation 
(2-4) 
7 -' /a-T pK a-F\ _ 1 f 1 a /. aY\ i a^Yi 
fW- df) Ylsinoaav ae/ siffiea^-j’ 
and since the two sides of equation (2’4) are functions of r, t and of 0, <p 
respectively, it is clear that each side must be a mere constant. If we write the 
constant in the form 7i{n+l), it is evident that Y must be a surface-harmonic of 
order 7i. 
Accordingly in problems (such as those with which we shall be concerned in the 
sequel) where the ivJiole of angular space is considered, the value of n must be a 
positive integer; for (except when n is an integer) there are no surface-harmonics 
which are everywhere continuous and single-valued. 
Thus we may reduce our solution to the form 
(2-5) 
U or V = 2F,{r, t)YM i>), « = 1. 2. 3. .... 
where F„ is a solution of the equation 
( 2 - 6 ) 
and 
1 3“F„ n(7i+l) -p 
ar“ df 
cd = cV(mK). 
= 0, 
