484 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONIC 
We see that this equation is the differential equation satisfied by the hyper- 
geometric series F (a, (3, y, p), where a = -—, /3 = m + ^ + y = L ; we thus see 
that the differential equation (1) is satisfied by either of the expressions 
f n+m—l , 
(p* — lf m -2- (1 
f m—7i—2 
(p 2 — I )-’ il I U ~ (1 
' — m—n+l n—m 
— u) 2 ' (I — [x-'u)~ du, 
n—m ^ n+m+1 
— u)~*~ (1 — p~u)~' 2 du, 
when, as in the other cases, the integrals are taken along closed paths. We thus 
obtain a third class of definite integrals, by which the functions P n m (p), Q,™ (p) can 
be represented. It is unnecessary to obtain the exact expressions for the functions in 
four of these definite integral expressions, as all the results of interest may be obtained 
from the two classes which have been already considered. 
The existence of these three classes of definite integrals which satisfy the funda¬ 
mental differential equation (1) is equivalent to the result obtained by Olbricht, 
that the equation is satisfied by three distinct Riemann’s P-functions, 
r~ 
0 
00 
1 
P^ 
— n 
\m 
L 
— b n 
n fi- 1 
- 
0 
00 
1 
n 
11 
p^ 
2 
m 
~ 2 
n+1 
— m 
71+1 
2 
2 
0 
00 
1 
11 
m 
0 
p« 
~ 2 
9 
n + 1 
m 
1 
2 
2 
a 
/x + yZ/x- — i 
2 \/fj~ — 1 
-v 
Expansion of P/ (p), Q, “ (p) in powers of ^ 1 
28. In the formula 
Q n m (p) = ie <m ~ n)m . 2 "\ 
n ( m— ±) II (-£) 
47 r sin (n + in) 7 r 
i„, f (H- 0+ !-■”-) 
h“ +m 
(1-2 ph + hf 
i + i 
dll . (40) 
