450 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
c n m e nm € (n + 1, — n —m) 
^ + l n (n+r) 1 _ 
,p — 1/ ^ II (r) II (n — r) (1 — m) .. . (r — m) 
or 
c u m e nm G (rt + 1 , 
— n 
n, n -f- 1, 1 — rn, 
5 
where F is used with the ordinary notation, for the hyper-geometric series. 
In virtue of the property ( 4 )', we have € (n -f- 1, — n — m) = G (n + 1, m); and 
from ( 3 )' we have 
, X . . n (n)U(m — 1) 
G (n + 1 , m) = 4 sin nv sm . - n ^ + m) ; 
hence, whatever n and m may be, the expression (3) becomes 
. . . fa + l\ iOT II (n) n (m — 1 ) _, / , 1 — a 
c™. e nm . 4 sm rnr sm mn ( 1 ) ■— TT ^ —- F — n, n -j- 1, 1 — to, —— 
p, — 1/ II (n + m) 
4. In the case m — 0, we have, since II (— to) II (to — l) = 77 cosec m tt, 
r<M+> i+, m-> i-) 
(£ 3 — !)"(£ — ju,)“" 1 dt — c,?.e nn \ 477 sin ?i7r.F — n + 1 , 1 , —) > 
1-/ 
when mod. ^ (1 — p,) < 1; in accordance with usage we take the Legendre’s function 
P„ (/x) of the first kind to be given by P„ (p,) = F 
n, w + 1,1 ■ ^ hence, if we 
choose c, t ° equal to 
47r sin 7177 
, we have 
P. M = 
477 sin ?i7r 
+ , 1 + , M-, 1-1 1 
The integral on the right-hand side defines P M (p.) over the whole plane, the 
function represented in the domain of the point 1 by the series, being analytically 
continued over the whole plane. 
In order to obtain a definition of P„ w (p,), we shall first consider the case when m is 
a real positive integer, and shall then define P„ m (p,) for general values of m in such 
a way that the definition agrees with the usual definition for the special case in 
which to is a real integer. 
When to is a positive integer, we may define P,“ (/x) by means of the formula 
P„ OT (p,) = (p, 3 — l)* w ; thus in this case 
M = 
II (7? + to) 
477 sill 7177 II (ri) 
(F-1)*“ f' 
(fj. +, I +, f-L 1 “) ]_ 
(, t 3 —!)"(< — p,)“"“ m-1 
