446 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
ordinary case of integral values of n and m is the only one which has to be 
considered. 
Heine uses the symbols P m (n) ([x), a), D-J n) (p), which are connected 
with the symbols P/*(p), Q/ l (p) used in this memoir by the relations 
p,«w = p-.«m = (r - i)-‘-!p,“w = x ,; 2 g P *’M 
Q -"M = Q-»‘”L) = L 2 - i)-*-sv>(D = (- 1)*Q,”.(/»)• 
Thomson and Tait use the symbols &J m) (p), 3- n (m) (p), which are connected with 
Heine’s P J n) (p) by the relations 
(- 1)*»P.“W = e«w = 
Ferrers uses T„“ (p) for what is denoted here by (— l) im P n m (fi), except in the 
case of a real p lying between i 1, in which case T„ m (p) and P n m ([x) are identical. 
The Gaussian function II (x), which is equivalent to r(£c'+ 1), is used throughout 
the memoir. 
Definition of the functions P n m (/x), Q u m (/x) by means of definite integrals. 
I. If, in the differential equation 
cFV dV f 
-,}V = 0. . . . (1). 
— F J 
which is satisfied by Legendre’s associated functions, we substitute A’ = (p . 3 — l) im AV, 
then W satisfies the differential equation 
r 72 w r/W 
(1 — fi) — — 2 (m + 1) h + (n — m) (n + m + l) W = 0 . . (2). 
If, in the expression on the left-hand side of (2), we substitute 
W = | (t 2 — !)"(£ — ix)~ n ~ m ~ 1 dt, 
we find 
72 7 1 « 
(! — fi) dfj? — 2 (m + l)p — + (n — m) (n + m + 1) |j(£ 3 — 1)" (t — p,) - " - ”' -1 
= - ( n + m + ~ l Y +1 ( t - fi " m z ] 
dt. 
