470 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
the values P„ m (p + 0. i), P„ m (p — 0 . t) on opposite sides of the cross-cut for real 
values of p lying between the values d= 1. 
Referring to the expression (5), we see that in this case 
p " <?■+ 0 ■ *) = (r^r F (-».»+1.1 - V 
P*" P - 0 • ‘) = rT^V) (rrfF ( - » +1,1 - m. 1 - 
hence we have the relation 
e imm P n m (n + 0 . l) = e ~ hnm P n m (p — 0 . i) 
1 /I + p\* 
IT (- m) \1 - p 
F — /i, n -j- 1,1 — m, 
1 - 
(28). 
It is convenient to define the function P,* (p) for real values of p between + 1 and 
— 1, in such a way that its value shall be real for real values of m and n ; the 
definition which we give is that for such values of p, 
P n m (p) = e imn P/ 1 (p + 0 . t) = e “ P„* (p - 0 . i) 
n(-m)\l-p/ 7l » w + 1 > 1 2 ) ' ' ( 2 
From (27), we find in this case 
„ , . „ n + m 
P,™ (p) = 2 ; " COS-— 77 . 
n 
'n + m — 1 
n (—W 
r pi - P) ! " F ( m + ” + 1 , P 
, „ . n + m 
+ 2" ! sm —-— 77 . 
A 
n 
n + ?h 
„ , n — w — 1 . „ , _ 
n < - 2 -J n U 
/, <m„, m + % + 2 m - n + 1 3 „ 
1 x (1 - p-)‘* F (-y- , -y- , y, p 
when (1 — p 2 ) i,w denotes e 4mlog ' (1-MS > and logv(l — p 2 ) has its real value. 
We see from (2D) that when m is zero, or an even integer, the values of the function 
on the opposite sides of the cross-cut are equal, so that in this case the cross-cut is 
necessary, so far as the function P n m (p) is concerned, only from — 1 to — oo. 
18. Next, let us consider the values of Q„ m (p) on opposite sides of the cross-cut for 
values of p lying between d; 1 5 from (15) we have 
