514 DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
\ and —both the formulae (109) and (105) hold. In the special case m = 0, we find 
by adding the two expressions in (105) and (109), 
P„ (cos 6) 
2 
— COS 
7r 
( n + i )*\ 0 <2 
cosh (n + J) v 
(2 cos 6 + 2 cosh vf 
dv 
when the real part of n is between 0 and — 1. 
(no) 
A definite integral form for P u (g), when the real part of n is between 0 and — 1. 
49. Taking the formula 
p m 
n 
M = 
2itl 
n (m - i) 
n(- i) 
+ - i)*-f 
'(*+, l/z—) 
fci+m 
(1 - 2 fill + Jl 2 ) m+i 
dh. 
we see that, provided m is half a real integer, and also \ — m is positive, the path 
may be replaced by one which consists of a single curve enclosing both the points 
1 
In the first figure the initial phases at A are 277 — (3, for 1 — hz, and — (2n — a) 
for 1-— • In the second figure the phase of 1 — — is zero at C, and that of 
z 
1 — hz is 277 at D. The formula becomes 
IV M = 
1 ri (m - A) . „ .. d !+ ’ J / s+ > h n+m 
si 2 - ^ - 1 >f (i - ^ 
dh 
Now suppose the real part of n — m is negative, and that of n + m + 1 is positive ; 
we may replace the path by one round a circle of infinite radius, straight paths along 
