502 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
If the real part of n — m -j- 1 is positive, the integrals along the semicircles with 
p, as centre vanish when the radius is made indefinitely small. If the real part of 
n + m -f- 1 is positive, the integrals along the infinite semicircles vanish. We thus 
have, 
Q -u~r (g) 
n On) ,. 
_F_/__o;/ + l 
sin (n — to) irll(n — to) 
e nm . 2 cos mn | Xdt — e nm . 2 cos mr 
J V- 
f >4 
where in the integrals X commences with the phase it has at A initially. The phase 
of t + 1 at A is — (27r — y). 
From equation (8), we have 
PA (g) 
— e 
+ ?l7Tl 
47t sin (n — to) 7t IT (n — m) 
rr (n) i+, m-. >-) 
( ’ • 2" + 1 I (i 2 - (t - dt, 
where the phase of t -J- 1 in the integrand is y at A, and thus 
hence 
(f — l ) - " -1 (t — = e~ z,ini X ; 
PA (g) = 
— c 
n<» 
4c.Tr sin (n — to) tv II (n — m) 
Oil +1 
(n + , 1 + , (X — , 1 — ) 
X dt. 
oo 
Taking the path as in the figure, we have, provided the real parts of n ± m -fi 1 are 
positive, 
r(n+. 1+.M-, 1-) r°°' r“ r°° f*' 
Xefc = e 2 <»-**H Xdt- e^H X^ + e-(» + *)" Xcfc-| Xdt 
* j [A J IX <j IX J H 
-QO <■> 00 > 
= — e -2m7rt . 2i sin (n — m) to Xdt. -\-d- n ~ n,)irL . 2isin (n— m)TO Xdt, 
J u. • u 
