512 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
— f {e-(»-*+ 1 )* ,r . e -(*+*>. e 2 "( w -i) — e (*-»+i)" e -(»+i)» c 2 in(«-i) j ^ cosh v + 2 cos d?;, 
J 0 
or 
—e( m b 2m 2c sin (n — m) n I — 
JO (^ 
we thus obtain the formula 
g —{yi+V)v 
cosh v + 2 cos 0)^ 
i— ch; 
E — 777. 
P* w (cos 0) = 
n (ft — m) 
n (?i + ??i) 
2 sin m 6 
i cos mTT. P n m (cos 6) 
— - sin mft. Q n m (cos 0)1 
1 
ft J 
2«n (—-|) II (m — 4) 
+ cos (n + i — m) tt ( 
[ r cos [(ft + j) — (m + ±) 7-] 
(2 COS 6 — 2 COS 
0 —(n+i)v 
d<f> 
J o (2 cosh v + 2 cos 0)-' 
(105), 
which holds provided the real parts of m n — m + 1 are positive. If n — m is 
a positive real integer this becomes 
n (h — hi) r 2 
ri (ft + m j C0S m ' !T * ( cos ^ sin TKlTT . Q/ (cos 6) 
2 sin~ m 0 
31,1 
COS [(ft + i) 0 — (jft + J) ft] 
2 m IT ( —J) II (in — J) J e (2 cos 0 — 2 cos 
When m and n are both positive integers, and n > m, we obtain 
II (ft — m) -p, /_ m 2 sin - ™ 0 /' 7r sin (ft + J) <f> 
d(f> 
(106). 
n (?i + ?ft) 
P,/“ (cos 0) = 
Om 
_r 
II (— J) II (m — |) J 0 (2 cos 6 — 2 cos <£)i 
r; d<f> • (107), 
which becomes, when m = 0, 
t> / /i\ 2 r sin («+ j) 7 / 
P /( (cos 6) = - --—-—bn arf> 
v ’ 7T J fl (2 cos 0 - 2 cos </>) 4 r 
(108), 
which is the second expression given by Mehler for P„ (cos 6). The formulae (105), 
(106), (107) are therefore generalizations of the known formulae of Mehler and 
Dirichlet. 
48. Next suppose the condition that the real part of n — m + 1 is positive does 
not necessaifily hold, but that the real part of n + m is negative, and that of m + i 
is positive; we may replace part of the path of integration in the last Art. by 
straight paths from — 1 to — od along the real axis, and a circle of infinite radius. 
From — to — 1, the phase of 1 — 2fxh -fi Id is 77 — (f), where </> is initially equal to 9; 
from — 1 to z, the phase of 1 — 2 fih + Id is 3ft -f- (j>, where (f> is equal to 6 at the 
