464 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
In the particular case m = 0, we have 
P* (f) = 
tan mr II (n) 1 
2 n+l n (n + i) n (— fj. n+l 
+ L 
n + 1 
2 
, n + 
+ 2 " 
n (n - i) 
n(n) n(- i) 
F 
w F 
/l — % 
n 
~2 
(23). 
It will be observed that when n -fm is a positive integer, the expression (20) 
reduces to its second term, but not so when n + m is a negative integer, since 
sin (n m)u. II (n + m) is then finite. Heine gives* as the expression for P„ (p), 
when n is unrestricted, a formula which is equivalent to the second term in (23); his 
formula is, therefore, only correct when n is a real integer. 
Expressions for P„ m , Q,” 1 in ser ies of powers of p, when mod. p < 1. 
14. It will be convenient to obtain the expansion of Q„ m (p), first in powers of p, 
when mod. p < 1, and afterwards to deduce the corresponding series for P,/" (p). 
Taking the formula 
p—(n+\)ur TT ( v 4 - f(—1 + , 1—) 1 
QE p = f--(p 3 - lf m J (t 2 - 1)* (t - p)—*- 1 cfe. 
v 7 4t sin ?i7r x 7 II (n) } c 2 n v 7 v ^ 
Consider first the case in which the imaginary part of p is positive; the path of 
integration can be so chosen, as in the figure, that, for every point of it, mod t > mod p ; 
the term (t — f)~ n ~ m ~ x can then be expanded in ascending powers of p, and we thus 
find 
/>— ('/l+ 1) 17T 1 GO 
Q* w f) = T-v- (p 3 ~ lf m -* 
4 1 sm mr u 
2«noo,r 0 n (r) 
n (, ‘ + " + r) (i» -1)* A. 
p 
Let us now consider the integral, J (-1+il-) ( t 2 — l) H t?dt. 
c . 
-/ 
First, suppose n and p to be such that the real parts of n + 1, p + 1 are both 
positive, the path of integration may then be as in the second figure, the loops round 
* ‘ Kugelfunctionen,’ vol. 1, p. 38. 
