4G0 
DR. E. W. HOBSON ON A TYPE OE SPHERICAL HARMONICS 
r(-i+,i-) 
(t~ - 1 ) n (t — /x)"' 1 -™- 1 clt 
n-’lnm f 1 +, n —, 1 —) 
= if - (t - dt 
(/i + , -1 + , M —, -1 -) 
(,t 2 - 1)* (f - ix)- n ~ m - l dt L 
or 
^ g—? 7 r(m+re)t 
X the same expression.(14), 
according as /x is above or below the real axis. 
10. The relation (14) enables us to find the expression for Q n m (g) in series, for 
values of g which are such that mod. (1 + g) and mod. (1 — g) < 2. Using the 
formulae (5), (9), (13), we find at once 
QU M 
rrc 
mm 
1 
2 sin (to -f n) ir IT ( — to) 
R-Trt 
/ g + 1 
V-1 
— n, n -f- 1, 1 — m, 
g 
- 1 
g + 1 
F( — M, n + 1, 1 — m , 1 
(15), 
the upper or the lower sign being taken in e Tnm , according as the imaginary part of /x 
is positive or negative. 
When m is zero, we have 
«.»+ rr’U*)} (i6)- 
The particular case (1G) agrees, when /x is above the real axis, with the result 
obtained by Schlafli. 
If we use the relation (12) between Q„“(/x) and QU'* 1 (g), w r e can write (15) in 
the form 
Q» (g) = 
7r 
2 sin mr 
? T 717TI 
F — n, n + 1, 1, 
g 
-F(- 
\ 
7TC V 
n (» + to) 1 L^/g-iN 4 "*- 
sin(g — to)7t n (g — to) n (to) [ \g + 1 
) F^—g, n + 1, 1 + ???., 
- (urfiY F(-g,g-fl, 1+™, (U)- 
When g + m is a positive integer, the expression (10) shows that Q/ 1 (/x) has in 
general a finite value, hence we see from (15), that in this case 
5 T rnn 
Li + IN *” 1 / 
—jd T ( — n, g + 1, 1 — w, —- 
g 
1 ~ g\ _ /g “ / _ , , , _ 1 ±g^ ; 
\g + 1 
F ( — g, g + 1, 1 — ?g, —— 
