448 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
between d: tt, in order that a single value may be assigned to (/ d — 1 ) im ; by (rr')' is 
denoted d r ‘ llog{r ' ,J) where log (rr') has its real value; the value of (p~ — l) lm has thus 
been uniquely specified for all values of p, except those which are real and lie 
between + 1 and — oo . Next, in (t 2 — l) n = (t — 1)” (t -f 1)”, we shall suppose the 
phase of t — 1 to commence with the value (f> at C, where </> is the angle (between d: tt) 
the line joining C to + 1 makes with the positive direction of the real axis ; the phase 
of t -|- 1 at C we shall suppose to be </)', where (f>' is the angle (between d: tt) the line 
joining C to — 1 makes with the positive direction of the real axis; if at C, 
t — 1 = ke^, t -j- 1 = k'e^', the value of (t 2 — 1)" will be e Klog(U '°. where 
log (kk') has its real positive value ; after the positive turn round 1, (t 2 — 1)“ will have 
become e nlosa ' k ' ) . + ^ + 
The phase of t — p we shall choose to be such that it is zero when t passes through 
that point of the path for which t — p is a positive real quantity, thus the initial 
value of t — p at 0 is where xfj is the angle (between dz ^ which the line Cp 
makes with the positive direction of the real axis, hence (t — p) -(u+m+I) changes from 
p—(n+m+i) g(7r— <i<) (n+m+\)i £ 0 ^-(n+m+i) e -(»+m+i)(,r+'iMq going from C round the point p to C 
again, p -( ' i+ “ +1) denoting e -(™+ m + 1 ) 1 °g'p ) where log,, p has its real positive value. 
3. Let us now consider the value of 
(p2 _ f 
* i 
(M+, 1+,/*-)!-) 1 (J? — 1)« 
2" (t - p) 
n+m +1 
dt 
( 3 ), 
with the specifications of the phases just given, in the case in which p is such that 
mod. (1 — p) < 1 . We shall make the substitution t —- 1 — (p — 1) u ; it will be 
convenient to place the path so that C is on the straight line joining 1 and p, so that 
u has a real value less than unity when t is represented by the point C. 
The integral becomes 
nl+,0+,1—,0-) / 
C n m (p 2 - l) im j (p - l)- m u n (u - l)-*-™- 1 (1 + 
/* 
— 1 \ 
u j du 
where C' is the point corresponding to C. 
In this integral the initial phase of u at O' is zero, that of u — 1 is — tt, and 
( 1 + — ~ — u) has the value given by the Binomial expansion. 
On performing the expansion, we obtain 
r m 
^n 
/p + “ 
\P f/ r-0 
n (n) 
II,(r) II (n — r) 
p - 1 
K 
(L+,0+,1-,0-) 
U 
n+r 
(u - I)’ 
■n—m—l 
du. 
