454 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
a positive turn round the point — 1 is followed by a negative turn round the point 
+ 1* 
Consider thus the expression 
(-i + ,i-) i 
1 ) ,l (t — fi)-*-*- 1 dt, 
taken along the path as in either of the figures. The phase of t — p will be 
measured as before ; those of t — 1 , t + 1 , we shall take to be such that they vanish 
at the instant when t passes in the integration through the point A of the real axis, 
for which t — I, t + 1 are both real and positive; thus, in the second figure, the 
initial phrases of t — 1, t + 1 at C are tt and — 2tt respectively. 
Let t — p = (/x — t) e~ L7r , then the phases of p — t are such that at the point E, 
where the line joining p and 1 cuts the path, the phase of p — t is the angle 
(between dz v) the line makes with the positive direction of the real axis; the 
expression becomes 
fr (p 2 - 1 )* 
(-i+,+i-) i 
M_ e (» + » + iy«r ^2 _ iy t ^ 1 c ( t 
Suppose now that mod. p > 1, the path of integration can then always be so 
placed that mod. t is everywhere less than mod. p; expanding by the binomial 
theorem, the expression becomes 
fn m (A 2 “ l Y m • 
r= oo 
__ 0 ('ll + m + 1 ) ltt 
|(- 1 +, + 1 -) 
n ( n + m + r) 1 , 2 
II (n + m) II (r) ^^ m +\+r \ 
1)" t r dt. 
j \ lj i / 
To evaluate ( t 2 — 1)” V dt, we may place the path so that the two loops 
are exactly equal, C being half-way between the points 1 and — 1 ; it is thus seen 
that the integral vanishes when r is odd, and that when r is even and equal to 2s it 
is equal to 
1)" t~ s dt; 
making the substitution t' — t z , we see that t' — 1 , or (t — 1) (t + 1) is such that its 
phase increases from — tt to tt during the integration, we thus have 
