456 
DR. E. W, HOBSON ON A TYPE OF SPHERICAL HARMONICS 
which we shall take as the definition of the function Q n m (/x) for unrestricted values of 
m and n. 
When mod. (/x) > 1, Q/* (p) is represented by the expression 
Q* 
.. t™* U(n + m) IK-i) , g 
(d) On+l TT/V„_l1\ 
n (%+j) 
■ I) 4 * 
/X" 
tf + y/i+. 
F 
w + m + 2 m+to+1 
?? +W’W) ( 10 )- 
- r 
The uniform function obtained by continuing the function in (10) over the whole 
plane, with the exception of the cross-cut along the real axis from -f- 1 to — oo, is 
represented by the expression in (9). 
When n is such that the real part of n + 1 is positive, the definition (9) can be 
simplified, the integral being then reducible to one along a line joining the points di 1. 
The path may be as in the figure ; then, since the integrals along the loops round the 
points 1 and — 1 become indefinitely small when the loops are made indefinitely small, 
we have 
Or—*— < — 
-/ +/ 
*(-!+, 1-) 
(? — l) n (t — ix)- , ‘- m ~ l dt = (e nnL - e~ nm ) j (1 — t a ) u (t — dt 
= 2t sin mr | (1 — t~) u (t — dt ; 
hence, when n + 1 has its real part positive, we may substitute for (9) the definition 
o»* w = 
— (n+ 1) i7r 
II (n + m) 
O 2 - 1)'"' f__ (1 - D" (f - D-*-’- 1 dt 
2 , + i n(B) L 2 - iCf!., (1 -D'O 1 dt. . . . (ii). 
2 * n (n) 
e mi ’ 7r n (71 -f to) 
The integral may be taken along the real axis, (1 — t' 2 ) n denoting e" ,0Sil ‘9 where 
the logarithm has its real value. 
It will be observed that when n is a positive integer, the form (9) is undetermined 
(qo X 0) ; we can, however, in this case use the formula (11). When n is a negative 
integer, the value of Q n m (/x), as given by (9), is in general finite, since 
sin • n (») = - n { _W i> : 
if, however, n + m is also a negative integer, or if m is zero, the value of Q/' (p) is 
infinite, so that the factor II (n + m) must be rejected if we wish to obtain a finite 
solution of the differential equation. 
