OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
453 
As regards the formula (5), 
P■* (fo> = 
n (- m) \/x - 1 
fjL + 
n, n + 1, 1 — m, \ > 
we may remark that 
(a) When n is a real integer and m is not so, the series is finite, and therefore 
P»* (/x) is an algebraical function. 
(ft) When m is a real positive integer and n is not so, the formula may 
be written 
A" M 
1 II (n + m) 1 
2m IT (n — to) II (to) 
(m — n, n m -f- 1, m + 1, 
(y) When n and m are both positive real integers, and n > m, it falls under 
case (/3), the series being however finite since the first element m — n of 
the hypergeometric series is a negative integer, thus V,™ (g) is an 
algebraical function. 
(8) When n and m are both positive integers, and n < m, case (/3) shows 
that P,y (/x) is zero ; in order to obtain an integral of the differential 
equation we must take IT (n — rn) P u m (/x) which is finite. 
In defining the function P„ m (g) by means of a definite integral taken round a 
closed path, in which turns are made round the points p and 1, but none round the 
point — 1, it is necessary to specify the position of the path with reference to the 
point — 1. The figures (a) and (h) represent two distinct paths for the same value 
of /x, but the integrals obtained from them will be, in general, different in value, as 
one path cannot be brought by continuous deformation into coincidence with the 
other without crossing the point — 1, which is a singular point for the integrand. 
We shall consequently specify that the path by means of which P n m (/x) is defined in 
(4), is one which does not cut the real axis between — 1 and — co , or is, at all 
events, a path which can by continuous deformation be brought, without crossing the 
point — 1, into a path which does not cut the real axis between — 1 and — oo. 
6. Another closed path for the integrand ( t~ — 1 ) n (t — /x) -;< ~“~ 1 is that in which 
