OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
447 
It appears thus that the differential equation (2), is satisfied by 
W = j(£ 3 — 1 ) n (t — dt, 
for unrestricted values of n and to, provided the integral is taken along a closed 
path, i.e., one such that the integrand ( t 3 — 1 ) n (t — [x)~ n ~ m ~ 1 attains the same value 
when the path has been completely described, as that with which it commenced. 
The integrand has, in general, the four singular points t = + 1, t = — 1, t = g, 
t — co , and we shall see that it is possible to choose two distinct closed paths, 
defined with reference to these singular points, which will represent the values of W 
required for the two Legendre’s associated functions. 
2 . 
If the variable t, starting from a point C, describes a path in which a positive 
(counter-clockwise) turn is made round the point g, then a positive turn round the 
point 1, then a negative turn round g, and lastly a negative turn round 1, such a path 
will be closed, i.e., the integrand ( t 2 — l)"(£ — g)~ n ~ m ~ l will have the same value 
at C at the beginning and at the end of the path. In the first figure the path will 
be (Ca/3C, CySC, CfiaC, CSyC) ; in the second figure it will be (CD, D abD, DC, 
CfgC, CD. D&aD, DC, Cg/C). In Pochhammer’s notation, the value of V will be 
+> i +> —. i -) 
( t 2 — l) n {t — g)~ n ~ m ~ 1 dt, 
C 
which will satisfy the equation (1); it is necessary to specify precisely the values of 
the multiple valued functions in the integral, in order that the integral may have a 
definite value. 
First, to define the meaning of (g 2 — 1)*™, let g — 1 = re' 0 , g + 1 = re 10 ', and 
suppose g to have moved from a point in the real axis for which g > 1, along any path 
up to its actual position ; we shall suppose that 6 = 0, 6' = 0, when g is in the real 
axis and greater than unity, the value of (g 2 — l)* Ml at any point will then be 
( rr ')im j cos — (0 O') + i sin — (0 + 6') j, where 6, 6' are the angles the lines joining 
g and 1, g and — 1 make with the real axis; 6 and 6 must be restricted each to lie 
