L1, where L1 is the absolute
value of the real part of [lambda]1; the same is true for the
[sigma]-region representing the branch [sigma]. The condition that the
s-region shall not overlap itself requires, then, L1 = 1. But,
further, we may form an infinite number of branches [sigma] =
[stigma]e^(2[pi]i[lambda])1, [sigma]1 = e^(2[pi]i[lambda])1, ... in
the same way, and the corresponding regions in the plane upon which
y2/y1 is represented will have a common point and each have an angle
2[pi]L1; if neither overlaps the preceding, it will happen, if L1 is
not zero, that at length one is reached overlapping the first, unless
for some positive integer [alpha] we have 2[pi][alpha]L1 = 2[pi], in
other words L1 = 1/a. If this be so, the branch [sigma]_a-1 =
[stigma]e^(2[pi]ia[lambda])1 will be represented by a region having
the angle at the common point common with the region for the branch
[stigma]; but not altogether coinciding with this last region unless
[lambda]1 be real, and therefore = [+-]1/a; then there is only a finite
number, a, of branches obtainable in this way by crossing the barrier
(-[oo], 0). In precisely the same way, if we had begun by taking the
quotient

[stigma]' = (x - 1)^[lambda]2 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x)/F([lambda], [mu], 1 - [lambda]2, 1 - x)

of the two solutions about x = 1, we should have found that x is not a
single-valued function of [stigma]' unless [lambda]2 is the inverse of
an integer, or is zero; as [stigma]' is of the form (A[stigma] +
B)/(C[stigma] + D), A, B, C, D constants, the same is true in our
case; equally, by considering the integrals about x = [oo] we find, as
a third condition necessary in order that x may be a single-valued
function of [stigma], that [lambda] - [mu] must be the inverse of an
integer or be zero. These three differences of the indices, namely,
[lambda]1, [lambda]2, [lambda] - [mu], are the quantities which enter
in the differential equation satisfied by x as a function of [stigma],
which is easily found to be

x111 3^2x^211
- ---- + -------- = 1/2(h - h1 - h2)x^-1 (x - 1)^-1 + 1/2h1 x^-2 + 1/2h2(x - 1)^-2,
x1^3 2x1^4

where x1 = dx/d[stigma], &c.; and h1 = 1 - y1^2, h2 = 1 - [lambda]2^2,
h3 = 1 - ([lambda] - [mu])^2. Into the converse question whether the
three conditions are sufficient to ensure (1) that the [stigma] region
c