[lambda]1 is a positive integer, not zero, the second solution
about x = 0 persists as a solution, in accordance with the order of
arrangement of the roots of the index equation in our theory; the
first solution is then replaced by an integral polynomial of degree
-[lambda] or -[mu]1, when [lambda] or [mu] is one of the negative
integers 0, -1, -2, ..., 1 - [lambda]1, but otherwise contains a
logarithm. Similarly for the solutions about x = 1 or x = [oo]; it
will be seen below how the results are deducible from those for x = 0.

March of the Integral.

Denote now the solutions about x = 0 by u1, u2; those about x = 1 by
v1, v2; and those about x = [oo] by w1, w2; in the region (S0S1)
common to the circles S0, S1 of radius 1 whose centres are the points
x = 0, x = 1, all the first four are valid, and there exist equations
u1 =Av1 + Bv2, u2 = Cv1 + Dv2 where A, B, C, D are constants; in the
region (S1S) lying inside the circle S1 and outside the circle S0,
those that are valid are v1, v2, w1, w2, and there exist equations v1
= Pw1 + Qw2, v2 = Rw1 + Tw2, where P, Q, R, T are constants; thus
considering any integral whose expression within the circle S0 is au1
+ bu2, where a, b are constants, the same integral will be represented
within the circle S1 by (aA + bC)v1 + (aB + bD)v2, and outside these
circles will be represented by

[(aA + bC)P + (aB + bD)R]w1 + [(aA + bC)Q + (aB + bD)T]w2.

A single-valued branch of such integral can be obtained by making a
barrier in the plane joining [oo] to 0 and 1 to [oo]; for instance, by
excluding the consideration of real negative values of x and of real
positive values greater than 1, and defining the phase of x and x - 1
for real values between 0 and 1 as respectively 0 and [pi].

Transformation of the equation into itself.

We can form the Fuchsian equation of the second order with three
arbitrary singular points [xi]1, [xi]2, [xi]3, and no singular point
at x = [oo], and with respective indices [alpha]1, [beta]1, [alpha]2,
[beta]2, [alpha]3, [beta]3 such that [alpha]1 + [beta]1 + [alpha]2 +
[beta]2 + [alpha]3 + [beta]3 = 1. This equation can then be
transformed into the hypergeometric equation in 24 ways; for out of
[xi]1, [xi]2, [xi]3 we can in six ways choose two, say [xi]1, [xi]2,
which are to be transformed respectively into 0 and 1, by (x -
[xi]1)/(x - [xi]2) = t(t - 1); and then there are fo