ur possible
transformations of the dependent variable which will reduce one of the
indices at t = 0 to zero and one of the indices at t = 1 also to zero,
namely, we may reduce either [alpha]1 or [beta]1 at t = 0, and
simultaneously either [alpha]2 or [beta]2 at t = 1. Thus the
hypergeometric equation itself can be transformed into itself in 24
ways, and from the expression F([lambda], [mu], 1 - [lambda]1, x)
which satisfies it follow 23 other forms of solution; they involve
four series in each of the arguments, x, x-1, 1/x, 1/(1-x), (x-1)/x,
x/(x-1). Five of the 23 solutions agree with the fundamental solutions
already described about x = 0, x = 1, x = [oo]; and from the
principles by which these were obtained it is immediately clear that
the 24 forms are, in value, equal in fours.

Inversion. Modular functions.

The quarter periods K, K' of Jacobi's theory of elliptic functions, of
which K = [int] [0 to [pi]/2] (1 - h sin^2[theta])^-1/2 d[theta], and K'
is the same function of 1-h, can easily be proved to be the solutions
of a hypergeometric equation of which h is the independent variable.
When K, K' are regarded as defined in terms of h by the differential
equation, the ratio K'/K is an infinitely many valued function of h.
But it is remarkable that Jacobi's own theory of theta functions leads
to an expression for h in terms of K'/K (see FUNCTION) in terms of
single-valued functions. We may then attempt to investigate, in
general, in what cases the independent variable x of a hypergeometric
equation is a single-valued function of the ratio s of two independent
integrals of the equation. The same inquiry is suggested by the
problem of ascertaining in what cases the hypergeometric series
F([alpha], [beta], [gamma], x) is the expansion of an algebraic
(irrational) function of x. In order to explain the meaning of the
question, suppose that the plane of x is divided along the real axis
from -[oo] to 0 and from 1 to +[oo], and, supposing logarithms not to
enter about x = 0, choose two quite definite integrals y1, y2 of the
equation, say

y1 = F([lambda], [mu], 1-[lambda]1, x),
y2 = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x),

with the condition that the phase of x is zero when x is real and
between 0 and 1. Then the value of [sigma] = y2/y1 is definite for all
values of x in the divided plane, [sigma] being a