)1/[[phi]'([xi]1)]^2 = 0, and
if [alpha]1, [beta]1 be its roots, we have [alpha]1 + [beta]1 = 1 -
[psi]([xi]1)/[phi]'([xi]1) and [alpha]1[beta]1 =
[theta]([xi])1/[[phi]'([xi]1)]^2. Thus by an elementary theorem of
algebra, the sum [Sigma](1 - [alpha]i - [beta]i)/(x - [xi]i), extended
to the m finite singular points, is equal to [psi](x)/[phi](x), and
the sum [Sigma](1 - [alpha]i - [beta]i) is equal to the ratio of the
coefficients of the highest powers of x in [psi](x) and [phi](x), and
therefore equal to 1 + [alpha] + [beta], where [alpha], [beta] are the
indices at x = [oo]. Further, if (x, 1)m-2 denote the integral part of
the quotient [theta](x)/[phi](x), we have
[Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i) equal to -(x,
1)_m-2 + [theta](x)/[phi](x), and the coefficient of x^m-2 in (x,
1)_m-2 is [alpha][beta]. Thus the differential equation has the form

y" + y'[Sigma](1 - [alpha]_i - [beta]_i)/(x - [xi]_i) + y[(x, 1)_m-2 +
[Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i)]/[phi](x) = 0.

If, however, we make a change in the dependent variable, putting y =
(x - [xi]1)^[alpha]1 ... (x - [xi]_m)^[alpha] m[eta], it is easy to
see that the equation changes into one having the same singular points
about each of which it is regular, and that the indices at x = [xi]_i
become 0 and [beta]_i - [alpha]_i, which we shall denote by [lambda]i,
for (x -[xi]_i)^[alpha]j can be developed in positive integral powers
of x -[xi]_i about x = [xi]_i; by this transformation the indices at x
= [oo] are changed to

[alpha] + [alpha]1 + ... + [alpha]m, [beta] + [beta]1 + ... + [beta]m

which we shall denote by [lambda], [mu]. If we suppose this change to
have been introduced, and still denote the independent variable by y,
the equation has the form

y" + y'[Sigma](1 - [lambda]_i)/(x - [xi]_i) + y(x, 1)_m-2/[phi](x) = 0,

while [lambda] + [mu] + [lambda]1 + ... + [lambda]_m = m - 1.
Conversely, it is easy to verify that if [lambda][mu] be the
coefficient of x^m-2 in (x, 1)_m-2, this equation has the specified
singular points and indices whatever be the other coefficients in (x,
1)_m-2.

Hypergeometric equation.

Thus we see that (beside the cases m = 0, m = 1) the "Fuchsian
equation" of the second order with _two_ finite singular points is
distinguished by the fact that it has a definite form when the
singular points and the indices