- [xi])^-1 [psi]1 of the [eta] equation will
give an integral of the original equation containing log (x - [xi]);
if r2 - r1 be an integer, and therefore a negative integer, the same
will be true, unless in [psi]1 the term in (x - [xi])^(r1 - r2) be
absent; if neither of these arise, the original equation will have an
integral (x -[xi])^r2 [phi]2. The [eta] equation can now, by means of
the one integral of it belonging to the index r2 - r1 - 1, be
similarly reduced to one of order n - 2, and so on. The result will be
that stated above. We shall say that an equation of the form in
question is _regular_ about x = [xi].

Fuchsian equations.

Equation of the second order.

We may examine in this way the behaviour of the integrals at all the
points at which any one of the rational functions a1 ... an becomes
infinite; in general we must expect that beside these the value x =
[oo] will be a singular point for the solutions of the differential
equation. To test this we put x = 1/t throughout, and examine as
before at t = 0. For instance, the ordinary linear equation with
constant coefficients has no singular point for finite values of x; at
x = [oo] it has a singular point and is not regular; or again,
Bessel's equation x^2 + xy' + (x^2 - n^2)y = 0 is regular about x = 0,
but not about x = [oo]. An equation regular at all the finite
singularities and also at x = [oo] is called a Fuchsian equation. We
proceed to examine particularly the case of an equation of the second
order

y" + ay' + by = 0.

Putting x = 1/t, it becomes

d^2y/dt^2 + (2t^-1 - at^-2)dy/dt + bt^-4 y = 0,

which is not regular about t = 0 unless 2 - at^-1 and bt^-2, that is,
unless ax and bx^2 are finite at x =[oo]; which we thus assume; putting
y = t^r(1 + A1t + ... ), we find for the index equation at x =
[inifinity] the equation r(r - 1) + r(2 - ax)_0 + (bx^2)_0 = 0. If
there be finite singular points at [xi]1, ... [xi]m, where we assume
m>1, the cases m = 0, m = 1 being easily dealt with, and if [phi](x) =
(x - [xi]1) ... (x -[xi]m), we must have a.[phi](x) and b.[[phi](x)]^2
finite for all finite values of x, equal say to the respective
polynomials [psi](x) and [theta](x), of which by the conditions at x =
[oo] the highest respective orders possible are m - 1 and 2(m - 1).
The index equation at x = [xi]1 is r(r - 1) +
r[psi]([xi]1)/[phi]'([xi]1) + [theta]([xi]