[xi])^r1 [[chi]1 + [chi]2 log (x - [xi]) + [phi]1(log(x - [xi]))^2],
and so on. Here each of [phi]1, [psi]1, [chi]1, [chi]2, ... is a
series of positive and negative integral powers of x - [xi] in which
the number of negative powers may be infinite.

Regular equations.

It appears natural enough now to inquire whether, under proper
conditions for the forms of the rational functions a1, ... an, it may
be possible to ensure that in each of the series [phi]1, [psi]1,
[chi]1, ... the number of negative powers shall be finite. Herein
lies, in fact, the limitation which experience has shown to be
justified by the completeness of the results obtained. Assuming n
integrals in which in each of [phi]1, [psi]1, [chi]1 ... the number of
negative powers is finite, there is a definite homogeneous linear
differential equation having these integrals; this is found by forming
it to have the form

y'^n = (x - [xi])^-1 b1y'^(n-1) + (x - [xi])^-2 b2y'^(n-2) + ... +(x - [xi])^-n b_n y,

where b1, ... bn are finite for x = [xi]. Conversely, assume the
equation to have this form. Then on substituting a series of the form
(x - [xi])^r [1 + A1(x - [xi]) + A2(x - [xi])^2 + ... ] and equating
the coefficients of like powers of x-[xi], it is found that r must be
a root of an algebraic equation of order n; this equation, which we
shall call the index equation, can be obtained at once by substituting
for y only (x - [xi])^r and replacing each of b1, ... bn by their
values at x = [xi]; arrange the roots r1, r2, ... of this equation so
that the real part of ri is equal to, or greater than, the real part
of r_i+1, and take r equal to r1; it is found that the coefficients
A1, A2 ... are uniquely determinate, and that the series converges
within a circle about x = [xi] which includes no other of the points
at which the rational functions a1 ... an become infinite. We have
thus a solution H1 = (x -[xi])^r1 [phi]1 of the differential equation.
If we now substitute in the equation y = H1 f[eta]dx, it is found to
reduce to an equation of order n - 1 for [eta] of the form

[eta]'^(n-1) = (x - [xi])^-1 c1[eta]'^(n-2) + ... + (x-[xi])^(n-1) c_n-1 [eta],

where c1, ... c_n-1 are not infinite at x = [xi]. To this equation
precisely similar reasoning can then be applied; its index equation
has in fact the roots r2 - r1 - 1, ... , rn - r1 - 1; if r2 - r1 be
zero, the integral (x