a rational equation of lower order. From this it follows that if
an irreducible equation P = 0 have one solution satisfying another
rational equation Q = 0 of the same or higher order, then all the
solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we
can by differentiation express y^(k+1), y^(k+2), ... in terms of x, y,
y^(1), ... , y^(k), and so put the function Q rationally in terms of
these quantities only. It is sufficient, then, to prove the result
when the equation Q = 0 is of the same order as P = 0. Let both the
equations be arranged as integral polynomials in y^(k); their
algebraic eliminant in regard to y^(k) must then vanish identically,
for they are known to have one common solution not satisfying an
equation of lower order; thus the equation P = 0 involves Q = 0 for
all solutions of P = 0.

The variant function for a linear equation.

Now let y^(n) = [alpha]1y^(n-1) + ... + [alpha]_n y be a given
rational homogeneous linear differential equation; let y1, ... yn be n
particular functions of x, unconnected by any equation with constant
coefficients of the form c1y1 + ... + cnyn = 0, all satisfying the
differential equation; let [eta]1, ... [eta]n be linear functions of
y1, ... yn, say [eta]i = A_i1 y1 + ... + A_in yn, where the constant
coefficients Aij have a non-vanishing determinant; write ([eta]) =
A(y), these being the equations of a general linear homogeneous group
whose transformations may be denoted by A, B, .... We desire to form a
rational function [phi]([eta]), or say [phi](A(y)), of [eta]1, ...
[eta], in which the [eta]^2 constants Aij shall all be essential, and
not reduce effectively to a fewer number, as they would, for instance,
if the y1, ... yn were connected by a linear equation with constant
coefficients. Such a function is in fact given, if the solutions y1,
... yn be developable in positive integral powers about x = a, by
[phi]([eta]) = [eta]1 + (x - a)^n[eta]2 + ... + (x - a)^(n-1)n[eta]n.
Such a function, V, we call a _variant_.

The resolvent eqution.

Then differentiating V in regard to x, and replacing [eta]i^(n) by
its value a1[eta]^(n-1) + ... + an[eta], we can arrange dV/dx, and
similarly each of d^2/dx^2 ... d^NV/dx^N, where N = n^2, as a linear
function of the N quantities [eta]1, ... [eta]n, ... [eta]1^(n-1), ...
[eta]n^(n-1), and thence by elimination obtain a linear differenti