al
equation for V of order N with rational coefficients. This we denote
by F = 0. Further, each of [eta]1 ... [eta]n is expressible as a
linear function of V, dV/dx, ... d^(N-1)V/dx^(N-1), with rational
coefficients not involving any of the n^2 coefficients A_ij, since
otherwise V would satisfy a linear equation of order less than N,
which is impossible, as it involves (linearly) the n^2 arbitrary
coefficients Aij, which would not enter into the coefficients of the
supposed equation. In particular, y1 ,.. yn are expressible rationally
as linear functions of [omega], d[omega]/dx, ...
d^(N-1)[omega]/dx^(N-1), where [omega] is the particular function
[phi](y). Any solution W of the equation F = 0 is derivable from
functions [zeta]1, ... [zeta]n, which are linear functions of y1, ...
yn, just as V was derived from [eta]1, ... [eta]n; but it does not
follow that these functions [zeta]i, ... [zeta]n are obtained from y1,
... yn by a transformation of the linear group A, B, ... ; for it may
happen that the determinant d([zeta]1, ... [zeta]n)/(dy1, ... yn) is
zero. In that case [zeta]1, ... [zeta]n may be called a singular set,
and W a singular solution; it satisfies an equation of lower than the
N-th order. But every solution V, W, ordinary or singular, of the
equation F = 0, is expressible rationally in terms of [omega],
d[omega]/dx, ... d^(N-1)[omega]/dx^(N-1); we shall write, simply, V =
r([omega]). Consider now the rational irreducible equation of lowest
order, not necessarily a linear equation, which is satisfied by
[omega]; as y1, ... yn are particular functions, it may quite well be
of order less than N; we call it the _resolvent equation_, suppose it
of order p, and denote it by [gamma](v). Upon it the whole theory
turns. In the first place, as [gamma](v) = 0 is satisfied by the
solution [omega] of F = 0, all the solutions of [gamma](v) are
solutions F = 0, and are therefore rationally expressible by [omega];
any one may then be denoted by r([omega]). If this solution of F = 0
be not singular, it corresponds to a transformation A of the linear
group (A, B, ...), effected upon y1, ... yn. The coefficients Aij of
this transformation follow from the expressions before mentioned for
[eta]1 ... [eta]n in terms of V, dV/dx, d^2V/dx^2, ... by substituting V
= r([omega]); thus they depend on the p arbitrary parameters which
enter into the general expression