dPy/

by hypothesis the second factor of this does not vanish identically;
hence dP[Phi]/dPx = 0 identically, and [Phi] does not contain x; so
that F is expressible in terms of u, v only; as was to be proved.

_Part II.--General Theory._

Differential equations arise in the expression of the relations between
quantities by the elimination of details, either unknown or regarded as
unessential to the formulation of the relations in question. They give
rise, therefore, to the two closely connected problems of determining
what arrangement of details is consistent with them, and of developing,
apart from these details, the general properties expressed by them. Very
roughly, two methods of study can be distinguished, with the names
Transformation-theories, Function-theories; the former is concerned with
the reduction of the algebraical relations to the fewest and simplest
forms, eventually with the hope of obtaining explicit expressions of the
dependent variables in terms of the independent variables; the latter is
concerned with the determination of the general descriptive relations
among the quantities which are involved by the differential equations,
with as little use of algebraical calculations as may be possible. Under
the former heading we may, with the assumption of a few theorems
belonging to the latter, arrange the theory of partial differential
equations and Pfaff's problem, with their geometrical interpretations,
as at present developed, and the applications of Lie's theory of
transformation-groups to partial and to ordinary equations; under the
latter, the study of linear differential equations in the manner
initiated by Riemann, the applications of discontinuous groups, the
theory of the singularities of integrals, and the study of potential
equations with existence-theorems arising therefrom. In order to be
clear we shall enter into some detail in regard to partial differential
equations of the first order, both those which are linear in any number
of variables and those not linear in two independent variables, and also
in regard to the function-theory of linear differential equations of the
second order. Space renders impossible anything further than the
briefest account of many other matters; in particular, the theories of
partial equations of higher than the first order, the function-theory of
the singularities of ordinary equations not linear and the applications
to differential geometry, are tak