e independent, and we
have

(Xi Xj) = 0, (Xi U) = [delta]Xi, (Pi Xi) = 1, (Pi Xj) = 0, (Pi Pj ) = 0, (Pi U) + Pi = [delta]Pi,

where [delta] denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1,
... Xn be independent functions of x1, ... xn, p1, ... pn, such that
(Xi Xj) = 0, then U can be found by a quadrature, such that

(Xi U) = [delta]Xi;

and when Xi, ... Xn, U satisfy these 1/2n(n + 1) conditions, then P1,
... Pn can be found, by solution of linear algebraic equations, to
render true the identity dU + P1 dX1 + ... + Pn dXn = p1 dx1 + ... +
pn dxn; (3) Functions X1, ... Xn, P1, ... Pn can be found to satisfy
this differential identity when U is an arbitrary given function of
x1, ... xn, p1, ... pn; but this requires integrations. In order to
see what integrations, it is only necessary to verify the statement
that if U be an arbitrary given function of x1, ... xn, p1, ... pn,
and, for r < n, X1, ... Xr be independent functions of these
variables, such that (X_[sigma] U) = [delta]X_[sigma], (X_[rho]
X_[sigma]) = 0, for [rho], [sigma] = 1 ... r, then the r + 1
homogeneous linear partial differential equations of the first order
(Uf) + [delta]f = 0, (X[rho]f) = 0, form a complete system. It will be
seen that the assumptions above made for the reduction of Pfaffian
expressions follow from the results here enunciated for contact
transformations.

Partial differential equation of the first order.

Meaning of a solution of the equation.

We pass on now to consider the solution of any partial differential
equation of the first order; we attempt to explain certain ideas
relatively to a single equation with any number of independent variables
(in particular, an ordinary equation of the first order with one
independent variable) by speaking of a single equation with two
independent variables x, y, and one dependent variable z. It will be
seen that we are naturally led to consider systems of such simultaneous
equations, which we consider below. The central discovery of the
transformation theory of the solution of an equation F(x, y, z, dz/dx,
dz/dy) = 0 is that its solution can always be reduced to the solution of
partial equations which are _linear_. For this, however, we must regard
dz/dx, dz/dy, during the process of integration, not as the differential
coefficients of a function z in regard to x and y, but as variables
independent of x, y, z, the too great ind