then the 2n functions P1, ... Pn, X1, ... Xn are independent, and we have
(XiXj) = 0, (XiU) = δXi, (PiXi) = 1, (PiXj) = 0, (PiPj) = 0, (PiU) + Pi = δPi,
where δ denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1, ... Xn be independent functions of x1, ... xn, p1, ... pn, such that (XiXj) = 0, then U can be found by a quadrature, such that
(XiU) = δXi;
and when Xi, ... Xn, U satisfy these ½n(n + 1) conditions, then P1, ... Pn can be found, by solution of linear algebraic equations, to render true the identity dU + P1dX1 + ... + PndXn = p1dx1 + ... + pndxn; (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 (XσU) = δXσ, (XρXσ) = 0, for ρ, σ = 1 ... r, then the r + 1 homogeneous linear partial differential equations of the first order (Uƒ) + δƒ = 0, (Xρƒ) = 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.
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 Partial differential equation of the first order. 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 indefiniteness that might thus appear to be introduced being provided for in another way. We notice that if z = ψ(x, y) be a solution of the differential equation, then dz = dxdψ/dx + dydψ/dy; thus if we denote the equation by F(x, y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every solution, any solution such as z = ψ(x, y) will necessarily be associated with the equations p = dz/dx, q = dz/dy, and z will satisfy the equation in its original form. We have previously seen (under Pfaffian Expressions) that if five variables x, y, z, p, q, otherwise independent, be subject to dz − pdx − qdy = 0, they must in fact be subject to at least three mutual relations. If we associate with a point (x, y, z) the plane
Z − z = p(X − x) + q(Y − y)
passing through it, where X, Y, Z are current co-ordinates, and call this association a surface-element; and if two consecutive elements of which the point(x + dx, y + dy, z + dz) of one lies on the plane of the other, for which, that is, the condition dz = pdx + qdy is satisfied, be said to be connected, and an infinity of connected elements following one another continuously be called a connectivity, then our statement is that a connectivity consists of not more than ∞² elements, the whole number of elements (x, y, z, p, q) that are possible being called ∞5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then to be understood to mean finding in all possible ways, from the ∞4 elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of ∞² elements forming a connectivity; or, more analytically, finding in all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q and independent of F = 0, so that the three relations together may involve
dz = pdx + qdy.
Such a set of three relations may, for example, be of the form z = ψ(x, y), p = dψ/dx, q = dψ/dy; but it may also, as another case, involve two relations z = ψ(y), x = ψ1(y) connecting x, y, z, the third relation being