Operations necessary for integration of F = a.

In the foregoing discussion of the differential equations of a
characteristic chain, the denominators dF/dp, ... may be supposed to
be modified in form by means of F = 0 in any way conducive to a simple
integration. In the immediately following explanation of ideas,
however, we consider indifferently all equations F = constant; when a
function of x, y, z, p, q is said to be zero, it is meant that this is
so identically, not in virtue of F = 0; in other words, we consider
the integration of F = a, where a is an arbitrary constant. In the
theory of linear partial equations we have seen that the integration
of the equations of the characteristic chains, from which, as has just
been seen, that of the equation F = a follows at once, would be
involved in completely integrating the single linear homogeneous
partial differential equation of the first order [Ff] = 0 where the
notation is that explained above under CONTACT TRANSFORMATIONS. One
obvious integral is f = F. Putting F = a, where a is arbitrary, and
eliminating one of the independent variables, we can reduce this
equation [Ff] = 0 to one in four variables; and so on. Calling, then,
the determination of a single integral of a single homogeneous partial
differential equation of the first order in n independent variables,
_an operation of order_ n - 1, the characteristic chains, and
therefore the most general integral of F = a, can be obtained by
successive operations of orders 3, 2, 1. If, however, an integral of F
= a be represented by F = a, G = b, H = c, where b and c are arbitrary
constants, the expression of the fact that a characteristic chain of F
= a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] =
0, these three relations being identically true. Conversely, suppose
that an integral G, independent of F, has been obtained of the
equation [Ff] = 0, which is an operation of order three. Then it
follows from the identity [f[[phi][psi]]] + [[phi][[psi]f]] +
[[psi][f[phi]]] = df/dz [[psi][phi]] + d[phi]/dz [psif] + d[psi]/dz
[f[phi]] before remarked, by putting [phi] = F, [psi] = G, and then
[Ff] = A(f), [Gf] = B(f), that AB(f) - BA(f) = dF/dz B(f) - dG/dz
A(f), so that the two linear equations [Ff] = 0, [Gf] = 0 form a
complete system; as two integrals F, G are known, they have a common
integral H, independent of F, G, determinable