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 Operations necessary for integration of F = a. 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 [Fƒ] = 0 where the notation is that explained above under [Contact Transformations]. One obvious integral is ƒ = F. Putting F = a, where a is arbitrary, and eliminating one of the independent variables, we can reduce this equation [Fƒ] = 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 [Fƒ] = 0, which is an operation of order three. Then it follows from the identity [ƒ[φψ] + [φ[ψƒ] + [ψ[ƒφ] = dƒ/dz [ψφ] + dφ/dz [ψƒ] + dψ/dz [ƒφ] before remarked, by putting φ = F, ψ = G, and then [Fƒ] = A(ƒ), [Gƒ] = B(ƒ), that AB(ƒ) − BA(ƒ) = dF/dz B(ƒ) − dG/dz A(ƒ), so that the two linear equations [Fƒ] = 0, [Gƒ] = 0 form a complete system; as two integrals F, G are known, they have a common integral H, independent of F, G, determinable by an operation of order one only. The three functions F, G, H thus identically satisfy the relations [FG] = [GH] = [FH] = 0. The ∞² elements satisfying F = a, G = b, H = c, wherein a, b, c are assigned constants, can then be seen to constitute an integral of F = a. For the conditions that a characteristic chain of G = b issuing from an element satisfying F = a, G = b, H = c should consist only of elements satisfying these three equations are simply [FG] = 0, [GH] = 0. Thus, starting from an arbitrary element of (F = a, G = b, H = c), we can single out a connectivity of elements of (F = a, G = b, H = c) forming a characteristic chain of G = b; then the aggregate of the characteristic chains of F = a issuing from the elements of this characteristic chain of G = b will be a connectivity consisting only of elements of
(F = a, G = b, H = c),
and will therefore constitute an integral of F = a; further, it will include all elements of (F = a, G = b, H = c). This result follows also from a theorem given under Contact Transformations, which shows, moreover, that though the characteristic chains of F = a are not determined by the three equations F = a, G = b, H = c, no further integration is now necessary to find them. By this theorem, since identically [FG] = [GH] = [FH] = 0, we can find, by the solution of linear algebraic equations only, a non-vanishing function σ and two functions A, C, such that
dG − AdF − CdH = σ(dz − pdz − qdy);
thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx + qdy and constitute a connectivity, which is therefore an integral of F = a. While, further, from the associated theorems, F, G, H, A, C are independent functions and [FC] = 0. Thus C may be taken to be the remaining integral independent of G, H, of the equation [Fƒ] = 0, whereby the characteristic chains are entirely determined.
When we consider the particular equation F = 0, neglecting the case when neither p nor q enters, and supposing p to enter, we may express p from F = 0 in terms of x, y, z, q, and then eliminate it from all other equations. Then instead of the equation [Fƒ] = 0, we have, if F = 0 give p = ψ(x, y, z, q), the equation
| Σƒ = − ( | dƒ | + ψ | dƒ | ) + | dψ | ( | dƒ | + q | dƒ | ) − ( | dψ | + q | dψ | ) | dƒ | = 0, |
| dx | dz | dq | dy | dz | dy | dz | dq |
moreover obtainable by omitting the term in dƒ/dp in [p − ψ, ƒ] = 0. Let x0, y0, z0, q0, be values about which the coefficients in The single equation F = 0 and Pfaffian formulations. this equation are developable, and let ζ, η, ω be the principal solutions reducing respectively to z, y and q when x = x0. Then the equations p = ψ, ζ = z0, η = y0, ω = q0 represent a characteristic chain issuing from the element x0, y0, z0, ψ0, q0; we have seen that the aggregate of such chains issuing from the elements of an arbitrary chain satisfying
dz0 = p0dx0 − q0dy0 = 0
constitute an integral of the equation p = ψ. Let this arbitrary chain be taken so that x0 is constant; then the condition for initial values is only