z′ = z − p1x1 − ... − psxs (s = 1 or 2),
so that the solution becomes of a form z’ = ψ(x′y′), p′ = dψ/dx′, q′ = dψ/dy′, which then will identically satisfy the transformed equations F′ = 0, G′ = 0, H′ = 0. The equation F′ = 0, if x′, y′, z′ be regarded as fixed, states that the plane Z − z′ = p′(X − x′) + q′(Y − y′) is tangent to a certain cone whose vertex is (x′, y′, z′), the consecutive point (x′ + dx′, y′ + dy′, z′ + dz′) of the generator of contact being such that
| dx′/ | dF′ | = dy′/ | dF′ | = dz′/ ( p′ | dF′ | + q′ | dF′ | ). |
| dp′ | dq′ | dp′ | dq′ |
Passing in this direction on the surface z′ = ψ(x′, y′) the tangent plane of the surface at this consecutive point is (p′ + dp′, q′ + dq′), where, since F′(x′, y′, ψ, dψ/dx′, dψ/dy′) = 0 is identical, we have dx′ (dF′/dx′ + p′dF′/dz′) + dp′dF′/dp′ = 0. Thus the equations, which we shall call the characteristic equations,
| dx′/ | dF′ | = dy′/ | dF′ | = dz′/ ( p′ | dF′ | + q′ | dF′ | ) = dp′/ ( − | dF′ | − p′ | dF′ | ) = dq′/ ( − | dF′ | − q′ | dF′ | ) |
| dp′ | dq′ | dp′ | dq′ | dx′ | dz′ | dy′ | dz′ |
are satisfied along a connectivity of ∞¹ elements consisting of a curve on z′ = ψ(x′, y′) and the tangent planes of the surface along this curve. The equation F′ = 0, when p′, q′ are fixed, represents a curve in the plane Z − z′ = p′(X − x′) + q′(Y − y′) passing through (x′, y′, z′); if (x′ + δx′, y′ + δy′, z′ + δz′) be a consecutive point of this curve, we find at once
| δx′( | dF′ | + p′ | dF′ | ) + δy′( | dF′ | q′ | dF′ | ) = 0; |
| dx′ | dz′ | dy′ | dz′ |
thus the equations above give δx′dp′ + δy′dq′ = 0, or the tangent line of the plane curve, is, on the surface z′ = ψ(x′, y′), in a direction conjugate to that of the generator of the cone. Putting each of the fractions in the characteristic equations equal to dt, the equations enable us, starting from an arbitrary element x′0, y′0, z′0, p′0, q′0, about which all the quantities F′, dF′/dp′, &c., occurring in the denominators, are developable, to define, from the differential equation F′ = 0 alone, a connectivity of ∞¹ elements, which we call a characteristic chain; and it is remarkable that when we transform again to the original variables (x, y, z, p, q), the form of the differential equations for the chain is unaltered, so that they can be written down at once from the equation F = 0. Thus we have proved that the characteristic chain starting from any ordinary element of any integral of this equation F = 0 consists only of elements belonging to this integral. For instance, if the equation do not contain p, q, the characteristic chain, starting from an arbitrary plane through an arbitrary point of the surface F = 0, consists of a pencil of planes whose axis is a tangent line of the surface F = 0. Or if F = 0 be of the form Pp + Qq = R, the chain consists of a curve satisfying dx/P = dy/Q = dz/R and a single infinity of tangent planes of this curve, determined by the tangent plane chosen at the initial point. In all cases there are ∞³ characteristic chains, whose aggregate may therefore be expected to exhaust the ∞4 elements satisfying F = 0.
Consider, in fact, a single infinity of connected elements each satisfying F = 0, say a chain connectivity T, consisting of elements specified by x0, y0, z0, p0, q0, which we suppose expressed as Complete integral constructed with characteristic chains. functions of a parameter u, so that
U0 = dz0/du − p0dx0/du − q0dy0/du