is everywhere zero on this chain; further, suppose that each of F, dF/dp, ... , dF/dx + pdF/dz is developable about each element of this chain T, and that T is not a characteristic chain. Then consider the aggregate of the characteristic chains issuing from all the elements of T. The ∞² elements, consisting of the aggregate of these characteristic chains, satisfy F = 0, provided the chain connectivity T consists of elements satisfying F = 0; for each characteristic chain satisfies dF = 0. It can be shown that these chains are connected; in other words, that if x, y, z, p, q, be any element of one of these characteristic chains, not only is
dz/dt − pdx/dt − qdy/dt = 0,
as we know, but also U = dz/du − pdx/du − qdy/du is also zero. For we have
| dU | = | d | ( | dz | − p | dx | − q | dy | ) − | d | ( | dz | − p | dx | − q | dy | ) |
| dt | dt | du | du | du | du | dt | dt | dt |
| = | dp | dx | − | dp | dx | + | dq | dy | − | dq | dy | , |
| du | dt | dt | du | du | dt | dt | du |
which is equal to
| dp | dF | + | dx | ( | dF | + p | dF | ) + | dq | dF | + | dy | ( | dF | + q | dF | ) = − | dF | U. |
| du | dp | du | dx | dz | du | dq | du | dy | dz | dz |
As dF/dz is a developable function of t, this, giving
| U = U0 exp( − ∫ | t | dF | dt ), |
| t0 | dz |
shows that U is everywhere zero. Thus integrals of F = 0 are obtainable by considering the aggregate of characteristic chains issuing from arbitrary chain connectivities T satisfying F = 0; and such connectivities T are, it is seen at once, determinable without integration. Conversely, as such a chain connectivity T can be taken out from the elements of any given integral all possible integrals are obtainable in this way. For instance, an arbitrary curve in space, given by x0 = θ(u), y0 = φ(u), z0 = ψ(u), determines by the two equations F(x0, y0, z0, p0, q0) = 0, ψ′(u) = p0θ′(u) + q0φ′(u), such a chain connectivity T, through which there passes a perfectly definite integral of the equation F = 0. By taking ∞² initial chain connectivities T, as for instance by taking the curves x0 = θ, y0 = φ, z0 = ψ to be the ∞² curves upon an arbitrary surface, we thus obtain ∞² integrals, and so ∞4 elements satisfying F = 0. In general, if functions G, H, independent of F, be obtained, such that the equations F = 0, G = b, H = c represent an integral for all values of the constants b, c, these equations are said to constitute a complete integral. Then ∞4 elements satisfying F = 0 are known, and in fact every other form of integral can be obtained without further integrations.