in 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 dx dy\ d /dz dx dy\
-- = --( -- - p-- - q-- ) - --( -- - p-- - q-- )
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 dF\ dq dF dy /dF dF\ dF
-- -- + --( -- + p-- ) + -- -- + --( -- + q-- ) = - -- U.
du dp du \dx dz/ du dq du \dy dz/ dz

dF
As -- is a developable function of t, this, giving
dz
_
/ / t dF \
U = U_{0} exp( - | --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 = [theta](u), y0 = [phi](u), z0 = [psi](u), determines by
the two equations F(x0, y0, z0, p0, q0) = 0, [psi]'(u) = p0[theta]'(u)
+ q0[phi]'(u), such a chain connectivity T, through which there passes
a perfectly definite integral of the equation F = 0. By taking [oo]^2
initial chain connectivities T, as for instance by taking the curves
x0 = [theta], y0 = [phi], z0 = [psi] to be the [oo]^2 curves upon an
arbitrary surface, we thus obtain [oo]^2 integrals, and so [oo]^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 [oo]^4
elements satisfying F = 0 are known, and in fact every other form of
integral can be obtained without further integrations.