0;
\dx' dz'/ \dy' dz'/

thus the equations above give [delta]x'dp' + [delta]y'dq' = 0, or the
tangent line of the plane curve, is, on the surface z' = [psi](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 [oo]^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 [oo]^3 characteristic chains, whose
aggregate may therefore be expected to exhaust the [oo]^4 elements
satisfying F = 0.

Complete integral constructed with characteristic chains.

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
functions of a parameter u, so that

U0 = dz0/du - p0dx0/du - q0dy0/du

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 [oo]^2 elements, consisting of the aggregate of these
characteristic chains, satisfy F = 0, provided the cha