by an operation of order
one only. The three functions F, G, H thus identically satisfy the
relations [FG] = [GH] = [FH] = 0. The [oo]^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 [sigma] and
two functions A, C, such that

dG - AdF - CdH = [sigma](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 [Ff] = 0,
whereby the characteristic chains are entirely determined.

The single equation F = 0 and Pfaffian formulations.

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 [Ff] = 0, we have, if F
= 0 give p = [psi](x, y, z, q), the equation

/df df\ d[psi] /df df\ /d[psi] d[psi]\ df
[Sigma]f = - ( -- + [psi] -- ) + -