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 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 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.
Operations necessary for integration of F = a.
In the foregoing discussion of the differential equations of a
characteristic chain, the denominators dF/dp, ... may be supposed to
be modified in form by means of F = 0 in any way conducive to a simple
integration. In the immediately following explanation of ideas,
however, we consider indifferently all equations F = constant; when a
function of x, y, z, p, q is said to be zero, it is meant that this is
so identically, not in virtue of F = 0; in other words, we consider
the integration of F = a, where a is an arbitrary constant. In the
theory of linear partial equations we have seen that the integration
of the equations of the characteristic chains, from which, as has just
been seen, that of the equation F = a follows at once, would be
involved in completely integrating the single linear homogeneous
partial differential equation of the first order [Ff] = 0 where the
notation is that explained above under CONTACT TRANSFORMATIONS. One
obvious integral is f = F. Putting F = a, where a is arbitrary, and
eliminating one of the independent variables, we can reduce this
equation [Ff] = 0 to one in four variables; and so on. Calling, then,
the determination of a single integral of a single homogeneous partial
differential equation of the first order in n independent variables,
_an operation of order_ n - 1, the characteristic chains, and
therefore the most general integral of F = a, can be obtained by
successive operations of orders 3, 2, 1. If, however, an integral of F
= a be represented by F = a, G = b, H = c, where b and c are arbitrary
constants, the expression of the fact that a characteristic chain of F
= a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] =
0, these three relations being identically true. Conversely, suppose
that an integral G, independent of F, has been obtained of the
equation [Ff] = 0, which is an operation of order three. Then it
follows from the identity [f[[phi][psi]]] + [[phi][[psi]f]] +
[[psi][f[phi]]] = df/dz [[psi][phi]] + d[phi]/dz [psif] + d[psi]/dz
[f[phi]] before remarked, by putting [phi] = F, [psi] = G, and then
[Ff] = A(f), [Gf] = B(f), that AB(f) - BA(f) = dF/dz B(f) - dG/dz
A(f), so that the two linear equations [Ff] = 0, [Gf] = 0 form a
complete system; as two integrals F, G are known, they have a common
integral H, independent of F, G, determinable 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] -- ) + ------ ( -- + q -- ) - ( ------ + q ------ ) -- = 0,
\dx dz/ dq \dy dz/ \ dy dz / dq
moreover obtainable by omitting the term in df/dp in [p-[psi], f] = 0.
Let x0, y0, z0, q0, be values about which the coefficients in this
equation are developable, and let [zeta], [eta], [omega] be the
principal solutions reducing respectively to z, y and q when x = x0.
Then the equations p = [psi], [zeta] = z0, [eta] = y0, [omega] = q0
represent a characteristic chain issuing from the element x0, y0, z0,
[psi]0, q0; we have seen that the aggregate of such chains issuing
from the elements of an arbitrary chain satisfying
dz0 = p0dx0 - q0dy0 = 0
constitute an integral of the equation p = [psi]. Let this arbitrary
chain be taken so that x0 is constant; then the condition for initial
values is only
dz0 - q0dy0 = 0,
and the elements of the integral constituted by the characteristic
chains issuing therefrom satisfy
d[zeta] - [omega]d[eta] = 0.