dz = adz + bdy
on the hypothesis that the condition
da/dy + bda/dz = db/dz + adb/dz.
It is natural to inquire what relations among x, y, z, if any, are
implied by, or are consistent with, a differential relation adx + bdy
+ cdx = 0, when a, b, c are unrestricted functions of x, y, z. This
problem leads to the consideration of the so-called _Pfaffian
Expression_ adx + bdy + cdz. It can be shown (1) if each of the
quantities db/dz - dc/dy, dc/dx - da/dz, da/dy - db/dz, which we shall
denote respectively by u23, u31, u12, be identically zero, the
expression is the differential of a function of x, y, z, equal to dt
say; (2) that if the quantity au23 + bu31 + cu12 is identically zero,
the expression is of the form udt, i.e. it can be made a perfect
differential by multiplication by the factor 1/u; (3) that in general
the expression is of the form dt + u1dt1. Consider the matrix of four
rows and three columns, in which the elements of the first row are a,
b, c, and the elements of the (r+1)-th row, for r = 1, 2, 3, are the
quantities u_r1, u_r2, u_r3, where u11 = u22 = u33 = 0. Then it is
easily seen that the cases (1), (2), (3) above correspond respectively
to the cases when (1) every determinant of this matrix of two rows and
columns is zero, (2) every determinant of three rows and columns is
zero, (3) when no condition is assumed. This result can be generalized
as follows: if a1, ... an be any functions of x1, ... xn, the
so-called Pfaffian expression a1dx1 + ... + a_ndx_n can be reduced to
one or other of the two forms
u1dt1 + ... + u_kdt_k, dt + u1dt1 + ... + u_k-1 dt_k-1,
wherein t, u1 ..., t1, ... are independent functions of x1, ... xn,
and k is such that in these two cases respectively 2k or 2k - 1 is the
rank of a certain matrix of n + 1 rows and n columns, that is, the
greatest number of rows and columns in a non-vanishing determinant of
the matrix; the matrix is that whose first row is constituted by the
quantities a1, ... an, whose s-th element in the (r+1)-th row is the
quantity da_r/dx_s - da_s/dx_r. The proof of such a reduced form can
be obtained from the two results: (1) If t be any given function of
the 2m independent variables u1, ... um, t1, ... tm, the expression dt
+ u1 dt1 + ... + u_m dt_m can be put into the form u'1 dt'1 + ... +
u'_mdt'_m. (2) If the quantities u1, ..., u1, t1, ... tm be connected
by a relation, the expression n1dt1 + ... + umdtm can be put into the
format dt' + u'1 dt'1 + ... + u'_m-1 dt'_m-1; and if the relation
connecting u1, um, t1, ... tm be homogeneous in u1, ... um, then t'
can be taken to be zero. These two results are deductions from the
theory of _contact transformations_ (see below), and their
demonstration requires, beside elementary algebraical considerations,
only the theory of complete systems of linear homogeneous partial
differential equations of the first order. When the existence of the
reduced form of the Pfaffian expression containing only independent
quantities is thus once assured, the identification of the number k
with that defined by the specified matrix may, with some difficulty,
be made _a posteriori_.
Single linear Pfaffian equation.
In all cases of a single Pfaffian equation we are thus led to consider
what is implied by a relation dt - u1dt1 - ... - umdtm = 0, in which
t, u1, ... um, t1 ..., tm are, except for this equation, independent
variables. This is to be satisfied in virtue of one or several
relations connecting the variables; these must involve relations
connecting t, t1, ... tm only, and in one of these at least t must
actually enter. We can then suppose that in one actual system of
relations in virtue of which the Pfaffian equation is satisfied, all
the relations connecting t, t1 ... tm only are given by
t = [psi](t_s+1 ... t_m), t1 = [psi]1(t_s+1 ... t_m), ... t_s = [psi]_s(t_s+1 ... t_m);
so that the equation
d[psi] - u1d[psi]1 - ... - u_s d[psi]_s - u_s+1 dt_s+1 - ... - u_m dt_m = 0
is identically true in regard to u1, ... um, t_s+1 ..., t_m; equating
to zero the coefficients of the differentials of these variables, we
thus obtain m - s relations of the form
d[psi]/dt_j - u1 d[psi]1/dt_j - ... - u_s d[psi]_s/dt_j - u_j = 0;
these m - s relations, with the previous s + 1 relations, constitute a
set of m + 1 relations connecting the 2m + 1 variables in virtue of
which the Pfaffian equation is satisfied independently of the form of
the functions [psi],[psi]1, ... [psi]s. There is clearly such a set
for each of the values s = 0, s = 1, ..., s = m - 1, s = m. And for
any value of s there may exist relations additional to the specified m
+ 1 relations, provided they do not involve any relation connecting t,
t1, ... tm only, and are consistent with the m - s relations
connecting u1, ... um. It is now evident that, essentially, the
integration of a Pfaffian equation
a1dx1 + ... + a_n dx_n = 0,
wherein a1, ... an are functions of x1, ... xn, is effected by the
processes necessary to bring it to its reduced form, involving only
independent variables. And it is easy to see that if we suppose this
reduction to be carried out in all possible ways, there is no need to
distinguish the classes of integrals corresponding to the various
values of s; for it can be verified without difficulty that by putting
t' = t - u1t1 - ... - u_s t_s, t'1 = u1, ... t'_s = u_s, u'1 = -t1,
..., u'_s = -t_s, t'_s+1 = t_s+1, ... t'_m = t_m, u'_s+1 = u_s+1, ...
u'_m = u_m, the reduced equation becomes changed to dt' - u'1 dt'1 -
... - u'_m dt'_m = 0, and the general relations changed to
t' = [psi](t'_s+l, ... t'_m) - t'1[psi]1(t'_s+1, ... t'_m) - ... -t'_s[psi]_s(t'_s+1, ... t'_m), = [phi],
say, together with u'1 = d[phi]/dt'1, ..., u'm = d[phi]/dt'm, which
contain only one relation connecting the variables t', t'1, ... t'm
only.
Simultaneous Pfaffian equations.
This method for a single Pfaffian equation can, strictly speaking, be
generalized to a simultaneous system of (n - r) Pfaffian equations dxj
= c_1j dx1 + ... + c_rj dxr only in the case already treated, when
this system is satisfied by regarding x_r+1, ... x_n as suitable
functions of the independent variables x1, ... xr; in that case the
integral manifolds are of r dimensions. When these are non-existent,
there may be integral manifolds of higher dimensions; for if
d[phi] = [phi]1 dx_r + ... + [phi]_r dx_r + [phi]_r+1(c_1,r+1 dx1 + ... + c_r,r+1 dx_r) + [phi]_r+2 ( ) + ...
be identically zero, then [phi][sigma] + c[sigma]_,r+1 [phi]_r+1 + ...
+ c[sigma]_,n [phi]_n = 0, or [phi] satisfies the r partial
differential equations previously associated with the total equations;
when these are not a complete system, but included in a complete
system of r - [mu] equations, having therefore n - r - [mu]
independent integrals, the total equations are satisfied over a
manifold of r + [mu] dimensions (see E. v. Weber, _Math. Annal._ 1v.
(1901), p. 386).
Contact transformations.
It seems desirable to add here certain results, largely of algebraic
character, which naturally arise in connexion with the theory of
contact transformations. For any two functions of the 2n independent
variables x1, ... xn, p1, ... pn we denote by ([phi][psi]) the sum of
the n terms such as d[phi]d[psi]/dp_idx_i - d[psi]d[phi]/dp_idx_i. For
two functions of the (2n + 1) independent variables z, x1, ... xn, p1,
... pn we denote by [phi][psi] the sum of the n terms such as
d[phi] /d[psi] d[psi]\ d[psi] /d[phi] d[phi]\
------( ------ + p_i------ ) - ------( ------ + p_i------ ).
dpi \ dxi dz / dpi \ dxi dz /
It can at once be verified that for any three functions
[f[[phi][psi]]] + [[phi][psi]f]] + [[psi][f[phi]]] = df/dz
[[phi][psi]] + d[phi]/dz [[psi]f] + d[psi]/dz [f[phi]], which when f,
[phi],[psi] do not contain z becomes the identity (f([phi][psi])) +
(phi([psi]f)) + ([psi](f[phi])) = 0. Then, if X1, ... Xn, P1, ... Pn
be such functions Of x1, ... xn, p1 ... pn that P1 dX1 + ... + Pn dXn
is identically equal to p1dx1 + ... + pn dxn, it can be shown by
elementary algebra, after equating coefficients of independent
differentials, (1) that the functions X1, ... Pn are independent
functions of the 2n variables x1, ... pn, so that the equations x'i =
Xi, p'i = Pi can be solved for x1, ... xn, p1, ... pn, and represent
therefore a transformation, which we call a homogeneous contact
transformation; (2) that the X1, ... Xn are homogeneous functions of
p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous
functions of p1, ... pn of dimension one, and the 1/2n(n - 1) relations
(Xi Xj) = 0 are verified. So also are the n^2 relations (Pi Xi) = 1,
(Pi Xj) = 0, (Pi Pj) = 0. Conversely, if X1, ... Xn be independent
functions, each homogeneous of zero dimension in p1, ... pn satisfying
the 1/2n(n - 1) relations (Xi Xj) = 0, then P1, ... Pn can be uniquely
determined, by solving linear algebraic equations, such that P1 dX1 +
... + Pn dXn = p1 dx1 + ... + pn dxn. If now we put n + 1 for n, put z
for x_n+1, Z for X_n+1, Qi for -Pi/P_n+1, for i = 1, ... n, put qi for
-p_i/p_n+1 and [sigma] for q_n+1/Q_n+1, and then finally write P1, ...
Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following
results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1,
... pn, such that the expression dZ - P1 dX1 - ... - Pn dXn is
identically equal to [sigma](dz - p1 dx1 - ... - pn dxn), and [sigma]
not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are
independent functions of z, x1, ... xn, p1, ... pn, so that the
equations z' = Z, x'i = Xi, p'i = Pi can be solved for z, x1, ... xn,
p1, ... pn and determine a transformation which we call a
(non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify
the 1/2n(n + 1) identities [Z Xi] = 0, [Xi Xj] = 0. And the further
identities
[Pi Xi] = [sigma], [Pi Xj] = 0, [Pi Z] = [sigma]Pi, [Pi Pj] = 0,
dZ dXi dPi
[Z[sigma]] = [sigma]-- - [sigma]^2, [Xi [sigma]] = [sigma]---, [Pi [sigma]] = [sigma]---
dz dz dz
are also verified. Conversely, if Z, x1, ... Xn be independent
functions satisfying the identities [Z Xi] = 0, [Xi Xj] = 0, then
[sigma], other than zero, and P1, ... Pn can be uniquely determined,
by solution of algebraic equations, such that
dZ - P1 dX1 - ... - Pn dXn = [sigma](dz - p1 dx1 - ... - p_n dx_n).
Finally, there is a particular case of great importance arising when
[sigma] = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pn
be 2n + 1 functions of the 2n independent variables x1, ... xn, p1,
... pn, satisfying the identity
dU + P1 dx1 + ... + Pn dXn = p1 dx1 + ... + p_n dx_n,
then the 2n functions P1, ... Pn, X1, ... Xn are independent, and we
have
(Xi Xj) = 0, (Xi U) = [delta]Xi, (Pi Xi) = 1, (Pi Xj) = 0, (Pi Pj ) = 0, (Pi U) + Pi = [delta]Pi,
where [delta] denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1,
... Xn be independent functions of x1, ... xn, p1, ... pn, such that
(Xi Xj) = 0, then U can be found by a quadrature, such that
(Xi U) = [delta]Xi;
and when Xi, ... Xn, U satisfy these 1/2n(n + 1) conditions, then P1,
... Pn can be found, by solution of linear algebraic equations, to
render true the identity dU + P1 dX1 + ... + Pn dXn = p1 dx1 + ... +
pn dxn; (3) Functions X1, ... Xn, P1, ... Pn can be found to satisfy
this differential identity when U is an arbitrary given function of
x1, ... xn, p1, ... pn; but this requires integrations. In order to
see what integrations, it is only necessary to verify the statement
that if U be an arbitrary given function of x1, ... xn, p1, ... pn,
and, for r < n, X1, ... Xr be independent functions of these
variables, such that (X_[sigma] U) = [delta]X_[sigma], (X_[rho]
X_[sigma]) = 0, for [rho], [sigma] = 1 ... r, then the r + 1
homogeneous linear partial differential equations of the first order
(Uf) + [delta]f = 0, (X[rho]f) = 0, form a complete system. It will be
seen that the assumptions above made for the reduction of Pfaffian
expressions follow from the results here enunciated for contact
transformations.
Partial differential equation of the first order.
Meaning of a solution of the equation.
We pass on now to consider the solution of any partial differential
equation of the first order; we attempt to explain certain ideas
relatively to a single equation with any number of independent variables
(in particular, an ordinary equation of the first order with one
independent variable) by speaking of a single equation with two
independent variables x, y, and one dependent variable z. It will be
seen that we are naturally led to consider systems of such simultaneous
equations, which we consider below. The central discovery of the
transformation theory of the solution of an equation F(x, y, z, dz/dx,
dz/dy) = 0 is that its solution can always be reduced to the solution of
partial equations which are _linear_. For this, however, we must regard
dz/dx, dz/dy, during the process of integration, not as the differential
coefficients of a function z in regard to x and y, but as variables
independent of x, y, z, the too great indefiniteness that might thus
appear to be introduced being provided for in another way. We notice
that if z = [psi](x, y) be a solution of the differential equation, then
dz = dxd[psi]/dx + dyd[psi]/dy; thus if we denote the equation by F(x,
y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every
solution, any solution such as z = [psi](x, y) will necessarily be
associated with the equations p = dz/dx, q = dz/dy, and z will satisfy
the equation in its original form. We have previously seen (under
_Pfaffian Expressions_) that if five variables x, y, z, p, q, otherwise
independent, be subject to dz - pdx - qdy = 0, they must in fact be
subject to at least three mutual relations. If we associate with a point
(x, y, z) the plane