----- ( -- + 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.

Hence this equation involves dz - [psi]dx - qdy = 0, or we have

dz - [psi]dx - qdy = [sigma](d[zeta] - [omega]d[eta]),

where [sigma] is not zero. Conversely, the integration of p = [psi]
is, essentially, the problem of writing the expression dz - [psi]dx -
qdy in the form [sigma](d[zeta] - [omega]d[eta]), as must be possible
(from what was said under _Pfaffian Expressions_).

System of equations of the first order.

To integrate a system of simultaneous equations of the first order X1
= a1, ... Xr = ar in n independent variables x1, ... xn and one
dependent variable z, we write p1 for dz/dx1, &c., and attempt to find
n + 1 - r further functions Z, X_r+1 ... Xn, such that the equations Z
= a, Xi = ai,(i = 1, ... n) involve dz - p1dx1 - ... - pndxn = 0. By
an argument already given, the common integral, if existent, must be
satisfied by the equations of the characteristic chains of any one
equation Xi = ai; thus each of the expressions [Xi Xj] must vanish in
virtue of the equations expressing the integral, and we may without
loss of generality assume that each of the corresponding 1/2r(r - 1)
expressions formed from the r given differential equations vanishes in
virtue of these equations. The determination of the remaining n + 1 -
r functions may, as before, be made to depend on characteristic
chains, which in this case, however, are manifolds of r