1 remaining
variables. Thereby the investigation of the common solutions is
reduced to the same problem for r - 1 equations in n - 1 variables.
Proceeding thus, we reach at length one equation in n - r + 1
variables, from which, by retracing the analysis, the proposition
stated is seen to follow.

System of total differential equations.

The analogy with the case of one equation is, however, still closer.
With the coefficients c_[sigma]j, of the equations Q_[sigma]f = 0 in
transposed array ([sigma] = 1, ... r, j = r + 1, ... n) we can put
down the (n - r) equations, dx_j = c1_j dx1 + ... + c_rj dx_r,
equivalent to the r(n - r) equations dx_j/dx_[sigma] = c_[sigma]r.
That consistent with them we may be able to regard x_r+1, ... x_n as
functions of x1, ... x_r, these being regarded as independent
variables, it is clearly necessary that when we differentiate
c_[sigma]j in regard to x_[rho] on this hypothesis the result should
be the same as when we differentiate c[rho]j, in regard to x[sigma] on
this hypothesis. The differential coefficient of a function f of x1,
... xn on this hypothesis, in regard to x_[rho]j is, however,

df/dx_[rho] + c_[rho],r+1 df/dx_r+1 + ... + c_[rho]n df/dx_n,

namely, is Q_[rho]f. Thus the consistence of the n - r total equations
requires the conditions Q_[rho]c_[sigma]j - Q_[sigma]c_[rho]j = 0,
which are, however, verified in virtue of Q[rho](Q[sigma][f]) -
Q_[sigma](Q_[rho]f) = 0. And it can in fact be easily verified that if
[omega]_r+1, ... [omega]_n be the principal solutions of the Jacobian
system, Q_[sigma]f = 0, reducing respectively to x_r+1, ... xn when x1
= x1^0, ... x_r = x_r^0, and the equations [omega]_r+1 = x_r+1^0, ...
[omega]_n = x_n^0 be solved for x_r+1, ... x_n to give x_j =
[psi]_j(x1, ... x_r, x_r+1^0, ... x_n^0), these values solve the total
equations and reduce respectively to x_r+1^0, ... x_n^0 when x1 = x1^0
... x_r = x_r^0. And the total equations have no other solutions with
these initial values. Conversely, the existence of these solutions of
the total equations can be deduced a priori and the theory of the
Jacobian system based upon them. The theory of such total equations,
in general, finds its natural place under the heading _Pfaffian
Expressions_, below.

Geometrical interpretation and solution.

Mayer's method of integration.

A practical method of reducing the solutio