lopable about the values x1 = x1^0, ... xn=
xn^0, and that for these values the determinant just spoken of is not
zero. Then the main theorem is that the complete system of r
equations, and therefore the originally given set of k equations,
have in common n - r solutions, say [omega]r+1, ... [omega]n, which
reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are
respectively put x1^0, ... x_r^0; so that also the equations have in
common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an
arbitrary function [psi](x_r+1, ... x_n) which is developable about
x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1,
... [omega]_n). It is seen at once that this result is a
generalization of the theorem for r = 1, and its proof is conveniently
given by induction from that case. It can be verified without
difficulty (1) that if from the r equations of the complete system we
form r independent linear aggregates, with coefficients not
necessarily constants, the new system is also a complete system; (2)
that if in place of the independent variables x1, ... xn we introduce
any other variables which are independent functions of the former, the
new equations also form a complete system. It is convenient, then,
from the complete system of r equations to form r new equations by
solving separately for df/dx1, ..., df/dx_r; suppose the general
equation of the new system to be

Q_[sigma]f = df/dx_[sigma] + c_[sigma],r+1 df/dx_r+1 + ... + c_[sigma]n df/dx_n = 0 ([sigma] = 1, ... r).

Then it is easily obvious that the equation Q_[rho]Q_[sigma]f -
Q_[sigma]Q_[rho]f = 0 contains only the differential coefficients of f
in regard to x_r+1 ... xn; as it is at most a linear function of Q1f,
... Qrf, it must be identically zero. So reduced the system is called
a Jacobian system. Of this system Q1f=0 has n - 1 principal solutions
reducing respectively to x2, ... xn when

x1 = x1^0,

and its form shows that of these the first r - 1 are exactly x2 ...
xr. Let these n - 1 functions together with x1 be introduced as n new
independent variables in all the r equations. Since the first equation
is satisfied by n - 1 of the new independent variables, it will
contain no differential coefficients in regard to them, and will
reduce therefore simply to df/dx1 = 0, expressing that any common
solution of the r equations is a function only of the n -