identically.

Passing now to the case of three variables, suppose that the Jacobian
determinant of the three functions F, u, v in regard to x, y, z is
identically zero. We prove that if u, v are not themselves
functionally connected, F is expressible as a function of u and v.
Suppose first that the minors of the elements of dPF/dPx, dPF/dPy,
dPF/dPz in the determinant are all identically zero, namely the three
determinants such as

dPu dPv dPu dPv
--- --- - --- ---;
dPy dPz dPz dPy

then by the case of two variables considered above there exist three
functional relations. [psi]1(u, v, x) = 0, [psi]2(u, v, y) = 0,
[psi]3(u, v, z) = 0, of which the first, for example, follows from the
vanishing of

dPu dPv dPu dPv
--- --- - --- ---.
dPy dPz dPz dPy

We cannot assume that x is absent from [psi]1, or y from [psi]2, or z
from [psi]3; but conversely we cannot simultaneously have x entering
in [psi]1, and y in [psi]2, and z in [psi]3, or else by elimination of
u and v from the three equations [psi]1 = 0, [psi]2 = 0, [psi]3 = 0,
we should find a necessary relation connecting the three independent
quantities x, y, z; which is absurd. Thus when the three minors of
dPF/dPx, dPF/dPy, dPF/dPz in the Jacobian determinant are all zero,
there exists a functional relation connecting u and v only. Suppose no
such relation to exist; we can then suppose, for example, that

dPu dPv dPu dPv
--- --- - --- ---
dPy dPz dPz dPy

is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we
can express y and z in terms of u, v, and x (the attempt to do this
could only fail by leading to a relation connecting u, v and x, and
the existence of such a relation would involve that the determinant

dPu dPv dPu dPv
--- --- - --- ---
dPy dPz dPz dPy

was zero), and so write F in the form F(x, y, z) = [Phi](u, v, x). We
then have

dPF dP[Phi] dPu dP[Phi] dPv dP[Phi] dPF dP[Phi] dPu dP[Phi] dPv dPF dP[Phi] dPu dP[Phi] dPv
--- = ------- --- + ------- --- + -------, --- = ------- --- + ------- ---, --- = ------- --- + ------- ---;
dPx dPu dPx dPv dPx dPx dPy dPu dPy dPv dPy dPz dPu dPz dPv dPz

thereby the Jacobian determinant of F, u, v is reduced to

dP[Phi] /dPu dPv dPu dPv\
-------( --- --- - --- --- );
dPx \dPy dPz dPz