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 satisf