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<, t1 = [psi]1(t_s+1 ... t_m), ... t_s = [psi]_s(t_s+1 ... t_m);
so that the equation
d[psi] - u1d[psi]1 - ... - u_s d[psi]_s - u_s+1 dt_s+1 - ... - u_m dt_m = 0
is identically true in regard to u1, ... um, t_s+1 ..., t_m; equating
to zero the coefficients of the differentials of these variables, we
thus obtain m - s relations of the form
d[psi]/dt_j - u1 d[psi]1/dt_j - ... - u_s d[psi]_s/dt_j - u_j = 0;
these m - s relations, with the previous s + 1 relations, constitute a
set of m + 1 relations connecting the 2m + 1 variables in virtue of
which the Pfaffian equation is satisfied independently of the form of
the functions [psi],[psi]1, ... [psi]s. There is clearly such a set
for each of the values s = 0, s = 1, ..., s = m - 1, s = m. And for
any value of s there may exist relations additional to the specified m
+ 1 relations, provided they do not involve any relation connecting t,
t1, ... tm only, and are consistent with the m - s relations
connecting u1, ... um. It is now evident that, essentially, the
integration of a Pfaffian equation
a1dx1 + ... + a_n dx_n = 0,
wherein a1, ... an are functions of x1, ... xn, is effected by the
processes necessary to bring it to its reduced form, involving only
independent variables. And it is easy to see that if we suppose this
reduction to be carried out in all possible ways, there is no need to
distinguish the classes of integrals corresponding to the various
values of s; for it can be verified without difficulty that by putting
t' = t - u1t1 - ... - u_s t_s, t'1 = u1, ... t'_s = u_s, u'1 = -t1,
..., u'_s = -t_s, t'_s+1 = t_s+1, ... t'_m = t_m, u'_s+1 = u_s+1, ...
u'_m = u_m, the reduced equation becomes changed to dt' - u'1 dt'1 -
... - u'_m dt'_m = 0, and the general relations changed to
t' = [psi](t'_s+l, ... t'_m) - t'1[psi]1(t'_s+1, ... t'_m) - ... -t'_s[psi]_s(t'_s+1, ... t'_m), = [phi],
say, together with u'1 = d[phi]/dt'1, ..., u'm = d[phi]/dt'm, which
contain only one relation connecting the variables t', t'1, ... t'm
only.
Simultaneous Pfaffian equations.
This method for a single Pfaffian equation can, strictly speaking, be
generalized to a simultaneous system of (n - r) Pfaffian equations dxj
= c_1j dx1 + ... + c_rj dxr only in the case already treated, when
this syste]]>
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