dψ − u1dψ1 − ... − usdψs − us+1dts+1 − ... − umdtm = 0
is identically true in regard to u1, ... um, ts+1 ..., tm; equating to zero the coefficients of the differentials of these variables, we thus obtain m − s relations of the form
dψ/dtj − u1dψ1/dtj − ... − usdψs/dtj − uj = 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 ψ,ψ1, ... ψs. 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 + ... + andxn = 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 − ... − usts, t′1 = u1, ... t′s = us, u′1 = −t1, ..., u′s = −ts, t′s+1 = ts+1, ... t′m = tm, u′s+1 = us+1, ... u′m = um, the reduced equation becomes changed to dt′ − u′1dt′1 − ... − u′mdt′m = 0, and the general relations changed to
t′ = ψ(t′s+1, ... t′m) − t′1ψ1(t′s+1, ... t′m) − ... − t′sψs(t′s+1, ... t′m), = φ,
say, together with u′1 = dφ/dt′1, ..., u′m = dφ/dt′m, which contain only one relation connecting the variables t′, t′1, ... t′m only.
This method for a single Pfaffian equation can, strictly speaking, be generalized to a simultaneous system of (n − r) Pfaffian equations dxj = c1jdx1 + ... + crjdxr only in the case already treated, Simultaneous Pfaffian equations. when this system is satisfied by regarding xr+1, ... xn as suitable functions of the independent variables x1, ... xr; in that case the integral manifolds are of r dimensions. When these are non-existent, there may be integral manifolds of higher dimensions; for if
dφ = φ1dx1 + ... + φrdxr + φr+1(c1,r+1dx1 + ... + cr,r+1dxr) + φr+2( ) + ...