x = xo + t, y = yo + mt, and regard m as constant, we shall in fact be considering the section of the surface by a fixed plane y−yo = m(x−xo); along this section dz = dt(a + bm); if we then integrate the equation dx/dt = a + bm, where a, b are expressed as functions of m and t, with m kept constant, finding the solution which reduces to zo for t = 0, and in the result again replace m by (y−yo)/(x−xo), we shall have the surface in question. In the general case the equations
dxj＝c1j dx1 + . .crjdxr
similarly determine through an arbitrary point x1ᵒ, . . . xnᵒ Mayer’s method
of integration. a planar manifold of r dimensions in space of n dimensions, and when the conditions of integrability are satisfied, every direction in this manifold through this point is tangent to the manifold of r dimensions, expressed by ωr+1 = x0r+1, . . . ωn = xnᵒ, which satisfies the equations and passes through this point. If we put x1−x1ᵒ = t, x2−x2ᵒ = m2t, ... xr−xrᵒ = mrt, and regard m2, ... mr as fixed, the (n−r) total equations take the form dxj /dt = c1j + m2c2j + ... + mrcrj, and their integration is equivalent to that of the single partial equation
in the n−r + 1 variables t, xr+1, ... xn. Determining the solutions Ωr+1, ... Ωn which reduce to respectively xr+1, ... xn when t = 0, and substituting t = x1−x1ᵒ, m2 = (x2−x2ᵒ)/(x1−x1ᵒ), ... mr = (xr−xrᵒ)/(x1−x1ᵒ), we obtain the solutions of the original system of partial equations previously denoted by ωr+1, . . . ωn. It is to be remarked, however, that the presence of the fixed parameters m2, ... mr in the single integration may frequently render it more difficult than if they were assigned numerical quantities.
We have above considered the integration of an equation
dz＝adz + bdy
on the hypothesis that the condition
da/dy + bda/dz＝db/dz + adb/dz.
It is natural to inquire what relations among x, y, z, if any, are implied by, or are consistent with, a differential relation adx + bdy + cdx = 0, when a, b, c are unrestricted functions of x, y, z. This problem leads to the consideration of the so-called Pfaffian Expression adx + bdy + cdz. It can be shown (1) if each of the Pfaffian Expressions.quantities db/dz−dc/dy, dc/dx−da/dz, da/dy−db/dz, which we shall denote respectively by u23, u31, u12, be identically zero, the expression is the differential of a function of x, y, z, equal to dt say; (2) that if the quantity au23 + bu31 + cu12 is identically zero, the expression is of the form udt, i.e. it can be made a perfect differential by multiplication by the factor 1/u; (3) that in general the expression is of the form dt + u1dt1. Consider the matrix of four rows and three columns, in which the elements of the first row are a, b, c, and the elements of the (r + 1)-th row, for r = 1, 2, 3, are the quantities ur1, ur2, ur3, where u11 = u22 = u33 = 0. Then it is easily seen that the cases (1), (2), (3) above correspond respectively to the cases when (1) every determinant of this matrix of two rows and columns is zero, (2) every determinant of three rows and columns is 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 + ... + andxn can be reduced to one or other of the two forms
u1dt1 + ... + ukdtk, dt + u1dt1 + ... + uk−1dtk−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 dar/dxs−das/dxr. 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 + u1dt1 + ... + umdtm can be put into the form u′1dt′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′1dt′1 + ... + u′m−1dt′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.
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 Single linear
Pfaffian equation. 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 satisfied, all the relations connecting t, t1 . . . tm only are given by
t＝ψ(ts+1 ... tm), t1＝ψ1(ts+1 ... tm), ... ts＝ψs(ts+1 ... tm);
so that the equation
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( ) + ...
be identically zero, then φσ + cσ,r+1φr+1 + ... + cσ,nφn ≈ 0, or φ satisfies the r partial differential equations previously associated with the total equations; when these are not a complete system, but included in a complete system of r−μ equations, having therefore n−r−μ independent integrals, the total equations are satisfied over a manifold of r + μ dimensions (see E. v. Weber, Math. Annal. 1v. (1901), p. 386).
It seems desirable to add here certain results, largely of algebraic character, which naturally arise in connexion with the theory of contact transformations. For any two functions of the 2n Contact transformations. independent variables x1, ... xn, p1, ... pn we denote by (φψ) the sum of the n terms such as dφdψ/dpidxi−dψdφ/dpidxi For two functions of the (2n + 1) independent variables z, x1, ... xn, p1, ... pn we denote by φψ the sum of the n terms such as
It can at once be verified that for any three functions [ƒ[φψ]] + [φ[ψƒ]] + [ψ[ƒφ]] = dƒ/dz [φψ] + dφ/dz [ψƒ] + dψ/dz [ƒφ], which when ƒ, φ,ψ do not contain z becomes the identity (ƒ(φψ)) + (φ(ψƒ)) + (ψ(ƒφ)) = 0.Then, if X1, ... Xn, P1, ... Pn be such functions Of x1, ... xn, p1 ... pn that P1dX1 + ... + PndXn is identically equal to p1dx1 + ... + pndxn, it can be shown by elementary algebra, after equating coefficients of independent differentials, (1) that the functions X1, ... Pn are independent functions of the 2n variables x1, ... pn, so that the equations x′i = Xi, p′i = Pi can be solved for x1, ... xn, p1, ... pn, and represent therefore a transformation, which we call a homogeneous contact transformation; (2) that the X1, ... Xn are homogeneous functions of p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous functions of p1, ... pn of dimension one, and the 12n(n−1) relations (XiXj) = 0 are verified. So also are the n² relations (PiXi = 1, (PiXj) = 0, (PiPj) = 0. Conversely, if X1, ... Xn be independent functions, each homogeneous of zero dimension in p1, ... pn satisfying the 12n(n−1) relations (XiXj) = 0, then P1, ... Pn can be uniquely determined, by solving linear algebraic equations, such that P1dX1 + ... + PndXn = p1dx1 + ... + pndxn. If now we put n + 1 for n, put z for xn+1, Z for Xn+1, Qi for -Pi/Pn+1, for i = 1, ... n, put qi for -pi/pn+1 and σ for qn+1/Qn+1, and then finally write P1, ... Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1, ... pn, such that the expression dZ−P1dX1−...−PndXn is identically equal to σ(dz−p1dx1−...−pndxn), and σ not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are independent functions of z, x1, ... xn, p1, ... pn, so that the equations z′ = Z, x′i = Xi, p′i = Pi can be solved for z, x1, ... xn, p1, ... pn and determine a transformation which we call a (non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify the 12n(n + 1)
