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
| dφ | ( | dψ | + pi | dψ | ) − | dψ | ( | dφ | pi | dφ | ). |
| dpi | dxi | dz | dpi | dxi | dz |
It can at once be verified that for any three functions [ƒ[φψ] + [φ[ψƒ] + [psi[ƒφ] = dƒ/dz [φψ] + dφ/dz [ψƒ] + dψ/dz [ƒφ], which when ƒ, φ,ψ do not contain z becomes the identity (ƒ(φψ)) + (phi(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 ½n(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 ½n(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 ½n(n + 1) identities [ZXi] = 0, [XiXj] = 0. And the further identities
[PiXi] = σ, [PiXj] = 0, [PiZ] = σPi, [PiPj] = 0,
| [Zσ] = σ | dZ | − σ², [Xiσ] = σ | dXi | , [Piσ] = | dPi |
| dz | dz | dz |
are also verified. Conversely, if Z, x1, ... Xn be independent functions satisfying the identities [ZXi] = 0, [XiXj] = 0, then σ, other than zero, and P1, ... Pn can be uniquely determined, by solution of algebraic equations, such that
dZ − P1dX1 − ... − PndXn = σ(dz − p1dx1 − ... − pndxn).
Finally, there is a particular case of great importance arising when σ = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pn be 2n + 1 functions of the 2n independent variables x1, ... xn, p1, ... pn, satisfying the identity
dU + P1dx1 + ... + PndXn = p1dx1 + ... + pndxn,