unctions of
p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous
functions of p1, ... pn of dimension one, and the 1/2n(n - 1) relations
(Xi Xj) = 0 are verified. So also are the n^2 relations (Pi Xi) = 1,
(Pi Xj) = 0, (Pi Pj) = 0. Conversely, if X1, ... Xn be independent
functions, each homogeneous of zero dimension in p1, ... pn satisfying
the 1/2n(n - 1) relations (Xi Xj) = 0, then P1, ... Pn can be uniquely
determined, by solving linear algebraic equations, such that P1 dX1 +
... + Pn dXn = p1 dx1 + ... + pn dxn. If now we put n + 1 for n, put z
for x_n+1, Z for X_n+1, Qi for -Pi/P_n+1, for i = 1, ... n, put qi for
-p_i/p_n+1 and [sigma] for q_n+1/Q_n+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 - P1 dX1 - ... - Pn dXn is
identically equal to [sigma](dz - p1 dx1 - ... - pn dxn), and [sigma]
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 1/2n(n + 1) identities [Z Xi] = 0, [Xi Xj] = 0. And the further
identities

[Pi Xi] = [sigma], [Pi Xj] = 0, [Pi Z] = [sigma]Pi, [Pi Pj] = 0,

dZ dXi dPi
[Z[sigma]] = [sigma]-- - [sigma]^2, [Xi [sigma]] = [sigma]---, [Pi [sigma]] = [sigma]---
dz dz dz

are also verified. Conversely, if Z, x1, ... Xn be independent
functions satisfying the identities [Z Xi] = 0, [Xi Xj] = 0, then
[sigma], other than zero, and P1, ... Pn can be uniquely determined,
by solution of algebraic equations, such that

dZ - P1 dX1 - ... - Pn dXn = [sigma](dz - p1 dx1 - ... - p_n dx_n).

Finally, there is a particular case of great importance arising when
[sigma] = 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 + P1 dx1 + ... + Pn dXn = p1 dx1 + ... + p_n dx_n,

then the 2n functions P1, ... Pn, X1, ... Xn ar