m is satisfied by regarding x_r+1, ... x_n 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

d[phi] = [phi]1 dx_r + ... + [phi]_r dx_r + [phi]_r+1(c_1,r+1 dx1 + ... + c_r,r+1 dx_r) + [phi]_r+2 ( ) + ...

be identically zero, then [phi][sigma] + c[sigma]_,r+1 [phi]_r+1 + ...
+ c[sigma]_,n [phi]_n = 0, or [phi] 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 - [mu] equations, having therefore n - r - [mu]
independent integrals, the total equations are satisfied over a
manifold of r + [mu] dimensions (see E. v. Weber, _Math. Annal._ 1v.
(1901), p. 386).

Contact transformations.

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 independent
variables x1, ... xn, p1, ... pn we denote by ([phi][psi]) the sum of
the n terms such as d[phi]d[psi]/dp_idx_i - d[psi]d[phi]/dp_idx_i. For
two functions of the (2n + 1) independent variables z, x1, ... xn, p1,
... pn we denote by [phi][psi] the sum of the n terms such as

d[phi] /d[psi] d[psi]\ d[psi] /d[phi] d[phi]\
------( ------ + p_i------ ) - ------( ------ + p_i------ ).
dpi \ dxi dz / dpi \ dxi dz /

It can at once be verified that for any three functions
[f[[phi][psi]]] + [[phi][psi]f]] + [[psi][f[phi]]] = df/dz
[[phi][psi]] + d[phi]/dz [[psi]f] + d[psi]/dz [f[phi]], which when f,
[phi],[psi] do not contain z becomes the identity (f([phi][psi])) +
(phi([psi]f)) + ([psi](f[phi])) = 0. Then, if X1, ... Xn, P1, ... Pn
be such functions Of x1, ... xn, p1 ... pn that P1 dX1 + ... + Pn dXn
is identically equal to p1dx1 + ... + pn dxn, 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 f