dimensions
obtained by integrating the equations [X1f] = 0, ... [Xrf] = 0; or
having obtained one integral of this system other than X1, ... Xr, say
Xr+1, we may consider the system [X1f] = 0, ... [X_r+1 f] = 0, for
which, again, we have a choice; and at any stage we may use Mayer's
method and reduce the simultaneous linear equations to one equation
involving parameters; while if at any stage of the process we find
some but not all of the integrals of the simultaneous system, they can
be used to simplify the remaining work; this can only be clearly
explained in connexion with the theory of so-called function groups
for which we have no space. One result arising is that the
simultaneous system p1 = [phi]1, ... pr = [phi]r, wherein p1, ... pr
are not involved in [phi]1, ... [phi]r, if it satisfies the 1/2r(r - 1)
relations [pi - [phi]i, pj - [phi]j] = 0, has a solution z = [psi](x1,
... xn), p1 = d[psi]/dx1, ... pn = d[psi]/dxn, reducing to an
arbitrary function of x_r+1, ... xn only, when x1 = x1^0, ... xr =
xr^0 under certain conditions as to developability; a generalization
of the theorem for linear equations. The problem of integration of
this system is, as before, to put

dz - [phi]1dx1 - ... - [phi]_r dx_r - p_r+1 dx_r+1 - ... - p_n dx_n

into the form [sigma](d[zeta] - [omega]_r+1 + d[xi]_r+1 - ... -
[omega]_n d[xi]_n); and here [zeta], [xi]_r+1, ... [xi]_n,
[omega]_r+1, ... [omega]_n may be taken, as before, to be principal
integrals of a certain complete system of linear equations; those,
namely, determining the characteristic chains.

Equations of dynamics.

If L be a function of t and of the 2n quantities x1, ... xn, [.x]1,
... [.x]n, where [.x]i, denotes dxi/dt, &c., and if in the n equations

d / dL \ dL
--- (--------) = ----
dt \ dx_i / dx_i

we put p_i = dL/d[.x]_i, and so express [.x]1 , ... [.x]_n in terms of
t, x_i, ... x_n, p1, ... p_n, assuming that the determinant of the
quantities d^2L/dx_i d[.x]_j is not zero; if, further, H denote the
function of t, x1, ... xn, p1, ... pn, numerically equal to p1[.x]1 +
... + pn[.x]n - L, it is easy to prove that dpi/dt = -dH/dxi, dxi/dt =
dH/dp_i. These so-called _canonical_ equations form part of those for
the characteristic chains of the single partial equation dz/dt + H(t,
x1, ... xn, dz/dx1, ..., dz/dx_n) = 0, to which then the solution of
the origi