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.
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 d²L/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 original equations for x1 ... xn can be reduced. It may be shown (1) that if z = [psi](t, x1, ... xn, c1, .. cn) + c be a complete integral of this equation, then pi = d[psi]/dx_i, d[psi]/dc_i = e_i are 2n equations giving the solution of the canonical equations referred to, where c1 ... cn and e1, ... en are arbitrary constants; (2) that if xi = Xi(t, x^01, ... pn^0), pi=Pi(t, x1^0, ... p^0n) be the principal solutions of the canonical equations for t = t^0, and [omega] denote the result of substituting these values in p1dH/dp1 + ... + pndH/dpn - H, and [Omega] = [int] [t0 to t] [omega]dt, where, after integration, [Omega] is to be expressed as a function of t, x1, ... xn, x1^0, ... xn^0, then z = [Omega] + z^0 is a complete integral of the partial equation.
Application of theory of continuous groups to formal theories.
A system of differential equations is said to allow a certain continuous group of transformations (see GROUPS, THEORY OF) when the introduction for the variables in the differential equations of the new variables given by the equations of the group leads, for all values of the parameters of the group, to the same differential equations in the new variables. It would be interesting to verify in examples that this is the case in at least the majority of the differential equations which are known to be integrable in finite terms. We give a theorem of very general application for the case of a simultaneous complete system of linear partial homogeneous differential equations of the first order, to the solution of which the various differential equations discussed have been reduced. It will be enough to consider whether the given differential equations allow the infinitesimal transformations of the group.