g respectively to x1^0,
... xn^0 when t = t^0, are those determined by continuation of these
series. If the result of solving these n equations for x1^0, ... xn^0
be written in the form [omega]1(x1, ... xnt) = x1^0, ... [omega]n(x1,
... xnt) = xn^0, it is at once evident that the differential equation

df/dt + [phi]1 df/dx1 + ... + [phi]n df/dxn = 0

possesses n integrals, namely, the functions [omega]1, ... [omega]n,
which are developable about the values (x1^0 ... xn^0t^0) and reduce
respectively to x1, ... xn when t = t^0. And in fact it has no other
integrals so reducing. Thus this equation also possesses a unique
integral reducing when t = t^0 to an arbitrary function [psi](x1, ...
xn), this integral being. [psi]([omega]1, ... [omega]n). Conversely
the existence of these _principal_ integrals [omega]1, ... [omega]n of
the partial equation establishes the existence of the specified
solutions of the ordinary equations dxi/dt = [phi]i. The following
sketch of the proof of the existence of these principal integrals for
the case n = 2 will show the character of more general investigations.
Put x for x - x^0, &c., and consider the equation a(xyt) df/dx +
b(xyt) df/dy = df/dt, wherein the functions a, b are developable about
x = 0, y = 0, t = 0; say

a(xyt) = a0 + ta1 + t^2a2/2! + ..., b(xyt) = b0 + tb1 + t^2b2/2! + ...,

so that

ad/dx + bd/dy = [delta]0 + t[delta]1 + 1/2t^2[delta]2 + ...,

where [delta] = a_r d/dx + b_r d/dy. In order that

f = p0 + tp1 + t^2p2/2! + ...

wherein p0, p1 ... are power series in x, y, should satisfy the
equation, it is necessary, as we find by equating like terms, that

p1 = [delta]0 p0, p2 = [delta]0 p1 + [delta]1 p0, &c.

and in general

p_s+1 = [delta]0 p_s + s1 [delta]1 p_s-1 + ... + [delta]_s p0,

where s_r = (s!)/(r!) (s - r)!

Now compare with the given equation another equation

A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,

wherein each coefficient in the expansion of either A or B is real and
positive, and not less than the absolute value of the corresponding
coefficient in the expansion of a or b. In the second equation let us
substitute a series

F = P0 + tP1 + t^2P2/2! + ...,

wherein the coefficients in P0 are real and positive, and each not
less than the absolute value of the corresponding coefficient in p0;
then putting [Delta]r = A_r d/dx + B_r d/dy we obtain necessary