equations of the same form as before, namely,

P1 = [Delta]0 P0, P2= [Delta]0 P1 + [Delta]1 P0, ...

and in general P_s+1 = [Delta]0 P_s, + s1[Delta]1 P_s-1 + ... +
[Delta]_s P0. These give for every coefficient in Ps+1 an integral
aggregate with real positive coefficients of the coefficients in P_s,
P_s-1, ..., P0 and the coefficients in A and B; and they are the same
aggregates as would be given by the previously obtained equations for
the corresponding coefficients in p_s+1 in terms of the coefficients
in ps, p_s-1, ..., p0 and the coefficients in a and b. Hence as the
coefficients in P0 and also in A, B are real and positive, it follows
that the values obtained in succession for the coefficients in P1, P2,
... are real and positive; and further, taking account of the fact
that the absolute value of a sum of terms is not greater than the sum
of the absolute values of the terms, it follows, for each value of s,
that every coefficient in p_s+1 is, in absolute value, not greater
than the corresponding coefficient in P_s+1. Thus if the series for F
be convergent, the series for f will also be; and we are thus reduced
to (1), specifying functions A, B with real positive coefficients,
each in absolute value not less than the corresponding coefficient in
a, b; (2) proving that the equation

AdF/dx + BdF/dy = dF/dt

possesses an integral P0 + tP1 + t^2P2/2! + ... in which the
coefficients in P0 are real and positive, and each not less than the
absolute value of the corresponding coefficient in p0. If a, b be
developable for x, y both in absolute value less than r and for t less
in absolute value than R, and for such values a, b be both less in
absolute value than the real positive constant M, it is not difficult
to verify that we may take

/ x + y\-1 / t\-1
A = B = M( 1 - ----- ) ( 1 - - ),
\ r / \ R/

and obtain
_ _
| 4MR / x + y\-2 / t\-1 |1/2
F = r - (r - x - y) | 1 - ---(1 - ------) log (1 - - ) |,
|_ r \ r / \ R/ _|

and that this solves the problem when x, y, t are sufficiently small
for the two cases p0 = x, p0 = y. One obvious application of the
general theorem is to the proof of the existence of an integral of an
ordinary linear differential equation