F = P0 + tP1 + t²P2/2! + ...,
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 Δr = Ard/dx + Brd/dy we obtain necessary equations of the same form as before, namely,
P1 = Δ0P0, P2 = Δ0P1 + Δ1P0, ...
and in general Ps+1 = Δ0Ps, + s1Δ1Ps-1 + ... + ΔsP0. These give for every coefficient in Ps+1 an integral aggregate with real positive coefficients of the coefficients in Ps, Ps-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 ps+1 in terms of the coefficients in ps, ps-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 ps+1 is, in absolute value, not greater than the corresponding coefficient in Ps+1. Thus if the series for F be convergent, the series for ƒ 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 + t²P2/2! + ... 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 A = B = M[1 − (x + y)/r]-1 (1 − t/R)-1, and obtain
| F = r − (r − x − y) [ 1 − | 4MR | (1 − | x + y | ) | −2 | log ( 1 − | t | ) | −1 | ] | 1/2 | , |
| 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 given by the n equations dy/dx = y1, dy1/dx = y2, ...,
dyn-1/dx = p − p1yn-1 − ... − pny;
but in fact any simultaneous system of ordinary equations is reducible to a system of the form