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
/ x + y\-1 / t\-1 A = B = M( 1 - ----- ) ( 1 - - ), \ r / \ R/
and obtain _ _ | 4MR / x + y\-2 / t\-1 |½ 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 given by the n equations dy/dx = y1, dy1/dx = y2, ...,
but in fact any simultaneous system of ordinary equations is reducible to a system of the form
Complete systems of linear partial equations.
Suppose we have k homogeneous linear partial equations of the first order in n independent variables, the general equation being a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1, ... k, and that we desire to know whether the equations have common solutions, and if so, how many. It is to be understood that the equations are linearly independent, which implies that k <= n and not every determinant of k rows and columns is identically zero in the matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i = 1, ... n, [sigma] = 1, ... k). Denoting the left side of the [sigma]-th equation by P[sigma]f, it is clear that every common solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We immediately find, however, that this is also a linear equation, namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i - P[sigma]a[rho]_i, and if it be not already contained among the given equations, or be linearly deducible from them, it may be added to them, as not introducing any additional limitation of the possibility of their having common solutions. Proceeding thus with every pair of the original equations, and then with every pair of the possibly augmented system so obtained, and so on continually, we shall arrive at a system of equations, linearly independent of each other and therefore not more than n in number, such that the combination, in the way described, of every pair of them, leads to an equation which is linearly deducible from them. If the number of this so-called _complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0, leading to the nugatory result f = a constant. Suppose, then, the number of this system to be r < n; suppose, further, that from the matrix of the coefficients a determinant of r rows and columns not vanishing identically is that formed by the coefficients of the differential coefficients of f in regard to x1 ... x_r; also that the coefficients are all developable about the values x1 = x1^0, ... xn= xn^0, and that for these values the determinant just spoken of is not zero. Then the main theorem is that the complete system of r equations, and therefore the originally given set of k equations, have in common n - r solutions, say [omega]r+1, ... [omega]n, which reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are respectively put x1^0, ... x_r^0; so that also the equations have in common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an arbitrary function [psi](x_r+1, ... x_n) which is developable about x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1, ... [omega]_n). It is seen at once that this result is a generalization of the theorem for r = 1, and its proof is conveniently given by induction from that case. It can be verified without difficulty (1) that if from the r equations of the complete system we form r independent linear aggregates, with coefficients not necessarily constants, the new system is also a complete system; (2) that if in place of the independent variables x1, ... xn we introduce any other variables which are independent functions of the former, the new equations also form a complete system. It is convenient, then, from the complete system of r equations to form r new equations by solving separately for df/dx1, ..., df/dx_r; suppose the general equation of the new system to be