We may conveniently begin by stating the theorem: If each of the n functions φ1, ... φn of the (n + 1) variables x1, ... xnt be developable Ordinary equations of the first order. about the values xº1, ... xn0t0, the n differential equations of the form dx1/dt = φ1(tx1, ... xn) are satisfied by convergent power series
xr = xºr + (t − t0) Ar1 + (t − t0)² Ar2 + ...
reducing respectively to xº1, ... xºn when t = t0; and the only functions satisfying the equations and reducing respectively to xº1, ... xºn when t = t0, are those determined by continuation of these series. If the result of solving these n equations for xº1, ... xºn be written in the form ω1(x1, ... xnt) = xº1, ... ωn(x1, ... xnt) = xºn, Single homogeneous partial equation of the first order. it is at once evident that the differential equation
dƒ/dt + φ1dƒ/dx1 + ... + φndƒ/dxn = 0
possesses n integrals, namely, the functions ω1, ... ωn, which are developable about the values (xº1 ... xn0t0) and reduce respectively to x1, ... xn when t = t0. And in fact it has no other integrals so reducing. Thus this equation also possesses a unique integral reducing when t = t0 to an arbitrary function ψ(x1, ... xn), this integral being. ψ(ω1, ... ωn). Conversely the existence of these principal integrals ω1, ... ωn of the partial equation establishes the existence of the specified solutions of the ordinary equations dxi/dt = φi. 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 − x0, &c., and consider the equation a(xyt) dƒ/dx + b(xyt) dƒ/dy = dƒ/dt, wherein the functions a, b are developable about x = 0, y = 0, t = 0; say
a(xyt) = a0 + ta1 + t²a2/2! + ..., b(xyt) = b0 + tb1 + t²b2/2! + ...,
so that
ad/dx + bd/dy = δ0 + tδ1 + ½t²δ2 + ...,
where δ = ard/dx + brd/dy. In order that
ƒ = p0 + tp1 + t²p2/2! + ...