y = yºu + yº1u1 + ... + yºn−1un−1,
where u, u1, ..., un−1 are functions of x, independent of yº, ... yºn−1, developable about x = xº; this value of y is such that for x = xº the functions y, y1 ... yn−1 reduce respectively to yº, yº1, ... yºn−1; it can be proved that the region of existence of these series extends within a circle centre xº and radius equal to the distance from xº of the nearest point at which one of a1, ... an becomes infinite. Now consider a region enclosing xº and only one of the places, say Σ, at which one of a1, ... an becomes infinite. When x is made to describe a closed curve in this region, including this point Σ in its interior, it may well happen that the continuations of the functions u, u1, ..., un−1 give, when we have returned to the point x, values v, v1, ..., vn−1, so that the integral under consideration becomes changed to yº + yº1v1 + ... + yºn−1vn−1. At xº let this branch and the corresponding values of y1, ... yn−1 be ηº, ηº1, ... ηºn−1; then, as there is only one series satisfying the equation and reducing to (ηº, ηº1, ... ηºn−1) for x = xº and the coefficients in the differential equation are single-valued functions, we must have ηºu + ηº1u1 + ... + ηºn−1un−1 = yºv + yº1v1 + ... + yºn−1vn−1; as this holds for arbitrary values of yº ... yºn−1, upon which u, ... un−1 and v, ... vn−1 do not depend, it follows that each of v, ... vn−1 is a linear function of u, ... un−1 with constant coefficients, say vi = Ai1u + ... + Ainun−1. Then
yºv + ... + yºn−1vn−1 = (Σi Ai1 yºi)u + ... + (Σi Ain yºi) un−1;
this is equal to μ(yºu + ... + yºn−1un−1) if Σi Air yºi = μyºr−1; eliminating yº ... yºn−1 from these linear equations, we have a determinantal equation of order n for μ; let μ1 be one of its roots; determining the ratios of yº, y1º, ... yºn−1 to satisfy the linear equations, we have thus proved that there exists an integral, H, of the equation, which when continued round the point Σ and back to the starting-point, becomes changed to H1 = μ1H. Let now ξ be the value of x at Σ and r1 one of the values of (½πi) log μ1; consider the function (x − ξ)−r1H; when x makes a circuit round x = ξ, this becomes changed to
exp (-2πir1) (x − ξ)−r1 μH,
that is, is unchanged; thus we may put H = (x − ξ)r1φ1, φ1 being a function single-valued for paths in the region considered described about Σ, and therefore, by Laurent’s Theorem (see [Function]), capable of expression in the annular region about this point by a series of positive and negative integral powers of x − ξ, which in general may contain an infinite number of negative powers; there is, however, no reason to suppose r1 to be an integer, or even real. Thus, if all the roots of the determinantal equation in μ are different, we obtain n integrals of the forms (x − ξ)r1φ1, ..., (x − ξ)rnφn. In general we obtain as many integrals of this form as there are really different roots; and the problem arises to discover, in case a root be k times repeated, k − 1 equations of as simple a form as possible to replace the k − 1 equations of the form yº + ... + yºn−1vn−1 = μ(yº + ... + yºn−1un−1) which would have existed had the roots been different. The most natural method of obtaining a suggestion lies probably in remarking that if r2 = r1 + h, there is an integral [(x − ξ)r1 + hφ2 − (x − ξ)r1φ1] / h, where the coefficients in φ2 are the same functions of r1 + h as are the coefficients in φ1 of r1; when h vanishes, this integral takes the form
(x − ξ)r1 [dφ1/dr1 + φ1 log (x − ξ)],
or say
(x − ξ)r1 [φ1 + ψ1 log (x − ξ)];
denoting this by 2πiμ1K, and (x − ξ)r1 φ1 by H, a circuit of the point ξ changes K into