his value of y is such that for x =
x^0 the functions y, y1 ... y_n-1 reduce respectively to y^0, y^01, ...
y^0_n-1; it can be proved that the region of existence of these series
extends within a circle centre x^0 and radius equal to the distance
from x^0 of the nearest point at which one of a1, ... an becomes
infinite. Now consider a region enclosing x^0 and only one of the
places, say [Sigma], at which one of a1, ... an becomes infinite. When
x is made to describe a closed curve in this region, including this
point [Sigma] in its interior, it may well happen that the
continuations of the functions u, u1, ..., u_n-1 give, when we have
returned to the point x, values v, v1, ..., v_n-1, so that the
integral under consideration becomes changed to y^0 + y^01v1 + ... +
y^0_n-1 v_n-1. At x^0 let this branch and the corresponding values of
y1, ... y_n-1 be [eta]^0, [eta]^0_1, ... [eta]^0_n-1; then, as there is
only one series satisfying the equation and reducing to ([eta]^0,
[eta]^0_1, ... [eta]^0_n-1) for x = x^0 and the coefficients in the
differential equation are single-valued functions, we must have
[eta]^0_u + [eta]^0_1u1 + ... + [eta]^0_n-1 u_n-1 = y^0v + y^01v1 + ... +
y^0_n-1 v_n-1; as this holds for arbitrary values of y^0 ... y^0_n-1,
upon which u, ... u_n-1 and v, ... v_n-1 do not depend, it follows
that each of v, ... v_n-1 is a linear function of u, ... u_n-1 with
constant coefficients, say v_i = A_i1 u + ... + A_in u_n-1. Then

y^0v + ... + y^0_n-1 v_n-1 = ([Sigma]_i A_i1 y_i^0)u + ... + ([Sigma]_i A_in y^0_i)u_n-1;

this is equal to [mu](y^0u + ... + y^0_n-1 u_n-1) if [Sigma]_i A_ir y^0_i
= [mu]y^0_r-1; eliminating y^0 ... y^0_n-1 from these linear equations,
we have a determinantal equation of order n for [mu]; let [mu]1 be one
of its roots; determining the ratios of y^0, y1^0, ... y^0_n-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
[Sigma] and back to the starting-point, becomes changed to H1 =
[mu]1H. Let now [xi] be the value of x at [Sigma] and r1 one of the
values of (1/2[pi]i) log [mu]1; consider the function (x - [xi])^r1 H;
when x makes a circuit round x = [xi], this becomes changed to

exp(-2[pi]ir1) (x - [xi])^-r1 [mu]H,

that is, is unchanged; thus we may put H = (x - [xi])^r1 [phi]1,
[phi]1 being a function single-valued for pa