ifferential coefficient of z_[rho] which enters in
[Phi]_[sigma], say

d^(k1 + ... + kn) z_[rho]/dx1^k1 ... dx_n^k_n,

we have k1 < h_[rho] and k1 + ... + k_n <= h_[rho]. Let a1, ... an, b1,
... br, and b[rho]_(k1 ... kn) be a set of values of

x1, ... x_n, z1, ... z_r

and of the differential coefficients entering in [Phi]_[sigma] about
which all the functions [Phi]1, ... [Phi]_r, are developable.
Corresponding to each dependent variable z_[sigma], we take now a set of
h_[sigma] functions of x2, ... xn, say [phi][sigma], [phi][sigma]^(1),
..., [phi][sigma]^(h-1) arbitrary save that they must be developable
about a2, a3, ... an, and such that for these values of x2, ... xn, the
function [phi]_[rho] reduces to b_[rho], and the differential
coefficient

d^(k2 + ... + kn) [phi]_[rho]^(k1)/dx2^k2 ... dx_n^kn

reduces to b^kn_(k1 ... kn). Then the theorem is that there exists one,
and only one, set of functions z1, ... z_r, of x2, ... x_n developable
about a1, ... an satisfying the given differential equations, and such
that for x1 = a1 we have

z_[sigma] = [phi]_[sigma], dz_[sigma]/dx1 = [phi]_[sigma]^(1), ...
d^(h_[sigma]-1) z_[sigma]/d^(h_[sigma]-1) x1 = [phi][sigma]^(h_[sigma]-1).

And, moreover, if the arbitrary functions [phi]_[sigma],
[phi]_[sigma]^(1) ... contain a certain number of arbitrary variables
t1, ... tm, and be developable about the values t1^0, ... tm^0 of these
variables, the solutions z1, ... zr will contain t1, ... tm, and be
developable about t1^0, ... tm^0.

Singular points of solutions.

The proof of this theorem may be given by showing that if ordinary
power series in x1 - -a1, ... xn - an, t1 - t1^0, ... tm - tm^0 be
substituted in the equations wherein in z[sigma] the coefficients of
(x1 - a1)^0, x1 - a1, ..., (x1 - a1)^(h_[sigma]-1) are the arbitrary
functions [phi]_[sigma], [phi]_[sigma]^(1), ..., [phi]_[sigma]^h-1,
divided respectively by 1, 1!, 2!, &c., then the differential
equations determine uniquely all the other coefficients, and that the
resulting series are convergent. We rely, in fact, upon the theory of
monogenic analytical functions (see FUNCTION), a function being
determined entirely by its development in the neighbourhood of one set
of values of the independent variables, from which all its other
values arise by _continuation_; it being of course understood that the
coefficients in the differential equations are to be con