Colloquium publications.

194

THE MADISON COLLOQUIUM.

Theorem. Let the functions
(1) ~k(Ul, * *', JU; a, * * *. Ip), k = 1, ', l,
be analytic in the point (u; x) = (0; 0) and vanish there. Furthermore, let the I equations
(2) k(Ul, * *, Ul; 0,. ' ', 0) = 0, k, 1,
admit no other solution than (u) = (0) in the neighborhood of
this latter point. Then, to each point (x) in a certain neighborhood of (x) = (0), with the exception of those which lie on a
locus D presently to be considered, there will correspond N
distinct points, (u1i),..., u(j)), j= 1,..., N, and hence N
distinct points (ulj, * *, uz); xi,., * *, such that the equations
(3) 4k(U, * *, u; x1, *, Xp) = 0
are satisfied in these points. N is a fixed positive integer.
Moreover, these are the only points of the neighborhood of
(u; x)= (0; 0) in which these equations are satisfied and for
which (x) does not lie on D.
These points are determined as follows. ul is given by an
algebroid equation having no multiple factors,
(4) uzN+ Alul —' + *. + A- = 0,
where Ak(xl, x*, ) is analytic in the point (x)= (0) and
vanishes there. If (x) does not lie on D, the roots of (4) are all
distinct, and analytic in (x), and the further functions ul,..,
ul-l which enter to form the roots of (1) are also single-valued
and analytic on the analytic configuration, or configurations,
(4) except at most for points for which (x) lies on D.
The points of D are those whose co-ordinates satisfy at least
one of a finite number of equations
D1(x1, *', Dq(1x, Xp) - *, ( Xp) = 0,
where Dk is analytic in the point (x) = (0) and vanishes there,
and is irreducible.