Colloquium publications.

162

THE MADISON COLLOQUIUM.

origin, (0,, 0), the function can in general* be written in the
form:
Z + Al 1 nm-i +... Am,..,
(2) F(.zi, z.) = _- BI z-1 + *...+ B n)
where the coefficients A, B are functions of (zl,..., _Zn), each
analytic at the point (0, *, 0) and vanishing there, the two
polynomials in zn being prime to each other; and where, moreover,
U2 is analytic and not zero at the origin.
In every neighborhood of a pole there are other poles, their
locus being the (2n- 2)-dimensional analytic manifold or
manifolds
G(zl, *,Z) = 0.
But there are no other singularities in the neighborhood in
question. For the special case n = 2 the non-essential singularities of the second kind are isolated points, since two functions
G(w, z), H(w, z) which are prime to each other, like two polynomials having this property, can vanish simultaneously only
in isolated points. But when n > 2, there will be a whole
(2n - 4)-dimensional locus of singularities of the second kind,this locus consisting of a finite number of analytic configurations,
each of the dimension in question. In fact, the necessary and
sufficient condition that the numerator and the denominator of
the fraction in (2) vanish simultaneously is that their resultant
vanish. The latter is analytic in zi, *., Zn-_ and vanishes at
the origin; but it does not vanish identically.
As regards the poles which lie in the neighborhood of a singularity of the second kind, they are situated on the manifold, or
manifolds,
G(zl,..., Zn)=,
and they consist of the totality of such points with the exception
of those for which H also vanishes, i. e., the singularities of the
second kind.
* In any case, a suitable homogeneous linear transformation of zi,.*, z.
will yield a new function for which the statement is true; cf. IV, ~ 1. The
theorems of the paragraph just cited are assumed in the present paragraph.