Colloquium publications.

174

THE MADISON COLLOQUIUM.

stants, and where, moreover, r' < r < K), into the neighborhood
of (x', y'), but not to this point.
It will be observed that if the conditions of the theorem are fulfilled for a given K, then they are also fulfilled for any smaller K.
We can picture the loci x = x' = a, a < h, as surfaces which
form afield (in the sense in which this word is used in the Calculus
of Variations) in the 4-dimensional neighborhood of the origin.
If we make any analytic transformation of this neighborhood, of
the form
u= ((x, y), v = (x, y),
where (p and 4/ are functions of the complex variables (x, y)
analytic at the origin, and where the Jacobian of o( and 1/ does
not vanish, we can then state an obvious corollary of Lemma
2 for the surfaces in the (u, v)-space, into which the surfaces
x = a have been carried.
This is all the preparation Levi needs for his main theorem,
to which we now turn.
Theorem. Let E be a perfect set of points in the 4-dimensional
space of the complex variables x, y; and let 0 be a fixed point
of this space. Let r be the distance from 0 to a variable point
of E. If there be a point P of E for which r has a relative maximum,* then there cannot exist a function f(x, y) which is meromorphic in the neighborhood of P except for the points of E,
and in each of those points has an essential singularity.
More precisely stated, the conclusion is this. Consider the
continuum, T, exterior to the hypersphere through P with 0
as centre and interior to a small hypersphere with P as centre.
Then there cannot exist a function meromorphic in T and not
admitting meromorphic continuation at P.
Since Hartogst has proven Lemma 2 for the case that f(x, y)
is required to be analytic instead of being allowed to be mero* Maximum is here to be understood as meaning that r shall not, in the
neighborhood of P, take a larger value than at P; but it may attain that
value at other points of the neighborhood.
t ~ 6, first theorem.