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continuous correspondences on a Riemann surface, whether algebraic or not, uithout recourse to transcendental considerations.(d) Open manifolds. Here an adaptation of a reasoning due to Alexander leads to the solution of the question.(e) One-sided manifolds. They may be replaced by suitable two-sided open manifolds. (f) Conformal representation. Julia's theorem on the conformal representation of a plane region on a part of itself,3 together with Ritt's extension4 and other generalizations for functions of several complex variables can be readily obtained by our method.Consider a system S made up of a finite number of simple noninteresting closed curves located in real euclidean 3 space. The curves S may be arbitrarily knotted and linking, but we shall assume, in order to simplify matters as much as possible, that each is composed of a finite number of straight pieces. The problem will be to prove that the system S is always topologically equivalent (in the sense of isotopic) to a simpler system S', where S' is so related to some fixed axis in space that as a point P describes a curve of S' in a given direction the plane through the axis and the point P never ceases to rotate in the same direction about the axis. An application of this lemma to the theory of 3-dimensional manifolds will be given at the end of the communication.It will be convenient to visualize the system S by means of its projection S, upon a plane. By choosing the center of projection in general position, the projection S,. will have no other singularities than isolated double points at each of which a pair of straight pieces actually cross one another. Wherever a double point occurs, it will be necessary to indicate which of the two branches is to be thought of as passing behind the other, either by removing a little segment from the branch in question or by some equivalent device. The problem will then be to transform the figure S,, by 93 VOiL. 9, 1923

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