Secant functions, the Reiss relation and its converse

Generalizing a classical Euclidean theorem for the circle, certain meromorphic functions on ${{\mathbf{P}}_1}$ relating to the geometry of algebraic plane curves are shown to be constant. Differentiated twice, this gives a new proof of the Reiss relation and its converse. The relation of these functions to Abel's Theorem is discussed, and a generalization of secant functions to space curves is given, for which the Chow form arises in a natural way.