Harris Hancock and the Elliptic Realm

Owing to a theorem due to Liouville, we are able to show the real significance of the one-valued functions of position on the Riemann surface, viz., they are the general elliptic functions. Those one-valued functions form a "class of algebraic functions" or "a closed realm of rationality," since the sum, difference, product, or quotient of any two such functions is a function of the realm. This realm of rationality is of the first order, corresponding to the connectivity of the associated Riemann surface, the realm of the ordinary rational functions being of the zero order. The former realm is derived from the latter by adjoining an algebraic quantity, which quantity defines the Riemann surface. This latter realm, which we call the "elliptic realm," includes as special cases the natural realm of all rational functions, and also the realm of the simply periodic functions.