The present chapter is concerned with a theory of the rational functions and integrals, associated with an (irreducible) algebraic curve f(x, y) = 0, which began with the arithmetical work of Kronecker and Dedekind for the theory of integer numbers. Developed in detail the theory gives an alternative to much of what has preceded; what is given here however seems a desirable, if not a necessary, accompaniment of what has already been proved.

We have explained what is meant by a rational function belonging to the curve f(x, y) = 0; to fix the ideas we recall certain properties of a general kind for such functions. It was seen that, as a rule, there exists no rational function of the first, nor of the second order; there is thus, in general, a least order for which there exists a rational function, associated with the curve. But we cannot expect to be able to construct a function of this least order with its poles taken arbitrarily on the curve; for instance, when the equation f(x, y) = 0 is of the form y 2 − u = 0, where u is a polynomial in x, though there exists a function, of the form (x – a)−1, with two poles, these must be at places for which x has the same value. There is thus another least number, say k, such that, whatever k places be taken, a rational function exists with poles, all of the first order, at these k places, or at places chosen from these.