Introductory explanation. The present chapter deals with important fundamental conceptions in the theory of algebraic curves and functions, and, for greater precision, some familiarity with the elements of the theory of functions is assumed.

The preceding chapter incidentally brings out the want of definiteness in regarding a curve as denned by its points, each given by one set of values of the coordinates; for we have seen that a multiple point may be replaced, on another curve which is in (1, 1) birational correspondence with it, by several distinct points. In the present chapter we are led to consider a point of a curve as belonging to a definite range of points, lying on, and forming all the points of a so-called branch of the curve. For clearness, a point so considered, in association with a branch of the curve to which it belongs, will be called a place. A point may belong to several branches, but when this is so it is accidental, the distinction of the branches being the essential fact. The point of view which is reached will ultimately be found to be of great importance.