Introduction. In a birational transformation between two surfaces to certain curves of one of the surfaces may correspond simple points of the other. Such curves have been called exceptional curves. Exceptional curves have been subdivided into those of the first or second kind according as a point of the curve is not or is transformed into a curve of the other surface. The ones previously considered have either been irreducible or consisted of at most two components. A treatment of reducible exceptional curves with components all simple is found in (4, Chap. II, 6). We shall consider exceptional curves of the first kind with s irreducible components, each counted a certain number of times. This case arises when the birational transformation possesses infinitely near fundamental points on one or several irreducible algebroid branches. In the sequel we shall use some fundamental notions of linear systems of curves on an algebraic surface and the theory of singularities as developed by Enriques (3, pp. 327-399). Sections 1 and 2 of this paper are devoted to definitions and the derivation of the virtual degree and genus of an exceptional curve of the first kind; the presenitation here is parallel to but more complete than that found in (4, Chap. II, 6). In sections 3 and 4 we introduce s fundamental points of the birational transformation, mentioned above, on that surface where the given exceptional curve has been changed into a point, and limit ourselves to the case in which these fundamental points lie on a single irreducible algebroid branch. In 5 we consider the correspondence between the irreducible components of the given curve on the one surface and the immediate neighborhood of each of the fundamental points on the other surface. In 6 we determine the multiplicities of these fundamental points on branches of lowest order passing through the first a (ca,? s) of them. In 7 we describe the intersections of the irreducible components of the given exceptional curve, using extensively the notion of proximate points introduced by Enriques in his theory of singularities (3, p. 381) ; in 8 we study the classification of the s fundamental points as free points or satellites when the intersection numbers