I. On hyperjacobian surfaces and curves

Abstract

In a paper published in the ‘Mathematische Annalen’ (vol. iii. p. 459 Brill has discussed the question of curves having three-point contac with a doubly infinite pencil of curves, and in particular he has investigated some of the properties of the curve passing through all the point of contact with the individual curves of the pencil. In the same jouma (vol. x. p. 22.1) Krey, of Kiel, has applied a method similar to that o: Brill with partial success to the question of curves having four-poinl contact with a triply infinite pencil. Some formulæ, however, given my paper “On the Sextactic Points of a Plane Curve” (Phil. Trans 1865, p. 657) have proved to be directly applicable to both questions An application of them to Brill’s problem will be found in the ‘Comptes Rendus’ for 1876 (2nd semestre, p. 627), and a solution of Krey’s problem in the ‘Proceedings of the London Mathematical Society’ foi the same year (vol. viii. p. 29). The present subject was, in the first instance, suggested by the foregoing papers; and from one point of view it may be regarded as an attempt to extend the question to the case of surfaces; viz. to determine a curve which shall pass through the points of contact of a given surface IT with certain surfaces belonging to a pencil V, and to investigate some of its properties.