X. On hyperjacobian surfaces and curves

In a paper published in the ‘Mathematische Annalen’ (vol. iii. p. 459), Brill has discussed the question of curves having three-point contact with a doubly infinite pencil of curves; and in particular he has investigated some of the properties of the curve passing through all the points of contact with the individual curves of the pencil. In the same Journal (vol. x. p. 221) Krey of Kiel has applied a method similar to that of Brill with partial success to the question of curves having four-point contact with a triply infinite pencil. Some formulæ, however, given in 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 for 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 U with certain surfaces belonging to a pencil V, and to investigate some of its properties. From a slightly different point of view, however, it may be considered as an extension of two ideas, viz. first, that of the Jacobian surface, or locus of the points whose polar planes with regard to four surfaces meet in a point; and secondly, that of the Jacobian curve, or locus of points whose polar planes with regard to three surfaces have a right line in common. More particularly, commencing with the facts, first, that if a surface of the form aϕ + bψ + cZ touch a surface U, the point of contact is a point on the Jacobian, and secondly, that if a surface of the form aϕ + bψ touch a surface U, the point of contact is a point on the Jacobian curve, I have endeavoured to extend them to higher degrees of contact.