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On Hyperjacobian Surfaces and Curves. [May 17, 



May 17, 1877. 



Dr. J. D ALTON HOOKER, C.B., President, in the Chair. 



The Presents received were laid on the table, and thanks ordered for 

 them. 



The following Papers were read : — 



I. " On Hyperjacobian Surfaces and Curves." By William 

 Spottiswoode, M.A., Treas.R.S. Received April 23, 1877. 



(Abstract.) 



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 investi- 

 gated 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 formulae, 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 fore- 

 going 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. 



Prom 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 a0 + &ip + <?x touch a surface U, the point of contact 

 is a point on the jacobian, and, secondly, that if a surface of the form 

 a,(j> + b\p touch a surface IT, the point of contact is a point on the jacobian 

 curve, I have endeavoured to extend them to higher degrees of contact. 



