Egberts — On Binary Cubics and their Associated Forms. 521 



LXXIII. — A Geometeicai Eepeesentatiok of a Systeit of Two 

 Bi^J^AEY Cubics a^d theik Associated Foems. By "W. R. Westeopp 

 EOBEBTS, M.A. 



[Read, November 14, 1881.] 



I. 



The object of this Paper is to invest with a certain geometrical meaning 

 the algebraic forms arising in a system of two binary cubics, that is, 

 to construct geometrically points which shall represent the linear, qua- 

 dratic, cubic, and quartic covariants of the system, and to express the 

 Yanishing of invariants by geometrical relations connecting such points. 

 We may consider any binary quantic as derived from a system of three 

 surfaces by assuming 



equations which in themselves imply, by elimination of x^^ and X2, two 

 fixed relations between X, Y, Z, W, denoting a fixed curve in space, 

 w^hile the given binary quantic, equated to zero, enables us to obtain 

 a third such relation. The transformation here employed is one in 

 which ^1, <^2) ^3) 4>i ^^6 cubic functions of Xi and Xo. By linear trans- 

 formation this substitution is reducible to 



the fixed curve in this case being evidently a twisted cubic. The 

 equation of an osculating plane of the curve, the parameter of the 

 corresponding point of which is Xi : x^, being (Salmon's Geometry of 

 Three Dimensions, Art. 368) 



Xx.? - 3 Kpj^i^ + ^Zx^~Xo - xi^ 7F= 0, 



the parameters answering to osculating planes through any point 0, 

 the co-ordinates of which are X', Y', Z', W, are given by the 

 equation 



xiX' - 3 Y. x^x^ + 2>Z'. Xy^X2 - x^' W = 0, 



the points of contact lying in the plane 



XW -3YZ'+Z Y'Z- TFX' = 0. 



But this plane passes through 0, the given point. To any plane, 

 then, corresponds a point 0, the point of intersection of the osculating 

 planes at the points where it meets the curve — a point which plays an 



2X2 



