Mr Grace, Note on the Rational space curve etc. 27 



Note on the Rational space curve of the fourth order. By 

 J. H. Grace, M.A., Peterhouse. 



[Read 26 November 1900.] 



1. The relations between the geometry of points on a rational 

 quartic and the algebraic theory of a binary quartic have been 

 frequently discussed l , but the application of similar methods 

 to the line-geometry of the tangents to the curve has not, as far 

 as I know, been previously remarked. 



If we supposed the coordinates x, y, z, w of a point on the 

 curve to be expressed as quartic functions of a simple para- 

 meter A, so that 



#=/iO), y=f.(H z=f*(S)> w =fi( x l 



then the six coordinates of the tangent at any point are the six 

 Jacobians of the f's taken in pairs. For convenience I denote 

 these by 



"235 "31; "12) "14; "24; "34- 



2. Each of the four f's is apolar to one and the same 

 quartic f and thus the condition that four points should be 

 coplanar is that the quartic giving their parameters should be 

 apolar to f 



In like manner each of the sextics J is apolar to one and 

 the same sextic <f>, and the condition that the tangents at six 

 points should belong to the same linear complex is that the 

 sextic giving their parameters should be apolar to <p. 



3. Let us suppose that /= a* 4 , $ = c^, and that a, B, 7, h, e, £ 

 are the parameters of six points on the curve ; then it is well 

 known, it can be easily verified, and it is indeed obvious from 

 the simple properties of transvectants that a, @, 7, 8 are co- 

 planar if 



a a apttya$ = 0. 



Further, the tangents at the six points belong to the same 

 linear complex if 



C a CpC y C S C e C£ = 0. 



Hence just as f=a^ gives the points at which a plane 

 contains four consecutive points, so (f> = c x 6 gives the points at 

 which a linear complex contains six consecutive tangents to the 

 curve. 



1 See Mr Richmond's recent and comprehensive Memoir (Camb. Phil. Soc. Trans. 

 Vol. xix. Pt. 1.), in which full references will be found. 



