PURE MATHEMATICS 525 



conjugate hexads, defined b}^ six points such that the plane of 

 any three is conjugate to the plane of the remaining three, self- 

 conjugate pentads, five points such that the line joining any 

 two contains the pole of the other three, and self-polar tetrads, 

 four points such that each is the pole of the plane of the other 

 three, are introduced. The properties of twisted cubics passing 

 through the vertices of self-conjugate hexads are considered, 

 and it is shown that (i) if a twisted cubic be such that a self- 

 conjugate hexad, with respect to S, can be inscribed therein, 

 then an infinite number of such hexads can be inscribed therein ; 

 (ii) if a twisted cubic be such that a self-polar tetrahedron can 

 be inscribed therein, then an infinite number of such tetrahedra 

 and also of self-conjugate pentads and hexads can be inscribed 

 therein ; (iii) if a twisted cubic be such that a self-conjugate 

 hexad can be described therein, then self-conjugate pentads 

 and self-polar tetrads can be described therein. Similar pro- 

 positions are shown to be true in the case of quadrics. 



C. G. F. James (ibid., 150-78) discusses the analytical repre- 

 sentation of congruences of conies by means of matrices. 



F, P. White (ibid., 216-27) examines curves and surfaces 

 which are generated from projective systems of hyperplanes in 

 space of four dimensions. 



The fundamental properties of the twisted sextic curve 

 which is the complete intersection of a quadric and a cubic 

 surface have up till now only been investigated by means of the 

 periods of Abelian functions of genus 4, and it seems likely that 

 purely geometrical methods will lag a long way behind analytical 

 ones in problems of this kind. An important advance has, 

 however, been made recently by W. P. Milne (Proc. L.M.S., 21, 

 1922, 373-80), who establishes synthetically the properties of 

 the sextactic quadrics and tritangent planes of one of the 255 

 systems which Clebsch showed to exist for the quadri-cubic 

 curve. If the curve be the complete intersection of the quadric 

 fa and the cubic surface fa, and S^ be a sextactic quadric, then 

 he proves that the vertex of one of the cones of the pencil 

 Sg r2 lies on the twisted cubic through the six points of contact. 

 He then considers the four-nodal cubic surfaces which contain 

 the curve, and shows that the quadric cones, into two of which 

 the tangent cones to one such surface from any point of it break 

 up, form a complete set of sextactic cones of the same system ; 

 also the quadrics which contain the points of contact of any 

 two sextactic cones of the same system with the curve are the 

 pencils of quadrics determined by fa and the quadrics which 

 contain any two twisted cubics lying upon the four-nodal cubic 

 surface. The 28 surfaces of the system which break up into 

 pairs of tritangent planes are found to correspond to the 28 

 bitangents of a plane quartic curve, the section of the quartic 



