MAXIMUM AND MINIMDM VALUES OF A LINEAR FUNCTION, ETC. 69 



The cubic 



XryzB- = xyzS" + xziu/i- -\- xywy^ -{-i/zica", 



we shall represent briefly by o-'. In (57) both s and a are zero, hence 

 must 0-' = 0. The cubic surface* a-' = passes through the six edges of 

 the tetrahedron. It does not enter the tetrahedron, it cuts each face in the 

 edges in that face and nowhere else. The plane passing through an edge 

 which is tangent to o-' = is tangent all along that edge. The equation 

 to the tangent plane through a- = 0, y ^ being 



y- 6- 



with similar equations of the tangent plane along each other edge. Each 



edge being repeated twice gives 13 straight lines on the surface. The 



tangent planes through the opposite edges meet in three straight lines 



which lie on the surface and in the same plane, for 



X y ,2 w 



^- + ^ = and — + — =0 



clearly intersect on the surface 



X y z tv 

 and also on the plane 



a" p" y" o" 



Any plane through an edge cuts the surface also in a conic, in general 

 a hyperbola, which becomes two intersecting straight lines when the plane 

 becomes tangent. 



The cubic a' ^ cuts the sphere o- = in a line of intersection (spher- 

 ical cubic) passing through P, A, B, G. Hence P can be moved along 

 tliis spherical cubic to either vertex preserving the relation (56) for 

 constant values of a, p, y, 8. The ratios a-.p-.y.S must bear a fixed 

 relation to the dimensions of the tetrahedron, since at A, B, C respectively 



;8c + y6 — 8/ = 0, 

 ac + ya — ^g = 0, 



ah + Ba — S/i = 0, 



*This surface has been studied by Professor Cayley, M4moire sur les Courbes du 

 Troisi^me Ordre, Collected Works, I, 183. Journal de Math. Pure et AppliquSes 

 (1844), IX, pp. 285-293. 



Salmon counts each edge of the tetrahedron as four straight lines on this cubic 

 which together with the other three mentioned above give the 27 lines. An. Geom., 

 p. 501. 



