PROCEEDINGS OP SECTION A. 



309 



2.— ON STEIXER'S QUARTIC SURFACE. 



By EVELYy G. HOGG, M.A., Christ's College, Chrisiehurch, Xew Zealand. 



§ 1. The surface known as Steiner's quartic, whose equation is of the 

 form — 



^/ 



la 



In 



+ y!+/j+y:=^' 



Ic 



p " q 



is usually obtained as the reciprocal of a 'cubic surface, whose equation 

 is of the form — 



« + :^ + £ + 1^ = 



a ft ^l C 



by means of the auxiliary quadric — 



a' + ft' + 7' + ^- = 0. 

 The object of this paper is to describe another method of obtaining 

 the equation of the surface in question, and to point out certain results 

 which follow immediately from the method employed. 



§ 2. Let ABCD be the vertices of the tetrahedron of reference, and let 

 a point O whose quadriplanar co-ordinates are («o/^o7o''o) ^^ taken. Let 

 the lines AO, BO, CO, DO meet the opposite faces of the tetrahedron in 

 a, b, c, (I respectively. It may be easily shown that the lines of inter- 

 f?ection of the four pairs of planes BCD, bed; CDA, cda ; DAB, dab; 

 ABC, abc lie in a plane whose equation is — 



+ 



+ ^ + ^ = 0. 



7o "o 



This plane will, for the sake of brevity, be called the Steiner plane 

 P^ of the point {oofto^/o^^). 



Let a ft' 7' c be the co-ordinates of any point on the plane P^ ; the 

 equation of its Steiner plane will be — 



a' ^ ft' ^ 0/ ^ c 



To find the envelope of this plane, subject to the condition — 



= 0, 



+ /T, + " + 



7o '■q 



we have, by the method of undetermined multipliers, 

 a \ ft A, 7 X c X. 



7^ ~ ;;' ^ ~ /^' 7^ ~ ^' a^ "" a; 



whence we at once obtain — 



•^ = la , Ift. 



+ .A + ./i = o 



^/7c 



V 



as the " norm " equation of the envelope of the Steiner planes of points 

 lying in the plane T^. The surface S^ is Steiner's quartic surface, and is 

 of the fourth degree. 



