rROCEEDINGS OP SECTION A. 311 



From the first three equations we learn — 



1 1.1.1 A . A . V . N 



a p 7 c 



Where Aj Ao A3 A4 are the determinants obtained by omitting in turn each 

 of the columns, starting from the left-hand corner of the determinant — 



a /3 7 ^ 

 ai /?! 7i Oi 



The equation of the tangent planes is therefore — 



«o Ai ^o A2 7o A3 ^o A4 



which, as Ai A2 A3 A^ are each of the first degree in (0/^7 0), represents 

 an equation of the third degree. 



§ 5. Let two Steiner quartics 2, S2 be tal^en : since these surfaces are 

 the envelopes of the Steiner planes of points lying on the planes P, and 

 Pj, it follows that the Steiner [)lanes of the line of intei section of Pj and 

 P2 will be the common tangent planes of 2, and 2^, and also that, since 

 the line of intersection of Pi and Pj will meet the Steiner cubic surface 

 Sq in three points, three planes can be drawn through the point (ao^o7o^o) 

 to touch both 2i and 2^. 



§ 6. Any three Steiner quartics 2, 2^ ^3 have in general one common 

 tangent plane distinct from the faces of the fundamental tetrahedron, 

 namely, the Steiner plane of the point common to the Steiner planes 

 P,, P.^, P3, but if these Steiner planes have a common line of intersection 

 the Steiner planes of points lying on that line will touch each of the 

 surfaces Sj "Z.^ Sg. 



§ 7. If X. + /A -|- r + /} = 0, the co-ordinates of any point on P^ may 

 be taken to be (Xo^, /</3q, v^Iq, po^) and the Steiner plane of this point will 

 touch 2q at the point {X-a^, /ii'(3^, v^^i^, pb^) : any point on S^ may be 



^o /^^ To ^ 

 /t ' 1/ ' /J 

 S„ at this point will have for its equation — • 



«o Po 7o "o 



Hence it may be easily shown that the envelope of the Steiner planes of 

 points lying on 2^ is the surface whose equation is — 



\l^ + '/^ + \!"L + '/^- = 

 V „o -^ V ^„ v/ 7^ V 0^ 



and that the locus of points whose Steiner planes touch S^ is the surface 

 whose equation is — 



represented by the co-ordinates [^ •> ~;f'' ~f ' ~^ ) ^"^1 the tangent plane to 



/«o -1- y/^o -L /To 4. /!° = 



V^^\/^^^/7 ^ V 



= 0. 



