310 PRDC3EDIS^GS OF SECTI3N A. 



Intimately related to the quartic surface 2^ is the cubic surface, 

 whose equation is — „ 



S, E -° + ^° + ^^ + ^ = 0, 



a p 7 o 



which for the sake of brevity will be referred to as the Steiner cubic 

 surface Sq. it is at once seen to be the locus of points whose Steiner 

 planes pass through the fixed point ao/5o7o^o- 



§ 3. Two Steiner cubic surfaces S. and S2 determine by their inter- 

 section a curve of the ninth degree, which consists of a cubic curve and 

 the six edges of the fundamental tetrahedron. 



The Steiner plane of any point lying on this cubic curve evidently 

 passes through the line joining the points (a, /3i 71 «i), (a^ /Sa 72 ^2)- 



If a third point (03 ji^ 73 ^3) be taken on the line joining the two 

 above points its Steiner cubic Sj will pass through the cubic curve of 

 intersection of S, and S2: for let any point (a yS' 7' 5') be taken on that 

 curve, we then have — 



f^i I A I 7 1 I £1 _- 



a p 7 



a p ^/ c 



'1 — «2 1 ^1 — A I 71 ~ 72 I ^1 — ^ 2 _ A 

 / •" ->/ •" / •" f/ 



whence- 



/3' 7' _ ^' 



also since {a,i3i^/iCi) (^2/3272^2) ("3^3373^3) are collinear, 



« 3 — «i __ (^3 — ^ i _ 73 — 7 1 _. ^3 — ^1 



"1 — «2 ^1 — A 7l — 72 Oi — ^2 



and therefore it at once follows that — 



^ + § + ^ + 1=0, 



a p 7 c 



showing that S, S2 and S3 have a common curve of intersection. 



Any three Steiner cubic surfaces S, S2 S3 have in general one com- 

 mon point apart from the edges of the fundamental tetrahedron, namely,, 

 the point whose Steiner plane contains the points («] /3i 71 0,) (a^ ^2 72 ^2) 



§ 4. The curve of intersection of S, and S2 will in general meet the 

 plane P_, in three points, the Steiner planes of which will pass through 

 the line joining («, fti 7, ^,) and {a.^ /3j, 72 ^2) arid touch 2q. Hence through 

 any straight line three tangent planes can in general be drawn to touch 

 2q. To obtain the equation of these tangent planes let [a'/3'r/'S') be a 

 point of intersection of the plane Pq with the curve of intersection of 

 Si and Sj. Then — 



^' 



7 



'^' + § + ^^ + 5 = 



a p 7 c 



«2 ^ A + 72 + £2 ^ 



"o Po 7o ^o 



