Russell — Geometry of Surfaces derived from Cubics. 471 



and it obviously contains the line PP' as a generator ; the tangent 

 plane to this cone through PP' is obtained by writing down the polar 

 plane of P' with regard to it, and from its identity with the tangent 

 plane at T, we may define the litangent surface C as the envelope of 

 polar quadrics that are cones. 



9. The polar plane of the point T with "respect to the cubic is 



ax^ (IX' 



this is evidently identical with the polar plane of V, and is therefore 

 the tangent plane to the Hessian at V. 



"We can now locate the eight poles of the tangent plane to the 

 Hessian at V. Take the point U corresponding to it, and draw the 

 six lines through it which connect a pair of corresponding points ; the 

 six points T on these lines are six of the poles, the remaining two, of 

 course, coincide with F itself. 



From the above we see that the bitangent surface C may be 

 defined : — 



(1.) The locus of points whose polar planes with regard to the 

 cubic touch the Hessian. In this result the squai-e of the Hessian 

 would appear as a factor. If therefore we obtain the condition that 

 the Hessian may be touched by the plane 



ax'X +hfY+ c^'Z + dv" V+ew^Jr=0 



the result is CK~ = 0. 



The class of the Hessian being 16, it immediately follows that the 

 degree of C is 24. 



(2.) The envelope of polar quadric cones. 



10. The Degree and Class of C. — In Salmon's " Geometry of Three 

 Dimensions," Art. 510, it is proved that if ^ and v be the order and 

 class of a congruency, and M and N the order and class of the bi- 

 tangent surface ; then 



M- N=2{ix-v); 



putting Jf = 24, /A = 7, i/ = 3, 



we have iV= 16. 



"We shall also determine the value of N directly from geometrical 

 considerations (see Art. 13). 



