472 ■ Proceedings of the Royal Irish Academy. 



11. Since the polar cones having their vertices at P and P' touch 

 the surface C in two sets of six points that are respectively poles of 

 the planes P'VV and PVV, they are situated on the curve of inter- 

 section of the polar cones of Fand V ; but it has already been seen 

 (Art. 7) that these cones have the line PP' as a common gene- 

 rator ; therefore the twisted cubic which is the remaining portion of 

 the curve of intersection passes through the twelve contacts of the 

 cones having vertices at P and P'. 



12. The comflete intersections of the tivisted nibic and the line PP' 

 with C and the Sessian. — Through the line W can be drawn twelve 

 tangent planes to the Hessian distinct from the tangent planes at V 

 and V'\ the eighty-four poles of these fourteen planes that lie on 

 the surface C will therefore be situated either on the line PP' or the 

 twisted cubic referred to in Art. 10. Now since the equation of the 

 polar plane of any point on the line PP' is of the form 



%x[ax'^ ^Uo, 



there are always two points harmonic conjugates with regard to PP' 

 which have a given common polar plane passing through the line W . 

 The only exceptions are in the cases of the points P and P' . We can 

 now arrange the eighty-four poles of the fourteen planes in the 

 following table : — 



Of the eight poles of tangent plane at Ftwo are coincident at XT, 

 one at T, and five on the twisted cubic, and we can similarly account 

 for the poles of the tangent plane at V . 



Of the eight poles of the tangent plane at P two are coincident at 

 P'^ and the remaining six are on the twisted cubic, and similarly for 

 the tangent plane at P' ; finally, for each of the remaining ten 

 planes two coincident poles are on the Hessian, two on the line PP' ^ 

 and four on the cubic. These ten pairs of points and the two contacts 

 at T and T' is the complete intersection of PP' with C. 



In order to account for the points in which the bitangent surface 

 is met by the twisted cubic, it is only necessary to observe that the 

 polar cone having its vertex at TJ touches C in six points, one at T^ 

 and in five other points on the twisted cubic ; these six points being 

 the poles of the tangent plane to the Hessian at V\ we see therefore 

 that the twisted cubic touches C in ten points, meets it in two hexads, 

 and in ten tetrads, or seventy-two points in all. 



The same cubic intersects the Hessian in TJ, U', and the points of 

 contact of the ten planes in this article. 



