464 Proceedings of the Royal Irish Academy. 



The geometrical interpretation of this quadratic leads to the follow- 

 ing property of the surface C. 



Suppose the line PP' meets the Hessian in two other points, U, V , 

 the coordinates of either of these poiats are proportional to 



4> ^ 



ax by 



and in order that this point may lie on the Hessian^ we must have 

 _ 1 ^ X 



0, 



ax~ + (i> 

 a\x + ^ ' 

 \ ax 



the roots of which are 



(J3 = - 6i, and cfi = - Oo. 



It follows, therefore, that TU, and T' U' are divided harmonically 

 by PP' Hence we derive a construction for the surface C. Take a 

 pair of corresponding points P, P' in the Hessian, produce the line 

 joining them to meet it again in TI, V ; the harmonic conjugates of 

 UTJ' with respect to PP' are the points on the bitangent surface in 

 which it is touched by PP ' . This is exactly analogous to the property 

 of the Cayleyan of a plane cubic. 



3. The directions of the points near P and P', so tha consecutive 

 lines of the congruency may intersect, can be readily found. 

 Erom (3) we have, for the point consecutive to P, 



hence the directions are on the lines joining P to the points F, F', 

 whose coordinates are 



« _ _ y 



ax^ - 01 hy~ - ^i' ■ ' ' 



X y 



ax^ - 0^' by' - 6^' ' " 



These points F, V are on the Hessian, and they are the points 

 corresponding to U, U', respectively. The tangent plane to the 

 Hessian at F is 



