470 Proceedings of the Royal Irish Academy. 



with two similar equations for v and w ; adding these, we have 

 2{x + y) {axx' - hjy') 



= (^' + y') («-^^ - %^) ± 2 h'o{ax^ - %2)2 _^_^y(^^yf_^iy\^ [axx'- hyy'). 



When this expression is rationalised the result is of degree 16; it is 

 satisfied by putting axx' - hyy' = 0, and also by putting xy' -x'y = 0: 

 there are therefore 14 points common to the surfaces, and therefore 

 7 lines through x'y'z'v'w' which connect corresponding points. 



8. The tanyent planes to the surface C. — The plane of the two lines 

 which intersect in T will touch C at T', and the plane containing the 

 two that intersect in T' will touch it at T ; but we have seen in Art. 

 3 that the two directions at P, for which consecutive lines of the 

 congruency intersect in ^ and T', are those joining P to Fand V ; 

 we infer therefore that the planes PP' V and PP' V touch C at the 

 points Tand T'^ respectively. 



From the values of the coordinates of these four points, viz. : — 



P. . . X, y, z, V, w, 



P. . 

 V. . 



v. . 



by' cs' dv' ew 



hf-e^ c%^~e^ dv'-d^ etv-'-e^' 



e: if--e^ c%^-e^' dv-'-e^ 



we immediately deduce that the equation of the tangent plane at T is 



= 0, 



and of the tangent plane at T' is 



2-^ = 0. 



ax~ — 



It may here be observed, though it will afterwards appear more 

 naturally, that the tangent plane at T touches at T the polar quadric 

 (cone) of V with respect to the cubic, and a similar statement is true 

 for the tangent plane at T' . "We easily verify this by considering the 

 polar cone of V, its equation is 



