TRANSACTIONS OF THE SECTIONS. 29 



It will be simpler, and not less general, to take the centre of the inscribed circle 

 (1, 1, 1) as the origin of transversals ; the cubic then becomes 



^O-y)+L , (y_ o ) + f!(a-/3)=0. 



a /3 y 



3. The vertices of the quadrilateral (+ar, +»/, +z) constitute the first tetrad of 

 corresponding points, of which the centre (1, 1, 1) is their common tangential 

 point. A second tetrad of corresponding points is formed by the diagonal points of 

 the quadrilateral A, B, 0, and D, the centre (1, 1, 1); their common tangential 

 point (a; 2 , y", s 2 ) is the conjugate pole of the first tangential (1, 1, 1) with respect 

 to the system of conies (in § 2). 



4. A transversal FDD' intersects this cubic; the coordinates of P D D' are thus 

 connected. 



Let (/, g, h), (1, 1, 1), (\F, XG, XH) denote them ; they are thus related, 



F/_Ggr_HA 



F+f=G+g=Jl+h = 2r. 



By the first property, the two points are conjugate poles with respect to the 

 system of conies through the basis-points (+x, +y, +s). 



5. Transversals through a point in the cubic (P) pass through the points of the 

 second tetrad, A, B, C, D; the coordinates of the third points of intersections 

 A', B'. C. D' are 



F f ft*$/ f G,A}/ } ftH; F,G,H. 



(3. If these four new corresponding points be joined in pairs with the four former 

 points, the points of intersection are three, and correspond with P. 

 Thus 



A'D and AD', B'C and BO' intersect in (Q) (/, Ct, H), 



B'D and BD', AC and A'C „ in (R) (F, g, II), 



CD and CD', AB' and A'B „ in (S) (F, G, h). 



7. For the tangent to the cubic at a point (/*, g, h) } 



By the aid of the relations established in § 4, this becomes 



|r07-A)(/-F)+|(A-/)07-G)+|(/- ? )(A-H)=O. 



So the equation to the tangent at A'(F, g, h) is 



}(ff-ty(F-f)+^h-FXg-G)+l(F-g)(h-n)=0. 



This and the tangents at B', C, D' intersect in the common tangential point 



<f-F) fi(ff-G) y(ft-H) 



fF i f{F+f)-JiG-hg}-gG{g((i+g)-Fli-fh}-hll{h(ti+h)-FG-fgy 



In like manner the tangents at P, Q, R, S are concurrent in 



q(/-F) fi(ff-O) y(A-H) 



fF{f(F+f)-gH-Gh}-gG{g(G+g)-Fh-fiI}-hll{h(ll+h)-fG-Fgy 



