O'SuLi.iVAN — Points on the Curve of Intersection of two Quadrics. 145 

 Now, let the quadric (^i) be the cone (8). Then Eq. (3) becomes 



X 



V/3-7-A-a. F-»f7-±2/v/7-"->^-|3. y -^U 



+ 2v/« -/3-A.-T- l^-7f^=0, (5) 



which, when rationalised, is 



^^%x'{^-ff{\ - a)-[V - aUy 



- 2S//V (y -«)(«- /J) (A - /J 1 (X - 7) ( F - J3 (7) ( F - 7 U). (6) 



This system of surfaces constitutes the quadricuspidal surfaces of de la 

 Gournerie.* 



Let A also coincide with S. (5) now becoines 



xjL(V-aU) ± yjM{V-l5V) ± zJN{V - yU) = 0, 



which is the equation of the developable generated by tangents to UV. 

 In fact, the generators now become the generators of quadrics of the system 

 which lie in tangent planes to the cone (S). But we have seen (§ 3) that 

 these are the tangent lines to W. 



If A coincides with one of the other cones, say (a), the equation becomes 



yjj -a.a-[i. V-jiU + zJa'-'ji.a-y.V^U = 

 or 

 ?/ { (a - /3>'^ + (y - /3>2 + (g - |3)w3 ! + ^^ ( (« - jyx' + (|3 - y)i/ + (S - y)w^ j = 0, 

 which is identical with O. In fact, the generators now are those which lie in 

 planes passing through the vertices of the cones (§) and (a), or through the 

 line y = 0, z = 0. 



Section 9. — I'he canonical form of uv. 



If V, v be the roots of 



u^ = (/j + -y - a - S) A= - 2 (jSy - a?) A •+ ^y („ + S) - aS(|3 + y) = 0, 



since the quadratic factors A - a . A - S, A - /3 . A - -y, of /(A) are both 

 harmonic with ?<.,, we have the relations 



A 



v' - a V - S „ v' - B v' - y - 



+ 5. = 0, ^ + ^ = 0. ,,. 



V - a V - b V - p V - y (1) 



If we take as fundamental quadrics, instead of II and V, Ug and Fq, where 



u, = vu- r, r, = vu- V; 



and write 



X'' = {v- a)x\ F^ = (v - jd)f, &c., 



* De la Gournerie, " Beolierches sur les surfaces reglees-tetraedrales symetriques." 

 Paris, 1867. Compare also Matthews, P.L.M.S, 



