O'SuLLlVAN — Points on the Curve of intersection of ttvo Quadrics. lol 



From (12), putting u = wi, idz (03 successively, 



0i,„, = ijn (n ~ iTj, \ 



^jwi =Jn (m + n], 

 •^JtDi = J VI [m + 11). 



(16) 



\Pi<02= ei ll(n - il). 



(17) 



6\wz = - ijm . m - / 

 ^l«3 = -Jlim - I), 

 -fhh = -Jim. 



(18) 



Section 11. — Representation of Conoiectors hy Points on a Quartic Curve. 



It is known from the theory of the plane cubic that a 1-2 correspondence 

 exists between points on the Hessian of a cubic and tangents to the Cayleyan. 

 If we project on an arbitrary plane from a point on the curve UV, the curve 

 projects into a cubic which is the Hessian of three cubics, whose Cayleyans 

 are the envelopes of the projections of the connectors of the three systems. 

 It might be expected then that a correspondence of the same type would 

 exist between the connectors and the points of Wa.s exists between tangents 

 to the Cayleyan and points on the Hessian. We will now show how this 

 correspondence can be determined analytically. 



If (p, q, r, s, t, u) be the coordinates of the connector passing through 

 A (x, y, z, (ii), their values are (§ 6 (1)), 0, zx, - xy, 0, yw, ztc, and A lies 

 on the surface 



$ s (^ - a) ^x" + (j3 - §) y''vf + (/3 - a) xhf + (7 - S] s'w-. 

 Hence q, t, r, n satisfy the equations 

 r - 2 {qt + ru) =0, - r^ [y - a) q" + (/3 - g) (1= + (/3 - a) r- + [y - l)iiP = 0. (1) 



The coordinates of a connector then are four quantities connected by two 

 quadric relations, just as the coordinates of a point on the curve UV are, so 

 that if r and r be regarded as two quadrics, their curve of intersection Vr is 

 a quartic curve, the points on which may be taken to represent the connectors 

 in the first correspondence. 



The discriminant of XT - r is 



(A^- - M) (\' + N) = A (A) = 0, (2) 



the roots of which are 



+ JJ/, +J^^, or ±2im, ±2in. (§9(5) 

 Since it is in the canonical form, P and ;■ are a pair of Vossian quadrics of 

 the system (§ 9), and their equations can be reduced to the canonical form 



