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VIII. 



COEEESPONDING POINTS ON THE GUEVE OF INTEESECTION 



OF TWO QUADEICS. 



By a. C. O'SULLIVAN, M.D., S.F.T.C.D. 



Eead Juxe 25, 1923. Publisliid Decembek 17, 1924. 



Section 1. — The curve discussed in this paper is the quartic curve of the 

 first class, namely, the curve of intersection of two quadrics. ' 



For the sake of reference we give the characteristics of this curve, assum- 

 ing that the two quadrics do not touch. 



m = 4, n = 12, -/• = 8, « = 16, /i = 



x = l&, y = 8, g = S8, h=2. 



It will be useful to bear in mind that every generator of a quadric containing 

 this curve is a " line through two points," and consequently it follows that, 

 since through any assumed point a quadric containing the curve can be drawn, 

 the two generators of that quadric which pass through the point are two 

 "lines through two points." 



This curve has attracted the attention of mathematicians from time to 

 time, and each investigation sheds a fuller light upon the beautiful geometry 

 of the curve. 



Section 2. — Let the quadrics be 



f/ = 3;- + y- + ST + v/, V = ax- + \iif + yz- + Svf. 



Then any quadric \U - V passes through their curve of intersection, 

 which we will call the curve II V. The discriminant of \JJ - V is 

 /(A) = X - a . X - jS . A - 7 . X - 8, and the four cones of the system are 

 those quadrics for which A has the values a, /3, 7, S. 



For brevity we will write Xa for A - a, X^y for A - /3 . A - 7, &c., and 

 will call the quadric \U - V the quadric (A) . 



Also let Z = i3 - 7 . n - 8, if = 7 - a . /3 - S, A^ = « - /3 . 7 - S. 



n = LMN, the product of differences of the roots of /(A) = 0. 



R.I. A. PROC, VOL. XXXVI, SECT. A. [14] 



