O'SuLLivAN — Points on the Curve of Intersection of tvjo Quadrics. 147 



Section 10. — Elli2Mc Parameters. 



{'^ dX 

 Let u = - and let Si^ = 4 (< - Ci) (t - e^) (t - 63) be the reduc- 



J5v7(A) 



ing cubic of/(X). The I'oots of Qt, regarded as a quartic with an infinite 



root, are homographic with those of /(A), since the anharmouie ratio of both 



is the same. If A, t be correspondents in this homography, (A, «, (3, y, g) = 



{t, e-i, 62, C3, cc) ; 



X- a.j6 - S t - Bi . 62 -00 _ _ («! - e,) (/3 - S) A - a 



•■• A-S.j3-a ° t - a> . e, - ei ' ''• " ''' " « - j3 A^S 



Similarly, 



1/ «w,a ./'(S) A-j3 , 1,^ ,, f{S)X-y 



and 



d\ dt r-^ ^A 



J/ (A) Jo, J 5 J/ (A) J.JOe 



Hence 



«^ 4 (pi^ - fii) ,?/- 4 (pn - e-,) z- 4 (pic - e.,) 



w' y-a.a-ft' vfi a - ^ . (5 - y' W- f5 - y . y - a 



is the parametric representation to the coordinates of UV. 

 Applying this representation to the canonical form (§9 (6)) 



X- 2^!i - Ci Y' p2C - e-i Z- ^j;« - 63 



W^ vm ' W'^ n{m-n)' W'^ m (m - n) 

 Let 



Jjm - e, = — = eu, J2m - e, - — = fu, Jpu - e, = -- = ^u, 



flU o tv (T to 



we may take the equations 



X - iOit T (i>u Z i4iu 



(1) 



fV Jm7i ' W J71 [m - n)' W Jm (m - n)' 



as the parametric representation of the coordinates in the canonical form. 

 The following relations connect the fnnctions 9, tj>, -ip. If I ^e^ - c,, 



^2 - (j)- = l\ }p'' - 0- = m\ (f- - B- = n-, m(f' + n\p- = {m + n)(d^ + mn) 



m<j>^ - n\p- = (in - n) (0- - mn). (2) 



Also, since ^'u = - 20(p\l/, 



dO , d'l, „ , d-iL . .„, 



du-'^^' dir-'^' £ = -^*- (') 



R.I.A. PBGC, VOL. XXXVI, SECT. A. [15] 



