O'SuLLiVAN — Points on the Curve of Intersection of two Quadrics. 133 



two generators of opposite systems whose parameters are 6 and ^ are given by 

 kJXoi .x=6+<p, kJX0 .y = 6ip-l, ikjj . s = S^ + 1, U-Js . to = - <p, 



where ;,. = —— ,y. 



s/Xb ■ y - -y^y • s- ^ ' 



Hence any two generators of a quadric (A) which have the same parameter 

 meet on the plane v- = 0, and conversely. 



The parameters chosen here have special relation to the grouping TZ, 

 xvj of the coordinate planes. 



The tangent line to UV at tlie point x'y'z'vj' , parameter X, is the inter- 

 section of the tangeiit lines to U and V, viz. Ixx = 0, ^uxx' = 0. Hence 



omitting the common factor 



il 



_ v/5 _ 



P=jLX.s, q=J.UXBS. r=jN\^s, (9) 



S=JZX$y, t=J.yXya., V.=jNXaB. 



Comparing these values witli those in (da) we find that if be the para- 

 meter of the tangent, 



1-6' M .— , ah-- J^^ 



^, = - ICf =J^> say, whence = / . 



1 + 0- < N V 1 + JX (10) 



Hence the parameter of a tangent to Z7F is a constant. 



We see from (8) that the tangent at a point whose parameter is A is a 

 generator of the quadric (A). 



Section 3. — If x, y, z, v: be the coordinates of a point A on the curve UV, 

 the seven other points derived from A by varying the signs of the coordinates 

 also lie on the curve, and as they have the same parameter X (§ 2;, the 

 tangents at them to the curve are all generators of the same quadric (A), 

 four of one system and four of the other. We will call a generator of one 

 system, without regard to its parameter, a generator Ai, one of the other 

 systems a generator Aj. 



These eight points form two groups of cotangential points. Any pair 

 belonging to one of the groups is called a pair of corresponding points,, 

 i.e. corresponding points on the curve are defined as those whose tangent 

 lines are generators of the same system of the same quadric. A pair 

 belonging one to each group may be called a pair of collinear points, because, 

 as we shall see, they are collinear with one of the vertices of the tetrahedron 

 of reference. 



[U*] 



