O'SuLLiVAN — Points on the Curve of intersection of tv>o ^uadrics. 135 



"quadruples" meet in sixteen points wliicli lie in funis on the coordinate 

 planes. In particular the tangent lines at four cotangential points meet the 

 tangent lines at their collinear points in sixteen points which lie in fours on 

 the coordinate planes. 



If four points ABCD lie in a plane, AB, CD are generators of opposite 

 systems of the same quadric (A). But AB, A'B' are generators of the same 

 system of (A) ; .'. A'B', GB are generators of opposite systems. .-. A'B'CD are 

 coplanar. Thus, if four points lie in a plane, any two of thcni are coplanar 

 with a pair of correspondents of the otlier two of the same kind. By the 

 same reasoning, since A'B'CD are coplanar, so are A'B'CD'. Again, since 

 AC'BD' are coplanar, so are AC'B"D"' , since B"D"' are the correspondents of 

 the second kind of BD' ; and in general any four points whose affixes are all 

 different are coplanar. Thus if ABCD lie in a plane we have three types of 

 coplanar points : [a) those wliose affixes are all the same, {b) those in which 

 they are all different, (c) those in which there are two points with one affix 

 and two with another, (a) gives four planes, (6) twenty-four, and {c) thirty- 

 six. Hence the sixteen points consisting of four coplanar points and their 

 cotangentials lie in sixty-four planes, four points in eacli plane, and sixteen 

 planes passing through each point. 



If we take two points A (xyzw) and B [x'y'z'ia) on the curve, and if 

 (jj, q, r, s, t, w) be the coordinates of the cliord {AB), those of A'B' are 

 {-p, q, r, - s, t, u), of A"B" (p, - q, r, s, - t,u), of {A'" B'") {p,q, - r,s, t,-a). Hence 

 from (§ 2 (3)) if the parameter of the AB be 0, those of A'B', A"B", A'" B"' 



11 . . /I - f^2\2 



are - ^, -, , and the anharmonic ratio of the four is 



i) «' \l + df (1) 



Hence the A.E. of the four chords which join two sets of cotangential points 

 is independent of the parameter of the qnadric of which they are generators, 

 and depends only on the parameters of the chords themselves. 



In particular, if A and B coincide, the parameters of the tangents at 



A, A', A", A'" are 0, -li,\,~ \, wiiere 6 = / — '^£ ((lOj § 2) Hence 



^ ^ V 1 + j/r 



the A.E. of ° the tangents at any set of four cotangential points is 



Also, since the tangent planes through any chord Aj to the curve are the 

 planes joining the chord to the four tangent lines Xj, the A.E. of the planes, 

 being the same as that of the tangent lines, is K. 



The A.E. of the four points in which any line (jjqrsin) meets tlie coordinate 



nlanes is - — . Hence (tj 2 (9)) the A.E. of the points in whicli any tangent 

 ^ ru 



