O'SuhiAVAN— Points on the Curve of Intersection ot ttvo Quadrics. l-il 



Sectiok 6. — If A (>'/, //, z' , v:') and A' (3/, - y', - z , iv') be a pair of corre- 

 sponding points of the first kind, the coordinates of the line A A' are 



^j = 0, 2' = sV, r = - :cy' , s = 0, t = y'v:' , u = z'v/. (1) 



Hence all such lines meet the two opposite edges yz, xxv, of the tetrahedron 

 of reference, and the line throngh a point A on C^F" which meets tliese edges 

 passes through the corresponding point Al , since only one line can be drawn 

 throngh a point to meet two non-intersecting lines. 



Hence, if a quadric F have these two edges as generators of one system, 

 it meets UV in four pairs of corresponding points which lie on four generators 

 of the other system. 



If j/, 2' . . . be the coordinates of any transversal of AA' , US', CC\ the ratios 

 c/ : t' : r : u' are constant. For they satisfy three linear relations expressing 

 that the line meets three lines for which f and q vanish. Hence, the 

 equation of F is qyvj + t'zx + r'zv} = vfxy, where p', g' . . . are the coordinates of 

 any generator of F of the same system as the lines yz, xw. For it is of this 

 form, since yz, xv: are generators, and the coordinates of the line AA are 

 ci = z'x', r = - x'y, t = y'v/, u = zw , so that the equation is satisfied for the 

 points A, B, G. 



If we form the discriminant of I; !\U - V) - 2F, it is easily seen that 6 = 0, 

 0' = for any quadric i*' and any quadiie (A), so the two systems are doubly 

 apolar. 



If four points ABCI) lie in a plane, and if we take two triads of points 

 ABC", A'RC", the planes BC"A"D, B'C"'A"I); C"AB"D, C"'A'B''D; 

 ABC'B, A'B'C'B are cut by any plane in two triangles whose sides intersect 

 in pairs in tlie points in which the plane meets the lines A"B, B"D, CD. 

 But A"B"CD lie in a plane (§ 3) .". the triangles are in perspective. Hence 

 the lines joining corresponding vertices pass through a point, i.e. the lines 

 AA',BB', C'C' meet a line passing throngh D. Hence AA', BB', C'C", 

 DD' are all generators of the same quadric F. Hence, if four points lie in a 

 plane, the lines joining any three of them to their corresponding points, with 

 the Hue joining the cotangential corresponding points of the fourth, are 

 generators of the same quadric F. 



Section 7. — We will call the line joining a pair of corresponding points 

 1 and r, whose coordinates are {xi, yi. z^, «•,), (x,, - yi, - z„ u-i), the "connector " 

 {1, I'j. The coordinates of any point [Jc, I) on this connector are 

 X = {k + I) .i'j, y = {k - I) y-^, z = {k - I) z^ 10 = [k + I) u\. Since x^, y^, z^, w^ 

 satisfy the equations U = 0, J' = 0, x, y, z,vj satisfy the equations 



ro^ + vfi f- + £- _ ax"- + l ie- \if- + 7;.^ 



w^if ^ (k - If " "' Kk + ff ^ u^ - If ' 



