140 Proceedings of the Royal Irish Academy. ' 



Section 5. — The squares of the cooidiuates of points on vi- are systems of 

 four quantities connected by two homogeneous linear relations, just as the 

 coordinates of points on a line are. Hence the same algebraic relations hold 

 among the two systems. Thus, for example, if 1, 2, 3, 4 be any four points on 

 UV, oa-i- = fei^ + IXi-, 7/3- = Iqji^ + lyi', &e. 



Hence {y-^z^- - y.^z{-) (ar-J'wi' - .Cj-w/) = (x'j'^'-'^' - ^^i^^') (j'a'-'i' - 2/4V) (1) 



If the four points lie in a plane, ps' + f's = 0. (§ 3.) 



i.e. (;yiS2 - 2/2*1) (■V-'i - ^'i^'-'s) = " (■^I'^'-'-z ' •^2«'i) (.%«4 - Vih)- (2) 



Dividing (1) by (2) 



Adding and subtracting (2) and (3) 



y-^z^^w^ + y^v^iW^ + y^^x^w^ + y^Sg-rjWa = 

 with two other pairs formed from the groupings s:r, yv) \ xy, zv:, giving 

 6 equations which hold between the coordinates of four coplanar points on V V. 



In general, the condition tliat four points shall lie on a plane is the 

 vanishing of the deteiminant {.r^j^i.^vj^. This delerniinant is formed of six 

 blocks, each containing four terms, all affected with the same sign. If the 

 points lie on UV, each of these blocks vanishes separately. 



A more symmetrical relation between the coordinates of four coplanar 

 points is 2/'(o) ■ x^x.^x^x^ = 0. 



It may be obtained as follows : — 



If four points 1, 2, 3, 4 lie in a plaiie, the chords (12), (34) are generators 

 of the same quadric (A) ; hence, (I, 2) and (3, 4) are conjugates with respect 

 to that quadric, and '2(X - aj^r^.r^ = 0, 2 (A - (O-'s^^ = ov the plane 

 P = 2(A - a)x contains the points x-^^ — x-^i-^, and x^^ = x^x^. This plane touches 



■ X _ 2 

 the quadric 2/'(a) . x^ = S, for all values of A since ^ ' ' = 0. Hence it 



/(a) 

 contains two genera.tors of ^S'. One of these is the intersection of the planes 

 3« = 0, 2«* = 0, and is fixed. The other varies with A. 

 If the parameters of 1, 2, 3, 4 be ju^, fi^, jug, /i^, 



If (a) .a^i2- = ^ y/, X ^ 0, ana 2/ («) . x,J = 2 ^ „/' ^ 0. 



Hence S contains the points r^, x.^^. These points must lie on the variable 

 generator common to jS and P, since they do not satisfy the equations 2rr = 0, 

 S(i:r = 0. Hence they are conjugate with respect to <S, and 



'^f{»)^v^u ^ '^f\'^)^'^-zH^^ = 0. 

 This condition is satisfied if the chords (12) (34) are generators of the same 

 quadric (A), whether they be of the same, or of opposite systems. 



