of lines by projection from four dimensions 141 



It is not then allowable to modify further these symbols by multi- 

 plication with numbers, except the same multiplier for all. And, 

 the six points being supposed not to lie in a three dimensional 

 space, there is no further syzygy connecting them. 



Each of the points P, Q, R of our diagram (§ 2) is then ex- 

 pressible linearly by two of these six points, P hj B and C , etc., 

 while P, Q, R, being collinear, are themselves connected by a 

 syzygy. This becomes then a syzygy for A, B,C, A', B' , C", which 

 must be the same as the fundamental syzygy. Whence it appears 

 at once that, by absorbing proper numbers in the symbols P, Q, R, 

 we can write 



P = B + C', Q = C + A', R = A + B'. 



By a similar argument we deduce for the points P', Q' , R' of our 

 diagram (§ 2) 



P' = B' + C, Q' = C' + A, R' = A'-\-B. 

 From these two sets of equations we have however 

 P + P'^-{A + A'); ^ 



there is then a point lying both on the line PP' and on the line AA' , 

 namely these lines intersect, say in the point L. So also M, N are 

 the points B + B', C + C respectively. And the identity 



A + A' + B + B' + C+C' = 



shows that L, M, N lie on a line, e. 



A plane meeting the lines a, b, c can evidently be defined by 

 three points, one on each of them, represented by symbols of the 

 form 



yB + zC, zC + xA', x^A + yx^'. 



In order that the plane should also meet the line d, it is necessary 

 and sufficient that numbers A, /x, v, p, a exist for which there is 

 the syzygy 



A {yB + zC) + ijl{zC+ xA') + v {x^A + y^B') + pP' + aQ' = 0; 



if herein P' , Q' be replaced respectively by B' + C, C + A, it 

 must reduce to the fundamental syzygy. Thus we find at once 

 that Xx = X and y-^ = y, and any one of the oo ^ planes meeting 

 a, b, c, d is the join of points respectively on the lines a, b, c, 

 given by 



yB + zC, zC + xA', xA + yB'. 

 This plane however contains the point represented by 



X {yB + zC) + y{zC + xA') -\-z{xA + yB'), 

 which is yzP' + zxQ' + xyR' , 



and is therefore its intersection with the line d. And the plane 

 contains the point represented by 



{yB + zC) + {zC + xA') + {xA + yB'), 



