of lines by projection from four dimensions 143 



dimensions, are subject to two syzygies, whicli, witliout loss of 

 generality, may be written 



A + B + C + A' + B' + C = 0, 



aA + bBi-cC + a' A' + b'B' + c'C = 0. 



The points P, Q, R, respectively in syzygy with {B, C), with 

 (C, A'), and with {A, B'), are themselves in syzygy. Thus, not 

 only is P given, save for a multiplier, by an expression fjuB + v'C, 

 and Q by vC + X'A', and R hy XA + [x'B', but the multipliers 

 fx, v', V, A', A, jLt' may be chosen so that we have 



liB + v'C + vC + X'A' + XA + fji'B' = 0. 



This must then be a consequence of the two fundamental syzygies; 

 or, for a proper value of p, we must have 



A : [x : V : X' : jx' : v' = a + p :b + p \ c + p : a' -\- p -.b' + p : c' + p. 

 Thence P, Q, R may be expressed by 



P = bB + c'C +p {B + C), 

 Q = cC + a' A' + p (C + A'), 

 R^aA + b'B' +p {A -\- B'). 

 Similarly P' , Q', R' are expressible by symbols 

 P' = b'B' -{-cC +p' {B' + C), 

 Q' = c'C +aA +p' {C + A), 

 R' = a'A' + bB +p'{A' + B). 



Suppose now that one of the two transversals, say ?/, of the 

 four lines a, b, c, d, intersects one of the two transversals, say u, 

 of the four lines a', b', c', d'. Then as before (§ 1), remarking that 

 the two quadric surfaces determined respectively by the triads of 

 lines (6, c, u), {b', c', u') contain respectively also the lines {a', d', ii'), 

 {a, d, u), we infer that the points A, A', P, P' are coplanar. And 

 by similar reasoning, also because m, u' intersect one another, we 

 infer that B, B', Q, Q' are coplanar, and also that C, C, R, R' are 

 coplanar. Conversely let us assume only the first fact, that 

 A, A', P, P' are coplanar. We can then prove that the numbers 

 p, p', occurring respectively in the expressions of P, Q, R and of 

 P' , Q' , R', are equal. For the fact that A, A', P, P' are coplanar 

 involves the existence of a syzygy of the form 



m [bB + c'C + p{B + C)] 



+ m' [b'B' + cC + p' {B' + C)] + pA+p'A' = 0; 



for suitable values of m, m' , jj, p' and of another number k, this 

 must be the same as 



{a + k)A + {a' + Jc) A' + {b + k) B + {b' + k) B' 



+ {c + k)C+{c' + k)C = 0; 



