PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 61 



Since [F <j)] and [0 0] are zero this gives 



— X [((/). l-i) {p-(l)-pXo) + (<^.T2) (p-pxycl)) + (PiPXi-pXz)] 



{Fpxi) (Fpxi) 



^ — X { (<A -^-'i) (0 •^•2) + (0 '^'2) U'l </>) S + (y a:i Xa) 

 (Fpxi) {Fpx2) 



^ (p -^1 -^2) 



((A.vi) (0x2) 



Hence the correlation changes .rj, .r2 into two planes |i, ^0 such that 



^1 ^2 = Xi X2. 



In particular if X = 0, the correlation between x and ^ is the null 

 system determined by the complex p. The distance between any 

 two points is therefore equal to the angle between their polar planes 

 with respect to the complex p. 



12. Angl^ between two lines. We define the angle between two 

 lines r, s as such a function r s of their coordinates that, one of them 

 being fixed and the angle constant, the other satisfies a linear relation 

 {i. e. belongs to a linear complex) and for lines r, s, t oi a, plane pencil 



rs-{-st^tr = 0. 



By the same argument as for distance between two points we find 



fi (r, s) 



r s = 



1-2 {r, sY 

 where 



/i {r, 5) = - /i {s, r) 



and fo (r, s) factors into a linear function of r times the same linear 

 function of s. Hence 



{a r) (b s) — (a s) {b r) 



r s = 



(c r) (c s) 



where a, b, c are matrices of two rows and aba dyadic setting up a 

 correspondence between lines or complexes. The numerator of rs 

 can be written in a different form. In fact 



U Br) {CDs) -{CD r) {A B s) = 



h\[ABC-r-Ds] - [ABD-r-Cs] + [BCD-r-As] - [ACD-r-Bs]}, 



as is seen by expanding the right hand member. The expression 



