PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 63 



In particular any line of the congruence cutting D' C and C D has 

 the required property. We may use instead of C, D any two points 

 of the line. If then C D and C D' do not intersect this gives us an 

 infinite number of congruences generating the complex to which s 

 belongs. 



13. For a general coUineation a A these lines r, s making with 

 each other zero angles have an interesting geometrical interpretation. 

 It is well known that a general coUineation whose linear invariant 

 (a A) vanishes has a system of tetrahedra A, B, C, D such that each 

 point is carried by the coUineation into a point of the opposite face. 

 Two opposite edges A B and C D oi such a tetrahedron determine a 

 zero angle. For in this case since C, D' are in the planes A B D and 

 ABC, the lines C D and C D' cut A B. 



Conversely if A B and C D are two non-intersecting lines making 

 with each other a zero angle and those lines are not left entirely 

 invariant by the coUineation we construct a tetrahedron upon them 

 as follows. Join A' and C , the correspondents of A, C through the 

 coUineation a A, to CD and A B respectively and let these planes 

 determine on A B and C D respectively the points B and D. Then 

 B will pass into a point B' such that ^1 B' cuts C D (i. e. a point of 

 A CD). Similarly for D. Thus, with the possible exception of 

 fixed lines, the entire system of non-intersecting lines making with 

 each other a zero angle consists of the opposite edges of these particular 

 tetrahedra associated with the normal coUineation a A. 



If P, Q, R, S are any four points it is seen on expanding the right 

 side that 



(aP) {AQRS) = Uo.PQ-ARS-aPR-AQS+ aPS-AQR] 



Hence if x is any point and ^ any plane (a .i-) {A ^) is expressible as 

 a sum of terms of the form a r-A s. Under any coUineation leaving 

 all angles invariant this last expression must be covariant. Hence 

 the form (a x) {A ^) must also be covariant. 



CoUineations leaving angle invariant must then leave the complex 

 c invariant and the coUineation a A fixed. We wish these angles 

 to be invariant under the group of transformations that leave distance 

 fixed. In that case c must coincide with />. There is a transforma- 

 tion of this group changing any distance x y into any equal distance 

 X y'. Since to x there can correspond through a A only one point y, 

 this point must be fixed under all the coUineations. Therefore to 

 each point x corresponds the point F. Hence 



aA = ^F 



