brewster : collineations of space. 289 



§6. Type 1. 



19. Fundamental group of first class.— The group G 2 (ABCD)r-(- r' 

 1, of type I, second class, leaves invariant a tetrahedron and a sys- 

 tem of quadric surfaces, of which AC, BD, AD and BC are generators, 

 while CD and AB are reciprocal polars.* The one-dimensional trans- 

 formations are all loxodromic, while the plane transformations are of 

 type I. Here, again, we concern ourselves with but one of the quadric 

 surfaces, and shall therefore designate this group as ^(lrinm'F). 

 This group has as its independent parameters the cross-ratios k and k'. 

 Hence, it is a two-parameter group and contains go 2 collineations. 



There exists one such two-parameter group for each set of four 

 generators, chosen two from each system on the quadric. Therefore, 

 there are oc 6 collineations of space of type I which leave a given 

 quadric invariant. 



20. Higher groups. — Let us form the resultant of two transforma- 

 tions ^(ll'mm'F) and ^(U'mm'iF), where m' varies so as to remain 

 always a generator of the system ^. 



Express the two tetrahedrons as ABCD and ABi CDi, respectively . 



Substituting the relation of cross-ratios necessary for the presence of 



an invariant quadric, .viz., r + r' =1, we have our cross-ratios as 



follows : 



X T==AB BC CD DB AC AD, 



k k r k 2rl k 1 -' k lr k r 



1 T, 



The abov 



loxodromic transformations with two common invariant points com- 

 bine according to the law k 2 ==kki. Along AD 2 and AB 2 , where the 

 transformations have but one common invariant point, the same law 

 holds true. Moreover, through A, in the plane ADC, the loxodromic 

 one-dimensional collineations have two invariant rays in common, and 

 hence, combine according to the above law, and along any line cutting 

 that pencil of rays the new cross-ratio is k 2 2rl . The two remaining 

 cross-ratios may be determined by taking the product of cross-ratios 

 around the various triangles, this product around any given triangle 

 always equaling 1. From this it is readily seen that the group 

 property holds throughout. Hence, there is a three-parameter group 

 ^(ll'mF). 



♦Kansas University Quarterly, vol. X, p. 38. 



