BREWSTER \ COLLTNEATIONS OF SPACE. 293 



that the values of the cross-ratios of the one-dimensional transforma- 

 tions in each case are taken perfectly general and with no fixed rela- 

 tion to each other. Our next task will be to investigate the result of 

 choosing space collineations in which the one-dimensional transforma- 

 tions bear definite relations to each other. By proper choice of par- 

 ameters, we may resolve a collineation of any type into components of 

 almost any other type. This is the foundation principle of group 

 structure ; for it is only by this means that we may obtain groups of 

 one type within groups of another type. 



C. — Group Structure. 



§1. Type XI. 



28. For special values or specially related values of the parameters 

 of a group, the resulting transformation may degenerate into one of a 

 lower type. For the most part these will be only indicated where 

 they are easily seen. It has been shown already that the group 

 11 Gi(Z/aF) contains only transformations of that type. Hence we 

 may state in symbolic language as its structure, 



G 6 (F)Z2 oc 1 ]1 Gi(^F). 



§2. TypeX. 



29. Two transformations of type X, having reciprocal values of k, 

 the same invariant system, but only one other common invariant gen- 

 erator, result in a collineation of type XI ; since along a line two loxo- 

 dromic transformations, with reciprocal values of k and but one in- 

 variant point in common, result in a parabolic transformation. Hence, 

 by proper choice of components within the group 10 G<}(1 p F) we may 

 get all the collineations of the group n Gi(7 /iF). Combining cc 1 of 

 the groups 10 G,(UF),we get "'G^F); but the oo 1 n Gi(Z/*F) do 

 not form a two-parameter group of type XI. Symbolically expressed, 

 we have : 



10 Gi(ZZVF)= = 10 GK(ZZ>F) 



10 G-2(/Z> F) ^cx 1 10 Gi(Z FiiY) + n Gi(Z/xF) 

 10 G 3 (^F)= = oo 2 10 G 1 (TZ T /aF) + oc 1 u Gi(TftF) 

 G 6 (F) Z 2 10 G 3 ( ^ F) == 2 oc 2 w Gi(T T> F) + 2 oo 1 J1 Gi(7 /TF). 



§3. Type IX. 



30. In article (10) it was shown that the invariant generators of a 

 transformation of type IX divided harmonically the line of invariant 

 points and the axis of invariant planes. Therefore, as t varies, the 

 axis of planes must also vary, so that when t, the axis of invariant 



