BREWSTER : COLLINEATIONS OF STACK. 299 



with the same invariant figure, would give an identical transforma- 

 tion, while if their invariant figures differed in any part the resultant 

 would necessarily be of type I. 



§2. Type VI. 



36. The groups of type VI of the first variety correspond very 

 closely to those of type I, second class. If we choose k= -1, the 

 cross-ratios become : 



AB BC CD BD AC AD 

 + 1 -1 +1 -1 -1 1, 



which we see to be an involutoric transformation of type X, with the 

 invariant axes reciprocal polars with regard to the quadric surface. 

 Such transformations, we have seen, cannot form a group. More- 

 over, in each (; Gi(H'mtn'/m'F) there exists but one such transforma- 

 tion — i. e., the one for k = — 1. 



In building up the higher groups of type VI of the first variety 

 these singular transformations remain throughout, but when com- 

 bined with other types which may occur in the group of type VI, 

 they produce no new singular transformations. But in G G2(APpsF) 

 we have a group of collineations of type IX, each of which, when 

 combined with the singular transformations of type X, gives a re- 

 sultant of type IX, but with a new line of invariant points, there- 

 fore, not belonging to the group 9 Gri(lmp£F) — i. e., we have singular 

 transformations. Moreover, these will remain as singular transforma- 

 tions in 6 G3(PsF), since the cc 1 axes of invariant points included in 

 the oc 1 ! 'G(lmp£F) necessarily lie in a cone, while those of the singular 

 transformation may take any position in space. 



§3. Type III. 



37. The only singular transformations left for consideration are 

 those of type XI and type IX occurring in groups of type III, second 

 class. Our fundamental group here is 3 Gi(H'mF) a , in which there 

 exists the relation of parameters, k = a 1 where a is a constant. It has 

 been proved* that we may so choose two transformations with cross- 

 ratios k = a l and ki=a f , that the resultant may be k2 = a 1 + t i = l, 

 where t -f- ti is not 0. Such a resultant would give an identity along 

 m, and a parabolic transformation along each generator of system A. 

 This is a collineation of type XI. These singular transformations of 

 type XI in the group 3 G-2(lmF) a may combine with the collineation of 

 9 Gi(lmp£F), resulting in singular transformations of type IX. 



Therefore, we find : 



3 Gi(ll'mF)aZ 11 S.T.(^AF) 

 3 G 2 (lmF)aZ 11 S.T.(^AF)+ 9 S.T.(lmpTF). 



♦American Journal of Mathematics, vol. XXIV, No. 2, p. 159. 



