BREWSTER: COLLINEATIONS OF SPACE. 295 



§5. Type III. 



32. If from the group 3 G-i(H'mF) we choose two transformations 

 with reciprocal values of k. then along m the result is identical, and 

 we have the invariant figure (w.AF), that of type XI. For every 

 pair of reciprocal values of k, we have a new transformation of type 

 XL with this same invariant figure; hence the group n Gi(y»AF). 

 Similarly we determine the presence of the group 10 Gi(7/VF). 



In combining cc 1 of these two-parameter groups of type III, we find 

 that we have not ex 1 distinct groups u Gi(mXF), but only the same one 

 occurring each time. There now appears another group n Gi(Z^F), 

 each tranformation of which is obtained by choosing two transforma- 

 tions from different two-parameter groups 3 G>(irmF) and 3 G 2 (H"mF), 

 with t = — ti, and k = -^-> when our transformation along 1 becomes 

 identical, and that along m parabolic. Having these two groups, 

 H Gi(//*F) and n Gi(-;» A.F), on different systems of generators, the 

 presence of y G 2 (lmpF) is recognized from the previous work (art. 30), 

 and we have, finally : 



3 G 2 (U'mF) = 3 G 2 (U'mF) + 10 Gi(ZZ>F) f u Gi(mAF). 

 3 G 3 (lmF)-oci *G,>(17'mF) -f «>G 2 (7/xF)+ 9 G 2 (lmpF) + u Gi(^F) 

 + n Gi (mXF). 



3 G4(mF)==oc 2 ^(TT'mFj+^GaC/xFj + oc 1 !, G,.(Tm^F) 

 -j-cx 1 1J G 1 (7^F)+ 11 Gi(wAF). 



It is to be noted that in 3 G4(mF), while there occurs ex 1 n Gi(l^ F), 

 the group n Gi(m A.F) occurs only once. 



Second class. — The structure of the second class involves but one 

 type other than itself, save where transformations of such types occur 

 as singular transformations. If we consider the resultant of two 

 transformations of type III, with 1 and m as the only common invariant 

 generators, we have the resultant one-dimensional transformations 

 parabolic ; i. e., the resultant space collineation is of type IX. 

 3 Gi(ll'mF)a = 3 G 1 (ll'mF) ;i + S. T. 

 3 G-2(lmF)a==oc 1 3 Gi(17'mF)„-K''G 1 (lmp*F)+ S. T. 



§6. Type I. 



33. First class.— The groups of this most general type include 

 all of the groups of the types heretofore considered. The method of 

 determining the presence of the various types is the same as that em- 

 ployed before. The list, while long, is very simple, if we bear in 

 mind that, having once determined the presence of one group, we 

 have necessarily determined at the same time the presence of all of 

 its component parts. Thus, for example, in ] G 3 (ll'mF). we may de- 



21-Kan. Univ. Sci. Bull., No. 11, Vol. I. 



