92 KANSAS UNIVERSITY QUARTERLY. 



fore c, 9, and t. It is convenient to replace 6 by nt and thus have 

 c, n, t as the three parameters. 



Let T and Ti be any two collineations of the group eG3(ABCl) 

 for which the i^arameters are c, n, t and ci, ni, ti, respectively. Let 

 the values of the parameters of the resultant be co, n>, U. We have, 

 therefore, t2=tH-ti, n2t2=nt + niti, and c-2n2t-.>=:cnt + ciniti. 



One- and two-^tarametef subgroups of eG-i{ABCl). — If c and n 

 remain constant and only t varies, we get a one-parameter subgroup 

 of eGs. If c is fixed and n and t vary, or n fixed and c and t vary, there 

 result two-parameter subgroups. Thus we have two distinct singly 

 infinite systems of two-parameter subgroups and cc'^ one-parameter 

 subgroup of eG3(ABCl). The path curves of the one-parameter 

 subgroup are, except in very special cases, transcendental curves ; in 

 these special cases the subgroups are of other types than type II. 



Special subgroups of eGi{ABCl). — If t=0 and c and 6 vary, the 

 transformation along the line Al is identical and there remains a two- 

 parameter elliptic subgroup of type VIII. When c=go, n=0, 

 cn±:0, the transformation along BC is identical, and there results a 

 two-parameter subgroup of type IX. When c=go, n=0, and cn=0, 

 the transformation in the plane ABC is identical, and there results a 

 one-parameter subgroup of type VII. When t^O, c=oo, n=0, and 

 cn=0, i. e., when the conditions for a two-parameter group of type 

 VIII and type IX are simultaneously fulfilled, the transformations 

 along both Al and BC are identical, and there results a one-parameter 

 subgroup of type X. The elliptic group eG3(ABCl) has no real 

 subgroups of types VI or XL 



Theorem 8. The group eG3(ABCl) contains one two-parameter 

 group of type VIII and one of type IX ; also one one-parameter sub- 

 group of type VII and one of type X. 



The group eG3(ABCl) contains one other two-parameter subgroup 

 worthy of special notice. When c=0 the path curves of the one- 

 parameter group of collineations in the plane ABC are conies having 

 double contact at B and C. This group derives its importance from 

 the fact that, in case the plane ABC is at infinity and the points B 

 and C are the circular points in the plane, it becomes the group of all 

 screw motions about the line 1 as an axis. 



