NOV ^7 1G95 



Kansas University Ouarti-.rly. 



Vol. IV. 



OCTOBER, 1895. 



No. 2. 



Continuous (iroups of Projective Transform- 

 ations Treated Synthetically. 



Prof. Soplms Lit-, in his Vor/rsu;i^i:^('ii i/r/x-r Coiitiniiii-rliclir Gruppen. 

 Abteilungen i and 2, develops in detail an analytical theor}^ of 

 continuous groups of projective transformations in one and two 

 dimensions. In the domain of Projective Geometry, anal3'tical ami 

 synthetic methods are of approximately equal power, and are 

 always mutually supplementary. The object of this paper is to 

 develop a synthetic theory of these groups, based on geometric 

 construction. In the course of this development all of Lie's chief 

 results are reached, and many new relations are seen when the 

 subject is approached from this new point of view. 



In this paper only groups of projective transformations in one 

 dimensional space will be treated. The development of the theory 

 for the projective groups of the plane will appear later. 



>^1. Construction of Projective Transformations. 



Let 1 and 1' be 

 two lines intersect- 

 ing in O and mak- 

 ing any convenient 

 angle with on e 

 another; and let 

 OX be the bisector 

 of the angle lOl'. 

 A projection of the 

 line I on 1' is de- 

 termined by choos- 

 ing three points on 

 1 and their three 

 corresponding points on 1.' Thus let x, y, z be three points on 1 

 and X,' >•.' 7/ the three corresponding points on 1.' (Fig. i ). Draw 



(Tl) KAN. UNIV. ^U.\K., V(JL. IV. NO. :.', uCTOUliU. lS9-">. 



Fig. 3. 



