AUG 3 1896 



Kansas University Quarterly 



Vol. v. JULY, 1896. No. i. 



Projective Groups of Perspective Collineations 

 in the Plane, Treated Synthetically. 



[A Dissertation presented to the Faculty of the University of Kansas to attain the 

 decree of Doctor of Philosophy.] 



BY .\RNOLD EMCH. 



Preliminary. 



In presenting this paper the author has attempted to consider 

 the subject of Perspective Collineations from the modern point of 

 view of the theory of groups. Hitherto collineation has always 

 been treated either by means of descriptive geometry, or analytic 

 geometry, and without reference to group properties. 



Sophus Lie's " Vorlesungen iiber Continuierliche Gruppen," 

 Ayhich will be chiefly referred to, treats of the projective transforma- 

 tions of the plane of which collineation is only a special case. 

 There are five types of projective transformations in the plane 

 under which all the groups can be subordinated, and it has been 

 the aim of Professor Newson to enumerate the projective groups 

 of the plane by means of the synthetic method, and to classify 

 them according to those five types. 



This paper treats in the same manner the two types of trans- 

 formations that are known as perspective collineations. 



For suggestions in the treatment of the subject the author is 

 thankfully indebted to his collaborator. Prof. H. B. Newson. 



§1. Representation of Perspective Collineation.* 



If the corresponding lines a, a^ ; /?, fi^; y, y^; 8, 8^; intersect 

 each other on the line of intersection of the two planes tt and tt^, 

 and fulfill the further condition that all the connection-lines of 



*Perspective Collineation has the same meaning as the German Centrische Collinea- 

 tion (Fiedler), or the French homoUxjie. 



(1) KAN. CNIV. QUAR., VOL. V, NO. 1, .JULY, 189G. 



