32 KANSAS UNIVERSITY QUARTERLY. 



D. CONGRUENCE. 



(</). Two-TERiviED.^ — Invariant points of the line at infinity. 

 (/'). One-termed. — Invariant points of the line at infinity and 

 invariant rays of a pencil of parallel rays. 



E. PSEUDO-PERSPECTIVE COLLINEATIONS. 



(i). Two-termed pseudo-group. 

 (2). One-termed pseudo-group. 



In type V we have elations and under these the following groups: 



A. TWO-TERMED. 



(t). Invariant line-element, (24). 



(2). Invariant points of a straight line, (29). 



(3). Invariant points of a pencil, (30). 



B. ONE-TERMED. 



Invariant points of a straight line and invariant rays of a pencil on 

 the straight line. 



Finally we will add a formal representation of groups of per- 

 spective collineations in the plane. 



The whole plane contains 00^ perspective collineations .(combina- 

 tions of the 00^ straight line and the 00- points of the plane = 00*, 

 and 00^ values of the characteristic anharmonic ratio. Designating 

 a general perspective coUineation by the index c, an n-termed 

 group by Gn and its dualistic by Tn , a selt-dualistic group by Hn , 

 dilation by the upper index d, corresponding equal areas by a, 

 similarit}' by s, congruence or translation by t, elation by e, and 



the dualistic interpretation of the group G^^^ by T.,, the following 

 symbolic equations between the different groups and sub-groups 

 of collineations exist: 



oo^C^oo-G-g+ooSr^ + oo^* H^* + iG';+ooirJ+i Ht 

 Gc=oolG';; + lGd + lGe+lGa 



H^^ = ooiG.^+ooirc+ I Re 



GCz=ooiHj^+iG;^+ I Re 

 G| = oo^G|+ iHt 



* These 00^ groups are all equivalent, especially 00' groups can always be made 

 identical by a simple translation. 



