emch: projective groups. 31. 



Gruppen," page 510 to 512, to which the groups belong, and the 

 Arabic numerals in brackets denote the numerals of the groups in 

 Lie's table of projective groups of the plane (pages 288 to 291). 



A. THREE-TERMED. 



(i). Invariant line-element, III, (14). 



(2). Invariant points of a straight line, III, (21). 



(3). Invariant rays of a pencil. III, (22). 



B. TWO-TERMED. 



(4). Invariant line-element, V, (24). 



(5). Invariant points of a straight line and another invariant 



straight line, III, (32). 

 (6). Invariant ra3's of a pencil and another invariant point, 



III, (33)- 

 (7 and 8). See two-termed groups of type V. 



C. ONE-TERMED. 



(g). Invariant points of a straight line and invariant rays of a 



pencil being not on the straight line. III, (38). 

 (10). Invariant points of a straight line and invariant rays of a 

 pencil on the straight line, V, (39). 

 As groups of the special cases of collineations we have within 

 the type III, the following: 



A. DILATION. 



(a). Three-termed. — Invariant rays of a pencil of parallel rays. 



{/'). Two-termed. — (i). Invariant rays of a pencil of parallel 

 raj's and another invariant point. 

 (2). Invariant point of a straight line. 



(<;■). One-termed. — Invariant points of a straight line and invari- 

 ant rays of a pencil of parallel rays. 



B. CORRESPONDING EQUAL AREAS. 



{(i). One-termed. — Invariant points of a straight line. 



C. SIMILARITY. 



(if). Three-termed. — Invariant points of the line at infinit}'. 



(/'). Two-termed. — Invariant points of the line at infinit}' and an- 

 other invariant straight line. 



(("). One-termed. — Invariant points of the line at infinit}' and 

 invariant ra}s of a pencil. 



