28 KANSAS UNIVERSITY QUARTERLY. 



Theorem 22. TJie system of similarities leaving the points of the line 

 at infinity iniHiriant forms a three-termed groi/f. 



If the centres C, C^, C", are confined to a straight line the 

 theorem follows: 



Tlieorem 2j. The system of similarities leaving the points of the line 

 at infinity and on another straight line invariant forms a two-termed 

 group. 



If the axis at infinity and the centre are kept fixed, the similarities 

 differ according to their characteristics. A, A'; A', A" being two 

 pairs of corresponding points on a ray through the centre, A, A" 

 will be the corresponding pair of the resulting similarity, and there 

 is evidentl}^: 



CA CA' _ CA 



CA' ■ 'CA'^"~"CA'~'' 



k . k,=k", i. e., 



the third similarit}' belongs to the same system. As there are co i 

 such similarities we have 



Theorem 24. The system of similarities leaving the line at infinity 

 (axis of similarity) and the rays through a point invariant forms a one- 

 termed group. 



The same is true for the system of similarities leaving the rays 

 through a point and another point (on the line at infinity) 

 invariant. 



In central symmetry to the similarity is added the condition 

 k= — I. Central symmetry as well as oblique and orthogonal 

 symmetry does not form a group. 



If the centre of similarity is at infinity we have congruence in 

 which k=-j-i. Taking A, A' and A', A" as corresponding pairs, 

 in two congruences, A, A" will be a corresponding pair in a third 

 perspective collineation, resulting from the first two. There is 

 ooA ooA ooA 



ooA' ooA" ooA" 

 As there are coS such collineations we have the 



Theorem 2j. The system of eongruenecs leaving the points of the line 

 at infinity invariant forms a two-termed group. 



Supposing A, A', A" in a straight line, the above relation still 

 holds. The system of congruences has in this case a fixed centre 

 at infinity and consists of ooi congruences. Hence 



Theorem 26. The system of congruences leaving the poifits of the line 

 at infinity and the rays of a pencil of parallel rays invariant for7ns a 

 one-ternied group. 



