lO KANSAS UNIVERSITY QUARTERLY. 



It is also obtained by revolving one half-plane of oblique symme- 

 try into the other. Corresponding points lie always on the same 

 side of the axis and are, as in the previous case, equidistant from 

 the axis. 



(d). As the previous cases, ((c), (V^), (r), were characterized by 

 the assumption of the centre C being at affinit}', there remains to 

 consider the collineation with an infinitely distant axis. Obviously 

 the conies K and K^ become coaxial parabolas which intersect 

 each other either in two finite real, or two imaginary points 

 (Fig. 6). There is 



k=:(;Coo AAi)=CA : CAi=(c o. aai); 

 the distances of corresponding points from the centre have a con- 

 stant ratio and form similar ranges. . 



Corresponding straight lines are parallel, and corresponding 

 ranges similar, so that their constant ratio is k. 



Fio 6 



The collineation thus characterized is termed similarity of 

 systems in similar position. According as k is positive or 

 negative, real or imaginary intersection-point of the parobola, 

 the similarity is said to be direct or inverse similarity. 



(e). If to the former case the further condition k= — i is added, 

 the relation becomes 



k-=(C 00 AAi) = (c 00 aa"i)=— I, or 

 CA=CAi. 



The conies K and K^ have two imaginary intersection-points 

 and are equal (K and Ri in Fig. 6.). Corresponding points are 

 in opposite directions and ai equal distances from the centre. 



