emch: projective groups. 27 



Theorem ig. The system of dilations leaving the rays of a parallel 

 pencil of rays invariant forms a three-termed group. 



Theorem 20. The system of dilations leaving the centre of dilation 

 and another finite point invariant forms a tiuo-termed group. 



The one-termed dualistic group of dilation is the same as the 

 original one; this group is self-dualistic. 



In the oblique and orthogonal symmetry, which is dilation 

 with k=: — I, there is no group, for two symmetries applied one 

 to the other give a collineation with the characteristic k=-{- i, 



(-iX-i= + i). _ _ 



In the case of dilation with k^-|-i the centre is at infinity in the 

 direction of the axis. As is known from §3, this is the relation of 

 corresponding equal areas. Two points, A, A', on a ray parallel 

 to the axis determine the relation 



k=.=^ =+i. 

 <xA' 



Taking A', A" as a corresponding pair in another collineation of 



this kind, 



00 A' 

 k , ^=. — ~- — = -t- 1 , we obtain 

 00 A ' 



<» A cx) A ' ooA 



I, or 



o=A' odA" odA" 



(+i)X( + i)= + i. 

 Thus the 



Theorem 21 . The system of perspective collineations characterized 

 hy corresponding equal areas and leaving the points of a straight line 

 invariant forms a one-termed group. 



In discussing the relation k"=k. k, it was pointed out that there 

 are groups with the characteristic k:^-|-i. This assertion is now 

 proved; but we yet shall find other groups with the characteric 

 k=+i. 



Instead of the line joining the centres we can suppose the line 

 I of a system of perspective collineations to be at infinity. In this 

 •case we have similarity which is expressed by 



CA C,A' CA 



. 1 — , or again 



CA' CiA"~CA" " 



k"=k. k,. 

 Just as in the case of dilations, to each similarity can be found 

 its inverse belonging to the same system. There are co^ such simi- 

 larities, therefore 



