l8 KANSAS UNIVERSITY QUARTERLY. 



straight line invariant. There are, however, oo2 dilations in which 

 the rays through the centre and a finite point in one of these rays 

 are invariant. 



If k= — I dilation becomes oblique and orthogonal symmetry. 

 Obviously there are ooi such symmetries leaving the points of a 

 straight line invariant. On the other hand there are oo3 (straight 

 lines of the plane) symmetries leaving one and the same point at 

 infinity invariant. There is only one symmetry belonging to a 

 line-element. 



Revolving one system of axial symmetry through i8o°, k be- 

 comes -|-i, and the centres move to infinity in the direction of the 

 axis. The relation is that of figures of equal areas, and the num- 

 ber of such collineations is coi. For, taking a line parallel to the 

 axis, or, connecting two corresponding points A and A^, one and 

 the same collineation is determined by any two points A and A' 

 which include the same length AA^. This gives odI different 

 collineations leaving the points of a straight line invariant. 



In the case of similarity the axis is at infinity, the centre finite, 

 and k ranges from — co to -(- °°5 thus including central symmetry. 

 The 00 2 positions of the centre together with c»i values of k give 

 oo3 similarities leaving the points of the line at infinity invariant. 



The centre of similarity may lie on a finite straight line and since 

 each finite point of the plane represents ooi similarities, there are 

 c»2 similarities leaving the points of the line at infinity and another 

 straight line invariant (line-element with its point at infinity). 

 Taking the point of the line-element in the finite part of the plane, 

 there are just ooi similarities leaving this line-element, or also a 

 point invariant. 



All similarities with k= — i are, as it is known, central symme- 

 tries. As k= — I their number is ooS 



Assuming the centre of similarity at infinity, k becomes +i and 

 the collineations are congruences. A congruence is determined 

 by the direction of its centre and two corresponding points which 

 subtend the same length. As there are oo^ directions and coi sects 

 in the plane, the number of congruence in the plane is « 2. AH of 

 them leave the line at infinity invariant. The number of perspec- 

 tive collineations belonging to a certain direction, or leaving the 

 centre at affinity invariant, is ooi. 



The pseudo-collineations which are characterized by k^o, co,- , 



o 



have as their numbers 002^ oo2, coi, respectively. In all three cases 



the axis is the invariant element. 



