EMCH: PROJECTIVE GROUPS. 1 7 



If the straight line is a priori supposed to be the axis, the state- 

 ment: there are perspective coUineations leaving the points 

 of a straight line invariant is not entirely logical. Making this 

 assumption it is self-evident that the points of the axis are invari- 

 ant. The same may be said in regard to the centre of perspective 

 coUineations. The reason for putting the above theorem into this 

 form is to have conformity with the statements in the cases of 

 general collineation. See for this remark Lie's " Vorlesungen," 

 table of groups, pages 288-291. 



After having found the number and invariant properties of the 

 general perspective collineation the special perspective coUineations 

 as classified in the previous chapter can be subjected to the same 

 investigation. As before, only one system out of the ooS of the 

 plane shall be taken into consideration. Following the division 

 given in the classification, §2, we have first the involution. It is 

 determined by the centre and axis, since k=: — i and, hence, there 

 are just as many involutions belonging to a straight line as an axis, 

 as there are points (centres) in the plane, i. e., ooS. Hence there 

 are oo3 involutions leaving the points of a straight line invariant 

 and dualistically 00- involutions leaving the rays through a point 

 invariant. The same numerical result is found in regard to a line- 

 element. But there is only one involution leaving the points of a 

 straight line and the rays through a point invariant. What has 

 been said about the involution holds in general for a whole sys- 

 tem of coUineations with a constant characteristic anharmonic 

 ratio. 



Dilation is characterized by an infinitely distant centre and a 

 characteristic k varying from — 00 to -|- c», and includes oblique and 

 orthogonal symmetry (k= — i). The centre may be one of the 00 1 

 points of the line at infinity, and since k can assume 00 ' values, 

 there are oo3 dilations which leave the points of a straight line 

 invariant. The dualistic interpretation of dilations leaving a point 

 at infinity invariant, respectively a pencil of parallel rays, does not 

 lead to the same numerical result. 



A certain direction, i. e., the centre at infinity, being given, the 

 co2 straight lines and oo^ values of k may be combined to form 

 dilations. Hence, there are co'J dilations leaving a point at infinity 

 invariant. Taking a point of the axis and a ray through it as a 

 line-element, it follows that there are co2 dilations leaving a line- 

 element invariant. 



By the same reasoning as in the general case we find that there 

 are 00I dilations leaving the points of a straight line and another 



