6 KANSAS UNIVERSITY QUARTERLY. 



Thus, any pair of corresponding points in the perspective 

 collineation has a constant relation to the counter-points, and we 

 have the well known 



Theorem 2. Eaeli pair of eorrespoiidiug points on a rax i/i rough the 

 centre forms a constant anharmonic ratio witJi the intersection-point of 

 the ray with the axis of perspective collineation. 



Any point M on the axis 1 may be connected with the points C, 

 L, Q, R, 00, A, Ai, B, B', and designating these rays by small 

 letters, there is obviously 



(claa') = (clbb^)=const. 



This fact can be stated as the dualistic of the above theorem, viz: 



Theorem j. Each pair of corresponding rays through a point on 

 the axis forms a constant anharmonic ratio with the ray through the 

 centre and the axis of perspective collineation. 



We call this constant the characteristic anharmonic ratio of the 

 perspective collineation and designate it by k.* By aid of it a 

 classification of the perspective collineation can easily be made, 

 and so far as it will be of avail for our further consideration we will 

 discuss the different cases of perspective collineation from this 

 point of view. Among all the oo i values of k the special case 

 k= — I deserves the greatest attention, and it shall be considered 

 first, because it enables us at once to draw important conclusions 

 from its combination with particular positions of the center, the 

 axis, and the counter-axes of the perspective collineation. 



From the assumption k= — i follows: 



CQi CR 



^^- = — = — I, or 



LQi LR 



CQi=— LQi, and CR=— LR; i. e., 



the counter-points and therefore also the counter-axes are midway 

 between C and L, and, therefore, coincide. For every pair of 

 corresponding points the relation exists: 



(CLAA')^-i= f-. 



CA CA^ CAi CA 



LA LAI LAI LA 



From this follows 



(CLAAi)==— i=(CLA^A) and 



in a similar way: 



(claai)= — i=(cla^a), i. e., 



in this collineation the points and rays of each pair are inter- 

 changeable. The collineation whose characteristic anharmonic 

 ratio is k= — i is therefore involutoric. 



*This constant was first introduced into Geometry by Fiedler. 



