44 KANSAS UNIVERSITY QUARTERLY. 



We need a convenient and expressive notation for these five types 

 of transformations. We shall designate a transformation of the 

 first type by the single letter T. The hyperbolic and elliptic sub- 

 types will be designated by jjT and ^T respectively. When we 

 wish to call attention to the triangle which remains invariant we 

 shall use the notation T(ABC). Transformations belonging to 

 types II and III will be designated by T' and T" respectively. 

 Transformations belonging to type IV and V will be designated by 

 S and S' respectively. 



§1. Type I; Its Properties and Normal Form. 



2. ThreeCross-Ratios Whose Product is Unity. We shall now 

 consider in detail the most general case (type I) whose invariant 

 figure is a triangle. Let the vertices of the triangle be represented 

 l:)y A, B, C: and the opposite sides by x, y, z, respectively. 

 Suppose in the first case that the triangle is real in all its parts, so 

 that our transformation belongs to the hyperbolic sub-type. By 

 means of a transformation jjT the line x is transformed into itself 

 in such a way that the points B and C on it are invariant points of 

 the transformation. Now we know that the one-dimensional 

 transformation of the points in a line, which leaves two points of 

 the line invariant, is characterized by the constant cross-ratio of 

 the invariant points and any pair of corresponding points.* 



Let kx be the characteristic cross-ratio of the one-dimensional 

 hyperbolic transformation along the line x. In like manner we 

 have hyperbolic transformations of one dimension along each of 

 the invariant lines of y and z. We sliall call their characteristic 

 cross-ratios ky and k^ respectively. In reckoning these cross- 

 ratios the points will be taken always in the same order around the 

 triangle. Thus we see that every projective transformation of 

 type I in the plane determines three characteristic cross-ratios 

 along the three invariant lines. It is also evident that the pencil 

 of lines through the vertex'A of the invariant triangle is transformed 

 into itself in such a way that the rays AB and AC are invariant rays 

 of the transformation. Also the cross-ratio of the invariant rays 

 and any pair of corresponding rays of the pencil is constant for all 

 pairs of corresponding rays; this cross-ratio is equal to kx, the 

 characteristic cross-ratio along the side x opposite A. Similar con- 

 siderations apply to the pencil of rays through the invariant points B 

 and C. We shall now proceed to show that these three cross-ratios 

 are not independent, but are connected by a very simple relation. 



*K. U. Quiirlfily. Vol. iv, IW)."), p. 74. 



