iSIEWSON: CONTINUOUS GROUPS. 83 



group of tJie general projeetive group. (See "Cent. Gruppen," 

 page 113.) 



By means of this theorem many of the sub-groups of the general 

 projective group can be readily determined. 



It will be convenient to have separate symbols to designate each 

 of the five types of transformations referred to above. We shall 

 represent the five types of transformations whose invariant figures 

 are 



! —a—k--^-% — *— —A — *—•—■*— * — 



by T, T', T'', S, S', respectively. 



We shall now consider more in detail these different types of 

 transformations, beginning with the most general case (^type i) 

 whose invariant figure is a triangle. Let the vertices of the tri- 

 angle be represented by A, B, C; and the opposite sides by x, y, z, 

 respectively. B}' means of a transformation T the line x is trans- 

 formed 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 on a line, which 

 leaves two points of the line invariant, is characterized by the 

 constant anharmonic ratio of the invariant points and any pair of 

 corresponding points. (Kansas University Quarterly, Vol. IV., 

 page 74.) Let k^ be the characteristic anharmonic ratio of the 

 one-dimensional projective transformation along the line x. In 

 like manner we have projective transformations of one dimension 

 along each of the invariant lines y and z. We shall call their 

 characteristic anharmonic ratios ky and k^. respectivel}^ In reck- 

 oning these anharmonic ratios the points will be taken always in 

 the same order around the triangle. Thus we see that every pro- 

 jective transformation of the kind T in the plane determines three 

 characteristic anharmonic 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 anharmonic ratio of the invariant rays and any pair of corres- 

 ponding rays of the pencil is constant for all pairs of corresponding 

 rays; this anharmonic ratio is equal to k^ , the characteristic 

 anharmonic ratio along the opposite side x. Similar considerations 

 apply to the pencils of rays through the invariant points B and C. 

 We shall now proceed to show that these three anharmonic 



