newson: projective transformations. 



«5 



In the first place it is not difficult to determine the number of 

 such one-termed groups. There are oo' points on the line 1; by tak- 

 ing each of these points with all the others, including itself, we can 

 form 00^ pairs of points. Each of these pairs of points can be 

 made the invariant points of a one-termed group G , . Thus the 

 00"^ projective transformations T)^ of tlie points on a line fall readily 

 into 00- one-termed groups, Gj, each consisting of 00* transforma- 

 tions which leave two fixed points invariant. Among these 00^ 

 groups Gj are 00^ groups of the kind Gj . The coincident invariant 

 points of these groups are obtained by taking each point of the 

 line with itself. All the one-termed groups considered above are 

 included in the formula oo'^T== 00- G,- cxj'Gj. 



We can readily see from the mode of formation of these one- 

 termed groups that ever}' transformation T belongs to some one of 

 these groups, and that no two of these groups can have a trans- 

 formation in common, (except possibly the identical transformation 

 which may be considered the same for all the groups). 



77u'o/-ci/i 8. — /'//(■ svstci/i oj 'ji'-'' project ii'c tra)jsfor)iiatio)is of a li)ic 

 falls apart into 00- oin-tcr/iicd groups. Eacli proper traiisfonnation of 

 the line belongs to one and onlv one of these groups. There are two 

 !^enej-al types of these groups, G, ajiel G ,' . 



We shall now examine the question of two-ternted groups of pro- 

 jective transformations on a line. There are oo'^ projective trans- 

 formations of the points on a line, and onlv 00' points on the line. 

 Hence any point can be transformed into an}' point on the line in 

 00- different ways. In other words there is a system of 00- trans- 

 formations which leaves any point A on the line invariant. We 

 shall proceed to show that this system of transformations has the 



fundamental property 

 ri?: LI. J^-*^^^"^ °^ ^ group, i. e. that 



the combined effect of 

 any two or more of the 

 transformations of the 

 system is equivalent 

 to some single trans- 

 formation of the same 

 system. 



To show this we 

 take a range of points 

 on the line 1 (Fig. 4) 

 and project it by 

 means of a conic K into a second ranire on the line 1.' Rexolve 1' 



