116 KANSAS UNIVERSITY SCIENCE BULLETIN. 



numbers. His geometrical interpretation is by points on a line, both 

 points and line being, in general, imaginary. 



Klein,* Poincar6f and others have extensively investigated the same 

 transformation, but chiefly with respect to its discontinuous groups. 

 They have interpreted their results geometrically by points in the 

 complex plane. In addition to the results of Lie and Klein, the prin- 

 cipal results of two former papers by the writer are incorporated in 

 the present memoir. These are entitled "Continuous Groups of 

 Projective Transformations Treated Synthetically," J and "Continuous 

 Groups of Circular Transformations." § In the first of these papers 

 is developed geometrically the elements of this new theory of pro- 

 jective transformations in one dimension. In the second paper the 

 continuous groups of the transformations given by ( 1 ) are deter- 

 mined and interpreted as transformations of the points of a complex 

 plane. 



PART I. 



§1. General Properties of One-dimensional Transformations. 



The resultant of two transformations. — A projective transforma- 

 tion should be looked upon as an operation which, when applied to a 

 finite set of points on a line or to the totality of all points on the line, 

 has the effect of rearranging and redistributing them in such a way 

 that the new set or range is projective with the old. If we carry out 

 in succession two projective transformations on a set of points, the 

 result is equivalent to a single projective transformation. This may 

 be shown as follows : 



Let T and Ti be two transformations whose equations are re- 

 spectively 



Xl = j^±£ an d x 2 = ^±K 



1 ex + d ' dx, + d x 



The first transforms the point x into xi, and the second transforms 

 Xi into X2. We suppose the operations are carried out in the order in 

 which the equations are written. If we eliminate xi from the above, 

 we get 



(da + d,c) x + (db + d t d) 



It should be observed that ( 2 ) is of the same form as ( 1 ) and 

 differs from it only in the value of the coefficients. Equation ( 2 ) 

 therefore expresses a projective transformation T>, which transforms 

 the point x directly to X2, and is equivalent to the successive applica- 

 tions of T and Ti in the order named. The transformation T2 is 



*See Modulfunctionen, Band I, S. 163-207. 



fSee Acta Mathematica, Tome I, pp. 1-62. 



% Kansas University Quarterly, Vol. IV, pp. 71-93. 



§ Bulletin Amer. Math. Soc, 2d Series, Vol. IV, pp. 107-121. 



(a t a+ b,c) x + (a t b + bid) /n\ 



X2 ~ ~ (r.n 4- fl.rt) x 4- (n.h 4- d.dV \ ' 





