DEC 



8 



T?P7 



Kansas University Science Bulletin. 



Vol. I, No. 5. 



SEPTEMBER, 1902. 



i) Whole Series, 

 \ Vol. XI, No. 5. 



PROJECTIVE TRANSFORMATIONS IN ONE DIMENSION 

 AND THEIR CONTINUOUS GROUPS. 



BY H. B. NEWSON. 

 INTRODUCTION. 



The writer of this paper is engaged in establishing and developing 

 a new theory of collineations and their Lie groups. A recent memoir 

 in the American Journal of Mathematics, Vol. XXIV, p. 109, entitled 

 "A New Theory of Collineations and their Lie Groups," treats of 

 collineations in a plane. A series of papers applying the theory to 

 collineations in space has been published in the Kansas University 

 Quarterly, Vol. X. This theory of collineations in two, three and 

 higher dimensions is based upon, and presupposes, a corresponding 

 theory of one-dimensional projective transformations. 



In the present paper the theory is developed for the range of points 

 on a line, but it applies equally well to all three one-dimensional 

 primary forms of projective geometry, viz., the range of points on a 

 line, the pencil of lines through a point, and the pencil of planes 

 through a line. The object of the paper is to collect and set forth the 

 principal facts of one-dimensional transformations, and to build 

 thereon a comprehensive theory of their continuous groups. The pa- 

 per is sufficiently complete to serve as a foundation on which to build 

 a consistent theory of collineations in two, three and higher dimen- 

 sions. In form and content it is suitable for the beginner, and will 

 serve as an introduction to the general theory. 



The chief sources are as follows: In his "Vorlesungen iiber Con- 

 tinuierliche Gruppen," Kapitel 5, Lie gives a detailed account of his 

 theory of one-dimensional projective transformations and their con- 

 tinuous groups. His equation is, 



ax +b 



Xi = 



(1) 



ex + d 



where all the variables and coefficients are assumed to be complex 



(115) 



