Kansas University Quarterly. 



Vol. VIII. 



APRIL, li 



No. 2. 



The Five Types of Projective Transformations 

 of the Plane. 



1!Y HENRY I!. NEWSON. 



I. It was shown in this Quarterly, Vol. IV, 1896, pp. 243-49, 

 that there are five types of projective transformations of the 

 points of a plane and that each of these types is characterized by 

 its invariant fiu'ure. 



Fig. 1. 



The present paper is devoted to a fuller discussion of these five 

 types. The special or characteristic properties of each of these 

 five types will be brought to light and the analytic expression for 

 the transformation of each type will be reduced to a normal form. 



We shall assume throughout this paper that the coefficients of 

 the equations of the transformation are always real so that the 

 transformations are always real. In this case the invariant figures 

 of the different types are real in all their parts, except sometimes 

 in type I. The invariant triangle of type I is either real in all its 

 parts or has one real and two conjugate imaginary vertices. If the 

 invariant triangle is real in all of its parts the transformation is 

 said to be hyperbolic; in the second case with the invariant triangle 

 partly imaginar}' the transformation is said to be elliptic. 



(i:{) KAN. UNIV. QUAR., VOI,. Vlir. NO. :'. APR.. IS'.m, SERIES A 



