Involutoric Collineations in the Plane and 



in Space. 



BY ARNOLD EMCH. 



In No. 2, Vol. 4, of this Quarterly I have treated the involutoric 

 transformation of the straight line and made an attempt to show 

 the usefulness of the elementary conceptions of the theory of groups 

 in the solution of geometrical problems. In this article I propose 

 to do the same thing with regard to the plane and space, and shall 

 confine myself to transformations with real variables and co- 

 efficients. 



1. The Plane. 



I. Among the collineations of the plane invol lit ions occur only 

 in the perspective collineation. Referring to cartesian coordinates 

 their equations may be written in the general form 



ax 



y 



bx + cy — a 



_ ay 

 ^ bx-|-cy — a 



The invariant elements are: the origin (x=y=:o), every ray y:=mx 

 through the origin, and every point of the straight line 



bx-]-cy — 2a =0. 



The origin and this line are known as the center and the axis 

 of the perspective involution (collineation). To the points (Xj, }', ) 

 of the line at infinity corresponds the so-called counter-axis, in 

 German Gegenaxe, 



bx-j-cy — a=o. 



The other counter axis, or the line whose corresponding points 

 (x, y) are at infinit}', is 



bx^ +cyj — a=:o 



The counter-axes of an involution in a plane coincide, as is seen 

 from their equations.* 



*See W. Fiedler, Darstellende Geometrie. Vol. III. i ed, pasje r)7:{-rT. 



I'JO.'i) KAN. I'M. QUAR. VOL. IV. NO. 1. APRIL. 18!ll). 



