EMCH: I\V(M,U'1'()RIC COLLINEATIONS in plane and in space. 211 



It may be well to add the remark that every linear involution 



Xj^.— ax-(-by, 



y,=cx+ay, 

 with a24-bc=i 



is a perspective involution with its center at infinity, or an axial 

 symmetry. The axis of symmetry is 



X, 



b 



and the center is at infinity in the direction of the pencil of parallel 

 rays 



y=: — ^^ x + m, 



where m is a parameter. 



7 



Space. 



3, In space there are two different kinds of involutions. The 

 first is comparable to the one in the plane and the theorems found 

 there maybe generalized for space; their nature is the same. The 

 equations are 



ax 



bx-j-cy-j-et — a 

 and do not change their character by the three-termed group of 

 elations that leave the origin invariant, i. e., 



a,x^ 



x„ = 



biXi+Ciyi+CiZj+ai' 



^ ^_xU ^ 



- b^Xj+Ciyi+e^z^+a/ 



2 b,Xj+c,yj+ejZi+a^ 

 4. The second kind may be represented by the formula* 



'W. Fiedler loc. cit. pa^e .5T3-TT. 



ax+by 



(5) 



