EMCH: INV(jLUT()RK; •rRAXS|-()R.MA'i'10NS OF S'J'KAICHT LINE. II3 



In the case of involution 



ax + b 

 ex — a 

 The operation of a general projective transformation, 



x„ = -i-JLi__L (9) 

 " CjXj-fdj 



gives: 



^ _ (a,a-hb,cjx+(a,b-b,a ) ,^^^ 

 (c,a — djC)x-|-(c,b — d^a) 

 i. e., a general projective transformation. Obviously this trans- 



■ b 



Substituting these expresions for d, and Cj in (g) the projective 

 transformation which does not alter the involutoric transformation 

 becomes 



a.x, -f b. 



formation is involutoric if d, a,, and bjC-- — c^b, or c^ 



Xo 



b,c 



, (12 



or II we i>ut - =k, 



a, 



bx,— bk 



X,= — i (II) 



kcXj-(-b 

 In this case equation (lo) assumes the form 

 _(ab — bck)x--(b2-^abk) 

 ' (ack — bc)x — ( ab ~bck) 

 and, in fact, represents an involutoric transformation. If this 

 equation represents all involutions it must be possible to choose 

 three arbitrar}' real numbers A, B, and C, such that for constant 

 values of a, b, and c, and an}' real value of k between two limits, 

 however large, 



ab — bck— A, 



b~^L-abk=B, (13) 



ack pbc:=C. 

 These three equations cannot exist together. We ma\' choose any 

 of the three numbers at random, but then the value of k is deter- 

 mined, and so are the values of the two other numbers. The 

 system of involutions represented by equation ( 1 2) consists, there- 

 fore, of only a special class of involutions. This is also seen from 

 the fact that there occurs but one parameter in (12), while the 

 general involution has two. The system of transformations ( 1 1) 

 has one parameter, so that we may conclude 



T/wrc arc do ^ projective traiisforinaiioiis wliicli fra/isjcrni a certain 

 i/n'oii/tioii into x ^ oflier inro/utio/is. 



