2o6 KANSAS UNIVERSITY QUARTKRIA'. 



The effect of the general projective transformations in the plane 

 X _„ a,x^+b^y,+c^ ^ 



upon the perspective involution (i) is the new transformation 

 ^ __ (aja + Cjb)x + (b^a + c^c)y— c ,a 

 ^ (a3a+C3b)x+(b3a+C3c)y— Cga' 

 _ (a2a+C3b)x+(b3a+C3c)y— c,a 

 ^ (a3a+C3b)x+(b3a+C3c)y— Cga ■ 

 This will be an involution if 



bj =a3=Cj=;C2^o, 

 and aj=b2=C3. 



In this case the transformations which do not change the character 

 of an involution are (we change the indices 3 into i): 



a , X , 



Yi 



t>iXi+Cjy,+aj 



a,y, 



^ b,x,+c,y,+a, 

 The resulting involution is 



a, ax 



¥2 



(bja-f ajb)x-|-(c^a-(-ajC)y — a^a 

 ajay 



^ (bja+a,b)x-|-(Cja-f ajC)y — a^a 

 Putting in (2) b j ^c^=b2=:C2:=:o, a2^:a,, the perspective colline- 

 ation leaving the center invariant arises. Each ray through this 

 center C intersects the axis of coUineation in an invariant point S, 

 such that for two corresponding points A and A^ on this ray the 

 so-called characteristic aniian/ioiiic ratio of tlic perspective col/i/ieatioti^ 

 is 



A=(CSAA,=-^^ 



Taking an other transformation of the three-termed group of per- 

 spective collineations, vs'ith the characteristic anharmonic ratio 

 l\^, the combined effect of the two collineations is a coUineation 

 of the same group with the characteristic anharmonic ratio 



In the involution A = = — i, while in the transformation 



— a 



a. 



(3) Ai= — ^=+i» so that, in fact 

 a , 



*The first time introducod into geometry by W. Fiedler. 



