EMCH: INVOLUTOKIC I'RANSKORMATION.S OF SlKAU.Ui' LINE. II5 



Fig. 1. 



Now let us represent ' — and — — bv the points I and I,, 



c c " 



respectively, and draw a circle with 11^ as diameter. Each circle 

 normal to it and having the center on the straight line II ^, 

 intersects this line in a pair (P, P.-,) of the involution, so that 



P0XP,0= ^ , or xx,==-- 

 c c 



Making OP,T=OPg (Fig. i), the elliptic involution 



b 



XX, =^ 



' c 



results, and its points are obtained by the intersection of all the 

 circles passing through F and F'. 



To conform with the representation in Prof. Newson's article, 

 draw any two lines through the original O and symmetrical to the 

 axis II J. Let these lines be J and J'. With O as a centre and OP, 

 OPj, and OPj as rad'i describe circles intersecting the rays J and J' 

 in the points A, A'; Aj,A^'; and Bj, B^' respectively. A and Aj', 

 or A^ and A' are corresponding points in the hyperbolic involution, 

 while the pair A, Bj', or A', Bj belongs to the elliptic involution. 

 It is easy to prove that the rays connecting corresponding points. 



