The Imaginary i-n Geometry 



x'x" + y'y 1 



a- a 



29 



(2) 

 (3) 



where k is. real. Moreover, since the equations give 



(a'*+k)x" = a'a"x'±y'yk(k + a n +a" a ), 



*(a ,s + k)y" = a'a"y' =p x'y/k(k + a' s +a"*), 

 k is positive. 



When the black point of a red vector lies upon (1) the blue 

 point (x'-\-x", v' + y") lies upon 



x 2 + y 2 =(a' + a") 2 + 2k, 



while the vector has the length given by (2). 



Fig. 29 represents x 2 -{-y 2 =(i — i) 2 . For £ = 0, 1, 3, 8 the 



Fig. 29. 



radius of the black circle is respectively 1, V 2 , 2 < 3 while that of 

 the corresponding blue circle is c, V 2 > V6, 4- Eventually, when 



29 



