The Imaginary in Geometry 49 



The F's are all elements of the ellipse; the Q's, of the supple- 

 mentary hyperbola. 



In fig. 45 the supplementary curve of fig. 44 is made funda- 

 mental. The /-ray through G x is tangent in P 3 , the /-ray through 

 B-l in .P 5 . The Q's are now elements of the supplementary 

 ellipse. 



If the fundamental curve is 



a 2 b 2 *' 



then it has in the first place for elements red vectors radial 1 from 

 the origin joining points on 



to 



-,±K=k' 2 . 

 a" b" 



a b 



In the second place, it has the elements obtained by multiply- 

 ing by i the y-coordinates of the radial elements of the supple- 

 mentary curve 



x y 



a b 



These elements belong to the companion curve 



x 2 y 2 



a 2 (k' + ik"Y ^ ¥(k"~^i¥) = h 



for which we can take 



x 2 y 2 



a 2 k' 2Z¥ b 2 k" 2 ~ 1 ' 

 and 



x 2 y 2 



1. 



a 2 (k + k"Y- b 2 {k' — k") 



as black curve and blue curve respectively. The equation of the 

 companion curve, notice, is the same as that of the fundamental. 



1 That is, on real lines through the origin. 



49 



