44 



Ellery W. Davis 



different, the change is even more marked in 38, 39 and 40. 

 Observe also that not only the red vectors but the involution rep- 

 resenting the line is changed, since it is — XI" Y J". 



Bearing in mind the possibility of such generalizations let us 



Fig. 40. 



turn to the representation of conies and, for convenience, let us 

 use ordinary rectangular coordinates. 



The Conic 



Any conic which by a real shift and turn of axis can be brought 

 to the form 



x 2 v 2 

 a 2+ b 2 ~* 



where a and b are real we call an ellipse. The axes of the ellipse 

 are 



ak = c and bk = d 

 where c and d are similar, i. e., 



' d' 



d" 



= 0. 



The supplementary curves when k 2 is real (for convenience let 

 k==i) are hyperbolas whose pairs of asymptotes are pairs of 

 conjugate diameters of the ellipse. These asymptotes form invo- 



44 



