46 Ellery IV. Davis 



The modification for the imaginary ellipse 



x y 



-1 + ■« = ~ 1 



is but slight. Fig. 42 shows it. Note that the real foci are out- 

 side while the imaginary foci are inside of the blue ellipse. 

 Fig. 43 shows the four foci G t , G 2 , G, G of 



V 



Fig. 42. 



x 2 —y 2 = i. 



The figure also shows two supplementary curves ; the circle 



x 2 +y 2 =i, 

 and the ellipse 



(my-\- x) 2 + (y — mx) 2 — m 2 — 1. 



Just as in the first, multiplying y by i changes the equation to 

 that of the hyperbola, so in the second, multiplying {my -f- x) by 

 i produces the same effect. Obviously the blue hyperbola is the 

 envelope of the totality of blue ellipses. 



If the figure, exclusive of the coordinate axes, be turned through 

 a right angle it becomes the figure for 



the conjugate hyperbola. 



•y 



46 



1, 



