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Ellcry W . Davis 



This gives for x" = 0, y"=>i\/(i + .r' 2 ) so that (V, y") lies 

 on a rectangular hyperbola but the red elements are now parallel 

 to the real axis of that hyperbola. As before, revolution about 

 the center gives us the other red vectors, in particular, all the red 

 radii of the blue circle enveloped by the hyperbolas. To any real 





X / 



■X 



Fig. 27. 



point P there is an imaginary point of tangency represented by 

 a red vector, say P 2 . The black point on this is inverse to the point 

 .P with respect to the imaginary circle. As in the other case the 

 arrow joining two inverse points represents an imaginary point 

 on the circle, the same point namely as that represented by the 

 red vector whose black point is the mid-point of the arrow and 

 which lies to the right of one looking along the arrow toward the 

 barb. As before the circular rays through the center are tangent 

 at / and / and the Q's are elements of them (see fig. 27). 

 Consider the more general case 



equivalent to 



x 2 + y 2 = a 2 , 



(0 



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