On the Connexion of the Circle and Hyperbola. 89 



totes (from the perspective used the axis of imaginaries ap- 

 pears to coincide with one of the asymptotes), and C I A J is 

 the inscribed square of the asymptotes. The dotted lines 

 (representing a circle as commonly drawn) are the lines on 



the one plane, while the black ones (representing the common 

 form of the hyperbola) are those on the other plane : hence 

 if the black lines be considered as those on the real plane, the 

 dotted circle will be on the imaginary plane, and the figure 

 will be the locus of the hyperbola : if, on the other hand, the 

 dotted lines be considered as being on the real plane, the 

 black ones will be imaginary, and the figure will be the locus 

 of the circle. 



4. If we take any point P on the hyperbola (regarding for 

 the present the black lines as those on the real plane) deter- 

 mined by the co-ordinates CT=.r, TP=y, and draw the line 

 PR parallel to CL, then by the property of the hyperbola we 

 have 



area 1APR _ ] CR 



areaCIAJ S CI ; 



but IAPR=ACP, CIAJ = -i 



CI 



\/i ; 



.■.2(ACP) = log(CRxv / 2) (a.) 



The equation of the hyperbola being y = V ' x 1 — 1, and of 

 the asymptote y — x, we have 



