810 



ANALYIK //,') 



Mil. 



;irding equations (1) and ('!) as simultaneous, the . 

 and y involved RM the eoordinates ot the point /' : more 

 over, since the point Tsfx^y^ is on the hyperUla, then -for 



Kliminatiiitf ./^ and t/ { between e-piations H ).(-), and ( :1 ) 



gives 



(r -f y 2 ) 2 = a-(x* J/ 2 ), . . . (0 



which is, therefore, the equation sought. 



The definition of the lemniscate, as well as the equation 

 just derived, shows that the curve is symmetru-al with 

 regard to both coordinate axes; that it lies wholly IM -t 

 the two lines whose equations are x = a and x = + a : 

 it passes through the origin and the two points (a, 0) and 

 (-ha, 0); and that y is never larger than x : IH -n< tin? 

 lemniscate is a limited closed curve as represented in Fig. 128. 



NOTE 1. The polar equation of the lemniscate is easily derived from 

 equation (4) if the x-axis be chosen as initial line and the origin as 



