389] 
ON A LOCUS DERIVED FROM TWO CONICS. 
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which has a pair of imaginary asymptotes parallel to the axis of x, and a like pair 
parallel to the axis of y, or what is the same thing, the curve has two isolated points 
at infinity, one on each axis. 
The next critical value is A. = ^(m + 1) 2 ; the curve here reduces itself to the four 
lines 
and it is to be observed that when A exceeds this value, or say A>5(??i+1) 2 , the 
curve has no real point on either axis; but when A = oo, the curve reduces itself to 
(x 2 + y 2 — a) 2 = 0, i.e. to the circle x 2 + y 2 — a. = 0 twice repeated, having in this special 
case real points on the two axes. 
It is now easy to trace the curve for the different values of A. The curve lies 
in every case within the unshaded regions of the figure (except in the limiting cases 
after-mentioned); and it also touches the two ellipses and the four lines at the eight 
points k, at which points it also cuts the circle; but it does not cut or touch the 
four lines, the two ellipses, or the circle, except at the points k. Considering A as 
varying by successive steps from 0 to oo; 
A = 0, the curve is the two ellipses. 
A < '— the curve consists of two ovals, an exterior sinuous oval lying in the 
four regions a and the four regions b ; and an interior oval lying in the region c. 
C. VI. 
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