1 
T" 
374 
and thence 
and 
A MEMOIR UPON CAUSTICS. 
[145 
x 2 + y'* — a = — 
4A 
A + x 2, (cc - to) = 0: 
this equation will have three real roots if A < and only a single real root if 
A > ; for A — , the equation in question will have a pair of equal roots. It 
is easy to see that there is always a single real root of the equation which gives 
rise to a real value of y, i. e. to a real point upon the curve; but, when the equation 
has three real roots, two of the roots may or may not give rise to real points upon 
the curve. 
XXXVI. 
It is now easy to trace the curve. First, when m = 0, or the directrix passes 
through the centre of the dirigent circle, the curve is here an oval bent in so as 
to have double contact with the directrix, and tying on the one or the other side of 
the directrix according to the sign of A. See fig. a. 
Fig. a. 
Fig. b. 
Fig. c. 
Fig. d.