ME. A. CAYLEY’S MEM;0IE UPON CAUSTICS. 
307 
XXXVI. 
It is now easy to trace the curve. First, when m=0, or the directrix passes through 
the centre of the duigent circle, the curve is here an oval bent in so Eig. a. 
as to have double contact with the directrix, and lying on the one or 
the other side of the directrix according to the sign of A. See fig. a. 
Next, when the directrix does not pass through the centre of the 
dirigent circle, it will be convenient to suppose always that m is 
positive, and to consider A as passing first from 0 to oo and then 
from 0 to — oo , ^. e. to consider first the different inside curves, and 
then the different outside curves. Suppose 05>-|-, the inside curve 
is at first an oval, as in fig. where (attending to one side only of 
the axis) it will be noticed that there are three tangents parallel to the 
axis, viz. one for the convexity of the oval, and two for the concavity. 
For A=-^ the two tangents for the concavity come together, and 
give rise to a stationary tangent {i. e. a tangent at an inflection) 
parallel to the axis, and for A > the two tangents for the con- 
ca\ity disappear. The outside curve is an oval (of course on the opposite side of, and) 
bent in so as to have double contact with the directrix. 
Eig. 1. 
Next, if a= 
4m^ 
Eig. c. 
2 -, the inside cun^e is at first an oval, as in 
fig. c, and there are, as before, three tangents parallel to the 
axis: for A=-^, the tangents for the concavity of the oval 
come to coincide with the axis, and are tangents at a cusp, 
4 / 71 ^ 
and for A > the cusp disappears, and there are not for the 
concavity of the oval any tangents parallel to the axis. The 
outside curve is an oval as before, but smaller and more com- 
pressed. 
Next, a< then the inside cmwe is at first an oval, as in fig. c^, and there are, 
as before, three tangents parallel to the axis ; when A attains a 
• • • 47 ? 1 ^ 
certain value which is less than the curve acquires a double 
point ; and as A further increases, the curv^e breaks up into two 
separate ovals, and there are then only two tangents parallel to 
the axis, viz. one for the exterior oval and one for the interior oval. 
As A continues to increase, the interior oval decreases ; and when 
• • • 47 ? 2 ^ 
A attains a certain value which is less than the interior oval 
2 s 2 
