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A MEMOIR UPON CAUSTICS. 
375 
Next, when the directrix does not pass through the centre of the dirigent circle, 
it will be convenient to suppose always that to is positive, and to consider A as 
passing first from 0 to oo and then from 0 to — qo , i. e. to consider first the 
different inside curves, and then the different outside curves. Suppose a > , the 
inside curve is at first an oval, as in fig. 6, 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 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 concavity 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. 
Next, if a =, the inside curve 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 con- 
cavity of the oval come to coincide with the axis, and are tangents at a cusp, and 
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for A > —y- 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 compressed. 
Next, a < ~- > to 2 , then the inside curve is at first an oval, as in fig. d, and 
O 
there are, as before, three tangents parallel to the axis; when A attains a certain 
value which is less than , the curve acquires a double point; and as A further 
increases, the curve 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 A attains 
a certain value which is less than , the interior oval reduces itself to a conjugate 
point, and it afterwards disappears altogether. The outside curve is an oval as before, 
but smaller and more compressed. 
Next, if the directrix touch the dirigent circle, i.e. if a — to 2 . Then the inside 
curve is at first composed of an exterior oval which touches the dirigent circle, and 
of an interior oval which lies wholly within the dirigent circle. As A increases the 
interior oval decreases, reduces itself to a conjugate point, and then disappears. The 
outside curve is an oval which always touches the dirigent circle, at first very small 
(it may be considered as commencing from a conjugate point corresponding to A = 0), 
but increasing as A increases negatively.