376 
A MEMOIR UPON CAUSTICS. 
[145 
Next, when the directrix does not meet the dirigent circle, i.e. if a < m 2 . The 
inside curve consists at first of two ovals, an exterior oval lying without the dirigent 
circle, and an interior oval lying within the dirigent circle. As A increases the 
interior oval decreases, reduces itself to a conjugate point and disappears. The outside 
curve is at first imaginary, but when A attains a sufficiently large negative value, it 
makes its appearance as a conjugate point, and afterwards becomes an oval which 
gradually increases. 
Next, when the dirigent circle reduces itself to a point, i.e. if a = 0. The inside 
curve makes its appearance as a conjugate point (corresponding to A = 0), and as A 
increases it becomes an oval and continually increases. The outside curve comports 
itself as in the last preceding case. 
Finally, when the dirigent circle becomes imaginary, or has for its radius a pure 
imaginary distance, i.e. if a is negative. The inside curve is at first imaginary, but 
when A attains a certain value it makes its appearance as a conjugate point, and 
as A increases becomes an oval and continually increases. The outside curve, as in 
the preceding two cases, comports itself in a similar manner. 
The discussion, in the present section, of the different forms of the curve is not 
a very full one, and a large number of figures would be necessary in order to show 
completely the transition from one form to another. The forms delineated in the four 
figures were selected as forms corresponding to imaginary values of the parameters by 
means of which the equation of the curve is usually represented, e.g. the equations in 
Section xxviii. 
XXXVII. 
It has been shown that for rays proceeding from a point and refracted at a 
circle, the secondary caustic is the Cartesian; the caustic itself is therefore the evolute 
of the Cartesian; this affords a means of finding the tangential equation of the 
caustic. In fact, the equation of the Cartesian is 
(¿c 2 + y 2 — a) 2 + 16A (cc — m) = 0 ; 
and if we take for the equation of the normal 
+ Ytj + Z — 0, 
(where £, r) are current coordinates), then 
X : Y : Z = — y (x 2 + y 2 — a) 
: x (oc 2 + y 2 — a) + 4 A 
: 4 Ay,