308 
ME. A. CAYLEY’S 3IEMOIE LTOX CAUSTICS. 
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 diligent circle, i. e. if a—trf. 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. 
Next, when the directrix does not meet the dirigent circle, i. e. if a<ra^. The inside 
curve consists at first of two ovals, an exterior oval lying without the diligent chcle, 
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, ^. 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 cuiwe comports 
itself as in the last preceding case. 
Finally, when the dirigent circle becomes imaginary, or has for its radius a pime 
imaginary distance, t. 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 pomt, and as A increases 
becomes an oval and continually increases. The outside curve, as in the preceding tivo 
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 larger number of figures would be necessary in order to show completely 
the transition from one form to another. The forms delineated in the fom- figmes 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 
— m) — 0 ; 
and if we take for the equation of the normal 
X?+\ ;?+Z=:0, 
