ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
295 
XXIII. 
The equation 
a^y\x^ ■Y'f— — 0 
becomes when a=\ (i. e. when the radiant point is in the circumference), 
{3f+(x-lX3x+l)Y-27f(f+x^-lf=0; 
it is easy to see that this divides by (^—1)^; and throwing out this factor, we have for 
the caustic the equation of the fourth order, 
27f+18f(3x^-l)-\-(x-l)(3x-j-lX=0. 
XXIV. 
The equation 
{ ( — I )(x^ — 2ax— a^}^—27 -\-y^ — a?f = 0 
becomes when a= oo [i. e. in the case of parallel rays), 
(4^2_^4^2_q^3_27^2^0, 
which may also be written 
64.r®+48.rW-l)+lM4/-l)'+(%'+lW-l)=0. 
XXV. 
It is now easy to trace the curve. Beginning with the case «=oo , the curve lies 
wholly within the reflecting cu’cle, which it touches at two points; the line joining the 
points of contact, beiug in fact the axis of y, divides the curve into two equal portions ; 
the curve has in the present, as in every other case (except one limiting case), two 
cusps on the axis of x (see flg. 6). Next, if be positive and >1, the general form of 
the curve is the same as before, only the line joining the pouits of contact with the 
reflecting circle divides the curve into unequal portions, that in the neighbourhood of 
the radiant point being the smaller of the two portions (see flg. 7). When a=l, the two 
Eig. 6. fl=oo . Eig. 7. a>\. 
points of contact with the reflecting circle unite together at the radiant point ; the curve 
throws ofl!, as it were, the two coincident lines x—\, and the order is reduced from 6 to 4- 
The curve has the form flg. 8, with only a single cusp on the axis of x. If a be further 
diminished, the curve takes the form shown by fig. 9, with two infinite 
branches, one of them having simply a cusp on the axis of x, the other having a cusp 
on the axis of x, and a pair of cusps at its intersection with the circle through the 
radiant point, there are two asymptotes equally inclined to the axis of x. In the case 
