ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
293 
It is easy to see that the curve passes through the circular points at infinity, and that 
these points are cusps on the curve ; the two points of intersection with the axis of x are 
cusps (the axis of x being the tangent), and the two points of intersection with the circle 
— are also cusps, the tangent at each of the cusps coinciding with the tangent 
of the circle ; there are consequently in all six cusps. 
XXII. 
To investigate the position of the double points we may proceed as follows : write for 
shortness V={^a^~V)[x^-\-y^)—1ax—a^, Q=o^j/S, ^=x^-\-'if—a^; the equation of the 
caustic is 
P^-27Q^=0. 
Hence, at a double point, 
p2^_18Q^=0 
ax ax 
one of which equations may be replaced by 
<?P dQ, 
dx dy dy dx 
= 0. 
Now 
^^=2{(4a^-l)^— «}, ^=2(4a^-l)2/ 
dx 
dy 
~ =«(a^+3/-«^)=a(S+2/). 
Substituting these values in the last preceding equation, we find 
or reducing 
{4a^—\)x—a 2xy 
{4a^—l)y ii + 2y^'‘ 
(4:a^—l)x—a=^^ ; 
and using this to simphfy the equation 
dx 
we have 
t. e. 
and therefore 
— 1 8a^S . 2«ir^ = 0, 
9a^S=0, 
p2 
Q/y — — 
^X— ^g2 
Multiplying by P and writing for P* its value 27a^^^S*, we have 
P^=3«^^ 
