294 
]\IE. A. CAYLEY’S IMEMOIE UPOX CALSTICS. 
and thence 
whence 
P=^, P=^=27ayS= 
^ ^5 3. ^5 C 1 ’ 
and substituting in the equation 
a 2y^\ 
we find 
X-. 
'Aa^ 
and rationalising 
4iax^— {( 4 «^— 1 )^— «}^= 0 , 
or, what is the same thing, 
The factor 4a^— 1 equated to zero gives x=-^ from which y may be found, but the 
resulting point is not a double point, the other factors give each of them double points ; 
and if we write 
x=a{a-\-i \/ 1 — 
we find ^ 
y 2aH[a + i \—a^Y 
(3fl — 2 v' 1—a^)^ 
values which, in fact, belong to one of the four double points. It is easy to see that the 
points in question are always imaginary. 
It may be noticed, by way of verification, that the preceding values of x, y give 
+ ' 
— 4fl® 
{^a^~l){x^-\-y'^)— 2ax-~a^= — 4a^— Iff^'v^I— a‘^) 
x^-{-y^—a^ 
y 
~ l+8fl2' 
and if the quantities within ( ) on the right-hand side are represented by A, B, C, then 
whence we have identically. 
A \ 3 n 
b) =B- orA»=B‘C, 
by means of which it appears that the values of x, y satisfy, as they should do, the 
equation of the caustic ; and by forming the expressions for ( 4«^ — I )x — a and x^ + oy^ — a~, 
it might be shown, a posteriori., that the point in question was a double point. 
