284 
ME. A. CAYLEY’S JklEMOIE UPOY CAUSTICS. 
where 
A'=(^+/) {(1 - f./)V+(l 
i. e. 
fco 
I I . , 1 + ft® . 
]x=±iy, x=±Yz:p>^y, 
or the curve meets the hue oo in foui’ points, each of them a point of triple intersection ; 
two of these points are the circular points at oo . 
To find where the circle x^-\-y^=\ meets the curve, this gives — if, and thence 
B=:4-/ 
C=/a^+2, 
and the equation becomes 
{^V-4)+4(l+2^.y}^+27^X4-2/W-216(^/-^+2)y 
+6W+2)/(4-/){f^V-4)+4(l+2f<.')/} = 0. 
which is only of the eighth order ; it follows that each of the circular points at oo (wliich 
have been already shown to be points upon the curve) are quadruple points of intersec- 
tion of the curve and circle. The equation of the eighth order reduces itself to 
the values of x corresponding to the roots y— +yj are obtained without difficult}', and 
those corresponding to the other roots are at once found by means of the identical 
equation 
(^-_4)’+27;«,‘+(l-;t.')(ft’+8f=0; 
we thus obtain for the coordinates of the points of intersection of the curve with the 
circle ^-1-^=1, the values 
roo ^x=+\/l — iJj^ 
\x=±iy, \y=±(^, 
(|x® + 8) v'l-ia® 
S'v/Sft® 
each of the points of the first system being a quadruple point of intersection, each of the 
points of the second system a triple point of intersection, and each of the points of the 
thii’d system a single point of intersection. 
Next, to find where the circle x^-\-y'^—\ meets the curve ; writing g — y^, w^e obtain 
fX* 
for y an equation of the eighth order, which after all reductions is 
(y vy +(i - V)’ l = 0, 
and we have for the coordinates of the points of intersection. 
