485] 
PROBLEMS AND SOLUTIONS. 
555 
The general equation of a conic having double contact with each of these conics then is 
n 2 z 2 — 2n (ya' + y'a) yz — 2n (y/3' + y'/3) zx — 4tnyy'xy + [(/3y — /3'y) x — (ya' — y'a) yf = 0, 
where n is arbitrary: and, having double contact with this conic, we have (besides 
the above-mentioned two conics) two new conics each passing through the angles of 
the triangle; viz. writing for greater convenience 
W - /3'y) (y a ' - y'a) 
K — yy' 
or K = ry' + (fty - ffv) (t«' - 7«) 
n 
then the equations of the two new conics are 
y'a yz + y/3' zx + Kxy = 0, ya' yz + y@ zx -1- Kxy = 0. 
In fact, writing the equation under the form 
[xz + (/3y — /3'y) x + (ya! — y'a) yf 
— 4 (/3y — /3'y) (ya! — y'a) xy — ^nyy'xy 
— 2n (/3y' — /3'y) xz — 2n (/3y' + ¡3'y) xz 
— 2n (ya! — y'a) yz — 2n (ya! + y'a) yz = 0, 
we at once see that this is a conic having double contact with the conic y'ayz+yl3'zx+Kxy=Q, 
the equation of the chord of contact being nz + ((3y — /3'y) x + (ya' — y'a) y = 0: and similarly 
it has double contact with the conic ya' yz + y'/3 zx + Kxy = 0, the equation of the chord 
of contact being nz - (/3y' — /3'y) x — (ya' — y'a) y — 0. 
[Vol. v. pp. 99, 100.] 
1554. (Proposed by Professor Cayley.)—Show that, in the ellipse and its circles 
of maximum and minimum curvature respectively, the semi-ordinates through the focus 
of the ellipse are 
For the circle of maximum curvature y x = a (1 — e) (1 + 2ef, 
for the ellipse 
for the circle of minimum curvature 
y 2 =a( 1 — e 2 ), 
a {(1 — e 2 + e 4 )$—e 2 
V 3 = 
(1 - e 2 f 
and that these values are in the order of increasing magnitude. 
[Vol. vi., July to December, 1866, pp. 18, 19.] 
1931. (Proposed by Professor Cayley.)—Find the stationary tangents (or tangents 
at the inflexions) of the nodal cubic 
x(y - z) 2 + y (z- x) 2 + z(x-y) 2 = 0. 
70—2