367] ON A TRIANGLE IN-AND-CIRCUMSCRIBED TO A QUARTIC CURVE. 
491 
or, substituting for f, 7], £ 2 — 1, and rf — 
cI> 4 + 4 0 2 — 1 
40 (0 2 +1) 
, their values, and throwing out a 
the equation becomes 
)-40№-l)(r 
2 V0 0 4 + 40- - 1 
^ =04 + 60 2 + 1, 
V2(0 2 +l) 40 (0 2 +l) 
or, what is the same thing, 
- 80 2 {X (0 2 -1) - VI (0 2 + 1)} - (0 2 - 1 ){Y. 80 V0 - VI(0 4 + 40 2 -1)} 
= VI (0 2 + 1) (0 4 + 60 2 + 1) ; 
that is 
(0 2 - 1) (- 80 2 X - 80 V0 Y) 
= VI (0 2 + 1) (0 4 + 60 2 + 1) - VI (0 2 + 1) . 80 2 - V2 (0 2 - 1) (0 4 + 40 2 - 1), 
= V2 (0 2 + 1) (0 2 - l) 2 - VI (0 2 - 1) (0 4 + 40 2 - 1), 
= V2 (0 2 - 1) {(0 4 - 1) - (0 4 + 40 2 - 1)}, 
= - 4 VI (0 2 - 1) 0 2 , 
whence, finally, 
X V0 + F= V ^ 0, 
which is the required equation. 
It may be remarked that for 0 = 1, the equation of the curve is (of — 1 ) 2 + (V 2 —i) 2 = 
which is the binodal form a 2 > 1?, c 4 = a 4 . We have in this case £=0, rj = V£, and the 
curve and triangle are as shown in the figure, viz. the base ce of the triangle, instead 
c 
of being a proper tangent, is a line through the node D. For any other value of 0, 
the curve consists of an exterior oval (pinched in at the sides and the top and bottom) 
and of an interior oval ; the angles a, c, e lie in the exterior oval, the sides ac, ea 
touch the interior oval, and the base ce touches the exterior oval. 
62—2