342 On a Triangle in- and-circumscribed to a Quartic Curve, 



whence, finally, 



X4+Y= •£<£, 



which is the required equation. 



It may be remarked that for <£ = !, the equation of the curve 

 is (x 2 — l) 2 -f- (^ 2 — J) 2 = l,whichisthebinodalform« 2 >-Z> 2 , c 4 = « 4 . 

 We have in this case £ = 0,?? = \/ \, and the curve and triangle are 

 as shown in the figure, viz. 

 the base c e of the triangle, 

 instead 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 in- 

 terior oval ; the angles a, c, e 

 lie in the exterior oval, the 

 sides ac, ea touch the interior oval, and the base c e touches the 

 exterior oval. 



If, to fix the ideas, we assume <j> > 1, then we have always 

 c 4 >■ a 4 < a 4 -\- b 4 : for = 1 we have, as appears above, b 2 — \, 

 which is < a 2 ; but for a certain value of cj> between 3 and 4, 6 2 

 becomes =# 2 , and for any greater value of <f> we have b 2 >d*. 

 The condition for the equality of a 2 and W- is 



04+4£ 2 -l=40(£ 2 + l), or 4 -4</> 3 + 40 2 -~40-l=:O; 



this equation maybe written 2(f>((f>— 2)(</> 2 -f 1) = (6 2 — I) 2 , and 

 we thence obtain 



16<£ 2 (<f+l)' 



i«--2) s 



or the equation of the curve is (a? 2 — 1) 2 + (?/ 2 — 1) 2 = 1 -f^(</> — 2) 2 , 

 where <$> is determined by the equation just referred to. The 

 curve is in this case symmetrical in regard to the two axes ; and 

 there are in fact four triangles, each in- and-circumscribed to the 

 curve. 



Cambridge, June 16, 1865. 



