492 ON A TRIANGLE IN-AND-CIRCUMSCRIBED TO A QUARTIC CURVE. [367 
If, to fix the ideas, we assume </> > 1, then we have always c 4 >a 4 <a 4 +6 4 : for 
</> = 1 we have, as appears above, b 2 = \, which is < a 2 ; but for a certain value of <£ 
between 3 and 4, b 2 becomes = a 2 , and for any greater value of cf> we have b 2 > a 2 . 
The condition for the equality of a 2 and b 2 is 
<£ 4 + 4(f> 2 — 1 = 4cf) {(f) 2 + 1), or <£ 4 — 4(f> 3 + 4t(f> 2 — 4<f) — 1 = 0; 
this equation may be written 2cf> (cf> — 2) (<fi> 2 + 1) = (<f> 2 — l) 2 , and we thence obtain 
(4> 2 -l) 4 
16</> 2 (<tr + l) 2 4 
(</>- 2) 2 ; 
or the equation of the curve is {x 2 — l) 2 + (y 2 — l) 2 = 1 + | — 2) 2 , where <f> 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.