On a Trianylc in- and- circumscribed to a Quartic Curve. 341 

 to be verified may be written 



<£ 4 + 4(/> 2 -iy_(<£ 4 + 6<£ 2 +l) 2 





4</>(c/> 2 + l) J 16<£ 2 (<£ 2 + 1) 2 ' 

 and we have 



* 1_ 2(<£ 2 + l) 2 ' V " 4^(^ + 1) " 40(<^ + l)* 



so that the equation becomes 



(<fr 4 + 6(fr 2 +l) 2 (j) 2 -l) 2 ((/) 4 +60 2 +l) 2 _ (<ft 4 + 6<ft 2 + l) 2 . 

 4(<^+l) 4 + 16<£ 2 (<£ 2 + 1) 4 ~ 16<^(<£ 2 +1) 2 ' 



that is 



40 2 +(f 2 -l) 2 = (<£ 2 + l) 2 , 



which is right. 



Next, the equation of the tangent at the point (£, rj) is 



(^-a*)(%X-a*) + ( V *-b*){ V Y-b*)-c 4 = 0; 

 that is 



IS A)tfA lj + ^ 40^+x)^ 1 4tf# 2 + l)>/ 



(<£ 4 + 6<£ 2 + l) 2 . 



16<£ 2 (</> 2 + P 2 



^ 

 ;' 



or, substituting for f, ?;, f 2 — l, and t; 2 — ^.. J ., , their 



J 4 i f* 19 I "I 



values, and throwing out a factor ^ g ^ — ^ tne equation 

 becomes 



-^^T,-0-W-.»(v^_ r ^^) 



= <£ 4 + 6<£ 2 + l, 

 or, what is the same thing, 



-8<£ 2 {X(<£ 2 -1)- */2(</> 2 +l)|- 

 -(<£ 2 -l){Y.80^- 4/2(£ 4 + 40*-l)}= \/2(^ 2 + l)(0 4 + 69 2 + l); 

 that is, 



(<£ 2 -l) (-8<£ 2 X-8<£ VfY) 



= \/2(0 2 + l)(0 4 4-6^+1)- ^2(<£ 2 + 1) .8<£ 2 - •2(^IX^ + 4#»-l] 



= ^/2(<£ 2 + l)(<£ 2 -l) 2 - */2(<£ 2 -l)(tf> 4 + 4<£ 2 -l) 

 = a/2(<j6 2 -1){(</> 4 -1)-(0 4 + 4^> 2 -1)} 

 = -4v / 2(0 2 ~l)^) 2 , 



