496 Transactions. 



hence the centres of perspective determine a triangle inscribed in the 

 Steiner ellipse and circumscribed to the maximum inscribed ellipse of the 

 triangle ABC. 



The Steiner ellipse of the triangle PiPaPs is 



ax bi/ cz ax , by cz ax by cz 



-7--t- — + - — + — + -r — + -r + - 

 L m n m n L n I m 



which reduces to bcyz + cazx + abxy = o ; i.e., to the Steiner elHpse of 

 the triangle ABC. 



The equations of the parabolas having their foci at Fg, Fg are, when 

 expanded, mnty^ + nkC^ + Iviv'^ = o 



mnv^ _[_ nlw"^ + Imii^ = o, 

 where u^by -\- cz, v^cz + ax, id z=ax -\- by ; 



hence 7nn : nl : Im = ?t^ — vho'^ : v* — lohi^ : w* — ic^v^. 



The locus of the intersections of inscribed parabolas which have their foci 

 at the vertices of a triangle inscribed in the circumcircle and circum- 

 scribed to the Brocard ellipse of the triangle ABC is — j 1 — — o, 



U V IV 



where u ee (by + czy — (cz + axY {ax + byy. 

 The condition that an inscribed parabola 



/- — / — — / V? v^ vo^ 



S = \lax + Vmow + \ ncz — o, i.e., -j H — = o. 



U 1j 77? ^ 



may pass through the fixed point (x^yiZ^) is -!- -f — H — - = o. Hence 



L m n 



the equation of the two parabolas which may be drawn through the 

 point (XiyiZ^) is 



1 1_ 1- 



and the equation of the two directrices is 



+ . .._ ^ ' . + . ' -^ = 0. 



y cos B — z cos C z cos G — x cos A x cos A. — y cos B 

 The line joining the foci of the two inscribed parabolas has for its 



2^ 1 ., 2^ 1 ,„2? _ 



equation ?^^- + ■Ui r + ^i = o. 

 ^ a b c 



If H be any point on the line fx + gy + hz = o, then writing this line 



in the form pu 4- qv -\- no = o, where ri^ — - -\-% A — (7=;i_£_| — 



a b c a b c' 



and r= -+ ^ , we easily find that the line joining the foci of the 



two inscribed parabolas which pass through the point H envelops the 



. aifi bq'^ cr^ 



conic ~ — I — — •] = 0. 



X y z 



