328 Transactions. 



If the poles of two sub-polar triangles be the circular points at in- 

 finity, conic (i) takes the form 



a' + /3' + y' + 2;8y cos A + 2ya COS B + 2a/8 cos C = 0. 



If the two points Oj, 0.> be the extremities of a diameter of the circle 

 ABC, then conic (i) takes the form 



sin A cos A sin B cos B sin C cos C 



+ — — - (u tan B + 1' tan C — X tan A) 



^ sin B Fin C ^'^ ' 



'VCt 



+ (v tan C + X tan A — yx tan B) 



sin L sin A 



+ (X tan A -f «, tan B — v tan C) = o, 



sin A siti B 



where \ -{■ ij. + v = o. 



Since the equation of any diameter of the circle ABC is 



'k + — ^ + — 7, = 0, 



cos A cos B cos O 

 if the diameter pass through the symmedian point 



A tan A + /A tan B + '^ tan C = o, 

 and the above equation reduces to 



sin (B-C) .^ sinJC-A) sin(A-B) ,, ^ 2 sin A sin (B-C) 



sin A sin B sm G sin B sin C 



2 sin B sin (C-A) 2 sin C sin (A-B) 



— - — ya — : ap ^ 0, 



sin C sin A sm A si i B 



a conic which passes through the symmedian point of the triangle ABC. 



If Oi, 0-2 be the extremities of a diameter of the Steiner ellipse 



111 



~ + >fl+ ~ = ^' 

 aa. op cy 



then the conic (i) reduces to 



A(aV - 26c/3y) -{■fx{b'l^' - 2caya) + , (cV - 2a6a/5) = o, 



where X -\- /i + v = o. 



§ 2. The condition that conic (i) should be a rectangular hyperbola is 

 i- + J + _L + cos A (^ + -i-U eos B f-i- + i) 



iia2 /3ig2 7i72 V/8i72 ^sYi/ V7i«-2 72 ai/ 



+ COS C ('J_ + J-") =0. 



Hence, if Oi(ai/5iyi) be fixed, the locus of O.^ will be the circum-conic 

 1 /I . cos C I cos B\ . 1 /cos (J 1 1 I '-"OS A\ 

 a Vai ^1 71/ ^ V «! /3i 7i J 



, 1 /C :3 B , C0> A , 1 N ... 



+ - + — -- + - = (iv). 



7 \ ai Pi 71/ ^ ^ 



If Oi be the centroid 01 the triangle ABC, then the conic (iv) reduces 

 to the circle ABC. 



aii 



