336 Transactions. 



will be perpendicular to the equi-con jugate diameters of the 

 ellipse S. The directions of the axes of that conic may there- 

 fore be determined. 



If 6 and c' are the eccentricities of S and S' respectively, 

 we have 



2R 2R 



£ 2 _ £ I1 _ 



p + R ' p' + R 



Eliminating p and p' by means of the relation pp' = R 2 , we 

 have 



■2 I '2 O 



9. The foci of any conic inscribed in the triangle of refer- 

 ence are isogonal conjugates. If the trilinear co-ordinates of 

 one focus be (a' o /3 y o ), then the co-ordinates of the other focus 



/k 2 k 2 k 2 \ 

 will be ( — , "5-> — ] where k is the semi-minor axis of the come. 

 \a Po 7o/ 



The conic may be regarded as the envelope of a variable 

 line la -\- mft + ray = o, which moves so that the product of 

 the perpendiculars on it from the foci is equal to k 2 . The 

 relation between I, m, n is easily found to be 



mna ©! + nl($ a ® 2 + lniy ® 3 = o, 



where © 2 = pY 2 + y ' 2 + 2^ y cos A 



© 2 = y 2 + a 2 + 2y a COS B 

 © 3 = a 2 +/3 2 +2a /3 COsC, 



and the equation of the inscribed conic is 



Vaa ® 1 + Vp(3 ® 2 + v'yyo©,, = 



If D be the focus (a /3 y ), then ©^ ® 2 , ©; are respectively 

 (DA sin A) 2 , (DB sin B) 2 , (DC sin C) 2 . 



10. If we take D to be the incentre of the triangle ABC, 



then ©j = 4r 2 cos 2 -x , ® 2 = 4r 2 cos 2 - , © 8 = 4?' 2 cos 2 -^ , and we 



obtain the equation of the incircle, viz., 

 * -p n 



cos -^ v 'a + cos -g v 7 ^ + cos - v~ = o 



In a similar manner the equations of the ex-circles may be 

 at once determined. 



Let D be the centre of the circle ABC, then 

 ©! ©■. ®, _ R , 



sin'-A sin 2 B siirC 

 and we find the equation of the inscribed conic having its foe 



