NOTES ON TRILINEARS. 335 



Let I = sin 6, so that 6 is the angle between the line and the 



side a. Then we have 



— m ■=■ sin (C -\- 6) ■=. sin G cos 6 + cos C sin 6, 



n = sin {B — 6) = sin jB cos ^ — cos £ sin ^, 



and, eliminating cos 0, sin 6, from these equations, we obtain, 



m'^ + n'^ + 2 mn cos A = sin** J.. 



This is one form of the relation sought for; but another form, 



iovolving all the qnantities symmetricallj, can readily be deduced 



by eliminating the product onn by aid of the first relation. Thus : 



2 he ran = aH"* — h^m"^ — c^n^, 



and, substituting this value, 



a* cos A l^ + h (c — b cos A) m^ + c (a — c cos B) «* 



= he sin'' A, 

 or, 



sin 2 AI^ + sin 2 B. m^ + sin 2 C. n^ = 2 sin ^ sin B sin C. 



It is proposed to employ the above equation in the examination of 



the conic to which the triangle of reference is self-conjugate; viz.: 



ua^ + vfi^ + ivy^ = . (1). 



1. To find the conditions that the above conic may be a circle. 



Cutting the circle by the line 



« -/ _ /^ — ff _ y — ^ _ 

 I m ~ n ~ ' 



the segments of the line intercepted between (fi g, h) and the circle 

 are the values of r in the equation. 



{uP + vm"^ + ton'') r^ + ( ) r -\- uf^ + vg^ + wh^ = 0. 



• (2). 



If {fy g, h) be a fixed point, then, by a property of the circle, the 

 rectangle under the segments is constant for all values of I, m, n ; 

 and, therefore, 



ul^ + vm^ + wn^ = const ; 

 but, 



sin 2 A.l^ + sin 2B.m^ + sin 2 C. «" = 2 sin A sin B sin C ; 

 and, these being satisfied identically, we have 



u V w 



sin 2 A ~ sin 2^ ~ sin 2 C 



and these are the conditions sought for. 

 Vol. X. w 



