336 NOTES ON TRILINEARS. 



2. To find the radius (B) of this circle : 

 Let the point (/' y, h) be the centre. Then 



(ul'^ + vm" + tvn^) B^ + uf^ + vg^ + wh^ = 0, 

 or, 



mZ« 4- vm^ + wn^ = — — (uf^ + vg^ + wl^)-, 

 but. 



sin 2 yl ~ sin 2 ^ "" sin 2 C> 



~ sin 2 A.l^ + sin 2 B. f/<^ + sin 2 C. n^ 

 — _ J:_ «/•" + fy' + u>h* 

 ~ i^a ' 2 sin J. sin B sin (7* 

 Again, (f, g, h) being the centre, we have 

 uf vg voh 

 a h c 



_ uf^ + ty" + wh^ 

 ~ 2 A 



Dividing the terms of the former equalities by these respectively,, 

 we have, 



1 2^A _ __a h c 



~ B^ ' 2 sin J. sin ^ sin C ~ / sin 2 A ~ gsm2B ~ ^ sin 2 C 



o2 Ja c^ 



= sin 2 yl "^ ^YB "^ sin 2 C 



¥a 



and, therefore, 



1 2 sin A sin .g sin C / a^ ^i" c" \ 



~ ;B^ = (2 A)=' \sin 2 J. + sin 2 5 + sin 2 C/' 



which expression is easily reduced to either of the following forms : 



= — ;- (a sec A + b sec B + c sec C) 



aba ^ ' 



= (tan A + tan B + tan C) 



2 A ' . 



= jT^ — (tan A tan ,S tan C) 

 . 3 A 



3. To find the condition that the conic may be a rectangular 



hyperbola. 



