390 



ON THE AXES OP A CONIC IN TRILINEARS. 



and multiplying these respectively by «, b, c, and adding, 



H- 



sin 2 A 



2 sin ^ sin ^ sin G r^ 

 or by an obvious reduction , 

 tan A 



2 sin A sin BsmC . 



H 



+ anal -}- = 



+ anal + ... = 0,, 



•(7) 



sm 2 A 



a quadratic in r^ which gives the squares of the semi-axes. 

 To determine P, we have, since (/, g, h) is the centre, 



a b c 



«/ + bg + ch 

 ^ 2ct.(/,g, h) __ -_P 



2 A A ' 



Hence P is found by eliminating/, g, h from the equations 



aF 

 ^' (/) + — = 0, 

 A 

 bP 

 ^' iff) + — •■= 0, 

 A 

 cP 

 <f>' (A) + _. = 0, 

 A 

 together with af + bg + ch = 2A', 



and therefore P is given by the determinant 

 , , aP 



= 0,. 



.(8) 



or if we expand the determinant 



wm'2 -f t)»'2 -f. u}w'^ — uvw — 2u'v'w' 

 ^ ^'a^{vw-u^)+ anal + ... + 2hc (v'w — uu) + anal + ... 



