252 NOTE ON TRILINBARS. 



will, when the centre is pole, be identical with 



a a + b (3' + C y = 0, 

 and therefore 



1 <?<^ _ 1 d(ti _ \ dcfi 



a da h d^ c dy 



determine the coordinates of the centre. 

 This result may he obtained independently as follows : 



4. The centre of a conic. 

 Let the conic ^ (a, |8, y) = be cut by the line 



g— " _ ^—P _ y— y _ ^^ 



I m n ' 



Then for the points of section 



da dp dyl 



and, if I 



da d/S dy 



the two values of r are equal and opposite, and the point (a, /8, y) is 

 the centre of the chord. If then the above condition be satisfied for 

 all values of I, m, n, consistently with the condition 

 al + bm + en = 0, 



nil the chords through (a, ft, y) are bisected by it, and (a, /8, y) is 

 the centre. Comparing the two conditions, we have 



ld^_ld^_ld^ 



a da h 6?/3 c dy 



for determining the centre. 



Cor. If the conic be such that the triangle of reference is self- 

 conjugate with regard to it, its equation is 



M a2 + « yS» + «> y' = 0, 

 and the centre is given by 



a b c 



If the conic be a circle, then 



M _ r _ w 

 a cos A h cos B c cos C 



