598 REV. HUGH MARTIN ON CENTRES, FAISCEAUX, 



that is to say, the curve cuts the side of the triangle in points where (3 = ± 7, and 

 (3 : 7 : : m : n. And similarly for the intersections with the other sides. The 

 geometrical interpretation is, that the curve cuts the sides of the triangle in the 

 points where the bisectors of the interior and exterior opposite angles cut them ; 

 as also, in the points determined by drawing straight lines from the angles to the 

 opposite sides respectively, through the points /, m, n ; that is, the pole of the line 

 { la + m(3 + ny — } in which the inverse centre of homology moves, while the 

 direct centre describes the circumscribing conic. Considering the triangle as a 

 curve of the third order, the 3 2 = 9 points, in which it intersects the third-order 

 locus thus found, are in this manner somewhat elegantly determined. 



XI. If the centre of homology move in a conic touching two sides of the 

 triangle where the third side meets them, required the locus of the intersection of 

 the direct and inverse lines of homology. 



Here we have to eliminate from the three equations, 



f - tfgh = 0, ) 



«/ + £.7 + yh = 0, ( (49) ; 



agh + Bhf + yfg = 0. ) 



and the result is, 



(^7T_^+7* 2 1_ r490 . 



a curve of the fourth order, which is not altered by interchanging j# and 7, — as 

 it evidently ought not to be ; nor by inverting k,— which evidently also it ought 

 not to be, since while the centre of homology moves in 



Pa 2 = /3 7 , 



the inverse centre moves in 



I: 



= /3 7 , 



and if these are interchanged, the lines of homology are simply interchanged, and 

 the result of elimination ought to be unaltered, as we see is the case. 



XII. Required the locus of the intersection of the direct and inverse lines of 

 homology, when the centre of homology moves in a conic with respect to which 

 the triangle is self- conjugate. 



Taking the equation of the curve in which the inverse centre moves, the 

 square of the equation of the direct line of homology, and the equation of the 

 inverse line cleared of fractions, we have to eliminate /, g, h, from the three 

 following equations : — 



u = If + mg 2 + nh 2 = . . (50); 



v = a 2 / 2 + B 2 g 2 + y 2 h? + 23 7 gh + 2 7 ahf + 2aB/g = . . (51); 



w = ugh + Bhf + yfg = . . (52). 



