604 . REV. HUGH MARTIN ON CENTRES, FAISCEAUX, 



and inverse circumscribing conies of homology, when the centre of homology 

 describes a given circumscribing conic. 



[It may be noted here, that the direct and inverse circumscribing conies of 

 homology are two members of a faisceau of courbes pivotantes, passing through 

 the 2 2 == 4 points, namely, the three angles of the triangle and the point whose 

 locus is now sought. If the direct circumscribing conic is 



the inverse is 



? + 9 Q + h =0 (81), 



a p y y ' 



i + 4 + r = (82) ; 



fa, gf3 hy 



and any member of the faisceau pivotant to which they belong will be repre- 

 sented by 



-^ + -p-- + V =0; 



where X may have any value from to ± oo , the direct conic being represented 



when X = 0, and the inverse conic when X = ± oo ]. 



To find the locus required, we have to eliminate /, g, h, between (81) and 



(82) and 



If + mg + nh = (83). 



Now, these variables, the parameters, are involved in these three equations pre- 

 cisely as in (42), (43), (44) ; and the two sets of equations are identical, if for 

 a, (3, 7 we read their inverses. Hence inverting a, /3, y in (47), we get the locus 

 required, namely, 



mn 



®-*&) + "fr-*r) + , <?-*?) 



+ I 2 + m 2 + n 2 = (84) ; 



a curve of the 6th order, and which circumscribes the triangle. 



XXVIII. Required the locus of the same point when the centre of homology 

 moves in the bi-tangent conic k 2 a 2 = j3y . Here again the elimination is the 

 same, as in (48') and a, /3, y have to be inverted in (49). This gives 



(& - 7 y p + y 2 2 1_ 



(85), 



and which is the same curve if the bi-tangent conic were not k 2 a 2 = fiy, but 

 a 2 = }?Py. 



XXIX. Required the locus of the same point when the centre of homology 

 moves in a self-conjugate conic. 



