AND ENVELOPES OF HOMOLOGY. 595 



triangle, the linear faisceau of homology will revolve round a point whose co- 

 ordinates are in the ratios of the sides of the triangle. 



This is an immediate corollary from the former, the equation of the circum- 

 scribing circle being, as is well known, 



^4-1 + ^ = (22). 



VII. If the centre of homology describe a straight line, the inverse centre will 

 describe a conic circumscribing the original triangle ; the linear faisceau will 

 envelope a conic touching its three sides, while the inverse linear faisceau will 

 revolve round a fixed point. 



This singular system of movements is merely a combination of propositions 

 I. and V., — it being remembered that the co-ordinates of the two centres are 

 mutually inverse. 



VIII. If the centre of homology describe a conic touching two sides of the 

 triangle in the points where the third side meets them, the inverse centre will 

 describe a second conic similarly situated ; the linear faisceau will envelope a 

 third conic similarly situated, and the inverse linear faisceau a fourth conic, also 

 similarly situated. 



The equation of the conic which, by hypothesis, the centre describes is, 



Fa 2 = /3 7 (23) ; 



or, substituting the co-ordinates of the centre, 



k*gh-f = (24). 



Also fa + gP + hy = = u (25). 



Differentiating as before, multiplying by an indeterminate coefficient p, &c, we 

 have, 



f - ~2 > 9 - ~ h % . Il ~ h 2 • • • W- 



And substituting in (25), we find, as the equation of the envelope of the direct 

 linear faisceau, 



Fa 2 = 4/3 7 (27). 



Farther, while the centre moves in k 2 a 2 = 0y, it is evident, inverting the 

 co-ordinates, that the inverse centre moves in 



t 



" =j8y • • • • (28), 



and that, by the portion of the proposition already proved, the inverse faisceau 

 must envelope 



p = 4/3 7 (29); 



which completely proves the proposition. 



VOL. XXIV. PART III. 7 Y 



