AND ENVELOPES OF HOMOLOGY. 601 



XV. If the centre of homology move in a straight line, the faisceau of circum- 

 scribing conies 



1 1 I 



fa g»$ k A y 



will envelope the tri-tangent conic 



± (Pa)* ± (m 3 /3)^ dz (»»/)* = . . . . (67). 



XVI. If the centre of homology move in a straight line, the faisceau of cir- 

 cumscribing conies 



/#« #3/3 % 



will envelope the self-conjugate conic, 



Pa 2 + m 3 ,8 2 + nY = .... (68). 



XVII. (2.) Required the envelope of a faisceau of circumscribing conies, when 

 the centre of homology moves in a given circumscribing conic. 



In this case, instead of (64), we have 



lf+ mg + nh = (69). 



If, therefore, instead of /, g, h, we take as variables their inverses, and consider 

 s as negative, our new variables will be involved precisely as in (62), and (64) and 

 (Go) will represent the envelope we now seek, provided for s we substitute — s, 



which gives 



i i i 



(fa)'" 1 + (m'jS)'- 1 + (rfy)'" 1 = . . . (70). 



Therefore, giving to s the appropriate values 2, 3, f ; we find as follows : — 



XVIII. If the centre of homology describe a circumscribing conic, the faisceau 

 of circumscribing conies 



■» 



« p y 

 will envelope the straight line 



Pa + m 2 /3 + n-y = (71). 



XIX. If the centre of homology describe a circumscribing conic, the faisceau 

 of circumscribing conies 



f z o 3 A 3 



a p y 



will envelope the tri-tangent conic 



± {Pa)± ± (m 8 /3)*± (n 3 7 f = . . . (72). 



XX. If the centre of homology describe a circumscribing conic, the faisceau of 

 circumscribing conies 



/* 9 f h * 



« P 7 



