AND ENVELOPES OF HOMOLOGY. 605 



The same consideration as that used in the last two propositions shows that 

 this locus is to be got by inverting a, /3, y in the determinant (61). 



Section III. — The Faisceau of Homology being Self-Conjugate Conies. 



XXX. (1.) Required the envelope of a faisceau of self-conjugate conies, when 

 the centre of homology moves in a straight line. 



Here we have 



fa? + g^ 2 + /t-y = = u . . . . (86), 

 and -, + — + -r = (87). 



Proceeding as before, the envelope is 



i i i 



(Z s « 2 ) s + 1 + (m']8 a / + 1 + (ny 2 )' + 1 = . . . (88). 



Giving s appropriate values, namely, 1 ; — 3 ; 3 : we have as follows : — 



XXXI. If the centre of homology move in a straight line, the faisceau of 

 self-conjugate conies 



fa, 2 + gj3 2 + hy 2 = 



will envelope, or inscribe, the quadrilateral represented by 



± (l)"a ± (m)=/3 ± (nfy = . . . (89). 



XXXII. If the centre of homology move in a straight line, the faisceau of 

 self-conjugate conies 



/ 3 g 3 + h 3 ~ U 



will envelope the circumscribing conic, 



F m^ ra 1 = (90). 



a fi y 



XXXIII. If the centre of homology move in a straight line, the faisceau of 

 self- conjugate conies 



fa? + g 3 (3 2 + h 3 7 2 = 



will envelope the tri-tangent conic 



± ffa? ± (m f /3)* ± (n l yf = ° " " ' ( 91)> 



XXXIV. (2.) Required the envelope of a faisceau of self-conjugate conies when 

 the centre of homology moves in a circumscribing conic. 



On the same consideration as before, s in (88) must be read — s, which gives 



the required envelope 



i i i 



©-' + (ft- 1 + (£T= o 



