AND ENVELOPES OF HOMOLOGY. 613 



LXVII. (4.) Required the envelope of a faisceau of bi-tangent conies when 

 the centre of homology moves in a tri-tangent conic. The envelope is 



-■ = (t?) *Vr < 127) 



LX VIII. (5.) Required the envelope of a faisceau of bi-tangent conies when 

 the centre of homology moves in the bi-tangent conic 



a 2 = k 2 B 7 . 



The envelope is 



« 2 = k 2s l3 7 (128). 



If the centre of homology move in this conic, the envelope is 



a 2 = Am3 7 (129). 



If in this, the envelope is 



And the f h envelope is 



«2 z=jB*Py (130). 



a 2 = k^By (131). 



If we equate the indices (inverted when necessary) of (124) and (126) ; (125) 

 and (126) ; (124) and (127); (125) and (127) ; (126) and (127); we get the follow- 

 ing somewhat elegant propositions :— 



LXIX. Whether the centre moves in a straight line or in a self-conjugate conic, 

 the faisceau of bi-tangent conies 



envelopes the same bi-tangent conic 



(4mrc) 3 « 2 = l e By (132). 



LXX. Whether the centre moves in a circumscribing or in a self-conjugate 

 conic, the faisceau of bi-tangent conies 



ghu 2 = f 2 By 



envelopes the same bi-tangent conic 



Pa 2 = AmnBy (133). 



LXXI. Whether the centre moves in a straight line or in a tri-tangent conic, 

 the faisceau of bi-tangent conies 



fa? - Vgh . 0y 



envelopes the same bi-tangent conic 



la 2 = 2*/mn. By (134). 



