AND ENVELOPES OF HOMOLOGY. 611 



will envelope the self- conjugate conic 



± y ± 5 ± S=° ,IH) - 



LVIII. (4.) Required the envelope of a faisceau of tri- tangent conies when 

 the centre of homology moves in a bi-tangent conic, say 



Here we have 



±/ s a* ± g s /B» ± fry* = 0, 



and f* — k*gh = 0. 



Proceeding as before, the envelope is found to be 



(g ) « 2 = % • • ' " ( 115 )- 



another bi-tangent elegantly related to the former. 



LIX. If the centre of homology move in this bi-tangent, the bi-tangent envelope 

 will of course be 



If in this, the envelope will be 



(jpFfw) " 2 = h ■ • (H7). 



If in this, the envelope will be 



/k Ssi \ 4 



\jT 8* + 4fi+2s + l ) " 2 = P7 ■ • * ( 118 )' 



And so on, en suite, the law being obvious,— the t tb envelope being 



(2_s)< » 4 



* V'rf = 07 (118'). 



!-(->») ' I 

 2 1-* / 



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

 centre of homology moves in a self-conjugate conic, say la 2 + mfi 2 + ny 2 - 0. 



By a similar consideration to that used in LIV., in (101') for s read ^ s, and 

 we have the envelope required, namely 



i i i 



±(Z'«)*+ 2 ±(m'j8) ! + 2 ±(»is7) s+2 = . . . (119). 



Hence, giving to s appropriate values, namely, — 1; — f ; —3: we find as follows : — 



VOL. XXIV. PAET III. 8 C 



