596 



REV. HUGH MARTIN ON CENTRES, FAISCEAUX, 



It is also evident that if the centre of homology now move in the direct enve- 

 lope of homology (27), the direct linear faisceau of homology will now envelope 



Pa 2 = I6/37 (30). 



the inverse centre will move in 



and the inverse linear faisceau will envelope 



fc 



i = ICjSy 



(31), 



(32); 



and so on, en suite, while the resulting envelopes are successively taken as the 

 curve in which the centre moves, — the n th envelope of the direct faisceau being, 

 of course, 



fc 2 a 2 = (4) B .jS y 



and of the inverse faisceau, 



p = (4)".0y • 

 IX. If the centre of homology move in the curve 



(33), 



(34). 



I 2 m 2 n 2 mn nl Im 







the linear faisceau will envelope a conic section. 



We have, l 2 f 2 + m 2 g 2 + n 2 h 2 + mngh + nlhf + Imfg = 



and, fa. + ^7/3 + \y— = u 



(35), 



(36), 

 (37). 



Differentiating (35) and (37), and neglecting the indeterminate coefficient, as its 

 square will divide out in the final result, we have 



21/ + mg + nh — j, 



If + 2mq + nh — — 

 If + mg + Ink = — , 



(38). 



Eliminating g and h, and neglecting the numerical coefficient, which would 

 divide out as merged in the indeterminate multiplier, we have, 



or, 



Similarly, 



And, 



hy 



3a 



I 



— ; 



T ~~ 



m 



n ' 



3 a 2 

 ~1 2 



a/3 

 Im 



_ 71 



nl' 



3/3 2 



TO 2 



mn 



a/3 

 Im 



V 



yet, 



_ h 



n 2 



nl 



mn 



(39, 



' / 



