600 REV. HUGH MARTIN ON CENTRES, FAISCEAUX, 



These, with u = ; v = ; w = Q, give 6 equations in/ 2 , g 2 , h 2 , gh, hf,fg : whence 

 we have the following determinant : — 



I, «.\ 0, 3«.Z(/3 2 -y 2 ), ,3.(7-m)* 2 + i(/3 2 -y 2 y, y .(»-/)«* + '(P 2 -* 2 ) =° ( G1 



m, /3 2 , 0, «.(^-wj)/3 2 + m(y 2 -« 2 ), 3/3.wi(y 2 -« 2 ), y . (»i-n),3 2 + m(y 2 - * 2 ) 



n , y 2 , 0, a. . (n-Z)y 2 + n(<* 2 -», /3 . (w-H)y 2 + »i(* 2 -/3 2 ), 3y . n (a 2 - /3 2 ) 



0, 2/3y, «, 0, 2y . (m-«)/3 2 + m)y 2 -« 2 ), 2/3.(»ft-«) v 2 + n( a 2 - / 3 2 ) 



0, 2y«, /3, 2y . (m - 0* 2 + ?(/3 2 -y 2 )7 0, 2* . (>i-Z)y 2 + n(« 2 -/3 2 ) 



0, 2«/3, y, 2/3.(Z-wi)* 2 + Z( / 3 2 -y 2 ), 2a . (J-»n)/3* + »n(y s - a 2 ) , 0, 



which is the equation of the locus required, and which I have not yet calculated 

 out. (See Postscript, — where I 2 , m 2 , n 2 , are written for /, m, n.) 



Hitherto we have proceeded on the supposition of the faisceaux being curves 

 of the first order, — that is, straight lines. We shall now suppose them to be 

 conies, and we shall take the four special forms, which for brevity we shall call 

 the circumscribing, the self-conjugate, the tri- tangent, and the bi- tangent conies. 



Section II. — The Faisceau of Homology being Circumscribing Conies. 



XIII. (1.) Required the envelope of a faisceau of circumscribing conies, while 

 the centre of homology moves in a straight line. 

 Let the circumscribing conic of homology be 



/ s a s h s 



J- + 9 + "l = o = u , 62). 



« P 7 



and the straight line in which the centre moves 



la + »ij3 + ny = (63), 



i.e. - + —+-. r = (64). 



/ 9 h 



Differentiating (62) and (64), introducing an indeterminate coefficient, adding, 

 making the coefficient of the differentials to vanish as before, and substituting in 

 (62) the values thus found for/, g, h, we have, as the envelope required, 



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



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

 cumscribing conies 



J_ JL A- -o 



fa + <f/3 + h?y ~ 



will envelope the straight line 



Pa + w 2 /3 + n-y = (66). 



