AND ENVELOPES OF HOMOLOGY. 593 



Third, the conic with respect to which the triangle is self-conjugate, viz., 



u 2 a 2 + v*p + w 2 y 2 = (3); 



And, fourthly, the conic touching two sides of the triangle of reference in the 

 points where the third side meets them, viz., 



h 2 a 2 = jSy (4). 



Section I. — The Faisceau of Homology being Straight Lines. 



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

 (being straight lines) will envelope a conic touching the three sides of the triangle 

 of reference. 



Let the centre of homology move in the straight line 



la t- m/3 + n y = (5). 



Retaining the co-ordinates of the centre in the form in which they appeared in 

 the former paper,/" 1 , g~\ h' 1 (co-ordinates of PJ, and substituting these in (5)> 

 we have 



If' 1 + mg- 1 + nh- 1 = (6). 



The equation of the straight line of homology is, 



fa + g& + hy = = u (7). 



Now, by the theory of envelopes, 



du = = adf + (3dg + ydh ..... (8). 



Differentiating (6), = - if-' 2 df - mg~ 2 dg - nh-' 1 dh .... (9). 



Multiplying (8) by an indeterminate coefficient p, and adding (9), we have 



(pa - If- 2 ) df + (pj3 - mg- 2 ) dg + (py - nhr 2 ) dh = . . . (10). 



Now, as p is indeterminate, and as it is the ratios merely of the three para- 

 meters/, g, h, we are concerned with, reducing them virtually to two — with which 

 the fact that they are homogeneously involved accords — we are at liberty to 

 make two suppositions. Let these be that the coefficients of df and dg in (10) 

 shall vanish. This makes the coefficient of dh also vanish ; and we have, 



f=±Jh ° = ± Jw' h=± Jry ■ ■ (11) - 



Substituting these values in (7) gives, 



± (la? ± (m/3)* ± (nyf = . . . . (12) ; 



which is the equation of a conic touching the three sides of the original triangle. 



II. If the centre of homology move in a conic touching the three sides of the 

 original triangle, the linear faisceau of homology will envelope the curve, 



± (la)i ± (»ij8)» ± (ny)> = . . . . (13). 



