22 On Cones which pass through a given Curve of the Third Order. 



And the equation of the cone is 



1^^nfh^lmg'-2lnh^ng'^-\-m%^Q. 

 Or substituting and expanding, 



-7(/98-/)a«rf+iS(«r-iQ^)flrf' + 3y(/e7-a8)62c_3/3(/^7-a8)^»c2 

 -\-h{py-uh)abc-a[^y-uh)dbc-k-{oLyh-^^rf^<^^h)abd 



Now putting 



bd—c^ssp, bc^ad=q, ac—b^—r, 



and in like manner 



^8-/=P, /37-aS=Q, «7-/82=R, 



and reducing, the final result may be expressed in the form 



P{ — 2a6p + (7a— 2/SZ>)^+ {8a— 7c)r} 



+ Q{ - acp-{-{rib - ^c)q -\- Ur) 



■^B>{-{ud-^c)p-(^d-2yc)q 4- 28cr} 



=0, 



where a, b, c, d are current co-ordinates, and p, q, r are quadratic 

 functions of a, b, c, d. The equation is (as it should be) satisfied 

 by the equations (/?=0, ^' = 0, r = 0) of the given curve; it is 

 also satisfied per se when P = 0, Q = 0, R = 0, 2. e. when the 

 vertex is a point on the curve ; this indicates a change in form 

 of the equation, and in fact the cone is in this case of the second 

 order only. Suppose that the co-ordinates of the vertex are in 



this case given by 7^ = — = ^ = — {a an arbitrary quantity), it 

 p 7 6 o" 



may be easily shown that the equation of the cone is 



or at full length, 



{bd-c^) + a(bc-ad) -\- a^ac-b'^) =0. 



In fact this equation is evidently that of a surface of the second 

 order passing through the curve ; and there is no difficulty in 

 showing that it is a cone. 



2 Stone Buildings, 

 MavM856. 



