Roberts — On Binary Cubics and their Associated Forins. 525 

 wMle the co-ordinates of a tangent line are 



a = Xx'x^, f= Zxx'X'i, 

 h = - 2x^X2, g = Ixx x^, 

 c = a;/, h = Xo^. 



Forming then the well-knoTSTi condition that these lines should meet, 

 we find 



Xi^{aoai - oofli) + 2Xi^Xn{aoa2 - a^ao) + x^xi[aQaz - OzOc + 3(fl«ia2 - Oiffo)} 



+ 2x{ai ai - tti a^) Xi x.? -^{ana^-a^ a,) a?2* = 0, 

 or, 



{aa) a^a^ = 0. 



Hence, to the Jacobian of /and ^ correspond the four points on the 

 curve, the tangents at which meet the line/, ^. 



It is to be observed that these four lines also meet the line corre- 

 sponding to the line/ <^. The Jacobian is thus geometrically shown to 

 be a combinantive covariant, since it depends only on the position of 

 the line / ^. 



YI. 



In addition to the forms / and <^, and their cubic covariants, there 

 are two cubic forms, 



(«V)«^^V-.> (aA)a^-A^, 



which we now discuss. 



Since the cubic covariant of the binary cubic is the evectant of its 

 discriminant, it follows easily that the cubic covariant plane is the 

 polar plane of the point with regard to the developable. 



By taking the polar plane of a point on the line 0' we find it to 

 be of the form 



\^ Q + X-jxq^ Xfx-k + ix^K= 0. 



Hence we have two new planes, q and k, giving rise to two cubic 

 covariants in the binary system, the leading terms of which are 



dQ dQ dQo dQ^ 



g-O = do -3- + tti -— - + Oo -y-^ + as -J-, 



dao acii da- aUi 



dK^ dKo dKo dK 

 Q/Oq aai ci(X2 ao-i 



where 



^0 = Co'^z - 3<ro ai a. + 2^^ , 



£o= CLQ-a^— Soq 01102 + 2o.i'. 



