NEWSON: LINEAR GEOMETRY. 89 



This readily reduces to the form 



(ax+by) \ (k+w)p-r(k + w2)p'^H \ -y] (k + w2)p^ _(k + \v)H2 \^o. 



This is a linear equation and represents some point on the line. 

 By giving to k different values this equation may be made to represent 

 all points on the line. By giving to k in succession the values i, o, 

 and CO, we obtain the three linear factors of the cubic, each multiplied 

 by a constant. The product of these three factors Lj, Lg, and Lg, is 

 L,L3L3=a3p2^«C. 



By giving to k in succession the values — w and — w^, we get for 

 the linear factors of the Hessian 



p"^'] (ax-|-by)H — pyj-=o, and ] (ax-|-by)p-t-H'y [=0. 

 The product of these factors fj and fg is 



pifjf2=p'*a2Hx; or fjfg^^paSHx. 

 I 

 By giving to k the values — , 2, and — i, we have 



2 



11 1 



fg — p3f^=o; fg — wp3fj:=o; and fg — w^pSf^^^o. 



3 3 

 The product of these three factors is pf — f^^a^p'-^Gx- 



This last equation gives an expression for the cubi covariant in 

 terms of the factors of the Hessian. 



The equation of relation among the invariants of P and C, viz: 



3 ^ ^1 

 ^ E2=E'' -(-4E becomes, when — is substituted for , 



2 



y 



COVARIANTS OF THE SYSTEM OF THE POINT AND CUBIC. 



In the investigation of the covariants of the system of the point and 

 cubic I shall make use of a principle which it is not necessary for me 

 to stop to prove. It is this: the relation of pole and polar in geome- 

 try of one dimension as well as in geometry of two and three dimen- 

 sions is unaltered by projection. The successive polars of the point 

 P with respect to the cubic C and its covariants are unaltered by 

 projection, and hence are covariants of the system. And moreover 

 the complete set of such polars is identical with the complete list of 

 fundamental covariants of the system. 



Assuming these principles as sufficiently well known, it is easy to 

 write down a complete list of the covariants of the system. 



(i). Tzuo cubic covariants, viz: the cubic itself, C; and the rubi- 

 covariant (ix- These written in full are C=ax3-^3bx2y-f-3cxy2-|-dy3. 

 Gx=(a2d— 3abc + 2b3)x3 J-3(abd + b2c— 2ac2)x2y 

 4-3(2b2d— acd— bc2)xy2 4-(3bcd — ad2— 2c3)y3. 



