88 NiavsoN: Linear gkomeirv. 



Expanding this and observing that p^ — H^^: — Gp, and 

 p* : H6 = p,(G2^2H3) we have 



P2(D-^G— 3a,DHG ■ 6a, D^Hso-a^Hs )^o. 

 It may now be easily verified that the last ciuantity in parenthesis is 

 equivalent to a^E^; where E., is the eliminant of P ami the Cubi- 

 covariant of C. This gives us another fundamental invariant of the 

 system P and C, and completes the list. We have four fundamental 

 invariants of the system, viz: /^^, E, E, and E^. These four quan- 



2 3 



titles are connected bv the relation /\E2--E - 4E" . 



i_ 21 



INVARIANTS AND COVARIANTS OF A CUBIC. 



It is an easy matter by a very simple device to obtain by the above 

 method a complete list of the invariants and covariants of a cubic. 

 But we must first give a geometric definition of a covariant of a binary 

 form. A covariant of one or more groups of points on a line is a 

 group of points which bears to the given group or groups some rela- 

 tion unaltered by any projection whatever. 



Now instead of representing the point P by the linear equation 



a,x-|-b,y r o, we may say that (x, y) are the coordinates of a movable 



point on the line. Solving the eiiuation a,x ; b,y _o, we have the 



X b, 



ratio — -^r^-- . If we substitute in the formula given above for k, 



y a. 



X b, 



in place , the expression for k gives us a homogeneous equa- 



y a, 



tion in x and y. And since the value of k is unaltered by any projec- 

 tion whatever, we have, by giving different values to k, the ecjuations 

 of an infinite number of covariants of the cubic. The fundamental 

 covariants* are found as before by giving to k its critical values. But 

 instead of repeating the above operations we may readily find the 

 fundamental invariants and covariants of the cubic by substituting 



X bj 



- - for — — in A, E, E,, and E„. The results are ^, the discrim- 

 y a, - 



inant; C, the cubic itself; Hx , the Hessian; and Gx, the cubi- 



covariant. It is necessary, however, in order to bring to light some 



of the fundamental properties of these covariants to go back to the 



x b, 



formula for k. Making the proposed substitution of for — - in 



y a, 



the expression for k we get 



2 1 2 



_ (pS-fwH*) i (ax+by)p^— y(wp^— w2H) {- 

 (p^+H) -;(ax-rby)p^-y(p^-H);- 



