newson: linear geometry. 9 I 



ax^ -i-3bx2y-p3cxy'^ -f-dy-'^o, as above, the ground points of the 

 system of binary coordinates are not in any special relation to the 

 points of the cubic. But any two points on the line may be chosen 

 for ground points. If by choosing for ground points two points on the 

 line specially related to the points of the cubic, we can simplify the 

 equations of the cubic and its covariants, we are at liberty to do so. 

 The most convenient points to choose for ground points of our system 

 of binary coordinates are the two points constituting the Hessian of 

 the cubic. The equation of the Hessian in full is 



(ab— c2)x2^(ad— bcjxy4-(bd— c=^)y2— o; 

 and the ecjuation of the ground points is, of course, xy — o. We 

 observe that if b and c are put equal to zero, the Hessian reduces to 

 adxy=o; and hence becomes identical with the ground points. Choos- 

 ing, therefore, the points of the Hessian for ground points, the equations 

 of the cubic and its system of covariants are all reduced to their sim- 

 plified or canonical forms by putting b=o and c--o. 



The canonical forms of the above quantics are as follows: 



(i). C-^ax^-pdy'^; Gx -^ad(ax^ — dy ^ ). 



(2). Qj=:=ab,x- — ajdy"-; Q2=ad(ab jX-- - a,dy- ); H=:=adxy. 



(3). P=ajX + bjy; Pj^^abjx + a^dy; P2^ad(ab;|x— a^dy; ) 

 p3=ad(ajX— bjy). 



The invariants of the system ^, E, E,, E^, reduce to 



(4). ^i^a^d^; E=ab' — da ; Ej :r_-adajbj ; E2=ad(ab — da' )• 

 We shall hereafter make use of these canonical forms in studying 

 the geometric and other properties of the cubic. 



INVARIANTS OF THE QUARTIC. 



The invariants of a quartic may be determined by means of the 

 anharmonic ratio of the four points constituting the quartic, in a 

 manner entirely similar to that which we used for determining the 

 invariants of the system of point and cubic. Let the general equation 

 of the quartic be written in the form 



ax*-j-4bx'*y-]-6cx-y2-f"4'^-'^y ■* ~ ey*=o. 

 Let the four factors of the quartic be represented by 

 x-i-Rjy; x + Rgy; x — Rgy; and x~j R^y. 

 b b 



where Rj== yV + yl-r} r, Rg^^ TlP— l q + v''» 



a - a 



b b 



R3= hlP + l q-ir, R^--^ 1 p-i q — I r. 



a a 



In these expressions p^^a^Zj — H, q-^=a-Zo H, v=^a.'^z^ — H; where 



|a b 

 H=j and Zj, z.,, z^ are the three roots of the reducing cubic of the 



la b ' ■ 



