238 THE EEV. GEOEGE SALMON OX QTJATEEXAET CEBICS. 
The simplest contravariant of the third order is the evectant of the invariant A. 
which is of the seventh order in the coefficients. It is 
2cW(a-5)(«-/3)^-2a5c(ZV(2«-/3-7)(2/3-7-aX27-« ■/3). 
The next is of the fifteenth, 
'^a^lP'&d?e^{hcde){a— h){u—y){a— B)(a — s). 
After the Hessian, the simplest covariant quartic is 
cdlfc^d?-e\ + h^y* + (fz‘^ + dhd + e-v*). 
We have already mentioned the simplest contravariant quartic, viz. 
S — ^bcde(Gc — /3)(a — 7 )(a — b)(a — s). 
The reciprocant of the cubic is 64S®=T^ where T is the contravariant sextic, 
T = Ic^d^eXa - /3)® - 2'l.hcd^e\o, - /3)^(a - yf 
-Y21<abcde^{{a — y){(i — l) — {a — l){y—^)]{{a — l){y—^)~{tt,—^){l—y)) 
{{a — ^){l — y) — {a — y){^ — l)]. 
I have already mentioned the covariant quintic 
<^—ahcde^{ahx^ifz)^ 
which, however, may be replaced by 
abcde{ cdx^ -\-¥y^-\-c^z^-\- dhd + , 
the two differing only by the product of the original cubic and the simplest covaiiant 
quadric. 
The simplest covariant of the fifteenth order in the coefficients and fifth in the vari- 
ables, is what may be called the canonizaiit a^b^c^d^e\vyzuv, since it represents the five 
planes of the canonical form. 
The only covariant of higher order which we shall notice is that of the ninth order, 
which we shall call the covariant 0, viz. the determmant. 
d^U 
d^XJ 
d^XJ 
d^R 
dR 
dxdij 
dxdz 
dxdco 
dx 
d^XJ 
d’^X] 
d^XJ 
dydx 
dr/ 
dydz 
dydeo' 
dy 
d^V 
d^X] 
d^^X] 
d"‘R 
dR 
dzdx 
dzdr/ 
dz^ ’ 
dzdeo 
dz 
d^U 
d^X] 
d^X] 
d^R 
dR 
dwdx 
doody 
diudy 
dm 
dR 
dR 
dx 
dy’ 
dz’ 
dm 
The importance of this covariant consists in this, that the twenty-seven right lines on 
the cubic are determined as the intersection with the given cubic of the smTace of the 
ninth order, 0=4HO, where O is the covariant of the fifth order just mentioned. 
This can be verified with little labour by takmg the general equation of a surface passing 
