THE EEV. GEOEGE SALMON ON QUATEENAEY CUBICS. 231 
Let us now examine Avhat the latter function becomes when the given quantic is in 
the form 
ax^ -\-hy--\-cz^-\- du--\-ev^. 
Eliminating v by the help of the relation connecting y, z, u, v, the coefficients of 
id become a-fe, b-\-e, c-\-e, d-\-e respectively, while every other coefficient 
becomes e. Substituting these values in No. 2, the contravariant is 
cd {bcd-\-{hc-\-cd-\-dh)e} 
{cda-\-{da-\-ac-\-cd)e} 
+ '•Z {dab {ah -\-hd-{-da)e} 
+ {abc-{-{bc-\-ca-{-ab)e} 
— 2^ { adfiy -j- bdya -{- cda^ -|~ bcab -f- caj3^ -j- abyb } . 
Lastly, write in the form just found a — s, j3 — s, y— £, s for a, /3, y, and it becomes 
bcd{a — zf-\-cda{\i — df + dab{'/—zf + abc(^ — sf + bce{a — bf + cae{j3 — 
-\-abe{'y—'bf-{-ade{(3 — '/y-}-bde{a — 'yy-{-cde{a — (By, 
or, as it may be written, (No. 2 bis) 
2cde{a — l5y. 
I have given this work at length, in order fully to illustrate the process employed in 
other cases. The discriminant may be obtained either by substituting in No. 1, a-^-e, 
b-\-€^ c+e, d-\-e for a, i, c, d^ or by substituting in No. 2 bis, ditferential symbols for 
a, /3, y, h, £, and operating on U. In either case we get — 
(No. 1 bis) 
bcde-\- cd ea + d eab + eabc-\- abed, 
labed. 
If the given quadric had been given as a function in its most general form of five 
variables connected by the relation x-\-y-\-z-\-u-\-v, its discriminant would have been got 
by taking the bordered Llessian of the quadric (the variables being considered as inde- 
pendent), and then writing cc=(i = 'y='h=i=l. And in general it is evident that any 
invariant of a function of n variables connected by the relation 
ccx -j— j3 y -|- yz -|“ ^ ii “h ^ ^ “h . — 0 , 
is a contravariant of the same form considered as a function of n independent variables, 
a, j3, y, &c. being the contragredient variables. 
The reciprocant and discriminant are the only functions concomitant to a quadric. 
Let us proceed, then, to the quaternary cubic 
U = axd-\- bif + cz^ -f- did -J- eid. 
Its Hessian is the discriminant of any polar quadric, 
axx'‘^ -j- byij^ -f- czz'^-{- duu'’‘ evd^. 
