232 THE EEV. GEOEGE SALMON OX QLATEEXAEY CUBICS. 
therefore, by No. 1 bis, is immediately, as Mr. Sylvestee has also proved. 
hcdeyzuv-\-cdeazuvx-\-(leabuvxy-\- eahcvxyz-\-abcdxyzu, 
or 
labcdxyzu. 
We also get immediately from No. 2 bis a mixed concomitant, \iz. the bordered Hes- 
sian, which may be written 
• 'l,cdezuv(a—(if. 
If in this last we substitute ditferentials for a, /3, &c. and operate on U, we get the 
Hessian. If we operate on the Hessian, we get a covariant of the fifth order, to which 
I shall afterwards refer as O, 
<I) = ahcde'S^abafy^z. 
The reciprocant of the given quantic being, as is knoAvn, of the form 64S^=T-. pre- 
sents us with two other contravariants, S and T, the consideration of which is very fertile 
in results. If the quantic were to reduce itself to the form 
ax'^ + + cz^ + du^, 
it is easy to s,ee that the reciprocant reduces to 
6 4i{abcda^ybf — { — 2 } v 
From this, then, we can readily anticipate the form of S in the general case, which, how- 
ever, I have verified independently. It is of the fomdli order in the variables and in 
the letters, and is 
^■=.^abcd{ci — s)(j3 — s)(y — s)(^ — s) ; 
that is to say, 
bcde{Gc, — /3)(a — y)(a — ^)(a — s)-^cdea(i3 — a)(j3 — 7)(/3 — ^)(/3 — s) 
-j-deably — a)(y — f3)(y — ^)(y — s)-}-eabc(b — a)(^ — j3)(S — 7)(5 — s) 
-i-abcd(s — a)(s — ^)(s — y)(s—b). 
From this we can derive several other concomitant functions. For instance, operating 
on this with U, we expect to get a contravariant of the first degree in the variables, and 
the fifth in the coefficients, but this will be found to vanish identically. Operating with 
S on the Hessian, we get the simplest invariant of the cubic, to which I shall refer as 
Invariant A, 
A = 'Vf&dre^ —labcde^abc. 
Symbolically A is (1235)(I246XI347)(2348X5678)^ 
Again operating with S on U^, we get the simplest covariant quadric which is of the 
sixth order in the coefficients, and is 
abcde{ ax^ + by'^ -f cz^ -fi (Zzr + . 
Symbolically this covariant is expressed (I234)(1235)(1456)(2456). Again operating 
with this on S, we get a contravariant quadric of the tenth order in the coefficients ; and 
