THE EET. GEOEGE SALMON ON QHATEENAEY CUBICS. 233 
again operating with this last on U, we obtain a covariant of the first order in the 
variables and the eleventh in the coefficients. But it being sufficiently obvious that new 
invariants, covariants, and contravariants can be formed ad libitum by the process 
already explained of combining a covariant and contravariant already found, I consider 
it needless to enter into further detail as to the steps by which particular covariants are 
obtained, and proceed to form into tables the most important results. 
I write, for brevity, 
a-\-b-ic-\-d-\-e—f^ 
ab-\-ac-\-ad-\-ae-ibc-ibd-\-be~\-cd-\-ce-\-de—q^^ 
cde-\-bde-{-bce-\-hccl-\-ade-\-ace-\-accl-{-abe-\-ahd-\-abc=7\ 
bcde -\-cdea-{-deab-\-eabc-\-abcd=s, 
abcde=t. 
I commence with the invariants of the cubic, which appear to be all reducible to the 
five following fundamental invariants of the 8th, 16th, 24th, 32nd, and 40th orders 
respectively : — 
A=s^—^rt. 
B 
C=fs. 
D=fq. 
E=f. 
Whence also C*— AE = 4^V. 
Since any invariant must be a symmetric function of «, <?, d, e, it can be expressed 
in terms of p, q, r, s, t, and therefore in terms of A, B, C, D, E. We can form, however, 
precisely as in the case of binary qirintics, skew irrvariarrts which cannot be expressed 
rationally in terms of the five fundamental invariarrts, but whose squares can be expressed 
as rational functions of these quarrtities. The simplest invariant of this kind is of the 
hundredth degree in the coefficierrts, and for the canoirical form is multiplied by the 
product of all the differences betw^een airy two of the quantities a, b, c, d, e. The 
following is the expression for the square of this invariant F in terms of the simpler 
invariants ; — 
256F^= 800000 E^+240000 E^DA-640000 E^CB+ 36000 Em^-128000 E^B^A 
+ 3600 E^BA^-27 EW+576000E^D^B+13200 E=’DW+272000 E^DC=^ 
+131200 E*DCBA+2520 E^DCA^-409600 E^DB^+8960 ETO^A^-72 ETOA^ 
+ 30400 E^ChA+ 115200 E^C^B^+5520 E'eBA^+108 E'CW-10240 E^CB^A 
-96 E*CB^A^+ 65536 E^B^-2048 E-^B^AH-16 E^B^A^-230400 E^D^C 
+40960 E^D3AB-64 E^D'A^- 02240 E^D^C^A+261120 E^DW 
+5696 E^D^CBA^-90240 E^DeB-9216 E^D^B"A+16 E^D"B*A^-5256 E^DC'A^ 
MDCCCLX. 2 I 
