236 THE EEV. GEOEGE SALHOX OX QUATEEXAET CTBICS. 
discriminant of a cubic. In fact, in this case S is the product of the four deteiminants 
of the system 
a , 
b, 
c , 
d 
a! , 
b', 
c', 
d' 
a”, 
b\ 
c", 
d" 
and therefore its vanishing expresses the condition that some one of the quadi'ics of the 
system X\J -{-vJ]” shall be a perfect square; or geometrically, it expresses the con- 
dition that it shall be possible, through the intersection of two conics of the system, to 
draw a conic having double contact with the third. In this case the Jacobian of the 
system breaks up into a right line and conic. 
But we cannot, in like manner, conclude that the result of elimination between four 
quaternary quadrics is expressible as the result of clearing of radicals the sum of five 
square roots, because the quadrics cannot in general be expressed as four functions each 
of the form 
du^ + ev^. 
We proceed now, from the consideration of the invariants of a quaternary cubic, to 
that of its covariants. To the number of these, however, there is, as far as I am 
aware, no limit, and I must therefore confine myself to enumerating some of the simplest 
and most important. Of covariants of the first order in the variables, the follo\\'ing may 
be regarded as the four fundamental forms. The simplest, L, of the eleventh order in 
the coefficients, 
L = a^’^c^d^e^ {ax-\-hy-\-cz-\- du -\-ev] 
of the nineteenth order in the coefficients, 
L' = cv'lf‘c^d?e^ { hcdex-\-cdeay deohz-\-eahcu -j- ahcd v } ; 
of the twenty-seventh order, 
L" = d^e^ { a^x -\-¥y -\-c~z-\- clhi + eh’ } . 
There is no covariant of the first order in the variables and 35th in the coefficients 
which may not be expressed in terms of the precedmg three and the fundamental 
invariants ; but we have a fourth of the 43rd order, viz. 
L'" = aJ'h^c^d^e^ { a^x -\-lfy-\-c^z^ dhi -j- eh ' } . 
Every other covariant can be expressed in terms of these four. The resultant of these 
four covariants, that is to say, the condition that they shall represent four planes meet- 
ing in a point, is the invariant F of the lOOth degree already noticed. The cubic itself 
can be expressed as a function of L, L', L", L'" having invariant coefficients, which expres- 
sion M. FIekmite calls the form-type of the function. I have not thought it worth while 
to undertake the labour of finding the actual values of the coefficients of the function 
so transformed. 
The simplest linear contravariant is of the 13th degree, that of the 5th vanishing, as 
