468 
ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN 
[150 
this of course gives as the function to be equated to zero, the discriminant of the 
quartic. 
6. In order that there may be two pairs of equal roofs, or that the system may 
be of the form 22, the simplest type to be considered is 
14.24.34; 
this gives the function 
2 (a - g)(/3 - 8) (7 -8)(x- ay)\x - ^y)\x - 7y)\ 
which being a covariant of the degree 3 in the coefficients and the degree 6 in the 
variables, can only be the cubicovariant of the quartic. 
7. In order that the quartic may have three equal roots, or that the system of 
roots may be of the form 31, we may consider the type 
13.14.23.24, 
and we obtain thence the two functions 
20-7) («-8)03-7)03-8), 
2(a - 7)" (a -1)(3 - 7,)(/3 - if, 
which being respectively invariants of the degrees 2 and 3, are of course the quadrin- 
variant and the cubinvariant of the quartic. If we had considered the apparently more 
simple type 
12.34, 
this gives the function 
2(*-m7-S) 2 , 
which is the quadrivariant, but the cubinvariant is not included under the type in 
question. 
8. Finally, if the roots are all equal, or the system of roots is of the form 4, then 
the simplest type is 
12; 
and this gives the function 
2 ( a ~ £) 2 0 “ 72/) 2 0 - 8 y) 2 > 
a covariant of the degree 2 in the coefficients and the degree 4 in the variables; this is 
of course the Hessian of the quartic. 
Considering next the case of the quintic: 
9. In order that a quintic may have a pair of equal roots, or what is the same 
thing, that the system of roots may be of the form 2111, the type to be considered is 
12.13.14.15.23.24.25.34.35.45; 
this of course gives as the function to be equated to zero, the discriminant of the 
quintic.