SYSTEMS OE EQTJAEITIES AMONG THE EOOTS OP AN EQUATION. 73] 
And dmding out by {x-\-y) and reducing, the equation is at once seen to be identically 
true. 
11. In order that the quintic may have three equal roots, or that the system of roots 
may be of the form 311, the simplest type to be considered is 
12.13.23.45; 
this gives the function 
which being an invariant, and being of the fourth degree in the coefficients, must be 
the quartinvariant of the quintic. The same type gives also the function 
2 (« - y)\7- - i)\x—ly)\x - iy)\ 
which is a covariant of the degree 4 in the coefficients and the degree 4 in the variables ; 
and it must vanish when fi5=J=c=0, this can only be the covariant 
3(No. 20)-2(No. 14)^ 
which it is clear vanishes as required. 
12. In order that the quintic may have three equal roots and two equal roots, or that 
the system of roots may be of the form 32, the simplest type to be considered is 
12.13.14.15, 
which gives the function 
2(a-^)(a - 7)(a- ^)(a- 
a covariant of the degree 4 in the coefficients, and the degree 12 in the variables ; and it 
must vanish when a=5=c=0, e=f—^ ; this can only be the covariant 
3(No. 13)^(No. 14)-25(No. lb)\ 
which it is clear vanishes as requhed. 
13. In order that the quintic may have four equal roots, or that the system may be 
of the form 41, the simplest type to be considered is 
12.34, 
which gives the function 
a covariant of the degree 2 in the coefficients, and of the same degree in the variables ; 
this can only be the covariant (No. 14). 
14. Finally, in order that all the roots may be equal, or that the system of roots may 
be of the form 5. the type to be considered is 
12 ; 
and this gives the function 
a covariant of the degree 2 in the coefficients, and the degree 6 in the variables, and 
this can only be the Hessian (No. 15). 
It will be obseiwed that all the preceding conditions are distinctive; for instance, the 
covariant which vanishes when the system of roots is of the form 311, does not vanisli 
when the system is of the form 221, or of any other form not included in the form 311. 
