470 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN SYSTEMS, &C. [150 
which is a covariant of the degree 4 in the coefficients and the degree 4 in the 
variables; and it must vanish when a = b = c = 0, this can only be the covariant 
3 (No. 20) - 2 (No. 14) 2 , [= 3H - 25 2 ], 
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 con 
sidered is 
12.13.14.15, 
which gives the function 
2 (a - ¡3) (a - 7) (a - S) (a - e) (x - /%) 3 (sc - 7y) 3 (x - 8y) 3 (x - ey)\ 
a covariant of the degree 4 in the coefficients, and the degree 12 in the variables; 
and it must vanish when a = b = c = 0, e—f= 0; this can only be the covariant 
3 (No. 13) 2 (No. 14) - 25 (No. 15) 2 , [= 3A 2 B - 25<7 2 ], 
which it is clear vanishes as required. 
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 
2 (a - ^) 2 (7 - S) 2 (x - ey)\ 
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), [= B\ 
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 
2 (a - ¡3f (x - 7y) 2 (x - %) 2 (x - ey)\ 
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), [= G]. 
It will be observed 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 
vanish when the system is of the form 221, or of any other form not included in 
the form 311.