,' : V ■ 
150] SYSTEMS OF EQUALITIES AMONG THE ROOTS OF EQUATION. 469 
10. In order that the quintic may have two pairs of equal roots, or that the 
system of roots may be 221, the simplest type to be considered is 
14.15.24.25.34.35.45; 
a type which gives the function 
S (a — S)(a - e)(/3 - S)(/3 - e)(y - 8)(y - e)(8 - e) 2 (x - ay) 3 (x - fty) 3 (x - yy) 3 . 
This is a covariant of the degree 5 in the coefficients and of the degree 9 in the variables; 
but it appears from the memoir above referred to, that there is not any irreducible 
covariant of the form in question; such covariant must be a sum of the products 
(No. 13)(No. 20), (No. 13)(No. 14) 2 , (No. 15)(No. 16) (the numbers refer to the Cova 
riant Tables given in the memoir), each multiplied by a merely numerical coefficient. 
These numerical coefficients may be determined by the consideration that there being 
two pairs of equal roots, we may by a linear transformation make these roots 0, 0, oo, oo, 
or what is the same thing, we may write a=b=e=f= 0, the covariant must then 
vanish identically. The coefficients are thus found to be 1, — 4, 50, and we have for a 
covariant vanishing in the case of two pairs of equal roots, 
1 (No. 13)(No. 20) 
- 4 (No. 13) (No. 14) 2 
+ 50 (No. 15)(No. 16) 
[or in the new notation AH — 4AB 2 + 50CD], 
In fact, writing a = b = e=f= 0, and rejecting, where it occurs, a factor x 2 y 2 , the several 
covariants become functions of cx, dy; and putting, for shortness, x, y instead of cx, dy, 
the equation to be verified is 
1.10(# + y)(6^ + 8 x 3 y + 2Sx 2 y 2 + 8 xy 3 + 6y 4 ) 
— 4.10(¿c + y)(3x? + 2xy + Sy 2 ) 2 
+ 50 (Qx 2 + 8xy + 6y 3 )(oc? -I- x*y + xy 3 + y 3 ) = 0; 
and dividing 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 
2(a-/3) 2 (/3-y) 2 ( 7 -a) 2 (8- e ) 4 , 
which being an invariant, and being of the fourth degree in the coefficients, must be 
the quartinvariant of the quintic [that is No. 19, = 6r]. The same type gives also the 
function 
2 (a - /3) 2 (£ - y) 2 (y - a) 2 (8 - e) 2 (x - 8y) 2 (x - eyf,