730 ME. A. CAYLEY ON THE CONDITIONS EOE THE EXISTENCE OF GIYEN 
this gives the function 
“(«— Wr— 
which is the quadrinvariant, 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 
42 , 
and this gives the function 
'2{cc-^f{x-y2/f(x—hjy, 
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. 
10. In order that the quintic may have two pair's 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 
This is a covariant of the degree 5 in the coefficients and of the degree 9 in the vai-iables ; 
but it appears from the memoir above referred to, that there is not any ii'reducible 
covariant of the form in question ; such covariant must be a sum of the products 
(No. 13)(No. 20), (No. 13)(No. 14)^, (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, cc, oc. 
or what is the same thing, we may write a=b=e—f=0, the co variant must then 
vanish identically. The coefficients are thus found to be 1, —4, 50, and we have for a 
covariant vanishuig in the case of two pairs of equal roots, 
1 (No. 13)(No. 20) 
- 4 (No. 13)(No. 14)- 
+ 50 (No. 15)(No. 16). 
In fact, writing a=b=:e—f=0, and rejecting, 'where it occurs, a factor the se^eral 
covariants become functions of cx, dtj ; and putting, for shortness, x, y instead of cx, r///, 
the equation to be verified is 
1 . 10(.i'+y)(6.ri+8.ri^+28.ri/+8.r/+6/) 
— 4.1 0(a;+3/)( + 2xy + oy-f 
+ 50(6a’"+ 8.t^+ 6/)(.ri+.riy +A’/+y) = 0. 
