46 
ME. A. CAYLEY ON THE EQUATION EOE A EUNCTION 
quintic equation, one of these covariants is, in regard to the coefficients, of the degree 6, 
which exceeds the limit of the tabulated covariants, the covariant in question has there- 
fore to be now first calculated. The covariant equations for the cubic and the quartic 
might be deduced from the formulae Nos. 119 and 142 of my Fifth memoir on Quantics* ; 
they are in fact the bases of the methods which are there given for the solution of the 
cubic and the quartic equations respectively ; and it was in this way that I was led to 
consider the problem which is here treated of. 
1. The notation ^ (a, /3, y . .) is used (after Professor Sylvester) to denote the pro- 
duct of the squared differences of (a, (3, y . .), and the notation ^(a, j3, y . .) to denote 
the product of the differences taken in a determinate order, viz. 
((3-y)((3-l) . . 
(y—h).. 
2. The product of the differences of the roots of an equation depends, as akeady 
noticed, on the square root of the discriminant ; and in order to fix the numerical 
factors and signs, it will be convenient, in regard to the equations 
{a, b, cjv, 1 )"= 0 , 
(a, b, c, djy, 1 )' = 0 , 
{a, b, c, d, djv, 1)'‘=0, 
{a, b, c, d, e,fXy,lf=0, 
to write as follows : — 
A 7 )= ^2\/— (27ff'*<Z-+4ac“-l- . .) 0 , 
^^(a, (3, y, ^)=— ^>/256aV— 27a"c?"+ . . □, 
(3, y, e)=— ^\/3125ay‘"-f-266aV+ . . = — 
where it is to be observed, for example, that writing in the last equation g=0, and 
therefore /=0, we have ^^(a, (3, y, 0)=—~^256aV+ , which agrees with the 
equation ^^(a, f3, y, 0) = a/3y^^^(a, f3, y, b)~^^^(oc, (3, y, h), if for ^^(a, (3, y, h) we sub- 
stitute the value given by the last equation but one. 
For the cubic equation (a, b, c, d^y, 1)^=0; 
3. We have to find the equation for (3)=u—(3; the roots are 
6i=^—y, 6^z=y—cc^ ^3=a— j3. 
* Philosophical Transactions, vol. cxlviii. pp. 415-427 (1858). 
