THE THEOET OF EQUATIONS OF THE FIFTH OEDEE. 
265 
memoir on the Equation of Differences was therefore applicable, and by means of it, the 
equation, it will be seen, is readily obtained. The auxiliary equation gives rise to a 
corresponding covariant equation, which is given at the conclusion of the memoir. 
1. 1 will commence by referring to some of the results obtained by Mr. Cockle and 
Mr. Haklet. 
In the paper “ Eesearches on the Higher Algebra,” Mr. Cockle, dealing with the 
quintic equation 
r®— 5Qf;+E=0, 
obtains for the Eesolvent Product the equation 
+ 2 QE5’^‘ + 2 Q*E^5 - ( 5 8Q® — E^)E^ + 5 = 0 ; 
and he remarks that this equation may be written 
{6^ + 5=QE^ + 5'Q^)^ = 5'“(1 08Q^E- E-^)^, 
so that ^ — d is determined by an equation of the sixth order, involving the quadratic 
radical \/E(E^— 108Q®), which is in fact the square root of the discriminant of the quintic 
equation. 
2. Mr. Harley, in his paper “ On the Symmetric Product, &c.,” makes use of the 
functions 
r = ^,^ 2 + 4- ^ 3 -^ 4 + *^’ 4 ^ 5 + -^5-2^1 ( = 12345), 
T = r .o-a (=24135), 
and he obtains for the form — 5Qy^+E=0, the relation ^=5rr', which, since here 
r-|-r'=0, gives ^=— 5r^ 
Hence r 
is the root of an equation of the sixth order involving the radical 
\/E(E^— 108Q*), and which is in fact s/ — the equation 
f+5QEf+v/E(E^-108Q^)f-5Q^=0, 
given in Mr. Harley’s paper “ On the Theory of Quintics.” 
3. And in the same paper there is given a system of equations 
f 1 f 3 + CC + = a’, ( 3 'Q — rf ), &c. , 
connecting the five roots of the given quintic equation with the combinations 
C^3+^/5+C^6) &c. 
of the roots of the equation in t. 
4. I quote also, Avith a slight change of notation, the following results from the paper 
“On the Symmetric Product, &c.,” viz. considering the quintic equation under the form 
(a, 5, c, c, d, 1)'=0, 
we have 
fcofa,^=^X-^-\-7{aj + 
2 0 
MDCCCLXr. 
