268] 
THE THEORY OF EQUATIONS OF THE FIFTH ORDER. 
311 
It is proper to notice that, combining together in every possible manner the six 
roots of the auxiliary equation, there are in all fifteen combinations of the form 
</>i</> 2 + <^3</>4 + $ 5 </> 6 . But the combinations occurring in the above-mentioned equations 
are a completely determinate set of five combinations: the equation of the order 15, 
whereon depend the combinations </>+ <£ 3 $ 6 , is not rationally decomposable into 
three quintic equations, but only into a quintic equation having for its roots the 
above-mentioned five combinations, and into an equation of the tenth order, having 
for its roots the other ten combinations, and being an irreducible equation. Suppose 
that the auxiliary equation and its roots are known; the method of ascertaining what 
combinations of roots correspond to the roots of the quintic equation would be to 
find the rational quintic factor of the equation of the fifth order, and observe what 
combinations of the roots of the auxiliary equation are also roots of this quintic factor. 
The direct calculation of the auxiliary equation by the method of symmetric functions 
would, I imagine, be very laborious. But the coefficients are seminvariants, and the 
process explained in my memoir on the Equation of Differences, [262], 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. I will commence by referring to some of the results obtained by Mr Cockle 
and Mr Harley. 
In the paper “ Researches on the Higher Algebra,” Mr Cockle, dealing with the 
quintic equation 
v 5 — 5 Qv + E = 0, 
obtains for the Resolvent Product 6 (=/w/w 2 /« 3 /« 4 ) the equation 
6 R + 2QE 5 5 6> 4 + 2Q 4 5 7 6 3 + Q°-E 2 o 10 6 2 - (58Q 5 - E s ) E0 + 5 14 Q 8 = 0; 
and he remarks that this equation may be written 
(i9 3 + 5\QE6 + 5 7 Q 4 ) 2 = 5 10 (108 Q 5 E - E*) 0, 
so that — 0 is determined by an equation of the sixth order, involving the quadratic 
radical VE (E 3 — 108Q 5 ), 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 
t X]0c<2 -j- -|- x z x A -j- X4X5 -p XfrX] (= 1234o), 
r = x x x 3 + x 3 x 5 + x 5 x 2 + #2#4 + «4«! (= 24135), 
and he obtains for the form v 5 — 5Qv 2 + E — 0, the relation 6 — 5tt, which, since here 
T + t = 0, gives 6 = — 5t 2 .