128 RESOLUTION OF SOLVABLE EQUATIONS 
SKETCH OF THE METHOD EMPLOYED. 
s 
§2. Let 71, 72, 73, 174, 15, be the roots of the solvable irreducible 
equation of the fifth degree wanting the second term, 
F (a) = @ + poo + p3 a2 + pyx + ps = 0. (1) 
It was proved in the “ Principles ” that 
$ 5 3 3 
m= $(4) + 45 + 43 + 4); 
where 4;, 42, 43, 44 are the roots of a biquadratie equation 
auxiliary to the equation # (x) = 0. It was also shown that the 
root can be expressed in the form 
47 ee ee 
rm = $ (4) + ay A; ater A, + hy 4; ) (2) 
where a, ¢, 41, involve only surds occurring in 4; and nosurds 
oveur in d; except (hz + h / 2) and its subordinate ,/ 2; z being 
equal to 1 + ¢? , and hand ¢ being rational. As in the “ Principles,” 
1 1 1 1 
Ae AIS Ae sy A 
we may put du, = 4? , 5g = A? , 5ug = 4} , Sug = 4. Then 
7] = U1 + Ug + Ug + MH. (3) 
Let ‘Sy be the sum of the roots of the equation 7 (x) = 0, S the sum 
of their squares, and so on. Also let 
2 2 2 2 2 
Y (uy uz) = uy ug + uz my + uzuy + Up u2, , 
Bae 3 3 3 a peas 
“(uy Uz) = U1 Ug + U2 Ug + UZ U1 + U4 2I3, 
: 2 2 2 2 22 ein 2 2 
2 (uy U3 U4) = Uy U3 Ug + Ug Uy UZ + UZ Ugue + UsuZmM; 
5 5 5 5 5 
Then = (ui) = uy + u2 + wa + us; r (4) 
So = 10 S3 = 15 JS (ui us); 
2 = 10 (ur wy + re us ), S3 = 15 } XY (ui us );, 
: ( 3 2 
4 —r 4b) 12 (wr U2 yt -- 35 (S82) —- 60 Uy U2 UZ U4 
5 2 2 
Ss = 5 }E(ui)} + 2 (S2 Ss) + 50 {E(u wg wa)}- 
§3. It was proved in the “ Principles” that 1 ws and w ug are the 
roots of a quadratic equation. But 
25 uy Us = hy 41, and 25 uy ug = ay €; 4. 
Therefore, because a, e,, 4, , involve no surds that are not sub- 
1 
ordinate to Ae , “¥ zis the only surd that can appear in wy w4 and 
2uU3;. Consequently we may put 
