MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
173 
x 4 = — -T + s/ a + co ^ -f - co 4 \/ y + <« 2 \/ & 
#5 — r- + a/ 4 \/ a -f- co 3 \/ -j- co 2 ^ / y + co § ; 
the formulae for x 3 , x 4 , x 5 , x 1 are derived from the formula for x 2 , by writing succes- 
sively co 2 , co 3 , &> 4 , cu° instead of co. 
oc — oc! -j- \/ (3 r -j- \/ y 1 -f- \/ h 1 
(d — oc! -j- \/ j3' — \/ y' — \/ V 
y — a! — v' (3 ; -j~ V y' — V 
§ = a! — V (3 r — v' y' + s/ cf, 
such being the forms of the roots of a biquadratic. 
Again, the expressions for (3', y', h' as roots of a cubic, are 
p = a” + y p" + 4/ r" 
y = cc" + 6 4/ (3" + IP 4/ y" 
i 1 = <*" + p 4/ ,3" + 6 4/ y. 
Lastly, |3", y" as roots of a quadratic, are expressed by the following formulae : 
(. 3 " = oc!" + y' a IV 
y" = a " 1 — « 1V . 
The quantities — j, u', a", a'", a lv are the constituent or essentially distinct parts 
of the roots x x , x 2 , x 3 , x 4 , x 5 , and the analysis of their formation is to be sought by 
observing all the conditions under which each may vanish. 
35. If for - y we put -- + x<i + ^ - ~h Xf > 3 it is obvious that the system of five 
equations for the roots is equivalent to one of only four, viz. 
5 \/ a = x 4 -{- aft x 2 + ^ X z -f co 2 X A -f- co x 3 
5 (3 = x x + co 3 x 2 -f- co x 3 -j- <w 4 x 4 + ^ ■% 
5 \/ y — Xj + x 2 + x 3 + m x 4 + x 5 
5 \/ 'h = x 4 + CO X 2 + co 2 x ? -f- co’’ x 4 + ■% 
the right hand members of which equations differ from each other only by the par- 
ticular fifth root of unity in use, which considered as co in the first, will be co 2 , co 2 , co 4 
respectively in the second, third, and fourth. 
36. Suppose « IV = 0, then by art. 33 two roots of the biquadratic must be equal. 
First let y = b, which can only happen under the five following relations, 
\/ y = \/ s, 7 = co y = u 2 <$, y = co 3 &, y = ^ -y i, 
which furnish five factors, linear functions of the differences of the roots, and since 
