172 MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
32. Collecting’ the results of the last three articles they give us 
d’=~.¥(~h) 
a'" — . p' (a?i) • <p' (x 2 ) . ©' (x 3 ) . p' (x 4 ), 
all of which may be expressed easily in terms of the coefficients. 
33. Theorems deduced.— The root of the biquadratic oc 4 -\-ax i -\-bx 2 -\-cx-\-d — 0 
being expressed by 
^1 = — { a> + \/ i a " V' tt") "b \/ (&" — V' to'")} 
+ / {a! + 6 Z/ {to! 1 + s/ «"') + G 2 s/ (to." — s a!")} 
-| - */ {ot! Q 2 («" -J- a/ a!") -j- 6 \/ (a" — */ a"’)}, 
where 6, G 2 are the imaginary cube roots of unity, then the condition a!" — 0 denotes 
the existence of two equal roots in the proposed equation. 
The condition a" = 0 denotes the following relation of the roots 
Oi ff 4 ) ( x 2 — %) + (a?i - %) ( x 2 — x i) = °- 
The system of coexisting conditions a!' = 0, a" 1 = 0 are necessary and sufficient 
for the existence of three equal roots. 
The condition a' = 0 denotes the following relation of the roots, 
2 (aq — x 2 ) 2 = 0. 
The simultaneous system of conditions u! = 0, a" = 0, od" — 0 essentially and suffi- 
ciently express the coexistence of four equal roots. 
The rational part of the root — only vanishes with the sum of the roots. 
34. We now proceed to determine the constituent parts of the roots of equations 
of the fifth degree by the conditions of their evanescence. 
6, G 2 represent the imaginary cube roots of unity. 
co, co 2 , to 2 , a 4 the imaginary fifth roots of unity. 
aq, x 2 , x 3 , x 4 , x 5 are the five roots of the proposed equation of the fifth degree, viz. 
x 5 -j- a x A -j- h x 3 -f- c x 2 + d x -f- e — 0. 
x i = — j + \/ to -\- \/ /3 + v^ y ^ 
x 2 = — ~ co \J a -f- co 2 \/ (3 -j -a 3 \f y co 4 \/ h 
x 3 = — co 2 \/ a -J- co 4 \/ ft -\ - m \/ y + co 2 & 
