MR. MURPHY ON THE ANALYSIS OP THE ROOTS OF EQUATIONS. 169 
when 6, 6 2 are the imaginary cube roots of unity, and (3 (3" being the roots of a qua- 
dratic are of the forms 
f 3 ' = a" -f v' oi"' 
(3 Set!". 
The three other roots of the biquadratic are 
x 2 = 2 S a — (^ + -|-) = — + v' a — S (3 — S y 
x 3 = 2 S(3 — 4 |~) = — — v' a + S (3 — S y 
x i — 2 ^ y — («i + y) = — - j - -/a — /3 + ✓ y. 
The constituent or essentially different parts of the roots are a', a", a"', which we 
proceed to analyse by the conditions of their evanescence. 
25. Suppose a!" = 0, then by art. 23 two of the roots of the cubic are equal, or 
(3 — y, from whence we have x 3 = x 4 , therefore x 3 — x 4 is a factor of a'", and forming 
all the other symmetrical factors, we have 
a'" =: k (a?j — x 2 ) 2 Oj - x 3 ) 2 (x 4 - x 4 ) 2 (x 2 — x 3 ) 2 (x 2 — x 4 ) 2 (x 3 — x 4 ) 2 , 
k being a numerical multiplier. 
26. Next suppose a" = 0, then by the properties of the roots of the cubic already 
demonstrated we have 2 « = (3 + y, or a — (3 = y — a, whence 
( */ a + S 8) (S a — S (3) = (S y -j- S «) (V y — </ a), 
therefore 
fa — x 4 ) (x 2 - Xo) = (x 1 — x 3 ) (x 4 — x 2 ) ; 
one factor of a" is found thus to be 
Oi — ( x 2 — x z) 4 fa “ x :>,) ( x 2 — X i) ■ 
The two remaining symmetrical factors are 
fa — x 2 ) (a? 3 — x 4 ) + (x x — j? 4 ) (a? 3 — x 2 ) 
fa — a? 3 ) fa “ a? 2 ) + (®i — * 2 ) (a? 4 ~ * 3 ) 5 
and a" is the product of all three multiplied by a numerical factor k'. 
2/. Again, suppose a! = 0, then by the article above referred toa + /3-j-y = 0; but 
/3 + 7 = — * 2 ) 2 4 (% “ a h) 2 
« 4 7 = (^1 — ^ 3 ) 2 4 (* 2 ~ x ^ 
a (3 — (x x # 4 )“ 4 C x 2 * 3 ) 2 , 
the sum of which being per se symmetrical, shows that a! has no other but a nume- 
rical factor ; therefore 
a! - k" {(a?j — x 2 ) 2 + fa — x ? ) 2 + (x x - x 4 ) 2 + fa — x 3 ) 2 + fa — x 4 ) 2 + fa — .r 4 ) 2 }. 
MDCCCXXXVII. Z 
