MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
163 
9. Let us now extend the same views to equations of the third degree, and let 
x 1 x 2 x 3 be the three roots of the cubic x 3 + a x 2 + b x + c — 0. Put 
*1 = “ T + V a ' f 
* 2 =-y + + 
= ~ y -1- 1/ « + 04/ 
The numbers 0, are the imaginary cube roots of unity, and we may observe that 
the formulae for x 3) x l differ only from that for x 2 in having 6 2 , ft respectively instead 
of &. 
10. The quantities a, (3, which are obviously similarly involved, are the roots of a 
quadratic, and of the forms 
« = ot! + V a" 
(3 = cc' — V a " ; 
the three quantities — -y, a' and a" are the constituent parts of the roots of the cubic 
in the sense in which those words have been used ; the first is the same as Xx + + X& , 
and the other two can be found from the conditions of their evanescence as follows. 
11. Suppose a" = 0, the theorem of art. 8. gives us a = /3, whence we find x 2 = x 3 ; 
now x 2 — x 3 vanishing with a" is a factor of it ; and the other symmetrical factors 
k being a number, we must therefore 
have 
u" = k (a?j — x 2 ) 2 (x 1 — x 3 ) 2 (x 2 — x 3 ) 2 . 
12. To find a! in like manner, suppose a' = 0 the theorem of art. 8 before referred 
to, in this case makes a = — (3, and the three roots of the cubic accordingly are 
changed to the following : 
a 
*1= “T 
% = — -j — (0 — Q 1 ) V «; 
whence we readily find that 2 x l = x 2 + x 3 , therefore 2 x x — x 2 — x 3 is a factor of a', 
and the other symmetrical factors are 2 x 2 — x 1 — x 3 , and 2 x 3 — x x — x 2 • hence if k' 
be a numerical factor 
a' = Ic' (2 x x — x 2 — x 3 ) (2 x 2 — aq — a? 3 ) (2 a? 3 — aq — a; 2 ) ; 
these symmetrical functions can be expressed by the given coefficients of the equa- 
tion. 
