MU. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
167 
and therefore 
5 3 _ _ 2 a ? 2 _ A ? _ (±* _ 2 i if* 
it will follow that <p (fj) or ij 3 -f « ii 2 + & l\ + c > is the same as A % l + B, putting 
/2 a 2 2 b\ 
A = - It - t } 
a b 
Hence 
B = c — 
P (Ii) • <P (i 2 ) — (A + B) (A | 2 + B) 
and since ii i 2 — y an d Ii + l 2 — 
«" = H (a 2 . -| 
— rr , therefore 
O 
2 a 
. AB 
-f B 2 ) 
, 2 ab ,4 a 3 , o s S*\ 
H w - — • c + *7 • c + w ~ ~rr) 
Again since X = ^ therefore 
Suppose a and b to vanish, then v' oi — c s/ H, a" = K . c ; therefore 
= \ 3 / c { K + V H} + \/ c {K — -v/ 11} — \J — c : hence 
A* -f- H = — 1 K — v'H = 0; 
therefore 
thus the transformation is affected from symmetrical functions to given coefficients. 
23. Expressions for a', a" having been found by various methods, we have also 
the following properties with respect to their evanescence, 
= — Y + \/ (a' + «') + (a' — V' a"). 
When the quantity («") under the quadratic and cubic radicals vanishes, the pro- 
posed equation has two equal roots. 
When the quantity (a') under the cubic but not under the quadratic surd vanishes, 
two of the three differences of the roots are equal, or one root is one half the sum of 
the other two. 
When both these quantities (a', a") vanish conjointly, the given equation has three 
equal roots. 
Conversely when the proposed has two equal roots the equation of condition is u" = 0, 
when it has three equal roots the two equations of condition are a! = 0, a" = 0. 
