170 MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
values ai ready 
28. The numbers k, k', k" may be found by the 
of a cubic, by which we have 
« = «' + </ («" + ✓ «"') + V («" - ✓ «"') 
and 
therefore 
a, 
« + ft + y 
K" = 
2 . 3 
— • (2 a — /3 - y) (2 [3 — « — y) (2 y 
■P) 
as 
and the factors into which k' is multiplied are the same 
. (2 a — (3 — y) . (2 (3 — a — y) . (2 y — a — (3) 
therefore 
Lastly 
But since 
and 
therefore 
k' = 
04 3 3 ‘ 
— — oir^ • (« — PY 0 ■" y) 2 (P - y) 2 - 
« + y = (#1 - x 3 f + (a? 2 - x±) 2 
P + y = ( x i ~ ^ 4 ) 2 + O2 — -^3) 2 
a — (3 = 2 a? 4 — ■% — ^2 ^4) 
= 2 (a?j — a? 2 ) (a? 4 - x 3 ). 
- ^ 2 ; V X 4 “ x i)‘ 
Similarly a — y and (3 — y are expressed, and comparing the expression for 
thus arising with that found before, we have 
2 2 
2 4 
3 3 a'" — 2 6 . or A: = — 33 . 
29. Let us next seek the same quantities a', a", a 1 ", by the theory of elimination. 
When a'" = 0 , the proposed equation which it will be convenient to express by 
<p (x) = 0 has two equal roots, the condition for which is also obtained by eliminat- 
ing x between <p (x) = 0 and the first derived <p' ( x ) = 0 , the function of the coeffi- 
cients arising from this elimination is of the same dimensions, and expresses the same 
condition as the constituent quantity k" 1 , and therefore only differs from it by a nu- 
merical multiplier. 
This quantity in a symmetrical form relative to the roots is therefore 
u'" = h.<p' (a?!) . <p' (x 2 ) . <p' 0 3 ) . 0 (# 4 ) 
and since 
¥ (*1) = (*i — x 2 ) Oi - •%) i x \ — X D 
we have the same result as by the former method and h = k. 
30. When u" vanishes jointly with cd", then since a = (3 = y we have also 
