162 
MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
x Y — x 2 vanishing- with a is a factor of it ; and since the roots must be symmetrically 
involved in a, the other factor is x 2 — x 1 , whence 
a = k (x l — x 2 ) 2 , 
where k is simply a number. 
Put for - — its expression in terms of the roots, and we thus have 
whence 
then 
+ S4 k, 
X 0 — 
x i ~i~ x i ___ £1 x i ^ ^ J, 
x x — x 2 = (x l — x 2 ) V' 4 k, or k = — ; 
a = T ~ x z) 2 ’ 
which is a symmetrical function, and therefore easily expressed by the coefficients. 
7. From this it follows that « = 0 is the condition that the equation may have two 
equal roots ; but if the proposed quadratic be represented by <p — 0, and its derived 
equation by <p' — 0 , which is the same as 2 x -f- a ~ 0 , the condition for two equal 
roots is obtained by eliminating x between these equations, which by the theory of 
elimination gives as the sought condition 
< p ' oo . < p } 0 2 ) = 0 5 
we have thus 
cc = k' p' (aq) . <p ' (x 2 ), 
k' being numerical. 
Now <p ( x ) = (x — aq) (x — x 2 ), therefore <p' (x) = (x — aq) + (a? — x 2 ), whence 
<p f (a? x ) = a?j — x 2 , <p' (a? 2 ) = x 2 — aq, 
consequently k' = — ~ ; and converting the sum and product of aq, x 2 into the co- 
efficients, we obtain 
8. The constituent parts in the roots which have been the objects of investigation 
were — ~ and a, and with respect to their evanescence we have the following theorem. 
The vanishing of that part of the root of a quadratic which is under the radical 
sign implies the existence of two equal roots. 
The vanishing of the other part which is unaffected by that sign signifies that the 
roots are equal, but with contrary signs. 
By the aid of this theorem we shall be able to find two of the three constituent 
parts of the roots of a cubic equation. 
