166 MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
18. When od' = 0 we have seen that the equation <p (#) = 0 has two equal roots. 
But when <p (x) = 0 has equal roots, is expressed by making- the result of eliminating 
x between <p (x) = 0 and <p' (x) = 0 to vanish, and by the theory of elimination this 
result is <p' (x^ . <p r (x 2 ) . <p' (x 3 ) ; hence we have ( h being a number), 
a" = h<p' Oq) . <p’ (x 2 ) . <pl (x 3 ). 
19. Suppose now the joint evanescence of od and a", the equation has then three 
equal roots x t = x 2 = x 3 , the system of equations od = 0 a" — 0, the first of three 
dimensions relative to the roots, the second of six, are therefore the two conditions 
necessary for the existence of three equal roots. 
The results of the elimination of x between <p (x) = 0 <p" (a?) = 0 and <p (x) = 0 
<p' ( x ) = 0, give also the two conditions for three equal roots of the same dimensions 
as the above, with which this system is identical, we have thus, h! being numerical, 
a 1 = h! <p" fa) . <p" (x 2 ) . f (a? 3 ). 
20. Since 
<p (x) — (x — x x ) (x — x 2 ) (x — x :i ) 
<p' (x) = {x — x 2 ) (x — x 3 ) -f- (x — (x — ^ 3 ) + (x — x 3 ) (x — x 2 ) 
<p" (x) = 2 (x — x 3 ) + 2 (x — .r 2 ) + 2 (x — xj ; 
therefore 
<p' ( X l) • <?' M • 0 ( x z) = — ( x 1 — X 2 ) 2 — X z) 2 ( X 2 - x s) 2 
p" ( x i) • <p" M • f Oa) = 8 (2 x l — x 2 — x 3 ) (2 x 2 — x 1 — x 3 ) (2 x 3 -x l - x 2 ) ; 
the values of od od' are therefore conformable to those before found, and 
j _ , ] , , _ i_ 
” " 02 33 'L y £4 ^3 * 
21. In the elimination of a quantity between two equations into which that quan- 
tity enters rationally, it is in general indifferent which of the two equations is selected 
that its roots may be substituted for such quantity, for the dimensions of the product 
is the same whether we substitute the n roots of an equation of n dimensions in one 
of m, or the m roots of the latter in the former of n dimensions, and take the pro- 
ducts ; these products are not only of the same dimensions but imply the coexistence 
of the same system of equations, and can only differ from each other by numerical 
multipliers, when the numerical coefficient in the one differs from that of the other. 
In the present instance, if f 2 be the roots of the equation <p' = 0, and X that of 
p" = 0, the values of a 1 od' may also be expressed in the following form, 
«" = H <p &) . (i 2 ) 
a" — K ,<p (X) 
the factors H, K being numbers. 
22. Now if we observe that by the equation <£>' = 0, 
