Resolution of Algebraic Equations. 287 
that we can be enabled, a priori, to determine the formula of 
any direct resolution of this case. 
2 1 . Let us then try to trace some relation which may convert 
some or all of the roots, or some regular function of them, into 
equal quantities; when, the equation of that function having 
equal roots, of course those roots will be separately deducible, 
as shewn in Art. 11, 12. In Art. 14, we may remember, two 
particularities were mentioned to belong to three quantities, 
viz. that their differences were so related as to be every two 
of them equal to the third ; and that, if the quantities themselves 
have their sum equal to nothing, two of them also must equal 
the third, and their magnitude be respectively the same, whether 
they are taken simply, or summed in pairs. To avail ourselves 
of both these properties, let us suppose the second term to be 
expunged from the given equation, (which we know may always 
be effected,) its form will then be (x 3 — qx -f- r = o),* and the 
sum of its roots equal to nothing. Let ( a ) and (6) be two of 
its roots, the third will therefore be (—a — 6); take their' 
sums by two ( — a, — b, a -{• b) ; take their differences 
(2 a b, a -{- 26, a ~b) and their negatives, which may be 
divided into two sets whose sum is nothing, like that of the 
a — b , a -f- 26, — 2 a — 61 
a + 6, — a — 26, sa -\- b J 
So that, from the given equation we derive three others, which 
make a set of four exactly similar. 
roots, viz. 
* Besides expunging (/>), the sign of ( q ) has been changed ; because, in cases of 
real roots, it will invariably become negative upon destroying the second term. Vide 
note in p. 275. 
Pp 2 
