Resolution of Algebraic Equations. 291 
other; and, as the sums of both equal nothing, when added to- 
gether their sum will still be nothing; so that no new second term 
can arise , as in Art. 22. If we now return to the considerations 
in Art. 2 1 , where we shewed how to derive from every cubic 
equation ( x 3 — qx-\-r — o) wanting the 2d term, a similar 
equation (x 3 — gqx -f s/ ^q 3 — 27ri = o), being the equation 
of three of the differences of the roots of the former, so arranged 
as to want the second term also, we may perceive that, to ren- 
der the third term the same in both, we need only divide the 
roots of the latter by (\/3)> or > which is the same thing, mul- 
tiply them into the (x/y)- For, the equation (a: 3 — gqx -f- 
\/^q 3 — 27^] = o), when its roots are multiplied by the (v/y)> 
becomes [x 3 — qx -j- ^ = oj.* If, by the same rea- 
son, they had been multiplied by the it would be 
(jc 3 -f- qx - f- ’ ' = o) ; where the sign of the coeffi- 
cient of [x) is opposite to that of (<?) in the given equation. 
Therefore, the roots of the equation (x 3 — qx -j- r = o), and 
that of its differences, multiplied into the imaginary surd s/ — -f, 
viz. [x 3 -j- q x -j- = o), will, by being added toge- 
ther, (according to the method in the last Article,) lead to a re- 
duction of that equation to a pure cubic ; i. e. the equation 
formed from their addition will have three roots, whose cubes 
are the same. 
25. The analysis of the pure cubic gives us the following 
general properties , belonging to any set of those equations whose 
sum is nothing. 
Viz. 1st. If three such quantities (a, b, — a — b) be added 
* Vide San de rson’s Algebra, Vol. II. p. 6S8 ; and Hale’s Analysis, p. 146. 
