Resolution of Algebraic Equations. 289 
therefore, in destroying the third term, the second would be 
revived, so that nothing would be gained. 
23. To understand how this difficulty is ever removed, let 
us examine particularly some equation that wants both second 
and third terms, and observe accurately the constitution of its 
roots. The simplest of the kind is the pure cubic (x 3 — 1 ), whose 
roots are (1, ——— 2 - 3 - ] ; but, to avoid fractions in the roots, 
let us take ( x 3 = 8 ), whose roots are (2, — 1 =±= V 7 — 3). 
Distinguishing the real and the imaginary parts, the real are 
(2, — 1, — 1) ; the imaginary are (=±= — 3 or =±= 3 x */ — §), 
which are the differences of the real parts, multiplied by the ima- 
ginary surd (v/ — i). It appears, therefore, that the roots of 
a pure cubic are compounded of the roots of some affected 
cubic, added to their differences drawn into the imaginary surd 
(v/ — ■§■), The real parts (2, — 1, — 1) are the roots of the 
cubic equation (x 3 — gx-f- 2 = 0). The imaginary, of the 
equation (x 3 -}- 3 x -j- * = o), or the roots of the similar equa- 
tion of the differences of the former, viz. (x 3 — 9X + * = o), 
drawn into the (^Z — -§-) ; and, from their addition are formed 
the roots of the pure cubic (x 3 = 8). In constructing which, it 
is material to observe, each root of the first equation is joined to 
the difference of the remaining pair; but it may be remembered, 
that three quantities whose sum is nothing, are the same when 
summed in pairs, i. e. each is (in quantity) the sum of the other 
tv/o, therefore, each difference is in fact added to the sum of 
the same quantities ; and, if the question were proposed to re- 
duce the equation (x 5 — 3X -f- 2 = 0) to a pure cubic, the rule 
furnished by this example would be, to find the equation of 
the sum of its roots in pairs, which, by the last Article, is 
