542 PHILOSOPHICAL TRANSACTIONS. [ANNO 17QQ. 



imaginary. But it is clear, from what has been before observed in art. 9, that tis 

 reduction is not generally possible, since it supposes 1 contiguous intermediate 

 terms to vanish together, which real roots do not admit of: it must therefore be 

 effected by means of some imaginary assumption. Those who are conversant in 

 the use of impossible quantities, will at once perceive, that the addition or subtrac- 

 tion, (which in surd quantities is always the same thing, as they are equivocal in 

 sign,) of the imaginary surd */ — \q to each root of the equation, will infallibly 



cause q to vanish, but the new roots -J , " . " •»' — a — b + */ — t?> so formed, 



would not have their sum equal to nothing ; and therefore, in destroying the 3d 

 term, the 2d would be revived, so that nothing would be gained. 



23. To understand how this difficulty is ever removed, let us examine particu- 

 larly some equation that wants both 2d and 3d terms, and observe accurately the 



constitution of its roots. The simplest of the kind is the pure cubic x 3 = 1, 



— 1 "^~ / — 3 

 whose roots are 1 , ; but, to avoid fractions in the roots, let us take 



x 3 = 8, whose roots are (2, — jr y/ — 3. Distinguishing the real and the ima- 

 ginary parts, the real are 2, — 1, — 1 ; the imaginary are + </ — 3 or ± 3-/ — -£-, 

 which are the differences of the real parts, multiplied by the imaginary surd y/ ' — •£. 

 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 

 y^ — \. The real parts, 2, — 1, — 1, are the roots of the cubic equation 

 #3 _ . 3x + 2 = 0. The imaginary, of the equation x 3 -f 3x -+- * = O, or the 

 roots of the similar equation of the differences of the former, viz. x i — gx -j- * = O, 

 drawn into the y/ — 4- ; and from their addition are formed the roots of the pure 

 cubic x 3 = 8. In constructing which, it is material to observe, that each root of 

 the first equation is joined to the difference of the remaining pair ; but it may be 

 remembered, that 3 quantities, whose sum is nothing, are the same when summed 

 in pairs, i. e. each is, in quantity, the sum of the other 2, therefore each difference 

 is in fact added to the sum of the same quantities; and if the question were pro- 

 posed to reduce the equation x 3 — 3x -f 2 = O 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 x 3 — 3x — 2 = O ; to find the similar equation of 

 their differences, x 3 — gx -{- * = O ; and, to find the equation produced by the 

 quantities formed from the addition of the roots of the one to those of the other 

 multiplied into the imaginary surd */ — 4 . The equation last found would how- 

 ever, be of the dimension of the 9th power, at least : for, the addition of each root 

 of the 2d equation to every separate root of the first, produces a separate quan- 

 tity : thus, 



2,-1, - J T2 + -1 + 0, -1+0, "\ 



being ^rootsj,nh^l 3 St...J 2 + 3^_, j _ j+ 3^_,, . 1+3 ,_jf 



those of the 2d *■ 2 — Zs/ — f, -* 1 — 3V— -,, ►•• 1 — 3^— -J- J 



