VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 543 



will be the 9 quantities formed by their addition. But we have a decisive clue to 

 distinguish some from the rest ; for we know, that if we find the equation of the 

 cubes of those quantities, it must have 3 equal roots ; for every time the sum of 2 

 of the roots of the 1st equation meets its own difference, it will constitute a cube 

 root of 8, and therefore the equation x 3 — 8 = will be 3 times contained in the 

 resulting equation of cubes. That equal root being discovered by the method of 

 finding equal roots, so often alluded to before, reduces the equation x l — 3x -f- 2 = O 

 to the pure cubic # 3 = 8. 



24. The instance in the last article, of the reduction of the equation x 3 — 3x 

 -f. 2 = O to a pure cubic, by means of the equation x 3 -\- 3x = O, evidently de- 

 pends on the co- efficient of the 2d term vanishing ; and also, that of the 3d term 

 being the same in both, but of opposite signs. For, the roots of the one, in their 

 combinations by 2, producing — 3, and those of the other + 3, of course destroy 

 each other ; and as the sums of both equal nothing, when added together their 

 sum will still be nothing ; so that no new 2d term can arise, as in art. 22. If we 

 now return to the considerations in art. 21, where we showed how to derive from 

 every cubic equation x 3 — qx -f- r = O, wanting the 2d term, a similar equation 

 x 3 — 3gx + v/(4<7 3 — 27r 2 ) = O, being the equation of 3 of the differences of 

 the roots of the former, so arranged as to want the 2d term also, we may perceive 

 that, to render the 3d term the same in both, we need only divide the roots of the 

 latter by \/3, or/which is the same thing, multiply them into the V"-^. For, the 

 equation x 3 — 3qx -j- s/ (4q 3 — 27'" 2 ) = O, when its roots are multiplied by the 



*S i, becomes x 3 — qx -\ 9 ~ — -) = O *. If, by the same reason, they had 



/do 3 — 27r^ 



been multiplied by the /— 4-. it would be a? + qx -| ~ — - — - = o ; where 



the sign of the co-efficient of x is opposite to that of q in the given equation. 

 Therefore the roots of the equation x 3 — qx -f- r = 0, and that of its differences, 



multiplied into the imaginary surd s/ — .i, viz. x 3 -f- qx -| 3 7 = O, will, 



by being added together, 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 3 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 3 such 

 quantities, a, b, — a — b, be added in pairs, and 3 of their differences be also 

 taken so as to have their sum nothing, a — b 3 a + 2b, — 2a — b ; if then each 

 sum be formed into a binomial, by joining to it its correspondent difference, mul- 

 tiplied by the imaginary surd \f — -f , the quantities so formed, viz. a -f- b -f- 

 (a - b) V— £) and - a + (a + 2b) V - i K and - b - (2a - b) V - $ will 

 have the same cube. 



Now let the equation of the 3 quantities a, b, — a — b, be x 3 — qx -f- r = ; 



* Vide Sanderson's Algebra, vol. 2, p. 688 3 and Hale's Analysis, p. 146.— Orig. 



