544 PHILOSOPHICAL TRANSACTIONS. [ANNO 1799* 



then, by the construction of equations, this equation may be reduced to a pure 

 cubic of this form a? 3 = 4 X (— r -\ ~- — ~- — ), which, when cleared of its 



irrational quadratic surd, becomes x 6 + 8rar 3 + l6Y* = ~ * — ; - or x 6 -f- 8rx 3 , 



-J — = O ; or, dividing its roots by 2, to reduce it still lower, a 6 -f. rx 3 -f -- 



= O, the common reducing equation obtained by Cardan's rule. 



Example id. Let 1,2, — 3, the roots of the equation a? 2 — Tx -\- 6 = o, be 



so 

 taken ; the quantity 24 — - — - is the common cube of the 3 binomials. 



Exam. 3d. Let — 1, — 4, + 5, the roots of the equation x 3 — 21a? — 20 = O, 

 which are the differences used in the last example, be next taken ; the cube comes 

 out — y -+- 24 \S — 3, or exactly the reverse of the former. Now the cube 

 24 — V ^ — -t> when its equation x 3 = 24 — e T V — -r is made rational, gives 



the quadratic formed equation of the 6th degree, x G — 48x 3 p- 576= — 6 4-t° » 



or, transposing all the terms to one side, and dividing it by 2, to reduce it, as be- 

 fore, x 6 — 6x 3 -\- Vr = ° > tne same equation that results from the common 

 methods. 



2dly. The differences of the 3 differences a — b, a -j- lb, — 2a — b, are 3a, 

 3b, 3 (a + b), or merely 3 times the original quantities. Therefore, had the dif- 

 ferences themselves been taken as original quantities, and binomials being formed 

 from them, according to the directions before observed, those binomials, and the 

 ultimately resulting cubes, would differ from the former, in nothing essential but 

 the place of the surd. The differences which were affected with it before, would 

 now be clear ; and the quantities themselves, or, which is the same thing, their 

 sums in pairs, be affected with it. However, as these latter are to be multiplied 

 by 3, that multiplication will destroy the fraction when they come again to be mul- 

 tiplied by the surd 1/ — -i-, since 3 X V — -£■ = </ — 3. Therefore the same end, 

 as to reducing the equation, will be obtained, whether, after adding the sums of 

 the roots in pairs to their respective differences, we multiply the sums by 1/ — 3, 

 or divide the differences by it. 



3dly. If any cubic equation, wanting the 2d term, be transformed into the equa- 

 tion of that function of its roots, formed of the cubes of the binomials arising 

 from joining the sum of each pair of roots to its correspondent difference drawn 

 into the imaginary fractional surd </ — tj or eacn difference to its correspondent 

 sum drawn into the surd </ — 3, the transformed equation will have among its 

 roots 3 equal cubes ; by finding which, according to the methods of finding equal 

 roots, the equation is reduced to a pure cubic. 



4thly. The roots of a cubic equation may be all real ; or only one of them real, 

 and the remaining 2 imaginary. If only one be real, they will be of this form 



a — 



a ±b</- 1 

 2 



; and, by taking their sums and differences according to the rule, 

 and multiplying the latter into the / — t> one of the resulting binomials will be 



