546 PHILOSOPHICAL TRANSACTIONS. [ANNO 1799. 



into equality at some known period. They also fail to account for the most striking 

 part of the result; the irreducibility happening uniformly in cases where it has 

 been supposed least to be expected, i. e. when the roots are real; which they refer 

 to a particular limitation in one of the steps taken, when it is, in truth, of much 

 deeper origin than any particular method, being the necessary consequence of the 

 constitution of the cube power. 



27. The result of these observations on cubic equations shows, that directly 

 they are not resolvable, i. e. they cannot, like quadratics, be always brought to a 

 mere extraction of their correspondent root : that however, by means of the pe- 

 culiarities inseparable from the number of 3 quantities, a relation is discoverable, 

 which inevitably gives equal roots to the equation of the cubes of a particular 

 function of them; but that, that function involves sometimes a quadratic surd 

 which was not in the roots themselves, but arose from the form necessary to be given 

 them ; that the equal relation not taking place in any case, till the cube of that 

 function, and in some cases not being rational, till the square of that cube, the 

 equation is not lowered in degree, by the operation, but rather increased. 



28. Let n = 4, and the equation becomes the general biquadratic x 4, — px 3 -f- qx* 



— rx -\- s = O, the number of differences are 12; we cannot therefore hope to 



obtain a direct simple resolution. But, in art. 14, two peculiarities belonging to 



sets of 4 quantities were pointed out, from which it is easy to obtain a reduction 



of the equation to a cubic form. The first peculiarity there mentioned, was shown 



to subsist among the sums of the combinations of the roots in pairs. If a, b, c, d, 



be supposed the roots of the given equation, and their combinations by two, ab, 



ac, ad, be, bd, cd, be summed in pairs, though the number of quantities so 



formed are no fewer than 30, yet there is an evident distinction observable among 



them ; for in some, (the first 6,) no letter occurs twice. If, therefore, instead of 



simply requiring the sums of the combinations of the roots in pairs, that function 



of the roots had been required, consisting of the sums of these combinations, 



into the forming of which no root enters twice, only 6, out of the whole number 



of combinations of the kind, would answer that condition ; and those six would be 



the same 3 repeated, for ab + cd, and cd -f- ab &c. are the same quantities. So 



that the 3 quantities ab -J- cd, ac -f- bd, ad -f- be, would be the functions required, 



and all of the kind that can be made. Now there is no proposition in the theory 



of equations more certain, than that the equation of any regular function of the 



roots may always be found by means of the known values of the co- efficients*. 



As there are but 3 functions in this case, the resulting equation must consequently 



be a cubic; and, by taking the several combinations of the quantities ab -j- cd, 



ac + bd, ad + be, we may obtain their equation, viz. x 3 — qx* -\- (pr — As) 



x — p"*s -f Aqs — r 2 = 0. Therefore the finding the equation of that function of 



the roots of a biquadratic which arises from its combinations by 2 summed in pairs, 



so however that no root shall occur twice hi any such sum, reduces the biquadratic 



to a cubic. 



* Waring's Med. Algeb. cap. I, p. 1, et infra. — Orig. 



