VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 547 



29. Another peculiarity of 4 quantities is also given in art. 14, i.e. that if taken 

 originally so far as to have their sum equal to nothing, the 6 quantities formed 

 from their sums in pairs, will be the same 3 quantities caken both affirmatively and 

 negatively. Then we know, by the reasoning in art. 11, that the equation of 

 those quantities, though of the 6th degree, will want every alternate term, or be 

 of a cubic form ; accordingly the equation of the function of the roots formed by 

 summing them in pairs, is {x — ^p) 6 — {2q + -fp*) X {x — 4-/>) 4 -J- (f 2 — 4* — qp* 

 -L. p r _|_ -Jt^p 4 ) x {x —'■kpy — -rP* — -kp* + r)* =0*, which when p is supposed 

 to vanish, becomes x 6 — 2qx 4 -\- (q* — 4s) x 1 — r 2 = O. 



30. These 2 methods, one applying to the biquadratic equations complete in 

 their terms, and the other to those from which the 2d term has been expunged, 

 are all that have yet been discovered; and, notwithstanding the number of different 

 methods attributed to different writers, which from their manner of setting out 

 appear distinct, they will all be found to resolve themselves, in principle, into one 

 of these. Dr. Hutton's Mathematical Dictionary, under the article Biquadratic 

 Equations, gives 4 methods; viz. Ferrari's, Des Cartes's, Euler's, and Simpson's; 

 to which may be added another by Dr. Waring-f~, and perhaps many more. They 

 proceed on a variety of different contrivances; but when analysed, and the real 

 object gained is viewed apart from the process that led to it, Ferrari's, which is the 

 oldest, and does not require the extinction of the 2d term, will be seen to produce 

 the cubic x 3 -— qx 2 + {pr — As) x — p*s + Aqs — r 2 = 0; and Des Cartes's, which 

 supposes the 2d term to be first destroyed, terminates in the cubic-formed equation 

 of the 6th degree, x 6 — 2qx 4 + {q* — 4s) x* — r* = O. The rest produce cubics, 

 or cubic-formed 6th powers, whose roots are some parts or multiples of this last; 

 except Waring's method, which does not expunge the 2d term, and therefore 

 produces a cubic whose roots are part of the first. But, whether the resulting 

 equation be that of the function, formed by summing the combinations by 2 of 

 the roots in pairs, or summing the roots themselves in pairs, or the equation of 

 the halves, or quarters, or doubles, trebles, &c. of those functions, is immaterial; 

 no new function is employed, no other principle put in action, than what is de* 

 rived from the general properties of this degree of quantities here explained. 



31. Biquadratics being generally thus reducible to cubics, of course, by resolving 

 those cubics, distinguishing what function their roots are of the roots of the ori- 

 ginal biquadratic, they may all be found; and, for practical utility, there is no 

 preference to be made of either of the 2 methods; for the first, though a real 

 cubic, being formed from products of the roots, it requires a quadratic equation 

 to obtain them after the cubic is resolved; whereas the 2d, though an equation of 

 the 6th power, being formed from simple addition of the roots, gives them at 

 once. But as both these cubics necessarily have all their roots real, when those of 

 the biquadratic are so, and the resolution of cubics is in that case imaginary, it 



* Waring's Medit. Algeb. p. 138. f Ibid. p. 133 ; and the Appendix to Dr. Hutton'* 



Dictionary. — Orig. 



4 A2 



