respecting Equations of the Fifth Degree. 211 



or X3, x^f or^; the value of 3 ft/ (p^ + p^ o/) being, In the first case, unity; 

 in the second case, p^ ; and, in the third case, p^ ; while, in every case, the 



value of bj is to be —3^, and that of 5/ b^' (b^ -\- b^ a,') is to be Cj. All these 



suppositions are compatible with the conditions assigned before ; nor is there 

 any essential difference between the three cases of arrangement just now men- 

 tioned, since the passage from any one to any other may be made (as we have 

 seen) by merely multiplying the coefficient &/, which admits of an arbitrary 

 multiplier, by an imaginary cube-root of unity. We have, therefore, the following 

 irrational and irreducible expression for the root x of the general cubic, as a 

 function of the second order, 



— h" — ~"' J_ "" _i_ ScaCPo + PiOi') 



x-o --g-i- 3(^„+^,„;^+ all 5 



in which it is to be remembered that 



«/' ' = 27 {p, -Yp, alf |c. + -iV (/>/ - ft)-^'} , 



and that 



0/*= — 108ft^(c,^-c/); 



c, and Cj having the determined values above referred to, namely 



^i = ~-si (2«i' — 9a, a^ + 27 03), c^ = l (a,^ — 3a,), 



and /93 being an imaginary cube-root of unity, but J and p^, />,, being any arbi- 

 trary rational functions of o,, a,, O3, or even any arbitrary numeric constants ; 

 except that b must be different from 0, and that p^, p^ must not both together 

 vanish. (In the formulae of the earlier articles of this essay, these three last 

 quantities had the following particular values, 



* = tV(/'3'— />3), i>o = i» i'i = 0-) 

 By substituting for the cubic radical a/' the three unequal values a,", p^ a,", p^ a,", 

 in the general expression, just now found, for x, we obtain separate and unequal 

 expressions for the three separate roots x^, x^,x.^; these roots, and every rational 

 function of them, may consequently be expressed as rational functions of the 

 two radicals a/ and o," ; and therefore it is unnecessary and improper, in the 

 present research, to introduce any other radical. But these two radicals a,' and 

 a," are not essentially distinct from those which enter into the usual formulae 



