REPORT ON CERTAIN BRANCHES OF ANALYSIS. 309 



This mode of effecting the solution of a cubic equation would 

 altogether fail if the original equation possessed all its terms : 

 and though the absence of the second term of a cubic equation 

 cannot be said, in a certain sense at least, to affect the gene- 

 rality of its character, yet it would lead us to expect that the 

 method which we had followed was of so limited a nature as not 

 to be applicable to general equations of a higher order. Thus, 

 if it was proposed to find two quantities, u and ?j, the sum of 

 whose n*-^ powers was equal to 2 r, and whose product was equal 

 to q, we should find 



u = {r + ^(r^ - ?")}»; 



i 1 



u + V = [r + s/ (r2 - ^")}« + T, 



{r + -v/ if' — ?")}» 

 where m + t? is the root of the equation 



n 9 . '^ (^' — 3) 9 „_. W (« — 3) (« — 4) ., „ fi 



x"" — nq x"-^ H ^j — ^ q^ x"^ ^^-j — ^f^ • 5^ x"'^, 



+ &c. = 2 r*. 



The form of this equation is of such a kind as to prevent its 

 being identified with any general equation whatever, beyond a 

 cubic equation wanting the second term ; a circumstance which 

 precludes all further attempts, therefoi'e, to exhibit the roots of 

 higher equations by radicals f of this very simple order : but 

 it is possible that there may exist determinate functions of the 

 roots of higher equations (not symmetrical functions of all of 

 them, which are invariable as far as the permutations of the roots 

 amongst each other are concerned,) which may admit of triple 

 values only, and which will be expressible, therefore, by means 

 of a cubic equation, and consequently by the general formula 

 for its solution. 



Thus, if Xy, Xq, Xq, x^, were assumed to represent the roots of 

 a biquadratic equation 



* This equation was first solved by Demoivre in the Philosophical Trans- 

 actions for 1737, and it was readily derived from the theorem which goes by 

 his name. It was afterwards shown to be true, by a process, however, not al- 

 together general, by Euler, in the sixth volume of the Comment. Acad. Petrop., 

 p. 226. See also Abel's " Memoire sur une Classe particuliere d'Equations 

 resolubles algebriquement," in Crelle's Journal, vol. iv. 



t Abel has used the term radicality to designate such expressions. To 

 say, therefore, that the root of an equation is expressible by radicalities, is 

 the same thing as to say that the equation is solvable algebraically. It is 

 used in contradistinction to such transcendental functions, whether of a known 

 or unknown nature, as may, possibly, be competent to express those roots, 

 when all general algebraical methods fail to determine them. 



