364 On the Theory of Equations of the Fifth Degree. 



A. There is a criterion of solvability of a quintie which can be 

 put under the form 



F*{'^)-Pf(e/y)(!?)p)(^)=0. 



B. The pair of equations 



conducts us to De Moivre's form, i. e. to 



C. Either of the pairs 



F,(^^)=o, r,(«(!?)(!')(-^?)=o 



leads to Euler's form, i. e. to 



B2 = 0, 5B34 - 52B43 + 53B3B4B5 = 0. 



D. The general equation of A leads to a resultant in which P 

 enters to twenty-four dimensions. 



E. The functions Fp satisfying the equation 



Mefg):=¥,,{efg)(?.), 

 the general relation of A takes the form 



Consequently, the interchanges being independent, the factors 

 of the general resultant occur in pairs, which we may represent by 



F.3-i3F'.3, Piz-i%z, or F-i^G, G-i^F. 



F. The decomposition of the general resultant of the twenty- 

 fourth degree into biquadratic factors depends upon an equation 

 of the sixth degree. 



33. Judging from the analogies afforded by Mr. Jerrard's 

 discussion of De Moivre's form, we shall, prior to inquiring into 

 the possibility of solving this equation of the sixth degree, have 

 to take into consideration the forms of the functions denoted by 

 t, u and V. Into this somewhat laborious and complicated in- 

 quiry I am not at present prepared to enter, and it may weU 

 form the subject of a separate investigation. 



76 Cambridge Terrace, Hyde Park, 

 February 24, 1857. 



