356 Sir William Rowan Hamilton on Equations of the Fifth Degree. 



(103) 



it ought from its form to be divisible by all of them, which is immediately seen 

 to be impossible. The conclusion of the twenty-fifth article is, therefore, con- 

 firmed anew ; and we see, at the same time, by the theory of biquadratic equa- 

 tions, and by the meanings of e, tj, i, that the denominator of the fraction which 

 is to be squared, in the form (81) for H4, may be expressed as follows : 



5.r'* + 3px" + 2qx' -\-r = {a;, — x.,) (or, — x^) (^, — x,) (x^ — x,) 

 =z (5ay - 6e (5a)* + 8r] (5a) — 3 (e^ - 4^ ; 



a result which may be otherwise proved by means of the relations (15). 



28. The investigations in the preceding articles, respecting equations of the 

 fifth degree, have been based upon analogous investigations made previously with 

 respect to biquadratic equations ; because it was the theory of the equations last- 

 mentioned which suggested to Professor Badano the formulas marked (a) and 

 (b) in the seventeenth article of this paper. But if those formulae had been sug- 

 gested in any other way, or if they should be assumed as true by definition, and 

 employed as such to fix the meanings of the quantities h which they involve ; 

 then, we might seek the values and composition of those quantities, h„ . . . h^, by 

 means of the following converse formulas, which (with a slightly less abridged 

 notation) have been given by the same author : 



H3 + Vh, = 2V (V345 + ^'^453 + o^^mT ; 

 H, - v^He = 2V (V345 + ^^53 + ^^34)' ; 



and 



H, - ^/H, = ^ (v3,4 + v^3 + V43J ; 



(c) 



H5 + \/He = 2V (^3*4 + ^v,« + e'\,^y 



(d) 



Let us, therefore, employ this other method to investigate the composition of h^, 

 by means of the equation 



54 ^/H4 = (V34, + 0^4,3 + ey,^y - (v3,4 -f 6%,, + ev,,,y ; (104) 



determining still the six functions v by the definition (33), so that each shall still 

 be the mean of four of the twenty-four functions t ; and assigning still to these 

 last functions the significations (32), or treating them as the fifth powers of 



