Sir William Rowan Hamilton on Equations of the Fifth Degree. 335 



x'*-\-px'^-\-qx' -{-r-O, 

 so that the sum of the four roots vanishes, 



we may consider x", x'", x"', as roots of the cubic equation 



x"^ + x' x'" + {x'^ -\-p) x" + x'^ ^px'-\-q = 0; 



and this may be put under the form 



(^x" — ay — 3rj (x" — a) - 2e = 0, 



of which the roots (by the theory of cubic equations) may be expressed as fol- 

 lows : 



x"=a-\-p-\-y, .r'" = a + e/3 + eV x"" = a -{■ e'p-\- By, 



/3, 7, and 6, being such as to satisfy the conditions 



^3 _j_ ^3 _ 2e, j8y = »;, 02 4- e -I- 1 = 0. 



Comparing the two forms of the cubic equation in x", we find the relations 



x'= — 3a, x'^ -\- p = 3 (a-" — 7]), x" -\- px' -\- q = — a' -\- 3ari — 2e; 



which give 



a=-^x', ri = - ^ (2x'' -^ 3p), e =-^(20x" + I8px' + 27q). 



Thus, any rational function of the four roots of the given biquadratic can be ex- 

 pressed rationally in terms of a, j3, 7 ; while a, ^, and /3' -\- 7^ are rational func- 

 tions of x', p, q ; and the function x'* -\- px"^ -\- qx' may be changed, wherever it 

 occurs, to the given quantity — r. 



1 1 . With these preparations it is easy to express, as follows, the function 



{x' - x" + xf" - x'y, 



which the general theorems of Lagrange, already mentioned, lead us to con- 

 sider. Denoting it by 4^, we have 



z = (— 2a+ ep + 6^7)2 = a' + els' + ey ; 

 in which 



a' = 4a? -\- 2/^7, /3' = 7^ - 4a]3, 7 = ^* — 4a7 : 



and the three values of z are the three roots of the cubic equation 



