180 Sir William R. Hamilton on the Argument of Abel, 



which expresses a root of the general biquadratic equation, may be developed as 

 follows : 



b"'= S . (5" . a/" . <" ) 



/3,"'<2 V pr.pr ^ 



j8,"'<2 



in which 



T // 1 1 // __ 1 1 // 1 



"o »o — 4 > "i >o — ^J "o , 1 — ^» 



and 



64 64 64 



J,". . = 



And this last coefficient h", „ which is itself a function of the second order, may 

 , be developed thus : 



6/',. = b" = S . Tb' . o/' )=B;+B,'a,"+B,'a,"^ 



i3,"<S V i3." / 



m 



which 



''» - 64' ^' - ^4 ' ""^ - 6401"^ - e,{e,+a,') " 646/ ' 



So that, upon the whole, these functions h" and b'", which express, respectively, 

 roots of the general cubic and biquadratic equations, may be put under the 

 following forms, which involve no radicals in denominators : 



and 



^" = ^ + < + («.-a/)(^T 



6" = =f- + <" + <" 



+ \ {^3+ p! «/' + Pa (« -<) ©} «.'" < ^ 



