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



in which 



a' = - 27 a (a^ - ,3=), /3'^ = _ 27 ^^ (9a^ - ^f. 



But a and /3' are rational functions of x' and p ; and substituting their expres- 

 sions as such, we find corresponding expressions for a! and ^^ namely, 



a' = ^-x' {x'- +;,), ^"^ = ^ (Sy^ + 4p) (3a;- + pf. 



9. Finally, or' is such that 



x'^-\-px' = — q; 



and it is found on trial to be possible by this condition to eliminate x' from the 

 expressions for a' and j3'^, obtained at the end of the last article, and so to arrive 

 at these other expressions, which are rational functions of p and q : 



a'^-Y*?. r = ^(27?^ + 4/). 



In this manner then It might have been discovered, what has long been other- 

 wise known, that the function ^ is a root of the auxiliary quadratic equation 



(t'y-{-2lq (f)-27p'=0. 



And because the same method gives 



(y + ex" + e'x'") (x' + e' x" + ex'") = ga^ + 3^ = — Sp, 



we should obtain the known expressions for the three roots of the cubic equation 



x" -\- px' -\- q - 0, 

 under the forms : 



•^-3 ?'^"-3~r-^-3 T' 



which are immediately verified by observing that 



't\3 



'^ = >. ©-(?)=-'■ 



The foregoing method therefore succeeds completely for equations of the third 

 degree. 



10. In the case of the biquadratic equation, deprived for simplicity of its 

 second term, namely. 



