256 



Sir William R. Hamilton on the Argument of Abel, 



ADDITION. 



Since the foregoing paper was communicated, the writer has seen, in the 

 first Part of the Philosophical Transactions for 1837, an essay entitled " Analysis 

 of the Roots of Equations," by a mathematician of very high genius, the Rev. 

 R. Murphy, Fellow of Caius College, Cambridge ; who appears to have been 

 led, by the analogy of the expressions for roots of equations of the first four 

 degrees, to conjecture that the five roots x^ x^ x^ x^ x^ of the general equation of 

 the fifth degree, 



x'^ax''-\-bx''-\-cx''-\-dx-\-e = 0, (1) 



can be expressed as finite irrational functions of the five arbitrary coefficients 

 a, h, c, d, e, as follows : 



X, 



—a 



— a 



+ « v'a + «* v' ^ + w^ v^7 + «)* v'2, 



X3 = -^ + «'''v'« + «V/3 + «'v'7 + w'v'2, 



—a 



X^ = -f- + a?^a + w^^ + w' ^y+u?^l. 



—a 



(2) 



w being an imaginary fifth-root of unity, and a ^ y I being the four roots of an 

 auxiliary biquadratic equation, 



a = a' + v/i3' + /7' + /2', 

 /3 = a'+/^'-/7'- /S', 

 7= a' - //3' + /y- /B', 

 8=a' - //3'- /y+v/S'; 



(3) 



in which j8' 7' S' are the three roots of an auxiliary cubic equation, 



P' = a"+^/3" + iJ'7", 1 



7' = a"+0^r + eV7", \ (4) 



8'=a"+e^^^"+0^7"; -1 



