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



and that if the development of the n'* power of this function t be reduced, by 



the help of the equation 



a":=l, 



(and not by the equation a"~' -f- &c. = 0,) to the form 



r = ^'°' + a^ + a'l" + . . . + a"-' ^"-", 



then this power f itself has only - different values, and the term ^°' has only 



— T-^ — rr such values, or is a root of an equation of the degree 

 n{n — \) ^ *' 



1.2.3....m 



m\"' 



^(7^_l)(l.2.3...-J 



of which equation the coefficients are rational functions of the given coefficients 

 A, B, c, &c. ; while ^', ^", . . . ^'"~" are the roots of an equation of the degree 

 n — 1, of which the coefficients can be expressed rationally in terms of ^''" and 

 of the same original coefficients A, ... of the given equation in x. 



2. For example, if there be given an equation of the sixth degree, 



x^ — KX^ 4" B.r'' — cj;^ -j- Yix"^ — eo: + f := 0, 



of which the roots are denoted by x', x", x'", x'^, x^, x"^, and if we form the 



function 



t-x'^ax"\ a^x'" + a^x'''-\- a' x" -{- a? x"", 



in which a = — 1 ; we shall then have 



ni = 6, 7^=2, /x = ^ = 20, ^ = 10, , ^ ,, = 10; 

 3b n n{n — \) 



and the function t will have twenty different values, but its square will have only 

 ten. And if, by using only the equation a^ ■=. 1, and not the equation o = — 1, 

 we reduce the development of this square to the form 



f = ^o' + ar, 



the term ^°* will itself be a ten-valued function of the six quantities x' , . . . x''; 

 and ^ will be a rational function of ^'"^ and a, namely, 



r = A^ - 1^°). 



