190 Sir William K. Hamilton on the Argument of Abel, 



so that they must represent some three unequal roots x^, x^, x^, and consequently 

 all the three roots of the cubic equation proposed. 



[7.] Combining the result of the last article with that which was befoi-e ob- 

 tained respecting the isolating of a term of a development, we see that if any 

 root X of any proposed equation, of any degree s, in which the s coefficients 

 A,, . . . A^ are still supposed to be rational functions of the n original quantities 



(m) 



flp . . . a^, can be expressed as an irrational and irreducible function h of those 



. («) 

 original quantities ; and if that function h be developed under the form above 



assigned; then every term ^ .. of this development may be expressed as a 



rational (and indeed linear) function of some or all the s roots a;,, a'^, • ■ • x^ of 

 the same proposed equation. 



For example, when we have found that a root x of the cubic equation 



x^\a^x'^-\-a^x-\-a^-=-Q 



can be represented by the irrational and irreducible function already mentioned, 



a; = 6" = 6; + hi a," + &>/' ^ = C'+ K' + K'. 



(in which 6/ = 1,) we can express the separate terms of this last development as 

 follows, 



- ■ C = V = ^(^i + ^'2 + -2^3)» 



—1 —2 



^," = 6,' < = ^ (oTj-f p3 OT^+ft 0:3), 



—2 —4 



C = ^2' <" = ^ (^1+ ft X^-\-Pz OC^; 



namely, by changing &„", 5,", h^' to .r,, x^^ x^ in the expressions found before for 



/ " f" i" 



In like manner, when a root x of the biquadratic equation 

 x^\- a^x^ -\- a^x^ ■\- a^x -\- a^ = 

 is represented by the irrational function 



X = b'" = 6;'., + b:\, a/"+ b,"„ <"+&,",, a/" <" 



— /'" 4-/'" _1_/"' 4-/"' 



— '■0 ,oT^n .on^t'o an^*'! ,i' 



