174 Sir William R. Hamilton on the Argument of AbeU 



b"^J^a,h"''^a.,V'^a.,= Q 

 is satisfied, without any restriction being imposed on the three coefficients a^, 



For n = 4, that is, for the case of the general biquadratic equation 

 x^ -|- a, x^ -\- a^ x"^ -\- a^ X -{- a^ =L 0, 



it is known in like manner, that a root can be expressed as a finite irrational 

 function of the coefficients, namely as the following function, which is of the 

 third order : 



X = V" =/" («/", <", a,", a/, a„ a„ a,, a,) 



wherein 



— — 4- (2 '" 4- a '" -4- —^ 



<"'=/■"«' <' «1' «2. «3. «4) =e3+< + ^. 





<"=/2"«'» «/' «i' «2' «3' a,) = e,-^ p,a," -\- ^^„ 

 a/'^ =//(«,', a„ Oj, Og, a^)■=e,-\-a^', 

 <" =/i («i» «2' «3» «4) = «!* — 62' 5 

 ^4> ^3> ^2' ^1 denoting for abridgment the following rational functions: 



^4 = fV (— «i' + 4«i 02 — 8 03)* 

 «3 = i;ff(3«i'-8a2). 



«2 = tI:¥ (— 3a, 03 + «2' + 1204). 

 e, = ^(3e,e3 — e/ + 

 = F^ (27«>4 - 9a. o, 03 + 202' - 72a, a, + 27a/), 

 and /J3 being a root of the numerical equation 



/'3'+P3+l=0- 



It is known also, that a root x of the same general biquadratic equation may 

 be expressed in another way, as an irrational function of the fourth order of 

 the same arbitrary coefficients a^, a^, a^, a^, namely the following : 



