232 Sir William R. Hamilton on the Argument of Abel, 



which may also be thus written, 



(m — ^Oj)'— 48 e^ (w — ^ aj — 128 e, = 0. 



Reciprocally if w„ u^, u^ be known, by the solution of this cubic equation, or in 

 any other way, we can calculate a,' and a/', without any new extraction of radi- 

 cals ; since if we put, for abridgment, 



t^-Uy-u^- (x, - x^) {x^ - X4), 



^3 = W, U.^— {Xi — X4) (^2 ~ X.J, 



we have 

 and 



O, — Pi, Cj '3 > 



Oi' = (Po+PibtJ^t,)(u, + p,^u.,-\-p,u,). 



Again, it is important to observe, that if any one of the three quantities <,, t^, t^, 

 such as <,, be given, the other two, t^, t^, and also m,, u^, u^, can be deduced 

 from it, without any new extraction ; because, in general, the difference of any 

 two roots of a cubic equation is sufficient to determine rationally all the three 

 roots of that equation : it must therefore be possible to express the radicals a,' 

 and a," as rational functions of #, ; and accordingly we find 



0/ = 6^1 (144^2 - ^,*), 

 and 



< = {p. +P.t '. (M4 .. - ,.')} (fl^ t, + ,^.) ; 

 while ^1 may reciprocally be expressed as follows. 



Hence the most general irrational function of the second order, 



which can enter into the composition of b , and in which &„', b^, b^ are functions 

 of the first order, and of the forms 



may be considered as a rational function of ^, , 



