respecting Equations of the Fifth Degree. 205 



of the unsymmetrlc function f/, and consequently being = 2; so that the 

 radical a/ must be a square root, and must have two values differing In sign, 

 which may be thus expressed : 



+ o/ = f/ (a;,, Xj), — a/ = F/(a;„a?,). 

 But, In general, whatever rational function may be denoted by f, the quotients 



2 ^^ 2{xi-Xi) 



are some sjmimetrlc functions, a and h ; so that we may put generally 



F (^u x^) = a-\-b (Xi — x^), 



therefore, since we have, at present, 



F,'(^2» ^.) = — F.' (^.. a;,), 

 the function f/ must be of the form 



the multiplier b being symmetric. At the same time, 



< = ^ (^. - ^2)r 

 and therefore the function^ is of the form 



/ (a„ a,) = a/^ = ¥ {x^ - x.f = ¥ (a.;- - 4 o,), 



so that the radical a/ may be thus expressed. 



in which, b Is some rational function of the coefficients a,,a2. No other radi- 

 cal a^' of the first order can enter into the sought irreducible expression for x ; 

 because the same reasoning would show that any such new radical ought to be 

 reducible to the form 



< = c (x^ - x^) = ^ o/, 



c being some new symmetric function of the roots, and consequently some new 

 rational function of the coefficients ; so that, after calculating the radical a/, it 



