of the Fifth Degree. 115 



In which, then, of these two ways will the coincidence take 

 place ? 



8. We shall see that it is the second system (<r 2 ) which must 

 generally obtain*. 



In effect, if, in the formation of the groups, we suppose the 

 roots to become fixed in the order x e , x a , Xq, x , x s , there will 

 arise a new system of equations, 



D(V F , V G , V H ) = V(* e ), ^ 



D(VF(«/3), V G («/3), VH(« J 8))= 2 K* a )> I . , (£) 



D(V F (« e) , VG(« f ), v h(*6)) = 5 r( x 8> ;J 



wherein V-, 2 r, . . 5 r are also expressive of rational functions. 

 Accordingly, on comparing (2) and (2), we shall find 

 hix a ) = l r(x e ), 



*r(x 6 ) = 5 r{x s ). 



But the roots of the general equation of tbe fifth degree do 

 not admit of being inseparably linked together in any such 

 system of rational expressions. It is certain therefore that the 

 two sets of functions, V(o?J, *r{x$ } . ■ 5 r(x e ), and l r{x 6 ), ~r(x K ), 

 . . b r{x s ), must merge into V(0), 2 r(0), . . 5 ;-(0). 



9. Thus the equation (o") will become 



(V 3 + V, (0) V 2 + V 2 (0)V + V 3 (0)) x 

 (V 3 + 2 r,(0)V 2 + 2 r 2 (0)V + 2 r 3 (0)) x 

 > . . (o" 



(V 3 + V, (0) V 2 + V 2 (0) V + V 3 (0)) 



= 0; J 



and will consequently be resolvable into five cubic equations, the 

 coefficients of which will be known rational functions of A,, A& 

 . . A 5 . A result in perfect accordance with the one which we 

 proposed to verify — " that there will be an equation of the third 



* I thought, indeed, at one time, while viewing the subject at a greater 

 distance, that there was no way to escape from the first system (sec my 

 remarks in this Journal for January 18-1'i); but 1 implicitly assumed, what 

 I beli< \ ( has never been contested, the universality of M. Cauchy's theorem. 

 It will, however, appear from what follows, that the theorem in question 

 cannot he safely rested on, but must yield its place to another theorem 

 consisting of two parti or branches, which, exclusively of particular cases, 

 are- distinct, and incapable of coincidence when v is equal to 5 or to any 



higher number. 

 b 12 



