116 On the Resolution of Equations of the Fifth Degree. 



degree with given coefficients simultaneous with the equation 

 V 1S + C 1 V ,4 + .. =0." 

 The verification would indeed have been even more striking 

 had we taken into account the equations analogous to (aa). 

 10. It is obvious that generally 



Vf(«k) + Vg(«u) + Vh(«i;) = — Bj ; 



and that therefore 



1~ —2,. _3,. _4,. _5 r . 



r 1 — ; i — / j — r x ~ >i . 

 if, then, the roots of a particular equation of the fifth degree be 

 so related to each other as to cause the equation 



V is + Ci vi4 + <#= o 

 to take the form 



((v+r(°))y= o > 



the system (er,) will in that case coexist with (o" 2 ). 



11. With respect to the theorem of M. Cauchy, it fails to 

 apply to a system of congeneric expressions such as 



D(V F , V G , V H ) = V(0), 

 D(V F(a ^ Vg ( ^ ) ,Vh(« / 3))= 2 K0), 



(SO 



D(VF (af)j VG(« f ), VH (a6 ))=M0). 



12. Every difficulty is therefore at an end. We are thus a 

 second time brought to the conclusion, which (guarded as it now 

 is and fenced round on every side) must soon approve itself to 

 the minds of mathematicians : that the roots of the general equa- 

 tion of the fifth degree admit of being expressed by finite combi- 

 nations of radicals and rational frictions. 



Long Stratton, Norfolk, 

 January 13, 1852. 



