On the Algebraic Equation of the Fifth Degree, 341 



great, and the expense of observing probably nothing at all. 

 If your government will co-operate, I think the Smithsonian 

 Institution will undertake the organization for the United 

 States. 



With much respect I remain, 



Yours truly, 



Elias Loomis. 



LI v. On the Algebraic Equation of the Fifth Degree. 

 By the Rev. Brice Bronwin*. 



I 



T appears that the resolution of equations of the fifth and 

 higher degrees into factors, one of which is of the second 

 degree, depends upon the solution of the proposed equation 

 itself. This circumstance appears to me deserving of notice, 

 as it seems to indicate the impossibility of solving such equa- 

 tions in finite terms. Suppose 



x^+Ka;'^-\-Bx'^+Cx-\-Ti={pir^-\-ax'^-\-bx + c)'\ ,^. 



{x^-ax+f) = 0. J 



Multiplying the two factors, and comparing the result with 

 the first member, we find 



Eliminating b and c from these, we have 



2fl/^-(a3 + A« + B)/+D=0 

 /3-(a2+A)/2 + C/+Da=0. 

 From these we easily deduce 



/2 _ (3^2 _^ A)/+ a'l + Aa2 + Ba + C = 



(«3 + A« - B}/2 - (2Ca - D)/- 2Da^ = 0. 



Eliminating /2 by 2a/^=(fl3 + Aa + B)/— D, we shall have 

 two equations, in whichywill be only of the first degree; 

 and then, by eliminating it from these, there results an equa- 

 tion in a of the tenth degree ; and it is obvious that/, c, and 

 b will be determined from a by simple equations. 



Now let a„ ag, &c. be the roots of the equation in a, and 

 Oi'i, a^g, &c. those of (1.); then, since x'^—ax-\-f=-0 must con- 

 tain two of the last, we shall have 



^5 = ^2"T""^3» <'^6"^^'*^2'^"'^4' ^7 ^=^ "^a "^ "^S* ^8 ^^ "^S "I" "^4' ( * \ ') 

 ^=^3 + '2?6j «]o = a?4 + T5. -J 



* Communicated by the Author. 



