Equation of the Fifth Degree, 343 



the roots being 



Xj = rt6, A2=a«, A3=a5, X^-a\. 



Here we shall find, as before, putting for a^^ Cg* &c. then* 

 values in w^, Xc^ &c., that 



■\-5x^{xl^x\ + x\ + x*^+xl + x\ + xl + xlz=z~27x\ 



We may find h, k, and / in terms ofg, as we found n, p, and 

 r in terms of ?w ; and as S(^j), 2(a:^), &c. are known functions 

 of A, B, &c., we shall have g, h, &c. functions of x^. The 

 determination of these therefore may be said to depend upon 

 the solution of the given equation. If otherwise found, as 

 they may be by finding the equation on which g depends, it 

 must be by an equation of the fifth degree not reducible ; for 

 the five values of Xy, .r^, &c. being distinct, there will be as 

 many distinct values of ^. 



It may be observed that if we make A any other integer 

 function of a, not passing the fifth degree, we shall still have 

 an ultimate equation to solve of the same degree. 



To give two very simple examples of the equation in g, let 

 x^ + Kx+B — O. 

 Then 



2:K) = 0, 2:(a;?)=0, 2(^t)=-4A, 2(^J) = -5B; 

 and 



g=27:rf + 20A.ri4-5B. 



Eliminating x^ between this and a:J + Aa*j + B=0, we hav 



{g + 22B)5 + 74A^(^ + 22 B) - l^M'B = 0. 

 Again, leta?^ + Aa;^ + B = 0. In this case 



and 



g=z'21x\ + iiOAx\+5^. 



Eliminate -Tj from this and x\-{- Ax\->r^ = 0, and there results 



{g + 22B)* + 3W(g + 22B) - 3*A^^B = 0. 



By making g + 22B = i; in the first of these examples, and 

 ^-l-22B = t;^ in the second, the equations in v are similar to 

 those in .r, and are no way in a more solvable form. 

 Let us now take the equation of the sixth degree, 



a^ + ha^ +^x^ + Cx^ + Dx + ¥.=: {af^ ^ aa^ -{■bx'^ + ex + d) 



{x^—ax+f)=0. 



