242 Sir WiLLiXM R. HxuihTOJi on the Argument of Abel, 



the coefficients being all symmetric, and being determined through the elimina- 

 tion of all higher powers of x^ than the fourth, by means of the equations 



or/ + a, xj' + a^ x^ + a^ x^ + a^x^ + a^- Q, 



Xa + «l-^a' + «2 V + «3 a^a' + ^4 «„' + ^s 37^ = 0, &C. ; 



and it would always be possible to find symmetric multipliers c,, c.^, c,, c<, which 

 would not all be equal to 0, and would be such that 



c, h, + c, 5/^' + C3 5.^^' + c, 6,'^' = 0, 

 c. &3 + c, ^^'^^ + C3 63''' + c. K*' = 0, 

 c,&, + c,6/-^' + C3 5/'' + c,6/'=0; 



in this manner then we should obtain an equation of the form 



c, a/ + c, a,'* + C3 a,'* + c, a/ * = 



c, b, + c, 6„'*' + c, h^'' + c, &;"' + (c, 5, + c, 6/^' + C3 5.'^) + c, 6/^0 a:„, 



in which it would be impossible that the coefficient of x^ should vanish, because 

 the five unequal values of a/ could not all satisfy one common equation, of the 

 fourth or of a lower degree; we should therefore have an expression for x^ of the 

 form 



x^=d^-\- rf, a/ + d^ a/' + d^ a/^ + d, a^\ 



the coefficients rf,,, . . . d^ being symmetric ; and for the same reason we should 

 have also 



^/j = t^o + «^i Pi «/ + <^2 Pi «i" + d^ p,^ a/ =• + c?4 Pi* 0/ ^ 

 a;^ = c?„ + «?, />/ 0/ + c?2 P5* «/^ + d^ p., a," + d, /»/ o/^ 

 j;^ = «?a + rfi /)/ o,' 4- c?2 /»5 «.' "* + «^3 /Ss" «." + d^ />/ o/", 



^, = <^0 + <^I Pi «/ + «^2 /Js" «.' ' + C?3 P5' «/ ' + C?4 Ps «,'*> 



■r„, a;^, or^, ar^, or^ denoting, in some arrangement or other, the five roots x^, x^, 



