respecting Equations of the Fifth Degree. 



259 



products x^ OTj" ^3 and x^^ x^ x^^ are, respectively, — 6n and — 4n ; they are 

 therefore unequal, and a^^ is not a symmetric function of .ri, oTj, x^, x^, x^. 



The same defect of symmetry may be more easily proved for the case of the 

 function a', by observing that when x^ and or^ are made = 0, the expression 



4 . 5* . a' = {x^J^ U) x^-\- u)"^ x^-\- w^x^ -\- to* x^'' 



-|- {x^-\- w^ X^-{- m X;^-\- w^ X^-\- w"^ X^^ 



-\- (a;, -\-w^x^-\- w^ x^-^-uy^Xi-^-w x^y 



(19) 



becomes 



(x^ + WX3 + w^ x^y + (^2 + «^ 0^3 + tt»* a; J* 



+ (X^ + a)^Xj + a)X^y + (x^ + w* iCg + oP X^y 



=z 4a;/— So:-/ (x^ + x^) — lOx/ (x^^ + 2X3^^ + x^) 



— 10a;/ (X3' + SoTj^ a;,— 12a:3^/ + a;/) 



— 5a-2 (a;3*— 16x3^ a:^ + 6X3^ a;/ + 4a;3 a;/ + a?/) 

 + 4a;3^— 5a;3* x,- lOx^'x^^—lOx^^ x^^—Sx^x,'' + 4ar/, 



which is evidently unsjonmetric. 



The elegant analysis of Mr. Murphy fails therefore to establish any conclu- 

 sion opposed to the argument of Abel. 



(20) 



The Reader is requested to make the following Corrections : 

 In article [7.], the biquadratic equation ought to be vr* 4- ^<;' 



In the first page of article [14.], read a,'"' =/, (a,, a,, a,), and a/=: f/ (:r,, x^, x,). 

 In the enumeration, in article [19.], of the cases in which the twelve-valued function (II.) has a 

 six-valued square, insert : 

 or of the form 



^ = (^a + ^li — ^y — XS) ^ {Xa + ^fi — ^y — ^S-^a — ^fi' ^y — ^s)- 



