respecting Equations of the Fifth Degree. 



257 



6 being an imaginary cube-root of unity, and /3'' 7" being the two roots of an 

 auxiliary quadratic, 



And, doubtless, it is allowed to represent any five arbitrary quantities x^ x^ x^ 

 x^ x^ by the system of expressions (2) (3) (4) (5), in which a, w, and 6 are such 

 that 



a=z — {x^-'tX.■^ + X^+ X^ + X^), (6) 



a,' + w'+w^+w + l = 0, (7) 



02 + + 1 = , 



(8) 



provided that the auxiliary quantities a /3 7 8 a' ^' 7' 8' a" /3" 7" a'" a^^ be de- 

 termined so as to satisfy the conditions 



^■^ a■=X^•\- U)^ X^ + u? X^-\- (O^ X^ + wX^, 



5 ^ ^ = x^ + to^ x^ + ti)X^-\- w^ x^ + (o^ x^, 

 b-^y=.x^ + m^X^ + w*X,-¥toXi + ui^x^, 

 5 V' S ■= X^-\- wX^-\- w'^X^ + tO^Xi-^ W^X^, 



(9) 



4a' = a+p-|-74-S, 

 4/^' = a-l-/3-7-8, 

 4/7' = a-/3-|-7-8, 

 4 v/g' = a-/3-7+5, 



3a"=Z|8'-f-7'-f-g', 



3 ^^" = P' -1-0^ 7' -I- 08', 



3^7"=/3' + 07'-f-02 8', 



2a"' = /3"-f7", 

 2/a^'' = ^"-7". ] 



1 



(10) 



(11) 



(12) 



But it is not true that the four auxiliary quantities a', a", a'", a^'', determined by 

 these conditions, are symmetric functions of the five quantities x^, x^, x^, x^, x^, 

 or rational functions of a, b, c, d, e, as Mr. Mokphy appears to have conjectured 

 them to be. 



2m 2 



