respecting Equations of the Fifth Degree. 209 



the rational function f/' admitting of 2 a/' values, and consequently the exponent 

 a/' being = 3, (since It cannot be = 2, because no function of three variables 

 has exactly four values,) so that we must suppose the radical a," to be a cube- 

 root, of the form 



a," = v^6o + 6.<, 



Jg and 6, being rational with respect to o,, a^, a^. But In order that a six- 

 valued rational function f/'> of three arbitrary quantities x^y x^, x^, should have 

 a two-valued cube, it must be of the form 



F," (a;,, x^, X,) = (p^ +p^ a;) {x^ -t- p^^ x^ -f p^x^); 



in which p^ and p^ are symmetric, a/ has the form recently assigned, and p^ is a 

 root of the numerical equation 



P3* + P3 + l = 0; 

 we must therefore suppose that 



«." = {Po +i>. O {^. + 9^ ^, + ft ^,), 

 and 



h + b^a: = 27(p,+p^a:y {c. + t:V(p,^- />,) y }' 



Cj retaining here its recent meaning ; so that the radical a/' may be considered 

 as the cube-root of this last expression. If any other radical a^' of the second 

 order could enter into the composition of b , it ought, for the same reasons, 

 to be either of the form 



< = (^o + ?i <) (^. + ft' -^^a + ft ^3). 

 or else of the form 



O2" = (q^ + qi a/) (or, -{- p,x^-\- p^ a;,), 



pi being here the same root of the numerical equation p^ -\- p^-^-lzi 0, as in 

 the expression for o/' ; we should therefore have either the relation 



or else the relation 



U _ ; 7- a, , 



Po+Pidi 



„ _ 9 Ca (Po + JPi «i) (yp + gi "iQ 



2f2 



