respecting Equations of the Fifth Degree. 177 



For, inasmuch as h is, by supposition, a rational function f of all the 

 radicals a , it is, with respect to any radical of highest order, such as a , a 



t i 



function of the form 



/ (m)\ 



b = 



M (a ) 



M and N being here used as signs of some whole functions, or finite integral 

 polynomes. Now, if we denote by p any root of the numerical equation 



a 

 a— I a— 2 a—Z 2 



P +/" +/> +--. + P +P +1=0, 



a a a a a 



80 that p is at the same time a root of unity, because the last equation gives 



a 



a 



P =1; 



a 



and if we suppose the number a to be prime, so that 



2 3 o— 1 



p , p , p , . . . p 



a a a a 



are, in some arrangement or other, the a — 1 roots of the equation above as- 

 signed : then, the product of all the a — 1 whole functions following. 



M 



(paj.ufp a) . . . u (p aj =h (a), 



is not only itself a whole function of a, but it is one which, when multiplied by 

 M (a), gives a product of the form 



a 



L (a) . M (a) = K (a ), 



K being here (as well as l) a sign of some whole function. K then we form 

 the product 



Jim)_ J 

 .,f W^ f 2 {m)\ ( "• {m)\ ( (m)"\ 



Mlp al.Mlp a. )...Mlp,, a''l = L(a I, 



i i ■ ' i 



2b2 



