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tic equation of which the coefficients are rational functions 
of the coefficients of the given quintic equation 
# 5 4 ~ bx* + cx z + dx? + ex-\ -f— 0 , 
while Ui, u 2 , u 3 , are the roots of a quartic equation of 
which the coefficients can he expressed rationally in terms 
of u 0 and of the coefficients b, c, d, e, f of the quintic, yet we 
are really no nearer the solution of the quintic ; for how are 
we to solve the reducing sextic ? 
Lagrange has shown generally that the function ( a,x ) 
has only 
\n 
n 
V 
different values, and that if the development 
of ( a,x) p he reduced hy the help of the equation a p =l (and 
not a v ~ l a p ~ 2 . . . +1=0), to the form 
^zn , Wf)+ nU] -\-ci~Uo • • • +ct^ ^Up 
I n 
then this power ( a,x) p has only — k — _ values, and the term 
Pi 
u 0 has only 
\n 
n(n — 1)/ 
n\d 
P J 
values, or is the root of an equation 
of the degree 
\n 
n (n — 1)( 
of which equation the coefficients are rational functions of 
the given coefficients b, c, d, &c. ; while u v u 2 ,...u, p _ 1 are the 
roots of an equation of the degree p — 1, of which the coeffi- 
cients can he expressed rationally in terms of u 0 and of the 
same original coefficients b, c, d, &c., of the given equation 
in x. 
Although the method fails when applied to the quintic 
and higher equations, yet it is interesting on account of 
its simplicity, directness, and uniformity. “It is of great 
