DETERMINE THE FORM OF THE ROOTS OF ALGEBRAIC EQUATIONS. 735 
Formation of the Differential Equation. — General finite integral. 
2. Proposition. — If u represent the mth power of any root of the algebraic equation 
y n -xy n ~ 1 -1=0, 
then u, considered as a function ofx, satisfies the linear differential equation 
[D]"m+ 
in which e e =x, and D=-^* 
And the complete integral of the above differential equation will be 
w= C$r+C 2 y” • • +C n yZ, 
y„ y 2 , ..y n being the n roots of the given algebraic equation. 
Representing y n by z, we may give to the proposed algebraic equation the form 
n-1 
z=b-\-xz n , . . . (1) 
in which 5=1. Hence by Lagrange’s theorem 
m m n-1 J / m\ J / 2(n-l) J m\ r 2 
|s-)^+&c„ .. ... (2) 
the general term of the expansion being 
(®) 
d « 
5 * db 6 ’ 
1.2 ..r 
(3) 
which, on effecting the operations indicated, becomes 
m + r(n — l) 
» J 
w[r] r 
We see then that w is expanded in a series of the form 
w 0 -f-w 1 ;r+w 2 # 2 + &c. ad inf. 
in which, since 5=1, 
(4) 
|- m + (ra— l)r ^ 
x(l)- 
n [r] r 
(5) 
and this expression will represent the first term as well as the succeeding coefficients 
of the Lagrangian development, provided that we interpret the form \_p~f by 1, and 
1_ ' ~ i 
As 1« admits of n distinct values, the above development may be made to represent 
the wth power of any one of the n roots of the given algebraic equation. In particular, 
