DETERMINE THE EORM OF THE ROOTS OF ALGEBRAIC EQUATIONS. 739 
unvarying in its type. I have thus verified some other particular forms obtained by 
Mr. Harley. 
Solution of the Differential Equation by Definite Integrals. 
5. On account of the difficulty of the investigation, I propose to employ two distinct 
methods leading to coincident results. 
First Method. — Operating on both sides of the given differential equation (I) with 
{[D]”} -1 , we have 
u+ 
r?=i D+ “_ 1 i-Y5_! 
L n n J \ n 
[DT 
e n6 u= C 0 + CA. +C n _f 
. . (1) 
C 0 , Cj , . . C n _! being arbitrary constants. Let us represent 
L n n J \n n J n 
[Dr 
e ne U, 
whatever the nature of the subject U, by pU, then the differential equation becomes 
u+g u=C 0 +C 1 e e • • +C n _ 1 e (n ~ 1)0 , 
or 
(l+ ff >=C 0 +C^.. +C n _f n -" e ; 
«=( i +? )-*{C 0 +C/. • +C n _ 1 d n -'»}=%C i (l+g)-'e i9 , 
Now 
But if 
we have 
m 
L n n J \n n ) 
[or 
ge ie =<p(D)e n9 e id , 
? V 9 =^(D>”V(I>> ( ” +! ' ;0 =^(H)^(D-w)e (2re+! r 
fe ie = <p( D)p(D — n) . . <p (D — ( q > — 1 )n}e {pn+i)9 . 
But from the form of <p(D) it is easily seen that 
V n ~ 1 -q , m _ D m ^l 2 
tiPMP -n)= -L^ "~^ yn 
and generally 
f(D)p(D— n). .<p(D-(^-l>) = 
[?=l D+ “_ll 
n 1 n 
H 
D m 
Jn ^ J 
LU 
i]p» 
•. g p e ie = 
n— 1 ^ m 1 ‘jpC»- 1 )rD m ^ 
n ' n J j n n 
'• [Dp 
( 2 ) 
5 F 
MDCCCLXIV. 
