DETERMINE THE EORM OF THE ROOTS OF ALGEBRAIC EQUATIONS. 745 
which falls under the general type 
u+cp(D)e n0 u=\J, (1) 
U being a function of 6 when the inverse operation { ^(D)} -1 0 has been performed. 
The theory of equations of the above type has been discussed fully in my memoir 
“ On a General Method in Analysis.” In particular it is there shown that the above 
equation can be converted into another of the same type, 
by assuming 
where 
v-\-^{D)e n6 v—Y , 
u — p ? ( P) v 
u — r *t(P) v ’ 
( 2 ) 
p <p(D) <p(D)<p(D— n)<p(D — 2n) . . ad inf. 
~\J/(D)4/(D— ra)\f/(D - 2n) . . ad inf. ' 
This theory I shall apply here, not to the ordinary finite solution, but to the solution 
by definite integrals of the differential equation (I). In doing this I shall give to U and 
V the particular values 0. We are justified in doing this by the canons relating to the 
arbitrary constants which are laid down in the memoir ; but it will suffice here to direct 
attention to the fact that while the processes employed are strictly speaking particular, 
they lead to a solution involving the requisite number of arbitrary constants, and at 
the same time of the proper form, as manifested by the succession of the indices in its 
development. 
Giving then to (I) the form 
u-\-- 
~n— 1-q 
n n 
r-V2 _“- A 
- 1 V" " 
assume as the transformed equation 
[D]» 
e n9 u=0, 
w +[Dp^=0- 
Then by (2) 
Now 
( 2 -^- 1 )}, 
since representing ij by <p(D), the first term in the factorial expression 
of <p(D—n) will so follow the last term in that of <p(D) as to leave the law of factorial 
succession unbroken. Again, if Afi 19 be any term in the development of v, we have, 
i being a positive integer, 
=A.cr (==*<+=)* 
