246 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
provided that q x , q 2 .., q n are roots of the equation 
q 4- «i q n ~ 1 + « 2 q n ~ 2 - . + «„ = o, 
and that N 1? N 2 ,..N B are of the forms 
1 (fl'l-ffaXS'l *«/ S1AX* -qt)“{qn — q»- 1)’ 
Let (1 — g r 1 f) _1 U=M 1 , (i — ? 2 f) _1 U=w 2 5 an( l so on ? then 
Wj — = or u x — q x <p(D)i e u x =XJ, 
whence 
« = NjMj + N 2 m 2 ...-|-N b m b , 
( 24 .) 
wherein u l u 2 ..u n are determined by the system of equations, 
u \ ~ QiQi D)s*m 1 =U,'' 
w 2 -^( d ) £ S=U, 
( 25 .) 
The forms of <p(D) which render the above system integrable will hereafter be deter- 
mined. The most important of these is obviously 
which reduces the proposed system of equations to the first order. 
For the particular form p(D) = (D) -1 , the equation above considered will represent 
the general linear differential equation with constant coefficients ; for every other 
form of ®(D) it will represent an equation with variable coefficients. 
dj^u 
Ex. Let the given equation be (a^+ma^+w# 4 ) 
du 
(2bx + (a-\-b -\-'2)mx 2 [2a-\-A)nx i )-^-\-{b{b— \)-{-(a-\-\)bmx-\-(a-{-'2)(a-\-\)nx 2 )u = Q 
Putting x—i\ and reducing to the symbolical form, we have 
( 26 .) 
Here q v q 2 are the roots of the equation q 2 -\-mq-\-n — 0, whence 
( 1 \ u i |_ [h u \ £§^ 
<h-<h 
u x and u 2 being given by the equations 
