MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
233 
• d n u 
For multiplying by x n , and considering first the expression (a-\-bx-\-cx 2 ..)x n j ^, -, 
let x=i e , we have 
(a+^+cg 2 ^..)D(D— 1)..(D— n-{-\)u, 
= «D(D — 1)..(D — n -\- 1 )u, 
-f-6(D — l)(D — 2)..(D — n)z 6 u, 
+ c(D — 2)(D — 3)..(D— w— l)s 2 ^. 
In like manner may every term in the first member of the original equation be 
reduced; also the second member, which is a function of x, will become a function 
of a"; the aggregate of these results will produce an equation of the form (VIII.). 
More generally, let it be supposed that we have a system of linear differential 
equations, total or partial, the dependent variables being u 1 v v . the independent va- 
riables x Y x 2 .. whereof the second members of the equations are functions ; if we 
assume x^fi, x 2 =&.. the transformed equations may be so written as to satisfy the 
following conditions. 
1st. Every term involving u shall be of the form <p(D l3 D 2 ..)z r ^ +r ^u, and similarly 
for every term involving v. 
2nd. The second members shall be functions of 
Let us now consider the expression fQ(Tfuff(T))z 6 u-\-f 2 (D)t 26 u.., and let us therein 
assume u — 'Xu m e m6 , then passing the symbols /j/D), /^(D)... within the sign of sum- 
mation, collecting the coefficients of z m6 , and observing that fff))z m6 =ffrn)z m6 , &c., 
we have 
/ 0 (D)M+/ 1 (D)2^+/ 2 (D)g 2 ^... = 2{(/o(m)«< w -l-/ 1 (m)M TO _i+/ 2 (m)M m _2...)s w } . . (IX.) 
which is a particular form of the fundamental theorem of development. 
To any aggregate of terms of the form <£>(D l5 D 2 ..)g r A+V 2 --w the same analysis is 
applicable, whence our fundamental theorem, viz. 
If U='Zu m , n ... S mf i+ n ^", 
then <p(D l ,D 2 ..)s r A+ r &-ni=2{<p(m,n..)u m -r 1 ,n-r 2 J m * 1+,t< ’ i --'}' ...... (X.) 
In applying this theorem to an expression consisting of many terms, the sign 2 
must be affixed to the aggregate in the second member, as in (IX.), and not to each 
term separately. 
The relation which the first member of (IX.) bears to the linear differential equa- 
tion, is the same as is borne by the coefficient of in the second member to the 
linear equation of finite differences. This analogy extends to the fundamental 
theorem, which may be defined as a general relation connecting any linear differential 
equation, or system of linear differential equations, with a corresponding equation or 
system of equations in finite differences. 
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