248 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
••• /(D) v + p(D)/(D — r) s ri v = U by (V.), 
, <XD)/(D— r) 
/(D) 
i r6 v= {/(D)} _1 U (27.) 
Comparing this with the equation v-{-^(D)s r *v=Y, we have 
<KD)/(D— r) 
J( D) 
/( D )=|)By/( D -’-)i (as.) 
lienee /(D— r)=-| |g_^ /(D— 2r), 
and so on, wherefore the value of /(D) will be represented by the infinite product 
(}/'(D)4;(D— r)4/(D— 2r) — 5 w ' liC ^ we shall express under the form Pr-Dj^’ in accordance 
with Sir John IIerschel’s notation for the integrals of equations of finite differences 
of the first order, of which, in fact, (28.) i an example. Hence (27.) becomes 
with the relations 
v -j- (D) z r6 v = V, 
u — p $(D) tj — p 4(D)y 
lr 4/(D)^ 5 U— Fr 4/(D) V - 
As the above Proposition is of great importance in the solution of differential 
equations, we shall devote some attention to the circumstances which attend its 
application. 
That the expression of P,. 
4(D) 
*(D) 
may be finite, it is sufficient that for every elementary 
factor ^(D) occurring in the numerator, there should correspond a similar factor 
%(D + /r) in the denominator, i being an integer, and vice versa ; for 
p X(P) _ x(D)x(D~ r)x(D — 2r).., 
r x(D-H>) %(D + ir)x(D+ (*—!)/■)... 
1 
X(D + ir)%(D + (i — l)r) . -%(D + r)* 
which is a finite expression. Again, 
p *(D) _ x(P)x(P— *•)••• 
r X(P — * r ) %(P-* r )x(P-/ + 1 ) r )-” 
= X(D)x(D—r) . .%(D— (i— l)r), 
which is also finite; the product of any number of such expressions is finite also. 
If %(D) is any elementary factor of <p(D), it may be converted into %(D+*‘r) ; for 
let <p(D)=%(D)/(D), and let -/(D) =%(D+ir)% I (D), wherein %/D) denotes the product 
of the remaining factors, then 
p 4(D) .... p * (P) 
^(P) _lr x(D ±ir)’ 
which is finite. 
