MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
265 
Effecting the integrations, and determining the constants by comparison with the 
original series, we have 
u 
— 4~ x2 ~ ~ (t “ 2x ' + T x2 ) 1( °S ( 1 “ x ) ■ 
When, as in the above case, the factors of the denominator of f(m) are equidifferent, 
the value of a may be determined by the solution of an equation of the first order. 
Thus : 
/( 0) , /(l) „ , /( 2) 
Ex. Let u— 
1.2 ..n r 2.3..(rc+l) 3 A..(n + 2)' 
wherein f(m) is of the form a-\-bm-\- cm 2 -}- &c., being finite and not involving any 
negative or fractional indices. 
f(m) 
-x 2J r &c., ad inf., 
Here 
Um — 
tin 
(»J + l)(m + 2)..(m + ») 
f(m) m 
, whence 
f{m — 1 ) m -j- n Um 1 
u- 
/(D) D 
Assume 
/(D — 1) D + n 
D 
e’u - 
A Q) 
1 . 2 ..//* 
D -\-n 
dv=V, 
we find u=f(T>)v, ~ ^=/(D)V, whence V=yJ-^, and substituting 
D 
D + n 
i°V- 
' 1.2. .7*’ 
(D-f-n)y — l)v. 
{l-S* 
T. 2 ..(w-l) — r(re) 
V ~ T(n)e» 
“=/( D) i» 
i r> e n6 t 
( 1 — e ^- 1 r 
1 . / d\(l — x) n ~ l (mp-'dx 
: Y(n)J \ X dx) A J n (1—i 
o U-*r 
a result always finite when n is an integer. 
The theorem (XVIII.) maybe extended to series involving any number of variables. 
Let u=lij\m l m 2 ..)x 1 m w 2 m z..,f being a function of invariable form, and the summation 
2 extending to all positive integer values of then 
»=/(D 1 D 2 -) ( |. <|) 1 |^ -, (63.) 
wherein fr=x l3 &—x 2 , &c. The performance of the operation /(D 1 D 2 ..) will involve 
differentiation, or the solution of a partial differential equation with constant coeffi- 
cients. 
2 m 2 
