264 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
If we wish to sum a finite portion of the development, let f(p)xP be the first term of 
the series as before, and f(p')x p ' the first term of the remainder of the series, that is, of 
the portion following that which is to be reduced to a finite form, then 
u p s _ f p’ 6 
u 
=AD) 
: r6 
1-e' 
d n , x— x t+l 
x dx n l—o? 
cl \ n X n X ^ n 
=x 
] — X 
Ex. 4. Let u—l n x J r 2 n x 2 -\-3 n x i . . . -j -t n x f . 
Here f(m)=m n , whence 
w=D 
. e - e (t+W 
1 — s e 
( d\ n x—x l 
\ dx) 1 —a 
Ex. 5. Let u—l + (cos y):r+(cos2r?)o: 2 + &c. to t terms. 
Here f(m) — cos(mv), wherefore 
w=cos(t?D) 
1 — d 
=-L( 
2 V 
g t>DV-l_|_ g -«A>V-lj 
1 — 8« 
~ l ( 
£ t(6— v-J— 1) 
~ 2 V 
1 — o? COS V — o^cos (tv) + x t+ 'cos (t — \)v 
Ex. 6. Let u 
Her e/(m) = 
4x 3 5x 4 
~1. 2.3 “r" 2.3.4 "*"3.4.5 
m + 1 
1 — 2o? cos + a? 
6x 5 
See., ad inf. 
m(m — - \)(m— 2) 
D + 1 
, whence 
= 3 6 
u D(D — 1)(D— 2) i—d’ 
= {|d-1-2(D-1)-'+|(D-2)-> 
= j ^D- 1 -2^D-L-'+|- CUD-h-v [ 1 J - 
= 2 -J 7 I^*+V 2 J 
1 /~i xddx ^ n xdx . 3 „ /» dx 
=-5- / , 2 x r — +-w^ 2 / 1 — 
2 J l—o? J l—o? 1 2 J 1 — X 
Z6 
J ? 
J’ 
(62.) 
Ex. 3. Let m= L2..wo?+2.3..(w+ l)o? 2 +3.4..(w-j-2) l x’ 3 . . . (t-\-n~\)x t . 
¥Levef(m) = m(m-\- 1 ) . . ( m-\-n — 1), therefore 
j U+ \)0 
w=D(D+l)..(D+ra-l) e 
