262 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
Here the only values of m, whereof account is to be taken, are p and If 
7>i=p, we have, under the sign of summation, the expression {u p —<p{p)u p _ r )i^, but 
u p _ r not being a coefficient of the original series is to be rejected, so that we have 
simply u p iP 6 . Assuming we have under 2 the expression {u t+r —<p(t-\-r)u t ) 
z {t+r)e , from which u t+r being rejected, leaves —<p(t-\-r)u t , wherefore 
U — <p (D) i r6 ll — UpiV 6 — $(t -\-r)u t z {t+r)e . 
Since by (56.), 0, the above equation may be written under the 
somewhat simpler form, 
u — <p(D)z r 6 u~u p z'P 6 — u t+r z < ' t+r 6 (5 7.) 
If the series is infinite, 
u — <p(D „ \ (58.) 
Let u=u p x p -\-u p+1 x p+1 -\-u p+2 x p+ . 2 ...-{-u t x t , and let the law of derivation of the coeffi- 
cients be 
WVi+?2( ffl )VF 0. 
Here by the theorem, 
u -f «p 1 (D)g^« + P 2 (D)s 2 'm = 2 { tii m +<p x (in)u m _ j + p, (m)u m _ 2 y^ } , 
the values of m to be considered being p, p-\-\, /-fl, £-f2. 
Whence the second member gives 
■M P e^+(Mp+i + ?i(/»+lK)s (,+lW + (?>!(#+ l)w f +?) 2 (<+ l <p 2 (if+2> ( ‘ +2) '. 
This expression also may be simplified, as in the preceding case, for 
u p+l +<p(p+\)u p = — <p 2 (p + l> P -i, 
( Pi(^+ 0^+i — ~ u t+ 1- 
Wherefore finally, 
u^(p l (JJ)i 6 u^<p 2 (J})z i hi—u p i^—<p 2 ( y p^\)u p+l z {p+l)6 — ii t+l z it+1)6 ^ < p 2 {t-\-2)ufi {t+2)6 . . . (59.) 
Ex. 1. Let u= 1 — 
» 2 (ft 2 — 2 2 ) . ra 2 (w 2 — 2 2 )(?z 2 — 4 2 ) 
1.2. 3.4 
1.2. ...6 
# 6 +&c. 
Here u = 
n 
2 (pyi 
wherefore by (57-), 
it- 
(1 
(D-2) 2 — rc 2 ^ 
z 16 u= 1. 
D(D-l) 
rf 2 w du 
^dS*-“%+ nu = 0 - 
u — c x cos ( n sin '#) -f- c 2 sin (wsin - ’#) . 
Determining the constants by comparison with the original series, we find 
fjT 7T 
u=cos(nw), wherein w is that value of sin -1 # which lies between — ^ and y 
n 2 js r n 2 1 2 V^ 2 3 2 ) r 1 
Similarly for the series # — yyr# 3 + - — \ 2 34 5 — ; # 5 — &c., we find sin {nw). 
The correctness of these results will be shown by substituting Poinsot’s expansions. 
Ex. 2. To sum the remainder of Taylor’s series, viz. 
d n Q(a) x n d' l+x §{a) x n+1 . c 
da n 1.2../P du n+1 1.2..?/ + I - *" &C ‘ 
