MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
271 
the above product. Hence 
* i w n(n + \)..(n + r- 1) ,, , n n(n + l)..(n + r) 
A , — IX 12 r v Y X l.2..r + 1 
n(n + 1 n(n-\-\)..(n + r -\-\) 
+ ~T.2~ X 
1.2../-+ 2 
-j' r + 4 -j-&c. ad infinitum. 
Put v 2 =t, then 
K=t^z:{u m n, ( 77 .) 
.. n(n + \)..{n + m — 1) n(n+ \)..{n-\-r + m— 1) 
wherein generally u m = ■■ n 0 X 
1.2 ..vi 
1 .2 ..(r + ?«) 
the law of derivation being 
(n + m — \){n + r + m— 1 ) 
U — 7 ; — . 'u — 0 . 
'« ?h(?k + r) 1 
Hence, if 2u j"‘ = u , and if t=s 6 , then 
(D + ra — l)(D + r + rc— 1) t n{n + \)..{n + r—\) 
U ~ D (D + r) iU ~ 1.2 ..r “ U * 
To integrate this equation, assume 
Then 
„ (D + rc— 1)(D + r + n— 1) 
u = hq tstys v v. 
D(D + r) 
— (D+n— l)(D+w — 2)..(D+ I ).(D + r-f-»~ l)..(D-fr+ 1 )v, 
U=(D+w-l)(D+w~2)..(D + l).(D + r+»-l)..(D+r+l)V. 
Hence determining V, we have 
6 n[n + \)..{n+r—\) 1 
V iV ~ 1.2. .r X 1.2..(rc-l).(r+l)(r + 2)..(r-fra-l)’ 
( 78 .) 
Now 
but 
and 
lienee 
But 
— { 1 .2 ..(n - 1 ) }*’ “ { I» } 2 ’ 
1 
w=:(D+r+w— l)..(D+r+ l).(D + w — 1)..(D+ l)t>, 
(D+r+n-l)..(D+j-+l)=r’-(j ( )”"V + *- 1 , 
(D+n- 1)..(D+ 1 ) = (s) 
/ d\ n 
**- 
dtj {r(»)j 2 (i-/) 
' d \ i /d\ n ~ l 1 
j t j = ( ^ ) (this may be shown by expansion) 
1.2. .w—l r(w) 
~Ti— 0" (i — 
2 N 
MDCCCXLIV. 
