274 Mr. Babbage’s new method of investigating the 
less than 2, consequently this expression will be finite or in- 
finite, according as the series. 
-Lx''-^+ - Nx‘i^Z + & c . 
w * ir w 1 
is so or the contrary: the product 1.2 ...n may be expressed 
by means of a definite integral, thus : 
L*...n=Sdv( l o g ±y [— ] 
These products being replaced by the integral, we have 
~fdv{— L(^) 2 (log -i -) 3 + M(A ) 4 (log i )5_N(-i) 6 (log -i ) 7 + See . } 
Io s v ) 2 + M (v 1 o st) 4 — N (v Io s vf+ &c -} 
Now in order to determine the sum of this series, which evi- 
dently depends on the function f(.r), let us assume 
f(x) a B 
2k 
. zk 
2k— 2 
&c. — K = %(#) 
then we have 
%(*) = Lz*-{- N.r 6 -f- &c. 
And if we put — — - log 
instead of x , it becomes 
x{ X ^°g vl = - L (f '°S v) 2 + M (v log t) 4 - N(^ log^) 6 + &c. 
And the sum of the series in question is 
icgv-M^osv) [:=;] 
If therefore this definite integral is finite, the theorem (A) 
will give correct results. A more convenient form for inte- 
gration may however be obtained by the following conside- 
ration ; the series 
L-^-. 1.2.3 + M-£ 1...5 + N 1...7+ &c. 
will always be finite, if the following series 
A + B (Jj ) 2 ,. 2 . 3 + c (y) 4 i. 2 .. S + D (“} 6 i *2.. 7 + &c. 
is finite, because this series when prolonged to the terms 
