272 Mr. Babbage’s new methods of investigating the 
we turn to the process employed in its investigation, we may 
remark, that the vertical column Lx 2 (iW+ 3 a - 4 *+ &c.) 
has been neglected, because the series which enters into it as 
a 'factor is equal to zero; so also the vertical column Mjc 4 (i 4 — 
&c.) is neglected for the same reason, and similarly 
for all the remaining vertical columns. Now, although it 
would be perfectly correct to omit any one, or even any finite 
number of these vertical columns, as being multiplied by a 
factor equal to zero, yet it is not legitimate to neglect an 
infinite number of terms, each multiplied by zero, unless it 
can be proved that the sum of all the terms so multiplied is 
not an infinite quantity: this, then, is the latent cause of the 
false results at which I arrived at the commencement of these 
enquiries. I shall now explain how they may be obviated, or 
rather how to assign the condition on which the truth of the 
theorems just deduced depend. We have considered the 
series of terms 
Lx 2 (1 2 — 2 2 + 3 2 — &c.) + M<r 4 (i 4 — 2 4 -j- 3 4 — &c.) -j- 
+ Mu ; 6 ( l 6 — 2 e -f- 3® — &cc. ) + &c. 
as equal to zero. Any one of the series which here multiply 
the powers of x, may be considered as arising from the series 
1 2 *y — 2 2 ”/ -f 3 &c. 
W hen y = 1, call this series K n (y), and instead of making 
y = 1 , let y = 1 +0, which differs from unity by the infinitely 
small quantity o ; then we shall have 
K «( 1 + °)= : c z ,n° + c 3,«° a -f“ &c - 
Where 
-or- i 2 M — 2 2 w -|-3 2 m -&c. ^ 2)W = i 2M + i - — &e. 
by substituting this value of K w ( 1 + o) in the series we had 
neglected, we shall find 
