XIII. On the Determination of the General Term of 
a New Class of Infinite Series. 
By CHARLES BABBAGE, Esq. M.A. 
FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH, 
AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. 
[Read May 3, 1824.] 
Tue subject of investigation on which I have entered in 
the following Paper, had its origin in a circumstance which is, 
I believe, as yet singular in the history of mathematical science, 
although there exists considerable probability, that it will not 
long remain an isolated example of analytical enquiries, suggested 
and rendered necessary by the progress of machinery adapted 
to numerical computation. Some time has elapsed since I was 
examining a small machine I had constructed, by which a Table, 
having its second difference constant, might be computed by 
mechanical means. In considering the various changes which 
might be made in the arrangement of its parts, I observed an 
alteration, by which the calculated series would always have its 
second difference equal to the unit’s figure of the last computed 
term of the series; other forms of the machine would make the 
first or the third, or generally any given difference equal to the 
unit’s figure of the term last computed ; and a further alteration 
would make the same difference equal to double, or generally to 
(a) times the digit in the unit’s place: or if it were preferred, 
Vol I. Part I. Ke 
