260 Mr. Babbage on the General Term 
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, the digit 
fixed upon might’be that occurring in the ten’s place, or ge- 
nerally in the zth place. I did not, at that time, ‘possess the 
means of making these alterations which I had contemplated, 
but I immediately proceeded to write down one of the series 
which would have been calculated by the machine-thus altered ; 
and commencing with one of the most simple, I formed the 
series, Series. Diff: 
ne 
Dn 
OP ro D OS vo 
Hf u, represent any term of this series, then the equation which 
determines w_ is 
Au, = unit’s figure of w_, 
an equation of differences of a nature not hitherto considered, 
nor am | aware that any method has been pointed out for the 
determining u. in functions of z from such laws. I shall now 
lay before the Society, the steps which I took for ascertaining 
the general terms of such series, and of integrating the equa- 
tions to. which they lead. I shall not, however, commence 
with the general investigation of the subject, but shall simply 
point out the paths through which I was led to their solution, 
conceiving this course to be much more conducive to the pro- 
gress of analysis, although not so much in unison with the 
taste which at present, prevails in that science. 
If we examine the series, and its first differences, it will be 
perceived, that the terms of the latter recur after intervals of 
four, and that all the changes in the first differences, are com- 
prised in the numbers 2, 4, 8, 6, which recur continually, and 
the series may be written thus: 
Series. Diff. 
2 2 
4 4 
8 8 
16 6 
5 22 = 20+ 2 2 
24=20+ 4 4 
