264 Mr. Babbage on the General Term 
solely on that of the one which occupied the same place in the 
preceding term, it will appear that the same digit must re- 
appear in the course of ten terms at the utmost, since there 
are only ten digits, and that it may re-occur sooner. The 
same reasoning is applicable to the case of series whose first 
difference is equal to any multiple of the digits found in the 
unit’s place of the corresponding term, or to those contained 
in the equation : 
Au, =a x (unit’s figure of w.), 
as also to those in which this is increased by a given quantity, as 
Au. =a (unit’s figure of wu. )+ b. 
Ifthe second difference is equal to some multiple of the figure 
occurring in the unit’s place of the next term, as in the series 
2, 2, 4, 10, 16, 
already given, since the unit’s figure must always depend on 
the same figure in the first term of the series, and its first dif- 
ference 2 0 . 
g 2 
4 6 
10 6 
. . 
whenever those two figures are the same, a similar period must 
reappear : now as there are only two figures concerned, they can 
only admit of 100 permutations, consequently, this is the greatest 
limit of the periods in such species of series.—In the one in 
question the period is comprised in ten terms. This reasoning 
may be extended to other forms of series in which higher 
differences are given in terms of the digits occurring in the 
unit’s, ten’s, or other places of uw, or w, 41 0F elsewhere, but I 
am aware that it does not in its present form present that de- 
gree of generality which ought to be expected on such a sub- 
ject: probably the attempt to solve directly that class of equa- 
tions to which these and similar inquiries lead, may be at- 
tended with more valuable results. 
As the term “ unit’s figure of” occurs frequently, it will be 
convenient to designate it by an abbreviation; that which I 
shall propose is the combination of the two initials, and I shall 
write the above equation of differences thus 
Au, = aUFu,.......... (a). 
This may be reduced to a more usual form by the following 
method. If S, represent the sum of the xth powers of unity, 
divided by ten; then 
