262 Mr. Babbage on the General Term 
Series. 1 Diff. 
15 408 60 = 40 + 20 
468 68 = 40 + 28 
536 74 = 40 4+ 34 
610 74 = 40 + 34 
684 78 = 40 + 38 
20 762 SOS tOPE Se, 
842 82 = 80+ 2 
924 {a LORS Si als: 
1010 86 =, 60) 16 
1096 O2'==, “80.7 12 
25 1188 100 = 80 + 20 
1288 108 = 80 + 28 
1396 114 = 80 + 34 
1510 114 = 80 + 34 
1624 LS: ==" 480 =" 38 
30) «1742 120 = 120+ 0O 
1862 122 =120 + 2 
In this series it may be observed, that ~, when z is less 
than 10, is equal to the sum of the first differences of all the 
preceding terms ; and if z be greater than 10, it will be com- 
posed of four terms,viz. first the sum of the ten first terms of the 
first difference, multiplied by the number of tens contained in 
z; secondly, of the sum of the series 40 + 80 + 120 + to as 
many terms as there are tens in z, this must be multiplied by 
10, as each term is ten times added; and thirdly, of the num- 
ber 40 multiplied by the same number of the tens, and also 
multiplied by the digit in the unit’s place of z; and fourthly, 
of the sum of so many terms of the series as is equal to the 
unit’s figure of z; this being expressed by (@) signifying the 
number opposite a in the previous table. These four parts, if 
z= 100 + a, are thus expressed, 
1st 1808, 
b.b—1 
2 
34 40 ba, 
ede (9G 
These added together produce 
u, = 206(106 + 2a — 1) + (4). 
This value of ,, , if diminished by 2, is equal to the sum of z—1 
term of the series which constitute the first difference. 
This inductive process for discovering the mth terms of such 
series, might be applied to others of the same kind; but it does 
not admit of an application sufficiently general or direct, to 
render it desirable that it should be pursued further. 
and 40 
10, 
If 
