124 Mr Babbage on the Theoretical Principles 
0S+9S + 8S + ... 
V * v V 4. J v+2 
IS 
represent the units figure of the number v . Now, if we put 
u + x instead of v in the above equation, we have 
Am =0S + 9S + 8S+... IS 
n u W+ 1 tf- f2 ?l~h 9 
«+ 1 n +l n +i n +i 
an equation whose mode of solution is as yet quite unknown. 
Finding the difficulty of a direct attempt so considerable, I em- 
ployed two other processes ; one was a kind of induction, and 
the other was quite unexceptionable. From these I have de- 
duced the following formula ; 
v n + (a) + 205 (106 + 2a — 1) + 2, 
where a is the units figure of n ; b is the number n , when its 
unit figure is cut off ; and (a) represents whatever number is 
opposite to it in the subsidiary table below : 
If a — 0 
1... 0 
2.. . 2 Example: Required the 27th term 
3.. . 8 of the series here, a = 7, and 
4.. . 14 5 = 2: hence, 
5.. . 26 105 + 2a — 1 = 20 + 14 — 1 = 33. 
6.. . 46 
7.. . 74 
8.. .108 
9.. -142 
then 
(a) or the number op- 
posite 7, is - 74 
205 (106 + 2a — 1) = 1320 
2=2 
1396 = u , or the 27th term, 
27 
similarly if n = 1121 a = 1 b — 112 
then u = 251106. 
1121 
