The equation of finite differences from which it is produced is
A 2 u = units fiff . of u
which is one of a class of equations never hitherto integrated. I succeeded in
transforming this equation into a more analytical form : but still it presented
great difficulties ; I therefore undertook the investigation in a different man-
ner, and succeeded in discovering a formula which represented its nth term.
It is the following :
Table
0 2
12
2 4
3 10
w = (T) + 206(106 + 2a-l) 4 16
- 5 28
6 48 7 76 8 110 9 144 where (a) represents the number opposite a in the annexed subsidiary table, and a is the figure in the unit's place of s, and b is that number which arises from cutting off the last figure from z. Example : let the 17th term be re- quired, then 2=17, and a=7> 6=1 ; the number opposite 7 in the table is (7) = . • • 76 106 + 2a- 1 = 10+14- 1 = 23 206 = 20 206 (106+2a-l) = 460 536 = u 17 VOL. I. PART II. 2 T
