Page:Memoirs of Royal Astronomical Society Volume 01.djvu/336

Rh 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 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
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