Infinite Series . 
169 
— = — = — ==- 2 -~ j5 ~ — * X m + 2 . — U x -if . . „ 
n- 3 • h — 5 • n — j . » — 9 . . . 4x2 2 3 
— — - — = o, if n be an odd number* 
»— 2«2 — 5 
Many more arithmetical theorems may be deduced from the 
fluents of thefe and other fluxions by fimilar methods, which 
cannot, without fome difficulty, be found from the common 
methods of finding the fums of feries. 
This method of finding approximations to the areas and 
lengths of curves, fluents of fluxions, and fums of feries, &c. of 
which the equations, to their increments, fluxions, &c. are given 
from the areas of curves, fluents, &c. of which the equations 
to their increments, &c. differ by very fmall quantities from 
the given equations was publifhed in the Meditat. Analyt. 
near twenty years ago. 
I fhall conclude this Paper with two theorems of fome little 
life in the dodtrine of chances. 
THEOREM 
"H = a + bxa + b~ 1 . a + b -2 . a + b~ 3 ... a + b — n+ 1 
»*— ci • ci i . # — 2 • • ci n ! -\-n . a . ci — i • ci — 2 • • ci n -|- 2 
X b n • X ci • a 1 u & — 2, * * , d - — n -f- ^ X b • b — ■ 1 -}- 71 * 
71 — 1 n— 2 
+ n 
a . a — 1 . a — 2 . • a *~n -f 4 x ^ X ^ ~ 1 . />- 2 + 
ti— 1 n — -2 n — 3 
n-l - f 1 
a-3 .. .a-n + l+i xb »b - 1 . b-z ... b r l+ 1+ .. .+■ 
You LXXXI. ‘ . A a - - . ■ 
* « e 
