of Logarithmic Tables. 
22 
much less than the preceding factor for let ~j=i — 
— 1- — and p, and p/ being small fractions ; the new 
factor (IS( I ~+8') = S= 1 —(^—1^') near] y : and consequently 
differs less from unity than the factor which was last before. 
4. These equations, being put into logarithms, give a series 
of converging expressions for the logarithm of x. We have 
successively, 
1st. L . a? == L (x-\-n) -f L (^) 
2d. L . oc = L(a? “f* ti} -j— L ( oc 11 ) — L ( oc «-!■= n 11 ! ) -J- 
T { x z +(n+n')x \ # 
\x x -\-(n-\-ri)x+nn j 9 
but before we put the third equation into logarithms, it will 
be better to simplify it ; one of the most obvious ways of 
doing which is to make n^n'=n ,J ; then 
3d. L.x=L(a:+«)4=L(a;+?i / )-J-L(a:-f 2^ ,/ )— L(x-f-;^4-w // ) 
— L(jr+» / +wO 
+ r / - i* + 3 n"x z -j-(nri + znf^x \ 
U ’ 3 + 3 »v + {nil 1 + 2 n" z )x ] ' 
This may be still farther simplified by making n=n', con- 
sequently n"=z 2 n, then 
L . x — 2L . (x + n) — 2L (a: -f- 3n) -j- L (x -f* 4«) 
, J I X 3 -\-6nx z + gn z x \ 
‘ \x 3 -\-bnx*--t-gn z x +^n i j 
If now we change n into — i, and x into x-\-2, we shall fall 
upon the elegant formula of Mr. Bqrda.* 
* If any one shall attempt to calculate a Table of logarithms by means of diffe- 
rences, each logarithm being got, not from the next, but from the next but one 
below it, he will fall upon the series of Borda: for A 3 . L(x — 1) L . (;r + 2) 
— 3 L 0 r + 0 + 3L . x—L(x— 1); A 3 - L(x — 2)— L(#+ 1)— 3L.2 + ^L(x— 1)— L(x— 2), 
by adding which, a 3 - L (x— 1) + A 3 . LCr— 2)— L f ( x — l\ ) which gives the 
\(x—z){x+i) i ) 
series we are speaking of. This remark will be exemplified in one of the following 
Propositions. 
