'336 NEW SERIES for the 



6 {x — i) 4 8.9.10 



1 _ / _, _J_ X+I2X' — 13 I X'+ I2X'~ 13 



lo g 4 * ^ 3-i6 2 X + 4 X'+ 3 "*" 3.16 3 X' + 4 X^F3~ 



j_ X-+12X-+13 

 L ^ 3.16 4 X// + 4 X'" + 3 ^' 



The terms of this feries approach continually to thofe of a 

 geometrical feries, of which the common ratio is j- : and 

 hence it follows, that the fum of all the terms after any align- 

 ed term, approaches the nearer to ^- of that term, according 

 as it is more advanced in the feries. 



6$. Besides the foregoing, our method furnifhes yet another 

 kind of expreilion for the logarithm of a number, namely, a 

 product confuting of an infinite number of factors, which ap- 

 proach continually to unity. Such an expreflion may be invef- 

 tigated as follows. From the identical equation 



X— i = {X* — 1) (X* + i). 



Let there be formed the feries of equations 



X I — 2 (X 2 i) ■ > 



2{X- i) — 4 (x 4 i) ^ I > 



A J- vT±T 



f- V 



2 



™ (x™ - 



2 K 



1 

 1 — 



"1 1 — 

 v f m >. X -f- I 



- 1) z= m {x — 1) ! — , 



2 



here 



