344 



NEW SERIES for the 



Example VI. To ftiew that the feries investigated in this 



Paper are applicable in every cafe, whether the number whofe 



logarithm is required be large or fmall, let it be required to 



calculate the common logarithm of the large prime number 



1243 to feven decimal places, by the fecond feries, (Art. 63.). 



x — 1243 X" — I «4 2 356i48 



X = 621.50040225 X IV = 1.10080913 



X'= 17.64228446 X v = 1.0248925 . 



x *= 3-°53°5457 X VI = 1.006204. . 



=: .000 805 801 



XT = -083 333 333. 



Sum of pofitive terms, .084 139 134 



= - 55 794 8i3 

 = .007 914766 

 == .000682688 

 tz .000 046 861 

 =2 .000 003 001 

 = .000 000 189 



4 



2 X'+ 1 



1 



* 



— I 



4 



5 X" 



+ 1 



1 



x ff/ 



— I 



4 4 



x //; 



+ 1 



1 



X IV 



— I 



4 s 



X IV 



+ 1 



1 



X v 



— I 



4 6 



x v 



+ 1 



1 



X VI 



— I 



,ooo 000013 



47 X VI + 

 R > .000000012,6 

 R < .000000012 

 Sum of negative terms, .064 442 331 



= .019696 803 



:3*= 



Nap. log 2 x 



1 

 Nap. log x 



= .140 345 300 



Common log of 1243 = ^^345300 = 3^4 47"- ■ 



No. IX. 



