342 NEW SERIES for the 



Example IV. The reciprocal of Napier's logarithm of 

 10, (which is the modulus of the common fyftem), calculated by 

 the fecond feries for logarithms. (See Art. 63.). 



x = 19, and hence X = 5.05 X IV =3 1.010 373 154 20. . 



X' = T.739 252 713 092 7 X v = 1.002 589 934 6. . . 



X' = 1.170 310 367 614 6 X TI zz 1. 000 647 274 .. . . 



X'"= 1.041 707 820748. X VI = 1. 000 161 805 ... . 



\ x —iy ~ * 123 4s6 79 ° I23 5 



tt = -083 333 333 333 3 



Sum of pofitive terms, .206 790 123456 8 

 1 X'— 1 



4 4 X'+i 



1 X" — 1 



zz .016 867 116475 8 

 = .001 226 137 760 6 



4 ' X # + 1 



.000 079 796 518 o 



I X"—! _ 



V X" + 1 



JL X' v — 1 = >000 005 038 882 6 

 45 x iv + 1 



1 X v — 1 



4°"X V + 1 



— .000 000 315 745 2 



4? 1^+1 = ,0 °° °°° ° 19 74<5 9 



T Y VI1 I 



_ _ : zz: .000 000 001 2 34 4. 



4 8x TII 4-i **•* 



R > .0000000000822,97 Tj ««««~,-n™««Q 



-.'•< ^ > R ==: .000 000 000 082 3 



R <. 0000000000823, 1 J 



Sum of negative terms, .018 178 426 445 8 



Difference of the pofitive -> i = . l88 6lI 697 01I 



and negative terms, j log 10 



—1-- =.434294481903. 

 Example 



