334 NEW SERIES for the 
we put - («+ =) = X, it follows, that CS == 
beret; I 
In like manner, putting * ie +=) = X’, and i(#-) 
4 xe 2 ¢ 4 
1 a 1 2 
w?—I x4—I 
u" x’ — I xe 
= X”, we have | — a= _ and 4 begee ee 
at I eh PP 
x? ty 
&c. Again, becaufe X'= : (#44). therefore 2X* = 
x? 
I 
I € +1) ERs (: 4 =) =X, therefore 2X" =X +4, 
2 in 2 x 
and X’ i — In like manner, it will appear, that 
yy = (eee &c. 
63. From the preceding inveftigation it appears, upon the 
whole, that our fecond feries for the calculation of a logarithm 
may be expreffed as follows. 
Purrinc «x for any number, let a feries of quantities X, X’, 
X’, X”, &c. be found fuch that 
Xai fet te x NATE ie &e. 
Then will 
log’ « 
