482 ON THE SUMMATION OF SERIES, 
5 log. # dv= log. a (log. x —log. n-z 
oy A) 
1 i 1 1 1 1 1 1 
(iz iy Sees tae }s nad = is }+ &e., 
En I Hg PRES Se) AN Rand © ee mTOR aE Ye (16) 
This series is better adapted for computation, 
and converges much faster, than the series given 
by Professor De Morgan (see his Diff. and Inte- 
gral Calculus, pages 313, 314). 
If we caleulate | log. w dx, between the limits — 
y=l1, and x=20, we shall have as follows :— 
9 
20 
\ ~ log: a dx=s 
li l 
1 1 1 l 1 ] 
+79 GF -3)- 3034 % 0) &e., Xe. 
—24.8333086 —.2989185 +.0034090 —.0000017 
=24.5377974. which is true to the seventh place 
of decimals. 
1 
0 
. log. e—— (log. 20 —log. 11) 
: 2 
It will be seen that the above near approach to 
p 
20 
the actual value of | log. vdx has been obtained, 
1] 
by summing only three terms of the complement 
