IMPERIAL STANDARD TROY POUND WEIGHT. 
491 
The formula (3) becomes, after having put for q its value b (3, 
R 3 r3 
M = m-j-mb(3 — — m b (3—, 
which, it must be remembered, is only an approximate formula, the exact formula 
being 
T?3 ^ 
M = ?n + M& |S — — m b (3 Y- 
R 3 
In general the logarithm of mb (3— will be identical with the logarithm of 
R3 
M b (3 -£■ when we use logarithms with five decimals, which give even more accuracy 
than the weighings can pretend to : but should log M (M being found by the first 
equation) differ in the fifth decimal from log m, we must use the value of M obtained 
R 3 
by the first equation and put it in the latter, in the term M b (3 — , in order to obtain 
a result for M as exact as may be found with logarithms of five decimals. 
Reduction of the weighings of K. 
U supposed of Brass. 
log m = 3 - 76042 
log b = 1*47197 
(Table I.) log a = 4*70566 
c = 0*00033 
9*93838 
mb 6— = 0*86772 
K A 
log m = 3*76042 
log b = 1*47197 
log a = 4*70566 
9*93805 
mb a — 0*86706 
- m5a = 0*86772— 0*86706= +0*00066 
m =5760*03389 
M=5760*03455 
The logarithm of M is 3*76042, the same as that of 
m, so that it is not necessary to repeat the calculation 
with log M. 
U supposed of Copper. 
, fl R s , f log m = 3*76042 
mb 1 — as before ° 
A log b =1*47197 
= 0*86772 (Table n.) log (3 = 5*60829 
(Table IV.) log r* = 0*00041 
log4-= 9*05611 
0 
9*89720 
mbf 3 y = 0*78922 
mb 3— - mbS^- — 0*86772—0*78922= +0*07850 
m = 5760*03389 
M = 5760*11239 nearly. 
The logarithm of M is 3*76043, being by one unity 
in the fifth decimal greater than log m. Therefore, by 
using log M instead of log m, we obtain 
log M5/3^ = 9*93839 Mb 3^ = 0*86774 
■R3 r 3 
Mb 3— - mb3l~ =0*86774-0*78922= +0*07852 
A o 
m = 5760*03389 
Correct value ofM = 5760*11241 
3 R 2 
