REDUCTION OF PENDULUM OBSERVATIONS. 
123 
for one oscillation. The following are the results by the 
different methods : 
s 
I. 
0.00000177 
II. 
173 
III. 
176 
IV. 
172 
The near agreement of the first with the others is due to 
the small amplitude of the arc. Now it is evident that 
when the corrections differ so immaterially as in these four 
cases that method should be chosen which requires the least 
labor. The first one, or that of Borda, is therefore to be 
recommended for differential gravity work. 
The coefficient of expansion of brass is about 18 millionths 
for 1° centigrade, and since the time of oscillation varies as 
the square root of the length of the pendulum, the time of 
one oscillation will be changed by 9 millionths per degree. 
Hence it appears that when the range of temperature is 
not more than one-tenth of a degree and where therefore we 
can assume to know the mean temperature of the pendulum 
within this amount, we should not on account of erroneous 
temperature expect to find discrepancies of more than one 
or two units in the sixth place. But an examination of the 
Lick observatory observations shows that we do actually 
find differences several times as large where the temperature 
changes during a swing much less than 0°.l. Then, ad¬ 
mitting that the coefficient of expansion (not that derived 
from linear measures, but that deduced from oscillation); 
admitting that this is correct to two places, we must turn 
elsewhere for an explanation of the apparent discrepancies. 
On one of the mountain stations where the daily range of 
temperature was 11° in the pendulum house, we found differ¬ 
ences of 40. This would require, temperature alone being 
considered, an uncertainty of at least 2° in the mean tem¬ 
perature of the pendulum for any one swing, which is 
entirely inadmissible. Therefore we are forced to the con¬ 
clusion that, although the temperature has usually been 
14—Bull. Phil. Soc., Wash., Vol. 11. 
