120 
PRESTON. 
mathematical and physical conditions of the case, leaves 
nothing to be desired in point of accuracy. Its use, however, 
in differential measurements of gravity is hardly necessary, 
as the accuracy attained in the final result is far from being 
commensurate with the additional labor entailed. In fact, 
for differences of gravity, and where the descent of arc is 
not abnormal, the results are practically identical by the 
two methods. This is especially true for modern observa¬ 
tions, because the accuracy in the construction of the instru¬ 
ments makes it possible to swing in arcs whose half ampli¬ 
tude does not much exceed one degree; so that the second 
term of the development is almost insensible. In order to 
compare the different methods, some observations made in 
1887 at the Lick observatory have been reduced in four ways, 
and it will be seen that the corrected time of one oscillation 
is practically the same for all. Any method is accurate 
enough for differential gravity until we can better control 
the temperature and clock rates. 
The following are the four ways : 
I (Borda). Assuming the arcs to decrease in a geometrical 
ratio, the quantity to be added to n oscillations in order to 
get the number made in an infinitely small arc, during the 
same time, is given by— 
M n sin {p + <p') sin (pp — <p') 
32 log sin p — log sin ip' 
where p and p' are the initial and final arcs, and M the 
modulus of the common logarithmic system. When the 
arc falls from about 40' to 5' this is equivalent to a correc¬ 
tion of 0 S .‘00000177 for one oscillation. 
II (Peirce). Taking the equation 
dp , 2 
- di = -b<p-e<p 
to represent the relation between the arc and time, the con¬ 
stants are first determined, and then the correction for arc ? 
by integrating the expression YtJ'p 2 dt By limiting the 
differential equation at the second power of the arc, the 
