MATHEMATICAL SECTION. 91 
D,g=— 5 {(e + 8) — 400+ (9 +1) /@ FO dae} 
ie Se bY 43 ace’ 
aah ht oe gee tte 
The correction to the time of oscillation (4 De 2”) involves 
the logarithm of ¢ — e, and is not very simple in practical applica- 
tion. 
The second convenient method is the one by which the residuals 
in the last column of the table above given were calculated. In 
this the rate of diminution is supposed proportional to g1+”, n be- 
ing a proper fraction. Hence, 
¢ (t—e) =a, and ip 9 =——___,_ =-— 
(2—n) (t-e)n* 2—n 
This formula is very simple, and the table shows its agreement 
with observation to be fair for the larger amplitudes—those of chiet 
importance. In this calculation n = 4, e = —10716°, and a= 
89400. Better results would have been obtained by using a slightly 
smaller value of n, say 0.44; but in practice the nearest tenth or 
reciprocal of a whole number is sufficient. In reducing the obser- 
vations given by Prof. Oppolzer, n was taken equal to 0.28; but 
one of the residuals exceeded 1’, though two others were as high as 
0’.9. The observations at low pressures, above referred to, indi- 
cated a much smaller n. By using the value 0.04, however, the 
agreement of formula and observation was perfect. thus appears 
to be nearly proportional to the square root of the atmospheric 
pressure; but when very small, it may be supposed to vanish, and 
g” replaced by the logarithm of g. In this case e will of course 
be the time of unit-amplitude, instead of that of infinite amplitude 
as in former cases. 
No two observations of the diminution of amplitude of the same 
pendulum will in general be found to be copies of each other, for 
differences in atmospheric conditions and in friction on the support, 
imperceptible otherwise, will manifest themselves in a changed rate 
of diminution. Even in calculating the correction for different 
parts of one extended swing, it is advisable to adopt different values 
of one or other of the constants found. By so varying the quan- 
