330 
Mr V/atts’’ Remarks on Captain Eatery's 
These observations being premised, I shall now proceed to 
demonstrate the formula which ought to be employed for deter- 
mining the correction due to the amplitude of the arc of vibra- 
tion. And for this purpose I shall employ the following me- 
chanical principle, which has been confirmed by experiments ; 
that is to say, that when the pendulum performs its oscillations 
in air, the amplitude of the arcs of vibration decreases in geome- 
trical progression, while the times increase in arithmetical. 
If, therefore, we call I the length of the equivalent simple 
pendulum, t the time of one of its oscillations, w the ratio of the 
circumference to the diameter, and g the force of gravity repre- 
sented by double the space which a heavy body describes in the 
first second of its fall, then we shall have, by the theory of the 
pendulum. 
+ 
b / 1.3\^ / by 
Kyi) 
/1.3..5...(2ra— 1)\® / 6 Y 
V2.4.6 2 TO 
1 . 3.5 
■*" 2 . 4.6 
&c. 
-j~. . • 
When the arc of vibration is small, all the terms of the series 
after the second may be neglected, on account of their small- 
ness, and then we shall have, 
Let the length of the equivalent simple pendulum be repre- 
sented by unity, then the time of a complete oscillation will be, 
but h 
, very nearly, because 6 ^ 
24 "^720 
&c. a be- 
ing the arc of which h is the versed sine to radius 1 ; therefore. 
The second term of this value of t is the correction due to 
the amplitude of the arc of vibration. This correction varies 
with the arc described, when the pendulum vibrates in air ; and 
it is in this respect that the resistance of the medium has a small 
influence upon the duration of the oscillations. 
