the Times of Vibration of Watch Balances. 127 
These are expressions for the time of a semivibration, whatever 
may be the figure of the balance, the other conditions remaining 
the same as they have been above stated. If the balance should 
be a cylindrical plate, it is known that the distance of the centre 
of gyration from the axis is to the radius as 1 to v/i~; where- 
fore in this case g 1 = -T; and the time of a semivibration, or t~ 
• The balances of watches are usually of such a form as to place the centre of gy- 
ration nearly at the same distance from the axis, as if the figures were cylindrical 
plates of uniform thickness and density. If it should be required to obtain from 
theory the time of a balance’s vibration precisely exact, it would be necessary to 
calculate rigidly the position of the centre of gyration from the dimensions of each 
part of the balance, and whatever vibrates with it. But in cases merely illustrative of 
the general theorems for ascertaining the times of vibration, it is unnecessary to enter 
into prolix and troublesome calculations depending on the form of any particular 
balance ; since by assuming it as a cylindrical plate, the time of a vibration will not 
differ materially from that which would be the result of the correct investigation. 
Being desirous of comparing the time of vibration, as deduced from the theory of 
motion, with the actual vibration of a watch balance, I requested Mr. Earns h aw (the 
excellent performance of whose time-keepers is well known) to make the experiments 
from which the necessary data for this calculation are derived. These experiments were 
made on the balance of a watch constructed by Mr. Kendal, on Mr. Harrison’s 
principles, and is the instrument which Captain' Cook took out with him during his 
last voyage to the South Seas. The results are underneath : 
Diameter of the balance - m 2 x inches. 
Weight of the balance, and parts which vibrate with it 42 grains. 
Weight applied to the circumference of the balance, which counterpoises 
the force of the spiral spring when the balance is wound through an angle 
of,8 °° 48 grams. 
The weight which counterpoises the spring’s force when the balance is 
wound to 90 degrees from quiescence is - . - 24 grains. 
These determinations give the following substitutions in the expression for the 
time of a semivibration l~ ^ / w P 3 r 
x 31P r 
