the Times of Vibration of Watch Balances. 133 
nution of the semiarc of vibration from B O to I O, will be 
■— 1. 
H" * ■!£; 
To apply this proposition, let a case be assumed ; suppose 
a watch to be regulated to mean time when the semiarc 
of vibration is 135 0 , and let this semiarc be diminished 8°, 
so as to become 127°; let the ratio of the spring's elastic 
* From this general expression it appears, that when n — i , that is when the 
spring’s elastic force varies in the precise ratio of the angular distances of the balance 
from the quiescent position, the alteration of the daily rate in consequence of a dimi- 
nution of the arc of vibration is = o ; because in that .case 
BO 
IO 
2 — 1, and 
2 —1=0. When n is less than I, or when the force varies in a less ratio 
B O I 
I O [ 
than that of the distances from quiescence, the rate will be accelerated, because in that 
, t — n , 1 — n 
' d r\ I- . 
— 1, will be a positive quan- 
case 2 be greater than t ; and -y-— j 
tity : but when n is greater than i, or when the force varies in a ratio greater than that of 
the distances from quiescence, the rate will be retarded, because in that case 
will be less than i, and 
BO 
IO 
1 becomes negative. The converse of these pro- 
positions is likewise derived from the general theorem. 
Whenever therefore it is found, by observing the rate of a time-keeper, that a di- 
minution of the arc of the balance’s vibration causes an acceleration of the daily rate, 
it is necessary to conclude, that the elastic force of the spring in this case varies in a 
ratio less than that of the distances from the quiescent position. In like manner, when 
a diminution of the arc of vibration causes a retardation of the rate, it is certain that the 
spring’s elastic force varies in a higher ratio than that of the distances from quiescence. 
It appears, indeed, from some experiments, that the weights which counterpoise a spiral 
spring’s elastic fSrce, when wound to different distances from the quiescent position, 
are in the ratio of those distances ; but it is shewn from this proposition, and the an- 
nexed table, that the differences between the weights, by which the ratio of the dis- 
tances, and a ratio a little less is indicated, although far too small to be discoverable 
by experiment, are yet sufficient to create a material alteration of the daily rate. 
