154 Mr • Atwood's Investigations for determining 
plied to ascertain what alteration of the daily rate is occasioned 
in consequence of any deviation from coincidence of the points 
of quiescence of the balance spring and auxiliary springs, (fig. 
8, and fig. 4 and 5.) that may have taken place either through 
a casual derangement of their position, or from a purposed ad- 
justment. This inquiry offers three cases, which are to be se- 
parately considered; the first is, when the points of quiescence 
of the auxiliary springs coincide with the point of quiescence 
of the balance spring. 2dly. When the point of quiescence Q 
of the auxiliary spring u is situated in the first semiarc of the 
vibration, or between B and O (fig. 4 and 8.). 3dly. When Q 
is situated in the latter semiarc of vibration between E and O 
(fig. 5.) ; the like cotemporary position is to be understood in 
respect of the other auxiliary spring. 
The consideration of these cases requires that the accelera- 
tive* forces of the balance spring and auxiliary springs, when 
at a given tension, should be known. This point will not be 
difficult to ascertain, if the proportion of the spring s forces at 
the same tension is given. By information received from Mr. 
Mudge, it appears that the force of each auxiliary spring is 
about J- part of the force of the balance spring. Assuming this 
proportion, let the points of quiescence of the auxiliary springs 
be supposed to coincide with the point of quiescence of the 
balance spring, and let the radius of the balance be taken at 
one inch ; the watch being adjusted to mean time, when the 
balance makes five vibrations in a second, if the force of gra- 
vity is _ 1} the force which accelerates the circumference at an 
* The accelerative force being always proportional to the absolute forces of impulse 
(as measured or counterpoised by a weight) divided by the mass moved equivalent to the 
resistance of inertia, the latter quantity remaining the same in the case of a balance s 
vibration, the accelerative forces will be in the same ratio with the forces of impulse. 
