the Times of Vibration of Watch Balances, 141 
This solution is confined to that case in which the point of 
quiescence Q (fig. 4.) of the auxiliary spring is situated in the 
first semiarc of the vibration, that is, between B and Q. Ano- 
ther case still remains to be considered, which is, when the point 
of quiescence O of the auxiliary spring deviates from O by the 
given angular distance O Q, but is situated in the latter semi- 
arc of the vibration (fig. 5.), between O and E, instead of be- 
tween O and B, as in the former solution. According to this 
condition, making B Q = c, BO = b, and the other notation 
remaining as before, it appears from an investigation no ways 
differing from the preceding, that the time in which the ba- 
lance describes the semiarc B O will be 
t=xV 2 a ■ x into a circular arc, of which the sine is 
l /x n + 1 
\V b x n + - 1 - expressed in parts of a second. 
2 b + 2?IC r 1 
This result expresses the time in which the balance de- 
scribes the semiarc BO, (fig. 5.) by the accelerative force of 
two springs, namely, the balance spring, of which the point 
of quiescence is O, and an auxiliary spring, of which the 
point of quiescence is Q. On considering this case more fully, 
when applied to the actual vibrations of a balance, it will ap- 
pear evident, that the action of a third spring on the balance 
while it is describing the semiarc BO, must be taken into the 
calculation, in addition to the two springs already mentioned, 
in order to obtain a solution entirely correspondent with the 
circumstances of the case, when the points of quiescence of the 
auxiliary springs are situated in the latter semiarcs of the vi- 
brations. 
To state this more clearly, it is to be observed, that when 
the points of quiescence of the balance and auxiliary springs 
