the Times of Vibration of Watch Balances. 167 
than the larger arcs of vibration : and secondly, from the posi- 
tion of the points of quiescence of the auxiliary springs. But it 
is evident from the preceding considerations, that although the 
balance spring should not be isochronal, yet the regularity of the 
time-keeper will not be at all affected, however the points of qui- 
escence of the auxiliary springs may be situated in respect of the 
point of quiescence of the balance spring, as long as the semiarc 
of vibration continues unchanged ; and if the semiarc of vibra- 
tion should be liable to increase or diminution, Mr. Mudge's 
construction affords an effectual remedy against this cause of va- 
riation in the rate, since the arms projecting from the auxiliary 
springs may be so adjusted, as to place their points of quiescence 
either in the first or latter semiarcs of vibration, according as 
the balance spring, when acting singly, causes the shorter or 
longer vibrations to be described in the least time. 
On revising; the preceding pages a few observations have occurred, which may be 
here inserted. 
Note to page 122, line 13. — The elastic force of the spiral spring when at a given 
tension to turn the balance, is here assumed to be the same, whether the balance is at 
rest or in motion; being in both case$ equal to the weight by which the spring’s force 
at the given tension would be counterpoised. 
Note to page 131. — It is not necessary to add constant quantities to the fluents of 
— x —y 
the fluxions , ■ t — r ; because when the entire fluents 
y/ b n + — X r ‘ + y/ C n + — y" T 1 
are taken; they are precisely in the same proportion, whether the constant quantities 
(or corrections as they are sometimes termed) are added or omitted. 
Addition to the note in page 140. — If the points of quiescence are in the first semi- 
arc of vibration, and c — o, the point B will coincide with Q_(figt 4.), from which point 
the vibration will commence ; in this case the expression for the time of a semivibra- 
— — — x an arc of which the sine is — -, or an arc of qo° — 
2 If d y 
'ap 
; which agrees entirely with the solution in page 126 ; for in 
this case the auxiliary spring not acting on the balance while it describes the arc Q _0 N, 
the balance will vibrate by the force of the balance spring only ; of which the force at 
the distance a or 9o°is —f, and consequently by the theorem investigated in page 126, 
tion will become t — . 
