the Times of Vibration of Watch Balances. 163 
this case, the balance making 5 vibrations in a second when 
adjusted to mean time, the daily rate is = o (p. 146). 
Secondly, suppose the arms G O and I O to be so affixed to 
the axes TR, FS, (fig. 8 and 4.) that instead of just touching 
the rods L M and N W when quiescent, they are inclined* to 
that position at any angle O G Q = O I N : the situation of 
the pallets p and q not being altered : in this case the point of 
quiescence of the auxiliary spring u, will be separated from the 
point of quiescence of the balance spring by the angular dis- 
tance OGQ = OCQ. In dike manner the point of quiescence 
of the auxiliary spring v will deviate from the point of quies- 
cence of the balance spring by the angular distance O IN = 
O C N. Suppose that the arc GO — ON is = i° : the point 
of quiescence Q of the auxiliary spring u is in the first se- 
miarc of vibration, that is, between B and O, (fig. 4.) while 
the balance vibrates from B to E ; and the point of quiescence 
of the auxiliary spring v, is in the first semiarc of vibra- 
tion between E and O, while the balance vibrates from 
E to B ; on these conditions, the semiarcs of vibration B O 
and OE being 135 0 , the daily rate of the time-keeper will be 
19".40 slower than mean time, (p. 157) and if the semiarcs BO, 
OE, should be diminished from 135 0 to 125°, the daily rate 
will be 2,0". qsz slow (p. 158) ; the retardation of the daily rate 
in consequence of the diminished semiarc of vibration being 
To consider the remaining case ; if the arms GO, I O, 
(fig. 8.) are adjusted so as to press equally by the force of the 
auxiliary springs against the rods L M, W Z, when quiescent, 
the equal and contrary pressures prevent any apparent effect 
* The force of the main spring not being supposed to act on the balance wheel, or 
pallets^ andg. 
