142 Mr. Atwood’s bivestigations for determining 
are coincident ; the auxiliary spring commencing its action 
from the extremity B of the arc BO, (fig. 6.) continues to 
accelerate the balance till it arrives at the quiescent posi- 
tion O, at which point the action of the auxiliary spring en- 
tirely ceases ; on this account it is plain that another auxi- 
liary spring, equal and similar to the former, having also the 
point of quiescence coincident with O, must be applied to act 
by retardation on the balance while it describes the arc O E, 
in order that the times of describing the arcs BO and OE, as 
well as these arcs themselves, may be equal. According, there- 
fore, to this disposition of the auxiliary springs, the balance 
will describe each semiarc of its vibrations precisely in the 
same manner as if it was impelled by one spiral spring only, 
the strength of which is equal to that of the balance and 
either auxiliary spring, when wound to the same tension. 
Suppose the balance to vibrate from B to E, and, for the sake 
of distinction, let the auxiliary spring which accelerates the 
balance from the extremity of the arc B, in the direction 
B O, be called u ; and let the other auxiliary * spring which 
retards the balance in the semiarc O E be denoted by v ; con- 
sequently, when the balance vibrates in the contrary direc- 
tion from E to B, the auxiliary spring v will accelerate the ba- 
lance from E to O, and the other auxiliary spring u will retard 
it from O to B. In respect, therefore, to the spring u , B O is 
the first semiarc, and O E is the latter semiarc of vibration ; 
and on a similar principle in respect to the spring v, EO is 
the first semiarc, and OB is the latter semiarc of vibration. 
* The circular arcs which are drawn interior to the circumference of the balance in 
the figures 4, 5, and 6, are intended to represent those portions of the balance’s vibra- 
tion in which the auxiliary springs respectively act. 
