the Times of Vibration of Watch Balances. 161 
acting on the balance while it describes the arc from N to k, is 
merely a mechanical expedient for supplying the power which 
is lost by friction, and as it is thus wholly employed and coun- 
teracted, it appears requisite, in making calculations of the 
times in which the balance vibrates, that the entire forces of 
the springs are to be taken into the calculation in the same 
manner as if friction did not exist, and no mechanical contri- 
vance was necessary for compensating the loss of motion in 
consequence of it. But since, in point of fact, no retarding 
force acts on the balance while it describes the arc N O, ac- 
cording to the conditions under consideration, it may be satis- 
factory to calculate the time of a semivibration in the arc 
B O on this ground also, that is, on a supposition that the 
entire semiarc B O is described by the joint accelerative forces 
of the balance spring, and the auxiliary spring u only ; it will 
appear that this slight variation of the conditions does not 
at all affect the conclusions deduced from the preceding calcula- 
tions, and very little alters the results themselves. 
The time of describing the semiarc BO, on these conditions,, 
will be determined by referring to the theorem (page 141% 
and making the following notations. 
a = 1.5707 963 = an arc of 90° to radius = 1 
b == 2. 356 1945 = an arc of 135 0 
c = 2.3736478 = an aYc of 136° 
d = 0.0174533 = an arc of a°. 
/= 0.9562754, 
n — To' 
The time of a semivibration is = ■*/ ~ z a x into a cir- 
V If XH + i 
MDCCXCIV. 
Y 
