the Times of Vibration of Watch Balances. 129 
mean time when the balance makes 5 vibrations in a second, 
the time of a semivibration will in this case = part of a se- 
cond : the substitution of T ~ for t being made in the preced- 
ing equation, the force which accelerates the circumference of 
the balance, when at any given angular distance c° from the 
quiescent position, will be determined for all time-keepers ad- 
justed to mean time, in which the balances make 5 vibrations 
in a second. Suppose the given angle c°— go° ; then making 
c = g o°, p = 3.14159, &c. / = 193, t ~ the accelerative 
force at the angular distance from quiescence go° or F = 
8 ~~~ x l8o o = r x 1.0040892b. We have therefore arrived at 
the following conclusion : if the radius of the balance is equal 
to 1 inch, and the time-keeper is adjusted to mean time when 
the balance makes 5 vibrations in a second, the force which 
accelerates the circumference of the balance at the distance of 
90° from the quiescent position, is = 1.0040892b, the accele- 
rative force of gravity being = 1. And if the radius of the 
balance is greater or less than 1 inch, the force by which the 
circumference is accelerated at the distance of go° from quies- 
cence, will be greater or less than 1.0040892b in proportion 
to the radii. 
According to the principles assumed in the preceding solu- 
tion, the spring's elastic force is supposed to vary in the pro- 
portion of the angular distances from the quiescent position, 
and on this condition, the vibrations are shewn to be isochro- 
nous, whether they are performed in longer or shorter arcs ; 
but if the spring's elastic force at different distances from 
quiescence should not be precisely in the ratio here assumed, 
the longer and shorter arcs may be described in times differing 
in any proportions of inequality. If, for instance, the spring's 
MDCCXCIV. S 
