on the rate of chro?iometers . 41 1 
than unity ; or, which is the same thing, we may suppose 
the elastic force of the spring to vary in a less ratio than 
that of the angular distances from the point of quiescence ; 
and which hypothesis will produce different results, accord- 
ing to the values assigned to a ' ; for if the time-keeper be 
placed in condensed air, so as to make a 1 less than a, a. posi- 
tive value will be given to the function, or the chronometer 
william. If, on the contrary, the time-keeper be placed in 
rarefied air, so as to make a' greater than a , still preserving 
the magnitude of n, the numerical value of the formula will 
be negative, and the chronometer lose. Cases to illustrate 
both suppositions are recorded among the experiments. 
Thirdly, we may suppose the elastic force of the spring to 
vary in a greater ratio than the angular distances from the 
point of quiescence, and in which case n must be greater 
than unity. If then we suppose the chronometer to be 
placed in condensed air, the value of a ' becoming in such a 
case less than a , the numerical value of the function will be 
negative , and a retardation of rate will be the necessary 
result. But if the time-keeper be placed in rarefied air, so 
as to make a' greater than a, preserving the same value of 
n, the numerical result of the formula v ill be positive, 
and the time-keeper must gain. And this, according to 
the foregoing experiments, is by far the most general con- 
dition of chronometers. It may also be inferred from the 
same circumstance, that time-keepers are more frequently 
constructed with the elastic forces of their springs, varying 
in a greater ratio than the angular distances from the point 
of quiescence, than the contrary. ** 
The preceding suppositions will therefore explain why 
