4io Mr. Harvey on the effect of the density of air 
change in the arc of vibration, has been demonstrated by 
Mr. Atwood to be the following function. 
in which a denotes the primitive arc of vibration, a' the 
arc resulting from the action of a disturbing force, and 
which, in addition to the case assumed by the abovementioned 
philosopher, may be regarded as either greater than a , or 
less than it ; and n, the exponent, denoting the ratio be- 
tween the elastic force of the spring, and the angular dis- 
tances from the point of quiescence. 
Supposing the primitive arc constant, the above function 
will undergo different modifications, according to the values 
assigned to the elements a ' and n. 
In the first place, we may conceive the elastic force of the 
spring to vary directly with the angular distances from the 
point of quiescence, the exponent n being, in this case, de- 
noted by unity. This supposition, by causing the whole 
function to vanish, will indicate a perfect isochronism ; and 
that therefore, whatever magnitude be attributed to the arc 
a f , in consequence of the action either of rarefied or con- 
densed air, no alteration of rate will result. And it is re- 
markable that, during the whole course of experiments, I 
have not met with an instance to illustrate this case. Every 
chronometer examined exhibited a tendency either to gain 
or lose, when the density of the air was changed ; and 
which therefore proves, that perfect isochronism is seldom 
if ever attained in the construction of a time-keeper. 
Secondly, we may assign' to the exponent n, a value less 
