Magnetism on Chronometers. 201] 
keeper, when the magnitude of the are of vibration is changed, and 
which is 
w{(5)F 
a 
where a denotes the primitive arc of vibration, a’ that produced 
by the action of a disturbing force; and which, according to the 
direction of its action, may be either greater or less than a; and x 
the exponent dependent on the peculiar ratio existing between the 
elastic force of the spring, and the angular distances from the 
point of quiescence. 
If we suppose the primitive are constant, and the other elements 
a’ and x of the formula variable, the entire function, as Mr. Atwood 
observes, will be susceptible of different modifications. Suppose, 
for example, we attribute to x a less value than unity *, a condition 
which corresponds to that of the elastic force of the spring, varying 
in a less ratio than the angular distances from the point of quiescence ; 
it is manifest, that different values will be communicated to the func- 
tion, according to the value assumed for a. If the supposition alluded 
to in an early part of the paper, of the attracting force passing through 
the thermometer-pieces be referred to, and in which the are of 
vibration would be shortened by its operation, the value of a’ must 
necessarily become less than a; and a positive value being thus 
communicated to the function, the time-keeper will gain. 
In the next place, if the attracting force be conceived, as in the 
second supposition, to pass through the centre of the balance and 
the limit of the semi-are of vibration, and which application will 
necessarily occasion a’ to become greater than a, the numerical 
value of the formula will be negative, and the chronometer will lose. 
If again we suppose n to be greater than unity, or the elastic 
force of the spring to vary in a greater ratio than that of the 
distances from the point of quiescence, the first of the preceding 
suppositions with respect to a’, will give to the function a negative 
value, indicating a retardation of rate in the time-keeper. 
* If wesuppose z = 1, the whole function will vanish, in indicating a per- 
fect isochronism ; so that whether the arcs of vibration be increased or dimi- 
nished by the action of a disturbing force, no alteration of rate will take place. 
