Bappicker—On the Influence of Magnetism on the Rate of a Chronometer. 31 
artificial magnet or that of the earth; in the latter case the situation which 
the balance will try to regain will be that in which it would repose if detached 
and freely suspended. 
In order to explain why ‘similar changes in the are of vibration phontd be 
frequently attended in different chronometers with opposite alterations of rate,” 
Harvey says, “that it may be questioned, if ever a chronometer existed in which 
the vibrations of the balance were perfectly isochronous, or, in other words, in 
which the adjustments of the spiral spring were such as to admit of its elastic 
force varying precisely with the arcs of vibration.” And, according to a Paper of 
Atwood in the Philosophical Transactions for 1794, even the most minute and 
most imperceptible deviations from perfect isochronism may be sufficient to 
produce a sensible alteration of rate in twenty-four hours. Now it is obvious 
that “the elastic force of the spring may vary either in a less ratio than the 
angular distances from the point of quiescence, or in a greater,” and this 
accounts for the apparent anomalies alluded to. 
This is made evident by a discussion of Atwood’s formula for the daily 
aberration of a time-keeper with changes of the are of vibration— 
{G4 
where @ is the primitive are of vibration, a that produced by the disturbing 
force, and n the exponent dependent upon the peculiar ratio existing between 
the elastic force of the spring and the angular distances from the point of 
quiescence. Thus, if @ = const., and 
1. » <1 (or the elastic force varying in a /ess ratio than the angular distances), 
the function will become positive, or the time-keeper will gain; if the ares of 
vibration are shortened by external influence, or a becomes less than a, the 
function will become negative, or the time-keeper will dose, if a’ becomes greater 
than @ (or the ares are lengthened). 
2. > 1 (or the elastic force varying in a greater ratio than the angular 
distances), we shall have in the corresponding cases— 
ad <a, a negative value, or retardation ; 
a’ >a, a positive value, or acceleration. 
Thus the ‘varieties of imperfect isochronism” account for the different 
effects of the same disturbing force upon different instruments. 
The above deduction is a concise mathematical illustration of Barlow’s 
remark.(p. 16): its value lies in its showing that and why it will be impossible 
to give a general law for the effects of magnetism upon a chronometer. For 
this reason it is to be regretted that this Paper of Harvey’s has completely 
escaped the attention of all the following experimentalists. 
