the Times of Vibration of Watch Balances. 165 
diminished, an acceleration or retardation of the daily rate will 
take place^ according to the situation of the points of quies- 
cence of the auxiliary springs referred to the semiarcs of vi- 
bration in which these springs respectively act. If the points 
of quiescence of the auxiliary springs should be situated in 
the first semiarcs of vibration, a diminution of these arcs will 
cause a retardation of the rate ; but if the points of quiescence 
should be situated in the latter semiarcs of the respective vi- 
brations, the daily rate will be accelerated. 
By means of Mr. Mudge's construction, we may apply the 
principles deduced from the preceding investigations to correct 
such alterations in the daily rate as may arise from a dimi- 
nished or increased arc of vibration. For if it should be 
known, from any satisfactory mode of trial, that the properties* 
of the balance spring are such as cause the longer arcs of vi- 
bration to be described in less time than the shorter arcs ; 
whenever the arc of vibration is diminished, the time-keeper 
will lose ; and this error in the rate would be corrected by ad- 
justing the points of quiescence of the auxiliary springs in the 
-respective latter semiarcs of vibration at such a distance O N 
* Different opinions have been entertained respecting the times in which a balance, 
vibrating freely by the action of a spiral spring, describes the longer and shorter arcs of 
vibration. Mr. Harrison says, “ large arcs are naturally performed in less time than 
“ small ones.” — Notes taken at the Discovery of Mr. Harrison’s Time-keeper, p. vii. 
In this opinion he is followed by M. Berth oud ; “ J’ai appris par des experiences sures, 
“ que les grand arcs et les petits arcs d’un balancier ne sont pas isochrones, et qu’en 
“ general, dans un balancier libre, les grands arcs sont plus prompts que les petits.” 
Mr. Ludlam, in his report addressed to the commissioners of the board of longitude, 
and intituled, a Short View of the Improvements made or attempted in Mr. Harrison’s 
Watch, has the following remark : c( The principle on which Mr. Harrison forms 
“ the alteration of the third part (before described) is, that the longer vibrations of .a 
" balance moved by the same spring are performed in less time. This is contrary to 
“ the received opinion among philosophers and workmen.” 
These contradictory opinions may possibly have arisen from experiments, in which'. 
