VOL. LXXXIV.] PHILOSOPHICAL TRANSACTIONS. 387 



and the time of describing the semiarc lo, will be the entire fluent of this 

 fluxion, while y decreases from c to O. Though the fluents of tbe fluxions 

 ——. ; — r and — - — ; — - — r-T caunot be expressed in general terms, yet 



the exact proportion of the said fluents may be assigned, which will be the pro- 

 portion of the times in which the balance vibrates in the 2 semi-arcs bo, io; the 



multiplying quantity ^— — tt-^- being common to both fluxions; and since the 



entire fluent of — - — ; — r— r is to the entire fluent of — - — ■ — ; — - as 



^(6" + I — ^r" + •) V(c''+ ■ — y''+ ') 



\ — n \ — n 



h ^ is to c * , it follows, that the time of a semivibration in the arc bo is to 



I — a 



• IO, 2 



the time of a semivibration in the arc lo, as i ^ toe * . or as 1 to ( — ") 



Suppose a watch to be adjusted to mean time when the semi-arc of the balance's 

 vibration is = bo, and let this semiarc be afterwards diminished to lo; the time 

 shown by the watch in any given portion of mean time I, when the semi-arc of 



vibration is lo, will be =r < X ( — ) ; and if t be put = 24^, the alteration of 

 the daily rate, in consequence of the diminution of the semiarc of vibration from 



I 



/BOv 



BO to IO, will be 241^ X ((^) - 1).* 



To apply this proposition, let a case be assumed: suppose a watch to be regu- 

 lated to mean time when the semi-arc of vibration is 135°, and let this semiarc be 

 diminished 8°, so as to become 127°; let the ratio of the spring's elastic force 

 deviate from that of the distances from the quiescent position by a small differ- 

 ence of -T-oTTTr P^rt, so that the spring's force shall be in the -jW-o- power of the 

 distances, instead of in the entire ratio of the said distances from the quiescent 

 position. The alteration in the daily rate of the watch will be obtained from the 

 preceding theorem by making the following substitutions, bo = 133°, lo = 



* From this general expression it appears, that when n = 1, that is, when the spring's elastic force 

 varies in the precise ratio of the angular distances of the balance from tlie quiescent position, the al- 

 teration of the daily rate in consequence of a diminution of the arc of vibration is = 0. Whenever 

 therefore it is found, by observing the rate of a time-keeper, that a diminution of the arc of the ba- 

 lance's vibration causes an acceleration of tlie daily rate, it is necessary to conclude, that the elastic 

 force of tire spring in this case varies in a ratio less than that of the distances from the quiescent posi- 

 tion. In like manner, when a diminution of the arc of vibration causes a retardation of the rate, it 

 is certain that the spring's elastic force varies in a higher ratio than that of tlie distances from quies- 

 cence. It appears indeed, from some experiments, that the weights which counterpoise a spiral 

 spring's elastic force, when wound to different distances from the quiescent position, are in the ratio 

 of those distances ; but it is shown from this proposition, and the annexed table, that the differences 

 between the weights, by which the ratio of the distances, and a ratio a little less is indicated, though 

 far too small to be discoverable by experiment, are yet sufficient to create a material alteration of the 

 (iaily rate, — Orig. 



3 ]J 2 



