386 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794. 



the quiescent position, and on this condition, the vibrations are shown to be iso- 

 chronous, whether they are performed in longer or shorter arcs; but if the 

 spring's elastic force at different distances from quiescence should not be precisely 

 in the ratio here assumed, the longer and shorter arcs may be described in times 

 differing in any proportions of inequality. If, for instance, the spring's force, 

 instead of varying in the ratio of the aforesaid distances, should vary in the 

 ■i°o°o''o power, or -fH-^ power of the distances, it does not ap[)ear from the pre- 

 ceding solution what alteration in the daily rate would be caused by this change 

 in the law of the force's variation, when the semi-arc of vibration is increased or 

 diminished by a given arc. To ascertain this point fully, other researches will 

 be necessary, by which it may be known, what alteration of the daily rate of a 

 time-keeper is occasioned by a given increase or diminution of the arc of vibra- 

 tion, when the spring's elastic force varies in a ratio of the distances from the 

 quiescent position, the general index or exponent of which is any number or 

 fraction n. 



The force which accelerates the balance being assumed in that power of the 

 distances, the exponent of which is n, let bo = Z; (fig. 15) be the semi-arc of 

 vibration when the time-keeper is adjusted to mean time; let do =. a; the acce- 

 lerating force on the circumference at the distance from quiescence od = f; sup- 

 pose the circumference to have described the arc bh from the extremity of the 

 arc b; and let ho = x: then the force by which the circumference is accelerated 

 when at the angular distance from the quiescent position oh = ^; let zi be the 

 space through which a body falls freely from rest by the acceleration of gravity, 

 to acquire the velocity of the circumference when it has described the arc bh; 

 the principles of acceleration give this equation : 

 u = ~ Y "*^ : taking the fluents while x decreases from b to x, u = ^ ' ' ~ ^■'^'' 



(" + 1) X a- ' 



and / being I93 inches, the velocity acquired by the circumference after describ- 

 ing BH, will be 



(« + 1 ) X a" ^ ' 



let T be the time of describing the arc bh ; therefore 



(« + 1) X a" - i 



The time of describing the arc bh will be the fluent of this fluxion, while x 

 decreases from b to x, and the time of describing the semi-arc bo will be the en- 

 tire fluent of the same, while x decreases from b to 0. 



Now let the balance commence its vibration from any other point i, and let 

 10 = c; suppose the circumference to have described the arc ik, and make ok 

 = 7/; let i be the time of describing the arc ik ; then by proceeding in the same 



manner as in the former case, it is found that t = </ — — 77 X —r, — 7-^ tt. 



