VOL. LXXXIV.] PHILOSOPHICAL TRANSACTIONS. 385 



It is observable that the semiarc of vibration bo = b, does not enter into these 

 expressions for the time of a semivibration ; if therefore, instead of the semi-arc 

 BO, an arc of any other length lo, terminating in the point of quiescence o, 

 fig. 13, should be substituted in the preceding investigation, the time of describ- 



(ID 7} iC 



ingr LO would be still =^-7- or >>/——- tt equal to the time of describinfr the 



° 8/f 8/f X 180° T ° 



other semi-arc bo ; consequently, whether the balance vibrates in the largest or 

 smallest arcs, the times of vibration will be the same. 



Since watches and time-keepers are usually adjusted to mean time, when the 

 balance makes 5 vibrations in a second, the time of a semivibration will in this 

 case = -rV part of a second: the substitution of -^ for t being made in the pre- 

 ceding equation, the force which accelerates the circumference of the balance, 

 when at any given angular distance c° from the quiescent position, will be deter- 

 mined for all time-keepers adjusted to mean time, in which the balances make 

 5 vibrations in a second. Suppose the given angle c° = 90°; then making c = 

 gO°, p = 3.14159, / = 193, t = -J^, the accelerative force at the angular dis- 

 tance from quiescence 90 or f = -r^ — — -^ = r X 1 .00408926. We have 

 therefore arrived at the following conclusion : if the radius of the balance be 

 equal to 1 inch, and the time-keeper be adjusted to mean time when the balance 

 makes 5 vibrations in a second, the force which accelerates the circumference 

 of the balance at the distance of gO° from the quiescent position, is = 1 .OO4O8926, 

 the accelerative force of gravity being =: l . And if the radius of the balance 

 be greater or less than 1 inch, the force by which the circumference is accelerated 

 at the distance of 00° from quiescence, will be greater or less than 1 .OO408926 

 in proportion to the radii. 



According to the principles assumed in the preceding solution, the spring's 

 elastic force is supposed to vary in the proportion of the angular distances from 



Kendalj on Mr. Harrison's principles, and is the instrument which Capt. Cook took out with him 

 during his last voyage to the South Seas. The results are as below : 



Diameter of the balance 2r = 2| inches. 



Weight of the balance, and parts which vibrate with it w = 4-2 grains. 



Weight applied to the circumference of the balance, which counterpoises the force 

 of the spiral spring when the balance is wound through an angle of 180° 48 grains. 



The weight which counterpoises tlie spring's force when the balance is wound to 

 90 degrees from quiescence is p = 2i grains. 



I = 193 inches; p = 3.14159. 



These determinations give die following substitutions in the expression for the time of a semivibra- 



v/p'r 



parts of a second. 



tion t = V^ = 0.099-t 



32p« 



The balance, when adjusted to mean time, makes 5 vibrations in a second; tlie actual 

 time of a semivibration is therefore 0.1000 



Difference between the actual time and the time by the calculation only O.OOO6 



— Orig. 



VOL. XVII. 3 D 



