384 PHILOSOPHICAL TEANSACTIONS. [aNNO 1794. 



surface describe in one second of time, the velocity of the circumference, when 

 the extremity a of the index ca has arrived at the point h, will be = 



a ^ 



Let t represent the time in which the circumference describes the arc bh ; then 

 will i = v/-^ X ^ (p _ j .n '> 3"d i = ^— X into a circular arc, of which the 

 cosine =: - to radius = 1, which is the time of describing the arc eh expressed 

 in parts of a second ; when x = 0, that is when the circumference has described 

 the entire semi-arc bo, the circular arc of which the cosine = - is a quadrant of 

 a circle to radius = 1. Let p = 3.1415g. The time t, of describing the semi- 

 arc BO = ^-x ^- -Z,-^. 



In this expression for the time of a semi-vlbration, the letter a denotes the 

 length of the arc od, (fig. 13) ; if this arc should be expressed by a number of 



degrees c°, a will then = '^ ; and this quantity being substituted for a, the time 

 of a semi-vibration will be t = \/ ^'\^^o ? if instead of f, its value — be sub- 



o/F X loL) ^S 



stituted in the equation t = ^ s/f i8Q° ' ^'^^ ''""^ °^ ^ semi-vibration will be 



8pW X ISO" 



Let the given arc c° be = QO ; in this case t = v^-^^ . These are expres- 

 sions for the time of a semivibration, whatever may be the figure of the balance, 

 the other conditions remaining the same as they have been above stated. If the 

 balance should be a cylindrical plate, it is known that the distance of the centre 

 of gyration from the axis is to the radius as 1 io \/ 1 •, therefore in this case 



g^ = -ir ; and the time of a semi-vibration, or t =y'^^-'.* 



* The balances of watches are usually of such a form as to place the centre of gyration nearly at 

 the same distance from the axis, as if the figures were cylindrical plates of uniform thickness and 

 density. If it should be required to obtain from theory the time of a balance's vibration precisely 

 exact, it would be necessary to calculate rigidl) tiie position of the centre of gyration from the dimen- 

 sions of each part of the balance, and whatever vibrates with it. But in cases merely illusti-ative of 

 the general theorems for ascertaining the times of vibration, it is unnecessary to enter into prolix and 

 troublesome calculations depending on the form of any particular balance ; since by assuming it as a 

 cylindrical plate, the time of a vibration will not differ materially from that which would be the re- 

 sult of the correct investigation. 



Being desirous of comparing the time of vibration, as deduced from the theory of inolion, witli the 

 actual vibration of a watch balance, I requested Mr. Eanisliaw, the excellent performance of whose 

 time-keepers is well known, to make the experiments from which the necessary data for this calcula- 

 tion are derived. These experiments were made on the biilance of a watch constructed by Mr. 



