PENDULUM. 



plication of this curve to the vibration of 

 pendulums designed for the measures of 

 time, the source of errors even greater 

 than those which by its peculiar proper- 

 ty it is intended to obviate, and it is now 

 not used. 



Although the times of vibration of a 

 pendulum in different arches be nearly 

 e'qual, yet if the arches differ very con- 

 siderably, the vibrations will be perform- 

 ed in different times, and the difference, 

 though very small, will become sensible 

 in the course of one day or more. In 

 clocks for astronomical purposes, the arc 

 of vibration must be accurately ascer- 

 tained, and if it be different from that 

 described by the pendulum, when the 

 clock keeps time, a correction must be 

 applied to the time shown by the clock. 

 This correction, expressed in seconds of 

 time, will be equal to the half of three 

 times the difference of the square of the 

 given arc, and of that of the arc de- 

 scribed by the pendulum when the' clock 

 keeps time, these arcs being expressed 

 in degrees ; and so much will the clock 

 gain or lose, according as the first of 

 these arches is less or greater than the 

 second. Thus, if a clock keeps true 

 time when the pendulum vibrates in an 

 arch of 3, it will lose 10 J seconds daily 

 in an arch of 4, and 24 seconds in an arc 

 of 5, for 423* Xj=7x| = 10i: 

 and generally P,2_A2 x 4 gives the 

 time lost or gained. See Simpson's Flux- 

 ions, vol. ii. prob. xxviii. 



In all that has been hitherto said, the 

 power of gravity has been supposed con- 

 stantly the same. But, if the said power 

 varies, the lengths of pendulums must 

 vary in the same proportion, in order 

 that they may vibrate in equal times ; for 

 we have shewn, that the ratio of the 

 times of vibration and descent through 

 half the lengths is given, and conse- 

 quently the times of vibration and de- 

 scent through the whole length is given ; 

 but the times of vibration are supposed 

 equal, therefore the times of descent 

 through the lengths of the pendulum are 

 equal. But bodies descending througli 

 unequal spaces, in equal times, are im- 

 pelled by powers that are as the spaces 

 described, that is, the powers of gravity 

 are as the lengths of the pendulums. 



Pendulums' length in latitude of Lon- 

 don, to swing 



Inches. 



Seconds 39.2 



^ Seconds 9.8 



A Seconds 2.45 



Length of Pendulums to vibrate Seconds 

 at every Fifth Degree of Latitude. 



Rule. To find the length of a pendu- 

 lum to make any number of vibrations, 

 and vice -versa. Call the pendulum, mak- 

 ing 60 vibrations the standard length ; 

 then say, as the square of the given num- 

 ber of vibrations is to the square of 60, so 

 is the length of the standard to the length 

 sought. If the length of the pendulum 

 be given, and the number of vibrations it 

 makes in a minute be required ; say, as 

 the given length is to the standard length, 

 so is the square of 60, its vibrations in a 

 minute, to the square of the number re- 

 quired. The square root of which will 

 be the number of vibrations made in a 

 minute. 



The greatest inconvenience attending 

 this most useful instrument is, that it is 

 constantly liable to an alteration of its 

 length, from the effects of heat and cold, 

 which very sensibly expand and con- 

 tract all metalline bodies. See HEAT, 

 PrnoMETEn, &c. 



To remedy this inconvenience, the 

 common method is by applying the bob 

 of the pendulum wich a screw ; so that it 

 may be at any time made longer or short- 

 er, according as the bob is screwed down- 

 wards or upwards, and thereby the time 

 of its vibrations kept always the same. 

 Again, if a glass or metalline tube, uniform 

 throughout, filled with quicksilver, and 

 58.8 inches long, were applied to a 

 clock, it would vibrate seconds for 39.2= 

 f of 58.8, and such a pendulum admits 

 of a twofold expansion and contraction, 

 viz. one of the metal, and the other of the 

 mercury, and these will be at the same 

 time contrary, and therefore will correct 

 each other. For by what we have shewn, 

 the metal will extend in length with heat, 

 and so the pendulum will vibrate slower 

 on that account. The mercury also will 

 expand with heat, and since by this ex- 



