Pendulums. 221 



i 



Suppose t' equal to 1", and that the arcs described on each 

 side of the vertical are 5. The versed sine of 5 is 0,0038053, 



radius being 1. Consequently, - = 0,0004757. With res- 



9h* 

 pect to the term ^, it is less than a unit of the sixth place. 



The error in each oscillation will, therefore, be 



i' = 1" X 0,0004757 = 0",00047i>7. 



Thus, if a body descend by the action of gravity along a circu- 

 lar curve, and describe arcs infinitely small on each side of the 

 lowest point in a second of time, the duration of each oscillation, 

 no allowance being made for friction or the resistance of the air, 

 would differ only 0",0004757 from that of an oscillation through 

 an arc of 5 on each side of the lowest point, so that in a day or 

 during 24 X 60 X 60 = 86400" vibrations, the difference would 

 amount to 86400 X 0",0004757 or 41". Thus a pendulum of 

 the length required to vibrate seconds, and performing its oscil- 

 lations through arcs of 5 on each side of a vertical, would lose 

 only 41" a day, when compared with one vibrating in arcs infi- 

 nitely small. 



If the arcs described on each side of the vertical were only 

 1, the versed sine of which is 0,0001 523, the daily loss would be 

 only 1",64, that is, If nearly, and for half a degree, the loss would 

 be 0",41 or | of a second daily. 



Of Pendulums. 



344. What we have said is particularly applicable to pendu- Fiff 16 

 lurns. By a pendulum, is to be understood a rod or thread sus- 

 pended at one extremity from a fixed point, and supporting 

 at the other extremity one or several bodies. It is called a sim- 

 ple pendulum when it is supposed to consist merely, of a mass or 

 weight sustained by a thread or rod without gravity, and when 

 at the same time this mass is of a diameter very small relative 

 to the length of the pendulum. We shall speak for the present 

 only of the simple pendulum. 



