f> K N 



w probable, however, that the parallax of a star of the second magnitude 

 is not more than of a second ; and of a star of the sixth magnitude, 

 iiot more than or of that quantity. 



PENDULUMS, oscillation of, %c.(Woof], Whev-cU, &>: 

 1. Let T ~ time of vibration of a simple pendulum iu a cycloidal arc, 

 L length, F = accelerating force, g = force of gravity 3aV 6 feet, 

 x =. 3.14159, &c., n number of vibrations in a given time T', then 



71 '= ij 



" , or in case of gravity T = \- --^~- 



I jp 'JVg / o- 1'/2 



and n V , T , or in case of gravity n V ft r . 



* L IT* Li 



Cor. Hence if x = space fallen through by gravity in 1" in any lati- 

 tude, and L ~ length of a seconds pendulum, then if .v be given, L 



; and if L b given, x = -^- . 



5T2 



By help of this last formula x is found more exactly than can be done 

 by direct experiment. In the latitude of London L 39.126 inches, 

 hence x = 16.09 feet. 



2. To find the vibration of a pendulum in a circular arc, let a ver. 

 sin. of % arc of vibration, r radius of the circle ; then 



t- x , + _<L. + + & , =) whm a or 



the arc is very small, V time of vibration in a cycloid. Hence 



the formulae above given are applicable to bodies vibrating in very small 

 circular arcs. 



3. If a pendulum vibrating in a circular arc keeps true time whilst os- 

 cillating through degrees on each side of the vertical ; then when it os- 

 cillates through D degrees, the seconds lost in 24 hours, if D is greater 

 than 5, 



or if D is less than J, time gained 



