338 Expression for the Vibration of a simple Pendulum, 



marked that the setting sun had a very unusual appearance, 

 being; as he described it, of a pale dirty yellow : and a letter 

 which I have received from Cromer in Norfolk, mentions on 

 that evening " a very beautiful setting sun, quite out of the 

 common way ;" but it is not stated in what its peculiarity con- 

 sisted. 



The latitude of Che'sfield Lodge is 51° 56' 15", and its 

 longitude about 43 seconds in time west from Greenwich. 



I am, Sir, your obedient servant, 

 Chesfield Lodge, Oct. 14, 1828. Henry Kater. 



LVIII. Expression for the time of Vibration of a simple Pen- 

 dulum in a Circidar Arc. By A Correspondent. 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 

 T TAKE the liberty of inclosing you an expression for the 

 -*■ time of vibration in a circular arc, which does not appear 

 to have been noticed by any writer on Dynamics. 



I am, Gentlemen, your humble servant, 



J. W. L. 



Let \J/ be the angle which the pendulum makes with the 

 vertical, a. the arc through which it moves, ;• the radius of 

 the circle, g the force of gravity, t the time ; then 



dt = xA 



d^ 



2g js/cos^P — cos a 



Let 1 —cos 4/ == (1 — cos a) sin 2 cr 



at- FL - === / 



*/ S n/ (l-cos«) . , V 



'J 



— - — ■ smV v 1— sin 9 - surV 



y sin 4 4~ + cos 4 4-+ 2sin 3 4-cos 2 ~ cos 2cr 

 4 4 4 4 



or, 



1 + tan 4 4- + 2tan 2 4" cos 2 <r 

 4 4 



^1 + cot 4 4-+ 2 cot 2 -4- cos2cr 

 4 4 



Expanding these series by the known methods, and integra- 

 ting 



