140 Mr Sang on the Manner in which 



On supposing A — <p exceedingly minute, the value of the 

 above time becomes w sj — 1 —~ : but the time of the cycloidal 



g sec <p J 



oscillation, were there no friction, would be sr *J — , so that if v 

 denote the true time of a cycloidal oscillation, we have 



t, = t v cos | i +{\y{ sin ^^y + etc } 



In the very same way we would find 



T 2 = r \/ cosp j 1+ ( - ) ( sin——-— ) + etc i Or generally 



T fi== ,V^{l + (i) 2 (si n ^^L.^) 2 + etc} . 



The method of determining the duration of a beat of the re- 

 versing pendulum, is to draw it aside, and, allowing it to oscil- 

 late freely, to contrast its motions with that of a pendulum kept 

 going by clock work : the rate of the clock being known, and 

 the entire time X, + T 2 + .... T„ of a given number of oscillations 



X + X + X ♦ 

 of the experimental pendulum, the average — 1 2 ' n gives 



the first approximation to the duration of the beat. Denoting 

 this average by T, we have 



_ r „ + ( 1 ) {( sm __J )+ (sm ._L_if)} 



+ ( 2A ) \ ( Sm ~T- ) + ( Sm 2^ ) J 



+ etc 



And, to complete the investigation, it only remains to obtain the 

 sums of the series indicated in the second member of the equa- 

 tion, in such a form as to give a ready computation. 



In all practical cases, the value of <p is exceedingly small, and 

 the number n very large ; so that q> being regarded as the dif- 

 ferential of ^, and nq> being the difference between £A and |A«, 



A 

 2 



we have 



i(A-AJT 



-f- etc . 



