140 Mr Sang on the Manner in which 



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

 above time becomes tt ^ : but the time of the cvcloidal 



g sec ip J 



oscillation, vi'eie there no friction, would hev ,J -, so that if ^r 



9 



denote the true time of a cycloidal oscillation, we have 

 T, = T sJ'Z^ I 1+ (2) (sin^^)' +etc} 

 In the very same way we would find 



T^= r V cos^ 1 1+ ( _ ) ( sin^=^— )' + etcj Or generally 



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 T, + T„ + . . . . T„ of a given number of oscillations 



of the experimental pendulum, the average — ^ — '-^-^ — - gives 



the first approximation to the duration of the beat. Denoting 

 this average by T, we have 



__,= ,+ (_) |(s,,__)+ (,„ 2—^)1 



+(2:4) |(^i»-^)+ (^"^ — 4 — -)\ 



+ 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 ail practical cases, the value of <p is exceedingly small, and 

 the number n very large ; so that <p being regarded as the dif- 

 ferential of -^j and n<p being the difference between ^A and ^A«, 

 we have 



*-^^4<-\H(^)"{/(-t)'4-/(s4)''l} 

 -(^)*(/(-l)*'^/(''4")''l} 



+ etc . 



