44 MEASUREMENT OF OSCILLATIONS. 



Let n be the order of the initial elongation in the second series, 

 or the difference of the orders of two corresponding elongations. 

 We know whether this number is even or odd, and the approximate 

 value of the time of oscillations deduced from the first series enables 

 us to determine it without possible error. 



The difference of the times of corresponding elongations in the 

 two series is successively 



We obtain thus five values 



n\ 2 2 / n\ 2 2 / 



of the time of oscillation relative to the mean period comprised 

 between the two series. This time might be reduced to that of 

 infinitely small vibrations by making the correction for the angle of 



deviation in the times - , - in each series, or by introducing 



into the mean duration -( --- -) that of the mean of the 



n\ 2 2 / 



elongations wriich correspond to each other in the two series. 



* The reduction of each time of vibration to infinitely small angles 

 makes the calculations longer, but it has the advantage of furnishing 

 a continuous control of the observations. 



697. The logarithmic decrement deduced from the initial 

 amplitude a, and the amplitude b of the nih oscillation, which follows 



i a 



the preceding one, is A. = - /. - . 

 n b 



If we consider the first amplitude a as known exactly, and that 

 the probable absolute error is the same for all the following, we may 

 choose that number n of oscillations which gives the value of A with 

 the closest approximation. If d\ is the error corresponding to an 

 error db in the last amplitude , we have 



dX_ db 



A. , ci 



