26 MEASUREMENT OF OSCILLATIONS. 



time T which Would correspond to oscillations without any retarding 

 force. The ratio of the two times is 



T n 



The period t of any elongation defined by the equation satisfies 

 the ratio 



y y 

 sin t=-== = t. 



Substituting this value in equation (16), we obtain for the angle 

 of deviation 



As the times of the elongations increase in arithmetical progression, 

 it follows that the deflections diminish as the terms of a geometrical 

 progression, the rule of which is e~* T . If we call the successive 

 deflections a v a 2 . . . , a n , we have 



The amplitudes, each of which is equal to the arithmetical sum 

 of two successive angles of deflection, vary according to the same 

 law. If we represent by a lt a 2 ---- , a n the successive amplitudes, 

 and put 



we shall have 



A-/. - /,- 



# 2 2 # 3 n - 



The number X is called by Gauss the logarithmic decrement of the 

 oscillations. 



Observing that the first angle of deflection a x corresponds to the 

 time / p given by the least root of equation (17), 



y y 



7/ x = arc tan - or c/j = arc tan - ; 



