ISOCHRONOUS OSCILLATIONS. 



we get from this 



e 

 n 



-- arc tan 



If we express the constants of the motion as a function of the 

 time of oscillations r, of the logarithmic decrement A, and of the 

 initial angle of elongation a v which are quantities directly ob- 

 servable, we get 



(20) 



7 = 



7T / A 2 



V^rV. 1 ** 



- arc tan 



7T A. 



When the damping is not rapid, the ratio is small, and we replace 



7T 



the exponential in the value of w by the first terms of its develop- 

 ment in a series. We obtain thus a simpler expression, which 

 could be arrived at directly, observing that between the first angle 

 of deviation a 1 and the third a 3 there have been four semi-oscillations, 

 during each of which the deviation has undergone virtually the same 



loss as during the first. This loss is therefore 



By adding it 



to the first deflection we shall have virtually the deflection which 

 would have been obtained without damping. We may then write 



(21) 



Conversely, if we find experimentally that the successive ampli- 

 tudes of a vibratory motion decrease in geometrical progression, we 

 may conclude that some retarding force is at work proportional to 

 the velocity. 



Borda, for instance, had found that the amplitudes of the small 

 oscillations of a pendulum in air decrease slowly in geometrical 



