166 Dr. E. Warburg on the Deadening of the Sounds 



very accurately represented by a geometrical series — a law of 

 decrease which Gauss and Weber had already observed for thin 

 metal and silk threads, and which by the author's experiments 

 is extended to caoutchouc threads 1 sq. nnllim. in section. When 

 the exponent of the series (whose logarithm is the logarithmic de- 

 crement) was suitably deduced from a Dumber of observed deflec- 

 tions, and by its means another number of deflections was calcu- 

 lated, the differences of the observed and calculated values were 

 throughout within the limits of possible errors of observation, 

 which, corresponding to one division of the scale, amounted to 

 -j4g of the smallest and ? }g- of the greatest elongation. 



Hence the motion of the system is represented by the formula 



w = A . e~ et co$ nt, 

 in which 



x is the elongation from the position of equilibrium in angular 



degrees, 

 A the elongation at the time £ = 0, 

 n the number of vibrations in the time 2tt, 

 € is a magnitude which is a constant for each experiment. 



! In this formula the magnitude e is inversely proportional to 



the time within which the amplitude is reduced from a to — a, 



and is the measure of the deadening. The law above-mentioned, 

 as is known, presupposes a deadening force proportional and 

 opposite to the velocity ; the measure of the deadening force referred 

 to the unit of velocity is the product eM, in which M is the 

 moment of inertia. Now, as in these experiments the length 

 only of the threads was changed, the balance with its belongings 

 being always the same, the moment of inertia was sensibly con- 

 stant in all the experiments. Hence the magnitude e may be 

 considered a measure of the actual deadening, as well as a mea- 

 sure of the deadening force. 



The relative magnitude of the deadening force (to ascertain 

 which was the object of these experiments) might be deduced by 

 directly observing the time within which the amplitude of a is 



reduced to - a. That magnitude may more certainly be deduced 



from the logarithmic decrement by dividing it by the duration of 

 vibration. In this manner the author determined the magni- 

 tude e. 



It was the first aim of the author to investigate the dependence 

 of the deadening on the duration of vibration ; and with this 

 view observations were first made with the space filled with air. 

 These experiments gave the following result : — For caoutchouc 



