422 M. I. Pupin — Electrical Oscillations of 



where r = —a 



2 



L 



= ratio (approx.) of f fictional loss during any half period 

 to the amplitude of the kinetic energy during the same 

 half period. I shall call it the dissipation ratio. 



It follows therefore that whenever the dissipation ratio is 

 smaller than ^ then T„ differs from T 1 by less than -^ of one 

 per cent. But since on the other hand 



R It 



e = e 



It follows that when the dissipation ratio r = \ then the 

 pendulum will be practically reduced to rest after 16 com- 

 plete oscillations. This simple calculation shows, therefore, 

 that even in very damped oscillations the period can and in 

 most cases will he practically independent of the frictional 

 resistance. 



The following observations are too well understood to need 

 a mathematical commentary : — a. If a periodically varying 

 force is applied to a torsional pendulum the oscillations will be 

 free oscillations if the period of the force is the same as the 

 natural period of the pendulum, that is if the force and the 

 pendulum are in resonance to each other. When this resonance 

 does not exist the oscillations are forced. 



b. Of two periodically varying forces of the same mean 

 intensity the one which is in resonance with the pendulum will 

 produce the largest maximum elongation. The maximum 

 elongation is reached when the work done by the resonant 

 force during a complete period is equal to the frictional losses 

 during that time. 



c. The torsional force of the suspension varies periodically, 

 its period being the same as that of the impressed resonant 

 force, but differing from it in phase by a quarter of a period. 

 The amplitude of the torsional force can be much larger than 

 the amplitude of the impressed force, especially when the 

 frictional resistances are small, the moment of inertia large 

 and the oscillations rapid, that is the torsional coefficient 

 large. For in this case that part of the work of the impressed 

 force which is stored up in the kinetic energy of the pendu- 

 lum will become large before the maximum elongation has 

 been reached. But since this large kinetic energy has to be 

 stored up in the potential energy of the torsional forces once 

 during each half oscillation it is evident that a large torsional 

 force will be called into action. The amplitude of the tor- 



