Low Frequency and their Resonance. 421 



Let the torsional force be as ordinarily assumed proportional 

 to angle of displacement and the frictional resistance to angu- 

 lar velocity. An impulse having set the pendulum in motion 

 it is required to describe the motion. The differential equa- 

 tion of motion is obtained by writing down the symbolical 

 statement of the principle of moments, viz : 



Rate at which the moment of moraen- ) ( Moment of all the 

 turn about the line of suspension > = -j forces about the 

 varies ) ( same line. 



That is 



d( dd\ dd /ja 



-jA l dt) = a j t +' st> •••••• < l > 



I df +a J t + ^ = ° < 2 > 



Certain well known conditions being fulfilled the following 

 integral is readily obtained : 



a 



d = Ae 2I sin — t (3) 



where T = natural period of the pendulum = 



The arbitrary constant A depends on the energy of the im- 

 pulse and can be easily determined by well known rules. 



When jp is small in comparison to y then 



T = 2 Vlj (*) 



that is, the natural period of the pendulum is independent of 

 the frictional resistance. 



I venture to discuss briefly this rather familiar mechanical 

 problem ; for, the discussion seems to throw a strong light 

 upon some of the electrical problems which form the subject 

 of this paper. 



Let Tj= natural period calculated by (3) 

 T 2 = " " " " (4) 



By a simple transformation it is easily shown that 



T, = T,(1 + JL .*-.... +..)'.. (5) 



