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XXI. Note on Damped Vibrations. 

 By H. S. RoWELL *. 



IT is well known that the space time curve for free un- 

 damped vibrations may be derived from the projection 

 of a rotating vector, the end of which describes a circle, 

 and it is fairly well known that for vibrations which are 

 resisted by fluid friction proportional to the velocity, the 

 space time curve may be projected (as remarked by P. G. Tait) 

 from a rotating vector, the end of which describes an equi- 

 augular or logarithmic spiral. 



The vibration of bodies when resisted by a constant 

 frictional force — say solid friction — is of great importance 

 in practical work and does not appear to have been adequately 

 treated. The results obtainable are, moreover, in themselves 

 of much interest. 



If F is the constant force of friction the equation of 

 motion is 



m'x+c 2 x±¥ — 0, 



wherein the sign of F depends on the direction of motion. 



Substituting a? = X+ F/c 2 , we have 



F 

 X = x + -j = A cos 



\\/m ' 



which gives a series of harmonic vibrations about alternating 

 centres distant F/c 2 from the equilibrium position when 

 * Communicated by the Author. 



