162 Mr. A. Stephenson on the Maintenance of 



where X is a positive quantity proportional to the velocity- 

 rate of change of the f rictional force at the value correspond- 

 ing to the relative velocity in equilibrium, v say. The position 

 of equilibrium is therefore unstable if X is greater than k, 

 and any slight disturbance is secularly magnified in free 

 period. 



The equation continues to represent the motion for larger 

 amplitudes if X is taken to represent the mean velocity-rate 

 of change in the f rictional force between and k. So long 

 as x is less than w, Xis positive ; but when there is no relative 

 motion it is indeterminate, having any value from that given 

 by continuity down to a large negative limit. The motion 

 therefore increases until x reaches the limit v, if k is small 

 enough, and the steady motion is determined by the fact 

 that x — v when 



x = (X — /c)v/n 2 , 



where X has the greatest statical value, which is proportional 

 to the quotient by v of the difference of the coefficients of 

 friction at the relative velocities zero and v. Starting from 

 this configuration the motion is approximately simple and of 

 slowly increasing amplitude, so that the velocity v is again 

 attained when x has a negative value numerically greater 

 than that given above : the static friction then prevents any 

 further change ot velocity until the initial state is reached. 



The magnifying action of solid friction was observed by 

 "W. Froude in the case of a pendulum suspended from a 

 rotating shaft *. 



2. The instability of the position of equilibrium under 

 friction holds evidently when the system has more than one 

 degree of freedom. The general problem of determining the 

 steady motion would be difficult and of little interest, but 

 happily in the important case of the violin string simplicity 

 is introduced through the infinitude of freedom, and a possible 

 steady motion may readily be determined f . 



In the absence of damping any free motion is possible in 

 which the bowed point has no relative motion with the bow 

 and constant speed against it, the mean position being that 

 produced by a constant force equal to the friction in the 

 backward swing. This is evident from the fact that the 

 maintenance of the static state at the point of contact during 

 the forward swing necessitates the constancy of the friction 

 throughout the whole motion. 



* Cited by Lord Rayleigh, ' Sound,' i. § 138. 



t The kinematics of the motion were long ago established by Helm- 

 holtz in his well known experimental examination. 



