246 Mr. H. Tomlinson on some 



to which, at equal distances from the wire, are two cylinders 

 of equal mass and dimensions, and the whole system be set 

 in torsional oscillation, the amplitude of the vibrations will be 

 found to diminish more or less gradually until finally rest 

 ensues. This diminution of amplitude is almost entirely due 

 to two causes, namely, the friction of the air, and the internal 

 friction of the metal. The internal friction of the wire may 

 be measured by the logarithmic decrement of arc for a single 



vibration, or by — - — , where A is the initial arc, and 



B is the arc after n vibrations. By the aid of Prof. G. Gr. 

 Stokes's mathematical formulae *, and an experimental deter- 

 mination by myself f of the coefficient of viscosity of air, I 

 have been able to eliminate the resistance of the air, and to 

 compute the damping effect due to the internal friction of the 

 metal. Where the deformations produced are sufficiently 

 small, I have proved the following laws respecting the loga- 

 rithmic decrement of arc { : — 



1. It is independent of the amplitude ; 



2. It is independent of the vibration-period. 



These laws only hold good when the wire has been allowed 

 to rest for a considerable time after any change has been 

 made in the arrangements, and when there has been a large 

 number of oscillations executed previously to the actual 

 testing. 



What is the nature of this so-called internal friction of the 

 metal ? It cannot resemble fluid friction ; because for such 

 velocities as we have here the friction of fluids is proportional 

 to the velocity. Neither can it resemble altogether the ex- 

 ternal friction of solids ; because the latter is not nearly so 

 independent of the velocity as is the internal friction, nor 

 would the logarithmic decrement be independent of the am- 

 plitude. Some experiments by Prof. G. Wiedemann§ throw 

 light on the subject. Let A B represent the original posi- 

 tion of equilibrium of the axis of the bar to which the wire is 

 attached, and let a torsional couple be applied so as to bring 

 the bar to A l Bj. On reducing the torsional stress gradually 

 to zero the bar will not come back to A B , but remain in a 

 new position A 2 B 2 , however small may have been the angle 

 of torsion A l A . Again, if the bar be twisted by an equal 

 torsional stress in the opposite direction to A 3 B 3 and the stress 

 be then reduced to zero, the bar will remain permanently 



* Camb. Phil. Soc. Trans, vol. ix. no. x. (1850). 

 t Phil. Trans. 1886. 



X « The Internal Friction of Metals," Phil. Trans. 1886. 

 § Wiedemann's Annalen, 1879, No. 4, vol. vi. ; Phil. Mag. January 

 and February 1880. 



