M. G. Wiedemann on Torsion. Iu3 



also be of especial importance for the theory of torsion oscil- 

 lations &c* 



21. Torsion Oscillations. — Only after the study of the static 

 conditions into which a body is brought by alternately directed 

 forces, does it appear possible to discuss with more exactitude 

 what takes place in its elastic oscillations, since evidently pre- 

 cisely the same phenomena must prevail in these as in the 

 former — only with this difference, that in them the temporal 

 course of the elastic afteraction is not attended to. 



As according to the preceding investigations a wire twisted 

 hither and thither behaves quite otherwise on being twisted 

 toward one side of the existing position of equilibrium than on 

 being twisted toward the other, the hypothesis according to 

 which the diminution of the amplitudes of oscillation in tor- 

 sion oscillations, for example, is to be referred to an internal 

 friction which, for the entire course of the oscillations, is a 

 function of the velocity, can no longer be maintained. Just 

 as little can the elastic afteraction alone suffice for the expla- 

 nation if, for example, it be assumed that the force impelling 



d 2 x 

 the wire which undergoes an elongation as, namely -^- 2 , 



does not depend simply on x (for instance is not = ax), but, 

 since the wire at the time t has not yet attained the position 

 of equilibrium, is equal to a(x—p), p being a function of x and 



t (for example, -j- = — ap)f . 



The chief cause of the diminution of the vibration-ampli- 

 tudes lies rather, according to the above observations, in the 

 shifting of the position of permanent torsion at the end of each 

 oscillation, beside which the slow alteration of it in the course 

 of time, the elastic afteraction, plays a secondary part. Let 

 us first neglect the latter, and consider a wire "accommodated" 

 by frequent oscillations hither and thither within certain limits, 

 the procedure at the alternating oscillations may be figured in 

 the simplest form in the following manner : — 



In the absolute first position of rest of the wire, before any 



* In reference to the theories on the elastic afteraction compare espe- 

 cially, besides those cited mother places in this memoir, the following : — 

 O. E. Meyer's papers in Pogg. Ann. cli. p. 168 (1874), and Borchardt's 

 Journ. lxxvii. p. 130, lxxx. p. 315 ; Neesen, Pogg. Ann. cliii. p. 498 

 (1874) ; Warburg, Wied. Ann. iv. p. 232 (1878); Boltzrnann, Wiener 

 Ber. lxx. (Oct. 1874) ; Wied. Ann. v. p. 430 (1878); and the reflections 

 of Butcher (based upon the views of Clerk Maxwell), Proc. Lond. Math. 

 Soc. viii. pp. 110-112 (1878), Wied. JBeibl. ii. p. 625. The former papers 

 at the same time treat the causes of the diminution of the amplitudes of 

 vibration of bodies, e. g. in torsion oscillations. 



f Compare Neesen, Berl. Monatsber. 1874, Feb. 12. 



