﻿Torsional Oscillations of Wires. 51 



If, in equation (4), we put ra = 2, equation (5) takes the form 



0=e o e- 



which expresses the above law. This law therefore follows 

 from our theory if we suppose that n decreases in value to 

 the limit zero ; and the theory shows that the decrement of 

 energy per swing then follows the compound-interest law — 

 just as the decrement of angle does. 



Explanation of After-action. 



When a wire is held in a state of torsion under a constant 

 couple, some of the less stable molecular groups will in time 

 break down, and so the strain slowly increases. If it be held 

 in a given state of strain, this gradual rupture of groups 

 necessitates a slow diminution of the couple. On the removal 

 of the couple, the wire remains in a state of internal stress 

 because of the set. Consequently the gradual rupture of 

 groups produces a slow diminution of set ; for the strongest 

 groups remain unbroken in the original deformation, and, in 

 any ordinary experiment, the groups which break form a 

 small fraction of the whole. This is Maxwell's explanation. 



The after-action takes place with comparative rapidity at 

 first: afterwards it goes on more slowly. It takes place more 

 and more completely the longer the strain is continued, and 

 requires proportionally longer maintenance of an equal reverse 

 strain to undo it. Hence, if a wire be twisted first to the 

 right through a given angle for a long time, then to the left 

 through an equal angle for a short time, and be then gradually 

 put into the position of set, we should expect that the set 

 would change (as it does) first in the sense of recovery from 

 the second strain, and finally in the sense of recovery from 

 the first strain. 



Conditions of Maximum and of Zero Resilience. 

 From equation (3) we obtain 



rl~V 

 - L -^ =-k0 + mp0 m - 1 , 



or, say, = -kd + k'6 n+1 , 



and -^ = -* + *'(n + l)^/ 



Hence we see that there is angle of maximum resilience 

 given by 



^=1(^1), • < 6 ) 



E2 



