TORSIONAL OSCILLATIONS OF WIRES. 441 



When, in the inward motion, the zero is reached, every group which has broken 

 breaks and re-forms into its initial condition, so that the oscillation proceeds, as formerly, 

 on the other side of the zero, but with less initial energy, — so giving rise to the lessening 

 of amplitude. 



Now, as a given increase of maximum range decreases the inward force at any stage 

 of the inward motion more and more as that range is greater, the time of inward motion 

 increases when the range increases. But the form of (3) shows that the time of outward 

 motion is less when the range of oscillation is small than when it is large. Therefore 

 the period of complete oscillation is greater for large oscillations than for small. This 

 was shown in the first paper. Kupffer pointed it out first in 1853. 



The result that the zero of oscillation is a point at which groups re-form into their 

 orioinal condition explains the fact of the constancy of that zero which was found to 

 obtain as oscillations proceed (see Second Paper). 



The expression (7) vanishes when 



/ , 5N V / 5N \ 2 aW 



This is, according to the theory, the relation which connects the angle of set with the 

 angle of maximum twist, provided that the former does not exceed half the latter, 

 and provided also that v is constant — a condition which seems to hold, as we have 

 seen, when the wire is greatly fatigued. This equation represents an ellipse whose 

 semi-axes have a ratio of about 13 to 10, and would imply that the wire would flow round 

 under the action of continued stress when the set equalled about ten-thirteenths of 

 the distortion, if we could apply the equation to sets beyond half distortion (see Note). 

 If the inward motion were stopped just short of the zero, and the wire were then 

 given an outward motion, the conditions differ from those in the first outward motion. 

 When the angle reaches a value \|/-, equation (6) gives the inward force due to unbroken 

 groups if <p be replaced by \|/-. With the same substitution, (5) represents the outward 

 pull due to groups which broke first between \J/- and 0. So also, \J/- being substituted 

 for d, (1) gives the inward pull due to groups which broke between and \J/-. Hence, 

 the expression in (1) being referred also to distance a from the axis, the total inward 

 force in this case is 



^J,Na 2 (aif;) - ±7r£ra 2 (a 2 ^ 2 ) 2.i - *■*»** (« 2 ^ 2 ) ( 9 ) 



This differs from the expression (7) in the multiplier of the middle term. The 

 value of 2.i is very closely 5/3 and that of fi is closely 2/3. 



1 ! * 



The expressions (3) and (9) have identical values when ^ = 0, after which, the 

 angle 6 not being exceeded, the inward motion again obeys the law of force given 



VOL. XXXIX. PART II. (NO. 14). 3 X 



