214 Proceedings of Royal Society of Edinburgh. [sess. 
down and instantaneously re-form. The loss of energy in oscilla- 
tions of the system is due to the rupture of such groups. 
If the assumption that the breaking limit of distortion is evenly 
distributed among groups be made, the theory indicates that the 
relation y (x-{- a) = b holds between range of oscillation and number 
of oscillations. This is the relation which was found to hold under 
great fatigue. Thus the effect of fatigue may be to produce this 
even distribution. In this case Hooke’s Law is not obeyed — a 
term involving the square of the distortion appearing ; so that, in 
an outward oscillation, the distortion is an elliptic function of the 
time. The theoretical law is shown to accord very closely with 
the results of observation. 
In the second half of the inward motion, according to the 
theory, the inward acceleration is less than that at the same stage 
in the outward motion by a constant amount. It is always less 
at any stage. Thus we deduce the experimentally found relation 
between the times of out and in motions over a given range. 
Further, if the wire, after its first positive distortion, be stopped 
just short of the zero and be again distorted in the positive 
direction, the theory shows that, while the same law connecting 
distortion with distorting force holds, the co-efficient of the term 
which involves the square of the distortion is greatly reduced. 
Thus Hooke’s Law is much more nearly obeyed in the second 
distortion. This is in complete agreement with Wiedemann’s 
observations {Phil. Mag., Jan. 1880). On the other hand, if the 
zero be passed, the wire is reduced to its original condition. 
If the assumption that the number of groups which have a 
definite breaking distortion is constant for all distortions be 
replaced by the assumption that it is proportional to a power of 
the distortion, the term, in the expression for the distorting force, 
which represents the deviation from Hooke’s Law involves a 
power of the distortion different from the second. Thus the 
assumption, made in the first paper on the subject, that the defect 
of the potential energy, from the value that it would have in 
accordance with Hooke’s Law is proportional to a power of the 
distortion, is deduced as a consequence of a certain, uneven, distri- 
bution of number of groups in respect to the extreme limit of 
sustainable distortion. 
