If, in equation (4), we put m=2, equation (5) takes the form 0=00€-at,

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 lawjust 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

Hence we see that there is angle of maximum resilience

As the torsion of a wire is increased, the set and the difference between the angle of torsion and the angle of set increase. This goes on until the angle is reached. As the twisting couple is further increased, the set increases at a greater rate than the torsion. The stronger configurations now break down, and the removal of the twisting couple is followed by small recoil. If the twisting couple be maintained in excess of the value required to overcome the maximum resilient couple, work is done constantly in breaking up molecular groups, and the material of the wire flows steadily, the angle of torsion and the set increasing at constant equal rates. Under that constant couple there is also constant resilience. The condition corresponding to the theoretical angle 。 is attainable under a finite couple of moment k'0n+1 +1=ke。. Thus the theory indicates that the melting-point is conditioned by shearing-stress.

0

Ө

The flow will of course commence at the surface of the wire. The angle ẽ might also be called the Angle of Plasticity, and the couple ke, might be termed the Couple of Fluidity.

Relation between Torsion and Set.

If we assume that the torsional rigidity is not sensibly altered by set, the quantity k is constant, and we may write (3) in the form

·k(0—a)2=k02 —p✪”,

where a is the angle of set. This gives

To test this expression I have used Wiedemann's statical observations given in Table I. p. 4, Phil. Mag. 1880, vol. ix. I find that the equation

a=0(1-√1-896(10)-163-548)

corresponds to a remarkable extent with his observations. The curve in fig. 2 represents this equation with values of as ordinates and values of a as abscissæ; and the points on or near it represent Wiedemann's results. No stronger confirmation of the theory need be desired.

Period of Oscillation.

The potential energy of the system is

V = 11⁄2 k02-— p0TM.

The kinetic energy of the system oscillating as a whole is

where I is the moment of inertia; and the second term in the expression for V represents kinetic energy of molecular motion. So the total kinetic energy at the angle is

which shows that the motion outwards is simple harmonic motion as reckoned from the origin; but it is only so in virtue of the condition that the defect of the potential energy from the value that it would have in accordance with Hooke's Law is due to its transformation into a kinetic form. The periods of the outward swing from zero and of the inward swing to the position of set, on the assumption that k does not change, are each equal to

Wiedemann's statical experiments show that after the few preliminary applications of the maximum twisting couple necessary to fix the set, 0-a varies almost in accordance with Hooke's Law, and that the slight difference is in the direction of too great magnitude as the torsion increases: and Tomlinson has shown that great permanent torsion decreases the torsional elasticity. These facts may indicate that k is slightly decreased at the greater torsions, in which case the period of oscillation will slightly increase as the range is increased.

Concluding Remarks.

The experiment A was not the first made with the given wire, though it was the first made with it under the stated conditions of length &c. Thus, in A the wire was in a fatigued condition relatively to its condition in the experiments R and S.

It has been found by Kelvin and Tomlinson that, in the case of small ranges, the rate of decrease of range per oscillation is practically constant for all periods of oscillation in

the less viscous metals and increases with the period in other metals. According to the above theory this is due to the fact that a given state of stress is continued longer, so that the molecular configurations have more opportunity to break down.

If the theory were pushed to the extreme in its application to Wiedemann's results on torsion and set above quoted, we should find that =2400, and that the couple necessary for zero resilience was fully double the maximum couple employed by Wiedemann (that corresponding to 0=1725).

Various deductions might be drawn from the theory in connexion with the observed values of the constants in the empirical equations. I do not think that such deductions would be of any value except in connexion with a much wider experimental basis than that furnished above. I hope soon to be able to communicate the results of further observations.

IV. On the Mechanism of Electrical Conduction.-Part I. Conduction in Metals. By CHARLES V. BURTON, D.Sc.*

1.

THE

HE view of electrical conduction which it is here my object to explain receives general support from more than one consideration; for it leads to the conclusion that deviations from Ohm's Law must be quite inappreciable in the case of metallic conductors, and it goes far to explain, I think, why metals are so much less opaque than thei ordinary conductivities would lead us to infer. But it is no alone on such considerations that we have to rely, for, as i seems to me, the main conclusions are capable of exact demonstration; and accordingly it would appear most convenient to commence with a few simple theorems, seeking afterwards to account for known phenomena by means of our definite results.

2. THEOREM I.

not

In a region containing matter, there may be (and probably always are) some parts which are perfect insulators and some parts which are perfect conductors; but there can be no parts whose conductivity is finite-unless every finitely conductive portion is enclosed by a perfectly conductive envelope.

Before proceeding to the proof of this theorem, it may be remarked that the presence of the last clause in no way modifies any application of our result, since the space within a perfectly conductive envelope is completely shielded from

* Communicated by the Physical Society: read April 13, 1894.