The details of the calculation are as follows: let p, and p_ be the densities of positive and negative electricity, respectively.
The condition that a free electron in the matter shall be in equilibrium is Since is given in terms of D by (7), equations (54) and (i5) serve to determine a and (3 in terms of G and p. The last named quantity becomes determined as the value ne cessary to give rise to the earth's magnetic field as a result of its rotation. The necessary value of p is Tempting as the foregoing view may seem, it is not without its difficulties. For, it turns out that if we desire to formulate a complete scheme of electrodynamics in such a manner as to conform to the requirements of the theory of relativity, we must not allow ourselves to modify the law of force between statical charges without modifying that between moving charges as well, and in the same ratio. Now the effect of motion of a charge on the force exerted on that charge by other charges is precisely the effect which we recognized as that of the magnetic field caused by those other charges; and, when everything is taken into ac count, it turns out that the magnetic field caused by the rotation of the positive electricity is different in magnitude as well as in sign from that caused by the rotation of an equal distribution of negative electricity, and the difference is just such as to cancel the magnetic field resulting from the excess of one kind of charge over the other"). Thus, the net result is that no magnetic field is obtained as a result of the rotation of the system ; and, the only way in which to secure a magnetic field on these lines is to sacrifice conformity with the requirements of the restricted theory of relativity.
where the electromotive forces necessary to maintain such cur rents could come from, and here we encounter a rather curious phenomenon. The electrical inertial considerations (self induc tion) in the case of a body of the earth's size become of enormous importance, so that currents once generated take a long time to die down after the source of electromotive force which caused their generation has been removed. Thus H. Lamb") has shown that currents generated in a copper sphere of the size of the earth would take about 3 million years to die to one-third of their value after the source of electromotive force was removed. A somewhat similar situation exists in relation to a sudden destruc tion of an initial state of permanent magnetization. Thus if a sphere of the size of the earth were originally magnetized and the state of magnetization were destroyed, for example, by a sudden rise of temperature, the change of magnetic flux attend ing the destruction of the magnetization would set up induced currents which would tend to perpetuate the field to such an ex tent that in the case of a conductivity comparable with copper, about 3 million years would elapse before the field due to the sphere at a point had sunk to a value comparable with one-third of its original value"). Suggestive as are these considerations in tempting us to see in them the origin of currents which would account for the earth's magnetism we have to realize the great improbability of a conductivity of the earth's interior sufficient to provide for the necessary requirements. Data as to the con ductivity of the earth's interior drawn from considerations based on the theory of the diurnal variation suggest a value so small as to insure that such current as the above would fall to one-third of their original values in two or three days.