(1) Mathematical theory. The differential equations of the field are solved by a method first given by H. Lamb. The induced currents are found to flow in concentric shells; to an observer moving with the sphere they would be equivalent to two sets of steady currents revolving in the sphere with frequencies equal respectively to the sum and the difference of the frequencies of the impressed alternating field and of rotation of the sphere. The currents may be supposed to diffuse from the surface inward in the usual manner; at higher frequencies the penetration is smaller, resulting in a "skin effect" for high frequencies. Equations are also derived for the resulting magnetic field and for the torque on the sphere and for the logarithmic decrement in the case of an oscillating conducting sphere. (2) To test the theoretical results, measurements of the logarithmic decrement of a solid metal ball oscillating about a vertical axis in an alternating field produced by a long horizontal solenoid were made, and the results for three balls were found to agree with values computed from the theoretical expression for the torque within the limit of experimental error, which was not greater than a few tenths per cent.

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