XII. The oscillations of a rotating ellipsoidal shell containing fluid

Abstract

In a paper published in 'Acta Mathematica,’ vol. 16, M. Folie announces the fact that the latitude of places on the earth’s surface is undergoing periodic changes in a period considerably in excess of that which theory has hitherto been supposed to require. This result has been confirmed in a remarkable manner by Dr. S. C. Chandler, in America ( vide ‘ Astronomical Journal,’ vols. 11, 12), who, as the result of an exhaustive examination of almost all the available records of latitude observations for the last half-century, has assigned 427 days as the true period in which the changes are taking place. The old theory, based on the assumption that the earth was rigid throughout, led to a period of 305 days, and M. Folie proposes to account for the extension of this period by attributing a certain amount of freedom to the internal portions of the earth. The earth he supposes to be composed of “a solid shell moving more or less freely on a nucleus consisting of fluid at least at its surface.” The argument advanced by M. Folie in favour of this constitution of the earth, namely, the independence of the motions of the shell and the nucleus, appeared to me to be unsatisfactoiy, and I therefore proposed to myself to test the validity of it by examining a particular case which lent itself to mathematical analysis, namely, that in which the internal surface of the shell is ellipsoidal and the nucleus consists entirely of homogeneous fluid. The principal axes of the shell and of the cavity occupied by fluid are assumed to be coincident, and the oscillations are considered about a state of steady motion in which the axis of rotation coincides with one of these axes. It is clear that a steady motion will be possible in this case, and that such a motion will be secularly stable in the event of the axis of rotation being the axis of greatest moment for both the shell and the cavity.