300 Oscillations of a Rotating Shell containing Fluid. [Feb. 7, 



problem in hand, enables us to determine an expression for the fluid 

 pressure at all points on the boundary in terms of the disturbances 

 communicated to the shell. From this, the couples on the shell, due 

 to the fluid pressure, are estimated and introduced into the dynamical 

 equations of motion of the shell. A frequency equation of the 

 6th degree is derived, apparently involving three fundamental modes 

 of oscillation. Tlie equation, however, is found to be satisfied by the 

 frequency of rotation of the system, and the corresponding oscillation 

 is shown not to be real but to arise (analytically) only in consequence 

 of the motion of the axes of reference. We are left with two funda- 

 mental modes. 



The case where the inertia of the shell is negligible, compared 

 with that of the fluid, is of analytical interest, and can be approxi- 

 mately realised physically by means of a liquid gyrostat ('Nature/ 

 vol. 15, p. 297) mounted in such a way that its centre of gravity is 

 held at rest. When the axis of rotation is an axis of symmetry, the 

 roots of the frequency equation will be real, and the motion therefore 

 stable, either when this axis is the least axis, or when it exceeds 

 three times the equatorial radius. When, however, the figure is not 

 one of revolution, the analytical conditions of stability are not so 

 simnly expressible, but they will always be satisfied when the axis of 

 rotation is the least axis, or when it exceeds three times either of the 

 other axes. 



On taking into account the inertia of the shell, the discussion is 

 confined to the case where the ellipsoid is approximately spherical, 

 and the solutions of the frequency equation then assume a simple 

 form. Of the two modes of oscillations, the motion of the shell in 

 one is analogous to the motion of a rigid body when slightly dis- 

 turbed from a motion of rotation about a principal axis, but the 

 period is found to be shorter than it would be were the fluid solidi- 

 fied ; the other exists only in consequence of the contained fluid. 



The former of these presents the greater interest. It has been 

 supposed that if the axis of rotation of the earth were displaced 

 from its axis of figure, an oscillatory motion would ensue which 

 would give rise to a variation in the latitude of places on the earth's 

 surface in a period of 305 days. Becent observations (vide Chandler, 

 ' Astronomical Journal,' vols. 11, 12) have proved that such an oscil- 

 lation is taking place, but that the theoretical estimate of the period 

 is considerably too short. This paper was undertaken with the object 

 of investigating whether the extension of the period could be ex- 

 plained by supposing that the earth possessed a fluid interior, in 

 accordance with a suggestion made by M. Folie (' Acta Mathematical 

 vol. 16). It is shown that the hypothesis of a fluid interior leads 

 to a result directly opposite to that which observation requires, and 

 that, therefore, the discovery of the variations of latitude so far from 



