a Rotating Ellipsoidal Shell containing Liquid. 549 



until for a certain thickness it becomes infinite (the speeds of 

 the forced and free oscillations then coinciding) ; then the sign 

 changes again, and as the thickness is still further increased 

 the disturbance decreases in magnitude, but i's always greater 

 than if the whole were rigid, and is of the same sign. It 

 also appears that if a/en be small the disturbance is nearly 

 the same as if the whole were rigid whatever be the sign 

 of <7. 



3. If the mass of the shell be neglected and e and ea/n be 

 small, the denominator in (13) and (14) is by (16) approxi- 

 mately Ce?i 2 / (a + en) . If the whole were rigid the corre- 

 sponding denominator would be C'n — AV, in which, if cr/n be 

 small, the latter term may be neglected. Comparing these 

 results, it appears that the disturbance of the shell may be 

 obtained from the disturbance it would experience if the 

 whole were rigid by multiplication by 1 + a/en. It should be 

 noted that a may be either positive or negative. This result 

 agrees with Lord Kelvin's for a negative value of a only; he 

 appears to have overlooked the distinction in sign. 



4. We will now consider the application of the two hypo- 

 theses to the case of the earth ; it may be well to repeat, 

 however, that it is not intended to suggest that such an appli- 

 cation is of much practical interest. In this case the plane 

 of the ecliptic will be supposed fixed and Z its pole. If ZC 

 produced through C intersect the equator in A', and W be the 

 first point of Libra (B' lying between A and B), the dis- 

 turbing couples are more readily specified with respect to 

 the axes OA', OB'. They consist, of course, of the sum of 

 a number of terms each of which varies as the sine or cosine 

 of an angle increasing uniformly with the time; we may take 

 as typical terms the couple about OA' to be — L / sin st, and 

 that about OB' to be M' cos st, the origin of time being properly 

 chosen. Replacing these couples by equivalent couples whose 

 axes are OA, OB, that about OA is therefore 



— L' sin st cos <\> + M' cos st sin <f>, 

 or 



L' + M' . , / .v L'-M' . , 



^ sin {st— <p) - sm (st 4- <p) ; 



and that about OB is 



L' sin st sin <p + M' cos st cos <£, 

 or 



x — cos (st — <£>) g — cos (st + <f>). 



