1878.] of Viscous and Semi-Elastic Spheroids. 423 



objections which may be raised to it, I came to the conclusion that the 

 investigation was still of some value. 



The equations of flow of such an ideal solid have been given (with some 

 assistance from Professor Maxwell) by Mr. Butcher* and they are such 

 that, if the body be incompressible, and if inertia be neglected, they 

 may be written in exactly the same form as the equations of flow of a 



purely viscous fluid, the" coefficient n/-+-^\ merely replacing the 



coefficient of viscosity. Hence it follows that the solution previously 

 found may be at once adapted to the new hypothesis. 



In the application to the tidal problem, if the tide- generating poten- 

 tial be as before, wr 2 S cos (vt + y), and if tan \jr = vt, tan x=vt(l+ 



\ 2gwa/ 



and tan e=tan x~ tan \p~, it appears that the bodily tide raised by this 

 potential is equal to the corresponding tide of a perfectly fluid sphe- 

 roid multiplied by and the tide is retarded by a time Also 

 cos \jr v 



the equilibrium tide of a shallow ocean overlying the elastico-viscous 

 nucleus is equal to the corresponding tide on a rigid nucleus mul- 

 tiplied by cos" % tan e, and there is an acceleration of the time of high 



water equal to — — -. 



If t be taken as zero, whilst n is infinite, but nt (the coefficient of 

 viscosity) finite, the solution becomes that already found for a purely 

 viscous spheroid. If, on the other hand, t be infinite, the solution is 

 that of Sir W. Thomson's problem of the purely elastic sphere. This 

 hypothesis is therefore intermediate between those of pure viscosity and 

 pure elasticity. 



Sir William Thomson worked out numerically the bodily tides of 

 elastic spheres with the rigidities of glass and of iron ; and tables of 

 results are here given for those rigidities, with various times of re- 

 laxation of rigidity, for the semi-diurnal and fortnightly tides. 



It appears that if the time of relaxation of rigidity is about one 

 quarter of the tidal period, then the reduction of ocean tide does not 

 differ much from what it would be if the spheroid were perfectly 

 elastic. The acceleration of high tide, however, still remains consider- 

 able ; and a like observation may be made in the case of pure viscosity 

 approaching rigidity. This leads me to think that one of the most 

 promising ways of detecting such tides in the earth, would be by the 

 determination of the periods of maximum and minimum in a tide of 

 long period in a high latitude. But I am unfortunately unacquainted 

 with practical tidal observation, and therefore cannot tell how far it 

 would be possible to carry out this suggestion. 



* Proc. Lond. Math. Soc, Dec. 14, 1876, pp. 107-9. 



