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Mr. G. A. Darwin on the Bodily Tides [May 23, 



the earth's surface, that the belief is not yet extinct that we live on a 

 thin shell over a sea of molten lava. It appeared to me, therefore, to 

 be of interest to investigate the consequences which would arise from 

 the supposition that the matter constituting the earth is of a viscous 

 or imperfectly elastic nature. In this paper I follow out these hypo- 

 theses, and it will be seen that the results are fully as hostile to the 

 idea of any great mobility of the interior of the earth as are those of 

 Sir W. Thomson. 



I begin by showing that the equations of flow of an incompressible 

 viscous fluid have precisely the same form as those of strain of an 

 incompressible elastic solid, at least when inertia is neglected. Hence, 

 every problem about the strains of the latter has its analogue touching 

 the flow of the former. This being so, the solution of Sir W. Thom- 

 son's problem of the bodily tides of an elastic sphere may be adapted 

 to give the bodily tides of a viscous spheroid. Sir W. Thomson, how- 

 ever, introduces the effects of the mutual gravitation of the parts of 

 the sphere, by a synthetical method, after he has found the state of 

 internal strain of an elastic sphere devoid of gravitational power 

 The parallel synthetical method becomes, in the case of the viscous 

 spheroid, somewhat complex, and I have preferred to adapt the solution 

 analytically so as to include gravitation. 



The solution is only applicable when the disturbing potential is 

 capable of expansion as a series of solid harmonics, and it appears 

 that each harmonic term in the potential acts independently of all 

 others ; it is thus only necessary to consider a typical term in the 

 potential. 



It is shown finally that if p, vr, v, be the velocities of the fluid (at 

 a point whose polar co-ordinates in the sphere are r, 0, 0,) radially, 

 and along and perpendicular to the meridian ; /(, the coefficient of 

 viscosity, tor*Si the disturbing potential, a solid harmonic of degree i; 

 a the mean radius, w the density of the homogeneous spheroid, and 

 g gravity; r — a + a % the equation to the bounding surface of the 

 spheroid ; then 



^(. + 2)^-/(^-l )r 2 r ,_ 1T . 

 H 2(t-i)[2(i+l)*+l> 



2(^-l)[2(^+l) 2 + l> do' 



v— the same X , 



sin d(p 



where surface harmonic of order i. A dif- 



\ 2i+l a 1 J 



ferential equation is then found, which gives the form of the free 

 surface at any time under the action of any disturbing potential 



