72 Mr Love, On Sir William Thomsons estimate [April 28, 



(3) On Sir William Thomson's estimate of the Rigidity of the 

 Earth. By A. E. H. Love, M.A., St John's College. 



(Abstract.) 



In Thomson and Tait's Natural Philosophy (part II. art. 834 sq.) 

 there is given an estimate of the rigidity of the earth derived from 

 a consideration of tidal phenomena. For the purpose of obtaining 

 such an estimate the earth is regarded as a homogeneous elastic 

 solid sphere, of gravitating matter and incompressible, which is 

 supposed to be strained by its own gravitation and by the action of 

 external disturbing bodies such as the moon. The amount of the 

 tidal distortion is measured by the ratio of the spherical harmonic 

 deviation of the disturbed surface from the mean spherical figure 

 to the radius of the mean sphere, and this is compared with the 

 corresponding ratio in case the rigidity is annulled, i.e. in the case 

 of the earth regarded as a homogeneous fluid sphere disturbed by 

 the same system of forces. We may for shortness speak of the first 

 dynamical system as a ' solid earth,' and of the second as a ' fluid 

 earth.' It is shown that the ratio of the amount of tidal distortion 

 in a * solid earth ' of the same rigidity as glass to that in a ' fluid 

 earth 5 is about "612 or nearly f, while if the rigidity be taken to 

 be that of steel the ratio is about "321 or nearly i In a perfectly 

 rigid 'solid earth' the ratio would be zero. The height of the tide 

 as given by the observable rise and fall of the sea relatively to the 

 land is the difference of the spherical harmonic deviations of the 

 ' fluid earth ' and the ' solid earth ' by which the actual earth can 

 be most approximately replaced. It is concluded that in case the 

 rigidity were that of steel the height of the tide would be reduced 

 to about | of what it would be in case the earth were perfectly 

 rigid, and that if the rigidity were that of glass the height of the 

 tide would be reduced to about § of what it would be in case the 

 earth were perfectly rigid. From the observations that have been 

 made Professor G. H. Darwin, whose method and conclusions are 

 given in the same volume, deduces that the actual amount of the 

 observable fortnightly tide cannot be much less than §-, and cer- 

 tainly cannot be nearly so small as f of that calculated in the 

 ordinary equilibrium theory in case the earth is regarded as per- 

 fectly rigid, thus confirming Sir William Thomson's estimate of the 

 ' tidal effective rigidity ' of the earth, that it is much greater than 

 that of glass and probably about that of steel. 



It appeared to me to be worth while to try to find out what 

 would be the effect of supposing the material of the solid replacing 

 the earth to have finite compressibility as well as rigidity. For 

 most solids that have been tested by experiment the two elastic 

 constants, the resistance to compression, k, and the resistance to 

 distortion, n, are connected very nearly by the relation 3k — bn, so 



