Rigidity of the Earth. 499 



The ellipticity of this surface relative to the earth is 



1-353^-^ = 0-703^. . . , . (80) 



*"£? "if 



If the earth were perfectly rigid e x would be zero and the 



relative ellipticity would be exactly ^-. Thus the relative 



ellipticity of the equipotential surface when the earth is 

 about as rigid as wrought iron is 0*703 of what it would be 

 if the earth were perfectly rigid. 



With the same rigidity and the second expression, viz. the 

 one in (58), for the density the relative ellipticity is found to 

 be 



0-7055? (81) 



which scarcely differs from the previous result. It appears 

 then that variations in the law of density do not greatly 

 affect the relative ellipticity, provided the mean density and 

 the density at the surface are kept the same. 



Assuming the conditions given in (63) and (64), the latter 

 condition being that the rigidity is about half that of wrought 

 iron, we find for the relative ellipticity of the equipotential 

 surface 



0-569^ (S2) 



2g ' 



It is proved in Thomson and Tait's * Natural Philosophy ' 

 (Art. 840) that the relative ellipticity, assuming the earth 

 incompressible and of uniform density. 5*5 w, is 



19 71 



11 gaw ha /QQ . 



~19» • V {bi} 



11 gaw 



the values of which are 0*685 ^ and 0*520 77- when n has the 



% 2 9 



values 8 X 10 6 and 4 x 10 6 grams per square centimetre 

 respectively. The former value of the ellipticity should be 

 compared with (80) and (81), and the latter with (82). 



Comparison with Observations. 



For a given position of the tide-producing body the devia- 

 tion of the plumb-line from its mean position is proportional 

 to the relative ellipticity of the equipotential surface and the 



