Rigidity of the Earth. 483 



1010, it will be found that this also gives about the same 

 difference from the result for sin incompressible sphere. 



In the following calculations I assume that if the tidal 

 force did not act the density of the sphere at distance r from 

 the centre would be a function of r only. When the tidal 

 force acts it is assumed that each naturally spherical shell is 

 deformed into a spheroid of small ellipticity e, which is a 

 function of r. The additional assumption of incompressibility 

 enables me to express the strains in terms of e, r, and 

 differential coefficients of e with respect to r. Then 

 there are two equations of equilibrium of an element 

 of the material, either of which, it might appear, ought 

 to give a differential equation for e. But since the cubical 

 dilatation has been assumed zero and the bulk-modulus 

 infinite, the product of these two quantities, which appears 

 in each of the equations of equilibrium, is an unknown 

 quantity and has to be eliminated from the equations. The 

 eliminant is the differential equation for e. 



In the differential equation for e the variation of the 

 density with r is involved and it becomes necessary to assume 

 an expression for the density. If p is the density, a the 

 external radius of the earth, I have used two expressions for 

 the density, viz, 



P=(l0-7-5'%, (a) 



and p=(l4'5-12^\r, (£) 



where w is the density of water. 



It will be found that both these densities give a mean 

 of 5*5 w at the surface where r = a. 



Taking the modulus of rigidity to be 800 x 10 6 grams per 

 square centimetre, about the value for wrought iron, I find 

 that the ellipticities of the surface are proportional to 0*325 

 and 0'321 corresponding to the densities (a) and (@) 

 respectively. Then usinor the density (a) I recalculate the 

 ellipticity taking the rigidity to be half that of wrought iron, 

 that is, 400 x 10 6 grams per square centimetre, and I find in 

 this case that the ellipticity is represented by 0473. These 

 results are afterwards used to calculate the quantity measured 

 by Dr. Hecker's pendulums. 



The ellipticities calculated on the assumption of uniform 

 density corresponding to the two moduli of rigidity mentioned 

 above are represented by the numbers 0"394 and 0*600. 

 When these are compared with the ellipticities for variable 



