486 Mr. J. Prescott on the 



Now by equations (5), (fi), and (9) 



= - Los 6 sin 6 ^(i*e)+2re cos sin 6. . (10) 



Let k be the bulk-modulus and n the rigidity-modulus o£ 

 the sphere. These are assumed to be constant throughout 

 since average values are all we can hope for from theoretical 

 investigations. 



The equations of equilibrium in the present case, where 

 everything is symmetrical about the axis from which 6 is 

 measured, are (see Love's ' Tiieory of Elasticity/ Art. 125), 



l/j+^n)^ (q, s1i1 — p-^— = 0, . (11 



\ 6 J or rsmV 0? or 



(i . 4 ^ S _l9 3 / ^ ^ W n n^ 



^ + rvp +2 ^ (rft,) "^^ =0 ^ • • ■ (12) 



where W is the potential of the external forces on the 

 element whose coordinates in the unstrained state are r, 0, 

 and p is the density of the element. 



Now the assumption of incompressibility makes $ zero 

 and k infinite, but it leaves kS finite, since this is the negative 

 of hydrostatic pressure. The best thing to do is to eliminate 

 kB from (11) and (12). _ 



Differentiating (11) with respect to and (12) with respect 

 to r and then subtracting corresponding sides of the resulting 

 equations we get 



Now making use of the fact that p is a function of r and 

 not of we can write this last equation in the form 



r'^0\sm0^0 K y J dr 2V ; 00 dr 



.... (13) 



When we have substituted for 2rco and W in this equation 

 we shall have the differential equation from which e is to be 

 found. 



