Forces in opposing Distortion of an Elastic Sphere. 439 



magnitude as the harmonic forces. Professor Love does 

 not neglect this difference, but he allows for it in a wrong way. 

 Instead of modifying the bodily forces throughout, he assumes 

 that the difference is properly accounted for by treating the 

 bodily forces on the matter displaced outside the original 

 bounding surface as if it were a surface-traction. This is a 

 consequence of regarding the coordinates which occur in the 

 equations of equilibrium as the space-coordinates of the 

 particles of the strained body; whereas, as I have pointed 

 out, they are the space-coordinates of the particles of the 

 unstrained body. Surface-tractions are forces applied at the 

 boundary of the body, and not forces applied at that surface 

 which was the boundary in the unstrained state. 



When the harmonic forces do not act, the radial force acting 

 on the spherical shell whose new radius is r x , has a potential 

 f{i\.) But when the spherical shell is strained to the surface 

 whose equation is 



r=r 1 -}-(7 



the radial force acting at different points of this shell is 

 derived from the potential 



=/O'i) + 0/'( r i), nearly. 



The first term of V is the one that causes radial strain, and 

 it has no effect whatever on the value of a. Since, in the 

 rest of the paper, we shall only be concerned with the de- 

 viation of the shells from the spherical form, we can ignore 

 this term both in the differential equations and in the boundary 

 conditions. 



But the second term in V causes displacements from the 

 mean spheres ; and if its magnitude is of the same order as 

 that of the harmonic forces, it must be added to the potential 

 of these forces. 



We shall assume that 



where e n is a small coefficient and Q n+i is a spherical solid 

 harmonic of order ?i-f-l. 



We shall write r, x, y, z instead of r 1} # l3 y 1} z x , to save 

 labour in writing wherever no ambiguity can arise. 



In problems concerning the tidal action of the sun and 

 moon on the earth and the action of " centrifugal force " due 

 to the earth's rotation /(r) is proportional to r 2 ; and therefore 



