176 PROFESSOR A. E. H. LOVE ON. THE 



where A is the dilatation expressed by the formula 



.3_wch^3M> 



= 3* ty ~Sz' ' ' ' ( 8 



Further, we may put 



V = V +W ', (9) 



where W is the additional potential due to concentration of density at internal points, 

 and to displacement of mass across the initial bounding surface. We may neglect 

 terms of the type /3 A3W/3a:. When we substitute for X Z) ... from equations (1), and 

 make these simplifications, the equation (6) becomes 



On omitting the terms which cancel each other in virtue of equation (3), we have the 

 first of the three equations (10) below. The remaining two of these equations 

 are obtained in the same way. Thus we have the equations of vibratory motion in 

 the forms 



%> "> 







In addition to these equations we have the equation connecting the potential with 

 the density in the form 



V 2 W = 47ry/> A ........... (11) 



The system of equations (10) and (11) are the differential equations of the problem.* 

 4. Besides satisfying the differential equations (10) and (11), the additional 

 potential W and the components of displacement u, v, ^v must also satisfy certain 

 conditions at the surface r = a. Let U denote the radial component of displacement, 



so that 



Ur = xu+yv+zW) ... ...... (12) 



and let U denote the value of U at r = a. The potential W is that due to a volume 

 distribution of density p A, together with that due to a superficial distribution /3 U a 



* In the problem as formulated by JEANS, when the self-attraction of the body is balanced by an 

 external field of force, the equations of vibratory motion differ from those which are obtained here by the 

 omission of the terms such as ^iryp^xA. In Lord KAYLEIGU'S paper already cited, the equations given by 

 JEANS are discussed in accordance with the analysis which was developed by LAMB in the paper on the 

 vibrations of a sphere. 



