Solids under Body Forces. 323 



2. Returning now to the equations for displacements (1), 

 let us put , D 



so that ^-.o . f - . X BA _a 



and fjLS7 2 u 2 + pX = 0, 



Let further A = V 2 </>, 



where </> is a function of x, y, z to be determined. 



Then j t 4Ml + (X+ At )|*=0&c, 



as a particular solution, as also 



Again, since 



^ 4tt(\ + 



and :*/ lN i 



v *(*:) =2 _W ,&c. 



we get 



and 



Hence finally 



The result was originally obtained by Lord Kelvin. His 

 method was to find u, v, w for a distribution of body 

 forces through the volume of a sphere as a potential problem 

 and then reduce the sphere to a point. [See also Love's 

 4 Elasticity/] 



3. In the case of uniform distribution of body forces through 



