Elastic Solids at Rest or in Motion in a Liquid. 2-41 



(17). 



a /x = Ply = - 1^2 { n + gp'(i + «) + ] 



Instead of (4) we have 



Also u = d 2 '{\dt\ 



thus, if d' 2 y/dt 2 be omitted, we have 



d 2 C 



or = constant + lglZJ?t 2 (18). 



-/? + */> 



This is, of course, only the well-known result, that the dynamical 

 action of the liquid may be regarded as adding to the mass of the 

 sphere that of a hemisphere of the liquid.* We may suppose the con- 

 stant in (18) to be zero, suitably interpreting II. 



As in the first case considered, the existence of t 2 in ( and, conse- 

 quently, in a, /3, y, makes a supplementary solution necessary. The 

 stresses of the supplementary solution must satisfy the surface equa- 

 tions (10) as well as the following body stress equations : 



\ dx dy dz ) I \ dx dy dz j I " 



dxz dp dzz\ I _ l-2>? 2fpp( p-p) (m 

 dx + dy + dz]/ * ~ E 2 P + p f V '' 



It will be observed that the retention of the term in u in the pres- 

 sure has only modified (reduced) the acceleration without altering the 

 type of the supplementary solution. It will thus suffice to record the 

 complete expressions for the displacements, viz., 



1 - 2^7 

 E 



(20). 



(1 - 2 V ) 2 g 2 pp( P - p) _ 



* Cf. Lamb's 1 Hydrodynamics,' Art. 91 ; or Basset's ' Treatise on Hydro- 

 dynamics,' Art. 182. 



