Elastic Solids at Rest or in Motion in a Liquid. 237 



On looking more closely into the matter an inconsistency manifests 

 itself. Supposing for mathematical simplicity that the solid is 

 isotropic, of bulk modulus Jc, we find that the displacements answer- 

 ing to (3) are given by 



a/x = Ply = -{n + gp'(t+z)}l3k, 



y = - [II* + g P '{z (f + z)-\ (a* + + z*)}]/3k 



The inconsistency consists in the fact that, by (6), a, /3, y contain 

 terms in £, and so by (5) terms in t 2 , while above it was assumed 

 that d 2 a,/dt 2 , &c, vanished. It thus appears that the solution embodied 

 in (3) and (6) is valid and complete only when f does not vary as t 2 , 

 i.e., only when the solid is at rest or moving with uniform velocity in 

 the liquid. 



Though thus restricted, the solution is notable from its simplicity 

 and generality, as applicable to any homogeneous solid (free from 

 cavities) at rest in a liquid of equal density. 



The values (3) for the stresses apply irrespective of the species of 

 elasticity. The displacements are given by (6) only when the material 

 is isotropic, but corresponding expressions are immediately obtainable 

 for materials of greater complexity. If for instance we have material 

 symmetrical with respect to the co-ordinate planes, we have 



a= - x{U + gp' ((+z)}(l - rj 12 - ^ 13 )/Ei , 



/?= - y {II + gp' (f+ - mi - ^ 23 )/E 2 , 



y ... (7). 



7 = -z{IL + gp' (f + iz)}(l - 7)31 - *?32)/E 3 



+ hp I^C 1 -^- ^is) + |j(i-^2i- ^23) I 



Here Ei, E 2 , E 3 are the three principal Young's moduli, while 

 ^12 5 viz, &c, are the corresponding Poisson's ratios, 



§ 2. Presently we shall consider the equilibrium problem in greater 

 detail. Meanwhile, in the case of uniformly accelerated motion, we 

 shall obtain a self-consistent solution for a sphere, or any form of solid 

 ellipsoid, under the conditions assumed in § 1. 



The procedure to be adopted is the same for all species of elastic 

 material. If for definiteness we suppose the material symmetrical with 

 respect to the three co-ordinate planes, we first assume that the 

 stresses (3) and displacements (7) form part — but only part — of the 

 complete solution, ( being given by (5). Then substituting from (7) 

 in the body stress equations (1), we find that the stresses of the 

 supplementary solution, as we may call it, must satisfy 



s 2 



