Elastic Solids at Rest or in Motion in a Liquid. 



243 



when the retarding action of viscosity neutralises the acceleration due 

 to gravity, it is worth considering. The hydrodynamical solution 

 really assumes the velocity to be small, and the ellipsoid to be so 

 remote from the surface and other boundaries as to be practically in an 

 infinite liquid. 



It is not very difficult to deduce from the formulae in Lamb's 

 ' Hydrodynamics,' Art. 296, — though I have not seen the result noticed 

 — that the viscous surface action reduces to a force fur per unit surface, 

 opposite to the direction of motion, w being the perpendicular from the 

 centre on the tangent plane, and / a constant. The recognition of 

 this fact saves us from the labour of considering the general expressions 

 for the hydrodynamical pressures, which are of a very complicated 

 nature. 



As the motion is steady, the body stress equations are 



dxz dxy dxz dxy dyy dyz dxz dyz dzz _ 



while the surface equations are — a, b, c being the semi-axes of the 

 ellipsoid — 



a~ 2 xxx + b- 2 yxy + c~ 2 zxz = - a,- 2 x{IL + gp'(t + z)}, ) 

 a- 2 xx}+b- 2 yw + c- 2 z^z = - 6-M n W(?+*)}> * ( 24 )- 

 a~ 2 xS+ b- 2 y^z + c~ 2 z?z = - c~ 2 z{ U + #/(f + *)} -f , 

 The surface equations are satisfied by 



7x = -TL-gp f (C + z) + (a 2 lc 2 )fz,) 

 ?z= -IL-g P '(C+z)-fi, \ (25). 



3 = o 3 



xz/x - yzjy = -f 



The values (25) also satisfy the body stress equations (23), pro- 

 vided 



-3f+g( P -i>') = (26). 



As 



[[/WEI = Sf.irrabc, 



when the integral is taken over the surface of the ellipsoid, (26) is 

 simply equivalent to the condition that the motion is not accelerated, 

 or that 



C = lit- 



