240 Dr. C. Chree. 



a 'A* = P'/y = y'/z 



= 9! % n (1 " *> pT {(3 " 7?) ft2 " (1 + ? >> r2} - *] ~ (13) - 



At the very beginning of the motion, the expression inside the square 

 bracket is positive for all values of r ; but as t increases it changes 

 sign, first at the surface, last close to the centre of the sphere. If £ a , 

 (o represent the distances fallen when the expression vanishes at the 

 surface and at the centre respectively, we have 



d>/15£,^| 

 >>/30& J 



Ca/a = (l-y])g(p-p')allbk, 



.(H). 



Unless a is enormously large, f a and f must be extremely small for 

 any ordinary elastic material. 



In reality, in order to be instantaneously at rest, the sphere would 

 require to be supported or acted on by some suddenly suppressed force, 

 or to be in the act of reversing some previously impressed motion. 

 The elastic strains and stresses might initially retain the impress of 

 the pre-existing state of matters, and there are thus special sources of 

 uncertainty affecting the applicability of (14) to actual conditions, 

 which should not be lost sight of. 



§ 4. The problem just considered has been advanced as showing how 

 under a consistent dynamical system, producing uniform acceleration 

 in a straight line, there appear elastic strains and stresses which simu- 

 late the action of self-gravitation in the material in motion. The 

 conditions postulated do not answer exactly to what happens when a 

 real solid moves through real liquid at the earth's surface. Under 

 such circumstances the action between solid and liquid is not fully 

 represented by the hydrostatic pressure. If the fluid be "perfect," 

 ordinary hydrodynamical theory* gives for the pressure p on the 

 surface of the sphere, supposing u the velocity, 



p = n+^xt+«)+p'(i^i+KP»-W) ( 15 )> 



where Pi, P2 are zonal harmonics, whose axis is the vertical diameter. 

 We shall now consider this case, on the hypothesis that the velocity is 

 so small that terms in u 2 are negligible. Instead of (3) and (6) we 

 find for the stresses and displacements, the material being supposed 

 isotropic, 



xx = yy == zz = - II - gp(£+z) - Ju/>' 



(16); 



xy = xz = yz — 



} 



* Cf. Lamb's ' Hydrodynamics,' Art. 91. 



