236 



Dr. C. Chree. 



If a, /3, y represent the elastic displacements, xx, xy, &c, the stresses 

 in the notation of Todhunter and Pearson's 1 History of Elasticity,' 

 the body stress equations are of the type 



dxx dxy dxz 

 ~dx~ + ~dy + ~dz 



df- 



= 0, 



dxy dijy dyz d 2 /3 

 lx~ + 1$ + ch~ p W = ' 



dxz d^ diss r <P{£+y)\ 



where p represents the density of the solid, g the acceleration of 

 gravity. 



The equations treat x, y, z as constants for each element of the 

 solid, and so assume that the motion, if motion takes place, is 

 purely translational. 



If X, /x, v be the direction cosines of the outwardly directed normal 

 at a point x, y, z } the surface equations are 



(Xxx + [xxy + vx:)!X = (Xxy + fiyy + vyz)/p. = (Xxz + pyz + vzz)/v 



= - IL-gp'(C+z) (2). 



The equations (2) are satisfied by the assumption 

 yy = S= -IL-gp'(£+z), 

 • try = xz = yz = 

 Also the values (3) satisfy the body stress equations (1), provided 



We can satisfy (4) by assuming 

 d' 2 y 



(3). 



cPcc 



dp 



d?p 



9 



gp 



jr- 



o, 



{ = const, + hg ^— ^ f- 

 P 



(5). 



For brevity, the constant in (5) will be supposed to be zero. 



The result (5) is of course that given by ordinary elementary 

 methods for the accelerated motion of a solid rising or falling in a 

 liquid of different density. 



