Elastic Solids at Best or in Motion in a Liquid. 239 



P~P 



xx = - II - gp (z + \g 



P 



fp'(p-p'){l-2r ] ) 



+ IOE^-t?) 

 5J= -U-gp'iz + yP—t 



p 



Q 2 P(P-P')(1-21) 

 + 10E(1-t/) 



zz = -n-gp'(z + igP—£-t*) 

 P 



g 2 p\p-p)0~-2i) 



10E(l-^) 



xyjxy = xz/xz = yz/yz = - g 2 p(p -p)(l- 2>/) 2 -r {5E (1 - 77)} 

 a/* - p/y - - i^{n + ^ + ^^0} 



E 



fp'ip-p'Kl-W 



10E 2 (1-?/) 



{(3 -7/)ft2_.(l + ^) r 2 



{H* + fl/*(s + \g p -f- t 2 ) - igp'r*} 



10E 2 (1-^) 



{(3-7;) a 2_ ( l + ^) r 2 } 



(12). 



The terms in g 2 constitute what has been called above the supple- 

 mentary solution. In the case alike of the stresses and of the dis- 

 placements they are exactly the same as if the sphere were under a 

 self-gravitative force which followed the ordinary gravitational law, 

 and which had for its accelerative value at the surface of the sphere 



9 2 p'(p-p) l-2g 

 p E 



a . 



This imaginary gravitative action represents attraction or repulsion 

 between elements of the solid according as p - p is negative or posi- 

 tive. It is thus an attraction when the sphere rises in a heavier liquid, 

 a repulsion when it sinks in a lighter. The smaller 1 - 2rj, or in 

 general the less compressible the solid, the smaller is the effect of this 

 imaginary gravitative force relative to that of the hydrostatic pressure 

 II + gp'(z + ; on the other hand its relative importance increases 

 rapidly with the size of the sphere. 



Bepresenting by dashed letters the parts of the displacements 

 depending on p - />', we have 



