Elastic Solids to Metrology. 535 



must be in the same vertical as the centre of gravity ; it thus 

 lies on the axis of z at a height of, say, f above the C.G. In 

 the present case Z= — gp, thus by (2) 



SkSv = -^gpzdxdydz + jJ(Fff + Gy + H.z)dS + fjjHdS', . (7) 



where d& is a portion of the infinitesimal surface where the 

 suspending cord is attached, while dS is an element of the 

 general surface of the body. By elementary statics, 



jjHdS' = tension of string= W-W, 



where W is the weight of the solid in vacuo, W the weight 

 of the displaced medium. Again, if \, //,, v be direction- 

 cosines of the outwardly drawn normal at d$, we have 



F/X=G//,= H/r=-(p-^), 



and X# + fiy + vz=iv, 



where fa is the perpendicular from the C.G. on the tangent 

 plane to the surface at #, y, z. Also, as the origin is at the 

 C.G., the triple integral in (7) vanishes ; thus we have 



M$v = -pjjWS + gp'$«rzd& + f ( W - W) . 



But ^WS = 3v, 



and jJ OT ~^ = 0> 



as follows from the consideration that the C.G. lies in the 



plane 3=0. 



Thus we have 



s "=i(( w - w ')?-3^}' •':<■*>. 



which may be put in the alternative form 



Sviv=-Sp/p=y(p-p')i;-Sp}i3k . . .(8') 



The change of volume, or mean density, may thus be regarded 

 as composed of two distinct parts, the first representing the 

 influence of the ' f apparent weight " of the solid in increasing 

 the volume, the second the influence of the pressure of the 

 surrounding medium in reducing the volume. The second 

 pirt is the same as if the pressure in the medium w r ere 

 everywhere the same as at the level of the C.G. of the solid. 

 This result, it will be noticed, is true irrespective of the shape 

 or absolute size of the solid. 



If the solid, instead of being suspended, had been supported 

 on a horizontal plane, at a depth $' below the C.G., we 

 should have got in place of (8) and (8'), p having the same 



2N2 



