Elastic Solids to Metrology. 541 



first, we can put this in the form 



But we have 

 ypv = W, the weight of the solid, 



<#> 'im 1 + -r> + -^rr) -"= W', the weight of the liquid displaced, 

 Z/2= £ the depth of the C.G; of the solid, 



2 V 1 + Mf) = ^ " " " " liquid dis P lacecl > 



Il-f gpj al 1+ "77 + tS" ) = ^j the mean value of the pressure 

 ZK * 6/d J between z = and z=Z. 



Thus (19) is equivalent to 



8 w = (W?-W , £ / -3pt;)/3ifc. . . . (20) 



§ 7. As an example of discontinuous variation of pressure, 

 take the case of a cylinder or prism supported on its base 

 2 = 0, surrounded to the height z = l L by a liquid of density p 1 , 

 and throughout the remainder l 2 of its height by a liquid of 

 density/a''. If II be the pressure in the liquid when 2 = 0, 

 then the pressure elsewhere is given by 



p = U—gp'z between z = and z = l u 



p = U—gp'l 1 —gp ,, (z — li) between z = l x and z — l 2 . 



Employing these values we easily find from (2) for the 

 change in volume of the solid 



»/•- - {gp 1 ^ +^-'2g{p'h+p"k)-g(p'-f")hkl{k+k)\iu. (2i) 



For given values of l l9 l 2 , pi, and p 2 , the reduction in the 

 volume is least when — as is necessary for stable equilibrium — 

 the heavier liquid is at the bottom. 



Aeolotropic Solids. 



§ 8. As an example of an aeolotropic solid, take the case 

 of a right prism or cylinder of density p, whose shape is 

 symmetrical with respect to three planes of elastic symmetry, 



