Elastic Solids to Metrology. 539 



density of the medium uniform. If, however, we assumed 

 that p', instead of being constant, varied according to the law 

 (10), we should clearly add to the right-hand sides of (8') 

 and (9') only terms involving the product (1/&) X (1/&), and 

 unless kfjk were very small, the retention of these terms 

 would be open to criticism. Under ordinary conditions we 

 may utilize (8'), supposing the solid is suspended, and replace 

 (12) by 



W'=^ 'fo{ 1+ (U'+gpJQ^ - l)+ff(Po- Po ')m}- (120 



Here % p , p ' refer to the ideal state of absolute freedom 

 from pressure ; but inside the square bracket the distinction 

 between p and p , or p' and p ' is negligible to the present 

 degree of approximation. In general it would be more 

 convenient in practice to have a standard atmospheric 

 pressure II. If v, p, and p' refer to this, then (12') becomes 



W=g P 'v 1 1 + (lL'-Ii+ffp'S)(± - ~j+g{p-pl)m^} ■ (13) 



For the weight W" of air displaced in a weighing in air, 

 we mav take 



where v is the volume of the solid, unaffected by gravity, 

 under standard pressure II, while II" is the atmospheric 

 pressure, and p" the density of the air at the time of weighing. 

 II" and p' should be measured at the level of the C.G. of the 

 body, which is supposed suspended. In general, the term 

 —gp"Q'dk inside the bracket in (14) might be neglected. 



§ 6. When the variation in density of the liquid is taken 

 into account, the integrals appearing in the expression for 

 the change in volume of the solid would probably prove 

 tractable only in particular cases. The case of a right circular 

 cylinder or prism with its axis vertical may be treated as 

 follows, the law of variation of the density with the depth 

 being supposed continuous. 



Let the origin of coordinates be at the upper end of the 

 cylindrical axis, z being measured downwards ; and let the 

 density of the liquid follow the law p'=f(z). 



W 



Let 



«M s r*M* ( 16 ) 



