Elastic Solids to Metrology. 547 



i. e. according as the length of the immersed portion of the 

 cylinder is greater or less than the barometric height as given 

 by a barometer containing a liquid whose density is p — p'. 



If the whole of the cylinder be immersed then the liquid 

 must not have access to any part of the upper surface, other- 

 wise the stress could not differ from the liquid pressure at 

 that depth. 



If p be less than p' the cylinder may be held down either 

 partly immersed, or fully immersed but with its upper surface 

 not exposed to the liquid, or it may float with a length h! 

 emergent. ^ 



In the latter event zz over z = h/2 must represent the 

 pressure exerted on its base by a cylinder of density p and 

 height h\ whose upper surface is exposed to atmospheric 

 pressure. Thus, neglecting the variation of atmospheric 

 pressure throughout the height h f , we have 



-gph'^gph-gp'h, 



or 



p(h + h')=p'h, 



in accordance with elementary Statics. 



Case (ii.) in like manner includes various sub-cases. The 

 first thing to notice is that unless p=p f the solution assumes 

 that the liquid cannot reach any part of the base ; otherwise 



zz could not have the value assigned to it when z= —A/2. 

 This explains why the solution does not in itself introduce 

 the restriction p > p'. In practice, a solid lighter than a liquid 

 in which it is immersed almost invariably rises to the surface 

 of the liquid, but if the lower surface of the solid and the 

 bottom of the vessel holding the liquid be both perfectly 

 smooth, this need not happen. Usually liquid would force its 

 way under the base, so that in practice the solution would 

 apply only to a cylinder cemented to the bottom or projecting 

 up through it like a pile. When a cylinder or prism rests 

 on, or is supported from, a series of isolated supports, the stress 



zz is not uniform over the cross section, and the previous 

 solutions do nor, strictly apply. If, however, the height h be 

 great compared to the largest horizontal dimension, the 

 solutions will be very approximately true except in the imme- 

 diate neighbourhood of the supports. 



Flask containing Liquid. 



§ 11. In obtaining (8) and (9) we tacitly assumed that 

 there were no cavities in the solid, or that the liquid pressure 

 was the same at all points in the same horizontal plane. 



