Elastic Solids to Metrology. 545 



The equations (29) and (30) are satisfied by 



zz=gpz+G (a constant), y (31) 



xy=xz=yz = 0. J 



As to the conditions over the ends : — 



(i.) If the cylinder has its upper end held — whether sus- 

 pended or pushed down — the base being freely exposed to 

 the liquid, 



7 = when z = h/2, for, say, sc=y = 0, 



zz= — {p + gp'h/2) when z= —h/2. 



The latter condition gives 



and the complete solution will be found to be 



xz=yy=-(p-gp'z), ~) 



h ' 



_ ^ _ 



xy=xz=yz = ; J 



«/#=/%,= -(1/E) [(l-2 V )p + vg(p-p') $+ff3{vp-(l-v)p'\\ 



y= ^l^^ { l-2 v )p- g j{Sp--2(l + v )^}-yz(p-2 V p^] J. (38) 



+ ll$(*?+t/ 2 ){vp-(i-v)p'l 



The force per unit area exerted at the top of the cylinder 

 or prism, i. e. the value of zz when z = h/2, is 

 gph—ip+gp'hfi). 



It is thus — as follows from ordinary Hydrostatics — a pull 

 or a push according as the weight of the cylinder is greater 

 or less than the pressure exerted by the liquid on the base. 



The elastic increment in the height is deducible from (33), 

 whence 



Mlh=g( P -p')(hl2E)-{l-2v)plE (34) 



(ii.) If the cylinder or prism is supported on its base, the 



