Elastic Solids to Metrology. 551 



the supporting sulid surface underneath. On either alterna- 

 tive, the conditions in the immediate neighbourhood of z=0 

 are determined partly by the flat base or the supporting flat 

 solid ; and if the cylinder be short compared to its diameter, 

 more especially if its vvalls be thick, the solution we have 

 obtained above will not be satisfactory. If, however, the 

 walls be thin, as in an ordinary glass beaker, the solution 

 will, I think, prove in general sufficiently accurate. 



As illustrating the nature of the solution, suppose the outer 

 cylindrical surface free of stress, whilst inside there is liquid 

 of density p to a height h. Then, if II be the atmospheric 

 pressure at the surface of the liquid, the pressure at a height 

 z above the base of the cylinder is Ti. Q +gp(li— z). As this 

 equals by hypothesis IF + Cz, we have 



n'=ii +<7/oA 3 a =-0o. 



The law of pressure holds only between the values and A 

 of Zj and the solution must be similarly restricted. In such 

 a case we should take 



l=h, 



7r(a' 2 — a'*)Z = weight of tube above level of liquid. 



§ 13. The internal and external pressure systems in (10) 

 and (41) are quite independent, but for brevity we shall 

 regard them as coexistent and applicable up to the same height 

 h above the base. Let p and p be the external and internal 

 pressures at the height h/2 above the base, and H + A/2 the 

 total height of the cylinder, so that 



II+iCZi=/y, -j 



n' + ic/,^/, I m 



Z +\<iph=gpH ) 



The volumes enclosed by the outer and inner surfaces up to 



a height A in the original unstrained state are 



v—inrh, r. — ira'%. 



The elastic increment in the volume within the outer 

 surface is given by 



Bv —27ra\ u r = a dz + 'nd i { v >s=h—Wz=o}r=a \ 



and the corresponding increment hi\ of the volume within the 

 inner surface is given by an analogous formula. After some 



2 2 



