182] HOLLOW SPHERE AND CYLINDEE. 477 



In Oersted's piezometer the internal and external pressures are equal, 

 so that 6 = 0, and the sphere receives a homogeneous strain, which 



is of the same magnitude, 6 = ^~> as if the sphere were solid. A 



second practical application is found in the correction of thermometers 

 due to the pressure of the mercury causing the bulb to expand, the 

 amount of expansion being found from 66). 



In treating the case of a very long hollow cylinder, we proceed 

 in precisely the same manner, except that the problem is a two 

 dimensional one. We will number the equations in the same manner, 

 with the addition of an accent. The formulae have an application 

 in finding the pressure able to be borne by tubes and boilers. 



57') w = ' V== -' w==0 > 6 



59') 



d fp (a . , b \ x 



u = -^~ = ( r + - ) - 



dx \2 r/r 



dm /a , b\ y 



v = -2- = ( r + I 



dy \2 r) r 



du a b 2bx z du 2bxy 



d^Y + r*" ~r^' J~y = ~^~ 

 dv _ 2bxy dv a b 

 Jx = ~7^" ; ~d = + T 2 



v fi o ( a . b 2bx*\~\ , x 4:bxy 



X n = \la + 2fi (- + T2 -- -H cos (nx) -- -f- cos (ny), 



62'") 



' ^ bxy , r, la . b 2bx*\~] , ^ 



Y n = - -^ cos nx + \la + 2^ (- + - 2 - -^jj cos (ny), 



63')' l^Ol+fOa-^' 



64 f ) X w = p cos (wa?), TJ, =._p cos (ny), 





66') 



-,- 



_2 f~ \ 



