Elastic Solids to Metrology. 555 



Cylinder under Pressure a Quadratic Function of z. 



§ 15. The method of § 6 would enable us to calculate the 

 change in the total volume of a solid vertical cylinder of any 

 form of section when exposed to a pressure which is a 

 quadratic function of the vertical coordinate z. When the 

 section is circular we can obtain, however, what is in most 

 cases a satisfactory complete solution for this law of pressure. 

 As we have already dealt with the case when the pressure is 

 a linear function of z, we need consider in addition only the 

 elastic displacements, strains, and stresses due to a pressure 

 varying as z 2 . Employing r, z as in § 12, and making use 

 of my solution * in ascending powers of r and z, I find for 

 the axial and longitudinal displacements due to a pressure 

 (/z 2 on the cylindrical surface r=a. 



w=(q'IE){i(l- V )z(^-a') +§v»} ) 



The corresponding stresses are 



)2) 



> ■ (53) 



rcf) = rz = <$>z = 0. 



This solution is not altogether complete, supposing no 

 forces to act except on the curved surface ; for, instead of 

 making zz vanish at every point of the flat ends, it only gives 



According, however, to St. Venant's theory of equipollent 

 systems of loading, this defect is very trifling in a long 

 cylinder except in the immediate neighbourhood of the ends. 



As an example, suppose the pressure arises from a liquid 

 whose density follows the law p\l — 2qz), where z is measured 

 upwards from the base of the cylinder. This includes the 

 case of a compressible liquid treated in § 5. The pressure p 

 in the liquid follows the law 



r = 11 -gp':(l -<;;), (54) 



where 11 is the pressure at the level of the base of the 



cylinder : thus in applying the solution just obtained we 



replace <f by </ p'</. The complete values of the displacements 



* Camb. Phi). Trans, vol. xiv. /. c. 



