SPHERES AND CYLINDERS 131 



If the ends of the cylinder are free from restraint, or if the cy Under 

 is subjected to a uniform longitudinal stress, the longitudinal defor- 

 mation must be constant throughout the cylinder. The longitudinal 

 deformation, however, is due to the lateral action of p r and p h , and is 



of amount -^- L - + -*2- , or - (p r + p h ), in which m denotes Poisson's 

 tnK in a 'niEi 



constant. Therefore, if this expression is constant, p r + p h must 

 be constant, and hence 7 



Pr + Pk = *> 



where k is a constant. Consequently,^ = k p r , and substituting 

 this value of p k in equation (74) and multiplying by r, it becomes 



krdr = 2 rp r dr - 

 which may be written , 



Integrating, ^ 



in which C v is the constant of integration; whence 



Also, since p k = k - p rJ 



Now suppose that the cylinder is subjected to a uniform internal 

 pressure of amount // p-r unit of area, and also to a uniform external 

 pressure of amount w e per unit of area. Then p r = w e when r = a, 

 and p r = w f when r = b. Substituting these values in equation (75), 



k C k Ci 





Tl i .-ref ore, substituting these values of C l and k in equations (75) and 

 <76), they become 



/'r = - ^Ta TT- TIT 



(77) 



