122 RESISTANCE OF MATERIALS 



this expression may be constant, p r + p h must be constant. Denoting 

 this constant by &, we have 



(170) P r + Ph = * 



Now eliminating^ between equations (168) and (170), we have 



As the radius r increases, the stress p r increases or decreases ac- 

 cording to whether the constant C is positive or negative ; that is, 

 whether the internal pressure is greater or less than the external. 

 Since the sign of C has no effect on the result, we may say that for 

 a point at a distance r-f-Ar from the axis the radial stress is of 

 amount p r Ap r , such that 



S implifying this expression, it becomes 



(k - Pr) r* + 2 r&r (k - p r ) - 

 and subtracting from it the original relation, namely, 



we have 2 rAr (k p r ) r 2 Ap r = 0,* 



whence *& = 2r(*-j>,) = g. ; 



Ar r r 



C 

 and, since p h = , this becomes 



T 



Substituting equation (171) in equation (169) and making use 

 of equation (170), we have 



, 2(7 2C 



Ph=Pr + -^- = k -P h + -jT> 



whence 



("2) * = ! + 



and therefore, from (170), 



