SPHERES AND CYLINDERS 129 



Hence the integral becomes 



i 



_ MQ M Q \Wa 



wa wa >J/ 



or 



At the upper end of the quadrant B the conditions are I = ^ and 



- = 0. Substituting these values in the first differential coefficient 

 dl 



obtained from equation (71), namely, 



dU _ MQ 



dl wa 

 we have 



sin 

 whence 



\\hcre X is an arbitrary integer. Choosing the smallest value of X, 

 namely 1, this condition becomes 



. _ a 



1 */ 2 " ' 



whence 



If the thickness of the tube is denoted by h, then, for a section of 

 unit length, /= > and formula (72) becomes 



(73) w = - t 



4 



Formula (73) gives the critical pressure just preceding collapse ; that 

 is to say, it gives the maximum external pressure w per unit of area 

 which a cylindrical tube of thickness h can stand without crushing. 



Problem 132. What is the maximum external pressure which a cast-iron pipe 

 18 in. in diameter and \ in. thick can stand without crushing ? 



