44 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



a plate of nitre treated in the same way. That these films are in a state of 

 constraint may be proved by heating them slightly, when they recover their 

 original dimensions. 



As all these permanently compressed substances have passed their limit of 

 perfect elasticity, they do not belong to the class of elastic solids treated of in 

 this paper ; and as I cannot explain the method by which an uncrystallised body 

 maintains itself in a state of constraint, I go on to the next case of twisting, 

 which has more practical importance than any other. This is the case of a 

 cylinder fixed at one end, and twisted at the other by a couple whose moment 

 is .1 / . 



CASE II. 



In this case let 80 be the angle of torsion at any point, then the resistance 

 to torsion in any circular section of the cylinder is equal to the twisting force M. 



The resistance at any point in the circular section is given by the second 

 Equation of (14). 



_m dS0 

 qt ~~2 r ~fa- 



This force acts at the distance r from the axis ; therefore its resistance to torsion 

 will be q t r, and the resistance in a circular annulus will be 



_ , , , 



q t r2irrdr = mtn 3 -? dr 



and the whole resistance for the hollow cylinder will be expressed by 



,, ftnt dW , 



T 7E"fa *fl 



88 

 720 M 



m = 



ir* n 



In this equation, m is the coefficient of linear elasticity ; a, and o, are the 

 radii of the exterior and interior surfaces of the hollow cylinder in inches ; M is 

 the moment of torsion produced by a weight acting on a lever, and is expressed 



