LAWS OF TORSION. 55 



Coulomb have shown that within pretty wide limits these oscil- 

 lations are virtually isochronous. It follows that the moment of 

 the couple which tends to bring the twisted wire to its position 

 of equilibrium is proportional to the angle of deflectiftn. 



If the total torsion is 0, the expression for this couple is CO. The 

 constant C represents the moment of the couple which would be 

 necessary to twist the wire through unit angle, if the law of pro- 

 portionality held for all angles ; we shall call it the coefficient of torsion 

 of the wire. Experiment shows that it is virtually independent of the 

 weight of the suspended body in other words, of the tension of the 

 wire. 



If the wire is cylindrical and circular in section, of length / and 

 diameter d, we have, according to Coulomb, 



(2) C= /4" 



The coefficient of a wire is therefore inversely as its length, and 

 proportional to the fourth power of its diameter, or to the square of 

 its section, the factor /* only depending on the nature of the wire, and 

 on the temperature. 



708. If in this formula we make d = i and /= i, we have C = /*. 

 The coefficient /x is then the numerical expression of a couple which 

 can twist through unit angle a cylinder the length of which is a 

 centimetre, and the base a centimetre in diameter, and which would 

 act upon one of the bases, the other being rigidly fixed. The 

 coefficient /u, is not really the moment of a couple ; it represents the 

 quotient of a force by a surface, and only differs by a numerical factor 

 from what is usually called the rigidity or second modulus of elasticity. 



Let us consider, in fact, a straight cylinder of section S, and 

 suppose that while one of the bases is rigidly fixed the other is acted 

 upon at each point by a tangential force, constant in magnitude and 

 direction, equal to F for unit surface ; each section parallel to the 

 bases is displaced parallel to itself without deformation, and by a 

 quantity proportional to its distance from the fixed base; the 

 cylinder is then turned through an angle a, independent of its height, 

 and proportional to the force F, and we may write 



a = -F, or F = a$. 



9 



The coefficient 9 is the second modulus of elasticity ; it represents 

 physically the force which would be necessary to incline through an 



