ISOTHERMAL AND ADIABATIC CHANGES. 305 



The consideration of the ratio y is especially important in this case 

 since the torsional resistance of wires is so frequently used to measure 

 small torques, the torque being deduced from the angle of twist and the 

 time of vibration of the system acted on. Now the angle of twist may 

 be produced slowly and the condition of the wire may be isothermal ; the 

 vibrations, on the other hand, may be rapid and the conditions may be 

 somewhere between isothermal and adiabatic conditions. If the adiabatic 

 and isothermal elasticities differed seriously, error might arise in such 

 experiments as that of Cavendish on the constant of attraction. 



The error would be the greatest if the conditions in vibration were 

 truly adiabatic and those in steady deflection by the torque truly 

 isothermal. 



We may show that 



7 = T 



2 



For steel A. is not far from 3 . Suppose a wire 1 mm. radius and 



10 4 



1 metre long is turned through 1 radian at its lower end, then the 



extreme value of e is -; n is about 10 12 . 

 ICr 



Substituting the numerical values we get 



r = T^lcF 



and the ratio will be still more nearly 1 for less shear. This implies 

 that the two rigidities are for all practical purposes identical for steel 

 wire. 



If we have any other type of stress and strain we have the results 



( + if the effects of rise of temperature and increase of stress are opposite) 



1 

 and y 1-/32E,0 



where ft is the change of strain under constant stress for 1 rise in 

 temperature, E* is the isothermal modulus of elasticity, K P is the specific 

 heat under constant strain, and p is the mass in the volume so strained. 

 Thus in a spiral spring we may deal with unit length measured along the 

 axis of the spiral, when p will be the mass of the spring per unit length. 

 The reader will find that for such a spring y is practically 1, as might 

 be expected when we remember that the strain is chiefly torsional. 



