304 HEAT. 



Young's modulus. We must give the negative sign, since the lower 

 isothermals are higher in the diagram. 



In the case of india-rubber f3 is negative, for if a weight be hung on 

 to a rubber cord, the weight rises if the cord be heated. Hence dd is 

 positive for a sudden increase of stretch. This may be verified by 

 suddenly pulling out an india-rubber band and applying it to the upper 

 lip a sensitive thermoscope when we can easily detect the warming. 

 Keeping the band extended, it soon falls to the temperature of the air. 

 Now allowing it to contract suddenly, it is very appreciably cooled. The 

 ratio of the specific heats under constant stress and constant strain is 

 easily seen to be equal to the ratio of the adiabatic and isothermal 

 moduli, and we can obtain as the formula corresponding to that on 

 p. 289 



For small stresses /3 = a + \s may be generally assumed equal to a, the 

 coefficient of expansion. If we take as the values of the constants for 

 steel 



P = ^ : Y, = 2 x 10 12 : P K V = 4. 2 x 10 7 x 7.8 x 0. 1 12 

 we get 7 = 1.002 



or the adiabatic elasticity is greater than the isothermal by about 1 

 in 500. 



The term Xs would only seriously affect this result when of the order 

 10~ 6 . Since A. is of the order 10~ 4 , this requires a stretch of the order 

 10~ 2 , which cannot be given without permanent set, when our assump- 

 tions are all inadmissible. 



But though not seriously affecting y the existence of As implies that 

 the adiabatic elasticity increases with the strain. It will follow that 

 when longitudinal waves travel along a wire, the elasticity at the points 

 of greatest tension will be greatest, and if we represent the waves by the 

 curve of tension, the crests in the curve will move most rapidly and will 

 therefore gain on the hollows. 



Similar formulae can at once be worked out for the case of shear 

 strains and stresses involving the rigidity modulus. If abscissae repre- 

 sent angles of shear and ordinates tangential stresses per unit area, areas 

 on the diagram will represent work done with unit cube of the sub- 

 stance. We shall not now have anything corresponding to the ordinary 

 coefficient of expansion a, but if we put for the modulus of rigidity 



n = n (l - kt) 



a shear e at constant stress will increase by Xe for a rise of 1, so that 

 we must use A.e instead of ft in the last case. 



The adiabatic change of temperature for change de in shear is 



Xen+Ode 



that is, there is a cooling for increase of shear. But in general it is of 

 the second order, since both A and e are usually small. 



