214 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



Moreover, since p must vanish in (452) when / y = r 3 , we have 



2e + 4/r 2 +&r =0. (456) 



From the three last equations may be obtained the values of e, f, h t 

 in terms of r , V y and R ; viz., 



h=-tR-V. (457) 



The quantity r , like J? and V, is a function of the temperature, the 

 differential coefficient $ representing the rate of linear expansion 



of the solid when without stress. 



It will not be necessary to discuss equation (443) at length, as the 

 case is entirely analogous to that which has just been treated. (It 

 must be remembered that r] T , in the discussion of (443), will take the 

 place everywhere of the temperature in the discussion of (444).) If 

 we denote by V and R' the elasticity of volume and the rigidity, 

 both determined under the condition of constant entropy, (i.e., of no 

 transmission of heat,) and for states of vanishing stress, we shall 



have the equations : 



* 



, (458) 



(459) 







2e' + 4/V 2 + feV = 0. (460) 



Whence 



S=*r,K-*r,r, /'=^^. h'=-%K-V. (461) 



In these equations r , R', and V are to be regarded as functions of 

 the quantity T/ V >. 



If we wish to change from one state of reference to another (also 

 isotropic), the changes required in the fundamental equation are easily 

 made. If a denotes the length of any line of the solid in the second 

 state of reference divided by its length in the first, it is evident that 

 when we change from the first state of reference to the second the 

 values of the symbols e V '> ^v> ^v> H are divided by a 3 , that of E 

 by a 2 , and that of F by a 4 . In making the change of the state of 

 reference, we must therefore substitute in the fundamental equation 

 of the form (444) a^ T) a*E, a*F, o?H for ^ T , E, F, and H, 

 respectively. In the fundamental equation of the form (443), we 

 must make the analogous substitutions, and also substitute a B r] T for 

 7/v'- (It will be remembered that i', e', f, and h' represent functions 

 of jj v >, and that it is only when their values in terms of 7/ V ' are 

 stituted, that equation (443) becomes a fundamental equation.) 



