STRAINED ELASTIC SOLIDS 481 



the solid during the infinitesimal strain is as usual 



Xx'daii + XY'dai2 . . . + Zz-dazz. 



This is of course equal to the work of all the surface forces 

 during the variation of strain. These surface forces may be 

 regarded as due to the pressure p on all the faces (a hydrostatic 

 pressure) together with additional forces on four of the faces. 

 The work of the hydrostatic pressure is —'pdv which is equal to 



— p(Aii^aii + Avidan . . . + Azzdaz^. 



Hence by subtracting this from the increase of energy of strain 

 we obtain the work of the additional forces and this is seen to be 

 equal to the right hand member of our [404a], and becomes the 

 right hand side of [404] when Gibbs' special geometrical con- 

 ditions are assumed. 



10. Commentary on Pages 201-211. Expression of the Energy 

 of a Solid in Terms of the Entropy and Six Strain-Coefficients. 

 Isotropy. Having discussed the conditions of equilibrium Gibbs 

 proceeds in the subsection on the Fundamental Equations for 

 Solids to consider the problem of expressing the functional re- 

 lationship between the energy per unit volume, the entropy per 

 unit volume and the nine strain-coefficients. If ck- is expressed 

 as a function of -qv, an, an, . . . azz, or i/t' is expressed as a function 

 of t, an, an, . . . azz, we can by differentiation obtain, as we have 

 already pointed out in this article, the stress-strain relations, 

 which will be nine of the eleven independent relations referred to 

 by Gibbs on page 203 . He opens the subsection with some rather 

 involved considerations on a special point, which we pass over 

 for the moment, and then briefly touches on the fact that the 

 energy or free energy functions must have a special form in the 

 nine strain-coefficients, inasmuch as the strain of an element is 

 capable of only six independent variations. This we have 

 already explained in our discussion, where we chose the six 

 quantities /i, f^, ... /e to represent the displacements arising 

 from pure strain, as distinct from possible additional dis- 

 placements involved in the nine coefficients an, an, . . ■ azz, which 

 are the result of a pure rotation and produce no distortion of the 



