STRAINED ELASTIC SOLIDS 477 



these that Xx, Xy, • • • Xz must individually maintain constant 

 values throughout the solid, and that the strains, therefore, 

 being definite functions of these, would also be uniform in value 

 throughout ; but the conclusion is unwarranted, as the equations 

 do not assert that each of the nine differential coefficients is zero. 

 The torsion of a bar by gripping in the hands and twisting is an 

 instance of heterogeneous strain under surface forces, which 

 will be familar to all readers who have a special acquaintance 

 with text-books of elasticity. 



9. Commentary on Pages 1 97-201 . The Variations of the Tem- 

 perature of Equilibrium with Respect to the Pressure and the 

 Strains. The Variations of the Composition of the Fluid. At 

 the bottom of page 197, Gibbs begins an argument leading 

 to equations [407] and [411]. Equation [407] is the analogue 

 of the well-known equation, first discovered by James Thom- 

 son, giving the alteration in the melting point of a solid due to 

 the increase of pressure on the surface. Perhaps if we put the 

 analysis in a more general form than in the text it may assist 

 the reader. We make no special arrangement about axes. 

 The unit cube in the state of reference becomes in general, in 

 the state of strain, an obhque parallelopiped whose volume has 

 changed to y^/, which as we have seen is equal to the determinant 



A pair of opposite faces of the cube are in contact with the fluid in 

 the state of reference and in the state of strain, so that one of the 

 principal axes of stress is normal to this pair of faces of the 

 oblique parallelopiped, the assigned homogeneous state of strain 

 being maintained by suitable surface constraints on the remain- 

 ing pairs of faces. Let there be an infinitesimal change to a new 

 condition of equilibrium; this will involve changes of the strains 

 to an + dan, an + dan, ■ ■ ■ ass + dazs, of the fluid pressure to 

 p + dp, of the temperature to t + dt, of the potential ni to 

 Ml + dni, and of the energy and entropy of the soHd to e + de 



