STRAINED ELASTIC SOLIDS 457 



volume vv in the state of strain. (See Gibbs, I, 188, line 27.) 

 This quantity is, as we proved in the exposition, the determinant 

 of the Urs coefficients, which is denoted later in Gibbs' discussion 

 by the symbol H. If we multiply the differential equation 

 written above by vv we obtain 



dev' = tdijv + H ZXrdfr. 



Also, the fr coefficients are defined in the exposition as certain 

 functions of ei, ... ee) i.e., of ai, ... ae which are in their turn 

 functions of the nine coefficients an, so that any differential 

 dfr can be expressed as a sum of the differentials dara, such as 



<f>ndaii + 4>i2dai2 • • • + ^zzda^z, 



where ^n, <i>n, ... ^33 are functions of an, a^, . . . a^. In this 

 way we arrive at Gibbs' expression [355], where Xx', Xy', . . . Zz' 

 are functions of Xx, ■ • • Zz, an, • ■ • 033- The actual func- 

 tional forms we have already developed in the exposition and 

 given the actual linear relations which connect Gibbs' stress- 

 constituents with the usual stress-constituents. 



On page 187 we have an expression for the variation of the 

 energy of the solid body if an infinitesimal amount of material is 

 added to it. Again we must carefully distinguish between the 

 variational symbol 8 and the differential symbol D, and interpret 

 correctly the use of the accents. Thus an element of the 

 surface of the body in the state of strain is represented by 

 Ds. If by crystallization from a surrounding fluid, for example, 

 the body increases in size, the surface is displaced normally 

 outwards by an infinitesimal amount which we represent by 

 8N. This might be regarded as having a constant value every- 

 where on the surface, giving a uniform thickness for the addi- 

 tional layer. But this is not so of necessity; 8N in general is 

 regarded as a function of the position of the center of the element 

 Ds, a function obviously infinitesimally small in value. Indeed 

 8N could be regarded as some ordinary function (t>{x, y, z) of the 

 coordinates of a point on the surface multiplied by an infinitesi- 

 mal constant. A sign of integration, of course, refers to the 

 differential Ds. For example f8NDs is the increase in volume 



