492 RICE AHT. K 



in an aeolotropic body the principal direction of stress and those 

 of strain do not in general coincide, and if we carried out the 

 conceptual experiment suggested above of rotating a spherical 

 body keeping the forces and their points of application "in situ" 

 in the frame of reference, the strains and stresses would not in 

 general be same in an element as they were previously in the 

 element which originally was situated in the same place in the 

 frame of reference ; for the orientation of the two elements would 

 be different although their relation to the external forces would 

 be the same, and that would be a significant change for an 

 aeolotropic element, even although the two elements were 

 homogeneous in nature. Hence the rotation would in general 

 involve an entire alteration in the general state of stress and 

 strain and a change of strain-energy. Thus one of the premises 

 of the argument would collapse. 



We have already referred to the arguments by which Gibbs 

 justifies the use of the determinant H (with a positive value) 

 instead of G for expressing the energy of an isotropic material. 



11. Commentary on Pages 211-214- Approximative Formulae 

 for the Energy and Free Energy in the Case of an Isotropic Solid. 

 The approximative formulae given by Gibbs in [443] and 

 [444] are just examples of the expansion of a function in series 

 by the use of Taylor's theorem, neglecting powers higher than 

 the first. For small strains ri, r2, rz differ little from unity. By 

 [439] E differs little from 3, F from 3, and G or H from unity. 

 Writing E' for E - 3, F' for F - S, and H' for i^ - 1, we can 

 express any function of E, F, H asa, function of E', F', H'. We 

 can expand this function as a series by Taylor's theorem, say 



k-}-aE' + bF' + cH' + higher powers and products of E', F' , W . 



For small strains the higher powers and products are negligible 

 compared to the terms involving the first power. So to the 

 first approximation the function will be 



1 + aE + hF -]- cH 



(where Z = fc — 3a — 36 — c), which has the form of [443] 

 or [444]. 



