488 RICE ART. K 



the principal axes of strain are oriented alike in all elements; 

 that will also be true of the principal axes of stress if in addition 

 the body is homogeneous in nature. But it will naturally occur 

 to the reader to inquire whether the principal axes of strain are 

 coincident with those of stress, and indeed this query and its 

 answer is just the matter at issue at this point in Gibbs' text. A 

 few lines before, Gibbs refers to the now familiar fact that the 

 state of strain (as distinct from rotation) is given by six func- 

 tions of the strain-coefficients an, a^, . . . ass, choosing, for 

 reasons now fully discussed, ai, . . . ae as these functions (or 

 A,B, C, a, b, c, as he styles them) and points out that for any 

 material, homogeneous in nature or not, isotropic or not, the 

 energy per unit volume will be a function of the entropy per 

 unit volume and the six strain-functions. This we have 

 already discussed in the present article. For isotropic materials, 

 however, there is a certain simplification, three functions of the 

 strain-coefficients being sufficient for this purpose. Gibbs 

 derives this result from the sentence at the end of the short para- 

 graph referred to above, namely the sentence: "If the unstrained 

 element is isotropic" (the italics are the writer's) "the ratios of 

 elongation for these three lines must with rjv determine the 

 value of €v'." Now this is hardly obvious without some 

 further consideration of the meaning of isotropy in this con- 

 nection. Space does not permit us to discuss the matter fully, 

 but the central idea can be indicated. The essential character 

 of an elastically isotropic solid is embodied in two facts. 



1. For any system of external forces the principal directions 

 of stress in any element are identical with the principal direc- 

 tions of strain. 



2. The number of elastic constants required to express the 

 relations between stress and strain for small strains is two. 



Thus if we take the axes of reference to be parallel to these 

 principal directions, we have the extremely simple stress-strain 

 relations (in the conventional text-book form) 



Xx = X3 + 2/xeii, 



Yy = X8 + 211622, 



Zz = X5 4" 2^1633. 



