STRAINED ELASTIC SOLIDS 417 



set [439]. Again pursuing a line of argument such as led to 

 (18) we obtain 



Ka = K'iAncx' + ^12/3' + A,,y'),] 



K^ = K'iA^icc' + A22^' + A2,7'),\ (27) 



Ky = K'iAncc' + ^32/3' + AW),] 



where K and a, j8, y are the area and direction cosines of the 

 normal after the strain for a bounded plane surface (referred to 

 OX, OY, OZ) whose area and direction cosines are given hy K', 

 a, j8', y' in the unstrained state (referred to OX', OY', OZ'). 

 As already stated these results are of importance on pages 192, 

 193 of Gibbs' discussion, 



3. Heterogeneous Strain. In the discussion just completed 

 X, y, z have been considered as linear functions of x' , y', z' , with 

 the result that the Ors quantities (i.e., dx/dx', etc.) are constants 

 throughout the system, and the same remark applies to the 

 Ars quantities (viz., (dy/dy') (dz/dz') - (dy/dz') (dz/dy'), etc.). 

 If, however, the displacements of the points from the un- 

 strained to the strained states have such values that x, y, z 

 are not linear functions of x', y', z', then the quantities denoted 

 by Gts are functions of x', y', z' varying from point to point, and 

 the same is true for the quantities denoted by Ars and also for 

 the determinant denoted by the symbol H in Gibbs. (The 

 flexure or the torsion of a bar are examples of heterogeneous 

 strain.) As far as interpretation is concerned these functions 

 still determine the various ratios of enlargement, with the 

 understanding that the values of these functions at a given 

 point give the necessary data for calculating the conditions of 

 strain in a physically small element of volume surrounding the 

 point. In short, we regard the strain as homogeneous through- 

 out any physically small element of volume, giving the various 

 Qri and Ars quantities the values throughout this element which 

 they have at its central point. 



4. Analysis of Stress. In using such a phrase as "the system 

 in its unstrained state" we implicitly assume that we shall take 

 this state as one in which the internal actions and reactions 

 between any two parts of the body shall be regarded as vanish- 

 ing. When we begin to consider if such actions are really zero, 



