402 RICE 



ART. K 



replace the differential coefficients, so we shall write these 

 equations as 



r' = enr + eW + Cut',] 



v" = 621^ + 6227?' + e23r,(' (3) 



r" = 631^ + 632^' + e33f'J 



dx" 

 ^ dz'' 



djT 



dz'' 



dz^ 

 dz'' 



(4) 



2. Homogeneous Strain. In order to grasp most readily the 

 physical interpretation of these "strain coefficients" which are 

 denoted by the symbols e„, let us consider the case in which 

 u, V, w (and therefore x", y", z") are linear functions of x' , y', z'. 

 Under such a limitation, the quantities e^ are uniform in value 

 throughout the body; in other words the strain is homogeneous. 



Now it is very important to remember at this juncture that 

 it is not so much the actual displacements of the various points 

 which determine the strain, as the differences between the dis- 

 placements of the various points. In Fig. 3 P' is displaced 

 to P" and Q' to Q" ; but to obtain a clear idea of the strain in the 

 part of the body surrounding P' , we must imagine the whole 

 body translated without change of shape and without rotation, 

 i.e., as a rigid body, so as to bring the point P" back to its 



* We are of course using the well-known notation of the "curly" d for 

 partial differentiation. When Gibbs wrote his paper this device for 

 indicating a partial differential coefficient had not established itself 

 universally, and many writers used the ordinary italic d to indicate total 

 and partial differential alike, relying on the reader's own knowledge to 

 make the necessary distinction in each situation. But as, of course, the 

 differential coefficients in [354] and in subsequent equations are partial, 

 we venture to make this small change in Gibbs' notation in view of the 

 universal practice adopted in these matters nowadays. 



