438 RICE 



ART. K 



above, which is the usual treatment. It arises from his en- 

 deavor to make the foundation of his arguments as wide as 

 possible. He lays down no restriction that the state of 

 reference shall be so near to that of the state of strain that a 

 rectangular element is but little strained from that form in the 

 changes which take place between the two states. His only- 

 proviso is that the differential coefficients dx/dx', etc. shall 

 not alter appreciably over molecular distances, i.e., that the 

 strain is homogeneous within a physically small element of 

 volume. Let us retrace the ground covered by the argument 

 which we followed when deahng with the energy of strain. 

 The rectangular element of volume in the state of strain has its 

 center at a point P whose coordinates are x, y, z with reference 

 to the OX, OY, OZ axes; this element was, in the state of 

 reference, an obhque parallelopiped whose centre was at the 

 point P' whose coordinates are x', y', z' with reference to the 

 OX', OY', OZ' axes. Let the edges of the element in the state of 

 strain be parallel to OX, OY, OZ, and following the course we 

 used earlier let us call the mid -points of the faces perpendicular 

 to OX, Q and U, so that the local coordinates of Q with reference 

 to local axes of x, y, z at P are ^, 0, 0, and of U are — ^, 0, 0. 

 Those of Q', the center of the corresponding face of the un- 

 strained element, for the local axes of x', y', z' at P' are ^', -q', f ' 

 where, by equations (21), 



k = an^' + anv' + Qisf', 

 = a.i^' + 0227?' + a23^', 



= 031^' + ^321?' + Ossr'., 



(33) 



Now let the slight increase of strain take place which we 

 considered above when we treated this problem in a more 

 restricted manner; the point P is displaced to a neighboring 

 point Ps, say, while Q and U are displaced to neighboring 

 points Qs and f/g. The strain-coefficients are now an -\- 8an, 

 etc. The local coordinates of Qs with reference to local axes of 

 X, y, z at P5 are ^ + b^, 8r}, 8^ where 



