STRAINED ELASTIC SOLIDS 401 



Hence the coordinates of the point in its displaced position, 

 viz., P", are given by 



x" =x' + u{x', y', z'), y" = y' + v{x' , y' , z'), 



z" = 2' + w{x', y', z'). (1) 



Consider a neighboring point whose undisplaced position is 

 Q' with the coordinates 



x' + r, y' + V, z' + r. 



After the displacement, the coordinates (of Q") are 



a:' + r + u{x' + r, y' + V, 2' + r), 



and two similar expressions. Neglecting as before and for the 

 same reason the differential coefficients higher than the first, 

 these become x" + ^", y" + r,", z" + f ", where 



du , du , du , 

 dv , dv , dv , 

 dw dw dw , 



(2) 



(For convenience and brevity we drop the bracketed coordinates 

 after the symbols u, v, w; but it must not be forgotten that u 

 is to be understood as the function u{x', y', z'), etc). 



It will be convenient to introduce single letter symbols to 



the strain to a different set of axes OX, OY, OZ. The two sets of axes are 

 not necessarily identical, but he regards them as "similar, i.e., capable 

 of superposition" ; so that if one set is orthogonal, then also is the other. 

 At the outset, however, there is an element of simplification in keeping 

 the same set of axes; but in order that there may be no confusion later 

 when we adopt Gibbs' wider analysis we are now referring to the co- 

 ordinates of the displaced point as x", y", z" instead of x, xj, z, thus 

 keeping the latter triad of letters to represent, as Gibbs does, the coor- 

 dinates of the displaced point with reference to a second system of axes. 



