398 RICE ART. K 



where the coordinates of the point P after the strain are x, y, z. 

 Let Q' be a point adjacent to P' whose coordinates in the state of 

 reference are x' + ^', y' , z'\ the coordinates of Q, i.e., the position 

 after the strain, are 



a;' + £' + u{x' + r), 2/, 2, 



where u{x' + ^') is the same function of the argument x' + ^ 

 that w(x') is of x' . Hence the hnear element P'Q' has been 

 altered from a length |' to a length ^ + u{x' + ^') — w(a;'), 

 besides of course experiencing a bodily translation which is of no 

 importance in discussing the strain. Thus the alteration in 

 length of the linear element is 



u{x' + r) - u{x'), 

 which by Taylor's theorem is equal to 



du ^ ^ d^u 



„/ ^ + 2 j„/2 ^ "T 



If the differential coefficient du/dx' does not vary in value 

 appreciably over a range within which we choose the value of ^', 

 we may neglect the terms in ^'^ etc. (Thus if P'Q' is a range of 

 length extending over a few molecules in the actual body this 

 proviso is the same as that referred to by Gibbs on page 185, 

 line 20.) Under these circumstances the length of P'Q', viz., ^', 

 is altered to ^' (1 + du/dx'), and hence du/dx' is the fraction of 

 elongation of the body at P', viz., the ratio of the change in 

 length to the original length. Gibbs in his discussion actually 

 uses the differential coefficient dx/dx', but it is readily seen 

 that this is just 1 + du/dx', i.e., the ratio of elongation, or the 

 "variation" of the length in the strict meaning of "variation," 

 viz., the ratio of the varied value of a quantity to its previous 

 value. If u(x') is a linear function of x' so that du/dx' is con- 

 stant over the whole body, the elongation has the same value 

 everywhere, and the strain is homogeneous. Otherwise du/dx' 

 varies from element to element of the body, and is in fact a 

 function of x' itself, so that the value of du/dx' depends on 

 where the point P' of the element is situated in the body, 



