436 RICE 



I 



ART. K 



ing pull hkXz across a Z-face or a shearing pull klZx across 

 an X-face. One way yields hklXzdfi for the work done; the 

 other yields hklZxBf^ for it; these are the same quantity since 

 Zx = Xz, but we must not count both or we shall obtain twice 

 the correct value, and this is just what we would be doing if we 

 added all the terms obtained above. In this comparatively 

 simple way we can reasonably assume a result which can be more 

 rigorously established by other methods, viz., that when the strain 

 of a solid is varied from a state in which the strain coefficients 

 are en, • . . ^33, to one in which the coefficients are en + 5en, ■ . . 

 633 + 8633, the increase in energy in an element of volume is the 

 product of the volume of the element and 



Xi8f, + X25/2 + X35/3 + X45/4 + X,8f, + X,5U (32) 



This expression takes the place of the expression —p8v for a 

 fluid in the formulation of the variation of the internal energy of 

 a solid body in any general change of temperature and state. 

 That the expression (32) degenerates to this in the case of a 

 fluid can be readily demonstrated, for we have seen earlier that 

 in the case of a fluid X4, X5, X 6 are zero, and Xi = X2 = X3 

 = —p; hence (32) becomes 



-p5(/i+/2+/3), 



and, since unit volume expands in this case to 



(1 + 6/0 (1 + 5/2) (1 + 5/3), 



or practically 



l+6(/i+/2+/3), 



it follows that 8v is equal to the original volume of the element 

 multiplied by 8(fi + /2 + /3). 



The whole of the argument so far has avoided any considera- 

 tion of changes of temperature arising from strain and assumes 

 all the energy to be mechanical. In so far as this is allowable 

 the expression X16/1 ... + X&Sfe must be regarded as the 

 variation of a function of the six quantities fi, ... /e, so that 



