STRAINED ELASTIC SOLIDS 501 



Similar chaDges will have to be made in [459] and [461], if the 

 writer's emendation of [449] is correct. 



Before leaving this subsection we shall revert for a moment to 

 the special point passed over at the beginning of the com- 

 mentary on this part. Pages 201 and 202 are rather involved 

 but the point appears to be as follows. It has been implied 

 hitherto that no particular physical properties are imposed 

 on the state of reference. In ordinary elementary discussions 

 in the text-books it is taken as unstressed, i.e., without any 

 strain energy. Thus if a relation is given between ev and 

 rjv', duy ^12, . • • ^33, then ev is the intrinsic energy of the 

 state of strain; but if no such restriction is imposed on the 

 state of reference then, since the coefficients an, an, ... ass 

 express a relation between the state of strain and the state of 

 reference, the function ev will give the excess of energy in the 

 former state over the latter for the material occupying unit 

 volume in the latter. Provided the state of reference is at all 

 events one of homogeneous strain, this introduces no difficulties 

 since the energy in any element of the solid in the state of 

 reference is the same as that in any other, and therefore ev 

 differs from the intrinsic energy in the state of strain (per unit 

 volume of the state of reference) by a constant amount, (i.e., 

 the same for all elements of volume). But if, as Gibbs suggests, 

 it happens that in some cases it is impossible to bring all ele- 

 ments in the state of reference simultaneously into the same 

 state of strain, this means that in the state of reference the 

 energy in an element depends on its position in the state of 

 reference, i.e., on the coordinates of the point which it surrounds. 

 We can, however, take some particular element in the state of 

 reference as being in what we may call a "standard state." 

 The condition in any other element in the state of reference can 

 be stated in terms of the strain-coefficients which give the relation 

 between the state of this latter element and the standard state, 

 and the energy in this element in the state of reference will, 

 apart from a constant, be a function of these latter strain-coeffi- 

 cients. Thus ev will now be a function not only of the strain- 

 coefficients ail, ai2, ... ass (connecting the state of strain with 

 the state of reference) but also of other strain-coefficients con- 



