362 J. TK Gibhs — Equilibrium of Heterogeneous Substances. 



£y, = R fv„, r/v, = ^ V\»- (414) 



If, then, we liave ascertained by experiment the value of fy, in terms 

 of 7/y,, -— , ... y-, , and the quantities which express the composi- 

 tion of the body, by the substitution of the values given in (412)- 



, „ -, . . „ dx dz dx" dz" 



(414), we shall obtam £^,„ m terms ol //v„, ^,, . . . ^" ^" • • • ^, 



and tlie quantities which express the composition of the body. 



We may apply this to the elements of a body which may be varia- 

 ble from point to point in composition and state of strain in a given 

 state of reference {x\ y\ z"), and if the body is fully described in 

 that state of reference, both in respect to its composition and to the 

 displacement which it would be necessary to give to a homogeneous 

 solid of the same composition, for which fy, is known in terms of 7/v,, 



-J—, , . . . ^-, , and the quantities which express its composition, to 

 djx dz 



bring it from the state of reference (a;', y\ z) into a similar and 



similarly situated state of strain with that of the element of the non- 



dx" dz" 



homogeneous body, we may evidently regard -^, . . . —, as known 



for each element of the body, that is, as known in terms of x", y", z". 



dx dz 



We shall then have Sy,, in terms of ?/y„, y— „ , . • . -tt/ ? 8"", y",^"; and 



since the composition of the body is known in terms of x", y" , z" , and 



the density, if not given directly, can be determined from the density 



of the homogeneous body in its state of reference {x', y\ s'), this is 



sufficient for determining the equilibrium of any given state of the 



non-homogeneous solid 



An equation, therefore, which expresses for any kind of solid, and 



with reference to any determined state of i-eference, the relation 



dx dz 



between the quantities denoted by fvM 'Am -t-, -, • • • -r, ■, involvino; 



"" dx dz 



also the quantities which express the composition of the body, when 



that is capable of continuous variation, or any other equation from 



which the same relations may be deduced, may be called a fmida- 



mental eqimtion for that kind of solid. It will be observed that the 



sense in Avhich this term is here used, is entirely analogous to that in 



which we have already a|)plied the term to fluids and solids vvdiich 



are subject only to hydrostatic pressure. 



-i^T, 1 P T 1 -1 dx dz . 



When the lundamental equation between fy, , ;/v,, — , , . . . — , is 



