202 



EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



x", y", z" the co-ordinates of the same points in the second state of 

 reference, we shall have necessarily 



dx dx dx" . dx dy" , dx dz" 



' 



and if we write R for the volume of an element in the state (x", y", z") 

 divided by its volume in the state (x' } y', z'\ we shall have 



(413) 



. (414) 



If, then, we have ascertained by experiment the value of e v > in terms 

 of J/ V '> -T-, >> - -T-?J and the quantities which express the composition 



of the body, by the substitution of the values given in (412)-(414), 



,,,,,. . dx dz dx" dz" , . 



we shall obtain e v m terms of ^ v , -7-77, . . . -^-7,, -j-r, . . . ^-^-, and the 



dx dz dx dz 



quantities which express the composition of the body. 



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

 variable 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 e v is known in terms of T/ F , 



dx dz 



-7-7, . . . -j ft and the quantities which express its composition, to 



bring it from the state of reference (x' } 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 -7-7 , . . . -r-r 



as known 



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



dir el z 



We shall then have e v in terms of ;/ v , -7-7,, . . . -7-77, x", y", z" ; 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', z'), 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 reference, the relation 



