J. W. Gibhs — Eqiiilihrhiii) of Heterogeneous Subsfatwes. 301 



Fundamental Equations for Solids. 



The principles developed in the preceding pages show that the 

 solution of problems relating to the equilibrium of a solid, or at least 

 their reduction to purely analytical processes, may be made to de- 

 pend upon our knowledge of the composition and density of the solid 

 at every point in some particular state, which we have called the 

 state of reference, and of the relation existing between the quantities 



doc dsc dz 



which have been represented by f,,, ?;,.», ^" ^" • • • ^ ' ''■' y'' 



and z. When the solid is in contact with fluids, a certain knowledge 

 of the jiroperties of the fluids is also requisite, but only such as is 

 necessary for the solution of problems relating to the equililirium of 

 fluids among themselves. 



If in any state of which a solid is capable, it is homogeneous in its 

 nature and in its state oi strain, we may choose this state as the state 



of reference, and the relation between fy,, 7/v, , -y^, , . . -^ , will be 



independent of x', y\ z'. But it is not always possible, even in the 



case of bodies which are homogeneous in nature, to bring all the 



elements simultaneously into the same state of strain. It would not 



be possible, for example, in the case of a Prince Rupert's drop. 



dii* ffz 



If, however, we know the relation between e^,, , //y, , -'— , . . . -7-^, 



«.>■ dz 



for any kind of homogeneous solid, with respect to any given state of 



reference, we may derive from it a similar relation with respect to 



any other state as a state of reference. For if x, y\ z' denote the 



co-ordinates of points of the solid in the first state of reference, and 



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 H 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 



B 



(413) 



