■ X. On the Equilibrium of Heterogeneous Substances. 



By J. WlLLARO GlHBS. 



{Continued from page 248). 



THE CONDITIONS OP INTERNAL AND EXTERNAL EQUILIBRIUM FOR 



SOLIDS IN CONTACT WITH FLUIDS WITH REGARD TO ALL POSSIBLE 



STATES OP STRAIN OF THE SOLIDS. 



In treating of the physical properties of a solid, it is necessary to 

 consider its state of strain. A body is said to be strained when the 

 relative position of its pai'ts is altered, and \)y '^t^ state of strain \^ 

 meant its state in respect to the relative position of its parts. We 

 have hitherto considered the equilibrinm of solids only in the case in 

 which their state of strain is determined by pressures having the 

 same values in all directions about any point. Let us now consider 

 the subject without this limitation. 



If x', y', z' are the rectangular co-ordinates of a point of a solid 

 body in any completely determined state of strain, which we shall 

 call the state of reference, and x, y, z, the rectangular co-ordinates of 

 the same point of the body in the state in which its properties are the 

 subject of discussion, we may regard x, y, z as functions of x , y\ z' , 

 the form of the functions determining the second state of strain. 

 For brevity, we may sometimes distinguish the variable state, to 

 which ic, y, z relate, and the constant state (state of reference), to 

 which x\ y\ z' relate, as the strained and the unstrained states ; but 

 it must be remembered that these terms have reference merely to the 

 change of form or strain determined by the functions which express 

 the relations of x, y, z and x', y', z', and do not imply any particular 

 physical properties in either of the two states, nor prevent their 

 possible coincidence. The axes to which the co-ordinates .>', y, z, and 

 x', y\ z' relate will be distinguished as the axes of X, Y, Z, and 

 A"', Y\ Z'. It is not necessary, nor always convenient, to regard 

 these systems of axes as identical, but they should be similar, i. e., 

 capable of superposition. 



The state of strain of any element of the body is determined by the 

 values of tlie differential coefficients of x, y, and z with respect to 

 ic', ;y', and z'; for changes in the values of x, y, 2, when the differential 

 coefficients remain the same, only cause motions of translation of the 



Trans. Conn. Acad., Vol. III. 44 May, 1877. 



