EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 185 



the state of reference, and x, y, 0, 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 

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

 x', y', z' relate, as the strained and 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 y, z and x', y', z', and do not imply any particular physical 

 properties in either of the two states, nor prevent their possible coin- 

 cidence. The axes to which the co-ordinates x, y, z and x', y', z' relate 

 will be distinguished as the axes of X, Y, Z and X', 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 the differential coefficients of x, y, and z with respect to 

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

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

 body. When the differential coefficients of the first order do not 

 vary sensibly except for distances greater than the radius of sensible 

 molecular action, we may regard them as completely determining the 

 state of strain of any element. There are nine of these differential 

 coefficients, viz., 



dx dx dx 



dx~" djj" dz 7 ' 



dy dy dy 



dx" dy" dz" 



dz dz dz 



dx" dy" dz 7 ' 



(354) 



It will be observed that these quantities determine the orientation of 

 the element as well as its strain, and both these particulars must be 

 given in order to determine the nine differential coefficients. There- 

 fore, since the orientation is capable of three independent variations, 

 which do not affect the strain, the strain of the element, considered 

 without regard to directions in space, must be capable of six inde- 

 pendent variations. 



The physical state of any given element of a solid in any unvarying 

 state of strain is capable of one variation, which is produced by 

 addition or subtraction of heat. If we write ey and rj y , for the energy 

 and entropy of the element divided by its volume in the state of 

 reference, we shall have for any constant state of strain 



06y = 



