186 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



But if the strain varies, we may consider e v / as a function of q v , an( i 

 the nine quantities in (354), and may write 



(355) 



where Z X ', ... Zy denote the differential coefficients of e V ' taken with 



doc dz 

 respect to -^ n ...^ f . The physical signification of these quantities 



aX Q/Z 



will be apparent, if we apply the formula to an element which in the 

 state of reference is a right parallelepiped having the edges dx', dy', dz', 

 and suppose that in the strained state the face in which x' has the 

 smaller constant value remains fixed, while the opposite face is moved 

 parallel to the axis of X. If we also suppose no heat to be imparted 

 to the element, we shall have, on multiplying by dx f dy' dz', 



Now the first member of this equation evidently represents the work 

 done upon the element by the surrounding elements; the second 

 member must therefore have the same value. Since we must regard 

 the forces acting on opposite faces of the elementary parallelepiped as 

 equal and opposite, the whole work done will be zero except for the 



dx 

 face which moves parallel to X. And since S-T,dx' represents the 



distance moved by this face, X^dy' dz' must be equal to the com- 

 ponent parallel to X of the force acting upon this face. In general, 

 therefore, if by the positive side of a surface for which x f is constant 

 we understand the side on which x f has the greater value, we may say 

 that Z x / denotes the component parallel to X of the force exerted by 

 the matter on the positive side of a surface for which x' is constant 

 upon the matter on the negative side of that surface per unit of the 

 surface measured in the state of reference. The same may be said, 

 mutatis mutandis, of the other symbols of the same type. 



It will be convenient to use 2 and 2' to denote summation with 

 respect to quantities relating to the axes X, Y, Z, and to the axes 

 X', Y', Z f , respectively. With this understanding we may write 



This is the complete value of the variation of e V ' for a given element 

 of the solid. If we multiply by dx' dy' dz', and take the integral for 



