EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 211 



the positive or toward the negative direction of the axis of Z. In 

 the first case, H is evidently positive ; in the second, negative. The 

 determinant H will therefore be positive or negative, we may say, 

 if we choose, that the volume will be positive or negative, according 

 as the element can or cannot be brought from the state (x, y, z) to the 

 state (x' y y f , z') by continuous changes without giving its volume the 

 value zero. 



If we now recur to the consideration of the principal axes of strain 

 and the principal ratios of elongation r t , r 2 , r 8> and denote by U ly U 2 , 

 U 3 and U^, U 2 , U 3 ' the principal axes of strain in the strained and 

 unstrained element respectively, it is evident that the sign of r v 

 for example, depends upon the direction in U l which we regard as 

 corresponding to a given direction in U^. If we choose to associate 

 directions in these axes so that r x , r 2 , r s shall all be positive, the 

 positive or negative value of H will determine whether the system of 

 axes U lf U 2 , U s is or is not capable of superposition upon the system 

 //, U 2 , U 3 ' so that corresponding directions in the axes shall coincide. 

 Or, if we prefer to associate directions in the two systems of axes 

 so that they shall be capable of superposition, corresponding directions 

 coinciding, the positive or negative value of H will determine whether 

 an even or an odd number of the quantities r lt r 2 , r 3 are negative. 

 In this case we may write 



(442) 



It will be observed that to change the signs of two of the quantities 

 r i r z> r s ls simply to give a certain rotation to the body without 

 changing its state of strain. 



Whichever supposition we make with respect to the axes U lt U 2 , U 3 , 

 it is evident that the state of strain is completely determined by the 

 values E, F, and H, not only when we limit ourselves to the consider- 

 ation of such strains as are consistent with the idea of solidity, but 



also when we regard any values of -r ,, ... -j-> as possible. 



Approximative Formulce. For many purposes the value of e V ' for 

 an isotropic solid may be represented with sufficient accuracy by the 

 formula 



6y , = i' + e 'E +fF+ h'H, (443) 



where i', e, /', and h' denote functions of q v > \ or ^ ne value of i/r V ' by 

 the formula 



VT V , = i + eE+fF+ hH, (444) 



