456 IX. DYNAMICS OF DEFORMABLE BODIES. 



The gp's will in general be functions of the coordinates of the point, 

 but if the body is homogeneous, that is alike at all points, they will 

 be constants. We shall assume this to be true. If there be a natural 

 state of the body or one in which the body is in equilibrium under 

 the action of external forces, so that the stresses vanish for this 

 state, it is convenient to measure the strains from the natural state. 

 Then the stresses and strains vanish together, so that the terms 

 ^P 01 , . . . qp og vanish. For such a body there are accordingly thirty -six 

 constants qp, the so-called coefficients of elasticity. In the case of a 

 gas there is no natural state, for a gas is never in equilibrium, 

 unless kept so by an envelope, so that every portion of the gas 

 always experiences pressure, consequently we cannot measure the 

 strains from any natural state. 



We have now the theory of elasticity as it was left by its 

 founders, Navier and Cauchy. The idea is due to Green 1 ) of supposing 

 the elastic forces to be conservative and accordingly due to an energy 

 function of the strains. If we call the function &(s x> s y , s z , g x , g y , g z ] 

 we have for the total potential energy due to any strain 



124) 

 The work done in changing the strain is then 



125) 



Comparing this with equations 121 122) we find 



If then the stresses are to be linear functions of the strains, 

 > must be a quadratic function, and, if we measure from the natural 

 state, a homogeneous quadratic function. A homogeneous quadratic 

 function of six variables contains twenty -one terms, so that instead 

 of thirty -six elastic constants for the general homogeneous body we 

 have only twenty -one, that is, the determinant of the qp's in equa- 

 tions 123) is symmetrical, fifteen coefficients on one side of the 



1) Green, Mathematical Papers, p. 243. 



