of Electrons in an Elastic Solid u^Etlier. 133 



an element of the boundary in the positive direction round a 

 near d%. Jn (15) d<? is an element of an arbitrary volume, 

 and d% an element of the boundary in the direction which 

 points away from the volume bounded. 



Th 6. The bodily equation of motion of any elastic body; 

 solid or fluid; with strains which are large or small; with 

 density in the standard state varying from point to point or 

 constant; with w varying in form from point to point or 

 constant in form; is 



^=(to) 1 Vi (16) 



Bodily forces other than those due to the stress are not 

 here included, merely because below they are not required. 

 They may, however, be included in the following without 

 any difficulty. 



(16) and the corresponding surface traction equation are 

 obtained from the dynamical principle ML<7£=0, where t is 

 the time and L the Lagrangian function. In our case 



fc=-J3J(w+*)*> (i7) 



= - {d/dt) jjj nS^oV* + Iff (n$' n 6 V + SfyidwViy? 

 = - (d/dt) jjj nS^Ms + jj S&KItw*2 



+J5ysa9[»5-(a«) 1 Vi]*. 



Th. 7. If v is any term in w, whether or not a mere 

 function of v, either of the following two equivalent rules 

 suffices to determine G> the corresponding part of dw. 



First rule. — Express dv in the form S</x£V/>?, where </> is a 

 vector linity. Then Qv is — <f>. 



Second rule.— Express dv in the form S<fyi</>Vi> where <b 

 is a vector linity. Then G.v is — (p. 



For instance, if the term 



occurs we put 



d v = -2Sede= -2S^ 1 VeVi, 

 and deduce 



<lra> = 2Vea>, (a^Vi 358 — 2VV«- 



Th. 8. Apply (11) and (15) to the statements that JV = 



