STRAINED ELASTIC SOLIDS 



425 



parallelopiped we represent by the symbol F, estimated per unit 

 volume, so that the a;-component of this on the element of volume 

 we are considering is Fx • S^v^. Since the medium is in equilib- 

 rium, the sum of the components in any direction of all the 

 forces on an element of volume (including those due to influences 

 external to the medium and those arising from the part of the 

 medium surrounding the element) is zero, and therefore 



dXx dXy dXz 

 dx dy dz 



In just the same manner we can prove that 



dYx dYy dYz \ (29) 



dx dy dz 



dZx dZy dZz 

 — +— +— +Fz = 0. 

 dx dy dz 



The equations [377] constitute a particular case of these; for 

 the forces arising from gravity have no horizontal components 

 and, since in Gibbs OZ is in the vertically upward direction, 

 Fz is his —gT. 



If at the surface there are external forces in the nature of 

 thrusts or pulls on it, and if at any point such an external force is 

 represented by F estimated per unit area (regarded as positive if 

 it is a pull), then at the surface we also have the equations 



aXx + iSXr + yXz = F., 

 aVx + ^Yy + yYz = Fy, 



aZx -h pZy -j-yZz = F,,j 



(29a) 



where a, jS, y are the direction cosines of the outwardly directed 

 normal to the surface at the point. This follows from the 

 consideration that a thin layer of matter at the surface of the 

 body exerts on the matter in the interior a stress-action per unit 

 area, whose component parallel to OX is aXx + fiXy + yXz, 

 etc. Hence the interior matter exerts on this thin layer an 

 action whose a;-component per unit area is — {aXx + ^Xy + 



