STRAINED ELASTIC SOLIDS 



421 



are a, /?, 7 cut them in the points Q, R, S. Let K be the area of 

 the triangle QRS; then Ka is the area of the triangle PRS, Kfi of 

 PSQ and Ky of PQR. The portion of the medium within the 

 tetrahedron PQRS is in equilibrium under the body forces on it 

 and the stress actions on it across the four triangles mentioned. 

 Let us enumerate the latter first. Parallel to OX we have a 

 force across PRS of amount —KaXx. (We are assuming 

 that Xx is positive if it is a tension, and negative if a pressure; 

 also that the tetrahedron PQRS lies in the positive octant, i.e., 

 the octant for which the local coordinates ^, tj, f are all positive). 

 Also parallel to OX we have a force —K^Xy (a tangential shear- 

 ing force) across PSQ, and across PQR a force —KyXz (also 



KaX^ 4 





Y <, 



9 KiaK^-^px^i-yX^) 



Fig. 5 



shearing). In considering the equilibrium we can, if we 

 gradually reduce the size of the tetrahedron, neglect the body 

 forces on it in comparison with the surface forces just enumer- 

 ated. The point involved is the same as that introduced in 

 elementary treatises on hydrostatics when proving the uni- 

 formity of fluid pressure in all directions, and will doubtless be 

 known to the reader, or easily looked up. (Actually it only 

 requires us to remember that the body forces involve the 

 product of a finite quantity and the volume, while a surface 

 action involves the product of a finite quantity and an area. 

 As the size of the tetrahedron diminishes, the magnitude of the 

 volume becomes very small in comparison with the magnitude 

 of the surface, since the former involves the cube of a small 



