16 MR CLERK MAXWELL ON 



responding element of surface in the diagram of stress, then the resultant stress 

 on the whole closed surface will vanish ; for the corresponding surface in the 

 diagram of stress is a closed surface, and the resultant of a uniform normal 

 pressure p on every element of a closed surface is zero by hydrostatics. 



It does not, however, follow that the portion of the body within the closed sur- 

 face is in equilibrium, for the stress on its surface may have a resultant moment. 



Theorem 2. — To ensure equilibrium of every part of the body, it is necessary 

 and sufficient that 



*~ dx V dy ^~ dz' 



where F is any function of x y and z. 



Let us consider the elementary area in the body dy dz. The stress acting 

 on this area will be a force equal and parallel to the resultant of a pressure p 

 acting on the corresponding element of area in the diagram of stress. Resolving 

 this pressure in the directions of the co-ordinate axes, we find the three com- 

 ponents of stress on dy dz, which we may call p xx dy dz, p^ dy dz, and p xz dy dz, 

 each equal to p multiplied by the area of the projection of the corresponding 

 element of the diagram of stress on the three co-ordinate planes. Now, the 

 projection on the plane yz, is 



/dr,c]X _ dr L dZ\ ( l h 

 \dy dz dz dy) ^ 



Hence we find for the component of stress in the direction of x 



lxx ~ 2 \dy dz dz dyj' 

 which we may write for brevity at present 



Similarly, 



Pxy = pJ(£, Ziy,z) Pxz = pJ(£, 77 ; y , z) . 



In the same way, we may find the components of stress on the areas dz dx 

 and dx dy — 



Pyx = PJ(V > £; * > x ) Pyy = p2(£> £ > z > x ) Py; = pJ (£ > V, * , %) 



Pxz =pJ(v>%\ x >y) Pzy =^J(?;^; ®,y) Pzz =p3{%,y, x >y) ■ 



Now, consider the equilibrium of the parallelopiped dx dy dz, with respect to 

 the moment of the tangential stresses about its axes. 



The moments of the forces tending to turn this elementary parallelopiped 



about the axis of x are 



dz dx p yz . dy — dx dy p zy . dz . 



To ensure equilibrium as respects rotation about the axis of x, we must have 



Pyz = Pzy • 



