166 General Analytical Theorems [CH. vn 



Let W be the total energy of the body A in the field of force from 

 B, B f , .... Then W= Sw, and therefore 



i.e. the sum TF=2w, satisfies Laplace's equation, because this equation is 

 satisfied by the terms of the sum separately. It follows from this equation, 

 as in 52, that W cannot be a true maximum or a true minimum for any 

 values of x, y, z. Thus, whatever the position of the body A, it will always 

 be possible to find a displacement i.e. a change in the values of x, y, z for 

 which W decreases. If, after this displacement, the electricity on the con- 

 ductors A,B, B', ... is set free, so that each surface becomes an equipotential, 

 it follows from 189 that the energy of the field is still further lessened. 

 Thus a displacement of the body A has been found which lessens the energy 

 of the field, and therefore the body A cannot rest in stable equilibrium. 



STRESSES IN THE MEDIUM. 



193. Let us take any surface S in the medium, enclosing any number 

 of charges at points and on surfaces S lt S 2 , 



Let I, m, n be the direction-cosines of the normal at any point of 

 S lt $ 2 , ... or S, the normal being supposed drawn, as in Green's Theorem, 

 into the space between the surfaces. 



The total mechanical force acting on all the matter inside this surface 

 is compounded of a force eR in the direction of the intensity acting on every 

 point charge or element of volume-charge 0, and a force 27rcr 2 or ^o-R per 

 unit area on each element of conducting surface. If X, Y, Z are the com- 

 ponents parallel to the axes of the total mechanical force, 



where the surface integral is taken over all conductors S lt S 2 , ... inside the 

 surface S, and the volume integral throughout the space between S and these 

 surfaces. Substituting for p and a-, 



1 ffff&V 9 2 F 8 2 F\ 9F , 

 x = - hrv + -0-5- + ^r 5~ dxdydz 



4>7rJJJ\dx 2 dy z dz* J dx 



1 v ff 

 fT"2 I 



4?r JJ 



-^- +n-^-\ ^ dS ......... (108). 



dy dz 



