226 THEORY OF NEWTONIAN FORCES. [FT. I. CH. V. 



That is : The mutual energy of any two Newtonian distributions 

 differing infinitesimally from each other, and supposed co-existing, 

 is less than the sum of their individual energies. This theorem 

 is probably true for all repulsive forces. We shall make use of 

 it in 180 to deduce the laws of dielectric and magnetic actions. 



120. Potential of Polarized Bodies. Double Distribu- 

 tions. If a particle of mass dm be placed at a point where the 

 field-strength is F, it experiences a force of amount Fdm in the 

 direction of the field. If any distribution m be placed in a field 

 which is uniform, that is, for which at all points F is constant in 

 value and direction, the force P experienced by the whole mass is 



P = (({Fdm = F (((dm = Fm. 



= F [I j dm = 



Certain bodies exist in nature which, on being placed in a uniform 

 field, experience no tendency to move in any direction, so that P 

 is zero. Accordingly for such bodies m must be zero, or their 

 density must be in some points positive and in others negative. 

 Such bodies experience a couple when placed in a uniform field, 

 although the resultant force vanishes. Not only does the above 

 property hold for the whole body, but if it be broken into any 

 number of parts the resultant force on each part is zero. A 

 magnet is the most familiar example of such a body, for placed in 

 a uniform field of magnetic force it experiences no resultant force, 

 no matter into how many pieces it may be broken. In such a 

 body then the mass of any part, however small, must be zero. Let 

 us consider how such a condition is possible. 



Let us suppose that any portion of space T is occupied by a 

 body A of constant density p and that occupying identically the 

 same space is a second body B of constant density p. The two 

 bodies will then completely neutralize each other's action in every 

 way, and when placed in a uniform field would experience neither 

 force nor couple. Now suppose that the first body A is displaced by 

 an infinitesimal amount, so that every point in it moves a distance 

 h in the same direction. The effect will 

 be that while in the space occupied in 

 common by the two bodies the densities 

 neutralize each other, there is a space on 

 one side filled with positive matter, and on 



