and of a cube 4 1 3 



For if we suppose the electricity on the square rendered immoveable, and 

 if we cut off portions from two sides of the square and place them on the other 

 two sides so as to form a rectangle, we are carrying electricity from a place 

 of higher to a place of lower potential, and are therefore diminishing the energy 

 of the system. 



If we now make the electricity moveable, it will re-arrange itself on the 

 rectangle and thereby still further diminish the energy. Hence the energy of 

 a given charge on the rectangle is less than that of the same charge on the 

 square, and therefore the capacity of the rectangle is greater than that of the 

 square*. 



NOTE 23, ARTS. 288 AND 542. 

 On the Charge of the Middle Plate of Three Parallel Plates. 



The plates used by Cavendish were square, but for the purpose of a rough 

 estimate of the distribution of electricity between the three plates we may 

 suppose them to be three circular disks. 



First consider two equal disks on the same axis, at a distance small com- 

 pared with the radius of either. 



If the disks were in contact, the distribution on each would be the same as 

 on each of the two surfaces of a single disk, and it would be entirely on the 

 outer surface. 



If the distance between the disks is very small compared with their radii, 

 the force exerted by one of the disks at any point of the other will be nearly 

 but not quite normal to its surface. The component in the plane of the disk 

 will be directed outwards from the centre, so that the density will be greater 

 near the edge than in a single disk having the same charge, but as a first ap- 

 proximation we may assume that the sum of the surface-densities on both sides 

 of any element of the disk is the same as if the other disk were away. 



But the density on the outer surface of the disk will be increased, and the 

 density on the inner surface diminished, by a quantity numerically equal to 

 the normal component of the repulsion of the other disk divided by 477, and the 

 whole charge of the outer surface will be increased, and the whole charge of the 

 inner surface diminished, by a quantity equal to the charge of that part of the 

 other disk, the lines of force from which cut the disk under consideration. 



* [Approximate results may be readily obtained for such problems by the 

 principle that the potential energy of a system is stationary in the neighbourhood 

 of a position of equilibrium. Thus the energy of a given charge on a square plate 

 is very nearly the same as on a circular plate of equal area: therefore the capacities 

 of the two plates are nearly the same according to Maxwell's result the capacity 

 is greater for the square plate, in the ratio 1-0027, i n agreement with the final argu- 

 ment in the text. So also a cube may be reduced to a sphere of equal volume. 

 The degree of error may be elucidated by comparing the known results for elliptic 

 and circular plates of equal area, or for ellipsoidal and spherical bodies of equal 

 volume. 



