﻿Cubic Crystals with Theoretical Atoms. 765 



formulae (23), (24), and (25) show that when a remains the 

 same and the direction cosines X, Y, and Z are each re- 

 versed, each component and, therefore, the total force 

 changes sign. The force of each and every pair of atoms 

 situated at equal distances along any diagonal line upon the 

 central atom is therefore zero ; and this completes the 

 demonstration of equilibrium of the whole for tran^lational 

 forces. 



Stable Equilibrium for Translational Forces. 



Moreover, the central atom is in stable equilibrium for 

 small displacements. It has already been shown to be stable 

 for the six adjacent face-centred atoms. The restoring force 

 per unit of mass and distance for one such atom may be 

 found by differentiating (40) with respect to v, giving 



dFv 

 dv 



= + t^T ^-4PP7^tr B + ll-4iT 7 2m 2 2rA ; (42) 

 Kai L p P' J 



and substituting the equilibrium distance (41) we get 



'•=+3-8T f 4- T Sm»2»»; . . . (43) 



dv Ka i v 



r 



For the opposite face-centred atom we get the same value, 

 for a small displacement of the central atom toward the one 

 face-centred atom and away from the other makes the force 

 of each toward the original position, changing sign when 

 passing through the origin. For a pair of such atoms we 

 must then double (43). 



It is different with two atoms along a diagonal of the 

 cube, for example. In the case where the axes are all in 

 the same straight line, the atoms being of different kinds, 

 the formula where a = ir applies, showing a repulsion at all 

 distances. By differentiation of the force equation for <x = 7r 

 and addition of those for two adjacent atoms, we show that 

 the rate of change of the force is zero for each pair of such 

 atoms, producing a uniform field of force of zero value at 

 the central atom. This reasoning applies to all the other 

 atoms except the six face-centred adjacent atoms ; and we 

 have, therefore, completed the demonstration of stability 

 with the exception of the non-translational forces, which 

 produce only turning moments to control the directions of 

 the axes. These will now be considered. 



