﻿Cubic Crystals with Theoretical Atoms. 767 



Stable Equilibrium for Turning Moments. 



The stability of the equilibrium may be demonstrated by 

 considering the nearest face-centred atoms a, b, c, d, e, and/. 

 Suppose the axis of A is slightly displaced in the plane 

 Aeto, so as to become more nearly parallel with e and /. 

 The moment of force due to e and / is diminished and may 

 be represented by the arrow An', a little less than An but 

 in the same direction. The moments due to the other pairs 

 <x, b, c, and d are scarcely affected in magnitude, as the 

 angle between their axes and that of the central atom A 

 is only changed by a comparatively slight amount. The 

 direction of Ah is slightly rotated out of the plane of the 

 hexagon toward the front side, and of A s similarly rotated 

 toward the back side. The resultant of Ah -{-As then main- 

 tains approximately the same magnitude and direction as 

 before, namely A/. The resultant of the moments of the 

 six atoms a, b, c, d, e 9 and /after displacement of A is then 

 the sum of Al and An', namely AM in the direction of Al. 

 The resulting moment AM tends to turn the atom A in the 

 counter-clockwise direction when viewed from /, and hence 

 acts against the direction of the small displacement which 

 was in the opposite direction, so as to make A more nearly 

 parallel with e and/ The original position of the axis of A 

 is, therefore, one of stable equilibrium for such a displace- 

 ment. Were it displaced in any other direction we would 

 arrive at a similar result. A similar process of reasoning 

 may be applied to the twelve atoms 7i, i, j, k, I, m, n, o, p, q r 

 r, 5, arriving at a similar conclusion. 



This completes the proof of stability both of the directions 

 of the axes of rotation and the transnational position of each 

 and every atom in the whole structure in a cubic crystal. 

 Before giving numerical values of the distances between 

 atoms it seems best to consider a special case, that of the 

 hydrogen molecule. 



The Hydrogen Molecule. 



Assuming that the two atoms in the hydrogen molecule 

 are alike, each having a single electron revolving at the 

 same speed, we have to find the average force between two 

 synchronously revolving electrons, resolved along the line 

 joining centres. This has been done in a former paper* for 

 the case where the axes of rotation are parallel. This elves 



* Loc. cit. equations (42) and (44). Note an omission in the co- 

 efficients, corrected in a note at bottom of page ol'5, in a subsequent 

 paper, Phil. Mag 1 , vol. xxix. Feb. 191o. 



