Electron Theory of Chemistry to Solids. 745 



which satisfy the ordinary conditions of valency, its investi- 

 gation is of exceptional importance. 



Stability of the arrangement. 



Take the origin of coordinates at the corners of one of the 

 <jubes, the axes parallel to the sides of the cube. Let 2d be 

 a side of the cube. Then the coordinates of the corners of 

 the cubes are given by 



x = pd, y = qd, z = rd, 



where p, q, r are even integers. 



The coordinates of the centres of the faces are given by 



x = pd, y = qd, z = rd 7 



where two out of the integers p, q, r are odd and the third 

 even. 



The coordinates of the atoms are given by 



x = pd, y = qd, z = rd, 



where p, q, r are all odd. 



Consider a displacement of the electrons parallel to the 

 axis of z, such that all the electrons of one type have the same 

 displacement. Let p l9 p 2 , p s be the displacement of the 

 electrons forming the corners of the cube, the centres of 

 the faces parallel to ocy, and the centres of the other faces 

 respectively. 



The force tending to increase p u due to the displace- 

 ment p 2 , is 



e 2 / 1 3r 2 \ 



(P1-P2) Ja 2 ((p» + j» + r s)^a";(p» + 0* + r*)6/*J 



for all odd values of p and q and even values of r. 



The summation I find to be 1*1. The force tending to 

 increase pi may thus be written 



a{Pi — p2), 

 where 



e 2 

 a = 1'1-to. 

 d 6 



Let us now consider the force tending to increase p 1 

 due to the displacements p 3 . If p 2 is equal to p s , then 

 the symmetrical system formed by the whole of the 

 electrons at the centre of the faces will not produce any 

 force tending to increase pi ; hence the force due to the 



