Electron Theory of Chemistry to Solids. 727 



Stability of the Distribution of Electrons, 



Hitherto in considering the possible distribution of elec- 

 trons we have only taken into account geometrical con- 

 siderations ; it is, however, of fundamental importance to 

 consider the stability of the distribution, as no distribution 

 is of any use for our purpose unless it is stable. 



We shall begin with the case of a monovalent element 

 when the electrons are at the corners of cubes and the 

 atoms at the centre. 



Take the origin of coordinates at one of the electrons and 

 the axes parallel to the sides of a cube ; let 2d be the length 

 of a side of the cube. 



Then the coordinates of the array of electrons are given 

 by the equations 



x=.2pd, y = 2qd, z=2rd, 



those of the atoms by 



x = (2p + l)d, y = (2q + l)d, z = (2r+l)d, 



where p, q, r may have any positive or negative integral 

 values. 



If the electron (/?, q, r) has a vertical displacement p pqr , 

 the force due to this displacement tending to increase p , the 

 displacement of the electron at the origin is 



(P0 — pP,q,>-) Bpqr, 



where 



r> I ( L 3r * ^ 



° pgr ~ (2dy V( p 2 + q 2 + r 2 f' 2 (f + q 2 + r 2 fl 2 ) ' 



When the attraction between an atom and an electron 

 separated by a distance r is expressed by 



if the (p, q, r) atom has a vertical displacement w v , q , r , the 

 force tending to increase p Q , the displacement of the electron 

 at the origin is 



— (j)Q — CO pqr ) Cpq r , 



where 



_ e 2 f 1 3r 2 c/ 1 4r 2 \ > 



Cl (. -TXr>o r " IXpqr it \\Xpq r J&pqr / ) 



when 



\>pq r -Hpqr «* \- i V2 r J-vpqr 



R n ,.= (2p + iy + {2q + iy + (2r + 1) ! 



