the Molecule and Chemical Combination. 513 



exerted by the other electrons along OP is increased by 



the attraction exerted by the central positive 



If **g) is 



charge on the electron at P, then when P is displaced the- 

 attraction is increased by 



e Br 



•**(;) 



but unless this increase in the attraction is greater, or at any 

 rate not less than that in the repulsion o£ the electrons, the 

 electron when displaced will not return to its original 

 position and the equilibrium will be unstable : hence for any 

 symmetrical distribution of electrons to be stable 



i+M>&*(&8.-£9}- ■ ■ (1) 



As the number of electrons increases the angle between 

 an electron and its nearest neighbour diminishes and the 

 right hand side of this expression becomes greater. It 

 becomes exceedingly large when the electrons are so 

 numerous that 6 is a small quantity, and so the equilibrium 



will be unstable unless -r^> (-) is also very large. Now 



whatever be the law of force, we may assume that at the 

 same distance from the centre both the magnitude of the 

 force and its rate of increase with the distance will be pro- 

 portional to the magnitude of the central positive charge ; 

 this central charge is proportional to the number of 

 electrons. We can see that if we disregard stability we 

 can find a position o£ equilibrium even though the number 

 of electrons is very large, for as we increase this number 

 we increase the repulsive force £ 2 S w /4r 2 exerted by the 

 electrons, but the increase in S„ is comparable with the 

 increase in the number of the electrons and therefore 

 comparable with the increase in the central attractive force. 

 Thus as the number of electrons increases, the attractive and 

 repulsive forces can keep pace with each other, and it will be 

 possible to find a position o£ equilibrium for a wide range in 

 the number of electrons. When, however, we come to 

 consider the stability of the arrangement the case is very 

 different. We see that the right-hand side of (1) contains 

 the term X(ljsm 3 s ) and this term when S is small increases 



