720 Prof. H. A. Wilson on the 



~F 2 ct ds/§7r, and if the tube is slightly distorted the volume of 

 each element and the value of Fa remain unchanged, so that the 

 change in the energy in the element will be due to the change 

 in F. The energy will be a minimum when the tube is in 

 equilibrium, so that F will be as small as possible and therefore 

 a as large as possible. This means that the tubes tend to 

 become as short as possible, their volumes remaining constant. 

 The effect of this will evidently be to make the field round 

 each electron tend to become as nearly spherical as possible 

 with the electron in the middle. 



Consequently, to determine approximately the distribution 

 of the n electrons in the positive sphere, it is sufficient to find 

 how the sphere can be divided up into n equal volumes all as 

 nearly spherical as possible and to put an electron at the 

 centre of each of the n volumes. When n is large it is easy 

 to see that this requires the electrons to be arranged like the 

 centres of the shot in a pile of shot. Thus with thirteen 

 electrons we should expect to have one in the middle and 

 twelve arranged round it all at the same distance from it. 



It is easy to see from considerations of symmetry that the 

 electrons will arrange themselves on nearly spherical surfaces 

 concentric with the surface of the positive sphere. The 

 condition that the fields of the electrons shall be as nearly 

 spherical as possible evidently requires the distances between 

 the successive surfaces to be all equal. The fields of the 

 electrons on the surface of a sphere will form a layer the 

 cube of the thickness of which will be approximately equal 

 to the volume of the field of one electron. 



According to Sir J. J. Thomson's theory each element in 

 a series of similar elements, such as fluorine, chlorine, bromine, 

 iodine, is derived from the one before it in the series by the 

 addition of a spherical layer of electrons together with the 

 necessary amount of positive electricity to keep the atom 

 neutral. 



Let w 1? n 2 , n 3 , etc., denote the numbers of electrons in the 

 atoms of a series of similar elements and let A 1? A 2 , A 3 , &c. 

 denote their atomic weights. Then, if we assume that the 

 number of electrons in an atom is proportional to its atomic 

 weight, we can write /3A 1 z=n A , /3A 2 = ?? 2 , &c. where /3 is 

 a constant. 



Let ri, r 2 , &c. denote the radii of the positive spheres and 

 let v — c\p be the volume of the field round each electron. 



Then we have 



4 



~7rrl = n m v = f3vA m , 



3 



*'* + .=n m+1 V = ffvA, 



+ 1 ' 



