370 WILSON— CONSTITUTION OF THE ATOM. [April 22. 



tube is i/47r so that its volume is i/^irp. The total volume of all 

 the 47rr tubes is therefore e/p. Thus the tubes of force starting 

 from the electron occupy a volume c/p and this is true in any case 

 whether other electrons are near or not. Also since every tube of 

 force must end on positive electricity it is clear that the volume c/p 

 can only contain the one electron from which the tubes start. Thus 

 when any number of electrons are present each one will be sur- 

 rounded by its own field which will occupy the volume e/p. The 

 positive charge in the volume c/p is equal to c, so that if the sphere 

 has a positive charge equal to the total negative charge on the n elec- 

 trons in it, it will be divided up into ;; equal volumes, each contain- 

 ing one electron. 



The energy in an element of a tube of force is equal to 

 F-ads/S-rr, and if the tube is slightly distorted this element will still 

 have the same volume and also (Fa) will 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 pos- 

 sible, 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 11 electrons in the positive sphere it is sufficient to find how the 

 sphere can be divided up into .V equal volumes, all as nearly spher- 

 ical as possible and put an electron at the center of each of the n 

 volumes. 



When Ji is large it is easy to see that this requires the electrons 

 to be arranged like the centers of the shot in a pile of shot. Thus 

 with thirteen electrons we should have one in the middle and twelve 

 arranged around it, all at the same distance from it. 



Suppose the volume of the field of one electron is z', and let 

 «i, »2, "3, etc., denote the number of electrons in the atoms of a 

 series of similar elements. Each element is formed by the addition 

 of a spherical layer to the one before it and it is clear that all the 

 layers must be of nearly the same thickness if the fields of all the 

 electrons are to be nearly spherical. Consecjuently if i\,i\,r^, etc., 



