Number of Electrons in the Atom. 719 



is equal to the positive charge on the sphere in a neutral 

 atom. 



The object of the present paper is to show how to obtain 

 an approximate solution o£ the problem of the distribution of 

 n electrons in a positive sphere and how to deduce the 

 number of electrons in any atom from the atomic weights of 

 the elements. 



Consider an electron having a charge e inside a sphere of 

 positive electricity of uniform density of charge p per c.c. 



Close to the electron the electric field is of strength -^ , where 



r is the distance from the electron, so that kire tubes of 

 electric force come out of the electron, if the number of tubes 

 per sq. cm. is taken to be equal to the field strength. Con- 

 sider one of these tubes of force and let ds be an element or! 

 its length and a its cross-section at ds. The charge in the 

 length ds is pads, so that 



j- (Fa) = ±7Tpa, 



where F is the electric force along ds. Integrating along 

 the tube this gives 



FiOti — Fa. — -lirp J a ds, 



where F^ denotes the value of Fa at the surface of the 

 electron. This shows that as we go along the tube Fa 

 diminishes and when 



F^i = 4:7rp\a ds 



it will be zero and the tube will end. Now F 1 = e/a 2 , where 

 a is the radius of the electron, and a 1 = a 2 /e, so that F 1 « 1 = l, 

 hence 4:irp\ads from the surface of the electron to the end of 

 the tube is equal to unity. Thus the volume of each tube is 



- — and the volume of all the 4zwe tubes is therefore el p. 

 Airp r 



Thus the tubes of force starting from the electron occupy a 



volume ejp, 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 e\p can only 



contain the one electron from which the tubes start. Thus 



when any number of electrons are present each one will be 



surrounded by its own field which will occupy the volume e p. 



The positive charge in the volume e\p is equal to e, so that if 



the sphere has a positive charge equal to the total negative 



charge on the n electrons in it, it will be divided up into // 



equal volumes each containing one electron. 



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



