114 Dr. H. Stanley Allen on the Angular Momentum 



No proof of McLaren's theorem appears to have been 

 published, and it may be o£ interest to show that the result 

 can be obtained in a comparatively simple way. 



Let the position o£ any point in the aether be referred to 

 cylindrical coordinates, taking the axis of symmetry of the 

 magneton as the axis of z. Since each Faraday tube is 

 rotating about this axis, the motion of any portion of a tube 

 is at right angles to its length, and the equivalent mass 

 per unit volume of a tube is 4ztt/jlN 2 , where N is the electric 

 polarization or displacement at the point. The angular 

 momentum for unit volume of the tube is consequently 

 4z7r/jiN 2 r 2 co, where co is the angular velocity with which the 

 system is rotating about the axis. The motion of the 

 Faraday tubes produces a magnetic field in a direction at 

 right angles to their length and to the direction of motion 

 of magnitude H = 47rN? , o). Hence the angular momentum 

 for unit volume of the tube can be expressed in the form 



/L6H 2 /47rft) or 



2 /*H 2 



CO 07T 





Thus the total angular momentum 



2 v H2 i ,i 



= - ^~—, where the 



CO 07T 



summation extends over the whole space external to the 

 magneton. But ^jaW/Sit represents the amount of energy 

 associated with the magnetic field, and it is easy to show 

 by considering the energy as distributed in the magnetic 

 tubes that this is equal to ^Li 2 , where L is the coefficient of 

 self-induction of the magneton and i the current flowing- 

 round it. Thus the total angular momentum =~Li 2 /co. 



But Lt=s.N OT , the number of magnetic tubes linked with 

 the magneton, and i\co = el2ir = ^ e l2ir, where e is the charge 

 on the magneton. So we obtain finally as the total angular 

 momentum of the magneton 



The difference between this expression and that given by 

 McLaren arises from the fact that he employed rational 

 units. 



It is easily shown from a consideration of the dimensions 

 of N TO and N e that the product has the same dimensions as 

 angular momentum both in the electrostatic and in the 

 electromagnetic system. 



The proof given being perfectly general applies to any 

 magneton of the type considered, whatever may be the shape 



