548 Prof. J. W. Nicholson on the High-frequency 



equilibrium, all the electrons in any ring having the same 

 velocity, and a normal acceleration aco'\ to which Newton's 

 Second Law is applied. When there are two rings, the par- 

 ticles cease to move in a circle, and we can see at once that 

 no dynamical equilibrium with a constant velocity for each 

 electron can exist consistent with ihe principle of universal 

 constancy of angular momentum per electron, unless every 

 electron is describing a circle, whether the angular momenta 

 and velocities of every electron are the same or not. For it 

 would involve, for any electron, of polar coordinates (r, 6), 

 where A and B are constants, 



(D* +**=*' ^ =B ' 



or . . B 2 . _ „ , C rdr 



unless r = B/A, a special solution, and finally 



?> 2 A 2 = B 2 + A 2 (£ + Const.) 2 , 



so that the electron must proceed to infinity. But a non- 

 circular path, permanently described with constant angular 

 momentum in an atom, with variable velocity, may always 

 be regarded as part of a steady state, according to Bohr's 

 theory, where the consequent radiation does not have the 

 value calculated by the ordinary electrodynamics as corre- 

 sponding to accelerated motion. 



Returning to the ordinary electrodynamics, the only 

 solutions of the general problem of orbits which can be 

 obtained, subject to the condition of interference of radiation 

 or the equivalent condition of a zero vector sum of accelera- 

 tions, require that the electrons shall be arranged at every 

 instant on circles coaxial with each other, as proved by Schott. 

 Two coplanar rings could satisfy this condition, but it would 

 involve further a regularly preserved phase relation among 

 the electrons in each circle, and between the two circles. 

 We must imagine that the electrons in any one ring are all 

 equidistant from the nucleus at any instant, and if the rings 

 are rotating with different angular velocities, the electrons 

 cannot be distributed round their rings at equal angular 

 intervals, but must be oscillating tangentialJy in relation to 

 the general motion of the ring. 



Apart from the improbability of such phase relations being 

 preserved — for it cannot be conclusively proved that they 

 are impossible in all cases, — there is a decisive way of settling 



