Faraday \s " Magnetic Lines " <#s Quanta. 527 



5. On the hypothesis now put forward the atomicity 

 required by any form of the quantum theory is an atomicity 

 of tubes of magnetic induction, the unit tube being defined 

 as the ratio of Planck's constant, h, to the charge, e, of an 

 electron. One important consequence of this hypothesis 

 must be noticed. The electrokiuetic energy associated with 



a single "quantum tube" may be expressed as ^(^ — ) Ads, 



w 



here yu,H 2 /87r is the energy in unit volume, A is the area of 

 cross-section, and ds an element of length in the direction 

 of the tube. But, since the magnetic induction is the same 

 for all cross-sections of the tube, /xIIA is constant and equal 

 to h/e for a quantum tube. Thus the energy for the tube is 



— XUds. If we suppose the unit tube is linked with the 

 hire 



magneton or with an electron orbit, XHcfo = 47n = 47r<?v. 



Hence we obtain the simple result that, in this case, 



the electrokinetic energy of the unit tube —-\iv. (I) 



II. Magnetic Tubes in the case of an Elliptic Orbit. 



6. In the original form of Bohr's theory the electron was 

 supposed to be revolving round the positive nucleus in a 

 circular orbit. As we have seen, it is easy to show that in 

 this simple case the quantum integer which determines the 

 steady motion may be taken to represent the number of 

 quantum magnetic tubes passing through the orbit. The 

 extension of Bohr's theory to the case in which the orbit is 

 elliptical has been made independently by Sommerfeld * and 

 W. Wilson f . It is found that the size and shape of the 

 ellipse now depend upon two integers n and n', the first 

 introduced by the application of the quantum theory to the 

 angular motion, the second by the application of the theory 

 to the radial motion. In the following pages, in which 

 Sommerfeld's notation is employed, I propose to show that 

 the sum of these integers represents the number of quantum 

 magnetic tubes passing through the elliptic orbit. It is the 

 sum, n + n', which determines the frequency of the emitted 

 radiation in accordance with the equation 



Here N is Rydberg's constant, «t, m' refer to the initial. 

 n, n' to the final orbit. 



* A. Sommerfeld, Ann. d. Thysik. vol. li. p. 1 (1916). 

 t W. Wilson, Phil. Mag. vol. xxxi. p. 161 (1916). 



