54 Dr. L. Silberstein on the Quantum 



principles, according to what was said above, immediately 

 after formula (11). This will be done exactly as in that part 

 of Sommerfeld's paper {Ann. tier Phys. -vol. 51) which is 

 based on simple Newtonian mechanics, i. e. without rela- 

 tivists refinements and, of course, for the undisturbed system 

 (nucleus = point-charge). Namely, let $ be, as in (5), the 

 angle between a variable and a fixed meridian plane (say, 

 that passing through the line of nodes), so that the kinetic 

 energy is 



2-T 1 f 1 ' Q . 9 • >> , 9 • 9 J 9 I 



and let us quantize with respect to </>, 77, r. Thus 



p^d<j> = n 1 li, 1 p rj dv = n 2 Ji, }p r dr=\ ]^ C ^dO = 7iJi } 



where m • • • v / 



j) ( j ) = m l r 2 sin 2 rj . (£, p n = m'r 2 rj, p r — rtir. 



Remembering that p=mV 2 #, it will be seen at once that 



m = p cos i, 

 Pp being simply the projection of p from the orbit plane upon 

 the equatorial plane. Thus, both p and i being constant, the 

 first of (19) gives simply p cos i = n l h/27r. The second and 

 the third of (19) become at once, in virtue of the orbit equa- 

 tion (15), 27rp(l — cos i) = n 2 h, and 27T/>[(1 - e 2 )"- — 1] = n z h, 

 as in Sommerfeld's paper, the only difference (not affecting 

 the value of the integral) being that 6 is replaced by 6 — co. 

 Thus the quantized values of the three elliptic elements are, 

 in terms of the three independent integers appearing in (19), 



. . . (20) 

 These are to be substituted in W as well as in the 

 expression (IS) for F. The sum of W and J' 1 will give 

 the required W, as in (11). 



Now, W is easily found to be equal to «¥n/(l- e 2 )/2p 2 , 

 and therefore, by (20), and with the previous meaning of R, 



TTr htR ch 



(»i + "2 + "3) 

 while (18), with b as in (17), becomes, in virtue of the first 

 of (20), 



