Theory of Spectrum Emission. 47 



so on, generally rce, where k is an exact (positive) integer. 

 Now, in the case of helium and, a fortiori, of more complex 

 atoms, it is certainly inadmissible to treat the nucleus as 

 a point-charge. But even in the case of hydrogen there is 

 no reason (besides a tendency to mathematical simplifi- 

 cation) a priori to assert that its single nucleus-charge 

 has a radially symmetric distribution. Under such circum- 

 stances it is scarcely necessary to justify an investigation 

 into the spectral behaviour of differently shaped nuclei. 



2. No matter what the shape of a nucleus may be, 

 we will call its total charge * zee, and we will denote by — e 

 and m the charge and the mass of the electron moving 

 around it. No account will be taken of the perturbations 

 due to other electrons or planets belonging to the same 

 system. Again, the relativistic complications (already 

 studied by Sommerfeld f) will be disregarded, and the 

 electronic mass m will, therefore, be treated as a constant. 

 In short, the system will be treated on Newtonian prin- 

 ciples, as far, of course, as the " stationary "- orbits are 

 concerned. It is well-knowm that under these circum- 

 stances, and if the nucleus be a simple point-charge, the 

 negatived total energy W belonging to any one ot the 

 stationary orbits is given by 



z w =^ w 



ch ii A 



where n is an integer, c the light velocity 



A Planck's constant, and R the Bohr expression of 



Uydberg's constant, i. e. 



„ 2ir 2 e A m' . Mm 



K — yr; , 7)1 = r— (9 ) 



ch d ' M+m W 



M being the mass of the nucleus, and therefore, practically, 

 the whole mass of the atom. If v be the reciprocal wave- 

 length or (what is improperly called) the frequency, the 

 corresponding spectrum is, by Bohr's fundamental assump- 

 tion, v =(W n - W n )/eh, i. e., by (1), 



" = ^(i"i) O) 



where n' is a fixed, and n a variable integer. In fine, 



* In usual electrostatic (irrational) units. 



t A. Sommerfeld, Annalen der Physik, vol. li. 1916, pp. 1 et seq. 

 especially Part II. p. 44 et seq. The latter is open to some serious 

 objections which will be pointed out in a later publication. 



