I0 THE QUANTUM [i. 6 
mental hypothesis of the generalized form of the quantum theory 
is expressed in equations of this form : 
p pdq =nh i: 8 
where q and p are Hamiltonian positional and impulse co 
ordinates, and the symbol p denotes that the integration is 
extended over a complete period corresponding to the co-ordinate 
q considered. 
By employing these principles with relativistic extensions 
taking into account the dependence of the mass of the electron 
upon its velocity, Sommerfeld * * * § has been able to explain and 
even to predict the fine structure of the lines in Balmer’s series 
of hydrogen, and also in the enhanced helium spectrum. 
On the basis of this theory an explanation has been given 
of the resolution of the hydrogen lines brought about under the 
influence of an electrostatic field. This was observed indepen 
dently by Lo Surdo f and by Stark, j 
The corresponding case where the resolution is due to a 
magnetic field—the well-known Zeeman effect (Chapter XV)— 
has been discussed by Sommerfeld, Debye, Bohr, and others, 
and although a complete explanation of all the complexities 
observed has not yet been given, the cases normally occurring 
have been accounted for. “ Here, the result of the quantum 
theory is identical with that from classical mechanics, and an 
inspection of the quantum equation shows that the quantum 
occurs on both sides and so divides out. This suggests that 
many phenomena which at present are thought to be satis 
factorily explained by dynamics are really quantum phenomena, 
but that the quantum has divided out from the equations.” § 
6. Molecular Rotations 
If the principles of the quantum theory be accepted, the 
important question arises as to how they are to be applied in 
connection with rotations, particularly rotations of molecules. 
Ihe application of the theory to molecular rotations has been 
based, in some cases, on assumptions which render the results 
less secure than those derived in connection with other pheno 
mena, but it seems certain that the clearer knowledge gained 
as to the generalized form of the theory will remove such 
ambiguities as still remain. In this connection the most 
important subjects are, first, the specific heats of gases having 
* Sommerfeld, Ann. d. Physik, vol. 51, pp. 1, 125, 1916. 
t Lo Surdo, Accad. Lincei, Atti, vol. 22, p. 664, 1913. 
+ Stark, Preuss. Akad. Wiss. Berlin, vol. 40, p. 932, 1913. 
§ C. G. Darwin, Brit. Assn. Report, Section A, 1921.