692 BELL SYSTEM TECHNICAL JOURNAL 



Here, as there, this necessity imposes of itself a certain condition upon 

 the coefficient of \l/ in the right-hand member, which is tantamount 

 to this condition imposed on E: 



E = n{n+\) g^ = (n + iy-^ + const., w = 0, 1, 2, 3 • • •. (192) 



The energy of the rotator is thus by the mere fact that the variables 

 are cyclic Hmited to a single sequence of permitted values, furnishing 

 incidentally another example of half-quantum-numbers. 



The Eigenwerte, the permitted energy -values, are thus for the rotator 

 determined by an exceptionally lucid condition ; yet the complications 

 of the General Equation already appear on the horizon. Equation 

 (190) differs from the wave-equation which I have hitherto used by 

 virtue of the substitution of moment-of-inertia A for mass m. This 

 replacement seems sensible enough ; one might rely on intuition in this 

 particular case; but strictly it is caused by the form preassumed for 

 the General Wave- Equation and by the specific form of the kinetic- 

 energy-function for this specially restricted kind of rotator. If now 

 we remove the restriction, and permit the rotator to spin about its 

 axis-of-figure at the same time as it whirls about some axis normal to 

 that — if we pose the general problem of the rigid rotator unrestricted 

 save by the conditions which the wave-equation imposes, it is neces- 

 sary to invoke the General Equation with the non-Euclidean geometry. 

 The problem is soluble, and has been solved ; ^'^ the utility of the results 

 for the interpretation of band-spectra gives valuable support to the 

 form selected by de Broglie and Schroedinger for the General Equation. 



The polarization of a gas by an electric (or magnetic) field may be 

 treated by supposing that each atom is an electric (or magnetic) 

 doublet. The treatment is simplest if we may assume that the electric 

 (or magnetic) axis of the doublet coincides with the axis-of-figure of 

 a dumbbell-molecule, not allowed to spin around its axis-of-figure. 

 Let M stand for the moment of such a doublet, and suppose the field 

 H to be parallel to the direction from which the angle 6 of the fore- 

 going paragraphs is measured. The field supplies the potential- 

 energy term to be added to the left-hand member of equation (190); 

 this new term is: 



- Vyp = iMHcosd)rp. (193) 



It is easy to see that the wave-equation has Eigenwerte, so that the 

 atoms are in effect limited to certain "permitted" orientations in the 



^'^F. Reiche, ZS. f. Phys., 39, pp. 444-464 (1926); R. de L. Kronig, I. I. Rabi, 

 Phys. Rev. (2), 29, pp. 262-269 (1927). 



