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transitions between two states, one (or both) of which belong to the multitude of 

 states in which the electron possesses sufficient energy to remove to infinite distance 

 from the nucleus. While the electron in the states of the type first mentioned may 

 be said to be "bound" by tlie nucleus to form an atom, it may in the slates of the 

 last mentioned type be described as "free". In order to account for the appear- 

 ance of the continuous spectrum, it is necessary to assume that the motions 

 in the latter states are not restricted by extra-mechanical conditions of the type 

 holding for tlie former states, but tliat all motions, which are consistent with the 

 application of ordinary mechanics, will represent physically possible states. This 

 assumption would also seem to present itself naturally from the point of view on 

 the principles of the quantum theory, taken in the present paper. ') Thus it will in 

 the first place be observed that any attempt to discriminate between the different 

 states of the type in question, by means of considerations of the mechanical stabihty 

 of stationai'y states for slow transformations of the external conditions, would fail 

 on account of the essentially non-periodic character of the motion, which is irrecon- 

 cilable with the idea of invariance of extra-mechanical conditions for such trans- 

 formations. Next, with reference to the formal analogy between the quantum theory 

 and the ordinary theory of radiation, it will be seen that the fact, that the motion 

 of a free electron, in. its hyperbolic orbit cannot be resolved in a sum of harmonic 

 vibrations of discohtinuouslj' varying frequencies but can only be represented by 

 a Fourier integral extended over a continuous range of frequencies, suggests before- 

 hand that the free electron may pass, under emission or absorption of radiation, to 

 any one among a continuous multitude of other states corresponding to a continuous 

 multitude of values for the energy of the system. From the preceding considerations 

 we may infer, by application of (1), that the complete spectrum emitted by the 

 hydrogen atom .will, besides the series spectrum and the continuous ultra-violet 

 spectrum mentioned above, which corresponds to transitions from a state in which 

 the electron is free to a stationary state characterised by n = 2 in (41), contain a 

 set of continuous spectra, corresponding to transitions from free states to other sta- 

 tionary states, and each extending in the direction of larger frequencies from one 

 of the values of the frequency, given by (35) if we put ;!' ^ co. Moreover, we may 

 expect the presence of a weak continuous spectrum, extending as a continuous back 

 ground over the whole region of frequencies, which will correspond to transitions 

 between two different states in both of which the electron is free. The relative intensities 

 of these different continuous spectra, and the laws according to which the intensity 

 is distributed within each of them, may be expected to vary to a large extent 

 according to the different conditions under which the radiation is excited. Thus, 



') A view contrary to this has been taken by Epstein, who in a recent paper (Ann. d. Phys. L, 

 p. S15 (1916)) has made an attempt to obtain an explanation of certain observations on the photo- 

 electric effect of hydrogen occluded in metals, by applying conditions of the same type as (22) to 

 states of the hydrogen atom in which the electron describes a hyperbolic orbit, and lias tried in a 

 similar way to develop a theory of the characteristic /î-ray spectra of radioactive substances. 



