330 



46 



in a simple way in terms of the values of this frequency in the two states. With 

 reference, however, to the fact, that it is possible, as shown in the preceding, to 

 represent the frequency u of the emitted radiation in a simple way as the mean 

 value of the mentioned frequency, taken over the continuous multitude of mechani- 

 cally possible states characterised by //, = n',' - /An[— /?;') (k = 1, 2 . . . .s) where / 

 takes all values between and 1, the expectation lies at hand that it might also 

 be possible to obtain an expression for the probability in question by comparing 

 the emitted radiation with the intensity of the radiation emitted on ordinary 

 electrodynamics by an electron performing a simple harmonic vibration which 

 may be represented by 



= ^ C cos 2 - 'J t, (108) 



where C is equal to a suitably chosen mean value of the amplitude C; of the 

 vibration of frequency (/»'^ — /j") +....( /»s — nl')oj,. occurring in the motion in 

 the different states characterised by different values for ') The value for the pro- 

 bability A', for the spontaneous occurrence of the transition in question would then 

 be given by | c^v^. The exact determination of A^,, however, is at present a quite un 

 solved problem which involves fundamental difficulties. But, even if the exact value 

 of A'„ was known, a calculation of the intensities would moreover require the know- 

 ledge of the number a' of atoms which in the initial state are present in the vacuum 

 tube; the determination of this number, which will obviously vary to a large extent 

 with the experimental conditions (pressure, voltage, etc.), is in general a difficult 

 problem in itself. 



' Among the possible expressions for sucii a mean value, an expression of the tj'pe 



C = eil^^li^^'^' (109) 



offers itself naturally, since, with this definition of C, the expression Cei~i'>i, of which ( 1 08 1 forms the 

 real part, appears directly as the logarithmic mean value of the expression 



e2T.i[ («; - „'[) OH+.... - ,/;) to, ) i , 



the real part of which represents the corresponding harmonic vibration which occurs in the motion of 

 the system in the states characterised by the different values of X. It follows from the well known 

 properties of such logarithmic mean values that it makes no difference whether we take the mean 

 values of the squares of the amplitudes or the squares of their mean values. It may moreover be 

 remarked that in the special case where the relative intensities of the components into which a given 

 hydrogen line is split up are asked for, — and in which, as mentioned in the text below, it is possible to 

 obtain a direct test for a formula representing a theoretical estimate of the relative values for the a-priori 

 probabilities of transition between the different pairs of stationary states, — the above mean value pos- 

 sesses the advantage that we sliall obtain the same relative values for the estimate for these probabilities, 

 whether for C we take the amplitude (or the „relative" amplitude introduced on page 52) of the vibra- 

 tion itself or the "amplitude" of the corresponding velocity, or acceleration; a point the importance of 

 which will be understood when it is remembered how small our actual knowledge of the mechanism 

 of radiation is. In § 8, however, it will be shown, in connection with the theory of the Zeeman effect, 

 that mean values of the type C-', as defined by (109), can never represent an exact expression for the 

 relative intensities of the components, because they do not satisfy the fundamental condition that 

 small external forces can only produce small changes in the intensity distribution of spectral lines. 



