Dispersion in Relation to the Electron Theory. 233 



would thus add to the difficulty of detecting selective effects, 

 in such cases the values of \ 1? X 2 > & G -> can 0U ^J ^ e obtained 

 from the dispersion curve itself. In cases where the dis- 

 persional bands lie in the near ultraviolet (*4yu, to 2 /u), and 

 are therefore amenable to direct experimental investigation, 

 it is still desirable to determine the dispersional frequencies 

 from the dispersion curve itself. For according to the 

 theory of Lorentz-Planck the periods \ 1? \ 2 , . . . . do not 

 coincide with the free periods of the resonators responsible 

 for the absorption but are sensibly shorter than the latter ; 

 and the constants a 1? a 2 , .... acquire a meaning different 

 from that ascribed to them in the original elastic-solid and 

 earlier electromagnetic theories. In the latter « x , a 2 , ... 

 are proportional respectively to the number of resonators of 

 each period in the unit volume. In the Lorentz-Planck 

 formula we may write 



3.9 



««-i±£ and A i 2 =T^> 



where g is proportional to the number of resonators in unit 

 volume and \ is the wave-length corresponding to the free 

 period. Thus 



n »-i- 1 -g 



The well-known labours of Rubens, Nichols, Paschen, 

 Martens and others have provided ample experimental con- 

 firmation of the application of formulae of the type (1) or (2) 

 above so far as natural dispersion is concerned. The 

 observations can, however, seldom be pushed far enough 

 into the ultraviolet to determine the differential effect of two 

 bands in the Schumann region, and the dispersional period 

 deduced is in most cases an effective mean value only. 

 Denoting this by X 1 we may write 



a{K 2 



A- ~ * Aii 



where a is the contribution made by electrons whose 

 frequency is so high that their effect on the dispersion is too 

 small to be detected experimentally, and — c\ 2 represents 

 the small residual effect of infra-red resonators in the region 

 of the spectrum to which the formula applies. As, however, 

 there are here four adjustable constants, the formula acquires 

 a somewhat empirical character, and it is therefore desirable 



l = «o+^F^-cX 2 , .... (3) 



