an Electric Source, and Line Spectra, 793 



Generalities on Distribution of Spectral Lines. 

 As we have repeatedly remarked, the position of the lines 

 of the spectrum emitted from a source of high permittivity 

 K is given, with more than sufficient accuracy, by the roots 

 themselves of the transcendental equation tanw = w, 



u — Ui, u 2 , w 3 , &c, 

 where 2™ SK 



X being the wave-length reduced to vacuum, and a the radius 

 of the source or atom. To abbreviate let us write, as before, 



47r 2 a 2 K = k ; (41) 



then & = XV (42) 



Now, let the permittivity be any function of X, k = h(\), 

 representing what we have previously called "the intrinsic" 

 or "the atomic dispersion/'' If the function k(X) be given, 

 we can draw, with A. 2 as abscissa and u 2 as ordinate, the curve 



M '=^ = /(^), say, .... (43) 



and a series of parallels to the axis of abscissae, u 2 = u 2 , 

 u 2 = u 2 2 , &c, in general, u 2 = u^. Notice that the equation 

 tanw = w has certainly no purely imaginary roots *, so that 

 all these parallels will be contained in the upper part of the 

 plane A, 2 , u 2 . The position of the spectral lines will be deter- 

 mined by the points of intersection of these parallels with 

 the curve (43), the corresponding wave-lengths being given 

 by the square roots of the abscissas of these points. Accord- 

 ing to the nature of the curve (and therefore of the dispersion 

 law) there may be one or more spectral lines, or none, 

 corresponding to the i-t\\ root, u { . Should any branch of 

 the curve lie in the lower part of the plane X 2 , ?r, there will 

 nevertheless be no points of intersection and, therefore, no 

 spectral lines corresponding to this branch f. 



* In fact, g(u) = sin u/u— cos u can be written 

 2 4 6 



^)=o T « 2 - 5lW l -f- 7! H G - 



Now, if u 2 were negative, all the terms would be negative. Thus, there 



is no negative u 2 satisfying the equation y = 0. 



t It will be remembered that the physical meaning of the real part 



c 2 

 of K is, in general, not c 2 /v 2 but -3 (1 — a 2 ), where a is the " coefficient 



of extinction." Thus a negative real part of K would correspond to 

 o->-l, *. e. to very strong extinction. 



Phil. Mag. S. 6. Vol. 30. No. 180. Dec. 1915. 3 F 



