342 Dr. Ketteler on the Dispersio?i of Light in Gases. 



But continued and very comprehensive calculations have de- 

 termined me to replace this formula by the following : — 



w - i= «-V (III) 



If we here also put /3=X =VT , where V stands for the velo- 

 city of light in a vacuum, and T for the time of vibration, and 

 if l=v T be regarded as the internal wave-length connected with 



y 

 X by the index n = -, equation (III) takes the following form : — 



V 





X 



X 



n- 



n ~ 



-1 

 -1 



X 



"o 







\>" 



"7° 



or 



(>-f)(¥-v)-g-T)' 



and since, for X=oo , we get n^ — l = a, 



X X r 



v"" x ) '- v /t " _/t oo; — jT • -f> 

 or 



V-v v T 2 



If in equation (III) we put - for I and solve for n } developing 



the resulting exponential quantities by the binomial theorem and 

 making the proper reductions, we obtain 



n — 1 fl 2 8* 



_i=l + (l + «)g r +(l + «)(l + 2«)^ + ...*. 



On the other hand, if X be replaced by nl 3 we obtain the analo- 

 gous series^ 



!LzI_t . __L_ £ 2 , _!_ ff 4 , Lz* £ . 



a ^l-f-*' Z 2 "* 1 "(l+a) 8 ' / 4 (1 + a) 5 / 6 " 



Hence we get for the square of the reciprocal of the index of 

 refraction 



1 2u /3 s «(3«-2) /3 4 3«(l-4«-|-a 2 ) /3 6 



' (l + «) 2 (1 + *) 4 ' I 2 + (1+a) 6 ' / 4 (l + «) 8 ' l 6+ " 



an equation which, when the values of the constants have certain 

 ratios, may be reduced to the first two terms. 



* Writing n 2 for n, this brings us back to the equation first proposed. 



