SPECTRAL LINES 451 



Thus the frequency of the emitted radiation differs from the 

 natural frequency by an amount 



dn = — 27rNae 2 /mn o 



and the corresponding change of wave length is given by 



$X dn _ Nae*\* 



X n — 27rmc a 



since X = 27rc/n. 



Now the number of electrons per unit volume may be 

 assumed proportional to the density and therefore to the 

 pressure ; and so when the pressure is increased there is an 

 increase in the wave length of the emitted light which is pro- 

 portional to the increase in pressure and also to the cube of the 

 wave length. The increase is moreover independent of the 

 nature of the gas surrounding the arc. These results are all in 

 accordance with experiment. 



In the more general case in which the gas emits a number of 

 spectral lines, corresponding to electrons with different free 

 periods, P is given by 2Ner, the summation being with regard 

 to the different free periods. If light of frequency n is emitted, 

 the equation of motion of an electron becomes 



m(n* - n 2 )r = 4?raeP 



and using the relation P = 2Ner to eliminate P, the frequencies 

 of the emitted light are given by the equation 



4?rNae* 

 2 m(n;-n*) =I 



For the value of n near n only the term in the summation 

 which contains (n 2 — n*) in the denominator need be retained as 

 a first approximation, and this value of n is thus the same as 

 that first obtained. It should be noted that now N will probably 

 vary from line to line, and one cannot expect to deduce any 

 general law of variation of displacement with wave length, but 

 other things being equal, the result points to a variation pro- 

 portional to the cube of the wave length. This law may be 

 expected to hold for lines which have a common origin. 



The effect of a change of density upon the positions of the 

 absorption lines may be treated in a somewhat similar manner. 

 If the electrons are set into vibration by the periodic electric 

 force E of frequency n (varying as e int ), in an advancing light 

 wave, the typical equation of motion becomes, in the usual way, 



m(n* - n' + ihn)r = e(E + 47raP) 



