:H 



«!l! 



268 Dr. L. Silberstein on Fluorescent Vapours 



it is enough to note that pi is proportional to | b p 

 therefore, by (5), 



an( 



Pi 



(&)'■ 



Otherwise the coefficients b, b u and the following ones are 

 of no interest in the present connexion, since we are con- 

 cerned here with the emitted frequencies only. Substituting 

 the second approximation in the term as? in (4) we have 



(be im + b ie ™ pt ) p = b p e ipm +pb p -\e in2p - l)i 



Thus, as a third approximation, 



iS(Sp-2)t 



+ . 



= be im + he ip ™ + h 2 e i{2p - l)m + h 



$e 



i(3p-2)m 



(6) 



(6 a) 



where b, b l are as above, and b 2 , & 3 , etc., are easily determined. 

 With regard to these coefficients it will be enough to remark 

 here that as long as | bi | contains a positive power of c /&N, 

 the corresponding line will be strong enough to be visible ; 



now, if 



p> 



1 



(7) 



wnere r is a positive integer, the last condition will be still 

 fulfilled for the rth line of the spectrum (the fundamental 

 line being the zeroth line). Thus, if p satisfies (7), we can 

 have, theoretically, as many as r lines, not counting the 

 fundamental one. 



Returning to (6 a), we see that the third approximation 

 gives a series of lines whose frequencies are 



n = N, ni =pN, n 2 = (2^-l)N, n 3 =(3p— 2)N, etc. (8) 



Pushing the process to the fourth approximation we should 

 obtain, in addition to (8), such frequencies as _p 2 N, etc., 

 satellites of the above, whose interest will be shown in a 

 future communication. For the present we shall confine 

 ourselves to the third approximation, i. e. to the series of 

 frequencies (8). 



This series can, obviously, be written 



n,-=N— t(l— ;>)N, 1=0,1,2,3,..., . . . (9) 



so that its members should succeed one another in constant 

 frequency-intervals, 



8n = (l-p)N, (9 a) 



the series extending from the fundamental line either towards- 



