266 Dr. L. Silberstein on Fluorescent Vapours 



Keeping this in mind, we can mathematically describe 

 the excitation and emission of fluorescent spectra by saying 

 either 



1st, that the atoms of the vapour behave so as if each 

 contained a Hookean resonator under the simultaneous 

 action of forces of all the frequencies n = N, %, ?i 2 , n 3 , etc., 

 i. e. by writing 



2+jfei + lPa?=^ <!r ' + c 1 «* ,l *+ V * , »*+..., . . (1) 



or, 2nd, that each atom contains an appropriate non-Hookean 

 resonator, acted upon by c Q e %m only, i. e. that 



x 2 + kx + Wx+f(x) = c e im , .... (2) 



where f(x) is some non-linear function of the displacement. 



The equivalence of the two methods of treatment is 

 manifest. In fact, the non-Hookean resonator will be the 

 "appropriate" one when, and only when, the supplementary 

 term —•/(#) ultimately reduces to cie mit + c 2 e ln ' 2t + ... 



If the frequencies n l5 n 2i etc., accompanying the funda- 

 mental one N, and the ratios c x : c , etc., are all assumed to 

 be known (from experience), then we do not require the 

 form (2) at all, and we can study the properties of each line 

 of the spectrum separately by writing x = x + x 1 + ... and 

 splitting (1) into 



%i + kXi + W3Bi=Cie in i\ i=c,l,2, .... ... (3) 



and introducing into each of these equations, and of the 

 analogous ones for y^ £,-, the well-known supplementary 

 terms due to a superposed magnetic field. It is only when 

 we wish to make a guess as to the law of sequence of the lines 

 in the spectrum that we require the form (2). This will 

 occupy our attention for a few moments in the next section, 

 in which we shall assume hypothetically a particular form 

 o£/(#), and test it experimentally; but in all the remaining 

 sections of this paper we shall avail ourselves only of the 

 equation (3) and of ordinary electromagnetism, so that all 

 results obtained in these sections will be entirely independent 

 of that hypothesis. 



2. Law of Constant Frequency Intervals. 



The simplest case of a non-Hookean resonator is well- 

 known from acoustical problems in which use is made of a 

 quadratic supplementary term, const. Xx 2 . Such a term 

 would, obviously, not answer our purpose, since it gives, in 



