REVERSED AND NON-REVERSED SPECTRA. 25 



If both the difference of wave-length and wave-velocity are considered, 

 we should have for the first spectrum v and n, and for the second spectrum 

 v and n'. The conditions would be left unchanged, if the second velocity is 

 taken equal to the first and the frequency n'(v'/v) replaced by n'. From this 

 it follows that the number of beats N is nearly 



If 5X is considered negative, if ^ = A B/\ 2 and the multipliers ju and ju 2 be 

 neglected, 



which is the difference of the two cases above computed. As the first is very 

 large compared with the second, the visibility of the phenomenon is not 

 changed. 



The theory of group waves usually introduces a factor 2. Thus if Xi, Vi, HI, 

 be the group wave-length, velocity, and frequency, 



or, 



or with the above data 



results otherwise like the above and without bearing here. There is a possible 

 question whether differences of wave-length due to velocity and not to period 

 can be treated as dispersion. 



The occurrence of forced vibrations has also been looked to as an explana- 

 tion. Though here again, even if the spectra are almost always of unequal 

 intensity, the reason for the preponderance of one would have to be stated. 

 True, equal mean strength is not equivalent to equal instantaneous strength. 

 In the case of forced vibrations, however, if the harmonic forces of one spec- 

 trum are F = A cospt (forced, T = 2Tr/p), of the other F = A'cosqt, (free, 

 T = 2ir/q) and there is no friction, the resulting harmonic motion will be 

 given by 



r-t 



Now if we regard the case of figure 15, on one side of the line of coincidence 

 Xo, q 2 >p 2 ', on the other side, p 2 >q 2 . Hence, whenever a brilliant line flashes 

 out due to coincident phases, there should also be a black line due to opposi- 

 tion; and, in fact, when the phenomenon is produced under conditions of 

 perfect symmetry of the component beams, this seems to be its character; 

 i.e., the enhanced line cuts vertically across the breadth of the spectrum. 

 The case q 2 = p 2 , being of infinitely small breadth, would not be visible. It is 

 not to be overlooked, however, that in certain adjustments, particularly in 



