796 Dr. L. Silberstein on Radiation from 



molar optics, has in the present connexion a remarkable 

 analytical property. In fact, by (44) and (42), the equation 

 determining X 2 , that is the position of the lines of the 

 corresponding spectrum, is 



W=* + 2 x2 J 2 , 



that is an equation of the k + 1 order in X 2 , say 



X 2(k+1) + A k X 2k + . . . A : X 2 + A = 0. 

 Now, by multiplying out, 



Ao=-(-irVv-7 < s f4-4' 



Thus, when (45) is satisfied, A vanishes, and the equation 

 for X 2 is, with the obvious rejection of the root X = 0, 

 reduced to the /e-th degree. Now, since each y t gives rise to 

 a convergence-point with its series of lines, it is obviously 

 desirable to have, in the case of dispersion with k terms, 

 /c and not /e-f 1 lines to correspond to each root u=Ui. And 

 this is precisely what is secured by the relation (45). If we 

 desire to obtain the more general dispersion law (44), with 

 any k 0l we have only to introduce a (« + l)st convergence- 

 point 7^»co with such a coefficient b that /3 = b/y 2 should 

 have the required finite value; thus, the relation (45) is 

 ultimately no real restriction. 



There is no need of expanding here the expressions of the 

 remaining coefficients A, in general. It will be enough to 

 write down, for the sake of illustration, the formulae for 

 k = 2. Thus, when the dispersion is 



*-*"+ X 2 - 7l 2 + X 2 - 7 / ~ X U 2 -7i 2 ^-7/J ' 



we have for the determination of the spectrum-lines the 

 quadratic equation 



-A_+ ft =u , 



Ai —y l Xi —y 2 



or 



— aix + a = 0, x = \ 2 , 



«o = 7iV+ ^t(A72 2 + A7i 2 ) 5 



Ui 



(46) 



giving for every root ui (i==l, 2, 3, . , ,) two different roots 

 x l and, in as far as both of these are real and positive, two 



