Refractivity and Atomic Interaction. 125 



showing a peculiar regularity of structure. Developing it, 

 we obtain 



1 q 97 



D = l-Zg^+ l9 ' +i g^- u g\ . . (46a) 



The interesting part of our problem consists not so much 

 in constructing the coefficients B^ of the final dispersion 

 formula (43), as in actually finding the "new" free fre- 

 quencies, vV? V7", etc. These will belong to the 

 molecule in absence of E, that is for Gr 13 = 0, etc., when 

 the equations (44) become 



^3=^13 — i^is), etc.; r d2 =g(r 12 — Jr 13 ), etc., . (44 o) 



defining the ratios of the six displacement components in 

 the molecular plane. The required free frequencies will be 

 given by the roots g', g", etc. of the equation D = 0, where 

 D is as in (46) or (46 a). This is an equation of the sixth 

 degree, containing not only g 2 , (g 2 ) 2 , (<7 2 ) 3 > but also g 3 . 

 At first, the task of solving it rigorously seemed hopeless. 

 But the form of (44 o) suggested that it would be worth while 

 to try whether these equations can be satisfied by assuming 



n2 = ^3=^3i = ^ say, 

 and 



*'21= **32=?'l3 = y> 



This has turned out to be the case. In fact, on substitution 

 in (44 o) we find that these six equations are all satisfied, 

 provided that x/y=(l + ^g) : g = g : (1 + |#), and therefore, 

 g 2 — ±g=%, that is to say, either 



q=g / = 2, and y = x ") 



[. . . . (47) 

 g=g 1 =-§, and y=-x) 



This gives at once two roots of D = Q. The first of these 

 might have also been derived directly from the tabular 

 form (46). In fact, 1- -^g is the sum of the elements of 

 each row (or of each column) of that determinant, and 

 therefore a factor of D. At any rate, we have in (47) two 

 of the six roots and, at the same time, the corresponding 

 fundamental modes of vibration, which are shown in fig. 3. 



