122 Dr. L. Silberstein on Molecular 



Thus the six equations of motion, (38), become 



" i / \ '711.9 / -^ l 



r23—a 2 r 21 + y 2 (r 23 — a 2 r 2i ) /o 3 r l3 = G 23 , etc. 



m 2 



^32 — «3^31 + 7a(^32 — «3^0 l /Vl2=Gr32, etc., 



991>3 



where Gr 23 , etc. are homogeneous linear functions of the 

 components of E, the electric vector in the incident wave. 

 Therefore, for monochromatic light of frequency n= ^7, 



(72 — 7)0*23 — Wsi)- -Vs r 13 = Gr23, etc. , 



[•• (41) 



(73 — 7)(^32-^'3l)-~ /Wl2=G-32, etc. J 



These are six linear equations for the six displacement 

 components r 23 , etc. Resolving them and substituting r 23 , etc., 

 as linear functions of the components of E, into the right- 

 hand member of equation (10) projected upon the molecular 

 plane, we shall have K p (and therefore also co p and JS T P ) as a 

 linear vector operator in that plane. Since -p^K^Ep has to 

 represent the (density of) electric energy, K p , and therefore 

 iV p , should be self-conjugate operators. Thus S p will have, 

 in the molecular plane, two orthogonal principal axes, and 

 two principal values, say, N Pl and N P2 , generally differing 

 from one another. Now, the determinant of the system (41) 

 is of the sixth degree in 7. The roots, 7', etc., of that deter- 

 minant equalled to zero will be the squared free frequencies 

 of the molecule, produced by atomic interaction. Whatever 

 the shape of the molecule, i. e. of the triangle Oi0 2 O s , all of 

 these roots will be real and positive, provided that the inter- 

 atomic distances i? z - do not fall below certain values marking 

 the limit of optical stability *. Thus, within the range of 

 stability, there will be six real free frequencies, in general 

 different from one another, belonging to the oscillations in 

 the molecular plane, besides the three original atomic ones, 

 which belong to oscillations normal to that plane. These 

 will be shortly referred to as the normal oscillations and 

 frequencies. 



In the case of equal orientation of all molecules, therefore, 

 the resulting substance will be a pleochroistic crystal with 



* In other words, 3>, in (40), will be a positive quadratic form, 

 provided that pi, p 2 , p* are small enough. 



