usual theory of dispersion (Lorentz, loc. cit. p. 139), the 

 external force acting on our z-th electron will be 



Refractivity and Atomic Interaction. 101 



>n (Lorent 

 >ur z'-th ele< 



^(E + aP), 



where a is Lorentz's l + s, and may for the present be 

 simply taken equal to 4, and P, the electric polarization of 

 the body, is the average of ffl{e l Y 1 + e 2 r 2 + ... +e K r K ) taken 

 over a physically small volume, but containing a sufficiently 

 large number of molecules, i. e. 



P = Slfari + e 2 T 2 + . . . . + e K x K ) = 9&eft* . . (7) 



Now, D being the (macroscopic) dielectric displacement, we 

 have D = iTE = E + P, where, for periodic oscillations, 



K=fi 2 ; 



the " dielectric constant " or the permittivity K being in 

 general, for anisotropic bodies, a linear vector operator, and 

 for isotropic bodies an ordinary scalar. Thus the external 

 force on the z-th electron becomes 



*[l + a(2r-l)]E, 



and with a = 3, L e., strictly for regular cubical distribution 

 of molecules *, and approximately for various other cases, 



Dividing this force by m* and introducing it on the right 

 hand of (6) we have the k equations 



*+v*~i**-lS;V+*>* • - .- («) 



For monochromatic incident light of frequency n = 27rc/\ 

 (in 2-7T seconds) the electric force E and all the vectors r are 

 proportional to e tnt , i = \/ — 1» s0 that r= — ryr, where y=n 2 . 

 Thus we obtain, for a substance composed of ^-atomic 

 molecules, 



fa_ 7)r< _ i, 4 = |J(^+2)E ; i=h 2, ...«•• (9) 



* This concerns the distribution of points, of which each is the 

 representative of a molecule as a whole, and does not necessarily imply 

 parallelism of orientation of such lines as O x 2 or 2 O s , etc.; the latter 

 will be the case only in (optically) crystalline bodies. On the other 

 hand we may have, say, a strictly cubic arrangement, but with a hap- 

 hazard distribution of directions of O x 2 , etc. ; then the body will be 

 optically isotropic. 



