Refractivity and Atomic Interaction. 99 



within the same molecule, constitute a rigid system, without 

 in the least inquiring into the forces keeping them together. 

 Let, in an isolated atom Ai, the displacement r» be counter- 

 acted by the usual restitutive force —772^7^, so that w t == V7; 

 is the free frequency (in 2ir seconds) of the atom when 

 unmolested by foreign interference. In other words, let 

 the differential equation of motion for the i-th isolated 

 atom be 



r; + Y;r;=0, i = l,'2, etc (3) 



Thus each atom by itself will be optically isotropic, since 

 <y i= m 2 is an ordinary scalar. 



We shall not introduce explicitly any " friction "-terms, 

 keeping solely in mind that such terms are to be brought in 

 whenever the period of the external agents approaches a free 

 period. 



Next, as to the interaction of atoms, in order to proceed 

 any further, we have to make some definite assumption about 

 the electrical properties of the centres 1? 2 , etc. We shall 

 choose a possibly simple one, viz., that when the electron 

 — ei is displaced, the centre Oi behaves as if it were the seat 

 of a point charge +^-, so that an electric doublet is created, 

 of moment efc *. xlnd so for every atom. (This does not 

 imply that there are no other charges in the atom, or that 

 the whole atom is electrically neutral. It means only that 

 we are concentrating our attention exclusively upon the 

 dispersive electron and its position of equilibrium.) Again, 

 having once assumed that the centres O l5 2 , etc. are fixed 

 with respect to one another, there is no need of contem- 

 plating the mutual forces between the charges + ei, + e 2 , etc. 

 appearing in O l5 2 , etc. Further, the action of Oi upon 

 its own electron can be considered as already contained in 

 the restitutive force — m^r;. Thus, the electron of the i-th 

 atom will be acted upon only by all the doublets produced 

 in the remaining atoms of the molecule. (The influence of 

 other molecules being disregarded, of course.) Of this 

 action we shall assume that it is ordinary electrostatic action, 

 which will certainly be the case if the velocities of the oscil- 

 lating electrons are but small fractions of the light velocity 

 in vacuo. Finally, we shall assume that the amplitudes of 

 these oscillations, i. e. of r z -, are small as compared with the 

 mutual distances Ra of the centres Oi, Oj. 



Under these conditions the force on the i-th electron due 



* If the reader desires to have a picture of these conditions he may 

 look upon Oi as the centre of Sir J. J. Thomson's "sphere of positive 

 electricity." But this is by no means necessary. 



H2 



