of Emission and Absorption Limes in a Gas-Spectrum. 275 



essential point is that the positive electricity is rigidly 

 connected with the atom. 

 Now consider 



Pz= 2, ex 



taken for all the electrons and positive charges in the manner 

 indicated. Let (.#, y, z) be the mean position o£ the positive 

 charge in the element of volume at any instant, then, since 

 the atom was originally neutral, this is also the mean position 

 of the negative charges before they are displaced relatively 

 to the positive charge. Thus 2 ex for the positive charges is 



x%e, 



and for the negative charges it is 



x 2 e + 2 ex, 



where (x, y, z) is now the displacement of the negative 

 electron e from its neutral position of equilibrium. 



But (2 e) for the positive charges is equal but opposite in 

 sign to (2 e) for the negative electrons, and thus we have 



P x = 2 ex, 



2 being now taken only for the negative electrons, (x, y, z) 

 denoting the component displacements of that electron from 

 its neutral position of equilibrium. 



We have thus expressed the polarization in terms of the 

 negative electrons alone, and it is in this form that we shall 

 make use of it. 



We shall now follow the procedure usually adopted in 

 these theories. It is admirably set out by Lorentz in his 

 book on electrons, and I follow him closely and often 

 verbally. 



§ 3. Equations of motion of the electron. — The position of 

 the electron, or better its displacement (x, y, z), and there- 

 fore the polarization, depends on the forces acting on the 

 electron. These forces are of four possible types. 



(a) Quasi-elastic forces. — If the electron vibrates about 

 any one position of equilibrium in a motion with a definite 

 period, it must be acted on by some elastic resilience, tending 

 to drag it into this position. We shall suppose a force of 

 this kind exists, directed towards the position of equilibrium 

 and proportional to the displacement from that position. If 

 the displacement is small we can easily imagine the force to 

 have its origin in the mutual electrical actions in the atom. 

 If we denote by k a certain positive constant which depends 

 on the properties of the atom, which may, however, be 



T2 



