SYSTEMS WITH NON-LINEAR REACTANCE 439 



being parallel to the direction of vibration. Suppose now that at some 

 point along the string we apply an alternating mechanical force, acting 

 normal to the string, through the medium of a mechanical structure, 

 which as viewed from the string may be represented by a linear me- 

 chanical impedance. This structure prevents motion of its point of 

 attachment to the string, in the direction of the string. 



If now we analyze the forces and motions into their components in 

 the direction, x, of the motion of the vibrating member and that, y, of 

 the applied force, we find that, to a first approximation, the relations 

 connecting them are identical with those used above for the condenser, 

 provided we identify the force and velocity of the vibration in the x 

 direction with those of the condenser plate and those of the point of 

 attachment of the string, in the y direction, with the electromotive 

 force and current in the electric circuit associated with the condenser. 

 Such a structure can therefore produce oscillations of the sort described, 

 provided the mechanical impedance of the driving structure has the 

 proper values at the sum and difTerence frequencies. 



Suppose now we have a molecule which we assume to be rigid with 

 the exception of one atom, which is bound to it by a pair of electrons. 

 Let the attached atom correspond to the plate, the relatively heavy 

 molecule to the support and the electrons to the point of application 

 of the driving force. Let the forces of electrostatic attraction between 

 the electrons and the atom, and between the electrons and the center of 

 the molecule, correspond to those due to the tense strings. Let the 

 other static forces between the atom and the molecule correspond to 

 the stiffness of the plate. For small displacements these forces may 

 be assumed to vary linearly with distance, and so be capable of repre- 

 sentation by constant coefficients of stiffness which correspond to the 

 elasticities in the mechanical system. The applied external force is 

 that exerted on the electrons by that component of the incident light 

 which is normal to the line through the centers of the undisplaced 

 particles. The mechanical impedance of the electrons for motion in 

 the direction of the applied force corresponds to that of the structure 

 through which the force is applied. This impedance includes the 

 effects of any elastic constraints the rest of the molecule may exert on 

 the electrons in this direction ; of the electromagnetic mass of the elec- 

 trons, which may be affected by the reactions of neighboring molecules; 

 and of the dissipation of energy as radiation or by transfer to neighbor- 

 ing molecules. 



Unlike other classical models of the Raman effect, this one provides 

 for the persistence of the difference line, and the disappearance of the 

 sum line, at low temperatures. It also provides that the intensity of 



