108 PROCEEDINGS OF THE AMERICAN ACADEMY. 



prime facts or hypotheses underlying the analysis. These supposi- 

 tions are as follows: 



1°. The oscillator executes in the main a simple harmonic motion. 

 This leads to the statement of the (approximate) equation 



i+««-ft • a) 



where x is the coordinate defining the state of the system. 



The basic reason for this assumption is that it is desirable for the 

 oscillator in its function of emitter of energy to be as simple as possible, 

 that is, to emit monochromatic radiation of a definite frequency; 

 for our customary analysis resolves heterogeneous radiation into 

 monochromatic elements. 



(If we wish to assume that the oscillator consists of a widely spread 

 uniform positive charge at rest and of a concentrated negative charge 

 vibrating to and fro through the center of the positive charge, we may 

 obtain a model of the oscillator which satisfies the condition that the 

 force on the moving particle shall be proportional to the displacement ; 

 for if a particle moves within a uniform sphere and if the elemental 

 attraction follows the Newtonian law, the resultant force is proportional 

 to the distance of the particle from the center of the sphere.) 



2°. The motion of the oscillator is not affected by frictional re- 

 sistance. This leads to (or corresponds with) the omission of a possi- 

 ble corrective term proportional to the velocity dx/dt in (1). 



Perhaps the best reason for omitting frictional terms is that there 

 seems to be no necessity for complicating the mathematical equations 

 or physical hypotheses by admitting friction. In the Hertz radiator 

 there is frictional dissipation of energy due to the resistance of the 

 circuit; in the idealised oscillator the resistance would naturally be 

 omitted. 



3°. The motion is damped by the emission of energy. If the form 

 of (1) appropriate to the principle of energy be used, that is, the 

 integrated form, the undamped motion would be characterized by the 

 equation 



'dx^ 



I) 1 ***-' . - i 



d t> + ™ 



= o, 



and the damping would then be inserted by a change to the form 



d 



(!)'+„_ c _ 2 jr; M 



or — 

 dt 



dx\ 2 



= - 2F, (2) 



where F is the rate of radiation. 



