ON THE THEORY OF THE RECTILINEAR OSCILLATOR. 1 



By Edwin Bidwell Wilson. 



Introduction. 



Planck's rectilinear oscillator is of great importance in theoretical 

 physics. To grant this it is not necessary to assert a belief in the 

 physical existence of the oscillator, much less to assert Planck's 

 belief in such existence. The Carnot engine may not exist; but the 

 concept has proved so fruitful as to be fundamental in the general 

 field of thermodynamics. The (perfect) semipermeable membrane 

 may be a figment of the imagination; but, in special fields in thermo- 

 dynamics, reasoning based on the membrane is of crucial impor- 

 tance. And in that branch of thermodynamics which deals with 

 radiant energy, the oscillator has held the position of cornerstone. 

 Moreover, as in the case of the Carnot engine and semipermeable 

 membrane, there does exist in physics a (more or less imperfect) 

 prototype of the oscillator — the radiator or resonator of Hertz and 

 his followers in radiotelegraph}-. 



In addition to the thermodynamic importance there is an interest 

 attaching to the oscillator as the simplest model of the atom. Hypo- 

 thetical models of the atom vary greatly; some are founded on 

 statical configurations, others on dynamic equilibrium, some call for a 

 number of corpuscles per atom comparable with the atomic weight, 

 others for a number far greater than this, some represent the positive 

 charge as widely spread out with the negative corpuscles imbedded in 

 it, others concentrate the positive charge at a point. It is not neces- 

 sary to take the oscillator very seriously as a model of an atom; all 

 that is useful is to observe that by its means energy is absorbed and 

 emitted and that thus it performs the functions of matter in establish- 

 ing a temperature equilibrium in a closed field of radiant energy. 



The object of this article is to make some comments on the equation 

 which defines the motion of the ^oscillator. These comments are 

 obvious at first sight to any one who examines the differential equa- 

 tions used in the theory of the oscillator, but as I cannot find them in 

 the literature, I venture to print them at this late day. 



1. The differential equations. Let us consider first some of the 



1 Read to the Academy at its Stated Meeting, March 11, 1914. Received 

 June 16, 1914. 



