126 PROCEEDINGS OF THE AMERICAN ACADEMY. 



tion of the second order with constant coefficients, and we may deter- 

 mine the logarithmic decrement of this equation so that the average 

 loss of energy shall be equal to that determined by the theory of 

 Hertz. 12 This is frankly phenomenological and seeks not to go too 

 far below the superficial appearances ; but it is probably better than to 

 delve so deeply that we must sacrifice either our skin or our conscience 

 in getting back to the surface. 



We have thus far confined our attention to the oscillator, but simi- 

 lar results may be obtained for other systems. For instance, J. J. 

 Thomson 13 has proposed as a model of the atom a center repelling a 

 corpuscle with a force varying as the inverse cube of the distance (and 

 furthermore attracting as the inverse square if the corpuscle lies 

 within a tube issuing from the center). To treat the motion under 

 the repelling force he writes 



d 2 .v ce 

 at- x 6 



integrates the equation and then calculates the radiation by 



const J (MJ dt 



The motion is therefore obtained by ignoring the reaction of the radia- 

 tion. 



If, however, we write the equation for rate of change of energy at 

 each instant, we have an equation of the type 



*(* Vu-«f£Y 



dt \dt- r :i J \df 



or 



■ f 2k 2k ^ + 7T- 



For approach to the center, v < 0,/ > 0; but the acceleration becomes 

 imaginary when — vr 3 — 4k, and the motion has a point d'arret. 

 Motion away from the center does not have this peculiarity. 



The real interest of our work lies, however, not in the critique of 

 current "demonstrations," nor in a tour de force integration of a 



12 See, for instance, J. Fleming, Principles of Electric Wave Telegraphy, 

 chaps. 3, 5 (1906). 



13 J. J. Thomson, On the Structure of the Atom, Phil. Mag. (6) 26, pp. 792- 

 799, Oct. 1913. 



