WILSON. — RECTILINEAR OSCILLATOR THEORY. 115 



is a special case in which the particle just comes to rest at the origin, 

 remaining there in a sort of unstable equilibrium.) 



The consequences of this discussion of the second type of equation — 

 the equation (6) founded on the radiation formula generally adopted — 

 may be formulated in the unsatisfactory statement that, except for 

 two special possibilities, the motion runs into a point d'arret where at 

 a certain time and in a certain position, with a definite velocity and 

 acceleration neither of which vanishes, the motion suddenly becomes 

 imaginary. This is very disconcerting; it is at direct variance with 

 our accepted notions of what happens in mechanical or electrodynamic 

 systems. 



It is also a consequence that the oscillator governed by equation (6) 

 cannot oscillate. 



Unfamiliar as this result may sound to those who have a feeling 

 that a slight change in the differential equation introduces only a slight 

 change in the motion, we must nevertheless admit the result and de- 

 velop the contrary feeling that a slight change may entirely alter the 

 type of motion. This fact has been brought out by Borel in a paper 

 just printed 7 wherein it is shown that the equation ir + x 2 = or 4- A/, 

 no matter how small the constant X, does not define an oscillatory 

 motion, nor does the motion approach the oscillatory type as X ap- 

 proaches zero. Any one might admit that, in the course of a long 

 time, \t would become important and the type of motion be changed; 

 but this is not the true explanation, for, no matter how small the 

 constant X, the motion departs from the simple harmonic type from 

 the time t = r/2, that is, after executing a quarter oscillation. 



4. The integrals of (6). We shall now turn to the integration 

 of the equation 



The first thing we shall treat is motion from a position of rest. In 



7 Borel, Annali di Matematica (3), 21, 225, (1913). The equation of 

 Borel corresponds to a constant radiation of energy. The solutions of the 

 equation do not exhibit the point d'arret, but are continuous. It has seemed 

 to me, notwithstanding the publication of Borel's paper, and perhaps even on 

 account of it, that the printing of my own results would be of interest. The 

 comments on Borel's paper, which he prints as a footnote to a brief account of 

 his results in his new book, Introduction Geom6trique a Quelques Theories 

 Physiques, 107 (1914) appear less applicable to the results of our present 

 work. 



