114 PROCEEDINGS OF THE AMERICAN ACADEMY. 



unit of time is the fraction \/(2irc) of the second. This equation 

 differs very minutely from (5') or (7), but there is no reason to believe 

 that the motion denned by the equation should resemble very closely 

 the motion defined by those. As the integration of the non-linear 

 equation of the second order is not simple, we shall first examine the 

 equation to find certain elementary qualities of the solution. 



A. When the velocity vanishes, the acceleration also vanishes. 



B. When the acceleration vanishes, either the velocitv or the 

 displacement vanishes. 



C. In the neutral position x = 0, either the acceleration vanishes 

 or the acceleration and velocity satisfy the relation kj + v = 0. 



D. Arbitrarily assumed (initial) values of the acceleration and 

 position determine the velocity, and similarly values of the accelera- 

 tion and velocity determine the position, but assumed values of the 

 velocity and position allow the acceleration either of two values 



j- v 1 ,— — 



Ik Ik 



E. Values of v and x which make v' 2 — 4kvx negative are not ad- 

 missible. Hence if x is positive, v must either exceed 4/a- or be nega- 

 tive; and if x is negative, v must be positive or less than 4.kx (alge- 

 braically) . 



F. The case of x positive and v greater than 4kx leads to a point 

 d'arrM. For under these conditions / is negative, no matter which 

 sign is taken before the radical, and the velocity must be decreasing as 

 x increases; when v becomes equal to 4/cr, the acceleration becomes 

 imaginary, and so does the motion. The same arises under the case 

 of x negative and v less than 4/c.r. 



G. The case x positive and v negative also leads to a point d' arret. 

 For under these conditions if the negative sign is taken with the 

 radical, then / is negative and the magnitude of v is increasing as x 

 diminishes toward zero, the particle will reach the origin with a definite 

 velocity and pass through to the negative values of x where it falls 

 under the case F; and if the positive sign is taken with the radical, 

 so that / is positive, the particle is slowing down, and we have to 

 consider two possibilities, first where the particle reaches the origin 

 before its velocity vanishes and where the argument just given will 

 then again be applicable, second where the particle comes to rest 

 before reaching the origin only to start on again with a negative / and 

 fall under the first supposition of this case G. A similar discussion 

 may be given for the hypothesis x negative and v positive. (There 



