WILSON. — RECTILINEAR OSCILLATOR THEORY. 119 



and smaller values of v than the latter motion for the same value of 

 to — t. When the (+) motion passed the origin, the acceleration was 

 zero; when the (— ) motion passed the origin, the acceleration was 

 negative and twice as large as at the cusp. The (+) motion may 

 be traced back to a point where the particle comes to rest for a certain 

 negative value of x; this corresponds to the solution (9), and the 

 motion may be traced still further back, the velocity and retardation 

 increasing all the while. The (— ) motion may likewise be traced back 

 through the origin x = 0, the velocity and retardation increasing. 

 The curve which shows roughly the relation between x and r is 

 given in Figure 1. If we call the curve which corresponds to the posi- 

 tive sign x — « < S ) ( + )(r) and the curve which corresponds to the negative 

 sign x = a$(_)(r), we may at once write the complete solution of the 

 differential equation (6'), namely, 



x = Ci$ 1+) (t - C 2 ), x = C& ( - } (t - C 2 ), 



with the singular solutions x = C. The first solution could have 

 been written at once from (9) ; the particle is at rest at x = C\ when 



T = Co. 



The way in which the constants of integration enter into the solu- 

 tion (which could be foreseen from the differential equation itself 

 inasmuch as the equation admits the two-parameter group x' = x + a, 

 r — t + a) shows that the motion, though not periodic, has definite 

 intervals of time independent of the constants of integration associated 

 with it — the time from rest to the origin and the times (two possi- 

 bilities) from the origin to the point d'arret. 9 



5. Numerical Calculations. To ascertain these periods of time 

 we must integrate the equation (6')- Let 



o 



or 



Suppose x = a < 0, v = 0, / = are the initial conditions. 

 Then u = 0. Moreover, as / becomes positive, du/dr—u 2 becomes 



9 Those latter would not appear in the case Borel treated 



