WILSON. — RECTILINEAR OSCILLATOR THEORY. 121 



As k is in the neighborhood of 2 10 -8 , the difference between r and t/2 

 is about 4 10 -7 . 



We desire next to follow the motion through the origin and on to 

 the point d'arret. 10 Construct the Riemann surface for the integrand 

 in (12). The surface has branch points at u = and u = — 4k, and 

 a junction line between them. We have let u trace the real axis from 

 to oc ; now let u turn through a large circuit in the upper half-plane 



and come down to the other end of the axis of reals. No time elapses 

 during this process, for approximately dr = du/u 2 . The variable x 

 merely swings around the origin on an infinitesimal semi-circumference 

 and is ready to start off on positive values as u comes in along the 

 negative real axis. That the branch point u = — 4x corresponds to 

 the point d'arret for the motion is readily surmised, and may be 

 proved from the defining equations (11). 



The time from the origin to the point d'arret is therefore 



/•- 4. &i 



T'= / . u -1 ,- . 



J- co u 2 — — V M 2 + 4 KW 



But with the substitution (13) the integrand becomes again the same : 





4k dy 



8 K y + {i - y*) a - y y 



Hence 



r = K(1 -,,)[-,o g L^ + 2 ^-ta-„'!) 



10 The simplest method seems to be one dependent on the theory of functions 

 of a complex variable; for it enables us easily to pass around the singularity 

 at the origin. 



