710 



PROCEEDINGS OF THE AMERICAN ACADEMY. 

 TABLE II. 



Showing the motion of any point, .r, of a string bowed at the aliquot point 1/k. 

 The table describes a half vibration, from equilibrium ascending to equilibrium 

 descending. The ascending velocities alternate until (j+1) ascents are accom- 

 plished ; then the descending velocities alternate until equilibrium is reached ; 

 then the vibration is continued below equilibrium in such a way as to make the 

 displacement an odd function of the time. The last ascent is always with the 

 smaller of the ascending velocities; similarly the first descent is with the smaller 

 of the descending velocities. The table describes (k + \) straight lines, but the 

 first and the last of them are halves of longer stretches in the complete curve 

 which thus consists essentially of 2 k straight lines. This shows why the duration 

 of the first velocity is given in the table as half that of the third ; the other half of 

 the interval is thought of as belonging to the preceding half-vibration. 



