706 PROCEEDINGS OF THE AMERICAN ACADEMY. 



replaced by V = l/k, T' — T/k, H' = Hjk~. That is, at the time t = 0, 

 each point in the first &th of the string would be ascending through its 

 position of ecpailibrium with a velocity 



4H> l> -x _1H/1 _x\ 



and this motion would continue until the time 



the turning point coinciding with that of the major vibration. It would 

 then descend during an interval centred at t = T/k with a velocity 



, _ ilT x _ \Hx 

 9i - T , J,— T j ' 



then ascend during an interval centred at <= 2 T/k with the velocity// 

 as before, and so on indefinitely. 



The complete motion, in the actual case, of any point x, < x < l/k, 

 is, therefore, an asceut from its position of equilibrium, during the inter- 

 val < t <t, with a velocity 



J - h - T I T \k I ) 



'~^T~k~ 



followed by a number of descents, the first of them during an interval 

 centred at t — T/k, with a velocity 



ff-ffi' = o, 



the second during an interval centred at t = 2 T/k, with a velocity 



rl AHx Aff/1 x\ 

 9+A' = -TT + -T\k-l) 



4c Hx . AH (\ x\ AH I 



~Tk 



the third during an interval centred at t = 3 T/k, with a velocity as 

 before, and so on. At the time t= T, the point is descending through 

 ecmilibrium with a velocity which is either AH/kT or 0, according as 

 k is even or odd. The other half of a complete vibration is most easily 

 determined by the fact that both components are odd functions of the 

 time. The whole curve y = (f> (t) for any point in the first £th of the 

 string is, therefore, like a flight of steps with one steep asceut and k — 1 



