ELASTIC VIBRATIONS 
379 
Hence the period (6) differs from the period the same system has 
when there is no damping, § 1, (3), by a small quantity of the second 
order, referred to k/w as of the first order: 
— = — + (' K ^ I — -+- a small quantity 
v n \nj [ 4n 
The amplitude of the oscillation dies down, owing to the expo 
nential factor, and approaches 0 as its limit. 
4. Forced Vibrations. The phenomenon of forced vibrations is 
familiar to the race through varied manifestations. A regiment of 
soldiers, in crossing a bridge, is commanded to break step. The 
chances are that it is unnecessary to do so. But if the natural note, 
or period, of the bridge should be about the same as the beat of 
their steps, serious consequences might ensue, for the bridge could 
be brought into violent vibration. 
We are told, too, how the piper fiddled down the bridge by strik 
ing the note of the cables, and the walls of Jericho are reported to 
have fallen in a similar manner.* We have all had the experience 
of sneezing in a room where there was a banjo, and then hearing 
the banjo sneeze, too. 
The tides form another example, for they are due to the attraction 
of the sun and the moon. 
One of the cheerful recollections of my school days is that of 
shaking the room in the old Rice Grammar School in Boston. A 
child, sitting at his desk, with the ball of the foot on the floor could, 
by causing the leg to move up and down with a period nearly equal 
to the natural period of the floor, produce vibrations most disturb 
ing to the lady school teacher. 
* Joshua vi. 20. It was Mr. Fulton Cutting who called my attention to this 
fact years ago in Mathematics 5.