1887.] E. Sang on Oscillations of Uniform Flexible Chain. 299 
with its radii opposed to their former directions, and thus we have a 
new series of configurations ; at the same time, owing to the incom- 
mensurability of the motions, the whole of these configurations 
undergo a gradual change. 
When the chain divides in three, its simple oscillations take the 
form shown in (C) of figure 4 ; while ( D ) and (. E ) show the forms 
when the chain is divided into four and five oscillating parts. The 
respective lengths of the corresponding pendulum, the periodic 
times, and the frequencies of oscillation are given in the following 
table, in which the whole length of the chain is taken as the linear 
unit, and the periodic time of a pendulum of that length as the unit 
of time. 
No. 
Pendulum. 
Periodic Time. 
1 
•691 6603 
•831 6512 
2 
•131 2712 
•362 3137 
3 
•053 4138 
•231 1143 
4 
•028 7686 
•169 6132 
5 
•017 9427 
•133 9102 
6 
•012 2488 
•110 6742 
Frequency. 
1- 202 4128 
2- 760 0391 
4- 326 8640 
5- 895 7672 
7-465 4589 
9-035 5320 
1-557 6263 
1-566 8249 
1-568 9032 
1-569 6917 
1-570 0731 
Here the almost uniform increase of the frequencies is remark- 
able, the difference approximates to J7 r, and this is in accordance 
with what has been said of the successive roots of the equation 
x — 0. On comparing the frequencies themselves with tt, we find 
for 1, J 7r x 3*0619 ; 
for 2, -g- 7r x 7*0234 ; 
for 3, Jttx 11-0183; 
for 4, tv x 15-0136 ; 
for 5, -Jttx 19-0106; 
for 6, |tt x 23-0088 ; 
and we are tempted to conclude that, when the chain vibrates in a 
great number (N) of parts, the frequency of its oscillation may be 
denoted by ^7 r x (4IST - 1). 
The closeness of these results to a simple arithmetical progression 
