1887.] E. Sang on Oscillations of Uniform Flexible Chain. 297 
pendulum having the whole length L in the ratio of Jl *44580 :1 , 
that is, as 1 *20241 : 1. The chain makes almost exactly 101 oscil- 
lations, while the simple pendulum makes 44. 
The second oscillation, that in which the chain has one node, is 
represented by the part EBO of the first figure ; and on dividing 
PO of the second figure in the ratio of EB to BO, we get B', the 
node of the actual chain ; on making OQ' also in proportion, we 
have the length of the pendulum swinging in the same time as the 
chain. This length is ^ -- or L x *13127 ; and consequently 
these second oscillations are more frequent than those of a pen- 
dulum whose length is L in the ratio of v /7’61782 to 1, that is, of 
2-76004 to 1. 
When two oscillations, represented by the (A) and the (B) of 
figure 2, are coexistent, the character of the compound motion 
results from the ratio of their periodic times, that is, of 1 -20241 to 
2-76004. On examining this ratio by the method of continued 
fractions, we find the successive quotients, 2, 3, 2, 1, 1, 2, 13, 2, 7, 
&c., which give the approximating fractions — 
1_ 3 7 10 17 44 o 
2 ’ 7 5 16 ’ 23 ’ 39 ’ ToT ’ 
Taking the second of these for the sake of illustration, while (A) 
has made three complete oscillations, (B) has made seven, and the 
chain is (nearly) in the same position as at first, so that the same 
phases would be repeated. But the periodic times are incom- 
mensurate, and so the same phase can never be accurately repro- 
duced. 
The two sets of oscillations may or may not be in one plane ; 
when they are in planes inclined to each other, the path of a point 
in the chain is analogous to the curve produced by the vibration of 
a straight wire whose periodic times are in the same ratio; only in 
the present case the figure is not necessarily circumscribed by a 
rectangle. 
In order to form some idea of the compound movements, let us 
draw AB, figure 3, to represent the extent of the oscillation (A), 
and BC, inclined to it, to show that of the simple oscillation (B). 
Then, having described a semicircle on each of these, we divide the 
