1887.] E. Sang on Oscillations of Uniform Flexible Chain. 291 
approximating to the ratio of A to B, is identical with that of the 
coefficients of A and B in the expressions for the successive deriva- 
tives ; in fact, the excesses above found are the very derivatives 
themselves, with the alternate signs changed ; and thus it appears 
that in no case can the progression of derivatives become divergent. 
Proceeding now one step forward, the computations for z = 2 are 
found to give x— -*19655 ; lz x= -*28928, so that the curve 
must cross the axis between z = 1 and z = 2. The exact place of 
crossing may conveniently be reached from either side ; it is at 
z 1 = 1*4-4580. Thus it appears that a chain having the length 
1*44580 will perform a simple oscillation in the same time as will 
a pendulum whose length is 1*00000. Or, conversely, that a 
chain of the length unit will perform its slowest simple oscillation 
along with a pendulum having *69166 for its length. These pro- 
portions are shown in the figure, OB being the length of the chain, 
OQ that of the pendulum oscillating along with it. 
Proceeding onwards in search of the second crossing, we find it 
to lie between z — 7 and 2 = 8, from either of which an easy 
approximation gives us 2 2 = 7*61782, rather more than five times 
the preceding ; this is the OE of the figure, the curved line 
EDCBA representing in a most exaggerated way the character of 
the oscillation. The second simple oscillation of the chain is thus 
isochronous with that of a pendulum *13127 long, the length of 
the chain being unit. 
In continuing the search for the remote crossings, the labour of 
the trial calculations increases greatly, and we seek to lessen the 
toil by watching the progress of the distances ; and, to our consider- 
able relief, find that the second differences are almost, though not 
quite, constant, as is seen in the subjoined table for six crossings. 
24 = 1*44579 64903 
6*17201 90958 
z 2 = 7*61781 55861 
4*93191 70158 
11*10393 61116 
^3 = 18*72175 16977 
4*93438 33025 
16*03831 94141 
2; 4 = 34*76007 11118 
4*93468 53785 
20*97300 47926 
£5 = 55*73307 59044 
4*93475 75309 
25*90776 23235 
z 6 = 81 *64083 82279 
