296 
Proceedings of Royal Society of Edinburgh, [june 20, 
We have seen that the area of the curve represented by Jxdz or 
by - lz oc, is the product of the abscissa 2 by l2 x, the derivative of 
the ordinate ; hence the area of the portion AOB is 
1-44580 x -43175 = -62423. 
But the derivative at C is zero, wherefore the area BcC on the sub- 
tractive side must balance AOB on the additive side ; its value must 
also be -62423. The derivative again becomes zero at F, and con- 
sequently the areas CcE, E/F balance each other, each of them being- 
given by the product of the abscissa OE into the derivative at E, 
—that is by 7-61782 x -12328 or -93914. 
The same law continues all along, the areas increasing, but more 
and more slowly as we proceed upwards, as is seen from the sub- 
joined list — 
•62423 
•31491 
•93914 
•23540 
1-17454 
*19599 
1-37053 
•17196 
1-54249 
•15374 
1-69623 
-•07951 
- -03941 
- -02403 
- -01822 
— from which, however, we can form no idea as to whether the 
increase be or be not confined to within some definite limit. 
Hitherto we have been considering the form of an indefinitely 
long chain, whose oscillations are performed in a fixed time, namely, 
that of a pendulum whose length is unit. We shall now proceed to 
investigate the forms and times of oscillation of a chain having a 
determinate length. 
The simple oscillations of any chain, PO, are easily got from the 
preceding investigations : thus the slowest oscillation, that in which 
the whole chain swings from one side of its mean position to the 
other side, is represented by the part BO of the first figure, the 
ordinates of the curve being, in imagination, reduced so as to be 
scarcely perceptible. If L be the total length of the chain, ^ ^ 
= Lx *69166 is the length of the pendulum oscillating synchron- 
ously with it, and its oscillations are more frequent than that of a 
