1887.] E. Sang on Oscillations of Uniform Flexible Chain. 301 
Let us first consider the case when n is We get 
f 2 a Z 4 a 2 z 2 8a 3 Z 3 16a 4 z 4 „ } 
^ 2 = A ) 1_ TT + IT3 172” 1.3.5 lT273 + 1.3.5.7 1.2.3.4 + ’ &C ' J ' 
Since the powers of a accompany those of 2 , its value serves 
merely to fix the scale on which the z’s are to he measured, and 
does not at all affect the generality of the formula. Let us then 
assume the generic condition 
~ife = biz<f> z + z ‘2z&’ 
and we at once get 
<j>z = A 
z z 2 
1 . 2 + 1 . 2 . 3. 4 
z 3 
O 
+ 
If, having only this formula to guide us, we inquire as to the 
shape of the curve represented by it, our course is to compute the 
values of the ordinate corresponding to various values of the 
abscissa. The series converges much more rapidly than that for 
the oscillating chain, and the points of crossing are much more 
remote; thus the first would be found at 2 = 2-46740, the second 
at 2 = 22*20661, and the third at 2 = 61-68503. These numbers 
are exactly in the ratios of 1, 9, and 25, and the first of them is 
just \tt 2 . Moreover, in the course of this arithmetical quest, we 
should find that the ordinate varies within the limits + 1 and - 1, 
unlike the preceding, where the limits decrease. 
In the present instance, however, we readily perceive that on 
writing v 2 for 2 , our series becomes 
V 2 V 4 V 6 
1_ rT2 + TT7i - l7^6 + &c ‘ 
the well-known representative of cos v ; and that thus our function 
may be written 
4>z = A . cos Jz . 
But this formula defines only one part of the curve, namely, that 
on one side of the origin of co-ordinates ; for the other part we 
must have recourse to catenarian functions, and write 
c />2 = A . rod J - z. 
