1887.] E. Sang on Oscillations of Uniform Flexible Chain. 287 
in which n may he any integer number, positive or negative ; and 
thus the solution of our problem will virtually contain that of a 
whole series of allied ones. The particular case, when n = 0, merits 
notice, it becomes 
- a . <f>z — z . 2z <f)Z 
, a 
or 2 t cf>z = 
and therefore represents the movements of a body actuated by a 
a 
spring whose stiffness — becomes lessened in inverse proportion to 
z 
the elapsed time. 
The coefficient a in these formulae regulates the scale on which 
the abscissas 2 are measured, and if it be taken as unit, the periodic 
time of the chain’s oscillation will be that of a simple pendulum 
having the linear unit for its length ; so the generality of the results 
will not be impaired by the assumption a= 1. Let us then seek to 
determine the relation of x to the primary 2 from the equation 
ry> I a; /y» 
lAJ 1 gAs & • QyX' • 
Naturally we try whether it be possible to represent a; by a series 
of terms involving the powers of the variable 2 . We shall suppose, 
then, 
x = A + Bz + C z 2 + Ds 3 + . . . Ms n_1 + Ns” + &c., 
which gives, on being differentiated, 
1? a; = B + 2Cz + 3Ds 2 + 4E2 3 4 - nNz n ~ 1 + ,&c., 
z. 2z x = 1. 2Cz + 2. 3D2 2 + 3.4Ez 3 + 1)^N2 W_1 + , &c., 
wherefore, equating the terms containing the like powers of z, 
- A = B, -B — 4C, - C = 9D, -D = 16E, and 
in general - M = n 2 N , so that 
A A 
B = - 
Whence 
x — A 
2 > 
C= + 
l 2 . 2 2 
1 + 
Z 2 
D= - 
2 3 
1 2 .2 2 . 3 2 
and so on. 
z* 
&c. | 
12 (1 . 2) 2 (1.2. 3) 2 (1.2.3.4) 2 
where the multiplier A depends on the extent of the oscillation and 
on the particular instant of time. For the present we may assume 
A also to be unit and confine our attention to the equation 
1 2 
X A l 2 
2 2 
2 2 . . . . 3 2 
n 1 - 
, &c. 
VOL. XIV. 
11 / 11/87 
