1887.] E. Sang on Oscillations of Uniform Flexible Chain. 293 
From this we see that x is zero when the arc represented 
by — 'j- is zero, or is any multiple of the half circumference 7r ; that 
is, when 2 = 0, z = tt Jw, z = 2tt Jw, &c., so that the curve must 
cross its axis at points separated by the uniform distance i r Jw. 
If now we imagine a second string having its tension greater than 
that of the former by the weight of this length, or altogether 
■tv + tt Jw ; the distance between its cusps, so as to keep the same 
time of oscillation, must be 
7T J(W + 7 T Jw) , 
and the increase, analogous to our second difference, becomes 
TT JW 
V { J(w + 7 r Jw) - Jw} = 7 T- 
= 7T‘ 
Jw + J(w + TT Jw) 
l 
i+v(i+-^V 
V Jw/ 
How when w increases, the fraction —j— decreases, and the 
Jw 
denominator of this fraction becomes more and more nearly equal 
to 2, so this analogue of our second difference approaches to J7 r 2 . 
Having thus determined the points of crossing, we proceed to 
consider the extreme distances to which the curve reaches on either 
side, as at the points c, F, I of the figure. Thereat the curve is 
parallel to the axis and the derivative lz x is zero ; we have no other 
way of discovering these points than by the solution of the trans- 
cendental equation 
Z Z 2 Z 3 
I " R2 + 1 2 .2' 2 .3 
&C. 
which we manage exactly as before — that is by calculations arranged 
according to the fundamental law known as Taylor’s theorem. The 
results, with the corresponding values of x, are 
