Solution
= Solution
For $C=1/2$,
$$
(W')^2=W^2+2W=(W+1)^2-1,
$$
so
$$
\boxed{W(Z)=\cosh Z-1,
\qquad A(Z)\propto\frac1{\cosh Z-1}.}
$$
The thread thins exponentially for large $Z$, and its tensile stress is proportional to $\sinh Z$.
= Solution
For $C=1/2$,
$$
(W')^2=W^2+2W=(W+1)^2-1,
$$
so
$$
\boxed{W(Z)=\cosh Z-1,
\qquad A(Z)\propto\frac1{\cosh Z-1}.}
$$
The thread thins exponentially for large $Z$, and its tensile stress is proportional to $\sinh Z$.