\(\ \)\vspace{-1cm}
\noindent
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\par
The above diagram represents a suspension bridge. A heavy uniform
horizontal roadway is attached by vertical struts to a light flexible
chain at points \(A_{1}=(x_{1},y_{1}),\) \(A_{2}=(x_{2},y_{2}),\ldots,\)
\(A_{2n+1}=(x_{2n+1},y_{2n+1}),\) where the coordinates are referred
to horizontal and vertically upward axes \(Ox,Oy\). The chain is fixed
to external supports at points
\[
A_{0}=(x_{0},y_{0})\quad\mbox{ and }\quad A_{2n+2}=(x_{2n+2},y_{2n+2})
\]
at the same height. The weight of the chain and struts may be neglected.
Each strut carries the same weight \(w\). The horizontal spacing \(h\)
between \(A_{i}\) and \(A_{i+1}\) (for \(0\leqslant i\leqslant2n+1\))
is constant. Write down equations satisfied by the tensions \(T_{i}\)
in the portion \(A_{i-1}A_{i}\) of the chain for \(1\leqslant i\leqslant n+1\).
Hence or otherwise show that
\[
\frac{h}{y_{n}-y_{n+1}}=\frac{3h}{y_{n-1}-y_{n}}=\cdots=\frac{(2n+1)y}{y_{0}-y_{1}}.
\]
Verify that the points \(A_{0},A_{1},\ldots,A_{2n+1},A_{2n+2}\) lie
on a parabola.