$6\mathrm{.}1$ Consider the continuous string of $(6\mathrm{.}7)-(6\mathrm{.}10)$ as the continuum limit of a beaded string with $W$ beads as $W.$ Write the analog of $(6\mathrm{.}8)$ and $(6\mathrm{.}10)$ for finite $W.$ Show that the limit as $W$ yields $(6\mathrm{.}10).$ Hint: This is an exercise in the definition of an integral as the limit of a sum. But to do the first part, you will either need to use normal coordinates, or prove the identity

${\displaystyle \underset{k=1}{\overset{W}{}}}sin(\frac{nk}{W1})sin(\frac{n^{\text{'}}k}{W1})$

$=\{\begin{array}{ccc}b& if& n=n^{\text{'}}0\end{array}$

$0ifn{n}^{\text{'}}$ and $n,n^{\text{'}}>0$

for a constant $b$ and find $b.$