Glauert's, G ( ) = ( ) / ( 2 l U ) =

Glauert's, $G\left(\varphi \right)=\Gamma \left(\varphi \right)/\left(2lU\right)={\sum }_{i=1}^{m}2A_i\sin i\varphi$, series can be written with Multhopp's distribution for $\varphi$ as ${\varphi }_n=\frac{n\pi }{m+1},n=1,2,\dots ,m$. , to obtain better resolution of the circulation at the wing tips. Show that with the Multhopp's distribution coefficients, $A_i$, with integrating for varible $\phi$ reads as.

(i) $\text{Extra left or missing right}$.

(ii) The induced velocity at station, $n=j$, is

$$\frac{1}{4\pi U}{\int }_{-l}^l\frac{d\Gamma }{d\eta }\frac{d\eta }{y-\eta }=\frac{1}{m+1}\sum _{n=1}^m{G}_n\sum _{i=1}^m\frac{i\sin i{\varphi }_n\sin i{\varphi }_j}{\sin {\varphi }_j}$$
(iii) the $j$ th station induces the velocity on to itself bjj Gj and $n$ th station on $j$ induces with, bjn Gn, the total of $b_{jj}{G}_j-{\sum }_{n=1}^m{b}_{jn}{G}_n$.
$$neq1$$
Here, using the definition in (ii) show that $b_{jj}=\frac{m+1}{4\sin {\varphi }_j}$ and $\text{Extra left or missing right}$.
(iv) Finally, show that unknowns, $Gj$, in terms of the angle of attack at $j$ are given by $$\text{Extra left or missing right}$$

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a.c 2.8m 7m 1.4m