Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/306

The most important application of the foregoing mathematical theory is found in the case where $V$ is fixed by external conditions and where the area ($A$) is the variable; this application gives rise to pressure values greater than those given in Table X., the difference being dependent upon the extent to which aerofoil area is “penalised” by the direct effect of its weight in causing additional resistance.

We start the present continuation with Equation (3) of § 171:—

Substituting the constants in full we have—

$V^{4}={\frac {C_{1}\ W_{1}^{2}}{C_{3}\ \xi \ L^{4}}}+$ ${\frac {(2-q)\ C_{1}\ W_{1}\ k}{C_{3}\ \xi \ L^{4-q}}}+{\frac {C_{1}\ k^{2}\ (1-q)}{C_{3}\ \xi \ L^{4-q}}}.$

whence

We now require to substitute for $C_{1}$ and $C_{3}.$ These values were not investigated in § 171; they are obtained as follows:—

We know that—

$W=\rho \ \kappa \ A\ V^{2}\ (\epsilon +1)\ \beta$

and

$y={\frac {\rho \ \kappa \ A\ V^{2}\ (1-\epsilon ^{2})\ \beta ^{2}}{2}}$

we require $y$ in terms of $W$ eliminating $\beta ,$ hence—

${\frac {y}{W^{2}}}={\frac {1-2\ \epsilon }{2\ \rho \ \kappa \ A\ V^{2}\ (1+\epsilon )^{2}}}$ $={\frac {1-\epsilon }{2\ \rho \ \kappa \ A\ V^{2}\ (1+\epsilon )}}$

or

$y={\frac {1-\epsilon }{2\ \rho \ \kappa \ (1+\epsilon )}}\times {\frac {W^{2}}{A\ V^{2}}}$

and if we define $L$ as being $={\sqrt {A}},\quad C_{1}$ is defined by expression $y=C_{1}\ {\frac {W^{2}}{L^{2}\ V^{2}}}.\quad$ (§ 171.)

∴

$C_{1}={\frac {(1-\epsilon )}{2\ \rho \ \kappa \ (1+\epsilon )}}.$

Now $C_{3}$ (see § 171) is defined as such that the quantity $C_{1}\ \xi \ V^{2}\ L^{2}$ is the skin friction on the aerofoil, but we know that

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