Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/280

are supposing that the ah- that reaches the plane $z\ z$ between $b\ b$ and $c\ c$ has received momentum pro rata with that within the region bounded by $b\ b\ b_{1}\ b_{1},$ that is to say:—

	$M_{\alpha }\left/{\frac {M_{1}}{2}}=n\right/2\ \alpha \ \kappa ,$
or	$M_{1}=M^{\alpha }\ {\frac {4\ \alpha \ \kappa }{n}}$
that is, by (1)	$\epsilon ={\frac {M_{\alpha }}{M_{\alpha }\ \left(1+{\frac {4\ \alpha \ \kappa }{n}}\right)}}$
or	$\epsilon ={\frac {n}{n+4\ \alpha \ \kappa }}.$
We may take a constant $\epsilon$ to represent $4\alpha ,$ and our expression is—
	$\epsilon ={\frac {n}{n+e\ \kappa }}.$

§ 180. $\kappa$ and $\epsilon$, Plausible Values.—We are now able to find an expression for $\kappa$ in terms of $c,\ C,$ and $n$, for we already have the equation—

	$\epsilon ={\frac {c\ C}{\kappa }}-1,\quad$ ∴ $\quad {\frac {n}{n+e\ \kappa }}={\frac {c\ C}{\kappa }}-1.$
or	$c\ C={\frac {2\ \kappa \ n+e\ \kappa ^{2}}{n+e\ \kappa }}$
∴	$c\ C\ n+c\ C\ e\ \kappa =2\ \kappa \ n+e\ \kappa ^{2}$
∴	$e\ \kappa ^{2}+2\ \kappa \ n-c\ C\ e\ \kappa =c\ C\ n$
or,	$\kappa ^{2}+\left({\frac {2\ n}{e}}-c\ C\right)\ \kappa ={\frac {c\ C\ n}{e}},$
whence—
	$\kappa =$ $\pm \ {\sqrt {\left\{{\frac {c\ C\ n}{e}}+\left({\frac {n}{e}}-{\frac {c\ C}{2}}\right)^{2}\right\}}}$ ${\boldsymbol {-}}\ {\frac {n}{e}}+{\frac {c\ C}{2}},$

the rest is a matter of choosing such a value of $e$ as fits in best with experience. The author has taken $e=3.3,$ and this is the basis on which the following Table of plausible values of $\kappa$ and $\epsilon$ is founded.

§ 181. Best Value of $\beta .$ Least Value of $\gamma .$—Assuming the values given in the Tables (I., II. and III.), we are now in

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