Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/266

§ 171 {|
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 * Let || $$x$$ || = direct resistance, of which—
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 * || $$x_1$$ || is that due to body resistance, and
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 * || $$x_2$$ || is that due to aerofoil area.
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 * „ || $$a_1$$ || = a normal plane area whose resistance is the equivalent of $$x_1 ,$$ and
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 * „ || $$a_2$$ || = a normal plane area whose resistance is the equivalent of $$x_2 .$$
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 * „ || $$V$$ || = velocity of flight.
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 * „ || $$\xi$$ || = coefficient of skin friction.
 * }

$$k$$ and $$q$$ are constants, as in preceding section, and $$C_1\ C_2\ C_3$$ are further constants.

It can be shown that— $$y = C_1\ \frac{W^2}{L^2\ V^2} ,$$ and $$\ x = C_2\ a_1\ V^2 + C_3\ \xi\ V^2\ L^2 .$$

We require to know the minimum value of $$x + y ;$$ or, we require to solve for minimum value the expression— $$C_1\ \frac{W^2}{L^2\ V^2}$$$$\ + C_2\ a_1\ V^2 + C_3\ \xi\ V^2\ L^2 .$$

Now $$W = W_1 + W_2 ,$$ and $$\ W_2 = k\ L^q ,$$ so that expression becomes $$C_1\ \frac{(W_1 + k\ L_q)^2}{L^2\ V^2}$$$$\ + C_2\ a_1\ V^2 + C_3\ \xi\ V^2\ L^2 .$$

or $$C_1\ W_1^2 \frac{1}{L^2\ V^2} + C_1\ W_1\ 2\ k\ \frac{1}{L^{2 - q}\ V^2}$$$$\ + C_1\ k^2\ \frac{1}{L^{2 - 2q}\ V^2}$$$$\ + C_2\ a_1\ V^2 + C_3\ \xi\ V^2\ L^2 .$$ where $$L$$ and $$V$$ are variables.

Making a temporary substitution of constants in order to abbreviate, we have— $$\frac{a}{L^2\ V^2} + \frac{b}{L^{2 - q}\ V^2}$$$$\ + \frac{c}{L^{2 - 2q}\ V^2}$$$$\ + c\ V^2+ f\ V^2\ L^2 .$$

Differentiating in respect of $$L$$ and $$V$$ gives simultaneous equations as follows:—