Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/308

§ 195 for least resistance $$= \frac{39}{8.76} = 4.45$$ as against $$3.70$$ as given in Table X.

We can calculate the corresponding value of $$\beta$$ from the expression $$\frac{P_2}{V^2} = \rho\ \kappa\ (\epsilon + 1)\ \beta$$ (§ 185), or $$\beta = \frac{P_2}{\rho\ \kappa\ ( \epsilon + 1 )\ V^2}$$ which in the present example gives $$\beta = \frac{.0585}{\rho\ \kappa\ (\epsilon + 1)} = \frac{.0585}{.164} = .356 ,$$ or $$20.4$$ degrees against $$16.8$$ degrees according to Table IV.

It would thus appear that unless the aerofoil weight in the above example has been greatly exaggerated, its influence on the conditions of least resistance is a fact that should certainly be taken into account. At present the accuracy with which the fundamental data have been ascertained would scarcely justify the preparation of Tables to include the influence of this factor.

§ 196. The Relative Importance of Aerofoil Weight.—The importance of the present branch of the subject evidently becomes greater with any increase in the size of the aerodrome, for the necessary proportionate weight of the aerofoil will be greater on a large aerodrome than on a small one; this fact is almost self-evident, but is in any case easy of proof.

Let the weight of the aerofoil, as in the preceding section, be represented by $$W_2 ,$$ and as before let $$W_1$$ be the essential load; then we have seen that we can represent W2 by the approximate expression—$$W_2 = k\ L^q .$$ We assume that the weight of the aerofoil itself does not materially add to its stresses, being supported directly.

Now let us suppose, as is the case for similar bodies, that $$W_1$$ varies as $$L^3 ,$$ then $$W_2$$ will vary as $$W_1\ k\ L^q ,$$ that is, as $$k\ L^{3 + q} ,$$ or the relation of $$W_2 / W_1$$ will be represented by $$\frac{L^{3 + q}}{L^3} \times$$ constant, that is, $$L^q \times$$ constant. Now, if the index $$q$$ were as low as 1 (and it is improbable that it is lower) the relative weight of the aerofoil $$W_2 / W_1$$ will increase as