Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/243

Rh more than a certain proportionate thickness the skin-friction disappears and the total resistance may be assessed as edge effect.

Amongst the former may be classified laminae of mica (such as used by the author) of an inch or a few inches breadth and 1/1000 inch or 3/1000 inch thickness. To the latter might be said to belong a plane of the proportions of a common floor-board. Probably the planes employed by Langley, about one square foot area, and of various proportions, by 1/10 inch thickness, would be intermediate, where edge effect and skin-friction give a total greater than either, but far less than their separately computed sum.

§ 159. Planes at Small Angles.—It commonly happens in physical problems that the conditions are greatly simplified when limited to the case of some particular angle being small, that is to say, within the range for which the angle (in circular measure), its sine, and its tangent, are sensibly equal to one another. In a case such as the present, where a high degree of accuracy is not important, and not attainable, such a range may be said to extend to as much as ten or fifteen degrees, and thus include practically the whole range of angle that can be usefully employed in the application of the aeroplane to aviation. It is consequently of importance to examine the extent to which simplification is possible under these restricted conditions.

It has been shown in the case of the square plane that Duchemin's formula: $$P_\beta = P_{90}\ \frac{2\ \sin \beta}{1 + sin^2 \beta}$$ does not greatly differ from the results of direct experiment, and we know that for small values of $$\beta$$ the quantity $$\sin^2 \beta$$ may be neglected, so that the expression becomes: $$P_\beta = P_{90} \times 2\ \sin \beta ,$$ or neglecting the difference between $$\beta$$ and $$\sin \beta$$ (which for 10 degrees is less than 2 per cent.), we have: $$P_\beta = P_{90} \times 2\ \beta ,$$ where $$\beta$$ is expressed in circular measure.

The same form of expression is found to apply to planes