Frederick William Lanchester

“ Examination of Hypothesis.—According to the hypothesis on which the foregoing propositions are founded, it is supposed that a constant weight is sustained at a varying velocity by an aerofoil of constant area, so that on the one hand the resistance due to skin-friction for any stated velocity undergoes no change, and on the other that the law shall be applicable, this law being that ascertained as pertaining to an aeroplane for small angles, and deduced generally in § 160 from the hypothesis of constant “sweep.” ”

“ Therefore the correct area has been given to the aerofoil when its aerodynamic resistance is equal to its direct resistance.
Thus, if for the given conditions of weight and velocity the aerofoil be made too large, the skin friction will be in excess of the aerodynamic resistance; if insufficient surface is provided, the aerodynamic resistance will be in excess; in neither case will the energy required for a given distance be the least possible. ”

“ III.—The constancy of demonstrated in the present proposition has for an immediate consequence the constancy of the gliding angle (if be ignored) , that is to say, the thrust required to maintain an aerodrome in flight will be constant for a given value of and if this thrust be supplied by a component of gravity (Fig. 110) , and if be the angle of descent, we have:—
that is to say, is constant. If we take account of the body resistance , we shall find that the value of will increase the higher the velocity. ”

“ II., prop, v., that the velocity for which an aerodrome is correctly designed to cover the greatest distance on a given supply of energy is not the velocity at which it will actually cover the greatest distance, unless the body resistance is zero. ”

“ If the necessary data to any aerodrome are known we can thus ascertain the velocity of least resistance and prescribe the correct “sail area.” It is not always, however, that the general solution of the problem is desired—in fact, more frequently than not the value of is prescribed by considerations external to the aerodynamics of the subject, when the problem becomes to determine the area of least resistance corresponding to the stated value of In this case the differentiation in respect of is all that is necessary, and we fall back on Equation (1) , being a constant. ”

“ In most game birds and other fast fliers the proportion would work out very much higher. In the above illustration we have assumed that the bird has been correctly designed by Nature on the basis of oar present hypothesis. There is an item of some importance which we have hitherto neglected and which will be subsequently taken into account, i.e., the influence of sail area on total weight. ”

“ Now we know that an expression consisting of the sum of a number of quantities of the form may be approximated over a moderate range by a simple expression of like form, and the approximation is greater the less the variation in the value of in the component terms. ”