Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/259

Rh moment so proportioned to the former as to result in the least possible resistance (in accordance with prop, iv.), then the total resistance of the aerodrome will be constant in respect of velocity and the energy required to pass from any one point to any other point will be constant, no matter what the speed may be.

Cor. I.—If the body resistance ($$x_1$$) be taken into account, the total resistance may be taken as composed of two parts, the one part which includes the $$x_2$$ and $$y$$ of the equations and which is constant, and the other part $$x_1$$ which varies approximately as the square of the velocity, and results in making flight at high speeds, distance for distance, less economical than at low speeds.

Cor. II.—The conditions of greatest economy for a given aerodrome as enunciated in prop. i. will not be those of best value of area $$A,$$ as laid down in prop, iv., unless the aerodrome have zero body resistance, for, the influence of body resistance being always to make low velocities more economical than high velocities, the velocity of least energy (per unit distance) will be less than that for which the aerodrome is correctly proportioned. This is the explanation of the apparent paradox of § 165. If we imagine an aerodrome designed for a given velocity, so that $$x_2 = y,$$ then we could reduce its expenditure of energy, for given distance, by reducing its velocity till $$x_1 + x_2 = y$$ (that is, $$x = y,$$ prop, i.), then by re-designing its area till once more $$x_2 = y$$ we can again render it more economical; this could be repeated ad infinitum, the economy increasing at each step, the net result, however, merely being the saving effected by transferring the “body” less rapidly through the air.

Cor. III.—The constancy of $$x_2 + y$$ demonstrated in the present proposition has for an immediate consequence the constancy of the gliding angle (if $$x_1$$ be ignored), that is to say, the thrust required to maintain an aerodrome in flight will be constant for a given value of $$W,$$ and if this thrust be supplied