Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/265

simple expression of like form, and the approximation is greater the less the variation in the value of $q$ in the component terms. Now in the present case in any reasonable design $q$ is found to lie for every structural component between 1 and 2, so that we shall be justified in assuming the expression $W_{2}=k\ L^{q}$ as approximately applicable.

§ 171. The Complete Equation of Least Resistance.—In prop. i. of the present chapter we investigated the conditions of least resistance in the simple case of an aerodrome of fixed weight and sail area. In prop. v. (cors. i. and ii.) we have dealt with the influence of a body resistance independent of the aerofoil area. In the present section it is proposed to generalise and include in the investigation the influence of the weight of the aerofoil as a variable, assuming the form of expression deduced in the preceding section.

It has been shown that the effect of body resistance is to make the resistance at high speeds greater than that at lower speeds; but we know that at low velocities the sail area requires to be increased and that consequently the weight becomes greater and the resistance will be increased on this account, and when the velocity becomes less than a certain value the increase of resistance from this cause will more than compensate for the decrease due to the reduction in the direct body resistance. We may therefore anticipate that the resistance has a minimum value at some definite velocity at which $A,$ and consequently $L,$ will have some definite ascertainable value.

Let	$W$	= total weight.
„	$W_{1}$	= constant essential weight.
„	$W_{2}$	= variable weight, dependent upon $L.$
„	$A$	= aerofoil area.
„	$L$	= a, linear dimension which we may take to be ${\sqrt {A}}.$
„	$y$	= aerodynamic resistance, of which—
	$y_{1}$	is that due to $W$ and
	$y_{2}$	is that due to $W_{2}.$

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