Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/267

	${\boldsymbol {-}}\ {\frac {a}{L^{2}\ V^{3}}}\ {\boldsymbol {-}}\ {\frac {b}{L^{2-q}\ V^{3}}}\ {\boldsymbol {-}}\ {\frac {c}{L^{2-2q}\ V^{3}}}$$\ +e\ V+f\ V\ L^{2}=0.$	(2)
By (1)	$V^{4}=$$\ {\frac {a}{f\ L^{4}}}+{\frac {(2\ {\boldsymbol {-}}\ q)^{b}}{2\ f\ L^{4-q}}}+{\frac {(1\ {\boldsymbol {-}}\ q)^{c}}{f\ L^{4-2q}}}.$	(3)
By (2)	$V^{4}=$$\ {\frac {a}{(e+f\ L^{2})\ L^{2}}}+{\frac {b}{(e+f\ L^{2})\ L^{2-q}}}$$\ +{\frac {c}{(e+f\ L^{2})\ L^{2-2q}}}.$	(4)
Or, eliminating $V$, we have—
	${\frac {a}{(e+f\ L^{2})\ L^{2}}}+{\frac {b}{(e+f\ L^{2})\ L^{2-q}}}$$\ +{\frac {c}{(e+f\ L^{2})\ L^{2-2q}}}$$\ ={\frac {a}{f\ L^{4}}}+{\frac {(2\ {\boldsymbol {-}}\ q)^{b}}{2\ f\ L^{4-q}}}+{\frac {(1\ {\boldsymbol {-}}\ q)^{c}}{f\ L^{4-2q}}}.$
Simplifying and substituting for $a,\ b,\ c,\ {\mbox{etc.}},$ we obtain—
	${\frac {C_{3}\ \xi }{C_{2}\ a_{1}+C_{3}\ \xi \ L^{2}}}=$$\ {\frac {W_{1}\ {\boldsymbol {-}}\ (q\ {\boldsymbol {-}}\ 1)\ k\ L^{q}}{L^{2}\ (W_{1}+k\ L^{q})}}.$	(5)

This is the solution in its most general form, and gives the condition of least resistance. All the quantities except $L$ are known to the designer of the aerodrome; the value of $L$ determined from the equation gives the value of $V$ from either Equation (3) or (4); it also immediately defines the area. The form of Equation (5) is such that it can only be solved by plotting.

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 $V$ 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 $V.$ In this case the differentiation in respect of $L$ is all that is necessary, and we fall back on Equation (1), $V$ being a constant.

The practical application of the present investigation and the employment of the foregoing equation are discussed in the subsequent chapter.

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