Aerodynamics (Lanchester)/Chapter 7

by Frederick William Lanchester

Chapter VII.

 

The Economics of Flight.

 

§ 163. Energy Expended in Flight.—There are certain general propositions relating to the Economics of Flight that may now be demonstrated, and which are essential to the further development of our subject.

The energy expended in flight is utilised in two directions: firstly, in the renewal of the aerodynamic disturbance, or wave necessary to the support of the weight, that is, the energy expended aerodynamically; secondly, the energy expended in overcoming the direct resistance, i.e., that due to skin friction and eddy making, which varies approximately as the square of the velocity. Let the latter be denoted by the symbol $x,$ and let $y$ be the aerodynamic resistance.

Now we have seen that the aerodynamic resistance varies approximately in the inverse ratio of the velocity squared (§§ 159 and 160), for any given weight sustained, so that if we take the case of an aerodrome supporting a given load (inclusive of its own weight) we have the relation, $y\propto {\frac {1}{V^{2}}},$ and if we further assume the factors which give rise to direct resistance to undergo no change, we have, $x\propto V^{2}.$ And the total resistance $=x+y,$ ∴, the energy expended in flight per unit distance $=x+y,$ and energy per unit time, or power $=V\times (x+y).$

 

§ 164. Minimum Energy. Two Propositions.—Taking the achievement of flight for granted, the problem of least energy presents itself in two forms:

(1) To determine the conditions under which the greatest distance may be covered on a given supply of energy that is to say, the conditions of least resistance;

(2) To remain in the air for the longest possible time on a given supply of energy, that is, to determine the conditions of least horse-power.

Prop. I.—We have—

By § 157, $x\propto V^{2}$ and by § 159 (5), $y\propto {\frac {1}{V^{2}}}$ or $x\propto {\frac {1}{y}}$

∴

${\frac {dx}{dy}}={\boldsymbol {-}}{\frac {x}{y}}$

(1)

Now conditions are fulfilled when $x+y$ is minimum, that is when $dx={\boldsymbol {-}}dy,$ or ${\frac {dx}{dy}}={\boldsymbol {-}}1,$  ∴  by (1), $\ {\boldsymbol {-}}{\frac {x}{y}}={\boldsymbol {-}}1,$

Therefore, under the conditions of hypothesis, an aerodrome will travel the greatest distance on a given supply of energy when its aerodynamic and direct resistances are equal to one another.

Prop. II.—We have—

$xV$ power expended (energy per second) in overcoming direct resistance.

$yV$ power expended (energy per second) in overcoming aerodynamic resistance.

Then $xV\propto V^{3}$ and $yV\propto {\frac {1}{V}}$

Denote $xV$ by $X$ and $yV$ by $Y,$ we have $X\propto {\frac {1}{Y^{3}}}$ or, ${\frac {dX}{dY}}={\boldsymbol {-}}3{\frac {X}{Y}},$ and when $dX={\boldsymbol {-}}dY$ we have $Y=3\ X$

Therefore, an aerodrome will remain in the air for the longest possible time on a given supply of energy, that is to say, its fight will be accomplished on least horse-power, when the resistance due to aerodynamic support is three times the direct resistance.

On the foregoing propositions a third may be founded as follows:—

Prop. III.—To determine the relation of the speed of greatest range to the speed of least power.

Now $x=kV^{2}$ and $y=n\ {\frac {1}{V^{2}}},$ where $k$ and $n$ are constants.

When $x=y$  let $\ V=V_{1}.$

When $3\ x=y$  let $\ V=V_{2}.$ It is required to find the relation of $V_{1}$ to $V_{2}.$

When $x=y$  we have $k\ V_{1}^{2}={\frac {n}{V_{1}^{2}}}$  or $\ k={\frac {n}{V_{1}^{4}}}.$

When $3\ x=y$  we have $\ 3\ k\ V_{2}^{2}={\frac {n}{V_{2}^{2}}}.$

Substituting for $k$ we have—

$3\ {\frac {n}{V_{1}^{4}}}\ V_{2}^{2}={\frac {n}{V_{2}^{2}}},$  or $\ 3={\frac {V_{1}^{4}}{V_{2}^{4}}}$  or $\ {\frac {V_{1}}{V_{2}}}=3^{\frac {1}{4}}=1.315$

That is to say, the speed of greatest range is 1.315 times the speed of least power.

Corollary to Prop. III.—For a plane aerofoil the change in value of the angle $\beta$ involved in the change of velocity from $V_{1}$ to $V_{2}$ can be immediately deduced.

$V_{1}=3^{\frac {1}{4}}\ V_{2},$ but by § 159, $V^{2}\ \beta \propto W$ (for small angles). Consequently $V_{1}^{2}\ \beta _{1}=V_{2}^{2}\ \beta _{2}$ where $\beta _{1}$ and $\beta _{2}$ are the angles appropriate to the velocities $V_{1}$ and $V_{2}$ respectively. Therefore—

$\beta _{1}\ (3^{\frac {1}{4}})^{2}\ V_{2}^{2}=\beta _{2}\ V_{2}^{2}$

Thus calculations of $\beta$ values for least resistance require to be multiplied by ${\sqrt {3}}$ to give appropriate values for least horsepower. We may thus anticipate that birds whose object in flight is to fall as slowly as possible (as birds whose habit is to be sustained on an upcurrent, and so to take advantage of the least upward velocity possible), will have wings of hollower form than those whose object is to get from point to point.

§ 165. 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 $y\propto {\frac {1}{V^{2}}}$ 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.”

So long as the foregoing hypothesis applies it is not important whether the direct resistance is entirely due to the skin friction of the aerofoil or whether it is in part due to the resistance of the “body” of the aerodrome, i.e., that part that may be supposed to constitute or contain the load. If we require to concern ourselves with changes of aerofoil or “sail” area, it becomes necessary to distinguish between these two kinds of resistance, the total resistance $x$ being supposed to be divided into two parts, the one $x_{1}$ being defined as independent of the sail area and the other $x_{2}$ as dependent and as directly proportional thereto. In all cases the approximate assumption is made that this class of resistance is proportional to velocity squared, the error that may result from this assumption being considered later.

Prop. IV.—To determine the conditions controlling the aerofoil area for an aerodrome of given weight travelling at a specified velocity.

Let $A=$ area, then, since $x_{1}$ is fixed by the conditions, $x_{2},y,$ and $A$ are the variables with which we are concerned:—

and as in prop. i. we have the minimum condition fulfilled when $x_{2}=y,$

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.

It may be noticed that this result is in appearance out of harmony with prop, i., for there it was shown that the most favourable velocity at which to run an aerodrome in order to cover the greatest distance on a given quantity of energy is that at which the aerodynamic resistance is equal to the total direct resistance, that is $x;$ whereas according to the present proposition the most economical conditions are met with when $y=x_{2},$ which is only a portion of the total.

The explanation of this apparent paradox will be given in the light of the subsequent proposition.

§ 166. Velocity and Area both Variable.—Prop. V.—Given that $x_{1}=0,$ then, for an aerodrome of given weight, with $V$ and $A$ both variable, find the velocity at which a given flight (distance) can be accomplished with least energy.

By § 159 (5a)	$y\propto {\frac {1}{AV^{2}}}$
and,	$x_{2}\propto AV^{2}$
Now for $x_{2}+y$ minimum, we have $x_{2}=y$ (by Prop. IV.), or,—
	$AV^{2}\propto {\frac {1}{AV^{2}}}$
∴	$(AV^{2})^{2}=$ constant.
hence,	$y=$ constant.
and,	$x_{2}=$ constant.
∴	$x_{2}+y$ is constant,

or the resistance is independent of the velocity.

That is to say, if for an aerodrome of given weight the velocity be supposed to undergo continuous variation, and the “sail area” also undergo corresponding variation, so that the latter is at every 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 by a component of gravity (Fig. 110), and if $\gamma$ be the angle of descent, we have:—

${\frac {\mathit {Thrust}}{\mathit {Weight}}}=\sin \gamma ,$

that is to say, $\gamma$ is constant. If we take account of the body resistance $x_{1}$, we shall find that the value of $\gamma$ will increase the higher the velocity. This effect is more fully investigated in the subsequent section.
Fig. 110.

§ 167. The Gliding Angle, as affected by Body Resistance.—Let $\gamma ,$ as before, stand for the theoretical (constant) gliding angle when the body resistance is zero.

Let $\gamma _{1}$ be the gliding angle when the body resistance is $x_{1};$ then:—

When the total resistance is $x_{2}+y$ gliding angle $=\gamma ,$ and when total resistance is $x_{1}+x_{2}+y$ gliding angle $=\gamma _{1},$ the weight $W$ being the same in both cases.

Consequently we have for small angles ${\frac {\gamma _{1}}{\gamma }}={\frac {x_{1}+x_{2}+y}{x_{2}+y}},$ which for a correctly designed aerodrome, when $x_{2}=y,$ becomes—

${\frac {\gamma _{1}}{\gamma }}={\frac {x_{1}+2\ x_{2}}{2\ x_{2}}},$

Taking as an illustration the case of a bird, and estimating the relation of $x_{1}$ to $x_{2}$ on the basis of skin-friction alone (which is probably near the truth), we find by measurement of different species that the body surface is at least $\scriptstyle {\frac {1}{3}}$ of the wing surface, that is to say, we may take it that $x_{2}=3\ x_{1}$ and we have—

${\frac {\gamma _{1}}{\gamma }}={\frac {7}{6}};$

that is to say, under the most favourable circumstances in bird flight the gliding angle is increased about $\scriptstyle {\frac {1}{6}}$ by body resistance above what it would be were such resistance absent. 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. We have so far assumed that the weight is constant and that the sail area may be increased or diminished at will, whereas in reality a part of the total weight is due to the wings themselves, and the total weight should be represented as some function of the area (F) A, plus a constant. This extension of the subject will be left for later investigation; for the present we will continue to exhaust the problem from the present standpoint.

§ 168. Relation of Velocity of Design to Velocity of Least Energy. It has been pointed out in respect of props, i. and iv., in Cor. 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. Let us put the matter in the form of a further proposition:—

Prop. VI.—Given the relation of $x_{1}$ to $x_{2},$ determine the velocity of least resistance in terms of the velocity for which the aerodrome is designed for least resistance.

Let us represent the designed velocity by the symbol ${\mbox{V}},$ and let $V_{1}$ (as in prop, iii.) represent the velocity of least resistance, that is, when $x=y$ (prop. i.). Then at velocity ${\mbox{V}}$ we have $x_{2}=y,$ and at velocity $V_{1}$ we have $x_{1}+x_{2}=y,$ where $x$ and $y$ are variables.

Let $a_{1}$ and $a_{2}$ represent normal areas that will give rise to resistances equal to $x_{1}$ and $x_{2}.$

Then,	$x_{1}=n\ a_{1}\ V^{2}$
	$x_{2}=n\ a_{2}\ V^{2}$ where $n$ is a const.
and,	$y=m\ {\frac {1}{V^{2}}}$ where $m$ is a const.
When	$V={\mbox{V}},\quad x_{2}=y,$

∴	$\quad n\ a_{2}\ V^{2}=m\ {\frac {1}{V^{2}}}\quad$ or $\quad a_{2}\ V^{4}={\frac {m}{n}}$	(1)
When,	$V=V_{1},\quad \quad x_{1}+x_{2}=y,$
∴	$n\ (a_{1}+a_{2})\ V_{1}^{2}=m\ {\frac {1}{2}}\quad$ or $\quad (a_{1}+a_{2})\ V_{1}^{4}={\frac {m}{n}}$	(2)
By (1) and (2) we have—
	$(a_{1}+a_{2})\ V_{1}^{4}=a_{2}\ {\mbox{V}}^{4}$
that is,	${\frac {V_{1}^{4}}{{\mbox{V}}^{4}}}={\frac {a_{2}}{(a_{1}+a_{2})}}={\frac {x_{2}}{(x_{1}+x_{2})}}$
or,	${\frac {V_{1}}{\mbox{V}}}={\sqrt[{4}]{\frac {x_{2}}{(x_{1}+x_{2})}}}.$

The signification of this result is that if an aerodrome be designed to travel at a velocity ${\mbox{V}},$ its “sail area” being such as will involve the least total resistance at that velocity, such an aerodrome will experience its least resistance when its velocity is reduced to—

${\mbox{V}}\times {\sqrt[{4}]{\frac {x_{2}}{(x_{1}+x_{2})}}}.$

As an example we may, as before, assign relative values $x_{1}=1,\ x_{2}=3,$ we have velocity of least resistance,

$V_{1}={\sqrt[{4}]{{\frac {3}{4}}\times {\mbox{V}}}}=.93\ {\mbox{V}}.$

If we take $x_{1}=x_{2}$ we shall have—

$V_{1}={\sqrt[{4}]{\frac {1}{2}}}\times {\mbox{V}}=.84\ {\mbox{V}}.$

Least Horse-power.—If we require to know the velocity of least power $V_{2}$ we have by prop. iii.: $V_{2}={\frac {V_{1}}{3^{\frac {1}{4}}}}=.76\ V_{1},$ or in terms of ${\mbox{V}}$ we have—

$V_{2}={\mbox{V}}\times {\sqrt[{4}]{\frac {x_{2}}{3\ (x_{1}+x_{2})}}}.$

In the case of the values given above,

When $\ x_{2}=3\ x_{1}$	$\quad V_{2}=.706\ {\mbox{V}}.$
When $\ x_{2}=x_{1}$	$\quad V_{2}=.638\ {\mbox{V}}.$

§ 169. Influence of Viscosity.—The influence of viscosity in the resistance of bodies is to cause a departure from the V-square law. It has been shown (Chap. II.) that the resistance in a viscous fluid can be expressed as a power of the velocity whose index must be less than 2, this form of expression not representing a definite law that holds good over any wide range, but rather defining the rate of change in the quantities round about the values for which the index value may have been determined (§ 40).

Adopting this approximate form of expression, we shall have in prop. i. $x\propto V^{n},$ and assuming (as we are probably justified in assuming) that viscosity has only a negligible influence on the aerodynamic resistance, we have:—

$x\propto V^{n}\quad y\propto {\frac {1}{V^{2}}},$  ∴ $\ x\propto \left({\frac {1}{y}}\right)^{\frac {n}{2}}$

∴ differentiating, we have—

${\frac {dx}{dy}}={\boldsymbol {-}}{\frac {nx}{2y}}.$

Now $x+y$ is minimum when $dx={\boldsymbol {-}}dy,$ that is—

${\boldsymbol {-}}{\frac {nx}{2y}}={\boldsymbol {-}}1,$  or $\ nx=2y,$  or $\ x={\frac {2}{n}}y.$

This is the solution to the equivalent of prop, i., on the modified hypothesis.

Thus if $n=1.75,$ that is to say, if $x=V^{1.75},$ we have the minimum total resistance when $x={\frac {8}{7}}y.$

The necessary modifications to generalise the further propositions in respect of the index of $x$ may be easily effected; the matter, however, has not been pursued further in the present work, the approximate assumption of $n=2$ being deemed sufficient for the needs of the practical application of the theory.

§ 170. The Weight as a Function of the Sail Area.—It has been pointed out, in § 166, that we cannot, strictly speaking, regard the weight of an aerodrome as constant in respect of the value of $A$, for the supporting members themselves must possess weight, and such weight must be some function of the area. This consideration will not affect the results given by props, i., ii., iii., for in these the hypothesis does not contemplate any change in the value of the “sail area” $A$.

We may regard the total weight $W$ of an aerodrome as consisting of two parts, $W_{1}$ and $W_{2},$ of which $W_{1}$ is constant being the weight of the essential load, and $W_{2}$ as the weight of the aerofoil which must vary in some way with the area $A,$ or $W_{2}=(F)\ A$ where the nature of the function must depend upon the conditions of design and construction.

Before any attempt can be made to investigate the influence of the matter under consideration, some assumption must be made as to the form of the function in question. The basis on which we have to found our assumption is that of some probable constructive method; thus we might suppose that as the aerofoil undergoes change of area its geometrical form is preserved and remains constant. If we take $L$ as representing the linear dimension of the aerofoil ($L$ may be chosen as any linear dimension so long as it is the same in all cases); then if the weight of the aerofoil per unit area were constant, we should have $W_{2}$ varies as $L^{2};$ or suppose we base our relationship on an assumption of constant geometrical form, but all scantlings of appropriate strength, investigation gives $W_{2}$ varies as $L.$ The actual relation, whatever it may be, depends upon the exigencies of design and can be established for any set of conditions empirically by designing aerofoils of different area and plotting an $L:W^{2}$ curve.

In detail we find that the weight of each element of the aerofoil structure may be represented by the simple expression $k\ L^{q}$ where $k$ and $q$ are constants which are different for the different functional elements. Now we know that an expression consisting of the sum of a number of quantities of the form $k_{1}\ L^{q_{1}},\ k_{2}\ L^{q_{2}},\ k_{3}\ L^{q_{3}},\ {\mbox{etc.}},$ 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 $q$ 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 $W_{2}.$

Let	$x$	= direct resistance, of which—
	$x_{1}$	is that due to body resistance, and
	$x_{2}$	is that due to aerofoil area.
„	$a_{1}$	= a normal plane area whose resistance is the equivalent of $x_{1},$ and
„	$a_{2}$	= a normal plane area whose resistance is the equivalent of $x_{2}.$
„	$V$	= velocity of flight.
„	$\xi$	= coefficient of skin friction.

$k$ and $q$ are constants, as in preceding section, and $C_{1}\ C_{2}\ C_{3}$ are further constants.

It can be shown that—

$y=C_{1}\ {\frac {W^{2}}{L^{2}\ V^{2}}},$  and $\ x=C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.$

We require to know the minimum value of $x+y;$ or, we require to solve for minimum value the expression—

$C_{1}\ {\frac {W^{2}}{L^{2}\ V^{2}}}$$\ +C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.$

Now $W=W_{1}+W_{2},$  and $\ W_{2}=k\ L^{q},$ so that expression becomes

$C_{1}\ {\frac {(W_{1}+k\ L_{q})^{2}}{L^{2}\ V^{2}}}$$\ +C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.$

or

$C_{1}\ W_{1}^{2}{\frac {1}{L^{2}\ V^{2}}}+C_{1}\ W_{1}\ 2\ k\ {\frac {1}{L^{2-q}\ V^{2}}}$$\ +C_{1}\ k^{2}\ {\frac {1}{L^{2-2q}\ V^{2}}}$$\ +C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.$

where $L$ and $V$ are variables.

Making a temporary substitution of constants in order to abbreviate, we have—

${\frac {a}{L^{2}\ V^{2}}}+{\frac {b}{L^{2-q}\ V^{2}}}$$\ +{\frac {c}{L^{2-2q}\ V^{2}}}$$\ +c\ V^{2}+f\ V^{2}\ L^{2}.$

Differentiating in respect of $L$ and $V$ gives simultaneous equations as follows:—

${\boldsymbol {-}}2\ {\frac {a}{L^{3}\ V^{2}}}\ {\boldsymbol {-}}\ (2\ {\boldsymbol {-}}\ q)\ {\frac {b}{L^{3-q}\ V^{2}}}$$\ {\boldsymbol {-}}\ (2\ {\boldsymbol {-}}\ 2q)\ {\frac {c}{L^{3-2q}\ V^{2}}}$$\ +2\ f\ L\ V^{2}=0.$

(1)

	${\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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