THEORY OF THE AEROPLANE.
THEORY OF THE AEROPLANE.
A RESUME OF THE PRINCIPLES EVOLVED
BY PAST EXPERIMENTS.
Major B. BADEN-POWELL
(Late Scots Guards),
F.R.A.S., F.R.Met.S., &c.
Vice-President Aeronautical Society of Great Britain.
27, CHANCERY LANE, LONDON, W.C.
It was my original intention to compile a complete
book dealing- in concise and simple manner with such
of the principles of Aerodynamics as are applicable to
the practical construction of flying machines. The sub-
ject, however, is one which calls for a great deal of
time and attention, and, so far, I have not been able to
devote sufficient of these to the matter so as to complete
But having published some portions as articles in
" KNOWLEDGE," I find numerous applicants asking for
elseif (getClientWidth() > 430)
copies of what has appeared, and craving for further
information. I think, therefore, that it may be of some
assistance to those now working on the subject to bring
out the first portion as a complete (if small) volume,
and, following the good example of Mr. Lanchester,
leave the second part to be published at some future
time when I am able to complete it.
This second part will deal with such matters as the
behaviour of models in the air; details of actual flying
machines; propellers; and natural bird flight.
Since most of this book had actually been written,
Mr. Lanchester's two great volumes have been pub-
lished. While his copious comments and arguments
are of the greatest value to the student, they need, as
a rule, hardly be referred to in this book, wherein theory
is gone into as little as possible.
GENERAL PRINCIPLES ... ... ... ... ... i
RESISTANCE TO A PLANE SURFACE MOVING THROUGH
THE Am 7
VARIATION OF PRESSURE DUE TO SIZE AND SHAPE
OF PLANE 18
THE PRESSURE OF THE AIR ON INCLINED PLANES ... 22
LANGLEY'S LAW 33
THE LEE SIDE OF A PLANE 39^
SUPERPOSED PLANES , 42
COMPOUND PLANES AND AEROCURVES 44
FRICTION OF THE AIR 47
THE RESISTANCE OF VARIOUS SHAPED BODIES ... 51
THEORY OF THE AEROPLANE.
THE subject of aerial navigation has recently been
prominently brought before the public. A wide interest
has been aroused, and people generally are beginning to
see what a vast future there is open to a machine able
to traverse, surely and safely, the realms of blue.
Although I am one of those who always prefer fact to
theory, and though most of the important inventions
which have aided human progress have not sprung from
the mathematician's brain, I quite realise that a certain
amount of study of the principles underlying any such
subject is most necessary to one who would add any
important work towards the conquest of the atmo-
The air, then, and the effects of its pressure on bodies
moving through it, demands our earnest attention.
Air may seem a light, subtle fluid. If we pass our
hand through it we notice very little resistance to the
motion, and we may wonder how it is possible to
utilise this very yielding medium to support the heavy
weight of a human body or metal machinery against
the force of gravity. From a mechanical point of
2 PRACTICAL AERODYNAMICS.
view it is just the same whether a body be pushed
against the air, or the air blows against a stationary
body. Yet we all know what air, when in motion at a
great speed, may effect. We know that if the wind be
blowing with the force of a gale perhaps 60 or 80
miles an hour it is capable of exerting a very great
pressure, especially on .suitably disposed surfaces. We
know well enough that when out on a windy day, an
umbrella held, even with its convex side to the wind, is
sometimes most difficult to hold, and that directly it is
turned so as to present a concave surface it is immedi-
ately blown inside out, or if made strong enough to
resist this action, would pull with such force as to be
almost impossible to hold. This enables one to realise
what may be effected by making an apparatus to travel
very rapidly through the air.
It seems probable that an ordinary umbrella (suitably
strengthened) held so as to let a very strong wind strike
underneath it, would pull so hard as to be almost
capable of lifting a man off his legs, momentarily at
least. This fact hardly seems extraordinary, yet if we
imagine a flying apparatus only as big as an umbrella
progressing at 40 or 50 miles an hour through the air,
it would surprise most of us to think that it was capable
of raising a man.
This enables one to realise that if only we can get
the power, properly applied, a very small apparatus
may be sufficient for our purpose and, if a very largfc
aeroplane be used, what great lifting 1 power is to be
derived from it.
This subject, though likely, as already intimated, to
become one of very great importance, yet is one that
has received but comparatively little attention among
Langley, in the introduction to his book, " Experj-
GENERAL PRINCIPLES. 3
ments in Aerodynamics," published in 1891, says : " In
this untrodden field of research ... I think it safe
to say that we are still, at the time this is written, in a
relatively less advanced condition than the study of
steam was before the time of Newcomen."
No complete treatise on the subject exists.* All the
information that is available has to be extracted from
works dealing with aeronautics (mostly historical),
hydrostatics, and pneumatics, and from the various
technical papers which have been compiled on certain
definite branches and on results of particular series of
experiments. The following is a general review of the
whole subject gathered from these sources. It does
not pretend to be complete or exhaustive, but it is
hoped that it may be of assistance to those anxious to
get an idea of the science, and who are unable to wade
through the various sources of information enumerated.
I propose treating of the subject in the following
order. It will be necessary first to briefly refer to* the
theory of the balloon, and ascent by reaction of a fluid,
and then to get on to the main subject of aeroplanes
and apparatus working on kindred principles.
This latter subject must again be subdivided into air
pressures acting perpendicularly on a plane surface,
air pressures on inclined plane surfaces, the effect on
the back of such planes, and pressures on curved sur-
faces moving through the air.
Finally, to consider the combined effects on various
shaped bodies in practice, the flight of birds, and the
action of aerial screw propellers.
In considering the different methods possible for the
attainment of artificial flight which is practically
synonymous with means of overcoming the force of
* Since this was written (in Feb., 1907) several works have been
published, notably those by Lanchester, Maxim, Moedebeck, and
4 PRACTICAL AERODYNAMICS.
gravity there are three principles to be taken into
(1) Displacement. By displacing- a bulk of air by a
body of less total weight than that air. Under this
head would be included hot-air balloons, gas balloons,
and the thepretical, if impracticable, vacuum balloon.
(2) Downward Reaction. By the reaction of a fluid
driven forcibly downwards. Such is the principle of
(3) Sub-Pressure. Deriving support from the pressure
of the air on the under surface of a body driven through
it. This would include not only what is understood by
the term " Aeroplane," but also revolving aeroplanes
or lifting screws, and wings and paddles striking the
air downwards. Under this heading, too, must come
the wind-borne soaring birds and thistledown.
As regards the first of these methods we need but
briefly go into it, since the subject of ballooning is
rather beyond our present scope.
If a given volume of air be displaced and the space
filled by a vessel inflated with some substance lighter
than air, such as hydrogen, coal gas, steam, or air
rendered less dense by being heated, then, if the con-
taining vessel is not too heavy the whole will rise in
the air. This is in obedience to well-known laws. The
heavier particles of air will slip under the lighter body
and buoy it up, just as water when poured into a basin
would slip under and buoy up a cork lying in the basin.
That air has definite weight can easily be proved by
carefully weighing a bottle which has been exhausted of
air, and weighing it again when air is admitted to it.
In this way air is found to weigh .076 Ib. per cubic
foot, or 1,000 cubic feet will weigh 76 Ibs. (at 60 F.
Bar. 30 in.).
Hydrogen gas can be weighed in the same manner,
GENERAL PRINCIPLES. 5
and is found to be .005 Ib. per cubic foot, or 5 Ibs. for
1,000 cubic feet. Coal gas varies, but may average
about 35 to 40 Ibs. per 1,000 cubic feet. Steam, which
has actually been applied to ballooning, varies accord-
ing to its temperature. As regards heated air, what is
known as Charles's law shows that a given volume,
under constant pressure, increases with temperature
.00367 times its bulk per degree Centigrade, or .002
(5^0) P er degree Fahrenheit. If, then, the air in a
balloon can be raised by 100 F., one-fifth of its weight
will be expelled; that is, each cubic foot will then weigh
f of .076, or .06 Ib., or 1,000 cubic feet will weigh 64
Ibs. instead of 76.
These principles are often overlooked by unscientific
inventors, who suggest adding a small balloon to aid
in lifting their apparatus, or who> anticipate a hope of
finding a gas lighter than hydrogen.
One Francis Lana, in 1670, was probably the first to
suggest the idea of a machine on this principle, but his
suggestion was to exhaust the air from large copper
globes, ignoring the practical fact that the pressure of
the atmosphere would crush in any such vessel as soon
as a very small quantity of air had been extracted from
The second method, though interesting as a specula-
tive suggestion, seems hardly likely to> prove of prac-
tical utility, for a man-carrying machine.
Rockets are well known. They are practically useful
for many special purposes, but are extremely wasteful
of fuel, and, therefore, short-lived. Steam jets striking
downwards have been suggested.
Mr. H. Wilde, F.R.S., conducted a number of ex-
periments at one time* in order to ascertain what force
* " On Aerial Locomotion," by Henry Wilde, F.R.S. Vol.
xliv No. II. of the " Memoirs of the Manchester Literary and
Philosophical Society." 1900.
6 PRACTICAL AERODYNAMICS.
could be practically applied with this idea. He tried
high pressure steam and compressed air discharged
through orifices of many various forms, also explosions
of gas mixed with air and ignited by electric sparks.
He, however, sums up the matter by saying 1 : "The
results of all these experiments on the discharge of
elastic fluids, made with a view to the possibilities of
aerial locomotion, were purely negative, and proved
decisively that the solution of the problem was not to
be found in that direction." It occurs to me, though
I have not actually tried the experiment, that liquid air
might be used in this connection. A vessel of liquid air
in ordinary atmospheric circumstances is practically
equivalent to a vessel of water placed in the middle of
a furnace. The liquid air in the one case and the
water in the other are boiling hard and rapidly
evaporating into air or steam respectively. So that by
employing this method we practically have a steam
boiler exposed to a comparatively very high temperature
(that is the difference between that of the liquid and
that of the surrounding atmosphere), yet without any
fuel or apparatus for burning fuel. A great pressure
may thus be obtained with but little weight, and it
could, therefore, be made to ascend. It is true that
this action may be very wasteful and would not last
long. Still, as an experiment, it might be interesting
to see a vessel rise in the air by this novel means.
It may be added that though a continuous stream
issuing from a jet may, theoretically, be wasteful of
power, it would probably not be difficult to make the
jet intermittent, or, by progressing rapidly in a hori-
zontal direction, to cause it to act continually on fresh
The third principle, which promises the most practi-
cal results, and is a much larger subject, is treated of in
the following chapters.
RESISTANCE TO A PLANE SURFACE.
RESISTANCE TO A PLANE SURFACE MOVING
THROUGH THE AIR.
ANY body which is being rapidly driven through the
air, whether it be the main body or structural parts of
the apparatus or the blades of the screws, wings, or
other propelling appliances, is acted upon by three
different forces (in addition to gravity), which tend to
retard its speed. These are : First, the head resistance,
caused by the inertia of the particles of air which have
to be displaced in order to make way for the body.
Second, the negative pressure o>r suction due to the
partial vacuum which is usually formed behind the body,
and the air which has been displaced taking time to
flow back to fill the space which it originally occupied.
Third, the side pressure or skin friction, sometimes re-
ferred to as tangential force.
As Lord Kelvin has said, " In Nature every fluid
has some degree of viscous resistance to change of
shape," which also accounts for these opposing forces.
It will readily be understood that the pressure on a
body being pushed against the air, or falling vertically
through it, is exactly the same as if the body Were held
stationary and a steady current of air be driven against
it. It must, however, be noted that wind is not always
In considering head resistance we will first take the
case of a plane surface propelled perpendicularly to the
line of advance. The problem to be decided is, what
is the force opposing the progress of the plane as com-
pared to the speed and area? This is a most important
consideration, as it is on this that all our calcula-
8 PRACTICAL AERODYNAMICS.
tions must be based. When we come to investigate the
pressure developed on inclined or curved surfaces, it will
be seen that these are but a certain definite proportion
of the pressure that would be imparted to a plane sur-
face of similar area moving" at right angles.
Supposing- we find that to propel a given mass at a
certain speed it is necessary to apply a steady push of
one pound, it is evident that in order to increase that
speed it will be necessary to apply more force. But the
question is, how much more force must be applied in
order to develop, say, double the speed.
Newton, by noting the time taken by spheres in fall-
ing from the dome of St. Paul's Cathedral, concluded
that the resistance of the air on the body is proportional
to the square of the velocity. All later experiments have
shown this law to be approximately true. This is to
be expected, since if we imagine a plane moving
against the air at a given rate, if its speed be doubled
it will strike the air twice as hard, but it will also pass
over double the distance in the time, and will, therefore,
strike twice as many particles of air, hence the pressure
or force required will be four times as great.
There is, however, still some doubt as to whether
this law applies when the body is travelling at com-
paratively high velocities. Experiments made on the
resistance offered by the air to projectiles moving at
speeds of 2,000 or 3,000 feet per second tend to prove
that the resistance increases in a greater ratio than at
rates below a hundred miles an hour (146 feet per
second).* But it is only the latter that we need now be
To get at a proper working formula for computing
the resistance of the air, we put P = Kv 2 ; that is, the
pressure in pounds per square foot equals the square
* Prof. Greenhill, Britt. Ass. 1883.
RESISTANCE TO A PLANE SURFACE. 9
of the velocity in miles per hour multiplied by a certain
constant K, which has not yet been very exactly deter-
A number of separate experiments have been made to
solve this point, but the results have not been in per-
fect agreement. It may be desirable, considering- how
important the results are to the subject, briefly to re-
capitulate what has been done in this line, and the con-
clusions come to, since, as far as I know, no connected
account of these has hitherto been published.
The resistance of the air was first carefully investi-
gated by Robins* in 1742. His experiments were
chiefly directed on the investigation of the resistance
offered to bullets fired from a musket, which at that
time was a matter almost entirely ignored. Later on
he constructed a " whirling machine," consisting of a
light arm rotated by means of a weight unwinding" a
cord wound round its support. On the end of the arm
was mounted a sphere to represent a cannon ball. But,
before considering the effect of air pressure on bodies of
varying forms, such as spheres, etc., we may take the
case of a plane surface at right angles to the motion.
Smeaton, who had been investigating the force ob-
tainable by means of windmills, shortly afterwards pub-
lished a table of wind pressures! which had been
communicated to him by Rouse. This was compiled on
the supposition that K = .oo5. The table, though pro-
duced in so uncertain a way over 150 years ago, never-
theless was accepted as authoritative, and is often
quoted intact to this day in books dealing with engineer-
ing and wind effects. Subsequent investigations, how-
ever, show that these deductions were altogether
* " New Principles of Gunnery," by Benjamin Robins, F.R.S.,
new edition, to which is added " Subsequent Tracts." 1805.
| " Philosophical Transactions " for 1759, p, 165.
fo PRACTICAL AERODYNAMICS.
Nor does there seem to be any record of exactly how
the figures were arrived at. Smeaton, in his paper on
1 'The Natural Powers of Water and Wind" (Phil.
Trans. Vol. LI., 1759), merely says :
" Some years ago Mr. Rouse, an ingenious gentle-
man of Harborough, in Leicestershire, set about trying
experiments on the velocity of the wind and force
thereof upon plane surfaces and windmill sails; and
much about the same time Mr. Ellicott contrived a
machine for the use of the late celebrated Mr. B. Robins
for trying the resistance of plane surfaces moving
through the air. The machines of both these gentle-
men were much alike, though at that time totally un-
acquainted with each other's inquiries." And later in
the same paper he quotes the table, prefacing it with
these remarks : " The following table, which was com-
municated to me by my friend, Mr. Rouse, and which
appears to have been constructed with great care from
a considerable number of facts and experiments."
As almost all the more recent practical tests, which
have been made with all care and with the best ap-
pliances of modern science, have shown pressures con-
siderably less than those in this table, it is not unreason-
able to assume that Rouse's figures are incorrect. But
now another fact must be referred to.
A number of writers o<n the subject have endeavoured
to get at a theoretical basis for ascertaining the
pressure. It has been laid down (Rankine, &c.) as a
law of fluids that the magnitude of the pressure of the
stream against any surface bears to the weight of the
fluid the same ratio that the velocity of the current
bears to the velocity generated by gravity. That is to
say, for air, the pressure is to the weight of a cubic foot
of air (.076 Ib.) in the same proportion as velocity is
to 32 feet per second. Or, working out the equation,
RESISTANCE TO A PLANE SURFACE. n
P = 2 =.00237 v. in feet per second, or just about
.005 miles per hour. It is, therefore, noteworthy that
this theoretical pressure gives exactly Rouse's equiva-
lent for K.
This theoretical computation probably does not, how-
ever, take into account the conical cushion of air which
is formed in front of the plane, or other factors which
tend to lessen this pressure. Nor, of course, is the lee
suction on the back of the plane here considered.
It is evident that in computing the pressure of air, due
allowance must be made for the density of the air, but
this will be dependent both on the temperature and
barometic pressure. When the air is warm and ex-
panded the pressure it is capable of exerting is bound
to be less than when it is cold and dense.
The following formula is given by Kempe (Engineer's
Year Book] to get the velocity of wind in feet per
second, when the air moves from a state of greater
density H to one of less density h, t being the tempera-
v = 1347-4 V ^(i +0-02088 t.)
In 1876, Lord Rayleigh deducted theoretically a value
of .00225 for K.
In 1876, Mr. A. R. Wolff* published, in New York,
a theoretical method of determining the pressure corre-
sponding to a given velocity of wind. He compiled a
complete table giving velocities in miles per hour and
pressures at temperatures from o to 100 F. This was
based on the theoretical value of K at .005.
Professor C. A. Carus- Wilson computed the wind
pressure theoretically as being the weight of air (in Ibs.
per cubic foot) divided by twice the accelerating force
* "The Windmill as a Prime Mover," by A. R. Wolff, New
i-2 PRACTICAL AERODYNAMICS.
of gravity, and the result multiplied by the square of
the velocity. This would imply a value for K of .00254.
Meanwhile different conclusions have been come to
as the result of practical trials. In 1809 Sir George
Cay ley* came to rather different conclusions as the re-
sult of his own experiments. The latter were conducted
with an apparatus in which a surface of one square foot
was mounted upon an arm about five feet long and
rotated by weights over a pulley. He found after
" many carefully repeated experiments, that a velocity
of 11.538 feet per second generated a resistance of 4
ounces, and that a velocity of 17.16 feet per second gave
8 ounces resistance," which would give a value of .004
and .0034 respectively for the symbol K.
Dr. Huttomf continued the experiments of Robins
using the same or a precisely similar machine (which is
still preserved in the model room at the Royal Academy,
Woolwich). His investigations were more extensive
and precise, and will again be referred to.
In compiling a table of wind pressures his figures
differed slightly from Smeaton's table, giving less force
for a given velocity. In these K would work out at
just about .004.
Capt. Thibault, in 1826, made various experiments
on the resistance of plane surfaces, and found
K = 0.00475 v 2 .
In 1869, Capt. Murray, R.N., wrote a letter to the
Aeronautical Society of Great Britain in which he said :
"As at present the Society and its members are
groping in the dark for want of any knowledge of the
fundamental relation between pressure and velocity of
air, they should offer a substantial prize of not less
than 100 for a complete and general solution, on
* Nicholson's Journal, November, 1809.
t " Mathematical Tracts," Vol. 3, by Dr. C. Hutton. 1812.
RESISTANCE TO A PLANE SURFACE. 13
rig-id mathematical principles, of the following problem :
' Required, the relation between the velocity of a cur-
rent of air and its pressure on a plane surface of given
size, shape, and inclination.' " The result of this com-
munication was that a subscription was raised, ap-
paratus was constructed by Mr. Browning, and experi-
mented with by Mr. Wenham in 1871. But as these
were chiefly directed towards obtaining data on inclined
surfaces, we will refer to them again later on.
Hagen, in Berlin, in 1874, made a series of careful
observations on wind pressures, which gave materially
different results to those formerly adopted. His value
for K was .00359.
According to the experiments of Du Buat, K would
Ritter von Loessl, experimenting in Vienna in 1881,
found greater resistances, K working out at about
Colonel Renard, who conducted numerous experi-
ments at the military establishment at Chalais-Meudon,
as well as on the Eiffel Tower, made K about .0035.
Probably the most extensive and careful experiments
which have been undertaken in this line were those
conducted by Prof. S. P. Langley* in 1888-90 at the
Smithsonian Institution. These must frequently be re-
ferred to hereafter, as they embraced a number of tests
made with different forms of apparatus. The " Roll-