Flight Without Formulae

THE

LIBRARY

OF

THE

UNIVERSITY

OF

CALIFORNIA

THE

MECHANICS

OF

THE

AERO-PLANE.

A

Studyofthe Principles of

Flight. By

COMMANDANT

DUCHENE.

Translatedfromthe_{French by}JOHNH.

LEDEBOER,B.A.,andT.O'B.HUBBARD.

With 98 Illustrations and Diagrams.

8vo. 8s.net.

FLYING

: Some Practical _Experiences.

ByGUSTAV

HAMEL

and CHARLES C. TURNER. With72 Illustrations. 8vo.

I2s.6d. net.

LONGMANS,

GREEN,

AND

CO., LONDON,NEWYORK,BOMBAY.CALCUTTA,MADRAS

FLIGHT

WITHOUT

FORMULA

SIMPLE

DISCUSSIONS

ON

THE

MECHANICS

OF

THE

AEROPLANE

BY

COMMANDANT

DUCHENE

OFTHE FRENCHGENIE

TRANSLATED FROM THE FRENCH BY

JOHN

H.

LEDEBOER,

B.A.

ASSOCIATE FELLOW,AERONAUTICALSOCIETY; EDITOR

"AERONAUTICS";JOINT-AUTHOROF"THEAEROPLANE"

TRANSLATOROF"THE MECHANICSOFTHE AEROPLANE"

SECONDEDITION

LONGMANS,

GREEN,

AND

CO.

39

PATERNOSTER

ROW,

LONDON

FOURTH AVENUE

& 30 STREET,

NEW

YORK

BOMBAY, CALCUTTA,

AND MADRAS

1916

FirstPublished . . _{July 1914}

TL

S" JD

I (>

TRANSLATOR'S PREFACE

and

_{equations are necessary} evils; they

repre-sent, asitwere, theshorthandofthemathematician

and

the

engineer, forming as they do the simplest

and

most

con-venient

method

of expressing certain relations between

facts

and

_phenomena

which _{appear complicated}

when

dressed ineverydaygarb. Nevertheless, it istobe feared

thattheir _{veryappearance}is _forbidding

and

strikes terror

tothe heartsof

_many

readers_{not possessed}ofamathematical turnof mind.

However

baseless this_prejudice

_may

be

as indeedit is the factremainsthatit exists,

and

hashi the past deterred

many

fromthestudy ofthe_{principles of} the aeroplane, which is

playing a part of ever-increasing

importanceinthelifeofthe_community.

The

_present

work

forms_{an attempt}to cater for this class

ofreader. It has_{throughout been}written in_{the simplest}

possible language,

and

contains in its whole extent not a

single formula. It treats ofeveryone of the_{principles of}

flight

and

of every one of the _problems involved in the mechanics of the aeroplane,

and

this _{without demanding}

fromthe reader

more

thanthe_{most elementary knowledge}

of arithmetic.

The

chapters on stability should prove of

particular interesttothe pilot

and

the student, containing

as_{they do} several

new

theories ofthe highestimportance here_{fully set}outforthefirsttime.

In conclusion, I have to thank Lieutenant T. O'B.

Hubbard,

my

collaboratorfor

_many

_{years, for his}kind

and

diligent perusal of the proofs

and

for

_many

_helpful

suggestions.

J. H.L.

CONTENTS

CHAPTER

I

PAGE

FLIGHT IN STILL AIR SPEED 1

CHAPTER

II

PLIGHT IN STILL AIR POWER 16

CHAPTER

III

PLIGHT IN STILL AIR POWER _(concluded)

....

35

CHAPTER

IV

FLIGHT IN STILL AIR THE POWER-PLANT . . . . 53

CHAPTER V

FLIGHT IN STILL AIR THE POWER-PLANT (concluded) . . 70

CHAPTER

VI

STABILITY IN STILL AIR LONGITUDINAL STABILITY . . . 90

CHAPTER

VII

STABILITY IN STILL AIR LONGITUDINAL STABILITY

(cotl-cluded)

...

. . . . 115

CHAPTER

VIII

STABILITY IN STILL AIR LATERAL STABILITY . .

.142

CHAPTER

IX

STABILITY IN STILL AIR LATERAL STABILITY _(concluded)

DIRECTIONAL STABILITY, TURNING . . . 161

CHAPTER X

Flight

without

Formulae

Simple

Discussions

on

the

Mechanics

of

the

_Aeroplane

CHAPTER

I

FLIGHT

IN STILL

AIR

SPEED

NOWADAYS

everyone understands something of the

main

principles of aeroplane flight. It

may

be demonstrated

in the _{simplest possible}

_way

_by

_plunging the

hand

in

water

and

trying to

move

it at

some

speed horizontally,

after first _{slightly inclining} the _palm, so as to meet or

"

attack" the fluid at a small "_angle of incidence." It

will be noticed at once that, although the

hand

remains very nearly horizontal,

and

though it is

moved

horizon-tally,thewaterexerts

upon

itacertain

amount

of _pressure

directed nearly vertically upwards

and

tending to lift the hand.

This, in effect, is the _{principle underlying the flight} of

an _aeroplane, which consists in drawing through the air wings or planes in a position nearly horizontal,

and

thus employing, forsustaining the weightofthewholemachine,

the _vertically

_upward

_{pressure exerted}

_by

the air onthese

wings, a pressure which is caused

_by

the very forward

movement

of_{the wings.}

Hence, the sustentation

and

the forward

movement

of

an_{aeroplane are absolutely interdependent,}

and

the former 1

2

FLIGHT

WITHOUT

FORMULAS

can_onlybe_produced,instillair,

by

thelatter,outofwhich

itarises.

But

the entire _problemof _aeroplane flight isnot solved

merely

by

obtainingfromthe "

relative"aircurrentwhich meets _{the wings, owing} to their forward _speed, sufficient lift to sustain the weight of the machine; an aeroplane,

in addition,

must

_alwaysencounter therelative aircurrent

inthe

same

attitude,

and must

neither upset norbethrown

out of its _path

_by

even a slight aerial disturbance. In

other words, it is essential for an _{aeroplane to} remain in

equilibrium more, instable_equilibrium.

This consideration clearlydividesthestudyof _aeroplane

flightincalmairintotwobroad, natural parts:

The

studyoflift

and

the_studyofstability.

These

two

aspects will be dealt with successively,

and

willbefollowed

_by

aconsiderationof_flightindisturbedair.

First

we

will _{proceed to}examinetheliftofan_aeroplane

instillair.

Following the exampleofabird,

and

inaccordance with

theresultsobtained

_by

_{experiments with models, thewings}

of an_{aeroplane are given a span}five or six _{times greater}

than then1

fore-and-aft dimension, or "chord," while they arealsocurved, so thattheirlower surface isconcave.* It

is desirable to give the wings a _{large span} as _compared

to the chord, in order to reduce as far as _possible the escape or leakage of the air

along the sides; while it

has the further advantage of _{playing an important part} in_{stability. Again, the}camberofthe_wingsincreases their lift

and

at the

same

time reduces their head-resistance or "

drag."

The

angle of incidence of a wing or plane is the anerle.

* In English this curvature of the_wing is _generallyknownas the "

camber." Onthe whole,it_{would perhaps be more}accurate to describe theuppersurface as being convex, since highlyefficientwings have been

designedinwhichthecamberisconfined totheuppersurface,thelower surface being perfectlyflat. TRANSLATOR.

expressed in degrees,

made by

the chord of the curve in

profilewith thedirection ofthe aeroplane'sflight.

As

stated above, the pressure oftheair ona wing

mov-ing horizontally is _nearly vertical, but only nearly. For, though it lifts, a wing at the

same

time offers a certain

amount

of resistance

known

either as head-resistance or drag* which

may

well bedescribed asthe _{price paid} for

thelift.

As

the result of the research

work

of several scientists,

and

ofM.Eiffel in particular,withscalemodels, unitfigures,

or "coefficients," have been determined whichenable us to

calculate the

amount

of lift _possessed

_by

_{a given surface}

and

itsdrag,

when moving

throughtheairatcertainangles

and

at_{certain speeds.}

Hereafter the coefficient which serves to calculate the

lifting-power of a _plane will _{be simply termed} the _lift,

whilethatwhereonthe calculation ofits_dragisbased will

be

known

as the_drag.

M. Eiffel _{has plotted the} results of his _experiments in

diagramsorcurves, whichgive, for each typeof wing, the values of the lift

and

_{drag corresponding to the various}

anglesofincidence.

The

following curves are herereproduced from M.Eiffel's work,

and

relate to:

A

flat_plane(fig. 1).

A

_slightly cambered plane, a type used

by

Maurice

Farman

(fig. 2).

A

plane of

medium

camber, adopted

by

Breguet

(fig. 3).

A

deeply cambered plane, used

by

Bleriot on his

No. XI. monoplanes,cross-Channel type(fig. 4). *_{The word}"

drag

"

is_{here adopted,}inaccordancewithMrArchibald

Low'ssuggestion, in_{preference to the}moreusual "drift," inorder to prevent confusion, and so as to preserve for the latterterm its more

general,and certainlymoreappropriate meaning, illustratedinthe

ex-pression "

thedrift ofan aeroplanefrom itscourse inaside-wind," or

"

4

FLIGHT

WITHOUT

FORMULA

These _diagrams are so _simple as to render further

explanationsuperfluous.

.0.00.

0.02 001 000

Drag. Drag.

FIG.1. Flatplane. FIG.2. MauriceFarmanplane.

The

calculation of the _{lifting-power}andthe

theairata_{given angle} ofincidence

and

at_{a given speed,} is_{exceedinglysimple.}

To

obtain thedesired resultallthat J0.08

0.00 0.00

0.02 :01

Drag.

FIG.3. Br6guetplane.

O.QO 02 0.01 0.00

Drag.

FIG.4. BleriotXI.plane. is needed is to _multiply either the lift or the _drag

6

FLIGHT

WITHOUT

FORMULA

by the area of the plane (hi square metres, or, if _English

measurements_{are adopted,}hi_squarefeet)

and by

the square

ofthe_{speed, himetres per second}(ormiles per hour).*

EXAMPLE.

A

Bleriot _{monoplane, type} No. XI., has an area_of15 _sq. m., and_flies at 20m, per second at an_angle

of incidence of

7.

(1)

What

weight can its _{wings lift,} and

(2)whatisthe_{powerrequired}to_propelthemachine?

Referringtothecurveinfig. 4,thelift_{of this particular}

typeof_wingatan_angleof7 is0-05,

and

its_drag0-0055.

Hence

T-f. . Squareof Lift. Area. the Speed. 0-05

x

15

x

400

givesthe required valueofthe_{lifting-power,}i.e.300_kg.

Again

Drag. Are, * 0-0055

x

15

x

400

givesthe valueoftheresistance of_{the wings,} i.e.33 _kg.

Let usfor _{the present only consider the question}of lift,

leaving thatof_dragon oneside.

From

the

method

of calculation

shown

above

we

_may

immediately proceed to

draw some

highly important de-ductions regarding the speed of an _aeroplane.

The

fore-and-aft _equilibrium of an _aeroplane, hi fact, as will be

shown

_{subsequently,} isso_{adjusted that the aeroplane} can

only fly at one fixed angle of incidence, so long as the

elevator or stabiliser remains untouched.

_By

means

of the

elevator, however, the _angle of incidence can be varied

withincertainlimits.

In _{the previous example,} let the Bleriot monoplane be taken to have been_designed to _flyat

7.

It _{has already}

been

shown

that this machine, with its areaof 15 _sq. m.

and

its_speed of 72

km.

per hour, will _{give alifting-power} equalto300_kg.

Now,

if_{this lifting-power}begreaterthan

*

Throughout this work the metric _system will henceforward be

the weightofthe machine, thelatterwilltendtorise; ifthe

weight beless,itwilltendtodescend. _{Perfectly horizontal} flightat aspeed of 72

km.

_per houris_{only possible}if the aeroplane weighsjust 300kg.

In other words, an _aeroplane of _{a given weight}

and

a given plane-areacan onlyflyhorizontallyata given angle of incidence at one _single speed, which

must

be that at which the _{lifting-power} it _produces is _{precisely equal} to

the weightof_{the aeroplane.}

Now

it _{has already} been

shown

that the lifting-power for _{a given angle} of incidence is obtained

by

multiplying

theliftcoefficient_{corresponding}to this_angle

_by

_{the plane}

areaand

_by

the squareofthe_{speed. This, therefore,}

must

also give us the weightof the aeroplane. It is clear that

this is _{only possible for} one definite speed, i.e.

when

the

squareof the speedis_equal to_{the weight, divided}

_by

the area multiplied

by

the inverse of the lift.

And

since the weight of_{the aeroplane divided}

_by

itsarea_givesthe

load-ingonthe planes persq. m., the followingmost important

and

_{practical rule}

_may

belaid

down

:

The _{speed (in} metres per second) ofan aeroplane, flying

at a given angle of incidence, is obtained _{by multiplying}

thesquareroot_ofits_{loading(in kg. per}

sq. in.}bythe_square

root _of the inverse _of the _{lift corresponding} to the _given

angle.

At

first_sighttherule

_may

_appearcomplicated. Actually

it is_{exceedinglysimple}

when

_applied.

EXAMPLE.

A

Breguetaeroplane,withanareaof30sq.m. and weighing 600 kg., flies with a lift of 0-04, equivalent

(according to the curve infig. 3) to an angle of incidence

ofabout4.

What

isits_speed?

600

The

loading is

=20

_kg.per

sq. m. Squarerootof _{the loading} 4'47. Inverseoftheliftis

=25.

0-04

FLIGHT

WITHOUT FORMULA

The

speed required, therefore, in metres per second 4-47

X 5=22-

3 m. _{per second,} or about 80

km.

_{per hour.}

But

if a different angle of incidence, or a different _figure for

the lift which is _equivalent, and, as will be seen

here-after,

more

usual be taken, a different _speed will be obtained.

Henceeach_{angle of incidence}hasits

own

_{definitespeed.}

For instance, if

we

take the Breguet aeroplane already

considered,

and

calculate its _speed for a whole series of

anglesof incidence,

we

obtain theresults

shown

inTableI.

But

before_examining theseresults in_greaterdetail, so far

astherelation between_{the angles} of incidence, or thelift,

and

_{the speed}isconcerned,afewpreliminary observations

may

beuseful.

TABLE

I.

In the first place, it should be noted that

when

the Breguetwhig has no_angle of incidence, when,thatis, the

wind meets it _{parallel to} the chord, it still has a certain

a cambered _plane. While a flat _{plane meeting} the air

edge-on has no lift _whatever, as is _evident, a cambered plane striking the air in a direction parallel to its chord

still retains a _{certain lifting-power} which varies _according

to_{the plane}section.

Thus,inthose conditionsa_Breguetwingstillhas aliftof

0-019,

and

if_figs. 4

and

2 areexamineditwillbeseenthat

at zeroincidence theBleriotNo. XI.would _similarlyhave

a lift_{of 0-012,} butthe Maurice

Farman

of _only 0-006. It

follows thata cambered_planeexerts noliftwhatever_only

when

thewindstrikesit_slightlyonthe _uppersurface. In

other words,

by

virtue of this _{property, a} cambered_plane

may

be regarded as_{possessing an imaginary chord} if the expressionbeallowed inclinedata_{negative angle(that}is,

inthedirection_opposedto_{the ordinary angle}of incidence)

tothechordofthe_profileofthe planeviewedin section.

If _{the necessary experiments} were

made

and

the curves on the _{diagrams were} continued to the horizontal axis, it

would be found that _{the angle} between this "_imaginary

chord"

and

the actual chord is, for the Maurice

Farman

planesectionabout 1,forthatof theBleriotXI.

some 2,

and

forthatofthe_Breguet

4.

Let it be noted in passing that in the case of nearly every planesectiona variationof1 inthe angleofincidence

is

roughly equivalenttoa variationinliftof 0-005, at

any

rate forthe smaller angles.

One

may

therefore generalise

and

_{say that} for

_any

_{ordinary plane} section aliftof 0-015 corresponds toan_angleofincidenceof 3 _{relatively to}the "

imaginarychord," aliftof 0-020toanangleof

4,

alift of0-025to5,

and

so forth.

Turning

now

to the _{upper portion} of the curves in the diagrams, it will be seen that, beginning with a definite

angleof incidence,usuallyinthe_{neighbourhood}of 15,the

liftof_{a plane} nolonger increases.

The

curves _{relating to} the_Breguet

and

theBleriot cease at 15, but the Maurice

Farman

curve clearly shows that for _angles of incidence greaterthan15 thelift_{gradually diminishes.} Suchcoarse

10

FLIGHT

WITHOUT

FORMULA

angles, however, are never used _{in practice, for} a reason

shown

in _{the diagrams,} which is the excessive increase in

the drag

when

the angle ofincidence is _greater than 10. In aviation the anglesofincidence that areemployed there-fore_onlyvarywithinnarrowlimits,the variationcertainly

not surpassing10.

We

may now

returntothe

main

objectforwhich TableI.

was compiled, namely, the variation in the _speed of an aeroplane according to the angleofincidenceofits_planes.

First, it is _{seen that speed}

and

_angle of incidence _vary

inversely, which is obvious enough

when

itis

remembered

that in orderto _supportits

own

weight,which necessarily

remains constant, an aeroplane

must

fly the faster the smaller_{the angle at}whichitsplanesmeettheair.

Secondly, it will be seen that the variation in _speed is

more

pronouncedfor_{the smaller angles}ofincidence; hence,

by

utilising a small lift coefficient _{great speeds} can be

attained. Thus, for a lift _equal to 0-02, at which the

Breguet wing would meet the air _along its _geometrical chord, the speed of the aeroplane, according to Table I.,

wouldexceed 113

km.

anhour.

If an _{aeroplane could fly} with a lift coefficient of O'Ol,

thatis, if_{the planes}

met

theairwiththeiruppersurface

the_{imaginary chord would then have anangle}ofincidence

ofno

more

than2 the

same

method

ofcalculationwould giveaspeedofover 160

km.

_{per hour.}

The

chief reasonwhich_{in practice places a}limiton the

reductionof thelift is, aswill be

shown

subsequently, the rapidincrease inthemotive-power requiredtoobtain high speeds with small angles of incidence.

And

further, there

isa considerable elementofdangerinundulysmall_angles.

Forinstance, ifan_aeroplanewereto_flywith aliftof O'Ol

so that the _{imaginary chord}

met

the air at anangle of

only2 aslightlongitudinaloscillation,onlyjustexceeding

this_{very smallangle,would be enough}toconvert thefierce

air current _{striking the aeroplane}

_moving

atan enormous speed from alifting force intoone provoking a fall. It is

true that the machine would for an instant _preserve its speed owing to inertia, but the least that could _happen would be a violent dive, which_{could only}end in disaster ifthemachine wasflyingnear the ground.

Nevertheless there are certain_pilots, to

whom

the

word

intrepid

may

be justly applied,

who

deny the _danger

and

argue that the disturbing oscillation is the less _likely to

occur the smaller the angle of incidence, for it is true, as

will be seenhereafter, thata_{small angle} ofincidenceisan

important condition of stability.

However

this

may

be,

there can be no question but that flying at a very small

angleofincidence

_may

set

_up

excessivestrains inthe frame-work, which, in _consequence, would have to be _given enormous_strength.

Thus

,ifitwerepossible foranaeroplane

toflywith aliftcoefficientof 0-01,

and

if, owingtoawind gustor toamanoeuvrecorrespondingtothesudden"

flatten-ing out

"

practised

by

birds of_prey

and by

aviatorsatthe conclusion of a dive,the planesuddenly

met

theairatan angle of incidenceatwhich thelift reaches a

maximum

that is, from0-06to 0-07_accordingtothetype of_plane

the machine would have to _{support, the speed remaining}

constantforthetime _being

_by

reasonofinertia, a pressure

sixor_{seven times greater}thanthatencounteredinnormal flight,orthanits

own

weight.

In_{practice, therefore,various considerations place a}limit

on the decrease of the _angle of incidence,

and

itwould accordinglyappear doubtfulwhether hithertoanaeroplane has flown with aliftcoefficientsmallerthan0-02.*

It is _{easy enough} to find out the value of the lift

co-efficient at which exceptionally high speeds have been attained from a few

known

_{particulars relating} to the machine_{in question.}

The

particularsrequired are:

The

velocity of the _aeroplane, which

must

have been

carefullytimed

and

correctedforthe speedofthewind;

The

total_weightof_{the aeroplane}fullyloaded;

The

_supportingarea.

12

FLIGHT

WITHOUT

FORMULA

The

lift

_may

then be found

_by

_{dividing the loading} of

the planes

by

the squareofthe speedin_{metres per second.}

EXAMPLE.

An

aeroplane with a plane areaof 12 sq. m.

and

weighing,fully loaded, 360 kg. has flown ata speed of

130km.or 36-1m.persecond.

What

wasits_{liftcoefficient}?

Of*(\

The

loading=

=

30_kg.persq. m.

12

Squareofthe_speed=1300.

Ofi

Requiredlift=-?"-

=about

0-023.*

Table I. furthershows that

when

the angle ofincidence reaches the _{neighbourhood} of 15 _{(which cannot,} as has

been seen, be employedin practical flight)the liftreaches

its

maximum

value,

and

the speed consequentlyits

minimum.

* Atthetime of_{writing (August 1913) the speed record, 171}'7 km.

perhouror 47'6m.per second,isheld _bytheDeperdussinmonocoque

with a_{140-h.p. motor,weighing}525kg.withfullload,and with aplane areaofabout12sq.m. (loading,43'7 kg. persq. m.). Another machine ofthesametype,but witha100-h.p.engine,weighing 470kg.inall,and

with anareaof11sq.m., hasattainedaspeedof 168km.perhouror 46'8m.per second. According totheabovemethodof calculation, the

flight in bothcaseswas made withalift coefficient of about 0'0195.

AUTHOR.

Sincethe above was written, allspeed records were broken during

the last Gordon-Bennett race in _September 1913. The winner was

Prevost,on a 160-h.p.GnomeDeperdussin monoplane,whoattaineda

speedofa fractionunder 204km.perhour; while Vedrines,ona

160-h.p.Gnome-Ponnier monoplane,achievedcloseupon 201 km.perhour.

The_{Deperdussin monoplane, with an}area of 10_{sq.m., weighed,} fully

loaded, about 680 kg.; the Ponnier, measuring8 sq. m., weighed

ap-proximately 500 kg. Adopting thesame method of calculation,it is

easily shown that the lift coefficients worked out at 0'021 and 0'020

respectively. Itisjustpossible that thesefigureswereactuallyslightly

smaller,sinceitisdifficulttodeterminethe weightswith any

consider-able degreeof accuracy. However,theerror,if therebe_any,isonly

slight,andtheresultonly confirms the author's conclusions. Since that

timeEmileVedrinesisstated tohave_{attained, during}an officialtrial,

a speed of 212 km. per hour, on a still smaller Ponnier monoplane measuring only 7sq. m. in area and weighing only450kg.in _flight.

This_{would imply} alift coefficientof0'0185, afigure which cannot be

If _{the angle surpassed 15 the}liftwould diminish

and

the speed againincrease.

A

_{given aeroplane,} therefore, cannot in fact _flybelow a

certain limit speed,whichinthe caseofthe_{Breguetalready}

considered,for instance,isabout63

km.

per hour.

It will be further noticed that in Table I. one of the columns, the second one, contains_{particulars relatingonly}

tothe_{Breguettypeof plane.} Ifthiscolumn wereomitted, the whole table would give the speed variation of any aeroplane with a loading of 20 kg. per sq. m. on its

planes, for a variationin the lift coefficient ofthe _planes.

It wasthisthatledtotheaboveremark,

made

in passing,

thatit was

more

usualtotake theliftcoefficientthanthe angle of incidence; for the former is independent of the

shapeofthe_plane.

The

_{speed variation}ofan_aeroplaneforavariationinits

liftcoefficientcaneasilybeplottedhiacurve,whichwould

havethe_shape

shown

infig. 5,whichisbasedonthefigures inTableI.

The

previous considerations relate

more

especiallyto a

studyof_{the speeds}at which agiventype of_aeroplanecan

fly. In order to compare the speeds at which different types of aeroplanes can fly at the same lift coefficient,

we

need_{only return to the} basic rule_already setforth _(p. 7). .

It then becomes evident that these _speeds are to one anotherasthe squareroots oftheloading.

The

fact that only the loading comes intoconsideration

in _{calculating the speed} of an _aeroplane shows that the

speed, for a given lift coefficient, of a machine does not depend on the absolute values of its _weight

and

its _plane

area,butonlyontheratio oftheselatter.

The

most_heavily

loaded aeroplanes yet built (those of the French military trials in 1911) were loaded to the extent of 40 _{kg. per}

sq. m. of plane area.*

The

square root of this

number

being6-32,anaeroplaneof this type,driven

by

asufficiently

*_The

140-kp. Deperdussin monocoque had a _loading of 43-8 kg. persq.m.

14

FLIGHT

WITHOUT FORMULA

powerful engine to enable it to _fly at a lift coefficient of

0-02 (the square root of whoseinverseis _7-07),couldhave

attainedaspeed equalto 6-32

x

7-07,thatis,it couldhave exceeded44-5m._{per second}or160

km.

per hour.

Itistherefore evident that thereare _only

two means

for

increasing the speedof anaeroplane either to reduce the

30 20 10 00 Lift 0.0/0 0.020 0.030 0,0*0 0.050 0.060 0.076 FIG. 5.

lift coefficient or to increase the _loading.

Both

methods

requirepower;

we

shallseefurtheron whichof thetwo

is

the

more

economicalin this respect.

The

former has the disadvantages contested, it is true

which have already been stated.

The

latter _requires

exceptionally strongplanes.

In

_any

case,it_{would appear}that, inthe present stageof

aeroplane construction, the speedofmachineswill scarcely

could onlyhave beenachievedwith the aidof_goodengines developing from 120 to 130 h.p.* Sothat

we

are still far

removed fromthe speedsof200and even300

km.

perhour which were _prophesied on the

morrow

of the first advent

ofthe aeroplane,f

In concluding these observations on _{the speed}of

aero-planes,attention

_may

be

drawn

toarulealreadylaid

down

in_{a previouswork,! whichgivesa rapid}

method

of

calculat-ing with fair _{accuracy the speed} of an _average machine whose_weight

and

_{plane area are}known.

The speed ofan average aeroplane, in metres per second,

is _{equal to five} times the _square root _of its _loading, in kg.

persq.m.

This rule_{simply presupposes that the average aeroplane}

flies with a lift coefficient of 0-04, the inverse of whose

square root is 5.

The

rule, of course, is _{not absolutely}

accurate,but has the meritof_{being easy}to

remember and

to_apply.

EXAMPLE.

What

is the _{speed of} an aeroplane weighing 900kg.,andhavinganareaof36sq.m.?

900

Loading

=

r

=

25kg.persq. m. OO

Squarerootofthe loading

=5.

Speed

required=5x5=25

m. _{per second} or 90

km.

per hour.

* _In

previous footnotes it has already been statedthat the

Deper-dussin _{monocoques, a} 140-h.p. and a 100-h.p., have already flown at

about170km.per hour. Butthesewereexceptions, and,onthe whole, the author's contentionremains_{perfectly accurate}even_to-day. TRANS-LATOR.

fThereference, ofcourse,isonlytoaeroplanesdesignedfor_everyday

use,andnotto_{racing machines.} TRANSLATOR.

CHAPTER

II

FLIGHT

IN STILL

AIR

POWER

IN the first _{chapter the speed} of _{the aeroplane} was dealt

with in its relation to the constructional features of the

machine, or its characteristics _(i.e. the _weight

and

_plane

area),

and

to itsangle of incidence. It

_may

seemstrange

that, inconsidering thespeedofamotor-drivenvehicle,no

account shouldhave been taken ofthe one elementwhich usually determines the speed of such vehicles, that is, of

the motive-power.

But

the

_anomaly

is

only apparent,

and

wholly due to the _unique nature of _{the aeroplane,} which alone _{possesses the faculty} denied to terrestrial vehicles

which are _compelled to crawl _along the surface of the

earth,or,inother words,to

move

hibut

two

dimensions of

being free to

move

_upwards

and

downwards, in all three dimensions, thatis,of space.

The

subject of this _chapter

and

the next will be to

examinethe part played

by

the_motive-powerin_aeroplane

flight,

and

itseffectonthevalueofthe speed.

Inall that has _gonebeforeithasbeenassumed that,in order to achieve horizontal _flight, an aeroplane

must

be

drawn

forward ata_speed sufficientto cause the weight of

the whole machine to be balanced

_by

the _{lifting power} exerted

_by

the _planes.

But

hitherto

we

have left out of

consideration the

means

wherebythe aeroplane is

endowed

with the speedessential for_{the production}of_{the necessary}

dealwith the head-resistance_{or drag,} whichconstitutes,as alreadystated,the_{price tobe paid}forthelift.

This pointwill

now

beconsidered.

Revertingtothe concrete casefirstexamined,thatofthe horizontalflightatanangleof incidenceof 7 ofa Bleriot

monoplane weighing 300kg.

and

possessing a wingarea of 15 _sq. m., it has been_{seen that the speed}of this machine

flying at this angle would be 20m. per secondor 72

km.

per hour,

and

that the drag of the _{wings at the speed} mentionedwould

amount

to33_kg.

Unfortunately, though alone producing lift in an

aero-plane, the planes are not the only portions productive of

drag, for they have to

draw

_{along the fuselage,} or

inter-plane connections, the landing chassis, the _motor, the occupants,etc.

Forreasonsof simplicity,it

_may

beassumedthatallthese together exert the

same

amount

of resistance _{or drag as}

that offered

_by

an imaginary plate placed at right angles

tothe wind, soas tobe struckfullintheface, whosearea istermedthedetrimental surface ofthe aeroplane.

M. Eiffel has calculated from _experiments with scale

models that the detrimental surface of the _average

single-seater monoplane

amounted

to between _f

and

1

sq. m.,

and

that of an _{average large biplane} to about

1| sq. m.*

But

it is clear that these calculations can

only havean approximate value,

and

that the detrimental

surface of an aeroplane

must

always be an uncertain

quantity.

But

in

_any

case it isevident that _{this parasitical effect} should bereducedtothe lowest possible limits

by

stream-lining every part offering head-resistance,

by

diminishing

exterior stay wires to the utmostextent compatible with

safety, etc.

And

it will be

shown

hereafter that these

measures

become

the

more

important the greater the speedof_flight.

The

drag or passive resistance can be _easily calculated

* _These

figureshavesincebeen undoubtedlyreduced.

18

PLIGHT

WITHOUT

FORMULA

for a _given detrimental surface

_by

_multiplying its area

in _{square metres}

_by

the coefficient 0-08 (found to be the averagefromexperiments withplatesplaced normally),and

by

the squareofthe speedin_{metres per second.}

Thus, taking once again the Bleriot _monoplane, let us suppose it _{to possess} a detrimental surface of 0-8 _sq. m.;

its_dragat_{a speed}of 72

km.

_perhouror20m. per second willbe:

n ? * Detrimental Squareof Coefficient.

gurface>

^

gpeed>

0-08

x

0-8

x

400

=26

_{kg. (about).}

As

the _drag of the _planes alone at the above _speed

amounts

to 33 kg., it is _necessary to

add

this _figure of

26 kg., in order to find the total resistance, which is

therefore _equal to 59 _kg.

The

_principles of mechanics

teach that to overcome a resistance of 59 _kg. at a _speed

of 20m._{per second,power}

must

beexerted_{whose amount,} expressedinhorse-power, isfound

_by

_{dividing theproduct}

of the_{resistance (59 kg.)}

and

the speed(20 m.per second)

by

75.*

We

thus obtain 16 h.p.

But

a motor of 16h.p.

wouldbeinsufficienttomeet_{the requirements.}

I^or the _propelling plant, consisting of motor and

pro-peller, designed to overcome the drag or air resistance of

the aeroplane,islike_{every otherpiece ofmachinerysubject}

to losses of energy. Its _{efficiency, therefore,} is _only a

portion of the _power actually developed

by

the motor.

The

efficiency of the power-plant is the ratio of _useful,

power thatis, thepowercapableofbeing turnedto effect

aftertransmission tothe motive power.

Thus, in order to _produce the 16 _{h.p. required} for

horizontal _flight in the above case of the Bleriot

mono-* Thisis easilyunderstood. Theunit of power, or horse-power, is

thepowerrequired toraisea_weightof 75 kg.to _{a height}of 1 in. in 1second, sothat,toraisein thistime aweightof59kg.to_{a height}of

59x20

20m., werequire =^ h.p. Exactlythesameholds_goodif,instead

of _overcoming thevertical forceof _gravity, we have to overcomethe horizontalresistance oftheair.

plane,it would _{be necessary to possess} an_engine

develop-ing 32 h.p. ifthe _efficiency is _{only 50 per} cent., 26-6 h.p.

foran_{efficiency of60 per}cent.,etc.

But

if_{the aeroplane}wereto_flyatanangleofincidence other than 7 which, asalready stated, would depend on

the _position of the elevator the speed would _necessarily

be altered. If this _primary condition were modified, the

immediate result would be a variation in the drag of the

planes, in the head-resistance of the _aeroplane, in the

propeller-thrust,whichis_equaltothe_{total drag,}

and

lastly,

intheuseful_{powerrequired}for_flight.

Each

valueof _{the angle}of incidence

and

consequently

of _{the speed} therefore has only one _{corresponding value}

oftheuseful_powernecessaryforhorizontalflight.

Returning to the _{Breguet aeroplane weighing} 600 _kg. with a plane areaof30sq.m.,on which TableI.wasbased,

we

_may

calculate the valuesof the useful_powers required

to enable it to fly along a horizontal path for different

anglesofincidence

and

fordifferentlifts.

The

detrimental

surface

_may

be assumed, for the sake of simplicity, to be

1-2 sq. m.

The

values of the _{drag corresponding} to those of the

lift will be obtained from the polar diagram

shown

in

fig.3.

Table II., p. 20, summarises the results of the calcula-tion required to find the values of the useful _powers for

horizontal_flightat differentliftcoefficients.

Various

and

_interesting conclusions

_may

be

drawn

from

the_{figures in}columns8

and

9 of thisTable.

In the first _place, it will be noticed from the _{figures in} column 8 that thepropeller-thrust (equivalent to the drag

of_{the planes}addedtothe head-resistanceof the machine,

i.e.column6

and

column7)hasa

minimum

valueof 91 kg.,

correspondingtoa liftcoefficientof 0-05,

and

to _{the angle}

6.

This _angle, which, in the case under consideration,

is that _{corresponding} to the smallest _{propeller-thrust,} is

20

FLIGHT

WITHOUT

FORMULA

TABLE

II.

When

the lift coefficient is small, the _requisite thrust,

it will beseen, increases very rapidly,

and

the

same

holds

goodfor_highliftcoefficients.

Secondly, the figures in column 9

show

that, together with thethrust,theuseful_powerrequiredfor_flightreaches a

minimum

of 23 h.p., corresponding to a lift value of

0-06.

The

angle of incidence at which this

minimum

of

useful_{power can be}achieved,about10 inthe presentcase,

can be termedthe economicalangle.

This angleis_greaterthanthe

_optimum

_angle, which can be _explained

by

the fact that, though the thrust begins

to increase again, albeit very slowly,

when

the angle of

incidenceisraisedabovethe

optimum

angle,the speedstill

continues to decreasetoan_appreciableextent,

and

forthe time _being this decreasein _speed affects the useful _power

more

stronglythantheincreasingthrust;

and

the

minimum

value of the useful _power is, consequently, not attained until, as the angle of incidence continues to _grow, the

increase in _{the thrust exactly balances the decrease} inthe

The

_{figures in}column9again

show

the great expenditure

ofpowerrequiredforflightatalowliftcoefficient. _Thus, the _{Breguet aeroplane already} referred to, driven

_by

a

propellingplant of_{50 per}cent, efficiency,flying ataliftof

0-05 thatis,ata speedof72

km.

perhour onlyrequires

an_{enginedeveloping 46h.p.} ; butitwould need a136-h.p.

engine to flywith a lift of O02, or at about 113

km.

_per

hour. Itis_{mainly on}thisaccountthat, as

we

havealready

stated,the useoflowliftcoefficientsis_{strictly limited.}

The

variations in _{power corresponding} to variations in

speed (andinlift)can beplottedina_{simple curve.}

Fig. 6isof_{exceptional importance,}forit

_may

besaid to

determine the character of the machine,

and

willhereafter

bereferred to astheessential_aeroplanecurve.

After these _preliminary considerations on the _power required for horizontal _flight,

we

_may

now

proceed to

examine the precise nature of the effect of the motive-power onthe speed, which will lead at the

same

time to

certainconclusions_{relating to gliding}flight*

Forthis,recourse

must

behadtooneofthemost

elemen-tary principles of mechanics,

known

as the composition

and _{decomposition offorces.}

The

principle is one which

isalmostself-evident,

and

has, infact,alreadybeenusedin

these pages,

when

at the beginning of _Chapter I. it

was

shown

thatinthe_{air pressure,}whichisalmostvertical, on

a plane

moving

horizontally, a clear distinction

must

be

made

betweenthe_principal partof this pressure, which is

strictly vertical (the lift),

and

a secondary part, which is

strictlyhorizontal(the drag).

And, conversely,it isevident that forthe actionof

two

forces_workingtogetheratthesametime

_may

besubstituted

* _There is_reallynoexcuse for _{the importation into English} of the

French term "vol plane," and still less for the horrid anglicism "

volplane," since"glidingflight

"

is_{a perfect English equivalent}ofthe French. TRANSLATOR.

22

FLIGHT

WITHOUT FORMULA

that of a _single force, termed the resultant of these

two

forces. This proceeding is

known

as the _composition of

forces. _So, in _compounding the vertical reaction

con-stituting the lift,

and

the horizontal reaction which forms

71) 40 3* id -10 Flying Speeds(m/s). FIG. 6.

Thefigureson the curveindicate thelift.

thedrift, oneobtains thetotalair_pressure,whichis_simply their resultant.

Both

the _composition

and

_{decomposition} of forces is

accomplished

by

way

of projection.

Thus

(fig. 7),theforce

Q,* whichis inclined, can be decomposed intotwo forces,

*

_A

_forceis_{representedby}a

straightline,drawninthe directionin which the force operates, and of a _length just proportional to the

F

and

r, vertical

and

horizontalrespectively,

by

projecting

inthe horizontal

and

vertical directionstheend_point

A

on

two

axes _starting from the point 0, where the forces are

applied. Conversely, these two forces

F

and

r

_may

be

FIG. 7.

FligU-Path. ._

FIG. 8.

compounded

intooneresultant Q,

by

drawingthe diagonal

of the parallelogram or rectangle of _{which they form}

two

of the sides.

We

_{may now}

return to the _{problem under}

consideration.

If

we

_{take the aeroplane as a whole, instead} of dealing with the _planes alone, it will be readily seen that the

24

FLIGHT

WITHOUT FORMULA

machine is _equal to the drag of the planes added to the passiveor head-resistance,the while thevertical_component

remains _{practically equal} to the bare lift of the _planes,

since theremaining parts of the structureof an_aeroplane

exertbut _slight lift, if atall.*

The

entire_pressure ofthe

air on a complete aeroplane in flight is therefore farther

inclined to_{the perpendicular}thanthatexertedon_{the planes}

alone.

If (seefig. 8) the aeroplaneisassumedto_{be represented}

by

a_singlepoint O, inhorizontal flight, theairpressure

Q

exerted

_upon

it

_may

be decomposed into

two

forces, of

which the lift

F

is _equal

and

_{directly opposite} to the weight P,

and

the drag r, or total resistance to forward

movement,

which

must

be _{exactly balanced}

_by

the thrust tofthe_propeller.

But, supposing the engine be _stopped

and

the _propeller consequently to _{produce no} thrust (fig. 9), the aeroplane will assume _{a descending flight-path such that the planes}

still retain the _{single angle} of _7, for instance, which

we

have assumed, so _{long as the elevator} is not moved,

and

such that the air _pressure

_Q

on the planes becomes absolutely vertical, in order to balance the weight of the machine, insteadof_remaininginclinedas heretofore. This

is_gliding

flight.

Relatively to the direction of flight, the air pressure

Q

still retains its

two

components, of which r is _{simply the}

resistance ofthe air _opposed to the forward

movement

of

the _glider.

The

second

_component

F

is identical to the lifting powerin horizontalflight,

and

itsvalue isobtained

by

multiplying the lift coefficient _{corresponding} to the angle 7

_by

_{the plane}area,

and by

the squareofthe speed

ofthe aeroplaneonits

downward

_flight-path.

Fig. 9showsthat,

by

the veryfact ofbeinginclined, the

force

F

is_slightlylessthan_{the weight}ofthe machine,but,

since the _{gliding angle} of an _aeroplane is _{usually a slight}

* _For

the sakeof simplicity,we may considerthat the tail _plane,

one,the_liftingpower

F

_may

stillbe

deemed

tobe_equalto

the weightofthe aeroplane.

Clearly, therefore,every considerationinthefirst_chapter

which related to _{the speed} in horizontal _flight is _equally

applicable to _{gliding flight,} so that it

_may

be said that

FIG. 9.

when

an _{aeroplane begins} to _glide, without _changing its

angle,the speed remains the

same

as before.

In fact, horizontal flight is _simply a glide in which the

angle of the flight-path has been raised by mechanical

means.

On

_comparingfigs. 8

and

9itwillbe seen thatthisangle

26

FLIGHT

WITHOUT

FORMULAE

angle is _represented, as in the case of

_any

_gradient, in

termsof a decimalfraction, itwill befound to_{depend on}

the ratio which the forces r

and

F

bear to one another.

Hence,the followingrule

_may

bestated:

RULE. The gliding angle assumed at a given angle of

incidence _{by any aeroplane} is equal to the thrust _required

for its horizontalflight at the same angle, divided by the

weightof themachine.

Thus

the Bleriot _monoplane dealt with in the first

instance,which requires for horizontal flight at an angle

of 7 athrustof 59kg.,

and

weighs 300kg.,would assume

on its glide, at the same angle of incidence, a _descending 59

flight-path equal to , or 0-197, which is equivalent to

oOO

nearly 20 cm. in _every metre (1 in 5).

The

Breguet aeroplaneon whichTables I.

and

II.were based, weighing 600kg.,would assumeatdifferentangles(orliftcoefficients)

the_glidingangles

shown

inTableIII.

TABLE

III.

Itwill

now

be seen that thebest glidingangleisobtained

angle of the aeroplane.

The

latter, therefore, is the best

from the_{gliding point} of view, so far as_{the length} of the

glideisconcerned.

Infig. 10,startingfroma point 0, are

drawn

dottedlines

corresponding to the gliding angle given in column 6 of

Table III.,

and

on these lines are

marked

off distances proportionaltothe speed valuessetoutincolumns 3or 4;

the _diagram will then _give, if the _points are connected

into a curve, the _positions assumed, in unit time,

by

a glider,launchedatthe various anglesfromthe point 0.

It will be observed in the first _place that

_any

_given

gliding path, such as

OA,

for instance, cuts the curve at

two

_points,

A

and

B, thus showing thatthis gliding path

couldhave beentraversed

_by

_{the aeroplane}at

two

different

speeds,

OA

and OB,

corresponding to the

two

different

anglesof incidence, 1

and

15 in_{the present}case.

Only for the single gliding path

OM,

corresponding to

the smallest _{gliding slope}

and

the

_optimum

_angle of inci-dence,dothese

two

pointscoincide.

But

it isnot

_by

_{followingthis glidingpath}thatan

aero-planewilldescendbestinthevertical_{sense during}a_given period of time; for this it will only do

by

following the path

OE

correspondingto the highest point onthe curve,

and

the angle of incidence to be _adopted to achieve this

result is none other than the economical angle.

But

the

difference in therate of fallis

only slight for theexample

in question.

Itwillbe notedthatas_{the angle}ofincidence diminishes,

the _{gliding angle rapidly} becomes _steeper. If the curve

were extended so as to take in _very small _angles of

incidence, it would be found that at a lift coefficient of

0-015 the _{gliding path would already} have

become

_very

steep, that this steepness would increase very rapidly for

the coefficient 0-010,

and

that at 0-005 it _{approached a}

headlongfall.

The

fall,infact,

must become

vertical

when

the lift _disappears, that is,

when

the plane meets the air

In these conditions, a slight variation in the lift there-fore brings about a very large alteration in the gliding

angle, and this effect is the

more

intense the smaller the

lift coefficient.

The

glide becomes a dive.

Hence

it is

clear that this is another danger of _adopting a low lift

coefficient.

Thisbriefdiscussionon_{glidingflight,interestingenough}

in itself, was necessary to a proper understanding of the

part played

by

power in the horizontal _flight of an

aero-plane, for

we

can

now

regard the latter in the light of

a _glide in which the _{gliding path} has been _artificially

raised.

And

this raising ofthe_{gliding path} is dueto thepower

derivedfromthe_{propelling plant.}

Thiswillbebetter understoodif

we

assume_{that, during}

the courseof a_glide, thepilotstarted

up

his_{engine again}

without altering the _{position of} the elevator, so that the planes remained at the

same

angle as before; the gliding

path would gradually be raised until itattained

and

even surpassed the horizontal, while_{the aeroplane} (as has been

seen) would approximately maintain the same speed

throughout.

Hence

it

_may

be said that

when

the angle ofincidence remains constant, the_{speed of}an _aeroplaneis not _produced

by its motive _power, as in the case of all other _existing

vehicles, since, when the motor is _stopped, this speed is

maintained.

The

function of _{the power-plant} is _simply to overcome

gravity, to prevent the aeroplane from yielding, as it

in-evitably

must

doincalmair,tothe attractionofthe earth;

inother words,to_governitsvertical_flight.

In the case

now

underconsideration,the speedtherefore is_{whollyindependent}ofthe power,since, ashasbeenseen,

it is _entirely determined

_by

the angle of incidence,

and

if

thisremains constant,asassumed,

any

excessofpowerwill simply cause the aeroplanetoclimb, whilealackof power

30

FLIGHT

WITHOUT

FORMULA

willcauseittocoinedown,but without_anyvariation in the speed.

But

this

must

not be taken to_implythatthe available

motive-power cannot be transformed into speed, for such,

happily, is not the case. Hitherto the elevator has been assumed to be immovable so that the incidence remained constant.

As

amatteroffact,the incidenceneed_{only be diminished} through the action of the elevator in order to enable the aeroplane to _adopt the _{speed corresponding} to the

new

angle of planes,

and

in this

_way

to absorb the excess of

powerwithout climbing.

Nevertheless

and

the point should be insisted

_upon

as

itis oneof the_{essential principles of aeroplane} flight the

angle of incidence alone determines the _speed, which cannot

be affected by the _{power save through the intermediary} of

theincidence.

Hitherto

we

have constantly alluded to the different

speeds at which an _aeroplane can fly, as if, in practice,

pilots were able to drive their machines at almost any

speed they desired. In actual fact, a given aeroplane usually only flies at a _single speed, so that

we

are in

the habit of referring to the

X

_biplane as a 70

km.

per hour machine, or of _stating that the

Y

_monoplane does 100

km.

per hour. This is _simply because

_up

to

now,

and

with_{very fewexceptions,}pilotsruntheirengines

attheirnormal

number

ofrevolutions. Intheseconditions

it is evident that the useful power furnished

_by

the propelling plant determines the incidence,

and

hence the

Thus, referring once again to Table II., it will be seen

for example that, if the _{Breguet biplane} receives 29 _h.p.

in useful _{power from} its propelling plant, the pilot, in

ordertomaintain horizontalflight, willhavetomanipulate

his elevator until the incidence of _{the planes} is

approxi-mately 4, whichcorrespondstothelift0-040.

FLIGHT

Experience teaches the pilot to findthecorrect position of the elevator to maintain horizontal flight. Should the engine runirregularly,

and

if_{the aeroplane} is tomaintain itshorizontal _flight,the elevator

must

be _slightlyactuated

inordertocorrect thisdisturbinginfluence.

Horizontalflight,therefore, implies a constant mainten-ance of equilibrium, whence the designation equilibrator,

which is _{often applied} to the elevator, derives full

justifi-cation.

But

if_{the engine}is_{running normally, the}incidence,

and

consequently the speed,of an_aeroplaneremain _practically constant,

and

these constitute its normal incidence

and

speed.

Generally the engine is _running at full _{power during} flight,

and

so in the ordinary course ofevents the normal speedofan_aeroplaneis_{the highest}itcanattain.

But

there is a _growing tendency

among

pilots to

reserve a portion of the _power which the _engine is

capable of _developing,

and

to throttle

down

in normal flight. In this case the reserve of power available

_may

be savedfor _{an emergency,}

and

be used the case will be

dealt with hereafter for _climbing rapidly, or to assume

a higher speed for the time being. In this case the

normal _speed is, of course, no longer the highest possible speed.

In the _examplealready considered, the Breguetbiplane would fly atabout 80

km.

per hour, if it _possessed useful

power amountingto29h.p.

But by

throttling

down

the engine so that it _normally

onlyproducedareduceduseful_powerequivalentto24_h.p., the normal _speed of _{the machine, according} to Table II., would_onlybe72

km.

perhour(thenormalincidence being

6|

and

thelift_0-050).

The

pilot wouldtherefore have at his disposalasurplus

of power amounting to 5 _h.p., which he could use, by, opening thethrottle, either forclimbing orfor_temporarily

32

FLIGHT

WITHOUT FORMULA

Although, therefore, an aeroplane usually only flies at onespeed, which

we

call itsnormal_speed, itcan _perfectly

well_flyatother_{speeds, as}was

shown

in_ChapterI. _But, in order to obtain this result, it is essential that on each occasion the engine should be

made

to_{develop the precise}

amount

of power required

by

the _speed at which it is

desiredtofly.

Speed variation can therefore _only be achieved

_by

simultaneously varying the incidence

and

the power, or,in

practice,

by

operating the elevator

and

thethrottle together.

This

_may

be _accomplished with greater or less ease according to the typeof motor in use, but certain pilots

practise it most _cleverly

and

succeed in achieving a very

notable _speed variation, which i? of _{great importance,}

especially in the case of high-speed aeroplanes, at the

moment

of alighting.

As

_{has already}beenexplained, the horizontalflightofan aeroplane

may

be consideredin the _{light of gliding}

flight with the_gliding angle artificially raised.

From

this point

ofviewit is_{possible to calculate in}another

_way

the_power

requiredforhorizontal_flight.

For instance, if

we

know

that an _aeroplane of a _given

weight, such as 600kg., has, fora given incidence,a

glid-ing angle of 16 cm. per metre (approximately 1 in 6) at which its _speed is 22-3 m. _{per second,}

we

conclude that

in 1 second it descended 0-16

x

22-3=3-58 m. Hence,

in order to overcome its descent

and

to preserve its

hori-zontal flight, it would be _necessary to expend the useful

power required to raise a weight of 600 kg. to a height

of 3-58 m. in 1 second. Since 1 _h.p. is _{the unit required}

to raisea weightof 75kg. toaheight of 1 m.in 1 second,

the desiredusefulpower

=

about 29h.p. This,

as a matter of fact, isthe

amount

_given

_by

Table II. for

the Breguet biplane which complies with the conditions

given.

horizontalflight ofan aeroplaneflying at a given incidence,

andhenceatagiven speed,multiplythe_weightofthemachine by this _speed and _by the _{gliding angle corresponding} to the

incidence,anddivide_by75.

By

asimilar

method

one

_may

_{easily calculate}theuseful

powerrequiredtoconvert horizontalflight intoaclimb at

any

angle.

Thus, if_{the aeroplane already} referredto

had

to climb, alwaysatthe

same

speedof22-3m._{per second,}atanangle

of 5cm._permetre₍₁in20),itwould be_necessaryto_expend

the additional_power

0-05x600x22-3

,

=about

9 h.p.

75

Of course, this _expenditure of surplus power would be

greater the smaller the efficiency of the propeller,

and

would be 12 h.p. for 75 per cent, efficiency,

and

18 h.p. for _{50 per} cent, efficiency.

Clearly, this

method

of

_making

an_{aeroplane climb}

_by

increasing the motive power can only be resorted to if

there isa surplus of_power available, thatis, if_{the engine} is not _{normally running} at full_power, which until

now

is

the exception.

Forthis reason,when,asis_{generally thecase,the engine}

is _running at full _{power, climbing} is effected in a

much

simpler manner, which consists in increasing the angle of

incidenceofthe planes

by means

oftheelevator.

Let us once

more

take ourBreguet biplane which, with motor working at full _power, flies at a normal _speed of 22-3 m. _{per second (80-3}

km.

_{per hour)} at 4 incidence (or

a lift coefficient of _0-040).

The

useful _{power needed} to

achievethis_speed(seeTableII.)is29_h.p.

Assume

that,

by means

of his elevator,thepilot increases

the angle ofincidenceto 10 (liftcoefficient 0-060). Since

horizontal _flight at this _incidence, which

must

inevitably

reduce the speed to 18-2 m. per second or 65-6

km.

_per

hour, would only require 23 h.p., there will be an