136 MONTHLY WEATHER REVIEW. APRIL, 1897
-
Time bB8-p.m. cloud. 8.011.
&=@-a8 148.1 A=a-@ @.lo aa.P 18.60
dl=4-ao 1od.W
2
~ ~~ ~ ~
azimuth and angular altitude, a, and h,, of the cloud are im- mediatelymeasured with a theodolite; then the azimuth and angular altitude of the sun, a, and h , are measured ; these are to be corrected when necessary for the slight change in the position of the cloud and sun during the time elapsing be- tween the first and second observations ; then the azimuth, a,,
and distance, b, of the cloud shadow are measured, as seen on
the ground when the first observation was taken. The dis-
tance is taken from a map, of which we have a number, de- tailed, accurate, and on a large scale for the region around Blue Hill.
A moments consideration will show that the measurement of the azimuth and angular altitude of the sun from Blue Hill takes the place of an observation of the sun and cloud with a theodolite a t the position of the cloud shadow. For if an observer were at that point and looked toward the cloud he would find it exactly in the direction of the sun, and since the rays of the sun are practically parallel, he would get the same angular altitude and azimuth as that measured by ob- serving on the sun from Blue Hill. Hence, by making the
measurements of a, and h2 from Blue Hill alone, we get the same results as if we had observers looking at the cloud simultaneously from the two ends of a base line, and the formulse for calculating the results are the well-known trig-
onometrical relations :
A, = a, - a,;
A, = (id, - a,; z,= e + z,= b sin A, cosec A tan h,
z, = b sin A, cosec A tan h,
A =a ,-a ,; z,=+ (z,+z,-c)
This formula gives the mean altitude of the cloud above Blue Hill (zm) from the results calculated from the two ends of a base line drawn from the lower station or shadow to
a point below the upper station; c is the differeuce in level
between the stations, and must be read off from the map by means of the contour lines, if the country is not quite
level. The relations of the angles to each other in the above formulre, and the agreement of the calculated heights z, and
2,- c, furnish criteria for determining the accuracy of the
observations and for preventing errors which might arise, for example, from taking the angular measuremente on one
cloud, while by mistake using the shadow of another. The following table gives the results of measurements made on a cumulus cloud on May 19,1896. The heights in
the second to the fifth columns were determined parallac- tically by two observers with theodolites observing simulta- neously a t the ends of the base line. The mean of their four
results is given in the sixth column. The height given in
the seventh column was determined a few minutes later (at
I Theodolltemethod. I Merm .I Cloud 6bad0~
logtauhl 9.(9986
1ORsInd~ 9.5880
logslndl 9.9688
log 2=%i+c 8.1638
log b a.BLI1li
logtanhq 9 .m
a. in8
log owe0 d 0.1737
~~ ~
rime ........................... ........ 8b 611
Calonlated height in meters..
8:51 a. m.) by means of cloud shadows; evidently the two methods give practically the same results. The base of the nearly uniform stratus or nimbus, meas- ured by kites, is, I think, lower as a physical reality than the broken sheets of the same clouds, the only kinds which per-
mit of measurement with theodolites.
latmnpk 57hWating t7M mstliod of mpoltalion by tha oboe fomurks.
Dlstanoe of shedow b-4.488 meters; shadow below Blue Hill c=44 meters.
1,648 meters 44 meters
1 498 meters 1 :~ metm 14 meters 1,498 m-6
~ ~~ ~~ ~~
!lHEl MEU-US AND EQUILIBRJUM OF KITES.*
A monograph prepared by C. F. Marvin, Professor of Meteorology, U. 8. Weather Bureau, submitted with the ap- proval of Prof. Willis L. Moore, Chief of Weather Bureau, in competition for the Chanute Prize offered by the Boston Aeronautical Society.
ANNOUNCEMENT OF PRIZE.
Octave Chanute, Esq., ex-president of the American Society of Civil En neem, generously offers the prize herein described. &der date of Ma 97 1896, he writes to the Society 88 follows - I herewith endie $100, and authorize the Boston Aerona~tid Society to offer this sum as a s ecial prize for the beet monograph on tpe kite, giving a full theory o f i h mechanica and stability, with quan- tatijtive computations appended. This qnze to be awarded by judges appointed by the Society. It may be withheld one year in caae no suffiuently complete monograph is handed in. TO explain the latter reservation, it may be suggested that the fol- lowin in& need oonsideration :
rrl.%e resolution o! all the forces acting upon an. ordinary kite with a tail ; 5. d., the wind pressure upon 1% surface, its ml, and its string and the weight (gravity) of these vmous parts. The resultin equilibrium, or the divi spinning round or glancing sideways, an8 how the forces act wh% restore the dance. State the pition of the center of gravity, center of pressure, and beet point of attnchment for the strin , with numerical exam le. 3. Give &e same elementa for &e tailless kite, distinguishing be- tween the Malay, the Japneae or Chinese, the Bi-polar, the Hargrave, and the Fin (Boynton) kites. Indicate also what are the neral prin- ci lee upon which each group of the taillese kites depeng for its sts- bfit
c What effect is produced b changes in the point of attachment of the string, and what is the pulf thereon with various positions and with various strengths of wind.
I r 4. What is the difference in effect between the kite string and the attraction $ gravity on the mass of a soaring bird.
Competitors for tzis prize are requested to have their essays type- written, and to send them in on or before November 15,1896. Addrees Secretary of the Boston Aeronautical Society, P. 0. Box 1197, Boston, Mass., U. 8. A.
Seotlon. Introduction .................................................. I Definitions and axiomatic statements .......................... I1 General statement.. .......................................... 111
Forces acting on kites in general .............................. IV Circumstances of equilibnum and motion.. .................... V Explanation of the flight of kites ............................. M Abnormal flight of.kites.. ..................................... VI1 ElTecta of changes in wmd and ition of kite.. ............... VI11 Conditions of stability and a t e g e e s of kitexi.. ................ Ix Numerical data ............................................... X The mechanics of the kite string .............................. XI Properties of the catenary .................................... XI1
Since our purpose is to present in full a theory of the mechanics and stability of kites in general, it will be well at the outset to clearly define the essential and fundamental concep- tion that we consider to beconveyed or suggested by the word
kite. Furthermore, the behavior of kites, however diverse in character and detail, results from the action and reaction
of a small number of well known natural forces. Some of the forces that we must consider act upon the kite itself, while others, in a wholy independent manner, act upon the string
or line employed to restrain the kite. Very important limi- tations to the attainments of which kites are capable arise wholly in consequence of the action of forces upon the string. Under theee circumstances it will be most logical, after de- fining the kite, to consider without regard to any particular or specialized form of kite, those general and fundamental principles of physics and mechanics that underlie the action of all kites and study that relation of forces which is essential
Sincere1 youm, 0. CHANUTE.
CONTENTB.
1.-INTRODUCTION.
*The above monograph, by Professor Marvin, was awarded the Chanute Prize by the jud ea ap ointed by the Boston Aeronautical Society, which society has $80 su%mitted it for publication in the MONTHLY WEATHEB REVIEW.
APRIL, 1897. MONTHLY WEATHER REVIEW. 137
to their flight and stability. I n these studies we shall be led
also, to develop separately the results of the action of tht forces on the stripg. In order, finally, to give due consideration to the pointr specified under items 1 to 4, inclusive, in the announcemenl
of the Chanute prize, it will suffice to apply the general theory that we shall first preaent to the several individual
cases, pointing out the particular application and developing any special features that may arise.
11.-DEFINITIONS AND AXIOMATIC STATEMENTS.
Kite.-Fundamentally a kite is a surface or series of sur. faces, either flat or curved, which are provided with a re- atraining'attachment having the nature of a string or line, The surfaces and liue are arranged in such relations that when the kite, held in restraint by the line, is exposed tu
masses of air in motion, the surfaces are subjected to wind pressures which sustain the kite in mid-air, where it as-
~umes a position of equilibrium. Ideal and actual kites dietingui8hed.-Occasion will arise in
our study of the mechanics of the kite to consider the results that follow when various dtrfects and imperfections, generally found in kites, are absent. It will be desirable, therefore, to clearly distinguish at all times between what we shall call the perfect or ideal kite, from which all defects and imperfec- tions are wholly absent, and the actual or material kite, which is always unavoidably trammeled by greater or less defects and imperfections. If the kite is a surface against which it is designed the wind shall press, then the ideal kite is that surface ; the actual kite is a material substance having thick- ness, weight, edges, possibly a tail, etc. The string, or kite line, or simply line, as we shall generally call it, is a subordinate accessory to the kite, and is unes- sential in so far as the mechanics of the kite is concerned. The specific function of the line is to hold the kite in re- straint, and while doing 80 to assume without opposition any
direction essential to the kite's equilibrium. It is conceiva- ble that all the phenomena incident to the stability of a kite can be exhibited without the use of a string, as, for example, by securing the kite to the top of a suitable pole or mast by
means of a connector in the nature of a universal joint. The only object in presenting this thought is to emphasize the essential independence of the kite and the string, so far as
the mechanics of the former is concerned. The ideal kite string is a flexible, inextensible, mathemati- cal line, which has unlimited etrength and is without weight
or thickness. The actual material kite line is, in general, sufficiently flexible, but its weight and thickness, and some- times its roughness are objectionable, and its strength is limited. Bridle.-The bridle is a subordinate, often not an essential
accessory of the kite. Its purpose is to provide a point for the attachment of the restraining line to the kite, such that certain relations are permitted between the restraining forces aoting in the line and the forces of the wind pressure and
gravity acting upon the kite. The bridle may also be made
to serve a secondary purpose, as follows : The resultant pull,
due to all the forces acting upon the kite, is accumulated and transmitted to the line through the bridle. Iu addition,
therefore, to serving its normal function, as stated above, the bridle may incidently be made to distribute the strains upon aertain members of the framework of ths kite in such a man-
ner that they can safely sustain greater strains than might be the case without the bridle. f i l l , l i f t , drift, tensim.-In referring to the strain upon the kite line which tends to tear it asunder we shall employ the words pull or tension, and when so need the words will be
considered synonymous. Lift is the vertical and drzft the horizontal component of the pull. We may logically deal
with the pull, lift, and drift, not alone at the kite, but like- wise at any point along the line. If at a given point the inclination of the kite line to the horizontal is the angle 8,
and the tension is t , then the lift equals t sin 0, and the drift equals t cos 0.
Tail.--Some kites can not preserve their equilibrium with- out the assistance of a tail; with others it is unnecessary,
although a tail may he applied to any kite. When present, the kite is subjected to the action of one more force than
would otherwise be the case. This force is the resultant of the action of gravity and the wind upon the tail.
III.4ENERAL STATEMENT.
The flight of any conceivable kite and the m.otions it may execute-including, for example, the general case of a
faulty and insubordinate member of B tandem, which, being held at a comparatively fixed point high up, in the free air, can not end its erratic flight in a precipitate dash to the
ground, but must go on and execute any orderly or disorderly movements the circumstances may demand, even if it be broken and distorted-a11 possible evolutions of such a kite
or of any other, under any and all circumstances, find their full explanation in the application of the following general proposition of mechanics, the full demonstration of which is to be found in the ordinary text books :
Any systant of forces acting upon a rigid body ntay always be reduced to a single resultant force, R,, having a dejnite and de- terminate podtion a d a resllltaitt couple, Z,, the forces of which
act i n a plane perpendicular to the force R,. The force and couple can, together, produce ezactly the same efects a8 the system. The kite is a body which is rigid within the present mean- ing, and, when flying, is acted upon by a complex system of forces. The conditions of equilibrium or motion are these :
When the couple Z,=O, and when the string pulls in a
direction opposite to the force R,, and exactly in line with it, then the kite will be in equilibrium. If, however, the couple 2, is finite, while the string constantly neutralizes the force
R,, then the kite will spin around upon the string as an axis of rotation, the kite as a whole remaining fixed in one posi- tion. If on the other hand, the pull of the string does not balance the force R,, but the couple 2, still remains zero, then the kite will be translated in some direction without rota- tion. Finally, if 2, be finite and the force R, is unneutral- ized by the restraint of the string, then the kite will be both translated and will also rotate about the string as an axis.
All the evolutions any kite may execute are but combinations of the foregoing cases, as will be more fully discussed here- bfter. While we may thus, by the fundamental propositions of mechanics, pass a t a single step from the conception of the highly complex system of forces due to the action of wind md gravity upon every point and particle of the kite to its 3xceedingly simple equivalent, and establish all the circum-
stances of motion or rest, by doing so we omit from considera- tion many relations between the forces of the system that are
If great practical importance and utility, especially in the lesigning and constructing of kites to perform any assigned
hty. Before discussing, therefore, in full, the specific causes of ;he various motions kites are observed to execute, we will
rnalyze in detail the forces acting upon kites in general.
1V.-FORCES ACTING ON KITES IN GENERAL.
In flight, the forces acting upon a kite and ita line are:
1) The total of all the wind pressures upon the whole struc- ;um, including pressures upon not only the sustaining sur-
hces but upon every part of the framework, also all friction !ffects of the wind gliding over the surfaces, e t a ; (2) The ittraction of ravity for the kite; (3) The tension of the itring at the f ite, that is, the restraining pull of the line;
138 APBIL, 1897
~~
(4) The attraction of gravity for the string; (6) The pres- sure of the wind upon the string; (6) The action and reac- tion of the forces a t the reel that restrain the whole system. If the kite is provided with a tail, one more composite force acts in addition to the six mentioned above, namely, (7 the resultant of the forces of the wind ( W) and gravity (a upon the tail. When a kite, or tandem of kites, is employed
to lift and sustain objects in mid-air, such as meteorological instruments, photographic apparatus, etc., these may be at- tached to the kite frame itself, in which case the pressure of the wind against the object and its weight may logically be considered with the forces (1) and (2) above. It is not, how- ever, customary, as a rule, to carry such loads in this way. The apparatus is generally suspended from the line at some point below the top end. In this case the forces of wind (a)
and gravity (b) acting upon the attachments are properly considered in connection with the other forces (4), (6 ), and
(6). acting upon the string. If the forces (wind and gravity)
acting upon the short piece of string or line employed to suspend the instruments are not inappreciable, then those forces, also, if not included in (a) and (b) above, must be
treated separately. Under the circumstances cited all of these forces are con- cerned in determining the positions of equilibrium assumed by a kite flying freely in mid-air. In studying the mechanics of the kite proper, however, we are concerned only with the equilibrium of the forces (l), (2), and (3), including (7), if the kite has a tail. We need to consider the string and the forces acting thereon only when we wish to know how high a given kite can fly, or the position it will assume when carry- ing a given load with a certain length of line, etc. Having thus called attention in a general way to all the forces upon which the action of any kite depends, we will
omit from present consideration the string and its forces and proceed first to develop the relations upon which the equi- librium of the kite itself depends, that is, the relations between the forces (l), (2), (3), and (7). In order that the reader may form a mental picture of just what we now desire to consider, he may imagine a kite in mid-air under conditions of free exposure to the wind but held in restraint by a comparatively short piece of string or line. The action of gravity and the wind upon this short line being inappreciable, it will therefore be sensibly straight, The direction in which this string is pulled; the inclination of the surfaces of the kite to the wind; the action of the tail, when present; the variations of the several forces with changes in the character of the wind ; the process by which the kite once in equilibrium is able to reestablish equilibrium
under constantly changing conditions of the wind-are the questions with which we are now concerned. The force (2) is wholly and simply a gravitation effect, and
is, therefore, perfectly constant in amount and direction. The tension in the short piece of restraining line and its direotion constitute the restraining force (3) which always acts through the point a t which the line is fastened to the bridle or kite stick, as the casemay be. This force (3) is not an independent force ; it exists and undergoes variations only
as a result of the action and variation of the other forces
specified. Of the remaining forces, (1) is wholly and (7) partly a wind effect of comparatively complex character and
subject to variations of great frequency and very consider- able magnitude. In order to set forth fully the effects of the wind pressure upon the kite, as we now imagine it flying upon
a short straight string, it will be necessary to analyzein some detail the nature and composition of the forces (1) and (7), which together embrace the total effect of the wind upon the
kite and its tail. We are particularly concerned with (1) which includes by far the most important forces with which
we have to deal. For the purpose of this analysis it will be
1
convenient to claasify the several portions of the kite stma-
tare, as follows : sustaining surfaces ; framework; edges; neutral surfaces ; and finally the tail, which consistently be- longs within this classification, notwithstanding that the forces acting thereon have already been specified under (7). In some specialized and uncommon forms of kites perhaps
portions of the structure may not fall within this classifica- cation; it will suffice, however, for our present purposes, as
we seek to show only the ultimate effect resulting from the pressureof the windupon the entire structure of the kite. We embrace under  sustaining surfaces  all those extended surfaces of cloth, paper, or similar material whose normal function is to sustain the kite as a result of the pressure of the wind thereon. In many forms of kites, especially of oriental types, we find the most remarkable diversities in the shapes of the supporting surfaces. The logical inference we may draw from this diversity of itself is that the mere shape of the surface is of little importance. As we shall see, the
continued flight of kites depends upon the action of certain forces, one of which is the pressure of the wind against sur-
faces. There is scarcely any limitation to the form of sur- face that may be used. The necessary force is produced and can be made to act in proper relation to the other forces with almost any surface whatsoever. The framework, in the present classification, includes all the sticks, struts, ties, braces, and those members of the struc- ture that spread out the sustaining surfaces and give form to
the kite. The group entitled  edges, includes principally any exposed edges of the cloth or covering material. These are usually reinforced by a hem, or otherwise thickened, often
with the addition of a cord within the hem. While the wind pressure upon such edges  mill generally be relatively un- important, yet it is very proper to recognize them in our analysis. Finally, an example of neutral surfaces is found in the fin or keel of the Bopnton kite, and in the lateral surfaces of the rectangular cells of a Hargrave kite. Primary characteristics of wind pressures.-It is well known that the pressure experienced by any object exposed to the action of the wind is due not alone to the direct impact of the air on the front or windward surface, but also to the diminution in the static pressure over the back surface or lee side of the object. For our present purposes we need not push the analysis so far as to separate these effects, and we will in all cases regard them as combined into one resultant pressure exerted against the front side of the object under consideration. In dealing with surfaces exposed to pressure
we wholly disregard the edges. The effect of pressure upon the edges is reserved for separate consideration. Whenever the wind encounters a perfectly emooth surface which causes a change in the direction of motion of the cur- rent, the surface experiences a pressure which acts exactly normal to every element affected. This is a fundamentel and well known principle of hydrodynamics and finds im-
portant application in the theory of the kite. I n the case of slightly roughened, fuzzy surfaces, such as
the cloth used in kites, the surface can not be regarded as perfectly smooth, and there will in consequence be a friction effect resulting from the flow of the particles of air over such rough surfaces. The air may be regarded as pressing
against the minute projections and irregularities which con- stitute the roughness of the surface, and a relatively slight force is thereby developed which tends to urge the surface along in the direction in which the streams of air are flowing over it. This effect of skin friction combined with the pres- sure normal to the surface gives the total effect of the wind
on the supposed roughened surface. Obviously, this total effect is a pressure which is not quite normal to the surface but will be inclined thereto more or less in proportion to the
relative magnitude of the friction effect. Having thus recog-
APRIL, 1897. MONTHLY WEATHER REVIEW. 139
~~
nized and assigned the proper place to the effects of skin friction, it will scarcely be necessary to give it further con- sideration, since, as is well known, the effect is so small, relative to the principal forces acting on kites, as to be quite unimportant. Without further mention, therefore, we will
hereafter consider that the slight effect due to friction is included with the normal pressure against the surface. Fur-
thermore, we will still designate this combined effect as the noma1 pressure since in practical cases it will be sensibly per- pendicular to the surface. Pressure upon sustaining surfaces.-In accordance with the characteristics of wind pressures, as enunciated above, the whole effect of the wind's action upon the sustaining sur- faces, either flat or curved, of any form of kite whatever, con- sistsof a pressure sensibly normal to the surface at every point. Flat surfaces.-If the sustaining surfaces are flat, the nor- mal pressures are parallel to each other, and the total effect of all the individual pressures may be represented by a single force or pressure acting sensibly normal to the supposed flat
surface and at a point commonly called the center of pressure, which is a point through which the resultant force must act to produce the same effect as the individual pressures. Curved OT arched surfaces.-The sustaining surfaces of kites, being generally formed of yielding materials, such as cloth or paper, will, whether designedly or not, form curved surfaces when pressed by the wind. This curvature will sometimes be wholly in one direction, either coincident with the direction of the flow of the particles of air across the surface or at
right angles thereto ; whereas in many cases the surface will be curved in all directions. In any case the individual pres- sures at elementary points upon such surfaces must still be regarded as sensibly normal a t each point, but the direction of the resultant pressure can rarely or never be fully assigned by any of the principles of hydrodynamics thus far estab-
lished. The direction may be partly predeterniined by known laws in some cases, but, in general, i t can be fully established only by aid of experimental investigations. This is especially the case when the surface is curved in the direction in which the particles of air flow across it. Data of this character are comparatively scanty and incomplete, often ohtained by erro- neous methods, so that we can not, even from experimental results, definitely assert more than a few general conclusions relative to the pressure of wind upon arched surfaces. Much has been written upon this subject, and especially upon the asserted property of " aspiration," by virtue of which the pressure of a horizontal wind upon a properly disposed arched surface is able not only to sustain it, but also to propel
it forward. Such a remarkable performance is undoubtedly in direct violation of the fundamental laws of nature, and, in the numerous cases where it is claimed not only that birds in the free air have been certainlyobserved to exhibit aspira- tion, but that these effects have likewise been fully reproduced with artificial free-flying models, it is easy to show that these claims are wholly unsupported by any evidence that the wind
was strictly in the horizontal motion virtually assumed. Lack of proof on this point alone is fatal to the claim that any case of real aspiration has been observed or reproduced. Lilien- thal, Langley, Maxim, Wellner, F. von Loessl, and others have conducted extensivd experimental investigations upon aero- dynamic problems, but, as far as known to the writer, the results of several of 'these investigations have been only partlypublished up to the present time. The most com- pletely published results of wind effects upon arched surfaces is found in Lilienthal's book' and in a pamphlet by Wellner.'
Der Vogelflug als Grundlage der Fliegekunst. Von Otto Lilienthal. Berlin, 1889. * Versuche iiber den Luftwiderstand gewiilbter Fliichen in Winde und auf Eisenbahnen. Von Georg Wellner. Zeitachrift f i r Luftschif- fahrt. Beilage zu Heft S. Berlin, October, 1893.
REV^
The conclusions reached independently by these investi-
gators are practically the same as regards the main features. Their results show : (1.) That the resultant pressure of the wind upon arched surfaces inclined at moderate add small angles of incidence is from two to three times the pressure upon an equal area of flat surface similarly inclined. (The
inclination of the arched surface here refers to the inclina- tion of the plane in which the chords of the arch are assumed
to lie.) (2.) The action line of this resultant pressure, when the surface is placed at certain favorable angles of incidence, ranging from about Oo to 25O, was found to intersect the chord of the arch at an angle yreater than 90, and in such a sense as to produce a .folrunrd lwopclliqy coayioneiit; that is, if A C By Fig. 1, is a section of an arclied surface by a vertical plane parnllel to the lines of flow of the wind across the surface, and P C the resultant wind pressure, then the angle A 0 P
is found to be greater than 90.
"
8 'P
FIQ. 1.
The second of these results is of special importance in de-
signing kites intended to attain very great elevations. As yet, however, it can not be admitted that these conclusions
are fully established. A. v. Obermayei has called attention to a serious source of error in the methods employed by both Lilienthal and Wellner, and has shown that especially the
second conclusion cited above is by no meaiis proven. The real effects of the pressure of the wind upon arched surfaces are, in fact, but very imperfectly known at the present time. The writer has reason to believe that much valuable experi- niental data on the subject has been obtained which is not yet published. Center of pressure.-The foregoing discussions relate to the direction of action of the resultant pressure upon flat and arched surfaces respectively; we will next consider the po- sition of the point through which this resultant pressure acts, namely, the center of pressure. Here, again, the general laws of hydrodynamics, so far as known, do not suffice to defi-
nitely locate the center of presmre, except, perhaps, in certain simple cases. Its position must, therefore, be sought by means of experimental investigations. Jobsel, Kummer, Langley, Lord Rayleigh, and others have contributed to this question. Johsel's results were published in 1870. Those of Kunimer in 1875 and 1876. The work of Kummer comprises experiments with paper bodies modeled to resemble heavy shot and cannon projectiles. The position of the center of pressure was also determined for square and rectangular planes formed of sheet tin. These, however, he found were bent by the pressure of the wind, thus causing erroneous results, so that afterwards he was led to repeat his obser- vations on plane surfaces, which, for this purpose, were made of thin plates of glass. While his results are strictly ap- plicable only to small bodies (planes 90 by 180 mm. aud less) moved at moderate velocities (less than 18 miles per hour), his experiments are quite as comprehensive as any. Langley's investigations are of more limited scope (confined to a plate 1 foot square), but the results, so far as they go,
bear internal evidence of high accuracy. Lord Rayleigh,
Wber die Wirkung des Windes auf Schwach gewijlbte Fliichen. Von A. v. Obermayer. Siteungaberichte der Kaiserlichen Akademie der Wissenschaften. Heft VIII. Vienna, October, 1895.
la0 MONTHLY WEATHER REVIEW. APBIL, 1897
from purely theoretical considerations, has deduced a formula, stated below, giving the position of the center of pressure on plane surfaces. There is a remarkable agreement between the theoretical formula of Lord Rayleigh and the experi- mental results by Langley. If d is the distance from the
front, or windward edge of a rectangular plate to the position of the center of pressure, and if 1 is the length of the plate in the direction of the flow of the air particles across it, and i is the angle of incidence to the wind, then, according to Jobsel :
and, according to Lord Rayleigh :
d = 1 (0.2 + 0.3 sin i ) (1)
d = l(0.6 - - 4 4 +r s i n i
The formulse given above for the position of the center of pressure are strictly applicable only to simple rectangular
f i t surfaces. Kummers work conclusively demonstrates that even a slight curvature gives rise to very considerable differences in the position of the center of pressure. It therefore results that the general laws of hydrodynamics, and even the results of any direct experimental investigations thus far known, will be of little or no assistance in correctly locating the position of the center of pressure of the wind upon ordinary kites, because their surfaces, even when of the simplest form, will be so different from those employed in experimentation or considered in theoretical deductions that laws thus determined can not apply. Much looseness prevails in the use of the term ceibter of
pressure, and it is very important that a clear idea be formed of its exact mechanical significance. When we think of the pressure of the wind upon an infinitely thin and rigid smooth plane we are dealing with a system of parallel elementary pressures acting a t every particle of the surface, and there is then a very close analogy between the center of pressure, that is the point a t which the resultant may be conceived to act and the center of gravity, for example. This center, whether of gravity or wind pressure, can, however, have a real exist- ence only when the forces are unaffected by angular altera- tions in the position of the body. This is regarded as true in our ordinary dealings with terrestrial gravitation, but obviously when we deal with wind pressures upon material objects, even of the simplest form (and much less so with the complex devices we call kites), the forces of the system are
no longer parallel, and even if they were the slightest modifi- cation of the angular relations between the body and the wind would change in a corresponding manner the whole sys- tem of forces. Under these circumstances no real mechanical significance can be attached to the so-called center of pres- sure, except that it is s m point on the action line of the re- aultant of the whole system of forces. We may, for example, imagine the center of pressure to be the point where the action line of the resultant intersects the surface of the body, but we may just as logically imagine it to be a point within the interior of the body, or any other point on the line of the re-
sultant. In fact, in the mechanics of the wind pressure upon
material objects there is no such thing as the center of pres- sure. The thing which does, however, have a real existence is a central nxis. It is a line, not a point, we are to think of
in this connection as having some mechanical significance, and when we wish to limit our consideration to some specific point of this central axis as the origin of a force its signifi- cance is simply that of an assumed point of application of
the force. Summary.-The foregoing analysis of the presslire of the wind upon sustaining surfaces of kites leads us to the follow- ing conclusions : ( 1) The resultant pressure (including skin friction) is sensibly normal to flat surfaces, and under favora-
ble conditions the resultant pressure upon arched surfaces
may possibly be so inclined that a component will act in a
direction forward of the normal to the chord of the arch.
(2) When the edges of thin rectangular flat surfaces are pre- sented respectively perpendicular to and in the same plane with the direction of the wind then the central axis of the system of pressures intersects the surface at a point (center of pressure) which is given with a close degree of approxi- mation by Lord Rayleighs formula, equation (2) above. For flat surfaces not rectangular in form or presented to the wind otherwise than specified and for the complex and multiple flat and arched surfaces usually found in kites the position of the cedral azis can not be located, a priori. (3) The in-
tensity of the pressure upon slightly arched surfaces at small angles of incidence, such as those at which kites are ordinarily flown, is, upon the authority of Hargrave, Lilienthal, and others, considerably greater than upon an equal area of flat
surface a t the same inclination. Pressicre ripon the framework.-The framework of the kite
contains a variety of surfaces which are presented to the wind in a great diversity of ways. Some are completely sheltered
behind the sustaining surfaces, others are partly, and many, especially in cellular kites, are fully exposed to the winds action. The general characteristics of wind preesures already enunciated will fully suffice for analyzing in detail the effect of the wind upon the framework. Such an analysis, however,
is not now required. Whatever the preesure of the wind may be on the individual surfaces its total effect may be repre- sented by a single line of appropriate length and direction. The point of action of this total effect can not,in general, be exactly located. This, however, is not of special importance, since the whole force is small compared to the main ressures
disposition of a considerable portion of the surfaces of the framework the pressure of the wind thereon will be partly beneficial in character, that is, it will have at least a slight lifting tendency and, as a result the line representing this total effect will have an upward inclination. Pressures upon edges.-At least portions of the exposed edges of the covering material of kites tare subjected to pres- sures which are not treated of under either of the foregoing topics. These pressures are now considered, and, as in the case of the framework, the whole pressure may be summed up into a total effect, which can be represented by a certain line. The point of action of this total force also can not be located
with accuracy, but this is of very slight consequence. Any unbalanced upwardor downward pressure upon the edges may be regarded as a pressure belonging properly to those upon the sustaining surfaces themselves, and should be included in the resultant pressure thereon. As a consequence of this the
action line of the resultant pressure on the edges will be hori-
zontal. Presmre upon neutral surfaces.-These surfaces are designed
to be neutral under conditions of normal flight and should,
therefore, experience only equal pressures upon the opposite sides ; the pressure ppon the exposed edges belongs properly with the class considered in the preceding paragraph. Owing toimperfections in the kite structure and lack of perfect symmetry of corresponding parts, it will probably never hap- pen that the neutral surfaces are really neutral as regards the equality of pressure upon opposite faces, so that even during steady flight there will he a slight excess of preesure upon one side of such surfaces. The real function of neutral
surfaces is to steady the kite during variations in the wind force, and this action will be considered when discussing the stability of the kite under variable winds. Forces of wind aid gravity upon tui1s.-As ordinarily made the tail consists of long strips of cloth, often with a bushy tassel a t the free end. A better form consists of tassel-like
bunches of paper or cloth tied together a t intervale on a
acting upon the sustaining surfaces. Owing to the P avorable
APBIL, 1897. MONTHLY WEATHER REVIEW. 141
length of string. Probably the most efficient device for thf tail consists of a aeries of light, hollow, cloth or paper cones strung together upon a string or wire and presented with theii
bases to the wind. All such tails are fastened to the kite by
a piece of string or equivalent flexible attachment, and they will therefore draw away from the kite in a direction which iE
the action line of the resultant of all the forces affecting thc
tail at the kite. The direction of action and the intensity of this resultant force is all that now concerns us in OUI
study of the mechanics of the kite. To fully determine these
data experimentally for specific cases, it will suffice to detach from the kite the tail whoee constanta are desired, and fasten
it to a suitable dynamometer, Fig. 2, placed in neutral equilib- rium as regards both wind and gravity, and provided with a
graduated arc for measuring the angles of deviation from the
vertical aseumed by the tail when exposed freely to the action
Fro. 2.
of the wind. By this means the resultant force exerted by a given tail in winds of different velocities may be fully estab- lished, which data are the constants of that particular tail. Kites may be provided with other appendages than de- scribed above which answer some of the purposes of a tail. These may resemble the tails of birds or have rudder-like
effects. Such devices are, in fact, a part of the kite structure
itself, being more or less rigidly connected therewith. Ap- pendages of this character are generally very pliable and easily flexed by the pressure of the wind, and their effcacy will often result from vibratory, fluttering motions which they acquire under the action of the wind, and which are dis- cussed in the following paragraph :
E'ecte of fluttering, wavinesa, etc.-Throughout the analysis of the actionof the wind upon the structure of the kite it has been virtually mumed that all portions are quiescent relative to each other. If the action of the wind causesflut- tering and produces a more or less permanent system of waves over the pliable material of which kite surfaces are generally made, or, if the kite structure is provided with epecial mem- bers which are designed to bt, set in vibration by the wind for the purpose of producing musical sounds, etc., as is the case, for example, in some of the ingenious oriental kites, then whenever such effects are present there will be called into ac- tion an additional force, not thus far considered, and which will result from the action of the wind upon the wave fronts,
etc. By reason of this action the surface so affected will tend to be pushed along in the direction in which the streams of air flow across it, just as a flag, for example, with its surface formed into a multitude of waves tugs at ita halyards with much greater force than if the waves were wholly absent. We may, therefore, represent the effect of waviness and flut- tering by a line parallel to the general flow of the air over the kite surfaces.
whirls, or eddy efect8.-There is another circumstance which may produce an effect not thus far considered. In mme forms of kites a greater or less portion of the whole current of air affected by the presence of the kite is broken up into numerous whirls, or eddies. These may be formed
when the air flowing against the kite is suddenly stopped, or when its movement is abruptly changed and diverted to a
new direction. Angles and changes in the continuity of the surfaces such as formed by the preeence of the cross stick in the malay kite, for example, and other causas that prevent
the air from flowing easily and by smooth changes of motion
over and past the kite will give rise to eddies. Whirls of
marked character exist over the leeward surfaces of the kite. Strong eddies may thus be set up at numerous points adja-
cent to the body or surfaces of the kite. It is possible, and indeed quite probable, that some of them may remain nearly stationary in certain favorable spota. Such eddies, or whirls, in a certain sense, may have much the same effect as obstruc-
tions to the flow of the air. Quite as much of an obstruction may be thus formed as if an excrescence of rigid material
were placed on the kite a t one of the points in question. In cellular kites generally the cells are virtually short tubes
through which large streams of air must flow. Pronounced eddy formations within these tubes have much the same effect
BS real obstructions by which the flow of the air is, as it were,
choked up. We perceive, by the aid of the comprehensive principle of the conservation of energy, that the power re- quired to form these eddies and maintain the air within them in rapid motion must be derived by reaction from the kite and its string. The necessary reaction can be derived from the kite only when the resultant forces acting thereon sxperience some modification, depending upon the presence D f the eddies. The nature of this modification must be equivalent to a force which tends to cause the kite to move
in the direction of the general current of air. The eddy 3ffect may, therefore, be represented by a horizontal line. Combination of all wind e$ects.-We have now separately snalyzed the action of the wind upon the several more or less
msential members present in all forms of kites. We have
dso shown the general characteristics of the resulting forces
30 far as they are of importance in the theory of the kite, and
:alled attention to the effects of waves and eddies. Let us aext combine these several elementary effects, and thus aacer- tain the general total effect due to the action of the wind
ipon the whole structure of the kite. This total effect of the wind is the force we have designated (1) above. In assigning a magnitude and direction to any of the several wind effects i t is to be observed that the wind is not constant
,ither in force or direction, nor is it even homogeneous. We ;herefore assume that the value of all those wind effects whose nbrrelations we wish to study are momntay vnlws, sa'mdta-
aemaly takm. Furthermore, owing to this momentary char-
lcter of the forces, the kite is constantly obliged to shift its 3osition in order to adapt itself to the ever changing condi- ;ions. At any assigned moment the kite is, therefore, doubt- ens alreadyin motion as a result of the relations between ,he forces of a previous condition. Recognizing this circum- rtance, which is of great importance in the dynamics of the light of birds and flying machines, but yet of only passing nterest in the statics of kites, we assume that, in view of the imall mass of the kite, in relation to the large forces acting ,hereon; a condition of rest promptly ensues whenever equi- ibrium exists between the forces, and that the movement xecuted by the kite a t an instant when equilibrium does not mist is modified in only an unimportant degree by reason of ,he momentary velocity the kite may then possess. The several partial effects of the wind upon the different nembers of the kite structure, namely: the resultant normal masure of the windupon the sustaining surfaces, N; the total 3ffect of the wind upon the framework, f ; the total pressure
jffect of the wind upon the edges, e ; the excess of pressure ipon one side of neutral surfaces, n ; the total effect due to Raves and fluttering, w ; and, finally, any effect due to the 3resence of eddies or vortex motions, o, can all be combined
142 MONTHLY WEATHER REVIEW. APBIL, 1897
in a simple manner by aid of the graphic methods employed in mechanics. These depend upon the following fundamental propositions,.namely : Any system of forces is equivalent in ef-
fect and may always be redwed to a single ,force and a couple, and the .force may be made to act throrbgh any point. Also :
When a system of .forces has been redwed to a single force and a cotbple there is but one position qf the .force possible in which the axis of the couple will be parallel to the direction qf the force. This position of the force is called the central axis of the tystem.
SI' I
I
I I =L
s: FIO. 3.
Let S S, Fig. 3, be the central axis of the system of mind pressures acting upon the sustaining surfaces of any kite. Then by the above cited proposition of mechanics our so-
called normal resultant pressure, N, will act in this line. (I t may be remarked that if the sustaining surfaces consist
of flat surfaces in parallel arrangement then the central axis
will be sensibly normal thereto, but in general there will be
a complex set of surfaces inclined to each other and probably curved; in this case S S, and with it the so-called normal resultant, N, will no longer be normal to the sustaining sur- faces; furthermore, N is designated a normal resultant simply because it is the resultant of a system of elementary wind pressures, each sensibly normal to its corresponding elementary surface.) Let C' be the assumed point of nppli- cation of N. I n single surface kites we may take C' where
'the central axis intersects the surface, but in cellular and in other kites we may take some other point. C' may, in any
case, be taken at any point on the caiatral axis and rigidly connected with the kite. Let C' N' represent the magnitiide of the resultant, N. It is not to be supposed, as is generally done, that a force like C' N' represents the total effect of the wind upon the sustaining surface of the kite. We must also recognize a possible couple To, which, in this case, as required by the proposition cited above, acts in a plana perpendicular
to C' N'. (When a force and a couple are in perpendicular planes the couple will be designated with a subscript zero.) The possible existence of a couple of this character is but rarely or never recognized in the discussion of wind pressures upon surfaces, but, as we shall see hereafter, it is a factor of vital importance in the mechanics of a kite. Let the couple be indicated on the drawing by the 2-formed character, thus :
,z appended to the action line of the force with which
the couple is supposed to be associated. The arrow points
indicate the direction the couple turns. As we shall presently
find, it is necessary to deal with couples which do not act in
a plane perpendicular to the force with which they may be associated, we shall adopt the convention of indicating this
fact by modifying the symbol, thus:
In proceeding further to represent in Fig. 3 the remaining
partial effects, e, f, n, v, and w, of the wind upon the kite, it
'
is to be noticed that each one of these effects, for example, the resultant pressure of the wind upon the framework, is the resultant of a complex system of forces and according to our fundamental principle each system is reducible to a single. resultant force and a couple in a plam perpendicular to the force. These resultant forces are not necessarily in the same
plane as the principal wind pressure, N, nor even parallel to such a plane. Moreover, we can not assign, a priori, any fully logical relation between the position and magnitude of anyone of these resultants and those of another or the resultant N. But this is not of any consequence, as will be shown. We are, however, able to affirm something as to the direction in which the forces act. This was done in a general way when the several effects were separately discussed. Obvioualy, the tendency of all these effects is to force the kite leeward. The pressures upon the framework, for example, may have a slight supporting component; an excess of pressure upon one side of a fin or neutral surface may push sideways at a high angle, but all the forces trend to leeward. If this be not so, then we are confronted with the absurd or impossible con- sequence that the wind, blowing against a body of assignable form, but uninfluenced by any other forces, as for example is the case if the body is cast free in the wind and has the exact density of the ambient air, is able to cause this body
to move stdadily to windward. "Aspiration," in its eeeence, is
not more nor less than this impossible consequence. The projected direction of the wind is shown at Wand in conformity with the foregoing we have shown, in Fig. 3, the projected positions, chosen at random, of the several partial
wind effects, each associated with a couple. We do not affirm that any of these couples necessarily have finite values. We need not say more than that each of the several forces is the result of the action of a complex system of forces applied at innumerable points of the kite structure and the principles of
mechanics require that the possible existence of these couples be recognized. As we have said, the relative magnitude of the forces can
not be accurately assigned. Compared with N the.others are all small, much smaller than shown, as the lines'in the diagram are made longer than logically proper simply for - the sake of clearness. The force, N, is by far the largest and most important force acting upon a kite, and each of the other forces may be regarded as a small disturbing influence superposed upon the primary effect N, which is due to the relatively simple pressure of the wind upon the sustaining sur- faces. We have thus shown by groups in Fig. 3 all the forces
due to the wind that may act upon any kite. The combined effect of these constitute the force we designated (1) in our original category. It will be remembered, too, that other forces were epumerated as acting on the kite, namely, gravity
(2) and the pull of the tail (7). We will not a t once combine all these forces into a single resultant, as we might do, but we will first combine only those shown in Fig. 3, in order that we may thus ascertain the total effect of the wind, and at the same time be able to show how the several small modify- ing and disturbing forces and their changes affect the final result.
By the principle of mechanics that any system of forces is reducible to a single force, acting through any point and a couple, we obtain for the system of forces shown in Fig. 3 the single force, R, acting through C', and the couple shown at 2'. The single equivalent force is found graphically by means of the well-known principle of the polygon of forces applied as indicated by joining to C' N' the broken dotted line made up of parts parallel respectively to the forces, e, f ,
n, v, and w, thus giving the resultant R'. Thus far we have not affirmed anything of the plane of the diagram, except that it contains the force N, we now assume that it was so chosen as to contain also the resultant R . It is to be noted,
APRIL, 1897. MONTHLY WEATHER REVIEW. 148
a couple among the forces acting upon a kite may be the
however, that the other forces are not necessarily in this plane nor parallel to it. The couple, Z', not only represents the effect of the several original couples, but also includes the couple resulting from the combination of the system of forces, N, e, .f, n, v , and w. The full identity of this couple
can not be determined a priori. We know, however, that its axis need not necessarily be parallel with the resultant R . The fact that the possible existence of a couple is recognized is sufficient for our present purposes and its whole effect will be taken into account when we come to establish the condi- tions neceesary for equilibrium. The following analysis brings out the effects of the small
forces we have called disturbing influences. Suppose all these forces are so small that they may be neglected, the resultant,
R', will then be sensibly coincident with and equal to N. No one of these partial effects tends to neutralize that of another. All combine to increase the angle included between Nand
R'. The magnitude of R' will depend, in a secondary and un-
important manner, upon the disturbing forces. The result-
ant, R', will be exactly the same as shown, no matter what positions may be chosen for the partial effects, provided their magnitude and direction remain the same. We have shown that the direction of action of all these forces mwt trend to leeward, and whatever finite values and rational directions may be assigned to these forces, it is clearly demonstrated that, by reason of their disturbing influence, the action line of
the total wind ef'ct on any kite is deficted away to leeward from
the direction of the remltant presmre upon the suetaining sur-
faces. Furthermore, the fact that these several forces can exist in nonparallel planes is sufficient to produce a final re- sultant couple, and, as we shall see hereafter, the presence of
This force, R, and the couple, Z, are the final desiderata in
~~
we will designate by Z". Let the two couples, Z' and Z", be :ompounded by the methods of mechanics into the single resultant couple, 2, which will be omitted from the diagram io avoid confusion of lines. In Fig. 4 the plane of the dia- gram is so chosen as to contain both forces, R' and R.
source of much mischief.
The complex nature of the forces resUltiW from the action of the wind upon all the parts Of the kite has necessarily in-
Si
I
our analysis of the action of the forces upon kites. We have
aimed to include in the derivation of this result every possible force that can in any way affect the position assumed by a
0
T
7%-
__ ____ . ____ ...______
~
-~ __ __._ _____ __ __ the direction in which the kite will tend to move under the
combined influence of all the forces. The function .of the
string is to restrain the kite and prevent this motion. The magnitude of the force R is the measure of the force with which the kite will "pull." The relation which R bears to the elementary and partial effects of the wind, viz, the partial effects, N, e, f , n, v , w, and the important forces, (2) and (7), has been fully shown in connection with the diagrams, Figs. 3 and 4. 2 is a couple which tends to turn the kite about an axis not yet determined, but not necessarily parallel to R. Having thus fully established, in the most general manner, ,the character of the combined effect of all the independent forces acting upon any kite, we will proceed at once to indi- cate the conditions that must be satisfied in order to produce
equilibrium. .
V.--CIRCUHSTANCES OF EQUILIBRIUM AND MOTION.
The problem presented for present consideration is : Given the force R and the couple Z, which repreeent the combined and total effect of all the forces due to wind and' gravity that may act upon any kite, to find how the string or kite line shall be attached in order to produce equilibrium and to explain the various movements of a kite. Let C' 0, Fig. 6, represent the resultant force R as found
in Fig. 4. Also, let Z represent the couple resulting from the combination of Z' and Z".
We have added to the diagram a pictorial representation of a common form of kite, in order to assist the mind in grasping the general relations we seek to establish, but this can not in any way limit or confine our conclusions to that specific form of kite, for the whole analysis of the forces has,
from the first, been conducted upon the most general linea possible, and the conclusions apply equally to all kites.
set forth the real character of that action. Like all systems of forces this system (l), as we have found, is reducible to a force, R (which in the present case we have arbitrarily chosen shall pass through the point C'), and a couple, Z'. Let us next combine all the forces acting upon the kite, namely, (l), (2), and (7). Conabination of all the forccs.-In Fig. 4, let C' M represent the total pressure effect, R , of the wind upon the entire structure of the kite. (The deviation of the force, R , in Fig. 3, from the normal resultant force, N, was unduly exaggerated
for the sake of clearness in that diagram. The line, R , in Fig. 4 is given a less pronounced deviation.) Let (2) repre- sent the projected force due to the weight of the kite, acting at its center of gravity, and (7) the force exerted a t the kite and due to the influence of wind and gravity upon the tail. The forces (2) and (7) are not necessarily in the same plane
as R . On the average the tail will generally dispose itself iu
a vertical plane, and the forces (2) and (7) might, therefore, be regarded as in the same vertical plane; but there is no advantage in thus specializing our analysis, and we will, there-
fore, regard the forces (2) and (7) as in different planes. T and g are the projected positions of the points of attachment of the tail and the center of gravity, respectively. All the independent forces that in general may act upon any kite are now fully recognized and represented in the diagram,
Fig. 4. The restraining pull of the line, as already pointed out, is not an independent force, but exists as a result of the
combined action of the other forces. This pull of the kite line is the force that is to put the whole system of forces in equilibrium. The combined effect of the forces, R , (2), and (7), may, as we have seen, be reduced to the single force, R (obtained by aid of the polygon of forces, as indicated), and a couple, which
144 MONTHLY WEATHER REVIEW. APRIL, 1897
The point C', it will be remembered, is the point of appli- cation of the resultant normal wind pressure upon the s w -
taining siwfaces. It is intended to show the point, C', dis- placed slightly from a perfectly symmetrical position with re- spect to the figure of the kite. The reason for this choice is
that in all kites unavoidab1e"defects of construction and
inequality of effects will causeIthe position of C' to be more or less eccentric.
f 0
/
FIG. 5.
I f now we assume that the couple Z=O, it is plain that the entire effect of wind and gravity on the kite and its tail is represented by the eingle force R, and obviously all that is necessary in order to hold the kite in equilibrium is that the string shall be so fastened to the kite that its action line (that is, the string prolonged) shall be able to coincide with
R. This is the condition shown in Fig. 6, and the point C', it will be remembered, is the point where the central axis of
the system of mataining pressures intersects the kite. It is also the point at which the central axi8 for the entire system,
including every force, intersects the kite. We thus recognize that Z=O affords a special case of possible equilibrium, yet, owing to the necessarily rare occurrence of such a condition, we give it no further notice. Equilibrium whea Z is finite.-Equilibrium is possible, but with some limitations, when Z is finite, and provided further
that its axis is not parallel to R. I n Fig. 6 let the axis of
the couple be inclined to the direction of R. By the niethode
.
FIQ. 6.
of mechanics this couple can be resolved into two component couples, one having its axis parallel and the other perpen-
dicular to R. Let the parallel component be designated Z,,
and the other one P. The forces of the latter couple will be
in a plane parallel to R. Now transform the couple P into one whose forces +R, and -R, are each equal to R. The
arm of the couple will then be x=P+R. Move the couple parallel to its plane nnd turn it about it.s axis until one of ita forces, --h?,, is directly opposite to R. These forces thus neutralize each other, and the whole systeni of forces has thus been reduced to the couple 2, and the single force R,, parallel and equal to R, and acting in the same direction but at a perpendicular distance, x=P+R, from it. The action
line of R, now coincides with the central nxia of the whole sys-
tem of .forces, including gravity and the tail. The string can now hold the kite in equilibrium only when Z,=O, and when the bridle or other device for fastening the string is so
arranged that the action line of the latter can coincide with
22,. These are the general conditions of equilibrium. The point C, a t which this action line of the force R, cuts the sur- face, is often eaid to be a center of pressure, but had we chanced to show in our picture a Hargrave kite, for example,
all our conclusions would hold just the same, and it is possi- ble that the action line of the force R, might then have failed
to intersect any of the actual surfaces. A center of pressure which has specific, definite, and assignable properties can not exist in the present connection, and we believe that on account of the obscure and indefinable conception of center of pressure generally entertained by experimentalists, inves- tigations upon the position of the so-called center of premsure, and especially measurements of the pressure of the wind upon arched surfaces, have often led to erroneous and even anoma- lous conclusions. In the preceding pages we have given an exhaustive and classified analysis of both the principal and the subordinate
or modifying forces which act upon any kite. We have en- deavored to show the general character and relative import- ance of the complex effects resulting from the action of the wind upon the whole etructure exposed to it. Finally, all the forces have been combined and the general conditions of equilibrium eetablished, as explained in connection with Fig. 6. Our final conclusions are not, however, dependent upon this elaborate analysis of the action of the forces, nor are
they affected in any way by faulty or questionable assump- tions therein unavoidably made. It is obvious, moreover, that the presence or the absence of any one or all of the sev- eral disturbing wind effects we have considered is not nece8-
sary to the final results. Any one or all may be infinitesimal and the resultant R and Z', Fig. 3, may be determined for the the forces that remain, just the same. So again, thepresenceof the tail is not essential. The force (7) may be made zero in Fig. 4 and the tail vanishes from the mechanics of the prob- lem. The resultant R of the remaining forces is found just the aame however. The very general character of the fore- going theory of the kite is thus exhibited. We have said that the forces which we deeignated e, f, n, v,
and w, were in the nature of disturbing influences that modify the results which would ensue if we had to deal only with the main pressure of the wind N upon the sustaining surfaces.
It is also a parent that the effects of the weight (2) and of
the tail (71 as shown in Fig. 4, are closely analogous in char-
acter to thoeeof the above mentioned disturbing effects. The force (2) in a very direct manner tends to diminish the mag- nitude of the resultant R, while the presence of the force (7) causes an angular deviation of the resultant R away from R and Nand in a leeward direction. In general, it is desired that what we may call the upward going tendeny of a kite shall be the greatest possible, while the tendency to go to leeward shall be the leaet possible ; that is, that the lift shall be a maximum and the drift a minimum. Now with the sustaining surfaces of our kite set in a particular attitude to the wind we get a force N having a certain lift and drift, whereas, owing to the
APRIL, 1897. MONTHLY WEATHER REVIEW. 146
presence of numerous disturbing effects, due to the action oi
the wind on the framework, etc., and these further aggravated by the effectsof the weight (2) and the tail (7), we are able t c realize out of this original and primary force N, only a modified resultant force R, which has a less lift and a greaterdrift thar
N. The kite is, therefore, by reason of the presence of thest disturbing forces, including the weight and the tail, less eflec. tive than it would be if there were nothing present but sua. taining surface. In the ideal kite we imagine all these dis. turbing causes absent. Such a kite is one, therefore, withoul
weight or tail, and which is made up wholly of mcstailtiy surf aces. Finally, the general conditions of equilibrium developed in the foregoing analysis are identical with those which, a1 the beginning of this paper, were shown to necessarily follow
as the result of the fundamental principles of mechanics.
VI.-EXPLANATION OF THE FLIGHT OF KITES.
In general, kites are restrained by only a single line fastened
either to some form of bridle or directly to some point of tht framework. More lines than one are, however, sometimer used, as, for example, in the case of dirigible kites. With two independent flying lines, either of which may be paid oul
or wound in at will, an operator is able to control, withiu certain limits, the position of his kite in either altitude 01
azimuth. With three independent strings fastened to the kilx at points not in the same line, the operator may cause thc kite to ascend or descend as well as fly to the right or thc left of the lee point. Such systems of lines are simplg equivalent to the well-known bifilar or trifilar suspensions, At a given instant of time, with lines of fixed length and foi
a certain position of the kite, the forces acting in the several Rtrings are always equivalent to a possible couple and a single force which acts through a determinate point and in a definite direction, which will be parallel to the axis of the couple, If two strings are used the determinate point will be in the liiie joining the points a t which the strings are fastened ta the kite or its bridles. With three strings the equivalent force will pass through a determinate point located in the
plane containing the three points at which the strings are made fast to the kite. If more than three strings are used the separate forces in them will be indeterminate in relation
to the force R,, but their resultant, when equilibrium pre- vails, must be in line with and opposed to R,. The use of more than one line becomes impracticable in many cases and limits the flight of a kite to moderate eleva- tions. The explauation of the phenomena of flightgiven below pro- ceeds upon the assumption that the kite is held in restraint
by a single line only. On page 147 will be found some further reference to effects resulting from the use of two or more strings. Let the reader imagine any kite he pleases launched into the ever-changing wind under the restraint of one string, its behavior is developed in detail as follows :
At each instant, and in each position, the whole effect of the wind and gravity is reducible to a single force R, and a
couple Z,, whose forces act in a plane perpendicular to R,. The kite at first will probably not be in equilibriuni because
2, is probably finite, but more particularly because the man- ner of attaching the string to the kite so limits and restricts the possible relations these two may assume t.hat coincidence between the action line of the string and that of the force,
R,, is impossible. Let the condition of affairs a t a given in- stant be as shown graphically in Fig. 7. R,, assumed to act a t C, is the equivalent force; 2, the resultant couple of the
system ; F i s a point through which the strhg constctntly pulls.
The bridle shown in the diagram makes the point F, bear a
fixed relation to the kite, yet that circumstance does not limit
Dur conclusions. I n whatever manner the string is fastened
to the kite, F is to be regarded as simply a point through which the action line of the string always passes, and it may be either fixed in relation to the kite structure, as in the illus- tration, or it may be capable of moving in some prescribed manner in relation thereto. When a bridle is made of a sin- gle bight of string, as in malay kites and those of other Forms, the point F is not fixed but is constrained to a circu- lar arc in a plane perpendicular to the midrib.
FIQ. 7.
Without affecting the system of forces in any way we may ~pply a t F the two equal and opposite forces, +R, and -Ro,
jach equal and parallel to Ro a t C. The whole system now :onsists of the  pull, or tension of the line, and the force
t R ,, both acting a t F, also two couples, namely: 2, aud ;he couple Z, consisting of -R, a t F and R, a t C. The iiovements it is possible for the point, F, to execute are lim- ted and are constrained by the string to be in a surface, the
wigin of which is at the fixed end of the string aiid the radius
rector qf a poiiit the catenary formed by the strilag. <f the ength and curvature of the string are iiivariable the eurjace of
:onstmint will be spherical. To find how the point F will nove, we resolve the force + R, a t F into two components, )ne, -t, acting in a direction along the string, the other, M, ,angent to the above-described sltrfnce of constraiiit, and in
he plane containing the string, and the force, FRO. M, then epresents the moving force and shows the direction in which ,he kite will be moved as a whole; -t, is a measure of the ,1111 on the line.
The force R,, acting a t F, has, it will be remembered, been eplaced by the two equivalent forces 31 and -t. This lattar orce calls into existence the tension in the string, that is,
bur force, (3). The depeiident character of this force (3) is
hus shown. The tension in the line, t, and the force, -1,
being equal and opposite, neutralize each other, and there-
ore, may be dropped from the system. There remain, then,
he force If, acting at F, and the two couples, Z and 2,. Let he latter be combined into a single equivalent couple, 2. In mder to understand clearly the ef ects of the couple 2, we hall find it best to diflerentiate the kinds of angular move-
nents the kite can execute. For this purpose let us imagine he point F, and the string near the kite to be stationary. hree distinct kinds of movements still remain possible to
he kite. For example, it may rotate about an axis passing hrough F, and parallel to the force, R,. Secondly, it may ip longitudiually, that is, in a fore and aft sense, about the
Ioint F. This motion corresponds to the pitoh  of avessel
APRIL, 1897
amid waves. Finally, the kite may rock sidewise, that is, careen or tilt laterally, either about the point, F, or about the line joining the fixed extremities of the bridle in case the kite i s bridled like a malay kite, for example. This rocking movement corresponds to the LLroll of a vessel. The axes of the two couples Z, and Z are already perpen- dicular to each other, hence the one couple can not have any component in a direction parallel to the other, and it will be most convenient to consider the effects of these two couples separately rather than to discuss the action of the equivalent
single couple, 2, as a whole. The axis of the couple Z, is, it will be remembered, parallel to the force R,. Let this axis be moved parallel to itself until it passes through the point F. In this case the axis will coincide with the line R, F-R,, Fig. 7. Now the couple Z, will make the kite turn its head
, either to the one side or the other about this line as a11 axis. At the same time the couple Z will impart to the kite a movement which in general will consist of both a tip, in the fore and aft sense, and a certain amount of careen or tilt
to one side.
It is very important to notice, however, that the movement of the kite, under the action of the couple Z, tends to bring the point C and the force R,, acting through C, directly in line with the point F. As soon as this occurs the whole couple Z vanishes from the system ; that is to say, its moment arm is zero. The kite will, therefore, tip and careen only a limited amount, and thus assume an attitude to the wind in which the action line of the force R, passes through the point F. It is equally important to notice that no matter how the kite may tip or teeter about the point F, this in itself can have no power to suppress or nullify the turning action of
the couple 2,. The manner in which the action of this couple is controlled will be discussed more fully hereafter. As a result of the foregoing resolution of the forces acting upon a kite a t a given instant of time, the following general theorem is demonstrated. When the system qf force8 acting ripon a kite is not i n equilibrium and is reduced to a single force, M, acting at the point F (the top end qf the string), and the
couple Z, then the ,force M will move the kite in a determinate direction tangent to a dejaite swface of constraint, and under the action of the couple Z the kite will, 1 :~ general, turn its head
to the right or left, will careen laterally to some exteiit, and tip longitudinally. When, therefore, any conceivable form of kite is cast free in the wind and is not in equilibrium under the restraint of its string, incipient motion of the kind defined in the above
theorem will ensue. At each moment and a t each point of its path, however, the kite is subjected to a new set of forces which differ from the forces of the preceding moment, not only because the wind may have changed, but also because the movement of the kite itself has placed i t in a new attitude with respect to the wind, in which new position the forces are cor-
respondingly different, even if the wind be constant. As the kite moves forward and rotates under the action of the couple Z it will, in general, unless it be a very poor kite, promptly assume an attitude in its path such that the couple Z becomes zero and continues sensibly so ; that is, its values, wheii finite,
succeed each other rapidly with alternate + and -signs, which means that the kite will execute partial turning move- ments in opposite directions, and these are often perceptible
as rocking, wiggling, tipping, pendulous motions. Moving along it.s path iu this manner the kite preseutly attains what
we will call ita steady positioa, which may be defined to be a position in which Y has also become zero, or, more exactly, a position in which the values of M are small and rapidly suc- ceed each other in widely diverse directions, while the values
of the couple Z,, if finite, chango signs rapidly. The kite does not, therefore, move far in any one direction, but is con-
.
stantly shifting ita position as a result of ite efforts to adapt ita position to the changing conditioris of the wind. In its steady position the relation of forces shown in Fig. 6 is ap- proximately established aud we are to imagine that the point
C a t which the action line of the string chances to intersect the surface of the kite, and which is often thought of as a
L L center of pressure, is constantly shifting from point to
point within n limited space in definite correspondence with the movements of the kite and the changes of the wind. I n the meantime the couple of the system takes on constantly changing values, but with quickly alternating algebraic signs, which means that the kite rocks and teeters on its bridle strings, turuing its head to the right and left as circumstances may require. Buch is the progress of a well-arranged kite from a starting point to its steady position, and such are the actions going on in that steady position which will become one of perfect eqiiilibrimi when the wind is perfectly constant in force and direction, provided further that Z,= 0.
A well-arranged kite is able to glide gracefully up to its
steady position simply because, owing to its symmetry and the corresponding equality of the pressures upon the lateral halves of the kite, the force M falls approximately in a ver- tical plane parallel to the wind direction, to which plane motion of translation must also be confined. The application of the general propositions of mechanics is sufficient to thus explain every phase of the movementa any kite may execute, nor does the theory make any distinc-
tion between kites that are good and those that are bad. Any object blown by the wind and restrained by a string is a kite within its meaning, and the darting and gyration, the diving and spinning, the downward as well as the upward or sagging flight a kite may execute when flown from a point well up in the free a i r -a l l its evolutions are equally explained and ac- counted for by the relations that are possible between the force Mand the couple Z. A study of the conditions that contribute to the stability and steadiness of a kite is a most important application of its mechanical theory. Before entering upon such a discus- sion, however, we will first trace out the conditions and caiises of the more characteristic evolutions of flight, and thus be the better able to understand how the flight may be con- trolled.
VI1.-ABNORMAL FLIGHT OF KITES.
Assuming the flow of the wind to be in straight lines and steady, we may regard normal flight to obtain when the kite
Bssumes and holds a steady position which is dead to leeward, that is, when the kite and the string are in a vertical plane parallel to the direction of the wind. Normal flight in a
steady position results when the force R, is in the vertical plane of the wind and 2, = 0. One of the primary and ultimate causes of abnormalitiee in kite behavior is to be found in the general and funda- mental weakness of Bite structures. The idea is popularly
current that yielding, flexible construction is of great d-
vantage. There is little or nothing in mechanics to sup- port such a claim, and there is much against it. Elastic bending aud yielding of a framework is much more likely
to take place un~ymmetricalZy than otherwise, and distortion of the prescribed figure of a kite is fatal to normal flight. Many of those sudden, unexpected, and unaccounted for evo- lutions of kites that sometimes occur result wholly from distortions made possible by too frail a construction. When
the conditions that are essential to stability have been ob- served. and are rigicllypresemxl in a given structure it will By steadily enough in all winds its strength is capable of re- sisting. Swerve, rlcaiate, sag, are terms we may use to designate cer- tain mild kinds of anonialous flight in which the kite tends
~PBIL, 1897. MONTHLY WEATHER REVIEW. 147
to steadily and continuously fly off to the right or left of the
direct lee point. This behavior is, to a moderate degree, char- acteristic of all kites, and is not of serious consequence unlese the tendency is greatly exaggerated and the kite flies far off
to one.side with a downward tendency, in which case the kite is said by some to '' sag."
F1a-8.
These actions all result from the same general conditione which are primarily a want of symmetry. Let us imagine we have a perfectly normal kite and that its steady position is dead to leeward. Let Fig. 8 represent the kite as seen from
a point in front (to windward) and in the plane of the kite. In normal flight the string and the force R, will be in a ver-
tical plane and appear as shown. Now suppose that instead of being perfect, some irregularity, of density for example, causes the center of gravity to be laterally eccentric, as at g.
All the wind forces on the kite must remain exactly as they were before, since a change of density in a portion of the kite can not affect them. The resultant of all the forces, except- ing gravity, will still be represented in the figure by a line in
the direction of R,. Let the force (2 ) at g now represent the gravity of the kite. This force combined with the force act- ing along R, will give rise to acouple and a resultant, still acting, as it appears projected on the drawing, along the direc- tion of R,. The couple will turn the kite, thus changing its
attitude with respect to the wind and introduce a whole new syetem of forces leading finally to a new steady position for the kite, which can not possibly be in the vertical plane of the wind.
By similar reasoning it can be shown that a lateral shift- ing of the point F, or a change in the point of action of the tail, or a lateral displacement of the point of application, C, of the resultant, R,, will cause the kite to fly to one side or
I
the other of the normal plane. The converse of this is also demonstrated, namely, that if a kite flies, moderately at least,
to one side of the lee point, it can be made to execute normal flight by a proper alteration of the position of the center of gravity, or a lateral shifting of the point F, if the bridle is of the rigid type, or by a lateral displacement of the action
line of the tail, if the kite has a tail. It does not necessarily follow that bad cases of deviation can always be corrected in this manner.
Sagging generally results from pronounced distortion of the prescribed figure of the kite when subjected to consider- able strain from the wind, or from original unsymmetrical construction and possible shifting of the loosely-fitted cover. The defect may not always be apparent or discoverable when the kite is hauled in.
Darting, diving, gyrating, pinning, are other terms that
may be used to describe evolutions of the kite. These differ from each other in the main only in degree.
Darting will be considered as referring to erratic and approx- imately periodic shiftings of the kite from side to side, but without losing elevation; a dart becomes a dive when the kite pursues a downward course for a time, but recovers and
R E V 4
ascends without executing a complete yyration. Spinning is suggestive enough of the pin-wheel evolutions a kite may execute while standing pretty steadily in one position. When the resliltant force, R,, of the system and the action line of the string are sensibly in the relation that corresponde
to equilibrium, as shown in Fig. 6, there will be little or no tendency for the kite to move. If now, however, the couple
2, has a finite value the kite will turn about an axis sensibly coincident with the string. This turning alters the attitude of the kite surfaces to the wind and results in a change of
position. If slight motion of this sort does not cause the couple 2, to change its value the movements of turning and '
translation will, of course, go on. Now darting results when
a considerable movement of translation must take place be- fore the couple has vanished, but it so happens that at the point in the path of the kite at which the couple 2, vanishes
the moving forceM has a Jtnite value, and the motion of trans- lation continues without any rotation for a short time. A new value of 2, with opposite sign and a different attitude of the kite with respect to the wind, however, soon results and causes the kite to retrace its previous movements in a similar but opposite attitude. Thus is produced the continuoue pe- riodic movement we call darting. M and 2, in this case are
never &imultaneowrly inappreciable, which is the condition for steadiness. A dive is the result of nearly the name condition of forces
as a dart, except that at the turning point, Mand 2, are nearly '
simultaneously zero. Dives often result from the eudden overthrow of equilibrium due to peculiar inequalities of the
wind and temporary distortions of the figure of the kite. The dive becomes a gyration simply because the algebraic sign of
2, does not change throughout the entire evolution ; if the kite does not attain the conditions corresponding to a steady position a t one gyration, it follows that 2, is still finite and has the same algebraic sign and that further gyrations are in-
evitable. The radius of the gyratory orbit is wholly dependent upon the relation between the speed of rotation produced by the couple 2, and the velocity of translation under the force M. When M produces a considerable translation during the time that 2, turns the kite but a little, then the orbit of gyration will be one of great diameter. When the power of the couple to turn the kite is great the orbit of gyration will be small. In the special case of M- 0 gyration obviously reduces to the evolution we call spinning, in which M is sensibly zero, while
Zo is considerable in magnitude and constant in algebraic sign. Thus is fully explained, from the mechanical point of view, every conceivable movement that any kite restrained by a
single line can execute, for any evolution must in character
be within the movements classified and explained above. Our discussion has throughout been a study of the action of certain specified forces, and the reader may use his own choice as to what particular kind of kite he conceives to be
the origin of the forces in question. When the conditione among the forces we have specified exist, the evolutions we have described will be executed.
Use of two or more lines.-Very little need be added to the foregoing general explanation of the flight of kites to adapt
it completely to the case of kites held in restraint by more than one string. The three degrees of freedom of angular movement possi-
ble to a kite restrained by a single string were fully explained
on page 146. Two strings attached at points either side of a
fore and aft middle line prevent all sidewise tilting or careen- ing of the kite, and hence only two degrees of freedom of angular motion remain, that is, the kite may still tip longi-
tudinally and may rotate under the action of the couple 2,.
Even this latter freedom, however, is under some constraint, due to the fact that the system of parallel strings will resist
148 MONTHLY WEATHER REVIEW, a, lW7
bein twisted up to a certain point; that is, any tendency 01
therewith, will be opposed by a greater and greater resistance until the two strings become crossed, whereupon their efficacy
to nullify the couple 2, practically vanishes. If a kite with two or more strings succeeds in executing one or two complete gyrations the strings will be twisted around each other, and much, if not all the efficacy of the system vanishes until the strings can be untwisted. We have found that two flying lines not only may serve ta suppress and prevent all lateral tilting of the kite surfaces, and place such movements under the control of an operator, but the strings also exert an opposition to free rotation of
the kite under the action of the couple 2,. This counter- couple action of the strings is not of sufficient power, in
general, to completely nullify the couple 2, under all circum- stances, but it is important to notice that it can contribute
to this effect. With two strings the kite will, for these rea- sons, be more steady in its motions than with one.
We pointed out, on page 146, that in its steady position a
kite must constantly vary its position and attitude with changes of the wind, and that a so-called center of pres- sure, C, constantly shifts from point to point within the
surface of the kite.. Now, when two strings are used, as shown in Fig. 9, it is obvious that the action line of the re- sultant force, R,, may take many different positions and still constantly pass through the line, F F, connecting the ex- tremities of the two strings. All these changes of R, (and
of the point C) can occur without any change whatever in the position of the kite, provided the direction of the force, R,,, remains unchanged. Changes within these limitations may often occur, and therein lies another reason why ti kite with two strings will fly more steadily than with one.
the f ite to turn in its own plane, or one sensibly agreeing
FIQ. 9.
When three or more strings are employed only one degree of freedom of angular motion remains to the kite, namely, it may turn its head to one side or the other under the action of the couple Z,, but even this, as already explained, is under some constraint from the strings. Furthermore, the action line of the resultant, R,, may now take any position whatever, provided that it always passes within the figure which in- closes the system of strings, and the kite will be under no
necessity of changing its position, except in those cases and only to the extent that the force, R,, changes its direction. We thus see that the use of two or more strings properly applied renders changes in the position of a kite far less ex-
tensive than they are likely to be with a single string. While the greatest possible degree of stability is thus obtained, yet the steadiness with which the kite flies will depend, as always,
upon the control that is exercised upon the couple 2,. No matter how stable the equilibrium may be, if the couple 2, can acquire appreciable finite values the kite is likely to fly in a restless, unsteady manner. For the distinction between the words  stability ) and steadiness, as here used, consult page 163.
The efficacy of dual or triple strings depends wholly upon
their not crossing each other. With. long, deeply sagging strings necessarily under different degrees of tension it be- comes impracticable to prevent crossing, and their use is feasible only in special cases and under considerable tension.
In order to discuss completely the conditions which can contribute to the equilibrium of a kite, it is necessary to un- derstand thoroughly the character of the changes which take place in the systems of applied forces when the kite changes its position or when variations arise in the force or direction of the wind. In this study we must consider both the force,
R,, and the couple, 2,.
VII1.-EFFECTS OF CHANGES IN WIND AND POSITION OF KITE.
a/
r Is
RO. 10.
Relation of forces.-The relations which the force R, bears
to the primary forces of wind, gravity, and action of the tail have been fully set forth on pages 142-144. These .may be briefly shown again, as follows : If S 5, Fig. 10, is the central axis of that important system of forces due to the pressure of the wind upon the sustaining mflaces of any kite and C an assumed point of application of the equivalent force, N, of the system, then the total pressure of the wind upon the entire kite structure, excluding the tail, will be equivalent to and may be represented by a force R and a couple Z, as shown ; the former acting through C, but inclined to N at a slight angle. Again, the total effect of all the forcee.of wind, gravity, and tail will be equivalent to a couple 2 and a single force repre-
sented by the line R, also acting through C. Finally, by resolving the couple 2 into components with axes respec- tively parallel and perpendicular to the force R, and then combining the perpendicular component with R, we get the force R, parallel and equal to R, but a t a greater or less dis- tance from it. We also get the couple 2, whose forces sat
in a plane perpendicular to R,. These resultante represent
the total effect of all the forces acting upon the kite. The force R, coincides with the central axis, So So, of the whole system of forces. It is possible, but exceedingly improbable, that in actual kites the forces N, R , R, and R, may all fall
APBIL, 1897. MONTHLY WEATHER REVIEW. 149
within one and the same plane. We will consider, however, only the more general case, that the forces are in different planes.
To assist in the further study of the results of changes i n the force#, we have added to the diagram the line A B and the
forces (2) and (7) of gravity and the tail, respectively. The line A B represents simply an edge view of a reference plane parallel, for example, to the longitudinal axis of the kite,
It may also be imagined to represent an edge view from the side of a single flat-surface kite. The diagram is, however, representative of any kind of kite if we regard A B as simply
a reference plane. The point C', the center of gravity, g, and the point of attachment, T, of the tail are, for simplicity,
all supposed to fall within this plane, but this is not essen- tial. If A is the forward portion of the kite, then C will be
nearer A. than g; at least, this is a more favorable relation than otherwise for the mont satisfactory flight, as will more fully appear hereafter. Having thus before us a clear picture of the relations be- tween R, and the forces of wind and gravity, we may formu- late the changes in R, corresponding to variations of the wind and alterations in the position of the kite, as follows: Variations of wind force.-Mere changes in the force of the wind will have little other effect than simply to alter the magnitude of the force N without changing either the posi- tion or the direction of the central axis S S. Furthermore, the force R' will experience a proportionate change of magni- tude, but without any important change in direction. These results will follow modcrate changes in mere wind force, be- cause, from what is known of the laws of wind pressure, its effects upon the surfaces of the kite, the framework, edges,
etc., and even the effects of waves and eddies, are, for slight changes, all proportional to the same power of the wind veloc- ity. For considerable changes of velocity the eddy and wave
effecta may possibly vary at a higher power of the velocity then simple pressure effects, and if pronounced waves and eddies nre present in a given kite, then the direction of the force R will probably deviate from the direction of N by greater and greater amounts with higher and higher veloci-
ties. Such kites are actually blown down and fly lower with very strong winds. The force (7) due to the tail is partly a wind effect, and for an increase of velocity (7) will be both increased in magni- tude and its direction made more nearly horizontal, and vice
versa. The weight of the kite (2) is constant, and obviously it follows from the above that even if the force R changes only in magnitude with variations of wind force, the total resultant
R will, in general, take a more horizontal direction and di- minish in magnitude with a falling off in wind force, whereas, with increased wind velooity, the resultant R will be greater in magnitude and deviate less from R. R, is always equal and parallel to R, but the distance, 2,
between them depends upon the couple 2. We will refer to the variations of x presently. Variations of the couple 2.-It will be remembered that the couple 2 is the total resultant couple of the whole system of forces when that system is reduced to the force R acting
through C'. The existence of this couple results principally from the action'of the forces of gravity (2) and of the tail (7), but there is also included in Z a small and very important couple effect, resulting from the action of all the wind forces. These form such a complex system of elementary pressures,
acting at every conceivable point of the kite, that it is utterly impracticable to produce that exact condition of symmetry required in order that the resultant couple of so complex a
system shall necessarily be zero. Or, rather this is only pos- sible by the introduction of certain forces which automatic- ally control the values of 2,. While this small couple effect
is included in 2, yet all the variations of 2 that are of im- portance from the present standpoint depend upon the moments of the forces (2) and (7) about the point, C', which,
as we have found above, is not changed in position by a mere change of wind force, per 8e. I t follows from this latter cir- cumstance and the constant magnitude of (2) that the mo- ment of the force of gravity remains constant under changes
of wind force as long as the kite holds its position. The couple Z will, therefore, also be sensibly constant, provided the kite has no tail. With a tail, the important variations of
the couple Z result from changes in the action of the tail cor- responding to changes in the force of the wind. Methods of determining, once for all, the magnitude and direction of action of the tail force (7) were fully described at page 141.
If the action of gravity on the tail is very small as compared with the wind effects, then the changes of the force (7) will be mostly changes in magnitude, the direction of action of the tail remaining nearly, constant and nearly horizontal, except
for the very lightest winds. If, however, t.he weight of the tail is relatively considerable, then the direction of action of the force (7) will change considerably with different inten- sities of the wind. It results from the foregoing that the values of the couple
2, corresponding to the possible values of the ,tail force (7), may vary considerably with mere changes of wind force;
thus, when the action line of (7) passes through C' the mo- ment is zero; if the tail blows out horizontally the action line in general will pass below C', and the moment will then have the opposite algebraic sign from the moment due to gravity acting at g ; while with the tail in the position shown in the figure, its moment and that of gravity hnve the same sign. We, therefore, see that, in general, even slight changes in the wind force alone can give rise to considerable and important changes in the couple 2. We are now prepared to consider further how changes in the couple Z affect the force R,. When 2 is resolved into two components, 2, and Z,, of which the latter is combined with the force R, in the manner already explained on page 144, we get the force R,, the position of which will be forward of
R; that is, nearer the A end of the kite than R. This is neces- sary because the moment of R, about an axis through the
point C', perpendicular to R, must be equivalent to the couple 2,. The perpendicular distance, x, between R, and R must, moreover, be x = 2, i R. Even supposing now that
2 remains constant under variations of wind force (and this can be the case with the tailless kites) it is obvious that z varies inversely with R. When Z is not approximately con-
stant its values, in general, will tend to increase, rather than otherwise, with diminution of wind force, because the force
(7) then takes a more nearly vertical direction, as shown at
T T, tending, on account of the rapid increase of leverage,
to produce a greater moment about the point C than before. The combination of circumstances that can cause the couple,
Z to diminish when the wind force diminishes is an improb- able and therefore not an important or general combination for our present purposes. From the foregoing it results that, in general, when the wind force diminishes, not only does the resultant R, diminish in magnitude and change the inclina- tion of its action line, but the whole axis, So So, shifts to a
new position nearer the forward end of the kite. To avoid confusion the distance x is greatly exaggerated
in the fi ure. R in general, is a very large force compared
with (2f and (71 and as a result the numerical value of x
and its variations will be small, except in very light winds.
It must be noticed that thus far we have considered only the changes that take place instantly in the relations between the forces whenever the force of the wind changes. The kite
itself it still supposed to retain its original position.
Hswd of remlts.-Briefly stated, what happens among the
160. MONTHLY WEATHER REVIEW. Amk, 1897
forces acting upon a kite when the wind pressure changes is this : (1) The pressure Nupon the sustaining surfaces simply changes magnitude, but retains eensihly i t 0 original action line and position ; (2) The total resultant R, not only changes in magnitude directly with the wind force, but when the wind force diminishes its direction deviates more from the vertical and its position shifts toward the forward end of the kite, and wice weraa. Finally, the couple Z,, which is sensibly zero when equilibrium prevails, in general will not necessarily undergo any important change. The couple may take on a
small value which may be either positive or negative, or it may remain zero. Any one of these resulk may occur in spe+al cases, and even in the same kite under diverse cir- cumstances.
FIG. 11.
Fig. 11 shows diagrammatically the relation of the forces
C', N, and R, with the kite in equilibrium under the restraint of the etring acting through the point F. A reduction in the wind force gives rise to the condition shown by the forces
C', W, and R,,. which are no longer in equilibrium with the pull of the string. Before we discuss the motions that must ensue let us first follow the effects of a change in directioit of the wind. Variations of wind direction.-Any change in the direction of the lines of flow of the air, whether to the right or left, or in their upward or downward trend, has the same effect upon the *nd forcee as if the wind were constant and the attitude of the kite had changed in a corresponding manner. These effects are identical only to the extent of the wind forces; the gravitation and tail effects will be changed in re- lation to the other forces if the attitude of the kite is changed. Any change whatever in the direction of the wind is accom- panied by such an alteration of the system of wind forces acting upon the sustaining surfaces of the kite that the cen- tral m.8 S S of that system changes its position, and pos- sibly its direction, depending upon the nature of the kite. .If the kite contains but a single flat plane, the axis 8 S will remain sensibly parallel to its original direction, but, in gen- eral, with kites whose surfacee are more or less curved the axis will not only shift its position, but will change elightly
its direction at the same time. If the change in wind direc- tion is wholly confined to a vertical plane, the point C', Fig.
11, will move toward the forward end of the kite when the angle of incidence of the wind upon the surfaces diminishes, and vice veraa. If the change is in the azimuth of the wind the
point C' will, in general, shift laterally as well a0 forward or
~~ ~ ~~ ~
lackward. In kites having automatic lateral stabilit (page
L certain longitudinal line, as explained more fully hereafter.
We find, therefore, that with a change of wind direction, not mly must the axie 8 8 shift its position and, in general, :hange it0 direction slightly, but the magnitude of the force
V increase0 and decreases according ae the angle of incidence )f the wind upon the sustaining surfaces ie increased or di- ninished by the change in direction. The distance that the ioint C' will move for a given change of incidence of the wind :an be computed in certain came by Lord Rayleigh's formula,
3quation (2), page 140. Its applicability, however, is exceed- ngly limited and restricted to single, narrow, rectangular, )lane surfacee presented to the wind in a special manner. rhe formula is utterly inapplicable to the general case of kites.
3imilarly, the relative magnitude of the force N correspond- ng to a certain incidence can, under certain conditions, be :omputed by Duchemin's formula, but in this case, likewise, ;he proper application of the equation is so exceedingly lim- ted as to be of little practical use in the present connection. rhere does not seem to be any thoroughly complete data wailable by which to determine, a priori, the exact changee in ;he position of (I' or in the direction and magnitude of N :orresponding to changes in the incidence of the wind.
We have already shown fully the effects on R, of a change
n wind force. Not only will these reeults follow when the
Kind changes its direction, but there will be added to them iimilar and additional effects resulting from the movement )f the point C' and the change in the direction of N. More- mr, important eecondary effects will arise from the pressure
If the wind represented by the series of disturbing forces we
lave designated e, f, n, PI, and w. Especially is R, the pree- m e upon the neutral surfaces when present, likely to expe- rience very considerable changes. The tail, also, will be down off in a different direction, and while the force (7)
m y retain the same numerical magnitude, the rnmnt of the rorce may change very greatly and in such a manner that the :ouple Z,, which is sensibly zero when the kite is in equi- ibrium, takes on a considerable plus or minus value, espe- :idly if the wind has changed its direction in azimuth. Following a line of reasoning similar to that employed on page 149 in establishing the effects on the couple Z and the iistance x, resulting from a change in wind force, it may be ihown that with a change of wind direction, R,, will be changed
n magnitude, direction, and distance, x, from the force R, which itse1f;shifts:its position. If the incidence of the wind
164) the point C' must always remain ve y nearly at 9 east in
apon the kite surfaces is smaller as a result of the change in wind direction, then R, will become maller and both Rp and the point C' will move toward the front edge of the kite to iuch positions, for example, a0 shown a t R', and PI, Fig. 12.
APRIL, 1897. MONTHLY 'WEATHER 'REVIEW. ltii
Except for the changes in Z,, these effects are identical in gen- eral character with those which result from a dimintbtion in wind force. Similarly, a change in direction which causes the incidence of the wind upon the surface to iwcrease can be shown to produce changes in R,, which are similar to those resulting from an imrense of wind force. The tendency of
the kite in the two cases to move will, however, be entirely different because the couple 2, is but little affected by a mere change of force, but may be greatly affected by a change of
direction. It must not be forgotten that the several forces shown in the diagrams 10, 11, and 12 are not necessarily in the same, or even in parallel planes. If R, is in the plane of
the diagram, then the other forces are in all probability not in that plane nor parallel thereto. It is not even probable
that R,, is in the plane of the diagram, because in that case the force would be in a plane perpendicular to the reference plane A B, and this could be the case under only the most perfect conditions of symmetry of the kite.
FIQ. 13.
We have now fully set forth the general character of the changes in the resultant force R, and the couple Z,, corm sponding to changes of either the force or direction of the
wind. These results may be shown graphically, as in Fig.
13, in which the force R, is shown directly in line with the kite string passing through the point F. The kite is, there-
fore, in equilibrium. Several lines, R,, R',, etc., show other possible values of the resultant force R, such as may logically arise from changes of the wind. While the disparity between
these forces is exaggerated for the sake of clearness of the diagram, yet nothing impossible is there shown, and it must be remembered that no two of these forces need necessarily be in the same plane. Moreover, these are momentary states
of the resultant R,, corresponding to sudden changes of the wind and obtain only until a change takes place in the position of
the kite. Theimportant feature brought out by our analysis, and to be noticed here,, is that when the magnitude of the resultant R, diminishes its action line takes a more forward position and a direction more inclined porn the vertical. On
the other hand the opposite effects ensue when the magnitude of R, increases. Speoial cases may arise where simultaneous changes of both direction and force may tend to compensate and offset each other, respectively; but these are rare and unimportant and not general effects. The ability of a kite to recover its equilibrium when dis- turbed by a change of wind conditions producing results such as shown in Fig. 13 depends upon the ability of the kite string to align itself with the resultant R,, whatever be the position and direction it may assume. Now, no two of
the possible values of R, necessarily pass through a common point. I n Fig. 13 there seems to be a tendency for the forces
to oross each other behind the line A B, but when we remem-
ber that these forces are not necessarily in the same plane, we see that it is improbable that even any two of the forces intersect. Since the action line of the force exerted in the
kite string must always pass through the point F, where the string is made fast to the kite, it is obvious that no matter how the string may change its relations to the kite (which we still consider to remain fixed, in posit.ion), or how the bridle may be arranged, it is impossible for the string, being in alignment with one value of R,,. to align itself with any other value, except on rare and improbable occasions, such as referred to above. These results will follow under the condition that the position of the kite remains fixed, and it is important that they be fully recognized. Thus, if a
kite were fixed in mid-air so that it could not change its posi- tion, a string could be tied to the kite in such a way that the string alone, without assistance from other means, could neu- tralize the action of the resultant force R, for one particular value only, but no matter how the lower end of the string were shiftedabout, if still fastened to thesame point at the kiteend,
it would, in general, be impossible for the string to neutralize
any other than one value of R, depending upon one condition of the wind. We thus see that mere changes in the angular
relations between the kite and its string tied at a given point, F, are, in general, each and all inadequate to reestablish an equilibrium that has been disturbed by changes in the condi- tions of the wind. Only by a complete change in the atti- tude of the kite to the wind can equilibrium be again estab- lished. Changes in attitude of the kite.-Whenever a state of equilib- rium is disturbed by any cause and in such a manner that the resultant fopce is diminish.&, the action line will, as we have shown above, make a smaller angle with the general plane of the kite and its position will be farther forward, as
shown in Fig. 14, in which R, is the resultant before and R, that after the equilibrium is disturbed; the gravity (2) and the tail force (7) are also shown. If the string is attached at F then the kite will be moved in the direction indicated by the force at M, which can be fully determined by the methods already described on page 145. The force M in this case and the motions that ensue are of such a nature that the kite in its onward movement assumes a greater angle of incidence to the wind. This causes the resultant pressure of the wind upon the sustaining sur- faces to increase and, as a further effect of the increase of the angle of incidence, the point of application of this force at C', moves away from the front edge of the kite to such a po-
FIQ. 14. FIQ. 15.
sition as C',, Fig. 16, which is supposed to represent the kite Bfter it has moved to some new position and has changed its incidence to the wind. The pressure of the wind upon the sustaining surfaces before moving and after equilibrium was disturbed is shown at C', N'. The pressure after the kite has moved to its present position will be represented by a line, C', TI, nearly parallel to C1 N' but farther from the front
162 MONTHLY WEATHER REVIEW. APBIL, 1897
edge of the kite. The forces C1, N', and R',, in Fig. 16, art shown in their original relation to the reference plane A B The absolute direction of these forces in the first or origina position of the kite WM, of course, more nearly vertical. The change in the position of the kite effects the followin# changes in the relations of forces :
(a) The pressure of the wind upon the sustaining surface1 is increased in magnitude and its action line cuts the refer. ence plane A B a t a point, C',, farther from the front edge oj
the kite, and, in general, C', will be nearer the center oj
gravity, g, of the kite. (b ) The forces (2) and (7), after the change in posi. tion, will be parallel and equal to their status before tht change. (e) In consequence of (b) and the shorter distance betweer
C', and g, the moments of the forces (2) and (7) will bt smaller than before the change of position.
(a ) Finally, M a general result of all the foregoing altera. tions the total resultant R", will be greater in magnitude, itr direction will be more nearly a t right angles to the reference plane A By and its position will be farther from the front edgt of the kite than that of the force R which is the state oi the total resultant just after the equinbrium was disturbed and before the kite changed its position. A similar line of reasoning may be employed to bring oul
the exactly opposite results that ensue when the equilibrium has been disturbed, because the resultant R,, Fig. 14, has a
greater value than R,. It follows, as a natural consequence of the above results that when the string is tied to the kite at some point, F, i n front of, that is, on the windward side of the kite, the force
M, whenever it has finite values, will tend to move the kite t c a psition in which the string and the total resultant R', will exactly align with each other, as shown in Fig. 15. Referring again to Fig. 13 it was shown in connection therewith that as long as the kite retained a fixed position the possible values of the total resultant R, were in such re. lation to each other that no two values could be neutralized by the string when attached to the kite at a single point, I? When, however, we permit the kite to move and change itf attitude to the wind and consequently its relation to the action line of gravity and the tail force, it is possible to find such a position of the kite corresponding to each possible force and direction of the wind that the resultant will alwayf pass through the point F.
3%
FIQ. 16.
Fig. 16 shows numerous values of the total resultant R, each corresponding to a certain force of the wind and a cer-
tain position of the kite. In the figure we can conveniently show the kite only in one position. The forces are, there- fore, shown only in their relation to the reference plane of
the kite,.and we must imagine the kite to take as many dif-
~ ~~~~~~
ferent positions as there are forces shown, each for each. For each force and for each position the string can align itself with the force, and thus neutralize it, because all the forces are shown passing through the common point F. These forces are not necessarily confined to one plane. The whole problem of the ability of a string to neutralize the total resultant of all the forces of wind and gravity acting upon a kite depends upon the circumstance that a relation, such as shown in Fig. 16, is possible between the values that
the total resultant R, can assume, no matter what the changes in conditions may be. The analysis is not limited to those circumstances that must prevail when the kite must fly up- ward or else dash itself against the ground, but finds its full application in the case of a kite string fixed to a point high up ill the free air, in which case the kite can assume without
interference any position the forces impose upon it.
We have thus fully set forth the general character of the changes among the forces acting upon any kite corresponding
to such changes as may occur in the wind effects. The gen- eral conditions under which kuilibrium may be established have also been shown. Center of pressure.-We have already taken occasion to point out the indefinite, unphiloeophical character of the expression L'center of pressure" as applied to wind pressures upon kites. The idea that the "center of pressure'y is a tangible concep- tion is so generally prevalent that a few additional remarks upon the same subject may not be amiss. The sustaining surfaces of a kite are the origin and generating cause of the principal wind pressures acting upon the structure. When these surfaces are simple and consist of but a single plane,
as indicated in Fig. 6, and if we think of the pressures upon this surface only, we may logically imagine all the elementary pressures concentrated and acting at a single point, as at C', which we may then regard as "a center of preamre." The point, however, as we have shown, has no other mechanical significance than that i t is an assumed point on the central
azis of that system of pressures which act upon the suetaining
surfaces. If any or all of the small disturbing forces which
we have designated e, f, n, eto., have finite values, then tmme other point, C", not shown, will be ('a center of presmre" for all the wind forces upon the whole kite. Again, if we include in our thought the attractionof gravityand the pullof a , possible tail among the forces acting upon the kite, then another point, C, not necessarily coincident with either C' or
c", becomes " a center of pressure," or, more logically, a center of forces. Finally, when the structure of the kite is complex, with superposed and separated supporting surfaces, we are quite unable any longer to follow out the analogy of a center of pressure or assign its location.
If the "center of pressure'y of all the forces acting upon the kite is the desideratum, then it is the point C, for example, in Fig. 6, or it is any one of the points in Fig. 13 where the forces R intersect the reference plane A B, or it may be any one of tke points in Fig. 16 where the forces there shown in- tersect a reference plane which may be chosen anywhere at random. None of these points can logically be designated as centers of presmre because they depend upon the action of the tail of the kite, and upon grbvity quite as much as upon wind pressure. The nearest approach to a center of pressure is tho point we have designated C', CYl, C',, etc., Figs. 6,10,11, 12,14, and 16, but even this is nothing but an assumed point of action of the resultant pressure upon the wtaining s ~r - faces. If anyone point of the kite is worthyof a special name on account of the distinctive properties possessed by it, then it should be the point F, whose properties are those of a real center of forces; that is, a point through which resultant forces constantly act. This is seen in Fig. 16. In another Bense, however, F is simply a point of support of a body sub-
jected to forces acting in an-upward-direction.
APIIIL, 1887.
IX.--CONDITIONB OF BTABILITY AND STEADINEBB OF KITE&
Before we attempt to show how to impart stability to a kite, that is to establish in general those conditions that will en-
abh the kite to maintain its equilibriiim under variations of
the wind and changes in the position of the kite, it is im- portant to clearly point out upon what stability depends and show when the terms stability and steadiriesa may have dis- tinct applications. When employed in the technical sense, stability has refer- ence to a state of stable equilibrium, and, as is well known, the
equilibrium of a body is stable whenever, if displaced from its
condition of rest, the body tends to resume its original attitude. When we deal with the equilibrium of ordinary objects sup- ported against gravity, or of vessels floating upon liquids, or of objects sustained in the air, as in the case of balloons, for example, the whole quality of stability results whenever a
certain relation is established between two forces, namely, the force of gravityacting at the center of gravityof the body and the resultant of the system of supporting forces. This relation is such that the applied forcea will always re- store a displaced body to its original attitude, whenever the disturbing cause is removed or will automatically establish a new position of equilibrium when natural changes occur in the system of applied forces. The case, however, is very different when we consider the equilibrium of a kite. Here,
as has been fully shown in the preceding pages, we must deal with two separate and distinct conditions of equilibrium, that is, not only must the restrainingpull of the string, which is the supporting force, neutralize exactly the resultant R, of the complex system of the applied forces, but the resultant couple 2, of that system must allso be zero at the same time. The entire independence of these two conditions of equilib- rium must be fully recognized. All the kite string can do toward bringing about a condition of equilibrium is to neu- tralize the force R,. Except for the very feeble and wholly in- sufficient resistance it may offer to being twisted or untwisted, the string is utterly powerless to oppose, in any way, the
action of the couple 2, which, when the kite is otherwise in
equilibrium, tends to produce rotation around the string as an axis. It therefore happens, in the case of a kite, that we may establish between the string and the resultant R,all those relations that in ordinary cases are sufficient for a state of stable equilibrium, but the kite may, nevertheless, dart about in the most restless and erratic manner imaginable, simply because the couple 2, may have finite values when R, is neutralized, and vice versa. We thus observe that the two conditions essential to the equilibrium of kites give rise
to twoindependent sets of conditions that must be separ- ately established to produce stability. One of these is the set of conditions, in consequence of which the string can neutralize the resultant force R, under all variations of the wind and changes in the position of the kite. The other is the set of conditions that must be provided in every kite, and by virtue of which the couple Z,, whenever it temporarily acquires finite values because of changes in either the wind or the position of the kite, is immediately reduced to zero in consequence of the incipient rotation resulting from its own action. The complete stability of the kite results only when both the above mentioned sets of conditions are fully established. Even under these circumstances, and notwithstanding that its equilibrium is perfectly stable in every detail, the kite
may not poise in that comparatively motionless manner which the word stability, when applied to kites, is intended
to imply. This paradoxical result will arise whenever the force R,, is neutralized with the kite in one position while the couple 2, becomes zero only with the kite in a different posi- tion. The kite will then execute those rythmical, periodic,
pendulous movements already referred to in the paragraphs on darting on page 147. Hereafter, therefore, if a kite poises in a nearly motionless attitude we shall say that it flies steadily, or that it is steady.
On the other hand, it will be an unsteady kite if i t constantly shifts about in a pronounced and restless manner.
It should be noticed that if the words are used in strict ac- cord with their technical meaning, the slightest disturbance will completely overthrow the equilibrium if it is unstable,
and an unstable kite would be wholly unable to fly at all. If, however, its equilibrium is stable the kite is bound to fly somehow, and it will fly steadily whenever the wind is steady and the force R, and the couple 2, are simultaneously neu- tralized. An important step is taken toward reaching aclear un- derstanding of how to control the flight of kites when the
two independent conditions for which stability must be es- tablished are thoroughly differentiated and when the dis- tinction between stability and steadiness is recognized. We need very badly two good words that shall be demrip- tive of the two independent sets of conditions by virtue of which stability results. In the one case stability is established wholly with respect to the action of the resultant force R,, and in the other with respect to the couple 2,. Recognizing that R, tends to produce translation only and that the couple
2, produces only rotation, the expressions, translational stabil- i t y and rotational stability, suggest themselves, and, in the ab- sence of better terms, these will be used. Translational stability.-Under this topic we shall discuss simply those conditions which will produce states of stable equilibrium with respect to the action of the resultant force R,. .
Whatever the force or direction of the wind may be there is always a corresponding position and attitude of the kite in which the total resultant R, of all the forces acting upon it will pass through a point F, Fig. 16, at a greater or less dis- tance in front of the kite surfaces. F may have many dif-
ferent positions with respect to the kite, each giving rise to its own corresponding positions of steady flight.
We have already fully shown that whenever the point F is at some distance in front of the surfaces of a kite the action of the forces when not in equilibrium will always tend to move the kite into a position and attitude in which the re- sultant force R, is fully neutralized by the wtion of the string. This being the cam, the equilibrium of the kite with respect to the force R, is, therefore, stable; that is, a condi- tion of translational stability is established simply by giving
the point F a position well in front of the kite surfaces. On
the other hand, the equilibrium will be w~table when F is
near to or behind the kite surfaces. A true state of neutral equilibrium is not possible in the case of a kite. As the point
F approaches nearer and nearer to the surface of a single plane kite the equilibrium presently becomes only partly stable ; that is, the equilibrium is stable for some conditions, but unstable for others; whereas, when F is quite behind the kite as a whole, the equilibrium is unstable for all conditions. Considering only the resultant force R, of the system of forces acting upon a kite, we' may therefore say that the sta- bility of the kite is assured whenever the point F (which may be defined to be the point at which the string ends and the kite actually begins) ie fixed, or at least constrained, in reference to the kite structure and is in front of the kite. Stability with bridles of diferent types.-The manner of at- taching the string to the kite, therefore, wholly determines whether the translational equilibrium will be stable or not. In a single surface kite F must be in front of that surface, but it need not necessarily bear a fixed relation to that sur- face. In kites with superposed surfaces F may be within or even behind the lowermost surfaces without wholly impairing
164 MONTHLY WEATHER REVIEW. APBIL, 1897
the quality of. stability. I n these cases an actual bridle may not be employed at all, but the string may be fastened di- rectly to some stick of thd kite, and the point F will then, in general, bear a fixed relation to the kite structure. Most other methods of attaching tbe string are generally reducible
to a condition that the point F is free to move in a circular arc, the plane of which is perpendicular to a line which itself bears a fixed relation to the kite and also contains the center of the arc. The bridle on a malay kite is a
typical bridle answering these conditions, but obviously other forms are possible. In whatever manner the string may be fastened to the kite the point F is either fixed in reference to the structure, or it is permitted to move in rela- tion thereto only in some constrained manner. Peculiar and rather complex cases of coDstraint arise in the case of the use of dual and triple strings. Different methods of brid- ling and restraining the kite give rise to different conditions under which stability is possible. The conditions of sta- bility and equilibrium with dual and triple restraining lines were presented on page 146. Other cases are briefly discussed
as follows: (a ) When F is a $.xed point.-When the string is tied to
a fixed point of the framework, as may be the case in cellular kites, or to bridles such as shown in Fig. 6, then, as we have found, the equilibrium will be stable, because a reasonable disturbance of the relation of the forces is accompanied by a
tendency to restore equilibrium. In the case of the kite this tendency may be nullified and made ineffectual because of the presence of a disturbing couple, but the equilibrium is of
a stable character nevertheless. (b ) When F is not ajzed point.-When a bridle is made of
a single bight of string, the point F will no longer bear a fixed relation to the kite, but will be free to move in a circular arc, as described above. For the equilibrium of the kite to be stable under these circumstances, it is necessary that the central axis of the whole system of forces shall always pass
through the fixed line joining the extremities of the bridle and fall between those extremities. This will be understood
from Fig. 17. The bridle is fastened at A B and the string at
FIQ. 17.
F. If the central axis of the system of forces pasees at one side of the line A B, as, for example, at C, or beyond either extremity of that line, even if on the line, as at C', then it is impossible for the kite to assume a flying attitude, simply because the action line of the string can not pass through the point C or C', as is necessary for equilibrium. In order that a kite with a bridle of this type may fly at all, i t is necessary to employ some artifice that will cause the
central axis of the whole system of forces to always pass through the line A B and within its extremities. We will discuss more fully the nature of such an artifice in the next
paragraph. This essential condition being established and
the point F being in front of the kite, then, as before, the equilibrium will be stable, because the forces will both move the kite and alter its attitude to the wind until the string and the resultant force Ro come into alignment, and thus produce equilibrium. That quality of the kite by virtue of which the resultant force Ro is made to always pass through the line A B, joining the extremities of the bridle, is generally known as its lateral stability. The lateral stability becomes a question that requires sepa- rate consideration only when bridles are employed which do riot restrain the point F laterally. When F is rigidly fixed in reference to the kite structure, then both lateral and longi- tudinal stability are equally provided for. Whenever F is fixed in onlya longitudinal sense, as in the present case, then longitudinal but not lateral stability is assured.
If it were customary to bridle a kite with a single bight of string having its two ends made fast to the framework at two lateral points, then we would need to separately consider longitudinal stability. This, however, is not the case, and we will only remark further in this connection that both lateral and longitudinal stability are simply component parts which together make up that quality which we have called transla- tional 8tabiGty.
A simple and comparatively effectual method of securing lateral stability in single surface kites is to dispose the lateral halves of the kite in such a manner that the surfaces form a
dihedral angle with each other. Jn this cam, when one side temporarily receives agreater pressure than its fellow the kite is tilted laterally, and the side moves into a position of lese inclination to the wind, the intensity of pressure thereon be- ing thus diminished, whereas, the opposite side is placed by the tilting in an attitude of greater inclination to the wind and receives a correspondiiig increase of pressure. A bal-
ance between the opposing forces on the lateral halves of the kite is thus preserved. In the malay, Eddy,.and many differ- ent forms of oriental kites the dihedral angle is produced mostly by springing the cross stick or sticks of the kite into.
a bow of slight depth with the convexity presented to the wind. The slight angle thus made possible between the lab era1 halves of the kite is, however, in general, made more
effectual by the bagging of the covering, which ie often iu- tentionally fitted somewhat loosely. In man cases the cross sticks are not originally bowed, but are su ciently flexible
to bend when strained by the wind, and thus form the neces- sary dihedral angle.
Even a t the best kites of this type can acquire by meansof the dihedral angle resulting from the bowed cross stick and the loosely fitted covering only slight degrees of lateral sta-
bility. Th9 idea is current with some that flexible, yielding frames, per se,give rise to remarkable degrees of stability. This is, however, hardly the case. It is true that the stability of a flat, or nearly flat surface, will be greater, provided the
lateral halves of the surface when strained can bend back- ward. The increased stability, however, never results from the mere yielding of the surface under strain, but comes about wholly from the dihedral angle effect made possible by the bending. If the surface were originally made rigid and of
the exact form assumed by the bent pliable surface then the stability of the two would be exactly the same with the further advantage in favor of the rigid surface that when the force of the wind falls off and the bent surface relaxes, i t loses part, at least, of its stability, while that of the rigid form is
fully preserved. In any case, all kites which depend upon such expedients for lateral stability are only slightly stable
flyers. Vastly better methods of contributing stability are well known and will be mentioned farther on. The mechanics of lateral stability is readily understood
from Fig. 18. Let BD, at (a ), Fig. 18, represent an edge view
APBIL, 1897. MONTHLY WEATHER REVIEW. 166
from the front of a kite composed of but asingle plane surface. Let the two extremities of the bridle be tied at points A A, both represented a t A, one of the points being directly be- hind the other. We may, therefore, regard A as the end view of a longitudinal axis about which the kite is perfectly free
to tilt laterally. A slight inequality in the amount of sur- face in the lateral halves of the kite, or a temporary intensity of pressure of the wind upon one half greater than on the other will cause the resultant R, of all the forces to act through a point C at one side of A. The force R,, therefore,
@J R
YIQ. 18.
acts upon a short lever arm, C A, and tends to rotate the kite around A as an axis. Any tilting of a plane surface in a strictly lateral sense, that is, about an axis such as A, which is considered to be in a plane parallel to the lines of flow of the
wind-any such tilting affects all parts of the surface alike, and the resultant force R, will continue to act through sen- sibly the same point, C, whatever the position of the plane may be. Lateral stability of a perfectly flat surface sup- ported in the manner now supposed is, in fact, impossible.
If, however, the lateral surfaces are inclined to each other, 80
as to form a slight dihedral angle, as at (b ), Fig. 18, then,
whenever the resultant 5, acts at some point C not in the axis A, the force R, will still tend to rotate the surfaces about
A, as before, but, when the surfaces have moved to some new position, as shown by the dotted lines, the portion B A will have a less and A D a greater inclination to the wind than previously, and in consequence, there is always some attitude of the surfaces in which the resultant R, will pass through
the axis A. In the meantime, or subsequently, the kite, as
a whole, will be translated to a position as shown at (e),
Fig. 18, where the plane of the bridle, A F, contains also the force R,, onecondition essential to equilibrium. Other means than the dihedral angle may be resorted to for the purpbse of
producin lateral Rtability. Some are indicated in Fig. 19,
(a ) to (9 i
FIG. 19.
Surfaces set edgewise to the wind, but perpendicular to the sustaining surfaces, have been called LL neutral surfaces  in the preceding pages. These, in general, contribute lateral sta-
bility as well as serve other functions. At (a), Fig. 19, we have shown a sort of keel attached to the sustaining plane B D. Under normal conditions the pressures on the opposite sides of such surfaces will be equal, but if the structure is tilted about a longitudinal axis, as at A, then one side of the neutral surface begins to experience an excess of pressure that tends
to oppose the tilting, and a condition of attitude of the sur-
REV^
faces will, in general, be possible in which the cerrtral crzis of the entire system of forces passes through the axis of support A A. There is no essential difference in the mechanics of the question for the case in which the neutral surface is
made in the form of a dorsal fin, as shown at (b). Similarly, neutral surfaces at the sides of the supporting plane B 0, as shown at (c ), also contribute lateral stabili
the addition of a single surface, B D, as shown at (d t ;fz duces one of Hargraves rectangular cells. It is not at all remarkable that the cellular kites should be so ve y stable
when we notice the relatively great extent of neutral surfocee in a cell and compare the considerable power of resistance these surfaces can offer to lateral tilting of the kite with the corresponding feeble opposition possible in the case of a very flat dihedral angle or the even slighter resistance possible in
those kites which depend upon the uncertain pliability of lateral wings. It is but a single step from the dihedral angle
at (e ) to the diamond cell a t (f ), and the trapezoid at (9 ) is another obvious oombination, which, by the way, ~OEWESEM
remarkable stability. Each of these forms possesses the
quality of lateral stability, because whenever the resultant force R, does not act through the longitudinal axis of sup- port, as A A in (d ), (f ), (g), Fig. 19, the kite will be tilted and a new force called into action, which, with a certain amount of tilting, will be just sufficient to restore equilib- rium; that is, to cause the total resultant of all the forces to pass through the axis of support, A A. A single cell, bridled in the manner shown at (d), in steady winds is a
perfectly stable structure, both laterally and fore and aft. It will he 80 even if the string be tied at some single point be- tween A and A. In general, however, a single cell can not be made to fly well, or even at all, but this is chiefly because the resultant couple 2, of the system of forces is generally of finite value, and there is little or nothing in the relation
of the forces of the system to automatically neutralize this
couple. We have thus fully analyzed and discussed first the effects
of changes in the wind and position of the kite upon the ap- plied forces, and then shown in detail all the essential con- ditions and circumstances under which the equilibrium of the kite can be stable with respect to the resultant force R,. To this end it is necessary only that the point F at which the string ends and the kite begins shall be in front of and not too near the supporting surfaces of .the kite. If F is not Bxed in a lateral sense, then we must establish conditions of automatic lateral stability, as described in detail above. By these methods the so-called translational stability of any kite may be fully and certainly established. It now remains to show how the resultant couple 2, may be neutralized when
it attains finite values owing to changes of the wind, etc.
Rotational stability.-When kites are in equilibrium the aotion line of the string is then the central a2$s of the whole system of applied forces, and, ordinarily, stands roughly at right angles to the general plane of the kite surfaces. The
Bxis of the couple 2,is not only necessarily parallel with this central axis, but in kites coincides with that axis iteelf because the kite is free to turn only thereon. Our purpose is
now to discuss the means that may be employed to prevent
or to properly regulate and control this turning, that is, to establish rotational stability. If it were possible to make a kite such that the resultant couple 2, of all the forces would always be zero, no matter
how the conditions of wind might change or how the kite might be turned about the string as an axis, then we might say that the rotational equilibrium of such a kite is perfectly neutral. On the other hand, if the couple 2, retains a finite value, no matter how much the kite may turn under the action of the couple, then we should say its rotational equi- librium is unstable. If, however, as the turning of the kite pro-
166 MONTHLY WEATHER REVIEW. A P ~, 1897
ceeds under the action of the couple Z,, the latter, as a neces;
sary result of the turning, is reduced in magnitude and made zero, then the rotational equilibrium is stable. Function of the tail.-The whole useful function of the tail is to suppress and govern the action of the couple 2,. In
every other respect the tail is a useless and even objectionable appendage to the kite, but if it is properly adapted to its pur- pose the couple Z,, when momentarily of finite value, is im- mediately reduced to zero, as a result of the action of the tail under the incipient rotation which ensues. This is readily
understood from Fig. 20.
P
: =\. ;
ypy!+ --<-..
pJp@
: It I --n
D L Y z A B
FIQ. 20.
A t A is shown a kite as it may appear when viewed ex- actly in line with the string, which intersects the kite at C. The force (7), due to the tail and the couple Z,, are also shown. There being nothing initially to oppose the action of the couple, when its value beconies finite, the kite turns a little, but immediately that this occurs the action line of the tail force, (7), passes more and more to one side of the central axis of the system of forces, as shown at B, Fig. 20. The
force is thus made to act a t the end of a short lever, or mo- ment arm (I D aDd if the tail be adequate for its purpose, a
slight turning of the kite causes such new relations between the forces, including that of the tail, that the resultant
couple Z,, which had momentarily attained a finite value, is reduced to zero. The action of the tail is in these cases
equivalent to a couple as may be easily shown by introducing at C two forces, + (7) and - (7), equal and opposite to each other, and equal and parallel to the tail force (7). The lat- ter force and the force -(7) at C, constitute a couple with the moment arm C D. The change in the position of the kite, resulting from the turning, has quite disturbed the equilibrium of the force R,, but this is of no consequence, since the relations of this force to the string are those of
stable equilibrium and a new position of rest will be quickly assumed. The important point is that in consequence of the action of the tail the couple 2, is prevented from maintain- ing a finite value for any considerable time. It must be
noted, furthermore, that the couple-action of the tail does not have an independent existence. It is called into action by the general couple 2, of the whole system of forces. I f
the latter couple increases or diminishes, or changes sign, so
does that due to the tail. The existence of the couple 2,
results, in the main, from imperfections in the symmetry of the kite and from temporary inequalities in the wind forces at different points over the kite ; also, from changes in wind direction. 2, is therefore, not permanently a very powerful couple, and, in consequence, a slight shift of the kite, of the nature shown at B, will, as a rule, efYect its elimination, pro- vided the force (7) is relatively considerable. The tail pro-
duces the maximum possible couple when in the position shown at D, but such an extreme turning of the kite is, of couree, fatal to satisfactory flight. The usefulness of the tail depends upon its power to neutralize the couple -2, with only very small angular turnings of the kite. Gravity may
also assist the tail in causing a couple to vanish. This is
generally the case, because in most kites the centra2 axis is nearer the forward edge than is the center of gravity. The
action of gravity in this connection, and some further facta
concerning the effect of the tail will be understood from an examination of E, Fig. 20, which shows the side view of the
kite and etring. It is a well-known result that gravity, act- ing at a point, g, on a body like A B, free to rotate about an axis such as S S, will make the body take a position about that axis such that g, when the kite is in its normal dying
attitude, will be the lowest possible, or very nearly EO. A couple such as Z,, which tends to produce rotation about the axis, S S, must, in turning the kite, carry the point g to a higher position, and to this extent the turning will be op-
posed by gravity. The opposing power of gravity has, in gen- eral, however, only a relatively feeble moment of little im- portance. Our only object has been to call attention to its existence, as the principle may have great importance in the flight of birds and flying machines of considerable weight.
It should be noticed that, in general, the direction in which the force (7) due to the tail acts makes an angle below the hori- zontal, as shown at E, Fig. 20. In considering the effectiveness of the tail to nullify a couple, the force (7) must be resolved into two components, one parallel to the axis S S, and the other in a plane, m n, perpendicular to S S. The moment of this latter force with respect to the line S S is the only effective part of the force (7). The most effective tail is, therefore, one which would dispose itself in the plane m n, in which case the full power of the force (7) is effective in sup-
pressing rotation. However necessary and useful the tail may be in contributing to the rotational stability of a kite, it is very plain that its force, (7), as may be seen in Fig. 4, teuds to very strongly deflect the resultant R away from the normal N, and' thus impair the effectiveness of the kite. If the pull of the tail could be upwardly inclined i t would be
much better, as there would then be exerted an advantageous lifting tendency. But this is, of course, impossible as a steady effect, unless, for example, the tail was rendered buoyant by inflation with gas.
Moat eflcient tade.-Now that we know the exact function of the tail we are prepared to state the conditions for maxi- mum effectiveness. In the first place the necessity for a tail arises from the fact that, owing to some organic, permanent,
and constant defect in the kite, the structure is uasymmetri- cal, and in consequence the couple 2, still exista when all the
conditions of equilibrium are otherwise established. This couple may also assume relatively large temporary values when the wind suddenly shifts its direction in azimuth, as
explained on page 150. The power of the couple will, on the average, be constant for a given wind force, although its monientary values will change more or less with inequalities
of wind pressure on different parts of the structure. The power of the couple will also irlcrease and decrease with the wind velocity. Now a tail which depends largely upon ite weight for the force (7) will exert a nearly constant pull. In light winds it will pull the kite down and be unnecesaary,
while in heavy gusts it will be inadequate to overcome the increased moment of the couple 2,.
The most efficient tail, therefore, is one without weight and
whose pull depends upon the pressure of the wind. When properly adapted to a given kite for one wind velocity i t should answer equally for other velocities, at least within
comparatively wide ranges. Tails made of very light pervious cloth in the form of Erustrums of conee distended at the barn by a large and a t the'summit by a relatively quite small ring of aluminum
wire or rattan are seemingly the most efficient tails. Their weight may be reduced to the minimum and the pull they sxert is nearly proportional to the wind force and nearly horizontal in direction. Tails thus afford a means of controlling and regulating the magnitudeof the resultant couple 2, of the system of ap-
plied forces. The rotational stability may, however, also be
APRIL, $897. MONTHLY WEATHER REVIEW. 167
established without the use of a tail, and this brings us tc
the consideration of a new class of kites. Tailless kilea.-From the mechanical point of view thc presence or absence of a tail in the case of a kite is simply I
question of the magnitude of the force (7). Thus far we haw supposed the force (7) to have a finite magnitude; we wil
next consider those cases in which (7)=0. This suppositior that (7)=0 is all that is necessary to adapt the whole fore going theory to tailless kites. Referring to Fig. 10, showing the relation of the forces or
a kite, it will be recalled that the central axis, So Bo, of thc whole B stem of forces is parallel to the force C' R, and sep
the couple Z. If now the force (7) is zero, then the line (
R of the polygon of forces, which line is equal and paralle:
to (7),will vanish and the resultant of all the forces that re.
main will be represented by a line C' Q instead of C' R. Thf couple associated with the new resultant, C' Q, will, of course have a value depending upon the forces of wind and gravitj
on the kite alone, exclusive of the tail. Likewise, the neM
central axis will be parallel to C' Q and a change will OCCUI
in the distance, x, corresponding to the difference betweex
the new and the original couple Z. Finally, the resultani muple Zo acquires a new value and may even change its sign
We thus find that if we omit the tail from a kite, othei things remaining the same, the total resultant of the applied forces may be greater in magnitude, will assume a mor
nearly vertical direction, and will shift its position. Suppos. ing the kite is able to fly either with or without a tail, it fol. lows from the above that, if the incidence of the kite surfaces
to the wind and other things be kept the same, the kite will often pull harder and always fly higher without than with
the tail. Removing a tail from a kite or adding it theretc will, in general, cause the kite to assume a different angle 01
incidence when equilibrium is attained. This may be offset by a corresponding change in the bridle or it must at least be taken into consideration if we wish to compare the be- havior of the kite in the two cases. Such are the general relations between the forces acting
upon a kite with and one without a tail. Our particular pur- pose at present, however, is to show what contributes to the rotational stability of tailless kites. The rotational equilibrium of all flat surface kites, of which the form shown in Fig. 6 may be regarded as representative,
is always likely to be quite unstable without a tail. The couple Z, when finite is free to turn the kite about the string
as an axis and generally to an indefinite extent. It is con- ceivable that the symmetry of such a kite may be so perfect that the couple Zooan not have permanent finite values under steady winds, or that the values may be so small that the feeble opposing couple resulting from the action of gravity
may suffice to prevent rotation. These are not probable cases, and while a very slight degree of stability may thus be gained, yet, in fact, as every one knows, a kite pf this type with insuffi- cient tail is likely to dart about in the most unsteady man-
ner and, if without, any tail whatever, will often spin about the string as an axis a t a very high velocity.
It is poseible to contribute a slight degree of rotational sta- bility to a flat surface tailless kite by adding weights to the aftermost end of the kite, and this method has sometimes beeu adopted. It increases thegravity of the kite, and shifts the center of gravity to a point farther aft, and therefore, as
a rule, farther from the axis of the couple Z,, which is the cen- tral axis of the system of forces. Both these effects increase the relatively feeble opposition that the action of gravity on
the kite offers to its being rotated out of a position in which the center of gravity is the lowest possible. Another method is more generally employed, which also
gives only a slight degree of stability. It consists in arrang-
arated T rom it by a distance, 2, dependingupon the nature oj
ing loose flaps of cloth or covering .material near the dter end of the kite. The kite itself is sometimes made flexible in the aftermost portion by shaving the sticks down thin and
attaching the bridle strings at some distance from the ends. In either case, these flexible portions flap and flutter in the wind to a greater or less extent, and thereby exert a small
force approximately in the direction in which the air is flow-
ing over the surfaces. Some mention has already been made of this point under the discussion of waviness and fluttering
on page 141. The force, in fact, is not unlike the regular tail force (7) and acts exactly the same, but in a less powerful manner to suppress the couple Z,, that is, to keep the kite headed to the wind. Loose fluttering appendages of this character are con-
spicuous in many different forms of Chinese, bird, and butterfly kites, and in still other forms of oriental kites. In many cases the flaps are so disposed that under the pres- sure of the wind the surface is formed into dihedral angles near the aftermost portion of the kite, and, even if the loose material does not flutter and vibrate, the action of the side
thrust of the wind on the slightly inclined surface at the back end of the kite tends to prevent rotation of the kite about the string. Such means as we have just discussed will suffice to control flat surface kites in moderate winds, but will generally prove inadequate in strong gusty winds, and in such cases tails must often be called into requisition. Dihedral angle.-We have already shown that a degree of lateral stability is imparted to a kite by placing the lateral halves of the surfaces at a dihedral angle with each other. This arrangement also produces a degree of rotational eta- bility. All those kites of which the malay and Eddy kites are typical, including many oriental tailless kites whose framework is pliable and yields under the strain of the wind
so that the lateral surfaces form a dihedral angle, acquire
a degree of rotational stability as a result of this disposition.
It is well known that these kites fly best when the covering material is pliable rather than stiff. It should be tense and
fit closely over the forward portion of the kite and be loose
and form concavities in the rear part. Under the pressure of the wind the loose and pliable covering forms a sort of keel, especially pronounced along the after portion of the midrib or upright stick of the kite. If the kite turns a little under the action of the couple Z, when ita value becomes finite, the keel must swing sidewise against the wind and a
very little turning under these circumstances reduces the couple Z, to zero. It is especially because the keel is deeper and more pronounced when the cloth is pliable and loose, that all these kites then fly better. In considering questions of rotational stability in kites of the malay or Eddy type i t should be noticed that the action line of the string, that is, the central axis of the system of forces and also the axis of the couple Z,, cuts the surfaces
a t a point only a short distance to the rear of the cross stick
3f the kite. This position is relatively close to the front
3dge of the kite, whereas, the keel-like disposition mentioned rtbove is most conspicuous well toward the after end of the kite, a relation which is most effectual in producing rota- tional stability. Boynton jn kites.-The use of genuine keels or fins, espe-
:ialIy if they are cut away forward and are present only in the zfter parts of the kite, are still more effectual in suppressing
thR couple of the system of forces, and they may easily be manged so that the kite to which they are attached can fly without a tail. The same remarks are equally true of the
bipolar kite.
Bi olar kites.-In these the central spine is extended aft be- yon I f the body plane and carries both a vertical and hori- tontal plane. The vertical plane, on account .of its consider-
188 MONTHLY WEATHER REVIEW. APBIL, 1897
able distance aft from the central axis of the-forces, afford6
a very powerful control upon the rotation of the kite undei the action of the couple 2,. Cellular kites.-We have already remarked that the greal stability of the Hargrave kite results from the relatively great extent of neutral surfaces composing the lateral sidec
of the cells. The presence of these not only establishes a
great degree of lateral stability, as already explained, bul the neutral webs in the aftermost cell also act in the mod advantageous manner to produce rotational stability, as i f
readily understood from what has been said above concern- ing the action of keels, fins, etc. Longitudinal stability re- sults both from the superposition and the separation of t h e
supporting surfaces and also from the use of a bridle, causing
the int F, at which the main line is attached, to fall in fronl
Steadiness.-The quality of steadiness in flight depends, ac
we have seen, upon the circumstance that the couple 2, and
the resultant R, are simultaneowrly neutralized. Very mang
forms of kites are deficient in rotational stability, and slighl inequalities of wind force very easily turn them considerablg about the string as an axis. This alters the attitude of the kite to the wifid, and thus causes the kite to shift its position
in order to reestablish translational equilibrium. The greatesl steadiness results when the rotational eqwilibriwn is the moa1
stable. This is why kites with an abundance of tail powex fly BO much more steadily than those with deficient tails. The foregoing exposition of the action of the forces 01
wind, gravity, and the string upon kites suffices, we believe,
to fully explain their behavior under all circumstances. The analysis of the conditions upon which stability and
steadiness depend embraces every t e of kite, not only the
which, in many cases, are only adapted to sport and often fly a t relatively low angles of elevation, but includes also the malay, the fin, the bipolar, and all forms of cellular kites.
We have aimed in our presentation of the mechanics of kites
to state the generic relation and interaction of the applied
forces. The principles thus laid down have a very wide and
comprehensive application. Numerical data.-In regard to the questions raised in Mr. Chanutes letter at the head of this memoir as to the positions of the center of gravity, center of pressure, and best point of attaahment for the string, we may repeat here a statement
already made that, in general, it is better to have the center of gravity fall farther aft than the so-called center of pres- sure, as gravity then assists in producing rotational stability. This position generally comes about without design on the part of the maker, but it is conceivable that a kite with a tail might fly with the center of gravity even forward of the center
The indefiniteness of this conception,  center of pressure,
has already been fully discussed. A central azis is the idea to be kept in mind in this connection, and if we include all the forces of wind, gravity, and tail, the central axis always coin- cides with the action line of the string when the kite is in equi- librium. As we have shown, the position and direction of this action line changes with every change in the force and direction of the wind and variation in the attitude of the kite. The central axis may be completely located in a given case
by simultaneous measurements of the incidence of the kite, ita angular elevation, and the inclination of the kite line at
the reel. The axis always passes through the point F, at which the string is fastened to the kite. Methods of making the simultaneous measurements for this purpose have been fully described by the writer in Kite Experiments at the Weather Bureau, pages 82 to 87 (reprinted from the MONTHLY WEATHER REK~EW, 1898. Vol. XXIV, p. 240), and it seems
unneoessary to -peat the descriptions here.
of t E e kite.
numerous odd forms constructed by t Kp e Chinese and Japanese,
of pressure.
The best point of attachment for the string depends partly
upon the use to which the kite is to be put. Changes in the point of attachment cause changes in the angle of incidence which the surfaces assume to the wind. For reaching great altitudes certain angles of incidence are better than others. This subjct has also been fully discussed in the pamphlet referred to above, and the results will be found on pages 98
to 103 (see MONTHLY WEATHER REVIEW, Vol. XXIV, p. 250).
To repeat these demonstrations here will greatly extend the limita of this paper, and appears unnecessary.
X.-NUMERICAL DATA.
The gathering of numerical data derived from actual obser- vations made by sound and accurate scientific methods is a
matter of considerable labor and difficulty. Work of this character done by the writer for the Weather Bureau, in order
to determine the efficiency and availability of kites for reach- ing high altitudes, has thus far extended only to various types of cellular kites, which, as a class, all things consid- ered, are probably the most stable and availableof all. Very complete numerical data respecting these kites .have been published in the  Kite Experiments, pages 66 and 87 (see
MONTELY WEATHER REVIEW, Vol. XXIV, p. 244). The fol- lowing additional example is cited from recent work :
Higmw mkgdw &.-Each cell 78 inches from side to side; 32 inches from top to bottom surfaces. The cambric cloth bands, includ- ing the flat, spruce sticks (t by # inch) in the form of rectangular frames! with which the cloth is ed ed all around, are 25.3 inches (2.1 feet) wide. The front cell has a %ird sustaining surface midway be- tween the upper and lowersurfacesof thecell. The two cells are connected by a longitudinal truss. The upper and lower spines are spruce sticks (4 by 4 inch) spread out by four columns and properly braced by wire ties. The extreme length over the cells is 79 inches. This kite contains 68.2 quare feet of sustainin surface and weighs 7.32 pounds viz, 0.107 pound per square foot. #he bridfe is made of asingle bight of string attached to the front and rear ed lower side of the front cell. The center of gravity is 1.6 inc!%%% Df the center of figure. By actual observations in a wind of from 12 to 16_milea per hour the antral neis of the s stem of forces waa found to make an angle of So with the general fane of the sustaining sur- faces, and intersected the lower surface of the front cell at a point 8 inches aft of the forward lower ed e. The angle of incidence of the kite averaged 24.8O, and the ull, wehich varied greatly, averaged about 25 pounds. The efficiency o! the kite waa 92 per cent.
The simultaneous observation of all thome elements of the flight of a kite that are necessary in order to accurately de- termine  the pull thereon, with various positions and with
various strengths of wind, in its broadest sense, is equivalent to the determination of the law of relation between wind velocity and the pressure upon the surface of the kite at
various angles of incidence. This is a matter which, if the resulta are to possess real scientific value, presents very seri-
DUS experimental difficulties which are greatly aggravated by the extreme variability of the wind. Simple methods that
may be imagined to give sufficiently approximate resulta are found upon trial and critical inspection to be exceedingly unreliable, and to yield widely misleading numerical values. rhe pull, if measured at the reel, must be reduced to its cor- responding value at the kite, and the simultaneous velocity
If the wind must also be definitely measured at tlre kite.
Due allowance must be made for the weight of the kite, and ;he fact that in cellular kites the pressure of the wind per init area upon the surface of the forward cell is much more ntense than upon that of the rear cell must be properly recognized if the final results are to be susceptible of useful
rpplication in other cases. Investigations have shown the
writer et least the real experimental difficulties of securing rccnrate data of this character, and he is not prepared at
)resent to cite resulta whose accuracy can be fully defended.
Thus far in the study of the behavior of kites, and in the malysis of the forces acting thereon, we have considered only
XI.---THE MECHANICS OF THE KITE STRINQ.
APBIL, 1897. MONTHLY WEATHER REVIEW. 169
____~ ~
the kite with a short piece of string or restraining line imme- diately next to it. We now wish to study the forces actin8
upon the line, with a view to clearly setting forth in whal manner and to what extent these forces limit the elevation attainable with a given kite and the position it assumes. If we could employ a string or wire having no weight, and
so fine that the pressure of the wind upon it would be wholly inappreciable, then, as more and more of this wire is paid oul
to it, the kite would pass outward and upward along the samc straight line, such as R K, Fig. 21, retaining always the sanic
FIO. 21.
angular elevation as seen from the reel. Provided the wind
continued unchanged in force, there would be no limit to the height to which a kite could be flown under such circum- stances. Unfortunately, however, we can not fly kites with
wire having no weight, and against which the wind will not press, and, in consequence, our actual kite behaves in a very different manner from that described above. Supposing, aa before, that the wind force is the same at all points, high or low, the results actually obtained with the kite above em- ployed will be something like the following: When but a
short length of wire is paid out to the kite, it will take its po- sition upon the same line, R K, as before ; that is, for example, at K,. When more wire is unreeled, the kite does not con- tinue upward on this line, but, instead, drifts gradually away
to leeward and amsumes, successively, such positions as at K,, K,, K,, etc., which positions lie on a curve identical with that of the line, but having the ende and sags reversed. An im-
portant feature, common to all of the positions the kite may assume, is that the portion of the wire next the kite remains always at exactly the saw inclination. The inclination is not only the Bame for all positions, but is the same as it originally was at R K,. Changes of the wind force and other influences may cause this inclination of the wire to change, but the mere reeling out or in of the wire itself has no effect on the inclina- tion. With a certain amount of wire out, the portion next
the reel become6 horizontal, and the limit of altitude is then reached. The kite can lift no more line. All these effects
have been brought about under the limitations imposed by the action of gravity and the wind upon the wire. We have
mentioned the wind equally with gravity as affecting the wire. It is probable that with moderate wind forces the pres-
E U ~ upon wire, owing to its fineness in proportion to ita weight and strength, is a smaller and less important force than gravity. With a given weight of line the maximum of strength and the minimum of wind effect are obtained by the use of tempered ateel mueio wire.
By the aid of well-known mathematical formulse we can de- termine in the most complete'and exact manner all the effecta
iue to the action of gravityon the wire. On the other hand, the effects of the combined action of wind and gravity are of
L very complex character, and but little known and under- h o d and can be mathematically represented only in a most general and imperfect manner. The effect of the wind pres- sure on the wire will be disregarded in the present treatment, md we will prooeed , to develop the properties. of the curve rssumed by the kite wire, as if i t were dependent upon gravity done.
The catena y, which is approximately the curve assumed ~y a kite line of steel wire, possesses many very remarkable tnd interesting properties that have a more or less important
iearing upon the art of flying kites. A comparatively full liecussion of these will be found in the published account of Weather Bureau work. We will reproduce here only a few )f the most important equations. The fundamental equations of the cntenary may be written n a variety of forms, depending upon the variables employed. Each equation expresses some interesting property of the
:urve. Some of the forms most convenient for use are the
X1I.-PROPERTIES OF THE CATENARY.
'ollowing : w
y =4 i T p -c
8+4- x = e nap log c
tan@=!= d y c d x
t = w (e + y )
(3)
(4)
In these equations x is measured horizon-dly and y ver-
;ically, and the origin of coordinates is taken at the point uhere the curve is horizontal; 8 is the length of the curve neasured from the origin; e is a constant; 6 is the angle of nclination of the curve to the horizon a t the upper end of a
lortion of length s; t is the tension at thie upper end, and w
s the weight per unit length of the material of which the
:atenay is formed.
P
A
1
FIO. 22.
In Fig. 22 let A 0 B represent a catonary. The curve has imilar branches on either side of 0 Y, but we are generally ioncerned with only a portion of the curve on one side. If
he wire at the reel is just horizontal, then the position of he reel will be represented by the point 0 in the diagram. :f the wire at the reel is inclined upward, more or less, then he position of the reel will be represented on the diagram by
ome such:point:aszR, at :which :point the curve is inolined
b t the same angle as-the-wire at-the reel.
180 MONTHLY WEATHER REVIEW. APRIL, 1897
18
9.08
82.00
8.1 86.9
8.10
40.8
8.65 45.40
Tension.-The quantity e, in the equation above, is given
by the expression c = h. That is, c is the length of a piece
of wire whose weight equals to, the tension in the curve at the lowest point, where x, y, s, and 8 are all zero.
If t, 8 and. t', 8' are, respectively, the tensions and inclina- tions of the curve a t any two points, then, from equations (l), (3 ), and (4), there results,
'IO
.aP
1.11 m.90
1.68 40.40
1.m 44.80
9.48 48.8
-__
(5)
Slack, per cent.. ...........
Slack, per cent.. ...........
Slack, per cent.. ...........
B=W { *t ...........................
B=15Bo {w ...........................
J=W{*r ..........................
FIQ. XI.
General equations .for partial ascensions.-Fig. 23 represents a partial ascension in which the reel is a t R and the kite at
K, with the origin of coordinates at 0. Letters designating
the coordinates of the catenary at the point representing the reel are distinguished by a superscript ('). The linear ele- vation of the kite is h = y - y' and the length of wire out is
If t' is the tension of the wire at the reel, then the linear
height of the point R above the origin of coordinates is given by the equation
8 - 8'.
8.B
8 8 .8
8.81 82.8
4.68 87.80
The length of the catenary from R to the origin is
8'' 41_ sin e'
Eliminating c from equation (1) by its value in terms of
t' and 8' and replacing 8 by its value 8 = I + -sin 8' we ob-
tain,
w
t'
20
Whence,
That is, when the length of wire out is known, together with the tension and inclination at the reel, we may deduce the
height of the kite, even wlien it is concealed from view, as by clouds, darkness, ita remote distance, etc. This results from
a general property of the cn'tenary and the equation is equally applicable to the case of either partial or complete ascen-
sions. Owing to great momentary variations that take place in the tension of the wire the calculations of elevations depend- ing upon the tension a t the reel will not, as a rule, be as accu- rate as those deduced by other methods, but equation (6) will undoubtedly prove useful in cases where other methods of
ascertaining elevation are not available. The elevation of an invisible kite deduced by equation (6) will be more accurate as the sag in the wire is greater.
If 8 and t are the inclination and tension of the wire at the kite, we may write,
t t' y = lo (I-cos e), and y'= - (l-cos e') W
whence, by equation (6 ), we get,
Slack in the wire in partial nseensions.-Let r be the length
of the chord of the catenary from the reel to the kite, and I'
the angular elevation of the kite from the reel, then,
T slack = I - r and percentage of slack = 1 - - 1
The ratio of any chord of a catenary to the corresponding Brc is given by the equation
T - COS e' -cos e -- I sin @' sin (e a ' )
which may be obtained from equation (7) by eliminating
t - in terms of 1.
The relation between @', e, and 8' is obtained from equa- tion (2 ) together with the value of h in (7), thus,
1U
tan I' =- h - - e - sec ti'
sec 8' + tan e'
.(9)
riap log [ sec 8 + tan 8 1
x - 2'
In this equation @' is the angular elevation of the kite, as Been from the reel. The following table contains a series of values of @' de- duced from equation (9) corresponding to such assumed values of 8 and e' as may occur in practice. With each value of @' is also tabulated the corresponding percentage of slack coniputed by means of equation (7). The results are rigor-
QUS representations of the properties of the catenary, and even though the wind effect has been omitted, the relations
of the quantities concerned are such that the wind effect on
the wire can not modify tAe percentage of SlUek, corresponding
to given values of @' and e', except by a quantity of second- my magnitude.
A N ~I M 6 h t h aid $hWWn&~6 Of 8 h k .
Inclination or wire at kite. -
I 8
Slack, per cent ............ 6.17 3=W{w ...........................
W=InclInatlon of wire at reel.
0.61 0.18 4l.Q 45.80
44.80 48.80 62.60 .......
1.11 0.50 0.18 ....... 41.8 61.40 56.9 .......
Eleaation attainable.-It is desirable to know bow much pull and what length of line is necessary in a given case to reach a certain elevation, that is, to sustain the wire from the kite down to the reel. The pull may be deduced by solving squation (7) for t. The length of line required may be found
by the equation,
t
W l = -(sin L) - cos 8 tan e')
APRIL, 1897. MONTHLY WEATHER REVIEW. 161
The steel wire used as kite line by the writer is 0.0286 of
an inch in diameter and weighs 2.166 pounds per 1,000 feet. The table below gives the pull and the amount of line re- quired to reach certain elevations, depending upon the inclin- ation 8, at which the kite pulls the wire and also the angle
e, to which the wire is permitted to sag a t the reel. The wind effect on the wire is neglected, and necessarily so from our present ignorance of the numerical magnitude of these effects. The results are, therefore, not exactly representative of actual kite ascensions. The discrepancies will be small in moderate winds, that is, under 20 miles per hour, but become more im- portant a t this and higher velocities.
Ewntid ebnwnir in kils aacmsion.
IU = 2.166 lbs. per 1,OOO ft.
cos 61
I 1
PoUnd6. 18.4 18.8
24.8 1 .8
8 .8 40.9
49.8 64.6
03.0
I. a
I
11.6 ham 18.8 18.4 6,180 18.8
ga.8 8,080 19.8 87.8 7.W 20.8
86.0 1o.m 1 .4 M.8 10,- 1 .1
4 .8 iaw 88.0 40.1 18,884 88.9
~
It is asked LL What is the difference in effect between the kite string and the attraction of gravity on the mass of a soaring bird?
The effects are very different in many respects. In the first place the string constrains its kite to move in a certain defi-
nite surface of constraint which, in general, is sensibly a spherical surface. Gravity does not exercise any similar con- straint upon the bird. Secondly, the tension or pull of the string may act in many different directions, and with a mag-
nitude which depends almost entirely upon the pressure of the wind upon the kite; that ie, if the wind suddenly stops blow- ing, the string will quite as suddenly slacken up entirely and
the feeble pull it exerts due to its direct weight will be in a
new direction, and will often be but a small fraction of its
original pull. The tension in the string is not a primary but a dependent force. The force of gravity on the bird is, however, a primary force, is always constant in amount, and
its direction of action is always vertical. We do not con- sider that there is any real mechanical analogy between the effects of the force of gravity on a bird and the pull of
the string of a kite. Without therestraint of the string the kite and the bird are similar in that both are subject to the attraction of gravity, and both exposecertain surfaces to the action of the wind. The kite completely lacks the intelli- gence or instinct of the bird, but stable forms of tailless kites cast adrift in the wind are carried longdistances before reaching the ground. Two large Hargrave kites broke away from the writer in a violent rainstorm when at a height of about 7,000 feet. The total supporting surface was about
80 square feet, and the weight 8 pounds plus the effects of rain and several pounds of wire. These kites drifted 16
miles before reaching the ground and descended entirely un- injured.
NOTES BY THE EDITOR.
SEASONAL FORECASTS FOR OREGON.
On the morning weather map of April 12, published at Portland, Oreg., Mr. B. S. Pague, Local Forecast Official,
says :
The first summer type of weather conditions appeared Saturday night and Sunday and continues this morning. The appearance of thle type marks the commencement of the summer or dry season and the pas- sage of the winter or wet season. From now on until the first ap- pearance of the winter conditions next autumn, fair weather is more frobable than rainy weather. Summer conditions do not mean a b - ute dr ness, but rather fewer rainy periods, more aunshine, and more warmti In 1895, the first appearance of summer conditions appeared on April 20; rain fell in that year from A ril30 to May 9 in showers, and again from May 20 to May 30, the totarbeing 3.43 inches. June had rain on two da s In I&, summer condihons did not ap ar until &e 13, and then, except a few sprinkles, no rain fell untirAuguat 1, when showers oc- curred for five days. Winter conditions appeared in 1895 on November 12, and in 1896 on October 30. Frosts west of the Cascades are not, as a rule, injurious after the appearance of summer weather conditions. Summer conditions are distinguished from winter by the movement of the areas of high and low pressures. In the summer the highs move from off the central California coast northward along the coast to about 60 north latitude, thence eastward. Cool weather a Fears when the high is between Ca Mendocino and the Columbia giver, and it is warmer when the hi$ is moving over the Coast and Cascade ranges. The warmest occura when the hi h is to the northeast of S Lane. Under the summer conditions, t%e low areas retreat tot% north and pass eastward about the latitude of Sitka; for this reason, aummer rams prevail in Alaska.
On a preceding page we have published Mr. H. H. Claytons modification of Feuesners method of computing the height of a cloud by observations of the location of its shadow.
May was cool and June had normal tem ratures.
ULOUD HEIGHTS-A PRO- FOR STUDENTS.
This modification consists essentially in applying the double or check computation first extensively introduced by Ekholm and Hagstrom into the so-called parallactic method. The formulse for the computation are essentially the same as were arranged a year ago by Professor Bigelow for use at tho Weather Bureau, in reducing simultaneous cloud observations made at the two ends of a fixed base line.
As the trigonometrical relations are vary simple it is to be hoped that many of our intelligent observers will interest themselves in determining frequently the actual heights and
motions of the clouds. There are many methods of doing
this and these, together with new ones that yet remain to be invented, constitute problems that may well interest every youth when studying trigonometry and its practical applica- tions. Such problems should be explained in our high
schools and colleges generally, as an interesting practical ex- ample of the use of trigonometry. It is customary for our universities to require special original work from those who are candidates for the higher degrees, such as B. S., M. S.,
D. S., C. E., A. M., Ph. D., etc. To those candidates who are interested in the atmosphere and terrestrial physics we earn- estly recommend the study of the clouds, their altitudes, motions, and laws of furmation as a thoroughly appropriate problem and one of the greatest importance to meteorology. The following special deduction of Ekholms formulse, as
needed in Feussners method, will interest such students. Nearly all the known methods are enumerated in the Editors Treatise on Meteorological Apparatus and Methods, Wash- ington, 1887, except that based on the use of the nephoscope a t sea, which has been elaborated since then by Finemann and the Editor. Doubtless other methods remain to be in-