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 moment’s 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 ~ ~~ ~~ ~~ !l’HEl 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. Kummer’s 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 Rayleigh’s 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 wind’s 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 resultan’t 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.