Aerodynamics (Lanchester)/Chapter 10

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Aerodynamics, constituting the first volume of a complete work on aerial flight (1906)
by Frederick William Lanchester

Chapter 10

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3417615Aerodynamics, constituting the first volume of a complete work on aerial flight — Chapter 101906Frederick William Lanchester

Chapter X.

Experimental Aerodynamics.

§ 221. Introductory.—Experimental aerodynamics must at present be regarded as in its infancy. The methods employed up to date have not yielded results of an exactitude comparable to that readily obtainable in other branches of physical science.

There are in the main three methods of investigation open. Firstly, experiments upon planes or other bodies propelled through still air, the subject of experiment and the measuring appliances being mounted either on the arm of a whirling table or on the front of a locomotive vehicle. Secondly, the measurement of the reactions produced by a fluid in uniform motion on a fixed body. Thirdly, by measurement and deduction from experiments in free flight.

The first method is that most generally adopted, admirable work having been done in this direction by Dines. Langley and others. The second method has been used to some extent by Dines, and the third (the method of free flight) has been developed to a certain extent by the author.

The earlier experimenters. Robins (1761), Hutton (1787), and Vince (1794-5, 1797-8), employed a primitive form of whirling table, the invention of which is attributed by Hutton to Robins, and no earlier record appears to exist of the employment of this device for the purpose contemplated. The whirling table as known to Hutton is represented diagrammatically in Fig. 140, in which a horizontal arm A is mounted on a vertical axis B, which is caused to rotate by the silk cord C and weight D; the body on which experiments are to be made is mounted at the extremity of the arm as at E, and the determination of its resistance is made by altering the weight D until a certain definite speed of rotation is attained. It is evident that, knowing the length of the arm, the diameter of the axle on which the cord is wound, and the weight employed, the calculation of the resistance is a matter of simple arithmetic.
Fig. 140. The resistance of the body under investigation only forms part of the total resistance, and a preliminary experiment is necessary to determine the Resistance proper to the apparatus itself. Precautions were usually taken to prevent so far as possible frictional resistance, the variations of which would otherwise give rise to error of sensible magnitude.

§ 222. Early Investigations. Hutton—Vince.—One of the objects on which considerable experimental attention was focussed at an early period was the investigation of the solid hemisphere in plane and in spherical presentation. This is probably due to the fact that this problem forms the basis of prop, xxxiv., Book II., of the “Principia” as touching the Newtonian Medium. It was evidently the aim of the early workers to ascertain the extent to which Newton's results would prove applicable to a real fluid.

In the case in question it was found by Hutton that—

(1) The pressure on the hemisphere in either presentation varies, according to the Newtonian law, that is as the square of the velocity; and,

(2) The resistance in plane presentation is approximately two-and-a half times as great as in spherical presentation, instead of only twice as great as demonstrated by Newton. This result was subsequently confirmed by Vince, whose relative figures were 1 to 2.46.

These results were considered at the time as a substantial confirmation of prop, xxxiv., or, rather, as showing that the behaviour of a real fluid does not greatly differ from that of the discontinuous medium; probably we have here the reason why many subsequent writers have been misled into assuming the applicability of the Newtonian sine² law in the case of the inclined aeroplane. In point of fact the coincidence, had it been far more complete, would be of no significance whatever; the system of flow in actuality bears no resemblance to the dynamic system of Newton.

The fallacy of the sine^ law was first clearly demonstrated by Vince in his paper (to which reference has already been made); he gives experimental data showing that the resistance in the line of flight varies, for small angles, as sine^1.73 of the angle; this, if we neglect the influence of skin-friction, corresponds to a pressure normal to the plane varying as the sine^.73. The fact that the index here is less than unity can be reasonably accounted for on the assumption that part of the resistance is due to skin-friction, which is roughly constant in respect of the inclination.

§ 233. Dines' Experiments.^[1] Method.—Coming now to the modern period, we have to examine two independent series of experiments, made almost simultaneously, respectively by Mr. W. H. Dines in England and by the late Prof. S. P. Langley in America.

To Dines we owe a particularly beautiful and original method of employing the whirling table for the determination of aerodynamic data. In all the modern applications of the whirling table,
Fig. 141. the determination of the resistance of the object of experiment is made quite independently of the propulsion of the table, some form of balance being employed mounted at the extremity of the rotating arm, the motion of the latter being maintained by the application of a power installation, and the speed of rotation accurately recorded by a chronograph. Dines conceived the possibility of balancing the aerodynamic reaction, which varies approximately as the square of the velocity, against the centrifugal force of an appropriately arranged weight, which varies in like ratio. By this means the measurement of the reaction is made virtually independent of the velocity of flight, so that a possible source of error is avoided and the conduct of experiments is greatly simplified.

The Dines apparatus is illustrated in the form of a rudimentary diagram in Fig. 141, in which A is an arm of the whirling table pivoted at and revolving about the point B, a bell crank lever C D, delicately centred at E, carries on its two arms respectively the pressure plane, F (or other body whose resistance it is desired to ascertain), and the bob weight G whose centrifugal force forms the measure of the pressure reaction. The condition of equilibrium is that the resultant of the two forces passes through the pivot centre E. It is evident that these two forces, in equilibrium at any one speed, will be in equilibrium for all speeds, for their relative direction undergoes no change, and they are each proportional to the velocity squared and so are proportional to one another. The bob weight D is made adjustable on the arm D and the condition of equilibrium is ascertained by trial. In one modification the instrument is made to perform automatically its own adjustment.

§ 224. — Dines' Method (Mathematical Expression).

Let,	$A=$ area of plane.
„	$V=$ velocity of the bob weight.
„	$c_{1}\ V=$ velocity of the centre of pressure of the plane.
„	$r=$ radius of the path of the weight.
„	$R=$ resistance (poundals) acting on the plane.
„	$F=$ centrifugal force of the bob weight (poundals).
„	$c_{2}=$ lever ratio, so that $R=c_{2}\ F.$
„	$M=$ mass of bob weight.
„	$P=$ pressure on plane (poundals per square foot.)
„	$C=$ constant in expression, $P=C\ \rho \ V^{2}$ (§ 134).

Then—

P=C\ \rho \ (c_{1}\ V)^{2}

and

R=A\ P

∴

R=A\ C\ \rho \ c_{1}^{2}\ V^{2}

But—

F={\frac {M\ V^{2}}{r}},

and

R=c_{2}\ F

∴

A\ C\ \rho \ c_{1}^{2}\ V^{2}={\frac {c_{2}\ M\ V^{2}}{r}}

or

C={\frac {c_{2}\ M}{A\ \rho \ c_{1}^{2}\ r}}

^[2]

all of which are known quantities; consequently the value of the constant $C$ is determined.

§ 225. Dines' Method (continued).—Of the practical working of the foregoing apparatus, which may not inaptly be termed a centrifugal balance, Mr. Dines says:—“It had been assumed in the above that the wind pressure varies as the square of the velocity. The experiments have proved this to be the case, for when upon a calm day equilibrium for any plate is once attained, it has been found impossible to disturb it by any alteration of the velocity of rotation, and since the centrifugal force varies as the square of the velocity the wind pressure must do so also. For the smaller planes the maximum velocity of which the machine is capable is about seventy miles per hour.”

We may consequently infer that the $V^{2}$ law of resistance holds good as a very close approximation over a very considerable range of speed, certainly as great a range as concerns the problem of flight. This is a result previously considered in doubt.

The arrangement of the centrifugal balance figured is not one altogether suited to planes of large size, owing to the fact that the different portions of the plane are situated at different distances from the centre of the whirler, and the position of the centre of pressure becomes uncertain. In such cases a modified design is adopted (Fig. 142), of which the description is given in § 227.

The obvious difficulty of observing the position of the plate when in rapid motion is overcome electrically, the range of motion permitted to the lever being limited by stops to about one degree, the stops forming at the same time electrical connections to alternative circuits.
Fig. 142. The two circuits are arranged in connection with a galvanometer so that contact with one stop causes a current in one direction and contact with the other causes a current in the opposite direction. When equilibrium is established the galvanometer needle either remains unaffected, or oscillates from one side to the other indicating that the current flows about equally in either direction.

§ 226. Dines' Results. Direct Resistance.—A considerable number of experiments on the resistances of planes and solids of various forms are found summarised in the following table, which is taken intact from the “Experiments on the Pressure and Velocity of Wind.” Many of these results have been already referred to in a previous chapter.

Table XVII.

Showing the pressure upon various plates at a velocity of 20.86 miles per hour. The values are reduced to the standard temperature and pressure. The flat plates were cut out of hard wood

\scriptstyle {\frac {3}{8}}

inch thick. Allowance has been made for the arm which carried them.

Plates.	Actual Pressure in pounds.	Pressure in pounds per square foot.	No. of Experi- ments.
A square, each side 4 in. A circle, 4.51 in. diameter, same area A rectangle, 16 in. by 1 in. A circle, 6 in. diameter A square, each side 8 in. A circle, 9.03 in. diameter, same area. A rectangle, 16 in. by 4 in. A square, each side 12 in. A circle, 13.54 in. diameter, same area A rectangle, 24 in. by 6 in. A square, each side 16 in. A plate, 6 in. diameter, 4 $\scriptstyle {\frac {3}{4}}$ in. thick A cylinder, 6 in. diameter, and 4 $\scriptstyle {\frac {3}{4}}$ in. long A sphere, 6 in. diameter A plate. 6 in. diameter, with 90 degrees cone at back The same, with cone in front A plate, 6 in. diameter, with sharp cone 30 degrees angle at back The same with cone in front A 5 in. Robinson cup, mounted on 8 $\scriptstyle {\frac {1}{2}}$ in. of $\scriptstyle {\frac {1}{2}}$ in. rod The same with its back to the wind A 9 in. cup, mounted on 6 $\scriptstyle {\frac {1}{2}}$ in. of $\scriptstyle {\frac {5}{8}}$ in. rod The same with its back to the wind A 2 $\scriptstyle {\frac {1}{2}}$ in. cup, mounted on 9 $\scriptstyle {\frac {3}{4}}$ in. of $\scriptstyle {\frac {1}{4}}$ in rod The same with its back to the wind One foot of $\scriptstyle {\frac {5}{8}}$ in. circular rod	0.17 0.17 0.19 0.29 0.66 0.67 0.70 1.57 1.55 1.56 2.70 0.28 0.18 0.13 0.29 0.19 0.30 0.12 0.28 0.12 0.82 0.28 0.13 0.05 0.09	1.51 1.51 1.70 1.47 1.48 1.50 1.58 1.57 1.55 1.59 1.52 1.45 0.92 0.67 1.49 0.98 1.54 0.60 1.68 0.73 1.75 0.60 2.60 1.04 1.71	4 9 7 7 8 12 4 7 14 6 6 5 4 8 4 4 4 4 8 4 3 3 3 3 9

In addition to the experiments tabulated, other resistance experiments are recorded, notably those on perforated plates dealt with in § 143, also the effect of a projecting lip on the edge of the normal plane, § 141.

§ 227. Dines' Experiments (continued). Aeroplane Investigations (Apparatus).—The description and results of these experiments are given in Mr. Dines paper on “Wind Pressure upon an Inclined Surface,” (Proc. Royal Soc., Vol. XLVIII.). The apparatus employed is a form of the centrifugal balance described in § 223, and the “planes” employed are of the triangular section already illustrated in Fig. 100 (a).

Two diagrammatic views of the centrifugal balance employed are given in Fig. 142, which is reproduced from Mr. Dines' paper. In this, like letters refer to like parts in the two views; and we have the pressure plane $P$ mounted on the arm $E\ F$ pivoted to turn about the axis $M\ N,$ which is arranged radial to the axis of the whirling table. We have the bar $A\ B,$ which is the mass whose centrifugal force is employed to measure the reaction, mounted slidably in a pivot piece free to turn about the vertical axis $C\ D,$ and the pivot piece and the pressure plane arm are geared together by a stud projecting from the former and engaging with the latter. This may be looked upon as equivalent to a single tooth of an imaginary pair of bevel gears. A counterpoise weight $K$ performs the double function of statically balancing the pressure plane, and of rendering the arm $E\ F$ symmetrical in respect of wind pressure.

The adjustment of the bar $A\ B$ by an ingenious device was arranged to take place automatically. This automatic arrangement consists of a windmill carried on the arm of a whirling table, arranged to drive whenever possible a crown wheel, by means of a long pinion engaging with the crown wheel on both sides. The crown wheel is carried on the pivot piece, and by means of a rack and pinion moves the bar $A\ B$ in the one direction or the other. When the pivot piece reaches either of its extreme positions (the total range being a few degrees), the crown wheel becomes disengaged from its pinion on the one side or the other, the windmill immediately begins to operate, and the bar undergoes displacement in the direction required to restore equilibrium. So long as the balance is perfect and undisturbed, the pivot piece remains in an intermediate position, and the crown wheel is locked by the double engagement of its pinion.

In this centrifugal balance, as in all apparatus of its type, the precise speed at which the whirling table is propelled is unimportant for the reasons already given. It is, however, necessary that the velocity should be steady, i.e., the whirling table should not be undergoing acceleration when the experiment is made. The reason of this precaution is that the pivot piece and bar possess moment of inertia, and change of speed of rotation involves a torque about the axis $C\ D$ foreign to the conditions. This effect could be minimised by concentrating the mass as a bob weight, and making the lever and pivot piece carrying it as light as possible. The difficulty could be entirely eliminated by arranging a duplex apparatus, in which two sliding bars are employed, having opposite rotation, their preponderating weights being arranged at opposite ends.

§ 228. Dines' Experiments (continued). Aeroplane Experiments.—In the determination of aeroplane data, other than in the special case of the normal plane, the Dines method presents certain difficulties. It may have been noticed from the mechanical disposition of the parts, that it is the moment of the pressure reaction about the axis $M\ N$ that is the quantity measured, consequently before the magnitude of the pressure reaction can be ascertained the position of the centre of pressure must be known. In order to avoid the necessity of independently determining the centre of pressure, observations are made with the adjustable arm in two complementary positions (Fig. 143 (a)), the angle of incidence being the same in both cases. It is evident that the arithmetical mean of the two readings will give the moment of the pressure reaction as if the total force were applied at the geometrical centre.

By investigating two further positions (Fig. 143 (b)), an attempt was made to compute the tangential force or skin-friction on the plane: this, however, proved abortive. Mr. Dines attributes the failure of this portion of his investigation to the existence of eddy currents subsequently discovered to have been set up by the frame of the machine.

Experiments with roughened surfaces and with the planes thoroughly wetted showed a diminished reaction, in both cases equal to about a 20 per cent, drop for the angle of maximum moment when compared with the same plane polished and dry.
Fig. 143.

Experiments are also recorded, made with the object of determining, in a rough and ready manner, the direction of the stream lines in the immediate vicinity of the surfaces of an inclined plane. A number of pins were driven into the face and back of the plane, and short lengths of coloured silk attached to act as weather-cocks from point to point, and show the local direction of the air currents. The results were drawn from observation; sample diagrams for a square plane at 45 degrees are given, for front and rear aspect, in Fig. 144. It is evident that we have strong evidence here of the centrifugal shedding of the “dead-water,” the influence of which has already been the subject of comment in connection with the theory of the screw propeller.

The quantitative results of Mr. Dines' pressure reaction experiments have been given in most part in Chap. VI.
Fig. 144.

§ 229. Dines' Experiments Discussed.—The simplification resulting from the employment of the centrifugal balance in the apparatus and observations necessary for the determination of data, either of direct resistance or lateral reaction, is very great, and the apparatus in question should certainly be of further service in the future. With a modification such as suggested in § 227, it would appear possible to make determinations with very great rapidity, perhaps even with but a single revolution of the whirler. If this should become possible, one of the chief objections to an indoor whirling apparatus—the residuary motion of the air—would be done away with, and the accuracy of the results greatly increased.

Incidentally the employment of the centrifugal method seems to have demonstrated that the velocity squared law is in all ordinary cases a nearly perfect approximation to the truth.

One of the most remarkable results brought out by these experiments is the peculiar “kick up” in the pressure curve (Figs. 93—101, Chap. VI.). This "kick up" seems to have been entirely missed until pointed out by Mr. Dines, although the matter has been investigated by other careful observers. It might be thought that the peculiarity of the Dines curve is related in some way to the triangular section “plane” employed in these experiments, but this hardly seems possible. It would certainly have been more satisfactory if the experiments had been repeated with “planes” of more usual form. For small angles it is highly probable that Dines' results are not accurate, but when once the motion is frankly discontinuous it is difficult to believe that the form of the back of the planes employed can account for so marked a departure of the curve as that observed. It is therefore most probable that the “kick up” is a real feature in the pressure-velocity curve that has escaped the notice of other experimenters.

Dines concludes from his experiments that the effect of skin friction is negligible; one experiment, especially directed as a quantitative test, gave an entirely negative result: no tangential component could be detected. If it were not for the extreme subtlety of the subject it would be difficult to resist the conclusion stated; the pitfalls connected with skin-friction are, however, numerous, and the evidence is inconclusive. It is fair to remark that on this point Dines is in agreement with the late Professor Langley and Sir Hiram Maxim.

§ 230. Langley's Experiments. Method.—The method of experiment adopted by the late Professor Langley resembles that of Mr. Dines in the employment of a whirling table driven by power, and in the use of independent measuring appliances to determine the resistances or reactions on the body subject to investigation.

The following account of these experiments is condensed from Professor Langley's Memoir, “Experiments in Aerodynamics,” published by the Smithsonian Institute, Washington, 1891.

The site chosen for these experiments was situated in the grounds of the observatory at Allegheny, Pa., U.S.A., some 1,145 ft. above sea level.

The whirling table, erected in the open air, consisted of a trussed cantilever beam, arranged to rotate about a central vertical axis, being driven by an underground horizontal shaft from a separate power house through the medium of bevel gearing. The total length of the beam is given as 60 ft., that is to say, the extremities describe a circle of 30 ft. radius. In construction the beam itself is figured as resembling a light ladder, laid horizontally and stayed from a point about 9 ft. above its centre by a vertical strut, and a number of wire guys taken out to various points along its length. Lateral stiffness is given to the structure by a pair of guys on either side stretched from each extremity to a central outrigger. Provision is made for obtaining at will peripheral speeds from 15 to 100 ft. per second, and for chronographically recording each quarter-revolution by electrical means.

The mode of employment of the whirling table above described involves the use of a number of distinct apparatus, each specially schemed and designed by Prof. Langley for the particular purpose in view.

It is impossible to altogether detach the description of the apparatus from the discussion of its employment and results. In the Memoir a chapter is devoted to each instrument, and in the present précis and discussion the author has followed a similar arrangement, a separate section being devoted to each chapter of Professor Langley's work.

§ 231. Langley's Experiments, “The Suspended Plane.”—This instrument consists of a square plane of thin brass, mounted “slidably” on anti-friction rollers in a frame and suspended by a spring, the frame being mounted on horizontal trunnions. The function of this device is not clear; in the words of the text, its object is “to illustrate an unfamiliar application of a known principle,” but the employment of the apparatus does not seem to lead to any results of importance.

§ 232. Langley's Experiments. “The Resultant Pressure Recorder.”—This instrument is designed, in the words of the Memoir, “for the purpose of obtaining graphically the direction of the total pressure on an inclined plane (in practice a square plane), and roughly measuring its amount.” The instrument consists of a beam (Fig. 145) hung symmetrically at its centre in gimbal joints, and carrying at its outer extremity the “wind plane,” that is, the plane under investigation, and at its opposite end a tracing point or pencil adapted to record on a sheet of diagram paper, arranged at right angles to the beam itself. A co-ordinate combination of tension springs is employed to hold the beam radial to the whirling table, and the whole is accurately counterpoised so that the plane is virtually weightless; thus, so long as the apparatus is at rest the pencil point is central or at the co-ordinate zero, but when the whirling table is in motion the total reaction on the plane is measured, both as to direction and magnitude, by the resulting displacement of the pencil point. In order to obviate friction the pencil is held away from the recording paper until the desired velocity is reached, when it is released by means of an electro-magnet. Due precautions are taken to ensure proper calibration; to this end the entire spring system is carried in a revoluble frame shown in the figure.

The method of employment is described as follows: “The wind plane is set at an angle of elevation

\beta ;

a disc of paper is placed upon the recording board and oriented so that a line drawn through its centre to serve as a reference line is exactly vertical. The whirling table is then set in motion, and when a uniform velocity has been attained a current is passed through

Fig. 145.

the electro-magnet, and the pencil records its position on the registering sheet. Since gravity is virtually inoperative on the counterpoised plane, the position of this trace is affected by wind pressure alone. Thus the instrument shows at the same time the direction and magnitude of the resultant wind pressure on the plane and for different velocities of the whirling table.

The results achieved with this instrument were as follows:—

(1) The confirmation of the law of pressure as: $P=k\ V^{2},$ and the determination of the value of $k$ for the normal plane.

(2) The determination of the pressure-angle curve for the square inclined plane, incidentally providing a substantial confirmation of Duchemin's formula.

In addition to the above, Langley claims to disprove “the assumption made by Newton that the pressure on the plane varies as the square of the sine of its inclination,” and elsewhere he states: "Implicitly contained in the Principia, prop. xxxiv., Book II." Now, whatever Langley's experiments prove or disprove, the assumption that he attributes to Sir Isaac Newton is one that he did not make, and nothing of the kind is “implicitly contained” in the proposition to which reference is made.^[3]

Prof. Langley further states that the experiments with this instrument “further show that the effect of the air friction is wholly insensible in such experiments as these.” Now, as bearing on this contention. Fig. 1 from the Memoir is here reproduced (Fig. 146), and shows the result of plotting a series of observations, with an averaging curve drawn to indicate the probable true values. It will be noted that this curve does not pass through the origin, but cuts the axis of $y$ at a point representing (on the ordinate scale) a matter of some 8 per cent.; or, if we take the possible extremes indicated by the observed points, this quantity will be something between 3 and 13 per cent.

The discrepancy is accounted for by Prof. Langley on p. 24 of the Memoir, and a “corrected” curve drawn accordingly. The explanatory paragraph is as follows:—

“The values in the Tables are subject to a correction resulting from a flexure of the balance arm and its support.
Fig. 146. It was observed that the trace of the plane, set at 90 degrees, did not coincide with the horizontal (i.e., the perpendicular to the vertical) line marked on the trace, but was uniformly 4 degrees or 5 degrees below it; so that the angle between the vertical and the trace of the plane did not measure 90 degrees, as had been assumed, but uniformly 94 degrees or 95 degrees, the average being 94.6 degrees. This result was found to be due to the bending backwards of the balance-arm and its support by the pressure of the wind, while the recording board and plumb line presented only a thin edge to the wind, and consequently remained relatively fixed. During motion, therefore, the plane actually had an inclination to the horizon about 5 degrees greater than the angle at which it was set when at rest. This flexure seemed to obtain for all angles of exi3eriment, but with indications of a slightly diminishing effect for the smaller ones; consequently the pressure ratios above given for angles of 45, 30, 20 degrees, etc., really apply to angles of about 50, 35, 25 degrees, etc. After making this correction the final result of the experiments is embodied in the line of Fig. 1 designated corrected curve.”

Now the author has determined the coefficient of “skin-friction,” and it has been shown in the present work that it is nowise a negligible factor; the value would (under the conditions of experiment) in all probability be about 2 per cent., and when the angle of the plane is sufficient to give rise to motion of the discontinuous type it will be in effect about half this amount; the value would require to be considerably less than this before it could be considered as negligible. It is a quantity of this order that Langley confidently asserts does not exist, because it has remained unrecognised in the results of an experiment of which he himself writes as in the paragraph quoted, and which has been subjected to a correction, on very doubtful grounds,^[4] many times greater than the quantity involved. The remedy for so serious an error was obviously to redesign the apparatus with a symmetrical frame; had this been done there is every probability that the effects of skin-friction would have been clearly recognised.

§ 233. Langley's Experiments. “The Plane Dropper.”—The plane dropper, as its name implies, is an apparatus in which the aeroplane is allowed to fall under the influence of gravity against the aerodynamic resistance encountered in its flight. In this instrument the “plane” is clamped to a “falling piece” arranged with friction rollers to slide freely on a vertical guide bar; a detent is employed to hold the falling piece in its top position until released by an electro-magnet. The angle made by the plane to the line of flight has a range of adjustment from horizontal up to 45 degrees. The total fall permitted is four feet, and the time of fall is registered electrically, both at the top and bottom, and later in the experiments at each foot of fall, the observatory chronograph being employed.

The experiments made with the plane dropper are numerous, and the results are highly instructive from a qualitative point of view; it would not seem, however, that the method is one that should be imitated by future experimenters: the results are in general deficient in quantitative value, except in the special case when the plane is recorded as “just soaring.” The weak point in this kind of instrument is the uncertainty that must prevail as to the existence or otherwise of a steady state. During the first portion of the drop there is acceleration taking place, that is to say, part of the weight of the paraphernalia is spent in overcoming its own inertia, and only a portion is supported aerodynamically; so that a considerable calculation is necessary before the results recorded can be made quantitatively available.

Admitting its defects, the method is one that appeals strongly to the imagination, imitating as it does many of the conditions of free flight, and in the hands of Prof. Langley it was shown capable of giving some valuable information. The chief points demonstrated were as follows:—

(1) That the time of fall of a horizontal plane in horizontal motion is greater than when no horizontal motion exists, and is greater the greater the horizontal velocity.

(2) The vital importance of the shape of the plane and of its aspect on the weight supported.

(3) The existence of a critical angle at which the aspect effect undergoes reversal.

(4) The fact that planes can be superposed in flight without sensible diminution of their individual supporting power, provided that they are separated by a certain minimum distance. For planes fifteen inches by four inches in pterygoid aspect the minimum distance between the superposed planes was found to be about four inches, or approximately equal to the “fore-and-aft” dimension.

It is curious that, although Langley in many places elsewhere in the Memoir has pointed out the failure of the sine² law (the law of the Newtonian medium) as applied to air, he apparently overlooks the fact that the falling plane is on this point actually the experimentum crucis, for it has been shown (§§ 145—50) that if the sine² law holds good in any fluid or medium, the rate of fall of a horizontal plane will be independent of and unaffected by its horizontal motion.

We find once more, in the chapter dealing with the plane dropper, the assumption that skin friction is negligible, resulting in much false inference. This time the error is tacitly assumed. No further proof is announced. The statements that are affected by this error are sufficiently numerous. The following example will put the reader of the Memoir on his guard:—

“The results of these two series of experiments furnish all that is needed to completely elucidate the proposition that I first illustrated by the suspended plane, namely, that the effort required to support a bird or flying machine in the air is greatest when it is at rest relatively to the air, and diminishes with the horizontal speed which it attains, and to demonstrate and illustrate the truth of the important statement that in actual horizontal flight it costs absolutely less power to maintain a high velocity than a low one.”

Later we find:—

“... For the former case this is 0.0156 horse-power, and for the latter case approximately 0.0095 horse-power—that is, less power is required to maintain a horizontal velocity of seventeen metres per second than of fourteen, a conclusion which is in accordance with all the other observations and the general fact deducible from them, that it costs less power in this case to maintain a high speed than a low one—a conclusion, it need hardly be said, of the very highest importance, and which will receive later independent confirmation.”

“Of subordinate, but still of very great, interest is the fact that if a larger plane have the supporting properties of this model, or if we use a system of planes like the model, less than one-horse power is required both to support in the air a plane or system of planes weighing 100 lbs., and at the same time propel it horizontally at a velocity of nearly forty miles per hour.”

§ 234. Langley's Experiments. The “Component Pressure Recorder.”—This is by far the most important of the appliances originated by Prof. Langley for use in conjunction with the whirling table, and is one that should receive careful study from future experimenters. In construction, the component pressure recorder somewhat resembles the resultant pressure recorder already described, but instead of measuring the magnitude and direction of the total reaction by a symmetrical spring combination, the reaction is resolved into its horizontal and vertical components, which are separately recorded, the former directly on a chronograph cylinder forming part of the instrument, and the latter by the condition that the “soaring speed” is reached. It may be remarked that, whereas in the resultant pressure recorder the plane is counterpoised so as to be virtually weightless, in the present instrument the weight of the plane, loaded to whatever extent desired, is used as a measure of the vertical component.

The drawings of this instrument as figured in the Memoir are

Fig. 147.

given in Fig. 147, in which a light stiff beam, built on the lattice principle, is mounted on knife-edge gimbals on a frame which is in turn free to rotate about a vertical axis. This beam, which is functionally comparable to the beam of a balance, carries at its outer end (i.e., the end remote from the axis of the whirling table) the “wind plane,” attached by a tubular arm to the divided circle $G,$ by which it may be set to any desired angle. The inner and outer limbs of the beam are symmetrical, and the whole is enclosed in a box or case to afford shelter from the wind. A dummy end is allowed to project at the inner end to balance (as to wind pressure) the attachment arm at the outer extremity.

A pencil arm (indicated as such in the figure) is provided, attached to the gimbal frame, to record direct on the chronograph drum. This pencil arm also serves to attach the spring by which the horizontal component is measured. Friction wheels $R$ are fitted at both ends of the beam to limit the vertical movement.

The delicacy of suspension was found to be greater than could be employed under outdoor conditions, and a brush $H$ was added to develop a certain regulated amount of friction.^[5]

The chief work accomplished with the component pressure recorder was the following:—

(1) The determinations for planes of different proportions, of the velocity of “soaring” corresponding to different values of angle and load. Incidentally, the existence of an angle of reversal, already mentioned in connection with the plane dropper, was clearly brought out, the previous result being confirmed.

(2) The determination of the values of $P_{\beta }/P_{90}$ for planes of different aspect ratio.

(3) The determination by direct measurement of the horizontal component of the reaction on planes at different angles and soaring speeds supporting a known weight.

Fig. 148 gives, plotted to a reduced scale, one of the soaring speed diagrams taken from the Memoir. The “reversal” is well shown in the case of the curves $A$ and $B,$ whilst the curve $C$ shows that for planes of this particular proportion and aspect the curve for small angles approximates very closely to the law $V^{2}\times \beta =$ constant. The theoretical curve is given by the dotted line showing the degree of approximation. This law has already been given in the form—

\beta \propto {\frac {1}{V^{2}}}

(§ 152 (e)).

Fig. 148.

Fig. 99 (Chap. VI.) is an example taken from the Memoir (Fig. 10), giving the $P_{\beta }/P_{90}$ relationship for changes in $\beta$ value, as determined by means of the present appliance.

Fig. 149 is also a plotting (Fig. 11 of the Memoir), showing the direct determination of the horizontal component. The small circles represent the actual observations, and the crosses give points calculated on the basis of a simple resolution of forces, assuming the reaction exactly at right angles to the surfaces of the plane—that is to say, neglecting skin-friction. The curve drawn is a so-called corrected curve, the basis of the correction being given in the Memoir, and which we will now proceed to examine.
Fig. 149.

Let us first draw two curves, from theoretical considerations alone, on the basis (a) that the fluid is frictionless, and (b) that there is a tangential force due to a coefficient of skin friction $\xi =$ .025 effective on one face only of the plane—that is, an effective value = .0125 (see § 182); it is also supposed that the tangential velocity of the air is proportional to $\cos \beta$ —that is, equal to $V\cos \beta .$ The two curves thus plotted are given in Fig. 150.
Fig. 150.

Now, it is evident that the curve (b) (dotted line) corresponds more closely to the actual observed values than the curve (a), and we have certainly here primâ facie evidence of a tangential force about equal to that which we have assumed, and which is deduced from the author's own experiments.

Let us see what Langley says on the subject. We find (p. 63) the story of the plotting told m a few words:—

“... These values have been plotted in Fig. 11, and a smooth curve has been drawn to represent them as a whole. For angles below 10 degrees the curve, however, instead of following the measured pressure, is directed to the origin, so that the results will show a zero horizontal pressure for a zero angle of inclination.”

It may be remarked parenthetically that here the complete assumption has been made of that which it should have been the function of the experiment to prove. The author of the Memoir continues:—

“This, of course, must be the case for a plane of no thickness, and cannot be true for any planes of finite thickness with square edges, though it may be and is sensibly so with those whose edges are rounded to a so-called fair form. Now the actual planes of the experiments presented a squarely cut end surface one-eighth of an inch 3.2mm. thick, and for low angles of inclination this end surface is practically normal to the wind. Both the computed pressures for such an area and the actually measured pressures, when the plane is set at degree, indicate conclusively that a large portion of the pressures measured at the soaring speeds of 2 degrees, 3 degrees, and 5 degrees, is end pressure, and if this be deducted the remaining pressure agrees well with the position of the curve. The observed pressures, therefore, when these features are understood, become quite consistent. The curve represents the result obtained from these observations for the horizontal pressure on a plane with fair' shaped edges at soaring speeds.”

The above argument appears to the present author to be excusable as an attempt to explain why the results of one experiment or series of experiments might differ from some other experiment or established fact, but it does not constitute a demonstration that skin friction is negligible. The fallacy of an argument on these lines has been already pointed out in Chap. VI., § 158. It is in any case very difficult to defend plausible reasoning of this kind, when the actual experiment with planes of “fair” form could have been tried for the expenditure of an additional few shillings and with hut little delay.

The inevitable statement as to the power expended in flight follows:—

“The most important conclusion may be said to be the confirmation of the statement that to maintain such planes in horizontal flight at high speeds, less power is needed than for low ones.”

“In this connection I may state the fact, surely of extreme interest as bearing on the possibility of mechanical flight, that while an engine developing one horse-power can, as has been shown, transport over 200 pounds at a rate of 20 metres per second (45 miles per hour), such an engine (i.e., engine and boiler) can be actually built to weigh less than one-tenth of this amount.”

§ 235. Langley's Experiments. The “Dynamometer Chronograph.”—This apparatus was devised for the measurement of the thrust and torque of screw propellers, as a means of practically testing trial models and ascertaining efficiency obtainable. For the details of the instrument reference should be made to the Memoir.

In the chapter on the use of this appliance. Prof. Langley explicitly states that the details of his investigations are reserved for future publication; certain particulars are, however, vouchsafed, including a sample determination, which is of considerable interest in view of the theory of the preceding chapter.

It would appear from the data given that the propeller employed, having a diameter of 30 inches, had an effective pitch of 1 foot approximately, that is to say, its radius was 1 ${\tfrac {1}{4}}$ times the pitch.

Referring to our efficiency diagram (Fig. 127), we see that this denotes the employment of rather more than the whole of the diagram given, so that the efficiency will vary over the length of the blade from 70.4 per cent, to about 40 per cent.; if we take the mean as a rough approximation of the efficiency value to be expected, we have 55 per cent. The actual efficiency obtained was 52 per cent., which is quite as near as could be anticipated.

Again, as to the number of blades, Langley found that two blades gave a better result that any greater number of blades. Now the rule laid down in § 218 can hardly be relied on in the present case: the design of this propeller is abnormal. We may fall back on § 211. In the propeller under discussion the thickness of the peripteral zone (§ 210) will be evidently nearly as great, if not quite as great, as the pitch, consequently we must be approaching the point at which one blade will interfere with itself, and two blades will certainly overlap to some extent. It is consequently quite evident that any increase on the number must be detrimental. Thus we again find substantial confirmation of the peripteral theory. The concluding words of the chapter on the Dynamometer Chronograph are singularly to the point in view of the conclusion in § 211 on the comparison in theory of the aerial and marine propeller. Professor Langley says:—

“... It may be said that, notwithstanding the great difference between the character of the media, one being a light and very compressible, and the other a heavy and very incompressible, fluid, these observations have indicated that there is a very considerable analogy between the best form of aerial and of marine propeller.”

§ 236. Langley's Experiments. The “Counterpoised Eccentric Plane.”—An apparatus devised for determining the variations in the positions in the centre of pressure, for varying angles of inclination of a plane to its line of flight.

This appliance follows on established lines, the point of suspension of the plane being fixed for each trial and the angle of equilibrium being experimentally recorded. The result of these experiments has already been given; in the main the work of previous investigators receives confirmation (Fig. 94, Chap. VI., § 148).

§ 237. Langley's Experiments. The “Rolling Carriage.”—This instrument is a highly specialised contrivance for investigating the law of pressure on the normal plane, and for determining with a greater degree of accuracy than previously the value of the constant relating to same.

The instrument consists of a frame beautifully mounted on friction rollers, and recording direct on a chronograph barrel. The wind plane is attached to the front end of a bar, carried forward from the frame and clamped thereto, the pressure on the wind plane being taken by a carefully calibrated spring and the deflection recorded on the chronograph drum.

The experiments made with this instrument proved disappointing, the results, owing to the open air conditions, being no more consistent than those previously obtained with the resultant pressure recorder. The value of $C,$ cited in Chap. VI., is that given by Langley as determined by the rolling carriage; the value, however, is substantially the same as that previously ascertained with the earlier instrument.

§ 238. Langley's Experiments. Summary.—Prof. Langley concludes his account in the Memoir with a summary. Much of this deals with the question of the power required for flight, where naturally the same error is made as elsewhere m the work, the energy necessary to support in a frictionless fiuid alone being taken into account.

It is from no wish to belittle the work of the late Prof. Langley that attention has so frequently been drawn to the point at issue. Langley's name will always stand as one of the most distinguished pioneers of experimental aerodynamics. The whole of the mis-statements to which attention has been directed hinge upon the one fundamental error, that of the assumption of the negligibility of skin friction ; and if the whole Memoir be prefaced by the words, “neglecting the influence of skin-friction,” Langley's position would be substantially regularised.

Professor Langley's work has, however, been widely read, and his statements, unqualified as they stand, have been commonly accepted, and it is therefore impossible in a work of this type to be too emphatic in denouncing the errors in question.

It would seem probable that the publication of the “Experiments in Aerodynamics” was unduly hastened; it would otherwise be difficult to account for the repeated misleading citation of Newton (pp. 4, 8, 15, 24, 25, 89, and 105), when a moment's reference to any reliable edition of the Principia would have prevented any such mistake. Newton dealt with a hypothetical medium clearly defined in the enunciation to prop, xxxiv., and not with air at all, and the proposition cited is perfectly sound.

Beyond this the mathematical analysis constituting Appendix “B” is scarcely convincing. Also the calculation forming the second footnote, p. 9, the details of which are not given, is manifestly conducted on insufficient premises. This calculation purports to be a theoretical proof of the negligibility of skin-friction as confirming the supposed experimental result.

So far as the experimental work itself is concerned, apart from inference, it is undoubtedly the most valuable contribution to our knowledge that has so far appeared, with the exception perhaps of the work of Dines already discussed. The general results of Langley's experiments are entirely confirmatory of the theory set forth in the present work, but the experiments suggest that we have in our theory carried the “small angle” hypothesis to about its limit, and that if we have to deal with angles greater than those tabulated in Chap. VIII. some correction or refinement of method may become necessary.

§ 239. The Author's Experiments.—The author has investigated experimentally many of the problems connected with aerial flight. The greater part of these investigations relate to the subject matter of the later volume, “Aerodouetics,” and only certain experiments having an immediate bearing on the aerodynamics of flight will be dealt with at the present juncture.

A method of experiment that the author has used to some advantage involves the employment of gliding models. Up to the present time the whole question of the stability of such models has been but little understood, and it is necessary to some extent to anticipate the conclusions of the later portion of the work.

It is currently believed that the equilibrium of a bird in flight is essentially maintained by the intervention of the brain and nerve centres, and that an aerodrome or aerodone, in order that it should possess stability, must be fitted with parts capable of ready and rapid adjustment, and furnished with some “brain equivalent,” or be immediately directed by an aeronaut. It may be stated at once that no such provision is necessary, and that a properly designed rigid structure is capable of maintaining its own equilibrium, and possesses complete stability within prearranged limits; and further, that such a rigid structure (or aerodone) may be designed to automatically simulate many of the apparently life-like movements of birds in flight or at will glide steadily at its natural velocity at a constant angle of flight path.

The above are mere bald statements of fact, that will be fully substantiated in the subsequent volume. The behaviour and stability of an aerodone in flight will for the present be taken for granted, an indication of the underlying principles having been given in § 162 on the Ballasted Aeroplane.

§ 240. Scope of Experiments.—The scope of the present series of experiments has been in the main confined to the determination of $\xi$ by a variety of methods.

This quantity, which has been frequently stated to be negligible, is (as has been demonstrated in the present work) one of very great moment in relation to the dynamics of flight, and the determination of $\xi$ with a reasonable degree of accuracy is therefore a matter of prime importance.

Incidentally, data are obtained from which other aerodynamic constants may be deduced, though in this respect the method of free flight experiment has not furnished as reliable data as may be expected in the future when more suitable apparatus has been elaborated.

The various methods employed by the author do not give results that are altogether in accord, but in view of the extent of the general disagreement in the work of other investigators, and in the difficulties of determining $\xi$ in particular, this is in no way surprising.

Experiment apart, there is a primâ facie case for the existence of a coefficient skin-friction of some considerable magnitude in the fact that the similar coefficient, as determined by Froude and others, is, in the case of water, a matter of one per cent, or thereabouts, and in the fact that the kinematic viscosity of air is fourteen times as great as that of water.

§ 241. Author's Experiments. Method.—Three modifications of the free flight method of experiment have been employed; these may be enumerated as follows:—

(1) The Added Surface Method.—In this an aerodone is first constructed on the lines laid down in patent specification 17935 of 1905 (Fig. 151), the auxiliary surface being made about the minimum necessary for stability. The natural gliding angle and velocity are very carefully measured from trial “glides.” The auxiliary surface is then increased by gumming extension laminae on to the fins, care being taken not to alter the total weight or the position of the centre of gravity. Further glides are then made and the angle and velocity are again measured; the added resistance is then calculated from the data obtained, and so the value of the $\xi$ is determined.

(2) The Total Surface Method.—An aerodone is constructed of

Fig. 151.

special design, Fig. 152, the supporting member being a plane whose centre of pressure is known from independent experiment. The tail plane is divided into two portions arranged so as to be as little as possible affected by the wake disturbance;
Fig. 152. this is essential on account of the fact that the angle between the supporting plane and the tail plane is assumed to be the angle made by the former by the line of flight. The computation of $\xi$ is made from gliding data, the whole surface being assumed as subject to skin-friction, or an allowance may be made in respect of the supporting plane on the lines laid down in §§ 182, 183 and 184.

(3) The Method of the Ballasted Aeroplane.—Reference has already been made to this method (§ 162). A number of planes are prepared of exactly the same size and total weight, but with their centres of gravity situated at different distances from their geometrical centres. Trial flights are made, and the resulting data give simultaneous equations from which the values of the constant may be deduced.

§ 242. Author's Experiments. Method (continued).—In addition to the free flight experiments enumerated above, an attempt has been made to effect the direct measurement of $\xi$ by means of a new instrument, which may be appropriately termed an aerodynamic balance.^[6]

The magnitude of $\xi$ as determined by the method of free flight suggested that, in spite of the failure of previous experimenters, it should be quite possible to effect a direct measurement of this quantity by the aid of a suitably designed appliance.

The aerodynamic balance, Fig. 153, consists of a horizontal arm or beam $A,$ pivoted about a vertical axis $B,$ the amplitude of motion permitted being regulated by the screws $C\ C,$ which also form electrical contacts.

For the determination of $\xi ,$ a normal plane $D$ is attached to one end of the beam and a friction plane $E$ to the other, the areas of the two being adjusted until they exactly balance. The instrument is used either by being exposed to the wind and held stationary, or fitted in front of an automobile vehicle in still air.

In either case the planes require to be carefully balanced about the vertical axis so that gravitation and inertia forces are inoperative. When the instrument is held stationary this precaution is unnecessary so long as the axis is exactly vertical, but it is more convenient to have the instrument properly balanced in any case. In spite of every precaution, when the instrument is mounted on a motor car the beam is found to be in a continual state of oscillation between its stops, probably due to slight rotational movements of the car body produced by the unevenness of the road. This difficulty was actually experienced to so great an extent that the employment of the instrument in its present form on a motor vehicle was abandoned.^[7]

The uses of the aerodynamic balance obviously are not

Fig. 153.

confined to the determination of $\xi ;$ the instrument may be used quite generally as a comparator of the resistance of planes of various shapes or of different solid forms.

§ 243. Method of Added Surface.—Mica Aerodone. Series C., No. 1, Fig. 154.

Weight (after adjustment of ballast) = .60 gram.

Aerofoil, elliptical, 4 ${\tfrac {1}{2}}$ in. $\times {\tfrac {3}{4}}$ in.; actual area	=	2.65	sq. in.
Tail plane, area	=	.50	„ „
Back-bone, surface ÷ 2 = equivalent area	=	.48	„ „
Fin area, (without added surface)	=	.14	„ „
		——
Total area, (without added surface)	=	3.77	„ „
		——
Added surface		1.06	„ „

Sept., 1905. Trial of model without added surface.

Launched from 7 ft. Altitude.

	Distance. Ft.	Time.^[8] Secs.	Velocity. Ft./Secs.
1 2 3 4 5 6 7	37 40 34 35 37 32 36	3.2 3.2 3.2 3.0 3.2 2.8 3.2	11.55 12.50 10.62 11.66 11.55 11.43 11.25
Total	251	21.8	80.56
Mean	35.8	3.11	11.50

Whence, $\gamma ={\tfrac {7}{35.8}}=$ .1955 or resistance in line of flight = .1955 $\times$ .60 = .1172 grams.

Sept., 1905. Trial of model, with added surface.

	Distance. Ft.	Time. Secs.	Velocity. Ft./Secs.
1 2 3 4 5 6 7 8	29 29 32 28 34 33 31 36	2.6 2.6 2.8 2.8 ? 2.8 2.6 2.8	11.15 11.15 11.44 10.00 11.80 11.80 12.84
Total	8) 252	7) 19.0	80.18
Mean	31.5	2.71	11.45

Fig. 154. Whence, $\gamma ={\tfrac {7}{31.5}}$ = .222 or resistance in line of flight = .222 $\times$ .60 = .1332 grams.

∴ Resistance due to added surface = .1332 – .1172 = .016 grams.

∴ resistance per sq. ft. added surface = ${\tfrac {.016\times 144}{1.06}}$ = 2.175 grams.

Now pressure per sq. ft. in pounds at 11.5 ft./sec. is given by expression

P={\frac {.7\ \rho \ V^{2}}{g}}=

{\frac {.7\times .078\times 11.5\times 11.5}{32.2}}=.224

which in grams becomes

.224\times 453.6=101.5\ {\mbox{grams}}.

∴

\quad \xi ={\frac {2.175}{101.5}}=.0214

The above example is one of several determinations made by this method. Generally speaking, the flight measurements showed greater variation than in the example given; the day of these experiments was exceptionally calm, and the aerodone used (No. 1) made a long series of good straight glides without mishap; a performance which it is not always easy to obtain. Flights of circular or otherwise curved path need to be rejected.

The results of different series of experiments were found to give values of $\xi$ varying from a trifle over .012 to nearly .030 as a maximum.

Using the value above determined ( $\xi =$ .0214) we may calculate the total skin resistance of the model employed.

Total area (without added surface)

= 3.77 sq. in. = .0262 sq. ft. or

resistance = .0262

\times

101.5

\times

.021 = .0558 grams.

But total resistance = .1172 grams, hence we may audit the resistance account for this model as follows:—

Frictional	=	.0558
Aerodynamic	=	.0614
		———
Total	=	.1172
		———

A result which appears to be quite consistent, as showing the model to be approximately designed for the conditions of least resistance within the limits of experimental error.^[9] (§ 164.)

§ 244. Method of Total Surface.—Mica Aerodone. Series E., No. I. Fig. 155.

This aerodone was one of a series of models specially designed for the purpose of measuring $\xi$ by the method of total surface.
Fig. 155. The supporting member is an aeroplane whose angle to the line of flight is determined by a pair of tail planes whose sole function is directive. It may be noted that the tail planes are so placed as to be influenced equally by the downward and upward components of the terminal vortices, so that no error shall be introduced by an allowance for the “downthrow current,” as would be necessary were the tail plane situated in the ordinary position (Fig. 151).

Weight			.24 grams.
Area	1.125
+	1.030
+	.490
	———
=	2.645	sq. in.
Angle of aeroplane ${\tfrac {1}{20}}.$

Flight Data.

	Distance. Ft.	Time. Secs.
1 2 3 4	30 33 28 21	2 2 1.8 ?	${\begin{cases}{\mbox{Launched}}\\{\mbox{from 10 ft.}}\\{\mbox{altitude.}}\end{cases}}$ ${\begin{cases}{\mbox{Launched}}\\{\mbox{from 7 ft.}}\\{\mbox{altitude.}}\end{cases}}$

Whence,

Mean $V=15.5\ {\mbox{ft./sec.}}$
Mean $\tan \gamma ={\tfrac {1}{3}}$ or $\gamma =18{\tfrac {1}{2}}^{\circ }$ or,
Resistance $=\sin 18{\tfrac {1}{2}}^{\circ }\times .24$	$=.075$
Aerodynamic resistance $=W\beta =.24\times .05$	$=.012$
∴ Skin resistance	——— $=.063$

And area $=2.645$ sq. in. or skin resistance per sq. ft. area—

{\frac {.063\times 144}{2.645}}=3.43\ ({\mbox{grams}}).

{\frac {3.43\times 32.2}{453.6}}=.243\ ({\mbox{poundals}}).

But normal plane reaction at $15.5\ {\mbox{ft./sec.}}$

=.7\times .078\times 15.5\times 15.5=13.1

∴

\quad \xi ={\frac {.243}{13.1}}=.0185

Fig. 156.

A further trial with this model, repaired after damage, gave a value of $\xi =$ .025. The cause of this divergence was not ascertained.

Model No. 2. Series E. (Fig. 156).—Trials with this model gave the result $\xi =$ about .03. This value is probably too high. The weather was unfavourable, and the weight of the model (1.66 grams) proved too great for the method of construction; frequent repairs had to be made in the course of a single series of experiments.

Model No. 3. Series E. (Fig. 152), Construction.— Aeroplane and fins, varnished cedar. Tail planes, mica plates. Body, cedar, ballasted with lead.

Weight = 46.3 grams = .102 lbs.

Area—

	Aeroplane	44
	Tail plane	13
	Fins	12.4
	Body	3.25
		———
		72.65 sq. in. = .504 sq. ft.

Angle of aeroplane $(\beta ),={\tfrac {1}{20}}.$

Preliminary Trial, Aerodone launched by hand, 13 ft. 6 in. altitude.

Flight Data.^[10]

	Distance. Ft.	Time. Secs.	Velocity. Ft./Sec.
1 2 3 4 5	39 42 40 42 42	1.4 1.2 1.4 1.2 1.2	28 35 28 35 35
Mean	5) 205	6.4	161
Mean	41	1.3	32

Whence $\tan \gamma ={\frac {13.5}{41}}=.33$ or $\gamma =18{\frac {1}{4}}^{\circ }\sin \gamma =.313.$

Resistance in line of flight
	$=W\sin \gamma =.102\times .313$	$=.032\ {\mbox{lbs.}}$
Aerodynamic resistance
	$=W\ \beta ={\frac {.102}{20}}$	$=.005\ \ {\mbox{„}}$
		—————
∴ Skin resistance	. . .	$=.027$

or per sq. ft. $={\frac {.027}{.504}}=.0536$ (pounds) $=1.725$ poundals.

But the normal plane reaction,

P=.7\times .078\times 32\times 32=55.8

(poundals)

or

\xi ={\frac {1.725}{55.8}}=.0309

It was evident in the course of the above trial that sufficient velocity was not being given to the aerodone, that is to say, its projected velocity was less than its natural velocity and that the necessary velocity could not be given by hand throwing without sacrificing accuracy.^[11] A further series of trials made with a catapult launching device gave data as follows:—

Aerodone launched from 20 feet altitude.

	Distance. Ft.	Time. Secs.	Velocity. Ft./Sec.
1 2 3	66 62 68	1.8 1.8 1.8	36.6 33.2 37.8
4	Model collided with tree and damaged.
	3) 196		107.6
Mean	65	1.8	36

Whence, $V=36\ {\mbox{ft./sec.}}\tan \gamma =.246$ or $\gamma =13^{\circ }-50'$ or $\sin \gamma =.239.$

Resistance in line of flight
	$=W\sin \gamma =.102\times .239=$		$.0244$
Aerodynamic resistance
		$=W\beta ={\frac {102}{20}}=$	$.0050$
∴ Skin resistance (lbs.)		=	——— $.0194$

or in poundals per sq. ft.

$={\frac {.0194\times 32.2}{.504}}=1.24$

But normal plane reaction at 36 ft./sec. (poundals),

$P=.7\times .078\times 36\times 36=71$

If we make an allowance in respect of the aeroplane in accordance with §§ 182, 183, and 184, deducting half the area, we have, 72.65 – 22 = 50.65 sq. in. = .351 sq. ft. in lieu of .504 as above. Or skin resistance in poundals per sq. ft.

$={\frac {.0194\times 32.2}{.351}}=1.78$

∴ $\quad \xi ={\frac {1.78}{71}}=.025$

In the case of an aerodone having a natural velocity as high as 36 ft./sec, it is impossible to be sure, in so short a flight as 65 feet, that the true natural velocity and gliding angle are recorded; in a short flight the launching velocity and angle have a serious influence on the flight path.

If the altitude of discharge could be increased to 50 feet or thereabouts, with a flight path of some 200 feet length, this difficulty would be overcome, or at least its importance would be reduced to a negligible quantity.^[12]

§ 245. The Method of the Ballasted Aeroplane.—The method of the ballasted aeroplane not only permits of the determination of the coefficient of skin-friction but simultaneously provides data from which the constant $c$ and the relation between the angle and centre of pressure of the aeroplane may be calculated.

The following examples will serve for the purposes of illustration. A standard form of aeroplane has been employed throughout (Fig. 109), measuring 8 in. by 2 in., and ballasted by a lead shot presenting a resistance taken as equivalent to .025 sq. in. of normal plane. The weights of different planes employed for any given series of experiments are all brought up to the same amount by gumming lead-foil in the region of the centre of gravity, the only difference between the different planes of a series being the position of the centre of gravity, and therefore the position of the centre of pressure, and consequently the angle of equilibrium.

The launching of the planes was in all cases effected by means of a launching stick, the aeroplane being placed on a small platten on the top of a straight stick, the lower end of which is held about shoulder high, the act of launching being accomplished by swaying the body so as to give an approximately parallel motion. A certain degree of skill is easily acquired, and a reasonable percentage of good straight flights may be obtained without difficulty.

Example,

Two planes, weight 5 ${\tfrac {1}{4}}$ grams (.372 poundals).

Launching data.

	Velocity.	7.5 ft. Altitude.
No. 1 No. 2	15 ft./sec. 12.5 ft./sec.	47 ft. mean glide. 35 ft. mean glide.

Or,	$V_{1}=15\ {\mbox{ft. /sec.}}$
	$V_{2}=12.5\ {\mbox{ft. /sec.}}$
	$\gamma _{1}={\frac {7.5}{47}}=.160$	^[13]
	$\gamma _{2}={\frac {7.5}{35}}=2.14$
	$x_{1}+y_{1}=.160\times 5.25=.84$	(1)
	$x_{2}+y_{2}=.214\times 5.25=1.12$	(2)
Now we know that $x=n\ V^{2}$ and $y={\frac {m}{V^{2}}}$ where $n$ and $m$ are constants,
or,	$x_{1}=15^{2}n,\quad y_{1}={\frac {m}{15^{2}}}$	(3)
	$x_{2}=12.5^{2}n,\quad y_{2}={\frac {m}{12.5^{2}}}$	(4)
By (1) and (3)	$15^{2}n+{\frac {m}{15^{2}}}=.84$	(5)
By (2) and (4)	$12.5^{2}n+{\frac {m}{12.5^{2}}}=1.12$	(6)
∴	$12.5^{2}n+{\frac {(.84-15^{2}n)\times 15^{2}}{12.5^{2}}}$
∴	$156\ n-325\ n=1.12-1.21$
or,	$169\ n=.09$
	$n=.000532$
or,	$x=.000532\ V^{2}\ {\mbox{grams.}}$

multiplying by ${\frac {144}{16}}$ to obtain grams per square foot, and by ${\frac {32.2}{453.6}}$ to convert to British absolute units, we have—

$V^{2}\times .000532\times {\frac {144}{16}}\times {\frac {32.2}{453.6}}$ $=.00034\ V^{2}$

But normal plane pressure is given by expression—

$P_{90}=C\ \rho \ V^{2}$

or, $\quad \xi ={\frac {.00034}{C\ \rho }}={\frac {.00034}{.7\times .078}}=.0062$

In the foregoing calculation no allowance has been made for the direct resistance of the ballast. Taking this as the equivalent of a normal plane area of .025 sq. in., or .000174 sq. ft., and multiplying by ${\frac {144}{16}}$ to bring up to a per sq. ft. basis, we have .00156, or resistance per sq. ft. due to ballast

$=.00156\ C\ \rho \ V^{2}$

or $\xi ={\frac {.00034-.00156\ C\ \rho }{C\ \rho }}$

$={\frac {.00034}{7\times .078}}-.00156$ $=.0062-.0015$

$\xi =.0047$

Again,

Two planes, weight 5.9 grams (.418 poundals).

	C.g. distance from front edge.	Velocity.	8 ft. altitude.
No. 3 No. 4	25% 30%	17 ft./sec. 13 ft./sec.	54 ft. 37.5 ft.

or,	$V_{3}=17{\mbox{ft./sec.}}$
	$V_{4}=13{\mbox{ft./sec.}}$
	$\gamma _{3}={\frac {8}{54}}=.148$
	$\gamma _{4}={\frac {8}{37.5}}=.213$
	$x_{3}+y_{3}=.148\times 5.9=.875$	(1)
	$x_{4}+y_{4}=.213\times 5.9=1.26$	(2)
	$x_{3}=17^{2}\ n,\quad y_{3}={\frac {m}{17^{2}}}$	(3)
	$x_{4}=13^{2}\ n,\quad y_{4}={\frac {m}{13^{.2}}}$	(4)
By (1) and (3)	$17^{2}\ n+{\frac {m}{17^{2}}}=.875$	(5)
By (2) and (4)	$13^{2}\ n+{\frac {m}{13^{2}}}=1.26$	(6)

∴	$13^{2}\ n+{\frac {(.875-17^{2}n)\times 17^{2}}{13^{2}}}=1.26$
∴	$169\ n-494\ n=1.26-1.495$
or,	$325\ n=.235$
	$n=.000724$
or,	$x=.000724\ V^{2}\ {\mbox{grams.}}$
or, in poundals per square foot

$000724\times {\frac {144}{16}}\times {\frac {32.2}{453.6}}=.000461$

$\xi ={\frac {.000461}{.7\times .078}}-.0015$

$=.0085-.0015=.007$

the deduction .0015 being made, as in the last example, for ballast resistance.

Determination of constant $c.$

$W=A\ P_{\beta }=A\ c\ \beta \ P_{90}$ $=A\ c\ \beta \ C\ \rho \ V^{2}$

or,	$\beta ={\frac {W}{A\ c\ C\ \rho \ V^{2}}}=1.26$
but,	$y=W\ \beta \quad$ ∴ $\quad y={\frac {W^{2}}{A\ c\ C\ \rho \ V^{2}}}$
and,	$y={\frac {m}{V^{2}}}$
∴	$m={\frac {W^{2}}{A\ c\ C\ \rho }}$
or,	$c={\frac {W^{2}}{A\ m\times .0546}}$

all quantities in absolute units.

Thus, for the determination of $c$ in any particular case the value of $m$ must first be obtained from the equations, the remaining quantities in the expression $A$ and $W$ being the area (sq. ft.) and weight (poundals) of the aeroplanes employed.

Example.—Planes 1 and 2.

Flight data as given.

By (5)	$m=15^{2}\ (.84-15^{2}\ n)$
where	$n=.000532$

or,

m=(225\times .84)-(225\times .1197)

$=189-26.9$

$=162.1$

This is the value of $m$ for $y$ expressed in grams; for $y$ in poundals this becomes—

$m=11.5$

and for the aeroplanes in question,

$A=.111$ and $W=.372$

∴ $\quad c={\frac {.372^{2}}{.111\times 11.5\times .0546}}=1.98$

This is about the value as determined directly by Duchemin, Dines and Langley for the square plane; it is probably too low for a plane of $n=4$ as used in these experiments.

Example.—Planes 3 and 4.

	$m=17^{2}\times (.875-17^{2}\ n)$
where	$n=.000724$
whence	$m=192.5$
or, when absolute units are employed, $m=13.6.$
	$A=.111\quad W=.418.$
∴	$c={\frac {.418\times .418}{.111\times 13.6\times .0546}}=2.11$

a result which is still probably less than the true value.

Calculation of $\beta .$

$\beta ={\frac {y}{W}}={\frac {m}{W\ V^{2}}}$

Taking planes 3 and 4.

Plane No. 3.

$\beta ={\frac {192.5}{5.9\times 289}}=.112\quad$ $\quad \beta ^{\circ }=6.5^{\circ }$

or,

Plane No. 4.

$\beta ={\frac {192.5}{5.9\times 169}}=.193\quad$ $\quad \beta ^{\circ }=11.1^{\circ }$

Plotting these results on the basis of § 148 for the known positions of centre of pressure for these planes, we have diagram Fig. 157.

The divergence shown in the above determinations is largely due to the temporary and insufficient character of the apparatus employed, and to the fact that for want of suitable accommodation the experiments were conducted out of doors.

It is further possible that the considerations raised in §§ 182, 183, to some extent invalidate the present method. So long as the type of fluid motion in the periptery of the aeroplane is frankly discontinuous the method will in theory give consistent results, but so soon as the live stream touches the upper surface
Fig. 157. of the plane, as it must do when the angle becomes very small, the area subject to skin-friction will increase in some way as an inverse function of the angle, and the equation $x=n\ V^{2}$ will cease to hold good. We may consequently anticipate that when the angle $\beta$ becomes less than some critical value the curve will cease to be of the form plotted in Fig. 112, and the present method will break down. It is principally for this reason that the author has confined his observations to the low velocity portion of the curve; it will be time enough to carry these observations further when better launching and measuring appliances have been developed.

§ 246. Determination of $\xi$ by the Aerodynamic Balance.—In the determination of $\xi$ by the aerodynamic balance, one arm of the beam A. Fig. 153, is furnished with a lead block D. Fig. 158, whose sectional form is an isosceles triangle, the base of which is formed by the face in presentation, being the normal plane the pressure on which constitutes a measure of the reaction on

Fig. 158.

the friction plane. The area of the normal plane can be extended at will by cementing a mica plate to its face, the edges of which are clipped until exact balance is obtained.

Fig. 159.

The opposite arm of the beam carries the friction plane E (Fig. 159); this is carefully formed of cedar of 6 mm. maximum thickness, the grain being well filled and served with a thin coat of varnish, or otherwise finished as may be required. The friction plane is carried on a holder formed by two pen steel plates J riveted to a shank K which fits into a socket in the balance arm A (Fig. 158) being secured by a set screw L.

In making any determination the area of the normal plane is adjusted until the beam is in equilibrium. The coefficient of skin-friction $\xi$ is then calculated from the relation of the areas of the normal and friction planes multiplied by their respective distances from the pivot axis.

Determination, June 19th, 1907, Cobley Hill, Alvechurch.

Wind velocity (estimated) 20 to 40 miles per hour.^[14]

Friction plane No. 1, cedar shellac varnished and roughly polished. In pterygoid aspect.

	Length.		Breadth.		Leverage.
Normal plane	2.5″	$\times$	.95″	$\times$	3.25	= 7.7
Friction plane	16.25″	$\times$	5″	$\times$	10.12	= 823

$\xi ={\frac {7.7}{823}}=.00936$

Determination, June 23rd, 1907.

High wind.

Friction plane No. 2, cedar filled and water gilt and burnished. In pterygoid aspect.

	Length.		Breadth.		Leverage.
Normal plane	2.25″	$\times$	.9″	$\times$	3.25	= 7.3
Friction plane	16.25″	$\times$	5″	$\times$	10.12	= 823

$\xi ={\frac {7.3}{823}}=.00888$

Friction plane No. 1 (polished cedar), substituted for No. 2 as above, showed no appreciable change of balance.

With width of normal plane increased to 1 in., both planes, Nos. 1 and 2, gave insufficient reaction to balance pressure on normal plane. It is therefore to be concluded that for a well varnished surface or for polished metal, under the conditions of experiment, the effective value of $\xi$ is approximately .09, with a probable error of less than 10 per cent, plus or minus.

Roughened surfaces. June 23rd, 1907 (later).

Wind, as before.

Friction plane, covered Oakey's No. 2 ${\tfrac {1}{2}}$ glass paper.

	Length.		Breadth.		Leverage.
Normal plane	2.5″	$\times$	1″	$\times$	3.25	= 8.1
Friction plane	12.25″	$\times$	4.6″	$\times$	8.12	= 458

$\xi ={\frac {8.1}{458}}=.0177$

On the face of it the method of the aerodynamic balance is so direct and straightforward as to leave no possibility of doubt as to the validity of the results. Under constant wind conditions it seems that a change of 10 per cent, can be readily detected, and it appears, therefore, fair to assume that this is the outside limit of experimental error.

If it had been found possible to conduct experiments in still air with the instrument in motion, it would have been difficult to resist the above conclusions; it is, however, by no means certain that, under the conditions of wind reaction experiment, the matter is quite as simple as it appears.

There seems to be some possibility that the normal plane and friction plane are not equally affected by the wind turbulence; it is even uncertain whether the influence of turbulence is in the same direction in the two cases. We know the effect of turbulence in the case of the normal plane is probably to increase the pressure reaction (§ 131), but it is by no means established that the effect is the same in the case of skin-friction. If the direction of motion of turbulence were confined to the line of motion of the main translation, it would certainly seem that the influence of turbulence would be to increase the frictional drag. If, as is actually the case, the motion of turbulence have a component at right angles to the friction plane, it is conceivable that it will give rise to discontinuity in the system of flow, which, on the principles discussed in §§ 182, 183, and 184, may actually result in a diminution in the tangential reaction.

§ 247. Author's Experiments. Summary.—The experiments described in the preceding sections are quite convincing from a qualitative point of view, although quantitatively speaking the results are inconclusive.

Any and all of the methods described should be capable of giving results of a reasonable degree of accuracy—far more so than at present achieved—and the results so obtained should be in closer accord, one with another, than the author has so far been able to demonstrate.

The deficiency in the present experiments is chiefly that of apparatus and opportunity. The launching of free flight models requires a suitable apparatus to be designed, by which the initial velocity shall be placed under definite control; beyond this it must be considered quite essential, if reliable results are required, that experiments should be conducted inside a building; the absolute calm necessary for aerodynamic determinations is so rare a phenomenon as to render outdoor experiment almost impossible. It is only those who have watched and waited for a really calm day who can fully appreciate its rarity. In repeating these experiments it would be well to arrange for the use of a large hall, well secured against draughts; the equilibrium of low velocity models, such as it is necessary to employ, is very sensitive; even the previous motion of a person across the line of flight will affect the gliding path. High velocity models, although possessing greater stability, are not well suited to the determination of aerodynamic data.

The author's conclusions as to the value of $\xi$ have been given in § 157. Some of the experiments here recorded have been made since these conclusions were formulated, but the differences are not such as to render revision necessary. In brief, it would appear that under all practical conditions the coefficient of skin-friction lies between the values .01 and .03, rarely being less than the former or greater than the latter.

It is, perhaps, of some interest to record the fact that for air in motion in a pipe the accepted resistance coefficient gives, on the present basis of computation (i.e., for a double surface in terms of the pressure on a normal plane of equal area), a value of $\xi =$ .016, which is in substantial agreement with the present conclusions, in spite of the totally different conditions that obtain.

The author considers that the method of the ballasted aeroplane has not at present had a fair chance of showing its capabilities. The method is one that demands considerable nicety of manipulation. In the absence of any mechanical launching device, it is quite easy to obtain faulty data if any but straight uniform glides are recorded, and if such data are utilised it is as likely as not the values of the constants deduced will be wide of the mark, even negative values being sometimes obtained. The method is one of which the advantages have only very recently impressed themselves on the author, and time and opportunity have been at present lacking to carry out more than a few rough preliminary experiments. The present publication, in this respect, must therefore be regarded more in the light of an exposition of method than a serious experimental demonstration.

↑ In the present account of the investigations of Mr. Dines, the following publications have been consulted:—"Some Experiments made to Investigate the Connection between the Pressure and the Velocity of the Wind" (Dines, Quart. Journ. Royal Met. Soc., Vol. XV., October, 1889), and "On Wind Pressure upon an Inclined Surface" (Dines, Proc. Royal Soc., Vol. XLVIII., 1890). Further particulars of Mr. Dines' experiments of aerodynamic interest will be found in the following:—"Mutual Influence of two Pressure Plates upon each other," and "On the Variations of Pressure caused by the Wind Blowing across the Mouth of the Tube" (Dines, Quart. Journ. Royal Met. Soc., Vol. XVI., October, 1890).
↑ This expression differs somewhat in form from that given by Dines, the difference being due firstly to the introduction of the area (A) of the plane, and the restriction of the use of the term pressure as being of the dimensions force divided area, and secondly to the difference in the units employed, the use of absolute units eliminating the gravitation constant.
↑ Vide “Principia.” (The author relies on the translation by Andrew Motte, 1803.)
↑ These words are fully justified. If the correction were required for the reason stated then it would be of many times greater magnitude when the plane is normal than when it presents a small angle to the line of flight. Langley says: “This flexure seemed to obtain for all angles of experiment, but with indications of slightly diminishing effect for the smaller ones.”
↑ A crude makeshift.
↑ A prototype of the aerodynamic balance was employed by Dines (see § 49).
↑ The difficulty could be overcome by re-designing the apparatus with two beams having opposite rotary movement. (Comp. § 227.)
↑ Taken by stop watch.
↑ We may compare the pressure relation of this model with the values laid down in Table IX., § 185.
Area (effective) of aerofoil on ${\frac {2\ L^{2}}{3\ {\mbox{n}}}}$ basis is
${\frac {20.25\times 2}{6\times 3}}$ $=2.25\ {\mbox{sq. in.}}=.0156\ {\mbox{sq. ft.}}$
or weight carried $={\frac {.6}{.0156}}\ {\mbox{grams per sq. ft.}}=38.5$ or ${\frac {38.5\times 32.2}{45.36}}=2.73\ {\mbox{poundals}}.$
Now $V=11.5$ or $V^{2}=132$ or $P/V^{2}={\frac {2.73}{1.32}}=.0214$ against $.0286$ given in Table IX.
This difference is not more than might be expected in view of the present state of knowledge.
↑ Flights of curved or irregular path, were not recorded. Time taken with stop watch reading to .2 second.
↑ Any rotational movement imparted at the moment of projection is most detrimental to the gliding path.
↑ The author had intended repeating these experiments under more favourable circumstances, but the difficulty of hitting the right weather conditions, at an appointed place, away from home, at a time that is otherwise convenient, has hitherto proved insuperable.
↑ In these experiments the hypothesis of the small angle is taken as applying to $\gamma$ values; $\gamma$ is expressed in radians.
↑ Determinations made with aerodynamic balance are approximately independent of velocity of wind; a rough estimate is sufficient for the purposes of record.

[1] In the present account of the investigations of Mr. Dines, the following publications have been consulted:—"Some Experiments made to Investigate the Connection between the Pressure and the Velocity of the Wind" (Dines, Quart. Journ. Royal Met. Soc., Vol. XV., October, 1889), and "On Wind Pressure upon an Inclined Surface" (Dines, Proc. Royal Soc., Vol. XLVIII., 1890). Further particulars of Mr. Dines' experiments of aerodynamic interest will be found in the following:—"Mutual Influence of two Pressure Plates upon each other," and "On the Variations of Pressure caused by the Wind Blowing across the Mouth of the Tube" (Dines, Quart. Journ. Royal Met. Soc., Vol. XVI., October, 1890).

[2] This expression differs somewhat in form from that given by Dines, the difference being due firstly to the introduction of the area (A) of the plane, and the restriction of the use of the term pressure as being of the dimensions force divided area, and secondly to the difference in the units employed, the use of absolute units eliminating the gravitation constant.

[3] Vide “Principia.” (The author relies on the translation by Andrew Motte, 1803.)

[4] These words are fully justified. If the correction were required for the reason stated then it would be of many times greater magnitude when the plane is normal than when it presents a small angle to the line of flight. Langley says: “This flexure seemed to obtain for all angles of experiment, but with indications of slightly diminishing effect for the smaller ones.”

[5] A crude makeshift.

[6] A prototype of the aerodynamic balance was employed by Dines (see § 49).

[7] The difficulty could be overcome by re-designing the apparatus with two beams having opposite rotary movement. (Comp. § 227.)

[8] Taken by stop watch.

[p375-9] We may compare the pressure relation of this model with the values laid down in Table IX., § 185.
Area (effective) of aerofoil on ${\frac {2\ L^{2}}{3\ {\mbox{n}}}}$ basis is
${\frac {20.25\times 2}{6\times 3}}$ $=2.25\ {\mbox{sq. in.}}=.0156\ {\mbox{sq. ft.}}$
or weight carried $={\frac {.6}{.0156}}\ {\mbox{grams per sq. ft.}}=38.5$ or ${\frac {38.5\times 32.2}{45.36}}=2.73\ {\mbox{poundals}}.$
Now $V=11.5$ or $V^{2}=132$ or $P/V^{2}={\frac {2.73}{1.32}}=.0214$ against $.0286$ given in Table IX.
This difference is not more than might be expected in view of the present state of knowledge.

[10] Flights of curved or irregular path, were not recorded. Time taken with stop watch reading to .2 second.

[11] Any rotational movement imparted at the moment of projection is most detrimental to the gliding path.

[12] The author had intended repeating these experiments under more favourable circumstances, but the difficulty of hitting the right weather conditions, at an appointed place, away from home, at a time that is otherwise convenient, has hitherto proved insuperable.

[13] In these experiments the hypothesis of the small angle is taken as applying to $\gamma$ values; $\gamma$ is expressed in radians.

[14] Determinations made with aerodynamic balance are approximately independent of velocity of wind; a rough estimate is sufficient for the purposes of record.

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