riiiinV,[y.,?r.f,4M">'"-iA s. iiiiiiiii ■iiiiiii 3,1822 01l3fi 0484 ^ LIBRARY ^ UNIVERS CALIFOR SAN DIE SI IfORNIA SAN DIEGO E 3 1822 01136 0484 University of California, San Diego Please Note: This item is subject to recall. Date Due t 1 CI 39a (4/91) UCSDLib. ah SSE RESISTANCE OF AIR RESISTANCE OF AIR BY LiEUT.-CoL. R. DE VILLAMIL R. ENG. (ret.) " I have always been, am, and propose to remain a mere schcSar. All that I have ever proposed to myself is to say, this and this I have learned ; thus and thus have I learned it ; go thou and learn better ; but do not thrust on my shoulders the responsibility for your laziness if you elect to take, on my authority, conclusions, the value of which you ought to have tested for yourself.' HUXLEV. 22 ILLUSTRATIONS AND 35 TABLES XonDon: E. & F. N. SPOX, Ltd., 57 HAYMARKET, S.W. i IRcw IJoik: SPON & CHAMBERLAIN, 120 LIBERTY STREET 191; BY THE SAME AUTHOR A B C of Hydrodynamics. With 48 illustrations, xi+135 pp. 8vo, cloth (1912). 6s. net. The Laws of Avanzini. Laws of planes moving at an angle in Air and Water. With 1 folding plates, 3 illustrations in text. 23 pp. Super royal 8vo, sewed (19 12). 2S. net. Motion of Liquids. With 86 illus- trations and 30 tables, xiv + 210 pp. 8vo, cloth (19 14). ys. 6d. net. F f'f F N SPON T rn 57 HAYMARKET, XL,. *^ r. i\. OI^V^iN, J-,lU.j LONDON, S.W. i TO THE flDcmor^ OF CHARLES HUTTON LL.D., F.R.S., ETC. LABORIOUS AND PAINSTAKING AUTHOR WHOSE EXPERIMENTAL WORK (much OVERLOOKED TO-DAY) LAID A FIRM FOUNDATION FOR A SOUND THEORY OF AIR RESISTANCE THIS LITTLE BOOK IS RESPECTFULLY DEDICATED CONTENTS Frontispiece — PORTRAIT OF CHARLES IIUTTON PREFACE . . CHAl'. 1. INTRODUCTION ...... 2. DYNAMICAL SIMILARITY .... 3. FORCE 4. PRESSURE — SCALAR AND VECTOR QUANTITIES — VISCOSITY — KINEMATIC VISCOSITY . 5. RESISTANCE OF FLUIDS EXAMINED TIIEORETI CALLY ....... 6. VALUES OF K . 7. RESISTANCE IN WATER SPHERES VALUE OF K FOR 8. RESISTANCE OF SPHERES FALLING IN \VATER 9. RESISTANCE OF BODIES IN CIRCULAR MOTION 10. THE RESISTANCE OF AIR .... 11. RESISTANCE OF AIR AT LO\V VELOCITIES . 12. RESISTANCE OF SHOT AT HIGH VELOCITIES (BALLISTICS) ...... 13. RESISTANCE TO WIND OF BODIES AT REST 14. BALLISTICS CONTINUED — ZAIIM, PIOBERT, DIDION HELIE, MORIN, PERRY .... 15. RESISTANCE OF BULLETS AT VERY HIGH VELOCI TIES — WORK OF MR A. MALLOCK . PAGE ix I 8 16 29 40 48 55 62 7^ 85 93 103 117 131 viii Resistance of Air CHAl'. HAGE 16. EXPERIMENTS WITH WHIRLING MACHINES: LANGLEY, DINES, COLONEL RENARD, COLONEL BEAUFOY, PROFESSOR W. C. KERNOT, FINZI AND SOLDATI 1 45 17. WIND TUNNEL EXPERIMENTS — STANTON, BAIR- STOW, EIFFEL 1 64 18. EIFFEL'S EXPERIMENTS — "DIMENSIONAL EFFECT " — "ADDED MASS" — CONCLUSION . . .174 INDEX . , . . . . . . .190 PREFACE M. LUCIEN POINCARE has said that, in science, the reahy essential thing is to have, as a guide through the unknown, a map which certainly does not claim to represent all the aspects of Nature, but which, having been drawn up accord- ing to predetermined rules, allows us to follow an ascer- tained road in the eternal journey towards Truth. Such a map I offer the reader. It can, perhaps, lay no claim to minute accuracy ; but it will, I trust, enable him to find his way about in the science of aerodynamics — the dynamics of air. In The Great Instauration it is said that it is wiser to engage in an undertaking that admits of some termination, than to involve oneself in perpetual exertion and anxiety about what is interminable. The ways of contemplation, indeed, nearly correspond to two roads in Nature, one of which, steep and rugged at the commencement, terminates in a plain ; the other, at frst smooth and easy, leads to huge rocks and precipices. Following this good advice, I shall not attempt to treat of the whole subject — which is, indeed, interminable, — but confine myself to a very small part of it : the lazv of the resistance, and the coefficie?it of resistance of flat plates of about one foot square. How the resistance is altered bv shape is outside my present purpose, with one exception, and that is, the coefficient of shape for a sphere. As I elect to take the "first road in Nature," it must be expected to be a little steep and rugged at the commence- ment ; but it terminates in a plain, where the reader will see that a good deal that appeared contradictory and con- fusing is far from unintelligible. He will be, I trust, well rewarded for his trouble. X Resistance of Air I rely very largely on the experiments made on the resistance of shot at very high velocities for acting as a check on my coefficients. When a small number will stand being multiplied by several niilliofis — some even by thousands of millions — and still give correct results, one can have considerable faith in its accuracy. This, indeed, is my touchstone. I have subjected all the experiments I am acquainted with to careful examination. If I have omitted any it was not from intention. I have not even left out many which are generally considered "unreliable" ; because, I can only suppose, they do not fit in at all well with the views at present in fashion. My advice to the non-expert reader is : read the book through from cover to cover — even if there are parts he does not understand. After an interval, read it through again. Many of the points which he could not grasp on the first reading will now appear quite simple. After another interval, read it chapter by chapter, and carefully follow each piece of reasoning. I specially commend to his attention the chapters dealing with dynamical similarity and dimen- sions. Once he has grasped this subject (and it is by no means difficult), he will find it very fascinating, besides being an excellent guide as to whether an argument is sound. 1 trust that, by this means, even the non-mathematical reader will be induced to say of mathematics that — as Artemus Ward said of "baked beans" — "They are a cheerful fruit when used tempritely. " As on previous occasions, I wish to acknowledge my great indebtedness to Mr Lewis R. Shorter, B.Sc. , for his great help and acute criticism on all the mathematical questions; to Colonel H. E. Rawson, C. B. , R. E., for many valuable suggestions ; and to Mr Charles Spon for the preparation of the Index, which adds so much to the value of any scientific book. R. DE VILLAMIL. Nov. 19 1 7. RESISTANCE OF AIR CHAPTER I INTRODUCTION In treating of the resistance of fluids, I first wrote the A B C of Hydrodynamics.^ This was not supposed to be a work reducing the subject to a kind of "A, B, C " : it was intended to be an introduction to the fundamental questions on which this science is based. The keynote of the book was the reason why the mathematicians and the engineers were so frequently at "loggerheads" on certain questions of the flow of liquids. I showed, I trust satisfactorily, that the reason was that the mathematicians always assume their imaginary liquid to have no free surface ; whilst the engineer, very ordinarily, finds that his liquid has a free surface. I pointed out that, in certain cases, water which has no free surface acts as the mathematicians say it should : whilst if it has a free surface, it behaves quite differently ; and, in some cases, in an exactly opposite manner. The reason for this being that the mathematician, by his assumptions, eliminates the action of gravity : whereas, when the liquid has a free surface, gravity must, generally, be taken into account. Since, during the last five years, no reviewer has attempted to contradict this — not one has even referred to this question at all, — I suppose I may assume that what I said was correct. This will act as a kind of bridge between the mathematicians and the engineers, enabling them to visit one another ; and, in the case of the latter, 1 E. & F. N. Spon, 191 2. 2 Resistance of Air will show them wlicn the mathematicians can be useful to them, and when not. In the JMotiou of Liquids ^'^ 1 continued the subject, drawing special attention to the difference between "static" and "non-static" liquids. The dominant note in this book is that the resistance of a body moving in water at rest is not, necessarily, the same as the resistance this same body, at rest, experiences to the flow of the same liquid, moving at the same relative velocity. One distinguished reviewer objected to my basing arguments on " Dubuat's hundred- year-old experiments." The reason is not obvious, since Dubuat was a very good observer. I cannot understand what is supposed to be the difficulty in understanding my point of view. It is well known, and accepted by everybody, that if a body is moving, in a liquid at rest, witli accelerated motion, the resistance is greater than it would be if it moved with uniform velocity. This has been accepted because Bessel (a mathematician) pointed it out — quite forgetting the fact that Dubuat had stated the same thing forty years before Bessel. If this is true — and nobody will contradict it — where is the difficulty in believ- ing that it is equally true when the conditions are reversed ? That is to say, when the body is at rest and the fluid is accelerated. Under these conditions, the resistance experi- enced by the body should be greater than when the "relative velocity" is uniform. Why, then, should one be surprised if experiment shows that such is the case ? To make the comparison, it is not only necessary that the relative velocity should be the same : the relative motion must also be the same. You cannot compare a uniform motion with an accelerated motion. It may be said that all the experimenters will insist, and be prepared to prove by X-f-Y, that the air they experiment with is not being accelerated — that its motion is uniform. It will be difficult, however, to show how the air is caused to travel through a wind-tunnel if it is not accelerated, by a fan or some other means. ^ E. & F. N. Spoil, 1914. Introduction 3 A static fluid may be defined as one which is ncitlier changing its velocity nor its shape. Of these two, the latter appears to be the more important. Experimenters do all they can to prevent the air from clianging its sJiape : they " comb the curl " out of it as much as they can, but it is not possible for them to prevent it changing its velocity. Very curiously, however, though they do all they can to prevent the air curling, they frequently remark that the air is in " eddy motion " — as if that were what they specially desired to experiment with. Doubtless, air that has been "well combed" can be made to act (just a^ can water) as if it were a static fluid : i.e. that it will cause impact with- out shock. Though it is absurd to say that any accelerated air is a "static" fluid, it may app7'oxiinate to it. In the present book I treat, essentially, of the resistance of air. As this is a very vast subject, I shall confine myself strictly to the law of the resistance and the coefficients to be used iox flat, square, or circular plates — w^ith one or two small exceptions. I shall again be guilty of basing argu- ments on the experimental work of Dubuat and Duchemin. In fact, so impenitent am I, that I shall do worse, and refer to experiments of the Chevalier de Borda, Mutton, and even Newton. I will not, however, omit to draw attention to the modern work (even up to the last decade), of which there is an enormous mass to choose from. Much of this is, indeed, quite undigested ; and there are times when it is difficult to " see the wood in consequence of the trees." Some of the experiments occasionally appear to contradict others ; so that frequently it is not easy to say which are right. It will be my endeavour to try and put some kind of order into these questions. I shall not go out of my way to seek contentious matter ; though I shall certainly not avoid it. I hope the reader will not find the commencement a little "dry." He may not, perhaps, see what "dynamical simi- larity " — whatever that expression may mean — has to do with resistance of air. He may also be inclined to think that I am, at times, quibbling about the exact meaning of words, when the importance of this is not self-evident. 1 4 Resistance of Air cannot do better than remind him of what Father Bayma ^ said : — "Scientific men would lose nothing, and gain not a little, by speaking more correctly than they often do. In any branch whatever of science, a terminology, which of its own nature leaves room for ambiguity, can only multiply the chances of error." The necessity for clear thinking (especially by students) cannot be overrated, and this can only be developed by the proper use of terms. The reader must remember that I am leading him by the "first road in Nature " of the great instauration^ and that he will find the advantages of this later. Someone has said that orthodoxy and intellect are in- compatible and mutually destructive. This is quite true if "orthodoxy" be considered as a kind of worship of authority. One should accept no authority but Nature — without examination, of course. There is no subject that I know which lends itself more easily to the propagation of absurdities than aerodynamics. If this is considered un- just, it is only necessary to refer to the very numerous books which speak of "soaring Alight," or " sailing, flight " {vol a voile) of birds. One author explains it by ' ' suction " — whatever that may be ; another that the bird abstracts energy from the air (not the wind, as he is careful to say) ; another says the bird can only "soar "when the sun shines, because the bird gets energy from the sun, and so on. Would it not be simpler to try mechanics ? Why, unnecessarily, assume a new property of the air called " soarability," which is neither explained nor defined, and which, on the face of it, appears rather childish ? The ordinary explanation of what is commonly called "auto- rotation" implies (implicitly) "perpetual motion" — another absurdity. How many reams of paper have been covered by mathematicians to explain the action of the screw propeller ? Every man his own theory. Does one of them understand how a propeller works} If he did (which I deny) he ought to be able to say why you only get some 70 per cent, efficiency out of it, ^ Molecular Mechanics, 1 866. Introduction 5 If the reader will have patience and follow the examina- tion of the results of all the experiments I shall refer to, I think 1 can promise him that he will have clearer ideas of resistance of air than he has ever had before : and he will find the subject is not so hopelessly complicated or difficult as he thought. My object is to attract young students. Such being my aim, I regret that I have had to introduce so much mathe- matics (not a popular subject to many readers) : since it was unavoidable, I must confess that I do not apologise for having done so. At the British Association Meeting in South Africa in 1905, there was a discussion on the "Teaching of Ele- mentary Mechanics," the report of which is very instructive, besides being distinctly amusing. One of the speakers, W. H. Macauley of King's College, Cambridge, made the following remarks : — " No one can master anything without thinking about it independently, and I should hesitate to smooth down any road in such a way as to discourage inde- pendent tliougJit. But there are certain things which we have all had experience of, as to which, when we come upon them by a laborious and circuitous route, we feel that, surely, if this be true, we might have been told it at once ; and I think cases of this kind occur in the study of dynamics. There is too much tendency to mask what is really known of the subject under a mass of somewhat complicated rela- tions between things which might, to a greater extent, be each hung on its own nail." It appears to me that such remarks would apply very aptly to "dynamical similarity." The whole subject is so enshrouded in unnecessary mystery that it is, at present, the special preserve of a few professors. There is no text- book on the subject — nor even one with the snare " ele- mentary " in it, — so it is not easy for the young student to understand what is, really, not such a very complicated subject. Surely it is one of those things of which he might, very justly, say, that he "might have been told it at once." If every text-book on dynamics began by ex- 6 Resistance of Air plaining the general principles of "similarity," even young students would never think of, or speak of, ' ' force " and ' ' pressure " as if they were almost interchangeable terms, as some of their elders occasionally do. Apart from their not having the same "dimensions," one is Sl vector qu3int'ity, whilst the other is a scalar. Such being my views, and this being a book on aero- dynamics — dynamics of the a/r, and not of a fancy fluid, which sometimes has contradictory properties : just as I call hydrodynamics the dynamics o{ water, — I will begin : 1. With a dissertation on "dynamical similarity." I hope the young reader will not be frightened by this rather terri- fying name. It is essential that he should understand this subject, so that he may be able to see what 1 mean when I say some equation violates dynamical similarity : to see, in fact, whether statements are nonsensical or not. I suppose I shall stand to be shot at ; and I, equally, suppose I shall deserve it, since I am doing my best to ' ' give the show away. " 2. I will then discuss the values of K. In all modern work — M. Eiffel's, for example — the resistance of air is assumed to vary as the square of the velocity, the sectional area and a constant K : R = KSV^ so that one may call a good deal of this work "the hunt for K." We have ample evidence that the resistance of the air does not vary as the square of the velocity ; it is also debatable if it varies as the area.^ Eiffel's work appears to show it does not : hence K is found to be a very variable coefficient. 3. The question of the resistance of bodies moving in circular motion will then be considered ; and how this differs from the resistance of the same body, in the same fluid, in rectilinear steady motion. That it is in excess of the latter, was well known to Thibault. It is well known at the present day ; but it is considered that, if the whirling-arm is long, the error is negligible. This is true : and it is not. In Langley's experiments with a 6-inch square plate, the error is probably only about 2 per cent. In the experiments with the 12-inch square plates, the error is probably nearer ^ By "area," I mean of the surface. Introduction 7 5 per cent. — certainly not a negligible quantity. All this part of the present book applies equally to both water and air. 4. The law of the resistance of air will then be discussed. This is generally believed to vary as the square of the velocity, for ordinary speeds. It is, further, commonly stated — chiefly, I believe, on the authority of Mr Lanchester — that the elasticity of the air is not a factor of importance. "This error does not become of sensible magnitude till velocities are reached considerably in excess of lialf the velo- city of sound" (F. W. Lanchester, Advisory Covimittee of Aeronautics, 1909-10) [italics added]. If by this is meant that the resistance varies sensibly as the square of the velocity up to this speed, I wish to combat this view, which is not supported by any experiments that I am aware of. Experiments will even be referred to showing that this view is incorrect. As will appear later, the "error" at, say, 650 feet per second — not so '■'■consider- ably in excess of half the velocity of sound " — may be over 40 and nearing 50 per cent. From Mr Bairstow's empirical formula for the resistance of flat plates (y^.^T.^., 1910-11), which is F = o-ooi 26 {vlf -\r O'ooo 000 1 {vlf the "error" would appear to be about }^6 per cent. — and I think even this is rather low. By this I mean that the formula gives a result which is ^^ per cent, greater than it would be if the resistance varied as the square of the velocity ; i.e. F = 0'00i 26 {vl)-. 5. I shall then conclude by references to papers by the best authorities ; always, of course, keeping within the definite limits I have previously laid down in this Introduction. REFERENCES Advisory Committee of Aeronautics, vols. 1909-10 and 1910-11. " The Teaching of Elementary Mechanics " lyBrit. Assoc. Meeting, 1905)- L. A. Thibault (Lieutenant de vaisseau), Recherches experi- ment ales stir la resistance de Pair, etc., 1S26. J. Bayma, S.J., Molecular Mechanics, 1866. CHAPTER II DYNAMICAL SIMILARITY What do we mean by "dynamical similarity"? Stripped of all unnecessary verbiage, it means that in all dynamical equations the terms, on both sides of an equation, shall be dynauiically similar : or, as a mathematician might say, the terms must be "homogeneous." So far, so good ; but what do we m.ean by their being "similar"? "That all dynamical equations must be such that the ' dimensions ' of the terms, added together, are the same in space, time, and mass " (Routh, Elenienta/y Rigid Dynamics). In other words, that the terms in the equation shall have the same "dimensions." The word dimensions is employed in a rather special sense, which will be quite apparent shortly. Dynamical similarity, or dynamical similitude, is a branch of the "principle of similitude." The definition I have given is sufficient to cover the subject as far as here dealt with. "The mathematical processes of addition, subtracting, and equating possess intelligible meaning only when applied to quantities of the same kind. We cannot add or equate masses and times, or masses and velocities, but only masses and masses, and so on. When, therefore, we have a mechanical equation, the question immediately presents itself whether the members of the equation are quantities of the sa)/ie kind \ that is, whether they can be measured by the same unit, or whether, as we usually say, the equation is homogeneous''' (E. Mach, TJie Science of Mechanics). Every schoolboy knows what is meant by "geometrical similarity." He would think it absurd to equate (say) Dynamical Similarity 9 10 cubic feet to so many poles or perches, plus so many yards of ribbon. The terms have not the same "dimen- sions," and so cannot equate one another. One may say volume,^ = volumCa + volume^ but it is ridiculous to say volume^ = volumej, + surface^ or volume^ = volume,, + displacementj. "In two geometrically similar systems we have but one ratio of similarity, viz. that of the linear ^ dimensions " (Routh). The ratio of similarity, or the "dimensions " of unit of "displacement" — space o^i one diviension — being [L] the " dimensions " of unit of " surface " — space of tzvo dimensions— \v'\\\ be . . . . . . [L^] and the ' ' dimensions " of unit of ' ' volume " — space of three dimensiojis — will be . . . . . [L^] It is unfortunate that L, instead of S, should have been employed for the dimensions of space. The letter L suggests "length," which is not a "dimension." Since, however, everybody uses it, I suppose it is rather late to suggest a change. It certainly makes the subject nti- necessarily confusing to the young student. I further suggest that these should be read L, L tzvo and L tliree ; and not L squared and L cubed : space of one, two, or tJiree "dimensions." If read as L squared, the student will think he is doing algebra, and that he can put Lg-f-L^L. To avoid this error it is customary to put the dimensions in square brackets ; as I propose alwa}'s doing. ^ A "number" has "no dimensions''; consequent!}- an}- ^ ? "spatial." Length is not a "dimension " (S.T.M.). - Stallo {Concepts of Modern Physics) says : "The practice of reading x^ and .r' as x square and x cube, instead of x of ttie secotid or third power, is founded upon the silent and express assumption that an algebraic quantity has an inherent geometric import. The practice is, therefore, misleading, and ought to be discontinued." [Italics added.] lO Resistance of Air term multiplied by any number will not have its "dimen- sions " changed. We may say, volumeA=2 volumeu+3 volumec without violating geometrical similarity. Clearly we can also say, , volume,, volumCc volume . = A 2 ^ 3 From this it will follow that, though "displacement" has dimensions of [L], "length" or "distance" — which are simply mivibers, i.e. numbers oi units of displacement — have no dimensions. "Area" is the number of units {in two dimensions) in a "surface" whose dimensions are [L'-^J. Also, "capacity" is the number of units {in three dimen- sions) of a volume : and has zero dimensions. It is very necessary to be careful about noticing the distinction between these expressions. Dynamical similarity is comparable to geometrical simi- larity (which is always postulated, when referring to it) in that, as was previously stated, the terms must all have the same "dimensions": but. whilst in geometrical similarity there is only one "dimension," viz. space [L], in dynamical similarity we have two more, viz. mass [M] and time [T], and all three have to be taken into account. What is the meaning of the word "mass"? As com- monly defined, even in modern text-books, "the word mass is used to denote ' quantity of matter'" (date 191 1). Dr Glazebrook says: — "If we consider the bodies with which we have to deal as composed of matter, then any body will consist of a definite quantity of matter. This quantity is usually called its mass" {Dynamics, 1913).-^ Thus "mass" is defined as the quantity of a supposed thing called "matter." What, however, is the meaning of the word " matter " ? Dr Glazebrook frankly says: — "We do not know what 1 I refer to this book several times ; not because it is specially remark- able, but because its date is 191 3, and because it is a text-book published by the Cambridge University Press. Dy7iamical Swiilarity i i matter is : it may be that it has no phenomenal existence apart from our conception of it " {Dynamics). Hence ** mass "also may have no "phenomenal existence": the definition gets very obscure,^ Schelling said : — " Matter is the general seed-corn of the universe, wherein everything is involved that is brought forth in subsequent evolution." This is very poetical, but it is rather too nebulous to be accepted as a good mechanical definition. The only complete definition that I have come across (1 suppose it must be complete from its length) is that of Hegel. "Matter is the mere abstract or indeterminate- reflection-into-something-else, or reflection-into-self at the same time as determinate ; it is consequently Thinghood which then and there is — the subsistence or substratum of the thing. By this means the thing finds in the matters its reflection-into-self ; it subsists not in its own self, but in the matters, and is only a superficial association between them, or an external bond over them " {The Logic of Hegel, translated by W. Wallace). I hope the reader will understand this definition : for myself — not having the brain of a Hegel — I am like the old Scotch fishwife, who was asked if she understood the meenister's sermon : " I wouldna ha'e the preesoomption." I am afraid we must fall back on Thomson and Tait's Eletnents of Natural Philosophy, and say : " \\"e cannot, of course, give a definition of matter which will satisfy the metaphysician ; but the naturalist may be content to know matter as tliat ivJucJi is perceived by the senses, or as that which can be acted upon by, or can exert, force " [italics added]. Even this explanation is hardly satisfactory, since, b\- this definition, ether must be "matter." If we believe that light (and why not heat ?) can exert pressure, it is clear that the ether must have mass — and so be "matter," — whilst it cannot be ' ' perceived by the senses, " We also have force ^ Many other authors refer to "mass "and "matter," but, generally, rather unsatisfactorily. 12 Resistance of Air referred to as a quasi-personal thing, which acts on matter. We shall see later, when referring to force, that the most modern idea is that, like Mrs 'Arris, "there ain't no sich person. " Since we cannot define "matter," we, clearly, cannot define "mass." I trust, however, that the reader will have a reasonable idea of what is meant by the word mass, from Thomson and Tait's definition of "matter." It will be sufficient here to say that mass has dimensions [M]. The reader who wishes to go more thoroughly into the question should read carefully Karl Pearson's Grammar of Science, where he will find that if a body A is acting on (accelerating) a standard corpuscle O, and is being acceler- ated by it, then c , acceleration of O due to A mass of A = ;, -, . 7 , -r- acceleration of A due to O Similarly, E. Mach {TJie Science of Mechanics) gives a definition of " equal masses " : — " All those bodies are bodies of equal mass ivhicJi, mutually acting on each other, produce in each otJier equal and opposite accelerations. "We have, in this, simply designated, or named, an actual relation of things. In the general case we proceed similarly. The bodies A and B receive respectively, as the result of their mutual action, the accelerations — ^ and +0', where the senses of the accelerations are indicated by the signs. We say then, B has 0/0' times the mass of A. If we take A as our unit, zve assign to that body the mass m which imparts to A ra times the acceleration that A in tJie reaction imparts to it. The ratio of the masses is the nega- tive inverse ratio of the counter-accelerations. That these accelerations always have opposite signs, that there are therefore, by our definition, only positive masses, is a point that experience teaches, and experience alone can teach. In our conception of mass no theory is involved ; 'quantity of matter ' is wholly unnecessary in it ; all it contains is the exact establishment, designation, and denomination of a fact. Dynamical Similarity 1 3 "When the negative inverse ratio of the mutually in- duced accelerations of two bodies is called the mass-ratio of these bodies, this is a convention, expressly acknowledged as arbitrary ; but that these ratios are independent of the kind and of the order of combination of the bodies is a result of mquiry. "To accomplish anything dynamically with the concept of mass, the concept in question must, as I most emphati- cally insist, be a dynamical concept. Dynamics cannot be constructed with quantity of matter by itself, but the same can at most be artificially and arbitrarily attached to it. Quantity of matter by itself is never mass, neither is it thermal capacity, nor heat of combustion, nor nutritiv^e value, or anything of the kind. Neither does ' mass ' play a thermal, but only a dynamical, role. " All uneasiness will vanish when once we have made clear to ourselves that in the concept of mass no theory of any kind whatever is contained, but simply a fact of experience. " If it should be found, ^ later, that mass is a function of velocity — as already some advanced thinkers are inclined to believe, — we may have to modify somewhat our views on the subject of mass. M. Lucien Poincar^ {The New Pliysics) says: — "We have been led to suppose that inertia depended on velocity and even on direction. If this conception were exact, the principle of invariability of mass would naturally be de- stroyed. Considered as a factor of attraction, is mass really indestructible ? " [italics added]. M. Gustave Le Bon {The Evolution of Forces) goes a step further, for he says : — "Not only does mass vary with the ^ I say " if," because I do not believe that such a very revolutionary idea in mechanics will be necessary : it would upset dynamical similarity. As will appear later, the theory of air resistance that I am here advanc- ing is, as if the density — and ergo mass — increased until the velocity. Such is not, however, the case ; though it may appear so at the first glance. 14 Resistance of Air velocity, but it has lately become a question whether it does not also vary with the temperature. " It will be unnecessary to define " time " : everyone knows what is meant by the word, and I am afraid that any attempt at a complete definition would only tend to confuse what is a very fairly clear idea. We might even say that time, space, and mass being irreducible, cannot be compared with anything, and are indefinable. We only know of them that which our common sense tells us. So soon as, in order to define these great entities, we endeavour to go beyond what is revealed by ordinary observation, we meet with inextric- able difficulties and end by acknowledging, as do the philo- sophers, that they are simply creations of the mind, and cover completely unknown realities. It is best to say frankly, with Sir George Greenhill {Notes on Dynamics for the Senior Classes of Artillery Officers, 1893), that the three indefinables in Nature are space, time, and mass. ^ The dimensions of time are [T]. Besides these, what are called " fundamental units," there are "derived," or compound dimensions. For example, velocity is defined as rate of displace7ne7tt : that is to say, the ratio between the displacement and the time. Its dimensions are therefore [L / T] or [LT~^]. Speed is the ratio between a lengtli and the numerical coefficient (or duration) of the time. It has, clearly, "no dimensions," and is simply a number. To prevent any possible misunderstanding, I would like to point out that I do not mean to say that velocity is space, or "displacement," divided by time. Such a statement would, I think, be absurd : what is implied is that the velocity is a measure of the ratio between the space, or displacement, and the time : that the velocity varies as the space, or displacement, and inversely as the time. You can no more divide a "space" by a "time" than you could divide a cow by an acre. What would be meant by "dividing 10 acres by five seconds"? You may divide the 1 Greenhill says " space, time, and fnatter.''' Dynamical Similarity i 5 numerical coefficient of the space, or displacement, by the numerical coefficient of the time, and you get the numerical coefficient of the velocity : but you cannot divide any dimension by any other. This c^uestion will be referred to again, later. "Acceleration" is, by definition, the rate of change of velocity ; velocity being the rate of displacement, or rate of change of position. We may describe it as velocity ^ time — understood as in the last paragraph ; conscc|uently its "dimensions" are LT~^-^ T = [LT"'^] : possibly even better as[LT-iT-i]. Density (p) is defined as mass per volume, or mass-^- volume {space of three dimensions). Its "dimensions" are, therefore [ML-^]. REFERENCES E. J. RouTH, Eletnentary Rigid Dyfiamics, 1897. E. Mach, T/ie Science of Mechafiics, 1907. R. T. Glazebrook, Dynamics, 1913. W. Wallace, The Logic of Hegel. Thomson and Tait, Elements of Natiiral Philosophy, 1885. Karl Pearson, The Grannnar of Science, 191 1. LuciEN PoiNCARE, The New Physics, 1907. GusTAVE Le Bon, The Evolution of Forces, 1 908. Sir G. Greenhill, Notes on Dynamics for the Senior Classes of Artillery Officers, 1893. CHAPTER III FORCE We will now discuss the dimensions of that much-abused word "Force": but, in the first jjlace, what do we mean by the word ' ' Force " ? Referring to Thomson and Tait {Elements of Natural Philosophy, part i, 1885) we find: — "183. Force is any cause which tends to alter any body's natural state of rest, or of uniform motion in a straight line. . . . Forces may be of different kinds, as pressure, or gravity, or friction, or any of the attractive or repulsive actions of electricity, magnetism, etc." [italics added]. In another part we also find: — "The standard or unit force is that force which, acting on a national standard unit of matter, during the unit of time, generates the unit velocity." This is known as Gauss's absolute unit. Here we find that force is, clearly, something objective — it is an agent. It is a ^'' cause" of motion, and it acts. I have quoted this definition, not because I think it is satisfactory, but because it is that adopted, implicitly, if not explicitly, in almost all, even modern, text-books. John Cox {Mechanics, Cambridge University Press, 1909) defines force as ' ' anytJiing which cJianges or tends to change a body's state of rest or motion." He, also, calls pressure ' ' a force " ; tension ' ' a force " ; etc. To my mind this is exceedingly confusing to any young student, since he will, of course, consider "Force" as an agent or thing — something objective, and which has a quasi- personality — which causes, or tends to cause, motion. He will, further, confuse force, pressure, and tension ; so as to i6 Force 1 7 think that pressure and tension are forces. Indeed, he is distinctly told that pressure and tension are "different kinds of Force." This leads to hopeless confusion at times. As will appear later, Force is only a concept of the mind and not 2. percept : it is purely subjective and not objective. In making this quotation from Thomson and Tait, I must hasten to confess that I have not acted quite fairly to the memory of Professor Tait. It is quite true that this was his view in 1885 ; but his later, and more matured, judgment was very different.^ In his Properties of Matter, 4th edition, 1899, we read : — " If for a moment we use the word Tiling to denote, generally, whatever we are constrained to allow has objec- tive existence — i.e. exists altogetJier ifidependently of our senses and of our reason, — we arrive at the following conclusions : — " A. In the physical universe there are but two classes of things — matter and energy. "II. The -word force must often, were it only for brevity's sake, be used in the present work. As it does not denote either matter or energy, it is not a term for anything objective. "... the great majority, even of scientific men, still cling to the notion of force as sometliing objective. "... Force is a mere phantom suggestion of our mus- cular sense. " " I have seen so much mischief done by this quasi-pcrsoji- ification of a mere sense impression that, even in an elemen- tary book, I am constrained to protest against it. I feel assured that the difficulties which are now everywhere felt as to the great scientific question of the day, the nature of what we call electricity, are in great part due to the way in 1 Even in 1876 he said Force was, "in fact, merely a convenient ex- pression for a certain ' rate '." 1 8 Resistance of Air which our modes of tlLinking have been, by early training atid subsequent habit, encouraged to run in this fatal groove " [italics added]. From the foregoing exceedingly powerful and clear state- ments, it will be seen that Professor Tait, towards the end of his life, did not consider Force as a thing; though he, unfortunately (probably from long habit), occasionally employs the word in a quasi-personal sense : and this, at times, detracts from clearness. However, "one must needs be a poor psychologist and have little knowledge of oneself not to know how difficult it is to liberate oneself from traditional views ; and how, even after that is done, the remnants of the old ideas still hover in consciousness and are the cause of occasional backslidings even after the victory has been practically won " (E. Mach, The Science of Aleciianics). It may be objected that Newton in the Principia employs the word Force in this quasi-personal manner : Force is " impressed," it ''causes motion," etc. etc. This is most unfortunately true. Profound, however, as may be our admiration for this extraordinary book, and the transcen- dent genius of its author, is it really necessary that we should, with our present knowledge, indulge in Newton- olatry ? Of all the delusions of man, perhaps the most difficult to cast forth is an "olatry." Whether a man loves his idols or fears his idols, for some mad reason he is as unwilling to test them as he is unreasoning in his worship. Dr E. Mach says : — "We join with the eminent physicists Thomson and Tait in our reverence and admiration of Newton. But we can only comprehend with difficulty their opinion that the Newton doctrines still remain the best and most philosophical foundation of the science tJiat can be given " {Science of Mechanics) [italics added]. We should ever remember what Bacon said : — "Credulity in respect of certain authors, and making them dictators instead of consuls, is a principal cause that the sciences are no further advanced. Let great authors, therefore, have their due, but so as not to defraud Time, which is the Force 1 9 author of authors, and the parent of Truth " {^Advance- ment of Learning). Is it quite certain, however, that Newton conceived " Force " as a ////«^ (something objective) which causes or tends to cause motion ? There are times when I have very serious doubts about this having been his view. I am not aware that Newton ever defined "Force": and his defini- tion of " impressed force " is that it was an action. In Def. viii he distinctly says : — " I . . . use the words Attraction, Impulse or Propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not Physically but Mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the mantier of any action, the causes or t/ie pJiysical reason tJicrcof, or that I attribute Forces, in a true and Physical sense, to certain centres {wJiich are only Mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers " [italics added]. We may, further, examine Newton's view of Force from another standpoint. Referring to his second law of motion, we see that: — "Newton's second law of motion is as follows : Change of momentum is proportional to force, and takes place in the direction in wJiich the force acts. " "Thus, according to Newton, a force always produces change of momentum. Hence there is no balancing of forces, though there may be balancing of the effects of Forces " (Tait, Properties of Matter).^ Since, according to Newton, a force always produces change of momcfitum, it is, logically, clear that if tw change of momentum is produced, there can be no force. Conse- quently, force cannot be objective. Modern text-books (Glazebrook's Dynamics, for example) change Newton's law to ''rate of change of momentum, etc.": and justify this by reference to Def. viii, which only treats of ''■centripetal force." No objection can be raised against this, beyond ' " Newton's notion is, if there is a force at all, it is ttoi/ft; soiiiel/iing"' (Tait, Lectures on some Recent Adva?iccs in Physical Scieticc). 20 Resistance of Air that it is not Neivton's wording. "Change of momentum" and '■^rate of change of momentum" are not exactly the same thing. Professor Karl Pearson {Gnimniar of Science), when referring to this second law of motion, says : — "This is a veritable metaphysical somersault. How the imperceptible cause of change of motion can be applied in a straight line, surpasses comprehension ; the only straight line that can be conceived, or, as some physicists would have it, perceived, is the direction of change of motion. We may assert that the imperceptible has this direction, but to postulate that the imperceptible will determine this direction for us seems to be pure metaphysics. We come down to our feet again, however, when we interpret this law as simply indicating that, physically, force is going to be ta/ceji as a iiieasure for some change in motioti. As to the exact meaning of change of motion taking place in a straight line, all the real difficulties as to what thing we are to suppose changing its motion, and what is the presence associated with this change of motion, i.e. the difficulties about the line joining two corpuscles, are concealed by talk- ing vaguely about force as an entity ' acting in a straight line. ' "The glib transition from force as a cause io force as a measure of motion too often screens the ignorance which it is as much the duty of science to proclaim from the house- tops as it is its duty to assert knowledge on other points " [italics added]. It may be said that, generally, " Matter is, in Newton's system, regarded as the plaything of Force ; submitting to any change of state that may be imposed on it. But rigor- ously persevering in the state in which it is left, 7intil force again acts on it " (Tait, Properties of Matter). There are great difficulties in accepting this view of Force, I remember, as a boy, being taught (as I should not have been ; but as, probably, most boys are still taught) that ' ' if anyone presses a stone with his finger, the finger Force 2 1 is pressed witJi the same force in the opposite direction by the stone " (Thomson and Tait, Natural Philosophy). This I could, of course, believe, since I had been taught that equal and opposite forces would balance, or neutralise, one another. I could draw two pointed arrows (to repre- sent these "forces") opposed to each other, and imagine one of them nullifying the action of the other. Knowing nothing about dynamical similarity, I believed a pressure was a ' ' force " : I was also ignorant that one was a scalar and the other a vector quantity. When, however, I was told that "a horse towing a boat on a canal is dragged backwards by a force equal to that which he impresses on the t Giving rope forwards'" (Thomson and Tait), I simply could not understand it. If the horse and the boat were doing a " tug-of-war," and WmSxx forces were equal, the boat and the horse ought both to remain immovable. Since, however, the horse pulled the boat along — and that is undisputed, — it appeared clear that the force of the horse must be greater than t\\^ force of the boat : they did not neutralise one another ; therefore the one was stronger than the other. Either equal and opposite forces balanced one another, or they did not. If they did, then, there was no residual force to cause the motioti of the boat. This is how I argued ; and the explanation of the mystery, given to me, appeared exceedingly unsatisfactor)-. If the reader will turn to the Principia (Motto's trans- lation), he will see that Newton did not say that "if anyone presses a stone with his finger, the finger is pressed zvith the same force in the opposite direction." What he said was, "the finger is also pressed by the stone.'^ " With the same force" is a gloss, not being in the Principia. It is, further, a gloss which makes nonsense of the phrase. Newton's evident meaning was that pressure and counter-pressure — stresses, not forces — are equal and opposite. The "horse and boat" story is taken (with modifications) from the Principia. If we refer to John Cox, Mechanics (published by the Cambridge University Press, 1909) — a book specially dedicated to E. Mach, though the teaching 2 2 Resistance of Air is by no means in accord with that of Mach — we find that "the earth exerts two forces on the cart (a cart is substi- tuted for the usual boat), {a) an attraction , . , and {U) an upward pressure. " Here we have a frank and clear state- ment showing that pressure is a force — which we know is incorrect. Later on Mr Cox speaks of the " forward thrust of the earth " ; and the general impression conveyed, eventually, is that the horse has nothing to do with pulling the cart : that it is really the earth which pushes the cart. Of course, this is not intended, but it is the kind of impression which is conveyed to the student. Let us see what Newton actually said. " If a horse draws a stone tyed to a rope, the horse {if I may so say) will be equally drawn back towards the stone ; for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone as it does the stone toivards the horse " (Motte's translation) [italics added]. We see that Newton never mentions the word "force." The horse is not dragged back dy a force : it is only " (if I may so say) drawn back " dy the rope. In other words, that the stress on the rope is the same in both directions. Since, however, the stress causes no change of relative motion, this stress — which is a tension — cannot be a force, since by the second law of motion a force always produces change of momentum. Dr Glazebrook {Dynamics) repeats the horse and stone story, but omits Newton's, bracketed, " if I may so say." He says (p. 92): "Rate of change of momentum Jias re- ceived a name, it is called Force," Yet at p. 116 he says : ' ' Rate of change of momentum is proportional to the im- pressed force." It is certainly confusing to be told that Force is (or means) rate of change of momentum, and then to be told it \s proportional to the same thing. How can a man, for example, be proportional to himself} Or, if a stone is allowed to fall, how can "Force" (a word, a name) be proportional to the falling stone ? Force is some- times "action": sometimes "measure of the action": Force 23 and, more frequently, "measure of the intensity of the action " : and at p. 92, as one single word (a name) synonymous with " mass acceleration." Professor Tait {Properties of Matter) refers to another difficulty when he says : — ' ' But if it [a force] were objective, what an absolutely astounding difficulty would have to be faced by one who tries to explain the nature of hydrostatic pressure ; and who finds that by the touch of a finger on a little piston he can produce a pressure (say) a pound weight on every square inch of the surface of a vessel, however large, if full of water, and the same amount on every square inch of surface of every object immersed in it, even if that object consists of hundreds of square miles of sheets of tinfoil far enough apart to let the water penetrate between them. All this, moreover, is found to disappear the moment he lifts his finger. " Exactly the same difficulty, and for the same reason, will be encountered (as I have pointed out elsewhere) in attempt- ing to explain the flow of water out of a hole in a tank, or out of a squirt. That exceedingly clear thinker and lucid writer, Professor Karl Pearson {TJie Gratninar of Science), says, in reference to the mutual acceleration between a body A and a body B, that acceleration of B due to A mass of A acceleration of A due to B mass of B " " Hence it follows that mass of A X acceleration of A due to B = mass of B X acceleration of B due to A. " We will, then, give a name to this product of niass into acceleration ; we will term tiie product of the mass of A into the acceleration of A due to the presence of B, t/ie force of B on A. This force will be considered to have the direction and sense of the acceleration of A due to B, whilst its magnitude will be obtained by multiplying the nu7nber of units in the acceleration of A due to B by the nmuhcr of 24 Resistance oj Air units ill the mass of A. Thus the proper measure of a force will be its number of ujiits of Mass-acceleration " [italics added]. 1 This very clear and masterly description produces a vivid mental picture of what is meant b}' the word Force. It is one single word which means "(vector) Mass-acceleration" ; or, if one employs Professor Perry's definition, " (vector) rate of change of momentum." Clearly, therefore, if there is no acceleration, or change of momentum, there can be no force. I shall always use "Force" and "Mass-acceleration" (employed vectorially, of course) as interchangeable terms. Hence, if I press my finger against a wall, since 1 do not accelerate the wall, there is no force. My action, however, causes a /r^j'i'?^;'^ : it is clear, therefore, that I do not con- sider that " pressure " is a y^irr^. What the "imperceptible cause of the change of motion, which can be applied in a straight line," is, does not interest us. We do not know why masses act upon one another. I will close the discussion of this question by giving abstracts of the opinions of two other great thinkers. "That which physicists call Force is, strictly, nothing else than the intensity of the action measurecC by the intensity or quajttity of viotion which in the given circumstances it is capable of communicating. "The force is not the action, but the intensity of the action ; and is measured by its dynamical effect " [italics added]. (Father Bayma, S.J., Molecular Mechanics.^ J. B. Stallo, in his Concepts of Modern Physics, says : — "Force is not an ijidividual thing or entity that presents itself directly to observation or to thought ; it is purely an incident to the conception of tJie interdependence of moving masses. ^ I wish the reader specially to understand that the views of Karl Pearson, E. Mach, Stallo, and others I have quoted, are not peculiar to them. They are the present views of all the leading thinkers. I have specially selected a few on account of their particularly clear enunciation. Force 2 5 " It is of the greatest moment, in all speculations con- cerning the interdependence of physical phenomena, never to lose sight of the fact that force is -a. purely conceptual term, and that it is not a distinct or tangible thing. " Most unfortunately, in nearly all modern text-books it will be found that the word ' ' Force " is employed in several senses, almost indiscriminately. I could point to one where it is used in three senses on the very same page. When the young student finds himself face to face with a word which means so many things, is it surprising that he finds the subject of mechanics so very difficult} Is it astonishing that Professor Perry should think that ' ' for one of them to learn mechanics is almost miraculous " ? or that, later, he should "dislike the thought of mechanical theory " ? This discussion of " P"orce " has been rather lengthy ; but I wished, by rigidly defining the word, to indicate, beyond all shadow of doubt, the sense in which I employed it. I should like to think that, in so doing, I have been "flogging a dead donkey"; but even in the discussion on the teach- ing of elementary mechanics I could perceive no indication that Professor Perry's definition of "Force" was generally accepted — though it was agreed that force was equal to, or proportional to, mass-acceleration, or rate of change of momentum. Rather was there a tendency to regard it as " an imperceptible cause of the change of motion " ; and that it could be "applied in a straight line." The young student who has thoroughly grasped the idea that " Force" is a single word — a name — which is employed for "(vector) mass-acceleration," or "(vector) rate of change of momentum," will never, in the future, fall into the error of speaking of a " pressure " as a "Force." Of course, "when we speak of a 'force of nature' (as we sometimes do popularly), we use the word force in a sense very different from that which it bears in mechanics. A ' force of nature ' is a survival of ontological speculation ; in common phraseology the term stands for a distinct and real entity. But as a determinate mechatiical functiofi force 2 6 Resistance of Air is simply tlie rate of cJimige of movieiituni — mathematically expressed, the differential of nioinejitum at a given instant of time" (Stallo, Concepts of jModern PJiysics).^ Knowing, now, that Force is Mass-acceleration, it is clear that its dimensions are MxLT-2 or [MLT-2]. Some of the foregoing may be put into a short summary : Space is "space" (undefinable) . " Displacement" is space of one dimension Its "scalar" — or footrule — is "length" " Surface " is space of two dimensions Its " scalar " is "area" "Volume" is space of three dimensions Its " scalar " is "capacity" [S] [Si] [o] [SJ [o] [S3] [o] One may speak of a "volume" of water, say: but one refers to the " capacity " (not volume) of a tank. Time is " time " (undefinable) .... [T] Its " scalar " is "duration" . . • [o] One commonly says, "The time was three seconds": to be strictly accurate, one should say, " The duration of the time was three seconds." " Velocity" is rate ^displacement [S / T] or [ST~^] Its "scalar" is "speed" . . . • [o] ' ' Momentum " is rate f?/ displacement of mass [MS / T]or[MV] ' ' Acceleration " is rate of change of velocity [V / T] or [ST-2] "Force" is rate of change of momentum, or "mass-acceleration" [MV / T] or [MST-^] The reader will observe that in this list there are several lacuna;. We have no scalars for momentum, acceleration, or Force. ^ Or, as Tait puts it : Momentum is the Time Integral of Force, be- cause Force is the rate of change of momentum. Force 27 If we adopt Clerk-Maxwell's notation {Matter and Motion) and call a "displacement" ^. point-vector^ then Velocity would be rate of change of point-vector. Momentum ,, rate of change of mass-vector. It would thus appear reasonable to call the scalar of momentum the rate of change of mass-scalar ; or, in other words, the "mass-speed." I would therefore suggest the name " mass-speed " for the scalar of momentum. I have been unable to propose any satisfactory name for the scalar of acceleration. Any suggestion would be grate- fully accepted. The word acceleration being "used to denote any change in the velocity, whether the change be an increase, a diminu- tion, or a change of direction''. . . "hence we say that the acceleration may be in the direction of motion, in the con- trary direction, or transverse to that direction " (Clerk- Maxwell, Matter and Motion) [italics added]. It is clear, therefore, that "rate of change of speed" is not, necessarily or generally, the same as " rate of change of velocity." We may (as in circular motion) have a uniform speed, and have, at the same time, an accelerated velocity.'^ I also suggest, tentatively, that pressure is the proper scalar for Force : pressure, of course, resolved in the direction of the Force. This will be referred to again in the next chapter. Philip E. B. Jourdain, M. A. (Cantab.), in his Notes on Maclis Mechanics, 191 5, refers to "the looseness of phrase according to which it is customary, in most treatises on mechanics and geometry, to talk about /, s, and z' simply as the time, the distance [ ? displacement], or the velocity, instead of the numerical measures of these quantities" [i.e. the "scalars "]. He gives an example " 'the velocities are as the times' cannot be misunderstood so readily as the equation v=gt. A beginner would think that we were equating the velocity 1 Tentatively, I would suggest the term "speed-acceleration" for the scalar of "acceleration" — which is, by definition, a '■'■ velocity-acceleration.'''' 28 Resistance of Air to the time multiplied by a certain constant, and most authors (even Mach) encourage this misunderstanding by their sacnfice of accuracy to brevity. The truth is, of course, that ' v ' does not stand for the velocity, but for the numerical measure, in terms of some unit of velocity" [italics added]. I would prefer to put it that the " ^' " stands for the "speed," in some arbitrary units. Equally, t is 7tot the Time, but the duration of the Time, in some units — sec: hour : year. REFERENCES John Cox, Mechanics (Cambridge Physical Series, Camb. Univ. Press, 1909). P. G. Tait, Properties of Matter, 1907. Newton, Prituipia (Motte's translation). P. G. Tait, Lectures on some Recent Advances in Physical Science, 1876. Father Bayma, S.J., Molecular Mechanics, 1866. J. B. Stallo, Concepts of Modern Physics, 1882. Philip E. B. Jourdain, M.A. (Cantab.), Notes on MaclCs Mechanics, 1915. J. Clerk-Maxwell, Matter and Motion, S.P.C.K., 1876. CHAPTER IV PRESSURE — SCALAR AND VECTOR QUANTITIES — VISCOSITY — KINEMATIC VISCOSITY It may be well here to enter a protest against an expression which is, not uncommonly, found, even in papers issued by the Advisory Committee of Aeronautics. I refer to such wording as "mean force per unit area," employed as equi- valent to "Pressure." It is, really, exceedingly confusing, and appears to be based on the idea that Force ^L- = Pressure. It will also give me a good nail on which to hang a short discussion on scalar and vector quantities.^ Now, if we substitute "mass-acceleration," or "rate of change of momentum," for the word force, we should have : — Pressure = mean mass-acceleration per unit area ! What idea does one get of an acceleratio7i per unit area ? Or, of a (vector) rate of chatige of viomeniuvi per unit area ? To me it sounds like scientific jargon, and as meaningless as saying cows ^ acres = yards. Before continuing this discussion, I must make a small digression. "As is well known, we distinguish, since Hamilton's time, two kinds of physical magnitudes — scalars and vectors. These two kinds of magnitudes are essentially different in their nature, and one can never be represented by tJie other'' (Professor Ostwald). A vector quantity is one in which some direction is im- plied. A scalar quantity does not imply any direction : it has no ' ' dimensions " : it is, as its name would lead us to ^ Many authors employ Force and Pressure as almost interchangeable terms. 29 30 Resistance of Air believe, nothing more, nor less, than a number.^ As previously stated, we cannot imagine a "displacement" without imagining a directioti. Without some "frame," such a word as "displacement" has no meaning whatever. If a body is displaced, it iimst be relatively to some other body or bodies : it is impossible to imagine anything else. Similarly, if we imagine a sphere, we mtist imagine a centre. If we then mark any spot on the surface of the sphere, this spot must have a definite direction relative to this ceritre. If we speak, however, of a "capacity," which we may call 20 cubic units (inches, feet, yards, etc.), we might be referring to any kind of volume. We are only speaking of a number. It is most iinportant to bear these facts well in mind. Force is, clearly, a vector magnitude, since its dimensions are [MST"^]. Pressure is a scalar quantity : if we speak of a pressure of 15 lbs. per square inch, no "direction" is implied. If we now return to our equation, we shall see that Pressure x Area = Force, (scalar) (scalar) (vector) which, on the face of it, is absurd. A distinguished Professor (an F. R. S. ), to whom I pro- pounded this conundrum, first asserted that area was a vector quantity. Subsequently, he said the proper equation was ,, Force Pressure = ^ . Surface I think this will only lead us into a worse morass. If we divide one vector by another, what do we get ? Certainly not a scalar: a "quaternion."^ In Kelland and Tait's Introduction to the Study of Quaterniojis we find : — "It appears that, generally, the. product or quotient of two vectors may be represented as i\\Q product of a tensor and a ' One might liken it to a footrule. - Also, in quaternions, to multiply A by D \s not the same thing as to multiply B by A. Pressu7^e — Scalar and Vector Quantities 31 vector. This product Sir \V. Hamilton names a quaternion. Cor. It is evident that a quaternion is also the suvi of a scalar and a vector'^ [italics added]. This only gets (as Alice would have said) " worserei and worserer. " I think the reason for all this confusion has been well summed up by Gustave Le Bon in TJie Evolution of Forces: — " What has given very great force to certain principles of physics and mechanics has been the very complicated mathe- matical apparatus in which they have been wrapped. Every- thing presented in an algebraical form acquires for certain minds the character of indisputable truth. Tlie most perfect sceptic willingly attributes a mysterious virtue to equations and bows to their supposed power. They tend more and more to replace, in teaching, reason and experiments. These delusive veils which now surround most simple principles only too often serve to mask uncertainties." There is too much tendency, when thinking, or writing, about "dimensions," to treat the subject as if it were ordinary algebra. It must be recollected that in algebra we only deal with letters which represoit numbers : we can do what we like w ith these. When in solving an equation we put .r^ the number of the cows, we are dealing with the numbers and not the cows. If we then put j/ = the number o{ acres, we may say x j j^' = the number of cows per acre : this is the ratio of the number of cows to the number of acres. When, however, we are dealing with " dimensions," we can- not divide one by another, as 1 have previously stated. You could not, in algebra, put x'^^d. cow, and then take the cube root of it. A cube root of a cow is simple nonsense. Even, however, in algebra, "the use of letters as algebraical symbols, i.e. as representatives of numbers, is in itself a serious (though perhaps an unavoidable) infirmity of mathe- matical notation. In the simple formula, for instance, ex- pressive of the velocity of a moving body in terms of space and time {v = s j t), the letters have a tendency to suggest to the mathematician that he has before him direct repre- sentatives of the things or elements with which he deals, and not merely of their ratios expressible in numbers. In c\-er\- 32 Resistance of Ai7' algebraical operation the use of letters obscures the real nature, both of the processes and of the results, and tends to strengthen ontological prepossessions'' (Stallo, Concepts of Modern PJiysics) [italics added]. "The distance c traversed in a second of time we call the velocity/ and obtain it by the examination of any portion of the distance and the corresponding time, by the help of the equation r=/ / /; that is, hy dividing the number which is the measure of the distance traversed, by the jiumber which is the measure of the ti^ne " (E. Mach, Science oj Mechanics) [italics added], ^ "The idea of speed as so many units of linear space described per unit of time is a complex one, involving both of the fundamental ideas. We express this by saying its dimensions ^ are [L / T], This implies that, in whatever proportion we increase our unit of length, the measure of speed is diminished in that proportion : while it is increased in the same proportion as that in which the unit of time is increased " (Tait, Properties of Matter). "It must be understood that it is only by arbitrary convention that a dependency is established between a derived unit and the fundamental units. The laws of numbers in Physics are often only laws of proportion. We transform them into laws of equation because we introduce numerical coefficients and choose the units on which they depend, so as to simplify as much as possible the formulas most in use, A particular speed, for instance, is in reality nothing else but a speed, and it is only by the peculiar choice of unit that we can say that it is the space * covered during the unit of time. In the same way a quantity of electricity is a quantity of electricity ; and there is nothing to prove that, in its essence, it is really reducible to a function of mass, of length,^ and of time. 1 More correctly " speed." — R, de V. ^ Even this is rather loose writing, c, the velocity, or speed, is a rate, and so cannot accurately be expressed as a " distance." ^ Has it any "dimensions" at all? No. — R. de V. * ? "distance" : length. '' ? "space."— R. de V. Pressure — Scalar and Vector Quantities 33 " Persons are still to be met with who seem to have some illusions on this point, and who see in the doctrine of the dimensions of the units a doctrine of general physics, while it is, to say truth, only a doctrine of metrology. The knowledge of dimensions is valuable, since it allows us, for instance, to verify easily the homogeneity of a formula, but it can in no way give us a?iy infonnation on the actual nature of the quantity measured " (Lucien Poincare, The New Physics) [italics added]. That great mathematician, Henri Poincare, even said : — *' Mathematics are sometimes a hindrance, or even a danger when, by the very precision of their language, they lead us to affirm more than we know " {Comptes Rendus de r Academic des Sciences, Dec. 17, 1906).^ "The mathematician who pursues his studies without clear views of this matter, must often have the uncomfort- able feeling that his paper and pencil surpass him in intelligence, 2 Mathematics thus pursued, as an object of instruction, is scarcely of more educative value than busying oneself with the Cabala. On the contrary, it induces a tendency towards mystery, which is pretty sure to bear its fruits " (E. Mach, Science of MecJianics). It will be clear now, I hope, that if we say Pressure = Forced area ; since "area" is a scalar — a number, in fact — with consequently no "dimensions"; the result will be that Pressure has the same '' dimensions''' as Force \ If, with a view to avoiding this absurdity, we say Pressure = P"orce -^- surface, we get into further, hopeless, entanglement. But, my F. R.S. friend said, "this" — one or the other — " is the definition of pressure." All I can say is that, if this is so, it is a very bad definition. I do not know any author who has defined pressure, though I have looked through a good many books. Everybody knows, generally, what is meant by "pressure": but I do not know a reasonable ' Les mathematiques sont quclquefois une gene, ou meme un danger quand, par la precision meme de leur language, elles nous amenent ;\ affirmer plus que nous ne savons. ''■ It is said that Euler sometimes felt this. 3 34 Resistance of Air definition for it. One 1 could suggest is, Pressure = energy -f volume. Here, again, my friend says "that is the same thing," since „ energy Force x space Force Pressure = — ^ — ^^ = — = — ,; — . volume surface x space surface I am afraid that we are here playing with dimensions as if they were all numbers. Using volume here as a space of three dimensions, the assumption is volume = surface x space. Here, again, I am sorry to say, we are talking nonsense. You cannot multiply a surface by a space (of one dimension) to make a solid, any more than you could multiply a ten- acre field by 50 yards in order to get a solid. You may add as many surfaces as you like together, but you will never get a solid. Yes, but, it may be said, "this is what we do in the Integral Calculus. We integrate [word almost as blessed as Mesopotamia] a surface between certain limits and get a solid." That is, exactly, what we do not do : you cannot integrate x^-, say : what you can, and do, integrate is x^dx, which is a very thin solid. You add — integrate, if you prefer the more grandiloquent word — a vast number of very thin solids to make a thick solid. From my definition, Pressure = energy^ volume (all the terms here are scalar), we see Boyle's law follows as a consequence : Pressure x volume = energy = constant. Many other curious things also follow logically. The reader will doubtless say that, in this argument, I have been using the word "volume" as a scalar quantity, whilst I had previously said it was a vector. This is quite true : the word I should have employed is ' ' capacity. " I used it, temporarily, in the sense in which it is employed in " Boyle's law." I thought it better to do so — though I agree it was wrong — so as to keep to the wording of Boyle's law, Pressure X volume = constant : in my terminology I should say Pressure x capacity = energy = constant. Pressure — Scalar and Vector Quantities 35 The clear-thinking reader (and I hope there will be more than one) may say that all this is "elementary" — every- body knows it. Certainly, everybody otigJit to know it : yet many, even professors, do not appear to have clear ideas on these subjects. How many use "surface" and "area" as if they were synonymous terms? How many speak of "velocity" when the context shows they really mean "speed"? Careless thinking is the cause of all this confusion. A friend of mine once said to me, " How can you expect English people — [he is English himself] — to have clear ideas, when they have a language which is frequently so obscure ? " I do not agree with this : I think it is putting the cart before the horse. The language is all right, if you employ it properly. As previously stated, you have "speed" and "velocity," which imply different conceptions. If you use "velocity" when you mean "speed," is that the fault of the language ? No ! I think books are frequently obscure because the writers do not think clearly : because, often, they do not know exactly what they mean. If they would define their words, use them in one sense — and in one sense only, — Dynamics, for example, would be vastly simpler than it is. But, it may be said, " What does it matter, if you can do the sum, and get the right answer out?" If the end, and aim, is to pass examinations, I quite agree. If the object, however, is to learn Dynamics, then the first aim should be to have clear ideas on the subject. It will now be evident w/iy I have commenced this book on Resistance of Air by a discussion on dynamical similarity, as well as vector and scalar quantities. But to return : so far I have discussed this question theoretically — I was almost saying mathematically. What does experiment tell us on the subject ? Does it support the statement that Pressure = Force -^ surface ? I think not. Let us suppose the vessels represented in fig. i to have the same sized, circular bases and to be of the same height : also filled with water to the same level. Now, taking the 36 Resistance of Air weight of the water as the "force," what should be the * ' pressure " on their bases ? I must here make another digression. I am using " Force" in a new sense, and I must justify it. I might say that all books say weight is Force ; and so, why should not I ? Weight, certainly, is Force. But, as it is not a "Mass-acceleration" in the ordinary sense of the expres- sion, I must add that I use it in the sense of a potential force : just as one speaks of kinetic and potential energy, so I shall speak of kinetic and potential Force. The dimensions of the two are the same, since weight is M^ : g being an acceleration due to what we call gravity. Fig. I. — Hydrostatic Pressure. We have, in fig. i, two "Forces" which are certainly not equal : the areas of the surfaces are equal, and experiment tells us the pressures are equal. We should have then an equation Force (variable) Pressure (constant) = Surface (constant) which, it need hardly be said, is absurd. I have been told that the " Forces" in the two cases are the same. 1 do not understand why. But if they are, the word "Force" must be used in a new and, to me, an un- known sense. The weights are certainly not equal. Let us take another case, where the " Force" (weight) is kept constant, whilst the pressure is made to vary. 1 refer to "Pascal's Paradox": it is certainly not as well known as it deserves to be. The vessel G, fig. 2, fixed to a separate support and Pressure — Scalar and Vector Quantities 2)7 consisting of a narrow upper and a very broad lower cylinder, is closed at the bottom by a movable piston, which, by means of a string, or wire, passing through the axis of the cylinders, is independently suspended from the extremity of one arm of a balance. Let G be filled with water, then, despite the smallness of the quantity of water used, there will have to be placed on the other scale-pan, to balance it, a very considerable weight. But if the liquid be frozen. Fig. 2. — Pascal's Paradox. and the mass loosened from the walls of the vessel, a very small weight will be sufficient to preserve equilibrium. Here ' ' Force " (weight) and area of surface are constant, whilst ihc press ?i re is variable. Force (constant) ,. , ■ i i x - — \ '— = Pressure (variable) Surface (constant) There does not appear to be much sense in this. We may also consider the case of a gas, under com- pression, in a cylinder whose diameter is verj' large (so that the area of the wall shall be negligible, when compared to that of the piston and end of the cylinder). If the gas be allowed to expand a little we shall have 38 Resistance of Air " Force " (constant) Area {nearly constant) Pressure {very variable') In these examples I have been referring to what is called "hydrostatic pressure": but, to understand what pressure really is, it were better perhaps to take a simpler case. Let us imagine a weight resting upon a scale pan. We have the weight acting as a potential force, in the direction of gravity. This is equivalent to it exerting a pressure in a vertical direction. We see, hence, that pressure is the ' ' scalar " of force : such pressure being, clearly, the total pressure., which may be anything per unit area — depending on the area of the surface the weight is resting on. The pressure is, of course, the scalar of the Force in the direction of the potential acceleration. Hence pressure has ?io dimensions, though its measure- ment may be obtained from the equation p^nilt-'^ and not from / = ;;//-!/- 2 F In Hering's Conversion Tables wq find/ = — = L"^MT"^ This, I maintain, is incorrect, though it is the form commonly employed for it. Dr E. Mach {The Science of Mechanics) puts his views as follows: — "We should say to-day that the force of per- cussion, the momentum, the impulse, the quantity of motion inv, is a quantity of different dimensions from the pressure/. The dimensions of the former are mlt~^, those of the latter mlt~^. In reality, therefore, pressure is related to momentum of impact as a line is to a surface. Pressure is/, the momentum of impact is//." If one remembers that pressure is the " scalar" of Force, this paragraph is quite intelligible. We have pt = mv, or mv 1 1 r 1 /= — : whence we may say that pressure may be defined as the rate of chajige of a mass-scalar. I said, previously, that speaking of a " mean-force-per- Kinematic Viscosity 39 unit-area" was "confusing." It appears to be so for the following reasons : — It rather suggests (without justification) that the resistance varies as L^ — qj- ^^e area of the surface, — which experiment shows is not always correct. It causes the L^ in the formula to disappear, and so leads to the inference that the resistance varies as some function of V, and withotit any refercfice to 'L : or it might lead to the in- ference that the resistance varied as L^ and witJiout atty reference to W. As will appear later, L and V are intimately related: the resistance varying, generally, as some function of (LV). This fact must not, however, be driven to death, as there appears to be a slight tendency to do at the present time. The dimensions of viscosity depend on experiment. Maxwell {Theory of Heat) says: — "The viscosity of a substance is measured by the tangential force on the unit of area of either of two horizontal planes, at the unit distance apart, one of which is fixed, while the other moves with the unit velocity, the space between being filled with the viscous substance." Hence the " dimensions " of /a, the coefficient of viscosity, are those of a Force, divided by a surface and by a velocit}', and multiplied by a space (of one dimension) : /. e. the " dimensions " are MT T-2 \ xL, or [ML-iT-i] = (;z> Again, the dimensions of kinematic viscosity (r) are those of (/a) divided by (p), tJie density : i. e. they are ML-iT-i^ML-3, or [L2T-iJ = (,/) This may, also, be conveniently expressed as [LV]. REFERENCES Kelland and Tait, Introduction to the Study of Quaternions. Henri Poincare, Comptes Kendus de VAcadcmie dcs Sciences, Dec. 17, 1916. Carl Hering, Conversion Tables, 19 14. CHAPTER V RESISTANCE OF FLUIDS EXAMINED THEORETICALLY Having now cleared the ground and defined all the terms we are going to employ, let us examine the case of the resistance of a body, moving at a very lozv velocity, in a viscous liquid — what is called the resistance due to viscosity. We first ask ourselves. Does the resistance due to viscosity cause change of momentum or not ? We know perfectly well that it does: hence it is a "Force." We may consequently equate /x 'L^V^p" = Force = M LT - ^ We know, from experiment, that this resistance is inde- pendent of the density ; we may therefore omit p ; or, if we prefer it, we may retain p, but put « = o, so that p shall have no "dimensions." Up to the present we do not know the values of -v, y^ and z. But in a dynamical equation the "dimensions" of eacJi ie)in and on both sides vmst ahvays be the same ; otherwise the equation is not homogeneous; i.e. ihe, sums of all the powers of M, L, and T must be the same. Expanding our equation dimensionally, MLT-2 = (ML-iT-i)-xL>'x(LT-i)= MLT-2 = M'L--^T-^xL^'xL=T-^ Equating now the powers of M, L, and T, we get I =x, for M 1= —x-\-y-^a or 7' + - = 2, for L and — 2——X — S or A'+;?=2, for T 40 Resistance of Fluids Examined Theoretically 41 Hence X— I, |'= I, and z— i and the equation may be put as R (resistance) = K/xLV where K is a constant which has no "dimensions." The equation might also be put as R = K^(LV) This is in accord with "Stokes' law" that at low velocities, where the resistance varies as the velocity only, it also varies as \\\q first power of the linear dimension, and not as the square : it is also in accord with experiment, u/> to tJie critical velocity^ where discontinuity of the liquid occurs. In Nature of March 18, 191 5, under the heading of "The Principle of Similitude," Lord Rayleigh made the following remarks : — " I have often been impressed by the scanty attention paid even by original workers in Physics to the great principle of similitude. It happens not in- frequently that results in the form of ' laws ' are put forward as novelties on the basis of elaborate experiments, which might have been predicted ii priori ^Sl^x a few minutes' con- sideration. However useful verification may be, whether to solve doubts or to exercise students, this seems to be an inversion of the natural order." These remarks appear to apply very aptly to Stokes' law. which "might have been predicted" from the principle of similitude. Let us next examine the case of the resistance of a liquid which has no viscosity ; or, if that be unimaginable, the resistance of a liquid of which the viscosity is so small as to be a vanishing quantity. As before, we may put MLT-2 = ;,xV'L>'V^ when, proceeding as before, we shall get R^KpL^V^ or Kp(LV)2 42 Resistance of Air This is in accord with the Principia, where Newton states that : — " In mediums void of all tenacity, the resistances made to bodies are in the duplicate ratio of the velocities " ; also : ' * and tJiat part of the resistance which arises from the density of the fluid is, as I said, in the duplicate ratio of the velocity " [italics added]. As far as such a thing is possible, it is also confirmed by experiment. At this point I can imagine the reader who has studied hydrodynamics (as defined in the Encyclopcedia Britannica, — the dynamics of an imaginary liquid, which does not, and could not, exist, since some of its properties are self-con- tradictory) in the classical school, saying that he was always taught that in a "perfect" liquid iJiere could be no resistance : that this could he proved niatliematically. That is quite true. But, if you grant the mathematicians' assumptions, they can prove you anything. Was it not the great mathematician Napier who {granted his assumptions) proved that seventeen swallows were as strong as a horse ? Of course he was quite right : the unfortunate thing was, however, that his assumptions were wrong. In the case under discussion, the mathematicians' assump- tions are (i) the liquid is inviscid, (2) it is incompressible, and (3) it is of infinite volume ; i.e. it has no free surface. As regards (i) we are told in books that it implies that the liquid has no elasticity.'^ It is also supposed to transmit pressure at an infinite speed: this implies that it has infinite elasticity. These assumptions contradict one another. As regards (2), it is difficult to iniagine how the liquid transmits pressure at all ! The mathematician gives us no idea of what the machinery for this is. The third assumption is a very fair one : but it must be remembered that by it gravity has been eliminated. I have gone into this question (at considerable length) in the A B C of Hydrodynamics. I would specially refer to the ^ ''''Elasticity^ in the correct use of the term, imphes that property of a body in virtue of which it . . . requires the continued appHcation of de- forming stress to prevent recovery, entire or partial, from deformation " (Tait, Properties of Matter). Resistance of Fluids Examined Theoretically 43 "principle of no momentum." Since, under the assump- tions, no inomentum is generated — or, perhaps more accur- ately, transferred, — there is no Force : and, consequently, no resistance. There is no contradiction in all this. We will now examine the case of the resistance to a body moving in a viscous liquid at a velocity above the ^^ critical." We should ^;t^^f^ that, in this case, since the resistance is partly due to viscosity and partly to inertia, the form the equation would take would be something like R = KMLV)-hK,/,(LV)2 We will assume, with the National Physical Laboratory, that the liquid (or fluid) is " sufficiently defitied by p and v." We may, as before, put MLT-2 = /y'L^VV' ,, =(ML-3)-xL-^x(LT-i)^"x(L2T-7' When it is clear, at a glance, that x=i . . . . (I) Equating the powers of L, 1= - ^x + y + ,■:;+ 2n or jF + ^-f2;/ = 4 . . . . (2) Similarly, equating the powers of T, — 2=—s — Ji or ,'::-{- n — 2 . . (3) Subtracting (3) from (2), J' -\-}i — 2 or y—2 — ji Also, from (3), z=2 — n and the equation may be put as R=Kp(LV)2-«xr'' or R = Kp(LV)-x(r / LV)« and, since we do not know the value of ;/, we ma\' say R = K/,(LV)y(. / VL) . . . (A) 44 Resistance of Air This equation may be considered the fimdaniental equa- tion of fill id resistance. It was presented in a " Note as to the Applicability of the Principle of Dynamical Similarity," by Lord Rayleigh {A.C.A., 1909-10), who adds that: "It is for experiment to determine the form of this function, or in the alternative to show that the facts cannot be repre- sented at all by an equation of form (A). It is known that somewhat approximately p is proportional to V', and again that it is independent of /. If either of these approxi- mations is supposed to hold good absolutely, it follows that f is constant, in which case / is independent oi v \ or con- versely, if / is independent of v, f must be constant" [italics added]. I have given the equation (A) worked out in detail, because the result is not so self-evident to an ordinary reader as it, doubtless, is to Lord Rayleigh. We know from experiment that the resistance does not vary exactly as the square of the velocity. We know, further, that it also does not vary exactly as the square of the dimension L. It follows, therefore, that /(jy / LV) is not a constant. Is it possible to find a form of this function which will satisfy experiment ? Since, in Lord Rayleigh's formula (A), ^(LV)^ has the dimensions of Force, it is clear that f{y / VL) must have no dimensions — it must be a number. The dimensions of \y\ = [VLJ : hence v / VL has no dimensions : and if, further, f(v I VL) is composed of more than one term, they must each be without dimensions. If we put f{y I VL) = (i/ / VL)-|-X.', we shall have a func- tion which will give us what I said we might expect would be the form which the resistance of a body, in a viscous liquid, would take. This will then be R = Kp(LV)2{(i; / VL)-f^'} and if this be expanded we shall get R = Km(LV) + K,^(LV)2 exactly the type which it would be reasonable to expect. Hoiv much of the resistance varies as the velocity, and how Resistance of Fluids Examined Theoretically 45 much as the square of the velocity, will depend largely on the values of K and K^. If the velocity is very low (below a critical velocity) Ki = o, and the resistance varies as (LV) only. At the critical velocity discontinuity takes place, suddenly, and the second term is suddenly " switched in." The resistance will now increase faster than as (LV) : we may say at some higher power than i. As Kj increases relatively to K, this power will continue increasing, though it can never reacJi 2 ; it will approximate to it indefinitely. Now, will this formula satisfy all the requirements of the resistance of air? It will not. This is not to say that Lord Rayleigh's formula (A) is wrong. It is the truth : but not the whole truth. All experiment teaches us that the resistance of air — 1. At very low velocities varies as (LV). 2. At moderate velocities it varies rougJUy (or somewhat approximately) as (LV)^ 3. At higher speeds the resistance varies at a liiglicr rate than as V^ ; though it does not appear to vary at a higher rate than as L^ 4. At speeds of the order of i 200 feet per second the resistance varies nearly as the cube of the velocity — up to about I 300 or i 400 feet per second. 5. At speeds above i 400 feet per second the resistance increases as the square of the velocity only. All the writers on ballistics, however much they may differ from one another on other points, are absolutely unanimous on this one. It is, perhaps, the otily point on which they are all agreed. It will not be uninstructive to put all this in a diagram (fig. 3), so that it can be seen at a glance. The diagram is not to any scale, and it has no pretensions to being any- thing but explanatory. At A there is a "critical velocity" where the stability of the fluid, w'hich is under severe stress, breaks down. Up to this point the resistance varies as the velocity, accurately. At A the resistance suddenly increases, one may say, to some higher power of V. This power of V 46 Resistance of Air continues to increase ; and at some point C, AB cuts the V^ line. It further increases up to B, where it nearly touches the V^ line. It is not pretended that AB is a straight line (as shown in the diagram), AB is only supposed to indicate an increase, and not the law of the increase. At B there is no fall in the resistance, as the diagram would appear to indicate. The fall is in the rate of the increase^ which falls abotj± no f.s. about, 1400 f.p.fi. VEL.O CITIES P'iG. 3. — Diagram indicating relation between Resistance and Velocity. from V^ to V^. The curve of resistance continues to rise, though at a less rate : the curve, in fact, becoming y^^/Z^r, The reader may want to know the position of A. This may be found by the use of Osborne Reynolds' formula ("Lubrication," Ency. Brit.), "UD.(p / ^)=K where K is a physical constant independent of units, which has a value between i 900 and 2 000. " Put into the notation employed here, we should get or VL . (i / iy) = 2 000 (say) VL = 2 000 V Resistance of Fhnds Exainined Theoretically 47 Mr Bairstow, of the National Physical Laboratory, has kindly informed me the i/ = 0'OOo 159 for a temperature of Therefore VL= -318. If L be taken as i foot, V= -318 feet per second : whilst if L= '5, V= '636 feet per second. Just as A is not a fixed velocity, so B is not a fixed point. It depends on the barometric pressure, temperature, and the density of the fluid. Our formula, in its present state, will not satisfy these very severe requirements. It will, clearly, require a third term in it ; which term must be a function of V, to enable us to get V^. How this is arranged will appear later. At this point the reader may say that I have made no mention of the resistance due to "fluid friction": a word which is so frequently employed in books on hydraulics. As I have said previously, and elsewhere, I do not know what is meant by the expression: I cannot " picture " it to myself. In case 1 should be thought very dense and stupid, I may plead, on my own behalf, that I do not think that miybody knows what it is. Lord Rayleigh, lately, during a Friday evening discourse on "fluid motions" at the Royal Institution, remarked that we did not know what it was, "but it wdiS di vciy convetiient expj'ession.'"^ I sup- pose it is "convenient," since it is very popular. I might, perhaps, describe it as a kind of hydrodynamical "dug- out" or "funk-hole": one can always retire into it when hard pressed in an argument. REFERENCES Lord Rayleigh, "The Principle of Similitude," Nature, March 18, 1915. Osborne Reynolds, "Lubrication," Encyclopcedia Brilattmca. 1 This remark was made in an "aside," and was not printed in the paper. It was so clear, however, that I could not have been mistaken. CHAPTER VI VALUES OF K In this chapter I propose to discuss the values of K, in the formulae for the resistance of bodies of rectangular and circular section, the edges of which are parallel to the direction of the motion : a thin circular plate will be con- sidered to be a cylinder whose length is zero. The values of K will, therefore, be strictly limited within the foregoing defined boundaries — unless the contrary is stated. It was customary, for a very long time, to confound the shock against the anterior face of a body with the resistance which the body experienced when it moved in a fluid at rest, or when it was exposed to the shock of the same fluid in motion. Dubuat appears to have been the first who considered these effects under their true aspect, when he pointed out that the real resistance is equal to the differeiice between the pressure on the anterior face and that on the posterior — both these pressures being, of course, measured tositively. This definition of the resistance, against which it does not appear possible to raise any reasonable objection, furnishes us with the means of obtaining the values of K for any parallelopiped or cylinder. With this view let us put — // = height due to the velocity u of the body ; or of the stream of fluid, if the body is at rest. (o = the area of the anterior or posterior face of the body. A = the density of the fluid, or the mass of unit volume of this fluid. \ = the relation between the height of the pressure on the posterior face to the height Ji. 4S Values of K 49 R = resistance of the body, or the effective pressure which the body experiences. K=i— (X / 2), the coefficient of this resistance: that is to say, the relation between the effective resistance and the pressure w^u'- on the anterior face. This value of K is constant for all similar bodies subjected to the same con- ditions : that is to say, it is the same for all similar plates, moving in a fluid at rest : or, for all similar plates at rest and being ^'■shocked" by the fluid. It must not, however, be assumed that the value of K is the same, whether the body is moving in the fluid at rest, or the body is at rest and the liquid flowing past it. There are cases where it is the same ; as there are cases when it is not. I have treated of this very disputed question, at considerable length, in Motion of Liquids. Professor D. Riabouchinsky very kindly promised me that he would repeat Duchemin's experiments, so as to settle the point. Unfortunately the war broke out, and he has not been able to redeem his promise. I shall therefore continue to assume that the values of K in the two cases are not necessarily the same. This settled, I have shown, in the Motion of Liquids, that when the body moves in the water at rest and when it is exposed to the "shock" of this fluid in movement, the height of the anterior pressure may be measured by 2J1 ; this pressure has consequently a measure of 2u}\gJi. But \Ji being the height of the pressure on the posterior face, this face is pressed by a weight 'Aw'^gh ; and we have, for the measure of the resistance which the body experiences. smce K=i— (X / 2), and 2gh=u^ Now, since the value of K depends on X, it is necessary to find the value of the latter. Table I will explain how Duchemin found its value. The first column gives the relation between the length of the square parallelopiped and the length of the side of its 4 50 Resistance of Air anterior and posterior faces. Col. 2, the measurement, by Pitot tube, of the velocity head of the current in metres. Table I. — Body at Rest. Relation between length and diameter. Value of "velocity head." Value of H', relative pressure. Value of H' / //. Value of H'" / h relative to velocity of filament. Value of A, 4/H' try o I 2 3 m 0-057 0-057 0-057 0-057 m -0-0339 -0017 5 -001 1 5 -o-oio 5 -0-594 -0-307 -0-202 -0-185 0-764 0-961 I -018 ro25 0-136 0-523 0-653 0-672 Col. 3, the pressure head at the centre of the posterior face, also in metres. Col. 4, the value of H' / h. (I have re- tained the same notation as in Motion of Liquids.) Col. 5, the velocity head of the filament moving towards the centre of the posterior side of the body. This is taken from table XV of Motion of Liquids. Col. 6 gives the value of X, which is f of the sum of cols. 4 and 5. Table II gives the values of X when the body is moving. The remarks about Table I apply strictly to Table II. Col. 5 is also taken from table xv of Motion of Liquids. Table II. — Body in Motion. Value of Value of Value of H'", relative to velocity of filament. Value of A, Length to diameter. /I, velocity head. relative pressure. Value of H' / /t. 4/H' try m. m. r o-o6i -0-034 ] f 0-059 0-053 0-055 -0-034 -0-028 -0-023 h -0-556 1 1-488 0-746 • \ 0-057 -0-025 h -0-435 1-333 0-718 y 0-049 -0-022 J { 0-052 -0-016 r -0-313 ^ \ 0-054 -0-018 1-180 0-694 \ 0-047 -0-014 5 ( 0051 - o-oo8 ) ' 1 0-047 0048 - 0-009 - 0-009 5 ^ -0-186 1-024 0-670 Values of K 51 A very cursory inspection of these tables will show that the values of X steadily increase as the body becomes longer, when the body is at rest ; whilst exactly the opposite occurs when the body is in motion in the liquid at rest. We can now find the values of K which we have been seeking. Let us first arrange the values of A when the body is moving. From Table II we have, values o{ b I c o i 2 3 X = 0746; 0718; 0-694; 0-670 K= I — iX = o-627 ; 0-641 ; 0-653 ; 0-665 In the search for an empirical formula which would give the values of K, Duchemin found, as follows : This formula gives the following figures : b I c= o I 2 3 K = o-627 0; 0*6412; 0-6529; 0-6626 which differ very little from the preceding. From the foregoing we see that when the body is moving, the value of K for a flat plate is 0-627, and that the value of K increases slightly as the length of the body increases. Let us next examine the case where the body is at rest and the liquid flows. Arranging the values of X as before, b I c= o I 2 3 X = 0'i36; 0-523; 0-653; 0-672 from which we get K=i-iX = o-932 ; 07385; 0-6735; 0-664 We see in this case that the values of K decrease as the length of the body increases ; they therefore move in the reverse direction to the preceding ones. Examined carefully, it will be seen that there is a remark- able analogy between these values of X and the values of the relation between what is called the effective and 52 Resistance of Air "theoretical" discharge of liquids through cylindrical "ajutages," or additional tubes. This is not perhaps extraordinary, since in one case it may be called the flow "inside the cylinder," whilst, in the other, it is the flow " outside" the same cylinder : we might expect an analogy. I have given this relation (taken from F. D, Michelotti's Speriuienti Idraulici) in table xxiv of Motioti of Liquids, but I will repeatit here in Table III, Table III.— Liquid flowing through a Tube. Relation I \ d, length of tube to diameter of tube. Relation ;«, effective to theo- retical discharge. o \ I T 2i o-Go9 6 o-6i6 9 0-767 I 0-8157 0-822 I 3 0-820 I 4 5 6 7 0-8179 0-809 5 0-807 0-803 2 8 0-799 7 From F. D. Michelotti's Sperinienti Idyaulici It will be seen that for the same values oi b j c (or, what is the same thing, I j d) we have A-0-I36 = 1776 m^ — (o -609 6)2 X and m being the numbers which correspond to the same values o{ d / c\ whilst the numbers 0'i36 and 0*609 6 are the respective values of X and in when b / c=o. From this relation we easily get X= I 776;/^2 — 0-523 5 and K=i-26i 8-o-888;;/2 (^) Values of K 53 In order to obtain the value of K corresponding to any value oi b I c, from this formula, it is necessary to know that of Jii. But the law connecting the values of in, or the relation between m and b j c, is not known. We must, therefore, fall back on Michelotti's experimental values in calculating Table IV. In the cases where Michelotti gives no value of in, K has been found by interpolation. The values of K, for the cases when the body is moving, have been calculated from the formula (y). The values of K are given for ratios from b / c = o up to b j c—?>. We will call these coefficients of resistance, since experimental results will show that they are really constants. Table IV. Value of K. Ratio of ^ 1 c, length to diameter of body. Body moving Body at rest (formula 7). (formula 5). 0"00 0-627 0-931 8 0-50 0-634 5 0-923 8 I '00 0-641 2 0739 3 1-25 0-644 4 0-7137 1-50 0-647 3 0-694 5 175 0-6502 0-680 5 2 -00 0-652 9 0-6709 2-25 0-655 5 0-664 8 2-50 0-6579 0-661 7 2-67 o-66o 0-661 I 275 o-66o 5 0-661 6 3-00 0-662 6 0-664 5 4 '00 0-670 8 0-667 7 5-00 0-6780 0-6799 6"oo 0-684 0-683 5 7"oo 0-689 3 0-6S8 9 8 -GO 0-694 0-693 9 It is not to be supposed that these are absolutely accurate. It is quite probable that later work may show that the values require slight alteration. They are, however, of the 54 Resistance of Air right order : and they will be found to give accurate results — as will appear later. These values of K are suitable for either air or water : but, in the case of the latter medium, the body must be completely^ and sufficiently, immersed. They are not in- tended to apply to partly immersed, or floating, bodies. These two series of the values of K show very curious results. They show that in col, 3 K has a minimum value of 0'66i i, which corresponds to the ratio h j c = 2^. After this ratio, the values of the coefficients of resistance are the same iji both cases. So that a cylinder, whose length is at least 2f times that of its diameter, will experience the same resistance when it is moving in water at rest as when it is exposed to the "shock" of the fluid in motion — all other things being equal, of course. But if the length is less than 2§ diameters, this is no longer true : for then the resistance when the body is at rest is sensibly greater than when the fluid is at rest and the body is moving. REFERENCES Colonel DuCHEMiN, Les lots de la resistance des fluides. F. D. MiCHELOTTi, Sperimenti Idraulici. CHAPTER VII RESISTANCK IN WATER— VALUE OF K FOR SPHERES Up to the present the whole subject has been treated theoretically. It will be well now to see how experiment agrees with theory, and to show how the resistance can be calculated. I will first begin with some very simple examples of the resistance of water, showing how Table IV is used. In these calculations I shall, systematically, ignore the resistance due to viscosity- as is the usual custom with all authors. It is clear that in the formula R = Kpft)V2{(v / VL)+KJ Since the first term (that due to viscosity) varies inversely as V and L, it becomes smaller and smaller, as V and L increase : we will therefore neglect it, and consider that R=Kpa)V2 If we refer, again, to Osborne Reynolds' formula, we have, when L and V are unity, LV / 1^ = 2000, or V I VL=i / 2000 = 0-0005 This is really very small compared to K^, which is unity : it is therefore usually neglected. Dubuat measured the resistance of a plate, a cube, and two parallelopipeds whose lengths were 3 and 6 times the length of the side of the square of the anterior surface respectively. All the bodies were at rest in a stream which had a velocity of 3 (Paris) feet per second. The resistance was measured by an ingenious balance, which it is un- necessary to describe in detail ; and the weights are in old 55 56 Resistance of Air French pounds {livrcs). The experiments were Nos. 258, 259, 260, and 261 {Principes d'Hydraulique). The resistance being defined as R = Kw^u^ where K is the proper coefficient which will be found in Table IV ; w the area of the anterior and posterior faces = 1 foot; A (density) = 70 / 30-2, where 70 livres is the weight of a cubic foot of water, and 30*2 is the gravity- constant ; u being, as before stated, = 3 feet : log 70= I -845 0980 log 30 -2 = I -480 006 9 log A = 0-365 091 I log ?(!2 = 0-954 242 5 I •319 333 6 w being i, we have log(oA«2=i-3i9 333 6 For the case of the plate, we find from Table IV that K = o-93i 8 : log K = 1-969 322 7 \ogu)^u'^= 1-3193336 logR= 1-2886563 and therefore R= I9'44 lbs. {livres). (Dubuat's experiment gave 19-455 lbs.) For the cube we find that K = 0*739 2 : log K = T-868 8207 logft)A2^2^ I -319 333 6 logR= I -188 1543 and 1^=15-42; against Dubuat's experimental value of 15-225. For the treble cube K = 0-664 5 : log K = T-822 495 o log w^u- =J[;319^33^ log R= I -141 8286 and R= 13-86 ; whilst the measured value was 13*875. Resistance in Water — Value of K for Spheres 57 Lastly, for the sextuple cube K = 0"683 5 : logK = T-834738 5 log«A«2^i-329 333 6 log R= I -154072 I and R= 14 "26 ; whilst the experimental value was 14 "270. I think the reader will consider such close agreement eminently satisfactory : the differences are quite within the limits of experimental error. Even in the case of the cube (where the difference is greatest), if we employ K = 0738 5, which is the value Duchemin obtained from his own experiments, R will become 15 "40, which gives a still closer agreement with 15 '225. I wish specially to draw attention to the fact that, though the reasoning is Duchemin's, the experiments were carried out by Dubuat — possibly before Duchemin was born. They are old — but good ! We will next examine a case where the body, itself, moves in a liquid at rest. In the Academic de Marine there is a memoire by M. de Marguerie ^ in which he gives the results of experiments on cubes moving through sea-water. In the first case the cube had a side of 3 (Paris) feet; whilst in the second the area of the surface of one side was 4^ square (Paris) feet. The cubes were completely immersed, and were moved in a direction perpendicular to one of their faces, by means of falling weights acting through suitably arranged pullies. They travelled almost uniformly a distance of 130 feet in the number of half-seconds entered in the second column of Table V, the numbers being the mean of two experiments. In this system of pullies it was found that two-thirds of the action of the motor weights was absorbed by the internal resistances of the arrangement. The resistance, as before, = KwAz^-. The weight of a foot cube of distilled water being 70 tivrcs, and that of a cubic foot of sea-water being i "026 3 of the weight of distilled water, we have ^ — {yo / 30'2)x i'0263, takin^ the ' I have been unable to find this up to the present. I rely on Duchemin's report and figures. 58 Resistance of Air (Paris) foot as the unit of linear measure. From Table IV we have K = o-64 in all the cases. From the data a) = 9 for the first body, and (0 = 4-5 ^oi" the second. z^= 130 / /, if we put t for the number of seconds in each experiment. Table V. Body experi- Half- seconds in travelling 130 feet. Motive weight. Resistance. mented on. Observed. Calculated. Cube of 9 sq. feet surface of face Cube of 4^ sq. feet surface of face / '75 I 346 r 127 I 170 lbs. 87 47 87 47 lbs. 29 15-667 29 15-667 lbs. 30-249 15-435 28-715 16-025 The agreement here again, between the observed and calculated results, is quite as near as one could reasonably expect. Let us next examine an experiment of Colonel Beaufoy. Resistance of a square flat plate moving in water at rest. Immersion 9 feet, area of plate 2 '971 8 square feet. Experi- ment was very similar to M. Marguerie's, just examined. R=Ka)A2^2 K = 0-627; (0=2-9718; A = 62 -3 1 / 32-18 By proceeding as before and putting u=c^ : u=\o : ti= 12 we get 90-2 : 360-8 : 519-55 . . . calculated 88-96 : 354-09 : 509-19 . . . observed The agreement is not so good — the differences being rather under 2 per cent. ; but the experiments of Colonel Beaufoy cannot be depended upon in the same manner that one can on those of Dubuat and the Abbe Bossut, To show that this is not an unfair statement, let us examine the results obtained for the same sized plate, on another day, when the immersion was only 3 feet. Resistance in Water — Value of K for Spheres 59 u= ^ : 11= 10 : u= 12 R = 8377 : 337"2i : 486-12 We may also examine the resistances recorded of two bodies which ought to have experienced about the same resistance to motion through the water. [P. 430] : a cube of the capacity of i cubic foot ; immersion 6 feet ; temp, 72° F. (temp, of dock 64° F.). [P. 452]: a cylinder, sectional area i square foot; length i foot; capacity i cubic foot; im- mersion 6 feet ; temp. 59° '5 F. (temp, of dock 59° F, ), u =t^ : u= 10 : ii= 12 R = 2y8S7 : 1 15 •61 : 167-64 . . . cube R= 24-753 '• 101-44 '■ 146-62 . . . cylinder Now, admitting that the resistance of the cylinder sJiould be rather less than that of the cube — independent experi- ments appear to have indicated this as being the case : there are, also, other theoretical reasons for this, that I cannot go into at present, — is it probable that the difference is as great as is here shown ? I think not : the differences here exceed 10 per cent. It must be admitted that we cannot place very great reliance on these experiments : not because they are 100 years old, but because the results are frequently very inconsistent with one another. We must be grateful for the work, carried out under great difficulties, and which was certainly not "scamped." We must not, however, accept it as being quite faultlessly accurate. A series of experiments carried out at the National Physical Laboratory would be exceedingly valuable : but we can only hope for these. All these calculations are very simple. In the next chapter I propose to proceed to something rather more difficult — the resistance of spheres moving in liquids at rest. Before doing so, it will be necessary to know how K changes for a spherical body. It is not my purpose to discuss how K changes, when the shape of the anterior part of a body is altered, I must, however, make one exception, and refer to the value of K for a sphere. I cannot now discuss it theoretically ; but I must take it for granted that the 6o Resistance of Air resistance of a sphere is two-fifths of that of a circumscrib- ing cylinder. Experiment will support this assumption. It is rather customary, in these cases, to say that Newton said that the resistance of a sphere was only half of that of a cylinder : and then to show that Newton was wrong ! Such discussion is worse than useless, because Newton was right, Newton was most careful to say that his well-known proposition had tiothing to do ivitJi air, or any other known medium. It only referred to his assumed, hypothetical, medium. But to return : the Chevalier de Borda made experiments on the comparative resistances of spheres, hemispheres, and the plane surface of a hemisphere of the same diameter of 4| (French) inches, when moving circularly in the air, by means of a whirling machine {Memoires de I'Academie des Sciences, 1763). The speed of their centres was 18 "39 (French) feet per second. In the case of the sphere and the hemisphere with the flat face leading, he found that the resistances were as i to 2*44, or as 2 : 4*88. This pro- portion differs very little from that I have assumed, since the value of K (before correction) would be 0'6\\ 2 for the sphere, and 0*634 5 for the hemisphere. This would give the proportion of 2 : 4 '95. The same philosopher performed the same experiments in water, the bodies having diameters of 59 lines (French measure).^ These experiments gave a ratio of 2 500 : i 000, or 5 : 2. He also caused the hemisphere to move with the plane, and convex, sides leading. The ratio of the resist- ances measured was as 2 525 : i 000, or as 5-05 : 2. Vince made similar experiments in water which gave analogous results. His hemispheres had a diameter of I "I inches: and he found that the ratio of the resistances of the plane and convex surfaces were as 8339 : 3400, or as 4*9 : 2.2 Hutton also, by the same method, measured the resist- ances, in air, of the plane and convex surfaces of a hemi- • Memoires de I'Academie des Sciences, 1767. 2 Philosophical Traftsactio?ts, 1798. Resistance in Water — Vahie of K for Spheres 6 1 sphere of 6| inches diameter. The ratio, he found, was as I 730 : 715, or as 4-84 : 2.1 From the foregoing it will be seen that the assumption made appears to be justifiable. The value of K for the circumscribing cylinder {when it is moving in Jluid at rest) being 0'64=i6 / 25, the value of K for a sphere, under the same conditions, will be lxlf = Y%"^: so that the resist- ance may be expressed as R = (32 / i25)(oA«^ where <« = sectional area, A = density of the fluid, and u the velocity of the sphere moving in the liquid at rest. REFERENCES Le Chevalier Dubuat, Principes d' Hydraidique. M. DE Marguerie, Academie de Marine. Colonel Beaufoy, Nautical Experiments. Le Chevalier de Borda, Mhnoires de V Academie des Scietices, 1763, also 1767. Rev. S. Vince, " Bakerian Lecture," Philosophical Transactions, 1798. Charles Hutton, Tracts on Mathematical and Philosophical Subjects. Colonel Duchemin, Les lois de la resistance des fluides. ^ 36th Tract, No. 42, " New Experiments in Artillery." CHAPTER VIII RESISTANCE OF SPHERES FALLING IN WATER We saw in the last chapter that the value of K for a sphere moving in liquid at rest might be taken as 32 / 125. It follows, therefore, that the resistance to a sphere falling in water by its own weight will be R = (32 / i25)tt(Az^^ This is the ' ' force " of the water on the sphere — the mass- acceleration caused by the water acting on the sphere. In other words, the acceleration produced in the mass of the sphere. Now, if we put S for the density of the sphere and c as its diameter, its mass will be %wcS: and if the "mass-accelera- tion " of the sphere is (32 / i25)a)A?^^ whilst its mass is IwcS, it is clear that the acceleration caused in the sphere by the water will be (32 / i25)a)A?^2^fa)^5 = (48 / \2^){^u'^ / cS). Since, however, the water retards the motion of the sphere, this acceleration is negative, = —a^Z^.u'^ / 125 cL Besides this, gravity is accelerating the sphere positively, and the value of this acceleration may be expressed as W i — ^ j : hence the actual acceleration of the sphere in falling freely in a liquid is a\ 48AZ/2 ^ 8) i2Sc6 and the equations of motion of this body are, at the end of a time /, during which it has acquired the velocity u and traversed the space e, du dt = a\ 48A2^2 8) \2^c8 62 Resistance of Spheres Falling iji Wate^'- 63 and u . du iie = -/ A\ 48A2^2 I — * Sj i2$c8 Let us next, to simplify, put gli-^\ = q and 48A / \2^cS = q / a'^ We get ,, rt'z^ a^ . du at = qu'^ qia^ — ti^) a^ Integrating this for the simultaneous values t — O and « = o, we get a , (a-\-u / = — . log, — — 2q \a — u^ Similarly, , u . du a'^u . du de = — qu"^ q(a'^ — u^') Integrating this for the simultaneous values of e = and u = o, we get q [u .du ^ . , — e= I h constant a^ ja^ — u^ = — h log, (a^ — u'^) + constant But e = O when 7c = o. constant = I log, rt- = log, « ••• -„e = \og,a-^\og,(a^-u^) a' = Hloge «^ - log. (rt^ - /^2)| and a^ , / a^ • '• f = — log, — 1 2q ^ \a^-u^ I have worked this out in detail, as it might give the young reader some trouble. 64 Resistance of Air We have from the first inte<.^ration a I a -Til t= locf or If we now put we shall have and hence a-\-u in = a — u from which we get 'in — I 2g \a — u 2qt . a-\-u ^- = logJ . a \a — u 2qt -^=— = loe, in log,w = log7'^ ^) \a — u u = a\ ^m + I Also from the second integration 2q \a^-u-^j q ^'Ja^-u^ W- a = — loo" ' v- in — I a' in+ I «2 , m+ I aP- . [ in+ I = — log* ,, -..--; ;„ = — l°r g J {in + I )2 - {in -if q \ 2 slin Now, when in is a very big number, we may put in in the place of w+ I, and we then get e=— (^log,7;^-log, 2) If we now put '^^— , the value of log^ in, in the equation, it becomes e = — — -log, 2 q\a J^ = L — log, 2 and ^ = ^e + log, 2 Resistance of Spheres Falling in Water 65 But and q=g[ i-^ a"' 12 5<;-t) \ and hence 48A 5 48A =v- 2 5^f(^-A) 48A ' ^ W=^, e + loCT^2 and, changing this into ordinary logarithms, ,^.Ull^M.log2]J 48A I 48A "^ J > 1 25^^(6 -A) The modulus M = 2 "302 6 very nearly. Newton, in the second volume of the Principia, gives the times, observed by him, of balls falling freely in water ; the details given in Table VI furnish us with data for an application of this formula. The measurements were made in inches and the weights in grains troy. The weight of a cubic foot of water being ']6 lbs. troy, this will give 16600 grains for the weight of a sphere of water of 5 inches diameter. If we now put W for the weight m vacuo of a sphere of this liquid having a diameter of c inches, and VV for the weight of a globe, of the same diameter, when plunged in this liquid, we shall have W'= 16600^-^ / (5)^ grains and ^ / A = (W + VV') / w where ^ = density of the globe, and A = density of the fluid. Putting these values into the equation given previously, 5 66 Resistance of Air we shall get the values in Table VI of the times of fall. It will be seen that the calculated and observed times do not differ very materially. TAliLE VI. Time of fall. Diameter Weight in water. Height of fall. of globes. Observed. Calculated. inches grains inches sees. sees. 0-842 24 n 1 12 4 4-ni o-8i2 96 5iVr 1 12 '5 14-88 0-998 68 ll 1S2 24I 25-0 I -000 \o 2li 182 i4f 14-49 roGO 10 79l 182 8 7-54 0-999 70 6A 182 25f 26-20 0-999 70 \^o\ 182 6i 5-69 1 -C09 90 \i 182 3'l 31-82 Newton's experiments on spherical bodies falling in water. Notwithstanding this very good agreement between the calculated and observed values of the time, it is necessary to point out that this application does not comprise all the circumstances of the problem. This formula for resistance is, in reality, only correct for uniform motion, whilst the spheres, in falling, were subjected to an accelerated motion. We know, further, that "when a sphere descends from rest in a fluid by the action of gravity, the motion will be the same as if a moving force equal to that of the sphere minus that of the fluid displaced ?LcX.Qd on a mass equal to that of the spJiere plus half that of the fluid displaced" (Stokes, vol. i). It is clear, therefore, tliat our formula should be modified to locf 2 48A i2Seg{S-^) = f , i25.-(o- + o-5A) 6 + 48A M . log 2}^- 48A 2SCO-(S-A) Resistance of Spheres Fallin<^ in Waaler 67 In comparing these it will be seen that the error is o-5i'5^M.log2J 48A _ This error for the eight globes experimented with is respectively o" '026 7 ; o""096 7 ; o"'i22 5 ; c""07i ; o"'036 7; o"'i28 3; o"'027 5 ; o""i6o, and these fractions of seconds are so small that they may be neglected, without sensible error, in the times calculated in the table. It must also be observed that in these calculations we have assumed that ni may be substituted for ni-\- i. This is justifiable, since vi — from the formula ^^^ = log^;// — is a a number of not less than 40 powers of 10. REFERENCES Sir George G. Stokes, Mathematical and Physical Papers, vol. i. Newton's Principia (Motte's translation). CHAPTER IX RESISTANCE OF BODIES IN CIRCULAR MOTION In experimenting on the resistance of bodies, either in air or in water, there are many different methods which may be adopted. Every method has its advantages, as well as its disadvantages. One of the most commonly employed, up to quite lately, was that of the whirling machine. It has the advantage that the velocity, as well as the resistance, can be measured with considerable accuracy. It is gener- ally said to have the disadvantage that the machine causes a circulation of the air, so that the latter cannot be con- sidered "at rest." This error should cause the resistance measured to be too small. It is well known, however, that the measured resistance is always too great. The Chevalier de Borda appears to have been unaware of this ; as were also Hutton and Vince.^ The first author that I am aware of who thoroughly grasped this fact was Thibault. He says the resistance in circular motion is ^'much greater than it is really in direct motion ; and it does not allow us to fix the intensity of the resistance in direct motion, by employ- ing circular motion, either continuous or oscillatory " {RecJiercJics Expa iuicntales siir la Resistance de I'Air).'^ Dubuat certainly makes a rather vague reference to this, and says we ought to expect that circular resistance should be greater than that in direct motion. It appears to be 1 Even, apparently, S. P. Langley, Experiments in Aerodyna/nics, 1891. 2 " Resistance circulaire est beaucoup plus gratidc qu'elle ne Test reelle- ment, dans le mouvement directe d'une surface : et ne permet pas de fixer Tintensite de la resistance directe, en employant le mouvement circulaire ; soit continu, soit oscillatoire." 68 Resistance of Bodies in Cii^cular Alotion 69 reasonable that this should be so, since we cannot have "steady motion" in a circle: it must necessaj'Uj' be an accelerated motion. True, the body in motion (assuming the angular velocity of the moving arm to be constant) is not being "spurted"; but it is undoubtedly being "shunted." The body is being constantly accelerated towards the centre of the circular path. M. Eiffel, in referring to this, says that "according to Duchemin " the measured results are too great. From this one might infer that Duchemin was the first who noticed it. Such is not, however, the case ; nor did he ever lay claim to any such pretension. Acceleration undoubtedly causes resistance in all cases. We know that a body moving steadily in an inviscid, in- compressible liquid, which has no free surface — this last condition must be specially noted — would meet with tio resistance. If, however, the body were being accelerated, there would be resistance. Duchemin was certainly the first, if not the only, author who has shown us how, knowing the resistance to circular motion, we can calculate what it would be in direct motion. M. Eiffel, in his small book La Resistance de V Air (Dunod et Pinat, 19 10), gives a formula which he calls " Duchemin's formula " for this correction. I regret to say that it is not in the least like Duchemin's real formula, since M. Eiffel in ' ' simplifying " it has quite altered it. Quoting from Les Lois de la Resistance des Fluidcs, \\q find it is where R is the resistance in rectilinear motion, when all the points of the body are animated by a velocity Ji ; Rj the resistance in circular motion, referred to the centre of figure of the greatest section (,) of the body, taken perpendicularly to the arc described : this centre having the same speed u as in the rectilinear motion ; /the distance between this same centre and the axis of rotation ; "JO Resistance of Air s the distance between tJiis same central point and the centre of gravity of that part of the section w which is on the side of the fixed axis of rotation. (The diagram (fig. 4) will explain this better than a great many words) ; K is the coefficient of the resistance taken from Table IV. In the case of a plate in motion, in fluid at rest, we see that it is 0"627 ; C is a linear quantity representing the thickness of the filaments deviated parallel to the anterior surface of the body, a thickness which is equal to \c.5\na when the body is one of revolution, and \ J w . sin a in the contrary case : c being the diameter of the section w, and a what is commonly called the " angle AXIS OP ^ ^ / >: ^-^'^ '^ rotation • i ! ' Fig. 4. of attack." Clearly, in the diagram, a = 90" and sin a= I. When the length of the arm is infinite, Rj = R. Let us take, for simplicity, the case of a plane surface m which is moving perpendicularly to its plane — i.e. where the angle of attack is 90°. The resistance of this surface, in rectilinear motion, being R=Kft)Az^2, we have, for the resistance in circular motion, Ri = K(oA?/2+ I "624 4Cft)A- f-s We see that CojA is the mass of the molecules of the fluid composing, at each instant, the filaments deviated in front of the anterior face, and that CwA - is the quantity of motion which would be impressed on them by (what is popularly and commonly called) the "centrifugal force," if this mass zvcre concentrated at a distance f—s from the axis of rotation and if this point had a speed u. Resistance of Bodies in Circular Motion 7 1 To render this point clearer, let us suppose that the plane surface &> is a rectangle abde (fig. 5), turning uniformly about an axis gJi situated in the prolongation of its plane, and parallel to its two sides. Let a be the breadth bd of this rectangle ; c the length ah ; f the distance (/" between the centre of figure of the rect- angle and the fixed axis of rotation ; f^ the distance ^y from the exterior edge to the same axis ; /i, the distance /yfrom the interior edge to the same axis ; "6=7^ //"the angular velocity of the circular motion, ?^ being the absolute speed at r, the centre of the surface. X the radius of the arc described by any point of the rectangle. hj We have now Fig. 5. /=/+!.- and /,=/-ir Hence the resistance experienced by the small area adx will be composed of : (i) the quantity Y^adx^x-'i"-, derived from the anterior and posterior pressures of the fluid ; and the moment of this, referred to the axis of rotation, will be Y^at^'6'^j^dx\ and (2) that of ^Zt^a .dx .x'i'^, due to the centrifugal force acting on the mass of the filaments CA^ . dx ; the moment of which is ACc?Aa-a--. ^/x, A being a constant, to be found by experiment. We should therefore have : Rj/"=Kr?A«-[ X or, integrating between these limits, R /= Kr^A«'3 v.^ -U) + ^(y? -m L4 3K ./,r+AC (7A«'- x-dx J/,, 72 Resistance of Au'- Putting for/J and/^, their values, viz. /.=/+^ and /„=/-^ we shall cfet Rj/= KrtA«2 4 3l^V 4 Putting /"-a" = 2/2 and ac = w we have, finally, r2 AC ^Cr2- Rj = KcoAz/- R,= R AC r2 ACr2 Therefore, if we give A the value of i "624 4, which satisfies, in formula (e), the results of experiment, the value of Rj of this formula will differ very little from the preced- ing one. Since C — \l\ s — \c, K = 0*627, and that, in the experiments, \ is the inaxiinuin value of c J /, we may take I -624 4C _ AC + ACc^ K(/-s) K/ ' 4/- 12K/3 From the foregoing it follows that (i) the last term of the second member of the formula (e) is in great part caused by the "centrifugal force"; (2) this formula, under a form which is simpler than the preceding one, accounts for the same circumstances of the circular motion ; and (3) that in this motion one may consider the mass of the molecules of the filaments which surround the body as having their centre of gravity at a distance /"—j from the ' Duchemin gives this ^Y . AC , 3^% ACr^ x K/ 8 ' I2K/' Init there appears to be a clerical error in this. - We have I -624 4C _ I -624 4C/ i,±,^, _. _ V /^ /•' AC . •6^2 ACf2 K/ 8/- ' 16K/3 Resistance of Bodies in Circular Motion 73 axis of rotation. This last remark will be found to be very useful later. Let us now make an application of this formula to a special case. Samuel Vince, in his " Bakerian Lecture" in 1798,^ describes the results he obtained by causing small plates to travel in circular motion in water. There were four plates whose areas were each 0*932 5 square inches, the centre of each of these plates being 7-57 inches from the axis of rotation ; and the speed of the centres of the plates was 0'66 foot per second, measured round this axis. The area w was therefore 373 / 144 square feet. The weights were taken in ounces troy. We have therefore, from Table IV, K = o-627, A = 9i2 / 32-18 (oz. troy); a cubic foot of water weighing ']6 lbs. troy = 9 12 oz. troy. The resistance in rectilinear motion would be R = Ka)Az/2 but for circular motion it would be ^^-^^"i^-'-S0^ Now ■^12 V 144 ^ 14 44 ' ■* 144 Putting these values into the equation, wc get R^ = 0"234 88 oz. by calculation; whilst the actual resist- ance, measured, was 0*232 i oz. troy. The accuracy of this measurement is extraordinary, if we consider that the machine (a whirling machine with arms under 8 inches in length) was little better than a very beautiful toy. If we wish to simplify Duchemin's formula, it may be put as ' Philosophical Transactions^ 1798- 74 Resistance of Air where c is the length of the side of the square plane and f is the distance from the centre of this plane to the axis of rotation. Clearly, (i-624 4X^) / 0"627=i-3, This form of the equation is only applicable to the measure- ment of the resistance of square plates, and when the ^^ angle of attack " is 90°. It may not be uninteresting, as bearing a little on this subject, to refer to a curious experiment carried out by Robins (A/athc^nalical Tracts'). He put a plate, measuring 8| inches X 4 inches, in a whirling machine in two different positions : — 1. With the 4-inch edge leading ; the "angle of attack" being 45°. 2. With the 8j-inch edge leading ; the " angle of attack " being also 45°. Referring to 2, he says : — " Now, instead of going slower, as might be expected, from the greater extent of part of its surface from the axis of the machine, it went round much faster." Experiments with a whirling machine, if they are tiot properly interpreted, frequently lead to fallacious deductions. The Chevalier de Borda (to name only one experimenter) said that his experiments showed conclusively that the resistance of plates increases — per unit area — very rapidly with increase of size. It will be seen later that, when properly interpreted, this statement is not justified. If de Borda's experiments are objected to, in consequence of being "150 years old," I will refer to Langley's Experi- ments in Aerodynamics (table xxviii). We find that— 1. For a plate 12-inch square K = 0"00i 80 2. ,, ,, 8 ,, ,, K = o-ooi47 3. ,, ,, 6 ,, ,, K = o-ooi59 If we omit 2 (the result of 7 experiments) and compare 3 (the result of 34 experiments) with i (the results of 27 experiments), we see that doubling the length of the side of the plate increases the value of K by about 12^ per cent. — say 16 : 18. Is it possible to believe that this is correct? Resistance of Jhniics in Circ2i/ar Alotion 75 Certainly the whirling arm was 30 feet long : but, all the same, the results require correcting. If we apply Duchemin's formula to these, some of the discrepancy can be accounted for ; though not the whole. Confining our attention to the 6-inch plate (where the error is less than with the 12-inch plate), I think the coefficient too high.^ The value of the coefficient for the 8-inch plate appears to be the most accurate : of this the reader will be able to judge for himself later. REFERENCES L. A. Thh^aui-t^ (Lieutenant de vaisseau), RechercJies Experi- mentates sur la Resistance de t'Air, etc., 1826. Samuel Vince, PJiitosopIiical Transactions, 1798 (Bakerian Lecture). S. P. Lanoley, Experiments in Aerodynamics, 1891. Charles Hutton, Tracts on MatJiematicat and Phitosopliicat Subjects, vol. iii, 1 8 1 2. Robins, Mathematical Tracts. ' Even if we also make allowance for temp. = 10°, as against the more usual one of 15° C. ^ Duchemin always refers to him as Thicbault : the spelling of his name, is, however, as above. CHAPTER X THE RESISTANCE OF AIR The unqualified statement that the resistance of air is the same as that of water, if allowance is made for the differ- ences of densities, has been responsible for much retardation of progress. To this has also been added another obstacle, viz. that the compressibility of the fluid can be ignored. It is perfectly true that the two fluids deviate in a similar manner round bodies, and that their absolute resist- ances only differ according to their densities. But if the density of water is the same in the state of rest as it is in that of motion, 1 this is not true in the case of air, since this fluid, by reason of its compressibility, which is one of its characteristics, becomes denser as the pressure is increased. To express this property of the air in the value of A in the formula, or the density of the fluid in contact with the body, Duchemin found that experimental results can be satisfied, both for small and great velocities, in the follow- ing manner : — Let S' be the density of the air at rest, V the speed with which this fluid, at tJiis pressure, would rush into a vacuum, and u the speed of the moving body. We have, for the density of the fluid filaments in contact with the body, A = o'( I -f — ), from u — o to u — Y; and, constantly, A =2^', when the speed n \'~, greater than V. It results from these values of the density of the fluid in 1 This is, of course, only approximately true : water, far from being incompressible, has had its bulk reduced to 8o per cent, by sufficient pressure. 76 The Resistance of Air yj contact with the body that the resistance of the air increases (as is well known) at a greater rate than as the square of the velocity. It is quite usual, in books treating on the resistance of the air, to refer to the "Newtonian law," and to say, implicitly if not explicitly, that Newton said that the resist- ance of the air varied as the square of the velocity. We find this, for example, in Lanchester's Aerodynamics, on p. 71, in fig. 31 : "Ro:V2, Newton." Now, what did Newton actually say ? In the Principia, Book II, he says: — "And when the resistance of bodies in non-elastic fluids is once known, we may then augment this resistance a little in elastic fluids, as our air" [italics added]. Also : — " Besides, the more swiftly the globes move, the less are they pressed by the fluid at their hinder parts ; and if the velocity be perpetually increased, they will at length leave an empty space behind them, unless the compression of the fluid be increased at the same time. For the compression of the fluid ought to be increased (by Props. 32 and 33) in the duplicate ratio of the velocity, in o)'der to preserve the resist- ance in the same duplicate ratio. But because this is not done, the globes that move swiftly are not so much pressed at their hinder parts as the others ; and by the defect of this pressure it comes to pass that their resistance is a little greater than in a duplicate ratio of their velocities. " We see that Newton never said anything of the sort ! We were first to find the resistance in non-elastic fluids — which he said, elsewhere, varied as Av-\-Bv^, — and then to aug/nent this a little. He does not say by how much ; but it ivas to be increased. This is exactly what 1 shall proceed to do, when we calculate some resistances of air ; when it will be seen that the resistance increases at a higher rate than as the square of the velocity. Robins, who was one of the earliest experimenters, was well acquainted with this ; his reasoning on this subject was also very sound, for he says : — "We shall understand an increase or diminution in the 78 Resistance of Air resisting power of the medium similar to ivJiat might be occasioned by incrcasijig or (iimi)iishing its density ; the principal purport of our present attempt being to evince that, according to the different compression of the medium, . . . such changes may arise in the resisting power of the medium " {New Principles of Gunnery) [italics added]. Robins found that the resistance, at high velocities, was three times what the law of the duplicate power of the velocities would give : this excess he put down to the com- pression of the air. As will be seen later, the resistance should only be twice this value for all velocities above about I 366 feet per second. Robins' experiments, which were made with lead balls, were tainted by reason of an error, which has since been recognised. The lead splashed from the pendulum, thereby causing increased resistance to be observed. He specially mentions that one should not stand near the ballistic pendulum, for fear of accidents from this cause. It is curious, therefore, that he should not have made allowances for this, since he gives "this simple axiom of mechanics : That if a body in motion strikes on anotJier at rest, and they are not separated after the stroke, but move on with one common motion, then that common motion is equal to the ^notion with which the first body moved before the stroke " {New Principles of Gunneiy). We must therefore rule out any deductions from Robins' experiments, the recorded results of which (certainly at velocities greater than i 000 or i 200 feet per second) are well recognised as being too great. There have been many rules proposed, showing how the resistance of the air increases at a greater rate than as the squares of the speeds, but none indicate, like Duchemin's values of A, the cause of this greater increase. It will not be contested by anyone that the pressure on the anterior side of a body moving in still air is greater than that of the static fluid at a distance. It is clear, therefore, that its density at this part will also be greater than that ot the fluid at a distance. Experiment proves incontestably that the increase in this density is proportional to V, from The Resistance of Air 79 u — o\.o 7it = V=4i6'34 m., in the ordinary conditions of the atmosphere ; ^ so that in the last hmit of the velocity we have After this limit, the speed u being greater than V, the relation a = ( Y>' would become A=2o+/— — -U', which would give an increase of density, above 20', pro- portional to « — V. It is necessary therefore that, for this increase not to take place, the lateral velocity of the filaments, in their passage between the moving body and the fluid not deviated, should be increased in the same ratio. But this is precisely what happens if we take V as the speed of flow of the air into a vacuum from a density S' ; for the fluid moving behind the body, not being able to follow it with a speed V, whilst the body is moving with a greater speed u, a vacuous space of the length u — V would tend to be formed at this part ; and the velocity of the lateral filaments must evidently increase by reason of the greater facility for flow resulting from this vacuous space — or pro- portionately to ti — V, Hence, whilst on the one hand the density of the fluid in front tends to an increase which varies as u — V, on the other hand it is diminished in the same manner, in consequence of the increase of velocity of the filaments : there is reason, therefore, to suppose that these two causes compensate one another, and that the density A remains constantly equal to 28' — as experiment appears to show is the case. Now the density S' of the air at rest varies with the temperature and barometric pressure occurring at the time of the experiments. According to Biot and Arago (I follow Duchemin closely), the weight of dry atmospheric air, at the temperature of melting ice and under the pressure 0760 m., is, for equal volumes, yfg- of that of distilled water. ^ This Duchemin takes as 6 = 760 mm.; T=i8^C. Modern experi- menters take T= I5°C., but I will follow Duchemin here : the results are much the same. 8o Resistance of Air Experiment has also shown that the same fluid expands 0"003 75 of its volume for each degree of the centigrade thermometer. 1 But atmospheric air, in its natural state, is not dry ; it contains water vapour which, less dense than this fluid, diminishes the density of the mixture ; and, further, as the air becomes warmer it tends to saturate itself with a greater quantity of vapour, the effect of which is to further increase the dilatation of its volume. That is why Duchemin, in accord with his contem- poraries, adopted, in questions relating to ballistics (i) the relation of i to 850 between the density of free air and that of water, the temperature being 18° centigrade and the pressure 760 mm.; (2) 0-004 for the dilatation of the volume of the air for each degree of the thermometer. Consequently, in any state whatever of the atmosphere indicated by the height B of the barometer, and the tem- perature T in degrees of the centigrade thermometer, we shall have, very nearly, in designating by D the density of water, B m. / 1 + o -004 X 1 8 \ (A) D o 76 m. X 8 50\ I -f o -004 X T It will be necessary here to make a small digression, and compare Duchemin's formula (X), for the correction of temperature, with that employed by modern authors — say M. Eiffel, for example. The critic will {and very rightly) object to the coefficient 0*004, unless I can justify it. If the corrections are wrong, no great reliance can be placed on the deductions I shall draw from experiments. The modern method of correction for temperature is to multiply by -^ ~ when reducing the density to the 273 + T standard temperature I5°C. If Duchemin's fraction is put 250-f 1 5 into the same form, it would be =. (I suppose the 250-J-T reduction made to 15°, so that the comparison may be correct.) The modern coefficient of expansion is, as stated ^ Regnault has since shown that this should be 0003 665 o, or ^tb- The Resistance of Air Si previously, 0003665; whilst Duchemin's is 0004. The former figure is, there can be very little doubt, correct for dry air : but is it quite certain that it is not too small for moist air ? The objection has been raised that any correction required for the modern coefficient, which is due to moisture, must be so small as to be negligible, since the amount of moisture in the air is very small. Tyndall says : — "The quantity of vapour is really small. Oxygen and nitrogen constitute about 99^ per cent, of our atmosphere, and of the remaining 0*5, about 0^45 is aqueous vapour. The rest is carbonic acid " {Heat a Mode of Motion). Zahm, in his Aerial Navigation, gives a table, computed by W. J. Humphreys for Moore's Descriptive Meteorology, of the composition of normal air : — Water vapour at 760 mm. — i "20 per cent. Carbonic dioxide Nitrogen Oxygen Argon Hydrogen O'O 77-08 2075 0-93 O'OI lOO'OO He also says: — "The vapour weighs 0*622 as much as dry air having the same volume, temperature, and pressure, or quite accurately | as much " {^Aerial Navigation). We find also in Ganot {Elemoitary Treatise on Physics): — "Tyndall has established the fact that ... if aqueous vapour be compared atom for atom with air, its power of absorption and radiation is more than 16000 times that possessed by air. Such facts as these are sufficient to show the importance of the small quantity of this vapour that exists in our atmosphere." The ' * elastic force " of water vapour is greater than that of air. Robins says : — " Water, when rarefied into vapour, is generally supposed to be near teti times more elastic than air equally heated." 6 82 Resistance of Air Further than that, the elasticity increases inucJi more rapidly than the temperature. Table VII gives the tensions of aqueous vapour from o° C. to 24° C. (Ganot's Physics). Table VII. Tempera- tures. Tension in millimetres. 4-600 4-940 5-302 5-687 6-097 6-534 6-998 7'49^ 8-OI7 8-574 10 1 1 12 13 14 '5 16 17 18 19 9-165 9-792 10-457 1 1-162 ' 1 1906 12-699 13-535' 14-421 '5-357 16-346 First differences. 20 21 22 17-391 18-495 19-659 20-888 22-184 0-340 0-362 0385 0-410 0-437 0-464 0-494 0-525 0-557 0-591 0627 0-665 0-705 0-744 0-793 0-836 0-886 0-936 0-989 1-045 1-104 1-164 1-229 1-296 Second differences. 0-022 0-023 0-025 0-027 0-027 0-030 0-031 0-032 0-034 0-036 0038 0-040 0-039 0-049 0-043 0-050 0-050 0-053 0-056 0-059 o-o6o 0065 0-067 ' ' This table shows that the elastic force increases much more rapidly than the temperature. The law which regulates this increase is not accurately known " (Ganot, Physics). Comparatively little appears to be known about how the expansion is altered by the addition of water vapour to air. W. H. Dines ("On Wind Pressure upon an IncHned ' In the original this is iro62. - In the original this is T3-635. The Resistance of Air 83 « Surface," Proc. Roy. Soc, 48) makes the following very suggestive remarks : — "No determinations of the dew-point have ever been made in connexion with the experiments, but I think that damp air is conducive to a high, atid dry air to a low, relative pressure. ^ "On some days the values will not vary to more than i or 2 per cent, throughout ; on others, under apparently pre- cisely similar circumstaticcs, variations of 10 or even i^ per cent, will occur witJiin a few minutes. "These variations give an immense amount of trouble, because it is imperative that an experiment should be re- peated many times before the mean value is considered correct " [italics added]. W. Ferrel ("Motions of Liquids and Solids," Math. Monthly, vols, i and ii, i860) says : — "46. The ratio of the density to the elastic force decreases -^\^ for every degree of Fahrenheit. But as a higher temperature is always accompanied by a , greater amount of aqueous vapour, the density of which is less than that of the atmosphere, the rate of decrease has been found to be -^\^ for every degree. " This value, for centigrade, is about t^l^ ; which is equi- valent to a coefficient of 0*004 "^^ly nearly. To continue this discussion would serve no purpose, since I know of no experiments on which one can base argument. It appears certain that -^\^ is too small a fraction ; though how much too small is uncertain. 1 will continue to employ the constant 0*004, since it appears to give very fairly correct results. It is, in fact, the coefficient used by Ferrel in his well-known formula in A Popular Treatise on the Winds, 1894. As regards the value of V, if this speed is that with which the air will flow into a vacuum, when this fluid has a density 0', the height to which this is due is that of a ' See note at end of chapter. 84 Resistance of Air column of air whose weight is equal to the barometric pressure. But the height of a column of water which corresponds to the pressure 0760 m. of the barometer, for a mean temperature, is 10 "395 m. ; we have therefore V= V2-xio-395x(D / <5') . . (m) where D is the density of water. In the majority of the experiments to which we shall apply these formulae of the resistance of the air, the state of the atmosphere corresponds to a pressure of 0760 m. and to a temperature of 18° centigrade ; we have therefore, by the formulas (\) and (/x), (5'=D / 850 and V = 416 -34 metres. This value of V is equivalent to i 282 French feet, or to I 366 London feet, and is thus found conformable to the evaluations which have been made by Lombard and Hutton. REFERENCES Robins, New Principles of Gunnery. John Tyndall, Heat a Mode of Motion. G. Eiffel, The Resistance of Air and Aviatio/i (Constable & Co.), London, 19 13. Ganot, Elementary Treatise on Physics (Atkinson's translation), 1873. W. H. Dines, " On Wind Pressure upon an Inclined Surface," Proc. Roy. Soc, 48. W. Ferrel, "Motions of Liquids and Solids," Mathematical Monthly, vols, i and ii, i860. W. Ferrel, A Popular Trratise on the JJ'in./s, 1894. A. F. Zahm, Aerial Navimtion. Note I have an "aerial tourbillon " which, ordinarily, works very well. I have observed, however, that on some very hot and dry days // 7vill not zvork at all. I am investigating this very curious question. When carrying out experiments with air, it does not appear to be sufficient to record the temperature and barometrical reading: the reading of the wet-bulb thermometer should also be noted. CHAPTER XI RESISTANCE OF AIR AT LOW VELOCITIES In this chapter we will examine the results given in experi- ments on the resistance of air at comparatively low velocities, so as to see how they agree with theory. We will first take the case of planes moving uniformly and normally in air at rest, and turning about a fixed axis: i.e. planes being moved in a whirling machine. The resistance in rectilinear motion would be expressed as R = 0"627odA^6ooo5' and 3000 times greater than the >naxii)utni Resistance of SJiot at High Velocities 95 have, for the equation of motion of the projectile, vdv = — (48Az^- / \ 2 sc^)dc or dv / 7^= -(48 A / \2^co)de Tai;le XII. Resistances. Velocity. Ratio — experi- Observed. By theory. ment : theory. oz. avoirdupois 50 0-676 0-51 1 1-32 100 278 2-046 1-36 200 11-34 8-18 1-39 500 74-4 51-2 1-46 600 110-5 73-6 1-50 7UO 156-0 100-2 1-55 800 2120 130-9 1-62 900 280-3 165-7 I 69 1000 362-0 204-6 1-77 1 100 456-9 247-6 1-84 1200 564-4 294-6 1-91 1300 683-3 3457 1-98 1400 811-5 40 10 2-03 1500 947-1 460-1 2 -06 1600 I 066-9 5237 2-08 1700 I 228-4 591-2 2-07 1800 I 368-6 662-8 2-06 1900 I 505-7 738-5 2-03 2000 I 637-8 818-3 2-00 Resistance in uz. avoirdupois. Barometer 30 inches. Temperature not given. But by the law.s of the resistance of air, which we are verifying, A = ( i + — )(5' for all speeds below V, and \—2o' for those above this same speed V. This last value of A is constant, and, for the present purpose, we may also suppose the first value constant, in taking the arithmetical mean between the value (i+— ^)5' which it has at the origin and that of ( i + - lo' which it has at the end of the value of A, it jis clear that we may neglect 0-5A in the formula, without causing any sensible error in the calculation. 96 Resistance of Air movement; or by putting a= iH Vf~ M tlurmg the 2V whole of the movement. By this hypothesis, which is justifiable, as will be seen later, the integral of the above equation, taken between the limits v = N ^^, e = and v = u, e = e is ,^g'V.\ 48A. or, by common logarithms, /V^X _48A5_ \u J 125^-oM where M is the modulus for changing Napierian into ordinary logarithms ; and, since 1 \T 1 48-^f log Vn — log // = ^ ^ " "^ i25r(5M then 48Ae I If we now put log ?( = log Vq i2ScSM 48Ae , = lOCf VI i25r^M we shall have u^V^ / m . . . . (i) The data in the following tables were as follows : — Temperature and pressure, not having been recorded, we will assume were about 18° C. and 076 m. ; consequently we shall have (5' = D / 850 and V= i ^66 feet. But, accord- ing to Lombard, the specific gravity of cast-iron balls is 7 166; therefore ^ = 609 1 • I ^'. In the first table, XIII, a// the velocities exceed i 366 feet ; consequently we may put A = 2(5'. The modulus M = 2 "302 6. Putting these values into the equation; we get, in Table XIII, the velocities at the different distances from the sero pendulum, which are found in the last column. The agreement between the cal- culated and observed velocities may be said to be perfect. Resistance of Shot at High Velocities 97 Table XI 11. —Mean Velocities, etc., at Several Distances. Distance i-pr. with 16 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet 60 120 180 240 300 360 ft. sec. 2 100 2 005 I 914 I 827 I 744 1665 I 590 ft. sec. 95 91 87 83 79 75 4 4 4 4 4 ft. sec. 2 100 2 005 1914-7 I 828 I 7457 I 666-8 I 591-6 Readings of barometer and thermometer not recorded. 144, Tract 34. In Table XIV we have similar data for the velocities obtained with 8 oz. of powder. As before, we may put A =2(5' for the first four distances (the speeds being, i.e., not less than i ^66 feet) ; whilst for the last two we must put A = i(3 4--:j^' since A = 2(5' for the initial velocity Vq, whilst A= i + — d' for the terminal velocity. All the remaining data are the same as in Table XIII. Here again the calculated velocities agree exceedingly well with the observed ones. Table XIV.— Mean Velocities, etc, at Several Distances. Distance i-pr. with 8 oz. of powder. from pendulum. Velocities observed. Velocity lost. Difif. Velocities calculated. feet 60 120 180 240 300 360 ft. sec. 1637 I 563 1493 1427 1365 I 306 I 250 ft. sec. 74 70 66 62 59 56 4 4 4 3 3 ft. sec. 1637 I 563 I 492-5 1425 I 361 I 303 124S Readings of barometer and thermometer not recorded. 144, Tract 34. 98 Resistance of Air In the next three tcables, XV, XVI, XVII, all the speeds are less than i 366 feet : we must therefore put and Here again the agreement between calculation experiment is eminently satisfactory. Table XV.— Mean Velocities, etc., at Several Distances. Distance i-pr. with 4 oz. of powder. from pendulum. Velocities observed. Velocity- lost. Diff. Velocities calculated. feet 60 120 1 80 240 300 360 ft. sec. I 360 I 300 I 243 I 189 I 138 I 090 1045 ft. sec. 60 57 54 51 48 45 3 3 3 3 3 ft. sec. I 360 I 300 1243 I 192 I 140 I 092 I 048 Readings of barometer and thermometer not recorded. 144, Tract 34. Table XVI.- -Mean Velocities, etc ., AT Several Distances. Distance i-pr. with 3 oz. of powder. from pendulum. Velocities observed. Velocity lost. Difif. Velocities calculated. feet ft. sec. ft. sec. ft. sec. 60 I 203 I 151 52 3 3 3 3 3 I 203 I 152 120 180 I 102 I 056 49 46 I 105 I 060 240 300 360 I 013 973 936 43 40 37 I 018 978 939 Readings of barometer and thermometer not observed. 144, Tract 34. It was said, previously, that it was justifiable to put the value of A as the mean between its values at the origin and at the end of the movement. It will be well, perhaps, to Resist mice of Shot at High Velocities 99 give the reasons for this assumption. Starting from the equation dv / v= —{d^'^^de / 125^^) if we put, in the place of A, its real value ( i + - ) 0' the equation becomes dv ^ 1 + V 48 (5 W^ Table XVII. — Mean Velocities, etc., at Several Distances. Distance i-pr. with 2 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet 60 120 180 240 300 360 ft. sec. goo 866 834 804 776 750 726 ft. sec. 34 32 30 28 26 24 to to to to to ft. sec. 900 866 83s 804 776 748 722 Readings of barometer and thermometer not recorded. 135, Tract 34. of which the integral, taken between the limits v — \ ^, e — O and v = u, e — e, is locr Vo 1 + V M^ + TT V 4SS'e from which we get, by changing the Naperian logarithms into ordinary logarithms, and putting 4SS'e / i25r^M = log n VoV .KVo + V)-Vo (2) This equation (2), in which the density A remains variable, gives the same calculated velocities as those given by equation (i) for the charges of 4, 3, and 2 oz. Hence, the lOO Resistance of Air employment of the mean and constant value of A produces no sensible error in the calculation of the remaining velocities, if, during the whole time of the motion, the Table XVIII.— Mean Velocities, etc., at Several Distances. Distance 3-pr. with 24 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet o 50 100 150 200 250 300 ft. sec. I 730 1682I I 636 I 592 I 550 I 510 I 471 ft. sec. 48 46 44 42 40 39 2 2 I ft. sec. I 730 1683 I 638 I 594 I 551 I 510 1470 Readings of barometer and thermometer not recorded. 166, Tract 34. Table XIX.— Mean Velocities, etc., at Several Distances. Distance 3-pr. with 16 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet 50 100 150 200 250 300 ft. sec. I 555 I 514 1474 1436 I 399 I 364 1330 ft. sec. 41 40 38 35 34 I 2 I 2 I ft. sec. I 555 I 513 1473 1433 1395 1358 1324 Readings of barometer and thermometer not recorded. 166, Tract 34. projectile is animated with a speed less iha?i V. When the speed exceeds V, it is admitted by all the authorities that the resistance varies strictly as the square of the velocity ; this is equivalent to A remaining constant. 1 1688 in original is evidently a misprint. Resistance of Shot at High Velocities loi Since the question is important, it will be as well to give a few more examples from Hutton's Tracts. Here are some for the 3-pounder gun. The diameter of the shot ^ = 0'23i 67 feet; all the other details are as in the examples of the i -pounder gun. The state of the atmosphere is assumed as normal, viz, 6 = 076 m. and T=i8°C. In the case of the 6-pounder the diameter of the shot ^ = 0-295 83. Table XX. — Mean Velocities, etc., at Several Distances. Distance 3-pr. with 12 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet SO 100 150 200 250 300 ft. sec. I 290 I 259 I 229 I 200 I 172 I 145 I 119 ft. sec. 31 30 29 28 27 26 ft. sec. I 290 I 257 I 226 I 197 I 167 I 140 I 120 Readings of barometer and thermometer not recorded. 166, Tract 34. Table XXI. — Mean Velocities, etc., at Several Distances. Distance 3-pr. with 8 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet 50 100 150 200 250 300 ft. sec. I 060 I 036 I 012 989 966 944 922 ft. sec. 24 24 23 23 22 I I ft. sec. I 060 1038 I 012 986 962 937 914 Readings of barometer and thermometer not recorded. 166, Tract 34. ro2 Resistance of Air Table XXII.— Mean Velocities, etc., at Several Distances. Distance t -pr. with 48 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet o 85 170 255 ft. sec. I 813 1748 1686 I 627 ft. sec. 65 62 59 3 3 ft. sec. I 813 1748 1686 1626 Readings of barometer and thermometer not recorded. 192, Tract 34. Table XXIII.— Mean Velocities, etc., at Several Distances. Distance 6-pr. with 32 oz. of powder. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet 85 170 255 ft. sec. 1676 I 618 I 562 I 508 ft. sec. 58 56 54 2 2 ft. sec. 1676 I 616 I 559 I 503 Readings of barometer and thermometer not recorded. 192, Tract 34. Table XXIV.— Mean Velocities, etc., at Several Distances. Distance 6-pr. with 2^ oz. of povvc er. from pendulum. Velocities observed. Velocity lost. Diff. Velocities calculated. feet 85 170 255 ft. sec. I 506 1454 1404 1356 ft. sec. 52 50 48 2 2 2 ft. sec. I 506 1452 I 401 I 351 Readings of barometer and thermometer not recorded. 192, Tract 34. REFERENCE C. Hutton, Tracts on Mathematical and Philosophical Subjects^ 1812. CHAPTER XIII RESISTANCE TO WIND OF BODIES AT REST With a view to a little variety, let us now examine some cases of the resistance of plates to the wind, where the body is at rest and the fluid is moving. Thibault measured the resistance of bodies to moving air the velocity of which was measured by means of anemo- meters. The data of his experiments, as well as the mean results obtained from a number of observations, are to be found in Table XXV. In the first column are the areas of the surfaces in square centimetres ; in the second, the velocity of air in metres per second ; in the third and fourth, the barometric and thermometric records ; in the fifth, the resistances observed, expressed in grammes ; whilst in the sixth, the results calculated by the formula R=Ka)Az'2, in which the coefficient K=0'93i8. TABLt . XXV. Resistances. Area of Velocity of air. Height of bar. Temp, cent. surfaces. Observed. Calculated. sq. cm. metres metres grms. grms. I 089 4-177 0-766 19 222-50 215-32 I 089 4-854 0-751 18 249-60 286-57 I 089 4-955 0746 15 257-66 300-06 2304 4-253 0-747 14-5 448-12 468-44 I 089 8-219 0-752 14-5 920-33 840-35 I 089 5-602 0-749 15 448-92 385-66 2304 1-829 0-736 9-4 93-00 86-54 I Difterences I -156-88 + 99-69 Thibault's experiments on the resistance of plates, at rest, to the wind. 103 104 Resistance of Air Here is an example of the calculation. In the first experiment we have (0 = 0-1089 square metres; v^/^-xyj m. ; B = 0766 m. ; T = 19° C. These values give (as explained previously) (5' = D / 8467, and V = 4i5-53 m. But the weight of a cubic metre of distilled water being , ^ I 000 000 . , I 000 000 grms. , we have, D= . And, smce g ^=9-808 8, we shall have . I 000000 / 4"I77\ ^ A = — — I + Z. — LL. =121-62 8467X9-8088V 415-53/ Putting these values into the formula, we find R = 2i5'32 grms. The other resistances are calculated in the same manner. The agreement in places is not, perhaps, as close as one might wish : the differences are, however, both positive and negative, and their sum is very small. One must consider the very great difificulty there is in measuring the velocity of the wind accurately. Smeaton, in his Experimental Researches on the Resistance of Water and Air,^ gives the results found in Table XXVI for the resistance which a 1 2-inch square plate experiences when it is exposed to the direct " shock" of wind animated by different velocities. The calculated resistances have been obtained by the use of the formula (^). In taking K = 0'93i 8 from Table IV, w—i, u — one of the velocities given in the first column in ,1 , M 62-312 q I / U \ the table, whilst A = ^ X - — i -\ -) ; notmg, for 32-18 825\ I 346/ ' ^' this value of A, that the weight of a cubic foot of distilled water is 997 oz. avoirdupois, or 62-3125 lbs. The baro- metric and thermometric readings are not given, but the force of the wind caused Duchemin to suppose 6 = 760 mm. and T=io°C. ; from which it results that S' = B / 825 and V = I 346 feet. The comparison of the calculated and observed resistances shows a very satisfactory agreement, more especially when ^ F/ii/. Trans., 1759. Resistance to Wind of Bodies at Rest 1 05 one considers the great difficulty there is in measuring the velocity of the wind. As will be seen, the totals of the resistances may be considered to be absolutely accurate. Table XXVI. Resistances. Velocity of the wind. Observed. Calculated. ft. per sec. lbs. lbs. I '47 0-005 0-004 731 09 2*93 0020 0-018 945 I 4-40 0-044 0-042 479 7 5-87 0-079 0075 688 8 Til 0-123 o-ii8 140' • 14-67 0-492 0-475 792 22 'OO rio7 1-07576 29'34 1-968 1-92366 36-67 3-075 302082 44-11 4-429 4-394 8 5 1 "34 6-027 5-984 I 58-68 7-873 7-8589 66-01 9-963 9-989 7 73-35 12-300 12-407 8 47-505 Diff.= 47-3908 0-114 Smeaton on pressure of air on 12-inch square plate. Let US next examine some details given by Colonel Beaufoy. In Thomson's Annals of Philosophy, vol. vi, 18 1 5, we read : — "When the wind blows at the rate of 12 geographical miles per hour, or 20 "29 feet in a second, the power of the wind on a plane i foot square at right angles to the current is equal to 13 "567 oz. avoirdupois; and the generality of vessels upon a wind blowing at this rate can barely carry top-gallant sails. " When the wind blows 24 geographical miles per hour, the force is 3-541 lbs. avoirdupois, and vessels are under close-reefed topsails. io6 Resistance of Air "When the wind increases to 31*16 geographical miles in an hour, vessels are under their courses, and the power of the wind is equal to 6 lbs. ' ' When the force of the wind is 8 lbs. on a square foot, its velocity is 35 '931 geographical miles in an hour, and may be denominated half a storm. "When the strength of the wind is 12 lbs. on a square foot, its velocity is 43*918 geographical miles in an hour, and may be called a full storm." Colonel Beaufoy advanced no theory : in fact, he continues : — " Whilst on this subject, I have subjoined some experi- ments on the resistance of air and water which prove how very erroneous the theory of resistance is, and the small advantage it has been to practical men " [italics added]. We may calculate the resistances in exactly the same manner as we did in the last example ; putting K = 0*931 8, (a— I, u one of the velocities given in feet per second, whilst A, as before, is = 62*312 5 I u The barometrical 32*18 85o\ and thermometrical readings not being given, I assume that the coefficient of A = - — 850 and V = I 166 feet. Putting these values into the equation, the calculated values will be found in col. 3, Table XXVII. Table XXVII. Resistances to free wind. Velocities. Observed. Calculated. feet per sec. lbs. lbs. 20-29 •848 ■^n 40-58 3-541 3'564 52-68 6-00 6-IO 60-75 8 -GO 8-14 74"257 I2-00 12-24 Colonel Beaufoy on wind-pressure. Readings of barometer and thermometer not given. Resistance to Wind of Bodies at Rest 107 The calculated results are exceedingly close to the observed ones. Of course it is probable that the state of the atmosphere was not ' ' normal. " B was probably less than 760 mm. ; and T was probably less than i8°C. A proper correction would, in all likelihood, have reduced the calcu- lated values by some small amount. Since these observa- tions of Colonel Beaufoy's were printed in 181 5, there can be no suspicion of collusion between us. I have not picked out his observations because they agree with my views, for I am examining all reliable experiments that are available, and by good authorities. These are the only experiments that I am acquainted with, and which are worthy of presentation. I give another one, but the results are not considered very reliable, even by the author Jiimself. Professor W. C. Kernot {Australian Association for the Advancement of Science, vol. v, 1895) made some experi- ments on "wind pressure" by the use of a fan, and got the rather extraordinary result of R = 0'007z^2. In vol. vi, where he continues the subject, he says : — " In my previous paper on wind pressure certain important points are left to some extent in doubt, my methods of experimenting not being su-fficiently perfected to give thoroughly reliable results " [italics added]. I only refer to this paper, since Professor Kernot was considered an authority, and was invited by the Institution of Civil Engineers to discuss Dr Stanton's paper on wind pressure in 1908. After stating that his whirling-machine experiments gave a value R = 0*003 3^^) he said : — " How the earlier experimenters came to exaggerate the pressure ^o per cent., it would be interesting to know." One might pertinently inquire how Professor Kernot "came to exag- gerate the pressure" more than \oo per cent.} — i.e. from 0*007 to 0*003 3 ("Corresp. on Wind Pressure," 1908). Professor Kernot in this paper draws attention to a thing which is not generally known. He measured the pressure of a current of air on a 3-inch square card and found it to be 0*15 lb. avoirdupois. He then took a io8 Resistance of Air 9-inch circular disc, which he placed behind the card, in the current of air. Without disc, pressure . =0"i5 lb. Disc 1 2-inch from card pressure = 0*14 ,, >) y>> )) )> ^^^>> ,, 6 ,, ,, ,, =0-09 ,, > > 3 > > ) > 11 =0 07 , , > > 1 ) > ) > > > =0 03 , , He also exposed a cylinder and a cone to a current of air, as in fig. 6. When the resistance had been measured, a plate AB was CYUINDER CONE A B A' A B B A' Fig. 6. placed close to the models as in fig. 6, when it was found that the resistance was increased by about 20 per cent. Let us now examine the resistance to the wind of very large plates. We have, up to the present, only considered the pressure on plates of about one foot square ; and it is not right to assume that the pressure, per square foot, will Resistance to Wind of Bodies at Rest 109 be the same for large as for small plates. The information on this subject is, unfortunately, very small, but Dr Stanton read a paper bearing on this subject at the Institution of Civil Engineers in 1908 ("Experiments on Wind Pressure"). This paper was a sequel to the one he had read in 1904, which dealt chiefly with the resistance of small plates : this will be referred to again, in more detail, later. Ur Stanton said that one of the chief objects of these papers was to determine, if possible, the effect of the form and dimensions of the plate upon its resistance, since it appeared to be commonly recognised that the geometrical form, or possibly actual dimensions, had considerable influence on the intensity of pressure. By an exceedingly ingenious arrangement Dr Stanton measured the pressure on three plates, 5x5, 5 x 10, and 10 X 10 feet, and found the value of K for miles per hour was 0*003 2. That is to say, R = 0*003 ^V^ in miles per hour. He was careful to point out that, for these sizes of plates, there was no "dimensional effect" apparent. This remark was qualified, later, by his saying that "for this range in size [25 to 100 square feet] any purely dimensional effect in the resistance, if it exists, is a very small one." Since he had, in his previous paper, given the values of the pressure on very small plates in a wind tunnel, and had found, as a result, R = 0"00i 26V^ for feet per second, or R = o*oo2 7V2 for miles per hour, the inference conveyed, certainly to one of the listeners, ^ was that the resistance offered to the wind by any structure in the open air was about 18 or 20 per cent, more than the resistance of the same structure in a uniform current of air. Such was not Dr Stanton's view, however, for he said, in reply, that he regretted that the description of the experiments given in the paper should have led Dr Shaw to suppose that they indicated any difference betweefi the pressure on a plate in the wind and the pressure on the same plate when in a unifortn current of air. He was satisfied that the pressures in wind 1 Dr W. Napier Shaw, F.R.S. (now Sir Napier Shaw). no Resistance of Air and uniform current were identical. The i8 per cent, to which Dr Shaw referred was entirely due to the size of the plates used in the two cases. What this very decided view was based on is not quite clear, since in his previous experiments on small plates (as will be seen later) there was no appearance of any "dimensional effect," although the lengths of the sides of the square plates varied from 0*442 inch to 2 inches, and the circular plates, of about the same areas, varied within the same limits (nearly ten to one). If there was no "dimen- sional effect" when experimenting with very small plates, and equally none with relatively very large plates, why should there be any when operating with medium-sized plates ? This was apparently the view of another of the audience,^ who said he could not agree in imputing this difference in the coefficients to any real dimensional effect, since he con- sidered it difficult to imagine that the physical properties of a gas could involve an absolute lengtJi of some inches. In this I think most people would thoroughly agree : the explanation, on such a slender basis, hardly appears satisfactory. Dr Stanton even drew attention to experi- ments of Mr Dines {Quart. Journ. Roy. Met. Soc, 16) which gave diametrically opposite results. Mr Dines had balanced two plates, one 42 square feet and the other 9 square feet, in a wind, and found that the pressure on the larger plate was only y^ per cent, of that on the smaller — a ratio of i : i "25. He also balanced a 3 x 3 foot plate against a 1 1 x \\ foot plate, and found the pressure on the larger plate was only 89 per cent, of that on the smaller one. To be quite fair, however, I must point out that, at this meeting at the Civil Engineers, Mr Dines agreed in thinking the conclusion that he had drawn from his own experiments was wrong, and that he considered that wind pressure was "strictly proportional to the area." Whether this was intended as implying that he did not believe in "dimen- sional effect" is not quite clear from the report of the meeting. 1 Mr A. Mallock, F.R.S. Resistance to Wind of Bodies at Rest 1 1 1 If we believe in dynamical similarity, we must equally, I think, believe in a small dimensional effect when dealing with very small plates. It is, however, very small, and should be most apparent with very small plates ; and, further, it would not seem to be large enough to satisfy Dr Stanton's requirements. Any dimensional effect, amongst plates that are not very small, would appear to cause a violation of dynamical similarity. There is, however, another effect, which will be discussed later, which appears to explain this difference of pressure per unit area occasion- ally measured on plates of varying sizes. It cannot be denied that, as one of the speakers ^ observed, though Dr Stanton's method was very ingenious, the sources of error were numerous. It would have been more satisfying to have measured the pressure on the boards in a more direct manner. There are, also, grave doubts on the differences between the readings of the "pressure," "velocity," or "dynamic" tube (by whichever name one calls it) and that of the " static" tube. Dr Stanton admits that at first the pressure in the static-pressure tube was by no means constant. A different type was then adopted and calibrated in the 24-inch experimental channel — but, again, by means of another Pitot tube. It was then compared with a Dines tube, calibrated on a whirling machine, and agreed within 2 per cent.. Is, however, calibration on a whirling machine calculated to give an accurate result ? It is certainly debatable. For experiments carried out by means of Pitot tubes to be thoroughly convincing, it would appear to be necessary — to me, certainly — that the energy in the air, both kinetic and potential, should be measured and recorded. When we find that the sum of these is always constant, we can be thoroughly satisfied that the observa- tions are probably accurate. To give an example of my meaning, I would refer to Bazin's superb work on the measurements of water flowing over a weir. In this case all the energy was accounted for. After balancing all the pros and cons in this discussion, it J Mr W. Gilbert. 112 Resistance of Air would appear that Dr Stanton's experiments show that the resistance on large plates appears to be about R = 0"003 2 V^. ^ Possibly the coefficient might be found to be more nearly 0*003 I. It has been shown, however, that Smeaton's, Thibault's, and Beaufoy's experiments, on plates of the order of about I square foot, gave values of K nearer 0*0048, or R = 0*004 8V2 in miles per hour. This is about 50 per cent. more than Dr Stanton's value for large plates. Sir Benjamin Baker {The Forth Bridge, 1884) made a series of classical experiments on the pressure of wind on very large plates. He also observed the pressure on plates i^ square feet in area, whilst the large plate had an area of 300 square feet. In his lecture on " The Forth Bridge," to the members of the British Association, Sir Benjamin Baker summarised the maximum daily readings of the three gauges (one small one revolved), and he showed that in all these cases the mean readings of the two small gauges were con- siderably higher than the corresponding mean readings for the large gauge. Taking the whole of the readings, the ratio of the revolving-gauge indications to those of the large-gauge indications is i '5 : i. We see from the foregoing that Dr Stanton's experi- ments with large plates give a value of the resistance which may be expressed as R = o*627a)Az^2 ({^^ ^^g notation em- ployed here), whilst Smeaton's, Thibault's, and Beaufoy's experiments would appear to give a value expressed as R = 0*931 8(oA?/^ Sir Benjamin Baker's experiments with large and small plates have all the appearance of confirming both these values. When we must consider all the experiments good, and beyond dispute, how are we to account for this difference ? It is not easy to say ; but I offer the following, as a ten- tative explanation. Let us imagine a body of air advancing as shown at A (fig. 7) with a speed V. Let us, further, assume it to be ' In miles per hour. Resistance to Wind of Bodies at Rest 113 rotating, as shown by the smaller arrows, with a speed at B and C (say) of v. It is clear that when this body of air strikes the small plates D and E, it will strike the former with a speed N -\-v normal to the plate, whilst it will only strike the plate E with a speed \ — v. If we take BC as 1 1 feet (in order to agree with Mr Dines' experiment, described in the Quart. Journ. Roy. Met. Soc, 20, 183), the gauge at D might easily register a speed measured by 1*5, whilst the gauge E only registered one measured by 075. Neither measure uwuld, however, be the real speed of the wind V. B ---f--- m mm — !- ^A .-' E \ .. We can easily understand that "from anemograph records ... it has been found that two similar instru- ments within 8 feet of one another may show a difference in velocity, for a second or two, of 100 per cent."^ Yes, but the reader may say, Why do I assume that the small plate will ahvays be opposite B ? It is not necessary to assume it. If we suppose the small plate to be opposite A, the wind would sweep circularly past the front of it, and so cause a great ificrease in the resist- ance over that which would be produced by a steady wind moving with the speed V ; the velocity being an accelerated one. I offer the suggestion for what it is worth ; but anyone 1 Colonel H. E. Rawson, C.B., il/^A'^n^/^ <,'■)' ("Aeronautics"). 8 114 Resistance of Air who has seen a tornado in the Tropics, and has seen a path swept through a forest of trees of, say, i8 inches to 24 inches in diameter — as by a gigantic reaping machine in a field of corn — will certainly be inclined to believe that such a thing could not have occurred unless tJiere had been a powerful cyclic motion ifi the air. I am certainly inclined to believe that some of the speeds of wind which have been recorded are excessive. It must be distinctly understood that by "speed of the wind" I mean the "speed V" shown in fig. 7 : the actual speed of translation of the body of air represented as A. I do not wish to question the actual speed (V + z') measured, normal to the small plate, as being inaccurate : I only question the deductions drawn from the observations. But to return to the large plates. Why should the pressure registered on them be only, approximately, about two-thirds of that on plates of the order of i square foot ? I think the reason is that with large plates there is a kind of "cushioning" effect (if I may so call it) which prevents, ordinarily y any impact with shock. The pressure on a very large plate would therefore be that produced by the real speed of the air (V in fig. 7) acting without shock : the value of K (in my notation) would then be 0'62'/ ; and the coefficient, in the ordinary notation, /or miles per hour, would be 0003 I. This would be in accord with Dr Shaw's observation, in the discussion on Dr Stanton's paper (here referred to),i that "experiments had been made with anemometers at Holyhead, where the exposuj'e was exceptionally good, and, as far as could be seen, the results were consistent with the supposition that the pressure could be derived from, the velocity by the use of a factor which did not differ much from 0"003" [italics added]. When, however, the wind is acting on plates of the order of 1 2 inches square, there is usually an increased pressure on the plate caused by the shock of impact, the excess appearing to vary up to 50 per cent. This is, of course, " Dubuat's paradox," applied to 1 And also with Dr Stanton's experiments with large plates. Resistance to Wind of Bodies at Rest 1 1 5 the air ; a subject I have dealt with at considerable length in Motion of Liquids. What I have called a "cushioning effect" is well recognised by aeronauts and meteorologists. In Colonel Rawson's Meteorology , when referring to air being driven against the scarped face of a cliff, we find him saying : " Such a mass of air will be compressed against the face of the cliff and act as an elastic buffer, throwing up over the cliff or round its edges the succeeding masses which are driven on it.'' Also, "On the windward side of a com- paratively small mass of rock, not exceeding 100 feet long by 50 feet high, complete shelter will be found in a stiff gale and not a breatJi of wind will be felt " (Meteorology) [italics added]. This is, of course, a matter of common observation. As I explained in Motion of Liquids, the pressure on a small plate is an oscillating one, varying from h, the " pressure head " due to the velocity, and 2h, the pressure due to impact : the mean, measured, pressure being i -5 h. From perusal of books on hydrodynamics or aerodynamics, it might be supposed that the curve of resistance is a nice smooth curve. Nothing could be more incorrect : the curve is an oscillating one. Experimenters employ all kinds of devices — dash-pots, etc. — to damp this curve, and so "smooth it out." It is only necessary to examine any meteorological record to see the very violent oscillations in the pressure. Captain G. Costanzi in his Alcune Esperienze di Idrodinaniica, gives an undamped curve, which I have reproduced in fig. 8. The upper line is a measure of the space travelled (2 '50 every interval). The line below it gives a measure of the time (in fifths of seconds). The straight lines indicate, as shown, the pressure corresponding to i 500, 2 500, 3 500, 4 500, 5 500 grms. The diagram must be read from the right, where the resistance is somewhere about 2 500 grms., with a tendency to increase. When the water was disturbed, the resistance instantly increased, but rapidly diminished again, and appeared to fall below the original pressure. ii6 Resistance of Air On giving a very sudden disturbance to the water the re- sistance again increased to about 5000 grms. , where, from the extinction of the oscillations of the balance, it appeared as if a stable regime had been attained. This fascinating question is, however, outside the boundaries 1 have laid down for this book. Fig. 8. — Costanzi's undamped curve of resistance. REFERENCES L. A. Thibault, Recherches Expiritnentaies sur la Rhistance de PAir, 1826. Smeaton, Experifnental Researches on the Resistance of Water and Air, 1759. Colonel Beaufoy, " On the Resistance of Air " (Thomson's Annals of Philosophy, 6, 1 8 1 5 ). Professor W. C. Kernot, " Experiments on Wind Pressure " {Australian Association for the Advancement of Science, 5 and 6, 1895)- Dr T. E. Stanton, " Experiments on Wind Pressure " {Inst, of Civ. Eng., 1908). W. H. Dines, Quart. Journ. Roy. Met. Soc, 16, 1890. Sir Benjamin Baker, The Forth Bridge, 1884. W. H. Dines, Quart. Jotirn. Roy. Met. Soc, 20, 1894. Colonel H. E. Rawson, Meteorology (the airman's vade mecum : " Aeronautics"). Captain G. Costanzi, Alcune Esperienze di J drodin arnica, 1913. CHAPTER XIV "BALLISTICS" CONTINUED — ZAHM, PIOBERT, DIDION, H^LIE, MORIN, PERRY Let us now return to the resistance of shot. I attach the greatest importance to these results, because (i) the experi- ments were very well made, (2) the results were obtained by the ve7y simplest means possible, and (3) the coefficients obtained from these experiments, at very high speeds, are much more to be relied upon than any obtained from experiments at comparatively very low speeds : errors, when multiplied by millions, become very perceptible. Let us therefore turn once more to Hutton's figures of the resistances of 2-inch shot (given in ounces) in Table XXVI II. The best authorities agree that at speeds of not less than i 400 feet per second the resistance varies exactly as the square of the velocity. The curve of the resistance is, in fact, a parabola ; and we can, by dividing the resistance by the square of the speed, get the parameter of this parabola. When I use the word ' ' parameter " here , what I really mean is what would usually be called the inverse of the parameter, the coefficient by which we multiply V^ to get the resistance. I shall (in this sense) employ the word parameter for brevity. [Table 117 ii8 Resistance of Air Table XXVIII. Resistances. Velocity. Differences. Observed. By theory. oz. oz. lOO 2-78 2-24 + 0-54 200 1 1 '34 9-6 + 1-7 500 74-4 71-8 + 2-6 600 1 10 4 108-6 + 1-8 700 1560 155-0 + i-o 800 2I2-0 211-87 + o-i 900 280-3 281-7 - 1-4 I 000 362-1 362-4 - o"3 I 100 456-9 456-3 + 0-6 I 200 564-4 577-2 -12-8 I 300 683-3 690-3 - 7-0 I 400 811-5 821-3 - 9-8] I 500 94yi 942-9 + 4-2 I 600 1 086-9 I 072-8 + 14-1 + 46-4 I 700 1 228-4 I 2iri + 17-3 .-45'4 I 800 1 368-6 I 357-8 + 10-8 10 I 900 1 505-7 I 512-8 - 7-1 2 000 1 637-8 I 6763 -28-5 ) Shot 2-inch diameter. Resistance in ounces. Hutton's parameter varies from 409 to 425, or about 4 per cent., i.e. ±2 per cent, probable error. Dividing the resistances at velocities i 400 to 2 000 feet per second, by (1400)^, (1500)2, . . . (2000)2, we get the parameters here given. The average of these we see is -000 4 19 074. If we next recalculate the resistances with this * ' mean " parameter, we shall get the corrected values in Table XXVI 1 1. We see that if we add up the differences between these and Hutton's figures, they are almost exactly zero. This rectifica- tion of the curve may be considered to be very good. •00041403 -00042093 •000424 57 -000425 05 •000 422 40 •000 4 1 7 09 •00040945 7)13352 •000 4 1 9 074 Now, since in this part of the curve the value of a = 2^', ''Ballistics'' C07iti7tued 119 it is clear that in the other part, where ^ — \ i+ — jo', we must halve the value of the parameter '000419 and increase the resistance found by |i + - |. Putting these values into the equation, we get what I may call a correction of Hutton's Table of Resistances (Table XXVIIl). It will be seen that the differences are very small, except for the value at 2 000 feet : there are independent reasons for believing that Hutton's figure for this speed is too low — compared with the others, of course. The temperatures are not recorded, so I have assumed them as normal, i.e. T= 18° C. : consequently V= i 366 feet. Let us now see what the parameter should be from theory only. Since ^2 I V 0)0 V\ I + — 125 V V 32 the parameter will, clearly, be - — wo'. We have for a 2-inch shot , 3T416 . . w = irr- = - — - — square feet 144 ^,^ D _ 997 850 (32-18)850 since i cubic foot of water weighs 997 oz. avoirdupois. It will be seen that the calculated parameter is O'ooo 203 57, as against Hutton's O'OOO 209 5. This is rather less than 3 per cent, below the value derived from Hutton's table : certainly not a very bad agreement, but there are reasons for believing that all Hutton's resistances are a little too great. Colonel Didion {Lois de la Resistance de r Air) draws attention to this, and attributes it to some torsion in the pendulum, caused by the shot not striking it centrally. This torsion caused the pendulum to register too low a I20 Resistance of Air velocity ; consequently a too great loss of velocity in the distance passed over, and therefore a too great resistance. There are a set of experiments, made later, by the Rev. Francis Bashforth which are considered more accurate, and by which we can check those of Hutton. If we refer to his ///j-/^;7r(^/ ^X^^/r/^ . . . of the Resistance of Air to the Motion of Projectiles {Cdimh. Univ. Press, 1903), we shall find his measures of the resistance of 2-inch shot, which we may compare with Hutton's values. These values will be found in Table XXIX, together with the values calculated according to the theory which has been here advanced. The resistances given by Bashforth are in pounds avoirdupois, and are only given for speeds of 900 feet per second and over. Hutton's resistances are there- fore divided by 16 to reduce them, also, to pounds for this comparison. Table XXIX. Velocity. Hutton, 1791. Bashforth, 1870. Theory. ft. sec. lbs. lbs. lbs. 900 I 000 I 100 17-5 22"6 28-6 12-5 17-5 25-0 17-0 22'0 27-6 I 200 I 300 I 400 35-3 427 507 32-9 40-3 48-2 35-0 41-9 49-86 Difif. + 1-6 I 500 I 600 I 700 59-2 67-9 76-8 56-2 64-9 737 57-24 65-13 73-5 + I-0 + 0-2 -0-2 I 800 I 900 2000 85-5 94-1 102-4 82-0 92-6 103-3 82-4 91-8 101-76 + 0-4 -0-8 -1-6 + 0-8 Shot 2-inch diameter. Resistance in lbs. Let us now take out the parameters of Bashforth's curve, exactly as we did Hutton's, for all speeds of i 400 feet " Ballistics " continued 121 per second and over. We shall get for lbs. ...... 0-000024592 We are justified in doing this, since Bash- 24 978 forth states (p. 28): " As I have found that 25 351 the resistance of air varies as (velocity)- -5 50i for velocities i 300 to 2780 feet per second, 25 309 I shall assume that this, the Newtonian 25651 law [which it certainly is not], holds good 25 825 for all higher velocities" (date 1903). In n\,^ 207 1870 he had said that, between the velo- " ^^ j. cities of I 100 and i 500 feet per second, the resistance varied as the cube of the velocity. We see that Bashforth's " mean parameter" is o '00002 5 315, which is not very different from the one found by the theory advanced here— o '000025 44. In fact, if we had omitted the first number — which is the only one sensibly less than 0*000025, and is therefore probably the worst — the agreement is absolutely perfect, viz. 0*000025439 to 0*000025 44. In Table XXIX will be found Hutton's measures of the resistance from 900 to 2 000 feet per second ; Bash- forth's measures of the same thing : and in the last column the values calculated by the theory advanced. The reader will, I trust, consider the agreement very satisfactory. From I 500 to 2 000 the calculated figures agree almost exactly with Bashforth's : at the lower speeds the agreement is better with Hutton's figures. Why this should be, 1 am unable to offer any suggestion. It may not be uninteresting to compare the probable accuracy of Hutton's and Bashforth's curves of resistance — comparison to be made as to regularity of the curves only. If we assume, with Bashforth, that the resistance at velo- cities not less than i 400 feet per second varies as the square of the velocity — this being, as I have stated previously, the one point on which all the best authorities are agreed — we find that Hutton's values of the parameter vary from 409 to 425. This is rather less than 4 per cent. : so we may say that the probable error is less than ±2 per cent. Bash- 122 Resistance of Air forth's values, for the same, vary from 246 to 258 : this is about 4 "8 per cent., or a probable error of ±2*4 per cent. If, however, we omit the value at i 400 feet per second — which we saw was probably an inferior one — then the variation will be from 250 to 258, or 3 "2 per cent., the probable error being zb i '6 per cent. Bashforth's writings show him as always trying to prove that the resistance of shot (certainly at velocities below I 400 feet per second) increases as the cube of the velocity. This is approximately true for such speeds as 1 000 to i 300 feet per second. Bashforth was not, however, the first to point this out — as he, apparently, thinks he was. This was definitely stated as early as 1857. It could also be logically inferred from Duchemin's book, which was published in 1842. It is clear that if we consider the resistance expressed as av^-\-dv^, since the first term increases as the square of the velocity, whilst the other increases as the cube of the velocity, at speeds of, say, i 200 feet per second, the value of the first term will be unimportant, compared with that of the second : and the resistance may then be said to vary nearly as the cube of the velocity. The reverend gentleman is, however, very angry at anyone suggesting that he was not the original discoverer of this fact. He rather loses his temper at times, and airs his grievances more than is necessary in the scientific discussion of a very interesting question. He is specially angry with Professor Helie for pointing out that M. le Capitaine Wetter had stated, in 1857, that '■'■la resistance de fair sur les projectiles sph^riques est s implement proportionnelle au cube de la Vitesse " (Professor Helie, Ballistique Experimentale, 1884). According to Bashforth, the resistance at speeds between 900 and I 100 feet per second varies as some "fancy power " of the velocity. In 1871 he said it varied as the sixth power of the velocity. In 1890 he said that, from i 000 to I 100 feet per second, it varied as the fourth power : the resistance between 900 and i 000 feet per second being ''Ballistics'' continued 123 reduced from the sixth to the third power ! How it is possible to make a ' ' dimensional formula " with the resist- ance varying as the fourth and sixth powers of the velocity, 1 quite fail to see. Mr Bashforth admits that his coefficients are "variable." He also says that * ' when the resistance of the air is expressed by the help of a variable coefficient of some power of the velocity, it is a mere question of convenience what power shall be used" [italics added]. This may be *' convenient," ibut it can hardly be said to be scientific. It would be simpler to say, at once, that the resistance ' ' will be found in my tables. " Why speak of a theory of the motion of projectiles, when he has no theory of any kind to offer ? Bashforth, however, says, most emphatically, in his latest book, that "we now know that the resistance of the air does not vary, even approximately, within practical limits, according to any single power of tJie velocity.'' He also says, in another place, * ' the resistance of the air does not vary strictly as the cube of the velocity. " I have not specially drawn attention to it before, but the reader will have, I hope, observed that the formula I am advocating does not violate dynamical similarity. v\ . . V . \-\-~\ has "no dimensions," since — is the ratio between two speeds : it is a number, and consequently has no dimensions. We see from Hutton's experiments, generally, and still more accurately from Bashforth's, that the coefficient we have taken for the "parameter" of the parabola, viz. O'OOi 428, when a square plate is moving in air at rest, is suffi- ciently accurate to allow of its being multiplied by four millions and still to give results which will agree with the (to my mind) most perfect experiments on the resistance of air which have ever been made. Some fifteen years ago some experiments wxre carried out in America: let us see how these agree with our theory. (A, F. Zahm, Ph.D., " Resistance of Air at Speeds below 124 Resistance of Air One Thousand Feet," Phil. Mag., 1901. Same title: Catholic University of America, 1900.) Frankly, I do not understand these two papers on the resistance of wooden bullets. In the first named we have the resistance of 4-inch bullets (3|^f to be exact) given as R = o -ooo oo8z^2 + o 'ooo 000 049 2v'^ A curve is given, and a table of the experimentally observed velocities and resistances. In the second paper we have the resistance expressed as R = o -ooo 8z^2 _|_ o -ooo 000 049 2v^ We have the same graph given; and beneath, "The velocity resistance curve, whose equation is R = o -ooo 8z^2 j^ o -ooo 000 049 2v'^ " Since the graph does not agree with this equation, I can only suppose the discrepancy between the two papers is caused by faulty editing, and that what is really intended is R = o -ooo ooSz^^ + o 'OOO 000 049 2V^ I shall, therefore, assume that this is a clerical error, and that the graph is correct. Now, as regards the units, we find {Phil. Mag.) that the mass of the ball is in grammes ; temperature, centigrade ; barometer, millimetres; h^, h^, h^, etc., in millimetres; velocity in feet seconds ; and resistance in pounds (presum- ably avoirdupois). Since Professor Zahm makes no corrections for barometer or thermometer — though some of the temperatures exceed 24° "4 ; whilst the average for the 3rd September 1897 was 24° "I, — the units in which these are expressed are of no importance. It is a little confusing, however, to have to jump from grammes to lbs. avoirdupois, and from milli- metres to feet per second. Putting Professor Zahm's formula into the shape pre- viously employed here, we get R = o -ooo Qo2,v\ I + o -006 I t)V) '''Ballistics'' continued 125 We see, at once, that the number 0"Oo6 15 is extremely — suspiciously — large. The coefficient o '000008 is also too small. Let us, however, compare the resistances obtained by this formula with those observed by Hutton. I admit that we are not, perhaps, justified in doing this for velocities of over i 000 feet per second ; but, as we shall see, these results really agree better than those at lower velocities. In Hutton's Table of Resistances, we have them given in ounces, whilst Zahm's results are given in pounds. To compare them we must consequently divide Hutton's figures by 16. On the other hand, Hutton's shot had a diameter of 2 inches, whilst Zahm's had one of 4 inches. We must, therefore, multiply Hutton's figures by 4. On the balance, therefore, we must divide Hutton's figures by 4, in order to get a correct comparison. I have calculated the values of R by Zahm's formula from 2;= 300 to ^^ = 2000, and the results are to be found in Table XXX. It will be seen that these figures do not agree, except at very high velocities. At somewhere about I 950 feet per second the results are the same : after which Zahm's figures would greatly exceed Hutton's. Now, admitting that Hutton's figures are rather too large, the error is only of the order of 2 or 3 per cent. ; nothing sufficient to bring them into line with Zahm's. I regret being unable to compare the latter's figures with Bashforth's ; but the latter only commence at 900 feet per second, where the resistance is given as 50 lbs. Let us next examine what I have called the "para- meter " of Zahm's curve. To compare it with Bashforth's we must divide it by 4 — the shot having twice the diameter. Dividing, therefore, 0"000 008 by 4, we get o '000002, compared with Bashforth's o 'ooo 01 2 65 or o 'ooo 0127. We see that it is not of the right " order" at all. Professor Zahm, I regret to say, like other authors, repeats that " Newton himself taught that the resistance is a quadratic function of the velocity " [italics added]. As I 126 Resistance of Air have previously pointed out, this is not in accord with what we find in the Principia. Table XXX. Velocity. Resistance of 4-inch shot in lbs. avoirdupois. ft. sec. Hutton. Zahm. 300 400 500 600 6-45 11-62 18-6 27-62 1-33 4-41 8-15 i3"5o 700 39-00 20-79 800 53-00 30'3i 9CXD I 000 I 100 I 200 70-07 90-5 1 14-2 141-1 42-34 57-20 75-16 96-63 I 300 I 400 170-8 202-8 I2r6i 150-68 I 500 I 600 I 700 236-8 271-7 307-1 184-05 222-00 264-84 I 800 1 900 2 000 342-1 376-4 409-4 312-85 366-43 425-60 Hutton's temperatures probably normal. Zahm's temperatures probably about 24° C. There are some other authors whose writings should not be overlooked. General Morin {Lecons de Mecanique pratique, i860), who collaborated with the generals Piobert and Didion in the experiments carried out at Metz, says : — "General Piobert proposed, to represent the law of the resistance of air to the motion of projectiles, the formula R = o-023AV2{i +0-002 3V}." Colonel (afterwards General) Didion, in his Lois de la Resistance de I' Air, 1857, gives certain formulae for calculat- ing the resistance of shot. His final one appears to have been R = o -027 V2( I + o -002 3V) or and more fully Ballistics'' continued 127 R = 0-027V2( l + — 435 R = 7rR2V2oo27( i + — 435 Some of his argument is vitiated by his assumption that the resistance of a sphere is half that of the circumscribing cylinder. He says, very clearly, that " the value of the co- efficient, by which the square of the velocity and the great circle of the projectile must be multiplied to give the resist- ance, increases with the velocity, at least within the limits of the experiments." He also points out the error of comparing whirling- machine experiments with those carried out when measuring the resistance of shot. Professor Helie {Ballistique Experimentale) gives a variety of formulae for expressing the resistance of the air to shot. His general formula is R = (A /^) a^fiv^v^, where A = density ; ^= gravity constant ; <2:^ = area of cross section of the shot ; and f{v) is some function of v, the velocity. This he expands into R = 0"i72 2(1 -|-0'002 3z^)Z. -v^ TT g and eventually R = o •027( I -|- o '002 3?y)t;2 It is hardly necessary to point out that this formula — as well as those of Zahm, Piobert, and Didion — violates dynamical similarity. The terms on both sides of the equation have not the same dimensions (making due allow- ance, of course, for the values of L^ and p, which are omitted). Professor Helie says that his f{v) "-becomes sensibly constant" for all speeds exceeding 400 or 500 metres per second.^ 1 In other words, that the resistance varies, sensibly, as the square of the velocity. 128 Resistance of Air He refers in his book to a very curious experiment which is well worth calling attention to. He was experimenting with a hemispherical headed shot, when a Colonel Treuille de Beaulieu (at Gavre, in i860) particularly asked him to try the effect, on the resistance of the shot, of putting a small open-ended tin cone on its nose. The shot presented the appearance in fig. 9. A short time ago I casually asked a friend what he thought the effect of such an addition would be. He replied that obviously there would be an increase in the resistance. Such was not the case, however. Professor Helie tells us that the initial velocity was about the same: 409*2 metres per second without the cone, and 409*5 metres per second with the cone. At a distance of 400 metres, however, the velocity of the shot without the cone was 379 'O metres per second ; whilst the velocity of the shot with the cone was 382 t metres per second. The experiment was repeated several times with the same result. Sir William White, in his Manual of Naval Architecture, says that Froude found the resistance of a plate of i square foot, for a speed of i foot per second, was I 7 R = — - — = o 'oo 1 7 lbs. 1 0000 . •. at 15 knots R= i "09 lbs. [assuming, of course, that the resistance a V-]. This is considerably less than the resistance of the wind found by Colonel Beaufoy, which I have previously referred to. Sir W. White does not say how the experiment was carried out, whether the plate tvas moving, or the wind was flowing past it. If the experiment was made on a whirling machine, with a not very long arm, this result is what one might expect. According to these formulae, the resistance at very high Fig. 9. ^^ Ballistics ^^ continued 129 velocities (over i 500 feet per second) would increase as the cube of the velocity, whereas experiment shows it actually increases as the square of the speed only. A very curious thing, which I cannot understand, as re- gards all the work of Generals Morin, Didion, and Piobert, as well as that of Professor Helie, is that all these generals belonged to the artillery. Professor Helie was also closely connected with the artillery : they all worked not very many years after Duchemin : they all probably had, or had easy access to, the Mem,orial d'Artillerie, where Duchemin's work was published : Duchemin was also a colonel of the artillery. Notwithstanding all this, not one of them ever mentions, or refers to, Duchemin's Recherches Expcrinicntales sur les lois de la Resistance des Fluides. It seems extra- ordinary that this book should have attracted so little attention. In the A B C of Hydrodynamics I stated, on the authority of Professor Perry {Applied Mechanics), that, at speeds greater than i 100 feet per second, the resistance of a shot increased as the velocity only. As this appeared to be incorrect, I wrote to Professor Perry asking him on what he based this statement. He kindly told me that he had thought a good deal about it, before writing this, but that he could not now remember on what he had based his views. ^ Sir George Greenhill (^Motion of a Projectile in a Resisting Medium, 1882) based his work, which is strictly mathe- matical, on the assumption that air resistance, certainly at ' ' ballistic velocities, " varies as the cube of the velocity. He relies on Bashforth's work, but refers to his Final Report on Experiments withtJie Bashfo)th Chronograph, which is dated 1880. I prefer to refer to his latest work of 1903. REFERENCES General DmiON, Lois de la Resistance de I'Air, 1857. ^e\.].^hSYivoYCs:H, Historical Sketch . . . of the Resistance of Air to the Motion of Projectiles (Camb. University Press, 1903). 1 Possibly it was on Mr Mallock's work, referred to in the next chapter. 9 130 Resistance of Air Professor Hklie, Ballistique Experimentale, 1884. A. F. Zahm, " Resistance of Air at Speeds below One Thousand Feet" {Fhii. Mag., 1901). Same title (Catholic Univ. Press of America, 1900). General Morin, Lecons de Mecatiique Pratiqjie, i860. Sir William White, Manual of Naval Architecture. Professor J. Perry, Applied Mechanics. Sir G. Greenhill, Motion of a Projectile in a Resisting Medium, 1882. CHAPTER XV RESISTANCE OF BULLETS AT VERY HIGH VELOCITIES — WORK OF MR A. MALLOCK I HAVE said that all the best authors were unanimously agreed that the resistance of shot at speeds exceeding about I 400 feet per second varied strictly as the square of the velocity. I must qualify this statement, since there is one exception. Mr A. Mallock, in the Proceedings of the Royal Society, 1904 and 1907, indicates views which are not apparently in accord with this. It would even appear as if this writer considered that the resistance experienced by bullets, at very high velocities, vaj-ied as the speed only. Since this work is very modern, and as Mr Mallock is an author whose writings will always carry great weight when they deal with mechanics, I propose devoting a special chapter to these papers. In the first place, it is necessary to see what Mr Mallock's assumptions are ; next, to examine the details of his experiments ; and, lastly, to judge of the deductions he draws from these experiments. The paper " On the Resistance of Air," 1907, com- mences: — "The great mass of work, both theoretical and experimental, which has been done on this subject may be divided into two classes, one of which comprises all the results having reference to such moderate velocities as can be obtained with artificially produced currents {i.e. up to about 100 feet per second), while the other is confined to velocities about and exceeding the velocity of sound. Between these two there is a gap in the experimental record extending over a range of i 000 feet per second." 131 132 Resistance of Air I am afraid that Mr Mallock has overlooked the work done by Charles Hutton, to which I have drawn attention previously. His Tracts are a mine of very valuable in- formation on this subject. Mr Mallock continues: — " I am not aware that any attempt has been made to connect theoretically the experimental results at high and low velocities, and it is the object of this paper to find an expression which will represent the resistance generally. "The expression which is arrived at does not pretend to be anything but an arbitrary foniiula, but it is, I think, useful as separating the resistance into its component parts'' [italics added]. With reference to the above, since the "arbitrary formula " does not appear to conform with dynamical similarity (certainly as defined by me), I am afraid that no scientific importance can be attached to it. The reader will, however, be better able, later, to judge for himself as to the correctness of this statement of mine. Dealing with the assumptions made by Mr Mallock, we read : — " At low velocities it is known that [i] the resistance varies as the square of the velocity ([2] as it would at all velocities in an incompressible fluid), but two distinct cases may be noticed depending on the shape of the resisting body. If it has what, in a ship, would be called a fine run, that is, if the 'after' part of the body tapers very gradually, so that the stream-lines follow its contours [3], any resistance experienced by it is due to surface friction only ..." [italics added]. With regard to (i), it is, I think, absolutely certain that the resistance does not vary accurately as the square of the velocity — except at the speed of about no feet per second. I need only refer to the examples previously given, as confirmation of this very emphatic statement of mine. (2) Neither is it correct to say that the resistance would vary as the square of the velocity at all speeds in an incom- pressible fluid. This would only be true, as Newton pointed out, if the pressure also increased as the square of the velocity. Resistance of Bullets at Very High Velocities 133 (3) This paragraph is certainly not clear, in consequence of the vagueness of the term "surface friction." There has been a great deal of confused writing on the subject of ' ' stream-line flow " and the reverse ; the truth being that in an incompressible fluid without a free surface — this condition is imperative — all motion of the fluid is stream-line. If, however, the fluid has a free surface, then no motion of the fluid is stream-line. When I speak of a fluid "having a free surface," I mean, of course, a free surface within a reasonable distance of the body moving in it. A body moving in the sea at a depth of 2 000 fathoms, say, would almost certainly cause "stream-line motion" of the fluid, although the water really has a free surface. It does not appear to matter whether the fluid is viscous or not, for Dr Hele-Shaw has even caused glycerine to flow in stream-lines round bodies of the most extraordinary shapes. I have reproduced some of his photographs, by his kind permission, in Motion of Liquids. The mathe- maticians have shown that in a non-viscous fluid all motion of fluids is stream-line. They have, however, assumed that their fluid had no free surface. Lord Kelvin, on the other hand, has shown how vortices would be produced in a "perfect" liquid, if such a thing existed. In his assump- tions, however, the liquid lias a free surface. He further promised to explain to the Royal Society how vortices, in a perfect fluid, would be dissipated ; though I regret to say that I have not been able to find out that he actuall}' re- deemed this promise. I have gone into this question at some length in the A B C of Hydrodynamics. When I use the word "vortices" here, I refer, of course, as did Lord Kelvin, to coreless vortices. The mathe- matician's vortex is imaginable to a mathematician : it is as unimaginable to a physicist as any other form of infinity. I also adopt Lord Kelvin's definition of a vortex : "I now define a vortex as a portion of fluid having an}' motion that it could not acquire by fluid pressure transDiitted through itself froj)i its boujidajy " [italics added]. It will be evident that 134 Resistance of Air Lord Kelvin's definition will include all forms of sinuous motion. In this sense all eddies are vortices. But to resume. After this follows a short dissertation on the resistance of a plane moving in the direction of its normal, and the solution of the problem (in two dimensions) by what is commonly known as the Kirchhoff-Rayleigh method. This so-called solution is a beautiful piece of mathematics, based on certain assumptions which are, I think, pJiysically impossible. It would appear to be in direct opposition to Newton's statement that the pressure must also increase as the square of the velocity, for the resistance to increase in that same ratio. The eminence of the authors has caused this theory to have assumed an import- ance which I think it hardly deserves. In the paper "Air Resistance, etc., 1904," we read: — "To determine the resistance of an ogival-headed shot from the resistance experienced by a flat-headed shot moving with the same velocity, the resistance of the latter must be multiplied by a coefficient, which is generally taken as \. ' ' If the circular edges of the flat-head are slightly rounded this value is nearly correct, but I have found by experiment that when the edges of the flat are quite sharp (as they were with the aluminium cylinders used) the coefficient is rather less than half . . . Probably it is not a constant for all velocities, but approximates to a constant as the velocity increases." This method of obtaining the coefficient of resistance for an ogival-headed shot appears a little arbitrary, more especially as no indication is given of the details of the ogive. The value of K (as employed by me) is in reality a coefficient of sJiape, which requires determination as much as any other of the data. Why Mr Mallock thinks a coefficient of shape should be a function of the velocity is not very clear. This value of K has all the appearance of being a constant : nor is there any apparent reason why it should not be so. In Mr Mallock's 1904 paper he says: — "The coefficient Resistance of Bullets at Very High Velocities 135 was taken as ~—^, and the curve [apparently the curve of the resistance of the flat-headed shot corrected by dividing the ordinates by 2t] represents, at any rate very approxi- mately, the resistance experienced by the ogival-headed shot." The foregoing is useful as showing the foundations on which Mr Mallock builds : I might almost say that it gives us an idea of how his mind is working ; a thing always very desirable to know when one is examining any man's work. Fig 10. — Growth of eddies behind plate moving in still water. {By courtesy of A. Mallock and the R.S.) Now, as regards experiment, he says: — "It must be noticed that the eddies in the wake [of the non-stream-line body] are not of the nature of vortices in which each fila- ment has the same velocity potential, but consist of a kind of sandwich-like structure, about half the fluid belonging to each eddy being derived from the interior of the wake and half from the general stream outside. In two dimen- sions the formation of the eddies is a discontinuous process. In fig. 2, a, b, and c [see fig. 10] illustrate their gradual growth ; in a the eddy is just beginning to form at the edge of the plane, in c the growth has proceeded until it 136 Resistance of Air impedes the forward flow of the wake which is necessary to feed it. When this stage is reached the full-grown eddy- breaks away and joins the procession of eddies forming the margin of the wake." Mr Mallock's fig. 2 is reproduced, by kind permission, in fig. 10. It gives a very excellent idea of the formation of the eddies when the plane is moving in fluid at rest. Such eddies can be produced very simply, as was shown by Lord Rayleigh at the Royal Institution on a Friday even- ing not long ago, by drawing a semicircular disc through water normally to its plane. The surface of the liquid having had sulphur dusted on it, showed the eddies very beautifully. Mr Mallock says that these eddies "require a continuous expenditure of work." In a sense one must agree with this ; but in another, not. No eddy could, of course, be formed witJiout the expenditure of work. Where we may disagree, however, is that this expenditure of work has taken place whe^i the fluid leaves the surface of the moving body. Kinetic energy has been generated in the fluid, and it is this kinetic energy which has to be paid for. Whether eddies are formed subsequently or not (as is imagined in the Helmholtz-Kirchhofif flow) would not affect the amount of work done in overcoming the resistance. This kinetic energy may be treated as a "waste product." If we imagine the body, for example, to have a hemi- spherical head, the kinetic energy is all generated when the fluid passes the great circle of this cross-section. If, now, a tail-piece be added to the body, some of this kinetic energy (which I have called a waste product) can be recuperated, with the result that the nett resistance is reduced. This may be called utilisation of a " waste product." Mr Mallock has found (as I quote later) that the negative pressure behind the edge of a plate in a current of air may be measured by — 2-5 //, where h is the velocity head of the current. This is equivalent to stating that the velocity of the filaments at the edge of the plate may be measured by I "6 z* ; or, that their velocity is 60 per cent, greater than that of the fluid at a distance. This is exactly what Resistance of Bullets at Very High Velocities 137 Duchemin observed in water, as I have pointed out in Motion of Liquids. Referring again to Mr Mallock's paper: — "In three dimensions the eddies may be produced continuously, being in different stages of growth at different parts of the perimeter of the body behind which they are formed. In this case theyeither appear in the wake as a spiral (see fig. 3) . . ." In fig. 3, which is here reproduced, by courtesy, as fig. 1 1, the action of the fluid flowing past a circular plate is very beautifully illustrated. It shows that the flow of the fluid is spiral, and not as it is ordinarily depicted in text- books. D. Riabouchinsky has observed this spiral motion in air, and has a very interesting paper bear- ing on this in the Bulletin de rinstitut Aerodynaniique de Koutchino, fascicule iv. Mr Mallock's experiments on the resistance at veloci- ties up to 4 500 feet per second were made with small aluminium flat-headed shot shaped as in fig. 12 (reproduced from two small cannelures were filled with equal parts of black lead and tallow. From the 1904 paper we see that velocities of the bullet were obtained by shooting into a ballistic pendulum from ranges varying from 12 up to I 000 yards." Mr Mallock has also kindly informed me that the light bullets were fired from a long-barrelled rifle, since the necessary charge could not be burnt in a service Fig. II. — Eddies formed by moving water behind a plate at rest. {By courtesy of A. Mallock and the Royal Society. ) his paper). The a composition of " the remaining 138 Resistance of Air rifle. The bullets were all fired into sawdust and subse- quently examined. The edges of the flat face were un- altered by firing, and the resistances were calculated from the actual area of the face of \}!\& fired bullets. The results of the experiments on these flat-headed shot are shown in fig. 13 (also reproduced, by courtesy, from Mr Mallock's paper), where the small crosses give the measured resistances at speeds of i 500, 2000, 2 500, and 3 000 feet per second. Scaling these off" as accurately as I can, I find the resist- ances 30, 5375, 85, and 121 lbs. per square inch res^pQciiYQly. Fig. 12. — Plan of Mr Mallock's aluminium bullet. If we treat these figures (which are all for speeds in excess of i 400 feet per second) exactly as we did Bashforth's and Hutton's results, and divide them by (1500)2, (2000)2, (2500)2, and (3000)2, ^e y^,[\\ get ^he following ' ' parameters " : — I 500 feet per second ^ 2000 2500 3000 o "ooo 0133 0-000013 44 o -ooo o 1 3 60 o -ooo o 1 3 44 4)181 o 000 01345 ^ I have taken this as i 500 : but from Mr Mallock's diagram it would appear to be slightly less than this — say i 495. This would make the " parameter" o"ooo 013 42. Resistance of Bullets at Ve^y High Velocities 1 39 (*) Lbs. per square 'ich. ■> / Re siata nee 1 1 1 I 1 of Flat-headed Projectile 1 1 1 1 I _ / = p/r/-^)-'^ ^/^ere ^ = rJ- ^' ^ / y-i 1 1 z " soty whei •e a= velocity 1 1 / -Th R= X z the ent. resu ICs / e cr assej 3 X shew experlm of 1 r 1 1 V / / / 1 / / / / I / / / / / / / / / x/ / / / / ; Y / / / / / / / / / / y i. X Y / ^ .^ /:> X ^ }^ y i^ ss=^^ ^ ■^ (»)? 1,000 ipoo {v'-a) Fig. 13. ( Copied from Mailock's paper. ) 3,000 FeeD per sec. 140 Resistance of Air and we see that the curve has every appearance of being a parabola ; i.e. that the resistance varies as the square of the velocity. These experiments appear, therefore, to agree with those of the other experimenters on this point. It must be noted, however, that this statement is based on four vieasurevients only: the curve has " every appearance " of being a parabola, but one cannot say, with certainty, that it is one. The 7-ate of the increase of resistance appears to be in accord with the theory here advanced. If we examine the coefficients, however, the actual resistances measured by Mr Mallock appear to be very considerably less than those measured by Hutton, Bash- forth, and others — making due allowance, of course, for the different coefficients of shape. Taking Bashforth's figures for the resistance of a 2-inch spherical shot at speeds of i 500 feet per second and 2 000 feet per second from Table XXIX, we find them to be 56-2 lbs. and 103 "3 lbs. Since these have a cross-section of 3'i 416 square inches, the resistance per square inch of the cross-section will be 17 '9 lbs. and 32-9 lbs. Mr Mallock's measured pressures for these speeds are given as 30 lbs. and 53 75 lbs. The coefficient of shape for Bashforth's shot was 32 / 125=0-256. This, also, is f of that of the circumscribing cylinder. The coefficient of shape of Mr Mallock's "cylinder" would therefore appear to be | of that of Bashforth's spherical shot, instead of #, which one would expect. PVom Table IV the coefficient of shape would appear to be about 0"647. Mr Mallock's line of argument is as follows: — "The behaviour of the ordinary cup anemometer shows that at moderate speeds the ratio of the head resistance of the convex, to that of the concave, hemisphere is of the order of I, and the coefficient \ seems to apply fairly to the relation between the head resistance of pointed and flat- headed projectiles at high speeds." "Using \ as the coefficient for the resistance of the pointed head, I find that the wave-making coefficient must be multiplied by i, owing to the change of shape, and that Resistance of Bullets at Very High Velocities 141 in a pointed projectile the resistance is fairly well expressed in C.G. S. units by the equation R=¥Poi*' + (i-Ar'5^+i-35x io-7(z;'-^)2 . (i) the corresponding equation for flat heads being R = ^glt;2 + (i _ Af5S4-j_7-5 X io-"(z^'-«)2 . (2) Where ^o = the density of the undisturbed air /'o = the pressure of the undisturbed air — I z;2\v/y- so that I — A is the resistance. z;'=the velocity of sound in the air just in front of the projectile, and is equal to v when v > a,'' [<2: = the velocity of sound.] "These curves are plotted in figs. 8 and 9 [reproduced, by courtesy, in figs. 14 and 13] respectively, and corre- sponding experimental values are marked by dots. "^ It will be observed that in neither of these equations (i) and (2) is there any p^. This appears curious ; but if we examine the equations on the graphs we see that R= . . ./„(i-A)-354 ... . (I) and (2) I presume, therefore, that these are correct, and that the error (?) in the text is due to the printer. Such being the case, then, clearly the sentence "so that I — A is the resistance " (at the end of the last quotation) should read " so that/'(j(i —A) is the resistance." Further, Mr Mallock says, referring to the graph given here in fig. 14: — " One of the most remarkable features of the resistance curve for pointed projectiles is, that for velocities ranging from that of sound up to 3 000 feet per second it is almost a straight line, which, if produced, would cut the axis of abscissae at a point where the velocity is 850 feet per second " [italics added]. This is, of course, equivalent to saying that the resist- ^ " In the diagrams the units employed aie pounds per square inch and feet per second." 142 Resistance of Air Resistance of a Pointed Projectile . -Y = jD^ f / -A)-^5^ where A^^( i-i^ ^,) f; Z = 1-69 Y. lo'^iv'-a) 3pOO FeeC per sec Fio. 14. {By courtesy of K.S.) Resistance of Bullets at Very High Velocities 143 ance varies as tJie velocity. Certainly this appears incon- ceivable : it would require a great deal more evidence than Mr Mallock has produced for such a result to be acceptable. It would be necessary to show how this is reconcilable with dynamical similarity : to show, in fact, how a resist- ance varying as the velocity could have the dimensions of a " Force." Since all the other authors have found that the resistance, at these speeds, varies as the square of the velocity, I am afraid we must consider Mr Mallock's results as certainly very questionable. No readings of barometer or thermometer are given, so we cannot say how these would affect the results obtained, the deductions also are based on nine observed points only. Further, if we are to attach any importance to Mr Mallock's equation — which he admits is an "arbitrary formula " — it is certainly not an equation to a straigJit line. It is more like an equation to a parabola — which has been " touched up." We may examine Mr Mallock's formula from the point of dynamical similarity. Assuming that the equations on the graph are correct, and that R = X + Y + Z, X has the dimensions of a Force : so also has Y, if we assume that — is the ratio between two speeds, and con- a^ sequently has no dimensions — is a number, in fact. Z may be abbreviated to c{v' — ay or c{v'^ — 2av' + a"^). I cannot see that either of these three terms can be equated to MLT~2. If we suppose that ^ is a function of p, the first term would be all right ; but this would not help the others, a (the speed of sound) is clearly a constant, and with no dimensions. These very interesting papers should be studied in the originals. I have referred to them in considerable detail, because it is most important to know if the resistance of a shot at high speeds varies as the square of the velocity — as I believe it does. I do not think that Mr Mallock's curves 144 Resistance of Air are evidence against this point : in fact, the resistances of the flat-headed shot appear to indicate this as being correct. The graph for the pointed shot is, I am afraid, too ragged to be satisfactory evidence on one side or the other. If, however (as quoted previously), the ordinates of the curve in fig. 13, divided by i / 2'i, are the ordinates of a curve which ' ' represents very approximately the resistance experienced by an ogival-headed shot," then the resistance of this shot would also appear to vary as the square of the velocity at speeds above i 400 feet per second. In the figures Nos. 13 and 14 the coefficient (instead of T / 2"i) appears to be about i / 2*7. If the resistance does vary as the square of the speed, it will be possible to draw the exact curve of the trajectory of a shot, since Daniel Bernoulli has shown how this curve can be calculated. As regards the actual values of the resistances at different speeds, there is no indication in Mr Mallock's paper of how these were calculated : it is not possible, therefore, to discuss them. Experiments with the ballistic pendulum are very diffi- cult to carry out satisfactorily, since they are notoriously subject to very considerable errors. REFERENCES A. Mallock, F.R.S., "Air Resistance encountered by Projectiles at Velocities up to 4 500 feet per second," Proceedings of the Royal Society, 74» 1904- " On the Resistance of Air," Proceedings oj the Royal Society, A, 79. '907- CHAPTER XVI EXPERIMENTS WITH WHIRLING MACHINES : LANGLEY, DINES, COLONEL RENARD, COLONEL BEAUFOY, PRO- FESSOR W. C. KERNOT, FINZI AND SOLDATI In this chapter I propose examining the work of different experimenters with the "whirling machine," I shall com- mence with S. P. Langley's Experiments in Aerodynamics ^ since this book is a classic. The whirling machine had an arm which was 30 feet long. Langley considers that, with this length, the "corrections for the effects of circular motion are negligible^ in relation to the degree of accuracy aimed at." He continues later: — "To show that these corrections are negligible in rela- tion to such degree of accuracy as we seek, we may advantageously consider such a numerical example as will present tJie uiaximum error of this sort tliat obtains under the most unfavourable circumstances . "Let this example be the use of a plate of the greatest length hereafter described in the experiments, viz. 30 inches, and let us suppose its center to be at the end of a revolving arm 30 feet in length which was that employed. "The first disturbing effect of circular motion to present itself to the mind of the reader will probably be centrifugal force ; but in regard to this he may observe that in all the pieces of apparatus hereafter to be described, the various parts are so disposed that the centrifugal force proper, viz. the outward thrust of the plane or model which is the subject of experiment, shall not disturb or vitiate the quantitative data which are sought to be observed. " 145 10 146 Resistance of Air Nevertheless, it is well known that air is driven outwards, i.e. from the axis of rotation ; and tJiis has to be paid for. The whirling machine acts as a kind of " centrifugal pump." After this follows a calculation showing that the error, in such a case, would be only 0"2 per cent., or one-fifth of I per cent. By Duchemin's formula I should make the error not less than 10 per cent. ! The experiments were made in the open air, and in as calm weather as possible : but, as Langley says, " calm days almost never came." The apparatus, we are told, "was designed to give approximations to the quantitative pressures, rather than as an instrument of precision." Errors caused by the wind were clearly recognised by Langley, since, on p. 96, he says that "if the velocity at the end of the arm be V and the velocity of the wind be v" the resistance measured is not that caused by V^, but that caused by 'V'^-\-v^, " since 2 There was, further, a bending in part of the machine, so as to give what may be called an " index error" of 5° '5 in the readings of the "setting of the plane." This error Langley (rather arbitrarily, I think) considers constant : so that he deducts 5° '5 from every angle of setting. The apparatus was well designed, so that the centre of the plate experimented with was always in the central line of the moving arm. I have called attention to these points with the special object of showing the amount of importance one can attach to the accuracy of Langley's figures. There is a tendency in this book to take the average of very widely divergent numbers for the correct reading : to average 58 and 98, 35 and ^6 — and even, in one case, O and 12 — appears hardly justifiable. On p. 99 Langley points out that "the resulting values of K,„ for the 6-, 8-, and 12-inch square planes are 7iot entirely accordant [they vary from O'Oio 27 to O'OO/ 70], as Experiments with Whirling Machines 147 the successive sets of observations with the 12-inch plane all give considerably larger values than those obtained with the smaller planes. I am not disposed, however, to consider this as a real effect due to an actual differe?tce in the pressure per unit area on these planes. The difference, if one exists, is in all probability quite small, and much within the degree of accuracy possessed by these experiments. The resulting differences in the mean values of K„, I consider, therefore, as discrepancies in the observations^ the cause of which has not become apparent " [italics added]. If we believed that these discrepancies were all caused by experimental errors, our confidence in the results would be seriously shaken. Such "discrepancies," as might be reasonably put down to a "dimensional effect," were, however, observed more than 100 years previously by de Borda, who said that the dimensional effect was undoubted. As has been explained previously, Langley's results require correction for circular motion ; in the experiments with the 12-inch plane the results are probably 4*4 per cent, too great ; the 6-inch plane experiments, in the same manner, require to have the results reduced by about 2 "2 per cent. : whilst the 8-inch plane results require reducing by about 3 per cent. These corrections will bring the coefficients nearer together : more correction, however, is required than this. Doubtless the wind increased the pressures, as explained by Langley. Further, it is well known that wind blowing across a plane will increase the resistance ; sometimes very largely. One of the chief causes why Langley's figures are so high is that he averaged bad experiments with good ones, giving them what he calls ' ' the same weight. " To justify this state- ment, let us examine some of his figures. On August 27, 1888, he experimented with a 12-inch plane: T = 2i°C. ; 15 = 736 mm. ; mean wind velocit)' 0-52 metre per second. He obtained a mean value of K,„ = ooo8 10.^ ' Gram/Hcs per square centijnctre. M. Eiffel would give thisaso-o8i kilogrammes per square metre. The notation is slightly different. 148 Resistance of Air On August 28 he repeated the experiments: T=i9°'4C. ; V) = 'j2f>6 mm.; mean wind velocity o 37 metre per second. The mean value of K,„ recorded was o'oo/ 94. Same experiments repeated on October 4 : but the results, being "inferior," were rejected. He therefore averaged the values of the other two days, with a result of K,„ = o-oo8. On the face of it, this appears quite fair : let us, how- ever, look at the details more closely. On August 27 there was a Hght wind : the wind was still lighter on the 28th. But, we see a remark at the bottom of the page of the results of the 28th : "During these experiments the slight breeze had almost died away " — one of the calms which "almost never came." It would appear, therefore, to have been fairer to have relied more on the later observations of the 2Zth than on those of the 27th : to have " weighted them," in fact, more heavily. This would have brought the average down a little. If this figure were, say, 0*00785 — slightly less than the inea7i for the second day — the figure would agree very closely with the theory advanced. Since the error for circular motion is about 4*4 per cent, for this sized plate, if we reduce the 000785 by this amount we get K,„ {for rectilinear motion) = 0'00j 5 : and this, in English measures, is = "00143, very nearly; which is not very different from the calculated value O'ooi 428. To make the comparison really accurate, it must be noted that Langley reduced all his observations to 10° C. and 735 mm., whilst my figure was derived from reductions to 18° C. and 760 mm. This would further reduce our imaginary oooi 43 still nearer to O'OOi 428. In the experiments with the "rolling carriage" Langley gives, in his table xviii, other values of K,„. For the 12-inch plane the "weighted mean value" of K,„ = 0"009 4 (varying from O'Oio 27 to O'OOS 59) ; for the 6-inch square plane the " mean weighted value " of K„, = 0"008 33 ; whilst for the 8-inch square plane the value is only o 007 70. He averages the lot, and gets a "general weighted mean" = 0"008 7 : or, in English measures, 0"00i 66. This is the Experiments with Whirling Machines 149 value which is generally quoted as " Langley's." ^ A more common method is to give the coefficient in miles per hour, instead of feet per second. Langley's figure would then become O'ooj 57. A good deal of this book is devoted to combating the * ' sin- law, " which Langley (as is usual) attributes to Newto7i. More than 120 years before this, de Borda showed that the resistance of planes at an angle did not increase as the square of the sine of the '^ angle of attack.''' Further than this, Newton never said it would, for air : though the law would, undoubtedly, be true for Newton's imaginary " medium" — a medium which would somewhat resemble a charge of small shot. Langley's values of K appear, therefore, to be too high. Properly corrected, they would appear to be nearer 0'003 i for miles per hour : which agrees very well with the theory advanced. Of course, as Newton said, we must "add a little to this " as the velocity increases. As stated by himself, Langley's value of K = 0'OOi66 feet per second: or 0'003 56 in miles per hour. The next most important experiments are those of W. H. Dines, "On Wind Pressure upon an Inclined Sur- face " {Proceedings of the Royal Society of London^ vol. xlviii, 1890). The whirling machine had a diameter of 56 feet : i.e. the whirling arm was 28 feet long. By a very ingenious arrangement the pressure on the plate experimented with was balanced by the centrifugal action of a bar at right angles to the central line of the whirling arm of the machine. This will best be explained by the diagram in fig. 15. Let AB represent the arm of the whirling machine. CDP is a frame pivoted about a vertical axis at B. The part of the frame BP carries the plate fixed to it, as shown in diagram. On the part of the frame lettered CD a known weight E can slide, and can be fixed at any position between B and D. The whirling arm moving, as shown by the large arrow, the resistance of the air on the ^ Quoted from p. 99 of his book. I50 Resistance of Air plate, as shown by the small arrow, will be opposed by the centrifugal action of the weight E, as shown by the arrow at G. It is clear that if the weight E is properly adjusted these two actions can be made to balance one another. For example, if the bar weighs 2 lbs. and its centre of mass be placed at a distance x from the axis of rotation, the moment is {2v'^ / gr)x (feet and lbs.). Mr Dines says : — " This arrangement renders any deter- mination of the velocity unnecessary during the experi- ments, since the wind pressure also varies as v-, and there- fore, as soon as ,r is known, the moment due to the wind pressure can be expressed in terms oi v.''- B a E- - P UA N . Fig. 15.- I) -Diagram of Dines' whirling machine. M. Eiffel says about this (La Resistance de l Air, Dunot et Pinat, 1910) : — "The essential condition to enable us to apply the preceding relation, is that the plate P (fig. 15) shall be exactly in tJie prolongation of tJie axis of tJie whij'ling arm. If it is not so situated, the centrifugal force which would be exerted on this plate would create a third couple and the foregoing relation^ would become inexact." There are two other assumptions implied : — I. That the centrifugal force of the weight E is normal to the arm of the bar BD : this is very nearly true, though not exactly so. 1 The assumption is here made that the resistance varies accurately as the square of the velocity: this we know is not correct. ^ - . — ;r=KSV^a; where a = distance from centre of the plate to the vertical axis B. Experiments with Whirling Machines i=;i 2. That the resistance of the air varies as the square of the velocity. We also see : — " The long arm of the whirling machine is not perfectly rigid, and gives way under the torsion produced by the wind pressure. Experiments made by hanging weights from the end of a lever showed that a force of 5 lbs. acting i foot from the axis caused a deflexion of 2°, and since this is about the moment -< <^ ■< ^^ -< ^^ N? 3 Fig. 16. — ^" Setting" of Dines' plates. caused by a velocity of 40 miles an hour, when the plate is exposed normally, the curves have been drawn on the supposition that a moment represented by 100 caused a deflexion of 2°, and that the other moment caused a proportionate deflexion," In this whirling machine the centre of the plate is Jiot in the axis of the whirling arm ; consequently, a centrifugal action causes most of the observations to be tainted with error. Let us examine the case when the plate is "at an angle " ; i.e. when the "angle of attack" is not 90°. The four 152 Resistance of Air typical positions are shown in fig. i6, and are referred to as positions i, 2, 3, and 4. The following extract from Professor Darwin's memor- andum of instructions for these experiments shows how the normal component of the wind pressure may be obtained, as well as indicating the assumptions which Professor Darwin made in drawing it up : — "It may be supposed that the couple due to the wind pressure upon all the moving parts except the plate may be eliminated, so that the couple necessary to hold the plate in position alone remains to be considered. "Consider the first and second positions of the plate or vane : since the wind meets the vanes at the same angles in both cases, the couples would be identical, if the centre of pressure were at the middle of the vane. But it is well known that the centre of pressure is nearer the forward edge, and hence the couples are unequal." An assumption is here made that the centre of pressure on the plate is nearer the forivard edge : this is un- doubtedly true if the vane or plate moves in a straight line. It is a pure assumption that the same would be true when the plate is niovinsr in circular motion. The log-ical deduction from this assumption would be that the couple caused by the resistance of the air to the plate in position i would be much less than the couple in position 2 — always supposing the plate to be rigidly fixed to the whirling arm. Morin, however, states {Lecons de Mecanique Pratique, i860) : — " When the axis of the whirling arm is in front of the plane [as in fig. 17], in regard to the direction of the motion, the resistance diminishes rapidly as the angle of inclination is 6 Fig. 17. Experiments with Whirling Machines 153 increased, and that at an inclination of 55° it is barely 0*571 5 of the perpendicular resistance ; whilst when the axis of the whirling arm is behind the surface of the plate [CD in fig. 17], the resistance increases up to the angle 55", when it is equal to i "229 3 times the perpendicular resistance" [italics added]. This is the reverse of Professor Darwin's assump- tion, and, that I have not misunderstood him, is clear from the sequel : — "If a be the distance from the centre of the plate to the axis of rotation, and if x be the unknown distance of the centre of pressure from the centre of the plate, and if P be the mean pressure estimated over the whole plate, and L^ and Lg the couples corresponding to the two positions, then it is clear that Li = P(«-;r), L2=?(a+x) from which we easily get P=i-(L,+ L,), X '^' ^' a and P^=|(L,-L,) Thus this pair of experiments gives two of the things to be measured." All this follows logically from Professor Darwin's assump- tions. Lj is less than Lg (which experiment shows is the reverse of the truth), and Ya is the arithmetical mean of Lj and \^^. Here we have another assumption implied, though not definitely stated — and that is that x in the position I is equal to x in position 2. Such an assumption is absolutely unjustified ; it might, or might not, be correct, but without experiment one has no right to say that it is so. What, however, do Dines' experiments show ? We see in fig. 18 that the resistance measured in position i is slightly less than that in position 2 (fig. 19). The curves are almost identical from 50° to 90°; but the "hump" at 55° is considerably higher in position 2 than in position i, ' The a is left out in the original : this is evidently a clerical error. Fig. i8. ammHi ^ m • Fig. 19. No^e. — Resistance in fourth position is greater than that in the third. By Darwin's assumption ?V should be less. {By courtesy of Mr W. H. Dines.) Experiments wit/i Whirling Machines 1 5 5 and the slope from 55° to o'^ is also steeper in the former case than in the latter. This ap|)ears to support the assumption. Here we come up against another assump- tion, viz. that "the couple necessary to hold the plate in position alone remains to be determined.''' If we look at fig. 16 we see that the mass of the plate (and that of the counterbalancing weight referred to) create a couple which in position i acts against the pressure of the wind, whilst in position 2 it acts with the pressure of the wind. Now, admitting that the arms are so clamped as to be unable to move, it is clear that this stress will tend to bend the arms, and so to introduce an error. In the case of the first position it is highly probable that the pressure measured was too low, whilst in the second position it was too high. Nor can we assume that the mean between these measurements is a correct reading. If we consider the third and fourth positions, which, we are told, "afford a redundant equation, and this may be expected to give a check on the consistency of the results with themselves," we see that Mr Dines laconically says, "An expectation unfulfilled." There does not appear to be any very close connection between the curves of positions I and 2 and those of positions 3 and 4 : the results appear opposed to each other. I do not think we can attach great importance to the accuracy of a mean between two measurements, both of which we have reason to believe are incorrect. Nor could we even if the measurements were accurate, since the mean of O" 571 5 and i '229 3 is not unity. If we compare Dines' apparatus with Langley's, we see how careful the latter was in keeping the centre of his plate in the line of the bar, whereas Dines' plate is always away from this line. His arrangement is exceedingly ingenious, but I cannot think the results obtained by it are absolutely convincing ; more especially when the reasoning applied to them appears to be unsound. One thing which appears very clearly from Dines' curves is the "hump" in them. Dr Stanton {Inst. Civ. Eng., 156 Resistance of Air I56> 1904), referring to it, says :■ — " It appears from Dines' experiments that when the angle between the direction of the current and the normal of the plate is about 45° ^ the normal pressure on the plate rises to a value consider- ably higher than its value for normal incidence of the current. This remarkable effect does not seem to have been noticed by Langley." It was pointed out at the meeting at which Dr Stanton's paper was read that it had been noticed previously by Wenham and Unwin. It is certainly not a very new discovery, since Morin and Duchemin implicitly refer to it. Many years before this, Thibault mentioned it ; and, even earlier, if we examine Beaufoy's work, the hump can be seen, as was pointed out by A. VV. Johns at the Institution of Naval Architects in 1904. It is not a peculiarity due to whirling-table apparatus, for M. Eiffel refers to it in his La Resistance de I Air et r Aviation, 191 1. Dines' critical angle appears to be about 55°, whilst M. Eiffel's angle appears to be about 52° ; 38°, if you measure it in relation to the surface of the plate and not the nonnaL There are reasons for thinking this angle may be nearer 'x,']" 30' : but, whatever the correct angle may be, I think it will play a very important part in calculations of resistance. Dines says very little about the normal pressure on the plate, but he gives the equation P = 0"002 9^''- for miles per hour. This appears suspiciously low. In his "Some Ex- periments made to investigate the Connection between the Pressure and Velocity of the Wind " {Quart. Journ. Met. Soc, 15, 1889) he gave the relation as P = 0'003 672^2 foj- jjiHes per hour. Since this value would require to be corrected for "circular motion," we must deduct about 5 per cent, from it ; which will give P = 0*003 5 very nearly. M. Eiffel, \n La Resistance de r Air, 1910, says: — "Accord- ing to the author himself these results are tainted with an error which, in our opinion, varies according to the size of the plate, and which one may value at 15 per cent." If we deduct 15 per cent, from 0"003 67, we shall get 0'003 i, ^ Clerical error : what was meant was 55°. Experiments with Whirling Machines 157 which agrees vvitli the theory here advanced and with the results of most other experimenters. Mr Dines said {Quart. Journ. Met. Soc, 16, 1890): — " Experiments upon the whirling machine at Hersham have led me to suppose that the greater the plate the /ess pro- portionately is the wind pressure upon it. " This is exactly the opposite of the result found by other experimenters. To be quite fair, however, I must point out that Mr Dines, in 1908 {Inst. Civ. Eng., 171, "Correspondence on Wind Pressure ") repudiated this, and said ' ' he was prepared to endorse fully the author's [Dr Stanton] statement that wind pressure was strictly proportional to area when the shapes of the two areas were similar, and he agreed with the reasons the author gave for thinking his own conclusions [those just referred to] to be wrong" [italics added]. On the next page of Mr Dines' paper, referred to, we find that he says : — " I was led to think, from the result of experiments, . . . that the pressure upon a plate 6 inches diameter was proportionately less than upon a foot plate, but I now believe tJiat my conclusion was wrong, the greater pressure upon the foot plate being probably caused by an eddy from the frame of the apparatus used for mounting the plates. The edge of the larger plate being nearer the frame, it seems reasonable to conclude that it would be more affected than the other" [italics added]. From consideration of the work of other authors, I think there can be no reasonable doubt that the pressure measured, by the use of whirling machines, is greater per unit of area on a large plate than on a small one. At the same time I do 7iot think that this is caused by anything which might be described as "dimensional effect" — in the proper sense of the expression. Colonel Renard {Comptes Rendus de fAcadc'mie des Sciences, May 2, May 16, and May 24, 1904) gives some results of experiments made by means of a kind of whirl- ing machine which he calls a " dynamometrical balance." This dynamometric balance, he says, "does not allow of the correct measurement of p [the coefficient of resistance 158 Resistance of Air to rectilinear motion], because it makes use of circular motion, the coefficients of resistance in such cases being always greater tJian those of the resistance in rectilinear motion " [italics added]. Colonel Renard found that the resistance of a circular plate, whose diameter was 0'200 m. , turning in a circle, whose diameter was 2-30 m. , gave a value K = 0-103, "notably superior to 0'085 : which shows the influence of circular motion." It is not absolutely clear from the paper, although it appears probable from the photograph, that the "diameter of the circle " is measured from the centres of the circular plates (as shown in fig. 20). K- ...2^30 -^ ¥lG. 20. — Diagram of Colonel Renard's "balance." If we assume this to be correct, the coefficient 0-103 should be corrected, by Duchcmin's formula, by about 24 per cent, to find the resistance to rectilinear motion. With the notation previously described the correction I ' ^ X S would be -^ ^, where 5 = 0-200 and /= 1-15. This will give the corrected value of K = 0-077 \ or, in English measures, K = 0-00147 — 'which is of the same "order" as 0-00 1 428, derived from the theory here advanced. Colonel Renard found that the resistance of a sphere (measured by his instrument) was only= i / 6-31, or O'l 58 5, of that of a circular plate. He says: "The coefficient of reduction of a sphere is notably much lower, i / 6-31, than was believed up to this." It is certainly considerably less than the coefficient found by any other experimenter, so I must doubt its accuracy. Dines {Quart, fourn. Met. Soc, 15, 1889) gives the ratio Experiments with Whirling Machines 159 as 145 -.^J for a 6-inch circular plate 4f inches thick, and a sphere of 6-inch diameter, or about i / 2-2. Other authors have found the ratio (as pointed out previously) as nearer I / 2*5. There are other theoretical reasons, which are foreign to my present purpose, for supposing that the pro- portion of 5 : 2 is the correct one. Table XXXI. Velocity, ft. sec. Resistances. oz. avoirdupois. 1st differences. 2nd differences. I 2 3 4 5 6 7 8 9 10 0-032 0-129 0-294 0-525 0-825 rigi 1-627 2-131 2-704 3-345 •097 •165 -231 •300 •366 •436 •504 •573 -641 •710 •779 -849 -918 •987 I 056 I-I27 1-195 1-266 1-335 •072 •066 •069 •066 •070 •068 •069 •068 •069 1 1 12 13 14 15 16 17 18 19 20 4-055 4-S34 5-683 6-6oi 7-588 8-644 9-771 10-966 12-232 13-567 ■069 -070 •069 -069 •069 -071 •068 •071 •069 Beaufoy's experiments with whirling machine. Length of arm 9-122 feet. Plate i foot super (? square or circular). Barometrical and thermo- metrical readings not given. Colonel Beaufoy (Thomson's Annals of Philosophy) made some experiments with a peculiar form of the whirling machine : it was a sort of revolving windmill. He gives the resistances, per foot super, experienced by a plate at velocities from 1-20 feet per second. The results are given in Table XXXI, where it will be seen that the first and second differences (which I have added) indicate that i6o Resistance of Air the work was very carefully carried out. He points out in a note that the resistance increased as V-''-"^ or rather -more rapidly than as the square of the velocity. If we take his coefficient 0*032, in ounces avoirdupois^ and divide it by 16, we shall get 0002 for resistance in pounds. Correcting this for circular motion, where y"=9"ii2 and S = 2 (there were four sails to the windmill, measuring 12 inchx 6 inch each ; and these were, of course, fixed for this experiment), we shall find that a correction of about 30 per cent, is required to give the resistance in rectilinear motion. Such reduction will change 0*002 to 0"00i 4, or about the coefficient found by the theory here advanced, viz. 000 1428. Since we know nothing about the temperature or the barometrical pressure; whether the experiments were made in the open air (with a possible wind); or, if made in a room, what the size of the room was, or how near the windmill was from the wall when passing it, it is impossible to discuss this coefficient further. It was, apparently, quite of the right order. Professor VV. C. Kernot {^Australian Association for tJie Advancement of Science, 6) gives the results of some whirl- ing-machine experiments in a table. His figures are rather ragged, and do not inspire any confidence. For example, exps. I, 2, and 3 are exactly the same ; the velocity being 12 miles per hour. The resistances given are '29 lb. per square foot, -47 lb., and '56 lb. He says the whirl- ing arm was a " strong radial arm," but he omits to give its length — the really important detail. He gives no baro- metric pressures nor temperatures. His remarks are dis- tinctly apologetic: "These results, though presenting occasional anomalies due to the defects of the apparatus and imperfect observation, correspond very fairly with the formula R = 0"003 3V2 . . . and are utterly irreconcilable with Smeaton's.'' Nor is this the whole story, for we find Professor Kernot saying later: "It should be mentioned that the observed pressures were all increased by 10 per cent. to allow for mit-wind, or the general air in the room due to the action of the apparatus and friction of the mechanism." Experiments with Whirling Machines i6i Since we know that Professor Kernot's experiments should have " overmeasured " the resistance to rectiHnear motion — they having been carried out by means of circular motion, — it is curious that he should have added lo per cent, to his observed results. Why lo per cent.? If he had added 8 per cent., the result would have been even closer to the theoretical results deduced from the theory here advanced. I do not think we can attach any more importance to these results than we can to those he derived from the experiments on resistance caused by a " wind." Dr George Finzi and Dr Nicholas Soldati carried out "experiments on the dynamics of fluids," the details of which are given in Engineering, March 17, March 31, April 14, 1905. They were also published as Sperimenti sulla Dinaniica dei Fluidi, in Milan, 1903. I have not seen the Italian publication, so I rely on the papers in Engineering. These papers deal largely with subjects which do not interest us here, so I will only refer to a very small part of them. The experiments were carried out by means of a whirling machine, the arm of which was 4 m. There was, however, an "extension piece," which made the arm "4 '5 or 4 "6" m. long: apparently the doctors thought the exact length was unimportant. As usual, they considered the whirling arm as "of such dimensions as to render of no account the inaccuracies due to the cu7'vature of the path travelled over by the body" [italics added.] The details will be found in Table XXXII, which 1 think requires no explanation. The figures (even when corrected for circular motion) look a little ragged : so much so that I would not like to rely on their accuracy. In the first place, it is well known that the value of K for a circle is rather less than that for a square. We find in the table that this is the reverse for the smaller plates, but is true for the larger plate. Most of the coefficients appear rather small. The corrections were made in the usual manner by the use of Duchemin's formula. I have assumed the II l62 Resistance of Air "whirling radius" (if I may so call it) as 4*5 m. If this radius was 4 '6 m. the correction I have made was probably 2 per cent, too great. As will be seen, the cor- rection for the 100 X 100 plate is a very heavy one. The velocities were from 5 to i 5 m. per second. Table XXXII. Description of plate. Size in cm. Value of K. Correction for circular motion. Corrected value of K. Remarks. ^ Length of Circular 10 30 o'076 0-082 - 0-002 - 0-008 0-074 0-074 0-076 arm of whirling >5 100 o'099 -0-023 machine Square lox 10 0-073 - 0-002 0-071 assumed as >> 30x30 o'oSo 5 -0-008 0072 5 J 4-5 ;//. 100 X 100 O'lOI -0-023 0-078 Rectangle . 10 X 30 o'oSo 5^ No data as to how these rectangles 10 X 100 0-094 2 \ were " set " in relation to the axis jj 30 X 100 0-095 S-* of the whi rhng arm. Barometrical and thermometrical readings not given. Finzi and Soldati, experiments with whirling machine. When we refer to the values of K for the rectangles, it is not possible to say what correction should be applied, since there is no indication of how they were "set" with regard to the axis of the whirling arm. The "set" is by no means immaterial, as these authors would appear to have thought. M. Eiffel, in referring to these experiments {La Resistance de I' Air, 1910), says : "They seem to indicate that the co- efficient K increases with the surface "; on which point I do not agree. The system of measuring the pressure would appear to be one which is open to inaccuracy. In any case, the figures do not inspire confidence. REFERENCES S. P. Langley, Experiments in Aerodynamics, 1891. W. H. Dines, " On Wind Pressure upon an Inclined Surface " {Froc. Roy. Soc. of London, 48, 1890). Experhnents with Whirling Machines 163 G. Eiffel, La Resistance de I'Air, 19 10. MoRiN, Lecons de A/ecatiiqiie Pratique, i860. W. H. Dines, "Some Experiments made to investigate the Con- nection between the Pressure and Velocity of the y^'\x\^" {Quart. Jojcrn. Roy. Met. Soc, 15, 1889). Colonel Renard, Comptes Rendus de VAcadhtiie des Sciences, May 2, May r6, and May 24, 1904. Colonel Beaufoy, Thomson's A?ifials of Philosophy, 6 and 7- Professor W. C. Kernot, Australian Association for the Advance- ment of Science, 6. A. W. Johns, Institution of Naval Architects, 1904. G. Eiffel, La Resistance de T Air, 191 1. FiNZi and Soldati, " Experiments on the Dynamics of Fluids " {Engineering, 1905) : also Sperinienti sulla Dinamica dei Fluidi, Milan, 1903. CHAPTER XVII WIND TUNNEL EXPERIMENTS — STANTON, BAIRSTOW, EIFFEL One of the most fashionable methods of measuring the resistance of the air, at the present day, is by means of wind tunnels. It has the great advantage that the resist- ance can be measured very accurately by an ordinary weighing machine. It is as easy as weighing a letter, and the method is very direct. Of course, there are dis- advantages, and these are: — (i) That the air moving through the tunnel is not travelling with uniform motion ; it is being accelerated. The fluid is not, in fact, a static fluid. Many precautions are taken to prevent it from changing its shape ; but it is changing its velocity. Nevertheless, as we have seen previously, the "added mass" question is, frequently, not a serious one, and we may perhaps neglect it. Many experimenters appear to assume that their currents are uniform ; but it is not easy by any stretch of imagination to suppose that the fluid is not being accelerated. (2) It is extremely diflicult to measure the velocity of the air at any part of the tunnel with any certainty that the measurement is accurate. I know this will be disputed ; but the fact remains that hardly any two tunnels give the same numerical results, and when they do there is great rejoicing amongst experimenters. (3) There is great danger of "boundary effects," which vitiate the results arrived at. I will commence by referring to the experiments of T. E. Stanton, "On the Resistance of Plane Surfaces in a Uniform Current of Air, " described in a paper read at the Institution of Civil Engineers, 1904. This paper, which 164 Wind Tunnel Experiments 165 is classical, is the first paper of the two read on aii resistance, the second of which ("Wind Pressure") I have already referred to. The paper covers a very wide field, but I will confine my attention to only a very small part of it. Dr Stanton makes no reference to theory, so I will only examine his coefficients ; and by coefficient I mean the amount the velocity squared has to be multiplied by to give the resistance. Dr Stanton assumes that the resistance varies as the square of the velocity, so that ^ = cv'^ where c is a constant. But, since v^-=2gh, we may put ^ = 2cgh, where h is what Stanton calls the "velocity head" of the current — in feet of air. R = resistance in ' ' units " of I / 100 000 lbs. avoirdupois. Clearly, c—^ / 2gh. Since the tunnel was very small, 24 inches x 24 inches, the experiments also were on a very small scale. From his Table III, for square plates ranging from 0*442 inch side to I 77 inches side, we find the values for R / /: for the first experiments. The average is 59 "8 — varying from 59*2 to 60 "3 : the plate having a side of i •/ inches. From the second experiments (side of plate i "33 inch) we get this mean value as 59'6 — varying from 54'2 to 62"3. From the third (side of plate o"886 inch) the mean value is 60*5 — • varying from 5975 to 62*5; whilst from the fourth the mean value is 58*5 — -varying from 56'5 to 61 "87. If we take the average as 60 — it is really 59 "6 — multiply it by 1^14, to reduce it to square feet \ then divide by 64*36, (2^) ; and, lastly, divide by 100 000, we shall get the value of c, when R is in pounds avoirdupois. This gives c = 0'00\ 34244. If we do exactly the same with the data in Table II, for small circular plates, we shall get the mean value of R / /2=59*5 — almost exactly the same as was obtained for the square plates. Dr Stanton tells us, however, that there was an error of 3 per cent, in his Pitot tube, which measured the velocities; so that we have further to divide the value of ^ by i '06. This will give the corrected value of the coefficient as 1 66 Resistance of Air 0"00i 26, and the equation of resistance will become R=i:0"00i 26^'2 (lbs., feet, seconds). This value appears suspiciously small. ^^ We have seen from Bashforth's experiments with 2-inch shot, as well as from the theory advanced, that the coefficient is nearer O'OOI 4. We also see from Ur Stanton's experiments that the resistance of circular plates is the same as that of square plates of the same area. Most experimenters have found that the resistance of circular plates is appreciably less. This result might also be deduced from Dr Stanton's own statement: "The ratio of the maximum pressure on the windward side, to the pressure on the leeward side of the circular plate, is found to be 2 "i : i ; whereas the ratio for the rectangular plate used was i "5 : i." Now, if the velocity of the air is the same in the two cases, the maximum pressure should also be the same in both cases : Dr Stanton says elsewhere that this is so. Taking the maximum pressure as unity, the resistance of the circular plate can be expressed as i+0'48; whilst the resistance of the square plate would be expressed by i+0'66. In other words, the resistance of the square plate should be about 10 per cent, greater than that of the circular plate of the same area. Dr Stanton's experiments, quoted, would appear to show a difference of only about one-sixth of i per cent. ; and even this is not so pronounced that we can consider it as definitely indicated. M. Eiffel's experiments (to be referred to later), which were carried out in a very similar manner to Dr Stanton's, appear to confirm them, on some points. For example, with plates 10x10 cm. (the smallest he appears to have experimented with), M. Eiffel found the value of K was = 0"c65. This, in English measures, would be O'ooi 241 5 — not very distant from Dr Stanton's o-ooi 26. ' Mr Mallock said about this : — " It is the lowest value found by any experimenter, too low in fact, since, if correct, it would indicate that there was scarcely any negative pressure on the down-stream side of the planes he used." M. Eiffel's value was slightly less. Wind Tunnel Experiments 167 We notice that in all Dr Stanton's experiments, here referred to, tJiere is no appearance of any ^^dimensional effect'' — the coefificient is the same for the smallest and the largest plates, although they vary in size, in a linear ratio of 4 : I. If there were any marked effect it ought, we should think, to be more apparent in very small plates than in those of larger size. In the experiments on wind pressure, when the plates were very large, the value of K was found to be 0'003 2 for miles per hour, as against 0"002 7 {also for miles per hour) for the very small plates. There was, also, no appearance of any "dimensional effect " in the experiments with the very large plates. Nevertheless, Dr Stanton attributes the great increase in the value of K to dimensions only. It is difficult to say on what grounds, since his experiments do not appear to support this view. If we believe that the resistance increases as the square of the speed, we are irresistibly led by the laws of similitude to believe that it also increases as the area — certainly for moderately low velocities. In the experiments with spherical shot we saw no sign of the resistance increasing otherwise than as the area of the great circle. Nor is it any conclusive argument that M. Eiffel's experiments show this dimensional effect very markedly, since these experiments were carried out in a very similar manner to those of Dr Stanton. One might also refer to experiments with whirling machines, which all appear to show this dimensional effect very clearly ; but when properly examined the effect disappears. That there is a dimensional effect for very small plates, and at low velocities, appears highly probable — it would follow logically from what has been said previously. Since the resistance, at very low velocities, varies as A(LV) ; whilst for larger plates, and higher velocities, it varies as A(LV) + B(LV)2, ^ye should certainly expect to find some dimensional effect, though it would doubtless be small. The remainder of this interesting paper does not come within the scope of my present purpose. It is a pity that the author should have employed such an unusual measure- 1 68 Resistance of Air ment for the barometric pressure of the air as pounds pet- square inch, instead of the more ordinary one of milli- metres, or inches, of mercury. In the discussion on this paper, Mr Mallock pointed out that he had observed that the negative pressure on the lee- ward side of a disc of lO inches diameter, and y^^ of an inch from the edge, was 2\ times the dynamic liead in front of the plate. This accords very accurately with Duchemin's observations on the pressure in the filaments of water passing the edges of a 12-inch plate, which was —2 •5/2 where h is the velocity head of the stream. This is equivalent to saying that the velocity of the filaments past this edge is about i "6 times the velocity of the stream "at a distance." It is another confirmation of the similarity between the flow of water and air. One thing should follow, logically, from Dr Stanton's experiments ; that is, that if O'ooi 26 is a correct co- efficient for very small plates (and the accuracy of this work is not to be questioned), and that the coefficient increases as the plates get larger — as Dr Stanton firmly believes, — then the value O'OOi 26 is absolutely useless for calculating the pressure on plates of a few square feet of area. One has no right to build any formula on this coefficient ; unless, of course, the law of increase of the coefficient is known and allowed for in the formula. Further than this, the dimensional effect must be reconciled with dynamic similarity. Such a formula has been circulated from the National Physical Laboratory. In the Report of the Advisory Committee on Aeronautics, 1910-11, we find in the Reports and Memoranda, No. 38, Mr Bairstow expressing the resistance of air as \ vl V To be accurate, the v in the last term is implied only, since I cannot find it in Mr Bairstow's paper. It is clear, how- ever, that without it the formula would violate dynamical similarity, and not be homogeneous. That my assumption Wind Tunnel Experiments 169 is a fair one is shown in Memorandum No. 33, by Dr Stanton, who gives this formula in full, and states that it "is precisely siiiiilar to that found by Mr Bairstow and Mr Booth in regard to the experiments on the normal re- sistance of flat plates of different sizes " [italics added]. In the appendix to A B C of Hydrodynamics I referred to this formula with approval. I now see that there are three important, and, I think, fatal, objections to it. 1. If the fluid were supposed inviscid, v would be zero, and the resistance would consequently be infinite. This is, of course, absurd. The classical text-books say the resistance would be zero ! 2. At high velocities of the order of i 200 or i 300 feet per second, since the last term would be dominant, the resistance would increase approximately as the cube of the velocity (as we know it does), and also as the cube of I. In other words, if two shots were fired at a velocity of, say, 1 200 feet per second, one of which had a diameter of 2 inches, whilst the diameter of the other was 4 inches, the resistance of the larger shot would be eight times greater than that of the smaller one. Experiment shows, as one would expect, that it is only four times greater. This objection appears to be quite fatal to the formula. 3. By this formula the resistance at, say, 2 000 feet per second would increase as the cube of velocity, whereas all the leading experts are agreed that it only increases as the square of the velocity. In this formula I presume that Mr Bairstow relies on the / in the last term for the "dimensional effect." Without attaching too much importance to this formula, we see that Mr Bairstow says, later: — "After some trial and error it appeared that a formula of the type Y = aiviy -\- b{vrf was a satisfactory empirical approxima- tion, a is given as 'ooi 26 from Dr Stanton's experiments, and b was then determined from fig. i [I have been unable to find fig. I ] to the nearest round figure, indicating the degree of accuracy possible, as o "0000007. We have seen that F = o'OOi 26(?7/)- + o-oooooo 7(zV)3 . . (4) 170 Resistance of Air is an apjjroximation to all the known experiments on square plates which can lay claim to sufficient accurac}' to be worth consideration, the values of vl in tJicsc experiments ranging from i to 350" [italics added]. This statement is a little sweeping, since it rules out some good experiments which do not happen to fit in well with the formula. In this empirical equation Mr Bairstow has dropped the first term {as is Jisual/y done), since at any but very low velocities it is quite negligible : the formula has only two terms, but, of course, it is understood that the first term is necessary for very low velocities. The formula violates dyiiamical similarity — •// is not homogeneous. We are told that the agreement of experiment with this "empirical formula is very close and within the errors of observation. It has therefore been concluded that the pressure on square plates can be expressed by a dimensional equation of the type suggested by Lord Rayleigh " [italics added]. This last statement is hardly in accordance with Lord Rayleigh's own view, since he says, in Memorandum No. 39, 191 1 : — -"The experimental evidence so far avail- able can hardly be said to support the general applicability of the form (d) \y = pvf{v / vl)] at all. According to Stanton and Eiffel, a small linear scale carries zvith it a di}}iinutio7i of P. If {d) is applicable, it follows that the same effect should accompany an increase in v — a paradoxical conclusion, as Mr Lanchester has remarked. Moreover, Dines' observations point in the opposite direction " [italics added]. I have referred to Dines' experiments previously : I will, later, pay attention to the work of M. Eiffel. That the resistance can be expressed in Lord Rayleigh's form {d) I do not think can be doubted — it is funda- mental. His formula, however, is homogeneous ; whilst Mr Bairstow's is not. That the resistance can be expressed by the formula under review, I very much doubt. Both the Wind Tunnel Experhnents 171 /3 and the v"^ in the last term, cause this term 7iot to have the diinensions of force. It is, confessedly, an empirical formula, based on experiments made by different men at different times and by different methods. The objections 2 and 3 made to the other formula apply equally to this one. To make it homogeneous the / in the last term must come out; whilst the v can be arranged for by changing the function to/"( — , — |, when the equation would be Let us, however, compare Mr Bairstow's formula with experiment. Will it satisfy the requirements of experi- ment ? Will his coefficient stand being multiplied by a few millions ? I shall assume, in this argument, that there is ojily one law for air resistance : that the sajnc law applies to both round and flat bodies — subject, of course, to some ' ' coefficient of shape. " Let us see how Mr Bairstow's formula fits in with the experiments on 2-inch shot up to, say, 2 000 feet per second. We are justified in doing this, since Mr Bairstow's limit for vl is 350 ; which is greater than ^ X 2 000. Let us first put the equation F ^ o -oo I 26(2>l)" + o -ooo 000 /(vl)^ into the form I have employed previously F = o-ooi 26(2^/)2{ I +0-000 5(z'/)} To apply this to a 2-inch shot we will put P (the sectional area)= i / 46 of a square foot, whilst we will put /= ^ of a linear foot. The formula will now become F = o'oooo27 4v^{ I -j-o'OOOOQz'} which will express the resistance of a 2-inch circular plate. But the resistance of a sphere is (certainly approximately) only f of that of a flat plate. The resistance of a 2-inch 172 Resistance of Air sphere will be expressed, by this formula, as F — o -000 o I o 96e'-{ i + o -ooo o()v } in feet, pounds, seconds. Since, however, I propose to compare this with Hutton's experiments, where the results are given in ounces, we must multiply by 16 to give the resistance in ounces also, F = o -ooo 1 7 5^-"^ I H 1 l II oooj It will be apparent, almost at a glance, that this formula will give results very much below those derived from experiment. I have calculated the values from 300 feet per second up to 2 000 feet per second, the results of which will be found in Table XXXIII contrasted with Hutton's Table XXXI II. Velocity. Experiment. Hutton. Calculation Bairstow's formula. feet 300 400 500 600 700 800 900 I 000 I 100 I 200 oz. 25-8 46-5 74-4 110-5 156-0 2I2-0 280-3 362-0 456-9 564-4 oz. l6-22 29-12 45-94 6678 91-75 120-96 154-50 192-5 232-92 27972 I 300 I 400 I 500 I 600 I 700 I 800 1 900 2 000 683-3 811-5 947-1 I 086-9 I 228-4 I 368-6 I 5057 I 637-8 331-24 387-59 448-90 515-00 58667 663-39 745-46 84000 Resistance of 2-inch spherical shot, calculated by Mr Bairstow's formula, and compared with Hutton's experimental observations. Wind Tunnel Experiments 173 observed values. Even if we admit that Hutton's resist- ances are a little too high, the two results are not of the same ' ' order " ; and if I compared this formula with Smeaton's experimental measurements, it would appear to still greater disadvantage. I do not admit that Smeaton's experiments cannot "lay claim to sufficient accuracy to be worth consideration." He was a very careful observer, his results are all consistent, and, as I have shown, they can be made to agree with the theory advanced. That they do not agree with the generally accepted theory of resist- ance — if, indeed, there is any theory — is unfortunate. It is no argument, however, to say that therefore his experi- ments must be bad. In the next chapter I will deal especially with the experiments of M. Eiffel. I have reserved these for the last, not because I consider them the least important — very much the reverse, — but because they are rather different (in the method of carrying them out), and also because the so-called "dimensional effect" is most pro- nounced in them. REFERENCES T. E. Stanton, " On the Resistance of Plane Surfaces in a Uniform Current of Air " {^Proceedings of the Inst. Civil Engineers, 1904). L. Bairstow and H. Booth, "The Principle of Dynamical Simi- larity in reference to the results of Experiments on the Resistance of Square Plates normal to a Current of Air " (Memo. No. ^Z, Report of the Advisory Committee for Aeronautics, 1910-11). Lord Rayleigh, Memo. No. 39, A.C.A., 1910-11. Smeaton, Fhil. Trans., 1759. CHAPTER XVIII EIFFeL'S experiments — "DIMENSIONAL EFFECT" — * ' ADDED MASS " — CONCLUSION M. Eiffel's work covers an immense field. I wish, how- ever, specially to concentrate attention on that part of it which bears on the "dimensional effect " (so called). Referring first to his Recherches Experiment ales sur la Resistance de I' Air executees a la Tour Eiffel^ we find that MM. Levy et Sebert made the following remarks when presenting the report on M. Eiffel's memoire to the Academie des Sciences in November 1908. Referring to the " V^ law," they say: "The coefficient K of the formula R = KSV^ representing this law, which should be constant if it were rigorously exact, has been found variable within rather wide limits, according to the methods by which the experiments were carried out and according to the shapes and dimensions of the bodies submitted to experiment, *' . . . preceding experimenters have found, admitting the exactitude of the square law, which is not rigorously certain, values of K . . . varying from O'O/O up to 0'I25, the greatest values found corresponding generally to the greatest surfaces and to the greatest velocities, which latter were obtained by means of whirling apparatus. " . . . in fact, the power of the velocity appears to increase continuously for plates, passing by the value of 2 for the velocity of about 33 m., but always remaining 174 Eiffel's Experiments 175 near enough to this value [for velocities 18 m. to 40 m. per second] that one may accept this ratio as very approximate. *'The coefficient K of the formula thus admitted was found constantly between O'O/ and 0'o8, for air reduced to a temperature 15° C. and a pressure of 760 mm. . . . The coefficient increases, gradually , with the area of the plate and with its perimeter" [emphasis added]. This gives an exceedingly good general idea of the views to be found in this book. Let us, however, refer to the original, and see what M. Eiffel himself says. Having given some details of an experiment, he says, on p. 34, under the heading " Coefficients of Resistance " : — "If, in the experiment which we have just studied, we calculate the relation between the resistance on unit surface to the square of the velocity, that is to say, ^ to V^, we S have the following values :- For H = 20, 40, 60, 80, 95 R / SV2 = o"o63, o'o65, 0-064, 0-062, 0-064 " The value of this ratio is sensibly constant, and differs little from the mean 0-064. "All our experiments lead to analogous conclusions, and give results sensibly conformable to tJie law of Newton : R = KSV^, in which K is constant for a detcnnincd surface : it is the coefficient of resistance of the surface, or its specific resistance. . . . " However, we shall see later that, for a given surface, K is not rigorously independent of the velocity ; in other words, the law of the square of the velocity is not absolutely true. At velocities attained by artillery projectiles the resistance becomes proportional to the cube, and even the fourth power of the velocity, and this reason alone would suffice to make us reject the theoretical demonstrations that one seeks to give to the Newtonian formula. " Which for- mula, as I have said previously, is not Newton's. 176 Resistance of Air In paragraph 8 he says the values of K,, (for T= 15° C. and B = 76o mm.) do not differ much from 0"073. In Exp. No. 3 square ^^ m.2 K = -070 7 circle | m. - K = -07 1 ,, 1 1 square \ m. ^ K = -07 1 ,, 21 circle | m.^ K = 0-074 25 square j m.^ K = 0-073 32 circle | m.^ K =0-078 39 : square | m.^ K = o-o8i 42 circle i m.^ K=oo8 (M. Eiffel averages rectangles with the squares : I have omitted them). From the above figures it would, at first sight, appear quite certain that K increases with the increase of the area of the plate : this, of course, in a sense, is true. M. Eiffel continues : — "The coefficient K increases with the surface, but the rate of this increase diminishes more and more as the surface increases: in other words, K appeals to tend towards a maxi- mum of about O'oSo" [italics added J. Nevertheless, on looking more closely, we notice a tendency of these coefficients to increase at low velocities. M. Eiffel gives a table, made from the values of 200 co- efficients, or 40 for each velocity, which I reproduce in Table XXXIV. The third column gives the sum of the coefficients for the velocity ; the fourth column gives the -mean value. Referring to this, M. Eiffel says : — "The variation of these coefficients represented as a function of the square of the velocity in fig. 21, where the ordinates are taken from their mean value 0*074, follows a regular curve. "This curve passes through a -minimum for the velocity of 33 m. (about). // is at this velocity that Newton's formula would be exact, since we find that in its neighbourhood K is constant. From the movement of this curve,, for velocities Eiffel's Experiments 177 178 Resistance of Air inferior to 19 m., the coefficient K appears to increase pretty rapidly. Table XXXIV. H. Mean velocity. Coefficient K. Total. Mean. 20 40 60 80 95 18-89 26-36 3173 36-01 38-60 Gene 3-026 2-954 2928 2-933 2940 ral mean . 0075 6 0-073 8 0-073 0073 3 0-073 7 0-073 9 Table made from 200 coefficients, or 40 for each velocity. (From M. Eiffel's book.) " For velocities other than 33 m. it would be necessary to adopt, as the power of the velocity, a number « more or less different from 2. In calculating it by means of the values in the preceding table, we get : — Velocities 18-89 to 26-36 m. 26-36 ,, 31-73 m. 3173 >, 36-01 m. ,, 36-01 ,, 38-60 m. «=i-93 11^ 1-95 n — 2-o\ 71—2 -08 " All the foregoing makes a very strong prinia-facie case for the "dimensional effect." Nevertheless, I am inclined to think that this effect can be explained in a simpler manner, and in one which docs not violate dynamical simi- larity — as the ordinary explanation appears to do : accelera- tion must be taken into account. We must first consider that in these experiments the bodies were falling almost freely, and were being accelerated. About this there can be no doubt, for M. Eiffel, on p. 2, refers to the experiments of Cailletet et Colardeau, where the falling bodies were so much retarded that the motion became uniform ; that is to say, that they had a "terminal '' Added Mass'' 179 velocity." "In our experiments, on the contrary, the plate . . . moved by a very heavy weight, and which offered a very small resistance to the air, was up to the end of the fall animated by a velocity ve?y nearly equal to that it would have had if it liad fallen in a vaeuum ; that is to say, attain- ing 40 m. per second after a fall of 95 m." [italics added]. We know that when bodies are being accelerated the resistance requires to be increased by what is commonly called the " added mass " — a mass of fluid which is supposed to be specially moved by the body. Dubuat was the first to point this out — though Bessel generally gets the credit for it, — and he concluded that the volumes of the " added masses " are proportional, for bodies geometrically similar, to the volumes of these bodies, and that they do not depend on the density of the medium, nor on the speed of displacement of the bodies (see Principes d'Hydraulique, t. ii). Stokes {^Mathematical and Physical Papers, 3, paper dated 1850) refers to this subject, and gives the coefficient, for a sphere, as about 0*5 of its volume. 1 have adopted this figure in reference to the fall of bodies in air, although I am inclined to think it should be nearer o'6. Values have been fovnid from 0'9 down to below 0*5, but the exact figure does not affect my argument, so I will retain the figure 0'5. It may be said that this applies to spheres. Is it equally true for flat plates ? Dubuat is the only one, that I know, who made experiments with plates. In his exp. 306 he employed a circular lead disc, whose diameter was 2*677 (French) inches, and thickness I'lO (French) lines, as the plumb-bob of a pendulum. The length of the pendulum was 32 feet, and the lead disc moved perpen- dicularly to its flat surface. This gave him a value of n for the added mass of 16 '80. When the pendulum was shortened to 19 feet the value of // was 22 "03. If we multiply I'l lines by 22*03, ^^'^ get 2 inches, or 079 of the diameter of the disc. Similarly, if we multiply I'l lines by 1 6 "80, we get a value of about 066 of the diameter. i8o Resistance of Air In exp. 307 a square plate was employed, 2 '37 (French) inches side of square, and thickness i '03 lines. With a pendulum 32 feet in length (about) n was found to be 18 '38. The pendulum being shortened to 20 feet, the value of 71 was 24 "63. From these we get, as before, the proportions to the side of the square as 0"66 and 0"86 respectively. From these experiments we may fairly judge that — 1. The "added mass" for a circular plate is a volume which bears some ratio (o"6 to 0*9, or thereabouts) to the volume of a cylinder whose height is equal to the diameter of the circle ; or, for square plates, to a cube whose side is the same as that of the square plate. 2. That «, the coefficient of the added mass, is greater as the length of the pendulum is diminisJied. 3. Dubuat's experiments also show that n decreases as the size of the globe increases. Since these experiments were made by means of oscillat- ing^ or intermittent, circular motion, it would be fair to suppose that the "added mass" during accelerated circular motion would be greater than during accelej'ated direct motion. D. Riabouchinsky, Bulletin de I'Institut Aej-odynaviique de Koutchino, fascicule III, refers to this "added mass" when discussing the " shock of an unlimited current on a plate." This is apparently, though he does not say so definitely, his explanation of "Dubuat's paradox." I cannot enter into his argument at present, further than to point out that he claims that the resistance varies as R = KS"V2 This I cannot agree with, nor with some other things he says : still, it is clear that the added mass is a volume — it has three dimensions. The surface of the plate has two dimensions only. If, therefore, the length of the side of a square plate be doubled, the surface is increased four times ; but the "added mass" is increased eight times. The resistance should therefore increase with the size. Now, without tying ourselves down to any definite figure " Dimensional Effect " 1 8 1 for the coefficient, we may say that the resistance R^ of a plate in accelerated motion is where R is the resistance in steady motion, O is the volume of the added mass, Oj the volume (real or fictitious) of the moving mass. In the case of a circular plate we may put it as the volume of a cylinder whose height is equal to the diameter. /"(Q / Qi) is at present an unknown function. It will now, I trust, be clear what I consider the real explanation of the increased values of K, which M. Eiffel found increased with the size of the plates. It is not really a dimensional effect at all : if it were, it would violate dynamical similarity. There is no increase in the "dimensions" — used in the strict sense. The function of Q / Qj has no dimensions : it is simply the ratio between two volumes, and so is simply a number — a scalar quantity. This increase in the value of K, according to the size of the plate, does not occur when the velocity is uniform. It is purely an effect caused by the acceleration, and by the more rapid increase in the size of the added mass when compared with the increase of the area of the plate. We know, from Dubuat's experiments, that n decreases as the size of the globe increases. It would appear probable, therefore, that this apparent "dimensional effect" (as it is called) would have a limit — as M. Eiffel has found that it has. Sir George Stokes says: — "Among the 'additional experiments ' in the latter part of Mr Baily's paper is a set in which the pendulums consisted of plain cylindrical rods. With these pendulums it was found that n regularly increased, though, according to an unknown law, as the diameter of the rod decreased. While a brass tube 1 1 inches in diameter gave n equal to about 2 "3, a thin rod or thick wire only 0'072 inch in diameter gave for n a value as great as 7*530."^ ^ This will be referred to again, in a little more detail, in the " Conclusion." 1 82 Resistance of Air We have seen in M. Eiffel's curve (fig. 21) that K increases, apparently rather rapidly, as V^ {and, therefore, probably IJ') decreases.^ This fact seems to fall into its proper place. Sir George Stokes, after referring to all the investiga- tions made by others, says : — " They all fail to account for one leading feature of the experimental results, namely, the increase of the factor ti with the decrease in the dimensions of the body." It is in consequence of the "added mass" — another variation on the same tune — that the resistance of a trellis is sometimes greater than that of a solid plate. In fact, it will be found that the added mass plays a very important part in determining the coefficient of sJiape ; for K, in the last analysis (certainly as I employ it) is a coefficient of shape. There is a very interesting function of -, where a is the longer side and b the smaller side of a plate and where the resistance is expressed as Rj = Ri i +/ , ) \- I must, however, quit this fascinating subject for the present. 1 have said that this apparent "dimensional effect" is not observed when the velocity is steady. As it is well not to make any statement without experimental verifica- tion', I must bring some evidence for what I have said. MM. Cailletet and Colardeau {Comptes Rendus de I' Academic des Sciences, 1892 and 1893) carried out some experiments on the resistance of plates falling from the Tour Eiffel. Their method was different from that of M. Eiffel, in that the plates fell freely, being accelerated by different weights attached to them. The weights were such that the plates attained a "terminal velocity" — after which the velocity was uniform. As one would expect, the results are quite conformable with the theory that the resistance varies exactly as the area of the surface. MM. Cailletet and Colardeau called attention to this in more than one place in their papers. 1 This is, of course, diametrically opposed to Dr Stanton's view. ''Dimensional E^ect'' i8- M. Eiffel, in his Resistance de I'Azr, 1910, refers to this work, and gives the values of K which 1 have put in Table XXXV. He also says: — "Although no correction was made to take account of the variations of the tem- perature and of the pressure of the atmosphere, these values must be considered ainotigst some of the best which have been obtained zip to the present. "They present no indication of any variation with the size of the surface experimented with^ nor with its shape " [italics added]. This agrees very well with what I have said previously, but there are M. Eiffel's other experiments in a (sort of) wind tunnel which confirm those carried out from the Tour Eiffel, and which must now be examined. Table XXXV. Shape of Area of Total Velocity. K=4- plate. surface. weight. V. m''. kg. m. Square . 0-022 5 0715 21-27 0-070 )> 0'022 5 0739 21-74 0-072 )) 0*045 ^ I "600 2272 0-070 )) 0-045 2-532 28-57 0-069 Circular 0-022 5 0-690 21-28 0-068 Triangular . 0'02I 2 0-688 21-74 o-o68 Cailletet and Colardeau experiments on plates falling from the Eiffel Tower. Barometer and thermometer readings not given. In La Resistance de I' Air et I' Aviation, 191 1, M. Eiffel has gone into this question of the "dimensional effect" at some length. We find, on p. 41, the diagram which I have re- produced in fig. 22. This diagram gives the results of all the experiments he has conducted with a view to decid- ing this question. The three smallest plates, 10 x 10 cm., 15x15 cm., and 25x25 cm., gave values of K = o-o65, 0-066, and 0-067 in the aerodynamical laboratory, which 1 have called a (sort of) wind tunnel. The remaining plotted results were obtained y>'^;;/ the Eiffel Toiver experiments just 1 84 Resistance of Air referred to. The latter I have remarked on already, so will only now consider the smaller ones. '—1 o crown 8vo, leaUier, gilt edges. Second ed. {New Yoyk, 191 i.) £1 4s. net. Spons' Architects' and Builders' Pocket Price-Book. Edited by Clyde Young. Forty-first ed., viii + 308 pp., green leather cloth. Published annually. (Size 6^ in. by 3|in. by |in. thick.) 2s.6d. net. RAILWAY ENGINEERING AND MANAGEMENT Practical Hints to Young Engineers Employed on Indian Railways. By A. W. C. Addis. 14 illus., 154 pp., i2mo. (1910.) 3s. 6d. net. Up-to-date Air Brake Catechism. By R. H. Blackall. Twenty-sixth edition, 5 coloured plates, 96 illus., 305 pp., crown 8vo. {New York, 1916.) 8s. 6d. net. Prevention of Railroad Accidents, or Safety in Railroading. By Geo. Bradshavv. 64 illus., 173 pp., square crown 8vo. {New York, 1912.) 2s. 6d. net. Simple and Automatic Vacuum Brakes. By C. Briggs, G.N.R. II plates, 8vo. {1S92.) 4s. net. Permanent -Way Material, Platelaying and Points and Crossings, with a few Remarks on Signalhng and Inter- locking. By W. H. Cole. Cr. 8vo, 288 pp., 44 illus., 7th ed. {1915.) 7s. 6^. net. Postage: inland, 3^.; abroad, 6d. Railway Engineers' Field Book. By Major G. R. Hearn, R.E., Assoc. Inst. Civil Engrs., and A. G. Watson, C.E. i2mo, leather, 230 pp., 33 illus. {1914.) 21s. net. Postage : inland, 3^. ; abroad, 6d. Locomotive Breakdowns, Emergencies and their Remedies. By Geo. L. Fowler, M.E., and W. W. Wood. Eighth ed., 92 illus., 266 pp., i2mo. {New York, 1916.) 5s. net. Permanent-way Diagrams. B}^ F. H. Frere. Mounted on hnen in cloth covers. {1908.) 3s. net. Formulae for Railway Crossings and Switches. By J. Glover. 9 illus., 28 pp., royal 32mo, 2s. 6d. net. Setting out of Tube Railways. By G. M. Halden. 9 plates, 46 illus., 68 pp., crown 4to. {1914.) los. 6d. net. 38 i.. & F. N. SPON, Ltd., 57, HAYMAUKET, LONDON, S.W. I Railway Engineering, Mechanical and Electrical. Bj' J. W. C. Haldane. New edition, 141 illus., xx + 583 pp., 8vo. {1908.) 15s. net. The Construction of the Modern Locomotive. By G. Hughes. 300 iilus., 261 pp., 8vo. {1894.) gs. net. Practical Hints for Light Railways at Home and Abroad. B}' F. R. Johnson. 6 plates, 31 pp., crown 8vo. {1896.) 2s. 6d. net. Handbook on Railway Stores Management. By W. O. Kempthorne. 268 pp., demy 8 vo. {1907.) ios.6d.net. Railway Stores Price Book. By W. O. Kempthorne. 490 pp., demy 8vo. {1909.) los. (bd. net. Railroad Location Surveys and Estimates. By F. Lavis. 68 illus, 270 pp., 8vo. {New York, 1906.) 12s. 6d. net. Pioneering. By F. Shelford. 14 illus., 88 pp., crown 8vo. {1909.) 3s. net. Handbook on Railway Surveying for Students and Junior Engineers. By B. Stewart. 55 illus., 98 pp., crown 8vo. {1914.) 2s. 6d. net. Modern British Locomotives. By A. T. Taylor. Second ed. 100 diagrams of principal dimensions, 118 pp., oblong 8vo. {1914.) 4s. 6d. net. Locomotive Slide Valve Setting. By C. E. TuUy. Illus- trated, i8mo. {1903.) IS. net. The Railway Goods Station. By F. W. West. 23 illus., XV + 192 pp., crown 8vo. {1912.) 4s. 6d. net. The Walschaert Locomotive Valve Gear. By W. W. Wood. 4 plates and set of movable cardboard working models of the valves, 193 pp., crown 8vo. Third ed. {New York, 1913.) ys. net. The Westinghouse E.T. Air-Brake Instruction Pocket Book. By W. W. Wood. 48 illus., including many coloured plates, 242 pp., crown 8vo, {New York, 1909.) 8s. 6d. net. SANITATION, PUBLIC HEALTH AND MUNICIPAL ENGINEERING Engineering Work in Public Buildings. By R. O. Allsop. jj illus., ix + 158 pp., demy 4to. {1912.) 12s. 6d. net. SCIENTIFIC BOOKS. 39 Public Abattoirs, their Planning, Design and Equipment. By R. S. Ayling. 33 plates, 100 pp., demy 4to. {1908.) 8s. 6d. net. Sewage Purification. By E, Bailey-Denton. 8 plates, 44 pp., 8vo. {1896.) 5s. nee. Water Supply and Sewerage of Country Mansions and Estates. By E. Bailey-Denton. 76 pp., crown 8vo. {1901.) 2s. 6d. net. Sewerage and Sewage Purification. By M. N. Baker. Second edition, 144 pp., i8mo, boards. {New York, 1905.) 2S. 6d. net. Housing and Town-Planning Conference, 1913. Being a Report of a Conference held by the Institution of Municipal and County Engineers at Great Yarmouth. Edited by T. Cole. 42 folding plates, 227 pp., 8vo. los. 6d. net. Housing and Town Planning Conference, 1911. Report of Conference held by the Institution of Municipal and County Engineers at West Bromwich. Edited by T. Cole, Secretary. 30 plates, 240 pp., 8vo. ids. 6d. net. Sanitary House Drainage, its Principles and Practice. By T.E.Coleman. 98 illus., 206 pp., crown 8 vo, {1896.) 3s. 6d. net. Stable Sanitation and Construction. By T. E. Coleman. 183 illus., 226 pp., crown 8vo. {1897.) 3s. net. Discharge of Pipes and Culverts. By P. M. Crosthwaite. Large folding sheet in case. 2S. 6d. net. A Complete and Practical Treatise on Plumbing and Sanitation. By G. B. Davis and F. Dye. 2 vols., 637 ihus. and 21 folding plates, 830 pp., 4to, cloth. {1899.) £1 10s. net. Standard Practical Plumbing. By P. J. Davies. Vol. I. Fourth edition, 768 illus., 355 pp., royal Svo. {1905.) ys. 6d. net. Vol. II. Second edition, 953 illus., 805 pp. {1905.) IDS. 6d. net. Vol, III. 313 illus., 204 pp. {1905.) $s. net. Conservancy, or Dry Sanitation versus Water Carriage. By J. Donkin. 7 plates, 33 pp., Svo, sewed. {1906.) is.net. Sev/age Disposal Works. By W. C. Easdale. 160 illus., 264 pp., Svo. {1910.) los. 6d. net. 40 E. & F. N. SPON, Ltd., 67, HAYMARKET, LONDON, S.W. 1 House Drainage and Sanitary Plumbing. By W. P. Gerliard. Eleventh ed., 6 illus., 231 pp., i8mo, boards. {New York, 1905.) 2s. 6d. net. Tlie Treatment of Septic Sewage. By G. W. Rafter. 137 pp., i8mo, boards. Third ed. (New York, 1913.) 2s. 6d. net. Reports and Investigations on Sewer Air and Sewer Ven- tilation. By R. H. Reeves. 8vo, sewed. {1894.) is. net. Sewage Drainage Systems. By Isaac Shone. 27 folding plates, 47 illus., 440 pp., 8vo. {1914.) 25s. net. Drainage and Drainage Ventilation Methods. By Isaac Shone, C.E. 7 folding plates, 36 pp., 8vo, leather. {1913.) 6s. net. Valuations. By S. Skrimshire, F.S.I. 2ou fully worked examples, 460 pp., 8vo. {1915). 10s. 6d. net. Postage: inland, 5^. ; abroad, 10^. The Law and Practice of Paving Private Street Works. By W. Spinks. Fourth edition, 256 pp., 8vo. {1904.) 12s. 6d. net. STRUCTURAL DESIGN {See Bridges and Roofs) TELEGRAPH CODES New Business Code. 320 pp., narrow 8vo. (Size 4f in. by 7I in, and | in. thick, and weight 10 oz.) {New York, 1909.) £i IS. net. Miners' and Smelters' Code (formerly issued as the Master Telegraph Code). 448 pp., 8vo, hmp leather, weight 14 oz. {New York.) £2 10s. net. General Telegraph Code. Compiled by the Business Code Co. 1,023 PP-. small 4to, with cut side index for ready refer- ence. {New York, 1912.) 63s. net. Postage: inland, 6d. ; abroad, is. 4^. Billionaire Phrase Code, containing over two million sen- tences coded in single words. 56 pp., 8vo, leather. {New York, 1908.) 6s. 6d. net. SCIENTIFIC BOOKS. 41 WARMING AND VENTILATION Hot Water Supply. By F. Dye. Fifth edition, new impres- sion, 48 illus., viii + 86 pp., 8vo. {1912.) 3s. net. A Practical Treatise upon Steam Heating. By F. Dye. 129 illus., 246 pp., 8vo. {1901.) los. net. Warming Buildings by Hot Water. ByF. W. Dye, Second Edition, revised, with 159 illus., xvi + 316 pp., 8vo. {1917.) IDS. net. Postage, Inland $d. ; Abroad, 8^. Charts for Low Pressure Steam Heating. By J. H. Kinealy. Small folio. {New York.) 5s. Formulae and Tables for Heating. By J. H. Kinealy. 18 illus., 53 pp., 8vo, {New York, 1899.) 3s. 6d. net. Centrifugal Fans. By J. H. Kinealy. 33 illus., 206 pp., fcap. 8vo, leather. {New York, 1905.) 12s. 6d. net. Mechanical Draft. By J. H. Kinealy. 27 original tables and 13 plates, 142 pp., crown 8vo. {New York, 1906.) gs. net. Theory and Practice of Centrifugal Ventilating Machines. By D. Murgue. 7 illus., 81 pp., 8vo. {1883.) 5s. net. Mechanics of Ventilation. By G. W. Rafter. Third ed., 143 pp., i8mo, boards, {New York, 1912.) 2s.6d.net. Principles of Heating. By W. G. Snow. New edition, 59 illus., xii 4- 224 pp., 8vo. {New York, 1912.) gs. net. Furnace Heating. By W. G. Snow. Fourth edition, 52 illus., 216 pp., 8vo, {Neio York, 1909.) 6s. 6d. net. Ventilation of Buildings. By W. G. Snow and T. Nolan. 83 pp., i8mo, boards. {New York, 1906.) 2s. 6d. net. Heating Engineers' Quantities. By W. L. White and G. M. White. 4 plates, 33 pp., folio. {1910.) 10s. 6d. net. WATER SUPPLY {See also Hydraulics) Potable Water and Methods of Testing Impurities. By M. N. Baker. 97 pp., i8mo, boards. Second ed. {New York, 1905.) 2S. 6d. net. Manual of Hydrology. By N. Beardmore. New impres- sion, 18 plates, 384 pp., 8vo. {1914.) 10s. 6d. net. Bacteriology of Surface Waters in the Tropics. By W. W. Clemesha. 12 tables, \aii + 161 pi\, 8vo. {Calcutta, 1912.) ys. 6d. net. 42 E. & F. N. SPON, Ltd., 67, HAYMARKET, LONDON, S.W. 1 Water Softening and Purification. By H. Collet. Second edition, 6 illus., 170 pp., crown 8vo. {1908.) 5s. net. Treatise on Water Supply, Drainage and Sanitary Appliances of Residences. By F. Colyer. 100 pp., crown 8vo. {1899.) IS. 6d. net. Purification of Public Water Supplies. By J. W. Hill, 314 pp., 8vo. {New York, 1898.) 10s. 6d. net. Well Boring for Water, Brine and Oil. By C. Isler. Second edition, 105 illus., 296 pp., 8vo. {1911.) 10s. 6d. net. Method of Measuring Liquids Flowing through Pipes by means of Meters of Small Calibre. By Prof. G. Lange. I plate, 16 pp., 8vo, sewed. {1907.) 6d. net. On Artificial Underground Water. By G. Richert. 16 illus., 33 pp., 8vo, sewed. {1900.) is. 6d. net. Notes on Water Supply in new Countries. By F. W. Stone. 18 plates, 42 pp., crown 8vo. (1888.) 5s. net. The Principles of Waterworks Engineering. By J. H. T. Tudsbery and A. W. Brightmore. Third edition, 13 folding plates, 130 illus., 447 pp., demy 8vo. {1905.) £1 is. net. WORKSHOP PRACTICE For Art Workers and Mechanics Alphabet of Screw Cutting. By L. Arnaudon; Fifth edition, 92 pp., cr. 8vo., sewed. {1913.) 4s. net. A Handbook for Apprenticed Machinists. By O. J. Beale. Third ed., 89 illus., 141 pp., i6mo. {New York, 1901.) 2s. 6d. net. Practice of Hand Turning. By F. Campin. Third edition, 99 illus., 307 pp., crown 8vo. {1883.) 3s. 6d. net. Artistic Leather Work. By E. Ellin Carter. 6 plates and 21 illus., xii 4- 51 pp., crown 8vo. {1912.) 2s. hd. net. Calculation of Change Wheels for Screw Cutting on Lathes. By D. de Vries. 46 illus., 83 pp., 8vo. {1914.) 3s. net. Milling Machines and Milling Practice. By D. de Vries. 536 illus., 464 pp., medium 8vo. {1910.) 14s. net. French -Polishers' Manual. By a French-Polisher. New impression, 31 pp., royal 32mo, sewed, {1912.) yd. net. SCIENTIFIC BOOKS. 43 Galvanizing and Tinning. By W. T. Flanders. 134 Illus., 350 pp., 8vo. {New York, 1010.) 14s. net. Postage : in- land, 6d. ; abroad, is. Art of Copper-Smithing. By J. Fuller. Fourth edition, 483 illus., 319 pp., royal 8vo. {New York, 1911.) 14s. net. Saw Filing and Management of Saws. By R. Grimshaw. Third ed., 81 illus., i6mo. {Neze; York, 1912.) 5s. net. Cycle Building and Repairing. By P. Henry. 55 illus., 96 pp., or. 8vo. (S. & C. Series, No. .43.) is. 6d. net. Turner's and Fitter's Pocket Book. ByJ.LaNicca. i8mo, sewed, yd. net. Tables for Engineers and Mechanics, giving the values of the different trains of wheels required to produce Screws of any pitch. By Lord Lindsay. Second edition, royal 8vo, oblong. 2S. net. Screw-cutting Tables. By W. A. Martin. Seventh edition. New imp., oblong 8vo. is. net. Metal Plate Work, its Patterns and their Geometry, for the use of Tin, Iron and Zinc Plate Workers. By C. T. Millis. Fourth Ed., New imp., 280 illus., xvi + 456 pp., cr. 8vo. {1912.) gs. net. The Practical Handbook of Smithing and Forging. Engin- eers' and General Smiths' Work. By T. Moore. New impression, 401 illus., 248 pp., crown 8vo. {1912.) 5s. net. Modern Machine Shop Construction, equipment and man- agement. By O. E. Perrigo. 208 illus., 343 pp., crown 4to. {New York, 1906.) £1 is. net. Turner's Handbook on Screw-cutting, Coning, etc. By W. Price. New impression, 56 pp., fcap. 8vo. {1912.) yd. net. Introduction to Eccentric Spiral Turning. B3' H. C. Robinson. 12 plates, 23 illus., 48 pp., Svo. {1906.) 4s. 6d. net Forging, Stamping, and General Smithing. By B. Saun- ders. 728 illus., ix -f 428 pp., demy 8vo. {1912.) £j is. net. Pocket Book on Boilermaking, Shipbuilding, and the Steel and Iron Trades in General. By M. J. Sexton. Sixth edition, new impression, 85 illus., 319 p])., royal 32mo. roan, gilt edges. {1912.) 6s. net. 44 E. & F. N. SPON, Ltd., 57, HAYMARKET, LONDON, S.W. 1 Power and its Transmission. A Practical Handbook for the Factory and Works Manager. By T. A. Smith. 76 pp., fcap. 8vo. {1910.) 2s. net. Spons' Mechanics' Own Book : A Manual for Handicrafts- men and Amateurs. Seventh edition, 1,430 illus., 720 pp., demy 8vo. {1916.) 6s. net. Half bound in leather, marbled edges, ys. 6d. net. Spons' Workshop Receipts for Manufacturers, Mechanics and Scientific Amateurs. New and thoroughly revised edition, crown 8vo. {1917.) 3s. 6(1. each net. Vol. I. Acetylene Lighting to Drying. 223 illus., 532 pp. Vol. n. Dyeing to Japanning. 259 illus., 540 pp. Vol, HI. Jointing Pipes to Pumps. 257 illus., 528 pp. Vol. IV. Rainwater Separators to Wire Ropes. 321 illus., 540 pp. Wire and Sheet Gauge Tables. By Thomas Stobbs, Sheffield. A Metal Calculator and Ready Reckoner for Merchants, and for Office and Shop use, in Sheet, Plate, and Rod Mills, and Forges. Svo, 95 pp. (1916.) 3s. 6d. net. Postage, 4d. Gauges at a Glance. By T. Taylor. Second edition post, Svo, oblong, with tape converter. {1917.) 5s. net. Simple Soldering, both Hard and Soft. By E. Thatcher. 52 illus., 76 pp., crown Svo. (S. & C. Series, No. 18.) {New York, 1914.) is. 6d. net. The Modern Machinist. By J. T. Usher. Fifth edition, 257 illus., 322 pp., Svo. {New York, 1904.) los. 6^. net. Practical Wood Carving. By C. J. Woodsend. 108 illus., 86 pp., Svo. Second ed. {New York, 1908.) 4s. 6d. net. American Tool Making and Interchangeable Manufacturing. By J. W. Woodworth. Second Ed. 600 illus., 535 pp., Svo. {New York, 1911.) 20s. net. USEFUL TABLES See also Curve Tables, Earthwork, Foreign Exchange, Interest Tables, Logarithms, and Metric Tables. Weights and Measurements of Sheet Lead. By J. Alex- ander. 32mo, roan. is. 6d. net. Barlow's Tables of Squares, Cubes, Square Roots, Cube Roots and Reciprocals, of all Integer Numbers from i to 10,000. 200 pp., crown Svo, leather cloth. 4s. net. SCIENTIFIC BOOKS. 45 Tables of Squares. Of every foot, inch and ^ of an inch from ^ of an inch to 50 feet. By E. E. Buchanan. Eleventh edition, 102 pp., i6mo. 4s. 6d. net. Land Area Tables. By W. Codd. For use with Amsler's Planimeter. On sheet in envelope with explanatory pamphlet, is. 6d. net. Or separately, tables on sheet is. net. Pamphlet, 6d. net. Calculating Scale. A Substitute for the SUde Rule. By W. Knowles. Crown 8vo, leather, is. net. Planimeter Areas. MultipHers for various scales. By H. B. Molesworth. Folding sheet in cloth case. is. net. Tables of Seamless Copper Tubes. By I. O 'Toole. 69 pp., oblong fcap. 8vo. 3s. 6d. net. Steel Bar and Plate Tables. Giving Weight per Lineal Foot of all sizes of L and T Bars, Flat Bars, Plates, Square, and Round Bars. By E. Read. On large folding card, is.net. Rownson's Iron Merchants' Tables and Memoranda. Weight and Measures. 86 pp., 32mo, leather. 3s. 6d. net. Spons' Tables and Memoranda for Engineers. By J. T. Hurst, C.E. Twelfth edition, 278 pp., 64mo, roan, gilt edges. {1916.) is. net. Ditto ditto in celluloid case, is. 6d. net. Wire and Sheet Gauge Tables. By T. Stobbs xx + 96 pp., cr. 8vo, leather cloth. [1916.) 3s. 6^. net. Postage, 4^^. Optical Tables and Data, for the use of Opticians. By Prof. S. P. Thompson. Second edition, 130 pp., oblong 8vo. {1907.) 6s. net. Traverse Table, showing Latitudes and Departure for each Quarter degree of the Quadrant, and for distances from i to 100, etc. i8mo, boards. 2s. 6d. net. The Wide Range Dividend and Interest Calculator, showing at a glance the percentage on any sum from ;^i to ;^io,ooo, at any Interest from 1% to 12^%, proceeding by J% ; also Table of Income Tax deductions on any sum from £1 to £10,000, at ()d., IS., and is. 2d. in the £. By Alfred Stevens. 100 pp., super royal 8vo. 6s. net. Quarter Morocco, cloth sides, ys. 6d. net. The Wide Range Income Tax Calculator, showing at a glance the tax on any sum from One Shilling to Ten Thou- sand Pounds at the Rates of 9^., is., and is. 2d. in the £. By Alfred Stevens. 8 pp., printed on stiff card, royal Svo. IS. net. 40 E. & F. N. SPON, Ltd., 57, HAYMARKET, LONDON, S.W. 1 Fifty-four Hours' Wages Calculator. By H. N. Whitelaw. Second edition, 79 pp., 8vo. 2s. 6d. net. Wheel Gearing. Tables of Pitch Line Diameters, etc. By A. Wildgoose and A. J. Orr. 175 pp., fcap. 32mo. 2S, net. MISCELLANEOUS Popular Engineering. By F. Dye. 704 illus., 477 pp., crown 4to. (1895.) 5s. net. The Phonograph, and how to construct it. By W. Gillett. 6 folding plates, 87 pp., crown 8vo. {1892.) 5s. net. Engineering Law. By A. Haring. Demy 8vo, cloth. (New York.) Vol. L The Law of Contract. 518 pp. {1911.) i8s. net. Particulars of Dry Docks, Wet Docks, Wharves, etc., on the River Thames. By G. H. Jordan. Second edition, 7 coloured charts, 103 pp., oblong 8vo. {1904.) is. net. New Theories in Astronomy. By W. Stirling. 335 pp. demy 8vo. {1906.) 8s. 6d. net. Inventions, How to Protect, Sell and Buy Them. By F. Wright. 118 pp., crown 8vo. (S. & C. Series, No. 10.) Second edition. {New York, 1911.) is. 6d. net. SPECIAL LISTS ISSUED BY E. & F. N. SPON. LTD. Post free to ant; part of the World on application. BUILDERS, ARCHITECTS and CONTRACTORS. CIVIL ENGINEERING. ELECTRICAL ENGINEERING. INDUSTRIES, TECHNOLOGY, MANUFAC- TURES and MINING. MECHANICAL ENGINEERING. S. & C. SERIES. TABLES FOR ENGINEERS, ACTUARIES AND ACCOUNTANTS. PAINTING AND DECORATING. METRIC POCKET TABLE BOOKS. SCIENTIFIC B00K6. 47 PRICE 6/- NET POST FREE 6/5 and Home Office SPON'S ELECTRICAL POCKET-BOOK A REFERENCE BOOK OF GENERAL ELECTRICAL INFORMATION, FORMULA AND TABLES FOR PRACTICAL ENGINEERS By WALTER H. MOLESWORTH, m.i.e.E.. m.i.m.e. fVitf) 325 Illustrations. 500 pp. Fcap. 8Vo, Cloth. SYNOPSIS OF CONTENTS Preface — Weights and Measures — British, metric, and of other countries Tables of British and metric .... Mensuration, areas and circumferences of circles, etc. . Logarithms, trigonometry, and temperature tables Moneys of British and other countries Conversion factors, graphic and international symbols. Units, laws, and terms ...... Heating, fuse wires, insulation, and resistance Wiring, wire gauges, and aluminium conductors. Copper conductors ....... Weight and calculated size of conductors (all systems) Regulations — " B.O.T." Overhead Lines and PubUc Supply Transmission — Systems and cost, and line calculations Transmission Line — Poles, insulators, etc. . Testing- Apparatus and sets, measurements, etc. Location of faults, etc. .... Magnetism and electro-magnetic circuits . Continuous Current — Circuits and machinery British Standard Rules for Electric Machinery Calculation of corona and corona losses, condensers, capacity coefficient of self-induction ..... Alternating-current Circuits — Impedance, reactance, capacity reactance, inductance, pov;er factor, synchronous machinery and power factor, frequency, skin effect, E.M.F. and current, harmonic waves, resonance, himtiug, surging, etc. ............. Alternating-current generators and motors ........ Alternating-current transformers and converting macliines ..... Generating costs, etc., lifts and other plant ....... Primary batteries and stationary accumulators ....... Accumulators for vehicles, battery vehicles and boosters ..... Electric furnaces and electro-chemistry ........ Illumination of streets, factories, houses, etc. ....... Incandescent and arc lamps .......... Traction — Regulations, " B.O.T." re Tramways and Trackless ..... „ MetropoUtan Motor Carriages ....... Speed conversion and gradient tables ........ Adhesion, acceleration, energy of rotation, tractive force and H.P., kinetic energy, time, speed, and energy, etc., calculations ...... Car services and stock required for services ...... Tramcars — Controllers, brakes, capacity and size, calculated size of motors, con- sumption, etc. ........... Tramways — Permanent way, poles, rails, bonding of rails, voltage drop in rails, over- head equipment, cost of systems, etc. ....... Trackless and other road motor vehicles ....... Railways — Conditions governing size of locomotives and rolling stock .... Tractive force, train resistance, and acceleration ...... Notes on practice, locomotive tj^pes and systems described ... Sub-stations and catenary construction, with costs ..... Cost of electrification and operation of lines ....... List of regulations, etc., engineering standards, and where obtainable . I.E.E. Wiring Rules, 1916 Index , c o . . . ....... charging current, and PAGLS Vll I to I I 12 .. 25 26 .. 32 33 ,. 41 42 ,. 4" 48 „ 63 b4 „ 73 74 „ 86 87 .. 94 <).=! „ 107 108 „ 116 118 , 137 138 . 143 144 .. 155 1.56 , 164 I<5.5 . '75 i7t> , 188 189 . 204 205 , 2n 212 , 221 222 . 231 232 , 241 242 . 257 258 , 264 26,5 , , 271 272 , , 280 281 , 289 290 . 299 300 , 306 307 , . 317 318 ■ 325 320 ■ 333 ^^4 , 345 346 , 351 352 , 365 ^66 , 378 379 . 381 382 . 389 390 . 403 404 . 421 422 . 430 43' . 435 4.3b . 438 439 . 465 472 . 492 48 E. & F. N. SPON, Ltd., 57, HAYMARKET, LONDON, S.W. 1 The Journal of the Iron and Steel Institute. Edited by G. C. Lloyd, Secretary. Published Half-yearly, 8vo. i6s. net. Carnegie Scholarship Memoirs. Published Annually, 8vo. The Journal of the Institution of Electrical Engineers. Edited by P. F. Rowell, Secretary. Issued in quarto parts. The number of parts are from 12 to i6 annually. Annual Subscription, 46s., payable in advance. Single numbers, 3s. 9^. post free. The Proceedings of the Institution of Municipal and County Engineers. Edited by Thomas Cole, Assoc.M.Inst.C.E., Secretary. Issued in monthly parts (fortnightly during April, May, June and July). Price is. 9^., post free, each part. Transactions of the Institution of Gas Engineers. Edited by Walter T. Dunn, Secretary. Published Annually, 8vo. los, 6d. net. Proceedings of the International Association for Testing Materials. Transactions of the American Institute of Chemical Engineers. Published Annually, 8vo. 30s. net. Transactions of the Paint and Varnish Society. Annual Subscription, Inland, 5s. /\d. ; Abroad, 5s. M., post free. Bound Volumes, each, 5s. net. Issued Annually. Bailee & TsEser Frome tsd LoetJcn liniSlllin^l REGIONAL LIBRARY FACILITY iiiillilllllllllllilllli^ A A 001 414 221 SCIENCE AND ENGINEERING LIBRARY University of California, San Diego DATE DUE IMN f^ n 1973 SE 16 -•-■'■ T ' '. -, i nil 2:^ 1Q7