MOTION OF LIQUIDS 
 
 R, DE VILLAMIL 
 
GIFT OF 
 MICHAEL REESE 
 
MOTION OF LIQUIDS 
 
BY THE SAME AUTHOR. 
 
 A.B.C. of HYDRODYNAMICS. 
 
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MOTION OF LIQUIDS 
 
 By 
 LIEUT.-COL. R. DE VILLAMIL 
 
 
 
 R. Eng. (Ret.) 
 
 " When a man applies himself and braces his faculties to an investigation of anything, 
 he first asks and ascertains what has been said about the subject by others ; then adds 
 his own meditation, and with much mental turmoil appeals to his own spirit and invokes 
 it to open its oracles to him." 
 
 BACON, Novum Organum. 
 
 EIGHTY-SIX ILLUSTRATIONS AND 
 THIRTY TABLES 
 
 Xonfcon 
 E. & F. N. SPON, LIMITED, 57 HAYMARKET 
 
 |?orfc 
 
 SPON & CHAMBERLAIN, 123 LIBERTY STREET 
 1914 
 
of. 
 
 TO THE 
 
 OF 
 
 LE CHEVALIER 
 COLONEL DUBUAT, 
 
 CORPS ROYAL DU GENIE, 
 
 WHOSE 
 
 PRINCIPES D'HYDRAULIQUE 
 
 is 
 
 UNFORTUNATELY MUCH NEGLECTED ; 
 ALSO 
 
 COLONEL DUCHEMIN, 
 
 ARTILLERIE ROYALE, 
 
 AUTHOR OF 
 
 THE REMARKABLE BOOK 
 LES LOIS DE LA RESISTANCE DES FLUIDES, 
 
 is >'">-' 
 
 ALMOST UNKNOWN IN ENGLAND, 
 
 : , -THli LITTLE; -BO^lt A 
 
 is 
 RESPECTFULLY DEDICATED. 
 
 3G0736 
 
LIST OF PLATES 
 
 PLATE I. FIG. 44 . .To face 
 
 II. 46 
 
 HI. 54 .-> i 
 
 IV. 68 . . . , 
 
 V. 84,85,86 . ;.' , 
 
 ' r> ** * } v *' *^ * m 
 
 * \ / r * %* * J ^ ** 
 
 116 
 
 118 
 
 133 
 171 
 199 
 
CONTENTS 
 
 CHAP. PAGE 
 
 DEDICATION . . . . ... v 
 
 PREFACE . . . . . . , ix 
 
 I INTRODUCTORY REMARKS MOMENTUM 
 
 ENERGY % . . . . ' . I 
 
 II THE "DIVIDE" . .. . 9 
 
 III MOTION OF LIQUID ROUND THE PLATE FLOW 
 OF LIQUID FILAMENTS IN FRONT OF THE 
 PLATE ... . . .. . . 23 
 
 IV SUBJECT CONTINUED, AND EXAMINED BY 
 REFERENCE TO EXPERIMENTS MADE PRE- 
 VIOUSLY TO DUCHEMIN THE " STATIC 
 LIQUID" . . . . ,.< . 35 
 
 V RELATIVE MOTION MOTION OF STREAM FILA- 
 MENTS IN FRONT OF A BODY AT REST, 
 
 EXPOSED TO A FLOWING STREAM " DU- 
 
 BUAT'S PARADOX" . . . . 51 
 VI DUBUAT'S PARADOX, continued . . . 63 
 
 VII MOTION OF THE LIQUID AT THE SIDE OF THE 
 
 PLATE : ALSO BEHIND THE PLATE . . 76 
 
 VIII WATER FLOWING IN JETS IMPACT . . 86 
 
 vii 
 
viii Motion of Liquids 
 
 CHAP. p AGK 
 
 IX JETS STRIKING A PLATE AT AN ANGLE DUCHE- 
 MIN'S FORMULA DORHANDT AND THIESEN's 
 
 FORMULA JOESSEL'S FORMULA M. DE 
 LOUVRIE'S FORMULA M. GOUPIL'S FORMULA 
 KIRCHHOFF - RAYLEIGH FORMULA 
 COLONEL RENARD'S FORMULA VON LOSSL'S 
 
 FORMULA . . . . . IOO 
 
 X EXPLANATION OF " DUBUAT'S PARADOX " 
 THE ADDED MASS OSCILLATORY PRES- 
 SURE . . ... 110 
 
 XI MOVEMENT OF LIQUIDS THROUGH APERTURES 
 IN THE WALL OF A VESSEL MOUTH- 
 PIECES . . . . . . 123 
 
 XII SUBJECT CONTINUED EXTERNAL MOUTHPIECES 
 THE SEVILLE PUMP THE BELLANGE 
 PUMP . . . . . . . 139 
 
 XIII RIVERS AND CANALS A BODY FLOATING IN 
 A STREAM MOVES FASTER THAN THE STREAM 
 " CORRAISON " OF STREAM^ . . 154 
 
 XIV NEGATIVE RESISTANCE IN LIQUIDS . . 177 
 
 XV CURVES OF RESISTANCE EXPERIMENTAL CON- 
 FIRMATION OF THEORY . . . IQ2' 
 
 INDEX . .. ." . . . . 207 
 
PREFACE 
 
 IN the A. B.C. of Hydrodynamics * I have dealt chiefly 
 with the fundamentals of the subject : especially examining 
 the assumptions on which the mathematical treatment was 
 based. In the present little work I have developed the 
 subject by practical application to definite cases of resis- 
 tance ; especially examining and studying the experi- 
 mental work carried out by Dubuat and Duchemin, to 
 whose memory I have dedicated this little book. 
 
 It is unfortunate that Dubuat 's Principes d'Hydraulique 
 is at present so exceedingly unpopular ; I am afraid that 
 this is largely due to the author having found out experi- 
 mentally, that the resistance experienced by a body moving 
 in water, was less than that which it exerted when it was 
 at rest in a flowing stream. This has caused his experi- 
 ments to be considered as " unreliable/' 
 
 Since the foregoing may tend to prejudice the reader 
 against this distinguished man's work, I will quote what 
 Sir George Stokes said about him. In his exceedingly 
 interesting papers On the Motion of Pendulums we read : 
 " I come now to the experiments of Dubuat, which are 
 contained in an excellent work of his entitled Principes 
 d'Hydraulique .... Dubuat justly remarks that the time 
 of oscillation of a pendulum oscillating in a fluid is greater 
 than it would be in a vacuum, not only on account of the buoy- 
 ancy of the fluid, which diminishes the moving force, but also 
 on account of the mass of the fluid, which must be regarded 
 as accompanying the pendulum in its motion : and even 
 determined experimentally the mass of fluid which must be 
 regarded as carried by the oscillating body in the case of spheres 
 and of several other solids. Thus Dubuat anticipated by 
 
 1 Spon, London, 1912. 
 ix 
 
X Motion of Liquids 
 
 about forty years the discovery of Bessel ; but it was not until 
 after the appearance of Bessel' s memoir that Dubuat's labours 
 relating to the same subject attracted attention." (Math, 
 and Phys. Papers Italics added.) 
 
 Also : " Dubuat's labours on this subject had been alto- 
 gether overlooked by those who were engaged in pendulum 
 experiments ; probably because such persons were not 
 likely to seek in a treatise on Hydraulics for information con- 
 nected with the subject of their researches. Dubuat had, in 
 fact, rather applied the pendulum to Hydrodynamics than 
 Hydrodynamics to the pendulum" (Italics added.) 
 
 Dr. Thomas Young, in his Summary of the most useful 
 parts of Hydraulics, says " another and a more practical 
 mode of attaining hydraulic knowledge has been attempted 
 by a distinct class of investigators, at the head of whom 
 stands Chevalier Dubuat." 
 
 Also in his article on Hydraulics in Napier's supplement 
 to the Encyclopedia Britannica, he adds : " It has happened, 
 from a combination of accidental circumstances, that 
 Dubuat has been deprived of a considerable portion of his 
 first merits, in favour of Mr. Eytelwein, without any kind 
 of voluntary plagiarism on the part of that very respectable 
 professor." 
 
 Dubuat's experiments are commonly supposed to violate 
 the " principle of relativity." Such is only apparently 
 the case : we must remember, however, that " If the princi- 
 ple of relativity does not hold good in any domain of actual 
 life, we must seek the cause in the material used, and not in 
 the principle of relativity." (Emphasis added.) 1 
 
 The great French Hydraulic engineer and experimenter 
 Bazin, whose work Recherches Hydrauliques is quite a 
 classic, refers to Dubuat as " cet habile experimenteur " : 
 to his work as " les experiences nombreuses et soignees de 
 Dubuat " : and he makes the following query, in reference 
 to some question, " comment cette action a-t-elle put echap- 
 per a I' esprit sagace et observateur de Dubuat ? " This is 
 high praise from one who was himself an extraordinarily 
 skilful and careful experimenter. 
 
 1 Paul Carus, The Principle of Relativity. 
 
Preface xi 
 
 It must be frankly admitted that the arrangement of 
 Dubuat's book laisse a desirer ; and I have endeavoured 
 to classify and arrange his experimental results, so as to 
 bring out the salient points. Whether his work is " re- 
 liable," or not, will, I hope, be shortly decided, since Pro- 
 fessor Riabouchinsky has authorised me to state that he 
 will take an early opportunity of repeating both Dubuat's 
 and Duchemin's experiments. The Koutchino hall mark 
 will enable us to judge what is gold from what is dross ; 
 for no one will question the accuracy of Riabouchinsky ! 
 I attach enormous importance to this, as it will open a new 
 and vast field of thought as regards the resistance of bodies 
 immersed in liquids. 
 
 Duchemin's book may be said to be quite unknown in 
 England, no copy is to be found in even the British Museum ; 
 it is, further, not an easy book to follow, though it well 
 repays careful study. 
 
 In America, Professor Langley refers to Duchemin, 
 " whose valuable memoir published by the French War 
 Department, Memorial de I'Artillerie No. V , I regret not 
 knowing earlier." Professor Zahm also speaks of it as, 
 " that remarkable book." 
 
 Duchemin repeated Dubuat's experiments (50 years 
 later than Dubuat) and obtained almost exactly the same 
 results. It may be also claimed for him that he was the 
 first to point out that when expressing the resistance of a 
 body moving in air, or other elastic fluid, in terms of the 
 velocity, it was necessary to add a third term, involving 
 the third power of the velocity ; or, algebraically 
 
 R=AV+BV 2 +CV 3 
 
 Where A, B and C are constants. 
 
 One very simple test of the reliance we can place on the 
 work of these authors, is the following. It is pretty certain 
 that neither of them knew anything about the doctrine of 
 the " conservation of energy "since it was only formulated 
 later. If we then examine the experiments from this point 
 of view, and if we find that the energy is properly accounted 
 for, it is prima facie evidence that the results may be accepted. 
 
xii Motion of Liquids 
 
 In the one case where there is an apparent excess of energy, 
 they neither of them offer any explanation of why it should 
 be so ; they satisfy themselves by saying we found it so. 
 
 In presenting the subject of the motion of liquids to the 
 reader, I have endeavoured to follow Locke's advice, in his 
 Conduct of the Understanding : "in learning anything, as 
 little should be proposed to the mind at once as is possible ; 
 and that being understood and fully mastered, to proceed 
 to the next adjoining part yet unknown, simple, unperplexed 
 proposition belonging to the matter in hand, and tending 
 to the clearing what is principally designed." I hope that, 
 by this means, the book will be well within the compre- 
 hension of any well educated youth of sixteen or seventeen. 
 
 I cannot refrain from making another quotation from 
 the same author. " The sure and only way to get true 
 knowledge, is to form in our minds clear settled notions of 
 things with names attached to those determined ideas. These we 
 are to consider, and with their several relations and habitudes, 
 and not amuse ourselves with floating names and words of 
 undetermined signification, which we can use in several senses 
 to serve a turn." (Italics added.) 
 
 Still another quotation, as a caution to the reader : 
 " Reading furnishes the mind only with materials of know- 
 ledge ; it is thinking makes what we read ours. We are of 
 the ruminating kind, and it is not enough to cram ourselves 
 with a great load of collections ; unless we chew them over 
 again, they will not give us strength and nourishment. ' ' (Italics 
 added.) 
 
 Starting from the assumption that when a body moves 
 in a liquid, the latter moves by some means or another, from 
 the front to the rear of the body which is a matter of 
 the very commonest observation I have endeavoured to 
 show (step by step) how the liquid moves, by reference to 
 suitable experiments. Further, by confining my attention 
 to flat plates chiefly, I have been able to neglect the resis- 
 tance caused by viscosity, and so to treat the fluid as if it 
 were in viscid. 
 
 In studying the works of distinguished authors I have 
 found what I can only call, " gaps," which required to be 
 
Preface xiii 
 
 bridged ; and I have endeavoured to supply the deficiency. 
 For example, I know of no author who treats of the difference 
 between a static liquid and one which is non-static : I even 
 very much doubt whether most authors have ever thought 
 much about this matter at all. The subject being of the 
 highest importance, I have devoted considerable space to 
 the consideration of this question, which, when properly 
 grasped, will show that " Dubuat's Paradox " is not so 
 absurd as some people seem to think. 
 
 Relative motion. It is generally assumed as self-evident, 
 that it is immaterial whether the body or the liquid moves, 
 provided that the relative motion is the same. Once the reader 
 will think, whether the liquid is static or not, he will at 
 once see whether the conditions are the same in the two cases. 
 
 An exactly parallel case may be quoted as regards solids, 
 for " It stands to reason that bodies in translation (in which 
 the entire system, as a whole, moves in the same direction 
 with the same velocity and without any internal change even 
 of its smallest particles] behave as if they were at rest, and so 
 the motion of a straight line cannot be observed so long as the 
 observer remains limited to his own system." 1 
 
 In treating of the passage of a liquid through a hole in a 
 thin partition I have dwelt a good deal on the great im- 
 portance of the " co-efficient of contraction ; " a subject 
 rather neglected in textbooks, and the importance of 
 which was not recognized, in some cases, by even the great 
 Bernoulli. 
 
 In the chapter on rivers and canals, I refer to many 
 curious points, which do not appear to be as well known as 
 they should be ; amongst others the frequently disputed 
 one of a body floating in a stream moving faster than the 
 stream. 
 
 Finally, I introduce a subject which is, as far as I am 
 aware, new ; and which I call " negative resistance." I 
 show that (much abused) " viscosity " is sometimes an 
 advantage, and actually causes a decrease in the resistance 
 of a body moving in a liquid : this being caused by the 
 
 1 Paul Cams, The Principle of Relativity. 
 
xiv Motion of Liquids 
 
 viscosity recuperating some of the kinetic energy, which would 
 otherwise be wasted. I do not remember having seen in 
 any textbook the idea of measuring resistance by the amount 
 of energy wasted : it is, however, a very useful way of examin- 
 ing the question. 
 
 I have borrowed freely from many distinguished authors, 
 and I trust that in all cases I have suitably acknowledged 
 the source of my information. 
 
 Understanding that it was more agreeable to the reader, 
 I have given a translation only of the quotations from foreign 
 authors in the text ; whilst I have relegated their actual 
 words to foot-notes. 
 
 Finally, I have endeavoured not to be dogmatic. I do 
 not ask the reader however young to accept anything 
 that I may have written without examination : rather would 
 I say, with Thomas Davis, " accept no opinion, or set of 
 opinions, without examination, no matter whether they be 
 enrobed in pomp, or holiness, or power ; admire the pomp, 
 respect the power, venerate the holiness ; but for the opin- 
 ions, strip them ; if they bear the image of truth, for its 
 sake, cherish them ; if they be mixed, discriminate them ; if 
 false, condemn them." I might go further and (with Lessing) 
 say to the readers, " Think wrongly if you will, but think 
 for yourselves " since " the scientific spirit is of more value 
 than its products, and irrationally held truths may be more 
 harmful than reasoned errors " (Huxley). 
 
 Now only remains the pleasing duty of again offering my 
 most sincere thanks to Mr. Lewis R. Shorter, B.Sc., for his 
 very keen criticism and most meticulous care in examining 
 disputed questions : not that I would wish in any way to 
 saddle him with the responsibility for certain points which 
 may be considered heretical. Where errors there are (and 
 I have done my best to reduce them to a minimum) I 
 accept full paternity. My best thanks are also due to Mr. 
 C. Spon for reading the proofs and also for making the index 
 which enhances the value of the book. 
 
 R, DE VlLLAMiL. 
 June, 1914. 
 
MOTION OF LIQUIDS 
 
 CHAPTER I 
 
 INTRODUCTORY REMARKS MOMENTUM ENERGY 
 
 IF a circular disc move in a liquid with its surface normal 
 to the direction of its motion, it is clear that the liquid has 
 to be displaced, so as to allow the disc to pass through it : 
 the liquid moves in some manner or another from the 
 front to the back of the disc. The resistance which a body 
 meets to its motion depends chiefly on the particular manner 
 in which the liquid is transferred from the front to the rear 
 of the body. 
 
 I have treated of the fundamentals of liquid resistance in 
 the ABC of Hydrodynamics, to which the reader is referred : 
 nevertheless, to make this book self-contained, it may be 
 advisable to enter into a few details, even if I have to repeat 
 what I may have said elsewhere. 
 
 Newton first pointed out that the resistance to bodies 
 moving in fluids might be divided into three parts. His 
 exact words are : ' The resistance of sphcerical bodies in 
 fluids arises partly from the tenacity, partly from the attri- 
 tion, and partly from the density of the medium. And that 
 part of the resistance which arises from the density of the 
 fluid is, as I said, in a duplicate ratio of the velocity ; the 
 other part, which arises from the tenacity of the fluid, is uni- 
 form, or as the moment of the time " (Principia). Put in 
 other words, the resistance arises from (i) the density, or 
 inertia, of the liquid, (2) from the " attrition/' or rubbing 
 between the liquid and the solid, and (3) that due to the 
 viscosity (Newton sometimes calls it the want of 
 " lubricity ") of the liquid. 
 
. . . t ^Motion af Liquids 
 
 This definition is quite complete, and Sir George Stokes 
 has treated the question mathematically and very thor- 
 oughly so thoroughly, that I fancy there is very little left 
 for any other writer to point out. 
 
 In dealing with water, we know that there is no motion 
 of the liquid against the solid, causing attrition always pre- 
 suming that the solid is properly " wetted " : for when the 
 body is wetted a layer of liquid remains permanently at- 
 tached to it. The result is that Newton's second term 
 becomes zero. The resistance of water is, therefore, com- 
 posed of two terms only, viz. : (i) that caused by the den- 
 sity which, it is unanimously agreed, varies as the square 
 of the velocity and (2) that due to the viscosity, or treacli- 
 ness, and which varies as the first power of the velocity only. 
 This last term is in accordance with Newton, who says, 
 " The resistance, arising from the want of lubricity in the 
 parts of the fluid, is, cceteris paribus, proportional to the 
 velocity with which the parts of the fluid are separated from 
 each other " (Principia}. It is clear, therefore, that this 
 can be expressed in algebraical shorthand, as, 
 
 R = AV + BV 2 
 where A and B are constants. 
 
 Since the first term depends on the coefficient of viscosity 
 and the second term on the density, we may put further 
 detail into our formula and express it as, 
 
 ju, and Q being coefficients of viscosity and of density, A and 
 B being also constants. 
 
 We may even enter into more detail and express this in 
 " Dynamical similarity " formula, for bodies of similar 
 shape, as 
 
 where / is the length of (any) one dimension. 1 It will be 
 clear from the above that the resistance experienced by a 
 
 1 In case this should be puzzling to the young reader, it may 
 be advisable to point out that this formula only applies to similarly 
 shaped bodies. For example, when comparing spheres, cubes, or 
 any other similar bodies, the cubic capacity varies as I 3 (where / 
 is any dimension), the surfaces as I 2 , and the lengths as I. 
 
Introductory Remarks Momentum Energy 3 
 
 body moving in an incompressible liquid, such as water, will 
 increase in a slower ratio than as the square of the velocities. 
 Equally, the resistance of circular discs, say, will increase at 
 a less ratio than as the areas. 1 
 
 If the fluid be compressible, then a third term must be 
 added to the formula which will then become 
 
 R =A[*(IV) +B e (JV) 2 +C(/V) 3 . 
 
 As Newton 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." 
 
 In this case, at moderately high velocities (C being a very 
 small number), the resistance in a compressible fluid, like 
 air, will increase in a higher ratio than as the squares of the 
 velocities : also the resistance of circular discs will increase 
 at a higher rate than as their areas. This is well known, and 
 the last statement has been shown, experimentally, to be 
 true both by Dines and Eiffel. 
 
 The reader may observe that I have made no mention of 
 " liquid friction." This is quite true : but then I do not 
 know what is meant by this expression. Further than that, 
 I have never come across anybody who was able to explain 
 to me what he meant by liquid friction or how it acted. It 
 appears to be some sort of resistance which increases as some 
 sort of power of the velocity, which may be anything between 
 17 and 2-0 ; an empirical formula for this is given as 
 
 R=aV". 
 
 This, so called, " empirical law " does not agree very closely 
 with experiment, and it can hardly be said to explain any- 
 thing. Besides, an empirical law is rather a contradiction 
 in terms : for law implies necessity, and we cannot feel that 
 a thing must be unless we know why it must be. Most 
 unfortunately a value of n can be found which will give (for 
 
 1 Sir George Stokes (Mathematical and Physical Papers, vol. I.), 
 referring to this question says that, according to the common 
 theory, " similar solids of the same material will descend [in a liquid] 
 with equal velocities. These results are utterly opposed to the com- 
 monest observation, which shows that large solids descend much 
 more rapidly than small ones of the same shape and material." Here, 
 since the accelerating force is the same, it is clear that the resistance 
 to the larger bodies must be less, proportionately. 
 
 B 
 
4 .-; jM&tion of Liquids 
 
 very small variations of the velocity) an approximate value 
 of the resistance. If the numbers are not very different, 
 their logarithms are still less different ; so, by plotting the 
 logarithms of the values of the resistances on squared paper, 
 they will be found to lie almost in a straight line. A fair 
 line is then ruled through these points, and all differences 
 are assumed to be due to experimental error. This is the 
 system of what is called the " logarithmic homologues." 
 
 But to return. If any one tries the experiment of moving 
 a small plate in a body of water -a bath, for example he 
 will see that the water is not pushed forwards, since there is 
 no disturbance of the liquid, at even a very short distance in 
 front of the plate : a small body floating in the water will 
 remain at rest until the plate gets very close to it, when it 
 will dart rapidly round to the rear, without apparently 
 touching the plate at all. Any one who has tried taking a 
 floating tea leaf out of a cup of tea will know that it is not 
 always quite easy to keep the leaf in even the hollow of a 
 tea spoon. It is necessary for me to insist on this point, 
 for so much is written, even at the present day, which is so 
 very misleading : such, for example, as the water " striking 
 the plate," or the liquid being " hurled back " by the pro- 
 peller, i 
 
 1 Or similarly the air being " shot downwards " by an aeroplane. 
 This last statement is from Sir Hiram Maxim's Artificial and Natural 
 Flight. It is fairly intelligible even if one does not believe it 
 but there is added, " the air also follows the exact contour of the 
 top-side, and is also shot downwards with the same mean velocity." 
 [Italics added.] Sir Hiram, however, states that " Even calcula- 
 tions made on this basis will not bring the lifting effect of the aeroplane 
 up to what it actually does lift in practice ; in fact the few mathe- 
 maticians who have made experiments themselves have referred 
 to the actual lifting effect of aeroplanes placed at a low angle and 
 travelling at a high velocity as being unaccountable.' ' [Italics added.] 
 
 No evidence has ever been produced in support of this statement 
 that the air of an aeroplane is " shot downwards " : very careful 
 observers have even stated that they have seen large birds flying 
 quite close to the surface of the Nile without producing the very 
 slightest ripple on the glassy surface of the water. It is certainly 
 true that, when a bird is " suspended " in the air it must compress 
 it : so that, with a sufficiently delicate barometer, one ought to 
 be able to register the increase of pressure caused by a flight of 
 birds overhead. - . -J- 
 
Introductory Remarks Momentum Energy 5 
 
 In the endeavour to be precise, I may ask, what do we 
 mean by the words push or pull ? "A tendency when a 
 body is strained to resume its original form ; a tendency in 
 a certain relative position of its parts to a certain relative 
 motion of its parts." (Karl Pearson, The Grammar of Science.} 
 Since there is, speaking generally, no tendency in a liquid 
 which has been strained, to resume its original form all 
 liquids taking what engineers call a " permanent set " at 
 once it is clear that it cannot be pushed or pulled. 
 
 I propose to examine the motion of water, step by step, 
 both as to direction and velocity of the stream lines : not 
 taking anything for granted, where it can be tested by experi- 
 ment, and not proceeding to the next step until the last one 
 has been solidly established, by reference to experiments 
 made in as many different ways as possible. This will be 
 a little tedious, but I must trust to the reader's patience in 
 following me. 
 
 I have said I would take " nothing for granted," but I 
 must qualify this by saying that I assume the " conservation 
 of momentum " as well as the " conservation of energy." 
 
 To explain myself still further ; if a plate is at rest and a 
 jet of liquid moves to meet it, the momentum of a unit 
 volume of the liquid will be mv, where ra=mass of unit 
 volume of the liquid and v =velocity. Since v units pass a 
 given point per second, the momentum passing that point 
 per second may be expressed by mv. 2 Now since, by assump- 
 tion, the plate is fixed, it would appear as if no momentum 
 had been imparted to it, and this would imply annihilation 
 of momentum. This is, of course, impossible ; the truth 
 being that though the plate is fixed, it is, in reality, fixed 
 to the earth : the momentum is therefore transferred to the 
 " earth-plate " system, so that the liquid moves the plate 
 and the earth at an exceedingly slow velocity ; and this 
 velocity is zero, only if referred to the earth. The velocity 
 would not be measurable on this earth, although it would 
 be measurable if referred to some other point in the solar 
 system. 
 
 The direction of the motion may be changed, but the 
 momentum, in the original direction of motion, must still 
 
 B2 
 
6 Motion of Liquids 
 
 exist. This is equally true if the liquid flows through a bent 
 tube, which may be real or imaginary : in the case of a 
 flowing liquid, the " liquid tubes " are imaginary. 
 
 As regards " energy ,7'the total energy per unit volume at 
 a given point in a stream-filament is composed of potential 
 energy, or pressure, and kinetic energy, which is measured 
 by the velocity : and the sum of these two is constant at all 
 points of the stream-filament. Expressed generally, per unit 
 volume of the stream-filament, 
 
 2 constant 
 potential energy -f kinetic energy = constant 
 
 where p ^pressure, v velocity, and Q =the density of the 
 liquid ; and this p+^gv* must retain the same value at all 
 points of the same stream line provided, of course, that it 
 always lies in the same horizontal planed. Or we may say 
 Ep + Ek =const ant, where Ep expresses trie potential energy 
 and Ek the kinetic energy : and this total energy is invari- 
 able. [Potential energy may be converted into kinetic 
 energy^jor vice versa, but the sum of the two must remain 
 constant, if we neglect that part which is transformed by 
 the viscosity of the liquid, to another form of energy, namely, 
 heat. 
 
 In short, whatever the motion of the liquid, there is never 
 any real loss of momentum or energy, though this may 
 apparently be the case if only isolated portions of the liquid 
 are considered. 
 
 It would appear here desirable that I should define 
 " energy " ; but this, I regret to say, I am unable to do, 
 since I know of no satisfactory definition. Mr. F. Soddy, 
 in Matter and Energy, although he says " this is the age of 
 energy," does not attempt to define what it is. It is true 
 that he says, " In physics work and energy are interchange- 
 able terms " ; but as, a little later, he speaks of " the amount 
 of work done, and the amount of energy spent in doing it," 
 it is clear that we could not change the position of these 
 terms and speak of the " energy done " and " the amount 
 of work spent in doing it." 
 
 " Energy is recognized in two forms, kinetic and potential, 
 
Introductory Remarks Momentum Energy 7 
 
 The first depends on motion, the second on the position of 
 the body under consideration, and the law of conservation 
 states that any loss of energy of motion is balanced by a gain 
 due to position, and vice versa " (F. Soddy, Matter and 
 Energy). Any body which is capable of performing work is 
 said to possess energy ; this, however, does not define what 
 the energy is. 
 
 " Since we cannot give a general definition of ' energy/ 
 the principle of the conservation of energy signifies simply 
 that there is some. 'king which remains constant. Well, 
 whatever may be the new ideas which future experiments 
 may give us of the world, we are sure, in advance, that there 
 will be something which will remain constant and which we 
 will be able to call energy " (H. Poincare, La Science et 
 I'Hypothese). 1 
 
 I have used the term " impact/' and it is well that I should 
 here define what I mean by the word. Most writers, I regret 
 to say, employ the term very loosely in fact, in two senses 
 and I have found great difficulty in avoiding doing the 
 same. I will therefore divide the expression into two, viz., 
 impact without shock and impact with shock. If a jet strikes 
 a plate I call this impact (generally) : but if the liquid flows 
 away without reflux, this is what I call impact without shock. 
 If there is reflux, then I call this impact with shock. If the 
 reader thinks this arbitrary, I must point out that I claim 
 the right to use a word in any sense I please, provided that I 
 define accurately the sense in which I employ it. 
 
 In order to try and explain, pictorially, what I mean : 
 when a body moves in a liquid at rest, it appears to form a 
 sort of " liquid prow " which divides the liquid and thus 
 acts as a kind of elastic buffer, in preventing the plate and 
 liquid striking one another suddenly. Reversing the action, 
 if the liquid is moved past the body, the " liquid prow " on 
 
 1 Comme nous ne pouvons pas donner de 1'energie une definition 
 generate, le principe de la conservation d'energie signifie simplement 
 qu'il y a quelque chose qui demeure constant. Eh bien, quelles 
 que soient les notions nouvelles que les experiences futures nous 
 donneront sur le monde, nous sommes surs d'avance qu'il y aura 
 quelque chose qui demeurera constant et que nous pourrons appeler 
 energie. 
 
8 Motion of Liquids 
 
 the body decelerates the liquid by slow degrees so that there 
 is no shock similarly to the manner in which a railway 
 buffer acts in preventing shock between two railway car- 
 riages. The momentum is transferred in very small doses 
 so as to prevent any sudden alteration of velocity. 
 
 When a liquid, however, flows past a body, it is continu- 
 ally changing its shape, and the liquid prow in front of the 
 body is only imperfectly formed : some of the liquid is not 
 properly deflected by it, with the result that there is some- 
 times impact with shock between the body and the liquid. 
 
 All this will be clear to the reader when he gets to the 
 chapter on jets. 
 
 Now since a circular disc in normal presentation divides 
 the liquid and this is equally true for any solid, meeting 
 the liquid at any angle the first point to be quite clear 
 about is how the liquid is divided : where does the " divide " 
 take place ? It appears clear that there must be some point, 
 or line, from which the stream filaments separate ; and that 
 this point, or line, must be on the anterior surface of the 
 circular plate. This will be the special point for examina- 
 tion in the next chapter. 
 
 SUMMARY 
 
 A body moving in a liquid divides it : the body cannot 
 and does not, push the liquid forwards. 
 
 REFERENCES 
 
 NEWTON'S Principia, vol. ii. (Motte's Translation). 
 
 Sir HIRAM MAXIM, Artificial and Natural Flight. 
 
 KARL PEARSON, The Grammar of Science. 
 
 F. SODDY, Matter and Energy. 
 
 H. POINCARE, La Science et I'Hypothese. 
 
 Sir GEORGE STOKES, Mathematical and Physical Papers. 
 
CHAPTER II 
 
 THE " DIVIDE " 
 
 IN the early part of the nineteenth century Colonel Duchemin 
 carried out some experiments with the object of finding 
 out how the stream-filaments meeting a plane surface 
 divided ; the results of which he published in his Recherches 
 experiment ales sur les lois de la resistance des fluides. His 
 experiments were made in water, and his method was as 
 follows. A plate of sheet iron, which had at first the form 
 of a circle of 0-422 m. (17 inches, nearly) diameter ; then 
 that of a square of 0-3 m. side ; and subsequently that of 
 a rectangle of 0-3 m. by 0-2 m., was pierced with sixteen 
 holes of 0-003 m. diameter. These plates could be fixed, in 
 the usual manner, in running water at any required depth 
 of immersion. By a very simple arrangement of cog-wheels, 
 a small vane (0-015 m. xo-04 m.), the axle of which worked 
 freely through any given hole, was able to indicate on a 
 dial, above the level of the water, the direction in which 
 the vane was turned by the moving water. It will easily be 
 understood that all the holes in the plate, but one, having 
 been stopped, the axle of the vane passing through the open 
 one, he could note the direction of the flow of the liquid at 
 this point. By repeating the experiment with all the other 
 holes, a " chart " of the flow of the liquid, all over the plate, 
 could be made. In the case of the circle it was found, as 
 would naturally be expected, that the flow on the front face 
 was strictly radial, as in fig. i. In other words, the " divide" 
 was a point, in the centre of the circle. 
 
 When the axle of the vane was reversed through the holes, 
 
 9 
 
TO 
 
 Motion of Liquids 
 
 FIG. i. 
 
 it was found that the flow at the rear of the plate was also 
 radial, from the centre to the circumference ; so that fig. i 
 
 will indicate the 
 lines of flow both 
 at the front and 
 at the back of the 
 circular plate in 
 " normal presenta- 
 tion." This, also, 
 is what a reader 
 of the ABC of Hy- 
 drodynamics would 
 naturally expect. 
 In both cases the 
 " divide " is at the 
 cent re of the circle. 
 It is necessary 
 to be very careful 
 not to confuse the 
 term " divide " 
 
 which is, of course, the point, or line, of greatest pressure 
 with the centre of pressure. In 
 this case they are the same, 
 but in a great many cases they 
 are not. It is well to avoid 
 anything like confusion. 
 
 Exactly similar experiments 
 having been carried out with 
 the square plate, the lines of 
 flow are indicated in fig. 2. 
 
 In this case, also, fig. 2 re- 
 presents the lines of flow across 
 both the front and the back 
 of the plate. There appears 
 to be a division of the stream, or " secondary divide " along 
 the diagonals of the square. Duchemin says, in reference 
 to this, that " it does not appear that there are [on the 
 posterior side] currents following these lines." [The dia- 
 gonals.] It seems more probable, however if not certain 
 
 FIG. 2. 
 
The "Divide 
 
 II 
 
 that there must be some flow from e to a, b, c, and d, but 
 that the vane, when pointing in these directions, was in 
 unstable equilibrium and so never took a fixed position along 
 either of the diagonals. 
 
 When the experimental plate is a rectangle the streams, 
 whether on the anterior or posterior side, still divide themselves 
 into four sheets of water, directed towards the perimeter ; 
 but the lines of division of the sheets appear to be curved, 
 and they seem to divide the angles into two equal parts, 
 as in fig. 3. Duchemin says, " on the anterior face there 
 are streams which follow the curved diagonals : analogous 
 ones have not been recognized 
 on the posterior face." [Italics 
 added.] The reason for this 
 would appear to be similar 
 to that given previously for 
 the square plate. 
 
 In the case of a rectangu- 
 lar plate the primary divide 
 is a line, and this is accom- 
 panied by secondary divides 
 in the direction of the corners 
 of the rectangle. 
 
 To explain the " divide " 
 by a parallel example ; if we 
 imagine rain to fall on a con- 
 ical hill the " divide " will be 
 the point of the hill. Similarly, if the hill be shaped like a 
 square pyramid, the " primary divide " will also be at the 
 point ; but along, what I may call, the spurs of the hill 
 there will be " secondary divides," which are lines along 
 the spurs. Again, if the hill be shaped like the hip roof of 
 a house, the " primary divide " will be a line along the ridge 
 of the roof, whilst there will be " secondary divides " which 
 are lines along the hips. 
 
 All these experiments were repeated when the plate was 
 moving in still water and with exactly similar results. From 
 this we may conclude that the flow of the water across the 
 faces of these plates was the same whether the plate was at 
 
 FIG. 3. 
 
12 Motion of Liquids 
 
 rest and the water was moving, or, conversely, if the plate 
 moved in the water at rest. I wish to draw special attention 
 to this, because most writers, assuming that action and 
 reaction are equal and opposite, and that everything depends 
 on the " relative motion " only which is perfectly true if 
 the conditions are exactly similar consider all this as quite 
 " self-evident." We should remember what d'Alembert 
 said : " Mathematics, which should only be the servant 
 of physics when they are working together, frequently com- 
 mands them. The more useful the application of mathe- 
 matics to physics, the more one should be circumspect in 
 this application. We must therefore know how to draw 
 the line where our ignorance commences and not think 
 that the words theorem and corollary become, by some 
 secret virtue, the essence of a demonstration ; and that in 
 writing at the end of a proposition which was to be demon- 
 strated one will have proved that which is not true." l It is 
 necessary to be cautious : theory may be in error, so we 
 must take nothing for granted, if the truth or error can be 
 experimentally verified. I will return to this subject later. 
 
 As it was interesting to know what the flow was like, when 
 the plates, always in normal presentation, were only partly 
 immersed in the water, Duchemin made the necessary experi- 
 ments. The level of the flowing stream at the spot where 
 the plate was fixed, was determined by the use of a very 
 thin plate placed edge-wise to the stream, and which did 
 not produce any sensible stoppage of the liquid. The exact 
 depth of immersion of the plate could thus be determined 
 with accuracy. 
 
 Let A B C D (fig. 4) represent the submerged portion of 
 the plate, A B being the undisturbed water level : the lines 
 of flow divide into four sheets (below the surface) exactly 
 
 1 La geometric qui ne doit qu'obeir a la physique quand elle 
 se reunit avec elle, lui commande souvent. Plus on peut tirer 
 d'utilite de 1'application de la geometric a la physique, plus on 
 doit etre circonspect dans cette application. II faut done savoir 
 s'arreter sur ce qu'on ignore, ne pas croire que les mots et de theo- 
 reme et de corollaire, fassent par quelque vertu secrette 1'essence 
 d'une demonstration et qu'en ecrivant a la fin d'une proposition 
 ce qu'il fallait demontrer, on rendra demontre ce qui ne Vest pas. 
 
The " Divide 
 
 as they did when the plate was wholly submerged, allowing 
 for the difference in shape of the submerged part ; that is 
 to say K is 
 half way be- 
 tween h and 
 /. The stream 
 filaments of 
 the sheet of 
 liquid A K B 
 do not, how- 
 ever, stop at 
 A B but rise 
 to the level 
 efg ; the flow 
 being as re- 
 presented by 
 the dotted 
 lines. They 
 then appear 
 t o curve 
 slightly back- 
 wards away from the plane of the paper flowing right 
 
 and left, and 
 so moving 
 past the 
 plate. 
 
 When the 
 posterior face 
 of the plate 
 is examined, 
 it is found 
 that the flow 
 here differs, 
 very mate- 
 rially, from 
 that at the 
 front. The 
 point K 1 
 (fig. 5) is sen- 
 sibly higher 
 FIG. 5. 
 
 Water 
 
 level 
 
14 Motion of Liquids 
 
 than K, whilst/ 1 is much lower than/ being, in fact, below 
 the level of the surface of the stream. The stream-filaments 
 divide into three sheets only, instead of four, whilst there are 
 streams flowing from e' and g' to /'. The movement of 
 the water is evidently ^quite different at the rear of the plate 
 from what it is at the front. The water is dammed up in 
 front, whilst there is a depression behind. 
 
 These experiments were also repeated when the plate 
 was moving in still water and with the same results within 
 the ordinary limits of unavoidable experimental error. 
 Duchemin, when commenting on these experiments, says 
 ':This phenomenon has been known for a long time, but 
 what was not known is that the liquid divides into sheets 
 on the anterior side of A B C D in the same manner as if 
 this part were isolated in the middle of the water." [Italics 
 added.] 
 
 Duchemin was evidently not aware that, about forty 
 years before, Avanzini claimed to have observed the same 
 thing. He said, referring to his experiments (Istituto 
 Nazionale Italiano, Tomo ii., Parte i). " In all these experi- 
 ments, since the axis of equilibrium will be found above the 
 centre of figure of the immersed part of the lamina, and since 
 the centre of resistance must fall in this same axis, it is easy 
 to see that whatever is the portion of the lamina which is 
 immersed, whatever its breadth, velocity and inclination, 
 the centre of resistance always falls in the centre of figure of 
 the aforesaid immersed portion of the lamina." 1 His exact 
 words are : "In tutti questi sperimenti trovandosi 1'asse 
 d'equilibrio sopra il centre di grandezza della parte im- 
 mersa della lamina, e nell' asse medesimo dovendo cadere 
 il centro di resistenza, e agevole il conoscere che qualunque 
 sia la porzione immersa della lamina, qualunque la sua 
 larghezza, velocita, e inclinazione, sempre il centro di 
 resistenza cade nell' centro di grandezza della suddetta 
 porzione immersa della lamina." 
 
 I am afraid the worthy abbot is a little mixed in this 
 
 1 By "axis of equilibrium," is meant the axis about which the 
 plate was free to turn, whilst it was in equilibrium. " Centre of 
 resistance " is clearly the reverse of the " centre of pressure." 
 
The "Divide" 15 
 
 statement : chiefly, I fancy, from employing one word to 
 express the " centre of pressure " and the " divide," or 
 point of greatest pressure. It is clear that if the " centre of 
 resistance " (or centre of pressure) was always found above 
 
 the centre of figure of the immersed part of the plate, how could 
 this centre of resistance always fall in the centre of figure 
 of the immersed portion of the same plate ? It seems probable 
 that what he meant was that the " divide " was in the centre 
 
i6 
 
 Motion of Liquids 
 
 of figure of the immersed part. In fact, I think the next 
 paragraph leaves no doubt that he recognized that the 
 
The "Divide" 17 
 
 centre of pressure of the immersed part of the plate was at the 
 centre of this immersed part. On the other hand, his dia- 
 gram does not show the "divide " at this point. Fig. 6 
 is a photographic reproduction of Avanzini's drawing, where 
 the divide is shown as above the centre of figure. Avanzini's 
 statement is also too broad and sweeping, for he says that 
 this is true for all angles of inclination : it appears pretty 
 certain that this is only true for " normal presentation." 
 
 Elevation* 
 
 Plan. 
 
 FIG. 
 
 Avanzini's own drawing of the plate moving at an angle 
 of which fig. 7 is an exact copy certainly dees not support 
 his own claim : since the divide is very considerably above 
 the centre of figure of the immersed part of the plate. In 
 any case it is curious to find that so much was known about 
 this subject so long ago. 
 
 Avanzini's curves of the vortices were worked out on 
 theoretical lines, and were then checked by observing the 
 
i8 
 
 Motion of Liquids 
 
 lines of flow, by the use of small balls of the same specific 
 gravity as the water and suspended in it. 
 
 Knowing now where the " divide " is on plates in normal 
 presentation, both wholly and partly submerged, the next 
 case that calls, naturally, for inquiry is how this divide 
 would be affected if the plate were at an angle to the stream. 
 
 45' 
 
 
 Elevation, 
 
 Plan, 
 
 FIG. 9. 
 
 To determine this Duchemin's rectangular plate was exposed 
 (vertically) at angles of incidence of 30, 45 and 60 to the 
 action of running water. The first trials showed that the 
 number of holes in the plate was not sufficient, and twelve 
 more were bored symmetrically. 
 The results are curious and were certainly unsuspected 
 
The "Divide" 19 
 
 by Duchemin : they are represented in figs. 8, 9 and 10, 
 which are copied from his own diagrams, d c represents 
 the plan of the plate which is vertically exposed, e d g h 
 representing, diagrammatically, a horizontal projection 
 of the column of water which is moving to meet the plate 
 d c obliquely. It is obvious that the stream lines do not 
 
 bend at these very sharp angles : it will, however, simplify 
 the explanation to imagine (for the present) that they do 
 so : the error can be very easily corrected later a bed 
 represents a vertical elevation of the anterior face of the 
 rectangular plane, with the " divide " and the directions 
 of the motion of the stream-filaments marked on it. 
 
20 
 
 Motion of Liquids 
 
 One of the first things one notices is that, as the angle of 
 incidence, or " attack," decreases, the "primary dividing 
 line " I k I, between the sheet of liquid moving towards a d 
 and the other sheet moving towards b c, approaches a d : 
 this line being, of course, the line of greatest pressure ; and 
 which, in normal presentation, passes through the centre of 
 the plate. 
 
 Fig. ii. 
 
 A very rough method for finding the position of K, and 
 its distance from the leading edge (or from the centre) of 
 the plate is to draw d g at right angles to the direction of the 
 flow of the stream (fig. n) ; bisect it in /and then draw/ K 
 perpendicular to d c. Drawing / h, parallel to g c, it is clear 
 that, a being the angle of incidence, or " attack," 
 
 W =?&) =i(<fc)cos a 
 also (Kh)=(fh)cos a=(dc)cos*a 
 
 and if we represent dc by I, then 
 
 (Kh) =llcos*a and (dK) =Jfein 2 a. 
 
 This method has no pretensions to accuracy. If one 
 wishes to find the correct position of the divide it is probable 
 
The " Divide 
 
 that one must employ the formula given by Lord Rayleigh 
 
 2(1 2cos a +cos 3 a) 
 
 sn a 
 
 47i sn a 
 
 where d is " the distance from the anterior edge of the point 
 where the stream divides, and where accordingly the pressure 
 attains its greatest value " * (Scientific Papers, Vol. I). 
 The values of d at different angles are given in this paper 
 as : 
 
 90 
 
 d 5000 x length ot plate. 
 
 70 
 
 ^=2676 
 
 50 
 
 ^=0981 
 
 30 
 
 ^=0173 
 
 20 
 
 ^=0040 
 
 10 
 
 ^=0004 
 
 s 1 - 
 
 T 
 
 FIG. 12. 
 
 In fig. ii, the dotted line does not, with an actual liquid, 
 strike the plate at an angle, at K, as shown. It really bends 
 round so as to meet the plate normally, at some point K 1 
 
 1 It is necessary to be careful to note the assumptions on which 
 this formula is based, 
 
 C2 
 
22 Motion of Liquids 
 
 as represented, diagrammatically, in fig. u ; the distance 
 between K and K 1 increasing as the angle of attack decreases. 
 
 The stream-filaments of the sheet of fluid alkld (fig. 10) 
 appear to be refluxed obliquely along the surface of the 
 plate. 
 
 The movement of the liquid at the posterior surface of the 
 plate is much more complicated ; and Duchemin was 
 unable to give any graphic description of it. Nevertheless 
 he was led to believe that the line of the " divide " of the 
 fluid filaments on the posterior side of the plate corresponded 
 to that on the anterior side : the general effect, he says, 
 showed him in a very positive manner that the motion of 
 the fluid on both these faces was from ihe interior to the 
 perimeter agreeing in this with Avanzini's drawing given 
 in fig. 12. 
 
 Knowing, now, how the liquid divides on both sides of 
 a plate, both at normal and oblique angles of attack, in the 
 next chapter I propose to examine the motion of the liquid 
 round the plate, more in detail. 
 
 SUMMARY 
 
 When the presentation of differently shaped plates is 
 normal, the " divide " appears to be similarly situated, 
 both on the front and rear surfaces : this is true, whether 
 the body is at rest and the liquid moves, or when the body 
 moves in liquid at rest. 
 
 When the " presentation " is acute, the exact position of 
 the divide on the posterior face appears to be a little doubtful v 
 
 REFERENCES 
 
 Colonel DUCHEMIN, Recherches Experimentales sur les lois de 
 la resistance des fluides. 
 
 D'ALEMBERT, Traite de l'quilibre et du mouvement des Fluides. 
 AVANZINI, Istituto Nazionale Italiano, Tomo II, Parte I. 
 LORD RAYLEIGH, Scientific Papers, vol. i. 
 
CHAPTER III 
 
 MOTION OF LIQUID ROUND THE PLATE FLOW OF THE LIQUID 
 FILAMENTS IN FRONT OF A PLATE 
 
 IN order to enable him to examine the direction of motion 
 of the stream-filaments round the plate, Duchemin fixed 
 sixteen pieces of strong fine wire to a square wooden plate, 
 so as to completely surround it. Each wire had three 
 small strips of brightly coloured silk fastened to it, at 
 different distances, like small flags. By noting the direction 
 that these pieces of silk took up, he was able to make a 
 very fair chart of the different eddies, etc. 
 
 When the plate was moving in normal presentation, in 
 still water being fixed to a boat he found that the shape of 
 the eddies behind the plate might be expressed diagrammat- 
 ically, as in Fig. 13 : what I may call the " depth " of 
 the eddies being somewhere about half the breadth of the 
 plate. This corresponds very closely with the drawing 
 given by Avanzini, and which he made after observing the 
 flow of small light balls of nearly the same weight as the 
 water. Fig. 14 is a photographic reproduction of Avanzini's 
 drawing ; and a comparison between this and Fig 13, which 
 is an exact reproduction of Duchemin's drawing, will show 
 the very close general agreement between the two obser- 
 vations. 
 
 Duchemin also remarks that the shape of the eddies 
 did not appear to change with any alteration of the velocity 
 of the plate. 
 
 When the plate was at rest, however, and exposed to a 
 stream of flowing water, the eddies were somewhat different ; 
 
 23 
 
24 Motion of Liquids 
 
 being of the same breadth, but the length (or " depth ") of 
 these moving eddies he found extended to, roughly, about 
 
 Plate moving : water at rest. 
 FIG. 13. 
 
 four times the breadth of the plate. The results did not appear 
 to be the same in the two cases ; though the flow in the 
 front of the plates appeared to be identical. 
 
 - T 
 
 Avanzini plate in normal presentation. 
 FIG. 14. 
 
 The next point Duchemin examined was the effect on 
 the flow caused by putting prismatic bodies behind the 
 
Motion of Liquid round the Plate 25 
 
 plate. Rectangular prisms of proportions i : i : 1,2 :i : i, 
 and 3:1: i single, double and treble cubes were 
 fixed behind the moving plate, and the results were -found 
 to be exactly similar to those produced behind the plate 
 alone except that the eddies in rear of the solid bodies 
 appeared perceptibly weaker than those produced behind 
 the plate alone. 
 
 When the bodies were at rest in the moving stream the results 
 were certainly curious. The eddies behind the cube occu- 
 pied a space whose depth was now only about one and a 
 half times the breadth of the plate instead of nearly four 
 times, as with the plate alone. Behind the body of propor- 
 tions 2:1: i the double cube the depth of the eddies 
 was further reduced to about three quarters of the breadth 
 of the plate ; whilst behind the treble cube (3 : i : i) the 
 depth of the eddies was only about half the breadth of the 
 plate or about the same size as the eddies behind the plate 
 when it was moving in still water or behind the treble cube 
 
 Plate at rest : water flowing past it. 
 FIG. 15. 
 
 moving in still water. The reasons for this very curious effect 
 will be referred to again at greater length after a more 
 detailed examination of the motion of the water in front 
 and around the plate. 
 
 From these experiments, the general impression that one 
 would carry away is, that when a plate is at rest in flowing 
 water, more eddies are formed i.e., more Kinetic energy 
 is generated than is the case when the plate itself moves in 
 still water. In the case of the cube this difference is still 
 apparent but much less accentuated', with a double cube 
 
26 Motion of Liquids 
 
 the difference appears to be small ; whilst with a treble 
 cube there appears to be no difference at all. This curious 
 effect requires to be pondered over. 
 
 Another fact that Duchemin noted was that " The mole- 
 cules of w r ater, deviated by the shock * of the anterior face, 
 are not only those comprised in the prism moved through 
 by this face, but also those in the neighbourhood, and the 
 deviation appears to extend in front to a distance of about 
 half the breadth of the surface." 
 
 " In the course of these experiments, which were repeated 
 several times, one had occasion to remark that the directions 
 of the filaments which were in the neighbourhood of the sur- 
 face remain the same when the velocity increases or diminishes 
 by insensible gradations. So that the filaments deviated 
 round the surface form a system with it, and are carried with 
 it (entraines] in its movement." All this is in accord with 
 Dubuat's observation that " when a body resists the action 
 of a current, the fluid is deviated at a certain distance in 
 front of it, and there is formed a sort of liquid prow, in which 
 the filaments, without losing all their velocity, have less than 
 the rest of the fluid in the direction of the general motion." 
 
 Having now a rough general idea of the direction of flow 
 of the different stream filaments, it is important to explore 
 the velocities of these stream-filaments, all round the plate, 
 or other body. The instrument used by Duchemin was 
 Dubuat's modification of the original Pitot tube. The 
 upright part of the tube was made of tinned iron and had a 
 length of 0-954 m. with an interior diameter of 0-032 m. 
 The lower part was curved and terminated (at right angles 
 to the upright part) in a cone with a small circular hole of 
 0-0035 m. in diameter : this small hole being in a plane 
 strictly normal to the direction of the branch of the tube, 
 so that it might be strictly normal, also, to the stream- 
 filament whose velocity it was desired to measure. In 
 the interior of the upright tube was placed a small hollow 
 cylinder, or float, of copper, of 0-065 m. in length and of 
 o-o2t3 m. in diameter. This float would clearly be able to 
 
 1 Duchemin believed in actual liquid shock to some extent : he 
 is a little obscure on this point. 
 
Flow of the Liquid in front of a Plate 27 
 
 move up and down the tube freely. On the top of the float 
 was fixed a small collar of 0-015 m. in length and 0-0035 m - 
 diameter, intended to receive a small rye straw. The 
 arrangement was probably as shown in fig. 16. 
 
 In the original Pi tot tube, one of the great objections was 
 the excessive oscillation of the level of the liquid in the 
 upright tube : it was also difficult to read this level with 
 accuracy. In Dubuat's modification, the diameter of the 
 
 Straw 
 
 Hollow 
 flooui 
 
 FIG. 16. 
 
 upright tube being ten times the diameter of the low r er 
 aperture, it is clear that the area of one being one hundred 
 times that of the other, the oscillations will be heavily 
 " damped." The use of the rye straw also allows of the 
 reading being made much more conveniently, and above the 
 level of the surface of the water. 
 
 Without entering into further details, the reader will 
 easily understand that the height of the level of the liquid 
 in the tube could be measured with a considerable measure 
 of accuracy by means of a scale, fixed close to the straw. 
 
28 Motion of Liquids 
 
 When the instrument was employed with a fixed plate in 
 running water, the thin plate (previously referred to) placed 
 edgeways in the stream allowed the level of the surface of 
 the stream to be measured accurately. 
 
 As to the properties of this instrument, it is well known 
 that if the orifice of the tube be placed below the surface of 
 running water, and perpendicularly to the direction of the 
 motion of any stream-filament, the elevation of the float 
 above its primitive position is equal to the height (or "head ") 
 due to the velocity of the current, when the liquid is moving ; 
 or the velocity of the tube, when this is moving and the 
 liquid is at rest. This height, which will be designated by 
 h, is commonly spoken of as the " velocity head." It is 
 more convenient to measure this than the velocity, which 
 
 can always be calculated by 
 the formula v=V2gh, where 
 v ^velocity and g =coefficient 
 for gravity, h is, clearly, the 
 head which would be required 
 to give a velocity v. 
 
 It is also well known that 
 if the direction of the orifice 
 be reversed the surface of the 
 FlG liquid in the upright tube will 
 
 descend below the level of the 
 
 surface of the stream by a quantity which is also equal to h, 
 or the height due to the velocity, whether the water or the 
 tube be in motion : it depends solely on the relative velocity 
 between the two. 
 
 It will be seen, therefore, that by fixing the tube in any 
 position near the plate at any given depth, if we know 
 the direction of flow of the stream-filaments at this spot, we 
 can measure the velocity of flow of the liquid in these 
 filaments with great accuracy. 
 
 Another observation, first made, 1 believe, by Dubuat, is 
 that it is indifferent whether the orifice of a tube be drowned or 
 not, to give a constant discharge, provided that the tube runs 
 full, and that the head is the same in both cases. That is to 
 say, if the liquid flows in at C (fig. 17) with a velocity head 
 
Flow of the Liquid in front of a Plate 29 
 
 h, the discharge would be the same (neglecting friction) 
 whether it takes place at B above the surface of the 
 liquid or at the drowned point A. This is, clearly, only 
 strictly true if the mouth of the tube is only just above or 
 just below the surface of the liquid, so that the head it is 
 discharging against is the same in both cases. 
 
 Referring again to fig. 4, the point K which separates the 
 stream-filament Kl, directed downwards, from the stream- 
 filament Kh, 
 directed up- 
 wards, is at 
 the centre of 
 the immersed 
 part of the 
 plate A BCD] 
 that is to say 
 at the middle 
 of Ih, and this 
 could only be 
 if the reac- 
 tions of the 
 molecules 
 which move 
 in these two 
 streams, or 
 small tubes, 
 were equal 
 
 and opposite ; or if the " heads " were the same at the two 
 orifices h and / of the imaginary liquid tubes. 
 
 To explain my meaning more clearly, let hKl (fig. 18) 
 represent a vertical section of the partly immersed plate 
 in fig. 4, hKl being the immersed part. The two imaginary 
 liquid tubes are shown, diagrammatically, as bending to- 
 wards h and /. The velocity of the flow in the tubes at h 
 and I will be the same, and the reactions caused will be equal 
 and opposite. The " velocity heads " at h and I will be 
 equal. But as the height hf to which the water rises above 
 the orifice h, of the imaginary tube, is clearly the height 
 due to the velocity of the molecules of water when they are 
 
 FIG. I/A. (Flo. 4.) 
 
Motion of Liquids 
 
 leaving this orifice, it will also, therefore, be the height due 
 to the velocity of the molecules of water which are leaving 
 the orifice /. It follows from this that by determining ex- 
 perimentally the height hf for different lengths of Khby 
 augmenting or diminishing the submerged part A BCD of the 
 plane surface one can obtain the " velocity head " of the 
 water at different distances along the fluid stream from the 
 point of inflexion K. 
 
 After a large number of experiments Duchemin found 
 that this velocity can be satisfactorily found by means of 
 the following empirical formula : 
 
 Level 
 
 I 
 
 FIG. 18. 
 
 where u velocity 
 of the undisturbed 
 stream in front of 
 K (fig. I 7 a). 
 h velocity head 
 
 of this stream- 
 
 : == filament. 
 
 v =velocity of 
 
 the liquid mole- 
 cules at the end 
 of the distance e, 
 which the mole- 
 cules have tra- 
 velled across the 
 surface of the 
 plate. 
 
 H velocity 
 head correspond- 
 ing to velocity v. 
 
 AC (fig. 4), half the depth of 
 
 e length Kh, equal to 
 1he immersed part of the plate. 
 
 K =0-25379 m -> a constant linear quantity of the same kind 
 as e. 
 
 This formula, which is a purely empirical one, is only 
 true within certain limits : it is clearly not true when e =o. 
 
 In the experiments which Duchemin made on the water 
 
Flow of the Liquid in front of a Plate 31 
 
 in movement meeting the plate at rest, the value of h, or 
 the " velocity head " of the current at the centre K was 
 taken with a Pitot tube before the plate was fixed in position, 
 the point of the tube not passing through the plate, but through 
 the point which was later the centre of the plate. A scale 
 marked on the plate enabled the height hf, or H, to be read 
 off directly. The results are given in Table I, annexed. 
 
 The first column of this table gives the breadth of the 
 plate exposed to the stream ; the second, the half depth of 
 the immersed part of the plate or the e in the equation the 
 immersion being measured below the ordinary level of the 
 surface of the stream ; the third, the velocity head of the 
 stream at the centre of the immersed part of the plate ; the 
 
 TABLE I. 
 BODY AT REST. 
 
 
 Values of 
 
 HeadH 
 
 Breadth of 
 
 
 
 Surface. 
 
 
 
 
 
 
 e 
 
 * 
 
 Observed . 
 
 Calculated. 
 
 
 m. 
 
 m. 
 
 m. 
 
 m. 
 
 ( 
 
 0-050 
 
 0-055 
 
 0-065 
 
 0-0658 
 
 
 o-ioo 
 
 0-055 
 
 0-079 
 
 0-077 
 
 0-300111 < 
 
 0-150 
 0-200 
 
 0-055 
 0-055 
 
 0-087 
 0-097 
 
 0-0875 
 0-098 
 
 
 0-250 
 
 0-054 
 
 0-107 
 
 0-107 
 
 V 
 
 0-300 
 
 0-054 
 
 0-118 
 
 0-118 
 
 0-150111 ! 
 
 0-150 
 0-300 
 
 0-055 
 0-055 
 
 0-089 
 0-118 
 
 0-088 
 
 0-120 
 
 o-2oom ! 
 
 0-150 
 
 0-055 
 
 0-087 
 
 0-88 
 
 I 
 
 0-300 
 
 0-054 
 
 0-I2I 
 
 0-118 
 
 fourth, the height of the centre of the " lip," * or eddy, hf 
 observed ; whilst the fifth column gives the value of this 
 " lip " calculated by Duchemin's formula (a). 
 It will be observed, on comparing the two last columns, 
 
 1 I know of no English word for this, so I have adopted " lip," 
 which is the translation of the Italian word " labbra " used by 
 Avanzini. 
 
Motion of Liquids 
 
 that the formula (a) gives results which are in very close 
 agreement with those observed. 
 
 Exactly similar experiments were made with the body 
 moving in still water, and the results are given in Table II. 1 
 
 This table, which is exactly similar to the last, shows also 
 the very close agreement between the observed values of 
 H and those calculated by the formula (a), notwithstanding 
 the fact that the velocities of motion have varied very sensibly. 
 
 One may therefore conclude from these experiments that 
 the height of the "lip," or eddy, H, against a plane in nor- 
 mal presentation partly immersed in a liquid, may be ex- 
 pressed by formula (a), whether the surface is at rest and 
 the water moves, or whether the plate is moving in water 
 at rest. 
 
 TABLE II 
 BODY IN MOTION 
 
 
 Values of 
 
 HeadH 
 
 Breadth of 
 
 
 
 Surface. 
 
 
 
 
 
 
 e 
 
 ft 
 
 Observed. 
 
 Calculated. 
 
 
 m. 
 
 m. 
 
 m. 
 
 m. 
 
 ( 
 
 o-ioo 
 
 0-062 
 
 0-087 
 
 0-085 
 
 0-300 J 
 
 0-200 
 0-300 
 
 0-056 
 0-051 
 
 0-098 
 
 O-IIO 
 
 o-ioo 
 
 O-III 
 
 ( 
 
 0-400 
 
 0-044 
 
 0-115 
 
 0-113 
 
 f 
 
 O-2OO 
 
 0-068 
 
 0-120 
 
 0-122 
 
 0-150 | 
 
 0-400 
 
 0-052 
 
 0-134 
 
 0-134 
 
 0-2OO -j 
 
 0-200 
 
 0-300 
 
 0-059 
 0-054 
 
 0-109 
 0-119 
 
 0-106 
 0-118 
 
 If this explanation of the manner of flow of the water 
 from the centre of figure of the immersed part of the plate 
 be the correct one, it should be true for all cases. Duchemin 
 
 1 For the future, for the sake of brevity (and simplicity) , I shall 
 only refer to the body, as being in motion or at rest. " Body at 
 rest " must be taken to imply that the liquid is flowing past it; 
 " Body in motion " that it is moving at uniform velocity in the 
 liquid at rest or what we, ordinarily, call " at rest." 
 
Flow of the Liquid in front of a Plate 33 
 
 made a slightly different experiment to test this, acting as 
 a kind of crucial test. 
 
 A square plate abed, length of side =0-250 m., was cut 
 out of a board, leaving a prolongation of one of the diagonals, 
 age as in fig. 19. This was immersed in a stream having a 
 velocity of 1-048 m. ; the velocity head, or h, of which being 
 0-056 m. The diagonal ac was vertical, and the angular 
 point a was at the primitive surface of the flowing stream. 
 
 The fluid rose 
 along the line 
 Ka to the height 
 af 0-093 m. 
 Now this eleva- 
 tion of the 
 water is, evi- 
 dently, due to 
 the velocity ac- 
 quired by the 
 molecules of 
 water, which we 
 have recognized 
 as following the 
 direction of the 
 diagonal ca of 
 the square. 
 Therefore if we 
 insert in the 
 formula (a) the 
 value of h = 
 0-056 m. and 
 that of e =Ka = 
 
 0-177 m -> we sna ll obtain H =0-095 m., a value which only 
 differs from the observed height by 2 millimetres, or about 
 2 per cent. From this it appears that af or H in Tables I 
 and II is the head corresponding to the velocity v, which 
 the fluid filament has acquired in travelling across the 
 surface of the plate from K to a, this velocity being in excess 
 of the velocity of the undisturbed stream at the point K. 
 In the next chapter we will examine this question further 
 
34 Motion of Liquids 
 
 by using the experiments of other and earlier observers 
 than Duchemin. 
 
 SUMMARY 
 
 The flow behind a plate does not appear to be the same 
 when the plate is moving in liquid at rest, as it is when the 
 plate is at rest and the liquid flows. More kinetic energy 
 appears to be generated in the latter case than in the former. 
 The same result appears behind a cube, and a double cube, 
 though the effect is not so marked. Behind a treble cube 
 no difference is apparent. 
 
 What Dubuat called the " liquid prow " appears to have 
 a length equal to half the diameter of the anterior face of 
 the body exposed to the liquid. 
 
 REFERENCES 
 
 Colonel DUBUAT, Principes d'Hydraulique. 
 
CHAPTER IV 
 
 SUBJECT CONTINUED, AND EXAMINED BY REFERENCE TO 
 EXPERIMENTS MADE PREVIOUSLY TO THOSE OF DUCHEMIN 
 
 IN the last chapter we examined the direction of motion and 
 the velocity of flow of the different liquid-filaments flowing 
 across the front of a plate, which was either (i) moving in 
 a liquid at rest, or (2) was itself at rest, with the liquid 
 flowing past it. We saw that Duchemin's formula (a) 
 enabled us to calculate the velocity of these filaments at the 
 edge of the plate with a considerable degree of accuracy. 
 It will have been noted that all the experiments referred to 
 had been executed by Duchemin himself ; and when one 
 sees that all the experiments accord very closely with any 
 theory, one is inclined, not perhaps unnaturally, to be a 
 trifle suspicious. In the very excellent programme of 
 instructions formulated by the Academic des Sciences, and 
 according to which Duchemin was working, it was laid 
 down that any empirical formulae which he might be able 
 to establish should be " subsequently compared with the 
 mass of experiments made previously on this subject." This 
 I propose now to do. 
 
 I know of no modern experimenters who have measured 
 this " lip," or eddy, formed in front of partially immersed 
 moving plates ; so that we must be satisfied with some 
 rather old measurements. The Abbe Bossut's experiments, 
 carried out for the French Government more than one 
 hundred years ago, are classical ; Colonel Dubuat, who was 
 a very careful observer, also experimented for, and at the 
 expense of, the French Government ; the Abate Giuseppe 
 
 35 D 
 
36 Motion of Liquids 
 
 Avanzini is the third and only other experimenter on this 
 subject that I am acquainted with. 
 
 Commencing with Bossut, the accompanying Table III 
 gives the results of experiments executed with moving 
 bodies. In this list of the results of thirty-one measurements, 
 column No. i gives the number of the experiment in Bossut's 
 report (Nouvelles Experiences sur la resistance des fluides] 
 No. 2 the details of the bows of the boats the measure- 
 ments were made on, No. 3 the time in half-seconds taken 
 in travelling 50 feet, 1 No. 4 the value of e half the depth 
 of immersion, whilst 5 and 6 are the values of H, as observed 
 and as calculated by the formula (a) : 
 
 where H = 
 
 when h =" velocity head " o the motion, e =half the depth 
 of immersion, and K =0-25379, a constant linear quantity 
 of the same kind as e. 
 
 The bodies experimented with were boats which had prows 
 of different shapes ; in experiments 855, 864 and 865 the 
 prows were flat and perpendicular to the direction of the 
 motion ; the prows of the 858 and 862 boats were angular 
 and formed angles of incidence of 63 26' in the former and 
 21 48' in the latter cases ; those of the last two boats were 
 semi-cylindrical surfaces, the axes of the cylinders being 
 vertical. If we put t to represent the number of seconds 
 
 employed in travelling the 50 feet, then the velocity u = , 
 
 u 2 (so) 2 
 from which it results that h = = ^-. This expression 
 
 gives, by employing the values of 2t given in the third 
 column of the table, the values of h, or the " velocity head " 
 of the body ; and putting these values of h and those of e, 
 given in the fourth column, in the formula (a) we obtain the 
 values of H in the last column. We see that the calculated 
 and observed values of H are as close as it would be reason- 
 able to expect, taking into account the very great difficulty 
 in carrying out these kinds of experiments. 
 
 ' 1 The feet are old French feet, divided into 12 inches, and each 
 inch into 12 lines. The "pied royal" = 1-06578 ft. (E) 
 
Experiments made before Duchemin's time 37 
 
 TABLE III 
 
 BODIES IN MOTION 
 
 No. of 
 Supt. 
 
 Breadth of 
 Surface. 
 
 \ seconds 
 per 50 ft. 
 
 Values of 
 e. 
 
 Head H. 
 
 Observed. 
 
 Calculated. 
 
 
 In. lines. 
 
 
 In. lines. 
 
 lines. 
 
 lines. 
 
 
 f 
 
 43-70 
 
 } 
 
 21'" 
 
 20-47'" 
 
 855 
 
 12" 
 
 38-37 
 
 f 6" oo 
 
 26" 
 
 26-55'" 
 
 
 ( 
 
 34-75 
 
 J 
 
 34'" 
 
 32-37'" 
 
 
 
 52-00 
 
 \ 
 
 15'" 
 
 14-66"' 
 
 
 
 46-05 
 
 1 
 
 18 
 
 1871'" 
 
 864 
 
 
 42-07 
 
 6" 2f" 
 
 20-5 
 
 22-42 
 
 
 
 37-25 
 
 ) 
 
 26-0 
 
 28-59 
 
 
 , 
 
 
 
 32-5 
 
 32-06 
 
 
 ( 
 
 50-75 
 
 \ 
 
 1 
 
 15-0 
 
 17-06 
 
 
 
 46-50 
 
 1 
 
 18-0 
 
 20-33 
 
 865 
 
 19" 8'" i 
 
 41-00 
 
 f 7" ii'" 
 
 24-0 
 
 26-15 
 
 
 
 36-50 
 
 
 33-o 
 
 34-55 
 
 
 \ 
 
 
 ; 
 
 39-0 
 
 38-73 
 
 
 Angular / 
 
 50-00 
 
 1 
 
 16-0 
 
 15-64 
 
 
 prow 
 
 43-50 
 
 I 
 
 21-0 
 
 20-66 
 
 858 
 
 Length 6" l 
 
 38-50 
 
 ( 6"-oo 
 
 27-0 
 
 26-38 
 
 
 Breadth 24" (^ 
 
 37-00 
 
 j 
 
 30-0 
 
 28-56 
 
 
 
 34-92 
 
 
 34-o 
 
 32-07 
 
 
 
 36-62 
 
 
 29-0 
 
 29-16 
 
 
 
 33-40 
 
 
 33-o 
 
 35-05 
 
 862 
 
 
 30-81 
 
 6"-oo 
 
 38-0 
 
 41-19 
 
 
 
 28-62 
 
 
 45' 
 
 47-73 
 
 
 { 
 
 27-00 
 
 \ 
 
 52-0 
 
 53-63 
 
 
 
 36-00 
 
 I 
 
 26-0 
 
 26-08 
 
 872 
 
 
 32-20 
 
 f 3" ii'" 
 
 33-o 
 
 32-60 
 
 
 
 29-60 
 
 j 
 
 40-0 
 
 38-58 
 
 
 
 
 27-40 
 
 
 46-0 
 
 45-02 
 
 
 ( 
 
 44-00 
 
 ^ 
 
 20-0 
 
 20-50 
 
 873 
 
 do. do. J 
 
 38-50 
 
 I 6" 2f " 
 
 26-O 
 
 26-77 
 
 
 1 
 
 35-40 
 32-69 
 
 J 
 
 32-0 
 38-0 
 
 31-66 
 37'!3 
 
 These experiments, besides giving results conformable 
 with Duchemin's formula (a) show another thing, and that 
 is that the same results are obtained whether the body is a 
 thin plate or a solid ; and whether the surface in presenta- 
 tion is a dihedral angle or the circular surface of a cylinder. 
 
 Dubuat executed many experiments, but I only know of 
 
 D2 
 
Motion of Liquids 
 
 one where he measured the "lip," or eddy, of water rising 
 against a plate. In his Principes d'hydraulique, No. 442, he 
 states that he placed a thin box in a running stream so that 
 its lower surface was 8 inches and 4 lines below the primitive 
 surface of the stream, and that a " lip " was formed with a 
 central height of 3 inches 7 lines. The velocity of the stream 
 at a depth of 4 inches and 2 lines was 42-1 inches per second, 
 corresponding to a velocity head of 2 inches 5-J- lines ; the 
 breadth of the box was 12 inches. We have, therefore, for 
 this experiment e=4 inches 2 lines, and h=2 inches 5^ 
 lines ; putting these values in the formula (a), we find H = 
 3 inches 6f lines, a quantity which only differs from the 
 observed amount by f of a line, or about i per cent. 
 
 FIG. 20. 
 
 Avanzini's experiments, though apparently very carefully 
 executed, give results which are not in accord with the fore- 
 going. I can only account for this by the fact that his 
 plates, moving in still water, were balanced on pivots, and so 
 were free to oscillate ; and although he says no oscillation 
 was perceptible, I ain inclined to think some must have 
 taken place. It is, further, by no means certain that his 
 plates moved strictly in normal presentation ; the eye is a 
 poor judge of this, and his method of measuring the angle 
 fairly accurate for small angles of attack is very defec- 
 tive when the angle approaches a right angle : the plates 
 when arriving at the end of their run being struck by a sharp 
 point, fixed horizontally, and so marked, the measurement 
 of the angle being made subsequently at leisure. The 
 
Experiments made before Duchemiris time 39 
 
 arrangement is shown diagrammatically in fig. o. It will 
 be clear that any small error in the horizontality of the 
 marking instrument might cause a large error. Even, 
 however, if the marking point were correctly adjusted, an 
 error of several degrees in the angle of the plate might easily 
 be made. 1 
 
 I propose to apply a kind of " cross-check " to this quer- 
 tion later, but meanwhile it is useless to blink the fact that 
 Avanzini's experiments are in disaccord with those of others. 
 It appears to be a pity that he did not carry out his experi- 
 ments with the plates clamped in position. 
 
 Since it is well, at times, to take stock of one's ideas, in 
 order to " scotch up " some line of thought, I will now enu- 
 merate the results we may be said to have arrived at from 
 the study of these experiments. 
 
 1. Whether the plane surface, in normal presentation, 
 and partly submerged, is moving in a direction normal to 
 this surface in water at rest, or whether the plate is at rest 
 and the liquid is flowing past it, the height to which the 
 water will rise, above its primitive level, against the centre 
 of the vertical anterior face, can be expressed by the formula 
 (a) ; and it is the same in both cases, all conditions being, of 
 course, the same. 
 
 2. This height is independent of the breadth of the plane 
 surface ; it remains constant even when the plane part of 
 the surface is reduced to a vertical line, as at the centre of 
 a cylindrical surface, or at the edge of a dihedral angle. 
 
 3. It is the same, under similar conditions, against a 
 simple plane lamina, or against a solid body which has a 
 similar plane surface for its anterior side. 
 
 4. This height depends entirely on the motion of the 
 molecules of the stream-filament moving from the centre 
 
 1 In The Laws of A vanzini I have given drawings of the apparatus 
 used ; these are photographically reproduced from the originals. 
 For the reader who does not know the arrangement I may mention 
 that the plate was free to oscillate about pivots, which were fixed 
 to a carriage moving above the surface of the water. When the 
 carriage had completed its run, the plate was struck by a fine point, 
 as in fig. 20, and so marked. The carriage being then brought back, 
 so that the point and mark corresponded, the angle of attack was 
 measured by means of a plumb-bob and a protractor. 
 
4 o 
 
 Motion of Liquids 
 
 K (fig. 4) of the submerged surface, to the point h on the 
 line of immersion, the velocity of these molecules at the 
 
 point h being expressed by v = u\J i -f- , where u =velocity 
 
 of undisturbed stream at the point K. 0=half depth of 
 immersion; 7 =0*25379, a constant linear quantity of the 
 same kind as e. 
 
 When an artist, in painting a picture, wishes to draw 
 
 attention to 
 some parti- 
 cular point, 
 he produces 
 his effect by 
 a sharp con- 
 trast of either 
 line, colour 
 or shadow. 
 Similarly, as 
 I wish to 
 draw special 
 attention to 
 this point, I 
 will make as 
 sharp a con- 
 trast as I 
 can between 
 
 (FlG ' 4 - ) the view here 
 
 advanced and that commonly taught in the schools. I will 
 select as my guide a very excellent modern text book, by 
 an author who is a member of several learned societies, and 
 so, not likely to teach anything that is not quite " correct." 
 Referring to this subject, the argument is somewhat as 
 follows : 
 
 Let the body have the form of a rectangular box of length 
 / and breadth b ; and let it be immersed to a depth d in the 
 liquid. When the body is moving forward with a velocity 
 V, the liquid immediately in front of it must also be moving 
 with a velocity V ; for this to be possible, the liquid must 
 
Experiments made before Duchemin's time 41 
 
 be heaped up in front of the body (fig. 21). To determine 
 the height h to which the liquid must be heaped up, let it be 
 supposed that a velocity, equal and opposite to that with 
 which the body is actually moving, is impressed on the body 
 and the whole of the liquid. Then if A is a point on the 
 surface of the liquid to which the disturbance created by 
 the body has not extended, the velocity of the particle of 
 the liquid at A is equal to V ; and if the particle has unit 
 
 V 2 
 
 mass, its kinetic energy is equal to . When the particle 
 
 reaches the point B at the summit of the ridge immediately 
 in front of the body, its velocity in the plane of the diagram 
 (fig. 21 ) has decreased to zero, and the particle has risen to 
 
 -A 
 
 FIG. 21. 
 
 a height h above the undisturbed surface of the liquid ; thus 
 the increase in its potential energy is equal to gh, and this 
 increase has been gained at the expense of the kinetic energy 
 which has disappeared. The foregoing will give a very fair 
 general idea of the line of argument and of the assumptions 
 on which it is based. 
 
 Now, apart from the (I think) rather clumsy way of ex- 
 pressing that the body is at rest, by imparting to it a velocity 
 +V and then superposing another velocity, V, this train 
 of reasoning is based on several very unjustifiable assump- 
 tions. It is assumed that the particle at A in travelling 
 towards B begins by forming a small wave, which gradually 
 increases in amplitude until the particle arrives at B. 
 Nothing is said about how this molecule of water gets away 
 from B, for it is clear that it cannot stick there. We are 
 
42 Motion of Liquids 
 
 not told if the liquid is a " static " one or if it flows like an 
 ordinary liquid. The student is told (somewhat unneces- 
 sarily it would appear) that the depth of immersion is d ; 
 whilst the explanation takes no account of this, thus leaving 
 him to suppose that the height of the " lip " is the same for 
 all depths of immersion. The author also produces no 
 " machinery " for the formation of the waves shown in 
 the diagram. Had he inserted the " stream lines," I fancy 
 we should have seen the fluid-filaments diverging from the 
 centre of the immersed part of the surface, and flowing along 
 the face up, down, right and left. As was pointed out by 
 Dubuat more than a century ago, " The lip which is formed 
 before the vertical face of a floating body is not composed 
 of stagnant water, but of molecules which tend to escape 
 at the surface of the current, as they do in all the other 
 directions. The invincible obstacle that they meet obliges 
 them to turn in gliding along the surface, to escape by the 
 edges. 1 
 
 The waves shown in the diagram (fig. 21) are partly 
 caused by the secondary action of the liquid " cascading " 
 backwards, after arriving at B, the motion being, more or 
 less, as shown in Avanzini's diagram (fig. 6). This is not, 
 however, the whole truth, for, as Riabouchinsky 2 has re- 
 marked, " the water rises and falls periodically " : this, if 
 I understand his explanation correctly, implies that these 
 undulatory movements necessitate an undulatory pressure 
 at the centre of the plate. As will be seenjater, and ap- 
 proaching the subject from an entirely different point of 
 view, I have arrived at the conclusion that such an undu- 
 latory pressure must exist, and I trust I have given a satis- 
 factory explanation of how it is caused. 
 
 The ordinary explanation has not even the merit of being 
 very simple and easily understood. Of course the h in the 
 
 1 Le remou qui se forme devant la face verticale d'un corps 
 flottant n'est pas compose d'une eau stagnante, mais de molecules 
 qui tendent a s'echapper a la superficie du courant, comme elles 
 font dans tous les autres sens. L'obstacle invincible qu 'elles ren- 
 contrent les oblige de se detourner en glissant le long de la surface, 
 pour s'echapper par les bords." 
 
 * Bulletin de I'Institut de Koutchino. Fascicule IV., 1912. 
 
Experiments made before Duchemin's time 43 
 
 V 2 
 
 diagram might be equal to (if the surface velocity hap- 
 
 2g 
 pened to bear the correct relation to the velocity of the 
 
 stream at the depth \ ; but again, it equally might not. 
 
 If the liquid were a " static " one, this could not be the 
 case, since the velocity at all parts would be the same. 
 
 V 2 
 
 The height h bears no immediate relation to : in fact, 
 
 2g 
 
 as long ago as 1763 (Memoir es de I' academic) de Borda found 
 that the " lip " against a moving cube did not much exceed 
 that due to the " velocity head " of the surface layer of the 
 liquid. 
 
 I propose now to examine this question from a slightly 
 different point of view ; acting as, what I may call, a " cross 
 check " on the views previously advanced. Before doing 
 this, however, it is advisable that I should give a description 
 of some of the " tools " I am going to employ. 
 
 I have had occasion to use the expression " static liquid," 
 or " static fluid." As this is not very commonly found in 
 books I do not remember ever having seen it in any text- 
 book it will, not improbably, be unfamiliar to the ordinary 
 reader ; quotations might even be given from several writers 
 who appear to be quite ignorant of the difference between 
 static and non-static liquids. 
 
 A static liquid is generally defined as one which is neither 
 changing velocity, nor changing shape. This definition, whilst 
 being strictly accurate and complete, will, however, I am 
 afraid, convey but a very blurred image to the mind of the 
 ordinary reader. To give a clear and distinct " picture " 
 it will be necessary to give a few examples of both static 
 and non-static liquids, and to point out some of the proper- 
 ties of static liquids which are implied in the foregoing 
 definition. 
 
 It might be imagined that the simplest form of static 
 liquid is one which is at rest. The expression " liquid at 
 rest/' however, without any further qualification, is mean- 
 ingless. We do not know whether any liquid is actually at 
 rest in fact, we have every reason for knowing that such 
 
44 Motion of Liquids 
 
 a thing does not exist and we most certainly have no 
 means of measuring whether it is so or not. Rest and 
 motion are only relative terms in physics. 
 
 The water in a pond, near London, which is at rest, re- 
 ferred to the earth, would be water moving at a velocity of 
 about 800 miles per hour, if referred to a point just outside 
 the earth. Besides this, the earth is moving some tens of 
 thousands of miles per hour round the sun ; and we do not 
 know the exact velocity of the sun. 1 It is clear, therefore, 
 that we require some " frame," and that we cannot speak 
 of any absolute velocity, but only of a " relative velocity " 
 referred to some points in our frame. When, therefore, 
 we speak of liquid " at rest/' it must be understood as at 
 rest relatively to the earth ; so also, if we speak of uniform 
 velocity of a liquid or of a liquid not changing its velocity 2 
 it is always to be understood as referred to the earth. Any- 
 thing else would be meaningless, as it is impossible to get 
 any other kind of uniform velocity, since in addition to its 
 rotational movement about its axis, the earth's transla- 
 tional motion is constantly changing its direction and mag- 
 nitude. 
 
 If the question of not changing its velocity is important in 
 the definition of a static liquid, it is vastly more important 
 that it should not be changing its shape. 
 
 To give an example of a static liquid : imagine a tank on 
 wheels on a railway, and filled with water. If this tank be 
 standirg still on the rails, it is a static liquid it is neither 
 changing its velocity nor shape. If the same tank be moving 
 at a uniform velocity of fifty miles per hour it is still a static 
 liquid, for the same reasons. 
 
 If, however, water be flowing in a canal it is not a static 
 liquid, for though its velocity may be uniform be not 
 changing it is perpetually changing its shape, in conse- 
 quence of the velocity at the surface exceeding that at the 
 bottom and sides. 
 
 1 Professor Dr. J. C. Kapteyn, in May, 1908, said at the Royal 
 Institution : " We have been able of late to fix with some precision 
 the velocity of this motion, which amounts to 20 kilometres per 
 second. 
 
 3 We might, perhaps, better say its " rate of velocity." 
 
Experiments made before Duchemin's time 45 
 
 In case the reader should not thoroughly grasp my mean- 
 ing, when I say a liquid is " changing shape," I will give a 
 very simple example. Suppose the liquid is an ordinary 
 viscous one, flowing at a low velocity, the motion is repre- 
 sented, in all textbooks, as in fig. 22 A represents, dia- 
 grammatically, a body of liquid divided into horizontal 
 laminae at a certain instant of time t ; B represents the 
 same body at a later instant of time ^. It is obvious that 
 this body has changed its shape. In the last analysis it can 
 be easily shown that all the molecules have rotated through 
 some angle 0. 
 
 I trust that the reader will now have a clear idea of what 
 I mean when I speak of a " static liquid." 
 
 
 1 1 
 
 
 1 1 
 
 
 1 , 1 
 
 A 
 
 B 1 
 
 
 1 
 
 
 1 1 
 
 FIG. 22. 
 
 A property, which is implied in the definition of a static 
 liquid, is that the pressure is the same in all directions at any 
 given point : this is a fundamental question on which a 
 great part of the science of Hydrodynamics is based, as well 
 as the whole of Hydrostatics. 1 
 
 We see, therefore, that if a liquid is moved past a solid, at 
 uniform velocity, the liquid is a " static " one. In short, a 
 
 1 The equations of Hydrostatics are founded on the principles 
 that the mutual action of two adjacent elements of a fluid is normal 
 to the surface which separates them, and that the pressure is equal 
 in all directions. . . . The same assumption is made in Hydro- 
 dynamics, and from it are deduced the fundamental equations of fluid 
 motion. But the verification of our fundamental law in the case 
 of a fluid at rest, does not at all prove it to be true in the case of a fluid 
 in motion, except in the very limited case of a fluid moving as if it were 
 a solid " (Sir George Stokes, Mathematical and Physical Papers, 
 vol. i. Italics added). The fluid moving as if it were a solid is[ 
 very clearly, a static fluid. 
 
46 Motion of Liquids 
 
 static liquid is one which is moving like a solid, its velocity 
 being either positive, or negative, or zero. It is clear, also, 
 that a gravitational flowing liquid is not a static liquid. 
 When the body moves past a liquid at rest (as defined pre- 
 viously) at a uniform velocity, the liquid is, in all cases, a 
 static one, but not if the body is being accelerated. 
 
 In referring to Pitot tubes it is very common for writers 
 to speak of the static and dynamic pressures as being recorded 
 by them. This is very confusing, for it is clear that if a 
 liquid is not a static one you cannot measure its " static 
 pressure " : it is further misleading as it suggests that the 
 pressure in the liquid is different in different directions. 
 This leads to a flat contradiction ; for, by definition, in a 
 static liquid the pressure must be equal in all directions. 
 
 In order to try and clarify the subject I may point out 
 that if the plane of the orifice of the tube is opposite, and 
 normal to the flow of any stream filament, the flow of the 
 liquid at the orifice is reduced to zero : so that the height 
 to which it rises is a measure of the " velocity head/' This 
 is commonly called the " dynamic pressure " ; but a little 
 reflection will show that it is a measure of the kinetic energy 
 of the stream line at this particular spot. I therefore prefer 
 to call it the kinetic pressure or kinetic head : and when 
 H is used by me it will always be intended to mean the 
 measure of the kinetic energy at the place where the orifice of 
 the tube is, i.e., the kinetic head. 
 
 Now, since E^, +E A constant, it is not difficult to calcu- 
 late the potential energy or pressure of the stream filament 
 at this point. The Pitot tube will, however, do this for us, 
 without any trouble ; for if we turn the plane of the orifice 
 through two right angles, so as to be directly away from the 
 line of flow, the liquid will descend as much below its primitive 
 level as it rises above it in the former case, when the orifice 
 is facing the stream. It is clear that this reading H' may 
 be used as a measure of the potential energy : so, as before, 
 we may call H' the " potential head," or pressure, of the 
 stream filament at this particular spot. It must be noted 
 that I always consider the values of H and H' as measured 
 in the same direction : H' is not the measurement below the 
 
Experiments made before Duchemin's time 47 
 
 primitive level of the stream, i.e., measured negatively, but 
 a positive measure above some arbitrary datum line. 
 
 Similarly, I shall always express the " velocity head " of 
 the undisturbed stream at this point by h. 
 
 Having clearly denned the ^ 
 
 exact meaning of these terms, 
 if we imagine fig. 23 to repre- 
 sent the values of H, h and H' 
 at some point near the plate 
 where the stream has been dis- 
 turbed, so that the stream- 
 filaments are flowing faster at 
 this point than the velocity of 
 the undisturbed stream ; they 
 are all measured from an ar- 
 bitrary line h units below the 
 upper extremity of h. 
 
 It is clear that since H is as 
 much greater than h, as H' is 
 less : 
 
 H h=hK' therefore 
 
 H + H' = 2/*; also 
 
 FIG. 23. 
 
 h 
 
 Now if we calculate H by formula (a) and measure H' we 
 
 should find that 
 
 h 
 
 = 2. 
 
 In order to verify this, Duchemin made a thin square 
 sheet-iron box of 0-300 m. side and 0-030 m. thickness. The 
 front face abed had five holes of 3 millimetres diameter 
 pierced in it : No. i at the centre ; Nos. 2 and 3 in the same 
 vertical line at distances 0-075 m. and 0-150 m. ; Nos. 4 
 and 5 on the diagonal at distances respectively o - io6 m. 
 and 0-212 m. There was a vertical tube to this box and a 
 float in it, as shown in fig. 24, so as to measure the level of 
 the water in the tube. This box was fixed on the front of 
 a boat, the velocity (or h) of which was measured by another 
 independent Pitot tube, which was quite clear of the box. 
 It will be evident that by closing all the holes but one, the 
 pressure at the open hole can be measured by the rise or 
 
4 8 
 
 Motion of Liquids 
 
 fall of the float. The results are shown in the annexed 
 Table IV, where the first column gives the number of the 
 
 FIG. 24. 
 
 TABLE IV 
 BODY IN MOTION. 
 
 No. 
 
 Value of 
 1, 
 
 Value of 
 
 H/ 
 
 Value of 
 
 Values of 
 
 rf 
 
 n 
 
 
 H' 
 
 
 Hole. 
 
 Velocity 
 Head 
 
 Potential 
 Head 
 
 IT 
 
 Q 
 
 H 
 
 () 
 
 
 
 
 
 
 h 
 
 h ) 
 
 
 m. 
 
 m. 
 
 
 m. 
 
 
 
 i 
 
 0-056 
 
 0-058 
 
 1-036 
 
 o-ooo 
 
 I -000 
 
 2-036 
 
 2 
 
 0-051 
 
 0-035 
 
 0-686 
 
 0-075 
 
 1-296 
 
 1-982 
 
 3 
 
 0-053 
 
 O-O22 
 
 0-415 
 
 0-150 
 
 1-591 
 
 2-006 
 
 4 
 
 0-057 
 
 0-034 
 
 0-596 
 
 0-106 
 
 1-418 
 
 2-014 
 
 5 
 
 0-049 
 
 0-007 
 
 0-143 
 
 O-2I2 
 
 1-835 
 
 1-978 
 
 H 
 
 The values of i- have been calculated by the formula (a). 
 
Experiments made before Duchemin's time 49 
 
 hole ; the second column the value of h, or the velocity head 
 of the boat ; column 3, H', the reading of the tube in the box 
 (what I have called the " potential head ") ; column 4, the 
 
 TT/ 
 
 relation ; column 5, the value of e ; column 6, the value 
 h 
 
 TT 
 
 of (H calculated by formula (a) ) ; and column 7, the 
 h 
 
 TT -4- TT r 
 value of - . It will be seen that the experiment 
 
 thoroughly confirms what was said previously, from a theo- 
 retical point of view, when the body is in motion and the 
 liquid at rest. 
 
 Dubuat in the Principes d'Hydraulique, No. 474, gives 
 the results of exactly similar experiments, detailed in Table 
 V, which requires no special explanation. The box was i 
 
 TABLE V 
 BODY IN MOTION 
 
 No. 
 
 Value of 
 
 h 
 
 Value of 
 H'' 
 
 Relation 
 
 Values of 
 
 of 
 Hole. 
 
 Velocity 
 Head. 
 
 Potential 
 Head. 
 
 H' 
 k 
 
 
 
 
 H 
 
 H + H' 
 
 
 
 
 
 
 h 
 
 h 
 
 
 lignes 
 
 lig- 
 
 
 pouces. 
 
 
 
 
 19-52 
 
 21 
 
 
 
 
 
 
 19-81 
 
 22 
 
 
 
 
 
 
 21-91 
 
 2 4 
 
 
 
 
 
 
 23-22 
 
 25 
 
 
 
 
 
 i 
 
 39-34 
 
 42 
 
 1-063 
 
 o 
 
 i-ooo 
 
 2-063 
 
 
 39-99 
 
 4 2 
 
 
 
 
 
 
 43' 11 
 
 45 
 
 
 
 
 
 
 48-29 
 
 50 
 
 
 
 
 
 
 50-00 
 
 52 
 
 
 
 
 
 ( 
 
 26-67 
 
 8-5 
 
 ) 
 
 
 
 
 2 
 
 51-97 
 
 16-5 
 
 \ 0-318 
 
 6 
 
 1-640 
 
 1-958 
 
 1 
 
 58-09 
 
 18-5 
 
 I 
 
 
 
 
 foot square (old French feet =12 polices =144 lignes). Hole 
 No. i was at the centre of the box ; hole 2 was at the edge 
 of the box (Fig. 24 will practically serve for both sets of 
 
50 Motion of Liquids 
 
 experiments). It will be seen that these experiments also 
 quite confirm what had been argued theoretically, when 
 the body was moving and the liquid was at rest. 
 
 They are further useful in deciding another point. The 
 assumption was made that there was no impact with shock : 
 since the conclusions agree with experiment it appears 
 reasonable to think that the assumption was correct there 
 was no impact with shock. 
 
 The experiments made when the body was at rest and the 
 water was flowing past it lead to some surprises ; and so 
 will be reserved for a separate chapter. 
 
 SUMMARY 
 
 Experiments made before the time of Duchemin, by 
 careful observers, confirm the accuracy of formula (a) 
 
 }i = h(i + \ : the sole exception being the experiments of 
 V KJ 
 
 Avanzini, which appear questionable, in consequence of 
 the plate having been free to oscillate. 
 
 Separate experiments of Duchemin and Dubuat where 
 kinetic energy was calculated and potential energy was 
 measured (the body moving in a liquid at rest) also confirm 
 the value of the formula. 
 
 REFERENCES 
 
 1'Abbe BOSSUT (and M. M. D'ALEMBERT et le Marquis DE 
 CONDORCET), Nouvelles experiences sur la resistance des Fluides. 
 Bulletin d& I'lnstitut de Koutchino, Fascicule iv., 1912. 
 Le Chevalier DE BORDA, Memories de I' Academic. 1763. 
 Professor Dr. J. C. KAPTEYN, Recent Researches in the Structure of 
 the Universe. 
 
CHAPTER V 
 
 RELATIVE MOTION MOTION OF STREAM FILAMENTS IN FRONT 
 OF A BODY AT REST, EXPOSED TO A FLOWING STREAM 
 " DUBUAT'S PARADOX " 
 
 THE reader who has not thought much about the matter, 
 will, perhaps rather naturally, presume that whatever is 
 true when a plate is in motion, in water at rest, will be equally 
 true when these conditions are reversed i.e., when the 
 body is at rest and the water flows past it. 
 
 Dubuat (Principes d'Hydraulique) says, in referring to this : 
 " All those who have studied the resistance of liquids, 
 have adopted as a first axiom, that the force capable of 
 keeping a body at rest in a liquid in motion, was equal to 
 the force necessary to move it with the same velocity in the 
 same liquid at rest. This principle, without being proved, 
 seemed at least so probable, that it was never doubted, and 
 it was never considered necessary to verify it by experiment : 
 but one often risks deceiving oneself, when one applies to 
 fluids the laws of motion which apply to solids. Is it not 
 possible to believe that, in a state of rest, water offers greater 
 facility in allowing itself to be divided, and consequently less 
 resistance than when it is in motion ? " * [Italics added]. 
 
 1 Tous ceux qui se sont occupes de la resistance des fluides, 
 ont pose pour premier axiome, que 1'effort capable de retenir un 
 corps immobile dans un fluide en mouvement, etait egal a la force 
 necessaire pour le mouvoir avec la meme vitesse, dans le meme 
 fluide en repos. Ce principe, sans etre prouve, paraissait du moins 
 si probable, qu'on n'en doutait pas, et qu'on n'avait jamais cherche 
 & le verifier par aucune experience ; mais on risque sou vent de se 
 tromper, quand on applique aux fluides les lois du mouvement qui 
 
 51 E 
 
52 Motion of Liquids 
 
 Mr. Lanchester (Aerodynamics) referring to this says : 
 "It is evident that the difference is merely one of relative 
 motion. The problems are identical " [Italics added]. 
 
 It is only fair to point out that, though he says this at 
 page 1 6, at page 181 his view is quite different. 
 
 " The experimental basis of the law of the normal plane 
 is two-fold ; test of wind pressure at known mean velocity, 
 and experiments on the resistance to motion of planes 
 through still air. At first sight there might appear to be no 
 fundamental distinction between these two methods ; the 
 difference might be thought to be merely one of relative motion ; 
 owing, however, to certain considerations that require to be 
 taken into account, the results obtained by the two methods 
 are strikingly different, and the discrepancy in the value of 
 the constant as given by different writers may be to a certain 
 extent explained " [Italics added]. It is not easy to see 
 how these two statements can be reconciled. 
 
 So also Professor A. F. Zahm (Scientific American Supple- 
 ment, July 20, 1912) : "An important function of theo- 
 retical aerodynamics is to determine the velocity and stress 
 of a fluid at every point of a medium when it flows past an 
 obstacle. . . . Equivalent results may be obtained if the 
 object is assumed to move against the fluid, since only the 
 relative motion is of consequence. This is regarded as self- 
 evident." [Italics added]. This remark is the more extra- 
 ordinary from Professor Zahm since he refers to and defines 
 a static liquid : Mr. Lanchester does neither, that I am 
 aware of. 
 
 The most emphatic writer, however, is M. Alexandre See 
 (Les lois experimentales de I' Aviation), for he says : 
 
 " In virtue of the principle of relative motion, these two 
 cases are identical, and the results found should be the same." 
 To this he adds, in a note, " There have been experimenters 
 who have not perceived this principle, who have believed it 
 necessary to make experiments in both cases, and who, won- 
 
 conviennent aux solides. Ne pourrait-on pas croire que, dans 
 1'etat de repos, 1'eau offre plus de facilite a se laisser diviser, et par 
 consequent moins de resistance que quand elle est en jnouvement, 
 (Principes d'Hydraulique.) 
 
Relative Motion 53 
 
 derful to relate, have found entirely different results. Such 
 is the case of Duchemin. . . ." 1 [Italics added]. In 
 parenthesis, I may add here that Duchemin was only follow- 
 ing out the programme proposed by the Academie des 
 Sciences.; in which it was laid down that the resistances, 
 etc., w r ere to be measured both when the body was at rest and 
 the liquid was in motion, as well as when the body moved in 
 still water. Apparently the authorities of the Academie at 
 that date were not prepared to assume that the conditions 
 were identical which they are not. 
 
 Such a remark from M. Alexandre See, is the more extra- 
 ordinary, seeing that, in the early part of his book, he gives 
 this very excellent advice : "In matters of aviation, nothing 
 is evident ; everything is unexpected and paradoxical. When 
 you read in a work the word ' evidently/ or ' it is evident 
 that/ be cautious and look carefully to see if the author is 
 not talking nonsense." 2 
 
 Now, following M. See's very excellent advice, let 'us 
 examine this (?) " self-evident " proposition. What is the 
 truth about this reversibility of the conditions ? I believe 
 the problems are identical if all the conditions are identical, 
 but not necessarily otherwise. 
 
 To explain my meaning further, if a tank be filled with 
 water the resistance encountered by a body moving through 
 this water will be the same if (i) the body is moving and the 
 tank is at rest, or (2) if the tank be moved past the body at 
 rest. All the conditions here are identical : the liquid in 
 both cases is a static liquid, and the relative motion is the 
 same. I can conceive no reason why it should not be so 
 
 1 En vertu du principe du mouvement relatif, ces deux cas sont 
 identiques, et les resultats trouves doivent etre les memes. 
 
 Note. 
 
 II sont trouves des experimentateurs qui n'ont pas aper9u ce 
 principe, qui ont cru devoir faire des experiences dans les deux cas. 
 Et qui, chose admirable, ont trouves des resultats entierement 
 differents. 
 
 Tel est le cas de Duchemin. . . 
 
 2 Rien n'est evident, en matiere d'aviation ; tout est inattendu 
 et paradoxal. Lorsque vous lisez, dans un ouvrage, le mot " evi- 
 demment," ou " il est evident que," mefiez vous, et examinez bien 
 si 1'auteur n'est pas en train de dire une betise. 
 
54 Motion of Liquids 
 
 mathematically it should be so though I am not aware that 
 the experiment has ever been tried. 
 
 If the water flows past the body at rest, the conditions are 
 different : the liquid is not a static liquid. The results 
 might, of course, be the same ; but I see no a priori reasons 
 for assuming that they ought to be so, and we must not be 
 surprised if we find that they are different. 
 
 With this, I hope, not unnecessarily long preamble, I pro- 
 pose now to consider the results of some experiments bearing 
 on this point. 
 
 In the last chapter we examined Duchemin's experiments 
 with the thin square box, shown in fig. 24, when it was fixed 
 to a boat. The same box was next fixed in a stream of 
 running water and the same experiments were repeated : 
 the results are given in the Table VI, which is exactly com- 
 parable with Table IV, of his experiments with the box fixed 
 to the boat. 
 
 The first thing that will strike the reader is that in Col. 
 
 W 
 
 4, at the central hole, is not i-o, but 1-5 50 per cent. 
 
 h 
 
 more than was the case when the body was moving in still 
 water. It will also be seen that the readings at the holes 
 
 TT/ 
 
 2 and 4 of the values of - - are also greater than those 
 
 found in Table IV, and that this excess is also 0-5. At 
 the holes 3 and 5, on the contrary, the readings are less 
 by 0-5. In consequence of this last, Duchemin added a 
 
 TT 
 
 unit to the value of , so as to bring the results " into line." 
 h 
 
 TT' 
 
 It is clear that if for these holes was less than that of 
 h 
 
 the others by -5 4-5 =i, it was necessary to add a unit 
 somewhere, if he wanted the results to all agree : it would 
 have been just as easy to have added it to the values of 
 
 tr 
 
 T 
 
 I am afraid one must confess that Duchemin's method 
 appears a little arbitrary and hardly to be justified. We 
 
Relative Motion 
 
 must, I regret to say, find the gallant Colonel " guilty " of 
 having cooked his Table VI. 
 
 TABLE VI 
 BODY AT REST 
 
 
 Value of 
 
 Value of 
 
 
 Values of 
 
 No. of 
 
 h. H 1 
 
 Value of 
 
 
 Hole. 
 
 Velocity \ Potential H ' 
 
 TT 
 
 H H'\ 
 
 
 Head. 
 
 Head. 
 
 h ' 
 
 
 - 1 -- 1 - 
 *' 
 
 / i 
 
 5\ h > 
 
 
 m. 
 
 m. 
 
 m. 
 
 
 i 
 
 0-054 
 
 0-082 
 
 1-519 
 
 o-ooo 
 
 i -ooo 2-015 
 
 2 
 
 0-054 
 
 0-065 
 
 1-209 
 
 0-075 
 
 1-296 2-004 
 
 3 
 
 0-054 
 
 0-005 
 
 -0-093 
 
 0-150 
 
 2-591* : 1-998* 
 
 4 
 
 0-054 
 
 0-058 
 
 1-074 
 
 0-106 1-418 1-994 
 
 5 
 
 0-054 
 
 -0-018 
 
 -Q-333 
 
 O-2I2 
 
 2-835* 
 
 2-OO2* 
 
 i_r 
 
 The values of have been calculated by the formula (a) 
 h 
 
 adding one unit to those marked with a star in consequence of 
 
 TT/ 
 
 being negative. 
 h 
 
 Neglecting, however, the last three columns of this table 
 and confining our attention to Cols, i, 2 and 3, it is clear 
 that the experiments show that, whatever is happening at 
 the central regions of the plate is not occurring at the edges 
 in fact, exactly the opposite effect is to be observed there. 
 The pressures, except at the edge, are all 0-5 h greater when 
 the plate is at rest than when the plate is moving, whilst 
 at the edges the pressures are 0-5 h less when the plate is at 
 rest than when the plate is moving. Thus, it appears that 
 whatever it is which is acting to cause increased pressure at 
 the central regions of the plate, ceases to act somewhere near 
 the edge : and then there is some other action which reduces 
 the pressure at the edge by 0-5 h. There is a very rapid, 
 fall in the pressure gradient near the edge of the plate, 
 when it is at rest, and this pressure gradient is much less 
 steep when the body moves. This rapid fall of pressure is 
 generally explained by that very inelegant term " suction." 
 This word suction, besides being inelegant, is likely to be 
 
5-> Motion of Liquids 
 
 misleading as it rather suggests the liquid being under 
 tension, which except in very special cases is absurd. 
 In the last analysis all pressures in liquids are positive ; 
 and if they sometimes appear to be negative, it is simply 
 because the datum line has been fixed too high no account 
 being taken of the pressure of the atmosphere, which is con- 
 siderable. Whilst, therefore, I do not agree with the artifice 
 employed by Duchemin, I have thought it best to leave his 
 Table as he gave it, and simply to explain my reasons for 
 disagreement. This particular question is of no special 
 importance to my argument. I am satisfied to point out 
 that the pressures at all the points, except at the edges are 
 5 h higher when the plate is at rest in flowing liquid, than 
 they are when the plate is moving in liquid at rest. 
 
 We also see in Duchemin's table that in the last column 
 
 TT 1 TT, 
 
 = 2-5 and not =2 as in the former cases. The 
 
 results are clearly not the same. 
 
 The greatest and most serious question which confronts 
 us is, however, that we appear to have here a violation of 
 one of our fundamental assumptions the conservation of 
 energy. It is clear that if H is a measure of the kinetic 
 energy, whilst H' is a measure of the potential energy, 
 and since, by assumption, ~E k +E p ^constant, per unit 
 volume of liquid, also H +H' =2h, when the body is 
 moving : we have an apparent generation of energy creation 
 of energy in fact, if it becomes =2 -5 h when the fluid is 
 moving which is, of course, absurd. All the other difficul- 
 ties are trivial compared to this ; and if it cannot be ex- 
 plained, the whole theory crumbles to the ground. I will 
 go into the question as thoroughly as I can later, but mean- 
 while we will examine the results of other experiments to 
 see if they agree with those carried out by Duchemin. 
 
 Dubuat carried out exactly similar experiments to those 
 of Duchemin : in fact, to state the case accurately, it was 
 Duchemin who carried out the same experiments as Dubuat ; 
 he copied Dubuat and only repeated work which had been 
 done half a century earlier. Dubuat 's box was also of 
 tinned iron, I foot square (old French measure, but nearly 
 
Relative Motion 
 
 57 
 
 0-3 m.) and 4 lines thick, as against Duchemin's 0-03 m. 
 It will be clear that the experiments are very similar : 
 Dubuat's box was thinner, 0-33 inches as against 1-18 inches ; 
 his holes were smaller, 0-083 inches, as against Duchemin's 
 o-nS inches. The holes were similarly arranged, so that 
 fig. 24 may be taken to represent both, and the method 
 of measurement was the same. The results are given in 
 
 TABLE VII 
 BODY AT REST 
 
 
 Value of 
 
 Value of 
 
 
 Values of 
 
 No. of 
 
 h 
 
 H 1 
 
 Value of 
 
 
 Hole. 
 
 Velocity 
 Head. 
 
 Potential 
 Head. 
 
 H 1 
 h' 
 
 e 
 
 H 
 
 F 
 
 4/H + Hi\ 
 
 5\ h ) 
 
 
 lines 
 
 lines 
 
 
 
 
 
 i 
 
 21-5 
 
 32-8 
 
 1-525 
 
 o 
 
 i-ooo 
 
 2-O2O 
 
 2 
 
 21-5 
 
 27-8 
 
 1-293 
 
 36 
 
 1-320 
 
 2-090 
 
 3 
 
 21-5 
 
 20-8 
 
 0-967 
 
 62 
 
 1-551 
 
 2-OI4 
 
 4 
 
 21-5 
 
 -5-5 
 
 -0-256 
 
 72 
 
 2-640* 
 
 I-907* 
 
 5 
 
 21-5 
 
 -8-5 
 
 -o-395 
 
 101-8 
 
 2-905* 
 
 2-008* 
 
 TT 
 
 The values of have been calculated by the formula (a) 
 
 h 
 adding one unit to those marked with a star, in consequence of 
 
 H 1 
 
 being negative. 
 
 h 
 
 Table VII, and are exactly comparable with those of Duche- 
 min. All the remarks made about the last table are strictly 
 applicable to this one, which may be said to speak for 
 itself. The holes 4 and 5 were at the edge of the plate. 
 This table, which is copied from Duchemin, also contains 
 the artifice for holes 4 and 5 which I have considered as 
 questionable. 
 
 Dubuat carried out an experiment which is somewhat 
 similar to the foregoing ; but there are differences which 
 are certainly very interesting, and it is well worth bringing 
 it to the reader's notice, as it confirms the results of the 
 other experiments in a rather striking manner. In this 
 case the box was only partly immersed ( 442) . 
 
Motion of Liquids 
 
 The box was 14 inches in height (French inches) and 12 
 inches in breadth. It was made of one inch boards and 
 measured 12" X4* over all. The top was open so that the 
 float could easily be seen (fig. 25). There were 13 holes 
 bored in the front, as shown in the figure, of 6 lines (-J inch) 
 
 n 
 
 Straw Je Gauge, 
 
 000 
 
 000 
 
 o o 
 
 000 
 
 V 
 
 FIG. 25. 
 
 diameter. Fig. 25 will show the arrangement generally, 
 and how it was fixed to the post driven into the bed of the 
 stream. The box was immersed in the liquid to a depth 
 of 8 inches 4 lines, and, as explained previously, when all 
 the holes were stopped, a lip was formed in front of the box 
 with a central height of 3 inches 7 lines, the lateral heights 
 being 2 inches 9 lines only. The results, with different holes 
 opened, are shown in Table VIII, which is strictly comparable 
 with the two preceding ones. It will be seen that, at the 
 
 TT> 
 
 central hole - - is sensibly less than 1-5 though consider- 
 
Relative Motion 
 
 59 
 
 ably in excess of i-o this I am inclined to attribute to the 
 size of the holes, which were, I think, too large to give the 
 best results. The pressures at the last three holes were 
 also lower than they would apparently have been, had the 
 holes been smaller. 
 
 The chief interest in this experiment is that it shows that 
 the flowing stream can sustain a very considerable head of 
 water in the box very much greater than is considered 
 possible, theoretically even when the holes are very large. 
 These holes are not " pin-holes/' but are of such size that 
 one can put a finger through them. 
 
 TABLE VIII 
 BODY AT REST 
 
 
 Value of 
 
 Value of 
 
 Value of 
 
 Values of 
 
 No. of 
 
 h 
 
 H 1 
 
 
 
 Hole. 
 
 Velocity 
 
 Potential 
 
 H 1 
 
 
 
 
 
 Head 
 
 Head 
 
 ' Ti~* 
 
 
 
 4 ( **- ~f~ " \ 
 
 
 
 
 
 
 h 
 
 5\ h ) 
 
 
 lines. 
 
 lines. 
 
 
 lines. 
 
 
 
 i 
 
 29-33 
 
 40 
 
 1-364 
 
 16-00 
 
 1-142 
 
 2-005 
 
 2 
 
 29-33 
 
 37 
 
 1-262 
 
 27-40 
 
 1-238 
 
 2-OOO 
 
 3 
 
 29-33 
 
 36 
 
 1-227 
 
 34-60 
 
 1-307 
 
 2-027 
 
 4 
 
 29-33 
 
 29 
 
 0-989 
 
 39-50 
 
 I-35I 
 
 1-872 
 
 5 
 
 29-33 
 
 18 
 
 0-614 
 
 78-90 
 
 1-702 
 
 I-853 
 
 6 
 
 29-33 
 
 25 
 
 0-852 
 
 44-00 
 
 I-39I 
 
 1-794 
 
 TT 
 
 The values of have been calculated by the formula (a). 
 
 This very curious and interesting point was first pointed 
 out by Dubuat, more than a century ago, and is commonly 
 spoken of as " Dubuat's Paradox." It is not commonly 
 accepted, at the present day possibly because it appears 
 like a violation of the law of the Conservation of Energy 
 but I am not aware that any one has taken the trouble to 
 verify the experiments of Dubuat and Duchemin. M. Eiffel 
 (La resistance de I' air) says in reference to it : " Dubuat 
 and even Duchemin, for example, have found by experi- 
 ments with water that the resistance was not the same 
 
60 Motion of Liquids 
 
 according to whether the body or the water were in motion, 
 This conclusion, which constitutes the paradox of Dubuat, 
 is actually rejected by the majority of experimenters." * 
 
 Before proceeding to examine the question further, by 
 the aid of other experiments, let us see what explanation, 
 if any, Dubuat and Duchemin give of this " Paradox." 
 
 Dubuat (Principes d'Hydraulique, 454) says somewhat 
 laconically " one sees that the head due to the pressure 
 against the centre of a plane surface, directly shocked, is equal 
 to one and a half times that due to the velocity ; and this 
 should be general, whatever be the size of the surface." 2 
 
 This can hardly be considered very satisfying ; for though 
 he says that this should be the same, whatever the size of the 
 surface he gives no explanation why it should be. If we are 
 told a thing should be, our first question is, naturally, why 
 should it be ? 
 
 There is a paragraph of Dubuat 's ( 425) which has some 
 bearing on this subject, and which appears to show that he 
 had some possibly indistinct idea why this should be 
 so. Referring to the liquid filaments being deviated in 
 front of the body at some distance away, he says : "If this 
 deviation occurred in all the filaments, at a distance, very 
 great, relatively, to the extent of the surface, the shock would 
 become insensible, which is not the case." 3 Dubuat has 
 an unfortunate habit of using the word " shock " in two 
 senses, and there are times when it is not clear in which 
 sense he is employing it. Dubuat was evidently well aware 
 that Bernoulli's law is only true when there is no " shock " 
 in the sense employed by me. 
 
 1 Dubuat et meme Duchemin, par exemple, ont trouve par des 
 experiences sur 1'eau que la resistance n'etait pas la meme suivant 
 que le corps ou 1'eau etait mis en mouvement. Cette conclusion, 
 qui constitue le paradoxe de Dubuat, est actuellement rejetee par 
 la plupart des experiment ateurs. 
 
 2 On voit que la hauteur de pression centre le centre d'une surface 
 plane, choquee directement, est egale a une fois et demie celle qui est 
 due d la vitesse du fluids , et ceci doit etre general, quelle que soit 
 la grandeur de la surface. 
 
 3 Si cette deviation avait lieu pour tous les filets, a une distance 
 tres-grande, relativement a 1'etendue de la surface, le choc devien- 
 drait insensible, ce qui n'est pas. 
 
Relative Motion 6t 
 
 Duchemin is equally vague, for he says : " The explana- 
 tions which I could give on this subject would be long 
 without being completely satisfying." 1 
 
 So far we have no explanation of any kind. The few 
 authors that ever mention Dubuat's Paradox are generally 
 satisfied by saying that Dubuat's experiments were inaccu- 
 rate ; in fact, there is a story that a distinguished English 
 civil engineer said at a meeting that he hoped Dubuat's 
 experiments might be " decently buried " : such arguments 
 are of the poorest kind. Dubuat was notoriously a keen 
 observer and a very good experimentalist. Why, then, 
 should all his experiments made with a body moving give 
 correct results i.e. in accord with accepted theory whilst 
 all his experiments made with a body at rest should show 
 an excess pressure at the centre of the body of 50 per cent. ? 
 If the experiments were bad ones we should expect some to 
 be 10 per cent., or 15 per cent., or some other percentage in 
 error. No ! they are all within the limits of experimental 
 error exactly 50 per cent. Even if we question the accuracy 
 of Dubuat's experiments, how can we account for the fact 
 that Duchemin, half a century later, repeated them ; and 
 got exactly the same results ? 
 
 I think we may say that the point which is considered as 
 " self-evident " (so self-evident that M. See considers it 
 absurd to question, much less to verify by experiment], is not 
 correct ! Thus there is a great deal to be said for the famous 
 " paradox of Dubuat." 
 
 In the next chapter I will refer to more experiments and 
 quotations bearing on this subject, before offering an ex- 
 planation of this curious question. 
 
 1 Lcs explications que je pourrais donner a ce sujet 
 longues sans etre completement satisfaisantes. 
 
62 Motion of Liquids 
 
 SUMMARY 
 
 When a body is at rest in a flowing stream the pressures 
 on the anterior surface are not the same as when the same 
 body is moving at the same velocity in a liquid at rest. 
 The pressures are considerably greater. There is evidently 
 something to be said for " Dubuat's paradox." Whenever 
 the liquid is a static one, the results obtained are always in 
 accord with theory : when the liquid is not a static one, 
 the results do not agree with theory. 
 
 REFERENCES 
 
 F. W. LANCHESTER, Aerodynamics. 
 
 A. F. ZAHM, Scientific American Supplement, July 20, 1912. 
 S. ALEXANDRE SEE, Les lois Experimentales de I' aviation 
 
 G. EIFFEL, La resistance de Vair, 1910. 
 
CHAPTER VI 
 
 " DUBUAT'S PARADOX " continued 
 
 THE question of " Dubuat's paradox " being a very interest- 
 ing, and a very important one, I must, even at the risk of 
 being tedious, refer to all the experiments that I know of and 
 which bear on the subject. The following is one of Duche- 
 min's. 
 
 If you place a Pitot tube with the orifice strictly normal 
 to the line of flow of any liquid filament of a stream, the 
 water rises in the tube to a level h (keeping to our old nota- 
 tion) which is called the " velocity head." If, however, 
 this orifice is placed in the centre of a small plate, the same 
 result is not arrived at : the liquid (provided the plate is not 
 too small) rises to a level of 1-5 h. Duchemin tried the 
 experiment by placing small round discs round the aperture 
 
 TABLE IX 
 BODY AT REST 
 
 Diameter 
 
 Value of 
 
 Value of 
 
 Value of 
 
 of the 
 
 h 
 
 H. 
 
 H 
 
 Discs. 
 
 Velocity Head. 
 
 Kinetic Head. 
 
 V 
 
 m. 
 
 m. 
 
 m. 
 
 
 0-003 
 
 0-057 
 
 0-057 
 
 I -000 
 
 o-oi 
 
 0-057 
 
 0-081 
 
 1-421 
 
 0-02 
 
 0-057 
 
 0-855 
 
 1-500 
 
 0-03 
 
 0-057 
 
 0-86 
 
 1-509 
 
 0-05 
 
 0-057 
 
 0-85 
 
 1-492 
 
 6.* 
 
64 Motion of Liquids 
 
 of the tube and the results are given in Table IX. The 
 orifice of the tube was 3 millimetres. 
 
 It will be seen that when this was placed in the centre of 
 
 TT 
 
 a disc, only one centimetre in diameter, the value of was 
 
 ri 
 
 1-421. I need not remind the reader that one centimetre 
 is less than T \ of an inch a very small disc. If the disc 
 
 TT 
 
 is increased to 2 centimetres, the value of increases to 
 
 ft 
 
 1-5, and, very curiously, further increase of the size of the 
 
 TT 
 
 disc does not increase the value of . 
 
 h 
 
 Dubuat ( 436) describes a somewhat similar experiment. 
 He made use of the flood gate of a mill race (Fig. 26) 
 which was raised 5 inches and 9 lines above the sill of the 
 race, this sill being nearly level on both sides of the gate : 
 the opening was about four feet broad, and the head of 
 water above the centre of the orifice was constantly 2 feet 
 i inch and one line. A small plate (16 lines in diameter), 
 and which had the point of a Pitot tube soldered into its 
 centre 'Similarly to Duchemin's small plates was pre- 
 sented to this running water, a little below the centre of the 
 opening, so that there was always a head of 2 feet one inch 
 and four lines above the centre of the plate, and about two 
 or 3 inches distance between the plate and the sluice gate, 
 so as to be about at the point of greatest contraction of the 
 fluid stream. In this position, the water rose in the tube 
 to a height of 2 feet 4 inches and n lines ; that is to say, 
 3 inches and 7 lines higher than the level of the reservoir (fig. 
 26). (Dubuat gives no diagram, but the arrangement 
 must have been somewhat as shown in fig. 26). This 
 result would, generally, be considered an impossibility. 
 
 A somewhat similar experiment is referred to by W. C. 
 Unwin (Encyclopedia Britannica Hydromechanics) : 
 
 " Pitot expanded the mouth of the tube so as to form a 
 funnel or bell mouth. In that case he found by experiment 
 
Dubuat's Paradox 
 
 " The objection to this is that the motion of the stream 
 is interfered with, and it is no longer certain that the velo- 
 city in front of the orifice is exactly the velocity of the 
 unobstructed stream." 
 
 The last paragraph is not quite clear ; for it appears 
 quite certain that the " velocity in front of the orifice " 
 was not the " velocity of the unobstructed stream." I 
 think that the real objection is that, theoretically, h cannot 
 
 v 2 
 exceed even when the fluid is at rest. Professor Unwin 
 
 2 a 
 
 does not give 
 sufficient details 
 of the experi- 
 ment. I have 
 not been able 
 to verify this 
 reference to 
 Pitot ; and I 
 am inclined to 
 think that he 
 was quoting 
 from memory 
 and confused 
 Pitot with 
 D ubuat whose 
 experiments I 
 will refer to later. 
 
 Whilst on the 
 subject of Un- 
 win 's article on Hydromechanics in the Encyclopedia Bri- 
 iannica, I should like to draw attention to an extract, 
 which has not, I think, received the attention it deserves. 
 ' * Resistance of a plane moving through a fluid or pressure 
 of the current on the plane. 
 
 ''' When a thin plate moves through the air, or through 
 an indefinitely large mass of still water, in a direction normal 
 to its surface, there is an excess of pressure on the anterior 
 face, and a diminution of pressure on the posterior face. 
 Let v be the relative velocity of the plate and the fluid, 
 
66 Motion of Liquids 
 
 Q the area of the plate, G the density of the fluid, h the 
 height due to the velocity, then the total resistance is 
 expressed by the equation 
 
 pounds =fGQh. 
 
 ' 
 
 Where / is a coefficient having about the value of 1-3 for a 
 plate moving in still liquid, and 1-8 for a current impinging 
 on a fixed plate ; whether the fluid is air or water. The 
 difference in the value of the coefficient in the two cases is 
 perhaps due to errors of experiment. There is a similar 
 resistance to motion in the case of all bodies of ' unfair ' 
 form, that is, in which the surfaces over which the water 
 slides are not of gradual and continuous curvature." 
 
 It will be seen that the formula ~R=fGQh, where/ =1-3, 
 when the body moves in still water, whilst it is 1-8, when 
 the body is at rest and the liquid flows past it, corresponds 
 to a difference of pressure of o-$h agreeing thus well with 
 Duchemin's and Dubuat's experiments of pressure on the 
 front of the plate. It appears to me to be rather a sweeping 
 remark to say that " the difference in value of the coefficient 
 in the two cases is perhaps due to errors of experiment." 
 Might this difference not be due to the different conditions 
 in the two cases ? one liquid being a static one, and the 
 other not. I may point out that Professor Unwin when 
 saying that " moving fluids, as commonly observed, are 
 conveniently classified thus/' . . . makes no reference to static 
 fluids at all. This appears to me to 'be a serious omission, 
 for if we do not recognize the difference between static 
 and non-static liquids, we cannot arrange our facts ; and 
 we are perpetually coming across apparent contradictions. 
 
 Dubuat in his experiment No. CCXIV which I cannot 
 help thinking is the one that Professor Unwin ascribes to 
 Pitot states that he made a funnel of 3 inches base, with 
 a tube bent at right angles to it of 16 lines internal diameter. 
 At the apex of the cone of the funnel there was fixed a 
 diaphragm which had a small hole of 3 lines diameter 
 pierced in it. When he exposed this normally to a stream 
 whose velocity was 36 inches per second, equivalent to a 
 " velocity head " (h) of 21-5 lines, the water rose in the 
 
" Dubuat's Paradox" 67 
 
 tube to a height of 34-4 lines, which is rather in excess of 
 
 The same experiment repeated with the same funnel, 
 but without the diaphragm, registered a pressure of 30-1 
 lines which is rather less than 1-5 h. In this latter case 
 the aperture into the tube was too large to get a satisfactory 
 result : it must be noted that 16 lines diameter is a hole 
 amply large enough for the insertion of a tea-spoon. 
 
 
 FIG. 27. 
 
 I will quote a statement of Professor Zahm's which will 
 make it clear why I have made special reference to the size 
 of these holes. In the Scientific American Supplement, 
 July 27, 1912, we find, when he is referring to the impact 
 of a jet of water against a plate closing a hole in a tank, that 
 he says : 
 
 " Obviously, the fluid in the second tank may be sus- 
 tained at any height whatever by force of the given jet, 
 if the given plate be made to close an aperture sufficiently 
 small. But in all cases it must be noted that, as formerly 
 calculated [by Bernoulli's Theorem] the unit pressure at 
 the centre of impact equals the pressure at the source t so that 
 
68 Motion of Liquids 
 
 if a pin-hole be made there in the plate, the level in the second 
 tank cannot exceed the level of the source " (Italics added). 
 
 This statement is, of course, equivalent to stating that 
 even if there were impact the level of the liquid could not 
 exceed h. Like many statements in Hydrodynamics this 
 may be said to be true and the reverse. If the liquid in 
 the jet is static, then it is quite true but not otherwise. 
 Jets very frequently behave as if they were static liquids ; 
 though they would not be considered to come within the 
 definition I have given previously. To give an example, 
 fig. 27, which may be seen daily in many sculleries. The 
 jet from the tap B, which has an " anti-splash " arrange- 
 ment adjusted to it, acts as a static liquid does it will not 
 splash. The jet from the tap A clearly, does not act as a 
 static liquid it will splash. 
 
 This question will, I trust, be clearer when I have treated 
 of jets and their resulting pressures. 
 
 Still another experiment of Dubuat. He made a small 
 thin box of sheet-iron i foot square, one surface of which 
 was pierced with 625 small holes, evenly distributed over it. 
 This box was fixed in a current whose velocity, when undis- 
 turbed, at the centre of the plate was 36 inches per second 
 corresponding to a velocity head (h) of 21-5 lines. Attached, 
 of course, was his modification of the Pitot tube, previously 
 described. When all the holes in the box were open the 
 liquid rose in the tube to a height of 28-3 lines : equivalent 
 to about 1-33 h. 
 
 The same box was fixed in front of a cube, and was exposed 
 to the same stream, also with all the holes open. The 
 mean pressure against all these holes was measured as 29-2 
 lines. 
 
 The same box, having been placed in front of a treble 
 cube," mean pressure recorded was 28-4 lines. 
 
 Vince in his " Bakerian lecture " (Phil. Trans., 1798) 
 referring to this, says : " The subject divides into two parts : 
 we may consider the action of water at rest on a body moving 
 in it, or we may consider the action of water in motion on 
 the body at rest." He found different results in the two 
 gases, but I am not inclined to attach too much importance 
 
" Dubuat's Paradox " 69 
 
 to some of his results (unless proper corrections are applied) , 
 for some were obtained by whirling table experiments where 
 the radius was very small 7-57 inches. As is well known, 
 the coefficients obtained in such experiments would be 
 very considerably in excess of what they would be if the 
 body moved in a straight line. If we can trust Duchemin's 
 formula, in some modern experiments even, a correction 
 up to 45 per cent, is sometimes necessary. 1 
 
 So confused is the state of knowledge on this subject that 
 it is not uncommon to come across passages like the follow- 
 ing : 
 
 " It will be seen that the value of K as determined by 
 moving the Pitot tube through still water differs .very con- 
 siderably from that obtained in running water. In the 
 latter case the pressure was considerably higher than in the 
 former, and it appears therefore, that K depends not only 
 upon the form of the tube but upon the pressure under which 
 it is working " (F. C. Lea, Hydraulics). 
 
 I cannot explain what this paragraph is intended to 
 convey, but it suggests (to me) that the Pitot tube is unre- 
 liable, and that its peculiarities are not at present explicable. 
 It is certainly very curious that the statement fits in with 
 what was said here previously ; and it is more curious that 
 such a very (?) unreliable instrument should have given 
 such very accurate results in the hands of Dubuat and 
 Duchemin. 
 
 Some writers go further than the above ; Mr. William 
 Kent, author of Kent's Mechanical Engineer's Pocket-Book, 
 says frankly, that the "time-honoured formula V =v 2gh 
 for the Pitot tube " is a perfectly correct theoretical formula 
 when the velocity is produced by the head, but it is incorrect 
 when the head is produced by the velocity." (The Pitot tube : 
 Its Formula, W. M. White.) 
 
 There are people who seem to think that a Pitot tube 
 requires " calibrating," for every series of experiments. 
 
 It is well known that M. de Grammont, when moving a 
 
 1 See Eiffel's references to Von Lossl's experiments (La resistance 
 de I' air, 1910). 
 
 F2 
 
Motion of Liquids 
 
 plane through still air, has obtained results which differ 
 materially from those observed by M. Eiffel, who fixes his 
 plates in a stream of moving air. The consequence is that 
 M. de Grammont's work does not meet with approval. 
 M. Eiffel, even, referred the question of " relative motion " 
 to M. Poincare, who gave the following very cautious reply : 
 " II n'y a pas de raison pour que les efforts exerces sur des 
 plaques par un courant d'air bien regulier [!!] different de 
 ceux que subirait cette plaque en mouvement dans un air 
 calme." 
 
 I trust that this very long list of experiments will not 
 have exhausted the reader's patience ; I shall have occasion 
 
 ^ to quote some other similar 
 experiments when treating of 
 jets. I think now it will be 
 clear that it is not immaterial 
 whether the body moves in a 
 .liquid at rest, or the body is 
 fixed in the midst of a flow- 
 ing liquid. All the conditions 
 are not identical, and experi- 
 ments have shown that the 
 results are not the same. I must 
 defer my offered explanation 
 of the reasons for this differ- 
 ence until after I shall have 
 
 treated of jets, so as not to distract the attention of the 
 reader, and will now continue the examination of the 
 motion of the liquid round the plate. 
 
 Having examined the motion of the liquid past the front 
 of the plate at considerable length, it is necessary to study 
 the velocity of the stream-filaments after they have left 
 the front of the plate, and have resumed a direction of motion 
 nearly parallel to their original undisturbed line of flow. 
 For this purpose a square plate, 0-3 m. side, was employed. 
 It was fixed in a flowing stream, the velocity of which, at 
 the point e, the centre of the plate, was expressed by 
 h =0-053 m. : this centre e being at a depth, below the undis- 
 turbed surface of the stream, of 0-3 m. (fig. 28). A Pitot 
 
 FIG. 28. 
 
" Dubuafs Paradox 
 
 tube was placed directly behind the edge be with the orifice 
 in a vertical plane 0-02 m. behind the front face of abed. 
 At the point b the water rose in the tube to a height of 
 0-105 m - At the point /the reading was 0-137 m. ; whilst at 
 c the rise was 0-104 m. These heights of the water in the 
 tube are those due to the velocity of the fluid-filaments 
 i.e., measures of the kinetic energy in these stream filaments. 
 If we compare them with h, the velocity head of the stream, 
 we see that they are about 2h for the points b and c, and 
 2-578 h for the point /. 
 
 Now if we calculate the velocity of the filament on arriving 
 at/by the formula (a), putting e =0-15 m., we find H =1-587 
 h. If, therefore, we express the head due to the velocity 
 of the filaments after they have left the surface by H", we 
 find H" =H +h. From which it follows that the head due 
 to the velocity of the filaments, after they have left the 
 edge of the plate, is increased by h, the velocity head of the 
 stream. 
 
 These experiments were repeated at the anterior face of 
 a cube and at that of a treble cube, or parallelepiped, having 
 a length of 0-9 m., and the results were almost exactly the 
 same as the preceding ; these are given in Tables X 
 and XI. In the case of the square plate and the cube 
 0-3 m. side, the body was moving in liquid at rest ; whilst in 
 the case of the " treble cube," the body was at rest in the 
 
 TABLE X 
 BODY MOVING 
 
 Description 
 of 
 Body. 
 
 Value of 
 h 
 Velocity 
 Head. 
 
 Dist. 
 from 
 edge of 
 Body. 
 
 Value of 
 H" 
 velocity 
 head of 
 filaments. 
 
 Value of 
 H" 
 compared 
 to h 
 
 Value of 
 H + A 
 
 h ' 
 
 
 m 
 
 m 
 
 m 
 
 
 
 Plane surface 
 
 0-045 
 
 0-020 
 
 0-116 
 
 2-586 
 
 2-587 
 
 Cube 
 
 0-042 
 
 O-O2O 
 
 0-109 
 
 2-588 
 
 2-587 
 
 do. 
 
 0-041 
 
 0-028 
 
 0-104 
 
 2-536 
 
 2-587 
 
 Square plate and cube 0-3111. side. H in last column calculated. 
 
Motion of Liquids 
 
 TABLE XI 
 BODY AT REST 
 
 Description 
 of 
 Body. 
 
 Value of 
 h 
 Velocity 
 Head. 
 
 Distance 
 from 
 Front 
 Edge. 
 
 Value of 
 H" 
 Velocity 
 head of 
 Filament. 
 
 Value of 
 H" 
 
 h ' 
 
 
 m. 
 
 m. 
 
 m. 
 
 
 Parallelepiped 
 
 0-053 
 
 0-02 
 
 0-138 
 
 2-604 
 
 0-3111 x 0-3111 
 
 0-053 
 
 0-32 
 
 0-134 
 
 2-526 
 
 X o-gm 
 
 0-053 
 
 0-58 
 
 0-132 
 
 2-490 
 
 
 0-053 
 
 0-88 
 
 0-131 
 
 2-472 
 
 Pitot tube placed o'O2m. from body at distances from the front 
 edge given in column 3, and always at an equal distance from 
 the top and bottom edges of the solid. 
 
 midst of a flowing liquid. It will be seen that the results 
 are the same in both cases, making allowance for small 
 experimental error which it is impossible to avoid in such 
 kind of work. In the experiments with the treble cube, 
 the distances of the point of the Pitot tube from the front 
 edge were gradually increased from 2 centimetres up to 88 
 centimetres ; the values of H" diminished with this increase 
 of distance. Evidently the velocity of flow of the stream- 
 filaments was being retarded ; and this retardation increased 
 as the rear of the body was approached. 
 
 Now, since the velocity increases as ^/h, we see that, 
 within the limits of speed experimented with by Duchemin, 
 the velocity of the water flowing past the edge of a plate, a 
 cube, or a parallelepiped, is roughly about 1-6 times the 
 velocity of the undisturbed stream. 
 
 It is important, next, to try and find out how far from the 
 body the water was disturbed. For this purpose a cube, of 
 0-3 m. side, was moved through still water, and the Pitot 
 tube was fixed in a plane 2 millimetres behind the front face 
 of the cube, but at different distances from the edge of the 
 cube, as given in Table XII. In all cases the Pitot tube 
 had its point situated exactly between the top and bottom 
 
" Dubuat's Paradox 
 
 73 
 
 edges of the cube : in other words, it was on a level witH 
 the centre of the cube. The velocity head was measured 
 by means of a separate tube, which was well clear of the body m 
 
 TABLE XII 
 BODY MOVING 
 
 Description 
 
 Value of 
 i 
 
 Distance 
 
 Value of 
 H" 
 
 Value of 
 
 of 
 Body. 
 
 Velocity 
 Head. 
 
 from edge 
 of Cube. 
 
 Velocity 
 Head of 
 Filaments. 
 
 H" 
 
 h ' 
 
 
 in. 
 
 m. 
 
 m. 
 
 
 
 0-046 
 
 0-02 
 
 0-118 
 
 2-570 
 
 Cube 
 
 0-043 
 
 0-05 
 
 0-098 
 
 2-268 
 
 o-yn. side 
 
 0-041 
 
 O-IO 
 
 0-071 
 
 1-730 
 
 
 0-045 
 
 0-15 
 
 0-046 
 
 1-022 
 
 
 0-043 
 
 0-25 
 
 0-042 
 
 0-980 
 
 Pitot tube placed 2 mm. behind the front surface of cube but at 
 distances from the edge given in column 3. 
 
 An examination of this table will show that at about a 
 distance of 0-15 m. about half the length of the side of the 
 cube the value of H", the velocity head of these filaments 
 
 TT* 
 
 (referred to the cube), fell to h, so that i. In other 
 
 words, at this distance the water was undisturbed. We 
 may, therefore, expect that, at that distance, in a static 
 liquid (at the velocities experimented with by Duchemin), 
 what are called " boundary effects " will cease to be appreci- 
 able. We may reasonably imagine that the " water-cube " 
 forms a complete " system " which travels in front of and 
 alongside of the cube, and at the same velocity. 
 
 If this be correct, we can very simply check the velocity 
 of the water past a plate, or the front edge of a cube, by 
 means of a little very simple arithmetic. 
 
 Let us, for example, assume the plate to be circular 
 and with a diameter =d. The sectional area of the disturbed 
 column of water having a diameter =2d, will clearly be 
 
74 
 
 Motion of Liquids 
 
 four times that of the area of the plate. The annular space 
 through which the water passes (fig. 29) has, therefore, a 
 sectional area of three times that of the plate. Since the 
 whole water in the cylinder whose diameter 2d has to 
 pass through this annular ring, its mean velocity will neces- 
 sarily be v, where v is the velocity of the undisturbed 
 
 3 
 
 stream. Now, if we assume this mean velocity to occur 
 at a point c, one-third of the distance from B to A ; the 
 
 FIG. 29. 
 
 pressure gradient, as measured by the Pitot tube, is very 
 nearly a straight line, so that the assumption made is prob- 
 ably not very far from the truth. 1 The velocity at A being 
 
 v, the velocity at d will be v, that at c will be -- v ; 
 
 o ,j 
 
 whilst the velocity at B will be ^ v =1-5 v. We see, there- 
 
 1 There are reasons for believing that there is a rather abrupt 
 fall in the pressure gradient quite dose to the plate ; this will not 
 materially affect the calculation, which is, at best, but -a very rough 
 approximation. 
 
Dubuafs Paradox 
 
 75 
 
 fore, that the velocity of flow past B should not be very 
 different from 1-6 v. 1 
 
 To borrow a simile from Dubuat, the lines of flow of the 
 stream-filaments will be not unlike those of a liquid flowing 
 out of a cylinder which 
 has a bottom as shown 
 in fig. 30. The simile 
 must not be pressed too 
 closely, however. 
 
 
 SUMMARY 
 
 Dubuat's Paradox 
 may be expressed some- 
 what differently. If a / / 
 body be exposed to a 
 static liquid, the pres- 
 sure on the centre of 
 the body (greatest 
 pressure) can never ex- 
 ceed h, the " velocity 
 head." If, however, 
 the liquid is not static, 
 this pressure generally 
 increases by -5 h, to 
 1-5 h. 
 
 It appears, further, to be true that if a body, a cube say, 
 moves through a static liquid, the cube and the water form 
 a " system " which appears to be steady. The " cube-water 
 system " is also much smaller than is frequently supposed. 
 
 REFERENCES 
 
 Rev. S. VINCE, Phil. Trans., 1798. 
 
 G. EIFFEL, La resistance de I' air, 1910. 
 
 F. C. LEA, Hydraulics. 
 
 A. F. ZAHM, Scientific American Supplement, July 27, 1912. 
 
 W. C. UNWIN, Encyclopedia Britannica. Hydromechanics. 
 
 W. M. WHITE, The Pitot Tube : Its Formula. 
 
 1 We know that in a non-viscous liquid, the velocity of flow 
 past the circumference of a sphere immersed in it, can be calculated, 
 mathematically, to be 1-5 v. 
 
 FlG - 
 
CHAPTER VII 
 
 MOTION OF THE LIQUID AT THE SIDE OF A PLATE : ALSO 
 MOTION BEHIND THE PLATE 
 
 KNOWING how the water flows past a rectangular parallel- 
 opiped it will now be interesting to examine the pressure 
 exerted by it on the sides of the prism. For this pur- 
 pose Duchemin adapted the small tin box, previously 
 referred to, to the sides of a body 0-3 m. Xo-3 m. xo-g m., 
 as shown in fig. 31, so that the hole which was opened 
 
 should be at right 
 angles to the axis 
 of the body, and at 
 suitable distances from 
 the front edge. These 
 distances are the same 
 as those given in the 
 Table XI, in the last 
 chapter ; the results 
 are therefore strictly 
 comparable. It must 
 be carefully noted that 
 in the Table XI, H" 
 was the " velocity 
 head " of the liquid 
 
 FIG. 31. 
 
 filament was, in fact, a measure of its kinetic energy 
 whilst in this Table XIII, H' is a measure of the potential 
 
 TT" 
 
 energy of these stream-filaments : - - is taken from Table XI. 
 
 h 
 
 The results are given in the Table XIII, where Col. i 
 gives distance from front edge of the prism of the reading 
 
 76 
 
Motion of the Liquid at the side of a Plate 77 
 
 of the Pilot tube. Col. 2, velocity head of stream. Col. 
 3, the value of H', the reading of the Pitot tube the measure 
 of the potential energy at this spot. It is clear that since 
 
 H' H" 
 
 is a measure of the decrease of potential energy, and 
 h n 
 
 is a measure of the increase of kinetic energy at this same 
 
 TTtf TT> 
 
 point : then + should be =zero. We see by the 
 h h 
 
 table that the pressure on a point on the side of the body 
 increases, as the distance of the point from the front edge 
 increases. An inspection of the last column will also show 
 that there was no lateral " shock." Since the pressures 
 on the opposite sides mutually balance one another, it is 
 clear that these pressures can in no way affect the resistance 
 of the body to motion in the liquid. 
 
 In order to be quite sure of what we are doing, it is advis- 
 able to see whether the same results would be obtained if 
 the conditions were reversed i.e., if the body were in 
 motion. Duchemin carried out two experiments in this 
 manner, with a cube of 0-3 m. side, and having the small 
 
 TABLE XIII 
 BODY AT REST 
 
 Dist. of 
 
 Value of 
 
 Values of 
 
 
 
 
 Orifice 
 
 h 
 
 H' 
 
 Values of 
 
 Values of 
 
 Values of 
 
 from 
 
 Velocity 
 
 reading of 
 
 H' 
 
 H" 
 
 H" H' 
 
 Front 
 
 head of 
 
 Pitot 
 
 h ' 
 
 h 
 
 h h' 
 
 Edge. 
 
 Stream. 
 
 Tube. 
 
 
 
 
 m. 
 
 m. 
 
 m. 
 
 
 
 
 0-02 
 
 o-055 
 
 0-142 
 
 -2-582 
 
 2-604 
 
 + 0*022 
 
 0-32 
 
 0-055 
 
 0-142 
 
 -2-582 
 
 2-566 
 
 0-016 
 
 0-58 
 
 0-055 
 
 -0-137 
 
 -2-491 
 
 2-490 
 
 o-ooi 
 
 0-88 
 
 0-055 
 
 -o-i35 
 
 -2-454 
 
 2-472 
 
 + 0-018 
 
 Parallelepiped 0-3 m. xo-3 m. xo-g m. 
 Axis immersed to a depth of 0-35 m. 
 
 thin tin box adapted to the sides. The results are given in 
 Table XIV, which requires no special explanation, since 
 
7 8 
 
 Motion of Liquids 
 
 what was said about Table XIII applies equally to this : 
 it is strictly comparable with it. In this case also, 
 there is no appearance of any " shock " having occurred. 
 
 TABLE XIV 
 BODY IN MOTION 
 
 Dist. of 
 
 Value of 
 
 Value of 
 
 Value of 
 
 Value of 
 
 Value of 
 
 Orifice 
 
 h 
 
 H' 
 
 
 
 
 from 
 Front 
 
 Velocity 
 head 
 
 reading of 
 Pitot 
 
 H' 
 
 h ' 
 
 H" 
 
 h ' 
 
 H* H' 
 
 h T' 
 
 Edge. 
 
 of body. 
 
 Tube. 
 
 
 
 
 m. 
 
 m. 
 
 m. 
 
 
 
 
 O-02 
 0-28 
 
 0-063 
 0-059 
 
 0-163 
 -0-149 
 
 -2-587 
 -2-525 
 
 2-588 
 2-536 
 
 o-oo i 
 
 O-OII 
 
 Cube 0-3 m. xo-3 m. xo-3 m. 
 
 We have now examined the motion of water past a body 
 immersed in it, in a very large number of ways. However 
 the subject was turned or examined, we invariably found 
 that, when the liquid was a static one, the sum of the kinetic 
 and potential energies was constant ; the errors being very 
 small and only such as are to be expected in experiments 
 of this kind. When, however, the liquids were not static, 
 frequently large departures from the assumed constant 
 values were apparent such differences as it would be absurd 
 to attribute to " experimental error." These departures 
 were quite constant : they were always in the same direction, 
 and the amount of the departure had all the appearance 
 of being subject to some law. 
 
 It only remains now to complete the examination by 
 inquiring how the liquid flows behind the plate, or body. 
 We have seen that the pressure on the anterior face of a 
 body moving relatively through a non static liquid was con- 
 siderably greater than when the liquid was static. We 
 cannot assume, without experiment, that therefore the 
 resistance to a body by a non-static liquid was greater than 
 that of a static one. The resistance can be measured only 
 by the differences of pressure on the anterior and posterior 
 
Motion of the Liquid at the side of a Plate 79 
 
 faces of the plate, say. If both pressures are increased 
 equally, the total resistance will not be altered. Just as 
 we found that the pressure on the front of a plate was not 
 the same when the plate moved, as it was when the plate 
 was fixed and the stream flowed past it as (I hope the 
 reader will say), when the liquid was " non-static " : so we 
 must not be surprised if the flow, and therefore pressures, 
 at the back of the plate differ in the two cases. The great 
 instability of the eddies at the back of a plate makes it very 
 difficult to measure velocities with any great accuracy. 
 Duchemin says he found the difficulties so great that he 
 contented himself with the measurements of the stream- 
 filaments directed towards the centres of the posterior surfaces, 
 the motion of these being stable as to direction. The results 
 of the experiments are given in Table XV : the first three 
 being for a thin plate, the next three for a cube, whilst the 
 remainder are for double and treble cubes, as shown in the 
 table. 
 
 TABLE XV 
 
 BODY MOVING 
 
 BODY AT REST 
 
 Relation! Value of 
 between h 
 length | Velocity 
 and dia- Head, 
 meter. 
 
 Value of 
 H ,,, 
 
 Velocity 
 Head of 
 Filament. 
 
 Value of 
 H'" 
 
 Value of 
 h 
 Velocity 
 Head. 
 
 Value of 
 H'" 
 Velocity 
 Head of 
 Filament. 
 
 Value of 
 H"' 
 
 h 
 
 h ' 
 
 m. 
 
 m. 
 
 
 m. 
 
 m. 
 
 
 ( 0-052 
 
 0-075 
 
 \ 
 
 0-055 
 
 0-043 
 
 \ 
 
 o 0-048 
 
 0-074 
 
 1-488 
 
 0-055 
 
 0-041 
 
 0-764 
 
 0-046 
 
 0-073 
 
 J 
 
 0-055 
 
 0-042 
 
 j 
 
 ( 0-042 
 
 0-056 
 
 ) 
 
 o-055 
 
 0-052 
 
 ) 
 
 i 0-049 
 
 0-065 
 
 1-333 
 
 0-055 
 
 0-052 
 
 \ 0-961 
 
 , 0-044 
 
 0-069 
 
 J 
 
 0-055 
 
 0-054 
 
 J 
 
 / 0-044 
 
 0-052 
 
 ) 
 
 o-055 
 
 0-056 
 
 
 2 0-042 
 
 0-051 
 
 \ 1-180 
 
 0-055 
 
 0-056 
 
 \ 1-018 
 
 ( 0-046 
 
 o-53 
 
 j 
 
 0-055 
 
 0-056 
 
 ) 
 
 f 0-0 4 5 
 
 0-044 
 
 ) 
 
 o-055 
 
 0-0565 
 
 } 
 
 3 ; 0-041 
 
 0-042 
 
 1-024 
 
 0-055 
 
 0-0565 
 
 \ 1-025 
 
 I 0-037 
 
 0-040 
 
 J 
 
 0-055 
 
 0-056 
 
 ) 
 
8o Motion of Liquids 
 
 In order to understand this table properly, it will be 
 necessary for me to make a small digression. 
 
 It will, I trust, be accepted that the effort necessary to 
 retain a thin plate in a stream moving normally to its surface 
 or, conversely, to move a thin plate through a liquid at 
 rest can be measured by the difference between the real 
 pressures which it sustains on its anterior and posterior 
 surfaces. These pressures are composed of what is com- 
 monly called " static " pressure (pressure due to the depth 
 of immersion ; called by Dubuat, rather aptly, the " dead 
 pressure ") which acts in all directions on the body ; and 
 what may be called the " live pressure," which is caused by 
 what is commonly referred to as the " shock," or impact (I 
 have carefully denned the sense in which I use the word 
 impact or shock). If both the fluid and the body are at 
 rest, the difference between these pressures is nil ; the static 
 pressure in front being balanced by that in rear. When 
 the fluid moves, the pressure derived from the impact, 1 
 of which we have determined the intensity, must be added 
 to the " dead-pressure " in front, and their sum will give 
 the real pressure on the anterior side. 
 
 It is well known that, within certain limits, lengthening 
 a body diminishes its total resistance : it was shown here, 
 previously, that lengthening the body did not affect the 
 anterior pressure. It is clear, therefore, that the diminution 
 in the resistance, when a body is lengthened, must, in some 
 manner or another, be due to alterations in the pressure on 
 the posterior face : that, in other words, this pressure must 
 increase with the increased length of the body, so that the 
 difference between this and the anterior pressure may be 
 diminished. 
 
 If, further, the liquid behind the plate " shocked " was 
 moving at the same velocity as if the plate did not exist ; as if 
 (Dr. W. Froude would have said) the plate was a " phantom 
 plate," it is clear that the pressure on the plate would be the 
 same both in front and in rear : the total pressure would 
 be the sum of the potential pressures in all the stream fila- 
 ments which strike the plate. 
 
 1 Impact is here used rather loosely. 
 
Motion of the Liquid behind a Plate 
 
 81 
 
 Even the direction of this motion is not very material, for 
 if we imagine a body of liquid A (fig. 32) flowing towards 
 a vertical plate whose plan is C D, at a velocity v, and then 
 disappearing into that the mathematicians call a " sink " ; 
 if we, further, imagine another body of liquid B, also with 
 a velocity v, flowing towards the plate in the opposite direc- 
 tion, and also disappearing into an imaginary " sink," it is 
 clear that the pressure will be the same on both sides of the 
 plate and that the liquid pressures will not tend to move 
 it in any direction. We 
 might even imagine the 
 plate to be formed like a 
 thin box, and acting as a 
 real sink, into which the 
 liquid on both sides was 
 free to flow without being 
 retarded in the least. 
 
 If any of the stream- c D 
 
 filaments move faster than 
 the general velocity of 
 the stream, since E p + E k 
 constant, it is clear 
 that the pressure, or poten- 
 tial energy, will be de- 
 creased, since the kinetic 
 energy has been increased. FIG. 32. 
 
 Conversely, if the velocity 
 
 of any of the filaments is less than the general velocity of 
 the stream, the potential energy or pressure will be in- 
 creased, since the kinetic energy has been decreased. 
 
 Exactly the same line of reasoning will apply if we imagine 
 the body to be moving. In the former case we referred all 
 velocities to the plate at rest ; whilst in this case we must 
 refer all velocities of the liquid to the plate in motion. What 
 we call a liquid " at rest," is, of course, a liquid at rest 
 referred to the earth, as explained previously. If we refer 
 the motion of a liquid at rest to the plate in motion, the liquid 
 clearly has a velocity towards the plate which may be 
 measured by h, the velocity head. In the same manner, a 
 
82 
 
 Motion of Liquids 
 
 filament of liquid following a plate at a velocity equal to 
 that of the plate through the water, will have a velocity zero, 
 when referred to the plate in motion. As before, if the fila- 
 ments of liquid behind the plate moved at the same velocity 
 (referred to the plate) as that of the general liquid in front, 
 the static pressures would balance one another, and the 
 resistance, if any, could only be caused by the " live pres- 
 sure." Also, as before, if the stream filaments move faster 
 than the general velocity of the stream (always referred to 
 the plate), the pressure, or potential energy, will be decreased, 
 since the kinetic energy has been increased. 
 
 Having, I hope, made this clear, let us now refer again 
 to Duchemin's Table XV. Referring to the case of a plate 
 moving in the water at rest, we see that in column 4, the 
 velocity of the central filament, compared to the velocity 
 
 head, or _, =1-488 ; .'. H'" =1-488 h. This being greater 
 than h, we should expect a negative pressure to be observed 
 
 TABLE XVI 
 BODY MOVING 
 
 Relation of 
 length to 
 
 Value of 
 h 
 
 Value of 
 H" 
 Pressure at 
 
 Value of 
 H" 
 
 Value of 
 H /// 
 
 z. 
 
 Diameter 
 of Body. 
 
 Velocity 
 Head. 
 
 Centre of 
 Posteri or 
 Surface. 
 
 * ' 
 
 n 
 from 
 Table XV. 
 
 
 m. 
 
 m. 
 
 
 
 ( 
 
 0-061 
 
 -0-034 
 
 ) -0-556 
 
 
 o 
 
 0-059 
 
 -0-034 
 
 
 1-488 
 
 I 
 
 0-053 
 
 0-028 
 
 j 
 
 wa 
 
 f 
 
 0-055 
 
 0-023 
 
 ) 
 
 i -i * 
 
 ' 
 
 0-057 
 
 0-025 
 
 Y 0-435 
 
 1-333 
 
 V 
 
 0-049 
 
 O-O22 
 
 j 
 
 
 f 
 
 0-052 
 
 0-016 
 
 ) 
 
 
 - 
 
 0-054 
 
 0-018 
 
 -0-313 
 
 1-180 
 
 \ 
 
 0-047 
 
 0-0145 
 
 j 
 
 
 t 
 
 0-051 
 
 0-008 
 
 1 
 
 
 3 
 
 0-047 
 
 0-009 
 
 -0-186 
 
 1-024 
 
 \ 
 
 0-048 
 
 0-0095 
 
 
 
Motion of the Liquid behind a Plate 
 
 at the centre of the plate and that this should be expressed 
 as 0-488 h. Duchemin made the experiments necessary 
 for measuring these pressures, the results of which are 
 given in Table XVI. It must be admitted that the figures 
 do not agree very closely, the negative pressures being, in 
 all cases, more than 10 per cent, too great. 
 
 Since Duchemin gives no details of his method of conduct- 
 ing these experiments, it is difficult to judge why the figures 
 disagree to such an extent. It appears probable, however, 
 that the velocity of the stream filament was taken at too 
 great a distance from the back of the plate. As this velocity 
 is being accelerated it appears probable, therefore, that the 
 real velocity of the filaments at the surf ace of the plate would 
 have been rather greater than that measured and given by 
 Duchemin in Table XV. One thing that comes out clearly 
 is that the velocity of these central filaments decreases with 
 increased length of the body, also that the pressure behind 
 the plate increases (measured positively, of course) as the 
 length of the body is increased. This being so, the resistance 
 of the body 'the difference between the anterior and posterior 
 pressures, both measured positively will decrease as the 
 length of the body increases. It must be remembered that 
 
 TABLE XVII 
 BODY IN MOTION 
 
 No. of 
 Experi- 
 ment. 
 
 Velocity 
 of 
 Body. 
 
 Value of 
 h 
 
 Velocity 
 Head. 
 
 Value of 
 H" 
 Negative 
 Pressure. 
 
 Value of 
 H" 
 
 h ' 
 
 Mean 
 Values 
 H" 
 
 h ' 
 
 
 inches. 
 
 lines. 
 
 lines. 
 
 
 
 244 
 
 33-52 
 
 18-62 
 
 9-0 
 
 0-483 
 
 0-483 
 
 245 
 
 34-32 
 
 19-52 
 
 9-5 
 
 0-486 
 
 0-486 
 
 246 
 
 44-11 
 
 32-24 
 
 -16-5 
 
 0-516 
 
 0-516 
 
 247 
 248 
 
 54-90 
 55-89 
 
 50-00 
 51-77 
 
 29-0 
 30-0 
 
 0-580 
 0-579 
 
 } -0-580 
 
 Body (not specified, but probably a plate) moving in water at rest 
 -approximately only as it was contained in a canal, but the velocity, 
 if any, must have been inappreciable. 
 
8 4 
 
 Motion of Liquids 
 
 this is only true up to a length of about three times the 
 diameter of the body : with a greater length than this, the 
 resistance increases again slightly. 
 
 Dubuat measured this pressure behind a body he does 
 not specify the kind of body, but it was probably the thin 
 box and his results, given in Table XVII, are, generally, 
 similar to those of Duchemin. 
 
 Another curious fact is shown in the Table XVII ; that 
 
 TT// 
 
 is that,- increases -- negatively with the velocity as 
 might, of course, be expected and at a rather high rate. 
 
 TABLE XVIII 
 
 BODY AT REST 
 
 Relation of 
 Length to 
 diameter 
 
 Value of 
 h 
 Velocity 
 
 Value of 
 
 H// 
 
 pressure on 
 
 centre of 
 
 Value of 
 
 31' 
 
 Body moving 
 Table XVI. 
 
 Value of 
 H// 
 
 of Body. 
 
 Head. 
 
 Posterior 
 
 h 
 
 
 
 
 
 Surface. 
 
 
 h 
 
 
 m. 
 
 m. 
 
 
 
 
 
 0-057 
 
 -0-0339 
 
 -Q-594 
 
 -0'55 6 
 
 2 
 
 0-057 
 
 0-0175 
 
 -0-307 
 
 -0-435 
 
 3 
 
 0-057 
 
 0-0115 
 
 0-202 
 
 ~o*3 I 3 
 
 3 
 
 0-057 
 
 0-0105 
 
 0-185 
 
 -oi86 
 
 
 
 1 
 
 
 Table XVIII (which is taken from Duchemin) may be 
 compared with Table XVI, also from Duchemin, and shows 
 the pressure in the centre of the rear of a flat plate, a cube, a 
 double cube and also a treble cube. Column 5 is taken from 
 
 TT/' 
 
 Duchemin 's Table XVI, and gives the values of - - when 
 
 the body is moving. 
 
 The accordance between these two sets of results is 
 as close as one could expect in this class of experi- 
 ment, with apparatus which was very rough. We see that 
 the pressure behind a moving body increases measured 
 positively as the body is lengthened. Why these pressures 
 
Motion of the Liquid behind a Plate 85 
 
 should increase and how this is caused by the viscosity of the 
 liquid, will be discussed at greater length when treating 
 of that curious question which I call negative resistance 
 when viscosity actually assists in producing motion, or, put 
 differently, reduces the resistance to motion : so that viscosity 
 may be said to act as a recuperative engine which utilizes 
 energy which would otherwise be wasted. 
 
 It will be necessary, in the next chapter, to make a small 
 digression and treat of " jets," the flow in which is some- 
 what different from that of a continuous liquid, which is 
 only partly bounded by a free-surface. After this, the 
 discussion of " Dubuat's paradox " will be closed. 
 
 SUMMARY 
 
 Experiment appears to show that there is no impact with 
 shock (as defined previously) on the sides of a body im- 
 mersed in a liquid, whether (i) the body moves in a liquid 
 at rest, or (2) the body is at rest and the liquid flows past it. 
 
 The pressure on the posterior surface of a body immersed 
 in a fluid and moving in it, appears to be about the same 
 whether the liquid is a static one or not. Since this pressure 
 is apparently caused in a different manner, this can only 
 be classed as a " coincidence," and not as a consequence : 
 more experimental results are required before this question 
 can be decided satisfactorily. 
 
 REFERENCES 
 
 As before. 
 
CHAPTER VIII 
 
 WATER FLOWING IN JETS IMPACT 
 
 " A JET is a stream bounded by fluid of a different kind." 
 (Unwin, Hydromechanics). Osborne Reynolds denned a 
 jet as a stream bounded by a free surface. (R. Inst., March 28, 
 1884). 
 
 When a stream issues from a hole in the wall of a tank, 
 say, and this jet strikes a plate, the pressure that will be 
 exerted on this plate will depend, very largely, on the 
 momentum in the jet and which is communicated to the 
 plate. If the jet has a cross-sectional area of one square 
 unit, its whole pressure against the plate will be @v 2 , always 
 provided that the plate be broad enough to stop all forward 
 velocity of the fluid in the jet ; and, further, that the plate 
 be sufficiently distant from the origin of the jet since, if 
 this distance be not sufficient, the pressure may, in certain 
 cases, be even negative. If the breadth of the plate be in- 
 sufficient, the transference of the momentum from the jet 
 to the plate may be incomplete and the pressure on the 
 plate may be less than @v 2 . It will be observed that this 
 total pressure exerted by the flattened jet, is just twice that 
 produced within a flowing stream, when its velocity falls 
 to zero : in this case the whole kinetic energy has been 
 converted into potential energy. Unwin (Hydromechanics) 
 is in agreement with this ; only he says : " But this pressure 
 
 v z 
 can in no case exceed , or h measured in feet of water , 
 
 2 
 
 or the direction of motion would be reversed and there would 
 be reflux. Hence the maximum intensity of the pressure 
 
 86 
 
Water flowing in Jets Impact 
 
 of the jet on the plane is h feet of water." This statement 
 is, I think, far too sweeping to be considered as accurate. 
 In the experimental investigation of this question, which 
 he refers to, the jet issued from a hole in the bottom of a 
 tank, and was allowed to fall on to a flat plate, perforated 
 with fine holes. Experiments showed that in no case was 
 the pressure, per unit of area, greater than the veocity head. 
 
 This might be considered conclusive ; but it is not so. It 
 is true under these 
 conditions ; but 
 it is not difficult 
 to devise condi- 
 tions when it 
 would not be so. 
 
 It will be re- 
 membered that 
 in Chapter VI, I 
 quoted Professor 
 Zahm as stating 
 much the same 
 thing, viz., that 
 " the unit pres- 
 sure at the centre 
 of impact equals 
 the pressure at the 
 source." In The 
 Pilot tube: Its 
 Formula, we find 
 
 the same remarks, and these statements are based on experi- 
 ments carried out in exactly the same manner as thore referred 
 to by Unwin. In no case was the pressure greater than the 
 velocity head h. In all these cases quoted, the liquid behaved 
 as a static liquid, and there was no " impact with shock " 
 (in the sense defined by me). 1 
 
 If, however, a turbulent jet had been selected the results 
 
 1 Dubuat described the liquid in such a jet as being " in equili- 
 brium " -" la Vitesse d'une veine, qui sort par un orifice mince, est la 
 meme sur toute sa largeur " this is equivalent to saying that the 
 pressure is the same over the whole breadth. 
 
 FIG. 33. 
 
88 
 
 Motion of Liquids 
 
 would have been different. To explain what I mean, if we 
 selected a jet of water issuing from a lead pipe in connexion 
 with the main, we should get a jet (fig. 33) which splashed 
 considerably. By putting an " anti-splash " arrangement 
 on the tap, this jet will be " tamed," and absolutely refuse 
 to splash at all. In the second case it acts as a static liquid, 
 whilst in the former it does not. The pressures caused 
 by the two are very different making every allowance 
 for, and measuring the velocities of flow as accurately as 
 possible. 
 
 |4 Since most of the experiments I shall refer to were carried 
 out so that the liquid acted as a static one, I am quite in 
 
 accord with Unwin 
 and Zahm ; but I 
 make the reservation 
 that their remarks 
 only hold with a static 
 flow, or when the 
 liquid acts as a static 
 liquid. 
 
 But to resume : 
 Duchemin says that 
 " amongst the causes 
 which experiment 
 has shown as lessen- 
 ing the intensity of 
 the shock of a jet of 
 water, the reduction 
 of the area of the 
 surface struck is 
 FIG. 34. that which produces 
 
 the greatest effect." l 
 
 If a jet issuing from a tank acts as a static liquid, the pres- 
 sure at the centre of any body exposed to it should never 
 exceed the " velocity head " at the source. Dubuat, as was 
 explained earlier, exposed funnels, small plates, etc., to a 
 flowing stream and invariably found that the pressure 
 
 1 " Parmi les causes que 1'experience a indiquees comme amoin- 
 drissant 1'intensite du choc d'une veine d'eau, la diminution de 1'aire 
 de la surface choquee est celle qui produit le plus d'effet." 
 
Water flowing in Jets Impact 89 
 
 exceeded the velocity head. The same experiments were 
 conducted with jets in the manner now described. A 
 glass tube of i-J- lines bore ( in. nearly) was bent at right 
 angles, and was employed to measure the pressure, the 
 different bodies experimented on being fixed to the, point. 
 Dubuat gives no diagram, but the arrangement must have 
 been something like fig. 34. The jet was 2 in. in diameter, 
 and issued from a circular hole made in a thin sheet of tin 
 nailed to a large barrel, which was kept constantly full by 
 means of boat pumps. The head of the water above the 
 centre of the orifice was i ft. 10 in. 9 lines. 
 
 A small plate, 16 lines in diameter (fig. 34), with a small 
 hole in the centre, was placed in front of the jet and a few 
 lines from the orifice of the barrel : the water in the glass 
 tube rose to a height of i ft. 10 in. n lines. This is 2 lines 
 above the level of the " source." Dubuat explains this by 
 drawing attention to the capillarity of the glass tube, which 
 would raise the level of the water in the tube rather too high. 
 
 A small cylinder, 6 inches long and half an inch in diameter, 
 was next substituted for the small plate in the last experi- 
 ment, when the water rose, as before, i ft. 10 in. n lines. 
 
 A double cone (size not given) with a small hole, one- 
 twelfth of an inch in diameter, pierced in one point, was 
 next employed when the water rose to i ft. n in. 3 lines. 
 If we here make the same allowance of 2 lines for capillarity, 
 there is a small excess of pressure of 4 lines (one-third of an 
 inch) ; this, as Dubuat explains, might easily have been 
 caused by the workmen having pumped a little more vigor- 
 ously than was necessary, and so caused the surface of the 
 water in the barrel to have been slightly convex thus in- 
 creasing the head more than it should be. The difference 
 is very small, being not much over one per cent. It is also 
 quite possible even probable that there might have 
 been a little air in the cone, which would have caused it 
 to act like a " ram." 
 
 Sundry other small bodies were tried and the result was 
 always the same ; the pressure was never in excess of that 
 caused by the " velocity head." 
 
 The double cone was presented to the jet, so that the 
 
go Motion of Liquids 
 
 point was sometimes in the centre of the jet and sometimes 
 at the edges ; the level of the water in the tube was the 
 same in all cases. I wish to draw special attention to this, 
 because there appears to be a difference of opinion on the 
 question, as to whether the water flows more rapidly at the 
 centre than at the edge of a jet, issuing from a hole in the 
 thin wall of a tank. I think Dubuat's experiments show 
 that the velocity was the same all over the surface of a cross 
 section of the jet. 
 
 The experiments w r ere then modified a little ; a 2 inch 
 pipe being soldered to the tin plate of the barrel so as to 
 make a continuation of the opening. When the small tin 
 plate, employed previously, was presented to the centre of 
 this jet, about J inch from the orifice, the water rose I ft. 
 10 in. ii lines or to the same height as before. 
 
 The same plate was next moved back to 3 inches from 
 the orifice (i-J diameters) , when the water only rose to i ft. 
 9 in. 3 lines. 
 
 The double cone was presented to the centre of the jet, 
 and subsequently at a small distance from the axis the 
 distance from the orifice was not observed and in both 
 cases the rise of the water was i ft. 9 in. 3 lines. 
 
 Having moved this double cone so that the point was 
 within about i line from the edge of the jet, the water was 
 found to rise to a height of only I ft. 3 in. 3 lines, thus show- 
 ing that the velocity of a jet issuing from a tube is greater 
 at the centre than at the edges. The importance of this 
 observation will be evident later on. 
 
 The experiments were still further varied by the substi- 
 tution of a tube of 7 lines diameter only ; the head of 
 water above the centre of the tube being 12 inches. 
 
 The double-cone being presented to the centre of this jet 
 and 6 lines from the orifice, the water in the tube rose to 
 12 in. 2 lines. 
 
 Dubuat observed that the level of the water in the tube 
 fell when the axes of the jet and of the double cone did not 
 coincide. 
 
 From the foregoing I trust the reader will have sufficient 
 information to enable him to see that a jet issuing quietly 
 
Water flowing in Jets Impact 91 
 
 from an aperture in a thin wall of a reservoir acts as if it 
 were a static liquid in fact, quite similarly to the manner 
 in which a liquid at rest acts, when a body is moving in it, 
 at a uniform velocity. 
 
 I may be thought to have somewhat laboured this point, 
 but I have done so intentionally. It is so commonly said 
 that Dubuat's experiments are " unreliable," and I wished 
 specially to show that all the experiments under the same set 
 of conditions agreed very closely ; and also all the experiments 
 under another set of conditions ; although the results in one 
 set of conditions were not in agreement with those in the 
 other. If one mixes up all Dubuat's experiments into one 
 set, the apparent inconsistencies and contradictions are 
 highly confusing to express it mildly. Classified as they 
 should be, the " inconsistencies " are capable of being easily 
 explained. 
 
 Having shown that the maximum pressure per unit area 
 of the surface of a plate struck by a jet does not, ordinarily, 
 exceed the pressure at the source I have specially said 
 " ordinarily," for there are cases where this is not true, 
 and where the pressure can, very much, exceed this initial 
 pressure we will next examine the total pressure caused 
 by the jet. 
 
 As was said, previously, if a jet of unit section is moving 
 at a velocity v, and the density of the liquid is A, then the 
 momentum of unit volume of the jet will be Av. But since 
 v units pass any point per second, the total momentum 
 passing this point per second will be expressed by Av z , and 
 the pressure on the plate can be calculated on the principle 
 that all this momentum has been transferred to the 
 plate. 
 
 The truth of the foregoing can be easily tested by allow- 
 ing a jet to issue from the bottom of a tank, kept constantly 
 full, and letting it fall on a flat scale pan ; the pressure 
 can be measured by putting suitable weights in the other 
 scale pan : W being =2coAgh, where CD =area of jet and 
 /=velocity head. In making the calculations it is neces- 
 sary, however, to make a trifling modification in this formula , 
 to get the real pressure of the fluid vein. If h is the velocity 
 
92 Motion of Liquids 
 
 head of the liquid issuing from the tank (fig. 35), then 1he 
 velocity of flow may be expressed by V2gh, during unity 
 of time. But if a is the distance between the orifice and the 
 
 FIG. 35. 
 
 The fig. shows the apparatus employed and explains itself. 
 Scale beam = 3$ ft. (pieds). Head A 6=4 ft. (pieds). 
 
 surface of the plate, it is clear that the velocity with which 
 the jet strikes the plate will be greater than this, being equal 
 to Vzg(h-}-a). Since, however, the mass of the liquid will 
 
Water flowing in Jets Impact 
 
 93 
 
 be measured by the velocity of the liquid issuing from the 
 
 orifice, or by V2gh, it follows that 
 
 W =2o)AgVh(h -fa) 
 where CD =section of jet, at the vena coniracta 
 
 A =density of liquid 
 andg=gravity constant. 
 
 The Abbe Bossut (Traite Theofique et experimental* 
 d' Hydrodynamique) made the experiment in this manner 
 and the results are given in Table XIX. It will be seen 
 that the calculated and observed results agree very closely. 
 
 TABLE XIX 
 
 Diameter 
 
 Value of 
 h 
 
 Values of p. 
 
 of 
 Orifice. 
 
 Velocity head 
 at Orifice. 
 
 By 
 
 Experiment. 
 
 By 
 
 Calculation. 
 
 Inches. 
 
 Inches. 
 
 Grains. 
 
 Grains. 
 
 0-8333 
 
 47 
 
 12608 
 
 12637 
 
 do. 
 
 23 
 
 6306 
 
 6285 
 
 0-500 
 
 47 
 
 4484 
 
 4549 
 
 do. 
 
 23 
 
 2243 
 
 2263 
 
 Old French measurements. 
 
 The apparatus is shown in fig. 35, which, I think, explains 
 itself. 
 
 This question being undisputed it is not, perhaps, neces- 
 sary to discuss it further. 
 
 When the jet is horizontal the above formula will, appar- 
 ently, not be found to be correct. In this case, to satisfy 
 the results of experiment, the weight W, which measures 
 the pressure against a vertical plane, must be represented 
 by the empirical formula 
 
 /! being the length of the lever at the extremity of which 
 the weight acts, / the distance of the axis of the vein from 
 the centre of rotation of the scale, and 0-0485 a constant 
 
Motion of Liquids 
 
 found by experiment. The sign " + " corresponds to the 
 position of the axis of the vein, when it is below the axis of 
 rotation, and the sign " 'to the opposite position. So 
 that for the same jet the pressure is either greater or less, 
 when measured horizontally, than it is when measured 
 vertically. The reason for this will be easily understood 
 if one considers how the fluid moves after striking the plate. 
 The liquid travels away in all directions from the centre of 
 impact : that part which moves upwards, falls back again, 
 
 and so tends to shift the 
 centre of pressure below the 
 axis of the jet ; and in so 
 doing reduces the length of 
 the leverage. Obviously a 
 smaller weight will be neces- 
 sary to balance the blow of 
 the jet (fig. 36). 
 
 If the jet strike the plate 
 below the axis of rotation, 
 the leverage will be increased 
 and a greater weight will be 
 necessary to balance the 
 blow of the jet ; for this 
 case imagine fig. 36 to be 
 inverted and reversed. 
 
 FIG. 36. 
 
 Now referring to the value 
 
 /I /I ' 
 
 which corresponds to the position of the jet below the axis 
 of rotation of the scale, the experiments of Michelotti 
 (Memoires de V Academic de Turin, 1784 and 1785) give the 
 results shown in Table XX, calculated by Duchemin. The 
 agreement between the observed and calculated values of 
 the pressure is very close especially in the experiments 
 with the circular jet. In the case of a square jet the measure- 
 ment of the vena contracta is not as easy as with a circular 
 jet. Fig. 37 gives, in diagrammatical form, the apparatus 
 employed: here A 6=7=30-305 inches. 
 
 C D=7i (as in the table). Distance between the orifice 
 
Water flowing in Jets Impact 95 
 
 of the jet and the plate =14-166 inches. Measures are old 
 French measures and Paris ounces. 
 
 TABLE XX 
 
 Area 
 of 
 
 Head H. 
 above the 
 
 Values of Values of W ' 
 
 Orifice. 
 
 Centre of 
 
 f X 
 
 
 
 
 
 Orifice. 
 
 
 Observed. 
 
 Calculated. 
 
 Sq. in. 
 
 Inches. 
 
 Inches. 
 
 Ounces. 
 
 Ounces. 
 
 1-0047 
 
 250-25 
 
 24-85 301-43 
 
 305-37 
 
 ^f 
 
 251-00 
 
 25-00 
 
 34-3 
 
 ,, 
 
 250-25 
 
 24-95 
 
 304-15 
 
 ,, 
 
 249-35 
 
 24-50 
 
 ,, 
 
 308-40 
 
 ,, 
 
 248-50 
 
 24-85 
 
 302-82 
 
 ,, 
 
 249-77 
 
 25-25 
 
 ,, 
 
 299-85 
 
 ,, 
 
 249-50 
 
 25-25 
 
 299-46 
 
 0-7854 
 
 250-00 
 
 20-05 
 
 ' ' 
 
 301-34 
 
 ,, 
 
 250-00 
 
 20-09 
 
 300-66 
 
 ,, 
 
 250-25 
 
 20-07 
 
 ,, 
 
 301-33 
 
 ,, 
 
 250-45 
 
 20-09 
 
 ,, 
 
 301-33 
 
 , , 
 
 249-5 
 
 20-01 
 
 301-21 
 
 " 
 
 248-50 
 
 19-93 
 
 300-90 
 
 Old French measures (Paris ounces). 
 The first seven experiments were with a square tube, 
 with circular tube. 
 
 Last six 
 
 J) 
 
 TV 
 
 A 
 plate. 
 
 FIG. 37 
 
96 Motion of Liquids 
 
 It is worth while drawing attention to the beautiful 
 simplicity of this apparatus : whilst A B is constant, C D 
 can be varied at will. 
 
 It will, therefore, appear that if the jet strikes below the 
 centre of rotation, the value of W will appear very consider- 
 ably greater than the real value. 
 
 Now for the opposite case where 
 
 IT/ A 7 // o-oAS^h\ 
 W = ZcoAghfJ- -Zr- ) 
 
 V i /i ' 
 
 which corresponds to the case where the jet strikes above 
 the centre of rotation. I know of only one experiment which 
 has been published, and that is by Vince in 1798 (Phil. 
 Trans., vol. 88). The object of his experiments was to find 
 the value of the impact at different angles, so I select the 
 value of W when the angle of impact is 90. The jet issued 
 from a horizontal tube and the discharge and velocity were 
 checked by two methods : 
 
 (1) The bore of the tube was carefully measured as 0-045 
 inches and the discharge was calculated accordingly. 
 
 (2) The water in the tank was kept at a constant level, 
 and the actual amount of the water flowing out during a 
 certain time was measured ; the two results agreed very 
 closely. 
 
 (3) The velocity was measured from the well-known pro- 
 perties of the parabola : the trace of the jet being marked 
 on a board placed near to it, when it was not difficult to 
 calculate the velocity of the jet. 
 
 I will not enter into any detailed description of the 
 apparatus beyond saying that the level of the water above 
 the nozzle of the tube was 45-1 inches ; and the distance 
 between this nozzle and the plate struck by the jet was 
 about 14 diameters ; /=i8 inches ; /i =18 inches. By 
 experiment it was found that W =i oz. 17 dwt. 12 grains, 
 or 1-875 ounces (Troy). By calculation W =1-880 ounces 
 (Troy), which differs but very little from the weight found 
 by observation. 
 
 Vince, in his paper, makes no allowance for this alteration 
 of the centre of impact, taking the apparent pressure as the 
 real one : this leads to an error of nearly 12 per cent. He 
 
Water flowing in Jets Impact 97 
 
 finds the pressure = - h, whereas by Duchemin's formula 
 
 it should be rather more than 2h the error being only about 
 one-quarter per cent. 
 
 The number of experiments referred to is small, as I know 
 of no others ; however, these are good ones : Bossut's and 
 Vince's, in particular, are classical. It may, therefore, be 
 considered as fairly well proved that when a jet strikes a 
 plate, normally, the pressure on the plate corresponds to 
 that of the whole forward momentum of the jet, having been 
 communicated to the " earth-plate " system. It may be 
 observed that the " correction " applied by Duchemin is, 
 in essence, a certain addition to or subtraction from the 
 weight a certain percentage, say which varies with the 
 velocity head : i.e. with the velocity. 
 
 So far, the jet has been supposed to strike a flat plate, 
 so that it has freedom to spread out laterally to any 
 extent. Suppose, now, that we limit this freedom of 
 motion : how will the pressure on the plate be affected ? 
 A little reflection will show that since the liquid will be 
 " refluxed," or sent backwards, momentum of an equal 
 and opposite amount will be generated, and that the pres- 
 sure on the plate should be doubled. Instead of the pres- 
 sure being represented by 2coAgh, it must be expressed 
 by 4a)Agh. 1 
 
 I know of no quantitative experiments on this point, 
 except a reference in d'Aubuisson's Traite d'Hydraulique 
 to some experiments of Morosi. He observed the action of 
 a jet on a horizontal plate, and found the pressures could 
 be balanced by weights of 5, 7 and 9 limes. He then fixed, 
 on the four edges of the plate, small borders which projected 
 0-014 m. above its surface. On repeating the experiments 
 he found the pressures caused by the jet had augmented so 
 that weights of n , 15 and 20 limes were required to balance 
 them : that is to say, more than doubled. This increase over 
 the theoretical amount may be explained, if the experi- 
 ments are to be relied upon, and if Duchemin's formula (a) 
 
 1 This is, of course, the fundamental hypothesis on which the 
 kinetic theory of gases is based, 
 
98 Motion of Liquids 
 
 is as true for jets, as it appears to be for the ordinary flow of a 
 liquid. 
 
 If we suppose fig. 38 to represent, diagrammatically, the 
 jet striking the plate at the point A, B and C being the small 
 projecting ledges ; it is clear that the velocity of the liquid- 
 filaments at B and C will not be v, the velocity of the jet, but 
 
 (e \ 
 i + J . The momentum generated backwards will 
 
 therefore not be measured by Av 2 , but by Av 2 /'i-f- \ 
 
 \ K/ 
 
 FIG. 39. 
 
 and the increased impactual pressure may be expressed as 
 
 and the total impactual pressure will be 
 
 There is also another reference to Morosi in Ruhlman's 
 Hydromechanik, where the experiment is somewhat different. 
 The plate struck was as in sketch (fig. 39), where the water 
 was refluxed on to a concave ring and then sent back again 
 on the plate. In this case he found the pressure was about 
 the theoretical amount should be 4coAgh, but 
 
Water flowing in Jets Impact 99 
 
 a great deal of energy must have been converted into heat, 
 and, apparently, lost. 
 
 SUMMARY 
 
 A jet issuing from a tank of still water appears to act as a 
 static liquid, in that the maximum pressure on any unit of 
 the surface it may strike will never exceed the pressure due 
 to the height of the source. 
 
 In measurements of pressure caused by jets in striking 
 plates, it is very essential that the plate receiving the pres- 
 sure should be large enough. The distance of the plate from 
 the nozzle of the jet must also be sufficient, since if it is too 
 close, the pressure caused by the jet may even be negative. 
 When there is reflux the pressure is doubled. 
 
 REFERENCES 
 
 JOSEPH T. MICHELOTTI, Memoir es de I' Academic de Turin, 
 1784 and 1784. 
 
 REV. S. VINCE, Phil. Trans., vol. 88. 
 D'AUBUISSON, Traite d'Hydraulique. 
 ZULIANI, Idraulica di Venturoli. 
 NAVIER, Architecture hydraulique dc Belidor. 
 
CHAPTER IX 
 
 JETS STRIKING A PLATE AT AN ANGLE DUCHEMIN's FORMULA 
 DORHANDT AND THIESEN'S FORMULA JOESSEL\S 
 FORMULA M. DE LOUVRIE'S FORMULA M. GOUPIL's 
 FORMULA COLONEL RENARD'S FORMULA VON LOSSL'SR 
 FORMULA 
 
 WHEN a jet of water, whose section, density and velocity 
 are CD, A and u, meets a fixed plane at an angle a, it is 
 generally admitted in works on Hydraulics that the normal 
 pressure supported by the plane may be expressed by 
 CD A u 2 sin a, that being the amount of momentum which 
 is supposed to have been transferred to the " earth-plane 
 system " in the direction of original motion. If the jet be 
 very small, experiment appears to confirm this. In the 
 experiments of Vince (Phil. Trans., 1798), previously re- 
 ferred to, the small jet was caused to strike a plate at differ- 
 ent angles ; the jet always moving horizontally and striking 
 the plate at angles measured horizontally. It was shown 
 previously that when the angle of impact was 90 the pres- 
 sure might be expressed by co A u 2 ; now, since the experi- 
 ments at all the angles are strictly comparable, it will only 
 be necessary to examine the weights put into the small scale 
 pan for different angles, to see if they vary as sin a. The 
 results, taken from Vince's paper, are shown in Table XXL 
 As Vince says : " It hence appears that the resistance varies 
 as the sine of the angle at which the fluid strikes the plane ; 
 the difference being only such as may be supposed to arise 
 from the want of accuracy to which the experiments must 
 necessarily be subject." To this must be added, provided 
 that the jet be very small ; without this qualification Vince's 
 
 joo 
 
Jets Striking a Plate at an Angle Forrmdce 101 
 
 TABLE XXI 
 
 Angle of 
 
 Comparison of Theory and Experiment. 
 
 Impact. 
 
 Experiment. 
 
 Theory (W,= 
 
 W 90 5W fl). 
 
 
 oz. 
 
 dwt. gr. 
 
 I j )zi i dv/t 
 
 gl j , .'" 
 
 90 
 
 i 
 
 17 12 
 
 ^ %/ if.' 17 
 
 12 
 
 80 
 
 i 
 
 17 o 
 
 
 , -,2-2 , , ^ 
 
 70 
 
 i 
 
 15 12 
 
 ..- AkJi'*k 
 
 o /', 5^' * ,' jj 
 
 60 
 
 i 
 
 12 12 
 
 I 12 
 
 II 
 
 50 
 
 i 
 
 8 ! 10 
 
 i 8 
 
 1 7 
 
 40 
 
 i 
 
 4 10 
 
 i 4 
 
 o 
 
 30 
 
 
 18 18 
 
 18 
 
 18 
 
 20 
 
 
 12 12 
 
 12 
 
 19 
 
 10 
 
 
 6 4 
 
 6 
 
 12 
 
 statement is certainly not justified. Because it is correct 
 for a very fine jet, is no argument for assuming that it would 
 be equally true for a large one. In cases like these you are 
 not justified in extrapolating, nor in building a general law 
 on the foundation of a special experiment. The reasons 
 for this statement will appear later. 
 
 If we examine the experiments made by the Abbe Bossut, 
 the results appear not to be in accord with Vince's. Refer- 
 ring to fig. 35 in the last chapter, we have Bossut's diagram 
 of his method of carrying out the experiments. The jet 
 was, as before, vertical, and struck the scale pan as shown ; 
 the experiments now referred to having been made at an 
 angle of 60. In order to make the results strictly com- 
 parable with those obtained when the impact was normal, 
 the beam of the scale was raised, so that when the pan was 
 inclined at 60 to the jet, the head of water was the same as 
 in the experiments previously referred to. When the beam 
 was horizontal, the weight required to balance the pressure 
 of the jet was 12,608 grains. It is clear that when the beam 
 is inclined at 60 to the vertical, while the weight is acting 
 vertically, its " leverage " will have been reduced from W 
 to W sin 60. Since the momentum of the jet is the same 
 as it was before, it is clear that if the normal pressure varied 
 1 1 8 dwts. in the paper : an obvious misprint. 
 
 H2 
 
102 Motion of Liquids 
 
 as the sine of the angle, it should now balance the W sin 60 ; 
 in other words, the same weight, 12,608 grains, should be 
 required in the scale pan to produce equilibrium at all 
 angles. Such was not found to be the case ; the weight 
 required to be placed in the pan was only 12,248 grains, 
 about 3 per cent, less than this theory would appear to require. 
 
 It might be thought that this experiment decided the 
 question ; but this is by no means the case. Although the 
 Abbe Bossxit took very great pains that the centre of the jet 
 should be exactly over the correct part of the scale-pan, the 
 centre of the jet will not, in this case, be the centre of pressure 
 of the jet ; and this has to be allowed for. If the formula of 
 Lord Rayleigh, previously referred to, is applicable to a jet, 
 the " divide " will have been shifted so as to reduce the 
 leverage by about 5 per cent. Per contra, however, the 
 centre of pressure will have been shifted " down hill," as 
 explained previously when speaking of jets striking vertical 
 plates horizontally. Until we know the exact amount of 
 each of these shifts it is not possible to say what this experi- 
 ment proves, or what it does not prove. 1 There are, how- 
 ever, other reasons for thinking that this " sin law " is not 
 strictly accurate, although at very small angles the error is 
 almost negligible. 
 
 Probably the first author who treated of this subject was 
 Duchemin. By an ingenious train of reasoning he came 
 to the conclusion that the sin a (for jets) should be modified 
 
 to ; but when applied to bodies moving in large 
 
 1 + sin 2 a 
 
 volumes of liquid, he modified this to ; , this latter 
 
 I +sm 2 a 
 
 formula being commonly spoken of as " Duchemin's for- 
 mula." This formula is very well known, being found in 
 most books, but I am afraid that the majority of the authors 
 are but imperfectly acquainted with Duchemin's Les lois 
 de la resistance des fluides ; I fancy their information is 
 
 1 If we examine it by Lord Rayleigh's formula for position of cen- 
 tre of pressure { x -=% - ), the decrease of the weight should 
 \ 4 i "" sin a / 
 
 ce about 4-2 per cent. 
 
Jets Striking a Plate at an Angh Formula 103 
 
 chiefly derived from Langley's great work. One author, 
 for example, speaks of it as an " empirical formula," which 
 it certainly is not. It agrees with the results of some experi- 
 ments notably with Langley's, as he pointed out but it 
 requires some very heavy corrections to bring it in line with 
 
 FIG. 40. 
 
 the results of others. It is not exactly satisfying, and I 
 am inclined to think that this, is chiefly due to the assump- 
 tion of the " sin 2 law." I propose, therefore, substituting 
 a modified type of this formula, based on the sarre train of 
 reasoning as Duchemin's, but without this " sin 2 " assump- 
 tion. 
 
IO4 Motion of Liquids 
 
 Now, following Duchemin's line of argument, let fig. 40 
 represent, diagrammatically, a jet of water striking a plane 
 ab at an angle a. dc is the centre of the jet, the projection 
 of whose section a> is ef, and gh the dividing line of the jet ; 
 eg that portion /?, of the section o), the molecules of which 
 pass along the slope ha of the surface ab ; gf, the other por- 
 tion y, of the same section o>, containing the stream-filaments 
 passing along the slope hb of the surface ab. We have there- 
 fore /? +y =co. 
 
 The pressure exerted, normally, on ah will be as before, 
 =f>Au 2 sin a (1) ; whilst the pressure on hb will be =yAu 2 . 
 The total pressure p on the plate will therefore be 
 p = ft Ail* sin a +yAu 2 . 
 
 Now since the reactions of these flows must be equal and 
 opposite, gh being the " divide," it follows that 
 
 /3Au 2 sm a=yAu z , 
 
 and since /?=co y, we may, by first eliminating /?, and 
 then y, get 
 
 co sin a , a) 
 
 y = _, and ft = 
 
 i +sm a i +sm a 
 
 From this we may express the total pressure on the plate as 
 
 - u sn a 
 
 i +sin a i +sin a 
 
 2 sin a 
 
 or fl= - coAu 2 . 
 i -fsm a 
 
 It will be apparent that this formula - 
 
 i + sin a 
 
 is of the same type as, and closely resembles, the Kirchhoff- 
 
 Rayleigh formula-^ -?: for it is only necessary to change 
 4 -\-n sin a 
 
 the unit in the denominator to for it to become 
 
 n 4 +n sin a 
 
 It will be clear that this formula is not an empirical one ; 
 nor can it be pretended that it is exactly new, for several 
 authrrs have employed it, with variations. For example, 
 Dorhandt and Thiesen used it in the form of 
 
 1 Duchemin in his reasoning assumes sin 2 a, instead of sin a ; this 
 is, I think, unsound. 
 
Jets Striking a Plate at an Angle Formula 105 
 2 sin i / 0-62 sin 
 
 i -f-sm i \ i -f sin ^ 
 
 where the " correction " is, as far as I am aware, purely 
 empirical. 
 
 JOESSEL'S FORMULA 
 
 Joessel modified the constants, I presume also empiric- 
 ally, until he got the form 
 
 sin i 
 
 ^ '~~ *39 +0-61 sinfc 
 
 The reader will have observed that in Duchemin's line 
 of argument, which I have here adopted, no allowance has 
 been made for " reflux " of the liquid. Since some reflux 
 actually occurs, the formula is, at best, but a first approxi- 
 mation to the truth. The formula for the pressure on the 
 plate must be changed from 
 
 p pAu 2 sin a -i-yAu 2 to 
 p =pAu 2 sin a -{-y~KAu 2 
 
 where X>>i, (when there is no reflux, X=i,) being a 
 measure of the increase of pressure due to the reflux. 
 By following the same calculations as on page 104 we get 
 
 2X sin a 
 
 p = - coAu 2 
 X +sm a 
 
 where Duchemin's formula is changed to 
 
 2X sin a 
 
 X + sin a" 
 
 M. DE LOUVRIE'S FORMULA* 
 
 M. de Louvrie, assuming that X might be represented by 
 (i +cos a), modified the formula to 
 
 _2(i +cos a) sin a 
 (i +cos a) -j-sin a 
 
 This is a curious formula which has a maximum at about 
 60 or 70. Such a maximum was actually observed by M. 
 de Grammont de Guiche on his plates moving through still 
 air. 
 
 1 I do not know this work of M. de Louvrie ; my authority is 
 Eiffel's Recherches expfr'imentales sur la Rtiistance de I' air. 
 
Io6 Motion of Liquids 
 
 M. GOUPIL'S FORMULA 
 This is generally found in the form of 
 
 99=2 sin i sin 2 2', 
 
 having all the appearance of being an empirical one. It 
 can, however, I think, be shown to be a rational one, where 
 the assumption has been made that the reflux is perfect : 
 that is to say, that X in the equation =2. 
 Inserting this we get 
 
 _2 X2 x sin i 
 
 2 +sin i 
 
 which is equal to 2 sin i sin 2 i, omitting all terms involving 
 higher powers of sin i than the square. 
 
 KIRCHHOFF-RAYLEIGH FORMULA 
 
 n sin a 
 
 This well known formula takes the shape of 
 
 4 -f-rc sin a 
 
 and is based on the rather arbitrary assumption that the 
 flow of the stream past the plate is the same as that of the 
 undisturbed stream at a distance. No ordinary fluid moves 
 like this : it would be dangerously near " perpetual motion." 
 
 COLONEL RENARD'S FORMULA 
 
 This, like Goupil's formula, looks like an empirical one ; 
 but it appears to have been derived from Duchemin's for- 
 
 fy ci T~| /i 
 
 mula - p . by assuming that the reflux was perfect, so 
 that X, =2. 
 
 Making the necessary change we get <p = : =2 sin i 
 
 2+ sin 2 * 
 
 sin 3 i, omitting all terms involving higher powers of 
 sin i than the cube. 
 
 There are several other formulae, but they are less well 
 known and are generally very empirical. There is one, 
 however, which is thought very highly of in Germany, 
 which is y =sin i, simply : this is the formula of Von Lossl. 
 
 VON Lo SSL'S FORMULA 
 
 Von Lossl verified the accuracy of this formula by a very 
 interesting direct experiment. 
 
Jets Striking a Plate at an Angle Formula 107 
 
 A rectangular wooden frame was mounted on the arm of 
 a whirling table so that it could turn freely about the whirl- 
 ing arm. It was i m. long, 25 cm. broad, and carried two 
 small planes of 500 cm. 2 area mounted symmetrically to 
 the axis the frame turns on. One of these planes was fixed 
 in the plane of the frame, whilst the other was free to move, 
 and to be fixed at any inclination to this plane. When the 
 whole system was set in motion, the frame assumed the same 
 angle to the horizon as that of the surface of the small plane 
 
 FIG. 41. 
 
 with the plane of the frame (fig. 41). This could clearly 
 only be the case if the moments of the pressures about the 
 axis were equal ; that is to say, 
 
 p. xd=P go xdsmi, 
 whence P^ =P 90 sin i, 
 
 P; being the normal pressure on the inclined plane, and P go 
 the pressure on the plane moving in normal presentation. 
 The experiment is highly interesting, but it can hardly be 
 said to give a conclusive result. If the centre of pressure 
 of the plane inclined to the line of motion were at the centre 
 of figure, it would be more satisfactory ; but such is not the 
 case, it being nearer the axis than the centre. There must 
 also undoubtedly be a shifting of this centre of pressure 
 " down hill," or away from the axis. Whirling table experi- 
 ments are further open to grave suspicion. The experi- 
 ments appear, however, to have been carried out with quite 
 especial care. 
 
io8 
 
 Motion of Liquids 
 
 Eyg shell 
 
 None of the foregoing formulae considered as rational 
 formulae can be applied to the resistance of plates moving 
 in a fluid, since they only refer to the pressure on the leading 
 face and do not apply, in any way, to the negative pressure 
 experienced at the back of the plate. The agreement 
 
 between Duchemin's formula 
 and Langley's experiments can 
 therefore only be considered to 
 have been accidental. 
 
 Even the particular part of 
 a narrow plate which a jet 
 strikes may affect the amount 
 of the resistance which is ob- 
 served. Vince, in his Bakerian 
 lecture, describes how he car- 
 ried out a series of experiments 
 with a narrow plate, but on 
 repeating them he got quite 
 different results. He repeated 
 them a third time, and got 
 different results again. At last 
 he found out that the differ- 
 ences were caused by the jet 
 having struck the plate at 
 the same distance from the axis 
 
 but sometimes at the centre, and sometimes near the " lead- 
 ing " or " trailing " edges. 
 
 Further, it must not be assumed that a formula which 
 gives true results down to 10 will give equally correct ones 
 at smaller angles. I believe that, at very small angles, these 
 formulae will all give incorrect results. Vince found in some 
 of his experiments of a jet striking a plate, that the pressure 
 exercised by the jet was actually negative. It is well known 
 that there is a mutual attraction between a jet and a solid. 
 If the solid is heavy the jet moves towards it ; whilst in 
 other cases, with light solids, the jet can actually attract them. 
 A well known example of a jet attracting a light solid is 
 to be seen in almost any shooting gallery at a fair, where 
 a small fountain jet supports a blown egg-sheil. The 
 
 Basin, 
 
 FIG. 42. 
 
Jets Striking a Plate at an Angle Formula 109 
 
 velocity of the liquid so reduces its pressure (according to 
 Bernoulli's law) that the egg is pressed by the atmosphere 
 against it. The egg tends to roll down the jet, and so 
 rotates whilst being drawn up it. 
 
 Exactly the same effect can be produced with an air-jet, 
 as was shown by Sir James Dewar at the Royal Institution 
 Juvenile Lectures, Christmas, 1912, with a ping-pong ball 
 and a powerful air-jet. That the jet attracts the ball, and 
 does not merely strike it, and so support it, can be shown 
 by moving the jet about, when the ball invariably follows this 
 movement. 
 
 SUMMARY 
 
 Pressure caused by the impact of a jet on a plate appears 
 to vary (when the jet is very small) as the sine of the angle of 
 impact. If the jet is large, the relation would appear to be 
 
 2X sin a 
 
 more nearly as where a angle of impact ; or 
 
 X +sin a 
 
 by some other formula of this type. 
 
 REFERENCES 
 
 S. P. LANGLEY, Experiments in Aerodynamics. 
 
 Lord RAYLEIGH, Scientific Papers. 
 
 H. S. HELE-SHAW, The Motion of a Perfect Liquid. 
 
 D. RIABOUCHINSKY, Institut de Koutchino III. 
 
 Von LOSSL, Luftividerstandsgesetz. 
 
 G. EIFFEL, La resistance de I'air. 
 
CHAPTER X 
 
 EXPLANATION OF " DUBUAT'S PARADOX " THE " ADDED 
 MASS "OSCILLATING PRESSURE " 
 
 WE have seen that the pressure on the face " in normal pre- 
 sentation " of a plate exposed to a stream, is greater when 
 the plate is fixed than when it is moving in a liquid at rest. 
 How can this be ? 
 
 Two explanations might be advanced. 
 
 (i) It might be due to what is well known as the " added 
 mass " : I will quote Professor Zahm to explain what I mean. 
 " Another general theorem asserts that when the above 
 hypothecated frictionless fluid moves past a solid with 
 accelerated speed . . . there is always a resistance, and this 
 is directly proportional to the acceleration. Commonly, the 
 fluid resistance in accelerated motion can be expressed by 
 merely endowing the moving object with a certain additional 
 mass." (Italics added.) 
 
 This description is hardly likely to convey a clear impres- 
 sion to the ordinary reader ; and as the subject, though 
 highly important, is not referred to in any textbook that 
 I am acquainted with, I will give an illustration in connection 
 with the oscillation of a pendulum in a liquid. 
 
 The properties of pendulums swinging in a vacuum are 
 treated of in every book on elementary mechanics. As is 
 well known, the time occupied by each swing can be ex- 
 pressed by the very simple formula T = 27i\/ ~, where T 
 
 o 
 
 time, Z=length of the pendulum, and g ^acceleration due 
 to gravity ; the time varies as the square root of the length 
 of the pendulum, and inversely as the square root of the 
 
 110 
 
" Dub^iat's Paradox " Oscillating Pressure in 
 
 force of gravity. Hence it is clear that the length of a pen- 
 dulum, corresponding to a given time of vibration, varies 
 directly as the force of gravity. If, therefore, a body be placed 
 at such a distance from the earth that its " gravity " or 
 weight be reduced by one-third the force of gravity having 
 been reduced by this amount the length of the pendulum 
 at this place would require to be reduced by one-third, for 
 the oscillations to take place in the same time as they would 
 at the surface of the earth. If, then, the body, remaining 
 on the earth, is of a density such that when it is plunged 
 in water it loses one-third of the weight which it would have 
 in vacuo, it will be the same case as if it were removed to the 
 distance from the earth, previously referred to. The 
 " mass " of the pendulum is the same in both cases, the 
 only variable being gravity. Hence if we put a to represent 
 the length of the pendulum which oscillates in any given 
 time in vacuo : / the length of the pendulum which oscil- 
 lates in the same time in the fluid ; p, the weight of the bob 
 in the fluid ; P, the weight of the fluid displaced by the 
 body ; P -\-p will express the weight of this body in vacuo, 
 
 P -\-i> 
 and will be the relation between the forces of gravity 
 
 P 
 
 in the two cases. We shall have, therefore, the equation, 
 
 P+p a . ap 
 
 - = ; and hence l~ 
 
 P+p 
 
 Now it might be imagined that this formula would enable 
 us to find the correct length of the pendulum. This is not 
 so, nor would it be correct if the liquid were a " perfect " 
 liquid, since its velocity, as well as the direction of its motion, 
 referred to the pendulum, is constantly changing. 
 
 It might reasonably be asked if the resistance which the 
 liquid opposes to the body would not disturb the isochronism 
 of the oscillations, and in destroying a part of the gravity, 
 necessitate the further shortening of the pendulum more 
 than is required by the preceding law. The mathematical 
 explanation of this would be beyond my ken, and certainly 
 beyond the scope of this small book. We may, however, 
 examine the question in a simpler manner and by reference 
 to experiment. 
 
112 Motion of Liquids 
 
 When a pendulum oscillates in vacuo, the accelerating 
 force diminishes positively during the descent to the lowest 
 point of the arc described by the pendulum, where it be- 
 comes zero. It then increases negatively during the ascent 
 on the other side, until the body has arrived at the same 
 height that it originally started from. But, if the oscillation 
 be executed in a fluid, the resistance will continually destroy 
 a part of this accelerating force, which will be reduced 
 to zero before the pendulum reaches the vertical. It is 
 there it attains its maximum velocity, and this point will be 
 the middle of the oscillation^ The whole oscillation will, 
 therefore, be diminished approximately by twice the distance 
 that the half oscillation was diminished. This loss of 
 amplitude indicates the effect of the resistance with- 
 out it being necessary that the time should vary from 
 this cause. In fact, if the resistance reduced the time of 
 the oscillations, the pendulums of great swing would oscillate 
 more slowly than those of small swing, which we know 
 is not the case, since experiment shows that the times arc 
 sensibly equal. We know, further, that water contained 
 in a syphon oscillates at a rate which depends strictly on 
 the length of the syphon, and in no wise with any reference 
 to the size of its cross-section, which would affect the resist- 
 ance. It may be laid down that the resistance which a body 
 experiences in oscillating in a fluid, and the time of these 
 oscillations, are two effects which have no immediate relation, 
 and that they do not depend on the same cause. 
 
 The preceding formula would give exactly the length of 
 a pendulum in a fluid, if the body in moving did not draw with 
 it a certain quantity of the fluid, which quantity appears to 
 vary very little at different velocities ; so the mass of liquid 
 in movement does not consist only of the mass displaced by 
 the body, but also of some " added mass " which the body 
 draws along with it, and which is practically constant for 
 different velocities. If therefore we take n, a constant 
 number, so that np may express in all cases the weight of the 
 fluid displaced, together with that of the fluid drawn along, the 
 
 1 The effect is, in fact, as if the direction of the action of gravity 
 had been shifted through a small angle. 
 
" Dubuat's Paradox " Oscillating Pressure 113 
 
 weight of the total mass in movement, i.e., its weight in vacuo, 
 will no longer be equal to p +P, but will be represented by 
 p +nP ; whilst its weight in water is always expressed by p. 
 It is necessary, therefore, to change the formula to 
 
 _ __^P _ f rom which we g e t 
 nP+p nP i * 
 
 P 
 
 Dubuat made very many experiments on this subject, 
 and found n practically constant for all diameters, weights 
 and lengths of pendulums, being rather more than 1-5. 
 The " added mass " in these cases was therefore equal to 
 about half the mass of the liquid displaced by the moving body. * 
 
 Although I cannot help thinking that there is some con- 
 nection bet ween this and Dubuat's paradox, I am unable to 
 consider the explanation quite satisfying. I am, even, 
 rather inclined to think that this may be putting the cart 
 before the horse. May it not be possible that "Dubuat's 
 paradox " might explain the " added mass/' The " 50 per 
 cent." addition gives serious cause for reflection ! 
 
 (2) It may be due to impact with shock of the fluid in 
 the sense previously defined by me, and implying reflux 
 such reflux being partial and not complete, as when a non- 
 static jet strikes a plate. 
 
 The cautious reader will at once say I am here contradict- 
 ing myself. Having so much insisted that there is no impact 
 with shock, that a liquid cannot strike a body immersed in it 
 and which completely surrounds it, I now wish to explain 
 a series of experiments by suggesting that the liquid can 
 strike it. What becomes of the theory that when the liquid 
 
 1 Sir George Stokes, referring to this, says that " added mass " 
 for a sphere of radius a, moving in an envelope of radius b, and 
 displacing a mass of fluid m will be 
 
 b 3 + 2 3 m 
 b*a s ' T 
 
 to which he adds : " Poisson came to the conclusion that in an 
 elastic fluid the envelope would have no effect " (Math, and Phys, 
 Papers. Italics added). 
 
H4 Motion of Liquids 
 
 is at rest the added pressure can be measured by the " velo- 
 city head " h ? 
 
 Most unfortunately, in Hydrodynamics there is hardly a 
 statement which can be made that is absolutely true in all 
 cases. It is best or I have thought it so to make general 
 statements, and later to try and correct any small errors, 
 in fact, to point out the exceptions. 
 
 In treating of the motion of liquids there are two ways 
 of approaching the subject. We may assume impact with 
 shock and follow what is commonly, though I think errone- 
 ously, called the " Newtonian system " ; or we may follow 
 what I may call " Euler's system," which leads us to " stream 
 lines," etc., etc. 
 
 The assumption of impact with shock leads to such absur- 
 dities that we may rule it out. We are therefore reduced 
 to the Eulerian system, on which all the mathematics of the 
 subject are based. 
 
 Now, what are the assumptions, or fundamental hypotheses, 
 in this theory ? (i) " Continuity " of the liquid ; this, if 
 not absolutely assumed, is provided for by supposing the 
 liquid as of infinite dimensions. It is quite evident that if 
 a liquid occupies infinite space any discontinuity is unthink- 
 able, since it would necessitate the liquid occupying more 
 than infinite space. I have dealt with this at considerable 
 length in the A B C of Hydrodynamics, to which the reader 
 is referred. 
 
 (2) The second assumption is that there is no impact with 
 shock. This also, if not stated in so many words, is pro- 
 vided for by the supposition that the fluid is of infinite 
 dimensions. A little reflection will show that impact with 
 shock is not possible if no particle of the liquid can move without 
 another particle moving away to make room for it. 
 
 Are we then to think that the Eulerian theory is not con- 
 firmed by experiment ? That, in fact, " the forms of flow- 
 that result from the assumption of continuity and the equa- 
 tions of motion bear in general but scant resemblance to those 
 we obtain in practice ? " (Lanchester, Aerodynamics.} By 
 no means ; but we must be careful that we do not put " peas- 
 cods " into the mathematical mill. We must not assume 
 
" Dubuat's Paradox " Oscillating Pressure 115 
 
 that what is true for a liquid which has no free surface is 
 equally true for one which has. We must not assume that 
 what is true for a static liquid is equally true for one which 
 is not static. In short, we must not extrapolated 
 
 We have seen, by experiment, that when a body of water 
 is static, the mathematical formula giving the relation 
 between potential and kinetic energy is verified ; but that 
 when a liquid, with a free surface, is flowing past a body at 
 rest, this appears not to be true. We found that the pres- 
 sure at the centre of the body could no longer be measured 
 by the velocity head h, but that it was increased, and was 
 equal to 1-5 A. This was found to be invariably the case in 
 all Dubuat's and Duchemin's experiments, for bodies of all 
 shapes, including funnels, hollow cylinders, thin plates, etc., 
 etc. The only exceptions being when the liquid flowed as, 
 what I may call, a quiet jet, when the liquid appeared to act 
 as a static liquid. It was pointed out in Chapter V that if 
 this increase of pressure at the centre of the plate were 
 caused by an increase in the potential energy of the liquid, 
 then energy must have been created. As this is unthinkable, 
 and is, in fact, barred by the assumption of the conservation 
 of energy, it is clear that the only explanation left is that 
 there must have been impact with shock. Since " reflux " 
 would, as we have seen previously, imply that the pressure 
 should be increased from h to 2h, it is clear that the reflux 
 must have been periodic, vibratory, or wave like, causing 
 intermittent " reflux " and " non-reflux " movement. 
 
 Now if we express the potential energy of the liquid at 
 
 1 Dr. Hele-Shaw has shown, experimentally, that in a viscous 
 liquid, without a free surface, all bodies are stream-line. Lan Chester 
 defines a " stream-line body " as " one that in its motion through a 
 fluid does not give rise to a surface of discontinuity.'' This definition 
 appears to be imperfect, for the facts seem to be, both for viscid and 
 inviscid liquids 
 
 (1) In a liquid having no free surface (the envelope being inextensi- 
 ble) all bodies are stream-line. 
 
 (2) At very low velocities, all bodies are stream-line under all cir- 
 cumstances. 
 
 (3) In a liquid with a free surface, the bodies moving at a reasonable 
 velocity and moderately near the free surface, no bodies are stream-line. 
 
 A compressible fluid is, clearly, similar to one enclosed in an clastic 
 envelope, 
 
 I 
 
n6 Motion of Liquids 
 
 the centre of a body as H', or h, since the kinetic energy 
 has here been reduced to zero and the extra pressure caused 
 by the reflux as h, it is clear that, when reflux is perfect, the 
 pressure at the centre of the plate will be H' +h, or 2h ; 
 whilst when there is no reflux it will be h only. As shown 
 in fig. 43, we may describe the pressure as h, 2h, h, 2h, etc. ; 
 so that the mean pressure at the centre of the plate will be 
 I -5 A, as was shown by experiment. At a distance from the 
 centre, the potential energy, still expressed by H', will be 
 less than h, and the pressure may be described as H', H' +h, 
 H', H'+A, etc. The mean pressure at this point being 
 
 TT/ I/TJ' I lj\ 
 
 =H' +-5A ; this is also in accord with Dubuat's 
 
 2 
 
 and Duchemin's 
 
 experiments. Near 
 
 the edges of the 
 
 plate this reflux 
 
 action appears to 
 
 cease, and the pres- - 
 
 sure is then mea- FIG. 43. 
 
 sured by H' only. 
 
 To make my meaning perfectly clear, I will explain myself 
 by giving different diagrams of the flow of liquids. 
 
 In fig. 44 is represented the flow of a static liquid past a 
 body, where there is no reflux. The reader who has not 
 seen this diagram before will probably think this is a drawing 
 of the flow of a perfect liquid past an obstacle. It is nothing 
 of the sort ; it is a photograph of the actual flow of that very 
 highly viscous liquid glycerine. This is reproduced, by per- 
 mission, from the Motion of a perfect liquid, and I have Dr. 
 Hele-Shaw's authority for stating that this photograph is 
 untouched and accurate. It is clear that, the conditions being 
 the same, this viscous liquid flows as the mathematicians 
 tell us a perfect liquid would flow : the curves are very 
 beautiful, and well worth a careful study. This flow, having 
 taken place between sheets of glass about -g^th of an inch 
 apart, is clearly one in two dimensions only ; and so, not 
 exactly the same as it would be in three dimensions. 
 For my argument, only the upper half in front of the 
 
PLATE I. 
 
 FIG. 44. 
 
 N.B. This is a reproduction from a reproduction hence the small 
 spots, caused by the grain of the paper. 
 
 To face p. 116.] 
 
 [de Villamil, Motion of Liquids. 
 
" Dubuat's Paradox " Oscillating Pressure 117 
 
 plate is to be considered ; there is no sign of any reflux. 
 
 Fig. 45 represents diagrammatically the flow of a non- 
 static liquid in front of a body at rest. As will be seen, in 
 the centre there is a small enclosed body of water thrown 
 into a kind of wave motion, alternately moving away from 
 and towards the plate. The reflux is here clearly indicated. 
 
 All this appears perfectly clear and simple ; but it is all 
 based on the assumption that there is reflux. Will experi- 
 ment support this view ? 
 
 FIG. 45. 
 
 I know of no author who has photographed stream lines 
 of water flowing past bodies except M. Marey. His photo- 
 graphs, however, are very small ; and though there appear 
 to be distinct signs of reflux in places, the evidence is not 
 such as to be very convincing. Dorn's photographs were 
 taken when the bodies were in motion in water at rest, and 
 so are of no use for my present purpose. The photographs 
 taken at the National Physical Laboratory, though also 
 showing signs of reflux, are very small and not quite satis- 
 factory. If, however, we refer to the research work done 
 on air, and if we assume as is generally done that the 
 movement of air is similar to that of water, we have plenty 
 of material to work on, 
 
 12 
 
n8 
 
 Motion of Liquids 
 
 Riabouchinsky (Koutchino, Fascicule III) made many ex- 
 periments with air flowing past bodies of very many shapes ; 
 the air having been rendered visible by the employment of 
 lycopodium powder, some very beautiful photographs were 
 taken of the stream-filaments. In every case there is dis- 
 tinct evidence of vibratory movement of the air in front of the 
 bodies. 
 
 Fig. 46 is a reproduction of a singularly fine photograph 
 by Dr. Hele-Shaw. The vibratory action is well indicated 
 in front of the body ; and the vortex, on the right, behind 
 
 FIG. 47. 
 
 the plate is particularly well shown. Even the vortex 
 action behind the plate appears as if it were of an oscillating 
 character. 
 
 It has been objected to me that since, in these experi- 
 ments, the plate of sheet iron on which the lycopodium was 
 resting was given a sharp blow with a hammer at the time 
 the photograph was taken, this vibratory movement may 
 have been caused by the blow of the hammer. I am not 
 inclined to attach much importance to this objection, since 
 the vibration caused by the blow was perpendicular to the 
 direction of motion of the air : the vibratory action is also 
 non-apparent behind the plate, or other body. The evidence 
 is by no means, however, confined to these photographs. 
 
PLATE II. 
 
 FIG. 46. 
 
 To face p. 118.] 
 
 \de Villamil, Motion of Liquids. 
 
" Dubuat's Paradox" Oscillating Pressure 119 
 
 Lilienthal studied this question, and his diagrams of stream 
 lines show this reflux most clearly. Von Lossl ((Luftwi- 
 derstandsgesetz) , who supported Lilienthal's view, gives the 
 same form of diagram of flow of air past a plate. Fig. 47 
 is a copy of Von Lossl' s diagram, which I may describe as 
 Lilienthal's simplified ; there can be no doubt that reflux 
 action is here indicated. 
 
 I might quote several other authors in support of this, 
 but I will confine myself to one whose evidence is incon- 
 testable M. Eiffel. In La resistance de I' air, 1911, we have 
 
 FIG. 48. 
 
 a diagram (fig. 48) of a stream of air meeting a plate at an 
 angle of 80, where the reflux action is shown most distinctly, 
 the stream lines having been drawn from actual observation. 
 
 On the next page of this monumental work we find a 
 schema of stream line flow past a plate in normal presenta- 
 tion (fig. 49). In reference to this, M. Eiffel says : " in the 
 two regions comprised between the dotted lines the eddies 
 are such that it is not possible to fix any mean direction." * 
 
 It will be clear that for air, certainly, and for water, 
 probably there is reflux when a stream of fluid meets a 
 
 1 " Dans les deux regions comprises entre les traits pointilles, les 
 remous sont tels qu'on ne peut fixer une direction moyenne." 
 
120 
 
 Motion of Liquids 
 
 body at rest. As shown in fig. 49, the ordinary stream lines 
 enclose the body of vibrating liquid, which thins out and 
 disappears at the edges of the plate. The pressure on the 
 plate is the potential pressure in the stream line transmitted 
 through the eddies, and periodically reinforced by the extra 
 pressure caused by reflux. We see, therefore, that there 
 is no violation of the principle of the conservation of energy. 
 At the edges, the stream filaments appear to " wipe-off " 
 the eddying masses of water, which thus have some cyclic 
 motion imparted to them. 
 
 FIG. 49. 
 
 How are we to picture this action as taking place ? If 
 we imagine the plate or body to have a kind of " liquid 
 prow "as it moves forward in liquid at rest, this prow 
 divides the liquid, and the whole " plate-water system " 
 moves forward at a uniform velocity. The liquid glides 
 along the prow and never strikes the plate. When the body 
 is at rest, however, the eddying liquid is not so easily deflected 
 ,by the prow ; so that some of it actually does strike 
 the plate, and so causes impact with reflux. It will be seen 
 thus, that not only does experiment show that pressure on 
 the anterior surface of the plate is not the same in both 
 
" Dubuat's Paradox " Oscillating Pressure 121 
 
 cases, but that there are good theoretical reasons why it 
 should not be so. 
 
 There is a statement made by Dr. W. Froude, and ap- 
 parently pertinent to this question, which has not, perhaps, 
 received the attention it deserves, though it is referred 
 to by Lord Rayleigh in one of his early papers. " As a 
 proof that an edgeways motion of an elongated body through 
 water is not without influence on the force necessary to 
 move it with a given speed broadways, Mr. Froude says, 1 
 Thus, when a vessel was working to windward, immediately 
 after she had tacked and before she had gathered headway, 
 it was plainly visible, and it was known to every sailor, 
 that her leeway was much more rapid than after she had 
 begun to gather headway. The more rapid her headway 
 became, the slower became the lee-drift, not merely relatively 
 slower, but actually slower. 
 
 " Again, any one might obtain conclusive proof of the 
 existence of this increase of pressure occasioned by the 
 introduction of the edgeways component of motion, who 
 would try the following simple experiment. Let him stand 
 in a boat moving through the water, and taking an oar in 
 his hand, let him dip the blade vertically into the water 
 alongside the boat, presenting its face normally to the line 
 of the boat's motion, holding the plane steady in that posi- 
 tion, and let him estimate the pressure of the water on the 
 blade by the muscular effort required to overcome it. 
 When he has consciously appreciated this, let him begin to 
 sway the blade edgeways like a pendulum, and he will at 
 once experience a very sensible increase of pressure ; and if 
 the edgeways sweep thus assigned to the blade is consider- 
 able and is performed rapidly, the greatness of the increase 
 in the pressure will be astonishing until its true meaning 
 has been realized. Utilizing this proposition, many boat- 
 men, when rowing a heavy boat with narrow-bladed oars, 
 were in the habit of alternately raising and lowering the 
 hand with a reciprocating motion, so as to give an oscillatory 
 dip to the blade during each stroke, and thus obtained an 
 
 1 Proceedings of the Society (sic) of Civil Engineers, vol. xxxii, in a 
 discussion on a paper by Sir F. Knowles on the Screw Propeller. 
 
122 Motion of Liquids 
 
 equally vigorous reaction from the water with a greatly 
 reduced slip or stern ward motion of the blade." 
 
 It is not exactly clear what Dr. Froude meant by the 
 expression " the edgeways component of motion " : it 
 conveys no mental picture to me, and it appears to be only 
 introducing an unnecessary complication into a subject 
 which is, already, sufficiently difficult. 
 
 Lord Rayleigh, referring to this, says : " It is not difficult 
 to see that in the case of obliquity we have to do with the 
 whole velocity of the current, and not merely with the 
 resolved part." This, also, does not appear to make matters 
 any clearer. 
 
 Following the previous reasoning, we see that when the 
 vessel has " headway " (is moving steadily) the liquid is a 
 static one ; whilst, when " tacking," it is not. The pressure 
 and therefore " leeway " should then be greater than 
 when the vessel is travelling at a steady velocity. 
 
 So, also, when the oar is kept steady, the liquid is a 
 static one ; whilst, when it is being made to oscillate, the 
 motion of the liquid referred to the oar is constantly 
 changing its direction. In fact, the result is just what we 
 should expect, on a priori reasoning. It is rather a strong 
 confirmation of Dubuat's Paradox. 
 
 SUMMARY 
 
 When a body is at rest and the liquid flows past it, the 
 pressure at the centre is, not the " velocity head " h, but 
 1*5 h. There are good theoretical reasons in favour of " Du- 
 buat's Paradox " being correct ; whilst all Dubuat's and 
 Duchemin's experiments, when carefully examined, confirm 
 it. 
 
 In all liquids, viscid or not, without a free surface (the 
 bounding envelope being inextensible) all bodies are stream-line. 
 
 REFERENCES 
 
 E. J. MAREY, Le Mouvement. 
 
 Dr. HELE-SHAW, "The Distribution of Pressure due to flow 
 round Submerged Surfaces " (Inst. Naval Architects, 1900). 
 
 Dr.W^FROUDE, Proceedings of the Institution of Civil Engineers, 
 vol. xxxii. 
 
CHAPTER XI 
 
 MOVEMENT OF LIQUIDS THROUGH APERTURES IN THE WALL 
 OF A VESSEL MOUTHPIECES 
 
 THE first author who investigated the problem of the flow 
 of liquids out of a hole in a vessel was Newton. In the 
 Prop. XXXVI, Book II of the Principia, he " defines the 
 motion of water running out of a cylindrical vessel through 
 a hole made 
 at the bot- 
 tom." The 
 form of flow 
 that he as- 
 sumed being 
 what is com- 
 monly known 
 as a " cata- 
 ract flow " : 
 such flow as 
 would occur 
 from the li- 
 quid obeying 
 the laws of 
 heavy bodies. 
 If A C D B 
 
 (fig. 50), be c E p D 
 
 a cylindrical FIG. 50. 
 
 vessel, A B 
 
 the mouth of it, C D the bottom parallel to the horizon, E F 
 a circular hole in the middle of the bottom ; then, in brief, 
 Newton's assumption is that the liquid layer A B, and the 
 whole body of water A M E F N B will descend vertically 
 
 123 
 
124 Motion of Liquids 
 
 and pass through the hole E F without disturbing any of 
 the rest of the water ; in other words, the flow will be such 
 that the parts of the water AMEC and BNFD will 
 remain at rest, the flow being entirely through the central 
 part in fact, as if the parts AMEC and BNFD were 
 frozen. 
 
 Liquids, clearly, do not flow like this and Newton's pro- 
 position is more interesting than useful in explaining the 
 motion of a liquid through an orifice. 
 
 The next author was Daniel Bernoulli, In his I'Hydro- 
 
 B 
 
 -r -*- 
 
 FIG. 51. 
 
 dynamique he proceeds to discuss the problem on the prin- 
 ciple of the conservation of energy called by him vis viva 
 and the assumption he made was as follows : 
 
 He assumed A CDB (fig. 51), to be a vessel with an 
 aperture at the bottom gh, and filled with water up to 
 the level ab. When the aperture gh was opened, he 
 assumed the water to move in horizontal layers, so that 
 the thin layer ab moved down to cd, the layer cd moving 
 down to a level just below it, and so on down to the bottom 
 of the vase ef. This, though nearly true for the layers at 
 the top, leads to the assumption that the liquid in the 
 
Movement of Liquids through Apertures 125 
 
 lowest layer ef must move from C and D to gh with an 
 infinite velocity. It need hardly be said that the idea, 
 though ingenious, very frequently leads to results which 
 are not in accord with experiment. 
 
 Later on we have the Chevalier de Borda (Academic des 
 Sciences, 1766), who made a somewhat different assumption. 
 
 If A C D B (fig. 52) represents the vessel containing the 
 liquid whose surface is A B. Dividing A B and E F into 
 an equal number of parts, the water is supposed to flow, as 
 shown by the stream-lines, to the opening E F at the bottom 
 of the vessel. 
 
 This form of flow, like all the preceding, violates what 
 may be considered one of the fundamental principles of 
 Hydrodynamics ; that is, that a liquid must flow from any 
 region of higher pressure to any region of lower pressure. It 
 is clear that in all of them there is supposed to be liquid at 
 rest at the parts of the vessel near C and D ; such liquid 
 being at rest, must, necessarily, be under greater pressure 
 than the liquid flowing out of the aperture. Per contra, 
 if it is under greater pressure it cannot be at rest. 
 
 To get a clear idea of how a liquid gets from the inside 
 to the outside of a vessel, it is necessary, first of all, to 
 
126 
 
 Motion of Liquids 
 
 recognize thoroughly that flow must take place, in any static 
 liquid, between any two points in the same horizontal 
 plane of the liquid, between which there is a difference of 
 pressure, or potential. This is recognized by meteorologists, 
 for the air ; but it does not appear to be seen ordinarily 
 that the same principle applies to all static liquids. 
 
 Having assumed this as beyond dispute, the following 
 theory of the flow of a liquid through an aperture is put 
 forward tentatively. 
 
 Let A B (fig. 53) represent the bottom of a large and deep 
 tank, which has a circular aperture E F in the centre of it, 
 
 A G 
 
 FIG. 53. 
 
 and which is temporarily closed. Let G, H, K, L, M and 
 N be any number of points all situated on any imaginary 
 hemisphere, whose centre is X the centre of the hole 
 EF. 
 
 Now, since we have assumed the tank to be deep, we may 
 assume the pressure at all these points, as well as at E and 
 F, to be sensibly the same : this assumption does not affect 
 the argument, and the error can be corrected later. 
 
 Let us next imagine the aperture E F to be opened. 
 What have we done by opening this aperture ? We have 
 created a region of lower pressure at E F, with the result 
 that all the molecules of water at G, H, K, L, M and N will 
 commence moving at an equal velocity towards the point 
 
Movement of Liquids through Apertures 127 
 
 X. All the molecules in the thin hemispherical layer, 
 moving at the same velocity towards the centre, will always 
 remain on some hemispherical surface, whose diameter 
 will gradually decrease. Let us now think of the pressures ; 
 when steady motion is established, the isobaric surfaces, or 
 surfaces of equal pressure, at this part of the vessel, will 
 also be hemispheres (like the different skins of an onion), 
 and these will also decrease from G, H, K, L, M, N towards 
 the point X. Following this train of thought, it is clear 
 that eventually the liquid enclosed in the hemisphere E Y F 
 will pass through E F into an imaginary tube E a /? F. 
 If we assume that there is no change of velocity inside and 
 outside of E F (not correct, of course, but this will be 
 allowed for later), then the liquid will flow into a cylinder 
 E a ft F whose height E a =X Y. It is evident that since 
 the volume of the hemisphere is only two-thirds of that of 
 the cylinder, the liquid in the former cannot fill the latter, 
 and that there must be some sort of contraction. The 
 sectional area of the vena contracta being two-thirds, or 
 0-66, of the area of E F. We see that on the assumptions 
 made the area of the vena contracta should be 0-66 of the 
 aperture. Newton, who first measured the size of this 
 
 vena contracta found it to be = = -7 nearly ; Newton's aper- 
 
 A/2 
 
 ture was only f of an inch in diameter, and the head was 
 20 inches, the hole having been made at the side of the 
 vessel, but he does not state if it was near the bottom or not 
 he says : " not to the bottom but to the side of the vessel." 
 If it was near the bottom his coefficient of contraction would 
 be larger than if away from it. He says, also, he took the 
 measurement of the vena contracta " with great accuracy 
 at the distance of about half an inch from the hole." Since 
 this was nearly one diameter from the hole, this would 
 perhaps account for his coefficient being greater than has 
 since been found by other authors. 1 
 
 The abbe Bossut found by measurement that the area of 
 this vena contracta was almost exactly 0-66. Subsequently, 
 
 1 Hachette says, however, that he found a coefficient of contrac- 
 tion for the smallest holes as great as 0-78, 
 
128 Motion of Liquids 
 
 from experiments made by the measurement of "discharges," 
 he came to the conclusion that for the " coefficient of dis- 
 charge " a more accurate relation was 5 : 8, or the area of the 
 contracted vein should be 0-625 of the area of the aperture. 
 
 Later experimenters appear to have found this to be 
 slightly too large, and that the coefficient of contraction, 
 " C c , is on the average 0-64 " (Unwin, Hydromechanics) ; 
 whilst the " coefficient of discharge " for a " sharp-edged 
 plane orifice, =0-97 x 0-64 =0-62 " (Unwin, Hydrome- 
 chanics), where the coefficient of velocity, C^o-gy. 1 
 
 Let us now examine the two assumptions referred to pre- 
 viously. It was assumed, temporarily, that the pressure 
 was sensibly the same at all points on the hemisphere 
 G H K L M N. This is clearly incorrect, for the pressure 
 at L must be less than that at any other part of the hemi- 
 sphere, and consequently the velocity of the molecule 
 starting from L will be less than that of any of the others. 
 This will necessitate the isobaric surfaces, or surfaces of 
 equal pressure, being changed from the surfaces of hemi- 
 spheres to those of oblate spheroids the figure G H K L M N 
 must be changed from a hemisphere to that of a half orange. 
 No alteration in the argument is necessary since the volume 
 of a semi-spheroid is equal to two-thirds of that of the cir- 
 cumscribing cylinder of the same height : in other words, if 
 Ea =XY, the volume of the spheroid E Y F = two-thirds of 
 that of the cylinder E a {$ F. It is clear, therefore, that 
 this assumption did not affect the argument in any way. 
 
 The other assumption, that there was no change of velo- 
 city inside and outside of E F, must now be examined. 
 This also is incorrect, since the velocity of the liquid is being 
 further accelerated by gravity. If we allow for this, it is 
 clear that E a will be greater than X Y : the volume of the 
 liquid in the semi-spheroid will therefore be flowing into a 
 cylinder whose volume is more than 50 per cent, greater ; 
 this will necessitate the coefficient of contraction being 
 less than 0-66. How much less will it be ? It is difficult to 
 say, since the coefficient is not a constant. 
 
 1 Fanning gives it as 0-974. 
 
Movement of Liquids through Apertures 129 
 
 A little reflection will show that the amount of acceleration 
 the relation between E a and X Y will depend on the 
 length X Y, which also varies as E F. We see, therefore, 
 that the coefficient of contraction will be smaller for a large 
 hole than it is for a smaller one. In other words, we may 
 express this in algebraical shorthand as 
 
 o-66 
 
 where D = diameter of the hole and /(D) is some undeter- 
 mined function of D, but varying directly as D. 
 
 Now, does experiment show that such is the case ? The 
 measurement of a vena contracta is exceedingly difficult, and 
 I am not aware that any one has ever made experiments to 
 see whether it is the same for all sizes of apertures, as well 
 as in all directions. Hachette (Annales de Chimie, 1816), 
 certainly found that the flow through very small apertures 
 was greater, proportionately, than it was through larger 
 holes it is not uncommonly supposed to be less and if the 
 flow was greater, it is clear that the contraction of the vein 
 must have been less. In their report on M. Hachette's paper, 
 MM. Poisson, Ampere and Cauchy say : "All conditions 
 being equal, the contraction of a vein which issues from an 
 orifice in a thin wall decreases with the dimensions of the 
 orifice. 
 
 " For diameters greater than 10 millimetres the contrac- 
 tion becomes almost constant and remains between the 
 limits of 0-40 and 0-37." * As stated previously, Hachette 
 found the coefficient of contraction for the smallest holes as 
 high as 0-78. 2 
 
 1 " Toutes circonstances etant d'ailleurs egales, la contraction de 
 la veine qui soit par un orifice en minces parois decroit avec les 
 dimensions de Torifice. 
 
 " Pour les diametres au-dessus de 10 millimetres, la contraction 
 devient presque constante, etreste comprise entre les hauteurs 0-40 
 et 0-37." 
 
 2 It will be noted that Unwin uses the term " coefficient of con- 
 traction " to mean that the cross section of the liquid jet is reduced 
 to -66 of the area of the aperture. MM. Poisson, Ampere, and 
 Cauchy employed the word " contraction " to mean that the section 
 of the jet is reduced by 37 or 40 per cent., i.e., to -63 or -60, of the size 
 of the aperture, 
 
130 Motion of Liquids 
 
 It would appear that de Borda suspected that such a 
 relation existed ; for when comparing Newton's coefficient 
 of contraction with his own measurements, he says, " per- 
 haps Newton made his experiments with too small an orifice ; 
 this reason alone suffices for explaining the difference ; in 
 fact, it is evident that the friction of the liquid against the 
 edges of the orifice should decrease the quantity of the 
 contraction, since it retards the parts which tend most to 
 contract the vein ; but this friction was proportionately 
 greater in Newton's experiment than in mine ; therefore, 
 for that reason alone, I ought to have found a greater 
 contraction." x 
 
 I am not aware that M. Hachette's paper was ever pub- 
 lished, so I rely entirely on the report on it made by MM. 
 Poisson, Ampere and Cauchy. 
 
 There is another way of looking at this question. Joseph 
 T. Michelotti, in 1783, first, I believe, pointed out that when 
 a liquid issues from an aperture in the side of a tank, the 
 pressure not being uniform over the area of the aperture, 
 there was a difference in the velocity of the different fila- 
 ments at different heights. If we imagine the orifice to be 
 square and to be divided into an infinite number of hori- 
 zontal areas, the velocity of discharge should be greater 
 through the lower areas than through the higher ones. 
 There is, nevertheless, a mean velocity by which we can 
 multiply the area of the orifice to find the total water dis- 
 charged ; there is also some horizontal sheet of water which 
 is issuing at exactly this mean velocity : the centre of the 
 line through which this sheet discharges may be called the 
 " centre of velocity." 
 
 Suppose, first, the level of the water in the tank to be 
 kept constantly to the top of a square orifice, the side of 
 
 1 " Peut etre vient-elle de ce que M. Newton a fait son experience 
 sur un orifice trop petit ; cette seule raison suffit pour expliquer 
 cette difference ; en effet il est evident que le frottement du liquide 
 contre les bords de 1'orifice doit diminuer la quantite de contraction, 
 puisqu'elle retarde les parties qui tendent le plus a contracter la 
 veine ; or ce frottement etait a proportion plus grand dans Texperi- 
 ence de M. Newton que dans la mienne ; done par cela seul je devais. 
 trouver une contraction plus grande," 
 
Movement of Liquids through Apertures 131 
 
 which is a, Q ^discharge of water, p the parameter (2g), 
 z the head of water above any selected sheet. The velocity 
 of discharge of this particular sheet will clearly be Vpz ; 
 and the discharge of water in this sheet will consequently be 
 adz Vpz ; the total discharge through the whole aperture 
 can be got by integrating z between a and o. 
 
 2 
 
 whence Q = - a z Vap . 
 
 O 
 
 Now if we call A the " compensated head " (Michelotti's 
 word) the head which would cause the mean velocity 
 then the discharge may be expressed as Qa*VAp. 
 
 .-. a*VAp= %a*Vap, or A/A = fV0~ 
 whence A=f0. 
 If we put = 1, then A become |- = 0-444, e *c. 
 
 Now if we call X the distance between the centre of 
 velocity and the centre of figure of the orifice, we have 
 still putting 0=1, 
 
 X = iHH T V = 0-0555, etc. 
 
 Let us next suppose the square orifice subjected to a con- 
 stant head of water b, above the top of the orifice. Following 
 the same line of argument ; the velocity of discharge of the 
 sheet which is situated at z, below the top of the orifice, will 
 clearly be V(b+z)p ; and the discharge of water through 
 this sheet will be adzV(b+z)p 
 
 C b + a - 
 
 whence Q=| adzV(b + z)p 
 
 = f^*{ &T7 -tf}. 
 
 Again, as before, A being the compensated head, which 
 produces the mean velocity, 
 
 from which we get A = 2 {-f- a 1 W } 2 . 
 
 The head which corresponds to the centre of figure being 
 
 K 
 
132 
 
 Motion of Liquids 
 
 X will be = 
 
 2 Q 
 
 X is given for different heights in Table XXIII, 
 
 TABLE XXIII 
 
 b: a. 
 
 A. 
 
 CL 
 b + 
 
 X. 
 
 b = a 
 
 1-485842 
 
 i'5 
 
 0-014158 
 
 b = 2a 
 
 2-491610 
 
 2-5 
 
 0-008390 
 
 b = $a 
 
 3-494036 
 
 3'5 
 
 0-005973 
 
 b = 4 
 
 4-495355 
 
 4-5 
 
 0-004645 
 
 b = 5# 
 
 5-496219 
 
 5'5 
 
 0-003781 
 
 6 = 6 
 
 6-496770 
 
 6-5 
 
 0-003230 
 
 & = 7 
 
 7-497236 
 
 7*5 
 
 0-002744 
 
 & = 8a 
 
 8-497630 
 
 8-5 
 
 0-002470 
 
 & = ga 
 
 9-497846 
 
 9'5 
 
 0-002154 
 
 6 = loa 
 
 10-498072 
 
 10-5 
 
 0-001928 
 
 6 = na 
 
 11-498131 
 
 n-5 
 
 0-001869 
 
 b = I2a 
 
 12-498361 
 
 12-5 
 
 0-001639 
 
 b = i3 
 
 13-498431 
 
 13-5 
 
 0-001569 
 
 & = 140 
 
 14-498546 
 
 I4'5 
 
 0-001454 
 
 6 = I5 
 
 15-498643 
 
 15-5 
 
 0-001357 
 
 & = i6a 
 
 16-498704 
 
 16-5 
 
 0-001296 
 
 b = ija 
 
 17-498791 
 
 17-5 
 
 0-001209 
 
 b = i8a 
 
 18-498840 
 
 18-5 
 
 0-001116 
 
 b = iga 
 
 19-498937 
 
 i9'5 
 
 0-001063 
 
 b = 2oa 
 
 20-498958 
 
 20-5 
 
 0-001042 
 
 The table shows that the difference, X, between A and 
 b+ , when 6=200, is only about o-ooi ; whilst, if b=a t 
 
 "2i 
 
 it is fourteen times greater ; and when b o t as was pointed 
 out previously, the difference between A and 6+ - -is 
 
 2 
 0-06. 
 
 If what was advanced previously is sound, the coefficient 
 of contraction should increase with the increase of head. A 
 reference to Fanning's Table (Unwin's Hydromechanics] 
 shows that this is the case with some apertures, whilst the 
 reverse is true for others. The results having, however, 
 been derived from a collection of different experiments, 
 
PLATE III. 
 
 ACTUAL FLOW (A). 
 
 To face p. 133.] 
 
 THEORETICAL FLOW (B). 
 FIG. 54. 
 
 [de Villamil, Motion of Liquids. 
 
Movement of Liquids through Apertures 133 
 
 carried out at different times and by different methods 
 and as they, further, appear to contradict one another 
 occasionally these tables can hardly be accepted as very 
 strong evidence. More experiment is required. 
 
 To resume : a little reflection will show that all the 
 " tubes of flow " of the liquid can only originate at the free 
 surface of the liquid ; a continual flow away from the boun- 
 dary walls is impossible, since, if it occurred, the space near 
 the walls would, be left empty. It is clear that, with the 
 same depth of liquid above the aperture, the shorter the mean 
 length of the tubes of flow, the greater will be the supply of 
 liquid to the aperture ; and consequently the greater the 
 coefficient of contraction and discharge. This will, perhaps, 
 be better understood by reference to fig. 54 (A), which repre- 
 sents the flow of a liquid out of a small hole in a wall of a 
 tank. This is not a fancy sketch, but from a photograph 
 made by Dr. Hele-Shaw of the flow of glycerine. As in 
 all the other photographs, it is a flow in two dimensions 
 only. 
 
 It may be compared with fig. 54 (B), which is from a 
 drawing of Dr. Hele-Shaw's of the theoretical flow of an 
 inviscid liquid : the resemblance is very startling. If, 
 therefore, the hole be at the bottom of the tank but 
 near one side, the discharge should be greater than if it were 
 well away from the side. Similarly, if the hole were near 
 two sides of the tank i.e., near a corner the discharge 
 and the coefficient of contraction should be still greater. 
 In the limiting case, when the aperture is near four sides 
 that is, approximating to the size of the bottom of the tank 
 the discharge and the coefficient of contraction should be 
 the greatest ; and the latter should approach to i-o. Ex- 
 periment shows that this is so in all cases. 
 
 A similar line of reasoning would lead us to expect that, 
 if the aperture be inside of the walls of the vessel i.e., 
 being the opening in a re-entrant tube the discharge should 
 be a minimum. This was first pointed out by de Borda : 
 the re-entrant tube being commonly referred to as de 
 Borda's " re-entrant ajutage." 
 
134 
 
 Motion of Liquids 
 
 MINIMUM COEFFICIENT OF CONTRACTION RE-ENTRANT 
 MOUTHPIECE OF DE BORDA 
 
 If a re-entrant tube was attached to the aperture, the 
 coefficient of contraction and ergo the coefficient of dis- 
 charge, was found by de Borda to decrease. 
 
 Unwin (Hydromechanics) referring to this, says : "In 
 
 ^B 
 
 FIG. 55. 
 
 one special case the coefficient of contraction can be deter- 
 mined theoretically, and as it is the case where the con- 
 vergence of the streams approaching the orifice takes place 
 through the greatest possible angle, the coefficient thus 
 determined is the minimum coefficient. 
 
 " Let fig. 55 represent a vessel with vertical [? flat] sides, 
 
Mouthpieces 135 
 
 O being the free water surface, at which the pressure is 
 P a . Suppose the liquid issues by a horizontal mouth-piece, 
 which is re-entrant, and of the greatest length which permits 
 the jet to spring clear from the inner end of the orifice, 
 without adhering to its sides. With such an orifice the 
 velocity near the points C D is negligible, and the pressure 
 at those points may be taken equal to the hydrostatic pres- 
 sure due to the depth from the free surface. Let Q be the 
 area of the mouth-piece, A B, co that of the contracted jet 
 aa. Suppose that in a short time t, the mass OOaa comes 
 to the position 1 O 1 a 1 ^ 1 ; the impulse of the horizontal 
 external forces acting on the mass during that time is equal 
 to the horizontal change of momentum. 
 
 " The pressure on the side O C of the mass will be balanced 
 by the pressure on the opposite side O E, and so far all the 
 portions of the vertical surface of the mass, excepting the 
 portion E F opposite the mouth-piece and the surface 
 AaaB of the jet. On E F the pressure is simply the hydro- 
 static pressure due to the depth, that is (P a +GA).Q. On 
 the surface and section AaaB of the jet, the horizontal 
 resultant of the pressure is equal to the atmospheric pressure 
 P fl acting on the vertical projection A B of the jet ; that 
 is, the resultant pressure is P fl ,Q. Hence the resultant 
 horizontal force for the whole mass OOaa is (P a -\-Gh)Q 
 P a jQ =GhQ. Its impulse in the time t is GhQt. Since the 
 motion is steady there is no change of momentum between 
 O 1 O 1 and aa. The change of horizontal momentum is, 
 therefore, the difference of the horizontal momentum lost 
 in the space O O 1 1 and gained in the space aaa^a. 1 
 In the former space there is no horizontal momentum. 
 
 " The volume of the space aaa l a l is covt ; the mass of 
 
 C C 
 
 liquid in that space is covt ; its momentum is ~a)V 2 t. 
 
 g g 
 
 Equating the momentum gained, 
 
 g 
 
136 Motion of Liquids 
 
 But v 2 = 2gh, and = C c 
 
 . . 
 
 a result confirmed by experiments with mouth-pieces of this 
 kind." 
 
 This proof, which was first given by de Borda, is, of 
 course, based on the assumption that the motion at C D 
 is negligible, or approximates to zero. Since there is un- 
 doubtedly some motion at C D it will be well to see how far 
 experiment confirms the fact that it is " negligible." De 
 Borda measured the coefficient of discharge x and found 
 
 it to be -_= 0-5149 ; " Bidone 0-5547 '* Weisbach 0-534 " 
 
 (Unwin, Hydromechanics). All these values are in excess 
 of 0-5, and if we divide these by the coefficient of velocity, 
 which Unwin says is generally about 0-97, the difference 
 is such as to indicate that the velocity at C D is certainly 
 not " insensible " ; that it can hardly be said that this 
 velocity is " negligible." 
 
 If we compare Unwin's solution of the coefficient of 
 contraction, with a re-entrant mouth-piece, with that 
 given by de Borda, we find that the fundamental assump- 
 tions are rather different. Unwin supposes the horizontal 
 mouth-piece to be " of the greatest length which permits 
 the jet to spring clear from the inner end of the orifice " ; 
 whilst de Borda supposes the tube to be " infinitely small 
 . . . and that the vein, after contraction, remains in the 
 state of contraction, and flows on a horizontal and perfectly 
 polished surface." 
 
 Unwin, clearly, made his assumption in order that the 
 tube should not be wetted : de Borda by his assumption of a 
 " perfectly polished surface " intends to postulate that the 
 flow shall not be interfered with by the surface of the inside 
 of the tube. If the liquid wets the tube, a secondary action 
 takes place, which alters and increases the flow ; as is very 
 well known. De Borda's conditions can be arrived at 
 
 1 The tube being at the bottom of the tank. 
 
Mouthpieces 137 
 
 by the employment of a liquid which does not wet the tube, 
 such for example as mercury flowing in glass. We find 
 in the report of MM. Poisson, Ampere, and Cauchy, on 
 Hachette's paper, previously referred to, that " perfectly 
 pure mercury flows in a mouthpiece of iron, as it would do 
 in an orifice in a thin wall ; when, on the contrary, the mer- 
 cury was dirtied by an alloy of tin, which tinned the interior 
 of the tube, the contraction of the vein disappeared." 1 
 
 The same authors also say : " In a mouthpiece coated 
 with perfectly dry wax, water escaped as through a thin 
 wall ; but it filled the tube as soon as it had wetted the wax, 
 for then the coating was, so to say, replaced by the first 
 layer of water which was attached to it." 
 
 J. T. Fanning (Hydraulic and Water Supply Engineering) , 
 when referring to the increased discharge caused by adding 
 a short tube to the hole in the thin wall, says that if the 
 tube be greased on the inside, increased discharge will not take 
 place. 
 
 I might quote de Borda himself on this question. In 
 giving a description of another experiment, he says : " I 
 lifted this weight [a cover to the vertical tube] perpendi- 
 cularly, so that the fluid should contract regularly in entering 
 the tube, and not touch the walls (for one must note that 
 when it touched them, the attraction of the walls, and after- 
 wards the exterior pressure of the atmosphere, deranged the 
 contraction and even sometimes forced the liquid to issue in 
 full bore)." 2 
 
 It is not by any means clear what de Borda meant by 
 the " exterior pressure of the atmosphere " causing an 
 increase of flow through the tube. This will be further 
 discussed in the next chapter. 
 
 1 " Le mercure parfaitement pur coule dans un ajutage de fer, 
 comme il le ferait dans un orifice en mince paroi ; quand au contraire 
 le mercure est sali par un alliage d'etain qui etamait 1'interieur du 
 tuyau, la contraction de la veine disparaissait." 
 
 2 " Je levais ce poids perpendiculairement a fin que le liquide se 
 contractat regulierement en entrant le tube et n'en touchat point les 
 parois (car il faut remarquer que lorsqu'il les touchait, 1'attraction 
 de ces parois, et ensuite la pression exterieure de I'atmosphere de- 
 rangeaient la contraction et forgaient meme quelque fois le fluide de 
 sortir a plein tuyau)." 
 
138 Motion of Liquids 
 
 SUMMARY 
 
 When a liquid issues through a hole in a thin wall of a 
 vessel, it contracts, and the discharge is less than might be 
 expected. This contraction appears to vary as some func- 
 tion of the diameter of the hole, i.e., being relatively less 
 through a smaller hole than through a larger. 
 
 A re-entering mouthpiece reduces the discharge to a 
 minimum, which appears to be rather greater than 0-5. 
 
 REFERENCES 
 
 Sir ISAAC NEWTON, Principia. 
 
 DANIEL BERNOULLI, I' Hydrodynamique. 
 
 Le Chevalier DE BORDA, Academic des Sciences, 1766. 
 
 M. HACHETTE, Annales de Chimie, etc., 1816. (Report by 
 MM. Poisson, Ampere et Cauchy.) 
 
 J. T. FANNING, Hydraulics and Water Supply Engineering. 
 
 JOSEPH THERESE MICHELOTTI, Memoir es de I' Academic de 
 Turin. 
 
CHAPTER XII 
 
 SUBJECT CONTINUED EXTERNAL MOUTHPIECES THE 
 SEVILLE PUMP BELLANGE PUMP 
 
 IN the last chapter the assumptions made by Unwin and 
 de Borda, when calculating the co-efficient of contraction 
 of a liquid issuing through a re-entering mouthpiece, were 
 compared ; and it was pointed out that, though they were 
 apparently very different, they resembled one another in 
 their object of eliminating the possibility of the tube being 
 wetted, or, in other vvords, of the liquid adhering to the walls 
 of the mouthpiece ; since, for some reason or another, if the 
 tube is wetted the discharge is increased. 
 
 It was observed that de Borda's explanation of this 
 action was by no means clear. If, however, de Borda's 
 explanation is not obvious, the same must be said for the 
 modern one. Fanning, referring to it, says : 
 
 " Increase of coefficient. There is an influence affecting 
 the flow of water through short cylindrical tubes, sufficient 
 to increase the coefficient materially, that does not appear 
 when the flow is through a thin partition. The contraction 
 of the jet still occurs as in the flow through a thin partition, but 
 after the direction of the particles has become parallel in 
 the vena contracta, a force acting from the axis of the jet out- 
 ward, together with the reaction from the exterior air, begins to 
 dilate the section of the jet and to fill the tube again. The tube 
 is, in consequence, again filled at a distance, depending upon 
 the ratio of the velocity to the diameter, of about two and 
 a half diameters from the inner edge of the orifice. The 
 axial particles of the jet not receiving so great a proportion 
 
 139 
 
140 Motion of Liquids 
 
 of this reaction from the edges of the orifice as the exterior 
 particles, obtain a greater velocity, a portion of their force 
 being transmitted to their surrounding films through diver- 
 gent lines, and the velocity of the exterior particles within 
 the tube is augmented, and the section of the jet is also 
 augmented, until its circumference touches the tube." 
 [Italics added to parts I wish to draw attention to.] 
 
 I am afraid that this paragraph conveys no impression of 
 any kind to my mind. What the writer means is certainly 
 not obvious. We see that de Borda's " exterior pressure 
 of the air " is brought in ; but what is meant by the " reac- 
 tion from the exterior air ? " The pressure of the air can 
 surely only retard the discharge, and not accelerate it. What 
 also is meant by the " force acting from the axis of the 
 jet," which behaves in this very curious manner ? We see 
 that the contraction of the jet still occurs ; and in another 
 paragraph we are told that there is " a vacuum round the 
 vena contracta." What causes this " negative pressure " to 
 change to a positive pressure ? In another paragraph we 
 find that if the inside of the tube is greased the " vacuum 
 does not appear." 
 
 I have only quoted this paragraph in order to show that 
 if de Borda's explanation is not satisfying, this modern one 
 is no improvement. 
 
 Unwin makes no reference to this at all ; he states the 
 facts and leaves the explanation severely alone. 
 
 The reader will have gathered from the foregoing, although 
 I have not perhaps stated it definitely, that if an exterior 
 mouthpiece, or " ajutage," as it is sometimes called, is fixed 
 to a hole in a thin wall, the discharge is increased, provided 
 that the walls of the tube are wetted ; but not otherwise. 
 
 Referring to this exterior mouthpiece Unwin says (Hydro- 
 mechanics) : " Let fig. 56 represent a vessel discharging 
 through a cylindrical mouthpiece at the depth h from the 
 free surface, and let the axis of the jet XX be taken as the 
 datum with reference to which the head is estimated. Let 
 Q be the area of the mouthpiece, co the area of the stream at 
 the contracted section E F. Let v p be the velocity and 
 pressure at E F, and v t pi the same quantities at G H. If 
 
External Mouthpieces 
 
 141 
 
 the discharge is into the air p l is equal to the atmospheric 
 pressure p a . 
 
 " The total head of any filament which goes to form the 
 jet, taken at a point where its velocity is sensibly zero, is 
 
 2* ; at EF the total head is^-f^; at G H it is 
 G 2g G 
 
 2g 
 
 FIG. 56. 
 
 " Between E F and G H there is a loss of head due to 
 abrupt change of velocity, which may have the value 
 
 " Adding this head lost to the head at G H, before equating 
 it to the heads at E F and at the point where the filaments 
 start into motion, 
 
142 Motion of Liquids 
 
 G 2g G 2g G 2g 
 
 But (ov =Dv lf and co =C C Q, if C c is the coefficient of contrac- 
 tion within the mouthpiece. 
 
 TT Q V t " 
 
 Hence v = v l = -?. 
 
 CO C c 
 
 " Supposing the discharge into the air, so that p 1 =p a : 
 
 G 2g G 2g 
 
 2g- 
 
 w. 
 
 where the first term on the right is evidently the coefficient 
 of velocity for the cylindrical mouthpiece in terms of the co- 
 efficient of contraction at E F. Let ,=0-64, the value 
 for simple orifices, then the coefficient of velocity is 
 
 T . -0-87 . . . (2). 
 
 " The actual value of C v found by experiment is 0-82, 
 which does not differ more from the theoretical value than 
 might be expected if the friction of the mouthpiece is allowed 
 for. Hence, for mouthpieces of this kind, and for the 
 section at G H, 
 
 C =0.82, C,=i-oo, c=o-82, 
 
 "It is easy to see from the equations that the pressure p 
 at E F is less than the atmospheric pressure. Eliminating 
 Vi we get 
 
 &U-**f* nearly ....... (3); 
 
 per square foot. 
 " If a pipe connected with a reservoir on a lower level is 
 introduced into the mouthpiece at the part where the con- 
 traction is formed (fig. 57) the water will rise in this pipe to 
 a height 
 
External Mouthpieces 
 
 143 
 
 KL = .' =& nearly. 
 G 
 
 " If the distance X is less than this, the water from the 
 lower reservoir will be forced continuously into the jet by 
 the atmospheric pressure, and discharged with it. This is 
 the crudest form of a kind of pump known as the jet pump." 
 
 I have quoted this in full, because it is somewhat clearer 
 than de Borda's 
 Problem IV, 
 being in more 
 modern nota- 
 tion. In neither 
 of the Pro- 
 blems, however, 
 has the length 
 of the mouth- 
 piece been taken 
 
 into account, ; . 
 
 and this is most 
 important. It 
 must be noted 
 that the con- 
 traction at E F 
 remains, - al- 
 though the 
 added mouth- 
 piece causes an 
 increase of dis- 
 charge. Clearly, 
 therefore, the FlG - 57- 
 
 velocity at E F 
 
 must be very considerably increased. If the velocity is in- 
 creased it is clear that the kinetic energy has been increased, and 
 that some of the potential energy must have been converted 
 into kinetic energy, according to Bernoulli's law. As the only 
 potential energy available for this is the atmospheric pressure, 
 the pressure in the vena contracta must be now less than the 
 atmospheric, as we see is the case in fig. 57. Also between 
 E F and G H there is an " abrupt change of velocity," and 
 
144 Motion of Liquids 
 
 consequent loss of energy (not annihilation, of course), result - 
 
 ( v _ v \ 2 
 
 ing in a loss of head, measured by - - . It is clear 
 
 2g 
 
 that the liquid must be here changing shape to account for 
 this loss. As was pointed out by de Borda and even Ber- 
 noulli before him Bernoulli's theorem only holds if the 
 change of velocity takes place by insensible degrees. If there 
 is any abruptness in this change, then there is a loss of vis 
 viva energy such as occurs when inelastic bodies strike 
 one another. It may not be uninteresting to give de 
 Borda's lemma on this latter point. 
 
 Let an inelastic body a (corps dur), having a velocity u, 
 strike another inelastic body A, which has a velocity V, 
 what is the loss of vis viva which will be caused by the shock ? 
 
 SOLUTION 
 
 -rt, x-u v XT, t, i flW 2 + AV 2 , 
 
 The sum of the vis vivce before the shock was = - 
 
 zg 
 
 ,, ,, , , ,, . a-}- A /flw + AV\ 2 , , 
 
 after the shock this sum = : ( - ) ; subtracting 
 
 2g \ a + A J 
 
 the latter from the former, it will be found that the loss of 
 vis viva energy 
 
 aA (u V) 2 
 ~ 2g~ C. Q. F. T. 
 
 De Borda, in his Problem IV employs m = instead of the 
 
 v 
 
 coefficient of contraction ; hence his formula, corresponding 
 to Un win's (i), becomes 
 
 Vi = / ; '===V2gh, and 
 vm 2 2 m + 2 
 
 Unwin's coefficient of velocity becomes 
 
 2m + 2 
 
 I think this, is perhaps, a more convenient form than that 
 given by Unwin. Some examples of the use of this formula 
 will be given presently, when the reader will see how impor- 
 tant it is to the comprehension of the flow of water into, as 
 well as out of, vessels. 
 
External Mouthpieces 
 
 145 
 
 To return to the question of the influence of the length of 
 the tube on the discharge, F. D. Michelotti (sperimenti 
 idraulici) gives the discharge for tubes of different lengths, 
 which will be found in Table XXIV. It will be seen that 
 the discharge increases steadily up to a length of about two 
 and a half diameters, after which it decreases slowly. It is 
 not a constant, as one might imagine from reading de Borda's 
 and Unwin's writings on the subject. 
 
 TABLE XXIV. 
 
 Relation ~. 
 
 Relation m 
 
 d 
 length of tube 
 to its diameter. 
 
 effective discharge 
 to (?) theoretical discharge. 
 
 
 
 0-6096 
 
 \ 
 
 0-6169 
 
 i 
 
 0-7671 
 
 2 
 
 0-8157 
 
 2} 
 
 0-8221 
 
 3 
 
 0-8201 
 
 4 
 
 0-8179 
 
 5 
 
 0-8095 
 
 6 
 
 0-8070 
 
 7 
 
 0-8032 
 
 8 
 
 0-7997 
 
 Venturi showed, further, that if a small hole is opened in 
 the tube opposite the position of the vena contracta " the 
 increase of expenditure diminishes or entirely ceases, and the 
 fluid is no longer continuous in the tube " (Prop. IV, Exp. 12). 
 
 I might here point out that de Borda showed that if the 
 orifice of the re-entrant mouthpiece was surrounded by a 
 circular plate the discharge was increased through the 
 tube, so that the flow was the same as if through a thin wall, 
 i.e., from about 0-5 to 0-62 or thereabouts. This might have 
 been suspected if one thinks of the different conditions 
 introduced. 
 
 De Borda carried out another very interesting experiment, 
 first suggested by Bernoulli. He made a tubejof tin 1-5 
 inches in diameter, and i foot in length (fig. 58), the tin, 
 
146 
 
 Motion of Liquids 
 
 being well polished and the edges of the tube very sharp. 
 This tube was immersed vertically in a large vessel of water, 
 the upper orifice having been closed, so that the contained 
 air prevented the water from rising in the tube. When the 
 upper orifice was suddenly opened, he observed the height 
 to which the water rose in the tube above the level of the 
 water in the vessel. After many experiments he found that 
 if the tube was depressed in the liquid 8 inches 0-5 line, the 
 water rose exactly to the top of the tube, i.e., 4 inches less 
 
 FIG. 59. 
 
 half a line. De Borda says that Bernoulli, who treated the 
 question theoretically and mathematically, had found that 
 the liquid should rise to a height of 8 inches ; he had, how- 
 ever, made no allowance for the reduction of discharge 
 caused by the re-entrant mouthpiece. 1 It is only fair, 
 however, to point out that Bernoulli cites other experiments 
 
 1 Bernoulli's argument being the same as that to be found in 
 textbooks, which is as follows : 
 
 " Let a vertical tube be connected with an orifice as shown in 
 fig. 59. In the first place, let the tube be empty, and let the orifice 
 be closed. On opening the orifice the liquid commences to rise in 
 the tube, and will continue to rise until the gravitational energy of 
 
External Mouthpieces 147 
 
 which are not in agreement with those of de Borda ; but the 
 tubes employed were too small for the results to be con- 
 vincing. 
 
 This experiment was again modified by de Borda, who 
 placed a flat circular disc, 12 inches in diameter, round the 
 lower orifice of the tube. He then found that if the tube 
 was submerged 7 inches i line, and, as before, the upper 
 orifice was suddenly opened, the liquid rose to the level of 
 the top of the tube. We have here the relation of 85 lines : 
 59 lines, which gives a " coefficient of head " (if I may coin 
 a word) =0-694, as against rather less than 0-5,, in the other 
 experiment. 
 
 Since, however, de Borda says the water rose in the tube 
 about 2 -5 lines before the upper orifice was opened, the depres- 
 sion of the liquid was thus only about 82*5 lines instead of 
 85 ; so that the real coefficient would be greater than 0-694, 
 
 viz. -52. =0-715. Making the same allowance in the other 
 
 o2'5 
 
 case the coefficient would be ^5 =0-5, almost exactly. 
 
 94 
 
 The reader will notice here the great difference between 
 the velocity of discharge when the aperture is opened sud- 
 
 the liquid within the tube is just equal to the work done on that 
 liquid before it issues from the orifice ; when this condition has been 
 attained, the liquid in the tube will be stationary for an instant. 
 Let a be the sectional area of the tube, and H the height to which 
 the liquid rises in it ; then the gravitational energy of the liquid 
 
 within the tube is equal to pH x , since the centre of gravity 
 
 TT 
 
 of the liquid in the tube is at a height above the orifice. The 
 
 work done on the mass paH before it issued from the orifice is equal 
 to paH x gh, and thus 
 
 and the liquid rises in the tube to a height A -above the free surface 
 of the liquid in the tank (fig. 59). The liquid then flows back into 
 the tank until the tube is emptied ; and in the absence of viscosity the 
 tube would be alternately filled to the height H, and then completely 
 emptied at regular intervals." (Edser, General Physics for Students}. 
 
 I 
 
148 Motion of Liquids 
 
 denly, and that when the velocity has become steady. In 
 the former case the " head " is zh, changing to zero, and 
 becoming 2h again. This oscillation is very rapidly damped, 
 when the constant head becomes h. It is a matter of the 
 commonest observation that if you are watering a garden 
 with a hose you can cause the water to go higher and further 
 if you open and close the cock suddenly, than it will when 
 the cock is left open and the velocity of the discharge becomes 
 steady. 
 
 If we now examine this question by de Borda's formula, 
 in Prop. IV, 
 
 m 2 2W + 2 
 
 where = coefficient of contraction. 
 m 
 
 Or by Unwinds formula (i) 
 GX = 
 
 when C c is the coefficient of contraction. 
 
 If m=2, or C c =% (as found theoretically for are-entrant 
 mouthpiece), then 
 
 AG AG 
 
 44+2 2 
 
 De Borda found, however, that the coefficient of contraction 
 for a re-entrant mouthpiece was 0-515, whence m =1-9425. 
 Putting this value in the equation, GX becomes almost 
 
 exactly equal to -' 
 
 2 
 
 When the plate was put round the orifice of the tube, the 
 contraction and discharge was as through a thin partition. 
 In this case, de Borda found the coefficient of contraction 
 =10. . Whence m =yf . Putting this value into the equa- 
 tion GX= AGXIO , which, by putting A G=825 lines, as 
 
 before, gives GX=6o-| lines, where experiment gave GX 
 = 59 lines only. This may easily be accounted for by the 
 resistance due to viscosity, which is small but appreciable. 
 
External Mouthpieces 
 
 149 
 
 The whole of this subject of the passage of a liquid through 
 a thin wall even a partition in the inside of a vessel is very 
 interesting, and most useful to engineers who wish to cal- 
 culate the time required 
 for emptying or filling any 
 vessel from the bottom. 
 The student who wishes 
 to study the mathematics 
 of the subject further is 
 referred to de Borda's 
 very interesting paper in 
 the Academic des Sciences. 
 It may be well to con- 
 sider the question of the 
 passage of a liquid up- 
 wards through a thin wall. 
 Unwin says (Hydrome- 
 chanics] that the liquid 
 " rises nearly to the level 
 of the free surface of the 
 liquid in the vessel from 
 which it flows " ; but, he 
 adds later, that the orifice 
 must have a suitable form. 
 In his diagram he gives a 
 very suitable concave form : 
 if the surface were convex 
 the same good result 
 would not be obtained ; 
 still less if there had been 
 a cylindrical mouthpiece 
 fixed to the aperture ; 
 worse still if the mouth- 
 piece had projected a 
 FlG - 6o - little inwards. 
 
 We have seen in the ex- 
 periments of de Borda referred to, that the liquid did not rise 
 in the tube as high as Bernoulli had calculated, theoretically, 
 that it should do. To explain, further, what I mean, I will 
 
150 Motion of Liquid 
 
 refer to the well known Hero's fountain (fig. 60), not uncom- 
 monly used as a scent fountain ; the discharge, being through 
 a re-entrant tube, A B, will not attain the height of the 
 difference of level between the two water surfaces. If a flat 
 collar were placed round the mouth of the tube at B, the 
 flow would be increased ; and, a fortiori, if the opening at 
 B were made funnel-shaped, or if the opening at A were 
 contracted, the water would rise more nearly to the " theo- 
 retical height." In this problem allowance has to be made 
 for the coefficient of contraction ; but this is not usually 
 done. 
 
 When there is a narrow jet at A, and this jet is removed, 
 it is very commonly said that " the pressure has been taken 
 of!." This is not strictly correct, for the pressure is the 
 same as before ; the volume of water passing in at B is not 
 sufficient, and there is a loss of energy caused by the con- 
 traction. 
 
 THE SEVILLE PUMP 
 
 As will be found stated in every elementary textbook, 
 a column of water of about 34 feet will balance the atmo- 
 spheric pressure, so that, in an ordinary " lift pump," the 
 suction pipe must be of less length than about 30 feet. So 
 well is this known, that if almost any professor or engineer 
 were asked how to make a pump which would lift water 55 
 to 60 feet, he would probably think the inquirer was an 
 ignoramus. At the same time, this same professor, or 
 engineer, would be perfectly well aware that there are many 
 trees that lift their sap well over this height, not to mention 
 the Sequoia, the height of which is colossal. How this is 
 done, the inquirer would be told, is " one of the unsolved 
 problems of nature." * One way of solving the problem 
 was discovered about 150 years ago in Seville, and the pump 
 
 1 Francis Darwin said (1906) that the upward flow of sap in plants 
 was still unexplained, as the fluid is, in fact, lifted in tall trees to 
 a greater height than is compatible with the pumping action caused 
 by the creation of a vacuum ; and all efforts to trace a succession of 
 pumping stages have failed, 
 
The Seville Pump 
 
 is called the " Seville Pump." As the origin of this pump 
 is very little known, it may not be uninteresting to repeat 
 the story ; though the pump can hardly be recommended as 
 very economical. 
 
 Some 150 years ago a Seville tinsmith entered into a con- 
 tract to make a pump to supply a terrace, about 55 feet 
 high, with water for some flowers. Apparently the tin- 
 smith only knew of one kind of pump, which he duly made, 
 and, needless to say, it would not work, since the water 
 would not rise to the height of 55 feet in the suction tube. 
 Very exasperated, the smith threw his hammer at this tin 
 pipe, and struck it so violently, at a height of about 10 feet 
 from the ground, that he made a hole in it of about j 1 ^ mcrj 
 in diameter. His rage did more than his genius, for imme- 
 diately the water rose to the 55 foot level and came out of 
 the pump. The experiment was repeated in Spain several 
 times, and always with the same result. 
 
 When the report reached Paris, the Abbe Nollet wrote to 
 M. Cat asking him, " since when had the laws of nature 
 changed ? " M. Cat's reply was, " La nature n'a pas change 
 ses lois, mais elle nous en cache encore quelques-unes, et elle 
 a mis des conditions a celles qu'elle nous a laisse voir." 
 
 The explanation is, of course, very simple. The water 
 was standing in the pipe the diameter of which was rather 
 less than i inch at a height of rather under 30 feet. When 
 the hole was made in the tube, air entered, cutting the water 
 column in two. The lower 10 feet of water descended, whilst 
 the upper 20 feet was forced up, by the pressure of the air, 
 to the top of the tube. It appears hardly necessary to add 
 that, since only the 20 feet of water was raised, it was neces- 
 sary, before another supply could be procured, to close the 
 hole and recommence pumping. 
 
 M. Cat made an improvement in this pump by putting a 
 small tap where the hole in the tube was ; this worked very 
 well, if not exactly economically. 
 
 Later, M. Bellange, a goldsmith of Paris, made a modifi- 
 cation, by which a constant supply of water could be pumped. 
 The principle was slightly different. His suction tube was 
 10 lines in diameter and about 55 feet in length. At about 
 
Motion of Liquids 
 
 i foot above the level of the water he made a small hole in 
 the pipe of about half a line (% inch) in diameter. When 
 the pump was worked, air rushed into the tube rythmically, 
 so that in the " rising main " there were alternate layers of 
 water and air all the way up the tube. The air occupying 
 more space than the water, the pump was only actually 
 lifting a column of considerably less than 30 feet of water. 
 This pump also worked well, though the " efficiency " was 
 not such as would satisfy a modern engineer. 
 
 M. 1'abbe Nollet, in his report on these pumps (Academic 
 des Sciences, 1766) showed, by means of a lecture table ex- 
 periment, how the Seville pump worked. He got a baro- 
 meter tube of about four feet in length, one end of which 
 was sealed up. At about 9 or 10 inches from the other end 
 he had a hole drilled, of sufficient size to admit a large pin, 
 this hole being stopped with wax. The tube having been 
 filled with mercury, and inverted in a vessel of the same 
 liquid, the level of the mercury in the tube came down to 
 about 27-J- inches (Paris measure) from the lower surface. 
 The liquid being at rest, the wax was removed from the 
 small hole, when the lower 9 or 10 inches of mercury fell, 
 whilst the upper portion ascended and struck the top of the 
 barometer tube with great violence. So great was the 
 velocity that the Abbe says the experiment had to be carried 
 out carefully for fear of breaking the top of the tube by the 
 shock. I wish specially to draw attention to this, as a very 
 little reflection will show that the mercury could have been 
 raised many times four feet. 
 
 In the Bellange pump, it must not be imagined that any 
 sized hole, placed anywhere, will give a satisfactory result. 
 The exact size of the hole and the exact position were found, 
 after many trials, as giving the best results. 
 
 Now, without pretending to say that this is the way tall 
 trees get the sap up to their leaves, there can be, I think, no 
 doubt that this is a way they might get it. 1 
 
 1 Sir Oliver. Lodge in his Presidential address to the British 
 Association at Birmingham, 1913, said : "To attribute the rise of 
 sap to vital force would be absurd ; it would be giving up the pro- 
 blem and stating nothing at all. The way in which Osmosis acts to 
 
The Seville Pump 153 
 
 The action taking place in a mouthpiece will be resumed 
 again when I treat of, what I call, " negative resistance." 
 
 SUMMARY 
 
 A tube attached to the outside of a hole in a thin wall of 
 a tank increases the discharge through this hole, provided 
 that the tube is properly wetted. This discharge increases 
 with the length of the tube up to a certain maximum ; when 
 the length of the tube exceeds this maximum, the discharge 
 decreases again slowly. 
 
 Any sudden change of velocity in a liquid causes an appar- 
 ent loss of energy, i.e., some of the energy is converted into heat. 
 
 It is not true, as is commonly taught in some books, that 
 liquid issuing vertically from an orifice will necessarily rise 
 to the level of the source : the defect, which is found experi- 
 mentally, being attributed to viscosity. The height depends 
 primarily on the contraction of the vein caused by the orifice. 
 Viscosity has very little to do with the defect, the shape of 
 the orifice being all important. With the same liquid, and 
 therefore the same viscosity, different forms of orifice can be 
 caused to increase or decrease this height. 
 
 REFERENCES 
 
 FRANCESCO DOMENICO MICHELOTTI, Sperimenti idraulici. 
 M. 1'abbe NOLLET, Academic des Sciences. 
 LE CITOYEN J. B. VENTURI, Recherches Experiment ales, 
 etc. 
 
 produce this remarkable and surprising effect is discoverable, and 
 has been discovered." 
 
 Not knowing what experiments Sir Oliver based this statement 
 on, / wonder if he was extrapolating 1 
 
CHAPTER XIII 
 
 RlVERS AND CANALS A BODY FLOATING IN A STREAM MOVES 
 FASTER THAN THE STREAM CORRAISON OF STREAMS 
 
 A RIVER, or canal, may be defined as a stream partially 
 bounded by a free surface. 
 
 The simplest form of canal we can imagine is one which is 
 perfectly straight, very broad, in order to eliminate boundary 
 effects at the sides, and properly designed, so that the slope 
 of the surface of the water is strictly parallel to the slope of 
 the bed of the canal. Now, it is clear that the slope of the 
 surface is the only effective cause which can generate motion 
 in the water ; since, without this slope, the liquid would 
 not be in movement. 
 
 If this water met with no resistance, its motion would be 
 accelerated indefinitely, according to the law of bodies sliding 
 down an inclined plane. We know that the velocity of flow 
 is uniform ; so the accelerating forces rriust be equal and 
 opposite to the retarding forces, since they are balancing one 
 another. 
 
 The accelerating force on a prism of the liquid may be 
 expressed as cobAg sin 9?, where CD is the area of the cross 
 section of the prism taken perpendicularly to the slope of the 
 surface, b an arbitrary length of the water parallel to the sides 
 of the canal measured along the surface, A the density, or 
 mass of the liquid per unit volume, g=gravity constant, 
 and 9? =the angle of slope of the surface with the horizontal. 
 
 As regards the retarding forces or resistance encountered 
 by the liquid in the canal, Coulomb first, I believe, showed 
 experimentally, that liquid resistance can be expressed by 
 the simple formula 
 
 R-AV+BV 2 
 
 154 
 
Rivers and Canals 155 
 
 where R =resistance, and V is the velocity of the stream, 
 A and B being constants. M. de Prony correlated these 
 formulae as, 
 
 Q / u 2 \ 
 
 sin a) = ( 0-000024^ + 0-003585 ; ) 
 co \ g / 
 
 where Q is the " wetted perimeter " and u =mean velocity 
 of the stream. This formula has been frequently verified, 
 and has been found to give correct results. 1 
 
 We may accept this as an empirical formula, if we please, 
 without committing ourselves to any theory, for the present. 
 All I wish to insist upon is that the resistance must consist 
 of two terms, one involving the simple velocity whilst the 
 other involves the square of the velocity ; and that Coulomb, 
 by means of a well-known experiment which I have de- 
 scribed elsewhere, showed that the term involving the first 
 power of the velocity was entirely due to the viscosity of 
 the liquid. 
 
 In the first chapter, referring to the Principia, it was 
 pointed out that all resistance in incompressible liquids could 
 be divided into 
 
 (1) Resistance due to the density of the liquid. 
 
 (2) That due to " attrition/' or the rubbing of the liquid 
 against the solid. 
 
 (3) That due to the viscosity, or " treacliness," of the 
 liquid. 
 
 We know that the term A V is solely due to the viscosity 
 of the liquid ; therefore the term B V 2 can only be due to 
 the density of the liquid and to the " attrition." 
 
 The resistance due to " attrition " may be described as a 
 " solid-liquid friction " (I am sorry I cannot call it " liquid 
 friction," but that term has been already appropriated 
 for another kind of resistance) ; such a resistance would, 
 clearly, vary as the pressure, and would not be affected by 
 the velocity, in other words, it would be a constant. If a 
 river were composed of mercury flowing in a glass bed, this 
 item would have to be taken into account. We know, 
 
 1 These coefficients, based on experiments made on very small 
 canals have since been shown to be incorrect for larger ones ; no 
 great importance should therefore be attached to them. 
 
156 Motion of Liquids 
 
 however, that there is no rubbing between a river and its 
 bed ; so that in the case of water there is no " attrition," 
 or resistance caused by it (there have been authors, however, 
 who thought liquid resistance might be expressed as, 
 R=C+BV 2 , where C is a constant). 
 
 It is clear, therefore, that the resistance varying as the 
 square of the velocity is due to the inertia of the liquid. 
 This is in accord with the mature view of Sir George Stokes, 
 who said : " Except in the case of capillary tubes, or, in 
 case the tube be somewhat wider, of excessively slow motions, 
 the main part of the resistance depends upon the formation 
 of eddies. This much appears clear ; but the precise way 
 in which the eddies act is less evident " [Italics added]. 
 
 Dubuat, Duchemin, and many others, even down to 
 modern times, believed that the resistance in rivers was 
 caused by the viscosity and the rubbing (frottement] on the 
 bed. Even at the present day it is commonly taught that 
 the resistance is caused by " fluid friction " of the river 
 against the solid boundary. 
 
 "According to this common theory the water ought to 
 flow with uniformly accelerated velocity ; 
 For even the supposition of a certain friction would be of 
 no avail, for such friction could not be transmitted 
 through the mass." 
 (Stokes, Math, and Ph. Papers. Emphasis added.) 
 
 I might point out some of the difficulties in the way of 
 believing the " fluid friction " theory such, for example, 
 as increase of wetted surface resulting in less resistance : or 
 the difficulty of imagining how a resistance caused by rubbing 
 could increase as the square of the velocity but if such resist- 
 ance " cannot be transmitted through the mass," it would 
 be only waste of time to discuss the question. Stokes gave 
 the explanation of the above-quoted statement, in an earlier 
 paper, in the following manner : " When a ball pendulum 
 oscillates in an indefinitely extended fluid, the common 
 theory gives the arc of oscillation constant. Observation, 
 however, shows that it diminishes very rapidly in the case 
 of a liquid, and diminishes, but less rapidly, in the case of 
 an elastic fluid. It has been attempted to explain this diminu- 
 
Rivers and Canals 157 
 
 tion by supposing a friction to act on the ball, and this 
 hypothesis may be approximately true, but the imperfection 
 of the theory is shown from the circumstances that no account 
 is taken of the equal and opposite friction of the ball on the 
 liquid " (Math, and Ph. Papers. Italics added). The 
 whole mechanics of this subject appear to be in a very 
 nebulous state. 
 
 Before discussing, in detail, the flow of a river, I may refer 
 to a point which is, perhaps, more curious than important ; 
 but which, when carefully considered, will throw much 
 light on the theory of resistance here advanced. It was, 
 as far as I am aware, first pointed out by Dubuat, though 
 he says it had been long known to boatmen accustomed to 
 navigating rivers, that a body floating freely on the surface 
 of a stream of uniform velocity, should acquire, and does, 
 in fact, acquire, a uniform velocity greater than that of the 
 central filament of the stream. This fact, which reflection 
 will show to be almost self-evident, has been denied by many 
 people, and so is worth referring to at greater length. 
 
 Bazin (Recherches Hydrauliques] was acquainted with 
 this fact, for he says : " It has been long known to the boat- 
 men of the Rhine and to our pontoniers that a loaded boat, 
 having a heavy draught of water, descends more quickly 
 than the water which supports it, or the bodies floating on 
 the surface." He, however, uses this as an argument that 
 the maximum velocity of a stream is below the surface', 
 which is sometimes true and sometimes not. 
 
 When any kind of body floats on a stream, having a slope 
 
 which may be expressed as , the body is situated on an 
 
 inclined plane, and consequently is subjected to an acce- 
 lerating force equal to the product of the weight of the 
 
 volume of water displaced by the fraction . This force 
 
 b 
 
 tends to make it descend the slope, and would accelerate the 
 velocity of its descent indefinitely, if the body were experi- 
 encing no resistance. Therefore, if we suppose the body, 
 initially, to be only moving with the velocity of the liquid 
 surrounding and supporting it, it will be at rest, relatively 
 
158 Motion of Liquids 
 
 to this fluid, and so will be meeting with no resistance from 
 it. Therefore the accelerating force will continue to act 
 and will impress more velocity on it ; until the excess of its 
 velocity over that of the fluid causes such a resistance as will 
 neutralize the accelerating force : when this occurs the 
 motion will be steady, the accelerating force being balanced 
 by the resistance. 
 
 The greater the volume of water displaced by the body, 
 the greater will be the accelerating force ; and the greater 
 also will be the excess of velocity of the body over that of 
 the stream. Conversely, the less the volume of water dis- 
 placed, the less will be this excess of velocity ; and, finally, 
 when this volume of water is equal to the size of a molecule, 
 all excess of velocity will disappear. 1 
 
 It may be asked, why should the body be more accelerated 
 than would be the volume of water it displaces ? The 
 accelerating action is, of course, the same in both cases ; 
 and if this liquid were enclosed in a weightless envelope- 
 or one whose density was the same as the water this 
 " body of water " would flow faster than the rest. Since no 
 change of form could occur in this box of water any more 
 than it could in the floating body, which might even be a 
 mass of ice it is clear that, one form of resistance being 
 removed, the body is less retarded than is the liquid sur- 
 rounding and supporting it. 
 
 If one looks at a barge going down with the stream one 
 will invariably see that the bargeman is steering a thing he 
 could not possibly do if the barge had no " way "on. 
 
 A corollary which follows from the foregoing is that a 
 body floating freely in a stream will always place itself in 
 the middle of the stream, and will follow the line of the 
 central filament. This it will do because it will there meet 
 with the least resistance in gaining the greatest uniform 
 
 1 It is clear that, generally, the viscous resistance to motion of 
 similar bodies floating in a stream is proportional to the square of 
 their linear dimensions ; and the resistance to change of shape, eli- 
 minated by replacing the water by the floating bodies, is proportional 
 to the cube of their linear dimensions ; hence the limiting velocity 
 excess is greater for large than for small floating bodies of similar 
 shape. 
 
Rivers and Canals 159 
 
 velocity which it is capable of acquiring ; the greater the 
 amount of liquid displaced, the greater will be the tendency 
 to retain this position, and to drive away, out of its course, 
 any bodies smaller than itself. 
 
 Another corollary which follows from this is, that the 
 better the shape of the body immersed i.e., the less the resist- 
 ance it offers to motion the greater will be its excess of 
 velocity over that of the stream. 
 
 Still another corollary : if a stream is charged with a 
 number of floating bodies, its velocity must be increased, 
 and become greater than if these bodies were replaced by 
 equal parts of homogeneous fluid. Hence, rivers carrying 
 ice after a thaw, or which are covered with floating logs 
 of timber, must have a greater velocity than during their 
 natural, uncharged condition. 
 
 It may be stated here, definitely, that the chief resistance 
 to the flow of a stream is caused by the fluid changing shape : 
 viscosity, undoubtedly, assists, but is only responsible for a 
 part of the resistance. 
 
 This statement may appear very dogmatic, but I think 
 sufficient evidence can be produced in support of it ; whilst 
 I am unaware of any evidence that can be brought against 
 it. 
 
 A further corollary to all this is that if a river were com- 
 posed of a frictionless liquid a liquid so " improved " 
 that the viscosity was a vanishing quantity it would flow 
 exactly like water, eddies and all, only faster. 
 
 Now, how does a river flow ? Let us first imagine the 
 slope of the surface and of the bed, parallel to it to be 
 exceedingly small ; and the breadth indefinite, so as to elimi- 
 nate any boundary effects at the sides. The top layer of 
 molecules will slide over the next layer, and so on down to 
 the bottom. 
 
 By assuming the slope of the surface to be very small we 
 can consider all the resistance to change of shape to be 
 " viscous resistance " resistance due to viscosity, and which 
 varies as the velocity, only. 
 
 Following Unwin's line of reasoning (" Hydromechanics," 
 Encyclopaedia Brit.} let fig. 61 represent a vertical longi- 
 
i6o 
 
 Motion of Liquids 
 
 tudinal section of a portion of the stream, and let O A and 
 O * A * be the intersections with this of two transverse sec- 
 tions at a distance apart /. Let ab, cd be the traces of two 
 planes parallel to the free surface, or the bed, and let us 
 consider the equilibrium of the layer abed of width unity. 
 Let Oa =y, ac =dy, and let v be the velocity of the particles 
 comprised in abed, v being the function of y which is to be 
 determined. Taking the components of the forces acting 
 upon abed, parallel to OO 1 , the pressures on ac, bd, being 
 
 FIG. 61. 
 
 proportional to the depth from the surface are equal and 
 opposite : also any viscous resistances on the lateral faces 
 of the prism are zero, since in a wide stream there is no relative 
 sliding between abed and the layers on each side. There 
 remain only the resistances on the upper and lower surfaces, 
 and the component of the weight. 
 
 If G=weight of a unit mass of the fluid, the weight of 
 the layer is Gldy ; and if i is the slope of the stream, the com- 
 ponent of the weight, parallel to O O 1 , is Glidy. The vis- 
 cous resistance on the face ab is proportional to its area, 
 and to the difference dv of velocity between that of the face 
 ab and that of a parallel face immediately above it and 
 
Rivers and Canals 
 
 161 
 
 inversely proportional to dy, the distance between these 
 
 two faces. The resistance is therefore K/ (where K is a 
 
 dy 
 
 constant), the negative sign being employed because, if v 
 
 increases with y, is positive, while the action of the 
 dy 
 
 o 
 
 Hi 
 
 Vm^cc 
 
 FIG. 62. 
 
 layers above ab is a retarding action. The resistance on 
 
 the face cd is similarly Kl -f- Kid . The resultant of the 
 
 dy dy 
 
 action of the layers above and below is, therefore, Kld~ . 
 
 dy 
 
 When the motion is uniform 
 
 dy 
 
 Gi 
 
 . . 
 
 or _-=: _; integrating, we get 
 dy* Jv 
 
162 Motion of Liquids 
 
 dv Gi . 
 _=-_3- + 
 
 an equation which gives the velocity v at any depth. 
 
 If on the vertical line O A representing the depth of the 
 stream (fig. 62) the values of v are set off horizontally, a 
 parabolic curve is obtained, termed the vertical velocity 
 curve for the section considered. The constant v is evi- 
 dently the surface velocity, being the value of v, when y =O. 
 The parabola has a horizontal axis corresponding to the 
 position of the filament of maximum velocity. If there is 
 no resistance at the surface of the stream, like that at the 
 bottom and sides, the maximum velocity should be at the 
 surface, and then C=O, and the equation becomes 
 
 - - 1 2 (2} 
 
 Assuming this for the present, the mean velocity is 
 vdy_ _ 1 GL 2 ,, 
 
 The bottom velocity, corresponding to a depth, h, is there- 
 fore, by equation (2) 
 
 v = v i- G V 
 
 and therefore v m =(2v v b ) (4). 
 
 Now assuming the general equation (i) 
 
 v will have the maximum value V, for a value, h', of y t 
 dv_ 
 dy 
 
 which makes zero. That is 
 
 and the maximum velocity is 
 
Rivers and Canals 163 
 
 Therefore, ^V- 
 
 - 
 
 Unwin adds : " It is now understood that the motion in 
 a stream is much more complex than the viscous theory just 
 stated assumes. The retardation of the stream is much 
 greater than it would be in simple motion of this kind. . . . 
 Nevertheless, the viscous theory may probably be so modified 
 as to furnish ultimately a true theory of streams " [Italics 
 added]. 
 
 There can, I think, be no reasonable doubt that the 
 general flow of a sluggish stream is like this. Experiments 
 on the Mississippi (probably the most accurate ever made 
 on any large river) showed that the vertical velocity curve 
 was a parabola, the axis of which was at a variable distance 
 below the surface such distance depending partly on the 
 wind, but also, and probably to a greater extent, on a separate 
 action which will be explained later. Methods of gauging 
 rivers have been based on this property. 
 
 So far we have been discussing the general flow of a 
 stream ; but, as all the retardation was viscous, the resistance 
 to this flow varied, only, as the velocity. This is not the 
 whole truth ; we must superpose on this another resistance 
 varying as the square of the velocity. 
 
 We originally assumed our stream to have a very small 
 slope. Let us now suppose the slope to be increased : the 
 stream will, clearly, flow faster, and the bottom velocity 
 become greater. At a certain " critical velocity " of this 
 water along the bed or more correctly, the water attached 
 to the bed there will be discontinuity, and eddies, or vor- 
 tices, will start from the bottom of the stream. I have ex- 
 plained at considerable length in the ABC of Hydrodynamics 
 (to which the reader is referred) how these eddies are formed 
 by great differences of pressure across the stream filaments. 
 These eddies will increase in size and spread upwards, until 
 they occupy a considerable part of the body of the stream. 
 Fig. 63 will give an idea of my meaning. These eddies will 
 contain considerable kinetic energy, and their formation 
 will cause a resistance which varies as the density of the 
 liquid and as the square of its velocity. It must be remem- 
 
 M 
 
164 
 
 Motion of Liquids 
 
 bered that in the case where all the resistance was viscous 
 there was no generation of kinetic energy in the stream, 
 all the energy required to overcome viscosity being directly 
 converted into heat. 
 
 With a greater slope the eddies will spread quite up to the 
 surface and the stream will become a boiling torrent. 
 
 We have now a clear " general idea " of how a stream 
 flows and how it gradually changes into a torrent. We 
 see also why and how the resistance =AV+BV 2 . 
 
 It is clear also that if the liquid of the river were inviscid, 
 
 FIG. 63. 
 
 the vortices would still be formed, and that the resistance to 
 its flow would vary as the square of the velocity only. 
 
 It will now be apparent, how this " secondary action " 
 tends to lower the position of the stream filaments having 
 the maximum velocity. The vortices formed at the bottom 
 of the stream roll on this bottom, so that the velocity at the 
 part nearest the surface of the stream will be about double 
 that of the mean velocity of the vortices along the bed. This 
 will clearly cause an accelerating action on the layers of fluid 
 above, instead of a retarding one ; and this will lower the 
 
Corraison " of Streams 
 
 165 
 
 dv 
 
 position of the stream filame nts where - is zero. The axis 
 
 dy 
 
 of the parabola will consequently be lowered. 
 
 Let us continue the subject and see how a river deepens 
 its bed. 
 
 E. C. Andrews (Journal of the Royal Society of N. S. 
 Wales, 1909) in his paper on Corraison of Gravity streams, 
 commences by saying : "no stream moves as a rigid bolt, 
 except possibly when falling freely." He shows that when 
 a stream deepens its bed, it does not do so like a plough ; but 
 rather with a kind of " scratching back " action : that it 
 deepens it as a rabbit would deepen its hole ; by scratching 
 
 FIG. 64. 
 
 backwards, and driving the debris backwards. When the 
 velocity of the stream is checked, by some sudden diminution 
 in the slope of the bed, corraison takes place. The action 
 is as shown in fig. 64, which is copied from Mr. Andrews' 
 paper. It will be seen that the stream is " scratching back- 
 wards." In this paper we find two laws of stream corraison. 
 
 " (i) All other things being equal, the greater the stream 
 velocity the more will the headward profiles of the cuts formed 
 below base level, by corraison be inclined to the vertical." 
 
 " (2) The more freely does the force of gravity act, the 
 more nearly will the basin head approach a vertical form." 
 
 These rules, which appear to be variations of one another, 
 
 M2 
 
1 66 Motion of Liquids 
 
 imply, of course, that in fig. 64 the slope at A B is steeper 
 than the slope at C D. Any one who has strolled about in 
 the bed of a dry, or even nearly dry, torrent can see for 
 himself that where the stream has dug a hole for itself, the 
 up-stream side of the hole is always steeper than the down- 
 stream one. There are occasions where the difference is 
 very small and the slopes appear to be nearly the same : 
 the reverse, however, is never the case, or if it is, I have never 
 seen it. The small stones are forced out of the bed 1 into 
 the stream, in consequence of the pressure in the stream 
 by reason of its velocity being considerably less than the 
 pressure in the liquid below the stones, which is at rest. A 
 very little simple arithmetic will show what enormous force 
 a difference of four or five pounds per square inch can exert. 
 The velocity of the stream passing A B is in excess of that 
 passing C D ; consequently there will be more erosion there, 
 and the slope at A B will be steeper than that at C D. 
 
 In fig. 64 we see a rapid change of slope : the surface 
 water is being retarded ; and, as we have seen, when this 
 occurs the position of the maximum in the velocity curve of 
 the liquid will be lowered. This will also increase the velo- 
 city of the water near the bed, and the liquid will commence 
 digging a hole. All this follows logically from what has 
 been said previously. 
 
 Suppose, next, that the bed is of such hard rock that the 
 water cannot lift it (speaking, popularly, as if the stones 
 were raised by suction) what happens then ? Supposing 
 that the stream is carrying hard and sharp sand which is 
 generally the case, under such circumstances the vortex 
 will act like an emery wheel and grind the surface. 
 
 The reader may ask, why should it act like an emery 
 wheel ? Why should the grinding material be on the rim 
 of the vortex rather than in any other part of it ? The 
 reply to this is to be found in the Principia, Prop. LI II. 
 Theorem xli. : "A solid, if it be of the same density with the 
 matter of the vortex, will move with the same motion as the 
 parts thereof, being relatively at rest in the matter that sur- 
 rounds it, // it be more dense, it will endeavour more than 
 
 * By suction. 
 
" Conaison J> of Streams 167 
 
 before to recede from the centre ; and therefore overcoming 
 that force of the vortex, by which, being as it were kept in 
 equilibrio, it was retained in its orbit, it will recede from 
 the centre, and its revolution describe a spiral, returning no 
 longer into the same orbit. And by the same argument, if 
 it be more rare it will approach the centre " [Italics added]. 
 
 In this case the " polishing " is undoubtedly caused by 
 " attrition " but it is that of solid against solid and not 
 liquid against solid. 
 
 In " hanging valleys " this " corraison " is very well 
 seen : the small streams (having their surface water re- 
 tarded) cut their beds back and so form waterfalls. This 
 action continuing, the stream keeps cutting the base of the 
 fall away and so working the fall backwards. 
 
 FIG. 65. 
 
 The question may be examined from another point of 
 view which is of interest to engineers. A stream meeting 
 the abutment of a bridge may produce this excavating 
 action at the front of the abutment ; and in certain cases 
 this danger has to be provided against. There have been 
 piers of bridges in India whch have been undermined so 
 that they fell up stream \ 
 
 Fig. 65 from Willcock's & Craig's Egyptian Irrigation 
 clearly explains the action against a partly submerged 
 obstacle. 
 
 " In some of the large canals in India, the bed upstyeam 
 of bridges has been scoured for miles to a depth of, perhaps, 
 two feet below the masonry floors of the bridges which are 
 left standing up, and forming, in fact, submerged weirs " 
 (E. S. Bellasis, Hydraulics. Italics added). 
 
 Sir Arthur Cotton, who was certainly a past master in 
 
168 
 
 Motion of Liquids 
 
 the art of dealing with large bodies of water, used to say that 
 you could take your protection out vertically, or horizontally. 
 Most engineers take it out vertically, by very deep founda- 
 tions, down to solid rock : Sir Arthur Cotton, however, 
 built weirs across large rivers having deep sandy beds, 
 without, practically, any foundations at all : he took his 
 protection out " horizontally," by preventing the stream 
 attacking the sand near the weir. 
 
 Mr. Andrews, in his 
 paper referred to, says 
 that the " conditions 
 favouring the forma- 
 tion of deep basins 
 (below the associated 
 /MR base levels) with flat- 
 tish floors and steep 
 sides are 
 
 " (a) Great stream 
 depth as compared 
 with the width of the 
 channel base. 
 
 " (6) Great local 
 increase of stream 
 velocity." 
 
 In certain cases, 
 where there is an 
 obstruction on one 
 side of a torrent, the 
 horizontally, as in fig. 66. 
 is nearly always circular ; 
 the water, circling round, being accelerated by the 
 stream which is shown as moving past it. There is inter- 
 change of liquid between the hole and the straight part of 
 the stream, but the interchange appears to be only partial. 
 There are cases also where the stream acts vertically up- 
 wards, and I have seen very fine hemispherical domes cut 
 out in this manner. 1 
 
 1 The Gorges de Sierroz on the Fier river, not far from Lake 
 Annecy, the waters of which flow into the Fier. 
 
 stream is caused to circle 
 When this occurs, the hole 
 
" Corraison " of Streams 169 
 
 There is one other point about rivers which is worth refer- 
 ring to, since it does not appear to be generally known ; that 
 is, that the surface of the water, from one bank to the other, 
 is not ordinarily terminated by a straight line, but by con- 
 cave curves : that is to say, the centre is higher than the 
 edges. 
 
 That such must be so is almost self-evident, since it follows 
 as a logical deduction from the famous law of Bernoulli. It 
 is well known that the velocity of flow is greater at the middle 
 of the stream than it is near the banks : the pressure in 
 the liquid is, therefore, less at the centre than at the edges. 
 It is clear, however, that the total pressure across the stream 
 must be the same at all parts, or there would be flow from 
 the edges to the centre. This, we know, is not the case ; there- 
 fore the pressure, or potential energy in the liquid, must be 
 supplemented by some "energy of position," in order that 
 equilibrium may be maintained. 
 
 The curves, which clearly must be parabolas, are usually 
 so flat that this concavity is not noticeable by the unaided 
 eye ; the small " hump " in the centre of the stream can, 
 however, be frequently observed, if one looks for it. Bazin 
 says : " there is always in the axis of the current, a protuber- 
 ance which is higher than along the edges " (Recherches 
 hydrauliques l ) . This is very clearly shown in his Plate XXII 
 figs. 2 and 8, even in the very small channels he was experi- 
 menting with. During the flood of some torrents, the differ- 
 ence in level is quite appreciable, being as much as 9 inches, 
 or more. 
 
 It now remains to examine the question of how a river 
 gets round a corner. 
 
 It is surprising that that very acute observer Dubuat 
 should have fallen into the error of supposing that the edges 
 of a stream, in cross section, should be level when the stream 
 is turning round a corner. Referring to a stream flowing 
 round a bend, he says, " if one supposes, as is natural (?) 
 that the water should be at the same level at the correspond- 
 ing points B, C : E, M : H, I (fig. 67), one sees that the 
 
 1 II existe toujours dans 1'axe du courant une protuberance plug 
 eleve que le long des parois. 
 
170 
 
 Motion of Liquids 
 
 slope of the water following C M I is greater than that 
 followingBEH . . . etc." The deduction is strictly logical, 
 but he had started from false premisses. Streams do not 
 flow like that. 
 
 The only other author that I know, who has treated of 
 this subject is Professor James Thomson (Proc. Roy. 
 Society, 1876). I will quote the commencement of his 
 paper, in extenso, as there are parts I do not agree with. 
 
 " In respect to the origin of the windings of rivers flowing 
 through alluvial plains, people have usually taken the rough 
 
 FIG. 67. 
 
 notion that when there is a bend in any way commenced, the 
 water just rushes out against the outer bank of the river at the 
 bend, and so washes that bank away, and allows deposition 
 to occur on the inner bank, and thus makes the sinuosity 
 increase. But in this they overlook the hydraulic principle, 
 not generally known, that a stream flowing along a straight 
 channel, and thence into a curve, must flow with a diminished 
 velocity along the outer bank, and an increased velocity along 
 the inner bank, IF we regard the flow as that of a per feet fluid." 
 [Italics added]. 
 
 Now, it requires some audacity to contradict such an 
 eminent authority ; but it is a matter of the commonest 
 observation that a real river does not flow in the very least 
 like that. I have done some " river training " and I know 
 that the river Indus did not flow like that. The maximum 
 
PLATE IV. 
 
 FIG. 68. 
 
 To face p. 171.] 
 
 [de Villamil, Motion of Liquids. 
 
Corraison " of Streams 
 
 171 
 
 velocity of the stream was always nearer the outer than the 
 inner bank. 
 
 I am aware of what James Thomson calls the " Hydraulic 
 principle " ; and I quite agree that, if the liquid had no free 
 surface, the velocity near the inner bank would be in excess 
 of that near the outer bank whether the liquid were viscous 
 or not. Water flows like that in a pipe, as is shown in fig. 68, 
 which is copied from one of Dr. Hele-Shaw's photographs. 1 
 Having a free surface, however, changes the conditions. 
 
 FIG. 69. 
 
 But to resume : " For any lines of particles taken across 
 the stream at different places, as A! Bj, A 2 B 2 , etc. (fig. 69), 
 and which may be designated in general as A B, if the line be 
 level [which it is not], the water-pressure must be increasing 
 from A to B, on account of the centrifugal force of the particles 
 composing the line, or bar of water ; or, what comes to the 
 same thing, the water surface of the river will have a trans- 
 verse inclination rising from A to B." 
 The paper is curious and, like everything by this eminent 
 1 This is a flow of air, but water flows in the same manner. 
 
172 
 
 Motion of Liquids 
 
 author, well worth reading ; but it is singular how, having 
 started from such erroneous fundamental assumptions, he 
 should yet have arrived at correct conclusions. 
 
 Now, what are the facts about the flow of an ordinary 
 river ? " At a bend there is a ' set of the stream ' towards 
 the concave bank, the greatest velocity being near that bank." 
 (E. S. Bellasis, Hydraulics). " It will generally be found 
 that the straight forward movement of the water across to one 
 side of the bend implies a portion of slack water opposite. 
 Thus the river, moving as shown in the diagram (fig. 70), 
 tends to strike the bank between A and B and to leave slack 
 water at C." 
 
 " Scour is therefore heaviest at bends, and as the move- 
 ment of the water be- 
 comes more complex 
 there, turbulence is 
 favoured, and the 
 scoured material is more 
 readily transported " 
 (Willcocks & Craig's 
 Egyptian Irrigation) . 
 
 Let us now follow 
 the motion of the water 
 in the river and see 
 how this comes about. 
 We have seen that in 
 
 the straight reach the central filament was moving with the 
 greatest velocity, the surface had a parabolic cross section, 
 and the level of the water at the banks was the same ; the 
 river had, what the French call, a regime. This is shown, 
 diagrammatically, in fig. 71, where the curve is, of course, 
 very exaggerated. A A 1 cuts the centre line of the stream, 
 and is the axis of the parabola. The surface is represented 
 by CAB, the ordinates of the curve (above CB) being 
 measures of the excess of velocity of the liquid at different 
 parts of the surface of the stream. 1 
 
 Arrived at the bend, the water moves on in a straight 
 
 1 Just as one may say the height the water rises in a Pi tot tube is 
 a " measure of the velocity " of the liquid at the point of the tube. 
 
 FIG. 70. 
 
Corraison " of Streams 
 
 line in consequence of its inertia. The " axis " of the 
 stream will shift to D D 1 and the surface of the water will 
 now be represented by C * D B *, being heaped up against 
 the concave bank and lowered at the convex bank. It is 
 clear that equilibrium has now been destroyed ; the pres- 
 sures at the opposite banks are no longer equal. There is 
 a defect of pressure at C 1 , whilst there is an excess of pressure 
 at B 1 . 
 
 FIG. 71. 
 
 Let us first deal with C 1 : there is, what Professor James 
 Thomson calls, " a fall of free level " here. The result of 
 this is that water will flow from all regions of higher pressure 
 to this region of deficient energy, so as to give it " energy of 
 position " : the flow being, of course, chiefly along the bed. 
 Referring now to fig. 72, we see the currents, represented 
 by arrows, flowing along the bottom to C 1 . This water has, 
 initially, comparatively little velocity ; but in its travel 
 towards C 1 it is gaining " energy of position " and so will 
 be losing some kinetic energy its velocity will be decreasing. 
 " Here, in general, some of the water becomes supersaturated 
 and deposits its silt " (Willcocks & Craig). 
 
 Let us now trace the action at B 1 : there is a " rise of 
 free level " here an excess of " energy of position/' The 
 water will descend, losing " energy of position " and gaining 
 kinetic energy. Besides this it has, what I may call, more 
 
Motion of Liquids 
 
 than its "fair share " of longitudinal velocity. 1 The result 
 is that it excavates the bed, as described previously, only 
 in a diagonal direction, as shown in section in fig. 72. This 
 excavating action will undermine the bank, and so cause it 
 to fall ; the silt being carried away by the eddies. 
 
 Let us now see how this would appear on plan. If we 
 represent a bend of the river by fig. 73, with the flow as 
 shown by the arrows, the direction of the different currents 
 along the bottom of the stream are indicated as flowing 
 
 FIG. 72. 
 
 from certain spots marked by stars. It will be seen that 
 they all cross the bed of the river and flow towards the 
 bank where there is a defect of energy. Professor James 
 Thomson showed experimentally that such currents actually 
 occur. 
 
 The nett result of all this is that " the water-level at the 
 concave bank is slightly higher than at the convex bank." 
 (E. S. Bellasis, Hydraulics). 
 
 That the surface is convex 2 is ordinarily true ; but there 
 
 1 The surface velocity being slightly retarded by the bend in the 
 river, the position of maximum velocity will be lowered, the bottom 
 velocity being consequently increased. 
 
 2 By " convex " I here mean that the general surface of the water 
 is above a horizontal line joining the banks. " Concave " meaning 
 the reverse. 
 
Corraison " of Streams 
 
 175 
 
 are cases where it is the reverse, where the surface is con- 
 cave. This will occur when a river is flowing into the sea 
 during a flood tide. The tide flowing more rapidly in the 
 centre of the river than near its banks will retard the velocity 
 of the stream more there. Following the argument in the 
 
 FIG. 73. 
 
 previous case, the " energy of position " will be transferred 
 to near the banks. 
 
 Since the surface is sometimes convex and sometimes 
 concave in cross section, it is clear that there must be cases 
 when it is level, and that is why I carefully said the section 
 was, ordinarily, convex. 
 
176 Motion of Liquids 
 
 SUMMARY 
 
 Resistance in rivers is not caused by the liquid rubbing 
 against their beds : it is almost entirely caused by the liquid 
 changing shape, and so forming " eddies." 
 
 A body floating down a stream will travel faster than the 
 water surrounding and supporting it. The excess of velo- 
 city will depend on : (i) the shape of the body, and (2) its 
 displacement in the liquid. If the body is not larger than 
 a molecule of water, there will be no excess of velocity. 
 
 A river carrying ice after a thaw, or logs of timber, will 
 flow faster than it would in its uncharged state. 
 
 The surface of the cross section of a flowing stream is not, 
 ordinarily, a straight line, but is a double concave curve : 
 the curvature depending on the difference between the 
 velocity of the water in the middle of the stream and that 
 near the banks. In some cases, however, the curvature 
 may be the reverse. 
 
 REFERENCES 
 
 E. C. ANDREWS, " Corraison of Gravity Streams " (Journal 
 of the Royal Society of N.S.W., 1909). 
 
 Professor JAMES THOMSON (Proceedings of the Royal Society, 
 March 14, 1876). 
 
 WILLCOCKS and CRAIG, Egyptian Irrigation. 
 
 E. S. BELLASIS, Hydraulics. 
 
 H. BAZIN, Recherches Hydrauliques. 
 
CHAPTER XIV 
 
 NEGATIVE RESISTANCE IN LIQUIDS 
 
 IN discussing any scientific subject it is very difficult to 
 avoid repeating oneself, and I cannot even plead that I 
 have done my best to avoid doing so. I am not even quite 
 sure that it is a disadvantage ; indeed, I cannot help think- 
 ing that many writers would make their books more intel- 
 ligible if they would insist more on certain important points, 
 instead of simply stating them 1 ; would, in fact, rub them 
 well in. When a musician writes a symphony, he starts 
 with certain " themes," and it is not considered wrong for 
 him to return to them periodically. Why should this not 
 be permissible when treating what is admittedly a complex 
 subject ? 
 
 However, to return : I have stated previously that all 
 forms of liquid resistance could be expressed by the formula 
 R AV+BV 2 subject to the condition that the solid is pro- 
 perly wetted. 
 
 This is as old as Newton, and was the view held by Lord 
 Kelvin that Newton of the nineteenth century as will be 
 seen by reference to Thomson and Tait. It was also held 
 by Sir George Stokes, who was no mean authority. 
 
 Latterly, this equation has gone out of fashion, and the 
 idea more generally taught now is that R=AV M . This 
 expression is purely empirical, and fits experiment very 
 
 1 " Redundancy is never necessary in logic, but it is often necessary 
 to the correct working of our process of understanding. We know the 
 value of judicious repetition when teaching." Philip E. B. Jourdain 
 (The Principle of Least Action). 
 
 177 
 
178 Motion of Liquids 
 
 indifferently ; there is, however, sign of a renaissance of the 
 old rational formula, as may be seen in the papers issuing 
 from the National Physical Laboratory. 
 
 It may, naturally, be asked, why was the original view 
 abandoned ? What is there to be said against the dictum, 
 " the resistance of a real liquid . . . may be expressed as 
 the sum of two terms, one simply as the velocity, and the other 
 as the square of the velocity ? " (Thomson and Tait, 
 Natural Philosophy. Italics added.) I have not been able to 
 trace the history of this change of view ; but most probably 
 it was because the term involving the first power of the velo- 
 city was frequently found to be negative. To explain my 
 meaning by an example, I will quote from a paper on the 
 resistance of water in pipes (A. C. A., 1910-11) which is 
 
 expressed as F = QV Z (A- + K\ ; or the total resistance 
 may be rewritten as F 1 = @v 2 / 2 /'A - + KY when, by sub- 
 stituting J for v, the kinematic viscosity, we have 
 
 ) -\-Q~K(vT) 2 ; which is an equation of the same 
 form as that given by me. 
 
 But, referring to this, it is added : " It is evident that 
 in the case of the rough pipe the frictional equation 
 
 ~F = QV 2 (A -{-K\ will not hold unless A becomes negative 
 
 with increasing roughness, for which there does not appear 
 to be any sufficient reason, so that to include the cases of 
 rough and smooth pipes an expression of the form 
 
 vl 
 
 appears to be necessary, in which K depends only on the 
 roughness of the pipe." 
 
 It appears clear that what is here meant is that if A 
 became negative, it would imply that viscosity was reducing 
 the resistance and that there is no reason to "believe that this 
 is possible. The question of A becoming negative is cer- 
 tainly a difficulty ; but I cannot think that the addition of 
 
Negative Resistance in Liquids 179 
 
 a third term to the equation, involving v 3 , appears to afford 
 a satisfactory way out of it. It seems to be only confusing 
 the question still more, by adding another difficulty in the 
 attempt to evade the first. That A does become negative, 
 at times, is a fact that has to be faced and explained ; 
 it is the expression of a physical action, which I shall proceed 
 to describe. It is a case of what I call " negative resistance." 
 
 The subject being curious, and not well known, it may not 
 be uninteresting if I explain how the idea first came to me. 
 Some eighteen or twenty years ago Mr. Yarrow very kindly 
 gave me some very fine "speed-horse-power "curves of two 
 torpedo boat destroyers he had just built. Being desirous 
 of rinding an equation which would satisfy these curves, I 
 very carefully measured the ordinates at every half knot, 
 and made a table of them. I then took out the first differ- 
 ences, as well as the second differences, and from these data 
 I made formulae which would always give me the right answer. 
 
 There was no theory implied in these formulae, which 
 were, frankly, worked backwards, without any assumptions 
 of any kind beyond the accuracy of the curves. On 
 examination I found that up to the critical velocity what 
 naval architects call the " hump "they were typified 
 by the R=AV+BV 2 formula; whilst, beyond that velo- 
 city, the resistance might be expressed as R' =C +A'V, 
 where C was a constant. This form of equation did not 
 surprise me at all, as it accorded with my views of the resist- 
 ance of liquids. What did astonia^ me, however, was that 
 I found that in the first formula A was negative \ It was 
 clear to me that to accept this was to admit that viscosity 
 was actually assisting the boat by reducing the resistance. 
 Such an idea appeared, not perhaps unnaturally, absurd ; 
 I therefore put the notes aside and thought no more of them. 
 Some years later, when studying Colonel Beaufoy's experi- 
 mental results, I again frequently came across the same 
 idea. I eventually arrived at the conclusion that there 
 were certain cases where viscosity was actually an advantage, 
 and that very many curious facts connected with Hydraulics 
 could not be explained without this assumption. 
 
 Although no one, that I am aware of, has explained this 
 
 N 
 
i8o 
 
 Motion of Liquids 
 
 " Paradox," other writers have struck against it, without 
 perhaps recognizing it. Mr. Lanchester (Aerodynamics) 
 says, referring to the fluid at the back of the plate, " the 
 dead water [sic] will become the seat of a lively circulation 
 as indicated by the arrows (fig. 74) , the motion of the fluid 
 in the vicinity of the plane being in the direction of the flight, 
 and that in the vicinity of the free surface being in the 
 opposite direction. Now the result of this will be to produce 
 a tangential drag in a forward direction ; in fact, any skin 
 friction experienced on the upper face or " back " of the 
 
 Line of 
 
 FIG. 74. 
 
 plane will be of negative sign." Having said this, he leaves 
 the subject. 
 
 Dr. Stanton, as the preceding quotation showed, appears 
 not to have considered it possible for A to become negative, 
 since this would imply that viscosity was actually reducing 
 the resistance ; he therefore thought it necessary to add 
 a new term to his equation. 
 
 M. Soreau (Etat actuel de V aviation), is, however, very 
 emphatic, and says that this " counter-resistance must 
 play an important role in aviation." He refers to a curious 
 experiment of M. Goupil as proving its existence (La Loco- 
 
Negative Resistance in Liquids 
 
 181 
 
 motion Aerienne, I884) 1 " a frame covered with canvas of 
 i square metre, weighing 070 kilogs. and loaded with a 
 weight of 3 kilogs. was attached by two cords m and n and 
 presented against different currents of air. When the 
 velocity of the current was 4 m.p.s. the frame took an 
 inclination of 45 ; at velocities of 5 and 6 metres per second 
 the inclination became -J- and J respectively ; at 7 m.p.s. 
 the apparatus took, to the great astonishment of the author, 
 
 Plan, T\ 
 
 FIG. 75. 
 
 the position indicated in fig. 75 : the spring-balance fixed 
 to one of the cords marked no traction, and even the surface 
 was drawn towards the post and against the wind (chassait 
 sur ses amarres contre le vent}. At greater velocities the 
 surface again caused a considerable pull on the spring 
 balance." M. Soreau adds in a note that " this pheno- 
 menon has been denied, by affirming, inaccurately, that 
 
 1 I have been unable to see this paper yet, and so to verify this 
 reference. 
 
1 82 Motion of Liquids 
 
 its application to sailing flight would lead to perpetual 
 motion." 
 
 If it were asserted that the frame remained permanently 
 without strain on the spring balance ; or if it were pretended 
 that this curious result had occurred with a static fluid, 
 something might be said for this argument. The experi- 
 ment, as described by M. Soreau, does not, however, appear 
 to contain anything which is a priori improbable. We know 
 so very little about the laws of resistance, of the why such 
 a thing should, or should not happen. M. Eiffel has lately 
 shown (Comptes rendm, 1913) that, with spheres, there is a 
 " critical velocity " beyond which there is a sudden decrease 
 in the resistance caused by a stream of air flowing past them ; 
 so that the resistance is, sometimes, actually (not relatively) 
 less at a slightly increased velocity beyond the critical one. 
 
 Captain G. Costanzi (Alcune Esperienze di Idrodynamica) 
 has also shown that the same thing occurs when spheres 
 are exposed to a current of water. Although these experi- 
 ments give much food for reflection, I have not found any 
 explanation of them which I consider at all satisfactory. 
 
 The term " negative resistance "is, admittedly, rather a 
 barbarous one ; but since M. Soreau's " counter-resistance " 
 is certainly no improvement, and as I know no better one, 
 I prefer to retain " negative " as it expresses the fact that 
 A in the formula is negative. 
 
 In case the reader should be afraid that I am going to 
 try and land him in the morass of " perpetual motion," I 
 will hasten to reassure him by saying that I consider, in these 
 cases, that viscosity acts as a kind of engine for recuperating 
 kinetic energy and converting it into work. It acts as a 
 utilizer of a waste product ; just as, by suitable means, the 
 waste heat of a furnace can be recuperated and converted 
 into work through some form of heat engine so viscosity 
 can, and does sometimes, recuperate some of the wasted 
 kinetic energy. 
 
 To make my meaning clearer, let A B (fig. 76) represent 
 the section of a plate exposed to a current of liquid. Since 
 the liquid flows past A and B, in the directions A C and BD 
 at a velocity considerably in excess of that of the stream, it 
 
Negative Resistance in Liquids 
 
 183 
 
 is clear that kinetic energy has been generated there; 
 some of the potential energy has been converted into kinetic 
 energy. Alt this energy is wasted, being gradually converted 
 into heat and so dissipated. 
 
 I may put the matter in another way. The flowing liquid 
 exerts a pressure on the front of the plate ; which pressure 
 is not counterbalanced by an equal and opposite pressure at 
 the rear of the plate. In other words, the liquid is doing 
 work. To enable this to be done some of the energy in the 
 
 FIG. 76. 
 
 liquid must be expended. This is brought about by some 
 of the potential energy being converted into kinetic energy ; 
 which kinetic energy is subsequently converted into heat, 
 and so wasted. We may, therefore, consider the resistance 
 as being measurable by the amount of kinetic energy generated, 
 and wasted. 
 
 This method of considering resistance energy wasted 
 may be new to the reader. If the " Helmholtz-Kirchhofi 
 flow " be examined in this manner it will be seen that work 
 is being done, whilst no energy appears to be expended. This 
 
184 Motion of Liquids 
 
 is why I object to the assumption made in this flow. 
 
 Now, can any of this wasted energy be recuperated, 
 and, if so, how ? We know that eddies, or vortices, are 
 produced behind the plate, as shown in the diagram ; if 
 we place a body like A E F B behind the plate so as to 
 produce surfaces for these vortices, or eddies to " brush 
 against " ; there will, to a certain extent, be a retardation 
 of the cyclic motion in these vortices or eddies, in consequence 
 of the viscosity, stickiness, or " treacliness " of the liquid 
 on the surface A E F B. Some of this cyclic momentum 
 will be converted into longitudinal momentum, in consequence 
 of the tangential drag ; this momentum being transferred 
 to the body A E B F. We have here, clearly, a recuperation 
 of waste energy. Now if we consider the resistance of a 
 body as being measured by the amount of energy wasted, l 
 it will be clear that if less is wasted which is the case if 
 some is recuperated the resistance of the body A E F B 
 should be less than that of the plate without a stern-piece ; 
 in other words, the resistance of the box should be less than 
 that of the plate. We know perfectly well that such is 
 the case. An examination of the curve of resistance of a 
 long box will show that it is satisfied by the equation 
 R =BV 2 A'V. There is, what I call, " negative resistance." 
 As is commonly taught, the resistance caused by " fluid 
 friction " increases as the wetted surface ; whereas we see that, 
 in this case, it is exactly the reverse. 
 
 To continue ; if we increase the length of A E F B, we 
 shall increase the surface for the vortices to " brush against." 
 If the view here advanced is correct, we ought to recuperate 
 more energy ; less ought to be wasted; i.e., there should be 
 still less resistance. This also is well known to be the 
 case ; but, as is also very well known, it is only true up to 
 a certain length. If the plate is circular it is true up to a 
 length of nearly three diameters, beyond this the resistance 
 increases slowly. If the shape of the plate be not circular, 
 then this law appears to hold good up to a length of nearly 
 3v/S, where S is the area of the surface in presentation : 
 
 1 Neglecting the resistance due to viscosity. 
 
Negative Resistance in Liquids 185 
 
 more experimental work is required, however, before this 
 can be asserted with any confidence as to its accuracy. 
 
 Colonel Beaufoy said, " among the conclusions suggested 
 by the tables [tables of his experiments] one of the most 
 curious is, that increasing the length of a solid, of almost 
 any form, by the addition of a cylinder in the middle, 
 exceedingly diminishes the resistance with which it moves, 
 provided the weight in water continues to be the same, a 
 fact which I apprehend cannot be easily explained.'' Nature 
 certainly hides some of her laws very cunningly ; but I think 
 that the foregoing is a satisfac- 
 tory explanation, and it has 
 the further merit of being ex- 
 ceedingly simple. 
 
 A very curious experiment in 
 the reverse direction was tried 
 in Washington, and is referred 
 
 to by D. W. Taylor in Speed and (3_ (O, (O. 
 
 Power of Ships. With a view (^T (^ (^T 
 
 to reducing " skin friction/' air 
 
 was pumped around a model 
 through a number of small holes 
 near the bow. The results 
 showed that the resistance was 
 always materially increased ! 
 Here again we find that reducing FIG. 77. 
 
 the wetted surface increased the 
 
 resistance : less kinetic energy was apparently recuperated. 
 Let us now turn the experiment " inside out " and see 
 if it is equally true when the liquid flows inside the solid', 
 let us imagine the water flowing through a tube. We have 
 seen that if a liquid flows out of a thin wall of a reservoir 
 the discharge is very considerably less than if a short tube be 
 attached outside the hole. Let A B in fig. 77 represent a 
 hole in the thin wall of a reservoir, and A B C D a short tube 
 attached to A B. Without the tube, the velocity of dis- 
 charge appears to be nearly the same at all parts of the aper- 
 ture. When the tube is attached, this is no longer the 
 case provided that the liquid wets the tube since the walls 
 
1 86 Motion of Liquids 
 
 of the tube retard the flow and cause the liquid to change 
 Us shape. It now moves in a cyclic manner, as shown, 
 diagrammatically, in fig. 77. If the tube is very small 
 capillary the flow is not increased by this action, and the 
 resistance varies as the velocity only. If the tube be not 
 very small, eddies will be formed, as is well known. 
 
 As in the previous case, the viscosity tends, in retarding 
 this cyclic motion, by tangential drag, to transfer momentum 
 to the tube. Since, however, the tube is fixed, this cannot 
 be done ; so this momentum is transferred to the liquid 
 instead, and the flow of discharge is increased. Clearly, 
 therefore, since the discharge is increased the resistance to 
 this flow must have decreased}- 
 
 I might point to the somewhat parallel case of a locomo- 
 tive. The engine in turning the wheels tends (provided that 
 there be sufficient adhesion between the wheels and the 
 rails) to drive the rails backwards. Since the rails are fixed, 
 and as something has to move, the locomotive goes forward. 
 
 To return to the tube : it will not be necessary to insist 
 much on the fact that, if this line of reasoning is correct, 
 lengthening the tube should further increase the discharge, 
 within the limitations pointed out in the previous case. As 
 has been shown, experiment confirms this. We might also 
 reasonably expect that if the tube were very slightly corru- 
 gated longitudinally, since the surface of contact between the 
 water and the tube would be increased, the discharge should 
 be slightly increased. I am not aware that this experiment 
 has ever been tried. 
 
 In all these cases we find that viscosity is an advantage, 
 in consequence of the resistance caused by it acting in a 
 negative direction. Conversely, as we have seen previously, 
 if arrangements are made so that the liquid shall not wet 
 the walls of the tube so that the viscosity shall not act the 
 increased discharge does not take place ; the flow being as 
 through a thin partition. Following our parallel of the 
 
 1 As was previously pointed out, if a small hole is made in the 
 tube, opposite the vena contracta, the increased discharge does not 
 take place. 
 
Negative Resistance in Liquids 187 
 
 locomotive, if there is no adhesion between the wheels and 
 the rail, the engine will not move along the railway track. 
 
 In November, 1912, Captain G. S. Macllwaine, R.N., read 
 a paper on " The Corrugated Ship " at the R.U.S. Insti- 
 tution. From his personal observations made during a 
 voyage on the Hiltonia,he came to the conclusion that she 
 consumed about 10 per cent, less fuel than an ordinary 
 ship of the same size and with similar engines and propeller. 
 If these observations can be relied on and it does not appear 
 probable that the author was mistaken, whilst his paper is 
 also most convincing this tends to confirm all that has been 
 said here previously. It is a case of " negative resistance." 
 
 Captain Macllwaine offers no explanation of why the 
 boat should be faster, though he says in the most approved 
 classical style " To the lay mind the unavoidable conclusion 
 would appear to be that the resistance offered by the ship 
 has been reduced, but we know that this is impossible ! for 
 the wetted area has been increased, and by long established 
 theory resistance increases as wetted area ! " [Italics added]. 
 
 The reader will not be surprised if I say that I consider 
 that reduction of the resistance is not impossible, even 
 though the wetted area has been increased. I am, further, by 
 no means satisfied that Captain Macllwaine intended this 
 paragraph to be taken quite seriously, for I observe two 
 " notes of admiration," which make me rather suspect that 
 it was " rote sarkastical." Is it not possible that when 
 the gallant Captain penned this paragraph he had his 
 tongue in his cheek ? 1 
 
 All these experiments show results which are diametrically 
 opposed to the " fluid friction " theory ; we add wetted 
 surface result, less resistance ; we increase this wetted 
 surface further result, still less resistance, instead of the 
 reverse. 
 
 We may examine this question in another way, by varying 
 the experiment. Referring to the case of the liquid flowing 
 out of a short tube, it was said that the viscosity tended 
 to drive the tube backwards. If this is true, we should be 
 
 1 This paper is highly interesting, and should be studied in extenso. 
 
i88 Motion of Liquids 
 
 able to cause it actually to do so, if we arrange that the tube 
 shall be free to move. In fig. 78 is shown a thin wall in a 
 tank, in which is firmly fixed a very short glass tube ; I 
 suggest glass because it is smooth. Inside this small tube 
 insert a longer glass tube, whose length L = 3 diameters, 
 and which just fits loosely in the fixed tube. If we then 
 smear the inside of the short tube and the outside of the longer 
 tube with kerosine to prevent wetting by the water on 
 filling the tank with water and allowing this to flow through 
 the longer tube, it will be found that it actually is drawn 
 in. In conducting this experiment it is necessary to hold 
 the tube until a regular flow has commenced, as, otherwise, 
 it may be driven outwards by the 
 sudden rush of the water. It is, fur- 
 ther, necessary to release it gradually, 
 as, if done too suddenly, the tube will 
 be drawn completely into the tank. If 
 the experiment is carried out properly 
 the tube will move inwards for a cer- 
 tain distance and then remain in a 
 position of stable equilibrium. When, 
 however, the level of the water falls 
 to not much above the top of the 
 tube, the latter is slowly pushed out 
 again. 
 
 The reader may ask, why should 
 the tube go in a certain distance and 
 no further ? Why should it not con- 
 tinue moving, until it falls into the 
 tank ? The explanation can, I think, 
 FIG. 78. be given by a reference to " Hamil- 
 
 ton's principle." This is defined by 
 M. Poincare (La science et I'Hypothese] as follows : 
 
 " If a system of bodies is in the situation A at the period 
 t ot and in the situation B at the period t lf it always moves 
 from the first situation to the second by a road such that the 
 mean value of the difference between the two kinds of energy, 
 in the interval of time which separates the two epochs t 
 and tit shall be as small as possible. 
 
Negative Resistance in Liquids 189 
 
 " This is the principle of Hamilton, which is one of the 
 forms of the principle of least action/' 1 
 
 Applying this to the present case ; the liquid and the 
 tube form a " system," in which the whole energy is initially 
 potential. To satisfy this principle it is clear that for the 
 position to be stable the amount of kinetic energy generated 
 must be a minimum ; there must be the least amount of 
 energy transformed. When the liquid flows and the tube 
 commences to move, it is clear that it will form a "re- 
 entrant ajutage," and, as we have seen previously, the 
 discharge will be reduced ; this will continue until the dis- 
 charge becomes a minimum ; beyond this the tube will cease 
 to be drawn in, the position being a stable one. 
 
 Still another example of this very curious action : what I 
 may call another song to the same tune. If water is dis- 
 charging over a weir, the amount of this discharge will 
 depend, to a great extent, on whether air is, or is not, freely 
 admitted beneath the sheet of water. 
 
 " Bazin . . . has investigated very fully the effect upon 
 the discharge and upon the form of the nappe, by restricting 
 the free passage of the air below the nappe. He finds that 
 when the flow is sufficient to prevent the air getting under 
 the nappe, it may assume one of three distinct forms, and 
 that the discharge for one of them may be 28 per cent, greater 
 than when the air is freely admitted, or the nappe is free." 
 (F. C. Lea, Hydraulics}? 
 
 This statement appears to require a little amplification 
 and explanation : it is not by any means clear how the flow 
 of water could prevent the air from getting under the nappe. 
 What is probably meant is that the water absorbs the air, so 
 that if the supply of this air is not sufficient, the water is 
 
 1 Si un systeme de corps est dans la situation A a 1'epoque t et 
 dans la situation B a 1'epoque t lt il va toujours de la premiere situa- 
 tion a la seconde par un chemin tel que la valeur moyenne de la dif- 
 ference entre les deux sortes d'energie, dans I'intervalle de temps qui 
 separe les deux epoques t et t lt soit aussi petite que possible. 
 
 C'est le principe de Hamilton, qui est une des formes du principe 
 de moindre action. 
 
 2 I have been unable to verify this reference. 
 
i go 
 
 Motion of Liquids 
 
 forced nearer to the weir, and that it will eventually touch 
 it, and wet it. 
 
 If we imagine fig. 79 (i) to represent a body of water 
 discharging over a weir, and falling clear of it, A being a 
 space filled with air ; since the water, in falling, will suck 
 air out of A, a strong draught will be set up from the ends 
 of the weir to supply the deficiency of air in A. This " wind" 
 is very well known at Niagara Falls and so requires no further 
 comment. Suppose, now, that the ends of the fall are closed, 
 so that air cannot get behind the nappe ; it is clear that the 
 water will eventually touch the front of the weir and so 
 wet it (fig. 79) (2). When this occurs the same action takes 
 
 place as was described 
 in the case of the liquid 
 flowing through the tube : 
 the discharge is much in- 
 creased. 
 
 It is curious, though it 
 may only be a coinci- 
 dence, that Bazin states 
 that this increase is 28 
 per cent., whilst the in- 
 crease in the case of a 
 tube is given by Dubuat 
 as about 30 per cent. 
 Dubuat was, of course, 
 well aware of the increase 
 of discharge through an 
 " ajutage," though he 
 said : " It is very diffi- 
 cult to understand why 
 an additional tube, of a 
 
 few inches in length, compels the liquid to follow its 
 walls, and increases the discharge, through the thin wall, in 
 the proportion of 10 to 13." 
 
 In the next chapter we will examine the curves of these 
 resistances, so as to get a clear idea of the law which governs 
 them. 
 
 (2) 
 
 FIG. 79. 
 
Negative Resistance in Liquids 191 
 
 SUMMARY 
 
 There are cases where viscosity appears to be an ad- 
 vantage ; since, by acting in a negative direction, it can 
 actually reduce resistance to motion. 
 
 Viscosity acts, in these cases, by recuperating what would 
 otherwise be wasted kinetic energy. This is found to be true 
 in cases of " long bodies " moving in a liquid at rest : water 
 flowing out of pipes, even when slightly roughened ; also 
 in certain cases where water flows over weirs. 
 
 The resistance in such cases can be expressed by the 
 formula. 
 
 where the coefficient A of V has a negative value. 
 
 REFERENCES 
 
 R. SOREAU, tat actuel de Variation. 
 
 Dr. STANTON, A. C. Aeronautics, 1910-1911. 
 
 G. EIFFEL, Comptes Rendus, [913. 
 
 Capt. COSTANZI, Alcune esperienze di Idrodynamica. 
 
 D. W. TAYLOR, The Speed and Power of Ships. 
 
 Philip E. B. JOURDAIN, The Principle of Least Action. 
 
 Capt. G. S. MAC!LWAINE, R.N., The Corrugated Ship (R.U.S. 
 Inst., Nov. 1912). 
 
 H. POINCARE, La science et I'hypothese. 
 
CHAPTER XV 
 
 CURVES OF RESISTANCE EXPERIMENTAL CONFIRMATION OF 
 
 THEORY 
 
 LORD KELVIN said somewhere that " when you can measure 
 what you are speaking about, and express it in numbers, 
 you know something about it ; but when you cannot 
 measure it, when you cannot express it in numbers, your 
 knowledge is of a meagre and unsatisfactory kind ; it may 
 be the beginning of knowledge, but you have scarcely in 
 your thoughts advanced to the stage of science/' 
 
 In order to get a thoroughly clear knowledge of the law 
 of resistance of liquids, it is imperatively necessary that we 
 should be able to express it quantitatively, either numeri- 
 cally or by means of curves, if we wish to advance to " the 
 stage of science." 
 
 It is clear that the equation R=AV+BV 2 is the equa- 
 tion of a parabola, the axes of which have been shifted : 
 the resistance therefore is strictly parabolic. Up to the 
 " critical velocity/' however, the resistance varies only as 
 the first power of the velocity or R' =AV ; and it is only 
 beyond this critical velocity that this parabolic resistance 
 commences. These two curves do not cut one another 
 I use the term curve in its widest sense, since R'=AV 
 is the equation to a straight line and they only have one 
 point common to the two, which is at the origin, where 
 V=O (fig. 80). It is clear, therefore, that in order that the 
 resistance, which is following the line O C V, shall shift to 
 the parabola M O B C 1 , it is necessary that there should be 
 a sudden change in the resistance, such as from C to B, 
 
 192 
 
Curves of Resistance 
 
 193 
 
 It is very well known that such sudden change does occur 
 at what is called the critical velocity. The curve of resist- 
 ance is therefore, as shown in fig. 80 (which is reproduced 
 from the A B C of Hydrodynamics) and follows O C a 
 
 straight line to C, which is the " critical velocity " ; the 
 resistance then takes a sudden leap to B, after which it 
 follows the parabola to C 1 . C 1 is a superior " critical velo- 
 city " which I will not refer to at present, having done so 
 elsewhere. On examination of fig. 80 it will be easily seen 
 
IQ4 Motion of Liquids 
 
 that the parabola is one whose equation would be R" =BV 2 , 
 if its vertex were at O. The vertex has, however, been shifted 
 from O to M, which is to the left and below O ; the shift 
 being such that O C V is a tangent to the parabola at O. 
 To obtain the curve of resistance of R=AV+BV 2 , it is 
 only necessary to trace the parabola R" =BV 2 ; then lay 
 off the line R' =AV ; and then shift the parabola (iryotation- 
 ally) until the line O C V is a tangent to it at O. 
 
 All this I have shown previously, but since the book went 
 to press, Mr. Shorter pointed out to me and proved to 
 me mathematically that the locus of the vertex of the para- 
 bola, during this shift, is the same parabola turned upside 
 down. This is a very curious point ; but I do not propose 
 to give the mathematical proof, because I think it can be 
 explained, much more simply, from the principle of " rela- 
 tive motion " ; Mr. W. Child, having pointed out to me 
 later, that not only the vertex, but every other point in the 
 parabola must necessarily describe this same parabola. 
 
 To explain this, let B O C (fig. 81) represent a parabola, 
 one of the thin wooden ones, such as are sold in mathe- 
 matical instrument shops. It is clear that if this is placed 
 on a sheet of paper and a pencil be run round it, the pencil 
 will describe a parabola. Let us next suppose the pencil to 
 be fixed and the parabola to travel past it irrotationally ; 
 the axis of the parabola must always be parallel to O Y ; 
 then the vertex of the parabola will trace its own reflection 
 B * O C 1 . The simplest way to carry out the experiment 
 is to fix a pin at O to represent the pencil whilst the pencil 
 is held to the vertex of the parabola in order to trace the 
 locus of the movement on the paper. A very litt]e reflection 
 -and, vastly better, a little experiment will show that 
 every point of the wooden parabola moves in the same manner. 
 The reader is very strongly recommended to devote a 
 couple of hours, or so, to tracing such curves, employing 
 thin bodies of any shape ; circles, ellipses, or irregular 
 shapes, when he will find that they always do this. 
 
 I have said trace " their reflections " ; this is not 
 strictly accurate, for if there be an irregularity on the curve 
 C (fig. 81) it will not be shown on the curve O B 1 (which 
 
Curves of Resistance 
 
 195 
 
 is the strict reflection), but on O C 1 . The images are not 
 only inverted, but also reversed. 
 
 Let us next try and get a clear image of the curve of 
 resistance represented by the equation R=BV 2 AV. 
 
 FIG. Si. 
 
 I have said, previously, that A in the curve of resistance 
 sometimes becomes negative. This, also, is not strictly accurate, 
 since, when A changes its sign, it also changes its significance : 
 the A in the two equations do not represent the same thing, 
 they are not in any way related to one another. To be accu- 
 
196 
 
 Motion of Liquids 
 
 rate, therefore, it is necessary to say that the curve 
 R=AV+BV 2 may become R'= A'V+BV 2 , where A 
 and A' are not dependent variables, and where they may, 
 or may not, be equal to one another. 
 
 FIG. 82. 
 
 With this correction, let us now examine the curves of 
 resistance of a " long body " the equations of resistance of 
 which are R =AV (viscous resistance only) and 
 R' = A'V+BV 2 . 
 
 In fig. 82 I have retained the same lettering as in fig. 80. 
 It is clear that to trace this curve it will first be necessary to 
 trace the parabola R"=BV 2 , with its vertex at O. Next 
 lay off the line O C V, whose equation is R =AV, and lastly 
 the line O V 1 , the equation to which is R"'= A'V. 
 The parabola must now be shifted down and to the right, 
 until it is tangential to O V 1 at 0, The curve of resistance 
 
Curves of Resistance 
 
 197 
 
 of our " long body " will then be O C (a straight line) and 
 then C C 1 , the curve of the parabola. 
 
 We have now arrived at such a point that we see that 
 " short bodies " experience a resistance which may be 
 expressed as R=AV+BV 2 where R is greater than BV 2 ; 
 in the case of " long bodies," this resistance may be expressed 
 as R' = _A'V+BV 2 where R' is less than BV 2 . It is 
 clear, therefore, that there ought to be some " medium- 
 
 length bodies " where the resistance would be exactly 
 R"=BV 2 . It is well known that there are. Even at the 
 risk of tiring the reader, let us see what form the curve of 
 resistance would take for these bodies. 
 
 Retaining the same notation, fig. 83 will represent the 
 curve. We here lay off the parabola whose equation is 
 R" =BV 2 with its vertex at O ; and then the line O C V 
 whose equation is R AV, The curve of resistance will 
 
 02 
 
198 Motion of Liquids 
 
 then be O C straight line continuing along C C 1 , the 
 parabola. 
 
 It will be observed that in the curve in fig. 80 there is 
 necessarily an abrupt change of resistance ; whilst in figs. 
 82 and 83 the change is not necessarily abrupt : the law 
 changes, but there is no necessary sudden change in the 
 resistance. Whether there is, or is not an abrupt change, 
 will depend on whether the " critical velocity " would, or 
 would not, put the point C outside the parabola. If C is 
 outside, some abrupt change is, clearly, necessary. 
 
 To follow the question to its logical sequence, if the liquid 
 has no viscosity or if it be so small as to be a vanishing 
 quantity if it be what is called a " perfect " liquid ; then 
 clearly, A in the formula R AV becoming zero, the resist- 
 ance will follow O X (fig. 83) to a point C" (the critical 
 velocity) when it will suddenly change to C and follow the 
 parabola C C 1 . The resistance will be zero up to the critical 
 velocity ; after which it will be parabolic. To explain more 
 clearly what I mean ; if the Atlantic Ocean were composed 
 of an inviscid liquid, a body moving in it at a depth of 
 immersion of, say, 50 feet might meet with no resistance, 
 up to a velocity v ; whilst at a velocity v -\-dv it might 
 encounter a very considerable resistance in consequence of 
 the sudden formation of vortices. 
 
 If any reader raises the objection that he has always 
 been taught that in a perfect liquid there could be no resistance, 
 I must refer him to the A B C of Hydrodynamics, where I 
 have discussed this question at considerable length. I 
 must here content myself by saying that this was the view 
 held by Lord Kelvin ! 
 
 A corollary which follows from what has been here said 
 is that, in a " perfect " or inviscid fluid a short body would 
 always experience less resistance than it would in a viscous 
 one of the same density ; a " medium length " body, at all 
 velocities exceeding the critical velocity, would meet with 
 exactly the same resistance in both liquids ; whilst a " long 
 body " at velocities exceeding the critical, would actually 
 experience more resistance in the inviscid than in the viscid 
 liquid ! 
 
PLATE V, 
 
 FIG. 84. 
 
 FIG. 85. 
 
 FIG. 86. 
 
 To face page 199.] 
 
 [de Villamil, Motion of Liquid. 
 
Curves of Resistance igg 
 
 This statement will, doubtless, take many readers by 
 surprise ; it is, however, only a logical deduction from what 
 has been said previously. If we accept the premisses we 
 must, of course, accept all logical deductions. I do not deny 
 that under certain conditions resistance in an inviscid liquid 
 is impossible ; I have devoted many pages to the explanation 
 of this question elsewhere. 
 
 As previously stated, in an incompressible liquid (viscous 
 or not) which has no free surface, and whose envelope is 
 inextensible, all bodies moving in it with steady motion are 
 stream-line. I give two examples from Dr. Hele-Shaw's 
 Distribution of pressure due to flow round submerged surfaces. 
 (Inst. Nav. Architects.) 
 
 Fig. 84 is that of a plate at an angle meeting a moving 
 fluid. This is an extraordinarily beautiful picture, almost 
 resembling a line engraving. Unfortunately, it is only a 
 reproduction of a reproduction ; the first one having been 
 done on very inferior paper. 
 
 Fig. 85 is the same as the last, but with the tubes of flow 
 much larger. 
 
 Fig. 86, a very irregular figure, which would certainly 
 not be called a " stream-line " one. We see that the fluid 
 flows as the mathematicians say it should. 
 
 All these figures represent the flow of glycerine, in two 
 dimensions, and without a free surface. 
 
 It will be well now to see how far experiment confirms 
 what has been said previously. I will select a number of 
 examples from Beaufoy. The bodies were parallelopipedons. 
 Length, 21-099 feet, breadth and depth, 1-219 feet, immersed 
 6 feet and suspended from a " conductor " by iron bars ; 
 the conductor being accelerated by weights. 
 
 In Table XXV, the figures of which are copied from Beau- 
 foy, the ist column gives velocity in feet per second ; the 
 2nd column the resistances measured less that of the con- 
 
 T? 
 
 ductor and bars. The 3rd column gives values of ' 
 
 whilst the last column gives the values of the resistance 
 calculated by the formula 
 
 R = 73V +i 76V 2 . 
 
200 
 
 Motion of Liquids 
 
 TABLE XXV 
 
 BEAUFOY'S EXPERIMENTS. Page 187. 
 PARALLELOPIPEDON. Square Ends. Immersion 6 ft. 
 
 V 
 
 in ft. 
 per sec. 
 
 Observed 
 Resistance. 
 
 R . 
 
 V 2 ' 
 
 Calculated 
 Resistance. 
 
 i 
 
 1-4453 
 
 1-445 
 
 I-Q3) 
 
 2 
 
 6-1191 
 
 1-529 
 
 5-58 ' 
 
 3 
 
 14-165 
 
 1-574 
 
 13-65 ) 
 
 4 
 
 25-649 
 
 1-603 
 
 25-24 
 
 5 
 
 40-613 
 
 1-624 
 
 40-35 
 
 6 
 
 59-084 
 
 1-672 
 
 58-98 
 
 7 
 
 81-079 
 
 1-655 
 
 81-13 
 
 8 - 
 
 106-642 
 
 1-666 
 
 106-80 
 
 9 
 
 13576 
 
 1-675 
 
 135-99 
 
 10 
 
 168-45 
 
 1-684 
 
 168-70 
 
 ii 
 
 204-75 
 
 1-692 
 
 204-93 
 
 12 
 
 244-62 
 
 1-699 
 
 244-68 
 
 It will be seen that the calculated and observed resistances 
 agree very closely for all velocities above 3 feet per second. 
 The reason why the results are apparently incorrect at very 
 low velocities will be referred to later. 
 
 Table XXVI gives the results of the experiments carried 
 out with the same body with a fine tailpiece. Here the 
 resistance may be measured by R=i -5257V 2 , where we see 
 that A has become =0, whilst B has been reduced from 
 1-76 to 1-526 ; the resistance with the tailpiece being less 
 than without it. 
 
 These bodies employed by Beaufoy are too long to enable 
 us to find out the amount of kinetic energy generated at the 
 head some of which has been recuperated and transferred 
 to the body and so the examples must only be considered 
 as " illustrations " of cases where the term involving the 
 first power of V becomes negative. 
 
 1 Refer to Fig. 82. OCV cuts the parabola; resistances at low 
 velocities are greater than calculated by parabolic formula. 
 
Curves of Resistance 
 
 201 
 
 Table XXVII gives the results from the same body, but 
 moving " tail first." 
 
 The resistance is considerably reduced. Possibly the amount 
 of kinetic energy generated was considerably less than in the 
 last case ? This is a conjecture only. 
 
 TABLE XXVI 
 
 PARALLELOPIPEDON. Page 192. 
 Immersion 6 ft. Angle 9 35' 40*. 
 
 
 ^^ # ^ | 
 
 
 
 V 
 
 Observed 
 
 R 
 
 Calculated 
 
 ft. per 
 
 Resistance. 
 
 V2~" 
 
 Resistance. 
 
 sec. 
 
 
 
 
 i 
 
 2 
 
 6-0741 
 
 I-5034 
 I-5I85 
 
 I-5257) 
 6-1028 V 1 
 
 3 
 
 13-712 
 
 I-5236 
 
 I3-73I3 J 
 
 4 
 
 24-412 
 
 I-5257 24-4II2 
 
 5 
 
 38-168 
 
 I-5267 38-1425 
 
 6 
 
 54-974 
 
 I-5270 
 
 54-9252 
 
 7 
 
 74-819 
 
 I-5269 74-7593 
 
 8 
 
 97-712 
 
 I-5267 97-6448 
 
 9 
 
 123-63 
 
 I-5263 I23-58I7 
 
 10 
 
 152-58 
 
 I-5258 I52-57 
 
 ii 
 
 184-54 
 
 I-525I 
 
 184-6097 
 
 12 
 
 219-54 
 
 I-5246 
 
 219-7008 
 
 Table XXVIII, results of the same body, moving in the 
 same direction, but with a tailpiece, blunter than the point, 
 
 R = -52IV+-878V 2 
 
 Here, apparently, the viscosity, acting on the tailpiece, has 
 recuperated some more energy. 
 
 Table XXIX, same body as the last, but direction of 
 motion reversed 51 
 
 Apparently, here the kinetic energy generated was less; 
 in any case the resistance is appreciably less. 
 
 1 The observed and calculated values are very close. See Fig. 83. 
 
202 
 
 Motion of Liquids 
 
 TABLE XXVII 
 
 PARALLELOPIPEDON. Page 201. 
 
 Immersion 6 ft. Angle 9 35' 40*. 
 
 I 
 
 y 
 
 4+ ^ OT . Observed 
 
 R 
 
 Calculated 
 
 it. per 
 sec. 
 
 Resistance. 
 
 V 2 ' 
 
 Resistance. 
 
 i 
 
 7105 
 
 710 
 
 M52 ) 
 
 2 
 
 3-1380 
 
 784 
 
 2-842 * 
 
 3 
 
 7-404 
 
 822 7-170 ) 
 
 4 
 
 13-568 
 
 848 13.438 
 
 5 
 
 21-662 
 
 865 21-640 
 
 6 
 
 3i-7i4 
 
 881 31-782 
 
 7 
 
 43'739 
 
 892 43-862 
 
 8 
 
 57-762 
 
 902 57-880 
 
 9 
 
 73-78 
 
 911 73-836 
 
 10 
 
 91-82 
 
 918 9I-730 
 
 ii 
 
 111-89 
 
 924 111-562 
 
 12 
 
 I33-98 
 
 -930 
 
 I33-332 
 
 TABLE XXVIII 
 PARALLELOPIPEDON. Page 217. 
 Immersion 6 ft. Rear Angle 14 28' 39". 
 Front Angle 9 35' 40". 
 
 V 
 ft. per 
 sec. 
 
 Observed 
 Resistance. 
 
 R 
 V' 
 
 i 
 
 6674 
 
 667 
 
 2 
 
 2-9013 
 
 725 
 
 3 
 
 6-793 
 
 755 
 
 4 
 
 12-383 
 
 774 
 
 5 
 
 19-697 
 
 788 
 
 6 
 
 28-748 
 
 -799 
 
 7 
 
 39-549 
 
 807 
 
 8 
 
 52-122 
 
 814 
 
 9 
 
 66-47 
 
 -820 
 
 10 
 
 82-59 
 
 -826 
 
 ii 
 
 100-5 
 
 831 
 
 12 
 
 120-18 
 
 835 
 
 R=- 5 2iV+-8 7 8V 2 . 
 
 Calculated 
 Resistance. 
 
 357 
 2-470 
 
 6-339 
 11-964 
 
 19-345 
 
 28-482 
 
 39-375 
 
 52-024 
 
 66-429 
 
 82-590 
 
 100-507 
 
 120-18 
 
 1 Calculated values less than observed. Refer to Fig. 82. 
 
Curves of Resistance 
 
 203 
 
 Table XXX, the original body with a fine nose and a fine 
 tailpiece 
 
 R=. 3 8V+-634V 2 . 
 Again a reduction in the resistance. 
 
 TABLE XXIX 
 BEAUFOY. Page 225. 
 Bow Angle of Incidence = 14 28' 39". 
 Stern Angle of Incidence =9 35' 40". 
 
 
 <L * * y 
 
 
 
 V 
 
 Observed 
 
 R 
 
 Calculated 
 
 ft. per 
 sec. 
 
 Resistance. 
 
 V 2 ' 
 
 Resistance. 
 
 i 
 
 0-8814 
 3-4128 
 
 881 
 853 
 847 
 
 1-07^ 
 3-62 p 
 777 J 
 
 4 
 
 13-188 
 
 824 
 
 13-40 
 
 5 
 
 20-368 
 
 814 
 
 20-55 
 
 6 
 
 29-053 
 
 807 
 
 29-22 
 
 7 39-22 
 
 800 
 
 39-41 
 
 8 
 
 50-86 
 
 794 
 
 51-12 
 
 9 
 
 63-96 
 
 789 
 
 
 10 
 
 78-52 
 
 785 
 
 79-10 
 
 ii 
 
 94-52 
 
 -781 
 
 95-37 
 
 12 
 
 in-94 
 
 777 
 
 113-26 
 
 R = . 3 iV+. 7 6V'. 
 
 It is unnecessary to multiply examples, since doing so 
 would not lead us any nearer to the " stage of science " ; 
 the foregoing should suffice, however, to show that in all 
 cases the resistance follows a parabolic path, although we 
 have not sufficient information to enable us to fix the values 
 of A and B in the formula. We do not know how much 
 kinetic energy was generated, and consequently cannot tell 
 how much was recuperated by the viscosity. 
 
 It only remains now to consider why, at the very low 
 velocities, the formulae do not give results corresponding with 
 the experimental figures. It is clear from figs. 82 and 83, 
 
 1 Calculated values greater than observed. Refer to Fig. 80. 
 
204 
 
 Motion of Liquids 
 
 TABLE XXX 
 
 PARALLELOPIPEDON. Page 209. 
 Immersion 6ft. Both angles 9 35' 40" 
 
 V 
 
 ft. per 
 sec. 
 
 Observed 
 Resistance. 
 
 R 
 V 2 ' 
 
 Calculated 
 Resistance. 
 
 i 
 
 7610 
 
 7610 
 
 I-OI \ 
 
 2 
 
 2-9399 
 
 7349 
 
 3-29 I 1 
 
 3 
 
 6-475 
 
 7194 
 
 6-84 J 
 
 4 
 
 n-333 
 
 7083 
 
 n-66 
 
 5 
 
 17-493 
 
 6977 
 
 I7-75 
 
 6 
 
 24-937 
 
 6927 
 
 25-10 
 
 7 
 
 
 6866 
 
 33-72 
 
 8 
 
 43-622 
 
 6816 
 
 43-61 
 
 9 
 
 54-83 
 
 6769 
 
 54-77 
 
 10 
 
 67-29 
 
 6729 
 
 67-20 
 
 ii 
 
 80-97 
 
 6691 
 
 80-89 
 
 12 
 
 95-87 
 
 6588 
 
 95-85 
 
 = . 3 8V+-63 4 V 2 . 
 
 that the resistance at very low velocities, varying as R =AV 
 will be in excess of those obtained by the parabolic formula. 
 
 In the cases which refer to Fig. 80, the observed re- 
 sistances are, and should be, less than those calculated by 
 the parabolic formula. 
 
 The values in Col. 2 cannot be accurately called 
 " observed " ; they are interpolations from observed results. 
 
 There are several other forms of motion of liquids that I 
 might have referred to, but, as Montesquieu said, " quand 
 vous traitez un sujet, il n'est pas n6cessaire de I'dpuiser ; 
 il suffit de faire penser," and I hope I shall not have failed 
 in this. 
 
 1 Again, calculated values are greater than those observed, 
 to Fig. 80. 
 
 Refer 
 
Curves of Resistance 205 
 
 SUMMARY 
 
 The relation between the resistance caused by, and the 
 velocity of a liquid may be expressed by a parabolic curve. 
 The vertex of the parabola is not, however, generally at 
 the intersection of the X and Y axes ; nor does the axis 
 of the parabola, generally, pass through this point either. 
 
 If any flat body, of any shape, be caused to move irrota- 
 tionally round a fixed point, every point in the body will 
 describe a figure which is an inverted reflexion of the body 
 moving. That this should be so is almost self-evident ; 
 probably few people, however, are aware of it. 
 
 REFERENCES 
 Colonel BEAUFOY'S Experiments. 
 
INDEX 
 
 " Added mass," 112 
 
 Poisson on, 113 
 
 Stokes' formula, 113 
 
 Aeroplane, action on air, 4 
 
 Air cannot be " shot downwards " 
 
 by aeroplane, 4 
 Andrews on Corraison, 165 
 Angle of plate, relation to " divide," 
 
 20 
 Apertures, effect of exterior 
 
 mouthpiece, 140 
 re-entrant mouthpiece, 138 
 
 flow through (Hele-Shaw), 133 
 
 formula for discharge through, 
 
 131 
 
 movement of liquid through, 123 
 Avanzini, curves of vortices, 17 
 
 experiment on velocity head, 
 
 38 
 
 on " divide," 14 
 
 Bazin on floating bodies, 157 
 
 on weir discharge, 189 
 Beaufoy's experiments, 199 
 Bellange pump, 151 
 
 Bellasis on flow of streams round 
 
 curves, 172 
 Bernoulli, movement of liquid 
 
 through apertures, 124 
 
 on rise of water in a tube, 146 
 Bidone, coefficient of discharge, 
 
 136 
 Bossut, experiments on jets, 93, 
 
 101 
 on velocity head, 36 
 
 on vena contracta, 127 
 
 Canals, flow of water in, 155 
 
 Cat, on Seville pump, 157 
 
 Centre of pressure of an immersed 
 
 plate, 17 
 Circular plate,. position of " divide," 
 
 9 
 Coefficient of contraction, 134 
 
 de Borda, 134 
 
 Unwin, 134 
 
 of discharge, 136 
 
 Coefficient of discharge, de Borda 
 
 136 
 
 -- through greased tube, 137 
 -- through wetted tube, 137 
 Conservation of momentum, de- 
 
 fined, 5 
 
 Corraison of streams, 165 
 Corrugated ship (Macllwaine), 1 86 
 Coulomb's formula for liquid 
 
 resistance, 154 
 
 Critical velocity (Eiffel), 182 
 Curves of resistance, 192 
 
 d'Alembert, on use of mathe- 
 
 matics, 12 
 de Borda, coefficient of contrac- 
 
 tion, 134 
 
 coefficient of discharge, 136 
 
 coefficient of discharge through 
 
 apertures, 144 
 
 experiment on tube in liquid, 
 
 145 
 
 movement of liquid through 
 
 apertures, 125 
 
 on rise of water in a tube, 147 
 
 on velocity head, 43 
 
 re-entrant ajutage, 133 
 
 de Grammont, experiments with 
 
 plates in air, 69 
 de Louvrie's formula, 105 
 Definitions, conservation of energy, 
 
 -- of momentum, 5 
 
 dynamic pressure, 46 
 
 Hamilton's principle, 188 
 
 impact, 7 
 
 jet (Osborne Reynolds), 86 
 -- (Unwin), 86 
 
 kinetic head, 46 
 
 potential energy, 46 
 
 push or pull, 5 
 
 shock, 80 
 
 static liquid, 43 
 
 static pressure, 80 
 
 stream line body (Lanchester), 
 
 velocity head, 28, 46 
 
 207 
 
208 
 
 Index 
 
 Density, effect on resistance, 2 
 
 Deviation of water in front of 
 plate, 26 
 
 Dewar's experiment with ball and 
 air jet, 109 
 
 Discharge through aperture, for- 
 mula for, 131 
 
 " Divide," Dubuat on, 9-14. 
 
 position of, at different angles, 
 
 21 
 
 Avanzini, 14 
 
 on circular plate, 9 
 
 on rectangular plate, n 
 
 on square plate, 10 
 
 relation to angle of plate, 20 
 Dubuat, experiment on funnel, 
 
 with and without diaphragm, 
 66 
 
 on the " liquid prow," 26 
 
 on velocity head, 38 
 
 with body at rest, 57 
 
 with jets, 88 
 
 with partially immersed 
 
 body, 58 
 
 Dubuat's paradox, 51 
 explained, no 
 
 Pitot tube experiment, 68 
 Duchemin and Dubuat's experi- 
 ments compared, 48, 49 
 
 experiments on Pitot tube, 63 
 on potential and velocity 
 
 head, 47 
 
 on stream filaments, 26 
 
 on velocity head, 33 
 
 with body at rest, 55 
 
 flow of water past side of 
 
 body, 76 
 
 formula, 102 
 
 for velocity head, 30 
 
 on deviation of water in front 
 
 of plate, 26 
 
 on motion of stream filaments 
 
 round plate, 23 
 
 on plates at an angle, 18 
 
 on position of divide, 9-14 
 Dynamic pressure defined, 46 
 Dynamical similarity formula, 2 
 
 Eddies, affected by length of body, 
 24 
 
 formation of, in streams, 163 
 
 shape of, behind plate, 23 
 not altered by change of 
 
 velocity, 23 
 
 Eiffel on " critical velocity " in air, 
 182 
 
 on Dubuat's paradox, 60 
 
 streamlines, 119 
 
 Empirical law of resistance, 3 
 Energy, cannot be defined, 6 
 
 conservation of, 5 
 
 relation of potential and kinetic, 
 
 6 
 Eulerian theory, 114 
 
 Fanning on coefficient of dis- 
 charge, 139 
 
 on discharge through aperture, 
 
 132 
 
 Floating bodies, Bazin on, 157 
 velocity of stream increased 
 
 by, 159 
 
 position of, in rivers, 158 
 velocity of, 157 
 
 Flow of a stream, 163 
 
 of liquid behind plate, 70 
 
 of rivers, 159 
 
 of water at side of body, 76 
 
 in rivers and canals, 155 
 
 past edge of plate, 72 
 
 Formulae, centre of pressure of 
 
 jets (Rayleigh), 102 
 
 de Louvrie's, 105 
 
 discharge through aperture, 131 
 
 Duchemin' s, 102 
 
 dynamical similarity, 2 
 
 empirical, for resistance, 3 
 
 Goupil's, 1 06 
 
 Joessel's, 105 
 
 Kirchhoff-Rayleigh, 106 
 
 length of pendulum, in 
 
 liquid resistance (Coulomb), 154 
 
 pressure of jets, 93, 96 
 
 Rayleigh's, for position of 
 
 " divide," 21 
 
 Renard's, 106 
 
 Stokes's for added mass, 113 
 
 velocity head, 48 
 
 velocity head (Duchemin), 30 
 
 von Lossl's, 106 
 
 Froude, on leeway of a ship, 121 
 
 on vena contracta, 129 
 
 Goupil's experiment, 180 
 
 formula, 106 
 
 Hamilton's principle defined, 188 
 Hele-Shaw's experiments, 199 
 
 experiment on flow through 
 
 an aperture, 133 
 
 on bodies in a viscous liquid, 115 
 Hero's fountain, 150 
 Horizontal jets, 93 
 
 Impact defined, 7 
 
Index 
 
 209 
 
 Jet defined, 86 
 
 pressure of, on a plate, 86 
 
 striking a plate, effect of, 86 
 Jets, Bossut's experiments, 93, 101 
 
 formula for pressure of, 93, 96 
 
 Dubuat's experiments, 88 
 
 horizontal, 93 
 
 maximum pressure per unit 
 
 area, 91 
 
 Michelotti's experiments, 94 
 
 Morosi's experiments, 97 
 
 refluxed, pressure of, 97 
 
 static and non-static, 68, 88 
 
 striking plates at an angle, 100 
 
 Vince's experiments, 96, 100 
 Joessel's formula, 105 
 
 Kent on Pitot tube formula, 69 
 Kinetic energy, relation to potential 
 
 energy, 6 
 
 Kinetic head defined, 46 
 Kirchhoff-Rayleigh formula, 106 
 
 Lanchester on relative motion, 52 
 
 on stream line body, 115 
 Lea on Pitot tube, 69 
 Leeway of a ship, 121 
 
 Length of body, effect of, on 
 
 eddies, 24 
 
 effect on resistance, 83 
 
 Lilienthal, diagram of stream lines, 
 
 119 
 
 Liquid friction indefinable, 3 
 " Liquid prow " of Dubuat, 26 
 
 Macllwaine, corrugated ship, 186 
 Marey on stream lines, 117 
 Maxim on action of aeroplane 
 
 on air, 4 
 Michelotti, experiments on jets, 
 
 94 
 
 on influence of mouthpiece on 
 
 discharge, 145 
 
 on movement of liquid through 
 
 an orifice, 130 
 Mississippi, vertical velocity curve, 
 
 163 
 
 Momentum, conservation of, 5 
 Morosi's experiment on jets, 97 
 Mouthpieces increase discharge, 143 
 
 Negative resistance in liquids, 177 
 Newton on movement of liquid 
 through apertures, 123 
 
 on vena contracta, 127 ij 
 
 on resistance of fluids, i 
 Nollet, report on Seville pump, 152 
 
 Non-static jets, 68, 88 
 
 liquid, reflux in, 117 
 
 Osborne Reynolds, definition of a 
 jet, 86 
 
 Parabolas, experiments with, 196 
 Partially immersed plate, 12 
 Pearson (Karl), definition of push 
 
 or pull, 5 
 Pendulums, properties of, no 
 
 swinging in liquid, in, 112 
 Pitot tube, character of pressures 
 
 recorded by, 46 
 
 Duchemin experiment on, 62 
 
 Dubuat's modification, 26 
 
 effect of bell mouth, 64 
 
 effect of plate round orifice, 
 
 63 
 
 formula, Kent, 69 
 
 properties of, 28 
 
 Unwin on, 64 
 
 Poincare, on energy, 7 
 
 on Hamilton's principle, 188 
 
 on relative motion, 70 
 Poisson on " added mass," 113 
 Potential energy, relation to 
 
 kinetic energy, 6 
 
 head, defined, 46 
 
 Duchemin's experiments, 47 
 
 Pressures behind a moving plate, 82 
 Prony's formula for liquid resis- 
 tance, 155 
 Push, defined, 5 
 
 Rayleigh, centre of pressure of 
 jets, 102 
 
 formula for position of divide, 
 
 21 
 
 Rear pressure, increased by length- 
 ening of a body, 84 
 
 on a body, 85 
 
 Rectangular plate, position of 
 
 " divide," n 
 Re-entrant ajutage (de Borda), 133 
 
 mouthpiece, effect of adding 
 
 plate to, 145 
 
 Reflux in non-static liquid, 117 
 Refluxed jets, pressure of, 97 
 Relation of potential and kinetic 
 
 energy, 6 
 Relative motion, 51 
 
 Poincare, 70 
 Renard's formula, 106 
 Resistance, curves of, 192 
 
 decreased by increase of length 
 
 of body, 185 
 
 empirical law for, 3 
 
2IO 
 
 Index 
 
 Resistance of a stream, 163 
 
 of liquids composed of two 
 
 terms, 2 
 
 Coulomb's formula, 154 
 
 (Newton), i 
 
 Prony's formula, 155 
 
 reduced by viscosity, 179 
 
 Stokes, 156 
 
 Riabouchinsky on undulatory 
 
 pressure, 42 
 
 stream lines, 118 
 Rivers, flow of water in, 154 
 
 manner of flow of, 159 
 
 position of floating bodies in, 158 
 
 S6e, on relative motion, 52 
 Seville pump, 150 
 Shock, denned, 80 
 Soddy on energy, 7 
 Soreau on counter resistance, 180 
 Spheres, critical velocity with, 182 
 Square plate, position of divide, 10 
 Static and non-static jets, 68, 88 
 Static liquid defined, 43 
 
 flow of, without reflux, 116 
 
 Stokes, 45 
 
 pressure defined, 80 
 
 Stokes' formula for added mass, 113 
 
 on liquid resistance, 156 
 
 on static liquid, 45 
 
 proportionate resistance of 
 
 bodies, 3 
 Stream-filaments, direction of flow 
 
 of, 23 
 
 Duchemin's experiment, 26 
 
 motion round plate, 23 
 
 velocity of, 26 
 
 Stream-line body defined, 115 
 bodies (Hele-Shaw), 199 
 
 lines, Eiffel, 119. 
 
 Hele-Shaw, 116, 118 
 
 Marey, 117 
 
 Riabouchinsky, 118 
 
 von Lossl, 119 
 
 Streams, action on submerged 
 obstacles, 167 
 
 formation of basins in, 168 
 of eddies in, 163 
 
 nature of flow of, 163 
 
 position of maximum velocity 
 
 in curves, 171 
 
 surface curvature, 169 
 Surface curvature of streams, 169 
 Syphon, oscillation of water in, 112 
 
 Thomson, Prof. J., on flow of 
 
 streams round curves, 170 
 Tubes of flow, 133 
 
 Undulatory pressure, Riabouchin- 
 sky, 42 
 Un win, coefficient of contraction, 134 
 
 definition of a jet, 86 
 
 on exterior mouthpieces, 180 
 
 on flow of rivers, 159 
 
 on Pitot tube, 65 
 
 on pressure of a current on 
 
 a plane, 65 
 
 on the pressure of a jet, 86 
 
 on resistance of a plane moving 
 
 through a fluid, 65 
 
 on vena contracta, 128 
 
 Velocity curve of Mississippi, 163 
 
 head, Avanzini's experiment, 39 
 
 Bossut's experiments, 36 
 
 -de Borda, 43 
 
 defined, 28, 46 
 
 Dubuat's experiments, 38 
 
 Duchemin's experiment, 33, 
 
 47 
 
 of floating bodies in streams, 
 
 159 
 
 of stream filaments, 26 
 Vena contracta, Bossut, 127 
 explained, 127 
 
 Hachette's experiment, 129 
 
 Newton's experiments, 127 
 
 Unwin, 128 
 
 Vince, experiment on jets, 96, 100 
 
 experiment on planes at an 
 
 angle, 108 
 
 whirling table experiments, 69 
 Viscosity, effect on resistance, 2 
 
 reduces resistance, 179 
 
 recuperative action of, 191 
 von Lossl, experiment on planes 
 
 at an angle, 107 
 
 stream lines, 119 
 
 von Lossl' s formula, 106 
 
 Vortices, curves of, Avanzini, 17 
 
 Water, not pushed forwards by 
 
 body, 4 
 
 Weir, discharge of water over, 189 
 Weisbach, coefficient of discharge, 
 
 136 
 
 Wetted tubes, 137 
 Whirling table experiments (Vince) , 
 
 69 
 Willcocks, on flow of streams round 
 
 curves, 172 
 
 Zahm on impact of a jet of water, 
 67 
 
 on pressure of a jet, 87 
 
 on relative motion, 52 
 
 Butler & Tanner Frome and London 
 
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SCIENTIFIC BOOKS. 15 
 
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SCIENTIFIC BOOKS. 17 
 
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SCIENTIFIC BOOKS. 19 
 
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SCIENTIFIC BOOKS. 21 
 
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SCIENTIFIC BOOKS. 23 
 
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SCIENTIFIC BOOKS. 37 
 
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SCIENTIFIC* BOOKS. 39 
 
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SCIENTIFIC BOOKS. 41 
 
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SCIENTIFIC BOOKS. 43 
 
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SCIENTIFIC BOOKS. 45 
 
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SCIENTIFIC BOOKS. 47 
 
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 for Testing Materials. 
 
 (English Edition.) 
 
 Proceedings of 5th Congress, i8s. net. Proceedings of 6th 
 Congress, 305-. net. 
 
 Transactions of the American Institute of 
 Chemical Engineers. 
 
 Published Annually, 8vo. 305. net. 
 
 Printed by BUTLER 6- TANNER, Froine and London. 
 

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 MAY 18 1079 
 
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