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 'K
 
 HYDROSTATICS 
 
 AND 
 
 ELEMENTARY HYDROKINETICS 
 
 MINCHIN
 
 £on&on 
 
 HENRY FROWDE 
 
 Oxford University Press Warehousr 
 
 Amen Corner, E.C. 
 
 112 Fourth Avenue
 
 HYDROSTATICS 
 
 AND 
 
 ELEMENTARY HYDROKINETICS 
 
 BY 
 
 GEORGE M. MINCHIN, M.A. 
 
 professot; of applied mathematics 
 in the royal indian engineering college, coopers hill 
 
 O;cforb 
 AT THE CLARENDON PRESS 
 
 1892
 
 0):fot6 
 
 PRINTED AT THE CLARENDON PRESS 
 BY HORACE HART, PRINTER TO THE UNIVERSITY
 
 
 PREFACE 
 
 In this work no previous acquaintance with the nature 
 and properties of a fluid is assumed. As in my treatise on 
 Statics, I have begun with the very elements, and, as- 
 suming- that the student's reading in pure mathematics 
 is advancing simultaneously with his study of Hydrostatics, 
 I have endeavoured to lead him into the advanced portions 
 of the subject. It will be noticed, however, that the way 
 in w^hich the reader is introduced to the notion of a perfect 
 fluid is very different from that which is usually adopted 
 in similar treatises. A definition of a perfect fluid founded 
 upon the elementary facts and principles of the theory 
 of strain and stress is not calculated to produce the impres- 
 sion of simplicity, more especially when the symbols of 
 the Difierential Calculus are employed in the process. 
 
 I maintain, however, that in such a presentation of the 
 basis of the subject there is really nothing which a beginner 
 who is familiar with the elements of Geometry, Algebra, 
 and Trigonometry cannot readily understand. The preva- 
 lent view that the fundamental notions of the Differential 
 Calculus are a mystery which the beginner should not dare 
 to approach, and which cannot be unveiled until great 
 experience in mathematics has been attained, has long 
 
 b
 
 \i Preface. 
 
 peoinotl to me to be a most unfortunate fjillacj. Indeed, I 
 think that, with the aid of a few simple illustrations borrowed 
 from ordinary experience, the process of (Vifferentiafioii and 
 the notion of a limiiing value towards which a ratio tends 
 will be found much more easy of attainment and compre- 
 hension than are the initial simple processes of Algebra 
 themselves — namely, those of subtraction, multiplication, 
 and transformation. 
 
 Hence I have not hesitated to define the particular kind 
 of body which forms the ideal basis of Hydrodynamics — 
 viz., a perfect jlidd — with reference to the nature of its 
 stress and strain ; and such a definition is, I think, the only 
 one that is really direct and scientific. 
 
 The critical reader will probably notice the complete 
 absence of reference anywhere in this work to a certain 
 term which is very familiar to students of Hydrostatics — 
 namely, the wfiole pressure of a fluid on a curved surface. This 
 term is one of the unfortunate misconceptions of Physics — 
 not, indeed, so fatally charg-ed with mischief as that other 
 venerable illusion, centrfugal force, by the balancing' of 
 which with centripetal force, in the science of previous 
 generations, the planets are enabled to revolve round 
 the Sun. This notion of rvltole pressure on a curved surface 
 has served as a peg from which to hang very many 
 visionary problems of pure mathematics ; but it has done 
 more than this ; for, the author of one of our elementary 
 text-books of H}drostatics tells his readers that, when 
 a cylindrical vessel is filled with water, the 'whole 
 pressure ' of the litpiid on its curved surface is the measure 
 of the total strain to which the vessel is subjected ! So far
 
 Preface. vii 
 
 as I am aware, there is no reality in Physics corresponding 
 to this notion of whole pressure on a curved surface, and 
 certainly none is indicated by those writers who use the 
 term. There is, indeed, in the theory of Electrostatics, and 
 in that of Newtonian gravitation in general, the result 
 that the surface-integral of the normal component of force- 
 intensity taken over any closed curved surface is pro- 
 portional to the quantity of attracting matter contained 
 within the surface ; but, apart from this, the arithmetical 
 (non-vectorial) addition of a number of forces which are 
 not parallel is a process devoid of physical meaning. 
 
 Some long controversies have recently taken place 
 between the representatives of the practical engineers, 
 on the one hand, and those who may be termed the 
 scientific reformers, on the other, with regard to the way in 
 which we should speak of the common gravitation measure 
 of force. The former speak of ' a force of one pound/ for 
 example, while the latter prefer ' a force of one pound 
 weight.' As in the last edition of my Statics, I have 
 everywhere throughout the present work adopted the latter 
 mode of speaking, because I regard the distinction between 
 the mass and the tveight of a body as absolutely funda- 
 mental — the former being an essential and invariable, 
 while the latter is a purely contingent and variable, 
 attribute of the body. Indeed, were it not for the fact 
 that the Earth is nearly spherically symmetrical as regards 
 shape and density about its centre, it seems scarcely pos- 
 sible that men, communicating with each other over long 
 distances, could ever have adopted the former mode of 
 speaking. 
 
 b2
 
 viii Preface. 
 
 Tljroug'hout the treatise I have made considerable use of 
 the Principle of Virtual Work. 
 
 It will also be noticed that in several instances in which 
 there is a conflict between the algebra and the physics 
 of the subject, or a difficulty or misconception which is very 
 likely to arise in the student's mind, I have been at pains 
 to emphasise and enlarge on the difficulty, and to invite an 
 attemi)t at explanation on the part of the student. This I 
 have done because I am convinced that more than one-half 
 of the efficiency of the teaching of any subject consists 
 in the anticipation and removal of difficulties which are 
 certain to occur to the mind of the student, and which, 
 if left unnoticed, (like uncaptured fortresses in the rear of 
 an advancing army) will greatly hinder progress and 
 I)erhaps necessitate the beginning of the work all over 
 again. I have no sympathy with the view which com- 
 monly prevails in the writing of scientific treatises, that the 
 reader, so long as he is supplied with the bare truths and 
 dry facts of the subject, has all that he wants and has no 
 right to complain. 
 
 In the English language there are at present very few- 
 treatises dealing with the advanced portions of theoretical 
 Hydrostatics ; and among these the treatise of Dr. Besant 
 has long held the foremost place. The present work is not 
 intended to supplant that of Dr. Besant, but rather to sup- 
 plement it. Some portions of the subject will, I think, be 
 found treated at greater length in the one treatise and some 
 in the other ; so that there is, I hope, ample room for both. 
 
 As regards the second portion of this work — that devoted 
 to elementary Ilydrokinetics — some explanation is necessar}'.
 
 Preface. ix 
 
 The critical reader will, I hope, remember that it is meant 
 to be strictly elementary, and to be merely the necessary 
 complement of that portion of the Hydrostatics which can 
 be studied by those who desire to attain a useful working- 
 knowledg-e of the subject without attempting the applica- 
 tion of the higher pure mathematics. 
 
 Three treatises dealing specially with Ilydrokinetics have 
 recently appeared in English — namely, Lamb's Treatise 
 on the Motion of Fluids^ Lord Rayleigh's Theory of Sound, 
 and Basset's Hydrodynamics. These treatises are practically 
 exhaustive, and, in all probability, wnll continue to be 
 so for many years to come. Hence it is certain that every 
 student who desires to carry his study of Hydrokinetics 
 to the limits of our existing knowledge will turn to these 
 works for information. The attempt, therefore, to treat at 
 length of this part of the subject is unnecessary, and it 
 would have resulted in nothing better than a mere copy of 
 the works of these authors. Probably a Chapter dealing 
 with the generalities of Hydrokinematics and Hydroki- 
 netics would have been found useful as an introduction 
 to the higher treatment of these subjects ; but this has been 
 reserved for consideration and, perhaps, a future edition. 
 
 In the passage of the work thi'ough the press I have 
 had the advantage of the assistance of my colleague Professor 
 Stocker, to whose practical knowledge of the subjects of 
 Gases and Capillarity, in particular, I am indebted for very 
 useful criticism and information. 
 
 GEORGE M. MINCHIN. 
 
 Coopers Hill, 
 September, 1892.
 
 TABLE OF CONTENTS 
 
 -♦♦- 
 
 CHAPTER I. 
 
 PAGE 
 
 General Properties of a Perfect Fluid i 
 
 CHAPTER II. 
 
 Centre of Parallel Forces (Elementary Cases) 27 
 
 CHAPTER III. 
 Liquid Pressure on Plane Surfaces 37 
 
 CHAPTER IV. 
 
 General Equations of Pressure 77 
 
 CHAPTER Y. 
 
 Pressure on Curved Surfaces m 
 
 CHAPTER YI. 
 Gases ^^5 
 
 CHAPTER VII. 
 
 Hydraulic and Pneumatic Machines 273 
 
 CHAPTER YIII. 
 
 Molecular Forces and Capillarity 298 
 
 CHAPTER IX. 
 Steady Motion under the action of Gravity 3^8 
 
 CHAPTER X. 
 
 Wave Motion under Gravity (Simple Cases) 394 
 
 Index "^^^
 
 HYDROSTATICS 
 
 AND 
 
 ELEMENTARY HYDROKINETICS. 
 
 CHAPTER I. 
 
 iT 
 
 GENEKAL PEOPEETIES OF A PEEFECT FLUID. 
 
 1. Strain and Stress in a Body. Whenever any material 
 body is suLjected to the action of force, whether throughout 
 its substance or merely at certain points of its siu-face, the 
 natural distances between its molecules become altered — 
 even though it be only to an infinitesimal extent — and the 
 body is said to be in a state of strain. 
 
 For example, let A B (Fig-, i) represent an elastic string 
 with the end A fixed while the end B is 
 pulled by any force. Consider the state of 
 aflairs at any point, P, in the substance of the 
 string. If at P we imagine a very small plane 
 having the position 7; q (represented in the 
 left-hand figure) perpendicular to the direction ^p': 
 of the string, it is clear that the molecules of 
 the body at the under side of this element- 
 plane experience an upward pull from the 
 portion of the body at the upper side of 2^ '?• 
 This pull is called a stress, and it results from 
 the strain of the substance at P — a strain which, in this 
 
 i' B 
 
 Fiff. I.
 
 2 Hydrostatics and Elementary Hydrokinetics. 
 
 instance, consists of tlic increase of natural distances between 
 molecules. 
 
 At the same time the molecules at the upper side of 7; q 
 experience a downward pull, exactly equal in magnitude to 
 the jjreviously named pull. 
 
 The stress on the element-plane has, therefore, two 
 aspects : it is an uj^ward force when considered wath re- 
 ference to the molecules on its under face, and a downward 
 force in reference to the molecules on its upper face. 
 
 This double aspect is a characteristic of every stress, and 
 of every force in the Universe, however exerted, whether 
 within the substance of what we call a sing-le body or 
 between two bodies influencing" each other by attraction 
 or repulsion. The double aspect is just as necessary a 
 feature of every force as it is of every surface, which we 
 are compelled to recog-nise as having" two sides. 
 
 Let us now, in imag'ination, consider a little element- 
 plane having" the position rt^ and separating' molecules 
 rig-ht and left. A moment's reflection shows that the 
 molecules at the rig-ht experience no force (or only an in- 
 finitesimal force) from those at the left side. Practically 
 we may say that the stress on the element-plane r t is zero. 
 
 In the same way, if we consider an element-plane at P 
 having a position m n, intermediate to p q and r t, the force 
 exerted on the molecules at the under side by the substance 
 on the upper side is an upward pull whose direction is 
 oblique to the plane. 
 
 licnce in this case of a stretched string- we see that — 
 on the element-plane 7; ^ the stress is normal tension, 
 „ „ m n „ oblique tension, 
 
 „ „ rf „ zero. 
 
 If we imagine AB to represent an iron bar, we have 
 exactly the same results when the end A is fixed and the 
 end yy is pulled ; but if the end A rests on the ground.
 
 General Properties of a Perfect Fluid. 3 
 
 while a load is put on 7), it is evident that the previous 
 state of stress is exactly reversed in sif,m ; that, for ex- 
 ample, the molecules on the under side of;?<7 experience a 
 normal pressure from those on the upper side, and that the 
 stress on m n is oblique pressure. This reversal of tension 
 into pressure could not practically take place if the body 
 A B were a perfectly flexible string- ; in other words, the 
 body would at once collapse if we attempted to produce 
 pressiire at B. 
 
 Ao-ain, if AB is an iron column whose base A is fixed 
 on the ground while a great horizontal pressure is exerted 
 from right to left at the top B, the column will be slightly 
 bent and its different horizontal sections have evidently a 
 tendency to slip on each other : in other words, the mole- 
 cules at the under side of an element-plane having the 
 position pg, experience a force from right to left in their 
 own plane from the substance above pq. 
 
 The stress, therefore, on an element-plane at any point 
 inside a solid body such as iron may have any direction in 
 reference to the plane — it may be normal pressure, normal 
 tension, or force wholly tangential to the plane, according 
 to the manner in which the body is strained by externally 
 applied force. Inside a body such as a flexible string the 
 nature of any possible stress is, as we have said, more 
 limited, inasmuch as it cannot be normal pressure. 
 
 We shall now imagine a body in which the nature of 
 any possible stress is still more limited — namely, a body 
 in which the stress on every element-plane, however ima- 
 gined at a point, can never be otherwise than normal. 
 Such a body is a perfect Jlnid ; and then the stress is, in 
 all ordinary circumstances, pressure — such in the sequel we 
 shall assume it to be — although in some recent experiments 
 large tensions, or negative pressures, have been observed in 
 some fluids. Formally, then, we define that a perfect fluid 
 
 B 2
 
 4 Hydrostatics and Elementary Hydrokinetics. 
 
 is a hody such that, whatever forces act npon it, this pro^ 
 ducing strain, the stress on every element-jdane throughout it 
 is normal. 
 
 2. Intensity of Stress. If we take any element-plane, 
 mn^ at a point and take the whole amount of the stress 
 exerted on either side of the plane, and then divide the 
 amount of the stress by the area of the element-plane, we 
 obtain the averag-e stress on the little plane. Thus, if the 
 area of mn is -ooi square inches, and the stress on either 
 side is -02 pounds' weight, the rate of stress on the plane 
 
 is ■ , or 20 pounds' weight per square inch. The stress 
 
 •001 
 
 on the plane is not uniformly distributed ; but the smaller 
 the area of the element-plane, the less the error in as- 
 suming the stress to be uniformly distributed over it. 
 Hence, according to the usual method of the Differential 
 Calculus, if we take an element-plane of indefinitely small 
 area, Is, and if 8/ is the amount of the stress exerted on 
 either side, the limiting value of the fraction 
 
 Is 
 when Is (and therefore also hf) is indefinitely diminished, is 
 the rate, or intensity, of stress at P on a plane in the 
 direction m n. 
 
 It is obvious from what has been explained that such an 
 expression as ' the intensity of stress at a point P in a 
 strained body' is indefinite, because different element- 
 planes at the same point F may have very different in- 
 tensities of stress exerted upon them. We shall presently 
 show, however, that in the case of a perfect fluid the above 
 exjjression is perfectly definite, because all element-planes 
 at the same point, if they have the same area, experience 
 normal pressures of exactly the same amount ; but for any 
 other body than a perfect fluid such would not be the case,
 
 General Properties of a Perfect Fluid. 5 
 
 and we i^huuld be ol)lii;'ed to speak of ' tlic intensify of stress 
 at P on a plane havin<^- such or such particnlar position.' 
 
 Intensity of stress, then, is measured in pounds we'ujU 
 per square inch, or kilogrammes weight per square centimetre, 
 or fh/nes per square centimetre, or g-enerally, in tmits of force 
 pier unit area. 
 
 3. Principle of Separate Equilibrium. The following 
 l)rinciple is very lar<^'ely employed in the consideration of 
 the equilibrium or motion of a iiuid, or, indeed, of any 
 material system : — 
 
 We may alivays consider the equilibrium or motion of any 
 limited portion of a system, apart f row. the remainder , provided 
 we hnagine as applied to it all the forces which are actually 
 exerted on il tjy the parts imagined to he removed. 
 
 Thus, suppose Fig. 2 to 
 represent a fluid, or other 
 mass, at rest under the action 
 of any forces, and let us trace 
 out in imagination any closed 
 surface enclosing- a portion, 
 J/, of the mass. Then all the Y\g. 2. 
 
 portion of the mass outside 
 
 this surface may be considered as non-existent, so far as M 
 is concerned, if we supply to each element of the surface of 
 J/, the stress which is actually exerted on it by the mass 
 outside it. The stresses exerted on the elements of surface 
 of J/, when the body is a perfect fluid, are represented by 
 the arrows in the figure. 
 
 The portion M is, then, in equilibrium under the action 
 of these pressures and whatever external forces (gravity, &c.) 
 also act upon it. 
 
 Again, it is evident that, having traced out in imagina- 
 tion any surface enclosing a mass, i/, of the fluid, we 
 might, without altering anything in the state of this mass
 
 6 Hydrostatics and Elementary Hydrokinctics. 
 
 .v. replace the imaQ-ined enelosinof surface by an actual 
 material surlace, and then remove all the fluid outside this 
 surface ; for the enclosing material surface will, by its 
 rigitlity, supply to M at each point the pressure which is 
 exerted at that point by the surrounding- fluid, 
 
 4. Equality of Pressure Intensity round a point. 
 We shall now prove that the intensity of pressure is the 
 same on all planes. ])q, mn, rt (Fig", i), at the same point, 
 P, in a perfect ihiid, to whatever system of external forces 
 
 the fluid may be subject. Let any 
 two planes, ACca and ABba (Fig-. 
 fJ^/C 3), of indefinitely small equal areas 
 ir" ^ be described at P, the figure of each 
 
 F . ■ . . . 
 
 pj _. area being taken, for simplicity, as 
 
 that of a rectangle^ the side Aa 
 being common to both, and P its middle point. These 
 areas being indefinitely small, each may be assumed to be 
 uniformly pressed, and the resultant pressure on it acts at 
 its middle point. 
 
 Now isolate in imagination the fluid contained within 
 the prism CBAalc. This prism of fluid is kept in equi- 
 librium under the influence of five pressures and the 
 resultant external force ; and for the equilibrium of these 
 forces we shall take moments round the line A a. Of the 
 five pressures those on the faces ABC and ahc are parallel 
 to Aa, and they have, therefore, no moments about Aa. 
 The pressure on the face BhcC intersects Aa, and gives no 
 moment. There remain the pressures on ABba and ACca, 
 which are represented by f, f, at the right — where the 
 figure is a middle section of the prism through F — and the 
 component, F, of the external force which acts in this 
 middle section. 
 
 Let AB = AC = m, Aa = n, and let e be the per- 
 pendicular from F on the line of action of F.
 
 General Properties of a Perfect Fluid. 7 
 
 Now if ]j is the intensity of pressure on the element- 
 plane ABba, the force./' = m7i-p ; and if 7/ is the intensity 
 of pressure on the face A Cca, the force /''= m^i .7/ ; while, 
 if r is the external force per unit volume acting- on the 
 fluid at P, the component F will be a fraction of the 
 quantity r x (vol. of prism) ; say F = k . ni^u . r, where /c is 
 some finite number less than unity. 
 
 Hence the equation of moments about Aa is 
 
 fx --fx~+e.F=o, 
 
 or w,2 .;^ ^ p —p) + 2 X: e m^ n .r = o, 
 
 or 7;— //+2/-e.;- = o, . . . (i) 
 
 in which p, p\ k, r are all finite and e alone is infinitely 
 small. Hence diminishing- the size of the prism in- 
 definitely, or, in other words, putting- e = o in the last 
 equation, we have 
 
 I>=P\ (a) 
 
 so that the intensity of pressure on the plane ABba is the 
 same as that of the pressure on every other plane at P. 
 
 The reason, then, why the bodily force does not interfere 
 with the fundamental result (u) is that the pressures on 
 the faces of the prism are finite quantities multiplied by 
 infinitesimal areas, while the bodily force is a finite 
 quantity multiplied by an infinitesimal volume, and, when 
 diminishing the size of the prism indefinitely, its volume 
 vanishes in comparison with the areas of its faces. 
 
 The j)roposition of this Article is a particular case of a 
 general result in the theory of the Stress and Strain of 
 any material body whatever. (See Statics, vol. ii. p. 396.) 
 
 This theorem is so imj)ortant that we reproduce it here. 
 
 At any point, P, (l^'ig. 4), inside any body which is 
 subject to the action of external forces, and which, there-
 
 & 
 
 8 Hydrostatics and Elementary Hydrokinctics. 
 
 fore is strained throughout its substance, let two very small 
 element-planes, s, /, of equal area, be placed in any two 
 positions, how^ever different ; let Pn, Pu be their normals 
 drawn at P. On the upper surface of s, as seen in the 
 fio-ure. let the resultant stress exerted bv the substance in 
 its neighbourhood be represented in magnitude and line 
 of action by Pf; and similarly let the stress on 
 '',{,'"' *' be represented by Pf. From ./' let fall fi' 
 perpendicular to Pn', and from/" let fall./"/ 
 perpendicular to Pn. Then Pr is the compo- 
 nent of the stress on s along the normal to /, 
 and Pr is the component of the stress on / 
 along the normal to .v ; and the important 
 general theorem to which we refer is that — whatever he the 
 nature of the body, whether solid., perfect Jltdd, or impeifect 
 "fluids 
 
 Pr = Pr', (2) 
 
 if, as supposed, the area of the element-plane s = that of /. 
 If these areas are unequal, the projections of the stress 
 
 Pf Pf 
 
 intensities, and -^> along the normals Pii and Pn 
 
 s s 
 
 are equal, i. e., p^ _ p/ 
 
 We may designate this remarkable theorem as the Theorem 
 of the projections of stress-intensities, and the following 
 simple proof of it may be given. 
 
 At the point P in the substance (Fig. 5) let xx'h'h be 
 any element-plane whose boundary is a rectangle of area s ; 
 let hU c c be another element-plane inclined at the angle B 
 to the former, its boundary being also rectangular, and 
 such that c'cxx' is a plane perpendicular to the plane 
 xx'h' h. The figure therefore is that of a small rectangular 
 prism, which we may imagine to be closed by the terminal
 
 General Properties of a Perfect Fluid. 
 
 t.nan<4"n];ir f:iccs hxc and h'x'c. Consider llic st'iiavate 
 equiliLiium of the substance enclosed within this prism. 
 Since the areas of all the faces are very small, the stress is 
 uniformly distributed on each of them, and the resultant 
 stress on any face acts, there- 
 fore, at its centre of area 
 ('centre of g'ravity'). Now 
 we aim at showino^ that the 
 theorem of projection holds 
 for the two faces xx h'h and 
 hh' c c. 
 
 Let n be the centre of area of the face xx' c c, and express 
 the fact that the sum of the moments of all the forces 
 acting" on the prism about the line mn, parallel to xx\ is zero. 
 
 To this sum of moments nothing* will be contributed by 
 the stress on the face xx c' c, since this force acts at n. 
 Resolve the stress on each face into three components 
 parallel to Tx, Py, and F z. No force parallel to Px will 
 give any moment, and it is easy to see that the sum of 
 moments contributed by the two faces hxc and h' x' c will 
 be an infinitesimal of the fourth order, the linear dimen- 
 sions of the prism being- infinitesimals of the first order. 
 For, draw Ty and Fz parallel to xc, and xh^ and let 
 Fx = a, Py = /3j Fz — y ; also let the components, in 
 these directions, of the intensify of stress on the plane zFy 
 be P, Q5 -^\ each of these being a function of the co- 
 ordinates of the point P, the co-ordinate axes being' sup- 
 posed to be taken at some fixed origin parallel to P«, P^, 
 IPz. The two latter components for the face hxc are 
 
 Q + a —r— and it + a - , - ; 
 ax ax 
 
 and those for Ij'x'c' are 
 
 (IR. 
 
 (e-a^)a„d-(A'-a^;
 
 lo Hydrostatics and Elementary Hydrokiiictics. 
 
 These are stross-ifi/cmsifies, and therefore each of them 
 must be multiplied bv the area, idy, of the faces on which 
 they act. Taking- moments about mn, the first two give 
 
 and last two g-ive a moment of opposite sign in which, 
 moreover, a is chang-cd to — a ; hence the whole sum of 
 moments for these terminal faces is 
 
 a/3y . ^ttR dQ. , x 
 
 This, we shall see, is infinitesimal compared with the 
 moments of the stresses on the faces hh'xx and hh'c'c, 
 which act, respectively, at the middle points of Pz and z}/. 
 For clearness the mid-section, zP>/, of the prism is repre- 
 sented in Fig-. 5 with the arrows representing the forces 
 on the faces above named, the components parallel to Px, 
 perpendicular to the plane of the figure, not being* repre- 
 sented. If i\^ and T are the components of stress intensity 
 on hij'x'x, the forces rN and rT are, respectively, N .s and 
 T.s, whose sum of moments round the point n in the 
 
 previous sense is ^ 
 
 --{yN+m^) (5) 
 
 If Y, Z are the components of stress-intensity on the face 
 // l/cc, the forces represented by ^ 1' and 2:)Z in the figure are 
 / . 5 sec and Z .s sec 0, whose sum of moments about u is 
 
 -.yJsec^ (6) 
 
 Now since s is ay, we see that the moment (4) is of 
 the fuurth order of small quantities, while (5) and (6) are 
 each of the third, and therefore (as in p. 7) the first 
 would disappear ultimately in any equation involving all 
 three. 
 
 Again, it matters not whether the substance of the 
 prism is acted upon by external force or not ; for this force
 
 General Properties of a Perfect Fluid. ii 
 
 would be proportional to the volume \afty, and its moment 
 about VI u would involve the product of this volume and 
 another infinitesimal of the first order; hence the moment 
 of the external Ibrce would be of the fourth order, and it is, 
 therefore, to be ne.^lected in comparison with (5) and (6). 
 The equation of moments, then, is simply 
 
 yYsQcO = yiy+l3T, 
 .-. Y= NcosO + TainO, .... (7) 
 
 which asserts the truth of the theorem, because the rif,-ht- 
 hand side of (7) is the projection of the stress-intensity on 
 the plane xjc b'b along- the normal to the plane hh' c c, and 
 the left the projection of the stress-intensity on iVc'c along- 
 the normal to the first plane. 
 
 To see how simply this shows that the stress-intensities 
 on all element-planes at a point P in a perfect fluid are 
 equal, let us recur to our former definition of such a body 
 (p. 3). Assume, then, the strained body to be such that 
 the stress Pf (Fig-. 4) acts in the normal nT andP/' acts 
 in the normal n'F. If (^ is the ang-le between the normals, 
 P;- = P/. cos(/ , and Pr - Pf . cos^ ; therefore (3) gives 
 
 Pf Pf 
 
 / ' 
 
 that is, p = p, 
 
 where jj and // are the intensities of pressure on the two 
 element-planes. Hence the result that if a hody is such 
 that the dresses on all element-planes at a jmnt are normal to 
 these planes, the intensity of the stress is the same for all. 
 
 This principle is sometimes loosely spoken of as the 
 principle of the ' equality of fluid pressure round a point.' 
 The equality is thus shown to be a mathematical deduction 
 from the normality. 
 
 Of course at any other point, Q, in the fluid the pressure-
 
 12 Hydrostatics and Elementary Hydrokinetics. 
 
 intensity may be very different from what it is at P, this 
 (liHerence always depending on forces which act hodllij on 
 the fluid mass ; forces which act only at the surface of the 
 fluid would not, as will be seen, produce different pressure- 
 intensities at different points. 
 
 Hence, then, at each point, P, in a perfect fluid acted 
 upon by any forces there is a certain pressure-intensity, 7?, 
 which has reference simply to the point itself and not to 
 any direcilon at the point ; in other words, if (a?, y, z) are 
 
 the co-ordinates of P, 
 
 p = <b{x,y,z\ (/3) 
 
 i. e., ;j is some function of the position of P, depending", of 
 course, on the nature of the forces acting- on the fluid ; and 
 no such simple result, independent of direction, charac- 
 terises the strain and stress of a natural solid. 
 
 Hence also the difference, dp, between the pressure- 
 intensity at P and at any very close point, P', is a pofect 
 differential; i.e., if dx, dy, dz, are the excesses of the 
 co-ordinates of F' over those of P, we must have neces- 
 sarilv some such result as 
 
 dp = Ldx + Mdy + Ndz, . . . . (y) 
 
 and we see from the above that L, 31, i\^are the differential 
 coefficients of one and the same function, 4>, with respect 
 to X, y, z. 
 
 Such a result, for example, as dp = ydx — xdy could not 
 hold for a perfect fluid. 
 
 5. External Bodily Forces and Surface Forces. A 
 mass of fluid on the surface of the Earth or any planet is 
 acted upon by the attractive force of the planet in such a 
 way that each particle of the fluid experiences this external 
 force, which is called the weig-ht of the particle. Some 
 iluids are of a mag-netic nature, i. e. each particle of them 
 feels the attractive or repulsive force of the pole of a
 
 General Properties of a Perfect Fluid. 13 
 
 mao-nct. Forces which, proccedini'- in iliis \vii\' from ex- 
 ternal bodies, are felt by the separate particles of the fluid 
 are called bod 1/1/ forces. If we ima<>-ine the Huid taken out 
 into interstellar space, at a practically infinite distance 
 from every star and mag-net, its molecules would have no 
 weig-ht and would experience no force of any kind from 
 external bodies ; and if we imag-ine further that we accom- 
 pany the fluid to such a reg-ion, supplied with nothing- but 
 a vessel fitted with pistons, we should be unable to in- 
 fluence the internal portions of the fluid in any other way 
 than by producing- pressure on various portions of its 
 bounding" surface. 
 
 External forces thus produced merely at places on the 
 bounding- surface are called su) face forces. 
 
 6. Principle of Pascal. Jf a perfect fluid is acted upon 
 ly no other than surface forces, the inte7isity of pressure is 
 constant all over the surface and at all points in the interior 
 of the mass. 
 
 Let a perfect fluid be contained within the contour 
 ABC D (Fig-. 6), and suppose pressure 
 to be applied over its surface so that 
 the intensity of this pressure at A is ]> 
 pounds' weight per square inch. At A 
 take a very small area, s square inches, 
 represented by A A\ and on this little 
 area erect a rig-ht cylinder, A P, of any 
 leng-th. Now consider the separate 
 equilibrium of the fluid contained 
 within this cylinder. This fluid is held in equilibrium by 
 the force j*; . s pounds' weight acting* on A A', a pressure on 
 the base at P, and a series of pressures all over its curved 
 surface. Resolving forces in the direction AP, we have 
 j),s = the pressure on the base at P, since the pressures on 
 the curved surface are all at right angles to AP. But the
 
 14 Hydrostatics and Elementary Hydrohinetics. 
 
 area of the base is also .« square inches, therefore if ;/= in- 
 tensity of pressure at P, 
 
 p .s ^= p'.s . . . . ( I ) 
 . • . p = [/. 
 
 Ao^in. the pressure intensity at every point on the sur- 
 face is also p. For, let the base at P be turned round 
 throuo-h any angle, and on its new position construct a 
 right cylinder cutting the surface obliquely at P. Let 9 
 be the angle between the normal to the surface at P and 
 the axis, F JS, of the cylinder; let j/ be the intensity of 
 pressure exerted by the envelope at P on the fluid, and 
 consider the separate equilibrium of the fluid in the 
 cylinder P B. The area of the normal cross-section of the 
 cylinder being ft, the area cut off from the surface at P is 
 .v sec 6, and the total pressure on this is j/. -s- sec 0. Now 
 resolving along the axis PB for the equilibrium of the 
 enclosed fluid, we have 
 
 p ..s = p . s sec 6 . cos 6, 
 
 ••• /=1^- 
 
 This may also be seen by constructing on the area s at J 
 a tube, AA' BB', of any form whatever and of uniform 
 normal cross section. The fluid inside this tube is kept in 
 equilibrium by the terminal pressures on A A' and B B^ 
 together with the pressures of the surrounding fluid which 
 are all normal to the sides of the tube. Hence (except 
 that the terminal forces at A and P are pressures and not 
 fensious) this fluid is in the same condition as a flexible 
 string stretched over a smooth surface and acted upon by 
 two terminal forces only, in addition to the continuously 
 distributed normal pressure of the smooth surface ; and it 
 is obvious, by considering the equilibrium of each ele- 
 mentary length of the tube, that the forces per unit area 
 at ^ and P are equal.
 
 General Properties of a Perfect Fluid. 15 
 
 In the same way the intensity of pressure at any point, 
 Q, in the mass can bo proved to lie ;; ; for Q and F can be 
 connected by a slender riyht cylinder havin*^' equal and 
 ])arallel plane bases at Q and P. 
 
 This method of proof shows that if the stress on every 
 element-plane in the substance were not normal, its in- 
 tensity would not be tlie same at all points. For, if the 
 stresses on the different elements of the curved surlace of 
 the cylinder APA' were ol)lique, they Avould furnish a 
 component parallel to AP, and the equality (1) would 
 cease. Hence in a viscous fluid, i. e. one in which there is 
 friction between neig-hbouring' molecules, the pressure- 
 intensity is not necessarily the same at all points. 
 
 If the area A A' is an aperture, fitted accurately by a 
 piston, in a vessel ABC I) containing" the fluid, the 
 pressure at A may be produced by loading* this piston. 
 Suppose the area of the base A A' of the piston to be 
 2 square inches, and the total load on the piston to be 
 40 pounds' weight ; then even/ element of the surface of 
 the containing- vessel will experience pressure at the rate 
 of 20 pounds' Aveight per square inch, every element-plane 
 in the interior will also experience this intensity, and the 
 pressure will be uniform all over every plane area imagined 
 in the fluid, however great its area may be. 
 
 Here, however, a caution may be given. If ABC I) is 
 a vessel filled with water, and if a piston at A produces an 
 intensity of pressure of 20 pounds' weight per square inch, 
 we shall not find the intensity at such a point as C to 
 be 20, but something" notably greater if C is at a lower 
 level than A ; and at a point of the surface hig'her than A 
 the intensity would be found to be less than 20 pounds' 
 weight per square inch. But the reason of this is suffi- 
 ciently indicated in our formal enunciation of Pascal's 
 Principle — viz., that on the surface of the Earth water is
 
 i6 Hydrostatics mid Elementary Hydrokinetics. 
 
 acted upon by a bodily force (gravitation). If, however, 
 we could take the vessel of water into interstellar space 
 and apply pressure at A by a piston, we should actually find 
 the same intensity at all points of the containing- vessel. 
 
 We may, indeed, regard the Pascal Princi])le as ahoayts 
 holding in a perfect fluid — even when the fluid is acted 
 upon by gravity or other bodily force — but the evidence of 
 the Principle will be masked by a second cause of pressure, 
 viz. bodily force. If, however, the bodily force were re- 
 moved, the undiminished intensity of surface pressure pro- 
 duced at any point would at once evidence itself. 
 
 If the fluid were hydrogen, or any light gas, the Pascal 
 Principle would, even on the surface of a planet, be almost 
 accurately verified within such a moderate volume as a few 
 cubic feet of the gas, because the bodily force (weight of 
 an element volume of the gas) is too small to generate any 
 appreciable pressure. In accordance with the Principle of 
 Pascal, loe may ahoays regard a perfect jluid, even when acted 
 upon by gravitation, as a machine for transmitting to all points, 
 in icndiminihhed amount, any intensity of pressure produced 
 at any point of its siirface. This invariable transmission 
 of surface pressure will proceed, pari ptassn, with increase or 
 diminution of pressure produced by gravitation; but the two 
 causes of pressure can be kept mentally quite distinct. 
 
 Thus, for example, at a depth of lOo feet in a fresh 
 water lake, the intensity of pressure due to the weight of 
 the water is about 43^ pounds' weight per square inchj as 
 will be seen later on. But at the top of the lake there is a 
 pressure intensity of about 15 pounds' weight per square 
 inch produced by the weight of the atmosphere, and the 
 water acts as a machine for transmitting this latter 
 unaltered to every point below, so that the actual pressure 
 intensity at a point 100 feet down is 58^ pounds' weight 
 per square inch.
 
 General Properties of a Perjcct Fluid. 17 
 
 7. The Hydraulic Press. A machine the action of 
 which ilhistrates the Princii)le of Pascal is the Hydraulic 
 Press, repreKented in Fig-. 7. 
 
 It consists of a stout cylinder, A, in which a cast iron 
 piston, or ram, P, works up and down. Tliis piston has 
 a strong- iron platform fixed on the top ; on this platform 
 is placed a substance which is to be subjected to great 
 
 Fig. 7. 
 
 pressure between the platform and a strong- plate, B, fixed 
 to four strong- vertical pillars. The pressure is applied 
 at the bottom of the piston P by a column of water which 
 is forced into the cylinder A throug-h a tube, t, which 
 communicates with a reservoir of water, B. The water is 
 driven out of i? by a force-pump whose piston, /;, has 
 a much smaller diameter than that of the piston B\ 
 
 c
 
 1 8 Hydrostatics and Elementary Hydvokinetics. 
 
 the piston ;j is worked up and down by means of a lever i/, 
 
 and the cylinder in which p works terminates inside the 
 
 vessel ^ in a rose, r, the perforations in which admit 
 
 water while preventing the entrance of foreig-n matter. 
 
 It is easy to see what an enormous multiplication of 
 
 force can be produced by this machine. If F is the force 
 
 applied by the hand to the lever L, n the multiplying ratio 
 
 of the lever, and s the area of the cross-section of the piston 
 
 p, the intensity of pressure produced on the water in the 
 
 nP . . 
 
 vessel B is — ; so that if S is the area of the cross-section 
 s 
 
 of the piston P, the total force exerted on the end of 
 
 this piston bv the water in A is 
 
 n£ .- • 
 
 s 
 
 Thus, if /S = I00 5 and the ratio, n, of the long to the 
 short arm of the lever i/ is 5, the upward force exerted on 
 the piston P is 500 F, so that if a man exerts a force of 
 100 pounds' weight on the lever, a resistance of nearly 
 50000 pounds' weight can be overcome by the piston. 
 
 In order to prevent the intensity of pressure in the 
 vessel B from becoming too great, a safety-valve closed by 
 a lever loaded with a given weight, JF, is employed. 
 
 The Hj^draulic Press remained for a long time com- 
 paratively useless, because the great pressure to which 
 the water was subject drove the liquid out of the cylinder 
 A between the surface of the piston P and the inner 
 surface of the c^dinder. This defect was remedied in a 
 very simple and ingenious manner by Bramah, an English 
 engineer, in the year 1 796. In the neck of the cylinder A 
 is cut a circular groove all round, and into this groove is 
 fitted a leather collar the cross-section of which is repre- 
 sented at c in the form (f^. This collar is saturated with 
 oil, in order that it may be water-tight, and it will be seen
 
 General Properties of a Perfect Fluid. 
 
 19 
 
 that it presses with its left-hand and upper portion ag-ainst 
 the cylinder A, while its rig-ht-hand portion is ag-ainst the 
 piston P. When, by pressure, the water is forced up 
 between the surface of the piston and the surfiice of the 
 cylinder, this water enters the lower or hollow portion of 
 the inverted U-shaped collar and firmly presses the leather 
 against both the piston P and the surface of the g-roove, 
 thus preventing any escape of water from the cylinder. 
 
 In consequence of this great improvement in the machine, 
 it is very commonly called BramaJts Press. 
 
 In order to prevent the return of the water from the 
 cylinder A on the upward stroke of the piston ;;, there is a 
 valve, represented at I in Fig-. 7, and shown more clearly at 
 
 Fig. 8. 
 
 i in Fig-. 8, which is a simple sketch of the essentials of the 
 force-pump. When the piston p rises, a valve, e, in the 
 pipe dipping- into the reservoir B, opens upwards and 
 allows the water to fill the cylinder J and to flow through 
 to the valve i. When /; descends, the water closes the 
 valve e and is forced to open the valve i which is pressed 
 
 c 2
 
 20 Hydrostatics and Elementary Hydrokineties. 
 
 down by a spiral si)rino^. When the piston p moves 
 u]) wards, the water which has passed the valve i into the 
 cylinder xL cannot return into the cylinder / because it 
 obviously assists the spring- in closing- the valve i. The 
 safety-valve is represented at v in Fig-. 8. 
 
 The piston p works in a stuffing-box in the upper part of 
 the cylinder /, this stuffing--box playing- the same part as 
 the leather collar round the ram — i. e., preventing- leakag-e. 
 The piston must not fit the lower part of the cylinder / 
 tightly, because when p in its downward motion passes o, the 
 cylinder *vould be burst if the water above the closed valve 
 e could not escape round the piston and out throug-h the 
 valve i. 
 
 Another machine depending- essentially on the same 
 principles and illustrating- the Principle of Pascal is the 
 Ihjdrodaiic Bellows, which is formed by two circular boards 
 connected, in bellows fashion, by water-tig-ht leather, the 
 boards being the ends of a cylinder the curved surface of 
 which is formed by the leather. One of these boards being- 
 placed on the g-round, the other lies loosely on top of it. 
 A narrow tube communicates with the interior of this 
 cylinder. If this tube is a long- one and held vertically, 
 when water is poured into it at its upper end, the upper 
 board of the bellows, and any load that may be placed on 
 it, will be raised by the pressure of the water, the intensity 
 of which pressure depends (as will be subsequently ex- 
 plained) on the height to which the narrow tube is filled. 
 
 8. Liquids and Gases. An absolutely incompressible 
 perfect fluid is called a liquid; but the term liquid is also 
 applied to fluids which can be compressed, but which re- 
 quire very great intensity of superficial pressure to produce 
 even a small compression. 
 
 The cubical cowj)ressihility of any substance (see Statics, 
 vol. ii., Art. 387) is thus measured : take any volume, v, of
 
 General Properties of a Perfect Fluid. 21 
 
 the substance (for simplicity a spherical, cubical, or cylin- 
 drical, volume) and let its surface be all over subject to 
 pressure of uniform intensity, hp ; measure the decrement, 
 — hv, of volume produced ; theu the fractional compression 
 
 is ; and if we divide the intensity of pressure which 
 
 produces this by the fractional compression we obtain 
 
 the measure, viz., 
 
 hp dp 
 r- ^ OV — V -—, (a) 
 
 8v' dv ^ ' 
 
 V 
 
 of the modulus of cubical compressibility of the substance, 
 this modulus being- evidently a force per unit area. 
 
 Thus, if the volume and its decrement are measured 
 in cubic centimetres, while the intensity of the pressure 
 is measured in dynes per square centimetre, we obtain 
 the modulus of compressibility in absolute C. G. S. units. 
 
 If k is this modulus, we have 
 
 -4, = ' W 
 
 If k is a constant, we have the case of a homogeneous 
 solid or a liquid — extremely large values of k characterising 
 a body of the latter kind. 
 
 If k varies sensibly with the intensity of pressure, p., 
 in any way, we have bodies of various physical natures, 
 according to the mode of dependence of k on p. 
 
 If, for instance, k is equal to p, the body is a perfect gas. 
 Putting k = p \u (/3) and integrating, we have 
 
 pv = constant, 
 the well-known equation expressing the law of Boyle and 
 Mariotte for a perfect gas whose temperature remains 
 unaltered while its volume and intensity of pressure vary. 
 Hence for a gas compressed at constant temperature t/ie
 
 22 Hydrostatics and Elementary Hydrokinetics. 
 
 modulus of cuhical conqnessihility is eqical to its intensity of 
 2)ressure. This modulus is sometimes called the ' resilience 
 of volume ' ot" the strained substance. It may be expressed 
 in terms of the density instead of the volume ; for if p is 
 the density of the substance inside the volume v, since the 
 mass remains unaltered, we have 
 
 vp = v^^Pq = constant, 
 where Vq and p^ are the volume and density of the element 
 considered before strain. Hence (/3) becomes 
 
 dp 
 
 p~ = k. 
 dp 
 
 (y) 
 
 Employing the units of the C. G. S. system (forces in 
 dynes, &c.) the following- is a table of resiliences of volume 
 for various liquids * : 
 
 
 Temp. 
 Cent. 
 
 Coefficient of 
 Resilience of Vol. 
 
 Distilled Water .... 
 
 Sea Water 
 
 Alcohol 
 
 Ether 
 
 >) 
 
 Bisulphide of Carbon . . 
 Mercury 
 
 15 
 
 17-5 
 
 
 15 
 
 
 14 
 14 
 15 
 
 2-22 X 10^'' 
 2-33 X io^° 
 
 I-2I X IO^° 
 
 I-II X 10'" 
 •93 X 10'" 
 
 •792 X 10^'' 
 
 i-6o X io^° 
 54.20 X io^° 
 
 9. Specific Weight. By the term specific loeight of any 
 homogeneous substance we shall understand its weight per 
 unit volume. 
 
 If w is the weight of any homogeneous substance per 
 unit volume, a volume V will have a weight given by the 
 equation 
 
 w = r. w. 
 
 * Taken from Everett's Uiiits and Physical Constants.
 
 General Properties of a Perfect Fluid. 23 
 
 It will be useful to remember that 
 
 I cubic foot of water has a mass of about 6%\ lbs. 
 
 I „ „ „ » jj » ,; 1000 ounces. 
 
 I „ inch of mercury „ „ -491 lbs. 
 
 These numbers are, of course, only approximate, because 
 the mass of a cubic foot of any substance depends on the 
 temperatm*e at which it is. 
 
 A gramme is defined to be the mass, or quantity of 
 matter, in i cubic centimetre of water when the water is at 
 its temperature of maximum density ; this temperature is 
 very nearly 4° C. 
 
 A term in frequent use is the specific gravity of a sub- 
 stance, which oug-ht, apparently, to sig-nify the same thing 
 as its specific weight; but it does not. The specific 
 gravity of any homogeneous solid or liquid means, in its 
 ordinary employment, the ratio of the weight of any 
 volume of the substance to the weight of an equal volume 
 of distilled water at the temperature 0° C. Thus, for 
 example, in the following table of specific gravities : 
 
 srold 
 
 gUlLi . . . 
 
 • • ^y:J 
 
 silver . . . 
 
 • . IO-5 
 
 copper . . 
 
 . . 8-6 
 
 platinum . . 
 
 . . 22-0 
 
 sea- water . 
 
 . . 1-026 
 
 alcohol . . 
 
 . . -791 
 
 mercury . 
 
 • • ^3-596 
 
 the number opposite the name of any substance does not 
 tell us the weight of a cubic foot, or of any other volume, 
 of the substance ; it merely tells, with regard to platinum, 
 for example, that a cubic foot of it, or a volume V of it, is 
 22 times as heavy as a cubic foot, or a volume V, of dis- 
 tilled water. A table of specific gravities is a table of 
 relative weights of equal volumes. In the C. G. S. system,
 
 24 Hydrostatics and Elementary Hydrokinetics. 
 
 since the unit of wcig-lit is that of i cubic centimetre of 
 water, and since water is the substance with which in a 
 table of specific gravities all solids and liquids are compared, 
 the number (specific gravity) opposite any substance ex- 
 presses the actual mass, in grammes, of i cubic cm. of the 
 substance. 
 
 If s is the specific gravity of any substance and to the 
 actual weio'ht of a unit volume of the standard substance 
 (water), the weight of a volume V of the substance is given 
 
 bv the ec| nation 
 -^ W= Vsw. 
 
 The term density is also used to denote the mass, per unit 
 volume, of a substance. Thus if mass is measured in 
 grammes and volume in cubic centimetres, the density of 
 silver is 10-5 grammes per cubic centimetre ; the density 
 of mercury is 13-596 grammes per cubic cm. If mass is 
 measured in pounds and volume in cubic inches, the density 
 of silver is '3797 lbs. per cubic inch and that of mercury 
 •491 lbs. per cubic inch. These latter numbers are, of 
 course, proportional to the former. 
 
 The term (Jensiti) has no reference to gra\'itation. If 
 silver and mercury are taken from the Earth to a position 
 in interstellar space in which there is felt no appreciable 
 attraction from any Sun or Planet, it is still true that silver 
 has a mass of 10-5 and mercury a mass of 13-596 grammes 
 per cubic cm. Neither would,- in this position, have any 
 specific tveifjht, since there is no external force of attraction 
 acting on them ; but the moment they are taken to the 
 sui'face of any Sun or Planet, each acquires weight, and the 
 ratio of the Aveights of equal volumes of them is the ratio, 
 10-5 : I3'596, of their densities. If, for example, they were 
 carried to the surface of the Planet Jupiter, the weight of 
 a cubic cm. of each would be nearly 2\ times as great as it 
 is on the surface of the Earth ; but a table of relative
 
 General Properties of a Perfect Fluid. 25 
 
 weii^hts of substances on Jupiter would be exactly the same 
 as a ta])le of relative weig-hts on the Earth. 
 
 If any given volumes of a num])er of homog-eneous sub- 
 stances are mixed tog-ether in such a way as to make a 
 homog-eneous mixture whose volume is the S7(ni of the 
 volumes of the separate substances, the specific weig^ht of 
 the mixture is easily found. For, let v^ and w^ be the 
 volume and specific weight of the first substance ; v^ and 
 Wg those of the second ; and so on. Then if w is the 
 required specific weight of the mixture, since the weight of 
 the mixture is equal to the sum of the separate weights, - 
 
 (i'l + v., -\-v..+ ...)io = v\w^ + v^w^ + fgWg + . . . ; 
 
 to = 
 
 l,v 
 
 Such a mixture is called a mechanical mixture — as, for 
 instance, a mixture of sand and clay. But when a chemical 
 combination takes place between any of the substances, the 
 volume of the mixture is not equal to the sum of the 
 volumes mixed — as when sulphuric acid is mixed with 
 water. If for any chemical mixture V (which must be 
 specially measured) is the volume of the mixture, it is evi- 
 dent that we have, as above, 
 
 w = 
 
 V 
 
 Example. 
 
 A cask A is filled to the volume v with a liquid of specific 
 weight w ; another cask, B, is filled, also to the volume v, with 
 
 another liquid of specific weight s ; - is taken out of A and — 
 
 also out of B, the first being put into B and the second into A , 
 and the contents of each cask are shaken up so that the liquid
 
 26 Hydrostatics and Elementary Hydrokinctics. 
 
 iu each becomes homogeueous. The same process is repeated 
 again aud again : find — 
 
 (a) the specific weight of the liquid in each cask after m 
 such operations ; 
 
 (6) the volume of the original liquid in each cask. 
 
 Ans. \i w — s is denoted by d, and if w^, s^ are the specific 
 weights of the liquids in A and B, respectively, after m opera- 
 tions, 
 
 
 and the volume of the original liquid in either cask is 
 
 l-(-Dl 
 
 V 
 2 
 
 [N.B. The liquids are assumed not to enter into chemical 
 combination.]
 
 CHAPTER II. 
 
 B(n) 
 
 CENTRE OF PARALLEL FORCES (ELEMENTARY CASEs). 
 
 10. Points and Associated Magnitudes. Let A and J^, 
 Fig. 9, bo any two jjoints, ri' any plane whatever, A3I and 
 SN the perpendiculars from A 
 and JJ on the plane, and G a 
 point on the line AB dividing- 
 it in the ratio of any two magni- 
 tudes, m and n, of the same kind, 
 
 so that -p7^, = - ; then the per- 
 
 pendicular, GQ, from G on the plane is given by the 
 
 equation 
 
 m.AM + 7i.BN I . 
 
 Gq= 
 
 m + n 
 
 For, draw Ast parallel to 3IN, meeting GQins. 
 GQ = Gs + sQ = Gs + AM. 
 
 Then 
 
 _, Gs n 
 
 But -j^, = ; 
 
 Bt m + n 
 
 Gs = 
 
 n 
 
 m + u 
 
 {BN-AM). 
 
 Substituting this for Gs in the value of GQ, we have (a). 
 
 The figure represents m and n as associated with A and B, 
 respectively, and G divides AB so that the segments next 
 A and B are inversely as the magnitudes associated with 
 these points. 
 
 If one of the magnitudes, m. and «, associated with either 
 point is negative, the dividing point G is to be taken 
 on AB produced or on BA produced, according as n or m is 
 numerkaUy the greater (irrespective of sign). Thus if the
 
 28 Hydrostatics and Elementary Hydrokinetics. 
 
 mag-nitude associated with B is — w, the distance of the 
 point G from the plane PP is 
 
 m.AM — n.BN 
 
 (/3) 
 
 m—n 
 
 The result (a) is well known in the composition of two 
 parallel forces of like sense acting at A and P, while (/3) 
 api^lies to the case in which the parallel forces are of 
 unlike sense. 
 
 If parallel forces w^hose magnitudes are vi and n act 
 in the same sense and in anjj common direction at A and B, 
 equation (a) gives the distance of their ' centre ' from any 
 plane ; while (^) gives the distance of the centre of parallel 
 forces of unlike sense. 
 
 The results (a) and (/3) have not, however, been restricted 
 to the case in which m. and n axe forces. These quantities 
 may be, as said before, any two magnitudes of the same 
 kind — e. g., two masses, two areas, two volumes, &c. The 
 case in which one — say n — is negative may also be repre- 
 sented by supposing in and — n to be a positive and a 
 negative charge of electricity. 
 
 When m and n are quantities of matter, the point G is 
 called their centre of mass (see Statics, vol. i., Art. 90). 
 
 When m and n are positive, the magnitude rii-\-n is 
 associated with G ; and if the magnitude associated with 
 B is —n, the magnitude m — n is associated with G. 
 
 As ^ Let there be any number of 
 
 '-f ^ As given points, A^, A^, A^,... 
 
 ^^^ ^n ^123 ^"'^' (Fio". 10) with which are asso- 
 
 A' -' . . 
 
 (m'^) ciated any given magnitudes 
 
 Fig- 10- %, Wo, m.y.. respectively, and 
 
 take the centre, ^^g' ^^ ^^^ magnitudes m^, w/^, then take 
 the centre, g-^o?,^ of the magnitude m^ + m^ at ^1^ and the 
 magnitude m.^ at ^3, then the centre of m^ + m^-\- m^ at ^j23
 
 Centre of Parallel Forces {Elementary Cases). 29 
 
 and W4 at A^, and so on. In this way we arrive at a final 
 point, 6', wliieli is the centre of the whole system of magni- 
 tudes. It is required to express the distance of this point 
 from any plane in terms of the "•iven mag-nitudes and the 
 distances of their associated points from the i)lane. 
 
 This is easily done by (a). For, if the distances of 
 A■^^, Ao, Ar^,... from any plane are, respectively, z^, z^, z^,... 
 and Cjo is the distance of ^12 fio"fi the plane, we have by (a), 
 
 W, Z-i + Wo Z., 
 
 ^12 = ; — ^-~"' • 
 
 Also if 2'i23 is the distance of ^123 from the plane, 
 _ (n?^ + m.^z^.^ + m^z.^ 
 
 _ m-i z-^ + m2Z2 + W3 2-3 
 
 Hence, by repeated applications of the simj^le result (a), 
 if z is the distance of G from the plane, 
 
 m-i z^ + m^z^ + m^z^ + m^z^+ ... 
 
 z = 
 
 • • • (y) 
 
 Imz 
 
 =Y^ (^^ 
 
 The plane of reference, PP, may be such that some of the 
 points are at one side of it and some at the other side. In 
 this case some of the ^'s are positive and some negative, the 
 side of the plane which we take as positive being a matter 
 of choice. 
 
 If the points A^, A.^,... are not all in one plane, to de- 
 termine the position of G, we shall require to find its 
 distances from some l/iree planes of reference. If the points 
 ^1, A.^,... all lie in one plane, it will be sufficient to find 
 the distances of G from any two lines in this plane. In 
 this case PP (Fig. 9) may be supposed to be a mere line in 
 the plane of the given points, although, strictly speaking,
 
 30 Hydrostatics and Elementary Hydrokinetics. 
 
 it represents, in this case, a plane perpendicular to the 
 plane of the points. If the points A^^ A^,..,^ lie on 
 a right line, G lies on this line, and its position will be 
 known if its distance from any other line is known. 
 
 When w/j, ^2 5 ^%v are masses, G is their centre of mass, 
 and equation (y) expresses the Theorem of Mass Moments^ 
 the product mz^ of any mass, w;, and the distance, ■?, of 
 its centre of mass from a plane being called the moment of 
 the mass with respect to the plane. 
 
 Cor. The sum of the moments of any masses with respect 
 to any plane passing through their centre of mass is zero. 
 
 Even when m-^^,m^, m^,... are not masses, but any magni- 
 tudes of the same kind (forces, areas, &c.) we shall refer to 
 (y) as the Theorem of Mass Moments. 
 
 When Wj, m^, 7/i.^,... are the magnitudes of a system of 
 
 parallel forces acting at A^, A^, A^,... in any common 
 direction, the point G is called the centre of the si/stem 
 of jiarallel forces. 
 
 It is evident that the distances z^,z^.,... need not be per- 
 pendiculars ; they may be oblique distances — all, of coui'se, 
 measured in the same direction. 
 
 The work of practical calculation is often facilitated by 
 
 forming tables of masses, distances, and products, in columns, 
 
 as in the following example. 
 
 Examples. 
 
 1. At the vertices, A, B, C (Fifj. 1 1) 
 of a triangle and at the middle 
 points, a, b, c, of the opposite sides 
 act parallel forces whose magnitudes 
 and senses are represented in the 
 figure; find the position of the centime 
 of the system. 
 p. The position of the centre will be 
 
 known if its distances from any two 
 sides, AB and AC, are known.
 
 Centre of Parallel Forces {Elementary Cases). 31 
 
 To find its distance from AB, let the length of the perpen- 
 dicular fi'oni C on AB he p; and form a tal)le of forces and 
 distances of their jwints of ajjplication from AB as in the ibl- 
 lowing scheme, taking as positive those forces which act in the 
 sense of that at A : — 
 
 Forces. 
 
 Distances 
 from A li. 
 
 Products. 
 
 Distances 
 from AC. 
 
 Products. 
 
 10 
 
 -16 
 
 8 
 
 - 6 
 
 — 12 
 
 14 
 
 
 P 
 
 \r 
 
 
 
 
 
 
 -8^> 
 8;; 
 
 -2P 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 -3 5 
 — I2q 
 
 iq 
 
 — 2 
 
 
 -zp 
 
 
 -Sq 
 
 The sum of tlie first column answers to 2 m, the denominator 
 of (S), p. 29, while the sum of the third column answers to 21 mz, 
 the numerator, so that the perpendicular distance of the centre 
 from -45 is 
 
 -3P 
 
 or 
 
 — 2 
 
 h- 
 
 Drawing, then, a line parallel to AB at a distance f p (above 
 C), we know that G lies somewhere on this line. 
 
 Denoting the perpendicular from B on A G by q, forming a 
 table (column 4) of distances from AG, and a column (number 
 5) of corresponding products, and dividing the sum of these 
 products by the sum of the forces, we have the distance of G 
 from A G equal to 
 
 -8<7 
 
 "" 4 q. 
 
 or 
 
 ■2q 
 
 Hence G lies on a line to the right of B distant 4 q from AC. 
 The i)oiut of intersection of this with the previous line is G. 
 
 2. From a solid homogeneous triangular prism is removed a 
 portion by a plane parallel to the base cutting off - of the axis 
 measured from the vertex ; find the distance of the centre of
 
 32 Hydrostatics and Elementary Hydrokinetics. 
 
 mass of the remaining frustum from the base. Let ABC be the 
 middle section of the prism perpendicular to its vertex, edge, 
 and let PQ rejiresent the cutting jjlane. 
 
 Now the volume of the whole prism is to the volume of the 
 removed prism as the area ACB is to the area PCQ ; and since 
 the areas of any similar plane figures are proportional to the 
 squares of corresponding lines in them {Euclid, Book VI. 
 prop. 20), 
 
 area PCQ = -^ (area ACB). 
 
 Hence if V is the volume of the whole prism, the volume of 
 
 the small prism = —^ , and that of the frustum = ( 1 ^^ V. 
 
 ^ IV- ^ w/ 
 
 Now let 7i be the length of the j^erpendicular from C on xiB, 
 
 and equate the mass-moment of the whole prism with respect to 
 
 the base AB i^ the sum of the mass-moments of the frustum 
 
 and the small prism. Since the centre of mass of a prism is 
 
 distant by \ of the height from the base, the distance of the 
 
 centre of mass of PCQ from AB is i — [- ( i \\ or 
 
 h. If, then, x is the distance of the centre of mass of 
 
 the frustum 
 
 3 ^ n J rr 3 n 
 
 {n—i){n+2) 
 .-. x=i- 7 -—./i. 
 
 3. From a solid homogeneous cone is removed a portion hy a 
 
 plane parallel to the base cutting off — of the axis measured 
 
 from the vertex ; find the distance of the centre of mass of the 
 remaining frustum from the base. 
 
 (The volumes of similar cones are proportional to the cubes 
 of corresponding linear dimensions. Hence if V is the vol. of 
 
 the whole cone, the vol. of the small cone = -^ V.) 
 
 Ans. The distance = ^ '-\~^ ,— . h. 
 
 4n{n +n+i)
 
 Caitrc oj Parallel Forces [Elenienlary Cases). 33 
 
 4. From the middle point of one side of a triangle is drawn 
 a porpondicular to the hase; find the distance, from the base, of 
 the centre of area of tlie quadrilateral thus formed. 
 
 Ans. If h is the height of the triangle, ^j ^• 
 
 5. Find the position of the centre of area of a trapezium. 
 Ans. It is on the line joining the middle points of the two 
 
 parallel sides, and if the lengths of these sides are a and h, and 
 /( the perpendicular distance between them, its distance from the 
 side a is 26 + a , 
 
 6. Prove that the distance of the centre of area of a triangle 
 from any plane is one-third of the algebraic sum of the distances 
 of its vertices from the plane. 
 
 (The centre of area of any triangle is the same as the centre 
 of mass of three equal pai'ticles placed at its vertices.) 
 
 7. Prove that the distance of the centre of area of a plane 
 quadrilateral from any plane is 
 
 where ^z is the sum of the distances of its vertices, and ^ the 
 distance of the point of intersection of its diagonals, from the 
 plane. 
 
 11. Continuously Distributed Forces. We shall now 
 find the centre of a system of parallel forces distributed 
 continuously over a plane area, in a few simple cases which 
 do not require the application of the Integral Calculus. 
 
 (i) If normal pressure acts all over any plane area 
 in such a way that its intensity is the same at all points, 
 the resultant pressure acts at the centre of area (' centre of 
 ccravity,' so called) of the figure. For, if at any two 
 points, A, B, in the given figure we take any two elements 
 of area, the pressures on them are directly proportional 
 to the areas themselves, and the resultant of these forces 
 acts at a point in the line AB dividing it into segments 
 inversely as the forces, i. e., inversely as the areas ; hence 
 
 D
 
 / 
 
 / 
 / 
 / 
 / 
 / 
 ^L „ 
 
 34 Hydrostatics and Elementary Hydrokinctics. 
 
 it acts at the centre of area of the two elements of area. 
 The process, therefore, of finding- the point of application 
 of the resultant of the whole system of pressures acting- on 
 the indefinitely g-reat number of elements of area into 
 which the g-iven fig-ure can be broken up is precisely the 
 same as that of finding- the centre of area of the fig-ure. 
 
 A (2) If parallel forces act at 
 
 \ all points of a rig-ht line, 
 
 \^ AB^ Fig-. 12, in such a way 
 
 \ that the force at any point 
 
 p _\f P is directly proportional 
 
 Q \ to the distance, VA, of P 
 
 \ from one extremity, A^ of the 
 
 \ line, the resultant force acts 
 at the point on AB which is 
 ^'^- "• I of the length AB from A. 
 
 For, imagine AB to be broken up into an indefinitely 
 great number of small equal parts, PQ ; describe any 
 isosceles triangle, MAN, having AB for height, and from 
 all the points, P, Q, ... of division of AB draw parallels to 
 the base MN, thus dividing the area MAN into an in- 
 definitely great number of narrow strips. The area of 
 any strip, QF, is simply proportional to the distance, PA, 
 of the strip from A. Hence the areas of the strips are 
 exactly proportional to the giv'^en system of parallel forces ; 
 but the centre of area of the strips, or centre of area of the 
 whole triangle, is | AB from A. This point is, then, 
 the centre of the given system of forces, 
 
 (3) If parallel forces act at all points of a right line, 
 AB, Fig. 12, in such a way that the force at any point P is 
 proportional to the scpare of the distance, BA, of P from 
 one extremity. A, of the line, the resultant force acts at a 
 point on AB which is |- of the length AB from A. 
 
 For, imagine AB to be broken up, as before, into equal
 
 Centre of Parallel Forces {Elementary Cases). 35 
 
 elements, such as PQ ; describe any solid cono havinq- AB 
 for its axis, and let this cone be represented by j\rAN. 
 From all the points of division of AB draw planes parallel 
 to the base iO^of the cone, thus dividing- the cone into an 
 indefinitely g-reat number of thin circular plates. The 
 volume of the plate at P is itPF- x PQ, and since the 
 thicknesses of the plates are all equal to PQ, the volume 
 of the plate is proportional to PF^, i. e., to PA^. Hence 
 the volumes of the plates vary exactly as the forces of the 
 given system, and therefore the centre of volume of the 
 plates is identical with the centre of the force system ; but 
 the former (centre of volume of the cone) is f A.B from A ; 
 therefore, &c. 
 
 (4) If parallel forces act at all points of a rig-ht line 
 AB, in such a way that the force at any point P is propor- 
 tional to the product of the distances PA, PB of P from 
 the extremities of the line, the resultant force acts at the 
 middle point of AB. 
 
 For, taking- a point, P', w^hose distance from B is equal 
 to that of P from A, the forces at P and P' are evidently 
 equal ; their resultant therefore acts at the middle of AB. 
 Hence the system of forces from A to this middle point 
 is the same as the system from B to this point ; the re- 
 sultant, therefore, of the whole system acts at the middle 
 point of AB. 
 
 (5) If each infini- 
 tesimal element of any 
 plane area is acted upon 
 by normal pressure pro- 
 portional conjointly to the 
 magnitude of the area and ^^--~.-____- — ^ 
 to the distance of the t'ig- ^3- 
 
 element from a given plane, the wagnitude of the resultant 
 pressure will be proportional to the product of the whole 
 
 D 2
 
 36 Hydrostatics and Elementary Hydrokinetics. 
 
 plane area and the distance of its centre of area from the 
 wiven plane. 
 
 For, let yOx, Fi"-. 13, represent the g-iven plane, and 
 ABCD the g-iven plane area. Take any point, P, in this 
 area, and round P describe a very small closed curve whose 
 area is s. Let Pi\^, the perpendicular from P on the plane 
 yOx, he denoted by z\ then, by hypothesis, the amount 
 of force,/, on the element at P is g-iven by the equation 
 
 where yt is a given constant. If /, /',.,. are any other 
 elements of area whose distances from the g-iven plane 
 are /, /',.., the resultant pressure, being- equal to the sum 
 o^f,f',f'\ ... of the individual pressures on the elements, 
 is equal to , , , , „ „ v 
 
 But if ^ = the area of the whole plane fig-ure, and z is 
 the distance, G^Q, of the centre of area, G, from yOx, 
 
 we have ._ , , „ „ 
 
 Az = SZ + s z + s z" + . . . . 
 
 Hence if P is the resultant pressure, 
 
 P = k.A.z (a) 
 
 The student must be careful to observe that the resultant 
 pressure does not act at G, but evidently at some such point 
 as /, whose distance from the plane yO:c is g-reater than the 
 distance of G from the plane. 
 
 In this case, then, the mean intensity of pressure on the 
 area is that which exists at G.
 
 CHAPTEE III. 
 
 LIQUID PEESSURE ON PLANE SURFACES. 
 
 Elementari/ Cases. 
 
 12. Intensity of Pressure produced by Gravity. Let 
 ACB, Fig. 14, be a vessel of any shape containing water 
 or other homogeneous 
 liquid. Then at each 
 point, P, of the liquid 
 the action of gravity pro- 
 duces a certain intensity 
 of pressui'e, the magnitude 
 of which we proceed to 
 find. At P draw an in- 
 definitely small horizontal element of area — s square inches, 
 suppose — and on the contour of this area describe a ver- 
 tical cylinder, PN. Consider now the separate equilibrium 
 (Art. 3) of the liquid in this cylinder. 
 
 If PN is z inches in length, the volume of the cylinder 
 = z .8 cubic inches, and if the specific w^eight of the 
 liquid is iv pounds' weight per cubic inch, the weight 
 of the cylinder = wzs. This cylinder is acted upon by 
 a vertically upward pressure on the base 5 at P and a 
 system of horizontal pressures round its curved surface, 
 in addition to its weight — omitting, for the present, the 
 surface pressure at N produced by the atmosphere or any 
 other cause.
 
 38 Hydrostatics mid Elementary Hydrokinctics. 
 
 If p pounds' weight per square inch is the intensity 
 of pressure at P, the upward pressure on the base * is j*; . s. 
 Resolving- forces vertically, we have, then, 
 
 p . s =^ wz . s \ 
 . • . p — wz^ (a) 
 
 which gives the required intensity of pressure. 
 
 If the surface intensity of pressure is p^ pounds' weight 
 per square inch, this will be added to the value (a), by 
 Pascal's princii)le ; hence the complete value of 7; is given 
 by the equation 
 
 p^wz^lh \P) 
 
 Observe that we have not assumed the bounding surface 
 AB to be horizontal. 
 
 Without any reference to the shape of the surface AB^ 
 we can see that the intensity of pressure is the same 
 at all points P, Q, ... which lie in the same horizontal 
 plane. 
 
 For, draw PQ ; at P and Q place two indefinitely small 
 equal elements of area, s, perpendicularly to PQ ; form a 
 cylinder having PQ for axis and these little areas for 
 bases, and consider the separate equilibrium of the liquid 
 enclosed in this cylinder. The forces keeping it in equili- 
 brium are its weight, a system of pressures all round 
 its curved surface, and the pressures on its bases at P and Q. 
 Resolving forces along PQ for equilibrium, neither the 
 weight nor the system of pressures on the curved surface 
 will enter the equation ; therefore the pressure on the base 
 5 at P = the pressure on the (equal) base « at Q ; that is, 
 the intensity at P = the intensity at Q. 
 
 From this it follows that the bounding surface AB 
 on which at all points there is either no pressui'e, or 
 pressure of constant intensity, must he a horizontal plane.
 
 Liquid Presstirc on Plane Surfaces. 
 
 39 
 
 For, take any two points, P, Q, in a horizontal plane, 
 and let their vertical distances below ylB be z and /. 
 Then by (/3), we have 
 
 ivz' + Pf^ = jvz + p^; 
 
 » » z — z^ 
 
 that is, all points in the same horizontal plane are at 
 the same depth below the surface AB — which proves AB 
 to be a horizontal plane. 
 
 It is usual to speak of the surface, AB, of contact 
 of the liquid wdth the atmosphere as ihe free surface of 
 the liquid. It is simply a surface at each point of which 
 the intensity of pressure is constant, the constant being 
 the atmospheric intensity. 
 
 The result at which we have arrived may be also stated 
 thus — all points in a heavy homogeneous liquid at which the 
 intensiiy of pressure is the same lie in a horizontal p)lane ; 
 and from this it follows that if a mass of water partly 
 enclosed by subten-anean rocks, &c., has access to the 
 atmosphere by any number of channels, the level of the 
 water will be the 
 same in all these 
 
 aj_|_j/; _H AJi-d. 
 
 4JLi/ 
 
 Fig. 15- 
 
 channels. It is 
 
 to be carefully 
 
 observed that z 
 
 in (a) and (/S) is 
 
 the depth of the 
 
 point P (Fig. 15) 
 
 below the free surface — not the distance, PD, of the point P 
 
 from the roof of the cavity in which the water is partly 
 
 confined. 
 
 We may here, if we please, consider the separate equili- 
 brium of a small vertical cylinder of the liquid terminating- 
 at i>, and we shall have simply the result (/3) in which,
 
 40 Hydrostatics and Elementary Hydrokinetics. 
 
 however, p^ would now mean the vertical downward com- 
 ponent of the pressure intensity of the roof of the cavity at 
 D on the water. But the result (3) holds for the intensity 
 of pressure at P if PII is put for r, where // is the foot of 
 the perpendicular from P on the plane of the free surfaces 
 ab, cd, ef of the water ; for, nowhere in the liquid will the 
 state of affairs be altered if we imagine the roof of the 
 cavity to be removed, and the space bl)c to be filled with 
 water up to the level he. In this way we shall have a 
 vertical cylinder, PH, unobstructed by the roof, and term- 
 inating on the free surface. 
 
 It is usual to illustrate the fact that all parts of the free 
 surface of a liquid lie in a horizontal plane by taking 
 a vessel, ABC, of any shape and fitting into it tubes 
 or funnels of various forms, and then pouring water in 
 through any one of these tubes, the visible result being 
 that the water stands at the same level in all the tubes. 
 This is, indeed, nothing more than the principle of separate 
 equilibrium (see end of Art. 3) ; for, these variously shaped 
 funnels may be supposed to have been surfaces traced out 
 in imagination in a large vessel of water whose free surface 
 was af\ these imagined surfaces being then replaced by 
 material tubes, and the outside liquid removed. The level 
 of the liquid in each tube would still be af. 
 
 13. Superposed 
 Liquids. If in a vessel, 
 AOB, Fig. 16, several 
 liquids be placed as 
 layers, one on top of 
 another, there being no 
 chemical combination 
 between them, the 
 common surface of each 
 pair of liquids is a horizontal plane. Let the specific
 
 Liquid Pressure on Plane Surfaces. 41 
 
 weights of the liquids be w^, Wg, W3, ... . The free sui-face, 
 AB, has been ah-eady proved to be a horizontal jilane 
 (Art. T 2) ; and the same process will prove CD, the surlace 
 of separation of n\ and w^, to be a horizontal plane. For, 
 in the licpiid ic, take any two points, Q, Q' (as in Fi^-. 14, 
 p. 37), in the same horizontal plane. Then by taking a 
 slender horizontal cylinder having QQ' for axis, we prove 
 that the intensity of pressure at Q = that at Q'. Now 
 taking a vertical cylinder Qtnn, at Q, considering its 
 separate ecpulibrium, we find that if p is the intensity 
 of pressure at Q, and Qm = x, nm = y, 
 
 Similarly if Q' m n' is the vertical line at Q\ and 
 
 m =. X , m n ■= ij , 
 
 p = w^x' + w^/. 
 
 Hence 
 
 w^{x-x) = w\{/-i/) (1) 
 
 But Q)i = Q!n\ i.e., x^-y = x'+y\ .'. x-x' ^y -y ; 
 so that unless x — x'—o and y —y = o, equation (1) will 
 give Wj = ?<'2, which is not the case, by hypothesis. 
 
 Hence we must have 
 
 Qm = Q^n/, and mn = m'n', 
 
 and since this holds for all points Q, Q' in the same 
 horizontal plane, all points, tn, m,.,. in the surface CD are 
 at the same height above the same horizontal plane ; 
 therefore CD is a horizontal plane. Similarly, by taking 
 two points in the same horizontal plane in the liquid W3, 
 we prove that EF is a horizontal plane. 
 
 If // J and Ji.^ are the thicknesses of the layers %\ and w.^ , 
 and if 11 is a point in iv.^ at a depth z below the surface, 
 EF, of w^, the intensity of pressure, p, at R is given by 
 the equation 77, /^\
 
 42 Hydrostatics and Elementary Hydrokinetics. 
 
 to which, if atmospheric (or other) pressure acts on the 
 uppermost surface AB, must be added /;q, the intensity of 
 this surface pressure, so that 
 
 2^ = p^ + w^ h^ + 10. Ji.^ + ti'^z (3) 
 
 Similarly for any number whatev^er of superposed layers. 
 
 Each layer of liquid, in fact, acts as an atmosphere, 
 producing- an intensity of pressure on the next layer below 
 it equal to 
 
 t^'h, (4) 
 
 where w is the specific weight of the layer and h its 
 thickness. 
 
 If the ^'s are measured in centimetres and the ?^'s in 
 grammes' weight per cubic centimetre, the above equations 
 express^ in grammes' weight per square cm. 
 
 The method of regarding any layer of liquid, even token 
 v/ there is only one liquid in question, as an atmosphere pro- 
 ducing an intensity of pressure given by (4) 07i the layer 
 on 2ohick it rests, this intensity being then transmitted tin- 
 altered to all points below (by Pascal's principle) is one 
 which we shall frequently employ in the sequel. 
 
 From the general principle [Statics, vol. i., Art. 121) 
 that, for stable equilibrium, any system of material particles 
 acted upon by gravity only must arrange themselves 
 into such a configuration that their centre of gravity 
 occupies the lowest position that it can possibly occupy, it 
 follows that in a system of superposed liquids of different 
 densities they must arrange themselves so that the density 
 of each liquid is greater than that of any one above it. 
 
 Again, if ABC, Fig. 17, represents a vertical section of a 
 vessel of any shape into which are poured two different 
 liquids, AB and BC, which do not mix, the system will 
 settle down into a position in which the centre of gravity 
 of the whole mass occupies the lowest position that it can
 
 Liquid Pressure on Plane Surfaces. 43 
 
 occupy, and the vertical heijni-hts, h, It, of the free surfaces, 
 A and 6', above the common surface, B, of the liquids will 
 be inversely propor- , 
 
 tional to the specific \ \ f ^^^—^^ 
 
 weio-hts of the li- ^ ^^ j k /~y ^ 
 
 quids. ^^ I '■^^^ 
 
 To see the latter, ^^s^ ^S^^^^^^ 
 
 we may take any ^^5^;^^:.;=:-=-^'^ 
 
 point (most con- -^'o- '7- 
 
 veniently a point in the common surface B) and equate 
 the intensity of pressure produced there by everything 
 at one side of the point to the intensity of pressure pro- 
 duced by everything at the opposite. Thus, let w and to' be 
 the specific weights of the liquids AB and BC, respectively ; 
 select a point, P, in the common surface B. Then if 
 h is the difference of level between P and A, the intensity 
 of pressure produced at P by the liquid AB and the over- 
 lying atmosphere at A is 
 
 wh +Pq. 
 
 Also, // being the difference of level of P and C, the 
 intensity of pressure at P produced by the right-hand 
 liquid and the atmosphere above C is 
 
 There is only one intensity of pressure at P ; hence these 
 
 must be equal ; 7/7/ 
 
 ^ . • . %o .11 — 70. h , (5) 
 
 which shows that the heights of the free surfaces above the 
 common surface, B, of the liquids are inversely as their 
 specific weights. 
 
 Thus, if AB is mercury and BC water, the surface C 
 will be 13-596 times as high above B as the surface A is. 
 
 As an example, let two liquids, AB, BC, Fig. 18, be 
 poured into a narrow circular tube held fixed in a vertical
 
 44 Hydrostatics and Elementary Hydrokinetics. 
 
 plane, the leng-ths of the arcs occupied by the liquids being 
 assigned ; it is required to find their positions of equi- 
 librium. 
 
 The figure of equilibrium will be 
 defined by the angle, Q, which the 
 radius. OB^ to the common surface 
 of the liquids makes with the verti- 
 cal, 01). 
 
 Let the angles, AOB, BOC, sub- 
 tended by the liquid threads at the 
 centre of the circle be a, a ; let 
 their specific weights be w, w\ re- 
 spectively ; and let r be the radius of the circle. 
 
 Equate the intensity of pressure produced at B by the 
 one liquid to that produced by the other. The difference 
 of level between B and ^ is r [cos ^ — cos (^ + a)}, and this 
 multiplied by w is the pressure intensity at B due to 
 the first liquid. The difference of level of B and C is 
 
 r {cos Q — cos {a — &). 
 Hence 
 
 w {cos^ — cos((? + a)} =?/ {cos 6) — COS (a'— 6)}, . (6) 
 
 a 
 
 w sm" 
 
 tan Q = ^ 
 
 • w sin- - 
 
 2 
 
 w' sin a + to sin a 
 
 (7) 
 
 The equation (6) is easily seen to express the fact that 
 the centre of gra\aty of the system of two liquid threads 
 has, in the position of equilibrium, the greatest vertical 
 depth below that any geometrical displacement of the 
 two liquid threads could give it. For, the centre of graxdty 
 of the thread AB lies on the radius bisecting AB and is at 
 
 . a 
 sm - 
 
 a distance ir from [^Statics, Art. i66); its ver-
 
 Liquid Pressure on Plane Surfaces. 45 
 
 . a 
 sm- 
 
 iical depth below is therefore 2r cos ( — \- O), and 
 
 ^ o ^2 ^ 
 
 the weig-ht of the liquid AB is proportional to raw. Hence 
 
 if is the depth of the centre of gravity, G, of the two 
 
 liquids, we have, by mass-moments, 
 
 (a?(/' + aw'^ - 
 
 = 110 sin - cos ( — h Q\-\-iw' sin — cos ( 6^ . . (8) 
 
 If we make Q such that 5 is a maximum, by equating to 
 zero the differential coefficient of the right side of (8), we 
 have the result (6). Of course it follows from the elements 
 of Statics that G is in the vertical radius 01). (The 
 reactions of the tube all pass through 0, &c.) 
 
 14. Pressure on a Plane Area. Let A BCD, Fig. 13, p. '^^^ 
 represent a plane area occupying any assigned position in 
 a heavy homogeneous liquid whose free surface is xOi/. 
 
 Then if w (pounds' weight per cubic inch, suppose) is 
 the specific weight of the liquid, and z (inches) is the 
 depth, FN, of any point below xO?/, the intensity of pres- 
 sure at P, due solely to the weight of the liquid, is tvz. 
 Hence (case 5, p. '^$) the resultant pressure on one side 
 
 of the area is , _ / x 
 
 A . z . w, (a) 
 
 where A (square inches) is the magnitude of the area, and z 
 is the depth, GQ, (inches), of its centre of area below the 
 free surface. 
 
 If on the free surface, xOy, there is intensity of pressure 
 (atmospheiic or other) of p^^ (pounds' weight per square 
 inch), this pressure will produce its resultant, Aj)q, at G, 
 and the total pressure on one side of the area is 
 
 A{zw-{p^) (^)
 
 46 Hydrostatics and Elementary Hydrokinetics. 
 
 As before pointed out (p. 36) the pressure (a) due to the 
 liquid does not act at G, but at some point lower down. 
 
 If a plane area, S, Fig*. 16, p. 40, occupies an assigned 
 position in a liquid on the surface of which are superposed 
 g-iven columns of other liquids, the resultant pressui-e on 
 the area is easily found. For, if z is the depth of the 
 centre, 6", of area heloio the surface EF, of the liquid w^, 
 the pressure of this liquid is Azw^, where A is the magni- 
 tude of the area. Also the column AB produces a resultant 
 pressure equal to Ak-j^ Wj , where h■^^ is the thickness of the 
 column ; the second column produces Ak^^ w^ ; so that the 
 total pressure on S is 
 
 A {Ji^ v\ + h.^ w.., + z ?/".) ; (y) 
 
 and similarly for any number of liquids, the resultant 
 pressure will be 
 
 A {7i^ w^ + //^ u-2 + /'3 w^+ ...+z w^), ...(b) 
 where z is the depth of G helow the surface of the liquid, w^, 
 in which the area lies. 
 
 Examples. 
 
 1. If a plane area, occupying any position in a liquid, is 
 lowered into the liquid by a motion of translation unaccom- 
 panied by rotation, show that the point of application of the 
 resultant pressure on one side of the area rises towards the 
 centre of area, G, the more the area is lowered. (See Fig. 20.) 
 
 Draw the horizontal plane CD, touching the boundary 
 of the area at its highest point, and consider the pressures 
 due separately to the layer between CD and the free surface, 
 AB, and to the mass of liquid below CD. Since there is no 
 change in the position of the area relative to the liquid below 
 CD, this latter pressure will always act with constant magni- 
 tude and point of application, /^ ; but the pressure of the super- 
 incumbent layer, always acting at G, increases in magnitude 
 with X, the distance between AB and CD. Hence of the two 
 parallel forces — at /^ and G — the first remains constant, while 
 the second continually increases ; their resultant, therefore, gets 
 nearer and nearer to G as CD is lowered.
 
 Liquid Pressure on Plane Surfaces. 47 
 
 2. A triangular area of 100 square feet has its vertices at 
 depths of 5, 10, and 18 feet below the surface of water; find 
 the resultant pressure on the area, the atmospheric intensity 
 being 15 pounds' weight per square inch. 
 
 Ans. 127-12 tons' weight. 
 
 3. Find the depth of a point in water at which the intensity 
 of the water pressure is equal to that due to the atmosphere. 
 
 Ans. About 34^ feet. 
 
 4. A rectangular vessel i foot high, one of whose faces is 6 
 inches broad, is filled to a height of 4 inches with mercury, the 
 remainder being filled with water ; find the total pressure 
 against this face, the atmospheric intensity being 15 pounds' 
 weight per square inch. 
 
 Ans. About 11172 pounds' weight. 
 
 5. Into a vessel containing mercury is poured water to a 
 height of 8 inches above the mercury. If a rectangular area 
 6 inches in height is immersed vertically so that part lies in the 
 mercury and part in the water, find the length of the area im- 
 mersed in the mercury when the fluid pressure on this portion 
 is equal to that on the portion in the water. 
 
 Ans. Nearly 1-39 inches. 
 
 6. Into a vessel containing a liquid of specific gravity p is 
 poured water to a height a. If a rectangular area of heiglit h 
 is immersed vertically, part in the water and part in the lower 
 liquid, find the length of the area in this liquid when the fluid 
 pressures on the two portions are equal. 
 
 V {2 a — hf + h{2 a—h) {j+p) — {2a — h) 
 A ns. • 
 
 7. A beaker containing liquid is placed in one pan of a 
 balance, and is counterpoised by a mass jdaced in the other jjan. 
 If a solid body sus^jended by a string held in the hand is then 
 immersed in the liquid, what will be the effect on the balance 1 
 
 If the string sustaining the solid is attached to the arm from 
 which the pan containing the beaker is suspended, and the 
 system counterpoised by a mass in tlie other pan, will the state 
 of the balance be the same whether the body is immersed in the 
 beaker or not ?
 
 48 Hydrostatics and Elementary Hydrokinetics. 
 
 BC, are at ^'^^^'^ T", ^Thr^^ BC inclined at the an.^le a 
 point B downward ^""f^'^^l^'^'^Xiy^^^ of specific wei,d.t w 
 to the horizon ; mto i5 ^M^o^^^d a iiq l ^^,^^^ ^ j. j 
 
 the leu-th of the column being , luto^o i 
 
 wl — w I tan a^ 
 of specific weight w is -;;7(74:iiir^ ' 
 
 the position of equilibrium. . the angles 
 
 . Tf 4-l.P m^ecific weights are w,, ^2- ''.''s' •■• » 
 
 i^is. If the specmc w ^ ^ | o^ liquid a^,a^,a„..., 
 
 ?o the free extremity of the liquid ..„ 
 
 tan , \ Po« f fj + a ^ + (^^'4 - ^'^s) cos{a^ + a^+a^ 
 
 - (^,,_^,)sina, + K ^3; V. ^ ^^^^^^ ^^^^^^ .^ 
 
 10. A rectangular area, L'^^^'\^,^^^^^^i,^, one side, L3f, 
 
 in the free surface of water ; 
 show how to divide the 
 area, by horizontal lines 
 into n strips on each ot 
 which the water pressure 
 shall be the same. 
 
 Let LM^ci, LR = h 
 
 w = specific weight ot 
 
 liquid, and measure the 
 
 Fig. 19- depths, x^,x^,x^, . . • of the 
 
 ,. • • 1 ^ 1 rm 1 m ..• each from the 
 
 successive lines of division, Z,m„ l,m,, l,m„ 
 
 surface. i ^ ^\ . 
 
 The- tl,e pressure on the rectangle Z», = - (p-ssure on LS) . 
 
 Lvi., = - ( 
 
 n 
 
 •' n 
 
 „ );
 
 Liquid Pressure on Plane Surfaces. 49 
 
 and so on. Thus, instead of calculating the pressures on the 
 
 separate strips, Lii\, L^m.,, L^vi^, ... and equating them to - 
 
 n 
 
 of the pressure on LS, we take successive rectangles each having 
 
 one side L3I in the free surfoce. Tliis is simpler. 
 
 h 
 Now the pi'essure on LS is ah . — .w; the pressure on Lm^ is 
 
 aajj . — .to; pressure on Lm^ is ax^ . — . w ; hence 
 
 03,^ = - /r, . • . X, = h 
 
 — J 
 
 ^ n ' ' ' \/ n 
 
 X.?' = -h"^, . • . x, = h/\ / -i 
 
 x,? = -h^, . • . X- =i h A. / -' 
 n 'V ** 
 
 11. A triangular ai-ea, ABC, has its vertex A in the surface 
 of water, its plane vertical, and its base BC horizontal ; divide 
 the area by horizontal lines into n strips on which the water 
 pressures shall be equal. 
 
 Ans. The depths of the successive lines of division are 
 
 h(^, h(^, h(^f,...h(^,.... 
 
 ^01'' ^n^ ^H'' ^H'' 
 
 12. A trapezium whose i^lane is vertical has one of the 
 parallel sides in the free surface of a liquid ; divide the area by 
 a horizontal line into two parts on which the liquid pressures 
 are equal. 
 
 Let a, h be the parallel sides, the former lying in the surface ; 
 let h ■= height of trapezium ; let c = h — a, and x = deptli of 
 the required line ; then x is given by the equation 
 
 2c{2x^ — h'^) + ■^ah[2X- — h-) — o (i) 
 
 The root of this equation which is relevant lies between 
 
 — and — , which, respectively, correspond to the cases of 
 
 2' 23 
 
 examples 7 and 8.
 
 o Hydrostatics and Elementary Hydrokinctics. 
 
 By putting - ■=k and 7 = 2/> ^^^^ depriving (i) of its 
 
 a , ,05 
 
 c 
 second term, we have 
 
 4^^ — 3^-^2; + P— 3^;— 2 = o. .... (2) 
 
 where 2 = y + | ^•. Now (2) can always be solved by either of 
 the well known results, 
 
 4 cos^0— 3 COS0 — cos 3 = o, .... (3) 
 
 or 4 cosh"' ^ — 3 cosh — cosh 3 ^ = o (4) 
 
 Thus, putting z in (2) equal to ^ cos 0, we have, to deter- 
 
 mine0, 2 + 3 ^-F , 
 
 cos 3^ = -^^ (5) 
 
 But if the numerator of (5) is greater than the denominator, 
 we must put z = P cosh Q, and Q is to be found from the equa- 
 
 cosh 3^ = -^^^^;^ (6) 
 
 In either case 3 is known from a table of circular or hyper- 
 bolic cosines, and thence z, &c. 
 
 If the horizontal line is to be drawn so that the pressure on 
 
 the upper trapezium = — of the pressure on the whole area, 
 
 the equation for x is 
 
 27100? ■\- -^naJi-ji^ ■=■ (3a-l- 2c) A', 
 
 which can be solved in exactly the same way. 
 
 When b ^ o, or the area is a triangle with its base in the 
 
 surface and vertex down, the values of z in (2) are o, + — ^ j 
 
 2 
 
 the first of which alone is relevant to the problem, since the latter 
 two give, respectively, a value of a; which is > h, and a negative 
 value of X — both of which are physically impossible. 
 
 13. A cube is filled with a liquid, and held with a diagonal 
 vertical ; find the pressures on one of tlie lower and one of the 
 upper faces. 
 
 2 I 
 
 Ans. — = W and -— : 17^, where W = the weight of the 
 
 , , V3 V 3 
 
 liquid in the cube.
 
 Liquid Pressure on Plane Surfaces. 51 
 
 14. A circular area is immersed in a liomogeueous liquid, a 
 tangent to the circle lying in the free surface, A being tlie 
 highest point of the circle : draw a chord, BC, of the circle 
 perpendicular to tlie diameter through .1 so that the pressure 
 on the trianfjle ABC shall be a maximum. 
 
 A ns. The distance of BG from J, is |^ of the diameter. 
 
 15. A triangular area, ABC, occupies any position in a liquid; 
 find a point, 0, in its area such that the liquid pressures on the 
 parts BOC, COA, and AOB shall be proportional to three given 
 numbers. 
 
 Let a, ^, y be the depths of A, B, C below the fi'ee surface ; 
 let the ratios of the pressures on the above areas, respectively, 
 to the pressure on the whole triangle ABC be p^, p.,, p^; let z 
 be the depth of 0, and put ic for z+a + jB + y; then x is deter- 
 mined from the cubic 
 
 _ei_ + _e?_ + _A_ = '' . . . (i) 
 
 03— a x — i3 x—y a + ^^i + y 
 
 Assuming a>^>y, the value of x in this equation which is 
 >a is the only one relevant, because the values which are 
 between a and /3 and between ^ and y give negative values of z. 
 
 The position of is completely defined by its areal co- 
 ordinates, i.e.. by the ratios of the areas BOC, COA, AOB to 
 the area ABC. If these ratios are I, m, n, respectively, the 
 
 equations ai-e , , ^ ^ , ^ 
 
 ^ l{z + ^ + y) = p,{a^^^y). .... (2) 
 
 and two similar, where z is la-\-in^-\-ny. When z is known 
 from (i), I is found from (2) ; &c. 
 
 16. A triangle has its base. BC, in the free surface of a liquid, 
 and its vertex. A, down; find a jjoint, 0, in its area such that 
 the pressures on BOC, COA, AOB shall be proportional to three 
 given numbers. 
 
 Ans. If the pressures on these areas are to the pressure on 
 ABC in the ratios Pi'. p^'- Ps> ^^^ ^ *=^ the depth of A, the point 
 is the intersection of a horizontal line at a depth h \/^, with 
 a line drawn from A to a point, P, in BC such that 
 
 V. 2
 
 52 Hydrostatics and Elementary Hydrokinetics. 
 
 15. Centre of Pressure. Hitherto we have been occu- 
 pied with the calculation of the magnitude of the resultant 
 pressure on one side of a plane area. We have now 
 to consider the point of the area at which this resultant 
 pressure acts. Except in the case in which the plane of 
 the area is horizontal, this point — which is called the centre 
 of j^i'eS'Wre — is alwa3^s lower in the area than G, the cen- 
 troid, or ' centre of gravity,' of the area. 
 
 The position of the centre of pressure on a given area 
 varies with the position (depth, orientation, &c.) of the 
 area in the fluid ; and before determining its position in a 
 few simple and frequently occurring cases, we shall lay down 
 a general principle, founded on the remark near the end of 
 Art. 13, which is often of great assistance in calculation. 
 When a plane area — or, indeed, any surface whatever — 
 occupies any position in a liquid, we may draw any hori- 
 zontal plane whatever in the liquid and consider the column 
 of liquid above this plane as playing the part of an atmo- 
 sphere — i. e., as producing at all points below the plane 
 a constant intensity of pressure, which is transmitted in 
 virtue of Pascal's Principle. The most convenient hori- 
 zontal plane for this purpose is one through the highest 
 point of the given area. 
 
 y. Thus, for example, if nrm, 
 
 ^V— = -7 ^ Fig. 20, is any plane area 
 
 d^Lzz :z^^:z:zp^=~=^^--^y ^ whosc plauc is vertical in a 
 
 \-— ^ ^y ^-^ ^y liquid, and we wish to find the 
 
 VzS^:==:^z^ magnitude and point of action 
 
 ^ ^^zs ^ of the resultant i)ressure on 
 
 Fig. 20. one side of this area, we may 
 
 draw a horizontal plane, CD, touching the contour of the 
 
 area at its highest point, n, and then consider separately 
 
 the pressures due to the layer of licpiid between AB and CD 
 
 and to the body of liquid below CD.
 
 Liquid Pressure on Plane Surfaces. 53 
 
 With reg-ard to the layer ACDB, if x is its thickness, we 
 know that it produces at all points on CD and at all points 
 below (Art. 13) an intensity of pressure equal to 
 
 w . x\ 
 
 and since this pressure is uniformly distributed over the 
 area 7ir7n, its resultant is (case i, Art. 11). 
 
 Awx acting at G^, (5) 
 
 where G is the centre of area of urm and A the magnitude 
 of the area. 
 
 Hence, if we knew the magnitude and point, 1q, of 
 application of the pressure of the liquid below CD, we 
 should have the magnitude and point, /, of apjilication of 
 the pressure of the whole liquid below AB on the area by 
 a simple composition of two parallel forces acting at G and 
 Iq. This we shall presently illustrate by a few simple 
 examples. 
 
 Thus we obtain the fol- a § b 
 
 lowing construction for the /| 
 
 centre of pressure, /,on a plane / ■ 
 
 area (Fig. 21) occupying any 
 
 position in a liquid : through 
 
 the highest point, n, on the 
 
 contour of the figure, draw -pj^ 21 
 
 a horizontal plane, CI), the 
 
 free surface of the liquid being AB ; from the centroid, 6", 
 
 (or ' centre of gravity ') of the figure draw a vertical line 
 
 meeting these planes in P and Q ', suppose Iq to be the 
 
 (known) position of the centre of pressure if the surface of 
 
 the liquid w^ere CD ; draw QIq , and from P draw PI 
 
 parallel to QIq, meeting GIq in /. This point / is the 
 
 required centre of pressure on the area. We shall presently 
 
 proceed to illustrate this method by some simple examples.
 
 54 Hydrostatics and Elcmcntmy Hydrokinetics. 
 
 Special Cases of Centre of Pressure. 
 
 (i) To find the position of the centre of pressure on a 
 plane parallelogram, whose plane is vertical, with one side in 
 the free surface. 
 
 Let ABCD, Fig. 32, be the parallelogram. Let the 
 area be divided into an indefinitely great number of in- 
 definitely narrow strips, of 
 which mnsr is the type, and 
 let E and F be the middle 
 points of the sides AB and 
 CD. Then the middle point 
 of every strip lies on the line 
 JEF. Also if w is the depth 
 of the strip ms below AB, 
 
 Fig. 22. 
 
 and w the specific weight of the liquid, the intensity of 
 pressure is the same at all points in the strip and (Art. 12) 
 equal to wx, and the resultant pressure on the strip acts at 
 its middle point, i. e., at the intersection,/, of ?//s with FF. 
 Hence the resultant pressure on the whole parallelogram 
 acts at some point on FF. Also, since the areas of the 
 strips are all equal, the series of joressures on them are 
 simply proportional to their distances from AF ; therefore 
 (case 2, p. 34) the point of application of the resultant 
 pressure is | of FF from F. Denote this point by T. Then 
 
 FT = iFF (a) 
 
 If // is the height of the parallelogram, and p the 
 perpendicular distance of the centre of pressure, T, from 
 the surface ^ _ 2 7, /^/x 
 
 If the plane of the parallelogram is not vertical, the 
 same point, T, will still be the centre of pressure. For, if 
 the area is inclined to the vertical at any angle, 6, and x is 
 the perpendicular distance of/" from AF, we have 
 
 X =fF . cos 6 ;
 
 Liquid Pressure on Plane Surfaces. 55 
 
 and as 6 is the same for all the strips, the pressures 
 on them will still be proportional to their distances /!£", &c. 
 
 (2) To Jiud the position of the centre of pressure on a 
 plane triangle having one side in the free surface, and vertex 
 doivn. 
 
 Let ABD be the triang-le. Divide the area, as before, 
 into an indofinitel}^ q-reat number of strips, of which ts is 
 the type. Let x be the perpendicular distance of this strip 
 from the base AB. Now compare this with another strip, 
 t' s\ whose perpendicular distance from I) is also x. Let h 
 be the heig-ht of the triangle, a — AB, k = the indefinitely 
 
 ft PC 
 
 small breadth of each strip. Then tn = —, — . a ; so that 
 (Art. 14) the pressure on this strip is 
 
 — a; (/; — «;) ?^ (i) 
 
 But this is also the pressure on the second strip, t' s' . 
 
 For, fn' = J a, and the depth of ('-';/ h h — x ; therefore (1) 
 
 is the pressure on this strip. Since each strip is pressed at 
 its middle point, and since all the middle points lie on EB, 
 the resultant acts at some point on EB. Also we have just 
 seen that the pressures along- EB are equal at two points 
 such that the distance of one from E = the distance of the 
 other from D. Hence (case 4, p. '^^) the resultant pressure 
 acts at the middle point, M, of EB ; that is, 
 
 EM=\EB (/3) 
 
 If ji; is the perpendicular distance of M from the surface 
 
 I>=\1> O') 
 
 Also, whatever be the angle of inclination of the plane 
 of the triangle to the vertical, the same point, J/, is 
 the centre of pressure.
 
 56 Hydrostatics and Elementary Hydrokinetics. 
 
 (3) To find the position of the centre of pressure on a 
 plane triangle leaving a vertex in the free surface and its hase 
 horizontal. 
 
 Let ACB be the triangle. Then a combination of the 
 two results just proved will enable us to find Q, the centre 
 of pressure. For, complete the parallelogram ABDC. Then 
 the pressure on the parallelogram is the resultant of the 
 pressures on the two triangles. But since all the narrow 
 horizontal strips into which the given triangle ACD can be 
 divided have their middle points (centres of pressure 
 for each of them) ranged along AF, the resultant pressure 
 on the triangle acts somewhere on AF. Join the point, M, 
 of application of one of the two parallel forces to the point, 
 T, of application of the resultant, and produce MT to meet 
 AF in Q. Then Q is the point of application of the 
 pressure on ACD. 
 
 OF FT 
 Now |l7=f^— '' ••• QF=IE3I, 
 
 • ■• AQ=IAF (y) 
 
 If h is the height of the triangle, and p the perpendicular 
 distance of Q from the//'^e surface, 
 
 p = T^^\ • {y) 
 
 and, as before, the point, Q, of application of the resultant 
 pressure is the same wdiatever be the inclination of the 
 plane of the triangle to the vertical. 
 
 The result might have been deduced directly from case 3, 
 p. 34. For, if the area be divided into strips, we have 
 
 CG 
 
 mt = - fl, where a = CD, and x is the perpendicular from 
 
 A on mt. Hence the pressure on the strip mt is j wa^, so 
 that the pressures along AF are j^roportional to tbe squares
 
 Liquid Pressure on Plane Surfaces. 57 
 
 of the distances of their points of application from A. The 
 resultant, therefore, acts at a point f of the way down 
 along- AF. 
 
 These three simple cases, combined with the principle 
 (see p. 42) of regarding- any column of liquid as an atmo- 
 sphere, producing its resultant pressure at the centre of area, 
 will suffice for calculations concerning the centres of pres- 
 sure of many plane polygonal and other ligm-es occupying 
 any positions in a liquid. 
 
 Thus, let the area be nrm, Fig. 20, p. 52 ; and suppose 
 that, if all the liquid above the horizontal plane CB is 
 removed, we know the depth, p^ , of the centre of pressure, 
 /(,, of the remaining liquid below CD. Then, if Zq is 
 the depth of G below CD, A = magnitude of the area, 
 w = specific weight of the liquid, the pressure, ij, at /q is 
 
 AzqW. 
 
 Let X = the thickness of the column AD. Then the 
 pressure due to this column = A xw, and it acts at G. The 
 resultant pressure (at 1) is of course the sum of these forces ; 
 and if p is the depth of 1 below AB, we have, by the theorem 
 of moments, 
 
 p = (p^+x)^^ +«;, (8) 
 
 Zq + X 
 
 the point /dividing ]^,G so that 
 
 ■^ = - (0 
 
 IG z, • ' ^^ 
 
 (4) To fncl the jiosition of the centre of pressure on a 
 plane triangle occupying any position in a liquid. 
 
 Let ABC, Fig. 23, be the triangle ; let A be its area, 
 and a, /3, y the depths of its vertices below the free surface 
 of the liquid. 
 
 We shall calculate the distance of 7, the centre of pressure,
 
 58 Hydrostatics mid Elementary Hydrokinetics. 
 
 Fig. 23. 
 
 from a side BC of the triang-le. Let 7; be the perpendicular 
 
 from A on BC 
 
 If through C we draw a horizontal plane, the column of 
 
 liquid above this plane pro- 
 duces a pressure equal to 
 Ayw 2A1 the centre of area of 
 the triang-le ABC, i.e., at 
 the point whose distance 
 from BC is ^p. Hence 
 
 represent this force and the 
 distance of its point of application from BC. 
 
 Consider now the effect of the liquid below this hori- 
 zontal plane through C. From B draw the horizontal plane 
 cutting the area ABC in the line B7i, and consider separ- 
 ately the pressure of this liquid on the aveas BnC and 
 BnA. 
 
 [a) If, for convenience, we let x and y be the perpen- 
 diculars from A and B on the horizontal plane through C, 
 
 the area BnC =■ A—, and the perpendicular from 11 on BC 
 
 Uu 
 
 is /? — . Kow the pressure of the liquid below C on BnC 
 
 acts at a point three-fourths of the way down Cm, where m 
 is the middle point of Bn, and the distance of this point 
 
 from BC is \p'—' Hence the j)ressure on BnC and 
 
 the distance of its point of application from BC are re- 
 presented by 
 
 (h^'~''W-;) 
 
 X 
 
 8 
 
 X' 
 
 [h] Consider now the pressure on BnA due to the 
 liquid below C. Treat the column above Bn as an atmo-
 
 Liquid Pressure on Plane Surfaces. 59 
 
 sphere. It produces pressure at the centre of ai'ea o{ BnA ; 
 and we easily find that this force and the distance of 
 its point of application from BC are represented hj 
 
 There is, finally, the pressure on BnA due to the liquid 
 below Bn. This acts at the middle point of Am ; so that 
 for this force we have 
 
 The sum of these four forces is, of course, 
 
 ^- A{a + l3 + y)7v; 
 
 and if the perpendicular from / on BC is denoted by p, we 
 have, by moments (Art. 10) with reference to BC, 
 
 ^{a + i3 + y}p 
 
 = TV;'(2a + /3 + y), 
 since x = a — y, ?/ = j3 — y. Hence 
 
 , 2a + ft + y 
 
 ^ '^ a + li + y 
 
 with similar values of <] and 7; the distances of / from AC 
 and AB. 
 
 Now this shows that / is the centre of gravity of three 
 particles whose masses are proportional to 
 
 2a + 13 + y, a+2f3 + y, a + ,3 + 2y, 
 
 placed at A, B, C, respectively. Or, at A we can imag-ine
 
 6o Hydrostatics and Elementary Hydrokinetics. 
 
 two particles (a, s) ; at B two particles (/?, s) ; at C two 
 particles (y, «), where 5 = a + /3 + y. 
 
 Hence the following simple construction for /: — 
 Find the centre of gravity, G\ of three particles whose 
 masses are proportional to a, (3, y, placed at A, B, C respec- 
 tively ; join G' to G, the centre of area of the triangle ABC, 
 and on GG' take ^-,7- i f-,p' 
 
 It is also easily seen that / is the centre of gravity 
 of three particles whose masses are proportional to a, /3, y, 
 placed at the middle points of the bisectors of the sides BC, 
 CA, AB, drawn from the opposite vertices. 
 
 And it is evident that we have also the following con- 
 struction for I : at the middle points of the sides of the 
 triangle imagine particles whose masses are proportional to 
 the depths of these points ; then their centre of gravity is 
 the centre of pressure of the triangle. 
 
 The preceding method of finding / has been given for 
 the purpose of illustrating the principle (Art. 15) that 
 any column of liquid above an area may be treated like 
 an atmosphere. For the following much more elegant 
 investigation the author is indebted to Mr. W. S. M^'Cay, 
 F. T. C. D. 
 
 Take any point, 0, in the area ABC ; let the perpendi- 
 cular from on the surface of the liquid be C the perpen- 
 diculars from on the sides BC, CA, AB being x, y, z \ 
 also denote the areas of the triangles BOC, CO A, AOB 
 by Aj, A^,, A3, respectively. Then evidently the line CO, 
 if produced, divides the side AB into segments proportional 
 to A^ and A 2 inversely, while BO divides AC inversely as 
 Aj to A3 ; hence is the centre of mean position of the 
 points A, B, C for the system of multij^les Aj, A^, A3. If, 
 then, A is the area ABC, \ve have 
 
 A . C= Aj . a + A^ . /3-|~A3 . y ; . . . (1)
 
 Liquid Pressure on Plane Surfaces. 6i 
 
 or, if ;;, q, r avo the ijcrjocndicalars from A, B, C on the 
 oi)posite sides, a 3 y 
 
 Now the intensity of pressure at \% C. . w, so that this 
 can be considered as a superposition of intensities 
 
 a fi y 
 
 - ivx, — wy, - wz, 
 
 i. e., of the intensities of pressure which would be produced 
 if the sides BC, CA, and AB were placed in the surfaces of 
 
 liquids whose specific wcig-hts are - w, - w, and - w, re- 
 spectively. Now the first of these liquids would produce a 
 resultant pressure equal to 
 
 pa . , * 
 A X — X — tv, I.e., ^Aaio, 
 
 3 P 
 acting at the middle point of the bisector of BC drawn 
 from A (p. 5S)- Similarly for the other two liquids ; so 
 that the actual pressure on the triang-le ABC is the result- 
 ant. of forces x Aa.-, ^ A/3.., J Ay.., 
 
 acting- at the middle points of the bisectors ; and its point 
 of application is, of course, the same as the centre of g-ravity 
 of particles whose masses are proportional to a, (3, y placed 
 at these middle points ; and this is the second of the con- 
 structions which we have given above for the point /. 
 
 For simplicity and elegance this proof of our construction 
 leaves nothing to be desired.
 
 62 Hydrostatics and Elancntary Hydrokinetics. 
 
 Examples. 
 
 1. A trianrjular area whose height is 12 feet has its base 
 hoiizoiital and vertex uppermost in water ; find the depth to 
 which its vertex must be sunk so that the difference of level 
 between the centre of area and the centre of pressure shall be 
 8 inches. 
 
 Ans. Four feet. 
 
 2. Find the depth of the centre of pressure on a trapezium 
 having one of the parallel sides in the surface of the liquid. 
 
 Ans. If the side a is in the surface and h below, h being 
 the height of the trapezium, the depth of the centre of pressure 
 
 2 2 6 + (t' 
 
 and it lies, of course, on the line joining the middle points of a, 
 and h. 
 
 3. In the last example find the position of the centre of 
 pressure by geometrical construction. 
 
 (Break up the area into a parallelogram and a triangle, or 
 two triangles.) 
 
 4. The plane of a trapezium being vertical, and its parallel 
 sides horizontal, to what depth must the upper side be suidv in 
 a liquid so that the centre of pressure shall be at the middle 
 point of the area ? 
 
 Ans. The parallel sides being a and 6, of which the upper, 
 
 a, must be the greater, the required depth = r /i, where h = 
 
 height of the trapezium. ^~ 
 
 5. Show how to find, by geometrical construction, the position 
 of the centre of pressure on a plane quadrilateral occupying any 
 position. 
 
 6. Find the depth of the centre of pressure of a plane quadri- 
 lateral in terms of the depths of its vertices and the depth of 
 the point of intersection of the diagonals. 
 
 Ans. If a, /3, y, 5, ^ are the dej)th of the vertices and inter-
 
 Liquid Pressure on Plane Surfaces. 63 
 
 section of diatronals, and if .s = a + /^ + y + S, and ^a^ denotes 
 (J? + '^' + y- + 6", the depth is 
 
 4 «-C 
 
 7. A rectangular area of height h is immersed vertically in a 
 liquid with one side in the surface; show how to draw a hoii- 
 zontal line across the area so that the centres of pressure of the 
 parts of the area ahove and below this line of division shall be 
 equally distant from it. 
 
 Ans. The line of division must be drawn at a depth 
 
 2 
 
 8. Supposing a rectangular vessel whose base is horizontal to be 
 divided into two water-tight compartments by means of a rigid 
 diaphragm moveable round a horizontal axis lying in the base of 
 the vessel ; if water is poured into the compartments to different 
 heights, find the horizontal force which, apj^lied to the middle 
 point of the upper edge of the diaphragm, will keep this dia- 
 phragm vertical, and iind the pressure on the axis. 
 
 Ans. Let a be the length of the axis, c the height of the 
 vessel, A and K the heights of the water in the compartments 
 
 {h>h') ; then the required force is — (]i^—h'^) i«,and the pressure 
 on the axis is ^ a {Tr — h'') w — —- {¥ — h'^) w. 
 
 16. Lines of Resistance. Supposins: Fig-. 24 to re- 
 present a vertical transverse section, A J^CD, of an embank- 
 ment which is pressed by water on the side AB (assumed 
 vertical), if we take any horizontal section, liQ, of the 
 embankment and consider the equilibrium of the portion, 
 RQAD, above this section, we see that it is acted upon by 
 its weig-ht and also by the water pressure which is a 
 horizontal force acting at a point two-thirds of the way 
 down AQ. Taking- the resultant of these two forces, its 
 line of action,7jA^, intersects the section llQina point A'^, the
 
 64 Hydrostatics and Elementary Hydrokinetics. 
 
 calculation of the position of which is of great importance 
 in the construction of reservoirs. 
 
 As the section UQ, varies in position, the point X 
 describes a curv^e which is called a line of resistance. For 
 the simple case in which the embankment is formed of 
 homoo-eneous material and the transverse vertical section is 
 a trapezium, we proceed to g-ive a simple rule for tracing 
 
 .f>' 
 
 /! 
 
 / 
 
 / ^'o 
 
 I 
 
 / 
 
 ' '/! 
 / //! 
 
 / 
 
 ! /n/ r 
 
 "~--.D' 
 
 
 ■■-- — > '"-^^ 
 
 "'---- ^-^ 
 
 ' ' / 1 
 
 " — ~--, ^->. 
 
 fh-/ g/ 
 
 " — --*_ '^'^^ 
 
 ~~'~-~ "'*■.» 
 
 
 r?L- J :::r?^ 
 
 
 *' / '// 
 
 
 / / /// 
 
 
 ' / '' // 
 
 
 r/ /'X 
 
 Q 
 
 
 / // 
 
 
 / // 
 
 
 / r" 
 
 
 
 
 / J 
 
 
 cL 1^ 
 
 B 
 
 Fig. 24. 
 
 the positions of the point X for any number of horizontal 
 sections. 
 
 To calculate the forces acting on the portion EQAB, 
 produce Jil) and QA to meet in 0, and consider its weight 
 as the weight, W, of JIQO acting through the centre 
 of gravity, G, of this triangle accompanied by an upward 
 vertical force, //'^q, the weight of DAO acting at //, the 
 centre of gravity of the triangle DAO. 
 
 Consider also the pressure against J Q as the pressure, P, 
 of water against OQ, the level of the water reaching to
 
 Liquid Pressure on Plane Surfaces. 6^ 
 
 — this force actinq- two-thirds of the way down OQ, 
 and therefore throug-h G — aeeomimniod by the reversed 
 pressure, 7^^, of the water ag'ainst OA, this force acting- 
 tliroiio-h f) ; and also the reversed water pressure ag-ainst 
 A(l due to a constant head OA, this force acting- towards 
 the rig-ht of the figure throug-h m, the middle point 
 oiAq. 
 
 Let OQ = 9/, 0A= y„, 7v = specific weig-ht of the w^ater, 
 7v'z= specific weig-ht of embankment, m = tan BOA, I = 
 leng-th (perpendicular to the plane of the paper) of the 
 embankment. 
 
 Then 
 
 2 
 
 • 
 
 /r= \mlwy, P =1 l?af, JF^ = hmlioy^^, P^= I hoy, 
 
 If R is the resultant of W and P, R^ the resultant of W^^ 
 and i^, we have 
 
 
 — 
 
 I 
 
 2 
 I 
 
 
 
 
 '•'Jo 
 
 acting 
 ', acting 
 
 at 
 
 ■ at 
 
 
 R 
 
 \^W^ 
 
 + m^ 
 
 to"', 
 
 G, 
 
 ^0 
 
 Vw- 
 
 '■ + 711^ 
 
 w'2 
 
 a. 
 
 and R is parallel to R^ , each making with the horizon an 
 
 r 
 
 angle whose tangent is — tan 1)0 A. To construct this di- 
 
 rection, taking DA to represent w, produce BA to B' so that 
 AB' represents w' ; draw OB' ; then R is perpendicular to 
 0B\ and is therefore of constant direction for all sections 
 RQ. 
 
 The resultant of the two unlike parallel forces R ai G and 
 Pq at ff is part of the force acting on the portion RQAB. 
 Suppose this resultant to act at the point i on the line gG 
 (which, of course, passes through and through the middle 
 points, i, 71, of AB and BC). 
 
 F
 
 66 Hydrostatics and Elcuicntary Hydrokinctics. 
 Then 
 
 tg~^ y^ ~^G0^ ' 
 
 '"• Wo ~ Gg X GO' ' 
 
 gO being produced to 0' so that^O = OC/. Hence 
 
 tG.GO'=gO', (a) 
 
 and this shows that t can be found by drawing- a line 
 J-y" parallel to 0)/. at a distance equal to g 0, erecting a per- 
 pendicular at G to On meeting j^' in/", and then drawing 
 ft perpendicular to fO\ Hence Ii — L\ acts in the line tr 
 which is perpendicular to OlJ\ 
 
 With this force must finally be coupled the horizontal 
 force Iw . OA . AQ acting at m in the sense mil. Denote 
 this force by H. Now B-H^: 11= OH : AO ; for 
 
 = I hv sec /3 . (y- — ^o")' where ^ = L A OD', 
 = Iw . AQ. Om sec ,3 
 = lw.AQ.OIL 
 
 Hence in the triangle AH the sides AO and OH are 
 perpendicular and proportional to the forces H and B — R^; 
 therefoi'e the resultant, I] of these forces is perpendicular 
 and proportional to the side AH; and therefore if we 
 produce the lines ;/ and Ht/i to meet in j), the line of action 
 of -t] the resultant force to which the portion IxQAl) 
 of the embankment is sul)ject, is the perpendicular from 
 ]) on AH. 
 
 Moreover F = Iw .AQ . AH, and therefore this force is 
 equivalent to the weight of a column of v-ater having AQ for 
 a horizontal base and having a height AH. 
 
 As the section RQ varies, the locus of X — i. e. the line of
 
 Ljqui'd Pressure on Plane Surfaces. 67 
 
 resistance — is a liy])erljola, as is easily proved thus. Take 
 the horizontal line thron<j;"h as axis of a?, and OB as axis 
 of _y, and let QA = x. Now express the fact that the sum 
 of the moments of the water pressure ag-ainst ACl (acting 
 two-thirds of the way down AC^ and the weight of the 
 portion JiQAB of the embankment about A' is zero. 
 
 The. latter force may be resolved into the weig-ht o^ JRQO 
 acting- downwards at G and the (negative) weight of 
 DAO acting upwards at ff. Our equation is then 
 
 1 Itv {// -?f^f - \ Iw'wf {x-\, VI, ij) + \ homy^- {x-\ wy^) = o. 
 
 The left-hand side contains^— ^o as a factor. Expelling 
 this, and denoting tan ^ by t, 
 
 is the equation of the locus, which is therefore a hyperbola. 
 This hyperbola passes through i, the middle point of 
 AD, and it has for one asymptote the line y -Vy^ = o, i. e. 
 a horizontal line at a distance OA above 0. The equation 
 of the other asymptote is 
 
 ^fx-(i+mt)y + 2yo = o,. . . . (i) 
 
 and this line can be easily drawn by means of its intercepts 
 
 on the axes of x and y. 
 
 V AO'^ 
 The intercej)t on the axis of x is — -y, or — r-,, so 
 
 that if (Fig. 25) OL is drawn per})endicular to OL', this 
 intercept is —AL; therefore if J/ is drawn parallel to OL 
 meeting Ox in /, the asymptote passes through /. The 
 
 intercept on the axis of y is -=^ — - -- ; but 
 
 I OA^ 
 
 i+mt OA' + l/A.AD' 
 now if a semicircle is described on DlJ' as diameter, it will 
 
 F 1
 
 68 Hydrostatics and Elementary Hydrokinctics. 
 
 cut AO at h so that L'A . AD = AP, and if AK is taken 
 
 ^ ^^'' I OA- 
 
 = ~^-^ = cos- A OK ; 
 
 j+jid OK- 
 hence by drawing As perpendicular to OK and sk perpen- 
 
 iP 
 
 ■/T-v: 
 
 ,' i~N 
 
 ^-J) 
 
 Fig. 25. 
 
 dicular to OA, the intercept on OB is 3 Ok, or OT; the 
 asymptote is therefore the line TJ. The point Tmay also 
 be found by producing OK through A' to S, so that OS =
 
 Liquid Pressure on Plane Snr/nces. 
 
 69 
 
 3 Os, and drawing" SV horizontal. Taking- Uie vertical 
 length OT = OA, the other asynii)tote is the horizontal 
 line TK, meeting- f'J in J'J, which is the centre of the 
 hyperbola. 
 
 Hence we have Loth asymptotes and one point, /, of the 
 curve, from which data the curve is easily constructed 
 
 T 
 
 E 
 
 7" 
 
 -}K. 
 
 ::iv,D' 
 
 H"- 
 
 Fig. 26. 
 
 by means of the property that if any line cuts the hyperbola 
 and the asymptotes the intercepts between the curve and 
 the asymi)totes arc equal. The relevant portion of the 
 curve is represented by the full line passing- through i. 
 
 If the face AB of the embankment is not vertical, 
 the line of resistance is still a hyperbola.
 
 70 Hydrostatics and Elementary Hydrokinctics. 
 
 As before, take AB to represent w, and Alf to represent 
 w ; from draw Or perpendicular to AD ; take Or as 
 axis of y, and the horizontal line Ox as axis of x; let 
 Or = t/Q, m = tan DOr, n = tan ylOr, ^^ = tan D'Or; then, 
 hy the method of moments before used, the equation of the 
 locus (deprived of the factor _y—.y„) is 
 
 Sid {iy + {t — 2 7i) t/q} — { [m — n)t— run — ?r + i } x 
 
 (/+yo5'+yo^)-3 (^^'- 1)^0^-3'^ Vo' = o- • (y) 
 
 The curve passes, as before, throug-h the point i ; the 
 equation of one asymptote, TE^ found by equating to 
 zero the coefficient of the highest power of x, is 
 
 t — 2n r . 
 
 y = ^••^0' • (2) 
 
 so that the asymptote is found by producing rA through A 
 to A' so that rA = AA', then drawing a parallel A'O' to 
 I)'0, and measuring 07' = 00'. 
 
 The equation of a line through parallel to the tangent 
 to the curve at i is 
 
 4{f~7/)x = {{m-n)f-2)im]^. . . . (3) 
 
 In any actual case all the constants, w, n, t, y^ , will be 
 given, and therefore the lines (2) and (3) can be accurately 
 drawn. A construction, independent of the numerical 
 values of the constants, can be given for the line (2), 
 and therefore for the tangent at i. Let a circle round the 
 triangle D'Oi cut Or in k\ and let a circle round AOB cut 
 Or in k ; then. Ok' being supposed > Ok^ measure off 
 a length kk' along Ox towards the left-hand, and along Or 
 measure off a length 1 Alf \ the diagonal (throuy-h O) of 
 the rectangle determined by these lengths is parallel to the 
 tangent at /. 
 
 A point on the other asymptote can now be easily found ; 
 for, let the tangent at i cut the asymptote TE in c ; then
 
 Liquid Pressure on Plane Surfaces. 71 
 
 take, iiloni;- this tang-ent, ic = ic, and we have the point c 
 on the second asymptote. 
 
 The direction of this asymptote is that of the line 
 
 '^tx—{{m — n)t — mn — n--\-i]^ = o,. . . (4) 
 
 as is obvious from the terms of the second degree in (y). 
 Hence wlien the constants arc numerically assig-ned, the 
 direction of this asymptote is easily constructed, and there- 
 fore (since it passes through e') the asymptote itself can be 
 drawn. 
 
 The circles above described may be utilised for drawing 
 this asymptote. For, if a perpendicular to OA at A meets 
 Or in I, we see that (4) is equivalent to 
 
 X Jo + ^ ^^' — '''^ ~ ^"^ 
 y~ 3 • ^•^' 
 
 so that if we measure the length i (^o + ^ rk'—rk — rl) from 
 along Or and the length yi>' along xO produced through 
 0, and draw the diagonal, through 0, of the rectangle 
 determined by these lines, the required asymptote is per- 
 pendicular to this diagonal. 
 
 The construction of the curve then proceeds exactly 
 as in the first case. 
 
 The resultant force to which any portion, RQAD, of the 
 embankment, cut off by a horizontal plane KQ is subject is 
 found by exactly the same construction as before — viz. 
 bisect AQ in m; draw 7/iJI horizontal, to meet OJD' in If; 
 then the resultant force is perpendicular to AH nnd is equal 
 to the weight of a column of water having- AH for height 
 and for base the vertical projection o? AQ. 
 
 If we adopt the method of fictitiously completing the 
 embankment and raising the level of the water to 0, as in 
 the first case, we shall have the forces Ji and ^q acting at 
 points G' and / which both lie on a fixed line passing 
 through ; /is obtained by taking the intersection of the
 
 72 Hydrostatics and Elcmcntmy Hydrokindics. 
 
 vertical throug-h g with a perpendicular to OA through 
 a point two-thirds of OA from 0. The resultant, R — Rq, 
 of R and -^0 ^^^'^ ^^ ^ point, t\ on the line Og'G' which is 
 found exactly as t was before found. The forces R and R^ are 
 each perpendicular to OJj' ; and, omitting* / for simplicity, 
 
 R =z \ v:ip- sec /3, Rq = \ loy^ sec /3, where /3 = //• Oh'. 
 
 Hence the first method of finding the line of resistance, 
 by tracing out the locus of X (Fig". 24), applies here also. 
 
 Let us now consider the case in which a section of the 
 embankment consists of two trapeziums, ABCl) and BEFC, 
 Fig. 27, the level of the water being I) A. 
 
 ^Ye may suppose IE to represent any horizontal section 
 across the second trapezium, the distance between EF and 
 BC being y ; and we shall calculate the resultant force, F.,, 
 acting on BEFC by first supposing CB to be the level of 
 the water, and then taking account of the weight of ABCB 
 and the effect of the column of water between A and B. 
 
 If CB is the water level and everything above is neg- 
 lected, the line of resistance through BEFC is a hyperbola 
 starting from the middle point of BC, and the point h in 
 which the resultant force, /a' on BEFC cuts EF is easily 
 found. This force f^ is found in line of action as before 
 explained — that is, by producing EB and FC to meet in (/, 
 taking BC: BC = w : w, and drawing a horizontal line, 
 nH\ through the middle point, n, of BE to meet CfC in 
 
 ^'' ^^^^ f, = y-u • BH' (5) 
 
 Now the efiect of the portion ABCB and the water 
 column between A and B is twofold : firstly, the water 
 column produces on BE a pressure, 7;, acting normally at n, 
 
 •i^^ p^n-h^.BE, (6) 
 
 or =wy . BI, (7) 
 
 where 7i^ is the height of the upper trapezium and / is the
 
 Liquid Pressure on Plane Surfaces. 
 
 73 
 
 \H 
 
 ,--"/o 
 
 
 Fig. 27.
 
 74 Hydrostatics and Elementary Hydrokinetics. 
 
 point in which EB cuts AI) ; secondl}', the upper structure 
 and \vatt'r cohimn will produce a force, 7'"^ , acting- at the 
 point a in BC, this point being- that in which BC is inter- 
 sected by the hyperbola before described. If m is the 
 middle point of AB, and if, as before, AB : AD' = w : to' , 
 and the horizontal line mil cuts OB' in //, we have 
 
 F^ = wh^.AE. (8) 
 
 Hence we have only to find the resultant of the forces 
 
 f<i,p^ and F^. This resultant will pass throug-h the point 
 
 of intersection of F^ and the resultant of f^, 7; ; and also 
 
 through the point of intersection of/, and the resultant of 
 
 Now taking the value (7) of;;, since 
 
 j/; = vy . BI, and p is perpendicular to BI, 
 f^ = ?ry . BH' , and/2 is perpendicular to BIF, 
 it follows that if the resultant of j) and f., is denoted by 
 
 {Pi fi) = ^'y -^7/', and is perpendicular to IJI', . (9) 
 and this resultant intersects F^ in s. 
 ^g'ain, ;) = w//^ . BE, 
 
 7-; = «•//, . ./77, 
 
 so that if BL is drawn equal and parallel to IIA, we have 
 for the resultant ( /;, Fi) 
 
 (p, F-^) = p-//i . EL and is perpendicular to EL, . (10) 
 
 and this resultant intersects /"g ^^ ^• 
 
 Hence the line of action, rs, oi F.^ is known. 
 
 If EN is drawn parallel and equal to II'B, we have 
 
 {P, ^'i) = vh^.EL, (ji) 
 
 f., = vy.EN, (12)
 
 Liquid Pressure on Plane Surfaees. 75 
 
 hence if T is the point dividing- NL so that 
 
 NT: TL = h^ :?/, 
 the resultant of the forces (ii), (12) is given by the equa- 
 ^'°" F,=w{K + >/)-J^T (13) 
 
 (see Sfaf/c-t, \o\. I, Art. 23). The line /-'?' is, of course, 
 perpendicular to rs, so that the point T may be found by 
 drawing- this perpendicular. 
 
 Observe that //j +y is the depth of the line EF below the 
 surface, DA, of the water, so that we have the same rule for 
 the resultant force on the section JJLFC as that for the 
 force on the upper section — viz. it is the weight of a water 
 column having for base the vertical projection of IF and 
 height FT. 
 
 If below FF there is another section of the embankment 
 in the form of a trapezium, the force F.> and the depth of F 
 below AB play the same part in the calculation of the 
 resultant force on this lower section as that which was 
 played by F^ and the depth of F in the calculation just 
 given ; so that this process can be employed for the com- 
 plete construction of the line of resistance through any em- 
 bankment the section of which can be broken up into suc- 
 cessive trapeziums. 
 
 In Masonry Dams for reservoirs the vertical section of 
 the upper portion, ABCD, Fig. 24, has often the simple 
 form of a rectang-le. If in this case the level of the water 
 reaches to AD, the top of the dam, the line of resistance is 
 a parabola whose equation referred to the vertical line in as 
 axis of y and the horizontal ID as axis of x is 
 
 y"^ =. 6 AD' . X, 
 
 so that in is the tangent at the vertex, and the directrix is 
 a vertical line to the right of ^ at a distance from i equal to 
 5 AD\ The line D'H (Fig-. 24) being the vertical through
 
 76 Hydrostatics and Elementary Hydrokinetics. 
 
 B\ the resultant force on any section 7?Q is perpendicular 
 and proportional to AH, where // is the point (as in the 
 fig-ure) in which the horizontal line inR through m meets 
 the vertical line throug-h iJ' \ the resultant force 
 
 = w.AQ.AIL 
 
 If the level of the water does not reach to the top of the 
 dam. let the top line of the dam be A^ I)^ ; then the force 
 on any section ^Q is found by tracing- the parabola and 
 combining- the force v) . AQ . AH perpendicular to AH with 
 the weight of the portion ADH^A^. The result is this: 
 from Aq draw ^,^ -^ parallel to QJj' and meeting AU in Z; 
 then the resultant force is perpendicular to ZH and is equal 
 to tv.AQ.ZH. 
 
 (For simplicity I has been omitted in the calculations 
 from p. 72. This omission amounts to considering the 
 forces B, i?(,, p, &c. as those exerted per unit length 
 of the embankment.)
 
 CHAPTER IV. 
 
 GENERAL EQUATIONS OF PRESSURE. 
 
 {This Chapter rnajj he omitted orifrd reading.) 
 
 17, Equation of Equilibrium of a Eluid under Gravity. 
 If in the case of a fluid acted upon solely by i^-ravity 
 we imagine the density not to he the same at all points, 
 the expression (a), p. 38, for the intensity of pressm-e will 
 no longer hold. For in Fig. 14, p. '^'], the weight of the 
 cylindrical column PiV^ will not be wzs, since to varies from 
 point to point of its depth. But if 
 w is the specific weight at any depth 
 z, the weight of this cylinder is 
 sfwdz, the limits of z being o and 
 NP ; and, as before, this weight 
 must be equal to the upward pressure 
 on the base at P, viz. /; . s. Hence 
 p = f%vdz 
 
 dp 
 
 Fig. 28. 
 
 dz 
 
 = w. 
 
 (0 
 
 If, for example, the density varies directly as the depth, 
 we have lo — kz, and (i) gives 
 
 ^ _ 1 1,2. 
 
 Equation (i) could have been obtained by considering 
 the equilibrium of a veiy small rectangular parallolopiped,
 
 78 Hydrostatics ami Eltincntary Hydrokinctics. 
 
 PQ, Fig. 28, described with vertical and horizontal sides at 
 P. For, if -f = area of each horizontal face, and if Q is 
 a point vertically below P so that PQ = (h, the weight of 
 the element is 7v . sfh, where w is the weight per unit 
 volume of the fluid at P. Also the downward j^ressure on 
 the horizontal face at P is p . s, where /; is the pressure 
 intensity at P ; and since the pressure intensity at Q is 
 
 /; + -y- (h, the upward pressure on the horizontal face at Q is 
 
 fp + ~j- dz\ -w Considering the separate equilibrium of 
 
 this elementary parallelopiped, and resolving forces ver- 
 tically, we have 
 
 (p + j- <lz^ '1 = p .s + w . sdz, 
 ftp 
 
 - - ^ 7V 
 
 as before. 
 
 If we are measuring force in pounds' weight and length 
 in inches, p will be in pounds' weight per square inch, 
 z being the depth of the point in inches, and f the weight 
 per cubic inch of the liquid at P in pounds' weight — 
 in other words, iv is the number of pounds mass of the 
 liquid per cubic inch at P. 
 
 If force is measured in poundals, the weight per cubic 
 inch of the liquid at P is about 32-2 iv, where w is still 
 the number of pounds mass per cubic inch at P. 
 
 It is usual to denote the number of units of mass 
 per unit volume by p. If then force is measured as a 
 multiple of the weight of the unit mass, the equation 
 
 for p is 
 
 dp
 
 General Equations of Pressure. 79 
 
 But if force is measured in absolute units, the numVjor 
 of these in the weii^ht of a unit mass beino- g (i. e. 32-2 
 poundals or 9H1 dynes, according as the " British Absolute " 
 or the C. G. S. system is used), ?/' in (i) is pg, and the 
 equation forjo is 
 
 With this form of the equation, and the C. (i. S. system 
 of units, the student must observe that — 
 ' ;; is in dynes per square centimetre, 
 
 ^ „ linear centimetres, 
 
 p ,, grammes per cubic centimetre, 
 
 // ,, centimetres per second per second (about 981). 
 
 If the fluid is of constant density, (i) g-ives^j = toz, the 
 result which in the previous chapters we have employed 
 in the case of water. 
 
 In the case of a g-as p, or ^o, is proportional to p : in a 
 subsequent chapter we shall prove that 
 
 T 
 
 p = 2926-9 -p, (4) 
 
 where ;; is the intensity of pressure in g-rammes' weight 
 per square centimetre, T is the absolute temperature of the 
 gas on the Centig-rade scale, s is the specific g-ravity of the 
 gas referred to air, and p is the mass of the g'as in grammes 
 
 per cubic centimetre. 
 
 T 
 
 Using equation (2), and denoting 2926-9 — by I; we have 
 
 s 
 
 dp _(lz 
 
 y-y ^''^ 
 
 z being measured vertically downwards. If z is measured 
 vertically upwards, we have
 
 8o Hydrostatics and Elementary Hydrokinetics. 
 
 Integrate this, assuming- T constant throughout the 
 g-as, and suppose that when z ■= o the vakie of /; is 2^^ ; 
 then _-' 
 
 p = Pq^ '' y'f 
 
 This g-ives the intensity of pressure at a height of r 
 centimetres in the atmosphere, on the assumption of con- 
 stant temperature,-? being unity, and^^Q being- the intensity 
 of pressure at the g-round. 
 
 Suppose any g-as contained in a pipe, vertical or not, 
 closed at the upper end. Let be the point at which 
 = o and p is i^o ; let P be any point above ; let the 
 pipe at be open to the aii-, so that j:)^ is produced by the 
 atmosphere in contact with the g-as at 0. At the point P 
 in the pipe the intensity of pressure of the enclosed g-as is 
 given by (7), and at P just outside the pipe the intensity 
 of the pressure, ]\ , of the air is given by the equation 
 
 z 
 
 A=A^'*'' (^) 
 
 where k^ = 2926-9 T, the gas and the air being assumed to 
 be at the same temperature. 
 
 Now if the gas is lighter than air — sujopose hj-drogen or 
 coal gas — k is > /("j , and therefore 
 
 i. e. the gas would escape into the surrounding air if an 
 aperture were opened in the pipe at P. At the gas does 
 not rush out of the pipe, although a communication is 
 established with the air ; the gas at would diffuse into 
 the air, but we may suppose that the pipe at contains a 
 piston which restrains the gas and on the top of which the 
 atmosphere presses. 
 
 The higher the point P in the pipe, the greater the 
 ratio of;? to j^Jj , and therefore the more rapid the escape of 
 the gas when a communication is opened. Hence the gas
 
 General Equations of Pressure. 8i 
 
 lii^hts at the top of a house are, if the taps are ojiened 
 to the same extent, bri^-hter than those at the Itottom ol' 
 the house ; and, in consequence of this, it is commonly 
 said that ' the pressure of the gas at the top of the house is 
 greater than that at the bottom ' — a thing which couUl not 
 possibly be true, since gravitation must diminish the 
 jjressure as the height increases. It is not the pressure 
 of the gas that is greater at the top but the velocity of its 
 escape. 
 
 When a balloon ascends, the neck is^ for safety, left open 
 to the air, so that the intensity of pressure of the gas at the 
 neck is that of the atmos]ihere at this point ; the gas 
 does not rush out at the neck ; but if a valve is opened at 
 the top of the balloon, the gas will escape for the reason 
 already given — viz. that the intensity of pressure of the 
 enclosed gas at this point is greater than that of the 
 adjacent air. 
 
 If the gas in the pipes were heavier than air, p would be 
 < Pi, and the reverse of the above would be true. 
 
 When density is measm*ed in pounds per cubic foot, 
 intensity of pressure in pounds' weight per square foot, 
 and T is 460 + 1, the absolute temperature on the Fahren- 
 heit scale, rp 
 
 P = 53-30222 - p, (9) 
 
 and at a height of z feet in the column of gas we have (7), 
 in which k has the value given in (9). 
 
 Since T will usually be a large number, if z does not 
 
 z 
 
 _ o* 
 
 exceed one or two hundred feet, we may take e * = i _ -, 
 
 fC 
 
 and we have , x 
 
 •^(^-*) / \ 
 
 53"3°222x7' ^ ' 
 
 for the excess of gas pressure in the pipes over that of the 
 
 G
 
 82 Hydrostatics and Elementary Hydrokinetics. 
 
 outside air at a height of z feet ; and in this equation the 
 pressures mav be estimated in any units whatever. 
 
 18. General Equations of Equilibrium. If the forces 
 acting- on the Ihiid are any assigned s^^stem, let the force 
 per unit mass at P have for components parallel to any 
 three rectangular axes the values X, Y, Z, so that on an 
 element of mass din these forces will be Xdm, &c. At 
 F draw a small rectangular parallelopiped, with edges Pa, 
 Plj, Pc, or c/x, dy, dz, parallel to the co-ordinate axes. 
 Then, if p is the density of the fluid at P, the mass con- 
 tained in this parallelopiped is pdxdr/dz. 
 
 Consider the separate equilibrium of 
 "Z-d/n this fluid. If;; is the pressure intensity 
 
 at P, the pressure on the face IPc is 
 •^dm p . dijdz, and since the pressure intensity 
 
 k 
 
 sz 
 
 T 
 
 ^, on the opposite face is ;? + -r- . dx, the 
 
 ■p. pressm'e on the face is 
 
 {p + Yx ^^^) '''^'^^' 
 
 For the equilibrium of the element, equate to zero the 
 component of force acting on it parallel to the axis of x, 
 and we have 
 
 Xdm-\-p . dydz— (p+-j- dx^ dydz = o, 
 Similarly dp 
 
 rr'"^' ''^ 
 
 I--. (3) 
 
 by resolving forces parallel to the other axes. 
 
 Each of these equations is a particular expression of the 
 general result that at every point in t/te fluid the line-rate of
 
 General Equations of Pressure. 
 
 83 
 
 Fig. 30. 
 
 variation of the intensiii/ of pressure in aui/ direction is equaf 
 to the component in this direction of the external force per 
 unit volume of the fluid — a result which is easily seen 
 thus. 
 
 Let PQ, Fig". 30, be an element, ds, 
 of leno^th of any curve through P ; 
 round this as an axis describe a cylinder 
 of small uniform cross-section, a ; con- 
 sider the sejiarate equilibrium of the 
 fluid contained within this cylindrical 
 element of volume. If F is the external 
 force per unit mass exerted on the fluid 
 in the neig-hbourhood of P, the force on the enclosed 
 fluid is F. pads ; if p is the pressure intensity at^J and^' 
 
 r which is p + J- ds^, the pressure intensity at Q, the forces 
 
 on the ends of the cylinder at P and Q are pa and^/rr. In 
 addition to these there are side pressures which are all at 
 rig-ht angles to the axis PQ. 
 
 Resolving forces along PQ, we have 
 
 pa —p'a + F. pads . COS = O, 
 where 6 is the angle between i^ and PQ; and this is 
 
 the same as «?« . , 
 
 -i- = pF con 9 (4) 
 
 as 
 
 The equations (i), (2), (3) lead to a certain condition 
 which the components X, Y, Z, of external force intensity 
 at each point must satisfy in order that the equilibrium of 
 the fluid may be possible. 
 
 For, we have 
 
 G 2 
 
 (5)
 
 84 Hydrostatics and Elementary Hydrokinetics. 
 
 Multiplyino- both sides of the first by Z, of the second by 
 X, of the third by 1\ and adding", we have 
 
 ^^.dY flZ. ,..(1Z dX. „.dX dJ\ ,,, 
 
 which is the necessary condition of equilibrium. 
 
 This condition may be thus expressed : for the equilibrium 
 of any f.uid under external hodily force it is necessary that at 
 all points the resultant force and its curl should be at right 
 angles. (This term ctirl is due to Clerk Maxwell.) Re- 
 garded analytically, (6) expresses the fact that the expression 
 Xdx + Ydy + Zdz, if not already a perfect differential, must 
 be capable of being* rendered so by means of a factor, 
 the factor being-, as we see, the density of the fluid at each 
 point at which the expression Xdx-\- ... is taken. 
 
 Since at each point 7; is some function of «?, y, z, we have 
 
 ^ dp J dp -. dp 
 
 Hence from (i), (2), (3), 
 
 dp = p [Xdx + Ydy + Zdz\ .... (a) 
 
 from which p is known by integ-ration. 
 
 So far, w^e have assumed only that the body is a perfect 
 fluid, so that our equations all hold for compressible fluids 
 as well as for liquids. 
 
 If the fluid is a gas, p = hp, and (a) becomes 
 
 ^ = \{Xdx^Ydy^Zdz). ... (7) 
 
 Now in all cases in which equilibrium is possible the 
 expression 
 
 p{Xdx-\-Ydy-\-Zdz) 
 
 must be a perfect difTerential — since it = dp (see p. 12); 
 and (6) expresses the condition that this should be the
 
 General Equations of Pressure. 85 
 
 case; but in many eases Xdx+ Yd// + Zdz is a perfect 
 difterential, i. e. tlie external forces have a Potential. 
 
 Assuming- that the forces have a potential, /', (n) becomes 
 
 dp = pdF. (,S) 
 
 If the fluid is a homog-eneous liquid, (i), (2), (3), give 
 
 vv^ = pv2r, (9) 
 
 , d^ r/2 r/2 
 
 where V'^ = -j-., + -7^, + -nr 
 aur d//" dz" 
 
 In all cases in which the external forces have a potential, 
 
 their level surfaces, or equij)otential surfaces [Statics, 
 
 vol. ii, Ai-t. 327) are also surfaces of equal pressure of the 
 
 fluid ; for (8) shows that in passing from a point P to 
 
 another close point such that df= o, we have dp = o. If 
 
 the density of the fluid is variable, it will be constant 
 
 all over a level surface of the external forces ; for, since 
 
 in (8) the left side is a perfect differential of a function 
 
 of X, y, z, the right side must be so, and this requires that 
 
 p is some function of / , i. e. 
 
 p=f{n M 
 
 so that at all points for which V is constant p is also 
 constant. 
 
 For a slightly compressible fluid, whose resilience of 
 volume is k (Art. 8), equation (8) becomes 
 
 ic'^f,=dr, (1.) 
 
 and for a gas, since 7; = Ap, where A is a constant, 
 
 x'^=dF. (12) 
 
 P 
 
 If the temperature of the gas varies, p = c p (\ + a t), 
 
 where c is a constant, I is the temperature at any point in
 
 (14) 
 
 86 Hydrostatics and Elementary Hydrokinetics. 
 
 the g-as, and a is a constant. (This will be explained more 
 fully in a subsequent chapter.) Hence, in general, 
 
 dp Xdx + Ydy + Zdz , , 
 
 ;; c{i +at) ^ ^' 
 
 and if the applied forces have a potential, V, 
 
 dp _ dV 
 
 p " c[i+aty 
 
 so that, since the left-hand side is a perfect differential and 
 the rig-ht side must also be one, it is necessary that t should 
 be some function of V; in other words, t is constant along 
 each equipotential surface of the external forces. Hence 
 for a gas subject to any conservative system of forces 
 (i. e. forces having a potential) each level surface of the 
 forces is at once a surface of constant pressure intensity and 
 a surface of constant temperature. 
 
 19. Non-conservative Forces. If Xdx + Ydy + Zdz is 
 not a perfect differential, and if at any point, P, in the 
 fluid we describe the surface of constant pressure, whose 
 equation is p = const., (i) 
 
 this surface will not coincide w4th the surface drawn 
 through P along which the density is constant, i. e. the 
 surface whose equation is 
 
 p =■ const (2) 
 
 These tw^o surfaces will intersect in some curve, which is 
 called the curve of constant pressure and constant density 
 at the point P. 
 
 We propose to find the direction of this curve at P. 
 
 Let I, m, n be the direction-cosines of the tangent to the 
 curve at P ; then if ds is the indefinitely small element 
 of its length between P and a neighbouring point Q ; 
 and if ;j =/" (.r, y, z) at P, the value oi p at Q is 
 f{x-\- Ids, y + mds, z -}- nds),
 
 General Equations of Pressure. 87 
 
 / t/p dp dp X 
 
 LC. y; + (/ -; + w/^ + 11 ) ds. 
 ^ ax d/j dz^ 
 
 Hence, since there is no change in the vahie of 7?, 
 
 , dp dp dp , . 
 
 I -f +m-j-+ 71^ =0, . . . . (3) 
 ax dy dz 
 
 or, by the general equations of equilibrium, 
 
 lX+mY+7iZ — o, (4) 
 
 so that PQ is at right angles to the direction of the 
 resultant force at P. 
 
 Similarly, since there is no change in p from i'* to Q, 
 we have 4 j ,/ 
 
 and therefore from (4) and (5) we have 
 
 I : m -.n ■= 1 Z -j- : Zi X ~r- : X —. J-p.(o) 
 
 dz dy doa dz dy dx 
 
 Now, denoting by A, /x, v the components of the curl of 
 the force, i. e. f]% ^jy 
 
 dy dz 
 
 _ dX dZ 
 ^-~dz~Tx\ (") 
 
 _dY (IX 
 dx dy 
 equations (5) of Art. 1 8 are 
 
 4;-4>^^=°' • • • • (^)
 
 88 Hydrostatics and Elementary Hydrokinctics. 
 
 These last show that (6) become 
 
 I : ?// : n = \ : fx : V, (lo) 
 
 and the differential equations of the curve are 
 
 dx dy (Iz , , 
 
 -r = — =— ' (^0 
 
 A [X V 
 
 from which, by integration, the equations of these curves are 
 found. 
 
 Hence the direction of any such curve at any point 
 coincides with the direction of the curl of the external 
 force. 
 
 If the fluid is a gas whose temperature varies from point 
 to point, we have p = cp[i +ai), where t is the temperature 
 at P, and c and a are constants. Now the previous result 
 is absolutely general, whatever be the connection between 
 p and p ; and Up and p are both constant along* any curve, 
 t must also be constant along- the curve. 
 
 When in any case the components of the external force 
 per unit mass are assigned — of course satisfying the neces- 
 sary condition (6), Art. i8 — there will be several laws 
 of density which permit the fluid to be in equilibrium. In 
 fact, p may be any of the integrating factors of the ex- 
 pression Xdx + Ydy + Zdz. We shall illustrate this in some 
 of the following examples. 
 
 Examples. 
 
 1. If a mass of fluid is acted upon by attractive forces 
 directed towards any number of fixed centres, find the equations 
 of the surfaces of equal pressure. 
 
 Let the fixed centres be A^, yl.,, ... ; let the distances of any 
 point, P, in the fluid from them be r^, rj, ... ; let the forces per 
 
 unit mass at unit distances from them be /^t, , \i^^ Then 
 
 {Statics, Vol. II, Art. 332) the forces have a potential, V, given 
 by the equation ,, ,, 
 
 7=^+'^ + ...,
 
 General Equations of Pressure. 89 
 
 and, the eqiiipotential surfaces Ijcing also surfaces of equal 
 pressure, the o(|iiation of iuij' surface of constant pressure is 
 
 In the case in which there are only two centres, if one of the 
 forces is repulsive, one of the series of surfaces is a sphere, viz. 
 
 that for which C = o, since if -— ^ = o, each ])oint, P, on 
 
 '1 '2 
 
 the surface is such that the ratio PA^ : PA^ is constant, and the 
 
 locus is well known to be a sphere having for diameter the line 
 joining the points which divide the line A^A^ internally and 
 externally in the ratio //j : /x.^. 
 
 2. If a fluid is acted upon by force whose components, per 
 unit mass, are at any point proportional to 
 
 2/- + v/c + z^, z'^ + zx-\- x^, x^ + xy^ y^, 
 
 determine a law of variation of the density of the fluid, the cor- 
 resf)onding surfaces of constant pressure, &c. 
 
 If / denotes a constant force intensity and c a constant 
 length, the components of force per unit mass will be of the 
 
 forms -^ (2/^ + 2/0 +s^), &c. 
 c 
 
 c" 
 
 Hence 
 dp 
 
 f 
 = p{y' + y^ + ^^)dx + p{^ + zx + x'^)dy + p{x''" + xy + y^)dz. . . (i) 
 
 It is to be observed at the outset that the force and its curl 
 are at right angles, so that equilibrium is possible, by (6), 
 Art. 18. 
 
 The prol)lem is to determine p, as an integrating factor, so 
 that the right-hand side of (i) shall be a perfect differential. 
 The analytical mode of ^''I'ocedure is to consider z at first as 
 constant, and to find an integral of the equation 
 
 (y- + yz + s^)dx+ {z^ + ^-v + x") c/y = o. 
 
 This is at once found to be 
 
 2 , _^2X + Z , 2 , 27/ + Z 
 
 — tan ^ — - H — tan = C, 
 
 >/3 zVs ^^3 ~ V3
 
 90 Hydrostatics and Elementary Hydrokinetics, 
 or {z being merely a constant) 
 
 ^ — 2xy — zx — zy 
 Now take the function U, where 
 
 u^ , °'+^+' , (.) 
 
 z^ — 2 xy — zx— zy 
 
 and diflferentiate both sides with respect to x, y, and z, consider- 
 ing all of them variable. Then, using D for the denominator of 
 the right-hand side of (2), 
 
 dU-jy^[ iy'^ + yz + z^) dx + ^z' + zx + x") dy 
 
 + \{^ + y--^-2XZ—2yz)dz].. . (3) 
 Using (1) to simplify this, we have 
 
 '=P{j'^-f-\{^^+y^-Ydz^ • . . (4) 
 
 y^,dp-U^dz,hy{2) (5) 
 
 dL 
 
 »2 
 
 -2 + '^^=7r7]2TH'^i'' (^) 
 
 Hence dU 2c^ 
 
 IT' '^ filFu 
 
 and since the right side must now be a perfect differential, we 
 must have pD^U'^ a constant or any function of^. Since by (2) 
 DU ^x-\-y + z, we have, then, 
 
 p{x-\-y + zf = Tc, (7) 
 
 where k is a constant. Hence the density at any point varies 
 inversely as the square of the distance of the point from the 
 plane x + y-\-z^ o. 
 
 From (6) we have, then, 
 
 ^^ooy + yz + zx 
 
 c' ' x + y-\-z ' 
 
 which shows that the surfaces of constant pressure are hyper- 
 boloids. 
 
 The direction-cosines of the line of constant density and con- 
 stant pressure are proportional to the components of the curl of
 
 General Equations of Pressure. 91 
 
 the force, and are, therefore, proportional to y — z, z — x, oc — y ; 
 and the differential equations of these lines are 
 
 dx dy dz 
 
 - -^ - (9) 
 
 y — z z — x x — y 
 
 To integrate tliese, put each fraction equal to 6, where is 
 unknown, 
 
 Tlien dx = 6{y-z), (10) 
 
 dy = 6{z-x), (11) 
 
 dz = e{x-y), ....... (12) 
 
 and from these, by addition, 
 
 dx + dy + dz^ o, 
 whose integral is 
 
 x + y + z = a, (13) 
 
 where a is any constant. Hence the curves in question are 
 plane curves lying in jihuies obtained by varying a in (13). 
 Also multij^lying (10), (11), (12) by x, y, z and adding, 
 
 xdx + ydy + zdz =: o, 
 whose integral is 
 
 x' + f + z' = b', (14) 
 
 where b is any constant. Hence the curves lie on spheres 
 obtained by varying b; and as they are plane curves, they are 
 circles. 
 
 Of course they lie on the surfaces of constant pressure. 
 
 But, as previously pointed out, this is only one special law of 
 density which we have investigated. Many other laws may be 
 found thus. 
 
 From (5) we have 
 
 dU+U'dz=j^,dp (15) 
 
 Now if is any integrating factor of the left-hand side, 
 — 2 == constant will exjiress a possible value of p. But any 
 
 pD 
 
 iunction of the form 
 
 h^^^'-ij)
 
 92 Hydrostatics and Elementary Hydrokinetics. 
 will render the left-hand side a perfect differential ; hence 
 
 where ^ is any function, will express a jiossible value of p. In 
 particular, choosing (p (v) = —;;- , we have 
 
 v 
 k 
 
 {xy + yz + zxj 
 
 2 ' 
 
 (17) 
 
 which gives another simple law of density. The surfaces of 
 constant pressure remain the same as before ; for, when (15) is 
 
 multiplied across by jjr^ <p (z — -^ j , if c/p = o, we have still 
 
 I (J v 
 
 z — — = C. The actual value of 2^ is, of course, different ; for 
 
 p will now be proportional to I (f) (z — y j . d (z — —j , or 
 
 ^//■(s — yr), SU^Dpose. 
 
 3. If the components of force per unit mass are proportional 
 
 {l/ + »Y, cz, -c{y + a), 
 
 find a law of variation of density, the corresponding surfaces of 
 constant pressure, and the curves of constant density and con- 
 stant pressui'e. 
 
 Ans. p = r , p ^ k(x I, showing the surfaces 
 
 of constant pressure to be hypei'bolic paraboloids ; the curves of 
 constant density and constant pressure are right lines. 
 
 Also p = ; .(b\x ) : so that 
 
 P = 
 
 {x{y + a) — cz}^ 
 will also be possible. 
 
 4. If tlie components of force per unit mass are proportional 
 
 x — a, — \/h'^ — z'^ — {x—aY, z, 
 
 find the same things as in the last problem.
 
 General Equations of Pressure. 93 
 
 k 
 
 the curves are circles determined by the intersections of phmes 
 y = C with cylinders {x — ay^-:? = C . 
 
 Generally, P = ., ,\ ==, • 4> ( ^^lr-z'-{x-af-y). 
 Vh^—sr — {x—a)- 
 
 5. The components of force Leing proportional to 
 
 cy—hz, az—cx, hx—ay, 
 show that the surfaces of constant pressure are planes passing 
 
 ni* 01 2^ 
 
 through the line '- = ^ = - , and the curves of constant pressure 
 ° a G 
 
 and constant density are right lines parallel to this. 
 
 6. In a spherical mass of homogeneous liquid, self-attracting 
 according to the law of nature, find the pressure int ensity at 
 any point. 
 
 Ans. If y is the constant of gravitation, at any point, P, 
 distant r from the centre, V = zny p{a} — ^r-), where a is the 
 radius of the sphere (see Statics, Vol. IT, p. 299, 4th ed.). 
 
 jy= 2T:yp'{ii'-lr'). 
 
 [The homogeneity of this is thus verified : if m denotes mass, 
 
 m^ 
 
 / denotes force, and I denotes length, we know that y .-- — f; 
 
 also p = ^ ; hence }> is of the nature r^ , as it ought to be. 
 
 Since the pressure intensity in a homogeneous sphere is thus 
 proportional to the square of the density, we see the nature of 
 the assumption made by Laplace {Statics, vol. i, Art. 174) in 
 the case of the Earth— that in passing from stratum to stratum 
 the change in the pressure intensity is proportional to the change 
 in the square of the density.] 
 
 20. Equations of Equilibrium in Polar and Cylin- 
 drical Co-ordinates. Let 1\ Fi','-. 31, be any point in a 
 fluid, at wiiich the components of force acting- on the fluid, 
 per unit mass, are J, Y, Z parallel to the rectangular axes,
 
 94 Hydrostatics and Elementary Hydrokmetics. 
 
 Fig. 31- 
 
 Ox, Oy, Oz ; and let the position of P be defined by the 
 usual polar co-ordinates, viz. the radius vector OP(=r), 
 
 the colatitude, PO- {= 6), and the 
 lono-itude, xO/i (= ^), this last 
 being- the ang-le between the plane 
 xz and the plane containing" P and 
 the axis of z. The arcs in the figure 
 are those determined on a sphere 
 whose centre is and radius OP by 
 the axes and the line OP. 
 
 Sometimes it is convenient to 
 consider the resultant force per unit 
 mass at P as resolved into three 
 rectangular components corresponding to radius vector, 
 latitude, and longitude, i. e. components along OP, along 
 the line at P perpendicular to OP in the plane POz, and 
 along the tangent at P to the parallel of latitude. 
 
 Producing the great circle zPn to T so that nT= zP = 6, 
 the second of these directions is parallel to OT ; and since 
 the third is at right angles to the plane POz at P, if 
 we produce the arc xi/ to Q so that //Q = xn = (p, the 
 line OQ is parallel to the tangent at P to the parallel of 
 latitude. 
 
 Let R be the component of the force-intensity (i. e. 
 force per unit mass) at P in the direction OP ; let be its 
 component in the second and 4> its component in the third 
 of these directions. 
 
 Now, the axis of x being in any direction, we have 
 proved the equation 
 
 dp 
 
 dx 
 
 = p^, 
 
 so that if ds is the element of length of a curve drawn 
 in any direction at P, and S the force-intensity along the
 
 General Equations of Pressure. 
 
 95 
 
 tano-ent to this curve at P in the sense in which (h is 
 measured, it follows that 
 
 djj 
 
 Takinj^ (Is alono- OP, we have ds = dr ; takin<:»" ds along- 
 the meridian cP at P, we have ds = rdO ; and takin* 
 ds alono- the parallel of latitude at P in the sense OQ, 
 we have ds = r sin 6 d(f), since the radius of the parallel 
 of latitude is r sin 0. Hence the equations of cciuilibrium 
 are , 
 
 = p.P, 
 
 dp 
 
 dr 
 
 dp 
 
 dd 
 
 dp 
 
 d4> 
 
 = pr.Q, 
 
 = prsinO . *. 
 
 (I) 
 
 Equations of eqidlibr'mm in Cylindrical Co-ordinates. By 
 the cylindrical co-ordinates of P are meant the distance, z, 
 of P from the plane xy, the perpendicular distance, C of P 
 from the axis of z, and the longitude, </j, i. e. the angle 
 between the plane xz and the meridian plane zOP. Hence 
 if Z, Z^ , (l> denote the components of force-intensity at P 
 parallel to Oz, perpendicular to Oz, and along the tangent 
 to the parallel of latitude, 
 
 dp 
 
 dz = '^' 
 
 ^P - r 7 
 
 dC-'^^' 
 
 -^ = p ;• sm d 
 
 (2) 
 
 (^.
 
 96 Hydrostatics and Elementary Hydrokinetics. 
 
 Examples. 
 
 1. If a homogeneous liquid i.s acted upon by gravitj', and each 
 particle is, in addition, acted upon by a force emanating from a 
 vertical axis proportional to the distance of the particle ii'om 
 that axis, find the intensity of pressure at any point. 
 
 Adopting the C. G. S. system, measuring forces in dynes 
 and masses in grammes, Z = —g (the axis of Z being drawn 
 
 C 
 vertically upwards), Z^= f .- , where / is a constant force in 
 
 dynes and a a constant length in centimetres, and f^ = o. 
 Hence 
 
 dj) dp C <Jp I \ 
 
 ± = -9r. j-^=pf--^ j^ = o; ... (3) 
 
 and since dp =^ -f- dz + ^r}.dC + -rr d(b, we have 
 ^ dz dQ ^ d(j) 
 
 dp = -pydz + pf.^d{^; (4) 
 
 where C is a constant. If on the free surface p = ^>„ at all 
 points, and the origin is taken at the intersection of the axis of 
 z with this surface, we have 
 
 P=P^-pil^+TZ-0^ (6) 
 
 which shows that all tlie surfaces of constant pressure are para- 
 boloids of revolution round the axis of z, their parameters being 
 
 2. If the fluid is compressible and follows the law that p is 
 proportional to ]) at each point ; then, supposing that if at each 
 point the intensity of pressure were ^q, the density would be p^, 
 
 we have p =: p^ . — , and (4) becomes 
 Po
 
 General Equations of Pressure. 97 
 
 the integral of which is 
 
 p=Ce''^^ ' '" ^ (8) 
 
 where C is a constant. If the origin is taken as before, C is the 
 constant intensity of pressure on the free surface, and the 
 surfaces of constant pressure are still paraboloids of revolution. 
 
 3. If a mass of homogeneous liquid surrounds a sphere of 
 uniform density, and is subject to the attraction of this sphere 
 as well as to a force emanating from a fixed axis through the 
 centre of the sphere and proportional to the distance, as above, 
 we may employ either the general equations of Art. 18 or cy- 
 lindrical co-ordinates, or we may take as co-ordinates r and C, 
 simply. Thus, if the attraction of the sphere (not represented 
 
 in the figure) per unit mass at P is represented hj k -^, where 
 
 c is a constant length in centimetres and k a constant force in 
 dynes, we have 
 
 and since dp = ~ dr + ~dC, we have 
 dr dQ 
 
 c2 r 
 dp=-pk^dr + pf^dC, (10) 
 
 .-. p= pk^+flC' + C, (11) 
 
 where C is a constant. If Oz cuts the free surface where r=R, 
 and if 2>o is the pressure intensity on the free surface, 
 
 C = p^-pk-' 
 
 4. Suppose a homogeneous sphere at rest surrounded by an 
 atmosphere whose particles attract each other and are attracted 
 by the sphere according to the law of nature; it is required to 
 find an equation for the intensity of pressure at any point of the 
 atmosphere. 
 
 The surfaces of constant density in the atmosphere will be 
 spheres concentric with the nucleus. Let F be a point in the 
 
 H
 
 98 Hydrostatics and Elementary Hydrokinetics. 
 
 atmosphere distant r from the centre, 0, and II = mass of 
 nucleus. Then, if y is the constant of gravitation, with the 
 units of the C. G. S. system the attraction at F per unit mass is 
 
 - — due to the nucleus, and in addition to this there is the 
 
 attraction of that portion of the atmosphere contained within 
 the spliere of radius OP : the portion outside this sphere, since 
 its layers of constant density are spherical shells with centre 0, 
 exerts no attraction at P {Statics, vol. II, Art. 319). 
 
 To find the attraction due to the portion of the atmosphere 
 within the sphere of radius OP, describe a sphere of radius x, 
 less than OP ; let p' be the density (grammes per cubic cm.) on 
 its surface, and describe another sphere of radius x + dx; then 
 the mass of the shell contained between these spheres is 
 
 X dx 
 
 ^Tip'x^dx, and its attraction per unit mass at P is Anyp' — — 
 
 [Statics, ibid.). *" 
 
 Hence the equation for p si P is, since the resultant force is 
 directed from P to 0, 
 
 _ldp^yM^^r (,) 
 
 p dr r^ 'y la 
 
 where a is the radius of the nucleus. 
 
 To form a differential equation for p, multiply both sides by 
 r^, and then differentiate both sides with respect to r. Now 
 observe that p' is some function of x, and that we are differ- 
 entiating the integral at the right-hand side with regard to its 
 upper limit, so that the result of this differentiation will be the 
 function under the integral sign with the upper limit, r, substi- 
 tuted for X (see Williamson's Integral Calculus, Art. 114, or 
 Greenhill's Dif. aiid Int. Cal, Art. 207). 
 
 Hence , , , 
 
 d /r' dps „ , . 
 
 * (7 *) = -"'"''"•• <'* 
 
 rp 
 
 in which p = kpy where k= 2926-9 — , as will be seen in a 
 subsequent Chapter; therefore 
 
 ±(r^dp. 4jry ,_ . 
 
 The integral of this equation can be presented in a series. If
 
 General Equations of Pressure. 99 
 
 J TTV 
 
 -^^ is tlenoted by [i, a particular value of p is given by the 
 
 ^ 2 
 
 equation 7) =. — - . Now assume 
 
 P = ^i, (4) 
 
 where (f) is an unknown quantity, and change the independent 
 
 variable from r to - • 
 
 Denoting - by x, and putting log (p ^ yp, equation {3) be- 
 comes ** 
 
 • ^'S^ + ^(^'-^) = ° ^5) 
 
 Let - be denoted by c, and expand -^ in powers of x—c 
 
 according to the formula 
 
 where \//n, f^) , ... are the values of yl/ and its differential 
 
 coefficients at the surface of the solid nucleus, i. e. where x = c. 
 In calculating the successive differential coefficients 
 
 VdxVo' \dx'K' '" 
 by successive differentiations of (5) in terms of the unknown 
 and arbitrary constants yj/^ and (-r--) , it will be found con- 
 venient to take e^' = vl and ( -^) = - i5, so that A and B 
 
 are the two arbitrary constants which belong to the integral 
 
 of (5)- 
 
 It will be seen that all the coefficients in (6) at the right-hand 
 side vanish for the particular values A = i, B = o, and the in- 
 
 2 
 tegral then obtained is \// = o, i.e. ^ = i, or p = — j« 
 
 fJ.T 
 
 H Z
 
 loo Hydrostatics and Elementary Hydrokinetics. 
 
 21. Centre of Pressure. Hitherto in finding' the position 
 of the centre of pressure on a plane area we have confined 
 our attention to areas of simple forms, such as triang-les, 
 quadiilaterals, &c. We shall now consider a plane area 
 of any form occupying* an assig-ned position in water, 
 or other liquid, subject to the action of gravity only. Let 
 the area be mm, Fig. 20, p. 52. If we draw at a depth z 
 below the surface AB of the liquid a horizontal line, and 
 another line parallel to this at the infinitesimal distance dz 
 below it, denoting the length of the line intercepted by the 
 area rn m by y, the pressure on the strip contained by these 
 close lines is wzydz, where w = specific weight of the 
 liquid ; and this pressure acts at the middle jooint of 
 the strip, therefore its moment about AB is wz^t/dz. Now 
 the whole pressure on the area mm is Aziv, where A is the 
 magnitude of the area and z the depth of G, its centre 
 of area ; hence if ^ is the depth of the centre of pressure, 
 
 p=^' « 
 
 in which y is a known function of z when the form and 
 position of the area are assigned. 
 
 The position of the centre of pressure may be otherwise 
 expressed thus. At any point in the surface of the liquid 
 draw two rectangular axes, Ox, Oy, and draw the axis of 
 Oz vertically downwards ; break up the area mm into 
 infinitesimal elements ; let x, y, z be the co-ordinates of 
 any point, P, in the area at which the element of area is dS. 
 Then the pressure on this element is wzdS. Hence the 
 co-ordinates of the centre of pressure, /, are 
 fxzdS fyzdS fz'^dS 
 fzdS ' 'jVd'S' /zdS' 
 
 If at all points on the contour, mm, of the area we draw 
 vertical lines terminated by the surface of the liquid, the
 
 General Equations of Pressure. 
 
 lOI 
 
 centre of gravity of the cylinder of fluid enclosed by these 
 lines and the area mm lies on the vertical throug-h /, 
 midway between /and the surface. For, \i dcr is the i)ro- 
 jection of d S on the free surface of the liquid, the elementary 
 cylinder stiinding- on dS has for volume zda, and the co- 
 
 ordinates of its centre of gravity are x, i/, - ; hence the 
 
 co-ordinates of the centre of gravity of the whole cylinder 
 are fxzda fyzda ^fz^da 
 
 JzdcT f^d'T ' fzdcr 
 and if 6 is the angle between the plane of the area r n m and 
 the horizon, it is evident that da =z cos 6 . dS, so that 
 in the numerator and denominator of each of these last 
 expressions we may replace da hy dS, and the result is 
 then obvious. 
 
 The position of the centre of pressure on a plane area can 
 be very easily expressed with reference to the principal 
 axes of the area at 
 
 its centre of gra- b n ^^ a 
 
 vity, G. Thus, let 
 CBE (Fic^. 32) be 
 the plane area, its 
 plane being' in- 
 clined to the ver- 
 tical at any angle, 
 d ', let G A and GB 
 be its principal 
 
 axes at G, intersecting the surface of the liquid in A and 
 £; let GN(= A) be the perpendicular from ^'(in the plane 
 of the area) on the line AB, and let GjV make the angle 
 a with GA. 
 
 The equation of AB with reference to GA and GB as axes 
 of X and y, respectively, is 
 
 X cos o +y sin a —h = o ; 
 
 Fig. 32.
 
 I02 Hydrostatics and Eleinentary Hydrokindks. 
 
 and if P is any point in the area at which the element of 
 area d& is taken, the perpendicular Pm from P on AB is 
 ^— a; cos a — j/sino, if x,y are the co-ordinates of P with 
 reference to GA and GB. Hence the perpendicular, Bt^ 
 from P on the surface of the liquid is 
 
 {Ji — X cos a —y sin a) cos ^, 
 
 and the pressure on cl8 is 
 
 w{li — XQ,0'aa—y^va.a)Q.Q)%Q.dS, ... (a) 
 
 where w is the weig-ht of the liquid per unit volume. Now 
 take the sum of the moments of the elementary pressures of 
 which (a) is the type about GA and equate it to the 
 moment of the resultant pressure, A x GT . w, where A is 
 the area and GT the perpendicular from G on the surface. 
 If (^, rj) are the co-ordinates of /, the centre of pressure, 
 we have 
 
 icAh cos 6,7]= to cos 6f[h—x cos a —y sin a)ydS . (2) 
 = — iv cos 6 sin a/y~dS, 
 
 the other integrals vanishing" since the principal axes at 
 the centre of area are those of co-ordinates. 'Now yy^dS 
 is the moment of inertia of the area about GA, which 
 we shall denote by A . ^^^, k^ being- the radius of g-yration 
 of the area about GA. Hence, finally, 
 
 »? ^-ycosa; (3} 
 
 and in the same way, equating the moment of the whole 
 pressure about GB to the sum of the moments of the 
 elementary pressures of the type (a), we have 
 
 k^ 
 ^ = - X sin « (4) 
 
 Thus the co-ordinates are independent of the inclination 
 of the given plane area to the vertical, as we have previously
 
 General Equations of Pressure. 103 
 
 pointed out, so that if the area were turned round the line 
 AB in which its plane intersects the surface of the liquid 
 throug-h any ang-le, the centre of i)ressure, /, would continue 
 to be absolutely the same point in the area. 
 
 The expressions (3), (4) lead at once to an obvious geo- 
 metrical interpretation, viz. — construct the ellipse whose 
 equation with reference to GA and GB is 
 
 -^> + ^ = I ; 
 
 take the pole, Q, of the line AB with reference to this 
 ellipse ; the co-ordinates of Q are ( — f, — >?), so that if 
 the line QG is produced throug-h G to / so that GI = QG, 
 we arrive at /, the centre of pressure. 
 
 These expressions (3), (4) give tis at once some simple 
 results concerning the motion of the centre of pressure 
 produced by various displacements of the given area. 
 
 Thus, if the area is rotated in its own plane about G, 
 while G is fixed, the only variable in the values of ^, ^ is a ; 
 and if this is eliminated from (3), (4), we have 
 
 l.t-l_ (r) 
 
 which is the locus described in the area by the centre of 
 pressure — viz. an ellipse. 
 
 To find the locus described in this case by the centre 
 of pressure with reference to fixed space, refer its position 
 to the line GN and the horizontal line through G in the 
 area as axes of x' and 7/\ respectively. If {x', ij) are the 
 co-ordinates of / with reference to these axes, we have 
 ^ =■ x' cos a — y sin a, 
 ■q ■= x' sin a -^-y' cos a. 
 
 Substituting- the above values of ^, 77, and eliminating a, 
 we have k,^ + k.K^ ,2 A^-^\^ /^
 
 I04 Hydrostatics and Elementary Hydrokinctics. 
 
 which shows that / describes a cii-cle in fixed space, the 
 
 centre of the circle being- on the vertical through G. 
 
 Ao-ain. if the area is lowered into the liquid without 
 
 rotation, h is the only variable in (3) and (4), by eliminating 
 
 which we have „ /. 2 
 
 ^ = iL,tana, (7) 
 
 which shows that / describes a right line in the area ; and 
 
 it describes also a 
 A ^ ^ -^ right line in space, 
 
 ^ ^ I ^ ^^'^ ^ for (7) gives a 
 
 linear relation be- 
 tween x' , y\ and 
 constants. 
 
 Let us next sup- 
 pose the plane of 
 the area to have 
 any position what- 
 ever, which we 
 shall define in the 
 usual way by the 
 
 Precession and Nutation angles ^, 0, \//. 
 
 Take the vertical Gz as axis of /, and any two rectangular 
 
 horizontal lines, Gx' , G/, as axes of x' and /. Let Gx, Gy 
 
 be the principal axes at G in the plane of the given area, 
 
 while Gz is the axis perpendicular to this plane. 
 
 Then, since the direction-cosines of Gz' with reference to 
 
 the axes of a?, y, z are 
 
 — sin -^ sin Q, cos y\t sin 0, cos Q, 
 
 the length of the perpendicular from any point {x, y, 0) in 
 the given area on the free surface AB is 
 
 // + a; sin \// sin 6 —y cos \// sin Q. 
 
 This multiplied by lo ds gives the pressure on the element
 
 General Equations of Pressure. 1 05 
 
 of area, and the total moment of pressure aljout Gx is 
 — Ak^^w cos y\f sin 6. Hence, as before, 
 
 ^= ^- siny/^sin^, (8) 
 
 k ^ 
 
 ri= ^ cos \// sin (9) 
 
 A 
 
 Suppose the area to be rotated, like a rigid body, round 
 
 any line, GL, fixed in space, the direction-cosines of this 
 
 line being- /, m, n with reference to the space axes Gx\ Gy\ 
 
 Gz'. Then the angles between GL and the principal axes 
 
 Gx, Gi/, Gz are all constant, and if we denote their cosines 
 
 by A, IX, V, respectively, we find, by eliminating 6 and \// 
 
 from (8) and (9) by means of the constants A, fx, v, the 
 
 equation 
 
 P ri^ I /AA ur] nJ' I ,^^ 
 
 which gives the curve described in the area by the centre 
 of pressure 
 
 This agrees with (5) for the case in which the plane of 
 the area is always kept vertical ; for in this case 
 
 6 = - , xj/ =—a, V = 1, n = o, k = IX = o. 
 
 If the line GL is any one in the plane of the area v = o, 
 and the locus described by / in the area is a right line, 
 
 ^y 4. i—i J — = o II 
 
 k.]" ^ k^' ^ h ^ ' 
 
 Examples. 
 
 1. Find the position of the centre of water pressure on a 
 circular urea whose plane is vertical and whuse centre is at a 
 given depth. 
 
 Take as the principal axes of reference at G the vertical and
 
 io6 Hydrostatics and Elcmenta)y Hydrokinetks. 
 
 horizontal diameters, and let li be the depth of G. Then in (3) 
 and (4) we have a = o and h^ ■= h^ z=\ r^, where r is the 
 radius of the circle. 
 
 Hence ^ r* 
 
 4/t 
 
 o 
 7— 
 
 so that 7 is on the vertical diameter at a depth h ■\ — - from 
 the free surface. ^ 
 
 If the area is just immersed, h = r, and the depth of / is 
 
 4 ' • 
 
 In the case of an elliptic area whose centre is at a depth h, 
 and whose major axis makes an angle a with the vertical 
 
 .a' 6« . 
 
 t = r cos a, 71 = r- sm a, 
 
 4/1 4 /i 
 
 where a and b are the semi-axes. 
 
 2. Find the pressure on a plane vertical area whose position 
 is given in a slightly compressible liquid. 
 
 Supposing, definitely, the units of the C. G. S. system 
 adopted, we have for the intensity of pressure at any point P, 
 
 dp , . 
 
 where z is the depth of P in centimetres below the free surface. 
 
 , , „ . . dp k dp 7.1 -T 
 
 Also from (y), p. 22, ~ =z - —- where k is the resilience of 
 
 volume. ^^ ^ ^^ 
 
 These give by integration 
 
 where p^ is the density at the surface of the liquid. Now since 
 k is very great, we may neglect 7^ , and we have 
 
 which indicates a uniform density superposed on a density vary- 
 ing directly as the depth.
 
 General Equations of Pressure. 107 
 
 Substituting this in (i), we have 
 
 i' = mG~ + 7^0 (3) 
 
 Let A be the magnitude of the given area, and z the depth of 
 its centre of area. 
 
 If dS is the clcmetit of area at P, the whole pressure is fpdS; 
 and \i fz^dS, which is the moment of inertia of the area about 
 the line AB (Fig. 32) in which its plane intersects the surface, 
 is denoted by A A^, while fzdS = Az, the resultant pressure is 
 
 in dynes. Dividing this expression by g, we have the pressure 
 in grammes' weight. 
 
 3. Find the position of the centre of pressure on any vertical 
 area which is symmetrical with respect to its principal axes at 
 its centre of area, when immersed in a slightly compressible 
 fluid. 
 
 AVith the notation of p. 103, and denoting the radius of gyra- 
 tion of the area about the line AB (Fig. 32) by A, we have 
 
 4. Assuming the resilience of volume of sea water to be, in 
 C. G. S. units, 2-33 X 10'" (see p. 22), and that i mile= 160933 
 centimetres, find the fractional increase in density at a depth of 
 I mile in the ocean. 
 
 Ans. From the equations 
 
 ^P 10 
 
 Pj-= 2-33 X 10'" 
 
 ^P o 
 
 £ = »**""■ 
 
 we have ,0 c?p o j 
 
 2-33 X io'° -^ = 981 rfs, 
 
 P-Po 981 Po^ 
 
 2-33 X io'"-98i/),;
 
 io8 Hydrostatics and Elementary Hydrokinctics. 
 
 where p^ is the density at the surface. Taking p^ = 1-026, we 
 find at the depth of a mile 
 
 '-^=-^^^ nearly. 
 Po M2-8 
 
 5. Assuming the resilience of volume of sea water to be con- 
 stant at all dej^ths, find what the depth of the ocean should be 
 at a point where the deubity of the water is double the surface 
 density. 
 
 Ans. Nearly 71-92 miles. 
 
 6. Represent graphically the densities of sea water at points 
 on a vertical liue drawn downwards from the surface. 
 
 Let be the point on the surface, OA the vertical line drawn 
 downwards to the point, A, at which the density would be 
 doubled ; produce OA to C so that OA=i AC; draw a hori- 
 zontal line, CE, through C. Then if the densities at various 
 points on OC are represented by ordiuates drawn at these points 
 perpendicularly to OC, their extremities trace out a hyperbola 
 whose centre is C and asymptotes CO and CH. 
 
 7. If the density of a fluid varies as any given function of the 
 depth, find the depth of the centre of pressure on a plane 
 vertical area. 
 
 Ans. If p = f\z), the depth of / is f^flj^g ' 
 
 8. A rectangular area, ABCD, has one side AB in the surface 
 of water and its plane vertical. If it is rotated about a hori- 
 zontal axis at A, find the curve described in the area by the 
 centre of pressure so long as the whole area continues immersed. 
 
 Ans. If AB = 2(1, AD = 26, the centre of pressure traces 
 out a right line in the area. This line is thus constructed : let 
 G be the centre of area of the rectangle, m and n the middle 
 points of DC and CB respectively; take a point p on Gm such 
 that Gp = ^ Gm, and a point q on Gn such that Gq = ^ Gn ; 
 then the liue joining ^; to q is that described in the area by the 
 centre of pressure. 
 
 9. A plane area of any form occupies a vertical position in 
 water. If it is rotated in its plane about any point in the area, 
 find the curve traced out in the urea by the centre of pressure.
 
 General Equations of Pressure. 109 
 
 Ans. Let be the point about whicli the area turns, G the 
 centre of area, a, j-i the co-ordinates of with respect to tlie 
 principal axes at G, h = tlio perpendicular from on the sur- 
 face ot the lifjuid, the radii of gyration being as in p. 103 ; then 
 the equation of the locus referred to the principal axes at G is 
 
 (a k,' w + l3k,'^j + k,' k,y = Ir (AV x' + V f), 
 
 which will be a hyperbola, an ellipse, or a parabola according as 
 
 GO >h, GO < h, or GO = h. 
 
 If is in the surface of the liquid, so long as none of the area 
 is raised out of the liquid by rotation about 0, the locus is a 
 right line. Thus if a plane polygon of any shape has a corner 
 in the surface of the \u[uid round wliicli the polygon is turned, 
 the centre of pressure describes a right line so long as the whole 
 area remains immersed. 
 
 10. Find the position of the centre of pressure of a semicir- 
 cular area whose diameter is in the surfiice of water. 
 
 Ans. If r is the radius of the circle, the centre of pressure 
 
 is at a distance ^ r from the horizontal diameter. (The centre 
 
 of gravity of a semicircle is — from the centre.) 
 
 o 
 
 11. If the diameter is horizontal and at a depth h, find the 
 depth of the centre of pressure. 
 
 Ans. ^-^- below the horizontal diameter. 
 
 4 4r + STTh 
 
 12. Find the position of the centre of pressure on a semicir- 
 cular area whose bounding diameter is vertical with one ex- 
 tremity in the surface of water. 
 
 . 4r 
 Ans. Its distance from the vertical diameter is — , and its 
 
 depth is ^ r. 
 
 (The point is on the vertical through the centre of gravity, G, 
 
 of the area, since this is one of the principal axes at 6-'.) 
 
 13. Find the position of the centre of pressure on a semicir- 
 cular area completely immersed in water, the bounding diameter 
 beini- inclined at an an^le a to the horizon and having one 
 extremity in the surface of the water.
 
 no Hydrostatics and Elementary Hydrokinetics. 
 
 Ans. The distances of the centre of pressure from the 
 bounding diameter and the diameter pei-peudicular to it are, 
 respectively, 
 
 r q TT cos a + 1 6 sin a , r q tt sin a 
 
 - • and - . '- -. — • 
 
 4 4 cos a + 3 TT sin a 4 4 cos a + 3 tt sin a 
 
 14. An elliptic area is immersed vertically in water; if it is 
 displaced by rolling along the surface of the water, find the locus 
 described in the area by the centre of pressure. 
 
 Ans. A similar, similarly placed, and concentric ellipse 
 whose axes are each \ of the corresponding axes of the given 
 ellipse.
 
 CHAPTER V. 
 
 PRESSURE ON CURVED SURFACES. 
 
 22. Principle of Buoyancy. If any curved closed 
 surface, M (Fig*. 2, p. 5), be traced out in imag-ination in 
 a heavy fluid the pressures exerted on all the elements 
 of this surface by the surrounding* fluid have a sing-le 
 resultant, which is equal and opj)Osite to the weight of the 
 fluid enclosed by M. 
 
 This is evident^ because the fluid inside 31 is in equili- 
 brium under its own weight and the pressure exerted on 
 its surface by the surrounding- fluid ; hence this pressure 
 reduces to a vertical upward force equal to the weight 
 of the fluid inside M and acting through the centre of 
 gra\dty of this fluid. 
 
 This is obviously true whatever be the nature of the 
 fluid — liquid or gaseous, homogeneous or heterogeneous. 
 
 If the curved surface M is not one merely traced out 
 in imagination in the fluid, but the surface of a solid body 
 displacing fluid, the result is the same — 
 
 the remltant pressure of a heavy finid on ihe surface of any 
 solid hody M is a vertical ujnvard force equal to the weight of 
 the fluid ivhich could statically replace M, and this force 
 acts through the centre of gravity of this replacing fluid.
 
 112 Hydrostatics and Elementary Hydrokinetics. 
 
 Fig. 34- 
 
 Let Pig. 34 represent the solid body, which we may 
 imag-ine to be a mass of iron, the suiTounding" fluid being 
 water, aii-, or any fluid acted upon by gravity. The body 
 is represented as held in its position by cords attached 
 to fixed points, C,D, ..., and the arrows represent pressures 
 exerted on its surface by the fluid at various points. 
 
 Now it is quite clear that if the iron body were replaced 
 
 by one having exactly 
 the same surface and 
 occupjdng exactly the 
 same position, the pres- 
 sure on each element of 
 its surface would be 
 identically the same as 
 before, of whatever sub- 
 stance the new body 
 maybe. If the new body 
 were of wood, or instead of being solid were a thin hollow 
 shell, it might be necessary to keep it in the position re- 
 presented by means which prevent its rising up out of the 
 fluid ; but we are not at all concerned with the forces which 
 keep the body M in equilibrium ; our object is merely to 
 ascertain the resultant, if any, of the jitiid jiressures exerted 
 in the given position on its surface. 
 
 In general, a number of forces acting in various lines 
 which do not lie in one plane have no single resultant : 
 their simplest reduction is to two forces whose lines of 
 action do not meet [Statics, vol. ii., chap. xiii.). But it 
 is remarkable that the pressures exerted on the various 
 elements of any closed surface by a heav)^ fluid Jiave a single 
 resultant ; and the truth of this we see by imagining the 
 place occupied by M to be occupied by a portion of the fluid 
 itself, placed in the vacancy without disturbing any of the 
 surrounding fluid.
 
 Pressure on Curved Surfaces. 113 
 
 With rco;-ard to tliis ro])l:icini^ fluid, observe two tliinf^s : 
 firstly, it is in iMjuiiibrimu ; secondly, it is kept so by its 
 own wei<i-ht and the very same system of i)ressures as that 
 which acted on the body 31, since this body and the re- 
 placing- fluid present identically the same surface to the 
 surrounding" fluid. Hence, then — 
 
 the system of pressures has a single resultant which is a 
 vertical vjnvard force erpial to the weight of the statically 
 replacing fluid and acting through the centre of gravity of this 
 fluid. 
 
 The centre of gravity, //, of the replacing* fluid is called 
 the centre of buoyancy ; and, so far as the general principle 
 of buoyancy is concerned, there is no relation between H 
 and the centre of gravity, G, of the body ; nor is there any 
 relation between the weight, W, of this body and the 
 weight, L, of the displaced fluid. 
 
 If the fluid is water, or any homogeneous liquid, the 
 resultant pressure is the weight of the liquid which would 
 flow into the vacant space if i)/ were removed ; but if the 
 density of the fluid is different in dill'erent layers, we must 
 not imagine the replacing fluid to be that which would 
 flow in when M is removed, but rather to be a continuation 
 of the surrounding fluid placed in the vacancy without any 
 disturbance of the external fluid, and having the same 
 surfaces of equal density as this fluid. The distribution of 
 this replacing fluid is unique and determinate, as will be 
 subsequently proved. 
 
 Cor. I. Pressure of uniform intensity exerted over an}- 
 closed surface produces no resultant. 
 
 For, imagine the closed surface to be one traced out in 
 a perfectly weightless fluid — or a very light gas — whose 
 surface is subject to any pressure. 
 
 The intensity of pressure will be uniform throughout the 
 whole fluid, and therefore over the given surfiice ; and by 
 
 I
 
 114 Hydrostatics and Elementary Hydrokinetics. 
 
 what has just been said, the resultant pressure over this 
 surface is equal to the weig-ht of the enclosed fluid — that is 
 to say, zero. 
 
 This result can, of course, be proved mathematically; for, let 
 dS be an element of the surface at any point, P, (x, ?/, z), let p 
 be the constant intensity of pressure, and let the origin of co- 
 ordinates be any point inside the surface. Then there is no re- 
 sultant force parallel to the axis of x ; for, if the element dS be 
 projected orthogonally on the plane yz by a slender cylinder, and 
 the sides of this cylinder be produced through the plane yz so as 
 to meet the surface again in an element of area dS^ at a point 
 P', the projections of dS and d&" on the plane yz are numeri- 
 cally equal but of opposite signs ; hence the pressures 2><^^'^' and 
 pdS' at P and P' neutralize each other in the direction of the 
 axis of X ; and in the same way the pressures at the ends of all 
 other cylinders parallel to the axis of x neutralize each other in 
 this direction, so that there is no force parallel to this axis; and, 
 similarly, no force in any direction. 
 
 Thus the area of the projection of any closed surfiice on any 
 plane must be considered as zero. In symbols, fjdydz taken 
 over any closed surface is zero. 
 
 In fact, in several investigations of mathematical physics, 
 each element, dS, of a surface may usefully be represented by a 
 vector, i. e. a directed magnitude, by drawing at the mean point, 
 P, of the element d^ a normal to the surface from the surface 
 into the surrounding space — always from the same side, or 
 aspect, of the surface into this surrounding space — taking a 
 length on this normal proportional to the area of the correspond- 
 ing element, dS, and marking the end of this normal length by 
 an arrow head. The orthogonal projection, or component, of 
 this marked line along any line, L, will then represent tlie pro- 
 jection of d8 on any plane perpendicular to L. Some of these 
 marked lines, or vectors, will have comi^onents along L in one 
 sense and others will have components in the opposite sense ; 
 and thus we understand more clearly the mathematical result 
 that the sum of the projections of the same aspects of the ele- 
 ments, dS, ... of any closed surface on any plane is zero. 
 
 In the same way there is no resultant moment of the pres- 
 sures round any line. For, if ffj, o-j, o-, are the jn'ojections of 
 dS at P on the planes y«, zx, xy, respectively, the components
 
 Pressure on Curved Surfaces. 115 
 
 of the force pdS parallt'l to the axes of x, _?/, z are 7>o"i, 7>o"2, l^^^'i 
 
 and the moment of this force about the axis of x is 
 
 i'(yo-3-co-2) (a) 
 
 Now if tlie pcrppiidiculiir from P on tlic 2)lam' yz is prochiced 
 through tliis phme to meet the suriace again in P'", the element 
 of area, dS"\ cut off at P'" hy the slender cylinder descriljed 
 on the contour of dS at P parallel to the axis of z, will have for 
 projection on the plane »•;/ tlu; value — o-g, and the co-ordinate 
 ij being the same for P'" as for P, the moment round the axis 
 of X' of the pressure at P"' will supply the term —ycr^, which 
 destroys the first part of («). The second part is similarly 
 destroyed by the pressure at P'\ the point in which the parallel 
 from P to the axis of y meets the surlace again. 
 
 Hence there is no moment of the pressure about any line. 
 
 Analytically this result is expressed thus : 
 
 ff {y dx dy — zdz dx) = o 
 for any closed sui-face. 
 
 The result of this Corollary may also be thus stated: 
 given any closed curve, plane or tortuous, in space ; if 
 a surface of any size and shape be described having* this 
 curve for a bounding' edge, and if pressure of uniform 
 intensity be distributed over one side of this surface, the 
 resultant of this ])ressure is the same whatever the size and 
 shape of the surface. 
 
 Hence if the curve is plane, the resultant pressure on any 
 surface having it for a bounding edg-e is the same as the 
 resultant pressure on the plane area of the curve. 
 
 Cor. 2. The principle of Archimedes. 
 
 The particular case in which the solid body M which 
 displaces lluid is in equiUbrium soleh/ under the action of its 
 oirn iceifjht and the fluid pressure over its surface furnishes the 
 Principle of Archimedes, 
 
 The resultant of the system of fluid i)ressures must then 
 be exactly equal and opposite to the weight of the dis- 
 placing- body. 
 
 I 2
 
 ii6 Hydrostatics and Elementary Hydrokinetics. 
 
 Fig. 35. 
 
 Thus, let Fi"-. "^^y represent a heavy body whose centre 
 of gravity is G^ floating- in equilibrium in a heavy fluid. 
 
 The surface over which 
 the fluid pressure is ex- 
 erted is ABB which is 
 not a closed surface ; but, 
 as there is no pressure due 
 to the fluid exerted over 
 the free surface, BM, of 
 the fluid, w^e can suppose 
 the immersed surface 
 ABB to be closed by the 
 section of the body made by the horizontal plane AB. 
 Hence the resultant of the pressures is the weight of the 
 fluid that would fill the space ABB ; and if H is the centre 
 of gravity of this fluid, the resultant pressure acts up 
 through II, so that G and // must be in the same vertical 
 line. Hence there are two distinct conditions of equi- 
 librium of a body floating freely in a heavy fluid, viz. 
 
 1°. ike weight of the body must he equal to the weight of the 
 fluid which it displaces; and 
 
 2°. the centre of gravity of the hody and the centre of gravity 
 of the jiuid that icould statically fll its place {centre of 
 hiioyancy) must he in the same vertical line. 
 
 We have hitherto supposed that the only fluid displaced 
 by the body is that reiDresented in the vessel below the 
 surface BM ; but if above this there is air, whose weight is 
 considered, there is also displaced a volume of air repre- 
 sented by ACB, and the resultant effect of the air is to pro- 
 duce an upward vertical force, even though (as in the figure) 
 the air pressm-e exerted by the air actually in contact with 
 the displacing body should be a doiomoard force ; for, we 
 must remember that the surface LM of the lower fluid is all 
 subject to air pressure which is (by Pascal's Principle)
 
 Pressure oil Curved Surfaces 117 
 
 transmitted undiminished all throu<4'li Uiis fluid, so that the 
 lower part, ADB, of the surface of the body is really acted 
 upon all over by air pressure of constant intensity. Now 
 by Cor, i, the resultant of this system of air pressures on 
 the curved surface ADB, is the same as if the pressure was 
 applied over the lower side of the plane area AB in which 
 the surface Ij]\[ cuts the body. The resultant air pressure 
 is, therefore, an upward force equal to the weig"ht of the air 
 that would statically 1111 the S])ace ACB, and it acts throug-h 
 the centre of g-ravity of this air. 
 
 The case of a balloon floating in the air is also an instance 
 of the principle of Archimedes ; the force of buoyancy is the 
 weig-ht of the air that could statically replace all the solid por- 
 tions of the balloon and the g-as which it contains. It must not 
 be supposed that, since the balloon is a comparatively small 
 body, the intensity of the air pressure is constant all over 
 its surface — a not unnatural error ; for, if this air pressure 
 were of constant intensity all over the surface, its resultant 
 would be absolutely zero, as we have already seen, and there 
 would be no force of buoyancy. If the medium surround- 
 ing- a body is ever so slig-htly acted upon by gravitation, 
 its intensity of pressure cannot be constant, and hence the 
 densities of the air at the top and at the bottom of the 
 balloon are not the same. 
 
 23. Introduction of Fictitious Forces. In the case in 
 which a body is partiall\' immersed in a fluid, or a part of 
 the body is in one fluid and the remainder in another, it is 
 often very convenient to introduce fictitious forces of buoy- 
 ancy in one part of the calculation and to take them away 
 in another. 
 
 Thus, suppose Fig-. ^^ to represent a body of which the 
 portion ABB is immersed in water while the portion ACB 
 is in vacuo. Then the actual force of buoyancy is due to 
 the volume ABB of water ; but we can complete the volume
 
 ii8 Hydrostatics and Elementary Hydrokinetics. 
 
 of the dis])laced water by supposing- the portion ACB to be 
 also surrounded by water, and then supposing- that there is 
 a dotvmoard force, in addition, due to the action of this por- 
 tion ACB of water taken negatively. Thus the actual force 
 of buoyancy — viz. an upward force at //equal to the weight 
 of the volume ABB of water — can be replaced by an ^qncard 
 force equal to the weight of the whole volume ADBC of 
 water acting at the centre of gravity of the homogeneously 
 filled volume ADBC (not G, the c. g. of the body, unless the 
 body is itself a homogeneous solid), together with a down- 
 ward force equal to the weight of the volume ACB of water 
 acting at the centre of gravity of the homogeneously filled 
 volume ACB. 
 
 In the same way, if the portion ACB is in a liquid of 
 specific weight 2i\, and ABB in one of specific weight Wg* 
 we may regard the force of buovancy as consisting of an 
 upward force equal to the weight of the whole volume 
 ABBC of the liquid iv^ together with a downward force 
 equal to the weight of a fictitious liquid of specific weight 
 w., — 2i\ acting at the centre of gravity of the homogene- 
 ously filled volume ACB. 
 
 24. Resultant Pressure on an unclosed curved surface. 
 
 Suppose BCBA, Fig-. 36, to repre- 
 = sent any unclosed surface in a heavy 
 iluid, and suppose its bounding edge 
 to be a plane curve so that the sur- 
 face can be closed by a plane base, 
 represented by AB. It is required 
 to find the resultant of the fiuid 
 Fig. 36. pressures exerted on one side of 
 
 the unclosed surface. 
 Closing the surface by means of the plane base AB, the 
 resultant of the pressures all over the outside of the com- 
 pletely closed surface is the vertical ujiward force, L repre-
 
 Pressure on Curved Surfaces. 119 
 
 sented by tlie line JIL drawn ihrong-h the centre of <2^ravity, 
 //, of the fluid which would fill the volume. But if P is 
 the resultant fluid pressure on the plane base AB, actin^i;- at 
 the centre of pressure, /, the force L is the resultant of P 
 and the resultant pressure over the unclosed part. This 
 latter force, Q, is therefore found by producin<»- the lines of 
 action of L and P to meet — at 0, suppose, — and drawin^i: 
 On and Om to represent L and P, respectively ; then the 
 required force Q is represented by the line OQ which is 
 equal and parallel to 7nn. 
 
 If the fluid is a homogeneous liquid of specific weig-ht w, 
 if A is the area of the plane base AB, z the depth of the 
 centre of area of AB below the free surface, and V is the 
 volume of the closed surface, 
 
 P = Azw, and L = J'lo. 
 Hence if 6 is the inclination of the plane base AB to the 
 horizon, ^ ^ ^^ VV-'^2rAzco^e + A^z'' ; 
 horizontal component o^ Q = Aztv sin 6, 
 vertical component of Q = (Az cos 6— F)iv. 
 
 Examples. 
 
 1. Suppose a right cone whose axis is vertical and vertex 
 downwards^ to be filled with a liquid ; find 
 the resultant pressure on one-half of the 
 curved surface determined by any plane 
 containing the axis. 
 
 Let ACB, Fig. 37, 1)6 the vertical 
 plane of section, and ACDB the half of 
 tlic curved gurfiice on which wc desire 
 to find the resultant \n\m<\ pressure. 
 
 Consider the separate equilibrium of 
 the fluid contained between tliis curved 
 surface and the triangle ACB. It 
 is kept at rest by its weiglit, the Fig. 37. 
 
 pressure of the remaining fluid on the 
 area ACB acting at /, the centre of pressure on this triangle, 
 
 )-p
 
 120 Hydrostatics and Elementary Hydrokinctics. 
 
 and by the pressure of tlie cun-ed surface ACDB. The weight 
 acts tiirough G, the centre of gravity of the semi-cone ; and if 
 on the diameter, OD, which is perpendicular to AB, we take the 
 
 AT 
 
 point n such that On ^ — , where r is the radius of the base, 
 
 this point n is the centre of area of the semicircle ADB, so that 
 G lies on nC and Gn = iCn {Statics, Vol. I, Art. 163). The 
 point / is half way down OC (Ait. 15). If F is the pressure on 
 the triangle A CB, h =■ height of cone, 
 
 P = ^ rh-w ; and W = ^ Trr-Jiw, 
 
 where W = weight of liquid. The lines of action of P and W 
 meet in a point c, whose position is thus completely known ; 
 and by drawing cP and c TF to represent P and W on any scale, 
 the diagonal through c of the rectangle thus determined will 
 represent Q, the resultant pressure of the curved sui-face on the 
 fluid in the semi-cone. The line cQ is drawn to represent this 
 pressure, and this force reversed is the pressure of the fluid on 
 the surface. 
 
 2. If the cone is closed by a base, and the axis is held hori- 
 zontal, find the resultant pressure on the lower half of the 
 curved surface. 
 
 Aois. If X and Y are the horizontal and vertical com- 
 ponents of the resultant pressure, 
 
 X = (- + -)r'w, 
 
 r = (i+ ).v. 
 
 and the line of action of the pressure jiasses thi'ough a j)oint 
 whose distances from the base and the axis of the cone ai'e 
 
 h 77-1-8 T r 377+16 
 
 — • and - • 
 
 4 77-1-6 4 377 + 4 
 
 (see example 10, p. 109). 
 
 3. If a hollow cylinder is filled with liquid and held with its 
 axis vertical, determine the magnitude and line of action of the 
 resultant pressure on one half of the curved surface cut ofi" by a 
 vertical plane through the axis.
 
 Pressure on Curved Surfaces. 121 
 
 Ans. It is a horizontal force equal to r^hw acting in a line 
 
 - from tlie base. 
 3 
 
 4. If the cylinder is closed at both ends and held with its 
 axis horizontal, find the resultant pressure on the lower lialf of 
 the curved surl'ace. 
 
 Ans. A vertical force = [2 Jt -\T"h 
 
 IV. 
 
 I 
 
 5. In example 2 find the magnitude and line of action of the 
 resultant pressure on the upper half of the curved surface. 
 
 Ans. If A" and Y are the hoi'izontal and vertical com- 
 ponents of the pressure, 
 
 Y ■=\\ — -~\ r^hw, 
 
 and the line of action of the resultant passes through a point 
 whose distances from the base and the axis of the cone are 
 
 h 8 — ■7T - r 16 — 377 
 
 - • and - • • 
 
 4 6-77 4 37^-4 
 
 6 A spherical shell is fdled with liquid ; find the magnitude 
 and line of action of the resultant pressure on the curved surface 
 of either hemisphere cut off by any vertical central plane. 
 
 Ans. The line of action passes through the centre of the 
 sphere; the horizontal component is irr^iv, and the vertical 
 ^ 77 r^ w. 
 
 7. A spherical shell is filled with liquid ; find the magnitude 
 and line of action of the resultant pressure on each of the hemi- 
 spheres into which the sphere is divided by any diametral 
 plane. 
 
 Ana. If d is the inclination of the plane section to the hori- 
 zon, the pressure on one hemisphere is the resultant of two 
 forces Tir^io and f^Tir^io, respectively perpendicular to the ])lane 
 section and vertical, the lines of action of these forces including 
 an angle 0, while tlie pressure on the other curved surface is the 
 resultant of the same forces including an angle Ti — 0; and both 
 pass through the centre.
 
 122 Hydrostatics and Elementary Hydrokinetics. 
 
 8. If a hole is made in tlie top of the shell and fitted with a 
 funnel, find the height to which the funnel must be filled with 
 the liquid in order that the resultant pressure on one of the 
 hemispheres shall be a horizontal force, 
 
 Ans. The height = r (§ sec a— i). 
 
 25. General Principle of Buoyancy. In Art. 22 we 
 have enunciated the principle of buo^'ancy in the case in 
 which the buoyant medium is acted upon by the attraction 
 of the earth — that is, the principle has been applied in the 
 case of a heavy fluid of uniform or variable density, com- 
 pressible or incompressible. It is evident that the same 
 principle holds in general for any medium the particles of 
 which are acted upon by any system of external forces, 
 electric, magnetic, or other. 
 
 Thus, in Fig. 38, let AB be a closed surface traced out in 
 imagination in a medium of any kind the particles of which 
 
 are acted upon by any sys- 
 tem of forces, and let the 
 resultant of these forces on 
 the particles contained 
 within the surface AB — if 
 they have a single resultant 
 at all — be a force repre- 
 sented by OF. Then, con- 
 sidering the separate equilibrium of the portion of the 
 medium within AB^ we see that this portion is kept in 
 equilibrium by the force OF and by the pressures (repre- 
 sented as normal, though not necessarily so) exerted by the 
 suiTounding pai-t of the medium on the various elements of 
 the surface AB. It follows that these pressures have a 
 single resultant, 0F\ exactly equal and opposite to OF. 
 
 Hence if AB were the surface of a foreign body the par- 
 ticles of which are subject to the same external forces as 
 those acting on the medium, though the resultant, B, of 
 
 Fig. 38. 
 
 Fig. 39-
 
 Pressure on Curved Surfaces. 123 
 
 these on the body AB may now be different both in mag-- 
 nitude and in direction I'rom OF, the resultant of the pres- 
 sures exerted on this body by the surrounding- medium will 
 still be OF'; and in order that this body should be in equi- 
 librium without the aid of further forces (tensions of cords, 
 &c.) it must take such a position in the medium that R is 
 exactly equal and opposite to OF' — a condition which it 
 may or may not be possible to fulfil. 
 
 Again, in Fig. 39, let M be any foreign body immersed 
 in a medium — air, suppose, — and let its molecules be subject 
 to the attractive and repulsive forces of two magnetic poles, 
 N and S ; suppose the resultant action of these poles on the 
 body to reduce to two equal and opposite forces, P, P, form- 
 ing" a couple ; and, to eliminate the effect of gravitation, 
 suppose the body supported by a cord attached to its centre 
 of gravity, G. Then the body is also acted upon by the 
 pressures of the surrounding medium on its various elements 
 of surface. What is the resultant action of these pressures ? 
 To answer this question, we must imagine the place of the 
 body occupied statically by a portion of the medium itself. 
 If the magnetic forces do not produce any effect on the 
 particles of the medium, the pressures on the elements of 
 surface of the replacing medium are simply equivalent to a 
 vertical upward force acting through the centre of gravity 
 of this portion of the medium and equal to its weight — 
 which would usually be a very small force ; but if the 
 medium is affected by magnetic forces, and if the resultant 
 action of these forces consists of a couple, Q, Q, the result- 
 ant action of the surrounding medium must consist — in 
 addition to the previously mentioned small gravitation 
 force of buoyancy — of a couple equal and opposite to the 
 couple Q, Q. Hence, neglecting gravitation forces, if the 
 foreign body is in equilibrium, it must place iti«elf in such 
 a position that the magnetic couple, P. P, produced on it ;
 
 124 Hydrostatics and Elementary Hydrokinetics. 
 
 too:cther with the mao-netic couple of buoyancy, ~ Q, — Q, 
 are in sfaUe equilibrium with the couple produced by the 
 torsion of the suspending- cord. 
 
 If the moment of the couple P, P, is g-reater than that 
 of the couple Q, Q of buoyancy, the position of stable equi- 
 librium of the body M will be different from that assumed 
 on the contraiy supposition. Thus, if 3f is a bar of iron, 
 the couple P, P is g-reater than the couple Q, Q, and the 
 bar will set axially, i. e. in the line NS joining the two 
 mag-netic poles ; but if 31 is a bar of bismuth, Q, Q is 
 greater than P, P, and the bar will set equatorially, i. e. at 
 rig-ht angles (or inclined) to the line NS. 
 
 Such is, in a general way, the explanation of the beha- 
 viour of diamagnetic bodies in a magnetic field, or of mag- 
 netic bodies placed in media more strongly acted upon by 
 mao-netic forces than are the bodies themselves. 
 
 Examples. 
 
 1. A solid homogeneous right cone floats in a given homo- 
 geneous liquid ; find the position of equilibrium, firstly, when 
 the vertex is down and base up ; and, secondly, wlien the base is 
 down and the vertex up. 
 
 Let to', V, h be the specific weight, volume, and lieight of the 
 cone ; let iv be the si^ecific weight of the liquid, and x the length 
 of the axis immersed when the vertex is down. Then since the 
 volumes of similar solids are proportional to the cubes of their 
 corresponding linear dimensions, the volume of the displaced 
 
 liquid = '77 V. Hence, equating the force of buoyancy to the 
 
 weight of the cone, 
 
 '"' Vw = Vw', 
 
 x"" 
 
 
 lo- 
 
 in the second case, if x is the length of the axis above the
 
 Pressure on Curved Surfaces. 
 
 12 = 
 
 liquid, the volume of the displaced liquid = (i — j^i) ^) and we 
 have 
 
 w' \it 
 
 7 f *'^ V 
 X = k\i ) 
 
 ^ to / 
 
 10 
 
 2. A solid homogeneous isosceles triangular prism floats in 
 a given homogeneous liquid ; find the position of equilibrium in 
 each of tlie two ]n-evious cases. 
 
 If X is the deptli of its edge below the surface, h the height of 
 the isosceles triangle which is the section of the prism by a 
 plane pcr2)endicular to the edge, and A the area of tliis section, 
 since tiie areas of similar figures are as the squares of their 
 
 corresponding linear dimensions, j^ A is, the area of the face of 
 
 the immersed jirism in the first case, and if Z = length of 
 edge, the volume of the prism is lA, so that the volume of the 
 
 X" 
 
 immersed prism is j^ V- Hence 
 
 X 
 
 ^, V^o = Vw^, 
 
 .-, X = hi — ) 
 In the second case, 
 
 .-. x = h(i ) 
 
 w\i 
 
 3. A uniform rod, AB, of small normal section and weight W 
 has a mass of metal of small volume 
 
 and wcio'ht - W attached to one ex- 
 ° n 
 
 tremity, B ; find the condition that 
 
 the rod shall float at all inclinations 
 
 in a given homogeneous liquid. 
 
 Let AB = 2 a, let vi be the middle 
 
 point of AB, Fig. 40, G the centre uf 
 
 gravity of the rod and the metal, z«' 
 
 the specific weight of the rod, w that 
 
 of the liquid, and s the area of the normal section of the rod. 
 
 Fig. 40.
 
 [26 Hydrostatics and Elementary Hydrokinetics. 
 
 n 
 
 Then TF= 2asio', and BG = — — a. Also G must be the 
 
 w+ I 
 
 centre of buoj-ancy if the rod floats in the oblique position repre- 
 sented, and the length, BC, of the displaced column of liquid 
 
 =: a, so that if the weight of this column = ( i H — ) H', 
 
 n+i ' ° ^ n^ 
 
 both conditions of equilibrium will be satisfied, whatever be the 
 
 inclination of the rod. Equating the weight of the body to the 
 
 force of buoyancy, 
 
 / i\ , 2?^ 
 
 ( I + ) zasw = ■ asiv; 
 
 \ «/ n+i 
 
 {n -\- if w' =^ V? w, 
 
 which is the relation required between the specific weights. 
 
 4. A solid homogeneous cylinder floats, with its axis vertical, 
 partly in a homogeneous liquid of specific weight z^j and partly 
 
 in one of specific weight w.,, 
 the former resting on the 
 latter ; find the position of 
 equililirium. 
 
 Let h be the height of the 
 cylinder, A the area of its 
 base, w its specific weight, 
 and c the thickness of the 
 upper liquid column. 
 
 Then if we assume the top, 
 
 A, of the cylinder to project a 
 
 distance x above the upjier 
 
 surface of the upper liquid, as 
 
 in Fig. 41, and equate the weight of the cylinder to the sum of the 
 
 forces of buoyancy due to the displacements of the liquids, we 
 
 liiive hw = cw^ + {h — c — x)io.2, (i) 
 
 ,. ^,,.= A(,_-)_,(x-^). ... (2) 
 
 Figs. 41, 42. 
 
 If c(i ^) > hfi ), we must write 
 
 , ^ w./ ^ wj 
 
 --H-:::")-"(-5)l'.-: 
 
 and it would a})pear that the position of equilibrium is one in 
 
 (3)
 
 Presstire 07i Curved Surfaces. 127 
 
 whicli, as in Fig. 42, A is below tlie upper surface of the upper 
 liquid by the distance 
 
 o(,--i)-;,(,_»), (,) 
 
 ^ V\'' ^ 111,/ ^ ' 
 
 in virtue of the usual interpretation of a negative co-ordinate in 
 algebra. 
 
 To take a numerical case, suppose c ■= \h, and 
 
 tv : tf J : ^2 = 5 : 2 : 6 ; 
 
 then x = —l-h, and it would appear that A is i/t below the 
 
 upper surface of the upper fluid. 
 
 Now if we had originally assumed A to be, as in Fig. 42, at 
 
 an unknown distance, x, below the surface, our equation would 
 
 have been 7 ., /„ \ , /? , \ /.v 
 
 hw= {c — x)w^ + {h — c + cc)ic.,, .... (5) 
 
 :. X = c—li — ^ > • • (6) 
 
 which disagrees with (4), and which in the particulai- numerical 
 case gives x = ^h, instead of a- = i/t, which we had been led to 
 expect by interpretation of the negative value (3). 
 
 Why the disagreement 1 Because the continuity of the values 
 of variables in algebra and algebraic geometry finds no corre- 
 sponding characteristic in the hydrostatical conditions. In fact, 
 the supposition that the negative value (3) harmonises with the 
 physical assumjitions leading to the first solution is untrue; for, 
 in this solution we assume that, whatever be the unknown posi- 
 tion of equilibrium of the body, the whole column of the upper 
 liquid is operative in producing its force of huoijancy, as is 
 evident from the first term, cw, , at the right-hand side of (i); 
 whereas the sujjjiosition that A is below the upper surface of 
 this liquid is an explicit assumption that the whole column of 
 the liquid may not be so operative. Hence we ought not to 
 expect the two solutions to agree. 
 
 In the case, therefore, in which the value of x in (2) is nega- 
 tive, the correct result is (6) and not (3). 
 
 5. A heavy uniform bar, AB, of small cross-section is freely 
 moveable round a horizontal axis fixed at one extremitv. A. at a 
 given heiglit above the surface of a homogeneous lii|uid in wliich 
 the rod partly rests ; find the position of equilibrium and the 
 pressure on the axis.
 
 128 Hydrostatics and Elementary Hydrokinetics. 
 
 Let AB = 2a; let h be the height of A above the liquid ; let 
 s = area of cross-sectiou of the rod ; let w^ aud lu be the specitic 
 
 weights of the rod and the liquid ; 
 and let 6 = the angle between AB 
 and the vertical. 
 
 Then if BC is the part immersed, 
 the centre of buoyancy, H, is the 
 middle point of i^6'. If Tr= weight 
 of rod, W =: 2asw\ also 
 
 BC = 2a— h sec 6, 
 
 .'. the force, L, of buoyancy 
 
 = {2a— h sec0) sw. 
 
 The rod is in equilibrium under the action of L, W, and the 
 pressure at A, which last must be vertical aud = W—L. 
 Taking moments about A for equilibrium, 
 
 W.AG smd = L. AH sind, (i) 
 
 and if we reject the factor sin 6, i. e. omit the consideration that 
 sin 5 = o gives one position of equilibrium (the vertical one), we 
 
 Fig- 43. 
 
 have 
 
 4a''iv' = {^a^—h? sec^ 6) iv, 
 
 cos 
 
 h 
 
 2 a ^iv — iu ^ 
 
 (2) 
 (3) 
 
 The oblique jiosition re- 
 quires w to be greater than 
 w' aud also 
 
 V} > — 5 — 7-2 . w ; 
 
 dfO^ — W- 
 
 so that, for examj^le, if the 
 bar were of metal and the 
 liquid water, the only posi- 
 tion of equilibrium would be 
 the vertical one. 
 
 Fig. 44. 
 
 6. A uniform square board, 
 ABCD, is moveal)le in a ver- 
 tical i^lane about a smooth 
 horizontal axis fixed at the corner ^ , at a given height above 
 the surface of a liquid; find an equation for its position of
 
 Pressure on Curved Surfaces. 129 
 
 equililiriiim, assuming the ligure ol' the iiiiinorscd portion to Ik: 
 a tiapi'zivim. 
 
 Lt't lie {lie inclination of AJ> to the vertical, ^15= 2a, 
 height ol' A above li(|ui(l = 2//, lu' =1 specific weiglit ol" hoai'd, 
 w = specific weight of liquid ; let PQ be tlie line of lloatation, 
 and draw Qli jjarallel to BC. 
 
 Then we must equate the moment of the weight of the board 
 about A to the moment of the i'oi-ce of buoyancy about A. But 
 we may consider the force of buoyancy as consisting of two 
 forces, viz. that due to the weight of the portion QJiCB acting 
 upwaids through ni, the centre of gravity of this parallelogram, 
 and that due to the weight of the triangular portion PQli acting 
 upwards through n, the centre of gravity of this triangle. 
 
 Now the area of QRCB := 4a (a — A sec ^), and the distance 
 of «( from the vertical through A = half difference of distances 
 of G and Q = a (cos (^ — sin d) — h tan 6. Also the area 
 
 PQR= 2 a- tan ^, 
 
 and distance of n from vertical through A 
 
 = I (dist. of P + dist. of i^-dist. of Q), 
 
 by ex. 6, p. 33, and therefore 
 
 dist. of « = f {a (cos0 + sec0) — 3/i tan d\. 
 
 Also distance of G from vertical through A is a (cos (^ — sin (J). 
 Hence the equation of moments about A is 
 
 4a{a — h sec 6) {a(coa^ — sin ^) — A tan^} w 
 
 + ^ a^ tan 6 {a (cos + sec 0) — '^h tan 6] w 
 
 = ^a^ w' (cos 6 — sin 6), 
 
 or (3 cos^0— 2 sin^ cos^ (J + sin 6) a'— 3«A + 37t2 sin ^ 
 
 = 3 a- — cos^ 6 (cos — sin 0). 
 
 w 
 
 7. Solve the previous problem on the assumption that the 
 unimmorsed portion is triangular. 
 
 Let r, Q be now the points in which AD and AB, respectively, 
 cut the surface of the liquid. Then wo may consider the force 
 of buoyancy, i- «• the weight <jf PQBGD, as consisting of the 
 weiijht of the whole ABGD (of specific weight lu) acting up- 
 wards at G, together with a doumcard Ibrce equal to the weight 
 
 K
 
 130 Hydrostatics and Elcnicutary Hydrohindics. 
 
 of a triangular portion FAQ acting at the centre of gravity of 
 this triangle. Hence the forces which give equal and opposite 
 moments about A are 4a'"(w — xv) acting upwards at G and 
 4/r 10 cosec 2 Q acting downwards at the centre of gravity of the 
 triangle PAQ. Now (ex. 6, p. 33) the distance of this point 
 i'rom the vertical through ^ = f /i (cot — tan ^). Hence, on 
 expelling the factor cos f — sin ^, the equation of moments be- 
 comes ^ ^^3 (^^ _ ^^/) g;j^2 2 ^ _ ^ 7,n ^ (cos + sin 5) = o, 
 
 or if we take (f) as the inclination of the diagonal AC to the ver- 
 
 tical, since ^ = 0, we have 
 
 4 _ 
 
 3a^(w — to') cos^ 2(|) — 4^^ tw V 2 cos (/) = o. . . (a) 
 
 The factor expelled indicates the vertical position of -4 C — a 
 position which is a priori evident. 
 
 26. Equilibrium of a Body in Heterogeneous Fluid. 
 If the fluid in which the body CAlJB, Fig-. '^^^ is floating 
 under the influence of oravitv alone is one of variable 
 density, the positions of equilibrium are still found by the 
 principle of buoyancy — viz. that the wei,i»-ht of the body is 
 equal to the weig-ht of the fluid displaced, and that the 
 two centres of gravity are in the same vertical line. 
 
 Thus, supposing the specific weight, to, of the fluid 
 at any depth, z, to be any given function of r, draw a 
 horizontal section of the body at depth z, and let A be the 
 area of this section. Drawing another horizontal section at 
 depth z + dz, the volume between these sections is Adz, 
 so that, if IF = weight of body, 
 
 r= f'wAdz, 
 • 
 w^here c is the depth of the lowest horizontal tangent 
 plane to the sm'face of the body. This equation will 
 determine c. 
 
 We shall confine our attention to a few simple examples, 
 as the question is one of small importance.
 
 Pressure on Curved Surfaces. 131 
 
 Examples. 
 
 1. If a luiilnnn solid cone floats, veitex down, in a fluid in 
 which the density is proportional to the depth, find the lengtli 
 of the axis immersed. 
 
 Let w' be the specific weight of the cone, and let the specific 
 
 weight of the fluid at depth z be represented by w' — , where /3 
 
 is some constant length. Ti\king a section of the cone at a dis- 
 
 tance x from the vertex, A = -r^-x^; therefore 
 
 h 
 
 X-nr^hw' ■= -— - w' I a?' (/i' — x) dx, 
 
 ^ l^h? Jo ^ 
 
 h' being the length of the axis, h, which is immersed. From 
 this we have K={^^h')i. 
 
 2. If the cone in tbe last is replaced by a solid cylinder, find 
 the length immersed. 
 
 Ans. h'={2f3h)^. 
 
 3. Find the height at which a spherical balloon of given 
 weight and radius will rest in the atmosphere, assumed of uni- 
 form temperature, and the variation of gravity being neglected. 
 
 Ans. If r feet = radius, h = height of centre of balloon, iv^ 
 the mass in pounds of a cubic foot of air at the ground, B = 
 mass of balloon, and k has the value given in (9), p. 81, the equa- 
 tion which determines h is 
 
 h 
 
 Tik^ Wq [r cosh t — ^ sinh y] e * = B. 
 
 4. Find the position of the centre of buoyancy of the balloon 
 on the same suppositions. 
 
 Ans. Its distance below the centre is 
 
 T T 
 
 (»*^+ 3^') sinh - — 3^r cosh r 
 r cosh J — k sinh 
 
 K 2
 
 132 Hydi-Qstatics and Elcuicntary Hydrokinctics. 
 
 27. Stability or Instability of Equilibrium. In de- 
 termining- the jjosilions of equilibrium of a system in 
 Hydrostatics, and in Statics g-enerally, we are not, so far, 
 concerned with the nature of the equilibrium. A particular 
 poisition which we determine by the ordinary statical 
 equations (those of resolution and moments of forces) may 
 be one which exists only theoretically and could not exist 
 in practice, because it may be one from which if the system 
 is displaced, ever so sli.ohtly, and then abandoned to the 
 forces in action, the disi^lacement becomes greater and 
 g-reater^ instead of being- corrected and destroyed. If a 
 position [A) is one in which the conditions of equilibrium 
 are all satisfied, and if the system is displaced by any 
 means^ or imag-ined to be displaced, into a position [A') 
 very slig-htly differing- from the first, and in this new 
 l)Osition abandoned to the forces in action, then, unless the 
 forces are such as to drive the system back from [A') to [A), 
 the equilibrium is unstable and is practically useless. Thus, 
 a pin resting- a horizontal plane of g-lass will theoretically 
 be in equilibrium if it stands vertically on its point ; but 
 this position does not practically exist, because it is unstable 
 for the slig-htest displacement. A uniform rod, AB, freely 
 moveable round a smooth horizontal axis at the end A 
 is theoretically in equilibrium if it is j)laced vertical with 
 the end B uppermost, but, again, this position does not 
 practicall}- exist. 
 
 Before discussing- particularly the criteria of stability or 
 instability, we may take a few simple examples in which 
 the principle to be employed is obvious. 
 
 Thus, to find whether the oblique position of the rod AB 
 in example 5? P- 128, is stable or unstable, imagine the 
 rod displaced into the position A'B\ such that the small 
 angle CAC'=. h6 ; then if the sum of the moments about A 
 of the forces acting on the rod in this position is in
 
 Pressure on Curved Surfaces. 133 
 
 the sense opi)osed to that of tlie an<^'nlar displacement, 
 these forces will drive the rod back into the position AH, 
 and the eciuilibrium will be stable. 
 
 Generally, without assaming- that yiB is a position of 
 equilibrium, let M be the sum of the moments of the force 
 of buoyancy and the weig-ht about A in a clockwise sense. 
 Then we have 
 
 M= -sin d {4a^{w-7v')-h^io&QQ'^6], 
 
 and the positions of equilibrium are determined by putting- 
 J7 = o. These have been determined in the example 
 referred to. Now if when 6 is changed to Q-k-hQ, the 
 increment of il/ is hM, and if 8 J/ and IQ have the same 
 sign, the newly introduced forces produce a moment which 
 is in the sense of the displacement and the equilibrium 
 is unstable. In other words, if in any position of equilibrium 
 
 — — is + , the equilibrium is unstable, 
 
 -^^— is — , the equilibrium is stable. 
 dd ^ 
 
 Examining first the vertical position, = o, we have 
 
 do 2 
 
 so that this position is stable when 1r w > 4a' [w — w'), i.e. 
 when the oblique position does not exist. 
 
 Examining the oblique position (when it exists), we have 
 
 — — — sli^ 10 sin'^ 6 see'' 6, 
 do 
 
 which value is necessarily negative ; therefore the oljlique 
 position, when it exists, is stable.
 
 [34 Hydrostatics and Elementary Hydrokiuctics. 
 
 Examples. 
 
 1. Find whether the equillhrium of the hody in example 6, 
 p. 128, is stahle or unstable in the positions determined. 
 
 Calculate the sum, M, of the moments of the forces about A, 
 in a clockwise sense, in any position, 6 ; and supposing 6 to be 
 increased by hO, let hM be the change in the value of i/. Since 
 the sense of M is opposed to that in which 6 increases, if hM 
 and hd have the same sign, the restoring moment will be in- 
 creased by the displacement, and the equilibrium will be stable. 
 
 In other words, the equilibrium will be stable if -y— is positive 
 in the position of equilibrium. 
 
 Now we find, as in the example quoted, if I is the thickness 
 of the board, 
 
 M= — (cos - sin 6) [3 a? (w—iv^) — 4h^w{cos 6 + sin 6") cosec^ 2 9]. 
 
 We may, for simplicity, omit I, or consider it to be the unit 
 of length, since its actual value is immaterial to the discussion. 
 
 It will be more simple to take (f), the inclination of AC to the 
 vertical as the variable ; then 
 
 M = sin^ [■^a^{w—to') — 4 ^f 2!^" w cos^ sec^ 2 <^] ; . (i) 
 
 and in any equilibrium position the equilibrium will be stable 
 
 dM . 
 or unstable according as -j-r is negative or positive. 
 
 Examining the symmetrical position, (^ =r o, we have 
 
 dM 4^2 
 
 \^a^{w — wf) — \ \/2A'i<;], 
 
 d^ 3 
 
 which shows that in all cases in which uf > vj the equilibrium 
 is stable ; and, in fact, the equilibrium is stable if 
 
 w' 4 -J 2 W 
 
 — > I — ■ -3, 
 
 w 3 a 
 
 , w' . 
 and unstable if — is less than this. 
 w
 
 Pressure on Curved Surfaees. 135 
 
 Whon the s}niimetiical position, (/> = o, is stable, the inclined 
 position does not exist, as is seen thus : let 
 
 w' 4 V2 /t ' 
 
 — «= I s(i-^'"). 
 
 w 3 a 
 
 where k is any number. 
 
 If A; is positive, the syninietrical position is stable, and if/: is 
 negative, this position is unstable. Then (i) becomes 
 
 jJ/= ^- /i,^w sin (|)[i — X; — cos(/) sec- 2 0], . . . (2) 
 
 and the inclined iK)sition is given by the equation 
 
 COS(/) 
 
 cos'' 29 
 
 Now this equation cannot be satisfied l)y any admissible value 
 of ^ if k is positive, because the least value of the left-hand side 
 is I, which it has when c/j = o. Hence, unless k is negative, 
 (3) cannot l)e satisfied by any admissible value of </). 
 
 When the symmetrical position is unstable, the inclined is, of 
 course, stable. 
 
 2. Find the conditions of equilibrium of a solid body floating 
 in a liquid of greater density than the solid ; and sIkjw that 
 when it is in stable eciuilibrium the height of the common centre 
 of gravity of the solid and liquid is a minimum. 
 
 If a body be floating in a liquid contained in a cylindrical 
 vessel and be pressed down through a small distance, iSz, show 
 tliat the common centre of gravity of the body and the liquid 
 will be raised through a height 
 
 B {^zf 
 
 where A and B are the areas of the cross-sections of the cylinder 
 and the body in the plane of floatation, and h is the height of 
 this plane above the base of the cylinder. {Mathematical 
 Tripos, 1878.) 
 
 Tlie first part is evident from the general principle that the 
 centre of gravity of any material system (consisting, in this case, 
 partly of a solid and jjaitly of a li(|uid) under the action of 
 gravity and of forces wliich do no work in disi)lacenient (pres- 
 sures exerted by the sides of the containing vessel) assumes the 
 lowest position possible.
 
 136 Hydrostatics and Elcmcntaiy Hydrohindics. 
 
 Let z be the lieiglit of the centre of oravity of the body and ^ 
 tliat of tlie centre of gravity of the liquid that woukl occupy the 
 place of tJio immersed portion above the base of the cylinder; 
 W =z weight of body, V = volume of liquid displaced, ?o = spe- 
 cific weight of liquid. Let CB be the original surface of the 
 liquid, and let the line of floatation be marked on the body and 
 denoted by F^^^, let C^D' be the new surface of the liquid, and 
 let CD' be at a height Ax- above CD. 
 
 Then, since the volume of the liquid is unaltered, 
 
 BAz = {A-n)Ax (i) 
 
 Now take the sum of the mass-moments of the body and the 
 liquid about the base in the new position, and we have 
 
 W{z-Az) + lA 10 {h + Axf-Vw (C- Az) 
 
 — lBw{Ax+Az) {2h+Ax — Az). 
 
 Subtracting from this the mass-moment before displacement, 
 and observing that W = Viv, there remains 
 
 lAw{2hAx+Ax-)-lBiv{Ax+Az)(2h + Ax-Az), . (2) 
 
 of which, by (i), the terms of the first order in the small quan- 
 tities cancel — as they must by the principle of virtual work ; 
 so that (2) becomes 
 
 hAw. Ax- + \Bw (Az-'-Ax-), 
 
 which by (i) is ^ ABw 
 
 ^a^b'^-^- 
 
 Dividing this by the sum of the weights, Ahw, of the body 
 and liquid, we obtain B A"'^ 
 
 for the amount by which the centre of gravity of the compound 
 system is raised. 
 
 28. Principle of Virtual Work. The g-encral principle 
 of Virtual Work may be thus stated : if any material system 
 18 in eqml'thmitn under the action of forces applied at given 
 pjoints in the system, and if we imagine these p)oints to receive 
 any small displacements whatever in space tmt not in the 
 material system itself, the sum of the quantities of icork done
 
 Prcssinr on Curved Surjaccs. 137 
 
 for »nch flisjihicements hy all the act ivy forces, exicrnal ami 
 inlerual lo l/ie si/alem, is zero, or, al viosl, an infinitesimal of 
 the second order if the greale-\t displacement in the system 
 is regarded as an infniUsimal of the frst order. 
 
 It is fully explained in Sialics, Vol. I, Chap. VII, and 
 Vol. II, Chap. XV, that the internal forces may or may 
 not enter into the equation of Virtual Work, accorflinii;- 
 to the nature of the dis])]acements imag-ined. Thus, if an 
 internal force* consists of the tension of a strin'j;' connecting- 
 two i)oints, A, B, of the system, this force will not enter 
 into the equation if in the displacements of A and B the 
 distance between them is unaltered. Also if the internal 
 force is the reaction at a smooth joint, A, connecting- two 
 bodies in the system, its virtual work will be zero if in the 
 disj)lacements the bodies are still represented as connected 
 by the joint. If the connection were represented as severed, 
 the stress would do ^\■ork which must be included in the 
 equation. 
 
 Now in Hydrostatics we have often to deal with forces 
 whose points of application in the material system would 
 be dis])laced not merely in space but in the material system 
 itself (and whose magnitudes would also be slightly altered 
 — thoug-h this is of no real consequence) if the system were 
 actvally displaced into any close ])osition ; and in such 
 cases a danger of error arises in applying the equatioii of 
 Virtual Work. 
 
 We shall illustrate this by solving the problem of 
 example 5, p. 128, by the principle of Virtual Work and 
 determining whether the equilibrium of the rod is stable or 
 nnstal)le. 
 
 * More properly sirvsst — with a view to the double aspect of every 
 intiTnal forcu in a system. What we call a force e.ttcnud to a j,Mveii system 
 would also be a stress (with equal and opposite values) if both the Ijodies 
 between which it is exerted were included in the system.
 
 138 Hydrostatics and Elementary Hydrokinetics. 
 
 For any small ang-ular displacement of the rod, from an 
 equilibrium position, about the axis A, the sum of the works 
 of the force of buoyanc}^ and the weig-ht IF vanishes. 
 
 In applying^ this condition it is not impossible that 
 the student would proceed thus : let ^ be the depth of // 
 below the surface and z the depth of G ; then the equation 
 of virtual work is 
 
 -L .hC+ W.h: = o. 
 But L = (2a — //sec6)s?o and ^=flcos^ — ^^, 
 bC = —asind . bO, while z = acos6 — //, 
 .'. bz = —a sin 6 .bO, 
 
 so that the above equation gives L = IF, a result which is 
 known to be false. 
 
 The fallacy involved in this solution is the following-. 
 
 In applying" the equation of virtual work to characterise 
 a position of equilibrium, we must imagine the points 
 of application of all forces to remain the same in the body ; 
 we contemplate simply displacements of these points in 
 space — not in the body as well ; and, strictly speaking, we 
 do not contemplate new forces introduced by the displace- 
 ment, although if such newly caused forces are of infini- 
 tesimal magnitude and their points of ajiplication receive, in 
 the displacement, only infinitesimal motions, the introduction 
 of such forces will not influence the validity of the equation. 
 
 The above expression for bCis not the one proper to the 
 equation of virtual work. 
 
 1 or, if the imagined new position of the rod is AB', the 
 new centre of buoyancy, 7/', is the middle point of the 
 immersed portion C^£\ and the 8^ above is the difference of 
 level of H and //' ; whereas if we take AJ = All, J is the 
 contemplated displacement of the original point, //, of 
 application of the force L, and the value of 6C proper 
 to the equation is the difference of level of // and /.
 
 Pressure on Curved Surfaces. 139 
 
 Now it is easily seen that the ( of J is 
 ^_ (,, -i_ \ // sec 0) sin (? 8 61, 
 
 so that the proper vahie of 8 C is —{a + \// sec 0) sin <? . 8 0, 
 and the equation of virtual work is 
 
 L(a + l/iseee)sin6b0-ira&m0be = o. 
 
 which gives the correct value of 6. 
 
 To find the work done in forcing- the rod down into the 
 lluid from one inclination, a, to any other, (f), take it in any 
 position, AB, and let us find the amount of work done by 
 the force of buoyancy and the weight in reaching the 
 position AB', the angle BAB' being 8^. The work of the 
 force of buoyancy is the work of L in the displacement of 
 // to J plus the work of a small force of buoyancy corre- 
 sponding to the submergence of a small element of length 
 above C, and this latter work is an infinitesimal of the 
 second order. Hence if 8 F is the work done by the forces 
 acting on the body, 
 
 in which bC has the second of the above values. Thus 
 . j'^ f. r*{4a2(w_w')-//2wsec-(9} sin 6d 9 
 
 2 J a 
 
 =: _ { —^a^ ^20 — w') + //^ w see a sec ^} (cos </>— cos a). 
 
 As regards stability or instability, a position of equilibrium 
 is one of stability if the amount of work done by the forces 
 in reaching it from a given position is a maximum ; for, in 
 that case, any other forces ai>plicd to disturb the system 
 would have to do positive work. Hence any position of
 
 140 Hydrostatics and Elementary Hydrohinctics. 
 
 equililninm uill Le stable or nnstahlo according- as it makes 
 
 d-r . .,. 
 
 -T-Tz neg-ative or positive. 
 do- '^ ^ 
 
 Assuming that the oblique position given by (3) exists, 
 
 we have for it 
 
 (1-V 
 
 -—-r = — sich- sec ' Q sin" Q, 
 
 (16^ 
 
 which is essentially negative ; therefore the oblique position 
 is stable. For the vertical position 
 
 = - {^a^ (w — iv') — Ii 
 
 17 1 1 
 
 which is positive when the oblique position exists, and 
 in this case the vertical position is therefore one of in- 
 staltility. When the oblique position does not exist, the 
 above expression on the right is negative, and the vertical 
 position is (as is evident a priori) stable. 
 
 The value of F could also be calculated from the moment, 
 3f, of the acting forces about A ; for the work done by 
 a couple of moment M for a small displacement of the body 
 to which it is aj)plied is M . b9 ; hence 
 
 bF= Il.bd 
 
 = ^ sin d [ao" {10 - w) - /2 m sec^ 0) .U, 
 
 which is identical with (a). 
 
 As another example of the application of the principle of 
 virtual work, consider the case of two thin uniform rods, 
 AB, BC (Fig. 45), each of specific weight ?//, freely jointed 
 together at the common extremity B, and resting partly 
 immersed in a homogeneous liquid of specific weight v), the 
 rod AB being freely moveable round a horizontal axis fixed 
 at A at a given dej^th, 2J1, below the surface of the liquid. 
 It is required to determine the position and nature of the 
 equilibrium.
 
 Pressure on Curved Surfaees. 
 
 '4' 
 
 Lot AB — ia, BC = 20, s = area of normal section ol" 
 each rod, 6 = inclination of AB and = inclination of BC 
 to the vertical. Let G, G' be the centres of j^-ravity of the 
 rods, and II, IV their centres of buoyancy, II beinp: the 
 middle point of the immersed portion Am, and //' the 
 middle point of Cn. 
 
 The i)ositions of equilibrium can be easily found l»y 
 elementary principles. Thus, considering- the sei)arate 
 
 Fig. 45- 
 
 equilil)rium of the rod BC, we see that the reaction of AB 
 on BC is a vertical upward force at B equal to W' — L'\ 
 and since the moments of W and L' about B are equal and 
 opposite, we have at once 
 
 = ikh, suppose. 
 
 (0 
 
 Thus, L' is known : and then taking- moments about ./ 
 for the separate equilibrium of xiB, we have 
 
 Ar,rrr.^ka{{a^oJj)[i-ic)-2h\-, . . (2) 
 
 and these two equations determine the position of the 
 system. 
 
 To obtain the equation of virtual work, imagine to
 
 142 Hydrostatics and Elementary Hydrokmetics. 
 
 increase by 8^ and ^ by 8(/), and calculate the vertical 
 descents of the points //, G, G\ IV , the points // and H' 
 being" supposed not to shift their positions in the rods. If 
 the vertical descents of these points are, respectively, 8^ ^•^j 
 8/, 8^', the virtual work, 8 F, done by all the forces is 
 given by the equation 
 
 hr=-UC^-Whz+W'hz-L'hC, . . . (3) 
 
 and this virtual work must be put equal to zero for the 
 position of equilibrium. 
 
 Now AH = h sec Q, and if AB turns round A through 
 8^, the vertical descent of H is AH .bO . sin6, so that 
 b( z= /i tan . hd. To get the vertical descent of H\ ob- 
 serve that if -S did not move, the vertical descent of //' due 
 to an increase of (f) would be —£H'.h({).siii^', but B 
 descends throug-h a distance 2a sin ^ . 8^ ; hence 
 
 bC= 2asine.h6-BH'sm(f)bci). 
 
 But BH^= h-\-{a cos Q — h) sec ; and we have 
 
 8C = // tan^. hQ\ hz = a sin ^.8^, 
 
 8 / = ia sin Q .hO — h sin ^ . 8 (/>, 
 
 ci' — ia sin Q .'hQ—\b-\-{a cos Q — li) sec(/)} sin <^ .h<\>. 
 
 Also W= 2asw\ W=2hsiv\ L = 2/iswsec9, 
 
 X'= 2SW [b — (a cos 6 — //) sec (j)] . 
 
 Substituting- these values in (3), and collecting the co- 
 efficients of the independent variations bd and 8^, we have 
 
 bV = 2s {{a'^ + 2ali) w' —2ahio—hhv sec? Q 
 
 -\-2a\a cos 6 — /i)iv sec <})] sin 6 . bd 
 + 2s {P{tv — w') — [a cos 6 — Ay w sec^ } sin . 8 . (4) 
 
 Now since the position of equilibrium is obtained by 
 putting bF = o, we must have the coefficients of 8 ^ and 
 b(f) each equal to zero; and thus we obtain both 6 and (p.
 
 Pressure on Curved Surfaces. 143 
 
 Equal in "f to zero the coefficient of 6 0, we ol)tain at once 
 the result (1). 
 
 It is manifest that the expression (4) for 8 ^ is a perfect 
 diirerential, since 
 
 dV 
 
 -— : = 2 .* ( {(r -\- 1 ah) to' — lahw — ]fi tv sec'"^ Q 
 
 4- la (a cos 6 — //) w sec </) } sin Q, . (3) 
 
 dF 
 
 — = 2s \1P- {io — io')—{a cos Q — 1if to sec'^ 0} sin 0, . (6) 
 
 and it is evident that we get identical results by dilFeren- 
 tiatino- ("z^) with respect to f/) and (6) with respect to 0. 
 
 To find V, the potential work of the forces, in any 
 position, integ-rate (6) with respect to <^ and add an unde- 
 termined function of 6. Thus 
 
 V= —2s{ U^ [lo — w') cos 4- (fl cos — Ii-Y 10 sec } 
 
 + ^s.f{Q). . . (7) 
 
 Taking" -— of both sides of (7) and ecpiating- the result to 
 (16 
 
 ^ in (j), we find 
 
 f (6) = {{a'^ + 2a/j} w'— 2. a fj 10- /rw sec- 6} sin 6, . (8) 
 
 ,-. f(^d) = — {{a'^ + 2ah') w' — 2 alio] cosd — /rw scc9 + 0, 
 
 where C is a constant. Hence 
 
 jr 
 
 — = — f(a-+2alj) to' — 2abto ] cos 6 — W^io sec Q 
 2.y ^^ ' 
 
 — /j-(w — 1(/) cos (p — {a co^O — ^)^ to sec (}j + C. . . (9) 
 This may be put into a more simple form by putting- 
 Am = 2x and Bn = 2t). 
 
 mi ^ ^1 /^ (^ — ^) 
 Ihus cos = - : cos = — ^ , 
 
 X ^y 
 
 V 
 
 and if we put j- = L, and 
 
 ^ 2 //s
 
 144 Hydrostaiics and Eloncntary Hydrokiiictics. 
 
 {a~ + lah) w' —lahw = A, li- [ro—io') = B, 
 
 TT -^ a — x^B N , , 
 
 U = —■\■v•T^ ( yvy). . . . (lo) 
 
 X X ^ IJ ^ ^ ' 
 
 II once 
 
 (lU If, /7? n) (IV a—X/r Bx , , 
 
 f/x X- ( ^ J/ ^) dij X ^ 1/-^ ^ ' 
 
 For the position of equilibrium -— = o ; and --— = o; 
 
 ax ay 
 
 B a 
 
 therefore y =z (~^ ; wx^ = yl+2a[B2v)^, . . (12) 
 
 ■7V 
 
 which agree with the previous results, and the first shows 
 that B must be positive, i. e. w must be > ?v\ 
 
 For the stahiUty of equilibrium, V must be a maximum, 
 and for a maximum or minimum (see Williamson's Differ- 
 ential Calculus, Chap. X.) 
 
 (PV (T-r .(PF.^ 
 dx- fly^ ^dxdy' 
 
 dHJ 2 ( , fB o drU iBa-x 
 
 dxdy x~ ^ y'^^ 
 
 and since in the oblique position (distinct from (/> = o) this 
 
 value of vanishes, and -y-g is essentially positive, the 
 
 condition is simply that -— - must be positive ; i. e. 
 
 fl — a? must be +, [i'^) 
 
 and this is necessarily the case. 
 
 d'^U . d^r . 
 
 Since -7-7, is +, —rr, is — , and the value of F which we 
 dx'^ dx-
 
 Pressure on Curved Surfaces. 145 
 
 are discussinf^ is therefore (Williamson, above) a maximuin 
 and not a niiniimim. Hence from (12), 
 
 ■ica- > A + 2a{I^w)K 
 Reptorinq- the values of A and i?, this g-ives as the condi- 
 tion for stability ,/ a^ + 4ab 
 
 — < Ji- - . 
 
 w (a + 2 hf 
 
 29. Positions of equilibrium of a freely floating body. 
 A given body, provided that its weight is less than that of 
 an equal volume of water, may be placed in several positions 
 of e(iuili])rium in the water. 
 
 We shall lay down a few definitions of terms in common 
 use with veg-ard to freely floating- bodies. 
 
 The section of floatation of a floating body (Fig-. 47) 
 is the section of the body made by the surface, LM, of the 
 liquid. In Fig*. 35 the section of floatation is represented 
 by the horizontal line AB. 
 
 The area of floatation is the area of the section of floata- 
 tion. 
 
 The (Jisplacement is the volume of the displaced liijuid, 
 which is the volume included between the section of floata- 
 tion and the surface of the immersed portion of the bodv. 
 
 In Fig. '^j the disjilacement is the volume represented 
 in projection by the curve ABB. 
 
 In all possible positions of equilibrium of a given body 
 floating freely in a g-iven liquid the displacement is constant. 
 For, if //' is the weight of the body, / the displacement, 
 and 70 the specific weig-ht of the li(iuid (i. e. the weig-ht 
 jier unit volume), the first condition of Cor. 2, Art 22, gives 
 
 r70= IF, 
 
 .: rJL, 
 
 w 
 which shows that the displacement is constant. 
 
 L
 
 146 Hydrostatics and Elcinoitary Hydrokinetics. 
 
 Hence, without any reference to the second necessary 
 condition of Cor 2, Art. 22, all possible positions of equili- 
 brium are exhausted by describing' planes so as to cut off a 
 
 IF 
 
 volume — from the body. All planes which cut off this 
 
 volume are (without reference to the second condition) 
 possible planes of floatation ; that is, if we mark the ex- 
 terior surface of the body along the curve in which it is 
 cut by any such plane, and we then place the body in the 
 liquid so that this curve lies wholly in the free surface, LM, 
 of the liquid, we shall obtain a position in which the body 
 will float when left to itself, provided the second condition of 
 Cor. 2, Art. 22, is fu If lied in this position. Of course, as a 
 rule, this condition ^^^ll not be fulfilled, so that of the 
 (infinite) number of possible positions, as above defined, only 
 a small number will satisfy both of the conditions that 
 must hold. 
 
 All the planes which cut off the constant volume 
 
 W 
 
 — from the body envelop a surface called the surface of 
 
 floatation, while the corresponding centres of buoyancy 
 trace out a surface called the surface of huoyancy. 
 
 Now it is evident that, in order that a plane cutting 
 
 W 
 
 off the volume — should determine an actual area of floata- 
 
 tion, the right line joining G, the centre of gravity of the 
 body, to the corresponding centre, iZ, of buoyancy must be 
 at right angles to this cutting plane, because in a position 
 of equilibrium of the body the line GH is vertical, while 
 
 W 
 
 the section of floatation Twhich cuts off the volume — ) is 
 
 V w ^ 
 
 horizontal. 
 
 We shall now show that in every j^osition of equilibrium 
 the line GH is normal to the surface of buoyancy at li.
 
 Pressure on Curved Surfaces. 
 
 147 
 
 Let AH (yiix. 46) be a plane cntting" oiF a volume ylUB 
 from a homog-eneous body ; let 7/ be the centre of gravity 
 of this volume ; let A'B^ be 
 another plane difFerinf'' slig-htly 
 in j)Osition from AB and cnttinq* 
 off an equal volume, A'JJB'; and 
 let ir be the centre of gravity 
 of this new volume. Then the 
 line nil' is ultimately parallel 
 to the plane AB. For, regard 
 the volume ADB as consistins: 
 of the portion A'DB and the 
 thin wedge ACA'; and also re- 
 gard the volume A'BB' as con- 
 sisting of the portion A'BB and 
 
 the thin wedge BCB\ Let n be the centre of g-ravity of 
 the portion A'BB which is common to both volumes, g the 
 centre of gravity of the first wedg-e and fj that of the 
 second. Then to find K we join (j io n and. divide gn at II, 
 
 Fig. 46. 
 
 so that 
 
 similarly 
 
 Hence ^, = 
 Hn 
 
 gll _ volume of yi'i) J? 
 Rn volume of wedge ' 
 
 g'H' _ volume of A'BB 
 H'n volume of wedge 
 
 ^rn— 3 and therefore the line IIII' is parallel 
 H n ^ 
 
 to gg', and therefore when the wedges ai'e both made inde- 
 finitely thin, the line gg' then lying in the plane AB, the 
 line Illi', w hich then becomes a tangent at IT to the locus 
 of II, is parallel to the plane AB. Since this is true what- 
 ever be the oneniation of the new cutting plane A' K , the 
 assemblage of lines IIIV which touch the surface of buoy- 
 ancy at II form the tangent plane to this surface at //, 
 which plane is therefore parallel to the cutting plane AB. 
 
 L 2
 
 148 Hydrostatics and Elementary Hydrokinetics. 
 
 It now follows that all positions in tchich a given bodi/ can 
 float freely in a Jwrnogeneous liquid are ohtained by drawing 
 normals, GH^, GH^, GH^, ... from the centre of gravity, G, 
 of the body to the surface of buoyancy, and placing the body so 
 that any one of these normals is vertical. For, the line Gil 
 must be vertical — that is, it must be perpendicular to the 
 plane of floatation ; and as the tang-ent plane at II to 
 the surface of buoyancy has been proved to be i^arallel to the 
 plane of floatation, the line GH must be the normal to the 
 surface of buoyancy at H. 
 
 When the contour of the floating body is a surface of 
 continuous curvature, the surface of buoyancy is, of course, 
 a surface of continuous curvature ; but when the contour of 
 the body is not of continuous curvatm'c (as in the case of a 
 ship with a closed deck when all geometrically possible dis- 
 placements — involving" the submersion of the deck, the keel 
 being above the surface of the water — are considered) the 
 surface of buoyancy will be a broken or discontinuous sur- 
 face. 
 
 As an example of a surface of buoj^ ancy of discontinuous 
 curvature, take the case in which the body is a triangular 
 jirism, the vertical section of which through its centre of 
 gravity is a triangle ABC (Fig. 47), and consider all pos- 
 sible displacements of the body in the plane of this triangle. 
 
 If, for definiteness, we assume the specific weight, iv, of 
 the fluid to be to the specific weight, v/, of the body as 16 
 to 9. the volume submerged will be -^^ of the volume of the 
 body. We are therefore to draw all possible lines, such as 
 1/31, across the face of the triangle ABC cutting ofi* areas, 
 LBCM, equal to jV of the area ABC. 
 
 The immersed area wull sometimes be triangular, and 
 sometimes (as in the figure) quadrilateral. To trace out the 
 locus of the centre of buoyancy when the vertex A alone is 
 above the liquid, the line LM is to be di-awn so as to cut ofl"
 
 Pressure on Curved Surfaces. 
 
 ^49 
 
 the trian ovular area LAM = ^^^ ABC \ hence if /i3/ = y, 
 AL = z, AC = h, AB = c, we have 
 
 Let the first position of the cutting- line pass through 7i, 
 and lot the line revolve clockwise so that it assumes the 
 position LM in the figure ; then it will reach another posi- 
 tion in which it passes through C. When the line passes 
 through B, we have z = e, .'. y = xV b = AQ, suppose ; the 
 
 Fig- 47- 
 
 cutting- line is then BQ, and the immersed area is the tri- 
 angle QCB, whose centre of gravity is the point q. 
 
 In the second extreme position ?/ = 0, .-. c = ^V c = AR, 
 suppose; the cutting line is RC, and the immersed area 
 is the triangle RCB, whose centre of gravity is Ij^. The 
 centres of gravity of all the intervening quadrilateral 
 areas will lie on a cui-ve, e^Mb^, between the points (?, and 
 b^. This curve is easily proved to be a portion of a hyper- 
 bola having asvmptotes parallel to AB and AC. For, A31 
 and AL being- !/ and z, the co-ordinates of the centre of 
 gravity of the triangle LA3I with reference to AC and AB
 
 150 Hydrostatics and Elementary Hydrokinetics. 
 
 are (3 ^, 3-), ^vhile those of ABC are (P, \c) ; and if (r?, C) 
 are the co-ordinates of H, the centre of gravity of the 
 quadrilateral LBCM, the Theorem of Blass Moments (Art. 10) 
 gives ^^+.^^1^0^^ (2) 
 
 9C+i^=-^^, (3) 
 
 which by ( i ) give 
 
 7 
 
 (M^_9,)(M._9C) = ^5., ... (4) 
 
 showing that the locus of -ff is a hyperbola whose centre is 
 at the point (|f- h, \^ c). 
 
 Let the cutting line still revolve clockwise from the posi- 
 tion RC, so that R moves towards A ; then the immersed 
 area will be a triangle whose vertex B is submerged, and 
 the locus of the centre of gravity of this triangle will be a 
 portion, h^ b^, of another hyperbola whose centre is B and 
 asymptotes BA and BC, the point d^ being- the centre of 
 gravity of the triangle ABP cut off when the revolving 
 line is in the position AP, the point JP being such that 
 BP = ^ BC. 
 
 Thus there is an abrupt transition from one curve to 
 another at the point b-^. 
 
 As the cutting line still revolves clockwase from the posi- 
 tion AP, the locus of i/ will be a portion, b^a^, of a hyper- 
 bola whose asymptotes are parallel to CA and CB, the 
 immersed area being a quadrilateral until the line reaches 
 the position Q'B, such that CQ' = yV^- After this the 
 immersed area will be a triangle with the vertex A im- 
 mersed, the locus of // being a portion, a^ a^, of a hyperbola 
 whose centre is A and asymptotes AB, AC, and the im- 
 mersed area will continue triangular until the line reaches 
 the position CR' such that ylR' = yg- c ; and so on. 
 
 Hence the vertical section of the surface of buoyancy 
 consists of portions of six different hyperbolas whose points
 
 Pressure on Curved Surfaces. 151 
 
 of intersection are ('^Jj^Jj^^d-i, ••• • The number of normals 
 that can ])e drawn from G, the centre of g-ravity of the 
 l)rism, to this broken locus of If will determine the number 
 of positions of equilibrium of the prism. 
 
 Examples. 
 
 1. A rectangular block of specific wiiirlit i«' floats in a liquid 
 of specific weight iv with one lace vertical ; fiud the curve of 
 buoyancy and the popitions of equilibrium, the same face being 
 always kept vertical. 
 
 Ans. Let the sides of the vertical face be 26. 2 c, and sup- 
 pose that in the initial position the side 2c is vertical; then, so 
 long as the upper edge 2 h is out of the liquid and the immersed 
 portion a quadrilateral, the curve of buoyancy is a parabola, 
 concave upwards, whose equation with reference to the hori- 
 zontal and vertical hues through the initial centre of buoyancy 
 as axes of x and 3/ is 
 
 The initial position is one of equilibrium (which may or may 
 not be stable), and other positions are obtained by drawing nor- 
 mals from the middle of the face to this parabola, provided that 
 these normals fall within the relevant portion of the parabola. 
 Now the relevant portion terminates at the point whose co-or- 
 dinates are (- j — • -), this point being the centre of buoyancy 
 
 when the immersed area begins to be triangular. In order that 
 it should be possible to draw a normal within the limits the y of 
 
 the point at which it is normal must be < — • - , and hence 
 
 - > 6 — (i )• 
 
 To this portion of a parabola succeeds a portion of a hyper- 
 bola which is the curve of buoyancy so long as the immersed 
 area is triangular; this, in turn, is succeeded by another portion 
 of a parabola ; and so on. 
 
 2. Find the surface of buoyancy in the case of a right circular 
 cone immersed in a homogeneous liquid with its vertex down- 
 wards.
 
 152 Hydrostatics and Elementary Hydrokinetics. 
 
 Ans. So loug as uo part of the base of the cone is sub- 
 merged, the surface of buoyancy is a hyperboloid of revolution. 
 
 Let a be the semi-vertical angle of the cone, i> the perpendi- 
 cular from the vertex, V, on any plane cutting the cone, and co 
 the angle which ^ makes with the axis of the cone. Then the 
 plane through p and the axis will cut the cone in two lines, VA, 
 VB, which intersect the given cutting plane in the points A, B, 
 which are tlie extremities of the major axis of the ellipse. If 
 VA = i\, VB = r.,, we have 
 
 VA= ^ -, VB = 
 
 co3(co + a) cos(co — a) 
 
 p sin 2 a . . 
 
 hence AB = —\ v-5 — ; and the semi-mmor axis 
 
 cos^a — sm-co 
 
 = (F^l.F5)*sina = 
 
 p sma 
 
 (cos^ a — sin^ 00)^ 
 
 If V is the volume of the displaced liquid, F= - x area of the 
 
 ellipse cut off. ^ 
 
 „ TT «' sin a sin 2 a ^.i • -^ r n j.i j. i.i. 
 
 Hence F= — ; ^ . From this it follows that the 
 
 6 (cos^ a — sin^co)- 
 
 product VA . VB is constant, and since the co-ordinates of the 
 centre, C, of the ellipse are the halves of the sums of those of A 
 and B, if we I'otate the cutting plane so as to confine the 
 motions of 2^ to one plane, the locus of C is a hyperbola having 
 tlie generators VA, VB for asymptotes. Hence for all possible 
 positions of the plane, the locus of C is a hyperboloid generated 
 by the revolution of this cui've al)out the axis of the cone. But 
 if H is the centre of buoyancy in any position, H lies on the 
 line VC, and VII = |FC; hence the locus of // is a similar 
 hyperboloid. 
 
 If I, m, n are the direction cosines of /> v;ith reference to any 
 two rectangular axes of x and y through the vertex and the axis 
 of the cone, we have l'^ + m^ = sin^ co ; and if x, y, z are the co- 
 ordinates of C, VA = rj, VB = r^, we have 
 
 I sin a 
 
 X— KTy — r^) — -. 1 
 
 ^ ' 2 sm fo 
 
 , in sin a 
 
 •^ ^ * -'2 sin (o
 
 Pressure on Curved Surfaces. 
 
 T53 
 
 2 = (^1 + r,) 
 
 cos a 
 
 Also, since ;; = ^-(008^0 — sin'^co)^, where k is a given con- 
 stant, it follows that the locus of G is 
 
 sos'^a 
 
 sin" a 
 
 = k\ 
 
 Fig. 48. 
 
 30. Geometrical Theorem. In connexion with the 
 question of the stability of floating- bodies the followin*,^ 
 theorem is important. 
 
 A volume AKB, Y\g. 48, being- 
 cut off" from a solid body by a 
 plane section ALBL' , any other 
 plane, A'LIYL\ making- a small 
 angle with the first plane and 
 cutting off" an equal volume, 
 A'KB', must pass through the 
 centroid (or ' centre of g-ravity '), 
 C, of the area ALBL'. 
 
 For, at any point, P, in the plane section ALBL' 
 describe a small element of area, (IS ; let the perpendicular, 
 Pn, from P on the line, LL\ of intersection of the two 
 planes be denoted by x ; let 8 ^ be the angle betwx-en the 
 two planes ; and round the contour of dS draw perpen- 
 diculars to the plane of (IS, these forming a prism which 
 intersects the plane A'LB'L' in a small area at Q. Then 
 Z Q?iP=hO, QP = xb6, and the volume of the small 
 prism is very nearly xdS.hO. Ilonce the new volume 
 A'KB' = vol. AKB + b6/x(lS, the prisms in the wedge 
 L'LBB' being- taken positively, while those in the wedge 
 L'LA'A are taken negatively. If the two volumes cut off 
 are the same, we must have 
 
 fx(lS=0, (a) 
 
 the integration including all the elements of area of the
 
 154 Hydrostatics and Elementary Hydrokinetics. 
 
 plane section ALBL' . Now, by the theorem of mass- 
 moments the rig-ht-hand side of (a) is Ax, where A is the 
 area of the plane section, and x the distance of its centroid 
 from the line LL' ; hence x — o, i. e. the centroid of the 
 area must lie on LL'. 
 
 A visible representation of this fact is obtained by 
 holding- in the hand a tumbler partly filled with water and 
 imparting- to it small and rapid oscillations which cause the 
 surface of the water to oscillate from right to left ; the 
 planes of the successive surfaces of the water can then be 
 seen to pass always through the centre of the horizontal 
 section. 
 
 Fig. 49. 
 
 31. Small Displacements. Metacentre. Suppose a 
 body, ACB, Fig, 49, floating in equilibrium in a homo- 
 g-eneous liquid to receive any small displacement; it is re- 
 quired to find whether the equilibrium is stable or unstable.
 
 Pressure 07i Curved Surfaces. 155 
 
 Every displacement can be regarded as consisting of two 
 kinds of displacement — viz. a vertical displacement of 
 translation, upwards or downwards, w^hich diminishes or 
 increases the volume of the displaced liquid, and a rotatory 
 or side displacement which leaves the volume of the dis- 
 placed li<iuid unaltered. 
 
 II' the displacement is small, these component displace- 
 ments can be treated separately, and it is evident that 
 equilibrium for the first kind of displacement is stable. 
 
 We shall confine our attention, then, to displacements of 
 rotation which leave the volume of the displaced liquid, 
 and therefore the magnitude of the force of buoyancy, un- 
 altered. 
 
 Let G be the centre of gravity of the body, AxBx' the 
 section of floatation before displacement, H the centre of 
 buoyancy (i. e. the centre of volume of the immersed 
 portion ACB) before the displacement, and let this section 
 of floatation be supposed to be marked on the surface of the 
 body. 
 
 The body is represented as slighty displaced, the new 
 section of floatation being A'xB'oa ; the new centre of 
 buoyancy (centre of volume of A'CB') is W, which must 
 be somewhere very close to H, the centre of volume of 
 ACB. 
 
 If the body, having been displaced to the new position, 
 is then left to itself, it wall be acted upon by two forces, 
 viz. its weight, W, acting through G, and the force of 
 buoyancy, L, (w hich is equal to W, since the volume of the 
 displaced liquid is constant,) acting vertically upwards 
 through W. These forces form a couple. 
 
 Now it is clear that if the line Il'L cuts the line 67/ 
 above G, in a point M, the body will be acted upon by a 
 couple which tends to destroy the displacement ; while, if 
 M is below G, the moment of the couple, being in the
 
 156 Hydrostatics and Elementary Hydrokinetics. 
 
 sense of the displacement, will cause the body to fall 
 farther from the position of equilibrium, which is therefore 
 unstable. 
 
 It may happen, on account of the shape of the body and 
 the position of the axis, ogx, of displacement, that the 
 vertical line through H' does not intersect the old line 
 GR of centres of gravity. At present we shall confine our 
 attention to cases in which it does intersect GH, and sub- 
 sequently we shall find the condition that such intersection 
 shall take place. 
 
 Manifestly if G is below H, the equilibrium will be 
 stable, and the consideration of this ease may be dismissed. 
 The case in which G is above H is very important inas- 
 much as it is the case of ships generally, and especially 
 that of large ironclads, in which so much of the mass is in 
 the upper portion. 
 
 If p is the length of the perpendicular from G on the 
 line K'L^ the moment 
 
 L-P . ■ ^ (i) 
 
 of the new force of buoyancy about G is called the moment 
 of stability. 
 
 To calculate 7), or the position of the point M — which 
 point is called the mefacenfre — replace the actual force of 
 buoyancy due to liquid A'CB' by a force of buoyancy con- 
 sisting of three components, viz., 
 
 (fl) an upward force due to liquid ACB, 
 {b) an upward force due to liquid Bxx'B\ 
 {c) a downward force due to liquid Ax'xA'. 
 
 The two sections of floatation intersect in the line xx\ 
 and since the volumes ACB and A'CB' are equal, this line 
 xx' must pass through the centroid, 0, of the section of 
 floatation (Art. 30). 
 
 Since the volume of the wedge Bxx' B' = the volume of 
 the wedge Ax'xA', the forces (b) and (c) being the weights
 
 Pressure on Curved Sitrfaees. 157 
 
 of those wedges of liquid, form a couplo, each actin<^ 
 throng'h the centre of g-rtivity of the correspondiny- \vedi,'-o: 
 while force {a) is L acting- up throuoh If. 
 
 Also L . 2i = t\\e sum of the moments of these forces 
 ahout the axis through G perpendicular to the plane of 
 displacement. Now since the forces (h) and {(■) form a 
 couple, the sum of their moments about all parallel axes is 
 the same, and hence the sum of their moments about the 
 horizontal axis through G = the sum of their moments 
 about xx', which latter we shall take. The wedges may 
 be broken up into an indefinitely great number of slender 
 prisms perpendicular either to the plane A'xB'x' or to the 
 plane AxBx'. Taking the latter mode, at any point P in 
 the area jLxBx' describe an indefinitely small area (IS, and 
 round its contour erect perpendiculars which will cut off a 
 small area at Q on the plane A'xBx'. Let AOB be the 
 diameter at perpendicular to x'x ; take Ox and OB as 
 axes of X and ij ; let the perpendicular Pn from F on x'x 
 be y, and let 6 be the small angle, QnP, through which the 
 body is displaced round x'x. Then the volume of the 
 prism PQ is Oi/dS; its weight (reversed for buoyancy) acts 
 at the middle point of PQ, and may be resolved into the 
 components ^^r^/f/xS cos parallel to PQ Viudi 6 in/ dS sin 6 
 parallel to Pn ; i.e. into components 6ioi/dS and &-wi/dS in 
 these directions. 
 
 The moment of the latter, being of the second order in 6, 
 may be neglected^ while the moment of the former is 
 
 eivfdS (2) 
 
 By integrating (3) throughout both wedges, we obtain 
 the sum of the moments of the forces {b) and (c), since they 
 both give moments of the same sign about xx'. 
 
 Hence the moment of buoyancy due to the wedges is 
 
 en-ffdS, (3)
 
 158 Hydrostatics and Elementary Hydrokinctics. 
 
 the integration extending: all over the section AxBx' of 
 floatation. 
 
 If A is the area of this plane section, and k its radius of 
 gyration about the axis xx' of displacement, (3) is 
 
 eio.Ak- (4) 
 
 The moment of the force (a) about G, in the sense of the 
 previous moment, is 
 
 -L.GII.6 (5) 
 
 Hence, if V is the volume of the displaced liquid, since 
 L = T to, the moment of the whole of the forces of buoyancy 
 about the horizontal axis throug-h G perpendicular to the 
 plane of displacement is 
 
 dtv(Ak^-F.GH); (6) 
 
 and, equating- this to (i), we have, since ^j = 6 . G3I, 
 
 T.G3I= Ak^-V.GR, (7) 
 
 ••• i^^l/= p-, (8) 
 
 which determines the position of the metacentre, M. 
 For stability, therefore, 
 
 ^>HG (9) 
 
 Displacements of constant volume may take place round 
 any diameter of the section, AxBx', of floatation proAaded 
 that the diameter passes throug-h the centroid of this section 
 (Art. 30) ; and since for all such displacements both A and 
 V are constant, equation (8) shows that the metacentre will 
 be highest when the displacement takes place round that 
 diameter about which the moment of inertia of the section 
 of floatation is g-reatest, and lowest if it takes place round 
 the diameter about which the moment of inertia is least. 
 These two diameters are the principal axes of the section of
 
 Pressure on Curved Surfaces. 159 
 
 noatation at its centroid. If /^ and /•, are the o-reatcst and 
 least radii of yy ration of the section of floatation al)ont its 
 principal axes, and j\f._,, Jlf^, the corresponding- nietacentres 
 for displacements round them, 
 
 IIM, = ^^-^, HM, = ^ (10) 
 
 The equilibrium will, then, be least stable when the dis- 
 placement takes place round the diameter of least moment 
 of inertia, which in the case of a ship is the line from stem 
 to stern. 
 
 Since Fw = JF, (8) can be written 
 
 HM=-^ ^''^ 
 
 32. Experimental determination of Metacentre. The 
 heig-ht of the metacentre above the centre of g-ravity of a 
 ship can be found experimentally by means of a plumb-line 
 and a moveable mass on the deck. Suppose one end of a 
 long string fastened to the top of one of the masts and let 
 a heavy particle hang from the other end of the string-. 
 Now if a considerable mass, P, be shifted from one side of 
 the deck to the other, the ship will be tilted through a 
 small angle which can be measured by means of the pen- 
 dulum if the bob of the pendulum moves in front of a 
 vertical sheet of paper on which the amount of displacement 
 of the bob can be marked. If I is the length of the string 
 and s the distance traversed on the paper by the bob while 
 
 the mass P is shifted across the deck, - is the circular 
 
 measure of the whole angle of deflection of the ship. 
 
 Let G be the centre of gravity of the ship, //'the weight 
 of the ship and moveable mass together, 2 b the breadtii of 
 the deck, a the perpendicular from G on the plane of the
 
 i6o Hydrostatics and Elementary llydrokinetics. 
 
 deck, and lQ the whole angle, - , of deflection. Then, on 
 
 account of the symmetry of the ship, we can in Fig. 49 take 
 the line IIG as passing through 0. 
 
 Let the mass P be at B, and take moments of the forces 
 acting about G ; then 
 
 W.GM.e =r F{lj + a6), 
 
 The value of a is usually much smaller than - , so that, 
 "with sufl^icient accuracy, we have 
 
 where = — 
 
 Thus, in a ship of 10,000 tons the breadth of whose deck 
 is 40 feet, if a mass of 50 tons moved from one side to the 
 other causes the bob of a plumb-line 20 feet long to move 
 over 10 inches, the metacentric height is about 4^ feet. 
 
 The metacentric heights of large war vessels vary from 
 about 2 1 feet to 6 feet. 
 
 Examples or the Metacentre. 
 
 1. A uniform rectangular block, of specific weight w', floats, 
 with one of its edges vertical, iu a liquid, of siieciiic weight w; 
 find the relation between its linear dimensions so that the equi- 
 librium shall be stable. 
 
 Let 2 a, 2b be the lengths of the horizontal edges, and 2 c the 
 length of the vertical edge, and let b < a. Then the equilibrium 
 is most unsafe Avhen a displacement is made round the longest 
 diameter of the section of floatation. If x is the length of the 
 vertical edge immersed, 
 
 v/ 
 
 X = 2C — } 
 
 w
 
 Pressure on Curved Surfaces. i6i 
 
 and til ere fore „_, / ii/\ 
 
 HG = c ( I ) • 
 
 Also k'^ = 3^^ round the axis of most dangerous displacement, 
 
 W 
 and V = , where W = Sabcio' = weight of body. Hence 
 
 10 
 
 IIM=-? — -,' 
 
 ocw 
 
 so that for stauility ,; — > > c ( i ) , that is 
 
 6C1« V 77) / 
 
 6 
 - > 
 
 c 
 
 /6f!^(,--'). 
 
 2. If the floating body is a solid cylinder, floating with its 
 axis vertical, find the condition for stability. 
 
 Ans. If r is the radius of the base and h the height, 
 
 h \/ w ^ w ^ 
 
 3. If the floating body is a solid cone, floating with its axis 
 vertical and vertex downwards, find the condition for stability. 
 Ans. If r is the radius of the base and h the height, 
 
 r I (^ Y — 
 
 4. If the floating body is a solid isosceles prism whose base is 
 uppermost, find the condition for stability. 
 
 A71S. If 2b is the length of the shorter side of the base 
 and h the height of the prism. 
 
 b I / -w s, I 
 
 I. 
 
 5. If the cone in example 3 floats with its vertex uppermost, 
 find the condition for stability. 
 
 Ans. l> . /(— ^— ,y 
 
 I. 
 
 M
 
 i62 Hydrostatics and Elementary Hydrokinetics. 
 
 We have assumed that //', Fig*. 49, lies in the plane of 
 displacement, and we can easily see that this will not be the 
 case unless the axis, x'x^ of displacement is a principal axis 
 of the section, AxBx\ of floatation. For, if we seek the 
 co-ordinates of 11' (which is the centre of volume of the new 
 volume A'CB') we may regard, as before, the volume A'CB' 
 as 'resolved' into the orig-inal volume ^C-5, the positive 
 wedge B'xBx', and the negative wedg-e A'xx'A. Hence if 
 X is the distance of the point P from the line OB and ^the 
 distance of //' from the vertical plane containing OB, we 
 have r.^=dfx7/dS, 
 
 since the volume of the prism PQ is 6ji/(J S, and its volume- 
 moment about OB is QxydS, the integration extending all 
 over the area AxBx' . 
 
 This shows that ^ = o only when Ox and OB are prin- 
 cipal axes at 0. In the case of a square or circular section 
 of floatation, every axis through is a principal axis, and 
 hence //' always lies in the plane of displacement. 
 
 In general, therefore, a small angular displacement round 
 a diametfr of the section of floatation produces a moment 
 of the forces not only round this axis but also round the 
 perpendicular axis in the plane of floatation, the eflPect of 
 which would be to produce small oscillations of the body 
 about this axis. 
 
 The question of stability, however, is not afTected by this 
 consideration, since any small angular displacement, Q. 
 round an axis xx could be resolved into two sejmrate small 
 angular displacements 
 
 Q cos a and Q sin a 
 
 round the two pmicipal axes at 0, where a is the angle 
 made by afx with one of these principal axes ; and if the 
 equilibrium were stable for small displacements round the 
 most dangerous of the principal axes, it would be so for the
 
 Pressure on Curved Surfaces. 163 
 
 «»-iveT) displncoment round x'x. On the other hand, if tlic 
 oquililn-inm is unstable round one of the principal axes, it 
 will be unstable round all axes in the section at 0, unless 
 these axes are inclined at indefinitely small anq-les to the 
 other principal axis — supposing- the equilibrium to be stable 
 for dis2)lacemcnts round this axis. 
 
 33. Siirfaees of Revolution. When the fig-ure of the 
 Hoating- body is that of a surface of revolution, take the 
 origin, 0, of co-ordinates at its lowest point, the axis ol" 
 X being- vertically upwards and that of// horizontal. ThcTi 
 \{ [x, 1/) are the co-ordinates which determine the surface ol' 
 floatation in the erect position, and [x', y) those belonging- 
 to any other parallel section, we have 
 
 ^^°^^ HM^-^ (.) 
 
 Aj/'clx 
 
 Also, by mass-moments, 
 
 OH X 77 f'' f-dx = 77 f'x'f^dx'- 
 
 Jo Jo 
 
 0M= ^-^^ (2) 
 
 Jo 
 
 Thus, to determine the figure of the fioatingf body when 
 HM is of constant length whatever be the depth of im- 
 mersion, let 7/J/ = VI in (i). 
 
 -Ij*=m ij'-dx'. 
 
 
 
 HI 2
 
 164 Hydrostatics and Elementary Hydrokinctics. 
 
 Differentiate both sides wdth respect to x ; then (see 
 ^Yilliamson's Integral Calculus, Chap. VI) 
 
 3 ^^/ 
 
 .•. y^ = 2mx, 
 
 which shows that the g-enerating* curve is a parabola ; hence 
 when a paraboloid of revolution floats in a liquid the heig"ht 
 of the metacentre above the centre of buoyancy is constant 
 for all depths of immersion. 
 
 Example. 
 
 Find the nature of the generating curve so that for the surface 
 of revolution and for all depths of immersion the height of the 
 metacentre above the lowest point shall be any assigned function 
 of the co-ordinates of the section of floatation. 
 
 Let OM = {x, y) in (2) ; then, writing (^y instead of 
 (^(x, y) for shortness, 
 
 ^^y'+r x'y''dx'=^r y'-'dx' (i) 
 
 Differentiating with respect to x, and putting ^ for -^ » 
 
 pf+xf = y'4> + (^^+/^)fJ/'dx.. . . (2) 
 
 Dividing out and again differentiating. 
 
 This is the differential equation of the required generating 
 cui've. 
 
 If, for instance, the metacentre is at a constant height, a, 
 above the lowest point, we know that the curve is a circle, 
 and this appears at once from (2), since (/) = a, 
 
 .■ . l^y + X z= a, .-. ydy + {x — a)dx = o, .-. {x — aY + y'^ = a\
 
 Pressure on Cttrved Surfaces. 
 
 16^ 
 
 34. Metacentric Evolute. Suppose a body of f^ivoii 
 mass to float in a li(pii(l ; then if we consider all })ossibl(' 
 displacements — and not merely small displacements — in 
 which the volume of the displaced liquid is constant, the 
 lines of action of the forces of buoyancy ^vill envelope a 
 certain surface fixed in the body. This surface is called the 
 ■metaceninc evolute for the g-iven dis])laccd volume. 
 
 As a particular case, consider the displacements of a 
 square board, ABCl), Fig-. 50, floating- in a liquid of double 
 its own specific 
 
 weig-ht. The dis- a, iB 
 
 placement is al- 
 ways half the 
 volume of the 
 board ; and when 
 the board floats 
 with AB hori- 
 zontal, the centre 
 of buoyancy is //, 
 the metacentre be- 
 ing- M, such that 
 7/iI7 =■ ^a, where 
 2a = AB. In this 
 position the equi- 
 librium is unstable. The curve of buoyancy for positions 
 intermediate to those in which the surfaces of floatation 
 are DB and CA is the portion /'/// of a parabola whose 
 parameter is f «. The lines of action of the forces of buoy- 
 ancy are always normals to this parabola, and their envelope 
 is the evolute, Q^IQ', of the parabola. The positions in 
 which JJB and CA are in the surface of the fluid are ])0si- 
 tions of stable equilibrium, the metacentric heig-hts GQ and 
 
 GQ' being each — a. 
 
 Fig. 50.
 
 i66 Hydrostatics and Elementary Hydrokinetics. 
 
 In g-eneral, for the displacements of any body in one 
 plane — the volume of the displaced liquid being- constant — 
 the mttacentric evolute is the evolute of the cui've of buoy- 
 ancy in the plane. 
 
 35. Stability in two Fluids. Let LAOB, Fig. 51, re- 
 present a body floating- partly in a homog-eneous fluid of 
 
 specific weig-ht lo' and partly in one 
 of specific weig-ht ic, the latter being- 
 the lower, and suppose the position 
 of equilibrium to be found. We 
 may evidently imag-ine the volume, 
 BAB, of the upper fluid completed 
 by adding- the portion AOB, and all 
 the forces in play will be those due 
 to an immersion of the whole 
 volume in a fluid of specific weight 
 iv' and an immersion of the portion 
 A OB in one of specific weight tv — tv\ 
 
 Let G be the centre of gravity of the body; G' its centre 
 of volume, i. e. the centre of gravity of the whole volume 
 supposed to be homogeneously filled ; 1£ the centre of 
 volume of the portion in the lower fluid before displace- 
 ment ; 31 the metacentre corresponding to this lower fluid 
 (of specific weight w — w') \ V the volume of the lower and 
 / ' that of the upper fluid displaced. The position of Ji is 
 
 given by the equation H31 = —pr • 
 
 For simplicity we have assumed G, G' and 11 in the 
 original position to lie on the same vertical line ; but the 
 method of investigating any case in wliich they are not 
 thus simply situated will be readily understood from the 
 simple case supposed. 
 
 The points 6", 6', M, H may in any individual ease have
 
 Pressure on Curved Surfaces. 167 
 
 relative positions dili'crent I'roiu those re])resented in the 
 fig-ure. 
 
 We may evidently suppose the displacement to be made 
 round some diameter of the section AB throuf^^h its centroid, 
 in which case the wedg-e forces of buoyancy at the section 
 are equivalent to a couple, whose moment in the present 
 instance is Ak- [iv — w'). 
 
 The equilibrium will be stable if the sum of the moments 
 of the forces acting- on the body in its position of displace- 
 ment round an axis perpendicular to the plane of displace- 
 ment, drawn through G or throug-h any other convenient 
 point, is in a sense opposed to that of the displacement. 
 
 Now, \{ W = weight of body, the forces in action are W 
 acting- down through G, together with V [iv — w') acting- uj) 
 through M, and (^' +?')?/ up throug-h G'. The sum of 
 their moments about // in the sense opposed to the angular 
 displacement is 
 
 Q { r{w - ?/) . IIM + ( r+ V) xc'. JIG' - W. 11 G) , 
 
 ;ind if the expression in brackets is positive, the equilibrium 
 is stable. 
 
 It is sometimes more convenient to take the restoring- 
 moment about the lowest point, 0, of the axis of the body. 
 
 In the above expression we may put // = Viv+ V 7o'\ 
 and it is evident that if the centre of gravity, G, of the 
 body coincides with its centre of volume, G\ the condition 
 becomes simply IIM>IIG — as is evident a priori. 
 
 EXAMFLK. 
 
 A circular cone the length of whose axis is 20 inches is 
 formed ui' two substances whose speciHc gravities are 3 and 8, 
 the denser forming a cone whose axis is 12 inches long and the 
 other forming the frustum which completes tlie whole cone ; it 
 is immersed, vertex downwards, in a liijuid whose specific 
 gravity is 14, on top of which rests a liquid whose specific
 
 1 68 Hydrostatics and Elementary Hydrokinctics. 
 
 gravity is i, the whole cone being immersed; show that for stable 
 equilibrium the radius of the base must be greater than 11-639 
 inches. 
 
 36. Floating Vessel containing Liquid. Suppose a 
 vessel, represented in I'ig-. 52, to contain a given volume of 
 
 liquid of specific weig-ht to and to 
 float in a liquid of specific weight 
 w. If the vessel receives a small 
 angular displacement, there will 
 be a force of buoyancy due to the 
 external fluid acting upwards 
 through its metacentre M ; the 
 line of action of the weight of the 
 contained fluid acts through its 
 new centre of gravity and it in- 
 ^g- 52- tersects the line GM in vi, the 
 
 metacentre of this contained fluid. 
 This force acts downwards, and W, the weight of the vessel 
 acting through its centre of gravity, G, also acts downwards. 
 The weight of the internal fluid may assist either in pro- 
 moting stability or in promoting instability according to 
 the position of m. If, as in the figure, m is below G', this 
 force promotes stability. If V = volume of displaced 
 external fluid, / '= volume of internal fluid, the restoring 
 
 moment is „, -rr ^,i,r >-,-, y^^T\ 
 
 e{toV. GM^wT'. GM). 
 
 Examples. 
 
 1. Find the least height to which a uniform heavy cylindri- 
 cal vessel of negligible thickness can be filled with water so that 
 when it is placed with its axis vertical in water the equilibiium 
 may be stable. 
 
 Ans. Let h be the distance of the centre of gravity of the
 
 Pressure on Curved Surfaces. 169 
 
 vessel from the base, W the weifrlit of the vessel, ami A ilie area 
 of the base; thtn the least height to wliich it can be tilled is 
 
 2 A IV 
 
 2. If the cylinder contains a liquid of specific weight v/ and 
 floats in a liquid of specific weiglit w, with its axis vertical, find 
 the condition of .stability. 
 
 A'/is. Let 11/ = n . iv, W=Aciv, and x = the height to wliich 
 the cylinder is filled ; then, for stability, the expression 
 
 2 71 (w— i)x'' + 4ncx+ 2c' — (n— i)r" — 4c/t 
 
 must be positive. 
 
 3. If a uniform hollow cone of negligible tliickness contains a 
 liquid of specific weight iv' and floats in a liquid of specific 
 weight IV with its axis vertical and vertex downwards, find the 
 condition of stability. 
 
 Anft. If X is the length of the axis occupied by the internal 
 fluid, 1/ the length occupied by the external fluid, h the whole 
 length of the axis, I the distance of the centre of gravity of the 
 cone from the vertex, r = radius of base, w''= mo, ir= weight 
 and V = volume of cone, and if IF := ?/i . Vw, we have 
 
 y^ — nx^ = mli^, 
 
 and for stability the expression 
 
 3(1 + ;^) (y* - nx') - 4 m h' I 
 must be po.sltive. 
 
 4. A thin vessel in the form of a surface of revolution contains 
 a given quantity of homogeneous liquid and rests with its vertrx 
 at the highest point of a rough curved surface, find the condi- 
 tion of stability for small lateral displacements. 
 
 Ans. Let TFbe the weight of the vessel (without the liquid), 
 /i the distance of its centre of gravity irom the vertex, Ttlie 
 volume of the liquid, w its specific weight, z the distance of its 
 centre of gravity from the vertex, A the area of the free surface 
 of the liquid, k the radius of gyration of this area about its dia- 
 meter of displacement, p and // the radii of curvature of tin-
 
 lyo Hydrostatics and Elementary Hydrokinetics. 
 
 vessel and the fixed surface in the phuie of displacement; then 
 tlie restoring moment is propoi'tional to 
 
 {Jw^ Tr)/5-(i + 4) {Vwz^Ali-w^ TIVi), 
 
 and if this expression is positive, the equilibrium is stable. The 
 restoring moment is equal to this exjjression multiplied by 
 
 fa 
 
 where Q is the small angular displacement of the vessel. 
 (See Statics, vol. ii., Art. 279, 4th ed.) 
 
 5. In the last examj)le find the position of the metacentre. 
 Ans. If H is the centre of gravity of the contained fluid, 
 
 V Vw^ p + p'y 
 
 6. If the vessel is a paraboloid of revolution resting on a hori- 
 zontal plane, the weight of the liquid being P and the latus 
 rectum of the parabola 4 a, the condition for stability is 
 
 W(2a-h) >|P( ) • 
 
 37. Stability in Heterogeneous Fluid. We shall now 
 suppose that a body floats in a fluid of variable density 
 which is subject to the action of gravity. The level sur- 
 faces of the external force being- horizontal planes, these 
 planes will also be surfaces of constant density. Hence if 
 iv is the specific v^^eight of the fluid at any point whose 
 depth below the free surface, LiY, Fig-. ^^, of the fluid is C, 
 we have ^/^x /,-> 
 
 Suppose the dotted curve to represent the original 
 ])Osition of the floating body, and that the full curve ACB 
 represents its position when it has received a slight angular 
 displacement, 0, round any assigned horizontal line Ox — 
 which we suppose to be perpendicular to the plane of the 
 paper.
 
 Pressure on Curved Surfaces. 
 
 171 
 
 Take the vertical plane tluoun-h llic oiij^inal line joiiiiiif^ 
 (1 and H, the centres of gravity of the body and ol" buoy- 
 ancy, \\hich is perpen- 
 dicular to Oa? as plane 
 of yz, the point, 0, in 
 which this ])lane cuts 
 Ox beinii" taken as 
 origin, the vertical Oz 
 as axis of z, and the 
 horizontaliine, Oi/,\)qv- 
 ])endicular to Ojc as 
 axis of }/. Thus the 
 displacements of all 
 points of the body take 
 place in })lanes parallel 
 to the plane o'( i/z. 
 
 The section of floata- 
 tion of the body in the 
 displaced position is 
 represented by A' B'. 
 
 Suppose AB to be the section of the body made by the 
 j.lane LN in the original position ; and in this position let 
 I) be the distance between the line 67/ and the axis 0:. 
 Let // be the height of above LN. 
 
 The equation of the plane A' B' is z — /i — o. and this was 
 the equation of AB in the original position ; but by rotation 
 in the sense indicated in the figure the equation of AB in 
 its displaced position becomes z-Qij-h = o, and therefore 
 the old andnewpositions of the plane ^ii^ — and of every plane 
 horizontal section of the body — intersect on the axis 0:. 
 
 Suppose B' to be any point in the body whose original 
 position was B (the latter point being sup] osed to be 
 marked in fixed space and not in the body) ; and let .?',//, .- be 
 the co-ordinates of B with reference to the fixed axes at 0, 
 
 rig. 53-
 
 172 Hydrostatics and Elementary Hydrokinetics. 
 
 Then the co-ordinates of P' are {x,i/ — 6z, z + 6f/), so that 
 the density of the iluid which Mould exist at P' if the body 
 were removed would be, by (i), 
 
 f{z + e^-J<),ovf{C-^6^),oxw + 6/~, . . (2) 
 
 since the depth of F' below the surfoce LN is Z + O1/ — J1. 
 Hence when the element of volume clxch/dz at P is carried 
 to P', it will experience a force of buoyancy 
 
 (^w + 6i/'j^)dxi]i/(lz; (3) 
 
 and since the points P are all those included within the 
 original volume, BCA, immersed, the corresponding forces 
 of buoyancy will omit the wedge B' rB and include the 
 wedge ArA' — the latter not, in reality, contributing any 
 force of buoyancy at all in the displaced position, while the 
 former does. We must therefore specially include the 
 wedge B^rB and exclude ArA'. 
 
 Let ?/„ be the specific weight of the fluid at the surface 
 LN \ let (IS be the area of any element of the surface AB 
 (such as that represented at P in P'ig. 49) ; then if i/q is 
 the distance of this element from the line through r parallel 
 to Ox, the volume of the small cylinder standing on (IS, as 
 in Fig. 49, is Oj/^dS. Also let x^ be the x co-ordinate of 
 the element dS, and let c be the original depth of 6' below 
 the horizontal plane x Oy. Then we have, in their new 
 positions, 
 
 the co-ordinates of P x, y — Qz^ z->r 6y, 
 
 „ G o, /j-Oc, c + e/j, 
 
 ,,'^S x^,yQ-0//, // + 0y^^. 
 
 Now we shall calculate the sum, L, of the moments of 
 the forces of buoyancy round the horizontal axis through (J 
 ])arallel to Ox in the sense ojjposite to that of the displace-
 
 Pressure on Curved Surfaces. 173 
 
 ment, i.e. oounterclockwise as we view the fif^ure. If ;i 
 force havin*:;- components X,Y,Z acts at the point (.r,//, r), 
 its moments round axes throng-h the point (a,/:{,y) parallel 
 to the axes are if(^ — ^)— 7(~ — y), and two similar ex- 
 pressions [Statics, vol. ii., Art. 202). 
 
 In the present case only the ^;-component of force exists, 
 and this at P' is the expression (3) with a nei^i-ative sigri, 
 while at the new position of the surface element dS it is 
 
 -dw^yJS. (4) 
 
 Hence we have 
 
 L = [ff{w^e/-^^)[y-b-6[z-c)] dxtlydz 
 
 + 
 
 d^^'oj^oij/o-^)^^^- • (5) 
 
 Now observe that we neg'lect 9 ^ ; also if W is the weight 
 of the volume ACB of fluid originally displaced, 
 fffioydxdydz = W . h, 
 
 since the y of H was originally b. Hence the term inde- 
 pendent of 6 in (5) disappears, as it must, of course ; and 
 we have 
 
 L = dfff{y^-hyf2'^xdydz-6J[[iv{z-c)dxdydz 
 
 + eioJ{y,'-by,)dS. . (6) 
 
 ^oA^o^-^o) 
 
 Obsei-ve also that tv is a function of c alone, so that the 
 first triple integral can be written in the form 
 
 \/Ar-W^'^^]j/^'^ .... (7) 
 
 and if A denotes the area of any section for which : is 
 constant (i. e. any section of the body parallel to AB), k 
 the radius of gyration of this section round the line in its 
 
 /[
 
 174 Hydrostatics and Elementary Hydrokmetks. 
 
 plane parallel to Ox, at the point where Oz cuts the section, 
 and 1/ the distance of the ' centre of o^ravity ' of the area 
 from this same line, the double integral in the brackets in 
 
 (7)i^ A{k'^-l]j), (H) 
 
 so that the first integral in (6) is 
 
 6JA{Jc^-llj)'^dz (9) 
 
 The second integral in (6) is -OWJIG, and the last is 
 6iP(^Ao{h^-f^^o\ where X-q is the radius of g-yration of the 
 section, AB, of floatation (whose area is A^) round the line 
 through ;• parallel to Ox, and ^o i^ ^^^ distance of the 
 ' centre of gravity ' of the section from this line. Hence 
 (6) becomes 
 
 1= rA(P-i^)'^fjz+woMh'-^^o)-'ff'''^^G' • M 
 
 For stability this must be a positive moment ; and in 
 the particular case in which w is constant and the displace- 
 ment is made round a diameter of the section ylB, it is 
 obvious that we get the same condition as in Art. 31. 
 
 But the forces of buoyancy will also, in general, produce 
 a moment round the horizontal axis througli G parallel to 
 Oy, i.e. a moment tending to turn the body across the 
 plane of displacement. If 31 is this moment, w^e have 
 
 M = fff{ to -^^/jz) xdx(l7/(h + 6 w'o ^A'oj'o dS. . ( 1 1 ) 
 
 Let P denote the product of inertia, ffxi/flxd)/, of any 
 section round axes in its plane parallel to Ox and Oi/ at the 
 point where the section is cut by Oz ; then
 
 Pressure on Citrved Surfaces. 175 
 
 This moinont will not exist if P is zero for all seetions, 
 or if the iluid is honiog'cneous and P is zero lor the surface 
 of iloatation. 
 
 Let us now calculate the tvork done in the disjilacement 
 of the body round Ox. 
 
 The work which would be done on a material system by 
 force the components of whose intensity at [x,i/,z) are 
 A', Y,Z{ov any small displacement whose typical components 
 are 8 x, 6y, 8 ^^ is 
 
 f{Xbx+Ybi/ + Zbz)dm; . . . . (13) 
 
 and if the displacement is produced by small rotations, 8^, , 
 8(^25^^35 i"oi^^^d tli6 axes of co-ordinates, we have hx = yhd^ 
 —xhd^, with similar values of hi/ and hz. Hence, if 
 7/, M, N are the typical moments of the force intensity 
 about the axes, the work is 
 
 Lh6^ + Mhe^ + Nhe. (14) 
 
 In the present case the only rotation is that about Ox. 
 .-. bdo = bd.^ = o. Consider the moment L as that of the 
 forces of buoyancy in the displaced position ACP, and 
 calculate the element of work done by these forces in any 
 further small displacement by which the angle Q is increased 
 bv dQ. Then the infinitesimal element of work done in 
 this further displacement is 
 
 Lde (1.3) 
 
 But (taking- the forces of buoyancy alone), 
 
 L = -ffJ{n' + d/lf^){i/-ez)dxdj/dz-9?v,ft^,^dS. (16) 
 
 = -Wb-e\fAk^'^^dz-W{c + nG)-\-w,A^kA .(17) 
 
 = — 7/7/— iT^, suppose ; l'"^) 
 
 and the integral of this expression I'lom — o to =
 
 176 Hydrostatics and Elementary Hydrokinetics. 
 
 expresses the work done by the forces of buoyancy in the 
 displacement from the initial position of the body (re- 
 presented by the dotted contour) to that represented by 
 ACB. Hence the work is 
 
 -WhQ-\K&^ (19) 
 
 The work done by the weig-ht of the body is simply 
 /A'8 5, in which hz must be accurate as far as 6^; i.e. 
 hz =■ hQ — \ c6^. Hence the work done by all the forces is 
 
 -- \/A/k''^dz-Jr.IlG + iv,AjA, . . (20) 
 
 and this, with reversed sign, is the work which must be 
 done against the forces to produce the displacement. 
 
 Examples. 
 
 1. If a solid homogeneous cone float, vertex down, in a fluid 
 in which the density is directly proportional to the depth, find 
 the condition of stability. 
 
 Ans. If, as at p. 131, h' is the length of the axis immersed 
 and h is the height of the cone, the equilibrium will be stable if 
 
 cos^ a < — ^ , where a is the semivertical angle of the cone. 
 5/i 
 
 2. Determine the condition of stability of a solid homogeneous 
 cylinder under the same circumstances. 
 
 Ans. If r is the radius, h the height of the cylinder, and lif 
 the length of the axis immersed (see p. 131), the condition of 
 stability is r-" > h\h-%h'). 
 
 3. If a spherical balloon of weight B is held at a given height 
 by a rope made fast to the ground, find the work done in dis- 
 placing it about the ground end of the rojje through a small 
 angle. 
 
 Ans. If h is the height of the centre of the balloon and W 
 the weight of the displaced air, the work is 
 
 ie''{W-B)h, 
 
 where Tl" has the value given in example 3, p. 131.
 
 Pressure on Curved Surfaces. 177 
 
 38. Green's Equation. Let ABC, Fig-. 54, Ix- anv 
 closed surfiice ; let U and V be any functions of x, 1/, z, 
 the co-ordinates of any point P at which 
 an element of volume il il is taken ; and 
 let V^ stand for the operation 
 
 then if we take the integ-ral 
 
 fUVTdQ. 
 
 throug-hout the volume enclosed by ABC, the result can be 
 txpressed in terms of another volume-integral taken through 
 the same sjoace and of a surface-integral taken over the 
 bounding- surface ABC. Thus, let Q be any point on the 
 surface, at which an element of area dS is taken and let dn, 
 be an element of the normal at Q drawn outwards into the 
 surrounding space (in the sense of the arrow). Then we 
 have (see Statics, vol. ii, chap, xvii, Section iv.) 
 
 Juv'-F.dil= fu'^.dS 
 
 r.dUdF dUdV dVdV. ,^ 
 
 In exactly the same way, if is any other function of 
 X, 1/, z, we have 
 
 ^,.d dV d dV d dr. , 
 
 ^dx dx di/ ^ d)j dz ^ dz^ 
 
 r _ r JV dV dUdV dU dV. 
 
 f 
 
 Cj.^ y ,^ r ,dU dV dV dV dU dV. , , , 
 
 The first of these is known as Green's equation ; the 
 second is a modification made by Sir W. Thomson. 
 
 By assig-ning- to U various values (such as a constant 
 value, the value V, &c.) we obtain (as shown in Stalicx 
 
 N
 
 178 Hydrostatics and Elementary Hydrokinctics. 
 
 above) various remarkable theorems with physical appli- 
 cations. 
 
 A most remarkable consequence of (2) is this. If and 
 V are any two functions satisfying the equation 
 
 ^^ d)— — d>— — d) — - ( ) 
 
 dx dx dy dy dz dz • ' \^ I 
 
 at all points within a closed surface, ABC, and if the value 
 of V is assigned at every point, Q, on the surface itself, its 
 value at each internal point, P, is determinate. 
 
 For, if possible, let there be two different functions, viz., 
 
 r =f{x, y, z), 
 
 r =f{x, y, z) ; 
 
 each satisfying (3) and such that F= V at each point, Q, 
 on the surface, while V is, of course, not equal to V at each 
 internal point P. Denote J — V^hy ^; then ^ satisfies (3). 
 Now employ (2) for the volume and surface of ABC, and. 
 moreover, choose for U the value f. Then 
 
 r^fd d$ d ^d$ d ^r/A .^ 
 J ^dx ^ dx dy ^ dy dz dz^ 
 
 But each term under the integral on the left-hand side 
 vanishes, and the surface-value of ^ which enters into each 
 term of the first integral on the right also A^anishes ; there- 
 fore the second integral on the right vanishes ; but since 
 each term of this integral is a square, we must have each 
 term equal to zero, i. e., 
 
 Tx^""^ dy^""^ Tz^""^ 
 
 must hold for all points inside ABC ; and this requires that
 
 Prcssitir oil Curved Surfaces. 179 
 
 ^ is constant for all iiilirnal jxtinis, and .*. zero, since it is 
 zero at the surface ])oints. 
 
 Ilt'ncc there cannot be two functions, /', / ', satisfvin<^ 
 (3), ai,''rot'ini,'' at each surface ])oint, while dilfcrinf^ at 
 internal iioiuts. If, therefore, any one function,./'(j', 1/, :), of 
 the co-ordinates is known to satisfv (3) and to have at each 
 l)oint on the surfjice an assij^ned ])articular value, it is the 
 only one applicable to the points enclosed by the surface. 
 
 The a])i>lication of this result to the case of lluid pressure 
 is obvious. If at each point of any lluid-mass the external 
 forces satisfy the equation 
 
 dX fir (IZ 
 
 -7- + -1- + 7- = o, 5) 
 
 <lx di/ <lz 
 
 i.e., if the external forces have a potential, equations (i). 
 (2), (3), p. 82, g-ive 
 
 (1 \ (In (I 1 (Jp (1 1 (In ,.. 
 
 . _ ^/- _j . _ _A .^ . ;- = O. . . (0) 
 
 fix p fix fly p fly (Iz p (Iz 
 
 Hence. if ABC is the surface of a forei^^n body immersed 
 in the fluid, the distribution of the lluid which could, under 
 the influence of the o-iven external forces, statically replace 
 the body is determinate since the value of the pressure in- 
 tensity is assi<4-ned at each surface jioint, Q. At each 
 internal i)oint, 1\ the jiressure intensity is determinate, and 
 if p is, for the <^ivcn fluid, a <,'-iven function of 7^ — say /(y>) 
 — the value of 7; at P is given by the equation 
 
 ^'-, = ^h (7) 
 
 where /' is the i)otential function of tlu- external forces, 
 involving: the co-ordinates of I*. This is the result referred 
 to in p. 113. 
 
 N 2
 
 i8o Hydrostatics and Elementary Hydrokinctics. 
 
 Example. 
 
 If in the midst of a mass of fluid which is not self-attract- 
 ing there is a solid body which attracts its own particles and 
 those of the fluid according to the law of inverse square of dis- 
 tance, and if the surface, A, of this body is one of constant po- 
 tential, prove tliat the intensity of jjressure, j), of the fluid at any 
 point, P, is less than the intensity of pressure, p,,, at any point 
 on A by an amount given by the equation 
 
 ^' = ^'--;^-/^^'<'"' 
 
 where y is the constant of gravitation {Statics, vol. ii., Art. 315), 
 M is the mass of the solid body, p is the density of the fluid at 
 any point at which the attraction per unit mass due to the body 
 is H, dQ, is an element of volume, and the integration extends 
 over the space included between the surface A and the equi- 
 potential surface, S, described through F. 
 
 In Green's equation (i) for U choose p—;;^^ and let V be the 
 jjotential at any point due to the solid body. Then we have 
 
 in which the surface-integral on the right is taken over the sur- 
 face A and over the surface S, and tlie element of normal (hi is 
 drawn into the space outside the volume enclosed by A and S; 
 this space is therefore the interior of the solid body and the ex- 
 terior of S, so that dn in the integration over A is measured 
 towards the interior of M. 
 
 Now w^e know that [Statics, vol. ii., Art. 329) V'V =: 4777//, 
 where p^ is the density of the attracting matter (to which V is 
 due) at the point to which V applies ; and as there is none of 
 this attracting matter at any of the points w'ithin the volume 
 (that included between A and S) included in the integration, 
 V'- V = o. Again, at every point on the surface of A we have 
 2)—P(, = o, therefore the part of the surface-integral on the right 
 which relates to the surface A is zero. Further at every point 
 on aSp is constant; hence the surface-integral is simply 
 
 (l>-po)J ] 
 
 ^^--
 
 Pressure on Curved Surfaces. i8i 
 
 and is confined to tlie suri'iico *S'. Moreover, at every point in 
 the fluid -— ■ = pX, &c., and X = - ; licnce (i) liceomes 
 
 o = (r-2\)fffy^ds-J,j]r-dn (2) 
 
 Now {Statics, ibid. ) 
 
 j,,dS=-47ryJ/, (3) 
 
 so that tlie required result follows at once from (2) and (3). 
 
 /: 
 
 39. Line-Integrals and Surface-Integrals. Tf any 
 directed nni^'nitude, or vector, has for components w, v, w 
 alony- three fixed rectang-nlar axes, the mag-nitudc which 
 has for components A, //, v along- these axes, where 
 
 (I) 
 
 ....*.. (2) 
 
 (3) 
 
 dw 
 dy 
 
 dv 
 
 du 
 
 div 
 
 dz 
 
 dx ~ ^ 
 
 dv 
 
 du 
 
 doe 
 
 ''} - '■■ 
 
 lias been called the ' curl ' of the given vector hy Clerk 
 Maxwell. (In the theory of Stress and Strain, and in the 
 motion of a fkiid, it is convenient to define the curl as 
 having the hakes of the above components.) 
 
 Any vector and its curl possess the following fundamental 
 relation : He line-integral of the tangeiitial component of am/ 
 rector along any closed curve is equal to the surface-integral 
 of the normal component of its curl taken over any curved 
 sxirface having the given curve for a bounding edge. (See 
 Statics, vol. ii, Art. 316, a.) 
 
 If /, 7n, n are the direction-cosines of the nornnil at any 
 point of such a surface, and dS the area of a superlicial
 
 i82 Hydrostatics and Elementary Hydrokinetics. 
 
 element at that point, while ih denotes an element of 
 length of the bounding- edge of the surface, the theorem is 
 expressed by the equation 
 
 J{lk + m„ + n v) dS ^j{u '!^ + v'^^+w '£) ds. . (4) 
 
 Now we are often given the components of curl, A, //, r, 
 and from these we require to determine the vector from 
 which they arise. In view of such a problem, the following 
 fact is useful. If we can find, by any means, some par- 
 ticular values, Mq, Vq , w^, of the components of the required 
 vector which will satisfy the equations (i), (2), (3), the 
 general values of u, v, to are simply 
 
 , ^^0 ft) 
 
 v = r,+'^, (6) 
 
 d(f) („\ 
 
 dz 
 
 where ^ is any function whatever of x, y, z. This is 
 evident, because if we substitute n^ , r^, , w^ for «, v, w in 
 (i), (2), (3), we have, by subtraction, 
 
 d{w-w^) ^ d{v-L\^ 
 dy dz 
 
 and two analogous equations ; and these signify that the 
 
 expression 
 
 {ji — 2(q) dx + {v — r^) dy + {to — w'q) dz 
 
 is a perfect differential of some function of x, y, z. If this 
 function is denoted by </>, we have the results (5), (6), (7). 
 
 Of course it is a necessity from (i), (2), (3) that any pos- 
 sible components of curl of a vector should satisfy the 
 identity dk ^d\x dv _ /gx 
 
 dx dy dz ~~ '
 
 Pressure on Curved Surfaces. 183 
 
 Thus, it is not i)Ossiljle to (k'tci'inine a vector the eoin- 
 ponents of whose curl are a\ y, z ; ])iit it is possible to 
 determine one whose components oi" curl iiic x, //, —2.-. 
 The values v^^ = ^i/z, v^ =— \ ~x, ?<?„ = \xj/ will g-ive these 
 latter components of curl; so will the values u^-=^yz, 
 i\ — —zx, Wj = o. But it is obvious that if c/) denotes the 
 function | xj/z, these two sets of components are related thus 
 
 ^'« = ''^ + Z^' ''o='^ + 7y ^'- 
 
 In the linc-inte<2:ral along- the closed curve the vector 
 
 d(t) d(l) d(f) . 
 
 whose components are -^ 5 — j -~ may be rejected, it 
 
 is a single-valued function. 
 
 Stokes's method of determining values of n, v, iv from the 
 given values of A, fj., v will be found in Lamb's Treatise on 
 the Motion of Fluids, p. 150. 
 
 Examples. 
 
 1. Given any unclosed curved surface in a heavy lioniogeneous 
 liquid, is it possible to express the total component of pressure, 
 on one side of the surface, parallel to — 
 
 (a) a horizontal line, 
 
 (h) a vertical line, 
 
 by an integral taken along the bounding edge of the surface ? 
 
 Ans. The first is possible, but not the second. If a hori- 
 zontal line is drawn at the surface of the liquid, which is taken 
 as the plane of a;, y, the component in the first case is flzdlS, 
 
 ^ -— ds. This result is evident from elenien- 
 ds 
 
 tary principles ; becaus^e, if through the edge of the surface we 
 
 describe a horizontal cylinder whose generators are parallel to 
 
 the axis of x-, and take a section of this cylinder ])Cipt'ntlieular 
 
 to its axis, the horizontal component uf the pressure on tlu' 
 
 curved surface is equal to the pressure on this plane section, 
 
 and is therefore independent of the shape and size of the given 
 
 surface.
 
 184 Hydrostatics and Elementary Hydrokinctics. 
 
 2. Given any unclosed curved surface in a heavy homogeneous 
 liquid, is it possible to express the sum of the moments of the 
 pressures, on one side of the surface, about — 
 (rt) a horizontal line, 
 (6) a vertical line, 
 by an integral taken along the bounding edge of the surface 1 
 Ans. The second is possible, but not the first ; for, 
 
 j{mzx-lyz)dS=\jz'(x'^£ + y^) ds; 
 
 and the result also follows from elementaiy principles, by closing 
 the surface with a fixed cap described on the bounding edge, and 
 then imagining the given surface to vary in size and shape, -while 
 retaining its bounding edge.
 
 CHAPTER VI. 
 
 GASES 
 
 40. Definition of a Perfect Gas. "When defining- the 
 modulus of cubical compressibility of a substance (Chap. I) 
 the law which reg-ulates the compressibility of gases was 
 g-iven. We may accept this law as an adequate deiinitif)Ti 
 and say that — 
 
 A pcifcrt (/as is a jinid whose resilience, or compressihililii, 
 of volnvie, when its temyeraUi^re is constant, is numericaJlij 
 equal to its intensity of jwessure. 
 
 It was shown (Art. H) that the expression for the re- 
 silience of volume of any substance is 
 
 dp 
 dv 
 
 and that if this is equal to p, we have by integration 
 
 pv = const (a) 
 
 If the volume v was r^ when the intensity of pressure was 
 Pq, we must therefore have 
 
 J}v = poVo {S) 
 
 If we take a given mass of gas and assume that at no 
 point in it is there any intensity of pressure due to the 
 weight of the g-as, the pressure-intensity must be reg-arded 
 as the same at all points and equal to whatever value 
 it has at the surface. Let us sup})ose the gas contained in 
 a cylindrical tube fitted with a gas-tig-ht piston which can l)e
 
 i86 Hydrostatics and Elementary Hydrokinctks. 
 
 weiy-hted with various loads ; then the volume of the g-as 
 will vary with each pressure intensity ; and if its tem- 
 perature remains unaltered, 
 we can graphically represent 
 its various states as expressed 
 in the fundamental equation 
 (/3), thus : draw any tw'o 
 rectangular axes, Ov, Op., and 
 let the volumes assumed by 
 the g-as be measured, on any 
 scale, along Oi\ while the in- 
 tensities of pressure are mea- 
 sured on any scale along OP. 
 If, on these scales, 03/ and 
 ON represent respectively 
 pj„ (.- any volume and the corres- 
 
 ponding intensity of pressure, 
 the point, P, whose co-ordinates are OJ/ and ON will 
 graphically represent the state of the gas ; and all points, 
 such as P, whose co-ordinates satisfy (;S) will be found on 
 a rectangular hyperbola passing through P and having the 
 axes Ov and 0/j for asymptotes. 
 
 Thus, then, the curve of traii-sforniatiou of a given mass of 
 gas at constant temperature is a recta?ig?dar hyperhola. Such 
 transformation is called an isothermal transformation. 
 
 The figure exhibits the fact that when the intensity of 
 pressure is infiniteh" increased the volume of the gas 
 becomes infinitely small, and that when the intensit}^ 
 of pressure is infinitely i-educcd, the volume becomes 
 infinitely great. 
 
 The first result would be strictly true for a substance 
 whose transformations strictl}' follow the law (a) for all 
 values of p ; but it will be readily understood that there 
 exists no gas for which (a) holds indefinitely, and that when
 
 Gases. 
 
 187 
 
 the intensity of pressure is very p^-reatly increased the ^-as 
 may approximate to, and actually hccomo, a li([ui(]. 
 
 The law (a) was arrived at exi)erimental]y by Hoyle and 
 by INTariotte, and was not deduced as a mathematical conse- 
 quence of the physical definition of a g-as which we have 
 given at the beg-inning- of this Article. It is genera 11 v 
 known in this country as l/ie law of Boi/le and Mariutte, and 
 it may be formally enunciated as follows — the temperalKrc 
 remaining constant, the volume of a given mass of gas varies 
 inversely as its intensity of pressure. 
 
 The experimental veriHcation is as follows. Let IIABK 
 (Fig-. 56) be a bent glass tube of uniform section — at least 
 in the leg AH which is closed at the top. 
 Let the gas to be experimented upon be 
 enclosed in the branch AH by means of a 
 column, ALB, of mercury, the branch 
 LBK of the tube being open to the atmo- 
 sphere. Suppose matters arranged so that 
 when the gas in AIL is in ecjuilibrium of 
 temperature with the surrounding- air, 
 after the pouring in of the mercury has 
 ceased, the surfaces A and B of the mer- 
 cury are at the same level in both branches. 
 Then the intensity of pressure at any point 
 in the surface A is equal to that at any 
 point in i? ; so that if 7;^ is the atmospheric intensity of 
 pressure, /Jp is also the intensity of pressure of the gas in AJf. 
 
 For simplicity denote p^^ by the height of the barometer 
 at the time of the experiment. 
 
 Let this height be h (inches or millimetres), and let 
 t"o (cubic inches or cubic millimetres) be the volume of the 
 gas AIL If w is the weight of a unit volume (cubic inch 
 or cubic millimetre) of the mercury, we havey^^ = ''' • ^'- 
 
 Let us now, by pouring mercury slowly into the open
 
 1 88 Hydrostatics and Elementary Hydrokinetics. 
 
 branch at A', reduce the volume of the gas in AH to half 
 its value. If 67/ = \ AH, the mercury is to be poured in 
 until its level in the closed branch stands at C after all 
 disturbance and heatino- effect due to the pouring- in of the 
 mercury have subsided. If we now read the difference 
 of level between C and the surface of the mercury in the 
 branch LK, we shall find it exactly equal to h, the heig-ht 
 of the barometer. Equating- the intensity of pressure at C 
 due to the imprisoned g-as to the intensity of pressure due 
 to the mercury and the atmosphere, we see that the former 
 must be equal to p^^wh, i. e., the new intensity of pressure 
 = 2])^, while the new volume is \Vq. 
 
 Again, let EH = | AH, and let us pour mercury in at K 
 until the volume of the imprisoned gas is EH, i. e., ^ t'o. 
 AVe shall then find that the difference of level between E 
 and the surface, E, of the mercury (not represented in the 
 figure) in the oj^en branch is 3 times the height of the 
 barometer, i. e., 3 /i, so that the intensity of pressure of the 
 gas in EH is p^ + 3 ivA, or 47^^ . 
 
 Hence we have the following succession of volumes and 
 pressure intensities for the gas, its temperature being 
 the same all through. 
 
 If, in the same way, the volume is reduced to -Vy, the 
 
 difference of level of the mercury in the two branches is 
 found to be {n— i) h, so that the new intensity of pressure 
 is np^ ; and from these results we see that in each case the 
 volume of the gas is inversely proportional to its intensity 
 of pressure, as expressed by equation (^). 
 
 The law of Boyle and Mariotte may also be verified 
 in the following simple manner by means of a single 
 straight tube, about 2 mm. in diameter. 
 
 Let AH be a tube of uniform section closed at the end A
 
 Am. 
 
 Gases. 189 
 
 and open at I) ; let a portion, .//)' of ilio tube be lilU'd wilb 
 air or other g-as, and let a thread oi" niereury, JJC\ of len<;tli 
 /, separate this o^as 
 from the external air. rs' a" 
 
 When the tube is 
 held horizontal and all 
 disturbance has sub- 
 sided, let the volume, 
 r^, of the g-as AB be 
 read ; its intensity of 
 pressure is the same 
 
 B" 
 
 c" 
 
 ^ D' 
 
 as that at C, i. e., p^, Fig. -7. 
 
 the atmospheric in- 
 tensity. Now let the tube be held in a vertical position 
 with the closed end A' downwards and let the g-as occupy 
 the volume A'£\ or v\ Its intensity of pressure is now 
 equal to that at B' due to everything- above -S', i. e.,pQ-\-7c/, 
 where w = weight of unit volume of mercury. If /^ is the 
 heig-ht of the barometer during the experiment, /^^ = 7c//, and 
 if p' is the intensity of pressure in yl'B\ 
 
 p'— w [h +/). 
 Finally, let the tube be held vertically with the closed 
 end A" uppermost, and let the volume of the gas be A"B", 
 or v'\ If its intensity of pressure is p'\ the intensity 
 at 6"' is p'^ + wl due to everything above C" ; but ;;„ is 
 also the intensity of pressure at C" since that is a point in 
 the external air. Hence 
 
 jj" = v: (^h — I). 
 Hence, as regards volume and intensity of pressure, we have 
 the succession of states 
 
 {t'o, wh], { V, w {h + 1)], {v'\ w {h - 1), 
 and we find on trial that 
 
 v,,h = v'{h + l)= v"{li-l\ 
 according to the requirements of Boyle's law.
 
 190 Hydrostatics and Elementary Hydrokinetics. 
 
 Reference has already been made to the chang-e of a g^s 
 into a liquid by compression. To this we shall subsequently 
 return. At present Ave shall merely remark that Boyle's 
 law is not accurately obeyed by any known g-as, but 
 the approximation is very close in the case of all g-ases 
 when thev are not near the state in which, either bv 
 increase of pressure or by diminution of temperature, lique- 
 faction begins. When any gas is near the state of 
 liquefaction, its volume decreases more rapidly with in- 
 creased pressure than it would if it followed Boyle's law. 
 'When it is actually at the point of condensation, the 
 slightest increase of pressure condenses the whole of it into 
 a liquid.' (Clerk Maxwell's Theory of Real, Chap. I.) 
 
 41. Law of Dalton and Gay-Lussac. The volume of 
 a given mass of gas may be altered by heat as well as 
 by pressure. The law relating to this change was dis- 
 covered independently by Dalton in 1801 and by Gay- 
 Lussac in 1802 ; and, apparently, it was discovered fifteen 
 years previously by INI. Charles, although not pultlished 
 by him. It is this — 
 
 The intensity of pressure heing constant, the volume of a 
 
 given mass of gas, when its temperature is raised from the 
 
 freezing to the hoiling point of water, increases by a fraction of 
 
 the volume at the frst temperature, tohich fraction is the same 
 
 for all gases. 
 
 In short, the law is that all gases have the same coeffi- 
 cient of expansion, and that this is indej)endent of the 
 magnitude of the (constant) intensity of pressure under 
 which they expand. 
 
 The fraction in question is, with certain reservations 
 to be mentioned presently, 
 
 so that if we measure degrees of heat by the Centigrade 
 thermometer, this is the fractional increase of volume for
 
 Gases. i n i 
 
 100°. The rate of expansion ])er deo-ree is also found 
 to lie uniform, and is -00366^,, which we shall use in 
 the form 1 *, 
 
 Hence if r„ denotes the volume of a jjfiven mass of anv 
 «)fas at o" C, and v its volume at f C, we have 
 
 T + ), (1) 
 
 273; ^ ) 
 
 whatever be the intensity of ju-essure ; and if / is its 
 volume at f, we have 
 
 v=v,,{ 
 
 273 +?; 273 + ^ ^ ' 
 
 If the point from which the temperature is reckoned on 
 the Centigrade thermometer is removed 273^ below the 
 ordinary zero, i. e., the point at which water freezes when 
 its surface intensity of pressure is that due to a standard 
 atmos])here (indicated by a mercurial column 760 mm. in 
 heig-ht), the expression 273 + t indicates the newly measured 
 temjwrature, and is always denoted by '1\ and called the 
 absolute temperatnre of the substance, the new point of 
 reckoning- being- called the absolute zero of temperature. 
 
 If it were possible to have T = o, that is t = — 273°, for 
 the gas — supposing the substance to remain a gas at 
 all temperatures with constant coefficient of ex])ansion. 
 ■^\^ — equation (i) would give 
 
 V = o, 
 
 i.e., the gfis would be reduced to zero volume. As the 
 substance does not satisfy the al)Ove supposition, but alters 
 its state in the process of lowering the temperature, the 
 
 * The fraction is more ."xccurately — , but the above is usually taken 
 
 2^2.05 
 
 for simplicity. Clerk Maxwell {Theory of Heat) gives various values ; tlius, 
 
 and , the latter deduced from experiments of Tliomson and 
 
 2734 2737' 
 
 Joule.
 
 192 Hydrostatics and Elementary Hydrokinetics. 
 
 consequence is not realised, and it would thus appear that 
 the notion of an absolute zero of temperature at — 273° C is 
 a g-ratuitous error. Indeed, if the conception of absolute 
 1 emperatiire rested on no other foundation, we mig-ht simi- 
 larly arg-ue from the coefficient of expansion of i)latinum, 
 
 for instance, that since for this body t; = r^ T i ■\ — j > 
 
 nearly, where v^ is its volume at zero and v its volume 
 at f, if we make t = — 37699 we shall arrive at the absolute 
 zero of temperature. 
 
 The truth is that the measure of absolute temperature 
 rests on quite another basis, that it is intimately connected 
 with the coefficient of expansion of a perfect gas, and that 
 273 + i^ is proi^erly to be regarded as measuring- the absolute 
 temperature of a body whose temperature indicated by 
 a Centigrade thermometer is t. This will be shown later on. 
 
 Adopting- absolute temperature, then, equation (2) gives 
 
 V V 
 
 Of coTirse in the expression of the law of Dalton and 
 Gay-Lussac it is not necessary to sig-nalise the particular 
 temperature corresponding to the freezing of 7oafer as pos- 
 sessing- any special reference to the expansion of gases. 
 
 The law may be stated thus : all gases expamt, per degree, 
 Ij/j the same fraction of their volumes at any common tempera- 
 ture. This is obvious because their volumes at anij tem- 
 perature, r, will all be the same multiple of their volumes at 
 0°, and a constant fraction of the latter will give a constant 
 fraction of the former. 
 
 In symbols, for any gas let u be the volume at t, 
 r that at t^ Vq that at zero, and a the coefficient of expan- 
 sion with reference to the volume at zero ; then 
 
 V = l■^^{l+at)^, u = v^,{l+aT);
 
 Gasfs. 193 
 
 I 4- « / 1 -f nr -f rt (/ — t) 
 
 anil therefore v= u = // 
 
 I + ciT 1 4- Q T 
 
 = 11 {i+i3{/-r)l, 
 
 where /3 = > so that /3 is obviously the rate of expan- 
 
 I +ar 
 
 sion of the g-as reckoned us a fraction of the vohiiiic // ; and 
 
 if a is the same for all <2^ases, so is /3. 
 
 It is remarkahle that a is the same for all g-ascs when far 
 removed from their condonsinij;' points, i. e., from the liquid 
 states, and that it is indejiendent of the intensity of 
 pressure under whicli the expansion takes place. 
 
 Clerk Maxwell {Tlieort/ of Ileaf) points out that if i fie lav 
 nf Ballon anil Coif-Lusftac i.t Irnr for ani/ one intcnaiti/ of 
 })rex!<nri\ and if l/ie lair of Boyle holds, it follows that the 
 former lav holds for all intensities of pressure. 
 
 This is very easily proved thus. Let r\ be the volume 
 of a p^iven mass of g-as at (o°,;j), i. e., p is its intensity 
 of pressure, and let the law of Dalton hold for this pressure 
 intensity ; then if v is its volume at (/, p) 
 
 r= r,(i+a/). 
 
 Now, keepinpf t constjxnt, alter p to // ; tlien by Boyle's 
 law the new volume, v, is <^iven by the equation 
 
 " = '0(1 +"0-^7 • 
 
 But if '•„ at (o", /)) were altered by keepinji; its temperature 
 zero and chan<,^in<^ its intensity of pressure to //, its value, 
 
 j/„, would be t'„. -^by Boyle's law; so that the last equa- 
 tion fjives „ _ ^Ji+n/^, 
 
 and therefore Dalton's hiw liolds for //' if it holds for /). 
 

 
 194 Hydrostatics and Elementary Hydrohinetics. 
 
 With reg-ard to the accuracy of the law of Palton and 
 Gay-Lnssac, !M. Reii'iiault has found that, a heiug the 
 coefficient of expansion per deg-ree Centigrade, 
 
 for Carbonic acid, a = -003710, 
 
 „ Protoxide of Nitxogen ,. = -003719, 
 
 „ Sulphurous acid, „ = -003903, 
 
 „ Cyanogen „ = -003877 ; 
 
 the last two of which are notably greater than the coeffi- 
 cient of expansion of air ; but these are precisely the gases 
 that can be most easily liquefied, while it is found that for 
 all gases which can be liquefied only with great difliculty, 
 a has very nearly the same small value, -003665, that it has 
 for air. Hence M. Regnault modifies the law of Dalton 
 and Gav-Lussac bv saving that the coefficients of expansion 
 of all gases approach more nearly to equality as their 
 intensities of pressure become more feeble ; so that it is 
 onlv when erases are in a state of great tenuitv that thev 
 have the same coefficient of expansion. 
 
 42. General Equation for the Transformation of a 
 Gas. (liven the volume, v, of a mass of gas at the 
 temperature t, and pressm-e intensity p, find its volume 
 at i' and ;/. 
 
 First let the temperature be altered from t to i' , the 
 pressure intensity remaining 7; ; then the volume v becomes 
 
 «, where ->--? _(.^' 
 
 11 = V ^^^ — 7 , 
 ^73 + f 
 l.>v equation (2) of last Art. 
 
 Now keep the temperature constantly equal to f and 
 alter p to p' ; then ft becomes r', whore 
 
 P 
 by Boyle's law. Hence we have
 
 Gases. 1 95 
 
 ,273 + ''' ;> 
 
 r = r 
 
 273 + /! ;> 
 
 / > 
 
 (0 
 
 
 / / 
 
 or ^J7- - ^T' \^) 
 
 where 7' and T' are the absolute temperatures of the i,-as. 
 
 Hence, wliatever changes of pressure and tenii)crature 
 may be made in a given mass of yas, we have the result 
 
 -^/- = constant (/3) 
 
 between its vohime, pressure intensity, and absohite tem- 
 j)erii1ure. 
 
 This most important result is the <i;-cncral equation for 
 the transformation of a yiven mass of g-as. 
 
 43. Formula in English Measures. Since the freezin<r 
 
 jioint of water is marked 32° on Fahrenheit's thermometer. 
 
 and the boilinq- point 212°, the fractional expansion ol" 
 
 .066') I 
 
 grasper dcg-ree Fahrenheit is 'V, ' , or alwut , of the 
 
 "^ ^ *^ 180 49i'i3 
 
 volioiie at 32°. This fraction is usually taken as :j^jj ; and 
 
 this, as will presently be seen, would ])lace the absolute zer<» 
 
 oftvmjterature 460 Fahrenheit de<>^rees below the zero of the 
 
 Falircnheit scale. The experiments of Joule and Thomson 
 
 indicate —460-66 as the position of the absolute zero; but 
 
 for all practical purposes we can take —460. 
 
 If a yiven mass of g^as has a volume n at 32° F, and its 
 
 temperature is mised to /, wc have, if the intensity of 
 
 pressure is unaltered, 
 
 r = W ( I + ~- ) ' 
 
 ^ 492 ^ 
 
 O 2
 
 196 Hydrostatics and Elementary Hydrokinetics. 
 
 Hence v = u — \ and if v' is its volume at /!', we 
 
 have 
 
 V v' 
 
 460 + f 460+ f 
 If the intensity of pressure, estimated in any way, 
 alters from /; to y/, 
 
 '^ = ''^ (a) 
 
 460+?! 460 + r ^ ^ 
 
 It thus appears that if the temperature of the gas were 
 reduced to — 46o°r, its volume would vanish, supposing- 
 that it obeys the laws of a gas during the whole process. 
 
 If we denote by T the absolute temperature, 460 4- f, 
 of the gas, we have the general equation of transformation 
 of a o-iven mass 
 
 Y~~7^~ constant (^) 
 
 44. Law of Avogadro. One of the fundamental laws 
 of gases is known as the Law of Avogadro. It is the 
 following : equal volumes of all stibstances token hi the state 
 of j^eifect gas, and at the satne temperature and intensity 
 of pressure, contain the same number of m.olecules. 
 
 This law enables us to find the relative molecular weights 
 of all substances by converting these substances into vapours, 
 and then measuring the weights of known volumes of the 
 vapours at known temperatures and intensities of pressure. 
 Thus, it is found that a cubic foot of oxygen weighs 
 16 times as much as a cubic foot of hydrogen under like 
 conditions of temperature and pressure ; hence we conclude 
 that the mass of each molecule of oxygen is 16 times that 
 of a molecule of hydrogen. 
 
 45. Air Thermometer. A long capillary glass tube, 
 AC, terminating in a bulb, B, is filled with air, and a short 
 thread, w, of mcrcurv is inserted into it, the end of the tube
 
 G 
 
 uses. 
 
 197 
 
 (beyond C) iR'iiiy- 0)1011. In order to (ill the biillj and jior- 
 tion of the tul)e 1 o t he left of in with air deprived of moisture, 
 the tube and Imll) are lirsl lilled witli mereurv which is 
 boiled in the l)un). The open end is then inserted into a 
 cork fittin<4- into the niek ol' a tube, 1), lilled with chloride 
 
 Fig. 58. 
 
 of calcium, which has the property of absorbing* aqueous 
 vapour from air, a fine platinum wire havino- been inserted 
 into the stem CA throug-h the tube I). If the instrument 
 is supported in a position slightly inclined to the horizon 
 on two stands and the })latinum wire is agitated, air enters 
 through the chloride of calcium, and gradually dis2)laces the 
 mercury from the bulb and stem, the process being* stopped 
 \vhen only a very short thread of mercury is left. 
 
 The air in the instrument may now be considered to be dry. 
 
 Detach the stem from the drying* tube 1>, and place it in 
 a vertical position with the bulb J^ in a vessel filled with 
 melting- ice. Suppose the barometer to stand at 760 mm., 
 thus indicating the standard atmos])heric pressure. Then 
 when the air has assumed the temperature of the melting 
 ice, mark o on the stem AC at the under limit of the 
 mercury index >ii. 
 
 If then the instrument is ])liiced vertical with the bulb 
 H surrounded by the steam of boilingf water — close to the 
 surl'ace of the water, but not in it — the index ;// will move 
 up towards C, and at its lower limit let lOO be marked on 
 the stem. Thus the two standard Centig*rade temperatures 
 are marked on the steu), and the intervening sjjace is to be 
 divided into 1 coequal pai'ts if the stem has previously been
 
 198 Hydrostatics and Elementary Hydrokinetics. 
 
 ascertained to be of imiform bore. The graduations may 
 be carried then below zero and beyond too°. 
 
 If the tube of the air thermometer is made cylindrical all 
 through — so that the bulb -5 is simply a uniform continuation 
 of the stem — and we continue the graduations to 273 parts 
 below the zero, we shall here reach the bottom, B, of the tube. 
 
 Hence the definition of the absolute temjjeratMre of a body, 
 which we are so far justified in giving, is simply, in the 
 words of Clerk Maxwell, its temperature reckoned from tha 
 tjottom of the tube of the air thermometer. 
 
 The upper end of the stem of an air thermometer neces- 
 sarily remains open to the atmosphere, otherwise the index, 
 m, would not move or would scarcely move at all : if the end 
 were closed and the air uniformly heated, m would not move. 
 
 Hence the air thermometer cannot be used to indicate 
 temperature except in conjunction with the barometer. 
 If the latter stands at p instead of j';^ , the standard height 
 (which we have above supposed to be 760 mm.) and the 
 temperature indicated by the index m is t, the real reading 
 is not t but that at which the index would stand if the 
 intensity of pressure were altered to 7^^. To find the point at 
 which the index would stand in this case, let s be the area 
 of the cross-section of the tube, c the length of the tube 
 between two successive degrees, and B the volume of the 
 bulb and tube up to the zero mark. Then when the index 
 m stands at the mark t\ the volume of the gas is B-\-cst'. 
 But since at the absolute zero the volume of the gas would 
 vanish, B = 273 c* ; hence 
 
 V = (273 + 0'^*'^ 
 
 and this is at the intensity of pressure p, its true tempera- 
 ture being t. If p were altered to p^ without any change 
 in the true temperature, the index would stand at t and the 
 volume would be (273 + ;^)^*, Now since these volumes
 
 199 
 
 Gases. 
 are inversely as the intensities of pressure, we have 
 
 273 + ^ = (273 + T' 
 
 Po 
 
 which <j^ives the true reading-. 
 
 46. Work done in Expansion. If a <;fas is contained 
 in an expaiisil)k' envel()j)e, the j)rt'ssure of the i^as on each 
 element, (/S, of area of the envelope is continually drivinf,'' 
 the element (/S outwards along the normal to it, and hence 
 the pressures on the various elements perform a certain 
 amount of work in the increase of the total volume of 
 the g-as from one value to another. 
 
 Thus if 7; pounds' weight per 
 stpiare foot is the intensity of 
 pressure of the gas at an}^ instant 
 and we take (/S square feet at 
 any point, P, Fig. 59, of the 
 envcloj)e, ABC, the force acting 
 on dS is pdS. If the jioint I' 
 is moved by the pressure to 
 a close position, P', along the 
 normal, the work done by this force is 
 
 juISxPP'. 
 
 Now if the new position of the envelope is A'B'C, 
 ilSxPP' is the volume of the small cylinder standing 
 on (IS and terminated by the new surface A'B'C ; and the 
 sum of the works done by the pressures on nil the elements 
 of ABC in moving this surface to A'B'C is 
 
 :i: {pdS X PP'), or ;> . 2: {,IS x PP'), 
 
 since p has the same vnlue at all points of ABC. But 
 1{(lSxPF) is the sum of all such small cylinders as that 
 above described, i. e., it is the volume of the whole space 
 between ABC and A'B'C. If, then, v cubic feet is the 
 volume of ylBC, and v + dv the volume of A'B'C, tho 
 
 I" 
 
 ^'d- 59-
 
 200 Hydrostatics and Elementary Hydrokinetics. 
 
 volume included is tlv, and the work, dW, of expansion 
 from ABC to A'B'C is given by 
 
 dW=p(h (i) 
 
 Hence the work done by the pressure of the gas on its 
 envelope in expanding from an initial volume v^ to any 
 final volume, v-^ , is given by the equation 
 
 JFz 
 
 : / pdv (2) 
 
 If p is measured in dynes per square centimetre, and v 
 in cubic centimetres, the work will be in ergs. 
 
 The amount of work may be represented in a diagram 
 by describing the curve (such as PQ in Fig. 55) whose 
 abscissae represent the volumes of the gas and whose ordi- 
 nates represent the corresponding intensities of pressure. 
 
 In the particular case of isothermal expansion, j^v =2^^^'^Q, 
 so that „ J 
 
 ' vo 
 
 V 
 
 = /'o^'olog,-S (3) 
 
 ^0 
 
 from which it appears that the "work done by the pressure 
 of the gas in expanding isothermally from one given 
 volume to another is independent of the temperature. 
 
 The work done by the pressure of an expanding gas 
 on its envelope may or may not be equal to the work done 
 against any external pressm^e which acts on the surface 
 of the envelope. Thus, if the gas is contained in a hori- 
 zontal cylinder and kept in by means of a piston on which 
 the atniOHphere presses, when the piston is released the 
 work done by the pressure of the gas on the piston is 
 equal to the work done against the atmospheric pressure 
 plus the gain of kinetic energy of the piston.
 
 Gases. 
 
 20 1 
 
 Examples. 
 
 1. A circular cone, hollow but of great weight, is lowered 
 into the sea by a roi)e attached to its vertex ; find tlie volume 
 of the compressed air in the cone 
 when the vertex is at a given depth 
 below the suriuce. 
 
 Let Fig. 60 I'epresent a section 
 of the cone ; let c be the depth of 
 the vertex below the surface, LN, 
 of the water, h = height of cone, 
 V= its volume, t = the tempera- 
 ture of the air at the surface, t' = 
 temperature of the water, and there- 
 fore of the air in the cone ; let F 
 be the surface of the water within 
 the cone, and let k be the height 
 of a column of sea-water in a water 
 barometer. 
 
 If these quantities are in English measure, we may regard the 
 lengths as measured in feet, and the temperature as Fahrenheit ; 
 then k will be about 33 feet. 
 
 Now if X is the depth of F below A, the volume of the air in 
 
 the cone is V -^ • The intensity of pressure of this air measured 
 
 by a column of water is k + c + x. Hence the following diagrams 
 represent the history of this mass of air as regards volume, 
 temperature, and intensity of pressure : 
 
 Fig. 60. 
 
 in which 7' and T' are absolute temperatures. 
 From Art. 42 or Art. 43 we have, then, 
 
 Vk_ Vx'{k + c + x)
 
 202 Hydrostatics and Elementary Hydrokinetics. 
 
 r 
 
 :. a;* + (X- + c) x^ -Icl^ ^ = O- 
 
 from which x can be found. 
 
 A vessel used in this manner is called a diving bell. The 
 above is a conical diving bell. 
 
 2. If in the above position of the cone it is desired to free 
 the interior of water completcl^y bj- pumping the air above the 
 surface iuto the cone, find the volume ot tliis surface air that 
 will be required. 
 
 Let U be the volume required, and h the height of the 
 cone ; then suppose the cone to be wholly filled with air of the 
 temperature t' of the surrounding water, and write down the 
 histoi-y of this air, thus : 
 
 {V+U)k _ r(k + c + h) 
 
 T ~~ r ' 
 
 (It is not improbable that the student will fall into the error 
 of supposing that U can be calculated as the volume of the 
 surface air which is required to occupy the lower portion of the 
 cone in Fig. 60, i.e., the portion occu2:)ied by water.) 
 
 Of course the result is the same whether the vessel is conical 
 or of any other figure. 
 
 3. If a conical diving bell of height h feet contains a mercurial 
 barometer the column of which stands at p^, inches when the 
 bell is above the surface of the water, and at a height p when 
 below, infer the deptli of the top of the bell below the surface. 
 
 4. Deduce the depth for a cylindrical or prismatic bell.
 
 Gases. 203 
 
 5. Fintl the tension of the suspending chain in a diving 
 hell which occupies any position in water. 
 
 Ans. The woight of the bell and its appurtenances di- 
 minished by the weight of the water which is displaced from all 
 causes. 
 
 (The water is displaced by the chain, the thickness of the bell, 
 and the air within the bell ; tlie weight of this water is tlie 
 force of buoyancy. In strictness, the weight of the contained 
 air should be added to that of the bell.) 
 
 6. If at the bottom of a river 40 feet deep, when the tem- 
 perature is 40" F, a bubble of air has the volume — ^ of a cubic 
 
 inch, what will be its volume on reaching the surface where the 
 temperature is 50'" F, and the height of a water barometer is 
 34 feet ? 
 
 Ans, 1. cubic inches. 
 
 10" 
 
 7. If an open vessel (such as a tumbler) made of a substance 
 whose specific gravity is greater than that of water is forced, 
 mouth downwards, into water, show that its equilibrium becomes 
 unstable after a certain depth has been reached. 
 
 (If the volume of the solid substance of the vessel is v, and 
 in any position of the vessel if X is the volume of its compressed 
 air, the downward force, P, required to hold it in equilibrium 
 is given by the equation 
 
 P= Xw — v(w' — w), 
 where to = specific weight of water, w= specific weight of 
 substance of vessel. 
 
 Hence when X is so far diminished by forcing the vessel 
 down that Xiv = vl^io —w), the pressure P vanishes, and after 
 this an upward pull would be required.) 
 
 8. If V is small (i. e., if the thickness of the vessel is small), 
 and if V is the volume of the interior of the vessel, prove that 
 when the position of instability is reached, the depth of the top 
 of the vessel below the surface of the water is approximately 
 
 (v[w —w) 3 
 where k is the height of a water barometer at the surface.
 
 204 Hydrostatics and Elementary Hydrokinetics. 
 
 9. If V cubic inches of the external air at the absolute tem- 
 perature T are inserted into the Torricellian vacuum of a 
 unifunn cylindrical barometer tube, calcuhitc the depression 
 produced in the column of mercury if the absolute temperature 
 changes to T' . 
 
 Ans. Let li inches be the height of the barometer at first, 
 a = length of Torricellian vacuum, s squai'e inches = area 
 of cross-section of tube, x = length of tube finally occupied 
 by the air ; then ^" ,^\ 
 
 x(x-a) = - .— . 
 
 10. A diving-bell of any shape occupies a given position 
 below the surface of water ; the bell has a platform inside ; if a 
 large block of wood falls from the platform into the water, prove 
 that the water will rise inside the bell, but that the bell now 
 contains less water than before. 
 
 Let the depth of the top be c, let h be the height of a water 
 barometer at the surface, put h =■ c-{- h, let B = volume of the 
 block of wood, w' its specific weight, iv = specific weight of 
 water, V= whole volume of the interior of the bell, let x be the 
 depth of the water in the bell below the to]) of the bell, and let 
 A' be the volume of the interior of the bell above this surface. 
 Then (X-B) (x + k) = Vh (i) 
 
 When the wood falls the volume of it which remains above 
 
 the surface is B ( i — ) • Let x' be the new depth of the water 
 
 surface in the bell below the top of the bell, and A'' the volume 
 of the interior of the bell above this new surface. Then 
 
 [^'-J'^(^-'^)}i-' + ^)=^'^- 
 
 (-^) 
 
 Now since X obviously increases with x, we must have x'<x, 
 since in the opposite case each of the factors at the left-hand 
 side of (2) would be greater than the corresponding factor 
 in (i), and the equations would be inconsistent. 
 
 Hence the surface of the water rises. 
 
 Again, if 12 = the first volume of water in the bell, and il' 
 the second ^2 = V-X, 
 
 .w' 
 
 ir= V-X'-B—; 
 w
 
 Gases. 205 
 
 a-ii'^B'^-ix-x') (3) 
 
 Now from (i) ;ui<l (2) 
 
 10' Vh{x-x') 
 
 
 vj {x + k) (x' + k) 
 h (x — x') 
 
 which shows that ii' is less than D,. 
 
 47. Weight of Gas. It is obvious th<at the weio-ht 
 of a cubic-loot of air, or any other g-as, is not the same when 
 its temperature is 20° or 100°, as when it is 0°, supposing- 
 the intensity of ]iressure the same. In other words, the 
 weight of a cubic foot of air depends on the temperature 
 and pressure intensity at which it is taken. 
 
 Taking- the units of the Metric System, let us enquire 
 what is the weight of v litres (i. e., cubic decimetres) of dry 
 air when its temperature is fC and its intensity of pressure 
 denoted by a column of mercury p millimetres high. 
 
 Supposing that we knew the weight of 1 litre of air 
 when its temperature is 0° and its intensity of pressure that 
 of a standard atmosphere, denoted by a column of mercury 
 760 mm. high, we could answer the question by finding- 
 the number of litres which would be occupied by the g-iven 
 V litres if its state were changed from {p, f) to (760, 0°). 
 But by (i) or (2) of Art. 42, if we put t'^o, i'=']6o, 
 we have in^ vp . . 
 
 '•» = 7-|o-| ^'^ 
 
 Now M. Regnault found that the mass of 
 I litre of dry air at (760, 0°) = 1-293187 grammes . (2) 
 
 Hence the mass of v litres at (p, t) is Vq multiplied by 
 this number. Denoting the mass in grammes by W, we 
 
 have then r v 
 
 7f^=.4645^^ (")
 
 2o6 Hydrostatics and Elementary Hydrokineties. 
 
 in which, be it remembered, T is the absolute Centig-rade 
 temperature of the air, p its pressure intensity estimated in 
 millimetres of mercury, v its volume in litres, and /Fits 
 mass in grammes. 
 
 For any other gas, if its specific gravity at (760, 0°) is 
 denoted by s, the mass of a litre of it in this state is 
 1-293187 X5 grammes, and evidently if //'is the mass of v 
 at (/), ^), we have simply 
 
 Tir ^ v.p.s ,. 
 
 JF=-4645—^ (/3) 
 
 The specific gravity of a gas is above assumed to be the 
 ratio of the weight of any volume of the gas to the weight 
 of an equal volume of dry air at (760, 0°) ; but it is easy to 
 see that we get exaetl}^ the same result by taking the ratio 
 of the weight of a volume of the gas at {p, t) to the weight 
 of an equal volume of air also at {p, t), whatever the pressure 
 intensity, ;;, and the temperature, t, may be, if it be true 
 that all gases have the same coefficient of expansion ; for, 
 equal volumes, v, of the two gases at {p, t) will become 
 equal volumes, Vq, at (760, 0°) since 
 
 V .p I 
 " J+at 760 
 
 and a is the same for both gases. 
 
 48. The equation^ = kp. From (/3) we see that if for 
 
 W 
 
 — we write p, where p is the mass, in grammes, of the gas 
 
 V 
 
 per litre, we have 
 
 P= — 7 — • - ■ Pi (0 
 
 .4645 s ' 
 
 p being measured in millimetres of mercury. 
 
 Let 7; be measured in grammes' weight per square 
 centimetre, and let p be the mass, in grammes, per cubic 
 centimetre ; then, if we have i cubic cm. of the gas at
 
 Gases. 207 
 
 (/;, t), and this becomes x cubic cms. at zero and an intensity 
 of pressure of 76x13-596 grammes' weig-lit per sq. cm., 
 we have a; x 76x13-596 _ijcp 
 
 Now the mass of i cubic cm. at the latter temperature 
 
 and pressure being- — ^ s grammes, the mass of x cubic 
 
 * 1000 ^ 
 
 cms. is obtained by multiplying this by the value of x in 
 
 (2), and this mass is p, the density of the gas at (f, p), i. e., 
 
 the mass of i cubic cm. Hence we have 
 
 T 
 
 P = 2926'()-p, (3) 
 
 where j) is in grammes' weight per sq. cm. and p in 
 grammes per cub. cm. Hence if we write the relation 
 between p and /) in the form p = kp, we see that 
 
 T 
 
 k = 2926-9 — • 
 s 
 
 If /3 is measured in dynes per sq. cm., we must multiply 
 
 this value of Jc by the value of ^ in dynes, i.e., by 981 
 
 (about). In this case, then, 
 
 p = 2926-9 - p (4) 
 
 (Observe that here s is the specific gravity of the gas 
 referred to air.) 
 
 If V is the volume of the gas at [p, T), and to its mass, we 
 
 have, by multiplying both sides of (3) by v, 
 
 w 
 pv = 2926-9 -.T. (5) 
 
 It is sometimes useful to express p in kilogrammes' 
 weight per square decimetre, v in cubic decimetres, and 10 in 
 kilogrammes ; in which case (5) becomes 
 
 IV 
 
 pv= 292-69 - . 7* (6)
 
 2o8 Hydrostatics ami Elementary Hydrokinctics. 
 
 It is usual to write (5) or (6) in the form 
 
 pv = RivT, (7) 
 
 where R stands for the constant ~ ~ in the first case. 
 
 s 
 
 49. Formulae in English Measures. The equation 
 
 connecting" the vohmies, &c., of a given mass of gas in 
 
 English measm-es is 
 
 vp ^ v'p' 
 
 460 ^-t 460 + r ^ ' 
 
 To obtain the formula for the mass of air, analogous 
 to (a). Art. 47, we may either convert the metric formula 
 into English measiu-es, or deduce a formula from special 
 observations on the mass of a given volume of air under 
 standard conditions. Dr. Prout found that the mass of 100 
 cubic inches of dry air at the temperature 60° F at an 
 intensity of pressure indicated by 30 inches of mercury in 
 a barometer tube is 31-0117 grains; in other words, 
 the mass of i cubic foot at (60°, 30''') is -0765546 pounds, (o) 
 
 Now if we have v cubic feet of dry air at {t°,p), where ^ 
 is in inches of mercury, this would, by (i), become 
 
 3 460 + 1 
 
 cubic feet at (60°, 30), and multiplying this by the 
 number (a), we have 
 
 r= 1-326946 -^^, .... (2) 
 
 for the mass, in pounds, of the given v cubic feet at {f, p), 
 the intensity of pressure, p, being supposed taken in inches 
 of mercury. 
 
 For a gas of specific gravity s (referred to air), 
 
 r= 1.326946 ^^f^- (3)
 
 Gases. 209 
 
 If /3 is estimated in pounds' weight jier square foot, and 
 460 + iJ is denoted by T, we have 
 
 w= — 1-_ . !^* (4) 
 
 53-30222 T ^^^ 
 
 T 
 
 ••• P= 53-30222-./,, (5) 
 
 where p is the density of the g-as in pounds per cubic foot, 
 p is its intensity of pressure in pounds' weight per square 
 foot, &c. 
 
 To obtain the analogue of (6), Art. 48, multiply both 
 sides of this equation by v ; then, with sufficient accuracy, 
 
 P'>^=53'3-T, (6) 
 
 in which 7v is the mass of the gas in pounds ; and if we 
 write this in the form 
 
 jw = RwT, (7) 
 
 R stands for ^^ • 
 s 
 
 50. Barometric Formula. We are now in a position to 
 deduce a formula for the height of a mountain, by neg- 
 lecting the variation 
 of gravity between the 
 base and the summit, 
 and by assuming the 
 temperature of the air 
 to be constant within 
 theselimits. The latter 
 assumptionwould often 
 be far from the truth ; 
 but we shall presently 
 see how it can be corrected. 
 
 Let A, Fig. 61, be a point at the base of the mountain 
 where the height of the barometer is jo^ inches ; let P be a 
 
 p 
 
 Fig. 61,
 
 2IO Hydrostatics and Elementary Hydrokinetics. 
 
 point at a heig-lit z feet above A, and at P let the height of 
 the barometer be p inches ; let Q he a point very near P, 
 the difference of level between P and Q being dz feet; 
 and let t° F be the temperature of the air at P. 
 
 Imagine a horizontal area of i square foot at P ; then 
 the atmospheric pressm'e on this area is the weight of the 
 column of air standing on it and terminated by the limit 
 of the atmosphere. Hence the difference of the pressures 
 on this area at P and Q is the weight of the vertical column 
 of air between P and Q standing on i square foot^ i. e., the 
 weight of dz cubic feet. 
 
 But '\i p — dp is the height of the barometric column at 
 Q, the difference of the pressui'es on i square foot at P and 
 at Q is the weight of a column of mercury standing on 
 I square foot having the height of —dp inches. Now 
 the mass of i cubic foot of mercury = 848 1 pounds at 
 the temperature of melting ice ; but if the temperature 
 of the mercury is t° F, this requires correction. The 
 coefficient of absolute expansion of mercury per degree 
 Fahrenheit is very nearly -^^t \ hence, if w is the weight 
 of a cubic foot at 32°, the weight of a cubic foot at t° is 
 
 , or %o{\ ^ ) J nearly; so that the weiijht of 
 
 9915 
 the column of mercury corresponding to the barometric 
 
 fall between P and Q is 
 
 _84875. _1^2),, 
 12 ^ 9915^ 
 
 If, then, p is the height of the barometric column at the 
 temperature f F, the corrected /i eight is 
 
 ^ V 9915 >> 
 
 To save a multiplicity of symbols, we shall suppose that
 
 Gases. 21 1 
 
 the hci<>ht.s of the mercury at the stations A, 2\ Q,... are 
 thus corrected ; in other words, we shall suppose in the 
 subsequent work that p, p^^, dp, &c. are coned ed heig/iix. 
 With this understanding-, if we equate the weig-ht of the 
 column of air obtained by writing- dz for v in (2) of last 
 Article to the weig-ht of the column of fall of the barometer 
 l)etwccn P and Q, we have 
 
 ^ ^ 460+ / 12 ^ ^ ^ 
 
 (If p is not a corrected height, the temperature coefficient 
 which mulliplics /; in (a) must be considered as part of the 
 variable dp in the right-hand side of this equation, so that 
 it does not disappear by division, unless the temperature is 
 constant all the way up the mountain.) Hence 
 
 ^^^ = -53-3022(460 + /).^', .... (2) 
 
 which is the differential relation between z and p, from 
 which the relation between them can be obtained only 
 by taking the temperature t constant in the term 460 + /. 
 If we do this, and integrate from A to P, we have 
 
 r r^ dp 
 
 / ^/^- = -53-3022 (460 + /) / --, 
 
 '■■ ^ = 53-3022 (460 + /) log/X ... (3) 
 Now logg n = log'io u ; hence (3) becomes 
 
 ^(feet)= i22.73(46o + ^;)log,o^°, . . (4) 
 which is sometimes put into the form 
 
 z= {60383+ 122-73 (''-32)} log.o^. • (5) 
 
 The constant value of / in tliis expression is usually 
 
 taken to be the arithmetic mean, — — , between the air 
 
 P 2
 
 212 Hydrostatics and Elementary Hydrokinetics. 
 
 temperatares t^ at A and t at P. In the case of a very 
 high mountain, several observations might be made at 
 different levels, taking the temperature constant in each 
 successive stage and equal to half the sum of the tempera- 
 tures of the air at the beginning and end of the stage, 
 and then adding the calculated heights of the successive 
 stages together. 
 
 The residt (5) can also be deduced from the general 
 equation of equilibrium, (2) or (3) of Art. 17. If 7^ is the 
 intensity of pressure of the air at P iu pounds' weight per 
 square foot, and p is the mass of the air in pounds per cubic 
 foot, observing that z is measured ujawards and the force 
 p pounds' weight doicmcards, 
 
 l = - (^) 
 
 and by (5) of last Art., 
 
 i5 = 53-3C222r.p, (7) 
 
 .-. f/2 =—53-30222 J— > .... (8) 
 
 which is (3) above. 
 
 If the mountain is so high that there is a sensible varia- 
 tion of gravity between the base and the summit, and iijj 
 is still measured with reference to the weight of a pound on 
 
 the earth's surface at the sea level, the force acting on 
 
 2 
 
 p pounds mass at the height 2 is p ( A pounds' weight 
 
 at the sea level, where r is the radius of the earth ; so that 
 (6) is to be replaced by 
 
 *- - '^ (9) 
 
 —P\ 
 
 dz (^ + 2) 
 
 while (7) still holds. Hence 
 
 dp 
 
 <^2;=— 53.30222^— 5 • • • • (10) 
 
 {r+zf^ ^JJ jj
 
 Gases. 2 1 3 
 
 Assumino: the station J to be at the height r^ above the 
 sea level, and that the intensity of pressure at A is />, 
 pounds' weight per square foot. The integral of (ii) is 
 
 In this equation jj and j)^ are measured in pounds 
 weight per square foot ; but they must be inferred from 
 the reading:s of the barometer at the two stations : and 
 if the heights of the barometer at the stations are ^ and 
 ^1 and the weights of a unit volume of mercury at these 
 stations are ir and fr^ , respectively, we have /' = }r/t and 
 
 /;, = ic, L, therefore^ = -1 . ^ = J-f -) ; hence (12) 
 
 becomes ' ^ 
 
 - ijh, - ^y --3 ^{io?/^+ ^>og,,;-±i-|, (.3) 
 
 in which the barometric heights may, of course, be measured 
 in inches, millimetres, or any other units of length. 
 
 Observe that - and — are verv small fractions whose 
 r r 
 
 squares may be neglected, so that M z — z^ is denoted by ^. 
 since logj^ (1 + -) is approximately equal to -4343 - ; we 
 have 
 
 A = 122.73 r(i+'^0|l«?rx.4+-86S6^|. . (14) 
 
 Let A = 122-73 ^logio - and B = 12273 x -8686 J; 
 
 then, pxitting . = A + -j, we have
 
 2T4 Hydrostatics and Elementary Hydrokinetks. 
 
 If the variation of gravity were neglected, we should 
 have A = ^, as in (4), and this value may be put for 
 A in the terms of (13) which involve r, so that finally 
 
 A = ^(i+ -)^ . . . (16) 
 
 an equation which gives the difference of level between 
 the top and base of the mountain when the height of the 
 base above the sea level is known. 
 
 It is understood, as before explained, that in this expres- 
 sion 2h ^^^ P ^^6 the corrected heights of the barometer at 
 the two stations. 
 
 51. Metric Formulae. By the same method — viz., the 
 equating of the weight of the vertical column of air between 
 P and Q, Fig. 61, standing on a horizontal square decimetre 
 to the weight of the column of fall of mercury standing on 
 the same area — w'e obtain the height of a mountain in 
 metric measures. 
 
 Thus, neglecting the variation of gravity, since a litre is 
 a cubic decimetre, if z is the height of P above A in 
 decimetres and the corrected barometic height at P is p 
 millimetres, the weight of the fall of mercury is — 1 35*96 ^^P 
 grammes' weight ; hence from (a) of Art. 47, 
 
 - 1 35-96 r/;. = .4645^^^, (0 
 
 ••• ^ = 673-962(273 + 0^0^10^' 
 
 r being in decimetres. If z is taken in metres, this 
 becomes f ^ 
 
 z = 18399.2(1 + -^ ^^^-10 y • • (2) 
 
 If the variation of gravity is taken into account, let ;j be 
 measured in grammes' weight per square centimetre, &c.,
 
 Gases. 2 1 5 
 
 as in (3) of Art. 48 ; then the equations are 
 
 dp 
 
 r^ 
 
 dz 
 
 '(r^zf 
 
 P = 
 
 2926-9 Tp, 
 
 ' ) = 
 
 r^-z' 
 
 673-962 7'lo^,o 
 
 > /inn T>n 
 
 p^ _h^ .r + z 
 
 in which a^ain we can put — = -f ( ) , and deduce a 
 
 result similar to (14). 
 
 Examples. 
 
 1. If a cubic inch of water is converted into steam at 2i2"F, 
 find the volume of the steam. 
 
 Arts. 1696 cubic inches. Hence it is approximately true 
 that I cubic inch of water yields i cubic foot of steam. 
 
 2. Oalcidate the mass of air in a room whose dimensions are 
 18, 18, and 10 feet, the temperature being 60^ F and the 
 barometer standing at 30 inches. 
 
 Ans. 248 pounds. 
 
 3. If I pound of water is converted into steam at 2i2°F 
 under the intensity of pressure of 15 pounds' weight per 
 square inch, prove that it will yield 26-66 cubic foet. 
 
 4. If any volume of water is converted into steam at the 
 temperature f F under the intensity of pressure p pounds' 
 weight per square inch, prove that the ratio of the volume 
 of the steam to that of the water from which it has been 
 formed is about 460 + < 
 
 ^' p 
 
 [This is called the relative volume of steam at the given tem- 
 perature and pressure.] 
 
 5. At the foot of a mountain the temperature of the air is 
 66°F, and the height of the barometer 29-35 inches ; at the top 
 the teniiwrature is 50'' F, and the barometric lieight 24-81; find 
 the height of the mountain, assuming the coefficient of expansion 
 of mercury to be xo^oo P^r degree Fah. 
 
 Ans. About 4590 feet.
 
 2i6 Hydrostatics and Elementary Hydrokinetics. 
 
 52. Nature of Gas Pressure. According- to the Kinetic 
 Theory of Gases, when a g-as is contained in a vessel, the 
 pressure exerted by the gas on each element of area of the 
 vessel is due to the incessant impacts of the molecules 
 of the gas on the element of area. 
 
 These molecules are, of course, extremely small : at each 
 instant a certain number of them will be in actual contact ; 
 but there will be a certain average distance between them, 
 and it is assumed that this distance is vastly greater than 
 the diameter, or greatest linear dimension, of a molecule. 
 Thus we are to imagine the space inside the vessel as being 
 comparatively void of gaseous matter. Nevertheless, this 
 space is what we mean by the volume of the gas, which, 
 therefore, is something very different from the sum of the 
 volumes of its material particles ; i. e., from the aggregate 
 number of the cubic centimetres occupied by its material : 
 the volume of the gas is the volume of the space within 
 which the excursions of all its molecules are confined. 
 
 Again, when the gas has settled do^^^l to constant 
 conditions of temperature and pressure, it is clear that, 
 in every respect, the state of affairs is the same at any one 
 instant as at any other. Not only so, but if we imagine 
 any area — say one square centimetre — placed at any point, 
 P, inside the vessel and occupying any position (orientation) 
 at this point, the number of molecules passing throug-h this 
 area in any time — say one second — whether from right 
 to left or left to right, is always the same. 
 
 It is easy to calculate the intensity of pressure produced 
 at each point of a vessel containing a system of molecules 
 moving in this manner ; but we shall confine our attention 
 here to the consideration of a very simple case from which 
 the state of affairs in the general case may be inferred. 
 The discussion of the general case will be found in Watson's 
 Treatise on the Kinetic Theory of Gases. The development
 
 Gases. 217 
 
 of the subject is, in the first instance, mainly the work of 
 Clerk Maxwell. 
 
 Imag'ine a cylinder whose base is AB, Y\^. 62, to contain 
 a larg-e number of molecules, the mass of each being- /x 
 g-rammes, all moving in lines parallel 
 to the axis of the cylinder, each p ?' 
 with a velocity of v centimetres per 
 
 second. Then if we measure a length ^ ^ ^ 
 
 AA' equal io v ./\t and draw a plane Fi„. 62. 
 
 A'B' parallel to AB, all the molecules 
 
 within the space ABB' A' which are moving* in the sense 
 A' A will in the time A t strike the base AB, and each will 
 be reflected with a velocity which is assumed to be equal to 
 V, i.e., the coefficient of restitution for each particle and the 
 base AB is assumed to be unity. Let the area of the base 
 AB be S square centimetres. 
 
 Now if there are n molecules in each cubic centimetre, 
 there are SnvAt within the space ABB'A', and since there 
 are as many moving" in the sense AA' as in the sense A' A, 
 the number striking- the base AB in the time AHs 
 
 \ SnvAt. 
 
 Hence the momentum incident on AB is \SnfMV^At; 
 and since the same quantity of momentum is g-enerated in 
 the opposite sense by impact on the base, the total chang-e 
 of momentum g-enerated by impact on the base in the time 
 
 ^^^^ Snixv-'At. 
 
 But if P dynes is the mean value of the pressure exerted by 
 the base on the column of molecules, P A ^ is the impulse of 
 this force in the time At, and this is equal to the chang-e 
 of momentum in the same time. Hence 
 
 P = Sntiv^ (i) 
 
 Of course n is enormously great and /x indefinitely small ;
 
 2i8 Hydrostatics and Elementary Hydrohinetics. 
 
 each is unknown, but n\i'\% the mass, in grammes, of one 
 cubic centimetre of the g-as. Denote this by p, and let p 
 
 P . . 
 
 denote -^ , the intensity of the pressure ; then 
 
 P = pv' (2) 
 
 This expi-esses p in dynes per square centimetre. If we 
 divide it by 981, or rather by ff, the acceleration of a freely 
 falling- body in centimetres per second per second, we have 
 
 p = '-^, (3) 
 
 which gives the intensity of pressure in grammes' weight 
 per square cm. 
 
 If V is in feet per second, p in pounds per cubic foot, and 
 ff in feet per second per second, (3) gives the intensity 
 of pressure in pounds' v; eight per square foot. 
 
 This formula w^ould apply to the case of a waterfall 
 which strikes the ground, the water not being' reflected by 
 impact, as the student will easily see on re-examining the 
 details. If // is the height of the waterfall, the velocity of 
 the molecules striking the plane is given by the equation 
 t;2 z= 2^//, so that p = 2pti, (4) 
 
 which equation gives p in gravitation measure — either 
 grammes' weight per sq. cm., or pounds' weight per square 
 foot ; for the latter units p is about 62\ pounds per cubic 
 foot. Hence (4) shows that the intensity of pressure on 
 the plane is twice as great as that produced by a statical 
 column of water of the same height. 
 
 Eut the case of a waterfall is in other respects different 
 from that of a gas ; for, in the latter case the molecules in 
 any column are not all moving in the same direction, 
 so that the value of 7; in (2) or (3) must be greatly superior 
 lo that of the intensity of pressure actually produced by
 
 Gases. 
 
 2T9 
 
 a g-as contained in the cylinder. We must remember that 
 V in these equations is the mean value of the velocity 
 in a fixed direction ; and that, if v^, v.,, v^ are the com- 
 ponents of velocity of a molecule in three fixed rectano-ular 
 directions, and v is the resultant velocity of the molecules, 
 
 This shows that when we consider an indefinitely great 
 number of molecules, if v'^ is the mean value of the squares 
 of their resultant velocities, 
 
 ^=3^-.^ (5) 
 
 where vj^ is the mean value of the square of velocity in 
 a fixed direction. 
 
 Now v^ in (5) must take the place of f- in (2) or (3). 
 
 Hence, then, for a s^^stem of molecules, moving- in all 
 directions inside a vessel, 
 
 P= \pv'^ (6) 
 
 P = \~^ (7) 
 
 according as p is measured in absolute units (dynes or 
 poundals) or in gravitation units (grammes' weight or 
 pounds' weight). 
 
 This elementary method of treating- the question is not 
 satisfactor}\ The following is more thorough and scientific. 
 
 Imagine the molecules of a gas contained within any 
 vessel divided at any instant into groups, the velocities 
 of all those in the same group being- nearl}^ the same 
 in magnitude and nearl}^ the same in direction. No one of 
 these groups is localised in a definite portion of the volume 
 of the vessel — the grouping- is not with reference to ^y/ac*? 
 but with reference to the clinracteri-sfics of vehcUy, so that 
 each group occupies the whole space within the vessel. 
 
 Now we can g-raphically represent the velocities in any
 
 220 Hydrostatics and Elementary Hydrokinetics. 
 
 group thus. Take any fixed orig-in, 0, and draw a line OP 
 to represent in mag-nitude and direction any velocity q ; 
 let Ox, Oi/, Oz be any three rectangular axes at 0, and 
 let the direction of OP be expressed by means of the usual 
 ang-les of colatitude and long-itude (see Statics, vol. i., Art. 
 176), i. e., let 6 be the angle zOF and let ^ be the ang-le 
 between the plane zOP and the plane oi xz; describe 
 a sphere with centre and radius OF ; let the axis Oz meet 
 this sphere in ^ ; on the surface of this sphere take a point 
 Q near P, its colatitude ZOQ being- 6 + d6, and its long-itude 
 <f) + d(f); through P draw a parallel of latitude Pli meeting 
 the meridian zQ in P, and through Q draw a parallel 
 of latitude QS meeting the meridian zP in S ; we thus 
 obtain a small spherical quadrilateral, PRQS whose area is 
 q^ sin 6 d6d({) ; produce OP to P' so that PP' represents 
 dq, and with OP^ as radius describe another sphere ; produce 
 OP, OQ, OS to meet the surface of this second sphere 
 in P\ Q', y, respectively ; then we obtain a small element 
 of volume between the two quadrilaterals PRQS and 
 P'R'Q'S', and the volume of this element is 
 
 q^ sinO dqdd d(f), or d£l (8) 
 
 The number of molecules in the group whose velocities 
 are represented by radii vectores drawn from to points 
 contained within this elementary volume will be propor- 
 tional to the product of this element and some function of 
 q, 6, (f). But since any given speed is found in all directions 
 indifferently within the vessel, it is clear that if from to 
 points contained within any element, d£l, of volume radii 
 be drawn and taken to represent a velocity group of nearly 
 the same speed, q, and nearly the same direction, the 
 quantity by which dQ, must be multiplied to give the 
 number of molecules in the group must be a function of q 
 alone and not of its direction (6, (p.)
 
 Gases. 221 
 
 Hence the number of molecules within the vessel and 
 contained in the selected group is 
 
 f(q).q^smedqdddc}>, (9) 
 
 where f(r/) is some unknown function of q. 
 
 It is not impossible that the following" objection will be 
 raised by the reader to the assumption that the number of 
 molecules in the group is f(q) .d£l: this expression vanishes 
 if d£l is zero, i. e., if we consider the number of molecules 
 moving- with exactly the same velocity, represented as above 
 by the radius vector OP, whereas every molecule in the 
 vessel might be moving with one and the same velocity — 
 as in the case of the elementary example first treated, 
 viz., that in which a stream of particles moved along the 
 axis of a cylinder. 
 
 To this the reply is as follows : the fundamental assump- 
 tion of the Kinetic Theory is that all possible sjieeds, from 
 o to oc, m all jwssible directions characterise the state 
 of affairs within the vessel, and therefore that the concep- 
 tion of the existence of only one speed among the molecules 
 is wholly inadmissible. Hence, there being an enormously 
 great number of molecules, the number of those moving 
 with the same speed — whether in the same direction or not 
 is immaterial — is relatively zero. 
 
 The number of the group (9) contained within a unit 
 volume of the space within the vessel is obtained by dividing 
 the expression (9) by the number of units of volume in the 
 vessel. We may, then, assume that (9) expresses the 
 number of the group per unit volume. 
 
 To calculate the intensity of pressure at any point, M, in 
 the gas, take a small plane area, dS, at the point perpen- 
 dicular to the direction of the axis Oz; on dS describe 
 a cylinder the length of whose axis is q cos 6 . dl^ where q 
 expresses the velocity characterising any group, and dt
 
 222 Hydrostatics and Elcnicntarv Hydrokindics. 
 
 is an infinitesimal element of time. Then all the molecules 
 within this small cylinder, whose volume is g co?, 6 dSdt, 
 will impinge on the plane dS within the time dt. Now it 
 matters not whether dS is a small material plane from each 
 face of which the imping-ing- molecules are reflected, theii' 
 normal velocities being- all restored to them in the reverse 
 sense, or an imagined plane area throug-h which two streams 
 of molecules pass in opposite senses ; for in either case the 
 quantity of momentum which in the time dt travels from 
 the plane in the same sense is the same. 
 
 The number of molecules of the (^, Q, 0) g-roup contained 
 within this cyKnder is 
 
 f[q).q-sv[iddqd6d(pxqco&6dSdt^ . . (lo) 
 
 half of them moving- towards the plane and half from it. 
 Let M be the mass of each molecule ; then, since the velocity 
 of each of these perpendicular to the plane is g cos 6, the 
 quantity of momentum passing- from the plane in one sense 
 within the time dt is 
 
 fxf (g) .q* sin 6 cos- 6 dqd 6 dqj . dSdt. . . (ii) 
 
 If p is the mean value of the force exerted, per unit area, 
 on the plane, the impulse of the force exerted by the plane 
 is pdSdt, and this must be equated to the integ-ral of (ii) 
 for all possible velocity g-roups. All such groups are 
 included by taking- the variables between the limits in- 
 dicated thus : 
 
 Hence 
 
 /(^)-?^/?/ sinOcos-ede d(j) 
 
 00 
 
 
 TT 
 
 
 3 
 
 6 
 
 ) 
 
 4> 
 
 
 
 
 

 
 Gases. 223 
 
 To see how this is connected with the mean square 
 of velocity of the molecules, let u be the number of molecules 
 per unit volume ; then from (9) 
 
 = 4^rf{q)-q''(Jq ('3) 
 
 To g-et the mean square of velocity, v~, multiply (9) by 
 q^, integrate and equate the result to n v^ ; then 
 
 '^=4^rf{q).q'dq (14) 
 
 Jj = I U IX . V^ 
 
 = ip'^\ (15) 
 
 as before obtained. (Of course _p is here expressed in 
 
 absolute units ; in gravitation units p = t — > ^s belore 
 explained.) 
 
 To get the mean velocity, v (which, of course, is not 
 
 equal to the square root of t;^) multiply (9) by q, integrate, 
 and equate the result to n v ; then 
 
 n 
 
 '0 
 
 Hence 
 
 '^4-f'^f{q)-f^k (16) 
 
 Thus we see that the result (15) is true whatever maybe 
 the form of the function f{q). In the Kinetic Theory 
 of Gases this function must have a pai'ticular form in order 
 that the incessant collisions between the molecules may 
 render the state of aflairs statistical/ 1/ the same at all times. 
 It does not belong to our present purpose to find the form 
 of this function ; its determination involves only the most
 
 224 Hydrostatics and Elementary Hydrokinetics. 
 
 elementary principles of the collision of elastic spheres, and 
 it will be found in detail in Watson's Kinetic Theory of 
 Gases. The statistical permanence of the state of the 
 molecules requires that 
 
 f{q) = Ae~~\ (17) 
 
 where // is a constant, and c is also a constant of the 
 nature of velocity. 
 
 Thus n = 4TTAj e ''\q~dq, .... (i8) 
 
 71 V 
 
 nv 
 
 
 
 2 
 
 Jr — — 
 e "''.q'^dq, .... (19) 
 
 
 TOO -t. 
 
 = 4ttA j e "^.q^dq (20) 
 
 Now it is known (see Williamson's Integral Calculus ^ 
 Art. 116, or Price's Injin. Cat., vol. ii., chap, iv) that 
 
 L 
 
 00 1 
 
 —hx" ■■ •■ / \ 
 
 e-"'^- dx = \ (^^ (21) 
 
 '0 ^ l^ 
 
 Hence differentiating both sides with respect to k 
 
 f 
 
 e-^^\x^dx = ^(^y. . . . (22) 
 and again differentiating this with respect to k, 
 
 e-^-^a;Va;=|(J)^ . . . (23) 
 
 £ 
 
 f 
 
 Also it is obvious that / e-^'^'.xdx^ — =, and bv dif- 
 
 Ja lk ^ 
 
 ferentiating this with respect to k, 
 
 r" I 
 
 / e-*'^^ x^dx = —p. • . . . . (24) 
 Jo 2F ^ '
 
 Gases. 225 
 From these results we have 
 
 n = At!^ c^, (25) 
 
 ^=|c2, (26} 
 
 v= ~^c (27) 
 
 From the last two we have 
 
 ^ = ^W (28) 
 
 which shows the relation between the mean square of the 
 velocities and the square of the mean velocity, the former 
 being" the g-reater. In terms of the latter, the intensity of 
 pressure is given by the equation 
 
 P = ^P(^')^ ori? = ^ ^ (v)2, .... (29) 
 
 according as p is taken in absolute or in gravitation 
 measure. 
 
 Example. 
 
 It is found that the mass of i ctibic foot of hydrogen at o'^C 
 when its intensity of pressure is 2116-4 pounds' Aveight per 
 square foot is -005592 pounds, the value of g being 32-2 feet 
 per second per second ; find the mean velocity of the hydrogen 
 molecules. 
 
 From the second value of p in (29) we have 
 
 IT -005592 
 
 .•. V = 5570-8 feet per second, nearly. 
 
 The vahie of \/v-, which is that velocity whose square is 
 equal to the mean square uf the velocities, and wliich is called 
 the relocity of memi square, is found from (15) to be 6046-5 feet 
 per second, nearly.
 
 226 Hydrostatics and Elementary Hydrokinetics. 
 
 The kinetic energy of the molecules in a unit volume is 
 obtained by multiplying- (9) by — ^ • If E denotes this 
 kinetic energy (in gravitation measure) we thus find 
 
 Thus we have 
 
 (30) 
 
 P = %E, (31) 
 
 expressing the intensity of pressure in terms of the kinetic 
 energy per unit volume. 
 
 53. Mixture of Gases. When two or more different gases 
 are present in the same space^ eack of them prochices exactly 
 the same intensity of j)ressure as if none of the others were 
 presetd. This fact is a result of the kinetic theory — ' when 
 several different sets of spheres are present together in the 
 region under consideration, the distribution of the centres 
 and of the velocities of the spheres of each set is independent 
 of the coexistence of the remaining sets.' (Watson's 
 Kinetic Theory of Gases, Prop. VI.) The a priori possibility 
 of such a result is manifest if we remember that the void 
 spaces in a vessel which contains even several gases ai'e 
 very much greater than the occupied spaces. 
 
 The result may be proved thus : suppose two masses of 
 gas whose volumes, intensities of pressure, and absolute 
 temperatures are represented in the following figures,
 
 Gases. 227 
 
 to be mixed and contained within a q-iven volume F; and 
 when the mixture becomes homog-eneous, let its intensity 
 of pressure become i"* and its absolute temperature T. Then 
 if we take the first g-as and alter its intensity of pressure 
 to jOg and its absolute temperature to T^, its volume will 
 become „ f 
 
 and in this state if it is added to the second gas, the volume 
 of the mixture will be 
 
 at {p.,, 1\). Denote this volume by U. Then if (£/, P2, ^2) 
 is altered to (T, P, T), 
 
 VP Up, 
 
 T ~ T. 
 
 o 
 
 " T ~ T^ '^ T^ ^^ 
 
 Now if the first gas filled the volume V alone, at the 
 absolute temperature T, its intensity of pressure would be 
 
 and if the second occupied F alone, its intensity of pressure 
 would be V T 
 
 and the sum of these is 
 
 which is precisely the value of P given by (i). 
 
 If several gases of given volumes, iutensities of pressure, 
 absolute temperatures, and specific gravities, are mixed together
 
 228 Hydrostatics and Elementary Hydrokinetics. 
 
 in a vessel of given volume at a given absolute temperatnre, 
 jxnd the intensity of iiressure and specific gravity of the 
 mixtitre. 
 
 Let the specific gravities of the gases be Sy, s^, s^, ... , 
 the specific gravity of the mixture S, its absokite tempera- 
 ture T, and its v^olume V; then by the laws of Boyle 
 and Dalton, we have 
 
 VV _VyPy v^jh, v^p^ . /^x 
 
 ■f" n^ T" 7; T ••• 5 • • \~'} 
 
 T ~ Ty ^ 1\ ' 2\ 
 
 3 
 
 and since the weight of the mixture is equal to the sum of 
 the weights of the constituents, 
 
 FTS _ v^ py s^ v^p, s^ v^sS^ ,. 
 
 m m "I m "f" rp + 1,3/ 
 
 54. Vapours. Many liquids, such as water, mercury, 
 ether, the various alcohols, and acetones, gradually pass 
 into a gaseous condition — i. e., continuously give off vapours 
 — at ordinary temperatures. Some of these liquids are 
 much more volatile than others. For example, a small 
 quantity of alcohol if left in an unclosed vessel will dis- 
 appear in a short time, whereas the same mass of water 
 would, under like circumstances, take a very long time 
 to pass away as a gas. 
 
 These vapours are essentially the same as the bodies 
 which we have defined as gases. In fact, all gases can be 
 regarded as the vapours of liquids, although the liquids 
 from which some of them come can be obtained or produced 
 only with extreme difliculty. Thus, it is now known that 
 even hydrogen and oxygen are the vapours of two liquids ; 
 and it was known a long time ago that nearly all the gases 
 with which we are familiar could be converted into liquids, 
 and even solids. 
 
 Every vapour — whether it be the gas which we call
 
 Gases. 229 
 
 hj'drogen or the vapour of alcoliol or of any ordinary" liquid 
 — obeys the typical law of a g-as (tliat of Boyle and 
 Mariotte), provided that the vapour, as regards temperature 
 or intensity of pressure, is not near its liquid state ; and, of 
 course, none of these bodies obey this law when they are 
 near this state. 
 
 The first fundamental characteristic of a vaj)our which 
 we shall sig-nalise is this : at a given temperature, there is 
 a limit to the intensity of pressure which a given vajwiir can 
 exert, or which can ly any means he exerted tijion that vapour. 
 
 For example, if we take the vapour of water at the 
 temperature of ioo°C, or 2I2°F, we cannot produce on it a 
 greater intensity of pressure than about that of 15 pounds' 
 wcig-ht per square inch. If we attempt to exceed this 
 limit, some of the vapour will at once become water. 
 Again, if we take the vapour of water at the tem^ierature of 
 107° F, it cannot sustain a greater intensity of pressure than 
 that of about 1 pound weight per square inch. 
 
 The result is different for different liquids. Thus, if we 
 take the vapour of mercury at the temperature of 100° C, 
 the greatest intensity of pressure that can be exerted 
 u})nn it is that of about -^g of a pound weight per square 
 inch. 
 
 On the other hand, the vapour of bisulphide of carbon 
 at the temperature of 100° C can sustain an intensity 
 of pressure of about 65-7 pounds' weight per square inch ; 
 and the vapours of alcohol and ether, at this temperature, 
 can exert intensities of about 33^ and 97 1 pounds' weight 
 per square inch, respectively. 
 
 To verify experimentally the above characteristic of 
 a vapour, take a long glass tube (baronietric tube), ABC^ 
 Fig. 60^. dipping into a vessel, LE, of mercury ; let A be 
 the point to which the mercury in the tube reaches, the 
 Torricellian vacuum being the portion of the tube above A.
 
 230 Hydrostatics and Elementary Hydrokinetics. 
 
 D 
 
 
 Then AC is the height of the mercurial barometer and 
 marks the intensity of atmospheric pressure. 
 
 Suppose now that a drop of ether is 
 inserted into the tube through the open 
 end below the mercury in the vessel. The 
 ether will rise up through the mercury in 
 the tube and reach the Torricellian vacuum, 
 where it will be at once converted into 
 vapour whose pressure will depress the 
 surface of the column of mercurv in the 
 tube. Let another drop of ether be simi- 
 larly inserted ; then a further depression 
 of the mercury column takes place. Add, 
 again, another drop of ether, and so on, 
 until the mercury column refuses to be 
 further depressed. It is an experimental 
 fact that such a limit to depression is 
 reached ; and if we continue to insert 
 drops of ether into the tube, they are not 
 converted into vapour on reaching the top, 
 but remain in the liquid state on the top, 
 B, of the column of mercury. It is supposed that the 
 point B marks the depression of the mercury when the 
 mercury refuses to be further depressed, and when the 
 ether ceases to be further converted into vapour. 
 
 Now it is found experimentally that the position of this 
 limiting point B depends simply on the temperature of the 
 surrounding air, i. e., on the temperature of the ether 
 vapour at the top of the tube. For example, if the tem- 
 perature of the ether is 
 
 — 20° C, the depression AB = 68 millimetres, 
 
 Fig. 63. 
 
 60° „ 
 
 „ = 183 
 „ = 1728
 
 Gases. 231 
 
 Supposing- now that for any given temperature, f, the 
 point Ji is the lowest to which the mercury is depressed, 
 the tube AJiC being held fixed, let us consider what 
 happens — 
 
 I**, if the tube is forced down farther into the vessel DE, 
 2°. if the tube is raised. 
 
 In the first case we find that the effect is to liquefy some 
 of the ether in JJ^ and to keep the height, BC, of J^ above 
 the level of the mercury BE constant. 
 
 In the second case, if there was any of the liquid ether 
 resting on the mercury at B, some of this would be 
 vapourised, and we should still observe that the height BC 
 is constant. 
 
 Now if the heights AC and BC are estimated in milli- 
 metres, the intensity of pressure of the vapour is measured by 
 
 AC — BC millimetres of mercury. 
 
 Thus, if the temperature of the vapour is 60° C, the 
 distance AB being 1728 millimetres, the point B will be 
 968 millimetres below C, assuming the height, AC, of 
 a barometric column to be 760 mm. 
 
 A vapour which, at a given temperature, t, cannot 
 submit to increase of pressure (becoming in part liquid if an 
 attempt to increase the pressure is made) is called a, saturated 
 ■vapour at the temperature t. Its intensity of pressure can, 
 of course, be diminished, and then it ceases to be a saturated 
 vapour. Thus we may define the saturation intensity of 
 pressure of any vapour at the temperature t as the greatest 
 intensity of pressure that the vapour at this temperature can 
 bear toithout liquefying. 
 
 As has been said, the vapours of different liquids, all 
 at the same temperature, have very different intensities of 
 pressure con*esponding to their saturation states. There is
 
 232 Hydrostatics and Elementary Hydrokinetics. 
 
 no known mathematical formula which g-ives the value of 
 the intensity of pressure, p, corresponding- to the state 
 of saturation of a given vapour at a g-iven temperature, t. 
 
 In the case of aqueous vapour, i. e., the vapour of water, 
 it is most important to have a knowledge of the maximum 
 intensities of pressure corresponding to a long range of 
 temperatures. M. Regnault by a simple method of experi- 
 ment w^as enabled to observe these saturation pressures for 
 temperatures ranging from below zero to above ioo°C; 
 and from the table which he compiled an empirical formula 
 connecting p and t may be constructed. One such formula 
 will be given presently. 
 
 It will be useful to regard the matter from a different 
 point of view, and, as it were_, to deduce the liquid from 
 the vapour instead of the vapour from the liquid. 
 
 Thus, suppose that we propose this problem : given 
 ether vapour at — 20°C, at o°C, at 60° C, and at 100° C, 
 successively, how shall W' e convert it into liquid ether ? 
 
 The answer would be : produce on it an intensity of 
 pressure of about i^, 3 J, 34, and 97 1 pounds' weight per 
 square inch, respectively. 
 
 It looks as if in the case of an}- gas whatever at auij 
 assigned temperature the answer would be the same kind — 
 i. e., simply produce a certain intensity of pressure ; but w^e 
 shall subsequently see that in the case, for example, of 
 hydrogen at any ordinary temperature this process would 
 not suffice. 
 
 55. Mixture of Gas and Vapour. Tf a given space is 
 saturated wii/i the vapour of any tiquid, at a given temperature, 
 the intensity of piressure of the vapour is the same whether this 
 space is a vacuum or contains any gas. 
 
 This follows from the result which has been already 
 given for the mixture of any gases in a given space, 
 viz.j that each gas behaves as a vacuum to all the others ;
 
 Gases. 
 
 233 
 
 K 
 
 and it may be experimentally verified by the following 
 method. 
 
 A glass tube IID, Fig. 64, is attached to another, LB, 
 both being vertical, the system being jirovided with a 
 stopcock s. The tube LB is fitted 
 with another stopcock b, and ter- 
 minates at A where the funnel, F, 
 or a glass globe^ G, fitted with a 
 stopcock, c, can be screwed on. At 
 first the two tubes are filled with 
 mercmy, the tube LB being filled 
 up to h and the stopcock s closed. 
 The globe G contains dry air, or 
 any other gas. 
 
 Let the globe be screwed on at 
 A and the stopcocks c, h, s opened, 
 and let a little mercury flow into 
 a vessel V so that dry air fills the 
 top of the tube LB. Close s, and 
 pour merciuy into the tube LIB 
 until the level of the mercury is 
 the same in both tubes. The tube 
 LIB being ojien to the atmosphere, 
 when the level is the same, the 
 air at the top of LB is at the atmospheric pressure. 
 Let the common level of the mercury at this stage be BB. 
 Now remove the globe from A, and screw on the funnel, F. 
 This funnel is fitted with a stopcock, a, \\hich is not 
 perforated but has a small cavity at the side, as seen in the 
 figure. Let some of the liquid whose vapour we are con- 
 sidering be poured into i^ and let a be turned so that 
 liquid falls down to h ; and let h be turned so that the li({uid 
 falls into the air space h B and becomes vapourised, a\ hen 
 its pressure, added to that of the air, depresses the level of 
 
 Fig. 64.
 
 234 Hydrostatics and Elementary Hydrokinetics. 
 
 the mercury below B. Repeat this operation until the 
 level of the mercury ceases to sink ; and let the final 
 level be L ; then the space hL vs, saturated with the 
 vapour. 
 
 Now pour mercury into the tube DH until its level 
 in the other tube is restored from L to B. This will 
 liquefy some of the vapour, but will leave its intensity of 
 pressure unaltered. Suppose that the level of the mercury 
 in BH is now at H. Then if li is the height of the baro- 
 meter during the experiment, the intensity of pressure of the 
 mixed gases in ^^ is represented by /^ + HB ; but as the 
 air in IB has been restored to its original volume, its 
 intensity of pressure is the same as it was at first, i.e., it is 
 represented by a column of mercury of height h ; hence 
 the intensity of pressure of the vapour is represented 
 by HB. 
 
 But if we now take a barometer tube, such as that repre- 
 sented in Fig. 6'^^^ and insert a few drops of the liquid 
 in question into the Torricellian vacuum until the space 
 becomes saturated, we shall find that the depression of the 
 mercury column is equal to BB^ which shows that the 
 intensity of pressure of the saturated vapour of the given 
 liquid at the given temperature is uninfluenced by the 
 presence of air with the vapom-. Hence in a given volume 
 which is saturated with any vapour there is the same mass 
 of vapour whether the given space is a vacuum or contains 
 any gas or gases. 
 
 56. Moist Air. It follows from Art. ^o, that when the 
 atmosi)liore contains aqueous vapour the intensity of pres- 
 sure which is observed by a barometer is the sum of the 
 intensities due to the air itself and to the vapour which 
 it contains. 
 
 If, then, h (inches or millimetres) is the height of the 
 barometer, and if the intensity of pressure of the vapour
 
 Gases. 235 
 
 present in the air is measured by / (inches or millimetres) 
 of the liquid which the barometer contains, the intensity 
 of pressure of the air itself is measured by 
 
 The weig-ht of a given volume of moist air is affected by 
 this consideration. 
 
 Thus, assuming" the specific gravity of aqueous vapour to 
 be -622, or f nearly, the weight of the air in a volume 
 
 of V litres, with the notation of Art. 47, is '4646 m * 
 
 vf 
 and that of the vapour in this volume is '4645 :^ x | ; so 
 
 that the weight of the whole is 
 
 .4645^^^' (.) 
 
 Similarly, formula (2) of Art. 49 will be replaced by 
 
 r= 1.326946^^%=^^ (2) 
 
 ^ ^ 460 + ?! ^ ' 
 
 The accurate measurement of the intensity of pressure of 
 the aqueous vapour present in the air at any temperature 
 is a matter of difficulty ; but we shall presently see how 
 it can be approximately effected by means of a hygro- 
 meter. 
 
 57. Vapour of Water. A knowledge of the maximum 
 intensity of pressure of the vapour of water for any given 
 temperature is important ; and, as said before, this know- 
 ledge must be derived from experiment. Water may 
 be made to boil at any temperature whatever by producing 
 a suitable intensity of pressure on its surface. 
 
 If any given intensity of pressure is by any means 
 ]>roduced on the surface of a liquid, and heat is continuously 
 applied to the liquid, the liquid will loil when the intensity
 
 236 Hydrostatics mid Elementary Hydrokinctics. 
 
 of pressure of its vapour becomes equal to the intensity of 
 pre-^snre on the surface of the liquid. 
 
 The given intensity of pressure on the surface is then 
 the maximum intensity of pressure of the vapour at the 
 temperature of the liquid. 
 
 This principle is the basis of the experiments of Reg-nault 
 for the determination of the maximum intensities of pres- 
 sure of water vapour at various temperatures. 
 
 His method was to make water boil under a continuous 
 series of surface pressures and to read the corresponding" 
 temperatures of boiling. 
 
 The figure of the apparatus used will be found in 
 treatises on Experimeotal Ph3^sics (see, for example, Ganot's 
 Fhi/sics, Art. 3.51). 
 
 Experiments with the same object had a short time 
 previously been carried out by Dulong and Arago, at the 
 request of the French Academy, and the results were 
 exjn-essed by an empirical formula kno^oi as the French 
 Commissioners" Fornmla. If t is the centigrade temperature 
 at which water is made to boil by adjusting the intensity 
 of pressure on its sui-faee, this intensity is exjiressed by 
 the equation t-100.^ 
 
 n 
 
 in which n denotes the number of atmospheres, of 760 mm., 
 under which the water boils ; that is, the intensity of 
 pressure on the surface (and therefore of the va])0ur) is 
 represented by a column of mercury 760 ?^ millimetres 
 high. 
 
 When compared with the tabulated results of M. 
 Regnault, this formula is fairly accurate for pressures 
 ranging from i atmosphere to 24 atmospheres ; but for 
 very low pressures the error in the formula is great. 
 
 The following table gives a few temperatures below
 
 Gases. 
 
 237 
 
 100° C at winch water is caused to Loil, tog-ether M-ith tlie 
 corresi)oncling' intensities of pressure on its surface (or of its 
 vapour), measured in mm. of mercury, as calculated from 
 the formula and as observed by Reg-nault : 
 
 / 
 
 imn. 
 calculated 
 
 mm. 
 observed 
 
 0" 
 
 1-42 
 
 4-60 
 
 20° 
 
 IO-88 
 
 17-39 
 
 60° 
 
 140-90 
 
 148-79 
 
 70° 
 
 227-14 
 
 233-09 
 
 80° 
 
 351-21 
 
 354-64 
 
 90° 
 
 524-39 
 
 525-45 
 
 Thus the disag-reement at low temperatures is very 
 marked. 
 
 From (i) we can deduce a formula in Eng-lish measures. 
 Let intensity of pressure be measured in pounds' weight 
 per square inch, and temperature on the Fahrenheit scale. 
 Putting- -|(?! — 32) for f in (i), the number of standard 
 atmospheres becomes 
 
 (1 + ^-Jl^^\' or 1- 39-^43 +^ '. 
 V 251-643^ ' V 251-643 ^ ' 
 
 and taking- a column of mercury 760 mm. high as equi- 
 valent to 14-697 pounds' weight per square inch, the 
 intensity of pressure of the vapour in pounds' weight per 
 square inch is the product of this expression and 14-697 ; 
 
 that is (— ) ; or, as it is usuallv taken 
 
 ^ 147-oci / ' ' -^ 
 
 i^ 
 
 ^ 147 ^ 
 
 (2)
 
 238 Hydrostatics and Elementary Hydrokinetics. 
 
 If intensity of pressure is measured in pounds' weig-ht 
 per square inch, while temperature is measured on the 
 Centigrade scale, the formula gives 
 
 ^ = (1^'/ (3) 
 
 The height of a mountain may be deduced from the 
 temperatures at which water boils at the base and at 
 the summit, on the usual assumption of a constant mean 
 temperature of the air, without the aid of a barometer ; for, 
 from the observed temperatures of boiling-, we can deduce 
 the corresponding atmospheric pressures by formula (i) or 
 (2), and make use of these pressures in the barometric 
 formulas of Arts. 50 and 51. 
 
 Examples. 
 
 1. If at the base and the summit of a mountain water boils at 
 212° and 190^, Fahrenheit, respectively, and the mean tempera- 
 ture of the air is 40°, find the height of the mountain. 
 
 Ans. About 12 172 feet. 
 
 2. At the base of a mountain 28000 feet high the atmo- 
 spheric intensity of pressure is 14 pounds' weight per square 
 inch ; assuming the temperature of the air to be uniformly 
 32*^ F, find the temperatui'e at which water boils at the summit. 
 
 Ans. About i6i°F. 
 
 58. Principles of Thermodynamics. It is now an 
 accepted principle that a quantity of heat is the same 
 thing as a quantity of kinetic energy — that, in fact, heat 
 is the kinetic energy of molecular motions in a body. Now 
 as kinetic energy is the equiv^alent of work, and can be 
 measured in ergs, foot-pounds' weight, metre-kilogrammes' 
 weight, and many other analogous forms, it follows that a 
 quantity of heat can be expressed as so many ergs, or so 
 many foot-pounds' weight, &c. 
 
 For a long time this was not the mode adopted for
 
 Gases. 239 
 
 expressing a quantity of heat, because until the experiments 
 of Joule were made, it was not recognised that heat and 
 ivork are equivalent things. To take an example, when by 
 burning coal under a vessel containing i pound of water 
 the temjjerature of this water was raised 1° F (supposing 
 none of the imjjarted heat to be radiated from the water) 
 it was said that a quantity of heat called one thermal unit 
 was imparted to the water. This mode of speaking is still 
 adopted : it merely amounts to a definition, and indicates 
 one particular way of measuring quantities of heat. But 
 at the present time we should also describe the result 
 thus — a quantity of kinetic energy equivalent to about 772 
 foot-pounds weight has been imparted to the pound of water. 
 The measurement of quantities of heat in thermal units 
 and in foot-pounds' weight may be compared with the 
 measurement of areas in acres and in square yards. 
 
 The British thermal unit is defined as the quantity 
 of heat {or molecular kinetic energy^ which must be itnparted 
 to one pound of water at its temperature of maximum density 
 [about 39° F) to raise this temperature i°F. 
 
 The Metric thermal unit is defined as the quantity of 
 heat (or molecular kinetic energy) tvhich mzist be impiarted to 
 one kilogramme of loater at its temperature of maximum density 
 [about 4°C) to raise its temperature 1° C. 
 
 It is not quite true that the quantity of heat necessary 
 to raise by one degree the temperature of a given quantity 
 of water is the same at all temperatures. Thus, it is found 
 that the heat necessary to raise the temperature of one 
 pound of water from 32° F to 212'', instead of being 180 
 times that required to raise it from 32° to ^'^°, is 180-9 
 times as great. The quantity of heat necessary to raise 
 the temperature by one degree increases very slightly 
 as the temperature of the water increases ; and this fact 
 may be otherwise expressed thus : the specific heat of
 
 240 Hydrostatics and Elementary Hydrokinctics. 
 
 water increases slig-htly as the temperature of the water 
 increases. 
 
 The Metric thermal unit is very frequently called a 
 calorie^ and the quantity of kinetic energ-y in it is about 
 425 metre-kilogrammes' weig-ht (usually called kilogramme- 
 metrei). 
 
 In the C. G. S. system, the thermal unit is tlie quant.ity 
 of heat (or molecular kinetic energ-y) ^c^/^'c/^ miist he imparted 
 to one gratmne of water at its temperature of maximum density 
 to raise its temperature 1° C, and it amounts to about 
 42 X 10*' erg-s. 
 
 In any of these systems the number of work-units 
 which is equivalent to the thermal units is called the 
 dynamical equivalent of heat, and it is usually denoted by / 
 (with reference to the name of Joule). Thus, in the British 
 system / = 772, in the Metric gravitation system /= 425, 
 and in the C. G. S. system / = 42 x lo*", all these numbers 
 being only approximate. 
 
 Let us now consider what happens when a quantity 
 of heat is communicated to any body. The result can be 
 comprised under two heads — 
 
 i". Kinetic energy is imparted to the molecules of the 
 body (i. e., the body is heated) and at the same time work 
 is done against the attractive forces of these molecules 
 (i. e., a certain amount of internal static energy is generated 
 in the body) ; 
 
 2**. The body at its bounding surface does work against 
 external pressures. 
 
 Thus the first heading comprises both heat in the body 
 and internal loork. The sum of these two is called the 
 energy of the body, and is denoted by the s3'mbol U. 
 
 Suppose now that a small quantity of heat, dQ, which we 
 shall estimate in dynamical units, is communicated to a 
 body, and that the effect of this is to generate a small
 
 Gases. 
 
 241 
 
 quantity, (III, of cnci'o;'}' in the l)ody, and to make the body 
 ])erform a small quantity, dlF, of external work. Then we 
 iiave the equation 
 
 dQ = dU+dTr. (i) 
 
 As it is not our object to consider the principles of 
 thermodynamics in their application to every kind of body, 
 we shall suppose the body in question to be a g-as or 
 a vapour. 
 
 Then there are three thing-s which determine the state 
 of such a body — viz., its temjperatnre, T, its vohime, v, and 
 its intensify of pressure, p. These three are not inde- 
 pendent ; for, in the case of a given mass of gas, we have 
 
 vp 
 the equation -^ = a constant, or as in Art. 48, 
 
 vp^wR.T; (2) 
 
 so that the state of the body at any instant may be con- 
 sidered as depending on any Uoo of the quantities, v, p, T. 
 Now the internal energy, U, of the body is alwaj^s the 
 same in the same state of the body. Hence we can 
 consider either 
 
 U=f, {v,p) or U =f, (v, T), or U=f, (;;, T), . (3) 
 
 where. /i, 7^0, y;5 are symbols of functionality. 
 
 Now, confining our attention to gases, the following is a 
 fundamental result : 
 
 tke energy, U, of a given mass of gas is a funcfion of its 
 temperature, T, alote, i. e., it does not depend on either 
 V or p. 
 
 This result was verified by the following experiment of 
 Joule. 
 
 A and B (Fig. 6^) are two copper vessels which com- 
 municate with each other by a tube in which is a stopcock. 
 c. These vessels are placed inside a vessel, CJ), which is in 
 
 11
 
 242 Hydrostatics and Elementary Hydrokinetics. 
 
 Fig. 6= 
 
 the form of two nearly closed cylinders communicating by 
 means of a narrow passage between two walls. The 
 
 reserv'oir A is filled with air, or any 
 gas, at a high intensity of pressure 
 (22 atmospheres in Joule's experi- 
 ment), while ^ has been exhausted 
 of air. \Vater is poured into the 
 vessel CD, and since A and B have 
 nearly the capacity of the cylinders 
 in which they are placed, the 
 quantity of this water is small, 
 and thus any alteration of its tem- 
 perature would be the more easily observed. 
 
 Now let the stopcock c be suddenly opened. The gas 
 rushes out of A, increasing its volume and diminishing its 
 intensity of pressure ; and when the whole mass of this gas 
 is considered, we see that it has altered its state without 
 doing any external work. At the same time no heat has 
 been communicated to it. 
 
 If the water is kept stirred the result of the experiment 
 will be that the water neither rises nor falls in temperature. 
 Actually, it was found that if the water was at rest, the 
 temperature of that surrounding A was lowered and the 
 temperature of that surrounding B was raised ; but, on the 
 whole, these local alterations of temperature balanced each 
 other, and the body (i. e., the mass of gas) underwent 
 change without development of heat. 
 
 Now, applying to this case equation (1), we see that 
 dQ = o and dJr= o, therefore dll ■= o \ and since both 
 V and p have altered, but not T, we see that U cannot be a 
 function of v or 7;. Hence for a gas 
 
 u=An ...... (4) 
 
 i. e. the energy of the gas is a function of its temperature 
 alone.
 
 Gases. 243 
 
 This amounts to the statement that the molecular forces 
 in a gam are zero ; for, if they existed, and were attractive^ 
 they would jjcrform negative work \\hen the volume ex- 
 ])anded ; and if they were rej^iihice, they would perform 
 positive work in the same case. 
 
 Subsequent experiments by Joule and Thomson revealed, 
 however, small changes of temperature, showing- that the 
 internal forces are not wholly evanescent ; but these chans^es 
 became smaller, for any gas, as its temperature was hig-her; 
 and the changes of temperature which, under the above 
 circumstances, take place are smaller the more nearly the 
 gas approaches to the condition of a perfect gas. Thus, 
 for air and for hydrogen they are much smaller than 
 for carbonic acid gas. Such a result would naturally be 
 expected ; and hence Ave are to regard the result (4) as 
 holding accurately in the case of a ])erfect gas only ; for 
 other gases the result is approximately true. 
 
 Supposing, then, that we are dealing with a perfect gas 
 only, equation (i) can be written 
 
 (iq = ^,dT^dW. (5) 
 
 When the gas does external work by overcoming resist- 
 ance applied over its bounding surface, the intensity of this 
 resistance being equal to the intensity of pressure, p, the 
 element, dll\ of work done during a small increment, dv, 
 of volume of the gas, is given by the equation (Art. 46) 
 
 d}F=pdv (6) 
 
 Hence (5) becomes 
 
 dQ = '^,dT+jxlv (7) 
 
 If w is the mass of the quantity of gas with which 
 we are dealing, we have equation (2), v/hich enables 
 
 R 2
 
 244 Hydrostatics and Elementary Hydrokinetics. 
 us to express dv in (7) in terms of r/Z^and dp, thus 
 
 = '"JldT-'-dv, (8) 
 
 p p 
 
 hence (7) becomes 
 
 dq = (~-Vtvn)dT-vdp (9) 
 
 It will be observed that we have been all through 
 measuring- heat in work-units. 
 
 We now lav down two definitions : 
 
 [a) The specific heat of any gas at constant volume is 
 the limiting value of the ratio of an infinitesimal quantity 
 of heat to the infinitesimal rise of temperature produced, 
 when this heat is imparted to a unit of mass which is not 
 allowed to expand. 
 
 [h) The specific heat of any gas at constant pressure is 
 the limiting value of the ratio of an infinitesimal quantity 
 of heat to the infinitesimal rise of tem2)erature produced, 
 when this heat is imparted to a unit of mass which is 
 allowed to expand under constant pressure. 
 
 Thus (a) is the value of — j^j-. when v is constant, and if 
 
 ^ '' w dl 
 
 this is denoted by c, (7) shows that 
 
 1 dU 
 '=wdT ^^°) 
 
 Also (b) is the value of — jjp, when p is constant, and if 
 
 ^ -' w dl 
 
 this is denoted by C, (9) shows that 
 
 C=--j^,-\-R (11) 
 
 w dl ^ '
 
 Gases. 245 
 
 It is evident u priori that C > c, because when the gas is 
 allowed to expand, it uses some of the imparted heat to do 
 work aii^ainst external resistance, so that all of the heat 
 does not go towai'ds raising- the temperature, whereas, 
 wlun no expansion is allowed, all this heat is used in 
 raisinii^ the temperature. 
 
 Now one of the fundamental laws of a perfect gas is that 
 for any given gas each of these specijic heafs is a constant at 
 all temperatures. Of course, since 
 
 C^c + T?, (12) 
 
 if either of them is constant, the other must be constant. 
 This law was proved experimentally by M. Regnault. 
 
 Since the specific heats are constant, we may say (as is 
 usually done) that the specific heat of a g-as at constant 
 volume is the quantity of heat necessary to raise its tem- 
 l)erature one deg-ree when the heat is imparted to the unit 
 of mass at constant volume ; or the quantity of heat 
 necessary to raise its temperature by any number of degrees, 
 if we divide by the number of degrees. 
 
 We have supposed that Q is measured in work-units : if 
 Q is measured in thermal units (whether English or 
 -Metric), and / is the corresponding work-unit value of 
 the thermal unit, the fundamental equation (i) becomes 
 
 Jciq^dU+dW- (13) 
 
 also, c being measured with reference to thermal units. 
 
 f = — ;„, , when V is constant ; so that 
 10 (IT 
 
 ■■■ J{C-c) = R (15) 
 
 Thus, Regnault found that for dry air C = -2375, and if
 
 246 Hydrostatics and Elementary Hydrokinetics. 
 
 we use Eng-lish measures, U = S^-^ (Art. 49), •/= 772 ; 
 
 Q 
 
 hence c = -1685, which gives — = 1-409. 
 
 From (10) we have the vahie of the internal energy of 
 a given mass, w, of a perfect gas at the absolute tempera- 
 ture T, viz., , , 
 
 U = civT (16) 
 
 in units of work if c is expressed with reference to these 
 units; or jj^ j^ioT (17) 
 
 if c is expressed with reference to thermal units. 
 
 It appears, therefore, that when a gas is expanded 
 isothermally by the continuous application of heat, all the 
 heat supplied is utilised in doing work against external 
 resistance. 
 
 59. Adiabatic Expansion. Garnet's Cycle. If a mass 
 of gas is contained within a cylinder closed by a moveable 
 piston, the cylinder and the piston being both impermeable 
 to heat, so that no heat can be communicated to the 
 gas from without and no heat can escape from the gas 
 itself, while the gas may drive out the piston before it 
 or may be compressed by means of the piston, the transfor- 
 mation is called adiabatic. If the gas is compressed by 
 means of the piston, heat is generated in the gas ; but 
 no heat is gained or lost by conduclion or radiation in an 
 adiabatic transformation. 
 
 We propose to find the relation existing between the 
 volume and the temperature, and the volume and intensity 
 of pressure, of the gas during this transformation. 
 
 These are found by using (7) and (9) of last Article and 
 putting r/Q = o. 
 
 Hence cwdT+pdv = o, (i) 
 
 CwdT—vdj) = o, (2) 
 
 in which the specific heats c, C are in work-units.
 
 Gases. 247 
 
 From these we derive the equation 
 
 Cpdv + cvdp = o, ..... (3) 
 which gives by integration 
 
 jiv" = constant (4) 
 
 It has been found that — = 1.408, nearly, for air. De- 
 
 noting^ by fi the value of this ratio for any g-as, if (/?(,, "^'o) 
 are the intensity of pressure and volume when the adiabatic 
 transformation beg-ins, the equation of the curve which is 
 analog-ous to the hyi)erbola in Fig-. ^^, and which exhibits 
 the relation between p and v in the adiabatic transfor- 
 mation,is pv^'=PoV,^ (5) 
 
 which, combined with the fundamental equation 
 
 gives the relation between p and T and between v and T. 
 
 It has been already pointed out (Art. 40) that the 
 curves of isothermal transformation are rectang-ular hy- 
 perbolas. 
 
 Fig-. 66 exhibits two isothermals and two adiabatics. 
 Thus, suj^pose the two rectangular lines Ov and Op to 
 be taken as axes of volume and pressure, respectively ; let 
 the gas be contained in a cylinder the base alone of which 
 is a conductor of heat — and we shall assume this base to be 
 a peifect conductor, i. e., we can imagine it to be made 
 of very thin cop])er ; let the initial state of the gas be 
 represented by the point A in the figure, i. e., its intensity 
 of pressure, p^, is represented by the ordinate Aa, and its 
 volume by the abscissa Oa, while its ahsolute temperature 
 in this state is T^.
 
 248 Hydrostatics and Elementary Hydrokinetics. 
 
 Kow let the base of the cylinder be placed on a perfectly 
 nonconducting- stand, and let work be done on the gas 
 by forcing down the piston. The result will be a rise 
 of temperature of the gas and, of course, an increase in its 
 intensity of pressui-e, the relation between its volume and 
 
 Fig. 66. 
 
 intensity of pressure being represented by the abscissae 
 and ordinates of the adiabatic curve AB^ whose equation 
 
 is (5). 
 
 Let this adiabatic transformation be stopped when the 
 
 absolute temperature becomes T^ and the state of the 
 
 gas is represented by the point B. Then the work
 
 Gases. 24Q 
 
 (lone on the gas is represented by the area Aa IB, whose 
 vahie is /-,.i 
 
 - / i^^^^' (7) 
 
 i\ being the final vohime Ob. 
 
 Substituting for ;j its value in terms of v from (5), 
 we find this area to be 
 
 
 or 
 
 
 (8) 
 
 Now since for the given mass of gas, w, we have (2) of 
 Art. 58, this expression is 
 
 — m-n) (.) 
 
 P'rom (i) it appears that the work done on a gas when 
 compressed adiabaticallv is also 
 
 ^^(n-n), (10) 
 
 where 1\ and T^ are the initial and final absolute tem- 
 peratures, and, as before stated, c is in work-units. 
 
 The co-ordinates, v^ , p^ , of the point B are easily found 
 to be 1 
 
 ^i=^o(^p (n) 
 
 n 
 
 7* n-1 
 Jh=Po{7JT) ...... (12) 
 
 When the gas has reached the higher temperature, 2\, 
 let the base of the cylinder be removed from the noncon- 
 ducting stand and placed in contact with a large reservoir 
 of heat of the temperature 7\. 
 
 The external pressure on the piston being removed, the
 
 250 Hydrostatics and Elementary Hydrokinetics. 
 
 piston will be driven outwards by the pressure of the g-as, 
 and this entails a fall of the temperature of the g-as ; but 
 instantly heat Hows into it from the reservoir, keejjing' the 
 temperature constantly equal to 7\ as the gas continues to 
 expand. The curve of expansion is a rectangular hyperbola, 
 BC^ whose equation is 
 
 />v = j5j Vj or = -Kw2\, (13) 
 
 and if the transformation is stopped at the point C, the 
 work done hy the gas on the piston is represented by 
 
 the area BCcb, whose expression is / ;;^/r, where v.^ is the 
 
 J t'l 
 
 volume Oc. 
 
 This work is -n m ^ '^'9 / \ 
 
 R^^i^og-' (14) 
 
 '1 
 
 Having reached the point C in this isothermal transfor- 
 mation, let the cylinder be removed from the reservoir and 
 again placed with its base on the nonconducting stand. 
 The pressure of the gas will continue to drive out the 
 piston ; but now the gas does this work at the expense of 
 its own heat, and therefore its temperature falls, the 
 lelation between pressure and volume being given by the 
 adiabatic curve CD. When the absolute temperature has 
 fallen to the original value, T^^, let this transformation 
 cease (the point 1) is supposed to represent the state of the 
 gas when the temperature Tq is reached), and let the 
 cylinder be placed \Yii\i its base in contact with a large 
 reservoir of heat whose absolute temperature is Tq. Now let 
 the piston be forcibly driven in, and therefore the gas 
 compressed by work done on it, until the original volume, 
 t'o , or Oa, is again reached ; then the final curve DA is a 
 hyperbola corresponding to the temperature T^^, and since 
 there is work being continuously done on the gas during 
 this isothermal compression, its temperature makes a con-
 
 Gases. 
 
 251 
 
 tinnous effort to rise ; but, on account of the perfect 
 conductivity of the base, the moment such a rise takes 
 place, heat flows from the gas into the reservoir, and 
 by this continuous flow of heat the temperature of the g-as 
 is steadily kept at 2\. 
 
 In the adiabatic expansion CD, the g-as has done work 
 of the amount 
 
 Po Vo »q Vr, 
 
 if the co-ordinates of C and D are [v.^^p.^) and {v.,p^) 
 respectively. 
 
 But since C belongs to the hyperbola corresponding to 
 7',, and i> belongs to the hyperbola corresponding to 2\, 
 we have v.ji, = llwl\ and v^p^ = llwl\ ; hence the work 
 CBdc is jp 
 
 ^,(T,-U (>6) 
 
 which has been proved to be also the work represented 
 by BAab, so that the work done 07i the gas in the adiabatic 
 compression AB and bi/ it in the adiabatic expansion CI) 
 cancel each other. 
 
 The work done on the gas in the isothermal compression 
 I) A is represented by the area Del a A, and is equal to 
 
 /^^'olog-e'-'OrT^^^^Z'.log '-?. 
 
 Now we can show that 
 
 
 (17) 
 
 for p^v\ = BpjT^, 
 
 A- «— 1 7- n— 1 
 
 '2 — '3 
 
 t:
 
 252 Hydrostatics and Elementary Hydrokinetics. 
 
 r-2 = V, (^) 
 
 
 which, by (i i); g-ives the result (17)- 
 
 Hence the total work done 6^ the gas in the whole 
 series of transformations, which is represented by the area 
 ABCLA, is 
 
 Rw{T,-T,)\og.' (18) 
 
 A series of transformations of a gas whereby, starting- 
 from a given state as regards volume, temperature, and 
 intensity of pressure, submitting to changes of these by 
 absorbing heat, doing work against external resistance, and 
 having work done upon it by external agents, the gas 
 is finally brought back exactly to its original state, is called 
 a c^cle of operations. When, as in the case just discussed, 
 the chano-es consist of two isothermal and two adiabatic 
 transformations, the operation is called a Carnot cyclt^ 
 because such a simple cycle was first discussed by Carnot in 
 the investigation of some of the fundamental principles of 
 Thermodynamics. 
 
 Such a cycle it would be impossible to realise in practice, 
 because perfect conductors and perfect nonconductors of 
 heat do not exist, although isothermal and adiabatic trans- 
 formations of a gas can be approximated to. For examj^le, 
 there is a common experiment which consists in igniting a 
 piece of tinder, or cotton moistened with ether or bisulphide 
 of carbon, at the bottom of a glass tube filled with air 
 and tightly fitted with a piston, by very suddenly forcing 
 the piston down the tube. In this case the transformation 
 of the imprisoned air can be assumed to be adiabatic, 
 because, during the time of the experiment, no heat can 
 enter or leave the tube. This instrument is sometimes 
 called 2, pneamatic syringe.
 
 Gases. 253 
 
 The order in which the clians;'es in a Carnot cycle 
 are made can be varied. Thus, we may start tlie operations 
 from the point 7i and work round throu<5'h C, I), and A to 
 B ag-ain. If we call the reservoir at the hij^her absolute 
 temperature, 2\, the boiler^ and the reservoir at the lower 
 temperature, T^, the condenser, the two main features of the 
 cycle are the abstraction of a certain amount of heat, 
 represented by the area BCch, in work-units, from the 
 boiler, and the transference to the condenser of another 
 amount of heat, represented by the area DdaA, the amounts 
 of heat generated within the working- g-as itself in the 
 abiabatic transformations neutralising each other. 
 
 Carnot, under the influence of the erroneous notion 
 prevailing in his time, supposed that, since the working 
 substance returns to exactly its original state in all respects, 
 the quantity of heat which it receives from the boiler must 
 be equal to that which it gives out to the condenser, 
 because heat is an indestructible substance. But the fiict 
 remains that, although the gas has returned to its original 
 state, a certain quantity of work, represented by the area 
 BCBAB, has been done by the engine — if we use the term 
 engine to denote the gas, cylinder, and piston. How, then, 
 did Carnot explain the doing of this work, since (according 
 to his view) the engine gave out, as heat, to the condenser 
 all the heat that it received from the boiler ? 
 
 Simply by saying that the letting down of the heat from 
 the higher temperature, or heat level, T^ , to the lower 
 level, ?') , constituted the doing of this work — just as the 
 fall of a stone from a higher to a lower gravitation level 
 constitutes the doing of work. 
 
 The view that heat is a substance (which used to be 
 called caloric) is now abandoned, since the experiments 
 of Joule and others ja-ove that it is kinetic energy. 
 
 Of the quantity of heat which a Carnot engine receives
 
 254 Hydrostatics and Etcmcntary Hydrokinctics. 
 
 from the boiler, how much is converted into work done by 
 the engine ? If H.-^ is the heat received from the boiler, 
 and //q the heat which the engine transfers to the con- 
 denser, the amount //, — //(, is converted into work. 
 Now 
 
 . ^i~-^^0 - i_ ?o (19) 
 
 The ratio of the quantity of heat converted into work to 
 the quantity received from the boiler is called the efficiency 
 of the engine, and we see that, unless T^^ o, i. e.,the tem- 
 perature of the condenser is absolute zero, the efficiency is 
 always < i. Thus, if the boiler is at the temperature 
 of M'ater boiling at the ordinary pressure, and the condenser 
 at that of melting ice, T^= o^'jo^, T(^= 273, and the efficiency 
 is only 1 — fyf, that is about -268. 
 
 This is only a small amovmt, and yet such an imaginary 
 engine (using a perfect gas as the working substance) has 
 a much greater efficiency than any actual engine. 
 
 The cycle of operations with a Carnot engine is 7'eversible, 
 i. e., if we start from the point A in Fig. 66, we can work 
 from A to D, then to C, then to B, and finally back to A. 
 
 In this way we should have to do work on the gas, this 
 w^ork being represented by the area ABCBA, and the 
 result of this would be the transference of a certain amount 
 of \\Q2d,from the condenser to the holler. 
 
 In the common steam-engine the various operations in a 
 Carnot cycle with a perfect gas are roughly approximated 
 to in the following way : 
 
 I. The isothermal exjiansion BC corres^Jonds to the 
 evaporation of the water in the boiler.
 
 Gases. 255 
 
 2. The adiiiLatic expansion CB corresponds to the 
 expansion of the steam in the cylinder. 
 
 3. The isothermal compression DA corresponds to con- 
 densation in the condenser. 
 
 4. The adiabatic compression AB corresponds to the 
 forcing- of the water into the boiler. 
 
 (See Cotterill's T/ie Steam Engi7ie considered as a Ileal 
 Engine, Chap. V.) 
 
 Examples. 
 
 1. The air column in a pneumatic syringe 10 inches long 
 is suddenly comj)ressed into a length of i inch ; its original 
 temperatui-e was 1 5° C ; find its final temperature. 
 
 Ans. 463'^-8. 
 
 2. Air is contained in a vertical cylinder, closed at the lower 
 end and open at the top, the area of whose cross-section is 
 2 square inches ; the air is compressed, so tliat it occupies 
 a length of 4 inches of the cylinder, by means of a piston whose 
 mass is i jiound, the intensity of ])repsure of the air being 
 150 pounds' weight per s(|uai'e inch; the intensity of atmo- 
 spheric pressure is 15 jiounds' weight per square inch. If 
 the piston is suddenly released, find its velocity when it is 
 I foot from the bottom of the cylinder, assuming the tempera- 
 ture of the air to be kept constant. 
 
 Ans. 75-55 feet per second. 
 
 3. In the above find the position of the piston when its 
 velocity is a maximum. 
 
 Ans. 38-7 inches from the lower end. 
 
 4. Find how high the piston ascends before coming to rest 
 for an instant. 
 
 Ans. About 142^ inches from the end. 
 
 5. If the area of the cross-section of the cylinder is A square 
 inches, the mass of the piston w pounds, the original intensity of 
 pressure of the compressed air F pounds' weight per square
 
 256 Hydrostatics and Elcmcntmy Hydrokindics. 
 
 inch, that of the atmosjihere f^, and the piston is originally 
 c inches from the bottom, find its velocity when it is x inches 
 from the bottom, taking (7^32 feet per second per eecond. 
 
 16 ( a; ) 
 
 Ans. 7;-= — \APc\oq.„ {Apf,-\-w){x—c)\ •> 
 
 3w; ( *- c ' ^ \ 
 
 where v is feet per second. 
 
 6. If in example 2 the air in the cylinder expands without 
 receiving or losing heat by conduction or radiation, find the 
 velocity of the piston when it is i foot from the bottom. 
 
 Ans. 65-9 feet per second. 
 
 7. If in example 5 the air expands adiabatically, find the 
 velocity of the piston. 
 
 where n = 1-408. 
 
 8. If a pound of air does 390-6 foot-pounds' weight of work 
 without receiving heat or lo&iug it by conduction or radiation, 
 find its fall in temperature. 
 
 Ans. 3°F. 
 
 9. If a pound of air at 60° F and intensity of pressure 15 
 pounds' weight per square inch is compressed without gain 
 or loss of heat by conduction or radiation until its temperature 
 is 200^^ F, find its new pressure intensity. 
 
 Ans. About 34-1 pounds' weight per square inch. 
 
 1 0. The temperature of a given mass of air is observed to fall 
 from 540" to 290° F when expanding to double its volume 
 without gain or loss of heat by conduction or radiation ; and at 
 the same time the external work done is observed to be 32600 
 foot-pounds' weight per pound of air ; find the specific heat of 
 air at constant pressure. 
 
 Ans. We have from the data c= 130-4 work-units [see 
 
 (10), Art. 59], and 2^~"= 3, where n = - \ .-. = 184-5 
 
 c 
 
 woi'k-units.
 
 Gases. 
 
 257 
 
 11. Air contained in a cylinder at the atmospliei'ic ]n-essuro 
 is adia])atically compressed to an intensity of pressure of m 
 atmospheres, and is then allowed to cool at constant volume to 
 the temperature of the snrroundini^ air ; if it is then allowed 
 to ex])and adiahatically until it reaches its oricinal intensitv 
 of pressure, find the efficiency of this method of storing work, 
 the work done on the air by the atmospheric pressure in the 
 compression and against it in the expansion being deducted. 
 
 1 1 
 
 A T^jxL ■ Wi"- 1 -»i(m»'-— i) 
 Ans. Jilticiency= ^ — -^ , where n = 1-408. 
 
 m- 1 —n (m" — i ) 
 
 60. Absolute Temperature. It has been already pointed 
 out (Art. 41) that the notion of an absolute zero at a point 
 273 Centigrade deg-rees below the temperature of melting 
 ice is not properly founded on the formula vp = Hw (273 + i) 
 which has been deduced for gases from experiments at 
 ordinary temperatures, and from which it would follow that 
 if t =—273, the volume of the gas would be zero. The 
 position of the absolute zero — conceived to be such that at 
 this temperature there would be no molecular motion 
 in any body — is deduced from the principles of Thermo- 
 dynamics as applied to the action of reversible engines 
 (such as Carnot's) when we imagine the working substances 
 in these engines to be any substances whatever that 
 can undergo such a complete cycle of changes as that of 
 Carnofc. 
 
 Thus, we can imagine the cylinder to contain, as a 
 working substance, a quantity of water and its vapour, 
 by the expansion of which work is done on the piston. 
 
 It is not the aim of this work to discuss the details 
 of Thermodynamics, which the student will find in treatises 
 on this subject, such as Cotterill's Sleani Engine, Clerk 
 Maxwell's Heat, The Mechanical Theory of Heat by Clausius, 
 Tait's Heat, Parker's Thermodynamics, and many other
 
 258 Hydrostatics and Elementary Hydrokindics. 
 
 works ; and therefore we shall assume the following- pro- 
 position, which is fundamental in the sul)jcct, and the 
 proof of which will be found in any of these works : — 
 
 if a reversible engine tvorks between any tico fixed tem- 
 peratnres^ its working subHance undergoing a complete cycle of 
 operations indicated by two isothermals and two adiatjatics, 
 and if it receives a fixed quantity of heat from a reservoir at 
 the higher temperature, it will convert the same fraction of this 
 heat into external work, whatever he the nature of its working 
 substat/ce. 
 
 Of course the isothermals and adiabatics of the substance 
 may be any curves whatever, and not the simple curves 
 whose equations are pv = const., and pv^^ = const., which 
 belong- to a perfect g-as. 
 
 It may occur to the student as an objection that, 
 inasmuch as we are now seeking* for a means of measuring- 
 temperature, it is not permissible to speak of the working- 
 substance in Carnot's cylinder as receiving- heat from 
 a reservoir at the hig-her temperature and transferring some 
 of it to the reservoir at the lower temperature. There is, 
 however, nothing- illog-ical in this, because, althoug-h we 
 may not be able to measure temperature numerically, 
 we are able to say when two bodies are at the same tem- 
 perature. A mercurial or any other thermometer will tell 
 us when two bodies are at the satne temperature. We can 
 say, for example, when the upper reservoir, or boiler, is at 
 the temperature of water boiling- at the normal pressure, 
 and when the lower, or condenser, is at the temperature of 
 melting ice. 
 
 Suppose, then, for definiteness, that the boiler is at the 
 temperature of water boiling under the normal pressure, 
 and that the condenser is at that of melting ice ; and with 
 any given substance in Carnot^s cylinder let a quantity, //, 
 of heat be taken isothermally from the boiler in a transfer-
 
 Gases. 259 
 
 malion such as that represented by the enrve BC in 
 Fig-. 66, and let the quantity 11^ be transferred to the con- 
 denser in the transformation corresponding- to DA. Then 
 our assinnption is that, whatever be the substance in 
 the cylinder, the ratio 
 
 -ry is constant. 
 
 When we speak of two reservoirs of heat each at a certain 
 temperature, we mean two bodies so larg-e in comparison 
 witli the dimensions of the Carnot cylinder that when the 
 working- substance in this cylinder takes heat from them 
 or gives heat to them, their temperatures do not fall or 
 rise. Let us imag-ine the cylinder to be small enoug-h, 
 and we can drop the term reservoirs and simply speak of 
 two lodics, in connection with which the cylinder is 
 supposed to work in a Carnot cycle. Denote the two 
 bodies by B^^ and B^ , and imag-ine any number of Carnot 
 cylinders C, C", C", ... containing- diflerent working- sub- 
 stances. 
 
 Suppose, then, each of these cylinders successively to 
 be put in connection with the body Bq, and each of them 
 to be worked until it takes a quantity of heat, Hq , from it, 
 this quantity being- the same for all the cylinders ; let each • 
 cylinder be worked in a Carnot cycle in which the second 
 body (that to which heat is transferred) is B^. Then all 
 the cylinders transfer the same quantity of heat, 7/, , to this 
 body, and this quantity de])ends simply on the temperature 
 of the body B^. Hence B^ can be taken as a measure 
 of the temperature of the body. On the same scale, of 
 course, H^ would be the measure of the temperature of the 
 body Bq. We may, if we please, imag-ine Bq to be a body 
 absolutely deprived of molecular motion, i. e., a body at the 
 absolute zero of temperature. If Bq is not such a body, 
 
 S 2,
 
 26o Hydrostatics and Elementary Hydrokinetics. 
 
 let its temperature above the absolute zero be ^^ , and let 
 that of By be T^ ; then, on the scale adopted now, we have 
 
 H T 
 
 This definition g-ives only \h^Q ratio of the absolute 
 temperatures of the two bodies, and makes it independent 
 of the working- substance in the Carnot cylinder, so that 
 we may select the most convenient substance that we 
 can find. 
 
 Now, supposing- B-^ to be water boiling at the normal 
 atmospheric pressure, and B^ to be melting- ice, and taking- 
 air as the most convenient substance, Thomson and Joule 
 (Tait's Heat, chap, xxi) found that 
 
 «F = i-365> (2) 
 
 -"0 
 
 T 
 ••• yr= 1-365 (3) 
 
 The magnitude of each degree is still undefined. Let the 
 magnitude be such that there are 1 00 deg-rees between these 
 two temperatures ; then T^-^Tq-^- 100, and (3) g-ives 
 
 ^0= 7757 = 274, very nearly (4) 
 
 Thus if the interval between the temperatures of melting 
 ice and water boiling- under the normal atmospheric pressure 
 is divided into 100 equal parts, the freezing point is 274 
 of these degrees above the absolute zero of temperature. 
 Hence the absolute zero coincides practically with that 
 suggested by the air thermometer. 
 
 61. Total Heat of Steam. If a kilogramme of water is 
 taken at the temperature 0° C and heated until it is 
 completely converted into saturated vapour at the tern-
 
 Gases. 261 
 
 peratnrc f C, tlicre will be two stag-es to note in the 
 process : 
 
 1. The water has its temperature raised from o" to 
 the boiling- point, showing" a continuous rise of a thermo- 
 meter placed in it. 
 
 2. At the boiling" point the water is converted into 
 saturated vapour, heat being" continuously absorbed, but no 
 rise of the thermometer being observed until all the water 
 is converted. 
 
 M. Regnault found the total quantity of heat thus 
 absorbed by I kilogramme of Avater, measured in metric 
 thermal units, to be 606-5 + -305?! (0 
 
 Of course the quantity of heat necessary to convert 
 w kilogrammes is w times this. 
 
 Now the number of thermal units absorbed in heating 
 the kilog-ramme of water from 0° to f is t ; and as the 
 quantity (i) exceeds t, it follows that the excess has been 
 absorbed in overcoming" external pressure and the molecular 
 forces of the water, converting it into steam, w^hich shows no 
 rise of temperature above / although heat is being" applied. 
 This heat, which is not indicated by the thermometer and 
 whose function is to perform internal and external work, 
 is called the latent heat of the saturated steam. If we de- 
 note it by i/, and denote the total heat used by //, per 
 kilogramme of water, we have 
 
 77= 6o6-5 + -305/^, (2) 
 
 L = 6o6-s--6i)5t (3) 
 
 If instead of a kilogramme of water at 0° we start with 
 a kilog'ramme of ice at zero, and apply heat, we find that, 
 though heat is being" continuously absorbed by the body, 
 a thermometer indicates no rise of temperature until the 
 whole of the ice is melted. Hence during- this process the
 
 262 Hydrostatics and Elementary Hydrokinetics. 
 
 absorbed heat is being' used to perform the work of dis- 
 integrating- the ice, and this heat is called the latent heat 
 of water, or the latent heat of fusion of ice. Its amount 
 per kilogramme is very nearly 80 thermal units. 
 
 If after the water has been completely converted into 
 steam of maximum intensity of pressure (saturated steam) 
 at the given temperature, t, heat is still applied, and the 
 gas expands at constant pressure, its temperature will, 
 of course, rise ; and the further amount of heat absorbed 
 when the temperature reaches B° is -4805 (^ — 1\ the number 
 •4805 being the specific heat of aqueous vapour under 
 constant pressui'e. 
 
 The expression analogous to (2) in English measures is 
 deduced thus: the equation (2) can be expressed in the form 
 
 total heat for any mass . , 
 
 - — ^ — ,- i — -^ — ^^ = 6o6-5 + -^o5/5 
 
 heat requu-ed to raise it i C 
 
 = 606.5 + .305 x|(/!'-32), 
 
 where t' is the Fahrenheit temperature corresponding 
 to f C. 
 
 total heat for any mass 
 
 heat required to raise it |°F 
 
 606.5 + . 305 X !(/{'- 32), 
 
 total heat for any mass ^ ^ , » , , /.' ^\ 
 
 ••• iT-^ -"XI • V .°p = ^°^'-5 X -5- + 'Z'^S (^ -32). 
 
 heat required to raise it i l 
 
 If mass is measured in pounds, the left-hand side is the 
 heat per pound, in the new units ; and we have 
 
 //= 108 1-94 + .305 i^, (4) 
 
 in which we have removed the accent from t' . This is the 
 number of thermal units required to raise the water from 
 the temperature of melting ice, i. e., from 32" F to t, and 
 completely evaporating it at t. In this process «! — 32 
 thermal units have been consumed in raising the tempera-
 
 Gases. 263 
 
 lure of the water to tlie evaporating* point, and hence 
 
 //—(/ — 32) thermal units have been absorbed in doing- the 
 
 work of conversion ; so that if L is the latent heat of 
 
 the steam, -r ^ 
 
 L= II 13-94 -.695/! (5) 
 
 Thus, in evaporating i pound of water at the ordinar\- 
 boiling temperature, 2 1 2° F, 966-6 thermal units are absorbed 
 while the water is being- changed into steam. The dyna- 
 mical equivalent of this is about 746215 foot-pounds' 
 weight. But this heat is employed in doing two things — 
 [a) disintegrating the water, i. e., converting it into 
 vapour, or doing internal work ; 
 
 {b) overcoming the resistance of the atmospheric pres- 
 sure, i. e., doing external work. 
 
 It is easy to calculate each of these quantities in general. 
 Let 1 pound of water be converted into steam at the 
 temperature f F. Then, when the conversion into steam 
 begins, the temperature and the intensity of pressure 
 of the steam remain constant until the conversion is 
 complete. Let the intensity of pressure be measured in 
 pounds' weight per square foot and the volume of the 
 steam in cubic feet ; then from (4), Art. 49, we have 
 
 _ I vpx '622 
 ~ 53-30222 ' T ' 
 .-. pv = 85-6gT. (6) 
 
 Now since for a small expansion of a gas the element of 
 work which it does against the external pressure is pdv, 
 and since here p remains constant, the work done from 
 volume v^J to volume v is 
 
 But ?'y is the volume occupied by the water, and v 
 the volume of the steam when evaporation is complete, and
 
 264 Hydrostatics and Elementary Hydrokinetics. 
 
 moreover v is vastly greater than r„, so that Vq may be 
 ne^-lected. Hence jw is the work done by the steam. 
 If this external work is denoted by /F, we have, then, 
 
 ^= 85-69 ^- (7) 
 
 The number of thermal units in this is obtained by 
 dividing by 772 ; so that if 11^ is the heat absorbed 
 in this external work, 
 
 ir^= .iiir=5i.o6 + -iii(^. . . . (<S) 
 
 Now if H^ is the heat used in the internal work (that 
 done in overcoming the molecular forces of the water), 
 
 L = H, + n^; (9) 
 
 .-. //;.= 1062-88 --806^ (10) 
 
 Thus, then, of 966-6 thermal units constituting the latent 
 heat of I pound of steam formed at 212°, there are 74-59 
 used in doing- external work. 
 
 Example. 
 
 A cylinder contains — 
 
 (a) water and its saturated vapour at (t, j)) in contact with 
 a hot body ; 
 
 (6) a perfect gas at {t, ]>) in contact with a hot body : 
 
 the volumes of both being the same, v, or occupying the same 
 length, h feet, of the cylinder ; calculate the distances through 
 which they must, respectively, drive the piston along the 
 cylinder in order that each sul)stance may take in the same 
 quantity of heat, II thermal units, from the hot body. 
 
 Ans. If X is the distance for the steam and x' for the 
 perfect gas, 
 
 - 85-69 (460 J- <) H 
 
 X = h • ;: J 
 
 pv 1113-94 — -695 < 
 
 772 7/ 
 
 x'=h{e "' -I);
 
 Gases. 
 
 265 
 
 t' 
 
 or, takin^f the licat absorbed and tlie laioiit lieat in work-units, 
 and denoting by H',. the external work done in the process 
 of evaporating the unit mass of water at t, 
 
 H 
 
 We II , ,7. 
 
 X = h — • '- : x'= h (e — 1). 
 pv L ' 
 
 62. Hygrometry. Various methods have been adopted for 
 determining' the pressure intensity (and hence the quantity) 
 of aqueous vapour present in the air. 
 The hvs'i'ometers of Daniell and a b 
 
 Reg-nault aim at lowering- the tem- 
 perature of the air so much that the 
 vapour in it just saturates it, and 
 therefore just l)eg'ins to de})osit as 
 dew on the siirface of the h^'g-rometer. 
 If at this instant the temperature of 
 the hygrometer is read, the pressure 
 intensity of saturated aqueous vapour 
 corresponding- to this temperature is 
 assnwed to tje the pressure intensity 
 of the aqueous vapour present in the 
 air. I'he point of temperature at 
 which the vapour is just thrown down 
 on a surface as dew is called the dew point. 
 
 Another method is that of the wet aiul dry bulb thermo- 
 meters, Fig. 67. ^ is a thermometer, fixed on a vertical 
 stand, and indicating" the temperature of the air; 11 is 
 a thermometer, fixed on the same stand, whose bulb is 
 covered with g-auze which is kept moist by a bunch 
 of threads which dip into a small vessel of v/ater. 
 
 If the air is still, we may assume that the air surrounding- 
 the bulb of B is always saturated ; but the temperature, t' , 
 indicated by B does not directly tell us the dew point or 
 the pressure of vapour present in the air remote from 
 the bulb of B. 
 
 Fig. 67.
 
 266 Hydrostatics and Elementary Hydrokinetics. 
 
 Let t be the temperature of the air remote from this 
 hulb, as indicated hj A ; f the intensity of pressure of the 
 vapour present ; ;; the total intensity of pressure indicated 
 by a barometer (/. 7;— /'is that of the air alone); /" the 
 maximum intensity of pressure of aqueous vapour corre- 
 sponding- to the temperature t'. 
 
 Consider a volume V of the air remote from the bulb of 
 B to come to this bulb, and to fall in temperature from t to 
 t'. Now in falling- it will g-ive out a certain quantity 
 of heat which will be sufficient to evaporate a certain mass 
 of the water at the wet bulb. 
 
 But V contains both air and vapour, and this heat will 
 be contributed by each of them. 
 
 Without assuming- specially either English or metric 
 measures, let to and w' be the weights of the air and 
 the vapour in V ; then 
 
 w = k ^'^/^ w' = k-4ir^ • • • • (0 
 
 .-. W =2V— _> (2) 
 
 where s = •622 = sp. g-r. of vapour. 
 
 When F reaches the bulb and falls in temperature, it will 
 be saturated by vapour, and fall to volume v ; then if w" is 
 the weig-ht of the vapour which it contains, 
 
 w = k^-^-^; w —k-^ (3) 
 
 :. w"- w j—-, ; (4) 
 
 hence the weight of the vapour which has been formed 
 
 at the bulb is , ^, ,.
 
 Gases. 267 
 
 Now if 7/ is the latent licat of a unit weig-ht of aqueous 
 vapour at f, in thermal units (the unit wei<jht being- 
 that involved in iv), the heat required to produce the 
 
 and it is assumed that this is the sum of the quantities of 
 
 heat given out by the air 7v and the vapour w - — -.in F in 
 
 falling- from t to t\ If c is the specific heat of air (-237) 
 and c' that of aqueous vapour (-48), the sum of these 
 quantities of heat is 
 
 Equating- (6) to (7), we have 
 
 Now cs — c = -061 which we may neg-Iect, so that 
 
 /=/'-'-if^>(«-0 («) 
 
 in which p is usually employed instead of ^ —f\ so that 
 
 f^f-'^xt-n (9) 
 
 from which / is found when f is read from a table of 
 pressures of saturated vapour. With English measures, we 
 may take L' equal to 1080, and with metric units equal 
 to 600. 
 
 Various objections have been urged to the assumptions 
 involved in this (which Clerk ]\Tax\vell calls the convection 
 theory of the wet and dry bulb thermometers) ; thus, it
 
 268 Hydrostatics mid Elementary Hydrohinetics. 
 
 may happen that when the air T comes to the wet bulb it 
 jn-oes away from it without being" saturated ; and, indeed, 
 this must happen if the air is in motion. 
 
 It is usual to assume a formula of the form (9), thus 
 
 f=f-k.p{t-t'\ (10) 
 
 where h is a constant which must be experimentally deter- 
 mined in each locality. 
 
 63. Liquefaction of Gases. All g-ases can be liquefied 
 by compression provided that the temperature is below a 
 certain limit, which limit is different for different gases. 
 
 Suppose that in a cylinder we have a volume of water 
 vapour (steam) at the temperature 21 2°, and at an intensity 
 of pressure of 10 pounds' weight per square inch. If now 
 we g-radually increase the pressure (keeping the temperature 
 constant) and draw an indicator diagram, such as that 
 in Fig'. ^^, representing- the volumes assumed during- the 
 compression and corresponding- to the various increasing- 
 pressures, when p reaches the value of about 15 pounds' 
 weig'ht per square inch, the steam begins to liquefy ; and 
 as we continue to diminish the volume, p remains at this 
 value until the whole of the g-as is liquefied ; so that at the 
 point where j»v = 15 the curve of pressures and volumes 
 becomes a rig-ht line parallel to Ov. Hence the isothermal 
 of steam for 2i2°F consists of a portion of an approxi- 
 mately hyperbolic curve and a right line parallel to the 
 axis of volumes. If instead of taking the water vapour 
 at 212°, we take it at 107° and an initial intensity of pres- 
 sure of, say, •! pound weight per square inch, and trace the 
 isothermal as we gradually increase the pressure, we shall 
 obtain a similar result : the curve will at first be hyper- 
 bolic, and when p reaches the value of about i pound 
 weight per square inch, the vapour will beg-in to liquefy, 7? 
 remaining- constantly equal to i, until the whole becomes
 
 Gases. 269 
 
 liquid ; so that at the point at which p = 1 the curve will 
 ehan(>-e to a n<j;-ht line. 
 
 The same result would he obtained with water vapour at 
 all temperatures, except for tlune above a certain very high 
 value (about 773° F). The mass of the steam in the cylinder 
 bein<4' constant, the rectilinear portion in w^hich every one 
 of the isothermal curves ends becomes shorter as the tem- 
 jierature at which the steam is compressed is hio-her ; but 
 if the steam has a temperature hig-her than the above value, 
 there is no rectilinear portion whatever : the curve is con- 
 tinuous, and, no matter how g-reat a pressure we apply, we 
 shall not see any distinction of vapour and liquid in the 
 cylinder such as that which must occur at lower tem2)era- 
 tures. The vapour of water is not a g-ood g^as, therefore, 
 for observing- the absence of an abrupt chang-e from the 
 g-aseous to the liquid state, because the temperatures above 
 which the change is absent are so hig-h. 
 
 Carbonic acid gas is much more manageable for this 
 purpose, because the temperature above which the chang-e 
 is absent is only about 30°-9 C, or 87-7 F. 
 
 The examination of the changes of this g'as under in- 
 creasing pressures and at various temperatures was made by 
 Dr. Andrews about the year 1862; and by him the tem- 
 perature atjove which a given gas cannot lie liqnejied Ijy any 
 amount of pressure — or, rather, above which the distinction 
 between the liquid and the gaseous states is never empha- 
 sised — has been called the critical temperature for the g-as. 
 That there is such a temperature for each gas had previously 
 been asserted by Cag-niard de la Tour. 
 
 The changes of carbonic acid gas are represented in Fig. 
 68, which is copied, on a diminished scale, from the pajjcr 
 of Dr. Andrews in the Philosophical Transactions of the 
 Royal Society, 1869, p. 583. The g-as experimented upon 
 was contained in a strong- glass tube of very small bore ; its
 
 270 Hydrostatics and Elementary Hydrokinetics. 
 
 100 
 
 temperature was kept successively at the constant centi- 
 g'rade values i3°-i, 2i°'5, 3i°'i, &c., and at each tempera- 
 ture it was powerfully compressed. 
 
 In Fig-. 68, Op is the axis of pressures, these being 
 
 measured in atmo- 
 spheres, and Ov the 
 axis of volumes ; the 
 initialpressuremarked 
 at is 47 atmo- 
 spheres, and the final 
 ITO atmospheres. 
 
 Now taking" the 
 g-as at 13°- 1 C, or 
 ^^•6 F, when the 
 pressure is below 47 
 atmospheres, the rela- 
 tion between^ and v 
 is shown by a curve, a 
 small portion of which 
 is shown to the right 
 of the point a. At 
 the pressure of 49 at- 
 mospheres liquefac- 
 tion begins, and 7; 
 remains equal to 49, 
 Fig. 68. from v= Oa to v=: 0^, 
 
 the gas liquefying all 
 the while, and the liquefaction being complete when 
 V = Oh. When the body has become liquid completely, 
 the further diminution of v requires enormous values of 
 J-;, and the upward continuation of the curve is so nearly 
 a right line parallel to Op^ that the diagram will not show 
 the difference. 
 
 Again, taking the g-as at 2i°-5C, it remains gaseous
 
 Gases. 
 
 271 
 
 lony-cr than in the previous case, liquefaction beo-innin^^ 
 when V = Oa\ and p = about 60 atmosj)licres, so that 
 the indicator diag-ram remains a rij^^ht line pai-allol to 
 Ov until tiie whole is liriuid : the volume of this liquid 
 is somewhat ii-reater than Ob, as is seen in the fio-ure. 
 
 In oacli of these experiments the eye could observe when 
 liquefaction lieo-an, and until the process was comi)lete the 
 gas and the li(juid could be clearly distinguished from each 
 other in the tul)e. 
 
 Again, taking the gas at 3i°-i C, it was found that no 
 part of the isothermal consisted of a right line during 
 any part of the process of compression, and moreover 
 the substance in the tube never exhibited the distinction of 
 liquid and gas, as it did at the lower temperatures of 
 the previous experiments : there was a continuit}^ of state 
 all through ; but the isothermal had not its curvature 
 at the same side all through ; there was a part (where the 
 mark 31-1 appears in the diagram) curved downwards, 
 and this part took the place of the right line in the previous 
 figures. 
 
 At the temperature 32''-5 C, the curve was of the same 
 nature as that last described, but the suggestion of a recti- 
 linear portion is less marked. It is still less marked 
 at the temperature o^^-^ ; while the isothermal for 48''-i C 
 loses all trace of discontinuity, being curved the same way 
 all through, and resembles the isothermals of a perfect gas, 
 although it is not really one of these curves. Dr. Andrews 
 found that the temperature at which if the gas was com- 
 pressed it never exhibited the distinction of Ii<juid and gas 
 was 3o°-92 C, or Sy"-; F. 
 
 The critical temperatures for ordinary gases are ex- 
 tremely low, and hence the great difliculty of liquefying 
 them ; because, until we get below this temperature for any 
 gas, there is no use in endeavouring to liquefy it by
 
 272 Hydrostatics mid Elementary Hydrokinetics. 
 
 compression ; and, moreover, the pressure at which lique- 
 faction begins is always extremely hii^-h. 
 
 The principle adopted by M. Cailletet, and independently 
 byM.Pictet, for liquefAing-oxyg-en.hydrog-eii, and other gases 
 which had previously escaped liquefaction, is that of en- 
 closing- the g-as in a stout tube surrounded by a freezing 
 mixture, then powerfully compressing- the g-as (500 atmo- 
 spheres were required in the case of oxyg-en, 650 for hydro- 
 g-en), and isuddenly liberating the gas under the great pressure. 
 The effect of this sudden expansion, (which maybe reg-arded 
 as an adiabati(' transformation) whereby the gas (or a 
 portion of it) is made to do work against enormous pressure, 
 is (Art. 59) to lower its temperature still more and liquefy 
 a part of it. 
 
 Thus for the liquefaction of any g-as two distinct con- 
 ditions must be satisfied — 1°. the temperature must be 
 below a certain point ; 3°. the pressure must have a certain, 
 value. 
 
 For the vapours of ordinary liquids, on the contrary, the 
 critical temperatures are very high. Thus, for water vapour, 
 the critical temperature is 773° F ; for alcohol vapour about 
 497" F, and at this point the intensity of pressure required 
 is about 119 atmospheres ; for bisulphide of carbon vapour 
 the temperature must be below 505" F ; at this temperature 
 liquefaction would require about 6"] atmospheres. Of 
 course, if these vapours are at ordinary low temperatures, 
 they can be liquefied by verv small pressures : the theory 
 of critical temperature merely asserts that, if they are atjove 
 these individual high temperatures, no amount of compres- 
 sion will liquefy them.
 
 CHAPTER VII. 
 
 HYDEAULIC AND PNEUMATIC MACHINES. 
 
 64. Water Pumps. The Coynmon Pump. Let DB, Fio-. 
 69. represent a vertical section of an iron cylinder termin- 
 ating- in a much narrower cylinder 
 or pipe, BA, which dips into a well 
 from which water is to be raised. 
 In the cylinder BB, or barrel, works 
 a piston having- a valve, v, which 
 opens upwards, the piston rod, r, 
 being' connected at c with a lever, 
 ///, working about the fulcrum/. 
 
 At the top, B, of the suction pipe 
 is a valve which also opens upwards. 
 
 Let the piston be at the bottom 
 of the barrel, the level of the water 
 in the well being A, and the pipe 
 and barrel both completely filled 
 with air. 
 
 When the piston is raised by 
 means of the handle, //, of the lever, ^ysfy^'^^y^ 
 the valve v remains closed, and the zrzz^T-j-v-i^— 
 pressure of the air in the pipe opens Fig. 69. 
 
 the valve at B, the air in AB rising 
 
 and partly filling the barrel, its intensity of pressure also 
 diminishing. As a result of this diminution of m-essure 
 
 T 
 
 ==P
 
 274 Hydrostatics and Elementary Hydrokinetics. 
 
 below its orig-inal (atraosj^heric) value, the atmospheric 
 pressure on the water forces some of the liquid into the 
 pipe. Let P be the level of the water in the pii)e at the end 
 of the upward stroke of the piston. On the downward 
 stroke of the piston the valve B closes and v opens allowing 
 the air in the barrel — which the downward motion of 
 the piston tends to compress — to escape through the piston 
 into the atmosphere, until the piston again reaches the 
 bottom of the barrel. On again raising the piston, the 
 valve V closes and that at B opens, thus allowing the air in 
 BB to expand and to diminish its intensity of pressure ; 
 and, in consequence, more water is forced up into the pipe, 
 and perhaps into the barrel. This process being continued, 
 the water ultimately reaches the level of the spout and 
 is thus driven out. 
 
 To find the height to which the water is raised in the pipe by 
 the first stroke of the piston, let A = area of cross-section 
 of barrel, a = area of cross-section of pipe, I = length of stroke 
 of piston, c = length BA , h = height of a water barometer, and 
 AP = X. The volume of air in the suction-pipe before the 
 stroke is ac, and its intensitj^ of pressure is represented by h. At 
 the end of the stroke the volume of this mass is a{c — x)-\-lA, 
 and its intensity of pressure is represented \>y h — x. Hence 
 
 (h — x) \a {c — x) + lA \ = ach, 
 
 .". ax"' — (ac -j- ah + 1 A ) x + lhA ■= o, 
 
 which determines x. 
 
 When the water is flowing out of the spout, there will 
 be a tension in the piston I'od on an upward stroke, which 
 is found thus. Let z be tlie vertical height of the piston above 
 the level A, and let z' be the height of the column of water 
 (which we may suj)pose to reach to any point, D, of the barrel) 
 above the piston. The valve at B being open, there is a contin- 
 uous water communication l)ct\veen A and the bottom of the 
 piston, flence if w is tlie weight of a unit volume of water, the 
 intensity of the upward pressure exerted by the water on 
 the bottom of the piston is w {h — z) ; and the intensity of down-
 
 Hydraulic and Pneumatic Machines. 275 
 
 ward propRuro exerted on the top of the piston is w{li-\-z'^\ 
 therefore the total downward pressure on the piston is w{z-\-z)A . 
 This is eqiial to 1\ the tension of the rod, if we neglect any 
 acceleration of the piston. Hence, approximately, 
 
 T=w.A .Bii, 
 
 which shows that the tension of the rod is the weight of a 
 vertical column of water having the piston for base, and for 
 height the difference of level of the water in the barrel and that 
 in the well. 
 
 On the downward sti'oke there is a pressure in the rod, 
 which is approximately equal to the weight of the column 
 of water above the piston. 
 
 When the water is flowing out, the force required at // 
 
 fc 
 to work the piston on the upward sti'oke is T x ~jj 3 where T 
 
 has the above value. 
 
 It is obviovis that, for the working of the pump, the length 
 AB of the suction-pipe above the well must be less than 
 the height of a water barometer, i.e., about 34 feet ; and, owing 
 to imperfect fittings. AB must be considerably less than this — 
 say about 25 feet. 
 
 In the middle ages a curious modification of the common 
 pump, called the helJoivs fump, was employed in Europe. Instead 
 of a piston worked by a lever, f J/, (Fig. 69) a large bellows was 
 attached firmly to the top of the barrel, and the nozzle of 
 the bellows was the spout through which the water was forced. 
 
 The top of the baricl fitted into the interior of the bellows 
 through a hole in the lower boai'd of the bellows ; there was no 
 valve in the top board, but there was one ojDening outwards fixed 
 in the nozzle. The action was, of course, the same as that in 
 our modern pump. 
 
 T/ie Forcing Pump. This is an instrument for raising 
 water to a great height. It differs from the previous pump 
 in having a completely solid piston. 
 
 To the barrel of the pump is attached the pipe through 
 which the water is to be raised. This delivery pipe is pro- 
 vided wdth a valve at D, Fig. 70, opening ui)wards, and 
 this pipe is represented as being provided with a spout and 
 
 T 2
 
 276 Hydrostatics and Elementary Hydrokinetics. 
 
 stop-cock by means of which the machine can be made to 
 act as a Common pump. 
 
 The action is the same as 
 in the previous case. 
 
 On the downward stroke 
 of the piston, the valve at 
 B closes and that at 1) 
 opens, and through this 
 latter the water is forced 
 out of the ban-el into the 
 delivery pipe, BV. 
 
 There is then a pressure 
 in the piston rod, ;•, equal 
 to the weig-ht of a column 
 of water having the piston 
 for base, and for height the 
 difference of level between 
 the piston and the water, 
 V, in the delivery tube. 
 '-. On the upward stroke 
 
 there is a tension in the 
 rod, whose value is the same 
 as in the previous pump. 
 
 The Fire Engine. This is simply a double forcing pump. 
 The figure (Fig. 71) represents the two barrels, P and Q, 
 as immersed in a tank, BE1\ full of water ; and from this 
 tank the pumps, which are both worked by the lever AB., 
 force the water through a hose connected with the chamber 
 C at h. The water is forced through this hose to the place 
 where the fire is to be extinguished. The action of the 
 valves is obvious in the figure. Such would be the arrange- 
 ment in a small fire engine, the tank BBI being filled 
 by buckets of water brought by hand. 
 
 When large fires have to be dealt with, a supply thus 
 
 r-'--^\ 
 
 
 Fig. 70.
 
 Hydraulic and Pneumatic Machines. 277 
 
 derived would be of no use, and the water which is pumped 
 throug-h the hose must be derived from a well or other 
 reservoir by means of a suction pipe. The fiji^ure represents 
 at c the place where such a pipe can be attached to the 
 engine. 
 
 The chamber C is partly filled with water and partly 
 with air, and is called an air cJiamher. Such a chamber 
 may be, and often is, fitted to forcing- and other pum})s, 
 the object being- to render the stream of water from 
 
 the delivery pipe at h continuous instead of intermittent ; 
 and this result is evidently secured by means of the 
 compressed air at the top of the chamber ; for, since 
 this air was originall}' at the atmospheric pressui-e (when 
 it filled the whole of the chamber) its intensity of pressure 
 after the chamber is partly filled with water is g-reater than 
 this value. This increased pressure is therefore continuously 
 exerted on the top of the water in the chamber and helps 
 to drive the stream through the hose. 
 
 Tfic lli/dmulic Screw. This is one of the most ancient 
 machines for raising water, and is still employed in some
 
 278 Hydrostatics and Elementary Hydrokinetics. 
 
 countries. It is often called the 8crew of Archimedes, be- 
 cause its invention is supposed to be due to the philosopher 
 of Syracuse. There is, however, reason to think that it 
 was first employed in Egypt, 
 
 As represented in Fig. 73, it consists of a pipe wound 
 spirally on an axis which is fixed in a position inclined to 
 the vertical, its extremities fitting into two solid supports, 
 
 Fig. 72. 
 
 //, B, the axis (and, with it, the screw) being able to revolve 
 freely. Both ends of the spiral pipe are open, and the 
 lower is immersed in the water which is to be raised to the 
 level U. 
 
 The revolution of the screw can be effected in various 
 ways : in the figure it is re])resented as produced by the 
 revolution of a shaft C fitted with a toothed wheel which 
 gears with another fitted to the top of the axis of the screw.
 
 Hydraulic and Pncnniatic Machines. '2-jc) 
 
 When the screw ia made to revolve so thut the lower 
 end comes towards us in the figure, water entering- at this 
 end continually drops to the lowest point of each j)art 
 of the spiral, and is thus carried continuously up to the top 
 where it is discharg-ed. 
 
 There is a certain condition tliat must be satisfied by the 
 inclination of the axis of the screw to the vertical and the "an<'le 
 of the spiral in order that the machine may be able to raise the 
 water. The condition is this : the inclination of the axis of tJw- 
 screw to the horizon must be less than the inclination of the tan<jent 
 line ufthe sjnral to the axis of the screw. To prove this, we may 
 put the matter thus : the axis of the screw must be so much 
 inclined to the vertical that it is possible to draw a horizontal 
 tangent to the spiral. This is obvious, because if we imagine a 
 single particle (suppose a small mar])le) entering the lower end 
 of the pipe, it would not drop down farther through the opening 
 unless there were in the pipe a place in which the particle 
 could rest under the action of its owu weight and the reaction of 
 the pipe on it ; and at such a place the tangent to the spiral 
 must be horizontal. 
 
 Let i be the inclination of the axis of the screw to the 
 vertical, a the angle of the spiral, i.e., the inclination of the 
 spiral to the axis (the spiral signifying the central helical axis 
 of the pipe), r the radius of the cylinder on which this helix lies, 
 and let m be used for tan a. 
 
 Then if we take as axis of ,:; the axis AB of the screw, and as 
 plane of xz the plane of the vertical line at Ji and tlie axis Bxl, 
 the axis of x being the line at B perpendicular, in this plane, to 
 BA, and the axis of y the perpendicular to this plane at B, the 
 direction-cosines of the vertical at B with reference to these 
 axes of co-ordinates are 
 
 — sin i, o, cos i ( i ) 
 
 Also if X, 1/, z are the co-ordinates of an_y point, l\ on the helix, 
 
 mz \ 
 X ■=. r cos — > 
 
 ' i (-) 
 
 . m,z I 
 2/ = rsm--.j
 
 28o Hydrostatics and Elementary Hydrokinetics. 
 
 If ds is an element of length of the helix at P, we have 
 
 ds = sec a . dz, and 
 
 dx . . inz \ 
 
 -— =: — Sin a sin — ? 
 ds r 
 
 dx . mz 
 
 — - = sin a cos — ? 
 ds r 
 
 dz 
 
 -— = cos a, 
 
 (3) 
 
 and the direction-cosines of the tangent to the helix at Pare the 
 expressions (3). 
 
 Supposing this tangent to be at right angles tothe vertical (i), 
 
 we have ^^ 
 
 sin i sin a sin \- cost cos a = o, 
 
 r 
 
 :. sm — = — cot^cota (,4) 
 
 r 
 
 which shows that cot i cot a must he < i, i. e., cot i < tan a, or 
 -—i < a, that is, the inclination of the axis, BA,oi the screw to 
 
 2 
 
 the horizon must be less than the inclination of the helix 
 to BA. 
 
 Tlie point P (which is now the lowest point in one turn 
 of the pipe) does not lie in the vertical plane through PA, 
 as might at first be supposed ; for (2) gives 
 
 y •= — r cot i cot a (5) 
 
 for this point. 
 
 To find the force necessary to turn the screw when a particle 
 of weight W is to be raised, let us consider the screw to be 
 either in equilibrium or in uniform motion. The forces acting 
 upon it are W, the reactions at the supports P, A, its own 
 weight, and the force applied to turn it. Of these the reactions 
 at P and A and the weight of the screw intersect PA. We 
 have, then, the result that the moment of IF about PA is ec^ual 
 to the niomciit of the api)lied turning force. (Hence an obvious 
 reason why the e(iuilibriuni ))Ositioii, P, of the ascending particle 
 cannot be in the vertical plane through BA.)
 
 Hydraulic and Pneumatic Machines. 281 
 
 Now tlie moment rouiul tlie axis of c; of a force whoso com- 
 ponents are A", Y, ^actinjj; at the point x, y, z h x Y—yX; iin<l 
 the ibrce at P is W, acting vertically clown, whose components 
 are ll'sin?, o, — IF cos ?: ; hence the moment of IF abont BA 
 is — Wy sin i, i. e., by (4) 
 
 Wr cos i cot a. 
 
 If a is the radius of the toothed wheel attached to the axis of 
 the screw, and /the force on it, we have then 
 
 f = W - cos ^ cot a. 
 
 If h is the radius of the toothed wheel attached to the driving 
 shaft (7, and if tliis shaft is turned by an effort F applied 
 at a distance A' from the axis C, we have/, b = F . R, and 
 
 vh 
 F = W~- cos i cot a. 
 all 
 
 If instead of a single particle which, always occujjying 
 the lowest point of the convolution in which it lies, ascends 
 to the top, we consider the whole tube full of water, the moment 
 
 of the weight of this water aboiit the axis is w sin i I yds, 
 
 , . , Jo 
 
 where w is the weight of water in the tube per unit length, 
 and I is the length of the tube. This is 
 
 o . . / Zsina\ 
 r" w sm t cosec a ( i — cos j • 
 
 The Hydraulic Screw is capable of a differential form. Suppose 
 the screw in Fig. 72 not to dip into water at its lower extremity, 
 but to receive into the upper end of the pipe at A a stream of 
 water from any source. Then, the screw being fixed exactly as 
 represented, would be driven by this stream in the direction 
 opposite to that in which it was caused to rotate under tiie 
 previous circumstances. 
 
 Now suppose, that it is, as before, desired to raise water from 
 a well at the extremity B to a position D, and su{)j)os(' that 
 there is available a stream of water at some lower level, repre- 
 sented by the arrow L. Let a second pipe be spirally wound 
 round the axis BA outside the tube re])re8ented in the figure and 
 in tfie reverse sense, its upper end terminating at the level L. It
 
 282 Hydrostatics mid Elementary Hydrokinetics. 
 
 the stream L is led into the upper end of this second pipe, it will 
 cause the whole machine to rotate in the sense required to rai^-e 
 the water from the well hy means of the internal spii-al pipe. 
 
 The idea of the Differential Hydraulic Screw appears to 
 be due to the iiifjenious Marquis of Worcester, who puldished 
 his notions on this machine and on many others in a work called 
 ^4 Century of Inventions, in the reign of Charles II. 
 
 Tlie Hydraidic Bam. This is a machine in which the 
 momentum of a stream of water is suddenly stopped, with 
 
 Fig. 73- 
 
 the result that a portion of the water is forced up to 
 a considerahle heig-ht. 
 
 Let AB, Fig. 73, be a stout iron chamber to which
 
 Hydraulic and Pneumatic Machines. 283 
 
 is attached a pipe F which admits a flow of water from 
 a stream or reservoir the level of which may be only 
 slightly hio-her than that oi AJi. The vessel AB is fitted 
 witii a support, E, for a valve, v, which can move freely up 
 and down. When this valve falls, there is a free communi- 
 cation between the interior of AB and the atmosphere, and 
 if AB is full of water, some of this water Hows away throug-h 
 the opening" at v, and is wasted. 
 
 To the top of the vessel AB is screwed a chamber, 6', 
 which has a valve c opening* upwards, and also a valve 
 s opening' inwards. This latter valve is attached to a rather 
 weak spring' fixed to a side pipe oi)ening' in to the chamber 
 C. Finally, a suj)ply pipe, S, is attached to the chamber C. 
 
 Imagine the whole machine free of water, so that 
 the valves c and v are down ; and then lot the stream flow 
 in at P. At first the water will rush throug-h the opening' 
 at V ; but soon the rush of water will close this valve, and 
 at this instant, the water being- suddenly checked, some 
 will be forced throug-h the opening at c. This valve will 
 then close and v will drop, allowing- an outflow again from 
 the vessel AB. The same process will be again and again 
 repeated until the water forced into C rises in the pipe 
 S. The upper part of the chamber C contains imprisoned 
 air, the pressure of which serves to keep the How up the 
 pipe S continuous. The valve v falling- and rising thus 
 regularly, the machine is self-acting. After a long time 
 the air in the air chamber C would be absorbed by the 
 water, and thus the advantage of an air chamber C would 
 be lost. The object of the snifting valve s is to prevent this, 
 and the air is renewed as follows. When the valve v drops, 
 the intensity of pressure immediately under it is that of the 
 atmosphere, and therefore the intensity of pressure of the 
 water at s is less than that of the outside air, so that 
 (as the spring which closes s is n weak one) this valve s is
 
 284 Hydrostatics and Elementary Hydrokinetics. 
 
 forced inwards, thus allowing some air to enter the water ; 
 and this air when the valve c is next opened will rise to 
 the top of C and replace the air absorbed by the water. 
 
 (It must not be supposed that the forcing- of water 
 by this self-acting machine to a height vastly greater than 
 that of the source whence the water was derived involves 
 any contradiction of the principle of "Work and Energy ; 
 for, it is by means of the kinetic energy generated by 
 gravity in the very large mass of water which flows into 
 and out of the ram that the comparatively small mass 
 of water is raised in the pipe.) 
 
 The Hydraulic Ram was invented in I772by "Whitehurst 
 of Derby, his machine, however, not being self-acting. 
 With him, instead of the self-acting valve Vy there was a 
 stop-cock through which the water flowed ; and it was on 
 the sudden closing of this stop-cock that the water was 
 forced through the valve c. 
 
 65. Air Pumps. The Common Air Pump. In Fig. 74, 
 _S is a cylinder or barrel in which works a piston with 
 a valve, c, opening upwards. The barrel is screwed, or 
 otherwise firmly attached, to a plate, B, through which 
 i-uns a groove which communicates with the interior of the 
 barrel through an opening which can be closed by a valve 
 a ; the other end, 11, of this groove opens up through a 
 large plate PQ, the upper surface of which is perfectly flat. 
 On this plate is placed a large glass vessel. A, called the 
 receiver, the mouth of which rests on the plate -PQ ; the 
 rim of the receiver is ground, and it fits the plate so 
 accurately that the junction is air-tight, especially as a 
 layer of grease is rubbed on the rim before it is placed on 
 the plate. 
 
 The object is to remove the air, partially or completely, 
 from the receiver, and therefore from any body or vessel 
 that may be placed under it.
 
 Hydraulic and Pneumatic Machines. 285 
 
 The manner in wliich tliis is cfrcctcd is obvious from the 
 fi<^uro. Su])pose the piston in the lowest position in the 
 barrel ; then when it is raised, 
 a vacuum tends to form above 
 a, so that the air of the re- 
 ceiver raises this valve and 
 fdls the barrel at the end of 
 the stroke. On the descent 
 of the ])iston, a closes and 
 c opens, and the air in the 
 bairel is thus expelled into 
 the atmosphere. This pro- 
 cess is I'epeated many times, 
 with the result that the air 
 in A is greatly diminished 
 in mass. 
 
 To calculate the degree of 
 exhaustion after n strokes of 
 the piston, let the volumes 
 
 of the receiver and barrel be A and B ; let the orig-inal 
 intensity of pressure of the air in A be 7^^, and let /^j, 
 JO2, p. ... be the intensities after i, 2, 3,... strokes. 
 
 Then after the piston has been raised the first time 
 the mass of air whose volume and intensity of pressure 
 were {A, p^) becomes [A -)- -^,7?^) ; hence 
 
 {A + B)2h = Apo (i) 
 
 When the down stroke is ended there is a different mass 
 of air in A, and it is denoted by {A,p^) ; and this same 
 mass of air is denoted by (A+ 7^,p.^) at the end of the 
 second upward stroke ; hence 
 
 (A + B)j>, = Ap, (2) 
 
 Similarly 
 
 {A + B}p.^ = Ap,, (3) 
 
 Fig- 74-
 
 286 Hydrostatics and Elementary Hydro kinetics. 
 Hence, by multiplication, 
 
 ••• P^=P^ij^^' ^"^ 
 
 which gives the final intensity of pressure. If /?^, is 
 the weig-ht of the air finally left, and W^ the original 
 weight, we have from (a) of Art. 47, 
 
 and a similar relation between the final density, /)„, and the 
 original p^. 
 
 I'or the very high exhaustions required in the globes of 
 incandescent electric lamps, and in the interior of vacuum 
 tubes, an air pump of this kind would be quite insufficient, 
 because, after the exhaustion has reached a certain limit, 
 the pressure of the gas is insufficient to raise the valve a. 
 
 Condensing Air Pumji. When it is desired to fill a vessel, 
 A, Fig. 75, with air or any other gas at a given intensity 
 of pressure, a condensing pump is emiiloyed. This machine 
 consists of a cylinder or barrel, fitted with a solid piston, 
 and having a valve, c, opening downwards. At the side of 
 the barrel there is attached a pipe having a valve, a, 
 opening inwards. When any other gas than air is to 
 be forced into A, the vessel supplying this gas is attached 
 to the pipe at a. The valve at a opens while the piston is 
 raised, and the barrel is filled with the gas. When the 
 piston is lowered, a closes, c is forced open, and if the stop- 
 cock, 5, fitted to the vessel A is properly turned, the gas 
 enters A. On the uj)ward stroke of the piston, c closes, 
 a opens, and the process is repeated. 
 
 Let A and B denote the volumes of the chamber A and 
 the barrel, and consider the gas which fills A after n strokes
 
 Hydraulic and Pneumatic Machines. 287 
 
 of the piston. Let p be its intensity of pressure, and let />„ 
 be the intensity of i)ressure of the gas which fills the 
 barrel : if // is being- filled 
 with atmospheric air, 7;,, is 
 the atmosi)heric intensity. 
 Then the ""as whose 
 volume and intensity of 
 pressure are [A, p) was 
 once represented by 
 
 {A + uB,/)^;), 
 
 supposing" that A contained 
 the gtis originally. Then 
 
 A._p = (A + ?iB)pf„ 
 
 The Geissler Pumj). 
 When very high exhaus- 
 tions are required, air 
 pumps with valves and 
 pistons are I'eplaced by 
 pumps in which a column 
 of mercury i)lays the part of a piston, 
 kind is that represented in Fig. 76. 
 
 To a certain extent, all air pumps are identical in 
 principle. In each of them a given mass of gas occupying 
 a volume V is made to occupy a larger volume, V+ U, and 
 then the portion occupying U is mechanically expelled. If 
 this process could be repeated indefinitely, an exhaustion of 
 any degree desired could be obtained ; but we have seen 
 that the raising of valves in the common air pumps puts a 
 limit to the process. 
 
 Fig. 75. 
 
 Of the latter
 
 288 Hydrostatics and Elementary Hydrokinetics. 
 
 The mercury pumps of Geissler and Sprengel are free 
 from this drawback. 
 
 AB is a g-lass tube of greater leno-th than the height of 
 the mercury barometer, having- part of the Torricellian 
 
 Fig. 76. 
 
 vacuum enlarged into a chamber, A, of large capacity. 
 Above A is inserted a two-way stop-cock, «, which 
 in the position represented establishes a communication
 
 Hydraulic and Pneumatic Machines. 289 
 
 between the chamber A and a side tube, f, fitted to the 
 tu])e BAd. This tube /has a stop-cock, c, wliich in the 
 position represented closes/; but if c is turned throug-h a 
 rig-ht angle it will establish a communication between 
 yand the external atmosphere at v. If a is turned through 
 a right angle from its present position, it will close 
 the communication of A with f, and open one between 
 A and a vessel /to which the portion d is joined as repre- 
 sented. The vessel J contains some sulphuric acid the 
 object of which is to remove aqueous vapour from air which 
 may pass over it ; and, by means of a stop-cock, t, J com- 
 municates with a very stout indiarubber tube, 7//;. which is 
 connected with the vessel G which is to be exhausted. 
 
 To the vessel / is connected a truncated manometer, 
 that is, a bent g-lass tube, mr, containing* mercury which 
 when the air in / is at atmospheric pressure quite fills the 
 leg- m. If the air in J is completely removed, the columns 
 of mercury in the leg's m and r will assume exactly the 
 same level. 
 
 To the end B of the tube BA is attached a stout flexible 
 tube T, which is also fixed to a large reservoir, C, of 
 mercury. 
 
 Suppose now that c is tunied so as to establish communi- 
 cation between f and the atmosphere at v, and that a is in 
 the position represented (i. e., closing* communication of A 
 with d) ; and let C be raised until the surface of the mercury 
 in BA reaches the stop-cock a, thus expelling- all the air 
 from A through fv. Then turn a and t so as to admit air 
 from G through J and d, and low^er C to its original 
 position. The air of G now occupies the volume G-\-A 
 together with the volumes of the communicating- tubes. 
 
 Turn t so as to break communication with the rarefied 
 air in C, and then turn a so as to establish communication 
 between A and the atmosphere at v. Again^ raise C until 
 
 u
 
 290 Hydrostatics and Elementary Hydrokinetics. 
 
 the mercuiy in BA drives out the g-as from A into the 
 atmosphere ; and repeat the process of estahlishing* com- 
 munication with G, &c. In this way, by repeated operations, 
 the air in G is exhausted almost completely. By this 
 pump the air left in G can be reduced to an intensity 
 of i^ressure represented by only -^^ of a millimetre of 
 mercury. 
 
 The Sprengel Pump. Fig-. 77 represents this pump, 
 in which, as in the Geissler, the vessel, G, to be exhausted 
 is made part of the Torricellian vacuum of a barometer 
 tube, H. 
 
 A funnel, F, prolonged into a narrow tube fitted with a 
 stop-cock, f, is supported in a vertical position (support 
 not represented in figure) and dips into a wide tube, B, 
 also supported. B is connected by indiarubber tubing 
 with a narrow vertical tube, C, above which is a large 
 chamber, A, open at the top, and fitted with a stopper, *, 
 the tube 1) being, like C, connected with the chamber. I) 
 is connected by indiarubber tubing with the vertical tube 
 7, which communicates freely with the very narrow tube //, 
 the top of which is connected with the vessel G, and the 
 bottom of which, curved up a little, dips into a vessel V full 
 of mercury. There is an overflow from Fto a trough T, 
 as represented ; and there is a clamp, c, by means of which 
 communication between B and I can be established or 
 broken. The vessel G is provided with a stop-cock, g. 
 
 The order of operations is as follows. The tubes being 
 all completely occupied by air, fix the clamp <?, remove the 
 stopper s, lower the system of tubes B, C, and pour mercury 
 into F and through its tube into B, until this mercury 
 completely fills the tubes C, B and the chamber A. Close 
 A -vvith the stopper s, and raise the system C, B to its 
 original position ; open the clamp c, and let the merciuy 
 run from B, C, B into / up to the top expelling air from //
 
 Hydraulic and Pneumatic Machines. 291 
 
 throuo-h the mercury in V. The mercury coming- over at a 
 
 into 11 will occupy a certain portion of H. Now let the 
 
 stop-cock .<7, be turned so that G is connected with the upper 
 
 part of //. INIercury may 
 
 be poured into /'' to keej) up 
 
 the flow from a throuii;"h //, 
 
 and the rate of supply of 
 
 this mercury can be reg^u- 
 
 lated by turning' the stop- 
 
 cockymore or less. 
 
 Now as each droji of 
 mercury fallsdown through 
 //, it forces the air in front 
 of it down through the 
 end of H ; and hence the 
 gas of G which keeps flow- 
 ing into the upper part of 
 H is perpetually driven 
 down and out by the suc- 
 cessive drops of mercury 
 which fall over from a. If 
 the mercury in A has fallen 
 down through B into /, //, 
 and F, the chamber ,4 is a 
 vacimm. 
 
 When the exhaustion of 
 G has not been carried 
 very far, the successive 
 threads of mercury falling ^'&- 77- 
 
 down II (and represented 
 
 in the figure) succeed each other comparatively slowly, and 
 they can be seen forcing the gas which reacts against their 
 fall ; but when the exhaustion is nearly complete, these 
 drops fall much more freely through the now exhausted 
 
 u a
 
 292 Hydrostatics and Elementary Hydrokinetics. 
 
 space ; and when there is only a very small quantity of gas 
 left in G, the drops falling from a on the top of the mercury 
 surface, //, produce a sharp metallic sound, like that of 
 a water hammer. This sound is an indication of a high 
 degree of exhaustion. 
 
 When the exhaustion is complete the surface of the 
 mercury in // will be at the barometric height above 
 the level in F, and the difference of the level of the mer- 
 cury in B and C will also be the barometric height. 
 
 The object of allowing the tube from F to dip into a 
 much wider tube, i?, is, partly, to let any air that may be 
 carried down with the mercury from F escape into the 
 atmosphere through the mercury in the wide tube, and 
 partly to avoid filling F a very great number of times ; 
 this incessant filling will not be necessary if the tube B is 
 very much wider than the other tubes. 
 
 The object of turning up the end of the tube H in 
 the mercury in V is to allow the gas (whatever it may be) 
 that is expelled from G through this end to be collected 
 in another vessel, a tube from which is led to the point 
 at the end of //. 
 
 The object of having the tubes C, B and the chamber A 
 is (when A is vacant of mercury) to catch in this chamber 
 any air that may have been carried over by the mercury 
 from F, so that the exhaustion in the tube II shall be 
 performed by mercury devoid of air. Hence the chamber 
 A is called an air trap. So far as the principle of the 
 Sprengel pump is concerned, we might dispense with 
 the tubes (7, B and the air trap. A, and connect B directly 
 with / — and the Sprengel pump is, in fact, usually so 
 represented. 
 
 66. Manometers. A manometer is an instrument for 
 measuring the intensity of pressure of a condensed or an 
 exhausted gas. The instrument has various forms, and the
 
 Hydraulic and Pneumatic Machines. 293 
 
 l)rineij)le of jvll will be easily understood from that repre- 
 sented in Fij*-. 78. 
 
 Let IIFEJJ be a vertical l)cnt glass tube, having- a portion, 
 ED, of one leg- enlarged into a capacious reservoir, and let 
 two necks C, D, i)roject from this reservoir 
 so that vessels may be connected with the 
 reservoir by means of indiarubber tubes 
 fitting- on C and JJ. The leg- IIF is closed 
 at the top. 
 
 Suppose the cross-section of this leg- to 
 be uniform, as also that of the reservoir 
 except near its ends. Let mercury be 
 poured into the instrument, and when the 
 air thus imprisoned in IIF assumes the 
 temperature and pressure of the surrounding- 
 atmosphere (which enters at C and I)) let 
 AB be the level of the mercury in both legs, 
 and let the number i be marked on the 
 leg IIF where the surface of the mercury stands, this point 
 being, of course, in the prolongation of BA. The number 
 indicates that when the air in IIF occupies the length ///, 
 at a given temperature, its intensity of pressure is i atmo- 
 sphere. Let c be the length ///. 
 
 Now sujipose that it is desired to fdl a globe, or other 
 vessel, with air at a great pressure and to measure the 
 intensity of the pressure. Let the vessel be connected with 
 the neck C, and let D be connected with a condensing 
 pump. When this pump forces air in through B, and 
 therefore through C into the vessel, thp surface of the 
 mercury falls in the reservoir EB and rises in the tube 
 HF. After the pump has been working for any time, let n 
 atmospheres be the intensity of pressure of the air in the 
 reservoir (and therefore in the vessel which was to be 
 lilled); let BQ be the dejn-essed surface in the reservoir, 
 
 Fig. 78.
 
 294 Hydrostatics and Elementary Hydrokinetics. 
 
 and let the surface of the mercury in IIF be at x, which is 
 c^ inches or millimetres above the level AB. If the 
 intensity of pressure of the air in the reservoir is n atmo- 
 spheres, the number n is to be marked at the level x on the 
 tube HF. Let the depth of PQ below AB be y inches or 
 millimetres ; a = area of cross-section of HF, A = that of 
 the reservoir ; and let // inches or millimetres represent 
 an intensity of pressure of i atmosphere. Then the dif- 
 ference of level between the point x and the surface FQ 
 being- x + f/, and the volume of the air in HF being now 
 a(c — x) with an intensity of pressure denoted by nh — x—y, 
 we have by Boyle's law 
 
 a{c — x)[n7t — x — y) = ach. 
 
 But evidently At/ = ax ; hence 
 
 a[c — x)\n/i — fi + -^\xl—aeA = o, 
 
 which determines the number, n, to be marked on any 
 
 part of the tube. 
 
 67. Hydrometers. These are instruments for the deter- 
 mination of the specific gravities of 
 liquids and solids. We shall describe 
 two only. 
 
 The Common Hydrometer, Fig. 79, is used 
 for finding the specific gravities of liquids. 
 It consists of a glass bulb, or cylinder, A 
 terminating at one end in a long narrow 
 graduated stem, and at the other end in a 
 small bulb, B, which contains a little 
 mercury, the object of which is to keep 
 the instrument vertical when it is im- 
 mersed in a liquid. 
 
 If when immersed in one liquid it floats 
 
 with a volume v immersed, and in another with a volume
 
 Hydraulic mid Pneumatic Machines. 295 
 
 v', the specific weig^hts of these liqviids being w and w\ 
 respectively ; and if W is the weig-ht of the instrument 
 itself, we have 
 
 to v' 
 vf ~ V ^ 
 
 (I) 
 
 so that the sj^ecific weights are inversely as the volumes 
 immersed. 
 
 The volume of the portion AB in-espective of the stem 
 can be found by graduating the stem (supposed of uniform 
 bore) into any number of equal parts, and then observing 
 the weight, W, of the instrument. Let masses 7;, 7/ be 
 successively attached to the top of the stem, and with 
 these let the instrument float in water up to the ^?.th and 
 «'th division, respectively. Then if V \s, the volume AB, 
 and a the volume per division of the stem, 
 
 {Y^-na)w = W+p ; {V+na) = lF+p\ 
 
 which determine V and a. 
 
 If when the hydrometer is immersed in two different 
 liquids the readings on the stem are n and n', we have 
 from(i) , 
 
 w' V-\-na 
 
 which shows that if a is very small, n and n' must be veiy 
 widely different, i. e., the instrument is exceedingly sen- 
 sitive to small differences of specific weight. 
 
 Sikes's Ilydz'ometer is a form of the above in Mhich 
 the stem is a very thin flat strip of metal, for which, 
 of course, a is very small. 
 
 NicJiohons Hydrometer. This hydrometer is employed to 
 measure the specific gravities of both solids and licpiids.
 
 296 Hydrostatics and Elementary Hydrokinctics. 
 
 Vgy 
 
 It consists of a hollow metallic cylinder, A, Fig-. 80, having 
 
 a very fine stem on which there is 
 a fixed mark, P ; the lower end of the 
 cylinder is connected by a wire with a 
 closed cone, J), which is heavy enough 
 to keep the stem vertical ; the base, C, 
 of this cone serves as a platform on 
 which a solid body can be placed ; the 
 stem terminates in cup, B, in which 
 solids can be placed. 
 
 To find the specific gravity of a 
 solid, place masses in B until the mark 
 P is just sunk to the surface of the 
 water ; then place the given body in 
 B : this will cause P to sink lower ; 
 remove weight from B until P again 
 reaches the surface ; if the weight removed is W, then W 
 is the weight of the given body. Now remove the body 
 from B to the platform C, and add weight, W\ to B until 
 P sinks to the surface ; then W is the weight of a volume 
 
 W . 
 of water equal to the volume of the body ; and -^ is the 
 
 required specific gravity. 
 
 To find the specific gravity of a liquid, let H be the 
 weight of the hydrometer itself; let the instrument be 
 immersed in the given liquid ; add weight, p, to B until P 
 sinks to the surface ; let m be the weight which must 
 be added to B to cause P to sink to the surface when the 
 instrument is immersed in water ; then evidently the 
 
 Fig. 80. 
 
 specific gravity of the given liquid is 
 
 ^+i^o
 
 Hydraulic and Pneumatic Machines. 297 
 
 Miscellaneous Examples. 
 
 1. A semicircular area is immersed vertically in water, with 
 its diameter in the surface ; show how to divide it into n sectors 
 all equally jiressed. 
 
 Ans. Divide the diameter into n equal parts, and draw 
 ordinates at tlie points of division : then the points in which 
 these ordinates cut the circumference determine the sectors. 
 
 2. If Wj, w^, Wg are the apparent weights of a given body in 
 fluids whose specific gravities are s^, s^, s^, show that 
 
 «^i («2 - ^3) + '^■2 («3 — «i) + ^3 («i - ^2) = o- 
 
 3. A ship sailing from the sea into a river sinks m inches, 
 and, on discharging P tons of her cargo, she rises n inches ; 
 calculate the mass of the ship assuming her sides vertical at the 
 water line, and that 9 is the specific gravity of sea water with 
 respect to fresh (about 1-025). 
 
 . ms „ 
 
 Ans. —. r . F tons. 
 
 n [8— I) 
 
 4. A uniform rod, AB, of length 2a is moveable round 
 a horizontal axis fixed at A which is in a liquid of specific weight 
 t/!o ; the end JS projects into a liquid of specific weight w, which 
 rests on top of the other liquid ; find the positions of equilibrium, 
 and determine whether they are stable or unstable. 
 
 If 2 A is the depth of A below the surface of the lower liquid, 
 6= inclination of AB to the vertical, iv = specific weight of 
 rod, the oblique position is given by the equation 
 
 sec^ e = 
 
 a^ w—w^ 
 
 and if it exists, it is stable.
 
 CHAPTER VIII. 
 
 MOLECULAR FORCES AND CAPILLARITY. 
 
 [Tkis Chapter may he omitted on first reading.) 
 
 68. Molecular Forces. Common observations on the 
 resistance which solid bodies oppose to any effort to elongate 
 or twist them have compelled physicists to assume the 
 existence of forces between the molecules of such bodies 
 other than the ordinary action of Newtonian g-ravitation. 
 
 Thus, let us fix our attention on any one molecule, m, 
 inside a body. It is surrounded by a g-roup of molecules, 
 and if we take all those molecules which lie within a sphere 
 of extremely small radius whose centre is m, there is 
 a special action exerted on m by each molecule within this 
 sphere, those molecules nearest to tn exerting* a more 
 powerful action than those near the surface of the sphere. 
 This holds, whatever be the sizes, the shapes, or the 
 distances between the molecules. 
 
 Beyond a certain distance, e, from m these special actions 
 are assumed to be insensible ; this length € is the radius of 
 the aforesaid sphere, called the sphere of molecular activity. 
 
 Now if dm and dm' are two elements of mass, the linear 
 dimensions of each being infinitely smaller than the length of 
 any line from the surface of the one to that of the other ^ it is
 
 Molecular Forces and Capillarity 299 
 
 assumed that these elements exert on each other a force 
 whose ma-nitude is fl^r).dmdm\ (i) 
 
 where r is the distance between the elements — i. e., the 
 length of a line drawn from any point on one to any point 
 on the other — and this force acts in tlie line joining- them. 
 
 If the elements dm and dm' were homogeneous spheres, 
 such a law of force as (1) could be assumed to hold, though 
 theii' dimensions were even large compared with the 
 distance between their centres, which distance would be 
 the value of r in (i); but if they are not spherical, such 
 a law could not be admitted (because it would be utterly 
 devoid of meaning) if the elements ai'e so close together 
 that their linear dimensions are of the same order of mag- 
 nitude as lines drawn from points on the surface of one 
 to points on the surface of the other. 
 
 Now there are several suppositions that may be made 
 with regard to the arrangement of matter in a body, such 
 as the following : — 
 
 1. The matter is absolutely continuous within the 
 volume of the body, there being no vacant spaces, however 
 small. 
 
 2. The matter consists of molecules (in the chemical 
 sense) which are packed very closely together, their linear 
 dimensions being great compared with the distances 
 between their surfaces. 
 
 3. The matter consists of molecules (in the chemical 
 sense) which are very distant from each other, so that the 
 space surrounding any molecule is comparatively void of 
 matter. 
 
 If the third supposition is made, it is clear that the 
 application of mathematical calculation becomes exceedingly 
 difficult, if not impossible. It is true that Lame in his 
 tllasticile des Corps Sulidcs objects strongly to the method
 
 300 Hydrostatics and Elementary Hydrokinetics. 
 
 applied by Navier and others in the theory of Elasticity, 
 because, in applying- the Integral Calculus to the deter- 
 mination of the action produced on a molecule of a body by 
 the neig-hbouring- molecules, they thus assume the con- 
 tinuity of matter, an assumption which Lame describes as 
 a ' hypothese absurde et completement inadmissible.' His 
 own method is a molecular one in which the existence 
 of vacant spaces between the molecules is admitted; and 
 the process of integration round a molecule is replaced by a 
 process of mere algebraic summation — which, no doubt, is 
 a much safer process, and should be adopted if it could be 
 legitimately applied. It is not, however, satisfactorily 
 applied by Lame, since he has no hesitation in assuming a 
 molecule to be wherever he wants one, and this assumption 
 is not essentially different from integration. 
 
 If the second of the above suppositions is adopted, the 
 matter surrounding a molecule, although not continuously 
 lining space with mathematical strictness^ may be assumed 
 to be practically continuous, and the method of integration 
 round a point becomes permissible as a very close approxi- 
 mation to the truth. The shapes of the molecules may 
 possibly be such as to allow of their filling space much 
 more effectively than if they were spheres. 
 
 But in adopting' this supposition when calculating the 
 forces produced on any molecule, m, by those within the 
 range, e, of molecular force, it will be necessary to imagine 
 m and any very close neighbour, m', as both divided into 
 infinitely smaller elements, of which dtn is the type for the 
 first and dvi that for the second, each of these elements 
 being now infinitely smaller than the distance between 
 them, and then assuming the force between them to be 
 given by the expression ( i ). Thus for a pair of molecules 
 so close that it is logically impossible to define anything 
 that could be called the ' distance between them ' we
 
 Molecular Forces and Capillarity. 301 
 
 must imagine a special process of integration performed 
 before we proceed to calculate the action of the more 
 distant molecules within the sphere of molecular activity. 
 
 Such a process it is, of course, quite impossible to follow 
 in detail because the form of / {r) and the shapes of mole- 
 cules are unknown ; nevertheless, on account of the 
 symmetry of arrangement of molecules round all points in 
 a homogeneous body, it is possible to represent the result of 
 such a process by a mathematical expression and to base 
 further calculation thereon. 
 
 Various forms for f{r) have been suggested, such as 
 
 A and e~"'': these are, of course, merely conjectural : 
 
 but it is conceivable that the observation of certain pheno- 
 mena measurable m the total might afford a clue to, if not 
 a necessary demonstration of, the law of this assumed 
 molecular force. 
 
 If, then, we admit the second supposition, with the 
 above notions, the first of our three suppositions becomes 
 unnecessary, and Lame's objection to the integi-ation 
 method loses its force. 
 
 In the study of the forms assumed by the surfaces of 
 liquids in contact with each other and with solid bodies it 
 is with these molecular forces that we have chiefly to deal. 
 Indeed, the curious forms of such surfaces become exjjlicable 
 on no other hypothesis than that of the existence of very 
 intense molecular forces having an extremely small range 
 of action. 
 
 Supposing that the force between two elements of matter 
 is given by the expression (i), its component along any 
 fixed line (axis of x) is 
 
 x' — X 
 
 f{r) dm dm,
 
 302 Hydrostatics and Elementary Hydrokinetics. 
 
 if the co-ordinates of dm and dm' are x and x\ so that the 
 total component force acting' on dm has for expression 
 
 dm \ f[r)dm', 
 
 if the integration is performed with reference to r, the 
 limits of r being o and e. Now, since the forces are zero 
 beyond the distance e, no error is introduced by assuming 
 ;• to extend to a; . so that such an expression is often 
 written in the form 
 
 
 dm 
 
 Some notion of the magnitude of e may be obtained 
 from experiments such as the following. Quincke covered 
 surfaces of glass with extremely thin layers of different 
 bodies, and on these layers then deposited drops of mercury 
 and other liquids. Now it will be seen presently that 
 there is a definite angle between the tangent plane to the 
 free surface of a liquid and the tangent plane to a solid 
 with which it is in contact ; this angle is constant all 
 round the curve in which the two surfaces intersect ; and it 
 matters not whether the solid is a millimetre or loo milli- 
 metres thick, the value of the angle does not alter. But if 
 the solid is, say, the millionth of a millimetre thick, the 
 angle alters. Covering the surface of glass with a layer of 
 sulphide of silver, Quincke found that there was no change 
 in the angle between the surface of the drop of mercury and 
 the plate until the thickness of the silver layer was reduced 
 
 to — g mm. ; and when the glass was coated with a layer 
 
 of iodide of silver, no change was observed until the 
 
 59 
 thickness of the layer was reduced to — ^ mm., or, say, the
 
 Molecular Forces and Capillarity. 303 
 
 wave leng-th of yellow lig"ht. These thicknesses, then, 
 indicate the order of magnitude of the diatances at which 
 molecular attractions are sensihle. 
 
 Granting" the existence of these molecular forces, it 
 follows veiy obviously that within a layer of a fluid just at 
 the surface, and of the 
 extremely small thick- 
 ness e, there is a 
 special intensity of 
 pressure which in- 
 creases in magnitude ™ _ 
 ^ Fig. 81. 
 
 as we travel from 
 
 any point P (Fig. 81) 
 
 along the normal Pb to the surface, AB, of the fluid, 
 
 towards the interior of the fluid. 
 
 For, consider a molecule of the fluid at P ; round P 
 as centre describe the sphere of molecular activity ; of this 
 only a hemisphere, abc, exists within the fluid, so that 
 the molecular forces acting on the particle P come from 
 the molecules of this hemisphere. Now it is obvious that 
 the symmetrical grouping of these molecules about the line 
 Ph results in their producing a resultant force on P inwards 
 along Pb. 
 
 Describe a surface, A' B', parallel to AB at the distance 
 e, or, Pb, from AB. 
 
 Consider now the molecular actions on a molecule Q 
 anywhere within this layer. Describe round Q the sphere, 
 ambnc, of molecular activity. Of this sphere the portion 
 ape does not contain any molecules of the fluid L, so that 
 the action at Q is due to the portion ambnc, and the result- 
 ant force will obviously be directed along the normal Qb 
 and will be less than the force at P, since there is some 
 component of force in the sense Qp. Finally, consider 
 a molecule at any point R on the inner surface A'B', and
 
 304 Hydrostatics and Elementary Hydrokinetics. 
 
 we see that since this molecule is completely surrounded 
 by attracting molecules, there is no resultant force what- 
 ever. 
 
 Now if F is the force exerted at Q per unit mass, and d/i 
 denotes an element of leng-th of the normal Qh at Q 
 measured towards b, while ct denotes the pressure intensity 
 at Q, due to the forces under consideration, we have 
 
 S-^ (^) 
 
 w being the density of the fluid. 
 
 Since as we travel alono- the normal Ph from P towards 
 ^, or from Q towards b, the force F constantly preserves the 
 sense Pb, although with diminishing value, we see that 
 
 -^ is constantly positive, that is, ot continuously increases 
 
 inwards until the surface A'B' is reached, when F vanishes 
 
 and -V- = o, i.e., ot becomes constant when we pass inwards 
 
 an 
 
 through A'B'. 
 
 Hence the intensity of pressure due to molecular forces 
 is constant throughout the interior of the fluid below A'B', 
 but it varies within the layer between AB and A^B'. 
 
 It is a matter of doubt with physicists whether we are or 
 are not entitled to assume in the case of a liquid that the 
 density within the layer contained between the surfaces AB 
 and A'B' is constant and equal to the density within the 
 main body of the liquid. M. Mathieu, following Poisson. 
 denies this constancy [T/ieorie cle la Capillar itS), but 
 arrives, by the method of Virtual Work, at results of the 
 same form as those obtained on the supposition of constant 
 density. 
 
 Whatever may be the nature of the molecular forces, 
 at any point close to the surface of separation of a liquid
 
 Molecular Forces mid Capillarity. 305 
 
 from anodicr niedimn. we can r(.'i)reseiit the iiiaj,'-nit.ii(]e of 
 the resultant molecuhir Ibree of the lif|uid on a molecule 
 /" hv the expression /-/ ^ 
 
 whore o) is the area of that part of the surface of the s[)here 
 of molecular activity which exists round vi within the 
 liipiid ; and this force vanishes when w = 41:^'^ Or we 
 mio-ht rei)resent this resultant force (alon-,'- the normal 
 to the surface) by /w \ 
 
 where z is the distance of the molecule m from the surface ; 
 and the force = o when c = f . Of course the form of thj 
 function /'' is unknown, but it is the same at all points 
 which are at the same normal distance from the surface, 
 and this fact is sufficient for the purjione of calcula- 
 tion. 
 
 69. Calculation of Molecular Pressure. Let AB (Fi..-. 
 «2) be the bounding surface of a liquid, P any point on the 
 surface, TN the 
 tani^-ent plane at 
 1\ A'B' the surface 
 parallel to. /YAvith- 
 in the liipiid at the 
 depth e ; take an 
 infinitely small ele- 
 ment, (T, of area at 
 P, and on the con- 
 lour of this area 
 
 describe a rit^ht cylinder, Pli, extending indefinitely into 
 the liquid. Consider now the action of molecular forces 
 only on the iicpiid contained within this cylinder PIi. 
 
 If •cr is the intensity of molecular ju-essure at /i, and the 
 cylinder is terminated at 7/ by a normal section, ct(t is the 
 pressure exerted on this section ; then, for the equilibrium 
 
 \ 
 
 A" 
 
 Fig. 82.
 
 3o6 Hydrostatics and Elementary Hydrokinctics. 
 
 of the fluid in the cylinder we see that oto- must be equal to 
 the integral of the molecular attraction exerted by the 
 whole mass of the fluid on the portion of fluid contained in 
 this slender canal. Now below the point P' there is no 
 chang-e in the molecular pressui'e, and there is no molecular 
 force exerted on the elements of liquid ; hence we mig'ht 
 have taken the slender canal as reaching* only to P'. 
 
 Let Q be any point in the canal between P and P' 
 and let dm be an element of mass at Q. We shall calculate 
 the attraction produced by the whole body of liquid below 
 AB in the direction QP, which we know to be the direction 
 of the resultant molecular attraction. 
 
 Now the total action on the canal FV (or PP) can be 
 calculated on the supposition that the liquid extends up to 
 the tangent plane TM, and then deducting- the attraction, 
 in the sense QP, which is due to the meniscus, ABMT. 
 of liquid thus added. The attraction of this added meniscus 
 is obviously in the sense QP, so that this must be added to 
 the attraction of the liquid terminated by TM. Let the 
 attraction of this fictitiously completed liquid on the canal 
 PP' be denoted by Ka where A" is obviously the same at 
 all points on the bounding surface AB. 
 
 Let the plane of the figure be a normal section of the 
 surface AB making the angle with the principal section 
 at P whose radius of curvature is P^, and let the radius of 
 curvature of the other principal section at P he R.^. 
 
 Let any point, iA', be taken on the tangent T3I; let 
 ]\'P = cc, and on the element of area xdxdd at iV" construct 
 the small cylinder I\L terminated by AB. 
 
 If IV is the mass per unit volume of the liquid, the mass 
 of this cylinder is jS\L . loxdxdd ; and if its distance, NQ, 
 from Q is /, the molecular force which it produces on 
 
 dmQ.tqis wdmxdxdef{;r).NL (i)
 
 Molecular Forces and Capillarity. ^Pl 
 
 Now if p is the radins of curvature of the section Ali^ 
 we have 2 
 
 NLz= — , nearlv ; 
 
 2p 
 
 and if PQ = z, the component of the force (i) along QP is 
 obtained by multiplying- it by - ; hence this component is 
 
 , zx^dxdd ... . . 
 
 warn fir) (2) 
 
 2rp '' ^ ' ' 
 
 Integrate this with respect to Q, keeping x and ;• con- 
 stant. 
 
 Now (Salmon's Geometry of Three Dimensions, Chap. XI) 
 we know that ^ _ ^^ sin^g 
 
 and the integration in from o to 2 tt makes (2) become 
 
 But since ;~ = a;- + ^^, xdx = rdr, and (3) becomes 
 
 Let 
 
 J{r)dr = -dct>{r\ (5) 
 
 then since f{r) rapidly diminishes with an increase of r, 
 </; (r) is a positive quantity. 
 
 The resultant action of the meniscus on dm is obtained 
 by integrating (4) from r =. z to r = c, or to r=ac. 
 Hence the rcsultiint is 
 
 X 3
 
 3o8 Hydrostatics and Elementary Hydrokinetics. 
 Now (30 ) = o, and 
 
 j-.r^d(f){r) = —r^(f){r) + 2 r(t>{r)dr, 
 
 (00 /"X> 
 
 -r^d(f>{r)= z^(p{z) + 2 r(\>{r)dr, 
 
 therefore (6) becomes 
 
 I I r** 
 
 t:w{-^ + ^)zdmj r4>{r)dr. ... (7) 
 
 Again, let 
 
 r(})(r)dr = -dx}j{r), (8) 
 
 where yjr (r) is ob^^ously positive ; therefore, since \f/ ( x) 
 is evidently zero, (7) becomes 
 
 '^'^{jf -^R')^'^^^)'^^ • • • • (9) 
 Now put dm at Q, equal to wadz, and (9) becomes 
 
 'nw^(T{^ + ^).zylr{z)dz,. . . . (lo) 
 
 and the integral of this from r = o to ^ = PP', or = oc , 
 is the total action of the meniscus on the canal PR. 
 
 (00 
 z \lf (z) dz, this action 
 
 IS 
 
 and hence equating cto- to the sum of (11) and the force 
 Kt, we have 
 
 ZT = K+TTw''H(^+ ^), . . . .(12) 
 which is the form obtained by Laplace for the molecular
 
 Molecular Forces and Capillarity. 309 
 
 pressure intensity at all points in thoIi(|ui(l below P". (See 
 the Mt'caniqne Celeste, Supplement to book X.) 
 
 We have supposed the surface of the liquid at J' Uv 
 be concave towards the liquid. If it is convex, 
 
 •CO- 
 
 = A-_..= //(i- + ^^) (,3) 
 
 Hence we have the following- obvious consequences. 
 
 1. If a licjuid is acted upon by molecular forces only 
 
 (no external forces) the quantity tt + tt must be constant 
 
 at all points of its bounding- surface ; for, otherwise we 
 should obtain conflicting values for the intensity of mole- 
 cular pressure at one and the same point in the body of the 
 liquid. 
 
 2. The molecular pressure at a point strictly oti its 
 bounding surface is zero ; for on the portion of liquid 
 contained within the canal FP' and included between F 
 and a point infinitely close to F the resultant force exerted 
 by the fluid is infinitely small (since the mass contained is 
 infinitely small). 
 
 3. The value of the intensity of molecular pressure at 
 a point within the body of a liquid is not a constant 
 related solely to its substance ; it depends on the cur- 
 vature of its bounding- surface. If this surface is plane, the 
 intensity is A'. 
 
 4. If (owing, as we shall see, to the action of external 
 forces) it happens that some parts of the bounding sm'face 
 are plane, others are curved and have their concave sides 
 turned towards the liquid, and others again have their 
 convex sides t^nvards the licjuid. the intensity of molecular 
 pressure just below the second kind of ])oints is i/reafer and 
 below the third less than it is at the plane portions. 
 
 . i
 
 3IO Hydrostatics and Elementary Hydrok luetics. 
 
 M 
 
 Fig. 83. 
 
 (We shall presently see how this is verified in the rise or 
 fall of liquid in capillary tubes when gravity is the ex- 
 ternal force.) 
 
 The constant K can be easily expressed in terms of the 
 function \^ thus : let CD, Fig. 83, represent the plane 
 
 surface of a liquid, the 
 
 c Q A p ^iq'^i^ lyi^g" ^elpw CD; 
 
 at any point A on the 
 
 surface take an infinitely 
 
 small element of area, o-, 
 
 and describe on it a normal 
 
 cylinder or canal, AMB, 
 
 extending- into the liquid 
 
 indefinitely ; then Ka is equal to the whole force produced 
 
 on the liquid within this canal by the whole body of liquid 
 
 below CD. 
 
 Take any point Q on CD ; let AQ = x, and take the 
 circular strip ZTixdx round A ; along this strip describe 
 normal canals (represented by QP) extending indefinitely 
 into the liquid ; take any point, P, in one of these canals ; 
 let PQ = ^ ; take any point, 31, in the canal at A, and let 
 A3I = z. Then if * = area of normal section of the canal 
 QP, the element of mass at P is swdy, and its action on a 
 mass m at Jf is m8wdi/f[r), where r = MP ; the component 
 
 of this along MR is mswdyfir) 
 
 y-z 
 
 or ms wf{r) dr ; 
 
 therefore taking the points P at a constant distance y 
 
 from CD all round the strip airxdx, we see that their 
 
 action on 7« is 7 ^/ \ 7 
 
 2'nwmxdx .J \r) dr. 
 
 Integrating this from r = MQ to r = 00, we have 
 (putting 3IQ = rj) 
 
 2-nivmxdx . 0(^i). 
 
 Now we must integrate with respect to x from o to qc ,
 
 Molecular Forces and Capillarity. 311 
 
 and observe that xdx = rj dr-y, so tliat the limits of i\ are 
 MA (or z) and 00 . Thus wc get 
 
 z 
 
 i.e., 2TTwm\l/ (z), 
 
 which is therefore the action of the whole mass of liquid on 
 the particle at M. 
 
 Also m ■=■ wcrdz ; therefore the action on the canal AR is 
 
 (00 
 
 This constant A' is called by Lord Raylcigh the intrinsic 
 pressure of the liquid, Philosophical Magazine., October, 
 1890. 
 
 70. Pressure on Immersed Area. Suppose a vessel to 
 contain a heavy homogeneous liquid of specific weight w, 
 and let P be a point in the liquid at a depth z below 
 the plane portion of the free surface. Then, as has been 
 shown in the earlier portion of this work, the intensity 
 of pressure at P due to gravitation is icz ; and, as has just 
 been proved, the intensity of pressure at P due to molecular 
 forces is A, the intrinsic pressure. Hence the total inten- 
 sity of pressure is ^^^ _^ j^ 
 
 Now it is known that A' is enormousl}'^ great : Young 
 estimated its value for water at 230CO atmospheres, while 
 Lord Rayleigh {Phil. Mag., Dec, 1890) mentions, with 
 ap])roval, a hypothesis of Dupre which leads to the value 
 i^ooo atmos2)heres for water. The hyi)otlu'sis is that the 
 value of A is deducible from the dynamical equivalent
 
 312 Hydrostatics and Elementary Hydrokinetics. 
 
 of the latent heat of water ; that evaporation may he 
 reg-arded as a jDroeess in which the cohesive forces of the 
 liquid are overcome. Now the heat rendered latent in the 
 evaporation of one gramme of water = 600 g-ramme-degrees 
 (about), or 600 x4a X lO*' ergs, and i atmosphere = 10'' 
 dynes per square centimetre ; hence K = 25000 atmo- 
 spheres (about). 
 
 If this is so, the question must naturally present itself 
 to the student : what becomes of our ordinary expressions 
 for the liquid pressure exerted on one side of an immersed 
 plane area? Instead of being merely Azw^ must it not be 
 very vastly greater — in fact A [ziv-^-K)? And moreover 
 it should always act practically at the centre of gravity 
 of the area. 
 
 We shall see, however, by considering closely the nature 
 of molecular forces, that this large pressm*e does not 
 influence in any way the value of the pressure exerted 
 by a liquid on the surface of an immersed body. Let 
 us revert to Fig. 82 and consider the result arrived at 
 in Art. 69. This result may be stated thus : at all points 
 on the surface which terminates a liquid — whether this be 
 a free surface or a surface of contact with any foreign body 
 — there is a resultant force intensity due to molecular 
 actions ; this diminishes rapidly as we travel inwards along 
 the normal to the surface, and vanishes after a certain 
 depth has been reached. 
 
 If we consider a slender normal canal of any length, FJ^ 
 (Fig. 82) at the boundary of a liquid, this canal will 
 experience/row? the surrounding liquid itself Sk resultant force 
 acting inwards along its axis PR ; this force is due to 
 molecular actions and the imperfect surrounding of points 
 near the liquid boundary (as explained in Art. 69), and its 
 effects are felt along only a very small length PP' (or e) of 
 'the canal. If the boundary of the liquid at P is plane, and
 
 M 
 
 Molecular Forces and Capillarity. 313 
 
 the canal has a cross-section a, this resultant inward mole- 
 cular force on the canal is Kcr. 
 
 Now let Fig-. 84 represent a vertical plane, AB, immersed 
 in a liquid having- a jjortion, at least, of its surface, LM, 
 horizontal, and let us con- 
 sider the pressure exerted y_ 
 per unit area on this plane 
 AB at P at the left-hand 
 side. At P take an infini- 
 tesimal element of area, (t, 
 and on it describe a hori- 
 zontal canal of any length Pi„ g, 
 PQ, closed by a vertical area 
 
 at Q,. Consider the equilibrium of the liquid in this 
 canal. 
 
 Now since AB is a foreign body, there is a termination 
 to the liquid along^ the surface AB, and hence there will be 
 resultant molecular force exerted by the liquid at points on 
 and very near AB. 'Hence if along the canal PQ we take 
 the length PP' = e, the liquid in PQ experiences a result- 
 ant molecular force from the sun-ounding liquid, of magni- 
 tude A . (T, this force acting from P towards Q and being 
 confined to the length PP\ In addition, the solid plane 
 AB exerts a certain attraction, a . <t, on the liquid in the 
 canal, together with a certain pressure, q ^o-, on it. Finally, 
 at Q the canal experiences the pressure [tvz + A") o- from the 
 liquid. Hence we have 
 
 {wz + K) <T — Kir + {q — a) a, 
 
 .-. q—a = xrz, 
 
 which shows that A' disappears. Now the action of the 
 canal on the jiiane at P is exactly {q — o.)(T in the sense 
 QP, and this action is that which is dcsciiljcd in ordinary 
 language as the pressure on the plane at P.
 
 314 Hydrostatics and Elementary Hydrokinetics. 
 
 If the immersed plane is inclined, as at CI), the resultant 
 action of the lic[uid on the plane at P on the element of 
 surface o- is seen in the same way to hawza, by considering 
 the equilibrium of a canal PQ, normal to the plane, PQ 
 being equal to 6, the radius of molecular activity. 
 
 Laplace is somewhat obscure on the subject of the action 
 between a liquid and an immersed plane (see Supplement 
 to Book X, Mecanique Celeste, p. 41). Thus he says : the 
 action experienced by the liquid in the canal PQ is equal, !», to 
 the action of the fluid on this canal, and this action is equal to 
 A' ; 2«, to the action of tlie plane on the canal ; 'but this action 
 is destroyed by the attraction of the fluid on the plane, and 
 there cannot result from it in the plane any tendency to move ; 
 for, in considering only reciprocal attractions, the fluid and the 
 plane would be at rest, action being equal and opposite to 
 reaction ; these attractions can produce only an adherence of the 
 plane to the fluid, and Ave can here make abstraction of them.' 
 He is considering the action experienced by the canal at the 
 extremity P where it touches the plane. But, in considering 
 the forces exerted on the fluid by the plane, it does not seem 
 allowable to balance any force exerted by 'the plane on the canal 
 by an opposite force produced on the flane by the fluid. 
 
 According to the view which we have taken, the action which 
 is commonly called the fluid pressure on the plane is, in reality, 
 a difference action — the diflTerence between a pressure proper 
 and a molecular attraction between the fluid and the plane. 
 
 71. Liquid in contact with a Solid. Admitting the 
 existence of molecular forces operative within infinitesimal 
 distances, the surface of a liquid near its place of contact 
 with a solid body must, in general, be curved, even 
 when gravity is the only external force acting throughout 
 the mass of the liquid. 
 
 For, let PAB, Fig. 85, represent the surface of a liquid 
 in contact at P with the surface PQ of a solid body. 
 
 Consider the forces acting on a molecule at P. We have 
 gravity in the vertical direction Pw ; also the molecular
 
 Molecular Forces and Capillarity. 315 
 
 forces exerted by the solid evidently produce a resultant 
 along Pn, the normal to the solid at P ; and the molecular 
 forces of the fluid molecules adjacent to P produce a 
 resultant, Pf, acting- somewhere between the tangent 
 plane to the liquid surface at P and the surface of the 
 solid. 
 
 Now in all cases the resultant force, due to all causes, 
 exerted on a molecule of a perfect fluid at its free surface 
 must be normal to the surface. Hence the resultant of the 
 forces Pn, Piv, and P/'will 
 determine the normal to the 
 fluid surface at P; and, in 
 general, this resultant will 
 not act along Piv, so that 
 the surface of the fluid at P 
 is not, in general, horizontal. 
 
 The form of the surface 
 remote from the solid body 
 is, of course, that of a hori- 
 zontal plane ; because at such 
 
 points as A and B there are only two forces acting, viz., 
 gravity and the molecular attraction, the latter of which is 
 normal to the surface, and if the resultant of it and gravity 
 is also normal, the force of gravity must act in the normal, 
 i.e., the surface must be horizontal. 
 
 72. Application of Virtual Work. When, under the 
 action of any forces whatever, a system of particles assumes 
 a configuration of equilibrium, tliis configuration is sig- 
 nalised by the fact that if it receives, or is imagined to 
 receive, any small disturbance whatever, the total amount 
 of work done by all the forces acting on the various 
 particles is zero. 
 
 We shall now a]q)ly this principle to the equilibrium 
 of a liquid contained within an envelope ACB (Fig. 86), the
 
 3i6 Hydrostatics and Elementary Hydrokinetics. 
 
 surface of the liquid being APB, and the forces acting 
 being — 
 
 1°. molecular forces between particle and particle of 
 
 the liquid, 
 
 2°. molecular forces 
 between the envelope 
 and the liquid, 
 
 3°. Any assigned 
 system of external 
 force. 
 
 Let m, m' denote in- 
 definitely small ele- 
 ments of mass of the 
 liquid at a distance r, 
 and assume the force 
 Fig- 86. between them to be 
 
 mm f{r) (i) 
 
 Let \i denote an element of mass of the envelope, and /// 
 any element of mass of the liquid very close to \x, and 
 assume the force between m and \x. to be 
 
 m 
 
 H-F{r). 
 
 (2) 
 
 The value of r in (i) must be < e, otherwise the force 
 between the elements of mass would be zero ; and r in (2) 
 must be < (', the radius of molecular activity for the solid 
 and the fluid. 
 
 The virtual work of the force (i) is —mm'f{r)clr. Now 
 if, as in Ai't. 69, we put 
 
 j' F{r)dr = yl,{r),
 
 Molecular Forces and Capillarity. 317 
 
 the total work done by the molecular forces for any system 
 of small displacements is 
 
 i6Sw»^'0(r) + 82»^/x^/^(r), • • • • (3) 
 
 the summations extending- to all pairs of elements between 
 which the molecular force is exerted, and one-half of the 
 result of the summation relating- to pairs of liquid elements 
 being taken, because this summation will bring- in each 
 term twice. 
 
 If V is the potential, per unit mass, at any point of the 
 liquid where the element of mass dm is taken, the virtual 
 work of the external forces is 
 
 ihd 
 
 \(7dm (4) 
 
 Hence the equation of virtual work for any system of 
 displacements of the liquid elements is 
 
 8 [/ Vdm + ^2 mm' <)b (/•) + 2»ijLt\/^ (r)] = o. . . (5) 
 
 It will be necessary, therefore, to calculate the functions 
 
 2 mm' <p (r) and 2miJt.^ (;•). 
 
 Since if we take any one element m of the liquid and 
 perform the summation I,m'(l)(r) round it, the process 
 is confined -^"ithin the sphere of radius e having- m for 
 centre, we may obviously put 
 
 2m'cf>{r) = L, (6) 
 
 L being- a constant throughout the whole of the fluid 
 contained in the vessel and bounded — not by the surface 
 AP£ but — by the parallel surface A'B' (see Fig-. 82) which 
 is at the constant distance e below APB, and also bv 
 a surface inside the fluid parallel to that of the vessel 
 at a distance e from the surface of the vessel ; for each 
 element m within this space is completely surrounded by a
 
 3t8 Hydrostatics and Elementary Hydrokinetics. 
 
 liquid sphere of radius e, while the liquid elements between 
 the surfaces AB and A'B\ and between the vessel and the 
 second surface named are not completely surrounded. Or. 
 if we please, we may imag-ine the summation L to extend 
 up to the bounding surface AB and to that of the vessel, 
 and subtract a summation relating* to a fictitious layer. 
 A'^B" , above AB of constant thickness e, included between 
 AB and A'B^' (^ig"- ^7)5 ^^d ^ fictitious layer outside the 
 surface of contact with the vessel, also of thickness e. 
 
 Hence if M is the whole mass of the liquid, the summa- 
 tion can be expressed in the form 
 
 M.L — crmm'^{r), (7) 
 
 in which o- denotes a summation confined within the super- 
 ficial layer, which is everywhere of the constant thickness 
 e, and which embraces the free surface and the surface of 
 contact of the liquid with the vessel. 
 
 As regards the summation 2 »«/xv|^ (r). it is obviously 
 confined between two surfaces each of which is parallel 
 to the surface, ACB, one inside the liquid and the other 
 inside the solid envelope, the distance between these 
 surfaces being 2 e'. 
 
 Hence equation (5) becomes 
 
 h{fVdm-^n- mm' <p (r) + '2mixy\f (;•)] = 0. . («) 
 
 Now we can easily see that the summation a is propor- 
 tional to the sum of the areas of the surface AB and that 
 of contact with the vessel ; for, if we draw any surface, 
 X7, Fig. 87, parallel to AB and A"B" within the fictitious 
 surface layer above indicated, and at any point Q on XY 
 take the element dm of mass, describing round Q a sphere 
 of radius f, the summation dm . am' cf) (r) will extend to the 
 volume of the sphere included between AB and A"B" ; and 
 if ^ is the normal distance, Qp, of Q from AB, the summa-.
 
 Molecular Forces and Capillarity. 319 
 
 tion 0- m' f/) (r) can obviously be written w n [z) ■ also if dS 
 is a small clement of area of AT at Q, we can take 
 
 dm = wdSdz, 
 where w is the mass 
 per unit volume of 
 the fluid ; hence we 
 have for this element 
 of the fluid the term 
 
 io^dS.U{z)dz. 
 
 Now we can make 
 a summation from ptoq along- a cylinder whose cross-section 
 is eveiywhere dS if the radii of curvature of the surfaces 
 AB, XY, A"B" are infinitely greater than e. The result 
 of this summation is 
 
 Fig. 87. 
 
 iv^dsf'n{z)dz, 
 Jo 
 
 the definite integral being- the same at all points, p, of AB. 
 If the definite integral is denoted by A, and we then sum 
 the results all over AB, we have AwS, where S is the area 
 of AB ; and similarly for the part of a which extends over 
 the surface of the vessel. 
 
 In the same way it is obvious that the term I,n/fx\j/(r) is 
 proportional to the ja-oduct of the densities of the envelope 
 and the liquid and to the area, f2, of their surface of contact. 
 We may therefore write (5) in the form 
 
 b^/rd,,-~.(s+a) + \.nl^ = o. . . . (^j 
 
 where i and A are constants which dej)ond on the densities 
 of the liquid and solid and on the laws of intermolecular 
 action. 
 
 A result of the same form will obviouslv hold if the
 
 320 Hydrostatics and Elementary Hydrokinetics. 
 
 density of the liquid varies both at the surface AB and at 
 the surface of contact with the envelope, provided that the 
 thickness of the stratum of variable density near AB 
 is everywhere the same, and the same at all points along 
 each surface, XT, parallel to AB ; and similarly for the 
 stratum near the envelope. 
 
 Hence, then, the work done by the molecular forces for 
 any imagined displacement is entirely superficial, and its 
 two parts are proportional to the small increments of the 
 area of the surface AB of the liquid and the surface of 
 contact with the envelope. 
 
 We shall now calculate 5 8 and 6 12. 
 
 In Fig. 86 the new surface, A'L'rB'N', of the fluid 
 (resulting from the imagined small disturbance of the fluid) 
 can be considered as consisting of two parts : firstly, the 
 portion bounded by the curve ci^r... which is formed 
 by the feet of the normals to this new surface or to the old 
 one (since they differ infinitely little in position) drawn at 
 all the points A, L, I, B, ... of the contour ALIBN ; and 
 secondly, the small strip included between the contour 
 A^L'B'N' and the curve cihr ... \ so that S-S is the area of 
 this strip plus the excess of the first of these portions over 
 the area of the old surface of the fluid. Also 8X2 is repre- 
 sented in the figure by the surface BILTtVB'N'SB minus 
 the surface ^i^i'^'-S'iA"^, each of these lying on the interior 
 of the vessel. 
 
 A simple geometrical investigation of the first part 
 aihS is as follows: at any point P (Fig. 86) on the old 
 surface of the fluid draw the two principal normal sections, 
 Pq, PJ (Fig. 88) of the surface, and the normal, PC, C^, to 
 the surface at P ; take the element of area, P QFJ, deter- 
 mined by the elements PJ = ds^ and PQ = ds.^ ; this is an 
 infinitesimal rectangle since dsi and ds^ are at right angles. 
 
 Let Cj and C^ be the centres of curvature of the principal
 
 Molecular Forces and Capillarity. 
 
 321 
 
 sections-', and PC, = 7»',,PC = Ji.^, the radii of curvature of 
 these sections. 
 
 Produce (lie normals to the surface 
 at the j)uints VQIJ to meet the new 
 surface of the fluid in P'Q't'J\ and 
 denote PP' by hn. Then we determine 
 the small rectangular area P'Q^F'J' on 
 the new surface, and the excess of this 
 above PQFJ when integrated over the 
 whole of the old surface is the first part of 
 hS. The figure assumes that the concave 
 side of the surface is turned towards the 
 liquid. 
 
 Jow 
 
 hn^ 
 
 FJ'^ (. + °i) PJ, 
 
 A- 
 
 P'Q'= (1 + ^) PQ 
 
 therefore if ^/^= area PQFJ, and dS+bdS= area P'Q'F'J', 
 we have 
 
 bdS={l +j-')^nclS. 
 
 Hence the first part of 8 'S is 
 
 .7^+ 7):,) «'"'«• ('°) 
 
 /(: 
 
 To find the second part of 8*9, i. e., the sum of all such 
 elements as iFB'b, let r/w be the clement of area Il'P'Ji of 
 the interior of the vessel, and let 6 be the angle between 
 the tangent plane to the surface of the liquid at B 
 and the tangent plane to the surface of the vessel at B, 
 i.e., the angle between the plane iVB'h and the plane 
 IVB'B; then 
 
 area il' B'b = qo^ 6 . dia (11) 
 
 Y
 
 322 Hydrostatics and Elementary Hydrokinetics. 
 
 Hence the second part of 8 iS is 
 
 f cos 6. (lea (12) 
 
 taken all round the curve of contact of the fluid and the 
 vessel. 
 
 Then, we have 
 
 bS = f(^ + ^)bndS + fcosdda>; . . (13) 
 and (9) becomes 
 
 + - I [%X — k — kcosd)(loy, (14) 
 
 since 8X2 is obviously the integral of all such elements as 
 II'B'B, i. e., f(U. 
 
 Now observe, however, that (14) is the equation of 
 Virtual Work irrespective of the condition that the volume 
 of the fluid remains the same after displacement as before. 
 The excess of the new volume over the old is obviously the 
 sum of such prismatic elements of volume as that contained 
 between the areas PQFJ and P'Q'rj' (Fig. 88) whose 
 expression is bndS, added to the sum all round the curve 
 ARLIBNA of such wedge elements as lihBB'V. The area 
 of this wedge is \ hn^ . cos Q dio, if hn^ denotes li, the normal 
 distance between the new and the old surface of the fluid at 
 any contour point, 7; and hence the sum of the wedges 
 will add nothing to the contour integral 
 
 /[iX — k — k cos 6) do) 
 
 in (14), since each element of this sum is an infinitesimal 
 compared with dot.
 
 Molecular Forces and Capillarity. 323 
 
 Ilcncc, the whole vohime of fluid being- constant, 
 
 fhn,lS=o (15) 
 
 We know from the )irinci))k^s of the Lagrano'ian method 
 [Sfnf'icft^ vol. ii., chap, xv.) that the condition of unchanf^-ed 
 volume combined with the princiido of Virtual AVork 
 is expresised by multipl\ ing- the left-hand side of (15) by an 
 arbitrary constant, //, and adding- it to the left-hand side of 
 (14). Hence, then, the complete equation is 
 
 + - {2\ — & — k COSd)(l(i} = O. (16) 
 
 We may finally simjilify the term 8./ Ffhn. It means 
 simply the variation of the potential of the external forces 
 due to changed configuration of the liquid ; and this varia- 
 tion is due merely to the two wedges BTLHrB'N'SB and 
 AlU/J'SNA, being- positive for one and negative for the 
 other. The type of the variation is the variation for the 
 element of mass contained in the small prism PP', Fig. 86, 
 that is whnclS ; so that if V is the potential of the external 
 forces (per unit mass) at any point P on the surface of the 
 liquid, the work of these forces for any small change of 
 configuration is .r/FbudS, (17) 
 
 and therefore (16) finally becomes 
 
 + - / (2 A - i-— k cos 0) (lo) = 0. (18) 
 The first integral is one extended over the surface of the 
 
 Y 2
 
 324 Hydrostatics and Elementary Hydrokinctics. 
 
 liquid, and the second is one relating only to its contour, 
 i.e., its bounding curve ALIBSNA. 
 
 Now, owing to the perfectly arbitrary displacement 
 of every point on the surface, each element of the first 
 integral must vanish, and hence at every point of the 
 surface we have 
 
 ''^*''--Si,*i) = ° <'9) 
 
 which is the equation of the surface. 
 
 Every element, also, of the contour integral must vanish, 
 and hence at all points of contact of the surface of the 
 liquid with the vessel 
 
 2k — /a , . 
 
 cos e = —7 — 5 (20) 
 
 which shows that i/ie liquid surface is inclined at the same 
 angle to the surface of the vessel all round. The angle 6 
 is called the angle of contact of the liquid and the solid, 
 which we shall definitely suppose to be the angle contained 
 between the normal to the liquid surface drawn into the 
 substance of the liquid and the normal to the solid drawn 
 into the substance of the solid. 
 
 If A > k, the angle of contact is imaginary^ and equih- 
 brium of the liquid in the vessel is impossible. 
 
 If the convex side of the surface is turned towards 
 the liquid, we shall have 
 
 Fj'= (. - ^;) pj, 
 
 ^'"■=-(i. + yT>'*'
 
 Molecular Forces and Capillarity. 325 
 
 and (19) is replaced by 
 
 If the density of the liquid is not constant (owinjo^ to the 
 variable molecular pressure) in the layers near the surface, 
 it will be the same at all jioints on a surface j^arallel to AB 
 (Fig-. 82) at a distance <e from AB\ and hence it is 
 obvious that the virtual work of the molecular forces 
 for any small displacement will still be proportional to the 
 variation, hS, of area of the surface, but the value of 
 the constant k, will not be the same as on the supposition 
 of constant density. The equation of Virtual "Work, will, 
 then, be still of the form (9), and the results (19) and (20) 
 will still hold. 
 
 If above the surface AB there is another fluid, virtual 
 work of its external and molecular forces will g-ive terms of 
 the same form as before, as will be shown in a subsequent 
 article. 
 
 If at each point of the free surface of a liquid there is an 
 external pressure whose intensity at a typical point on the 
 surfiiee is />„, the virtual work of this pressure must be 
 l)rouo-ht into equation (9) or (16). This virtual work 
 is obviously —f p^hndS, so that the equation (19) of the 
 fn-o surf:K'(- becomes 
 
 «,r-^„+/,_|(^^ + ^_) = o. . . . (22) 
 
 From (20) we see that the anf^-le of contact of a liquid 
 
 with a solid will be < " if A > ^, i.e., the surface of the 
 
 li([iiid will be convex towards the li(piid at the place of 
 contact. If the law of attraction between the liquid 
 elements themselves is the same as that of attraction
 
 326 Hydrostatics and Elementary Hydrokinetics. 
 
 between a liquid element and an element of the solid 
 in contact with it, (/) and v// in (5) differ only by constant 
 multipliers ; and we can state the last result thus : if the 
 attraction of the solid on the fluid is greater than half the 
 attraction of the fluid on itself, the surface of the fluid will 
 at the curve of contact be convex towards the fluid ; and if 
 
 A < -, will be > -, i. e., the surface of the fluid will be 
 
 2 2 
 
 concave towards the fluid. We shall subsequently see that 
 in the case in which a capillary tube dips into a liquid 
 which is under the action of gravity, the liquid must rise 
 in the tube in the first case, and fall in the second. These 
 results were first enunciated by Clairaut. 
 
 The experimental determination of the ang-le of contact 
 of a liquid and a solid has been made by means of the 
 
 measurement of a 
 large drop, OCyl), 
 Fig-. 89, of the liquid 
 placed on a hori- 
 zontal plane, AB^ 
 made of the solid. 
 If the drop is a very 
 large one, it is virtually a plane surface at its highest 
 point 0. Suppose the figure to represent a vertical section. 
 Then at any point P the two principal sections are the 
 meridian curve PO and the section made by a plane per- 
 pendicular to the plane of the paper through the normal to 
 the curve at P. The curvature of this section may be 
 neglected in the case of a large drop ; and if p is the radius 
 of curvature of the meridian at P, we have from (12) of 
 Art. 69 the intensity of molecular pressure at P equal to 
 
 T 
 K + —, where T is a constant. At the intensity of 
 
 P 
 
 molecular pressure is A', and if the depth, Fm, of P below 
 
 Fig. 89.
 
 Molecular Forces and Capillarity. 327 
 
 the tan<rent Ox is //, the intensity of pressure at P is also 
 A + uy, by transmission from 0. Hence 
 
 T 
 «'^ = p (23) 
 
 If the arc OP = s and 6 is the ang-le made by the tang-ent 
 
 at P with Ox,-= -—, and sin ^ = 4 > hence (li) becomes 
 p (18 (Is ^ ^' 
 
 %cydy = T sin 6(16, 
 
 .'. ivf = 2T(i~cosd) (24) 
 
 Let the angle of contact at Q be i, let Oy =. a \ then 
 
 xca^ = 2T{\-\-Q,osi) (25) 
 
 Let CB be the equatorial section of the drop ; then 
 at C we have ^ = -, and if the depth, h, of C below 
 is measured, we have 
 
 wf? = 2T. (26) 
 
 This last g-ives T, which is called (see Art. 76) the surface 
 tension of the liquid in contact with air ; and then (25) 
 g-ives ^2 
 
 cos/ = ^^.,-i, (27) 
 
 which determines the angle of contact. 
 
 The above arrangement is suitable in the case of a drop 
 of mercury. 
 
 To find the angle of contact between water and any 
 solid body, a somewhat similar method has been emi)loy(Hl. 
 Imagine Fig. 89 to be inverted, and suppose AB to be the 
 horizontal surface of a mass of water (which then occupies 
 the lower part of the figure). Along tiiis surface fix a plane
 
 328 Hydrostatics and Elementary Hydrokinetics. 
 
 of the given substance, and under this plane insert a large 
 bubble of air, QCOD, the lowest point of the bubble 
 being- 0. 
 
 Then, exactly as before, by measuring the thickness, O1/, 
 of the bubble, and the depth of C below AB, we obtain the 
 angle of contact at Q between the water surface and that 
 of the solid (the w^ater sui-face being bounded by air). 
 If a = O1/, and b = the vertical height of C above 0, 
 we have, as before 
 
 i wb'' = T, 
 
 I wa- = T{i + cos i), 
 
 i being the angle of contact CQA. 
 
 It is very difficult, if not impossible, to find a definite 
 value of the angle of contact between a given liquid and a 
 given solid, because any contamination or alteration of 
 either surface during the experiment will affect the result. 
 Thus, the angle of contact between water and glass is often 
 said to be zero, w^iile some experimenters quote it at 26°. 
 Again, it is known that in the case of mercury and glass 
 the angle varies with the time during which they are 
 in contact : at the beginning of an experiment the'angle 
 was found to be 143° and some hours afterwards 129°. 
 
 73. Analytical investigation of general case. The 
 expression (13) of Art. 73 can be analytically deduced from 
 the general theory of the displacements of points on any 
 unclosed surface. Thus if, as in Statics, vol. ii., Art. 291, 
 we denote the components of displacement of any point 
 {x, y, z) by «, v, iv, and if at any point, P, Fig. 86, on the 
 
 surface we put —-= p^ ~ = q, e = \/i+p''^ + q'\ we know 
 
 that the change, bdS, of the infinitesimal area dS at P is 
 given by the equation
 
 Molecular Forces and Capillarity. 329 
 
 Also {Statics, Art. 283) 
 
 dw (111 (Iv , , 
 
 (ho (hi civ , . 
 
 Hence 
 
 s c /T^ Si . ON f^fi / 9\ flv ,dv (hi. 
 
 Now, by the method of integration by parts, we have 
 the double integral equal to 
 
 / ~ {(^ +C2~)i((li/—pqnclx + {\ +p'^)v(1x—pqr(ly 
 
 +p7V(fy + qwdx} 
 d I + q^ d pq / d l ■\-p^ d pq 
 
 -//I" 
 
 ^dx € dy ^ > V/^ e dx t ^ 
 
 in which, of course, the single integral is one carried along 
 the liounding curve, ALJJjV (¥i<j;. 86) of the surface, wliile 
 the (loulijc integral is one carried over the surface itself. 
 
 Dealing with the double integral in (5) first, we easily 
 find that the coefficient of Ji is 
 
 I' ^8 , . . . ^o; 
 
 where r = , ,/! = --,«= ; • 
 dx dy dt/
 
 330 Hydrostatics and Elementary Hydrokinetics. 
 But (Salmon's Geometry of Three Dimensions, Chap, XI) 
 
 the multiplier of —p in (6) is ^ + — • Similarly the 
 
 jij -"2 
 
 coefficient of v in the double integral is — ^ (^ + ^, and 
 that of w is ^ + -^ ; so that the double integral in (5) is 
 
 ^Ti ^ r) (^'^ "^ ^^ ~ ^) ^^'^ ^^- ' • • (7) 
 
 But, u, V, IV being the components of displacement of 
 
 the point P, and -^-^ the direction-cosines of the 
 
 normal to the surface at P, the projection of the displace- 
 ment of P along the normal, which we have called 8«, 
 is given by 
 
 pu + qv-w 
 bn = ax ay 
 
 m 
 
 Hence (7) is 
 
 // 
 
 = (pu + qv — m) dS. 
 
 as before found. 
 
 Dealing now with the single integral in (5), and carrying 
 it continuously round the bounding curve, we see that the 
 sign of every term in dx must be changed, as is fully 
 explained in Statics, Art. 3160. 
 
 Of course this integral is one which we may consider as 
 carried round the projection on the plane xy of the bounding 
 curve. Hence the correct form of the sing-le integral is 
 
 / 
 
 — {{\ -\-(l^)iidy ■\-pqudx — [\ +jj^) vdx —pq v dy 
 
 ■\-jpwdy — qwdx], (8)
 
 Molecular Forces and Capillarity. 331 
 
 in which u, r, w are components of displacement of the 
 points on the surface, 12, of the solid body where it is 
 
 intersected by the surface of the liquid ; so that if ' \ — 
 
 are the direction-cosines of the normal to 12 at any point, 1, 
 we have now ^^ ^ ^y^^ ^ ^/^_ 
 
 Also, the projection of the element r/w of area of the 
 
 surfiice of the vessel at / on the plane of xi/ is udi/ — vdx, 
 
 so that , / / 7 7 \ 
 
 f/ot) = e [udj/ — V(lx). 
 
 And since dx, dy, dz are proportional to the direction- 
 cosines of the element IB of the bounding curve which is 
 at rig"ht ang-les both to the normal to the liquid and 
 to the normal to the solid, we have^J</i» + jfi(y — </;: = o and 
 
 p'dx + (/d)/ — dz = o, from which 
 
 (p—j/) dx + (q-q) dy = o. 
 
 Hence the terms multiplying — in (8) are equivalent to 
 
 (j)p^ -¥ ?/+ i) {udy — vdx), i. e., to € cos 6 do) ; so that (8) is 
 ^'"^V^y fcosddco, 
 
 as was found in (12), Art. 72. 
 
 (The alteration of signs in the terms of the single 
 integral in (5) which is rendered necessary by the carrying 
 of the integral by continuous motion in the same sense 
 round the bounding curve ALBN, or its projection on the 
 plane xy, is a circumstance which, perhaps, the student 
 would be very likely to overlook.) 
 
 74. Liquid under the action of Gravity. In the 
 particular case in which gravity is the only external force 
 acting on a liquid which has air or any other gas above its 
 surface, if z is the height of any point on the surface above
 
 332 Hydrostatics and Elcmentayy Hydrokinctks. 
 
 a fixed horizontal plane, V z=—z, and as ?r' is neg-lig-ible 
 compared with ^o, the equation (19) of the surface of 
 the liquid is 
 
 w'^ + - (-^ + ■^) = Constant, . . . (i) 
 
 from which it follows that if there are any points on the 
 surface at which the concavity is turned towards the liquid, 
 
 i. e., ^ + -yp is positive, those points must be at a lower 
 
 level than the points at which the surface is plane ; 
 
 and points where the surface is convex towards the liquid 
 
 must be at a higher level than the plane portions. 
 
 Thus, supposing- that two cylindrical capillary tubes, 
 
 BC^ FE, immersed vertically in a given liquid, and of such 
 
 different materials that the 
 surfiace of the liquid in one 
 is concave, and in the other 
 convex, towards the liquid, if 
 L is plane from which z is 
 measured and if B, B' are the 
 radii of curvature of the sur- 
 faces of the liquid within the 
 tubes at C and F, and C C 
 the heights of these points 
 
 A^^^SB? 
 
 :C:? 
 
 fe=S>iels=g€=SH 
 
 L 
 Fig. 90. 
 
 above L, we must have 
 
 k 
 
 where z is the height of the plane portion, AB, BE, Gil, of 
 the liquid above L. 
 
 A simple experiment with water serves to illustrate the 
 result that if in a continuous body of this liquid there 
 is a part of the surface plane and another convex towards the 
 liquid, this latter must be at a higher level than the former.
 
 Molecular Forces and Capillarity. 
 
 333 
 
 \ 
 
 f\i> 
 
 ft- 
 
 Fig. 91. 
 
 Let a hirf^v t^lass vessel be eonnoctod with a ca])il]arv tube. 
 
 /, ri<i;-. 91, and let water be poured ounlinuously into tlic 
 
 larye l)nineh. A stage will be reached at which the water 
 
 in I will just reach the to]) of this tube, and then the 
 
 surface of the water is aeO. If the •'[•lass is quite clean, this 
 
 surface will have its tangent planes 
 
 vertical round the rim a/j ; and the 
 
 level of the water in the large branch 
 
 will be at C, which is lower than a/j. 
 
 As water continues to be poured in, 
 
 the surface in I will become quite 
 
 horizontal and represented by the 
 
 right line ab, and the surface in the 
 
 other branch w ill be AB, which is at 
 
 the same level as ab. Continuing to 
 
 pour in water, the surface at the top 
 
 of f will become concave downwards, as represented by adb, 
 
 and then the level in the other branch is at D, which is 
 
 higher than d. 
 
 A side-figure at T shows how the surface of the liquid 
 can be adb. The horizontal edges at the top of t are of 
 appreciable breadth, and when the water lises above the 
 line ab, the surface of the glass is the horizontal rim of the 
 tube t (and the angle of contact being 0°) the surface of the 
 water at the rim lies horizontally. 
 
 75. Rise or fall of Liquids in Capillary Tubes. It is 
 a well known fact that if a tube of ver}- small diameter is 
 plunged into a mass of liquid contained in a vessel, the 
 level of the liquid in the vessel will not, in general, be the 
 same as its level in the narrow tube. AVhat is the cause of 
 this? To say that it is ' cajallary attraction' is to use an 
 expression which is at once inaccurate and vague. To say 
 that it is molecular attraction is to use an expression which 
 is true but vague. This was evidently in the mind of
 
 334 hydrostatics and Elementary Hydrokinetics . 
 
 Laplace when he said (Supplement to Book X, p. 5) that 
 the attraction of capillary tubes has no influence on the 
 elevation or depression of the fluids which they contain, 
 except in determining- the inclination of the surface 
 of the fluid to the surface of the tube along- the curve 
 of intersection of the free surface with the tube, and 
 thereby determining the curvature of the free surface. 
 That the angle of contact does determine the curvature 
 of the surface when this surface is inside a very narrow 
 tube is obvious. 
 
 For, suppose that the angle of contact for g-lass and 
 a certain liquid is 45°, and that the liquid is contained in a 
 vertical g-lass tube one-tenth of a millimetre in diameter ; 
 then it is evident that the free surface of the liquid within 
 the tube must be very much curved, because its tangent 
 planes where it meets the tube must all be inclined at 45° 
 to the vertical, while its tangent plane at its vertex must 
 be horizontal ; and in order that such a great amount 
 of change in the direction of the tangent plane should 
 be possible, the surface must be very much curved. 
 
 Now, great curvature of surface means great intensity of 
 molecular pressure, if the surface is concave towards the 
 liquid, and small intensity if the surface is convex towards 
 the liquid (Art. 69). 
 
 Hence, owing merely to the fact that within a very 
 narrow tube, the free surface of a liquid is curved — and not 
 to any special action due to the narroicness of the tube — 
 this liquid must rise or fall within the tube below the level 
 of the plane portions in any vessel into which the narrow 
 tube dips. 
 
 Let FE, Fig. 90, be a capillar}^ tube dipping into a vessel 
 containing a liquid such that the angle of contact (as 
 
 defined in Art. 72) for the liquid and the tube is < -•
 
 Molecular Forces ami Capillarity. 335 
 
 In this case the surface is concave upwards, and therefore 
 
 k 
 the intensity of molcenlar pressure at F is K— -, where 
 
 Jt is the radius of curvature of the liquid surface at the 
 
 lowest point of the surface at F (where the two radii 72, , 
 
 Ro are evidently equal), and the liquid must rise in the 
 
 tube until the intensity of pressure due to the weig-ht of the 
 
 column FE added to this molecular intensity produces 
 
 the intensity of pressure which exists along- I)F. If p^ is 
 
 the intensity of atmospheric pressure, and EF = z, the 
 
 intensity of pressure inside the tube at the level E is 
 
 k 
 Pq-^-vz + K—,,; and the intensity of pressure along the 
 
 plane surface BE is Pq + K ; hence 
 
 ' = ^ (') 
 
 determines the height to which the liquid rises in the 
 
 tube. 
 
 Let i be the angle of contact of the liquid with the tube 
 
 and r = internal radius of tube ; then li = r sec i, very 
 
 nearlv ; hence , 
 
 kcosi , , 
 
 z = , (2) 
 
 tor ^ ' 
 
 and the weight of the liquid raised in the tube above E is 
 
 Tifkco^i (3) 
 
 Equation (2) shows that the heights to which the same 
 liquid rises in capillary tubes of the same substance are 
 inversely proportional to the diameters of the tubes. 
 
 If the tube is such that the angle of contact is > -, the 
 
 2 
 
 free surface within the tube is concave towards the liquid, 
 
 and therefore the intensity of molecular pressure is greater
 
 336 Hydrostatics and Elementary Hydrokinetics. 
 
 than at the plane surface BI,. Hence the liquid must be 
 depressed in the tube, as represented in BC, and the amount 
 of depression is calculated as above. 
 
 76. Surface Tension of a Liquid. The amount of rise 
 or fall of a liquid, under the action of g-ravity, in a capillary 
 tube is usually calculated by means of the introduction 
 of the notion of surface tension. The free surface of the 
 liquid is considered to be in a state of tension resembling 
 that of a stretched surface of indiarubber — with, however, 
 this important difference, that, whereas the tension of the 
 indiarubber surface increases if the surface is further in- 
 creased, the tension of the liquid surface remains absolutely 
 constant whether the surface expands or contracts. 
 
 Let ABCB, Fig-. 92, represent a part of the boundino- 
 surface of a liquid ; let any line QPR be traced on it ; and 
 
 along- this line draw small 
 leng-ths Qq, Pj), Br into the 
 liquid and normal to the 
 surface. Consider now the 
 action exerted over the area 
 j-jo.^ Q2. Br])q(^B by the liquid at 
 
 the rig-ht side on that at 
 
 "»' 
 
 the left. 
 
 One part of this action will consist of molecular attraction, 
 towards the right ; and if the depth of the line qpr below 
 (^BR is nearly equal to e (Art. 68) or greater than e, 
 another part of the action will consist of pressure, towards 
 the left, in the lower parts of the area QqrB. If qpr 
 instead of being at an infinitely small depth is at any finite 
 depth, the molecular attraction exerted across any of the 
 lower portions of the area QqrR is exactly balanced by the 
 molecular pressure on such portions. But if qpr is at 
 a depth vcri/ much less than e, the molecular pressure 
 (towards the left) on any part of the area QqrR is negligible,
 
 Molecular Forces and Capillarity. 337 
 
 nnd wo may consider the portion of litiuid at the ri<ifht of 
 the line QR as sinijtly exertins;' a pull or tension on the 
 portion at the left. Observe that on the line QVR itself 
 the intensity of molecular pressure is zero, and that if qpr 
 is at a distance iniinitely less than e from Q7Vi', althou<ih 
 the intensity of pressure on any portion of the area (IqrR 
 is intinitely small, the force of molecular attraction exerted 
 across this area may be ]ari»-e. The value of the pressure 
 
 (i tZT 
 
 depends on the force by a differential relation -7— = wF 
 
 (see (2) of Art. 68), where /'' is the intensity of molecular 
 force in the direction of the normal to the surface of the 
 liquid at any point, P ; and we know that at P, where 
 /•' is "greatest, ot is zero. 
 
 Hence at points of the imai^ined surface QqrR of separa- 
 tion which are infinitely near to the surface we are to 
 imao'ine the stress to be merely tension ; at points whose 
 distances from the surface become comparable with e we 
 arc to imafji-ine this molecular pull, or attraction, as accom- 
 panied liy a contrary pressure ; and at points which are 
 at and beyond the distance e from the surface, the molecular 
 pull is balanced by the molecular pressure. 
 
 Hence, however far the imag-ined surface QqrTi extends 
 into the liquid, the whole stress exerted on the liquid 
 at the left by that at the rig-ht is confined to an action 
 which terminates at a curve, qpr, at the depth e, this 
 action bein'j;' a mixed one consisting- of molecular attraction 
 and an ojjposing molecular pressure which latter grows 
 in intensity from zero at the surface to a maximum value 
 at the depth «. 
 
 At any point P on ffie mrface the amount of thi,^ streas 
 
 (which is, on the whole, a tension) per nnit length of the 
 
 curve PQ i-f called the mrface tension exerted across the curve 
 
 PQ by the liquid at the right on that at the left of PQ. 
 
 z
 
 338 Hydrostatics and Elementary Hydrokinetics. 
 
 It is obvious that the amount of the stress per unit 
 leng-th is the same across all curves draAvn at P on the 
 surfoce, and is normal to these curves. This is another 
 point of diffierence between the surface tension of a liquid 
 and the stress of an elastic membrane in general ; for, 
 in the latter the stress exerted across any curve PQ at P is 
 not, in g-eneral, normal to PQ, nor is it of constant magni- 
 tude for all curves drawn in the membrane at P. 
 
 Now, although it is obvious that, if we grant the 
 existence of molecular forces, we must admit the existence 
 of this mixed surface stress (i. e., within a layer of thickness 
 e at the surface) no such stress has explicitly presented 
 itself in our investigation, by means of Virtual Work, 
 of the conditions of equilibrium of the liquid. This fact, 
 however, involves no difliculty or contradiction ; for, in 
 taking the molecular actions exerted between all possible 
 pairs of elements of mass, we are sure of having omitted no 
 forces that act ; but in this way surface tension (which is 
 obviously a resultant, and not a simple^ action) could not 
 have specially presented itself. 
 
 Knowing now of the existence of this stress, we can see 
 why the terms in the expression for the Virtual Work 
 of the molecular forces, (9), Art. 72, consist of constants 
 multiplied by the changes of the areas ^+X2, and i2. 
 
 For, if a surface of area A is subject to a tension T which 
 is the same at all points and of constant intensity in 
 all directions round a point, the work done by the stress in 
 an increase hA of the area is 
 
 -T.hA. 
 
 A liquid contained in a vessel, or resting as a drop 
 on a table, is sometimes spoken of as having a ' skin ' 
 within which the liquid proper is contained. 
 
 A drop of water hanging from the end of a tube and
 
 Molecular Forces and Capillar ily. 339 
 
 rcadv to fall is spoken of as being* contained in an ' clastic 
 biii^-.' Of course, if the surface of the liquid is oxidised, 
 contaminated Ijv forciy-n particles of any kind, or in any 
 way rendered diilercnt from the liquid helow the surface, 
 we may, if we jdease, say that the })ure licpiid is contained 
 within a surface which is not jnire liquid ; but even such a 
 contaminated surface is radically different from an elastic 
 Itasr, for the magnitude of the tension in a stretched baiir 
 increases with the stretch of the bag-, whereas the tension 
 of the bounding- layer of the liquid does not. In no case — 
 either that of a perfectly piu'e liquid or that of a liquid 
 with an oxidised or contaminated surface — is there any 
 skin or bag. In the case of a liquid with a pure surface 
 there is no material thing; at the surface which there is not 
 everywhere else in the liquid ; and we must not imagine 
 that, because we see a drop of water hanging, a globule of 
 mercury lying on a table, or a column of water with a con- 
 cave surface standing- in a capillary tube above the level of 
 the water outside, such conditions require bags for enclosing 
 the liquids or skins by means of which to catch hold of 
 thom. AVe can assure ourselves that molecular forces, 
 with special circumstances near the surftice (owing- to 
 incomplete surrounding of molecules, &c.) will amply 
 account for all such forms of equilibrium. 
 
 The height to which a liquid rises in a capillary tube 
 mav be calculated bv the introduction of surface tension. 
 
 For, in Fig-. 93, let the tube ABB' A' have any form (not 
 necessarily cylindrical) ; let / be the length of the curve of 
 contact of the li(piid surface at BB' with the surface of the 
 tul)e, let T be the surface tension of the liquid, and / 
 the angle of contact with the tube. Then consider the 
 equiliV)rivnn of the column in the tube above the level, OCx, 
 of the plane portion outside. 
 
 We may suppose the layer of particles round the tube at 
 
 z 2
 
 340 Hydrostatics and Elementary Hydrokinctics. 
 
 B which are in actual contact with the tube as exertinsr 
 the tension T per unit leng-th of the curve I on the particles 
 just outside them ; hence these supply an upward vertical 
 force Tlcos i on the column BOO'B'. If a is the area of 
 the cross-section of the tube, and z = heig-ht of B above 
 0, the weig-ht of the liquid is ivcrz. There is the downward 
 atmospheric pressure, Pq a-, at B, and an upward pressure at 
 consisting" of ^^ a and of a molecular part, Ka, and there 
 is finally a downward molecular attraction exerted on those 
 particles in the tube which stand on the area o- at 00' and 
 are contained within the distance e from a. 
 
 ^ B'_ 
 
 i 11 ni o 
 
 Fig. 93- 
 
 Now this downward attraction is precisely equal to K(t 
 
 (Art. 69), so that this force balances the upward pressure 
 
 K(j^ and we have for the equilibrium of the contained 
 
 column mi^ ■ „. ^ 
 
 i ^cos ^ = waz. 
 
 If the tube is cylindrical, 
 
 / = 2 7rr, (7 = wr^, 
 
 2 2' cos ( 
 
 and we have 
 
 wr 
 
 -:? 
 
 as in (2), Art. 75.
 
 Molecular Forces and Capillarity. 341 
 
 It is obvious that - in the g-eneral e(j nation (9), Art. 72. 
 
 is 1\ the surface tension. 
 
 By takini^ any element of area of the curved surface 
 lil'Ji', the ])rincii)al radii of curvature of this element beini; 
 /I'j and R.^, and consideriniif the equilibrium of the vertical 
 cylinder described round the contour of this element, we 
 at once deduce (1) of Art. 74; for, if dS is the area of 
 the element, the component, alon<if its normal, of the surface 
 
 tension all round dS is T (~ -f ^-\ dS, and the vertical 
 
 component of this is T (-j- + -^^da, if da is the horizontal 
 
 ^Jfj K./ 
 
 l)rojcction of dS\ also the weig-ht of the column is tczda ; 
 
 therefore 
 
 Let the ca])illary tube be replaced by two very close 
 parallel vertical plates, AB, A'B\ 
 
 Then, considering the equilibrium of the column BOO'B' 
 of unit thickness perpendicular to the plane of the figure, 
 we have 2 ^-^^^ ■ ^ ^^,^^^^ 
 
 where d is the distance 00' between the ])lates ; hence 
 
 _ 2 7'cos i 
 wd 
 
 which shows that the liquid rises twice as high in a 
 cylindrical tube as between two parallel i)lates whose 
 distance is equal to the diameter of the tube. 
 
 The existence of surface tension in a liquid may be shown 
 experimentally in many ways, of which we select two. 
 Take a rectangle forineil ul brass strips or wires, AB, BC, 
 CI), and EY (I'ig. 94) of which the first three form one
 
 342 Hydrostatics and Elementary Hydrokinetics. 
 
 Fig. 94. 
 
 rig-id piece, while the last, TjF, is capable of sliding up and 
 down on the bars AB and BC. The space, ahcil, enclosed 
 
 by the bars being- vacant, dip 
 the system into a solution of 
 soap in water, thus forming 
 a film (represented by the 
 shading) in this space. This 
 film attaches itself to all the 
 bars ; and if the moveable bar 
 EF is not restrained by the 
 hand, it will be drawn along- 
 the others by the film until it 
 reaches BC. If the system is held in a vertical plane, 
 EF being- below BC, the former will be raised, in opposition 
 to gravity, if it is not too heavy. 
 
 As a second example, take a circular brass wire, A, 
 Fig-. 95, dip it into the soap solution, thus covering its area 
 
 on withdrawal 
 with a thin film 
 (represented by 
 the shading); then 
 form a loop of a 
 piece of thread 
 and place itg-ently 
 on the surface of 
 the film. This 
 loop is represented 
 by ab. Now per- 
 forate the film inside the loop by a pin, and we shall 
 see the loop of thread instantly drawn out into a circle, 
 c, by the contracting* film. By the formation of this 
 circle the area of the remaining film is the least that it can 
 be — a result which is necessary because, the virtual work 
 of the tension being- ThA, the static energy of the film 
 
 F 
 
 %'• 95-
 
 Molecular Forces and Capillarity. 
 
 343 
 
 is proportional to its area, and every material system which 
 is subject to o-iven conditions assumes as a conli<^uration ol' 
 stable e([uilil)rium one in which the static energy of its 
 forces is least. 
 
 The following- table, taken from Everett's VnlU and 
 PJii/xical Conf{(anf.s,^x\QS the values of a few surface tensions 
 in dynes per linear centimetre at the temperature 20'' (/ : 
 
 
 Tension of surface separating 
 the liquid from 
 
 Air 
 
 Water 
 
 Mtrcury 
 
 Water 
 
 81 
 
 
 
 418 
 
 Mercury 
 
 540 
 
 418 
 
 
 
 Bisulpliide of Carbon 
 
 32-1 
 
 41-75 
 
 37-5 
 
 Alcohol 
 
 25-5 
 
 
 399 
 
 Olive Oil 
 
 369 
 
 20-56 
 
 335 
 
 Petroleum 
 
 317 
 
 27-8 
 
 284 
 
 from which it is seen how larw-e the surface tens-ion is for 
 mercury as compared with other li(iuids. 
 
 77. Two Liquids and a Solid. To illustrate further 
 the application of the principle of Virtual Work, take the 
 case in which two licpiids, w, w' (Fig-. 96), are in contact 
 with each other over a surface A and with a solid body. 
 The li(piid lo is contained within the space represented by 
 IB DC, and the second within CBV/ A' . Let S be the 
 area of the free surface of the first, and ii the area of 
 the solid in contact with this li({uid ; and -S" and 12' be the 
 correspondin'; thin^^s for the second li(|uid. 
 
 Then, exactly as in Art. 72, the Virtual ^Vork of all
 
 
 344 Hydrostatics and Elementary Hydrokinetics. 
 
 the forces acting- will reduce to the sum of terms relatin, 
 to the bounding" surfaces alone ; and since the whole 
 bounding surface of the liquid w is jy + A + Ii, the virtual 
 work of its own molecular forces will g-ive the term 
 
 -^8(5 + A + i2); 
 
 and we see that the equation of vii'tual work is 
 
 where the term ^8 A relates to the molecular forces exerted 
 
 at the surface A between par- 
 ticles of w and particles of w' ; 
 and jj-bQ, relates to the forces 
 between the jDarticles of the 
 liquid to and the solid. 
 
 Now, denoting- by bn, bn' 
 elements of normals at points 
 on S, S' drawn outwards from 
 the liquids ; bv an element of 
 normal of A drawn outwards from 
 the liquid iv ; dm an element of 
 il at the intersection of S and £i 
 (as in Ai't. 72), dy^r an element 
 of il at the intersection of i2 and A ; da/ an element of il' 
 at the intersection of H' and S', and by 6, x, 6' the 
 angles of contact with the solid at S, A, S\ we have, 
 exactly as shown in Art. 72, 
 
 8il =fdtii-\-fdx\i, 
 bil'= f ddi —/ d\l/, 
 
 Fig. 96.
 
 Molecular Forces and Capillarity. 345 
 
 hS = f(P + J, ) ofnlS-^-fcosOdw, 
 
 86"= r(J, + -^-,) 8«V5'+/cos 6'cW, 
 
 bA= l(--\ )bvdA+/cosxil^, 
 
 d/r dm = f VoudS+f FbvdA, 
 
 hf y'dm'= f rhn'd&-f rhvdA, 
 
 where p■^, p.^ are the principal radii of curvature at any 
 point on A. 
 
 To the left-hand side of (i) must be added the terms 
 
 ///b?i dS+ h'fh u'dS' + h" fhvdA, 
 
 which are rendered necessary by the constancy of the 
 volumes of the liquids in the supposed displacement of the 
 system. 
 
 As before, the coefficients of bn, bv, bn' must each be 
 zero, as also the coefficients of the terms relating' to r^w, 
 ^u)'. and d^\i. Hence, for example, wc have the equation 
 
 :it all points of S, and similar equations at all points of S^ 
 and A ; and at all jioints of meeting- of 8 and il 
 
 - {i+eosO) — pi = o, 
 
 which proves the constancy of the angle of contact at such 
 points ; a similar result holding for S' and il', while the 
 terms relating to dyj/ give 
 
 (X: — X:' — 2 /) cos X = 2 {iJL — fj.') — ^—/:',
 
 346 Hydrostatics and Elementary Hydrokinetics. 
 
 which proves the constancy of the angle of contact between 
 the solid and the common surface of the liquids. 
 
 78. Drop of Liquid on another Liquid. Let Fig^. 97 
 represent a drop of one liquid resting- on the surface 
 of another, the area of contact being- A, the free surface of 
 the drop being 6' and that of the supporting liquid S. 
 
 If the sides of the vessel in which this liquid is contained 
 are very distant from the drop, in considering a small 
 deformation of the system and applying the equation (i), 
 Art. 77, of Virtual Work, we may neglect terms relating to 
 12 ; so that the equation is 
 
 8/ Vdm + hf Vein/-- h (*S+ A)- - h {S' + A) + /8A 
 
 3 2 
 
 =0. (I) 
 
 1^'ig- 9/ 
 
 This equation will, as has already been seen, give equa- 
 tions satisfied at all points oi S, S', A, as well as equations 
 relating to their common bounding curve. Considering 
 merely the latter for the present, we may take 
 
 --8(5 + A)--8(y + A) + /5A = o, . . (2) 
 
 2 2 
 
 in which the variations of the surfaces are only those 
 portions at their common boundary curve ; or
 
 or ag-am 
 
 Molecular Forces and Capillarity. 347 
 
 \^s+^Us'+{^:±l-ly^=.o, . . (3) 
 
 Tbs+rbs^+r'h\ = o, .... (4) 
 
 where T, T, 7"' are the surface tensions in the surfaces 
 S, S', A. 
 
 Now when any surface 2 having* for bounding* edg-e 
 a curve C receives a very small deformation whereby it 
 becomes a surface 2£' having for bounding edge a curve C, 
 the Calculus of Variations leads (see Arts. 72 and 73) to the 
 result that the variation 2' — 2 is obtained by drawing" 
 normals to 2 all round the contour C, these normals being- 
 terminated by 2' and enclosing' a surface X2 on 2', and then 
 adding- to 12 — 2 a linear integ^ral taken all along the curve 
 C, the elements of this linear integral involving- the dis- 
 placements of points on C to their new positions on C. 
 The term 12 - 2 is 
 
 fii + i)"""' 
 
 while the term given by the line-integral along- C is (8) 
 of Art. 73. 
 
 Now take the case in which the figure of the drop 
 and that of its submerged part are surfaces of revolution 
 round the axis of r, and suppose Fig'. 97 to represent the 
 section of it made l)y the plane xz. Also let the displace- 
 ment of the point P be confined to the ])lane xz, and let its 
 components be k, ir. Hence in (8) of Art. 73 we are 
 to put f/ = o, V = o, and the terms of the linear integral 
 which relate to the disj>lacement of P are 
 
 -(u+pic)</i/,
 
 348 Hydrostatics and Elementary Hydrokinetics. 
 
 where (ty relates to P and a point on the curve (a circle) 
 which is the common bounding edg-e of the surfaces 5, 5', A, 
 this circle being represented in projection on the plane 
 of the figure by the right line 0I\ AVe may, then, 
 omit dy ; and the terms in (4) given by the three surface 
 tensions are 
 
 Now u and w are quite arbitrary and independent, 
 and the coefficient of every independent displacement 
 for each point involved in the line-integral must be zero. 
 
 Hence rp nv mff 
 
 - + - + -7> = O, 
 
 Tp Tj/ T'f 
 
 But if the tangent line to S in the plane of the figure 
 makes the angle Q with the axis of a?, we have 
 
 I . p 
 = sin Q and - = cos Q ; 
 
 similarly for the tangent lines to S and A ; so that these 
 become ^^.^ Q^r^m O'+T' sin 6"= o, 
 
 2' cos 6 + r cos 6' + T" cos 6"= o, 
 
 which plainly assert that three forces, T, T', T\ supposed 
 acting along the tangents in the senses represented have no 
 residtant ; in other words, if a plane triangle is formed 
 by three lines proportional to the surface tensions, the 
 directions of the distinct surfaces of the two liquids and 
 that of their common surface of contact are parallel to the 
 sides of this triangle.
 
 Molecular Forces and Capillarity. 
 
 349 
 
 Honce equilibrium of t.lie dro]) is inipossiljlo unless each 
 surface tension is less iJian the sum of the other two. 
 
 79. Liquid under no external forces. \\ hen a mass 
 of li([ui(l is in equilihiium under its own molecular forces 
 only, its surface can assume several forms. In this case (19) 
 of Art. 72 gives as the equation of the surface 
 
 ' (.) 
 
 I I 
 
 where a is a constant length 
 
 We shall confine our attention to surfaces which are 
 of revolution, and we shall suppose the axis of revolution to 
 he taken as axis of x. 
 
 Now if at any point, P, of the revolving curve (Fig. 98) 
 or meridian, PLE, 
 
 which by revolu- H- '^ 
 
 tion round the axis 
 AB generates the 
 surface, p is the 
 radius of curvature 
 and n is the length, 
 Pn, of the normal 
 terminated by the 
 
 axis of revolution, the i)rincipal radii of curvature of the 
 surface generated are p and ;/, so that (i) becomes 
 
 Fig. 98. 
 
 I I _ I 
 
 p n a' 
 
 (2) 
 
 Now let the tangent at P make the angle with the 
 
 axis of x, and let ds be an element of arc measured 
 
 (h 
 along the curve from P towards I) ; then p = — , and 
 
 11 =. y sec ; hence (2) becomes 
 
 de cos Q 
 da y 
 
 a 
 
 (3)
 
 350 Hydrostatics and Elemcntayy Hydrokindics. 
 
 (le . fie 
 or, since ^- = sm ^^ , 
 as dij 
 
 i.^^(^cos«) = l, (4) 
 
 ••• ycos^ = -^ + /^, (5) 
 
 where li is a constant. 
 
 We may observe that if the constant — is zero, (j) g-ives 
 
 as the property of the surface of the fluid that at every 
 point the two principal radii are equal and opposite ; 
 the two principal sections have their concavities turned 
 in opposite directions. If the surface is one of revolution, 
 this property at once identifies it with the surface generated 
 by the revolution of a catenar}^ round its directrix, and the 
 surface is called a catenoid. 
 
 Before proceeding- to integrate (5), we can show that 
 all the curves satisfying- it are g-enerated by causing' conic 
 sections to roll, without sliding-, along* the axis AB : the 
 curves satisfying- (5) are the loci traced out by the foci of 
 these rolling- conies. For, if Pti = n, (5) gives 
 
 / (1 - -1) = /, (6) 
 
 Now if /» is the perpendicular from the focus of an ellipse 
 on the tangent at any point, and ;• the distance of this 
 point from the focus, we have 
 
 a and h being the semiaxes. Comparing (7) with (6), we 
 see that P in Fig. 98 is the focus of an ellipse touching AB 
 at n, the semiaxes being a and -Jiah, and this ellipse
 
 Molecular Forces and Capillarity. 351 
 
 is (licrcrorc iiivan;il)lo wliatovcr l)e the jtosition of P on the 
 mcndiiin. 'Jlio locus of P when the rolling- eonic is an 
 ellipse is called the umhdoid, and is the locus PUE actuallv 
 represented in the fi<2fure. 
 
 If the rollinq- eonie is a parabola, the locus of the focus 
 is a catenary, which ^ives by revolution the catenoid. 
 
 If the rolling- conic is a hyperbola, the locus is a curve 
 having a series of loops, and the surface which it g-enerates 
 is called a nodouL 
 
 cly 
 
 Since tan 6 = -~ i we have from (5) 
 
 fix = ± —= ■' ^.(lij . . •. . . (8) 
 
 , f + 2a 7i 
 
 = H — ■ .ay . . (Q ) 
 
 f + 2afi ^ , . 
 
 = ± , ~ .dy, (10) 
 
 by putting- q- + /3^ = 40'^ — 4a//, and a^fl- = 4a^h~ ; so that 
 a and /3 are the greatest and least values of the ordinate. 
 
 Equation (10) is best integrated by expressing y, and 
 therefore x, in terms of a variable angle ; thus, let 
 
 y^ = a^ cos^ (p + ft' sin^ (j), ( 1 1 ) 
 
 .-. y ■= a \/ 1 — /t^ sin^ </) (12) 
 
 where h- — — ^ — . This gives, if A </> = \^ \—k^ sin'^c/), 
 
 ^o; = |a A (c/,) ± ^1^^ j ./0 {i^) 
 
 When (() = o, y = a, and when (p = - , y=^ ; so that if
 
 352 Hydrosfafics and Elcincntary Hydrokinetics. 
 
 2^ and E are the points of maximum and minimum ordinate, 
 all points between them on the curve are g-iven by values of 
 
 d) between o and — • 
 
 In the common notation of elliptic integrals 
 
 therefore we have the abscissa and ordinate of every point 
 on the curve expressed in terms of the variable by 
 the equations _ ,^(^^^^ ^^^^^ (.^'^ 
 
 y =aA(0) (l6) 
 
 In the unduloid the tangent can never be parallel to the 
 
 axis of y, i. e., -y- can never be zero, so that in (lo) the sign 
 
 + in the numerator belong-s to this curve, and therefore in 
 {^i^ the signs + belong-, respectively, to the unduloid and 
 the nodoid. 
 
 In the unduloid ^-- = o when tan (^ = f— ) , or ^= Va/^, 
 (hj-' ^^^ 
 
 and this gives the point of inflexion on the curve. 
 
 If s is the length of the arc between B and any point P, 
 we have .■? = (a + /3) 0. 
 
 When a = (3, the surface generated becomes a cylinder. 
 
 When a is very slightly greater than ^3, the generating 
 curve becomes, approximately, a airve qf/tmes. 
 
 The case of a liquid unacted upon by any external forces 
 was realised by M. Plateau by inserting a drop of olive oil 
 in a mixture of water and alcohol arranged so as to have 
 the same specific gravity as the oil. By seizing this drop 
 between two wires in the shapes of any closed curves, or by 
 allowing it to form round a solid of any shape held in
 
 Molecular Forces and Capillarity. 353 
 
 the watev-ak'oliol mixture, we can (il)tain a lar<>-e number of 
 liquid surfaces each satisfying- the common equation (i) of 
 such surfaces. 
 
 We shall subsequently see that the same surfaces can be 
 ])roduced by means of soap-bubbles instead of large masses 
 of liquids. 
 
 A full account of all such experiments will be found in 
 Plateau's celebrated work, Slatiqne Exj^erimenfale et Theorique 
 lies LiquiJt's souni'us aiix aevlcn Forces Moleculaires. 
 
 80. Liquid under action of Gravity. Taking- now 
 the case in which the only external force is g-ravity, 
 the equation of its surface is of the form 
 
 1 \ _z — h 
 
 \'^R.r"~^' ^'' 
 
 where // and a are constant lengths, and z is the height of 
 
 any point on the surface above a fixed horizontal plane : 
 
 T . 
 
 also a-— — > where T\.% the surface tension. 
 w 
 
 We shall begin by investigating the form of the surface 
 of a liquid in contact with a broad vertical plane, or wall. 
 
 Let this plane be supposed normal to the plane of the 
 paper, and let Fig. 93 represent the section of the plane and 
 the liquid surface made by the plane of the jiaper (supposed 
 also vertical), this section being far removed from the edges 
 of the immersed vertical plane BAG. Of the two principal 
 radii of curvature of the liquid surface at any point P one 
 will be inllnite, since one principal section at Pis the right 
 line through V perpendicular to the jilane of the jiaper, and 
 the other will be /o, the radius of curvature of the curve 
 AFC. 
 
 Taking- the axis of a; horizontal and the axis ofy vertical, 
 we replace *' in (i) by y, and the equation becomes 
 
 I y — J> 
 
 p a- 
 
 A a 
 
 (2)
 
 354 Hydrostatics and Elementary Hydrokinetics. 
 
 which shows that the curve AVC belongs to the class of 
 elastic curves, i. e., those formed by a thin elastic rod which 
 when free from strain was straig-ht, but under the action 
 of terminal pressures is bent. (See Statics, vol. ii., Art. 
 306.) 
 
 Since at a considerable distance from the wall the surface 
 is plane and p = oc, we see that if we measure 1/ from the 
 level the plane portion, the equation is 
 
 py = a- (3) 
 
 Let Ox be the plane level, which is now taken as 
 axis of a?. 
 
 Putting p = -J- ^ which in the figure is — the tangent of 
 
 (( tJu 
 
 the inclination of the tangent at P to the axis of x, we have 
 
 dp 
 
 P (i +p^f 
 and a first integral of (3) gives 
 
 — 2 ir/2 
 
 VI +/> a 
 
 where C is a constant. Since p = o when ^ = o, C = —2, 
 and then we have 
 
 ^;jz^ .di,=±,i. (5) 
 
 Putting j^ = 2 a sin </), (6) 
 
 ^^^^^" (^-2sin0)r/0 = +-. ... (7) 
 
 Now if the angle of contact is acute (as represented
 
 Molecular Forces and Capillarity. 355 
 
 in tlic li^ure}, - can never vanisli, and ■/ is continuonslv 
 (1^ ax 
 
 no<4;i1ive. Hence y is always < a^/ol^ and the neo-ative 
 sinn in (5) is the one to be used. In this case the inte^Tal 
 
 = 2 COS </) + log-g tan - (H) 
 
 This shows that the i)lane siivfoce of the liquid is in 
 realit}' asymptotic to the curve ATC, because when <\> = o, 
 X = oc. 
 
 dy 
 
 .-. y = as/l = OA. 
 
 If i is the angle of contact, — = -cote at A, and we 
 
 have from (4) ^ j /—/ ■ — ^ , , 
 
 ^^^ OA = a V2{i— Sim) (9) 
 
 Tt 
 
 — (i-sine), .... (10) 
 
 If llie ang-le of contact is zero, we have — = cc at A 
 
 ax ' 
 
 V 
 
 iv 
 
 which determines the heig-ht to which the liquid rises 
 against the plate ; and, if i is known, by measuring this 
 height the surface tension Tcan be found. 
 
 The equation (3) can be immediately deduced by elemen- 
 tary principles from the notion of surface tension. For, 
 let Q be a point on the curve indefinitely near 7^ ; draw 
 the ordinates Pw, Q/i, and consider the equililirinm of 
 the prism of liquid PmnQ, of unit breadth perpendicular 
 to the plane of the pai:)er. AVe ma}^ consider this prism as 
 kept in e(juililtrium by the surface tensions, each equal 
 to T, at P and Q, and its weigiit, the atmospheric ]>ressure 
 cancelling at the top and bottom. Now the vertical ujiward 
 comi)onent of 7' at P is T'sin 6, and the vertical component 
 
 at Q is r sin + (h . '^ (7' sin 6) ; hence 
 
 A a 2
 
 356 Hydrostatics mid Elementary Hydrokinetics. 
 
 — Tds . -J (sin 0) = wydx, 
 
 JQ dx 
 
 ds ^ ds 
 
 T 
 
 .-. - = wy, 
 
 ds fi or 
 
 since p = 5-;, and cos = -7- • This equation is the 
 
 dd ds 
 
 T 
 same as i'l), since — = a^. 
 
 The integration of (3) may be effected in another way 
 which gives the intrinsic equation of the curve. It can be 
 written ^iq ^ 
 
 -ds-a^ ^''^ 
 
 dn" a'ds oP- ^ ' 
 
 ,. (f) = c-^,cose (13) 
 
 ^ds ^ a^ 
 
 T do 2 
 
 Now = — - = o when ^ = o, therefore C = -„ > and 
 
 P ds cr 
 
 we have dd 2 . 6 , , 
 
 — =- - sm -, (14) 
 
 ds a 2 
 
 .'. log", tan- = f- C, 
 
 ®* 4 a 
 
 where C is a constant. Now at A we have 6 = - — i, 
 therefore 
 
 s 
 
 ^6 ~ n . /~ i 
 
 tan — 
 4 
 
 = e "tanQ--) . . . (15) 
 is the intrinsic equation of the curve.
 
 Molecular Forces and Capillarity. 357 
 
 Pass now to the case in which two larg-e parallel vertical 
 })latcs, BO, B'O' (Fiii:. 93)) firc immersed, very close toq-ether 
 in a liquid. Let BVB' be the curve in which the liquid 
 surface between them is intersected by the plane of the 
 tigure, V being- the lowest point, or vertex, of the curve. 
 
 One of the principal radii of curvature at every point, /;, 
 of this surface is still oc, and the other is p, the radius 
 of curvature of the curve BVB' at the point. Hence, 
 measuring the height of /; above the plane surface Ox, 
 we have still. py = a' ; 
 
 and if 6 is the inclination of the tangent at jt? to the 
 horizon and s = Vjp, 
 
 ds . ^ dy 
 
 therefore d'^d _ sin d 
 
 1^~ ~^' 
 
 dd ^2 /- 
 
 .'. ^- = — VC—Qosd, . . . (16) 
 as a ^ ^ 
 
 where C is a constant. Hence 
 
 i/ = a\^2 VC—cosd; . . . . (17) 
 and if // is the height of F above 00', 
 
 h =aV2 VC-i (i«) 
 
 An approximate value of h has been already found 
 (p. 341). If the abscissa of/? with reference to 7'as origin 
 
 is X, we have -^ = cos 6 ; therefore 
 ds 
 
 , a cos 6 do , 
 
 «ar = — p. — — (19) 
 
 >/2 VC-cose
 
 358 Hydrostatics and Elementary Hydrokinetics. 
 Substituting for C in terras of // from ( 1 8), we have 
 
 ?/ = 2af-^, +sin2-)" (20) 
 
 fh = -'—r, ^-r (21) 
 
 The value of a; can be expressed in terms of the ordinary 
 elliptic integrals by putting d = it -2(f). If then we put 
 
 2 
 
 X;-= — ^ — 77,, so that ^ is < I, we have 
 
 "-Tr-^^^-^(*)"*i' • •(-' 
 
 where, as usual, A (^) = \/i—P sin'^ (p. The limits of 
 being o and i, where i is the angle of contact, those of 
 
 are — and - + - ' The figure supposes the angle of 
 2 42 
 
 contact to be acute, as when water rises between two glass 
 
 plates ; if it is obtuse, as when mercury is depressed 
 
 between tw^o glass plates, the discussion proceeds in the 
 
 same manner. 
 
 Two plates close together in a liquid move towards each 
 
 other, as if by attraction, whether the liquid rises or is 
 
 depressed between them — as was first explained by Laplace. 
 
 In all such cases of approach between bodies floating on 
 
 a liquid the result is due to an excess of pressure on their 
 
 backs, or remote faces, over the pressure on their adjacent 
 
 faces. Thus, on the plate OB above JB the intensity 
 
 of pressure is the same on both sides, being that of the 
 
 atmosphere ; aUo below the intensities are the same, and
 
 Molecular Forces and Capillarity. 
 
 359 
 
 again at both !^icles of OA ; Init between A and B the 
 intensity of pressure on the left side is that of tlie at ino- 
 sphere while on the ri<i^ht it is less. For if on the surface 
 AB between A and B we take any point R, at a hei^ji-ht z 
 above Ox, the intensity of pressure exerted by the liquid on 
 the plane is p^ — wc, where p^^ = atmospheric intensity, 
 since it has been shown in Art. 70 that the intrinsic 
 pressure K disappears. Hence the total pressure on the 
 plane AB from left to rig-ht is less than that from right to 
 left ; and similarly for./'i/ ; therefore the planes a pj)roach. 
 The same result follows if (as in Fig. 90) the liquid is 
 depressed between both planes. 
 
 But if the liquid rises in contact with one plane and 
 is depressed in contact with the other, the two plates 
 are urged away from each other. 
 
 Suppose the liquid to rise in contact with the plate AB 
 (Fig. 99) and to fall in contact with the plate A'B' ; then 
 the level of the liquid 
 at the left of the first 
 nuist be higher than 
 that at the right ; and 
 the depression of the 
 li(iuid at the right of 
 the second plate is 
 greater than at the left. 
 Evidently, then, the 
 ]iortion AB of the first 
 plate experiences an ex- 
 cess of pressure towards 
 
 the left, while, the pressure at the left of B'A' being 
 greater than that of the atmosphere, the second plate 
 experiences an excess of pressure towards the right. Thus 
 the plates move away from each other. 
 
 This can be shown experimentally by placing on the 
 
 Fig. 99.
 
 360 Hydrostatics and Elementary Hydrokinetics. 
 
 surfoce of water two small pieces of cork, one of which 
 is clean and the other g-reased. 
 
 Next, taking* the case in which a liquid rises inside 
 a narrow vertical cylindrical tube, the free surface of 
 the liquid inside the tube will be one of revolution ; 
 and if y is measured from the line Ox, which is the 
 level of the plane surface, the equation of the surface 
 of the liquid is t t «/ 
 
 -+- = 4' (23) 
 
 where p is the radius of curvature of the meridian at any 
 
 point, and n the length of the normal between this point 
 
 and the axis of the tube. 
 
 If the horizontal line through the lowest point, F, 
 
 of the meridian (Fig. 93) is taken as axis of a?, the equation 
 
 becomes 
 
 I I // + y , . 
 
 - + - = -~ ; (24) 
 
 and if we put ^— and for - and — , this becomes 
 
 ^ as X p n 
 
 id. . ^+y 
 
 X dx 
 
 (^sin^) = — / (25) 
 
 a 
 
 This equation cannot be integrated accurately ; but 
 an approximate solution can be obtained by the following 
 method, which is, in principle, the same as that emplo^'ed 
 by Laplace (Supplement to Book X of the Mecanique 
 Celeste). 
 
 Take a circle having its centre on the vertical through V 
 and having a radius c ; and let us determine this circle 
 in such a way that its ordinate (for any abscissa) differs 
 very little from the ordinate of the point, ;?, on the curve
 
 Molecular Forces and Capillarity. 361 
 
 of the liquid whieli has the same abscissa. The ordinate of 
 the circle is given by the o(|nation 
 
 1/ = Ij— ^(P-—oc^^ 
 
 so that for any point on the curve we have 
 
 ^ = /._yc2_a;-^4^, (26) 
 
 where C is a very small quantit}^ This g-ives 
 
 dy _ X dC 
 
 + l-r.' (27) 
 
 dx ^c^^x^ <l^ 
 
 .-. a? sin = — + -5^ — - — '- ^, . . 
 c c^ ax 
 
 neglecting the square of -r- • Hence (25) becomes 
 
 (fOu 
 
 d (x"^ x(c^ — x^)^dO hx-\-xy 
 dx \c c^ dx) ~~ tt^ 
 
 (28) 
 
 li-\-h— ^/c'^ — x^ 
 
 .x, (29) 
 
 by neglecting C in the value of y. Integrating between 
 ^ = o and X = X, 
 
 x'- x(c^-x')^dC (/i + Ij)x^ Ic^-x'f C-' , , 
 
 - + — — T-^ -r = i + :7^ ; ' (30) 
 
 c c"^ dx ia^ 3 a- 3a- 
 
 d( _ yh + h 1 . c''x 
 dx~ ^ id' ~~ c^ (c'i-__Q,3.\^ 
 
 C^ C^ I
 
 362 Hydrostatics and Elementary Hydrokinetics. 
 
 6-=* 
 
 + —> log« [c + Ve-'-x') + C, (32) 
 
 where C is a constant to be determined. 
 
 Now this equation would make C = ^ when x = c, 
 which would, of course, be absurd. Hence we must have 
 
 Again, y = when x = o, .-. /j—c + C= o at V, and 
 (32) becomes 
 
 f=^-^ + ^log. ,, • . • ■ -(34) 
 
 SO that from (26) 
 
 + 7^^ 
 
 6'^ , (7+ VC^ — X 
 
 y = c- Vc''-x'+ ^log, ~ > . . (35) 
 
 which is the approximate equation of the curve when c is 
 known. Now we know that at the points £, B\ of contact 
 
 fly 
 
 with the tube -f- = cot i, and therefore if r is the radius of 
 ax 
 
 the tube. 
 
 cot i = — ^ ] I ^ s-^ ^\ . .(36) 
 
 which determines c ; and d is then known from [ss). 
 
 As a first approximation, (36) gives c = r sec i from 
 which, more accurately, 
 
 . ( r^ ^in.^ i(i—sini)) , . 
 
 ( 3«^ cos*? i ^-^'^
 
 Molecular Forces and Capillarity. 
 
 363 
 
 Finally, take the cabc in which liiiuid is contained 
 between two vertical plates which make with each other 
 a very small ang-le. Let the i)lates he Ay Ox, A'ljO.i: 
 (Fig-. 100), intersecting- in the vertical line Oy, and making- 
 with each other the very small angle e, or xOx . Let the 
 eur.ves in which the liquid surface intersects the plates 
 be yPQx, i/P'Q'x\ It is required to find the nature of 
 these curves. 
 
 Take any two indefinitely near points, P, Q, on one 
 curve ; let the corresponding points on the other be P', Q\ 
 the lines PP', QQ' being normal to 
 the plates and in the liquid surface ; 
 draw the ordinatesPw, Q«,&c.,and 
 consider the separate equilibrium of 
 the small prism PQ(m. 
 
 If Pm = y, Oni = X, the weight 
 of this prism is ewxj/dx, where 
 w« = (Ix, and it is balanced by 
 surface tension round the contour 
 PQQ'P. Let T be the surface 
 tension, and 6 the inclination of 
 the tang-ent to the curve PQ at P to Ox. 
 
 Then the amount of tension on PP' is T. ex, and its 
 vertical component is eTxsinO; therefore the vertical 
 component of the tension on QQ' is 
 
 Fig. 100. 
 
 cTxsinO+e I 
 
 t! (x sin 6) 
 
 dx 
 
 (1x. 
 
 Also the tension on PQ acts in the tang-ent plane to 
 the liquid surface and at right angles to the line PQ. Now 
 if I is the angle of contact (i. e., the ang-le between this 
 tangent plane and the plate i/Ox), we easily find that 
 the direction- cosines of the tang-ent ])lane are projiortional 
 to sin(^, cos 0, cot /, while those of the line PQ are cos 0,
 
 364 Hydrostatics and Elementary Hydrokinetics. 
 
 — sin 0, o. Hence the direction-cosines of the line of action 
 of the tension on PQ are sin Q cos z, cos Q cos e, sin i ; and 
 if PQ := ds, the vertical component of the tension is 
 Tfis . cos cos i, or Tcos i . dx ; while the tension on P^Q' 
 gives the same amount. Hence for the equilibrium of 
 the prism we have 
 
 m ■ m^H^ sin 6) , y. 
 
 which shows that the second term on the right-hand side 
 is of the order e-. 
 
 Neglecting it in comparison with the first, we have for 
 the approximate equation of the curve 
 
 iTcQ'&i 2 a- cos i , . 
 
 T 
 
 where, as previously, a^ = - • 
 
 The curve is, then, approximately a rectangular hyperbola 
 — a result which is commonly assumed in virtue of the fact 
 that the elevation of a liquid between two parallel close 
 plates varies inversely as the distance between them. 
 
 81. Liquid Films. The forms which can be assumed by 
 the surface of a Liquid which is under the influence of none 
 but molecular forces can be produced by means of thin 
 films of liquid, such as soap-bubbles. Imagine a thin film 
 of liquid in contact with au- at both sides of its surface, 
 the intensity of pressure of the air being, in general, 
 different on these sides. 
 
 Let ABC I), Fig. 10 1, be a portion of such a film ; let P 
 be any point on its surface ; let PQ, PS be elements of the 
 arcs of the two principal sections of the surface at P \ at Q 
 and S draw the two principal sections QR and SR. Thus 
 we determine a small area PQRS on the surface.
 
 Molecular Forces and Capillarity. 
 
 365 
 
 Let the normals to the surface at P and H intersect 
 
 Then PC'i =/i\, 
 
 in ^2 5 ^^^ those at P and Q in C^ 
 P(72 = i?2> where 7?^, P, 
 are the principal radii of 
 curvature of the surface at P. 
 Let PQ = ds^ , PS = cls^, 
 p = intensity of air pressure 
 on the lower or concave side 
 of the surface at P, and 
 jj(j = intensity of air pressure 
 on the convex side. Then 
 (p —Po) ds^ ds^ is the result- 
 ant air pressure on the area 
 PQRS in the sense C^P -, 
 and for the equilibrium of 
 
 the element this must be equal to the component of force 
 in the sense PC. g-iven by the surface tension exerted on 
 the contour of the element, assuming- that the film is so 
 thin that the action of g-ravity is negligible. Xow if 
 T is the tension per unit length along PS, the whole tension 
 on PS (which acts at its middle point perpendicularly to 
 PS) is T .ds^, and the component of this along the normal 
 
 Fig. lor. 
 
 to the surfiace is Tds2 . sin 
 
 \ovTds., 
 
 2l{, 
 
 The ten- 
 
 sion on QR gives a component of the same magnitude; 
 
 T 
 
 hence the sum of these is jr- ds^ds.,; similarly the sum 
 
 of the normal components of the tensions acting on the 
 sides PQ and SB is -^ ds^^ ds.^ ; so that the normal compo- 
 nent of the tension acting on the whole contour PQllS is 
 
 ^R^ ^ R, 
 
 ~) ds.d.^,. 
 
 (i)
 
 366 Hydrostatics and Elementary Hydrokinetks. 
 
 If the thickness of the film exceeds 2 e, where e is the 
 radius of molecular activity, there will be surface tension 
 exerted both at the upper and at the under side of the 
 surface, this action being confined (as explained in Art. 76) 
 to a layer of thickness e at each of these sides ; so that 
 we must replace T 'wx (1) by iT, and the equation of 
 equilibrium is 
 
 Hence, since p and 7;^ are the same at all points of 
 the film, the equation of its surface is 
 
 ^ + ^^ = const (3) 
 
 and the forms of these films are the same as those of 
 the surface of a liquid which is not acted upon by any 
 external force, i. e., the shapes of thin films are the 
 same as those of drops of oil in the water-alcohol mixture 
 of Plateau (see p. '>,S'^\ 
 
 The equation (2) can be otherwise deduced without 
 considering the separate equilibrium of an element of 
 the film. For, the intensity of pressure at any point 
 inside the film (beyond the depth e) due to the convex side 
 
 \s, pr,-\-K-\-T {^TT- + 7^) ; and the intensity of pressure at 
 -til ii-i 
 
 the same point due to the inner, or concave, side is 
 
 ''+^'-^0i + 7z)' 
 
 Equating these, we have (3). 
 
 For a spherical bubble B^ = Tl.^ = r, 
 
 4T={p-p,)r, (4)
 
 Molecular Forces and CapillarHy. 367 
 
 whicli shows that for all sizes of the bubbles the product 
 {p—p^r remains constant. 
 
 One possible shape of the bubble is that of a cylinder 
 closed by two spherical ends. If r is the radius of the 
 cylinder, / that of each end, we have 
 
 aT
 
 CHAPTER IX. 
 
 STEADY MOTION UNDER THE ACTION OF GEAVITY. 
 
 82. Steady Motion. When a fluid is in motion and 
 we confine our attention to any point, P, in the space 
 through which the fluid moves, it will be readily under- 
 stood that the magnitude and direction of the velocity 
 of the molecule which is passing through P at any instant 
 may not be the same as the magnitude and direction of the 
 velocity of the molecule which is passing through P at any 
 other instant. If these should be the same at all instants, 
 and if a like state of affairs prevails at all other points, 
 the motion is said to be stead?/. 
 
 It is obvious, for example, that if a vessel is filled from a 
 large reservoir of water, so that it is kept constantly full, 
 while the liquid is allowed to flow out from an aperture 
 made anywhere in the vessel, the motion at any fixed point 
 in the vessel will be the same at all times. 
 
 83. Methods of Euler and Lagrange. It is at once 
 obvious that the problem of the motion of a fluid acted 
 upon by given forces may be attacked by two different 
 methods. For, firstly, we may make it our aim to discover ^ 
 the condition of things — i. e., the magnitude and direction 
 of the resultant velocity, and the pressure intensity — at 
 each point, P, in sjmce at any instant of time and at 
 all times, and to do tlie same for all other points in the
 
 Steady Motion under the Action of Gravity. 369 
 
 space throuo-h whicli the fluid moves, and thus, as it were, 
 to obtain a map of the wliole re^-ion — or a series of maps, 
 if the motion is not steady — exhibiting- the circumstances 
 at each point as regards velocity and pressure. 
 
 Or, secondly, we may make it our aim to trace the path, 
 and other circumstances, of each individual molectde through- 
 out its whole motion. 
 
 The second olijcct is much more difficult of attainment 
 than the first, and, moreover, is not gcnorally so desirable. 
 
 The first method is sometimes called the stafidical, or 
 the method of Euler ; the second the historical^ or the 
 method of Lagrange. 
 
 84. Flow through a tube ; work of gravity. Suppose 
 a column of water to occupy at any instant a length AB of 
 a straight vertical tube of 
 uniform cross-section, and let 
 the end B of the tube be Ab 
 open. 
 
 If in a small element, A i^ 
 of time a mass, A ?/;, of water 
 flows out, what is the work 
 done by gravity on the water 
 during this interval ? 
 
 Divide the tube by a series 
 of very close horizontal planes, 
 P, Q, H,... into sections such 
 
 that the mass included between each adjacent pair is Awz. 
 If A « is the distance between the middle points of suc- 
 cessive layers, PQ, QP, &c., while A?« flows out the middle 
 point of each layer will fall through the height A.-?, and 
 the work done by the weight of this layer will be 
 
 A //^ . A 5 or ^ A m . A .?, 
 
 according as we use gravitation or absolute measure of 
 
 B b 
 
 Fig. 
 
 102.
 
 370 Hydrostatics and Elementary Hydrohinctics. 
 
 force. (If mass is measured in pounds and leng-th in feet, 
 the first expression g-ives the work in foot-pounds' weight, 
 the second in foot-poundals.) If we use the first, the work 
 is 2) Aw . A5, which can be written in either of the forms 
 
 Aw(A.94-A/+A/'+...), 
 
 ( A w^ + A m + A hi" + . . . ) A ,f , 
 
 in which, of course, the successive distances A*, A/, !S.s" ,... 
 are all equal, and the successive weig-hts Aw, A?;/, Aw'',... 
 are also all equal. 
 Hence the work is 
 
 A w X AB, 
 
 or M X A .f 
 
 where 31 is the weig-ht of the whole column. 
 
 The first expression shows that f//e toork done is the same 
 as if the mass Aw which jimvs out at B fell through the 
 height AB of the column. 
 
 Precisely the same result holds if the shape of the tube 
 is that represented in the rig-ht-hand fig-ure. Let it be 
 divided by close horizontal planes in the same manner 
 as before. If now A^; is the vertical distance between the 
 middle point of the layer, PQ, and the middle point of 
 the next layer, QB, the work done in the descent of the 
 first layer into the position of the second is Aw.Ar', so 
 that the work done by g-ravity on the whole tube of liquid 
 while the quantity A m flows out at I) is 
 
 A w X 2^ 
 
 in gravitation units, where z is the difference of level of 
 the ends C and IJ. 
 
 This is again the same as the work of carrying Aw from 
 C to i^.
 
 r? 
 
 N H 
 
 D j 
 
 1 jc 
 
 WM. 
 
 WM 
 
 Wmi 
 
 ''^^^3^ 
 
 l^^fe.-! 
 
 ^— ^iV3^^ 
 
 rig."T03. 
 
 Steady Motion binder the Action of Gravity. 371 
 
 85. Stream Lines. Tlic actual path of a particle of 
 a movinL'- iluid is called a stream line. If at anv point, //, 
 Fig-. 103, we describe a very 
 small closed curve and at each 
 point on the contour of this 
 curve we draw the stream line, 
 such as AP, and produce it in- 
 definitely, we obtain a stream 
 tube. When the fluid is a 
 liquid, the mass contained be- 
 tween the normal sections of a 
 tube at any two points, A, P, 
 must always be the same ; and 
 therefore the same mass of fluid 
 
 crosses every normal section of the tube per unit of time. 
 Hence if v is the resultant velocity of the liquid at P and 
 (T the area of the cross-section of the tube, the product 
 
 va- 
 is constant all along* the tube. 
 
 86. Theorem of Daniel Bernoulli. Consider at any 
 instant the liquid contained in the stream tube between the 
 normal sections at A and P, and suppose this liquid to 
 occupy the volume A'P' at the end of an infinitesimal 
 element of time; let t'o,7^, o-^ be the velocity, pressure 
 intensity, and cross-section of the tube at A ; let v, p, o- be 
 the same things at P ; let z^ and z be the depths of A and P 
 below any fixed horizontal plane ; let A.?o be the distance 
 between the cross-sections at // and A', A* being that 
 between those at P and P' ; and let w — vveight per unit 
 volume of the liquid. 
 
 Now ap})ly the equation of work and kinetic energ-y 
 to the mass of liquid between A and P in the tube. The 
 gain of kinetic energy in the small motion considered is 
 kinetic energy of ^'P'— kinetic energy of AP, 
 
 B b 2
 
 372 Hydrostatics and Elementary Hydrokinetics. 
 
 in which, as the motion is steady, the kinetic enerc^y 
 of the portion A'F is common to the two terms, and there- 
 fore disappears. Hence the g-ain of kinetic energ-y is 
 
 thatof PP'- that of ^^', 
 
 .2 • 2 
 
 or ^m ^ ~^" , (i) 
 
 where Aw? = weight of PP'= weight oi AA' . 
 
 The external forces doing- work on the cohimn of liquid 
 considered are 
 
 g-ravity, the pressure at A, the pressure at P. 
 
 The work of g-ravity is 
 
 /^m,{z-z^, (2) 
 
 by Art. 84. 
 
 The pressure at A '\% f^n^^ and its work — p^a^. l\s^\ 
 the pressure at T \s> pa, and its work = — ]_)(r . As. Hence 
 the work of the pressure is 
 
 Am . , ^ 
 
 - — '{P~Poh (3) 
 
 since a . As = a^. AS(,, i.e., the volume PP'= the volume 
 
 w 
 Equating (1) to the sum of (2) and (3), we have 
 
 1 z= 1 z^, . . . . (4) 
 
 ig w lg 10 
 
 in other words, since A and P are any two points along- 
 the stream line, 
 
 ^ + ^-.'=C (J) 
 
 lg w
 
 Steady Motion niidcr the Action of Gravity. 373 
 
 at every point of the stream line, C being- a constant ibr 
 tlie stream line chosen ; but this constant may have 
 tlillorent values as we pass from one stream line to another. 
 
 This result is the theorem of D. Bernoulli. 
 
 If at P we draw a vertical line, PQ, of such leng-th that 
 
 2) — w .FQ, 
 
 the height PQ is called the pressure head at V. If also Qlt 
 is drawn vertically of such leng'th that 
 
 QR is called the velocili/ head at P. Let AB be the 
 pressure head and BN the velocity head at A. Then (4) 
 ^ives PE = AN + z-z^ 
 
 where AL = r — r^ and is the perpendicular from A on the 
 horizontal plane throug-h P. 
 
 Since PL is horizontal, it follows that RN is horizontal. 
 Hence the theorem of Bernoulli may be expressed in these 
 words : if at each jjoiut along a stream line there be drawn a 
 vertical line ivhose length =. the pressure head -^ the telocity 
 head at the point, the extremities of all these vertical lines lie 
 in the same horizontal plane. 
 
 If the liquid has a horizontal surface, CB, at rest at 
 all points of which the intensity of pressure is constant 
 (e. g., that of the atmosphere), the extremities of these lines 
 drawn at all points of the liquid, and not merely along the 
 same stream line, will all lie jn the same horizontal plane. 
 If Cll is the pressure head on the surface CB (about 34 feet 
 if CB supports the atmospheric pressure), the extremities 
 7^,... of the vertical lines drawn at all points P,... lie in the 
 horizontal plane through IL.
 
 374 Hydrostatics and Elementary Hydrokinetics. 
 
 For a liquid in equilibrium, Q coincides with it, since 
 QK = o, and it has already been shown that the extremities 
 of all vertical lines representing* pressure heads lie in the 
 same horizontal plane. The theorem of Bernoulli is the 
 generalization of this result for a liquid in steady motion. 
 
 An approximate method of indicating the value of;?, the 
 pressure intensity at any point P in a moving liquid 
 consists in inserting a vertical glass tube, open at both 
 ends, into the liquid, one extremity of the tube being placed 
 at P. The liquid will rise to a certain height in this tube 
 and remain at rest. Thus, if the tube is so long that the 
 upper end is above the free surface CD, the liquid would 
 rise in it to the height PQ, the remainder of the tube being- 
 occupied by air. Such a tube is called a pressure (jauge ; 
 but it is evident that it does not strictly measure the 
 pressure, since the glass must, to some extent, alter the 
 motion of the liquid. 
 
 87. Flow through a small orifice. Let Fig. 104 re- 
 present a vessel containing a liquid whose level is LM 
 
 which flows out through a 
 small aperture made any- 
 where in the side of the 
 vessel, and let the thickness 
 of the side be so small that 
 the liquid touches the inner 
 edge, AB, of the orifice and 
 thence passes out without 
 touching the outer edge or 
 any intervening part of the 
 - aperture. 
 
 The curved lines in the 
 figure represent the stream 
 lines, or paths of particles, the forms and positions of which 
 cannot, however, be determined mathematically. 
 
 Fig. 104.
 
 Steady Motion under the Action of Gravity. 375 
 
 We are obliged to have l•eeoul•i^e to experiment for certain 
 facts concerning' the issuing" jet. Firstl\% it is found that 
 after leaving the orifice AB, this jet contracts to a minimum 
 cross-section, CI)^ beyond whicli, of course, the jet widens 
 out again. This minimum cross-section is called the vena 
 coiUracfa. 
 
 The ratio of the area of the vena contracta to that of the 
 orifice ^i^is called the coejlicieut of contraction. 
 
 For a circular orifice whose diameter is AB^ if CD is 
 the diameter of the vena contracta, it has been found 
 experimentally that 
 
 -AB=''^^ 
 
 so that if S is the area of the orifice and a that of the 
 vena contracta, 
 
 It has also been found that the distance, 70, between the 
 orifice and the vein is somewhere between -39 x AB and 
 •5 X AB, where, as before, AB is the diameter of the orifice ; 
 the uncertainty arising from the fact that in the neighbour- 
 hood of the minimum section the diameter of the jet varies 
 very little. All the streams which pass through the vena 
 contracta cut its plane perpendicularly. By consideration 
 of the general equations of motion, it will follow from this 
 fiict that the intensity of pressure is the same at all i)oints 
 in the vena contracta. At all points on the outer surface, 
 ACB, BBi\ of the jet the pressure intensity is, of course, 
 the same as that of the atmosphere, if the jet flows into the 
 atmosphere ; also the velocities at all points of the vein are 
 equal ; but in the interior of the jet this pressure intensity 
 does not exist, except at points in the plane of the 
 vena contracta.
 
 376 Hydrostatics and Elementary Hydrokinetics. 
 
 Of course at the orifice AB the directions of motion are 
 not all perpendicular to the cross-section of the jet, neither 
 are the velocities all the same at points in this section. 
 
 88. Theorem of Torricelli. In the case of a jet escaping- 
 into the air, the velocities of particles in the vena contracta 
 are expressed by a very simple formula. 
 
 lu (4) of Art. 86, let jy and z refer to a point. 0, in the 
 vena contracta while /;q, Zi^ refer to the point, iV, of the 
 stream line throug-h which is on the free surface of 
 the liquid. Then /j ■= p^, as we have said above, and 
 as the velocities at the surface LNM are all veiy small, we 
 may consider r^ = o. 
 
 = 2^/, (a) 
 
 Hence 
 
 where Ji, or z — z^^, is the vertical depth of the vena con- 
 tracta below the free surface LNM. 
 
 Hence when the particles reach the vena contracta, they 
 have the same velocity as if they fell directly from the free 
 surface. This is known as Torricelli's Theorem. Obviously 
 it holds with considerable exactness in the case of a small 
 orifice only. 
 
 Examples. 
 
 1. The Si/2)hon. We now take a few common practical illus- 
 trations of the application of equation (4), Art. 86, which 
 applies to the motiou <jf a liquid acted upon by gravity. The 
 first of the simple examples is furnished by the common syphon 
 which is employed for the purpose of raising a liquid out of 
 a vessel and lowering it into another vessel. The operation 
 miglit, of course, in many cases be directly performed by taking 
 the first vessel in the hand and i)ouring out the liquid bodily 
 into the second ; but if, as oiten happens, the liquid is 
 Sulphuric or Nitric Acid, which it would be most undesirable
 
 Steady Motion under tJic Action of Gravity, "yjl 
 
 to pour out with splashing, this metliod wouhl not answer, and 
 II syphon is uscih The syphon is a lient tuhe {usually of glass) 
 open at hotli ends, and with unequal branches. 
 
 Sup])oso M (Fig. 105) to be the vessel which it is desired to 
 empty into another (not represented in the figure), and suppose 
 the liipiid to be water. 
 
 A bent tube, DA EC, (the syphon) whose branch EG is longer 
 than the branch ED is first fdled with water, and the apertures 
 at D and C held closely by the fingers. 
 The end D is then inserted into the liquid 
 in the vessel M, the fingers removed from 
 D and C, and the tube held in the hand. 
 The result will be a flow of the liquid 
 through C until, if D is kept close to the 
 bottom of the vessel, nearly all the licjuid 
 is removed. 
 
 Let p„ be the atmospheric intensity of 
 pressure, which exists on the surface 
 of the liquid at A and also at C ; v^^, the 
 
 velocity of liquid on the surface A, is nearly zero ; if v = velo- 
 city of eflflux at G, z-=- depth of G below A, equation (4) of 
 Art. 86 gives, since AEG may be taken as a simjile stream line, 
 
 Fig. 105. 
 
 V 
 
 21/ 
 
 H z — — J 
 
 w w 
 
 V 
 
 = v 
 
 2gz, 
 
 so that the flow will continue all through if C is at a lower 
 level than D. Of course there will be a small residue of licjuid 
 in M, because when nearly all has flowed out, air will enter the 
 syphon at D. 
 
 If the li(|uid to be removed is an acid, as suli^huric or nitric, 
 the syphon must be filled with it at the beginning by first 
 inserting tiie end D into the vessel and then sucking the air 
 out througli C until the liquid rises in the syphon and falls in 
 the leg EG to a lower level than A ; and this suction may 
 be effected by joining another tube to the end G by means of a 
 short piece of indiarubber tubing which can be subsequently 
 removed. 
 
 2. Hero's Fountain. Hero of Alexandria (120, B.C.) constructed 
 a fountain, which is represented in Fig. 106. It consists of two
 
 378 Hydrostatics and Elementary Hydrokinetics. 
 
 glass globes, M and T, and a disli, DD. Each globe is partly 
 tilled with water and partly with air at the atmosiiheric pressure. 
 The globes are fitted with necks and are held together by 
 
 two glass tubes, A, B, each open at both 
 ends, which pass through necks fitted to 
 the o;lobes. The extremities of A are 
 in the air in the globes ; the lower ex- 
 tremity of B dips nearly to the bottom 
 of the liquid in M, while its upjier end 
 barely projects into the dish DD. A 
 third tube, C, open at both ends, passes 
 through the neck of N, its lower end 
 dipping nearly to the bottom of the 
 liquid in iV, while its upper end pro- 
 jects beyond the upper surface of the 
 dish. 
 
 In this state of afi'airs the water is at 
 rest in both vessels, the intensity of 
 pressure on both water surfaces being 
 p^ , that of the atmosphere. 
 
 If now water is poured into the dish, 
 it will fall through B into M and drive 
 some of the air into -tV where the surface 
 pressure on the water becomes greater 
 than p^^, and as a result the water fx'om 
 iV is forced up through the tube C into 
 the air. 
 
 To calculate the height to which it 
 
 Fig. 106. 
 
 rises, let z be the difference of level between the water in M and 
 that in JV ; let c = difference of level between that in D and 
 that in N ; and let h = the height of the top of the jet above 
 the water in J), 
 
 Then, since the pressure intensity of the air in N =z that in 
 M= tv (z + c) +.po ; since the velocity of the water at the surface 
 in N is nearly zero, and is also zero at //, where the pressure 
 intensity isp^, we have from (4) of Art. 86 
 
 10 ^ XV 
 
 '. h = z,
 
 Steady Motion under the Action of Gravity. 379 
 
 Eig. 107, 
 
 i.e., till' liciglit of the jot aliove tlio waler in D is c([ual to tlie 
 diflcrc'iice of level in M and N. 
 
 3. J/ariotte's Bottle. It is sometimes desired to produce 
 a narrow jet of water flowing for a considerable time with 
 constant velocity. Of course a very larf>e reservoir with a very 
 small aperture made in the side would ])roduce the result ; but 
 such a reservoir is not always at hand. 
 The result can also be produced by means 
 of a broad flask fitted with a stop-cock 
 near the bottom. Fig. 107 represents the 
 flask. The stop-cock (not represented) is 
 fitted at C, and the aperture is supposed 
 to be very small compared with the cross- 
 section of the flask. The flask is first 
 quite filled with water, the stop-cock 
 being closed. In the top of the flask 
 there is a neck fitted with a cork, and 
 into this is inserted a tube, HD, oj^en at 
 both ends, the tube also being quite filled 
 with water. 
 
 Now let the stop-cock be opened, and water will flow out, 
 because the atmosphere presses at // and at the outside of C, 
 and between C and H there is a column of water. The water 
 that first flows out comes from the tube IIJ) alone, the flask re- 
 maining filled to its upper surface ; and, moreover, the velocity 
 of efflux will be variable as the level sinks in JID. But when 
 the tube is emptied of water, some air will be forced through D 
 by superior atmospheric pressure, and it will rise to the upper 
 part of the flask, and will begin to force down the water of 
 the flask. 
 
 This being the case, the intensity of pressure at I) in the 
 water is p^, the atmospheric intensity, and we may assume that 
 Py is also the intensity all over the horizontal plane, LM, 
 through I), because the motion of particles in this plane is 
 very slow. The air at the top aided by the water above LJf 
 will keep the pressure intensity approximately equal to ^'o ^^ 
 points in LM other than D. 
 
 Now let c = CM =■ vertical distance of orifice below D, the 
 lower extremity of the tube ; let v^, p^^ in (4) of Art. 86 refer 
 to D, while v is the velocity at C. Then, since jp^ is also 
 the pressure intensity at C,
 
 380 Hydrostatics and Elementary Hydrokinetics. 
 
 2g w w 
 
 which shows that the velocity is constant whatever the position 
 of the upper surface, A B, of the water in the flask. 
 
 Tlie tube mu&t, of course, have such a position that D is 
 above the aperture. If the Avater, instead of escaping into the 
 atmosphere, escapes into a medium (gaseous or liquid) in which 
 the pressure intensity at C is p, we shall have 
 
 V w ^ 
 
 This vessel is known as Mariotte's Bottle. 
 
 89. Discharge from a small orifice. If a liquid devoid 
 
 of friction esea])es from a small orifice in a vessel in which 
 
 the free surface is maintained at a constant level, the 
 
 velocity in the vena contracta is given by the equation 
 
 (Art. 88) , 
 
 ^ ^ v= -J^gh (1) 
 
 In the case of water, however, it is found that the velocity 
 
 is not quite equal to this amount, but is very nearly a 
 
 constant fraction, ju, of the value g-iven by (i ). The fraction 
 
 \x is nearly equal to unity (about -97). We may therefore 
 
 put . 
 
 V = \i wig1i (2) 
 
 Again, if *S' is the area of the aperture, and c denotes the 
 coefficient of contraction (Art. 87), the area of the cross- 
 section of the vena contracta is c/S ; so that the volume of 
 water issuing from the vessel per unit of time is 
 
 cixSV2gh (3) 
 
 If the unit of leng-th is a foot and the unit of time
 
 
 D 
 
 ^ 
 
 m 
 
 ^-A~ 
 
 
 
 
 « 
 
 o 
 
 
 ^ 
 
 Steady Motion wider the Action of Gravity. 381 
 
 u second, this will be the discharg-e in cuhic feet per second, 
 and multiplying" it by v, the mass of the liquid per unit 
 volume (in this case 62^ pounds), we obtain the mass dis- 
 charged per unit of time. 
 
 Practically the product cii may be taken as -62. 
 
 90. Flow through a large orifice. The determination 
 of the discharge through a large orifice cannot be satis- 
 factorily accomplished by theory. 
 
 Suppose, for example, that the 
 
 orifice is a rectangle, ABCl), vn\h ^c ^ 
 
 vertical and horizontal sides, and [ 
 
 that i/iI/(Fig. lOcS) represents the j 
 
 level of the free surface in the 
 
 vessel, the flow being- supposed to 
 
 take place through the orifice 
 
 towards us as we look at the figure. Y\g. 108. 
 
 Divide the area of the aperture 
 into an indefinitely gi-eat number of narrow horizontal 
 strips, of which that between the horizontal lines m and n 
 is the type. 
 
 Let the depths below LM of the lines AD and BC be h^ 
 and h.,, respectively, those of the lines m and n being z and 
 z + dz. Let AD === h ; then, supposing that the aperture 
 between, m and n alone existed, the volume of the discharge 
 would be given by (3) of last Article, in which S = hdz. 
 
 Denoting the product c[i\,y ^^ ^^^ K^ '^^ ^^^ '^•'^^^ ^i^~ 
 charged per unit time through the strip, we have 
 
 dQ = kbio J 2CJZ .dz (i) 
 
 Now the assumption that /• is constant for all the strips 
 enables us to find Q, the total discharge ; but clearly this 
 assumption cannot be strictly correct, for each strip does 
 not discharge as if it alone existed as an aperture.
 
 382 Hydrostatics and Elementary Hydrokmetics. 
 
 Assuming' h to be constant, we integrate (i) from 
 i- = /5j^ to ^ = //gj ^^^ obtain 
 
 q = lHio ^/Tg{1i}--h-^) (2) 
 
 To calculate the energy per second which flows through 
 the orifice, if v is the velocit of the portion d Q, its 
 kinetic energy is 
 
 — . dO.. i. e. zd Q. 
 
 Energy per unit time is called Power. Hence if dV 
 is the power of this flow, 
 
 dF =. khw Vig . z^ dz, 
 
 .'. F = \khw */ig{1i.^ — 1i^). ... (3) 
 
 If in this expression length is measured in feet, time 
 in seconds, and w in pounds, since the unit of power called 
 a Horse-Power is 550 foot-pounds' weight per second, we 
 get the Horse-Power of the discharg-e equal to the right- 
 hand side of (3) divided by 550. 
 
 If ABC is small compared with the depth //j, and if 
 h is the depth of the centre of area of the orifice, we can 
 easil}^ find from (3) that 
 
 Q = kw8 V2gh, (a) 
 
 where S = the area of the orifice. For if AB = 2a, we 
 have /i.^ = /^ (i + j^, //^ = /i(^i — ^), and expanding in 
 
 powers of -^ 3 we see that the term y^ disappears, and (a) 
 
 is true if we neglect the small fraction yr • 
 
 As another example of the same kind, suppose the orifice 
 to be circular (Fig. 108).
 
 Steady Motion under the Action of Gravity. 383 
 
 Let h be the depth of 0, the centre of the orifice, below 
 the level, LM, of the water in the vessel or reservoir, 
 let r be the radius of the circle, and break up the area 
 by a series of indefinitely close horizontal lines. If F 
 is any point on the circumference of the circle, OA the 
 vertical diameter, and Z FOA = 0, the area of the strip at 
 F is 2 r- sin- 6 (/ ; therefore the discharge, dQ, through 
 this strip is g-iven by the equation 
 
 (IQ = 2kr^w \/2ff{/i-r cos e).&m^ Ode. . . (4) 
 
 Supposing r to be small compared with //, it will be 
 sufficient to expand the radical in powers of j as far as the 
 second. Then 
 
 dQ = 2h-^io V2gh (i — A cos6~~^ cos^ ^) sin^OdO. (5) 
 Integrating from ^ = o to (9 = tt, we have 
 
 ,.2 
 
 Q z= -nkr-w 'J2fjh (i ^) (6) 
 
 91. Fluid revolving about vertical axis. If a vessel, 
 represented in a vertical section by ACB, Fig. 109, and 
 containing a fluid, is set rotating round a vertical axis, Cr, 
 after a short time the fluid, owing to friction between its par- 
 ticles and against the surface of the vessel, will rotate like a 
 rigid body with the angular velocity w ; each particle, P, 
 will describe a horizontal circle with this angular velocity, 
 so that if FN is the perpendicular from F on the axis 
 of rotation, the resultant acceleration of the particle is 
 directed along FN from F towards N^ and is to'-. NF 
 in magnitude. Denote NF by r, and consider an indeli-
 
 384 Hydrostatics and Elementary Hydrokinctics. 
 
 nitely small jiarticle of mass dm at P ; then the resultant 
 mass-acceleration of this particle is 
 
 orr.chn, (i) 
 
 and this vector is directed from P towards K. [The reversed 
 
 mass-acceleration, — (jo'^rdm, is called the 
 
 z force of inertia of the particle, or its 
 
 ^ I g/ resistance to acceleration. It is most 
 
 jo_.^^^ important to understand that this force 
 
 =^^^^ of inertia is not a force acting on the 
 
 H^^f particle, but one exerted tjj/ it on the 
 
 Nr^^f surrounding- medium, or, g-enerally, on 
 
 i^^^T the agent or agents accelerating its 
 
 c motion. Thus, then, if o is the vector 
 
 Fig. loq representing the resultant acceleration 
 
 of a particle, dm, a force completely 
 
 represented by 
 
 — a . dm in absolute units, or 
 
 — . d/n in gravitation units, 
 
 is the resultant force exerted bj/ the particle ou the agents 
 acting upon it. 
 
 Now the fundamental principle of all Dynamics is this : 
 for eacA particle of any material system (whether rigid body, 
 natural solid, liquid, or gas) the resultant mass-acceleration 
 is in magnitude and direction the exact resultant of all the 
 forces acting upon the particle. These forces will, in 
 general, consist partly of pressures from the surrounding 
 particles, partly of attractions from these particles, and 
 partly of attractions from bodies outside the system. If we 
 consider the equivalence of the resultant mass-acceleration, 
 adm, to these forces under three separate heads, we deduce
 
 Steady Motion under the Action of Gravity. 385 
 
 three great i)rinciples of Dynamics. Thus, if we consider 
 that ad m has — 
 
 1. The same virtual work for any imag-ined displace- 
 ment of the particle, 
 
 2. The same moment about any axis, 
 
 3. The same component along- any line, 
 
 as the whole sj^stem of forces acting' on the particle, and 
 that this is true for (?t;(?ry particle of the system, we have at 
 once the principles of — 
 
 1 . Kinetic Energ-y and Work, 
 
 2. Time-rate of change of Moment of Momentum, 
 
 3. Motion of Centre of Mass, 
 
 for every material system. If the forces acting are 
 measured in gravitation units, their complete equivalent is 
 
 a 
 
 9 
 
 Suppose that at any point, P, Fig. 110, we take as the 
 element dm a very short and thin cylinder, abed, of the fluid 
 having its axis along the tangent at P to 
 any curve AB. Let the length Ic = ds ; let 
 (T be the area of the cross-section, ab, of the 
 cylinder ; let F be the external force per unit 
 mass exerted on the fluid at P, and there- 
 fore Fdm the force on the cylinder, not in- 
 cluding the pressure exerted on its surface by 
 the surrounding fluid ; let a be the resultant Fig. no. 
 acceleration of the particle ; and let j^ ^^ 
 the intensity of pressure on the face ah, and therefore 
 
 dp 
 p + -T- (I'S that on cd. 
 ds 
 
 Then dm = toads, if w is the mass of the fluid \)cx unit 
 
 c c
 
 386 Hydrostatics and Elementary Hydrokinctics. 
 
 volume at P ; and if we resolve forces in the direction 
 
 of the tangent at P to the curve, we see that wa-ds . - has 
 
 9 
 
 the same component along this tangent, in the sense PB, as 
 
 clt} 
 F. wa-ds and — j-ds.a, the length of the arc s being 
 
 measured from A. Hence if a, is the component of a along 
 the tangent at P and S is the component of F, we have 
 
 ?; = 5-if' (2) 
 
 g to lis ' 
 
 This equation connects the acceleration in any direction 
 with the force intensity and the rate of change of pressure 
 intensity in that direction. 
 
 Now suppose the external force to be gravity. Taking 
 the right line i\T (Fig. 109) as the direction of *, (2) 
 becomes ,„2„ ,/„ 
 
 10.— = -!^', (3) 
 
 g dr ^^' 
 
 and again, taking the vertical downward direction at P 
 as that of s, (2) becomes 
 
 dp 
 
 where z is the depth of P below any fixed horizontal 
 plane. 
 
 Now ji is a function of r and z only, so that 
 
 , dp - dp , 
 dp — ~- dr •\- -J- dz 
 
 = w{~dr+dz) (5) || 
 
 \ 2g J
 
 Steady Motion under the Action of Gravity. 387 
 
 where C is a constant, which may be determined from a 
 knowlcd,in;-e of 7; at some one point. If /v^, is the value of 7; 
 at 0, the point in which the free surftice cuts the axis 
 of rotation, and if is taken as orig-in, since r = z = o ut 
 0, we have C = p^ ; hence 
 
 2 *> 
 p=p, + tv[— + z) (6) 
 
 At all points on the free surface ;; = p^, therefore the 
 equation of this surface is 
 
 17+^ = ° (7) 
 
 showing' that the z of every point on it is neg-ative, i.e., all 
 these points are higher than 0. This equation denotes 
 
 a parabola whose latus rectum is -\ > and the free surface is 
 
 CO 
 
 therefore a paraboloid g-enerated by the revolution of this 
 surface round Oz. 
 
 If the vertical upward line Oz is taken as axis of x, 
 and a tangent at as axis of ^, the equation of the para- 
 bola is, in its usual form, 
 
 f=^-^ («) 
 
 If the fluid contained in the vessel is a g-as, equations 
 (2), (3), (4) still hold, and, in addition, p = ho (Art. 48) ; 
 hence (5) becomes 
 
 J) k^g 
 
 
 .'. p = Je ^ , (10) 
 
 where A is a constant to be determined either from a 
 
 c c 3
 
 388 Hydrostatics and Elementary Hydrokinetics. 
 
 knowledge of p at some point or from the given mass of the 
 fluid. Equation (10) shows that for a gas the free surface 
 and the surfaces of constant pressure intensity are still 
 paraboloids. 
 
 In the same way, if the vessel contains two fluids that 
 do not mix, their surface of separation is a paraboloid 
 of revolution. For if ii\ w are their specific w^eights, we 
 have (if they are liquids) 
 
 2 .2 
 
 p = rv ( \- z) + C for one, 
 
 \ 2ff ^ 
 
 2 2 
 
 y =w' (^^ + A + C for the other, 
 ^ y 2ff ^ 
 
 and since at all points on the surface of separation p = p', 
 we have the equation of a paraboloid of revolution, as 
 before. 
 
 The equation of the free surface can also be deduced 
 from the principle of Virtual Work thus. When the 
 position of relative equilibrium has been assumed, imagine 
 the bounding surface A OB to be very slightly displaced and 
 to become A'C/B' . 
 
 Then the variation contemplated in the equation of 
 Virtual Work may be confined to the surface particles 
 occupying- the volume between the surfaces AOB and 
 A'O'B'. As in Fig-. 86, p. 316, a part of this volume 
 will be positive and a part negative. The princijde of 
 Virtual Work applied to any system /)f particles in motion 
 is simply that the sum of the virtual works of the mass- 
 accelerations of the particles is equal to the sum of the 
 virtual works of the forces (in absolute measure) external 
 and internal acting on the system. Now the resultant 
 acceleration of a particle dm at a distance r from the axis 
 of revolution is co-y, towards the axis, and the resultant
 
 Sfradv Motion under flic Acfiou of Cravify. 389 
 
 ntiass-acceleration of this particle is ay^rflm, whose virtual 
 work is —uP-rdmhr. The external force actinj^ on the 
 particle is its weight, y^/w/, whose virtual work is ghz.dw, 
 if z is measured downwards from a fixed plane. 
 Hence 
 
 — oi^/r ^ r . dm = g fh z . dm, 
 
 + <)z] ondci= o, 
 
 /(' 
 
 where, as in Art. 72, 5« is the arbitrary normal displace- 
 ment of the element dS of the surface AOB. Hence 
 as hn is arbitrary at all points, we have 
 
 (ti^rbr ^ 
 
 ■ + bz = 0, 
 
 9 
 the integral of which gives the equation (6) of the free 
 surface before obtained. 
 
 Examples. 
 
 1. A cylinder contains a given quantity of water. If it is 
 rotated round its axis (held vertical), find the angular velocity 
 at which tlie water begins to overflow. 
 
 Let AOB represent the surface of the rotating liquid, the, 
 points A and B being at the top of the cylinder ; let r and /t be 
 the radius and height of the cylinder, and c the heiglit to which 
 the cylinder, when at rest, is filled. 
 
 Then since B is on the parabola, if f is the depth of below 
 
 ^^' 2(1 ^ 
 
 But the volume of the water remains unchanged, and it is the 
 volume of the cylinder minus the volume of the pai'aboloid AOB. 
 
 This latter is ^ . f I Hence 
 
 CO" 
 
 rVt_l,f = r^c, (2) 
 
 CO 
 
 ... ^='-^^SE^ (3)
 
 390 Hydrostatics and Elementary Hydrokinetics. 
 
 The above is on the supposition that the water begins to over- 
 flow before the vertex, 0, of the parabola reaches the base, C, of 
 the cylinder. In this case, with any angular velocity, to, if PQ 
 is the level to which the water rises, and LM is the level at 
 which the water stands when at rest, it is easily proved that 
 
 depth of below LM = height of PQ above LM. . (4) 
 
 Take now the case in which c is so small in comparison with 
 h that reaches C (or the base begins to get dry) before the 
 water begins to overflow. The angular velocity at which 
 
 reaches C is — . Let co = (i + w) ; then if 0' (below 
 
 r T 
 
 C) is the vertex of the parabola, we have 
 
 CO' =2n{i-\-n)c, (5) 
 
 the height of PQ (the water level) above the base is 2c{i-\-n); 
 and if the free surface cuts the base in R, we have 
 
 (6) 
 
 which is the radius of the dry circle on the base. The water 
 will begin to overflow when the height of PQ above the base is 
 A; i.e., ,- 
 
 0, = '*^/? (7) 
 
 which is quite different in foi m from (3) ; so that if c is in- 
 finitely small, i. e., if there is only an infinitely thin layer of 
 water put originally into the cylinder, it will not begin to over- 
 flow until 0) is infinitely great ; and in tliis case GR = r, as it 
 should be. 
 
 2. A heavy cylinder floats with its axis vertical in a liquid 
 contained in a vessel which rotates uniformly round a vertical 
 axis ; find the length of the portion of the cylinder immersed. 
 
 Let PQ, Fig. Ill, be the level of the liquid round the 
 cylinder, and PEDQ the immersed portion, the free surface 
 being APOQB, and the vertex of the parabola. 
 
 Now, by the same reasoning as that in Art. 22, it is obvious 
 that the resultant action of the liquid on the cylinder is the 
 same as that which the liquid would exert on the liquid which 
 would fill the volume POQDEP ; hence the weight, W, of the
 
 Steady Motion under the Action of Gravity. 391 
 
 cylinder must be equal to the weight of this volume of the 
 liquid. 
 
 Let r be the radius of the cy- 
 linder and Om the jjcrpendicular 
 
 from on FQ ; then Om = > 
 
 2g 
 and the volume of the displaced 
 liquid 
 
 = TTr''.PE-Yo\.POQ 
 
 
 ^irr'^PE- 
 
 
 hence U w = specific weight of 
 liquid, 
 
 PE = 
 
 W 
 
 + 
 
 w 
 
 fl^ 
 
 Tir^w 4g 
 
 Figr. III. 
 
 3. A vessel of given form containing water is set rotating 
 round a vertical axis, the vessel and the liquid being in relative 
 equilibrium ; find the greatest angular velocity of the vessel 
 which will allow all the water to escape through a small orifice 
 at the lowest point of the vessel. 
 
 Let the vessel be ACB, Fig. 109, C being its lowest point; 
 
 assume the free surface to pass through C, the latus rectum of 
 
 20 
 the parabola being -— ; then, taking C as origin, the tangent at 
 
 O) 
 
 C as axis of y and the vertical upward line as axis of x, express 
 
 20. 
 the condition that the parabola y'^ = —~ x mtersects the curve 
 
 ACB in no other point than C 
 
 Thus, if the vessel is a sphere (with another small hole at 
 
 the top) of radius a, the parabola will be completely outside the 
 
 sphere if co^ = -• 
 
 a 
 
 4. A cylinder whose axis is vertical is filled with a given 
 mass of gas and set rotating round its axis ; if the gas is 
 assumed to move in relative equilibrium with the cylinder, find 
 the intensity of pressui'e at any point.
 
 392 Hydrostatics and Elementary Hydrokindics. 
 We have, measuring z from the top of the cylinder 
 
 
 dp ■=■ wd( \- z), 
 
 . 1 /<u2r2 \ 
 
 k 
 Also if W is the weight of the gas put into the cylinder, we 
 ^^^e W = fwda, 
 
 where d Q, =^ element of volume at any point, F (Fig. 109). Now 
 if is the angle which the plane of F and the axis, Oz, of rota- 
 tion makes with any fixed vertical plane, 
 
 dil = rdedrdedz; 
 
 
 Integrating with respect to r, the limits of r are o and a, 
 where a is the radius of the cylinder, so that the integrations in 
 r and z may be performed independently, the limits of z being o 
 and h. We easily find 
 
 , . (u^a^ h 
 
 which determines A. 
 
 5. If the cylinder is replaced by a spherical shell rotating 
 about a vertical diameter, solve the previous problem. 
 
 6. A hemispherical bowl containing a given quantity of water 
 is set rotating about a vertical diameter, find the angular velo- 
 city at which the water begins to overflow. 
 
 Ans. If V is the volume of the water, a the radius of the 
 bowl, 
 
 Tia* ^^ ' 
 
 7. If in the last case the angular velocity is increased beyond 
 the value i — J , find how much of the bowl is dry.
 
 Steady Motion under the Action of Gravity. 393 
 
 2 (I 
 Ans. Tt IS dry to a vertical lici^ht n — \^ uliove the 
 
 lowest j)oint. "^ 
 
 8. If a hollow open cone with its axis vertical and vertex 
 downwards containing a given c|uantity of water is made to re- 
 volve round a vertical axis, discuss the question as to the pos- 
 sibility of emptying the cone by increasing the angular velocity. 
 
 9. A narrow hoi'izontal tube, BO^ has two open vertical 
 branches BA and CD, water being poured into the continuous 
 tulte, thus formed, to a given height. If this tube is set rotat- 
 ing round a vertical axis through a point in BC, find the 
 position of the liquid in its state of relative equilibrium. 
 
 Ans. If BO = m, OC = n, the difference of level in the 
 
 2 
 
 two vertical branches is — (m^—n^). 
 
 29
 
 CHAPTER X. 
 
 WAVE MOTION UNDER GRAVITY (siMPLE CASES). 
 
 92. General Equation of Motion. For a particle of 
 any shape which forms part of any moving- material system, 
 we have the fundamental law of motion that the mass- 
 acceleration of the particle is at each instant the exact 
 resultant of all the forces acting on this particle ; and from 
 this it follows that, when the particle belong's to a perfect 
 
 ""'• "-. = .-L* (0 
 
 (J Vj as 
 
 (p. 386) which may be considered as the equation of 
 motion of the particle in any direction (tang-ent to the 
 arbitrary curve AB at P, Fig. 110). 
 
 Now if the direction of the curve AB at P is such that 
 the intensity of pressure, jh does not vary from point to point 
 
 f/.Ti 
 
 along it, -7- will be zero, and we have 
 
 "as 
 
 J = '- (^) 
 
 which gives a rule for finding the direction of the surface 
 of constant pressure intensity at P, viz., — draw a vector PA, 
 Fig. 112, representing the resultant acceleration, a, of the 
 particle at P, and also a vector PF representing the force, 
 g .F, in absolute nnits, exerted on thejitdd per unit mass at P 
 by all causes other than the pressure of surrounding particles;
 
 Wave Motion under Gravity {Simple Cases). 395 
 
 (Is 
 
 (to) 
 
 Fig. 112. 
 
 then the plane through P perpendicjilar to the right line AF is 
 the tangent plane at P to the surface of constant pressure 
 intensity. For, if AF meets the perpendicular PT in r, the 
 vector Pr is at once the component of a along- PT and the 
 
 component of ^.-/^; hence ^7;^ = o alongf PI' at P. 
 
 If the external force is 
 simply g-ravity, we take 
 PA to represent a and 
 PF to represent g. 
 
 The student may 
 verify this construction 
 in the case of a fluid 
 revolving" about a verti- 
 cal axis (Art. 91). 
 
 Instead of using the 
 acceleration, PA, of the 
 particle we may use the 
 
 reversed acceleration, PA\ or force of inertia per unit mass 
 at P (Art. 91) ; and then the result is that t/ie direction 
 of the surface of constant pressure at P is at right angles to 
 the resultant, <t>, of the external force and the force of inertia. 
 (By the external force at P is meant, as before explained, 
 the resultant of all forces, excluding- pressures, which act 
 on the particle.) 
 
 The following is also an important result : if in the fluid 
 we describe a surface of constant pressure intensity, p^ , and 
 also a very close surface of constant pressure intensity, 
 /?2 (Pi and p,^ differing' by an infinitesimal amount), the 
 normal distance between these close surfaces at any point 
 P is inversely proportional to the magnitude of the force <t> 
 at P. 
 
 For in Fig. 112, (a) or (tj), take on the line P<P a point 
 Q indefinitely close to P, and let the surface of constant
 
 396 Hydrostatics and Elementary Hydrokinetics. 
 
 pressure throuo-h Q be described. Then taking the equation 
 (1) with reference to the direction PQ, denoting PQ 
 by i\n, we have 
 
 if 732 and ^j are measured in absolute units of force. 
 
 Now Og means the component of PA along FQ in the 
 sense PQ, and hence in Fig. (a) the right-hand side of (3) 
 is the projection of PF along P(t> minus the projection 
 of PA, which is obviously P4>, which we have denoted 
 simply by 4>. In Fig. (^) the projection of PA along P<1> 
 
 in the sense PQ is negative, so that the right-hand side of 
 (3) is the arithmetic sum of the projections of PA and PF, 
 which is, again, <I>, Hence 
 
 A«=L.^, (4) 
 
 and since at all points on the surface of constant pressure 
 through P w^e have p = p^ , and at all points on that 
 through Q, p = ^2^ ^'^ see that An, or PQ, the normal 
 distance between the two surfaces at any point, varies 
 inversely as <i>. 
 
 The case of a fluid at rest is, of course, a particular 
 instance. In this case 4> is simply the resultant external 
 force per unit mass, and the normal distance between 
 any two close surfaces of constant pressure varies inversely 
 as this force. 
 
 93. Definition of a Wave. Any disturbance w^hich is 
 communicated from point to point of a body whereby each 
 particle is displaced from its position of rest and the 
 relative distances and directions of the particles are altered 
 is called a wave. As a rule, in such disturbances the 
 displacement of each particle from its position of rest 
 in the body is small — or, at any rate, small comi^ared with
 
 PFave Mo/iofi itiuicr Gravity (Simple Cases). 397 
 
 otlier nia<^nitudes which arc involved in the parliculur 
 case concerned. Moreover, the motions of tlie individual 
 j)articles may be very complicated, or may be simple 
 oscillatory motions in small circles or other closed curves. 
 Thus, when in a long tube filled with air the air at one end 
 is disturbed by a sudden impulse along- the tube, the whole 
 air column will be set in motion of a to-and-fro kind, and 
 the disturbance will reach the far end of the tube, while no 
 particle ever departs far from its position of rest. 
 
 So likewise in the case of an iron bar which is sti'uck at 
 one end ; and so, again, in the well-known case of a long 
 stretched string one end of which is fixed while the other 
 end is agitated by the hand ; or both ends may be fixed 
 while the string is rubbed by a bow at any intermediate 
 point. In all these cases the disturbance which travels by 
 communication from particle to particle throughout the 
 medium is called a wave. 
 
 04. Trochoids. If a circle rolls without sliding along 
 a right line, any point ' carried ' by the cii'cle traces 
 out a curve called a trochoid. If the carried point is one on 
 the circumference of the rolling circle, the trochoid becomes 
 the common cycloid ; if the carried point lies outside 
 the circumference, the trochoid is a looped curve ; and 
 if it lies inside the circumference, the locus is devoid 
 of loops. 
 
 Thus, in Fig. 113, let the fixed line on which the circle 
 rolls be LM ; let E be the rolling circle having its centre 
 at and touching the line at B ; let A be the carried point 
 at a distance OA, or r, from 0, while OB = K. Then 
 as the circle rolls along LM, its centre describes the line 
 Ox, parallel to LM. Let C be the position of at any 
 instant ; then if Q is the angle rolled through, we have 
 OC = BI = RO, where / is the point of contact of the 
 rolling circle with LM. Now to find the position, P,
 
 398 Hydrostatics and Elementary Hydrokinetics. 
 
 occupied by the point A, we may measure off the arc Bi 
 equal to BI \ then BOi is the angle, Q, throug-h which 
 the circle — and every rig-ht line carried by the circle — has 
 revolved, the sense of the rotation being denoted by the 
 
 Fig. 113- 
 
 arrow a. Hence the line OA has revolved through this 
 angle, and therefore if w^e draw CB parallel to Oi and equal 
 to r, we obtain the position B to which A has come. 
 
 The trochoid traced out by A is the wavy curve ABQ^, 
 symmetrical at both sides of the line OA \ and obviously 
 the trochoids described by all other points on the circle of 
 radius OA will be merely the same curve in different 
 positions. 
 
 This curve will have a series of crests^ such as that at A^ 
 and a series of hollows or troughs, such as that at Q, — at 
 which point the moving point A reaches a maximum 
 distance, B + r, from LM. 
 
 We may put the case in another way. Instead of 
 imagining a single circle, B, to roll into successive positions 
 and to carry a point A with it, let us imagine at one and 
 the same instant a series of circles, B, B, . . . each of radius 
 B touching LM, and a series of points A, P, ... taken on 
 circles of radius r with centres 0, C, ... so that the angular 
 deflection of the lines OA, CB, ... from the vertical are pro-
 
 Wave Motion under Gravity {Simple Cases). 399 
 
 portional to the distances of the centres from ; then the 
 points A, P, ... all lie on a trochoid. If now these i)oints 
 A, P, ... are all revolving- with the same ang-ular velocity, 
 CO, they will all at each instant lie on a trochoid, which will 
 be merely the trochoid APQ in the fig^ure dis})laced in 
 a direction parallel to LJU. The trochoid will appear to 
 travel towards the right or towards the left of the figure, 
 so that there will no longer be a crest above 0, until all the 
 moving- points A, P,... have completed revolutions in their 
 circles, and then a complete wave length (distance between 
 two consecutive crests or two consecutive troughs) of the 
 curve will have travelled past 0, and past every other fixed 
 point. 
 
 The radius r is called the tracing arm of the trochoid. 
 
 Theorem I. If the radius of the rolling circle is taken 
 
 equal to '—j,, where w is the angular velocity of the points 
 
 ^, P, ... in their circles, the trochoid is a curve of constant 
 intensity of pressure of a liquid whose surface particles when 
 at rest under gravity are A, P, ..., and when set in motion 
 revolve in the smaller circles with constant angular velocity 
 (0, all these circles lying- in the same vertical plane. 
 
 (We assume for the present that such a motion of the 
 liquid particles is a possible one.) 
 
 For, assuming that the typical particle P moves with 
 constant velocity in the circle round C, its resultant acceler- 
 ation is (li^r and is directed in PC; also the dii-ection of y 
 is 76', therefore the direction of <i> is IP, because the re- 
 versed acceleration is cd". CP, and gravity is co^. /C, and the 
 direction of their resultant is IP, which is the normal to 
 the surface of constant pressure. But since / is the in- 
 stantaneous centre of the rolling- circle, IP is the normal to 
 the trochoid ; therefore, &c. 
 
 The trochoid APQ being- that occupied at any instant by
 
 400 Hydrostatics and Elementary Hydrokinetics. 
 
 the siu'face particles of the liquid, the constant intensity of 
 pressure along- it is^>„, the atmospheric intensity. 
 
 If BM and BO are taken as axes of x and y, respectively, 
 the co-ordinates of P are 
 
 y = R — r cos6. 
 
 Tlieorem II. As we descend vertically into the fluid, the 
 curves of constant pressure are also trochoids. 
 
 To get the indefinitely near curve, qNm, of constant 
 pressm'e p, we produce the normal IP throug-h P and on it 
 take a length given by (4) of Art. 92. 
 
 Let, then, PN = ^^^, Fig. 114. Now 4> = 0,2. 7P, 
 thereiore 
 
 wc^^IP 
 
 (0 
 
 h 
 
 1 
 
 
 
 L 
 
 wf- " 
 
 
 
 ^ 
 
 \. 
 
 i! 
 
 
 — 
 
 fc 
 
 kr^. 
 
 
 
 
 r 
 
 1 — 
 
 _-V 
 
 __ — 
 
 We shall now show that the locus of iV is a trochoid. 
 
 Let P' be a point on the radius CP^ 
 let PP' = — dr, and consider the 
 trochoid which is traced out by P' 
 as the circle of radius CI, or R, rolls 
 along L3I. Let this trochoid receive 
 a motion of translation P^F verti- 
 cally downwards. It will then, in 
 this new position, be the trochoid 
 generated by the rolling of a circle 
 
 of radius R along a horizontal line at a depth equal to 
 
 PT below B3I, and if GC'=Fr, the centre of the cii-cle 
 
 which generates this trochoid is C. 
 
 We shall now show that the lengths PP' and PTcan 
 
 be determined so that the trochoid last mentioned shall 
 
 pass through all the points N. 
 
 Let Z CIP = cjj, and ^ICP = 6, as in Fig. 114. Then 
 
 Fig. 114.
 
 JVaz'C Motion under Gravity {Simple Cases). 401 
 
 if PiV = sum of projections of PP' and PT along- IN, the 
 trochoid will pass throug-h N. Assume, then, 
 
 PN= ~ dr . cos (^ + <|)) + P' V. cos (/>. 
 
 Putting P'V=clr], and observnng that cos</) = ~^t^^ 
 
 , . r sin ^ , ^^ 
 
 and sin = — ^j^— , we have 
 
 PN. IP = rdr + Pr/77 - (P^// + n/7;) cos ^, . . (2) 
 
 so that PN . IP will be constant, as (l) requires, at all 
 points P of the surface trochoid, provided that 
 
 Pdr + rdr] = 0, (3) 
 
 and we can, of coui'se, choose dr and drj so as to satisfy this 
 condition. 
 
 As we descend vertically in the fluid the relation (3) is 
 that which holds between each trochoid and the next, so 
 that if we integrate (3), we obtain the relation between the 
 tracing arm and the depth, ij, below C, (or below the line of 
 centres of surface particles,) of the centre of the correspond- 
 ing- cii-cle. Denoting by r^y the tracing- arm of the surface 
 trochoid, we have, therefore, 
 
 V 
 
 r = r^e ^ (4) 
 
 Equation (2) then gives 
 
 P2 _ ,.2 
 PN.IP = 'l-j^.dr^, (5) 
 
 and (1) shows that, if^— j9q is denoted by dj), 
 
 W^ — r"- 
 dp — wuP- jT- .dr], (6) 
 
 from which and (4) the value of p is easily found in terms 
 of r]. 
 
 To show now that the assumed motion of the liquid is a 
 
 Dd
 
 402 Hydrostatics and Elementary Hydro'kinetics. 
 
 possible one, we sball adopt the method of Rankine (see 
 his Miscellaneous Scientific Paj)ers, p. 483). 
 
 If the motion of the particles in vertical circles, as 
 descrilied, is possible, it will be equally possible supposing 
 the whole liquid to receive a horizontal motion of transla- 
 tion equal to ^w, i.e., the velocity of the centre C of the 
 rolling- circle, in the sense LM ; and conversely. Now this 
 motion combined with the velocitv rco of P at rio-ht anHes 
 to CP will cause P to have a resultant velocitv along- the 
 tang-ent to the trochoid at P ; for, a velocity co . CP at 
 right angles to CP combined with a velocity w . IC 
 at rig-ht angles to IC will give a velocitv w . IP at right 
 angles to IP, by the proposition of the triangle of velo- 
 cities. 
 
 Now supposing the liquid at P to be moving along the 
 trochoid APCl and also the liquid at i\^to be moving along 
 the consecutive trochoid, mNci, so that the space between 
 the two trochoids is a channel of flow, the same quantity 
 of liquid must How across each normal section PN of this 
 channel in the same time ; and the fulfilment of this 
 condition is not only necessary but suflicient for the possi- 
 bility of the motion. 
 
 But the condition is obviously fulfilled ; for, the velocity, 
 V, of P along the trochoid is &> . IP. and IP . PN has been 
 proved constant, so that we have 
 
 and V . PN is proportional to the quantity which flows 
 across PN iper unit time ; this is, therefore, the same at all 
 points of the channel ; and hence the motion is possible. 
 
 Observe that the motion here shown to be possible does 
 not require the displacements of the particles from their 
 original positions to be small : the circles, each of radius r,
 
 Wave Motion midcr Gravity (Siuiple Cases). 403 
 
 (lescn])ed l)y thorn may be of any ma<,''nitnclo, with tlic sole 
 condition thut the depth of the litiuid must be very <rreat, 
 because at the bottom of the liquid mass the lifjuid must 
 either be at rest or move along- the bed ; and since all 
 throug-h the liquid the jiarticles describe circles, these 
 circles must become infinitesimally small at the bottom ; 
 biit since their radii are g-iven by equation (4). if /• becomes 
 infinitesimal, -q must be very g-reat. 
 
 This wave is know^n as Gersfner''s Trochoiilal Ware, having 
 been first discovered by Gerstner ; it was afterwards dis- 
 covered independently by Rankine. 
 
 The line, OCx, on which lie the centres of the circles 
 described by the particles of liquid forming the surface, 
 does not coincide with the surface of the liquid w-hen 
 it is at rest, but is vertically above this surface. Let SW 
 be the surface when the liquid is at rest ; let the tangent 
 to the trochoid at the low^est point, Q, meet the vertical 
 through A in 7/; let Ox be at a distance z above Sir, and 
 let the perpendicular from Q on SIF be Qn. Then the 
 volume APQIIA of disturbed liquid occupies the rectan- 
 gular volume Iln when at rest. But the perpendicular 
 from P on QH is r(i+cos^}, while the distance of J' 
 from AH is BO — r sin 6 ; hence the area APQIIA is 
 
 .;• / "(i +cose){P-rcosO)(/e, i.e., -r(P --), 
 
 while the area of the rectangle IIu is T:R{r — :) ; hence 
 
 z = — ^J 
 in 
 
 which is the height of the line of orbit centres above 
 the surface of the still liquid. 
 
 It is easy to show that at any instant after the regular 
 wave motion has been established all those particles which, 
 
 D d 2
 
 404 Hydrostatics and Elementary Hydrokinetics. 
 
 when the liquid was at rest, lay on a vertical line will lie 
 
 on a curve having this line for an asymptote. For, if we 
 
 draw the vertical line IC and produce it downwards into 
 
 the liquid, the position of the particle P is obtained by 
 
 drawing CP equal to i\ and making with IC the angle d, 
 
 00 
 or -^5- ; and if we take anv other point, 0\ on 10 and 
 
 ^ ' _!? 
 
 draw O'P parallel to CP and equal to r^e ^, where 
 
 r/ = OC , we obtain the position of a particle P' which 
 
 originally lay on the line 01. As the point 0' travels 
 
 down 10, the radius O'l" diminishes, according to the above 
 
 law, and hence the locus of P' approaches 7C asymptotically. 
 
 This curve possesses the simple property that if the 
 tangent to it at any point, P', meets the vertical line 10 in 
 1\ the leno-th CT is constant and = R. From this the 
 slope of the columns which were originally vertical is easily 
 determined. 
 
 The inclination of the surface of the wave ^PQ to 
 the horizon, being zero at A and at Q, is a maximum 
 at some intermediate point. In Fig. 114 it is obvious 
 that ^ is the inclination of the surface at P, and that since 
 10 = P, and OF = r, is a maximum when OF is per- 
 pendicular to PP In general 
 
 R — r cos 6 
 
 cot 9 = 1 • 
 
 It is obvious that when any particle, P, completes a 
 revolution in its circle, the wave has travelled horizontally 
 over one wave-length, so that if T is the time of travelling 
 
 ovex a wave-length, co = —^ ; and if t is the time since the 
 
 crest. A, of the wave has passed over the point 0, we have 
 
 t 
 e = 2 77^,-
 
 Wave Motion tinder Gravity {Simple Cases). 405 
 
 The wave-leng-th, A, is 2qII, or 2-nli, and the velocity, 
 
 r, of propag-ation of the wave is Rca, or since 11 = ^, 
 
 V = ^gli, and A = 2 7r7i'. 
 
 If T is the time taken by the wave to travel a wave- 
 leng-th, X =^ vT \ therefore 
 
 Momentum of the Wave. Consider the thin layer of 
 liquid between two consecutive trochoidal surfaces, APQ, 
 7nNq, and take an element of this at any point P, the 
 element being- contained between two consecutive values of 
 FN, the normal distance between the layers. If ds is the 
 element of length of the trochoid at P, the volume of the 
 element of liquid is PN.ds multiplied by its length per- 
 l)endicular to the plane of the fig-ure. This last dimension 
 we shall assume to be unity. Now ds = IP . dO, therefore 
 the weig-ht of the liquid element is 
 
 w.IP.PN.dd, 
 
 which at all points is simply proportional to dO, since 
 fP . PN is the same at all points on the sm-face of the 
 wave. 
 
 Now the velocity of P is cor, whose horizontal component 
 is wr cos ; hence the horizontal momentum of the element 
 is proportional to cos 6 dO, and the total horizontal com- 
 ponent for the half wave-length qll, or for a whole wave- 
 length, is zero. 
 
 This results from the fact that the particles near the 
 crest, A, are moving- backwards while those near the 
 trough, Q, are moving in the reverse direction, as seen 
 in the fig-ure.
 
 4o6 Hydrostatics and Elementary Hydrokinetics. 
 The horizontal momentum for a quarter wave is 
 
 ru).w .IP . FN / ^cos e (16, i, e., 
 Jo 
 
 rui.w.IP.PN. 
 Here substitute for IP . PN from (5}, and we have 
 
 v).o,^j^{R''-T'")dr^, or -w.oo{Pi^-r^)dr; 
 
 and if we integrate this from r = r^ to r = o, we obtain 
 for the whole mass of liquid 
 
 Energy of the Wave. Let us calculate the amount of 
 kinetic energy contained between the portion APCl of the 
 trochoid and the corresponding portion of the next con- 
 secutive trochoid. 
 
 The velocity of the particle at P being cor, the element 
 of kinetic energy is 
 
 w.IP. PN .dd. ^^', or -,v — de. (li- - r2) rdr, 
 whose integral for the half wave AP(l is 
 
 
 — WTi — (R'^ — r^) rdr : 
 2g^ ' ' 
 
 and the integral of this from r = r^ to ;• = o is 
 
 since ^ = R. 
 
 Calculate now the static energy of the layer APQ. The 
 height of the element at P above the level, SJf] of still
 
 Wave Motion under Gravity (Simple Cases). 407 
 liquid is /•cos^+ ,,, so that the clement of static energy 
 
 'J 
 
 is cv . IP . PNdO (r cos 6 + ^,) • Now the inte":ral of the 
 term in cos 6 for the half wave length vanishes, and since 
 
 
 ,. is the same as j the static energy of the layer is 
 
 equal to the kinetic energy above found. The energy 
 of the wave, then, is half kinetic and half static. 
 
 95. Small Displacements. Assuming that the particle 
 of liquid which, when at rest, occupies the point {x, y, r), 
 or jP, never moves very far from this position while the 
 liquid is in motion, the co-ordinates, x\y\ z\ of its position 
 P' at any instant may be denoted by 
 
 «^ + ^, y + ^. ^ + c 
 
 where ^, jj, Care small quantities each of which depends on 
 the time, t, as well as on the values of x^ y, z. 
 
 Confining om* attention to the motion of a liquid under 
 gravity, we shall assume the displacements of all particles 
 confined to the vertical plane x, y — i. e., we consider the 
 motion to take place in two dimensions only — the motion 
 being the same in all planes parallel to the plane x, y. We 
 shall take the origin of co-ordinates at a point on the bed 
 of the liquid, the axis of x horizontal, and that of y verti- 
 cally upwards. The displacement ( is zero ; and the 
 
 components of acceleration of any particle are — ^ ana -^ . 
 
 Hence for the motion of the particle the equations are, 
 
 by (i) of Art. 92, 
 
 dl^^_g_dp_^ ,. 
 
 cU^ w dx 
 
 W ^ wd/ ^ ^
 
 4o8 Hydrostatics and Elementary Hydrokinetics. 
 
 . . d (Ix d f d^^~'^ d 
 
 ^ ' dx dx dx ^ dx^ dx 
 
 dif ^ dij' dy 
 
 Also since the liquid is incompressible, ^ and y] must 
 satisfy the condition 
 
 . .7^+^ = ° (3) 
 
 These equations contain the theoiy of small displacements 
 of a liquid acted upon gravity only, irrespective of the 
 particular kind of motion which may exist in any case. 
 
 We shall now proceed to consider some particular kinds 
 of motion in which the displacements of all particles 
 are small ; and we assume, as above implied, that the axis 
 of X is in the direction in which the disturbance travels 
 along the surface of the liquid. 
 
 96. Oscillatory Waves. Assuming* the displacements 
 of the particles to be small, and also periodic, we shall have 
 
 where n is a constant ; for, it is well known that these are 
 the equations of oscillatory motion, whose integrals give 
 i = A cos (nl — a), where A and a are arbitrary constants, 
 with a similar value of ?; ; so that the values of ^ and i] 
 
 277 
 
 repeat themselves whenever t increases by — • 
 For such motion, then, the equations are 
 
 ''^-i% w
 
 Wave Motion under Gravity {Simple Cases). 409 
 
 W (I If ^ ' 
 
 (IS, (hi , . 
 
 i^^ir" ''' 
 
 Hence -7-, 7-7 = o : and since £ and r? are small, we 
 
 (hj dx V. / 
 
 can take -r-, = -t"? and -7-^ = 7,,. Therefore also 
 fto (W ay uy 
 
 r,-T, = ° <•'* 
 
 From (4) and (5) we have 
 
 ^ + ;^ = o,ory-£=o, .... (6) 
 
 , d-T] d'-rj , , 
 
 and — + -— = o, or V-/? = o (7j 
 
 Now from (1) the value of f is -4 cos {nt— a), where ^ 
 and a are independent of i! but may involve x and y. To 
 satisf)^ (6) we shall assume this form for $, with the further 
 assumption that the amplitude ^ is a function of 1/ alone, 
 i. e,, it depends on the depth of the disturbed particle only, 
 while the phase, a, is the same for all disturbed particles 
 which originally lay on the same vertical line. Hence the 
 general value of ^ will be given by 
 
 ^ = ^ cos {nt — mx) + B sin {nt — nix), . . . (8) 
 
 where A and B are functions of y only, and n and m are 
 indcj)endent of x, y, and t. 
 
 A similar expression gives the value of r? ; but the latter 
 is, of course, deducible from f by means of (4) and (5). 
 
 The periodic nature of a wave disturbance travelling
 
 410 Hydrostatics and Elementary Hydrokinctics. 
 
 throug'h a liquid, or throug-h any disturbed medium, will be 
 readily understood from Fig-. 115, in which AB represents 
 the surface of the li(|uid when at rest, and is the orig-in. 
 
 P'JP'T 
 
 Fiji;. 11: 
 
 Let P be the position of a particle when the surface is at 
 rest, AT being- .r, and V be its position at any time, t. 
 Then, considering* the value of ^ only (since tj runs throug'h 
 all its values in the same time as ^), we see that when t is 
 
 It: 
 
 increased by the time — , the position of P' will be exactly 
 
 % 
 
 the same as at the time I. Denote this interval by 1\ 
 so that 
 
 T = 
 
 377 
 
 n 
 
 (9) 
 
 Then T is ihe tme in which every imrticle completes its 
 osclliation. 
 
 Again, considering the disjjlaced positions of all particles 
 at the same time /, let P" be the position occupied at 
 this time by another particle, whose abscissa, along AB^ 
 was originally a' . 
 
 Then, if nt — mx' -=111 — wx—^tt, the displacements (f, tj) 
 of Q will be exactly the same as those of P. This gives 
 
 x' — x ■= , and this is therefore the distance, at each 
 
 ///
 
 Wave Motion under Gravity {Simple Cases). 411 
 
 instant, between all particles which are in the same state, 
 ov j}//ast', of disturbance. This distance is the wave-leng-th 
 of the disturbance. Denoting- it by A, we have 
 
 A = ^ (.0) 
 
 m 
 
 Of course the wave-length is also the distance between 
 any two successive crests, (/, C" . 
 
 It is, then, obvious that the thue, T, taken by a7ir/ jmrtide, 
 7^, to complete its oscillation is also lite time taken tjy the 
 disturbance to travel over a wave-lenyfh. 
 
 We now proceed to determine the values of A and B. 
 Substituting the value (8) of f in (6) and expressing the 
 fact that (6) is satisfied for all values of t, w^e have 
 
 d'^A , , rl^B ,^ , . 
 
 — iirA — o and -j-r — irrB — o, . . • ( 1 1 j 
 
 df df 
 
 which ffive 
 
 b' 
 
 A = Be'"" + 3Ie-'"", 
 
 B = i/'e"*.^ + il/'e-"'^ 
 
 where L, M, ... are constants, whose values must be deter- 
 mined from considerations not yet mentioned. 
 These values of A and B give 
 
 ^ = {Le"'" + Me-'"'^) cos [nt-mx) 
 
 + {L'e"'y + yl/ V""} sin {nt - 7»x). ( 1 2) 
 
 To determine 77, equate -r- to ^ 5 and then integrate ; 
 thus 
 ■<]=-[ Lt-"'y - J/e-'"") sin («/ - mx) 
 
 + (ZV"^ - i¥'e-'"^) cos {nt - mx) +f ( y), (13)
 
 412 Hydrostatics and Elementary Hydrokinetics. 
 where f{y) is an unknown function of y, to determine 
 which equate — to — 7- ' and we find that f\y) — o, 
 
 •'• fi^) is a constant, C. 
 
 We shall now assume the bed of the liquid to be a hori- 
 zontal plane, so that the undisturbed liquid is of uniform 
 depth, /i. Now the displacement of every particle at the 
 bottom takes place along the bed of the channel ; hence 
 ri = o when y = o, whatever t may be. 
 
 This g-ives a zero value to the above constant C, and 
 also makes j^j ^ j^^ M' = L', 
 
 so that we have 
 
 ^ ={,-'''« + 6-"^^) {Lco^{nt — mx) + L'Qva.[nt — mx)], . (14) 
 
 rj = [e^'^-e-"'^) { -L sin{?ii - mx) + L' cos (nt-mz)} . (15) 
 
 There is this further consideration, that on the free 
 surface of the liquid p is at all times equal to p^ , the atmo- 
 spheric intensity of pressure ; so that 
 
 dp 
 
 -77 = o at the free surface (16) 
 
 at 
 
 Now since , , 
 
 we have from (2) and (3) 
 
 I dp = n^ {^dx ^r]dy)-gdy\ . . . .(17) 
 
 in which, of course, the term ^dx + rjdy is a perfect differ- 
 ential ; and (17) g-ives 
 
 ^p= -(e"'« + e-'^'')(-Lsmcb + L'cos(t>)-(//+C, . (18) 
 where ^ stands for nt — mx, and C is a constant.
 
 Wave Motion under Gravity {Sinip/e Cases). 413 
 
 Since /=>/ + i], if we use the hyperbolic notation 
 c'"y + e-""" =. 2 cosh w/y ; e"*" — e" "•" = 2 sinh wy, (19) 
 we have 
 
 — n = 2 (i^ sin (|) — i^' cos (/j) f cosh mv+ff sinh »zv) 
 
 to ^ m ' 
 
 -gy^C. (20) 
 Hence the value of j'j at the surface will be independent 
 
 — cosh w?^ — fl' sinh ?;i/i = o, . . . . (21) 
 m 
 
 since 1/ = h for surface points. 
 
 If V is the velocity of propagation of the wave 
 
 K=^vT, (22) 
 
 so that (21) becomes 
 
 27r "^ A -^ 
 
 The velocity with which the wave is propagated will there- 
 fore depend on the depth of the liquid and on the length of 
 the wave. 
 
 Lo7ig waves in shallow water. If the depth of the liquid 
 is small compared with the length of the wave, 
 
 e '^ +e ^ =2, neglecting -j, and e ^ —e -^=471-; 
 and in this case ^ ^ y^^ (24) 
 
 which gives the well-known result that in shallow water, 
 all waves (provided their lengths are very much greater than 
 the depth of the water) are propagated with the same 
 velocity, which is that acquired by a body falling freely 
 through a height equal to half the depth of the water. 
 
 In this case it also appears from (14) and (15) that the 
 maximum horizontal displacements of all particles are much 
 greater than the maximum vertical displacements. In
 
 414 Hydrostatics and Elementary Hydrokinetics. 
 
 "•eneval, these equations show that the orbit described by a 
 disturbed particle about its position of rest is an ellipse 
 
 ^"' + ^W = 4(i' + i''), • (25) 
 
 eosh^^O sinh^(^0 
 
 whose horizontal axis in the case of long- waves in shallow 
 water is much greater than the vertical axis. 
 
 The value of ^ in this case (i. e., when - is small) shows 
 
 also that, to the second order of small quantities, it is true 
 that all particles which were orig-inally in the same vertical 
 plane perpendicular to the direction of propag-ation of the 
 wave, will at all times be in a common vertical plane 
 parallel to the original one : the vertical motions of these 
 points are, of course, different ; so that we can suppose the 
 motion to be produced by parallel vertical planes of particles 
 oscillating" backwards and forwards horizontally, while the 
 particles in these planes have small up and down motions 
 in the planes. The problem of such waves is, indeed, often 
 solved by starting with this assumption, and the equation 
 (4) is then used in an integrated form as follows : let the 
 vertical section Tp of the liquid at rest have abscissa x ; and 
 assume the particles in this section to occupy the plane 
 P7/ at any time when the liquid is in motion, the distance 
 between the planes P/j and V p being- f. 
 
 Let Q^ be a vertical section parallel to Pp, at distance 
 A a; from Pp, and let the particles in this plane occupy the 
 plane Q'/ when the liquid is in motion ; then, since 
 ^ z=.f{x), the displacement of Cl'q from its orig-inal position 
 
 is f{x + A a,-), i.e., ^ + 7- ^^ ; and the abscissae of P// and 
 
 QY ^'eing X \ $ and x + A,c + $+ — Ax, the distance be-
 
 IVnvc Motion under Gravity {Simple Cases). 415 
 
 tween these planes is f i + , ) A.x'. We liave now to cx- 
 
 ^ ax ' 
 
 press the foct that the vohime of li(ini(l contained between 
 the planes Fji and (^q is equal to that contained between 
 rp' and QV- 
 
 Let the elevation o{ P' above All be e ; then Vp =h + f, 
 and if the sides of the channel containing- the li(|uid are 
 vertical and parallel, i.e., if the channel is a rectangular 
 canal, its cross-section perpendicular to AB is of constant 
 area, so that we have simply 
 
 /iAa; = (// + e) (i + '--) Aa?, . . . (26) 
 
 it kAj 
 
 .-. //^+e = o (27) 
 
 ax 
 
 Now this equation is the equivalent of (4), because if we 
 integrate (4) with respect to y from the bottom to the top. 
 i.e., from p to P, we have (since ^ does not sensibly in- 
 volve j^) ^ rn^^^ + f^'^d =0 
 dx Jq J{) dji 
 
 or h - + e = o. 
 ax 
 
 Observe that this equation holds in all cases in which 
 the vertical motions are small compared with the hori- 
 zontal, and not merely in the case in which the motion is 
 oscillatory, so that we are now dealing with the general 
 equations (i), (2) of Art, 95 instead of the special equations 
 (2), (3) of Art. 96, which are limited to oscillatory motion. 
 
 Again, (2) of Art. 95 shows that if ^, is very small, 
 
 the value of p at any point in the disturbed liquid is 
 y,^ + 1^ iji + e ~y') where p^ is the value of p on the free
 
 4i6 Hydrostatics and Elementary Hydrokinetics. 
 surface ; hence from (i) of Art. 95, 
 
 dfi ^ dx 
 
 =^''0 (^«) 
 
 Now if V = -Jgh, the integral of this equation is well 
 knovv^n to be 
 
 ^= ^{x-~vt) + ^{x-\-vt\ .... (29) 
 
 where ^ and -^ are any symbols of functionality whatever. 
 
 If and t/^ are circular functions (sines or cosines) the 
 disturbance is periodic, or oscillatory, because the values of 
 ^ are reproduced after a constant interval, and then the 
 case becomes that which we have just discussed — viz., 
 oscillaiory motion in which the vertical displacement is very 
 small with respect to the horizontal. 
 
 In the general case now supposed — viz., motion in which 
 the vertical displacement is very small with respect to the 
 horizontal, while the motion is not necessarily of the oscil- 
 latory or periodic kind, the function ^{x — vt) denotes a 
 disturbance travelling' in the positive sense of x, while 
 y}/{x + vt) denotes one travelling in the oj)posite sense ; and 
 the velocity of each is v. 
 
 TFaves in very deep water. Applying now the results 
 obtained for periodic or oscillatory waves to the case in 
 which the depth, //, of the liquid is vastly greater than A, 
 the wave-length — as in the case of waves in the ocean — we 
 
 2jrA 7 
 2 77 A 
 
 have e ^ negligible, so that tanh — — = i, and (23) gives 
 
 .= ./^, (30) 
 
 SO that the velocity depends on the wave-length. 
 
 Also the ellipses described by the particles become circles.
 
 IVave Motion under Gravity {Simple Cases). 417 
 
 Channel of any uniform cross-section. In obtaining- e(ina- 
 tion (26) we have assumed the cross-section of the channel 
 to be rectang-ular. But if the cross-section has any form, 
 the corresponding- equation is easily obtained. Let A be 
 the area of the cross-section ; then, still expressing- the fact 
 that the volume between the planes F'p\ Q'q is equal to 
 that between Pp, Qq, if e is the elevation of P' above the 
 surface AB and /j is the breadth of the channel at the level 
 AP, the area of the cross-section P"/ is A + be, and we have 
 
 AAx={A + de)(^i+'^^jAx, . . . (31) 
 
 which shows that in the previous results for a rectang-ular 
 
 A 
 channel we have merely to replace /i hy j> since the 
 
 dynamical equations, (i), (2) of Art. 95, still hold. Hence, 
 
 in particular, the velocity of the disturbance is (-^)' for 
 
 lonff waves. 
 
 Channel with sloping sides. The determination of the 
 displacements of water contained in a channel of any cross- 
 section— even when this cross-section is the same at all 
 points along- the channel — has not yet been effected. 
 
 Moreover, the problem has not been solved even in the 
 simple case in which the cross-section is a triangle whose 
 two sides are equally inclined to the horizon, except when 
 the inclination is 45° or 30°. The solution for the former 
 case was obtained by Kelland on the assumption that 
 the motion of the water is irrolational, i. e., that the com- 
 ponents of displacement are the differential coefTicients 
 of a function of x, y, z. Thus, let the cross-Bection be 
 an isosceles triangle whose inclined sides are AB and AC, 
 its base BC being horizontal; let the horizontal line 
 
 E e
 
 4i8 Hydrostatics and Elementary Hydrokinetics. 
 
 through A be taken as axis of a?, the vertical upward 
 line throug-h A as axis of z, and the axis of ^ parallel to the 
 breadth of the channel. Let the small components of 
 displacement of the particle which at rest occupied the 
 point {x, y, z) be ^, 77, C Then the equations of motion are 
 
 en 
 
 w dx 
 
 \ 
 
 
 \ 
 
 ff_dp 
 zo dy 
 
 dt'^ -^ wdz' 
 
 With these must be combined the equation 
 
 dC 
 
 {?>3) 
 
 d^ d^ _„ _ 
 dx dy dz ' 
 
 (34) 
 
 which implies the ineompressibility of the liquid. 
 
 Now whatever values we determine for ^, 77, C, must 
 satisfy the boundary condition, viz., that the displacement 
 of every particle which, when at rest, lay on the side AB 
 must be along AB^ and the displacement of every particle 
 which lay on AC must be along AC\ i.e., the values of 
 f, 77, C, must be such that 
 
 («) 
 
 when y — zz=o we have rj — ^ = o, 
 and when y ■\-z ■=. o we have 77 + ^ = o. 
 
 Suppose f, 77, ^ to be the differential coefficients with 
 respect to x^ y, z, respectively, of some function ^, and 
 assume 
 
 (P =f(y,z). cos (fd-mx) (^s) 
 
 Then (34) gives 
 
 cPf . dY 
 
 dy 
 
 ^z + 
 
 dz^ '' 
 
 (36)
 
 Wave Motion under Gravity {Simple Cases). 419 
 
 To satisfy this assume 
 
 /(,y, z) = ./ (6-^" + e-^^) [e^- + e-'% . . . (37) 
 
 where A is a constant. 
 
 Then C^6) o-ives 
 
 F~ + P = w\ (38) 
 
 which will be satisfied by assuming 
 
 k = m cos a, I =■ m sin a. 
 Calculating- f] and ^from (37) with these values of k and 
 /, we find that the conditions (a) will be satisfied if a = - • 
 
 Hence the required function (^ whose differential coefficients 
 give the displacements is given by 
 
 m;i mil mz mz 
 
 <f) = A {e'^^ + e~'^-){e^^ + e~^-) COS {nf, — mx). (39) 
 
 The subject of Wave Motion will be found treated 
 at length in Basset's IL/drodj/namics, vol. ii., Greenhill's 
 JFave Motion in Hydrodynamics (reprinted from the American 
 Journal of Mathematics), Airy's Article on Tides and Waves 
 in the EncyclopiEdia Metropolitana, and the Mathematical 
 Papers of Stokes and of Green. 
 
 E e 2
 
 J
 
 INDEX 
 
 Absoldte temperatnre, 191, 257. 
 Adiabatic transformation, 246. 
 
 work done in, 249. 
 Air thermometer, 197. 
 Air, moist, weight of, 234. 
 Andrews's experiments on lique- 
 faction, 269. 
 Angle of contact of liiiuid and solid, 
 
 324- 
 Aqueous vapour, pressure of at boil- 
 ing point, 236. 
 present in air, 267. 
 Archimedes, principle of, 115. 
 
 screw of, 278. 
 Area, plane, pressure on, 45, 102. 
 not influenced by molecular forces, 
 
 311- 
 
 Atmosphere, any layer of liquid may 
 
 be treated as, 42. 
 pressure at any height in, 80. 
 Avogadro, law of, 196. 
 
 Balloon, pressure not uniform on, 
 
 117. 
 Barometric formula, 209, 238. 
 Bellows, hydrostatic, 20. 
 
 pump, 275. 
 Bernoulli, theorem of, 371. 
 Boiling point of water at various 
 
 pressures, 237. 
 Boyle and Mariotte, law of, 21, 186, 
 
 187, 189. 
 
 Bramah's press, 19. 
 Buoyancy, principle of, in, 113, 
 122, 179. 
 centre of, 113. 
 
 Capillary surface in front of vertical 
 plane, 353. 
 between two parallel planes, 357. 
 in cylindrical tube, 360. 
 between inclined planes, 363. 
 tubes, rise or fall of liquids in, 
 
 333- 
 Carnot's cycle, 246. 
 view of heat, 253. 
 Catenoid, 350. 
 Centre of pressure, 52. 
 
 position of on plane area, 53. 
 
 ,, ,, on parallelogram, 54. 
 
 „ „ on triangle, 55, 56, 57. 
 
 ,, ,, in general, 100. 
 
 ,, „ referred to principal 
 
 axes, 102. 
 change of by rotation of area, 
 
 104. 
 Compressibility, cubical, 20. 
 Compression of water in ocean, 107- 
 Condensing air pump, 286. 
 Co-ordinates, polar and cylindrical, 
 
 93- 
 
 Critical temperature of gas, 269. 
 
 Curl of force and vector, 84, 87, 
 181.
 
 422 
 
 Index. 
 
 Dalton and Gay-Lussac, law of, 190. 
 
 related to Boyle's, 193. 
 Density, 24. 
 
 laws of, wliich allow equilibrium, 
 
 88, 89, &c. 
 of water at any depth in ocean, 
 107. 
 Displacement, constant, theorem on, 
 
 153- 
 work done in, 175. 
 Diving bell, principle of, 201. 
 
 Efficiency of reversible engine, 254. 
 Energy of a gas, function of tem- 
 perature alone, 241. 
 Equilibrium, general equations of, 
 82. 
 necessary condition of, 84. 
 Equipotential sui-faces, 85. 
 Expansion, coefficient of for gas, 
 190. 
 
 Films, liquid, 364. 
 
 Fire engine, 276. 
 
 Floating body, positions of, 145. 
 
 Fluid, perfect, definition of, 3. 
 
 Forces, bodily and surface, 12. 
 
 fictitious, introduction of, 117. 
 
 gravitation and absolute measure 
 of, 78. 
 Free surface of a liquid, 39. 
 
 everywhere at same level, 39. 
 French Commissioners' formula, 236. 
 
 Gas, perfect, definition of, 21, 185. 
 pressure at top of house apparently 
 
 greater than at bottom, 80. 
 general equation of transformation 
 
 of, 195, 207. 
 weight of, 205. 
 
 pressure of, on kinetic theory, 2x7. 
 Gases, mixture of, 226. 
 
 mixture with vapours, 233. 
 
 Gases, liquefaction of, 268. 
 Gay-Lussac's law, 190. 
 Geissler air pump, 287. 
 Gerstner's trochoidal wave, 403. 
 Green's equation, 177. 
 
 Hero's fountain, 377. 
 Heterogeneous fluid, equilibrium of 
 body in, 130, 132. 
 
 buo^'p.ncy in, 179. 
 Hydraulic Press, 17. 
 
 Screw, 277. 
 
 Screw, differential, 281. 
 
 Ham, 282. 
 Hydrometers, 294. 
 Hygrometer, 265, 
 
 Irrotational motion, definition of, 
 
 417. 
 Isothermal transformation, 186. 
 work done in, 200. 
 
 Joule's equivalent, 239, 240. 
 
 experiment on energy of gas, 241. 
 
 Kinetic theory of gases, 216. 
 
 Laplace's formula for molecular 
 
 pressure, 309. 
 Latent heat of steam, 261. 
 Level surfaces, 85. 
 Level of free surface of water, 39. 
 Line and surface integrals, I Si. 
 Lines of resistance in masonry dams, 
 
 63. 
 Liquefaction of gases, 268, 272. 
 
 Manometers, 292. 
 Mariotte's bottle, 379. 
 
 law, 21, 186, 187, 189. 
 Mass moments, theorem of, 30. 
 M<=Cay's proof of theorem on centre 
 
 of pressure, 60.
 
 Index. 
 
 423 
 
 Metacentre, 154. 
 
 experimental determination of, 
 
 159- 
 Metacentric evolute, 165. 
 Moist air, weight of, 234. 
 Molecular forces, 298. 
 Moment of stability, 156. 
 Motion of fluid, equation of, 386, 
 
 394- 
 Mountain, lieigbt of, by barometer, 
 
 209. 
 
 height of, by boiling point, 238. 
 
 Nodoid, 351. 
 
 Orifices, flow through, 380. 
 Oscillatory waves, 408. 
 
 Parallel forces, centre of, 28. 
 Pascal's principle, 13, 42. 
 Pressure, intensity of, constancy 
 round a point, 6,11. 
 
 intensity of, <lue to gravity, 38, 77. 
 
 uniform on closed surface has no 
 resultant, 113. 
 
 molecular, 305. 
 
 intrinsic, of liquid, 311. 
 
 gauge in a stream, 374. 
 
 of water vaj^our at boiling points, 
 
 237- 
 Prism, floating, positions of, 149. 
 
 Pumps, water, 273. 
 
 air, 284. 
 
 Quincke, on range of molecular 
 forces, 302. 
 
 Regnault's formula for total heat of 
 
 steam, 261. 
 Relative volume of steam, 215. 
 Revolving fluid, 383. 
 
 Saturated vapour, 231. 
 
 Screw of Archimedes, 278. 
 Se|iarat6 eiiiiilibriuni, j)rinciple of, 5. 
 'Skin' of a liquid, 339. 
 Specific weight and gravity, 22. 
 
 heat of a gas, 244. 
 Sprengel pump, 291. 
 Stability of floating body, 139, 158. 
 
 in two fluids, 166. 
 
 of floating vessel containing liquid, 
 168. 
 
 in heterogeneous fluid, 170. 
 Steam, total heat of, 260. 
 Strain and stress, intensity of, i , 4. 
 Stream lines, 371. 
 Superposed liquids, 40. 
 Surface, closed, subject to uniform 
 pressure, 113. 
 
 unclosed, resultant pressure on, 
 118. 
 
 of floatation, of buoyancy, 146. 
 
 tension of a liquid, 336, 342. 
 
 of constant pressure in moving 
 fluid, 395. 
 Syphon, 376. 
 
 Temperature, critical, of a gas, 269. 
 Tension in a liquid surface, 336, 
 
 34^- 
 Thermal unit, 239. 
 
 Torricelli, theorem of, 376. 
 
 Transformation, reversible, 254. 
 
 Triangle of surface-tensions, 348. 
 
 Trochoids, 397. 
 
 Unduloid, 
 
 do 
 
 I. 
 
 Vapours, 228. 
 
 saturated, 231. 
 Vector, curl of, 181. 
 Velocity of mean square for a gas, 
 
 225. 
 Vena contracta, 375. 
 Virtual Work, 136.
 
 424 
 
 Index. 
 
 Waterfall, pressure clue to, 218. 
 Waves, velocity of, in shallow water, 
 
 413- 
 deep water, 416. 
 Wet and Dry Bulb Thermometer, 
 265. 
 
 Work done in lateral displacement 
 of floating body, 175. 
 
 expansion of gas, 199, 249. 
 
 external and internal in evapora- 
 tion of water, 263. 
 
 The End.
 
 Clarcnbon prcse, ©yforb* 
 
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