i 1077 ARTES 1837 VERITAS LIBRARY SCIENTIA OF THE | UNIVERSITY OF MICHIGAN | PLURIBUS-UNUM TUE BOR SI QUAERIS-PENINSULAM AMOENAME CIRCUMSPICE | E..P. Bouverie. Trin: Coll: Cambridge 1826 QA 901 W38 30269 268 THE THEORY OF THE Jones Erift EQUILIBRIUM AND MOTION OF FLUIDS. BY THOMAS WEBSTER, M.A. OF TRINITY COLLEGE. CAMBRIDGE: PRINTED AT THE PITT PRESS, BY JOHN SMITH, PRINTER TO THE UNIVERSITY. FOR J. & J. J. DEIGHTON, TRINITY STREÈT; AND JOHN W. PARKER, WEST STRAND, LONDON. M.DCCCXXXVI. PREFACE. IN the " Principles of Hydrostatics," published a few months ago, I dwelt in detail on the phenomena which occur in considering the mechanical properties of fluids, and on the principles to which they lead, and I illustrated those principles by their applications in various machines. In the present treatise, which may be con- sidered as a mathematical supplement to the former, taking these principles as established, I have endeavoured to develope them by the application of the Calculus. The two will, I hope, be found to contain the inductive and deductive reasoning which belongs to that depart- ment of natural philosophy of which they profess to treat. The present treatise is compiled principally from the writings of Poisson and Challis, the well-known work of the former having furnished most of the pro- positions in the equilibrium, as the various papers of the latter have done those in the motion, of fluids; and I have endeavoured to bring before the student what has hitherto been done in this department of science, and to point out the difficulties which present themselves to its further progress. These difficulties are purely mathematical, and I venture to hope, that when it is fully understood that this science and that of Light are at a stand because of the imperfect state of our analysis, は​11-3-37 iv 1 PREFACE. some vigorous efforts will be made by those who have time and talents for this pursuit, to remove this barrier, and to place these sciences in the same rank, as in- ductive and deductive sciences, with that of Gravitation. Much has been done in the last few years, much is almost within our grasp, but much still remains to be done. The importance of the theory of fluid motion in the present state of science is very great; for the physical and mathematical phenomena of this department present many suggestions in the theories of Light and Heat: thus the way in which the crests of one set of waves in water may be superposed over the hollows of another, producing a level instead of an undulated surface, is strikingly analogous to the interference of the vibrations of two musical strings producing a momentary silence, of two waves of light producing absolute darkness; a complete theory of the one may be the means of leading to a complete theory of the others, and all will advance contemporaneously. The obligations of this treatise to the published papers of Professor Challis are, as I have stated, con- siderable; but I am also deeply indebted to him for the assistance which he has afforded me on every occasion of difficulty throughout this work. When I commenced it, many points appeared to me involved in difficulty, and incapable of being explained in an elementary and distinct manner; the reverse is now however the case, as I hope the following pages, and especially Capillary Attraction, (which subject I had considered as hopeless, until he furnished me with the very simple and elementary propositions here given), will testify. PREFACE. V Notwithstanding that great care has been bestowed on the correction of the press, I can hardly hope that the errata will be either few or trivial; and as an author is generally the last person to detect them, I should be extremely obliged to any one who would forward to me or to the Publisher any which he may discover. TRINITY COLLEGE, February 1836. T. W. CONTENTS. CHAPTER I. ON THE GENERAL PROPERTIES OF FLUIDS......... Arts. 1-7. GENERAL Properties. 8-9. Equation of Virtual Velocities. CHAPTER II. ON THE GENERAL EQUATIONS OF EQUILIBRIUM..... 12. Remarks. 13—14. Con- Arts. 10. The General Equation. 12. dition of Integrability. 15_20. Surfaces of equal Pressure, Density and Temperature. CHAPTER III. THE APPLICATION OF THE GENERAL EQUATION......... PAGE 1 7 19 Arts. 24. and at Rest. The Atmosphere. 25—28. Mass of Liquid Revolving Law of Force as direct Distance. 29. CHAPTER IV. ON THE PRESSURE OF A HEAVY FLUID...... Arts. 30-34. Vessels. Pressure at any Point. 35-39. Communicating 40-45. Applications-Siphon, Barometer, Manometer. CHAPTER V. ON THE PRESSURE ON SURFACES......... Arts. 46–47. Pressure on any Surface. 47-53. Centre of Pres- sure and Applications. 54-57. Pressure on Curved Surface. Resultant of Horizontal and Vertical Forces. 58. Hydrostatic Balance. action Machines. 60. Total Pressure. 61. Flexible Vessel. CHAPTER VI. 59. Re- ON THE EQUILIBRIUM OF FLOATING BODIES...... Conditions of Equilibrium. Triangular Prism. El- Stability of Floatation. Moment of Fluid, Meta- Vertical and Angular Oscillations. Arts. 62-66. lipsoid. 67-73. centre. 74-76. 74-76. CHAPTER VII. ON THE APPLICATIONS OF THE BAROMETER.... Arts. 77-79. Weight, mass, and height of Atmosphere. 80. De- termination of Gravity. 81. Elastic force of a Gas. by Barometer and Thermometer. 82-84. Altitude 30 44 68 90 viii CONTENTS. CHAPTER VIII. ON CAPILLARY ATTRACTION Arts. 85-86. Preliminary Remarks. 87. Equilibrium of a Canal. 88. Equation to Capillary Surface. 89. Exs. Cylindrical Tubes, two Parallel Plates. 90. Angles of Contact. 91–93. Law of Ascent. 94. Actual Angle of Contact. 95. 95. Drop of Water in a Conical Tube. CHAPTER IX. ON SPECIFIC HEAT AND LAW OF COOLING. Arts. 95-96. Differential Equation for Quantity of Heat. 98. Increment of Temperature in Condensation. 99. Ratio of Specific Heats. 100. Integral of the Equation. 101-104. Law of Cooling. CHAPTER X. ON THE GENERAL EQUATIONS OF THE MOTION OF FLUIDS......... Point. Arts. 105-106. The Pressure at any Point. Equations. 109. The Character of the Motion. two Dimensions. 113–115. Motion in Space. CHAPTER XI. 107-108. 107-108. The Five 110–112. Motion in ON THE MOTION OF FLUIDS ON PARTICULAR HYPOTHESES... Arts. 116-117. Steady Motion. 118-121. Velocity of Efflux, Adjutages. 122–123. Variable Motion. 124. Steady Motion of Air. Arts. 125. Remarks. near Motion. Linearity. CHAPTER XII. ON SOUND....... 126. The Equation. 127-130. Rectili- Remarks. 131-133. Velocity of Propaga- tion. 134 136. Nature of the Transmitted Motion. 137. Reflexion 138. Propagated Wave. 139-141. Motion in Space. of Sound. 142-144. Velocity of Sound. cal Sounds. Nodes. Loops. Experimental Facts. Pitch. 145-150. Musi- PAGE 100 117 133 157 166 151–154. CHAPTER XIII. ON RESISTANCES... 203 156. Solid 158. Ascent of an Resistance of a Fluid. 155. Oblique Plane. of Revolution. 157. Descent of a heavy Sphere. Air Bubble. CHAPTER I. ON THE GENERAL PROPERTIES OF FLUIDS. 1. A FLUID may be defined to be a collection of particles which can be moved amongst each other in every direction by any assignable force. This definition will express the conception of a fluid mass as consisting of a collection of particles which have a connection with each other very different from the con- nection which subsists between the particles of a solid; and also, that while the particles yield to the least pres- sure, they yet require the exertion of some force to disturb their equilibrium, that is, to change their relative position, or the state of rest in which they exist. Hence it follows, that, 2. PROP. A fluid may be divided in any direction. For the obstacle which prevents the division of a solid mass in any direction, namely, the cohesion of its particles, does not exist here; hence a collection of particles con- stituting a fluid, may be considered as capable of division in any direction. 3. Any small or elementary portion of a fluid mass may be considered as consisting of a very great number of constituent particles; and into whatever elementary por- tions we conceive the fluid to be divided, the conditions for the equilibrium and motion of the whole fluid, and of this elementary portion, will be precisely the same, so long. as both retain their fluid character. A 2 THE GENERAL PROPERTIES OF FLUIDS. 4. In treating of the equilibrium of fluids, the trans- mission of pressure is the contradistinguishing property between them and solids. A fluid transmits pressure in all directions, a solid only in one, namely, in the direction in which the pressure is exerted. This characteristic property is involved in our con- ception of a fluid mass subjected to pressure and remaining in equilibrium; it may be considered as the necessary con- sequence of the application of pressure to such a collection of particles as constitute a fluid. For all the particles being equally free to move in all directions, if any number of these particles, that is, any portion of the fluid be sub- jected to pressure, the particles so acted on will be imme- diately put in motion, unless the action be counterbalanced by the action of the contiguous particles: this mutual action will extend throughout the whole fluid mass, that is, there will be a pressure transmitted in every direction. Hence it follows, that, 5. PROP. The transmitted action is equal to the original action. The action or pressure which is transmitted will be transferred to every point of the containing surface. At any point in the containing surface, the transmitted action will be balanced by the reaction of the surface. We have then two forces of precisely the same kind impressed on the fluid, namely, the original action and the reaction of the surface. Now this reaction, whatever be its magnitude, will, since it is a pressure impressed on a fluid, give rise to a transmitted action in all directions; and at the point at which the original action is impressed, the original action becomes the reaction; for to suppose it either greater or less, involves an absurdity. The original action then and the reaction at any point being thus convertible and equal, and the reaction being, by the general law of the equality of action and reaction, equal to the transmitted action, the transmitted action is also equal to the original action. THE GENERAL PROPERTIES OF FLUIDS. 3 Hence a pressure exerted on a fluid is transmitted equally in all directions; thus fluids press in all direc- tions, they also press equally in all directions. 6. The preceding is also true for fluids whose parti- cles have sensible tenacity or viscidity, the only difference being, that the pressure is not transmitted in all directions with the same velocity with which it is transmitted in the direction of the impressed action. This deviation is how- ever only instantaneous, and when the equilibrium is estab- lished the equality of pressure obtains. 7. The action exerted on any portion of a fluid evi- dently depends on the number of particles which are acted on, that is, the pressure is proportional to the area pressed. An area is measured by the number of units of area which it contains, or by the relation which it bears to that unit of area; hence the pressure at any point is most con- veniently measured by the pressure which is or would be exerted on a unit of area situated at that point. The quan- tity (p) is the symbol used to denote the pressure so re- ferred to a unit of area, and it is called the unit of pres- sure, and must be carefully distinguished from the pressure which is actually exerted on any portion of the surface. The quantity p does not represent any pressure actually produced by the fluid, but that which would be produced if the pressure at the point under consideration were uni- formly applied to a unit of area. When the pressure at any point in a fluid is simply the transmitted action from some pressure exerted upon it as at its surface, the pressure p will be the same at every point. But when, as is generally the case, the impressed forces are different for every point, p will vary from one point to another, that is, it will be a function of x, y, z, the co- ordinates of the point; and the determination of it for different impressed forces is the object of the following pages. 4 THE GENERAL PROPERTIES OF FLUIDS. The pressure exerted on any small elementary area (w) is represented by po; for we may conceive the unit of area sustaining throughout its extent the same pressure as this element; thus w being the area of this element, the product p will be pressure upon it. In elastic fluids the pressure bears a constant ratio to the density, the temperature being constant. This ratio is generally expressed by the quantity k, which depends on the nature and temperature of the fluid, being constant for the same fluid at the same temperature. If p be the density of the fluid, we have the equation p = kp. The pressure is in this case the measure of the elastic force of the fluid, and the same equation subsists. 8. PROP. PROP. Any forces being in equilibrium on a fluid, the equation of virtual velocities holds. A fluid may be considered as a machine which pos- sesses the property of transmitting equally in all directions the pressures to which it is subjected, hence the general conditions of equilibrium which apply to all other machines. must be expected to apply here also; and it will be found that the equation of virtual velocities is true for a fluid in equilibrium, and subjected to pressure. Now in the general proof of this principle, the absence of all change in the tensions or resistances of the parts of the system, is the supposition on which the whole demon- stration rests, and the analogous supposition in a system of fluid particles is, that the volume of the fluid is invari- able; for if the fluid change in volume, the mutual relation and dependence of the points of the system do not remain unaltered. Let any forces P, P', P",...applied to pistons whose areas are a, a', a",...be in equilibrium on a fluid mass. Let the points of application of the forces suffer a displacement, that is, let the pistons be moved through spaces h, h', h",... THE GENERAL PROPERTIES OF FLUIDS. 5 Then the new position is to be one of equilibrium, hence we must have as before the displacement, P = pa, P' = pa', P' = pa", &c……………..(1). Also the volume of the fluid must be the same as before the displacement, or we must have ah + a' h' + a"h" + 0……………….(2). These two conditions obtaining, multiplying (2) by p and substituting from (1), we have Ph + P'h' + P"h" + ... = 0.......(3); which is the general equation of virtual velocities. Thus the principle is true for all fluids compressible or incompressible by virtue of the equation (2), which is the condition of the system. If this condition do not obtain, the fluid mass is no longer the same machine, being a collection of particles related by different inter- nal forces. 9. The equation of virtual velocities, expresses the relation which may subsist between the external forces which act on a machine, independent of the internal forces or pressures. Assuming then this equation as a general truth, it may be worth while to apply it to a fluid. Let P, P', P",...be the forces which are in equilibrium on a fluid, and h, h', h",...the virtual velocities of the points of application of the forces, then, Ph + P'h' + P"h" + ... = 0 0... (1). But the equations of condition are in this case reduced to one, namely, the invariability of the volume of the fluid, which is expressed by the condition ah + a'h' + a″h" + 0..... …….(2) ; where a, a', a″ are the areas of the pistons by which the forces are impressed, or they may be considered as express- ing the points which suffer displacement. 6 THE GENERAL PROPERTIES OF FLUIDS. Let there be two forces in equilibrium on the fluid, then (1) Ph + P′h′ = 0 and (2) ah + a'h' = 0. Let a = a', then h=h', and P – P′ = 0, or P = P', - and since these forces may be applied at any part of the surface, the transmitted action is equal in all directions. Again, let (2) be multiplied by a quantity (q), then subtracting from (1), (P − qa) h + (P' — qa') h' + ... = 0, which cannot be satisfied in all cases unless P−qa = 0, &c.; whence P=qa, P' = qa', &c.; or the pressures are pro- portional to the areas pressed. CHAPTER II. ON THE GENERAL EQUATION OF THE EQUILIBRIUM OF FLUIDS. 10. PROP. To find the condition of equilibrium of a fluid mass, every particle of which is acted on by given forces. Let AB (Fig. 1.) represent any fluid mass in equili- brium, and let it be referred to rectangular co-ordinates whose origin is 0. Let the plane of ay be horizontal, and the axis of ≈ vertical; and let x, y, z be the co-ordi- nates of any point P in the interior of the mass. Let any elementary parallelopiped PQ of the fluid be taken whose edges are da, dy, dz, then, if dM be the mass of this element and ρ its density, dM= p dx dy dz. Let the impressed forces be resolved in the direction of the co-ordinate axes, and let X, Y, Z be the accele- rating force in the directions of x, y, z, respectively, at the point P. Then XdM, Yd M, ZdM, are the moving forces on the element d M in the direction of the axes. The mass d M is then pressed from without to within, on its six faces, by the surrounding fluid, and equilibrium must subsist between the internal forces and the external pressures. The pressure on the upper face of the element is pdx dy, (Art. 7.). But P is an unknown function of x, y, z, hence for the under face of the element, the co- ordinates of a point in which are x, y, z+dz, p will ر 8 THE GENERAL EQUATION dp dz. become p + • dz dp face (p + dz The pressure therefore on the under dz) dady, and the difference of the pres- sures on the upper and under face is (p + dp dz dz) dx dy – p dx dy = dp. dx dy dz. dz · The element d M is therefore pressed upwards by this force, and that it may move neither upwards nor down- wards, but may remain in equilibrium, we must have dp. dx dy d z = Zd M. dz But dM = p dx dy dx; when therefore there is equi- librium, we have dp dz pZ. Similary, if q, r be the pressures referred to a unit of area on the other faces, we have for the faces parallel to xx and y respectively, dq dr pr and dy dx pX. If the element d M were solid, there would be no neces- sary connection between the unit of pressure on the faces which are not parallel, but the element might be in equi- librium if the pairs of forces on any two parallel faces were in equilibrium; but the element being a fluid element, and consisting of an indefinite number of fluid particles, it transmits the pressure on any one face to all the others, and the equilibrium cannot subsist unless p three preceding equations become therefore = q = "; the dp = px, dp dp py, = p Z......(1); dy d x dz which are the general conditions of fluid equilibrium. OF EQUILIBRIUM. 9 Multiply these respectively by dx, dy, dz, adding and observing, that since p is a function of x, y, z, its complete differential is dp dx + dp dx dP dy dy dy + dP dz dz we have, whenever there is equilibrium, dp=p(Id +Ydy+Zdx).........(2), which is the general condition required, and from which the pressure at any point may be obtained. 11. In the preceding investigation we ought, in strict accuracy, to have taken notice of the moving force of the fluid contained in the element, which must be added to the transmitted pressure. If then y be the pressure due to this force which is exerted on the face dyd≈, we should have p dy dz+y for the whole pressure which takes place from within to without, or from the right to the left on this same face. Now the pressure arising from the surrounding fluid, and exerted from without to within, that is, from the left to the right on this face dy dx, has been represented by rdydz: this force is the resistance which the surrounding fluid opposes to the pressure transmitted from the interior, that is, to p dy dz+y; hence we must have rdy dz= pdy dz +7. But notwithstanding being unknown, we are certain that it must be a very small quantity of the third order, and may therefore be omitted in comparison with pdyd z; whence r = p, also q = p. The same conclusion would have been arrived at, if the element d M, instead of being taken rectangular, had been any polyhedron, indefinitely small in all its dimensions, and it had been shewn that the external pressure exerted perpendicularly on all its faces by the surrounding fluid, is proportional to their respec- tive areas, and independent of the moving force of the polyhedron. B 10 THE GENERAL EQUATION 12. The conditions of equilibrium which we have obtained, require that p should be a function of x, y, z, such as will satisfy the three equations (1) at once, or satisfy (2). Hence, that the value of p may be possible, the product of p and Xdx + Ydy + Zdz must be a com- plete differential of some function of the three independent variables x, y, z. Conversely, when this product is a complete differential of a functiou of these variables, the value of p can be found by integration, and the three equations (1) will be satisfied. This then being the case, we have p = f (Xda + Ydy+Zd )......(3). This integral may be taken with regard to any series of consecutive values of x, y, z. Hence, the integral taken with regard to any such series of values, or in other words, the pressure of every line of fluid particles or canal leading to the same point, is the same; this was assumed by Newton as the basis of the theory of the equilibrium of fluids, and would lead at once to the preceding conditions. The principle that we may integrate in regard to any line whatever of fluid particles drawn from the point to the free parts of the fluid admits of some important appli- cations, as will be seen hereafter. 13. That the equilibrium may subsist, we must have (Xd + Ydy+Zdx) P a complete differential, which it is when d. px_d.pl d. px d.pZ d.pY_d.pZ dy dx d X + X or p dy d Y p dz dx dz dp dp + Y dy da dx dp d Z dp +2 P da dx d Z dp + Z P dz dy dy d X +ď P dz dY P dz + ľ Y dz dy OF EQUILIBRIUM. 11. multiplying by Z, Y, X, respectively, and adding, dX dy Ꮓ + Y dy dx dz dX dx dz dY dZ + & dz dy = 0. Whenever then such a relation subsists between the forces that this condition is satisfied, the mass of fluid subject to these forces will be in equilibrium. If the fluid be homogeneous or incompressible, p is constant, and the conditions become d X d Y d X dz dY dZ > dy dic dz dx d z dy 14. The preceding conditions are satisfied whenever the impressed forces are some function of the distance from fixed or moveable centres. Let P be the law of force, and let it be directed to a centre whose co-ordinates are a, b, c, and its distance from the point x, y, z. Then V a X = P . Y = P. Y y - b > b, z=P. Z = P. 1° ༡་ 7' and r² = (x − a)² + (y - b)² + (≈ −c)²; dX dP dr x a X a dr then P. dy dr dy 2.2 dy dP y b x a 20 a y - b - P. dr dP dr Similarly, d YJd P dx dr 2° P) − (dp PJ (x - a) (y - b) プ ​P) (x − a) (y – b) the same quantity; therefore, ༡.ལཾ グ ​Z z (dx dY dy dx = 0. 12 THE GENERAL EQUATION And the same is the case with the other two terms, or the equation of condition becomes identically null, and therefore is satisfied for all laws of force which are a function of the distance. Hence, the laws of gravity and of centrifugal force are evidently such as may produce equilibrium, the one varying as the inverse square, and the other as the direct distance. 15. PROP. To find the equation to the surface of a free fluid, and to a surface of equal pressure. Substituting the co-ordinates of any point in the surface for x, y, z, in the value of p (3), we obtain the pressure which is exerted at this point on the side of the containing vessel; this pressure will always be destroyed by the re- action of the vessel, provided it be fixed and capable of sustaining it. But in those places where the vessel is open, or where the fluid is entirely free, there is no sur- face whose reaction can destroy the pressure p, conse- quently we must have this equal to nothing for all the points of the free surface of a fluid in equilibrium; whence Id + Idy +Zd x = 0.........(1), +Ydy+Zdx = is the differential equation to that surface. This equa- tion subsists also if a constant pressure be exerted at the free surface of the fluid. For at all the points where the pressure is constant, = dp 0; hence, the preceding equation obtains for a sur- face subject to a constant pressure. Also, if there be any series of points in the interior at which the pressure is constant, for all these points we have the same condition; hence (4) is the equation to a surface of equal pressure. If the pressure vary from one point to another at the free surface of a fluid, and the pressure referred to a unit of surface at any point x, y, z, be represented by ƒ (x, y, ≈), the value of p obtained from (4) would coincide for all OF EQUILIBRIUM. 13 points on the free surface with this given function; hence the differential equation of the surface would give p (Xd +Ydy+Zds) = d. f ( x, y, z). In the following articles the external pressure is always supposed either nothing or constant at all points of the surface of a fluid in equilibrium. The pressure p being proportional to the density in elastic fluids, it follows, that the pressure can never be nothing in an elastic fluid, unless the density also be nothing, that is, so long as the fluid exists and it has not lost by cold all its elastic force. An elastic fluid, then, cannot be in equilibrium, unless it is contained in a close vessel or acted upon at every point of its surface by pressures from without to within. 16. PROP. The resultant of the forces is perpen- dicular to the surface at all surfaces of equal pressure. At all surfaces of equal pressure d. +Ydy+ Z dã = 0. If now any curve be traced on this surface, and ds be the differential element of the curve, the cosines of the angle which the tangent line at any point a, y, z, makes with the axes of x, y, z, are respectively, dx dy dz ds ds ds Also, the cosines of the angles which the resultant R of the forces X, Y, Z makes with the axes x, y, z, are respectively, since R = √ X² + Y² + Z³, X Y Z R' R R' R* Hence, dividing the preceding equation by Rds, it becomes X dx Y dy Z dz + + R ds Rods R ds 0. 14 THE GENERAL EQUATION Y Let а, B, be the angles which the tangent line makes with the axis, and a', ß', ' the angles which the resultant R makes with the axis, then this equation becomes, cos a . cos a' + cos ß. cos ß' + cos y . cos y' = 0 B' which is the condition that two lines should be at right angles to each other. Hence, the resultant R is perpendicular to the tan- gent line, that is, it is a normal to the surface. This force will in general act from without to within, but when the external pressure is not equal to zero, it may be directed from within to without. 17. PROP. To find the equation to a level surface. Definition. The bounding surface of a free fluid, under whatever circumstances the equilibrium takes place, is a level surface; thus, in some cases, this surface may be ellipsoidal; in others, as in the figure of the earth, it will be spheroidal, or very nearly spherical; and, ana- lytically speaking, any surfaces which possess the same properties as a bounding surface, that is, all surfaces of equal pressure are levels or level surfaces. If we integrate the differential equation to a surface of equal pressure, and give to the arbitrary constant con- tained in the integral any particular values, the resulting equation will belong to as many surfaces as there are par- ticular values, each of which will have the same differential equation; and, consequently, will possess the properties of equal pressure at all its points, and of being at right angles to the resultant of the forces X, Y, Z. Those surfaces which are in the interior of the fluid, as determined by the value of the arbitrary constant, that is to say, those series of points within the fluid which are OF EQUILIBRIUM. 15 included in the integrated equation, some value being as- signed to the arbitrary constant, are level surfaces or levels, for they are surfaces of equal pressure. If the constant vary by very small quantities, the fluid mass is divided into a number of successive layers, or strata, each of which is comprised between levels; hence they are called level strata. The value of the constant which belongs to the surface depends in each case on the given volume of the fluid, so that the external pressure has no influence on the form of equilibrium. The equilibrium will not be disturbed by supposing any part to become solid, hence any constant normal pressures exerted from without to within on all the elements of the surface of a solid or fluid body are destroyed and cannot impress on the body any motion either of translation or of rotation. This equilibrium between the external pressures results from the character- istic property of fluids of transmission in all directions of all pressures exerted on their surface. 18. Let us now suppose the fluid which is in equili- brium to be homogeneous, and of uniform density and temperature throughout. The quantity p then being con- stant, we must have Xda + Ydy + Zds an exact differ- ential of the three independent variables; for this is a necessary condition of the equilibrium, and without it the equilibrium cannot take place, whatever form is given to the fluid mass. Now the condition of integrability is always fulfilled in all those forces which exist in nature, namely, attractions and repulsions; the intensities of which vary as some func- tion of the distance of the centres from which they proceed, (Art. 14). The equilibrium then of a homogeneous liquid subject to these forces is always possible, and that it may really take place, we must give to the fluid a form such that every point of its surface may cut at right angles the resultant of the attractive and repulsive forces. 16 19. THE GENERAL EQUATION PROP. The external surface and all internal surfaces of equal pressure are surfaces of equal density. Let the fluid be acted on by forces which are some function of the distance, that is, let Xdx + Ydy+Zdz be a complete differential dq. Then the equation (2) becomes dp = pdq. is variable, we must That this may subsist when P have the density some function of q; and conversely, when this condition is fulfilled there is always a value of p which satisfies the equation of equilibrium. Let p=$(9), then integrating and taking q' for the value of q at the external surface, p = 4 (q) − 4 (g'). Now this value of p must be the same whatever point on the surface is taken, that is, we must have (q) con- stant for all points on the surface; hence, at the external surface р also is constant, or the external surface is a surface of equal density. Since (g) is constant, we have q some function of p; hence pi is constant where p is constant; or surfaces of equal pressure are also surfaces of equal density. If the fluid be homogeneous, that is, if P be constant it is no longer a function of q; and the preceding condition that Ρ is the same where p is the same, does not hold. When the fluid is incompressible, p may be any func- tion whatever continuous or discontinuous of the quantity q; when this is given, we may obtain the value of p as a function of p by integration. P 20. PROP. In an elastic fluid, surfaces of equal pressure are surfaces of equal temperature. In an elastic fluid the density is connected with the pressure by a constant relation, and cannot be arbitrarily OF EQUILIBRIUM. 17 assigned as in a homogeneous or heterogeneous fluid. The relation betwixt them is expressed by the equation p = kp; hence, dividing dp = pdq, we have dp dq ..(5). Ρ k If the temperature be constant, k is constant, whence log p = q + C. k Р q Let p' be the value of p when q = 0, then log p' k q p' q .. p = p'e and k k P € .(6). k If the temperature varies from one point to another, k will not be constant, and it must be some arbitrary function of q. The temperature will also be a function of q, consequently the temperature must be constant throughout each surface of equal pressure of an elastic fluid in equilibrium. Hence, all levels are of equal tem- perature, and consequently strata of equal pressure are also strata of uniform temperature. This condition being included, we must replace equa- tions (6) by jaa p' jaa p = p'e k and k p: (7). k 21. When the mass ABCD (Fig. 1) is composed of many different gases, the conditions of equilibrium may be fufilled in two different ways; when the gases are per- fectly mixed so as to form a homogeneous mass, and when they are superposed in strata so that the bounding surfaces are levels. The former is the case with the atmosphere, which is found to consist of the same component gases at all heights. This state of perfect mixture is that in which C 18 THE GENERAL EQUATION OF EQUILIBRIUM. the equilibrium is the most stable; and when two different gases are superposed in a vessel closed at all parts, they must after a time become perfectly mixed, unless we can insure the containing vessel from the slightest disturbance. This however is not a sufficient explanation of the diffusion of gases through each other, but is a condition which must obtain if they are to remain superposed. CHAPTER III. ON THE APPLICATION OF THE PRECEDING THEORY. 22. PROP. A mass of fluid subjected to a constant pressure at its surface, is acted on by a force varying inversely as the square of the distance from a fixed centre; required the form of equilibrium and the pressure at any point. ű Let this fixed centre be taken as the origin of co- ordinates, let u be the intensity of the force at distance. unity from the fixed centre, then will be its intensity at a distance r; resolving it into the direction of the axes of co-ordinates, if x, y, z be the co-ordinates of the point acted on, X = μ 20 202 до Y = u y = Z = 2° 22 The equation to the surface of the fluid (Art. 15) becomes dư +ydy + dã = 0; which integrated, gives x²+y²+≈²=c², or r = a constant; that is, the surface is spherical, and the centre of the fluid mass is at the fixed centre of force. Again, supposing the forces to tend to this fixed centre, dp ρ μ xdx + ydy + zdz グ ​updr up .. p + C. " 20 THE APPLICATION OF At the surface of the sphere p has a constant value; let p₁ be this value of p when r = up Ρι + C; a and subtracting from the preceding p = p₁ + μp 1 a If the force tend from the fixed centre, that is, if it be repulsive instead of attractive, we have only to change the sign of u in this equation, and p = p₁ - μp { Suppose this fixed centre to be replaced by a sphere which acts on all the points of the fluid with a force varying as the inverse square of the distance from its centre. Let b be the radius of this sphere. The value of Ρ then would be given by the preceding equation for the points included between rb and r = a. If the sphere were repulsive, the least value of p would be that corresponding to r = b, namely, p = P₁ - μp {-} α If this expression become negative, the fluid will be detached from the solid sphere and be dispersed in space. Hence we must have p₁, that is the external pressure, greater than μρ a ab · b In general it is necessary, in the equilibrium of a fluid, that p have a positive value throughout the whole of the mass, so that the contiguous particles may everywhere be sustained one against the other, and the fluid be not sepa- rated. For wherever p becomes negative, it indicates a defect of continuity in the fluid. THE GENERAL EQUATION. 21 When the radius is large, the attractive forces directed towards the centre of the sphere may be considered to have their directions parallel; the surface of the fluid will then be plane and perpendicular to the direction of this fluid for a considerable extent. This is the case of the equilibrium of a heavy liquid, which we shall consider in the subsequent chapter. 23. When then a mass of fluid subject to a central force which varies as the inverse square of the distance from a fixed centre, is in equilibrium, it will consist of spherical layers concentric with this centre, and the re- sultant of the forces will be in the direction of the radius. If it be a heterogeneous liquid, it is a necessary condition of the equilibrium, that the mass be formed of spherical and concentric layers in which the density is constant throughout the same layer, but varies in any arbitrary manner from one layer to another. In the same manner, if any number of heavy liquids are contained in a vessel, it is a necessary condition of the equilibrium that each horizontal and indefinitely small slice contain only one fluid; and this condition will be fulfilled if the upper surface, which we suppose submitted to a constant pressure, and the surfaces which separate two consecutive liquids, are all plane and horizontal. The stability of the fluid moreover requires that the densities of the superposed liquids decrease from the lower to the upper liquid, so that the centre of gravity of this system of heavy bodies may be the lowest possible. 24. PROP. The atmosphere can never be in equi- librium. The centrifugal force and deviation from a spherical form of the earth being disregarded, the weight of the particles of air is directed towards the centre of the earth, and the level strata are spherical and concentric. In order therefore that the atmosphere may be in equilibrium, the 22 THE APPLICATION OF temperature must be everywhere the same at the same height above the surface of the earth, and vary only with the elevation of successive concentric strata. This, how- ever, is not the case, for the sun warms unequally dif- ferent points of the surface of the earth and of the level strata of the atmosphere. The temperature depending on the latitude is sufficient to prevent equilibrium taking place in the atmosphere, and produces permanent winds, such as are known to exist near the equator. Moreover, the condition of equilibrium of the atmospheric strata cannot give us any information respecting the variation of temperature in the vertical direction, for the equation (5) of the preceding chapter subsists, whatever function k be of q, and consequently whatever be the law of this varia- tion of temperature. 25. PROP. A mass of fluid revolves about an axis; required the form of equilibrium and pressure at any point. If a homogeneous or heterogeneous liquid turns uni- formly round a fixed axis, the preceding formulæ give us the necessary and sufficient conditions for its preserving a permanent figure, and moving as a solid. Let us take the axis of rotation for the axis of ≈, and let r be the distance of any point P from this line, then z² = x² + y². Let a be the angular velocity, which, since the motion is uniform, is constant and common for all points of the fluid mass; then ra is the absolute or linear velocity of the point P; and since it will describe a circle whose radius is r, the centrifugal force is ra². The tendency of this force is to increase r, and its components in the direction of the axis of m and y are raª.2, and raª.4, or a¨a, and a¨y, 2: THE GENERAL EQUATION. 23 which being added to the forces X, Y, Z, the general equation (2) becomes dp = p (Xd +Ydy+Zds +ả do tả gảy)….(a). α a The expression within brackets is an exact differential, namely, the differential of q increased by the differential of 1 a² (x² + y²), or a²r². Consequently, the form will be one of permanent equili- brium; and if the free surface sustain a constant pressure throughout, the equation common to this surface, and to all other level surfaces, is, a² (x d. +Ydy + Z đã tả (do +ydy) = 0......(6). Xdx In the case of a homogeneous liquid, the free surface will be determined by the integral of this differential equa- tion, the arbitrary constant being determined from the whole volume of the liquid, as we shall see presently. In the case of a heterogeneous liquid it must be com- posed of homogeneous strata, the forms of which will also be determined by the integral of this same equation, and which differ from the bounding surface only in the value. of the arbitrary constant. 26. PROP. A mass of liquid in an open vessel and subject to gravity revolves about a vertical axis; required the form of its surface and the pressure at any point. Let g be the force of gravity, and let the positive values of ≈ be measured upwards; let a be the angular velocity, then X = a²x, Y = a²y, Zg, and substitut- ing in (4), the differential equation to the surface is a² x d x + ydy − gdz = 0; whence integrating and adding an arbitrary constant, a (x² + y²) + c ; 2g 24 THE APPLICATION OF which is the equation to a paraboloid; hence the free surface of the fluid is that of a paraboloid whose axis is that of rotation, and whose latus rectum is 2g a² To determine the arbitrary constant c, let us suppose the vessel to be a vertical cylinder whose axis coincides with the axis of ≈ or of rotation. 2 Let a be its radius, and h the height due to the absolute velocity aa of the surface, so that a² a² = 2gh, and consequently 22 Z h a² 2 202 + C. Let b be the height of the water before the motion commences, then Tab is the whole volume of the liquid which does not change during the rotation; hence dividing the paraboloid into infinitely small cylindrical shells having the axis of x for a common axis, we shall have 2πrdr for the base, and 2πzrdr for the volume of the cylin- drical shell, whose radius is r and thickness dr. The total volume will then be found by integrating 2xrdr from 0 to r = a 0 to r = a; whence we may conclude, that π a² b = ["2πzrdr, or a²b = 2f"≈rdr. Now substituting for ≈ its value, ſzrdr = √(h 202 + + c) rdr h α h 2 p² + 1 / cr² + C ; 2 · · L = r d r = ↓ / a² + + ca² = 4 a² ( + c); 2 a² But we have seen that ab 2 zrdr, = f¶ .. 1 a²b = a² ( 1 + c ), or c = b − 4 h. - THE GENERAL EQUATION. 25 The equation then of the superior surface of the liquid is h p² + b − 1 h. a The least and greatest value of x, are those which cor- respond to r = 0, and r = a, hence calling them x, and x' respectively, we have ≈₁ = b − 1 h, x' = b + 1 h, Z whence it appears that the depression of the fluid at the axis, and its elevation at the circumference due to the rotation, are each equal to half the height due to the velocity of the circumference. To find the pressure on the side. Let p = 1, then the general equation becomes ·· 2 dịp = a (do +ydy) - gdx, p = ½ a² (x² + y²) − g≈ + C. Now at the part of the surface immediately in contact with the sides of the vessel, that is at the highest part of the surface, p = 0, and ≈ = b + 1 h ; · · 0 = ¦ a² (x² + y²) − g (b + 1 h) + C, which subtracted from the preceding gives p = g {b + } h − ≈}. The pressure on any elementary annulus of the side of the cylinder will be p × 2πads. Therefore the whole pressure on the sides = 1 · [2 π ag { b + } h − z } dz x} 2πag {b+1 h −1} ≈ + C which taken between the limits = b + 1 h, ≈ = 0, ≈ = mag (b + 1 h)². 27. When the forces whereof X, Y, Z, are the com- ponents, proceed from the attractions of all the points of D 26 THE APPLICATION OF the liquid varying as the inverse square of the distance, or according to other laws, the total values of X, Y, Z depend in general on the form of the liquid and its level surface, and conversely this form depends on the values of these components. This mutual dependence between the attractions of the fluid and its figure, renders the determination of the latter extremely difficult by means of the equation (6). When the liquid is homogeneous, the problem may be solved for the ordinary laws of at- traction, that is, of the inverse square of the distance, by supposing the centrifugal force very small, so that the form of the fluid differs but little from the spherical form which it would take if this force were nothing, that is to say, if the fluid were at rest. It may be shewn, that the form of the fluid is necessarily an oblate spheroid, the flattening of which at the poles is determined from the ratio of the centrifugal force at the equator to the attraction on the fluid at the same point. The investi- gation, however, cannot be given here*. 28. There is an essential difference between the level surfaces traced in the interior of a fluid, subject to the mutual action of all its points, and those which are de- scribed in a fluid subject only to extraneous force, that is to say, which are acted on only by attractions or repul- sions directed to or from fixed centres, and which are some functions of the distance. Let ABCD be the surface of a fluid in equilibrium, at rest or turning about a fixed axis. Let EFGH be any level surface, or surface of equal pressure in the interior, and let R be the resultant of all the forces which act in any point M of this surface. In both cases this force will be in the direction of the normal, but in the second case its magnitude and direction not depending on the action of the points of the fluid, will be perpen- dicular to the surface EFGH, if the strata of fluid com- * See Figure of the Earth, Encyc. Metrop. or AIRY's Tracts. THE GENERAL EQUATION. 27 prised between the two surfaces be removed, so that after this has been removed, the fluid bounded by EFGH will still be in equilibrium. But in the case of the action of the points of the system, the force R will depend on the action both of the internal fluid and of this inter- mediate stratum. It will change in general in magnitude and direction when the fluid comprised between ABCD and EFGH is removed, and the fluid bounded by EFGH will no longer be in equilibrium, which can only subsist by the surface becoming perpendicular at each point to the remaining force. The action of the external layer comprised between EFGH and ABCD, will be nothing on all points in the interior of the fluid and in the surface EFGH, when the mass of the fluid is homogeneous and differs but little from a sphere, and the points are only acted on by their mutual attractions, varying as the inverse square and by the centrifugal force. In fact, all surfaces of level are similar ellipsoids, and consequently the fluid comprised between ABCD and EFGH exerts no action on the fluid in the interior of EFGH, since the attraction of an ellipsoidal or spherical shell on a point in the interior is nothing. But this, that the action of a stratum terminated by level surfaces on the interior of the fluid is equal to zero, is not a condition of the equilibrium of fluids; for the forces being such as we have supposed, it is not zero when the fluid is heterogeneous; from which cause the surfaces of levels are dissimilar, but still elliptical, and such that the ellipticity of any surface EFGH de- pends on the thickness and constitution of the exterior layer. 29. Among the different laws of attraction, there is one which does not exist in nature, but possesses some remarkable properties; this law is that of a mutual action in the direct ratio of the distance, and the remarkable 28 THE APPLICATION OF property is that the resultant of the actions of all the particles of a mass on any point is independent of the form and constitution of the body, whether homogeneous or heterogeneous, and the same as if the whole mass were collected at its centre of gravity. Let x, y, z be the co-ordinates of an attracted point, attracting x', y', z′ and μ the mass of this second point, the distance between them, and ku the accelerating force in the direction from the first point to the second, where k is a constant. The components of this force in the directions parallel to the co-ordinate axes, are kμ (x' - x), kµ (y' − y), kµ (≈' — ≈). ль Hence, if X, Y, Z be the resultants of these attractions, X = kΣµx' - kxZμ, ΚΣμα Y = kΣuý - kyΣµ, Z = kΣuz' - kzΣμ, Σμπ the symbol Σ applying to the whole mass of the attracting body. If m be the whole mass of the body, and x,, Y,, Z,, the co-ordinates of its centre of gravity, Σμπ Σμ = m, Σµx' = mx,, Σµy, = my,, Σux' = mx,, Zu Σμ whence substituting in the preceding, ·X = km (x, − x), Y = km (y, − y), Ch Z = km (≈≈), or the forces are the same as if the whole mass were collected at its centre of gravity. Substituting these values of X, Y, Z in (b), and making a² 2 α km e, we have, THE GENERAL EQUATION. 29 S ( - ) dữ + (y - 3)dy+(−x)dx +e( dữ+ydy)=0, whence integrating and adding an arbitrary constant c, (x − x )² + (y − y)² + (≈ − x )² − e (x² + y²) = This equation is that of the levels of a fluid turning about an axis x, and attracted by a force varying directly as the distance; we can shew that all the surfaces are con- centric, and of the second degree. If the origin be transferred to their common centre, that is, to the centre of gravity of the fluid, the terms in- volving the first powers of x, y,, ≈, must disappear, or z, x = 0, y₁ = 0, ≈, = 0; . the equation becomes therefore, ≈² + (1 − e) (x² + y²) = c ; hence the levels are spheroids or hyperboloids, according as e is less or greater than unity, having in both cases the same axis, which is the axis of rotation. The volume of the fluid being given, the hyperboloid is not possible unless the fluid is contained in a vessel, and then the equation applies only to the free surface of the fluid. When then e is > 1, the permanent figure of a free liquid subject to the laws which we have supposed, is impossible. If e be <1, all the levels are spheroidal differing ac- cording to the value of c. To determine the value of this quantity corresponding to the external surface, we must equate the volume of the spheroid, which is the given volume of the liquid. 4 πc √ c C 3 (1-e) to It is remarkable that in this example the law of the densities of the strata has no influence on the external form, and on that of the levels. CHAPTER IV. Fig 3. ON THE PRESSURE OF FLUIDS SUBJECT TO GRAVITY. 30. IN the preceding chapters the conditions of the equilibrium of a mass of fluid subject to any impressed forces, have been fully examined; and we have now to consider the application of these principles where the only impressed force is gravity, and where different fluids are in equilibrium with each other. PROP. To determine the pressure at any point. Let any vessel (Fig. 3.) having its base AB on a hori- zontal plane contain a mass of liquid whose surface is PQ. If the liquid is at rest, the surface PQ is perpendicular to the direction of gravity, (Art. 16.), and consequently horizontal or parallel to the base AB of the vessel, since for small spaces the directions of gravity may be consi- dered as parallel. The equilibrium also of the surface will not be affected by the application of a constant pressure of any magni- tude (Art. 15.), and the pressure referred to a unit of surface being the same throughout each level (Art. 17.) when a fluid is in equilibrium, the pressure in this case will be the same throughout each horizontal section of the liquid mass. Let the surface of the fluid be taken for the plane of my, then the axis of will coincide with the direction of gravity. } THE PRESSURE OF FLUIDS, &c. 31 Hence in the general equation, (Art. 10.), putting we have X = 0, Y = 0, and Z = g, dp = gpdx; whence, if we consider p constant, we have by integration, p = gpx + C…………………….(1). Let the surface be subject to a constant pressure, and when x = 0 let p = p₁, substituting therefore in (1), · p − p₁ = gp≈, - P₁ = C'; or p = gpx + P₁ ………….. (2). p=gpx If the surface be subject to no constant pressure, P₁ or C = 0 and p = gpz. The equation (2) expresses the pressure on a unit of surface situated any where in the interior of the liquid, and it may be observed, that the pressure thus found is the sum of two pressures, whereof the one (gpx) is the weight of the superincumbent column of the fluid, and varies with every value of x, that is, for every point in the liquid, being in fact proportional to the depth of the point; and the other (p) is the same for every point, being transmitted equally in all directions throughout the fluid mass. 31. This latter pressure (p,) being the same at every point, may, for the sake of simplicity when we are con- sidering the pressure of the liquid at any point, be omitted, and the general expression is p = gp2 Epz. To find the pressure at any point in the base, since the whole base is at the same depth below the surface, putting x = h, we have p = gph. 32 THE PRESSURE OF FLUIDS Let A be the area of the base, then the whole pressure on the base = pA = gph A. But gph A is the weight of a vertical column of the fluid whose base is A and height h. Hence, the whole pressure exerted by the fluid on the horizontal base of any vessel containing it, is the weight of the superincumbent column of the fluid. Thus, the pressure exerted by any liquid on the base of the containing vessel, is independent of the forms of those vessels. Hence, if there be any number of vessels standing on the same horizontal plane and filled to the same height with the same liquid, the pressure on their bases if they be equal, or on equal portions of their bases if they be unequal, must be the same whatever be the shapes of the containing vessel, and all experiments shew most distinctly that this is the actual fact. 32. If several liquids be superposed one above an- other in the same vessel, the only condition requisite for the equilibrium is that the surface of each fluid must be a level, (Art. 17.), that is, in this case horizontal. Thus each fluid will exert a constant pressure on the surface of the one below it, which will be transmitted to all points below it, without in any way affecting the equilibrium of the lower fluids. 33. PROP. To find the pressure at any point in the bottom of a vessel containing any number of fluids lying one above the other. Let P'Q', P"Q", (Fig. 3.) represent the surfaces of fluids lying above PQ, these surfaces being all horizontal, are parallel to each other and to the base AB of the vessel. Then if h, h', h', be the thicknesses of the fluids, and p, p', p", their densities, the pressure on the base AB for the fluid whose surface is PQ, is P = gph. (Art. 30.) SUBJECT TO GRAVITY. 33 The pressure on the surface PQ for the fluid whose surface is P'Q', is p = gp'h'. The pressure on the surface P'Q' for the fluid whose surface is P"Q", is p= gp"h"; and so on, whatever be the number of layers. The pressure exerted on P'Q' is transmitted to every point of PQ; hence, the pressure at any point in PQ is p = gp"h" + gp'h'. The pressure exerted on PQ is transmitted to every point in AB, hence, the pressure at any point in AB is p = (gp"h" + gp'h') + gph have g (ph + p'h' + p"h"). Hence, whatever be the number of fluids, we should p = g(ph), the whole pressure on AB, if A be its area = P × A P× = g(ph + p'h' + ...) A or gAΣ(ph). Thus, whatever be the number of the fluids superposed above each other, the whole pressure which they exert on the base of the containing fluid depends on the magnitude of that base, the thickness and density of the different fluids. When the vessel is cylindrical and vertical, the whole pressure is equal to the weight of all the fluids, and the pressure will not change however the form of the vessel be changed, provided that the base of the vessel, and the thick- ness and density of each layer or stratum of fluid are all invariable. 34. Hence, when the same vessel contains different fluids, they are superposed in horizontal layers, and the pressure on the base is the product of its area and the sum of the thickness of each layer multiplied by its density. This result will hold when the thickness of each layer is E 34 THE PRESSURE OF FLUIDS indefinitely diminished, that is, when the density of the fluid mass varies continuously in the vertical direction; it is therefore true for compressible fluids. It is equally true when the weight varies from one stratum to another with the density, which is the case when the height of the fluid cannot be neglected, in comparison with the radius of the earth. The same conclusion is deduced immediately from the equation dp = pgdx, which applies to the equilibrium of all fluids compressible or incompressible, in which we may suppose that the force of gravity g and the density p are functions of the vertical ordinate z. 35. We have now to consider the equilibrium of a liquid contained in several vessels which communicate with each other, so that the liquid may run from one. to another. If the apertures be all closed at once the equilibrium will not be disturbed, but the surface of the liquid in each vessel will be horizontal; this condition is not, however, sufficient when the orifices are not closed, and there exists a certain ratio between the elevations of the liquid in the different vessels and their densities. The conditions of equilibrium under these circum- stances are determined in the following propositions. 36. PROP. When a liquid is in equilibrium in any system of communicating vessels the surfaces of the liquid must be on the same level, that is, when the vessels are near each other, in the same horizontal plane. Let AB, CD, (Fig. 4.) be the bases of two vessels standing on the same horizontal plane and communicating with each other, and containing the same liquid. Let the liquid stand at Pm in one vessel, and at Qn in the other, which are not in the same horizontal plane, but let Qn produced meet the other vessel in ab at a distance h below Pm. SUBJECT TO GRAVITY. 35 If the equilibrium can exist in this state, it will not be disturbed by replacing the open section Qn by a fixed plane, or by supposing the surface to become rigid. The fluid between Pm and ab exerts on ab a pressure which will be transmitted by the intermediate fluid and impressed on the rigid surface Qn. Let K be the area of Qn, then the pressure thus exerted on this surface from below = pK = gph K. The pressure upwards then on Qn being gphK, the fluid cannot be at rest unless this vanishes, which it can only do by h becoming zero. And the preceding demonstration is independent of the forms of the containing vessels, which may be supposed any whatever. Hence, when the vessels are contiguous, the surface of the liquid in each must be in the same horizontal plane. If the vessels are not contiguous, but at a considerable distance from each other, the preceding reasoning will apply, ab and Qn being taken on the same level, for all level surfaces are surfaces of equal pressure; it will follow therefore that the equilibrium is not possible unless Pm and Qn are on the same level. This proposition is also evidently true for any number of vessels, for since the surface in any two will be on the same level, the surface in all must be on the same level. 37. Prop. To determine the condition of equili- brium of several fluids contained in any communicating vessels. Let PQ, P'Q',...and P,Q,, PQ,,,….. (Fig. 5.) be the bounding surfaces of several liquids, which are contained in two vessels which communicate with each other. Let PQ be the level surface in which the two fluids bounded by P'Q' and P,Q, meet, and let PQ produced in- tersect the other vessel in mn. Then if a fixed plane be supposed at mn, or if the particles in the surface mn 36 THE PRESSURE OF FLUIDS become rigid, the transmitted pressure of the liquids super- posed above PQ will be sustained by the rigidity of this surface. Let K be the section of the vessel at mn, and let p', p"...be the densities of the liquid contained between PQ and P'Q', P'Q' and P"Q", &c. respectively, and h', h"...the thickness of the layers. Then the transmitted pressure on mn = pK = g(p'h') K. (Art. 33.) But this upward pressure may be counterbalanced by strata of liquids superposed above mn. , Let p, p,...be the density, and h,, h,,, the thickness of the successive strata of superposed fluids which effect this equilibrium, their surfaces being at P,Q,, P„Q,,... Then the pressure on mn = pK = g(p,h,) K, and the equilibrium will subsist if this equals the pressure on the under surface, that is, the condition required is Σ (p'h') = E(p,h,). If there are only two fluids the equation becomes p'h' = p,h,, that is, their elevations above the point in which they meet are inversely as their densities. 38. The following method of arriving at the same condition is given on account of the beautiful example it affords of the application of the principle of Virtual Velocities. Let any number of fluids whose surfaces are at P, P'... in one vessel be in equilibrium with the fluids whose surfaces are at P,, P, in another. (Fig. 5.) SUBJECT TO GRAVITY. 37 Let PQ be the horizontal plane in which the two fluids whose surfaces are at P and P, meet. Then if the fluid is in equilibrium, we may consider the pressures of the superposed strata as forces impressed on every particle in the surfaces at P and P, of the fluid machine. Then if k be the area of PQ, and K of mn, we have, using the same notation as before, the impressed force on PQ = pk = gkΣ(p'h'), PQ, = pK = gKΣ (p,h). Now let a small displacement consistent with the con- ditions of the system be given to the points of the appli- cation of the forces; that is, let the surface PQ and P,Q, move through the vertical spaces a, - ẞ, respectively; then by the principle of virtual velocities, - gΣ (p'h'). ka - gΣ (p,h,). Kẞ = 0. But the system must be invariable, that is, we must have ka - KB = 0, whence (p'h') - Σ (p,h) = 0, Σ or Σ(p'h') = (p,h). If there be but two liquids the condition becomes p'h' = p,h,, or their elevation above the horizontal plane in which they meet is inversely as their density. = If there is but one liquid, or p' = p,, then p' (h' − h) = 0, which can only be satisfied if h' h,, or the surfaces must be in the same level. 39. The general condition at which we have arrived for the equilibrium of several fluids in a system of com+ municating vessels, is Σ (p'h') = Σ (p,h,). 38 THE PRESSURE OF FLUIDS Either of the columns may be supposed to consist of indefinitely thin strata, whose density is uniform through- out the extent of any one stratum, but differing much from one stratum to another. Also it is quite immaterial, so far as the actual equi- librium is concerned, in what order the densities succeed each other; for if this equation be satisfied, the columns may be in equilibrium without any reference to the order of succession of the densities. Thus a heavier fluid may lie above a lighter, a liquid above a gas; such an arrangement, however, is not one. of stable equilibrium; for the equilibrium of a system of particles is only stable when the centre of gravity is the lowest possible. Such an arrangement then, being one of unstable equi- librium is theoretically possible, and also practicable, if any means be taken to prevent the least disturbance of any of the particles of the system. If, however, such a disturbance be not guarded against, the different fluids will pass into a position of stable equilibrium, in which the heaviest fluids will occupy the lowest place. Thus as we have seen (Art. 24.) the atmosphere can never be in a state of stable equilibrium, since the dif- ferent parts of all vertical columns are subject to great variations in density. 40. In applying the preceding condition where the atmosphere is one of the fluids, it will be convenient to assume Σ (p,h,) = ph. For the effect produced is the same as may be produced by a homogeneous column. If g be the accelerating force of gravity, the weight of the atmospheric column may in the same manner be represented by gph, the weight of a column of uniform density p, and given height h, and whose base is equal to the unit of area. SUBJECT TO GRAVITY. 39 41. If the surface Qn (Fig. 4.) be supposed rigid, it will sustain a column of fluid whose surface Pm is of any height, and the pressure on its under surface will be have pK=gph K. If W then be the whole upward pressure on Qn, we W = gph K, which may be increased indefinitely by increasing K and h, the unit of area remaining the same. But we may conceive Qn replaced by a piston so loaded as to equal W. There will then be equilibrium as before. Thus an enormous downward pressure may be sustained by the upward pressure, which is transmitted from the weight of a column of fluid of small section but considerable height; thus it is evident that an exceedingly small quan- tity of water may be made to sustain or raise a weight however large. 42. When water is contained in any vessel or in a system of communicating tubes or vessels, and exposed at its upper surface to the atmospheric pressure, it will be in equilibrium when the whole surface or the surfaces in the different vessels are in the same horizontal plane, whatever be the pressure to which their surfaces are sub- ject, provided it be the same for all the surfaces. the equilibrium may also subsist under certain circum- stances when the vessels are inverted, and the conditions requisite for equilibrium in these cases are supplied by the equation of the preceding articles. But Suppose a vessel, as for instance, a tumbler full of water to be inverted, then since we have only two fluids, air and water, the preceding equation of equilibrium reduces itself (Art. 40.) to p'h' = ph, 40 THE PRESSURE OF FLUIDS i where p' is the density of the water, and р of the air supposed homogeneous, and h', h are the elevations of the fluids above the horizontal plane in which the fluids meet; and as long as this equation is satisfied the equi- librium is possible. The equilibrium however cannot under these circum- stances actually take place, for this being an instance of unstable equilibrium, any disturbance which causes the least displacement in any part of the surface of the fluid will destroy the equilibrium. If a piece of paper be laid on the surface of the water, the vessel may then be in- verted and the water will remain suspended, the particles of the water being insured from any displacement by the rigidity of the paper. If the vessel be of small diameter, as a capillary tube, the molecular action will insure the particles at the surface from displacement. But the most usual way of effecting this in practice is to invert the heavy liquid over a basin containing the same liquid, which serves the double purpose of insuring the stability of the surface and of permitting the superincum- bent column to vary in altitude, the equilibrium still remaining stable. These conclusions will be sufficiently illustrated by an explanation of the siphon and barometer. 43. The Siphon. Let a tube ABC (Fig. 6.) be partly filled with water, and inverted; let P and Q be the surfaces of the water in the two branches, they will be in the same horizontal plane when there is equilibrium. Then if p' be the density of the water and h' the height of B above P or Q, the condition for the equilibrium of these columns with the atmospheric column is = p'h' ph. But the equilibrium will also subsist if ph be >p'h', since it is the same for both columns, that is, if the SUBJECT TO GRAVITY. 41 pressure exerted by the atmosphere at P and Q be greater than the weights of the columns BP or BQ of water. . Let p₁ be the atmospheric pressure, then gp'h' is the weight of the column of water whose section is unity; and we must have p₁ equal to or greater than gp'h'. If p₁ = gp'h', the pressure at B will be nothing. If p₁ be >gp'h', the pressure at B = p₁-gp'h'. — If p₁ be f(x + 1) y d x fxyda fyda' x or as (fa²yda) (fydx) is < or > (fxy dx)³, which expression being independent of l, it follows that the relative positions of these centres cannot change for different depths. 51. PROP. PROP. To determine the centre of pressure of a plane made up of portions whose centres of pressure are known. Let A be the area of a plane consisting of portions A1, A2, A3,...whose centres of pressure are known, then A = A₁ + A2 + A3 + ... 1 G 50 THE PRESSURE OF Let x1, x2...and y₁, y... be the co-ordinates of the centres of pressure of these portions. Let h1, h2...and k₁, k2...be the co-ordinates of their centres of gravity. Then a fx² y d x fxy dx between proper limits. Now fxy dx = A₁h₁, by the property of the centre of gravity; 2 .. fx²yda = А¸h₁x₁, 12 and similarly for all the other portions. But if X, Y be the co-ordinates of the centre of pressure of the whole surface, since the pressure on each portion may be con- sidered as a single force through its centre of pressure, we have by the general proposition for the centre of any number of parallel forces, X Σ fx²y dx Σfxy dx Y Σ 1 fx y² dx Σ fxy dx X A‚h‚æ‚ + A₂hqª½ + ... Y = 1/1/ A₁h, + Á₂h₂ + 1 A₁ k₁ y₁ + Aşk ₂Y ₂ + ... A₁ k₁ + A₂ ką + ... I which determine the centre of pressure of this compound area. 52. The formulæ in the preceding articles are suf ficient to determine the centre of pressure in all cases, in consequence, however of the difficulty of expressing y as a function of a in some cases, particular methods are more convenient than the application of the general for- mula, as will be seen in some of the following examples. When the plane is symmetrical about the axis of x, the equation (5) or (7) is sufficient to determine the centre FLUIDS ON SURFACES. 51 of pressure since it lies on the axis of x, the equation (6) or (8) being nothing in the case of a symmetrical plane. Similarly, if the plane be symmetrical about the axis of y, the equation (6) or (8) will be sufficient. 53. Ex. 1. A semiparabola at a given depth below the surface of the fluid with its axis vertical. The vertex being uppermost and the axis being taken for the axis of x, the ordinates y will be parallel to the intersection of its plane with the surface of the fluid. The formulæ here are X f(x² + lx) ydx and Y 12424 1ļ f(x + 1) y² dx f(x + 1) y d x j (x + 1) y d x In the parabola y² = 4mx; .:. f(x² + 1x)ydx = 2√m [(x² + læ³) dx 2 √ m (2 x + 3 1) x² ; 2√m f(x + 1) ydx = 2√ m [(x² + lx³) dx 27 2 √ m (3 x + ÷ 1) x³ ; f(x + 1) y³ dx = 4m f(x² + lx)dx = 4m ( x + ½ 1) x²; no correction being requisite since the integrals begin with X x and Y; // x + • V X, // x + 3 / 1 1 x + / / // Y = √ m 1 x + 3 / 1 10+1 = mx. // x + 3 / 1 3 1 / 2x + 1 ' whence the position of the centre of pressure is known. Let the vertex of the parabola be in the surface of the fluid, or 1 = 0, 0, then 52 THE PRESSURE OF X = x, and Y = ½ √ mx = 5 12 Y, as would have been determined at once from equations (7) and (8). then ty Let the parabola be sunk to a great depth, or l = ∞¹³, X = {x, Y = 3√ mx = & y, which are the expressions for the centre of gravity of a semiparabola, x and y being the extreme ordinates. Ex. 2. A rectangular flood gate its upper side co- inciding with the surface. Bisect the upper side, and taking this point for the origin of co-ordinates, equation (7) is sufficient. Let a be the vertical, and b the horizontal side of the gate, then a a f x²y d x fox dx ૨ X = > since y is constant, L x y d x Lxdx x d x || 23 a. — a² a The whole pressure on the gate = gp xx ab (Art. 46.) × 2 = ½ 8 pa²b. If then a force be applied at the centre of pressure in an opposite direction to the pressure of the fluid, and equal in magnitude to gpab, the gate will be kept at rest by this single force. Hence may be found the moment of the pressure to turn the gate about a given axis. The moment to turn the gate about a vertical axis at the side b gpab. 1 g p a²b². FLUIDS ON SURFACES. 53 The moment about an axis in the surface = ½ gpa² b × 2 a = gpa³b. The moment about an axis at the bottom of the gate = ½ gpa²b × { a = gpa³b. The flood gate may turn about a vertical axis on a single hinge at the depth of then a force equal to ds the height of the gate, and 1 gpa²b at the centre of pressure, 2 or gpa²b at the opposite side will keep the gate at rest. If it turn about an axis in the surface, then a force equal to gpa³b applied at the bottom of the gate will keep it at rest. If it turn about an axis coinciding with its bottom, the force which must be applied at the top is gpa³b. In the preceding reasonings the height of the water is supposed to be invariable. Ex. 3. The staves of a barrel held by a single hoop. From the preceding example it appears that the depth of the centre of pressure of any rectangle is equal to ds of its height. Hence if we conceive a barrel to be composed of a great number of similar staves, each of which differs insensibly from rectangular ones, they pre- sent a plane surface to the fluid, and the barrel when full of fluid will be kept together by a single hoop passing through the centre of pressure of all the staves. Hence the hoop must be at a distance equal to ds of the height of the barrel from the top. If the staves be prevented from revolving inwards by the bottom of the barrel, the hoop may be below the centre of pressure, but not above it; it will be best placed just below the line of pressure. 54 THE PRESSURE OF Ex. 4. A quadrant of a circle the radius coinciding with the surface of the fluid. Let a be the radius of the circle, and the centre be the origin, then by (7) and (8), since y = - a² - x², √ a² √x (a² – x²) d x √ x² √ a² − x² d x X Y = } 1 0 Sox √ a² - x² dx a 2 √ x √ a² - x². dx Now fo² Va²-xda may be found by the method of parts having multiplied and divided by va²², but the following substitution is remarkably convenient in integrals of this form, and a similar one may be adopted in all integrals which involve circular functions. Assuming x = a sin 0, then da = a cos 0 d0, and √ a²-x²= a cos 0, therefore 2 - - Ꮎ Ꮎ [“ x² √ a² − x² d x = a¹ f* sin² 0 cos² 0 d0. Now sin² 0 cos² 0 d0 = Ꮎ Ꮎ dᎾ sin² 20 d0 = √(1 − cos 4 0) do 4 (0 - sin 40) + C' ; π -- 1 π * sin? Ꮎ cos Ꮎ dᎾ = ㅎㅎ ​J = 2 And fa√ a² - x² dx = − ‡ (a² − x²)² + C; ... x* 2 · 'x√ a² − x² dx = ± a³. And fx (a²-x²) d x = ({ a²−4x²) x² + C' ; ... fox (a² − x²) dx = ‡ a¹ ; 1 a П 8 2 · X 3 - a³ 16 π a, Y = // 3a. a³ FLUIDS ON SURFACES. 55 Ex. 5. A right-angled triangle with its base in the surface. Let ABC (Fig. 10.) be the triangle, let AB=a, BC=b, then X So x²y d x 0 .a Y = 1/1/1 La x y d x Lexy² dx Lxy d x The relation betwixt y and x, or the equation to BC which cuts the co-ordinate axes at distances a and b from the origin is then substituting for y, Y + b a ३ । ɑ − da 1, 1 in³ C; Ja²y dx = fb (1 − 2) a² dx = b ( - ) + C = a 3 1 4 f" x²y dx = 1 a³b. Similarly fxy dx = 12 — a³b, [xy'dx = fb (1 − 2² + =) xdz fb² b2 a a² 2 x 1 x² 3 a + x²+C; 4 a² 2 a²b³ ·´`· L'"'x² ydw = b² (¦ − ¦ + ¦ ) a² = žœ²ð²; 3 4 12 .. X = { a = AN, Y = b = NP, whence P the centre of pressure is fully determined. Ex. 6. A sector of a circle, the centre being in the surface. Let ABC (Fig. 11.) be the sector, and let the side AB make an angle a with the surface of the fluid, and let BAC = ẞ and AB = a. Let AP=r be any radius vector inclined at an angle O to AB, and let PQ be the small element described by 56 THE PRESSURE OF dr, the radius vector r + dr having revolved through an angle do. Then the element PQ = dr × rd0 and p = gp· PN gpr sin (a + 0) ; therefore, the pressure on PQ = gpr²dr sin (a + 0) d0. Then, taking the moments about a vertical and hori- zontal line through A, we have if X, Y be the vertical and horizontal ordinates of the centre of pressure, since NP = r sin (a + 0) and AN = r cos (a + 0), Xƒª ƒ® r² sin(a+0)drdə = ƒ“ƒ®³r³ sin² (a+0) drd0, Ο Yƒ“ƒ³‚² sin(a+0) dr d0 = ƒ ƒ³ ³ sin (a+0) cos (a+0) drdə. .. 0 0 2.2 ffr³ sin² (a+0) dr d0 = ♫♫ª‚³ sin² (a+0)dr də ssa 0 0 √1 r² sin² (a + 0) d0 + C' ; a4 4 ƒ} {1 −cos 2 (a + 0) } d0 a¹ {0 − 1 sin 2 (a + 0)} + C' ; therefore, 20.3 ƒª Sª³r³ sin² (a + 0) dr də a Similarly, ƒª Sª³ r² sin² (a+0) dr də Also, a {ẞ+1[sin2 a − sin 2 (a + ẞ)]} · 8 a³ {cos a − cos (a +ẞ)}. 3 ƒ“ƒª³r³ sin (a + 0) cos (a + 0)drd0 = ƒ” * § sin2 (a + 0) d0+ C' 4 a 4 C − 1 cos 2 (a + 0)} a¹ 16 {cos2a-cos2 (a+ẞ) } ; FLUIDS ON SURFACES. 57 3 B+{sin2a-sin2(a+ß)} ... X 8 cosa - cos(a+B) 3 cos 2 a cos (a + B) Y 16 cosa cos (a + - B) a, whence the centre of pressure is fully determined. Let the radius AB be perpendicular to the surface of the fluid, then a = .. X π 2 3B-sin (+2ẞ) 8 cos (+3) - COS a 8 3 B+ sinẞ, cosß sin B B}a, 3 B -5 (sin/3+ cos Ba 8 3 COST cos (π + 2ẞ) Y 16 cos (+B) COS a 3 1 + cos 2 B a = 16 sin ß 3 sin² B 8 sinẞ a 3 == sin ß. a. Let the sector be a quadrant, or ẞ=; X 3 00100 8 π | 크 ​3 3 α = πα, Y = -a, 16 8 the same values as were obtained before, Ex. 4. Ex. 7. the surface. An oblique parallelogram with one angle in Let ABDC (Fig. 12.) be the parallelogram, and let AB = a, AC = b, make angles a, ß, with the vertical through A. H 58 THE PRESSURE OF Let X, Y, be the vertical and horizontal co-ordinates of the centre of pressure. Let x,, Y, ,y,, be the vertical and horizontal co-ordinates AN, NP, of any point P whose co-ordinates referred to the axes AB, AC are x, y. Then, x, = x cos a + y cos ß, Y = x sin a y sin B. The co-ordinates being oblique, an element dx dy at P = da. dy sin (a + B), and p = gpx, we have therefore, 2 X ƒ“ƒ³x,dx dy = ["f*x* dx dy, ь Y ƒª f' x, dx dy = ƒª fox, y, dx dy, whence X and Y may be determined. Now, ffx² dx dy = f(x² cos² a +2xy cosa cosẞ+ y² cos² ß) dx dy = = f(x² cos²a + xy cosa cosẞ+ y² cos² ß) y dx+C, the integration being performed for y, then 2 [[²x² dx dy = f(x² cos² a +xb cosa cosß + b² cos² ß) bdx b 2 (x²cos²a+xb cosa cosß+b² cos² ß) b x + C ; x² dx dy = (a²cos²a+ab cosa cosẞ+b² cos² ß) ab. Similarly, a fª L'x, dx dy = (a cosa + b cos ß) ab. FLUIDS ON SURFACES. 59 Also, ffx,y,dx dy = []{{a² sin2a+a y sin (a− ẞ) · -Lysin 2ẞ da dy = [ { ½ x² sin 2a + ay sin (a − ß) ... [[° x, y, dx dy = [{a² sin 2a+ab sin (a – ß) {a² sin2a+ab sin (a – ß) - y² sin 2ẞ} y dx; SS - b² sin² ß} bdx -b2sin 2ẞ} bx+C; -b² sin 2 ẞ} ab; .. ·· X = a² cos² a + ab cosa cosß + · ƒ“ƒ° x, y, dx dy = {a² sin 2a+ab sin (a – ẞ) Y 6 (a cosa + b cos ẞ) b² cos² ß a² sin2a+ab sin (a – ß) – b² sin 2ß J 1/ (a cosa +b cos B) Let the parallelogram be right-angled, the side AC being in the surface. Then a = 0, B: π a² X a, Y - - ab 1 a 12 ļa as we have already determined them, Ex. 2. The Pressures on Curved Surfaces. 54. The pressure on any portion of a curved surface is determined by resolving the normal force at each point into the directions of the three co-ordinate axes, and cal- culating by two integrations the total components in these directions. These components may always be reduced to two forces; these two forces however seldom admit of a sin- gle resultant. But when the pressure exerted on a curved surface is a fluid pressure, the pressures always admit of a single resultant whose direction and magnitude must be determined. To find then the resultant of the fluid pres- 60 THE PRESSURE OF sure on a curved surface, the pressure at any point being resolved into its components in the planes of the co-ordi- nates, it will be shewn that the horizontal pressures destroy each other, their resultant therefore is zero, and that the resultant of the pressures is therefore vertical. 55. PROP. The horizontal pressures on the surface of any body immersed in a fluid are in equilibrium with each other. Let APB (Fig. 13.) be any body immersed in a fluid; let the body be referred to three rectangular axes, the surface of the fluid being taken for the plane of xy. Let P be any point x, y, z, in the surface of the body, and let w be a small element of the surface at this point, then pw is the pressure on this element in the direction PG of the normal to the surface. The value of p will be the same for all points which are at the same distance from the surface of the fluid, that is, for all points which are in the same horizontal plane, whether the fluid be homogeneous, or whether it be composed of level strata of different densities. Let a, b, c be the projections of the element w on the co-ordinate planes yz, xz, xy, respectively; and let a, ß, y be the angles which the normal PQ makes with the axes of x, y, z. Then a, ß, y are also the angles which the tangent plane at the point P makes with the planes of y≈, xz, xy, respectively. Therefore, a = w cos a, b cosß, = wcos ß, c = w COS Y F and multiplying these equations by p, ра pw cosa, pb=pw cos ß, pe = pw cos ˜y. But pw cosa is the resolved part of the normal pressure pw in the direction of x. FLUIDS ON SURFACES. 61 Hence pa, pb, pc, are the components of the normal pressure pw in the directions of the axes of x, y, x; or the component in the direction perpendicular to any co-ordinate plane of the normal pressure on any portion of the surface, equals the product of that pressure and of the projection of the portion of the surface on the co-ordinate plane. Whatever be the nature of the body, there must always be a portion of its surface opposite to w, which will have the same projection on the co-ordinate plane. Let PP,M be drawn perpendicular to the plane of yz, meeting the side of the body opposite to P in P; and let w, be the element of the surface at this point, which has the same projection on the plane of yz. Then the pressure on @, resolved in the direction perpendicular to the plane of y≈, being by what has just been stated equal to the product pa, will be the same whenever p is the same, that is, for all points in the same horizontal plane. W Thus the pressures on any portion w in the direction of the axis x, destroy each other; and the same may be shewn for the direction of y, and for every point in the same horizontal plane. Hence it appears that the horizontal components of the pressures exerted on the elements of the surface of any body immersed in a fluid, destroy each other in each hori- zontal section; and therefore the horizontal forces on its whole surface destroy each other. There is then no force which can produce lateral motion, the resultant of the horizontal pressures being zero; it follows therefore that the only forces which are to have a resultant are the vertical pressures, and consequently that all the forces may be reduced to a single vertical force which is the resultant of the components perpen- dicular to the plane of wy, and which arises from the excess of the value of p for the lower parts of the body. 62 THE PRESSURE OF COR. If the pressure p arises from a pressure exerted on the surface of the fluid, its value would be constant for every point in the surface of the body; and the components would destroy each other in the vertical as well as in the horizontal direction. Whatever then be the form of a solid or fluid mass, a constant normal pressure impressed at all points of its surface cannot produce any motion either of translation or rotation. 56. PROP. To find the resultant of the vertical pressures acting on a body immersed in a fluid. From P draw a perpendicular to the plane of xy meeting the body in P', and let w' be the element of the surface at this point corresponding to w at P. Their projections on the plane of xy are the same and equal to c, but the value of p is different. Let p' be the value of p at P'. Then the vertical line of particles terminated by these two elements will be pressed vertically from below upwards, with a pressure pc – p'c. Let the fluid be homogeneous, then if PP' = l, p-p' = gpl and pc - p'c = gpcl. But gpcl is the weight of a column of fluid whose volume is lc, hence the vertical pressure equals the weight of a volume lc of the fluid, that is, it equals the weight of the fluid column whose place is occupied by that portion of the body; and the same being true for every other column of the body, the whole vertical pressure is the weight of a mass of the fluid equal in bulk to the body displaced. The resultant therefore of the fluid pressures is a vertical force applied at the centre of gravity of the body immersed in an opposite direction to gravity. When the body is homogeneous, the centres of gravity, of the body and of the fluid displaced, coincide. If the body be not wholly immersed, we have p′ = 0, and the resultant of the pressures is the weight of the FLUIDS ON SURFACES. 63 volume of the fluid displaced by the portion of the body which is immersed, and is applied at the centre of gravity of this portion. 57. The preceding results are also true when the fluid is composed of horizontal strata of very different densities; as will also be evident from the following considerations. The equilibrium once established will not be disturbed by supposing any part of the fluid to become solid, so that this part itself becomes a floating or immersed body. But in order that the normal pressures exerted on the surface of the body by the surrounding fluid may be in equili- brium with the weight of this solid part, their resultant must be a single force, and act in a direction contrary to the weight of the body; and if we replace the part of the fluid which is supposed to have become solid by another body having exactly the same surface, it is evident that no change can have taken place in the pressures of the surrounding fluid; consequently the pressures exerted on the surface of a body immersed, wholly or not, in a fluid at rest either homogeneous or heterogeneous, are always equivalent to a single force, which is equal to the whole weight of the successive strata of fluid whose place is oc- cupied by the body, and which is applied in a direction contrary to gravity at the centre of gravity of these strata. We may conclude then that a body totally immersed in a fluid will be in equilibrium when its mean density is equal to that of the fluid displaced, and when its centre of gravity and that of the fluid displaced are in the same vertical; which latter condition is always fulfilled when the body and liquid are both homogeneous. The equilibrium of bodies which are not wholly im- mersed, but float at the surface of a fluid will be examined in the following chapter. 64 THE PRESSURE OF 58. Hydrostatic Balance. The conclusion at which we have just arrived is generally enunciated by saying, that a body immersed in a fluid loses as much of its weight as is equal to the weight of the fluid displaced. Hence it is evident that to obtain the true weight of a body it ought to be weighed in vacuo. Two bodies weighed in air, or in water, or in any other liquid, and which are in equilibrium on a very exact balance, have really very different weights unless their volumes should be equivalent. The greater weight is that of the body which has the greater volume, because having experienced a greater loss in the fluid, it is still in equilibrium with the other. If the same body is weighed successively in vacuo and in water, and W be its weight in vacuo, and W' in water, W and W-W' will be the absolute weight of the body, and of a quantity of water of the same volume. But when the volume is constant the weight varies as the density, hence W and W-W' are as the densities of the solid and water. If then D be the density of the solid, that of water being unity, we shall have D W W-Wi It is thus that the densities or specific gravities of sub- stances which can be weighed in water without being dissolved, are ascertained by means of the Hydrostatic Balance. 59. The reasonings in the preceding articles apply equally to the pressures exerted on the sides of vessels containing fluid; and the same result would be obtained; namely, that the horizontal pressures exerted from within to without on all the internal surface destroy each other, that is, they consist of pieces of equal and opposite forces; whence if a vessel is set on a horizontal plane, the action FLUIDS ON SURFACES. 65 of the fluid which it contains cannot put it in motion: this result is also a necessary consequence of the conser- vation of the motion of the centre of gravity. But if an aperture be made in one of the sides of the vessel below the surface of the fluid, the fluid will run out, and the pressure being no longer exerted on that part of the surface which is removed, the pressure which is exerted on the opposite side will not be destroyed. In this case then the sum of the horizontal forces are not zero for these points, and consequently the vessel can be put in motion on the side opposite to the issuing fluid. This is the principle of all the machines whose motion depends on the reaction of a fluid, and which has been suggested by Bernouilli as applicable to the motion of vessels. The application of this principle is exhibited in the machine called Barker's Mill. 60. From the same reasoning it is also evident that the whole vertical pressure exerted on the bottom and on the sides of a vessel is always equal to the weight of the fluid contained and applied in the direction of gravity to the centre of gravity of the fluid. Each vertical line of the fluid which extends without interruption from the surface to any point of the vessel, exerts at this point a normal pressure, which is equal to the weight of this line: that line which is interrupted and meets the internal surface of the vessel in more points than one, as for instance, at a point in the bottom and in one of the sides, exerts at these two points pressures whose vertical components are in opposite directions. The component which belongs to the lower point is in the direction of gravity, and exceeds the other by a quantity equal to the weight of this line; and the same is true for all the points in which this line meets the containing surface; thus the excess of the pressures downward over those upwards is equal to the weight of the contained fluid. The resultant then of all the vertical pressures of I 13 66 THE PRESSURE OF these lines of fluid is precisely the same as the weight of the fluid in question. This pressure must be accurately distinguished from that which takes place simply on the bottom of the vessel, and which is only equal to the weight of the fluid when the vessel is a right cylinder. It is less than the weight when the vessel increases in size from the bottom to the top, as the frustrum of an inverted cone, because the vertical lines of fluid which extend from the surface, and are intercepted by the sides of the cone, do not press on the bottom of the vessel; on the other hand, it is greater than the weight of the fluid, when the vessel increases from the top to the bottom, because the vertical lines which extend from the bottom of the vessel, and are intercepted by the side, exert nevertheless the same pres- sure on the bottom of the vessel as if they extended to the surface of the fluid; the deficiency in the weight of each of these incomplete lines being made up by the re- action of the side by which they are terminated. 61. When fluid is contained in a flexible vessel, the pressure and the resultant of the tensions on a portion of any section of the vessel must be equal and opposite. Let PQ (Fig. 14.) be a portion of any section of a flexible vessel which is full of fluid. Let p be the pressure at any point, and t the tension, which must be uniform throughout each section. Draw normals at P and Q meeting in O, and tangents at P and Q meeting in T, join TO. Let Y be the radius of curvature at P, and 2ds the portion PQ of the curve, the pressure on PQ = 2pds. The tensions at P and Q compound a force in the direction TO FLUIDS ON SURFACES. 67 " nearly. = 2t. sin POT = 2t . ds Y Hence, since there is equilibrium, 2pds = 2t ds Υ t whence p " or t = py· Υ This equation will serve to determine one of the quan- tities when the other two are assigned. Other questions connected with the pressure of fluids, as the equilibrium of dykes, &c. the reader will find treated in Moseley's Hydrostatics, Arts. 49-54. CHAPTER VI. ON THE EQUILIBRIUM OF FLOATING BODIES. 62. WHEN a body is placed in a fluid, its density, if it be homogeneous, or its mean density, if it be not homo- geneous, being less than the density of the fluid, it sinks in the fluid until the weight of the fluid displaced becomes equal to the weight of the body; it then remains at rest, provided its centre of gravity and that of the fluid dis- placed are in the same vertical. For the only forces which exist are the weight of the body and the resultant fluid pressure. And it has been shewn that the resultant of the fluid pressures is a vertical force, the horizontal forces destroying each other. The weight of the body therefore, and the resultant of the fluid pressure, must be in equilibrium with each other; and they are parallel forces, and may be applied at the same point, or at different points in the same vertical, or at points in different verticals. In the first case, the body will either ascend or descend vertically, and when the weight of the fluid displaced equals the weight of the body, it will be absolutely at rest. In the second case, the body cannot rest but may have a motion both of translation and of rotation communicated to it. Now, these forces are applied at the centres of gravity of the respective masses; hence the conditions for the equilibrium which are both necessary and sufficient, are, 1. THE EQUILIBRIUM OF FLOATING BODIES. 69 That the weight of the fluid displaced be equal to the weight of the body. 2. That the line joining the centre of gravity of the body and of the fluid displaced, be vertical. 63. To find the positions of equilibrium of a floating body. P If V be the volume of the fluid displaced and its density, and V' the volume of the body and p' its density, then by the first condition Vpg = V'p'g*; .. Vp = V'p', :: V: V'p' p, or V: V-V: p: p' - p; :: : - that is, the body must be cut by a plane, so that the seg- ments shall have to each other a given ratio. Also, by the second condition the line joining the centres of gravity of the two portions must be vertical, or the cutting plane must be perpendicular to it. Hence the determination of the positions of equilibrium of a floating body is reduced to the following problem in Analytical Geometry:-"To cut any body by a plane so that the volume of one segment may be to that of the whole body in a given ratio, and that the line joining the centres of gravity of this segment and of the whole body may be perpendicular to the cutting plane." When the section of a body satisfying these two conditions has been de- termined, it must be placed coincident with the surface of the fluid; the segment whose volume has been con- sidered being the segment which is immersed in the fluid, and the other segment being above the fluid: this position will be one of equilibrium. These conditions may be expressed by equations, the complete solution of which will give all the positions of the equilibrium of a body; sometimes their number will be infinite, as is the case for a solid of revolution whose 70 THE EQUILIBRIUM OF axis is horizontal; but it would be difficult to demonstrate a priori that there is always a position of equilibrium what- ever be the form of the body. The method of proceeding in any case will be sufficiently illustrated by the following example. Ex. A triangular prism with its edges horizontal. The determination of the positions of the equilibrium of this body evidently reduces itself to the determination of the position of equilibrium of a triangle which is its generating section. Here two cases present themselves according as one or two angles are immersed; we shall first consider the case when one angle is immersed, and shew how one case may be reduced from the other. 1º. Let ABC and PQ (Fig. 15.) be the sections of of the triangular prism and of the surface of the fluid made by a vertical plane. Let a, b, c, be the sides of the triangle which are opposite to the angles A, B, C, and x, y the sides CP, CQ of the part which is immersed. Let s be the ratio of the specific gravities of the body and of the fluid, that is, the ratio of their densities. Let V be the area of the part immersed, and V' of the whole, then by the first condition, (Art. 62.) Vpg = V'p'g, or V = v¹l' = s V''. V' P But V = xy sin C, and V' - ab sin C, therefore xy = sab…………..(1). Again, let G and F be the centres of gravity of the triangles ACB, PCQ. Then if D and E be the bisections. FLOATING BODIES. 71 2 of AB and PQ, CG = 3CD and CF = CE; join GF, DE, DP, DQ. Then by the second condition (Art. 62.) GF is vertical, and therefore perpendicular to PQ. But DE is parallel to GF. It is therefore perpen- dicular to PQ; that is, the line joining the bisections of AB and PQ must be vertical. Hence also, since PE- EQ, and DE is perpendicular to PQ, DP = DQ. Conversely, if DP = DQ, the line DE will be perpen- dicular to PQ, and therefore its parallel GF will be perpendicular to PQ. Hence that the line joining the centres of gravity of the body and of the fluid displaced may be perpendicular to the surface of the fluid, it is necessary and sufficient that DP should be equal to DQ. Then if CD = h and a, ß be the angles DCA, DCB, we have DP² = h² + x² 2hx cosa, DQ² = h² + y² − 2hy cosß; a² – 2hx cosa = y² – 2hy cosẞ……….. ·(2). sab But from (1) y , eliminating y, we have X x¹ − 2 h cosa x³ + 2hsab cos ß æ s² a²b² 0…………..(3); whence having determined the four values of a, the cor- responding values of y are given by the equation Y sab X The equation (3) is of even dimensions, and has its last term negative; it must have, therefore, two real roots of contrary signs. The other two roots may be real or imaginary. If they are real, the rule of signs shews us that it has three positive and one negative root; for there must be three 72 THE EQUILIBRIUM OF changes and one continuation, whatever be the sign of the term which is wanting. The unknown quantities x and y, which are the sides of the triangle PCQ, can only be positive quantities less than the sides CA and CB respectively; the negative root, therefore, of the equation may be rejected as inapplicable. There are, therefore, at the most but three positions of equilibrium when one angle only is immersed. 2º. Let two angles, as A and B of the triangle, be immersed. Then if PQ be considered as the line of floatation, the centres of gravity of ACB and APQB must be on the same vertical, and we must have, as before, APQD : ACB :: p' p': p :: s : 1, whence PCQ : ACB :: 1 s: 1; .. xy= (1 − s)ab... (4). Hence, eliminating y between this and (2), the equation is the same as before, (1 − s) being in the place of s, or the equation required is ¿x¹ −2h cosa x³+2h (1 − s) ab cosẞ x − (1 − s)² a²b² = 0…….(5). And from the same reasoning as was applied to (3) it appears that there are at the most but three positions of a triangle floating with two angles immersed. There are, therefore, three positions of equilibrium of a triangular prism when one angle is immersed, and also three when that angle is the only one not immersed; that is, there are on the whole six possible positions of equi- librium for each angle, and therefore eighteen for the whole triangular prism. 64. Let the section of the prism be an isosceles tri- angle; then pursuing the same reasoning, we shall arrive FLOATING BODIES. 73 The at an equation which admits of immediate solution. equation of the preceding article may be adapted at once to this case. Let ab, then the triangles ACD, BCD, are right- angled and equal, whence, B = a, h² = a² - 1 c², a cosa = h, and equations (1) and (2) become ry = sa³, and x² – y² 4a² - c² 2 a - (x − y) = 0......(6). = This is satisfied by taking a ya√s, which is a possible value, since s is less than unity. Hence PQ must be parallel to AB, that is, AB is horizontal; and the same is true when C is out of the water. But there are other positions of equilibrium; for suppressing the factor (x − y) we have x + y = 4a² - c² 26. which combined with xy = sa" gives for the two values of r and y τα {4a² − c² ± √√/ (4a² – c²)² – 16s a¹} . Each of these being taken successively for a and y, if both be less than a, there are two new positions of equilibrium in which the base AB is out of the fluid. Substituting 1-s for s, there are two other positions when AB is immersed, provided both roots be less than a. When the two preceding roots are equal, the base AB is horizontal: these new positions ought to be identical with the former; in this case 4a² - c² = ± 4a² √√5, whence x = y = a/s, as before. K 74 THE EQUILIBRIUM OF 65. Let the section be an equilateral triangle, then a = b = c, and the equations of the preceding articles give for the unequal values of x and y, when one angle is immersed, a 3 ± √9 - 168}, 4 and when two are immersed, α 4 {3 ± √ 168 − 7} . ±√168 The value of the ratio s must be examined into, that these may be real and less than a. 9 1 6 7 If s be < and >, the first expression is real and less than a; and if s be < ½ and > 16, 1/ the second ex- pression is real and less than ɑ. 1 9 16 Hence when one angle is immersed, the limits of s are between and ; and when two angles are immersed, the limits are and; and between the values and 1% the prism has no oblique position of equilibrium. 9 6 7 1 6 Since all the angles are equal, there may sometimes be eighteen and sometimes only six positions of equi- librium. 66. Besides the horizontal positions of equilibrium which we have just treated of, prisms and cylinders may float in a vertical position with their bases parallel to the surface of the fluid; there are two positions of equilibrium for each body when they float in this manner, for there is one for each base of the solid. The line joining the centres of gravity of a vertical prism and of the part immersed must be perpendicular to the surface of the fluid; the ratio of their volumes is the same as that of their heights, and consequently the height of the part immersed is to that of the whole prism as the density of the body is to the density of the fluid; this one FLOATING BODIES. 75 condition then determines the depth to which the body sinks, and is the solution of the problem. Solids of revolution, and all bodies which are sym- metrical about a given axis, have two positions of equi- librium for each axis, which may be determined as in the following example. Ex. An ellipsoid with an axis vertical. Let a, b, c, be the semiaxes of the ellipsoid, and let the axis 2c be vertical. Let, be the distance of the plane of floatation from the centre of the ellipsoid, which will be positive or nega- tive according as this section is above or below the centre of the ellipsoid. Let K be the area of any section at a distance ≈ from the centre of the ellipsoid. The volume of the semiellipsoid=abc; the volume of the part between the plane of xy and the surface of the ellipsoid is "Kdz. Hence the volume of the part immersed π abc - "Kdz. 3 па 0 Hence when there is equilibrium } π a b c − ["Kd≈ = {πabe.s. Now the area K is an ellipse, and the equation to the ellipsoid being x² a² + ર y² + 1, b2 c² and the semiaxes of a section at a distance from the centre being obtained by putting y and x successively equal to nothing, are α b √e-2 c² - ² and √ c² - ~². C С 76 THE EQUILIBRIUM OF The area therefore required is παι K C2 (c² — x²) ; пав c² ... π abc - S (c² − x²) dz = ÷ ñπabcs. But f'(c² - x²)dx = (c² - 4x²)≈,, substituting and omitting the common factor πabc, we have z 001 20 3 1 c³ (C - 2,2)%, = 8, or ≈³ — 3c²x, − 2 (28 − 1)c³ = 0, whence the distance of the plane of floatation from the centre of the ellipsoid is known. This equation being independent of a and b, the values of ≈, are independent, or are the same for the ellipsoid, the spheroid, and the sphere. It must have one real root, which will lie between ±c, being positive or negative according as s is> or < 1. In the extreme cases when s = o and s = 1 this root is ≈ and ≈ Z C. C The other two roots will be found greater than c, and are therefore excluded. The Stability of a Floating Body. 67. The conditions that a body may rest in a fluid are, as we have seen, two; it remains now to consider what is the nature of the equilibrium in which it exists, that is, if the body be slightly disturbed from that posi- tion, by being moved through a small angle about some axis, and then left to itself, whether it will have a tendency to return to its original position, or to recede from it, or to move neither way, but rests in that new position. These distinctions in the nature of the equilibrium have, as is well known, received the corresponding terms FLOATING BODIES. 77 of stable, unstable, and indifferent or neutral equilibrium, they apply to the nature of the equilibrium of any system acted on by any forces and are therefore applicable to a floating body. The general proposition therefore that the positions of stable and unstable equilibrium recur alter- nately will obtain here. In general when a floating body is disturbed there will be a motion both of translation and of rotation about some axis; as these however may be independent, we shall consider only the motion of rotation, and suppose that the body is disturbed by being caused to revolve about some axis, and then left to itself; it is the mo- tion after the disturbance has ceased which we have to examine. It will always be supposed that the volume of the fluid displaced is unaltered by the disturbance, for the impressed force being the weight of the fluid displaced, if there be variation in its volume a motion of translation must any occur as well as a motion of rotation. It will also be supposed that the body is symmetrical about a vertical plane, and that the disturbance leaves the plane of symmetry vertical. Under these circumstances it will be found that in some cases, where the body is left to itself, its motions will be vertical and angular simultaneously. To illustrate this, we shall premise the following proposition. 68. PROP. If a body be turned through a small angle about an axis through the centre of gravity of the plane of floatation, the fluid displaced is unaltered. Let ADB (Fig. 16.), represent the body in its new position, having been turned through a small angle about an axis through C, the centre of gravity of the plane of floatation AB. Let the axis of rotation be taken for the axis of y, and let O be the angle of displacement. 78 THE EQUILIBRIUM OF 1 The wedges ACa, BCb may be considered as generated by the revolution of CA and CB, hence if dady be an element of the plane CA at a distance x, y, from C the elementary prism which is generated = x0.ddy, therefore, the wedge ACa = ffx0dx dy = 0ffxdxdy. Similarly, if da'dy' be an element in the plane CB at the distance a'y from C, the wedge BCb = 0 ffx'd x'dy'. But by the property of the centre of gravity ffxdxdy = ffx'dx'dy'; ..the wedge ACa = the wedge BCb. Hence a Db = ADB or the volume of the body im- mersed or of the fluid displaced is unaltered. 69. PROP. To explain the connection between the vertical and angular motions of a floating body. A body immersed in a fluid is acted on by two forces, its own weight applied at its centre of gravity, and the weight of the fluid displaced applied at its centre of gravity. Now the motion of the centre of gravity of a body will be the same as if these forces were applied at that point, and motion of rotation round the centre of gravity will be the same as if that point were fixed and the same forces applied. Suppose the body to have been disturbed from its position of equilibrium by being made to revolve about an axis through the centre of gravity of the plane of floatation, then as has been shewn the volume of the fluid displaced will not be altered. The centre of gravity of the body will describe a small circular arc which may be considered as a straight line. If the centre of gravity of the body be (in the position of equilibrium) vertically beneath the centre of gravity of the plane of floatation, this small straight line will be horizontal. There is no force at present tending to move the centre FLOATING BODIES. 79 of gravity of the body, and if the equilibrium be stable, so that the angular motion round the centre of gravity brings the body back to its original position of equili- brium, the centre of gravity of the original plane of floatation will remain in the surface of the fluid. In this case then the small angular motions will be unattended with any vertical ones, or there will be motion of rota- tion simply. But if in the position of equilibrium the centre of gravity of the body is not vertically beneath the centre of gravity of the plane of floatation, when the body is disturbed as before, its centre of gravity will be raised or lowered, and though there is no force in consequence of such a disturbance tending to produce a motion of translation in the centre of gravity but only of rotation about it, yet in consequence of this angular motion round that point the centre of gravity of the original plane of floatation will be raised above or lowered beneath the surface of the fluid. The volume of the fluid displaced will therefore be altered, and the weights of the body and of this fluid thus becoming unequal, a force will be gene- rated which tends to produce a vertical motion in the centre of gravity of the body. Hence there must be simultaneous motions of transla- tion and of rotation. In bodies which are symmetrical with respect to the vertical line through their centre of gravity, it is evident that the centre of gravity of the plane of floatation will be in this line. But there are cases in which it is not, as for instance, in a scalene triangle with one angle im- mersed. 70. The connection which thus subsists between the motions of translation and rotation, when the volume of the fluid displaced remains unaltered, having been shewn, we shall now suppose the floating body to assume a new position, and consider simply the force which exists in 80 THE EQUILIBRIUM OF consequence of this new position to move it about an axis through its centre of gravity. The body is supposed to be symmetrical about a ver- tical plane both before and after the disturbance, that is, in its old and new position, and the volume of the fluid displaced is constant. It will be convenient to premise the following proposition. 71. PROP. The intersection of the two planes of floatation is a line passing through their common centre of gravity. Let ADB (Fig. 16.), be the section of the floating body by the plane of symmetry and let the two planes of floatation AB and ab intersect in C, then C is their common centre of gravity. Since the fluid displaced is invariable, subtracting the common part a DB, the wedge AC a the wedge BCb. The wedges may be divided into elementary prisms the base of one of which dady and its height = x0, therefore, = the wedge ACa = [[x0dx dy = 0 ffxdx dy = Ah, if A be the area of the portion Ca of the plane of float- ation and h the distance of its centre of gravity from C. Similarly the wedge BCb = A'h', therefore Ah = A'h', for the wedges are equal. The distance of the centre of gravity of the plane ab Ah - A'h' from C= Á + A' 0, or is at C. The same reasoning would apply to AB, for the base dady of the elementary prisms may be taken in either plane, hence the centre of gravity of the two planes is in the line of their intersection and is therefore at C, since the solid is symmetrical about the vertical plane. FLOATING BODIES. 81 72. PROP. To determine the nature of the equili- brium of a floating body. This as we have seen depends on the tendency of its motion when left to itself after an angular disturbance; hence the moment of the impressed force, that is, of the fluid displaced about the centre of gravity of the body, is the quantity to be discovered. Let AB, ab (Fig. 16.), be the planes of floatation of the old and new position; they will intersect in C their common centre of gravity (Art. 71). Let G, H be the centres of gravity of the body and of the fluid displaced before the disturbance, and H' the centre of gravity of the fluid displaced in the new position. Through H' draw the vertical H'M and draw GN horizontal. The fluid displaced acts upwards in H'M, and if W be the weight of the body, that is, of the fluid displaced, the moment of the impressed force about G = W.GN and the equilibrium will be stable, unstable, or indifferent, according as this force diminishes or increases, or does not affect the angle GMH': to determine these different cases for any given body we must express GN in terms of assigned or assignable quantities. This may be readily effected by taking the moments of the whole body made up in two different ways. The moments may be taken with respect to any ver- tical plane; let them be taken with respect to the vertical through C. Let g and g' be the centres of gravity of the wedges, C'm and Cm' their horizontal distances; then, since AC a + a Db = ADB + BCb and the moment of each of these equals the moment of L 82 THE EQUILIBRIUM OF ADb, we have drawing the horizontals GE, HF, H'F', ACa.Cm + a Cb. H'F' = ADB.HF + ( − BCb.Cm'), or, V being the volume of the part immersed and observ- ing that HF - H'F' = GN + HL, V.(GN + HL) = ACa. Cm + BCb. Cm'.........(1). - But ACa.Cm = 0 ffxdx dy x Cm.........(Art. 71.) ffx² dx dy by the property of the centre of gravity. Similarly BCb. Cm' = 0 ffx' dx'dy' × Cm' 12 ffx'2 dx'dy'. And the sum of these two double integrals is the mo- ment of inertia of the plane of floatation ab about an axis through C, let this be I; .. V(GN+HL) = 01, or GN Ө I HL. V Let GH = λ, then HLλ sin 0 = 10 nearly; .. GN=0 I (= -x), whence GN is known for any given body. Ꮎ The moment of the impressed force = WO I V (1-x). When is greater than λ this is always positive, or the moment of the fluid displaced brings the body back to its original position; the equilibrium is therefore stable. I When λ the moment is zero, or the body has no V tendency to move; the equilibrium is therefore neutral or indifferent. FLOATING BODIES. 83 I. When is less than the quantity becomes negative, V or the moment evidently moves the body farther from its original position: the equilibrium is therefore unstable. This moment is a measure of the stability of the float- I ing body, and depends entirely on the quantity λ, that V is, on the moment of inertia of the plane of floatation, the quantity of the fluid displaced, and the relative position of the centre of gravity of the body and of the fluid displaced. Hence the equilibrium will always be stable if this is always positive, which will be the case if G is below H, for then, as will be seen at once by retracing the steps of the investigation, the term (-A) will be positive. The equilibrium then is stable, unstable, or in- I A) is positive, negative, or different, according as (= = x) zero. 73. PROP. To determine the metacentre of a float- ing body. The nature of the equilibrium evidently depends on the position of M the metacentre. Then, pursuing the investigation as in the preceding article, we have by (1), I GN + HL = 0· V But GN+HL = HM.sin 0 = HM.0, nearly; 時 ​I I or HM = v , y' .. HM.0 = 0 whence M is fully known. = Also, GN GM sin 0= GM.0, and HG = λ0. Let GM = μ, then I # λ, 84 THE EQUILIBRIUM OF according as G is above or below H. The determination of the nature of the equilibrium of a floating body depends. on the value of μ. If u be positive, the equilibrium is stable and M is above G. If μ be negative, the equilibrium is unstable and M is below G. If u = 0, the equilibrium is neutral and M coincides. with G. Hence the equilibrium is stable, unstable, and indif ferent, as M is above, below, or coincident with G. From an inspection of the figure, it is evident that when M is above, below, or coincident with G, the equili- brium will be of the character just assigned to it. Hence, if a body be ballasted so that M can never come below G, the equilibrium cannot be unstable. Ex. A cone floating vertically. The plane of floatation will be a circle, and let the cone float with its vertex downwards. Let x, y be the height and radius of the base of the cone which is immersed, and a, b of the whole cone. Then I = 1 πу¹, λ = 2 (a − x), V = } πy³x. 4 3 4 Making these substitutions in the value of u, we have 3 [y³ μ = (a - x) 4. 20 3 fy² a -1): X a } · 20 But the part immersed bears a constant ratio to the whole body, (1º Art. 63.), and the cones being similar solids are to each other as the cubes of their height or of the radii of their bases, hence, 3 x³ = sa³ and y³ = sb³ ; FLOATING BODIES. 85 y2 b? b 2² = st² and µ = {st ^ — (1 − s))} b, x α μ 4 − − − s³) a whence the stability for particular values of s and of the b ratio may be determined. α If the cone float with its base immersed, we must ex- press and in terms of the proper quantities, and replace s by (1 − s), as in 2º. Art. 63. The Oscillations of Floating Bodies. 74. In the preceding propositions, the conditions of the equilibrium and stability of floating bodies have been fully considered; we have now to consider the vertical and angular oscillations consequent on a body being disturbed and then left to itself. The body when left to itself will make oscillations about its original position of equilibrium, until by the action of the fluid it is reduced to rest. In a complete solution of this problem the motion of the fluid ought to be taken into the account, as this how- ever would be an investigation of extreme difficulty, we shall consider simply the vertical and angular oscillations about the centre of gravity of a body symmetrical about a vertical plane. In determining the time of an oscillation, no account need be taken of the resistance of the fluid, for this resistance is a disturbing force which affects the extent but not the time of each oscillation; and if the disturbing force ceased at any instant to act, the body would go on for ever oscil- lating in an arc of equal extent to that which it had the instant at which the disturbing force ceased to act *. Hence, when the oscillations are small the time found very accurately. may be * AIRY's Planetary Theory. Art. 104. 86 THE EQUILIBRIUM OF 75. PROP. To determine the time of the small ver- tical oscillations of a body floating in a fluid. The vertical motions of the body are the same as the motions of its centre of gravity, and the motion of the centre of gravity is the same as if the whole mass were collected in it, and the forces applied immediately to it. Now the resultant of all the forces acting on a body floating in a fluid, is a single force equal to the weight of the fluid displaced. If, therefore, the body floating in the fluid be depressed through any space and then left to itself, the force applied to the body will be the whole weight of the fluid displaced, the resultant of which being a single force in a vertical direction, the motion of the centre of gravity, and therefore the motion of the body will be wholly in a vertical direction. The body not being wholly immersed, let V be the volume of the fluid displaced when the body is at rest, and y' that of the whole body. Then (Art. 62.) Vp = V'p'......(1). Let the centre of gravity be the origin of co-ordinates, and a plane parallel to the surface of the fluid be the plane of xy. Let ≈ be the distance of the centre of gravity from its is the effective ď² z original position at any time (t), then dt2 accelerating force on the body to bring it back to its ori- ginal position. The impressed force is the weight of the fluid displaced by the motion of the body. Let U be the volume of the fluid displaced by the depression of the body, its weight = Upg; then since V'p' is the mass moved, Upg Ug by (1). the accelerating force V'P V FLOATING BODIES. 87 The impressed and effective forces are equivalent, therefore Ug ď² z V dt2 But since the motion is small, we may assume U = K≈ where K is the area of the plane of floatation; d² z Kg dz, dt2 V whence multiplying by 2dz and integrating dz 2 Kg = C - x2. dt V When the body is at its lowest point, that is, when the motion commences let x = a; Kg .. 0 C. a², V and subtracting from the preceding, dz dt 2 Kg (a³ − x·³), V whence the velocity is known. To determine the time a ! V \ dz t Va Kg V Kg this therefore is the time of the body returning to its ori- ginal position of rest, and it will go on oscillating till reduced to rest, the time of each whole oscillation being π √ V Kg > that is, the motions are isochronous with those of a cycloidal pendulum whose length is V K' 88 THE EQUILIBRIUM OF 76. PROP. To determine the time of the angular oscillations of a body about its centre of gravity. The impressed force acting on the body to turn it about its centre of gravity, is the moment of the weight of the fluid displaced. V Let be the fluid displaced, and p its density; then (Art. 72.) the I impressed force = Vpg · (==λ). Let k be radius of the circle of gyration, then if V' be the volume of body and p' its density, the moment of inertia about G = V'p' k². Dividing the impressed force by this, we have, since Vp = V'p', the effective force = (x) 0. g k2 And the equilibrium being stable, this tends to di- minish 0, therefore, the effective force being d20 - we dt have d 0 g πλθ. dt k FX) 0. Whence, multiplying by 2de and integrating and sup- posing that when t commences = a or that a is the am- plitude of the oscillation, we have 2 కొ I Ꮎ (do) m 干 ​λ) (a² - 0º), dt k² V k? for the square of the velocity, and FLOATING BODIES. 89 t مة k? 1 1 V I F x) ·√ε (1-x) g k² F 시 ​T 2 0 2 ᏧᎾ √ a² - 0 α for the time of the body's returning to its original position. Thus the body will perform isochronous oscillations, the k time of each of which is π I org 干 ​, and the length of the isochronous cycloidal pendulum is Ex. A cylinder floating vertically. k I 7 λ V Let a be its height and b the radius of its base. Then I b', k² = = π 11/12 a 2² + 1 b². (1-s) a, (Art. 62.); Also Vs. Tab², λ = a - sa = па 1 1/2 therefore the time of an oscillation π مع 1 12 1 62 4s a 1 a² + = b² 1 2 4 -0) (1 − s) a} s (a² + 3b³) a - g {3b2 - 6s (1 − s) a²} For a more general investigation, see Moseley's Hy- drostatics, (Art. 88.) M CHAPTER VII. ON THE APPLICATIONS OF THE BAROMETER. 77. FROM the explanation which has been already given of the barometer (Art. 44.) it appears that the atmospheric pressure is in equilibrium with the weight of the mercury in the barometer tube: hence, if m be the density of mercury, g the accelerating force of gravity, h the difference of level in the two branches of the tube, and p₁ the atmospheric pressure, we have mgh = P1. It must be supposed that the barometer is accurately filled, so that there is no sensible pressure above the mercury at the closed end of the tube. Now the open end of the tube may be considered as produced to the limit of the atmosphere; then p, is the weight of the vertical and cylindrical atmospheric column whose base is equal to the unit of area. This weight, as appears from the preceding condition, is equal to the weight of a column of mercury of the same base, and whose height, as ap- pears from observations on the barometer, to be very accu- rately 29.92 inches as its mean value. The pressure then of the atmosphere on each square inch of the earth's surface will be 29.92 × the weight of a cubic inch of mercury = 29.92 × 7.85 ounces 14.7 pounds avoirdupois. As we ascend above the surface of the earth, the height and consequently the weight of the superincumbent column of the atmosphere diminishes; the height therefore, and * Hydrostatics. Art. 68. Ex. 6. THE APPLICATIONS OF THE BAROMETER. 91 consequently the weight of the sustained barometric column must diminish also; there must then exist some relation between the height which has been ascended, and the height of the sustained column, and it is the object of the present chapter to ascertain this relation. 78. The mass of the atmosphere may be compared with the mass of the earth by the preceding article. Let S be the surface of the earth expressed in square inches, then the mass of the atmosphere may be considered as equal to m Sh. The mass of the earth (considered spherical) = pSr, if p be its mean density, and its radius. Then the mass of the atmosphere: mass of the earth :: m Sh 3 p Sr 1 : :: 3 mh pr : 1. But the mean density of the earth is about 5 times that of water, and the density of mercury is about 131 that of water at the same temperature and pressure; m ..=2.5, nearly; also r = 4000 miles, and h = 29.9 inches, P from which data it will be found that the mass of the atmosphere is a little less than one millionth part of the mass of the earth. 79. If the air had the same density throughout the atmospheric column, the height of this column, and the height (h) of the barometric column would be inversely proportional to the densities of the air and the mercury. Let H be the height of the atmospheric column, and its mean density, then P } 1 I m H: h :: .. H -h. P m p 92 THE APPLICATIONS m 13.58* But = 1045; P .001299 ... H = 1045 × 29.9 4.9 miles, nearly. = The atmosphere must evidently extend much higher than this, since the density and weight of its strata diminish as we ascend above the surface of the earth. We shall fix a limit to which it cannot reach by determining the point at which the centrifugal force is equal to gravity; for from that point the centrifugal force would disperse the molecules of air in space. This limit is less elevated at the equator than at any other place. At the equator the centrifugal force = g 289 At a height ≈ above the surface it becomes g (r + z) 289r and the intensity of gravity at that point is gr2 (r + x)² the limit proposed is given therefore by the equation 2+ ≈ 289r whence (1 + (1 + 2) gp2 (~ + z) * 3 289, ΟΥ 3 or ≈ = {√ 289 – 1} r, that is, about five times the radius of the earth. This might be near the truth were the temperature invariable as we ascend, but the repulsive power of the particles is so rapidly diminished by the cold, that a limit is soon fixed to the extent of the atmosphere. The general equation p = kp is true in all ordinary cases, but evidently fails in extreme cases, as when the condensation or rarefaction is extreme. The accurate equation must be of the form p = k (p − d), where d is a *Table of Specific Gravitics. + Figure of the Earth. OF THE BAROMETER. 93 very small quantity, which may generally be omitted, but at the limit of the atmosphere it is appreciable. 80. The force of gravity may also be measured by observations on the barometric column: for if the mano- meter (Art. 45.) be observed at different places on the earth's surface, the temperature and density of the air contained in the vessel C (Fig. 7.) remaining the same, the height of the mercury must vary inversely as the gravity, in order that the weight of the column may remain the same. In order to make these observations with accuracy, the variation in the volume of the air contained in the vessel at C, in connection with the height of the mercury in the closed tube must be taken into the account. Let g be the force of gravity, and h the height of the column at one station, the surfaces of the mercury in the manometer being at P and Q. When the mano- meter is transferred to another place, let g', h' be the values of g and h, the surfaces of the mercury being at P' and Q'. The pressures of the barometric columns in the two cases will be as gh and g'h'; these will be proportional to the density of the air in the manometer, and conse- quently in the inverse ratio of its volume. Let V and V' be its volume at the two stations, then gh gh' V' V Let k be the area of a horizontal section of the tube at P, then the volume of mercury contained between P and P' will equal (h'- h) k. But the mercury being in- compressible, this must be equal to the variation - V of the volume of the elastic fluid, therefore V' = V + (h' − h) k; 94 THE APPLICATIONS whence substituting for V' in the preceding مة امة Vh { V + (h' − h) k} h'' which gives the ratio of the intensities of gravity at the two places. This method is precarious, however carefully the observations are made, and the accuracy of the results cannot be compared with the accuracy of those derived from experiments with a pendulum. 81. It is found by experiment that the air and all other gases when subject to the same and a constant pres- sure dilate equally for equal increments of temperature, and this increment of bulk is found to be equal to th of its volume for each degree of Fahrenheit*. 480 If then the volume of any gas be constant, its elastic force will increase, and if the elastic force be constant, that is, if it be subject to the same pressure, its volume will increase for every increase of temperature. It is therefore of the greatest importance to connect these quan- tities by an equation. PROP. To express the elastic force of any gas as a function of its density and temperature. Let V be the given volume of a gas at the standard temperature, e its elastic force, and D its density. The elastic force e, that is, the pressure on a unit of surface remaining the same, let the temperature be in- creased by 0º, let V' be the volume, and D' the density of the gas, then if a be the increment of bulk for each degree of temperature, V' = V (1 + a0). But the density varies inversely as the volume; D' V 1 D .. D' = D VI 1 + a0 1 + a0 Hydrostatics. Art. 73. OF THE BAROMETER. 95 Now suppose that the pressure is changed, the tempera- ture remaining constant, namely, let p be the value of e, and P the value of D', then by Mariotte's law, P P pe e P P (1 + a0). e D' D' D e Let = k a constant quantity which expresses the D ratio between the elastic force and the density at a given temperature, therefore p = kp (1 + a0). This formula is applicable to all gases, vapours, or their mixtures. 82. PROP. To find the difference of the altitude of two stations by means of the barometer. The general equation between the pressure at any point of a fluid mass, and the impressed forces is (Art. 10.) dp = p (Xd + Ydy+Zd:). Ꮳ In the atmosphere, gravity being the only force X=0, Y = 0, and Z = gr² (1° + ≈)² for a point at a height ≈ above the surface of the earth, g being the gravity at the surface, and r the radius of the earth. Then since this force tends to diminish ≈ it is negative; the equation becomes therefore dp P gr² dz (~° + ≈) ² ° But p = kp (1 + a) (Art. 81.) hence dividing, dp gr² dz Ρ k (1 + a0) (r + *) ° ° 96 THE APPLICATIONS ≈ It is impossible to integrate this expression, since is an unknown function of ; and the exact law of the variation of the temperature being unknown, we shall con- sider as constant; integrating therefore on this hypo- thesis, 1 log p gp2 k (1 + a0) r + ≈ + C. In order to determine the constant, let p, be the value 0, that is, the pressure at the surface of of p when ≈ = the earth, then and subtracting log P, = grz k (1 + a 0) r 1 +C, gr Z 22 p log P k (1 + a0 (r + ≈) α (1). To apply this formula to determine the distance of any point above the surface of the earth. Let be the height of the upper station, and let p' be the value of p at that point. Let, be the number of degrees by which the tem- perature at the surface of the earth exceeds the standard temperature, and the number of degrees for the point at the height 2. Now the change of temperature as we ascend from the surface of the earth is gradual and nearly uniform for small elevations, hence there will be no great error T + T' in assuming the quantity 0 TI 2 Let h,, h' be the observed heights of the barometric column at the lower and upper station, then, since (Art. 77.) p, mgh,, p' = mgh', = p' h P h OF THE BAROMETER. 97 Making then these substitutions in the equation (1) and changing the sign, since log whence 212 Ρ log log P p h' k (1+0 a grø T +T Z Ρ we have .(2), (r + x') the height above the surface of the earth may be found, since all the other quantities are known, and the height of any other station being ascertained in the same manner, the elevation of one above the other is determined. 83. rections. The preceding equation will require several cor- 1º. The temperature will be different at the station whose height is required, and at the surface of the earth, and the mercury will be denser at the colder place than at the other, and consequently the same atmospheric pressure sustains a less column than it would have sus- tained had the temperature remained unchanged. Hence to compare the pressures at the station and at the surface of the earth, the barometric column must be reduced to the same density; and the column at the colder place must be increased by the quantity by which it would expand at the temperature of the warmer. Let ẞ be the coefficient expressing the change in bulk which each unit of volume undergoes for each degree of temperature. Then since (7,- T') is the difference of temperature of the two places (the upper being taken as the colder), each unit of bulk of the barometric column is diminished by ẞ(7,− 7'); and this correction may be considered as due to the height simply, no correction being necessary for the diameter of the column, since glass and mercury N UorM 98 THE APPLICATIONS expand and contract equally at ordinary temperatures. Instead therefore of using the observed height h', we must use h' {1 + ẞ (T, - T')}. ß Hence log h' is to be replaced by log h′ {1 + ß (7, − 7')} = log h' + log {1 + ß (†, − r')} = log h' + Mß (7, − 7'), nearly, where M is the modulus of the system of logarithms. 2º. The force of gravity varies with the latitude, hence g is not constant for all places on the earth's sur- face; and the general expression for gravity in terms of the latitude is g = E (1 + n sin² \)*, where E is the equatorial gravity, and n a known quan- tity. Then if G be the force of gravity at latitude 45º, G = E (1 + n), 1 + n sin² λ · g = G G {1 - 2 (1 − 2 sin² \)} nearly 1 + n 2 G (1 - 22 cos 2λ). 2 :: 3º. The coefficient a will require some correction, and also the constant k. In determining the values of these quantities, the air was either supposed to be dry, that is, not to contain any aqueous vapour, or that the quantity of that vapour is constant. But as the temperature increases, the quan- tity of vapour increases also in the atmosphere, and the elastic force of the vapour being added to the elastic force of the air, the increment of volume for a given volume of air must be greater for air which contains vapour than for dry air, hence a must be increased by a small quantity. Maou *Figure of the Earth. OF THE BAROMETER. 99 For the same reason k will require a small correction, since it expresses the ratio of the elastic force to the density at a given temperature of air that is dry, or contains a constant quantity of vapour. 84. For practical purposes an approximate value of the general equation (2), (Art. 82.), may be found. Multi- plying up it becomes k h x² - 1 (1 + a + + + ) log / 2 + 2 But g rx' r + z (1. jo -1 h' x' very nearly, in all cases to which the barometer can gene- rally be applied; k =*= (1 + a + ) log h' g k h M (1 + a log h'' g Z where M is the modulus of the common system of log- arithms. k Now M may be taken equal to 20117 yards, and g α 1 equal to 2 900 Whence -12 as mean values including the corrections. T I' h { 1 + 7, +++ log" /> = 20117 1 + 900 which will be found a convenient formula for determining in yards the elevation of one station above another, the temperatures being the number of degrees above 32º F. Uorm CHAPTER VIII. ON CAPILLARY ATTRACTION. 85. THE centres of the attractive or repulsive forces which act on a fluid mass, may be all the other points of the fluid. In this case, the components of X, Y, Z, of the accelerating force acting on the point P, will consist of an infinite number of terms; these may be certain functions of x, y, z, common to all points of the fluid, if we suppose that the principle of the equality of action and reaction obtain in their mutual attractions and re- pulsions, and that all the points are besides submitted to the same extraneous forces. In nature, these forces are of two kinds, the one varying according to the inverse square of the distance, and the intensities of the other are expressed by functions. which decrease with extreme rapidity and are not sensible except at insensible distances. The components of the former of these may be calcu- lated by dividing the fluid into small elementary masses, and obtaining by the integral calculus the sum of the attractive or repulsive forces in each direction. The other class of forces, which are molecular forces, and which are either attractive or repulsive according as the attraction of the ponderable matter is greater or less than the repulsive power which is due to heat, cannot be taken any account of in the calculation of the forces X, Y, Z, for any point in the interior of the fluid mass. For these molecular forces are those which produce the pressure p equal on all sides of the point, and which we have already considered in forming the equations of equilibrium. Mou CAPILLARY ATTRACTION. 101 It follows from this latter consideration, that the equa- tions (1), (Art. 10.) which we have obtained, are the necessary and sufficient conditions of equilibrium of all the forces, and that the molecular forces which act on any element of a fluid mass are comprised in them; so that the equilibrium most certainly subsists when there is a value of p which satisfies these equations for all the points of the fluid, which coincides with the value given directly of the pressure at a free surface, and which does not become negative at any point so long as the particles of the fluid remain contiguous. If the law of these molecular forces were given as a function of the distance, and we could deduce from these forces the expression for p as a function of the mean interval between the molecules, it might be substituted in the equations (1). One of them would determine the mag- nitude of this interval which exists in a state of equili- brium about the point P, and the other two would express the conditions of that equilibrium. The numerical value of p would be afterwards found from that of the mean interval or from the corresponding value of the density, and the method in which this pressure p may vary very much, for the very small variations of the density which we observe in fluids is explained by Poisson.* But the direct determination of the pressure p being im- possible, we are obliged to deduce its value from the con- ditions of equilibrium themselves, or from the equation (2) which results from these. When the point P is situated at the surface of the fluid or is distant from it by a less quantity than the radius of the action of the molecular forces, we must take account of these forces, and also of the rapid variation of the density at the surface, in the calculation of their com- ponents X, Y, Z, and consequently of the value of the pressure p deduced from (2). Thence there arises an * Journal de l'Ecole Polytechnique, 20 cahier. 102 CAPILLARY ATTRACTION. influence of molecular forces on the figure of a fluid in equilibrium, which is not in general sensible, and which cannot be so except in capillary spaces. 86. If a fluid be regarded as composed of atoms held in places of equilibrium by attractive and repulsive forces proceeding from the atoms, it will necessarily follow that every change of pressure is accompanied by a change of density, and that at their surfaces there will be a rapid change of density within a small, and, as experience shews, insensible extent, depending on the sphere of sensible ac- tivity of the molecular forces. In strictness, this super- ficial variation of density should, as we have just said, be taken into account in treating of capillary action, as Pois- son has done in his New Theory of Capillary Action; but as neither theory nor experiment has hitherto deter- mined to what degree it affects capillary phænomena, and, considering the great repulsive and feeble attractive molecular action of fluids, the effect is probably of small magnitude, we shall therefore neglect it in the following propositions, and suppose the fluid to be perfectly incom- pressible and to be acted upon, in addition to gravity, only by the molecular attraction of its own particles and of those of the solid with which it is in contact. The law of the attraction is unknown, but as experience teaches, must be considered sensible only at insensible distances from the attracting centres. With this limita- tion Problems in Capillary Attraction are to be treated as any other questions in Hydrostatics, with the modifica- tions that the peculiar nature of the forces introduces. 87. Let AB and CD (Fig. 17.) be the sides of a solid between which fluid is drawn up, and abc the capil- lary surface of the fluid. Let P and Q be points in the capillary surface and in the horizontal surface of the external fluid, both points being beyond the sphere of the molecular action of the particles of the solid. Let an indefinitely small canal be drawn from P to Q, its extre- CAPILLARY ATTRACTION. 103 mities being perpendicular to the surfaces at P and Q, and having every point beyond the sphere of the molecular attractions of the particles of the solid. PROP. To find the condition of equilibrium of this canal. The forces that sustain it are the molecular attractions of the surrounding fluid and gravity; the molecular attrac- tions of the solid being by the hypothesis laid out of the case, every point of the canal being beyond the sphere of these actions. If we find the resolved parts of these forces in the direction of the axis of the canal and equate their sum to zero since the canal is in equilibrium, we shall have an equation for determining the form of the surface abc. Take any element as R at a sensible distance from the extremities of the canal; the molecular attractions on this element must be equal in opposite directions, and therefore destroy each other. This will not be the case at the ex- tremity Q, where the attractions of the surrounding fluid on an element Qq downwards will not be counteracted by an upward attraction; and consequently, the canal will be urged in a direction from Q towards R. If a tangent plane be drawn to the capillary surface at P, the fluid below this plane will urge an element at that extremity of the canal in the same manner and to the same degree as the fluid below Q urges an element at that ex- tremity. These forces acting in opposite directions along the canal will destroy each other. The only remaining molecular attraction, is that of the fluid contained between the capillary surface a Pc and the tangent plane TP. Let the moving force on the canal due to this attraction for the present be considered equal to Ppk², where p is the density of the fluid, of the canal, and P is to be determined. this force is the action of gravity, tending to depress the part PS which rises above the external horizontal surface. the section Opposed to 104 CAPILLARY ATTRACTION. The action of gravity on the part QRS of the canal has plainly no tendency either to elevate or to depress the fluid. Now if ≈ be the vertical height of the point P above the horizontal surface, the action of gravity on PS resolved in the direction of its length is to produce a weight equal to that of a column of heights and base k². Hence Ppk² = gpzk², or P = g%. 88. PROP. To find an expression for P in terms of the principal radii of curvature of the surface. Let the tangent plane at any point O (Fig. 18.) in the surface be taken for the plane of xy, then Oz a normal to the surface will be the axis of x. Let a plane passing through the normal, and making an angle with the plane of xx, intersect the capillary surface and the tangent plane at 0 in OP and OT. Then Ox, xOy are the planes of maximum and minimum curvature. Let otr, pt = ≈, and R be the radius of curvature of Op P at 0, then R = = Ot 2pt' 2.2 whence ≈ = 2 R Let R1, R2, be the greatest and least radii of curva- ture; then, since 1 cos20 sin² 0 + R R₁ R₁₂ we have ર || 22 102 2.2 (cos² 0 sin² 0 + R1 R₁ R₁2 2 [cos³0 Let another plane be conceived drawn through the normal Ox, making an angle +de with the plane of xx; then we shall have a small pyramid or wedge whose four edges are the sections of the normal planes with X Z CAPILLARY ATTRACTION. 105 the capillary surface and the tangent plane. Let an elementary column of this pyramid be taken as at pt. The column thus taken is such as pbcd (Fig. 19.), where apbc is the portion of the capillary surface, and tdef of the tangent plane; whence it will be seen at once that the content of this column pt = zrdė dr. As the attraction is sensible only for very small dis- tances from the canal whose axis will coincide with Ox, the height of the column may be considered to be always very small, and its attraction to be the same as if collected at its middle point o. Let OG = %1, GO = r₁, and pkdz, be the element of the canal at G; the attraction then of the column at pt upon it is, if (r₁) be the law of the molecular attraction, zrd@drp(r₁) × k² pdz₁. Resolving this force in the direction GO, the part re- quired is k²pp(r₁) xr dr d☺ d≈₁ × ~ which, substituting the preceding value of x, becomes 1 cos² 0 sin20 + k²pp (r}) . _—_ . r³ dr. z¸dx, dz₁ + ᏧᎾ ; R₁ R₂ cos² 0 sin20 + 1° R₁ R2 2) d Ꮎ . 0 · ·. P = fff } $(~) — r³ dr≈, d z Integrating with respect to 0, from 0 to ◊ = 2π, 1 P = ↓ (2 + 2) [ƒr, þ (r,) r'dr.x,dx. R₁ 1 O 106 CAPILLARY ATTRACTION. 2 Now r₁² = x² + r², and as x, is to vary independently of r, r₁dr₁ = x₁d%₁ ; .'. 1 [[p(r₁) — ž₁dz₁r³ dr = [[p(r₁) dr₁r³dr. r1 Here the integration is to be performed with respect. tor considering r₁ as constant. Hence the limits of this integration must be from r=0 when GO = r₁_to_r=ri when GO = 0. 1 The integral therefore is equal to 4ƒp(r₁)r₁*dr, from r₁ = 0 to r₁ = ∞ty, as its value is not generally increased by increasing r₁, on account of the form of the function p(r). The last integration cannot be performed, since the form of the function is unknown. Let us assume, however, 4°$(r₁)r¸ªdr₁ = H. 1/ 1 Then P= P = (1 + 1) H; R₁ R 1 R2 + H = g≈......(A). R2 The equation thus obtained is the differential equation of the capillary surface, by integrating which and deter- mining the values of the arbitrary constants, the form of the surface will become known. We shall proceed in the following articles to apply the preceding theory to some of the known instances of capillary attraction. 89. Ex. 1. A cylindrical tube of small diameter. Let ACDB (Fig. 20.) be the section of the cylindrical tube, and abc the section of the capillary surface by a vertical plane through the axis of the tube. Since every point is symmetrical with respect to this axis, the capillary surface will be one of revolution. CAPILLARY ATTRACTION. 107 Let x, y be the vertical and horizontal co-ordinates MN, NP of a point P in the section abc. Thus the radii of greatest and least curvature at the point P are the normal and radius of curvature at that point; hence dy² 1 + dy² dx² R₁ = y √ 1 + R2 dx², dey dx² or, writing p and q for the first and second differential coefficients, and substituting in (A), we have 1 1 q H = gx, 2 \y (1 + p³)² (1 + p²)³) H Ρ УРЯ or 2g [(1 + p²)} (1 + p²) }) +p²)* xyp. Let Mb = h, bNx', then dy d'y x = h + x', p p = , 9 > dx' dx'2 and the preceding becomes H Р y p q 28 [(1 + p²)² - (14) } } = hyp + x'yp. 2g But the first side of the equation is the same as H d ย 2g dx √1+p² integrating, therefore, and adding an arbitrary constant, H Y g √ 1 + p² = hy² + 2 fx'yp + C. When the diameter of the tube is very small, a' will be small compared with h; hence, neglecting the term 108 CAPILLARY ATTRACTION. involving x', and supposing the integral to begin when y = 0, so that the constant becomes nothing, we have Y H 1 gh√1+ p² H Let gh = a; then 1 + p² y 2 a , y² p √ a² - y² and, integrating, x' a² and p² 1; 2 Y ydy or dx' √ a² - y² a² — y² + C. When y = 0, x = 0; 2 = = √ u² + C ; .. 0 = and, subtracting from the preceding, x' = a - √ a² - y²; that is, y² = 2ax' — x²², 12 the equation of a circle whose radius is a. surface is therefore very nearly spherical. , The capillary H Since a α = the radius of the capillary surface varies gh inversely as h. Ex. 2. Two parallel plates. Let AB, CD as before (Fig. 20.) be the sections of the plates, and abc of the capillary surface by a vertical plane perpendicular to the plates. Then BD is the dis- tance of the plates from each other. The capillary surface will here be cylindrical, and one of the radii of curvature will be infinite at every point. CAPILLARY ATTRACTION. 109 ty Hence if R₁ = ∞, and R₂ = the equation (4) becomes (1 + p²) 3 - q H q = ∞ = · h + x', 2g (1+ p²)! if Mb=h_and_bNx', as in the preceding example. Separating this into two terms by adding and sub- tracting p² in the numerator, 2 H q 2g ((1 + p²)³ p² q (1 + p² 2 Whence, integrating, H P = = h + x'. = h∞ + — x²² + C. 12 2g √1+ p² Now p = ∞ty when x = 0, whence, by subtraction, H C; 2g H p 12 = h x' + ½ x²² = (h + ½ x′) x'. 2g (1 + p²) ≤ Since is small compared with h, we may omit a compared with h, and assuming Whence 1 + I 12 1 P P H B, we have 2gh Ꮖ (1 + p²)³ ¯ ß 1+p² P B (B — x')³ ' x' B-x B or, 1 p В 2 2Bx-x¹2 (B - x')²' 110 CAPILLARY ATTRACTION. dy .. p or dx' and integrating, B-x' √2 ß æ 2Bx' - x2 12 y = √ 2ẞx' — x²², the equation to the section, no correction being requisite, since y and x' begin together. The section of the capillary surface is therefore a semi- circle whose radius is B. The whole surface is therefore cylindrical. We assumed H 2gh B, and in the preceding example we assumed α. It follows, therefore, that if h be H gh the same both for the parallel plates and for the cylindrical tube, 2ẞ= a. 90. In the preceding investigation every point of the canal was supposed to be beyond the sphere of the molecular attraction of the particles of the solid. Let AB, CD (Fig. 21.) represent the bounding sur- faces of the solid, and let A'B', C'D', drawn parallel to these through the points b, d, of the capillary surface, be the limits of the sensible molecular attraction of the sides of the solid. Draw tangents at these points, meeting the axis MN in m and n. The angle which the fluid in the capillary surface makes with the surface of the solid is called the actual angle of contact, or, the angle of actual contact. The angle which the fluid in the capillary surface makes with the line drawn at the limit of the molecular attraction of the particles of the solid at the point where its surface meets this line, is the angle of contact, or, the theoretical angle of contact. CAPILLARY ATTRACTION. 111 In the figure am N is the angle of actual contact, and bn N the angle of contact. 91. PROP. To determine the law of ascent of a fluid in different capillary tubes, or between parallel plates separated by different intervals. The equation (4) having been obtained on the hy- pothesis that every part of the canal was beyond the sphere of the molecular attraction of the particles of the solid, that is, that the canal was wholly without the por- tions of the fluid between the side AB and the line through b parallel to it, and between the side CD and the line through e parallel to it, cannot be applied to the fluid contained between these portions. The above equation applies therefore to the fluid bounded by a surface A'B'C'D', similar to the surface ABCD but not to the fluid which is included between these two surfaces. The angles in which the fluid meets these two sur- faces as well as the forms of the portions ab, cd, of the surface will depend on the law of the molecular attrac- tions, and their relative intensities for the solid and the fluid: they cannot therefore be determined since these elements are at present unknown: We may however assert, that, considering the small distances to which the mole- cular attractions are sensible, the portion ab, of the curve, and the angles which the tangents at a and b make with the vertical are no ways dependant on the diameter of the capillary tube, they will be the same for instance, in a tube one-twentieth of an inch in diameter, as when the fluid ascends against a plane surface. 92. Let us apply these considerations to the pre- ceding examples. Let be the centre of the circular arc bd, and w the angle of contact, that is, the angle which the tan- 112 CAPILLARY ATTRACTION. gent bn makes with the vertical through b. Let Ob = Od=r, and the chord bd = 2b. Then w = bn 0, b Nb, and since Obn is a right-angled triangle and bN is per- pendicular on its base, the angle ObN = the angle bnN; ... b = r cos w, or r= b sec w. Now since A'B' is exceedingly near to AB, bN in the case of a capillary tube differs by a very small quantity from the radius of the tube. But, it was shewn (Ex. 1.) that r = H gh H ; gh H cos w 1 = b sec w, or h = g b Consequently, as w is the same for tubes of different diameters; 1 пос that is, the height of ascent of the fluid in the capillary tube is inversely as the radius. And experiments con- firm this result. COR. As h may be taken for the mean height of ascent, the weight of fluid raised is bhp very nearly. This quantity, by substituting for h the above value, is πb Hp equal to COS W. Hence for a given tube, the g weight of fluid raised varies as cos w. Although the angle w is not affected by the magnitude of the radius. of the tube, it greatly depends on the matter of which it is composed, and the state of the internal surface as to polish or greasiness. The way in which the solid tube affects the height of ascent, is by determing the magni- tude of the angle w. The immediate action of the tube is on the aqeous cylindrical shell, included between the surfaces ABDC, A'B'D'C', and by the intervention of CAPILLARY ATTRACTION. 113 this, it supports the rest of the fluid. The vertical action of the tube on the aqueous cylindrical shell is very nearly the same as the vertical action of the shell on the rest of the fluid, since the weight of the shell by reason of its thin- ness is exceedingly small. If the latter action be calcu- lated and equated to the weight of fluid raised, this weight will be found to be proportional to cos w, in con- firmation of the result obtained above. This calculation however, which is given in Art. 18. of Poisson's Treatise, is too long to be inserted here. 93. When the fluid rises between two plates, 2b is very nearly the interval between the two plates, and H (Ex. 2.), r = Hence 2gh ... h H 2gh H cos @ 1 ∞ b sec w, 1 g 26 2 b' or the height of the fluid varies inversely as the interval between the plates, and is the same as in a tube where the radius is equal to the interval between the plates. If b be given, the height of ascent is greatest when @ = 0. W When is 90, the fluid is depressed below the level of the external fluid as is the case with mercury. It follows immediately from the law of ascent be- tween parallel plates above determined, that if two plates inclined at a very small angle be dipped in a fluid, with the line of their junction vertical, the fluid will ascend between them in the form of a rectangular hyperbola, the asymptotes of which are the line of junction and the in- tersection of either plate with the horizontal surface of the fluid. For any two opposite elements of the surfaces of the plates may be considered as parallel, and the rise between these elements will consequently be inversely pro- portional to their distances from each other, and therefore P 114 } CAPILLARY ATTRACTION. inversely proportional to their common distance from the vertical asymptote, which indicates that the boundary of the fluid surface will be a rectangular hyperbola. 94. PROP. To determine the angle of actual con- tact, with the capillary surface. The condition of equilibrium requires, that the re- sultant of the forces which act at any point of the sur- face should be perpendicular to the surface, and this will enable us to determine something about the angle of actual contact. We proceed to determine the direction of the resultant of the forces which act on a particle situated at a or c. Draw a tangent am, and let am N = 0. Conceive a plane perpendicular to the plane of the paper to pass through a, making an angle with AB, the dotted line representing its section. Let another plane be drawn through the same point, making an angle 0 + do with AB. Then do being indefinitely small, the attraction of the fluid between the planes on the particle at a will vary as do. Let it be equal to qd0. The parts of this, in the vertical and horizontal direction respectively, are qde cose, and qdo sin 0, hence, the total vertical attraction = q cos de = q sin, the total horizontal attraction = ♫ q sin 0 d0 = q(1 − cos 4). The total action of the solid, which will be wholly in the horizontal direction, will be found by putting q for q, and 180° for in the last expression, therefore the total action of the solid =2q'. The resulting attraction of the fluid between the sur- face ab, and the tangent plane am, cannot be calculated CAPILLARY ATTRACTION. 115 as the form of the surface is unknown. It will in general be small, and its direction will very nearly coincide with am; let its value be µ, then g being the force of gravity, we have, the total force in the vertical direction = g + q sin Ó + µ cos Ó, in the horizontal direction = 2q-q(1 cos) u sin o. − – The resultant of these is perpendicular to the surface at a, that is, to am, their ratio must equal tan ; or calling then X and Y respectively, and R their resultant, we have X = R sin o, Y= R cos p, whence g+ q sin 2q − q(1 + µ cos gcos + q sin - cos ) – µ sin cos 0 + μ cos² = ta tan ; φ; 2 q'sin - q sin 4 (1 − cos p) – µ sin²ė, or (2q′ − q) sin4 = g cos & + µ. Now, with respect to all fluids which are capable of hanging in drops of sensible thickness from the horizontal surface of the solid, 2q is greater than 2q, and both these quantities are exceedingly greater than gravity. Also, if were an angle of considerable magnitude, u must be exceedingly smaller than the terms on the left side of the equation. Hence this equation cannot in general be satisfied, except for a very small value of p. For mercury, which is not capable of suspension from a solid, that angle is not small. The smallness of this angle is a necessary condition, that a fluid may be capable of wetting a solid. In experiments with capillary tubes, it is usual to moisten the interior of the tube as much as possible be- fore the ascent of the fluid in them. In this case, the ascent is occasioned by the molecular attraction of the particles of the coating of fluid which lines the cylinder. 116 CAPILLARY ATTRACTION. To apply the preceding equation to these cases, we must put q'=q, and will still be a very small angle in consequence of the largeness of q in comparsion of g. The angle, will in this case be the same as that called w, and thus, as w will be very small, the capillary sur- face will be very nearly a hemisphere, and the height of ascent the greatest possible. = 0. μ = 0. If 2q'=q, = 90° and Hence also w = 90º, and h 0, or there is no ascent of the fluid. 95. PROP. A drop of water placed in a conical tube of very small vertical angle will run towards the vertex. Let abde (Fig. 22.) be a drop of water in a coni- cal surface, and aeb, cfd the bounding surfaces. Let ab 2b, and cd = 26', the capillary attraction at c will sustain a column, whose height equals C and at ƒ, will sustain a column whose height H cos w 1 org b H cos w 1 g b > and they act in opposite directions, that at ƒ acting to- wards c. Hence the resulting action towards C H cos w g مه (1 - }) H cos w b— b' b' b مع This is the force which causes the drop to run, which, when the drop is small, and b-b′ nearly con- 1 stant, varies as b² CHAPTER IX. ON THE SPECIFIC HEAT OF GASES, AND ON THE LAWS OF COOLING. 96. THE law of Mariotte, that the elastic force is proportional to the density, is true only on the supposi- tion that a fluid has had time after condensation or ra- refaction to return to its original temperature. If this be not the case, the temperature increases or diminishes with the density, and the elastic force increasing or de- creasing by reason of the increase or decrease both of the density and temperature, ought for the same fluid to vary in a greater ratio than the density simply. When the fluid is contained in a vessel whose sides are imper- meable to heat, it preserves all its caloric during conden- sation and rarefaction, and consequently, the temperature increases or diminishes. The same takes place when the variations in the density are so sudden that no transfer of heat can take place, that is, in the case of condensa- tion the heat has not had time to escape by radiation, or to communicate itself by contact to the neighbouring substances; and in the case of dilatation, the surrounding bodies have not had time to communicate to the fluid, either by radiation or contact, any sensible quantity of caloric. This is the supposition made, as will be hereafter seen in the case of the variations of density which take place in the waves of air which produce sound, the du- ration of these variations being some thousandths part of a second. In this and many other questions it is important to know the expression for the elastic force of a gas in terms of the density, and the corresponding elevation or 118 THE SPECIFIC HEAT OF GASES, depression of the temperature, the actual quantity of heat or caloric which the fluid mass contains remaining con- stant. In the present state of our knowledge however, we have not the requisite data for the complete solution of the problem, and the following chapter will contain what is principally at present known from calculation and experiment on this important subject. All gases expand equally for equal increments of temperature, and we have a relation subsisting between the elastic force, the density and the temperature, which is given by the general equation, p = kp(1 + a0)…………….(1), where α is the same for all gases, and k is different for different gases. The absolute quantity of heat which a given weight, as a pound of any substance, contains cannot be calcu- lated, but it is supposed to be inexhaustible, since ex- periment shews that all substances, however apparently devoid of heat, may be made to give some out; it is also supposed extremely great as compared with the quantities by which it is increased or diminished, when the body changes its density or temperature; it is these variations, that is, the quantities added and subtracted, which have to be compared together and submitted to calculation. This variation is evidently a function of the elastic force, the density, and the temperature, or of any two of them, by virtue of the equation (1), which sub- sists between these three quantities. 97. PROP. To express the variation in the quan- tity of heat. Let q be the excess of heat which a given quantity of any gas, whose elastic force is p, density p, and tempera- ture, contains above the quantity of heat which the same portion of gas contains at the standard pressure AND THE LAWS OF COOLING. 119 and temperature. Then q is a function of p, p, 0, or by virtue of the equation p = kp (1+a0), we have q=f(pp), where the form of the function must be determined. The specific heat of the fluid is the quantity of heat which must be added to raise its temperature one degree, or, it is the rate of increase of q with respect to 0, and dq will therefore be expressed by d Ꮎ Now two cases present themselves; first, we may consider the pressure constant, and that the gas has the liberty of expanding; and secondly, we may consider the volume constant, and that the pressure varies with the temperature. P In the first case p being constant, and the de- pendent, and the independent variable, we have from (1), (Art. 96.); dp 0 = k (1 + α0) + kap; de dp αρ d Ꮎ 1 + a0 In the second case p being constant, and p the de- pendent, and the independent variable, we have from the same equation, dp d Ꮎ - kap = αρ 1 + a0 Let e be the specific heat of the gas when the pres- sure is constant, and c, its specific heat when the density is constant, hence since is the general expression for d q do the specific heat, d q dq dp d Ꮎ dp de dᎾ d q dq d p and = do c= Ꮎ dp do 120 THE SPECIFIC HEAT OF GASES, Substituting from the preceding equations, dq C = αρ dp 1+ a0' d q αρ and c₁ = (2), α dp 1+a0 α whence, dividing so as to eliminate 1 + a0' dq c d q 0. P p d p + p cdp Let y express the ratio of the specific heat of the gas at a constant pressure to its specific heat at a con- stant volume, or y = C C d q 0, dq +YP dp p de The value of y can only be known by experiment, but it is evident that its value must be greater than unity, for it must require a greater quantity of heat to aug- ment the temperature of the gas and dilate it at the same time, than only to augment its temperature, with- out removing the particles from each other. We shall see hereafter the method of determining it. 98. PROP. To determine the increment of tempera- ture for a small condensation. Let be the temperature of the gas, and 0 + w be its temperature when the density of the fluid has been in- creased by a very sudden condensation in the ratio of 1 + s : 1, where s is a very small fraction. If the loss of heat during the compression is insensible, the increase of the temperature, corresponding to the increases of the density, is the quantity which has to be determined in the following experiment. AND THE LAWS OF COOLING. 121 For this purpose, suppose the atmospheric air to be the gas in question, and let it be contained in a closed vessel, the pressure, density, and temperature being the same both at the exterior and interior, which, for the external air, we shall suppose represented by p, p, O; during the whole experiment. Let a small portion of the atmospheric air be removed, and when the air has acquired its original temperature, let p', p' be its elastic force and density. Let a communication be again opened with the ex- ternal air; the elastic force, the density, and the tem- perature will increase together, so that in a very short. time the internal and external pressures will be equal to the external pressure. At this instant let the com- munication be cut off, and let p" be the density, and + the temperature of the internal air. Very soon, the increment w of the temperature will vanish, and with- out the density p" having undergone any change the pressure will be diminished to p" suppose. W The density of the internal air having passed very rapidly from p' to p", if we take s = and neglect p" - p′ > the small quantity of heat which is absorbed by the vessel during the small time of this passage, the increase w of the temperature is that which corresponds to the con- densation s, and is the quantity which is required. The change in the thermometer is too slow to indicate this increment of temperature, which exists only for a very short period, but its value may be inferred from the three pressures p, p′, p", as indicated by the heights of a baro- metric column at the time of the experiment. Now it is to be observed in the preceding experiment that there are two epochs, so to speak, at which the same temperature & corresponds to the densities p' and p", and to the pressures p' and p″. Q 122 THE SPECIFIC HEAT OF GASES, Hence, the temperature being constant, we have by Mariotte's law p" p" p" - p' p' whence the condensation s is known. Again, there are two epochs at which the same density p" corresponds to the temperatures ✪ + w, and 0, the pres- sures being p and p″”. Hence, since p = kp" {1 + a (0+w)}, p″ = k p″ (1 + ά0), 1+ a0 α Ρ 1 + a (0 + w) p" whence the value of s may be determined. (4), corresponding to the condensation Experimental Determination*. In an experiment made by Desormes and Clements, when the change of the density from p' to p" took place in less than half a second, they observed, p = 0.7665, p = 0.7527, p"-0.7629, whence s = 0.0133. The temperature was 12°5 C., and since a always = 0.00375, we deduce from equation (4), w = 1°·3173 C. Hence for a condensation 00133 without loss of heat, the temperature of the air is augmented by 1°.3173 C., or the temperature of the air would be raised 1º C. for * In these experiments I have retained the measures of the original experimenters. AND THE LAWS OF COOLING. 123 a condensation s = 0.01331 1.3173 = 0·0101. The increase of temperature may, as we shall see here- after, be deduced from the velocity of sound. 99. PROP. To determine the ratio of the specific heat of a gas at a constant pressure to its specific heat at a constant volume. Let us suppose, as in the preceding article, that the elastic force and temperature of any gas are p and 0; the condensation s may be equivalent to that which the fluid experiences when the temperature is slightly di- minished, the pressure remaining unaltered. Let e be this slight variation in temperature and p the value of p. Dividing the one by the other, we have, since p=p', αε P 1 + a (0 − e) 1 + a0 whence P = 1 + 1 + а0 αε 1 + αθ very nearly, and therefore, p' - p αε S. P 1 + а0 Let q' be the municated to the quantity of heat which must be com- given quantity of gas to raise its temperature from 0 to 0 without changing the pressure p, then if c be the specific heat at a constant pressure, -e € q' = cε. After this communication of heat let the fluid be sud- denly compressed so as to resume its former volume, it will then undergo a condensation s, and if there be no loss of heat, its temperature being augmented by w will become + w. Under these circumstances the pressure 124 THE SPECIFIC HEAT OF GASES, will be greater than p, but if without changing its volume the temperature be allowed to sink as far as ◊ – e, this pressure will diminish at the same time and become p. During this fall of temperature the gas will lose a quantity of heat proportional to the small diminution e+w of temperature, which may be expressed by c, (e + w), since c, is its specific heat at a constant volume. The volume, the temperature, and the pressure being all the same after this loss of heat as they were before the quan- tity q' of heat was communicated to the fluid, the loss c, (e+w) must be equal to q', hence cε = c, (ε + w) ; +w); C W 1 + C 6 C ω αε But Y and s = € 1 + a0 απ · · 7 = 1 + (5). (1 + a0) s p" p' p-p" Now s = , p' W - "-", (Art.98.) and from (4), 6-7: · · y = 1 + y=1+ (p - p") p' (p' – p') p' 1 + a0 p" which is an expression for y in terms of quantities capable of being observed. If we take the data furnished by the experiment detailed in the last example, we shall find Ρ 1 y=1+ 20″ " p" p' 1 +0.3482 = 1.3482, 1 for the value of the ratio of the specific heat of air at a constant pressure to its specific heat at a constant volume, AND THE LAWS OF COOLING. 125 By an analogous proceeding, Gay-Lussac and Walter have obtained y = 1.3748, and Dulong has obtained by a different proceeding y = 1·421 for air perfectly dry. These results differ but very slightly, and their small differences do not prevent us from considering y as con- stant. 100. Considering then y as constant, the integral of the partial differential equation d q dq P +YP dp dp is, if f be the form of the arbitrary function, 1 q=f ƒ (= p³). Hence, pl = pf¹(q), or p = p²p (q)………………….. (1), where is an inverse function of ƒ. But since p = kp (1 + a0), we have Ꮎ - = Р 1 1 1 ερ α α a p² - ¹p (q) = = = .(2). α If now q remain the same, and p, p, 0 become p', p', 0', respectively, 1 p'=p'¹p(q)…..(3), and ' 1 17-1 k $ (9) - —- ... (4). Eliminating (q) between (1) and (3), p' -p (2)", and eliminating it also between (2) and (4), a Ø' γ-1 1 + 0 Ꮎ α α These two equations express the laws of the elastic force and temperature of a gas compressed or dilated 126 THE SPECIFIC HEAT OF GASES, without any variation in the quantity of the heat; but it must be observed, that they depend on the fact of y being constant, which, from what has been said, may be considered as established for common air. The form of the arbitrary function may be determined by supposing that under a constant pressure a gas dilates equally for equal increments of temperature, as is shewn by Poisson*, to whom the reader must refer for further applications of this theory. On Cooling. 101. When a body cools suspended in air, the heat is transferred by conduction, that is, by transmission through particles in immediate contact, by convection, that is, by the motion of the warm particles which are replaced by colder ones, and by radiation. But when a body cools in vacuo it is by the latter method, namely, by radiation that the cooling takes place; and it is the laws of cooling as depending on this radiation that we are now about to consider. The principle arrived at by observation and which may be made the basis of the mathematical theory, is, that the temperature of a body is in the excess of the sensible heat which it gives out above that of the surrounding bodies, and the cooling of a body is the excess of its radiation above the radiation of the surrounding bodies †. PROP. To obtain the law of cooling in vacuo. If then be the excess of the temperature of the body cooling in vacuo, above the surrounding substances whose temperatures are 0, +0 will be the temperature of the body, and the velocity of cooling may be expressed by F († + 0) − F (0), where the form of the function F is to *Traité de Mécanique, Art. 639. ↑ Principles of Hydrostatics, Art. 147. AND THE LAWS OF COOLING. 127 be determined. If then v be this velocity of cooling, which will be nothing when is nothing, v = F (t + 0) — F (0). Now, Newton and all succeeding philosophers have from their observations been led to assign some geometric progression as expressing the velocity of cooling. Let us suppose, therefore, that the velocity of cooling may be expressed by (7) aº, where a is some constant and has to be determined. Then $ (7) a® = F (t + 0) − F (0) ; and expanding by Taylor's series, we have $(T) = F' (0) T ao 1 + F" (0) 7² ао + α 1.2 Now this equation must subsist for all values of 7, and since when is nothing, the temperature would be that of the surrounding bodies or constant, we must have in this case or when is small, T φ (τ) = n, some constant. But T F" (0) T + +... ao 1.2 $(T) F'(0) T T the right side of which when 7 is small is reduced to the first term. hence, if F' (0) n or F'0= naº. a Integrating and adding an arbitrary constant, n F (0) = log a aº + C = maº + C, n = m. Hence, log a F (τ + 0) = ma™+0 + C. 128 THE SPECIFIC HEAT OF GASES, Substituting then for these quantities, we have บ mat+o MAT+O __ maº maº (a¹ — 1). If then the theory of exchanges on which the preceding reasoning is founded be true, we arrive at the following law: “that when a body cools in vacuo in a vessel whose temperature is constant, the velocity of cooling for ex- cesses of temperature in arithmetic progression increases as the terms of a geometric progression diminished by a con- stant quantity." The experimental verification of this law is most remarkably exact, and the Memoir* of Dulong and Petit, from which the preceding is taken, is a most beautiful example of the plan that must be pursued in these and similar researches. The remarkable accuracy of the re- sults obtained from the preceding formula for all tempera- tures, removes all doubt respecting the truth both of the law and of the principles on which it is founded. 102. The total radiation of the surrounding medium is F (0), and its value is maº + C. But the point for the commencement of the absolute temperatures being arbi- trary, it may be chosen so that the constant will vanish; hence, the absolute radiation may be expressed by F (0) If then it were ma, simply without any constant. possible to observe the cooling of a body in vacuo, so that there was no interchange of radiation, that is, no portion of heat being restored from the surrounding bodies, we should have for the velocity of cooling, v mat+o =ma ат = Мат, if ma M, or the velocities of cooling would increase in geometric, the temperatures increasing, in arithmetic pro- gression. * Annales de Chimie, vır. 1817. See Encyc. Metrop. Art. Heat. AND THE LAWS OF COOLING. 129 The real velocity of cooling in vacuo in any case may then be expressed by v = M (a™ - 1), where M is the quan- tity to be taken from the terms of the geometric progression, and depends on the temperature of the surrounding bodies. When the temperatures are low V = Ma nearly, which is the Newtonian law. 103. PROP. To find the time of cooling in vacuo.. The time of cooling may be readily deduced from the velocity with respect to the time, for generally we have dr dr V dt dt in this case, since the excess of tempera- ture diminishes with the time; dr M (a™ − 1), and dt dt dr M (a™ − 1) dr .. t JM (27-1) = Moga J-6-7 (a™ 1 1 d.ar 1 t, M log a Mlg. When t=t, let 7 = 7,9 then T= log (1 − a¯¯₁) + C; 1 log (1 − a¯) + C. 1 1 J ατ . t − t - M log a When tť let T = T', log ……..(1). 1 а-т 1 ατι ť - t, log (2), M log a 1 a 1 and the coefficient being eliminated between these M log a two equations, the time of cooling is fully known. R 130 THE SPECIFIC HEAT OF GASES, 104. The laws of cooling in vacuo being known, it will be easy to deduce from them and observation the cooling which is due to the contact of any gas. For we have only to subtract from the actual velocities of cooling those quantities which would be the velocities of cooling if the body, cæteris paribus, were placed in vacuo. Thus we can determine the energy of cooling due to the sole contact of fluids, and such as would be observed directly if the body could be deprived of its property of radiating. From a series of most careful experiments, Dulong and Petit are led to infer that the state of the surface of the body has no influence on the quantity of heat which is carried away by the contact of the gas, and that the density and temperature of the gas do not affect the cooling, except by the variation which they cause in the elastic force of the gas. So that the cooling power of a gas may be considered as depending simply on its elastic force. The velocity of cooling of a body due to the contact of a gas depends on its excess of temperature and on the elastic force of the gas; and if v' be the velocity, the excess of temperature, and p the elastic force, we have as the result of experiment, v′ = mτ"pº, MT T where b is the same for all substances, c the same for all bodies, but varies for different gases; and m varies with the nature of the gas and with the dimensions of the solid. When a given body cools from the contact of any gas, mp is constant for that body and gas; hence we may have v′ = N7º. PROP. To determine the complete law of cooling. Let the body be suspended in air, then the velocity of cooling due to radiation is v M (a" − 1). = - AND THE LAWS OF COOLING. 131 The velocity of cooling due to the contact of the gas is v' = Nr. The total velocity of cooling then being the sum of these, is V = M (a˜ − 1) + N˜³. Now the velocity of cooling due to the radiation de- pends very much on the state of the surface of the body, but that due to the contact of the gas depends simply on the elastic force of the gas. Hence, if V' and M' be the corresponding values of V and M for a change in the surface of the body, V' = M' (a™ − 1) + N7'. T Then the ratio of the velocities of cooling is V M (a™ − 1) + N7° b V' M' (a-1)+NT Suppose M greater than M', that is, let M belong to the body which radiates best; and let the value of this ratio be ascertained for different values of T. Now when T = 0 or 7 = ∞ T= ∞ty, this ratio becomes which must be determined in the usual manner; hence, differentiating the numerator and denominator, V V' T a² M loga a + Nb7-1 M' loga at +Nor-1 M M' when = 0 or 7 = ∞ octy. Thus for very small or any large excesses of temperature, the ratio of the velocities of cooling depends simply on the nature of the cooling body. For other values of T we have Т V M + N ат 1 V' T M' + N a² – 1 132 THE SPECIFIC HEAT OF GASES, &C. Then, since is by hypothesis a ratio of greater M M' inequality, it is diminished by the additional term b N ; but the less diminished the greater becomes, ат - 1 so long as this quantity is a proper fraction. Thus it appears that for small excesses of temperature the velocities of cooling are less rapid for the surface which radiates most, and for large excesses of temperature are more rapid. Many other conclusions may be drawn from the pre- ceding equations and compared with experiment, but re- course must be had to the Memoir from which the pre- ceding propositions have been taken, or to the Article Heat, in the Encyclopædia Metropolitana. CHAPTER X. ON THE GENERAL EQUATIONS OF THE MOTION OF FLUIDS. 105. THE general equation of the equilibrium of fluids was obtained from the property which all fluids. possess of transmitting pressure equally in all directions, so that, it is impressed on every particle throughout its mass. This property is conceived by Poisson* to arise from the fact, that the particles of any fluid after compression or dilation return to a similar relative state, so that the fluid is a system of material points, similar to itself and existing on a smaller or a larger scale. The time of the fluid passing into a similar state, produces no influence on the laws of the equilibrium, which are only observed after it has obtained that state. But this time, however small, must influence the laws of the motion of fluids, so that the equal transmission of pressure does not obtain so accurately in the motion as in the equilibrium of fluids. Another distinction must be remarked with respect to Marriotte's law. This law requires that the temperature of the fluid should be the same before and after the com- pression or dilation. This distinction is of no importance in liquids, but in gases, where the vibrations of the parti- cles are very rapid, the equality of pressure is considerably modified. Journal de l'Ecole Polytechnique, 20 cahier. 134 THE GENERAL EQUATIONS These circumstances introduce conditions of great im- portance but of extreme difficulty, and in the following articles we shall suppose that the equal transmission of pressure obtains equally in a fluid at rest and in motion. 106. PROP. To find the pressure at any point of a fluid mass in motion. Let x, y, z, be the co-ordinates of any point P of a fluid mass at the time t, and let dM be any element- ary mass of the fluid at the same point. Let ρ be the density of the fluid at that point, and X, Y, Z, the impressed forces in the directions of the three co-ordinate axes. These quantities will be given functions of x, y, z, when the forces are directed to or from a fixed centre, and these functions will contain the time explicitly when the centres are moveable. When the centres are within the fluid they will be functions of x, y, z, t. Let u, v, w, be the velocity of the particle at the same time resolved in the same directions; these are un- known functions of x, y, ≈, t, because for the same value of t the velocity varies from one point to another, both in magnitude and direction, and for the same value of x, y, z, it changes from one instant to another. Now, d (u) d (v) dt dt d (w) dt , are the effective accelerating forces in the directions of the three axes at the time t. d (u) Hence X Y d(v), d (w) Ꮓ are the , dt dt dt forces lost during the time dt by the particle submitted to the action of the forces X, Y, Z. OF THE MOTION OF FLUIDS. 135 But by D'Alembert's principle the impressed and effective forces are in equilibrium with each other, or the forces lost are precisely such as would preserve the system in equilibrium. Hence the general equation of equilibrium will be satisfied by these forces, and we have V d(p) = p { ( Xx - d (u)) dx + (y = d(v)) dy + ( z −d (w)) dx}, d(p)=p{(X- dt +( Ꮓ dt dz dt or, d(p) _ X =Xd +Ydy+Z d d(u) d(v) dx+ dy+ dt dt dt d(1) dz) (1) P We have seen that u = f(x, y, z, t), and the incre- ments of x, y, z within the time dt will be udt, vdt, wdt, respectively; therefore u' = f(x+udt, y +vdt, z+wdt, t + dt). Whence, d(u) du du du du W + v + w + dt dx dy dz dt and similarly for the quantities v and w, d (v) d v dv u + v + d v w + dv dt dx dy dz dt d (w) dw dw dw dw Wu+ v + w + dz dt dt dx dy Before these quantities are substituted in (1), let us assume that Xdx + Ydy + Zdz is a complete differential of dP, and also that uda+vdy+wdz = d; we shall see hereafter to what circumstances these analytical facts have reference. Then, W = аф du " dx dx ď & du dx² dxdy d² p du dady' dz dxdx' z 136 THE GENERAL EQUATIONS (the quantities being written in the denominators in the order of the differentiation) and similarly for the other quantities. Then d (u) αφαφ ď o do đẹp độ аф + + + dt dx dx d x d y dy d x d z dz dx dt d (v) ď do do аф do do ď³p do ďo + + + dt d y d x d x dy* dy dydz dz dyd t d (w) ďp do do do do do аз ф + + + , dt dz dx dx dx dy dy dz dz dxd t these being multiplied respectively by dx, dy, dz, and added, and observing that z = ď² o do dεo аф do dx + dy + d² QdQdz = d x = + d ( x ) = dx² dx dy dx dx dz dx dx dx² and that there is a similar expression for the sums of the second, third, and fourth terms of these equations so multiplied, we have, by substitution in (1), = dp - + a. {(dp) dP d. d(p) αφ 2 2 + аф аф аф + d. dx dy dz dt P 2 ds do - dP - d. - d. , dt dt if ds be the space which the particle describes in dt; integrating, therefore, jd (p) 2 ds аф P - 1/ dt dt р The quantity (p) being properly determined, the f ρ pressure p at the point required will be known. This equation contains five unknown quantities; hence for the OF THE MOTION OF FLUIDS. 137 solution of the problem five equations will be necessary and sufficient. 107. PROP. To form the equations for the deter- mination of the five unknown quantities in the general equation of fluid motion. Three equations are supplied at once from the general equation (1), which has been obtained by applying D'Alembert's principle to pass from the general equation of equilibrium to that of motion. Let dp dp dp dx' dy dz be the partial differential of p with respect to x, y, z; then 1 dp d(u) 1 dp d (v) 1 dp d (w) X Y. ― 2- ...(4), ·(A), pdx d t p dy dt pdz dt where d (u) du du du du d (v) d (w) 2 + v + W + and dt d t dx dy dz d t dt dt have similar values. One equation, and in some cases two, may be formed out of the condition that the mass of any element of the fluid continues the same during the time dt; hence this equation is called the equation of continuity. It is formed as follows. Since during the motion the element of fluid will change both in form and density, but its mass is always to remain the same, the difference of the product of the volume and density at the time t + dt and at the time t will be zero. At the time t the co-ordinates of any point P are x, y, z, and the values of x for the two ends of the edge de of the element are x and x + dx. Let u, be the value of u for the point whose co-ordinates are a + dæ, y, ; then at the time + dt, these are z S 138 THE GENERAL EQUATIONS x+udt, and a+udt + dx; the length of the edge = (u, – u) dt + dx. Now u = f(x, y, z, t), then u, being the value of u for the variation only of x, u = f(x + dx, y, z, t) du = U + da, very nearly. dx du The length, therefore, of the edge dxdx+ dxdt. dx d v Similarly, the length of the edge dy = dy + dydt, dy dw and dz = dz + dødt. dz Again, the density is a function of x, y, z, t; ρ .. p' = f(¿x + udt, y + vdt, z+wdt, t+dt) dp dp dp dp udt + v dt + wdt + + = p + dx dy dz dt The new element will therefore du d v dw = p' (dx + dxdt) (dy + dydt) (dz + dxdt) dx dy dz du dv d w = p' (1+ dt) (1+ dt) (1 + dt) dx dydz, dx dy dt the change of the element from rectangular to oblique, introducing only quantities which may be omitted. Now the variation of this element is to be zero; hence, sub- tracting from this product pdx dydx, and omitting all terms above the fifth order, and all common factors, we have OF THE MOTION OF FLUIDS. 139 } do dx u + dp dy v + or dp dp w + +p dz dt d.pu d.pv + du d v + dw + 0.....(B), dx dy dz d.pw dp + + = 0. dx dy dz dt Another equation is p = kp, provided the motion is of such a nature that this equation can subsist; that is, if there be no change of temperature during the motion. Thus we have five equations which are sufficient to de- termine p, p, u, v, w, the five unknown quantities. 108. In the preceding article the fifth equation was furnished by Mariotte's law, but when the fluid is in- compressible this equation does not obtain. We have, therefore, only four equations. But in this case the equation of continuity will be sufficient. For let the fluid be incompressible and heterogeneous. d(p) Then 0, or, dt dp 26 + dx dp dp dp dy v + do 1. w + d t 0, which is one equation, and (B) becomes du d v + dx dy + dw 0, dz which is another equation. In this case, then, we have the proper number, or five equations. Again, let the fluid be incompressible and homogeneous. Then, &c. are all nothing, and (B) becomes dx du dx + d v dy dw -+ 0. dz 140 THE GENERAL EQUATIONS Here, then, we have but four equations; there are but four unknown quantities; hence we have in each case the proper number of equations. If аф аф аф be the partial differential coefficients dx' dy' dz with respect to x, y, z, derived from the supposition that uda+vdy+wdz is a complete differential, the preceding equation, substituting for u, v, w, their аф аф аф values dx dy dz ďo becomes ď o ď o + + 0, dx² dy² d22 a partial differential equation of the second order. Thus when the fluid is incompressible the equation of continuity (B) and the equations (4) are sufficient for the solution of the problem; and when the fluid is compressible, they furnish four equations, the fifth being given by Mariotte's law, which may be considered to hold if the temperature be the same throughout the whole mass during the motion as it was during a state of equilibrium. But variation in density is always attended with change in temperature; hence the pressure is no longer propor- tional to the density simply. When the motion is rapid, the development of heat gives rise to a great increase in the elastic force of the fluid, as will be seen in the theory of sound. But if the motion be slow, so that the variation in the density and elastic force is small, the equation. p=kp(1+a0) may subsist, and will furnish the fifth equation. 109. In the preceding investigations it was assumed that udx + vdy + wdz is a complete differential dø, and consequently the general equation obtained on this hypo- OF THE MOTION OF FLUIDS. 141 thesis can only be applied to the cases in which that condition obtains. Hitherto no general determination has been given of the cases in which uda+vdy+wdz is a complete dif- ferential. Particular cases have been indicated by Poisson*, and Professor Challis+ has shewn that when in incom- pressible fluids the motion at each point of any element. is directed to fixed or moveable focal lines, the equation du dv dw 0, depending on the above condition, is + + dx dy dz satisfied. It seems probable that ultimately it will be found that in every instance of fluid motion for which udx+vdy+wdz is a complete differential, the character of the elementary motions will be of the same description. When the parts of the fluid do not move inter se, that is, do not change their relative positions, but move as if they were rigidly connected, the quantity has no existence, and the pressure is determined by the equation P = jp(Xd + Ydy+Zd~), the forces X, Y, Z including those which arise from the rotation; as for instance, when a mass of fluid revolves about an axis without changing its form. It was also assumed that Xdx + Ydy + Zdz is a complete differential dP. This, as is well known, is the case whenever the forces are directed to fixed centres, or when they are directed to moveable centres ‡. The integrals of partial differential equations contain arbitrary functions, and the existence of these arbitrary functions in the equations of the motion of fluids is an analytical fact which shews that in their application to physical questions any motion whatever may be given to the particles, which is an evident consequence of the * Traité de Mecanique, Art. 654. + Camb. Phil. Trans. Vol. V. vi. ** Ibid. Vol. III. XVIII. xvш. 142 THE GENERAL EQUATIONS fundamental principle of the perfect mobility of the particles. In the following articles we shall proceed to illustrate fully the circumstances of the motion in space of two dimensions, and state what is at present known of motion in three dimensions. Motion in two Dimensions. 110. PROP. The motion being in space of two dimensions, to obtain an integral of the equation of continuity. Let the motion be in the plane of xy; then the equa- tion of continuity becomes Φφ αφ ΦΦ dx2 + 0. dy Let be a function of t, and a² + y², or ‚². do do dr dp x аф Then dx dr dx dr r d² pď² ď² p dr x αφι + dx² dr² d x r dr r do dr x dr dx r² d²p x² аф αφ (1 + A Φ dr² 22 d² o d² py do dr 2° 2013 аф Also + dy² dr p dr d² y² x² y² dp 2² + y dp (2 - 2+) -0, dr² 2.2 or + 心中 ​dr dp dp 1 dr² -+- dr r 0. 213 = OF THE MOTION OF FLUIDS. 143 раф d. dr аф, торф rd But + dr dr dr² раф d. dr = 0. rdr Then integrating and adding an arbitrary function of t, rdp dr = f(t), .. аф f(t) dr (1), and 4 = ƒ(t) log.r + F (t)…………….. (2). f(t) аф The velocity, or thus appears equal to dr The physical meaning of this result may be illustrated by supposing fluid to be contained in a cylinder capable of expanding in the direction of its radius, and a very slender cylinder of solid matter to be inserted with its axis coincident with the axis of the cylinder. The fluid particles, by the insertion of this solid cylinder, will be moved through spaces which vary inversely as the dis- tances from the axis. The distance of the point under consideration from the origin of co-ordinates being (r), we see by (1) that it is moving in such a manner that its velocity varies inversely as its distance from some point, and its motion is directed either to or from this point. 111. But it is of great importance to obtain also the integral of this equation of continuity independently of any hypothesis respecting p, and to shew the con- nection which subsists between this and the preceding integral. The following investigation is given by Professor Challis*. The usual method of finding the integral of * Camb. Phil. Trans. Vol. V. vII. 144 THE GENERAL EQUATIONS a linear partial differential equation of the second order between two variables leads in the present case to the integral $ = F(x + y√ − 1) + ƒ(x − y√√√ − 1). Φ To ascertain its general signification, let the forms of the functions F and ƒ be determined independently of any hypothesis respecting the mode in which the fluid was put in motion. The quantity condition douda+vdy, whence Then is subject to the аф dx u = F' (x + y √ − 1) + ƒ′ (x − y√ − 1), = 2, do dy v = √√ − 1 F' (x + y √ √ − 1) - √√ - 1 ƒ' (x − y√ − 1). V. First, it may be observed that u and v are not both possible for any values of x and y, unless the functions F and f' be the same. Again, as the form of F' we are seeking for is to be independent of all that is arbitrary, it will remain the same whatever direction we arbitrarily. assign to the axes of co-ordinates. Let therefore the axis of y pass through the point to which the velocities u, v, belong. Then y = 0, U = 2 F' (x), v = 0. If now the axes be supposed to take any other position, the origin remaining the same, u will be equal to 2x √ √ x² + y² F' (√ x² + y²). Hence 20 F' (x + y√ − 1) + F' (x − y√ − 1) F' (√ in² + y²), √ x² + y² a functional equation for determining the form of F". OF THE MOTION OF FLUIDS. 145 Let x + y√ - 1 1 = m, then 2 x = m + N, and a-y√- 1 = n; and √x² + y² = √mn. Therefore, m + n F' (m) + F' (n) F' (√mn) m mn √m n n F' (√mn) + F' (√mn). √m m n It is easily seen that if F(√mn) equation is satisfied. C = m n Hence аф C C 2 Cx + dx x + y √ = i - 1 x x − y √ - 1 x² + y² аф 2 Cy and dy x² + y² the and consequently the velocity at xy, or √ u² + v² 2C √ x² + y² For These results shew that the velocity is directed to or from the origin of co-ordinates, and varies inversely as the distance from it. But we must observe that this limitation as to the point to which the velocity is directed, is owing to the particular forms, a+y√ −1, x-y√−1, of the quantities which the function F involves. the differential equation is also satisfied by a more general value of these quantities, as is there shewn; and the result shews that the velocity is directed to a certain point, and varies inversely as the distance from it. And this result having been arrived at without considering any circumstances under which the fluid was caused to T 146 THE GENERAL EQUATIONS move, the inference to be drawn is, that such is the general character of the motion. Also the co-ordinates of the point to which the mo- tion is directed may be constant, or functions of the time and the given conditions of the motion. 112. The following considerations are added in con- firmation of the foregoing reasoning. In whatever man- ner the fluid is put in motion, we may conceive a line, commencing at any point, to be continually drawn in a direction perpendicular to the directions of the motions at a given instant of the particles through which it passes. This line may be of any arbitrary and irregular shape, not defined by a single equation between x and y. But it must be composed of parts either finite or indefinitely small, which obey the law of continuity. Consequently the motion, being at all the points of the line in the di- rections of the normals, must tend to or from the centres of curvature, and vary, in at least elementary portions. of the fluid, inversely as the distances from those centres. An unlimited number of such lines may be drawn through the whole extent of the fluid mass in motion. Motion in Space. 113. An integral of the equation of continuity for incompressible fluids may be obtained on the particular supposition that is a function of r and t, when 202 x² + y²+ x². Then, аф do dp dr афх dx dr dx dr r d² dx d² pď o dr x dr² dx r dp 1 + dr r d²x² αφ + d p² p² dr ( - ) аф X2 dr dx r² do dr x OF THE MOTION OF FLUIDS. 147 ď² d²p y² do (1 y2 dy2 dr² p² dr 203 d² x² аф αφ + 2 do dr² r dodo (1-5) dr And substituting in the equation of continuity, ď² © x² + y² + x² dp (3 x² + y² + x² + = 0. dr² 2.2 dr r 203 ďo + гаф d 12 r dr = 0. But d².rp + r dr² аф ф, гаф. dr² r dr ; rdr² ď².r Q = 0. Integrating and adding an arbitrary function of t, d.rp = f(t), dr and integrating again and adding another arbitrary func- tion, rp =ƒ (t) + F (t); F(t) F(t) p = f() + = f (t) + .. (1). √ x² + y² x² + y² + x² The velocity of the fluid ... the velocity 11 = √ u² + v² + w² dø² dø² dø² αφ dx2 F(t) аф + + ; dy do dr (2). 148 THE GENERAL EQUATIONS The preceding result may be illustrated as before by conceiving a spherical mass of fluid enclosed in an ex- tensible envelope, and that a small sphere being placed at the centre of this mass, the particles will be made to move from their original places through spaces which vary inversely as the square of their distances from the centre. 114. A general integral of the preceding equation of continuity cannot be obtained, and the general law of the motion of the parts of the fluid amongst each other so as to fill always the same space, cannot in this case be found in the same manner as in space of two dimen- sions by subjecting the general integral to a similar discussion, but by reasoning analagous to that contained in (Art. 112.), Professor Challis infers that the elemen- tary motions are every where directed to focal lines. But the reader must have recourse to the original paper*, where it is proved that the equation of continuity is satisfied by the kind of motion there supposed. Also the general conclusion arrived at by this reasoning is, that the law of variation or the velocity from any point to another indefinitely near in the direction of the mo- C tion at a given instant, may be expressed by r (r + 1) and r is where C and are constant at a given instant, the distance of the point under consideration from a line whose position is fixed at a given instant. C 2.2 If C=0, we have as a particular case the velocity which represents the law of the variation of the velocity under these particular circumstances, and agrees with the particular case of the integral just treated of. Since is ultimately in the direction in which the velocity V takes place, if a line commencing at a given Cam. Phil. Tran. Vol. V. Part 11. OF THE MOTION OF FLUIDS. 149 point be drawn constantly in the direction of the mo- tion, at a given instant, of the point through which it passes, dr may be considered as the increment of this line. Hence, if s be its length reckoned from the fixed point аф аф do = V. dr ds & Then integrating = [Vds + f (t), and differentiating un- der the sign of integration, аф d V ds +f' (t). dt dt Substituting this value in the general expression for p, p = [(Xd® + Ydy+Zds) fdv V ds − 1 V² – ƒ' (t). dt If V be always the same in quantity and direction at the same point, d V dt 0; : p = [(Xd +Ydy+Zds) – } V _ f'(t). x This equation, which may be considered as strictly deduced from the general equations of fluid motion, is the equation of steady motion, as we shall see presently. 115. When the fluid is compressible or elastic, the equation of continuity is not as we have seen, resolved into two, and the general equation is of too complicated a nature to be readily treated. In one case, however, when the motions of the parti- cles are small, and no extraneous force acts, the equation of continuity admits of simplification, and the general equation can be treated. 150 THE GENERAL EQUATIONS, &c. The motions being small, dp dp dx dy dp dz , omitted, and the equation of continuity becomes dp du d v dw +p dt + + dx dy dz may be 0, ď² O or + + = 0…………………..(1). dt dx² dy2 dx2 d.logp d² ď² o And since no extraneous force acts, the general equation becomes dp аф dt ρ dp Let p = a²p, then a² دینام dp a² fd.log p; P P аф ... a²fd.log p dt d.log p dep a² and substituting in (1), dt dt2 ď² o dt2 d² & d² p = α a² Φ + + dx² dy2 d se If as before, be supposed a function of rt, an in- tegral may be found, and the equation becomes d².rp dt2 d Q = a² dx2 which will be treated in the chapter on Sound. CHAPTER XI. ON THE MOTION OF FLUIDS ON PARTICULAR HYPOTHESES. 116. THE general equation of the motion of fluids is not readily applicable to practice, and in a case of such great difficulty as the present, recourse must be had to particular cases of the motion. Fortunately, indeed, the cases which most commonly occur in practice, are such as can be brought under the equation of motion, either directly or by particular hypotheses which conduct to results very nearly true. One large class of questions is where the motion is steady, and we have already seen that the general equa- tion admits of great simplification in this case; we shall shew how the same equation may be deduced at once from the general equation to which we are led by the application of D'Alembert's principle. 117. Steady Motion. Definition. The motion of a fluid is said to be steady, when the velocity at all points in space is constantly the same both in magnitude and direction; that is, when the accelerating force on each particle is the same as it passes through the same point in space. The motion being steady to find the pressure PROP. at any point. Let x, y, ≈ be the co-ordinates of any point in mo- tion at the time (t); X, Y, Z the impressed accelerating forces, and u, v, w the velocities in the direction of the co-ordinate axes. 152 THE MOTION OF FLUIDS d(u) d(v) d(w) Then are the effective accelerating dt dt dt forces, and d(u) d (v) d (w) X- Y- dt dt dt are the forces lost, hence the fluid would be in equili- brium if these forces acted on it, or we have as before (y_d(v) dp=p {(x - d (u)) dx + (y - d (v)) dy + (z_d(w) dt Y dt du d v dz d(w)) de}= dt d w dt .p=fp(Xd+Ydy+Zd ) [p dx+ dy+ dz dt dt This integral is generally to be taken between the limiting values of x, y, z, which belong to some point at the surface, and at some arbitrary point at which the pressure is required; t being constant, and a, y, ≈ being entirely independent, or having such relations as may be- long to any line of particles arbitrarily chosen betwixt the above limits. Let the integral be taken with regard to any line of particles which terminate in a given particle, as for instance, with regard to the line which this very particle has traversed in coming to the point under considera- tion, that is, to the point at which the pressure is re- quired. Then since each particle is moving with the same velocity as it passes through the same point in space, the particles in the line thus traversed will all be moving with the same velocity, or acted on by pre- cisely the same force at any given instant as the given particle was whilst traversing it; hence the values of d(u) d (v) d (w) for successive values of x, y, z, dt dt dt corresponding to successive values of t, will be precisely the same as the values of these quantities for the same ON PARTICULAR HYPOTHESES. 153 successive values of x, y, z corresponding to any one value of t; and therefore in the case of steady motion we may integrate the above equation considering x, y, x as functions of t, and we shall obtain the same value as if we integrated on the supposition of t being constant. But x, y, z being functions of t, we may write ď² x dt d (u) dx for dxx, dt and similarly for the other quantities; √(d(u) dx + dt d (v) dt d (w) dy + dz dt dx d² x + dy d² y + dx d² z jd a ď d t² ds 2 + C, dt if ds be the space described by the particle in the time dt, and C be the arbitrary constant which may be a func- tion of the time; then for convenience calling v the ab- ds dt solute velocity we have > p = f (Xdư + Ydy+Zds) – ấp +C, which is the equation of steady motion. To determine the value of the arbitrary constant, let v 1 V₁ be the velocity which the particle whose path has been considered had at the surface, and let p, be the value of p, then if the surface of the fluid be taken for the plane of xy, - P₁ = − 1 pv² + C, which being subtracted from the preceding, we have p - P = [p (Xd + Ydy+Zds) - đp (u – v)...(C). - U 154 THE MOTION OF FLUIDS Let gravity be the only force which acts, and let p = 1, then p - P₁ = gz - 1 (v² – v²). Let a be the ratio of the velocity at the upper surface to that of the issuing fluid, then 1 2 p-p₁ = gx - ¹ v ² (1 − a²). This equation is applicable to the issuing of water retained at a constant elevation in any vessel through any small orifice or adjutage fitted to the orifice. We shall proceed to illustrate its application. 118. PROP. To determine the velocity of a fluid issuing through a small orifice. Let the fluid be constantly supplied so that the surface is retained at a constant elevation, the motion will then be steady, and the equation of steady motion may be applied. Let k be the area of the orifice, and K the area of the surface of the fluid which will be constant, the fluid being retained at a constant elevation. The velocity of the fluid at the orifice, and the surface is inversely as these sections, for the fluid being incom- pressible, the quantity contained in the vessel is constant, hence the same quantity must flow out and in during a given time, that is, the product of the area of any section and of the velocity of the fluid passing through it is in- variable. Let v, V be the velocities of the fluid issuing through the sections k and K, then k V vk VK, or K V and the general equation is P - P₁ = gz - 1 v² (1. k? K2 ON PARTICULAR HYPOTHESES. 155 Now since k is small, the velocity at every point of the issuing stream which is in immediate contact with the air will be very nearly the same. But it appears from experiment, that when water issues through an orifice into the air, the stream converges for some distance, when it acquires a constant permanent form, neither converging nor diverging, which is called the vena contracta. This form of the issuing stream shews that the surface of the converging stream is moving with a greater velocity as far as the vena contracta than any point in the interior; hence the pressure at the surface, is as the equation shews, less than at any point in the interior, and consequently the pressure in the interior is greater than the atmospheric pressure. But at the vena contracta there is no tendency to diverge or converge, every point of the section is moving therefore with the same velocity, and the pressure is every where the same as the atmospheric. At this part then of the stream we have p we have p = P₁, and the equation becomes 0 = g≈ − 1 v² ( 1 ย - 1 2gx - い ​k² K2 ka K2 The section k then is the section of the vena contracta and not of the actual orifice, for it is through a section of the vena contracta that the efflux really takes place, since the stream has not acquired its greatest velocity before reaching this point, and the converging part of the issuing stream must be considered as a continuation of the containing vessel. The section of the vena contracta may be taken as equal to ths the actual orifice*. * See RENNIE's Report to British Association, 1834. 156 THE MOTION OF FLUIDS 119. If the constant surface of the fluid be large k compared with the orifice, the ratio will be small, and k2 K2 K exceedingly small, hence if h be the depth of the commencement of the vena contracta below the surface of the fluid, we have for the issuing stream v√√√2gh, or the velocity of the issuing fluid is that due to the height through which it has descended supposing it to fall freely. If the fluid be not supplied at the same rate as that with which it escapes, the surface is no longer stationary, and the hypothesis of the steadiness of the motion is not fulfilled. When, however, the orifice is exceedingly small, the true velocity will differ from 2gh by a quantity which is not assignable, and no appreciable error will be introduced by using this value. 120. PROP. To find the time of a vessel emptying itself through a small orifice in the base. At any time (t) let K be the area of the surface of the fluid, and ≈ the depth of the effective orifice k below the surface of the fluid, and v the velocity of the issuing stream. Then da being the descent of the surface in the time dt, we shall have kvd t Kdz. Kdz But v = √2gx; :. dt k√2gx 1 Kds and t k√2g. Let the vessel be prismatic, then K is constant, and 2 K t = {C = √x } . ર k√2g ON PARTICULAR HYPOTHESES. 157 When the motion commences let x = a; 2 K .. 0 {C - √ a}, k√2g and subtracting, 2 K t k√√2g {√ã-√x}. The whole time of efflux is 2 K√ a k√2g which is double the time in which the same quantity would run out, if the fluid were retained at a constant elevation. If the vessel be not prismatic, the surface K must be expressed as a function of x, and the time can be found as before. 121. The effect of adjutages in increasing the ex- penditure of a given orifice is known practically to be very considerable; we shall apply the preceding equation of steady motion to some of those which were employed by Venturi in his experiments on issuing fluids. Ex. 1. An adjutage consisting of a conical and cylin- drical tube of the form of the issuing stream. Let abge (Fig. 23.) represent this adjutage. Let k be the section at cd or the effective orifice, and h, h' the depths of ef, cd, below the constant surface of the fluid. The velocity of the fluid issuing into the air without the adjutage would be 2gh' 1 α 2 if a be the ratio of the effective orifice to the surface of the fluid. The velocity of the fluid issuing at eg A 2gh 1 − a² 158 THE MOTION OF FLUIDS fore The increase of velocity due to the adjutage is there- 2g 1 - a² (√h-√π). The pressure at cd = p₁ +gh' - 1 v² (1 − a²). 1 The pressure at the orifice = P₁ + g h − 1 v² (1 − a²), − hence the pressure at the commencement of the vena con- tracta is less than the atmospheric by g (h - h'), or the weight of the column in the cylindrical tube. This agrees very nearly with the results obtained by Venturi. Ex. 2. A cylindrical tube with its axis horizontal. When the fluid fills the tube, the velocity of rushing into the air will be 2gh a² 1 α , 2 and the expenditure will consequently be increased by the adjutage in the ratio of the area of the orifice to the section of the vena con- tracta in air; for in this case the actual orifice is rendered the effective orifice. Ex. 3. A horizontal adjutage converging to the vena contracta and then diverging. An adjutage consisting of two conical portions having their smaller ends united at the commencement of the vena contracta, was found by Venturi to give a large expenditure. The equation p = p, + gz − 1 v² (1 - a²) shews that as the velocity will decrease in passing from the minimum section towards the mouth of the adjutage, the pressure will increase; and this is confirmed by experiment. In such an adjutage as this, the stream is divergent when it leaves the adjutage and the velocity of that portion of it which is in immediate contact with the air, being nearly the same, it must consequently be less than the velocity in the interior of the stream at a small distance ON PARTICULAR HYPOTHESES. 159 from the aperture. At a small distance from the adjutage there must be a section where the stream ceases to be divergent, and at which consequently the velocity is the same for every point and the pressure equal to the atmo- be2gh spheric. The velocity at this point will be A α 2 1 - a²' and as this section, the stream having being divergent, is larger than the aperture, there must be a greater expen- diture from a conical diverging adjutage than from a cylindrical tube of the same aperture*. The Variable Motion of Fluids. 122. PROP. The motion not being steady, to deter- mine the pressure on the hypothesis of parallel sections. When the motion is not steady, returning to the general equation, we have p = fp (Xd® + Ydy+Zdx) d (w) - Sp dt d(u) dx + d (v) dy+ 2 dx}. dt dt dz Here since the particles are moving with different velo- cities as they pass through the same point in space, the quantities d(u) d(v) d(w) dt dt dt are no longer the same for successive values of x, y, z and successive values of t, as for the successive values of x, y, ≈ and the same value of t; but the integral must be taken at a given instant for a line of particles terminating in the given point, that is, the quantities must be integrated ex- clusively with reference to x, y, z. To effect the integration in the most general manner, without any special hypothesis respecting the direction of * See CHALLIS, Camb. Phil. Trans. XVIII. 1830. 160 THE MOTION OF FLUIDS the motions of the particles, let ds be the space described in time dt, by a particle whose co-ordinates are x, y, z, and let o be its velocity. Then is the effective accelerating force upon it in аф dt the direction of its motion, and du do dx d v do dy dw do dz dt dt ds dt dt ds dt dt ds Sp 10 du d v dw dx + ·dy + dt dt dt dx) = Sp = dx² + dy² + dx² do ds dt = = Sp dø ds. dt The quantity is a function both of the position of the whole mass of the fluid and of the given particle within it, or it is a function of t as well as of x, y, ≈; but since Sp аф dt ds is to be taken exclusively with reference аф to x, y, z, we must express in terms of the variables dt on which it depends, to effect which recourse must be had to some particular hypothesis. The one chosen is the hypothesis of parallel sections, which supposes, that if any section be taken at any instant perpendicular to the motion of any particle, then all the other particles are moving with the same velocities and in the same direction. This hypothesis amounts to supposing that a fluid, as water, descends in parallel slices, so that a portion which is at any instant included between two planes, will always be between planes parallel to these. It may be shewn from the general equations, that for some cases the motion is in strict conformity with this hypothesis*. * CHALLIS, Phil. Mag. Jan. 1831. ON PARTICULAR HYPOTHESES. 161 Then all the particles being supposed to move with the same velocity in the same section, let к be the section of the fluid, every particle in which is moving with the velocity . Also let k be any section of the vessel through which the particles are moving with a velocity v, then the fluid being incompressible, kv кф = kv, or ф ; whence, differentiating, K аф k dv kv dk ; dt K d t K 2 d t but since κ varies with the motion of the fluid, dk dr ds ds k v and Φ dt ds dt dt K аф k dv k v dκ k v • dt k dt K k² ds K аф dv ds or ds = dt k dt K ― k2 v2 dk K 3 ; Now v, and therefore d v is a function exclusively dt ds of the time, the position of k being given, and dê and K are functions of x, y, z; hence, integrating with respect to these quantities, аф d v ds = k dt dt jds k² v² + 1/13/1 K K Substituting and supposing that gravity is the only force and that p = 1, X 162 THE MOTION OF FLUIDS d v p = gx - k dt √ds - b k² v² K K² 1-124 2 + C, where C may be a function of the time. To determine the arbitrary constant, let us suppose that k=K_at_the surface of the fluid, let p,, ≈, be the values of p and ≈ at the surface, and let • ds N, then ༧] K d v - p − p₁ = g (≈ − ≈ ) − k N − 1 k² v ² dt K² Ꮶ K² whence the pressure at any point below the surface of the fluid may be determined. 123. The preceding equation may be applied to an issuing fluid, and the results compared with those which have been already obtained. PROP. To find the velocity of a fluid issuing through a given orifice. Let water issue into free air, then the pressure at the orifice will be the same as the atmospheric pressure, or p = p,· Let k be the orifice and v the velocity of the issuing fluid, then, since κ may be situated anywhere, let it coincide with the orifice, and let h be the depth of the orifice, then h = ≈ - ≈₁; dv 1 1 .. 0 = gh – k N - ½ k² v² dt k² K d v = gh – k N − 1 v² (1 − a²³), dt k if a = K Now since (1 - a) is essentially positive, we may assume it = B2. ON PARTICULAR HYPOTHESES. 163 Let the fluid be retained at a constant height, then h and N are constant; we have therefore dt 2 k Nd v 2gh-B v2 kN √2gh + ẞv :. t = log. B√2gh √2gh - ẞv No correction is requisite, since v and t begin to- gether. The quantities ẞ and √2gh may also be positive or negative. Taking them as positive, we have, assuming k N 1 B√2gh x √2gh + ßv √2gh - Bv λι βυ Elt 1 whence λι 2gh € Substituting for B its value, 2ght-1 v = k2 Eλt + 1 1 K 2 Ελί 1 1 -ε-λε As t increases, Elt t+1 1+6 1+€¯λt approaches rapidly to unity as its limit, and at this limit the velocity is independent of the time, or v = 2gh (六​) 1 the velocity when the motion is steady, as was before shewn. 164 THE MOTION OF FLUIDS It appears then that on the hypothesis of parallel sections, the motion can never be strictly steady; if how- k ever the ratio be small, and h be not exceedingly small, K λ is large, and the motion may be considered as steady, after a very small time. Steady Motion of an Elastic Fluid. 124. The variation of the temperature being neg- lected, which may be done without sensible error when the motions are not very rapid, our general equation of steady motion becomes, when applied to elastic fluids, if p = a²p, a² log p = gz − 1 v² + C. Ex. Let air be driven by a constant pressure through a small orifice out of a vessel into the atmosphere. The effect of gravity may here be neglected, and if P be the constant pressure and V the velocity where the pressure has this value, then a² log P = − 1 V² + C' ; subtracting from the preceding, 12/ .. a² log — = 1 (V² – v²) ; P hence the pressure is less as v is greater. Also, P Ρ v² = V² – 2 a² log P = V² + 2 a² log p P p or v² – V² = 2 a² log Let a be the ratio of the velocity at the surface sub- ject to the constant pressure, to the velocity at the orifice, then P 15² (1 − a³) = 2 a² log- P ON PARTICULAR HYPOTHESES. 165 2 When the orifice is very small, a may be omitted, and v2 a² log Р - P Р The equation a²log = (V² – v²), shews that points P · of equal pressure are also points of equal velocity, since p = P and v = V simultaneously; consequently, that as the pressure at every point of the surface must be nearly the atmospheric pressure, if the stream contract like water, as there is good reason to conclude it does, there will be a converging surface at every point of which the velocity is nearly the same and greater than the velocity at all points within it, so that the pressure within the surface is greater than the pressure of the atmosphere. CHAPTER XII. ON THE THEORY OF SOUND. 125. SOUND is caused by the vibrations arising from some disturbance to which the particles that constitute an elastic fluid have been subjected, and the theory of sound consists in applying the preceding general equations to the motion consequent on such disturbance. A single disturb- ance is not sufficient to produce the vibrations necessary for the production of sound, but they arise from the con- stant repetition of such disturbances, and the velocity of sound is the rapidity with which these disturbances are propagated through the elastic medium. The air is the fluid to which we shall now proceed to apply the general equations (Art. 106.); and we shall sup- pose that it is perfectly elastic and homogeneous, having, when at rest, every where the same density and tempera- ture; and also, that it is so slightly disturbed from its state of rest, that during the motion which arises from this disturbance, the velocities of the particles are exceed- ingly small, in consequence whereof the accompanying condensations and rarefactions will also be exceedingly. small quantities. Hence the squares and products of these small quantities may be omitted, the effect of which will be to render the general equation linear, and there- fore integrable under a finite form. This linearity of the equation is a point of the greatest importance, since, otherwise, the general equation is abso- lutely intractable; for it is evident that if no hypothesis be made limiting the extent of the motions of the particles THE THEORY OF SOUND. 167 from their points of quiescence, the case to which we should be about to apply the equations would involve all the possible motions of elastic fluids. We shall suppose also that no extraneous force acts, or that X, Y, Z, are all equal to zero, in which case the density will, in a state of equilibrium, be constant and uniform throughout, as we have already supposed it to be. 126. PROP. To form the equations for the small vibrations of an elastic fluid. Let D be the density of the air when at rest, and p its density at a point x, y, z, and after a time t from the com- mencement of the motion; then p = D(1 + s), where s is a small fraction either positive or negative. Let h and gmh be the height and pressure of the barometric column corresponding to the density D, m being the density of the mercury. Then when the fluid is in motion, the pressure p which corresponds to the den- sity p is gmh (1+s), provided the temperature remains invariable; this however is not the case, since the tem- perature is increased or diminished according as the density increases or diminishes, that is, according as the fraction s is positive or negative. Suppose then that we have σ p = gmh (1 + s + σ), where is a small quantity of the same sign as s, of which it is some function; and since s is small, let us assume σ Bs, where ẞ is a positive quantity and independent of s. We have then, making the substitutions and differ- entiating, dp = gmh (1 + ẞ)ds ; 168 THE THEORY OF SOUND. and dividing by p = D(1 + s) and representing the co- gmh (1+B) efficient by a², D dp ds dp a² a² log (1 + $); 1 + s P ρ no constant being requisite if the integral be supposed to vanish when s = 0. But log (1 + s) = s − 1/2 s² + ...s, very nearly; dp P a² s. But the squares of the velocities аф аф аф dx dy , , dz are to be omitted here; hence the general equation becomes S 1 аф a² dt (1). This together with the three equations. аф аф аф u = v พ , ... (2), dx dy dz gives four equations for determining the condensation and the velocity of the fluid at the time t and the point x, y, z, the function having been determined. The displacement of the particles of the fluid being аф аф аф small, the products of s and dx dy' dz > are to be omitted, and the equation of continuity (Art. 107.) be- comes, substituting for s its preceding value, ď² O dt2 a² ď² p dx2 + ď o dy² + Φ do 2) ... (8). THE THEORY OF SOUND. 169 These three equations are those of the theory of sound propagated in air of uniform temperature and density: they depend on the hypothesis that u dx + vdy + wdz is a complete differential, which it is in the cases to which we shall apply them. Rectilinear Motion. 127. Let a small quantity of air be enclosed in a cylindrical tube with its axis horizontal, and let the motion be in the direction of this axis; if then this axis be taken for the axis of x, we have v = 0, w = 0, and will be a function only of x and t; the equation (3) becomes. «Φ d t = a a² ď Q dx² This equation, which belongs to the simplest case of propagated motion, is the one which we shall employ. different one, which will also serve to determine the motion, may be obtained from the conditions of the problem in an independent manner, as we shall proceed to do in the following proposition. 128. PROP. To find the differential equation for a disturbance propagated in a small cylindrical column of air. Let the axis of the column be horizontal, and let it be supposed that the temperature is uniform throughout the motion, and that no extraneous force acts. Let the section of the column be unity, and PQ (Fig. 24.) an element da of the fluid at a distance x from the origin of co-ordinates. In the time dt let PQ be transferred to P'Q', let PP' = X and p, p' the densities of the air in PQ and P'Q'. Then, OP' = x + X, and since I will vary for da in passing from Q to Q', Y 170 THE THEORY OF SOUND. dx OQ'′ = x + dx + X + dx dx dX (1 + dx) x + x + ( 1 + dX .. PQ = (1+x) da. dx dx; But the mass of the fluid being the same, ρι pdx = p' (1+dx) dX da, or, p=p' (1+41) dx The motion of the element dx is owing to the dif- ference of pressures at P and Q; hence, if p be the pressure at P and p' at Q, dp dx p² = p + dx. The impressed moving force is dp p-p' = dx, dx and the mass moved is pdx, therefore the accelerating force But the effective accelerating force ď² X 1 dp dt2 p dx 1 dp p dx d² X dt2 is the general equation of motion, and 1 dp P must be ex- dic pressed differently, according to the nature of the fluid in question. THE THEORY OF SOUND. 171 Now the general law of elastic fluids is, that the pressure is proportional to the density; hence, since there is a change in the density, there will be a corresponding change in the elastic force. Let e, e' be the elastic force corresponding to the densities p, p, then the temperature being supposed constant, kp, and e' = kp'; e = e dX = 1 + ρ dx and e' = e (1 dX > dx dX the powers of above the first being omitted, since the dx motions are small. But e' is the pressure exerted at an instant t + dt, .. p p = = : e (1 d X dx , and e = kp; dp ď² X and e , dx dx² 1 dp ď X k p dx d x² and substituting this value, ď² X ď² X k ; dť dx² which is the partial differential equation required.. If the change of elastic force consequent on the sudden variations of density had been taken into account, a like equation would have been obtained with a² in the place of k. 129. The equations (1), (2), (3), become then in the case of rectilinear motion, S 1 аф a² dt , dx аф d² p dt2 do a² dx² 172 THE THEORY OF SOUND. The integral of this partial differential equation of the second order is Ф F (x − at) + f (x + at)………….. (4) where F, and f, are arbitrary functions; hence, u = F(x − at) + f (x + at) ......(5), and as F(xat) - f (x + at) ......(6). = The discussion of these equations will shew distinctly the nature of the motions which we have to consider. And first, the differential equation is linear. Now the linearity of this equation is a remarkable analytical fact, and arose from the omission of the terms which consisted of the products of the condensation and velocity of the particles; and this omission was allowable because of the hypothesis, that the motions of the particles from their state of quiesence were small. The physical fact in the propagation of sound through air is known to be in accu- rate accordance with this hypothesis, the agitation of each particle being so minute as not to move it sensibly from the state of rest; for when sounds are transmitted through a smoky or dusty atmosphere, there is no visible motion in the smoke or floating dust, unless the source of the sound be so near as to cause a wind*. If then the motions of the particles be exceedingly small, the differences of these motions for two consecutive particles, that is, the amount of condensation or rare- faction undergone, must also be exceedingly small; the products then or squares of these small quantities will be quantities of a higher order than those of the other terms, and may therefore be omitted in comparison with them. When motion therefore takes place in a medium under these circumstances and conditions, the equation will be linear. * Encyc. Met. Art. Sound, 54. THE THEORY OF SOUND. 173 Now it is the property of all linear equations of the first degree, that if any number of functions satisfy the equation independently, their sum will also satisfy it; and conversely, if the sum of any number of functions satisfy it, then each function will separately satisfy it. And this property of the linear equation of the first degree is a necessary consequence of its linearity; for the equation involves no powers of the differential coefficient but the first, and since any differential coefficient of the sum of any number of functions is the same as the sum of their differential coefficients, the substitution of a sum of func- tions is the same as their substitution separately and con- versely; therefore if the sum of any number of functions satisfy the equation, each function will satisfy it sepa- rately. This property then that the equation may be broken up into several parts similar to each other and to the whole, is an analytical fact and the necessary consequence of its linearity. 130. From equation (5) we have the velocity given by the sum of two functions, but from what has preceded it is evident that we may have 2 = F₁ (x − at) + F₂ (x − at) + 1 - 2 + ƒ₁ (x + at) + ƒ½ (« + at) + 2 If the disturbance be propagated only in one direction, there will be (as we shall see presently) but one of these two lines of functions to be taken, the form of each function being determined by the initial circumstances of the disturbance. If there be but one original cause of the disturbance not resolvable into component disturbances, there will be but one function as F, to be considered; but if there be several original disturbances, there will be one function corresponding to each, and the whole disturbance will be the algebraical sum of these functions. And it is par- ticularly to be remarked, that the whole disturbance thus 174 THE THEORY OF SOUND. found as the effect of all the original causes together, is precisely the algebraical sum of a number of disturb- ances, each of which would have been produced by one of the original causes acting separately. Hence, whenever a particle is affected by several dis- turbances simultaneously, the motion it receives is com- pounded of all the different motions it would have received had each disturbance acted separately. Thus the velocity at any point is the resultant of several velocities produced by different causes, and any given cause will have the same effect in producing velocity at a given point, whether or not other causes are operating to produce velocities at the same point. The preceding is the general theoretical proof of the co-existence of small vibrations in rectilinear propagation; an experimental confirmation of which may be derived from the well-known fact, that an ear is sensible of the effect of every instrument at a concert, which could not be the case, except on the hypothesis of the simultaneous transmission of different disturbances. It must also be remarked with respect to these equa- tions, that the origins of x and t are perfectly arbitrary; and that as the equations were investigated without any reference to the manner in which the particles were put in motion, all the results derived prior to any hypothesis about the mode of disturbance must be perfectly general; that is, these results must obtain whatever be the nature of the disturbance, or in whatever way the particles have been caused to move, provided always that u be small compared with a. 131. The determination of the velocity with which a disturbance is propagated in transmitted motion had been incorrectly treated by analysts, till Professor Challis directed their attention to it. In his Report* he expresses *Trans. Brit. Assoc. 1834. THE THEORY OF SOUND. 175 his doubts as to whether the arbitrary functions obtained by the integration of the differential equation can be im- mediately applied to any but the initial state of the fluid, and whether previous to their application at any subsequent epoch, the law of transmission must not be first deduced by means of the quantities which the arbitrary functions involve; that is, if the differential equations be applied to one, as for instance, to the initial state of the fluid, can they or can they not be applied to another state without first determining the law of transmission ? Now previous to an examination into the law of trans- mission, we cannot know whether the form of the functions may not change with the time; and if this be the case there will be an evident fallacy in determining the velocity of propagation, on the supposition that the same form which expresses the state of disturbance at a time (†) will also express it at the time t + t'. This determination of the velocity of propagation leads, however, to no erroneous results in the case of sound, because the velocity of propagation happens to be uniform; and the fact that the functions do not change form is the consequence of the uniform transmission. Whenever then the velocity of propagation happens to be uniform, this method leads to no erroneous results, because it rests. a supposition which implies the uniformity of pro- pagation. It could not, however be applied without error to an instance of propagation like that which obtains in waves at the surface of water, where the forms of the propagated waves, though dependent on the initial state of the fluid, are continually changing with the time. on The method of obtaining a general expression for the velocity of propagation is given in the following pro- position* 132. PROP. To determine generally the velocity of propagation in transmitted motion. * See CHALLIS, Ed. Phil. Mag. April 1835. 176 THE THEORY OF SOUND. Let y = a distance (x, t) express the state of the particles at from the origin at the time t. Suppose any given value of the ordinate (if we sup- pose the state of the particles to be represented by a curve) to be carried through space with the velocity v during the time t'; in general v will be a function of a and t, but it may be supposed constant during the small time ť, and for a small increment of a, that is, for a portion da of the axis of abscissæ. For on this supposition quantities of the order t² and t'do only will be omitted. Hence, for the small interval of time so far as it relates only to a portion de of the axis of abscissæ, the function may be considered invariable, consequently, $ (¿, t) = ¢ (x + vt', t + t') .. 0 : = v = аф аф $ (x, t) + vť + ť' +. dx dt аф dx v + аф very nearly, and dt | s|| dx which is a general expression for the velocity of propa- gation. This formula is of extensive application, and will serve either to find v in terms of x and t when F is given, or to determine F by integration when v is given. 133. To determine the velocity of propagation in a cylindrical column of air, and the nature of the motion. We have seen that either of the functions will satisfy the equation separately, then for the function F we have, since THE THEORY OF SOUND. 177 dF d F - a F' (x - at), and F' (x — at), dt dx a F' (x - at) = a. ..the velocity of propagation For f(x + at) in the same manner, the velocity of propagation F' (x − at) a. Thus it appears, that whatever be the initial disturb- ance, the velocity of propagation is constant; hence, we may consider that the ordinate of the curve representing the state of the particles is transferred with a uniform motion through space, and consequently the functions F and ƒ do not change with the time. It appears also that the functions apply to propaga- tions in opposite directions; the function F to propagation in the positive direction, and f to propagation in the negative direction, whether on the positive or negative side of the origin. Since the two kinds of functions F and f, which separately satisfy the differential equation, satisfy it con- jointly, the inference from this analytical fact is, that the most general character of the motion is such as results from two simultaneous propagations in opposite directions. The velocity and condensation of the par- ticles, whether at the instant of original disturbance or at any subsequent period, are such as are consistent with two motions transmitted in opposite directions with the uniform velocity a. 134. PROP. To determine the nature of the trans- mitted motions. Since each of the functions satisfy the equation, let us first consider F(x - at). This, as we have seen, refers to propagation in the positive direction with a Z 178 THE THEORY OF SOUND. uniform velocity a. The motion represented by one of the functions expresses a possible motion, but is not the most general which can obtain. Then u as = F(x − at). This equation shews that the velocity is always pro- portional to the condensation. Also, since u = as, and s = — ¹ F(x — at), a — if at any instant, that is, for any value of t, an ordinate, as determined from the equation s = F(xat), be 1 a erected at each point in any line taken as the axis of abscissa, these ordinates will be proportional to the condensation, and the bounding curve will give at once the law of the density and the velocity; the positive ordinates corresponding to velocities in the positive direc- tion ABE (Fig. 25.), or to condensations, and conversely; the negative ordinates to velocities in the negative direc- tion EBA, or to rarefactions, and conversely. The state and motion of the particles then at any time t, may be accurately represented by some curve, the exact form of which is of no importance. This being the state of things at a time t, let us enquire what is the state at a time t+t', that is, let t be supposed variable, and constant. Then u = as = F{x − a(t + t')} = F{(x-at) - at} which is of the = F(x-at), suppose, same form as F(x- at), that is, the state of the particles at a distance from the origin, and at a time t', is precisely the same as at a distance a THE THEORY OF SOUND. 179 from the origin and a time t; or we shall have the same velocities and condensations of the particles when a is constant and t becomes t+t', as when t is constant and x becomes x · at'. Hence the velocities and condensations which the par- ticles at a given point undergo during the time t' are the same as those which the particles in a space at measured from the given point towards the origin of co-ordinates are undergoing at the instant t' commences. The motion is therefore such as will be understood by imagining a curve, which gives the velocities and con- densations, to move without undergoing any alteration along the axis and from the origin; but it must par- ticularly be remarked, that this conceived transfer is a transfer of form, and not of matter, and that it is more properly expressed by saying that the particles at distance ~ +at' from the origin at the time t+t' are in the same relative state as were the particles at a dis- tance a from the origin at the time t. a The general conclusion then to which we are led by the preceding is, that each particle (taken successively in order of space) is successively, in order of time, in a similar state of displacement, which may be represented by conceiving a peculiar form of curve to advance from the origin with a uniform velocity. The same remarks apply to the function f, the only difference being that the propagation is in the opposite direction, or towards the origin. If there be several initial disturbances, there will be a function corresponding to each, and the ordinate ex- pressing the velocity and condensation at any point will be the algebraical sum of the ordinates which would obtain by virtue of each disturbance considered separately. Thus the corresponding modes of vibration, when co-exist- ing, will produce a compound curve very different from 180 THE THEORY OF SOUND. the curve which would be traced for any one of the disturbances acting singly. 135. PROP. To express the disturbance at any epoch in terms of the initial disturbance. Let t be dated from the commencement of the motion, and let the initial disturbance extend through the limits ±l, and let √(x) be the function which represents the initial values of the condensation, the velocity being sup- posed nothing when t = 0. Then, (Art. 129.), 0 = F(x) + f(x), †(x) = a F(x) − aƒ(x), and F = -f; therefore, since F(x) = − f(x), † (x) = 2F(x). In the same manner (x) = −2f(x); we have, there- fore, since the functions do not change with the time, Then F(x − at) = ½† (x − at), · 1 f(x + at) = − y(x + at). u = 1 4 (x − at) − ÷√(x + at), at = // √(x − at) + ½ √(x + at). x These equations apply to the motion for any values whatever of a and t, with the single limitation that the function must become evanescent for any values of x + at and ∞ - at not included between the limits ±l. It will hence appear that from the first instant of the motion the initial disturbance is divided into two equal propagations, one in the positive and the other in the negative direction, and that as soon as these two parts are completely separated from each other by the propaga- tion, the function (at) only applies to the former, and the function (x + at) only to the latter. THE THEORY OF SOUND. 181 The motion then at any point on the positive side of the disturbance commences when x - atl, and ends when x-at = -l, that is, it begins to move when t 20 α and ceases when t= fore, of its motion is of a particle. 21 a x + l a ; the duration, there- which is the time of vibration Now, since a is constant, the time of vibration depends simply on the extent of the disturbance; hence, if τ be the time of vibration and λ the length of the wave, we have, as in light*, the velocity of the wave = . 21 11 시사 ​= α, 21 a which is independent of the length of the wave, or the velocity of transmission is constant, as it was before determined to be. 136. The periodicity of the motions of the particles leads us to assign some trigonometrical form to the arbitrary function, and the simplest which suggests itself, from the analogy which subsists between the motion of the particles of air and the oscillatory motion of a pen- dulum, is n b sin - (x - at + C), a which will express a possible motion, but not the most general one. This function goes through all its values when nt increases by 2π, that is, when t increases by 2 п n is therefore the time of vibration of a particle. * AIRY'S Undulatory Theory, Art. 5. which 182 THE THEORY OF SOUND. But by the last article the time of vibration λ 2π n 2 T whence α n α λ a If, therefore, the origin either of t or a be so assumed that the arbitrary constant C is nothing, we have b sin 2π λ (x — at) - as the form of the function for a single disturbance; and if there be several such disturbances, the sum of a number of such functions will indicate a possible motion. 137. PROP. To explain the reflexion of sound. We have generally for the disturbance of a particle of air in a cylindrical column И = F(x − at) + f(x + at). Suppose that u=0 when x = l for all values of t; then f(l + at) - F(lat), and since this holds for all values of t, we have f(l + at') = − F(l — at'). - Let t and t' be so connected together that u x + at l+at'; .. 21x at l-at'; — = - and f(l + at') = − F(l - at'); .. f(x + at) = − F(21-xat); .. u = F(xat) F(21-xat) – = F(x − at) - F{21 − (x + at)}. I THE THEORY OF SOUND. 183 Hence it appears that when in a cylindrical column of air any particle at a distance from the origin is always at rest, the motion of a particle at any point less than will be such as results from two equal and opposite propagations having their origins equidistant from the point at rest, and commencing at the same instant. Now the effect will not be altered by supposing a rigid partition at any point, provided it be endued with the motion of the particles at that point. Let such a partition be placed at a distance from the origin, that is, at the point of rest; the partition will consequently be always at rest. Under these circumstances the air on one side of the partition does not act on the air at the other side; if one portion be removed, the motion will take place as before, but in this case the partition be- comes a rigid reflecting surface, thus a sound is reflected back again. 138. In a preceding proposition (Art. 135.) we sup- posed that the velocity was originally nothing, the condensation being expressed by a given function. We will now consider the case when both the initial velocity and the condensation are given by separate arbitrary functions. The equations for the velocity and condensation in a cylindrical column are u = F(x − at) +f(x + at), as = F(xat) − f(x + at). - Suppose that when t=0 we have u = 8 = x(x); then we have u(x) and ↓(x) = F(x) + f(x), and ax(x) = F(x) − f(x), whence, F(x) = {↓ x + ax(x)}, { f(x) = } {↓ x − a x(x)}; 184 THE THEORY OF SOUND. α •·. u= { {† (x−at)+4(x+at)}+~{x(x−at)−x(x+at)}, S 1 2 = = = = {†(x−at)−√(x+at)}+÷{x(x−at)+x(x+at)}. 2 a Let the initial disturbance extend through a dis- tance 7; then for all other values the initial functions are nothing; therefore (x) = 0 and x(x) = 0 from x = l to x = ty and from a - l to x = 2 If now x be >l, we shall have † (x + at) = 0, x(x+at) = 0, 1 - and u = {√(x - at) + ax(x − at)}, cty. S 1 2 a - {↓ (x − at) + ax(x − at)}; .. U = as. Again, if x be <-l, † (x−at) = 0, and x(x − at) = 0, and u = - = = { ↓ (x + at) − a x(x + at)}, 1 S - 2 a {↓ (x + at) − x(x + at)} ; '. U = as. Hence, beyond the limits of the primitive disturbance on each side of it, the condensations of the particles are proportional to their actual velocities, and the particles are in a state of condensation or rarefaction according as their motion is in the same or contrary direction to that of the propagation. This relation, which does not necessarily hold within the limits of the primitive disturbance, establishes a marked distinction between the primary and the pro- pagated waves, the former being subject to no law, but THE THEORY OF SOUND. 185 to the arbitrary one which we assign to the function, the latter being subject to this condition. Any impulse in which this condition is not satisfied will immediately di- vide itself into two pulses running opposite ways, in each of which the preceding condition holds, and so long as this condition holds, no subdivision takes place. Hence we see the reason why every propagated wave does not divide itself into two, but is propagated only in one direction. Suppose this condition to obtain in the primitive impulse, then, ¥ (x) = ax(x), and the preceding equations give u = 1 = \ (x − at), s = — \(x − at), a — whence it appears, that for all negative values of a greater than, we shall have u = 0, s = 0, which shew that on the negative side the motion is not propagated beyond the limits of the primitive disturbance. Whenever then in passing through a medium a wave receives from extraneous causes any modification, such as disturbs the preceding relation, it will be subdivided, and a portion reflected. Similarly, this portion may be again subdivided, and so on; on; this subdivision being always accompanied with reflection, will give rise to a continued series of repetitions of the original sound as echoes. Motion in three Dimensions. 139. Having fully discussed rectilinear motion, we shall proceed to apply the general equations to a mass of air of indefinite extent, and in which the disturbance extends itself in all directions from a centre. A A 186 THE THEORY OF SOUND. PROP. To find the propagated motion in a mass of air of indefinite extent. Let the centre of disturbance be taken for the origin of co-ordinates, and be the distance of any point x, y, ≈, at the time t, and its velocity, which will be in the direction of the radius r, and a function of r and t and of the condensation s; for during the whole motion. every thing must be symmetrical about the origin of co- ordinates. Then we have u = but x + y + X โย (૪ V ω r グ ​ལཾ :; ; xd + ydy + dã Z and udx+ydy+wdz = ¶ (xdx + ydy + xdx) = (dr; グ ​+ or, udx+ydy+wdz is a complete differential of some This function being the quantity function of r and t. $ determined from the equation (3) (Art. 126.), we have J = √ u² + v² + w² = dp y=√ dr as the resultant of the velocities u, v, w. with respect to x, y, ≈, we have Differentiating dp dødr афафар аф афар dx dr dx dy dr dy' dz dr dz differentiating again and substituting, d²x² ďo аф X 1 + 1 – dx² d202 202 dr ď²p x² do y² + * d 202 202 + dr 20.3 THE THEORY OF SOUND. 187 ď²p_d²py² Φ dø x² + x² аф + dy dr² p² dr 203 d²p d²p x² dp x² + y² + dx2 d p² p² dr 203 and the equation (3), becomes ΦΦ a² d t ( ď² o z dp аф , dre r dr which may be put under the equivalent form, ď².ro d².ro a² dt² dr² (4). The complete integral of which is But rp = F(r − at) + f (r + at). аф dr indicating therefore by the differential coefficients by accents, 1 } = = {F'(r−at)+f'(r+at)} = '__{F(r−at)+f(r+at)} ... (5). }= Also s = 1 аф a² dt ; 1 S= _—_ (r− {F' (r − at) − f'(r + at)}…………………..(6), ar and these formula will determine the velocity and con- densation at any instant when the functions F, F have been determined for all the positive or negative values of (r – at), and ƒ, ƒ' for all the positive values of (r + at). 140. The remarks in Art. 129, on the linear equa- tion for rectilinear motion, apply here also, and it is 188 THE THEORY OF SOUND. therefore unnecessary to repeat them. The complete discussion of this equation has not as yet been effected, and for what is at present known respecting it, the reader must have recourse to the researches of Poisson* and Challist. It appears, as in rectilinear motion, that the velocity and density are propagated uniformly, the velo- city of propagation being equal to a; that the function F applies to propagation from a centre, and f to pro- pagation towards a centre; and if the equations involve but one arbitrary function, they apply to a single dis- turbance. In this case, when r is very small, the second term of the equation (5), which involves in its deno- minator, may become much greater than that involving r; for expanding the functions, supposing to be very small, }= F'( − at) + F''( − at) + &c. r F'(-at) F" ( — at) - &c. F(-at) F'(- at) go 20 11 F( − at) nearly, ↓ (t) 202 When, therefore, the disturbance is made by a sphere of small radius, the motion is transmitted from its surface to other parts of the fluid nearly as if the fluid were incompressible. At a great distance from the centre of the disturb- ance we may neglect the term involving in (5) and 1 * Art. 660. Traité de Mécanique. + Camb. Phil. Trans. Vol. 111. and Vol. v. THE THEORY OF SOUND. 189 1 (6) in comparison with the term involving -; then during the whole motion y = as, as in rectilinear propagation. we have The velocity of the particles decreases in the inverse ratio of r, hence, since the intensity of sound is pro- portional to the square of its velocity, its intensity at a considerable distance from the centre of the primitive disturbance will decrease inversely as the square of the distance; and experience confirms this conclusion. 141. We may here also determine the manner in which the motion of the fluid is affected when the rect- ilinear transmission of an impulse tending from any centre is interrupted by a plane surface. For suppose two impulses tending from two centres to be of equal magnitude and in every respect alike; then if the straight line joining these centres be bisected at right angles by a plane, there will be no motion of the par- ticles contiguous to the plane in a direction perpendicu- lar to it, because the resultant of the velocities from the two causes must lie wholly in the plane. Hence, since the division of fluids may be effected without the ap- plication of any force (Art. 2), nothing will be altered if we suppose the plane to become rigid and to inter- cept the communication of the fluid on one side with that on the other. The motion on each side will then be re- flected, and the angle of incidence will be equal to the angle of reflection. 142. PROP. To determine numerically the velocity of sound. The velocity of propagation of a disturbance through an elastic fluid is a. Now by assumption (Art. 126.), 190 THE THEORY OF SOUND. gmh (1+B) α D and we must determine the value of B. From the same article, we have gmhp(1 + s + ẞs) gmhp p = gm h D D(1 + s) D { Bs 1 + 1+ p(1 + ẞs)............ (1) 1 + s very nearly. n Let be the increase of temperature corresponding to this value of s, so that the temperature which was when the fluid was at rest becomes + n at the time t when the fluid is in motion. At this instant, p, p, 0+n, being the simultaneous values of the elastic force, the density, and the temperature, we have the equation p = kp{1 + a(0 + n) } · But the fluid being at rest, we have p=gmh, p=D, n=0, and the preceding becomes hence, the fluid being in motion, gmh = kD(1 + a0) ; gmh 1+ 1 + a(0+ n) gmh p D P 1 + al DP (1+ an 1 + αθ and comparing this with the preceding equation (1) we have, an В = (1 + a0) s But the vibrations of the air being extremely rapid, so THE THEORY OF SOUND. 191 that the condensation s takes place without any loss of heat, we may substitute s and for and w in (5), (Art. 99.), hence, 1 + B = y; where y is the ratio of the specific heat of air under a constant pressure to its specific heat under a constant volume. The value of a becomes therefore a gmhy D Let be the density of the air under a constant pressure gmh at the standard temperature, then (Art. 81), and consequently D= A 1 + a0' a √&m gmhy A (1 + a0)……………....(2), the expression whence the velocity of sound may be cal- culated numerically. The value of y, as determined by experiment (Art. 99.), was considered as independent both of the pressure and temperature; it appears then, 1°, that the velocity of sound increases with the absolute temperature in the ratio of √1+a0 to unity; 2°, that it does not vary with the barometric column, since h and ▲ vary at the same time and in the same manner, so that their ratio is constant. The hygrometric state of the air must produce a slight influence on the velocity, this however may in ge- neral be omitted, since the total variation for the ex- tremes of dryness and of moisture will not amount to th of the velocity of sound. 1 250 192 THE THEORY OF SOUND. Numerical Determination. The values of the constants which Poisson* has taken, are m g = 9.80896, h=0".76, 10.462, D a = 0.00375, 0 = 15°.9 C., y = 1.3748, whence he deduces a = 337".07 = 1105 feet. This value is a little less than what he considers as its value according to the best observers, namely, a = 340m.89 = 1115 feet. We have seen (Art. 98.) that different experimentalists have assigned different values to y; if we take its larger value, namely, y = 1.421, we shall obtain, using the pre- ceding data, a = 342.69 = 1124 feet. Thus it appears that this value exceeds the observed value by nearly the same as the other falls short of it. 143. The determination of the ratio of the specific heats of an elastic fluid is a most important inquiry; we have already seen (Art. 99.) how this is to be determined experimentally, and that its value is essential to the nu- merical determination of the velocity of sound; and we shall now shew how its value may be determined from the observed velocity of sound. The circumstances under which sound is propagated, are far more favourable to the full production of the whole effect due to the cause in question, than the experiment with closed vessels; and the whole circumstances of the two cases are so widely different, that while a considerable deviation in the results would be insufficient to falsify the theory, a close agreement in the * Traité de Mécanique, Art. 664. THE THEORY OF SOUND. 193 results affords an evidence almost conclusive. Here then the results agree so nearly, that there can be no doubt of the truth of the hypotheses on which they rest. The observed value of the velocity of sound is α 340.89. Substituting in the formula a² A gmh (1 + a0) the data of the preceding article will give y = 1.4061, a most remarkable result, being nearly the mean of the smaller value of Gay-Lussac and Walter, and the larger one of Dulong. In comparing this value of y with the preceding ex- perimental ones, it must be remembered that the conden- sation and rarefaction were supposed to take place so rapidly, that the quantity of heat which the fluid con- tained had not time to vary sensibly. But in the propa- gation of sound in free air, it is possible that the heat may escape or return more readily by radiation than in the propagation of sound in confined air, as in a closed tube, where the heat of each stratum of air can vary but little except by contact with the sides of the tube, and the large value of y is the one which experiment assigns to the con- fined air. This remark may explain the difference of the two experimental results, and inclines to the larger value of Y as the more exact. 144. The velocity of sound as determined simply from the formula by neglecting altogether the change A gmh , D of temperature consequent on the alternate rapid conden- sations and rarefactions, is less by one-sixth than the BB 194 THE THEORY OF SOUND. observed velocity, and the accurate agreement of the the- oretical with the observed velocity can leave no doubt of the truth of this theory, which is due entirely to Laplace. The propagation of sound in the vapour of water at its maximum density is due to the same cause. If a vibration be excited in a close vessel full of vapour and not mixed with air, sound will be generated and pro- pagated without. But if the temperature of the stratum of vapour contiguous to the vibrating body was not aug- mented, the condensation consequent on the vibration would reduce the vapour to water, which would be precipitated on the surface of the vibrating body, since by the hypo- thesis the density is at the maximum, that is, the quantity of vapour is that which is due to the temperature under a given pressure. But heat being developed by the com- pression, the temperature of the condensed contiguous stratum is raised, and can consequently continue in a state of vapour. The condensation and increased temperature is propagated from stratum to stratum, and sound is pro- duced just as in a vessel of permanently elastic fluid. The rarefactions of the strata are accompanied with a diminution of temperature, but then the density being diminished at the same time, the vapour is not reduced to water, but descends to the maximum which is due to the relative temperature of the space it occupies. The preceding is an experimentum crucis for deciding on the validity of the explanation above stated as given by Laplace, of the excess of the observed above the theoretical velocity of sound as determined without any regard to the developement of heat. If the instantaneous condensations and rarefactions of an elastic fluid do (as is supposed in that explanation) give out and absorb heat, sound will be freely propagated in a saturated vapour, that is, in a vapour in contact with a THE THEORY OF SOUND. 195 liquid, or under a pressure it can just sustain. If not, no sound can be transmitted through it. The experiments are decisive*. 145. Water being considered as a fluid slightly com- pressible and elastic, sound will be propagated in it accord- ing to the same laws through any other elastic medium. The sound, when it reaches the surface of the water, will be partly transmitted to the external air and partly reflected; and the direction of the transmitted and reflected waves will follow the same laws as those of light. The velocity of the reflected sound will be the same as that of the direct sound, and the ratios of the intensities of the transmitted and reflected sound to the direct sound, will depend on the ratio of the velocities of propagation of sound in air and water. When a given column of water suffers condensation, there does not appear to be any development of heat, so that there seems reason to conclude that the velocity of sound propagated in water is not influenced by any varia- tion in temperature. Theory and observation give a velo- city of propagation about quadruple the velocity in air. On Musical Sounds. 146. The equations which express the nature of the disturbance produced in a cylindrical column of air have been discussed, and the sounds arising from the vibrations excited by blowing across the open end of a pipe or an aperture at its side, may be explained by the preceding equations. The current must be directed not into but across the aperture, so as to graze the opposite edge; a small portion will then be caught by the edge and turned aside down the pipe, thus giving an impulse to the con- tained air, and propagating along it a pulse in which the air is slightly condensed; this will be reflected at the end Encycl. Met. Art. Sound, 88. Mém. d'Arcueil, 11. 99. 196 THE THEORY OF SOUND. of the pipe as an echo, and return to the aperture where the condensation vanishes, since the density is the mean, that is, the same as in the undisturbed state. musical note is produced. Thus a PROP. To determine the note produced from a cylin- drical pipe. Let be the length of the tube closed at one end, the open end being the origin of co-ordinates. Then the equations are (Art. 137.) u = F(x − at) - F (21 − x — at), - as = F(xat) + F (21-xat). We must assign some values to the form of the arbi- trary function, let that value be taken which we have already seen (Art. 136.) may indicate a possible motion, then the preceding become, (x − at) – m sin (lat) sin 2π 2 п u = m sin (21 − x — at), λ 2 π 2π = 2m cos (1 - x) λ λ as = m sin 2π λ 2 п (21 − x — at). λ = 2m sin 2π λ M 2T (1 – at) cos Now in this case λ (1 − x). the condensation at the orifice is 2π (x − at) + m sin nothing, that is, s = 0 when ∞ = 0 ; 2πι 2 п ... u = 2m sin COS (l - at)………….. (1). λ λ 2 π l 2π 0 = 2m cOS sin (lat)......(2). λ THE THEORY OF SOUND. 197 These equations hold for all values of t, hence, when u = 0, we must have sin 2πι λ 0, and when (2) is satis- fied, cos 0; we shall proceed to discuss these cases. 2πι λ 147. Nodes. When a disturbance is propagated along a column of air, the column may at any instant be divided into several portions, in each of which the corresponding particles are in a similar state of displacement and motion. These portions are termed nodal sections, and the points in which the axis of abscissa would cut the curve which gives the condensations and rarefactions at any instant, are the nodes. At these points the velocities of the particles are nothing, hence their position is determined. by the equation 2πι 2π 0 = 2m sin COS (lat), λ λ 2πι which is satisfied when sin 0, that is, at a node λ 2πι λ = nπ, or /= n λ 2 where n is any term of the series 0, 1, 2, &c. Hence the interval between two consecutive nodes is half the length of a wave. The closed end of the pipe is a node, or the extremity of a nodal section, since at this point the velocity of the particles is nothing. Loops. Half way between two nodes the condensations are rarefactions, are evanescent, and the amplitudes of the molecular excursions are at a maximum; these points are called loops, and are given by the condition that s = 0; 198 THE THEORY OF SOUND. hence, in the preceding case the closed end being a node, we have a loop when (2) is satisfied, that is, when COS 2πι λ = 0. For a loop then, 2πι (2n + 1)—7, λ λ , or, l = (n + 1) 12, where n is any term of the series 0, 1, 2, &c. At any point, then, in a cylindrical column of air, at λ which 7 = n l is a loop. 2 λ there is a node; and 1= (n + 1) 1/2, there 148. When a musical note is produced from a tube whose length is closed at one end, by blowing across the open end, experience shews that when the lowest or λ the fundamental note is sounded, is equal to the length - 4 of the tube, hence n = 0 for this note; also the conden- sation at the orifice is nothing, which will be the case if the orifice be the place of a loop. If n = 0, then we have from 7 = (n + 1) 2012 l , or, λ = 41. 3λ 4/ Let n = 1, then l or λ and so on. 4 3 Hence, if the fundamental note be called 1, the others will be 3, 5, &c. being inversely as A, the breadth of the wave. It is found in fact, that 1, 3, 5, &c. are the only notes which can be sounded. In the case above considered, the existence of the nodes and loops depends on the reflection at the closed end. If THE THEORY OF SOUND. 199 the tube be open at both ends, and the disturbance be made as before, the same cause does not operate to produce nodes and loops; yet is found by experiment, that there are places of maximum and minimum velocity at regular intervals, and that the two ends are nearly positions of maximum velocity. Assuming the ends to be positions of loops, we shall proceed to apply the equations to this case. Let be the length of the tube, then as = F(xat) - F(21' = m sin ~=== (x − at) 2 п λ 2 π = 2m cos λ – xat) 2π m sin (2l' — x — at) λ 2 п (l' — at) sin~—— (l' − a). - λ And s = 0, when x = 0, and x = l'; Ωπ 2 П ... ·. 0 = cos (l' — at) sin l', λ λ 2πl' ηλ which is satisfied when = ηπ: ··· nl' λ But for the fundamental note n = 1; .. λ = 21'; therefore for the same note as in the preceding case we must have 21' = 41, or l' = 21, that is, a tube open at both ends gives the same note as one of half the length closed at one end; and experience confirms this result. note. 149. PROP. To find the time of vibration for any 1º. Let the tube be closed at one end and = 7, then 200 THE THEORY OF SOUND. 2π u = 2m cos λ 2 п (l – at) sin²™ (1 − x), λ 2π (lat). the period of which depends on that of cos EIN λ Now cos 2π λ (l - at) = cos 2 π (1-1) at 2n+1 at = COS 2 π 4 λ + cos (according as n is even or odd) Flo π 2 Tat nat) 2 which equals nothing when 2 Tat = ± sin + λ 2nat 1 1 λ 入 ​= nπ, or t = N a 2 that is, for any multiple of half a wave. Let n equal 1, 2, 3, &c. successively, and t₁, t₂, t3, &c. be the corresponding values of t. Then the time of vibra- tion is the interval tз — t₁, calling it 7, 1 λ 1 λ λ 4/ T 3 α 2 a 2 a a for the fundamental note. And the number of vibrations in 1" 2º. α T ᏎᏓ Let the tube be open at both ends and in length l' = 21, then 2п 2π u = m sin (x − at) + m sin (21x - at) λ λ 2 п 2 п = m sin (lat) cos (1 − x), λ λ THE THEORY OF SOUND. 201 2 п the period of which depends on that of sin (lat). λ Now sin 2π λ at (l - at) = sin 2 π λ N at sin 2π λ 2πατ sin λ 1 λ which is nothing when t a wave. Whence as before 2 for any multiple of half a 2 λ 21' 4/ T α Ɑ. Ο for the fundamental note. 150. Pitch. The pitch of a note is determined solely by the frequency of repetition of the impulse, so that all sounds, whatever be their intensity or quality, in which the elementary impulses occur with the same frequency, are pronounced by the ear to have the same pitch. The intensity of sound depends on the violence of the impulses, the quality on the greater or less abruptness of these impulses. The pitch then of a note depending on the number of waves which impinge on the ear during a given time varies inversely as the time of vibration of a particle; and the time of vibration λ a therefore the pitch a C c oik 202 THE THEORY OF SOUND. 151. u = 2m cos 2π λ (l - We have generally in the closed tube 2π — at) sin — (1 − x), λ 2 п which is nothing for all values of t, when sin ·(1-x)=0, λ λ λ or x = l l — n " 2 that is, when l − x = n If therefore be given, ponding to the values of x, ferent values to n. 2 we have nodal points corres- which arise from giving dif- 2π 2 П Also as = 2m sin (lat) cos (1 − x), λ λ and s = 0 for all values of t, when cos 2π (1 − x) = 0, λ or l − x = - λ 4 λ Therefore x = 1 − (n + (2n + 1) · − 1 ) 2 1) 212 gives the position of the loops, which evidently occur at equal intervals with the nodes. 2π The maximum vibration depends on 2 m sin 27 (1-0) λ which takes place when in the preceding value of u, π 2π (2n + 1); (1 − x) = (2n + 1) and its maximum value is 2m; and obtains at points where the condensation is nothing. For further information on this subject, see a paper by Mr Hopkins*. *Camb. Phil. Trans. Vol. v. p. 10. CHAPTER XIII. ON RESISTANCES. 152. WHEN a solid is moved through a fluid its motion is resisted, and this resistance arises partly from friction and the tenacity of the fluid, but principally from the inertia of the fluid, that is, from the force which the body moving through the medium, necessarily exerts in putting the fluid particles in motion. Hence it may be considered as the reaction of the fluid particles, and cæteris paribus, if the velocity be increased, the re- sistance also will be increased, for the body will strike more particles and with greater violence. The law, ac- cording to which this resistance varies with the velocity must be deduced from experiment, and the square of the velocity is the power according to which it appears to vary; but no formula has hitherto been discovered which expresses with sufficient accuracy the absolute amount of the resistance for different velocities. In the following propositions we shall see how the subject is to be treated theoretically on the hypothesis that the re- sistance is as the square of the velocity, and what con- clusions may thence be deduced. 153. Definition. The resistance of a fluid on a solid moving in it is the resultant of the excess of the pressure of the fluid on the solid in motion, above the pressure of the fluid on the solid at rest. This resistance being a pressure, is of the nature of a moving force, and may be represented by weight. Its effect then on the body, or the retarding force of the resistance, is the resistance divided by the mass. 204 ON RESISTANCES. 154. PROP. To find the resistance on a plane moving perpendicularly to itself with a given velocity in a fluid. Let us suppose that the plane and fluid are moving steadily with the given velocity v, the plane being im- mersed perpendicularly to the motion of the stream. Let the plane be stopped at any instant, then the motion of the, fluid is resisted, and the mutual action between the fluid particles and the plane is precisely the same as the resistance on the plane moving with the given velocity through the fluid at rest. For we may suppose a velocity v to be impressed on every particle of the system in the direction opposite to the motion of the stream, the consequence of which will be that the fluid is reduced to rest, and the plane moves through it with the given velocity. Now the pressure at any instant during the motion on a unit of the plane is p = gp≈ − 1 pv² + C. At the instant the plane is stopped, let p' be the value of p, then v = then v = 0, and p' = gpx + C' ; p' − p = ½ pv². But p' p is the resistance on a unit of surface, hence the resistance on an area A is pv² × A. Let h be the height due to the velocity v, then v²=2gh, …. the resistance on the plane = gphA, or the resistance on a plane moving perpendicularly to itself, is the weight of a column of fluid whose altitude is the height due to the velocity and base the area re- sisted. ON RESISTANCES. 205 COR. If both the solid and fluid are in motion, the resistance on the solid is as the square of the relative velocity of the plane and of the fluid. 155. In the preceding proposition the resistance de- pends simply on the equality of action and reaction at the anterior surface of the plane, and no account is taken of the variation in the pressure which results from the disturbance at the posterior surface of the plane. This is doubtlessly one source of the discrepancy between the results of theory and experiment. Again, no account is taken in the preceding of the fluid which collects in a quiescent state before the plane, the instantaneous effect will be such as is there stated, but the plane being moved through the fluid, the particles which have lost their velocity will constitute a conoidal mass of fluid, quiescent relatively to the plane, and bounded by a corresponding hollow conoid of moving fluid. The action between the surfaces of these two conoids will cause a pressure on the plane essentially different from the instantaneous action of the particles on the plane. These remarks are sufficient to point out the imperfections in the preceding theory. 156. PROP. A plane moves obliquely in a fluid, required the resistance in the direction of its motion. Let P (Fig. 26.) be any point in the plane, PA the direction of its motion, and PB perpendicular to the plane. Then we may either consider the plane as moving with a given velocity v in the direction PA, or the streami as impinging on the plane in the direction AP with the given velocity, the effect on the plane being in both cases the same; and the resistance on the plane in the direction of its motion is the same as the impelling force of the stream in the direction of its motion. 206 ON RESISTANCES. Let be the angle of incidence, that is, the angle APB, and R the resistance, or the force with which; the stream impels the plane. Let K be the area of the plane, and R the resistance upon it moving with the given velocity v, that is, the force with which the stream impels the plane in the direction perpendicular to the plane. Then the velocity of the stream resolved in the direc- tion PB = v cose; hence R = 1 pv² cos² 0 K. = And the part of R in the direction of the plane's motion, that is, in the direction of the stream, = R cose = pv² cos³ K. And the part in the direction perpendicular to the motion of the plane = R sine = pv² cos² sine K. Hence the resistance on any plane moving obliquely is as the cube of the cosine of the angle of incidence, that is, as the cube of the sine of its angle of inclination to the stream. 157. PROP. To find the resistance on a solid of revolution moving in the direction of its axis. Let BAC (Fig. 27.) be a solid of revolution moving in the direction DA of its axis. Let x, y be the co- ordinates of any point P, and PQ an element ds of the generating curve, and mn the corresponding element dy of the base, and the angle which the tangent at P makes with the axis; then v being the velocity, the resistance on PQ = pv² sin³ 0 × PQ, and the resistance on mn = 1 pv² × mn, by the preceding article; ON RESISTANCES. 207 .. resistance on PQ resistance on mn :: PQ sin³0: mn dy³ :: ds : dy dy² ds2 ds³ : 1. And the same holds for every element of the annulus whose breadth is PQ, and for the corresponding portion. of the base; it is therefore true for the whole annulus. The annular portion of the base corresponding to the portion of the surface = 2πydy, and the resistance upon it = 1 pv² × 2 πуdy; .. the resistance on the surface = fpv² ſydy dy ds2 The mass of the solid of revolution = πp'fy³dx, if p' be its density, and dividing, the retarding force of resistance P = v2 Sydy ds² dy² ρ Jy³dx Ex. Resistance and retarding force on a sphere. Let a be the radius of the sphere, the centre being the origin of co-ordinates; then the resistance = πрν² fуdу dy ds2. ds dx² y² α Now y² = a² - x², and √ 1 + √ 1 + dy dy x² dy² ds2 - 2/2 - fydy - fydy = y (1-2)dy 1 = (-2)² 4 a² y² + C', x² • 1 208 ON RESISTANCES. which, taken between the limits y = 0 and y = a, = + a²; therefore the resistance on the sphere πρυα the resistance on a great circle. And the mass of the sphere = πр'a³, if p' be its density ; ..the retarding force = 3 pv² 16 p'a 158. PROP. A heavy sphere descends vertically in a fluid, required its velocity. Let a be the radius of the sphere and p' its density, and ρ the density of the fluid. Then, as we have seen, the moving force of resistance upon a sphere is the resistance on one of its great circles moving perpendicularly to itself. The resistance on one of its great circles = рv²πа²; τρυπας; .. the resistance on the sphere = —πρа²v². And the mass of the sphere the retarding force of resistance = = p'a³; hence πρα; π ¯pa³v² πρα зр v2 16p'a kv, suppose. The force by which a body descends in a fluid is, neglecting the resistance, the excess of its weight above the weight of an equal bulk of the fluid. The weight of the sphere = πр'ga³‚ πρα, the weight of an equal bulk of the fluid = πpga”. ON RESISTANCES. 209 Hence, subtracting and dividing by the mass of the sphere, this force g = (1 − r)g, if r be the ratio of the density of the fluid to the density of the sphere. The whole accelerating force on the sphere = (1 − r)g - kv². Now generally, if ƒ be the accelerating force, v the velocity, and s the space, v dv=fds; .. vdv = {(1 − r)g - kv²} ds, or d. v² + v².2kds 2 kd = 2(1 − r)gds, which is a common linear equation, and will be rendered integrable by the factor eds. Multiplying and inte- grating, we have v2 €2ks = (1 − ¹)g 2ks + C…........(1). r) k To determine the arbitrary constant, let the sphere descend from rest, then v and s commence together; (1 − r) g ... ·· 0 = + C, subtracting and reducing, 1 v2 g(1 − e−2ks). k When s is large, the second term may be omitted; the velocity then becomes constant, or is the terminal velocity. Let V be this value of v, then D D 210 ON RESISTANCES. r V2 g, k and the preceding becomes v² = V² (1 − e−2ks ) e−2ks)…………. .....(2). 3r 3r The constant k .. 2k 16 a 8 a Let the sphere be double the density of the fluid, P then r = 1 =; and let the sphere have descended ρ 2 1 through a space equal to 16 diameters. Then s = 32 a ; ... k -2ks k = ε 1 = € 400, nearly. ... v = V(1), or, by v2 Hence v² V² (1 v² = √² (1 − + 1/1); 400. the time that a sphere of twice the density of the fluid has descended through 16 diameters, which, when the particles are small in an insensible space, the velocity is withinth of the terminal velocity, and may after that be considered as moving uniformly. The terminal velocity VA tuting the preceding values, V = (1 − r)g k and substi- √16 16ag which varies as , 3 the square root of the diameter of the sphere. Hence it appears that the smaller the sphere the sooner it acquires its terminal velocity, and the less that velocity is when acquired. If then any small spherical bodies, as small dust, descend in water, or condensed vapour, as very small rain, descend in air, the velocity will be uniform and almost imperceptible. 159. PROP. To determine the motion of an air- bubble ascending in a fluid. ON RESISTANCES. 211 The air-bubble will increase in magnitude as it ascends; and let it be supposed to start from a depth in the fluid at which its density is very nearly that of the fluid; let b be its radius at this instant, and a its depth below the surface of the fluid, and its density or that of the fluid. P After it has ascended through some distance, let y be its radius, x its depth, and p' its density. Then, since its magnitude is as the cube of the radius and inversely as its density, and the pressure, being propor- tional to the depth, is as the density, we have 1 1 y³ : b³ :: asb ; y x a X3 .(1). Also pp: px x : a; :· p = ...(2). a The accelerating force upwards of the fluid displaced ρ is -p g, which a g, -(-1)=(-1)8, by (2). g The retarding force of the resistance is 3 ai v² N 16b aš ?? 3pv2 which 16p'y by substitution from (1) and (2), if n be assumed therefore the whole accelerating force upwards a 2 1) g-n v² x 3 as 166 212 ON RESISTANCES. Then, since vdv=fds, and the force diminishes x, we have {(-1) 8-n Jda, g a v dv whence 2n a d. v² - v² dx 2 23 X -1)8 gdx, a linear equation, whence the motion may be determined in particular cases. P P B F N B C P C A H 8 C D. 7 B 14 T P a 21 n 77 N B B M D Id D P A B B 20 N b 15 B 3 A E M I G B a m M C 22 a m 16 A B M HL H D 过 ​B A 21 0 1' P N B 10 4 P 5 P & n P n m B B 12 11 N A B 17 M P M ·P A 18 a 16 b P T m 9. B 23 2.5 B D R S D 4 T 13 P G 19 P 27 9 B 26 B P m E བ D Nocle so g Burlagh St Stani UNIVERSITY OF MICHIGAN 3 9015 06708 1623 ? B 448364 DUPL 1 + i