M \ % JSfO ^ ^ 2^ Q^epartmerbt of 53^ P49 LIBRARY or llllm®iii IniBiIrful Wwiv ©f'sityf ^oohs are not to he tahen from the Library I^oom K ^ V N •- ■.• Vs.^t ELEMENTARY HYDEOSTATICS. Cambrttigc : PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS, For MACMILLAlSr AND Co. London: BELL AND DALDY. Oxford : J. H. AND JAMES PARKER. Edinburgh: EDMONSTON AND DOUGLAS. Dublin: WILLIAM ROBERTSON. Glasgow: JAMES MACLEHOSE. ELEMENTARY HYDROSTATICS. WITH NUMEROUS EXAMPLES. BY J. B. PHEAR, M.A., yEIJLOW AND LATE ASSISTANT TUTOR OF CLARE COLLEGE, CAMBRIDGE. ©ambrilige: MACMILLAN AND CO. 1857. ADVERTISEMENT. This Edition has been carefully revised throughout, and many new Illustrations and Examples added, which it is hoped will increase its usefulness to Students at the Uni- versities and in Schools. In accordance with suggestions from many engaged in tuition, solutions to all the Examples have been given at the end of the book. The Author wishes to acknowledge the assistance given him by his brother, the Rev. S. G. Phear, Fellow of Emmanuel College, in reading the proof sheets, adding many Examples, and furnishing the Solutions. Dec. I, 1857. Digitized by the Internet Archive in 2017 with funding from University of Illinois Urbana-Champaign Alternates https://archive.org/details/elementaryhydros00phea_0 CONTENTS. SECTION I. PAGE Preliminary Definitions and Explanations. . . i Definition of a jiuidf i ; explanation of the term fluid pressure and the definition of its measure, -2, 3, 4 ; a supposed solidification of a por- tion of fluid does not affect its action upon the remainder, 5 ; hydrostatic action of rigid surfaces upon fluids, 6 ; transmission of pressure by a fluid, 7 ; distinction between elastic and inelastic fluids, 8 ; measures of mass, density speciflc gravity, 9, 10, ii, 12 . . , . . . ir Examples .-....,,...11 SECTION II. Inelastic Fluids 12 Definition of vertical and horizontal ; pressure at a point in a heavy fluid, 13 ; pressure uniform in the same horizontal plane, 14 ; free surface horizontal, 15 ; common surface of two fluids which do not mix, 16 ; heights of the free surfaces above this, 17 ; total normal pressure on a surface immersed, 18 ; resultant of the same pressure, 19 ; conditions which a floating body satisfies, 20 ; stable, unstable and neutral equilibrium, 21 ; conditions of a body’s sinJcing, rising, or remaining stationary when immersed totally in a fluid, 22, 23, 24, 25; Hydrostatic Balance, 26; methods for finding the specific gravity of a body or fluid, 27, 28, 29, 30 ; Common Hydrometer, 31 ; Nicholson’s Hydrometer, 32 ; specific gravity of a mixed fluid, 33. Examples, 31 . . . . . . . . . *43 SECTION III. Elastic Fluids 44 Pressure in elastic fluids sometimes termed elastic force, 34 ; assumed to be uniform under certain conditions, 35 ; Barometer, 36 ; Boyle's law, 37; pressure varies as the density, 38; to find h for air, 38*; law of density in the atmosphere, 39 ; barometer employed to measure heights, 40 ; Air-Pump , 41 ; density after the nih. stroke, 42 ; causes limiting the exhaustion, 43; HawTcsbeds air-pump, 44; Smeaton’s do., 45; the Con- denser, 46 ; Common Pump, 47 ; tension of its piston rod, 48 ; Lifting Pump, 49; Forcing Pump, 50; Bramah's Press, 51; Siphon, 52, 53; Diving-Bell, 54 ; Atmospheric Steam Engine, 55 ; Watt’s improvements, 56 ; Double- Action Steam Engine, 57. Examples, 77 .......... 85 Vlll CONTENTS. SECTION IV. PAOH General Propositions 86 Virtual Velocities, 58 ; resolved parts of normal pressures upon the surface of a vessel containing heavy fluid, 59, 60, 61, 62 ; do, when the included fluid is elastic, 63 ; Center of Pressure^ 64 ; BarlcePs Mill, 65 ; free surface of a revolving fluid under certain conditions, 66 ; resultant pressure upon the surface of a body immersed in any fluid whatever, 67 ; tension at any point in a solid how measured, 68 ; tension at any point of a cylindrical surface containing fluid, 69, 69 * ; do. for a spherical surface, (69 a), (69 a*); mode of graduating a barometer, 70, 71, 72; Wheel Barometer, 73. Examples, 104 112 SECTION V. Mixture of Gases. — Vapour. . . *113 Boyle’s law holds for a mixture of gases, 74 ; gas absorbed by a liquid in contact with it, 7 5 ; relation between change of temperature and change of volume in all substances, 76 ; meaning of the term thermometer, 77; mercurial thermometer, 78; filling the thermometer; comparative expansions of substances, 79 ; temperature measured by the ther- mometer, 80 ; expansion of water, 81 ; proof of the formula^ = lcp{\-\-at), 82 ; relations between the heat absorbed, the resulting temperature and the mass, for a uniform substance, 83 ; algebraical expression of the same, 84 ; do. for a compound substance, 85 ; specific heat, 86 ; algebraical formula, 87 ; specific heat of gases, 88 ; all substances made to ex- perience the solid liquid or gaseous state by the application of correspond- ing amount of heat, 89 ; vapour, 90 ; saturation density, and its dew- point, 9 1 ; under what circumstances vapours follow Boyle’s law, or not, 92 ; gases, 93 ; the preceding results independent of the number of gases present, 94, 95 ; ebullition of water, 96 ; algebraical formula con- necting the pressures and temperature for a mixture of gas and vapour, 97 ; latent heat, 98 ; effects produced by absorption of heat in evaporation, 99 ; contraction produces the opposite results to dilatation, 100 ; hygro- metrical state of the atmosphere, 10 1 ; Clouds, 102 ; Rain, 103 ; causes producing these effects, 104 ; Snow and Hail, 105 ; Dew, hoar frost, &c., 107 ; Conduction, convection and radiation of heat, 108 ; cause of Dew, &c. 109; Dew-point, no; results of the law of expansion in water, in. General Examples. 13S Answers to the Examples 15^ HYDROSTATICS. SECTION 1. PRELIMINARY DEFINITIONS AND EXPLANATIONS. 1. Def. a fluid is a collection of material particles so situated in contact with each other as to form a continuous mass, and such that the application of the slightest possible force to any one of them is sufficient to displace it from its position relative to the rest. That part of Statics^ where a fluid appears as the princi- pal means of transmission of force, is termed Hydrostatics, The law of that transmission must, like the law of transmis- sion by a rigid body, by a free rod or string, or by contact of surfaces, &c. be established by experiment. The mutual forces called into action by the contact of surfaces are in Statics called pressures: this term is used in the same sense in Hydrostatics, where it is applied to denote the forces of resistance, which adjacent particles of the fluid exert, either upon one another, or upon rigid surfaces in con- tact with them. The nature of the reaction between a rigid surface and a fluid in contact with it might perhaps be arrived at by the aid of analysis from the above definition. But such an investigation, even if entirely satisfactory in itself, would 1 P. H. 2 PRELIMINARY DEFINITIONS be out of place in this treatise. It may here tlicrcforc be taken as the result of experiment that : WheM a fluid rests in contact with a rigid hody a mutual force of resistance is called into action at every point of the common surface of contact^ the direction of which force is normal to that surface. 2. If in the side of a vessel, containing fluid upon which forces are acting, a piston be placed, the pressure exerted upon it by the fluid particles with which it is in contact, would thrust it out, unless a force sufficient to counteract this pressure were applied to the back : this counteracting force is of course exactly the measure of the pressure of the fluid upon the piston. It is not difficult to conceive that, generally, the magnitude of this pressure would be different for different positions of the piston in the sides of the vessel ; inasmuch as the portions of the fluid which it would touch at those different places, would not necessarily be similarly circum- stanced, and would not therefore require for the maintenance of their equilibrium that the piston should exert the same force upon them : when, however, the pressure for every such supposed position of the piston, wherever taken, is the same, the fluid is said to press uniformly ; and when not so, its pres- sure is said to be not uniform. Again, it is clear that the pressure upon the piston in any given position must vary with the magnitude of its sur- face, and if this were reduced to a mathematical point the pressure upon it would be, strictly speaking, absolutely nothing, because the surface pressed is nothing ; but even in this case the conception of the pressure at the point is perfectly definite; it signifies the capability or tendency which the fluid there has to press, and which, if existing over a definite area, would produce a definite pressure ; and this AND EXPLANATIONS. 3 view of it leads us to the following usual definition of its measure. The pressure at any point of a fluid is measured hy the pressure which would he produced upon a unit of surface^ if the whole of that unit were pressed uniformly with a pressure equal to that which it is proposed to measure, 3. It is usual to represent this measure of the pressure at a point by the general symbol p ; and whenever it is said that a surface, in contact with a fluid and containing A units of area, is pressed uniformly with a pressure j 9 , it is meant that the pressure of the fluid at every point of it, measured as above defined, is equal to p units of force : hence if P be the pressure which the fluid exerts upon this surface since the pressure is uniform^ and therefore the actual pressure upon each unit is p, the pressure upon the A units must be A times or P—pA, It may be here remarked, that as P is of four linear dimensions, being the measure of a moving force, and A is of two, therefore must be of two dimensions, i, e, if the linear unit be supposed increased n fold, the numerical value of p for a given pressure will be increased fold. 4. In the foregoing explanation of the meaning of the term pressure at a point” in a fluid, the point has been assumed to be in contact with a rigid surface, which was supposed to be the subject of the pressure ; now if we con- sider any portion of fluid, within a larger mass and forming part of it, no force but that of resistance can be exerted upon it by the surrounding fluid; for we may imagine it to be isolated from the rest by an excessively thin enveloping film, which will manifestly produce no disturbance among the par- ticles of either portion of the fluid, because its existence neither introduces new forces nor destroys any of those which 1—2 4 PRELIMINARY DEFINITIONS are acting ; fnrtlier, this film may he supposed rigid, without affecting the relative positions or equilibrium of the particles forming the interior and exterior portions of fluid : but under this hypothesis the pressure at any point, of either the inte- rior or exterior fluid, which is in contact with the rigid film, acquires the meaning given above, and as the introduction of the film in no way alters the actions of the portions of fluid upon one another, we thus arrive at the conclusion that different portions of a mass of uniform fluid only press against each other in the same way as they would against rigid sur- faces of the same form, and therefore the term pressure at a point” means the same thing whether the point be within a fluid or be in one of its bounding surfaces *. 5. This last conclusion with regard to the action of different portions of the same fluid upon one another, which is of considerable importance in the solution of hydrostatical problems, does not rest solely upon reasoning analogous to that just given. It may be considered as a fact deduced from experiment, in the same way as all other physical laws (Art. 7 ), that: The statical action of any one portion of a fluid upon that which adjoins it^ is the same as if the latter portion were a rigid hody having the imaginary surface^ which divides the two portions^ as its surface of contact with the fluid. We are therefore justified, whenever it concerns us to in- vestigate the pressures exerted by a surrounding fluid upon an included portion, in replacing this portion by a conterminous solid. It is generally convenient to take for such a purpose * Tho analogy between ‘^pressure at a point” in a fluid, and ^Welocity at any instant” of a moving particle, and between their respective measures, is too striking to escape the notice of the student ; both terms are abstractions cmidoyed for the purpose of avoiding the constant use of the periphrasis, which is given once for all in tho definitions of their measures. AND EXPLANATIONS. 5 tlie solid which would be formed by supposing the constitu- ent particles of the portion of fluid, which it is wanted to replace, to become by any means rigidly fixed in their relative positions and to be still affected by the same external forces as before : it is manifest that such a solid could not differ from the fluid w^hich it replaces, as regards its action upon the surrounding fluid, for it would itself be identical with it in every way, were it not for the circumstance that its particles are supposed to be provided with an artificial check against moving from their relative positions, in addition to, or rather instead of, the mutual resistances which effect the same end in the fluid state. 6. If a surface opposed to a fluid be itself rough or capable of exerting friction, the particles of fluid adjacent to it would, as it were, adhere to it and thus form a sort of polished veneer, because the definition of a fluid, which states that the application of the slightest force is capable of dis- placing the particles, precludes all idea of the existence of any tangential action between the particles themselves ; and there- fore although there may be resistance to the tangential motion of the particles in contact with the surface, there can be none between them and their next neighbours. For the same reason whenever a portion of fluid is supposed to become solidified, its surfaces must be considered perfectly smooth. Hence in all cases the pressure of fluids is normal to the sur- faces pressed. (Art. 1.) It is true that very few fluids answer strictly to the defi- nition given above (Art. 1) ; there is generally a certain amount of friction or viscosity between their particles, and in all cases, where the instantaneous effect of forces upon a fluid is the subject of investigation, this mutual tangential action between the particles cannot be neglected. But it is found practically that when once equilibrium is established, the 6 PRELIMINARY DEFINITIONS particles have always assumed such a position inter se tliat no tangential action is called forth ; and therefore it is immate- rial to consider whether the capability for it exists or not. Thus if any semi-fluid such as honey or treacle be allowed to assume its position of equilibrium under the action of gravity, it will do so very slowly compared with water under the same circumstances, but in the end it will be found tliat its position is exactly the same as that of the water. Hence in Hydrostatics all fluids whatever may be assumed to be perfect fluids. 7. The law of transmission of pressure through fluids, which was alluded to above, may be stated as follows : A force applied to the surface of a fluid at rest is trans- mitted^ unchanged in intensity^ in all directions through the fluid. Like all other physical laws, this is experimental ; or rather it is suggested by experiment, and its proof is deduced from a comparison of the results of calculation based upon it, with those of corresponding observations : in* this sense the following experiment may be said to prove it. The annexed figure represents a vessel of any shape con- taining a fluid, which may be supposed to be acted upon by gravity, as must generally be the case, or by any other forces what- ever : into the sides of this vessel are fitted any number of pistons, represented by -4^, A^^,,,A^, and having plane faces whose areas are respectively sufficient forces are also supposed to be ap- plied to these pistons to keep them in their places ; in fact whatever be tlic forces whether only gravity or AND EXPLANATIONS. 7 any thing else which are acting upon the system, the whole is supposed to be in equilibrium. If now any additional force as be applied to the piston it is found that to preserve equilibrium additional forces P^,,,P^ must be also applied to the pistons A^..,A^ respectively such in P P P. , magnitude that — ^ = — ^ = &c. = — : this result clearly shews P that the application of a pressure upon each unit of area of the piston A^ has caused the same additional pressure upon every unit of area in each of the other pistons. In this way may the truth of the principle enunciated be verified. Con. It follows from this that the pressure at any point ivitJim a fluid mass is the same for all directions. For the action between any two adjacent portions of the fluid at any point would be the same as would exist if we suppose one of the portions to become rigid (Art. 5). In this case the pressure at the point would be caused by a rigid surface pressing on the fluid, and therefore, by the principle just enunciated, would be the same in all directions*. 8. Of fluids there are some, such as air, whose volumes or dimensions are increased by diminishing the pressure upon them and vice versa; these are commonly called elastic fluids, and all others inelastic. Inelastic fluids are also often dis- tinguished by the name of liquids^ while elastic are called either gases or vapours according as their state is one of permanent or temporary elasticity. It is probable that every fluid is compressible, when very great pressure is employed for the purpose, although within the limits of the forces with which we shall be concerned no appreciable error will be * The fact of fluid pressure at a point being equal in all directions, leads immediately to this: that the resultant pressure upon any indefinitely small surface passing through that point must be normal to the surface. (Art. i.) 8 rPvELlMINAUY DEFINITIONS committed by considering water, mercury, &c. wliicli con- stitute the inelastic fluids, or liquids, as incompressible. 9. The conception of the mass of any portion of a fluid is the same as that of a solid, and its measure is also the same: thus if M be the mass of a portion of fluid whose weight is TT, the accelerating force of gravity being we liave the relation, W = MF=P'(F-a,AP); 8 _V-a,AF F-a^AF’ hence the ratio between 8 and 8' is known when the numbers of graduations in AF and A F' are known. INELASTIC FLUIDS. 29 Nicholson s Hydrometer. 32. This instrument is adapted for finding the ratio between the weights of equal volumes of two fluids, or between the weights of a solid and an equal volume of fluid. It is represented in the an- nexed figure ; 5(7 is a hollow buoyant body of any symmetrical shape, ^ is a cup supported upon it by a rigid wire AB^ and i) is a similar cup suspended below by a wire CD. This cup is frequently capable of inversion, so as to hold down a body specifically lighter than the fluid. (1) To use this instrument for comparing the specific gravities of two fluids. Let W be the weight of the instrument, the weight which must be placed in A in order to make it sink in the first fluid to a point P in the stem AB^ the weight to be placed in A in order to make it sink to the same point in the second fluid: then the weight of the fluid displaced in the first case in the second JV -f TV^, and the volumes are the same in both, therefore if B and B' be the specific gravities whose ratio is required, B _ TL+ IF, WA (2) To compare the specific gravities of a solid and fluid : Let IF, be the weight required to be put in A in order to sink the instrument up to P in the fluid; remove this and place the solid in A ; and let IF^ be the weight which must in addition be put in A in order to sink the hydrometer in the fluid to the same point P; next place the solid in D and let 30 INELASTIC FLUIDf^. be tlic weight wliieli must now be put in A to sink the instrument to the same point ; then the weight of the solid is W^ — W^, also its apparent weight in the fluid is TF, — ; but this must be its real weight diminished by the weight of the fluid it displaees ; the weight of fluid displaced by it is since then this displaced fluid is equal in volume to the dis- placing solid, if S and 8' represent the respective specific gravities required, 8 ^ W^-TV, ~8' W,-W,' 33. When two or more fluids are thrown together in the same vessel, if they do not lie in masses superimposed so that the common surfaces are horizontal planes (Art. 16) they will become so intimately mixed as to form a new fluid. In this case, if the fluids be incompressible, the specific gravity of the compound will be known when that of each of the composing fluids is so. For let F^, be the volumes of the different fluids thus mixed together, 8^, their respective specific gravities; Fj+ Fg + ... + F^ is the volume of the resulting mixture, and if 8 be its specific gravity, since the weight of the whole must equal the sum of the weights of the parts (K+ F,+...+ K)8= V,8,+ K8,+...-h FA, and therefore Cr_ ^1^1+ F2>S2+...+ Ki^n K+K-h + EXAMPLES. 31 EXAMPLES TO SE^b5^v^i[L^’^‘^’^7'f ^ (1) Let ABC be a rigid pipe of small bore eating at C with a vessel D CB, whose top DE is moveable up and down by some means which allows of the vessel remaining water- tight: it may be a piston fitting closely to a cylinder, or it may be more simply a board connected with the bottom by leather sides. If the whole of this be filled with water, it is found that a man may easily support himself upon DE^ by merely closing the top of AB with his finger, or he may even raise himself by blowing into AB from his mouth. This phenomenon is sometimes called the Hydrostatic Para- dox: explain it. When the man applies his finger to a 4, he presses the surface of the water in the pipe with a certain force, which (by Art. 7 ), is thence transmitted through the fluid to every portion of surface in contact with it : if then a be the cross section of the tube at A and P the force he applies, a force equal to P will be transmitted to every portion of DE which is equal to a ; but if A be the whole area of DE^ it contains A — such portions ; therefore the whole force applied upwards A to DE=~P^ which maybe quite large enough to support the man’s weight, although P is small, provided the area a A be small compared with A, and therefore the fraction — a a very large number. 32 INELASTIC FLUIDS. If P be increased beyond this previously supposed value, by blowing or otherwise, the man will evidently raise himself. Ex. Find P that it may just support the man’s weight W. (2) The pressure at a point P within a body of water, under the action of gravity only, is 50 lbs., given that tlie 'weight of a cubic foot of water is 1000 oz., and that the unit of area is a square foot, find the depth of P below tlie surface. Let 2 ; be this depth in feet, then (by Art. 13) if p be the density of the water and^:) the pressure at P: wei 2 :ht of unit of volume of standard substance p=gpz'>i — ^ ; by the question, considering a foot as the unit of length and water as the standard substance, and .*. p = 1, 50 lbs. = ^ X lbs. 16 4 or the required depth is - ths of a foot. (3) A cylinder is immersed in water in such a way that its axis is vertical and its top is just level with the surface ; find the total normal pressure upon its bottom and sides. By Art. (16), this total pressure is equal to the vreight of a cylindrical column of water whose base equals the area of the given surface pressed, and whose height is equal to the depth of the center of gravity of this given surface below the surface of the water. EXAMPLES. 33 But if T be the radius of the base and li the height of our cylinder, the area of the surface pressed is = area of base + area of sides = Trr^ + 27rrA. Again, the depth of the center of gravity of this pressed surface below the top of the water iTT^ , h + ^irrh . ^ irr^ + 27Trh ^ the column of fluid whose weight is sought has a volume = x irrVi + 27rr — irr^ + 2irrh ^2 = irr^h + 2irr — ; 2 / /. the pressure required is gp f Trr^A + 27rr — = irgprh (r + A), p being the density of the water. (4) A triangle ABG is immersed in a fluid, its plane being vertical and the side AB in the surface : if 0 be the center of the cir- cumscribing circle, prove that the pressure on the triangle 0 CA : pressure on triangle OCB :: sin 25 : sin 2A. P. H. 3 34 INELASTIC FLUIDS. The pressures on the two triangles will be to each otlicr in the same proportion as the product of tlie area of cacli triangle into the depth of its center of gravity below AB] (Art. 18). But if g and g be these centers of gravity, they will divide the lines Ok, 01, drawn from 0 to the points of bisection of AG and BG respectively, in the same proportion; therefore they will be in a straight line parallel to that joining h and I, and therefore parallel to AB. Hence the pressures required will be as the areas only, i. e. pressure on OGA : pressure on 0GB :: area OGA : area 0GB :: sin A 0(7 : sin ^0(7 :: ^ui2ABG : sin25A(7. Q. E. D. (5) A regular hexagon is immersed vertically in a fluid, so that one side coincides with the surface ; compare the pressures on the triangles into which it is divided by lines drawn from its center to the angular points. (6) A cylinder whose height is 4 feet is sunk in water with the axis vertical till its upper face is 805 feet below the surface and the pressure on the top is found to be 35 lbs ; find the pressure on the lower face, neglecting the pressure of the atmosphere. (7) A square is just immersed in a fluid of density 8, with one side horizontal and with its plane inclined at 60® to the vertical : given that a cubic foot of the standard substance weighs 1000 ozs., find the side of the square that the pressure on it may be 216 lbs. EXAMPLES, 35 (8) A vessel containing water is placed on a table; supposing the vessel of such a shape that only half the fluid is vertically over its base, what is the pressure on the base? Is this the pressure on the table? Explain your answer. The reasoning of Art. (4) aided by a reference to the second figure of Art. (14) will explain how a rigid surface may supply the place of a vertical column of fluid. The rigidity is the result of internal forces and does not affect the pressure on the table. (9) The same quantity of fluid which will just fill a hollow cone is poured into a cylinder whose base equals that 4of the cone : compare the pressures on the bases, the axes of both vessels being vertical. If the cone and cylinder be resting on a horizontal plane state how the pressures on this plane will be affected, and explain the case fully. (10) Suppose a pound weight of a substance twice as specifically heavy as water to be hung into the water con- tained in a vessel, which is standing on a table, by a string not attached to the vessel, what would be the increase of pressure on the table ? (11) A cylinder of given radius, height, and specific gravity, is partially immersed with its axis vertical in water, being held up by a string which is attached by one end to its top, and by the other to a fixed point vertically above the cylinder: supposing the string to stretch 1 inch for every 5 lbs. which it supports, and that its unstretched length a feet, just allows the bottom of the cylinder to touch the water, and that a cubic foot of water weighs 1000 ozs., find the depth of immersion. 3—2 36 INELASTIC FLUIDS. Let tills depth he ^ feet : also let li he the height of the cylinder and r the radius of its hase in feet, c its specific gravity. Then the volume of water displaced hy the cylinder is irr^z cubic feet, and therefore the weight of it, which is the same as the resultant of the fluid pressures upwards upon the cylinder, must he (Art. 19) = irr^z 1000 16 Ihs. Also since the string is stretched ^ feet, its tension must hy question = 12^ X 5 Ihs. Now these two forces, each acting vertically upwards upon the cylinder through its center of gravity, and the weight of the cylinder itself acting vertically downwards through the same point, are the only forces which are acting upon the cylinder, therefore, for equilibrium it is only neces- sary that the sum of the first two equal the last, hut the weight of the cylinder = irr^h x cr x lbs. /. z ITT A 000 16 = TTGr^h X 1000 ^ 16 ’ or ^ = ircTT^h 7TT + 25 feet. It should be observed that all the symbols here used are necessarily by the statement numerical quantities. (12) What weight is just sufficient to hold down a bal- loon containing 2500 cubic feet of hydrogen gas (specific gravity .069 referred to air) supposing the weight of the EXAMPLES. 37 material inclosing the gas 4 lbs. and the weight of a cubic foot of common air 1.1 oz. ? (13) A cylinder which floats in water under an exhausted receiver has ^ of its axis immersed; find the alteration in the depth of immersion when air, whose specific gravity is .0013, is admitted. (14) A cone 7 inches in height and 2 inches in diameter at its base is attached to a hemisphere of equal diameter : the specific gravity of the cone is 1.5, that of the hemisphere is 1.75, find the specific gravity of the fluid in which this com- pound body will sink to a depth of 3 inches with the vertex of the cone upwards. (15) When 30 ozs. of an acid A whose specific gravity is 1.5 are mixed with 35 ozs. of an acid B whose specific gravity is 1.25, and with 35 ozs. of water, the specific gravity of the resulting mixture is found to be 1.35; find the con- traction of volume, assuming the specific gravity of water to be 1, and the weight of a cubic foot of its volume to be 1000 ozs. The volume of 35 ozs. of water = 35 1000 cubic feet, 30 A 30 1.5 X 1000 35 B 35 1.25 X 1000 Also the volume of the 100 ozs. of mixture 100 1.35 X 1000 cubic feet ; 38 INELASTIC FLUIDS. the loss of volume is , 80 , 35 100^ 1 TP, + rs + 05 - LssJ BoS (16) A man whose weight is 168 lbs. can just float in water when a certain quantity of cork is attached to him. Given that his specific gravity is 1.12, that of cork .24, and that of water 1, find the quantity of cork in cubic feet, assuming a cubic foot of water to weigh 1000 ozs. Let V be the required number of cubic feet of cork, then Fx .24 X + 168 is the weight of the man and cork to- 16 gether in lbs. By the question this must be just equal to the weight of the same volume of water; and the volume of the man in cubic feet is , ? because each cubic foot of him 1.12 X 1000 weighs 1.12 X 1000 ozs. (Art. 11), and his whole weight is given to be 168 lbs. ; 168 X 16 \ 1.12 X loooy 1000 16 lbs.= Fx .24 X ^ + 168^ lbs., lb y F(l-.24) 1000 168- 168 1.12 .^X 168 . 16x12x168 _ 16x12x1^ 112 X.76 X 1000 ~ 112x760 _6 X 168 _6 X 42_ “ 7 X 380 "7x95’ F= .37895 of a cubic foot. EXAMPLES. 39 (17) If in a circular to occupy 90° each, and if the diameter joining the two open surfaces be inclined at 60® to the ver- tical, prove that the den- sities are as Vs + l : Vs -1. Let A CBD be the tube; AD^ BD the por- tions of it occupied by the two fluids whose specific gravities may be repre- sented by p and p respec- tively : if the diameter A OB be drawn, it will be inclined to the vertical at an angle 60®. Let the horizontal line through i), the common surface of the two fluids, meet the verticals through A^ 0 and B respectively in JV, and P, and draw Af^ Bg perpendicular to ON. Then, by Art. 33, p _ BP _ ON Og _ OD sin 60® + OB cos 60® p AM ON — Of OD sin 60® — OA cos 60® _ tan 60® + 1 ~ tan 60® - 1 Vs + 1 Vs - 1 ’ (18) Equal volumes of oil and alcohol are poured into a circular tube so as to fill half the circle, shew that the common surface rests at a point whose angular distance from the lower point is tan"^ ; the specific gravities of oil and alcohol being .915 and .795. tube two fluids be placed so as 40 INELASTIC FLUIDS. (19) A body P weighs 10 lbs. in air and 7 lbs. in a fluid jI : if it be attached to a denser body Q and then suspended in another fluid jS, the apparent weight of both bodies is 5 lbs. less than that of Q alone j compare the specific gi’avitics of A and P. (20) A cone floats in fluid with its axis vertical, the ver- tex being downwards and half its axis immersed ; compare the specific gravity of the cone with that of the fluid. (21) The specific gravities of sea-water, olive-oil, and alcohol are 1.027, .915, and .795; the oil and alcohol have depths one inch and two inches above the water. Find the pressure on 3 square inches of a plane surface which is im- mersed horizontally at a depth of 5 inches below the upper surface of the oil : the weight of a cubic foot of distilled water being 1000 ozs. (22) If s be the specific gravity of a body whose bulk is n cubic inches and weight m ozs., then m X 1728 = 1000 x n x s, (23) The mean specific gravity of a plated cup is 7.6 ; that of the silver is 10.45; that of the unplated metal 7.3 ; compare the volumes and weight of the metals. (24) The specific gravity of zinc is 6.862, what is the weight of the water displaced by a portion of it, which when immersed weighs 5.862 lbs. ? (25) The volume between two successive divisions of the stem of a hydrometer is fh part of the bulk of the whole instrument; it floats in water with 20 divisions above the surface ; find the least specific gravity of a fluid in which it will float. EXAMPLES. 41 (26) A hydrometer that weighs 250 grains, requires 94 grains to sink it in water to the requisite point, and 8 grains in naphtha ; when a substance is placed successively in the upper and lower cup, 1^ grains and 14 grains are respectively sufficient to sink the instrument in naphtha to the requisite point ; required the specific gravity of the substance. (27) How many inches are there in the edge of a cubical mass of coal which weighs 2 tons, its specific gravity being I. 12 ; and what is the specific gravity of silver one cubic inch of which weighs 6.1 ozs. ; also what is the weight in ozs. of 30 cubic inches of mercury, its specific gravity being 13.6 ? (28) If be the apparent weights of a body when weighed in three fluids whose densities are respectively p^ pg, shew that ip, - P,) + ^2 ipl - Ps) + ^3 ip2 - Pi) = (29) Two metals of which the specific gravities are II. 22 and 7.25 when mixed in certain proportions without condensation form an alloy whose specific gravity is 8.72; find the proportion by volume of the metals in the alloy. (30) A small vessel when entirely filled with distilled water weighs 530 grains ; 26 grains of sand are thrown into the vessel^ and the whole then weighs 546 grains. Shew that the specific gravity of the sand is 2.6. (31) A crystal of saltpetre weighs 19 grains: when covered with wax (the specifie gravity of which is .96) the whole weighs 43 grains in vacuo and 8 grains in water. Shew that the specific gravity of saltpetre is 1.9. (32) 37 lbs. of tin loses 5 lbs. in water, 23 lbs. of lead loses 2 lbs. in water, a composition of lead and tin weighing 120 lbs. loses 14 lbs. in water, find the proportion of lead to tin in the composition. 42 INELASTIC FLUIDS. (33) A solid liemisphcre turning round a fixed liorizontal axis fits into a fixed hemispherical cup : shew tliat if the hemisphere he turned through any angle, and the cup then filled up with fluid of double the specific gravity of the solid, the solid will rest in that position. Let ADB be a section of the hemispherical cup made by the plane of the paper, perpendicular to the fixed axis about which the solid turns, C this axis, and HDBK' the solid turned through any angle A CH. If the part AHO of the cup be now filled up with fluid whose specific gravity is double that of the solid, equilibrium will be preserved. For it is manifest that equilibrium would obtain, if the space HGK were filled up with a portion of the same sub- stance as the solid hemisphere. Also since the centers of gravity of the two figures, HCA, A OK are necessarily in the same vertical line, the effect oiHCK in producing equilibrium must be the same as a uniform solid HCA whose weight is equal to the sum of the weights of HCA and ACK together, i. e. as a uniform solid II C A having double the specific gravity of the given solid. But since AG\^ horizontal no new circum- stances affecting the pressure on II G would be introduced by EXAMPLES. 43 supposing tins solid to become liquid, which would produce the proposed case: hence the truth of the proposition. (34) Is it advantageous to a buyer of diamonds that the weighing of them should be made when the barometer is high or when it is low, supposing their specific gravity to be less than that of the substance used as the weight ? (Art. 30.) (35) A sphere of weight W with its center of gravity bisecting a radius floats in a fluid: TV' is the weight of a volume of the fluid equal to the volume of the sphere, shew that if TV' > 8 TV, there is a point within the sphere where a weight may be placed so that the sphere may float in any position with half its volume immersed. (36) Find the vertical angle of an isosceles triangle in order that when floating with an angle at the base downwards in any fluid of greater specific gravity than itself, the opposite side may be horizontal. (37) A cylinder (s. g. a) floats with its axis vertical partly in one fluid {s, g. crj partly in another [s.g. (7, which is carried uj) or down by the ascent or descent of A. At II is a pipe whose orifice is covered by a valve opening into the large cylinder GH and which leads into a smaller cylinder LM: in this cylinder works a piston L by means of a rod and at the bottom is a pipe leading to a reservoir of water and covered by a valve M which opens into the cylinder LM, Suppose both cylinders to be filled with water and the valve M to be closed; if then a force be applied to the piston i, it will be transmitted through the fluid to all surfaces in contact with it, and therefore to the lower surface of by ELASTIC FLUIDS. 65 this means A will be pushed upwards, and any substance placed upon the platform may be pressed against a fixed framework DJ^^; when L has arrived at the bottom of the cylinder it can be drawn back, the water in GH will then be prevented from returning by the valve and water will also be pumped through M into the cylinder LM] L may now again be forced down, and therefore A be raised higher, until the substance between BC and jDiThas been sufficiently com- pressed. This machine may also be employed for the purpose of producing tensign, in rods and chains, &c., by rigidly at- taching a piston-rod to the lower part of A, which should pass through a water-tight collar in the bottom of the cylinder GF, and carry a ring at its outer extremity, to which the rod or chain to be strained may be connected. The pressure exerted by A may at any instant be removed by unscrewing a cock at (?, by which means the water is allowed to escape. Let W represent the force with which A is pressed up- wards, during a stroke of L downwards, made under the action of a force F] we may consider these forces as just balancing one another, and the pressure at all points of the water to be uniform, as the effect of gravity may be neglected in comparison with the pressure transmitted from F and W\ let this be represented byy>; if then r be the radius of the lower end of A^ r the radius of the surface of L, we have, W = pressure on end of A =p7rr^^ F— pressure L—pirr^] If, moreover, as is usually the case, F is produced by the aid of a lever, whose arms are represented by a and a\ and the power by P, p. II. 5 66 ELASTIC FLUIDS. F~ a’ P . L' W~ a P • This ratio may be rendered excessively small by reducing a' and r as compared with a and r, and the only limit to the enormous force which this machine may be made to exert, is put by the strength of the materials of which it is framed. It will appear, on considering the result of Art. 69, that the larger cylinder is required to be the stronger. The Si])lion, 52. If a bent tube, as ABC^ in the annexed figure, be filled with water, and then, both its extremities being closed to pre- vent the escape of the water, if it be inverted and one of its ends A immersed in a vessel of water whose surface is exposed to atmospheric pressure, while the other C remains outside the vessel at a level lower than the surface D of the water in the vessel, and if, when in this po- sition, the ends A and C be opened, the water will be observed to flow continuously from the vessel, along the tube and out at the extremity (7, until the surface D has been lowered to the level of Q or of A, if G be lower than A. A bent tube so employed is called a siphon. ELASTIC FLUIDS. 67 The explanation of this phenomenon is as follows: If in the side of a cylinder GH containing water, an orifice F be made below the C level of the free surface OF, the water will of course flow out; for the parti- cles of fluid inside at F are pressed outwards with a fluid pressure equal to the pressure of the atmosphere at the level GE together with that due to the height EF of water j while they are pressed inwards by the pressure of the air at F equal to the pressure of the atmosphere at the level E together with that due to the height EF oi air ; it is therefore on the whole pressed outwards with a force proportional to the depth EF^ the proportion being the difference between the specific gravities of water and air. The same would be true for a vessel of any shape, because, whenever a fluid is con- tinuous throughout a vessel, and is acted upon by gravity alone, the pressures at the same level below the surface are the same in every portion of the fluid. In the first figure of this Article, the water in the tube and in the vessel forms one continuous mass, hence the pressure at all points in the same level, wherever taken, must be the same, and therefore the fluid pressure in the tube at G must be the same as the fluid pressure in the vessel at 0\ G and G' being in the same horizontal line: if then G be lower than the free surface D, the water will flow out at (7: by the removal of each particle of water, all resistance to the motion of the next behind it, under the pressure to which it is subjected, disappears, and thus a continuous stream will be produced towards (7. If we further consider the pressure at different points in the tube AJBG, for all those which are in the leg EG helow the level of D it will, from what is said above, be greater than that of the outside atmosphere^ and therefore if an aperture were made at any such point, the water would flow out as it 5—2 E 08 ELASTIC FLUIDS. does at 0; for all those, however, whicli are above tliis level in either leg, as for instance in any level />’, it must be less than that at J) by the amount of the pressure due to the weight of the intervening column D'J) of water, while the pres- sure of the exterior atmosphere in the same level differs from that at D by the weight of the same height of air only: hence the pressure of the fluid in the tube at the level I)' is less than the corresponding pressure of the external air, and if therefore an aperture were made at any such point JX, the air would flow in and drive the water out alon a — b' (2) If a cylinder be full of air at atmospheric pressure, and a close-fitting piston be forced in through of the length of the cylinder by a weight of 10 lbs., shew that it will require an additional weight of 30 lbs. to force it through f of the length. If n be the pressure, estimated in pounds, of the air in the cylinder at first, i. e. be the pressure of the atmosphere, the pressure after the given compression into f of the volume is, by Boyle’s law, fll; therefore if A be the area of the piston, the pressure upon it upwards is f nJ., and by question this must balance the pressure of the external atmosphere together with lOlbs. acting downwards, hence fn^ = n^ + io, or UA = 20 lbs. ; and therefore -§ FI^ = 30 lbs. This pressure will be doubled, by the same law, when the compression is extended to another of the cylinder; there- fore the compressing force must be GOlbs.; hence 30lbs. must be added to that which is already acting, q.e.d. (3) A bubble of gas ascends through a fluid, whose free surface is open to the atmosphere, and whose specific gravity is . 9 ; supposing the bubble to be always a small sphere, com- pare its diameters, when at depths d and d\ having given that EXAMPLES. 79 the height of tlie barometer is h and the specific gravity of mercury cr. Since the bubble is always small, the pressure of the fluid upon its surface may be considered to be everywhere the same, and to be that which is due to the depth of the center of the bubble; also since this pressure is the only force limiting the volume of the bubble, it must be exactly equal to the elastic force of the air forming the bubble. Now the pressure of the fluid at the depth d is c^/^ + sd, d\.. Gh-\-sd' ] and the elastic force of the air in the bubble varies inversely as its volume, that is, inversely as the cube of the diameter; hence if a, a be the diameters corresponding to the depths of d, d' we must have _(y}i sd ah+sd'^ which gives the required ratio between the diameters. (4) An imperfect barometer is compared at two different times with a true one, and it is found that the readings are less than the true readings by the quantities e^, e.^ respect- ively. Shew that the true reading may at any other time be obtained by adding to the observed reading h the correction e^{k - h^) + - h) ‘ The error in the height of the mercury must be due to the presence of a certain quantity of air in the upper part of the tube, whose elastic pressure supplies the place of the weight of the deficient length of mercury; this pressure will not be con- stant, but will continually alter, for, the quantity of air re- maining the same, its volume diminishes as the barometer rises, and increases as it falls. 80 ELASTIC FLUIDS. Let ccj, a;.^, X be the length of tube occujiied by it wlien tlie barometer is at height \ \ and h respectively, p,, p its cor- responding densities, and h the proportion between tlie pressure and the density of air, (Art. 38 ), tlien, by question, tlie pres- sure of this air, when included in the length must be equi- valent to the weight of a length of mercury; hence if ; R = creoAA' cos (f). But acoAA' is the weight of the column of fluid AA', let it be represented by w, then P — cos (I). This result is true for all the columns parallel to AA' which have both their extremities meeting the sides of the vessel ; let aa' in the figure be a column, one extremity only of which, i. e. a, meets the side, the other a' is made by the surface of the fluid, and let the same letters, as above, be used with the same meanings in this case, then P' = 0, and 6' = ; GENERAL PROPOSITIONS. 89 /. i? = Pcos (n + o'a5)acos 0 = IV a cos 6' + crahco = Ua' cos ^ + crcoaa cos (f> = {UcV + w) cos(f> (II). Now the magnitude of the resultant required of the re- solved forces is the sum of all the P’s for which (I) holds, together with the sum of all the P’s for which (II) holds, i. e. it is the sum of all the quantities {w cos (/>), together with the sum of the quantities Hof cos as long as a' is the section of the cylinders made by HK\ but the sum of all these sec- tions is the whole area of the surface itself, let it be repre- sented by A, and the sum of all the ^(;’s is the weight of the fluid contained in the vessel, let this be W : we thus come to the conclusion, that the magnitude of the resultant of the resolved parts in directions parallel to AA' of the fluid pres- sures upon the sides of the vessel containing the fluid is (n^ + IT) cos r, but p — (jz\ 7 h ~ , z. But by the question t must be eonstant, and ar is so, h cc. z ] or the thickness must vary as the depth. (69 a). By a similar investigation to that of [Art. 69) we can find the tension at any point K of a spherical surface con-- taining fluid j the fluid pressure at that point being p. Take ah V a ^ a very small rectangular ]iortion of the surface having A as its middle point, and let 0 be the center of the sphere, the tension at A may be resolved into two, at right angles to each other, both in the tangent plane at A\ let that parallel to ah be represented by T, and that parallel to aa' by T\ these may be assumed to be equal on account of the symmetry of the surface about A : then the very small surface ah' may be considered to be kept in equilibrium by the normal pressures at every point which are approximately equal to y), the tangential tensions at every point of a a' and hh' approximately equal to and those at every point of ah and a'V approximately equal to T'. Now the resolved parts along AO o{ the tensions T at the points in a a' and hV GENERAL PROPOSITIONS. 101 ad . ah' AO approximately, by the last proposition, and similarly resolved parts along AO oi the tensions of ah and ah' Tdh' , ad AO approximately ; the sum of these = 2T aa ,ao ~AO approximately. Again, the resolved part of the normal pressure in the same direction =paa', dh' very nearly. But these two resolved parts must always counteract each other ; therefore, since the above expressions for them are true in the limit, AO’ or if, as before, r denote the radius A 0, T — \])T. ( 69 ^a). If y) be the same throughout the vessel, Tmay be found shortly by a method similar to that of Art. (69*). Suppose the sphere divided into two hemispherical shells by a plane through the center. Either of these is in equilibrium under the action of the tension throughout the annular band of junction, and the fluid pressures on its concave surface. If h be the thickness of the shell, r the radius, t the measure of the tension, supposed uniform throughout, andy) the pressure, we have, as in (Art. 69*), t . 27rr . h =y>7r. r^. , vr or t , k = ^ . A 102 GENERAL PROPOSITIONS. It must be remarked, that if the surfaee be pressed outside as well as within, by a fluid, the p whieh enters tlie above equations is the difference between these two normal pressures at the point A. It appears from these formula), that when p is constant, T and r are proportional to each other, or that the tension which a given normal pressure calls into action at any point of a cylindrical or spherical surface containing it is proportional to the radius. 70. In the description of the Barometer given in Article 36, no explanation was offered of any method of measuring the length of the column of mercury; a fixed scale of gradua- tions would manifestly not answer the purpose directly, as both extremities of the column necessarily shift at once: a descent of the mercury in the tube must cause the surface of the mercury in the basin to rise, and the sum of these two de- crements of length makes up the total decrement; and simi- larly in the case of an elongation of the column. Now a fixed scale could only mark the absolute alteration at one end, say the upper C\ of the column, but by a simple arrangement this may be made to indicate the total alteration. For, sup- pose G to sink d inches, as measured by a fixed scale; then, if a be the cross section of the tube, a volume da of mercury must thus descend into the basin, and the corresponding rise of the surface of the mercury in the basin must be due to this increase of volume ; or supposing this rise to be d’ inches, and the cross section of the basin to be -4, we must have da — d' A ; hence the total diminution of the column, being, as before said, d-{- d’ GENERAL PROPOSITIONS. 103 But it is clear that if the fixed scale he divided into equal parts, each of which is to an inch as 1 : q- , the length d inches will contain d(^ + ^ of these, and hence the number of such graduations observed upon the length d inches, through which C has sunk, will be the real number of inches by which the column has been shortened. It only needs that the real length of the column should be practically ascertained by measuring, when C is opposite any known graduation of this scale, and be there registered, in order that all future lengths be ascertained by inspection. 71. If the section of the tube be very small compared with that of the basin or lower vessel, will be a small ’ A fraction, and d-^d' will differ inappreciably from d\ this is the case with most barometers in common use, so that the graduations on their scales are made without respect to the considerations of the previous article. 72. In some barometers the bottom of the basin contain- ing the mercury {FG in fig. of Art. 36) is adjustable by means of a screw, and thus at the time when an observation is required to be taken, the whole mass of mercury can be raised or lowered until the surface DE is brought to a fixed level with regard to the instrument; an ivory pin, projecting from the side DF and pointing downwards, is generally used to mark this level, and the mercury can be easily brought into it by turning the screw until the image of the ivory point in the surface is made to coincide with the point itself: the fixed scale then measures upwards from this point. 104 GENERAL PROPOSITIONS. 73. The Wheel or Siphon Barometer differs slightly from those previously described: instead of tlie end A (fig. Art. 80) being plunged into a vessel of mercury it is bent round ; the second branch so formed is similar to BA in the fig. of Art. (37), and has its extremity open to the atmosphere; by this arrangement the column of mercury in the second branch, to- gether with the pressure of air on its surface, balances the column of mercury in the first; and therefore tlie difference between the lengtli of these columns is the same as the h of Art. (36). If the tube be uniform throughout, the variation of either column is exactly half the total variation of h ; it is therefore only necessary to observe the variation of the open one. This is done by attaching one end of a light string to a small body which is allowed to float on the surface of the mer- cury, and the other, after the string has been passed over a small pulley, to a weight less than that which would be suf- ficient to balance the float : then as the float rises and falls with the mercury, the pulley is turned round by the friction of the string, and an index needle fixed to it is made to traverse a sort of clock-face : if the circle of the face be large and carefully graduated, any very small motion of the float will be indicated and measured by the extremity of the needle. EXAMPLES TO SECTIOX lY. (1) A cylinder, the radius of whose base is 1 foot and whose weight is lOOlbs., is filled with water, a cubic foot of which weighs lOOOozs. : if the cylinder be inverted on a smooth horizontal table, find the greatest number of revo- lutions per second which the water may make about the axis of the cylinder consistent with no escape of water. EXAMPLES. 105 Let ABCD represent the cylinder inverted upon the smooth plane, and oc- cupied by fluid which is revolving about the dotted axis of the cylinder, say n times per second ; its angular velocity is therefore ^.27r, but we may for conve- nience call it 0 ). If the water were not revolving at all there would be no pressure upon BG\ and again, if BG were taken away, during the revolution the water would rise at the sides until the surface took a parabolic form ; BG must therefore supply some force sufficient to keep the water down, and it must be itself pressed upwards by the same : when this upward pres- sure becomes by reason of the magnitude of the angular ve- locity greater than the weight of the cylinder, i. e. 100 lbs., ABGD will be lifted up and the water will escape below its edges. We want then to find this pressure corresponding to the angular velocity co. Suppose the sides of the cylinder to be produced upwards as represented by the dotted lin^- ^ figure, and suppose 5(7 to be a rigid plat^ Capable of being made, to slide in and out: now let BG ^e drawn out while the water is revolving, and a sufficient qT^iantityJof water be added to make the curve free-surface /% take the position given In the figure where the vertex h of the parabola is exactly where tfi^ mid- dle point oi BG Avas: if when tfegLis the case 5 G ibe again pushed in, its presence can produce im disturbance in the revolving water either above it or below it ; but by this means the lower part is quite shut off from the upper, which may therefore be removed and neglected; and then the lower is only under the circumstances which were proposed by the problem. We thus see that the state of the revolving fluid ABGD (and therefore its pressure at every point, because it is 106 GENERAL rROl’OSITIONS. SO at one point li) is the same wliether we suppose tlie ])ortIon of fluid fBli Cg to be superincumbent, or wliether we sujipose BC to be stretched rigidly across : hence the pressure which BC exerts downwards and which we wanted to find, is ex- actly equal to the weight of the volume of fluid fBhCg, Call this volume V. Now the volume of the paraboloid fhg is ^ of that of the cylinder (7^, therefore V whicli is the remaining lialf of the cylinder = Also since the latus rectum of the parabola fh = , (Art. 66), Bf= ~ Bh^ ; y_'T^Bl± ~ ^ 0 TTft) 4 X 32.2 cubic feet, because Bh is 1 foot ; weight of r- - ais. This, as before explained, is the force tending to lift the cylinder, and therefore the greatest value that « can have, without escape of water taking place, is when this just equals tlie weight of the cylinder, or when 4x32.2 or 1000 16 10 = 100 , = 1 , 7^V=16 X3.22; .*. log n — \ (log 16 + log 3.22 — 3 log tt) = 1 (1.20412 + .50786 - 1.49145) = 1 X .22053 = .110265 = log 1.289 ; /. 71 = 1.289. EXAMPLES. 107 Hence the water can only make 1.289 of a revolution per second, or about 1^, without escaping. When would a cover, turning upon a hinge and unsym- metrically loaded, be raised? (2) An Indian-rubber ball containing air has a radius a when the temperature of the air is 0^ (centigrade). Supposing the tension of the Indian-rubber = /jl x (radius)^, shew that the radius r of the ball when the temperature is f will be given by the equation r® TT + 2 fir — o = I + eL a TT + 2/ia e being the ratio (Sect. V.) between the increase of volume and of temperature for air at a constant pressure, and tt being the pressure of the atmosphere. Let ^ and p' be the pressures of the included air, p and p' its densities when the temperature is f and 0^ respectively, then we have (Art. 82), {l+et). 1> P Also, since the densities are inversely as the volumes. T p __ a p but (Art. 69* a), — iT)r = fir'^ ] p = ir 2fir ; similarly, p =^7r + 2fia ; 108 GENERAL rROPOSlTIONS. by substitution r" TT + 2/ir — q — = 1 ~l~ c.t, if TT + 2^a (3) A spherical vessel contains a quantity of water wliosc volume is to the volume of the vessel as n : 1 ; shew tliat no water can escape through a small hole at the lowest point, if the vessel and the water in it liave an angular velocity about the vertical diameter not less than ( f! [r(l — n^)) r being the radius of the vessel, and g the accelerating force of gravity. Let ABBE be the spherical vessel revolving with an angular velocity a> about its vertical diameter A CB\ the surface of the inclosed 'water will be a paraboloid (Art. 66), whose latus rec- jy turn is ^ ; let the upper part of this surface meet the sides of the shell in D and E : by the question no water must run out at a small hole at A, hence there must be no water lying upon the hole, or in other words, the angular velocity is least possible when the lower part of the surface of the water just passes through A, Now because J? is a point in the j)arabola DAE, whose 2r/ latus rectum is —2 > EXAMPLES. 109 GE AG. Also because AJ is a point in tbe circle ABBE, whose dia- meter is AB, GE^ = BG.AG-, BG^\ Again, the volume of the contained water is evidently the difference between the volume of the portion DAE of the sphere, and the volume DAE of the paraboloid; but this difference is equal to the volume of a sphere whose diameter is vol. of water vol. of vessel containing it AC^ = 2^""^ question; or AB-BGy abA 1 k 1 = AB * This is easily apparent if the slice which would be cut off from the whole sphere by any two horizontal planes very close to each other be considered, say that cut off by KLM in the figure, and a plane parallel to it at a small distance d, for the portion which belongs to the paraboloid is irLM^. d, while the whole slice is TT KM^. d : the difference between these is Tr{KM^ — LM^) d = TT {AM. BM-AM. BC) d = w AM {BM-BC) d = Tr AM. MC.d; but if AK'C be a section of the sphere described upon as diameter AM . MO = MK'^, therefore the difference between the portion of the sphere BDKAEy and the portion of the paraboloid DAE cut off by these two parallel planes is equal to the slice of the sphere OK' A cut off by the same planes ; and this is true wherever the planes be taken, and hence the above result may be deduced. no GENERAL TROPOSITIONS. substituting from above, This is the value of the angular velocity which makes the surface of the revolving fluid pass through A ; all less veloci- ties would make the surface pass above A, and all greater would make the surface, produced, pass below A ; hence the truth of the proposition. (4) A bent tube of very small uniform bore throughout, consists of two straight legs, of which one is horizontal and closed at the end, and the other is vertical and open. If the horizontal leg be filled with mercury, and the tube be made to revolve about a vertical axis passing through the closed end, so that the mercury rises in the vertical leg to a distance d from the bend, shew that the angular velocity is _ f2g[k-\-d) I being the length of the horizontal leg, and h the height of the mercury in the barometer at the time of the experi- ment. Shew also that the mercury will not rise at all in the vertical leg unless the angular velocity be greater than V 2gli EXAMPLES. Ill Let ABC represent the tube revolving axis through the closed end P the height to which the mercury rises in the open vertical tube BC, and Q the distance to which it recedes from the axis in the horizontal one, then, since the tube is uniform, AQ = PB = d; also by question AB= I, Suppose the pressure of the atmospheric air upon P to be replaced by the weight of ad- ditional mercury poured into the open tube BC, and let the upper surface of this mercury be at P\ then PP' = 7^. about a vertical Now it is evident that since the pressure is nothing both at P' and Q, these two points must be in the free surface of the paraboloid, which would have been formed by the revolution about the vertical axis through of a sufficient quantity of mercury contained in an open vessel, with an angular velocity exactly equal to that of the tube in the given case; call this velocity &>, and let the dotted line P'QK be the supposed surface: draw P'A' perpendicular to the axis. Then, since the latus rectum of the parabola P QK is (Art. 66) equal to we hav^ AQ‘ = %AK, CO PA'^ = %AK-, CO P - Act = CO or 6 ) = -j- (7) 112 GENKPwVL rROPOSITIONS. The least value of co for wliicli tlie mercury will rise in tlic tube BG is clearly that just greater than tlie value which makes the parabola of the surface pass through P' and when F will of course be at i?; but under these circumstances r = % li, or w' = 0 )" ' Q.E.D. (5) Two hemispheres of equal radius are placed in close contact, so that their common surface is horizontal ; the upper one is fixed firmly and communicates with an air-pump : find the least number of strokes of the piston in order that a given weight may be suspended from the lower hemisphere. The internal air must be so rarefied that the difference between its normal pressure upon the lower hemisphere and that of the external air upon the same shall be great enough for its vertical resolved part (Art. 63) to be equal to the given weight, together wfith the weight of the lower hemisphere. Ex. Given the pressure of the atmosphere is 15 lbs. per square inch, the volume of the sphere is 20 times that of the pump, the area of a great circle of the sphere is 2 square feet, and the weight to be suspended, together with that of the lower hemisphere, is 19f cwt. Also log 2 = .3010300, log 2.1 - .3222193. (6) A vertical prismatic vessel, closed at the base and filled with fluid, is formed of rectangular staves held together by a single string passing round them, as a hoop. Find the position of the string. (Art. 64.) (7) A cylinder, into which water has been poured, re- volves uniformly about its axis, which is vertical, bubbles of air rise from the base; shew that these will all converge towards the axis in their ascent. (Art. 67.) SECTION V. MIXTUEE OF GASES. — YAPOUE. 74. Boyle’s law obtains for a mixture of gases as well as for a simple one; indeed, air, upon which the first experi- mental proof of it was practised, is made up of two gases, oxygen and nitrogen, in the proportion of 1 to 4 by weight : the experimental facts upon which the more general form of this law is based are the two following : (1) Whenever two gases of different densities are allowed to come into contact with each other, they very quickly inter- mix and form a compound which is of uniform density throughout and in which any equal volumes, wherever taken, always contain the same proportion of the two component gases. The rapidity with which the homogeneity is attained increases with the difference between the densities of the two gases. It may here be remarked, that no such intermixture ever results from the combination of inelastic fluids unless they be of equal densities; and if it be produced among them artificially, however complete it may appear, as in milk and tea, the fluids will in time separate themselves, and lie super- imposed upon one another in the order of stable equilibrium. The case of fluids, whether elastic or not, which act chemically upon one another is not here considered. (2) If a gas compounded of given quantities of two different gases be put into a closed vessel, the pressure at every point of it, or its elastic force (Art. 34), is the sum of the two pressures which would be respectively due by Boyle’s law to the given quantities of the two gases, if inclosed separately 8 P. H. 114 MIXTURE OF GASES. in the same vessel and at the same temperature as the com- pound: thus, let U be the capacity of the vessel, II the pressure of the compound occupying it, and let the quantities of the two gases, forming the compound, be such that the pressure of the first when occupying a volume V is and of the second in a volume V' is then the pressure of the V V' first in the volume U would be ^nd of the second , V V and therefore our result is 11 = P-^+ P' — which takes the simple form iiu=rv+F r ( 1 ). Dr Dalton has very concisely stated this fact by saying that One gas acts as a vacuum with respect to another.” From this formula it can be immediately shewn that Boyle’s law holds for the mixture; for supposing the same quantities of the two gases had been put into a space U' in- stead of P, then the pressure would have been given by the equation n'P'=PF+P'F'. Now it is manifest that this latter compound is the same as the former, and therefore n and TL' are its pressures corresponding to the volumes U and but nP=n'p' HU' or , the same result as that which Boyle’s Law gives. (Art. 37.) 75. A gas and a liquid, when in contact with each other, and of such a character as not to act chemically upon each other, generally intermix in a partial manner : if there be several gases present a portion of each penetrates the liquid and pervades its whole extent uniformly; the amount of this portion is quite independent of the number of gases which may be so penetrating, but is always such that if it occupied MIXTURE OF GASES. 115 alone the volume which the liquid does, its density would hear a certain proportion to that of the same gas, which is left outside and also supposed to occupy its space alone, this pro- portion depending upon the liquid and the gas together. This experimental fact may he stated more generally as follows : if the volume of a closed vessel he F+ Z7, and the part ?7he occupied hy a given liquid, and if into the remain- ing space V any numher n of gases he introduced in any quantities whatsoever, a portion of each of them will pene- trate the liquid, and the ratio which the quantity of each gas remaining in V hears to the quantity of the same which per- vades the liquid in U will he independent of n : thus if Vp he the one quantity. Up' the other for a particular gas, it is always found, that p' = ~ p whether there he only one, or whether there he fifty gases submitted to the liquid at once; as mentioned above, p. will differ with different gases, hut is constant as long as the gas and the liquid which are referred to remain the same. To take an instance: this proportion is found to he xV for oxygen and water, hut only for nitrogen and water; hence assuming common air to consist of 1 part of oxygen and 4 of nitrogen, their densities would he in the same proportion if they occupied space alone, and therefore by our rule, the amount of oxygen absorbed hy a given portion of water in contact with air would he, when compared with the amount of nitrogen absorbed hy the same portion of water, as is : io or as 1 : 2. 76. In all our previous investigations the effect of heat in modifying the action of fluids has been left out of consi- deration, or rather has been supposed to he invariable: this supposition holds true whenever the circumstances of the case presume the temperature to remain constant, hut if changes 8—2 IIG MIXTURE OF GASES. take place in it, there will generally he corresponding changes in the hydrostatical properties of the fluids. It would seem from experiment, that a free mass, whether solid or fluid, never experiences a change in its temperature without undergoing some alteration in its dimensions; and conversely. We may say generally, that additional heat im- parted to a body causes it to expand, while the withdrawal of heat is followed by its contraction; the converse of this is equally general, that the forcible compression of a body makes it give out some of the heat which was necessary for its more expanded state, and the extension of it in the same way obliges it to absorb heat. Whenever, too, the temperature of any body after suffering any succession of changes returns to any particular state, it is always observed that the dimen- sions of the body return also to a constant corresponding magnitude. 77. By accurate observation of effects of this kind it appears also that interchanges of heat always take place between portions of matter, whether in contact or at a distance from each other, until a speeies of equilibrium has been attained; when this is the case the different portions are said to have the same temperature. These facts lead us to the means of estimating and defining different stages of tem- perature and of measuring the amount of its increase or diminution. Any instrument constructed for this purpose is termed a thermometer ; of these there are many kinds, but it will be sufficient, for the purpose of illustrating the above- mentioned principle, to describe the mercurial thermometer. MIXTURE OF GASES. 117 The Thermometer. 78. The construction of this instrument is based upon the fact, that for low temperatures the expansion of mercury under the action of heat is such that the increase of its volume is always proportional to the increase of the heat. It is merely a glass tube, developed into a bulb at one extremity; this bulb, together with a portion of the tube, is occupied by very pure mercury, and a vacuum is preserved in the remainder of the tube by the extremity being hermeti- cally sealed ; the bore of the tube is uniform and extremely small, and therefore a slight expansion of the whole mass of mercury makes a great difference in the length of the tube occupied. By the aid of graduations along the tube, the increase or decrease in the volume of the mercury con- sequent upon an alteration of temperature may be readily observed, and therefore the difference between temperatures compared. If the difference between two known temperatures, or any part of it, be taken as the unit of measurement of tem- perature, this instrument affords us the means of expressing any other difference of temperature in terms of it. The two temperatures taken for this purpose are those of melting ice, and of the steam of water boiling under an atmospheric pressure of 29.8 inches of the mercurial barometer. In some kinds of thermometers, as the centigrade, the xJo of the difference between these temperatures is chosen for the unit ; in others, as in Fahrenheit’s, the part is taken. To gra- duate the thermometer accordingly,- the instrument must first be plunged in melting ice, and the level of the mercury in the tube marked; it must then be submitted to the steam of water boiling under a given atmospheric pressure, and the level of the mercury again marked; the volume of the tube between these two marks must be divided in the centigrade into 100, 118 MIXTURE OF GASES. and in Falirenlieit’s into 180 ccj^ual parts; cadi of tlicsc parts is termed a degree. These two kinds of tlicrmomctcrs also differ in respect to the division on the tube at wliicli tlic numbering of the degrees commences ; in the centigrade the reckoning starts from the freezing point, and tlierefore 100^ indicates the temperature of boiling water, while in Fah- renheit’s the initial point on the tube is 32® below the freezing point, thus making 212® the boiling water point. These distinctions must always be carefully regarded in the consideration of the temperature as given by either ther- mometer ; for instance, 40® centigrade indicates a temperature which is greater than freezing temperature by 40 degrees of that thermometer, (i. e. by iw of difference between freezing and boiling); but 40® Fahrenheit is only eight of its degrees (i. e. of difference between freezing and boiling) above freezing point : and generally, if and C® be the corresponding numbers of degrees upon Fahrenheit’s and the centigrade thermometers respectively, which in- dicate the same given temperature, they must belong to graduations which divide the distance between the boiling and freezing point, on each thermometer, in the same propor- tion. Now jF® of Fahrenheit denotes a graduation (i^— 32) degrees beyond freezing point, and therefore one which cuts F — 32 . off — _ of the distance between boiling and freezing points, 180 while the graduation on the centigrade whose number is C, is G at a distance from its freezing point equal to of the length between boiling and freezing ; therefore jF-32_ O '180 ”Tbb’’ or 5(F-32) = 9(7; a formula connecting the numbers of those graduations on the two thermometers which correspond to the same temperature. MTXTUKE OF GASES. 119 In a similar manner might be investigated a formula for any other two thermometers whose mode of graduation was known. The advantage of Fahrenheit’s thermometer over the cen- tigrade and others is, that the degrees are small, and therefore fractional parts of them are the less frequently requisite in ob- servations, and that the commencement of the scale is placed so low that it is seldom necessary to speak of negative degrees. 79. The filling and graduating of a thermometer is an affair requiring great skill and precaution. The object to be attained in filling the instrument, is to introduce a quantity of pure mercury, which shall occupy, at ordinary temperatures, the bulb and part of the tube, leaving a vacuum in the remainder. The method generally adopted is somewhat as follows. The instrument is held vertically with its open end upwards. A small funnel-shaped vessel containing mercury is placed over this open end. The flame of a spirit lamp is applied to the bulb, which, increasing the temperature of the included air, increases its pressure, and some of it forces its way in bubbles through the mercury. If the lamp be now removed, the temperature of the air in the instrument falls, its pressure diminishes, and some of the mercury is forced in by atmo- spheric pressure to occupy the space of the air expelled. By continually repeating this process all the air may be dislodged and the instrument filled with mercury. If now the funnel be removed, and the instrument heated, the volume of the mercury will increase so that some will flow out at the open end. The heat is raised to the highest temperature which the thermometer can be required to indicate, and then the hitherto open end is hermetically closed. As the mercury cools, it sinks in the tube, leaving a vacuum above it. In graduating the instrument it must be remembered that the temperature at which ice melts seems to be absolutely 120 MIXTURE OF GASES. constant nnder all circumstances, but that at which water boils is not so, unless the pressure and hygrometrical state of the atmosphere is also the same. It ought, perhaps, to be here remarked that the expansion of the mercury measured by the thermometer is not absolutely that of the mercury itself, inasmuch as the tube too expands ; it is therefore the difference between these two expansions which the graduations give us : but this fact introduces no difficulty, because the expansion of the glass, like that of the mercury, is proportional to the increase of heat, and therefore the difference between the two expansions must also be pro- portional to it. The expansion of solids is not so often made use of for the measurement of temperature as that of fluids, both because it is much smaller in amount and is less easily measured. 80. When a thermometer is brought into the neighbour- hood of a medium or substance whose temperature is desired, its presence alters that temperature by the interchange of heat which immediately takes place, and which is indeed essential to the use of the thermometer; it is this new temperature which is the subject of observation, and not the original ; under ordinary circumstances, however, when the mass of the thermometer is small compared with that of the substance around it (Art. 87), there is no appreciable difference between them. 81. It is a curious circumstance with regard to water, that although it decreases in volume as its temperature di- minishes down to about 40^ Fahrenheit, for a still further diminution its volume increases again*. Some most impor- tant results following from this fact will be noticed below, Art. 111. * Recent experiments seem to shew that this property belongs to many other substances as well as water. MIXTURE OF GASES. 121 82. It is discovered by experiment that the following very simple law regulates the expansion of gases under heat: — the ratio of the increase of volume to the original volume is for all gases in the same proportion to the increase of temperature^ provided the pressure exerted upon them remains unchanged^ whatever that pressure he; thus if Fbe the volume of a gas at a given pressure and temperature, F' its volume at the same pressure, but at a temperature increased by then where a is a number which is the same for all gases, but varies, of course, with the magnitude of the degrees in terms of which t is measured. It appears then from this law and from Boyle’s law (Art. 37) (each of them deduced from experiment) that the density of gas depends both on the temperature and pressure. To find the relation between these three, suppose 0, p^ p, t corresponding values of the pressure, density, and tem- perature for two conditions of a given gas. Then the change of density from p^ to p would be produced, by first changing the temperature from 0 to t, and then changing the pressure from To T- Let po become p by the first change, and let F^, F^ be the corresponding volumes of the gas. Then by the law above enunciated, we have, since the pressure is unaltered. F-F - = a^, = at^ Po or, 1 P = — (1 4 - cut). Po Vo 122 MIXTURE OF GASES. We liave now a pressure ^7^, a density p, and a temperature t. Now increase to ^ 7 , the temperature remaining t. Then the density becomes p, and by Boyle’s law (Art. 38), P=R Ro P = - (1 + at). Po But whatever was the original density p^, for a tempera- ture 0, it was connected with the pressure by the relation Po = hol (Art. 38) ^Po Po (1 + at), j)=zlcp (1 + d), the required relation. It should be remembered carefully, that h and a are quan- tities which must be determined by experiment. h is found to differ for different gases; a is the same for all. An explanation of a method for determining k for air has been given in (Art. 38*). 83. We are unable to ascertain the amount of heat actu- ally contained by a body when exhibiting a given temperature, but by careful thermometric experiments we can discover how much heat is absorbed or given out by it in passing from one known temperature to another; the amount of this heat so ab- sorbed or evolved being estimated by the number of degrees to which it will raise the temperature of a given mass of water. The results of such experiments are, that for the same substance: (1) The quantity of heat required to be absorbed or given out, in order to produce a given increase or decrease respectively of temperature, is proportional to the mass: MIXTURE OF GASES. 123 (2) The quantity absorbed or given out by a given mass is in a constant proportion to the consequent increase or de- crease of temperature. 84. This may be concisely illustrated by saying that if m and m be two masses of the same substance exhibiting temperatures t and i respectively, and if r be the uniform temperature which these two masses united will attain, then (m + m)T — mt-\- mt\ Thus if any volume of water heated to 70^^ of any ther- mometer, be mixed with twice the same volume of water at 100^, the mixture will be found to have the temperature of 90^ by the same thermometer. 85. When we come to the consideration of different sub- stances, we find the constant ratio of law (2) is different for each : if g represent the quantity of heat absorbed by a unit of mass of distilled water in order to increase its temperature by one degree of heat, and g the quantity required for the same purpose by a unit of mass of mercury, q and q are very different; of course their absolute magnitudes would, by both laws, vary in proportion to the magnitude of the unit of mass and the degree of heat; but their ratio, instead r of being unity, is ~= .033 nearly: if q refer to spermaceti This general fact may be got at by a variety of experi- ments. For instance, if equal weights of quicksilver and water be mixed, the first having a temperature 40^, and the second 156^, the temperature of the resulting mixture is found to be 152^.3; it thus appears that the water has lost heat while the mercury has gained some, and it cannot be doubted but 124 MIXTURE OF GASES. that these quantities are equal ; but then tliis quantity abs- tracted from the water only diminishes its temperature 3^.7, while it raises the temperature of the mass of mercury, to which it is added, and which is equal to the mass of water, 112^.3. From this it may be inferred, that for raising a given mass of mercury 1® it requires the heat which will produce the same result in an equal mass of water. Again, if a pound of water at 100'^ be mixed with two pounds of oil at 50^, the resulting temperature will be 75^; therefore the same lieat which will lower 1 lb. of water 25®, will raise 2 lbs. of oil the same amount. 86. These quantities, q and q, are generally denominated the specific heats of the respective substances, and are most commonly measured in terms of the specific heat of some one substance, taken as the unit; if this one be distilled water, as is usually the case, we should have q\ the specific heat of mercury, = .033; and similarly for all other substances. 87. To include these results under a general formula, let T® be the temperature resulting from the combination of a mass m, say of mercury, at a temperature ^®, with a mass m\ say of water, at a temperature f] and let q^ q^ as before, be the spe- cific heats of mercury and water respectively. Suppose t to be greater than and C, estimated in terms of the same unit as q and q\ the quantity of heat lost by the whole mass of mercury and therefore gained by the water; since then it would require a loss of heat = mq to reduce the temperature of the mercury one degree, its actual reduction is (7® — ; similarly the increase of the water’s temperature must be pio , therefore we have m q MIXTURE OF GASES. 125 t = ^ 4- -7-7 = T, mq m q from wliicli results by the elimination of (7, [mq -t- m'q')T = mqt + m'q'if (a), . 9.' mit-T ) •• q ~ the equation (Art. 84) is evidently only a particular case of (a). 88. The foregoing definition of specific heat is not ex- tended to gases. It is found more convenient in such cases to speak of two kinds of specific heat : the first refers to the quantity of heat required to raise by one degree the tempera- ture of a gas contained by the rigid sides of vessels of con- stant volume : the second^ to be the quantity required to increase by one degree the temperature of a gas contained by the ex- tensible sides of a vessel of such nature that the 'pressure is maintained constant. 89. It has been remarked (Art. 76) that an alteration of the temperature is always accompanied by an alteration of the dimensions of a body, unless some constraining force be in action : it may be further asserted that all solid bodies may by a sufficient increase of heat be rendered liquid, and by a still greater increase, changed into vapour. The converse seems also to be true, that all elastic fluids will upon a with- drawal of heat become inelastic, and, if the process be con- tinued long enough, at length solid. In short, alteration of temperature in any substance is always accompanied by either change of volume or change of chemical character. Also with elastic fluids increase or decrease of pressure produces the same effects as decrease or increase of temperature re- spectively. 12G MIXTURE OF OASES. 90. The term vapour is usually applied to tliosc clastic fluids which at ordinary atmospheric pressure and temperature lose their elasticity. The laws connecting their cliangcs of temperature, pressure, and fluid state, are very important, tlic more so as one of them, steam, is now of such extensive application as a motive power. 91. When an inelastic fluid occupies a portion of an inclosed vessel, the remainder being at first a perfect vacuum, a certain amount of vapour is disengaged from the fluid and passes into' the empty space, until the space will hold no more at that temperature; this space is tlien said to be saturated with the vapour and the vapour itself to be at saturating den- sity: the temperature at which a given density of vapour saturates space is termed the Dew Point of that density : the origin of this term will be explained in a later article (Art. 110). If the temperature be now increased, without any alteration in the volume of the inclosing vessel, the elastic force of the vapour increases also : and it does so to a much greater extent than accords with Boyle’s law; this arises from the circumstance that additional vapour is generated by the fluid, and therefore the space in its new state of saturation corresponding to the new temperature contains more vapour than before. The law which connects the density of a vapour with its dew point is not simple, and need not be introduced here. 92. If the space containing the vapour be shut off from all communication with the fluid, an increase or decrease of temperature in the vapour will cause an alteration in its pres- sure, accordant with the law of Art. 82, down to the point when the temperature is just sufficient to retain the given vapour in an elastic state, i. e. down to the dew point of its density; at this point and below it, just so much of the vapour will be deposited in its liquid state as will leave the remainder at the required saturating density. MIXTURE OF GASES. 127 Again, if the temperature remaining constant, the space above the liquid be increased, sufficient vapour will be disen- gaged to keep the space saturated. Generally, then, if vapour rise into a void space, it will, whether in contact with its own liquid or not, maintain a given density at a given temperature, a portion of it, if necessary, becoming condensed : excepting beyond the limiting tempera- ture when its quantity is not sufficient to effect this, and then it follows Boyle’s law. 93. There is every probability that what are usually termed permanent gases are vapours whose saturating densities are very great for low temperatures, and thence arises the difficulty of reducing them to the liquid state. 94. The results which have just been stated with respect to the quantity of vapour which rises at .a given temperature into a void space from a liquid exposed to it, are also true wffien the space is already occupied by any other gases or vapours; the only difference is, that in this case time is re- quired to complete the saturation. 95. When a space is occupied by any number of gases and vapours together (these last being supposed at saturation density, otherwise they are, as far as we are concerned, gases,) the laws of Art. (74) obtain for the mixture: it is uniform throughout, and the pressure at every point is the sum of those pressures which would be separately due to the indivi- dual gases or vapours if they occupied the space alone. 96. The curious phenomenon of ebullition which takes place when water is heated to boiling point admits of expla- nation by the aid of the foregoing principles. When the vessel is placed upon the fire the particles of water next its 128 MIXTURE OF GASES. sides become lieatcd, rise to tlic surface, and tlicre give off their vapour: the colder particles whicli take tlicir places become Iieated in turn, and by a continuation of tliis process the whole mass becomes gradually heated; but so far the operation of heating is quite tranquil. At length the lower layers attain such a temperature, that the pressure of tlieir vapour at the corresponding saturating density is greater than that of the superincumbent fluid: this vapour therefore ex- pands, forms itself into bubbles and rises towards tlie surface ; but in its progress it comes into fluid of a lower temperature and is consequently suddenly condensed; this produces the bubbling noise and disturbance in the water which precedes the boiling. These heated bubbles of vapour however greatly accelerate the equalization of the temperature of the whole fluid : it finally becomes uniform, and the bubbles of vapour generated in all parts of the fluid pass out to the surface at uniform pressure, which must manifestly equal that of the atmosphere; the disturbance arising from the passage of these bubbles continues, but the crackling and bubbling has ceased and the boiling is completed. Ebullition, then, which is the criterion of boiling, occurs as soon as the temperature of the interior particles of the fluid becomes such that the pressure of their vapour at the corresponding saturating density is greater than the pressure of the surrounding fluid. We see from this explanation that the pressure of the atmosphere upon the surface of water, must materially affect the temperature at which water will boil, and must therefore be taken into consideration in the graduation of the thermo- meter. The height of the barometer is about 30 inches when boiling water has the temperature 212° indicated by Fahren- heit’s thermometer. (Art. 78.) 97. The preceding laws connecting the pressures &c. of a mixture of a gas with vapour may be exhibited in an alge- MIXTURE OF GASES. 129 braical form. Suppose the gas to be always in contact with liquid affording the vapour, and its quantity to be given, then if Fbe the space it occupies at temperature and if p denote what would be its pressure if it occupied that space alone, by Art. (82), ^^=•^(1 +«o (i)> because the quantity of gas remaining the same its density varies inversely as its volume. Now by Art. (74) if P be the whole pressure of the mixture, F the elastic force or pressure of the vapour at saturating density, P=p + F (2), therefore substituting above, Y_P{X +o.t) ~ P-F (3). 98. From the preceding remarks it may be collected that, when bodies change their state and dimensions, a portion of heat is always either absorbed or given out by them, and that this portion is not all accounted for by the consequent altera- tion in their temperature. In fact one part of the heat absorbed or given out is employed in producing mechanical changes in the body itself, i. e. in making it larger or smaller, or in altering its chemical nature, the other goes towards increasing or decreasing the temperature. The definition above given of specific heat takes the whole of this heat into consideration, while the first part is usually distinguished by the name of latent or insensible heat. 99. The quantity of heat which becomes latent in the passage of an inelastic fluid into the gaseous state is very great, all of which is absorbed by the vapour at the moment of its generation either from its parent fluid or from any solid 9 P. H. 130 mixtukp: of gases. body with which it may then be in contact, such as would be the case if the evaporating liquid were percolating any sul)- stance. The vapour and liquid, like all bodies which arc in contact, and which readily impart heat to each other, arc always of the same temperature^ and therefore during the evaporation, if no extraneous heat be supplied them, they will both gradually cool, and thus, if the pressure above the liquid be diminished sufficiently to keep it boiling notwithstanding the lowered temperature, a freezing of the liquid will actually be the result of its boiling. With the aid of the air-pump many experiments may be made to shew this remarkable phenomenon: but perhaps the most instructive of all is the following: If two bulbs of glass be connected by a bent tube and one be filled with water and then heated so that the vapour rises sufficiently to drive all air out of the tube through an orifice in the empty bowl, and if this orifice be then closed, the pressure of the included steam will after cooling be reduced to that which is due to the temperature of the air: if now the empty bulb be immersed in a freezing mixture the steam will have its pressure so dimi- nished that the water in the other bulb will immediately boil very rapidly, and the consequent vapour will carry off from it to the freezing mixture enough latent heat to convert the remaining water into ice: this apparatus was invented by Wollaston and called by him the Cryophorus, or frost-bearer. 100. This explanation fully accounts for the well-known cooling effect of evaporation. A sudden dilatation of a gas produces the same result, as does also the reduction of a solid to a liquid state, such as the thawing of ice. On the contrary, physical changes of an opposite kind cause the body to evolve heat : thus when quicklime is slaked and a species of concre- tion or solidification of the water thereby produced, con- siderable heat is given forth. Also in some neat contrivances MIXTUIIE OF GASES. 131 for lighting matches the necessary heat is produced by a sudden forcible condensation of gas, or compression of a solid. 101. Many meteorological phenomena, such as clouds, fogs, rain, dew, &c. are caused by changes of temperature taking place in an atmosphere charged with moisture. We are now in a position to offer some explanation of them. The water at the surface of the earth in contact with the atmosphere is continually giving off vapour, but not with sufficient rapidity, compared with the compensating causes, to produce general saturation: the portion of air nearest the earth’s surface is found at mean temperature to contain about half the quantity of vapour required to saturate it. It may be here mentioned that the ratio which the quantity of vapour actually present at any time in a portion of air bears to the quantity which would saturate it at the temperature then existing, is taken as the measure of the hygrometrical state of the air at that time. The instruments, of which there are various kinds, used for ascertaining this ratio are termed hygrometers. 102. The vapour formed at the surface of the earth has a much less specific gravity than the air, it therefore rises rapidly, and becoming in consequence exposed to a diminu- tion of atmospheric pressure, expands; this expansion cools the surrounding air, which was probably at a lower tempera- ture already than that near the earth; and it thus happens that at a certain height the vapour is very nearly sufficient to produce saturation; here, therefore, if any reduction of tem- perature is by any means effected, vapour must be condensed. Now the first form of condensation of a mass of vapour is that which is exhibited by steam issuing into the atmosphere and which makes it visible: the vapour seems to condense into little hollow spheres, or vesicles, having a nucleus of air in 9—2 132 MIXTURE OF GASES. tlie center, which owing to the evolution of latent heat is of a temperature certainly not less than that which the vapour had previous to condensation; eaeh sphere, together with its included air, possesses a speeific gravity which need not he greater than that of the surrounding atmosphere, and thus the whole mass will float in a visible form, which we call a cloud. 103. A still further diminution of temperature causes the vesieles of the clouds to collect into solid drops, which being necessarily of greater specific gravity than air, immediately fall to the surface of the earth and produce what is called rain. 104. Changes of temperature of the kind here supposed may be often attributed to electrical causes, but generally, no doubt, they result from the passage of a current of hotter or colder air. It is possible that masses of air, eaeh charged with vapour, should by meeting form a mixture, whose reduced temperature would require for saturation less vapour than that whieh they all together brought with them; this would cause the instantaneous appearance of a cloud, and, perhaps, rain; and would very well account for a phenomenon which is by no means uneommon. Any mechanical means which would bring a quantity of air already saturated into a space where a temperature lower than its dew-point obtains (Art. 110), would produce a manifestation of vapour or rain ; it thus happens that the tops of mountains are very generally capped with clouds ; for the currents of air charged with moisture, which are carried along by the winds nearly parallel to the surface of a level country, slide up the sides of any hills which they meet with, and are thus raised by them, as by inclined planes, into an elevation, where the temperature perhaps, for reasons mentioned in Art. 102, is always lower than it is below\ MIXTURE OF GASES. 133 105. Snoio results from the freezing of vapour at the moment of condensation, while hail proceeds from the freezing taking place after the drops have been formed, and during their passage, in falling, through a portion of air having a very low temperature. 106. Fogs and mists are only clouds in different states of density actually in contact with the earth’s surface. 107. Dew^ and hoar frost, and night-fogs are phenomena of the same class as the preceding: although their imme- diately acting cause is peculiar. They are produced by the earth and the objects near its surface becoming colder than the superincumbent air, and then acting upon it as a refrige- rator : the layer next the earth may thus have its temperature so much reduced that the vapour contained by it is too much for its saturation, the superfluous quantity will then be de- posited in the form of Dew; at the same time, a little higher up, the temperature may be only just low enough to exhibit the condensed vapour in the shape of a fog, while above this the air will be clear. If, again, the temperature of the earth under consideration be as low as freezing-point, the vapour will freeze upon condensation, and an effect similar to snow will be produced on the surface of the earth; this is hoar frost. It is sufficiently apparent that the air in valleys, and above streams and pieces of water, will receive more vapour in the course of the day than that elsewhere, and will there- fore be the more nearly saturated; hence it is that these places are the most favourable for exhibiting fogs, dews, &c. 108. It remains now to explain how it is that the earth so often becomes colder at night than the air above, and thus produces the effects thus described. There are three distinct modes in which heat may be transmitted from body to body. Conduction, Convection, and Badiation. It is conduction when 134 MIXTURE OF GASES. tlie heat passes from one part of a body to another, tliroiigli the intermediate partieles, or from one body to anotlier whicli is 'in contact with it; thus if one end of a metal wire be lieated in a eandle, the other will beeome hot by conduction; and both the warmth experienced when the finger is immersed in boiling-water, and the cold, when it touches ice, arc the effects of conduction; in the first case it communicates heat from the water to the finger, and in the other it takes from the finger to give it to the ice. Some bodies are so much better con- ductors than others, as to produce very different sensations of heat upon being touched by the hand, although the real tem- perature of all may be the same; all metals arc so in com- parison with wood, &c.; and of this fact practical use is made when ivory handles are attached to teapots. Silk and wool are well-known non-conductors, and are therefore admirably adapted for clothes, to prevent the animal heat from escaping too rapidly. When any fluid, elastic or not, after receiving heat, passes to another place, and there gives it up again, the process of transmission is termed Convection, Radiation is of quite a different character; it is a term used to designate the process of interchange of heat which is continually going on between all substances at all distances from each other. Each body seems to throw out its own heat in all directions around itself, just as light radiates from a lu- minous point, and to absorb that Avhich comes to it in the same way from its neighbours. It is an ascertained fact, that when a portion of the surface of any body receives heat from a source which is at the same temperature as itself, it absorbs exactly the same quantity which it radiates ; and thus when equilibrium of heat is once established amongst bodies sur- rounding each other, it will be always maintained, unless dis- turbed by some extraneous cause ; but if the portion of radiating MIXTURE OF GASES. 135 surface be opposed to no source of heat whatever, it will lose all the heat which radiates from it, and the whole of the body of which it forms a part will cool by conduction. 109. Our promised explanation is now easy, for it so happens that the radiating power differs extremely with dif- ferent substances, and that with air, as with all perfect fluids, it is almost zero; at the same time air presents little or no ob- stacle to the free radiation of heat through it : thus at night, when the sun and all extraneous sources of heat are removed, the surface of the earth and the bodies upon it will entirely lose just so much of their heat as radiates through the air into empty space, supposing no such objects as clouds, trees, &c., exist in that direction to return it ; they may therefore cool sufficiently below the temperature of the air with which they are in contact, to effect by conduction the results described. This explanation is confirmed by the extraordinary differ- ence in the quantity of dew which is observed to be deposited in the same place, upon objects of different kinds ; thus grass, and the leaves of most vegetables, glass, chalk, and generally bodies with rough or dense surfaces, all of which have a power of rapid radiation, will be covered with dew at a time when metals and many bodies with smooth surfaces lie almost dry by the side of them. 110. If a solid body be introduced any where into the atmosphere, and if its temperature be just below that point at which the vapour in the atmosphere around it is sufficient to saturate it, some of this vapour must be deposited upon its surface in the form of dew. Hence if the temperature of such a body be observed, as it is gradually made to cool down till dew is seen upon it, and if also it be observed when it is reheated till this dew disappears again, the real temperature at which the vapour in that particular part of the atmosphere 136 MIXTURE OF GASES. saturates it, and wliicli must evidently be intermediate to tliese two, can be approximately ascertained. This temperature is called the dew-point (a term which we have already used) for that portion of atmosphere at that time, and from it can be obtained the actual quantity of vapour present, by the aid of tables which connect the saturating density of vapour with the corresponding temperature. 111. The heating or cooling of liquids is almost entirely effected by convection, as their conducting powers are slight. This has been already assumed to be the case in the explana- tion given of the boiling of water, and is confirmed by the circumstance, that to make water boil by heating it from above is a process requiring much time: the upper particles are in this case the first to have their temperature heightened, and they do not then, as the loAver would under like circum- stances, give place to others, because their diminished specific gravity is a sufficient reason for their remaining where they are. When liquids cool from above, i. e. when they lose the heat which radiates from their upper smffaces, each layer of particles as it becomes cooler than the rest, and therefore has its specific gravity by the contraction of its volume increased beyond that of the liquid below it, sinks, and is replaced by the adjacent portions of the same liquid, which will in turn undergo the same fate ; thus the whole mass has its tempe- rature uniformly reduced ; and if this reduction goes as far as the freezing point, the whole will be congealed. If such were the result of the cooling of water, it would be most disastrous in the present condition of our globe : all aquatic life would be destroyed every time that a severe frost occurred, and a lake or river when once converted into solid ice would never melt again in our climates, for the heated MIXTURE OF GASES. 137 water resulting from the partial thawing of the surface, would not, for the reasons just given, convey downwards sufficient heat for the thawing of the remainder. Fortunately for us the law given above, by which water contracts, prevents these unpleasant consecjuences : after a volume of water has been reduced down to a uniform temperature of 40*^, upon a con- tinuation of the cooling process, its upper particles will cease to descend, and will soon become a sheet of ice, which by its non-conducting power materially preserves the remaining water from the further effect of cold. GENERAL EXAMPLES. (1) A thin conical surface (weight W) just sinks to the surface of a fluid when immersed with its open end down- wards : hut when immersed with its vertex downwards, a weight equal to m W must he placed within it to make it sink to the same depth as before : shew that, if a he the length of the axis, h the height of the column of the fluid, the weight of which equals the atmospheric pressure, Ct g f ^ =m^l -hm. It is evident from the question that the weight of the fluid displaced in the second immersion is to the weight of that displaced in the first :: 1 + m : m. But the volume displaced in the first case is that occupied hy the compressed air within the shell, and is therefore a cone whose axis may he represented hy z, while that in the second case is the volume of the whole cone ; now the volumes of these cones are in the ratio of the cubes of their axes ; therefore, 7=1 ( 1 ). z Again, the pressure of the air compressed Into the cone whose axis is ^ must balance the pressure due to the depth 2 : below the surface of the fluid ; and before compression when GENERAL EXAMPLES. it occupied the whole volume of the cone, its pressure balanced a column of fluid equal to li ; therefore z -\-]i h * hence, substituting in (1) z Ji ( 2 ), = 1 -f m, or ^ = mh^ and therefore again from (1) j = 139 (2) A hollow conical vessel floats in water with its ver- tex downwards and its base on the level of the water’s surface : it is retained in that position by means of a cord, one end of which is attached to the vertex and the other to the center of a circular disc lying in contact with the hori- zontal plane upon which the water rests ; given the dimen- sions of the cone and the depth of the water, find the smallest disc which will answer the purpose, neglecting the weight of the cord, cone, and disc. As there is no fluid under the disc the resultant fluid pres- sure upon it is the same as the total pressure upon its upper surface: it is therefore equal to the weight of a cylindrical column of the fluid having the disc for its base and the depth of the fluid for its height : since the horizontal plane can only exert a force of resistance, the smallest disc is evidently that, upon which this downward fluid pressure is just sufficient to balance the tension of the string upwards, when the cone is in the given state of immersion: but the force required to be exerted by the string in order to hold the cone in this position is equal and opposite to the resultant of the fluid pressures upon the surface of the cone, i. e. to the weight of the fluid displaced by it. Hence we conclude that the smallest 140 GENERAL EXAMPLES. disc required will equal the base of a cylinder of tlic fluid, whose altitude is that of the fluid and whose volume is that of the given cone. (3) Find the depth in water at which the pressure is 140 lbs. assuming the atmospheric pressure to be 15 lbs. the square inch, and an inch the unit of length. (4) A vertical cylinder contains four cubic feet of water, of depth nine inches; find the pressure in lbs. at any point in the base, considering four inches the unit of length, and assuming one cubic foot of water to weigh 1 000 ozs. (5) A body in the form of an equilateral triangle floats in water ; determine the condition to be satisfied in order that one angular point may be in the surface of the water and the opposite side vertical. The centers of gravity of both the triangle and the fluid displaced by it may be easily proved to be in the same vertical line ; the condition referred to therefore concerns the specific gravities of the triangle and fluid. (6) A pyramid with a square base and with sides which are equilateral triangles is placed on a horizontal plane and filled with a fluid through an aperture in the vertex ; find the pressure on one of the sides. If the pyramid have no base find its least weight con- sistent with its not being raised from the plane. (7) The surface of a man’s body contains 14|- square feet ; find the pressure on it when at a depth of 20 fathoms in salt water whose specific gravity is 1.026. State also how the resultant of this pressure might be found. (8) If a cubie inch of distilled water weighs 253 grains, and the specific gravity of salt water be 1.026, what will be GENERAL EXAMPLES. 141 the pressure on a square inch at a depth of 20 feet below the level of the sea ? (9) The lighter of two fluids (s. g. 1 : 2) rests to a depth of four inches on the heavier, a square is immersed vertically with one side in the surface ; determine the side of the square that the pressure on the portions in the two fluids may be equal. (10) A hemispherical vessel with its base horizontal is filled with fluid through an orifice at its highest point ; prove that the whole pressure on the curved surface equals that on the base. (11) A cylindrical vessel with its axis vertical is filled with equal masses of two fluids which do not mix ; compare their densities, supposing the pressures on the upper and lower portions of the concave surfaces equal. (12) A cone with its axis vertical and base downwards is filled with fluid; find the normal pressure on the curved sur- face, and compare it with the weight of the fluid. (13) A cubical box filled with a fluid of a given weight Wy is supported in such a position that one of its edges is horizontal, and that one of its sides passing through this edge is inclined at an angle a to the horizon; shew that the sum of the pressures on the six faces is equal to 3 JV (sin a + cos a). (14) An isosceles triangle has its vertex in the surface of a fluid and base parallel to it; find the pressure and center of pressure. (15) A figure bounded by the arc of a parabola AP, the tangent at the vertex AB and the line PP parallel to the axis is immersed vertically in a uniform fluid, with A in the surface and BP horizontal ; find the depth of the center of pressure. 142 GENERAL EXAMPLES. (16) If the side of a rectangle he horizontal and at a given depth hclow the surface of a fluid, determine the wliolc pressure on the rectangle; and shew that the center of pres- sure lies below the center of gravity of tlic rectangle. (17) A hemispherical bowl is filled with fluid, and differ- ent sections of it are taken through the same tangent line to its rim ; determine the section upon which the pressure is the greatest. (18) A portion of a paraboloid, of density p, cut off by a plane perpendicular to its axis, floats with its axis vertical in a cylinder containing two fluids, of densities p and Sp, which do not mix. Having given that the radius of the cylinder, the latus rectum, and length of axis of the parabo- loid are all equal, find the volume of the upper fluid when the two ends of the axis of the paraboloid project equal distances above and below the surface of that fluid when there is equilibrium. (19) ABC is a right-angled triangular plate, and it floats with its plane vertical and the right angle G immersed in water ; prove that if its specific gTavity be to that of water as 2 : 5, and CB : GA = 5:4, GB is cut by the surface of the water at a distance from G = GA. (20) One end of a uniform rod is attached to a hinge fixed in a mass of fluid; to the other is attached by a free joint the vertex of a cone which floats in the fluid. Given tliat the volume of the cone is 9 times that of the rod and the specific gravity of the rod 20 times that of the cone, find the specific gravity of the fluid in order that the cone may float with f of its axis immersed. (21) A right cylinder of radius a and height 2/^ floats in a fluid of double its density with one of its circular ends GENEKAL EXAMPLES, 143 entirely out of the fluid; shew that it can rest with its axis inclined at a certain anrfe to the vertical if V2 (22) A piece of zinc (whose specific gravity is 6.9) weighs 59 ozs. in distilled water and 61 in alcohol; find the specific gravity of alcohol. (23) A lump of metal weighs 59 ozs. in water and 61 ozs. in alcohol whose specific gravity is .8 ; find its weight and specific gravity. (24) A uniform cylinder when floating with its axis vertical in distilled water sinks to a depth of 3.2 inches and when floating in alcohol sinks to a depth of 4 inches ; find the specific gravity of alcohol. (25) A vessel, of weight W times that of the cubic foot of water, in sailing down a river leaks V cubic feet of water and is observed to be immersed to a given depth. On reaching the sea F' cubic feet are pumped out: and after F" cubic feet of sea water have been leaked, the vessel is observed to be immersed at the same depth as before: find the specific gravity of sea water. Obtain a numerical result, taking specific gravity of fresh water = 1, weight of ship = 100 tons, F = 1000, F' = 500, F"=600. (26) Find the greatest amount of water displaced by the air in a cylindrical diving-bell. Also find the interior pressure then on the upper surface of the bell. (27) Given the heights of the barometer in a diving-bell before descent and at a certain depth respectively, find the depth. 144 GENERAL EXAMPLES. (28) Assuming that 100 cubic inches of air wcigli 31.0117 grains and a cubic foot of water weighs 1000 ozs., compare tlie specific gravities of air and water: and if 34 feet be tlie licight of a eolumn of water which the atmosphere wfill support, sliew that the height of the atmosphere considered homogeneous is about 5 miles. (29) The weight of a cubic foot of water being 1 000 ozs. and its specific gravity unity, determine the specific gravity of a substance whose bulk is m cubic inches and weight n ozs. (30) A cubic foot of water weighs 1000 ozs.; what is the speeific gravity of a solid of which a xiubic yard weighs 540 lbs.? (31) A cube of wood floating in water descends 1 inch when a weight of 30 ozs. is placed on it; find the size of the cube supposing a cubic foot of water to weigh 1000 ozs. (32) If the specific gravity of air be 5, that of water being 1, and if IF, IF' be the weights of a body in air and water respectively, shew that its weight in vacuo will be W+^ (W- W). 1—5 (33) A metal cylinder floats in mercury with one-fourth of its bulk above the surface ; find the specific gravity of the metal, that of mercury being 13.6. (34) A piece of wood weighs Gibs, in air; a piece of lead which weighs 12 lbs. in water is fastened to it and the two together weigh 10 lbs. in water ; find the specific gravity of the wood. (35) What weight of oil (specific gravity .75) must be added to a pound of fluid of specific gravity .5, that in the GENERAL EXAMPLES. 145 mixture a pound of a substance of specific gravity 4 may weigh 13.50ZS. ? (36) To a piece of wood which weighs 4 ozs. in vacuo a piece of metal is attached whose weight in water is 3 ozs. and the two together are found to weigh 2 ozs. in water; find the specific gravity of the wood. (37) A lump of silver weighs 550 grains in air and 506 grains in water, find the specific gravity of silver, and also the volume of the lump, having given that the weight of a cubic inch of water is 250 grains. (38) A cubical block of marble whose edge measures 2 feet and whose specific gravity is 2.7 has to be raised out of a river; determine its weight when entirely immersed and also when lifted out of the water. (39) Two bodies A and B in air weigh lOlbs. and 15 lbs. respectively; in mercury B alone and A and B together weigh respectively 9 lbs. and lib.: what is A’s specific gravity, that of mercury being 13.5? (40) A cubic inch of pure gold (specific gravity 16|^) is mixed with two cubic inches of mercury (specific gravity 13.6) ; find the specific gravity of the compound. (41) What weight of water must be added to a pound of fluid whose specific gravity is \ in order that the specific gravity of the mixture may be f ? (42) The apparent weight of a sinker, in water, is four times the weight in vacuum of a piece of material, whose spe- cific gravity is required : that of the sinker and piece together is three times the weight. Shew that the specific of the material = .5. P. H. 10 146 GENERAL EXAMPLES. (43) If the three weights used in Nicholson’s Hydrometer he 10, 12, and ISlbs., find the volume of the solid in inches; a cubic foot of the fluid weighing lOOOozs. (43*) If p parts by weight of a metal whose specific gravity is 5, when fused with p parts of a metal whose specific gravity is s form an alloy whose specific gravity is shew that -th part of the volume of the whole has been lost n ^ by condensation during the mixture where 1 _^ ss^p-Vj)) n S{ps -\-ps) ' (44) Find k, for air, that p = kp may give the pressure in ounces ; the barometer standing at 30 inches when the density of air referred to mercury is .0001 ; the unit of length being one inch, and a cubic foot of the standard substance weighing 1000 ozs. How will k be altered if the unit of length be increased to 6 inches ? (45) p^, p„ are corresponding values of the pressure, density, and temperature of the same gas ; shew that + h ^ Pi pJ = 0 . (46) The temperature at one place is 24® by the centi- grade and at another 52® by Fahrenheit; what is the differ- ence by Fahrenheit’s? (47) What is meant by the sensibility of the thermometer? What degree of a centigrade corresponds to 60 of Fahrenheit, and what degree of Fahrenheit’s to 60 of the centigrade? (48) Having given a eertain temperature in degrees ac- cording to Fahrenheit’s thermometer, find the number of de- GENERAL EXAMPLES. 147 grees indicating it on De Lisle’s thermometer, where the space between boiling and freezing point is divided into 150 degrees, and the boiling point is taken as the zero of the scale. (49) The point at which mercury freezes is indicated by the same number on the centigrade and on Fahrenheit’s scale : determine the number. (50) In a vessel not quite full of water, and closed at the top by a flexible membrane, a small glass balloon, open at the lower part, contains sufficient air just to make it float, explain the principle upon which the balloon sinks when the mem- brane is pushed in. (51) A weightless conical shell is filled with fluid and suspended by its vertex from a fixed point : it is then divided symmetrically by a vertical plane, and kept from falling asun- der by a hinge at the vertex, and a ligament at the base, coinciding with that diameter of the base which is perpen- dicular to the dividing plane : determine the tension of the ligament. (52) A closed vessel is filled with water containing in it a piece of cork which is free to move : if the vessel be sud- denly moved forward by a blow, shew that the cork wdll shoot forward relatively to the water. (53) A piece of cork is attached by a string to the bottom of a bucket of water so as to be completely immersed, and the bucket being placed in the scale of a balance is supported by a weight in the other scale ; if the string be cut, will the weight begin to rise or fall ? State your reasons. (54) A cylindrical vessel containing fluid revolves uni- formly about its axis with an angular velocity o), and a solid cylinder of less specific gravity than that of the fluid floats in 10—2 148 GENERAL EXAMPLES. it with its axis coincident with tliat of tlic revolving vessel ; find how deep it is immersed. (55) A transparent closed cylinder filled with fluid, in which there are extraneous particles, some lying at the bottom and some floating at the top, is set revolving about its axis : it is then observed that the floating particles all flow in towards the axis, while those at the bottom recede from it : explain this. (56) If a vertical cylinder containing heavy fluid revolves about a generating line with a uniform angular velocity, the depth to which the surface sinks below its original level : the height to which it rises above that level :: 3 : 5. (57) A circular tube is half full of fluid, and is made to revolve uniformly round a vertical tangent-line with angular velocity co : if a be the radius, prove that the diameter passing through the open surfaces of the fluid is inclined at an angle tan“^ to the horizon. 9 (58) If a spherical envelope, of thickness h and radius r, be formed of a substance, which, if made into a line having a section would bear a weight W: find the number of strokes of the piston after which this envelope placed under the receiver of an air-pump would burst. (59) A vertical cylindrical vessel, closed at the base, is formed of staves held together by two strings, which serve as hoops, and is filled with fluid ; shew that the tension of the upper string is to that of the lower :: 7^ — 3a : 2h — 3a, where // is the altitude of the cylinder, and a the distance of the upper and lower strings from the top and bottom of the cylin- der respectively. GENERAL EXAMPLES. 149 (60) In the case of the previous question, how much of the fluid must be withdrawn from the cylinder in order that the tension of the upper string may vanish ? (61) A cylindrical boiler, the interior radius of which is 10 inches, and the thickness of an inch, is formed of a material such that a bar of it, one square inch in section, can just support a weight of 10,000 lbs. without being torn asun- der ; find the greatest pressure which the boiler can sustain without bursting. (62) At 18^9 (centigrade) the weight of a cubic foot of distilled water is 997.84 ozs., and at 16^| its weight is 998.24 ozs. ; find the temperature at which it shall be 1000. (63) A cubic inch of water which weighs 252.458 grains will produce a cubic foot of steam at atmospheric pressure; find the specific gravity of steam. (64) A quantity of air under the pressure of mlbs. to the square inch, occupies n cubic inches when the temperature is f ; find its volume under a pressure of m lbs. to the square inch when the temperature is (65) Having given that mlbs. of steam at the boiling- point, mixed with nibs, of water at temperature t, produces m + 7ilbs. of water at the boiling-point, compare the latent heat of steam and the specific heat of water. (66) Upon what principle might the height of a moun- tain be approximately found by observing the temperature at which water boils at the top? (67) A vessel contains air at atmospheric pressure; find the force in pounds necessary to be applied to a piston of area A 150 GENERAL EXAMPLES. in the vessel to prevent its being forced out when the air is heated to temperature T above what it was at first. (68) A thermometer-tube, open at the top and filled with mercury, contains 1000 grains at 32® temperature; if the tube be heated till its temperature is 84^, find how many grains of mercury will be expelled. The expansion of mercury in volume between 32® and 212® being .018, and the linear expan- sion of glass between the same points .0008. ANSWEES TO EXAMPLES. SECTION I. (1) 2 feet nearly. (2) 2.5 seconds. (3) .25 of a second. (4) .016. SECTION II. (1) W. (5) 1 : 2 : 4 :5. (6) 352% lbs. (7) .6 of a foot. (8) Half the weight of the fluid. No, the pressure on the table equals the whole weight of the fluid and vessel. (9) 3:1; the same in both cases. (10) ^ pound. (12) 156 lbs. ^ ozs. (13) It rises of its height nearly. (14) 2 nearly. (19) 1 : 5. (20) 1 : 8. (21) 11.48 ozs. nearly. (23) Ratio of volumes =19 : 2; ratio of weights =17 : 11. (24) 1 lb. (25) .98. (26) .39. (27) 48 inches ; 10.5408 ; 236^ (29) 250 : 147. (32) 23 : 37. (34) When high. (35) 60". SECTION III. (7) 60 lbs. (9) 5 inches. (10) 3 lbs. nearly. (11) ^ of height from base. (15) 1 : 1083077. (16) 26000 feet. (17) 20^ inches. (18) ^ that of the atmosphere. (20) 10. (21) 35 ft. nearly. 152 ANSWERS TO EXAMl'EES. (22) When AB, BG are equally inclined to the vertical. (23) 28 ft. 34 in. (24) (25) Increase. (26) The depth of the surface of the water in the bell = h'. --j~ where h' is the height of the water-barometer, h the altitude of the cylinder of water of the same weight and trans- verse section as the bell, and a the altitude of the bell. (27) times the original quantity, where h is the height of the water-barometer, and 6? the depth of the lower rim of the bell. (28) Their apparent weights in the given positions must be equal. In the second case the water must be at the same level in both. (29) All the air would rise in bubbles, and the bell would sink. (30) 2528 tons nearly. SECTIOISr lY. (5) 15 strokes. (6) The string must pass round in a horizontal plane at one- third of the height of the prism from the base. GENEEAL EXAMPLES. (3) 288 feet. (4) 7flbs. (5) The specific gravities must be in the ratio of 1 : 2. /o (6) Pressure = x weight of fluid. The least weight of Jj the pyramid = 2 . weight of fluid. (7) 49 tons IGcwt. 25 lbs. 8 ozs. + pressure of atmosphere. The resultant pressure = the weight of water displaced. ANSWERS TO EXAMPLES. 153 (9) 2 (Vs + 1) inclies. W where a is the semi-vertical (8) 62298.72 grains. (11) p, :p,::l:3. 2 (12) Normal pressure = . ^ ^ sin a angle of the cone, and W is the weight of the fluid. (14) For the pressure, see Art. 18. The center of pressure is on the bisecting line at f of its length from the vertex. (15) Four-fifths of AB. For the pressure, see Art. 18. Inclination to horizon == 30®. T the paraboloid. (20) Six times that of the cone. .8. (23) 69 ozs.; 6.9. (24) .8. (16) (17) (18) ( 22 ) (2.5) W+ V w+ r- r+ r" (26) The amount displaced when the upper surface is level with the fluid, equals the weight of water displaced. (27) — where a is the specific gravity of mercury referred to water. (29) (33) (37) (39) (44) (48) renheit. 1.728 X - . (30) .32. 10.2. (34) .75. 12.25 ; .176 inches. 7.5. (40) 14.56. 173611.1; 36 fold. (31) The edge = 7.2 inches. (35) 11 lb. (36) .8. (38) 850 lbs.; 1350 lbs. (41) 2 lbs. (43) .032 feet. (46) 23.2®. (47) 15|; 140. 1060 6 where F denotes the number of degrees Fah- (49) -40. (50) The volume of air between the membrane and water is diminished, and therefore its pressure is increased. Hence the pressure throughout the fluid is increased, and therefore the volume of air in the glass balloon is diminished. Therefore the specific gravity of the balloon and included air, considered as one body, is increased, and, being at first just equal to, is now greater than that of the water, and the balloon consequently sinks. P. H. 11 154 ANSWERS TO EXAMPLES. > 3 ““ tcXTl^ ft (51) . JF where IF is the weight of the fluid, and a Att . tail ft is the half of the vertical an^le of the cone. (53) To descend: for, when the string is cut, the cork rises, and some heavier fluid takes its place; the center of gravity of the bucket and its contents descends; less force therefore is called into action at that end of the balance than was the case when this center of gravity was in equilibrium; hence the weight at the other end of the balance is no longer sujiported, and consequently begins to descend. (T 7* (54) —h ~ .r,. where r, is the radius, h is the height of ^ ^ a ig ^ 1 o the floating cylinder, and a- a are the specific gravities of the fluid and cylinder respectively. (55) Let V be the volume of one of the particles, p its density, p the density of the fluid, w the angular velocity of the cylinder, and r the distance of the particle from the axis. Then the par- ticle will move towards the axis or recede from it, according as the resultant force acting on it towards the axis be greater or less than p'y.coV. But this force equals p'voj^r (see Art. 67); therefore the particle will tend to the axis or away from it, according as p' > or < p ; i.e. the lighter particles will flow towards, whilst the heavier will recede from, the axis. (58) The number of strokes is the value of n given by the inequality n{l - (j^^g) } I greater than W, where A and B are respectively the volumes of the receiver and piston-barrel. (60) To within a distance Za of the bottom. (61) 100 lbs. per square inch. (62) 6.84 degrees centigrade. ANSWERS TO EXAMPLES. 155 (63) 1728 referred to water. 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