GIFT OF Harold Wheeler ^w l^ THE CAMBRIDGE COURSE OF ELEMENTARY NATURAL PHILOSOPHY. THE CAMBRIDGE COURSE OP ELEMENTARY NATURAL PHILOSOPHY BEING THE PROPOSITIONS IN MECHANICS AND HYDROSTATICS IN WHICH THOSE PERSONS WHO ARE NOT CANDIDATES FOR HONOURS ARE EXAMINED FOR THE DEGPJ^.i^ OP B.A. . > ' \'> ' \' UKIGINALLT COMPILED BY J. C. SNOWBALL, M.A. LATE FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE. FIFTH EDITION, REVISED AND ENLARGED, AND ADAPTED FOR THE MIDDLE CLASS EXAMINATIONS. BY THOMAS LUND, B.D. LATE FELLOW AND LECTUKER OF ST. JOHN'S COLLEGE, EDITOR OF WOOD'S ALGEBRA, &c. Cambrttge antJ Hontion. MACMILLAN AND CO. 1864. St PKINTED By C. 'j. CLAl, £I;A. AT THE UNIVERSITY PRESS GIFT OF ADVEETISEMENT. The following work, originally compiled, at my sugges- tion, by tlie late Dr Snowball, has been unconditionally transferred to me by liis Executors; and I have therefore not hesitated to re-write certain portions of it, and to make such alterations throughout as I conceived neces- sary. At the end of each chapter I have added a series of easy questions for the exercise of the Student. I have also carefully looked through all the papers of the Uni- versity Examinations for the last ten years; and have printed a Collection of the best of the Problems, with their Solutions^ as well as a large number of others with Ansivers only. I am disposed to hope that the work, as it now ap- pears, will be found suited to the wants, not only of University Students, but also of many others who require a short course of Mechanics and Hydro statics, and espe- cially of the Candidates at our Middle-Class Examina- tions. T.L. Morton Kectoey, near Alpreton, May 6, 1864. 968320 CONTENTS. MECHANICS. Chapter I. ARTS. 1 — 8. Definition of Force, Weighty Quantity of Matter, Denaity, Measure of Force. 10. Pressure another name for Force in Statics. 11. Definitions with respect to the action of Forces. 12. Forces properly represented by geometrical straight lines. Questions on Chap. I. Chapter II. — The Lever. 17. Definitions of a Plane, a Solid, Parallel Planes, a Prism, and a Cylinder. 18. Definition of Z^«7^r. 19. Axioms. 20. Prop. I. A horizontal prism, or cylinder, of uniform density will produce the same effect by its weight as if it were col- lected at its middle point. 21. Prop. II. If two weights, acting perpendicularly on a straight Lever on opposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum; and the pressure on the fulcrum is equal to their sum. 22. Converse of Prop. II. 23. Prop. III. If two forces, acting perpendicularly on a straight Lever in opposite directions and on the same side of the ful- crum, balance each other, they are inversely as their distances from the fulcrum; and the pressure on the fulcnun is equal to the difference of the forces. viil CONTENTS. ARTS. 25. Prop. IV. To explain the diflferent kinds of Levers. 26. Prop. V. If two forces, acting perpendicularly at the extremi- ties of the arms of any Lever, balance each other, they are inversely as the arms. 27. Prop. VI. If two forces, acting at any angles on the arms of any Lever, balance each other, they are inversely as the per- pendiculars drawn from the fulcrum to the directions in which the forces act. 28. Converse of Prop. VI. 29. Prop. VII. If two weights balance each other on a straight Lever when it is horizontal, they will balance each other in every position of the Lever. Questions on Chap. II. Chapter III. — Composition and Resolution of Forces. 30. Definition of Component and Resultant Forces. 31. Prop. VIII. If the adjacent sides of a parallelogram represent the component forces in direction and magnitude, the diagonal will represent the resultant force in direction and magnitude. 32. Prop. IX. If three forces, represented in magnitude and di- rection by the sides of a triangle, act on a point, they wOl keep it, at rest. Questions on Chap. III. Chapter IV. — Mechanical Powers. 36. Definition of Wheel-and-Axle. 37. Prop. X. There is an equilibrium upon the Wheel-and-Axle, when the power is to the weight as the radius of the axle to the radius of the wheel. 40. Definition of Pulley. 41. Prop. XI. In the single moveable pulley, where the strings are parallel, there is an equilibrium when the power is to the weight as 1 to 2. CONTENTS. IX ARTS. 42. Prop. XII. In a system in which the same string passes round any number of pulleys and the parts of it between the pulleys are parallel, there is an equilibrium when Power (P) : Weight ( TV) :: 1 : the number of strings at the lower block. 43. Prop. XIII. In a system in which each pulley hangs by a separate string and the strings are parallel, there is an equi- librium when P : W :: I : that power of 2 whose index is the number of moveable pulleys. 45. Prop. XIY. The Weight ( TV) being on an Inclined Plane, and the force (P) acting parallel to the plane, there is an equilibrium when P : TV :: the height of the plane : its length. 46. Definition of Velocift/. 47. Prop. XV. Assuming that the arcs which subtend equal angles at the centres of two circles are as the radii of the circles, to shew that, if P and TV balance each other on the TVheel- and-Axle, and the whole be put in motion, P : TV :: TV'& velocity : P's velocity. 48. Prop. XVI. To shew that if P and TV balance each other on the machines described in Propositions xi, xii, xiii, and xiv, and the whole be put in motion, P : TV :: TV's velocity in the direction of gravity : P's velocity. Questions on Chap. IV. Chapter V. — The Centre of Gravity/. 49. Definition of Centre of Gravity. 50. Prop. XVII. If a body balance itself on a line in all positions, the centre of gravity is in that line. 51. Prop. XVIII. To find the centre of gravity of tv/o heavy points ; and to shew that the pressure at the centre of gravity is equal to the sum of the weights in all positions. 52. Prop. XIX. To find the centre of gravity of any number of heavy j^oints; and to shew that the pressure at the centre of gravity is equal to the sum of the weights in all positions. 54. Prop. XX. To find the centre of gravity of a straight line. 55. Prop. XXI. To find the centre of gravity of a triangle. X CONTENTS. ARTS. 56. Prop. XXII. When a body is placed on a horizontal plane, it •will stand or fall, according as the vertical line, drawn from its centre of gravity, falls within or without its base. 57. Pkop. XXIII. When a body is suspended from a point, it will rest with its centre of gravity in the vertical line passing through the point of suspension. Questions on Chap. V. HYDEOSTATICS. Chapter I. 59. Definitions of Fluid; of elastic and non-elastic Fluids. Chapter II. — Pressure of non-elastic Fluids. 60. Prop. I. Fluids press equally in all directions. 61. Prop. II. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid. 62. Prop. III. The surface of every fluid at rest is horizontal. 63. Prop. IV. If a vessel, the bottom of which is horizontal and the sides vertical, be filled with fluid, the pressure upon the bottom will be equal to the weight of the fluid. 64. The pressure of a fluid on any horizontal plane placed in it, is equal to the weight of a column of the fluid whose base is the area of the plane, and whose height is the depth of the plane below the horizontal surface of the fluid. 66. Prop. V. To explain the Hydrostatic Paradox. 67. Prop. VI. If a body floats on a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body ; and it xjresses downwards, and is pressed upwards, with a force equal to the weight of the fluid displaced. Questions on Chaps. I. and II. CONTENTS. XI Chapter III. — Specific Gravities. ARTS. 69. Definition of Specific Gravity. 70. Prop. VII. If M be the magnitude of a body, S its specific gra-vity, and W its weight, W= MS. 71. 72. To find the relation which exists between the weights, mag- nitudes, and specific gravities, of two substances, and of a compound formed of them. 73. Prop. VIII. When a body of uniform density floats on a fluid, the part immersed : the wliole body :: the specific gravity of the body : the specific gravity of the fluid. 74. Prop. IX. When a body is immersed in a fluid, the weight lost : whole weight of the body :: the specific gravity of the fluid : the specific gravity of the body. 70. Prop. X. To describe the Hydrostatic Balance; and to shew how to find the specific gravity of a body by means of it, 1st, when its specific gravity is greater than that of the fluid in which it is weighed ; 2ndly, when it is less. 77. Prop. XI. To describe the common Hydrometer; and to shew how to compare the specific gravities of two fluids by means of it. Questions on Chap. III. Chapter IY. — Elastic Fluids. 70. Prop. XII. Air has weight. 81. Prop. XIII. The elastic force of air at a given temperature varies as the density. 82. Prop. XIV. The elastic force of air is inci-eased by an increase of temperature. 84. Prop. XV. To describe the construction of the Commoii Air- Pump, and its operation. 85. Prop. XVI. To describe the construction of the Condeiiser, and its operation. Xll CONTENTS. ARTS. 86. Prop. XVII. To explain the construction of the Common Barometer; and to shew that the mercury is sustained in it by the pressure of the air on the surface of the mercury in the basin. 87. Prop. XVIII. Tlio pressure of the atmosphere is accurately measured by the weight of the column of mercury in the Barom,eter. 92. Prop. XIX. To describe the construction of the Common Pump, and its operation. 93. Prop. XX. To describe the construction of the Forcing- Pump, and its operation. 94. Prop. XXI. To explain the action of the /S'^jt7Ao7^. 96. Prop. XXII. To shew how to graduate a common Ther- onometer. 97. Prop. XXIII. Having given the number of degrees on Fahrenheifs thermometer, to find the corresponding number on the Centigrade thermometer. Questions on Chap. IV. Examples and Problems with Solutions. Examples and Problems with Answers. University Examination papers. MECHANICS. [The explanatory matter, printed in small fppe,^rms no actual part of the University Course ; hut is iilastratice of the 2)articular Defirdtion, or Proposition, which it immediately follows ; and will he found useful for ansicering the Questims, and solving the Problems, which are usually given in the University Examina- tions!^ chapte:^ I. ■ -' ' ^ ' DEFINITIONS ; EXPLANATION OF STATICAL FORCES ; THE MANNER IN WHICH THEY ARE MEASURED, AND REPRESENTED. 1. Mechanics is the science, which treats of the causes that prevent, or produce, motion in bodies, or that tend to prevent, or produce, m,otion. It is divided into two parts. The one, which investigates the conditions fulfilled when a body is in a state of rest, is called Statics. The other, which treats of the causes and the eflfects of motion, is called Dynamics. Thus it is the province of Statics to shew how the roof of a building is supported by the beams and the walls. If the roof gave way, and fell, it would belong to Dynamics to account for the circumstances attending the fall — to explain why the motion took place in one direction rather than in another — to determine the time elapsed in falling — and the swiftness of the motion at any instant. The part of Mechanics treated on in the following pages is Statics. L. C. C. 1 2 MECHANICS. 2. Defi)u't ion of Force. Whatever the cause be wlilch produce??, or prevent:^, motiou, or which tends to produce, or to prevent, motion, in a body, it is called a Fokce. If a heavy body, as a stone, be laid on the open hand, experience sliews that, to prevent the stone from falling, the hand must make some effort. Again, to set a ball rolling along the ground require.^ some exertion. The effort, or exertion, is called in either case a FORCE ; and although the effect produced be not great enough to prevent entirely the fiill of the stone, or to communicate motion to the ball, yet it is still called a force. From the definition of Statics given in Art. 1, it will readily be understood, that in that branch of MECHAisrics the conditions are investigated which are fulfilled by those Forces only, which keep a body at rest. 3. Definition o/" Weight. All bodies^ if left to tbeaiselves, fall, or tend to fall, to- wards the 'earth's ceiitre, ttirough a power, which resides in the earth; of cOnstantl}' dravving' all substances towards it, called ^TiiE fOkce cp gka'yity. Consequently, if any body be reduced to a state of rest, it exerts a certain pres- sure downwards upon that which sustains it. And the 'precise amount of this pressure for any particular body is called the AVeight of that body. 4. The WEIGHTS of different bodies may be compared thus : — Let two bodies be successively attached in the same manner to a spring, so that they may act upon it by their weights in the same way. If they produce the same effects, (by bending the spring to the same extent,) the weights of the bodies are equal. Any other body, which j)roduces the same effect on the spring by its weight as both the former bodies when applied together do by their weights, has its weight clouhle of that of either of them. And by means of such a contrivance as this spring, bodies might be shewn to be three, four, or any number of, times the weight of a given body. 5. The WEIGHT of any body is measured thus : — The weight of a certain bulk of some particular substance is first fixed upon as :» standard. Thus the weight of a piece of lead of a certain size being called a pound, any other body, which by the force of gravity only, produces the same effect as four, or six, or ten, such pieces of DEFINITIONS, &C. 3 lead, will be four, or six, or ten, pounds in weighty as the ca.se may be. 6. i)e^?zz7/o?i 0/ Quantity OF Matter. The substance, material, or stuff, of which any body is made, is called Matter. And since all bodies have Weighty the property of having Weight is to be considered as necessarily belonging to Matter. Hence in the same ratio, or degree, that one body has more weight than another, it is concluded, that it contains more matter ; that is, the Quantity of Matter in a hody is proportional to its Weight. Thus, if a body A weigh one pound, and another body B weigh three pounds, the quantity of matter contained in A is said to be to the quantity contained in ^ as 1 to 3 ; or B is said to contain three times as much matter as A does. 7. The exact quantity of matter contained in any body may be measured by comparing its weight with the weight of some particu- lar body, which has been fixed upon for a standard. Thus, if a cubic inch of water be previously taken as the body by which to measure the quantities of matter contained in all other bodies, aiid the quantity of matter in this cubic inch of water be called 1, then the quantity of matter in any other body would properly be said to be 5, if the weight of that body were five times as great as the weight of the cubic inch of water. 8. Definition of Density. The Density of a substance, or body, is the degree of closeness, with which the matter composing it is, as it were, packed ; which closeness is measured ^ or compared in man- ner following: — Let equal bulks of two different substances be taken, suppose Water and Lead. .Then, if the bulk of loater which is taken weigh one pound, it will be found, that the piece of lead of equal size with it will weigh ll-[% pounds. There is evidently, therefore, ll^^r times as much heavy matter in a piece of lead as there is in an equal hulk of water ; and this fact is expressed, or described, by saying, that " The density of Lead is to the density of Water, as Uy^ is to 1" ; or by saying, " The density of Lead is \\~^^ times that of Water." 1^2 4 MECHANICS. If the Density of water be called 1, that is, if water be taken as a standard, to measure Density, then the Density of lead will be properly called lly^, or 11 •4. In the same manner as it has been explained how the Density of lead is estimated with respect to the Density of water, the Densities of any other substances, whether solid or fluid, may be determined "with respect to that of water. 9. Definition of " Measure of Force". In Statics a Force is measured by the weight wliicli it would support. In other words, the amount of a Statical Force is exj3ressed by stating the number of pounds it would support, if the Force were made to act directly oppo- site to the Force of Gravity. Thus, if the weight of a body were P pounds, and it were pre- vented from moving towards the earth's surface by a hand placed beneath it, the resistance ofiFered by the hand to the communication of motion (that is, the force exerted by the hand), would be P pounds ; and if this same pressure were produced by the hand in any other direction, it would be described in the same manner, by saying that it was ^'' equal to P pounds''', or that it was ^^ P pounds'. If, therefore, a Force be represented by P, it is meant that P is the number of pounds which the Force would support, on the supposi- tion that the Force is made to act directly opposite to the Force of Gravity. In other words, P is the number of pounds, which the Force is just able to lift. Forces, in Statics, also called Pressures. 10, tn whatever direction a Force tends to produce motion, its magnitude, as has already been stated, is measured by the weight of the body which would exert the same effect to produce motion downwards, as the Force under consideration exerts in the line in which it endeavours to produce motion. And that such a method of measuring Forces is allowable appears from this consideration, viz , that the effect produced by the weight of a heavy body* may be made to take place in any direction whatever ; horizontally, as in the case of a string being attached to an object lying on a table and kept stretched by a heavy body ( W), which hangs over the edge o * By ' a lieavy hody\ in Mechanics, is simply meant a body acted on by the Fmre of Gravity. DEFINITIONS, &C. 5 the table, as in fig. (1) ; or vertically upward.-, by passing the stiirg over a peg A, and at- tacliingf the end to a J3 IV (0 T\ r2) 6 ring B, so that BA may be vertical, as in fig. (2) ; or i?i any othei" direc-, Hon, as in fig. (3), by making the heavy body pull the string in the line BA, which is inclined at miy angle to -that {A W) in which it acts itself. 11. Definitions with respect to the action of Forces. (i) The point at which a Force acts upon a body is called the "point of appliccitiorC of the Force. (2) The line iu which a Force, acting alone, produces, or tends to produce, motion, is called "the line of the Force's action''^- and any line which is parallel to the line of a Forces action is said to be " in the direction of the Force's action'\ or " in the direction of the Force". When the direction of a Force's action, (or, as it is generally called, " the direction of the Force''' ^ is indicated by a line, either the very line is given in which the Force acts, or some line which is parallel to it. " The line of a Forces action", and " the direction of the Force", must by no means be confounded together. If the former be known, the latter is necessarily known also ; but if only the latter be given, the precise line in which the Force produces, or tends to produce, motion, is uncertain ; and all that can be said re- specting it is, that the line of action of the Force is either that given line, or some other line which is p><^i^^(dlel to it. (3) If two or more Forces be applied to a body, or at some point, and no motion is produced, they are said to " counteract', or to '"'' halance!\ one another, or to be "in eqaiiihrium" . 12. Forces properly represented hy geometrical straight lines. Since lines may be drawn of any length, and in any direction, from a point, the lines in which Forces act, and the ratios which the Forces bear to one another, may be represented by di'awing Hues, which coincide with the lines in which the Forces act, and vv^hose leugths bear to one another the same ratios that the Forces them- selves bear to one another. MECHANICS. Among other advantages which attend this method of expressing the magnitudes and directions of Forces, the addition and subtrac- tion of such Forces as act at a jioint in the same straight line are easily effected. Thus, if a certain Force act at A in the line AH^ and AB be taken to represent it, and another Force, half as great as the former, act at A in the same direction, and also tend to move the body from A towards H, then, by taking- BC equal to the half of AB, the line ^(7 will represent the whole pressure at A, both with re- a jj li c i ' J£ > ■ ■ ■ 11 spect to the magnitude of that pres- sure, and to the line in uchich it acts. And, in like manner, if a Force equal to ha^f the original Force AB act at A in tlie line AH, but tends to move the body at A from A towards K, half the pressure of the former Force will be coun- teracted by this new Force. Cutting off from the line AB, there- fore, a part BD equal to the half of AB, the effective pressure still remaining will be properly represented by AD, with respect both to its magnitude, and to its line of action. 13. jS'.B. It will be gathered from the above, that a Force AB applied to A has not the same effect as a force BA applied at that point ; for a Fo7xe AB would tend to move a body at A in the line A^/T towards H, but a Force BA would tend to move a body at A in the line KH from A towards K, It is not, therefore, indifferent whether the words " a Force AB\ or '•' a Force BA'\ be used ; since, though the two Forces represented by AB, and BA, are the same in magnitude, and also act in the same straight line, yet they tend to produce motions directly opposite to one another, the Force AB tending to move the body at A towards H., and the force BA tend- ing to move the body at A towards K. 14. The effect produced at a point hy any Force is the same at whatever point in its line of action the Force is applied, provided the latter point he supposed rigidly connected with the former. Thus, if a body P be suspended by a string CP, the q(j Force necessary to prevent P falling to the earth is found to be the same whether that Force be applied at A, or B, b or C; — the weight of the string being either neglected, or the weight of that portion of it which is supported along with the heavy body P, being counterbalanced. And al- though, in this case, the points A, B, G, are not, in fact, rigidly connected T\1th one another, and with P, the result i>i DEFINITIONS, &C. 7 is the same as if they were, but in certain other cases tlie rigidity of the system is necessary. 15. Def. If a string-, fastened at one end, be pulled by a Fotxe applied at the other end, the resistance to motion made by the string at any point in it is called the tension of the string at that point. If the string be supposed to be without weight, it will follow, from Art. 14, that the tension at every point of it is the same; namely, the Force by wliich the string is pulled. j> 16. To recapitulate the substance of this Chapter. (i) The precise amount of the Force, or pressure, with which any particular body endeavours to move towards the earth, is called the WEIGHT of that body. Art. 3. This WEIGHT of a body is measured by comparing the tendency of the body to move towards the earth with that of some other given body (of a certain size and formed of a certain material) which is taken for a standard, and to which the name of a grain, an ounce, a pound, or a ton is given, as the case may be. Art. 4. (2) Matter is the substance of which all bodies are composed. It is found, by universal experience, to have a tendency to move towards the earth. Art. 6. (3) The quantities of matter contained in different bodies (whether the bodies be gi-eat or small, rare like gas, or dense like lead) are proportional, (not equal,) to the weights of the bodies. Art. 6. (4) The densities of different substances are proportional, (not equal^ to the weights of equcd hulks of the substances. Art. 8. (5) Force, l Whatever be the cause which moves, or tends to move, matter, existing under any form whatsoever, it is called a FORCE. Forces which prevent motion taking place, — that is, statical FORCES, — are measured by the number of pounds they would support if they acted vertically upwards. Arts. 2, 9. II. The LINE OF a force's action is the actual line in wliich the force tends to produce, or to prevent, motion. Art. 11. And if a Force be said to be P, it is meant that it is equal to P pounds. III. The direction of a force is indicated either by the line of its action, or by any line which is pjarallel to the line of its action. Art. 11. b MECHANICS. IV. The magnitudes of Forces, and the lines (or the directions) in wliich they act, may be represented by means of straight lines. Art. 12. V. If a straight line AB represent a Force acting on a material point placed at A, and BA represent another Force acting on the same point, the two Forces AB and BA are equal, and tend to move the point in the same straight line, but in oj^posite ways from A. Art. 13. VI. A Statical Force produces the same effect at whatever point in its line of action it is applied. Art. 14. VII. To investigate the effects produced by a Force, there must be given : — 1st. The magnitude of the Force; — which is known, if the number of pounds be known which the Force would support. 2nd. T\\Q point 2^:, yi\^ck iho) Force \^ applied. 3rd. The line in which the Force acts, — or the direction; for knowing the direction of the Force, and the point it is applied at, the line in which the Force acts may be determined by drawing a line through the given point parallel to the given direction. Questions on Chap. I. (1) "What are the two gi'eat divisions of Mechanics called, and how are they separately distinguished ? (2) What is Force? Give an example. (3) Define Weight, and explain how it is reduced to numbers, so as to become the subject of calculation. (4) Is the Weight of the same body at the same place in- variable ? Is the Weight of a body changed by changing it's, Jig are '? (.5) How do you define Matter? and Quantity of Matter? (6) How is the Quantity of Matter in a body ascertained and 7neasured? (7) Which has the greater Quantity of Matter, a feather-bed weighing 50 lbs., or a child of the same weight ? (8) What is Density? How is it reduced to numbers, and measured ? (9) If the weights of equal hulks of two substances are in the ratio 3 : 1, what is the ratio of their Densities? THE LEVER. 9 (10) Is the Density of the same body at the same place invaria- ble, like Weight f Give an illustration. (11) How is Force measured in Statics? Does jonr measure of Force apply to the case of a Force acting vertically upwards ? also to a Force acting in any direction whatever ? (12) Before we can estimate the eflFect of a Force upon a body, what three things are required to be known % (13) Can a Force be represented by a straight line in magni- tude, as well as direction ? (14) If it be stated that two forces of 5 lbs., and 10 lbs., act upon a body, what more is wanting to enable us to determine the result ? (15) Is the ^^ direction of the Force'\ and the ^^line of action of the Forc^\ the same thing ? (16) If a Force be applied to a body by means of a string, the weight of which is inconsiderable, is the result aflected by the length oiiYiQ string^ CHAPTER II. THE LEVER. 17. Defs. (i) A PLANE, OR A PLANE SUPERFICIES, is that super- ficies, or surface, in which, H any two points be taken, the straight line joining them lies wholly within the superficies. Euclid, i. Def. 7. (2) A SOLID is that which hath length, breadth, and thickness. Euclid, xi. Def. 1. (3) Parallel Planes are such as do not meet one another, however far they may be produced. Euclid, xi. Def. 8. (4) A Prism is a solid bounded by plane rectilineal figures, oi which two that are opposite are equal, similar, and parallel to one another ; and the others all parallelo- grams. Euclid, xi. Def. 13. It is _^__^ further necessary, that the parallel ^ figures have their equal angles opposite each to each. 10 MECHANICS. Thus, let A BCD J and abed, be two equal and similar quadri- lateral figures, placed with their planes parallel, and with their isiniilar angles opposite, each to each ; and let all the figures, such as ABha, which are formed by joining the equal angles of ABCD and abed, be parallelograms ; then the solid included by these parallelo- grams, and by the ends, or hases, ABCD and ahed, is called a pristn. The length of the prism is any one of the edges Aa, Bb, &c.; Avhich lines, being the sides of adjacent parallelograms, are all equal to one another. If the parallelogTams be reetangles, the solid is a rectaiigidar prism. (5) A Cylixder is a solid described by jy c a rectangle, ABCD, revolving round one /^\ 7 ^ of its sides, AB, which remains fixed, (a/ t]3 J Euclid, xi. Def 21. ^^ ^-^ The side AB, which remains fixed, is the length of the cylinder, and is called its axis. The surfaces described by the two sides, AD, and BC, which are adjacent to the fixed side of the rectangle, are circles, and are parallel to each other; they are called the ends, or hases, of the cylinder, 18. Definition o/Leyer. A Lever is a rigid rod, moveaWe in one plane round a fixed, point in it called the fulcrum. The two parts into which the rod is divided by the fulcrum are called the Arms of the Lever. In the following Propositions the thickness of the rod is neg- lected, and the Lever is considered to be a geometrical line, without weight, but still rigid and inflexible. The Weights, or Forces, acting on the Lever, are supposed to act ^r* the same plane. > 19. The properties of the Lever are sometimes de- duced from the following principles, the' truth of which will be readily admitted, and which are therefore called Axioms : — Axiom I. Equal forces, acting perpendicularly at the extremities of equal arms of a Lever, exert the same effort to turn the Lever round. THE LEVER. 11 Axiom IT. If two weights balance each other upon a straight Lever, the pressure upon the fulcrum is equal to the sum of the weights, whatever be the length of the Lever. Axiom III. If a weight be supported upon a Lever, which rests on two fulcrums, the pressures on the fulcrums are together equal to the whole weight. Axiom I. is quite self-evident, since the forces are equal, and act upon the Lever in a manner perfectly similar. In Axiom II., the weights act in the same, i.e. in a vertical, direction, and therefore cannot in any degree counteract each other. Hence it is obvious, that the effective force on the Lever must be the sum of the weights, and this is supported by the fulcnmi, since there is equilibrium. Therefore the pressure on the fulcrum is equal to the sum of the weights. In Axiom III., there is notliing to support the weight but the fulcnims ; therefore the pressure on the fulcrums must be together equal to the whole weight. 20. Prop. I. A horizontal prism, or cifUnrler. of uniform density, will produce the same effect by its weight, as if it were collected at its middle point. Let AB be the axis of the given prism, or cylinder, supposed to be held at rest in a horizon- tal position ; G its middle point; DOE the vertical line through C meeting the outer surface of the prism, or cylinder, in D and E. Affix a string to the prism, or cylinder, at D ; and apply a force P vertically upwards equal to TF, the weight of the prism, or cylinder. Then, if the prism, or cylinder, be left free to move, by withdrawing the forces which first held it in its horizontal position, it is evident, that no motion will ensue. For, since P is exactly equal to W, by supposition, both which forces act upon the prism, or cylinder, in exactly o|)posite directions, there can be no motion upwards or downwards. Nor can there be any J) A\ \C M 12 MECHANICS. angular motion of the prism, or cylinder, round i), because the parts AD, BD, being exactly similar and equal, no reason can be assigned for an angular motion in one direc- tion, which would not be equally valid for an angular motion in the opposite direction, and therefore there will be no angular motion at all. Since, then, the whole effect of the prism, or cylin- der, to produce motion by its weight is exactly counter- acted by a single force P, equal to W, acting in the line BCD, that effect is exactly equivalent to a single force W acting at any point in the same line, and therefore at C the middle point (Art. 14). Cor. 1. Hence it follows, that any uniform rod, in a horizontal position, produces the same effect by its weight to turn the rod round any fulcrum in it, as if its whole weight were concentrated in its mi 'a die point, the rigidity of the rod being supposed to be still maintained. Cor. 2. Hence also it follows, that any horizontal prism, or cylinder, of uniform density, will balance on its middle point ; and the pressure on a fulcrum placed there will be the weight of the prism, or cylinder. And conversely, if such a prism, or cylinder, balance on a point in itself, that point is either its middle point, or in the vertical line which passes through its middle point. 21. Prop. II. If two weights, acting perpendicularly on a straight Lever, on ojjposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum; and the pressure on the fulcrum is equal to their sum. Let the two weights P, Q, acting perpendicularly at M -^- — ^1 — ^ — ^ '^ 35 and N on the straiglit Le:ver MCN, whose fulcrum is G, ^^ *^ balance each other, the Lever being in a horizontal posi- tion. It is required to shew, that P : Q :: CN : CM, In il/iNTtake a point E such, that ME : MN :: F : P+ Q, Euclid, vi. 12, ' then ME : MN-ME :: P : F+Q-P^ or ME : NE :: P : Q. THE LEVER. 13 Produce MN both ways to A and B, making MA equal to MEj and NB equal to NE, Then, since ME : NE :: F : Q, 2ME : 2NE :: P : Q, or AE : BE :: F : Q, Now suppose AB to be a uniform rod, whose weight is P+ Q. Then the weights of AE, and ^^, will be F and Q, respectively. And since F and $, acting at the middle points of the lines AE and BE, balance on G, therefore the portions of the rod, AE and BE, which may be sub- stituted for F and Q by Prop. I., will also balance round (7, that is, the whole rod AB balances itself on 0, and therefore 6' is its middle point. But P : Q V. AE^ : BE, :: AB-BE : AB-AE, :: 2BG-2BN: 2AC-2AM, :: BC-BN ',AG-AM, :: GN : GM. Also the Pressure on the fulcrum is not altered by sub- stituting the rod for the weights, and therefore = whole weight of the rod, = P+ Q, the sum of the weights. 22. Conversely, if two weights, or forces, acHng perpendicularly on a straight Lever, on opposite sides of the fulcrum, are inversely as their distances from, the fulcrum,, they will balance each other. To prove this, construct as in the foregoing Proposition, and assume that P : Q :: CN : CM; then from this it will be required to shew, that C is the middle point of the rod AB, and therefore that the rod will balance on C, by Cor. 2, Prop. I. The proof will stand thus, ME : NE :: P : Q, :: GN : GM, :. ME+NE : ME :: GN+GM : GN, or MN : ME :: MN : GN, :. ME=GN. Also MN '. NE :: MN : GM, :. NE^GM. 14 MECHANICS. But, by construction, MA = ME, and NB = NE ; .'. AC=MA + CM=ME+NE=MN, and BC=NB+CN=NE+ME=3fN; .'. AC=BC, and /. the rod AB will balance on C. Then, since the Aveight of AE=P, and weight of BE^Q, by Prop. I., P and Q will also balance on the Lever MCN. ^ 23. Prop. III. If tioo forces, acting inrpendicularly on a straight Lever in opposite directions and on the same side of the fulcrum, halance each other, they are inversely as their distajices from the fulcrum ; and the pressure on the fulcrum is equal to the difference of the forces. Let tlie two Forces P and Q, acting perpendicularly af M and N, on the straight Lever ^, MC, in opposite directions, and on | ^, A^' the same side of the fulcrum C, M | Ac balance each other. Then it is to Q^ yi be shewn, that P : Q. :: CN :^ Oil/; m^ 1 vc and [Q being the Force which is p| ^ | the nearer to the fulcrum) that the pressure on the fulcrum = Q — P. Suppose the fulcrum at G removed, and let its resist- ance [R] be supplied by a Force equal to R, and actin.s; perpendicularly to the Lever in the same direction as P. The equilibrium will not be disturbed. Then since P and R are exactly counterbalanced by Q, they must produce a pressure at N equal and opposite to Q. Let Q be removed, and its place supplied by 2^ ful- crum on the contrary side of the Lever to that on which Q acted, sustaining the pressure (namely Q) produced by P and R. The equilibrium is still maintained ; and the case is now that of two Forces acting perpendicularly on 02:}po- site sides of the fulcrum, and balancing each other ; and therefore (by Prop. II.), P : R :: CN : NM; .-. P : P+R :: CN : CN+j\M :: CN : CM. But Q, the pressure on the fulcrum which has been supposed to be placed at N, is equal to P + i^, by Prop. II., THE LEVER. 15 .-. P : Q :: CN : Ci¥. Aho since Q = P+B, r,Ii=Q-P, that is, the pressure on the fulcrum = the difference of the forces. 24. From the last two Propositions it appears, that if a straight Lever, which is acted on jyerpendicularly by two weiglits, or other Forces, P and Q, respectively applied at the distances CM and CN from the fulcrum C, be at rest, then, whether P and Q act on tho same side, or on diflferent sides, of the fulcrum, the proportion P : Q :: ON : CM is always true. Hence also PxCM=QxCN {Wood's Algebra, Art 237) is an equation which expresses the conditions of equilibrium in all such cases. There is no impropriety in multiplying a Force by a line, because both are expressed in numhers, when they become subjects of calcu- lation. Thus a force of 3 lbs. acting perpendicularly on a straight lever at a distance of 4 feet from the fulciimi will balance another foixe of 6 lbs. acting at a distance of 2 feet on the opposite side of the fulcrum and in the same direction, because 3x4 = 12 = 6x2. The product PxCM is sometimes called the moment of P about C ; and, similarly, QxCN is the moment of Q about C. Hence, hi the last two Propositions, the moments of P and Q are equal. Also, since if P : Q :: CN : CM, it is proved that P and Q will balance on C, therefore, conversely, if the moments of P and Q with, respect to C are equal, they will balance each other. When the Lexer is used to balance a given Force, Q, by the ap- plication of another Force, P, Q is usually called "the Weight'', and P "the Power". If C3I, the perpendicular distance from the fulcrum at which the Power acts, be greater than CN, the distance at which the Weight acts, the Power required to balan*ce the Weight is less than the Weight ; in this case ''^ force" is said to be '■^gained'' by the application of the Lever. But if CM be less than CN, the Power required to balance the Weight is greater than the Weight, and ^^ force'' is then said to be "lost". 25. Prop. TV, To explain the different hinds o/Levers. Levers are divided into three classes, according to the relative position of the points where the Power and the Weight are applied with respect to the Fulcrum, 16 MECHANICS. (1) Where the Power (P) and tlie Weight (Q) act on opposite sides of the Fulcrum {C), as thus (2) Where the Power and the Weight act on the same side of the Fulcrum, but the perpendicular distance from the Fulcrum at which the Power acts is greater than that at which the Weight acts, as thus (3) Where i\\e Power and the Weight act on the same side of the Fulcmim, but the perpendicular distance from the Ful- crum at which the Poicer acts is less than that at whicli the Weight acts, as thus U c ]sr 7^ ^P J^.P ^Q N C M ^Q ^^ .^ M I Of the FIRST class the poker, when used to raise the coals, is an instance ; the bar of the grate on which the poker rests being the Fulcrum, the force exerted by the hand the Power, and the resistance of the coals the Weight. In the common Balance, the Power and the Weight are equal Forces perpendicularly applied at the ends of equal arms. In the Steelyard, the Power and the Weight are perpendi- cularly applied at the ends of unequal arms. Pincers, scissors, and snuffers, are double Levers of this kind, the rivet being the Fulcrum. Since CM may be either greater or less than CN, the Power in Levers of this class may be either less, or greater, than the Weight, and consequently ^'- Force'' may be either ^^gained'\ or "lost^\ by using them. Of the SECOND class, a cutting Made, such as is used by coopers, moveable round one end, which is fastened by a staple to a block, and worked by means of a handle fixed at the other end, is an example. An oar is also such a Lever; the Fulcrum being the extremity of the blade (which remains fixed, or nearly so, during the stroke), the muscular strength and weight of the rower being the Power, and the Weight being the resistance of the water to the motion of the boat, which is counteracted and overcome at THE LEVER. 17 the rowlock. A pair of nutcrackers also is a double Lever of the second class. Here, since CM is greater than CN, the Power is always less than the Weighty or Foixe is "gained'' by using Levers of the second class. An example of the third class is the board which the turner (or knifegrinder) presses with his foot to put the wheel of his lathe in motion ; the Fulcrum being the end of the board which rests on the ground, the Power being the pressure of the foot, and the Weight being the pressure produced at the crank put on the axletree of the wheel. Fire-tongs and sugar-tongs are double Levers of this kind ; the Weight in either case being the resistance of the substance grasped. The limbs of animals are also such Levers : thus, if a weight be held in the hand and the arm be raised round the elbow as a Fulcrum^ the Weight is supported by muscles fastened at one extremity to the upper arm, and again attached to the fore-arm, after pass- ing through a kind of loop at the elbow. Here, since CM is less than CN, the Power is greater than the Weight, or Force -is '^losi'^ by making use of Levers of the third class. 26. Prop. V. If two forces, acting perpendicularly at the extremities of the arms of any Lever, balance each other, they are inversely as the arms. In this Prop, the ar7ns of the Lever are supposed straight, but joined together at the fulcrum so as not to be in the same straight line. Let the two forces P and Q, acting perpendicularly at the extremities of the straight arms, CM and CN, of any Lever whose fulcrum is C, balance each other ; then P: Q V. CN: CM. For, suppose the arm NC produced to M', so that CM'= CM; and suppose a force P', equal to P, to act perpendicularly at 21 ' on the L. c. c. 2 18 MECHANICS. Lever 31' CN; then, since F = F, and CM' = CM, by Axiom I., Art. 19, the effort of Pto turn the Lever MCN round C is equal to that of F on the Lever M'CN. But, by the supposition, P balances Q on the Lever MCN', therefore also P' balances Q on the straight Lever M'CN, Hence, as before proved in Prop. II, F : Q:: CN : CM'; but P=P', and (7i¥= CM', /. P: Q:: CN : CM. Con. Here again, as in the two preceding Propo- sitions, PxCM= Qy-CN 27. Prop. YI. If two forces, acting at any angles on the arms of any Lever, balance each other, they are inversely as the per^jendiculars drawn from the fulcrum to the direc- tions'^ in which the forces act. Let P and Q be two forces, wliicli, acting at any angles on the arms CA and CB of any Lever ACB, balance each other about the fulcrum (7; and let the perpendiculars CM and CN be drawn from the fulcrum C to the lines in which the forces act ; then P : Q '.'. CN: CM. For, since a force produces the same effect at whatever point in its line of action it is applied (Art. 14), the force Pmay be supposed to be applied at M; and in order that it may be so applied, let a rod, CMA, supposed without weight, be fastened to CA. In like manner, Q may be supposed to be applied at N perpendicularly to the part CNoi the rod CNB which is added to CB. And, since P acting at M jjerpendicularly to CM ba- lances Q acting at N perpendicularly to CN, .'., by Prop. V, P : Q :: CN : CM; and therefore also, when P and Q, acting at A and B in the lines AP, BQ, balance, P : Q :: CN : CM. * More correctly, "to the lines of action of the forces". See Art. ii. THE LEVER. 19 2S. Cor. Conversely, if P and Q be inversely as the perpendi- culars from the fulcrum upon their lines of action, they will balance eacli other. For suppose Q' to be the force which applied at £ in BN ba- lances Pat A; then, by what has been proved, P : Of :: CN : CM. But, by supposition, P : Q v. CN : CM', :. P : Q' ::P : Q, or Q^=Q- but Q! balances P, :. Q also balances P. In this Proposition the arms of the Lever may be either straight or crooked, since nothing in the proof is made to depend upon the particular /o>-w of CA, or CB. The rigidity, however, of the arms, whatever be their form, is a necessary condition. It may also be noticed here, that every possible case of two forces balancing on a Lever has now been discussed. In Prop. n. the Lever is straight, and the forces 2ict perpendicularly to the arms on opposite sides of the fulcrum. Prop. iii. is the same as Prop, it., except that the forces act 07i the same side of the fulcrum. In Prop. V. the forces still act perpendicularly to the arms, but the Lever is hent. In Prop. vi. the forces act at any angles to the arms, and the Lever is either straight or hent. But in every case PxCM=QxCN; from which equation, any three of the qnantities being given, the fourth may be found. 29. Peop. YII. If two weights halance each other on a straight Lever when it is horizontal, they will halance each other in every position of the Lever, Let P and Q be two weights, wliich balance each other round the fulcmm C on the straight Lever A CB, when it is horizon- tal. They will balance each other on the Lever, when it is made to take any other position, as ACB'. Produce QB' to cut AB in N, and A'P, if necessary, to cut AG in M, 2—2 20 MECHANICS. Since weights act perpendicularly to the horizon, A'P and QB'N are both perpendicular to the horizontal line ACB', .*. angle CMA' = right angle = angle CNB' , and angle A'CM= opposite angle B'GN, Euc. I. 15. .-. also angle C^'3/= angle 0^'^Y, and the triangles CA'M, GB'N, are equiangular, and therefore similar. Hence, by Euc. Yl. 4, CN : CB' :: CM : CA' ; alternando, CN : CM :: CB' : CA', :: CB : CA. But, smce Pand Q balance on A CB, CB : CA :: F : Q ; .-. CN : C2I :: P : Q. But CM and CN are the perpendiculars from C on the lines in which P and Q act, when thej are hung at A' and B' ; therefore, by Prop. VI. , Cor., P and Q will balance on A'CB' ; and since A'CB' is the Lever in tt?2?/ position, the above proof applies to every position of the Lever. Cor. 1. Hence, if two weights do not balance each other on a straight Lever, when the Lever is horizontal, they cannot balance each other in an inclined position of the Lever. For if they did balance in an inclined posiition, it would follow from Prop, vi., that P : Q :: CN : CM, :. P : Q :\ CB' : CA', by what has been proved, :: CB : CA] and .'. P and Q balance in the horizontal position of the levei, which is contrary to the supposition. Cor. 2. Hence, also, if two weights, acting freely, balance each other on a straight Lever in any one position of the Lever, except the vertical, they will balance in every other i)osition of the Lever. For the ratio CN : CM is independent of the angle at which the Lever is mclined ; therefore if it once satisfies the conditions of equilibrium, it yaW do so always. THE LEVER. 21 Questions on Chap. II. -> (1) What is a Lever? Is there any such Lever in practice as that which is assumed in this chapter ? ^ (2) What is the falcrum, and what arc the arms, of a Lever ? Must the arms necessarily He on opposite sides of the fulcrum ? (3) In Axiom, ii., if the Lever itself be supposed to have weighty how will the result be afifected % (4) In Axiom, iii., if the weight be placed exactly half-way between the fulcrums, what is the pressure on each ? (5) In Prop. I., if the jirism, or cylinder, were not horizontal^ how would the proof be affected ? (6) In Prop. I., if the prism, or cylinder, were not of imiform density, how would the proof be aff"ected ? (7) What is the meaning of '■^effecf'' in the enunciation of Prop. i. (8) In Prop. II., would P and Q balance, if they were to ex- change places ? Are there any other points between M and N at which they would balance 1 (9) In Prop. II., if Q were doubled, where must P act to main- tain the equilibrium 1 jc (10) If 2 cwt., acting at a distance from the fulcrum of 1 foot, is balanced on a horizontal straight Lever by a power of 28 lbs. acting perpendicularly, what is the length of arm, at which the power acts ? (11) In Prop. III., where P an/:! Q balance each other, acting on the same side of the fulcrum, would the equilibrium be disturbed, if P were doubled, and CM halved % Also would the pressure on the fulcrum remain the same ? ~> (12) There are three classes of Levers; what is it which distin- guishes one class from another ? (13) Is power '■''losV or '"'■ gained'''' in the use i^ti fire-tongs? (14) Where would you place the nut in a pair of luit -crackers to produce the greatest effect ; and why 1 (15) How does the contrivance of placing the row-locks outside the boat affect the efforts of the rower ? (16) In Prop. VI. is it necessary that the angles at which the forces act should be equal to one another ? If the forces once balanced acting at equal angles, would the same forces balance on the same Lever acting at any other equal angles ? 22 MECHANICS. CHAPTEE III. COMPOSITION AND KESOLUTION OF FORCES. j> 30. Definition of Component and Resultant Forces. It is found, by experiment, that a bodj which is acted on by two forces applied, at the same instant and in differ- ent lines, to the same point of it, instead of moving, or hav- ing a tendency to move, in either of the lines in which the forces act, moves, or has a tendency to move, in a line lying between them. Whence it appears, that the tv*^o original forces by their combined action produce the effect of a single third force, which third force is called, from the circumstance of its resulting from the actions of the original forces, their ^'Resultant'' with respect to them ; while they are called, with respect to it, its " Com2)onents''\ The Resultant (R), which produces the same effect as the com- pound action of the original forces P and Q appHed at the same point at the same instant, is said to be ''''compounded'''' of P and Q. This Resultant {R) also, if conceived to be the sole original force, may be supposed to be " resolved^' into the two forces P and Q ; since those two forces, acting in the manner described (namely, at the same point, and at the same instant), produce exactly the same effect on the body as the single force R does. Similarly, if there be more than two forces, acting at the same point, and at the same instant, the resulting action is found to be such as can be produced by a certain single force, wliich latter force is therefore called the Resultant of all the other forces, whilst those other forces are called the Components of such Resultant. 31. Prop. YIII. If the adjacent sides of a jparallelo- gram represent the component /orces in direction and mag- nitude, the diagonal will represent the resultant force in direction and magnitude. COMPOSITION AND EESOLUTION OF FORCES. 23 Let AB and AC re- present, in direction and magnitude, the two com- jyonent forces which act at A. Complete the pa- rallelogram ACDB, and draw the diagonal AD, Then AD will represent the Resultant of AB and AC, (1) indirection, and (2) in magnitude. From D draw DM and DN perpendiculars to AB and A C, produced if necessary. (1) Then in the triangles DBM, DCN, aDMB = a right angle = Z.DNC, and aDBM^aBAC, (since BD, AC are parallel, and MBA cuts them), = aDCN-, .*. the third angle, BDM, of the one triangle = the third angle, CDN, of the other; and the triangles are equi- angular and similar ; hence, CD : DK :: BD : DM, and alternately, CD : BD :: DN : DM. Now, if there be 2^. Lever AD whose fulcrum isi), which is acted on by the forces AB^ AC, applied at A, since Force in the line AM : force in the line AN :: AB -.AC, :: CD : BD, :: DN : DM, the two forces acting on the Lever AD are inversely as the perpendiculars from the fulcrum on their lines of action, and therefore the Lever will be kept at rest about D by them (Art. 28). Wherefore the Lever will also be kept at rest by the Resultant of those forces ; because that single force produces the same effect as they do, when they act at the same point and at the same instant. 24 MECHANICS. Tills Resultant therefore must act in the line AD, for it keeps the Lever at rest, which it could not do, were it to act at A and make anj angle with the Lever AD on either side of it. Hence the diagonal of the parallelogram represents the Resultant in direction. (2) Again : Having shewn, that the Resultant of AB and AC acts in the line of the diagonal, next to prove that the diagonal represents it in magnitude as well as in direction. Produce DA, and suppose a force AE to be taken in it equal and opposite to the Resultant of AB and A C. The joint effect of AB and A (7 will now be counteracted by AE; and the point A, which is acted on by the three forces AB, AC, and AE, will remain at rest, so that any one of them may be considered as the Resultant of the other two. ici'j^o,.. "^ • ;,* ■ -/•■- *- • "' Whatever, therefore, be the effect produced by the joint action of AE and AC, it is counteracted by AB; that is, AB must be equal and opposite to the Residtant of AE mxdiAC. Complete the parallelogram AEFC, and draw the dia- gonal AF. By the first part of the Proposition, AF is the line in which the Resultant of AE and AG acts; and since the force AB is equM and op- A posite to that resultant, AF must be in the same straight line with AB, and, therefore, it is paral- ^ lei to CD. Hence ADCFh sl parallelogram; and therefore AE=FC=AD. The Resultant, therefore, of AB and AC (which is equal and opposite to AE) will be properly represented in COMPOSITION AND RESOLUTION OF FOrX'ES. 25 magnitude by AD, tlie diagonal of the parallelogram of whicli AB and A C are the sides. This Theorem is commonly called " The Parallelogram of Forces''. > 32. Prop. IX. If three forces, represented in magni- tude and direction hy the three sides of a triangle, act on a point, they will keep it at rest. Let the sides AB, BC, 35 CA, taken in order''^ , of the triangle ABC, represent in magnitude and direction (see Art. 11) three forces which act on the point A ; they will keep ^ A at rest. Complete the parallelogram ABCD. Then ^Z> is parallel and equal to BC', and it will, therefore, represent in magnitude and in line of action (Art. 11) the force, which acts at the point A in the direc- tion BC and is represented in magnitude by BC, Now the forces AB and AD acting at A will produce a Residtant AC, by Prop. VIII. If, therefore, a force CA act at A, the force AC will be counteracted, and the point A will remain at rest. Wherefore, if three forces, represented in magnitude and direction by AB, AD, CA, — (or, which is the same thing, if they be represented by the three sides, AB, BC, CA, taken in order, of the triangle ABC) — act on the point A, they will keep it at rest. 33. Cor. 1. It appears from this proof, that if two sides of a triangle ABC, 2iS AB and BC, takeyi in order, represent in magni- tude and direction two forces which act at the same instant on the point A, the third side of the triangle, AG, not taken in the same order 2i^ AB and BC, represents their Resultant in magnitude and direction. * By the expression "taJcen in order'''' it is meant, that, if ABC he the triangle, and AB he one of the forces, BC (and not CB) is the next, and CA (not A C) is the third ; so that the forces are described in the same di- rection round the triangle, proceeding from A to £, from B to C, and from C to A again. 26 MECHANICS. 34. Cor. 2. Since the forces, which keep the point at rest, are represented by the sides of a triangle, it follows that the sum of any two of them must necessarily be greater than the third. Eucl. i. xx. Questions on Chap. III. (1) Have two or more forces, which act in the same straight line, a * Resultant ' ? If so, how is it determined ? -, (2) Can the Resultant of two forces in any case exceed the sum of the forces 1 Under what circumstances is it least i Can it ever be nil ? (3) If two forces, acting on a point at the same instant, are given both in magnitude and direction, their Resultant is readily found by Prop. VIII. Conversely, having given the Resultant, can you find the Components ? ^ (4) The Resultant of two forces is generally determined by the construction of a parallelogram; how would you determine the Resultant of three or nfiore forces 1 > (5) Can the Resultant in any case be equal to one of the Com- po7ients ? If so, what are the conditions ] > (6) Two forces, represented by 3 and 4, act on a point in direc- tions at right angles to each other, what is the numerical measure of the Resultant ? (7) In Prop. IX. how can three forces, represented in magnitude and direction by the three sides of a triangle, act on a point / Does direction always mean the line of action ? -• (8) Is it possible for three forces, represented by the numbers 3, 4, 7, acting on a point, to keep it at rest \ If not, why not % > (9) At what angle must two equal forces act, that their Re- sultant may be equal to each of them ? (10) If p and q represent two forces, acting on a point in directions at right angles to each other, what is the algebraical expression for the Resultant ? (11) The Resultant of two forces, which act on a point at right angles to each other, is given both in magnitude and direction, and tlie direction of one of the forces is also known. From these data can you determine geometrically both Components ? (12) Ii four forces, represented by the sides of a square, taken in order, act on a point, trace out the eflect, and say what it will be. MECHANICAL POWEKS. 27 CHAPTER IV. MECHANICAL POWERS. ;> 35. The 'Mechanical Powers' are certain simple machines by means of which power is, for the most part, gained in the application of force either to support weights, or to give motion to bodies. They are six in number, the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw. The Lever has been already treated of in Chap. II. The Wedge, and the Screw, are not included in this Course. "We proceed with the WHEEL AND AXLE. ^ 36. Definition of Wheel and Axle. The Wheel and Axle consists of a cylinder AB^ called the Axle, rigidly attached to another cylinder CD of greater diameter, called the Wheel. The two cylinders have a common axis, EF, about which the machine can turn. The Power (P) acts by means ^i of a rope, or chain, coiled round the Wheel, and fastened to it at one end; and the Weight (TF) acts in a similar man- ner by means of a rope, or chain, coiled about the Axle, but tending to turn the machine round in the opposite direction. If the axis EF be supported in an horizontal position, then P and W may hoth be represented by weights hanging vertically, as in the annexed diagram. ^^ 37. Prop. X. There is an equilibrium upon the Wheel and Axle, when the Power is to the Weight as the l. radius of the Axle to the radius of the Wheel. '^ The efforts of P and W to turn the machine round its axis will be the same in whatever plane they act perpendi- cular to the axis. •\v 28 MECHANICS. Suppose, then, the lines in which the Power and the Weight act to be in the plane which coincides with the junction of the Axle with the Wheel. Since tlie Wheel and Axle liave a common axis, this plane, or section, will present the appearance of two concentric circles, as AB, and CD^ Avith a common centre 0. Join with M and JV, the points in which the strings leave (and therefore touch) the circumferences of the circles. OM and ON are therefore perpendicular to MP and NW, the lines in which the Power and the Weight act, and the whole machine may be considered as a simple Lever, MON, with fulcrum 0, and the two forces, P and IF, acting per- pendicularly at M and N. Now there will be equilibrium, by Art. 28, in this Lever, and therefore in the case of the Wheel and Axle, when P: W:: ON: OM, :: radius of Axle : radius of JVheel. 38. Cor. 1. When P and W are both weights, since 03/ and ON are both perpendicular to the vertical lines in which P and W act, and also pass through the same point 0, MON is a horizontal straight line. 39. CoR. 2. If the thickness of the ropes, by which P and W act, be taken into account, and each be represented by It, then, since the action of the power and weight must be supposed to be trans- mitted along the axes of the ropes, there will be equilibrium, when P : W V. ONA-t : OM+t Hence it appears, that the ratio of the Power to the Weight is greater as the thickness of the ropes is increased ; for, if any quantity be added to the terms of a ratio of less inequality, that ratio is increased. (Wood's Algebra, Art. 227.) Wherefore, also, it is shewn, that the mechanical advantage in using the Wheel and Axle is diminished by increasing the thickness of the ropes or chains. For the ratio of P to W is increased, which means that a greater power is required to balance a given weight. MECHANICAL POWERS. 29 THE PULLEY. -> 40. Definition of FuLhEY, A Pulley is a small wheel moveable about an axis tlirough its centre, and having a groove of uniform depth along its outer edge to admit a rope or flexible chain. The ends of the axis are fixed in a frame called the Bloch. The Pulleij is said to hejixed, or moveable, according as the Block is fixed, or moveable. The Power acts by the rope, or chain, or string, which works in the groove ; and the Weight is fixed to the Bloch. In the following Propositions the groove of the Pulley is sup- posed perfectly smooth; and the weight of the machine is not taken into account. *> 41. Prop. XI. In the single moveable indley, ivhere the strings are parallel, there is an equilibrium when the Power is to the Weight as 1 to 2*. The annexed diagram represents the case here supposed. A power, P, acts by means of a &ixmg PAD BR, Rt> which passes under the moveable pulley and is made fast at R. A weight, W, is fixed to the block by a string, and the line of its action is (7TF, G being the centre of the section of the pulley made by a plane passing through PA -«^ and RB. By the supposition, PA, RB, and M-*d CW, RYQ parallel; and they are in one plane. 1 ^ Also PA, RB, are tangents to the circle at A and P; therefore, if AC, BC, be jomed, zP^4(7=a rlirht angle = aRBC, and .*. ACB is a straight line^ and (7 IF is at right angles to AB. Now, the whole machine being at rest under tbe action of these forces, we may consider it as a straight lever AB kept at rest round B, as a fulcrum, by the power P acting perpendicularly upwards at A, an I the weight W acting ■M A^' A • More correctly thus: — ^^when there is equilibrium, the Power is to tie Weisrht &c." 30 MECHANICS. perpendicularly downwards at C; but in this case, and therefore in the case first supposed, the forces are inversely as their distances from the fulcrum, that is, F : W :: BC : AB :: 1 : 2, ~:^ 42. Pkop. XII. In a si/stem in v^Mcli the same string passes round any number of pulleys, and the parts of it be- tween the pulleys are parallel, there is an equilibriu'rn when Bower (P) : Weight {W) :: 1 : the number of strings at the lower hlock'^. The annexed diagram represents such a case as is here supposed. The system con- sists of two blocks, having each two pul- leys, the upper block being fixed, and the lower one moveable. The string, by which B acts passes round each of the pulleys, as shewn in the figure — the several portions of it are parallel to each other, and to the line AW vcs. which the Weight {W) acts. Since the same string continuously passes round all the pulleys, its tension must be everywhere the same, otherwise motion will ensue, which is contrary to the supposition. But for the outer portion of the string, to which the power is immediately applied, this tension is P; therefore it is B through- out ; that is, each string at the lower block exerts a force P in opposition to W, And the forces are all parallel, therefore, since they balance, W is equal to the sum of those which act in the opposite direction, that is, is equal to P multiplied by the number of strings at the lower block, or P : W \\ 1 : N"". of strings at the lower b%3ck, whatever that number may be. 43. Prop. XIII. In a system in which each 'pulley hangs by a separate string, and the strings are parallel^ • More correctly thus : — " when there is equflibrium, the Power ia to the Weight &c" MECHANICAL POWERS. 31 F G It Q O Ap c there is an equilibrium when P : W :: 1 : that 'power of V 2 whose index is the number of moveable pulleys *. In tliis system the strlnf^s are fixed at jP, Gy H, &c. and pass round the moveable pulleys A, JB, C, &c. respectively, as in the figure, to the last of which the power P is ap- plied. The strings are all parallel, and in the direction in which the weight W acts. Now, (7 being a single moveable pulley with parallel strings, when there is equilibrium, by Prop. XI., the pressure downwards at C=2F; therefore the Tension of the string from C round the pulley P = 2P. Hence, the weight supported at the move- able pulley P= 2x(2P)=2^xP= tension of string which passes round the pulley A. So. weight supported at moveable pulley A = 2x(2'xP)=2'xP. Therefore, when there is equilibrium on the system of three moveable pulleys, as here represented, W= 2'xP. And so on, by the same mode of reasoning, if n be any number of moveable pulleys, it will appear, that, when there is equilibrium, TF= 2"xP ; wV or, P : W:: 1 : 2". J5k 44. Definition of Inclined Plane. An Inclined Plane, in Mechanics, means a plane inclined to the horizon, that is, neither horizontal, nor vertical. The plane is here supposed to be perfectly smooth, and rigid, and capable of counteracting, and entirely destroying, the effect of any force which acts upon it in a direction perpendicular to its surface. "When, therefore, a body is sustained on an inclined plane, by a Power directly appUed to it, the case is that of a body kept at rest • More correctly thua : — " when there Is equilibrium, P : W &c." 32 MECHANICS. by three forces, its own Weifjlit acting vertically, the power applied, and the resistance of the plane in a direction at right angles to its surface. The Power must also obviously act in the same x>lane as the other two forces, otherwise motion would ensue. It is evident, that the Inclined Plane will bear, or take ojBf, a portion of the Weight, how much is the question ; that is, we are required to find the ratio of P to W^ when there is equilibrium. 5*- 45. Prop. XIV. The weight ( W) heing on an Inclined Plane^ and the force (P) acting j)arallel to the jAane^ there is an equilibrium when P \ W v. the height of the ])lane : its length *. [The '■length^ of the inclined plane is the section of it made by the plane in which the forces act — the ^ height of the plane is the perpendicular let fall from the highest point in it to meet the hori- zontal plane through the lowest point — and the ' base ' is the distance from the lowest point to this perpendicular. Thus, in the annexed diagram, AB is the ' length' of the plane, BG its ' height\ and -4 C its 'hase\'] Let AB be the length of the Inclined Plane ; A G its horizontal base; and BC, perpendicular to AG, its height. Let the Weight (TF) be supported on b the plane at B by the Power ( P) acting ^ ^ in the direction BB parallel to the plane ; then, when there is equilibrium, P : W :: BG : AB, From B draw BE at right angles to AB^ meeting the base in E\ and from B draw ^P vertical, or at right angles to AE, meeting AB in P. Then in the triangles EFB and ABG, '.' EE, BG are parallel, angle PPP = angle ABG; and angle EBE— a right angle = angle BGA, ,'. angle PPP= angle CAB, and .*. the triangles are equiangular and similar. Hence BE : EE :: BG : AB. * More correctly thus: — "wJien there is equilibrium, P : W kc." MECHANICAL POWERS. 33 Now the body at D is kept at rest by three forces, the weight TF acting vertically, the reaction of the plane acting at right angles, and the power P acting parallel, to the plane. And these forces are respectively parallel to the sides of the triangle DEF, therefore those sides will repre- sent them in magnitude as well as in direction (by the con- verse of Prop. IX.). Hence F : W :: DF : FF, :: BC : AB, (by what has been proved.) 46. Definition 0/ Velocity. By the Velocity of a body in motion is meant the Degree of Swiftness, or Speed, with which the body is moving. And this "degree of swiftness" is described, or measured, by stating how long the line is that the body moves through, with uniform swiftness, in some given por- tion of time. Thus a clear notion would be conveyed of the Velonty of a coach, if it were said to be nine inlUs in an hour ; the space moved over by the coach being nine miles, and an hour being the portion of time during which the motion took place. If only the space that is moved through were mentioned, and nothing were said about the time of describing that space, — or if the time only were given, — it is evident that no idea could be formed of the degree of swiftness, that is, of the Velocity, of the coach. The Velocity of a body is ^neasured, for the. most part, in ma- thematical investigations, by the number of /^e^ passed over by the body, moving uniformly, in a second of time. Co a. Since the quicker a body moves the more space it will pass over in a given time, it will follow from the observations just made, that The Velocities of two bodies which move during any (the same) time, are in the ratio of the spaces which the bodies resjMC- tively describe — each with uniform swiftness — in that time. 47. Prop. XV. Assuming that the arcs which subtend equal angles at the centres of two circles are as the radii of the circles, to shew that, if P and W balance each other on the Wheel and Axle, and the whole he put in motion, P: Ww Ws velocity : P's velocity, L. C. C. 3 S4: MECHANICS. Suppose P and W to act (as in Prop, x.) at the circum- ferences of the Wheel and the Axle, in the same plane, perpendicularly to the horizontal radii OM, ON; and let the Avhole be put in motion round the axis*, so that m On may become the horizontal diameter, instead of MON, P will now act at m at right angles to Om, and W will act at n at right angles to On ; and the velocities of Pand TFwill be as Mm to Nn, since Mm, being the length of string unwrapped from the wheel, is the space through which P will have descended in the same time that IF has ascended through Nn, Kow, since P and W balance each other, by the suppo- sition, P : W :: ON : OM, (by Prop, x,) :: Nn : Mm, by the assumption, ' /. P : W :: TT^'s velocity : P's velocity. Cor. PxP's velocity = PFx^F's velocity. 48. Prop. XVI. To sliew that if P and W halance each other on the Machines described in Propositions XI., XII., XIII, and XIV., and the ichole he put in motion, P : W :: W's velocity in the direction of gravity : P's velocity. 1st. In the case of the single moveable pulley (Prop. XI.), let G, the centre of the pulley (see Fig.) be raised through any height, as an inch; TF will thereby be also raised through an inch, and each of the strings RB, AP, Avill have been shortened an inch; so that, if P continue to keep the string tight, it will have moved through two inches in the time that TF has been raised one inch. * Tliis motion is supposed to be given by the application of some extra- neous force, wliicb is removed as soon as the displacement of the machine is effected ; and then the system is in equilihrium in its new position. MECHANICAL POWERS. 35 But P : W :: I : 2, by Prop, xi., since thej balance each other, :: 1 inch : 2 inches, :: space described by W: space described in tlie same time by JP ; .', F : W :: TF's velocity : P's velocity. 2nd. In the system where the same string passes round all the pulleys, and the parts of it between the pul- leys are parallel, as in Prop. xii. (see Fig.), if the lower block be raised through any height, as an inch, each of the strings between the upper and lower blocks will be short- ened an inch, and therefore in the time that W moved through one inch, in order to have kept the string tight, P will have moved through as many inches as there are parallel strings at the lower block. But P : W :: 1 : No. of strings at lower block, since they balance each other, :: space described by W : space described in same time by P, .*. P ; W :: TF's velocity : P's velocity. 3rd. In the system where each pulley hangs by a separate string, and the strings are parallel (Prop. Xiii.), if W be raised through an inch, and P have also moved through such a space that the strings are kept tight, A will have been raised through one inch, and B through two inches (by the first case proved in this Proposition). And P having been raised through two inches, G (by the first case) will have moved through 2x2, or 2^ inches; and the next moveable pulley (the fourth) will have been raised through 2x2^, or 2^, inches. By the same reasoning, if n were the number of moveable pulleys, the highest of them will have moved through 2""^ inches, and the end there- fore of the string by which P acts, and therefore P itself, through 2** inches. 3-^2 36 MECHANICS. ButP: TF:: 1 : 2", by Prop, xiir., since tliej "balance each other, :: 1 inch : T inches, :: space described by W : space described in same time by P, /. P : TF :; TF's velocity : P's velocity. 4th. In the case of the Inclined Plane, let the weight ( TT") be kept at rest at B on the inclined plane AB by the power (P) which acts parallel to the j^lane by means of a string PP; then, if P be made ^ ^ to move through the space P/), TF will move through an equal space BG along the plane. Through G draw GH horizontal, and through P draw BB vertical. Then, TT^, by being moved through BG, has been raised vertically, that is, in the direction of gravity, through BB. Since, therefore, in the time that P moves, through a space equal to BG^ TF moves vertically through BE, BG is to BH as the velocity of P in the direction of its action is to the velocity of TF in the direction of gravity, Now, in the triangles GBH, ABC, since 6^P^is paral- lel to A C, angle B GH= alternate angle BA C ; and since BH, being vertical, is parallel to BG, angle GBH= alter- nate angle ABC; also angle P^6^ = a right angle = angle BCA, .*. the triangles are equiangular, and similar. But P : TT^ :: BC : AB, by Prop, xiv., since they balance each other, :: BE : BG, by similar triangles, .•. P : TF :: TF's vel. in direction of gravity : P's vel. in the direction of its action. Cor. In each of the above cases PxP's velocity = PFx (^F's velocity in direction of gi'avity. MECHANICAL POWERS. 37 Questions on Chap. IY. (1) In the Wheel and Axle, what is the difference between axle and axis ? Is the axis fixed, or rotatory ? (2) In the Wheel and Axle, is there any advantage in having the rope, which passes round the Wheel, thicker than that which passes round the Axle? ^ (3) In the Wheel and Axle, the radius of the Wheel being three times that of the Axle, and the rope on the Wheel being only strong enough to support a tension of 36 lbs., what is the great- est weight which can be lifted 1 (4) In Prop. X. how will the proof be affected if P and W, instead of acting vertically, act by means of strings in any other directions ? (5) What Mechanical Powers are employed in a Crane of the ordinary construction ? (6) In the single moveable Pulley, is any mechanical advantage gained, if the weight of the pulley be not less than the power? (7) "Why is it easier to move a heavy body when it is placed upon rollers, than to draw it along a rough horizontal plane ? (8) In Prop. XII., if the number of strings at the lower block be 6, what limit must be put to the weight of the lower block, so that any mechanical advantage may be gained by this system of pulleys ? (9) In Prop, xin., if a weight of 1 lb. be supported by 1 oz., what is the number of moveable pulleys ? Draw a figure to repre- sent this case. (10) In Prop, xiii., will the ratio P : W he increased or dimi- nished by taking into account the weights of the strings? (11) In the proof of Prop, xrv., where is it assumed, that the Inclined Plane is perfectly smooth ? (12) Why is it easier to push a heavy body up a smooth Inclined Plane than to lift it through the same vertical height ] % (13) A smooth Inclined Plane rises 3|- feet for every 5 feet of its length, what force must a man exert parallel to the plane, to prevent a weight of 200 lbs. from slipping down ? (14) A railway train travels over 150 miles in 5 h, 40 m., what is its average Velocity in feet per second \ 38 MECHANICS. (15) A race of 2 miles, 3 furlongs, and 62 yards, was run in 4 min. 12 sec. Wliat was the average Velocity in feet per second ? (1(,) A body moves uniformly through 40 feet in 3 seconds, and another body through 25 yards in 6 minutes. What is the ratio of their Velocities? (17) The earth's radius at the Equator is 3962*8 miles; and it makes a complete revolution about its axis in 23 h. 56 min. ; what is the Velocity of a point at the Equator in feet per second? (IS) In Prop. XV., are P and W supposed at first to balance each other ? If so, what puts the machine in motion ] Is the motion uniform ? (19) In Prop. xvL, Case 3, if W be made to descend with a given velocity {v\ what will be the Velocities of the ^Q\QX2iS. pulleys? (20) In Prop, xvr., Case 3, would the result be true, if the pul- leys were of different sizes? CHAPTER Y. THE CENTRE OF GRAVITY. P 49. Definition o/" Centre of Gravity. The Centre of Gravity of any body, or system of bodies, is that point upon which the body, or system, acted on only by the force of gravity, will balance itself in all posi- tions. This definition supposes all the particles of the body, or system of bodies, to be rigidly connected ; and the point, called the Centre of Gravity, is also supposed to be in rigid connection with all the parts of the system, while that point is itself maintained at rest. The Centre of Gravity of a body, or system, is, as it were, the fulcrum, round which the body, or system, when placed in any position, has, of itself, no tendency to turn, although the body be capable of being moved in any way about that fulcrum. 7 CENTRE OF GRAVITY. 39 50. Prop. XYII. If a body balance itself upon a line in all positions, the Centre of Gravity of the body is in that line'^. Let the body AB "balance Itself in all positions upon the straight line CD, the Centre of Gravity of the body shall be in CD, For, if not, let a point G, with- out CD, be the Centre of Gravity ; and, first bringing CD into an hori- zontal position, turn the body round CD until G is in the same horizon- tal plane with CD ; then draw GH perpendicular to CD., meeting it in H, and GF vertical. Xow, since, by Definition, the body will balance itseli on G in all positions, it will balance itself on G in this position, that is, the resultant of all the forces acting on the body passes through G ; and since these forces (being the pressures exerted upon the several particles of the body by the force of gravity) are all parallel and vertical, the re- sultant will also be vertical, and equal to the sum of them, \\z. the weight of the body. Eeplacing, then, all the forces acting on the body by their resultant, we have the case of a single force acting perpendicularly at G to turn the lever GH round the fulcrum H ( CD, and therefore H, being supposed fixed in position) ; which force is not coun- teracted by any other, and therefore will turn the body round H, that is, round CD, But, by supposition, the body balances itself upon CD in all jwsitions. Hence, the assumption that G lies anywhere without CD leads to an impossibility ; and therefore G can only be in CD, 51. Prop. XYIIT. To find the Centre of Gravity of two heavy points f, and to shew, that the pn^essure at the * That is to say — If there be a line round which, as an axis, a body can be made to revolve, so that, when the line is held in any position, the body, after being made to revolve round it into any position, remains at rest, the Centre of Gravity of the body is in that line. — It is evident, that the line must pass through the body. + By "heavy points", in this Proposition and the next, are meant exceedingly small material bodies, and not geometrical points. For a geome- 40 MECHANICS. Centre of Gravity is equal to the sum of the weights in all positions. Let P and Q .he the Aveiglits of two heavy points A and B, supposed to be connected by a straight rigid rod AB ivithout weight. In AB take a point (7, such that i?C : AB :: F:F-\-Q; then, m dividendo, BC :AB-BC ::P:'F+Q-F, >^ ovBC:AC::F: Q. ■Through C draw MCN horizontal ; and through A and B draw the vertical lines PAM and QNB ; these last are the lines in which the weights P and Q act. Then, the angles at M and iV being right angles, and the angle ^ (7J/ being equal to the opposite angle BCN, the angle CAM is equal to the angle CBN, and the tri- angles ^(7J/and BCJSfsiVe equiangular, and .*. similar. Now, P: Q'.'.BC'.AC, :: CN : CM, by similar triangles ; therefore, if ABC be considered as a lever with fulcrum C, since P and Q are inversely as the perpendiculars drawn from the fulcrum to the lines in which the forces act, by the converse of Prop. vi. (Art. 26), P and Q will balance each other on C. Also, if AB be turned round C into any other posi- tion, the same reasoning holds ; and therefore A and B will balance on C in all positions of AB. Hence, by Defi- nition, C is the Centre of Gravity oi A and B. Again, by what has been shewn, the weights Pand Q will balance on (7, when ACB is horizontal. But, in that case, by Axiom li. (Art. 19), the pressure on the ful- crum is equal to the sum of the weights. And in any other position of AB, as in the fig., P and Q will produce trical point does not possess len^h, or breadth, or thickness, and conse- quently can be of no weight, since it can contain no*mai = weight of a column of the fluid whose base is CD, and whose height is the vertical depth of CD below the horizontal plane of the surface of the fluid in the vessel. 65. From Art. 52 it appears, that the pressure on the Centre of Gravity of a system of bodies, considered as heavy pomts, is the same as if the weights of the bodies were collected at it. So long, there- fore, as the Quantity of Matter in the system remains the same, the amount of Pressure produced in the direction of gravity by the weights of the several f>arts of the system is the same, whatever be the manner in which they may be arranged. Now though, at first sight, it might appear probable that the pressure on the bottom or the sides of a vessel, filled with fluid, would depend (somehow or other) on the quantity of the fluid by which that pressure is produced, such is not the case ; for it is found, both from experience and by theory, that the pressure produced by a fluid, at rest, on the inner surface of the vessel containing it, is not dependent on the quantity of the fluid — a fact apparently so much at variance with the law governing the amount of pressure produced by solid bodies, that it has been called the Hydrostatic Paradox. 4—2 52 HYDROSTATICS. 6(j. Peop. V. To explain the Hydrostatic Paradox. The Hydrostatic Paradox is this : — " Any pressure, however small, may he made to counter- halance any other Pressure, however great, hy means of a small qiiantity offlidd^\ The top and bottom of a vessel AD are boards which are connected together by leathern sides. The vessel communicates with a vertical tube EF of small uniform bore, by means «jf a horizontal pipe CE, Let A be the number of square inches in the area of the board AB, and a the number of square ^V inches in the area of a horizontal section ^ A X7D of the pipe EF, B Let AB be held in a horizontal posi- tion, and water be poured into the tube until it just rises in AD to AB. The water will therefore rise in the tube to G, a point in the same horizontal plane as AB. Prop. ill. If now* there be a heavy weight of W lbs. laid upon AB, it is found, that it can be supported at rest by a small additional column EG, of water poured into the tube. To shew the reason of this ; Since the sides of the tube FG are vertical, the pres- sure on the horizontal section of the tube at G is the weight of the fluid column FG. Suppose w to be this Aveight, in pounds ; then, since the pressure at every point in the same horizontal plane of a fluid at rest is the same, and assuming the fluid to be made up of small equal par- ticles. Pressure on the area A oi AB \ pressure on the area a at 6^ :: N^ of particles in A : ls°. in a :: ^ : a ; or W : w :: A : a; » PrwESSUEE OF NON-ELASTIC FLUIDS. 53 If, therefore, W be given, and however great it may be, w may be made as small as we please by diminishing a, or increasing A ; that is, by adjusting the dimensions of -4 and a, any py^essure however small miay he made to balance any other pressure however great. Whether the pressure on the area a 2ii G he produced by the weight of the fluid column GF, or by means of a piston acted on by some force, the pressures on the areas, a at G, and A ^t AB, will still bear to one another the ratio of « to ^, a ratio which is wholly independent of the quantity of fluid contained in the vessel. 67. Prop. VI. If a hody floats on a fluid, it dis- places as much of the fluid as is equal in weight to the weight of the hody ; arul it presses downwards, and is pressed upwards, luith a force equal to the weight of the fluid dispAaced, Let ABCD be a body at rest that displaces the portion BEDCB of the fluid on which it floats. The weight of the floating body ij/'^i^St) produces a pressure which acts verti- ^^^ " " j^^z^ :. cally downwards. Therefore the pres- ^ ^^ 1^^=:^^^= sure of the fluid which keeps the body at rest must act vertically upwards, and be equal to the weight it balances. Now suppose the floating body to be removed, and the space BEDCB fllled with fluid of the same kind as the surrounding fluid ; the equilihrium of the fluid will not be disturbed ; neither will the pressure of that part of it which was formerly in contact with the surface of the float- ing body be altered, if the particles of fluid in BEDCB be supposed to become permanently connected with one another, and to form a solid. Let this take place ; then the pressure downwards of the part BEDCB of the fluid which becomes solid is its weight. And since this pressure is counteracted by the same sustaining power as that which balanced the weight of the floating body, the weight of the floating hody must he equal to the weight of the fluid it displaces. 54 . HYDROSTATICS. Questions on Chapters I. axd II. (1) What are the characteristic differences between a Fluid and a Solid? (2) If a Fluid be a ^material 'body\ according to the Definition, how can Steam be a Fluid? (3) If a 'material body' be compressible, is it necessarily elastic ? (4) Is Water elastic, or inelastic ? (5) If " Fluids 2yress equally in all directions'^ does this mean, that a Fluid, acted on only by the Force of Gravity, will press i(p- icard.s as well as downwards ? (6) In Prop. II. the pressure of the Atmosphere on the surface of the fluid is not taken into account ; is the truth of the proposition affected thereby % (7) Pressure is always the result of force; what then is the force suj^posed to be acting in Prop. ii. ? (8) If the pressure be different at different points of a fluid at rest, how can it be of ' uniform density'' ? (9) Is it assumed in Prop. iii. that the fluid is of uniform density? If so, where? (10) What/orc^s are supposed to act on the fluid in Prop. iii. ? (11) "What is meant by the ^surface' of a fluid in Props, ii., and III. ? (12) In Prop. IV., the pressure of the atmosphere on the surface of the fluid is not reckoned ; will not that greatly affect the pressure on the bottom of the vessel ? (13) How can the pressure on the bottom of a vessel filled with fluid, and acted on only by the force of gravity, be greater than the whole weight of the fluid ? (14) BramaKs p^ress, used by packers and others, is constructed on the principle explained in Prop. v. ; what are the practical limits to its power ? (15) Why is it necessary, that the fluid used in the Hydrostatic Paradox, and in BramaKs press, should be non-elastic? (16) If the floating body in Prop. vi. were wholly immersed, and at rest when left to itself, would Prop. vi. hold true ? (17) If a vessel be quite filled with fluid, and a solid body be put into it, which floats, and " displaces as much of the fluid as is equal in weight to the weight of itself", will the solid body increase the pressure on the bottom of the vessel, or not ? SPECIFIC GRAVITIES. 55 CHAPTEE III. SPECIFIC GRAVITIES. 68. The Bulk, or Volume, or Content, or Magnitude, of any body is measured by the number of times it contains that of some other body previously fixed upon as a standard of magnitude, or unit. A cube, wliose edge is an inch in length, is called ''a cubic inch". In the following pages a cubic inch will be taken for the unit of solid measurement; so that when it is said, that M is the Ijulk^ volume, content, or magnitude, of a body, it is meant that the num- ber of cubic inches in the body is the number of units in M. 69. Definition of Specific Gravity. The Specific G-ravity of any substance is tlie weight of a unit of its magnitude, or volume. If, as stated in the last Article, the magnitude of a body be measured by the number of cubic inches it contains, and the weight of one cubic inch be given in grains, then it will follow, that, S being taken to represent the Specific Gravity of any substance, S is the number of grains that one cubic inch of that substance weighs. The Tables, which are called " Tables of Specific Gkavities ", give the Ratios which the weights of bulks of various substances bear to equal bulks of water. In other words, they give the number of times that the weight of any bulk of each of the substances con- tains the weight of an equal bulk of water. It having been found by experiment that the weight of a piece of Iron : the weight of a bulk of Water of the sanfie size :: 7"8 : 1, and that the weight of a piece of Silver : weight of an equal bulk of Water :: 10"5 : 1, and so for other substances, Tables have been formed, in which the numbers 1, 7"8, 10*5, &c., are placed opposite the words, "Water", "Iron", "Silver", &c. By means of these Tables (as will be shewn), the weight of any bulk of any of the sub- stances so registered can be determined, if the weight be known of some particular bulk of any one of them. . The numbers given in these Tables are generally called the " Specific Gravities " of the several substances registered ; but the 5G HYDROSTATICS. enunciation of Prop. rii. Art. 70 will not permit them to be so called licre. The " Tajm.es of Specific Gravities " give — riatinuni 21-53 Gold 19-4 Mercury 13'6 Lead 11-4 Silver 105 Copper 8*9 Iron 7'8 Tin 7-3 Zinc 6-9 Diamond . . Sea Water . . AVater . . . Proof Spirit Pure Alcohol . Air, at the surface of the Earth — the average . . . . Ice 3-52 1-027 r 0-93 0-825 0-0012.J 0-9-26 Hence it may be shewn, that— i. The weights of any two substances, are of equal hulk which are in the ratio of the numbers given by the Tables as corresponding to the substances. For, ic, w\ w", being the respective weights of equal bulks of Water and of any two substances, as Iron and Silver, since, as explained above, u- w 7*8 : 1, and w w 10-5; therefore, compounding these proportions, w' : w" :: 7-8 : 10 5. So that, if it be required, for example, ii. To find the weight of a cubic foot of Iron, having given that the iceight o/lO cubic inches of Silver is 61 ounces nearly — •/ Weight of a cubic foot (or 12x12x12 cubic inches) of Silver = 12xi2xi2x~ ounces; Weight of a cubic foot of Iron 12x12x12x61 7-8 X - — oz. 10 10-5 12x12x12x61x7-8 105 oz. = 7830 ounces, nearlv. 70. Peop. VII. If M he the Magnitude of a hodij, S I'ts 8j)ecifc Gravity, and W its Weight, W=M8. Suppose the unit of the measurement of magnitude to be a cu'ic inch ; then il/= number of cubic inclies in the body. SPECIFIC GRAVITIES. 57 And the Specific Gravity (S) is the weight of one cubic inch ; .*. the whole weight of the body = 21^8, or, W=MS. Vl. To find the relation which exists heticeen the TVeights, Magnitudes, and Specific Gravities, of two substances and of a compound foriined of them. Let W,M,S, W\M',S', TF'', J/'', ;S"', be the Weight, Magni- tude, and Specific Gravity, of each of the two substances, and of the compound, respectively. Then, it being supposed that the portions of the substances •which are combined together lose neither bulk nor weight by being mixed, i. M"^M -^M', ii. W"=W^ W; :. by Art. 70, iii. 31" S", or {3f+3r)S''=3fS+M\S\ And it might be shewn, that similar relations exist between the Weights, Magnitudes, and Specific Gravities, of the several substances and the compound formed of them, whatever be the number of the simple substances. 72. Cor. Let c, o-', S^, w^hen the mag- nitudes of the bodies are the same, W^S\ .'. weight of P : weight of an equal bulk of fluid :: S. G. of solid : S. G. of fluid, or, W: W+W" :: S : S' ; W • IK+ IV" But TFand W" are known weights, and >S" is supposed to be given ; therefore S may be found. It will be observed that it is not requisite for the exact weight of the Sinker to be known. 77. Pkop. XI. To describe the Common Hydrometer ; and to shew how to compare the Specific Gravities of two fluids hy means of it. * The better way in practice is to restore the equiUbrium by removing a weight W" from the scale A ; but the method described in the text perhaps renders the demonstration easier to be understood. 62 HYDROSTATICS. The common Hydrometer consists of two hollow spheres attached to each otlier, and of a cylindri- cal slender stem, wliose axis, if produced, would pass through the centres of both the spheres. The upper sphere is empty ; and the loAver is tilled with lead or mercury, so as to make the instrument float steadily in a vertical position when put into a fluid. The stem is graduated by divisions of equal length. The Hydrometer is made lighter than an equal bulk of any of the fluids whose Specific Gravities it is employed to compare. Suppose the bulk of the portion of the stem in- cluded between every two graduations to be one ^ four-thousandth part of the bulk of the whole instrument. When the Hydrometer floats vertically in a fluid whose S. G. is S, suppose 20 divisions are above the surface ; and wdien it floats in a fluid whose kS. G. is B\ let there be 30 divisions out. Kow, the weights of the bulks which are displaced of the two fluids are the same, each being equal to the weight of the instrument, by Prop. VI. If M and M\ therefore, be the magnitudes of the fluids displaced, by Prop. Yii., Jf X S = wei ght of the Hydrometer = M' xS' ; .-. S : S' :: M' : M, :: 4000-30 : 4000-20, :: 3970 : 3980; and the ratio of the Specific Gravities of the two fluids is thus determined. 78. A mark P is made at the point in the stem to which the instrument sinks in a fluid called '■'■ Proof SpirW\ which is a mix- ture consisting of equal weights, — {not equal magnitudes), — of pure Alcohol and of Water. Alcohol being lighter than Water, if a mixture of these two fluids contain a greater weight of the former than it does of the latter, it "will be lighter than an equal bulk of Proof _ Spirit, and the Hydrometer therefore will displace a greater bulk of it than it does of Proof Spirit, that is, it will sink deeper in SPECIFIC GRAVITIES. 6 Q the mixture than in Proof Spirit. Wherefore the surface of such a mixture will rise to a higher point in the stem than P. In such a case the mixture is said to be ^'- above ijroof". But if the weight of the Water contained in the mixture be greater than that of the pure Alcohol, the Hydrometer will not sink so low as to the point P, and the fluid is then said to be " hslow proof'. Questions on Chap. III. (1) When M is said to be the magnitude of a body, what does it mean 1 Is it a number, or what is it ? (2) What is the difference between Gravity and Specific Gravity? (3) In Tables of Specific Gravities what is commonly taken for the unit, or standard? (4) Can the same substance have a different Specific Gravity under different circumstances ? Is water such a substance ? If so, how can it be used as a standard ? (5) If M be expressed in *cubic feet, and ^S* in terms of the Specific Gravity of Water, in terms of what must W be expressed, in order that the equality W—MS may be true ? (6) What is the datmn necessary for rendering the formula, W~MS, practically useful; so that, for instance, knowing the Sjye- cific Gracity of gold (19 4) you could apj)ly the formula to find the weight of a cubic inch of gold 1 (7) In Prop. VIII., where is it assumed, that the floating body is oluniforTR density? (8) In the definition of Specific Gravity of a substance, is it assumed, that the substance is of uniform, density? (9) In Prop. VIII., if the whole body be just immersed, and float there, what conclusion do you draw as to the Specific Gravities of the body and fluid? (10) Would Prop. VIII. apply to an empty ship constructed wholly of iron, and floating in smooth water ? (11) What becomes of the result in Prop, viii., if the Specific Gravity of the body be greater than that of the fluid ? (12) In Prop. IX. what is meant by 'weight losV ? Is the weight actually lost? In what case will a body ^lose' its whole weight in a MAI HYDROSTATICS. (13) Wliy docs a man learn to swim better in salt water than in fresh ? (14) In Prop. IX., will the result be affected by the greater or less depth^ to which the solid is immersed, below the surface of the fluid? (15) Can the Specific Gramties oi fluids, as well as solids, he determined by means of the Hydrostatic Balance ? If so, how ? (16) In certain specimens of milk, how would you be able to detect those, if any, which had been adulterated with water ? CHAPTER lY. ELASTIC FLUIDS. 79. Peop. XII. Air has Weight. This is proved by the following experiments : — The weio'ht of a vessel from which the air has been exhausted is found to be less than w^hen it was filled with air. Or, if into a vessel already filled w^th common air more air \)q forced, the vessel will then be heavier than it was before. Also, if a bladder be weighed in a vessel, from which the air has been exhausted, first when the bladder contains no air, and again after air has been forced into it, a greater w^eight is required to balance the bladder in the latter case. Whence it is concluded, that ^^ Air has Weight^\ The same conclusion seems to follow from the simj)le considera- tion, that Air is 2i fluid, which according to Definition (Art. 58) is a Tnaterial hody, and the property of having weight is considered as necessarily belonging to Matter. See Art. 6. 80. Air is a substance which, besides having weight, possesses the property of self-expansion, so that the matter of which it consists IS continually striving to occuj)y a greater space ; and this effort to expand, measured by the pressure required to counteract it, is called the Elastic Force of the air. ELASTIC FLUIDS. 65 81. Prop. XIII. The elastic force of air at a given temperature varies as the density. This is proved by experiment. Let ABCD be a glass tube, of uniform bore, 5, having the legs AB and CD vertical, — the end A of the tube open, and the other end D closed. A quantity of mercury is poured in at the open end A, so as to confine a quantity of air in the shorter leg CD ; the air is then extracted from the longer leg, by means of an instrument for that purpose, and the mercury stands at different heights, E, F, in the two legs. Through E is drawn the horizontal line Ee, meeting the longer leg in e. Then, the weight of the mercurial column Fe = pressure downwards at e on a surface h, by Prop. lY., = pressure upwards at e on a surface &, by Prop. I., = pressure upwards at -£/ on a surface 5, by Prop, ii., = pressure by the air in DE on the surface h of mercury with which it is in contact; because, the whole being at rest, the pressures upwards and downwards on the same horizontal plane must be equal. Similarly, when more mercury is poured in, if its sur- faces stand at G and H in the two legs, draw- ^^ ing Gg horizontally through G, it follows, that Weight of the mercurial column Hg — pressure by the air m DG on the portion h of the surface of mercury with which it is in con- D tact. I" Now, if the lengths ED, Fe, GD, Hg, be ^ measured, it is invariably found, however the e quantities of air and of mercury used in the j^ experiment be altered, that (the air retaining the same temperature during the experiment) DE:DG::Eg:Fe, Now, the Elastic Force of the air in DE is measured, as has been shewn, by the weight of the mercurial column L. C. C* O QQ HYDROSTATICS. Fe; and that of the air in DGhj the weight of the mer- curial column Hg ; therefore, since the density of the mer- cury is uniform, Elastic Force of the air m DG ', that of the air in DE :: volume of mercury in Hg : that of mercury in Fe, :: height Hg : height Fe, (since the bore is uniform,) :: DE : DG, (by what has been shewn,) :: content of tube DE : that of tube DG, But the same quantity of air is m DG which was in DE, and its density will be inversely proportional to the space which it occupies*, that is, Density of the air in i>6^ : that in DE :: Content of DE : Content of D G, .*. Elastic Force of air in DG : that in DE :: Density of air in DG : that in DE^ or Elastic Force of air oc its Density. 82. Prop. XIY. The elastic force of air is increased by an increase of temperature. This is proved by experiment. If a bladder partially filled with air be brought near the fire, the enclosed air expands as it becomes heated, and the bladder becomes fully distended. As the enclosed air cools down, the bladder becomes more and more flaccid. » * This has not yet been proved in this Course; but it may be she\\Ti as follows: — Assuming certain standard units of volume and quantity of matter, the Density of a body is most simply defined to be the quantity of matter in a unit of its volume. If, then, D be the Density, V the Volume, Q the quantity of matter, of any body or substance, j^ whole quantity of matter _ Q whole N*^ of units of volume V ' and if, while D and V vary, Q remains invariable, ^ 1 ELASTIC FLUIDS. 67 83. Defs. a Valve is a kind of door which fits an orifice, so that being pressed by a fluid on one side it opens and allows the fluid to pass through, but keeps the orifice tightly closed, if the fluid press on the other side. Valves are of various forms, — a flap of leather [A) fastened at one edge, — a frustum of a cone {B) made ^\ *v^ 15 , ol metal, — a sphere (O), — or a plate of ^^ ^ — 'ZZ1--C1 metal {D) with an axis passing perpen- ^^ dicularly through it. By any of these — v_^^^- contrivances the flow of a fluid upicards would be prevented only by the weight of the valve; but a rush of fluid from above would carry the vdve along with it, and keep the orifice which the valve fits completely closed. When the fluids employed are very rare, — like air or gas, — the valcss are generally made of flaps of oiled or varnished silk, which, being attached at two or three points to the surfaces in which the orifices are situated, are raised by very slight pressures, and so allow fluids cf exceedingly small densities to pass under them. 84. Prop. XV. To describe the construction of the common Aik-Pump, and its operation. Construction. The Air-Pump consists of a vessel A, called the Receiver ^ made to fit a table BC. so as to be air- tight. A tuhe^ DE connects the Receiver with a cylinder EF, called the Barrel. At the bottom of the b| Barrel there is a valve E opening upwards, and a piston F (also furnished with a valve open- ing upwards) plays within the Barrel. The instrument is used for pumping the air out of the Receiver A. Operation. Suppose the piston F at its highest point, and the instrument tilled with air the same as that of the surrounding atmosphere. When the piston is forced down, the air at first in the Barrel is condensed, and its elastic force therefore increased (Prop. XIII.) — the valve Eis kept closed, and the valve in F being pressed on the under surface more strongly than on 5—2 68 HYDROSTATICS. the upper, opens and allows the air in the Barrel to escape through it, until the piston readies the bottom of the Barrel, and the valve F closes by its own weight. Next, on raising the piston, the external air keeps F closed, and, there now being no air in EF, the pressure on the under surface of the valve at E will open that valve, and allow air from the Receiver and pipe to flow into tlie Barrel, until F has reached its highest point. Then the valve E closes by its own weight. (The figure represents the instrument during the ascent of the piston.) When the piston descends again, another barrelful of air escapes through the valve F, as before. And so on, until the air in the Receiver becomes so rare, that its pres- sure is insufficient to overcome the weight of the valve at^. Cor. Hence it appears, that although the air in the Receiver can be very much rarefied, it cannot be wholly exhausted. 85. Prop. XYI. To describe the construction of the Condenser, and its operation. Construction. The Condenser is a Barrel AB, furnished with a piston A, which has a valve in it opening downwards ] at the bottom of the Barrel there is a fixed valve (7, also opening downwards. The neck of the Bar- rel communicates with a strong air-tight vessel D, called the Receiver, Operation. Suppose the instrument filled with common air, and the piston at its greatest height. On the piston being forced down, the air in the barrel is condensed, and its elastic force being therefore increased (Prop, xiii.), it keeps the valve A closed, opens the valve (7, and is driven into the Receiver. (The figure represents the instrument during the descent of the piston.) On the piston ascending, the elastic force of the air iifi the Receiver closes the valve (7, and keeps it closed \ and there now being no pressure on the under surface of the ELASTIC FLUIDS. 69 valve A, tlie pressure of the external air opens tliat valve, and the Barrel becomes filled again with air, which may be driven into the Receiver by forcing down the piston, as before. And in this manner the condensation of the air in the Receiver may be carried on to any extent required, as far as the strength of the Barrel and Receiver will permit. The communication between the Barrel and the Receiver can be cut off at pleasure by means of a stop-cock at E; and the Barrel is made to screw off and on at a point above E. 88. Prop. XVII. To explain the construction of the Common Barometer, and to shew, that the mercury is sus- tained in it hy the 'pressure of the air on the surface of the mercury in tlie basin. The Barometer is an instrument for measuring the pressure of the Atmosphere. It consists of a glass tube (see Prop, xviir.) of uniform bore, closed at one end, and not less than 33, or 34, inches long. This tube is filled with mercury, and the open end, being first stopped with the finger, is placed below the surface of some mercury in a basin, when the tube being fixed in a vertical position and the finger being withdrawn, the mercury in the tube subsides, and stands at a height, above the mercury in the basin, which varies on different days from about 28 to 32 inches. A graduated scale is attached to tlie upper part of the tube, to mark the height at which the mercury may be standing at any time. That the column of mercury is supported in the tube hy the jjressure of the atmos2)he7'e appears from the experi- ment, that, when the whole is put into the Receiver of an Air-Pump^ the mercury in the tube sinks more and more for every barrel of air that is pumped out, until at length it is all but on a level with the surface of the mercury in the basin*. But on readmitting the air into tlie Receiver, the mercury in the tube rises to its original level. • It has been shewn, in Pi'op. xv., that it is not possible to pump all the air out of the Receiver. If it were so, the surface of the mercuiy ia the tube would then be on lirccUely the same level as the surface in the cistern. h 70 HYDROSTATICS. 87. Pnor. XVIII. The pressure of the atmosjyJiere ?V accuratdy measured hy the icelght of the column of mercury in the Barometer. Let the surface of the mcrcuiy in the vertical tube stand at P; and let tlie area of the section of the tube made by the horizontal surface of the mer- cury in the basin be represented by AB\ CD any portion of area in that surface criual to AB] then Weight of mercury contained in PB = pressure downwards on the area AB, by Prop. ly. = pressure upwards on AB, by Prop. I. a = pressure upwards on CD ; since AB and CD are equal areas situated in tlie same horizontal plane of a fluid at rest, by Prop. ii. ^^ ^ —I— But the fluid being at rest, the pressure — doivnwards on CD, (whicli arises solely from the pressure of the air in contact with it), is equal to the' pressure upwards on it. Wherefore the pressure of the atmosphere on the area CD is equal to the weight of the column of mercury sup- ported in the [vertical] tube of the Baroineter, of a base AB equal to that area ; or " the pressure of the atmosphere is accurately measured by the weight of the column of Mercury in the Barometer'\ 88. Cor. 1. By Art. 64, the pressure of a fluid on a horizontal plane immersed in it was shewn to be the weight of a cohunn of the fluid, whose base is equal to the area of the plane, and whose height is the depth of the plane below the siu'face of the fluid. Wherefore, the pressure exerted by the atmosphere on such an area, — being measured by the weight of the vertiail column of mercury (of equal section) which is supported in the Barometer, — is equal to the weight of a column of mercury whose base is equal to the plane acted upon, and whose height is the same as that of the column of mercury supported in the vertical tube of the Barometer. ELASTIC FLUIDS. 71 89. Cor. 2. The pressure of the air being measured by the weight of the cohimn of fluid -which it supports, the Barometer might be filled with any fluid whatever. But mercury being by far the heaviest fluid known, the column of mercury required to pro- duce a given pressure is very much shorter than if any other fluid were employed. For example, if it took 30 inches of Mercury to balance the pressure of the air, then since Mercury is 13' 6 times as heavy as Water (see Art. 69), it would take a column of Water 13'6 x 30 inches, or 34 feet high, nearly, to produce the same eflect. 90. Cor. 3. The mercury in the Barometer standing at 30 inches, the pressure of the air on a horizontal square inch of surface is equal to the weight of a column of mercury 30 inches long, and whose base is a square inch; i.e. to the weight of 30x1, or 30 cubic inches of mercuiy. To find how much this pressure amounts to. The weight of a cubic foot of Water is 1000 ounces avoirdupois, very nearly, .*. the weight of a cubic foot of Mercury = 1000 x 13"6 oz. (Art. 69.) cubic foot. cubic inches. cubic inches. oz. .-, 1, or 12 X 12 X 12 : 30 :: 1000 x 13'6 : weight (in ounces) i.«« t-. . 1 i? T u 13600x30 of 30 cubic mches of mercury, which .*. = -- — r- — ,— ounces = 236 oz. •" 12x12x12 nearly, or 14|lbs. And the pressure of the air on a square inch will be gi-eater or less than 14|lbs. according as the mercury in the Barometer stands at a greater or less height than 30 inches. It appears, then, that every square inch with which the air at the Earth's surface is in contact is subjected to a pressure of about 14 lbs. ; so that a page of a book 6 inches long by 5 inches wide sustains a pressure of about 6x5x14 lbs., or 420 lbs. The reason why the leaf is not torn by this enormous pressure is, that the pressure of the air on one side of it is counterbalanced by that on the other side. 91. Cor. 4. In Chapters r. and ii. the pressure of the Air on the surface of the fluid contained in an open vessel has been left out of consideration; in other words, the experiments there described 72 HYDROSTATICS. were supposed to be made in the exhausted Receiver of an Air- From Art. 89 it appears, that when wafer is the fluid employed, the pressure of the air on its surface is equivalent (when the mer- curial Barometer stands at 30 inches), to that which would be produced by a head of water 34 feet deep. In estimating, therefore, the pressure on any surface placed at a given depth below the surface of water, this large additional pressure, — amounting to more than 14 lbs. on a square inch of surface (Art. 90), must be taken into account. 92. Prop. XIX. To describe the construction of the Common Pump and its operation. Construction. In the Common Pump two bollow cylinders AB and BH, whose axes are in a the same vertical line, are connected toge- ther, and at their junction is fixed a valve B opening upwards. The upper cylinder AB is called the ''''Body of the Pump''' ; and in it a piston (7, containing a valve opening upwards, plays by means of a rod attached to the end ^ of a lever EFG, whose fulcrmn is F. A spont D is placed just above the highest point to which this piston ascends. The lower cylinder BH, which is called the '■^ Suction- Pipe' \ reaches below the surface ZT of a well of water. Operation. Suppose the piston to be at j/5, and the Suction-Pipe full of common air. As C is raised, the pres- sure of the external air keeps the valve at G closed, and a vacuum between B and C being consequently made, the air in HB, pressing against the under surface of the valve at B, opens it, and a portion of the air escapes into BC, The air, therefore, which, before the ascent of the piston occupied the space BH, now occupies the greater space CBH, and so, becoming less dense than before, has less elastic force, and exercises a less pressure on the sur- face of the water at Hy by Prop. xiii. Wherefore, since ELASTIC FLUIDS. 73 the external air continues to exercise the same pressure as before on the surface of the water in the well, it will force up water into the Suction-Pijye to a certain height K, such that the pressure of the air in CK, together with the weight of the column of water KH, produces the same effect on the section of the water in the Suction-Pipe at H, as the external air does on an equal area situated in the surface of the water in the well. When G has reached the highest point of its ascent, and equilibrium exists between the pressure of the external air on the surface of the water in the well on the one hand, and the pressure of the air in CK together with the weight of the fluid column KH on the other, the valve B, being equally pressed on its upper and its under surfaces, will shut by its own weight. The piston C is then pushed down; the air in CB is condensed, until its elastic force becomes greater than that of the external air, when it opens the valve in C and escapes. When C is raised again, the same circumstances recur. The water rises a little higher in the Suction-Pipe at every stroke of the piston, and at last flows through B ; and on G descending again, it raises the valve (7, passes through it, and on the next ascent of the piston, is brought up to the spout at D. As the average pressure of the atmosphere will not support a vertical column of water more than 34 feet high, if the valve B be more than 34 feet above the surface of the water in the well, the average pressure of the external air not being sufficient in that case to force the water up so high as B, the pump will not work. N.B. The figure represents the pump during an ascent of the piston ; when (air, or water, flowing through it) the valve at B is open, and that at C is shut, 93. Prop. XX. To describe the construction of the Forcing-Pump, and its operation. Construction. The Forcixg-Pump consists of a cylindrical ' BarreV AB in which a solid piston 6' works by means of a rod GG; BD is a '■ Suction-Pipe' reaching below the surface i) of a well of water ; BE a pipe con- G mm 74 HYDROSTATICS. necting BC wltli a vessel EF; at B and E valves are placed, opening upwards. Operation. Suppose the piston (7 to be at its greatest height, the pump full of air, and both valves closed. When C descends, the air in ^J5 is condensed, and its elastic force being increased (Prop, xiii.), the valve Ef] E is opened by it, and the greater part of the L air that was at first in AB is forced into EF. pt When the piston reaches its lowest point, the valve E closes by its own weight. On the piston reascending, the air in EB C he'mg now rendered less dense than that in EF and BE, the valve E is kept closed by the external air, the valve B is opened, and a portion of the air which was at first in BB rushes into the Barrel. The external air then forces some water a little way up the pipe (to ZT suppose), until the pressure on the horizontal section of the Suction-Pipe at D (a pressure which arises from the pressure of the rarefied air in HBG together with the weight of the column of water HD)^ is equal to the pressure of the external air on an equal area in the surface of the water in the well. Wlien this has taken place there is equilibrium ; and the valve B, being pressed equally on both its surfaces, closes by its own weight. The piston again descends, and the air in AB is driven through E into EF\ it ascends again, and more water is forced up the Suction-Pipe: and these operations are repeated until the Water rises above B, and is driven into EF by the next descent of the piston. As in the case of the common pump, unless BD be less than 34 feet, the pressure of the atmosphere u]3on the surface of the water in the well will not be sufficient to raise the water above the valve at B ; in which case the machine will not work. N.B. The Figure represents the Forcing-Pump during the ascent of the piston, when the valve E is shut, and the air (or water) is rushing from the Suction-Pipe into the Barrel through the open valve B. ELASTIC FLUIDS. 75 94. Prop. XXI. To exjyiam the action of the Si'PJioyi. The Siphon is a bent tube ABC open at both ends, and is often used for drawing fluids out of vessels. Let the tube be first filled with the fluid, and both ends be then closed. In- vert the tube, placing one end of it, A, in the vessel of fluid, and so that the other end, C, is heloiv the surface of the fluid in the vessel ; and let the plane of the sur- face of the fluid meet the legs of the Siphon in H and K, the vertical height of B, the highest point of the tube, being restricted to less than the height of a column of the fluid of which the pressure is equal to the pressure of the atmosphere. If now both ends of the tube be opened, the fluid in it will move in the direction ABC, will flow out at (7, and will continue to do so, until the surface of the fluid in the vessel sinks to A. The reason is this : — The moment A is opened, the pressure within the tube at H will be the atmospheric pressure acting ujnoards upon the column HB ; and the column HB, by its weight, which is less than the atmospheric pressure (by supposi- tion), will act in an opposite direction. Similarly, when G is opened, the atmospheric pressure acting upwards at G is opposed by the column BG acting downwards. But the column BH of itself will balance BK] therefore the Residtant of all the forces, acting on the system is the efl'ect of the column KG by its weight at G. As there is nothing to counteract this effect, the column of fluid KG will pass out through G', the column BK will follow by its own weight ; and the atmospheric pressure on the surface of the fluid in the vessel will prevent a vacuum from being formed at B by forcing up the fluid along HB, so as to keep the column ABG continuous, until the sur- face of the fluid in the vessel sinks to A, 95. If Water be the fluid employed, and the height of B above the surface of the fluid be greater than 34 feet, the pressure of the 76 HYDROSTATICS. column BH at H will be greater than the pressure upwards arising from the pressure of the atmosphere on the surface of the water in the vessel. In that case, therefore, when the end A of the siphon is opened, the column BH ^vill sink down until its extremity stands at a height above H in the pipe, such that the weight of the column of water in the tube is just balanced by the pressure of the atmo- sphere. If C be now opened, the water in CB will run out at i>, and (no water from the vessel flo^ving over B), the air -will rush up the i)ipe CB, and press on the upper end of the column of fluid sup- ported in BH. This column, therefore, being now equally pressed by the atmosphere at each of its extremities, will descend by its own weight into the vessel. The Siphon therefore will not act^ when WATER is the fluid used, if the vertical height of B above the surface of the fluid in tlie vessel be 7iot less than Z4cfeet. 96. Peop. XXII. To shew how to graduate a Com- mon Thermometer. The Thermometer is an instrument for comparing the temperatui'es (/. e. the intensities of heat) in solids or fluids. It consists of a slender glass tube of uniform bore, closed at the upper end, and terminating at the lower in a bulb. The bulb and part of the tube are filled with Mer- cury, Spirits of Wine, or any other fluid (not of a gaseous form) which expands on being heated, and contracts with cold ; the remaining part of the tube is a vacuum. If the fluid in the Thermometer occupy the same space when the instrument is plunged into two different fluids, the tempera- ture of those fluids must be the same. To graduate a Thermometer. Keeping the tube verti- cal^ plunge the bulb, and the part of the tube occupied by the mercury, into melting snow; and make a mark A at the point to which the mercury falls in the tube. (See Fig. Prop. XXIII.) This is called the freezing-point. Next plunge the bulb into boiling water, and make another mark B at the point to which the mercury rises*. This * The temperatures of melting snnw, and of boiling water, are taken to determine the fixed points A and B in the scale of the TJiermometer, because it is found, by experiment, that these temperatures are Jixed and invariable —the mercury cdu-ays faUing to A, when plimged into melting snow, ELASTIC FLUIDS, 77 is called the hoUing-point. The distance AB maj then be graduated by equal divisions. In the Centigrade Ther- mometer it is divided into 100 equal parts, called degrees — the freezing --point being marked O*', and the hoiUng-point 100**. In Fahrenheit'' s Thermometer the same length AB is divided into 180 parts — the freezing-point being marked 32'', and the boiling-point 212°. 97. Prop. XXIII. Having given the numher of de- grees on Fahrenheit's Thermometer, to find the correspond- ing number on the Centigrade Thermometer. In Fahrenheit! s gi'aduation of the scale of the Thermo- meter^ (which is that generally made use of in England), 32 is the number placed opposite to the freezing-point A ; and AB being divided into 180 equal parts (called degrees)^ the number placed opposite to the boiling-point -6 is 32 + 180, or 212. In the Centigrade thermometer, (which is that generally used on the Continent,) the graduation begins from A^ which is marked 0°, and AB ya divided into 100 equal parts, called degrees. Now let there be a TJiermometer, furnished on one side with a scale graduated according to Fahrenheit, and on the other w^ith a Centigrade scale. Let the mercury rise to any point D, and let F and C be the number of degrees respectively marked opposite to D on the scales of Fahrenheit and the Centigrade. Then, since the tube is uniform, X^ of degrees oi Fahrenheit in AB, (l80) N° AD, {F-S2) N° of the Centigrade in AB, (100) X° AD, {C); whether the snow be melting quickly or slowly, — and always rising to B, when plunged into boiling water, whether the fire applied to the vessel con- taining the water be great or small. 78 . HYDROSTATICS. .-. 180xC=100x(i^-32); the number of degrees in the Centigrade Thermometer cor- responding to F degrees Fahrenheit, Ex. If the mercury in Fahrenheit's thermometer stand at 77, then the corresponding number of degrees on the Centigrade will be ?x(77-32) = ^x45 = 25. 98. Similarly, if the number of degrees be given in the Centi- grade Thermometer, the corresponding number on Fahrenheit's Thermometer ^vill be found from the same equation. Thus 100x(i^-32) = 180x(7, .'. F=S2+C+tc o 99. There are other Thermometet^s besides the Centigrade and Fahrenheit's. In Reaumur's Thermometer the freezing-point is marked 0, and the boiling-point 80. In De Lislc's Thermometer the /r^^^i«^-point is marked 150 and the"~ boiling-point 0. Hence, if C, F, R, L denote the same temperature on the respective Ther- mometers, it is easily shewn, that 100. It may also be noted here, that although the temperature of melting snow is found to be the same under all circumstances, strictly speaking, that is not the case with boiling water. The tem- perature of boiling water varies with the atmospheric j^ressure; and this circumstance makes it necessary to observe the height of the Barometer at the time of graduating a Thermometer. See Hydrostatics hy Prof. W. H. MiUer, 4th Edition, Art. 92. ELASTIC FLUIDS. 79 Questions on Chapter IV. (1) If "rt/r has weight''^ why do we never take it into account in weighing articles with a common pair of scales ? (2) A bladder filled with air, being taken up in a balloon, hurst at a certain height. How do you account for this ? (3) Is not the density of air diminished by an increase of temperature ? If so, how does Prop. XIV agree with Prop. XIII ? (4) TVhen a tap is inserted in a full barrel of ale, why does the ale sometimes refuse to come out ? And in that case, what is the remedy ? (5) What is the use of the Vent-Peg in a beer-barrel ? Explain its action. (6) What practically limits the degree of exhaustion of air by the Air-Pump ? (7) A piece of wood floats in a cup of water under the Receiver of an Air-Pump ; will it sink deeper, or less deep, when the air is exhausted ? (8) What practically limits the extent to which condensation of air may be carried on by means of the Condenser? (9) Does the labour of working the Air-Pump with a single Barrel increase, or decrease, as the air is gradually exhausted ? And why % (10) Does the labour of working the Condenser increase as the operation goes on % And why % (11) In the common Barometer^ what is the object of having the surface of the mercury in the basin considerably larger than that in the tube ? (12) What would be the effect of allowing a small quantity of air to be admitted into the vacuum at the top of the tube of a Barometer ? (13) Why is Mercury used rather than any other fluid in con- structing a Barometer ? (14) What are the advantages and disadvantages, of a Water- Barometer ? 80 HYDROSTATICS. (15) A weight is sustained in air by a thread ; will the thread be more strained, or less strained, when the Barometer rises 1 (16) It sometimes happens, that a Common Pmnj), which will not work, is rendered effective by pouring water into it above the piston. What is the explanation of this ? (17) In the Common Pump, if the piston be not able to reach the fixed valve at the top of the suction-pipe, under what conditions will the piunp not work % (18) In the Common Pump, if the piston were nearly, but not quite, air-tight, how would this affect the action of the pump ? (19) Is the labour of working the Forcing-Pump affected by the size of the pipe, as to bore, up which the water is forced ? (20) Why is it necessary for the piston-rod of the Forcing- Pump to be made stronger than that of the Common Pump ? (21) When the Sijyhoii is in use, what would be the effect of making a small hole at its highest point % (22) How does the Siphon help to explain the phenomena of Intermittent Springs ? (23) What would be the inconvenience of having the bore of the tube of a Thermometer large % (24) Why is it necessary to note the height of the Barometer at the time of determining the Boiling-Point in a Thermometer ? (25) Can the heat of steam be made to exceed 212'' Fahrenheit ? " If so, how % EXAMPLES AND PEOBLEMS, WITH THEIR SOLUTIONS. In the Examination for the Ordinary Degree of B.A., (that is, of those who are not Candidates for Honours,) the University requires the attention of its Students to be directed not only to the Proposi- tions in the preceding Chapters, but, also to '■''such Questions, Ap2ylications, and Deductions, as arise directly out of the said Propositions ". The following Collection will give some notion of the sort of Questions, &c. which are to be expected, as well as of the manner in which they are required to be answered. No general Rides can be laid down for the solution of problems in Mechanics and Hydrostatics ; but the following suggestions will be found useful : — 1st. Let the Student make sure that he has a clear perception of the meaning of Ratio and Proportion (see Wood's Algebra, Art. 222,) in order that he may avoid the error of comparing things together which are not of the same kind. It is a fatal mistake to compare weight with money — length vsdth volume — time with super- ficial area—2iud so on. 2nd. In representing Force, Weight, Length, Volume, &c., by numbers, or by letters w^hich stand for numbers, let it be always distinctly expressed what is the unit of measure. Thus, if the weight of a body be represented by W^ let it be stated from the beginning, whether it be tons, or pounds, or ounces, &c. of which the number is W. Or if M be the magnitude of a body, let it not be left in doubt, whether the unit, of which the number in the body is M, be a cubic yard, or a cubic ybo^, or a cubic inch, or, &c. L. C. C. 6 82 MECHANICS, MECHANICS. 1. Forty cubic inches of a substance weigh 25 lbs., and two cubic inches of another substance weigh 1 lb. Compare the densities of the two substances. By Definition (Art. 8), dejisity of 1st. substance : density of 2nd. substance :: weight of 40 in. of 1st. : weight of 40 in. of 2nd. :: 25 lbs. : 20 lbs. 25 : 20 5 : 4. 2. A cubic inch of a substance weighs 280 grains ; and a portion of another substance, of twice the density, weighs 400 grains. Find the magnitude of the latter. Let X be the number of cubic inches required ; then - — - = weight of a cubic inch of latter substance, in grains, .*. density of 1st. substance : density of 2nd. :: 280 : — , or, by the question, 1:2:: 280 : , 40 5 , . . .'. ;i;=-^ = ^ cubic m. 3. Two weights of 6 lbs. and 9 lbs., respectively, balance each other at the extremities of a straight lever 10 feet long. Find the position of the /w^crz^m. Let X be the distance, in feet, of the fulcrum from that end at which the weight of 6 lbs. acts; then 10 — ^ is its distance from the other end; and, by Art. 21, exx=9x{10-x), or 6x = 90~9x, 15ar=90, .-. x = ^^ = 6. Hence the fulcrum is 6 feet from one end, and 4 feet from the other. EXAMPLES AND PROBLEMS. 83 4. Two men of equal height carry 4 cwt by means of a uniform pole, the ends of which rest on their shoulders. The weight is suspended at a distance of two-sevenths of the length of the pole from one of the men; how many pounds does the other man support ? Considering this as the case of a Lever, with the shoulder of the man who is the nearer to the weight as fulcrum^ we have, by Art. 23, 7 2 weight requiredx - = 4 cwt. x - ; 2 .*. weight required = 448 X" lbs. = 128 lbs. 5. A uniform cylinder, 4 feet long, and of which an inch and a half weighs 3 oz., is kept at rest in an horizontal position by a weight of 4 lbs. attached to one end of it. Find the position of the/M^crwm. Since 1^ inches weigh 3 oz., 1 inch weighs 2 oz., and the whole cylinder weighs 48x2 oz., or 6 lbs. Now, by Art. 20, the cylinder will produce the same effect by its weight as if it were collected at its middle point; therefore sup- posing it so collected, and taking x the distance, in feet, of the fulcrum from the end to which the weight of 4 lbs. is attached, we have the case of a straight Lever kept at rest by weights 4 lbs. and 6 lbs. acting in the same direction on opposite sides of the fulcrum at distances x feet, and 2—x feet, respectively. So then, by Art. 21, 4x;r=6x'2-a?), ^x^\2-QXy 10;»=12, .'. ^-i|= 1-2 feet. 6. A certain weight, when it is attached first to one end of a straight Lever of the first class, and then to the other, is balanced by half a pound, and 18 oz., respectively. Compare the lengths of the arms of the Lever. Let the given weight be P oz., and AB the Lever with fulcrum Cy BG being the shorter arm,; then P^AG=\SxBCy (1), ) C and P^BC= 8xAG, (2), j ^ ^ B 6—2 84 , MECHANICS. /. dividing (1) by (2), '£q=j^ -j^y ,,. , . , AC fACV 18 9 multiplying by -g^, (^^)= g—^; AGS or AC : BC :: Z : 2. 7. Two cylindrical rods, one of platina, the other of silver, have the same diameter, and being joined together with their axes in the same straight line, balance in an horizontal position". on a fulcrum placed at their point of junction. Given that the length of the platina rod is 9 inches, and that the density of silver is 0*48 times that of platina, find the length of the silver rod. Let AB, BC, be the lengths of the platina and silver cylinders, respec- ? ? l tively ; D and E their middle points ; and let BG be x inches ; then, by Art. 20, BD 2BI>_ wt.of si\y.cy].BC BE ' ^^ 1BE ~ wt. of plat. cyl. AB ' _ wt. of silv. cyl. BC wt. of plat. cyl. BC ~~ wt. of plat. cyl. length BC wt. of plat. cyl. AB ' but the weights of equal hulks of any two substances are proportional to their densities (Art. 8) ; therefore we have ?_i§_ ^ ^ 81x100 x-lOO^'d' ^^ ^~ 48 ' 9x10 45 15 ,- 4^3 .2^3 ^ 8. A uniform bent Lever, of which the weights of the arms are 3 lbs. and 6 lbs. respectively, when suspended on its fulcrum, rests with the shorter arm horizontal. What weight must be attached to the extremity of the shorter arm, that the Lever may rest with its longer arm horizontal ] Since the Lever is uniform, the lengths of the arms will be in the ratio of their iceights, that is, as 3 to 6, or 1 to 2. So, then, if ACB be the Lever w?th fulcrum C, GB is equal to twice AG, AC being the shorter ann. .... EXAMPLES AND PROBLEMS. 85 PNC D 1st. The Lever balances round C with the arm AG horizontal. Take Z> and ^ the middle points of AG, and GB, respectively ; and draw EM vertical to meet AG produced in M. Then, by Art. 27, 3x(7Z>=6xC7iltf, or GD = 2GM, ... (1). 2nd. Let ^Ibs., acting vertically at A, cause the Lever to balance with its longer arm CB horizontal ; then, constructing as in the annexed fig. (2), it will be easily seen, that the triangle CAP is equal in all respects to the triangle GEM in the former case, {for L.AGB, and /. its supplement, remains unaltered ; and GE = GA) ; hence CP ^GM, and cn=\gp, V cd=\ga. But XY^GP -h 3x GN^ 6 y.GE, or ;rKCJtf+^xa^-12x(7Z)-24xCJ/, by (1), 3 ;c + - = 24, .-, ^=22|. (1) B (2) 9. A uniform heavy rod is supported by two equal strings, attached to its extremities, and meeting together on a fixed peg, so that the rod and strings form ^n equilateral triangle. Compare tl^e tension of each of the strings with the weight of the rod. Let AB hQ the rod, and G the fixed peg ; from A draw AE perpendicular to BG ; and bisect AB in D. Consider the system as a Leter AB with fulcrum A-, kept in equilibrium by the weight of the rod ( W) acting at its middle point D at right angles to AD, and the tension of the string {T) acting in the line BG. Then S6 MECHANICS. T: W'.'.AD'. AE, (Art. 27,) :: IaB.CD, v AE^CD, :: IaB: s/AC'-AB'-, v.\aB:J AB^-\aB\ :: \AB.\^^Am, :: 1 : sjz, 10. Two forces of 4|lbs. and 3|lbs., applied at a point, have a Resultant of 8 lbs. In what directions do the forces act ? Since the Resultant is the sum of the component forces, these latter must act in the same straight line, and in the same direction ; in which line, and direction, the Resultant also will act. 11. Two forces, which are to each other as 3 : 4, act on a jwint in directions at right angles to each other, and produce a Resultant of 15 lbs. Find the forces. Let ^x and 4.1? represent the number of pounds in the two forces, which produce a Resultant of 15 lbs. ; then, since the com,ponent forces act at right angles to each other, they will be represented in magnitude by the sides of a right-angled triangle of which the hypo- thenuse is 15. Hence {2xf + {4.xf = {\5f, 9.^2 + 16^2=225, 25:^2 = 225, and the required forces are 9 lbs. and 12 lbs. 12. A string, passing round a smooth peg, is pulled at each end by a force of 10 lbs. ; and the angle between the parts of the string on opposite sides of the peg is 120'', that is, two-thirds of two right angles; find the pressure on the peg both in magnitude and direction. The pressure required is the Resultant of two eqiMl forces acting at a point. Let P^=P^ = 10 lbs. represent the two forces acting at P, lAPB=12QP'j complete the parallelogram PACB, and EXAMPLES AND PROBLEMS. 87 draw the diagonal PC; PC is the Resultant, or pressure on the peg, both in magnitude and direction. (Art. 3L) (1) For direction^ PC bisects the angle APB, since the forces are equal. (2) For magnitude, since L A PC= lPCB=60'^= L BPC ; therefore remaining angle of the triangle PBC= 60", and the triangle is equilateral. Hence PC- PA = 10 lbs, 13. A weight is supported by two strings, which are attached to it, and to two points in a horizontal line. If the lengths of the strings are unequal, shew that the tension of the shorter string is greater than that of the other. Let W be the weight ; WA, WB, the strings ; AB an hori- zontal line. Draw JVC ver- tical, meeting AB m C\ and draw CD parallel to WA^ meeting WB in D. Then W is kept at rest by three forces, its own weight acting in direction CW, tension of string WA, acting in direction DC, and tension of string WB, acting in direction WD. Therefore by converse of Prop. IX., Art. 32, the sides of the triangle WCD represent the magnitudes of the forces, respectively. And, therefore, tension of WA : tension of WB :: CD : WD. Now, by supposition, WB> WA ; .*. L WAB> L WBA ; but lBCD= l WAB, '.- CD is parallel to WA ; and BCW is a right angle; .*. L WCD, the complement of lBCD, is less than L CWD, which is the complement of L WBA. Hence WD < CD, (Euclid 1, 19,) i.e. tension of WA > tension of WB. 14. If the Velocities of two bodies are as 7 : 2, and the first moves through 42 feet in a second, what space will the other body describe in the same time ? Let the Velocities be 7v and 2v ; and let x be the space required, in feet; then, by Art. 46, Cor., 'Jv : 2v :: 42 : x, 88 MECHANICS. or 7 : 2 ::42 : ^; 2x42 84 ,^ •• ^=-=— = -^ = 12. 7 7 15. "What power will be suflScient to raise a weight of 2 tons, by means of a Wheel and Axle, if the radius of the Axle : radius of the WJied :: 1 : 10 ? Let P be the required Power ; then, by Art. 37, P : PT:: 1 : 10; „ ^ 2 tons 20 cwt /. P = — = = ■ = 4 cwt. 10 10 5 16. In the case of the single moveable pulley, will the mechani- cal advantage be increased or diminished by taking into account the weight of the pulley ? Def. When a power P balances a weight W on any simple ma- W chine, the number of times that W contains P, or the ratio -^ , is called the mechanical advantage of that machine. W In this case, when the weight of the pulley is neglected, -p = 2, (Art. 41.) But, if the weight (^c) of the pulley be taken into account, it may be supposed concentrated in the centre of the pulley, and acting in the same line as W, Then, reasoning as in Art. 41, P : W+w :: 1 : 2, or W^w^lP; .-. -p=^-p^ ^liich <2, and therefore the mechanical advantage is diminished. 17. In a system, in which the same string passes round any number of pulleys (Prop. XII), and the parts of it between the pul- leys are parallel, if the weight of the pulleys be regarded, under what circumstances will the mechanical advantage be reduced to nothing ? The weight of the upper block will have no effect, because it is supported, by supposition, from without. But the weight {w) of the EXAMPLES AND PROBLEMS. 89 loicer block will add to the weight Wj and the result, in case of equilibrium, will be P : W+w :: 1 : 71, the number of strings at lower block ; .'. W+w = nPy W w or -rp^n— rp, which becomes 0, when w = nP; that is, the mechanical advantage is nothing, when the weight of the lower block is equal to, or greater than, the power multiplied by the number of strings at the lower block. 18. In Prop. XIV, (Art. 45), find the inclination of the plane, when the pressure of W on the plane is equal to P. In the triangle DJEF, DF represents the power, FE the weight, and DE the pressure on the plane ; therefore the pressure on the plane is equal to the power, when DE=DF. But DE : DF :: AC : BC; therefore the j^ressure =P, when AC=BG, or lABG= lBAG, ' i. e. when the inclination of the plane equals half a right angle. 19. A weight of 20 lbs. is supported on an Inclined Plane by means of a string fastened to a point in the plane ; and the string is only just strong enough to carry a weight, hanging freely, of 10 lbs. The inclination of the plane to the horizon being gradually increased, find when the string will break. As long as there is equilibrium, the tension of the string is repre- sented by P in Prop. XIV ; and P : W \\ height of plane : its length. Now, as the inclination of the plane is increased, the height is increased, and P also ; and when P exceeds \ TV, or 10 lbs., then the string breaks, that is, when the height of the plane first exceeds half its length. 20. Prove that a body may balance in two positions round a line, which passes through it, without the Centre of Gravity (c. g.) being in that line ; but if it balance in three positions, the c. g. must be in that line. If the c. G. be not in the line, the body will nevertheless balance upon the line, when its c. g. is brought into the vertical plane, which passes through the line; and there are ttco such positions — one? 90 MECHANICS. ■when the c. g. is above, and the other, when it is below, the line. But if the body be brought into any otiier position than these two, it will have a tendency to move round the line (Art. 50), unless the 0. G. be actually in the line. 21. If, in any triangle ABC, whose c. g. is G, AG be joined, 2 and produced to meet one of the sides in F, shew that AG--AF^ o Take the fig. in Art. 55, omitting the line tfe, and join EF. Then, since BE :BA::li2 :: BF : BC, by Euclid VI. 2, EF is parallel to AC; and then it is easily shewn, that BFE, and BAG, are simi- lar triangles; and therefore EF : BF :: AG : BG, and alt EF : AG:'.BF:BG::1 : 2. Again, since EF is parallel to AG, it is easily shewn, that EFG, and AGG, are similar triangles, and therefore AG : AG :: FG : FE, and alt. AG : FG :: AG : FE :: 2 : 1; that is, AG^IFG, and there- fore ag=\af, o 22. The sides, which contain the right angle, of a right-angled triangle are Sin. and 4 in., find the distance of the c. g. of the tri- angle from the vertex of the right angle. Let ABG be the triangle, lACB a right angle, ^(7=3 in., ^(7=4in.; then ^^2^^1(72 + ^(72 = 9 + 16-25; .-.^^-Sin. Now bisect AB m D, and join CD; then, with centre D, and radius DA or DB, describe a semi-circle, and it will pass through G, because ACB is a right angle {Euclid, Bk, III, 31); therefore DG=DA = \AB=2lixi. 2 ^ And, if G be the c. g. of the triangle, it is in CD (Art. 55), and (7(y = ~C7Z>, by last Prob.; .-. CG=\^^=\lm. 23. Weights of lib., 2 lbs., and 4 lbs., being placed at the corners of a given horizontal triangle, which is itself without weight, find the point on which the triangle will balance. Placing the weights lib., 2 lbs., and 4 lbs. 2ii A, B, and C, respec- tively, and following the method in Art. 52, the c. g. of the two weights EXAMPLES AND PROBLEMS. 91 at A and 5 is at a point D in AB, such that AD : AB :: 2 : 3 ; then joining DC^ and supposing the t^YO weights from A and B to act together at D, the c. g. of the three weights will be in the line (7i>, and at a point G such that DG \ CD •.: 4: \ 3 + 4 :: 4 : 7. 24. Shew that the c. g. of three heavy points, of equal weight, and not in the same straight line, coincides with that of the triangle formed by joining the points. Let A,B,G, be the points ; then, by Art. 52, since A^B, the c. G. of A and B is in the line AB, at a point D equally distant from A and B. Join CD, and suppose A and B collected at i>, then the c. G. of A, B, and (7, will be in the line CD at a point G such that CG : DG '.: A+B : C .'.2 '.I, that is, CG^^CD, and therefore, o by Prob. 21, G is the c. g. of the triangle ABC. 25. From, a square A BCD, whose diagonals intersect in 0, the triangle AOB is taken away. Find the c. g. of the remainder. Bisect the sides AD, DC, CB, in E, Fy and H, respectively, and join OE, OF, OH. In OE take Oe equal to ? 0^; in OF take Of equal ^OF',m OH take Oh = l OH; and join ef, fh. Then e is the c. G. of the triangle AOD ; f is the c. g. of DOC; and h the c. g. of BOC; and the c. g. required will coincide with the c. g. of three equal bodies placed' at e, f, and h ; that is, with the c. g. of the tri- angle efh. It will therefore be in the line Of, and at a distance, OG, from equal io -Of. ^oy, 0f=\0F=^^AB; :. dist. of c. G. from 0=-AB=- side of square. "92 HYDROSTATICS. HYDEOSTATICS. [In the following Problems the pressure of the Atmosphere is not taken into account, unless the contrary be expressly stated. Also, it must be constantly borne in mind, that Water is the Unit of ' Specific Gravities'' ; that is, for example, if it be stated, that the s. g. of a particular substance is lOi^, the meaning is, that the s. G. of it : the s. g. of water :: lOi^ : 1, or, in ordinary language, that the particular substance is 10| times as heavy as Water. Again, it is a fact worth remembering, that, for all practical pur- poses, the weight of a Cubic foot of water may be taken at 1000 oz. avoirdupois^^ 1. Two rectangular planes, whose lengths are 4 in. and 6 in., and breadths 3 in. and 4 in., are placed horizontally, 4 feet and 2 feet respectively, below the surface of the same fluid. Compare the pres- sures on them. The pressures, (by Art. 64), will be in proportion to the superin- cumbent columns of fluid, that is (since the fluid is supposed to be of uniform density), to the magnitudes of those columns. And, the columns being rectangular prisms, their magnitudes will be as the base of the one x its height : base of the other x its height, :: 4x3x4 ; 6x4x2, that is, :: 1 : 1. Therefore the pressures are equal. 2. If a cubic inch of Mercury weighs 8oz., what will be the pressure on a square foot of the horizontal bottom of a vessel filled with Mercury to the depth of 6 in. ? The pressure required = weight of a prism of Mercury, whose base = 144 sq. in., and height =6 in., that is, weight of 864 cubic in, of Mercury. But 1 cubic in. weighs 8 oz. ; therefore the pressure = 864x8 oz. = 432 lbs. 3. Compare the fluid pressures on the top and bottom of a box, , 3 feet deep, which is sunk to a depth of 30 feet below the surface of smooth water. EXAMPLES AND PEOBLEMS. 9(31 The lottom of the box being supposed at the depth of 30 feet, and horizontal, and the areas of the top and bottom being the same, the press, on the top : press, on bottom :: depth of top : depth of bot- tom :: 27 : 30 :: 9 : 10. But if the top be at the depth of 30 feet, then the pressures are :: 30 : 33 :: 10 : 11. 4. In two uniform fluids the pressures are the same at depths of 3 in. and 4 in., respectively. Compare the pressures at depths of 7 in. and 8 in,, respectively. Let P^ represent the pressure in one fluid at depth 3 in. Pi the same 7 in. jt?4 the other 4in. Ps the latter 8in. , Then, by Art. 61, P^ : Pj :: 7 : 3, .*. i^-^i^. Also ^8 : ^4 :: 8 : 4 :: 2 : 1, .-. p^ = 2p^. But i3=i?4, by the question, /. Pj'.Ps.: \p^'. 2/?4::| : 2 :: 7 : 6. 5. Shew that the pressure on a square inch in the side of a vessel, filled with water, will be between 0324 lbs. and 0'288 lbs., if the depth of the water be 9 in., and one side of the square inch be in the bottom — ^the weight of a cubic inch of water being 0'036 lbs. The square inch in question may be in any position between vertical and horizontal. Suppose it vertical, then the highest point in it will be at a depth of 8 in. ; and in any other position of the square, every point in it will be deeper than 8 in. Therefore the pressure on it can in no case be so small as, (reckoning every point in it at 8 in. depth,) 0-036x8 lbs., or 0-288 lbs. Again, the square inch, being a part of the side of the vessel, can never be quite horizontal, for then it would become a part of the Ixise. So that the pressure on it can never be so great, as if each point in it were at a depth of 9 in. — that is, can never be as much as 0-036x9 lbs., or 0*324 lbs. The actual pressure, therefore, lies between 0288 lbs. and 0324 lbs. y: 6. Find the Specific Gravity (s. g.) of earth, when a cubic yard of it weighs a ton. 94 HYDROSTATICS. "wt. of a cubic foot of earth 1 « , ,^^^ s. G. = . — > — , . „ , -^ — , = „- of a ton -7- 1000 oz., wt. of a cubic foot of water 27 20x112x16 35-84 , ^_ . , = 27X1000 =-27--^^^7^"^^^^y- 7< 7. The s. g. of a certain fluid is 0"78, and of another 0*66. How much, in weight, of the latter fluid ^vill be of the same bulk as 91 lbs. of-the former 1 Generally, JV=3fS (Art. 70); and here, by the question, ilf is the same for both substances ; .". Wcc S. Wt. required _ 0^66 _ ^ _ H Hence - - ^.^^ " 78 ~ 13 ' .'. Wt. required = =7x11 = 77 lbs. y- 8. Twenty pints of a certain fluid weigh exactly the same as 22 pints of another. Compare the s. g.'s of the two fluids. Here, since IF=MS, and Wis the same in each case, .'. Jf^oc 1, or Sec ^r^. M ■ Hence s.g. of the first : s.g. of the second fluid ::—:—:: 22 :20 :: 11 : 10. "* 9. A quart of water is mixed with a gallon of milk; the s.g. of milk being 1*03, find the s. g. of the mixture. Let M be the number of cubic inches in a quart of water, then 4J^ will be gallon of milk ; and 5ilfwillbe \hQ mixture. But, since s. g. of water is 1, the weight of the water is Jfxl, and the weight of the milk is 4ilf xl-03. Therefore the whole weight of the mixture is iff +4il[fx 1*03. Hence s. g. of the mixture = weight of one cubic inch, iW+4-12xil[f 5-12 5Jf 5 1-024. 10. Five gallons of a fluid, of s. g. 0*98, are mixed with 6 gallons of a fluid of s. g. 1*6, without any weight being lost ; and the s. g. of the mixture is I "3. Has mlume been either gained or lost ? EXAMPLES AND PEOBLEMS. 95 Following the same method as in the last Prob,, and supposing neither weight nor bulk lost, C4.U ' ^ 5x0-98 + 6x1-6 s. G. of the mixture = — , _ 4-9 + 9-6 _:l4-5_ which, being greater than the actual s. g., shews that volume must have been gained. 11. By 20 oz. of a substance being sunk in a cylindrical vessel, whose base is horizontal, and which contains a pint of water, the depth of the water is increased from 4 in. to 4| in. The weight of a pint of water being 1 lb., find the s. g. of the substance immersed. The substance displaces a volume of water equal to its own volume, and this volume is equal to a portion of the cylindrical vessel half an inch in depth. Therefore, volume of water equal in size to the substance : 1 pint :: i : 4 ; and the weight of water is proportional to its volume, .*. weight of water equal in size to the substance : 1 lb. :: i : 4 ; .*. weight of water, &c. = — -^ = 2 oz. o 20 Hence s. g. required = — = 10. 12. A pieqe of wood floats with one-third of its bulk out of water ; and a stone, whose bulk is one-eighth that of the wood, when placed upon it, just sinks it below the surface of the water. Find the s. G. of the stone. Let M be the bulk of the wood in cubic inches ; then, by Art. 73, 2 - Jf : if :: s.G. of the wood : 1, (s. g. of water); o 2 /. s. G. of the wood = - . o Let s be the s. g. of the stone ; then, weight of the stone = - Ms, 8 weight of the wood= J/x- ; N 96 HYDROSTATICS. and, when the stone is placed on the wood, the weight of the fluid displaced = J^xl. But the two former weights together balance the latter, by Art. 75, .-. ^Ms + ' M = My o o 1,21 13. The s. G. of iron is 7'8, and of gold 19*4; find the weight in water of a substance composed of 1 lb. of iron and 1 lb. of gold. The masses of iron and gold being supposed to lose no hulk by- amalgamation, the whole weight lost in water by the compound will be exactly equal to the sum of the weights lost by the poimd of iron and the pound of gold separately considered. Now the pound of iron loses the weight of an equal bulk of water; but iron is 7*8 times the weight of water, therefore the pound of iron loses Ix— -lbs. So also the pound of gold loses 7*8 1 X — - lbs. Therefore the compound weighs in water 2 — ;r:^ — r^— lbs. ; that is, 2---— -2-5x^^-:::^-2-^^^-r*9^ibs 14. An ounce of silver, whose s. g. is 10|, is suspended by a string, so as to be wholly immersed in water. "What will be the tension of the string ? The tension of a string is always measured by the weight which it supports, which, in this case, would be 1 oz., if the body were supported in vacuo. But the body, by being immersed in water, loses the weight of an equal bulk of water, (Art. 75) ; and therefore the tension of the string is diminished to that extent. Now, the weight of the silver (1 oz.)=10^ times the weight of an equal bulk of water, .*. weight of an equal bulk of water = --j ^z. ; IO2 EXAMPLES AND PROBLEMS. 97 hence tension required = 1 — —i oz., IO2 9i 19 " = — - = --'■ oz. lOi 21 15. If the water, in the preceding Prob., be contained in a vessel placed on a table, will the pressure on the table be affected by sus- pending the silver in it ? And how much 1 The weight lost by the silver is the pressure upwards of the water upon it, and is equal to -—1 oz. IO2 But "fluids press equally in all directions" ; therefore the presence of the silver in the water causes an additional pressure downwards of —J oz. ; and this is communicated to the base of the vessel, and accordingly adds so much to the pressure on the table. 16. If, in the last Prob., we take an ounce of gold, whose s. g. is 19"4, instead of an ounce of silver, will the additional pressure remain the same ? Since the ounce of gold is only about half the hulk of the ounce of silver, and the additional pressure in question is the weight of an equal hulk of water in each case, it is plain, that the pressure on the table will be much diminished by using gold for silver. 17. A common Hydrometer floats in water with 43, and in milk with 100, divisions above the surface. The portion of the stem between each two successive divisions is one 2000th part of the bulk of the whole instrument. Find the s. g. of milk. Let S be the s. g. of milk, that of water being 1 ; the weight of fluid displaced by the instrument in both cases is the same, being the weight of the instrument ; therefore if M be the volume of milk displaced, and M' the volume of water, S:l : 2000-4.3 : 2000-100, : 1957 : 1900, 1900 L. C. C. Cy 1957 , ^„ , •••'5'-,,,, = 1-03, nearly. 98 HYDROSTATICS. 18. Find the weight of a Hydrometer, which sinks exactly as deep in a fluid, whose s.g. is 09, as it does in water, when loaded with 60 grains. Let M be the volume of fluid displaced, which, by the question, is the same in both cases ; then, W, (weight of the instrument) = il/x0'9, and W+ 60 = Mx 1, •/ s. g. of water is 1 ; •'•^^+^^ = 09=-^' 9JV +540 = 10 W, .'. TV= 540 grains. 19. In the Common Air-Pump, if the volume of the Barrel be one-fifth of that of the Receiver, shew that, after 4 strokes of the piston, the density of the air in the Receiver will be about one-half of what it was at first. (By a stroke of the piston here is meant a descent and ascent, so that the piston, after going to the bottom of the Barrel, returns to its original position at the top) Let B be the content, or volume, of the Barrel; then 5B is the content of the Receiver. And the same air, which occupied the space 6B at first, will occupy QB after the first stroke. Hence if d be the density of the air at first, and d^ the density after the first stroke, (by Art. 81, Note,) d^'.d :.5B:6B ::5 : 6, or d^ = ^^ d. Similarly, if d^^, d.^, d^, be the densities of the air in the Receiver after 2, 3, 4, strokes of the piston, respectively. 5 5 5 , 55, 555, 5555, '^^^6^6^^ = 6^6^6^^ = 6^6^6"6^' EXAMPLES AND PROBLEMS. 99 20. In the last Problem, supposing the pressure of the air at first to be 14 lbs. on the square inch, what is the pressure on a square inch of the inner surface of the Receiver after 4 strokes of the piston ? By Art. 81, the elastic force of air varies as its Density ; there- fore, in this case, since the density is about one-half its original density, the pressure will be about one-half the original pressure, that is, about 7 lbs. on the square inch. 21. Find the pressure of the air in the Receiver of a Condenser after 10 descents of the piston, the content of the Receiver being 10 times that of the Barrel. It is plain, that by 10 descents of the piston, the quantity of air in the receiver is Qxaidil^doubled ; therefore its density is doubled, and the elastic force, or pressure, will likewise be doubled. So that, if the instrument was filled with common air at first, of which the pressure is about 14 lbs. on the square inch, the pressure of the con- densed air will be about 28 lbs. on the square inch. 22. If the atmospheric pressure be 14 lbs. on the square inch, when the Barometer stands at 2S inches, what will it be, when the Barometer stands at 30 inches 1 By Art. 87, " the pressure of the atmosphere is accurately mea- sured by the weight of the column of mercury in the Baroinneter^\ And in the same Barometer the weight of the column of Mercury will be proportional to its height. Therefore here, atmospheric pressure in 1st case : atmospheric pressure in 2:id, :: 28 : 30, :: 14 : 15, or 14 lbs. : required pressure :: 14 : 15, .'. required pressure = 15 lbs. 23. The height of the mercurial Barometer being 30 inches, required the height of a Barom^eter, of which the column above the cistern contains equal weights of mercury (s. g. 13*6), and of proof spirit (s. G. 0-93). In any Barom,eter the weight of the column in the vertical tube is equal to the atmospheric pressure, which in this case is measured by 30 inches of mercury. In the coDipound Barometer in question, 7—2 100 HYDROSTATICS. lialf the column, in weight, is mercury; and therefore its height is 15 in. The length of the column of proof spirit is thus found : — The weight of the column being equal to that of 15 inches of mercury, and the thickness of the column the same, since W=MS always, and here 3/qc -^, the height of the column oc -^. Therefore, height of proof spirit : 15 in. :: 13-6 : 093, ,.,... . -x 15x13-6 5x13-6 6800 ••• ^^'^^^ ^f «P^"* = -(r93- ^ "0^ = "sT' -21911; /. height of 5«rome^^r= 15 + 21 9|i = 23411 inches. 24. Two bodies of different Specific GravWles balance in a com- mon pair of scales, when the Barometer stands at 28 inches. Will they balance also, when the Barometer stands at 30 inches % If not, which will preponderate ? The two bodies, being of different s. G., but equal in weight by the first trial, must be unequal in hulk; and the greater will be that which has the lesser s. g. Let M be the number of cubic inches in larger body, in smaller ... S the s. G. of air, when barometer stands at 30 in., , s 28 ... In either case the weight lost by the body is the weight of an equal bulk of air. The larger body loses 3f/S in one case, and 3Is in the other, the difference being therefore M{S-s). The smaller body loses mS in one case, and ms in the other, the difference being 7n{S—s). And since M is greater than m, the greater body suffers the greater loss of weight by being transferred to air of greater density. Therefore the lesser body will preponderate and the bodies will 7iot balance. 25. A Condenser at first is full of air, the same as that of the surrounding atmosphere ; and the content of the Barrel is one-tenth that of the Receiver. From a flaw in its construction, the Receiver will only sustain a pressure equal to three-fourths th?lt of the atmo- sphere. During what descent of the piston, and at what part of the descent, will the Receiver burst ? EXAMPLES AND PROBLEMS. 101 Let B^ and 10^, be the contents of Barrel, and Receiver, re- spectively ; P the pressure, and D the density, of the atmosphere, P, the pressure, and D, the density, of the air in the Receiver after x descents of the piston. Then, ^xZ) = mass of air forced in by each descent, and xxBxD= ^descents, /. 10-ffxi> + ^x^xZ>=mass of air in receiver after x descents, = 10BxD„ lOBD + xBD _ 10 + x •*• ^'~ lOB ~ 10 •^• P P 10 + ^ But ^ = jy (Art. 81), .*. P,= . P ; and, by the question, 3 7 the Conde7iser will burst, when P, exceeds P + , P, or - P, that is, 4 4 7 \0 + X as soon as-P>— — -^.P, or 70 > 40 + 4.2?, or ;^>75, that is, in the middle of the eighth descent. 26. Shew that 95*^ of Fahrenheit's TJiermometer denotes the same temperature as 35^ of the Centigrade. By Art. 97 180x(7=: 100x(P-32), where C and F are the numbers of degrees on the Centigrade, and Fahrenheit, Thermometer, respectively, for the same temperature. Here P=95, .-.(7-^^x63 = ^x63 = 35. 27. The sum, of the numbers of degrees indicating the same temperature on the Centigrade, and Fahrenheit, Therm,o'tneier being equal to 0, find the number of degrees on each. Here C+P=0, by the question, .*. C= —F. But 180x(7=100x(P-32), .-. 180xC=100x(-C-32), = -100xC7-3200, 280xC=-3200, 320^_80^_ 28 7 ^ And .-. P=-C=llf. EXAMPLES AND PEOBLEMS, WITH ANSWERS*. i. DENSITY, AND FpHCE. (Arts. 8... 16.) (1) Two bodies, each of uniform density, are of the same size. Shew that by weighing them it may be determined whether they are of the same density. (2) Eight cubic inches of a substance weigh 5 lbs., and one cubic inch of another substance weighs half a pound. Compare the densities of the substances. (3) A cubic foot of a substance weighs 5 cwt., and a cubic foot of another substance weighs 14 qrs. Compare the de7tsity of the latter substance with that of the former. (4) Half a cubic foot of water weighs 500 oz. avoirdupois, and a cubic foot of zinc weighs 7000 oz. Compare the densities of water and zinc. (5) A quart of water weighs 2 lbs., and fifty gallons of proof spirit weigh 372 lbs. Compare the density of proof spirit with that of water. (6) A rectangular log of wood is 16 feet long, 2| feet wide, Ij feet deep, and weighs 3000 lbs. Compare the density of the wood with that of a metal, a cubic foot of which weighs 480 lbs. (7) Equal bulks of two substances weigh 4 lbs. and 9 lbs. Find the ratio between two bulks of those substances, which contain the same quantity of matter. (8) Of alcohol 10 pints weigh 8 lbs. ; and of oil 10 quarts weigh 18 lbs. Compare the densities of alcohol and oil. * The Answers are placed together at the end of the Section. THE LEVER. * 103 (9) A body weighs four times stg much as another body which is thrice its bulk. Compare their densities. (10) Half a cubic foot of a substance weighs 2 cwt. ; what bulk of another substance that is six times as dense will weigh 3 cwt. ? (11) A mass of a substance weighs 8lbs. ; what will be the weight of a mass, twice its size, of a substance whose density is thrice as great? ^ i l^ i I ', ,' ^'> i- ' (12) A block of wood weighs 60 lbs'.; and a piece of iron, one- third its size, weighs 10 st. 10 lbs. Compare the^uaiiti^zef^ o/ inati^r in the two substances, and. their densities. ' * " ' " " / (13) If a force of 5 lbs. be represented by a line 1ft. Sin. in length, what force will a line 2 ft. in length represent ? (14) The measure of a force, when the unit of weight is 1 lb., is 3 ; what is the measure of the same force, when the unit of weight is 5 lbs. ? y^ ii. THE LEVER. (Arts. 17... 29.) [Note. — In the following Examples, unless the contrary he ex- pressly stated, the Lever is supposed to he straight, and to rest horizontally in equilihrium ; and the weights, or other forces, to he applied to it perpendicularly.] (1) Half a cwt. acts vertically at the extremity of one of the arms, 10 inches long, of a horizontal straight Lever. What number of pounds attached to the extremity of the other arm, 14 inches long, will balance it ? (2) Two weights, of 5 lbs. and 7 lbs. respectively, keep a hori- zontal straight Lever at rest. The length of the arm at which the larger weight acts is 2 feet 1 inch. Find the length of the other arm. (3) A weight of 14 lbs., suspended 2 inches from the fulcrum of a horizontal straight lever, is balanced by a weight of 8 oz. Find the length of the arm at which the latter weight acts. (4) A straight Lever is kept at rest by two forces acting per- pendicularly on it on opposite sides of the fulcrum, and at dis- tances from the fulcrum of 9 inches and 15 inches; the greater force being 6 st. 6 lbs., find the other. 104 EXAMPLES AND PROBLEMS. (5) Di\icle a cwt. into two portions that will balance on a hori- zontal straight Lever ^ whose arms are 8 feet and 6 feet. (6) If the length of a Lever be 28 in., and the weights, which balance each other at its extremities, are 3 lbs. and 4 lbs., find the position of the fulcrum. (7) On a straight horizontal Lever, 20 inches long, two weights, of 24 0?,. and of 6 Ibs^.,^ respectively, balance. Find the position of the fulcrum. ' ' (o) The anns of a horizontal straight Lever of the first kind are 12 inches and 18 inches, respectively, and the weight which acts at the extremity of the shorter arm is 3 lbs. Find the pressure on the fulcrum. (9) On a Lever of the first kind 12 lbs. is supported at a distance of 8 inches from the fulcrum. The pressure on the fulcrum being 14 lbs., find the Power, and the point of its application. (10) Two forces, of 3 lbs. and 5 lbs., respectively, balance when they act perpendicularly on a straight Lever in opposite directions at the distance of a foot from each other. Find the position of the fulcrum. (11) At one end of a horizontal straight Lever of a given length a force of 3 lbs. acts vertically upwards. At what point must a weight of 12 lbs. be placed to produce equiHbrium round a fulcrum situated at the other end 1 (12) On a Lever of the second kmd, the weight of 24 lbs. is suspended at a distance of 8 inches from the fulcrum : what must be the length of the arm at which the power, of 64 oz., acts to balance it ] (13) A force of 48 oz. acts on a Lever of the second class at a distance of a foot from the fulcrum, and is balanced by another of 24 lbs., both forces acting perpendicularly to the Lever. Find the distance from the fulcrum at which the second force acts ; and also the pressure on the fulcrum, and its direction. (14) Two, men, A and B, of equal height, carry a weight sus- pended from a pole, which rests on their shoulders. Where must the weight be placed, so that A may bear exactly twice as much weight as B does ? THE LEVER. 105 (15) A Lemr is held in a horizontal position by two supports that are 5 feet apart, and a weight of 10 lbs. is hung at the distance of 3^ feet from one of the supports. Find the pressure sustained by the other. (16) On a horizontal rod, 45 inches long, whose extremities are supported, where must a weight be placed so that the pressure on the supports may be as 5 to 4 ? (17) A Lever y 7 feet long, is supported in a horizontal position by props placed at its extremities. Where must a weight of 2 qrs. be placed on it so that the pressure on one of the props may be 8 lbs. % (18) The pressure on ^q fulcrum of a horizontal straight Lever of the first class is 32 lbs., and the difference of the lengths of the arms is 6 inches ; one-eighth, alio, of the pressure on the ful- crum is equal to half of one of the weights which balance each other. Find the weights, and the lengths of the arms at which they act. (19) Two forces, of 8 lbs. and 10 lbs., respectively, acting verti- cally balance on a horizontal straight Lever of the third order ; the difference of the lengths of the arms being 2 inches. Find the length of the arm at which the weight acts. (20) A heavy uniform rod, 12 feet in length, and weighing 24lbs., rests horizontally on two props distant 2 feet and 4 feet from the two ends. Find the pressure on each prop. (21) The arms of a false Balance are equal in length, but one scale is loaded. Find the true weight of a body by means of such a Balance. (22) If a body be weighed successively at the two ends of a false Balance^ whose arms are of imequal length, find the true weight of the body. (23) A weight, when it is attached first to one extremity of a straight Lever of the first class, and then to the other, is balanced by half a pound, and 18 oz., respectively. Compare the lengths of the arins of the Lever, (24) In a Lever, where the Power and the "Weight act perpen- dicularly on the same side of the fulcrum, the Weight is given, and its point of application, and there is no pressure on the fulcrum. Find the Power ^ and the point of its application. • 106 EXAMPLES AND PROBLEMS. (25) Explain to what class of Levers the handles and body of a Wheelbarrow belong-. (26) If a straight Lever ^ from the extremities of which weights are suspended, balance in any one position round a fulcrum, it will balance in every other position round that fulcrum. Prove this. (27) A uniform rod, 14 inches long, and 10 lbs. in weight, is joined so as to be in the same straight line with another uniform rod 16 inches long, and weighing 8 lbs. Find the point on which they will balance. (28) Two forces, acting perpendicularly to a straight Lever on the same side of the fulcrum, keep it at rest. The difference of the lengths of the arms is 7 inches, and their sum 63 inches. The greater force being 2^ cwt., find the number of pounds in the other. * (29) Two forces, 3 lbs. and 5 lbs., acting perpendicularly to a Lever on the same side of the fulcrum, balance. Half the sum of their distances from the fulcrum being 8 inches, find the lengths of the arms. (30) The difference between two weights which keep a hori- zontal Lever at rest is 5 lbs., and the pressure on the fulcrum is 31 lbs. The difference of the lengths of the arms is 4^ inches. Find the weights, and the distance between the points at which they are applied. (31) At one end of a Lever ^ 20 inches long, a weight of 4 lbs. is placed, and is balanced by a weight at the other end. The sum of the weights, together with the pressure on the fulcrum, being 30 lbs., find the second weight, and the distance of the fulcrum from it. (32) Two forces, acting perpendicularly and on the same side of the fulcrum, keep a Lever at rest. The distance between their points of application is 4 feet 8 inches, the difference of the forces 16 lbs., and the sum of the forces, together with the pressure on the fulcrum, is 60 lbs. Find the forces, and the distance from the fulcrum at which the lesser force acts. (33) A horizontal uniform cylinder, of which a portion 6 inches long weighs 8 lbs., is \\ yards in length. A weight of 18 lbs. being attached to one of its extremities, find the position qI \}ciQ fulcrmn upon which the whole will balance. THE LEVER. 107 (34) A heavy uniform rod rests horizontally between two smooth pegs, which are 1 in. apart. The length of the rod is 1 ft., and its weight is 4 lbs. Find the pressure on each peg, one peg being at one end of the rod. (35) Two cylinders, each of uniform density and of the same diameter, when joined with their axes in the same straight line, balance horizontally on a fulcrum placed at their junction. Their lengths are 9 and 16 inches respectively; and 5 inches of the shorter cylinder weigh 8 lbs. How many ounces will 10 inches of the other cylinder weigh ] (36) There is a cylinder 2 feet long, and whose base is a square inch. From one of its ends a hole is bored f of a square inch in area, its axis being the same as that of the cylinder. What must be its length, so that the solid may balance horizontally on a fulcrum placed under the point where the bore ends ? (37) There is a rectangular parallelepiped, 18 inches long, whose ends are squares of which a side is an inch. By taking shavings, each a hundredth part of an inch thick, from two-thirds of the length of one of the faces, the body is made to balance on the line from which the shavings begin. Find the number of the shavings. (38) Two uniform prisms, whose bases are similar and equal, are joined with their axes coincident ; and a fulcrum is placed at their junction. The length of one of the prisms is 8 inches, and 2 inches of it weigh 3 lbs. The length of the other is 13 inches, and 3 inches of it weigh 2 lbs. Shew, 1st, which of the arms will preponderate; and, 2nd, what length must be cut off that arm and joined to the extremity of the other, so that equilibrium may take place round the fulcrum. (39) The "Weight ( W) being given in each of the three kinds of Levers, shew within what limits the magnitude of the pressure on the fulcrum will lie. (40) Two cylinders of the same diameter, whose lengths are 1 ft. and 7 ft., respectively, and whose weights are in the ratio of 5 to 3, are joined together, so as to form one cylinder; find the position of the fulcrian, on which the whole will balance. (41) A uniform iron bar is 1 1 ft. in length, and 4 lbs. in weight. Find the weight, which acting at one end of the rod, will keep it at 108 EXAMPLES AND PEOBLEMS. rest iu an horizontal position, upon a fulcrum 3 in. distant from that end. (42) The directions of two forces, P and Q, which act on a bent Lever, and keep it at rest, make equal angles with the arms of the Lecer, which are 6 and 8 inches respectively. Find the ratio of ^toP. iii. COMPOSITION AND RESOLUTION OF FORCES. (Arts. 30... 34.) (1) Three forces, of 4 lbs., 2 lbs., and 3oz. respectively, act upon a point in the same direction, and in the opposite direction forces of S oz., and 5 lbs., act. What other single force will keep the point at rest ? (2) Shew that the Resultant of two forces, which act on a point in directions not in the same straight line, must be less than the sum of the two forces, and greater than their difference. (3) Three equal forces act on a point and keep it at rest. Find the angle at which one is inclined to another. (4) Find the magnitude, and the direction, of the Resultant of two given equal forces, which act on a point at right angles to each other. (5) If the sides AB, CA, of a triangle ARC represent in magnitude and direction two forces, which act on the point A, what line will represent the magnitude and direction of their Re- sultant f (6) If two sides, AB, BC, of a triangle ABC, represent in magnitude and direction two forces which act on the point C, what line will represent their Resultant in magnitude and direction ? (7) Can three forces, represented by the numbers 5, 6, and 12, keep a point at rest 1 (8) Two forces, of 5 lbs., and 12 lbs., act at right angles upon a pohit ; find the magnitude of their Resultant. (9) Forces represented in magnitude and direction by the sides AB, ACf BC, oi the triangle ABC, act upon the point C. Draw COMPOSITION AND KESOLUTION OF FOECES. 109 the line which will represent their Resultant in magnitude, and in line of action. (10) Shew that, if at a point A four forces be applied, which are represented in magnitude and direction by the sides of the quadrilateral figure ABCDA taken in order, the point will remain at rest. (11) Prove that, if three forces acting on a point keep it at rest, when their intensities are doubled they will still keep it at rest, if their directions be not changed. (12) Given the lines AB, AG, which represent in magnitude, and in line of action, the Resultant and one of the two forces (acting at A) of which it is compounded, draw the line which represents the other component force in magnitude, and in line of action. (13) Forces represented in magnitude and direction by the sides AB, GB, GA, of the triangle ABG, act on the point A. Required to draw the line which will represent the magnitude, and the line of action, of the force that will keep them at rest. (14) Let ABG be a triangle, and D the middle point of BG. If the three forces represented in magnitude and direction by AB, AG, DA, act upon the point A, find the ^magnitude and direction of the Resultant. (15) A force of 10 lbs., acting perpendicularly on the arm of a Lever, keeps it at rest. This force is removed, and the equilibrium is maintained by another force, applied at the same point in a line which makes half a right angle with \hQ Lever : find its magnitude, and shew what other effect it has, besides preventing the Lever from moving round the fulcrum. (16) If a weight be supported by two strings tied to it, which are pulled in directions at right angles to one another by forces of 3 lbs., and 4 lbs., find the weight. (17) Three forces acting on a point keep it at rest; and they continue to do so, when each force is increased by 1 lb., the direc- tions of the forces remaining as before. What is the inference to be drawn as to the magnitudes of the forces ? (18) A string passing round a small smooth peg is pulled at each end by a force of 10 lbs., and the angle between the two 110 EXAMPLES AND PROBLEMS. parts of the string is a right angle. Find the pressure on the peg, and the direction in which it acts. (19) A string, passing round a smooth peg, is pulled at each end by a force equal to the strain upon the peg. Find the angle between the two parts of the string. iv. VELOCITIES. (Art. 46.) [In the following Examples, unless it be intimated otherwisef the motion is supposed to be utii/orm.'] ' (1) A railway train goes at the rate of 30 miles an hour. What is its Velocity y estimated by the number of feet traversed in a second 1 (2j A railway train travels over 150 miles in 5h. 40 m. "What is its average Velocity in feet per second ? (3) Compare the Velocities of two bodies, of which one moves through 13 yards in 2J seconds, and the other through 260 feet in a minute. (4) A body moves with a Velocity of 30. With what Velocity must another body move, which starts from the same point 3 minutes after the former, and overtakes it in 10 minutes ? (.5) For 6 seconds a body moves with a Velocity of 10 feet; and for the next 9 seconds with a Velocity of 15 feet. What uniform Velocity would have carried it over the same space in the same time ? (6) In 12 minutes a man walks a mile. For the first 3 minutes his Velocity is 5, and for the last 5 minutes it is 10 ; what is his Velocity during the rest of the time ? V. MECHANICAL POWERS. (Arts. 35... 48.) (1) What is the greatest Weight, which can be supported by a Power of 40 lbs., by means of a ' Wheel-a7id-Axle\ when the diameter of the Wheel is 10 times that of the Axle ? (2) A Power of 56 lbs., by means of a ' Wheel-and-Axle^ sup- ports a weight of 10 cwt. ; and the diameter of the Axle is 6 inches : what is the diameter of the wheel 1 MECHANICAL POWERS. Ill (3) What is the ^Mechanical Advantage^ in the last Prob.? How does it differ from that of another machine, where the diameter of the Wheel is 20 in., and that of the Axle 1 in. ? (4) In Prop, XI., if the string, by which the Power acts, be carried over 2, fixed pulley, so that the 'power may be applied doicn- loards, will the '' Mechanical Advantage'' be affected] Will it be necessary, in this case, that the i^oicer be applied in a direction parallel to PA and RB ] (5) In a system of pulleys in which the same string passes round all the pulleys (see Prop. XII.), the ^Mechanical Advantage^ is expressed by the number 10; how many moveable pulleys are there ? (6) In a system in which each pulley hangs by a separate string (Prop. XIII.), the ^Mechanical Advantage^ is equal to 16; how many pulleys are there ? (7) In Prop. XIII., if there be three moveable pulleys, and the weight {w) of each be taken into account, shew that P-w : W-io :: 1 : 8. (8) In Prop. XIII., if there be three moveable pulleys, find the result of increasing P by 1 oz., and fF by 10 oz. (9) In Prop. XIII., if there be tico pulleys, equal in weight, shew that P is greater than it would be, if the pulleys were sup- posed without weight, by a quantity equal to three-fourths of the weight of either pulley. \In the folloicing Problems the 'Inclined Plane' is sitpjyosed perfectly smooth.] (10) The Weight sustained at rest on an Inclined Plane by a PoT^^r acting parallel to the plane is 16 lbs.; and the Pressure on the plane is 3 lbs. Find the Power. (11) In Prop. XIV., if the 'Inclined Plane' be capable of moving horizontally, what force will be required to prevent it ? (12) If a Weight of 10 lbs. be placed on an ' Inclined Plane', whose dase is 16 feet, and height 12 feet, and be attached by a string to an equal Weight hanging over the top of the Plane, find how much must be added to the weight on the Plane that there may be equilibrium. 112 EXAMPLES AND PROBLEMS. (13) In Prop. XIV., if P be three-fifths of W, what will be the relation between H^and the Pressure on the Plane? (14) Two ^Inclined Planes^ of the same height are placed to- gether, height to height, their lengths being 2 feet and 16 inches, respectively; and two Weights^ resting upon them, balance each other by means of a string passing over a pulley, the portions of the string being parallel to the planes. The weight supported on the longer plane is 42 lbs., find the other weight. (15) A ' Wheel-and-Axle^ is applied to sustain a weight upon 2^u ^ Inclined Plane\ the string being parallel to the plane. Find the condition of equilibrium. (16) A weight W is supported on a plane inclined to the horizon at an angle of SO'' (one-third of a right angle), by a force P acting parallel to the plane : find the relation between P and W. (17) A weight of 100 lbs. is suspended from the block of a single moveable pulley, and the string by which the power acts is fastened at the distance of 2 feet from the fulcrum of a straight horizontal Lever, 5 feet long, of the second class. Find the force to be applied perpendicularly at the extremity of the Lever to preserve equilibrium. (18) In a system of three pulleys, each of which hangs by a separate string (as in Prop. XIII.), the Power (P) is a heavy body weighing 2.5 lbs., which rests on an ' Jficlmed Plaiie\ with the string, passing round a smooth peg, parallel to the plane. The height of the plane being three-fourths of its base, find TV in case of equi- librium. (19) In the same system as the last, when P is half a cwt., and the height of the 'Inclined Plane^ is half its length, find W in case of equilibrium. (20) A heavy body is supported on an ' Inclined Plane', whose length is 10 times its height, by means of a string which is parallel to the plane, and is fastened to one end of a horizontal straight Lever, so that its direction is at right angles to the Lever; and at the other end of the Lever is placed 20 cwt. The lengths of the arms of the Lever are as 7 ; 1 ; find the weight supported on the plane. 9 CENTEE OF GRAVITY. ll'' vi. CEXTRE OF GRAVITY. (Arts. 49... 57.) - (1) Three weights of lib., 2 lbs., and 3 lbs., are placed along a straight line a foot ajDart. Find their c. g. (2) Three weights of 1 lb., 2 lbs., and 3 lbs., are in given posi- tions, but not in the same straight line. Find their c. g. (3) Two weights of 6 lbs., and 12 lbs., are suspended at the extremities of a uniform horizontal rod, whose weight is 18 lbs. Find the c. G. (4) Shew that the c. g. of a parallelogram is the point of inter- section of its diagonals. (5) Find the c. g. of four weights, lib., 2 lbs., 3 lbs., and 4 lbs., placed at the angular points of a given square. (6) Find the c. g. of a set of weights of 1, 2, 3, 4, and 5, lbs., placed at the angular points of a given regular pentagon. (7) Weights of 2 lbs. each are placed on two adjacent corners, and weights of 4 lbs. each on the other two corners, of a horizontal square, whose side is a foot in length. Find the distance of the c. g. of the whole from either of the lesser weights. (8) One side, and the c. g. of a triangle being given, construct the triangle. (9) On a given horizontal base construct a triangle, such that the vertical line through its c. g. shall pass through one extremity of the base. (10) Find, by construction, the c. g. of a triangle ABC, and of a heavy body placed at A, one of its angular points, the weight of the hea\'y body being 7 times that of the triangle. (11) One half of a given triangle is cut off by a straight line parallel to the base ; find the c. g. of the remaining trapezium. (12) One-third of a parallelogram is cut oflF by a line parallel to one of its diagonals ; find the c. g. of the remainder. (13) Shew that an isosceles triangle, when placed vertical with one of its equal sides on a horizontal plane will not fall over, what- ever be the length of its sides. (14) Why does a butcher's boy, in delivering meat, generally ride with one stirrup-leather shorter than the other 1 L. c. c. 8 114 EXAMPLES AND PROBLEMS. (Lj) If a quadrilateral be such, that one of its diagonals divides it into two equal triangles, shew that its c. Q, is somewhere in that diagonal. (16) If a triangle, of uniform density and thickness, be sus- pended from one of its angular points, shew that it will be divided into two equal parts by the vertical plane passing through the point of suspension. (17) If a triangle, of uniform material, when suspended from one of its corners, has its base horizontal, what is the form of the triangle ? (18) In the leaning tower of Pisa, the top overhangs the base by 12 feet ; why does it not fall ] HYDROSTATICS. vii. PRESSURE OF NOX-ELASTIC FLUIDS. (Arts. 60... 67.) [N.B. For practical purposes a cubic foot of water may he taken to weigh 1000 oz. avoirdupois^ (1) What is the pressure on the horizontal bottom of a cistern, filled with water, the area of the bottom being 10 square feet, and the depth of the water one foot ? (2) Find the pressure on the horizontal base of a vessel, whoso sides are vertical, and which is filled with fluid, to the depth of 6 inches ; a cubic inch of the fluid weighing an ounce and a half, and the area of the base being 64 square inches. (3) A cubic inch of Mercury weighs 8oz., what will be the pres- sure on the square inch in a vessel of Mercury at a de]3th of two feet below the surface ? (4) A vessel, with vertical sides and horizontal base, which contains fluid, is placed in the scale of a Balance, and is poised by weights in the other scale. A person, without touching the vessel, SPECIFIC GRAVITIES. 115 then holds his hands in the fluid. Shew whether this has any effect on the equiUbrium of the Balance. (5) If a solid be put into a fluid contained in a vessel, whose sides are vertical and bottom horizontal, and floats, explain what additional pressure (if any) there mil be on the base. (6) Explain the advantage of shallow cisterns over deep ones, cceier is paribus. viii. SPECIFIC GRAVITIES. (Arts. 68.. .78.) [/?z the following Examples, whenever a tnixture is supposed to he made of different substances,— fluid or solid, — unless it he expressly stated otherwise, the Weight and the Volume of the Com- pound are to he considered respectively equal to the sum of the Weights, and the sum of the Volumes, of the substances employed. Some substances, — Water and Alcohol for instance, — lose bulk when mixed together, and so form a mixture that is of a greater S. G. than if there had been no dim,inution of hulk. If equal weights of Water {S. G. 1) and of Alcohol (S. G. '825) were mixed together without bulk being lost, the S. G. of the m,ixture {Proof Spirit) would he ' 904, and not ' 93, the S. G. given in the Table, Art. 69. In the following Examples, if the S. G. of a substance he given in figures, the figures are the numbers given in the " Tables of Sp>ecific Gravities^'' explained in Art. 69.] (1) A cubic inch of iron weighs 4|oz., and a cubic foot of water 1000 oz., avoirdupois. Find the s. g. of iron. (2) Fourteen cubic feet of granite weigh a ton. Find the s. g. of granite, a cnbic foot of water weigliing 1000 oz. (3) If 9 cubic feet of a certain substance weigh 1000 lbs., find its s. G. (4) The weight of 540 cubic inches of a substance is 10 lbs., that of 360 cubic inches of another is 40 oz. The s. g. of the second substance being 3, what is that of the other ? (5) Ten cubic inches of a substance weigli 4 lbs. ; 4 cubic inches of another weigh 10 lbs. How often does the s. G. of the latter sub- stance contain that of the former ] 8—2 116 EXAMPLES AND PROBLEMS. (6) Find the length of an edge of a cubical block of a substance which -weighs 2000 tons, the s. G. of the substance being 1*12 times that of water. (7) The s.G. of tin being 7'2, find how many cubic inches of copper (s. G. 9) will weigh as much as a cube of tin whose edge is 4 inches. Find also the s. g. of the compound metal, formed by fusing the lumps of tin and copper together. (8) Ninety pounds of silver are of the same bulk as 75 lbs. of a mixed metal. How many times does the s.G. of silver con- tain that of the mixed metal ? (9) The s. G.s of two substances are as 45 to 81. The weight of 240 cubic inches of the former is 18 lbs. ; what will be the weight of 360 cubic inches of the latter ? (10) Six pints of fluid (s. g. 0'925) are mixed with two quarts of water. Find the s. g. of the mixture. (11) In 50 cubic yards of rock, whose average s.G. is 1*42, there enter 32 cubic yards of a substance whose s.G. is 1*24. Find the S. G. of the remainder of the rock. (12) Find the s.G. of a mixture composed of equal bulks of two fluids whose s. G.s are '9 and 1*2. (13) Five pints of a fluid (s.G. L04), 2 quarts of another (S. G. 0'97), and 4 gallons of a third (s.G. 11), are mixed together ; find the s. g. of the mixture. (14) Portions of two fluids, which are not of the same s. d., iona a mixture, of which the s. g. is half the sum of the s. g.s of the fluids composing it. Compare the volumes of the fluids employed. (15) If JV, W\ W", be the weights of two substances and of a compound formed of them, whose s. g.s, as given in "Tables of Specific Gravities ", are o-, a, a\ respectively, shew that W _ W W o-'' ~ o- "^ V • (16) Find the s. g. of a mixture formed by mixing equal weights of water, and of a fluid whose s. g. is 0*825. (17) Nine pounds of fluid (s.G. 1-05) are mixed with 7 lbs. of water. Find the ratio of the s. g. of the mixture to that of water. (18) Eight ounces of salt, mixed with 2 quarts of water, (with- out increasing its volume) make a brine in which an q^% just floats. The weight of a pint of water being 1 lb., find the s. g. of the ^gg. SPECIFIC GRAVITIES. 117 (19) There are mixed together 8, 12, and 20 lbs., of three fluids, respectively ; and the s. g.s of the first fluid, the second fluid, and of the mixture, are 16, OS, and 1*2, respectively. Find the s. g. of the third fluid. (20) A quart of fluid of s. G. 0-8 is mixed with a gallon of fluid of s. G. 105, but from a chymical union taking place the volume of the mixture is one hundredth part less than the sum of the volumes of the portions of the fluids composing it, though no weight is lost. Find the s. g. of the mixture to four places of decimals. (21) The s. G.s of platina, iron, and water being 21-5, 7'8, and 1, respectively, find the numbers to three places of decimals which will represent the s. G.s of platina and water, when the s. g. of iron is taken to be 1. (22) A vessel, whose sides are vertical and base horizontal, contains three quarts of fluid, the depth of which is 10 inches. When a piece of copper of s.g. 8-9 is immersed, the surface of the fluid rises to lli- inches above the base. Find the weight of the piece of copper, the weight of a pint of water being 1 lb. (23) A solid, whose weight is 6 lbs., floats ,on a fluid, so that the weight of the portion above the surface of the fluid is 2 lbs. Compare the s. g.s of the fluid and solid. (24) A cube of cork, whose edge is a foot, floating in water, ver- tically sinks to the depth of 2"88 inches. Find its s.g., that of water being 1. (25) A solid, of 15 cubic inches in volume, and whose s.g. is 06, floats on water. What is the content of the portion of it below the surface ? (26) A block of wood of s.g. 0-64 floats on a fluid with 5-9ths of it below the surface ; find the s. g. of the fluid. (27) A prismatic solid, 10 inches long, floats vertically with 3-5ths of its length immersed. Compare the s. g. of the solid with that of the fluid ; and if the solid float vertically after having been shortened 4 inches, find what length of it will then be above the surfiice of the fluid. (28) A solid, whose weight is 18 lbs, floats on a fluid, whose density is three times that of the solid. Find the weight of the portion of the solid which remains above the fluid. ¥ 118 EXAMPLES AND PROBLEMS. (29) A cubical block, whose edge is 3 inches, floats with a side 1 inch above the surface ; and if an ounce weight be put on the block, it becomes just even with the surface of the fluid. Find the weight of the solid. (30) The weight of a cubic foot of oak is 75*6 lbs., and when im- mersed in water it weighs 12'6 lbs. Find its s.g. (31) When to one of the faces of a cube of wood, whose edge is 1 foot, a plate of copper of the same area is attached, the whole is foun"*- to be of the same s.g. as water. The s.g.s of the wood and of the copper being 08 and 9, respectively, find the thickness of the plate. (32) Given that a cubic foot of water weighs 1000 oz., find how ma^ y nails, of 24 to the pound, must be driven up to their heads in a cubic foot of wood of s.g. 075, so that the whole may be of the same s. G. as water. (33) A piece of wood, which weighs 3 lbs., floats in water, and the S.G. of the wood : s.g. of water :: 3 : 4. What weight must be placed on the wood so as just to sink it? (34) The weight of a piece of wood is 45 lbs.; and when it is put into water 3 lbs. are required to sink it even with the surface. Find the s.g. of the wood. (35) A body floats with 3 lbs. of it above the surface of the fluid; and when another body, half its size, and of s.g. twice as great, is placed upon it, it sinks even with the surface. Find the weight of the former body. (36) A weight of 4 lbs., when placed on a piece of wood whose g.G. : S.G. of water :: 3 : 5, just causes it to be immersed. Find the weight of the wood. (37) A block of wood weighs 10 lbs. in air, and when put into a vessel of water it sinks, pressing the bottom of the vessel with a force of 2 lbs. Find what proportion of the block must be hollowed out, — the orifice of the aperture being closed with wood of the same kind, — so that it may float just even with the surface of the water, when it is held down with a force of 2 lbs. Find also the s. g. of the wood. " (38) A cylindrical vessel of horizontal base contains water. If a cubic foot of cork of s.g. 024 be allowed to float on the water, find the additional pressure on the base. SPECIFIC GRAVITIES. 119 (39) A body, whose volume is 250 cubic inches, weighs 6 lbs. in a fluid, 50 cubic inches of which weigh 2^ lbs. What will the body weigh out of the fluid '? (40) The edges of a rectangular block of marble are 4, 7, and 12, feet ; and a piece of the marble that weighs 8 oz. in air, weighs I } oz. in water. Find the weight of the block, it being given that a cubic foot of lead (s. g. 11-2) weighs 700 lbs. (41) 95 ounces of gold, and 99 ounces of lead, balance when weighed in water. The s. g. of gold being 19, find 1st the s. g. of lead, and 2nd. the s. g. of an ounce of gold and an ounce of lead melted together. (42) If the s. G.s of iron, and of gold, be 8 and 19 times, respec- tively, that of water, find the weight in water of a substance com- bined of 1 lb. of iron, and 1 lb. of gold. K (43) A piece of iron weighs 12 lbs. in water ; and when a piece of wood, which weighs 5 lbs., is attached to it, the two together weigh 9 lbs. in water. Find the s. g. of the wood. (44) A piece of iron weighs 10 lbs. in air, and when it is at- tached to a piece of platina (heavy enough to sink it) which has been immersed in mercury and balanced, the weight that restores the equilibrium is 7 lbs. 6 oz. The s.G. of mercury being 13'6, find that of iron. (45) A body rests, wholly immersed, between two fluids which do not mix, M cubic inches of it being surrounded by the lighter fluid (s. G. s) and N by the heavier (,s.g. /). Find the ratio M :M+ N, in terms of the s. G.s of the two fluids, and of that of the solid (/'). (46) In a mixture, formed of volumes or and y of two fluids, whose s. G.s are s and / respectively, a body floats with a volume M immersed ; but in a mixture formed of the volumes cc and y of the fluids whose s. g.s are / and s respectively, it floats with a volume N immersed. Find the ratio of x to y. (47) A cylinder of metal, of s.G. 12'4, and' 14 inches long' floats wholly immersed, with its axis vertical, in a vessel filled with mer- cury (s.G. 13*6) and water. Find the length of the portion. of the cylinder, which is surrounded by the water. 120 EXAMPLES AND PROBLEMS. (48) If TV, TV\ be the weights in air, and w, ic% the weights in water, of two substances, the former of which is of a greater density than the hitter, shew whether W: w be less, or greater, than W : w'. (49) Three masses, of gold, silver, and a compound of gold and silver, each weigh n ounces in air, and p, q, r, ounces, respectively, when immersed in water. Determine the weight of the gold in the compound. (50) Three masses, of gold, silver, and a compound of gold and silver, weigh respectively P, Q, and B, ounces in air, and p, g, and ;•, ounces in water. Find the ounces of gold in the compound. (51) Taking the quantities P, Q, R, and p, q, r, from the last question, shew what is the order of magnitude of the quantities p:P,q:Q,r'.R. (52) There are masses of two metals which weigh 51, and 39, oz. respectively, and a mass containing certain proportions of the metals weighs 45 oz. ; and each of the three bodies loses the same weight, when immersed in the same fluid. Find the number of omices of the first metal, which enters into the compound. (53) A vessel which contains fluid is balanced in a pair of scales. A heavy body is then suspended in the fluid as represented in the figure to Art. 74. What effect (if any) is produced on the equili- brium of the scales ? (54) A uniform cube floats with a face, of which the diagonal is 10 inches, horizontal, two weights being placed on the face, — one, of 8 lbs., being at one corner of it ; find the position of the second weight, which is 20 lbs. (55) A cubic foot of water, weighing 1000 oz., is put into a vessel which is of the form of two cylinders, j)laced with their axes verti- cal and coincident one upon the other, and communicating with each other. The areas of the hases of the lower and the upper cylinders are 54 and 24 square inches, respectively ; and the height of the lower cylinder is 18 inches. Shew that the pressure on the base of the vessel is 1546^ oz. (56) Two vertical tubes, of equal bore, are connected by a hori- zontal tube 2 inches long. Supposing 12 inches in length of mercury to be poured into one tube, and 26 of water into the other, find the altitudes of the water and the mercury in the two branches, the s. G. of mercury being supposed to be 13 times that of water. ELASTIC FLUIDS. 121 ix. ELASTIC FLUIDS. (Arts. 79... 100.) (1) Explain why a balloon rises, and why the higher it gets the tlower it rises. Why does it ever cease to rise ? (2) If the mercury in a Barometer stand at 30 inches, what is the greatest vertical length of the suction pipe of a Common Pump^ which will pump up Mercury ? (3) A Common Pump will raise water 33 feet ; how high will it raise oil of the s. g. 0*88 1 (4) If 13 inches of water be inserted in the tube of a Barometer upon the mercury, what will be the altitude of the upper surface of the water, when the Common Barometer stands at 30 inches, the 8. G. of mercury being 13 ? And how much will the top of the water fall, when the mercurial Barom,eter sinks 1 inch 1 (5) At what height will the Water Barometer stand, when the atmospheric pressure is 15 lbs. to the square inch % (6) The height of the mercurial Barometer being 30 inches, required that of a Barometer filled with equal lengths of mercury and proof spirit, the s. g. of mercury being 13'6, and that of proof spirit being 093. (7) Under what circumstances might the variation of the Baro- meter prevent the working of the Common Pump ? (8) If a cubic inch of Mercury weighs 78 oz., what will be the atmospheric pressure on a square inch, when the Barometer stands at 29^ inches 1 (9) A piston fits a hollow cylinder, whose height is 7 inches, and base 1 square foot. Supposing the cylinder originally filled with common air, the pressure of the atmosphere to be 14 lbs. on a square inch, and the piston, by its weight, to sink 1 inch in the cylinder, find the weight of the piston. (10) The tube of a mercurial Barometer is vertical, and of uniform bore. On a syringeful of air being introduced into the upper part of the tube the mercury falls 1 inch ; and it falls eight- tenths of an inch more when another syringeful is introduced. The mercury in the cistern being kept at the same level throughout, 122 EXAMrLES AND PROBLEMS. find the length of that portion of the tube which was originally a vacuum. (11) A cylindrical tube, 25^ inches in length, closed at one end, is immersed vertically in water, its closed end being in the surface of the water ; find the height to which the water will rise in the tube, assuming that a column of 32 feet of water measures the atmos- pheric pressure. (12) Find the pressure against the valve, which opens into the Receicer of a Condenser, after 15 strokes of the piston, when ^4= content of the Receiver, ^ = content of Barrel, and P— the atmospheric pressure. (13) If the content of the Receiver of an Air-Pump be three times that of the Barrel, what will be the pressure of the air in the Receiver after three strokes of the piston, the atmospheric pressure being 15 lbs. on the square inch ? (14) If a Sijyhon be used for drawing off Mercury, what is the greatest height at which the bend may be placed ? (15) Is the extreme depth, from which a Common Pump can raise water, always the same ? or, is it variable ? Upon what does it depend ? THERMOMETERS. (1) Find the degrees of the Centigrade Thermometer, which shew the same temperatures as 77°, 23°, and 32° below zero, of Fahrenheit. (2) Find the degrees of Fahrenheit, which shew the same tem- peratures as 50°, 110°,-10°,-20°-5, of the Centigrade. (3) Shew that 95° of Fahrenheit denotes the same temperature as 35° of the Centigrade. (A) "Temperate" is marked on Fahrenheit'' s Thermometer at 66°, find the corresponding number of degrees on the Centigrade. (5) What number of degrees of Fahrenheit corresponds with 4® below zero of the Centigrade^ THERMOMETERS. 123 (6) "What number of degrees of the Centigrade corresponds with zero of Fahrenheit ? (7) At what temperature will Fahrenheit and the Centigrade shew the same number of degi'ees ? (8) The temperature at one place is 35° by the Centigrade, and at another 64° by Fahrenheit. Find the difference of these tempe- ratures by Fahrenheit. (9) Find the temperature at which Fahrenheit shews as many degrees above zero as the Centigrade does below zero, (10) A Fahrenheit's Thermometer is not graduated higher than the '• Boiling Pointy and the length of the scale is one-fourth more than the distance between the '"Freezing'' and -Boiling'' points. Find the degrees of the Centigt^ide which correspond with the lowest graduation on the instrument described. (11) The ^Freezing' and ^Boiling' points on Be Lisle' s Ther- mometer are marked 150°, and 0°, respectively; at what temperature will the number of degrees on De Lisle's, and on Fahrenheit's, be equal ? (12) The sum of the numbers, which mark the same heat on the Centigrade, Bind Fahrenheit, Thermometer, is 102 ; find the number on each. (13) The sum of the number of degi'ees indicating the same tem- perature on the Centigrade, and Fahrenheit, Thermometer, being equal to 0, find the number of degrees on each. (14) How many times will the divisions on Fahrenheit's Ther- t^ mometer coincide with those on the Centigrade between Freezing point and 0° Fahrenheit? ANSWERS TO THE PEECEDING EXAMPLES AND PROBLEMS. i. Density, and Force. (1) Being equal in size, they will be equal in density, if they are equal in weight ; otherwise, not. (2) As 5 4. (3) As 7 : 10. (4) As 1 7. (5) As 93 : 100. (6) As 1 8. (7) As 9 4. (8) As 8 9. 1^ of a cubic foot. (9) As 12 : 1. (10) (11) 48 lbs. (12) (i) Ans. 2 : 5. (2) Ans. 2 : 15. (13) A force of 8 lbs. (14) f . ii. The Leyer. (1) 40 lbs. (2) 2 ft. 11 in. (3) 4 ft. 8 in. (4) 3 St. 12 lbs. (5) 48 lbs. and 64 lbs. (6) 16 in. from the 3 lbs. (7) 4 in. from greater wt. (8) 5 lbs. (9) 2 lbs., 4 ft. from fulcrum. (10) 18 in. from greater force. (11) At a distance from fulcrum of Jth the length of the Lever. (12) 4 ft. (13) (1) Ans. 11 in. (2) Ans. 21 lbs. (3) Ans. In the direction of the greater force. (14) One-third of the length of the pole from A. (15) 7 lbs. (16) 20 in. from either end. (17) 1 ft. from either end. (18) Weights 8 lbs. and 24 lbs. Arms 9 in. and 3 in. (19) 10 in. (20) 8 lbs. and 16 lbs. ANSWERS. 125 (21) Half the sum of the apparent weights, when the body is weighed in both scales suc- cessively. (22) The square root of the pro- duct of the apparent wts. (23) As 2 : 3. (24) P=TV, and is applied at the same point. (25) The 2nd. (26) See Art. 29. (27) ^in. from the junction, on the former rod. (28) 224. (29) 10 in., and 6 in. (30) (i) Ans. 13 lbs., and 18 lbs. (2) Ans. 27 9 in. (31) (1) Ans. 11 lbs. (2) Ans. 5-^ in. (1) Ans. 14 lbs. and 30 lbs. (2) Ans. 8 ft. 9 in. 21*6 in. from the wt. 20 lbs., and 24 lbs. 81. 16 in. 75. (1) Ans. The longer. / N A 25 . (2) Ans. ^-m. In 1st, between TV and any greater weight. In 2nd, between and TV. In 3rd, between and cc, that is, as small, or as great, as we please. (40) On the longer cylinder, 1 ft. from the junction. (41) 8 lbs. (42) 3 : 4. (32) (33) (34) (35) (36) i37) (38) (39) iii. Composition and Resolution of Forces. (1 ) 11 oz. acting in the direction of the latter forces. (3) One-third of 4 right angles. (4) If P be one of the forces, the Resultant = Pj^ ; and its direction bisects the angle between the two given forces. (5) CB. (6) AC, (7) ^0, (8) 13 lbs. (9) Produce AC to Z), so that CD = 2 AC, CD is the Re- sultant required. (12) Join BC, and complete the parallelogram ^^(7Z>, AD is the line required. (13) From A draw AD parallel to BC, and equal to 2BC ; AD is, the line required. (14) AD. (15) (1) Ans. 10>/2lbs. (2) Ans. A pressure of 10 lbs. on the fulcrum in direction of the arm. (16) 5 lbs. (17) The forces are equal to one another. (18) (1) Ans. 17-32 lbs. (2) Ans. Bisects the angle. (19) Four-thirds of a right angle. [26 ANSWERS. iv. Velocities. (1) 44. (4) .39 (2) 3S-8 nearly. (5) 13 (3) As4 : 1. (6) 5:^- (1) 400 lbs. C2) 10 ft. (3) (i)f . (2) No difference in theory ; but in practice the lat- ter machine will be wanting in strength. (4) (1) Xo. (2) No. (5) 5. (6)4. (8) W\nW.\)Q raised. (10) 15 711bs. Y. Mechajs^cal Powers. AG (11) P^-rjj, acting horizontally. (12) 6|lbs. (13) As 5 : 4. (14) 28 lbs. (15) P- JF:: rad-ofax!e ^ ^ ' " rad. of wheel ^ length of plane (16) P : TV:: (17) 20 lbs. (18) 120 lbs. (19) 2 cwt. (20) 32,000 lbs. its height ; 2. vi. Centre of Gratity. (1) Between 2 lbs. and 3 lbs. ; 8 in. from the latter. (2) A, B, C, being the positions of the weights, join AB, take BD one- third of ^^, join CD, and bisect it in G^; (r is the c. g. required. (3) Five-ticelfths of the length of the rod from the 12 lbs, (5) Proceed as in Art. 52. (6) Proceed as in Art. 52. (7) 10 inches. (8) Bisect the given side, AB,m C; join CG, and produce it to 2 Z>, so that CG=- CD ; ADB is the triangle required. ANSWERS. 127 (9) AB the base ; draw BC vertical ; Z> any point in BC; join ADj and produce it to E, so that AD = DE\ ABE is the triangle required. (10) Bisect BG in D, and AB in E. Join AD, CE, intersect- ing in F. In AF take G such that AG : AF :: 1 : 8 ; G^ is the o. g. required. • (11) G, g, being the c. g.s of the whole triangle, and of the tri- angle cut off, join gGj and produce it to i/, so that GH— Gg ; H is the c. G. required. (12) Draw the diagonals ACj BD, intersecting in E ; the cutting line parallel to AG; g, the c.g. of the triangle cut off, will bo inBD. In ^Z> take G such that EG^^Eg; G will be the c.g. required. (14) Because he has to lean over on the side opposite to his basket to preserve his balance. Otherwise the vertical through his c. G. would fall beyond his base, and he would fall off. (17) Isosceles. HYDROSTATICS. Pressure of Non-Elastic Fluids. (1) 625 lbs. (5) A pressure equal to the (2) 36 lbs. weight of the solid. (3) 12 lbs. (6) They are subject to less (4) The vessel will preponde- pressure, when filled with rate. fluid. Specific Gravities. (1) 7-77G. (7) (1) 51-2. (2) 2-56. (2) 8. (3) 1-778. (8) 1*2 times. (4) 8. (9) 48-6 lbs. (5) 61 (10) 0-955. (6) 40 feet. (11) 1-74. 128 ANSWERS. (12) 1-05. (39) 18^ lbs. (13) ro8. (40) 25,200 lbs. (14) The volumes are equal. (41) (1) 11. (16) 0-904. (2) 13if. (17) 112 : 109. (42) lilt lbs. (18) 1-125. (43) 0-625. (19) 1-5. (44) 7115 (20) 1-0101. (45) (21) s.G. ofplatma = 2-7.'(), /-s s. G. of water = 128. (46) Ms'-Ns '.Ns'-Ms. (22) 8-01 lbs. (47) \^ inches. (23) 3 : 2. (48) W : w is the lesser. (24) 0-24. (49) r-q n. oz. (25) 9 cubic inches. \ / p-q (26) 1-152. (50) p Qr-Rq Qp-Pq' (27) (1) 3 : 5. (2) 2'4 inches. (51) p \ P,r \ B,q :Q, the first (28) 12 lbs. ratio being the greatest. (29) 2oz. (52) 251 (30) 1-2. (53) The scale, in which the ves- (31) ^Q of an inch. sel is placed, will prepon- (32) 375. derate. (33) lib. (54) In the diagonal which passes (34) 0-9375. through the former weight, (35) 6 lbs. and at a distance of 7 in. (36) 6 lbs. from that weight. (37) (1) Two- fifths. (56) The altitudes of the water (2) 1-25. and mercury are 30 in. and (38) 240 oz. 60 in. Elastic Fluids. (1) It rises, because its whole weight, including the gas, is less than that of the air, which it displaces. But the pressure upwards diminishes, as the balloon rises, because the density of the air diminishes ; so that its ascent will be slower and slower, until the weight of the balloon exactly balances the weight of air displaced, and thcin the balloon will cease to rise. ANSWERS. Izy (2) Just less than 30 in. (9) 21 cwt., in vacuo. (3) 371 feet. (10) 6-2 in. (4) (l)'42in. (11) liin. (2) lin. (5) 34-56 feet. m^-j'^.P, (6) 56-16 in. (13) 6fi. (7) When the height of the (14) Height of Mercurial Ba- fixed valve above the sur- rometer. face of the well exceeds the (15) No; it varies according to height of a Water-Baro- the height of the Baro- meter. meter. (8) 14-38125. Thermo METERS. (1) 25, -5, -35|. (8) The former is the greater (2) 122, 230, 14, -4-9. by 49° Fahrenheit. (4) 131. (9) llf. (10) 25 below zero. (5) 24-8. (11) 96,V (6) 17f below zero. (12) 25, and 77. (7) -40^ (13) -llf, andllf. (14) Three times. L. C. C. UNIVEESTTY EXAMINATION PAPERS. The following are the Papers of Questions actually given in the Examinations for the ordinary degree of B.A.. at Cambridge in the years 1860, 1861, 1862, and 1863. MECHANICS AND HYDEOSTATICS. Saturday, May 26, 1860. FIRST DIVISION.— A, 1. Explain clearly what you understand by the Weight of a body. A cubic inch of lead is suspended by a spiral spring, and the consequent elongation of the spring observed. If the experiment were repeated at the Equator, would the elongation be the same or different ? Give your reasons. 2. If two forces, acting perpendicularly at the extremities of the arms of any Lever, balance each other, they are inversely as the arms. A grocer buys tea wholesale at the rate of 4^. a-pound, and in weighing it out to his customers uses a Balance, the arms of which are as 16 to 15; at what price must he profess to sell it per pound, in order that he may make a profit of 20 per cent. % 3. Enunciate the 'parallelogram of forces'; and assuming it to be true, so far as the direction of the Resultant is concerned, com- plete the proof. v - , Explain the nature and action of the forces, ' which are called into play, when a boat is towed down a river by men or horses pull- ing at a single rope. UNIVERSITY EXAMINATION PAPERS. 131 4. Investigate the conditions of equilibrium on the Wheel and Axle. An endless rope, sufficiently rough not to slip, is stretched round ^ two wheels whose radii are as 1 to 3 ; find the ratio of the radii of the corresponding Axles, in order that equal weights, susj>ended from strings wound round them in contrary directions, may produce equi- librium. 5. Find the Centre of Gravity of a triangle. A triangular plate of iron w^eighing 5 cwt. is carried horizontally by 3 men, one at each angular point ; find the weight supported by each. 6. Find the 'pressure at any point in a mass of fluid at rest, having given that the pressure of the air at the surface of the fluid is P. A town is supplied with water from a reservoir, the level of the water in which is 136 feet above the lowest point of the main supply- j pipe ; assuming that the height of the Water-Barometer is 34 feet, / and that the pressure of air on a square inch is 15 pounds, required the pressure which the main must be constructed to bear. ^ 7. Describe the Common Hydrometer ; and shew how to com- pare the Specific Gravities of two fluids by means of it. An Hydrometer, used for determining the Specific Gravity of spirits, is constructed in such a way that the zero point of graduation (at the top of the stem) is on the level of the fluid, when it is placed in proof spirit, and that the interval between two successive gradua- tions corresponds to loVo^^^ P^^'^ ^^ ^^^ bulk. If the reading of the instrument, when placed in spirit sold as proof, be 27, determine the amount of water unfairly introduced (Specific Gravity of pm*e alcohol -0-8). © 8. Describe an experiment for proving that the elastic force of air at a given temperature varies as the density. A Barometer contains a small quantity of air above the mer- cury; having given that the length of the tube measured from the constant level of the cistern is 32 inches, and that the mercury in it stands at 295 inches, when a standard Barometer is at 30, obtain a formula for obtaming the true height of the Barometer from the observed height generally. 9. Describe the construction of the Common Air-Pump, and its operation. 9—2 132 UNIVERSITY EXAMINATION PAPERS. By what contrivance may the degree of exhaustion of the air in the Receiver be exhibited to the eye ? Why cannot a perfect vacuum be produced by means of this instrument ? 10. Explain the method of filling, and graduating, a Common Mercurial Thermometer. How may the height of a mountain be roughly determined by means of the Thermometer ? FIRST DIVISION.— B. 1. Define the terms '^ Force^\ and " Weight"". Explain clearly the method of estimating and comparing statical forces. 2. If two weights, acting perpendicularly on a straight Lever on opposite sides of the fulcrum, balance each other, they are inversely as their distance from the fulcrum. A tobacconist buys tobacco wholesale at the rate of 3^. Ad. a pound, and in weighing it out to his customers uses a Balance, the arms of which are as 16 to 15; if he profess to sell it at 35. dd. a pound, what profit per cent, does he really make 1 3. Enunciate the ' parallelogram of forces'; and assuming it to be true so far as the Tnagnitude of the Residtant is concerned, com- plete the proof. It is found that if, on a rapid river, a ferry-boat be turned obliquely to the stream and prevented going down the stream by means of chains stretched across from one bank to the other, it will be carried across by the force of the stream alone. Explain this. 4. Investigate the condition of equilibrium when a weight W is supported on an Inclined Plane by a force P acting parallel to the plane. If the force P be the tension of a fine thread, which passes over a small fixed pulley, and is attached to a weight hanging freely, shew that if P be pulled down through a given space, the height of the Centre of Gravity of P and W will remain unaltered. 5. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line through its Centre of Gravity falls within or vrithout the ' base'. UNIVEIiSITY EXAMINATION PAPERS. 133 Why is the word horizontal introduced in the enunciation of this proposition ? With what modifications is it true of any plane ? 6. Shew that the pressure on the horizontal bottom of a vessel, filled with fluid, depends merely on its depth below the surface, and not at all on the quantity of fluid contained in it. A pipe carries rain water from the top of a house to a large tank, the surplus water in which escapes through a valve in the top v/hich rises freely. A weight of 2 libs, is placed on it, and it is foimd that the water rises in the pipe to the height of 20 feet before the valve opens. Required its area (assuming that the height of the Water-Barometer is, 34 feet, and the atmospheric pressm'e 15 lbs. on a square inch). (\ 7. Describe the Hydrostatic Balance ; and shew how it may be applied to compare the Si^ecijic Gravities of two fluids, by weighing the same solid in each. A piece of copper of Specific Gravity 8"85 weighs 887 grains in water, and 910 grains^ in alcohol; required the Specific Gravity of the alcohol. 8. Describe the construction of the Condenser^ and the mode of its operation. If the volume of the cylinder be one-fifth the volume of the Receiver^ find the pressure at any point of the latter after 20 strokes. 9. Explain the construction of the Common Mercurial Baro- meter. Having given that the S])ecific Gravity of mercury is 13"57, and that the weight of a cubic inch of water is 252-6 grains, find the pressure of the air on a square inch in lbs., when the mercury in the Barom,eter stands at 30*5 inches. 10. Describe the construction of the Common Pump. If the upward movement of the piston be stopped, when the water has risen to a given height (say 16 feet) in the supply-pipe, but has not yet reached the piston, find the tension of the piston-rod (the area of the piston being 4 square inches, and the atmospheric pressure on a square inch being known). 134 UNIVERSITY EXAMINATION PAPERS. SECOND DIVISION.— A. 1. If two weights P and Q, acting perpendicularly on a straight Lever on opposite sides of the fulcrum, balance each other, deter- mine the position of the fulcrum, and the pressure on it. The scale-pans of a Balance are of unequal weight, and its arras consequently also of unequal length, find the true weight of any sub- stance from its apparent weights, when placed in the two scale-pans respectively. 2. If two forces, acting at any angles on the arms of any Lever ^ balance each other, they are inversely as the perpendiculars drawn from thei fulcrum to the directions in which the forces act. Describe the Balance known as the 'bent-lever Balance^ (in which the weight (P) is always the same), and shew how to gra- duate it. 3. If three forces, represented in direction and magnitude by the sides of a triangle taken in order, act on a point, they will pro- duce equilibrium. Two forces, whose magnitudes are n/sxP, and P, respectively act at a point in directions at right angles to each other, find the magnitude and direction of the force which will balance them. ^ 4. In that system of Pulleys^ in which the same string passes round any number of pulleys, and the parts of it between the pul- leys are parallel, there is equilibrium, (neglecting the weights of the pulleys), when P : W v.\ : the number of strings (n) at the lower block. If n = 6, find the greatest weight which a man weighing 10 stone can possibly raise by means of such a system without any further machinery. 5. When a body is suspended from a point, it will rest with its Centre of Gravity in the vertical line passing through the point of suspension. Hence shew, how the Centre of Gravity of any plane figure of irregular outline may practically be determined. 6, Describe an experimental proof, that, if the pressure at any point of a fluid be increased, the pressure at all other points will be UNIVERSITY EXAMINATION PAPERS. 135 equally increased. By what short form of words is this property of fluid pressure sometimes described 1 In the common Hydraulic Press, are the fluid pressures and ten- dency to break uniform throughout the cylinders ? 7. If a body floats in a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body ; and it presses down- wards, and is pressed upwards, with a force equal to the weight of the fluid displaced. Does the above proposition contain all the necessary conditions of equilibrium of a floating body ? A uniform cylinder, when floating vertically in water, sinks to a depth of 4 inches ; to what depth will it sink in alcohol of specific gravity 079? ^ 8. Describe the Common Hydrometer. If the volume between two successive graduations on the stem of a Hydrometer be yq^qq th part of its whole bulk, and it float in dis- tilled water with 20 divisions, and in sea water with 46, above the surface, required the Specific Gravity of the latter. 9. Describe the construction of the Condenser, and the mode of its operation. A cylinder, filled with atmospheric air, and closed by an air-tight piston, is sunk to the depth of 500 fathoms m the sea ; required the compression of the air, (assume specific gravity of sea water to be 1*027, specific gravity of mercury 13*57, and height oi Barometer 30 inches). 10. Explain the action of the Siphon, A hollow tube is introduced into the bottom of a cylindrical vessel through an air-tight collar ; and a larger tube, of which the top is closed, suspended over it, so as not quite to touch the bottom, consider the effect of gradually pouring water into the cylin- der, until it reaches the level of the top of the inverted tube. SECOND DIVISION.— B. 1. If two forces P and Q, acting perpendicularly on a straight Lever in opposite directions, and on the same side of the fulcrum, balance each other, they are inversely as their distances from the 136 UNIVERSITY EXAMINATION PAPERS. fulcrum ; and the pressure on the fulcrum is equal to the difference of the forces. 2. If the adjacent sides of a parallelogram represent two forces acting at a point in direction and magnitude, the diagonal will repre- sent the Ilcsultant force in direction. Two forces act at a point in directions at right angles to one another, the magnitude of one is P, and the magnitude of the Re- sultant is 2P, find the direction of the Resultant, and the magni- tude of the other force. ^j 3. In that system of Pulleys, in which each pulley hangs by a separate string, and all the strings are parallel, there is equilibrium (neglecting the weights of the pulleys) when P : W v.l : that power of 2, whose index is the number of moveable pulleys. If the pulleys were heavy, and the weight of each - W[j find the additional force i^, which would be necessary to counteract the effect of their weights. 4. If P and W balance each other on the Wheel-and-Axle, and the whole be put in motion, P : W :: W^& velocity : P's velocity. Under what conditions is a similar proposition true of other machines? Shew that in all cases when it obtains, no increase in the work done is effected by the intervention of the machine, de- fining carefully the expression in italics, 5. If a body balance itself on a line in all positions, the Centre qf Gravity is in that line. Assuming the position of the Centre of Gravity of a triangle to be known, shew that the position of the Centre of Gravity of any trapezium may be practically determined by means of this propo- sition. f^, 6. Explain the so-called Hydrostatic Paradox. Describe the construction and method of application, of the Hy- draulic Press. If the area of the small cylinder be 1^ inches, and the diameter of the large piston 20 inches, find the lifting power of the machine under a pressure of 1 ton exerted on the piston of the small tube. 7. "When a body is immersed in fluid, the weight lost is to the whole weight as the Specific Gravity of the fluid is to the Specific Gravity of the body. UNIVERSITY EXAMINATION PAPERS. 137 Hence deduce the conditions of equilibrium of a floating body. A piece of cork {Specific Gravity '24) containing 2 cubic feet is kept below water by means of a string fastened to the bottom, find the tension of the string. p 8. Describe the Hydrostatic Balance; aiid shew how it may be employed to determine the Specific Gravity of a body heavier than water. Two bodies, whose weights are w^ and w.^ in air, weigh each w in water, compare their Specific Gravities. 9. Shew that, in the Common Barometer, the weight of that portion of the mercurial column, which is above the free surface of the mercury, accurately measures the pressure of the atmosphere. A Barometer is sunk to the depth of 20 feet in a lake, find the consequent rise in the mercurial column {Specific Gravity of mer- cury- 13-57). 10. Explain the action of the Siphon. A Siphon is placed with one end in a vessel full of water, and the other in a similar empty one, both of which are on the plate of an air-pump. As soon as the water has covered the lower end of the Siphon, 2l Receiver is put on, and the air rapidly exhausted, and then gradually readmitted ; describe the effects produced. Saturday, May 25, 1861. FIRST DIVISION.— A. 1. Shew that forces may be properly represented by straight lines. 2. If two weights, acting perpendicularly on a straight Lever, on opposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum. If the pressure on the fulcrum be equivalent to a weight of 15 lbs., and the diflFerence of the magnitudes of the forces to a weight of 3 lbs., find the forces, and the ratio of the arms at which they act. ^ 138 UNIVERSITY EXAMINATION PAPERS. 3. If the adjacent sides of a parallelogram represent two forces acting at a point in direction and magnitude, the diagonal will repre- sent the Resultant force in direction. At what angle must two forces, P, and 2P, act upon a point, that the direction of their Resultant may be at right angles to the direc- tion of one of the forces 1 4. Find the condition of equilibrium when a weight W is supported on an Inclined Plane by a force P acting parallel to the plane. If the force acts horizontally, there is equilibrium when P is to TV as the height of the plane is to its base. 5. Find the Centre of Gravity of three heavy points ; and shew that the pressure on the Centre of Gravity is equal to the sum of the weights in all positions. 6. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line drawn from its Centre of Gravity falls within or without the hase. A board, in the shape of a right-angled triangle, is placed in a vertical plane, with its right angle resting on a rough horizontal floor, and one of its acute angles leaning against a vertical wall per- pendicular to the plane of the board ; find its position when the pres- sure on the wall is the least possible. 7. If a body float in a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body. Why can a man swim on his back more easily than in any other position % 8. Define Specific Gravity. What is meant, when the Specific Gravity of a substance is said to be 0*00125 1 The Specific Gravity of sea water being 1*027, what proportion of fresh water must be added to a quantity of sea water, that the Specific Gravity of the compound may be 1"009 ] 9. When a body of uniform density floats on a fluid, the part immersed is to the whole body as the Specific Gravity of the body is to the Specific Gravity of the fluid. A solid sphere floats in a fluid with three-fourths of its bulk above the surface: when another sphere half as large again is at- tached to the first by a string, the two spheres float at rest below the UNIVERSITY EXAMINATION PAPERS. 139 surface of the fluid; shew that the Specific Gravity of one sphere is 6 times greater than that of the other. 10. Describe the construction of the Forcing Pump and its operation. "What ■will be the effect of making a small aperture in the Bar- rel? If the piston works uniformly up and down the length of the Barrel^ and a small aperture be made one-third of the way up the Barrel, how much more time than before will be consumed in filling a tank ? 11. Describe the method of filling and graduating a Common Therm.om.eter, Shew how to graduate a Thermometer, on whose scale 20" shall denote the Freezing-Point, and whose 80th degree shall indicate the same temperature as 80*^ Fahrenheit. FIRST DIVISION.— B. 1. Define Gravity, and Weight. How is Statical Force mea- sured ? 2. If two forces, acting perpendicularly on a straight Lever in opposite directions and on the same side of the fulcrum, balance each other, they are inversely as their distances from the fulcrum. If the pressure on the fulcrum be equivalent to a weight of 3 lbs., and the sum of the magnitudes of the forces to a weight of 15 lbs., find the forces, and the ratio of the arms at which they act. 3. If three forces, represented in magnitude and direction by the three sides of a triangle, when taken in order, act upon a point, they will keep it at rest. Two strings are respectively fastened by their upper extremities to two points A and B in the same horizontal line ; the string at A carries a weight W at its lower extremity which is passed through a ring attached to the lower extremity of the string at ^ ; if the dis- tance between the points A and B is so adjusted, that the strings rest at equal inclinations to the horizon, the tension of the string at B is equal to the weight W. 4. Find the condition of equilibrium on the Wheel-and-Axle. Why is the labour of di-awiiig a bucket of water out of a common 140 UNIVERSITY EXAMINATION PAPERS. well generally greater during the last part of the process than during the first ? 5. Find the Centre of Gravity of two heavy points, and shew that the pressure at the Centre of Gravity is equal to the sum of tlie weights in all positions. 6. When a body is suspended from a point, it will rest with its Centre of Gravity in the vertical line passing through the point of suspension. Two weights, W and 2 W, are connected by a rigid weightless rod, and also by a loose string, which is slung over a smooth peg : compare the lengths of the string on each side of the peg, when the weights have assumed their position of equilibrium. 7. The surface of every fluid at rest is horizontal. Shew how the inclination of a table to the horizon may be esti- mated by means of a tube bent into the form of the arc of a circle, and very nearly filled with fluid. 8. Explain fully the meaning of the equation, ' the weight of a body = its magnitude x its Specific Gravity^ If 2 cubic feet of a substance weigh 100 lbs., what is its Specific Gravity? 9. When a body is immersed in a fluid, the weight lost is to the whole weight as the Specific Gravity of the fluid is to the Specific Gravity of the body. The cavity in a conical rifle bullet is usually filled with a plug of some light wood. If the bullet be held in the hand beneath the sur- face of water, and the plug then removed, will the apparent weight of the bullet be increased or diminished 1 10. Describe the construction of the Condenser ^ and its ope- ration. If the direction in which the valves open were reversed, into what instrument would the condenser be converted ? Describe the effect of making a small aperture in the Barrel. 11. Explain the construction of the Common Barometer. Will the actual rise or fall of the mercury in the tube, observed by means of fixed graduations, accurately measure the increase or decrease of the pressure of the atmosphere ? UNIVEESITY EXAMINATION PAPERS. 141 SECOND DIVISION.— A. 1. Define force, and explain how forces are measured. Shew that they may properly be represented by straight lines. Construct a triangle, whose sides will represent the forces 2-i42857lbs., 2-009 lbs., and 2-009 lbs. 2. If two weights, acting perpendicularly on a straigiit Lever on opposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum; and the pressure on the ful- crum is equal to their sum. A heavy pole, weighing 12 lbs., whose Centre of Gravity falls at a distance of one-third of the pole from one end, is carried horizon- tally by two men, one at each end, find the weight supported by each man. 3. If three forces, represented in magnitude and direction by the sides of a triangle, act on a point, they will keep it at rest. ABCD is a quadrilateral inscribed in a circle; forces P, Q, R, acting in directions AB,AD, CA would keep a particle at rest, shew that F :Q r.CD : BC, 4. In a system in which each pulley hangs by a separate string, and the strings are parallel, there is an equilibrium, when p : w wl : that power of 2 whose index is the number of moveable pulleys. How may a boy, who can only lift 16 stone, be enabled to raise 1024 stone ? 5. Define Velocity. If p and w balance each other in the system of Pulleys., in which the same string passes round all the pulleys, and the whole be put in motion, shew that p -.w \\ w's velocity in direction of gravity : p'^ velocity. 6. "When a body is placed upon a horizontal plane, it will stand or fall, according as the vertical line, drawn from its Centre of Gravity, falls within or without its base. With what restriction is this true of any plane ? A sugar-lo:>f stands on an Inclined Plane, rough enough to pre- vent sliding, whose inclination to the horizon is 45*^, shew that it will fall over, if the height of the sugar-loaf be more than twice as 142 UNIVERSITY EXAMINATION PAPERS. great as the diameter of its base — the Centre of Gravity of a cone being distant from the base one-fourth of the height. 7. How is the pressure at a point in a fluid estimated ? Explain the Hydrostatic Paradox. If the area of the larger surface be diminished, the other cir- cumstances of the explanation remaining the same, what would be the effect? 8. Define Specific Gravity, and explain how it is measured. Explain how to find the weight of a body whose Specific Gravity is known. Find the weight of 36 cubic inches of cork, whose Spe- cific Gravity is 0*24. 9. "When a body of uniform density floats on a fluid, the part immersed : the whole body :: the Specifijc Gravity of the body : the Specifix^ Gravity of the fluid. A body, whose weight is 6 lbs., weighs 3 lbs. and 4 lbs. respec- tively in two different fluids, compare the Specific Gravities of the fluids. 10. Describe the construction of the Common Pump, and its operation. What limits the heiglit to which water can be raised by a Common Pump ? How is this obviated by a Forcing or Lifting Pum,p? 11. Shew how to graduate a Common Thermometer. One Thermometer marks two temperatures by 9*^, and 10'' ; an- other Thermometer by 12'^, and \^\ what will the latter mark, when the former marks 15*^ ? SECOND DIVISION.— B. 1. Define Force; how are forces measured? Shew that they may properly be represented by straight lines. Construct a triangle, whose sides will represent the forces 3-076923 lbs., 3*416 lbs., and 3127 lbs. respectively. 2. If two forces, acting perpendicularly on a straight Lever in opposite directions and on the same side of the fulcrum, balance each other, they are inversely as their distances from the fulcrum ; and the pressure on the fulcrum is equal to the difference of the forces. UNIVERSITY EXAMINATION PAPERS. 143 A heavy pole, weighing- 12 lbs., whose Centre of Gravity falls at a distance of one-fourth of the pole from one end, is fastened at this end by a rope to the ceiling, and the other end is raised by a man so that the pole is horizontal, what weight will the man sup- port? 3. If three forces, represented in magnitude and direction by the sides of a triangle, act on a point, they will keep it at rest. A BCD is a quadrilateral inscribed in a circle, forces P, Q, i?, act in directions AB, AD, CA, so that P -. Q \ R v.CD \BC : BD, shew that P, Q, R, form a system in equilibrium. 4. In a system of Pulleys^ in which the same string passes round any number of pulleys, and the parts of it between the pul- leys are parallel, there is an equilibrium when p \w v.\ \ the number of strings at the lower block. How may a boy, who can only lift 15 stone, be enabled to raise 90 stone ? 5. Define Velocity. If p and w balance in the system of Pulleys, in which each pulley hangs by a separate string, and the whole be put in motion, shew that p -.w w w's velocity in direction of gravity : jo's velocity. 6. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line, drawn from its Centre of Gratity, falls within or without its hase. With what restriction is this true of any plane ? A sugar-loaf, whose height is twice as great as the diameter of its base, stands on a table, rough enough to prevent sliding, one end of which is gently raised until the sugar-loaf is on the verge of falling over, when this is the case, find the inclination of the table to the horizon — the Centre of Gravity of a cone being distant from the base one-fourth of the height. 7. How is pressure at a point in a fluid estimated ? Shew that the surface of every fluid at rest is horizontal. In supplying a town with water, why is the locality of the reser- voir selected in the highest position possible ? 8. If a body floats on a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body ; and it presses down- wards, and is pressed upwards, with a force equal to the weight of the fluid displaced. 144 UNIVERSITY EXAMINATION PAPERS. A symmetrical box, weighing 8 lbs., with a weight on the top, floats just immersed in a fluid : liow heavy must the weight be, in order that, when removed, the box may float with only one-third immersed ? 9. Define Specific Gravity, and explain how it is measured. Shew how to find the weight of a body whose Specific Gravity is known. Find the weight of 54 cubic inches of copper, whose Specific Gravity is 8*8. 10. Describe the construction of the Common Air-Pump, and its operation. What advantage does Hawksl^ee's possess over the Common Air-Pump ? 11. Shew how to graduate a Common Thermometer. One Thermometer marks two temperatures by 8**, and 10^ ; another Thermometer by ll^, and 14^; what will the latter mark, when the former marks 16^1 Saturday, May 31, 1862. FIRST DIVISION— A. 1. If two weights, acting perpendicularly on a straight Lever on opposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum ; and the pressure on the fulcrum is equal to their sum. The pressure on the fulcrum is 12 lbs. and the distance of the fulcrum from the middle point of the Lever is J^^th of the whole length of the Lever, find the forces. 2. If the adjacent sides of a parallelogram represent the com- ponent forces in direction and magnitude, the diagonal will represent the Resultant force in direction and magnitude. If the Resultant force be represented in direction and magni- tude by a given straight line, give a geometrical construction for representing its two components, each of which is represented in magnitude by a given straight line. When is this impossible ? UNIVERSITY EXAMINATION PAPERS. 145 3. Describe the Wheel-and-Axle, and prove that there is equi- librium upon the Wheel-and-Axle, when the Power is to the Weight as the radius of the Aa;le to the radius of the Wheel 4. The weight ( W) being on an Inclined Plane, and the force (P) acting parallel to the plane," there is equilibrium, when P :W :: the height of the plane : its length. Find also the pressure on the plane. If the pressure on the plane be equal to the force P, find the in- clination of the plane, and express the pressure in terms of W. 5. Define Velocity. Assuming that the arcs, which subtend equal angles at the centres of two circles, are as the radii of the circles, shew that, if P and W balance each other on the Wheel-and-Axle ; and the whole be put in motion, P : W:: W^s velocity : P's velocity. If P's velocity exceed W's velocity by 7 feet per second, and the radius of the Wheel be 15 times that of the Axle; find the velocities of P and W. 6. jye^uQiYiQ Centre of Gravity. Find the Centre of Gravity of two heavy points ; and shew that the pressure at the Centre of Gravity is equal to the sum of the weights in all positions. Find the Centre of Gravity of weights 2, 4, and 6, lbs. placed respectively at the angular points of a triangle. 7. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid. 8. Explain the Hydrostatic Paradox. If a pipe, whose height above the bottom of a vessel is 112 feet, be inserted vertically into the vessel, and the whole be filled with water ; find the pressure in tons on the bottom of the vessel, the area of the bottom being 4 square feet, and the weight of a cubic foot of water 1000 oz. avoirdupois. 9. If a body floats on a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body. If a body, exposed to the pressure of the air, float in water, prove that it will rise very slightly out of the water as the barometer rises, and sink a little deeper as the barometer falls. L. C. C. 10 146 UNIVERSITY EXAMINATION PAPERS. 10. When the body is immersed in fluid, the weight lost : whole weight of the body :: the Specific Gravity of the fluid : the Specific Gravity of the body. 11. Describe the construction of the Common Pump; and ex- plain its mode of operation, and the conditions requisite to its working. Which of these conditions will be affected by the height of the Barometer? 12. Describe the Barometer, and shew that the pressure of the atmosphere is accurately measured by the weight of the column of mercury in the Barometer. If a Barom,eter were used by a diver under water, what change would take place in its height 1 FIRST DIVISION.— B. 1. If two forces, acting pei'pendicularly on a straight Lever in opposite directions and on the same side of the fulcrum, balance each other, they are inversely as their distances from the fulcrum ; and the pressure on the fulcrum is equal to the difference of the forces. The pressure on the fulcrum is 4 lbs., and the distance of the fulcrum from the middle point of the Lever is twice the whole length of the Lever; find the forces. 2. If three forces, represented in magnitude and direction by the sides of a triangle, act on a point, they will keep it at rest. Three forces whose magnitudes are 6, 8, and 10, lbs. respectively, acting upon a point, keep it at rest, prove that the directions of two of the forces are at right angles to each other. 3. In a system, in which the same string passes roimd any number of pulleys, and the parts of it between the pulleys are parallel, there is equilibrium, when power (P) : weight ( W) :: 1 : the number of strings at the lower block. If 6 strings pass round the lower block, and a man supports him- self by standing on it and holding the rope which passes round the pulleys with his hands ; find the tension of the rope. UNIVERSITY EXAMINATION PAPERS. 147 4. The weight ( W) being on an Inclined Plane, and the force (P) acting parallel to the plane, there is equilibrium, when P : TV :: the height of the plane : its length. Find also the pressure on the plane. If the pressure : power :: 3 : 4, express each of them in terms of TV. 5. Define Velocity, and prove that if P and W balance each other in the system described in the preceding question and the whole be put in motion, « P : TV :: TV's velocity : P's velocity. If P descends 12 feet, while PF ascends through 2 feet, find the number of strings at the lower block. 6. Find the Centre of Gravity of a triangle. Shew that, if the Centre of Gravity of three heavy particles placed at the angular points of the triangle coincides with the Centre of Gravity of the triangle, the particles must be of equal weight. 7. Define a Fluid, and distinguish between elastic and non- elastic fluids. Describe any experiment to shew that fluids press equally in all directions. 8. The surface of every fluid at rest is horizontal. Explain clearly why this proposition is not true for very large surfaces of water. 9. If a body floats on a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body. If a body float partly immersed in two or more fluids, state the conditions of equilibrium. If a body, floating in water and exposed to the atmospheric pres- sure, be placed under an exhausted Receiver, shew that the body will sink a little deeper. 10. Define Specific Gravity ; and shew how to determine by means of the Hydrostatic Balance the Specific Gravity of a sub- stance specifically heavier than water. 11. Describe the construction of the Common Air-pum,p and its operation. How is the degree of exhaustion limited in this pump 1 10—2 148 UNIVERSITY EXAMINATION PAPERS. 12. Describe the Siphon and explain its mode of action, and the conditions requisite to its working. Which of these conditions \vill be affected by the height of the Barometer ? SECOND DIVISION.— A. 1. Define the terms Force and Weight; and explain the term Mass of a body. Why is the Weight of the same body not constant at all points of the earth's surface ? Could this difference in weight be detected by a Common Balance ? 2. If two weights, acting perpendicularly on a straight Lever on opposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum ; and the pressure on the fulcrum is equal to their sum. If the Lever is in equilibrium, when weights P and Q are sus- pended from its extremities, and also when P is doubled and Q increased by 5 lbs, ; find the magnitude of Q, 3. Explain the meaning oi component and resultant forces. Enunciate the Parallelogram of Forces; and prove the proposi- tion, so far as the direction of the Resultant is concerned. If a given force be resolved into two equal component forces, prove that the extremities of these forces always lie on a certain straight line. 4. In a system, in which each pulley hangs by a separate string, and the strings are parallel, there is equilibrium, whenP : W :: 1 : that power of 2 whose index is the number of moveable pulleys. Supposing that there are 7i moveable pulleys, and that the weight of each of them is P, what must be the value of W, when there is equilibrium 1 5. Define Velocity, and prove that, if a weight W be supported on an Inclined Plane by a power P acting parallel to the plane, and the whole be set in motion, P : W :: Wa velocity : P'a velocity. UNIVERSITY EXAMINATION PAPERS. 149 6. "When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line, drawn from its Centre of Gravity, falls within or without its base. Shew that a rhombus will rest in stable equilibrium, when any side is placed on a horizontal plane. 7. Explain how fluid pressures are measured, taking as an illustration any familiar instance. If the pressure of the atmosphere be 14 lbs. on the square inch, what will be the pressure in tons on the square yard 1 8. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid. Water floats on mercury to the depth of 17 feet; compare with the atmospheric pressure the pressure at a point 15 inches below the surface of the mercury, taking into account the atmospheric pressure on the surface of the water, having given that the heights of the mercurial and water-barometers are 30 inches and 34 feet respectively. 9. Define Specific Gravity; and prove that when a body of imiform density floats on a fluid, the part immersed : the whole body :: the Specific Gravity of the body : the Specific Gravity of the fluid. 10. The elastic force of air at a given temperature varies as the Density. 11. Describe the Common Air-pum,p, and its operation ; and explain why a perfect vacuum cannot be obtained by this instrument. 12. Explain the method of filling and graduating a common mercurial Thermom^eter. If the difference of readings of a 7%i?nno7ng^^r, which is graduated both according to Fahrenheit's and the Centigrade scale, be 40, find the temperature in each scale. SECOND DIVISION.— B. 1. Define the terms Force and Weight; and enumerate the chief forces with which we have to deal in Statics. Explain the mode of representing and comparing Forces. 150 UNIVEESITY EXAMINATION PAPERS. 2. If two forces, acting perpendicularly on a straight Lever iu opposite directions and on the same side of the fulcrum, balance each other, they are inversely as their distances from the fulcrum ; and the j)ressure on the fulcrimi is equal to the difference of the forces. The Lever is in equilibrium under the action of the forces P and Q, and is also in equilibrium when P is trebled, and Q increased by 6 lbs : find the magnitude of Q,. 3. Explain the meaning of Resultant force. Enmiciate the Parallelogram of Forces; and assuming its truth so far as the direction of the Resultant is concerned, prove the pro- position completely. Find a point within a quadrilateral, such that, if forces be repre- sented by lines drawn from it to the angular points of the quad- rilateral, the forces shall be in equilibrium. 4. Describe the Wheel-and-Axle ; and prove there is equi- librimnupon the Wheel-and-Axle, when the Power is to the Weight as the radius of the Axle to the radius of the Wheel. 5. Define Velocity, and prove that if P and W balance in that system of pulleys, in which each pulley hangs by a separate string and the strings are parallel, and the whole be put in motion, P : W :: W's velocity : P's velocity. 6. Define the term Centre of Gravity; and prove that when a body is suspended from a point, it will rest with its Centre of Gravity in the vertical line passing through the point of susj)ension. How may we employ this proposition to determine practically the Centre of Gravity of any body whatsoever ? 7. Define a Fluid, and distinguish between elastic and non- elastic fluids. Describe an experiment to prove that fluid pressures are transmitted equally in all directions. 8. If a body floats on a fluid, it displaces as much of the fluid as is equal in weight to tho weight of the body ; and it presses down- wards, and is pressed upwards, with a force equal to the weight of the fluid displaced. What further condition is necessary for equilibrium ? Two bodies of unequal volume, when weighed in air, balance one another; what will take place if they be weighed (1) in vacuo, (2) when the Barometer rises ? UNIVERSITY EXAMINATION PAPERS. 151 9. "When a body is immersed in fluid, the weight lost : whole weight of the body :: the Specific Gravity of the fluid : the Specific Gravity of the body. Why will this proposition not be true, if the fluid be com- pressible ? 10. Shew that air has weight; and explain clearly why a balloon ascends. Why does it cease to ascend ? 11. Describe the construction of the Common Mercurial Baro- meter, and prove, that the pressure of the atmosphere is accurately measured by the weight of the column of mercury in the Barometer. 12. Explain the action of the Siphon, and the conditions under which it works. A Siphon, filled with water, has its ends inserted in vessels filled with water ; state what will take place when the vertical distances of the highest point of the Siphon above the surfaces of the fluid are both less, both greater, and one greater and the other less, than the height of the Water Barometer. MoKDAT, June 1, 1863. {A). 1. If two forces, acting perpendicularly upon a straight Lever in opposite directions and on the same side of the fulcrum, balance each other, they are inversely proportional to their distances from the fulcrum ; and the pressure on the fulcrum is equal to the diifer- ence of the forces. A uniform heavy rod AB {S inches long) is laid over two props (Cand Z>), which are 5 inches apart, and have their extremities in the same horizontal line, so that A is distant 2 inches from C. Find the pressures on the props ; and shew that, if a weight be laid upon the rod at B sufficient to remove all pressure from the prop C, the pressure on D will be ten tim^s as great as it was before. 2. Enunciate the Parallelogram of Forces, Assuming it to be true for the direction of the Resultant, prove it for the magni- tude of the Resultant, 152 UNIVERSITY EXAMINATION PAPERS. If the straight lines AB, AC, meeting in a point, represent two forces in direction and magnitude, the straight line joining A with the middle point of BC will represent half the Resultant of the two forces. 3. In a system of moveable pulleys, in which each pulley hangs by a separate string and the strings are parallel, there is equilibrium where P : ^ :: 1 : 2", the weights of the pulleys being neglected. If there arc three pulleys of equal weight, what must be the weight of each, in order that a weight of 56 pounds attached to the lowest may be supported by a power equal to 7 lbs. 14 oz. ? 4. On the Inclined Plane, where the power acts parallel to the plane, prove that P : W \: TV^s velocity in the direction of gra\dty : P's velocity. "What is meant by saying that in a machine " what is gained in power is lost in time" ? 5. Mention any causes which may make the weight of a body vary when it is estimated at different places. Define the Centre of Gravity of a body. Find that of a plane triangle. 6. "When a body is placed upon a horizontal plane, it will stand or fall according as the vertical line, dravm from its Centre of Gravity, falls within or without the base. 7. Define a Fluid. Describe an experiment by which it may be shewn that fluids press equally in all directions. A closed vessel full of fluid, with its upper surface horizontal, has a weak part in its upper surface not capable of bearing a pres- sure of more than 4| pounds on the square foot. If a piston, the area of which is 2 square inches, be fitted into an aperture in the upper surface, what pressure applied to it will burst the vessel ? 8. The surface of a fluid at rest is a horizontal plane. 9. If a body float in a fluid, it displaces as much fluid as is equal in weight to the weight of the body. 10. Explain what is meant by the Density, and by the Specific Gravity, of a body. What is meant by saying that the SpecifiA; Gravity of a substance is rs for instance? UNIVERSITY EXAMINATION PAPERS. 153 Prove the relation existing between the Weight, Magnitude ^ and Specific Gravity, of a body. Two metals are combined into a lump, the volume of which is 2 cubic inches. 1^ cubic inches of one metal weigh as much as the lump, and 2| cubic inches of the other metal weigh the same. What Volume of each of the two metals is there in the lump ] 11. State the law connecting the elastic force and the density of air at a given temperature. Describe the Barometer. If the weight of the column of mercury, which is above the exposed surface be an ounce, and the area of the transverse section of the tube 224 ^f ^ square inch, what is the pressure of the atmo- sphere on a square inch ? 12. Describe the action of the Siphon. (B). 1. If two weights, acting perpendicularly upon a straight Zet'^r on opposite sides of the fulcrum, balance each other, they are in- versely as their distances from the fulcrum ; and the pressure on the fulcrum is equal to their sum. A body, the weight of which is one pound, when placed in one scale of a false balance, appears to weigh 14 ounces. What will be its apparent weight, when placed in the other scale ? 2. Enunciate the Parallelogram of Forces, Prove it so far as the direction of the Resultant is concerned. Two forces, one of which is double of the other, act upon a point, and are such, that, if 6 lbs. be added to the larger, and the smaller be doubled, the direction of the Resultant is unaltered. Find the forces. 3. When a weight ( W) is supported upon a smooth Inclined Plane by a power (P) acting parallel to the plane, then P \W \\ height of the plane : its length. A weight W is supported upon an Inclined Plane by a string parallel to its length. The string passes over a fixed pulley, and then / 154 UNIVERSITY EXAMINATION PAPEES. under a moveable one without weight, to which a weight W is attached, and having the portions of the string on each side of it parallel. Prove that the height of the plane is half its length. 4. In the system of pulleys, in which each pulley hangs by a separate string, prove that P : FT :: W& velocity : P's velocity. "What is meant by saying that in mechanism " what is gained in power is lost in time" 1 5. Mention any causes, which may aflfect the weight of a body, when it is estimated at different places. Define the Centre of Gravity of a body. Shew how to find that of any number of heavy particles. 6. When a body is suspended from a point, about which it can swing freely, it will rest with its Centre of Gravity in the vertical line passing through the point of suspension. 7. Define a fluid. Describe an experiment, by which it may be shewn, that fluids press equally in all directions. A closed vessel full of fluid, with its upper surface horizontal, has a weak part in its upper surface, not capable of bearing a pressure of more than 9 lbs. upon the square foot. If a piston, the area of which is one square inch, be fitted into an aperture in the upper surface, what pressure applied to it will burst the vessel ? 8. Explain what is meant by the Density, and Specific Gravity, of a body. What do you mean by saying that the density of a sub- stance is 1*5 for instance? Prove the relation existing between the Mass, Volume, and Density, of a body. Two metals are combined into a lump, the volume of which is 3 cubic inches. 2i cubic inches of one metal weigh as much as the lump, and 3| cubic inches of the other metal weigh the same. What Volume of each of the two metals is there in the lump l 9. The pressure at any point of a fluid of uniform density at rest is proportional to the depth below the surface of the fluid. 10. When a body is immersed in a fluid the weight lost : whole weight of the body :: the Specific Gravity of the fluid : the Specific Gravity of the body. UNIVERSITY EXAMINATION PAPEES. 155 11. State the law connecting the elastic force and the density of air at a given temperature. Describe the Barometer. 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"A better introduction (Part I.) to Geometry can hardly be ima- gined ; and we are glad to find it is published at so reasonable a price, because it is a work that ought to be used extensively in our national schools of design, and widely circulated among our manu- facturing population." — A thenceum. 8. COMPANION TO WOOD'S ALGEBKA (for Stic- dents), containing Solutions of all the difficult Questions and Problems in the Algebra. Third Edition, 'js. 6d. boards. 9. A KEY for ScJioolmasfers to all the Questions and Pro- blems in Wood's Algebra, by Lund. -js. 6d. boards. 10. GEOMETEICAL EXEECISES, with Solutions, form- ing a Key to Parts I. and II. of the Geometet. Price 3s. 6d. This work contains a numerous collection of original easy 'Eiders' to Euclid, adapted to tbe Senate-House Examinations for B.A. Degree. 11. A KEY TO BISHOP COLENSO'S BIBLICAL ARITHMETIC. Second Edition, is. LONDON : LONGMAN, GREEN, LONGMAN, ROBERTS, AND GREEN. By J. C. SNOWBALL, M.A. LATE FELLOW OF ST JOHN'S COLLEGE, CAMBEIDGE. PLANE AND SPHERICAL TRIGONOMETRY, WITH THE CONSTRUCTION AND USE OF TABLES OF LOGAEITHMS. Tenth Edition. 240 pp. (1863). Crown 8vo. 7^. ed. In preparing a new edition, the proofs of some of the more important propositions have been rendered more strict and general ; and a considerable addition of more than Two hundred Examples^ taken principally from the questions in the Examinations of Colleges and the University, has been made to the collection of Examples and Problems for practice. MACMILLAN AND CO. 3lont)on anD ©am^ritigc. AN ELEMENTARY LATIN GRAMMAR, By H. J. ROBY, M.A. UNDEE-MASTER OF DULWICH COLLEGE UPPER SCHOOL, LATE FELLOW AND CLASSICAL LECTURER OF ST JOHN'S COLLEGE, CAMBRIDGE. 18mo. 2s. 6d, The Author's experience in practical teaching has induced an attempt to treat Latin Grammar in a more precise and intelligible way than has been usual in school books. The facts have been derived from the best authorities, especially Madvig's Grammar and other works. The works also of Lachmann, Ritschl, Key, and others have been consulted on special points. The accidence and prosody have been simplified and restricted to what is really required by boys. In the Sjntax an analysis of sentences has been given, and the uses of the difierent cases, tenses and moods briefly but carefully described. Particular attention has been paid to a classification of the uses of the subjunctive mood, to the prepositions, the oratio obliqua, and such sentences as are introduced by the English ' that.' Appendices treat of the Latin forms of Greek nouns, abbreviations, dates, money, &c. The Grammar is written in English. MACMILLAK AKD CO. Sontion anti ©ambntigc. May, 1869. LIST OF EDUCATIONAL BOOKS PUBLISHED BY MACMILLAN AND CO., 16, BEDFORD STREET, CO VENT GARDEN, ^0ntr0tT, w.c. CONTENTS. CLASSICAL Page 3 MATHEMATICAL 7 SCIENCE 17 MISCELLANEOUS ... 18 DIVINITY 21 BOOKS ON EDUCATION ... 24 Messrs. Macmillan & Co. beg to call attention to the accompanying Catalogue of their Educational Works, the writers of which are mostly scholars of emijience i?i the Universities^ as well as of large expei'ience in teaclmig. Many of the works have already attaijted a wide circulatioji in England afid i?i the Colofiies, and are acknowledged to be among the very best Educational Books on their respective subjects. The books can generally be procured by ordering them through local booksellers in town or cou?itry, but if at any time difficulty should arise, Messrs. Macmillan will feel much obliged by direct comnumication with the?n selves on the subject. Notices of eri'ors or defects in any of these works will be gratefully received and achwivledged. LIST OF EDUCATIONAL BOOKS, CLASSICAL ^SCHYLI EUMENIDES. The Greek Text, with English Notes, and EngHsh Verse Translation and an Introduction. By Bernard Drake, M.A., late Fellow of King's College, Cambridge. 8vo. 7J-. 6^. The Greek Text adopted in this Edition is based upon that of Wellauer, which may be said in general terms to represent that of the best manu- scripts. But in correcting the Text, and in the Notes,, advantage has been taken of the suggestions of Hermann, Paley, Linwood, and other com- mentators. ARISTOTLE ON FALLACIES; OR, THE SOPHISTICI ELENCHL With a Translation and Notes by Edward Poste, M.A., Fellow of Oriel College, Oxford, 8vo. 8j-. 6d. Besides the doctrine of Fallacies, Aristotle offers either in this treatise, or in other passages quoted in the commentary', various glances over the world of science and opinion, various suggestions on problems which are still agitated, and a vivid picture of the ancient system of dialectics, which it is hoped may be found both interesting and instructive. " It is not only scholarlike and careful ; it is also perspicuous.'' — G:iardiau. ARISTOTLE.— KN INTRODUCTION TO ARISTOTLE'S RHETORIC. With Analysis, Notes, and Appendices. By E. M. Cope, Senior Fellow and Tutor of Trinity College, Cam- bridge. Svo. I4J'. This work is introductory to an edition of the Greek Text of Aristotle's Rhetoric, which is in course of preparation, " Mr. Cope has given a very useful appendage to the promised Greek Text ; but also a work of so much independent use that he is quite justified in his separate publication. All who have the Greek Text will find themselves supplied with a comment ; and those who have not will find an analysis of the work. " — A thenceum. CATULLI VERONENSIS LIBER, edited by R. Ellis, Fellow of Trinity College, Oxford. i8mo. 3J. dd. " It is little to say that no edition of Catullus at once so scholarlike has ever appeared in England." — AthencE7i>n. " Rarely have we read a classic author with so reliable, acute, and safe a guide." — Saturday Review. EDUCATIONAL BOOKS. C/C^A^a— TPIE SECOND PHILIPPIC ORATION. With an Introduction and Notes, translated from the German of Karl Halm. Edited, with Corrections and Additions, by John E. B. Mayor, M.A., Fellow and Classical Lecturer of St. John's Col- lege, Cambridge. Third Edition, revised. Fcap. 8vo, 5J-. " A very valuable edition, from which the student may gather much both in the way of information directly commimicated, and directions to other sources of knowledge. " — A theficewn. DEMOSTHENES ON THE CROWN. The Greek Text with English Notes. By B. Drake, M.A., late Fellow of King's College, Cambridge. Third Edition, to which is prefixed yEscHiNES AGAINST Ctesiphon, with English Notes. Fcap. 8vo. 5J-. The terseness and felicity of Mr. Drake's translations constitute perhaps the chief value of his edition, and the historical and arch?eological details necessary to understanding the De Corona have in some measure been anticipated in the notes on the Oration of ^Eschines. In both, the text adopted in the Zurich edition of 1851, and taken from the Parisian MS., has been adhered to without any variation. Where the readings of Bekker, Dissen, and others appear preferable, they are subjoined in the notes. HODGSON.— MY TU01.0GY FOR LATIN VERSIFICATION. A Brief Sketch of the Fables of the Ancients, prepared to be rendered into Latin Verse for Schools. By F. Hodgson, B.D., late Provost of Eton. New Edition, revised by F. C. Hodgson, M.A. i8mo. 3J-. Intending the little book to be entirely elementary, the Author has made it as easy as he could, without too largely superseding the use of the Dic- tionary and Gradus. 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MARSHALL.— K TABLE OF IRREGULAR GREEK VERBS, Classified according to the Arrangement of Curtius's Greek Grammar. By J. M. Marshall, M.A., Fellow and late Lec- turer of Brasenose College, Oxford ; one of the Masters in Clifton College. 8vo. cloth, \s. MA F(97?.— FIRST GREEK READER. Edited after Karl Halm, with Corrections and large Additions by the Rev. John E. B. Mayor, M. A., Fellow and Classical Lecturer of St. John's College, Cambridge. Fcap. 8vo. 6^-. J/^FCt^.— GREEK for BEGINNERS. By the Rev. Joseph B. Mayor, M. A. With Glossary and Index. Fcap. 8vo. 4J-. 6^'. ^^/e/F^Z^.— KEATS' HYPERION rendered into Latin Verse. By C. Merivale, B.D. Second Edition. Extra fcap. 8vo. 3-5-. dd. FLA TO.— THE REPUBLIC OF PLATO. Translated into En- glish, with an Analysis and Notes, by J. LI. Davies, M.A., and D. J. Vaughan, M.A. Third Edition, with Vignette Portraits of Plato and Socrates, engraved by JeeNs from an Antique Gem. i8mo. 4^-. 6d. 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THRIN'G.—\N oxks by Edward Thring, M.A., Head Master of Uppingham School : — — A CONSTRUING BOOK. Fcap. 8vo. 2s. 6d. This Construing Book is drawn up on the same sort of graduated scale as the Author's Efig-iish GmiiDuar. Passages out of the best Latin Poets are gradually built up into their perfect shape. The few words altered, or in- serted as the passages go on, are printed in Italics. It is hoped by this plan that the learner, whilst acquiring the rudiments of language, may store his mind with good poetrj- and a good vocabulary. — A LATIN GRADUAL. A First Latin Construing Book for Beginners. Fcap. 8vo. is. 6d. The main plan of this little work has been well tested. The intention is to supply by easy steps a knowledge of Grammar, combined with a good vocabulary ; in a word, a book which will not require to be forgotten again as the learner advances. 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CHEYNE.—A^ ELEMENTARY TREATISE on the PLANET- ARY THEORY. With a Collection of Problems. By C. H. FI. Cheyne, B.A. Crown Svo. cloth. 6s. 6d. — THE EARTH'S MOTION of ROTATION. By C. H. H. Cheyne, M.A. Crown Svo. 3J-. 6d. CHILDE.—THE SINGULAR PROPERTIES of the ELLIPSOID and ASSOCIATED SURFACES of the Nth DEGREE. By the Rev. G. F. Childe, M.A., Author of "Ray Surfaces," "Related Caustics," &c. Svo. los. ()d. CHRISTIE— A COLLECTION OF ELEMENTARY TEST- QUESTIONS in PURE and MIXED MATHEMATICS ; with Answers and Appendices on Synthetic Division, and on the Solution of Numerical Equations by Horner's Method. By James R. Christie, F.R.S., late First Mathematical Master at the Royal Military Academy, Woolwich. Crown Svo. cloth, Sj. 6d. DALTON—ARYmMKYlCNL EXAMPLES. Progressively ar- ranged, with Exercises and Examination Papers. By the Rev. T. Dalton, M.A., Assistant Master of Eton College. iSmo. cloth. 2s. 6d. Z>^ K— PROPERTIES OF CONIC SECTIONS PROVED GEOMETRICALLY. 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Also a Collection of Problems, principally intended as Examples of Newton's Methods. By Percival Frost, M.A,, late Fellow of St. John's College, IMathematical Lecturer of King's College, Cambridge. Second Edition. 8vo. cloth, los. 6d. The author's principal intention is to explain difficulties which may be en- countered hy the student on first reading the Principia, and to illustrate the advantages of a careful study of the methods employed by Newton, by showing the extent to which they may be applied in the solution of prob- lems ; he has also endeavoured to give assistance to the student who is engaged in the study of the higher branches of Mathematics, by repre- senting in a geometrical form several of the processes employed in the Differential and Integral Calculus, and in the analytical investigations ol Dynamics. FROST and WOLSTENHOLME.—K TREATISE ON SOLID GEOMETRY. By Percival Frost, M.A., and the Rev. J. WoLSTENHOLME, M.A., Fellow and Assistant Tutor of Christ's College. 8vo. cloth, i8j-. The authors have endeavoured to present before students as comprehensive a view of the subject as possible. Intending as they have done to make the subject accessible, at least in the earlier portion, to all classes of students, they have endeavoured to explain fully all the processes which are most useful in dealing with ordinary theorems and problems, thus di- recting the student to the selection of methods which are best adapted to the e.xigencies of each problem. In the more difficult portions of the sub- ject, they have considered themselves to be addressing a higher class of students ; there they have tried to lay a good foundation on which to build, if any reader should wish to pursue the science beyond the limits to which the work extends. MATHEMATICAL. ii GODFRAY.—K TREATISE on ASTRONOMY, for the use of Colleges and Schools. 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Jones, M. A., and C. H. Cheyne, M.A., Mathematical Masters of Westminster School. New Edition. iSmo. cloth, 2s. 6d. This little book is intended to meet a difficulty which is probably felt more or less by all engaged in teaching Algebra to beginners. It is that while new ideas are being acquired, old ones are forgotten. In the belief that constant practice is the only remedy for this, the present series of miscel- laneous exercises has been prepared. Their peculiarity consists in this, that though miscellaneous they are yet progressive, and may be used by the pupil almost from the commencement of his studies. They are not in- tended to supersede the systematically arranged examples to be found in ordinary treatises on Algebra, but rather to supplement them. The book being intended chiefly for Schools and Junior Students, the higher parts of Algebra have not been included. 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