QC QUESTIONS AND EXEECMS STEWART'S LESSONS ELEMENTARY PHYSICS. -NRLF 110 BOS TO A X 1) 1! K A T II GIFT OF ENGINEERING LIBRARY QUESTIONS AND EXERCISES ON STEWART'S LESSONS IN ELEMENTAKY PHYSICS. GEOEGE A. HILL, 4 ASSISTANT PROFESSOR OF PHYSICS IN HARVARD UNIVERSITY. WITH ANSWERS AND OCCASIONAL SOLUTIONS. BOSTON: GINN AND HEATH. 1880. Entered according to Act of Congress, in the year 1874, BY GINN BROTHERS, in the Office of the Librarian of Congress, at Washington. GIFT OF ENGINEERING LIBRARY PRESS OF ROCKWELL AND CHURCHILL, 39 Arch St., Boston. t PREFACE. THE following pages have been drawn up with the aim of making Mr. Stewart's excellent work more useful in element- ary teaching. Part I. consists of questions upon the text of Mr. Stewart's book which are intended to be direct and exhaustive. Opin- ions differ as to the value of such questions. No doubt a thoroughly competent teacher will ask questions in his own way with the best effect ; but unfortunately such teachers, at least in scientific subjects, are not numerous. In all cases the questions will be found useful for review and examination purposes. Parts II. and III., which form the principal part of the work, have been written with two objects in view. First, to stimulate original thought on the part of the student, and to give the teacher the means of testing thoroughly the student's knowledge of principles. Secondly, to make certain needful additions to the felicitous but cursory sketch of Mechanics, Hydrostatics, and Pneumatics, contained in the first two chapters of Mr. Stewart's book. Molecular Physics is rapidly assuming the character of an exact science ; and in proportion as this takes place, the im- portance of a good knowledge of the general laws of Motion and Force, and of the ability to reason deductively, increases. Nothing can give training in deduction better than the study of Rational Mechanics. Training in the methods of induc- tion, which is so large a part of scientific culture, cannot, in 865704 iv PREFACE. our judgment, be imparted successfully by the study of text- books ; the place to receive it is in Physical Laboratories, which are happily becoming more and more common, or by observation and reflection in the vast Laboratory of Nature around us. The chief value, which the text-book study of Physics can be made to have, consists in disciplining the mind in scientific demonstration of the deductive kind. The Exercises are divided into two classes, as explained on page 69. A few of them are original ; the most have been selected from English works. Few of them require much numerical work, and many of them none at all. In preparing the Solutions, the author has been under obligations to the elementary writings of Professors Thomson, Tait, and Maxwell. It was found impossible to prepare solutions of the more difficult Exercises in small type in season for the present edition. In working out these Exercises, where aid is found necessary it should be obtained, if possible, from a competent teacher. The student is also strongly advised to consult special works which treat of the subjects covered by the Ex- ercises. By doing this, the student will not merely find the aid which he desires, he will be acquiring a habit of mind which is characteristic of the cultivated man and of all pro- ductive scholarship, the habit of consulting and carefully comparing the views which different minds take of the same subject, and of that originality in thought which comes from an independent use of many authorities. On page 69 will be found a list of elementary works which may be consulted with advantage. G. A. HILL. CAMBRIDGE, August 25, 1874. CONTENTS. PAGE PART I. QUESTIONS 1-68 II. EXERCISES AND PROBLEMS . . 69-114 III. ANSWERS AND SOLUTIONS . . 115-168 Answers ....... 115 Solutions 127 APPENDIX 169-188 I. English Weights and Measures . . . 169 II. United States Weights and Measures . . 173 III. Metric Weights and Measures . . . 175 IV. Mathematical Formulae . . . .179 V. Physical Tables 185 PART I. QUESTIONS ON STEWAET'S ELEIENTAEY PHYSICS. Introduction. 1. How do we become aware of the existence of objects outside of ourselves ? 2. What is the ground of our expectation that the sun will rise to-morrow ? In general, when is our expec- tation that a certain phenomenon will recur well grounded 1 3. What are characteristics of the knowledge of physi- cal laws which men acquire from every-day experience ? 4. How long since men first set themselves systemati- cally to the task of acquiring a knowledge of the laws of nature ? 5. What is the object of Physics ? 6. What do we learn from astronomy concerning the magnitude of the Universe ? 7. Explain the three-fold division of matter into sub- stances, molecules, and atoms. Illustrate by a familiar example. 8. What is the analogous three-fold division in astron- omy ? 9. What resemblance exists between the structure of the Universe and that of a body on the earth's surface, in consequence of which both may be called porous ? 10. Distinguish between physical and sensible pores. What proves the existence of physical pores ? Give in- stances of bodies having sensible pores. 1 A 2 QUESTIONS ON STEWART'S [INTROD. 11. Name the three states of matter. What are their chief characteristics ? Give examples of each state. 12. Explain relative motion by means of the motions- of the earth and the planets. 13. What have we strong reason to suppose to be the condition of any substance at rest, the molecules of a block of stone, for example 1 14. In view of our present scientific knowledge, what may be asset- te.d of -every body in the universe, lame or snvill ? 15. Illustrate by examples the meaning of the term ; forve. 16. Mention three of the most universal forces of na- ture.' What are their effects ? What consequences would ensue if these forces severally were to cease to exist ? 17. What must be the effect of a single force acting upon a body ? What may be the effect of two or more forces acting simultaneously on a body 1 Illustrate by the force of gravitation. LESS. L] ELEMENTARY PHYSICS. CHAPTER I. LAWS OF MOTION. LESSON I. Determination of Units. 18. What is the unit of time 1 19. Mention the two chief advantages of the Metric system of weights and measures. 20. What is the unit of length in the Metric system, and its value approximately in English inches ? 21. Enumerate the chief multiples and sub-multiples of the metre, and give their values in terms of the metre. 22. How are units of surface and of capacity derived from those of length ? Give examples of each kind. 23. What is the value in square metres of an are ? of a centiare ? of a hectare ? 24. What is a litre ? Name some of its decimal multi- ples and sub-multiples. 25. What is the ratio between two successive units of length, as the centimetre and the decimetre ? between the two corresponding units of surface ? between the two corresponding units of volume ? 26. How many square feet are there in 150 square inches ? How many square centimetres are there in 150 square millimetres ? 27. How many cubic yards are there in 93 cubic feet ? How many litres are there in 1789 millilitres 1 28. What superiority of the Metric system is established by such questions as 26 and 27 ? 29. What is the unit of mass in the Metric system, and how is it connected with the unit of length 1 30. Enumerate the chief derivative units of mass, and give their values in terms of the gramme. 4 QUESTIONS ON STEWART'S [CHAP. i. 31. Illustrate the meaning of velocity by the example of a railway train. How would you define the word ? 32. Show that the space passed over by a body moving for any time with a uniform velocity is equal to the velo- city multiplied by the time. 33. What is a convenient unit of velocity ? 34. How may the relative masses of bodies of the same kind be estimated 1 35. Why cannot weight be adopted as a fundamental method of measuring mass ? 36. What is the ultimate test that two different sub- stances have the same mass 1 37. What relation between mass and weight has been established which enables us to employ weight as a con- venient practical means of estimating mass ? 38. Define the unit of force. 39. It is true that it requires a double force to produce either (1) the same velocity in a double mass, or (2) a double velocity in the same mass ; show that one of these truths follows immediately from the definition of the unit of force? LESSON II. First Law of Motion. 40. What does the first law of motion assert 1 41. Explain how this law is apparently, but not really, contradicted by every-day experience. 42. What are the two great forces which tend to stop all motion on the surface of the earth ? Give illustrations of each. 43. What is the nearest approach to a perpetual motion with which we are acquainted ? 44. Explain the following illustrations of the first law of motion : 1. A man is on horseback, and the horse starts off suddenly. In what direction will the man fall ? 2. A man is on horseback, and the horse stops sud- denly. In what direction will the man fall 1 LESS, in.] ELEMENTARY PHYSIOS. 5 45. Show how the first law of motion serves to explain some of the common phenomena of rotation. LESSON III. Second Law of Motion. 46. State the second law of motion. 47. For the sake of clearness, what two cases may be considered separately under this law 1 48. Suppose that a ball is thrown upwards or sideways in a moving railway-carriage ; show that its motion rela- tive to the carriage is different from its motion relative to the ground, and that the motion relative to the ground is represented by the diagonal of a parallelogram, the sides of which represent the motions of the ball and of the car- riage respectively. 49. If I leap vertically upwards at the equator, I alight upon the place from which I sprang, although all places on the equator are moving, in consequence of the earth's rotation, at the rate of about one mile in three sec- onds ; explain this by means of the first and second laws of motion. 50. A balloon at the height of two miles above the earth's surface is totally immersed in, and carried along with, a current of air moving at the rate of 60 miles an hour. A feather is dropped over the edge of the car ; will it be blown away, or will it appear to drop vertically down 1 51. A ship is in rapid motion, and a stone is dropped from the top of the mast ; where will it fall ? 52. Examine the case in which a force produces motion in the same direction as an already existing motion, as when a ball is thrown directly forwards in a moving rail- way-carriage. 53. Discuss the following example of motion in a ver- tical direction : A movable chamber 4*9 m. high can be made, by machinery, to descend the vertical shaft of a mine with 6 QUESTIONS ON STEWARTS [CHAP. I. the uniform velocity of 9'8 m. per second. A ball is dropped from the top of the chamber, (1) when the chamber is at rest, (2) when the chamber is descending with the uniform velocity of 9*8 m. per second. 54. If a stone be dropped from the top of a cliff, what velocity will it acquire under the action of gravity in one second 1 in two seconds ? in t seconds ? in one quarter of a second ? 55. Explain with precision the statement that " at the end of one second a body falling freely will attain a ve- locity of 9*8 m. per second/' 56. What may be called the average or mean velocity of a falling body during the first second ? during the first two seconds 1 during t seconds ? 57. Prove that, in uniform motion, space passed over is equal to velocity multiplied by time. 58. Show that any case of uniform motion may be rep- resented graphically by the area of a rectangle. 59. Prove, by dividing a second into tenths, and sup- posing the motion uniform during each tenth of the second, that the space passed over by a body falling freely in the first second of its motion is 4'9 metres. 60. In general, what represents the space described by a body falling freely for any given time ? 61. If t = the whole time of fall, and s = the space passed over, show, from what has already been established, that s = 4-9 t z . 62. Comparing the results in questions 56 and 59 it appears that the space passed over in the first second of motion is equal numerically to the mean velocity during that second. Accepting this relation as true in general (which is the fact), find the space described during the second second of motion ; during the third second. 63. Suppose a projectile, as a bomb-shell, to be fired obliquely into the air ; prove that its actual path under the action of gravity will be a curve, bending farther and farther from the original line of impulse. What may this curve be shown to be ? LESS, iv.] ELEMENTARY PHYSICS. 7 LESSON IV. Second Law of Motion (continued). 64. Suppose a piece of iron to fall by the action of gravity, and also to be acted upon by a magnet so placed as to give it in one second a velocity in the same direc- tion as gravity of 9*8 m. per second ; find the velocity acquired and the space described in one second. 65. From the results in question 64, what may be in- ferred to be the proper measure of different forces applied to the same body ? 66. What have we found to be the measure of forces which generate the same velocity in bodies having differ- ent masses ? 67. In general, what product represents the magnitude of, or is the measure of, a force ? 68. What product measures the momentum of a moving body ? Define the measure of a force in terms of the mo- mentum which it will generate. 69. Give an instance of two forces acting in different directions simultaneously on a body at the same point, and determine the path which the body will take under the joint action of the two forces. 70. Explain how a straight line may be employed to represent the point of application, the direction, and the magnitude of a force. 71. What are the two chief results respecting the sec- ond law of motion reached, one in the last Lesson, the other in the present Lesson ? 72. Give examples of forces which act in such a way as to compel a body to remain at rest. 73. What really happens when a heavy body rests on the floor ? 74. What is the effect of a force which acts on a body without changing its state of rest or motion called ? 75. When must two pressures, or statical forces, be considered equal to each other ? 76. A man in a carriage supports a half-hundred weight in his hand. The carriage and all that it contains is now 8 QUESTIONS ON STEWARTS [CHAP. i. in the act of falling over a precipice. Will lie still con- tinue to feel the strain of the weight upon his arm ? 77. A weight equal to 100 kilogrammes rests upon a support, the weight of which support may be neglected. This support is not altogether prevented from falling, but, in virtue of the machinery with which it is connected, it is only allowed to acquire a velocity of 4*9 metres in one second. What will be the pressure on the support 1 78. What two ways are there of viewing the action of two forces acting simultaneously on a body in different directions ? , LESSON V. Forces Statically Considered. 79. What is the " parallelogram of forces " 1 80. Give a definition of the resultant of two forces acting along lines which intersect one another. 81. Distinguish between the resultant of two forces and what may be called their balancing force. 82. If two forces, P = 6, Q = 8, act on a point at right angles to each other, find the numerical value of their re- sultant. 83. Describe an experimental method of demonstrating the truth of the parallelogram of forces. 84. Suppose that we have two parallel forces in the form of weights acting at the ends of a straight horizontal rigid bar or lever, the whole resting on a fixed point or fulcrum between the forces. Neglecting the weight of the bar, under what condition will the system be in equilibrium ? What will be the pressure on the fulcrum 1 85. Define the moment of a force with respect to a point. 86. State the condition of equilibrium of a lever in the language of moments. 87. Give examples of levers where a comparatively small force at a great leverage produces a very great effect. What do you mean by a " great leverage " ? LESS, vi.] ELEMENTARY PHYSICS. 9 88. On a lever, the perpendicular distances of the lines of action of the forces from the fulcrum are often called the arms of the lever. What relation must exist between the lengths of the arms in order that a given force applied at the end of one arm may overcome a greater force ap- plied at the end of the other arm ? 89. What is the condition of equilibrium of any num- ber of forces applied in one plane to a body which is sup- ported on a fixed point or fulcrum ? 90. On a straight lever without weight we have on the right hand of the fulcrum two forces, namely, 8 grammes at a distance of 6 centimetres, and 12 grammes at a dis- tance of 8 centimetres ; while on the left hand we have 10 grammes at a distance of 10 centimetres. Which arm will tend to fall ? LESSON VI. Third Law of Motion. 91. In the third law of motion what is asserted of any force which alters the state of rest or motion of a body as a whole ? Give an illustration. 92. What does the third law of motion assert of the momenta generated in the parts of a body or system of bodies by the action of internal forces 1 Illustrate this truth by the example of firing a gun. 93. How is the third law of motion sometimes stated ? 94. Illustrate this law by the example of a stone fall- ing to the ground. 95. How does the discharge of a cannon which is firmly fixed to the ground furnish another illustration of the same law ? 96. According to this law of motion, what must take place whenever a man leaps upward from the ground ? 97. Suppose a bomb-shell flying along with a velocity of 200 m. per second explodes into two parts of equal weight, one of which is propelled forwards in the exact direction in which the shell is moving with an additional 1* 10 QUESTIONS ON STEWART'S [CHAP. i. velocity of 200 m. per second. Show, by means of the third law of motion, that the other half of the shell will be brought to rest in consequence of the explosion. 98. Explain the Eolipyle. 99. Explain the ascent of a rocket. LESS, vii.] ELEMENTARY PHYSICS. 11 CHAPTER II. THE FORCES OF NATURE. LESSON VII. Universal Gravitation. 100. Into what three groups may the forces of nature be divided ? 101. What is the distinction between molecular and atomic forces ] 102. Illustrate the general fact that some of the forces connected with molecules and atoms may be characterized as permanent while others are temporary and evanescent. 103. What is the most important and best understood force belonging to matter ] 104. What question respecting terrestrial gravity did Newton ask himself, and what answer did he find by ex- periment ? 105. What opinion on this subject was held by the fol- lowers of Aristotle ? 106. How did Galileo overturn the Aristotelian dogma ? 107. Describe the " guinea and feather " experiment, stating clearly what it proves. 108. What prevents us from making exact experiments on bodies falling freely ? 109. What effect would changes in the force of gravity have on the oscillations of a pendulum 1 110. Describe Newton's pendulum experiments, and show that they prove that the weight of a body is directly proportional to its mass. 111. Compare gravity with magnetism, as regards the relation between the acting force and the mass acted upon. 112. Show that the measure of the force of gravity which acts on one gramme is equal to 9*8. What will it be on 5 grammes 1 12 QUESTIONS ON STEWART'S [CHAP. n. 113. How is the, vertical direction defined ? How found by experiment ? 114. Why are plumb lines not strictly parallel ? What change in the direction of a plumb line is produced by travelling one mile on the earth's surface 1 115. What effect on the weight of a body would be produced by a change in the mass of the earth or attract- ing body ? 116. State the law of " inverse squares," or law which expresses the mathematical relation between the distance of two bodies from each other and the force of attraction between them. 117. Prove the law of " inverse squares " by the New- tonian method of comparing the force of the earth's attrac- tion at the moon with the same at the earth's surface. 118. Give the complete statement of the law of univer- sal gravitation. 119. Illustrate this law by supposing different numer- ical values for the attracting masses and their distance from each other. LESSON VIII. Atwood's Machine. 120. What is the object of Atwood's Machine ? Describe its chief parts. 121. Describe Experiment A, and state what it proves. 122. Describe Experiment B. Compare the results in ex- periments A and B, and state the law which they establish. 123. Describe Experiment C. What may be concluded, (1) by comparing together experiments A and C, (2) by comparing together the results in all three experiments, A, B, and C ? 124. Describe Experiment D. 125. Describe Experiment E. What truth do experi- ments D and E illustrate ? 126. Describe Experiment F. What is " the law of ve- locities " which is demonstrated by this experiment ? LESS, ix.] ELEMENTARY PHYSICS. 13 127. Describe Experiment G, and give the '-'law of spaces " which it proves. 128. Show that, if a body be projected vertically up- wards, the height attained is proportional to the square of the velocity of projection. 129. What is the relation in theory between the velo- city of projection of a stone, and the velocity with which it strikes the ground on its return ? How is this truth illustrated by Experiment H 1 130. Neglecting the weight of the pulley in Atwood's Machine, let the one box weigh 600 grammes, and the other 400 ; what will be the tension of the string during the downward motion of the heavier box 1 131. A body is projected vertically upwards with a velocity equal to 19 '6 metres per second ; what will be its velocity after it has risen 14'7 metres ? 132." Give a brief recapitulation of the facts connected with the action of gravity at the earth's surface. LESSON IX. Centre of Gravity, etc. 133. Show how the force which gravity exerts on a body may be resolved into a system of parallel forces, and from this point of view give a definition of the centre of gravity of a body.- 134. Describe a simple practical way of finding the cen- tre of gravity of a body. 135. If we have a heavy solid resting on a base, what condition must be fulfilled in order that it may remain at rest ? Prove the necessity of this condition. 136. Define stable equilibrium and unstable equilibrium, and give examples of each. 137. State a simple law which will always decide whether an equilibrium is stable or unstable. What grounds are given for the truth of this law ? Illustrate its application by the example of the egg. 138. Define neutral equilibrium, arid give an example. 14 QUESTIONS ON STEWART'S [CHAP. n. 139. A cone is placed on its apex on a flat horizontal surface ; determine the kind of equilibrium. 140. A uniformly heavy circular wooden disk has a piece of its substance taken out, and a piece of lead insert- ed instead. In. what position will it rest on a flat hori- zontal surface ? 141. How will a man rising in a boat affect its stability ? 142. Why is a cart loaded with hay more liable to be overturned from irregularities in the road than one loaded with the same weight of lead ? 143. Describe briefly the balance, and show that a sen- sitive balance enables us to ascertain with great exactness the weight of a body. 144. Can you determine what must be the position of the centre of gravity of a balance relatively to the centre of suspension in order that the balance may be very delicate ? 145. Explain the use of the pendulum, (1) in detect- ing changes in the force of gravity, (2) in regulating clocks. 146. What is meant by the isochronism of a pendulum, and how was it first discovered ? What is the length of a seconds pendulum ? 147. What is the law which expresses the relation be- tween the time of oscillation of a pendulum and its length 1 LESSON X. Forces exhibited in Solids. 148. Describe briefly, and illustrate, the chief attractive forces which are exhibited in bodies. 149. Describe briefly, and exemplify, the following re- sistances to deformation which are called into action by Various forces tending to alter the shape of a solid body : (1) Resistance to linear extension. (2) Resistance to linear compression. (3) Resistance to cubical compression. (4) Resistance to torsion. (5) Resistance to flexure. LESS, x.] ELEMENTARY PHYSIOS. 15 Which one of these resistances is also exhibited by liquids and gases ? 150. Explain what friction is, and define the coefficient of friction. 151. Prove that if the pressure remain the same, the friction is independent of the magnitude of the surface. 152. Give Rennie's laws upon friction. How may fric- tion be reduced to a minimum 1 153. What conditions are favorable to the formation of crystals ? Give an instance. 154. Give instances of crystals of great value which have not yet been formed artificially. 155. What conclusion about crystals is drawn from the behavior of many crystals, for example, those of Ice- land spar ? 156. What peculiarities of structure and what conse- quent properties are exemplified by such substances as wood and flax ? by wrought-iron ? by such substances as mica and oyster-shells ? 157. Give examples of solids exhibiting no apparent trace of structure. In general, what effect does time and vibration have on the structure of a solid 1 ? Give an example. 158. Define tenacity. How are experiments on tenaci- ty conducted 1 What law is established by the experi- ments ? What convenient measure of tenacity does this law point out ? 159. What do the results of Wertheim's experiments, as given in the table, show as to the effect of time on tenacity ? 160. What peculiarity is there in the tenacity of fibrous solids like wood ? According to Musschenbroeck, which is the most tenacious of woods 1 161. Instead of bodies suddenly giving way under forces tending to pull their particles asunder, what other form of rupture is common ? 162. What does ductility denote ? Give examples of bodies which possess it ? 16 QUESTIONS ON STEWART'S [CHAP. n. 163. What is malleability ? Which, metal is the most malleable ? 164. Explain brittleness. In what sense is a sheet of glass stronger, and in what sense is it weaker, than a sheet of paper ? 165. Define hardness. If we have three bodies A, B, and C, how are their proper places on a scale of relative hardness found ? 166. Which is the hardest of all known substances, and how is it cut ? 167. Explain the processes known as tempering and annealing. What are Prince Rupert's drops ? 168. What does the word elasticity denote ? What is meant by the limit of perfect elasticity ? 169. What condition should a solid structure such as a bridge fulfil ? 170. Give the laws which hold true, within the limit of perfect recovery, of resistance to linear extension. 171. What relation exists between the forces which will produce equal amounts of linear extension and linear compression respectively in the same body ? 172. How are the forces with which different sub- stances resist linear extension or compression compared together ? 173. How are experiments on torsion conducted ? What laws have been established ? 174. Mention some of the ways in which the force with which a solid resists any attempt to bend it is utilized. 175. What are the laws of flexure ? What follows from the last law as to the best form for a beam of given mass which is to be heavily loaded ? LESSON XI. Forces exhibited in Liquids. 176. What is the essential difference between a solid and a liquid ? What are proofs that cohesion is not entire- ly wanting in liquids ? LESS, xi.] ELEMENTARY PHYSICS. 17 177. Describe the state of liquidity called viscous. Give instances to show that time is an element of importance in determining the liquidity of a substance. 178. What is characteristic of the resistance to compres- sion offered by liquids ? What is the exact measure of the compressibility of water ? 179. State the law of liquid pressure discovered by Pascal, and illustrate it by the imaginary case of a hollow vessel full of water which is uninfluenced by gravity. 180. Show that, by employing pistons of different sizes, a fluid is capable of forming a very powerful mechan- ical arrangement. 181. What is the principle, and what are some of the uses, of Bramah's press ] 182. Prove that the surface of a liquid in an open ves- sel must be perpendicular to the force of gravity, that is, horizontal. What is the true character of the surface of a large body of water, as the ocean 1 183. Explain the construction and use of the water- level. 184. Explain the construction and use of the spirit- level. 185. Explain Artesian wells. 186. What is the measure of the pressure on the horizontal base of an open vessel full of a liquid ? Hence show what must be the relation between the pressure on any horizontal layer of liquid in an open vessel and (1) the depth of the layer, (2) the area of the layer. 187. Show by a simple experiment that the pressure of a horizontal layer of water is the same upwards as downwards. 138. A hollow cubic decimetre, open at the top, is filled with water ; what will be the pressure on the bottom and sides ] 189. A vessel contains water to the depth of a deci- metre, and one of the sides of this vessel is a rectangular surface, the bottom of which is one decimetre, while the 18 QUESTIONS ON STEWART'S [CHAP. n. side slopes at an angle of 45 ; what is the whole press- ure on this side ? 190. In what way is the pressure exerted by a liquid connected with the density of the liquid 1 191. Prove that a fluid buoys up a solid immersed in it with a force equal to the weight of the fluid displaced. 192. Apply the principle of buoyancy successively to the cases in which the density of the solid immersed is greater than, equal to, and less than that of the fluid. 193. A cube of wood, the density of which is equal to 0*8, is put into a vessel containing water ; what portion of its side will be immersed ? 194. Taking water at 4 C. as the standard of specific gravities or relative densities, how do you define the spe- cific gravity of any substance ? If, as in the Metric sys- tem, the density of water at 4 C. is equal to unity, what relation must exist between the specific gravity and the density of any substance 1 195. Explain a method of finding the specific gravity of a solid body. Suppose, for the sake of illustration, that a substance weighs in vacuo 120 grammes, and when immersed under water at 4 C. only 89 grammes ; find the specific gravity of the substance. Hence, what general rule may be given for obtaining the specific gravity of a solid substance 1 196. Explain a method of ascertaining the specific gravity of a liquid. As an illustrative example, let the loss of weight of a solid body in water equal 31 grammes, and its loss of weight in the liquid in question equal 28 grammes. Find the specific gravity of the liquid. 197. Describe some capillary phenomena which show, (1) what two kinds of capillary action exist, (2) what in- fluence the diameter of the tube has on capillary ascent and depression. 198. Mention other illustrations of the laws of capil- larity. 199. Describe endosmose and eooosmose. LESS, xii.] ELEMENTARY PHYSIOS. 19 LESSON XII. Forces exhibited in Gases. 200. In what respect does a gas differ from a liquid ? In what respects is it like all other substances ? 201. Describe an experiment illustrating that a gas has weight, and also that some gases weigh more than others. 202. How may a liquid be converted into a gas 1 203. True steam being invisible, how do you account for the visible cloud arising from a kettle or a railway engine 1 204. State and illustrate the distinction between gases and vapors. 205. What agencies tend to bring a gas into the liquid or solid state 1 What six gases have never yet been .li- quefied ? What substance has never yet been vaporized ? 206. What is the composition of the atmosphere 1 207. What effect on the air would the processes of respiration and combustion, if unbalanced, in course of time produce I Why does the composition of the atmos- phere in point of fact remain unchanged 1 208. Why is it that, although the air is exerting press- ure all around us, we seldom perceive any traces of it ? 209. What is the experiment of the Magdeburg hemi- spheres, and what does it illustrate 1 210. How was the ascent of water in pumps accounted for up to the time of Galileo 1 Give Torricelli's reason- ing, together with his celebrated and decisive experiment, upon this subject. 211. How was the truth of Torricelli's discovery veri- fied by Pascal ? 212. What remarks are made by the author in regard to the connection between the barometer and the weather ? 213. If we cork a flask full of air and then remove half of the mass of air within the flask, how much will the pressure on the interior of the flask be changed ? What is that statement of Boyle's law, which gives di- rectly the relation between the mass and the pressure of a quantity of gas 1 '^20 QUESTIONS ON STEWART'S [CHAP. n. 214. State the law of Boyle in the form adopted hy the discoverer, and verify its truth by a simple experi- ment. 215. Show that the two forms of stating Boyle's law come to the same thing. 216. What is the true explanation of gaseous pressure according to many philosophers ? Show, with the aid of a numerical illustration, that their hypothesis is in har- mony with Boyle's law. 217. Show, by an experiment, that gases as well as liquids possess buoyancy, and are subject to the principle of Archimedes. 218. What is the reason that a "balloon rises in the at- mosphere ? 219. Describe the construction and action of an air- pump. 220. Prove that the density of the air in the receiver diminishes in a geometrical ratio, i. e. that a constant fractional part of the mass of air remaining in the receiver, is expelled by each double stroke. Why could we never, even in theory, succeed in producing a perfect vacuum '? When is a practical limit reached ? 221. Describe the construction and action of the com- mon lifting-pump. What sets a limit to the height to which water can be raised by means of this pump ? 222. Describe a siphon, and its action, and explain why the flow of liquid from one vessel to the other is main- tained. When will a siphon once set in action cease to act ? What fixes a limit to the working length of the shorter arm of a siphon ? 223. Describe Graham's experiment on gaseous diffu- sion. 224. Mention substances which have the power of absorbing or retaining matter in the gaseous state. LESS, xiii.] ELEMENTARY PHYSICS. 21 CHAPTER III. ENERGY. LESSON XIII. Definition of Energy. 225. Adduce examples which illustrate the applica- tion, up to recent times, of Newton's law of action and reaction. What view did the old hypothesis take of the phenomena of collision and friction ? 226. Mention the considerations which probably led the way to a numerical estimate of work. 227. Define the unit of work, and show how to find the amount of work done in lifting a body to any given height. 228. Define energy, and estimate how much energy will be imparted to a stone weighing one kilogramme by projecting it vertically upwards with a velocity of 9 '8 metres per second. 229. Prove that the work which can be accomplished by a moving body is proportional, (1) to the square of its velocity, (2) to its mass. 230. Show that the work capable of being done by a body whose mass (in kilogrammes) is m, and whose velocity (in metres) is v, is represented by the expression m t<2 1 .. ~ . m i& . j^-. (The general form is - a ~.) 231. Estimate the energy of a body weighing 64 grammes projected vertically upwards with the velocity of 60 metres per second. 232. Show, by means of the force of gravitation, that two types or kinds of energy exist, which are mutually convertible. 22 QUESTIONS ON STEWART'S [CHAP. in. LESSON XIV. Varieties of Energy, 233. Give a summary of what was said about the two forms of energy in the last Lesson. 234. Illustrate the transformation of energy by an example, taken from chemistry, and point out the analo- gies to the case of gravitation. What appears from this example to be the true nature of heat 1 235. Name some of the varieties of visible or mechani- cal energy, both kinetic and potential. 236. How does the doctrine that heat is a form of energy explain the phenomena of latent heat 1 237. Trace the analogy between the mechanical world and tlje molecular world by means of the phenomena of sound and of radiant light and heat. 238. Show that electrical phenomena afford another illustration of molecular energy. 239. What great advantage as a source of energy does electricity in motion possess over water in motion or heat ? 240. Give a brief recapitulation of the various forms of energy. LESSON XV. Conservation of Energy. 241. What was formerly in many minds the great ideal of mechanical triumphs ? 242. What conclusive answer may now be made to the arguments in favor of perpetual motion ? 243. What is the principle of the conservation of energy, and what is the nature of the evidence in favor of its truth ? 244. Apply the principle of the conservation of energy to the case of a stone projected vertically upwards. 245. What becomes of the energy of a railway train when it is suddenly stopped ? or of a cannon-ball after it has struck the target ? or, in general, what becomes of the energy of visible motion when it has been stopped by per- cussion or friction ? LESS, xv.] ELEMENTARY PHYSICS. 23 246. What experimental evidence upon this point do we owe to Rumford and Davy ? What simple phenomena can you mention which furnish evidence in the same di- rection ? 247. Who were the first to point out the probability of a connection between the various forms of energy? Who established this connection on a scientific basis ? In particular, what was the result of the researches of Joule ? 248. Illustrate still further the connection between the various kinds of energy by what takes place in a galvanic battery. 249. Give an algebraic statement of the doctrine of the conservation of energy. 250. Illustrate the true function of a machine, by ap- plying the law of the conservation of energy to one of the ordinary mechanical combinations, such as a system of pulleys. 251. Apply the same law to the hydraulic press. 252. What law holds universally in machines, if fric- tion be left out of account, respecting the power, the weight, and the distances which they traverse 2 24 QUESTIONS ON STEWART'S [CHAP. iv. CHAPTER IV. VISIBLE ENERGY AND ITS TRANSMUTATIONS. LESSON XVI. Varieties of Visible Energy. 253. Mention some of the varieties of visible kinetic energy. 254. Why is the energy of a vibrating string classed among the forms of visible energy ? 255. Give examples of visible potential energy. 256. Suppose a rifle-ball weighing 20 grammes and moving with a velocity of 200 metres per second buries itself in a block of wood weighing 20 kilogrammes and hung by a string so as to form a pendulum. Find how much of the energy of the ball will reappear after the im- pact in the form of visible energy. What has become of the remainder of the energy of the ball 1 257. What distribution of momentum, and what con- sequent change of energy, take place during the passage of a ball through the air ? 258. What extension of the first law of motion are we now prepared to recognize 1 259. Examine the following case of direct impact of two inelastic solids, and find what part of the united energy of the two masses before impact is transmuted into heat : weights of the solids, 20 grammes and 10 grammes; velocity of the first, 20 ; of the second, 16 in an opposite direction. 260. What is the law of energy which applies to the case of the impact of two perfectly elastic bodies ? Sup- pose, as an illustrative example , that two perfectly elastic balls, weighing respectively 4 and 3 kilogrammes, moving in the same direction with velocities 5 and 4, impinge on each other. LESS, xvi.] ELEMENTARY PHYSICS. 25 261. Apply the laws of impact to the case in which one elastic ball impinges directly against the extremity of a row of elastic balls at rest. 262. Consider briefly the energy of a circular disk in rapid rotation. In general, wherever in nature a body revolves in a circle round a central force, what is true of its velocity and of its kinetic energy ? 263. Consider the energy of a body moving in an ellipse, and show that a definite part of its energy assumes alternately the kinetic and the potential forms. 264. Show that the doctrine of energy leads to the conclusion that the velocity of a body which has slid down a smooth inclined plane depends only on the height of the plane. Why does this proposition fail in case the plane be rough ? 265. Mention some instances of visible energy of posi- tion, and state how the energy in each case may be con- verted into energy of motion. 266. Show that the energy of an oscillating pendulum is alternately kinetic and potential. 267. A pendulum bob weighing one kilogramme is so swung that it is higher at the summit of its oscillation than at its lowest point by one decimetre ; what is its velocity at its lowest point 1 268. Describe Foucault's pendulum experiment, the pendulum being supposed, for the sake of simplicity, to oscillate at the north pole, 269. Consider the energy of a vibrating body, as the string of a musical instrument or a bell, and point out the analogies between the motion of such a body and that of a pendulum. 270. What effect does vibratory motion in the air usually produce before it assumes the shape of heat ? 271. Give a recapitulation of the various kinds of visible energy which have been considered in this Les- 26 QUESTIONS ON STEWART'S [CHAP. iv. LESSON XVII. Undulations. 272. Prove that for small vibrations of a pendulum the force which urges the ball in the direction of its mo- tion at each instant is proportional to the distance of the ball from its point of rest, i. e. is proportional to the dis- placement. 273. Explain the principle of isochronism. How far is it applicable ? 274. Upon what elements does the time of vibration of a body depend ? How may this be illustrated ? 275. Define wave motion, and give examples of it. 276. Define an up-and-down or transverse wave, and a wave-length; and illustrate by diagrams the manner in which an undulatory motion is propagated by an up-and- down wave. 277. If v denote the velocity with which the wave is propagated, t the time of a double vibration of a particle, and I a wave-length, show that I = v t. 278. What is the nature of waves of condensation and rarefaction, or longitudinal waves ? 279. What is meant by the phase of a vibrating par- ticle ? 280. What is an essential peculiarity of vibrating mo- tion as regards the phases of any two contiguous parti- cles ? 281. What is the amplitude of a vibration 1 282. Explain why the wave-length does not depend on the amplitude of the vibration. LESSON XVIII. Sound. 283. What is the definition of Acoustics ? 284. In what two senses is the word "sound" used] In which sense is it here used 1 285. Explain the difference between a noise arid a mu- sical sound or note. 286. What determines the pitch of a note 1 LESS, xviii.] ELEMENTARY PHYSICS. 27 287. What other characteristics of sound besides pitch are perceived by the ear ? 288. What is the nature of sound-waves 1 289. How may it be shown that sound is not prop- agated in vacuo 1 290. What are the laws of the reflection of sound? Illustrate them by means of a diagram. 291. What constitutes an echo ? When, for instance, is an echo more audible than the original sound ? 292. What condition must be fulfilled in order to pro- duce a distinct echo 1 What effect do whispering-galleries have upon a sound ? 293. What experiment upon sound may be performed with two conjugate reflectors 1 294. What is related of the Cathedral of Girgenti in Sicily ? 295. What power does a convex glass lens exercise upon rays of light ? 296. What was the experiment of M. Sondhauss, and the conclusion to be drawn from it 1 297. What is the velocity of sound in air, and by what method was it determined ? 298. What effect, if any, have pitch and intensity respectively on the velocity of sound ? 299. How does the velocity of sound in air compare with its velocity in hydrogen gas at the same pressure ? What reason can be given for this difference 2 300. Explain why the velocity of sound in a given medium does not vary w T ith its density. 301. Why does sound travel faster in warm than in cold air 1 302. What is the velocity of sound in water ? in wood 1 303. Prove that the intensity of a sound varies inverse- ly as the square of the distance from the source. 304. Why is it probable that the intensity of sound really diminishes somewhat more rapidly than is indi- cated by the law of " inverse squares " ? 305. What effect does a change in the density of the 28 QUESTIONS ON STEWART'S [CHAP. iv. medium have upon the intensity of sound ? How may this law be verified 1 What is the reason that it is true 'I 306. Why is sound much better heard when the air is calm and homogeneous ? 307. By what arrangement may the intensity of the sound of a musical string be strengthened ? LESSON XIX. Vibrations of Sounding Bodies. 308. State the laws which connect the vibrations of a stretched string -with the properties of the string. 309. Explain the mode of action of an organ pipe. 310. What is the reason that an organ pipe filled with any other gas than air yields an entirely different sound 1 What use may be made of this fact ? 311. What acoustical difference is there between a shut pipe and an open pipe of the same dimensions 1 312. Suppose we have rods of wood fixed at one end and free to move at the other ; w r hat two kinds of vibra- tory motion may be imparted to them ? 313. What are the laws which express the relations between the number of vibrations of a rod and its dimen- sions ? 314. What is the law which governs the vibrations of plates 1 315. What are nodal points or nodes on a vibratory cord, and how may they be produced ? What is a loop or ventral segment ? 316. In producing nodes along a cord, what condition must be observed in order that the vibrations shall not interfere Avith each other ? 317. How may the existence of nodes and ventral seg- ments be rendered evident by means of vibratory plates 1 318. What law governs the communication of vibra- tions from one instrument to another through the air 1 Mention instances in which this communication may be observed. LESS, xix.j ELEMENTARY PHYSICS. 29 319. Explain the action of Savart's machine for meas- uring the number of vibrations corresponding to a given sound. 320. Describe Lissajous's method of making vibrations apparent to the eye. 30 QUESTIONS ON STEWART'S [CHAP. v. CHAPTER V. HEAT. LESSON XX. Temperature. 321. What two kinds of energy are denoted by the word heat ? 322. What three-fold division of the study of phe- nomena connected with heat will be adopted in this chapter ? 323. When are two bodies said to be of the same tem- perature ? When is one body said to be of a higher temperature than another 1 How would you define tem- perature ? 324. What is the general law of expansion under the action of heat 1 Cite experiments which go to prove this law. 325. What is one remarkable exception to the above law of expansion ? 326. Why in general does expansion fail to furnish us with an exact measure of temperature ? How does the case of water illustrate this ? 327. Why is a substance near the point of changing its state not fitted to be used as a means of measuring tem- perature ? 328. Compare the relative merits of mercurial and air thermometers as regards accuracy and convenience. 329. What is the principle of the mercurial thermome- ter? Describe the process of filling the bulb and tube with mercury. 330. How may great delicacy be secured in a ther- mometer ? 331. What are the two fixed points of temperature on a thermometer ? Describe how they are determined. LESS, xx.] ELEMENTARY PHYSICS. 31 332. Explain how the stem of a thermometer is gradu- ated according to the centigrade scale. How do the Fah- renheit and the Reaumur scales differ from the Centi- grade 1 333. Deduce formulae for reducing Centigrade degrees to Fahrenheit, and vice versa. 334. Find the degree of Fahrenheit which corresponds to 45 Centigrade. 335. Find the degree Centigrade which corresponds to 86 Fahrenheit. 336. What degree Fahrenheit corresponds to 40 Centi- grade ? 337. Deduce formulae for reducing Centigrade degrees to Keaumur, and vice versa. 338. Explain the change of zero of a mercurial ther- mometer, and the corrections which must be applied to eliminate this source of error. 339. What is the effect of suddenly heating and cooling a thermometer ? What practical rule does this effect sug- gest with regard to the determination of the fixed points ? 340. Explain why it may be necessary to apply a cor- rection to the reading of a mercurial thermometer on account of the position in which the instrument is held. 341. Suppose a mercurial thermometer is placed under the receiver of an air-pump and the air exhausted, what will be the effect on the column of mercury 1 342. If the bulb of a thermometer, together with the lower part of the stem up to the zero-point, be immersed in boiling water, while the remainder of the stem from to 100 is exposed to air at the freezing-point, the temper- ature denoted will be 98'4 instead of 100. Explain this. 343. Why is mercury unsuited for measuring very low temperatures 1 What liquid is used for this purpose 1 344. Describe a minimum thermometer and its action. 345. Describe Phillip's maximum thermometer and its action. 346. Describe Leslie's differential thermometer and its action. 32 QUESTIONS ON STEWART'S [CHAP. v. LESSON XXI. Expansion of Solids and Liquids through Heat. 347. Explain (with, a diagram) how the comparatively small expansion of a solid rod due to heat may be rendered visible by mechanical means. What other method of magnifying expansion is also employed ? 348. Define the coefficient of linear expansion of a sub- stance. 349. Which is the most expansible of the metals men- tioned in the table ? which the least 1 350. What circumstances may modify the value of the coefficient of expansion of a substance ? Give an exam- ple. 351. Show that the cubical expansion of a substance is very nearly three times as great as the linear for the same change of temperature. 352. State the principle of one method of determining experimentally the cubical expansion of solids. 353. Illustrate this method by solving the following example : A solid weighs 600 grammes in vacuo, and only 400 grammes in a fluid at C., of which the specific gravity is 1*2, while it weighs 406 grammes in the same fluid at 100 C., for which temperature the specific gravity of the fluid is known to be 1*16 ; find the cubical expansion of the solid between these two temperatures. 354. What general relation between the coefficients of linear and cubical expansion of any substance is indicated by the table on page 161 ? 355. What notable exceptions are there to the follow- ing laws : (1) Solids expand through heat, (2) And expand equally in all directions. 356. What effect, in general, has the temperature of a solid on its rate of expansion ? 357. Explain the distinction between the apparent and the real expansion of a liquid. LESS, xxii.] ELEMENTARY PHYSICS. 33 358. Give an outline of the method of finding the real expansion of a liquid, called the method by ther- mometers. 359. Explain Matthiesson's method of determining the real expansion of a liquid. 360. Illustrate Matthiesson's method by solving the following example : A piece of glass, of which the linear expansion from to 100 C. is known to be 0.0009, loses at C. one gramme of its weight in the fluid in which it is weighed, while at 100 C. it only loses 0.96 of a gramme. Find the expan- sion of the fluid between and 100 C. 361. Give an outline of Regnault's method of deter- mining the expansion of mercury. 362. What peculiarity does water exhibit with respect to its expansion? 363. Explain Hope's method of ascertaining the point of maximum density of water. 364. What fact respecting the rate of expansion of water at different temperatures is shown by the table on page 165 ? 365. What reasons do we have for believing that the rate of expansion of very volatile liquids must be very great ? How is this conclusion verified in the case of liquid carbonic acid ? 366. Recapitulate the chief laws for the expansion of liquids. LESSON XXII. Expansion of Gases, Practical Appli- cations, 367. State and illustrate the law discovered by Charles, which expresses the relation between the temperature and the pressure of a gas, the volume remaining constant. 368. Prove that the same law expresses also the rela- tion between the temperature and the volume of a gas, the pressure remaining constant. 2* o 34 QUESTIONS ON STEWART'S [CHAP. v. 369. A bladder which at contains 900 cubic cen- timetres of air has its temperature increased to 30 C., the pressure under which the gas exists meanwhile remaining constant ; what will now be the volume of the gas in the bladder ? 370. What are some direct consequences of the fact that the coefficient of expansion is the same for all gases ? 371. Explain more fully than was possible in Lesson XX. the advantage of using an air thermometer. 372. At what temperatures are the metre and the yard respectively standards of length ? 373. Explain why it is necessary in all very accurate weighings to know the temperature of the air. 374. Show that, in the metric system, the weight (in grammes) of one cubic centimetre of any substance will denote at the same time its specific gravity. 375. Why is it necessary to fix upon a standard temper- ature in comparing the specific gravities of substances, and what is this standard 1 376. What is the standard pressure employed in com- paring the specific gravities of gases ? 377. Explain what effect change of temperature will produce upon the motion of a pendulum, and also upon the motion of the balance-wheel of a chronometer. 378. Explain Harrison's gridiron pendulum. 379. Explain the compensation balance for chronome- ters. 380. Mention other instances in which account must be taken of the expansion of bodies. 381. Mention instances in which advantage is taken of the fact of expansion. LESSON XXIII. Change of State and other Effects of Heat 382. What invariable rule holds in the production of changes of state through heat ? 383. Give examples of substances which differ from LESS, xxiii.] ELEMENTARY PHYSICS. 35 one another in the manner of their passage from the solid to the liquid state. What is this passage called 1 384. Give instances in which change of composition accompanies change of state. 385. What effect does pressure have on the melting- point of ice 1 What general law is found to hold true with respect to the connection between pressure and con- gelation 1 386. What is the reason that gold, silver, and copper coins cannot be cast in a mould, but must be stamped ? 387. Under what conditions may water be cooled be- low the freezing-point without becoming ice 1 What other instances of a similar phenomenon exist ? In all such cases how may solidification be immediately produced ? 388. What is regelation, arid what is Forbes's explana- tion of it ? 389. What is the difference between sublimation and vaporization 1 390. Distinguish between two kinds of vaporization. 391. What did Daltoii show with respect to the forma- tion of a vapor in a confined space 1 392. How far is the evaporation of a liquid modified by taking place in air instead of in vacuo ? 393. What conditions are most favorable to rapid evap- oration in the open air, and why ? 394. Explain the apparatus and process of distilla- tion. 395. What is ebullition, and how is it affected by the supply of heat ? 396. Enumerate the chief circumstances upon which the boiling-point of a liquid depends. 397. What are the boiling-points of ether and of mer- cury, as given in the table on page 1 78 ? 398. Describe two simple experiments which show that the boiling-point of a liquid depends upon the press- ure under which it exists. 399. Why is the boiling-point of water at the top of a mountain lower than at its bottom 1 How does this fact 36 QUESTIONS ON STEWART'S [CHAP. v. interfere with culinary operations, and how is the diffi- culty remedied ? 400. In what way is the fact that the boiling-point of water is lessened as we rise above the sea-level applied tor a practical use 1 401. How do glass and metal vessels compare with each other in their influence upon the boiling-point of water ? What is the effect of dropping iron-filings into the glass vessel 1 402. In what way was Donny able to raise the temper- ature of water to 135 C. without ebullition 1 403. What is the general effect of salts in solution upon the boiling-point of water ? 404. Give examples of the behavior of liquids in the spheroidal state. 405. In what way can the behavior of liquids in the spheroidal state be explained, and what experiment of Boutigny confirms the truth of this explanation ? 406. What was Faraday's experiment with ether, solid carbonic acid, and mercury, and how do you explain the effects which he observed ? 407. Andrews heated liquid carbonic-acid, under great pressure in a closed tube, to the temperature of 31 C,, or thereabouts. What phenomena were observed, and what inference has been drawn from them ? 408. What is sublimation, and what are instances of it '] 409. What is the effect of heating a strong solution of hydrochloric acid in water ? of heating chalk ? 410. Give an illustration of the great attraction which some gases have for water. 411. What six gases have never yet been condensed by the joint effect of cold and pressure 1 412. Explain what is meant by the maximum pressure of a vapor. 413. Show how, by means of the table on page 183, we may obtain the atmospheric pressure by observations of the boiling-point thermometer. LESS, xxiv.] ELEMENTARY PHYSICS. 37 414. What was the discovery of Gay Lussac in refer- ence to the densities of gases ? Illustrate it by the case of hydrogen and chlorine. 415. Give a recapitulation of the effects of heat which have already been considered. 416. In addition to the effects already discussed, enu- merate other ways in which heat influences bodies. LESSON XXIV. Conduction and Convection. 417. In what way do we derive our heat from the sun 1 418. Give an instance of the mode of distribution of heat called conduction. How does it differ from radiation ? 419. What example does this branch of the science of heat afford of the provision by nature for the welfare of the animal creation ? 420. For what two purposes may a bad conductor of heat be employed ? Give an instance of each. 421. Give an experiment which shows the difference between the conducting power of two different sub- stances. 422. Why is it that, when a metal bar has one of its extremities heated in the fire, the other extremity does not ultimately attain the same temperature? 423. Give Fourier's definition of the conductivity of a substance. 424. Suppose we have two bars of the same shape, size, arid conductivity, but unlike in material, the ends of which we heat by a spirit-lamp to the same extent ; and let the surfaces of both bars be covered with gilt. Now, if we observe the temperatures of both bars at equal distances from the lamp, we shall find them unequal or equal according as a short time or a considerable interval has elapsed since the lamp was first applied. Explain this. 425. If we take two precisely similar pieces, one of 'oismuth and the other of iron, and, coating one end of each with white wax, place the other end in a hot ves- 38 QUESTIONS ON STEWART'S [CHAP. v. sel, we shall find that the wax will melt first on the bis- muth, although iron is the best conductor. Account for this. 426. Explain the principle of Davy's safety-lamp. 427. What peculiarity do crystals exhibit in their con- ducting power, and how was this shown experimentally by De Senarmont ? 428. How may the bad conducting power of water be shown by experiment ? 429. Describe the process called convection. How may convection currents be rendered visible ? 430. Give an account of convection on the large scale, as exemplified by the freezing of a lake. What conse- quences would ensue if water had no point of maximum density and ice were heavier than water ? 431. Upon what two things does convection depend 1 How is this illustrated by the atmosphere of the sun 1 432. Explain the trade-winds. 433. Explain the land and sea breezes. LESSON XXV. Specific and Latent Heat. 434. Define the unit of heat ; and also the specific heat of a substance. 435. Explain the method of determining the specific heat of a substance called the " method of mixtures " by the aid of a numerical example. 436. What other methods of estimating specific heat have been devised 1 437. In general, what influence have temperature and density respectively on the specific heat of solids ? 438. Generally speaking, do substances have a greater specific heat in the solid or in the liquid state ? 439. What substance has generally been supposed to have the greatest specific heat 1 440. What two kinds of specific heat may be distin- guished in the case of a gas ? LESS, xxv.] ELEMENTARY PHYSICS. 39 441. What results did Kegnault obtain respecting the specific heats of gases 1 442. What law did Dulong and Petit discover with re- spect to the specific heats and atomic weights of simple substances ? 443. Under what circumstances does heat become la- tent ? How may we correctly describe the condition of water at 0., or of steam at 100 C. ? 444. How were Black's first experiments upon latent heat performed ? How were his subsequent experiments performed so as to measure the latent heat of one kilo- gramme of water ? 445. What is the value, in the metric system, of the latent heat of water ] of the latent heat of steam ? 446. What important parts do the facts that water has a greater latent heat than any other substance, and that steam has a greater latent heat than any other gas, play in the economy of nature ? 447. Viewing heat as a species of molecular energy, what twofold office does it discharge ? 448. What explanation of the phenomona connected with latent heat is furnished by the doctrine of energy ? 449. What is the principle of all freezing mixtures and processes ? 450. Explain the method of using the wet and dry bulb thermometers for estimating the hygrometric state of the air. 451. How did Leslie freeze water by means of its own evaporation ? 452. Describe Carre's apparatus for artificially pro- ducing ice ? 453. How did Faraday succeed in freezing mercury ? 454. Explain the fall of temperature which results from mixing snow and salt together. 40 QUESTIONS ON STEWART'S [CHAP. v. LESSON XXVI. On the Relation between Heat and Me- chanical Energy. 455. Give instances of the conversion of mechanical energy into heat. In what case is this conversion unde- sirable, and what means are taken to avoid it ? 456. How did Joule conduct his experiments on the relation between mechanical energy and heat 1 457. What is the mechanical equivalent of heat as de- termined by Joule ? 458. What was the nature of Mayer's method of calcu- lating the mechanical equivalent of heat ? 459. If we drop a weight into a large quantity of ful- minating powder, the result is the generation of a large amount of heat ; are we at liberty to suppose that all this heat is the mechanical equivalent of the energy of the weight 1 460. What question similar to the above may be asked in the case of the compression of a gas, and what answer to it is supplied by Joule's experiments ? 461. Give the mechanical explanation of the fact that a gas suddenly expanded becomes cooled. 462. Explain, with the aid of a diagram, how the alternating motion of the piston in the cylinder of a steain- engiiie is produced through the agency of steam. 463. What are the chief differences between high- pressure and low-pressure engines ? 464. What is the general law for the conversion of heat into mechanical energy ? How is this law exempli- fied by low-pressure and high-pressure engines respec- tively ? 465. State Carnot's analogy between the mechanical capability of heat and that of water? 466. What is the absolute zero of temperature, and its value on the centigrade scale 1 467. Under what conditions would it be possible to convert all the heat which passes through a heat-engine into mechanical effect 1 LESS, xxvi.] ELEMENTARY PHYSICS. 41 468. Suppose that the higher temperature of a heat- engine is 100 C., and the lower C. ; what proportion of the whole heat carried through the engine may be con- verted into mechanical effect ? 469. What is the general rule for finding how much of the heat carried through a heat-engine can be utilized ? 470. What difference in general is there between the theoretical and the practical limit of utilization ? 471. What three kinds of heat-engines are in extensive use } 472. Give a sketch of the history of heat-engines prior to the time of Watt. 473. How was Watt's attention drawn to the subject, and what were the three great improvements which he effected ? 474. Explain the advantages of Watt's arrangement for the condensation of the steam. 475. Explain the principle of double action introduced by Watt. 476. Explain the mode of expansive working. 477. How is the rate at which an engine performs work usually expressed, and what is the unit employed for this purpose called 1 What is its numerical value in this country ] 478. On reviewing the relations of heat and mechanical effect, what important difference in their mutual converti- bility is apparent 1 42 QUESTIONS ON STEWART'S [CHAP. vi. CHAPTER VI. EADIANT ENERGY. LESSON XXVII. Preliminary. 479. What is the velocity with" which radiant energy is propagated ? 480. Mention phenomena which naturally lead to a division of radiation into non-luminous and luminous, or into rays of dark heat and rays of light. 481. Define Optics. 482. What are the two hypotheses respecting the na- ture of light ? Why has the new hypothesis the better claim to be regarded as true ? 483. Define the following terms : a ray of light, & pen- cil of rays, divergent pencil, convergent pencil, pencil of parallel rays. 484. Into what two distinct classes are substances di- vided with reference to their effect upon light 1 485. What is the effect of allowing light to fall on a very thin slice of an opaque substance '? What does this show as to the distinction between opaque and transpar- ent substances ? 486. What two important exceptions are there to the law that light moves in straight lines 1 487. Explain (with a diagram) how Romer was able to determine the velocity of light from the eclipses of Ju- piter's satellites. 488. Explain Fizeau's method of measuring the velo- city of light. 489. Prove that the quantity of light which a surface receives from any source will vary inversely as the square of its distance from the source. LESS, xxvui.] ELEMENTARY PHYSICS. 43 490. Show that the intensity of the illumination of a plate or screen is proportional to the cross-section which it presents to the direction of radiation. What familiar facts respecting the power of the sun's rays are accounted for by this law ? 491. Prove that the intrinsic brightness of a luminous body does not vary with its distance ; meaning by bright- ness the light that would reach the eye by looking at the body through a long narrow tube, and supposing the tube to be always so narrow, and the source of light always so large, that in looking through the tube we should see nothing else but this light. Consider also the case in which the luminous body is so distant as to appear simply a luminous point like a star. 492. Explain Bunsen's photometer, and the method of measuring the intensity of light by means of it. 493. Suppose that one light causes the grease spot to vanish in a Bunsen's photometer when placed at the dis- tance of one foot in front of the screen, and another light when placed at the distance of two feet ; what is the rela- tive luminosity of the two lights ? 494. Explain the distinction between the illuminating power of a source of light and the inherent brightness or quality of the light. LESSON XXVIII. Reflection of Light. 495. State the law of reflection. 496. How may the truth of this law be rendered visi- ble to the eye ? 497. What is meant by a virtual image 1 498. Prove by the aid of a figure that the image of a luminous point lies as far behind the reflecting surface as the luminous point itself lies before it. 499. Explain by a figure the mode of determining the positions of the various points in the image of a luminous body which is in front of a plane mirror. 44 QUESTIONS ON STEWART'S [CHAP. vi. 500. What peculiar inversion is there in the reflection of the human figure in a vertical mirror ? Also, in the reflection of letters written from left to right on a wall in front of the mirror ? 501. When a ray of light strikes a curved surface, how may the direction of the reflected ray be found ? 502. Suppose a pencil of parallel rays strikes a con- cave spherical mirror ; prove that the focus of the rays is the point half-way between the centre of the mirror and the middle point of its surface. What is this focus called ? (See page 229.) 503. Explain how a concave spherical mirror pro- duces a circular image of the sun. Will this image be real or virtual ? 504. Show by a diagram that the focus of divergent rays proceeding from a point near a concave spherical mirror lies between the principal focus and the centre of the mirror. 505. What are conjugate foci, and what is meant by saying that conjugate foci are interchangeable ? 506. Prove the formula which gives the relation be- tween the conjugate foci of a concave spherical mirror. 507. Show that a virtual image will be produced if the luminous point be nearer the concave mirror than the principal focus. 508. Examine the five different cases to w r hich the 112 general formula - -f- = - is applicable. 509. The images produced by concave mirrors are in their nature either real or virtual, in their position rela- tively to the object either erect or inverted, in size com- pared with the object either magnified or diminished. Ex- amine as regards these particulars the image of an object such as a straight line placed beyond the centre of the mir- ror. 510. Examine as regards the above particulars the im- age of an object placed between the principal focus and the concave mirror. LESS, xxix.] ELEMENTARY PHYSICS. 45 511. An object is placed immediately in front of a con- cave mirror, and then gradually removed to a great dis- tance along the axis of the mirror ; trace the changes in the distance of the image from the mirror, and also in the nature, position, and size of the image. 512. Explain, with a diagram, the effect produced by a parabolic mirror upon a pencil of parallel rays incident on its surface. What advantage do parabolic mirrors pos- sess as compared with spherical mirrors ? On the other hand, what disadvantage I 513. What is the nature of the images produced by convex spherical mirrors ? LESSON XXIX. Refraction of Light. 514. Illustrate, by a diagram, the refraction of light by the surface of a transparent medium like glass, and state the law of refraction in general terms. 515. Instead of introducing the sines of the angles of incidence and refraction in the statement of the law of refraction, how can the law be expressed in purely geomet- rical language ? 516. Explain the case in which the ray of light, instead of passing from vacuo into a transparent medium, passes out from the medium into vacuo. 517. Explain the case in which a ray of light strikes the surface of a medium at right angles. 518. Explain how the truth of the laws of refraction may be illustrated experimentally. 519. Explain total internal reflection. What is the criti- cal angle of a medium ? 520. Explain the mirage. 521. If n v n 2 be the absolute indices of refraction of two media respectively, and ri be the relative index of refrac- tion for the two media, prove that n = . HI 522. Trace, by means of a figure, the path of a ray of 46 QUESTIONS ON STEWART'S [CHAP. vi. light through a glass prism. What is the angle of devi- ation ? 523. What is the condition of minimum deviation in a prism 1 524. What is the condition of total internal reflection in a prism 1 525. Why can we not employ for ordinary optical pur- poses a glass prism, of which the angle is greater than 84? LESSON XXX. Lenses and other Optical Instruments. 526. Describe the shapes given to the lenses in common use, and give the names of the lenses 527. From the action of a prism on a ray of light de- rive a rule for determining whether a lens is converg- ing or diverging ; and apply this rule to the six lenses mentioned in the book. 528. Discuss the formula which expresses the relation between the conjugate foci of a double convex lens, exam- ining the different cases which arise as the luminous point is supposed to move along the axis of the lens from an in- finite distance up to the lens. The formula is - + -j- = ~, in which p, p are the conjugate foci, and / the principal focus of the lens. 529. Show, by a figure, how to find the position and size of the image of a luminous body formed by a double convex lens, the luminous body being supposed to be farther from the lens than the principal focus. What is the law which determines the size of the image compared with that of the object 1 530. Show that if a luminous body be placed between a double convex lens and its principal focus, the image will be virtual, erect, and magnified. 531. Describe the two sets of appearances which may be seen in looking through a double convex lens, and state the conditions under which they are produced. LESS, xxxi.] ELEMENTARY PHYSICS. 47 532. Describe the camera dbscura and its use. 533. Describe the eye, regarded as an optical instru- ment 534. What power of adjustment does the eye possess, and under what circumstances is this power called into action ? 535. When is a person said to be short-sighted, and when long-sighted ? What are the remedies for these de- fects, respectively ? 536. What is the principle of the simple microscope ? What condition must be answered in order that the virtual image which is formed may be distinct 1 537. What are the essential parts of a telescope ? Ex- plain by a figure how a telescope forms a virtual and mag- nified image of a distant object. 538. What is the optical difference between the simple microscope and the telescope ? LESSON XXXI. Dispersion of Light by the Prism. 539. What great discovery did Newton make as to the nature of white light ? 540. Explain the dispersion of light by a prism. Give the seven principal colors of the spectrum in the order of refrangibility. 541. Why is it of great importance, in experiments upon the decomposition of light by prisms, to make use of a very narrow slit 1 542. What is the method, employed in the spectroscope, of multiplying the dispersions of rays of different refran- gibilities 1 Describe briefly the spectroscope of Gassiot. 543. Explain an optical method of recombining the various constituents of white light. 544. Explain a mechanical method of combining the various colors of the spectrum so as to form white light. 48 QUESTIONS ON STEWART'S [CHAP. vi. LESSON XXXII. Thermo-Pile. 545. In what way can we compare together the in- tensity of a ray of light and a ray of dark heat, and ob- tain a true measure of the energy of these rays, provided instruments of sufficient delicacy be employed ? 546. Explain the principle of the thermo-pile discov- ered by Seebeck. 547. In order to obtain by the use of thermo-electricity a very delicate instrument wherewith to measure radiant heat, what three objects must be accomplished ? 548. How may a strong thermo-electric current be produced ? Illustrate with a figure. 54'9. Describe Thomson's galvanometer and its action, explaining in particular how the magnetic force is over- come, and how by optical arrangements any small motion of the needle is very much magnified. 550. Explain the construction of the thermo-pile. 551. Suppose that when a source of radiant heat is placed before the pile, the luminous slit is made to move on the screen through twenty divisions of the scale ; sup- pose, again, that when a different source of heat is pre- sented to the pile, the index moves over forty divisions ; what is the relation between the heating effects of the two sources ? What is the general law ? 552. Explain how, by the joint aid of the spectroscope and the pile, we are enabled to analyze a beam of sun- light so as to estimate the heating effects of all the differ- ent rays in the solar spectrum. What weak point is there in this method 1 553. What source of heat did Leslie employ in his ex- periments on dark heat, and what was one of the results at which he arrived ? 554. What are some of the facts established by Melloni with the aid of the pile ? What is diathermancy ? 555. Explain, by the aid of a figure, the manner in which Melloni performed his experiment to prove that dark heat is capable of refraction. LESS. XXXIIL] ELEMENTARY PHYSICS. 49 556. What two facts in reference to dark heat were es- tablished by Forbes ? 557. Give a sketch of the manner in which we may explore experimentally the heat spectrum produced by a heated strip of coal, for example. 558. Suppose we begin by heating a strip of carbon to a heat below redness, and producing a spectrum of the radiation from the carbon by means of a rock-salt prism ; trace the changes in this spectrum as the temperature of the carbon is gradually raised to a very high point. 559. What is the position upon the spectrum of actinic rays as they are called, and what power do they possess ? 560. Give a graphical representation of the sun's visi- ble spectrum, locating the primary colors, and drawing the curve of intensity of light. 561. Give a graphical representation of the entire solar spectrum, and draw the curve of intensity of heat. 562. Where is the maximum luminous effect in the solar spectrum, and where the maximum heating effect ? 563. Explain the statement " the spectrum of carbon is a continuous one" In general, what bodies give continuous spectra 1 564. In what respect do the spectra of gases differ from those of solid bodies 1 What is the spectrum of ignited sodium vapor 1 of thallium 1 565. What chromatic phenomena will be observed if we ignite a piece of metallic sodium in a dark room 1 566. In what way has electricity been found service- able in spectrum-analysis ? LESSON XXXIII. Radiation and Absorption. 567. What great and striking difference between the spectra of solids and those of incandescent gases was made known in the last Lesson ? 568. Explain how it may be shown by experiment that at comparatively low temperatures, say 100 C., a 3 D 50 QUESTIONS ON STEWART'S [CHAP. vi. lamp-black surface, or one of glass or white paper, radiates much more than a surface of polished silver. 569. Explain how the relative absorbing powers at 100 C. of the substances mentioned in the preceding question may be determined experimentally. 570. On comparing two tables, one containing the radiating and the other the absorbing powers of a series of substances, what general law comes to view ? 571. In what important respect do surfaces differ as regards their absorbing powers for different rays ? What instances of this can you give ? 572. Describe three experiments illustrative of the radiation from bodies of high temperature. What gen- eral relation between absorption and radiation do these experiments tend to establish ? 573. What is found to be the behavior of transparent colorless glass as regards absorption and radiation ; also, of a film or stratum of air ? 574. Give a generalization of the conclusions to be drawn from the preceding experiments. 575. Explain what is meant by selective or partial ab- sorption by describing the behavior of white paper, and also of the glass bulb of a thermometer, at different tem- peratures. 576. What is it that makes the leaves of plants appear green ? In general, what is the physical cause of color ? 577. What familiar illustration of selective absorption is afforded by colored glasses ? 578. Describe an experiment in proof of the law that bodies when cold absorb the same kind of rays that they give out when hot. 579. Describe another experiment in proof of the above law. 580. Describe a third experiment in proof of the same law. 581. Suppose that we introduce into a chamber, kept uniformly at a white heat, transparent glass, polished plat- inum, coal, and black and white porcelain ; arid that, LESS, xxxm.] ELEMENTARY PHYSICS. 51 after leaving them until they have acquired the tem- perature of the walls of the chamber, as a first experiment we simply examine them through a small hole ; finally, suppose that, as a second experiment, we hastily withdraw the substances, and, without allowing them time to cool, examine them in the dark ; what will be the appear- ances presented, in the two experiments, and how may they be reconciled with one another 1 582. If we introduce red and green glass into a white- hot chamber, and then view them through a small open- ing, they will appear to have entirely lost their color. Explain this. 583. Give the grounds upon which black bodies have been selected as the standard or typical radiators. 584. What simple method may be employed to ascer- tain whether or not one body is hotter or colder than another ? 585. Explain how, by the aid of the spectroscope, we may learn the chemical nature of a substance. 586. Show how spectrum analysis has demonstrated that there are present in the sun, in the state of vapor, various substances well known on the earth, as sodium, iron, zinc, magnesium, etc. 587. What conclusion may we draw from the results of Professor Tyndall's investigations into the absorption of various gases for dark heat ? 588. Explain the part which the aqueous vapor of the atmosphere plays in relation to the heating effect of the. sun upon the earth's surface. 589. Show how the laws of radiation explain the depo- sition of dew. 590. Give some examples of the phenomenon called phosphorescence. Also give an instance of the similar phe- nomenon known as fluorescence. 591. What is Professor Stokes's explanation of the phenomena of phosphorescence and of fluorescence ] What is really the only difference between the two phe- nomena ? 52 QUESTIONS ON STEWART'S [CHAP. vi. LESSON XXXIV. On the Nature of Radiant Energy. 592. What two hypotheses regarding the nature of light were propounded by Newton and Huyghens, respec- tively ? 593. What crucial test between these two hypotheses has been found ? 594. What striking analogy is there between light and sound which leads to the belief that light must be a mo- tion similar to sound, that is to say, undulatory ? 595. On the undulatory theory, how does the eye dis- tinguish between rays of different wave-lengths ? What analogy is there in this particular between light and sound 1 596. Illustrate what is meant by the front of a wave, and give a general definition of the same. 597. In what direction, relatively to its front, does a wave always proceed 1 598. Deduce the law of reflection from the undulatory theory of light. 599. Deduce the law of refraction from the undulatory theory of light. 600. In the undulatory theory, what does the index of refraction of a substance represent 1 601. What reason may be assigned why the velocity of light should be less in glass, for example, than in vacuo 1 602. What analogy serves to aid the mind in perceiving why reflection and refraction accompany each other when light falls on a polished glass surface, for example ? 603. What well-known facts respecting shadows might lead us to imagine that light differs from sound in a fun- damental respect 1 What is the cause of the difference between sound-shadows and light-shadows 1 604. Give instances of the manner in which the beauti- fully colored appearances due to the interference of light may be produced. What is the general explanation of these appearances according to the undulatory theory 1 LESS, xxxv.] ELEMENTARY PHYSICS. 53 605. Explain Newton's rings. 606. Explain the colors of thin plates, such as those of a soap-bubble. 607. State an apparent objection to the undulatory theory, derived from the laws of energy, and show how this objection is entirely removed. 608. Explain why it is, that, when a sounding body is approaching the ear, its note is rendered more acute, while if it be receding from the ear, its note becomes more grave. 609. Show how Mr. Huggins has been able to make out the proper motions of several stars in a direction to and from the eye. LESSON XXXV. Polarization of Light. Connection be- tween Radiant Energy and the other Forms of Energy. 610. What two kinds of wave-motion are met with in nature ? 611. Which kind of vibrations is capable of assuming a particular side or direction, and how may this fact be illustrated ? 612. What is the meaning of the term polarization ? 613. Give an illustration to show how a mixture of vertical and horizontal waves may be sifted, so to speak, and deprived of the vertical components of the waves, or of the horizontal components, or of both. 614. Describe the action of tourmaline upon light. 615. What is the only possible explanation of the phe- nomena which are observed ? To whom are we indebted for this explanation ] 616. How is polarization by reflection effected, and what is meant by saying that the light is then " polarized in the plane of reflection " 1 617. Show that an ordinary ray of light may be made to disappear entirely by two reflections. 618. Explain the double refraction of light by a crystal 54 QUESTIONS ON STEWART'S [CHAP. vi. of Iceland spar. What is the appearance of a small body as seen through a piece of Iceland spar ? 619. Explain the general connection between radiant energy, mechanical energy, and the energy of absorbed heat. LESS, xxxvi.] ELEMENTARY PHYSICS. 55 CHAPTER VII. ELECTRICAL SEPARATION. LESSON XXXVI. Development of Electricity. 620. Mention two leading facts in the early history of electricity. From what is the word derived ? 621. What marked difference exists between metal and glass as regards their power to conduct electricity ? By what terms do we express this difference ? 622. Give the tables of the most important conductors and insulators. What is the character of the transition from the one class of bodies to the other ? 623. Why is it very desirable to make all experiments on electricity in a dry atmosphere 1 624. Show by an experiment that there are two kinds of electricity, and give their names. When do electrified bodies attract each other, and when do they repel each other ? 625. Explain the hypothesis of two fluids. 626. Give a table of twelve common substances in the order of their relative capacity for positive electrification. 627. Mention other modes of developing electrical sep- aration besides friction. 628. What appears to be an essential condition for the production of electricity by the mutual action of two bodies 1 629. What general connection is there between electri- cal separation and energy or mechanical work ? 630. Describe the electrical properties of tourmaline. What species of energy is spent in this case to produce the electrical separation 1 56 QUESTIONS ON STEWART'S [CHAP. vn. LESSON XXXVII. Measurement of Electricity. 631. Show how an electrical charge upon a metallic body can be subdivided. 632. Describe Coulomb's torsion-balance, and the experi- ments which demonstrate the law of electrical action be- tween two bodies, so far as it depends on the distance of the bodies from each other. 633. Explain how, by means of Coulomb's torsion-bal- ance, we may prove the law of action between two electri- fied bodies, so far as it depends on the quantities of electricity upon the bodies. 634. What is a convenient unit of electrical force 1 Find in terms of this unit the force exercised by 6 units of positive upon 4 units of negative electricity at the dis- tance 3. 635. Show, by an experiment, that electricity mani- fests itself only on the surface of bodies, and give the explanation of this fact. 636. In certain countries electrical manifestations are often produced by combing the hair, rubbing a silk dress, etc., while they are not observed in other parts of the world. How do you account for this ? 637. In what way does a charge of electricity distribute itself on a sphere 1 on a pointed conductor ? What accounts for the distribution in each case ? 638. Define the term electric density. Describe a body such that the electric density will be much greater at some parts than at others. 639. Show how the relative distribution of electricity over the surface of a body may be ascertained by means of the proof-plane. LESSON XXXVIII. Ekctrical Induction. 640. What will happen if we bring near together two insulated conductors, one charged with electricity, and the other not charged ? What is this kind of action called? LESS, xxxix.] ELEMENTARY PHYSICS. 57 641. Suppose tfcat the neutral conductor in the pre- ceding question be divided into two parts, what will be the electrical condition of each part 1 How may this fact be proved by experiment 1 642. Suppose that we slowly bring a conductor, charged with electricity, towards another conductor not charged, until they are very near each other ; explain the phe- nomena which will take place. 643. How may it be rendered evident that the induc- tive effect of electricity depends on the distance between the two conductors ? 644. What new light does electrical induction throw upon the fact that electricity only shows itself at the sur- faces of bodies 1 645. What important fact was discovered by Faraday in his researches upon electrical induction ? What is the inductive capacity of a substance 2 LESSON XXXIX. Electrical Machines, etc. 646. Of what two parts is every electrical machine composed ? 647. Describe the plate electrical machine, and explain its action. 648. Describe the simple experiments with an electri- caL machine which may be performed, 1. by holding the finger near the charged conductor ; 2. by placing an individual on an insulating stool ; and give the explanation of the experiments according to the two-fluid theory. 649. Describe the electrophorus, and explain its action. 650. Describe the gold-leaf electroscope, and explain how it enables us not only to detect the presence of electricity, but also to determine whether the electricity is positive or negative. 651. What difference between an electroscope and an electrometer is indicated by the derivation of the two words themselves ? 58 QUESTIONS ON STEWART'S [CHAP. vii. 652. Explain the method of measuring electrical charges employed by Sir W. Thomson in his electrom- eters. 653. Explain the accumulation of electricity by con- densers. If the condensing plates are separated, the pith- balls attached to them will diverge ; explain this. 654. Describe the Ley den jar. Show how it may be charged and discharged, and explain its mode of action. 655. If a Leyden jar be allowed to stand for a short time after being discharged, it is found that it has a small residual charge left in it ; what is the probable explana- tion of this ? 656. What is an electric battery, and how formed from its component parts ? 657. What knowledge of the nature of the electric spark has been obtained by viewing it through the spec- troscope, and what use has been made of this knowledge ? 658. What transmutation of energy do we have in the electric spark ? 659. Investigate the relation between the charge of a Leyden jar and the amount of heat produced by discharg- ing the jar, showing that the whole heating effect will be proportional to the square of the quantity of electricity divided by the surface of the jar. 660. How can it be shown, experimentally, that the duration of the electric spark is exceedingly short ? 661. How has Sir C. Wheatstone succeeded in measur- ing the duration of the electric spark ? What was the re- sult of his experiments ] What did he also find to be the velocity of electricity 1 662. Who first proved that lightning is only a manifes- tation of electricity on a large scale, and in what way did he prove that this is the case 1 What advantage has been taken of this knowledge ? 663. To what are the following phenomena due ? 1. The light which constitutes the electric flash. 2. The noise which accompanies the same. 3. Its destructive effect in rending substances. LESS, xxxix.] ELEMENTARY PHYSICS. 59 664. If we bring a hollow insulated brass ball near an electric machine in action, we shall get a spark, but it will be very feeble. If, however, we touch with our fin- ger that part of the conductor which is farthest from the machine, or make a connection between this conductor and the ground, the spark from the machine will be much more intense. Explain this. 665. Show how to obtain from an insulated conductor, near an electric machine in action, 1. a series of sparks or shocks ; 2. a continuous rush of electricity. 666. Explain the efficacy of lightning-conductors. 667. Discuss briefly the connection between electrical separation and the other forms of energy. 60 QUESTIONS ON STEWART'S [CHAP. vm. CHAPTER VIII. ELECTRICITY IN MOTION. LESSON XL. Magnetism. 668. What is the origin of the term Magnet ? 669. Describe some of the properties of a magnet. What are its poles, and how are they distinguished from each other? 670. What is the difference between magnetic and dia- magnetic bodies ? Enumerate the most important bodies of each class. In what respect does iron stand alone ? 671. Explain the behavior of magnetic and diamagnetic bodies when suspended midway between the two poles of a powerful magnet. 672. Explain the behavior of magnetic and diamag- netic bodies when suspended between the poles of a mag- net in a magnetic liquid instead of in air. 673. What is the law of the mutual action of magnetic poles ? 674. State the quantitative law of force in magnetic attractions and repulsions. By whom was this law dis- covered ? 675. Prove that, in consequence of this law, if we sus- pend a small magnet by a thread and cause it to approach the pole of a powerful magnet, the small magnet will exhibit a tendency to rush bodily to the large magnet ; and find the measure of this tendency. 676. Explain magnetic induction. 677. Describe the effect of breaking a magnet, and give a theory of the distribution of the magnetic fluids which will explain the properties of magnets, both when entire and when broken. 678. What difference is there between soft iron and LESS. XLI.] ELEMENTARY PHYSICS. 61 hard steel as regards susceptibility to magnetism 1 Ex- plain one mode of magnetizing a steel bar. What is the effect of heat on magnets 1 679. If we were to suspend a magnetic needle in such a manner that it was perfectly free to move in any direc- tion, how would it place itself ? 680. What are magnetic meridians ? 681. What facts are stated as showing that a magnetic needle will not everywhere and always point as it does in Great Britain at the present moment 1 682. At what places on the earth's surface is a mag- netic needle of no use to the mariner, and why 1 683. Explain why the effect of the earth's magnetism Upon a magnetic needle is merely directive. LESSON XLI. Voltaic Batteries. 684. What was the famous phenomenon first observed by Galvani in 1786, and how was it explained by Galvani and by Volta respectively ? 685. Explain the construction and mode of action of Voita's pile. 686. Describe the arrangement known as Volta? s crown of cups. 687. How did Volta explain the effect produced by the voltaic battery ? 688. Illustrate the manner in which the total effect produced by Volta's pile depends on the number of ele- ments in the pile. 689. Explain in what way the contact theory as held by Volta is inconsistent with the laws of energy. 690. What is the chemical theory of the action of the voltaic battery ? 691. What was the nature of the crucial experiment made by Sir W. Thomson in reference to the two theories of the voltaic battery, and what conclusions are to be drawn from it 1 62 QUESTIONS ON STEWARTS [CHAP. vin. 692. What is denoted by the term electromotive force ? 693. What results did Sir C. Wheatstone obtain in his experiments on the electromotive force in different com- binations of platinum, zinc, and potassium, and what gen- eral law do they illustrate 'I 694. Classify the metals according to their order in the electromotive series. 695. What two causes greatly enfeeble a single-liquid Battery after it has been in action a short time ? 696. Describe Daniell's constant battery, and its mode of action. 697. What are the advantages of amalgamating the zinc plates ? 6 9 a. Describe Grove's battery, and its mode of action. 699. How may the existence of a thermo-electric cur- rent be easily demonstrated 1 Why are the metals bis- muth and antimony generally used in thermo-electric combinations ? 700. Illustrate the application of the law of Art. 374 in thermo-electric combinations. 701. Within certain limits what is the strength of a thermo-electric current proportional to ? But what has Gumming shown in the case of copper and iron ? LESSON XLII. Effect of the Electric Current upon a Magnet. 702. When, and by whom, was the important dis- covery of the connection between an electric current and a magnet made 1 703. Explain the nature of Oersted's experiment. 704. State the rule which expresses the relation be- tween the behavior of the needle and the position and direction of the current, and apply this rule to the four distinct cases which are possible. 705. What is the object of a galvanometer ? Explain the construction and mode of action of a single-needle galvanometer. LESS. XLIIL] ELEMENTARY PHYSICS. 63 706. Explain the construction and mode of action of an astatic galvanometer. Describe the mirror arrangement for increasing the sensibility of a galvanometer. 707. Upon what law does the action of a current on a needle depend ? 708. Describe the tangent compass and its action. Why has the instrument received this name ? 709. Describe an electro-magnet. How do they com- pare in strength with natural magnets 1 710. What curious facts have been observed in the magnetization of soft iron bars ? 711. State the principle of the electric telegraph. What takes the place of a return wire in electric tele- graphs, and what advantages are gained by this substitu- tion ? LESSON XLIIL Action of Currents on One Another, and Action of Magnets on Currents. 712. What are the chief laws of the mutual action of electrical currents ? 713. Discuss the various cases which may arise under Law III. 714. Explain a case in which a continuous rotation of currents is produced by their mutual action. 715. Examine the action of the earth's magnetism upon a circular vertical current which is free to place itself in any position. 716. Explain the construction and behavior of a sole- noid. 717. State Ampere's hypothesis concerning nlagnetism, and show that it explains the known relations between magnets arid currents, and between magnets and magnets. 718. What instance of the conservation of energy do we have in the case of two similar voltaic batteries, each charged with the same amount of zinc, if one battery is made to do external work, while the other does no exter- nal work at all ? 64 QUESTIONS ON STEWARTS [CHAP. vm. LESSON XLIV. Induction of Currents. 719. State the laws of the induction of electric cur- rents. Who discovered current induction ? 720. Explain how magnets may be made to play the part of currents in the phenomena of induction. 721. Show that the phenomena of induction are in harmony with the laws of energy. 722. Describe a method, employed by Joule, for con- verting mechanical energy into that of induced currents, and from that into heat. 723. What two kinds of electrical machines are there which depend for their action on the laws of induction 1 724. What is the principle of a magneto-electrical machine ? What arrangement is employed in Clark's ma- chine ? What is the object of a commutator ? For w r hat purposes, among others, are these machines used ? 725. Describe RuhmJcorff's coil, and explain its mode of action. LESSON XLV. Distribution and Movement of Electricity in a Voltaic Battery. 726. Who first developed the laws regulating the mo- tion and distribution of electricity in a battery ? 727. State the laws which regulate the electro-motive force of a battery. 728. Investigate the subject of electrical resistance in a manner similar to that employed in studying thermal conductivity, and deduce Ohm's formula for expressing the relation between the intensity of the current, the elec- tromotive force, and the resistance of a galvanic circuit. 729. Upon what three things does the electrical resist- ance of a substance depend ? 730. Modify the fundamental formula of Ohm so as to express the intensity of the current in a battery of ten cells with a definite external resistance. LESS. XLVI.] ELEMENTARY PHYSICS. 65 731. Examine, by means of Ohm's formula, the effect of increasing the number of cells in a battery ; 1. when there is 110 external resistance ; 2. when the external resistance is small compared with the internal ; 3. when the external resistance is large compared with the internal. 732. Why is it necessary to have a large number of cells in order to produce the electric light ? 733. Why is it advantageous to multiply the number of couples in a thermo-electric current 1 734. Explain, by the aid of Ohm's formula, the effect of increasing the size of the plates in a voltaic battery. 735. What arrangement in a battery is preferable, when the battery is to be used to produce thermal effects ? Why? 736. What law, as to the intensity of the current in different portions of a circuit, is likewise due to Ohm ? 737. Explain a method of comparing the resistance (and hence the conductivity) of metallic wires by means of a galvanometer. 738. What points of resemblance have been observed between the electric and the thermal conductivities of substances ? LESSON XLYI. Effects of the Electric Current. 739. Compare, as regards quantity, tension and the resultant physiological effects, the Ley den jar battery, the voltaic battery, and a flash of lightning, 740. When an electric current is made to pass through a circuit, to what is the heating effect of the current pro- portional ? 741. Show that the increase of temperature produced by the passage of the same quantity of electricity through a wire will vary inversely as the square of the cross sec- tion of the wire. 66 QUESTIONS ON STEWART'S [CHAP. vin. 742. Deduce from the above law, that the heat gener- ated in a given time is proportional to the square of the intensity of the current. 743. If one part of a circuit be composed of a metre of silver wire two square millimetres in cross section, and another of five metres of zine wire four square millimetres in cross section, show that the relative heating effects of the current on these two wires will be as 1 : 8-62. Specific electric conductivity of silver, 100 ; of zinc, 29. 744. Looking at the subject from the stand-point of the doctrine of energy, in what consists the difference be- tween dissolving zinc by acid in an ordinary vessel and doing so by the voltaic arrangement 1 745* Describe how the electric light is produced. 746. Define the terms electrolysis, electrolyte. 747. Describe a voltaic arrangement which may be employed to decompose water. 748. What is the distinction between electro-positive and electro-negative elements ? 749. In the electrolytic decomposition of water, for example, the question naturally arises, Is the oxygen of each molecule which is decomposed carried bodily to the one pole, and the hydrogen to the other ] What was Davy's test experiment upon this point ? 750. Explain Grotthuss's hypothesis. 751. State the laws of electrolytic action discovered by Faraday. 752. Explain the principle of the electrotype process. 753. What is the effect of passing polarized light through glass subjected to the action of a powerful electro- magnet ? 754. Under what conditions are peculiar stratifications of light and colors produced by the current ? 755. Will a current pass through a perfect vacuum ? 756. What is the cause of the peculiar smell which is often noticed when an electric machine is in. action 1 LESS. XLVII.] ELEMENTARY PHYSICS. 67 CHAPTER IX. ENERGY OF CHEMICAL SEPARATION. LESSON XLVII. Concluding Remarks. 757. Why is it natural to expect that a definite amount of carbon will, when burnt, always furnish a definite amount of heat ? 758. Who have investigated the quantity of heat given out in chemical combination ? 759. To what general result was Andrew led by study- ing the heat given out during the mutual action of metals 1 760. What grounds are there for believing that the electro-motive forces are really those which cause heat when chemical combination takes place ? 761. What relation have we found to exist between the doctrine of the conservation of energy and the chimera of perpetual motion ? 762. In what way might a champion of perpetual mo- tion assent to the doctrine of the conservation of energy without absolutely giving up his cause 1 763. Give the outlines of the doctrine of the dissipation of energy, and show what bearing this doctrine has on the problem of perpetual motion. 764. Trace back the energy of our system through its various transmutations to its ultimate source. 765. What vast store of energy was provided by Nature in geological ages ? 766. Show that water-power and wind-power are really products of the sun's rays. 767. What single small exception is there to the state- ment that " all the work done in the world is due to the sun"? 68 QUESTIONS ON PHYSICS. [CHAP. ix. 768. What would seem to be the answer which we must give to the question. Will the sun last forever ? 769. In tine, to what ultimate conclusion does the prin- ciple of degradation conduct us 1 770. Enumerate the various kinds of energy which have been studied in this book. 771. Recapitulate various instances of the transmuta- tion of visible kinetic energy. 772. Give examples of the conversion of visible poten- tial energy. 773. Enumerate the instances of the transmutation of heat. 774. Give instances of the transmutation of radiant energy. 775. Mention examples of the transformation of the energy of electrical separation. 776. Give examples of the conversion of the energy of electricity in motion. 777. Give instances of the transmutation of the energy of chemical separation. PART II. EXERCISES AND PROBLEMS. THE Exercises in large type are, in the main, direct and simple appli- cations of, or deductions from, the principles of the text-book : in the cases in which special difficulties might arise or in which new definitions are introduced, hints or explanations will be found in Part III. (Answers and Solutions). These Exercises demand only a fair knowledge of the Elements of Arithmetic, Algebra, and Plane Geometry. The Exercises printed in smaller type are intended to be more difficult than the others, and some of them involve principles which are not ex- plicitly stated in the text-book. They are designed chiefly for use with advanced sections or as voluntary exercises. Those who wish to take them should have an elementary knowledge of Plane Trigonometry and of Analytic Geometry. In a few cases a knowledge of the Calculus may perhaps be serviceable, although it is not required. A summary of mathematical data and formulae is given in Appendix IV. When aid is required, it must be obtained from competent teachers, or from books. Appended is a list of elementary works which may be consulted with advantage, particularly the first two and the last two. THOMSON AND TAIT'S Elements of Natural Philosophy, Part I. (Lon- don & New York : McMillan & Co.) KERR'S Rational Mechanics. (Glasgow: W. Hamilton.) GOODWIN'S Elementary Statics, and Elementary Dynamics. (Cambridge, England : Deighton, Bell, & Co.) BESANT'S Elementary Hydrostatics. (Cambridge, England : Deighton, Bell, & Co.) HAUGHTON'S Manual of Mechanics. (London & New York : Cassell, Fetter, & Galpin.) TODHUNTEB'S Mechanics For Beginners. (London & New York : McMil- lan & Co.) GOODBYE'S Principles of Mechanics. (London: Longmans, Green, &Co.) BURAT, Precis de Mecanique. (Paris : Victor Masson et Fils.) BRIOT, Lecons de Mecanique. (Paris : Dunod, Editeur.) BRESSE ET ANDRE, Cours de Physique, les 2 premier fascicules. (Paris : Dunod, Editeur.) DESCHANNEL'S Natural Philosophy, Translated by EVERETT, Part I. (New York ; D. Appleton & Co.) 70 ELEMENTARY PHYSICS. [CHAP. I. INTRODUCTION. 1. Give an illustration, not mentioned by the author, of relative motion. 2. Give an illustration, not mentioned by the author, of force producing motion ; also, of force stopping motion. 3. Give an example of forces in equilibrium. 4. If you are running towards the North, and, as sud- denly' as possible, change the direction of your motion from the North to the East, do you think that force is expended in producing this change ? 5. Can you give an instance of a body which is not acted upon by any force whatever 1 6. Mention an object which is known to us through the medium of a single sense ; also, an object which is known through the medium of more than one sense. 7. Explain and illustrate the distinction between a phenomenon and a law of Nature. 8. What distinction can you draw between a body and a sub- stance. 9. Two steamers are moving with equal velocities in the same direction. A passenger on one steamer looks at the other from his state-room window ; how will it appear to him ? Suppose the other steamer suddenly appears to change its velocity ; in what two ways might this phenomenon be produced ? LESS, i.] EXERCISES AND PROBLEMS. 71 CHAPTER I. LAWS OF MOTION. LESSON I. Determination of Units. 10. How many square feet are there in 124 acres ? 11. How many square decimetres are there in 124 ares ? 12. Reduce 346768595 cubic inches to cubic yards. 13. Reduce 346768595 cubic centimetres to cubic metres. 14. How many litres are there in 2 steres 1 15. Reduce 6,000,000 grammes to tonnes. 16. Reduce 218*75 grains to grammes. 17. What ratio exists between a cubic centimetre and a cubic metre 1 18. What is the weight of 64 litres of water 1 of 64 cubic centimetres of water ? 19. Show that the number which expresses the volume in litres of a quantity of water also denotes the mass in kilogrammes. Examine also the case in which the volume of the water is expressed in cubic centimetres. 20. A rectangular trough is 12 metres long, 2 metres wide, and 80 centimetres deep. How many kilogrammes of water will it hold 1 21. Define density. What is the numerical measure of the density of a substance ? 22. Prove tkat the mass of a body is equal to the pro- duct of its volume and its density ; or, if V denotes the volume, D the density, and M the mass, that M VD. 23. Why is the density of water equal to unity in the Metric System ? 24. Find the mass of 74 litres of cork (density of cork, 0.24). 72 ELEMENTARY PHYSICS. [CHAP. i. 25. Prove that the densities of two bodies are propor- tional to the masses of equal volumes of the bodies. 26. Explain the distinction between mass and weight. 27. A ship sails 504 miles in a week. Find the average velocity in miles per hour. 28. Compare the velocities of two points which move uniformly, one through 5 feet in half a second, the other through 100 yards in a minute. 29. The daily rotation of the earth is uniform. Taking its circumference as 25,000 miles, determine the velocity of a point on the equator. 30. A body is moving with a velocity of 30 feet per second. With what velocity must another body move, which starts from a given point 3 minutes after the former and overtakes it in 10 minutes ? 31. For 6 seconds a body moves with a velocity of 10, and for the next 9 seconds with a velocity of 15. What uniform velocity would have carried it over the same space in the same time ? 32. Compare the velocities of two points, one of which moves uniformly around the circumference of a circle in the same time that the other moves along the diameter. 33. The height of a cylindrical cistern is 12 metres and its di- ameter is 6*5 metres. How many kilogrammes of water will it hold ? 34. One litre of a substance weighs 280 grammes, and a piece of another substance twice, as dense as the first weighs 400 grammes. Find the volume of the second substance. 35. Find the ratio of the kilometre to the nautical mile or knot. 36. Show that the proper measure of density is the mass of unit of volume. 37. If the unit of mass be increased a times, and the unit of volume be increased b times, how will the measure of density he altered ? 38. A cylindrical log of wood, a metres long, and b centimetres in diameter, weighs c kilogrammes. Compare its density with that of a substance the density of which is known to be d. 39. What are the dimensions of velocity in terms of the funda- mental units of length and time. 40. Define angular velocity. What is the unit of angular ve- locity, and what are its dimensions in terms of the fundamental units ? LESS, ii.] EXERCISES AND PROBLEMS. 73 41. Find the linear velocity with which a point must move on the circumference of a circle in order to describe one unit of angu- lar velocity per second. 42. What is the measure of the angular velocity of the hour hand of a clock ? of the minute hand '( 4-3. Prove that the linear and angular velocities of a point mov- ing on the circumference of a circle are connected by the equation, v r co, in which v denotes linear velocity, r radius of circle, and co angular velocity. 44. Two bodies begin to move uniformly at the same time along the same line, the first from a point A with a velocity v, the second from a point B with a velocity v f : (1) How far apart will they be at the end of t seconds ? (2) When will they be together ? (3) How far will they be from A when they are together ? 45. If a velocity be expressed by 6 when one second is taken as the unit of time, what would be its measure if one minute were taken as the unit of time ? 46. If v denotes a velocity in the metre-second system, prove that the same velocity will be denoted by , in a system in which the unit of length is m metres and the unit of time n seconds. 47. Prove that if two points move uniformly with any velocities in fixed directions, the line joining the points will always remain parallel to itself. LESSON II. First Law of Motion. 48. When we find a body moving uniformly and in one constant direction, what may we infer with regard to the total force that is acting upon the body ? 49. Illustrate the principle of Inertia by reference to the condition in which a person finds himself when stand- ing in a boat at starting or stopping. 50. Account for the practical rule which habit teaches us to observe in jumping from a carriage which is in mo- tion. 51. Serious accidents have sometimes happened by car- riages oversetting when moving along a sharp curve in 74 ELEMENTARY PHYSICS. [CHAP. I. the road. Explain the cause of these accidents, and show how they might have been prevented. 52. Show that the First Law of Motion contains the convention universally adopted for the measurement of Time. 53. What definition of Force does the First Law of Motion give us? 54. Review briefly the evidence in favor of the truth of the First Law of Motion. LESSON III. Second Law of Motion. Motion produced by Gravity. Kinematics. [In the problems upon the motion produced by gravity the resistance of the air is neglected, and g is to be taken as, equal to 32 '2 feet, or 9 '8 metres.] 55. Give an additional illustration of the action of a single force on a moving body. 56. State in general terms the rule for compounding two simultaneous motions or velocities in different direc- tions, a rule or proposition known as the Parallelogram of Velocities, and give a demonstration of the same. 57. If a man is rowing a boat directly across a river two miles wide at the rate of four miles an hour, and the current at the same time is taking the boat down stream at the rate of three miles an hour, find, (1) In what direction the boat will move ; (2) How far it will have gone when it reaches the opposite bank ; (3) How far the landing-place will be from the point directly opposite the starting-place ; How long the boat is in motion ; How long it would have taken to cross the river if there had been no current. 58. Explain the geometrical method of finding a single velocity equivalent to any number of simultaneous veloci- ties. LESS, in.] EXERCISES AND PROBLEMS. 75 59. Explain the geometrical method of resolving a ve- locity in a given direction into two component velocities in any given directions. 60. Prove that if velocities represented by the sides of a triangle taken in the same order be impressed simultane- ously upon a point it will remain at rest. 61. Prove that the resultant of velocities represented by the sides of any closed polygon whatever, taken all in the same_prder, is zero. 62. Show that the value,, or resolved part, or effective component of a known velocity, estimated along a given line, is the projection of the line representing the velocity upon the given line. Examine the case in which the given line makes a right angle with the line representing the velocity. 63. Suppose that six forces act simultaneously upon a body, such that separately they would impart to it the fol- lowing velocities : 4 feet per second towards the East, North, tt a West, u tt " South. Find the magnitude and direction of the resultant ve- locity. 64. If a cannon-ball were discharged from the rear end of an express-train, directly along the track, at the same rate as the train is moving forwards, what would be the motion of the ball relative to the ground 'I 65. Expose the fallacy in the following specimen of erroneous mechanical reasoning : "Let the ball be thrown upwards from the mast-head of a stationary ship, and it will fall back to the mast-head, and pass downwards to the foot of the mast. The same result would follow if the ball were thrown upwards from the mouth of a mine, or the top of a tower, on a stationary earth. Now put the 76 ELEMENTARY PHYSICS. [CHAP. i. ship in motion, and let the ball be thrown upwards. It will, as in the first instance, partake of the two motions, the upward or vertical A C, and the horizontal A B, as shown in Fig. 47 ; but because the two motions act conjointly, the ball will take the diagonal direction A D. By the time the ball has arrived at D, the ship will have reached the position B; and now, as the two forces will have been expended, the ball will begin to fall, by the force of gravity alone, in the vertical direction D B II ; but during its fall towards H, the ship will have passed on to the position S, leaving the ball at H, a given distance behind it." * 66. Bishop Wilkins, an English divine of the 17th century, and author of a Treatise on the Art of Flying, proposed the following " new and easy way of travelling." A large balloon was to be constructed and provided with apparatus to work against the varying currents of the air. The balloon, having been allowed to ascend to a conven- ient height, was to be kept practically at rest by working the apparatus, while the earth revolved beneath it ; and. when the desired locality came in view, those in the bal- * Earth Not a Globe, by " Parallax," pp. 64, 65. London : John B. Day. 1873. LESS, in.] EXERCISES AND PROBLEMS. 77 loon were to let out gas and drop down at once to the earth's surface. In this way New York, for example, would be reached from London in a few hours, or rather New York would reach the balloon at the rate of more than 700 miles an hour. Show the futility of any such method of travelling. 67. Define acceleration, uniform acceleration, variable acceleration. How is uniform acceleration measured ? 68. Find the average acceleration of a point the ve- locity of which increases from ten miles per hour to sixty miles per hour in two hours. If at the end of the first hour the velocity is fifty miles per hour, find the average acceleration during each hour. 69. A constant force acting thirteen seconds produces a velocity of four miles per hour. Find the accelera- tion. 70. A body falls to the ground from rest in 6 seconds ; find the space passed over. 71. Through what space will a body fall in the ninth second of its descent ? 72. A stone strikes the ground with a velocity of 98 metres per second. Find the height fallen through. 73. How long must % a body fall to acquire a velocity of 322, feet per second ? r 74. -A rocket begins to ascend vertically with a velocity of 161 feet per second. How high will it rise, and what time will be ocoupied in both ascent and descent 1 " 75. Deduce and explain the general formula for deter- mining the final velocity v of a body which, having an initial velocity a, is acted on by gravity for t seconds, viz. : v a + g t. 76. A body is thrown downward with a velocity of 160 feet per second ; find its velocity at the end of five seconds. 77. A body is thrown upward with a velocity of 49 metres per second ; with what velocity and in what di- rection will it be moving at the end of 7 seconds 1 78 ELEMENTARY PHYSICS. [CHAP. r. 78. If a body have an initial velocity , prove that the space passed over in t seconds under the action of gravity is given by the equation, s = at + \gt\ 79. From the formulae of Exercises 75 and 78 deduce the following, v' 2 = 2g-s + a 2 s = ^ (v + a) t 80. Prove that the time of ascent of a body projected vertically upward with the velocity a is ?, and that the 9 height ascended is 81. Prove that the times of ascent and of descent of a body thrown upward are the same. 82. Prove that if a body is projected vertically upward it returns to the ground with the same velocity as that with which it was projected. 83. A cannon-ball, fired vertically upward, returned to the ground in 20 seconds ; find the height ascended and the velocity of projection. 84. With what velocity must a stone be thrown down a well 100 metres deep, in order that it may reach the bottom of the well in one second ? 85. Analyze geometrically the action of the wind on the sails of a vessel, and explain how it is possible for a vessel to sail nearly against the wind. 86. Explain the action of the wind on a kite. 87. A man can row a boat at a certain rate, and the current of a river is flowing at a certain rate ; find the direction in which the boat should be steered in order that it may be rowed directly across the river. 88. Solve the problem of compounding two velocities by the trigonometrical method. Let u and v denote the velocities, w their resultant, $, a, ft, the angles between the direction of u and v, u and w, v and w, respectively ; then the problem is, given u, v, and , prove that, LESS, in.] EXERCISES AND PROBLEMS. 79 w 2 = u* + v z + 2 u v cos <. = sin <. w sin 8 = !i sin 0. 89. Deduce formulae from those in the preceding problem for resolving a given velocity (w) into two velocities (w, IT), making any assigned angles (a, /3) with the given velocity. 90. The value of two velocities are 36 and 60, and the angle between their directions is 54 ; find the resultant velocity, and the angles which its direction makes with those of the given ve- locities. 91. Two equal velocities are simultaneously impressed upon a body, one towards the north, the other towards the east. The resultant velocity is equal to 10. Find the two-velocities, and the direction in which the body will move. 92. Explain, on kiuematical principles, why the northern trade- wind appears to blow from a northeasterly direction, and the southern trade-wind from a southeasterly direction. 93. A person travelling due east at the rate of 4 miles an hour observes that the wind seems to blow directly from the south ; and that, on doubling his speed, it appears to blow from the southeast. Find the velocity and the direction of the wind. 94. A deer is running at the rate of 20 miles an hour, and a sportsman fires at him when he is at the nearest point, 200 yards distant ; what allowance should be made in taking aim, 'supposing the velocity of the rifle bullet to be 1000 feet per second ? 95. A particle descends vertically T&long the axis of a tube which at the same time is carried forward in the horizontal direc- tion, both motions being uniform ; find the inclination of the tube from the vertical line. 96. Show that, in the case of variable velocity, the equation v = (which expresses the definition of the average velocity for any time) is more and more nearly true as the interval of time is taken smaller and smaller. Thence obtain the true measure of variable velocity. 97. Prove that the dimensions of acceleration are - . 98. If g denote' an acceleration when the second is the unit of time and the foot is the unit of length, then, if we take m seconds as the unit of time and n feet as the unit of length, the same ac~ celeration will be denoted by 2!L g. n 80 ELEMENTARY PHYSICS. [CHAP. i. 99. Find the measure of the acceleration of gravity when one minute is taken as the unit of time. 100. A stone is thrown vertically upward with a velocity 3 g ; find at what time its height will be 4 g, and its velocity at this time. 101. A stone is dropped into a well, and is heard to strike the surface of the water after 4*5 seconds ; find the distance to the surface of the water, knowing that the velocity of sound is 340 metres per second. 102. A body is dropped from a height of 100 feet, and at the same moment another body is projected vertically from the ground: they meet half-way. What was the velocity of projection of the second body ? 103. A body is projected vertically with a velocity of 30 metres per second. A second later another body is projected vertically from the same point with a velocity of 40 metres per second. When and where will the two meet ? 104. With what velocity must a body be projected downwards that in n seconds it may overtake another body which has already fallen from the same point through a distance of a feet ? 105. Prove that, when we take into account the resistance of the air, the time of ascent of a body projected vertically upward is less than the time of descent ; and that the velocity on reaching the ground is less than the velocity on starting. 106. Prove that the velocity acquired in sliding down a smooth inclined plane is the same that would be acquired in falling freely through the vertical height of the plane. 107. Prove that the time of falling from rest down any chord of a vertical circle, drawn either from the highest or the lowest point of the circle, is constant. 108. Find the straight line of quickest descent from a given point to a given straight line. 109. If a be the base of an inclined plane, find the height in order that the time of descent may be a minimum. PROJECTILES. 110. Prove that a body projected in any direction not vertical, and acted on by gravity, will describe a parabola. 111. Find the velocity at any point of the path of a projec- tile. 112. Determine the position of the focus of the parabola de- scribed by a projectile. 113. Determine the latus rectum of the parabola described by a projectile. LKSS. IIL] EXERCISES AND PROBLEMS. 81 114. Find the maximum height reached by a projectile. 115. Find the whole time of flight of a projectile, and show that the times of ascent and of descent are equal. 116. Determine the range of a projectile on a horizontal plane - through the point of projection. 117. Prove that the maximum range along a horizontal plane for a given velocity of projection is obtained by making the angle of projection equal to 45, and is equal to the square of the velocity divided by g. 118. Find the horizontal range of a shell fired at an angle of 45 with a velocity of 500 feet per second. 119. Two bodies are projected simultaneously from the same point with different velocities and in different directions ; find their distance apart at the end of a given time. 120. Find the velocity and direction of projection in order that a projectile may pass horizontally through a given point. 121. Find the velocity with which a body must be projected in a given direction from the top of a tower so as to strike the ground at a given point. 122. A shell is to be fired from the top of a cliff 300 feet high with a velocity of 600 feet per second, so as to strike a ship at anchor 600 yards from the base of the cliff. What must be the elevation of the gun ? 123. Two bodies are projected from the top of a tower at the same instant, the one vertically upward with a velocity of 100 feet per second, and the other horizontally with a velocity of 60 feet per second ; find their distance apart at the end of 2 seconds. CURVILINEAR MOTION. 124. Taking acceleration in its expanded sense, viz., rate of change of velocity whether the change takes place in tlie direction of motion or not, illustrate what is meant by change of velocity, and show how a curve may be drawn which shall represent the direction of the acceleration of a moving point at every instant. 125. If a point describe a circle, of radius r, with the uniform velocity v, prove that the acceleration is directed towards the centre, and is equal to - . r 126. A stone is whirled round at the end of a string 2 metres long with a velocity of 12 metres per second ; find the acceleration. 127. If T denote the time of revolution or period of a point which describes a circle with uniform velocity v, prove that acceleration 4 *" 2 radius ~ T^ ' 4* P 82 ELEMENTARY PHYSICS. [CHAP. I. 128. If in which M denotes the mass of the sun, m and m' the masses of any two planets, T and T' their periods, and R and R' their mean distances from the sun. What inference may be drawn as to the masses of the planets ? 320. Give a summary of the evidence in favor of the Law of Universal Gravitation. 321. Investigate the effect of the centrifugal force due to the earth's daily rotation upon the weight of a body at the equator. 322. Find the time of revolution of the earth, which would cause bodies to have no weight at the equator. 323. Investigate the effect of centrifugal force at any place upon the weight of a body at that place. 324. "Explain how the centrifugal force due to rotation tends to alter the form of the rotating body. 325. Explain the total variation which is known to exist be- tween the force of gravity at the equator and at the pole. 100 ELEMENTARY PHYSICS. [CHAP. n. LESSON VIII. Attwood's Machine. 326. Why is it that a ball of lead and another of cork, precisely equal in volume, will not fall to the earth with equal velocities 1 327. In Attwood's Machine, what reason is there for having the wheels at the top as small as possible ? 328. Solve the following general exercise on Attwood's Machine : Two heavy bodies are connected by an inex- tensible string, which passes over a fixed pulley ; find the tension of the string and the motions of the bodies. 329. In Attwood's Machine, given P = 12 oz., Q 6 oz. ; find the pressure on the axis of the pulley. 330. In Attwood's Machine, given P = Q 18'6 oz.; what weight must be added to P in order that it may descend through 100 ft. in 8 seconds ? 331. In Experiment H with Attwood's Machine (see Stewart, pp. 51, 52), the boxes are observed to rise and fall alternately through diminishing distances for some time ; explain this. 332. A weight of two pounds hanging vertically draws another weight of three pounds up a smooth plane inclined at an angle of 30 to the horizon ; find the space described in 4 seconds. 333. Two scales are suspended by a string over a small pulley ; six equal bullets are placed in one scale and six in the other ; show that the tension of the string is greater with this arrange- ment of the bullets than with any other. 334. If two unequal weights connected by a string be allowed to fall, the string being vertical, what will be the tension of the string ? LESSON IX. Centre of Gravity. Pendulum. 335. Give an exact definition of the centre of gravity of a body. 336. Define and illustrate the terms homogeneous body, plane of symmetry, centre of figure or geometric centre. 337. Where is the centre of gravity of a homogeneous body, having planes of symmetry 1 LESS, ix.] EXERCISES AND PROBLEMS. 101 Locate the centre of gravity in the following cases : A straight line, circumference of a circle, area of a circle, area of a parallelogram, surface of a sphere, volume of a sphere, convex surface of a right cylinder., vdume of a right cylin- der, volume of a parallelopiped. 338. Show how to find the centre 'of gravity of 'any number of heavy points. .; " - 339. Weights of 1, 2, and 3 pounds are placed alo^g the same line a foot apart ; find their centre of gravity. 340. Examine the equilibrium of a body suspended from a fixed point. 341. Examine the equilibrium of a body resting upon a fixed point. 342. Examine the equilibrium of a body resting upon two fixed points. 343. Examine the equilibrium of a body resting on three or more fixed points. 344. Examine the equilibrium of a right cylinder rest- ing on a horizontal table, (1) on its base, (2) on its side. 345. Explain why a load of hay will overset on the side of a hill, when a load of iron of equal weight will pass along in safety. 346. Explain why it is difficult, if not impossible, for a person standing with his heels against a wall, to pick up a cent between his feet. 347. A balance is in equilibrium with horizontal beam, unloaded pans, and the points of suspension of the pans at a higher level than the fulcrum ; explain the effect produced by loading the pans. 348. Show how to determine the value of the accel- eration of gravity at any place by pendulum observations. 349. Given the length of a second's pendulum, find the length of a pendulum which will oscillate once a minute. 350. If a pendulum were taken to a place where the force of gravity was increased fourfold, how much would the length of the pendulum have to be changed in order that the time of oscillation should remain the same as before ? 102 ELEMENTARY PHYSICS. [CHAP. n. 351. A pendulum at Paris one metre long was found to oscillate in 1-00304 seconds ; find the value of g at Paris. ' :* 352. Sho\v how io find the centre of gravity of a triangle. &?3. A uniform v fcar, 4^ feet long, weighs 10 Ibs. ; and weights of 30 Ibs. and 40 Ibs. are^plaped on its two extremities ; on what poiijc -,\il ii; bailee ?- t f ' '354: Ho\v long a piece' mnsir be cut off from one end of a rod "of length 2 a, in order that the centre of gravity of the rod may approach towards the other end through a distance b ? 355. A beetle crawls from one end of a straight fixed rod to the other end ; find the consequent alteration in the position of the centre of gravity of the rod and beetle. 356. Two homogeneous spheres of equal density touch each other ; find the distance of their centre of gravity from the point of contact, the radii being respectively 8 inches and 12 inches. 357. How high can a cylindrically shaped tower of r metres radius be built without falling, if it be inclined from the vertical by an angle of B degrees ? 358. Show how the requisites of a good balance may be satis- fied. 359. A shopkeeper has correct weights, but a false balance; supposing that he serves out to two customers articles weighing W Ibs. by his balance, using first one scale and then the other, find whether he gains or loses on the whole, and how much. 360. Prove that the time of one small oscillation of a simple pendulum of length I is equal to 361. A pendulum, which would oscillate once a second at the equator, would gain 5 minutes a day at the pole ; compare equa- torial and polar gravity. 362. If two pendulums, of lengths I and I', at different points on the earth's surface make in the same time numbers of vibrations which are in the ratio m : m' \ find the ratio between the forces of gravity at the two places. 363. Show how the height of a mountain may be ascertained by pendulum observations. 364. A seconds pendulum is taken to the top of a mountain of height h ; find the number of seconds it will lose in one day. 365. A seconds pendulum, on being taken to the bottom of a mine, was found to lose 10 seconds a day ; find the depth of the mine, given that the earth's radius is equal to 4000 miles, and LESS, x.] EXERCISES AND PROBLEMS. 103 that in the interior of the earth gravity varies directly as the dis- tance from the earth's centre. 366. A pendulum, when taken to the top of a mountain, is ob- served to lose daily just twice as much as it does when taken to the bottom of a mine in the neighborhood ; show that the height of the mountain is equal to the depth of the mine. 367. The times of oscillation of a pendulum are observed on the earth's surface and at the bottom of a mine ; hence find the radius of the earth, supposed spherical. LESSON X. Forces exhibited in Solids. 368. Define and illustrate the terms coefficient of fric- tion^ angle of friction. 369. What are the forces acting on a body which stands at rest on the side of a hill ? 370. What are the forces acting on a ladder which stands with one end on a rough horizontal floor and the other end resting against a rough vertical wall ] 371. Prove that the coefficient of friction is equal to the tangent of the angle of friction (or to the ratio of the height to the base of an inclined plane the angle of in- clination of which is equal to the angle of friction). 372. A body will just rest on a plane inclined at an angle of 45 ; find the coefficient of friction. 373. The height of a rough inclined plane is to the length as 3 : 5, and a weight of 30 Ibs. is just supported on the plane by the friction ; find (1) the force of fric- tion, (2) the coefficient of friction. 374. Compare the strength of two beams, one of which is twice as long and twice as deep as the other, their breadths being the same. 375. Since, of beams of the same section, the deeper is the stronger, why are not beams made in practice exceed- ingly thin and deep ? 104 ELEMENTARY PHYSICS. [CHAP. n. LESSON XL Forces exhibited in Liquids. [In the exercises on this Lesson the pressure of the atmosphere is neglected. For specific gravities, see Appendix V.] 376. Define a perfect fluid. Define and illustrate vis- cosity. 377. State the chief differences between a solid and a fluid ; between a liquid and a gas. 378. What necessary consequence as to the direction of fluid pressure follows immediately from the definition of a perfect fluid ? 379. State Pascal's principle in a mathematical form. Why cannot the principle be completely established by direct experiment ? 380. Show that any force, however small, may, by transmission through a fluid, be made to support any weight, however large. 381. Show how, by the weight of a few ounces of water, the strongest cask may be burst. 382. Prove that in Bramah's press we have a direct verification of the general principle of Mechanics, that what is gained in force is lost in velocity. 383. A vessel full of liquid has two pistons, 3 and 18 centimetres in diameter respectively ; what pressure on the smaller will produce a pressure of 900 kilogrammes on the larger 1 384. A closed vessel full of liquid has a weak part in its upper surface, not able to bear a pressure greater than 9 Ibs. per square foot. If a piston the area of which is one square inch be fitted into an opening in the upper surface, what pressure applied to it will burst the ves- sel? 385. Prove that the free surface of a liquid at rest is normal at every point to the resultant of all the forces acting at that point. 386. Explain the form of the free surface of a liquid at rest on the surface of the earth. LESS. XL] EXERCISES AND PROBLEMS. 105 387. If we consider any horizontal plane in a liquid at rest under the action of gravity, the pressure on the plane is proportional, (1) to the area of the plane, (2) to the depth of the plane, (3) to the density of the liquid. On what grounds do these laws rest ] 388. Prove that the pressure on any horizontal sur- face in a liquid is given by the formula, . ( area (in c. m. 2 ) x depth (in c. m.) Pressure (in grammes) = j x d ^ nsity ^ 389. Prove that the pressure on any vertical surface in a liquid is given by the formula P = SH D where P denotes the pressure, H the depth of the centre of gravity of the surface (supposed uniform in density), and D the density of the liquid. Specify the units in which P, , H, and D respectively should be expressed. The pressure is equal to the weight of what column of water 1 390. Find the pressure at the depth of 30 metres in a lake. 391. What height must a column of water have which will exert a pressure of 1000 kilogrammes per square decimetre ? 392. Required the pressure on a rectangular vertical side of a tank full of water, the height of the tank being 4 metres, and its breadth being 80 centimetres. 393. Required the pressure on a vertical triangle im- mersed in water with the base in the surface, the base being 50 centimetres, and the altitude 30 centimetres. 394. Prove that the total pressure experienced by a cu- bical vessel full of water is equal to three times the weight of the water. 395. Two cubical vessels, the edges of which are as two to .one, are filled with water ; compare the pressures, (1) on their bases, (2) on their total interior surfaces. 396. Sketch the form of a vessel in which the pressure 5* 106 ELEMENTARY PHYSICS. [CHAP. n. against the sides shall much exceed the pressure on the base. 397. Taking the density of mercury as 13'6, find the total interior pressure in a cylinder full of mercury, the height of the cylinder being one metre, and the radius of the base being 16 centimetres. 39 Prove that the common surface of two liquids which do not mix must be horizontal. 399. Prove that the heights of two vertical liquid col- umns in communication are inversely as the densities of the liquids. 400. Distinguish between the centre of gravity and the centre of pressure of a vertical surface pressed by a liquid. 401. How will the pressure on the base of a vessel con- taining water be affected by dipping a piece of metal into the water, (1) when the vessel is just full of water, (2) when the vessel is not full ? 402. Prove that when a body is placed in a liquid, it will (1) sink, (2) remain at rest, or (3) rise to the surface of the liquid, according as its density (or specific gravity) is (1) greater than, (2) equal to, or (3) less than that of the liquid. 403. Find the weight of a boat which displaces 10 cu- bic metres of water. 404. Prove that, when a bod} 7 " floats on the surface of a liquid, the part of the body immersed is to the whole body as the density of the body is to that of the liquid. 405. Define and distinguish between density and spe- cific gravity, and explain how each is measured. 406. A body measuring 18 cubic centimetres floats in water with its whole bulk immersed ; find its weight. 407. Find the weight in water of 100 cubic centime- tres of iron. 408. Find the weight of one cubic centimetre of a body which floats in water with one fifth of its volume above the surface. 409. An irregularly shaped mass of granite weighs 182 kilogrammes in air arid 117 kilogrammes in water ; find its volume and its specific gravity. LESS. XL] EXERCISES AND PROBLEMS. 107 410. Two bodies differing in bulk weigh the same in water. Which will weigh the most in mercury and which the most in a vacuum 1 411. A man (specific gravity 1 : 12) weighs 70 kilo- grammes. What volume of cork will be required to just float him in water ? 412. A cylinder of wood floats in water with the^axis vertical. How much will it be depressed by putting a weight W on top of it ? 413. Suppose that a man exerting all his strength can just raise a weight of 100 kilogrammes ; find the weight of a stone (specific gravity 2 '5) which he can raise under water. 414. Show how to find the specific gravity of a mix- ture of gi\ en volumes of any number of given substances. 415. Show how to find the specific gravity of a mix- ture of given weights of any number of given substances. 416. Find the specific gravity of a mixture of equal volumes of water and alcohol, supposing no contraction to take place. 417. Find the specific gravity of a mixture of equal weights of water and alcohol, supposing no contraction to take place. 418. Three liquids, the specific gravities of which are respectively 1*2, 0*96, and 1*456, are mixed in the pro- portions by volume of 18 parts of the first to 16 parts of the second and 15 parts of the third. Find the specific gravity of the mixture. 419. What are the proportions of gold and silver in an alloy of these two metals which weighs 10 kilogrammes in air and 9*375 kilogrammes in water ? 420. If a diamond ring weighs 69 '5 grammes in air and 64-5 grammes in water, find the weight of the diamond in the ring. 421. Find the specific gravity of a piece of lead which weighs 47*48 grammes in air and 43*33 grammes in water. 422. Explain a method of finding the specific gravity of a solid lighter than water. 108 ELEMENTARY PHYSICS. [CHAP. n. 423. A block of wood, weighing in air 8 Ibs., is tied to a piece of metal weighing 6 Ibs. ; in water both together weigh 4 Ibs., while the metal alone weighs 5 Ibs. Find the specific gravity of the wood. 424. A crystal of salt weighs 6 '3 grammes in air ; when covered with wax, the whole weighs 8'22 grammes in air, and 3 '02 grammes in water ; find the specific gravity of the salt. 425. A glass ball, weighing 10 grammes in air, loses 3.636 grammes in water, and 2 -88 grammes in alcohol. What is the specific gravity of the alcohol ? 426. Prove that the volumes of different liquids dis- placed by the same floating body are inversely as the specific gravities of the liquids. How, by this principle, can the specific gravity of a liquid be ascertained ? 427. Prove that the entire interior pressure on a hollow sphere full of a liquid is equal to three times the weight of the liquid. 428. A rectangle is just immersed vertically in a liquid, with one side in the surface ; divide it by a horizontal line into two parts which shall be equally pressed by the liquid. 429. Divide a rectangle just immersed vertically in a liquid, with one side in the surface, by horizontal lines into n parts on which the pressures shall be equal. 430. Distinguish between total pressure and resultant pressure by the aid of illustrations (as a cubical vessel full of water, a tea- cup full of tea, &c. ), and prove that when a vessel of any shape is tilled with a liquid the resultant pressure is equal to the weight of the liquid. 431. A hollow cylinder (height h, radius of base r) is filled with a liquid ; compare the total pressure on its interior surface with the weight of the liquid. 432. A right cone (altitude h, radius of base r) rests on its base and is full of water; compare the entire interior pressure with the weight of the water. 433. Prove that the centre of pressure of a rectangular vertical area is on the vertical line which passes through the centre of gravity of the area at a distance of two thirds of the height of the area from the surface of the liquid. 434. Prove that, as a plane area is lowered vertically in a liquid, the centre of pressure approaches, and ultimately coincides with, the centre of gravity of the area. LESS. XL] EXERCISES AND PROBLEMS. 109 435. State the general conditions of equilibrium of a floating body, and define the states of stable, unstable, and neutral equi- librium, with illustrations. 436. Define the metacentre of a floating body and prove that the equilibrium of a floating body is stable, unstable, or neutral, according as the metacentre is above, below, or at the centre of ^gravity of the body. 437. A ship sailing into a river sinks 2 centimetres, and after discharging 12000 kilogrammes of her cargo, rises 1 centimetre ; determine the weight of ship and cargo, the specific gravity of sea- water being 1.026. 438. What quantities of zinc and copper must be taken to make an alloy weighing 50 grammes, and the specific gravity of - which shall be 8-2 '( 439. Given the weights and specific gravities of two bodies of different kind ; find the specific gravity of the compound formed by mixing them, (1) when the contraction is ~ih part of the sum of the component volumes, (2) when the expansion is th part of the sum of the component volumes. 440. Required the specific gravity of a mixture of 18 kilo- grammes of sulphuric acid and 8 kilogrammes of water, assuming that the contraction is -^. 441. Three masses of gold, silver, and a compound of gold and silver, weigh respectively, P, Q, and R grammes in air, and p, q, and r grammes in water; find the weight of gold in the com- , pound. 442. The weight W of a vessel full of air (specific gravity p) and the weight W of the same vessel full of a liquid (specific gravity sin 90. w = 86-22 : a = 34 15' 36" : = 19 44' 24". 91. Each velocity = 7 '07 ; each angle = 45. 93. From the southwest, with a velocity of 4 V~2 miles per hour. 94. He should aim to a point 17*605 feet in advance of the deer, or along a line making an angle of 1 40' 49'' with the deer's direction. 95. If u = velocity of tube, v = that of the particle ; then = sine of the angle of inclination. 99. 3600 g. 100. Times, 2 s and 4 s ; velocities, g and g. 101. 87*662 metres. 102. 10 ^~g feet per second. 103. 1-267 8 . 104. - -f- V/2 ga feet per second. 109. Height = a. ANSWERS AND SOLUTIONS. 117 In exercises on projectiles, let v denote the velocity of pro- jection, a the angle between the direction of v and the horizon. . v 2 sin 2 a _ 2 v sin a 114. -- . 115. - . 2 27 g 116. - sin 2 a. 118. 7764Jeet. 119. If u and v denote the velocities, and a /3 the angles of projection respectively ; then, distance at time t = ^[u 2 -+ v 2 2 uv cos (a j3)] t. 122. 4 54'. 123. 233;238 feet. LESSON IV. 144. 739. 24 kilogrammes. 145. (1) 480; (2) 48 '98. 146. 2 -3 nearly. 147. 0-45359 m ; m; '45359 m ; 3219m; 9-81 X '45359m. 148. 27 '734 feet. 149. The rate of change is mass X acceleration per second. 150. 772 800 000. 151. 10 '2 s ; 10 '2 times the first force. 157. (1) g is a number expressing the velocity produced in a falling body in unit of time. (2) g is a number expressing twice the distance through which a body falls in unit of time. (3) g is a number expressing the weight of unit of mass in dynamical measure. 160. ^ Ibs. 161. $ X J X 0* 1 = 49 ; v = | 0*. 162. s = -X^r;v = -gt. 163. 103-04 feet. m 2 ' m u 164. 17-96 Ibs. 165. pressure = m (l + ^ J . \ y ' IP~T . Time of revolution = 2 TT i / . V Qg 166. LESSON Y. 168. P f-. 172. 7 and 5. 173. No. 175. 17. a 118 ELEMENTARY PHYSICS. 176. The same and the opposite directions respectively. 178. Draw from any point, A, three lines representing the directions of the two forces and their resultant, and take, on the line which represents the direction of the given force, a length, A B, to represent the magnitude of this force ; from B draw a line parallel to the direction of the other force till it intersects the line of direction of the resultant at C ; then A is the magnitude of the resultant. 180. 27 and 120. 182. Each force = - 7 .1 2 . V/2 183. Their lines of action must form an equilateral triangle. 185. 80 Ibs. 186. 5 V~21bs. 187. Magnitude = 4 y 2 ; direction is perpendicular to one of the sides of the square. 191. 38 and 114. 206. (1)P+Q, (2)0 P, (3)P Q. 208. 4 metres from the end to which the weight of 9 kilo- grammes is attached. 209. 4 feet. 210. 7 m. from the given pressure. 213. 6 inches from the end supporting the weight of 5 Ibs. 214. 84J and 88| Ibs. 216. 40 Ibs. 221. 1792 Ibs. 222. 255. 224. 16 Ibs. OK 225.45. 229.^-. 235. 2 P. O 7T 236. 6'89 Ibs. ; angle of R with first force, 102 16'. 10 238. 67 "2 Ibs. 239. ~j^ kilogrammes. 240. Horizontal thrust =\W cot a, where a = inclina- tion of either rafter to the horizon. 241. (1) Pressure = JFcoseca, where W = weight of hat, a = semi-angle of the cone. (2) The pressure becomes infinite. 242. If P, Q, S, are the forces, their resultant, then, ANSWERS AND SOLUTIONS. 119 243. R* = (S P cos a) 2 + (S P cos /3) 2 + (S P cos 7) 2 S P cos a S P cos 8 S P cos 7 cos , cos ^ = ^ , cos a; . 244. It is necessary and sufficient for the equilibrium of any number of forces applied to a point, that the sums of the resolved parts of the forces in the directions of three intersect- ing lines not in one plane be separately equal to zero. 253. The point of attachment should be as far above the ground as the man is from the foot of the tree. JFsin/3 254. Pressure on 1st plane = - . sin (a H- /3) JPsina Pressure on 2d plane = . sin (a + j8) 255. Let = angle which a line drawn from W to centre of hemisphere makes with the horizon ; then, P y' (8 W* + P 2 cos 4 * = \ w 256. Let W = weight of each beam, = its angle with the horizon, then, horizontal thrust ^ W cot 0. 257. Let = angle of m with horizon ; then, m 2 -f- n 2 cos tan = - sin 258. Let W = weight of the beam, P = pressure of beam on the rail, R = pressure of beam on the wall, angle of beam with the wall, a = half the length of the beam, b = distance of the rail from the wall ; then, ; p = W cosec 6 ; R = Wcot 0. 259. Let W denote the weight of the ladder, m and n the segments into which its centre of gravity divides it, the angle which the ladder makes with the floor, P the horizontal force required ; then, P = ^ W cot 0. 120 ELEMENTARY PHYSICS. 260. (1) Between W and infinity. (2) " and W. (3) " and infinity. __ P _ product of the weight arms W product of the power arms ' 262. As 1 to 2. 264. Let P denote the power applied to the wheel, a and b radii of wheel and axle, a the angle of the inclined plane with the horizon, W the weight on the plane ; then, p _ W b sin a a W 267. 4- T/O of the pressure on the plane. 268. The weights are as the lengths of the planes. 269. TT inches. 270. 186 '86 Ibs. LESSON VI. 275. sfjHta f a foot from the i 376. 2 '8 feet per second. 278. By pushing against the air with the palms of his hands. 282. Let m, m f , denote the masses, v, v 1 , their velocities, then, m v -f- m 1 v f velocity alter impact = - . m 4- mf 283. 8 metres per second. 284. 15 metres per second. m m f .2 m m 1 289. Acceleration = - g ; tension = , g. m -\- m' r m-\-m 290. It will be nothing. ?7i sin a m 1 sin at 291. Acceleration = -. g ; m 4- m f m m f (sin a 4~ sin a') tension of the string g. m 4- m f ANSWERS AND SOLUTIONS. 121 298. Let u, v, denote the velocities before and after im- pact respectively, i t v, the angles of incidence and reflection ; then, tan i - e tan v, v 2 = V? (sin 2 i + e 2 cos 2 i). 299. b. LESSON VII. 304. 90. 307. 5. 308. The parts must be equal. 321. The centrifugal force at the equator diminishes the weight of a body by about YTyth part. 322. 17 times faster than it now revolves. 323. The component of centrifugal force diminishing gravity = ~~~m^~ cos 2 X, where R earth's radius, T = time of revolution, X = latitude. 325. The total variation amounts to yirth part of the weight of a body. In other words, a body which weighs 194 Ibs. on the equator would weigh 1951bs., very nearly, at the pole. Centrifugal force causes a part of this difference, and the variation in the earth's attractive force due to its spheroidal shape produces the remainder of the difference. LESSON VIII. 328. Let P and Q be the weights of the bodies, T the tension of the string, the acceleration ; then, ~ P+Q' 329. 16 oz. 330. 4 Ibs. 332. g. 334. Nothing. LESSON IX. 337. At, the centre of figure in each case. 339. At a point 8 inches from the 3 Ibs. 6 122 ELEMENTARY PHYSICS. 349. 3600 times the length of the second's pendulum. 350. Fourfold. 351. 9 '8098. 353. At a distance of 21 inches from the 40 Ibs. 354. 2ft. 355. , where I = length of rod, a its weight, J weight of beetle. 356. Let D = density of the sphere whose radius is 8 inches, D f = density of the other sphere ; then their common 540 D 1 centre of gravity is at a distance of &f~rii ^ rom t ^ ie o U ~T~ At JJi 8 -inch sphere. 357. 2 r cosec 0. 359. Weight lost = W ^~^. 361. As 144 to 145^| F . 362. As m 2 I to ml* I'. 9 h 364. , h being expressed in feet. 365. 0-94 of a mile. 367. Earth's radius = -^ %, where t l is the time of an ^2 *i oscillation at the surface, and t 2 at the given depth h. LESSON X. 372. 1. 373. 18 Ibs. ; 075. 374. Twice as strong. LESSON XL 383. 25 kilogrammes. 384. 1 oz. 390. 3 kilogrammes per sq. centimetre. 391. 100 metres. 392. 6400 kilogrammes. 393. 7500 grammes. 395. As 8 : 1, in both cases. 397. 348160 TT grammes. 403. 10000 kilogrammes. 406. 18 grms. 407. 725 grms. 408. 0-8 gramme. 409. 2'8. 410. The smaller, in mercury ; and the larger, in a vacuum. W 411. 9868 cubic centimetres. 412. 5. TTT 2 ANSWERS AND SOLUTIONS. 123 413. 166f kilogrammes. 414. Let Vi t V~z, V-z, & 416. 0-8975. 417. 0*8857. 418. 1*2. 419. 7511 grammes of gold, and 2489 grammes of silver. 420. 3'5 grammes. 421. 11 '2. 423. |. 424. 1-9 nearly. 425. 0.792. 428. The dividing line must be at the depth -7 , where h \/2 denotes the vertical edge. 429. Let h = the vertical edge ; then the lines of division are at the depths J \ -A,&c. Total pressure Weight of the liquid Total pressure 4;O4i. Weight of the liquid r 437. 947076 tonnesv 438. Zinc 17 '82 grammes, copper 32 '18 grammes. 439. If W,Wi and o-, _ * ^ ' W 124 ELEMENTARY PHYSICS. 440. 1-51 nearly. 441. P. Qr ~ E - q ~. 442. W * ~ W Qp Pq : h, or = 1. But experiments have shown that gases, except hydrogen, are more compressible than the law indicates, that is, for all VJi gases except hydrogen <^ 1. YJi But for hydrogen TFTT, > I- 470. 1728 times the initial pressure. 471. 6 -354 litres. 472. The air will flow out. ANSWERS AND SOLUTIONS. 125 473. More of it will be submerged. 474. The tension will be diminished. 476. Let h = depth descended in feet, Jc = height of a water barometer, in feet ; then, height ascended in the bell _ h total height of the bell h + k ' 477. 46-968 centimetres. 478. Let V = volume of the receiver, v = volume of the barrel, D = initial density of the air in the receiver, D n = the density of the air after n strokes ; then, 481, 6fi Ibs. per sq. inch. 482. As 5874 to 1. 483. It is equal to the weight of a column of water having i section equal to the area of the piston, and a height equal to the height of the column of water raised. 485. Not more than 76 centimetres, by the- suction prin- ciple. 436. The siphon would cease working and the water in the arms would flow out. 487. The rapidity of the flow, and the possible height of the short arm, would be diminished. 488. The siphon would work backwards, that is, the flow would be from the long arm into, and out of, the short arm. 489. 786'875 kilogrammes. 494. If d = diameter at depth a, d f = diameter at depth -, h = height of a water barometer ; then, df = 496. Let I = length of the tube, h = height of a perfect barometer, y = length of the air-column in the tube at the pressure h, x depression produced by the air ; then, 126 ELEMENTARY PHYSICS. 497 - a + T= (a '- a} - AQQ Initial density of the air capacity of barrel Final density attainable ~ untraversed space* 499. Let V, v, denote the volumes of the receiver and barrel respectively, v 1 the untraversed space, D the initial density of the air, D n the density after n strokes ; then, 500. Let A and B be the capacities of the receiver and barrel respectively, h the measure of the pressure of the at- mosphere ; then, tension after n strokes = - 7- . h. A 501. 350 strokes. 502. The general condition is, that the pump will not work unless the play of the piston be greater than the square of the distance from the surface of the water in the well to the highest position of the piston divided by four times the height of the water-barometer. ANSWERS AND SOLUTIONS. 127 II. SOLUTIONS. LESSON I. 19. The unit of mass in the Metric System (the kilo- gramme), which is, strictly speaking, the quantity of matter in a certain platinum weight kept in Paris, was intended to be, and may "be taken as, equal to the mass of one unit of volume (the litre) of pure water at 4 C. Hence, 2 litres of water weigh 2 kilogrammes, and in gen- eral a litres \veigh a kilogrammes. 21. The density of a body is the ratio of its mass to its vol- ume, or, in symbols, D denoting density, M mass, and V volume. The numerical measure of the density of a substance is ob- tained by taking the unit of volume, or putting V = 1, in which case D = M, or the measure of density is the number of units of mass in unit of volume of the substance. The value of the density of a substance evidently depends on the units of mass and volume adopted, but density is always the ratio of a mass to the cube of a length, or, as it has been expressed, the dimensions of density are -^ , M de- noting a mass, and L a length. 23. Because the unit of mass in the Metric System is the ~mass of unit of volume of pure water at 4 C. ; or, in other words, because, in the case of water, when V 1, M = 1, and therefore, 26. The word weighty both in scientific and common lan- guage, is usually employed to denote mass or quantity of matter. But the weight of a body, properly speaking, is the measure of the force with which the body is drawn towards the centre of the earth. This force is different in different places, 128 ELEMENTARY PHYSICS. being greater, for example, at the pole than at the equator, and greater at the level of the sea than at the top of a moun- v tain. The mass of a body is the quantity of matter which ^ it contains; this is invariable, and would remain the same even if the force of gravity did not exist, in which case bodies would have no weight. LESSON III. Representation of Velocities and Forces. A velocity (or a force) is said to be represented geometrically by drawing a straight line in the direction of the given velocity (or in the direction in which the given force acts) and making the line as many units long as there are units in the given velocity (or in the given force). For example, a velocity of 10 feet per second towards the north would be represented on paper by a line 10 inches long, drawn as in maps, &c., perpendicular to the upper edge of the paper. A force of 16 units acting toward the east might be represented on paper of moderate size by a line 4 inches long, taking for convenience one quarter of an inch to represent the unit of force, the line being drawn from a point where the force is supposed to act, towards the right- hand edge of the paper and perpendicular to that edge. An arrow -head may be added, as in Fig. 2, to show which of the two directions of the line is to be taken. When forces repre- sented by lines lettered at their extremities are referred to, the order of the letters indicates the direction of action. Thus, a force A B means a force acting from A toivards B. 56. The Parallelogram of Velocities may be stated as fol- lows : If two velocities be represented in magnitude and direction by two straight lines drawn from any point, the diagonal of the parallelogram constructed upon these two lines will represent the resultant velocity in magnitude and direction. This proposition is one of the utmost importance, inasmuch as the great majority of mechanical problems, whether of the Sractical kind which require solution in the operations of ivil Engineering, or of a purely theoretical nature, involve the action of several forces in different directions. Mr. Stewart has given, in Lesson IV., 24, a partial proof for the case of two simultaneous continuously acting forces. ANSWERS AND SOLUTIONS. 129 The proof there given is imperfect, because it is only shown that the body will be at the extremity of the diagonal at the end of the time considered, not that it will constantly be on this diagonal throughout the interval. This is a very natural inference, it is true, but it is an inference which can be de- duced legitimately from principles more general, if not more axiomatic in their character. In the example of composition of velocities selected by Mr. - Stewart, the velocities are generated by gravity and by mag- netic attraction, that is, by continuously acting forces, and consequently the velocities generated are not uniform but accelerated. In other words, Mr. Stewart's proposition is not the Parallelogram of Velocities but of Accelerations, a prop- osition equally true, but usually and properly considered a logi- cal deduction from the more simple case of uniform motion. Sir Isaac Newton, taking the case of uniform velocities, has given in his " Principia" the following simple and convincing proof of this famous proposition. It will be noticed that he obtains the result as a deduction from his First and Second Laws of Motion. Suppose that a body, in a given time, by the effect of a single force M impressed at A, would move with a uniform velocity from A to B ; and A ^ B suppose that the body, in the same time, by the ef- fect of another single force N impressed at A would move with a uniform ve- locity from A to C ; then if both forces act simul- taneously at A, the body will move uniformly in the given time along the YIG. 1. diagonal from A to D. For since the force N acts in the direction of the line A C parallel to B D, this force, by the Second Law, will not at all alter the velocity generated by the other force M by which the body is carried towards the line B D. The body, therefore,, will arrive at the line B D in the same time whether the force ^Vbe impressed or not ; and therefore at the end of that time it will be found somewhere in the line B D. By the same 6* I 130 ELEMENTARY PHYSICS. reasoning, at the end of the same time it will be found some- where in the line C D. Therefore it will be found at the point D where both lines meet. And it will move in a straight line from A to D by the First Law. It will be observed that Newton supposes that the two forces act instantaneously ; that is, are of the nature of blows, as thJtt of a bat upon a ball. Such a force communicates its effect in a time too small to be taken account of, and is in this respect totally different from a force like gravity or attractions of any kind which act continuously. Forces of the former kind are often called impulsive forces, and forces which act during finite periods of time, like gravity, are called continuous forces. Impulsive forces tend to produce uniform motion, being only prevented from so doing by the universal presence of retarding causes, such as friction, resist- ance of the air, &c. Continuous forces tend to produce accel- erated motions. In Newton's exposition, the uniform motions are supposed to be produced in the simplest possible way, by the action of impulsive forces, combined with the absence of retarding causes. But in nature no such instances can occur, because retarding causes always exist. Bodies may move with uniform velocities, but the uniform motion must arise, in every case, from the fact that several forces are acting on the body in such a manner that they neutralize each other's effects and leave the body free to obey the First Law of Motion. For instance, a horse is drawing a load along a road at a uniform rate ; here the muscular effort of the horse is just balanced by the friction of the wheels on the ground, and the resistance of the air, which latter is, of course, very trifling. In fact, the reference which Newton makes in his proof to force as the cause of the motion is unnecessary ; and, inas- much as it has been found of great advantage in Mechanics to treat many properties of "motion, displacement, and deforma- tion " independently of force, mass, &c., under the head of Kinematics, or the Geometry of Motion, we will present the solution of the Parallelogram of Velocities free from any ref- erence to force. Suppose that a body at A (Fig. 1) moves with uniform ve- locity from A to B, and that simultaneously the line A B moves uniformly, and parallel to itself, in the direction A C or B D. Suppose also that in the time required for the body to move from ANSWERS AND SOLUTIONS. 131 A to B, the line would move from the position A B to the position C D. Then if the line remained at rest, the body would be at B at the end of the time considered, but since by the motion of the line the point B takes the position D, and since both motions take place independently of one another, the body will be found at the end of the interval at the point D. This is true whatever be the time considered, provided both motions are uniform. If, then, we take one half the interval of time already considered, the velocities represented by A B and A C will each be reduced one half, and by draw- ing the dotted lines we obtain a parallelogram similar to the first one by Geometry. Therefore its diagonal is equal to A D ; that is, the time being halved, the space passed over by the body is halved, the direction of the body's motion being unchanged. The same reasoning may be applied to, any and every fractional part of the time considered ; therefore the body will move uniformly in a straight line from A to D. 58. Let the velocities be represented by the lines A, OB, C, and D in Fig. 2. (The line H is not employed in this Exercise.) FIG. 2. 132 ELEMENTARY PHYSICS. Find, by the Parallelogram of Velocities, the resultant E of the first two velocities A and B. Then compound by the same principle this resultant with the velocity C, obtaining F as the resultant of the first three velocities. Proceeding in the same way with the remaining velocity D, we finally obtain G as the resultant or single velocity equivalent to all the simultaneous velocities. This method is evidently applicable to any number of simultaneous velocities impressed upon a body at 0. It should be observed, that, speaking strictly, it is im- possible for a body to be at, or in, a point 0. Bodies are finite portions of matter and have finite magnitudes. And, in point of fact, if the several velocities above represented were simultaneously impressed on a body in directions having a common point of intersection within the body, N would not represent the actual resultant motion of the body (unless w r ere the centre of gravity of the body), on account of the connections of the parts of the body due to cohesion, arid the consequent mutual actions of those parts. The determination of the actual motion of the body is a problem of far greater difficulty than the simple operation of finding a resultant by the Parallelogram of Velocities. But this principle is a first and an essential step in the chain of reasoning which leads to the solution of any case of motion however complex. The first thing to be done, therefore, is to set forth the funda- mental principle in as simple a form as possible. Most writers on Mechanics do this by the aid of the conception of material points or particles, which are defined as bodies so small that their dimensions may be neglected. Bodies are regarded as composed of an indefinitely large number of parti- cles. It will be noticed that the conception of a particle is different from that of the molecule in Chemistry. If, then, we use language strictly, instead of speaking of a "body at 0," we should say a "particle at 0," or "a mate- rial point at 0," or, for shortness' sake, " a point at 0." 59. In Exercise 56 the object was to find a single velocity which was equivalent to two simultaneous velocities ; this is frequently called the Composition of Velocities. It is often necessary for the purposes of proof or illustration to perform the reverse operation, to substitute for a single velocity two velocities in assigned directions. ANSWERS AND SOLUTIONS. 133 For instance, a ship is sailing at the rate of 8 miles per hour in a direction 30 to the east of south ; at what rate is she moving towards the east and. towards the south respec- tively ? Questions of this kind fall under the general problem of the Resolution of Velocities ; this problem is solved geomet- rically as follows : Let C represent the velocity which we wish to resolve in the directions X and Y ; from C draw the lines C A and C B parallel to Y and OX Y respectively. Then, by the Parallelogram of Velocities, O C is the resultant of ve- locities represented by A and B, so that A and B are the resolved parts required. Resolution in di- rections making a right angle with each other is by far the most common and useful, being attended with this great and obvious ad- u "& X vantage, that the compo- FIG. 3. nents A and B are wholly independent of each other, so that each component represents the entire effect of the velocity C, estimated in its own direction. In the question above asked, it can be easily found that the ship is moving eastward at the rate of 4 miles an hour and at the same time moving southward at the rate of 4 V/3 miles an hour. 60. The resultant of the velocities A and B, in Fig. 4, is a velocity represented by the third side D of the tri- angle A D. This resultant would be neutralized by an equal and opposite velocity C impressed upon the partic'e at 0, so that A, OB, and C form a system of three velocities such that, if impressed simultaneously upon the particle at 0, the particle would remain at rest. Now taking the sides of the triangle O A D in order, A represents the first velocity, A D represents the second, because it is parallel and equal to B, and DO represents the third velocity O C, which is by supposition equal to D and has the same di- rection. 134 ELEMENTARY PHYSIOS. FIG. 4. 61. This is merely an extension of the process of proof in Exercise 60. Use Fig. 2 for illustration. 62. The term projection is one of great importance in cer- tain branches of study, especially Descriptive Ge- ometry and Mechanical Drawing. What we are here concerned with is the projection of one line on another ; this may be denned by reference to Fig. 5. Let A B be a line of given length. To _ find its projection on any ' line, XY, let fall from A and B lines perpendic- FlG. 5. ular to X Y and meeting X Y at the points M and N. Then M 1ST is the projection of ABonXY. Now it is clear (as was stated in the Solution of Exercise 59) that the effective component of a velocity in any direc- tion will be found by resolving the velocity into two com- ponents ; one in the given direction, which will be the effective component sought, the other in a direction perpen- dicular to the given direction, which component is wholly independent of the former. If we resolve A B into these two components by the aid of the Parallelogram of Velocities, we find them to be A C and A D, of which the former, A C, is the N ANSWERS AND SOLUTIONS. 135 effective component in the direction X Y, and is equal to M N, or the projection of A B on X Y. The examination of the special case referred to is left to the student. 67. A velocity may change, (1) in quantity, (2) in direction. We are here concerned only with the first kind of change, the motion being supposed to be rectilinear. For curvilinear motion, see Exercises 124-129. Acceleration is the rate of change of velocity. Velocity may change in quantity either by increasing or by diminishing, but the term " acceleration " may be extended to include both cases by applying to ve- locities and accelerations the algebraic convention of positive and negative signs to denote opposite directions. Acceleration of velocity may be either uniform or variable : it is said to be uniform when the point receives equal increments of velocity in equal times, and is then measured by the increase of velocity per unit of time. To illustrate, take the force of gravity, and let the direction of this force (towards the earth's centre) be considered positive. It is known that a body, let fall from a point above the earth's surface, acquires a velocity of 32 feet per second in the first second ; that is, at the end of the first second it is moving at such a rate that it would move over 32 feet during the next second, if its motion were uniform. But its motion is not uniform ; gravity exerts the same effect upon it during the second as during the first second, so that at the end of the second second its velocity is 64 feet per second ; and at the end of the third second it would be 96 feet per second, and so on. In this case the acceleration, or change of velocity per unit of time, is uniform, and is +32 feet per second in each second. Which direction shall be considered positive in any case is purely arbitrary, and is usually selected on grounds of convenience. In the example just given, where gravity was the only force concerned, and the motion pro- duced by gravity the only motion, the simple and obvious course was to call the direction of gravity the positive direc- tion. But take the familiar case of a stone thrown vertically upwards. This is an example of retarded motion. But the analysis of the case shows that there are two motions, simul- taneous, independent of each other, and in opposite direc- tions, one a uniform motion vertically upwards due to the muscular impulse of the hand, the other a uniformly acceler- 136 ELEMENTARY PHYSICS. ated motion vertically downwards due to the constant action of the force of gravity. We may take either direction as positive ; then the opposite direction must be considered neg- ative. It is rather more convenient, probably, to call the upward direction positive ; and it is certainly more natural to consider the direction of the actual motion as positive, and retarded motion as an instance of negative acceleration. Let this, then, be assumed, and suppose a stone thrown vertically upwards with a velocity of 96 feet per second. It has two simultaneous motions, one represented by the uniform velocity H-96, the other represented by the uniform acceleration 32. It is evident that at the end of 3 seconds the stone will have a velocity of 96 downwards, produced by gravity, which will just cancel the velocity +96, that is, the stone will come to reSt. The space passed over by the stone is quite another matter, and is investigated in Exercises 78 and 80. 68. The average acceleration during any time is the total velocity gained during that time divided by the time. 71. In 9 seconds a body will fall through 1 (9 '8) 81 metres ; in 8 seconds it will fall through \ (9*8) 64 metres. Therefore, during the ninth second it falls through \ (9'8) (81 64) metres = 4*9 X 17 metres = 83 '3 metres. 75. Here, and in 78 and 79, the notation is employed with special references to the most important and universal of all forces, that of gravity. The initial velocity a is supposed to be in the direction of gravity. The letter g is universally employed by writers on physical science, as in this Exercise, to denote the acceleration of gravity, which varies slightly with the locality, and is only expressible approximately by decimals not needed in most investigations, g, then, denotes the number of units of velocity generated by the force of gravity in one second. The force being constant, the number generated each second will be the same, and therefore at the end of t seconds the velocity acquired by the body will be g t units. This is the entire effect of the force of gravity, and this effect is by the Second Law of Motion the same whether the body be initially at rest or moving in any direction with any velocity. In this case the body is supposed to have initially a uniform velocity of a units, which velocity it w r ould continue to have forever, by First Law, were no forces to act upon it. It is evident, then, that as the initial and acquired ANSWERS AND SOLUTIONS. 137 velocities are in the same direction, the resultant velocity at the end of t seconds will be their sum, or a -+- g t. This formula includes the case in which the initial velocity a, is opposite to the direction of g simply by considering either a, or g negative, as pointed out in the Solution of Exercise 67. 78. We may employ here the principle of the Independence of Motions (or the Effects of Forces), stated by Mr. Stewart at the beginning of Lesson III., and there called by him the Second Law of Motion, although the Law as given by Newton contains, or at least implies, much more. (See Solutions of 135 and 138.) Where several forces act on a body, estimate separately the effect of each force in producing motion ; then combine these effects by addition or subtraction, if the forces act along one line, or by the aid of the Parallelogram of Veloci- ties, if they act along different lines. In this case we have two motions along the same line. The point at the instant when the time t begins to be reckoned is moving with a uniform velocity a due to a force (the impulse of the hand, say) which has ceased to act. From this instant the effect of gravity is also to be taken into account. In virtue of the uniform velocity a the point will describe the space a t in the time t. In virtue of the action of gravity, it will de- scribe the space %gt 2 in the time t, as shown by Mr. Stew- art in Lesson III. Therefore the total space described is a t H- J g t 2 . We may prove this theorem without reference to the Laws of Motion, as follows : the average velocity of a point during any interval of time is the space described during that time divided by the time. In this case, since the acceleration is uniform, the average velocity is tjie arithmetical mean between the initial and final velocities. For as the velocity increases uniformly, its value at any time before the middle of the inter- val is as much less than this mean as its value at the same time after the middle of the interval is greater than the mean. Here the initial velocity is a, and the final velocity a -4- g t. Therefore the average velocity = | (a -f- a + g t) = a-\-\gt. And space described = average velocity X time a t -f- ^ g t 2 . 80. Here the initial velocity is opposite to the direction of gravity. Consider the direction of projection positive, and that of gravity negative. Then the general formulae of 75 and 78 must be written, 138 ELEMENTARY PHYSICS. (1) v^a gt (2) s = at \g& The time of ascent is evidently the value of t in (1) when v = 0, or -. The height ascended is found by substituting this value of t in (2). 81. Put s = in equation (2) in the preceding solution, and solve with respect to t. We find t = 0, or t = . The first value corresponds to the instant of leaving the ground, the second to the instant of reaching it again. But we have already seen that the time of ascent = -. Therefore time of a 9 descent also = -. 82. If a = velocity of projection, the time of ascent = - = time of descent, by 80 and 81. But if a body falls freely during the time -, the velocity acquired = g X - = # y y LESSON IV. 135. "We will here give Newton's Laws of Motion, both in the original Latin of Newton, and as translated by Thomson and Tait in their Treatise on Natural Philosophy. Though called by Newton Laws of Motion, it would be more accurate to call them Laws relating to the Connection of Force with Motion. LEX I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare. LAW I. Every body continues in its state of rest, or of uni- form motion in a straight line, except in so far as it may be compelled by impressed forces to change that state. LEX II. Mutationem motus proportionalem esse vi motrici impressae et fieri secundum lineam rectam qud vis ilia im- primitur. LAW II. Change of motion is proportional to the impressed force, and takes place in tJie direction of the straight line in which the force acts. ANSWERS AND SOLUTIONS. 139 LEX III. Actioni contrariam semper et aequalem esse re- actionem : sive corporum duorum actiones in se mutub semper esse aequales et in paries contrarias dirigi. LAW III. To every action there is always an equal and contrary reaction ; or, the mutual actions of any two bodies are ahuays equal and oppositely directed. 136. Change of motion is determined in the, first place by the mode in which the quantity of motion of a moving body is measured. This measure Newton himself explicitly defines in the second of his Definitions which precede his Axiomata sive Leges Motus. The quantity of motion or momentum of a rigid body moving without rotation is, according to Newton, pro- portional to its mass and velocity conjointly. Thus with a double mass and equal velocity the quantity of motion is double ; if the velocity be also doubled, it is quadruple. Take, as unit of momentum, that of unit of mass moving with unit of velocity ; then the momentum of m units of mass moving with v units of velocity w$ll be m v, a result agreeing with the definition of momentum given by Mr. Stewart, 23. By change of motion, then, Newton meant change of momentum, and this change may arise either from change of mass or change of velocity. Mass is an element which is not con- sidered in the motion of material points ; in all other cases variations in mass simply produce proportional variations in momenta. What is meant by change of velocity is plain enough so long as the change takes place along the line of motion ; the change is to be added to the existing velocity if it takes place in the same direction, subtracted from this velocity if it takes place in the opposite direction. But sup- pose a velocity change in direction as well as amount ; how is this change to be measured ? We cannot discuss this ques- tion here, but Fig. 1, page 129, will serve to show the charac- ter of the answer. If the velocity A B be changed to A D, the change is represented in magnitude and direction by A C. In the example mentioned in the Solution of 59, if a ship be sailing due east at the rate of 4 miles per hour, and its course be suddenly changed so that it begins to move with a velocity of 8 miles per hour in a direction 60 south of east, the change has been 4 y 3 miles per hour due south. This represents w r hat has been added, so to speak, to the easterly motion to produce a velocity of 8 miles per hour in a direction 60 south of east. 140 ELEMENTARY PHYSICS. 137. This principle is given by Mr. Stewart at the beginning of Lesson 111. It is stated yet more generally by Thomson and Tait as follows : " When any forces whatever act on a body, then, whether the body be originally at rest, or moving with any velocity and in any direction, each force produces in the body the exact change of motion which it would liave 'produced if it had acted singly on the body originally at rest." The student should try to find illustrative examples of the truth of this principle additional to those given by the author. 138. The Second Law informs us that a force is propor- tional to the change of motion which it produces, and change of motion has been explained to be change of momentum in the Solution of 136. Force, then, is to be measured by the change of momentum which it produces. But as a force produces a continuous change of momentum, we can only compare forces with each other by comparing the changes of momentum produced in some one common interval of time. The simplest interval, and the one universally adopted, is the unit of time, or one second. Thus it appears that the mo- mentum generated in unit of time is the measure of a force, and this is equal to the product of the mass of the moving body and the acceleration, because the change of velocity per unit of time is by definition acceleration. 139. The general definition is a force which acting for a unit of time upon a unit of mass will generate a unit of velocity. 141. The statical unit of force (pound or kilogramme) is the weight of unit of mass, or the pressure whicli unit of mass exerts in consequence of the earth's attraction upon it. Now a unit of mass falling freely acquires a velocity of g units in one second. Therefore the dynamical measure of the force of gravity upon a unit of mass is equal to 1 X 9 or 9 units of force. In other words, the same force which is expressed in statical measure by 1 unit is expressed in dynamical measure by g units, or g dynamical units = 1 statical unit. 1 statical unit Therefore, 1 dynamical unit = . The derivation of the rules referred to is left as an exercise for the student. ANSWERS AND SOLUTIONS. 141 142. The first and fundamental use of the words "pound" and "kilogramme" is to denote units of -mass or quantities of matter equal to that contained in certain precisely defined stand- ards preserved in the Public Archives with the greatest care. The secondary use is as units of force or rather of pressure. "VVe are all brought into daily contact with the force of gravity as producing pressure if not motion, and Engineers, Architects, and many other practical men are constantly called upon to measure and compare pressures produced by gravity. It was very natural and very convenient, therefore, to adopt as a unit of force the pressure produced by the standard of mass, or "standard weight," as it is called, and to employ this unit in measuring all kinds of pressures, whether produced by gravity or by the action of other forces. Thus, a force of 20 Ibs. is a force just capable of sustaining against gravity a 20-lb. weight. "In all countries," says Prof. Maxwell, "the first measurements of force were made in this way, and a force was described as a force of so many pounds' weight or grammes' weight. It was only after the measurements of forces made by persons in different parts of the world had to be compared, that it was found that the weight of a pound or a gramme is different in different places, and depends on the intensity of gravitation, or the attraction of the earth ; so that for purposes of accurate comparison all forces must be reduced to dynami- cal measure." When great accuracy is required in expressing a force in statical or gravitation measure, it is necessary to speci- fy the locality where the observation is made ; thus, so many London pounds of force, so many Paris kilogrammes of force. 143. The spring balance is an instrument for measuriiig force ; for example, it measures directly the force of gravity upon a body ; if the same body be weighed by a spring bal- ance in different latitudes, it will have different weights, because the spring of the balance will be stretched more or less by the changes in the force of gravity. The common balance is an instrument for comparing quantities of matter or masses. However much the earth's attraction upon a given body may vary, it will vary in the same ratio upon the stand- ard weights employed. If, then, we put a body in one pan of a balance, and equipoise it by placing standard weights in the other pan, the equilibrium will be maintained in all parts of the earth, and everywhere the mass of the body will appear to be, as it really is, invariable. 142 ELEMENTARY PHYSICS. 150. Apply the formula given in Appendix V., at the end of Table II., remembering that in this formula force is sup- posed to be expressed in dynamical measure, whereas in the Exercise statical measure is used. This formula applies to every case of momentum generated by a constant force like gravity, and is true, since velocity = acceleration X time. The use of this formula for the Exercise shows at once that the mass of the train is not required in order to find the momen- tum. 153. Since the forces act on a point, mass is eliminated, and, therefore, the forces must be as their accelerations respectively. Moreover, they have the same directions as these accelera- tions, the direction of a force being defined by the direction of the motion which it tends to produce. The lines repre- senting the accelerations may then be considered to represent the forces also, and hence a single force measured by the resultant acceleration, and in its direction, will be the equiva- lent of any number of simultaneously acting forces. Now it can be easily shown that the propositions known as the Par- allelogram of Velocities and the Polygon of Velocities hold equally true of accelerations. Indeed, this may be at once inferred from the fact that acceleration is merely a change of component velocity in a given direction ; hence it is clear that its laws of composition and resolution must be the same as those of velocity. Therefore the laws of the composition and resolution of forces are the same. LESSON V. Definitions of Resultant and Balancing Force. "We will give here general definitions of these important terms, which are often mistaken for each other. I. Resultant. When any number of forces act upon a body, and are not in equilibrium, and when there is one force capable of producing the same effect as the system of forces, this one force is called the resultant of the system. - II. Balancing Force. When any number of forces acting upon a body are not in equilibrium, but are capable of being reduced to equilibrium by the application of a single force, this force is called their balancing force. The resultant and the balancing force of a system evidently form a pair of equal and opposite forces. ANSWERS AND SOLUTIONS. 143 167. The straight line should be drawn, from the point of application of the force, in the direction of the force, and containing as many units of length as there are units of force. 169. The principles here referred to are generally regarded as axiomatic when the terms in which they are expressed are distinctly understood. We shall so regard them, and shall now give them. They should be carefully learned, on account of their subsequent applications. I. Definition of Equal Forces. Two forces are said to be equal if, when ap2)lied to the same point in opposite directions, they balance one another, or are in equilibrium. Forces in equilibrium are often called pressures. This definition is only a modified form of the following more general principle, which is Newton's Third Law of Mo- tion applied to forces in equilibrium. II. Action and reaction are always equal and opposite. A table presses against a book just as much as the book against the table ; if this were not so, the book and table would move either downwards or upwards, according as the pressure of the book or of the table preponderated. III. If a material point or rigid body be acted on by a system, of forces, then the additional application of a system of forces in equilibrium will have no effect. IV. A force may be transmitted to any point in the line of its action, without altering its effect on a rigid body. V. The tension of a perfectly flexible, cord in contact only with perfectly smooth surfaces is the same throughout its length. 174. See 153, and Solution. 188. Two parallel forces are said to be like when they act in the same direction, unlike when they act in opposite direc- tions. 189. Let P and Q be the forces acting at A and B respec- tively, A and B being regarded as rigidly connected. The effect of the forces will not be altered if we apply two equal forces, S, S, at A and B acting along A B in opposite directions (Axiom III.). Compound them with P and Q respectively, and we obtain for the resultants X and Y. Pro- duce the lines of action of these resultants till they meet at D, and draw D C parallel to the lines A P and B Q, meeting 144 ELEMENTARY PHYSICS. A B at C. Transfer the resultants X and F to D (Axiom IV. ), resolve them along D C and a straight line through C parallel to A B ; each of the latter components will be equal to S, and they will act in opposite directions, and will balance each other ; the sum of the former components will be P -h Q. R' \ FIG. 6. Hence the resultant of the forces P and Q is P -f- Q, and acts along a line D C parallel to the lines of action of P and Q in the same direction ; so that it may be supposed to act at C. A force represented by C B/ equal and opposite to C R is the balancing force. ANSWERS AND SOLUTIONS. 145 To find the position of C. The triangles A P X and D A are similar, being equiangular with respect to each other, and the line A S = the line P X. TJTTvp *? ' "R P P B C Therefore - , = ^. Similarly - = ^. Therefore - = -^ 190. This theorem may be demonstrated in precisely the same way as the preceding, only the changed conditions re- quire a separate figure. The student should draw the figure and go through with the demonstration. Or, we may reason thus, using Fig. 6 and its conditions : The forces P, Q, and R' = P -f- Q. form a system of three parallel forces in equilibrium. Of these P and R* form a pair of unlike parallel forces held in equilibrium by Q, which is therefore by definition their balancing force (see beginning of solutions of exercises upon this Lesson) ; and a force equal to Q, and acting in the opposite direction at the point B, is their resultant. We have proved that P : Q = B C : A C. By theory of proportions, P -+- Q : P = (B C -h A C) or A B : B C, which proves the proposition. 193. For it is evident that the two preceding proofs hold equally good, whatever be the angle between the lines of action of the forces and the line drawn across them. In Fig. 6 this angle is a right angle, but this is not necessary. 194. Let P, Q, S, T, be the parallel forces acting at the points A, B, C, D, respectively. Join A B, and find a point L on A B such that P : Q = B L : A L. Then L is the n point of application of the resultant of P and Q, which resultant is equal to P + Q. Join L C, and in the same way find M the point of application of P + Q acting at L and S acting at C ; this re- sultant will be P -+ Q + S. Proceeding in the same way with T FIG. 7. we find N the point of application of P -h Q -H S 4- T = R, the resultant of the four 7* J 146 ELEMENTARY PHYSICS. parallel forces. If any of the forces act in the opposite direc- tion, the process of 190 must be used. 195 If the parallel forces in Fig. 7 were all turned through the same angular distance about their points of application in such a manner that their lines of action remained parallel, it can be easily seen that the point of application N of their resultant would remain unchanged, and the resultant would be turned about N through the same angular distance. This is true whatever be the number of the forces, and this remark- able constancy in position of the point of application of the resultant of any number of parallel forces has caused it to be called the centre of the system of parallel forces. 198. A couple is a system of two equal, unlike, parallel forces. Its arm is the distance between the lines of action of the two forces. Its moment is the product of either force into the arm, and is usually con- sidered positive or negative ac- cording as it tends to produce rotation opposite to, or the same as, that of the hands of a watch. Thus in Fig. 8, P, P, are the forces, A B the arm, P X A B the moment which in this case is negative. 199. The moment of a force with respect to a point is equal to the product of the force into the distance from the point to the line of action of the force. The distinction between positive and negative moments is the same as that between positive and negative couples. For example (Fig. 9), P is the force agting in the direc- tion P. Then its mo- ment with respect to the point A is P X A B ; its moment with respect to A' is P X A' B' ; its moment with respect to A" (or any PNl Fia 8. f point on its line of action) is P X Q == 0. FIG. ANSWERS AND SOLUTIONS. 147 The doctrine of moments holds a position of the first im- portance in the mechanics of rigid bodies, for the reason that a moment is the appropriate measure of the tendency of a force to produce rotation. 200. "The moment of a force round any axis is the mo- ment of its component in any plane perpendicular to the axis, round the point in which the plane is cut by the axis. Here we imagine the force resolved into two components, one par- allel to the axis, which is ineffective so far as rotation round the axis is concerned ; the other perpendicular to the axis, that is to say, having its line in any plane perpendicular to the axis. This latter com- ponent may be called the effective component of the force with respect to rotation round the axis. And its mo- ment round the axis may be defined as its moment round the nearest point of the axis, which is equivalent to the preceding definition." (Thom- son and Tait.) 201. Let (Fig. 10) P, Q, and R (= P + Q) be the forces acting at the points A, B, C, respectively. Choose any point in the plane of the forces ; join B, the line B cutting the lines of P and R at A' and C'. Then we have FIG. 10. x C' = (P + Q) x C', = P x (0 A' + A'C') + Q x OC', since, by 193, P x A' C' = Q x B C'. Therefore, P x A' -t- $ x OB P x C' = 0, the first side of which equation is the algebraic sum of the moments of the forces. 148 ELEMENTARY PHYSICS. 202. Let P t , P 2 , P 3 , &c., be the forces. Draw any line across their lines of action, and in it choose a point of refer- ence, 0. Let A x = a v A 2 = & 2 , A 3 = a 3 , &c. FIG. 11. First find by means of 193 the resultant of P, and P 2 ; if we denote it by IV, we have R 1 = P l + P 2 . Divide A t A 2 into parts inversely as the forces, so that P l x Aj Xj = 2 P x A 2 X r If we denote OX, by as' we have P i x (x' a,) = P 2 x (rt, a'), or, (P t + P 2 ) x' =P l a l + P 2 2 , that is, R' x' = P l a l + P 2 a v Similarly we should find the resultant of R* and Ps to be Rii = p t + P 2 + p s , and that P,'/ x" =Rfx' + P 3 ^ 8 = P t a, + P 2 , + P, r^ 3 . Hence, finally we have the two equations given in the state- ment of the Exercise. ANSWERS AND SOLUTIONS. 149 Negative forces or negative values of any of the quantities at, a, &c., are included in this method, provided the gen- eralized rules of multiplication and division in algebra are followed. 203. The necessary and sufficient conditions for the equi- librium of any number of parallel forces in one plane are the following : (1) algebraic sum of the forces = 0. (2) algebraic sum of the moments of the forces round any point in the plane = 0. If equation (1) does not hold, but equation (2) does, the forces have a single resultant whose line of action passes through the origin of co-ordinates. If equation (1) does hold, but equation (2) does not hold, the system reduces to a couple ; this is indicated by the fact that x is equal to infinity (since R x is a finite quantity, arid JR by hypothesis is zero) ; in other words, the point of appli- action of the resultant is at an infinite distance, which is the case with a couple. 204. In the first kind of Levers, the fulcrum is between P and W ; in the second kind, W is between P and the ful- crum ; in the third kind, P is between W and the fulcrum. 207. The mechanical advantage of any mechanical contriv- ance or combination whatsoever is expressed by the fraction W , P being the applied force or power, W the weight which is raised or pressure which is exerted, and friction, &c., being neglected. In the case of the lever this ratio is often called by workmen the leverage. There exists a popular impres- sion, that a machine can generate or create force. This im- pression arises from the well-known fact that by the aid of a machine an enormous weight can be raised, or resistance overcome, by the application of a very small power. But on examination it will be found that in the exact proportion in which a machine diminishes the power required to raise a given weight, it increases the distance through which this power must act in order to raise the weight through a given height. Take a lever the arms of which are as 10 to 1 ; we have seen that a pound weight hung at the extremity of the longer arm will balance 10 pounds at the end of the shorter 150 ELEMENTARY PHYSICS. arm ; the slightest addition to the pound weight, or rather, friction being neglected, the slightest exterior impulse, as a touch with the finger, will cause the pound weight to de- scend and raise the 10 pounds. But a little geometrical reflection will make it obvious that, in order that the 10 pounds may rise 1 inch, the pound weight must descend 10 inches. A more complex yet familiar instance is furnished by the mech- anism of a watch. Here the power communicated by the main-spring is applied to a train of wheels, and produces a much more rapid movement in the balance-wheel. But the slightest touch of the finger will check the balance-wheel, while the winding up of the main-spring requires a far greater force. And on examination it would be found that, making allowance for friction, the force with which the balance-wheel moved was precisely as many times less than that required to move the main-spring as its motion was more rapid. In fact, one grand and simple law is rigorously fulfilled by every ma- chine and mechanical combination, however complex. This law, if we neglect friction, may be stated as follows : In every machine the product of the power and the distance through which the power moves in its own direction is equal to the product of the weight (or resistance overcome] and the dis- tance through which the weight moves in its own direction. This is a modified form of the celebrated principle of Vir- tual Velocities ; it may now be more appropriately termed the Principle of Work. In fact, adopting the idea and measure of work, explained by Mr. Stewart in Chapter III., this prin- ciple is stated simply by saying that, friction being neglected, the work transmitted by a machine is unaltered in amount ; or more generally, friction being included, in every machine, the parts of which are moving uniformly, the work done by the power the work done against the resistance + the work done against friction. Thus it appears that a machine may increase or diminish the magnitude of the power in any given ratio, but that it cannot increase the product of the force acting at any part into the space through which this force moves, or, in other words, it cannot increase the work transmitted by the machine. In point of fact, this product is diminished by the effect of friction as the force is transmitted through the machine- The use of a machine is to apply more advantageously the force ANSWERS AND SOLUTIONS. 151 applied to it, to transmit force, and to change in a desired de- gree the direction and velocity of motion. 211. Let a and b denote the unequal arms, and x the true weight of the body. Put the body on the pan attached to the arm , and balance it by weights in the other pan ; call these weights W. Next reverse the operation by putting the body on the pan attached to the arm 6, and balancing it by weights W on the pan attached to the arm a. Then W and Wf are the two false weights. Now in the first operation we have, by 205, ax = Wb, and in the second operation similarly we have bx = W> a. Hence a b x 2 = IP or x 2 = W W. And x = \/ W W. 215. The Wheel and Axle is essentially a Lever of the first kind, the fulcrum being the centre of the axle, and the power and weight being applied at the extremities of radii of the FIG. 12. wheel and the axle respectively. This is easily seen by an inspection of Fig. 12, which represents a common Windlass. 152 ELEMENTARY PHYSICS. Here in place of a wheel we have what is equivalent, four arms or spokes, which are turned by hand. The proportion given in the statement of the Exercise can be readily obtained by the student. The Wheel and Axle possesses an important advantage over the simple Lever, in the fact that it allows us to raise a weight through any given height. With a simple Lever a weight can be raised only a small distance before it becomes necessary to place the Lever in a new position, and to support the weight by some other force while this change is being made. The Wheel and Axle is a practical arrangement for continuing the action of a Lever as long as may be required, the weight rising all the time. 217. One simple principle will explain the mechanical ad- vantage of any system of Pulleys, however complex, namely, the principle of the tension of a cord, given in the solution of 169. The single movable Pulley is shown in Fig. 13. There is only one cord, which, by the principle A"? L /CB referred to, has the same tension through- out (friction, &c., being neglected). The weight, W, is equally supported by the parts of this cord in contact with the Pulley at A and B. Therefore W = 2 X tension of the cord. But it is evident that P = tension of the cord. Therefore, W = 2 P. The single movable Pulley may also be regarded as a Lever with equal arms, and the fulcrum at the centre of the Pul- FIG. 13. l e y- The forces acting on equal arms must be equal, and therefore the press- ure on the fulcrum is twice the tension of the cord, or 2 P. But W causes this pressure ; therefore, W 2 P. 218. Fig. 14 represents the three systems of Pulleys referred to in Exercises 218-220. Of these systems (1) is the sim- plest, and also the most convenient in use. We will solve case (2). In this system there are in general n (here 5) mov- able Pulleys, each hanging by a separate cord as shown in the figure. The tension of the cord which passes under the highest movable Pulley A is by the principle of tension equal to the power P. By the same reasoning which was used in the case ANSWERS AND SOLUTIONS. 153 of the single movable Pulley, the tension of the cord pass- ing under the next lower Pulley is 2 P, of the cord passing under the third Pulley is 4 P = 2 2 P, of the next cord is 2 3 P, of the cord passing under the lowest Pulley is 2* P. The ten- sion of this last cord is by the same reasoning equal to ^ W. Therefore, W = 2 5 P. And, in general, W = 2 P. 223. The Inclined Plane is a plane inclined at any angle to the horizon. The principle of the Inclined Plane consists in tnis, that a weight W can be supported on the Inclined Plane by a power P which is less than W. It is a direct example of the Parallelogram of Forces. Conceive that the Plane is perfectly smooth, and likewise the body which is resting on the Plane. This body has its centre of gravity at G, so that, as will be explained later, its weight W may be regarded as entirely concentrated in a heavy point at G. Three forces are in action when the body is in equilibrium, the weight of the body W acting at G ver- tically downwards, the power P applied to keep the body at rest on the Plane and acting from G up the Plane, and the reaction R of the Plane arising from, the pressure of the body 154 ELEMENTARY PHYSICS. on the Plane and acting in a direction perpendicular to the Plane. Substitute for W its two components G M and G N, \ W FIG. 15. parallel and perpendicular respectively to the Plane. These components are opposite to P and R respectively ; hence they must be equal to them each to each, or, G M = P, and G N = R. The triangles ABC and G W N are similar ; whence it follows that and that P^ _ BC = height W~ AB length ' R AC AB length * 226. Let the student represent the conditions of this ques- tion by a diagram similar to that given in the solution of 223. He will see that the lines representing the three forces P, W 9 and R form a right triangle. This triangle may be shown to be similar to the large triangle the hypothenuse of which is the Plane. Hence the required proportion is easily obtained. 227. The Screw is a movable inclined plane wrapped round a right cylinder. The relation between the Inclined Plane and the Screw may be understood by reference to Fig. 16. ANSWERS AND SOLUTIONS. 155 Suppose that we unroll paper previously rolled round the right cylinder, A B K L, by causing the cylinder to revolve on its axis through exactly one revolution ; we shall obtain a rectangle, B A D F. Divide this rectangle into a certain number of equal rectangles by drawing lines parallel to the base, A D ; draw diagonals to these rectangles as shown in the figure ; and lastly, roll the entire rectangle again upon the cylinder. The diagonals will form a continuous curve called a helix, composed of spirals, one above another, equal in num- ber to the number of partial rectangles (in this case two). In rolling up the rectangle the line mp for example falls into the position M P, the point D falls, after one revolution, upon A, the point C falls upon E, and F upon B. If, now, we take a cylinder (of wood or iron) having on its surface, in place of a helix, traced in pencil or ink, a projecting thread making at all points the same given angle (called the pitch of the Screw) with the horizon, we have a Screw ready for use. Thus, the Inclined Plane A C forms one revolution of the helix, its base is equal to the circumference of the cylinder, and its height to the distance between two threads, a quantity of special importance in the theory of the Screw. 228. In order to adapt the Screw to use, a nut must be employed; this is a block pierced with an equal cylindrical aperture, upon the inner surface of which is cut a groove the 156 ELEMENTARY PHYSICS. exact counterpart of the thread of the Screw. It is evident that the Screw can only be made to move in the nut by revolv- ing about its axis. Suppose the Screw, with its axis vertical, to be contained in the nut without friction, and let a weight, W, be placed upon it ; then the pressure due to this weight will be transmitted to every point of the thread of the Screw. Suppose that there are n points on the thread, each point in contact with a corresponding point of the groove in the nut ; then each point is acted upon by a vertical force equal to . Now it is clear that under these circumstances the Screw would descend by revolving on its axis (that is, the various points of the thread would slide down the groove with a spiral mo- tion), unless prevented by some force ; call this force P v and let it act at some point on the circumference of the cylinder along a horizontal line tangent to the circumference. This force P. will likewise be distributed along the thread, each p point being acted upon by a horizontal force equal to . Con- sider now the equilibrium of any point A of the thread. It is P W kept at rest bv three forces, . , and the reaction of the n n thread, which is equal and opposite to the normal pressure on P W the thread arising from the combined action of and , n n and which need not be further considered in this investiga- tion. Bearing in mind the analogy between the Screw and the Inclined Plane, it is evident that this case is similar to that treated in 226, and considered as an instance of the Inclined P W Plane, we have, : = height of plane : base of plane. n n Simplifying the first ratio, and substituting for the terms of the second ratio their equivalents in terms of the Screw, we obtain P, : W = distance between two threads : circumfer- ence of the cylinder. Practically, the Screw is never used as a simple machine, the power being applied by means of a Lever passing either through the head of the Screw or through the nut. The Screw acts, therefore, with the combined power of the Lever and the Inclined Plane ; and in investigating the effect we must take ANSWERS AND SOLUTIONS. 157 into account both these Mechanical Powers. The proportion just given represents the effect of the Inclined Plane. To com- bine this with the effect of the Lever is left as an exercise for the student. The result is stated on page 91. FIG. 17. To produce pressure with the Screw either the Screw or the nut must be fixed ; whichever is free is then turned, and made to press against the resistance. It is evident that one revo- lution causes the Screw (or the nut) to advance by an amount equal to the distance between two threads. In the common Screw-press, represented in Fig. 17, the nut is fixed and the Screw movable. A is the Screw, enlarged below at C, where it is pierced with two holes at right angles for applying the Lever. B is the nut firmly fixed in the upper part of the press. By turning the Screw in, the nut it descends and pushes before it the plate D, which works in guides so that it can have only a vertical motion. The body is compressed by placing it on the fixed platform E, and lowering the Screw and plate D. 158 ELEMENTARY PHYSICS. In deducing the law of equilibrium of the Screw we have neglected the effect of friction for the sake of simplicity. But the amount of friction in every Screw is very great ; in fact, the Screw owes its utility to friction ; for if there were no fric- tion the Screw would overhaul, that is, turn backwards the in- stant the power was removed. To prevent this the friction must be sufficient to consume more than half the power applied. Note. All the Exercises upon the Mechanical Powers can be very readily solved by means of the principle of Work stated in the solution of 207. The student should first work out these Exercises by the methods already given or indicated, and then solve them by the aid of the principle of Work. The greater simplicity of this latter method is strikingly shown in the case of the Screw ; indeed, it is obvious that, friction being neglected, this beautiful principle may be employed with equal ease in any machine, whatever its nature and however complex its mechanism. LESSON VI. 271. For the Third Law of Motion, see Solution of 135. Newton gives three illustrations. 1. " If you press a stone with your finger, the finger is also pressed by the stone." This is an illustration of forces in equilibrium (see Solution of 169). 2. " If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone ; for the distended rope, by the same endeavor to relax or un- bend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the pro- gress of the one as much as it advances that of the other." 3. " If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities, but in the quantities of motion (momenta) of the bodies." 272. "Every force acts between two bodies, or parts of bodies. If we are considering a particular body or system of bodies, then those forces which act between bodies belonging ANSWERS AND SOLUTIONS. 159 to this system and bodies not belonging to the system are called External Forces, and those which act between the different parts of the system itself are called Internal Forces." (Maxwell.) The same force may play the part of an interior or exterior force according to the point of view adopted. Take, for in- stance, the motion of a body which falls to the earth ; the attraction of one of the particles of this body for a particle of the earth is an exterior force ; if, on the other hand, we consider the motion of the material system formed of the body and the earth, this same attraction becomes an interior force. The student should find other illustrations. 279. For this distinction, see the solution of 56. The student should give illustrations. The duration of the action of an impulsive force is too brief to be measured, so that we cannot measure the force by the momentum generated in one second. An impulsive force is measured by the total momen- tum -which it generates. 280. A body is said to be perfectly elastic if, when it im- pinges perpendicularly on a fixed plane, it will recoil back from the plane with equal velocity, or when the velocity of recoil is equal to the velocity of approach. An imperfectly elastic body is one whose velocity of recoil is less than its velocity of approach. A perfectly inelastic body is one whose velocity of recoil is zero. The ratio of the velocity of recoil to that of approach is called the coefficient of restitution. 281. The principle of conservation of momentum is stated in the enunciation of 292. In applying it to the case sup- posed, we have for the material system the two equal bodies ; and since their velocities are equal and in opposite directions, the algebraic sum of their momenta is zero. The collision does not alter this sum, for it brings both bodies to rest. LESSON VII. 306. Let m, m r denote the masses, d their distance apart ; then, mm) force = w~ d? The simplest unit of attractive force is defined by putting m = m f = d = 1. 160 ELEMENTARY PHYSICS. LESSON IX. 335. Regarding bodies as composed of particles, each, acted on by gravity, Mr. Stewart has given in 47 a definition of the centre of gravity of a body which may be stated in other words thus : A point in the body such that, if supported, the body will remain at rest, but, if not supported, the body will fall under the action of gravity. But this definition is an obvious consequence of the following mechanical defini- tion, which should be read in connection with the solution of 195 : the centre of gravity of a body is the centre of the system of parallel forces exerted by gravity upon the body. 336. A body is said to be homogeneous, from a mechanical point of view, when equal volumes of the body have equal mass'es. A plane of symmetry in a body is a plane such that every perpendicular line passing through it cuts the surface of the body in two points equally distant from the plane. A body may have more than one plane of symmetry ; in a sphere, for example, every great circle is a plane of symmetry. Similarly, if a line can be drawn through a body, such that every perpendicular drawn through the line cuts the surface of the body in two points equally distant from the line, this line is called a line (or axis) of symmetry, with respect to the body. The surface of the body is said to be symmetrical with re- spect to the plane, or the line, as the case may be. A point is called the centre of a surface when every straight line drawn through the point and terminated by the surface is bisected by the point. The centre of a surface is also the centre of the body which is bounded by the surface. Considered in this light, it is also called the centre of figure, or geometric centre. 340. In this, and the three next Exercises, the student should find and state the particular conditions of equilibrium for each case, and should define and distinguish between stable, unstable, and neutral equilibrium, and give examples of each kind. The general principle to be applied in all cases is that the centre of gravity must be supported in order to have equi- librium. 348. The principal use of the pendulum in Physics is to determine the value of the acceleration due to gravity. ANSWERS AND SOLUTIONS. 161 Taking the formula for the simple pendulum given in 360, we easily find that g = -j^~, whence it follows that the value of g can be found hy making a pendulum vibrate, and measur- ing I and T. But to measure accurately these quantities is by no means a simple matter. In the first place, the formula applies directly only to the simple pendulum, which is defined as a heavy particle sus- pended from a fixed point by an inextensible weightless thread. Such a pendulum can exist only in the mind, and is a concep- tion employed by the investigator for the sake of simplifying the inquiry after the laws of pendulum oscillations. In prac- tice we employ bodies which oscillate about a horizontal axis, called the axis of suspension ; these are termed, for the sake of distinction, compound pendulums. The length of a com- pound pendulum is the length of a synchronous simple pendu- lum, that is, of a simple pendulum which will oscillate in the same time as the given compound pendulum. This length can be calculated in a given case, when the pendulum is of regular form, by the aid of formulae which are given in higher treatises on Dynamics ; but its value is more easily obtained by what is called Kater's method. This method is founded on a remarkable property of a com- pound pendulum, discovered by Huyghens, called the converti- bility of the axes of suspension and oscillation. The axis of oscillation is an axis parallel to the axis of suspension in the plane containing it and the centre of gravity of the pendulum, and at a distance from it equal to the length of the synchro- nous simple pendulum. The body, in fact, oscillates as if its entire mass were collected on the axis of oscillation. Now the property discovered by Huyghens was this : that if we suspend a pendulum by its axis of oscillation, the former axis of sus- pension becomes the new axis of oscillation, and the pendulum oscillates in the same time as before. Kater constructed a reversible pendulum, which could be supported by either of two parallel knife-edges, one of which could be adjusted to any distance from the other. The length of this pendulum could be found by the method of repeated trials with a great degree of accuracy. In order to measure T, the method which naturally sug- gests itself is to count the number of oscillations which take 162 ELEMENTARY PHYSICS. place in a given time, and then divide the time by the number of oscillations. This is, however, far from an easy process, if accuracy in the results is aimed at. Borda devised a great improvement upon this method, by comparing the motion of the pendulum with the motion of the pendulum of an astronomical clock regulated' to beat seconds. By the use of Borda' s method, called the "method of coincidences," one can calculate the number of oscillations without being obliged to count them. Another method of finding, by the aid of a pendulum, the value of g consists in employing a seconds pendulum of regu- lar form ; in this case T 1, and I can be calculated in the manner already stated. 368. Let P denote the force or pressure perpendicular to two surfaces in contact, by which the surfaces are pressed to- gether ; and let F denote the least force parallel to the sur- faces in contact which is able to move one surface along the other : then, the ratio of Fto P is called the coefficient of fric- tion for the two surfaces. If a body be placed on an inclined plane whose angle of in- clination can be altered at pleasure, then that inclination of the plane for which the body is just about to slide is called the angle of friction for the materials of which the body and the plane are composed. 370. There are five forces to be considered : (1) Gravity, which may be regarded as acting entirely at the centre of gravity of the ladder, (2) and (3) the reactions of the floor and wall perpendicular to the surface in each case, (4) and (5) the forces due to friction, and which act parallel to the surface of the floor and wall. 371. Referring back to Fig. 15, and using the same nota- tion, it is plain that when the body is on the point of sliding, the friction F must be equal to the component of W 9 which is parallel to the plane, that is to W ,- rr-. The pressure R on the plane is equal to W -, r . Therefore we have ,, ~ . - f . , . F height the coefficient of friction = = y-r 3 . It base ANSWERS AND SOLUTION'S. 163 LESSON XI. 376. A perfect fluid is an ideal conception, like that of a rigid or smooth body ; it is defined as a body incapable of resist- ing a change of shape. Common liquids and gases fulfil this definition when in a state of rest, but no existing fluid fulfils the definition when it is in motion. "All actual fluids are imperfect, and exhibit the phenome- non of internal friction or viscosity, by which their motion after being stirred about in a vessel is gradually stopped, and the energy of the motion converted into heat." (Maxwell.} A viscous fluid is a substance such that the very smallest force applied to it will produce a constantly increasing change of form. The change of form may take place very slowly ; but if it takes place so as to be sensible, and continually increases with the time, the substance is viscous. "Thus a block of pitch may be so hard that you cannot make a dint in it by striking it with your knuckles ; and yet it will, in the course of time, flatten itself out by its own weight, and glide down hill like a stream of water." (Maxwell.} 378. It follows from the foregoing definition of a perfect fluid, that its pressure on any surface must be at all points perpendicular to the surface. The student should be able to give the reason. 379. Let S, S', denote the two surfaces, and P, P 1 , the pressures ; then S : S 1 = P : P. Pascal's principle cannot be proved direct- ly, on account of the action of the force of gravity. 385. The proof required is sim- ply this : if the resultant of all the forces which act on the liquid at any point of its surface were not normal (i. e. perpendicular) to the surface at that point, then re- solve the resultant into compo- nents perpendicular and parallel to the surface. The first compo- nent is neutralized by the resist- FIG. 18. 164 ELEMENTARY PHYSICS. ance to compression of the liquid, liquids being practically incompressible ; the other component would cause the particle on which it acts to move along the surface, which is contrary to the supposition that the liquid is at rest. Thus, in Fig. 18, let A C represent this resultant. Then A D and A B are the components referred to, and it is easy to see that the component A B being unresisted would cause motion. 386. The surface is everywhere termed horizontal. Small extents of the earth's surface may be considered planes, but large areas must be regarded as nearly spherical. 387. (1) follows from Pascal's principle, (2) , _ is self-evident, and (3) is likewise self-evident when the meaning of density is taken into account. 388. By 387 the pressure is proportion- al to area X depth X density. Now if the centimetre be used as a unit of length, this product is by 22 the weight in grammes of a volume of the liquid, having the given area and depth. Adopting the unit of weight as the unit of hydrostatic pressure, this product is the pressure in grammes. 399. Let d, d', de- note the densities of the~ two liquids, d) be- ing greater than d. Let C D and A B (Fig. 19) be horizontal lines FIG. 19. drawn through the sur- A/I B ANSWERS AND SOLUTIONS. 165 faces of the two liquids at D and B, the surface of the lighter liquid being at D. Let M N be a horizontal line drawn through the common surface of contact of the liquids at S, and let C A M be a vertical line. Finally, let s, s 1 , denote the areas of sections of the tube at S and N respectively, and put C M = h, and A M = h 1 . Since the liquid columns are in equilibrium, it follows from 387 (1) that the pressures ex- erted upon the surfaces s and s f by the superincumbent liquids must be as the areas of the surfaces. The pressure upon s = slid ; that upon sf = s 1 h' dt. Therefore shditfh'd* =s: h d = h' df h : h' = d' : d or that is, 400. The pressures against the various points on the ver- tical side of the vessel form a system of parallel forces, each force being proportional to the depth of the point to which it is applied. This condition of things is partially represented by the arrows drawn in Fig. 20. FIG. 20. 166 ELEMENTARY PHYSICS. A very little reflection is sufficient to show that the point of application of the resultant of this system must be some- where below G, the centre of gravity of the side. This be- comes evident by considering that all the pressures below G are greater than any of the pressures above G. Calculation shows that this point of application for a rectangular surface is at a point C, just two thirds the depth of the side. This point is called the centre of pressure of the side. 405. For the definition and measure of density, see Solution of 21. The specific gravity of a substance is the ratio of its density to that of some standard substance, usually water at its maximum density. The density of water in the Metric System being unity, it follows that in this system the density and specific gravity of a substance are numerically the same. 422. The method commonly employed is to attach to the body a sinker, that is, a body heavier than water, and large enough to cause both bodies to sink. The specific gravity of the sinker being supposed known, it is easy to determine that of the body ; the student should deduce a formula for this purpose. LESSON XII. 452. The three forms referred to are as follows : 1. By the number of statical units of force in the pressure on unit of area ; e. g. in pounds' weight for square inch, or in kilogrammes' weight per square centimetre. Pressures thus expressed are reduced to dynamical measure by multiplying by the value of g at the given locality. 2. By the height of a column of mercury at C. which would exert by its weight an equal pressure ; thus the average pressure of the atmosphere is described as a pressure of 30 inches, or 76 centimetres, of mercury. 3. In terms of a large unit which is nearly equal to the average atmospheric pressure at the level of the sea. This unit is called an atmosphere, and is used chiefly in measuring pressures in boilers, and in scientific experiments which re- quire very great pressures. These three measures are thus related : in the British sys- tem, one atmosphere pressure due to a height of 29 '905 ANSWERS AND SOLUTIONS. 167 inches of mercury at 32 F. at London, where the force of gravity is 32*1889 feet = about 14f Ibs. weight per square inch. In the Metric system one atmosphere = pressure due to a height of 76 centimetres of mercury at C. at Paris, where the force of gravity is 9*80868 metres = about 1033 grammes' weight per square centimetre. One British atmosphere = '99968 of a Metric atmosphere. 455. The volume of the mercury resting upon unit section of the tube at the bottom of the barometer is h, and its mass, and also its weight, is 13 '596 h. This is reduced to dynamical measure by multiplying by g. 460. In such questions as this the effect of any increase in the atmospheric pressure on the density of water is too small to be considered. 464. Let 2, Ds, &c., .... D n , the densities after 1, 2, 3, &c., . . . . n, strokes of the piston. When the piston is first raised the vol- ume V of air becomes increased to V H- v. Hence by Boyle's law, Do: J) l= = jr+v: V, A = A Similar reasoning shows that and thus finally we have 168 ELEMENTARY PHYSICS. 479. One limit is pointed out by Mr. Stewart at the close of 93. Since each stroke of the piston only removes a part of the air which remains, it is plain that the receiver can never be completely exhausted. This is indicated by the formula of the last Exercise ; as n increases, the value of D n diminishes towards zero as a limit, but this limit is reached only when n is made greater than any assignable quantity. There are other causes which tend to produce an actual limit after a definite and not very large number of strokes. In the first place, it is impossible to avoid the existence of a small space between the bottom of the barrel and the bottom of the piston, when the latter is in its lowest position. This space has been called untraversed space ; and it is evident that it contains a small quantity of air at the atmospheric pressure, when the piston is in its lowest position. "When the piston is drawn up, this air is rarefied ; but unless its tension becomes less than the tension of the air remaining in the receiver, no air can flow from the receiver into the barrel, and the pump ceases to produce any effect. A second cause is leakage, which exists in the best-con- structed air-pumps, and which increases rapidly as the tension of the air in the receiver and barrel diminishes. As the piston descends it expels a certain quantity of air ; as it ascends a certain quantity of air enters from leakage. If these two quantities are equal, the limit of rarefaction has evidently been reached. Lastly, perhaps the most serious cause is the absorption of air by the oil used in lubricating the piston. This oil is poured upon the top of the piston, where it is forced by the pressure of the external air between the piston and the sides of the barrel, and finally falls to the bottom of the barrel. Here it absorbs air which it gives out in part during the ascent of the piston. Hence arises another limit to the degree of rarefaction. APPENDIX. (Containing useful Data, Tables, and Formulae.) I. ENGLISH WEIGHTS AND MEASURES. THE fundamental units of Time, Length, and Mass, respec- tively, are the Mean Solar Day, the Imperial Standard Yard, and the Imperial Standard Pound Avoirdupois. The Mean Solar Day is the average interval between two successive passages of the sun across the meridian. The mean solar day is divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds, so that one second is s^iinr part of a day. For a great number of purposes the mean solar day is an inconveniently large unit, and the minute or second is therefore employed. The Imperial Standard Yard is, by Act of Parliament, the distance between two points in a certain bronze bar deposited in the Office of the Exchequer in London, the temperature of the bar being 62 F. (see 18 and 19 Viet. c. 72, July 30, 1855). The Imperial Standard Pound Avoirdupois is, by the Act above cited, a platinum weight marked "P. S. 1844, 1 lb.," deposited in the Office of the Exchequer in London. TABLE I. Measures of Length. in. ft. yd. rd. fur. m. Inch 1 Foot 12 1 Yard 36 3 1 Rod 198 16J 4 1 Furlong 7960 660 220 40 1 Mile 63 360 5 280 1 760 320 8 1 170 APPENDIX. NOTES. 1. In scientific investigations the foot and the inch are gener- ally employed, being more convenient than the yard. The inch is sub- divided decimally and also binarily (i. e. into halves, quarters, eighths, etc.). 2. The mile in the above table is the statute mile. The geographical or nautical mile (also called knot) is -fa of a degree of longitude on the equator or YTETfrTT f the whole equator, and is rather more than 1'15 stat- ute miles. 3. A hand = 4 inches ; 1 fathom = 6 feet ; 1 league = 3 miles. TABLE II. Measures of Surface. sq. in. sq. ft. sq. yd. sq. rd. R. A. sq. m. Square Inch 1 Square Foot 144 1 Square Yard 1296 9 1 Square Rod 39204 272* 30i 1 Rood 1 568 160 10890 1210 40 1 Acre 6 272 640 43560 4840 160 4 1 Square Mile 4 014 489 600 27 878 400 3 097 600 102 400 2560 640 1 TABLE III. Measures of Volume. cub. in. cub. ft. cub. yd. Cubic Inch 1 Cubic Foot 1 728 1 Cubic Yard 46 656 27 1 NOTE. 16 cubic feet of wood make 1 cord foot, and 8 cord feet' or 128 cubic feet make 1 cord. APPENDIX. 171 TABLE IY. Measures of Mass (Avoirdupois Weights}. gr- dr. oz. Ib. qr. cwt. T. Grain 1 Drachm 27-34375 1 Ounce 437'5 16 1 Pound 7 000 256 16 1 Quarter 196 000 7 168 448 28 1 Cwt. 784 000 28 672 1 712 112 4 1 Ton 15 680 000 573 440 34 240 2240 80 20 1 NOTES. 1. The pound is connected with the unit of volume as fol- lows : 1 cubic inch of distilled water weighed in air at 62 F. (bar. 30 inches) = 252*458 grains. 2. The Avoirdupois Pound of matter is equal to the mass of 27'7274 cubic inches of distilled water, weighed as above. 3. For many purposes it is sufficiently accurate to call the weight of 1 cubic foot of water 1000 ounces. Three other measures are much in use, viz., Liquid Measure, the base of which is the Imperial Gallon ; Dry Measure, the base of which is the Imperial Bushel ; and Troy Weight, the base of which is the Troy Pound. The Imperial Gallon is a measure of volume or capacity which w T ill contain 10 pounds, avoirdupois weight, of distilled water, weighed in air, at 62 F., the barometer at 30 inches. The Imperial Bushel is also a measure of capacity, and is equal to 8 imperial gallons. The Troy Pound is a measure of mass, bearing to the Avoirdupois Pound the ratio of 5760 : 7000 ; i. e. it contains 5760 grains' weight of matter. 172 APPENDIX. TABLE V. Measures of Capacity. Liquid Measure. Dry Measure. 4 gills = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 ruarts 1 gallon (gal.) 52^ gallons = 1 hogshead. 2 pints = 1 quart. 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bush.) NOTES. 1. Liquid Measure is used in measuring liquids, and Dry Measure in measuring solid matter consisting of small parts or pieces, as grain, fruit, salt, roots, ashes, &c. 2. The tun = 2 pipes = 4 hogsheads = 210 Imperial gallons. TABLE VI. Troy Weights. gr. dwt. oz. Ib. Grain 1 Pennyweight Ounce 24 480 1 20 1 Pound 5 760 240 12 1 NOTES. 1. Troy weight is chiefly employed in weighing gold, silver, and precious stones. 2. Apothecaries, in compounding medicines, divide the ounce (3) into 8 drachms (5) and the drachm into 3 scruples O), so that 1 scruple = 20 grains. 3. 480 minims = 1 fluid-ounce, 20 fluid-ounces = 1 pint. APPENDIX. 173 II. UNITED STATES WEIGHTS AND MEASURES. THE fundamental unit of time in the United States, as in England, is the mean solar day. It is also divided, as in Eng- land, into hours, minutes, and seconds. The United States standards of length and mass are copies of old English standards, arid are very nearly the same as the present Imperial standards of Great Britain. Careful comparison has shown that the United States actual standard yard (a brass scale made by Troughton of London, in 1813, and deposited in the Office of Weights and Measures in Washington) is equal to 1 '000024 Imperial yards, and that the United States actual standard of mass (a Troy pound of brass, made by Kater in 1827, and deposited in the United States Mint) is equal to '99999986 Imperial Troy pounds. The same ratio 0*99999986 also exists between the United States and Imperial Avoirdupois pounds. The United States Gallon, or standard unit of liquid measure, is the old wine gallon of England, and contains 231 cubic inches. The United States Bushel, or standard unit of dry measure, is the British Winchester bushel (formerly a standard in England), and contains 21 50 '42 cubic inches. The relative value of the Imperial and United States stand- ards of capacity is as follows : Comparative Values of English and TJ. S. Units of Capacity. 1 Imperial Gallon (277-274 cub. in.) = 1-2001 U. S. Gallons. 1 U. S. do. (231-000 cub in.) = 0-8331 Imperial Gallons. 1 Imperial Bushel (2218-192 cub. in.) = 1-0315 U. S. Bushels. 1 U. S. do. (2150-420 cub. in.) = 0-96945 Imperial Bushels. The ratio of the United States and Imperial gallons is very nearly as 5 : 6 ; and that of the two bushels nearly as 16 : 17. With one exception, the United States tables of Weights and Measures are the same as the English tables which have been 174 APPENDIX. given. This exception occurs in Avoirdupois Weight, in which the United States quarter is equal to 25 Ibs., and the United States ton therefore is equal to 2000 Ibs. NOTES. 1. The English (long or gross) ton of 2240 Ibs. is still used in estimating English goods in the United States Custom Houses, in selling coal at wholesale from the Pennsylvania mines, and in the wholesale iron and plaster trade. 2. In liquid measure 63 United States gallons = 52| Imperial gal- lons = 1 hogshead. 3. 1 United States fluid-ounce = 1 fluid-ounce and 20 minims, Imperial measure ; and 16 United States fluid-ounces = 1 United States pint. 4. The United States barrel contains from 28 to 32 United States gallons. 5. In dry measure the half-peck, sometimes called the dry gallon, contains 268' 8 cubic inches, so that liquid and dry measures, of the same name (for example, the quart), stand to each other in value as 231 : 268*8. 6. 'The bushel, heaped measure, contains about 2750 cubic inches, or rather more than 5 pecks. APPENDIX. 175 III. THE METRIC SYSTEM OF WEIGHTS AND MEASURES. The fundamental units of Time, Length, and Mass, respec- tively, are the mean solar day, the metre, and the kilogramme. The Mean Solar Day has been already denned (see Eng- lish Weights and Measures). All civilized nations, in fact, employ the mean solar day as the fundamental unit of time, and divide it into hours, min- utes, and seconds, as in England and the United States. Its universal use is due to the fact that it combines, in an unrivalled degree, the cardinal virtues of a fundamental unit of measurement. Its magnitude is determined, not by legis- lators or men of science, but by nature ; its duration can be easily ascertained to a high degree of precision ; it has not changed since the time of Hipparchus (150 B. c.) by so much a TOTT of a second ; and finally, the purposes of life and the actions of men are so largely dependent upon the position of the sun that a measure of time derived from his motion is vastly more convenient than any other. The Metre, or standard of Length, is the distance between the ends of a certain platinum bar made by Borda, and pre- served in the Archives de 1'Etat in Paris, the bar being at the temperature of melting ice. The metre was intended to be an exact ten-millionth part of a quadrant of the earth's meridian, but according to the most trustworthy measurements the actual standard metre ( Borda' s platinum rod) is less than its intended value by an amount not exceeding oVo of a metre, or about y^ of an inch. The Kilogramme, or standard of Mass, is the quantity of matter in a certain platinum weight made by Borda, and depos- ited in the Archives de 1'Etat in Paris. It was intended to be (and is very nearly) equal to the mass of one cubic decimetre of distilled water at the point of maximum density (about 4 C.). The metre and the kilogramme derive their authority as standards from a law of the French Republic in 1795. Of the prototype standards, kept at Paris, numerous copies have been taken, which, after having been compared with the originals with the utmost precision of which modern science is capable, have been made standards of reference and verification in the various countries ift which the Metric System has been adopted . 5* 176 APPENDIX. The Metric System of Weights and Measures is a system in the true sense of that word, and the simple mode in which all the other units are derived from the standards is in striking contrast with the want of system and troublesome numerical relations existing in English Weights and Measures. Nearly all the derived units in the Metric System are obtained by the application of two very simple principles, viz. decimal multiplication and division, and the use of squares and cubes of linear units as units of surface and of volume respectively. Hence each unit of length, and also of mass, is 10 times larger than the next smaller unit in order, each unit of surface 100 times larger, and each unit of volume 1000 times larger. Names for all the units are formed from the roots metre and gramme, by employing the Greek prefixes, deca (] 0), hecto (100), kilo ,(1000), for the multiples, and the Latin prefixes, deci (O'l), centi (0*01), milli (O'OOl), for the sub-multiples. In the meas- ures of surface and of volume the words square and cubic re- spectively are also employed. Thus, 1 centim-etre is equal to O'Ol of a metre, 1 cubic decimetre is equal to 0*001 of a cubic metre, 1 kilogramme is equal to 1000 grammes, &c. Four units have received distinct names, viz. : The Are, which is equal to 1 square decametre. The Stere, which is equal to 1 cubic metre. The Litre, which is equal to 1 cubic decimetre. The Tonne, which is equal to 1000 kilogrammes. The table opposite exhibits the names, abbreviations, and relative values of nearly all the measures in the Metric System. NOTES 1. The Are is employed as a unit of land measure ; the Stere as a unit for measuring cord- wood ; and the Litre as a unit for measuring fluids, and dry substances in small pieces, as grain, fruit, salt, &c. 2. Decimal multiples and divisions of the litre are in use, and are named by employing Greek and Latin prefixes in the same way as in measures of length ; thus, decilitre, kilolitre, &c. Similarly, 100 ares is called a hectare, YQTT f an are a centiare, lOsteres a decastere, -^o of a stere a decistere, 10000 metres a myriametre, 10000 grammes a myria- gramme. 3. Binary derivatives, as in England, are much employed on account of their practical convenience ; e. g. the demi-litre, the double hecto- gramme, &c., &c. 4. The cubic metre, when employed in France as a measure of the capacity of ships, is called a tonneau. The word tonneau also denotes, in the French navy, the weight of a cubic metre of water. 5. 100 kilogrammes is called a quintal. APPENDIX. 177 *^ * cc fcb bb bb ^ O "IJ. V r^ ^ o bb &JO q g fi ^>* 3- ^ ^ g fl 05 >^ ^ g o> i S 1 i 1:-* 1 1 1 111*111 3 1 1 1 1 1 1 1 O O o T 1 II * 11 fi IT i a IT a i o EH ^ ^3- r S g 1 I f i^ s^5^ ii it us i ^ ,3 o o ? o g i i fr rH O - OS CO CO rH !>. "* CM O t^COC -* OS CO CO OOOrH rH CO < O CO CM * rH OS CO CO O C t^ OO CO CO -Hi O OO VQ OS C TO O CO rH CO r I CO r- 1 1O < OS CO OS C^l J>- t> O CO J>- C CO OS OS C<1 O J> ^ rH 1O OO OSt^COCO OSCOO5 CO-*iO CO O rH O OS O J 1O 1O 1O OS CO OS OO C<1 rH COO5CO ''"^JS OCO *O l>- " rH OS rH O "* CO (N O O rH Q O d O CO ^H CO CO rH Q rH O O O s 1 II I. ^ S r2 a a s fl __ '-' D C o3 M ^ 5i S rs *-jj -^ o o a p) fl rt 3 ; *sS 11 CD f^ rO 8, 8,: 4 APPENDIX. 179 IV. MATHEMATICAL DATA AND FORMULAE. ALGEBRA. Arithmetical Progression. (1) Let a denote the first term, I the last term, d the com- mon difference, n the number of terms, s the sum of the terms ; then, I = a + (n 1)^; s = n - . 2 Geometrical Progression. (2) Let the same notation as above be used, except that in this case r denotes the common ratio ; then, rl a I =* a jT n\. s r l * GEOMETRY. (3) TT (ratio of circumf 'nee of circle to diameter) = 3*14159 (4) Circumference of a circle (radius r) = 2 irr (5) Area of a circle (radius r) = TT r 2 (6) Surface of a sphere (radius r) = 4 TT r 2 (7) Surface of ellipse (semi-axes a and b) = TT a b (8) Surface of right cylinder (height h, base TT r 2 ) = 2 TT rh (9) Surface of r,ight cone (height h, base irr 2 ) = TT r y/ r 2 + h 2 (10) Volume of a sphere (radius r) = | TT r 3 (11) Volume of ellipse (semi-axes a and b) = f TT a b c (12) Volume of right cylinder (height h, base irr 2 ) wr^h (13) Volume of right cone (height h, base TT r 2 ) = 5 irr' 2 h (14) Sum of the three angles of a triangle = 180 (15) Area of a triangle of altitude h and base b = %bh. (16) Two triangles, which have equal bases and equal alti- tudes, are equal. 180 APPENDIX. (1 7) Two triangles are similar ; (a) If they are equiangular with respect to each other. (b) If they have their homologous sides proportional. (c) If they have an angle of one equal to an angle of the other, and the sides including these angles propor- tional. (18) The perpendicular upon the hypothenuse of a right tri- angle from the vertex of the right angle, divides the tri- angle into two triangles which are similar to each other and to the whole triangle. (19) If a perpendicular be erected from any point on the di- ameter of a circle meeting the circumference at a point A, and chords be drawn from A to the extremities of the diameter ; then, (a) The perpendicular is a mean proportional between the segments of the diameter. (&) Either chord is a mean proportional between the di- ameter and the adjacent segment. (20) The square described upon the hypothenuse of a right triangle is equivalent to the sum of the squares de- scribed on the other two sides. (21) Similar triangles (or polygons) are to each other as the squares of their homologous -sides. (22) The sum of the plane angles which form a solid angle is always less than four right angles. (23) The area of the surface described by a line revolving about another line on the same plane as an axis is the product of the revolving line by the circumference de- scribed by its middle point. PLANE TRIGONOMETRY. Definitions of the Functions of an Angle. Let Ti denote the hypoth- enuse of a right triangle, a the perpendicular, b the base, A and B the angles opposite a and b respectively ; then, APPENDIX. A W sin A = T , , A w tan .4 = - , (24) (25) sec ^4 = 7 , o . COS A = 7 , h cot ^4 = - , Ob 181 h I* h cosec A = - . Values of the Functions of particular Angles. Angle. Arc. sin cos tan cot sec cosec 1 GO 1 OO 30 JIT 1 i/3 VI V/1J 2 V/l 2 45 JT V/l v/F 1 1 vd VT2 60 1* 1V3 i V/l V/I 2 2 V/l 90 1 1 OO OO 1 120 T isAs -i -Vi -VI 2 2 V/l 135 i* v/1 -VI 1 i -V/l V'lJ 150 l i -i\^ -v/1 -V^ -2 V/l 2 180 7T ^ OO 1 00 Useful Formulae, in which a and /3 denote any two Angles. (26) sin 2 a + cos 2 a = 1. sin a 1 tan a = : . cos a cot a sec 2 a = 1 + tan 2 a. cosec 2 a = 1 -f cot 2 a. (27) (28) (29) (30) versed sin a = 1 cos a. 182 APPENDIX. (31) sin 2 a = 2 sin a cos a. (32) cos 2 a = cos 2 a sin 2 a. (33) - tan 2 (34) sin (a /3) = sin a cos /3 cos a sin (3. (35) cos (a /3) = .cos a cos )S q= sin a sin /3. x , tan a tan /? to(a fl g (37) cot cot a (38) ' sin a + sin j3 = 2 sin J (a -f /3) cos | (a /3). (39) sin a sin f3 = 2 cos | (a + ) sin | (a /3). (40) cos a + cos j8 2 cos | (a -j- ) cos | (a /3). (41) cos a cos (3 = 2 sin \ (a -f p} sin | (a /3). In any plane triangle, let a,, b, c, denote the sides, A, B, C, the angles respectively opposite the sides, and let 5 = | (a + b 4- c) ; then, sin A = smB = sin 0' (43) a + b = tanH^ + ^) t ^ a 2 = b 2 4- c 2 2 b c cos ^4, (44) < b 2 =. c 2 -f- a 2 2 c a cos B f ( c 2 = a 2 -f- 5 2 2 a & cos (7. (45) (46) (47) APPENDIX. 183 ANALYTIC GEOMETRY. General Equation of the Straight Line. (48) Ax + By+C==Q. In the following equations of the straight line, a and b de- note the intercepts of the line on the axes of x and y respective- ly, m the tangent of the angle which the line makes with the axis of #, p the perpendicular on the line from the origin, a the angle which this perpendicular makes with the axis of x. (49) 1 + 1 = 1. (50) y = m x -f- b. (51) x cos a + y sin a = p. (52) Two straight lines, the equations of which are y = m x -{- b, and y = m 1 x + &', are parallel when m = m 1 , and are perpendicular when m m f -f- 1 = 0. General Equation of a Conic Section. (53) aa 2 + 2Az?/ + &7/ 2 + 2^ + 2/y + c = 0. If & 2 a b < 0, locus is an ellipse, Ifh 2 a b < 0, and a = b, locus is a circle, If h 2 a b > 0, locus is an hyperbola, Ifh 2 a b = 0, locus is a parabola. In the following equations of the conic sections, r denotes the radius of the circle, a and & the semi-axes of the ellipse or [~^2 fr2 hyperbola, e = .! - -^ , and is the measure of the eccen- tricity of the ellipse or hyperbola, p = the parameter of the parabola (that is, the double ordinate at the focus), m the distance from the vertex of a parabola to the focus (or focal distance). Circle. (54) Centre at point (a )8) ; (x a) 2 + (y p) 2 = r 2 . (55) Origin at centre ; x 2 + y 2 = r 2 . (56) Axis of x diameter, origin on circumf nee ; x 2 + y 2 = 2rx. (57) Tangent at point (x f y f ), origin at centre ; x x 1 -f- y y' = r 2 . 184 APPENDIX. Ellipse. (58) Origin at centre 2 + ^ = 1. (59) Origin at vertex y 1 = -^ (2 a a; x 2 ). (60) Pole at centre , p 2 = \ =-r 1 e 2 cos 2 (61) Pole at focus p = - -\- e cos 0) ' (62) Tangent at point ^ y ~+ ^ - 1. (63) Focal radii make equal angles with the tangent. Hyperbola. The equations of the hyperbola are the same as those of the ellipse, except that b 2 is negative. Parabola. (64) Origin at vertex y 2 = p x 4 m x. 2 IYI (65) Pole at focus . p = . 1 cos (66) Tangent at point x f y f . . . . 2 y y f = p (x + x'). (67) The subtangent is bisected at the vertex. (68) The subnormal is constant and equal to \p. (69) The point where any tangent cuts the axis of x, and its point of contact, are equally distant from the focus. (70) Any tangent makes equal angles with the axis of x and the focal radius vector. Analytic Geometry of Three Dimensions. If I, m, n, denote the cosines of the three angles which a straight line makes with three rectangular axes, or direction cosines ; then, (71) I' 2 + m 2 -f n 2 = 1. If denote the mutual inclination of two lines (I m n) (I 1 m 1 n f ) ; then,, (72) cos = I V + m m' + n n'. APPENDIX. 185 V. PHYSICAL DATA AND TABLES. TABLE I. Value of the Acceleration of Gravity. Place. Latitude. Value of g in Metres. Seconds Pendulum in Metres. Spitzbergen 79 49 58 N. 9-83141 0-99613 Stockholm 59 20 34 " 9-81946 0-99492 Konigsberg London 54 42 12 " 51 30 48 " 9-81443 9-81111 0-99441 0*99409 Paris 48 50 14 " 9*80979 0*99394 Isle Rawak 1 34 S. 9-78206 0-99113 Isle de France Cape of Good Hope Cape Horn New Shetland 20 9 23 " 33 55 15 " 55 51 20 " 62 56 11 " 9-78917 9-79696 9-81650 9-82253 0-99185 0-99264 0-99462 0-99523 The value of g is usually determined experimentally by pen- dulum observations. It may be calculated with sufficient accuracy by the following formula, g = G (1 0-0025659 cos 2 X) (\ 1-32 in which G- = value of g for the latitude 45 = 32*1703 feet '= 9-80533 metres, r = radius of the earth = 20 886 852 feet = 6 366 198 metres, and h height of the place in feet or metres above the level of the sea. TABLE II. Dimensions of Dynamical Units, T denoting a time, L a length, and M a mass. Dimension. Velocity. Acceler- ation. Momen- tum. Force. Energy. L T L ~T* ML T ML ML 2 T 2 T 2 mass X velocity = mass X acceleration X time = force X time. 186 APPENDIX. TABLE III. Specific Gravities of Solids and Liquids, referred to that of Distilled Water at 4 C. as a standard. Metals. Platinum, hammered 22-060 Roclcs. Granite 2*65 to 2 '75 Gold, 19-350 Sandstone 2 '25 to 2 "65 Silver, ^ Copper" " 10-510 8-900 Limestone 2*60 to 2 '70 Marble 2 '65 to 2 '75 Lead, cast . 11-350 Slate 2'84 Tin, cast . 7-290 Porcelain Clay 2 '21 Zinc . 7-190 Sand dry 1-42 Iron, cast 7 -250 Iron, wrought 7780 Various Substances Steel, soft . 7*830 Diamond 3 '530 Steel, tempered . 7*810 Glass flint 3 "000 Brass, cast .. 7-820 Brass, sheet . 8-390 Sulphur 2 "086 Bronze, statuary .. 8-950 Brick 2 '000 Bronze, gun metal .... .. 8-460 Coal, anthracite 1'800 Aluminum .. 2-600 Wax 0-960 Woods. Ice 0-918 Lignum Vitae .. 1-333 Box, Dutch 1-328 Liquids. Box, French 0'912 Mercury . . 13*596 Ebony, American 1-331 Bromine 2 "966 Oak, just felled 1-113 Sulphuric Acid, com'cial 1 '841 Oak, seasoned . 0-743 Nitric Acid, " 1'500 Mahogany, Spanish .. Beech .. 1-063 .. 0-852 Muriatic Acid " 1-200 Chloroform 1'492 Ash . 0-845 Milk 1 '031 Maple . . 0-790 Sea-water 1'025 Elm .. 0-673 Proof Spirit '920 Chestnut 0-565 Absolute Alcohol 0*795 Red Pine .. 0-657 Sulphuric Ether ' 7 1 5 White Pine .. 0-551 Olive Oil 0*915 Larch .. 0-530 Oil of Turpentine 0'870 Cork . 0-240 Naphtha 0'840 APPENDIX. TABLE IV. 187 Specific Gravities of Gases and Vapors, referred to that of Dry Air at the same Temperature and Pressure as the Standard. Oxvsren . . . 1-106 Vapor of Water 0-623 'Hydrogen 0-069 Carbonic Oxide 0-967 Nitrogen 0-972 Carbonic Acid . . 1-525 Chlorine 2-470 Ether 2-570 Ammonia 0-590 Marsh Gas 0-555 Nitrous Oxide 1-520 Coal Gas 420 to 0-520 Weight of 1 litre of dry air at C. and 76 centimetres press- ure at Paris = 1 '293 grammes. Weight of 1 cubic foot of dry air at 32 F. and 29 '905 inches pressure at London = 565 grains, or about 1 '25 ounces avoirdupois. BAROMETRIC FORMULAE FOR FINDING DIFFERENCES OF LEVEL. In these formulae X denotes difference of level, h height of barometer at lower station reduced to C., h' the same for higher station, t temperature at lower station, V the same for higher station (t and t? being expressed in Fahrenheit degrees in I. and in Centigrade degrees in II. ), X the latitude of the stations. The atmosphere is supposed to be in a mean hygrometric condition. The second and simpler value of X, given in both I. and II., will suffice for heights not exceeding 3300 feet, or 1000 metres. I. X(i (1 + 0-00256 cos2X) II. X (in metres) = 18400 log ^ ML + (1 + 0-00256 cos 2 X) X 188 APPENDIX. THE ASCENSIONAL FORCE OF A BALLOON. Lot V denote space occupied by the gas, v volume of the solid parts of the balloon, w weight of the same, w 1 weight of the aeronauts, a and a) weights of unit volumes of air and the gas respectively at C. and 76 centimetres pressure, h air-pressure at time of ascension, h f air-pressure at the alti- tude where the balloon is in equilibrium, P ascensional force ; then, changes of temperature being neglected, II. ( jr +v ) = r + + ,. This last formula, taken in connection with the barometric formulae before given, enables us to find roughly the size of the balloon necessary for reaching a given height with a given load. It is, however, of but small practical value. In the upper regions of the air, the pressure of the gas in the balloon may become so much greater than that of the external air, that the aeronaut will find it prudent to allow some of the gas to escape. Moreover, aeronauts are often obliged to throw out ballast under circumstances which cannot be foreseen. The changes of temperature, also, in different air- strata, which are very considerable, are not taken into account in the formula, and cannot be allowed for, being largely determined by the winds and atmospheric currents which happen to prevail at the time. THE END. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. 1946 ' SEE 27 I93b GUI xi ii M V 170ct'56GB DEC 27 i OC-f 1 *'-^ I r-k OCT a 1341M. V V * **^ JUL 4 t042 r ... n Af\ Jf% APR 23 1943 ifcil VA 01160 865704 s* THE UNIVERSITY OF CALIFORNIA LIBRARY