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THE 
 
 ELEMENTS OF F^HV^lC^S 
 
 BY 
 
 S. E. COLEMAN, S.B., A.M. (harvard) 
 
 HEAD OF THE SCIENCE DEPARTMENT AND 
 
 TEACHER OF PHYSICS IN THE OAKLAND (CAL.) HIGH SCHOOL 
 
 AUTHOR OF "a PHYSICAL LABORATORY MANUAL"; HONORARY MEMBER OF THB 
 
 SOUTHERN CALIFORNIA ACADEMY OF SCIENCES; PRESIDENT OF THB 
 
 PACIFIC COAST ASSOCIATION OF CHEMISTRY AND 
 
 PHYSICS TEACHERS, 1903-1905 
 
 BOSTON, U.S.A. 
 
 D. C. HEATH & CO., PUBLISHERS 
 
 1906 
 

 Copyright, 1906, 
 By D. C. Heath & Co. 
 
 r3?^ 
 
PREFACE 
 
 This book is intended as a text for a first course in physics. 
 It presupposes a knowledge of elementary algebra and plane 
 geometry, but requires no further mathematical training. While 
 complete in itself as a text-book, it assumes the support of a 
 systematic course of laboratory exercises essentially equivalent 
 to the author's Physical Laboratory Manual} The exercises 
 in this manual are referred to by number in the body of the 
 text, in connection with the topics which they serve to illustrate. 
 The text and the manual together present a single, coherent 
 course in elementary physics. 
 
 The subject-matter has been selected with reference primarily 
 to its value as a part of a general education, and includes an 
 unusual amount of information based upon the facts of our daily 
 experience, introduced as illustrations and applications of phys- 
 ical principles. This relieves the teacher from the necessity of 
 providing for and supervising a considerable amount of reference 
 reading, m order to rescue the subject from the uninteresting 
 barrenness to which it has been reduced in many of the more 
 recent text-books. 
 
 The sequence of topics, especially in mechanics, was chosen 
 with the view of presenting the serious difficulties of the sub- 
 ject gradually and in the easiest order. This purpose has led 
 to a wide departure from the conventional treatment of me- 
 chanics, which introduces a bewildering maze of abstractions 
 at the outset in connection with Newton's laws of motion. An 
 examination of the book will easily disclose the author's reasons 
 for the order of the major subdivisions. 
 
 1 Published by the American Book Company, 
 iii 
 
iv Preface 
 
 Classified problems, interspersed at frequent intervals, afford 
 opportunity for mastery of new ideas by their immediate use. 
 In addition to numerical examples, these problems present a 
 wide range of familiar facts and phenomena for explanation in 
 terms of established laws and principles. 
 
 The Teachers' Handbook accompanying the text contains 
 (i) brief comments on the subject-matter, including sugges- 
 tions and references for further experimental illustration ; (2) a 
 list of references to a few more comprehensive elementary 
 works, both general and special, for supplementary reading; 
 (3) answers to the problems. 
 
 The author's acquaintance with the many elementary text- 
 books of physics already published has, of course, been of 
 service in the preparation of the present work. This acquaint- 
 ance has afforded many helpful suggestions, for which no more 
 than a general acknowledgment is possible. The advanced 
 works and special treatises most frequently consulted were the 
 following: Atkinson's translation of Ganot's Physics, Everett's 
 translation of Deschanel's Natural Philosophy, Tait's Properties 
 of Matter, Madan's Heat, S. P. Thompson's Elementary Lessons 
 in Electricity and Magnetism, and Huxley's Lessons in Elementary 
 Physiology. Many facts and suggestions have been gathered 
 from these volumes, in addition to the quotations scattered here 
 and there throughout the book. 
 
 Special acknowledgments are due to Mr. P. T. Tompkins of 
 the Lowell High School, San Francisco, for reading the work 
 in manuscript, and to Dr. H. G. Thomas of Oakland for his 
 critical examination of the sections on the ear, the voice, and 
 the eye. 
 
 S. E. COLEMAN. 
 
 Oakland, California. 
 
CONTENTS 
 
 PAGE 
 
 Introduction i 
 
 CHAPTER I. — MATTER AND FORCE 
 
 I. The Three States of Matter 3 
 
 II. Force and Inertia 5 
 
 III. Measurement lo 
 
 CHAPTER II. — THE MECHANICS OF LIQUIDS 
 
 I. Pressure produced in Liquids by their Weight . . • 15 
 
 II. Transmission of Applied Pressure by Liquids ... 20 
 
 III. Buoyancy of Liquids 24 
 
 IV. Specific Gravity 26 
 
 CHAPTER IIL — THE MECHANICS OF GASES 
 
 I. Atmospheric Pressure 3^ 
 
 II. Boyle's Law 37 
 
 III. Applications of Atmospheric Pressure .... 41 
 
 CHAPTER IV. — STATICS OF SOLIDS 
 
 I. Concurrent Forces 49 
 
 II. Parallel Forces .... . • • • 55 
 
 III. Moments of Force 57 
 
 IV. Effect of Weight on the Equilibrium of Bodies . . 60. 
 
 CHAPTER v. — DYNAMICS 
 
 1. Motion 68 
 
 II. Falling Bodies 74 
 
 V 
 
vi Contents 
 
 PAGS 
 
 III. Projectiles • . . 80 
 
 IV. The Laws of Motion 83 
 
 V. Curvilinear Motion 93 
 
 VI. Universal Gravitation 98 
 
 VII. The Pendulum 104 
 
 CHAPTER VI. — ENERGY 
 
 I. Energy and Work Ill 
 
 II. The Simple Machines .• 123 
 
 CHAPTER VII. — MATTER 
 
 I. The Structure of Matter 137 
 
 II. Molecular Motion 141 
 
 III. Molecular Forces 147 
 
 IV. Surface Tension and Capillarity 151 
 
 V. Properties Due to Molecular Forces . . . .156 
 
 CHAPTER VIII. — HEAT 
 
 I. Heat and Temperature 161 
 
 II. The Transmission of Heat 164 
 
 III. Radiation 167 
 
 IV. Measurement of Temperature and Expansion . . .173 
 V. Calorimetry and Specific Heat 182 
 
 VI. Fusion and Solidification 186 
 
 VII. Vaporization and Condensation 192 
 
 VIII. Mutual Transformations of Heat and Other Forms of 
 
 Energy . ' '. . 208 
 
 CHAPTER IX. — SOUND 
 
 I. Origin and Transmission of Sound 214 
 
 II. Properties of Musical Sounds 227 
 
 III. Sympathetic and Forced Vibrations : Resonance . . 240 
 
 IV. The Ear and the Voice 249 
 
Contents vii 
 
 CHAPTER X. — LIGHT 
 
 PAGE 
 
 I. Nature and Transmission of Light 255 
 
 n. Reflection of Light 268 
 
 HL Refraction of Light 283 
 
 IV. Atmospheric Refraction 294 
 
 V. Lenses 296 
 
 VI. The Eye 306 
 
 VII. Optical Instruments 315 
 
 VIII. Dispersion and Color 322 
 
 CHAPTER XL — MAGNETISM 
 
 I. Properties of Magnets 339 
 
 II. Magnetization 342 
 
 III. The Magnetic Field 345 
 
 IV. Terrestrial Magnetism , 348 
 
 CHAPTER XII. — ELECTRICITY 
 
 I. The Voltaic Cell 353 
 
 11. The Electro-magnetic Field 362 
 
 III. The Electric Bell and the Telegraph . . . .365 
 
 IV. Electric Measurements 369 
 
 V. Heat, Light, and Power from Electric Currents . . 385 
 
 VI. Induced Currents 389 
 
 VII. The Dynamo and the Motor 396 
 
 VIII. The Telephone and the Microphone .... 404 
 
 IX. Chemical Effects of Currents 407 
 
 X. Electrostatics 411 
 
 Appendix .... * 427 
 
 Index 431 
 
or THF ^ 
 
 UNIVERSITY 
 
 OF 
 
 ELEMENTS OF PHYSICS 
 
 INTRODUCTION 
 
 1. Physics and its Place among the Sciences. — The knowledge 
 of the material universe is subdivided for convenience into several 
 branches, called the ?iatural sciences. The biological sciences 
 treat of living things ; the physical sciences deal with inanimate 
 matter in all its forms and with the changes and processes to which 
 it is subject. The physical sciences are chemistry, physics, as- 
 tronomy, geology, meteorology, and mineralogy. 
 
 Chemistry deals with the changes in matter which produce other 
 forms of matter. Thus the burning of wood is a chemical pro- 
 cess, for it causes a change of substance — after the wood is 
 burned it is no longer wood. Similarly, the burning of any sub- 
 stance, the rusting of iron, the decay of animal or vegetable 
 matter, and the manufacture of drugs and chemicals are chemical 
 processes, and their study falls to the science of chemistry. 
 
 Physics treats of changes and processes that do not result in 
 the formation of other substances. Its subdivisions are mechanics, 
 heat, sound, light, magnetism, and electricity. 
 
 Astronomy is the physics of the heavenly bodies ; geology, the 
 physics of the earth's crust ; meteorology, the physics of weather 
 and climate ; mineralogy, the physics and chemistry of rocks and 
 minerals. Thus all of the physical sciences are dependent upon 
 physics, being, in fact, the application of physics to special fields 
 of study. 
 
 2. The Place of Physics in Education. — Physics deals largely 
 with familiar natural phenomena,^ and is therefore of special interest 
 
 ^A phenomenon, as the word is used in science, is "any action, motion, 
 change, or occurrence of any kind," however simple and familiar it may be. 
 
 I 
 
2 Introduction 
 
 and profit as a part of a general education. But the value of the 
 study of physics does not consist alone in the acquisition of facts. 
 The pupil has already found this to be true of geometry. In this 
 subject the emphasis is laid not on the fact, but on the proof of it, 
 the main purpose being to train the pupil in the power and the 
 habit of accurate reasoning. The study of physics also affords an 
 excellent opportunity for valuable training. 
 
 The experimental work of the class room and the laboratory 
 furnishes the greater part of the material for this training ; but a 
 very important part of it is acquired through the experiences of 
 our daily life. As the subject is pursued, this miscellaneous body 
 of facts will be passed in review, assorted, compared, and classified, 
 and each fact traced to its causes and consequences ; in a word, 
 the whole will be organized into scientific knowledge. The pupil 
 is receiving scientific training only in so far as he learns to do this 
 sort of work for himself ; and it is only this that leads to scientific 
 abitity. The mere memorizing of facts and explanations from the 
 text-book results in little more than vaguely comprehended and 
 therefore generally useless information. 
 
CHAPTER I 
 
 MATTER AND FORCE 
 
 I. The Three States of Matter 
 
 3. Matter exists in three conditions or states, called respectively 
 the solicit the liquid^ and the gaseous state. All kinds of matter in 
 the same state possess certain properties or characteristics in com- 
 mon, which serve to distinguish them as a class from bodies in 
 either of the other states. 
 
 4. Solids and Liquids. — Liquids are distinguished from solids 
 by the fact that they tend to flow, and must therefore be contained 
 in vessels. Solids, on the contrary, have a definite form, which 
 they tend more or less strongly to preserve. Some solids, such as 
 stone and steel, offer great resistance to any change of shape ; 
 others, like soft clay and putty, can readily be molded into any 
 form. But even the small amount of resistance offered by the 
 latter distinguishes them from liquids. 
 
 5. Gases. — Air is the most familiar gas. Strictly speaking, it 
 is a mixture of a number of gases, principally nitrogen and oxygen ; 
 but in physics it is generally unnecessary to consider the separate 
 constituents. Although the air is everywhere about us, we are or- 
 dinarily unconscious of its existence unless it is in motion. When 
 it is in motion, we recognize it as a breeze or a wind, or, in smaller 
 quantities, as a current of air. We commonly call a vessel 
 " empty " when it is full of air ; and seldom stop to think that, 
 while the so-called empty vessel is being filled with any liquid or 
 solid, the air is at the same time being pushed out. It will help 
 toward clear thinking on this point to push an inverted tumbler 
 into a vessel of water (Ex/>.). The water does not rise to fill the 
 
 3 
 
4 Matter and Force 
 
 tumbler, being prevented from doing so by the confined air ; but, 
 when the tumbler is slowly inclined, the air escapes in a succes- 
 sion of bubbles, and water enters at the same time to take its 
 place. 
 
 The experiment shows that a body of air confined in any space 
 tends to keep other bodies out of that space ; and this is true of 
 all gases. But it is well known that, after a bicycle tire is fully 
 inflated, much air must still be pumped in to make it hard. Now 
 air can be forced into the fully inflated tire only by compressing 
 the air already in it into a smaller space ; and experience teaches 
 that the compression of the confined air can be carried just as far 
 as the strength of the tire or of the operator will permit. The 
 great compressibility of air can also be shown by a simple experi- 
 ment with a bicycle pump or other compression pump. The 
 piston can be pushed in a considerable distance although the 
 confined air is prevented from escaping by closing the outlet with 
 the finger. The force required to push the piston in rapidly 
 increases, the farther the piston is pushed ; and, when it is re- 
 leased, it is instantly pushed back by the expansion of the 
 compressed air (Exp.). All gases are highly compressible and 
 expansible, like air. When any quantity of gas, however small, 
 is put into an otherwise empty space, it expands so as to fill the 
 space completely. {^Experiments ivith air pump.) 
 
 All liquids and most solids are only very slightly compressible. 
 In fact, any change of volume with either increase or decrease of 
 pressure upon them is so slight as commonly to elude observa- 
 tion ; and, for all practical purposes, liquids are regarded as 
 incompressible. Hence gases are distinguished from solids and 
 liquids by their great compressibility and expansibility. Most 
 gases are colorless and invisible, but not all. 
 
 6. Fluids. — Since the parts of both liquids and gases move 
 over one another freely, or flow, they are both called fluids. The 
 fluidity of gases is very easily illustrated with carbonic acid gas ; 
 which will extinguish a lighted candle when poured upon it from 
 a vessel {Exp.). 
 
Force and Inertia 5 
 
 7. Summary. — Solids have both definite form and volume, 
 which they tend to preserve. 
 
 Liquids have definite volume ; but have no form of their own, 
 since their parts move readily over one another. 
 
 Gases have neither definite form nor volume. They are highly 
 compressible, and, with decrease of pressure, tend to expand 
 indefinitely. 
 
 II. Force and Inertia 
 
 8. Force. — The vf 0x6, force ^ as used in physics, means a push 
 or a pull. The following are familiar examples of forces : the pull 
 exerted by a horse upon a wagon ; the push or pull by which a 
 door is opened or closed ; the very brief but strong push exerted 
 by a hammer upon a nail in driving it ; the continuous pressure 
 of a book lying upon a table ; the pressure of the table upon the 
 book, by which the book is supported ; the pressure of a liquid 
 against the bottom and sides of the containing vessel. 
 
 9. Inertia. — We learn from our daily experience that a body 
 at rest remains at rest unless some other body exerts a force upon 
 it, or, in other words, that a body cannot acquire motion without 
 the action upon it of an applied force. For example, a ball is sent 
 flying through the air by the push of the hand in throwing it or by 
 a blow with a bat ; a high velocity is imparted to a rifle ball by the 
 pressure of the gases from the powder exploded behind it ; and a 
 loaded wagon is started by the pull exerted by the horses upon it 
 through the traces. 
 
 It is also a matter of common observation that moving bodies 
 come to rest more or less slowly after the forces that start them 
 cease to act. A book slides over a table when started with a sud- 
 den push, but quickly stops ; a ball can be made to roll a long dis- 
 tance over a smooth, level surface, as a sidewalk, but gradually loses 
 speed till it comes to rest ; and a wagon goes only a short distance 
 after the horses cease to pull. This behavior of moving bodies is 
 not due to any tendency of the bodies 'themselves to come to rest, 
 but is the effect of opposing forces developed by the rubbing of 
 
6 Matter and Force 
 
 surfaces as they move over one another. Such a force is called 
 friction. Friction acts as a resistance to motion, and tends to 
 bring moving bodies to rest. The smoother the surfaces are, the 
 less friction becomes ; hence a body slides farther on a smooth 
 surface than on a rough one. For example, after skaters are once 
 in rapid motion, they can go a long distance without further effort, 
 the friction between skates and smooth ice being very slight. 
 Rolling friction is in general much less than sliding friction ; 
 hence the use of wheels on vehicles of all sorts. Ball bearings 
 reduce the friction still further by substituting rolling for sliding 
 friction at the axle. 
 
 Another hindrance to motion is the resistance of the air. This 
 resistance is small upon a body moving slowly, but rapidly 
 increases with the velocity. For high velocities, such as those of 
 an express train or a rifle ball, it is very great. Bodies can, of 
 course, be stopped by other forces than friction. 
 
 The general truth illustrated by the above examples may be 
 stated as follows : A body remains at rest or, if in motion, con- 
 tinues with uniform motion in a straight line, unless it is compelled 
 to do othenvise by forces acting upon it from without. This is true 
 of all matter, solid, liquid, and gaseous; and the property of 
 passiveness thus exhibited is known as the inertia of matter. 
 
 The inertia of water can be shown by moving the hand quickly 
 to and fro in a tub of water. A strong push must be exerted by 
 the hand to impart motion to the water that is driven before it. 
 The inertia of the air is similarly shown by moving a large fan 
 rapidly to and fro, first flatwise, in the usual manner, then edge- 
 wise. Comparatively little effort is required in the second case, 
 as the fan cuts through the air, moving but little of it. (Try these 
 experiments.) Force is also required to stop a body of air or 
 water. The resistance offered by a house is sometimes insufficient 
 to stop or turn aside the air that strikes it during a violent storm, 
 and the house is blown down. Similarly, the inertia of a stream of 
 water from a fire hose or even from a garden hose is shown by its 
 power to break or overturn obstacles against which it is directed. 
 
Force and Inertia 7 
 
 10. Action of Forces with and without Contact : Weight. — All 
 the forces considered in the preceding article are exerted by direct 
 or indirect contact of the body exerting the force and the body 
 upon which the force is exerted. Thus a horse in drawing a 
 wagon pushes on the collar with his shoulders, and the collar 
 pulls on the traces, and the traces pull on the wagon. Certain 
 forces, however, act without visible or material connection between 
 the bodies concerned. The forces exerted by a magnet are of this 
 sort. Pieces of iron move toward a magnet from a greater or less 
 distance, depending upon the size of the pieces and the strength 
 of the magnet {Exp.). We know from the behavior of the iron 
 that it is acted upon by a force whose direction is toward the 
 magnet, although there is nothing whatever to show how this force 
 is exerted. 
 
 Similarly, the fact that a body falls unless supported indicates 
 that it is acted upon by a force whose direction is vertically down- 
 ward. This force is in some way due to the earth ; hence we 
 think of the earth as exerting a pull or a/traction by which it 
 tends to draw bodies toward its center. The attraction exerted 
 by the earth upon any body is called the weight of the body.* 
 
 11. Balanced and Unbalanced Forces. — A force acting alone on 
 a body always sets it in motion or changes its existing motion. A 
 stone moving through the air with a moderate velocity is a good 
 illustration ; for its weight is practically the only force acting on 
 it, the resistance of the air being inappreciable. The weight of 
 the stone causes a continuous decrease of velocity if the stone is 
 rising vertically, a continuous increase of velocity if it is falling 
 vertically, and a continuous change of direction if it is moving 
 obliquely. 
 
 Two or more forces acting on the same body at the same time 
 may neutralize each other in such a way that the body remains at 
 rest or, if moving, continues with uniform motion in a straight line. 
 Forces so neutralizing each other are said to balance each other or 
 
 1 The weight of a body is very slightly less than the earth's attraction for^it, 
 except at the poles, as explained in Art. 128. 
 
8 Matter and Force 
 
 to be /// equilibrium^ and are called balanced forces. The follow- 
 ing are illustrations : A cart remains at rest when two boys pull 
 equally on it in opposite directions. The weight of a body at rest 
 or in motion on a level surface is balanced by 
 the upward pressure of the surface ; the weight 
 of a body suspended by a cord is balanced by 
 the upward pull of the cord. In both cases the 
 sustaining force is equal and opposite to the 
 weight of the body. These examples illustrate 
 the simplest case of balanced forces : namely, 
 that of two equal forces acting in opposite di- 
 rections along the same line. An example of three forces in equi- 
 librium is illustrated in Fig. i. A body is supported by two cords. 
 Each cord pulls obliquely upward upon the body, and the two 
 forces together balance the weight of the body. 
 
 A single force acting on a body is always unbalanced ; two or 
 more forces acting together may be either balanced or unbalanced. 
 The study of balanced forces acting on bodies at rest and of both 
 balanced and unbalanced forces acting on bodies in motion con- 
 stitutes the greater part of the subject-matter of mechanics, and is 
 continued in the following chapters. 
 
 12. The Mutual Action of Two Bodies. — Whenever one body 
 exerts a force upon another, the second body exerts at the same 
 time an equal and opposite force upon the first. This opposite 
 action of two bodies upon each other is often evident from the 
 effects produced upon the bodies. For example, when one mar- 
 ble strikes another, the latter is set in motion, and, at the same 
 time, the motion of the first marble is stopped or checked by the 
 opposite force exerted upon it by the other marble. The mutual 
 action between a ball and a bat is a similar case. When a bullet 
 strikes a board, the force that it exerts makes a hole in the board ; 
 the equal and opposite force exerted by the board stops the 
 bullet. 
 
 The equality of the force exerted by each of two bodies on the 
 other can be illustrated by two hardwood or ivory balls of equal 
 
Force and Inertia 
 
 size, suspended as shown in Fig. 2. One of the balls is drawn 
 aside and released. It falls, strikes the other ball, and instantly 
 stops ; while the other swings out as far 
 (or very nearly as far) as the first ball 
 would have gone if its motion had not 
 been hindered. Since the balls are ex- 
 actly alike and the one loses as much 
 motion as the other gains, it follows that 
 the forces which they exert upon each 
 other are equal {Exp.). 
 
 The forces exerted between two bodies 
 at rest are also equal and opposite. Thus, 
 when the hand is pressed against a wall, 
 the wall exerts an equal pressure on the 
 hand. A book lying on a table exerts a downward pressure equal 
 to its weight ; the resistance offered by the table acts as an equal 
 upward pressure on the book. 
 
 To distinguish between the action of one body on another and 
 the action of the second body on the first, one is called the 
 " action " and the other the " reaction." Either may be called 
 the action, and the other will then be referred to as the reaction ; 
 but, if one of the bodies is at rest and the other in motion before 
 their mutual action, it is customary to say that the moving body 
 acts on the one at rest and that the latter reads on the former. 
 
 Fig. 2. 
 
 PROBLEMS 
 
 1. Describe and account for the motion of the occupants when a carriage 
 is started or stopped very suddenly. 
 
 2. In what direction is an inexperienced person likely to fall on alighting 
 from a rapidly moving car ? Why ? 
 
 3. What forces are acting on a wagon when drawn at a uniform rate 
 on a level road ? Are they balanced or unbalanced ? 
 
 4. What balanced forces are acting on a stone when at rest on the ground? 
 
 5. A boy exerts a lifting force of 75 lb. on a stone weighing 200 lb. 
 (a) Is this a balanced or an unbalanced force ? {b) What balanced forces 
 are acting on the stone ? • 
 
lo Matter and Force 
 
 6. Is it the forces exerted by or upon a body that affect its state of rest or 
 motion ? 
 
 7. Account for the " kick " or recoil of a gun. 
 
 8. Why cannot a boy lift himself by standing in a tub and pulling on the 
 handles ? 
 
 III. Measurement 
 
 13. Measurement and Units of Measurement. — Experimental 
 work ill physics consists largely in measuring the different kinds of 
 physical quantities, as length, surface, volume, force, velocity, mass, 
 etc. A quantity of any kind is measured by finding how many 
 times it contains a certain fixed amount of that kind of quantity. 
 This fixed amount is called a /////// and there are various units in 
 common use for measuring each kind of quantity. Thus there are 
 many units of length, among which are the inch, foot, meter, and 
 centimeter. 
 
 The amount of any quantity is expressed by naming the unit in 
 which it is measured, preceded by the number of times it con- 
 tains this unit, — as a volume of 3.7 cu. in. The name of the 
 unit in which a physical quantity is measured should not be 
 omitted either in oral or written statement. 
 
 On account of the great simplicity of the metric system of meas- 
 ures, it is almost exclusively used by scientists ; and it is the only 
 one that we need consider here. 
 
 14. Extension. — Extension is that property of matter by virtue 
 of which it occupies space, or has length, width, and thickness. 
 The amount of space occupied by any portion of matter is called 
 its size, hulk, or volume. 
 
 15. Units of Extension. — The fundamental unit of length in 
 the metric system is the meter. It is defined as the distance be- 
 tween two lines on a certain metallic rod preserved in the archives 
 of the International Metric Commission at Paris. It was origi- 
 nally intended to be one ten-millionth of the distance on the 
 earth's surface from the pole to the equator ; but it is not exactly 
 this fraction. The meter is now an arbitrary standard, just as the 
 
Measurement 1 1 
 
 yard is. Its advantage over the latter unit lies in the fact that its 
 subdivisions are decimal fractions. (See Table I of the Appendix.) 
 
 A decimeter (dm.) is a tenth of a meter (m.), a centimeter (cm.) 
 is a hundredth of a meter, and a millimeter (mm.) is a thousandth 
 of a meter. The centimeter is the customary unit of length for 
 scientific purposes, and is the only one that the pupil will ordina- 
 rily use in the laboratory. Thus a length of 3 dm. 5 cm. 7.5 mm. 
 is written 35.75 cm. It will be useful to remember that the meter 
 is equal to 39.37 inches, and the inch very approximately to 2.5 
 cm. (See Table II of the Appendix for exact relative values of the 
 English and metric units.) 
 
 The square centimeter (scm.) and the cubic centimeter (ccm.) 
 are the customary units of surface and volume respectively. Since 
 a square decimeter (sdm.) is 10 cm. in length and in width, it con- 
 tains 100 scm. ; and since a cubic decimeter (cdm.) is 10 cm. in 
 each of its three dimensions, it contains 1000 ccm. A cubic deci- 
 meter, when used as the unit of liquid measure, is called a liter. 
 It is slightly greater than a quart. 
 
 16. Weight. — The weight of a body (Art. 10) is constant at any 
 one place on the earth, but decreases slightly with increase of alti- 
 tude above the general level of the earth (as when a body is carried 
 up a mountain or up in a balloon) and also with increase of depth 
 below the surface (as when a body is taken down into a mine). A 
 body at the earth's center would have no weight, being attracted 
 equally in all directions by the earth. Weight also varies slightly 
 with latitude, increasing continuously upon any body as it is taken 
 from the equator toward either pole. The reasons for these 
 variations in weight will be considered later. 
 
 17. Mass. — If any two bodies at the same place have equal 
 weight, they are said to contain equal quantities of matter^ or to 
 have equal mass. Using the term " quantity of matter " in this 
 sense, the mass of a body may be defined as the quantity of matter 
 in it. 
 
 Although the weight of a body is affected by change of lati- 
 tude or altitude, its mass is not. A body would contain identically 
 
12 Matter and Force 
 
 the same matter, and would therefore have the same mass, if it 
 were transported to some region in space where it would be free 
 from the attraction of the earth on any other body, or to the sun, 
 where its weight (the sun's attraction) would be nearly twenty-eight 
 times as great as upon the earth. 
 
 18. Units of Mass and of Force. — The mass of a cubic centi- 
 meter of distilled water at the temperature of its greatest density 
 (4° Centigrade) was originally taken as the fundamental unit of 
 mass in the metric system, and is called the gram. As in the case 
 of the meter, the gram is now defined with reference to a standard 
 kept at Paris ; but for the purposes of elementary physics the above 
 definition is the only one of importance. The mass of a cubic 
 centimeter of water pure enough for domestic use, either cold or 
 tepid, differs so little from one gram that the difference may always 
 be disregarded in elementary physics. 
 
 The earth's attraction for the unit of mass affords a very con- 
 venient unit of force. From what has been said concerning the 
 variation of weight, it is evident that such a unit of force differs 
 appreciably at different latitudes ; but the variation is so slight as 
 not to be a matter of practical importance. 
 
 The unit of mass and the corresponding unit of force have the 
 same name. Thus any force equal to the earth's attraction for 
 a mass of 7 g. would be called a force of 7 g. To distin- 
 guish between the units, they are sometimes called the gram mass 
 and the gram iveight respectively ; but it is always possible to de- 
 termine from the context whether mass or force is intended. In 
 the English system the pound mass and the pound weight are the 
 fundamental units of mass and force respectively. 
 
 19. Measurement of Mass (Weighing). — The equal attraction 
 of the earth for equal masses at the same place is utilized in deter- 
 mining the equality of two masses by the familiar process of weigh- 
 ing with an equal-arm balance (Fig. 3). The horizontal bar of 
 the balance is called the beam, and either half of it is called 
 an arm. 
 
 When equal pressures are exerted upon the two pans, the beam 
 
Measurement 
 
 13 
 
 Fig. 3. 
 
 comes to rest in a horizontal position. Hence, when the beam 
 assumes this position under the pres- 
 sure of bodies in the two pans, these 
 bodies have equal mass. The reason- 
 ing is as follows : — 
 
 (i) The balancing of the beam in- 
 dicates equal pressures upon the pans. 
 
 (2) Since the bodies exert equal 
 pressures, they have equal weight; 
 that is, the earth attracts them equally. 
 
 (3) Since the bodies have equal 
 weight, they have equal mass. 
 
 The mass of a body is therefore 
 found by placing it in one pan and balancing it with standard 
 masses in the other. This process is called weighing, and the 
 quantity found is commonly called the weight of the body. The 
 standard masses are called weights. 
 
 20. Density. — The density of a substance is the mass of a unit 
 volume of the substance. In the metric system it is usually ex- 
 pressed as the number of grams in a cubic centimeter of it (g. per 
 ccm.) ; in the English system, as the number of pounds in a cubic 
 foot of it (lb. per cu. ft.). 
 
 The density of any substance is found by measuring the mass 
 and the volume of any convenient portion of it, and computing 
 from these measurements the mass of one cubic centimeter. 
 
 Laboraioty Exercises J and 2. 
 
 PROBLEMS 
 
 1. Would the " weight " of a body, as determined with an equal-arm bal- 
 ance^ differ in different latitudes and at different altitudes? Give the reason 
 for your answer. 
 
 2. Is the density of a body affected by its latitude or its altitude ? 
 
 3. Is a pound of iron heavier than a pound of wood ? What is implied 
 in the statement that " iron is heavier than wood "? Show that the statement 
 " iron is denser than wood " leaves nothing to be implied. The more definite 
 form of statement is to be preferred. 
 
 1 Cl 
 
14 Matter and Force 
 
 4. The volume of a stone is 630 ccm.; its mass is 1575 g. Find its den- 
 sity. 
 
 5. WTiat is the volume of 1000 g. of mercury ? of icxx) g. of brass ? of 
 1000 g. of aluminum ? (See table of densities in the Appendix.) 
 
 6. What is the mass of i cdm. of lead ? of i cdm. of marble ? 
 
 7. Find the densities of water, quartz, and gold in pounds, per cubic foot, 
 from the densities in grams per cubic centimeter given in the table. (See 
 also the table of equivalents in the Appendix.) 
 
 8. The density of (luartz is how many times that of water in the metric 
 system ? in the English system ? How do these answers compare ? "Why ? 
 
 9. From the known densities of ice and water, show whether water ex- 
 pands or contracts in freezing. 
 
CHAPTER II 
 
 THE MECHANICS OP LIQUIDS 
 
 I. Pressure produced in Liquids by their Weight 
 
 21 o Transmission of Pressure. — Suppose a number of books to 
 be placed in a pile, one above the other. Beginning at the top, 
 the first book presses upon the second with a force equal to its 
 weight. The second book transmits this pressure to the third, 
 and adds to it a pressure equal to its own weight. Hence the 
 third book sustains a pressure equal to the weight of the first two 
 books. Similarly, the third book exerts a pressure upon the fourth 
 equal to its own weight plus that of the books above it, and so on 
 to the last, the pressure of which upon the table is equal to the 
 combined weight of all. The entire pile is supported by the up- 
 ward pressure of the table, which is equal to the pressure of the 
 bottom book upon it (equal action and reaction) ; and any por- 
 tion of the pile, beginning at the top, is supported by the upward 
 pressure of the book next below. If we consider the leaves of 
 each book separately, instead of the book as a whole, we have an 
 illustration of pressure due to weight, increasing regularly and 
 continuously from top to bottom and exerted vertically up and 
 down, the upward and downward pressures at any depth being 
 exactly equal. 
 
 The weight of a pile of shot causes each shot to crowd in be- 
 tween its neighbors, thus exerting a pressure sideways as well as 
 upward and downward, as is shown by the tendency of the pile to 
 spread outward at the bottom. To make the sides of the pile 
 vertical, supporting surfaces must be provided to sustain the lateral 
 pressure. 
 
 15 
 
\ 
 
 u 
 
 1 6 The Mechanics of Liquids 
 
 Similarly, the weight of a liquid causes lateral and oblique as 
 well as vertical pressures within it. These pressures are more 
 fully developed in liquids than in a pile of shot, for the particles 
 of a liquid are free to move over one another, while in shot there 
 is considerable friction. Hence shot remains in a sloping pile, 
 and liquids do not. 
 
 22. Pressure in Liquids due to Weight. — The pressure produced 
 within a liquid by its weight can be studied experimentally with 
 various forms of apparatus. Cilass tubes of small 
 diameter and equal length (about 60 cm.), closed 
 air-tight at the top and shaped at the lower end as 
 shown in Fig. 4, serve very well for the purpose 
 of illustration, but are not adapted to exact meas- 
 urement. When one of the tubes is lowered into 
 a tall glass jar filled with water, the water enters 
 its lower end a short distance. This is due to 
 the compression of the confined air by the pressure 
 Fig. 4. that the water exerts upon it. The distance to 
 
 which the water enters the tube increases as the 
 tube is lowered, showing that the pressure increases with the 
 depth {Exp.). This is due to the fact that the pressure at any 
 depth is caused by the weight of the water above that level. 
 
 When the different tubes are inserted to the same depth, the 
 water enters an equal distance in all, showing that the pressure 
 of the water at a given depth is the same in the various directions 
 in which it enters the tubes {Exp.), Since each particle of water 
 is free to move in any direction, it would not remain at rest if it 
 were not pressed upon equally from all sides. 
 Laboratory Exercise j". 
 
 23. Laws of Liquid Pressure. — The facts concerning pressure in 
 liquids at rest are found by accurate measurement to be as follows : — 
 
 I. The pressure at any point in a liquid at rest is equal in all 
 directions. 
 
 II. The pressure of a liquid at rest is perpendicular to any surface 
 upon which it acts. 
 
Pressure in Liquids 17 
 
 III. At any point ill a liquid^ the pressure due to its weight is 
 proportional to the depth of the point below the free surf ace of the 
 liquid. 
 
 IV. The pressure is the same at all points in the same horizontal 
 plane. 
 
 V. At the sattie depth in different liquids, the pressure due to 
 weight is proportional to the density of the liquid. 
 
 These statements are known as the laws ^ of liquid pressure. 
 The first two hold for all pressures in liquids at rest, whether due 
 to their weight or to applied pressure, i.e. to pressure exerted 
 upon them in closed vessels. As stated in the preceding article, 
 the first law is a consequence of the freedom of movement of the 
 particles of a liquid. This is true of the second law ; for, if a 
 liquid pressed obliquely against a surface, it would move along the 
 surface instead of remaining at rest. The reasons for the other 
 laws are considered in the following articles. 
 
 24. Intensity of Pressure. — The pressure at any point in a liquicj 
 is defined as the pressure that the liquid would exert upon a hori- 
 zontal surface of ////// area at that depth in the liquid. The pres- 
 sure upon each unit area of a surface is sometimes called the 
 intensity of pressure or the rate of 
 pressure to distinguish it from the 
 /^/^/ pressure on the surface. Pres- 
 sure (/.<?. intensity of pressure) is 
 expressed in grams per square centi- 
 meter, pounds per square foot, pounds 
 per square inch, and other units con- 
 sidered later. ^,, , ^ 
 
 The rule for computing the pressure 
 at any depth in a liquid is derived as follows :• Let MN (Fig. 5 ) 
 
 1 Experience teaches that, in nature, whenever the same conditions are repeated 
 the same resuhs follow ; or, in other words, the same causes always produce the 
 same effects. Without this uniformity in nature science would be impossible. The 
 uniform behavior of matter under given conditions is called a natural law. A brief 
 descriptive statement of such uniform behavior in any given instance is also called 
 a law. 
 
1 8 The Mechanics of Liquids 
 
 be a horizontal surface upon which a liquid rests, and a any square 
 centimeter of it. The rectangular figure ac represents the portion 
 of the liquid lying vertically above a. We may think of this por- 
 tion as being separated from the remainder by imaginary bounding 
 surfaces, or as having become solid (without change of density). 
 The surrounding liquid presses perpendicularly against the sides 
 of this column, as indicated by the arrows. The pressures upon 
 opposite sides are equal and balance each other, but they do not 
 help to sustain the weight of the column. The pressure that the 
 column exerts on a is therefore equal to its weight. Let // denote 
 the height of the column in centimeters, and d the density of the 
 liquid in grams per cubic centimeter; then the volume of the 
 column is h ccm. and its weight hd g. Hence the rule : — 
 
 At any point in a liquid^ the pressure due to its weight is equal 
 to the product of the depth {of that point below the free surface) and 
 the density of the liquid. 
 
 25. Total Pressure upon a Surface. — Horizontal Surfaces. — 
 Since the pressure of a liquid is uniform over a horizontal surface, 
 the total pressure on such a surface is equal to the product of the 
 intensity of pressure and the area. 
 
 Oblique and Vertical Surfaces, — The pressure upon a vertical 
 or an oblique plane surface {MN, 
 Fig. 6) increases regularly from the 
 upper to the lower side. To find the 
 total pressure upon such a surface, 
 the average pressure (per scm.) upon 
 it is multiplied by the area. The 
 average pressure is equal to the actual 
 pressure at the center of the surface ; 
 and the latter is ^he same as the 
 pressure upon a horizontal area of i scm. at that depth, since 
 the pressure at any point is equal in all directions. 
 
 For example, the total pressure upon an oblique surface of 4 
 cm. by 12 cm., the center of which is at a depth of 10 cm. in 
 alcohol (density = .82 g. per scm.) is 4 x 12 x 10 x .82 g. 
 
Pressure in Liquids 
 
 19 
 
 Fig. 7. 
 
 The average pressure on the side of a cylindrical or rectangular 
 vessel containing a liquid is the actual pressure at half the depth 
 of the liquid. 
 
 26. Pressure Independent of Shape of Vessel. — One important 
 consequence of the laws of liquid pressure is that the pressure at 
 any point in a liquid is independent 
 of the shape of the containing vessel 
 {Exp.) . Suppose that vessels A, B, 
 Cy and D (Fig. 7) each have a bot- 
 tom 10 X 10 cm. and a height of 30 
 cm., and are filled with water. The 
 lower portion of B is 10 cm. high, 
 and is continued 20 cm. higher by 
 a tube. The pressure will be uniform over the bottoms of the 
 four vessels and will be equal upon all ; namely, 30 g. per scm., 
 or a total of 3000 g. upon each. This is evidently the case for A^ 
 since 3000 g. is the weight of the water it contains. 
 
 The pressure at the bottom of the tube in B is 20 g. per scm. 
 This causes an equal upward pressure (per scm.) against the top 
 of the lower portion of the vessel, as indicated by the arrows in 
 the figure. This upward pressure of the water is sustained by an 
 equal downward pressure of the top against the water. This 
 downward pressure of the top takes the place of an equal down- 
 ward pressure at the same level in A^ due to the weight of the 
 water above that level. Hence the pressures in the two vessels at 
 equal depths below this level are equal. 
 
 In C the weight of all the liquid not lying vertically above the 
 bottom is sustained by the sides of the vessel, which exert an 
 oblique upward pressure. The discussion of pressures for D is 
 similar to that for B. (State it.) 
 
 27. Equilibrium of a Liquid in Communicating Vessels. — If the 
 vessels represented in Fig. 7 were connected with one another by 
 tubes as indicated by the dotted lines in the figure, there would 
 be no flow to or from any one of them ; for the pressure would be 
 the same at lx)th ends of each of the tubes {Exp.). 
 
20 
 
 The Mechanics of Liquids 
 
 Illustrations of this fact are common. The liquid stands at 
 the same level in the spout of a teapot as in the vessel itself. 
 The pressure at the bottom of the spout must be the same on both 
 sides, otherwise the liquid would not remain at rest. Similarly, in 
 a pipe leading from a reservoir the water will rise to the level of 
 the water in the reservoir and no higher. This familiar behavior 
 
 of water is commonly expressed 
 by saying that " water seeks its 
 own level." This is, of course, 
 merely a statement of the fact, 
 not an explanation of it. 
 
 The flow of artesian wells is 
 explained by Fig. 8. If a por- 
 ous stratum of sand or gravel, 
 /I, lying between two imper- 
 vious strata and dipping under 
 a lower flat country, becomes filled with water above the level of 
 the groimd where a well is bored, an artesian or flowing well 
 will result. 
 
 
 FiC. 8. 
 
 II. Transmission of Applied Pressure by Liquids 
 
 28. Pascal's Law. — Pressure exerted upon any part of an 
 inclosed liquid is transmitted undiminished in all directions^ and 
 acts with equal force upon all equal surfaces ^ and at right angles to 
 them. This is known as Pascal's law, in honor of the French 
 mathematician and physicist, Blaise Pascal (1623- 
 1662), by whom it was discovered. 
 
 For example, if a stopper having an area of 
 2 scm. at the end be pushed into a botde full of 
 water until it presses with a force of 6 kg. upon 
 the water, this pressure of 3 kg. per scm. will be 
 added to the pressure due to the weight of the 
 liqtiid at every point within it, and will be trans- 
 mitted as an added pressure of 3 kg. upon each fig. 9. 
 
Transmission of Pressure 21 
 
 square centimeter of surface of the bottle. With the apparatus 
 
 shown in Fig. 9, a strong bottle can be easily burst by pushing 
 
 the rod a short distance through the stopper after the bottle has 
 
 been filled and closed tightly. In this way it would require only 
 
 a moderate effort to cause a total pressure of a ton on the interior 
 
 surface of the bottle. 
 
 Pascal's law is indirectly contained in the laws previously stated. 
 
 Thus, in vessel A of Fig. 7 it is clear 
 
 that the pressure at every point below 
 
 the level ae (see discussion in Art. 
 
 26) is 20 g. per scm. greater than it 
 
 would be if the water were removed 
 
 from above that level. That is, the 
 
 pressure exerted by the portion of 
 
 f, , . . J Fig. 10. 
 
 the water above ae is transmitted 
 
 throughout the portion below that level in accordance with Pascal's 
 
 law. 
 
 29. The Hydrostatic Press. — It will be seen from the fore- 
 going that water is a very effective instrument for transmitting and 
 increasing force. This is utilized in the hydrostatic (or hydraulic) 
 press, a machine invented over one hundred years ago. The 
 principle of the hydrostatic press is shown in Fig. 10. Two 
 cylinders of different diameters are connected by a tube and filled 
 with water. They are fitted with tight pistons,/ and /*, resting 
 upon the water. Let the area of the larger piston be fifty times 
 that of the smaller. A weight of i kg. placed on the smaller 
 piston will cause the water to exert a pressure of i kg. upon every 
 2irt2i equal to that of the smaller piston. Hence the total upward 
 pressure upon the larger piston will be 50 kg., and will sustain a 
 weight of 50 kg. 
 
 One form of hydrostatic press is shown in Fig. 11. ^ is a very 
 strong metal cylinder. In it there is a cast iron piston, /*, work- 
 ing water-tight in the collar of the cylinder. The top of the 
 piston carries an iron plate on which the substance to be pressed 
 is placed. The fixed upper plate, Q, is supported upon four strong 
 
22 
 
 The Mechanics of Liquids 
 
 columns. Water is pumped into the cylinder through the pipe, K^ 
 by means of a force pump, A (see Art. 52). When the piston of 
 the pump, /, is forced down, by the downward stroke of the 
 
 Fig. II. 
 
 handle My it exerts a pressure upon the water ; and this pressure 
 is transmitted through the water in the pipe and in the cylinder 
 to the lower end of P, forcing it upward. The force exerted on 
 the piston of the press is to the force exerted by the piston of the 
 pump as the area of the end of the first piston is to the area of 
 the end of the second piston. 
 
 The hydrostatic press is used for many purposes where great 
 pressures are required. It is used in compressing loose cotton 
 into bales, in bending iron plates, in forcing car wheels on to the 
 axles, in lifting heavy weights, etc. The more powerful presses 
 have pumps operated by engines and are capable of exerting 
 pressures of several hundred tons. 
 
Transmission of Pressure 
 
 23 
 
 PROBLEMS 
 
 1. The free surface of a liquid at rest is level. What is the difference 
 between a level surface and a horizontal plane surface ? Why is the dis- 
 tinction commonly disregarded ? 
 
 2. Why is the surface of a liquid at rest level ? 
 
 3. What is the pressure (a) at a depth of 20 cm. in water ? (d) at a depth 
 of 60 cm. in mercury (see table of densities in the Appendix) ? (f) at a depth 
 of 50 cm. in alcohol ? 
 
 4. What is the pressure in pounds per square foot (a) at a depth of 20 
 ft. in water ? (6) at a depth of 3 miles in the ocean? (Take 62.4 lb. per 
 cu. ft. as the density of pure water in all problems. The density of sea 
 water is 1.026 times this.) 
 
 5. What departure from the proportionality of depth and pressure would 
 result if liquids were appreciably compressible ? 
 
 6. A rectangular vessel 50 cm. long, 20 cm. high, and 35 cm. wide is 
 filled with a liquid whose density is 1.5 g. per ccm. Find the total pressure 
 (a) upon the bottom, (6) upon a side. 
 
 7. Find the total pressure in pounds against the side of a cylindrical tank 
 15 ft. in diameter and 12 ft. high, when filled with water. 
 
 8. A cask 2 ft. high and 18 in. in average diameter is fitted with an iron 
 pipe 40 ft. long, as shown in Fig. 1 2. The cross- 
 section of the pipe is .25 sq. in. Cask and pipe 
 are filled with water, (a) Compute the total 
 pressure against the side of the cask, (d) What 
 weight of water does the tube hold ? (c) How 
 would the pressure within the cask be affected 
 by using a pipe having a cross-section of 2.5 sq. 
 in.? 
 
 Note. — Pascal succeeded in bursting a very 
 strong cask in the manner here indicated, using 
 a slender tube 40 ft. high. 
 
 9. If the diameter of the piston in the pump 
 of an hydraulic press is 2 cm. and that of the 
 piston of the press is 30 cm., what will be the 
 total pressure upon the latter due to a force of 
 50 kg. upon the former ? 
 
 10. What is the total pressure against a vertical dam 100 ft. long, against 
 which the water stands to a depth of 12 ft. ? 
 
 11. A cylindrical vessel 20 cm. in diameter is filled to a depth of 15 cm. 
 with oil having a density of .85 g. per ccm. A piston fitting the vessel tightly 
 
 Fig. 12. 
 
24 The Mechanics of Liquids 
 
 is pushed down upon the oil with a force of lOO kg. What is the total pres- 
 sure upon the side of the vessel due to the weight of the liquid and the pres- 
 sure of the piston ? 
 
 III. Buoyancy of Liquids 
 
 30. Buoyancy. — It is a familiar fact that a liquid either wholly 
 or partly supports any solid placed in it. The supporting force 
 exerted by a liquid upon a body either wholly or partly immersed 
 in it is called buoyant force or buoyancy. If the buoyant force 
 upon a body is equal to its weight, the body will float ; if less, it 
 will sink, but its apparent weight in the liquid (measured by the 
 force required to support it while immersed) will be less than in 
 air. 
 
 Laboratory Exercise 6. 
 
 31. The Principle of Archimedes. — The law of buoyancy was 
 discovered by Archimedes, a celebrated Greek mathematician of 
 the third century b.c. It is known as iht principle of Archimedes j 
 and is as follows : A body either wholly or partly immersed in a 
 liquid is buoyed up by a force equal to the weight of the liquid dis- 
 placed by it. 
 
 The proof of this law in the case of a cylindrical or rectangular 
 body, wholly immersed and vertical, is as follows : Let a scm. be 
 the area of the base and // cm. the height of 
 the body (Fig. 13), and let its top be at a 
 depth of //cm. below the surface of a liquid 
 whose density is d g. per ccm. The pres- 
 sures upon opposite sides balance each 
 other ; but they do not tend to support the 
 body. The pressure upon the top is Hd %. 
 per scm., and the total pressure aHd g., or 
 " '^* the weight of the column of liquid lying 
 
 vertically above the solid. Since the bottom is {H-\-h) cm. below 
 the surface, the pressure upon it is {H-\-h)d g. per scm. ; and the 
 total pressure a (//-\-h) dg.^ or the weight of a column of the liquid 
 whose base is a and whose height is H-^h. The buoyant force 
 
Buoyancy of Liquids 
 
 25 
 
 is the difference between these upward and downward forces, that 
 is, a (^H-yh) —aHd or ahd g. ; which is equal to the weight of the 
 displaced liquid. 
 
 If the body is sunk to a greater depth, the pressures upon its 
 top and bottom increase by equal amounts, hence the buoyant 
 force remains constant. 
 
 The following proof of the law holds for a body of any shape. 
 Let A (Fig. 14) represent any portion of a liquid, 
 having any shape whatever. Since this portion 
 remains at rest, its weight is evidently balanced 
 by an equal buoyant force, due to the pressure of 
 the surrounding liquid. These pressures would 
 remain the same if A were replaced by any solid 
 having the same size and shape; hence the 
 buoyant force upon the solid would be the same 
 as it was upon the displaced liquid ; i.e, it would 
 be equal to the weight of the liquid displaced by the solid. 
 
 32. Buoyancy upon Floating Bodies. — If the density (or aver- 
 age density) of a body is greater than that of the liquid in which 
 it is immersed, its weight will exceed the buoyant force and it will 
 sink unless otherwise supported. If the density of the body is 
 less than that of the liquid, its weight will be less than the buoyant 
 force upon it when completely immersed ; hence, if not held down, 
 it will be pushed to the surface and will float partly immersed. 
 
 The buoyant force on a floating body is equal to the weight of 
 the liquid displaced by the immersed portion of the body ; it is 
 also equal to the weight of the body, since the two forces are in 
 equilibrium. Hence a floating body displaces its own weight of 
 the liquid in which it floats. 
 
 Fig. 14. 
 
 PROBLEMS 
 
 1. Will a ship sink to a greater or less depth on sailing from fresh into 
 salt water ? 
 
 2. From the fact that ice floats in water, what may b% inferred concerning 
 the change of volume of water in freezing ? 
 
26 The Mechanics of Liquids 
 
 3. From the table of densities in the Appendix, make a list of metals that 
 will float in mercury, and a list of those that will sink in it. 
 
 4. Since steel is many times denser than water, how can a ship made of 
 steel float ? 
 
 5. Experiments have shown the compressibility of sea water to be suohf- 
 that, at a depth of a mile, its density is j^^f greater than at the surface, 
 (a) Would a body that sinks in sea water at the surface be likely to sink 
 to the bottom of the ocean, however deep ? (^) Under what condition 
 would it not do so ? 
 
 6. The density of sea water is 1.026 g. per ccm. and that of ice is .917 g. 
 per com. What portion of an icel)erg is immersed ? 
 
 Suggestion. — The immersed portion of the iceberg is equal in volume 
 to the displaced water, and the weight of the displaced water is equal to that 
 of the iceberg. 
 
 7. Account for the relative position of two or more liquids that will not 
 mix (as oil, water, and mercury) when placed in the same vessel (^Exp."), 
 
 IV. Specific Gravity 
 
 33. Specific Gravity Defined. — The ratio of the density of a 
 substance to the density of distilled water at 4° Centigrade is 
 called the specific gravity (or specific density) of the substance. 
 Thus, if the density of a stone is 150 lb. per cu. ft., its specific 
 gravity is 150-5-62.4, or 2.44, 62.4 lb. per cu. ft. being the density 
 of pure water at 4° C. Hence to say that the specific gravity of 
 the stone is 2.44 means that the stone is 2.44 times as dense as 
 water. 
 
 Since the density of water is i g. per ccm., the density of the 
 stone in the above problem is 2.44 X i g., or 2.44 g. per ccm. 
 Thus, in the metric system, specific gravity and density are numer- 
 ically equal.^ This is also evident from the definition, since, in 
 the metric system, the numerical value of the second term of the 
 ratio is one. 
 
 Except where great accuracy is required in scientific work, it 
 is unimportant to take account of the very slight difference be- 
 
 1 The equality of density and specific gravity in the metric system is only numer- 
 ical ; the quantities are different in kind, as is clearly evident when English units 
 are used. A ratio is always an abstract number. 
 
specific Gravity 
 
 27 
 
 tween the density of fresh water at ordinary temperatures and the 
 density of distilled water at 4° C. ; hence the specific gravity of a 
 substance may be defined as the ratio of its density to the density of 
 water. For the purpose of laboratory work, it is useful to define 
 the specific gravity of a substance as the ratio of the mass (or 
 weight) of any volume of it to the mass (or weight) of an eqtial 
 volume of water. (Show that this definition is consistent with the 
 first ; i.e. that they are not independent and possibly contradictory 
 definitions.) 
 
 34. Methods of Finding Specific Gravity. — If the density of a 
 substance in pounds per cubic foot is known, its specific gravity is 
 found by dividing this number by 62.4. If the density is known 
 in the metric system, its specific gravity is known without further 
 computation. 
 
 There are several simple experimental methods for finding the 
 specific gravity of a substance without the necessity of determining 
 its density. The ones most frequently used are presented in the"^ 
 following articles. They depend, for the most part, upon the prin- • 
 ciple of Archimedes, and hold without modification for either Eng- 
 lish or metric units of mass and volume. 
 
 35. Specific Gravity of Solids. — Solids Denser than Water. — 
 The solid is weighed, then sus- 
 pended by a thread from a pan of 
 the balance, and again weighed 
 while hanging wholly immersed in 
 water (Fig. 15). The difference 
 between these weights measures the 
 buoyant force upon the body, and 
 is, therefore, the weight of an equal 
 volume of water. Hence the ex- 
 periment gives the masses of equal 
 volumes of the solid and of water. 
 The specific gravity of the solid 
 is the ratio of its mass to the mass of the displaced water. 
 
 Solids Less Dense than Water, — In this case a denser body, 
 
 Fig. 15. 
 
28 The Mechanics of Liquids 
 
 called a sinker^ is attached to the solid to keep it wholly immersed 
 when weighed in water. The relations involved will be understood 
 from the following example : — 
 
 \Veight of lK)dy in air = 40 g. 
 
 Weight of sinker in water = 35 g- 
 
 Weight of body and sinker together in water = 15 g. 
 
 It will be seen that the buoyant force upon the immersed body 
 sustains its entire weight and also 20 g. of the weight of the 
 sinker (35 — 15). That is, the buoyant force upon the body ex- 
 ceeds its weight by 20 g. Its value is, therefore, 40 + 20 g., or 
 60 g. ; and this, by Archimedes' principle, is the weight of water 
 equal in volume to the body. Hence the computations are as 
 follows : — 
 
 Amount by which buoyancy upon the body 
 
 exceeds its weight = 35 — 1 5 = 20 g. 
 
 Buoyancy upon the body = 40 + 20 = 60 g. 
 
 Specific gravity of the body = 40 -<- 60 = .667 
 
 Laboratory Exercise 7. 
 
 36. Specific Gravity of Liquids. — By the Specific Gravity Bottle. 
 — The masses of equal volumes of any liquid and of water can be 
 obtained by finding the weight of each that will completely fill 
 the same bottle. The bottle is weighed when empty, when filled 
 with the liquid, and when filled with water. The specific gravity 
 is computed from these three weights. A bottle having a glass 
 stopper should be used, as a cork or a rubber stopper would prob- 
 ably not be inserted the same distance each time. 
 
 By Buoyancy upon a Sinker. — The masses of equal volumes of 
 any liquid and of water can be obtained by finding the buoyant 
 force exerted by each upon the same solid when completely im- 
 mersed. The sohd is weighed in air, then in the liquid, then in 
 water. The solid must be dense enough to sink in either liquid. 
 A glass stopper will serve. 
 
 By Buoyancy upon a Floating Body: the Hydrometer. — The 
 same body displaces equal weights of all liquids in which it floats. 
 
Specific Gravity 
 
 29 
 
 (Why ?) The weight of the displaced Hquid is equal to the prod- 
 uct of its volume and density. Let v and V denote the volumes 
 of two liquids displaced by the same floating 
 body, and d and D their densities respectively. 
 Then vd = VD ; whence d'.DwVw. That 
 is, the volume of liquid displaced by a floating 
 body is inversely proportional to the detisity of 
 the liquid. Thus, if the second liquid is twice 
 as dense as the first, the floating body will dis- 
 place half the volume of it that it does of the 
 other. 
 
 This principle is utilized in the hydrometer 
 (Fig. 16). It consists of a closed glass tube, 
 with a bulb at the lower end filled with shot or 
 mercury to keep the hydrometer upright, and a 
 second bulb farther up, the purpose of which is 
 to cause the greater part of the displacement. 
 The hydrometer is provided with a paper scale 
 inclosed within the tube, and graduated so that 
 its reading at the surface of any licjuid in which 
 it floats is the specific gravity of the liquid. The reading of the 
 scale increases toward the bottom. (Why ?) 
 
 There are several instruments in common use similar to the 
 hydrometer, but having scales which are graduated with reference 
 to the special use for which each instrument is intended. The 
 alcoholi?neter, for determining the percentage of alcohol in liquors, 
 and the lactometer^ for determining the purity of milk, are exam- 
 ples. 
 
 Laboratory Exercises 8 and g. 
 
 Fig. 16. 
 
 PROBLEMS 
 
 1. A body weighs 500 g. in air and 300 g. in water. What is its specific 
 gravity ? 
 
 2. A stone weighs 4CX) g. ; its density is 2.5 g. per ccm. What will it 
 weigh in water ? 
 
30 The Mechanics of Liquids 
 
 3. A body weighs 200 g. in air and 150 g. in water. What will it weigh 
 in a salt solution the specific gravity of which is i.i ? 
 
 4. Find the specific gravity of a solid from the following data : — 
 
 Weight of solid in air =15 g. 
 
 Weight of sinker in water = 35 g. 
 
 W'eight of solid and sinker in water = 5 g, 
 
 5. A body weighs 600 g. in air and 250 g. in a liquid whose specific grav* 
 ity is 1.3. Find its density. 
 
 6. A stone weighs 1000 lb. ; its volume is 6 cu. ft. What is its specific 
 gravity ? 
 
 7. A body weighs 600 g. in air, 300 g. in water, and 250 g. in a certain 
 liquid. What is the sj>ecific gravity of the liquid ? 
 
 8. A body weighs 300 g. in air and 140 g. in alcohol, the specific gravity 
 of which is .82. What is the volume of the body ? 
 
 9. Find the specific gravity of a litjuid from the following : — 
 
 Weight of bottle = 40 g. 
 
 Weight of Iwttle filled with water = 120 g. 
 
 Weight of bottle filled with the licjuid = 150 g. 
 
 10. A piece of iron weighs 1000 g. ; its specific gravity is 7.2. What will 
 it weigh in water ? 
 
 1 1. A stone weighs 25 lb. in air and 16 lb. in water. What is its spe- 
 cific gravity ? 
 
 12. Find the weight of a cubic foot of lead from its specific gravity. 
 
CHAPTER III 
 
 THE MECHANICS OP GASES 
 
 I. Atmospheric Pressure 
 
 37. Weight of Air. — A vessel full of air is appreciably heavier 
 than the same vessel after the air has been partially removed by 
 means of an air pump {Exp.). The difference, although small, 
 is very noticeable if the vessel is of considerable size (a liter or 
 more) and the weighing is done on a sensitive balance. This 
 proves that air has weight. By accurate measurement (the neces- 
 sary allowance being made for the portion of the air remaining 
 after exhaustion), the weight of a liter of air at ordinary tempera- 
 tures is found to be very nearly 1.2 g. ( = .0012 g. per ccm.). 
 
 38. Pressure of the Atmosphere. — The weight of the atmos- 
 phere causes it to exert pressure, since each horizontal layer sus- 
 tains the weight of all the air above it and adds its own weight to 
 the pressure transmitted to the next lower layer. It is a curious 
 fact that, although the pressure of the atmosphere amounts to a 
 total of about 35,000 pounds on a person of average size, we are 
 unconscious of its existence under the ordinary circumstances 
 of life. The existence of atmospheric pressure is proved by the 
 following experiments. 
 
 39. Experiments proving the Existence of Atmospheric Pres- 
 sure. — A sheet of thin rubber is tied 
 
 over the top of an open receiver of 
 an air pump (Fig. 17). As the air is 
 exhausted from the receiver, the rub- 
 ber is depressed more and more, and 
 soon bursts with a loud report, caused 
 by the sudden entrance of the air 
 
 3» 
 
32 
 
 The Mechanics of Gases 
 
 {Exp.). Before the air is exhausted, it exerts a pressure equal 
 to that of the outside air, and the rubber remains flat. With 
 each stroke of the pump some air is removed, and the pressure of 
 the remaining air is correspondingly diminished. The rubber is 
 therefore pushed inward by the greater pressure of the atmosphere. 
 If the hand be placed over the receiver instead of the rubber, 
 it can be removed only with difficulty after a few strokes of the 
 pump, so firmly will it be held in place by the pressure of the air 
 {Exp.). The flesh of the pahii will be distended into the re- 
 ceiver, and will seem to be pulled in. This sensation is entirely 
 deceptive ; the flesh is pushed in by the pressure of the liquids 
 and gases within the body. Since the pressure within the body 
 sustains and balances the pressure of the atmosphere, the two 
 must be equal. Hence, when the external pressure upon any 
 part of the body is diminished, that part will be distended by the 
 greater pressure from within. 
 
 Two hollow brass hemispheres (Fig. i8), fitting air-tight, can 
 easily be pulled apart so long as none of the air has been removed 
 
 from the space they inclose ; but after 
 a considerable portion of this air has 
 
 rated only with great difficulty (/t;f/.). 
 
 Before the air is exhausted the pres- 
 FiG. 18. sure from within is equal to that 
 
 without and balances it; hence the 
 hemispheres are separated without hindrance from atmospheric 
 pressure. The removal of a portion of the air diminishes the pres- 
 sure from within proportionally, and the pressure upon the outside 
 pushes the hemispheres together with great force. This apparatus 
 was invented about the middle of the seventeenth century by Otto 
 von Guericke, burgomaster of Magdeburg, Germany. The large 
 copper globe that he used for the first trial was not strong enough 
 to withstand the great pressure upon it, and suddenly collapsed 
 with a loud report, terrifying all the spectators. The apparatus is 
 still known as the Magdeburg hemispheres. 
 
 '^1^ 
 
Atmospheric Pressure 33 
 
 40. Laws of Gas Pressure. — Pascal's law (Art. 28) holds for 
 gases as well as for liquids, and for the same reason — their 
 mobility. 
 
 The equal pressure of the atmosphere in all direc- 
 tions can be shown with the apparatus represented in 
 Fig. 19. Thin sheet rubber is fastened over the large 
 end of a thistle tube, and a rubber tube is attached 
 to the other end. On exhausting some of the air 
 with the mouth, the rubber is pushed in by atmos- 
 pheric pressure ; and, for a given exhaustion of the 
 tube, the depression of the rubber remains the same in whatever 
 direction the tube is turned. 
 
 The law of proportionality between depth and pressure (Law 
 III, Art. 23) does not hold for the atmosphere, since its density 
 increases rapidly with the depth, gases being very compressible. 
 The height to which the atmosphere extends is unknown, but is 
 variously estimated at from one hundred to two hundred miles. 
 It is known to extend above fifty miles ; yet the density decreases 
 so rapidly with increase of altitude that the pressure at a height 
 of 3.4 miles is only one half as great as at sea level. 
 From this we know that one half of the atmosphere lies 
 below an elevation of 3.4 miles above sea level. Men 
 have ascended to higher altitudes than this upon moun- 
 tains and in balloons. If the atmosphere were of the same 
 density throughout as at sea-level, it would extend only 
 to a height of about five miles. 
 
 41. Measurement of Atmospheric Pressure. — Figure 20 
 represents a tall U-tube about half full of mercury. The 
 mercury stands at the same level in the two arms under 
 the action of its weight and the equal pressure of the air 
 upon its two surfaces. If a rubber tube be attached to 
 one arm and some of the air exhausted by means of an 
 air pump or by applying the mouth, the pressure in that 
 arm will be diminished and the greater pressure of the 
 atmosphere in the open arm will force the mercury down in 
 
34 
 
 The -Mechanics of Gases 
 
 Fig. ai. 
 
 that arm and up in the other until equilibrium is restored 
 ^^^^ (Fig. 2i) {£x/>.). When the mercury comes to 
 
 ^rTfl rest, the portion of it below the level ac is in 
 
 m * equilibrium under the action of the downward 
 
 m pressure of the atmosphere on the surface a and 
 
 ■ the equal downward pressure at c. This pressure 
 
 I at r is the transmitted pressure of the air remain- 
 
 ■ ing above d, increased by the pressure due to the 
 
 weight of the mercury column //c. Hence the 
 pressure due to the weight of the column measures 
 the difference between the pressure of the atmos- 
 phere and the pressure of the air remaining in the 
 closed arm. For example, when the dilference 
 of level of the two surfaces is lo cm., the pressure 
 of the atmosphere exceeds the pressure in the 
 closed arm by lo x 13.6 or 136 g. per scm. 
 If all the air were removed from the closed arm, there would 
 be no pressure upon the surface of the mercury in it ; and the 
 mercury would rise in this arm until the pressure at r, 
 due to the weight of the column /fc, was equal to the J 
 pressure of the atmosphere. This important fact is util- 
 ized in the mercury barometer, an instrument for measur- 
 ing atmospheric pressure. Figure 22 represents a siphon 
 barometer, so called from its shape. The straight part of 
 the tube is 80 cm. or more in length and its upper end is 
 sealed. It is first completely filled with mercury so as to 
 expel all the air. When turned into an upright position, 
 the mercury falls until it reaches a position of equilibrium, 
 leaving an empty space {vacuum) at the top. The differ- 
 ence of level of the columns, he, is called the hei^t^ht of 
 the barometer. This height multiplied by the density of 
 mercury measures the pressure (per scm.) of the column 
 at c ; and consequently measures the equal pressure of the 
 atmosphere. Thus, when the height of the barometer is 76 cm., 
 the atmospheric pressure is 76 x 13.596, or 1033.3 g. per scm. 
 
 y 
 
 Fig. 22. 
 
Atmospheric Pressure 
 
 35 
 
 ^Z' 
 
 \ 
 
 Another form of barometer is represented in Fig. 
 straight tube about 85 cm. in 
 lenglh and closed at one end, is 
 completely filled with mercury, 
 and, with a finger held tightly 
 over the open end, is inverted, and 
 the open end inserted in a cup 
 of mercury. The height of the 
 column is measured from the sur- 
 face of the mercury in the cupt 
 (Why ?) 9 ? J 
 
 Atmospheric pressure is gener- 
 ally expressed in terms of the 
 height of the barometer, measured 
 in centimeters or inches; as a 
 pressure of 75.3 cm. {of mercury) 
 is understood. The pressure of 
 the atmosphere, in grams per l-i,: 
 
 square centimeter, is equal to the 
 
 weight of a vertical column of air one square centimeter in cross- 
 section, extending from the place where the pressure is taken to 
 the upper limit of the atmosphere. 
 
 Laboratory Exercises JO and li. 
 
 42. The Barometer. — The barometer was invented by Evange- 
 lista Torricelli (1608-1647), an Italian mathematician and scientist, 
 in 1643, some years before Guericke's celebrated experiments at 
 Magdeburg. The space above the mercury in a barometer is 
 called a Torricellian vacuum in honor of the inventor. 
 
 That the barometer column is sustained by the pressure of the 
 atmosphere was conclusively proved by Pascal, under whose direc- 
 tion the height of the barometer was determined at the foot and 
 at the summit of Puy de Dome, a high mountain in France. The 
 height of the mercury fell nearly 8 cm. during the ascent of about 
 900 m. The effect of decrease of atmospheric pressure upon the 
 height of the barometer can be more conveniently shown by 
 
 23- 
 
36 The Mechanics of Gases 
 
 exhausting the air from a tall receiver under which a barometer 
 tube has been placed. The mercury continues to fall as long as 
 the process of exhaustion is continued {£xp.). 
 
 Barometer tubes are mounted in a variety of ways, and pro- 
 vided with scales and other devices for convenience and accuracy 
 in reading. Before the tube is mounted, the mercury in it is 
 boiled to expel all air and moisture. 
 
 43. Uses of the Barometer. — It is a well-established fact that 
 different conditions of the weather are accompanied or preceded, 
 with considerable regularity, by certain changes of atmospheric 
 pressure, as determined by the height of the barometer. (Gener- 
 ally speaking, the barometer is low in stormy weather and high in 
 clear weather ; hence the approach of a storm is indicated by a 
 fall, and the approach of Hiir weather by a rise, of the barometer. 
 The difference between high and low barometer rarely exceeds 
 two or three centimeters. 
 
 This knowledge is used in predicting changes in the weather ; 
 but the forecasts made by the Weather Bureau are based on other 
 sources of information as well, including temperature, direction 
 and velocity of the wind, the course and progress of storms up 
 to the time when the forecast is made, and the existing state 
 of the weather ; all of which are reported to the central office by 
 the different stations distributed over the country. The problem 
 of forecasting the weather is thus a very complex one. 
 
 The barometer is also used for measuring altitudes. The 
 change of pressure due to a given change of altitude being known, 
 the height of a mountain can be computed from the reading of a 
 barometer at its base and at its summit. The height to which a 
 balloon ascends is determined in the same way. For moderate 
 altitudes above sea level, it is approximately correct to compute the 
 change of altitude at the rate of 900 ft. for a fall of the barometer 
 of one inch. 
 
 PROBLEMS 
 
 1. Explain the process of drinking through a straw. 
 
 2. When the mercury barometer stands at a height of 76 cm., what will 
 
Atmospheric Pressure 37 
 
 be the height of a barometer the liquid in which has a specific gravity of 
 1.6? 
 
 3. When the barometer stands at 76 cm., a hter of air at 0° C. weighs 
 1.293 g. At the same temperature and pressure, what will be the weight of 
 the air in a room 9 m. by 7 m. and 4 m. high ? 
 
 4. Compute the weight of i cu. ft. of air at 0° C. and 76 cm. pressure (sp. 
 gr. of air = .001293). 
 
 5. What weight of air at this temperature and pressure is contained in a 
 room 20 by 30 ft., and 12 ft. high ? 
 
 6. In ascending a mountain will the fall of the barometer during each 
 thousand feet of ascent be greater or less than for the preceding thousand 
 feet? Why ? 
 
 7. (a) The weight of the atmosphere is equal to the weight of an ocean 
 of mercury covering the entire surface of the earth to what depth ? (/v) What 
 would be the depth of water covering the entire surface of the earth and 
 having equal weight ? 
 
 8. The surface of the body of a man of medium size is about 16 sq. ft. 
 Assuming this value and also that the pressure of the atmosphere is 14.7 lbs. 
 per sq. in., compute the total pressure that a man sustains upon the surface 
 of his body. 
 
 II. Boyle's Law 
 
 44. The Elastic Force of Gases. — Let A (Fig. 24) represent 
 any portion of the air. It may be thought of as distinct from the 
 surrounding air, although no bounding sur- 
 face actually exists. The surrounding air 
 presses inward upon all sides of A, as indi- 
 cated by the arrows pointing inward. These 
 pressures are balanced at every point by an 
 equal outward pressure exerted by A upon 
 the surrounding air (equal action and reac- P^^' ^ 
 
 tion). This pressure exerted by A is not the 
 result of its weight, but of its tendency to expand; just as a com- 
 pressed spiral spring or a compressed piece of rubber exerts an 
 outward pressure that is independent of its weight. IVAy gases 
 tend to expand is a question that will be considered later (Art. 182). 
 The fact may be accepted for the present without explanation. 
 
 The pressure (per unit area) exerted by any body of gas is 
 
38 
 
 The Mechanics of Gases 
 
 called its elastic force. The English physicist, Robert Boyle, who 
 was among the first to study the mechanics of the air, called this 
 elastic force the spring of the air. The appropriateness of the 
 term will be evident if one suddenly pushes down the piston of a 
 small compression pump (such as a bicycle pump), at the same 
 time keeping the tube closed to prevent the escape of the air 
 {Exp.), The pressure exerted by the confined air rapidly increases 
 as the piston is pushed farther in, and this pushes the piston back 
 again when it is released. In fact, the confined air acts as a 
 spring would if put in its place. The experiment shows that the 
 elastic force of a gas is increased by compression. 
 
 At the same temperature and density, the elastic force of the 
 air in a closed vessel is equal to that of the atmosphere. (Which 
 of the preceding experiments of this chapter have shown this to be 
 true?) 
 
 45. Measurement of the Elastic Force of Gases. — A pressure 
 gauge, or manometer ^ is an instrument for measuring the elastic 
 force of a gas in a closed space. 
 
 An Open Manometer (Fig. 25) is commonly used for the meas- 
 urement of pressures only slightly greater or less 
 ^0j^ than one atmosphere. It consists essentially of a 
 
 ^T glass U-tube partly filled with water or mercury, 
 
 K t, with a rubber tube attached to one arm for mak- 
 
 ■ ing connections, and a scale for measuring the 
 
 height of the li(|uid in the two arms. On con- 
 necting such a manometer with the gas pipes and 
 turning on the gas, the liquid will be pushed 
 down in the arm in which the gas is admitted. 
 The pressure of the gas upon the surface a is 
 equal to the pressure at the same level, c, in the 
 other arm ; and the latter pressure is the sum of 
 the atmospheric pressure upon d, and the pressure 
 due to the weight of the column of liquid dc. 
 Hence the pressure due to the weight of l>c meas- 
 ures the dijference between the pressure of the gas and the pressure 
 
 Fig. 25. 
 
Boyle's Law 39 
 
 of the atmosphere. For example, if the liquid in the manometer 
 is water and the difference of level 8 cm., the pres- 
 sure of the gas exceeds that of the air by 8 g. per scm. 
 The Closed Manofneter. — A short siphon barom- 
 eter with a rubber tube attached to the open arm 
 (Fig. 26) is used to measure the pressure in partially 
 exhausted vessels. Since there is no air or other gas 
 in the closed arm, the mercury completely fills it 
 when under atmospheric pressure. While the air or 
 other gas is being pumped from a vessel to which a 
 closed manometer is attached, the mercury continues 
 to fill the closed arm for some time, if the original ^ 
 
 pressure was more than sufficient to sustain the full 
 height of the column. It is only after the mercury begins to fall 
 that the difference of level of the columns measures the pressure. 
 
 46. Units for the Measurement of Pressure. — The pressure of 
 gases may be measured in any of the following units : — 
 
 In grams per square centimeter or pounds per square inch. 
 
 In centimeters or inches of mercury or of water. 
 
 In atmospheres. An atmosphere is the pressure of a column 
 of mercury 76 cm. high. This unit is approximately the average 
 pressure of the atmosphere at sea level. It is constant and is not 
 to be confounded with the actual atmospheric pressure, which 
 varies from day to day and is different at different altitudes. 
 
 The pupil should be able to formulate rules for finding the 
 value of a given pressure in each of these units when its value in 
 terms of any one of them is known. 
 
 47. Boyle's Law. — The English physicist, Robert Boyle (1627- 
 1691), discovered a simple relation between the volume of a gas 
 and the pressure upon it. This relation, known as Boyle's law, 
 has been found to be approximately true for all gases. It is as 
 follows : — 
 
 The temperature remaining the same, the volume of a given body 
 of gas varies inversely as the pressure upon it. 
 
 For example, if the pressure upon any body of gas is doubled, 
 
40 The Mechanics of Gases 
 
 the volume of the gas will be decreased one half; or, if the pressure 
 is reduced to one fifth of its original value, the volume will be- 
 come five limes as great as at first. 
 
 If the volume of a mass of gas is Kj when the pressure upon it 
 is Pi (g. per scm.) and K when the pressure is F^y then, ac- 
 cording to the law, /j : /^ : : K^ : l\. From this proportion we 
 obuin Pxy\ = PiVt'i that is, at a constant temperature, the prod- 
 uct of the volume of a given body of gas and the pressure upon it 
 is constant. 
 
 It has been found that Boyle's law is not perfectly exact for 
 any gas ; but the departure from the law is so slight that it can be 
 detected only by very accurate measurement, unless the pressure 
 is so great that the gas is near the point of condensation. The 
 law does not hold if the change of pressure is accompanied by a 
 change of temperature ; for a rise of temperature will itself cause 
 an increase of pressure or of volume {Exp,). 
 
 Laboratory Exercise J J, 
 
 48. The Relation between the Density and the Pressure of a 
 Gas. — It follows from lioyle's law that — 
 
 The temperature remaining the same^ the density of a gas is pro- 
 portional to the pressure upon it. 
 
 Thus if the pressure upon a quantity of gas is increased three- 
 fold, its volume will be one third as great as at first ; and, since 
 the entire mass occupies one third its former volume, its density 
 will be three times as great as at first. The increase of the elastic 
 force of a gas with increase of density is well illustrated by the 
 effect of the air in a bicycle tire. After the tire is fully inflated, a 
 further supply of air causes a proportionate increase in its density; 
 and, as is well known, this makes the tire " harder." 
 
 Laboratory Exercise 14, 
 
 PROBLEMS 
 
 I. A cubical vessel 20 cm. in each dimension is full of air at a pressure of 
 one atmosphere. What is the total pressure exerted by the confined air upon 
 the walls of the vessel ? 
 
Applications of Atmospheric Pressure 41 
 
 2. Does the vessel support this pressure when it is surrounded by air 
 under equal pressure ? 
 
 3. What would be the total pressure tending to burst the vessel if it were 
 placed under a receiver from which half the air was exhausted (none of the 
 air being removed from the vessel) ? 
 
 4. Find the weight of air contained in the vessel, assuming that its den- 
 sity is .(X)i2 g. per ccm. 
 
 5. {a) At what depth in fresh water is the pressure due to its weight equal 
 to one atmosphere ? {b) At what depth in salt water ? 
 
 6. From what depth in water must a bubble of gas start in order that its 
 volume may be doubled on reaching the surface ? 
 
 7. {a) What is the pressure in pounds per square inch at a depth of 3 
 miles in the ocean ? {b) What is the total pressure at that depth upon a fish 
 the surface of whose body has an area of 25 sq. in. ? 
 
 8. A cubic decimeter of gas is under a pressure of 100 cm. of mercury. 
 What will be its volume at the same temperature under a pressure of 30 cm. 
 of mercury ? 
 
 9. A liter of gas is taken under a pressure of one atmosphere. What will 
 be its volume at the same temperature under a pressure of 100 cm. of mercury? 
 
 10. Two liters of gas under a pressure of one atmosjihere will have what 
 volume when the pressure is reduced to 900 g. per scm. ? 
 
 III. Applications of Atmospheric Pressure 
 
 49. The Air Pump. — A simple form of air pump is represented 
 in Fig. 27. The pump consists of a metal cylinder in which fits 
 an air-tight piston operated by the handle. There are two valves, 
 a and bj the former in the piston and the latter at the entrance of 
 the tube, at the bottom of the cylinder. The valves open in one 
 direction only, as shown in the figure. The simplest form of valve 
 consists of a piece of flexible leather, placed so as to cover the 
 hole and fastened at one edge. The valve closes the opening air- 
 tight when pressed against it, and leaves it open when pushed in 
 the opposite direction. The pump is connected by a tube to an 
 opening, (9, at the center of a flat metal plate, PQ, upon which 
 stands a receiver, R. 
 
 Suppose the piston to be at rest at the bottom of the cylinder. 
 Both valves will be closed, being held down by their weight. 
 
42 
 
 The Mechanics of Gases 
 
 During the up stroke of the piston, the small amount of air 
 beneath it expands and fills the increased space, and its pressure 
 
 Fig. 27. 
 
 decreases proportionally. The atmospheric pressure upon the 
 top of valve a being now greater than the pressure from beneath, 
 this valve is firmly closed. When the downward pressure upon d 
 is sufficiently diminished, the pressure of the air in the tube 
 beneath this valve lifts it, permitting some of the air in the 
 receiver to escape into the space below the piston. As soon as 
 the piston stops rising, the lower valve is closed by its own weight. 
 On pushing the piston down, the air beneath it is compressed. 
 This air cannot escape through the lower valve, since the increased 
 pressure only closes this valve more tightly. When the amount of 
 compression is such that the density of the confined air is slightly 
 greater than that of the atmosphere, the upper valve is forced 
 open, permitting the air to escape. 
 
 These processes are repeated with every stroke of the piston, 
 thus gradually removing the air from the receiver. The limit of 
 
Applications of Atmospheric Pressure 43 
 
 possible exhaustion is reached when the pressure of the air re- 
 maining in the receiver is insufficient to Hft the lower valve, or 
 when the quantity of air that enters the cylinder with the up stroke 
 is so small that it cannot be compressed enough to raise the upper 
 valve. 
 
 Pumps for obtaining a more nearly perfect vacuum are pro- 
 vided with metal valves or stoj)cocks, operated automatically by 
 a simple mechanism attached to the piston or to the piston 
 rod. 
 
 50. The Compression Pump. — If the valves of the pump repre- 
 sented in Fig. 27 were made to 
 open in the opposite direction, 
 the pump, when operated, would 
 force air into the receiver. A 
 pump made to force air or any 
 gas into a closed vessel is called 
 a compression pump. A pump 
 such as is represented in Fig. 28 
 may be used either for exhaust- 
 ing or compressing gases. On 
 operating the pump, air enters it 
 through A and leaves it through 
 C. Hence if a closed vessel be 
 
 attached to C, air will be forced into it ; if attached to Ay the air 
 will be exhausted from it. 
 
 A bicycle pump is a compression pump of very simple construc- 
 tion. It has but one valve, the entire piston serving this purpose. 
 The valve in the tube of the bicycle tire 
 takes the place of the outlet valve in the 
 pump. (Examine a bicycle pump and 
 explain its action.) 
 
 A bellows is a form of compression 
 pump. It is provided with two valves, 
 a and b (Fig. 29), the former opening inward, the latter outward. 
 (Explain its action.) 
 
 Fig. 28. 
 
 Fig. 29. 
 
44 
 
 The Mechanics of Gases 
 
 51. The Lifting Pump. — The lifting or suction pumpy used for 
 pumping water, is similar to an air pump in its construction and 
 action. The valves open upward, as shown in the figure. A pipe 
 extends from the cylinder or barrel of the pump to some distance 
 below the surface of the water in the well or cistern. The piston 
 is operated by means of a lever, called the handle. Starting with 
 the pump " empty," it first acts as an air pump to exhaust the air 
 from the pipe (see Art. 49). During this process the pressure of 
 the air within the barrel and the pipe decreases and the greater 
 pressure of the air upon the water in the well forces some of it up 
 
 Fig. 30. 
 
 Fig. 31. 
 
 Fig. 32. 
 
 into the pipe ; the pressure due to the weight of the column of 
 water thus sustained being equal to the difference between the 
 atmospheric pressure and the pressure of the air remaining in the 
 pump. 
 
 After the pump is filled with water, the water below the piston 
 follows it during the up stroke, being pushed upward through the 
 lower valve. When the piston begins to descend, the lower valve 
 closes, preventing the return of the water into the pipe. The 
 valve in the piston is forced open at the same time, and the water 
 flows through it into the space above. At the beginning of the 
 
Applications of Atmospheric Pressure 45 
 
 up stroke, the valve in the piston falls and the water above it is 
 lifted out. 
 
 Since the entire pressure of the atmosphere can sustain a col- 
 umn of water only to a height of about 10.3 m. (34 ft.), the lower 
 valve would have to be within that distance of the water in the 
 well even if the pump were capable of producing a perfect vacuum. 
 The actual limit of distance is about 28 or 30 ft. 
 
 52. The Force Pump. — In the force pump the second valve is 
 placed at the entrance to the discharge pipe, B (Fig. 32). There 
 is no valve in the piston. The action of the pump during the up 
 stroke of the piston is the same as in the lifting pump. (Which 
 valve is open ? Which closed ?) With the down stroke of the 
 piston the water is forced into the discharge pipe. The height to 
 which water can be forced in the discharge pipe depends only 
 upon the strength of the pump, being in no way affected by 
 atmospheric pressure. 
 
 Force pumps are generally provided with an air chamber, D, 
 connected with the discharge pipe. During the down stroke of 
 the piston the water is forced into the chamber, compressing the 
 air above it. The elastic force of the compressed air maintains 
 the flow from the air chamber during the up stroke of the piston, 
 making the flow continuous. The force pump is used to force 
 water to considerable heights, and to deliver it under great pres- 
 sure, as in fire engines. 
 
 53. The Siphon. — A bent tube or pipe for transferring liquids 
 over an elevation from a higher to a lower level is called a siphon 
 (Fig. 33). Either a rigid or a flexible tube 
 will serve the purpose. To start a small 
 siphon, it may be held with the bend down 
 and filled, then, with a finger over each 
 end, inverted and placed in position ; or it 
 may be placed in position and the air then 
 exhausted by applying the mouth to the 
 lower end. Siphons are generally provided with a suction tube 
 for this purpose, so that the liquid will not flow into the mouth. 
 
46 The Mechanics of Gases 
 
 The liquid will continue to flow as long as one end of the siphon 
 is covered by it, and the other end is below the level of its surface, 
 />. below ab in the figure ; but if the outlet of the siphon is also 
 immersed, the flow will cease as soon as the liquids in the two 
 vessels reach the same level. 
 
 To explain the action of the siphon we may suppose it to be 
 stopped by closing the outlet, r, with the finger. The liquid will 
 then be at rest, and the laws of pressure for liquids in equilibrium 
 will hold. At points a and b in the tube, on a level with the sur- 
 face of the liquid, the pressure is the same as that of the atmos- 
 phere. The pressure at c is equal to this plus the pressure due 
 to the weight of the liquid column be. Hence, when the finger 
 is removed, this pressure of the column be acts as an unbalanced 
 force u|X)n the liquid in the siphon, causing it to flow. The liquid 
 is held in a continuous column by the pressure of the atmosphere, 
 acting at the ends of the siphon ; otherwise the liquid would part 
 at the top and run out at both ends, leaving the siphon empty. 
 It is, in fact, the transmitted pressure of the atmosphere that forces 
 the liquid up the short arm. 
 
 Laboratory Rxereise 12. 
 
 64. Respiration. — In breathing, the size of the chest cavity is 
 alternately increased and diminished by muscular action. The 
 pressure of the air in the lungs causes them to expand so as 
 always to fill the space afforded them ; hence, when the chest is 
 raised and the diaphragm depressed in inhaling, the expansion of 
 the air already in the lungs diminishes its pressure and more air is 
 pushed into the lungs by the greater pressure of the outside air. 
 The familiar expression " drawing in a breath " is misleading in 
 that it implies a pulling force. When the chest is contracted in 
 exhaling, the air in the lungs is compressed and some of it is 
 forced out. 
 
 55. Buoyancy of the Air. — A body of considerable size and of 
 small specific gravity weighs appreciably more under a partially 
 exhausted receiver than it does in air. This fact may be illustrated 
 with the apparatus shown in Fig. 34. A hollow globe, closed air- 
 
Applications of Atmospheric Pressure 47 
 
 FIG. 34. 
 
 tight, is exactly balanced in air by a solid brass weight. When 
 the apparatus is placed under the receiver of an air pump and the 
 air exhausted, the globe descends, 
 showing that it is now heavier than 
 the weight {Exp.). 
 
 The experiment proves that air 
 exerts a buoyant force. The globe 
 and the solid body have equal 
 weight in air ; but the buoyant force 
 of the air is greater upon the globe, 
 since it is much the larger of the 
 two. Hence, with partial exhaus- 
 tion of the air in the receiver, there 
 is greater loss of buoyancy upon the 
 globe, and it therefore sinks. The 
 law of buoyancy for gases is the 
 same as for liquids and for the same reasons (Art. 31). 
 
 The amount of the buoyant force of the air upon solids and 
 liquids is relatively very small, and in the affairs of daily life may 
 be disregarded.^ The true weight of a body is its weight in a 
 vacuum ; its weight in air in called its apparent weight when it is 
 necessary to distinguish between the two. The difference between 
 the true and the apparent weight of a body is, of course, the 
 buoyant force of the air upon it. The buoyant force of the air 
 upon gases is relatively large. In fact, upon gases less dense than 
 air it exceeds their true weight. Such a gas tends to rise, just as 
 a cork does in water. The weight of a gas is always understood 
 to mean its true weight. 
 
 56. The Balloon. — A balloon is sustained by the buoyant force 
 of the air, the gas with which it is filled being lighter than air. 
 Hydrogen is best adapted to the purpose, being the lightest of 
 gases ; but illuminating gas is generally used, as it is cheaper and 
 more easily obtained. Hot air was used in the balloons first in- 
 
 1 The buoyant force of the air upon i ksj. (i liter) of water is the weight of a 
 liter of air, or about 1.2 g, ; upon i kg. of lead the buoyant force is about .1 g. 
 
48 
 
 The Mechanics of Gases 
 
 vented. A balloon will rise if the buoyant force upon it is greater 
 than its true weight, including the weight of 
 the gas with which it is filled and the weight of 
 the car and its load. 
 
 A balloon is not fully inflated at the start, 
 space being left for the expansion of the gas 
 as the atmospheric pressure upon it dimin- 
 ishes during the ascent. As long as this ex- 
 pansion continues, the buoyant force upon a 
 balloon remains constant, for the increase in 
 the volume of the displaced air offsets the de- 
 crease in its density. As a balloon rises after 
 becoming fully distended, the buoyant force 
 decreases until it is no greater than the true 
 weight of the balloon and all it carries. The 
 balloon then ceases to rise, unless lightened 
 by throwing out sand, a supply of which is 
 carried for that purpose. When the aeronaut 
 
 wishes to descend, he opens a valve at the top of the balloon and 
 
 some of the gas escapes. 
 
 Fig. 35. 
 
 PROBLEMS 
 
 1. Orer how great an elevation can water be siphoned ? Why ? Over 
 how great an elevation can mercury be siphoned ? Would a siphon work in 
 a vacuum ? Explain. 
 
 2. (a) At ordinary temperatures and under a pressure of one atmosphere, 
 a cubic meter of air weighs about 1.2 kg., a cubic meter of hydrogen about 
 .083 kg., and a cubic meter of illuminating gas about .74 kg. Assuming 
 these values, what is the buoyant force upon a balloon containing 500 cu. m. 
 of hydrogen ? (d) How great a weight will this buoyant force sustain 
 in addition to the weight of the hydrogen ? 
 
 3. With what volume of illuminating gas must a balloon be filled to rise, 
 if the empty balloon, the car, and the occupants together weigh 500 kg. ? 
 
 4. Will the true weight of a body be greater or less than its weight in 
 air when weighed on an equal-arm balance with brass weights (a) if the 
 density of the body is the same as that of brass? (^) if its density is less? 
 (<•) if its density is greater ? 
 
CHAPTER IV 
 
 STATICS OP SOLIDS 
 
 I. Concurrent Forces 
 
 57. Mechanics. — Mechanics is the branch of physics that treats 
 of the action of forces upon bodies. It is divided into statics and 
 dynamics or kinetics. 
 
 Statics is the mechanics of balanced forces (Art. ii) ; it treats 
 of the relations that must exist among the forces acting upon a 
 body at rest in order that the body may remain at rest. The 
 statics of fluids is the subject of the two preceding chapters. The 
 subject of the present chapter is the statics of solids. 
 
 Dynamics^ or kinetics, is the mechanics of unbalanced forces 
 (Art. 1 1 ) ; it treats of the effects of unbalanced forces in produc- 
 ing and changing motion. The dynamics of solids is the subject 
 of the following chapter. 
 
 58. Equilibrium of Two Forces. — The relations that must exist 
 among two or more forces in order that they may balance each 
 other are called the conditions necessary for equilibriutn^ or, simply, 
 the conditions of equilibrium. 
 
 The conditions of equilibrium for two forces can be studied by 
 means of two drawscales and a board supported upon three 
 
 Fig. 36. 
 
 marbles lying on a table (Fig. 36). Cords are attached to nails at 
 A and B. Horizontal forces are applied to the board through 
 
 49 
 
50 Statics of Solids 
 
 these cords and are measured by the drawscales. If these forces 
 are in equiUbrium with each other, the board will remain at rest ; 
 if they are not in equilibrium, it will move, since the friction is 
 inappreciable. By trial with the apparatus it will be found that : 
 ( I ) When equal forces are applied in opposite directions but not 
 along the same line, the board will not be in equilibrium, but will 
 rotate until the lines of action of the forces coincide (Fig. 37). 
 
 Fig. 37. 
 
 The board will then be in equilibrium. (2) When the applied 
 forces are opposite and have the same line of action, but are 
 unequal, the board will be pulled in the direction of the greater 
 force. (3) When the forces are either equal or unequal but 
 not opposite in direction, the board will not be in equilibrium 
 
 The experiment shows that fwo forces balance each other only 
 when they are equal in magnitude^ opposite in direction^ and have 
 the same line of action. 
 
 A and B are the points of application of the forces respectively. 
 A force produces the same effect when it is applied at any other 
 point in the same line of action. Thus, if either of the equal and 
 opposite forces be applied at C (Fig. 37) instead of at A or B, 
 they will still be in equilibrium. 
 
 59. Elements of a Force. — The effect of a force depends upon 
 its magnitude^ its direction, and \\.% point of application (or line of 
 action) . These are called the elements of a force. They must all 
 be considered in describing and comparing forces, and in discuss- 
 ing their effects. 
 
 60. Representation of Forces. — In studying the relations of a 
 set of forces to one another, it is often convenient to make use of 
 
>B 
 
 Concurrent Forces 51 
 
 a diagram in which each force is represented by a line. The 
 direction of the force is represented by 
 the direction of the line, with an arrow- 
 head placed on it ; the magnitude of the 
 force, by the length of the line; and its 
 point of application, by the point from 
 which the line is drawn. The method is 
 
 illustrated in Fig. 38, which represents two forces having a common 
 point of application, O, and differing in direction by a right angle. 
 The force represented by OB is twice as great as the other. 
 
 The magnitude of a force can be represented on any scale 
 desired. Thus i cm. may represent a force of 10 g., 100 g., 
 500 g., etc. But the same scale must be used for all forces in the 
 same figure. 
 
 61. Resultant and Components. — In many cases where two or 
 more forces act upon a body at the same time, a single force can 
 be found which, acting alone, would produce the same effect upon 
 the body as the given forces. This one force is called the result- 
 ant of the forces to which it is equivalent, and the latter are called 
 the cojnponetits {i.e. parts) of the resultant. 
 
 The resultant of any number of forces acting along the same 
 line in the same direction is their sum. Thus, if a boy pulls on a 
 cart with a force of 15 lb., and another boy pulls with him, exerting 
 a force of 25 lb., the effect upon the cart will be the same as that 
 of a single force of 40 lb. acting in the same direction. 
 
 The resultant of two forces acting in opposite directions along 
 the same line is their difference, and its direction is that of the 
 greater component. The resultant of two equal forces acting 
 in opposite directions along the same line is zero, since the two 
 forces exactly neutralize each other (Art. 58). The resultant of 
 any set of balanced forces is zero, for the same reason. 
 
 The process of finding the resultant of two or more given forces 
 is called the composition of forces. In the case of forces acting 
 along the same line, composition is effected by adding all the forces 
 that act in one direction, and subtracting all that act in the oppo- 
 
52 
 
 Statics of Solids 
 
 site direction. Other methods are required for forces acting at an 
 angle or along different parallel lines, as shown in the following 
 articles. 
 
 62. Equilibrium of Three Concurrent Forces. — ^ Forces whose 
 
 lines of action meet in a point are 
 called concurrent forces. 
 
 A simple form of apparatus for 
 studying the conditions of equilibrium 
 for three concurrent forces is shown 
 in Fig. 39» Three cords are tied to 
 a ring and a drawscale is attached 
 to each. The scales are adjusted so 
 that all exert a considerable force at 
 the same time. The ring will be in 
 equilibrium under the action of the 
 three forces, all of which lie in the 
 
 Fig. 39. 
 
 same plane. These forces are concurrent at 
 the center of the ring; their directions are 
 outward from this center in the directions of 
 the cords ; and their magnitudes are given by 
 the readings of the scales. In order to deter- 
 mine the relations that exist among the forces, 
 they are represented in magnitude and direc- 
 tion by the lines <j, b^ and r, respectively 
 (Fig. 40). When the experimental 
 work and the construction are ac- 
 curate, it will be found that the 
 diagonal R of the parallelogram con- 
 structed upon any two of these lines 
 as sides, is equal to the third line 
 and is in exactly the opposite direction from O, The conditions 
 necessary for equilibrium may therefore be stated as follows : — 
 
 In order that three concurrent forces may be in equilibrium, 
 their lines of action must lie in the satne plane, and the magnitudes . 
 and directions of the forces must be such that, if any two of 
 
 Fig. 40. 
 
Concurrent Forces 53 
 
 the lines representing them be taken as the sides of a parallelogram^ 
 the concurrent diagonal of this parallelogram will be equal and 
 opposite to, the line representing the third force. 
 Laboratory Exercise 75. 
 
 63. Resultant of Two Concurrent Forces : Parallelogram of 
 Forces. — If three concurrent forces are in equilibrium, the result- 
 ant of any two of them must be equal and opposite to the third ; 
 for they together balance the third force, and, by definition, their 
 resultant is the single force that would produce the same effect. 
 Thus, the resultant of the forces represented by a and b in Fig. 
 40 is represented by the diagonal R^ for we have seen that this 
 diagonal is equal and opposite to c. 
 
 This relation between two concurrent forces and their resultant 
 is known as the parallelogram of forces. It may be stated as 
 follows : If two concurrent forces are represented by lines drawn 
 from the same pointy the concurrent diagonal of the parallelogram 
 constructed upon these lines as sides^ will represent their resultant 
 in magnitude and direction. The numerical value of the resultant 
 is found from the length of this diagonal by applying to it the 
 scale of lengths chosen for the construction. 
 
 The resultant of two concurrent forces can always be found by 
 the above construction. It can also be computed by the rules of 
 trigonometry. In a few cases the resultant can be computed from 
 the relations established in plane geometry. The most important 
 case is that of two forces acting at an angle of 90°. In this case 
 the resultant is equal to the square root of the sum of the squares 
 of the components (since the square of the hypothenuse of £^ right 
 triangle is equal to the sum of the squares of the other two 
 sides). 
 
 64. Composition of More than Two Concurrent Forces. — The 
 resultant of any number of concurrent forces can be found by 
 combining the resultant of any two of them with a third, their 
 resultant with the fourth, and so on till each force has been in- 
 cluded once in the construction or computation, The last result- 
 ant is the resultant of all the forces. 
 
54 
 
 Statics of Solids 
 
 65. Equilibrant. — The single force that would balance one or 
 more given forces is called their equilibrant. The equilibrant of 
 any number of forces is equal and opposite to their resultant. 
 Either of two forces in equilibrium is the equilibrant of the other ; 
 and any one of three forces in equilibrium is the equilibrant of the 
 other two ( Fig. 40) . 
 
 66. Resolution of a Force. — It is frequently necessary in study- 
 ing the effects of a force to consider it as being replaced by two 
 or more concurrent forces which are together equivalent to it. 
 The given force, when so treated, is said to be resolved into its 
 components, and the process is called resolution. It is the reverse 
 of composition. A force can be resolved into any number of sets 
 of components ; but resolution into perpendicular components is 
 most frequently required, as in the following problem. 
 
 A ball whose weight is IK rests upon a plane, AB (Fig. 41), 
 
 inclined at an angle ^^Cwith the 
 horizontal. What portion of the 
 weight of the ball tends to cause 
 it to roll down the plane ? The 
 weight of the ball is represented by 
 OW. Its actual direction is verti- 
 cally down ; but it may be regarded 
 as being made up of two compo- 
 nents, OP and OF. The first, acting perpendicularly to the plane, 
 causes pressure upon it ; the second, acting parallel to the plane, 
 will, if unbalanced, cause the ball to roll down it. A force equal 
 and opposite to OF will hold the ball in equilibrium. 
 
 PROBLEMS 
 
 NoTK. — The numerical results called for in the following problems are all to be found 
 by computation. 
 
 1. A weight of 100 kg. is supported by two cords making equal angles 
 with the horizontal and an angle of 120° with each other. 
 What is the tension on each cord ? 
 
 2. What would be the tension on each cord supporting 
 the above weight if one made an angle of 30° with the 
 horizontal, and the other 60° ? Fig. 4a. 
 
 Fig. 41. 
 
Parallel Forces 
 
 55 
 
 3. What would be the tension on each cord if each made an angle of 45° 
 with the horizontal ? 
 
 4. A ball is placed on a plane inclined at an angle of 30°. What fraction 
 of its weight tends to cause motion down the plane ? 
 
 5. If the weight of the ball in the previous problem 
 is 2 kg., what pressure does it exert upon the plane ? 
 
 6. A body weighing 30 kg. is suspended by a cord as 
 shown in Fig. 43. The ends of the cord are fastened 
 
 at points A and B at the same height. The 
 portion AC ol the cord is 40 cm. long, and 
 the portion ^C is 70 cm. Angle ACB is a 
 right angle. What is the tension sustained by AC? hy BC ? 
 
 7. A weight of 120 g. is tied to a cord and suspended at O 
 (Fig. 44). A second cord is attached to the supporting cord at 
 Ay a distance of 100 cm. from 6>, and is pulled in a horizontal 
 direction till A is drawn 20 cm. from the vertical through O, 
 Find the tension upon AO and upon AF. 
 
 8. Find the resultant of three concurrent forces of 5, 16, 
 and II lb., respectively, the first two acting in opposite direc- 
 tions, and the third at right angles to them (Fig. 45). 
 
 9. A weight of ico lb. is suspended at the 
 middle of a rope, ACB (Fig. 46), 20 ft. long. 
 The ends of the rope are fastened at points 
 A and B at the same height. What is the 
 tension of the rope when CD is 3 ft. ? when CD is I ft.? 
 when CD is i inch ? 
 
 10. (a) A brick lies on the ground. What is the equilibrant of the weight 
 of the brick ? {b) What is the reaction of the pressure of the brick upon 
 the ground ? 
 
 11. How does the reaction of a force differ from its equilibrant ? Men- 
 tion examples to illustrate. 
 
 12. If any number of forces and their equilibrant together act upon a 
 body, what is their combined effect ? 
 
 r 
 
 Fig. 45. 
 
 II. Parallel Forces 
 
 67. Equilibrium of Three Parallel Forces. — Parallel forces are 
 forces having parallel lines of action. 
 
 It is found by experiment that if three parallel forces acting 
 upon the same body are in equilibrium, the following conditions 
 are always fulfilled : — 
 
56 Statics of Solids 
 
 I 
 
 /• 
 
 I. 754^ three forces ^f^^f^^ andf^ (Fig. 47), are in one plane. 
 
 2. The two outside forces act in the same direc- 
 T- tion and the inside force in the opposite direction, 
 j[=^ 3. The inside force is equal to the sum of the 
 other tuto, 
 
 4. The outside forces are ittversely proportional 
 to the distances * of their lines of action from the 
 Fig. 47. line of cu tion of the inside force ; that is, — < 
 
 fi : fi :: di I ^„ orfdi =f^t. 
 
 It will seem that the inside force is nearer the larger of the other 
 two ; but if the latter are equal, it is midway between them. The 
 points of application of the three forces need not lie in a straight 
 line. Any one of the three forces may be regarded as the equili- 
 brant of the other two. 
 
 Laboratory Exercise 16. 
 
 68. Resultant of Two Parallel Forces acting in the Same Direc- 
 tion. — When three parallel forces are in equilibrium, 
 the two outside forces together balance the third 
 force; hence their resultant would also balance it. t/j 
 This resultant {R, Fig. 48) must, therefore, have the 
 same line of action as the third force, and must be ^" ^ ' 
 
 equal to it in magnitude and opposite in direction; hence, — 
 
 Tlu resultant of two parallel forces acting in the same direction 
 is equal to their sum ; it acts in the same direction, and its line of 
 action divides the distance between them into parts inversely pro- 
 portional to the forces, 
 
 PROBLEMS 
 
 1. Two boys, A and B, carry a load between them suspended from a pole 
 5 ft. long. The load is 2 ft. from A's end. What fraction of it does A 
 carry ? 
 
 2. If the load weighs 161 lb., where must it be hung in order that A may 
 carry 92 lb. of it ? 
 
 1 The distance between two lines or from a point to a line is always understood 
 to mean the perpendicular distance. 
 
 R 
 
o 
 
 Moments of Force 57 
 
 III. Moments of Force 
 
 69. Tendency of a Force to cause Rotation. — A force applied 
 to a body may tend to cause it to turn round or rotate about some 
 line as an axis. The simplest case is represented in Fig. 49. A 
 slender stick (meter rod) is supported on a 
 horizontal axis, as a nail, through a hole so 
 situated that the rod will come to rest in a 
 horizontal position. A weight attached on 
 either side of the axis will cause rotation. '^' '^^' 
 
 Two weights, either equal or unequal, can be attached, one on each 
 side of the axis, at such points that the rod will remain in equi- 
 librium {Exp^. The experiment shows that the tendency of a 
 given force to produce rotation is increased by applying it farther 
 from the axis, and that the rod will be in equilibrium only when 
 the product of one force and its distance- from the axis is equal 
 to the product of the other force and its distance. If/i andy^ 
 denote the weights, and ^i and a^ their distances from the axis 
 respectively, then the condition for equilibrium will be expressed 
 by the equation 
 
 /i «i =/2 <h* 
 
 Replacing one of the weights by a drawscale, a measured pull 
 
 ^ can be exerted at any angle (Fig. 50). It 
 
 ^ «^\ will then be found that, as angle x increases. 
 
 I 
 
 a^.^ "V-* the force f^ required to maintain equilibrium 
 1^/, also increases {Exp.). Let a.^ now denote 
 ' the distance from the axis to the ime of action 
 
 ' ^ of /a ; then the condition for equilibrium is 
 
 still /i^i =/2«2- The products /i^i and /iCii therefore measure 
 the tendencies of the forces to cause rotation. The rod remains 
 in equilibrium when these tendencies are equal and in opposite 
 directions round the axis. 
 
 70. Moment of a Force. — The tendency of a force to produce 
 rotation about any axis is measured by the product of the force 
 
58 Statics of Solids 
 
 and the distance of its line of action from that axis. This product 
 is called the moment of the force with respect to the given axis. 
 The distance from the axis to the line of action of a force is called 
 the arm of the force. In the preceding article ciy^ is the arm of 
 the force /i with respect to an axis at O, and /i«i is the moment 
 of the force about that axis ; a^ is the arm of the force /^, and/a^^ 
 its moment about the same axis. 
 Laboratory Exercise ij. 
 
 71. Equilibrium about an Axis. — The conditions for the equi- 
 
 ik/j librium of two forces with respect to rotation 
 
 \ about an axis are that their moments must be 
 
 \ equal and must act in opposite directions 
 
 ^ \ \Zr ' ' round the axis. The condition of equality is 
 
 '^ expressed by the equation /,<z, —fja^. The 
 
 t/i Fig. 51. forces may act on the same side of the axis, 
 
 as in Fig. 51. The general law is as follows : — 
 
 Jn order that any number of forces may be in equilibrium about 
 an axis, the £um of the moments of the forces acting in one direc- 
 tion round the axis must be equal to the sum 
 of the moments of the forces acting in the 
 other direction round it. Thus in Fig. 52 the 
 condition of equilibrium with respect to an 
 axis at O is/,tfi =7,^2 ^/s^s- 
 
 When the line of action of a force passes 
 through the axis, it has no tendency to cause 
 rotation, for the arm of the force is then zero 
 and its moment is zero. This case is illustrated 
 
 byyiin Fig. 52. The pressure upon the body exerted 
 by the axis is also such a force; but although this 
 ^ 'q pressure has no moment, it is the equilibrant of all 
 
 the other forces acting upon the body. 
 
 72. Couple. — Two equal forces acting in opposite 
 directions upon the same body and having different 
 lines of action constitute a couple (Fig. 53). The dis- 
 FiG. S3. tance between their lines of action is called the arm 
 
Moments of Force 
 
 59 
 
 of the couple. The moment of the couple about any axis per- 
 pendicular to the plane of the couple is the product of its arm 
 and either force (=/^ in the figure). 
 
 (Prove that this is so for an axis at O, midway between the 
 points of application of the forces, and for an axis at Q, a point 
 anywhere in the plane.) 
 
 PROBLEMS 
 
 1. If in Fig. 49 /i = 90 g., a\ = 40 cm., and a^ = 25 cm., what is the 
 value of ^2 ? 
 
 2. If in the same figure yi = 250 g.,y^ = 125 g., and a^ = 50 cm., what is 
 the value of «i ? 
 
 3. How do two children of unequal weight balance each other in see- 
 sawing ? Explain how they make either end of the seesaw descend at will. 
 
 4. An object weighed with a steelyard (Fig. 54) is how many times heav- 
 ier than the weight upon the beam by which it is balanced ? 
 
 5. A man in moving a stone with a crow- 
 bar exerts a force of 50 lb. with each hand. 
 
 I 
 
 Fig. 54. 
 
 Fig. 55. 
 
 at distances of 100 cm. and 150 cm., respectively, from the point where 
 the crowbar is supported, and this point of support is 25 cm. 
 from the lower end of the bar. How great is the force exerted 
 at this end ? 
 
 6. In drawing a nail with a claw hammer a man exerts 
 a force of 50 lb. The point where the hammer presses against 
 the board is i in. from the nail and 9 in. from the point on 
 the handle where the force is applied. How great is the 
 pull upon the nail ? 
 
 7. If the arms of a balance (Fig. 3) are not of exactly 
 equal length, what error in weighing results when the body to be weighed is 
 placed in the pan on the longer arm? 
 
 Fig, 56. 
 
6o Statics of Solids 
 
 TV. Effect of Weight on the Equilibrium of Bodies 
 
 73. Gravity. — The earth's attraction for bodies is called gravity^ 
 when named in a general way without reference to its amount for 
 any particular body. Gravity acts on every particle of a body, 
 and these forces on the individual particles are all directed toward 
 the same point at (or near) the earth's center. Since this point 
 is at a distance of about four thousand miles, there is no measur- 
 able error in assuming that the forces of gravity acting on the parti- 
 cles of a body are parallel. 
 
 The line of action of gravity is called a vertical line. It is 
 the direction assumed by a cord from which a weight hangs at rest. 
 \ line or a plane perpendicular to a vertical line is horizontal. 
 
 74. Center of Gravity. — The weight of a body, regarded as a 
 single force, is the resultant of all the forces of gravity acting on 
 its particles. Since these forces all act in the same direction, 
 their resultant is equal to their sum and also acts in the same 
 direction — vertically downward. The point of application of the 
 weight of a body, regarded as a resultant force, is called the center 
 of gravity of the body. It is also called center of weight, center 
 of mass, and center of inertia. 
 
 It can be shown both experimentally and mathematically that, 
 
 so long as a body retains the same shape and distribution of its 
 
 parts, its center of gravity is a fixed point 
 
 relative to the body, however it may be 
 
 turned and in whatever condition of rest or 
 
 of motion it may be. Further^ under all 
 
 circumstances affecting the state of rest or of 
 
 motion of a solid, its weight may be regarded 
 
 as a single force applied at its center of 
 
 gravity ; for the effect of such a force would 
 
 be the same as that of the actual forces of 
 Fig. 57, 
 
 gravity acting on the particles of the body. 
 
 (A resultant force is always equivalent to its components.) For 
 
Equilibrium of Bodies 
 
 6i 
 
 example, let G (Fig. 5 7) represent the center of gravity of a stone ; 
 then, under all circumstances, we may regard its weight as a 
 single force applied at G and acting vertically downward. 
 
 75. Action of Weight on a Suspended Body. — If a body is free 
 to rotate about an axis from which it is suspended, it always 
 comes to rest with its center of gravity vertically below the axis. 
 The reason for this is shown in Fig. 58, which represents a flat 
 body, as a board, suspended at O. The weight 
 of the body, w, is regarded as a single force 
 applied at its center of gravity, C. The body 
 is evidently not in equilibrium in the position 
 represented in the figure, since the moment of 
 its weight is unbalanced and causes rotation. 
 When C is vertically below Oy the moment 
 of the weight of the body is zero, since its arm is then zero. In 
 this position the weight of the body and the pressure upon it at 
 the axis are equal and opposite and have the same line of action. 
 This is therefore a position of equilibrium. 
 
 76. Methods of Finding the Center of Gravity of a Body. — The 
 behavior of the center of gravity of a body when suspended affords 
 a simple method for finding it experimentally, as illustrated in 
 
 Fig. 59. The figure represents any flat body, as 
 a board, suspended at O upon a pin or a nail. 
 The vertical through O is determined by a small 
 plumb line suspended from the same axis. A line 
 indicating the position of this vertical is drawn on 
 the body. The center of gravity of the body is at 
 some point directly back of this line. The body 
 is then suspended at some other point, P, and a 
 second vertical, DP, determined, as before. Since 
 
 the center of gravity lies directly back of both verticals, it must be 
 
 the point that lies directly back of their point of intersection and is 
 
 midway between the surfaces of the body. 
 
 The center of gravity of any body may be similarly found by 
 
 suspending it successively at any two points about which it is free 
 
 Fig. 59. 
 
62 
 
 Statics of Solids 
 
 Fig. 6o. 
 
 to swing (Fig. 60). The only difficulty, if any, would be in 
 accurately determining the verticals through the 
 material of the body. 
 
 The center of gravity of a body of regular shape 
 and uniform density is its center of figure. The 
 centers of a sphere, a spheroid, an ellipsoid, a cube, a 
 parallelopiped, and a circular cylinder are examples. 
 The shape of a body may be such that its center of 
 gravity does not lie in any material part of it. This 
 is true, for example, of a ring or a hollow sphere. 
 
 77. States or Kinds of Equilibrium. — A body is in equilibrium 
 only when its weight is balanced by the other force or forces acting 
 upon it The equilibrium of a body in a given position is stable if 
 the body tends to return to that position after being turned or 
 tilted through a very small angle in any direction ; it is unstable 
 if the body tends to move still farther from its original position 
 after being thus disturbed ; it is neutral if the body remains in 
 equilibrium in any adjacent position. 
 
 Laboratory Exercise 18. 
 
 78. Stable Equilibrium. — A body suspended at a point other 
 than its center of gravity is in stable equilibrium when at rest with 
 its center of gravity vertically below the support ; for the unbalanced 
 moment of its weight turns it back to this position when it is dis- 
 placed in either direction (Figs. 58 and 59). The equilibrium of 
 a rectangular block standing on a horizontal plane 
 
 is also stable. When slightly displaced by tilting, 
 the moment of its weight about the edge on which 
 it is turned tends to bring it back to its former 
 position (Fig. 61). 
 
 When any body in stable equilibrium is inclined, 
 its center of gravity is raised ; and in returning to 
 the position of equilibrium, the center of gravity 
 falls. Every body at rest is in stable equilibrium 
 when its position is such that the slightest inclination in any direc- 
 tion raises its center of gravity. 
 
 c\ 
 
 w 
 Fig. 61. 
 
Equilibrium of Bodies 63 
 
 79. Unstable Equilibrium. — A body is in unstable equilibrium 
 when balanced on a point, an edge, or an axis with its center of 
 gravity vertically above the support. Figure 62 repre- 
 sents a rectangular block balanced in unstable equi- 
 librium on an edge. In this position of the block the 
 moment of its weight is zero ; but, with the slightest 
 displacement in either direction, the moment of its 
 weight tends to turn the block still farther in the same 
 direction. An egg standing on end upon a flat sur- 
 face would be in unstable equilibrium. The center of gravity of a 
 body in unstable equilibrium is higher than it would be in adjacent 
 positions of the body ; hence the slightest disturbance causes such 
 a body to fall. 
 
 80. Neutral Equilibrium. — A body supported on an axis pass- 
 ing through its center of gravity is in neutral equilibrium when at 
 rest in any position, since the moment of its weight is zero for 
 all positions of the body about the axis. When a body thus sup- 
 ported is set rotating, it will come to rest in any position in which 
 friction may chance to stop it. A perfectly balanced wheel is a 
 familiar illustration of neutral equilibrium ; but any body, however 
 irregular in shape, may be suspended in neutral equilibrium. The 
 equilibrium of a perfect sphere at rest on a horizontal plane is 
 neutral, since the vertical through its center of gravity passes 
 through the point of support in all positions. The center of 
 gravity of a body is neither raised nor lowered by any slight dis- 
 placement from a position of neutral equilibrium. 
 
 A body may be in different states of equilibrium at the same 
 time with respect to motion in different directions. Thus a cir- 
 cular cylinder at rest on its side upon a horizontal surface is in 
 neutral equilibrium with respect to rolling, and in stable equilibrium 
 with respect to motion by which either end would be raised. 
 
 81. An Important Property of the Center of Gravity. — Suppose 
 a body of irregular shape (Fig. 63) to be suspended in neutral 
 equilibrium at its center of gravity, C. Let c be the center of 
 gravity, and w the weight of the portion of the body lying on one 
 
64 
 
 Statics of Solids 
 
 side of a vertical plane through C, and c' and w' the center of 
 
 gravity and weight respectively of 
 the portion lying on the other side 
 of this plane. Also let a and a^ 
 be the arms of w and 7v' respec- 
 ^* tively about C as an axis. Then, 
 
 since the body is in equilibrium, wa = 7v^a'. This stated in words 
 means that a plane through the center of gravity of a body divides 
 the body into two parts whose weights have equal moments about 
 the center of gravity. 
 
 For irregular bodies the arms a and a' will, in general, be 
 unequal ; in which case 10 and 7v' will also be unecjual. Hence a 
 plane through the center of gravity of a body generally divides it 
 into parts having unequal weight. 
 
 82. Stability. — It is well known that a body is overturned 
 with greater difficulty when the area of the base upon which it 
 
 stands is increased. A rect- 
 angular block of wood provided 
 with a large base at one end 
 affords excellent illustration 
 (Fig. 64). It is in stable equi- 
 librium in either position shown 
 in the figure ; but its stability 
 is much greater when standing 
 upon the larger base. The rea- 
 
 4:71 
 
 r:d" 
 
 ic 
 
 
 
 7 
 
 C 
 
 Fig. 64. 
 
 son for this is that in the latter position the moment of the weight 
 of the body, wa, is much greater, and also that 
 the body must be turned much farther against 
 this moment before it will fall over. 
 
 The stability of a body standing on a given 
 base is increased by lowering its center of 
 gravity. This may be illustrated by a block 
 weighted with lead at one end. When stand- 
 ing on its heavier end, the block must be turned 
 
 farther before its weight acts to overturn it (Fig. 65). 
 
 Fig. 65. 
 
Equilibrium of Bodies 65 
 
 The two conditions for great stability are, therefore, a large base 
 and a low ce?iter of grain ty. So far as stability is concerned, the 
 base of a body standing on legs is the entire area inclosed within 
 straight lines connecting the legs. 
 
 Laboratory Exercise ig. 
 
 83. Equilibrium of Floating Bodies. — The buoyant force upon 
 a floating body is the resultant of the total pressure of the liquid 
 upon it. This resultant is equal to the weight of the body (Art. 32), 
 its direction is always vertically upward, and its point of applica- 
 tion is the center of gravity of the displaced liquid. This point 
 is called the center of buoyancy. We may therefore regard a float- 
 ing body as being acted upon by two resultant forces — weight 
 and buoyancy — which are always equal and opposite. A float- 
 ing body is in equilibrium when these forces have the same line 
 of action ; that is, when the center of gravity of the body and the 
 center of buoyancy lie in the same vertical line. This is illus- 
 trated by the first and 
 
 third parts of Fjg. 66, 1 
 
 in which C denotes ^!^^ \b 
 
 the center of gravity ^=-3 ^•:gj ^>5^^^^^fe j t^ F^^:=d 
 
 of the body and ^the ~^" ~ 
 
 / , Fig. 66. 
 
 center of buoyancy. 
 
 When a floating body is displaced from its position of equilibrium, 
 the lines of action of weight and buoyancy no longer coincide, and 
 the two forces constitute a couple whose effect is to restore equi- 
 librium, if the equilibrium was stable, or to carry the body still 
 farther from that position, if the equilibrium was unstable. When 
 the rectangular block represented in the figure is tilted from its 
 first position, the center of buoyancy shifts toward the deeper 
 displacement, while the position of the center of gravity remains 
 unchanged. This establishes a couple which tends to restore the 
 body to its former position. Equilibrium in the first position is 
 therefore stable. The third part of the figure represents the posi- 
 tion of unstable equilibrium {Exp.). (Illustrate this in a figure 
 showing the body slightly displaced.) 
 
 i 
 
66 Statics of Solids 
 
 The equilibrium of a floating body is always stable when the 
 center of gravity of the body is below the center of buoyancy ; 
 when it is above the center of buoyancy, the equilibrium may be 
 either stable or unstable, depending upon the shape and position 
 of the body, as illustrated above. The equilibrium is neutral if 
 the centers of gravity and buoyancy remain in the same vertical 
 line when the l)ody is disturbed, as is the case with a sphere or a 
 long cylinder with its axis horizontal {Exp.), 
 
 84. Stability of Floating Bodies. —The stability of a floating 
 body of given shape is increased by lowering its center of gravity ; 
 for this increases the arm of the couple, which tends to right the 
 body when displaced. It is for this reason that a vessel witho'ut 
 a cargo carries ballast. 
 
 PROBLEMS 
 
 I. In what direction does a person lean when carrying a heavy load 
 in one hand ? Why ? 
 
 2. Show that when a homogeneous hemisphere is 
 inclined {A^ Fig. 67), its weight tends to bring it into 
 the position shown in B. In what kind of e(|uilibrium 
 is it in the second position ? Is it in unstable equi- 
 librium in the first position ? Give reasons. 
 
 3. (a) Oil cans are made of the shape shown in Fig. 68, and are 
 weighted with lead at the bottom. Such a can rights itself when 
 tipped. Explain. (^) Does the can really rise or fall when it 
 rights itself? fiG. 68. 
 
 4. Why does a person always lean forward before attempting to rise from 
 a chair ? 
 
 5. A pencil with a knife attached can be balanced, as shown in Fig. 69. 
 Try it. ^^^lat is the evidence that the equilibrium is stable ? 
 Where is the center of gravity of the pencil and knife regarded as 
 one body ? 
 
 6. Show by means of figures that the .moment of the weight of 
 a sphere is zero upon a horizontal surface, but not upon an inclined 
 plane. 
 
 7. If a body that will not roll remains at rest when placed on 
 an inclined plane, three forces act to hold it in equilibrium. Two 
 
 Fig. 69. of these forces are its weight and the pressure of the plane. What 
 
Equilibrium of Bodies 67 
 
 is the third force, and in what direction does it act ? Draw a figure correctly 
 representing the direction and the relative magnitude of the three forces. 
 
 8. Two spheres weighing 50 kg. and 15 kg., respectively, are connected 
 by a rod so that the distance between their centers is 80 cm. Disregarding 
 the weight of the rod, where is the center uf gravity of the whole considered 
 as one liody ? 
 
 9. The average distance between the centers of the earth and the moon 
 is about 240,000 miles ; the mass of the earth is 80 times that of the moon. 
 How far is their common center of gravity from the earth's center? 
 
 10. Two men, A and B, carry a board 30 ft. long and of uniform cross- 
 section. A holds at one end ; where must B hold in order to carry .6 of the 
 load? 
 
 11. A boy weighing 40 lb. wishes to seesaw alone on a plank weighing 
 70 lb. The plank is 24 ft. long, and the center of gravity of the boy is i ft. 
 from an end of the plank. How far from that end must the plank be sup- 
 ported ? 
 
 12. Why is it an advantage to spread the feet when standing upon a sur- 
 face that is moving unsteadily, as the deck of a vessel ? 
 
 13. What would happen to the leaning tower of Pisa (Fig. 75) if the 
 vertical through its center of gravity fell without the base of the tower ? 
 
 14. Is the stability of a boat greater when the occupants are standing or 
 sitting ? Why ? 
 
 15. Why is it difficult to walk on stilts ? 
 
 16. A uniform stick oC timber 10 ft. long balances on an axis 3 ft. from 
 one end when a weight of 20 lb. is hung from that end. Find the weight of 
 the stick. 
 
 17. Why cannot one stand with hb heels against a wall and lean forward 
 without falling ? 
 
 18. Two boys, A and B, carry a uniform plank 24 ft. long, weighing 120 
 lb. A holds at one end and B 4 ft. from the other end. What load does 
 each carry ? 
 
CHAPTER V 
 
 DYNAMICS 
 
 I. Motion 
 
 85. Motion. — Motion is continuous change of position. The 
 line along which the center of gravity of a body moves is regarded 
 as the path of the body. The motion of a body is completely 
 known when we know its path and the rate of motion at every 
 point of the path, or the rate and direction of motion at every 
 instant during the motion. 
 
 Rate of motion is called speed. Velocity includes both rate and 
 direction of motion. The distinction in meaning between the two 
 words is frequently useful, but it is not strictly adhered to. Thus 
 the word velocity is frequently used to signify merely rate of 
 motion, its direction not being a matter of importance for the 
 purpose under consideration. 
 
 86. Uniform Motion. — The motion of a body is uniform if the 
 body passes over equal portions of its path in equal intervals of 
 time, however short these intervals may be. If the motion of a 
 body is imiform, its speed is constant, and is measured by the 
 distance that the body moves over in a unit of time. 
 
 Speed is measured in various combinations of units of distance 
 and of time, as centimeters per second, meters per second, feet 
 per second, miles per hour, etc. In the definitions that follow, 
 the second will be named as the unit of time, since it is the only 
 one that is used in scientific work. 
 
 The whole distance passed over by a body moving with constant 
 speed is equal to the product of the speed and the time occupied 
 in traversing the distance. Hence, letting d denote the distance, 
 
 68 
 
Motion 69 
 
 V the (magnitude of the) velocity, and / the time, we have for 
 uniform motion : — 
 
 d—vt'. also V = -, and /= -• (i) 
 
 87. Variable Motion. — The expression "the velocity of a 
 body " has no definite meaning unless the velocity is constant. If 
 it is variable, qualifying terms are required, as indicated in the 
 following definitions : — 
 
 T/ie velocity of a body at any instant (or at any point of its path) 
 is the distance that it would pass over during the next second (or 
 other unit of time) if its velocity continued unchanged from that 
 instant. Thus when we say that a train is running at the rate of 
 30 miles per hour, we mean that it would run 30 miles in an hour 
 if it continued at its present rate for one hour. 
 
 The average velocity of a body during any interval of time (or 
 between any two points of its path) is the uniform velocity that 
 would be required to pass over the same distance in the same 
 time. Average velocity is therefore equal to the distance divided 
 by the time. Representing average velocity by z>, its definition is 
 expressed by the formula, 
 
 d—vt) from which v = -* (2) 
 
 For example, if an automobile runs 108 mi. in 6 hr., its 
 average rate is 18 mi. per hr., since this is the uniform rate 
 required to run the given distance in the given time. The actual 
 rate may vary from o (during intervals of stopping) to 40 or 50 
 mi. per hr. 
 
 88. Representation of Velocities. — A velocity may be repre- 
 sented in both magnitude and direction by a b.^ 
 straight line, just as a force may be. Thus if 
 OA (Fig. 70) represents a velocity of 3 ft. per 
 
 sec. east, then OB represents a velocity of 2 ft. o' ■ ^^ 
 
 per sec. north. ^^''' 7°- 
 
 89. Composition of Velocities. — A body may have two or more 
 independent motions at the same tinae {Exp.). Thus a boat 
 
70 
 
 Dynamics 
 
 rowed across a stream has a motion imparted by the rowing, and 
 also a motion due to the current and equal to it. Suppose the 
 boat to be constantly headed directly toward the opposite shore, 
 and let O (Fig. 71) represent the starting point. 
 OB would be the path of the boat if there were 
 no current. OC is the distance the stream flows 
 while the boat is crossing. The actual motion 
 of the boat is the resultant of these two motions ; 
 its path is represented by OA, If OB and OC 
 be taken to represent the component ve/ocitieSt 
 then OAf the concurrent diagonal of the paral- 
 lelogram constructed on OB and OC as sides, 
 will represent the actual, or resultant, velocity upon the same scale. 
 Velocities are, in fact, compounded by the same rules as forces 
 (.\rts. 63 and 64). The construction is called the parallelogram 
 of velocities. 
 
 90. Resolution of a Velocity. — A velocity, like a force, can 
 be resolved into components in any chosen directions. The con- 
 struction is the same as for the resolution of a force (Art. 66). 
 For example, a vessel is sailing 30° north of east at the rate of 1 2 
 
 Fig. 71. 
 
 mi. per hr. At what rate is it advancing north- 
 ward and at what rate eastward ? It is proved in 
 geometry that in a right triangle having an acute 
 angle of 30° the hypothenuse is twice the shorter ^^^- 7a- 
 
 leg. Hence ON (Fig. 72), the northerly component of the 
 velocity, is 6 mi. per hr. ; and OE, the easterly component, is 
 
 Fig. 73. 
 
 V12* — 6* = 10.4- mi. per hr. 
 
 As a further illustration, let us consider how the 
 boat mentioned in the preceding article must be 
 rowed in order to reach the opposite bank at B 
 instead of at A. The resultant motion is now rep- 
 resented by OB, The component OC, due to the 
 motion of the stream, is the same as before. Hence 
 OB is the diagonal of a parallelogram of which 
 
Motion 71 
 
 one side is OC. The other component motion is therefore repre- 
 sented by OA^ (Fig. 73). This means that the boat must be con- 
 stantly pointed in a direction parallel to 0A\ and that it would 
 take as long to reach B as it would to row the distance OA' in 
 still water. (In the preceding problem of Art. 89 would more 
 time be required to cross to A than to cross to B in still water ?) 
 
 PROBLEMS 
 
 1. A ball rolls 53 m. in ii sec. Find its average velocity. 
 
 2. A train runs with an average velocity of 23 m. per sec. In what time 
 does it run a kilometer ? 
 
 3. From a train running at the rate of 9 m. per sec, a mail bag is thrown 
 at right angles to the track with a velocity of 4 m. per sec. Compute the 
 resultant velocity of the bag at the instant it leaves the hand, and draw a fig- 
 ure to show its direction. 
 
 4. From a train running at the rate of 12 m. per sec. a mail bag is thrown 
 so that its resultant velocity is equal to that of the train and at right angles 
 to it. What is the magnitude and direction of the velocity imparted in 
 throwing the bag ? 
 
 5. An arrow is shot directly backward from the rear of a train with a 
 velocity (relative to the train) equal to that of the train. What is the motion 
 of the arrow ? 
 
 6. The rotation of the earth carries its surface eastward at the rate of 
 about \ mi. per sec. (in temperate latitudes). When a ball is 
 thrown up, why is it not left behind (to the west) by the earth 
 in its rotation ? 
 
 7. Four boys, A, B, C, and D (Fig. 74), on the deck of a 
 moving vessel, pass a ball round in the order of the letters. 
 What allowance for the motion of the vessel, if any, must be 
 made by each of the boys in throwing ? Give reasons. ' '^ 
 
 8. A vessel sails due N.E. at the rate of 15 mi. per hr. Compute the 
 northerly and easterly components of its velocity. 
 
 9. A boat is rowed so that it crosses a stream 100 m. wide to a point directly 
 opposite to the starting point (Fig. 73). The stream flows .8 m. per sec, 
 and the boat is rowed at the rate of 1.2 m. per sec. in still water. How long 
 is the boat crossing the stream ? 
 
 91. Acceleration. — The velocity of a body is said to be accel- 
 erated when it is increasing, and retardedy or ?iegative/y accelerated, 
 when it is decreasing. The rate of change of velocity is called the 
 
72 Dynamics 
 
 acceUration. Thus, if a body starting from a state of rest has a 
 velocity of 3 m. per sec. at end of the first second, 6 m. per sec. 
 at the end of the second second, 9 m. per sec. at the end of the 
 third second, etc., its velocity increases 3 m. per sec. every second ; 
 i.e, its acceleration is 3 m. per sec. per sec. This is a case of 
 uniformly atceUraUd motion^ or constant acceleration^ the increase 
 of velocity being the same for each second. When the velocity 
 of a body decreases by the same amount during each second, 
 its motion is said to be uniformly retarded^ or to have a constant 
 negative acceleration. 
 
 It more frequently happens that the acceleration of a body is 
 variable. This is the case, for example, with a street car. As 
 its speed increases, the rate of increase diminishes. When its 
 speed is the greatest, the increase of speed, or acceleration, is zero. 
 There are, however, important cases of constant acceleration ; and 
 it is only these that we shall consider quantitatively. 
 
 Motion is accelerated when it changes in direction, even if the 
 speed remains constant. This, however, is reserved for later con- 
 sideration (Art. 119). 
 
 92. Formulas for Uniformly Accelerated Motion. — In the 
 case of uniformly accelerated motion in a straight line, the accel- 
 eration is measured by the constant change of speed that occurs 
 during each second. If a body moves with a constant accelera- 
 tion of a cm. per sec. per sec, starting from rest, its velocity at the 
 end of I sec. will be a (centimeters per sec), at the end of 2 sec 
 it will be 2 /7, at the end of / sec. it will be / a. This is expressed 
 
 by the formula 
 
 v^at, (3) 
 
 in which a denotes the constant acceleration and v the velocity at 
 the end of / sec. after starting. This is usually called the final 
 velocity. 
 
 Since the velocity increases from zero at a uniform rate, the 
 average velocity, z>, during the time /, is one half of the final veloc- 
 ity ; t^.v=—. The entire distance traversed by the body is the 
 
Motion 73 
 
 product of its average velocity and the time (Art. 87) ; hence, 
 letting d denote this distance, 
 
 . -, at ^ at^ 
 d=vt = —Xt = — ; 
 2 2 
 
 .0 / ^ 
 
 that is, d= — ; and /=\/ (4) 
 
 V 
 
 From equation (3), /=-. Substituting this value of / in equa- 
 tion (4), we have 
 
 2 2 a 
 
 zr' 
 
 that is, d= — -, and v^y/tad, (5) 
 
 2a 
 
 This formula expresses the relation between the constant accel- 
 eration of a body starting from rest, the distance that the body has 
 traversed, and its velocity at the end of that distance. 
 
 If any two of the quantities in formula (3), (4), or (5) are given, 
 the value of the third quantity can be found by substituting the 
 given values in the formula. 
 
 93. Laws of Uniformly Accelerated Motion. — The following 
 laws of uniformly accelerated motion in a straight line, for bodies 
 starting from rest, are contained in the above formulas : — 
 
 I. The velocity at any instant is proportional to the time during 
 which the body has been in motion. (Formula 3.) 
 
 II. The velocity acquired in a given time is proportional to the 
 acceleration. (Formula 3.) 
 
 III. The average velocity during the whole time is half the final 
 velocity. 
 
 IV. The distance passed over is proportional to the square of 
 the time. For, if the body traverses the distance d^ in t^ sec. and 
 the distance d., in /o sec, both measured from the instant of start- 
 ing, then, from formula (4), /?i = — ^, and 4= — ^. Dividing each 
 
 2 2 
 
74 Dynamics 
 
 member of the first equation by the corresponding member of the 
 second, we get . = tj » hence, 
 
 d,'d^.'. A* : i,\ (6) 
 
 V. Thf distance trm^ersed in a given time is proportional to the 
 acceleration. (Formula 4.) 
 
 VI. The acceleration is numerically equal to twice the distance 
 trax>ersed during the first second. 
 
 For, when /= i, formula (4) becomes d= -. 
 
 2 
 
 PROBLEMS 
 
 1. A ftreet car rant with a constant acceleration of 1.2 m. per sec. per sec. 
 for 8 sec. after starting, (a) What is its velocity at the end uf that time? 
 (^) What was its average velocity during the 8 sec? (r) How far does it 
 ran in the 8 sec.? 
 
 2. A stone falls with a constant acceleration of 980 cm. per sec. per sec. 
 In what time will it acquire a velocity uf 35 m. per sec. ? 
 
 3. A body moves with a constant acceleration a. (a) How far does it go 
 in the first second ? (^) What is its average velocity during the first second? 
 (r) What is the average velocity during the first 6 sec? (</) What is the 
 average velocity during the sixth sectmd? 
 
 4. A train, running with constant acceleration, goes 560 m. during the 
 first minute after starting. Find the acceleration in meters per sec. per sec. 
 
 5. A car runs with a constant acceleration uf 80 cm. per sec. per sec. for 
 a distance of 300 m. (<i) What is, then, its velocity ? (^) With what average 
 velocity did it ran that distance? {c) How long did it take to run this dis- 
 tance? 
 
 6. A ball rolling along the ground is uniformly retarded at the rate of 
 4 m. per sec. per sec. Its velocity at the start is 20 m. per sec. (a) How 
 long will it roll ? (^) How far will it roll ? 
 
 n. Falling Bodies 
 
 94. Relative Rate of Falling Bodies ; Resistance of the Air. — 
 An unsupported body falls because the attraction of the earth upon 
 it (i>. the weight of the body) is unbalanced. It is well known 
 that a feather or a sheet of paper falls less rapidly than a stone, 
 and that some bodies — a balloon, for example — rise instead of 
 
Falling Bodies 75 
 
 falling. From such familiar facts as these false conclusions are fre- 
 quently drawn ; but the truth can be gathered from a careful study 
 of a few simple experiments. 
 
 We know that a balloon rises because its weight is less than the 
 buoyancy of the air, leaving an unbalanced force acting upward. 
 Since, however, the buoyant force of the air upon solids and 
 liquids is relatively very small (Art. 55), it cannot appreciably 
 affect their rate of fall. 
 
 If we take two sheets of paper exactly alike and roll one of 
 them into a tight wad, it will be found, on dropping them simul- 
 taneously from the same height, that the wad falls much faster 
 than the open sheet {Exp.). The difference in their rate of fall 
 is due to the friction of the air, which is greater upon the open 
 sheet, since it has the greater surface exposed. Buoyancy is evi- 
 dently not the cause of the difference, since the buoyant force 
 upon the sheet is the same whatever its shape. 
 
 If two pebbles of very unequal size are held, one in each hand, 
 above the head at the same height, and dropped at the same in- 
 stant, they will reach the ground together. (Try it.) In this 
 experiment the friction of the air is too small to produce an 
 appreciable effect upon either body ; hence their observed motion 
 may be regarded as due to their weight alone. This experiment 
 illustrates the interesting fact that all bodies not appreciably 
 affected by the resistance of the air fall at the same rate, regard- 
 less of their weight. The pupil should test this further by compar- 
 ing in pairs the rates of fall of a number of different bodies. 
 
 Actual differences in the observed rates of fall may be assumed 
 to be due to friction of the air ; and the pupil should discover by 
 experiment the approximate effect of friction upon bodies of 
 widely different density, by trying simultaneously stone, wood, 
 cork, wad of paper, etc. The result of greater or less compact- 
 ness of form can be observed by dropping together a wad of 
 paper and an open sheet, a block and a very thin board or a leaf, 
 etc. The effect of the air for greater velocities is easily tested by 
 dropping the bodies out of a second or third story window. 
 
76 Dynamics 
 
 95. HistoricaL — Experiments similar to the above were first 
 tried, so far as is known, by Galileo Galilei (i 564-1 642), an 
 Italian mathematician and scientist. For two thousand years no 
 one had thought to question the doctrine of the Greek philoso- 
 pher Aristotle, who taught that the rate of fall of bodies was pro- 
 portional to their weight. Galileo, who more fully appreciated 
 the value of experiment than any of his predecessors, discovered 
 
 ^B^ the falsity of this doctrine, and proved the 
 
 J^^L correctness of his view to the citizens of Pisa 
 
 H^^B by dropping simultaneously a one-pound ball 
 
 ^^^^t and a one-hundred-pound ball from tlie top 
 
 ^^^^m of the leaning tower (P'ig. 75). The two 
 
 ^^^^H balls reached the ground together. 
 
 -^^^^K l'^^^ Any difference in the rate of fall of 
 
 .'^'^^^BHB^ bodies is due to the resistance of the air was 
 
 ^^ clearly proved, after the invention of the air 
 
 pump, by causing such bodies as a coin and a 
 
 feather to fall together from one end to the 
 
 other of a long glass tube from which the air 
 
 F»«- 7S« had been exhausted. The feather was found 
 
 to fall as rapidly as the coin. The experiment is known to the 
 
 present day as the guinea and feather experiment 
 
 Laboratory Exercise 2t. 
 
 96. Acceleration of Falling Bodies. — Our daily experience 
 teaches that the speed of bodies continually increases as they fall ; 
 />. bodies fall with accelerated motion. Since all bodies fall 
 equal distances in equal times, unless appreciably affected by the 
 resistance of the air, it is evident that gravity, acting alone, ac- 
 celerates all bodies equally. It can be shown by a number of 
 experimental methods that this acceleration is uniform. 
 
 One of the simplest and most accurate methods by which the 
 motion of a falling body can be studied is that known as Whit- 
 ing's pendulum method (Lab. Ex. 21). In this experiment the 
 distances </, and 4 through which a ball falls in measured inter- 
 vals of time /i and /, are determined, the ball in each case starting 
 
Falling Bodies 77 
 
 from a state of rest. The values obtained will agree (within the 
 limits of experimental error) with the relation 
 
 d,'.d,::t^'.t!', (6) 
 
 that is, the distance that a body falls from a state of rest is pro- 
 portional to the square of the time. Since this is one of the laws 
 of uniformly accelerated motion (Law IV, Art. 93), it proves that 
 the acceleration of a falling body is uniform. 
 
 97. Laws of Falling Bodies. — All the formulas and laws for 
 uniformly accelerated motion (Arts. 92 and 93) hold for falling 
 bodies, their motion being uniformly accelerated ; but in this case 
 the acceleration is denoted by g^ since it is due to gravity. The 
 motion of a body falling freely from a state of rest (and hence 
 falling vertically) is, therefore, completely expressed by the 
 formulas:- ^^^,. ^^^ 
 
 ^=f; (8) 
 
 ^=?; (9) 
 
 -.= il. (:o) 
 
 98. The Value of ^. — Since the weight of a body changes 
 slightly with a change of latitude or of altitude, the acceleration 
 that it causes also varies. Thus at the equator, where the weight 
 of a body is least, the value of^ is 978 cm. per sec. per sec, while 
 at the poles, where weight is greatest, it is 983 cm. For places 
 within the temperate zones 980 cm. or 32.15 ft. per sec. per sec. 
 is a very close approximation ; and this value is to be used in 
 solving problems. 
 
 99. Cause of the Constant Acceleration of Falling Bodies. — 
 The weight of a body is a constant force, which, so long as the 
 friction of the air is inappreciable, is wholly unbalanced during 
 free fall. The constant acceleration of a falling body is therefore 
 due to the continuous action of a constant unbalanced force. 
 
78 Dynamics 
 
 100. Effect of Friction of the Air. — The resistance to motion 
 due to the friction of the air increases rapidly with the velocity. 
 Bicycle riders are familiar with an excellent illustration of this. 
 For example, to ride on a level road at the rate of 10 mi. an hr. 
 in the direction of a wind of equal velocity requires but little 
 effort ; but to ride at the same rate against such a wind is very 
 difficult. The difference is due entirely to the resistance of the 
 air. In the first case this resistance is zero, since rider and air 
 are relatively at rest ; in the second case it is the same as if there 
 were no wind and the rider were going at the rate of 20 mi. per 
 hr. TTiat this resistance is very considerably is shown by the 
 increase of effort required to ride against the wind. 
 
 Falling bodies meet with a rapidly increasing resistance of the 
 air, which leaves a continually diminishing portion of their weight 
 unbalanced to cause further acceleration. Compact bodies of 
 considerable density, such as a stone, seldom fall far enough for this 
 resistance to become appreciable ; but bodies having a large sur- 
 face in proportion to their mass, as a sheet of paper or a leaf, do 
 not fall far before the friction equab their weight, and, as there 
 is then no unbalanced force, there is no further acceleration. It 
 is for this reason that the velocity of raindrops becomes constant 
 long before they reach the ground. 
 
 101. Canae of the Equal Acceleration of Falling Bodies. —The 
 equal acceleration of falling bodies is due to the fact that weight 
 is proportional to mass. For example, the earth attracts equal 
 portions of a large stone and a small one equally ; and the total 
 force on the larger (/>. its weight) is as many times greater than 
 that on the smaller as the mass of the one is greater than the mass 
 of the other. Thus the two forces cause equal accelerations be- 
 cause they are proportional to the masses of the bodies upon which 
 ihey act. 
 
 This important law holds for all forces. Thus, if the mass of 
 one street car and its load of passengers is twice that of another, 
 twice as great a force will be required to give the same accelera- 
 tion to it as to the other. It should be noted that weight is not 
 
Falling Bodies 79 
 
 involved in this illustration, for the entire weight of both cars is 
 supported by the track. 
 
 102. Motion of a Sphere on an Inclined Plane. — When a ball is 
 on an inclined plane but is otherwise unsupported, its motion down 
 the plane is due to the component of its weight acting parallel to 
 the plane (Art. 66). Since this component is constant during the 
 descent of the ball, the motion is uniformly accelerated (Art. 99) ; 
 but the acceleration is less than that of falling bodies, since only 
 part of the weight is effective. By diminishing the inclination of 
 the plane the acceleration can be made as small as desired, thus 
 making it possible to determine with considerable accuracy the 
 distances traversed by the ball in successive seconds {Exp.), It 
 will be found by trial that these distances are in the ratio of the 
 numbers i, 3, 5, 7, etc. That is, if x denotes the distance traversed 
 in the first second, the distances traversed during the second, third, 
 and fourth seconds are 3^, 5 -r, and 7 Xj respectively. The whole 
 distance traversed in i sec. is x, in 2 sec. 4 ;c, in 3 sec. 9 jc, in 4 
 sec. 16 AT, etc. It is evident that the whole distance is proportional 
 to the square of the time ; and a further analysis of these results 
 leads to all the laws of uniformly accelerated motion. 
 
 This method was first employed by Galileo in studying the laws 
 of falling bodies. 
 
 PROBLEMS 
 
 1. How far does a body fall during the first second? Account for the 
 fact that this distance is equal to half the acceleration. 
 
 2. (a) What is the velocity of a falling body at the end of the first second ? 
 (/J) How far does it fall during the second second? (r) Account for the 
 difference between these numbers. 
 
 3. What is the velocity of a falling body at the end of the fifth second ? 
 
 4. How far does a body fall («) in 5 sec? {b) in 6 sec? (<:) during the 
 sixth second ? 
 
 5. (a) \Vhat is the difference between the distance fallen during the sixth 
 second and the velocity at the beginning of that second? {b) Is this differ- 
 ence equal to that found in the second problem? Why? 
 
 6. A stone dropped from a cliff strikes the foot of it in 3.5 sec What is 
 the height of the cliff ? 
 
8o Dynamics 
 
 7. Why b it that the increased weight of a body when taken to higher 
 latitudes causes it to fait faster, while at the same place a heavy body falls 
 no faster than a light one? 
 
 8. When a train is leaving a station its acceleration gradually decreases 
 to zero, although the engine continues to pull with the same force as at the 
 start. Explain. 
 
 9. Would you expect the motion of equally smooth and perfect spheres 
 of different weight and material to be equally or unequally accelerated on 
 the same inclined plane? Give reason for your answer. Try the experiment. 
 
 m. Projectiles 
 
 103. Projectiles. — Any body moving through the air and hav- 
 ing a component velocity not imparted by its weight is called a 
 projectile. A bullet fired from a gun, an arrow shot from a bow, 
 and a ball that has been thrown or batted are examples of pro- 
 jectiles. 
 
 The velocity of a projectile at the instant when the force that 
 set it in motion ceases to act is called its initial velocity. Thus 
 the initial velocity of a bullet is its velocity at the muzzle of the 
 gun. The initial velocity of a ball when thrown is its velocity at 
 the instant it leaves the hand of the player. 
 
 104. Forces acting upon Projectiles. — The weight of a projec- 
 tile does not act as an unbalanced force until after the force that 
 imparts the initial velocity ceases. Thus, while a bullet is being 
 driven toward the muzzle of the gun by the pressure of the expand- 
 ing gases, it is constrained by the barrel of the gun to go in a 
 straight path ; but as soon as it leaves the muzzle, it is freed from 
 this restraint, and its weight acts to change its speed and direction 
 of motion. The* resistance of the air very appreciably affects the 
 motion of a swiftly moving projectile, as a rifle ball ; but in the 
 cases considered in this boolc it is disregarded. 
 
 It must be remembered that the force that imparts the initial 
 velocity is not in existence during the flight of a projectile. It 
 ceases at the instant the projectile is launched. Disregarding the 
 resistance of the air, the weight of the projectile is the only force 
 
Projectiles 8i 
 
 acting upon it during its flight, and is, therefore, the sole cause of 
 change of motion. 
 
 105. Effect of Unbalanced Weight. — The eff'ect of weight upon 
 the motion of projectiles is illustrated experimentally by releasing 
 two bodies simultaneously at the same height, — one with a con- 
 siderable horizontal velocity imparted by a sudden push or blow, 
 and the other without initial velocity, being dropped from a state 
 of rest (Fig. 76). The two bodies will always be found to reach 
 the floor at the same instant {Exp). The experiment illustrates 
 
 A 
 
 Fig. 76. 
 
 the fact that gravity causes the same acceleration in its own direc- 
 tion whether acting upon a body initially at rest or upon a body 
 already in motion. This is true whatever the direction of the 
 initial velocity may be ; but it is direcdy evident by experiment 
 only when the initial velocity is horizontal. 
 
 A projectile has, in fact, two component motions, namely: (i) 
 the initial motion, which is constant in magnitude and direction, 
 since there is no force acting to change it ; and (2) the uniformly 
 accelerated motion due to gravity. The direction of the first is 
 the direction of projection ; the direction of the second is always 
 vertically down. Neither of these component motions interferes 
 in the slightest degree with the other. 
 
82 
 
 Dynamics 
 
 "^ 
 
 a' 
 
 
 
 
 \ 
 
 b' 
 
 
 
 
 \ 
 
 c» 
 
 > 
 
 
 
 
 \ 
 
 Fig. 77. 
 
 106. Graphic Representation of the Path of a Projectile. — The 
 path of a projectile can be represented graphically by compoiind- 
 
 j B c D ^^^ its two motions as illustrated in Figs. 
 77 and 78. In the first case the direc- 
 tion of projection is horizontal, in the 
 second it is obliquely upward. In both 
 figures OA denotes the initial velocity 
 
 g 
 
 and Oa denotes - on the same scale. 
 
 2 
 
 The points a\ b\ c\ etc., represent the 
 position of the pro- y) 
 
 jectile at the end q^ 
 
 of successive sec- 
 onds. A smooth 
 ^' curve drawn through 
 these points repre- 
 sents the path of the projectile. 
 
 This construction indicates that if the 
 velocity of the projectile at any |K)int of its 
 path be resolved into two components, — 
 one in the direction of projection and the 
 other vertical, — the first component will 
 be the initial velocity, and the second 
 will be the same as that of a body start- 
 ing from rest and falling vertically for the 
 same time. This is illustrated in the fig- 
 ures at b\ 
 
 107. Projection vertically Upward. — While rising vertically, 
 the velocity of a projectile decreases at a rate equal to its rate of 
 increase in falling, provided the resistance of the air is not appre- 
 ciable. Hence the times of rise and fall are equal ; and both the 
 time and the distance can be computed by the formulas of Art. 97. 
 
 For example, if a stone is thrown vertically upward with an 
 initial velocity of 49 m. per sec, its velocity at the end of i sec. 
 is 49 — 9.8, or 39.2 m. per sec. ; and at the end of 2 sec. it is 
 
 Fig. 78. 
 
The Laws of Motion 83 
 
 49 — 2 X 9.8, or 29.4 m. per sec. Its time of rise is 49 -^ 9.8, 
 or 5 sec. ; for at the end of that time its velocity would be zero. 
 This is the time it would take to acquire a velocity of 49 m. per 
 sec. in falling. The distance that the stone will rise is found by 
 computing the distance it would fall in the same time, starting from 
 rest. 
 
 PROBLEMS 
 
 1. (a) What is the force that causes the initial velocity of an arrow? 
 (6) How long does it act? (c) How is it known that this force is many times 
 greater than the weight of the arrow? (</) Is the acceleration that it causes 
 greater or less than that due to gravity? (<?) How long does this accelera- 
 tion continue ? (/) What would be the motion of the arrow if it were not 
 acted upon by any force during its flight? 
 
 2. Two stones are thrown to the same height, one vertically, the other ob- 
 liquely. Is the time of flight the same for both? Explain. 
 
 3. A stone thrown to the height of a tree reaches the ground in 5 sec. 
 from the time of starting. How high is the tree? 
 
 4. A body is thrown horizontally, with an initial velocity of 100 ft. per sec, 
 from the top of a tower 150 ft. high. At what distance from the tower will 
 the body strike the ground? 
 
 5. An arrow is shot vertically up with a velocity of 42 m. per sec. (a) How 
 long will it rise? {d) How high will it rise? 
 
 6. A ball is thrown upward at an angle of 30° with the horizontal, with 
 an initial velocity of 35 m. per sec. (a) What is the time of its flight ? 
 (3) How high does it rise? (<•) How far from the starting point does it 
 strike the ground? 
 
 Suggestion. — Resolve the initial velocity into horizontal and vertical 
 components. The first component is constant ; the second is affected by 
 gravity, just as it would be if the first component did not exist. 
 
 rv. The Laws of Motion 
 
 108. Balanced and Unbalanced Forces. — The effect of a force 
 is always to cause motion or to change the existing motion of the 
 body upon which it acts, unless it is balanced by another force or 
 other forces whose tendency is to produce an equal and opposite 
 effect. A single force acting upon a body is wholly unbalanced. 
 When two or more forces act simultaneously upon a body, the 
 
84 Dynamics 
 
 unbalanced force is their resultant, since their combined effect is 
 the same as their resultant would produce if it were acting alone 
 {Exp.). If the resultant of all the forces acting upon a body 
 is zero, they do not affect its state of rest or of motion in any 
 way ; i.e. the body behaves as it would if no force at all were 
 acting upon it. 
 
 ITie general laws of dynamics, or, as they are generally called, 
 the laws of motion^ are concise and definite statements of the 
 behavior of bodies when acted upon by unbalanced forces and 
 when not acted upon by such forces. Throughout the following 
 discussion of these laws the word force^ when unqualified, must 
 always be understood to mean unbalanced or resultant force. 
 
 109. The Effect of a Constant Force. — Any constant {unbal- 
 anced) force acting upon any body causes uniform acceleration of 
 its motion in the direction in which the force acts. The weight of 
 a body, when acting alone, is such a force ; and it causes a con- 
 stant acceleration of 9.8 m. per sec. per sec. in its own direction 
 whether it acts upon a body initially at rest or upon a body hav- 
 ing any initial velocity in any direction (Arts. 105-107). 
 
 The acceleration of a given mass is proportional to the {unbal- 
 anced) force acting upon it. This law is illustrated by the motion 
 of a sphere on a plane when inclined at different angles. The 
 effective {i.e. unbalanced) component of the weight of the ball 
 varies with the inclination of the plane (Arts. 66 and 102) ; and 
 it will be found by trial that the acceleration is proportional to 
 this component (Exp.). Thus by doubling the height of the 
 plane {BC, Fig. 41), the unbalanced component of the weight is 
 doubled, and the ball will roll twice as far in the same time as 
 before, showing that the acceleration has been doubled (Law V, 
 Art. 93). 
 
 110. Effect of a Variable Force. — When the force acting upon 
 a body is not constant but varies from moment to moment, the 
 acceleration at any instant is proportional to the {unbalanced) 
 force at that instant. 
 
 Illustrations of this law in daily life are numerous. (See the 
 
The Laws of Motion 85 
 
 discussion of the motion of a falling leaf, second paragraph of 
 Art. 100, and of the motion of a street car, second paragraph of 
 Art. 91.) We can now understand why the speed of a street car 
 does not increase as long as the motor is running. The friction 
 (resistance of the air, etc.) rapidly increases as the speed increases, 
 leaving a constantly diminishing unbalanced force to accelerate 
 the motion of the car. When the total resistance of friction is 
 equal to the driving force due to the motor, the resultant force upon 
 the car is zero, and its speed is then constant (acceleration zero). 
 
 111. Relation between Force and Acceleration ; Mass Constant. 
 — The three laws stated in the two preceding articles are included 
 in the following : — 
 
 The acceleration of a body is always proportional to the unbal- 
 anced force acting upon it, and takes place in the direction in which 
 the force acts. 
 
 The law holds whatever the relative direction of the force and 
 the motion of the body may be. Three cases arise as follows : 
 (i) If the force acts in the direction of motion, the velocity will 
 be increasing but constant in direction; (2) if the force and the 
 motion are in opposite directions, the velocity will be decreasing 
 but constant in direction ; (3) if the force acts at an angle to the 
 direction of motion, the velocity will, in general, be changing both 
 in magnitude and direction. (What illustrations of the three cases 
 are afforded by falling bodies and projectiles?) 
 
 The law expresses the interesting fact that an unbalanced force, 
 however small, acting upon any mass, however great, will move it 
 or will change its existing motion. The change of motion may, 
 indeed, be very slow, but it will be none the less certain. On the 
 other hand, to impart a very great velocity to a body in a very 
 short time requires great force even though the mass of the body 
 be small. For example, the pressure of the expanding gases 
 behind a ten-pound cannon ball in the act of firing it is several 
 hundred thousand pounds. Since the acceleration is proportion- 
 ately great, the ball leaves the muzzle of the cannon with a very 
 high velocity. 
 
86 Dynamics 
 
 If two forces, acting upon the same mass or upon equal masses, 
 are denoted by /, and y^ and the accelerations that they produce 
 by ai and a^ respectively, then the law states that 
 
 /, : yi : : a, : <j^ (mass constant) (i i ) 
 
 Applying /^ and <?, to the case of a freely falling body, /^ is 
 the weight of the body, «/, and a^ is g. Hence in this case the 
 proportion becomes 
 
 /:w::a:g. (12) 
 
 From this proportion any one of the quantities/, 7a, and a can 
 
 be computed if the other two are known, g being taken as 980 cm. 
 
 or 32.15 ft. per sec. per sec. Thus, if a sphere weighing 800 g. 
 
 is placed upon a plane inclined at an angle such that the effective 
 
 component of its weight (allowing for the force necessary to 
 
 produce rotation) is 200 g., its acceleration will be given by the 
 
 proiK>rtion o o 
 
 "^ * 200 : 800 :: a : 980. 
 
 112. Relation between Force and Mass ; Acceleration Constant. 
 — Tke {unbalanced) force necessary to produce a given acceleration 
 is proportional to the mass of thf body upon which the force acts. 
 Expressed algebraically the law is 
 
 fx'.ft'.'. /«i : Wj. (acceleration constant) (13) 
 
 The equal acceleration of falling bodies (Art. 101) serves as the 
 simplest and best illustration of this law, since weight is always 
 proportional to mass. But experiments in which the acceleration 
 is not due to gravity are instructive, even if they do not prove the 
 definite relation expressed in the law. The following experiment 
 is of this sort. 
 
 Suspend a lead or an iron ball an inch or more in diameter by 
 a cord one or two meters long, and suspend a cork of about the 
 same size by a cord of equal length. With a swift horizontal swing 
 of the arm, strike the cork with the open palm. Strike the lead 
 ball in the same way, with equally rapid motion of the hand. (A 
 small board may be placed a foot or more beyond the ball to stop 
 
The Laws of Motion 87 
 
 it.) The force exerted upon the ball is very considerable, while 
 that upon the cork is almost inappreciable, although it is started 
 with equally accelerated motion. It is evident that the difference 
 is not due to the greater weight of the lead ball, for, in the vertical 
 position, its weight is entirely supported by the cord. Neither is 
 the difference due to friction or any other force. The lead ball 
 requires the greater force to start it because its mass is greater. 
 Having greater mass it has greater inertia, and this requires pro- 
 portionally greater force for an equal acceleration. 
 
 113. The Comparison of Masses by the Inertia Test. — It 
 follows from the law stated in the preceding article that, if equal 
 forces act upon unequal masses, the acceleration of the smaller 
 mass will be the greater. The acceleration will, in fact, be in- 
 versely proportional to the masses. Hence, if equal forces impart 
 equal accelerations to two masses, the masses are equal {Exp.). 
 
 This is the inertia test or acceleration test of the equality of 
 two masses. Masses that are equal by the inertia test are, of 
 course, equal by the usual weight test also. The inertia test is 
 more fundamental and scientifically more significant, — a fact that 
 can hardly be appreciated by students of elementary physics ; but 
 the weight test is more accurate and much more convenient, and 
 hence is used exclusively in scientific work as well as in daily life. 
 
 Laboratory Exercise 20. 
 
 114. The Element of Time in the Effect of Force. — Since 
 change of velocity is proportional to the time during which a given 
 acceleration continues {v = at), it follows that the- change of 
 velocity produced by a constant force is proportional to the time 
 during which the force acts. Some forces are very great, but act 
 for an extremely short time, as the blow of a hammer, or the 
 force exerted by a bullet in penetrating a board. The time is so 
 extremely brief in the latter case that a bullet fired through a door 
 standing ajar will scarcely disturb it, although it can be swung with 
 a light push of the finger. 
 
 These ideas are illustrated by a simple experiment with a small 
 coin and a calling card. The friction between them when the coin 
 
88 Dynamics 
 
 is placed on the card is sufficient to impart the motion of the card 
 
 to the coin when the card is moved slowly about ; but, when it is 
 
 very suddenly started, the coin is left behind. This is neatly 
 
 shown by placing the card and coin on 
 
 a finger (Fig. 79) and suddenly snapping 
 
 the card in a horizontal direction. If the 
 
 blow is successfully aimed, the card will 
 
 fly from under the coin, leaving it at rest 
 
 on the finger, friction being insufficient 
 
 to impart appreciable motion to the coin 
 
 in so short a time. (Try the experiment.) 
 Fig. 79. 
 
 PROBLEMS 
 
 1. The acceleration of any falling body is proportional to its weight in 
 different latitudes and at different altitudes (Art. 98) ; but all bodies at the 
 same place fall with equal acceleration, whatever their weight (unless re- 
 tarded by the air). Explain. 
 
 2. A bullet fired through a plate glass window will often make a smooth 
 hole without cracking the glass. Explain. 
 
 3. A nail can be driven by striking it with a hammer, but not by pressing 
 the hammer steadily against it. Explain. 
 
 4. Gravity upon the moon is one sixth as great as upon the earth. G)m- 
 pute the acceleration of a falling body upon the moon. 
 
 5. Gravity upon the sun is 27.6 times as great as upon the earth. Com- 
 pute the acceleration of a falling body upon the sun. 
 
 6. How far would a body fall during the first second (a) upon the moon ? 
 (^)upon the sun? 
 
 7. (a) Would the mass of a given body be the same upon the sun or 
 the moon as upon the earth? (6) Would its inertia be the same? 
 
 8. Would it take less powder to fire a cannon ball with a given velocity 
 upon the moon than it would upon the earth ? 
 
 9. Is it harder for horses to start a loaded wagon or to keep it in uniform 
 motion? Give reasons. 
 
 10. Why does a ball player move his hands quickly backward in the act 
 of catching a swift ball ? 
 
 11. An unbalanced force of 25 g. acts on a mass of 80 g. What is the 
 acceleration ? 
 
 12. WTiat force is required to impart an acceleration of 15 cm. per sec. 
 per sec. to a mass of 100 g.? 
 
The Laws of Motion 89 
 
 115. The Law of Inertia. — Since change of motion results only 
 from the action of an applied force, in accordance with the pre- 
 ceding laws, matter is said to be passive or inert, and the property 
 thus manifested is called inertia^ (Art. 9). The law of inertia is 
 as follows : — 
 
 Every body continues in its state of rest or of uniform motion in a 
 straight line unless compelled to change that state by an external 
 force. 
 
 This law follows as a corollary from the law stated in Art. 1 1 1 ; 
 for the acceleration must be zero when the unbalanced force is 
 zero, since the two are proportional, and, with zero acceleration, 
 motion remains constant both in magnitude and direction. It is 
 impossible to prove the law of inertia by direct experiment, since 
 no body can be freed from the action of all forces ; but the indirect 
 evidence of its truth is conclusive (see Art. 9). Astronomical 
 observations on the motions of the moon and the planets confirm 
 all the laws of motion as well as the law of gravitation. 
 
 116. The Law of Mutual Action. — To every action there is an 
 equal and opposite reaction ; or, the mutual actions of two bodies 
 are ahvays equal and in opposite directions. 
 
 Illustrations of this law were considered in Art. 12, and there 
 have been numerous applications of it in the study of fluids and 
 the statics of solids. The law holds for all forces, whether bal- 
 anced or unbalanced. Force exists only through the mutual 
 action of two bodies (or two parts of the same body, which amounts 
 to the same thing). Thus every force is one of a pair of equal and 
 opposite forces, exerted by each of two bodies on the other. The 
 two forces do not balance each other, since they act upon different 
 bodies ; but either or both may be balanced by other forces. For 
 example, the pressure exerted by a bat upon a ball in striking it is 
 unbalanced and imparts motion to the ball. The reaction of the 
 ball upon the bat is also unbalanced and checks the motion of the 
 
 1 The pupil should avoid the misconception that the inertia of matter is an 
 active agent opposing and, in some sense, neutralizing the effect of force. Inertia 
 is not force, nor does it ever balance a force. 
 
9© Dynamics 
 
 bat. ^Vhen a piece of iron placed on an anvil is struck with a 
 hammer, the blow of the hammer is balanced by the equal and 
 opposite pressure of the anvil, both acting on the piece of iron ; 
 hence the iron remains at rest. When a person jumps from a 
 boat, the reaction on the boat is unbalanced and pushes the boat 
 in the opposite direction from that in which the person jumps ; 
 but, in jumping from a rock, the reaction upon the rock is balanced 
 by the friction between it and the ground, and it remains at 
 rest. 
 
 It is often supposed that the motions of animals and self-pro- 
 pelling machines are independent of applied force, since they 
 *' make themselves go." The motion of all matter, animate or 
 inanimate, is in accordance with the same laws of motion. The 
 real difference between the conditions of motion of a stone and 
 an animal or an engine, is that the latter can cause the applied 
 forces that move them, and a stone cannot. For example, when 
 a boy jumps, he pushes vigorously with his feet downward and 
 backward, against the ground with a force much greater than his 
 weight. The ground reacts with an equal and opposite force upon 
 the boy, and it is this reaction that enables him to spring upward 
 and forward. A similar reaction of the ground takes place with 
 every step in running. The runner also leans forward, in which 
 position his weight pulls his body forward, as in the act of falling. 
 When one attempts to start, stop, or turn quickly while walking on 
 ice, friction is too slight to cause the necessary reaction upon the 
 feet, and this results in a fall. A bird in flying pushes the air 
 doAWiward and backward with its wings ; the reaction of the air 
 upward and forward sustains the bird in its flight. The mutual 
 action between the driving wheels of an engine and the rails is 
 different from that between the car wheels and the rails, as is 
 shown by the fact that the former sometimes slip, spinning round 
 and round, while the latter never do. A driving wheel exerts a 
 strong backward push on the rail, and slipping is prevented only 
 by friction ; the forward reaction of the rail on the wheel is essen- 
 tial to the motion of the engine, and is an external force. 
 
The Laws of Motion 91 
 
 117. Relative Velocities due to Unbalanced Action and Re- 
 action ; Momentum. — When the mutual actions of two bodies are 
 unbalanced, their accelerations are inversely proportional to their 
 masses, since the forces exerted by each upon the other are equal 
 (Art. 113, first paragraph) ; and, if the bodies are initially at rest, 
 their velocities will also be inversely proportional to their masses. 
 Thus when a man jumps from a boat that weighs three times as 
 much as himself, the boat is pushed back with a velocity one third 
 as great as the forward velocity of the man. A rifle " kicks " when 
 fired, because the gas from the burned powder presses back on the 
 rifle, as well as fonvard on the bullet, and with equal force. The 
 velocities of the rifle and the bullet are inversely proportional to 
 their masses. 
 
 Let nty and W2 denote the masses of two bodies initially at rest, 
 and Vx and v.^ their respective velocities imparted by mutual action, 
 the bodies being free to move ; then nti -.m^wv^: z/i, or tn-^Vx 
 «= W2?v The product of the mass of a body and its velocity is 
 called its momentum. Hence bodies initially at rest and free to 
 move acquire equal momenta in opposite directions as a result 
 of their mutual actions. Thus the momentum of a rifle in its recoil 
 is equal to the momentum of the bullet as it leaves the muzzle. 
 When a moving body strikes a body at rest, and their mutual 
 actions are unbalanced, the one loses as much momentum as the 
 other gains. 
 
 118. Newton's Laws of Motion. — The fundamental laws of 
 motion considered in the preceding articles are restated here for 
 convenient reference : — 
 
 I. Every body continues in its state of rest or of uniform motion 
 in a straight line^ except in so far as it is compelled by external 
 forces to change that state. 
 
 II a. The acceleration of a body is proportional to the unbalanced 
 force acting upon it^ and is in the direction of that force ; or, 
 
 f'.fi'.'.ai'.a^. (mass constant) 
 
 II b. The unbalanced force necessary to produce a given accelera- 
 
92 Dynamics 
 
 tion is proportional to the mass of the body upon which the force 
 
 €Uts ; or, 
 
 fx'fi'' '"i • '"s« (acceleration constant) 
 
 III. To nrr}' action there is an equa/ ami opposite reaction ; or 
 the mutual actions of two bodies are always equal and in opposite 
 directions. 
 
 The laws numbered I and III are known as Newton's first and 
 third laws of motion respectively ; Ila and lib are together equiv- 
 alent to his second law of motion. These laws constitute a com- 
 plete statement of the relation between matter and force. The 
 first and second were discovered by Galileo in studying the motion 
 of falling l>odies and projectiles. The third law was also known to 
 others before Newton. They are called Newton's laws because 
 he was the first to state them in their present form. 
 
 Note to the Teacher. — The second law of motion as stated by Newton 
 can be derived Irom II a and lib as follows: According to II a, /oc a (/« = con- 
 stant) ; according to II b, /ac m (a = constant). Hence /oc ma, and _/?oc mat. 
 When the initial velocity is zero, the latter may be written //« mv. The equality 
 of ft and mv requires the introduction of the dyne as the unit of force. With the 
 
 gravitational unit of force, the equations are /= — and ft = These 
 
 equations express the second law in mathematical form. Stated in words, it is : 
 Change of momentum is proportional to the appHed force and to the time during 
 which it acts, and takes place in the direction of the force. 
 
 PROBLEMS 
 
 1. Two boys, A and B, are pulling upon the ends of a rope. A pulls B 
 along. Is he pulling harder (/.^., with greater force) than B? Explain. 
 
 2. Two boats are afloat some distance apart, and at rest. A man sitting 
 in one of them hauls in a rope attached to the other. Describe and explain 
 the motions of the boats, assuming them to be {a) of equal mass (including 
 the mass of whatever is in the boats; ; (Z^) of uneciual mass. 
 
 3. Why does stamping remove mud from the shoes ? 
 
 4. Why does beating a carpet remove dust from it ? 
 
 5. Why can the handle be tightened in the head of an ax (a) by striking 
 the end of the handle against a log? (/') by holdingthe ax at rest and strik- 
 ing the end of the handle with a hammer ? 
 
Curvilinear Motion 93 
 
 6. Two battle ships are in an engagement, with one in pursuit of the 
 other. Does the reaction of the guns in liring aid or hinder the speed of the 
 pursuer? of the pursued? 
 
 7. How should a person handle his body to avoid a fall when alighting 
 from a rapidly moving car ? , Explain. 
 
 V. Curvilinear Motion 
 
 119. Cause of Curvilinear Motion. — We have learned that the 
 weight of a projectile causes its path to curve downward, unless 
 the motion is vertical. Any unbalanced force acting upon a body 
 at an angle to its direction of motion produces a similar effect ; 
 i.e. causes the path of the body to curve toward the direction in 
 which the force acts. A stone tied to a string and whirled in a 
 circle round the hand is a familiar illustration. The motion of the 
 stone in a circle is (Uie to the continued inward pull of the string 
 upon it. If the stone is released at any point of its path, it con- 
 tinues in the direction of its motion at the instant of release, 
 except in so far as its motion is then affected by its weight. 
 
 The following experiment affords a better illustration : A wooden 
 ball tied to a string is rolled round in a circle on the top of a large 
 table or on the floor. When released at any 
 point of its path, it continues in the direction 
 in which it was moving at that instant. Thus, 
 if released at A (Fig. 80), its path will be AB, 
 a line tangent to the circle at A {Exp.). The 
 law illustrated by the experiment is general. 
 A body moves in a curved path only when acted ^^' °* 
 
 upon by an unbalanced force directed toward the inside of the curve. 
 
 A force acting upon a body so as to change its direction of mo- 
 tion is called a centripetal force, because it acts toward the center 
 of the curved path (from the Latin centrum, center, ^n^petere, to 
 seek). A centripetal force may act at right angles to the direc- 
 tion of motion, or obliquely forward or backward. The three 
 cases are illustrated at M, N, and L respectively in Fig. 81, which 
 represents the path of a projectile. Strictly speaking, the centripe- 
 
94 
 
 Dynamics 
 
 / 
 
 Fig. 8i. 
 
 tal force at L and N is the component of weight acting at right 
 
 angles to the path, />. the component/. The tangential compo- 
 nent, Ty acts opposite to 
 # ^^ ^v,^^ the direction of motion 
 
 at Z, causing decrease of 
 speed, and in the direc- 
 tion of motion at W, caus- 
 ing increase of speed. 
 The centripetal compo- 
 nent at L and N and the 
 entire weight at M cause 
 change of direction only. 
 Uniform motion in a circle is due to a constant centripetal force, 
 
 which always acts toward the center of the circle and at right angles 
 
 to the direction of motion. 
 
 120. Laws of Centripetal Force. — It can be shown either by 
 experiment or by mathematical analysis based on the second law 
 of motion that centripetal force is (i) proportional to the. mass 
 of the body, (2) proportional to the square of the velocity, and 
 (3) inversely proportional to the radius of curvature of the path. 
 
 The effect of the mass of the body can be shown by whirling 
 unequal masses with equal rapidity, using strings of equal length ; 
 and the effect of velocity, by whirl- 
 ing the same body more and less 
 rapidly. (Try it.) 
 
 121. Illustrations of Centripetal 
 Force. — If a ball or other object 
 is suspended by a string from a fixed 
 support and started in a horizontal 
 circle (Fig. 82), it will continue to 
 revolve in a slowly diminishing circle 
 (more accurately a spiral) for sev- 
 eral minutes {Exp.). The decrease 
 in the size of the circle is due to 
 friction, chiefly of the air, and may Fig. 83. 
 
Curvilinear Motion 95 
 
 be disregarded. If all friction could be removed, the motion 
 would continue indefinitely without change. Disregarding fric- 
 tion, the ball is acted upon by two forces ; namely, its weight, 
 IV, and the tension, T, of the cord. The vertical component 
 of the tension, v, is equal to IV and balances it ; the horizontal 
 component, /, is unbalanced, and is directed toward the center 
 of the circle. The component / is the centripetal force that 
 causes the circular motion of the ball. Since this force acts at 
 right angles to the direction of motion, it has no effect on the 
 speed (Art. 119, last paragraph). 
 
 In rounding a curve a bicycle rider brings the necessary cen- 
 tripetal force to bear upon his body by leaning toward the inside 
 of the curve. Let C (Fig. S^) denote the cen- 
 ter of the curved path of the rider and wheel, 
 and OB the inclination of the wheel. The wheel 
 exerts an oblique pressure upon the ground in 
 the direction OB (as is shown by the fact that 
 when a wheel slips in turning a curve it always 
 slips outward). The reaction of this force is an 
 equal pressure of the ground against the wheel 
 in the direction BO, and is denoted in the figure fig. 83. 
 
 by OPy as if it were applied at the center of gravity. This 
 oblique inward pressure of the ground may be considerably 
 greater than the weight of the wheel and rider, which is denoted 
 by OIV. The vertical component of OP is equal to the weight 
 of the rider and wheel and balances it. The horizontal compo- 
 nent,/, is the centripetal force upon the rider and wheel. 
 
 122. Inertia shown in Curvilinear Motion. — The tendency of 
 moving bodies to move in a straight line, as stated in the law of 
 inertia, is shown by the fact that curvilinear motion continues 
 only so long as a centripetal force acts to maintain it. From the 
 instant that centripetal force ceases to act upon a body, it con- 
 tinues in a straight line, or, to use a familiar expression, it " flies 
 off at a tangent." Thus motion in a curve is explained by describ- 
 ing the centripetal force that causes it ; while " flying off at a 
 
96 Dynamics 
 
 tangent " is accounted for by noting the absence of centripetal 
 force, this behavior being merely a result of the inertia of matter. 
 
 To illustrate : When a carriage is driven round a corner, its 
 tend^cy is to continue in a straight line ; hence the wheels tend 
 to slip over the ground toward the outside of the curve. Ordi- 
 narily friction is sufficient to prevent the shpping, and this causes 
 the ground to react with an inward pressure on the wheels. But, 
 since the wheels cannot slide, the tendency of the carriage to 
 continue in a straight line results in a tendency to overturn out- 
 ward. This tendency is opposed by the weight of the carriage, 
 which, under ordinary circumstances, acts as a sufficient cen- 
 tripetal force to bring the carriage safely round the curve. If, 
 however, the motion is very rapid and the curve sharp, this cen- 
 tripetal force may be insufficient ; in which case the carriage will 
 overturn, not because of a force acting to oi'erturn it, but because 
 thf centripetal component of weight is insufficient to p?vduce the 
 necessary change of direction. The behavior of water on a rotating 
 grindstone is a further illustration. The water is held to the stone 
 by adhesion ; but when the speed reaches a certain value, the 
 adhesion is no longer sufficient to carry the water round in the 
 curved path, and it flies off. Mud flies from the wheels of a 
 rapidly moving carriage for the same reason. 
 
 123. Centrifugal Force. — When a stone is whirled at the end 
 of a string, it exerts an outward pull through the string upon the 
 hand, which is the equal and opposite reaction of the pull that the 
 hand exerts upon the stone. The reaction is called centrifugal 
 force, since its direction is outward from the center (from the 
 Latin centrum, ^xiA fugere, to flee). The centrifugal force is 
 exerted upon the hand, and tends to pull it (not the stone) 
 outward. 
 
 It is a common but wholly mistaken idea that centrifugal force 
 causes or tends to cause bodies to leave a curved path and fly off 
 or overturn. When a carriage rounds a comer, the only cen- 
 trifugal force in action is the outward pressure of the wheels upon 
 the ground. Centrifugal force does not act upon the body mov- 
 
Curvilinear Motion 
 
 97 
 
 ing in the curved path; hence it cannot under any circumstances 
 affect the motion of that body. The confusion of thought with 
 reference to centrifugal force arises from the fact that the term 
 was originally appHed to a fictitious force, and is still used in this 
 sense in unscientific language. This fictitious force is supposed 
 to act on the moving body and to be the cause of its tendency to 
 " fly off at a tangent." No such cause exists, and the assumption 
 that it does exist only leads to a misunderstanding of the whole 
 subject. Centrifugal force in the only sense in which the word 
 should be used, need not be mentioned in discussing curvilinear 
 motion, for, although it is a real force, it does not act upon the 
 body whose motion is under consideration. 
 
 PROBLEMS 
 
 1. Why is the curvature of the path of a projectile the greatest at its 
 highest point ? (See Fig. 8i.) 
 
 2. Explain what would happen if a bicycle rider failed to lean inward 
 sufficiently. 
 
 3. Why are curves in bicycle race tracks steeply inclined toward the 
 center ? 
 
 4. If a boy, while running, wishes to change his direction suddenly, as in 
 dodging, how does he handle his body ? Explain. 
 
 5. A bucket of water can be whirled in a vertical circle, the bucket being 
 inverted at the top of the circle, without any of the water spilling. (Try it.) 
 Explain. 
 
 6. Draw three figures similar to Fig. 82 ; one representing the case where 
 the deflection of the cord from the vertical is only a few degrees, one where 
 it is 40*^ to 50^ and one where it is 70° to 80°. The angles need not be 
 measured, but the forces are to be represented to the same scale in all. The 
 tension must always be taken of such magnitude that its vertical component 
 is equal to the weight of the ball. How does the centripetal force vary as 
 the cord becomes more nearly horizontal? Would it be possible to swing 
 the ball fast enough to bring the cord to a horizontal position? Give reason 
 for your answer. 
 
 7. A ball weighing 2 kg. is suspended from a cord 50 cm. long, and made 
 to revolve in a circle whose radius is 30 cm. Compute (^d) the centripetal 
 force upon the ball and {b) the tension upon the cord, {c) Draw a figure 
 representing the conditions, including the forces involved. 
 
98 Dynamics 
 
 VI. Universal Gravitation 
 
 124. Universal Gravitation. — The pupil is already familiar 
 with the fact that the earth attracts all bodies at and near its sur- 
 face, and that this attraction is directly evident as weight. There 
 is, however, no familiar evidence that all bodies attract one another 
 under all circumstances, yet such is the case. The attraction be- 
 tween masses of even several hundred pounds is exceedingly 
 small — so small, in fact, as to be far beyond any ordinary means 
 of detecting it.* Nevertheless a number of experimenters have 
 measured the attraction between masses of various substances 
 varying in weight from a fraction of an ounce to several hundred 
 pounds. These experiments not only prove the existence of 
 gravitational attraction between bodies, but afford a measurement 
 of it that is probably not in error by more than one per cent. 
 The methods by which such exceedingly delicate measurements 
 are carried out lie far beyond the range of elementary physics. 
 
 Sir Isaac Newton (1642-17 2 7), a noted English physicist and 
 mathematician, proved that the planets are held in their orbits by 
 the attraction of the sun, and the moon in its orbit by the attrac- 
 tion of the earth ; and that the motions of the planets are slightly 
 modified by their attractions for one another. He also discov- 
 ered that the attractions of the sun and the planets for one 
 another and the attraction of the earth for bodies upon its surface 
 are all in accordance with the same law. This law is called 
 Nev^'ton's law of gravitation, and is as follows : — 
 
 Every particle of matter in the universe attracts every other 
 particle with a force whose direction is that of the line joining 
 them, and whose magnitude is directly proportional to the product 
 of their masses, and inversely proportional to the square of the 
 distance between them. 
 
 1 Two spheres of cast iron each 1.8 m. in diameter would attract each other with 
 a force of i g. when placed close together. Such spheres would weigh about 22,000 
 kg. or 22 metric tous each. 
 
Universal Gravitation 99 
 
 Gravitation is the general term applied to the force with which 
 all bodies attract one another. The attraction of the earth for 
 bodies at and near its surface is usually called gravity. 
 
 125. Illustrations of the Law. — Newton proved that the 
 attraction between a sphere and any other body is the same as it 
 would be if the entire mass of the sphere were concentrated at its 
 center. Hence in considering the attraction of the earth for any 
 body upon its surface, the distance stated in the law is the earth's 
 radius. In considering the attraction between any two bodies of 
 appreciable size, the distance between their centers of gravity is to 
 be taken as the distance between the bodies. 
 
 Let/ denote the attraction between two masses m^ and m^, and 
 d the distance between their centers of gravity ; similarly, let F 
 denote the attraction between two masses J/j and Mc^, and D the 
 distance between their centers of gravity ; then, according to the 
 
 /./r.. __.___. (14) 
 
 The meaning of the law as expressed in this proportion will be 
 more readily understood from the following examples : — 
 
 1. What is the relation between the masses of two bodies and 
 their weights ? 
 
 Let m\ and M\ denote the masses of the two bodies, and / and F their 
 weights respectively. Since the second attracting body in each case is the 
 earth, wo = M^ = the mass of the earth and </ = Z> = the radius of the earth. 
 Hence the above proportion reduces to 
 
 /:F::mi : Mi; 
 
 that is, the weights of any two bodies are proportional to their masses — a 
 fact with which the pupil is already familiar. 
 
 2. How does the weight of a body upon the moon compare 
 with the weight of an equal mass upon the earth ? 
 
 Let mi and Afi denote the equal masses upon the moon and the earth, and 
 mo and Afo the masses of the moon and the earth respectively. Then d and 
 D denote the radii of the moon and the earth, and /and 7^ the weight of the 
 body upon the moon and upon the earth respectively. 
 
loo Dynamics 
 
 Snce mi = Mu the proportion reduces to 
 
 from which /h- /r= ^ X f - V- 
 
 The radius of the earth is 3960 mi., the radius of the moon 1082 mi., and 
 the mass of the moon ^ of the earth's mass. Substituting these values in the 
 equation, we get 
 
 f^F=i^x (3960 + io82)» = .1675 = \ (nearly). 
 
 3. How does the force of gravity at the distance of the moon 
 compare with its value at the surface of the earth? 
 
 The mean distance of the moon is almost exactly 60 times the earth's 
 radius. Let D denote the radius of the earth ; then </, the distance between 
 the centers of the earth and the moon, etjuals 60 D. Let Afx and /;/i denote 
 equal masses upon the earth and at the distance of the moon respectively. 
 J/j = wt = the mass of the earth. F is the force of gravity upon the given 
 mass at the earth's surface, and /its value at the distance of the moon. The 
 formula then becomes 
 
 / F ^»^^g A/iA/a 
 -^ ' • (60 Z?)« ' D^ * 
 
 from which /-*- F= 
 
 (60 Oy 6o5« 3600 
 
 Thus a mass of 3600 lb. at the distance of the moon would be attracted by 
 the earth with a force of one pound. 
 
 Since the masses involved fn this problem are the same at both distances, 
 a simpler and more direct solution is obtained by making use only of the 
 relation that the force varies inversely as the square of the distance. Thus 
 the ratio of the distances is 60, the square of this ratio is 36CX), and the recipro- 
 cal of this (taken because the relation is inverse) is 3^5. 
 
 126. Revolutioii and Rotation of the Moon and the Earth. — 
 Since gravity at the distance of the moon is ^^^ of its value at 
 the earth's surface, the acceleration due to gravity at that distance 
 is gg^QQ of 32.15 ft., or .1072 in. per sec. per sec. A body at the 
 distance of the moon, starting without any motion in the direction 
 of the earth, would fall one half of .1072 in., or .0536 in., toward 
 the earth during the first second. The earth's attraction, there- 
 
Universal Gravitation loi 
 
 fore, acting as a centripetal force, draws the moon out of a straight 
 course a distance of .0536 in. every second. As the moon's 
 velocity is about f mi. per sec, this deflection is very slight ; but 
 it is exactly what is required to keep the moon in its orbit. 
 
 The moon's attraction for the earth is, of course, equal to the 
 earth's attraction for the moon ; but since the mass of the earth 
 is 80 times that of the moon, the effect upon the earth's motion is 
 proportionately smaller. The earth and the moon, in fact, revolve 
 in the same direction round their common center of gravity ; 
 which, as it divides the distance between the centers of the two 
 bodies inversely as their masses (see problem 9 following Art. 84), 
 lies within the mass of the earth about 1 100 mi. below the surface. 
 (This motion of the earth has nothing whatever to do with its 
 rotation on its axis.) 
 
 The sun's attraction deflects the earth from a straight course by 
 about one ninth of an inch in a second, while the earth is traveling 
 nearly nineteen miles. The mass of the earth is so great that the 
 force required to produce even so slight a change of direction is 
 inconceivable, being no less than 3,600,000 millions of millions of 
 tons (36 with seventeen ciphers). 
 
 The effect of the sun's attraction for the planets is a continuous 
 change of direction of motion, not a change of speed. There is 
 no force acting to maintain the motion of the planets, and none 
 is necessary, since the heavenly bodies move through space with- 
 out friction or resistance of any kind. The same is true of the 
 rotation of the sun and the planets upon their axes. A spinning 
 top is brought to rest by the resistance of the air and the friction 
 upon the peg. The earth rotates without friction, hence its rate 
 of rotation remains constant without the action of any force to 
 maintain it. 
 
 127. Effect of the Earth's Rotation upon Its Shape. — If the 
 earth were fluid (as it undoubtedly once was) and were not rotat- 
 ing, the gravitation of its particles would cause it to assume the 
 form of a perfect sphere. The rotation of a fluid planet would 
 cause it to bulge at the equator and flatten at the poles, until the 
 
I02 Dynamics 
 
 distortion developed a centripetal component of gravity upon each 
 particle sufficient to overcome its tendency to fly off" at a tangent. 
 This is illustrated in Fig. 84, which repre- 
 sents a section of the earth taken through 
 the axis of rotation MN, (The departure 
 from the spherical shape is greatly exagger- 
 ated.) The centripetal force upon a particle 
 at A is directed toward C, the center of the 
 circle described by A about the axis. This 
 component tends to draw the particle toward 
 the pole, but is just sufficient to prevent 
 it from moving farther toward the equator. 
 
 The earth assumed its present form (disregarding inequalities 
 resulting in continents and oceans) while still fluid ; and, as a 
 result of its rotation, the polar radius is nearly 13.5 mi. less 
 than the equatorial. If the earth were to stop rotating, the 
 waters of the ocean would flow from equatorial regions toward 
 the poles, leaving the surface of the ocean truly spherical. The 
 Mississippi River would then flow toward the north, for its mouth 
 is farther from the center of the earth than its source. 
 
 128. Effect of the Earth's Shap^ and Rotation upon Weight. — 
 If a body were carried from either pole toward the equator, the 
 earth's attraction upon it would continually decrease, being at any 
 latitude inversely proportional to the square of the earth's radius 
 at that latitude. The total decrease of attraction between the 
 pole and the equator is about -^ of the whole. 
 
 All bodies on the earth must be acted upon by a centripetal 
 force to carry them round with the earth in its rotation. A certain 
 portion of the earth's attraction is thus employed in keeping 
 bodies from flying off", and only the remainder of this attraction 
 is sensible as weight. The centripetal force increases toward the 
 equator, being zero at the poles and -,^\^ of the whole attraction 
 at the equator. Since the centripetal force varies as the square 
 of the velocity (Art. 120), it follows that, if the earth rotated 
 17 times faster than it does, the centripetal force would be 
 
Universal Gravitation 
 
 103 
 
 289 times as great as it is (17- = 289), and bodies at the equator 
 would weigh nothing ; i.e. they would require no sustaining 
 force to keep them from falling, for the earth's attraction would 
 be just sufficient to keep them from flying off at a tangent as mud 
 flies from the wheels of a carriage. 
 
 These, therefore, are the causes of the difference between 
 weight and the earth's attraction referred to in the note to 
 Art. 10. As the result of the earth's shape and rotation com- 
 bined, the weight of a body would be very nearly ^Jr ^^^s at the 
 equator than at either pole. Thus a body weighing 191 lb. at 
 the pole would weigh only 190 lb. at the equator. 
 
 129. Cause of Gravitation. — The cause of gravitation is not 
 known, neither is there any generally accepted theory as to its 
 cause. It acts without visible or material connection between the 
 attracting bodies ; yet we must suppose that there is something 
 pervading all space by means of which and through which it is 
 exerted. It is inconceivable that two bodies not in contact 
 should be able to act upon each other with absolutely nothing 
 between them. Since gravitation acts undiminished in a vacuum 
 and beyond the limits of the atmosphere, it is clear that the 
 means, or medium^ for the transmission of gravitational force is 
 not a solid, a liquid, or a gas, an^ hence is not matter in any of 
 its ordinary forms. 
 
 PROBLEMS 
 
 1. («) Would the variation of weight at different latitudes be indicated by 
 any form of balance by which the object weighed is balanced by " weights " ? 
 (J)) Would it be indicated by an accurate spring balance ? Give reasons for 
 each answer. 
 
 2. What fraction of its weight would an object lose when taken from sea 
 level to a height of 4 mi.? 
 
 3. The diameter of Mars is 4230 mi. and its mass is approximately one- 
 ninth of the earth's mass. How does gravity upon its surface compare with 
 gravity upon the earth ? 
 
 4. What is the acceleration of a falling body upon Mars ? 
 
 5. Why does the atmosphere not offer resistance to the rotation or to the 
 revolution of the earth? 
 
1 04 Dynamics 
 
 6. Is the acceleration of a falling body due to the whole of the earth's 
 attraction or to the part that we call weight ? 
 
 7. What would be the subsequent motion of the moon and the planets if 
 gravitation should suddenly cease to act upon them ? 
 
 8. The average specific gravity of the whole earth is about 5.56. (a) How 
 would gravity compare with its present value if the average density of 
 the earth were equal to the density of water ? (fi) What would be the 
 acceleration of a falling body in that case ? 
 
 VII. The Pendulum 
 
 130. Simple and Compound Pendulums. — We have seen that, 
 when a suspended body is drawn asitle from a position of stable 
 equilibrium and released, its weight causes it to swing to and fro 
 about the position of equilibrium until it is brought to rest by fric- 
 tion (Art. 75 ) . Any body suspended thus and free to swing to and 
 fro, ox vibrate t is called di pendulum. The best form of pendulum 
 for experimental work consists of a small sphere of some dense 
 material, suspended from a fixed support by a slender thread. 
 The sphere is usually called the bob of the pendulum. The length 
 of such a pendulum is taken as the distance from the point of sus- 
 pension to the center of the bob. This is not perfectly correct, 
 but there is no appreciable error if the length of the thread is not 
 less than six times the diameter of the bob. A pendulum of this 
 description is usually called a simple petidulum ; although, strictly 
 speaking, a simple pendulum consists of a heavy particle having 
 no appreciable size, and suspended by a line without mass. This 
 is purely a mathematical conception, useful in the mathematical 
 study of pendular motion. 
 
 Any pendulum having an appreciable portion of its mass else- 
 where than in a compact bob at the end is called a compound 
 pendulum. Pendulums for other than experimental purposes are 
 always compound. 
 
 131. The Motion of a Pendulum. — After a pendulum has been 
 drawn aside and released, the bob is under the action of its weight 
 and the tension of the thread (friction being disregarded) . The 
 
The Pendulum 
 
 105 
 
 Fig. 85. 
 
 tension acting always at right angles to the path of the bob, causes 
 a continuous change of direction of the bob, but does not affect 
 its speed. The weight of the bob 
 may be resolved into two components, 
 / and /(Fig. 85), at any point of the 
 path. The component/, taken always 
 at right angles to the path, is balanced 
 by a part of the tension of the thread ; 
 the component/, taken along the tan- 
 gent, acts in the direction of motion 
 while the bob is descending, and op- 
 posite to the direction of motion while 
 it is rising. The motion is therefore p 
 accelerated to the lowest point and 
 retarded from that point to the end 
 of the swing. 
 
 It is evident that the tangential force / decreases toward the 
 lowest point of the path, where it is zero ; and that it has equal 
 values at equal distances on the two sides of this point. Hence 
 the bob would be in equilibrium in its lowest position, if at rest ; 
 but, if in motion, it would not be stopped by the action of its 
 weight alone until it had risen through an arc equal to that 
 through which it had fallen. If it were not for friction, therefore, 
 a pendulum, when once started, would vibrate indefinitely. It is 
 brought to rest by the friction of the air and the friction at the 
 point of support. 
 
 132. Definitions. — A complete swing of a pendulum in one 
 direction is called a vibration. The time occupied in making a 
 vibration is called the time of vibration or period. The period of 
 a pendulum is always measured in seconds. The rate of a pendu- 
 lum is the number of vibrations that it makes in one second. 
 The angle between the vertical and the direction of a pendulum 
 at the end of a vibration is called the amplitude (angle AOM, 
 Fig. 85). The length of a pendulum consisting of a small bob sus- 
 pended by a thread is (very approximately) the distance from the 
 
1 06 Dynamics 
 
 point of suspension to the center of the bob. The length of a 
 compound pendulum is defined as the length of a simple pendu- 
 lum having the same period. (It will be found by trial that this 
 is greater than the distance from the point of suspension to the 
 center of gravity of the pendulum and less than the length of the 
 body.) 
 
 Laboratory Exercise 22, 
 
 133. Effect of Amplitude on the Period of a Pendulum. — If 
 two pendulums of exactly equal length, having bobs of the same 
 size and material, are started together with unequal amplitudes, 
 any difference in their periods will evidently be due to the differ- 
 ence in their amplitudes.* Experiment shows that the periods are 
 equal if the greater amplitude does not exceed 5°, but that the 
 difference, though small, is readily appreciable if the greater ampli- 
 tude is above 20°. The period of a pendulum is constant for am- 
 plitudes less than j* / for larger amplitudes the period increases 
 very slightly with increase of amplitude. 
 
 It follows that, as the amplitude decreases, the average speed of 
 the bob decreases proportionally (for small amplitudes) ; other- 
 wise the period would not remain constant. The decrease of 
 speed is due to the fact that, as the arc through which the bob 
 swings grows less, the average value of the tangential force / 
 (Fig. 85) also grows less. 
 
 134. Effect of Mass and Material. — The effect of the mass or 
 the material of a pendulum is investigated by starting together two 
 pendulums of equal length, having bobs of unequal mass or of 
 different material, or both. Neither gains on the other ; hence 
 the period is independent of the mass and also of the material of 
 the pendulum. The reason for this is similar to that for the equal 
 acceleration of falling bodies (Arts. loi and 112), — the tangential 
 
 1 If two pendulums are started simultaneously, even a very slight difference in 
 their periods will be evident from the fact that, after swinging for a minute or less, 
 the ptendulum having the shorter period will reach the extremity of its arc in ad- 
 vance of the other. This test is at least ten times as delicate as that of counting 
 the vibrations of ^ ach pendulum for one minute. 
 
The Pendulum 
 
 107 
 
 component of the weight of a pendulum bob is proportional to its 
 mass. 
 
 135. Effect of Length on the Period. — If the length, /, and the 
 period, /, of different pendu]ums are accurately determined, it will 
 be found that the ratio V/ : / has the same value for all. That 
 is, letting /j and 4 denote the Jengths of any two pendulums and 
 A and /j their periods, then V4: A : : V^: 4, or, taking the ratios 
 of like quantities, which is preferable. 
 
 From which, by squaring the ratios, 
 
 /2 . /2 . . / . / 
 
 (15) 
 
 (16) 
 
 Stated in words, the period of a pendulum is proportional to the 
 square root of its length ; or, the 
 square of the period is propor- 
 tional to the length. 
 
 Figure 86 shows why the period 
 increases with the length. The 
 tangential force / acting on the 
 bob of the pendulum OA is much 
 less than that upon the bob of the 
 pendulum O^A at an equal dis- 
 tance from the vertical. Hence 
 the acceleration of the longer 
 pendulum is less and its period 
 greater than that of the shorter 
 pendulum. (This, of coarse, does 
 not establish the definite relation 
 stated in the law.) 
 
 136. Effect of Gravity. — When a pendulum is taken from one 
 location to another in which the force of gravity is different, its 
 rate is affected in the same manner as that of a falling body, and 
 for the same reasons (Arts. 98 and 1 13). If the force of gravity is 
 less in the new location, the tangential component of the weight 
 
 Fig. 86. 
 
io8 Dynamics 
 
 of the bob will be proportionately less and the period will be 
 increased. The period of a pendulum is inversely proportional to 
 tht square root of the acceleration of a falling body. Stated alge- 
 braically, the law is 
 
 tx'.t^w -<fgi\ Vgi. (17) 
 
 The effect of an increase in the force of gravity is illustrated by 
 means of a pendulum with an iron bob. By holding an end of a 
 strong bar magnet under and near the bob, its motion will be con- 
 trolled by its weight and the attraction of the magnet acting to- 
 gether, the latter force being equivalent to an increase of gravity. 
 With a small amplitude of vibration, the bob does not swing 
 beyond the strong attraction of the magnet and the period is con- 
 siderably shortened {Exp.). 
 
 137. The Pendulum Formula. — It can be shown by mathe- 
 matical analysis based upon the second law of motion that the 
 period of a pendulum, provided the amplitude be small, is given 
 by the formula 
 
 '=4i-> 
 
 (.8) 
 
 in which / denotes the period, and / the length of the pendulum, 
 g the acceleration of a falling body, and tt the ratio of the circum- 
 ference of a circle to its diameter (=3.1416). 
 
 The formula includes the four laws of the pendulum already 
 considered. The first two laws follow from the fact that the 
 formula contains no factor depending upon amplitude, mass, or 
 material. That the period is proportional to the square root of / 
 and inversely proportional to the square root of g is evident from 
 the formula as it stands. The formula holds for the compound 
 pendulum provided the length be taken as defined in Art. 132. 
 
 138. Uses of the Pendulum. — The principal use of the 
 pendulum is to regulate the motion of clocks. The mechanism by 
 which this is effected is represented in Fig. 87. The pendulum 
 rod passing between the prongs of a fork, A^ communicates its 
 motion to a rod, B, which turns on a horizontal axis, C. To this 
 
The Pendulum 
 
 109 
 
 axis is fixed a curved piece, called the escapement, which has a 
 projection at each end. These projections are alternately brought 
 into contact with the teeth of the escape7nent 
 wheel, D, by the motion of the pendulum. The 
 escapement wheel is the last of the train of 
 wheels in the clock and is driven by them ; 
 but the escapement permits only one tooth to 
 pass at a time, while the pendulum swings back 
 and forth. The control thus exercised on the 
 escapement wheel by the pendulum is com- 
 municated through the entire train of wheels to 
 the weight or spring that runs the clock and to 
 the hands. As the escapement wheel turns, its 
 teeth press upon the projections of the escape- 
 ment. These slight impulses are transmitted to 
 the pendulum and maintain its motion. 
 
 The rate of a clock is controlled by means of 
 a thread and nut at the lower end of the pendu- 
 lum. The bob is raised or lowered by turning 
 this nut. 
 
 A compound pendulum of special construc- 
 tion is used to determine the acceleration due 
 to gravity at different places. The length and 
 period of the pendulum are determined very 
 accurately, and their values substituted in the pendulum formula, 
 from which the value of ^ is then computed. 
 
 PROBLEMS 
 
 • I. How would the expansion of the rod of a pendulum in summer and its 
 contraction in winter affect the rate of a clock if the height of the bob were 
 not adjusted to compensate the expansion and contraction ? 
 
 2. What is the usual shape of the bob of a clock pendulum ? What is the 
 advantage of this shape ? 
 
 3. What is the length of a pendulum that beats seconds (/ = i) at a place 
 where the value of g is 980 cm? 
 
 SuoGESTioN.— Substitute the values of / and g in the pendulum formula, 
 and solve for /. 
 
 Fig. 87. 
 
iio Dynamics 
 
 4. Find the lengths of the pendulums whose periods are .7 sec. and 1.5 
 sec respectively. 
 
 Suggestion. — Substitute in the pendulum formula, assuming 980 cm. for 
 g ; or substitute in formula 16, taking /^ = i, for /j the length of the seconds 
 pendulum, and for t\ the given period. Solve for l\. 
 
 5. find the periods of pendulums whose lengths are 20 cm. and 250 cm. 
 respectively. 
 
 6. What is the length of a pendulum that makes 70 vibrations per minute? 
 
 7. What is the length of the seconds pendulum on Mars? 
 
 8. At what point or points in the path of a pendulum bob is its speed 
 greatest ? least ? increasing most rapidly ? constant ? Is the speed constant 
 for any appreciable distance ? Is the increase of speed constant in any part 
 of the path ? 
 
 Si'GGESTiov. — The answers are all to be determined from a knowledge of 
 the tangent^ force at any point of the path. 
 
CHAPTER VI 
 
 ENERGY 
 
 I. Energy and Work 
 
 139. Energy. — The word energy is used in science with a 
 definite meaning which i:an be understood only by a study of the 
 different forms in which energy exists. Energy is most generally 
 recognized by the ability of bodies possessing it to cause motion 
 in other bodies or to maintain their motion in opposition to friction 
 or other forces tending to stop them. Thus the energy of a bent 
 bow is shown by its ability to project an arrow, and the energy of 
 a coiled spring by its ability to run a clock. The energy of the 
 wind enables it to turn windmills, propel ships, uproot trees, etc. 
 The energy of coal, wood, and oil is utilized by means of the 
 steam engine in running mills, drawing trains, and propelling 
 steamships. Energy, in fact, is manifested in one or more of its 
 many forms in every phenomenon. 
 
 140. Kinetic Energy. — A moving body can impart motion to 
 other bodies ; it therefore has energy. The energy that a body 
 has by virtue of its mass and its velocity is called energy of motion 
 or kinetic energy, 
 
 141. Work. — The transference of energy from one body to 
 another is called work. The amount of energy transferred and 
 the amount of work done are equivalent expressions. Energy is 
 transferred from one body to another by means of the force that 
 each exerts upon the other. For example, when one ball rolls 
 against another, setting it in motion, the latter receives kinetic 
 energy by means of the pressure exerted upon it while the two 
 are in contact ; and, at the same time, the energy of the first ball 
 
1 1 2 Energy 
 
 is diminished by an equal amount by the reaction of the second 
 ball upon it in the direction opposite to its motion. The first ball 
 is said to do positive work upon the second, and the latter negative 
 work upon the first. The work by which a body gains energy 
 is called positive, and that by which it loses energy is called 
 negative. 
 
 To illustrate further : Positive work is done upon a train by the 
 pull of the engine. This, if it were the only force acting on the 
 train, would increase its kinetic energy ; but the retarding forces 
 of friction are at the same time doing negative work upon the 
 train. If the pull exceeds friction, positive work will exceed nega- 
 tive, and the excess of positive work wjll result in increase of 
 kinetic energy. If the two are equal (resultant force zero), the 
 positive and the negative work will be equal and the energy of the 
 train will be constant (the speed being constant) . After the engine 
 ceases to pull, the only work done upon the train (provided the 
 track be level) is the negative work due to friction, by which the 
 train loses its energy and is stopped. 
 
 142. Conditions Necessary for the Transference of Energy. — 
 Since one body does work upon another only by exerting force upon 
 it, we commonly say that the force does the work. Although work 
 is done only through the action of force, a force may act without 
 doing work. When a horse pulls on a load without starting it, no 
 work is done by the force exerted, for there is no transference of 
 energy to the load. If the load is started, the force acts through 
 a certain distance, and, in doing so, does work of acceleration in 
 imparting motion to the load and work against resistance in over- 
 coming friction.^ There is neither work of acceleration nor work 
 against resistance without motion of the body upon which the force 
 acts. 
 
 A* force may also act upon a moving body without doing work. 
 Whether work will be done or not depends upon the relative 
 
 1 Fricrion develops heat, and heat is a form of energy (Art. 148) ; hence to 
 maintain motion against friction involves the transference of energy, or the doing 
 of work. 
 
Energy and Work 1 1 3 
 
 direction of the force and the motion, as will be seen from the 
 following illustrations. The weight of a projectile rising vertically 
 constantly diminishes its kinetic energy by doing negative work of 
 acceleration. The weight of a falling body does positive work of 
 acceleration upon it, constantly increasing its kinetic energy. In 
 both cases the line of action of the force (weight) \^ parallel to 
 the direction of motion. As a pendulum vibrates, it is only the 
 tangential component of the weight of its bob that does work of 
 acceleration (Fig. 85), causing gain of kinetic energy during the 
 first half of each vibration and loss of kinetic energy during the 
 second half; and it is only this component that does work against 
 the resistance of the air. The other component of weight, which 
 acts at right angles to the direction of motion, does no work, for 
 it neither changes the speed of the bob nor helps to maintain its 
 motion against the friction of the air. The same is true of the 
 tension of the thread. 
 
 A force acting upon a body at right angles to its direction of 
 motion does no work ; for it neither 
 changes the speed of the body nor 
 helps to maintain its motion against 
 friction or other resistance. This 
 is true whether the force is bal- 
 anced or unbalanced. Thus, on 
 the whole,^ no work is done upon 
 the moon by the attraction of the 
 
 earth, or upon the planets by the 
 
 Fig. 80. 
 attraction of the sun. In the case 
 
 of oblique forces, as the weight of a pendulum bob, it is only the 
 tangential component of the force that does work ; that is, tiie com- 
 ponent whose line of action is parallel to the direction of motion. 
 
 1 Since the orbits of the moon and the earth are not quite circular but elliptical, 
 the attraction is not exactly at right angles to the path except at two points {A and 
 B, Fig. 88). As the planet movt-s from A to'B, the tangential component of the 
 attraction causes decrease of speed, while from i? to ^ it causes an equal increase. 
 Hence, on the whole, there is neither gain nor loss of kinetic energy. There is, of 
 course, no work against resistance, for the planets move without friction. 
 
114 
 
 Energy 
 
 Fig. 89. 
 
 143. Measure of Work. — The work done by a force is meas- 
 ured by the product of the force and the distance through which 
 the force acts, the distance being always measured parallel to the 
 bne of action of the force. 
 
 For example, suppose we wish to find the work done by the 
 weight of a ball while it rolls down an in- 
 chned plane, AB (Fig. 89). Let ^denote the 
 length of the plane, AB^ and h the height of 
 the plane, BC. The part of w that does work 
 is the component, /, acting parallel to the 
 plane. The distance through which it acts is 
 //, and the work that it does is measured by df. 
 From similar triangles, f \ w \\ h \ d ) hence 
 ■"c fd = wh. That is, the work done by the 
 weight of the ball is also measured by wh. 
 But, while the ball rolls down the plane the distance d^ it descends 
 through the vertical distance h. Thus the work done by w is 
 equal to the product of w and the displacement in its own direc- 
 tion. 
 
 In measuring the work done by an oblique force, it is immate- 
 rial whether we take ( i ) the product of the whole force and the 
 component of the displacement taken parallel to the line of action 
 of the force {wh in the above example), or (2) the product of the 
 whole displacement and the component of the force acting parallel 
 to it {fd in the above example). Both methods are fully included 
 in the above rule for the measure of work. The choice is a matter 
 of convenience, determined by the nature of the problem. In the 
 case where the applied force and the displacement have the same 
 or opposite directions, the work done is simply the product of the 
 two. If the force is variable, the average force exerted through 
 the distance considered must be taken. The work done by a 
 force, /, while acting through a distance, </, in its own line of 
 direction^ is expressed by the formula 
 
 Work=fd. 
 
 (19) 
 
Energy and Work 
 
 <i5 
 
 144. A Further Illustration. — The fact that the work done by 
 a force is equal to the product of the force and the displacement 
 in its own line of action, regardless of the actual path of the body, 
 is beautifully illustrated by the following experiment. A heavy 
 lead or iron ball is suspended by a cord 1.5 m. to 2 m. long so that 
 it will swing as a pendulum in front of a blackboard and within 
 a few centimeters of it ; and parallel horizontal lines are drawn 
 on the board a short distance apart to mark elevations. The pen- 
 dulum is started at an angle of about 30°, with the bob at the 
 height of one of the lines on the board. It swings to (about) the 
 same height on the other side (the effect of the resistance of 
 
 the air being nearly or quite inappreciable for a single swing). In 
 Fig. 90 BCD represents the path of the bob. A nail or a rod is now 
 driven into the blackboard at A, vertically below the support of the 
 pendulum, and a little above the line BD ; and the pendulum is 
 started from the same height as before. When it reaches the ver- 
 tical position, the cord is caught by the nail at A, and the bob 
 rises through the arc CE, having ^ as a center. The point E to 
 which the bob rises has the same elevation as B and D, and the 
 bob again rises to B in its return. This shows that the bob loses 
 equal kinetic energy in rising the same vertical distance by the two 
 
1 1 6 Energy 
 
 paths CD and CEy and gains equal kinetic energy in descending 
 the same vertical distance by the two paths. Since the gain or 
 loss of kinetic energy is equal to the work done by the weight of 
 the bob (the tension of the cord does no work in any case), it 
 follows that the work done is determined by the vertical displace- 
 ment and is independent of the path. If w is the weight of the 
 bob and h the height of B above C, the work done in either half 
 of the vibration is wh. The bob has at C the same kinetic 
 energy and hence the same velocity as it would have if it fell ver- 
 tically through the distance h. 
 
 146. Units of Work. — There are several units of work. We 
 shall define and use only three, liht f oof-pound (ft. -lb.) is the 
 work done when a force of one pound acts through a distance of 
 one foot. The kilogram- meter (kg.-m.) is the work done when a 
 force of one kilogram acts through a distance of one meter. The 
 gram-centimeter (g.-cm.) is the work done when a force of one 
 gram acts through a distance of one centimeter. 
 
 Examples. — i. The pressure exerted by the expanding gases behind a 
 cannon ball when it is fired, is exerted from the breech to the mouth of the 
 cannon. If this distance is 12 ft. and the average total pressure upon the 
 ball is 40,000 lb., the work done upon the ball is 40,cxx) x 12, or 480,000 
 ft.-lb. Hence, disregarding the comparatively small amount of work done 
 against friction, the ball leaves the cannon with 480,000 ft.-lb of kinetic 
 energy. 
 
 2. If a mass of 7 kg. falls a distance of 30 m., its weight does 7 x 30, or 
 210, kg.-m. of work upon it ; and this is also the measure of the kinetic energy 
 of the body at the end of its fall. 
 
 146. Measure of Kinetic Energy. — The kinetic energy of a body 
 is equal to the work that would be done by any unbalanced force 
 in imparting the given velocity to the body. The formula for 
 kinetic energy is most simply derived, however, by assuming that 
 the given velocity is imparted to the body by its weight in falling. 
 In this case the force,/, is numerically equal to the mass, ;«, of the 
 body. Let d denote the distance the body must fall to acquire the 
 given velocity, v, starting from rest, and g the acceleration. 
 
 The kinetic energy {K.E.) that would be imparted to the body 
 
Energy and Work 117 
 
 while falling is measured by the work that would be done upon 
 the body by its weight ; that is, 
 
 K.E.=fd. 
 
 Since 
 
 
 
 d = 
 
 I' 
 ' 2g 
 
 and 
 
 
 
 /= 
 
 m, 
 
 we have, 
 
 by 
 
 substitution. 
 
 K.E.= 
 
 fmP' 
 
 ^S 
 
 (equation 10, Art. 97) 
 
 (20) 
 
 That is, the kinetic energy of a body (whether a falling body or 
 any other) is measured by the product of its mass and the square 
 of its velocity, divided by twice the acceleration of a falling body. 
 Kinetic energy is measured in foot-pounds, kilogram-meters, or 
 gram-centi:neters according as the units of mass and distance 
 employed are the pound and the foot, the kilogram and the 
 meter, or the gram and the centimeter. In coming to rest, a 
 body will do work equal to its kinetic energy. 
 
 It is mass and not weight that is involved in kinetic energy. 
 The kinetic energy of a body moving with a given velocity would 
 be the same in regions so far from the earth or any other heavenly 
 body that it had no weight at all. 
 
 Examples. — i. Find the kinetic energy of a car weighing 12 tons, and 
 moving with a velocity of 40 ft. per sec. 
 
 „ _ 24000 X 402 
 
 K.E. — — — 597,015 ft.-lb. 
 
 2x32.16 ^^' ^ 
 
 2. If the velocity of the car is imparted by a constant pull of 800 lb. and 
 the friction is 150 lb., through what distance does the force act? 
 
 One hundred and fifty pounds of the pull does work against friction ; 
 only the remainder acts as an unl)alanced force to cause acceleration ; hence 
 650 d= 597,015. From which d = 918.5 ft. 
 
 3. How far would the car run before being brought to rest by friction 
 acting alone? 
 
 Since friction must do 597,015 ft.-lb. of negative work upon the car, the 
 distance d is found from the equation 150 d= 597»oi5« 
 
1 1 8 Energy 
 
 147. Potential Energy. — In bending a bow, work is done in 
 opposition to its elasticity, — a force whicii not only opposes the 
 distortion, but also tends to restore the original shape of the body 
 by bringing its parts into their former relative positions. The 
 bow, when released, is therefore able to exert a force through a 
 certain distance, or, in other words, it can do work. This is what 
 we have in mind when we say that a bent bow possesses energy. 
 It has the ability to cause motion in itself or otiier bodies, although 
 it is not actually in motion. Energy existing in this form is called 
 potential fntr^'. The energy of the bent bow is evidently the 
 result of the work done in bending it. 
 
 Other examples of bodies having potential energy are the coiled 
 spring of a watch, a raised clock weight, the compressed air in an 
 air rifle, and the lifted ram or hammer of a pile driver. The 
 coiled spring, like the bow, exerts a force in recovering from dis- 
 tortion, and the compressed air in expanding ; in both cases there 
 is relative motion of the parts of the body. The clock weight 
 exerts force, and hence does work, throughout its descent ; the 
 hammer of a pile driver does work only at the end of its fall. In 
 either case the ability of the body to do work is due to the fact 
 that its weight does work upon // during its descent. The poten- 
 tial energy of a body at any elevation is therefore measured by 
 the product of its weight and the elevation, and is independent 
 of the path by which the body descends to the lower level and 
 also of the time occupied in making the descent (Art. 144). 
 
 The energy that a body has by virtue of its position or the 
 relative position of its parts is C2i\\^d potential energy. This defini- 
 tion implies the existence of force tending to move the body or 
 to cause relative motion of its parts, and also the existence of room 
 for such motion to take place. 
 
 148. Heat is a Form of Energy. — Through our common ex- 
 perience we are familiar with many effects of heat ; yet these 
 afford no direct evidence as to what heat really is. This question 
 will be considered later ; for the present it is sufficient to recog- 
 nize heat as one of the many forms in which energy exists. 
 
Energy and Work 1 1 9 
 
 We have seen that the kinetic energy imparted to a body by 
 an unbalanced force is the equivalent of the work done by the 
 force. When work is done upon a body to maintain its motion 
 against friction, there is no increase of kinetic energy, but heat is 
 developed ; and the energy that appears in this form is the equiv- 
 alent of the work done (see note to Art. 142). Whenever one 
 surface moves upon another, heat is developed by the friction 
 between them ; but often so slowly that it is dissipated before 
 there is appreciable rise of temperature, and we fail to notice it. 
 With rough surfaces and rapid mo'tion, however, the rise of tem- 
 perature is often considerable. Many illustrations of this are 
 familiar. A match is ignited by the heat generated in striking it. 
 A piece of metal becomes very hot when ground on a dry grind- 
 stone (Exp.). When a cord or a rope that is grasped tightly is 
 drawn rapidly through the hands, sufficient heat is often developed 
 to cause a burn. 
 
 The friction by which a body is stopped changes or transforms 
 the kinetic energy of the body into heat at the same time. It is 
 for this reason that the brakes of bicycles, carriages, and cars 
 become hot when in use. The same transformation of energy 
 occurs when the motion of a body is suddenly stopped by impact. 
 Thus a piece of metal can be heated by hammering it, and the 
 hammer used also becomes warm {Exp.), ^^ullets are often partly 
 melted by the heat developed when they are stopped abruptly, as 
 in striking a steel target or a stone. • 
 
 149. Muscular Enargy. — All the movements of an animal are 
 due to muscular action, and in this action the muscles do work. 
 This work is accomplished by means of the potential energy of the 
 muscles themselves. The amount of energy stored in the muscles 
 is very great ; a horse, for example, can do about two million foot- 
 pounds of work per hour for several hours. It is evident, how- 
 ever, that the amount is not unHmited, for any animal becomes 
 exhausted after prolonged exertion. The renewed supply of energy 
 comes from the food eaten. 
 
 Muscular energy is available for doing work only through chem- 
 
1 20 Energy 
 
 ical changes by which the muscles are in part consumed, much as 
 fuel is consumed in a fire. Similar changes take place in all the 
 organs of the body while they are performing their special functions, 
 and some of the energy is always liberated as heat. It is this heat 
 that maintains the temperature of the body. 
 
 150. Other Forms of Energy. — Electrical energy, light, and 
 sound are other forms of energy ' which are considered under the 
 corresponding subdivisions of physics. The energy of a piece of 
 coal, as related to the oxygen of the air with which it unites in 
 burning, is an example of chefiiical potential energy. The study 
 of this form of energy belongs to chemistry. 
 
 Kinetic energy and energy of position are classed together as 
 mechanical energy ; and the work done in changing the kinetic 
 energy of a body or in maintaining its motion against friction is 
 called mechanical work. 
 
 151. The Transformation of Energy. — A number of instances 
 of the change of energy from one form into another have already 
 been mentioned (Arts. 148 and 149). The potential energy of a 
 drawn bow is transformed into kinetic energy in the act of shoot- 
 ing an arrow. The potential energy of the spring or the weight 
 of a clock is very slowly transformed into heat by the friction of 
 the moving parts, and a very small portion into sound in produc- 
 ing the " tick." A body thrown upward loses kinetic energy as it 
 rises, as the result of the negative work done upon it by its weight, 
 and, at the same time, gains an equal amount of potential energy 
 due to its elevation. During the descent of the body, its weight 
 does positive work and the reverse transformation takes place. 
 At the end of the fall the energy is all kinetic, and is equal to the 
 kinetic energy at the start, except for the resistance of the air, 
 which does negative work upon the body during both its rise and 
 fall. Muscular energy is transformed into kinetic in the act of 
 
 1 The words force and energy were originally used without a clear distinction 
 of meaning, and the different forms of energy were commonly called "the forces 
 of nature." This usage still persists in p>opular language ; but in science the word 
 force is restricted to the meaning with which the pupil is already familiar from 
 the previous work. 
 
universitV 1 
 
 Energy and Work \sAt 'rk^^^ 
 
 throwing a ball, into potential energy of position in carrying a hod 
 of bricks up a ladder, and chiefly into heat in sandpapering a board. 
 Other transformations of energy are considered under the different 
 subdivisions of physics. 
 
 152. The Dissipation of Energy. — Since the motion of all 
 bodies is opposed in a greater or a less degree by friction (except 
 in interplanetary space), motion necessarily involves the loss, or 
 dissipation^ of mechanical energy by its transformation into heat. 
 The energy thus transformed is said to be dissipated because it 
 is no longer available for doing useful work ; it is not lost in 
 the sense that it no longer exists. For example, if the hammer 
 of a pile driver weighs looo lb. and the friction that must be 
 overcome in lifting it amounts to loo lb., then, in raising the 
 hammer 20 ft., the lifting force of iioo lb. will do 20,000 ft.-lb. 
 of work against gravity and 2000 ft.-lb. of work against friction. 
 The work done against gravity is useful, since, as a result of it, the 
 hammer has 20,000 ft.-lb. of available potential energy. The work 
 done against friction is transformed into heat while the hammer is 
 being raised, and cannot be further utilized. The work done in 
 moving wagons, street cars, and trains from one place to another 
 (without change of elevation) is done against friction, including 
 the resistance of the air ; and the energy thus expended is dissi- 
 pated as heat. When the axle of a car wheel is not properly oiled 
 to reduce friction, the amount of heat generated is so great that 
 it causes a " hot box," and sometimes even sets fire to the car. 
 When bicycles came into general use, personal experience of the 
 waste of energy due to friction soon led to the invention of ball bear- 
 ings. 
 
 153. Power. — The rate at which work is done or the rate at 
 which an engine or other source of energy is capable of doing 
 work is called power. The customary unit of power is the horse- 
 power (H. P.), which is equal to 550 ft.-lb. or 76 kg.-m. of work per 
 second. Thus a twelve-horse-power engine working at three 
 fourths of its full capacity is doing work at the rate of 9 H. P., or 
 4950 ft.-lb., of work per second. 
 
122 Energy 
 
 PROBLEMS 
 
 1. Show from formula (20) that the kinetic energy of a body is propor- 
 tional (l) to its mass, (2) to the square of its velocity. 
 
 2. Two bodies have equal kinetic energ)', but the velocity of the second is 
 three times that of the first. How do their masses compare ? 
 
 3. A body is thrown vertically upward. (<?) What fraction of its initial 
 kinetic energy remains after it has risen to one half the height to which it 
 will ascend ? (A) What fraction of its initial velocity remains ? (<■) What 
 fraction of the initial kinetic energy and of the initial velocity remain after 
 the body has risen to three fourths of the total height ? 
 
 4. (a) A boy starting at rest coasts on a bicycle down a hill and up 
 another. If there were no friction, how far would he ascend the second hill 
 without pedaling ? (d) How would the result be affected if cither hill were 
 steeper than the other ? 
 
 5. An unbalanced force of 50 lb. acts through a distance of 4 ft. upon a 
 mass of 9 lb. (a) What is the work done by the force ? (/') What is the 
 kinetic energy of the body ? (<:) What is its velocity ? 
 
 6. A projectile weighing 500 lb. b fired from a cannon with a velocity of 
 3000 ft. per sec. (a) What is its kinetic energy? (^) What was the average 
 total pressure upon the projectile in firing it, if it moved a distance of 18 ft. 
 
 before reaching the mouth of the cannon ? 
 
 y""^ ^\ 7. In whirling a body round the hand at the end of a 
 
 string, the hand is moved in a smaller circle in advance 
 of tht body whirled (Fig. 91). Show how this imparts 
 kinetic energy to the body. 
 V / 8. (tf) Why is a brake not heated if it is applied with 
 
 such force that the wheel slides along the ground instead of 
 * * turning ? {b) Where will the heat then be generated ? 
 
 9. (<i) Is a person doing work upon a load that he carries over level 
 ground ? (^) What is the measure of the work done upon the load when it 
 is carried up hill ? 
 
 10. A stone weighing 1.5 lb. is thrown vertically upward by means of a 
 force of 20 lb. acting through a distance of 3 ft. How high will it rise 
 above the starting point ? 
 
 11. A body weighing 15 kg. falls vertically a distance of 20 m. What is 
 its kinetic energy ? 
 
 12. A bullet weighing 5 g. and having a velocity of 300 m. per sec. strikes 
 a log and penetrates it a distance of 10 cm. What average resistance did the 
 bullet encounter in penetrating the log? 
 
 13. (a) How much work is done in filling a reservoir that has a capacity 
 
 i^) 
 
The Simple Machines 123 
 
 of 1000 cu. m. if the water must be raised 12 m, to discharge it into the reser- 
 voir ? {b) How long would it take a twelve-horse-power engine to fill the 
 
 reservoir 
 
 14. What is the power of an engine that is capable of drawing a train at 
 the rate of 30 mi. per hr. against a resistance of 6250 lb. ? 
 
 II. The Simple Machines 
 
 154. Machines. — Any instrument or device the purpose of 
 which is to do work by transmitting energy from one body to 
 another is called a machine. In the general sense, as here defined, 
 the term includes not only the more complex machines, such as 
 the steam engine, the printing press, and the dynamo, but also the 
 simplest tools, as the nutcracker and the pocketknife. In many 
 machines the energy is transformed as well as transferred. The 
 dynamo, for example, transforms mechanical into electrical energy. 
 In the steam engine the chemical potential energy of the fuel 
 undergoes several transformations, and, in the end, is transferred 
 as mechanical energy to the driving wheels. 
 
 All machines, however complicated, are but modifications and 
 combinations of one or more of the six simple ' machines or 
 mechanical powers. These are the lever, the wheel and axle, the 
 pulley, the inclined plane, the wedge, and the screw. The study 
 of the simple machines will afford a review and application of 
 much that has already been learned concerning force and work. 
 
 155. The Lever. — A lever is essentially a bar or rod capable of 
 turning about some fixed support or axis, called the fulcrum. A 
 crowbar is used as a lever in moving heavy objects (Fig. 55). In 
 using a lever, a force, called the effort^ applied at some point of 
 it, causes it to exert a force at some other point upon the object 
 to be moved. The force exerted upon the lever by this object 
 is called the resistance. The resistance is generally, though not 
 always, due to the weight of the body upon which the lever acts. 
 
 1 The effort is commonly called the power ; but this use of the word is objec- 
 tionable since power also means the rate at which work is done. 
 
1 24 Energy 
 
 There are three classes of levers. In levers of the first class 
 (Fig. 92) the fulcrum O is between the points of application of 
 the effort / and the resistance Fy in levers of the second class 
 
 ^-^-^ n «. ?— ■ ^- V 
 
 \r Fig. 9a. 
 
 F Fig. 93. Fig. 94. 
 
 I 
 
 (Fig. 93) the resistance acts between the fulcrum and the effort ; 
 in levers of the third class (Fig. 94) the effort is applied between 
 the fulcnim and the resistance. The three classes are illustrated 
 in laboratory Exercise 17. (Refer to the exercise and identify 
 the illustrations.) 
 
 In all cases the effort and the resistance tend to turn the lever 
 in opposite directions about the fulcrum ; hence, when the lever 
 is in equilibrium, the moments of the effort and the resistance are 
 equal (Art. 71). If a and /^ denote respectively the perpendic- 
 ular distances from the fulcrum to the lines of action of/ and Fj 
 the condition necessary for equilibrium is 
 
 fa = FA^oi F'.f '.\a\A. (21) 
 
 When the resistance is a weight, it is commonly denoted by Wy 
 in which case the condition of ecjuilibrium is written/z = WA, 
 
 When a lever is in action, even with uniform motion, the 
 moment of the effort is very slightly greater than the moment of 
 the resistance, on account of friction ; but the small difference is 
 commonly disregarded, and in solving problems, the condition of 
 equilibrium is assumed to hold when the lever is in action. The 
 same assumption is made in studying the other simple machines. 
 
 166. Applications of the Lever. — The lever in different forms 
 is adapted to various special uses. In most cases it enables a 
 given force to overcome a resistance several times greater than 
 itself; for the effort and the resistance are inversely proportional 
 to their arms, and the arms may be taken in any desired ratio. 
 Thus a stone that requires a force of 600 lb. to move it can be 
 
The Simple Machines 125 
 
 moved with a crowbar by exerting upon the latter a force of 100 
 
 lb., when the fulcrum is placed so that the 
 
 arm of the effort is six times as long as the 
 
 arm of the resistance. The advantage 
 
 thus derived is referred to as a gain of Fig. 95. 
 
 force. The same advantage is afforded by forceps, pincers, wire 
 
 ^ cutters (Fig. 95), and nutcrackers (Fig. 
 
 ^^ ^"^ll^m 96) ; all of which are double levers 
 
 \ H^^ =^^^''^^^^^ having the arm of the effort longer than 
 
 _ „ _^ the arm of the resistance. 
 
 rIG. 90. 
 
 Other forms of the lever are designed 
 with reference solely to convenience or adaptability to the work 
 to be done, the relative value of effort and resistance being un- 
 important. Tweezers, coal tongs, sugar tongs, and scissors are 
 examples. With all except the last the effort is necessarily greater 
 than the resistance. (Why?) 
 
 In certain applications of the lever the end sought is a gain of 
 speed and distance ; that is, the conditions are such that the point 
 of application of the resistance moves faster and farther than the 
 point of application of the effort. This is well illustrated by the 
 movements of our bodies and the bodies of animals in general. 
 The movable parts of the skeleton are levers ; the joints are the 
 fulcrums. The muscles are attached 
 to the bones near the joints by means 
 of tendons. A muscle acts by con- 
 tracting or shortening. This causes 
 the bone to which it is attached to 
 move, and the farther extremity of _ ^ 
 
 the bone moves much faster and 
 
 farther than the point to which the muscle is attached. Figure 97 
 shows the action of the forearm in lifting a weight in the hand. 
 It is evident that a slight shortening of the muscle is sufficient to 
 raise the hand a foot or more. There is, however, a corre- 
 sponding loss of force ; that is, the effort is much greater than the 
 resistance. (Why?) 
 
1 26 Energy 
 
 A curved or a bent lever is often more convenient than a straight 
 one. A claw hammer used in drawing a nail is an example of a 
 bent lever (Fig. 56). The condition of equilibrium is the same 
 whatever the shape of the lever, it being understood that the arms 
 are always the perpendicular distances from the fulcrum to the 
 lines of action of the effort and the resistance. 
 
 157. Mechanical Advantage. — The ratio of the resistance to 
 the effort {F.f) is called the mechanical adi^antage. The princi- 
 pal problem in the study of a simple machine is to find the value 
 of this ratio in terms of certain dimensions of the machine. It 
 follows from the condition of equihbrium that the mechanical ad- 
 vantage of a ln>er is the ratio 0/ the arm 0/ the effort to the arm of 
 the resistance {a: A). This ratio is frequently called the leverage, 
 
 158. Work done with a Lever. — Suppose a constant effort/, 
 acting at the end of a lever of the first class (Fig. 98), to move 
 
 the lever through a certain angle against 
 ^' '^^ ^ " a constant resistance 7% applied at the 
 
 _ — ,_ _ _ , „,.^ — 
 
 "^-^^ ^i^^ other end. Suppose further that/" and ^ 
 ""-'' act at right angles to the lever through- 
 out the motion. I^t //denote the distance 
 through which the effort acts and Z> the distance through which 
 the resistance is overcome. Then fl is the work done by the 
 effort and FD the work done against the resistance ; or, we may 
 say, fl measures the energy transferred to the lever by the agent 
 producing the motion, and FZ> measures the energy transferred 
 by the lever to the object moved. Formula (21) is 
 
 F:f::a:A 
 and, by geometry, d: D::a: A; 
 
 hence, F:f::d:Df 
 
 or, fd=^FD. 
 
 This means that, neglecting friction, the lever transmits energy 
 from the agent to the body acted upon without gain or loss. 
 This is easily proved to be true for levers of the second and third 
 classes also; and it can be shown to hold in all cases, however the 
 
The Simple Machines 
 
 127 
 
 forces may vary in magnitude or direction. There may be a gain 
 of force by the use of a lever ^ but never a gain of etiergy. There is 
 always, in fact, a slight loss or waste of energy due to friction ; 
 that is,// is slightly greater than /^Z?. 
 
 PROBLEMS 
 
 1. In using scissors is greiater force required when the cutting is done near 
 the tips df the blades or near the handles ? Why ? 
 
 2. Classify the following levers, and state in each case whether the effort 
 is greater or less than the resistance : the wheelbarrow, oar, fishing rod, 
 equal-arm balance, steelyard, nutcracker. 
 
 3. Use a pencil as a lever of the first class to move a book ; also as a lever 
 of the second class. 
 
 4. In which class or classes of levers is the effort necessarily less than the 
 resistance ? greater than the resistance ? In which may it be either greater 
 or less ? 
 
 5. Prove that fit= ^D when a weight W is raised through a vertical 
 distance D by means of a force / acting ver- 
 tically through a distance </ (Fig. 99). 
 
 6. (rt) If a stone offers a resistance of 850 lb., 
 what leverage will be required to move it by 
 means of a force of 125 lb.? (Ji) If the stone is 
 moved by a crowbar 5 ft. long used as a lever 
 of the first class, the effort and the resistance 
 being applied at the ends, where is the fulcrum? 
 
 7. Show that when there is a gain of force by the use of a lever there is a 
 proportional loss of speed and distance ; and that when there is a gain of 
 distance there is a proportional loss of force. 
 
 159. The Wheel and Axle. — The wheel and axle (Figs. 100 
 
 and loi) consists of a 
 wheel and cylinder fas- 
 tened together and ca- 
 pable of turning on the 
 same axis. The effort is 
 applied at the circumfer- 
 ence of the wheel, and 
 the resistance at the cir- 
 FiG. 100. cumference of the axle. 
 
 Fig. 99 
 
128 
 
 Energy 
 
 The machine may be regarded as a continuously acting lever of 
 the first or the second class, depending upon whether the effort 
 and the resistance are applied on opposite sides or the same side 
 of the axis. The condition necessary for equilibrium is 
 
 /r= /ra, or IV: /: : r : ^, (22) 
 
 in which r and /^ denote the radius of the wheel and the axle 
 respectively. 
 
 TAf mechanical advantage of the wheel and axle Is, therefore, 
 the ratio of the radius of the wheel to the radius of the axle (r .• R) . 
 
 Laboratory Exercise 2j. 
 
 160. Work done with a Wheel and Axle. — During one com- 
 plete turn of the wheel the effort acts once round its circum- 
 ference, Cy and the weight is raised a distance equal to the 
 circumference of the axle, C. Whatever the angle may be through 
 which the wheel is turned, the distance d through which the effort 
 acts and the distance D through which the weight is raised are in 
 the same ratio as for one revolution ; that is, 
 
 d'.D=^c.C=r'.R. 
 But W:f:.r:R; 
 
 hence, W:f'..d:D, 
 
 or, //= WD. 
 
 That is, the wheel and axle, like the lever, transmits energy 
 
 without loss or gain, except in so 
 far as there is loss due to friction. 
 161. The Windlass (Fig. 102) 
 is a modified form of the wheel and 
 axle, and acts upon the same prin- 
 ciple. As the crank is turned, the 
 handle describes a circumference ; 
 hence the crank is equivalent to a 
 wheel having a radius equal to the 
 arm of the crank. 
 102. 162. The Fixed Pulley. — A pul- 
 
The Simple Machines 
 
 129 
 
 ley that turns on a stationary axis is called a fixed pulley, 
 be regarded as an endless or continuous lever 
 of the first class having equal arms. Since the 
 arms are equal, the effort and resistance are 
 also equal (/= IV). The same conclusion 
 follows from the fact that, except for a slight 
 possible difference due to friction, the parts of 
 the cord on the two sides of the pulley are 
 necessarily under the same tension. 
 
 It may 
 
 E 
 
 ^o^ 
 
 
 Fig. 103. 
 
 Fig. 104. 
 
 The only advantage gained by the use of 
 one or more fixed pulleys is a change in the 
 direction of the effort (Fig. 103). 
 
 163. The Movable Pulley. — Figure 104 represents the com- 
 bination of a fixed and a movable pulley. The fixed pulley serves 
 the same purpose as when acting alone. The weight, which is 
 applied at the axis of the movable pulley, is balanced by the two 
 equal upward pulls of the two parts of the cord that support the 
 pulley. These pulls are equal to the effort /, and their sum is 
 equal to the weight (including the weight of the supported pulley) ; 
 
 that is, 
 
 2/= rr, and IV :/=2. 
 
 The mechanical advantage of the movable pulley is therefore 
 two. 
 
 164. Systems of Pulleys. — A combination of several fixed 
 and movable pulleys is frequently used where great resistances are 
 
130 Energy 
 
 to be overcome. A variety of arrangements 
 is possible ; but we shall consider only the one 
 generally used, namely, that in which one con- 
 tinuous cord or rope passes alternately round 
 the pulleys in a fixed and a movable block, as 
 shown in the figures. 
 
 In finding the general condition of equi- 
 librium it is assumed that all parts of the cord 
 are under ecjual tension. The weight is there- 
 fore supported by as many equal parallel forces 
 as there are parts of the cord supporting the 
 Fic. 105. movable block. Let this number be denoted 
 by « (« is 3 in Fig. 105 and 6 in Fig. 106) ; 
 then, since the tension is equal to the effort, the con- 
 dition of equilibrium is expressed by the equation 
 
 nf = W, or W \f= n. (23) I lll'r 
 
 Tht mechanical advantage of such a system of pulleys ' ' ' 
 
 is therefore equal to the number of parts of the cord supporting the 
 movable block. 
 
 Laboratory Exercise 24. 
 
 165. Work done with Pulleys. — The ratio of the distance d 
 through which the effort acts to the distance D through which the 
 weight is raised is easily determined for any combination of pul- 
 leys, from the fact that the length of the whole cord remains con- 
 stant. With a single fixed pulley the part of the cord to which 
 the effort is a'pplied lengthens as much as the part to which the 
 weight is attached shortens {d = D) -, hence, since/ = /% we 
 have//= WD, 
 
 With any number of fixed and movable pulleys connected by 
 one continuous cord, when the weight is raised a distance D, 
 each of the n parts of the cord supporting the movable block is 
 shortened by an equal amount, and the length of the part to which 
 the effort is applied is increased by nD {i.e. d = nD) ; hence, 
 since /= IF-i- n (equation 23), it follows that/^/= WD, 
 
The Simple Machines 131 
 
 That is, neglecting friction, any combination of pulleys transmits 
 energy without gain or loss from the agent to the body acted 
 upon. 
 
 PROBLEMS 
 
 1. (a) The radius of a wheel is 40 cm. and the radius of the axle 12 cm. 
 Neglecting friction, what effort is required to raise a load of 150 kg.? 
 {d) Through what distance does the effort act in raising the load 35 m.? 
 (c) How much work b done by the effort ? (^) How much is done against 
 the weight of the load ? 
 
 2. Establish the condition necessary for the equilibrium of a single mov- 
 able pulley, regarding it as a lever of the second class. Draw a figure to 
 illustrate. 
 
 3. Draw a figure of a system of pulleys such that the mechanical advantage 
 is seven ; such that it is eight. 
 
 166. The Inclined Plane. — The principal use of the inclined 
 
 plane is to raise heavy bodies 
 
 that can be rolled. When 
 
 a barrel of flour is placed in 
 
 a wagon by rolling it up a 
 
 heavy plank, the plank is 
 
 used as an inclined plane. 
 
 The effort in such cases is 
 
 applied parallel to the ])lane, 
 
 and is sufficient to maintain 
 
 equilibrium if it is equal to the component of the weight of the 
 
 body acting parallel to the plane in the opposite direction (Fig. 
 
 107). It follows from this (see Art. 143) that the condition of 
 
 equilibrium is 
 
 lV:/::L:/f, (24) 
 
 in which Z denotes the length of the plane, AB, and H its height, 
 BC, 
 
 Hence, when the effort is applied parallel to an inclined plane ^ 
 the mechanical advantage is the ratio of the length of the plane to 
 its height. 
 
 Laboratory Exercise 25. 
 
 Fic. 107, 
 
132 Energy 
 
 167. Work done with an Inclined Plane. — Let D (Fig. 107) 
 denote the vertical distance through which a body is raised by 
 rolling it a distance, //, up an inclined plane {d being any 
 fraction of the length of the plane) ; then the work done by 
 the effort is fd and the work done upon the body (against 
 gravity) is WD. 
 
 From similar triangles d\ D\\ L\ H\ 
 hence, combining with (24), W\f\ \ d. D^ 
 or, //= WD, 
 
 That is, disregarding friction, the same amount of work is done 
 whether a body be rolled up an inclined plane or lifted verti- 
 cally to the same height. 
 
 168. No Machine creates Energy. — It has been shown that, 
 disregarding friction, the work done by any machine that we have 
 studied is exactly equal to the work done upon the machine. The 
 significant fact to be noted is that there is in no case a gain of 
 work, or a creation of energy. This is true of all machines without 
 exception. No machine has ever been invented that was an 
 original source of even the smallest amount of energy ; on the 
 contrary, there are the best of reasons for believing that no such 
 machine is possible (Art. 271). 
 
 169. Efficiency of Machines. — No machine, however perfect, 
 does useful work that is the full equivalent of the work done upon 
 it. The principal cause of waste is friction, by which a portion of 
 the energy transferred to the machine is transformed into useless 
 heat (Art. 152). Further waste of energy results from the bend- 
 ing and vibration of the parts of a machine and from the stiffness 
 of ropes and belts. 
 
 The ratio of the useful work done by a machine to the total 
 energy transferred to it is called the efficiency of the machine. 
 The efficiency of an ideally perfect machine would be unity or 
 100 per cent. A delicately balanced lever, such as the beam of 
 a sensitive balance, is the nearest approach to such a machine. 
 In general, the greater the complexity of a machine, the less its 
 
The Simple Machines 
 
 133 
 
 Fig. 108. 
 
 efficiency. The pupil doubtless observed in experimenting with 
 pulleys that friction becomes greater as the number of pulleys is 
 increased. With five or six pulleys in the combination, the effi- 
 ciency will hardly exceed 75 per cent and may be considerably 
 less. 
 
 170. The Wedge. — The wedge may be regarded as a movable 
 inclined plane. It is used for separating surfaces asjainst great 
 resistance, as in splitting logs and timbers. 
 The motion of a wedge is opposed by very 
 great sliding friction, and its efficiency is 
 correspondingly low; yet this friction is 
 useful and indeed necessary, for it keeps 
 the wedge from slipping out of place dur- 
 ing the intervals between the blows that 
 drive it farther in. The wedge utilizes 
 a great force acting for a very short time, 
 and is thus enabled to overcome a great 
 resistance through a small distance. Hence, though slow in 
 action, it is a very powerful machine (Fig. 108). 
 
 No definite law can be given that holds even approxi- 
 mately in the use of the wedge. The thinner the wedge, 
 however, the greater is its mechanical advantage. The ax, 
 the knife, and the chisel are forms of the wedge adapted to 
 special uses. 
 
 171. The Screw. —The screw is a modification of the 
 inclined plane, as will be seen by winding a paper triangle 
 about a pencil (Fig. 109). The ridge running spirally round 
 a screw is called the thread. The distance between suc- 
 cessive turns of the thread, measured parallel to the axis of 
 the screw, is called the pitch of the screw. The screw 
 turns in a block, called the nut, which is provided with a 
 spiral groove to receive the thread. 
 
 When the screw is used as a part of a machine, 
 
 the effort is generally applied to it by means di 
 
 Fig. 109. a lever or a wheel. The lifting jack (Fig. no) 
 
1 34 Energy 
 
 is an example of the first case, the copying press (Fig. iii) of the 
 second. During one complete turn of the lever or the wheel the 
 
 Fiii. na Fig. hi. 
 
 screw advances through a distance equal to the pitch of the 
 screw, p. The work done by the effort during one turn is 2 trrf, r 
 being the radius of the wheel or the arm of the lever ; and the 
 work done against the resistance, Fy is Fp. Hence, if friction 
 were negligible, we should have 
 
 2irr/=^Fp. (25) 
 
 This is sometimes given as the law of the screw, from which the 
 mechanical advantage is determined ; but the error is very large. 
 In fact, after the screw has been turned into any position, friction 
 alone is generally much more than sufficient to hold it in place 
 against the resistance. Friction is for this reason useful and nec- 
 essary in most applications of the screw, as it is for the wedge. 
 Without it, woodwork and the parts of machines could not be 
 held together by means of screws. 
 
 172. The Bicycle. — The bicycle is a familiar example of a 
 compound machine in which a number of the foregoing principles 
 are applied. The cranks and the large sprocket wheel are a modi- 
 fied wheel and axle. The motion of the larger sprocket wheel is 
 communicated by means of the chain to a smaller one attached 
 
The Simple Machines 135 
 
 to the rear wheel of the machine and turning upon the same 
 axle. Since the chain in turning passes by the same number of 
 sprockets on both wheels, the smaller one turns as many times 
 faster than the larger as the number of sprockets on the smaller is 
 contained in the number on the larger. If this ratio is three, for 
 example, the smaller sprocket wheel makes three complete turns 
 while the larger makes one. But the larger wheel turns at the 
 same rate as the cranks, and the smaller at the same rate as 
 the rear wheel of the machine. Hence, in the case supposed, the 
 rear wheel makes three revolutions while the cranks make one. 
 Thus the bicycle travels as far during one revolution of the cranks 
 as it would if they were attached directly (as in the old style 
 " ordinary ") to a wheel having a diameter three times as great. 
 The diameter of this equivalent wheel, measured in inches, is 
 called the^^^r of the bicycle. Thus, if the ratio of the sprockets 
 is three, the gear is 3 X 28, or 84, the diameter of the wheels of a 
 "safety " being 28 in. With this gear and cranks having a radius 
 of 7 in., the speed of the bicycle is 84 -h 14, or 6 times as great 
 as the speed of the cranks round the axle. 
 
 PROBLEMS 
 
 1. A block at rest upon a board i m. long begins to slide when the board 
 is inclined so that its higher end is 40 cm. above the other. The friction is 
 what per cent of the weight of the block ? 
 
 2. Find by geometry the mechanical advantage of the inclined plane when 
 the effort is applied parallel to the base of the plane, i.e. horizontally. 
 
 3. Assuming a loss of 20% (efficiency 80%), what force is required to haul 
 a load of 2 tons (including the weight of the wagon) up a grade such that the 
 ascent is one foot in a distance of 10 ft. ? 
 
 4. What effort will be required to raise a weight of 200 kg. with a wheel 
 and axle the efficiency of which is 90 % ; the radius of the wheel being 42 cm. 
 and the radius of the axle 14 cm. ? 
 
 5. The weight in the preceding problem is raised 25 m. Find : {a) the 
 work done upon the machine ; {b) the work done against gravity ; {c) the 
 energy wasted. 
 
 6. What do you gather from Arts. 168 and 169 concerning the possibility 
 
 of a ** perpetual motion " machine ? 
 
136 Energy 
 
 7. Is a sewing machine constructed for a gain of speed or of force ? By 
 what aicchanism is this gain accomplished ? 
 
 8. How long must a plank be in order that it may be used as an inclined 
 plane to raise a barrel weighing 150 lb. to an elevation of 4 ft. by a force of 
 40 lb., assuming an efficiency of ic» % ? 
 
 9. A lifting jack, the screw of which has a pitch of } in., is used to raise a 
 weight of 10 tons. The effort is applied at the end of a lever 2 ft. long, and 
 the efficiency is 40 %. Kind the efiort. 
 
CHAPTER VII 
 
 SOME PROPERTIES OF MATTER 
 
 I. The Structure of Matter 
 
 173. Properties of Matter. — The properties of matter are of 
 two classes — general or universal and specific or characteristic. 
 General properties are common to all matter. Among the most 
 important are extensioUy inertia, divisibility, porosity, compressibility ^ 
 elasticity, indestructibility. Specific properties are those possessed 
 by certain kinds or slates of matter, but not by all; such are 
 rigidity, fluidity, brittleness, hardness, tenacity, transparency, color, 
 odor, etc. It is only by means of differences in the properties of 
 substances that we can distinguish them from one another. 
 
 The laws of liquid pressure are direct consequences of the 
 characteristic properties of liquids, namely, their fluidity and their 
 very nearly perfect incompressibility. Similarly, the laws of gases 
 follow from the fluidity and the unlimited expansibility of gases. 
 In dealing with solids, we have assumed that the behavior of a 
 body under the action of forces is not affected by bending or other 
 distortion of the body, however the forces may be applied ; i.e. 
 solids were assumed to be perfectly rigid. Thus the mechanics 
 of solids, liquids, and gases is largely dependent upon the dis- 
 tinguishing properties of the three states of matter. 
 
 Extension and inertia have been sufficiently considered in pre- 
 vious chapters. A study of the other general properties mentioned 
 above suggests important inferences concerning the structure of 
 matter. 
 
 174. Divisibility. — Any substance can be divided into parts 
 or fragments ; and the division can be carried so far that the 
 individual particles are invisible to the unaided eye and are barely 
 
 137 
 
138 Some Properties of Matter 
 
 visible with the aid of a microscope. Suppose a grain of salt to 
 be pulverized till a microscopic particle is obtained. Subdivision 
 can be carried no farther by mechanical means ; but the particle 
 can be dissolved in a drop of water, and, although what happens 
 to it in this process is entirely invisible, many facts that cannot be 
 presented here point to the conclusion that, in dissolving, // is scp- 
 arattd into millions of parts. 
 
 The almost inconceivable divisibility of matter is well illustrated 
 by dissolving a minute grain of potassium permanganate or of aniline 
 dye in several quarts of water. The whole will be appreciably 
 colored, showing the presence of portions of the substance in every 
 drop {Ex'p.). Numerous illustrations might be given from daily 
 life. After some years of service a gold or a silver coin is found 
 to be considerably worn; yet no visible portion is lost at any 
 time. Similarly, a knife becomes dull from the gradual loss of 
 invisible particles at and near its edge. The tenth part of a grain 
 of musk will continue for years to fill a room with its odoriferous 
 particles, and at the end of that time will scarcely be diminished 
 in weight. 
 
 175. Molecules. — Since the divisibility of matter far exceeds 
 our power of vision, it is only by reasoning and inference based 
 upon other facts that it can be determined whether there is any 
 limit to divisibility. Many facts in chemistry indicate that there 
 is such a limit. In other words, there is such a thing as the 
 smallest possible particle of any substance. This smallest particle 
 is called a molecule. All molecules of the same substance are 
 exactly equal in size and weight, and are alike in every respect. 
 
 It is estimated that, in a cubic millimeter of any gas at atmos- 
 pheric pressure and at ordinary temperatures, there are some- 
 where about 20,000,000,000,000,000 molecules ; and that, in an 
 equal volume of a liquid or a solid, the number is hundreds or 
 thousands of times as great. 
 
 176. Porosity. — Openings within a solid are called sensible 
 pores whether large enough to be visible with the unaided eye, as 
 the holes in a sponge or a piece of bread, or so small as to be 
 
The Structure of Matter 
 
 139 
 
 visible only with a microscope, as the pores of blotting paper, brick, 
 and unglazed earthenware. The spaces or openings between the 
 molecules of a body are cdiWtdi p/iysical pores. The most powerful 
 microscope is incapable of rendering them visible ; but their 
 existence has been repeatedly demonstrated by experiment. 
 
 In 1 66 1 some academicians at Florence subjected to great 
 pressure a thin globe of gold filled with water. Their object was 
 to determine whether water was compressible ; they discovered 
 the porosity of gold instead, for " the water forced its way through 
 the pores of the gold, and stood on the outside of the globe like 
 dew." The experiment has been repeated with globes of other 
 metals, and always with the same result — the metals were proved 
 to be porous. A pressure of less than one atmosphere is sufficient 
 to force mercury through a thick piece of leather {Exp.) ; and, 
 with a pressure of 4000 atmospheres, mercury has been forced 
 through three inches of solid steel. 
 
 With the exception of glass and other vitreous bodies, similar 
 experiments have not failed to prove the porosity of any solid. 
 
 177. Porosity a General Property of Matter. — The fact that 
 the volume of any substance, whether solid, liquid, or gaseous, can 
 be changed either by change of temperature or change of pressure 
 is both a proof of the porosity of all matter and a consequence 
 of this property. Such changes of volume are considered in the 
 following paragraphs. Further striking evidence of the porosity 
 of certain liquids is afforded by the change of volume that occurs 
 in mixing them. Thus, when equal or nearly equal volumes of 
 water and strong alcohol are mixed, it is found that the volume 
 of the mixture is considerably less than the sum of the volumes of 
 the liquids before they are mixed,^ proving that the molecules 
 of water and alcohol fit together more closely than the molecules 
 of one or the other, or possibly both, of the liquids do among 
 
 1 This is readily shown by filling a long, slender test tube half full of water and 
 adding alcohol carefully, to avoid mixing, till the tube is nearly full. Mark the 
 height of the alcohol with a rubber band, close the mouth pf the tube with the thumb, 
 and shake. 
 
140 Some Properties of Matter 
 
 themselves {Exp.). The idea is iUustrated on an enormously 
 magnified scale by mixing together a quantity of sand and an equal 
 volume of coarse shot. A loss of volume occurs, due to the filling 
 of the spaces between the shot by the smaller grains of sand. The 
 loss of volume on mixing the alcohol and water is regarded as 
 conclusive evidence that there are spaces void of matter between 
 the molecules of the liquids. 
 
 A similar shrinkage occurs in mixing strong sulphuric acid and 
 water, and also when certain solids, as sugar or salt, are dissolved 
 in water. 
 
 178. Compressibility. — All matter is more or less compressible. 
 The compressibility of gases, as expressed by Boyle's law, has 
 already been considered. The compressibility of liquids and 
 solids, although extremely small, is of theoretical interest as an 
 additional proof of their porosity. It has been found that a pres 
 sure of one atmosphere diminishes the volume of water at the 
 freezing point by yi^i^nr 5 ^"^ ^^^^ under a pressure of 3000 
 atmospheres, its volume is diminished by -}j^. An equal pressure 
 diminishes the volume of ether by about J. The compressibility 
 of solids not having sensible pores is still less than that of liquids. 
 The compressibility of glass, for example, is about 1^ as great as 
 that of water. 
 
 There are two conceivable explanations of compressibility; 
 namely: (i) that, by compression, the molecules of a body are 
 crowded closer together, thus diminishing the void spaces between 
 them; (2) that the molecules themselves are diminished in size by 
 compression. If the latter were true, matter would be compress- 
 ible even if it were not porous. However, in addition to the 
 known porosity of matter, other facts, which are considered in 
 the next section, indicate that the first of the above suppositions 
 is the true explanation of compressibility. 
 
 It is evident from the very great compressibility of gases that, 
 under ordinary pressures, their molecules occupy only an exceed- 
 ingly small portion of the space allotted to them. The volume of 
 a gram of steam under a pressure of one atmosphere is 1661 ccm. 
 
Molecular Motion 141 
 
 — a volume 1661 times greater than it occupies as a liquid. 
 Even if we were to assume that, as a liquid, the molecules fill the 
 entire space, it would follow that the average distance between 
 the molecules of steam, under a pressure of one atmosphere, is 
 nearly twelve times the diameter of a molecule. Oxygen has been 
 subjected to a pressure of 3000 atmospheres, under which pres- 
 sure its density was greater than that of water, although it still 
 remained in the gaseous state. The compressibility of all gases, 
 when subjected to very great pressures, is less than that indicated 
 by Boyle's law, showing that a gas approaches the condition of a 
 liquid as its molecules are crowded together. Why the molecules 
 of a gas remain apart under enormous pressures is a question con- 
 sidered in the next section. 
 Laboratory Exercise 26. 
 
 179. Expansibility ; Effect of Heat. — While the effect of pres- 
 sure upon the volume of liquids and solids is far too small to be 
 demonstrated by simple experimental methods, the effect of heat 
 can very readily be shown, as illustrated by the experiments of the 
 preceding laboratory exercise. In general, the application of heat 
 to a body causes it to expand, and with a loss of heat it contracts. 
 With liquids and solids this change of volume is slight ; with gases 
 it is very much greater. When water is heated from the freezing 
 to the boiling point, it increases in volume by four per cent. 
 With the same change of temperature, the pressure remaining 
 constant, the volume of a gas increases over thirty- six per cent. 
 The expansion of a substance by heat is supposed to be due to the 
 wider separation of its molecules, not to any increase in their 
 size. 
 
 II. Molecular Motion 
 
 180. Diffusion of Gases. — When two or more gases are brought 
 in contact and left undisturbed, they quickly mix with one another, 
 even in cases where a difference of density would tend to prevent 
 such mixing, the denser gas being originally at the bottom. This 
 
142 Some Properties of Matter 
 
 spontaneous mixing of gases is called diffusion. The process may 
 be illustrated by the following experiments. 
 
 When illuminating gas or any other gas having a strong odor is 
 permitted to escape into a room, the sense of smell quickly reveals 
 the presence of the gas in all parts of the room. This 
 is well illustrated by a little ammonia poured on the 
 floor. The ammonia evaporates, and the odor of 
 ammonia gas fills the room {Exp.). 
 
 A bottle of oxygen and another of illuminating gas 
 are placed together, mouth to mouth (Fig. 112), the 
 one containing the oxygen at the bottom. After they 
 have stood thus for a few minutes they are separated 
 and a flame is quickly applied to the mouth of each. 
 There is an explosion in each 'case, indicating the 
 presence of a mixture of oxygen and illuminating gas 
 Fio. na. in both bottles. Note that, since the denser gas (oxy- 
 gen) is placed at the bottom, the mix- 
 ing is opposed by gravity {Exp.). 
 
 A porous cup of unglazed earthen- 
 ware is fitted with a cork and glass 
 tube, and supported, as shown in Fig. 
 113, with the lower end of the tube in 
 a tumbler of water. A large jar is held 
 over the cup and filled with illumi- 
 nating gas through a rubber tube. The 
 experiment has three stages. ( i ) Bub- 
 bles of gas immediately rise from the 
 end of the glass tube in the water, 
 indicating that the pressure within the 
 cup has been increased by the entrance 
 of some of the gas through its micro- 
 scopic pores. (2) On removing the 
 jar, the water immediately begins to rise ^^°' ^^3* 
 
 in the tube, indicating a decrease of pressure within, due to the 
 escape of the gas through the pores. (3) Presently the water 
 
Molecular Motion 143 
 
 begins to fall in the tube, and continues to do so until the level 
 is the same as in the tumbler. This indicates that air enters the 
 cup through the pores until the pressures within and without are 
 equalized ; and, moreover, suggests that the fuller explanation of 
 the first two stages of the experiment is as follows : Both the 
 illuminating gas and the air can diffuse through the pores of the 
 cup ; but the gas more rapidly than the air. In (i) the pressure 
 within increases because the gas enters more rapidly than the air 
 escapes; in (2) the pressure within decreases because the gas 
 escapes more rapidly than the air can enter. 
 
 181. Diffusion explained by Molecular Motion. — The phe- 
 nomena of diffusion, as illustrated by the above experiments, 
 indicate that the molecules of a gas are in constant motion. In 
 fact, all that is known about the properties and laws of gases sup- 
 ports the conclusion that the molecules of a gas are always in very 
 rapid motion, darting hither and thither in all directions at random, 
 like bees or gnats in a swarm. Each molecule, according to the 
 theory, moves in a straight line till it hits another molecule or the 
 wall of the containing vessel, when it is reflected and bounds off 
 in a different direction. 
 
 When two gases are brought in contact, the molecules of each 
 quickly find their way between those of the other, and in a short time 
 the entire space is filled with a homogeneous mixture of the two. 
 Molecular motion affords an explanation of diffusion through the 
 porous cup in the last experiment. A molecule that by chance is 
 moving toward one of the pores finds an easy entrance and a 
 passageway through to the other side. It must not be supposed 
 that diffusion ceases when equilibrium of pressure is established at 
 the end of the experiment. With the same gas (air) at the same 
 temperature and pressure on both sides of the partition, an equal 
 number of molecules find their way through it in both directions, 
 thus maintaining the equilibrium. 
 
 If two gases at the same temperature and pressure are of 
 unequal density, the molecules of the rarer gas are moving 
 more rapidly than those of the other. This accounts for the 
 
144 Some Properties of Matter 
 
 more rapid diffusion of the illuminating gas through the porous 
 cup. 
 
 Diffusion is a very different process from the flow of gases in 
 currents. In diffusion, the molecules move as individuals; in 
 currents, they move collectively as one body. Currents hasten 
 the process of mixing, and hence aid diffusion. Diffusion supple- 
 ments the action of winds in keeping the constituents of the air 
 uniformly mixed. 
 
 182. Molecular Motion the Cause of Gas Pressure. — The suppo- 
 sition that the molecules of a gas are in motion as described above 
 affords a complete explanation of gas pressure and of Boyle's law. 
 The force exerted upon the wall of the containing vessel when a 
 molecule strikes it is inappreciable ; but these blows are so numer- 
 ous that their combinetl effect is a continuous, constant pressure. 
 If a given mass of gas is compressed to half its former volume, its 
 density is doubled, and twice as many molecules strike a given 
 portion of the wall of the containing vessel every second ; hence 
 the pressure is doubled. 
 
 Molecular motion accounts also for the indefinite expansibility 
 of gases, and for the fact that the molecules remain separated 
 even when subjected to great pressure. 
 
 From the known density of a gas, it is possible to compute 
 what the average velocity of its molecules must be in order to 
 exert the observed pressure. The computation yields the sur- 
 prising result that the molecules of the air are darting about with 
 an average velocity of about eighteen miles per minute (varying 
 somewhat with the temperature). "Could we, by any means," 
 says Professor Cooke, " turn in one direction the actual motion of 
 the molecules of what we call still air, it would become at once a 
 wind blowing seventeen miles per minute, and would exert a de- 
 structive power compared with which the most violent tornado is 
 feeble." The velocity of hydrogen molecules is $.S times as great 
 as that of air molecules, or about sixty-eight miles per minute. 
 
 183. The Kinetic Theory of Gases. — The explanation of the 
 physicial properties of gases as consequences of the motion of 
 
Molecular Motion 145 
 
 their molecules is called the kinetic theory of gases. The theory, 
 in its complete form, applies the laws of dynamics to the indi- 
 vidual molecules, and accounts definitely {i.e. quantitatively) for 
 all the laws of gases ; but the mathematical treatment of the 
 theory belongs to advanced physics. 
 
 184. The Meaning and Value of a Theory. — An hypothesis, or 
 theory, is a suggested explanation of facts that cannot be traced 
 to any directly ascertainable cause. A theory is not necessarily 
 true even if it affords a complete explanation of all the known 
 facts ; for it is conceivable that the true cause may be very differ- 
 ent from the one supposed. In fact, it has happened repeatedly 
 in the history of science, that rival theories have been ably 
 defended at the same time by different scientists of recognized 
 authority. If, at any time, a new fact is discovered which is in- 
 consistent with a theory, the theory must be modified so as to 
 be in agreement with the fact, or, if this is impossible, it must 
 be abandoned. Newly discovered facts have often served to dis- 
 tinguish between a true theory and a false one. 
 
 Physical theories are of service in explaining properties of matter, 
 natural laws, and natural phenomena. Thus, as we have seen, the 
 molecular theory of matter accounts for the divisibility, porosity, 
 and compressibility of matter; and the kinetic theory of gases 
 fully explains the phenomena of diffusion, the pressure and expan- 
 sibility of gases, and Boyle's law. 
 
 185. Diffusion of Liquids. — If any two liquids that can be 
 mixed with each other are placed in the same vessel, the denser 
 at the bottom, and left undisturbed, they will mix by diffusion, 
 the process being similar to the diffusion of gases. The progress 
 of diffusion in liquids is visible in cases where it is accompanied 
 by a change of color, as in the following experiments. 
 
 A test tube or a tall jar is nearly filled with water colored with 
 blue litmus. A little strong sulphuric acid is then admitted at the 
 bottom through a long-stemmed funnel (Fig. 114). The acid is 
 considerably denser than the water and supports it, the surface 
 separating the two being distinctly visible. Since acid turns blue 
 
146 
 
 Some Properties of Matter 
 
 Fig. 114. 
 
 litmus red, the progressive change of color from blue to red, 
 which slowly takes place up the tube, indicates the height to 
 which the acid has risen by diffusion {Exp.). 
 
 A jar is half filled with water and a strong solu- 
 tion of copper sulphate is admitted at the bottom, as 
 the acid is in the preceding experiment. The progress 
 of diffusion is indicated by the very slow rise of the 
 blue color of the solution. The process requires 
 months for its completion {Exp.), 
 
 We see from these experiments that diffusion takes 
 place in litjuids, as in gases^ without the aid of cur- 
 rents and in opposition to gravity. The explanation 
 is therefore the same ; the molecules of a liquid are 
 in motion. T^he very slow rate of diffusion in liquids 
 indicates that the motion of the molecules is greatly 
 restricted as the result of their crowded condition. 
 The molecules of a liquid are always moving about among one 
 another whether a second liquid is present or not ; in the latter 
 case, however, there is no direct evidence of this motion. 
 
 186. Vibration of Molecules. — The molecules of a solid are 
 held together in fixed positions ; i.e. they have no motion of trans- 
 lation. Many facts, however, support the conclusion that the 
 molecules of all bodies, including solids, are in a state of rapid 
 vibration. We may think of the vibratory motion of a molecule 
 as something Hke the quivering of a piece of jelly when it is shaken. 
 
 The molecules of a gas are supposed to be vibrating freely while 
 moving in straight lines between successive collisions. A third 
 motion is also possible ; namely, rotation — like the spinning of a 
 top. In liquids these motions are modified and restricted by the 
 crowded condition of the molecules ; in solids molecular motion 
 is probably restricted to an irregular vibration. 
 
 187. Theory of Heat. — According to the kinetic theory, when 
 a gas is heated the velocity of its molecules is increased. This 
 explains the expansion of a gas when heated, or, if expansion is 
 prevented, the increase of its pressure. Further evidence that 
 
Molecular Forces 147 
 
 heat increases molecular motion is found in the increased rate of 
 diffusion, both of gases and of liquids, at higher temperatures. 
 The expansion of liquids aiid solids when heated is also attributed 
 to increase of molecular motion. The more energetic vibration of 
 the molecules enables them to push their neighbors farther away, 
 and the whole mass expands in consequence. 
 
 It was stated in Art. 148 that heat is a form of energy. We can 
 now understand what that form is. According to the theory uni- 
 versally accepted for more than half a century, heat is the kinetic 
 energy of molecular motion. When a bullet strikes a steel target 
 (Art. 148, end), the molecules of the bullet and of the adjacent 
 parts of the target are violently disturbed by the impact and set 
 in energetic vibration. The energy due to the motion of the bul- 
 let as a whole (molar kinetic energy) is t-ransformed by the impact 
 into energy due to the motion of its molecules and the molecules 
 of the target (molecular kinetic energy or heat). 
 
 III. Molecular Forces 
 
 188. Molecular Attraction and Molecular Pressure. — The mole- 
 cules of a body are, in general, subject to the action of two oppos- 
 ing forces, one of which tends to bring them together, and the 
 other to separate them from each other. The first is called molecu- 
 lar attraction, or cohesion. It is strong in some substances and 
 weak in others ; and, in the same substance, it grows weaker as 
 the distance between the molecules increases. Its action is very 
 different from that of gravitation, as will presently be seen. 
 
 The other molecular force is due to molecular motion, or heat, 
 and consists in the continuous and inconceivably rapid shower 
 of blows which each molecule exerts upon its neighbors. These 
 blows act as a molecular pressure, the tendency of which is to 
 drive the molecules farther apart. 
 
 189. Molecular Forces and the States of Matter. — It is the 
 mutual relation between molecular attraction and molecular pres- 
 sure due to heat which determines whether a substance will be in 
 
148 Some Properties of Matter 
 
 the solid, liquid, or gaseous state. In solids attraction greatly pre- 
 dominates, and the molecules hold together, or cohere^ in fixed 
 relative positions. The same relation holds in liquids but in a less 
 degree ; the molecules cohere, but are able to wander at random 
 among one another. In gases at ordinary pressures and tempera- 
 tures, the molecules are so widely separated that attraction is 
 inappreciable or wholly wanting, and molecular pressure is 
 balanced only by fxternai force. 
 
 When the molecular motion of a solid is sufficiently increased 
 by heating, cohesion is so far overcome by the increased molecu- 
 lar pressure that the solid is liquefied. By further heating, the 
 molecular i)ressure may become so great as to separate the mole- 
 cules beyond the range of their mutual attraction ; the liquid will 
 then be vaporized, or converted into a gas. 
 
 190. Strength of Cohesion; Tenacity. — To break a body or 
 tear it apart, a force must be exerted sufficient to overcome the 
 cohesion between the molecules lying on opposite sides of the 
 surfaces separated. Thus cohesion is the cause of tenacity 
 (the property of a body inconsequence of which it resists being 
 pulled or broken apart) ; and, conversely, the tenacity of a body 
 serves as a test of the strength of cohesion within it. 
 
 Steel is the most tenacious of all substances, a tension of 1 80 
 lb. being required to break a steel wire having a cross-section 
 of I sq. mm. For an equal cross-section, the tenacity of soft 
 copper wire is about 70 lb., of lead wire 5 lb., of glass 14 lb., of oak 
 (in the direction of its fibers) 15 lb. {Exp.). 
 
 Laboratory Exercise j. 
 
 191. Effect of Distance on Cohesion. — When the pieces of a 
 broken stick or stone are accurately fitted and firmly pressed to- 
 gether, the parts do not cohere with appreciable force ; for the 
 molecules upon the two surfaces cannot by such means be brought 
 within the range of their attractive power. Glue or cement, spread 
 between the surfaces, bridges the gap, and, on drying, serves as an 
 effective connecting link. When two pieces of clean plate glass 
 are firmly pressed together, they cohere with sufficient force to 
 
Molecular Forces 149 
 
 sustain the weight of one of them. The pieces are not held 
 together by atmospheric pressure, as the Magdeburg hemispheres 
 are, for there is still a thin layer of air between them, and, besides, 
 the experiment succeeds quite as well under the receiver of an 
 air pump from which the air has been exhausted. Two pieces of 
 common window glass do not cohere when pressed together. The 
 reason for the difference is that the surfaces of the plate glass are 
 quite accurately plane, and hence fit very closely together over 
 their whole extent ; while the surfaces of common window glass 
 are more or less wavy, and really come in contact at only a few 
 points. Two pieces of any metal, having surfaces accurately plane, 
 cohere like the pieces of plate glass. 
 
 These experiments afford some indication of the extremely short 
 distance to which cohesion can act. It is estimated that this dis- 
 tance is not greater than ^u-ffTny ''"^^- — about y^Vir ^^ ^^^ thick- 
 ness of the paper on which this is printed. Cohesion is weaker 
 between two pieces of plate glass than it is within them only 
 because the distance between the attracting molecules is greater. 
 
 192. Cohesion of Plastic Solids ; Welding. — Two surfaces of a 
 plastic solid, as soft putty or clay, warm molasses candy, or wax, 
 can be brought so close together by pressure that they will unite 
 perfectly. Even two pieces of lead will cohere quite firmly if 
 their freshly brightened surfaces are pressed together in a vise 
 
 In the process of welding, the two pieces of metal to be united 
 are rendered plastic by heating. Their surfaces are then brought 
 within the range of cohesion by hammering them together. Simi- 
 larly, two pieces of glass, rendered plastic by heating, will readily 
 unite, forming perfect union (£x/>.). 
 
 193. Adhesion. — There is no evidence of any difference in the 
 nature of the attraction between molecules of the same kind and 
 that between molecules of different kinds; but the former is 
 generally called cohesion, and the latter adhesion. The distinction 
 is convenient but unimportant. 
 
 There are many familiar examples of adhesion between solids. 
 
1 50 Some Properties of Matter 
 
 Mud sticks, or adheres, to any object with vexing facility ; butter 
 adheres to a knife and to the bread upon which it is spread. The 
 adhesion of metals is utilized in gold and silver plating. Ordi- 
 narily there is no appreciable adhesion between solids when brought 
 in contact, unless one of them is plastic ; but this is only because 
 their surfaces are not brought sufficiently close together. 
 
 Adhesion between solids and liquids is also familiar. In a large 
 majority of cases, when a liquid and a solid are brought in contact, 
 the liquid clings to the solid and wets it. This is because adhe- 
 sion between the two is greater than cohesion within the liquid. 
 Thus when the finger is dipped in water and removed, the layer 
 of water in contact with the finger is held by adhesion with suffi- 
 cient force to tear it away from the adjacent molecules of water. 
 In some cases a solid is not wet by a liquid. Water, for example, 
 does not wet a surface covered with grease or wax, and mercury 
 wets but few substances. In such cases the liquid clings together 
 in somewhat flattened drops upon the surface of the solid (Fig. 
 115). This behavior of the liquid does 
 not prove that adhesion is wanting. On 
 the contrary, there is direct experimental 
 iG. 115. evidence of adhesion ; and, between gla§s 
 
 and mercury, it is even very considerable (Laboratory Ex. 3). 
 The explanation is that, in such cases, cohesion in the liquid is 
 stronger than adhesion, whatever the strength of the latter may be. 
 Gases also adhere to solids, forming a very thin but dense layer 
 upon their surface. In setting up a barometer, air adheres to the 
 inner wall of the tube, and is driven off only by heating the mer- 
 cury till it boils. The small bubbles of air that gather upon the 
 side of a glass of water as it becomes warm are held there by 
 adhesion with sufficient force to overcome the buoyancy of the 
 water. 
 
 194. Cohesion and Gravitation Compared. — We know the law 
 of gravitation, but not its cause (Art. 1 29). We know neither the 
 law nor the cause of cohesion ; but it is evident that the law is 
 very different from that of gravitation, for cohesion acts only at 
 
Surface Tension and Capillarity 151 
 
 insensible distances, and within such distances it is enormously 
 stronger than gravitation between the same masses. Hence the 
 strength of bodies in general depends practically entirely on cohe- 
 sion, and gravitation becomes appreciable only in bodies of very 
 great size. The strength of the earth depends almost wholly on 
 gravitation. If we suppose the earth to be divided into hemi- 
 spheres by any plane through its center, the gravitational attraction 
 by which the hemispheres are held together is one hundred times 
 as great as cohesion would be if the earth were made of solid 
 steel. If there were a planet fifty miles in diameter having the 
 same density as the earth and the tenacity of sandstone, gravi- 
 tation and cohesion would be equally effective in keeping it 
 together. 
 
 IV. Surface Tension and Capillarity 
 
 195. Surface Tension. — A pin or a needle can be made to float 
 on water, if carefully laid upon the surface so that neither end 
 touches the water before the other. If the pin sinks and becomes 
 wet, it should be dried by rubbing it between the fingers or in 
 the hair, as this covers it with a coating of oil and diminishes the 
 adhesion of the water to it {Exp.). The floating pin lies in a 
 depression or trough which is several times larger than itself 
 (represented in cross-section in Fig. 116). It is, in fact, sup- 
 ported in much the same way as a person is 
 when lying in a hammock ; the cords of the 
 hammock are under tension, and this tension, ?^3§^^§Q 
 acting obliquely upward on all sides, constitutes ~-^-f= :^^^r^£3-iEf 
 the supporting force. Similarly, the floating of ^^^' ^^^• 
 
 the pin is explained by supposing the surface of the water to be 
 somewhat tough and in a state of tension. 
 
 The properties of the surfaces of liquids can be studied to 
 advantage by means of liquid films. It can be shown by a num- 
 ber of simple experiments that soap films and soap bubbles are in 
 
152 Some Properties of Matter 
 
 a state of tension^ (Laboratory Exercise 4). A soap bubble is 
 spherical, like a toy balloon, because the film of the bubble is in a 
 state of uniform tension throughout, just as the rubber of the bal- 
 loon is. Since a spherical surface is smaller than any other that 
 incloses an equal volume, the film of a bubble assumes this shape 
 in shrinking as much as possible. 
 
 A drop of any liquid, when freed from the distorting effect of 
 its weight, as in falling, is spherical. A drop of oil suspended in a 
 solution of alcohol and water of its own density is an excellent 
 illustration (Exp,). The spherical form of a drop is due neither 
 to the mutual gravitation of its particles nor to cohesion acting 
 throughout its mass, but to the tension of its surface. Its surface, 
 like the film of a bubble, contracts as much as possible, and when 
 thus contracted is spherical. 
 
 Laboratory Exercise 4. 
 
 196. Cause of Surface Tension. — Each molecule of a liquid 
 has hundreds, perhaps thousands, of neighbors near enough to 
 attract it by cohesion, all of which are inclosed within a spheri- 
 cal surface having the given molecule as a center and a radius 
 equal to the greatest distance to which cohesion can act (a dis- 
 tance of less than microscopic magnitude). Any molecule of a 
 liquid whose distance from the surface is greater than the radius 
 of such a sphere is attracted equally in all directions, since the 
 molecules within the range of cohesion are uniformly disturbed 
 around it. A molecule at the surface, however, is attracted inward, 
 but not outward ; and any molecule whose distance from the sur- 
 face is less than the range of cohesion is more strongly attracted 
 inward than outward, since the greater number of the molecules 
 that are near enough to attract it lie on the inside. The result is 
 that the molecules at the free surface of a liquid exert a strong 
 inward pressure, which tends to reduce the surface to the least 
 possible area, the effect being the same as if the liquid were 
 
 1 Recipe for a good soap solution : Put 2 oz. of Castile or palm-oil soap, shaved 
 thin, in i pt. of distilled or rain water. Shake, pour off the clear solution, add to it 
 \ pt. of glycerine, and stir. 
 
Surface Tension and Capillarity 153 
 
 inclosed in a stretched, elastic membrane. Thus the surface 
 tension of a liquid is due to the action of cohesion at its 
 surface. 
 
 197. Surface Tension of Different Liquids. — The surface ten- 
 sion of different liquids has been determined by experiment, and 
 it has been found to be greater for water than for any other liquid 
 except mercury ; hence the surface tension of water is diminished 
 by mixing any other substance with it. This is readily illustrated 
 by placing a drop of alcohol, ether, or oil on the surface of a tum- 
 bler of water beside a floating sliver of wood. The bit of wood is 
 quickly jerked away from the drop by the greater tension of the 
 pure water on the other side {Exp.). 
 
 198. Surface Viscosity. — The surface of most liquids becomes 
 more viscous than the interior mass after exposure to the air 
 for some time ; and it is to this superficial viscosity, rather 
 than to surface tension, that the strength of a liquid film is due. 
 Only very small bubbles can be formed on the surface of pure 
 water, and these quickly break, for the particles rapidly slip away 
 from the highest part of the bubble, leaving it too thin to hold. 
 The particles in a soap film move much more sluggishly, on 
 account of their greater viscosity ; hence a much longer time 
 elapses before any part of the film becomes so thin as to break, 
 although 'the surface tension of the film is less than that of pure 
 water. 
 
 199. Capillarity. — The combined action of surface tension and 
 adhesion, when a liquid and a solid are in contact, gives rise to a 
 class of phenomena, called capillary pheti07?iena because they are 
 most conspicuous in tubes of small bore (Latin capilliis, a hair). 
 Capillary action and capillarity are general terms for such 
 phenomena. 
 
 There are two types of cases to be considered, represented 
 respectively by water and mercury, each in contact with glass. 
 The surface of water in a clean glass vessel is turned sharply 
 up at the edge in a smooth curve (Fig. 117); while the sur- 
 face of mercury is curved downward at the edge (Fig. 118) 
 
154 Some Properties of Matter 
 
 (£x/.). In all cases where adhesion exceeds cohesion in the 
 liquid (cases in which the liquid wets the solid), the edge of the 
 liquid is drawn up against the surface of the solid ; in all cases 
 where cohesion in the liquid exceeds adhesion (cases in which the 
 solid is not wet by the liquid), the edge of the liquid is draw*?f'' 
 inward and away from the surface of the solid. 
 
 When small glass tubes of different bore are held vertically in 
 water, the water rises in each above the level in the vessel, and 
 stands higher in the tube of smaller bore {Exp.), The surface 
 
 i (f) if 
 
 mill 
 
 Fig. 117. 
 
 Fu;. lid. 
 
 of the water in the tube is concave (viewed from above), and is 
 curved throughout, forming a hemisphere, if the tube is not too 
 large. This curved surface, like that of a soap bubble, exerts a 
 pressure on the concave side, and thus partly sustains the pressure 
 of the air upon it. The water therefore rises in the tube till equi- 
 librium is restored. Atmospheric pressure, however, is not essential 
 to capillary action. The water would stand at the same level in 
 the tubes if the vessel and contents were placed in a vacuum ; for 
 the concave surface in the tube would exert the same upward force 
 as before, and would rise, carrying the column with it by cohesion. 
 The greater elevation of the water in the smaller tube is due to 
 
Surface Tension and Capillarity 155 
 
 the fact that the curvature of its surface is greater (radius of curva- 
 ture less) ; for it can be proved mathematically that the pressure 
 (per unit area) exerted toward the concave side by a curved sur- 
 face under a given tension is inversely proportional to the radius 
 of curvature of the surface. Thus if the diameter of the smaller 
 tube is one half that of the larger, the upward pressure exerted by 
 the water surface in it will be twice as great and will support a 
 column of water twice as high as that in the larger. 
 
 Mercury stands at a lower level in capillary tubes than it does in 
 the containing vessel, and the surface in the tubes is convex (Fig. 
 118) {Exp.). The pressure exerted by the curved surface is 
 toward the concave side, as with water ; but it causes depression 
 in this case, as the concave side is downward. The depression is 
 inversely proportional to the diameier of the tube, for the reasons 
 given in the case of capillary elevation. 
 
 Experiments with other liquids and tubes of other materials 
 would yield results agreeing with the cases considered, as expressed 
 in the following laws : — 
 
 I. If a liquid wets a capillary tube, its surface is concave and 
 it is drawn up ; if it does not wet the tube, its surface is convex and 
 it is depressed. 
 
 II. The elevation or the depression in a capillary tube is in- 
 versely proportional to the diameter of the tube. 
 
 200. Illustrations of Capillary Action. — The sensible pores of 
 solids serve as capillary tubes in absorbing liquids. The absorp- 
 tion of water by a sponge, of ink by blotting paper, and of coffee 
 by a lump of sugar, are familiar examples. The flame of a lamp is 
 fed by oil which is drawn up through the wick by capillary action. 
 In dry weather the moisture is drawn up from a depth of many 
 feet through the pores in the soil, and evaporates at the surface. 
 Cultivation of the soil increases the size of the pores, and conse- 
 quently checks the rise of water through the cultivated layer, thus 
 diminishing the loss by evaporation at the surface. 
 
 Capillary action would enable a short siphon having a capillary 
 bore to work in a vacuum. 
 
156 Some Properties of Matter 
 
 PROBLEMS 
 
 1. What is the distinction between a theory and a law? between a theory 
 and a fact ? Is a law a fact ? 
 
 2. What is the essential difference between a law of nature and a law 
 enacted by a legislative body? 
 
 3. If a gas is heated but not permitted to expand, how is the pressure that 
 it exerts affected ? Explain. 
 
 4. Capillary phenomena are sometimes said to be due to " capillary at- 
 traction." What b capillary attraction? 
 
 V. Properties due to Molecular Forces 
 
 201. Molecular forces give rise to different specific properties 
 in different substances. One of these, tenacity, has already been 
 considered (Art. 190). The list includes also elasticity, plasticity, 
 brittleness, malleability,- ductility, and viscosity. 
 
 202. Elasticity. — Elasticity is the property of matter in virtue 
 of which bodies resume their original form or volume when any 
 force that has altered their form or volume is removed. 
 
 Elasticity of volume is shown by recovery of volume after 
 compression. It is a general property of matter and is due to 
 molecular pressure. All fluids have perfect elasticity of volume ; 
 />. however great the compression, they always expand to their 
 original volume when the pressure is removed. Solids, however, 
 may be permanently diminished in volume to a slight extent by 
 the application of sufficient pressure. The rolling and stamping 
 to which silver is subjected in the process of coining causes a 
 decrease of volume amounting to about four per cent. 
 
 Elasticity of form is a specific property possessed only by cer- 
 tain solids. It is shown by recovery of form after distortion. An 
 elastic solid is able to recover from distortion whether caused by 
 compression, stretching, twisting, or bending. Rubber is a fa- 
 miliar example. Solids, such as soft putty or clay, which have no 
 power to recover from distortion, are called inelastic or plastic. 
 
 203. Limit of Perfect Elasticity. — An elastic solid has perfect 
 elasticity only for distortion within a certain limit, called the limit 
 
Properties due to Molecular Forces 157 
 
 of perfect elasticity, or generally, the limit of elasticity. When a 
 body is distorted beyond its limit of elasticity, it either breaks or 
 takes a permanent set, i.e: undergoes a permanent change of form 
 {Exp.). In the first case the solid is said to be brittle; in the 
 second case tough, malleable, or ductile, according to the manner 
 in which its form can be changed (Art. 207). It is a mistake to 
 regard brittle substances as inelastic. A long, slender strip of 
 glass or piece of glass tubing is quite flexible and springs back to 
 its former shape when released {Exp.) ; within its limit of elas- 
 ticity glass is perfectly elastic. 
 
 Substances differ widely in their limits of elasticity. A piece of 
 soft copper wire takes a permanent set before it is bent far, — its 
 limit of elasticity is small. A piece of steel wire can be bent many 
 times farther without a permanent change of form {Exp.). Rub- 
 ber is remarkable for its large limit of elasticity. There is no 
 limit to the elasticity of volume of fluids. However great the 
 pressure to which a fluid may be subjected, recovery of volume 
 is always complete when the pressure is removed. 
 
 204. Elastic Force ; Measure of Elasticity. — In common 
 speech we say that a body is very elastic or highly elastic if its 
 limit of elasticity is large (as rubber), or if it is highly compress- 
 ible (as gases). In scientific usage these terms have an alto- 
 gether different meaning. The elastic force of a body, or the 
 force with which it tends to recover from compression or distor- 
 tion, is the measure of its elasticity. 
 
 The elasticity of a gas is measured by the pressure (per unit 
 area) which it exerts upon the sides t>f the containing vessel, and 
 is increased by compression (Art. 44). The elasticity of liquids 
 is much greater than that of gases, and the elasticity of ivory, 
 glass, or steel is very great compared with that of rubber. 
 
 205. Elastic Impact. — When an elastic sphere (rubber, ivory, 
 steel, or hardwood ball, or glass marble) is dropped upon a smooth, 
 rigid surface, as a large, flat plate of stone or steel, it rebounds 
 nearly to the height from which it was dropped {Exp.). The 
 rebound is explained as follows : The force of the impact flattens 
 
158 Some Properties of Matter 
 
 the ball very slightly before it is stopped ; but, being elastic, it 
 instantly recovers its form, and, in doing so, // continues to press 
 against the plate^ much as a boy pushes against the ground in the 
 act of jumping, and with a similar result. 
 
 Although the ball recovers its form completely, the force with 
 which it does so is in all cases somewhat less than the distorting 
 force. If the recovery of force \yere complete, the ball would 
 rebound to the height from which it was dropped, but this is 
 impossible, as some of the energy of the ball is transformed by the 
 impact into heat and sound. The experiment proves the elas- 
 ticity of bodies that are often regarded as perfectly rigid. More- 
 over, by this test, the elasticity of glass is much more nearly 
 perfect than that of nibber. (How shown?) 
 
 206. Stress and Strain. — Any force or combination of forces 
 tending to change the shape or size of a body is called a stress. 
 A rubber band is stretched by two equal and opposite pulls, con- 
 stituting a tensiie stress. A tensile stress causes elongation. A 
 stress consisting of equal and opposite pressures causes compres- 
 sion. Any change of shape or size of a body, especially of a 
 solid, produced by the action of a stress, is called a strain. A 
 body in which a stress produces no appreciable strain is called 
 rigid ; but no body is perfectly rigid, i.e. absolutely unyielding to 
 stress. In the study of the mechanics of solids we have disre- 
 garded the strains produced by the forces under consideration. 
 
 207. Plasticity; Malleability and Ductility. — The specific 
 properties of bodies have no sharply defined limits ; on the con- 
 trary, they merge insensibly into one another in numerous instances, 
 and the names of such properties have, in consequence, a variable 
 meaning. This is the case with the term plasticity, which, in the 
 customary and narrow sense, is applied to bodies that can be 
 molded by moderate pressure into any desired form. Soft putty 
 and clay are typical examples. But gold and silver exhibit es- 
 sentially the same property under much greater pressure, but at 
 ordinary temperatures, when they receive the impression of the 
 die or mold in stamping coins ; and are therefore, in a broader 
 
Properties due to Molecular Forces 159 
 
 sense, called plastic. In this sense all elastic bodies that are not 
 brittle are plastic beyond their limit of elasticity. This form of 
 plasticity is, however, generally called malleability or ductility. 
 
 A substance is said to be malleable if it can be hammered or 
 rolled into sheets ; duciiky if it can be drawn out into the form of 
 a wire. In both cases ordinary temperatures are to be understood 
 unless otherwise stated. Thus we should say that glass is brittle, 
 not ductile ; but glass is very ductile when heated to redness 
 {Exp^, Gold, silver, and platinum are the most ductile metals, 
 gold the most malleable. Gold has been beaten into leaves so 
 thin that six hundred of them placed one upon another, would 
 have a thickness no greater than the paper upon which this is 
 printed. 
 
 208. Viscosity and Mobility. — A substance is said to be 
 viscous if its form changes more or less slowly under the action of 
 its own weight. Shoemaker's wax, pitch, and molasses candy 
 slightly warmed are examples of viscous solids. Such substances 
 are sometimes classed as liquids, since, when unsupported at the 
 sides, they slowly flatten out, or flow. Molasses and honey are 
 examples of viscous liquids. The term is commonly applied 
 only to those liquids that possess the property in a considerable 
 degree, and liquids that flow readily, as water and alcohol, are 
 called mobile. But all liquids and gases possess viscosity in 
 some degree. 
 
 Viscosity is that property, due to internal or molecular fric- 
 tion, in virtue of which liquids, gases, and some solids offer re- 
 sistance to an instantaneous change of their shape or of the 
 arrangement of their parts, although they offer no permanent re- 
 sistance to such change. The viscosity of water is shown by the 
 behavior of a bowl or tumbler of water that has been stirred till 
 it is rapidly rotating. If the motion of the different portions of 
 the water is made visible by means of sawdust or other floating 
 particles, it will be found that the rate of rotation steadily de- 
 creases, and at any instant is greatest at the center and least at 
 the sides of the vessel (^Exp.), The explanation is that friction 
 
i6o Some Properties of Matter 
 
 against the sides of the vessel retards the layer of water in 
 contact with it, and this layer in turn retards the more rapidly 
 moving layer next within. This retarding action — due to inter- 
 nal friction, as adjacent layers slip over one another in their un- 
 equal motion — extends throughout the entire mass, and finally 
 brings the whole to rest. 
 
 The dividing line between viscous liquids and plastic solids is 
 not clearly defined ; the two classes merge insensibly into each 
 other. Molasses candy, in cooling, passes continuously from the 
 stale of a viscous liquid to that of a plastic solid ; and, on cooling 
 further, becomes hard and brittle. 
 
 209. Hardness. — A body is said to be harder than another 
 if it can be used to scratch the latter but cannot be scratched by 
 it. Diamond is the hardest substance known. Pure metals are 
 softer than their alloys; hence gold and silver used for money 
 and jewelry are alloyed with copper to increase their hardness. 
 
 Steel is made very hard by sudden cooling after it has been 
 raised to a high temperature. The process is called tempering. 
 All cutting instruments are made of tempered steel. 
 
 PROBLEMS 
 
 1. The top of a river flows faster than the bottom, and the middle flows 
 faster than the sides. Why? 
 
 2. Write a list of all the general projierties of matter that have been con- 
 sidered in this or in previous chapters. Write a similar list of specific prop- 
 erties. 
 
CHAPTER VIII 
 
 HEAT 
 
 I. Heat and Temperature 
 
 210. The Caloric Theory. — Before the final acceptance of the 
 present theory of heat (Art. 187) about the middle of the last 
 century, heat was very generally supposed to be an invisible fluid 
 without weight, which could of itself pass from a hotter to a colder 
 body. This supposed fluid was called caloric (Latin calor^ heat). 
 With the overthrow of the theory, the word itself has become 
 obsolete ; but the root occurs in several words in common use, 
 among which are calorie, one of the heat units ; caloritnetry, the 
 art or process of measuring heat ; and calorimeter, a vessel in 
 which substances are placed in measuring their gain or loss of 
 heat. The use of these words does not imply any reference to 
 the caloric theory; but, unfortunately, the terms latetit heat 
 and radiant heat, which also owe their origin to the caloric 
 theory, are distinctly misleading, for neither is heat at all. 
 
 211. The Mechanical Theory of Heat: Historical. — " The first 
 prominent physicist who endeavored to overthrow the caloric 
 theory of heat was Benjamin Thompson" (1753-1814), a native 
 of Massachusetts. He is better known as Count Rumford, having 
 received that title from the Elector of Bavaria, whose service he 
 entered while still a young man. While engaged in the boring 
 of cannon for the Bavarian government, he was surprised at the 
 heat generated. " Whence comes this heat ? What is its nature ? 
 He arranged apparatus so that the heat generated by the friction 
 of a blunt steel borer raised the temperature of a quantity of 
 water. In his third experiment, water rose in one hour to 107° 
 
 161 
 
1 62 Heat 
 
 Fahrenheit; in one hour and a half to 142° ; *at the end of two 
 hours and thirty minutes the water actually boiled.' * It is difficult 
 to describe the surprise and astonishment,' says Rumford, ' ex- 
 pressed in the countenances of the bystanders, on seeing so large 
 a quantity of cold water (i8| lb.) heated, and actually made to 
 boil without any fire.' . . . The source of heat generated by fric- 
 tion * appeared evidently to be inexhaustible.' The reasoning 
 by which he concluded that heat was not matter, but was due to 
 motion, we can give only in part. He says, * It is hardly neces- 
 sary to add that anything which any insulated body, or system of 
 bodies, can continue to furnish without limitation, cannot possibly 
 be a material substance ; and it appears to me extremely difficult, 
 if not quite impossible, to form any distinct idea of anything ca- 
 pable of being excited and communicated in the manner in which 
 heat was excited and communicated in these experiments, except 
 it be motion.* 
 
 " Rumford 's conclusion regarding the nature of heat was vigor- 
 ously attacked by the calorists, but it was confirmed in 1 799 by 
 Sir Humphry Davy. By means of clockwork he rubbed two 
 pieces of ice against one another in the vacuum of an air pump. 
 Part of the ice was melted, although the temperature of the re- 
 ceiver was kept below the freezing point. From this he con- 
 cluded that friction causes vibration of the corpuscles of bodies, 
 and this vibration is heat." — Cajori's History of Physics. 
 
 Notwithstanding the work of Rumford and Davy, but few 
 physicists were convinced. In fact, the caloric theory was not 
 completely discredited until after J. P. Joule, an Englishman, 
 had proved, by a series of experiments extending over a period 
 of ten years (1840 to 1850), that the amount of heat which can be 
 generated by a given amount of mechanical work is invariable 
 (Art. 270). 
 
 212. Sources of Heat. — All other forms of energy are capable 
 of transformation into heat. Mechanical energy is transformed 
 into heat by friction and impact, as we have already seen, and 
 also by compression. Heat due to compression is appreciable 
 
Heat and Temperature 163 
 
 only in gases, and is familiar in the heating of a bicycle pump 
 when in use {Exp.). The heating is not due to the friction of 
 the piston, to any appreciable extent, but to the compression of 
 the air. The heat thus developed is, in fact, the exact equivalent 
 of the mechanical work done in compressing the air. 
 
 Chemical action is the most important source of heat, with the 
 exception of the sun. The burning of any substance is an exam- 
 ple. When a substance burns, its molecules are broken up, and 
 their parts (atoms) unite with oxygen from the air. New sub- 
 stances are thus formed, which are in most cases gases. It is not 
 surprising that, in this violent rearrangement of molecular struc- 
 ture, the motion of the molecules should be greatly increased. 
 Heat is also generated by other chemical processes. For exam- 
 ple, when water is poured on quicklime, the two unite chemically, 
 forming slaked lime, and the lime becomes very hot {Exp.). 
 Similarly, when strong sulphuric acid is poured into water, the 
 mixture becomes very hot {Exp.). 
 
 The transformation of other forms of energy into heat is con- 
 sidered in connection with other topics. 
 
 213. Temperature. — The relation between heat and tempera- 
 ture is similar to that between quantity of water and water level. 
 If two vessels containing water are in communication, we know 
 that the flow will be from the one in which the water stands at 
 the higher level to the other, whether the one or the other con- 
 tains the greater quantity of water. Similarly, the transfer of heat 
 between two bodies or between parts of the same body is always 
 from the hotter to the cooler, whether the quantity of heat in the 
 former be greater or less than that in the latter ; and this transfer- 
 ence will cease as soon as both are at the same temperature. 
 
 Temperature is therefore defined as the condition of bodies that 
 determines the direction in which the transfer of heat can take 
 place between them. Or, since the temperature of a body is 
 higher the greater the amount of heat it contains, temperature 
 may be defined as the ititeiisity or degree of heat (not the quantity 
 of heat). 
 
164 Heat 
 
 214. Temperature Sensations. — Our bodily sensations of heat 
 and cold afford direct but inexact and often misleading infor- 
 mation concerning the temperature of bodies ; they afford no 
 information whatever concerning the relative amount of heat in 
 different bodies (Art. 217). 
 
 PROBLEMS 
 
 1. What is ihe meaning of the adjective cold? of the noun cold? 
 
 2. A body {x>sscsses heat as lung as it is capable of becoming colder. Is 
 there any heat in ice? 
 
 3. (a) According to the theory of heat, what would be the molecular 
 condition of a body having no heat? {h) Why could not such a body be a gas? 
 
 4. Mention any familiar instances in which ccjual temperatures do not 
 cause equal temiierature sensations. 
 
 II. The Transmission of Heat 
 
 215. The transference of heat as heat from one place to 
 another takes place in two ways; namely, by conduction and by 
 convection. 
 
 Laboratory Exercise .?/. 
 
 216. Conduction. — Conduction is the transmission of heat from 
 hotter to colder parts of a body, or from a hotter to a colder body in 
 contact with it, without change in the relative positions of the parts 
 of the body. It is the only process by which heat travels in solids. 
 The heating of the farther end of a poker when one end is placed 
 in a fire, and the heating of the handle of a spoon placed in a cup 
 of hot tea are familiar examples. 
 
 The kinetic theory suggests a mental picture of the process of 
 heat conduction. When any part of a body is heated, its mole- 
 cules are set in more rapid vibration. These molecules jostle 
 their neighbors more violently, increasing the energy of their vibra- 
 tion. The disturbance thus spreads throughout the body without 
 change in the relative positions of the molecules themselves. In 
 conduction, therefore, molecular energy is transmitted without the 
 transmission of matter. 
 
 Substances differ widely in their conductivity, i.e. their power of 
 
The Transmission of Heat 
 
 165 
 
 transmitting heat by conduction. The metals are the best con- 
 ductors of heat ; other solids, with 
 few exceptions, are better conduct- 
 ors than liquids. Liquids, with 
 the exception of mercury and mol- 
 ten metals, are very poor conduct- 
 ors. Water can be boiled at the 
 top of a test tube for several min- 
 utes, while the greater part of it 
 remains cold (Fig. 119) (£x/>.). 
 But wood and paper, especially the 
 latter, are much poorer conductors 
 than water (see table below). Gases are practically nonconduct- 
 ors. In testing the conductivity of liquids and gases they must 
 be heated at the top to prevent currents (Art. 218). 
 
 The following table gives the conductivities of various substances 
 referred to silver as the standard : — 
 
 Fig. 119. 
 
 Tad/e of Conductivities for Heat. (^Approximations^ 
 
 Silver icxj 
 
 Copper 74 
 
 Brass 27 
 
 Iron 12 
 
 Lead 8.5 
 
 Mercury 1.35 
 
 Ice 0.20 
 
 Glass 0.20 
 
 Marble 0.15 
 
 Water 0.14 . 
 
 Wood 0.04 
 
 Writing paper 0.012 
 
 Wool 0.009 
 
 Air 0.005 
 
 217. Illustrations of Good and Poor Conductivity. — On a cold 
 morning the ^oox feels colder to the bare feet than the carpet or a 
 rug, and water that has been standing in the room over night feels 
 colder to the face than the air. These and other objects in the 
 room, which seem to be unequally cold, are all at the same tem- 
 perature as the air. The difference in the sensations is largely due 
 to the difference in the conductivities of the substances touched. 
 If the body touched is a good conductor, heat is rapidly conducted 
 to all parts of it from the hand \ and this continue^ a§ long as the 
 
1 66 Heat 
 
 hand remains in contact with it or until it becomes warm through- 
 out. This rapid and continued loss of heat makes the hand cold. 
 If the body touched is a poor conductor, but little heat is lost from 
 the hand in warming the part of it that is touched. Similarly, if 
 hot substances at the same temperature but of different conduc- 
 tivities are touched, the best conductors feel the hottest because 
 they conduct heat to the hand most rapidly, and hence make the 
 hand hotter than the poorer conductors do. (The temperature 
 sensation caused by a body depends also in part upon another 
 property, called specific heat, which will be studied later.) 
 
 The low conducting power of air is utilized in refrigerators and 
 ice houses, which have double walls filled between with charcoal, 
 sawdust, straw, or other loose, badly conducting material, which 
 hinders the circulation of the air. The warmth of fur, feathers, and 
 wool is partly due to the air entangled in them. 
 
 PROBLEMS 
 
 1. What is the advantage of the poor conductivity of wood in using 
 matches? 
 
 2. An overcoat is said to " keep out the cold." What is it that it really 
 does? 
 
 3. Why is woolen clothing warmer than cotton or linen? 
 
 4. Would it l>e better to wrap a piece of ice in a woolen or a cotton 
 cloth to keep it from melting? 
 
 5. What is the advantage of having the bottoms of tin teakettles and 
 boilers made of copper? 
 
 Laboratory Exercise 28. 
 
 218. Convection. — The transmission of heat in a liquid or a 
 gas by currents due to unequal temperatures in its different parts 
 is called convection , and the currents are called convection cur- 
 rents. 
 
 In heating liquids, the heat is applied to the lx)ttom of the 
 vessel. The liquid at the bottom receives heat from the vessel by 
 conduction and expands, thus becoming less dense. It is then 
 displaced by the denser liquid at the top, producing convection 
 currents. 
 
Radiation 167 
 
 Convection currents in air and other gases are due to the same 
 cause. The strong ascending current above a bonfire is indicated 
 by the leaping of the flames and the rapid rise of sparks and 
 smoke. The fire is fed by inward-flowing currents near the 
 ground. They occupy much more space than the ascending 
 current, and hence move more slowly and are less noticeable. 
 
 PROBLEMS 
 
 1. Would convection currents be caused by heating a liquid at the top? 
 by cooling it at the top? l)y cooling it at the bottom? 
 
 2. What would be the general direction of convection currents in a room 
 heated by a stove at one end of the room? Would the air be warmer near 
 the floor or near the ceiling? 
 
 3. What convection currents are set up when a door is left open between a 
 warm and a cold room ? 
 
 4. Why do openings at the top and bottom of a window provide better 
 ventilation than a single opening of the same total area at either top or 
 bottom? 
 
 5. A fire in a fireplace provides excellent ventilation for a room. Explain. 
 
 6. Docs an open fireplace provide as good ventilation whether there is a 
 fire in it or not? 
 
 7. How is the flame of a lamp provided with the constant supply of 
 oxygen necessary for combustion? 
 
 III. Radiation 
 
 Laboratory Exercise 2^ {l^ a and <5, and II, dr and b), 
 219. An Experiment on Radiation. — When the hand is held 
 close besiWe a hot flame, as that of a Bunsen burner, the side of the 
 hand turned toward the flame becomes hot. The hand is not 
 heated by convection, for it is in the path of the currents of cold 
 air moving toward the flame. When any object, as a sheet of 
 paper, is placed as a screen between the hand and the flame, the 
 hand instantly ceases to feel the heat, which shows that it could 
 not have been heated by conduction, for three reasons: (i) be- 
 cause the screen is at least as good a conductor as the air ; (2) if 
 the process were conduction, the air must have been at least as 
 warm as the hand and it could not instantly become cold on inter- 
 
1 68 Heat 
 
 posing the screen ; and (3) both sides of the hand would have 
 been warmed, for convection currents would prevent any consider- 
 able difference in the temperature of the air on the two sides. 
 The conclusion is therefore certain that, when the hand is beside 
 the flame with nothing interposed between, the side turned toward 
 the flame becomes much hotter than the air at that distance, and 
 that it is heated neither by conduction nor convection. 
 
 The conditions of this experiment are repeated on a large scale 
 when we stand near a bonfire in cold weather. Most people know 
 by experience of the last situation what it means to " roast on one 
 side and freeze on the other." 
 
 220. The Transformation of Heat into Radiant Energy and of 
 Radiant Energy into Heat. — All hot bodies lose heat by a pro- 
 cess called ratiiation, which is independent of conduction and con- 
 vection and is totally different from either. But radiation is not a 
 process by which heat is transmitted ; for the heat is transformed 
 at the radiating body into another form of energy, called radiant 
 energy^ and is transmitted as such until it meets some body capa- 
 ble of transforming it back into heat. 
 
 Tlie transformation of radiant energy into heat is called absorp- 
 tion. Absorption takes place at the surface of some bodies, as at 
 the surface of the hand in the preceding experiment ; in other 
 cases it takes place within the body through which the radiant 
 energy is passing. A body is heated only by the part of the radiant 
 energy that it absorbs. Air absorbs comparatively little, hence is 
 only slightly heated by radiation. The transformation of radiant 
 energy into heat by absorption may l>e compared to the transfor- 
 mation of the kinetic energy of a flying bullet into heat by impact 
 against a steel target. Radiant energy is as distinctly dififerent 
 from heat as is the kinetic energy of the bullet. 
 
 221. Luminous and Nonluminous Radiation. — Radiant energy 
 is often called radiation ; thus radiation may mean either a form 
 of energy or the process by which that energy is transmitted. 
 Radiant energy includes luminous radiation, or lights as well as the 
 Donluminous or dark radiation previously considered. The dis- 
 
Radiation 169 
 
 tinction between light and nonluminous radiation is a physiologi- 
 cal rather than a physical one, and is due to the fact that the 
 optic nerve is sensitive to the one and not to the other. The two 
 kinds of radiation are the same form of energy. 
 
 A hot body, as a piece of iron, gives out only dark radiation 
 below a certain temperature ; at " red heat " or above it radiates 
 light also. Light is due to the transformation of heat into radiant 
 energy at sufficiently high temperatures ; and, like dark radiation, 
 it is again transformed into heat by absorption. 
 
 The energy of dark radiation is greater than that of light ; hence 
 it causes greater heating when absorbed. Before heat was recog- 
 nized as a form of energy (Art. 210) it was supposed that dark 
 radiation was a form of heat and was essentially different from 
 light ; it was therefore called " radiant heat," and its transmission 
 was called the " radiation of heat." These terms are still com- 
 mon ; but their use should be avoided, at least by the beginner. 
 
 222. Radiant Energy can be transmitted in a Vacuum. — 
 Neither air nor any other form of matter that we have yet con- 
 sidered is necessary for the transmission of radiation. This is 
 evident from the fact that we receive radiant energy from the sun 
 over a distance of 93,000,000 miles ; through all of which space, 
 till the earth's atmosphere is reached, there is what we call a 
 perfect vacuum. The incandescent electric light affords another 
 illustration on a small scale. When the filament of the lamp is 
 heated by the electric current, it sends out both luminous and 
 nonluminous radiation, yet within the bulb there is a very nearly 
 perfect vacuum. 
 
 Hitherto we have found energy only in connection with matter ; 
 in fact, matter is often called " the vehicle of energy." How, 
 then, does energy travel in a vacuum ? The question is answered 
 by the theory of radiant energy ; which, together with the laws 
 of radiation, will be considered under the subject of light. One 
 law of radiation was illustrated when the screen was interposed 
 between the Bunsen flame and the hand : the radiation did not 
 pass round the screen. Radiation is transmitted along a straight 
 
170 
 
 Heat 
 
 path in a vacuum and in all homogeneous substances capable of 
 transmitting it. 
 
 223. The Radiometer. — The radiometer (Fig. 1 20) consists of 
 four light vanes of mica or aluminum attached to a vertical axis, 
 and inclosed in a glass bulb containing a small quantity of air 
 
 under very low pressure. One side of each 
 vane is bright, the other is coated with 
 lampblack. When the instrument is 
 placed in the sunshine or in the path of 
 other radiation, the vanes rotate with their 
 bright side in advance. 
 
 The rotation of the vanes is explained 
 as follows : The black surfaces absorb 
 more radiation than the bright, and hence 
 are warmer. The molecules of the rare- 
 fied air that strike the blackened surfaces 
 are heated, and rebound with greater veloc- 
 ity than those that strike the bright sides. 
 This causes a greater pressure upon the 
 black side of each vane than upon the 
 other ; hence the rotation. If the air in the bulb were not 
 highly rarefied, there would be no rotation, for the collisions 
 among the molecules would be so frequent as to equalize the pres- 
 sures throughout the bulb. 
 
 The rate of rotation of the vanes serves as a rough measure of 
 the energy of the radiation falling upon them ; hence the instru- 
 ment is very useful in the study of radiation (see Lab. Ex. 29). 
 
 224. Effects of Matter upon Radiation. — When radiant energy 
 falls upon any substance, it may be (i) absorbed at the surface, 
 (2) reflected by the surface, (3) transmitted through the sub- 
 stance, or (4) wholly or partly absorbed by the substance during 
 transmission. Generally two and often three of these effects 
 occur simultaneously with different portions of the radiation. The 
 study of these effects will be continued under the subject of light. 
 
 225. Reflection and Absorption at Surfaces. — In general, the 
 
 Fig. laa 
 
Radiation 171 
 
 surfaces of substances that transmit no radiation reflect part and 
 absorb the remainder ; they are heated only by the absorbed 
 radiation. Lampblack is the best absorber known ; it absorbs 
 practically all of the radiation that falls upon it, both luminous and 
 nonluminous. Any polished metal reflects much the greater part 
 of all radiation and absorbs the remainder. A piece of tin coated 
 with lampblack or painted black on the side turned toward a flame 
 will therefore become hot, while a piece of bright tin in the same 
 situation will be only slightly warmed. A white surface reflects 
 nearly all luminous radiation that falls upon it, a black surface re- 
 flects almost none, a colored surface reflects part and absorbs part. 
 
 226. Selective Absorption. — Substances that transmit d^irk 
 radiation, with but little absorption if any, are called diatherma- 
 nous (Greek therme, heat) ; those that transmit little or none are 
 called athennanous. The terms have the same meaning with 
 respect to dark radiation that transparent and opaque have with 
 respect to light. 
 
 The power of most substances to transmit radiation depends 
 very largely upon the temperature of the source of the radiation. 
 Some substances that transmit light also transinit dark radiation,^ 
 others do not. Clear glass transmits light and also a considerable 
 portion of the radiation from bodies nearly red hot, but absorbs 
 all or nearly all of the radiation from colder bodies. Thus solar 
 radiation enters a sunny room in large quantities through the 
 windows, and is absorbed by the objects upon which it falls. But, 
 as the radiation from these bodies cannot penetrate glass, the solar 
 energy is entrapped in the room, which may thus become con- 
 siderably warmer than the air outside. This explains the accu- 
 mulation of heat in greenhouses. Rock salt transmits all radiation, 
 being highly transparent and diathermanous. A solution of iodine 
 in carbon disulphide is perfectly opaque but diathermanous ; 
 water in its three states is transparent but highly athermanous 
 (Lab. Ex. 29). 
 
 The unequal absorption of luminous and dark radiation by the 
 same substance is called selective absorption. 
 
172 Heat 
 
 227. Relation of Radiating and Absorbing Powers. — If two 
 metal vessels of the same size and material, one highly polished 
 and the other coated on the outside with lampblack, are filled 
 with equal quantities of hot water at the same temperature and are 
 allowed to stand for some minutes, it will be found that the tem- 
 perature of the water in the blackened vessel is considerably lower 
 than that in the other {Exp.). The difference in the rate of cool- 
 ing is due to the more rapid radiation from the blackened surface. 
 The experiment illustrates the fact that good absorbers of radiation 
 are also good radiators, and poor absorbers poor radiators. 
 
 228. The Heating of the Atmosphere. — It is well known that 
 the atmosphere is colder at higher than at lower altitudes. Aero- 
 nauts always find intense cold at altitudes above three or four 
 miles, and the tops of high mountains are covered with perpetual 
 snow. The low temperature of high altitudes is due to the fact 
 that the dry and rarefied air absorbs but little of the enormous 
 quantity of radiant energy that passes through it. 
 
 As the radiation approaches the earth, the rate of absorption 
 rapidly increases, principally on account of the greater amount of 
 water vapor ; for experiments have shown that the absorbing 
 power of air containing the average amount of water vapor is 
 seventy-two times as great as that of perfectly dry air of the same 
 density. The absorption at lower levels is further increased by the 
 dust particles in the air. 
 
 But notwithstanding the loss by absorption on the way through 
 the atmosphere, it is estimated that from two thirds to three fourths 
 of the solar radiation reaches the earth. Much of this is absorbed 
 by the surface of the land and the ocean ; the remainder is re- 
 flected. The reflected radiation is partly absorbed on its way out 
 through the atmosphere again. The absorbed radiation warms the 
 surface of the land, and this in turn warms the air in contact with 
 it. This is especially noticeable on a hot day in summer, when, 
 if there is no wind, the air close to the ground is many degrees 
 warmer than at a height of a few feet. The heating of the air at 
 the bottom causes convection currents (winds), by which the heat 
 
Temperature and Expansion 173 
 
 is carried to considerable altitudes; but the temperature is, of 
 course, highest at the source of the heat^ i.e. at the earth's sur- 
 face. 
 
 The earth is cooled at night principally by radiation. The loss 
 is rapid on clear nights, especially when the atmosphere is very 
 dry ; but is checked in a large degree by moisture, especially by 
 clouds, which absorb the radiation before it has passed through 
 them, thus serving as a blanket to the earth. Hence clear nights 
 are, as a rule, the coldest At high altitudes, where there is but 
 little hindrance to radiation either by day or by night, sheltered 
 valleys are quickly warmed in summer by the early morning sun- 
 shine ; and a sudden chill immediately follows the disappearance 
 of the sun in the evening, the nights being often cold enough for 
 frost. 
 
 Thus we see that the atmosphere, or rather the moisture in it, 
 performs an indispensable function in moderating the intensity of 
 solar radiation by day and retaining the heat by night. 
 
 PROBLEMS 
 
 1. Why does snow melt more quickly when covered with a thin layer of 
 earth? 
 
 2. Why is Hght-colored clothing more comfortable in summer than black? 
 
 3. Why is the difference between the temperature in the sunlight and in 
 the shade greater upon the top of a mountain than at a low elevation? 
 
 4. Why must those who climb snow-covered mountains take especial care 
 to protect their faces?' 
 
 IV. Measurement of Temperature and Expansion 
 
 229. Measurement of Temperature. — Any instrument for meas- 
 uring temperatures is called a thermometer. In most forms of 
 thermometers the effect of heat in changing the volume of some 
 substance is utilized. Solids are rarely used, as their expansion is 
 small and they are otherwise inconvenient. It is important that 
 the substance chosen should have a uniform expansion ; i.e. equal 
 
174 Heat 
 
 quantities of heat should, cause equal increases of volume at all 
 temperatures. 
 
 Mercury fulfills this condition better than any other liquid, and 
 has the further advantage of remaining a liquid through a very 
 wide range of temperature. The apparent expansion of mercury 
 in a glass vessel (/.<•. the difference between the expansion of 
 mercury and glass) has therefore been adopted as the standard. 
 For temperatures below the freezing point of mercury, alcohol 
 thermometers are used, the freezing point of alcohol being — i3o°C. 
 The air thermometer, in which the expansion of air is utilized, is 
 adopted as the standard in the most accurate scientific work ; and 
 it can be constructed of such form as to be of practical use in meas- 
 uring temperatures above the boiling point of mercury, such as 
 the temperatures of furnaces. 
 
 230. The Mercury Thermometer. — The mercury thermometer 
 consists essentially of a capillary glass tube, called the stem, ter- 
 minating in a bulb (Fig. 122). The bulb and a part of the stem 
 are filled with mercury, and the expansion is measured by a scale 
 engraved upon the stem or attached to it. In making a thermom- 
 eter the mercury is heated to drive out all the air before the stem 
 is sealed at the top ; hence the space in the tube above the mer- 
 cury is a vacuum. 
 
 231. Determination of the Fixed Points. — Probably no two 
 thermometers have bulbs of exactly the same capacity and tubes 
 of exactly the same bore ; hence the readings of different ther- 
 mometers would be entirely inconsistent with one another if they 
 were provided with scales of equal length. The correct position 
 and dimensions of the scale must therefore be determined sepa- 
 rately for every thermometer. 
 
 The first step in this process is to determine the " fixed points," 
 called \k\^ freezing point diViA the boiling point. T\\^ freezing point 
 is the temperature at which pure water freezes ; but since this is 
 exactly the same as the temperature at which ice melts, whatever 
 the surrounding temperature may be, it is most conveniently 
 found by inserting the bulb of the thermometer in a dish of melt- 
 
Temperature and Expansion 
 
 ^75 
 
 ing snow or crushed ice. The ice is packed about the bulb and 
 stem, leaving the mercury just visible above it ; and a mark is made 
 on the stem at the top of the mercury column after it comes to rest. 
 
 The boiling point is the temperature at which pure water boils 
 under a pressure of one atmos- 
 phere. But, since the tempera- 
 ture of the water is subject to 
 appreciable changes from various 
 causes while that of the escaping 
 steam is constant, the thermom- 
 eter is adjusted so as to be sur- 
 rounded by the steam, as nearly as 
 possible to the top of the mercury 
 in the stem, and is not permitted 
 to touch the water (Fig. 121). 
 Since even a slight change in the 
 atmospheric pressure causes an 
 appreciable change in the tem- 
 perature at which water boils 
 (Art. 262), a correction must be applied to the observed height 
 of the mercury in the stem if tlie barometric pressure is not 
 76 cm. when the boihng point is determined. 
 
 232. Centigrade and Fahrenheit Scales. — The distance between 
 the fixed points is divided into equal parts called degrees. In the 
 Centigrade scale the number of these divisions is 100, the freezing 
 point being marked 0° and the boiling point 100°. The Centi- 
 grade thermometer is almost exclusively used in scientific work. 
 All temperatures referred to in this book are expressed in the Cen- 
 tigrade scale unless otherwise indicated. In the Fahrenheit- scale 
 the freezing point is marked 32° and the boiling point 212°, the 
 interval between them being 180°. The Fahrenheit scale is the 
 one in general use in this country. The scale of a thermometer 
 may be extended to any desired distance beyond the fixed points. 
 Temperatures below zero on either scale are indicated by the nega- 
 tive sign, as — 15° C. 
 
 Fig. lai. 
 
 or THr 
 
176 
 
 Heat 
 
 c 
 
 00 
 
 
 F 
 
 212 
 
 
 
 
 3? 
 
 
 
 ■a 
 
 FlO. 122. 
 
 Since the interval between the fixed points is loo Centigrade 
 degrees or i8o Fahrenheit degrees, it follows 
 that — 
 
 I Centigrade degree = f Fahrenheit degrees, 
 and 
 
 I Fahrenheit degree = f Centigrade degrees. 
 In changing a reading on either scale to the 
 equivalent reading on the other, allowance must 
 be made for the difference in the zero points. 
 Example : 50° C. means 50 Centigrade degrees 
 above the freezing point. This is equal to 50 x |, 
 or 90 Fahrenheit degrees a dove the freezing pointy 
 or to 122° F. 
 
 233. Linear Expansion of Solids. — With few 
 exceptions, none of which are important, solids expand when 
 heated and contract when cooled. Tiie total expansion or con- 
 traction of any body depends upon its size and material as well 
 as upon the change in temperature. The expansion of a solid 
 takes place, of course, in its three dimensions ; but, in most cases, 
 it is important only in the direction of its length. F^xpansion, 
 when considered in one direction only, is called linear expansion. 
 The expansion of a solid per unit of length when its tempera- 
 ture rises one degree is called its coefficient of linear expansion. 
 For example, the coefficient of linear expansion of oak is .000006, 
 which means that each centimeter of its length increases to 
 1.000006 cm. with a rise of temperature of one degree Centi- 
 grade. The coefficient of steel is .000012 — just twice that of 
 oak ; hence the expansion of a piece of steel for any change 
 of temperature is twice as great as that of a piece of oak of the 
 same dimensions for the same change of temperature. 
 
 The expansion of solids is so slight that some special device 
 must be employed in order to measure with any degree of accu- 
 racy the expansion of even a long rod for a considerable rise of 
 temperature. The methods by which this is done are best studied 
 in the laboratory. The computation of the coefficient from the 
 
Temperature and Expansion 177 
 
 data thus obtained is illustrated by the following example : A brass 
 rod 90 cm. long expands .13 cm. when heated from 23° to 100°. 
 Find its coefficient of linear expansion. The rise of temperature 
 is 77 degrees; hence, assuming the expansion to be uniform^ the 
 
 expansion of the rod for a rise of temperature of i degree is — 
 cm., and the expansion of i cm. for a rise of temperature of i de- 
 gree is — '—^ — , or .0000188 cm. 
 ^ 77 X 90' 
 
 The coefficient of linear expansion may equally well be re- 
 garded as the ratio of the whole expansion to the whole length for 
 a rise of temperature of one degree. 
 
 Coefficients of Linear Expansion 
 
 Zinc o.(xxx>294 
 
 Lead 0.0000286 
 
 Aluminum 0.000023 
 
 Brass 0.0000188 
 
 Copper ...... 0.0000172 
 
 Iron and Steel .... o.ooooi 22 
 
 Platinum 0.0000088 
 
 Glass 0.0000086 
 
 Wood, Dak 0.000006 
 
 Wood, Fir 0.0000035 
 
 Laboratory Exercise JO. 
 
 234. Effects and Applications of Expansion. — The force that 
 a body can exert in expanding or contracting with change of 
 temperature is equal to the force required to expand or compress 
 it to the same extent by mechanical means. This force is enor- 
 mous, and, under most circumstances, practically irresistible. If 
 a bar of malleable iron one square inch in cross-section were 
 placed between fixed supports, so as to make expansion impossible, 
 and its temperature then raised 40°, it would exert a pressure of 
 about five tons against the supports. 
 
 The rails of tracks are laid with a small space between their 
 ends, which provides room for expansion. A long steel bridge 
 changes in length several inches between winter and summer, 
 opportunity for this change being afforded by an expansion joint. 
 The tire of a wagon wheel is made just large enough to go on 
 
178 Heat 
 
 when hot ; it shrinks upon the wheel in cooling, making a very 
 tight fit. For a similar reason red-hot rivets are used in joining 
 the steel plates of tanks and boilers. A wooden rod is better than 
 one of metal for the pendulum of a clock, since its coefficient of 
 expansion is less. 
 
 235. The Expansion of Liquids. — In considering the expan- 
 sion of liquids and gases it is increase of volume, or cubical expan- 
 sion^ with which we are concerned. The coefficient of cubical 
 expansion of a solid or a liquid is its expansion per unit volume 
 for a rise of temperature of one degree. 
 
 As usually contrived, experiments on the expansion of liquids 
 give their apparent expansion, i.e, the difference between their 
 true expansion and the expansion of the containing vessel. The 
 tnie expansion of a liquid is the sum of its apparent expansion 
 and the cubical expansion of the material of the containing vessel. 
 In general, a liquid expands more rapidly as the temperature 
 approaches its boiling point. The following table gives the true 
 average expansion of a few liquids : — 
 
 Coefficients of Cubical Expansion 
 
 Ether 0.0015 
 
 Alcohol (5° to 6**) . . . . 0.00105 
 Alcohol (49° to 50**) . . . 0.00122 
 
 Acetic acid 0.00105 
 
 Petroleum 0.0009 
 
 Olive oil 0.0008 
 
 Turpentine 0.0007 
 
 Glycerine 0.0005 
 
 Water (5° to 6°) . . . . 0.000022 
 Water (49° to 50°) . . . 0,00046 
 Water (99° to 100") . . . 0.00076 
 Mercury 0.00018 
 
 It can be shown that the coefficient of cubical expansion of a 
 solid is three times its coefficient of linear expansion ; hence by 
 multiplying the values given in the table under solids by three, a 
 comparison can be made between the expansion of liquids and 
 solids. It will be found that the expansion of liquids, though 
 small, is considerably greater than that of solids. The total 
 expansion of water between 4° and 100° is a little over four per 
 cent. 
 
 Laboratory Exercise ji. 
 
Temperature and Expansion 179 
 
 236. Importance of the Irregular Expansion of Water. — The 
 expansion of water is curiously irregular. It contracts as its tem- 
 perature rises from 0° to 4° ; when heated beyond this point it 
 begins to expand, at first very slowly, then more and more rapidly 
 (see table). Hence the density of water is greatest at 4° C. 
 (about 39° F.). 
 
 This behavior of water is of the greatest importance in the 
 economy of nature. In winter the waters of lakes lose heat at 
 the surface by contact with the cold air and by radiation. As the 
 water at the surface cools, it becomes denser and sinks, displacing 
 the warmer water at the bottom. These convection currents con- 
 tinue until the water is cooled throughout to a temperature of 4° ; 
 beyond this point the water at the surface expands as it cools 
 and remains at the top. Thus the water at the surface freezes 
 while that below remains at 4°, even in the most severe winters, 
 — a temperature at which fishes and other inhabitants of the 
 waters are not destroyed. 
 
 Laboratory Exercise J2. 
 
 237. Expansion of Gases. — The effect of heat upon the vol- 
 ume of gases was first accurately investigated by the French physi- 
 cist, Jacques Charles, who discovered the law that bears his name. 
 
 Law of Charles : The volume of any gas increases under constant 
 pressure by -^-^ of its volume at zero (Centigrade) for each rise of 
 temperature of one degree. 
 
 The fraction ^|^, or .003665, is, according to the law, the co- 
 efficient of cubical expansion for all gases at all temperatures, 
 under any constant pressure. I^ter and more accurate experi- 
 ments have shown that this law, like that of Boyle (Art. 47), is 
 only approximately true ; though very nearly so indeed, unless the 
 gas is near the temperature at which it liquefies. 
 
 It can be shown by experiment or proven from the laws of 
 Boyle and Charles that, when a gas is heated without being per- 
 mitted to expand (volume constant), its pressure increases at the 
 same rate as the volume does when it is heated under constant 
 pressure. 
 
i8o Heat 
 
 It is instructive to contrast the identical behavior of all gases as 
 expressed by the laws of Boyle and Charles with the very marked 
 individual differences exhibited by solids and liquids in their 
 relation to changes of pressure and temperature. The gaseous 
 state is evidently very much simpler than the other two ; which is 
 explained by the fact that the molecules of a gas are separated 
 beyond the range of cohesion, 
 
 238. Absolute Temperature and Absolute Zero. — Let ?'„ be the 
 volume of a body of gas at o° C, and z^, its volume at any other 
 temperature /| under the same pressure. 
 
 The increase in volume is 7'i — rv. and this increase is — ^ of 
 the volume at 0** (law of Charles) ; that is, — ^^^ 
 
 From which Ti = ?'„( i H — ~ ). (i) 
 
 \ 273/ 
 
 Similarly, if r, be the volume of the gas at temperature /j, under 
 the same pressure, then — 
 
 .,=..(.+-4^). (.) 
 
 Dividing the members of equation (i) by the corresponding 
 members of equation (2), we have — 
 
 gi_ 273 
 
 273 
 
 which reduces to ^ = "^ ^^44 ' (s) 
 
 Vt 2734-/2 ^^^ 
 
 The relation expressed by equation (3) has led to the adoption 
 of a temperature scale whose degrees are the same as those of the 
 Centigrade scale but whose zero is at — 273° C. This scale of 
 temperature is called the absolute scale, and its zero the absolute 
 zero. The freezing point is 273° Abs. and the boiling point 373° 
 
Temperature and Expansion i8i 
 
 Abs. Any temperature on the Centigrade scale is changed to the 
 absolute scale by adding 273. If we let T denote temperatures 
 on the absolute scale, then 71 =273 + /!, and 7^2= 273 + 4, and 
 equation (3) becomes — 
 
 M- « 
 
 Expressed in words the meaning of this equation is : Under 
 constant pressure the volume of any body of gas is proportional to 
 its absolute temperature. This is but another (and the simplest) 
 way of stating the law of Charles. If this law held for all* tem- 
 peratures, it is evident that at absolute zero the volume of any 
 body of gas would be zero ; but, as before stated, the law fails to 
 express the behavior of gases when near the point of condensation, 
 and all licjuefy and even solidify before reaching absolute zero. 
 When near the point of condensation, the decrease of volume for 
 a given fall of temperature is less than that indicated by the law. 
 
 By reasoning based on the relation of heat to mechanical 
 energy, it is proved that the absolute zero is what its name 
 indicates ; namely, the temperature at which a body would pos- 
 sess no molecular kinetic energy, or no heat, — the molecules 
 would be at rest. No substance has yet been cooled to absolute 
 zero; but the temperature has been closely approached by the 
 evaporation of hydrogen after it has been liquefied (Art. 268). 
 By this means hydrogen has been cooled to a temperature 
 estimated at — 259°C. or 14° Abs., at which temperature it is 
 frozen. Air freezes at a considerably higher temperature and 
 boils at — 191° C. or 82° Abs. 
 
 PROBLEMS 
 
 1. (rt) The reading of a thermometer gives the temperaturt of the ther- 
 mometer. On what grounds do we assume that the reading of a thermometer 
 in a liquid gives the temperature of the liquid? (^) Why do we not take the 
 reading immediately on inserting a thermometer in a liquid to determine its 
 temperature? 
 
 2. The reading of a barometer is 76 cm. on a certain day when its tem- 
 perature {i.e. the temperature of the mercury in the barometer) is 0°. What 
 
1 82 Heat 
 
 would have been the reading of the barometer if its temperature had been 
 20^? 
 
 3. In accurate work the reading of the barometer must be " corrected for 
 temperature " ; 1^. the true height is taken as the heij^ht at which it 
 would stand if the temperature of the mercury were o*'. A barometer read- 
 ing is 75.6 cm. at a temperature of 22*^ ; find the true or corrected height. 
 
 4. The thinner a glass tumbler is, the less likely is it to break when hot 
 water is |)Oured into it. Why? 
 
 5. Why cannot an air thermometer be used for measuring the lowest 
 attainable temperatures? 
 
 6. What will be the volume, at 75", o( a body of air which, under the same 
 pressure, has a volume of 250 ccm. at 20*^ ? 
 
 SlUGESTlON. — Use equation (3) or (4) above. 
 
 7. A body of gas at 10*' and a pressure of one atmosphere is inclosed 
 in a vessel and heated to 300*^. What is the pressure at that temperature, 
 none of the gas being allowed to escape? 
 
 8. A quantity of gas is found to have a volume of 800 ccm. at 20" under 
 atmospheric pressure when the barometer reads 75 cm. What would be the 
 volume of the gas at 0° and a pressure of one atmosphere? 
 
 Si'G(;estion. — Find the volume at the given temperature and 76 cm. 
 pressure (Boyle's law), and from this the volume at 0° under the latter 
 pressure (law of Charles). 
 
 V. Calorimetry: Specific Heat 
 
 The Heat Unit. — Heat being a form of energy, it can be 
 measured in terms of any of the units by which mechanical energy 
 is measured (ft.-lb., etc.) ; they are not used, however, as there 
 are more convenient units for the purpose. Two heat tinits are in 
 common use : one, the calorie^ is the amount of heat required to 
 raise the temperature of one gram of water one degree Centi- 
 grade ; the other is the amount of heat required to raise the tem- 
 perature of 'one pound of water one degree Fahrenheit. The 
 calorie is almost exclusively used in scientific work and is the only 
 heat unit that is used in this book. 
 
 Example. — How much heat is required to raise the temperature of 70 g. 
 of water from 8° C. to 63"^ C? The rise of temperature is 55°; hence the 
 heat required is 55 calories per gram, and 70 X 55, or 3850 calories, for 70 g. 
 
Calorimetry : Specific Heat 183 
 
 The amount of heat required to raise the temperature of one 
 gram of water one degree is not exactly the same at all tempera- 
 tures, but the difference is too small to be of importance except 
 in the most accurate work. The numerical relation between the 
 calorie and the units of mechanical energy is considered in Art. 270. 
 
 240. Specific Heat. — It is found by experiment that only one 
 ninth as much heat is required to cause a given rise of temperature 
 in any mass of iron as is necessary to cause the same rise of tem- 
 perature in an equal mass of water. This ratio J, or .11, is 
 called the specific heat of iron. The term is also applied to the 
 number of calories required to raise the temperature of one gram 
 of iron one degree y which is evidently .11 calorie. 
 
 The specific heat of a substance is the ratio of the quantity of 
 heat required to raise the temperature of any mass of the sub- 
 stance one degree to the amount required to raise the temperature 
 of an equal mass of water one degree ; or, it is the number of 
 calories required to raise the temperature of one gram of the sub- 
 stance one degree (Centigrade). 
 
 ExAMi'LKS. — If the specific heat of a substance is .04, to raise the tem- 
 perature of 50 g. of it from 2^ C. to 6" C, would require 50 x 4 X .04, or 8 
 calories. The same body in cooling from 50° C. to 30° C. would give out 
 50 X 20 X .04, or 40 calories. 
 
 The specific heat of water is one, by definition ; it is very large 
 compared with that of most other substances, especially the 
 metals, and is exceeded only by hydrogen. In the following table 
 the substances are named in the order of their specific heats. 
 
 Table of Specific Heats 
 
 Hydrogen 3.409 
 
 Water i.ooo 
 
 Alcohol (0° to 50°) . . . .0.615 
 
 Ice 0.504 
 
 Steam 0.480 
 
 Afr 0.237 
 
 Marble 0.216 
 
 Aluminum 0.213 
 
 Glass 0.198 
 
 Iron 0.1 13 
 
 Copper 0.095 
 
 Brass 0.094 
 
 Mercury 0.033 
 
 Lead 0.031 
 
184 Heat 
 
 241. Measurement of Specific Heat. — The method generally 
 used for determining the specific heat of a substance is known as 
 the method of mixtures. It is illustrated by the following example : 
 A brass calorimeter weighing 100 g. contains 400 g. of water at 
 I8^ Into this is put a roll of sheet iron at 100°, weighing 190 g. 
 After stirring, the temperature of the water is 22°, and this is 
 assumed to be the temperature of the roll of iron and the calo- 
 rimeter. The specific heat of the calorimeter is given as .094. 
 Find the specific heat of the roll of iron. 
 
 Solution. — I^t j denote the specific heat of iron ; i.e. in this case, it is 
 the number of calories of heat given out by each gram of the roll of iron in 
 cooling one degree. 
 
 Rise of temp, of calorimeter and water = 22" — 18** = 4° 
 
 Heat received by the calorimeter = 100 x 4 X .094 = 37.6 cal. 
 Heat received by the water = 400 x 4 = 1600 cal. 
 
 Fall of temperature of the iron = 100° — 22° = 78° 
 
 Heat given out by the iron = 190 x 78 X .f = 14820 j cal. 
 
 Assuming that the transfers of heat take place only among the calorimeter 
 and its contents, it follows that the heat given out by the roll of iron in cool- 
 ing to the temperature of the " mixture " is equal to the heat gained by the 
 calorimeter and water in coming to the same temperature ; that is, 
 
 14820 J = 37.6 + 1600, 
 from which s — 1637.6 -4- 14820 = .110. 
 
 242. The Heat Equation. — The above example illustrates the 
 method of treating the experimental data in all experiments in 
 calorimetry. The following summary of the method will therefore 
 be of service now and later. 
 
 (i) Find numerical or algebraic expressions for the separate 
 quantities of heat involved in the equalization of temperatures. 
 
 (2) With these quantities of heat form the heat equation^ which 
 expresses the equality of heat lost and heat gained. 
 
 (3) The heat equation contains as an unknown quantity the 
 quantity sought (specific heat, heat of fusion, or heat of vaporiza- 
 tion). To find this quantity, solve the equation by the usual 
 algebraic processes. 
 
Calorimetry : Specific Heat 185 
 
 243. The Control of Heat in Calorimetric Experiments. — Any 
 transfer of heat between the contents of the calorimeter and the 
 surrounding air or other bodies during an experiment is a source 
 of error, and must be avoided in so far as possible. The calo- 
 rimeter is usually nickel plated and brightly polished to dimin- 
 ish radiation when it is warmer than the surrounding air, and to 
 diminish absorption when it is cooler. The calorimeter should 
 stand on a poor conductor (wood) and should be touched with 
 the hands as little as possible, to avoid conduction. 
 
 At the beginning of an experiment the water should be taken at 
 such a temperature that it (and the calorimeter) will be colder than 
 the air during a part of the time and warmer during a part, in 
 order that the gain of heat by conduction and radiation at the 
 lower temperature may be as nearly as possible equal to the loss 
 by the same means at the higher temperature. 
 
 Laboratory Exercise jy. 
 
 PROBLEMS 
 
 1. The specific heat of water is much greater than thai of rocks and soils. 
 How does this in part account for the fact that the change of temperature of 
 the land between day and night and between winter and summer is much 
 greater than that of the ocean ? 
 
 2. Are equal quantities of heat required to raise equal volumes of different 
 substances through equal changes of temperature ? (Consult table of densi- 
 ties and table of specific heats.) 
 
 3. What effect has the large specific heat of water on the sensation caused 
 by putting the hand in hot or cold water ? In general, how does the specific 
 heat of a substance affect the sensation of heat or cold caused by it when 
 touched (see Art. 217) ? 
 
 4. A roll of lead weighing 800 g. is heated to 100° and placed in a brass 
 calorimeter weighing 90 g. and containing 406.3 g. of water at 16.2°. The 
 final temperature is 21°. Find the specific heat of the lead. 
 
 5. A kilogram of mercury at 200° and a kilogram of water at 0° are mixed. 
 Find the resulting temperature, no allowance being made for the vessel. 
 
 6. A piece of aluminum weighing 60 g. is heated to 63°, and placed in a 
 copper calorimeter weighing 50 g. and containing 100 g. of alcohol at 8". 
 The temperature of the alcohol rises to 17°. Find its specific heat, taking the 
 specific heat of copper and aluminum from the ^ble. 
 
1 86 Heat 
 
 VI. Fusion and Solidification 
 
 244. Melting of Ice and Freezing of Water. — \\lien heat is 
 applied to ice at any temperature below o^, the temperature of 
 the ice rises, but melting does not begin until the temperature 
 has risen to o*. With the further application of heat the ice 
 begins to melt, but its temperature remains at o°. 
 
 When a vessel of water is surrounded by any substance whose 
 temperature remains below zero, the water loses heat and cools 
 U> ©•. With further loss of heat, the water begins to freeze ; but 
 its temperature remains at o° until it is all frozen. 
 
 Thus the wultimg point of ice and the freeung point of water 
 are exactly the same. Whether melting or freezing will take 
 place in a >*essel containing both ice and water depends upon 
 whether heat is passing into the vessel or from iL K the water is 
 losing heat, it will freeze ; if the ice is receiving heat, it will melt ; 
 if there is neither gain nor loss of heat, neither melting nor freezing 
 win occur. 
 
 Lmhoraifrx Exrrcisf jj. 
 
 945. lldting Points. — Every solid that can be melted has a 
 constant melting point, which is also the temperature at which it 
 freezes or solidifies. .Among fusible solids, some, like ice, change 
 abruptly from the solid to the liquid state. In such cases the melt- 
 ing point can be very accurately determined. Other solids, as seal- 
 ing wax and glass, gradually soften and pass by continuous change 
 into the liquid state. In such cases the melting point, although 
 constant, is nx>re or less indefinite. 
 
 Talflt of Melting Points 
 
 yrfc 
 
 4» 
 
 657 
 
 , 1050 
 
 Gla» 1000 to 1400 
 
 Iron, wrought . . 1500 to 1600 
 Flatinom 1775 
 
 -ijo C 
 
 Moc«y -39 
 
 Ice o 
 
 Batter n 
 
 Beeswax 62 
 
 Caae.ssgar 170 
 
 Solder, soft 225 
 
Fusion and Solidification 187 
 
 246. Change of Yolnme during Fusion and Solidification. — 
 Most substances expand in melting and contract in solidifying; 
 in many cases the change in volume is considerable. This is well 
 illustrated in cooling a dish of melted beeswax or paraffine : the 
 contraction leaves a depression at the center of the cake {Exp.). 
 Metals, with few exceptions, also contract in solidifying. Those 
 that do are not adapted for casting, as they would shrink away 
 from the surfaces of the mold. Cast iron and type metal, which 
 is an alloy of lead, tin, and antimony, are among the exceptions. 
 
 Water expands in solidif>ing, as is well known, the increase in 
 volume amounting to about one eleventh. As a result of ihb 
 expansion ice floats — a fact of great importance in nature. If 
 water contracted in freezing, ice forming at the surface of lakes 
 and rivers would sink. Freezing would consequently continue at 
 the surface throughout winter or until the lakes and rivers were 
 frozen solid, and all animal life inhabiting them would be de- 
 stroyed. 
 
 The enormous force exerted by water in freezing is shown in 
 the occasional bursting of water pipes in winter. The magnitude 
 of this force was strikingly shown by 
 some experiments of Major Wil- 
 liams, in Canada. " Having quite 
 filled a thirteen-inch bombshell with 
 water, he firmly closed the touchhole 
 with an iron plug weighing three 
 pounds, and exposed it in this state 
 to the frost. After some time the 
 iron plug was forced out with a loud 
 explosion, and thrown to a distance *^ 
 
 of 415 feet, and a cylinder of ice 8 inches long issued from the 
 opening. In another case the shell burst before the plug was 
 driven out, and in this case a sheet of ice spread out all round 
 the crack." (Fig. 123.) — Ganot*s EUnunts of Physics. 
 
 " Much of the destruction of rocks which is taking place on the 
 earth's surface is due to the same quiet but intensely powerful 
 
i88 
 
 Heat 
 
 action of freezing water. Rain sinks into the cracks and pores 
 which all rocks are liable to contain, and when it freezes there, the 
 crack is inevitably widened or the stnicture of the rock loosened. 
 Thus room is made for more water, which acts in the same way 
 when it freezes ; and so by degrees immense masses of rock and 
 earth are loosened from the mountainside, nor does the action end 
 until the material is reduced to the finest soil." — Madan's Heat, 
 Substances that expand in solidifying have a crystalline structure 
 in the solid state. The crystalline structure is plainly seen in the 
 ice that first forms when water begins to freeze, in the frost that 
 gathers on window panes, and in snow. In crystalline solids the 
 molecules are arranged in clusters of a definite shape, and hence 
 occupy more space than when they lie loosely side by side in the 
 liquid state ; just as a number of bricks would occupy more space 
 if arranged in patterns than if packed in layers. 
 
 247. Change of Melting Point produced by Pressure. — Experi- 
 ments have shown that increase of pressure upon a solid that ex- 
 pands in melting tends to prevent melting by raising the melting 
 point. The pressure evidently opposes melting because it opposes 
 the expansion that accompanies the process. 
 
 Similarly, pressure upon a liquid that expands in solidifying 
 tends to prevent solidification by towering the melting point, for 
 
 in this case, also, pressure 
 opposes the expansion that 
 accompanies the change of 
 state. While lowering the 
 melting point opposes solid- 
 ification, it aids melting. 
 Thus ice has been melted at 
 — 1 8° by a pressure esti- 
 mated at several thousand 
 atmospheres. The change 
 in the melting point of ice 
 Fig. X24. (jue to a pressure of one 
 
 atmosphere would escape detection by means of the thermom- 
 
Fusion and Solidification 189 
 
 eters used in elementary physics ; yet the effects produced under 
 certain conditions by small changes of pressure are very striking. 
 For example, a loop of fine wire to which weights are attached 
 slowly descends through a block of ice round which it has been 
 passed (Fig. 124) ; yet, after it has passed completely through, the 
 ice is in one solid piece as at the beginning {Exp.). The pressure 
 of the wire very slightly lowers the melting point of the ice immedi- 
 ately beneath it ; and the ice melts, receiving the heat necessary 
 for the purpose from the water filling the space just above the 
 wire. This water freezes in losing heat, since it is relieved from 
 pressure. The process is continuous : the water from the ice 
 melting below the wire, passes round and freezes above it. The 
 three stages of the process are (i) melting under pressure, 
 (2) change of position of the water, (3) regelation (refreezing) 
 under diminished pressure. 
 
 This explains the hardening of snow into a solid mass in making 
 snowballs, and the freezing of ice to flannel wrapped around it. 
 The flow of glaciers is supposed to be due to the same action under 
 great pressure. 
 
 248. Heat of Fusion. — We have seen that ice melts only while 
 receiving heat at 0°, and water freezes only while losing heat at 0°. 
 The quantity of ice melted or of water frozen is proportional to 
 the gain of heat in the one case, and to the loss of heat in the 
 other. This is true of any substance that has a definite melting 
 point. Since these transfers of heat during change of state take 
 place without change of temperature, it is evident that heat is lost 
 in the process of fusion and is recovered during solidification. In 
 what form does this energy exist in the liquid ? 
 
 A solid in melting must receive energy in the form of heat to 
 overcome (in part) the cohesion of its molecules (Art. 189). 
 After doing this work the energy no longer exists as heat, but as 
 potential energy in the changed molecular condition of the sub- 
 stance, — it is molecular potential energy. The process may be 
 illustrated by pulling apart two balls connected by a rubber band, 
 the balls representing molecules and the tension of the rubber 
 
190 Heat 
 
 band, cohesion. Work is done in separating the balls ; the result 
 is potential energy, which is recovered when the balls are per- 
 mitted to come together again. 
 
 The number of calories required to melt one gram of any sub- 
 stance is called its Juat of fusion ;^ this is also the number of 
 calories given out by one gram of the substance in solidifying. 
 Substances differ widely in their heats of fusion. The heat of 
 fusion of water is much larger than that of most other substances. 
 
 Table of Heats of Fusion 
 
 Caloribs 
 
 Ice . > 79.25 
 
 Wax 42 
 
 Zinc 28.13 
 
 Silver 21.07 
 
 Calories 
 
 Tin 14.25 
 
 Sulphur 9.37 
 
 Lead 5.37 
 
 Mercury 2.83 
 
 249. Determination of the Heat of Fusion of Ice. — The follow- 
 ing example illustrates the process of finding the heat of fusion 
 of ice by the method of mixtures : A quantity of dry, crushed ice 
 weighing 104 g. is placed in a brass calorimeter weighing 85 g. 
 and containing 220 g. of water at 44°. The temperature after the 
 ice is melted is 5.3°. 
 
 Solution. — Let /denote the heat of fusion of ice. 
 Fall of temperature of calorimeter and water = 44 — 5.3 = 38.7° 
 
 Heat given out by the calorimeter = 85 X 38.7 x .094 = 309.2 cal. 
 
 Heat given out by the water = 220 X 38.7 = 8614. cal. 
 
 Heat received by the ice in melting = 104 /cal. 
 
 Heat received by the ice water in warming to 5.3°= 104 X 5-3 = 551.2 cal. 
 104/+ 551.2 = 309.2 + 8514 ; 
 /= 79.5 calories. 
 
 Laboratory Exercise j8. 
 
 1 According to the caloric theory, heat must always remain heal, since it was 
 regarded as a form of matter ; and the heat that disappears in the process of fusion 
 (and vaporization) was called "latent" (meaning hidden), implying that it still 
 exists as heat, although its presence cannot be detected. The only form of energy 
 that is properly called heat at all was then called " sensible " heat to distinguish it 
 from " latent " heat. The terms have outlived the theory that gave rise to them ; 
 thus the heat of fusion of ice is often called the " latent heat of water " (Art, 210). 
 
Fusion and Solidification 191 
 
 250. Heat of Solution ; Freezing Mixtures. — Work is done in 
 overcoming cohesion in a solid when it is dissolved as well as when 
 it is melted ; and in many instances there is direct experimental 
 evidence that heat disappears in the process, proving that this 
 work is accomplished by heat.* Thus when ammonium chloride 
 or ammonium nitrate is dissolved in water, there is a fall of temper- 
 ature of several degrees ; for the heat required to dissolve the solid 
 is taken from the nearest available source; namely, the water. 
 Solution differs from fusion in that it can take place within a wide 
 range of temperature ; hence the temperature continues to fall 
 (unless heat is received from the outside) until all the solid is 
 dissolved or until the solution is saturated. 
 
 A mixture of one or more solids and a liquid, or of two solids, 
 is called 2i freezing mixture if the solution or the liquefaction of the 
 solids causes a fall of temperature below zero. The following are 
 examples of freezing mixtures : — 
 
 (i) One part by weight of ammonium chloride and one of 
 potassium nitrate or ammonium nitrate, powdered together and 
 dissolved in two parts of water. Fall of temperature about 20°. 
 
 (2) About 5 parts of strong hydrochloric acid and 8 parts of 
 powdered sodium sulphate. Fall of temperature about 30°. 
 
 (3) One part of table salt and 2 parts of snow or crushed ice. 
 Fall of temperature to about — 18°. The strong attraction of salt 
 for water causes the ice to melt rapidly. The heat required to 
 melt the ice and to dissolve the salt is taken first from the ice and 
 salt, then, by conduction, from surrounding bodies. This freezing 
 mixture is well known from its use in making ice cream. 
 
 (4) One part each of crystallized calcium chloride and snow 
 or crushed ice. Fall of temperature to about — 40°. The attrac- 
 tion of calcium chloride for water is stronger than that of table 
 salt, and hence causes the ice to melt more rapidly and at a lower 
 temperature. 
 
 1 When chemical action accompanies solution, it may result in a rise of temper- 
 ature, the heat generated by the chemical action being greater than the heat lost 
 in solution. 
 
192 Heat 
 
 PROBLEMS 
 
 1. How much heat is required to convert 750 g. of ice at — 20° into water 
 at 50^ ? 
 
 2. How many grams of ice at 0° can be melted by 500 g. of water at 60° ? 
 
 3. A kilogram of ice and a kilogram of water, both at o"^, receive heat at 
 the same rale. What will lie the temperature of the water when the ice has 
 all been converted into water at 0° ? 
 
 4. A piece of aluminum weighing 250 g. and heated to 100° is placed in a 
 dry cavity in a block of ice, and melts 63.1 g. of the ice. Find the specific 
 heat of the aluminum, taking the heat of fusion of ice as 79.25 calories. 
 
 5. What purpose is served by vessels of water placed in a cellar where 
 vegetables are stored or in a greenhouse on a frosty night ? 
 
 6. (o) Do freezing and thawing take place more or less rapidly than they 
 would if the heal of fusion of ice were less ? (d) Of what importance is this 
 in the economy of nature ? 
 
 VII. Vaporization and Condensation 
 
 251. Vaporization. — The change of a substance from the solid 
 or liquid to the gaseous state is called vaporization. Vaporization 
 may take place at the free surface of a licjuid, or within its mass 
 at the place where heat is applied. In the first case the phenom- 
 enon is generally called amporation ; in the second case, boil- 
 ing, A liquid that evaporates readily is said to be volatile. The 
 gaseous form of a substance that exists in the liquid or the solid 
 state at ordinary temperatures is generally called a vapor. 
 
 Evaporation takes place at the surface of most liquids at all 
 temperatures, but more rapidly as the temperature rises. It is 
 due to molecular motion. Some of the molecules of a liquid, 
 in their irregular motion, reach the surface with a sufficient up- 
 ward velocity to carry them into the space above, out of the range 
 of cohesion, where they exist as a gas or vapor. With a rise of 
 temperature the velocity of the molecules is increased, and more 
 of them are able to escape from the liquid in a given time. The 
 process is illustrated on the largest scale in the evaporation of 
 water from Ihe surface of the oceans, lakes, ponds, and streams ; 
 
Vaporization and Condensation 193 
 
 as a result of which, the air always contains a greater or less 
 amount of water vapor. 
 
 252. Disappearance of Heat during Vaporization. — It is well 
 known that evaporation is a cooling process. A room is appre- 
 ciably cooled by the evaporation of water sprinkled on the floor. 
 The skin is cooled by the evaporation of water or perspiration 
 from it. This is especially noticeable in a draft, which causes 
 more rapid evaporation by carrying the vapor away as fast as it is 
 formed. The rapid evaporation of highly volatile liquids, as alco- 
 hol and ether, causes much greater cooling. 
 
 The cooling effect of evaporation is explained as follows : A 
 liquid in vaporizing increases enormously in volume.* In this ex- 
 pansion work is done against cohesion and also against external 
 pressure, and heat is transformed into molecular potential energy. 
 In evaporation, as in solution, this heat is taken from the nearest 
 available sources — first the liquid itself, then adjacent bodies. 
 
 253. Vapor Pressure. — A vapor, like any gas, exerts 
 a certain pressure which is proportional to its density 
 and increases with its temperature ; but the behavior of 
 vapors differs from that of other gases in important re- 
 spects, as shown by the following experiment. 
 
 A barometer tube is filled with mercury and set up as 
 a simple barometer. Ether is introduced into the tube 
 at the bottom, drop by drop, by means of a dropping 
 tube or a pipette, care being taken to let no air enter 
 (Fig. 125). As the first drop rises to the top of the 
 mercury column, it instantly evaporates, and the pressure 
 that it exerts as a vapor causes a depression of the 
 column. The pressure of the vapor, expressed in centi- 
 meters of mercury, is measured by the amount of the - 
 depression. (Why?) Each drop of ether evaporates as HhMHI 
 it rises in the tube and causes a further depression of the ' ^^^^ ' 
 column. This continues, however, only to a certain fig. 125. 
 
 1 A cubic centimeter of v/ater forms 1661 ccm. of steam at 100° and a pressure 
 of one atmosphere. 
 
1 94 Heat 
 
 point, beyond which the liquid ether accumulates above the mer- 
 cury, and the column remains stationary. The ether vapor is now 
 saturatfd ; i.e. // cannot be made denser at its present temperature. 
 Before this condition is reached the vapor is unsaturated^ and 
 hence exerts a less pressure. 
 
 The pressure exerted by the saturated ether vapor is found by 
 subtracting the height of the mercury in the tube from the read- 
 ing of a barometer. When the tube is inclined, the space occu- 
 pied by the vapor becomes smaller; but the vertical height of 
 the mercury column remains the same as before, indicating that 
 the vapor pressure is unchanged. This is true even when the 
 tube is inclined so far that the space occupied by the vapor 
 almost disappears. In inclining the tube, the vapor is evidently 
 not compressed and made denser ; for in that case it would exert 
 an increased pressure. The fact is that a portion of the vapor is 
 liquefied, and the density of the remainder is unchanged. This 
 behavior agrees with the former statement that the vapor is satu- 
 rated, and cannot be made denser at its present temperature ; 
 and, since it cannot be made denser, it cannot be made to exert a 
 greater pressure. The pressure of the saturated vapor is therefore 
 the maximum pressure of ether vapor at its present temperature. 
 
 When the tube is returned to the vertical position and the ether 
 warmed by clasping the tube in the hands, the mercury descends 
 further, showing an increase of vapor pressure with rise of tempera- 
 ture. This is partly due to the heating of the vapor already in the 
 tube, but chiefly to the evaporation of more ether. The saturated 
 vapor is denser at the higher temperature. If there were no more 
 ether in the tube to evaporate, the heat of the hand would cause 
 expansion of the existing vapor, and it would become less dense 
 and unsaturated. 
 
 Similar results are obtained throughout when alcohol is substi- 
 tuted for ether in the experiment ; but they are all on a greatly 
 reduced scale, for the maximum pressure of alcohol vapor is much 
 less than that of ether at the same temperatures. With water the 
 effects are very slight. At 20° the maximum vapor pressure of 
 
Vaporization and Condensation 195 
 
 ether is 43.28 cm. (of mercury), that of alcohol 4.45 cm., and 
 that of water 1.74 cm. 
 
 Laboratory Exercise 34, Parts I arid II. 
 
 254. Laws of Vapor Pressure. — The above experiment illus- 
 trates the first three of the following laws of vapor pressure : — 
 
 I. At a given temperature there is a maximum density and 
 pressure for every vapor, in which condition the vapor is said to 
 be saturated. 
 
 Compression of a saturated vapor without change of tempera- 
 ture causes a portion of it to condense (liquefy) ; but the density 
 and pressure of the remainder are not changed. 
 
 II. At the same temperature the maximum pressures of different 
 vapors are unequal. 
 
 III. The density and pressure of a saturated vapor increase 
 with the temperature. 
 
 IV. The behavior of unsaturated vapors is approximately like 
 that of gases, as expressed in the laws of Boyle and Charles. 
 
 255. Mixture of Gases and Vapors; Dalton's Laws. — The 
 following laws relating to mixtures of gases and vapors are known 
 as Dalton's laws, from their discoverer. 
 
 I. The quantity of vapor which saturates a given space is the 
 same, at the same temperature^ whether this space contains a gas 
 or is a vacmnn. 
 
 II. The pressure of the mixture of a gas and a vapor is equal 
 to the sum of the pressures which each would exert if it occupied 
 the same space alone. 
 
 In a vacuum the evaporation of a volatile liquid is almost instan- 
 taneous. In the presence of air or any other gas, evaporation 
 takes place much more slowly ; but, as is implied in Dalton's 
 first law, it will not cease until any inclosed space above the 
 liquid contains as much of the vapor as it would if the gas were 
 not present. The kinetic theory of gases accounts at once for the 
 second law. The law holds for the mixture of any number of 
 vapors and gases, the most familiar example of which is the 
 atmosphere. 
 
1 96 Heat 
 
 256. Water Vapor in the Atmosphere. — The atmosi)here is a 
 mixture of several gases, principally nitrogen and oxygen ; the 
 only other constituents of importance are carbon dioxide and 
 water vapor. All of the constituents of the atmosphere except 
 water vapor are practically constant in amount ; the latter varies 
 from an inappreciable fraction to about 2 per cent of the whole, the 
 average amount being not far from i per cent. 
 
 The condition of the water vapor in the air with respect to 
 saturation is not in the least affected by the presence of the other 
 gases (Dalton's first law), and depends only upon its own density 
 and temperature (which, of course, is the temperature of the air) ; 
 yet common forms of expression seem to imply that the presence 
 and condition of the vapor are due to some action of the air. 
 Thus when the water vapor in the air is saturated, we say that the 
 air is saturated or that the air has all the moisture it can hold ; 
 although, strictly speaking, it is the space that has all the water 
 vapor that it can hold (at the given temperature) . There is per- 
 haps no objection to the use of such expressions when their true 
 meaning is understood. 
 
 The air is generally not saturated ; it is evidently not saturated 
 whenever further evaporation can take place. Nonsaturated air 
 may become saturated (i) by further evaporation, (2) by a fall 
 of temperature, (3) by the two processes combined. Saturation 
 results from a sufficient fall of temperature because the density of 
 a saturated vapor is less at lower temperatures (Art. 254, third 
 law). Consequently when the quantity of water vapor in the air is 
 less than that required for saturation at the existing temperature, 
 it is equai to the amount required for saturation at a definite lower 
 temperature (called the dew-point). 
 
 257. The Dew-point. — The temperature at which the water 
 vapor present in the air at any time would be saturated is called 
 the de7u-point of the air at that time. 
 
 When any body of air is cooled to the dew-point, condensation 
 of water vapor begins, and continues as long as the temperature 
 continues to fall. The moisture that gathers on the outside of a 
 
Vaporization and Condensation ' 197 
 
 pitcher of ice water is a familiar illustration. The moisture comes 
 from the surrounding air, which is cooled by coming in contact 
 with the cold pitcher, and begins to deposit moisture as soon as 
 the dew-point is reached. A fall of temperature several degrees 
 below this point generally occurs, causing a considerable deposit 
 which runs down the sides. (What error is implied in calling this 
 phenomenon "sweating" ?) 
 
 The dew-point may be determined experimentally by putting 
 water in a vessel on whose surface a thin film of moisture can 
 easily be seen (as a nickle-plated calorimeter), and slowly cool- 
 ing the water by means of ice or a freezing mixture till the first 
 trace of moisture appears on the vessel. The temperature of the 
 water when this occurs is the dew-point. 
 
 The dew-point varies between wide limits. In winter it is 
 often many degrees below zero. It is, of course, lower than the 
 temperature of the air unless the air is saturated at the time ; and 
 it approaches the temperature of the air as the air approaches 
 saturation. 
 
 Laboratory Exercise 34^ Part III. 
 
 258. Humidity. — The humidity (or relative humidity) of the 
 air at any time is the ratio of the amount of water vapor that it 
 contains at the time to the whole amount that would be required 
 to saturate it at the existing temperature. This ratio is usually 
 expressed as a per cent. Thus the humidity of the air is 75 per 
 cent when it contains three fourths as much water vapor as would 
 be recfuired to saturate it at the time. The humidity of saturated 
 air is 100 per cent by definition. 
 
 The dryness or dampness of the air depends not only upon the 
 amount of water vapor in it, but also upon the temperature ; in 
 other words, it is determined by the humidity of the air. This is 
 illustrated by the well-known fact that very damp, cold air in a 
 room becomes dry when the room is warmed, although there is no 
 less vapor in the room after the heating than there was before. 
 The air is drier because its temperature is farther above the dew- 
 point; i.e. its humidity has been diminished. For example: At 
 
198 * Heat 
 
 10° C. (50° F.) the maximum pressure of water vapor is .92 cm. ; 
 at 20** C. (68° F.) it is 1.73 cm. Hence when saturated air at 10° 
 is heated to 20°, without change in the quantity of vapor it con- 
 tains, its humidity falls from 100 per cent to about 53 per cent. 
 At 10° the air would be disagreeably moist ; at 20° it would feel 
 rather dry. 
 
 259. Laws of Evaporation. — The conditions affecting the rate 
 of evaporation of a liquid may be summarized as follows : — 
 
 I. TA^ rate of evaporation increases with a rise of temperature, 
 
 II. The rate of evaporation increases with an increase of the 
 free surface of the liquid. 
 
 III. The rate of evaporation of a liquid decreases as t/te space 
 around it approaches saturation by its own vapor ^ and ceases when 
 that space is saturated, 
 
 IV. The rate of ei^aporation in the open air increases with a 
 more rapid change of air about the liquid. Currents of air (winds) 
 carry the vapor away from the space about the liquid, and the 
 stronger the currents are, the farther will this space be from satu- 
 ration. 
 
 V. The rate of evaporation increases as the density of the air or 
 other gas surrounding the liquid is diminished; in a vacuum it is 
 almost instantaneous. Changes of barometric pressure are not 
 sufficient to materially affect the rate of evaporation in the open air. 
 
 It follows from laws I, III, and IV that the most favorable con- 
 ditions for the rapid evaporation of water in the open air are pres- 
 ent on a dry, hot, windy day. 
 
 260. Condensation of Water Vapor in the Atmosphere. — Water 
 vapor is ahuays invisible. The visible forms of moisture in the 
 atmosphere — as fog, mist, clouds, and the so-called " steam " 
 near the spout of a kettle in which water is boiling — consist of 
 minute particles of liquid water, and are the result of the conden- 
 sation that accompanies a fall of temperature after the dew-point 
 is reached. Dew, frost, rain, sleet, hail, and snow are forms in 
 which the moisture of the air is condensed and precipitated. The 
 conditions under which the different forms occur are as follows : — 
 
Vaporization and Condensation 199 
 
 Dew is condensed water vapor coming from the air immediately 
 surrounding the body on which it appears. It has been found 
 by numerous experiments that surfaces upon which dew is form- 
 ing are always at least 3° or 4° colder than the air or dewless 
 surfaces. (Cooling below the temperature of the air is due to 
 the rapid loss of heat by radiation.) When the air is nearly 
 saturated, it is cooled to the dew-point by coming in contact with 
 such surfaces, and moisture is condensed upon them. Dew forms 
 only at night, and most abundantly during the latter part of it; 
 when, by cooling, the air has become nearly saturated. It is 
 formed only on calm, clear nights ; for on clear nights cooling is 
 most rapid (Art. 228), and it is only on calm nights that any por- 
 tion of the air remains long enough in contact with the cold sur- 
 faces to be cooled to the dew-point. Dew forms most abundantly 
 on the coldest objects, which are in general the best radiators and 
 the poorest conductors. Grass, leaves, and boards are good 
 examples. A board lying on the ground will become wet with 
 dew when a stone pavement remains dry ; for the stone is a good 
 conductor and receives heat from the ground, which replaces that 
 lost by radiation, hence its upper surface is warmer than that of 
 the board. 
 
 When the dew-point is at or below zero, condensation takes 
 place in the form oi frosty under conditions otherwise the same as 
 are necessary for the formation of dew. The water vapor then 
 crystallizes in the solid state as it condenses, without passing 
 through the intermediate state of a liquid. (Is frost '•* frozen 
 dew"?) 
 
 At temperatures above zero the moisture of clouds is in the 
 form of fog or mist. As the individual particles grow by further 
 condensation and by uniting with one another, they may become 
 too large to be sustained in the air, and will then fall as rain. 
 A drop continues to grow by uniting with smaller particles that it 
 meets with in falling through the cloud. Sleet is formed by the 
 freezing of the raindrops as they fall through a layer of air whose 
 temperature is below zero. 
 
200 Heat 
 
 Snow is formed by the condensation of vapor in the atmospheie 
 at temperatures below zero. Snow and frost are formed under 
 the same conditions of temperature and humidity, and both have 
 a beautiful crystalline structure. 
 
 " I/aii is formed in violent storms, such as tornadoes and thun- 
 der storms, where there are strong, whirling currents of air. 
 Hailstones are balls of ice, built up by condensing vapor as they 
 are whirled up and down in the violent currents, freezing, melting 
 and freezing again as they pass from warm to cold currents. For 
 this reason they are often made of several layers, or shells, of ice." 
 — Tarr's Nov Physical Geography, 
 
 PROBLEMS 
 
 1. For what two reasons <locs a licjuid evaporate more rapidly in a wide 
 and shallow vessel than it does in an unstoppercd bottle ? 
 
 2. Why is a pan of water often kept on a heating stove ? 
 
 3. (/i) Why does the breath often form a visil)le cloud on a cold day ? 
 (^) Is it more likely to do so when the humidity Is high or low ? 
 
 4. The moisture from the spout of a kettle of boiling water is invisible for 
 a few inches beyond the spout, then for some distance farther some of it 
 forms a cloud ; still farther it is all invisible again. Account for these facts. 
 
 5. WTiy does frost form on the inside of a window pane but not on the 
 outside ? 
 
 6. Mow does the formation of frost and snow prove that water can exist as 
 a vapor at lower temperatures than it can as a liquid ? 
 
 Laboratory Exercise j^. 
 
 261/ Boiling. — When fresh water is heated, dissolved air is 
 given off in the form of minute bubbles, which begin to form on 
 the sides and bottom of the vessel as soon as the water has become 
 slightly warm. ITiese bubbles often rise to the surface in large 
 numbers, where the air that they contain escapes. After the 
 water has become hot, much larger bubbles begin to form at the 
 bottom where the heat is applied. These are bubbles of steam or 
 water vapor. They rise rapidly, but disappear before reaching 
 the surface, being condensed by the cooler water near the top. 
 It is the collapse of these first bubbles of steam that causes the 
 
Vaporization and Condensation 201 
 
 singing of a kettle of water shortly before it begins to boil. As 
 the temperature of the water approaches 100°, the bubbles rise 
 higher, until finally they burst at the surface, throwing the water 
 violently about. This formation and escape of the bubbles of 
 steam is called boiling. 
 
 262. Laws of Boiling. — The principal laws of boiling, as de- 
 termined by experiment, are the following : — 
 
 I. Boiling begins at a temperature which is invariable for each 
 liquid under the same conditions, 
 
 II. Whatever the intensity of the source of heat, the temperature 
 of a boiling liquid remains sensibly constant. 
 
 The conditions here are similar to those under which fusion 
 takes place : energy in the form of heat is required to produce 
 the change of state (Art. 266), and a more abundant supply of* 
 heat merely causes vaporization (boiling) to take plate more 
 rapidly. 
 
 III. 77ie pressure of the vapor given off by a boiling liquid is 
 equal to the pressure of the air {or other gas) upon its free surface. 
 
 This may be shown by measuring the vapor pressure at the 
 temperature of the boiling liquid by means of a pressure gauge 
 (I.ab. Ex. 35); and is directly evident from the fact that the 
 vapor makes room for itself within the liquid against the trans- 
 mitted pressure of the air, which it must therefore be able to 
 sustain. At any appreciable depth, the vapor must exert a slight 
 additional pressure to balance that due to the weight of the liquid. 
 A liquid cannot boil when the pressure upon its surface is greater 
 than the pressure of its saturated vapor at the existing tempera- 
 ture. 
 
 IV. An increase of pressure raises the boiling pointy a decrease 
 of pressure lowers it. 
 
 This is a necessary consequence of the preceding law together 
 with the third law of Art. 254. For an increase of pressure upon 
 the free surface of a liquid requires an equal increase of vapor 
 pressure to enable the vapor to form within the liquid, and an 
 increase of vapor pressure requires a rise of temperature. 
 
202 
 
 Heat 
 
 The lowering of the boiling point under diminished pressure is 
 
 readily shown by either of the fol- 
 lowing experiments : A glass of 
 water at 30° to 40° is placed under 
 the receiver of an air pump and the 
 air exhausted. When the pressure 
 is sufficiently diminished, the water 
 boils violently {Exp,), Water is 
 boiled in a round-bottomed flask 
 until the air has been expelled by 
 the steam. It is then quickly closed 
 with a tight cork, and inverted as 
 shown in Fig. 126. When cold 
 water is poured over the flask, the 
 waier within it boils violently. This 
 may be repeated till the water in 
 the flask is barely warm. The cold 
 water condenses some of the vapor, diminishing its pressure {Exp.). 
 As a result of the diminished atmospheric pressure at high alti- 
 tudes, the boiling p>oint of a liquid is considerably lower upon a 
 mountain than it is near sea level. On the summit of Mont Blanc, 
 for example, water boils at 84°. 
 
 263. Boiling Points. — A liquid is said to boil when bubbles of 
 vapor formed by vaporization within its mass are given off" at its 
 surface. ^Vhen no pressure is mentioned, the boiling point of a 
 liquid is understood to mean the temperature at which it boils 
 under a pressure of one atmosphere. The boiling point of a liquid 
 may also be defined as the temperature at which^the pressure of 
 its saturated vapor is equal to one atmosphere. 
 
 Tabli of Boiling Points 
 
 Ether 35° 
 
 Chloroform 61.2 
 
 Alcohol 78.4 
 
 Water icx) 
 
 Turpentine 160° 
 
 Glycerine 290 
 
 Mercury 357 
 
 Sulphur 448 
 
Vaporization and Condensation 203 
 
 The following table gives the pressure of saturated water vapor 
 (and hence also the pressure under which water boils) at a 
 number of temperatures, the pressure being expressed in centi- 
 meters of mercury in the first column of pressures and in atmos- 
 pheres in the last. 
 
 
 Temperature 
 
 Pressure 
 
 Temperature 
 
 Pressure 
 
 0° 
 100 
 
 .46 cm. 
 
 9.20 cm. 
 
 76.00 cm. 
 
 120° 
 
 140 
 
 160 
 
 1.96 atmospheres 
 3.58 atmospheres 
 6.12 atmospheres 
 
 
 Laboratory Exercise j6. 
 
 264. Distillation. — A liquid can be separated from impurities, 
 or from nonvolatile substances held in solution, by boiling it in a 
 closed vessel and condensing the vapor as it passes off through a 
 
 Fig. 127. 
 
 tube connected with the vessel. The process is called distillationy 
 and the apparatus a still. The process may be illustrated by dis- 
 tilling a solution of copper sulphate in water, using apparatus 
 similar to that shown in Fig. 127. The vapor is condensed by 
 
204 Heat 
 
 inclosing a portion of the tube through which it passes within a 
 larger tube in which it is surrounded by a continuous supply of 
 cold water. 
 
 Two or more liquids whose boiling points differ by several de- 
 grees can be separated from one another by distillation. When 
 such a mixture is slowly boiled, the vapor that passes off contains 
 a much higher percentage of the more volatile constituent than the 
 liquid mixture does. Some of the less volatile liquid also passes 
 off, and complete separation can be effecteil only by repeated 
 distillation. This process, which is generally known as fractional 
 distiliation^ is employed on a large scale in separating the constitu- 
 ents of crude petroleum and of coal tar, and in the manufacture 
 of distilled liquors. 
 
 265. Cooling by Expansion. — Work is done upon a gas in com- 
 pressing it, and by a gas in expanding against pressure. In tlie 
 first case, mechanical energy is transformed into heat and the gas 
 is warmed (Art. 212) ; in the second case, some of the heat of the 
 gas is transformed into mechanical energy (kinetic or potential) 
 and the gas is cooled, unless it receives an equal supply of heat 
 during the expansion. 
 
 The cooling of a gas by expansion is often beautifully illustrated 
 when the air is exhausted from the receiver of an air pump. As 
 part of the air is removed, the expansion of the remainder causes 
 a fall of temperature to the dew-point, and some of the water vapor 
 is condensed, forming a fog within the receiver. The fall of tem- 
 perature may be measured by a thermometer placed in the re- 
 ceiver {Exp.^. A further illustration is afforded by directing a 
 jet of air from a tank of compressed air against the bulb of a ther- 
 mometer held a few inches from the opening (Exp.). 
 
 266. Heat of Vaporization. — The Aea/ of vaporization of a 
 liquid is the number of calories required to vaporize one gram of 
 it at its boiling point. The heat of vaporization of water is 536 
 calories, and is greater than that of any other substance. Its value 
 for alcohol is 209 calories, for ether 90 calories, for mercury 
 62 calories. 
 
Vaporization and Condensation 205 
 
 Heat is lost during vaporization by transformation in part into 
 ( I ) mechanical potential energy in causing expansion against atmos- 
 pheric pressure (external work), and in part into (2) molecular 
 potential energy in overcoming cohesion during expansion (in- 
 ternal work). When the heat thus transformed is supplied by 
 the liquid itself and by adjacent objects, as is often the case in 
 evaporation, a fall of temperature results (Art. 252); in boiling, 
 the temperature remains constant, the amount of heat received 
 from the fire or flame being equal to the amount transformed 
 (Art. 262, second law). 
 
 The heat lost during vaporization is all transformed into heat 
 again when condensation takes place. 
 
 267. Determination of the Heat of Vaporization of Water. — 
 The determination of the heat of vaporization of water by the 
 method of mixtures is illustrated by the following example : A 
 brass calorimeter weighing 65 g. contains 200 g. of water at 5°. 
 Steam at 100° is passed into the water till the temperature rises 
 to 40°. It is found on again weighing the calorimeter and con- 
 tents that the weight of the steam condensed in it is 12.1 g. 
 
 Solution. — Let v denote the heat of vaporization of water, — in this case 
 the number of calories given out by i g. of steam in condensing to water at 
 
 lOO"*. 
 
 Rise of temp, of calorimeter and water = 40—5 = 35° 
 
 Heat received by the calorimeter = 65 X 35 X .094 =214 cal. 
 
 Heat received by the water = 200 X 35 = ycxx) cal. 
 
 Heat given out by the steam in condensing to water at 100° = 12.1 z/cal. 
 
 Heat given out by the water from the condensed steam in cool- 
 ing to 40° = 1 2.1 X 60 = 726 cal. 
 12.1 V -H 726 = 70CX) + 214; 
 z/ = 536.2 cal. 
 
 I 
 Laboratory Exercise jg. 
 
 268. The Condensation of Gases. — All substances that exist 
 only as gases at ordinary temperatures exist also as liquids and 
 even as solids at sufficiently low temperatures. Gases may be 
 condensed (i) by cooling, (2) by pressure, or (3) by the two pro- 
 
2o6 Heat 
 
 cesses combined. Sulphur dioxide (the gas formed by burning 
 sulphur) is easily liquefied under atmospheric pressure by a freez- 
 ing mixture of ice and salt, its boiling point being — 10.5°. It can 
 also be liquefied at 15° by a pressure of 3 atmospheres. Carbon 
 dioxide can be liquefied at 15® by a pressure of about 52 atmos- 
 pheres, or by cooling to — 80° under a pressure of one atmosphere. 
 
 Oxygen, nitrogen, air, and hydrogen can be liquefied only at 
 very low temperatures, however great the pressure. The neces- 
 sary reduction of temperature in such cases is effected by the 
 sudden expansion of a ix)rtion of the compressed gas, as in the 
 manufacture of liquid air, or by the evaporation of a gas that is 
 more easily liquefied. Thus in the liquefaction of hydrogen the 
 cooling is effected by the evaporation of liquid air (Art. 238, 
 end). 
 
 Following is a table of boiling points of certain gases under a 
 pressure of one atmosphere : — 
 
 Boiling Points of Liquefied Gases 
 
 C. Abs. 
 
 Hydrogen -243'' 30° 
 
 Nitrogen — 194 79 
 
 Air — 191 82 
 
 Oxygen — 184 89 
 
 Carbon dioxide — 78.2 194.8 
 
 Ammonia — 38.5 234.5 
 
 Sulphur dioxide — 10.5 262.5 
 
 269. Cooling by Evaporation : Applications. — The tempera- 
 ture of our bodies is largely regulated by the evaporation of the 
 perspiration. The process is continuous, although we are con- 
 scious of it only when the perspiration is formed more rapidly 
 than it can evaporate, and hence accumulates on the skin. An 
 average of about a quart of water is evaporated from the skin 
 daily ; and the heat that this requires is taken principally from 
 the body. The importance of more abundant perspiration when 
 the body is subjected to high temperatures and during active 
 
Vaporization and Condensation 207 
 
 exercise is evident. It is not the perspiration, however, but its 
 evaporation that takes heat from the body and thus prevents a 
 dangerous rise of temperature. Hence hot weather is especially 
 oppressive and dangerous when the humidity of the air is high. 
 In the very dry atmosphere of deserts there is comparatively 
 little danger of sunstroke even at a temperature of 100° F., for 
 evaporation is very rapid. 
 
 The use of ammonia (not ammonia water) and carbon dioxide in 
 the manufacture of artificial ice depends upon the absorption of 
 heat by these substances in vaporizing at low temperatures. 
 Ammonia is used on the largest scale. It is liquefied under pres- 
 sure in long pipes exposed to the open air, where the heat gener- 
 ated by the condensation is permitted to escape. It is thence 
 pumped into coils of pipes immersed in a large tank of strong 
 brine, where it vaporizes under a low pressure, cooling the brine 
 in the tank several degrees below zero and freezing cans of fresh 
 water placed in the brine. The vaporized ammonia is pumped 
 from the coils in the tank into the condensing pipes, where it is 
 again liquefied. 
 
 PROBLEMS 
 
 1. What is the pressure in grams per square centimeter exerted by and 
 upon a bubble of steam forming at a depth of 15 cm., when the barometer 
 reads 76 cm. ? 
 
 2. (a) Is rain water distilled water ? (3) Is it perfectly pure ? 
 
 3. Mention the important consequences of the great specific heat, heat of 
 fusion, and heat of vaporization of water, including the connection that any 
 of these properties may have with the effect of the ocean on climate. 
 
 4. A room 4 m. by 5 m. and 3 m. high is warmed by a steam heater. 
 Assuming no loss, what weight of steam must be condensed in the heater to 
 warm the room from 10° C. to 18° C. ? (Density of the air 1.25 g. per cdm.; 
 specific heat of air = .237.) 
 
 5. Water kept in porous earthenware jars in warm weather remains several 
 degrees below the temperature of the air. Explain. 
 
 6. What quantity of heat is required to convert 850 g. of ice at — 20° into 
 steam at ICX)° ? 
 
 7. How much heat is given out by 500 g. of steam at 100" in condensing 
 and cooling to water at 30° ? 
 
2o8 
 
 Heat 
 
 Fig. 12 
 
 VIII. Mutual Translormations of Heat and Other Forms of Energy 
 
 270. The Mechanical Equivalent of Heat. — The numerical re- 
 lation between heat ;>n<l mechanical energy was first determined 
 by Dr. Joule (Art. 211). The principle of his most successful 
 
 method is illustrated by 
 the simplified apparatus 
 shown in Fig. 128. A 
 set of paddles attached 
 to a vertical axle is made 
 to rotate in a calorimeter 
 filled with water, by the 
 fall of a weight attached 
 to a cord wound round 
 the axle. Stationary pro- 
 jections, extending inward 
 between the paddles from 
 the sides of the calo- 
 rimeter, prevent the water from revolving bodily with the paddles. 
 The work done by the weight in falling is converted into heat 
 within the calorimeter through the friction due to the agitation of 
 the water. The mechanical energy thus transformed is measured 
 in gram-centimeters by the product of th^ weight and the distance 
 through which it descends. The number of calories generated is 
 computed from the weight of the water and the calorimeter, the 
 specific heat of the calorimeter, and the rise of temperature. 
 Hence, after making necessary allowances for sources of error, 
 the equivalent of a certain number of calories is obtained in 
 gram-centimeters of mechanical energy ; from which the equiva- 
 lent of one calorie is computed. The latter is called the mechanical 
 equwaUnt of heat. 
 
 The value now accepted for this equivalent, after repeated 
 determinations by different experimenters, is 42,800 g.-cm.; i.e. 
 42,800 g.-cm. of mechanical energy will generate one calorie when 
 transformed into heat, and vice versa. 
 
Transformations of Energy 209 
 
 PROBLEMS 
 
 1. Compute the rise in temperature of the water in the calorimeter of 
 Joule's apparatus (Fig. 128) from the following data : Mass of the weight 
 used = 30 kg. Distance through which the weight descends = 25 m. 
 Weight of water in the calorimeter = 2 kg. Weight of the copper calo- 
 rimeter = I kg. 
 
 2. (a) A mass of iron weighing i kg. falls upon a stone from a height of 
 100 m. How much heat is generated, assuming that the energy is all trans- 
 formed into heat ? (d) If half of the heat is generated in the mass of iron, 
 what is its rise of temperature ? 
 
 3. From what height must a mass of iron fall in order that the heat gener- 
 ated when it strikes shall be sufficient to raise its temperature one degree, 
 assuming that the energy is all transformed into heat in the iron itself? 
 
 4. A lead bullet strikes a target with a velocity of 300 m. per sec. 
 Assuming that 90 per cent of its energy is transformed into heat in itself, 
 what is its rise of temperature? 
 
 271. The Conservation of Energy. — All the examples of the 
 transference and transformation of energy that have been con- 
 sidered serve to illustrate the fact that when energy is lost by one 
 body or in one form it always reappears in another body or in 
 another form. The experiments of Joule and other physicists led 
 to the view that the energy always reappears in equivalent amount. 
 In fact, since the middle^ of the last century, it has been an ac- 
 cepted principle that energy can neither be created nor destroyed. 
 This principle, which is of the greatest scientific and practical 
 importance, is known as the conservation of energy. No person 
 acquainted with modern science regards a "perpetual motion 
 .machine " as a possibility. 
 
 272. Transformations of Solar Energy. — The energy of winds 
 is directly traceable to the sun, for the winds are due to the un- 
 equal heating of different portions of the earth's surface. Water 
 power has the same ultimate source ; for it is by means of energy 
 from the sun that water is evaporated from the oceans and carried 
 through the agency of winds to the highest plateaus and moun- 
 tains. 
 
2IO Heat 
 
 The leaves of plants absorb carbon dioxide from the air, and, 
 under the action of sunlight, separate it into its constituents 
 (carbon and oxygen). The carbon unites with water sent up from 
 the roots, forming starch ; the oxygen is given off to the air again. 
 From the starch and various earthy materials absorbed with water 
 from the soil, the different substances are formed which are needed 
 for the growth of the plant. The work done in the leaves re- 
 quires an expenditure of energy. This is obtained from the sun- 
 light absorbed by the green coloring matter in the leaves (the 
 chlorophyll). Radiant energy is thus transformed and stored up 
 as chemical potential energy in the substance of the plant itself. 
 The energy of coal is also stored solar energv- ; for the coal beds 
 are the remains of immense forests that grew long before man ap- 
 peared upon the earth. 
 
 Animals, as already noted (Art. 149), derive the energy for all 
 their bodily activities from their food, and hence originally from 
 the sun, whether the food be of vegetable or of animal origin. 
 
 The sun is, in fact, directly or indirectly, the cause of nearly all 
 terrestrial phenomena. The only exceptions of any magnitude 
 are the tides and phenomena due to the slow shrinking of 
 the earth and to the heat of its interior, such as volcanic action, 
 earthquakes, the formation of mountains, etc. Although the 
 interior of the earth is intensely hot, the heat is conducted to 
 the surface so slowly that its effect upon the temperature of 
 the atmosphere is inappreciable. 
 
 273. Amount of Solar Radiation. — From a law of radiation it 
 is known that the amount of energy radiated from any portion of 
 the sun's surface in a minute is 46,000 times as great as that 
 received by an equal area of the earth's surface in the same time. 
 Since the latter has been approximately determined by experiment, 
 the rate at which the sun is giving out energy is known with the 
 same degree of accuracy. This amounts to over 100,000 horse 
 power per square meter of the sun's surface, acting continuously. 
 To maintain this rate of radiation by combustion " would require 
 the hourly burning of a layer of the best anthracite coal from 
 
Transformations of Energy 2 1 1 
 
 sixteen to twenty feet thick over the sun's entire surface, — a ton 
 for every square foot of surfice, — at least nine times as much as 
 the most powerful blast furnace in existence. At that rate the 
 sun, if made of solid coal, would not last 6000 years." ^ The earth 
 receives only about ^^nn^Ji^^T^T^ ^^ ^^^ ^^^^^ radiation from the 
 sun. 
 
 274. Source of the Sun's Energy. — The source of the sun's 
 energy is a question of the greatest scientific interest. A direct 
 answer not being obtainable, various theories have been suggested, 
 all of which recognize the principle of the conservation of energy. 
 The sun's heat cannot be maintained by combustion, for in that 
 case it would have been burned out long ago. " Nor can it be 
 simply a heated body cooling down. Huge as it is, an easy calcu- 
 lation shows that its temperature must have fallen greatly within 
 the last 2000 years by such a loss of heat, even if it had a specific 
 heat higher than that of any known substance."^ 
 
 The only theory that has met with general acceptance is Helm- 
 holtz's theory of solar contraction. This is that " the heat neces- 
 sary to maintain the sun's radiation is principally supplied by the 
 slow contraction of its bulk, aided, however, by the accompanying 
 liquefaction and solidification of portions of its gaseous mass. 
 When a body falls through a certain distance gradually, against re- 
 sistance, and then comes tp rest, the same total amount of heat is 
 produced as if it had fallen /r^^/^, and been stopped instantly. If, 
 then, the sun does contract, heat is necessarily produced by the 
 process, and that in enormous quantity, since the attracting force 
 at the solar surface is more than twenty-seven times as great as 
 terrestrial gravity, and the contracting mass is immense. . . . 
 Helmholtz has shown that under the most unfavorable conditions 
 a contraction in the sun's diameter of about two hundred and 
 fifty feet a year would account for the whole annual output of 
 heat."* At this rate, a period of 9000 years would be required 
 for a total contraction sufficiently great to be detected by measure- 
 ment with the best astronomical instruments. 
 1 Young's General Astronomy. 
 
212 
 
 Heat 
 
 275. The Steam Engine. — Many illustrations have been given 
 of the transformation of mechanical energy into heat. Such trans- 
 formations result in a loss of 
 available energy. The op- 
 posite transformation of heat 
 into mechanical energy is 
 the useful one, and is accom- 
 plished by various forms of 
 heat engines, of which the 
 steam engine is the most im- 
 portant type. Figure 129 
 represents a horizontal sec- 
 
 
 ^s 
 
 51 
 
 Fig. 129. 
 
 tion of the cylinder, r, and steam chest, //, of the stationary 
 engine shown in Fig. 130. Steam from a boiler enters the steam 
 chest through the pipe, /, whence it is admitted mto the cylinder at 
 the ends alternately, under the control of a sliding valve, v, which 
 
 Fig. 130. 
 
 is moved back and forth by the rod, r'. While the steam is 
 entering at either end of the cylinder, the other end is con- 
 nected by means of the sliding valve with an exhaust pipe at e, 
 
Transformations of Energy 2 1 3 
 
 through which the steam on that side escapes under reduced 
 pressure into the open air or into a condensing chamber, where 
 it is condensed by cold water. In the position shown in the 
 figure, the steam enters at the left end of the cylinder, and 
 pushes the piston and piston-rod, r, to the right. Before the 
 end of this stroke, the valve moves to the left, shutting off the 
 steam from the left end of the cylinder, and admitting it into 
 the right end as soon as the stroke is completed. The to- 
 and-fro motion of the piston rod causes the connecting rod, R 
 (Fig. 130), to turn the shaft S. The large fly wheel attached to 
 the shaft regulates the motion, which would otherwise be very 
 irregular in consequence of the varying magnitude and direction 
 of the force exerted by the connecting rod. The valve rod is 
 operated by a device on the shaft, to which it is attached. 
 
 A locomotive has two engines, one at each side of the boiler at 
 the front end. The piston of each of these engines is attached by 
 means of the connecting rod to the driving wheels on the same 
 side of the boiler. 
 
 276. Transformation of Energy by a Steam Engine. — The 
 valves of most engines are capable of adjustment so that the sup- 
 ply of steam can be cut off at any instant during the stroke. It is 
 generally cut off before the middle of the stroke, and often at 
 quarter stroke or even less. The stroke is completed by the ex- 
 pansion of the steam then in the cylinder. During this expan- 
 sion the temperature of the steam falls (Art. 265). Thus, if it is 
 admitted under a pressure of 6 atmospheres and discharged into 
 the air, its temperature falls from about 160° to 100° (see Table, 
 Art. 263). The heat lost by the steam in expanding is the equiva- 
 lent of the work that it does against the piston. 
 
 Only a comparatively small part of the energy of the steam is 
 available for doing the work of driving the engine. The heat re- 
 quired to convert the water supplied to the boiler into steam at 
 the temperature at which it leaves the cylinder of the engine is 
 all lost. The efficiency of an engine is increased by utilizing the 
 expansive force of the steam to the fullest possible extent. 
 
CHAPTER IX 
 
 SOUND 
 
 I. Origin and Transmission of Sound 
 
 277. Sounding Bodies. — It can be shown in a number of ways 
 that any sounding body is in a state of vibration. Although the 
 motion is always too rapid to follow with the eye, the fact of 
 motion is nevertheless often evident from the appearance of the 
 body. For example, the string of a violin or other stringed 
 instrument assumes a spindle shape and becomes indistinct when 
 sounding ; and the prongs of a tuning fork present a similar ap- 
 pearance. The vibrations can also generally be felt and, unless 
 very weak, can be shown by their mechanical effects. Thus a 
 shower of spray is thrown up when the prongs of a vibrating fork 
 are dipped in water {Exp.) ; and a pith ball or a bit of cork tied 
 to the end of a thread is driven away when suspended so as to 
 touch the edge of a sounding bell or glass jar {Exp.). 
 
 278. The Vibration of a Sounding Body. — As the free end of a 
 long, straight, steel spring is gradually shortened by clamping it 
 nearer the end in a vise, the vibrations become more and more 
 rapid, and finally produce an audible sound. As the vibrating 
 end is further shortened, — and the rapidity of vibration thereby 
 increased, — the pitch of the sound rises, i.e. the spring sounds 
 a higher tone {Exp.). Pitch will be studied later ; for the present 
 we merely observe that it is determined by the rate of vibration, 
 and that the more rapid the vibration the higher the pitch be- 
 comes {Exp. with siren). 
 
 As the length of the free end of the vibrating spring is increased, 
 its vibration does not change in character ; it finally ceases to pro- 
 duce an audible sound only because its motion is too slow. The 
 
 214 
 
Origin and Transmission of Sound 215 
 
 Fiu. 131. 
 
 character of sound vibrations can therefore be studied by adjust- 
 ing the spring to one or two swings per second {Exp.), It 
 will then be seen that the motion of 
 its end is similar to that of the bob of a 
 pendulum. Starting at D' (Fig. 131), 
 the motion is accelerated to Z), where it 
 is most rapid, and is thence retarded to 
 D". The motion is maintained by the 
 elastic force of the spring, which, like the 
 tangential component of the weight of a 
 pendulum bob, is greater, the greater the 
 displacement from the position of rest. 
 
 In the study of sound, one vibration 
 includes a swing both ways (from D* to 
 Z>" and back to /^'). The amplitude of vibration is the extent 
 of motion on either side of the position of rest. The rate of 
 vibration is independent of the amplitude ; the vibration is there- 
 fore regular or periodic. This can be shown by counting the 
 number of vibrations with different amplitudes when the motion 
 is sufficiently slow; but it is proved for all sounding bodies by 
 the fact that the pitch does not change as the sound becomes 
 fainter {Exp. with tuning fork). The rate of vibration is meas- 
 ured by the number of vibrations per second. 
 Laboratory Exercise 40. 
 
 279. The Vibration of a Tuning Fork. — A tuning fork is so 
 frequently used in sound experiments that it is important 
 to know definitely what its motion is when vibrating. A 
 quick blow upon one prong in the direction of the other 
 sets both prongs in vibration. Their motion is always 
 toward each other and from each other in succession 
 (Fig. 132). This transverse vibration of the prongs is 
 accompanied by a vibration of the stem in the direction 
 of its length {/ongitudina/ wihrsition) , which can be dis- 
 tinctly felt by placing the stem of a sounding fork against 
 
 Flo. 132. 
 
 the teeth. 
 
2i6 Sound 
 
 280. The Transmission of Sound. — Solid Media. — When the 
 stem of a sounding fork is pressed against the top of a table, the 
 sound becomes much louder {Exp.). The stem in vibrating 
 strikes a rapid succession of blows upon the table, and the impulses 
 thus imparted cause the table to vibrate in unison with the fork 
 — the table becomes a sounding body. 
 
 With the stem of a sounding fork against one end of a wooden 
 rod (meter stick), the sound instantly becomes loud when the 
 other end of the rod is touched to a table {Exp.). The rod 
 transmits the vibrations to the table, or, as we commonly say, the 
 sound travels through the rod. 
 
 Any substance through which sound travels is called a medium 
 for the transmission of sound, or a sound medium (plural media). 
 The above experiment succeeds equally well with rods of different 
 material; in fact, any highly elastic (rigid) solid is a good sound 
 medium. The sound of a distant train is plainly heard through 
 the rails, and the tread of a galloping horse can be heard for miles 
 through the earth by putting the ear close to the ground. On 
 the other hand, soft and yielding solids " deaden " sound, for they 
 transmit it poorly. Thus little or no sound will be heard when a 
 rubber stopper or a roll of cotton wool is placed between the 
 sounding fork and the table {Exp.). 
 
 Liquid Media. — Sound can be transmitted from a fork to a 
 table through a jar of water or other liquid ; but, in order that the 
 liquid may take up the vibrations with suf- 
 ficient intensity, the area of the vibrat- 
 ing surface in contact with it must be 
 rather large. The stem of the fork is 
 therefore firmly inserted in a hole in a 
 small block of wood, and the block touched 
 to the liquid {Exp.). Sounds made under 
 the surface of a large body of water 
 are transmitted through it over long dis- 
 tances. 
 Fig. 133. Gaseous Media. — The air is the usual 
 
Origin and Transmission of Sound 2 1 7 
 
 medium by which sound vibrations are transmitted to the ear. 
 That sound is not merely transmitted through the air but by it 
 is shown by the following experiment : A loud-sounding body, 
 as an electric bell (Fig. 133) or a metronome, is placed on a soft 
 cushion or suspended from wires under the receiver of an air 
 pump, and the air exhausted. The sound grows continually fainter 
 as the exhaustion proceeds, and becomes inaudible if a good 
 vacuum is secured. The sound is restored when air or any other 
 gas is admitted into the receiver (Exp.) 
 
 Sound, unlike radiant energy, is not transmitted through a vac- 
 uum ; it is transmitted only by elastic substances — solid^ liquid, 
 and gaseous. 
 
 281. Wave Motion. — A sounding body is the center of a peri- 
 odic disturbance consisting of impulses exerted in rapid succession 
 upon any body in contact with it, including the surrounding air ; 
 and these bodies serve as media for the transmission of the dis- 
 turbance. Since this disturbance is invisible, it will be helpful in 
 the study of its nature and the manner of its transmission to con- 
 sider briefly a few cases of visible motion that are in some respects 
 like it. 
 
 When a stretched rubber tube or a spiral spring, three or four 
 meters long, is struck a sharp blow near one end, a distortion is 
 produced which travels rapidly as a wave to the other end. By 
 tying strips of cloth to the tube at different points, it can be seen 
 that, as the wave passes any point, that point moves quickly out 
 in a direction at right angles to the length of the tube and returns 
 {Exp.). The curved form that we call the wave is, in fact, 
 passed from point to point along the tube by the transverse vibra- 
 tion of successive portions of the tube. A distortion of a different 
 character is started by stretching a portion of the tube near one 
 end either considerably more or less than the remainder and sud- 
 denly releasing that portion. The strips of cloth will now indicate 
 a to-and-fro or longitudinal vibration as the disturbance passes 
 {F.xfi.). 
 
 The waves that pass over a field of grain when a strong wind is 
 
21 8 Sound 
 
 blowing are due to the forward bending and springing back of 
 successive stalks of grain, a row of stalks at right angles to the 
 direction of the wind moving in unison. The vibration of each 
 head of grain is mainly longitudinal with respect to the direction 
 in which the waves travel. 
 
 The motion of the water in transmitting a water wave is mainly 
 a transverse vibration ; as is indicated by the rise and fall of any 
 floating object as the waves pass under it. (A chip in a tub of 
 water serves well for experiment.) A wave is not formed of the 
 same water as it travels ; it is the disturbance that travels, not the 
 water. When a pebble is dropped into a pool of still water, a train 
 of circular waves travels outward from the point where the pebble 
 strikes the surface. The waves are circular because the disturb- 
 ance is transmitted with equal velocity in all directions over the 
 surface ; they are concentric because they all originate at the same 
 point. Each wave consists of a crest and a trough. The length 
 of a wave is the distance from crest to adjacent crest or from 
 trough to trough, measured radially^ i.e. toward or from their 
 common center. The waves rapidly decrease in height as they 
 travel outward because their energy is transferred (radially out- 
 ward) to an ever increasing body of water. 
 
 282. Sound Waves. — Suppose a sounding fork to be held at 
 an end of a tube of indefinite length. As the nearer prong moves 
 taivard the tube, it pushes against the air immediately in front of 
 it. As this body of air is driven forward by the advancing prong, 
 it is slightly compressed, and hence expands on the farther side, 
 
 Fig. 134. 
 
 causing compression of the air in advance of it (Fig. 134). By 
 the repetition of this process between successive portions of the 
 air, the compression is rapidly transmitted along the tube. As 
 
Origin and Transmission of Sound 219 
 
 the compression passes any section of the tube, the air particles 
 in that space move forward in a body. Each particle starts for- 
 ward when the front of the condensation strikes it, and stops as 
 soon as the condensation has passed. 
 
 As the prong of the fork moves from the tube during the 
 second half of a vibration, the body of air on the side toward 
 the tube follows the fork up in its retreat, and expands in doing 
 
 ? 
 
 w 
 
 Fig. 135. 
 
 so (Fig. 135). This causes air from a greater distance to move 
 toward the fork in restoring the density and pressure in that re- 
 gion. Thus a rarefaction is transmitted along the tube by the 
 backward motion of the air particles. The rarefaction follows the 
 preceding compression (or condensationy as it is generally called), 
 and is itself followed by the next condensation (Fig. 136). 
 
 Fig. 136. 
 
 A conden g^M'nq and flr^j'^Cfi"^ mrpfhrfinn together ^Constitute. 
 a s ound _i pave. _A sound wave is transmitted, or propagated, by 
 a vibration of the air particles — forward in the condensation and 
 backward in the rarefiiction, as indicated by the arrows in the 
 figures. This vibration is in a straight line parallel to the direc- 
 tion of propagation of the wave, i.e. it is io?igifiidinal ; its ampli- 
 tude is very small — generally much less than that of the sounding 
 body. 
 
 A sound wave is represented graphically by a curved line as 
 in the lower part of Fig. 136. The curve above the straight line 
 
220 Sound 
 
 represents the condensation, the curve below the line the rare- 
 faction. This representation must not be regarded as a picture : 
 a sound wave does not consist of a crest and a trough as a water 
 wave does. 
 
 283. Sound Waves in the Open Air. — When a sounding body 
 is surrounded by an open body of air, the waves travel outward 
 from it in all directions. Usually a condensation starts out on 
 one side simultaneously with a rarefaction on another. In the 
 simpler case where the disturbance is the same at the same 
 instant all round the body (as when a firecracker is exploded in 
 the aiO, the waves travel outwar'l n< .-.,„.-.• „/;7V spherical shells^ 
 
 FIG. 137. 
 
 represented in section by Fig. 137. The waves are spherical 
 because they are transmitted radially with equal velocity in all 
 directions. 
 
 A wave front is the surface bounding the front of a conden- 
 sation. Under the conditions just considered it is a spherical 
 surface. A wave length is the distance, measured radially, be- 
 tween adjacent wave fronts or between any corresponding parts of 
 adjacent waves. 
 
 A sound is (i) a set or train of sound waves, or (2) the sensa- 
 tion produced by such a set of waves through the organs of hear- 
 ing. In physics the word is generally used in the first sense. 
 
 284. Energy of Sound Waves; Intensity of Sound. — A part 
 of the energy of a sounding body is transformed into heat by 
 molecular friction within the body itself; but most of it is trans- 
 ferred to the air or other medium in which sound waves are pro- 
 
Origin and Transmission of Sound 221 
 
 duced. The more rapidly the energy is thus transferred, the 
 greater will be the intensity (or loudness) of the sound and the 
 more quickly will the sound cease. 
 
 The rate of transference of energy from the sounding body to 
 the medium depends upon (i) the amplitude of vibration of the 
 body, (2) the area of the vibrating surface, and (3) the density and 
 elasticity of the medium. 
 
 285. Effect of Amplitude. — The gradual dying away of the 
 sound of a bell, a piano wire, a tuning fork, etc., is due to the 
 diminishing amplitude of vibration as the body approaches a state 
 of rest. With a decrease of amplitude the blows of the vibrating 
 surface against the surrounding air grow less vigorous and the 
 sound waves are correspondingly weaker. 
 
 286. Effect of the Area of the Vibrating Surface. — A narrow 
 vibrating surface cuts through the air, producing little effect ; the 
 air slips round it. A broad surface catches the air and carries it 
 bodily forward. This explains why the sound of a tuning fork is 
 very faint when it is held in the hand and loud when it is touched 
 to a table. In the latter case the vibrations are transmitted to 
 the air almost entirely by the vibrating table. The music of a 
 violin or guitar comes practically entirely from the body of the 
 instrument, and the music of a piano from the sounding board 
 on which the wires are strung. 
 
 287. Effect of the Density and Elasticity of the Medium. — In 
 the experiment with the sounding body under the receiver of an 
 air pump, it was observed that the sound grows fainter as the 
 exhaustion continues, /. e. as the density of the remaining air is 
 diminished. The reason is obvious : there is less matter in 
 motion in the wave of rarefied air, hence there is less energy — 
 kinetic energy being proportional to the mass of the moving body. 
 
 A number of experiments have already shown that sound is 
 louder when transmitted through elastic solids than when trans- 
 mitted through the air. A sounding fork held first in the hand 
 then against a table is an excellent illustration. The rigid wood 
 offers much greater resistance to the blows of the fork than the 
 
222 
 
 Sound 
 
 air does, and hence receives a correspondingly greater amount 
 of energy with each vibration.* Observe also that the fork conies 
 to rest much more quickly when held against the table (Exp.), 
 
 288. Effect of Distance on the Intensity of Sound. — It is well 
 known that a sound grows fainter with increase of distance from 
 the sounding body. Thfe principle of the conservation of energy 
 affords an explanation and a definite statement of the law of 
 decrease. 
 
 As a sound wave travels in the open air, its distance from the 
 source is the radius of its surface (the spherical wave front). 
 Now it is proved in geometry that the surfaces of two spheres 
 
 (or equal fractions of their 
 respective surfaces) are pro- 
 portional to the squares of 
 their radii. Hence, since 
 the length of a wave remains 
 constant, its volume and the 
 amount of matter in it are 
 proportional to the square 
 of the distance it has trav- 
 eled in the open air. This 
 f''°- '3^- is illustrated in Fig. 138, 
 
 which represents a section of a spherical wave at a distance d^ 
 from the source and again at twice that distance, or //g. If Z'l de- 
 notes the volume of the section at the first distance, v>i its volume 
 at the second distance, then z^g : z'l : • 4 • i, or z/j : Z'l : : //g* : di. 
 
 Since the energy of a sound wave is transmitted through the 
 medium with the wave, the intensity of the sound (or the 
 amount of energy per unit volume) is inversely proportional to 
 the volume of the wave. Hence, from the above proportion, the 
 intensity is inversely proportional to the square of the distance from 
 the source y when the sound is traveling in the open air. 
 
 1 That more work can be done upon the body that offers the greater resistance 
 is plainly evident to one who strikes out at something with his fist and only suc- 
 ceeds in " hitting the air." 
 
Origin and Transmission of Sound 223 
 
 289. Dissipation of the Energy of Sound. — We have assumed 
 that the total energy of a sound wave remains constant as it travels. 
 This is not strictly true, for the energy is more or less slowly trans- 
 formed into heat by friction in any medium. In the end it is all 
 dissipated in this manner. Hence the intensity of sound decreases 
 somewhat more rapidly than the law stated above indicates. 
 
 290. Confined Sound Waves. — Sound travels long distances in 
 elastic media with very little loss of intensity if the waves are pre- 
 vented from increasing in size. This is the principle of the speak- 
 ing tube, which is a long metal tube of small diameter, used for 
 communication between different rooms of a building or between 
 some room and the street door. The tube does not readily take 
 up the vibrations of the air, hence they are almost completely con- 
 fined within it. The intensity of the sound is slowly diminished 
 by friction. 
 
 The transmission of sound over long distances through the rails 
 of a track and through stretched wires and strings is due to the 
 same cause : the vibrations are largely confined to the solid 
 medium. This fact is utilized in the acoustic or siring telephone. 
 
 PROBLEMS 
 
 1. Are sound waves transmitted by the elasticity of form or of volume of 
 the medium ? 
 
 2. (fl) By what force are the waves transmitted along a stretched rubber 
 tube or spring ? (J)) By what force are waves transmitted over the surface 
 of water ? 
 
 3. How does the energy of sound differ from heat ? 
 
 4. Is the wave front of a water wave a line or a surface ? 
 
 5. {a) What angle does the direction of propagation of a water wave 
 make with the wave front ? (Jj) The direction of propagation of a soupd 
 wave ? 
 
 6. {a) Would a sounding body continue to vibrate longer in water or in 
 the air ? (^) In the air or in a vacuum ? Why ? 
 
 7. How does the intensity of sound at a distance of 5 m. from the source 
 compare with its intensity at 10 m. ? at 15m.? at 20 m. ? 
 
 8. At what distance is the intensity of sound one fourth as great as at 
 100 ro. ?. one half as great ? 
 
224 Sound 
 
 291. The Velocity of Sound in Air. — It is a familiar fact that a 
 distant phenomenon that is accompanied by a sound is seen before 
 it is heard. At a distance of a few hundred feet, the blow of an 
 ax is heard after the ax is raised for the next stroke ; a flash of 
 Hghtning is often seen many seconds before the thunder is heard, 
 although they are produced simultaneously ; the whistle of a dis- 
 tant locomotive may not reach the ear until the cloud of " steam " 
 has disappeared. Now the time required for light to travel terres- 
 trial distances is wholly inappreciable (the velocity of light being 
 1 86,000 mi. per sec.) ; hence the time that elapses between the 
 sensations of sight and hearing in such a case is the time occupied 
 by the sound in traveling from the sounding body to the observer, 
 and, from the measured time and distance, the velocity of sound 
 can be computed. 
 
 Observations have been repeatedly taken for this purpose by 
 firing a cannon at each of two stations several miles apart, and 
 noting the time between the flash and the report as observed at 
 the other station. By taking observations at each of the stations 
 alternately, the effect of the wind is eliminated. The average of 
 the best determinations is 332 m. or 1090 ft per sec. at 0°, At 
 20° the velocity is 344 m. or 11 29 ft. per sec. 
 
 That the velocity of sound is independent of its pitch and intensity 
 is proved by the fact that all the sounds produced simultaneously 
 by an orchestra are heard simultaneously at all distances. 
 
 Laboratory Exercise 41. 
 
 292. Effect of the Elasticity and Density of the Medium on the 
 Velocity of Sound. — Experiment shows that a wave travels more 
 rapidly along a stretched rubber tube when the tension is 
 increased, — a result to be expected, since the propagation of the 
 wave is due to the elastic force (tension) of the cord {Exp.). The 
 elasticity and velocity are not, however, proportional. 
 
 A wave travels more slowly along a stretched rubber tube that is 
 filled with shot or sand than it does along an empty tube of the 
 same size and under the same tension — a given force moves a 
 greater mass more slowly {Exp.). 
 
Origin and Transmission of Sound 225 
 
 It can be shown by mathematical reasoning based on the second 
 law of motion that the velocity of a sound wave is proportional 
 to the square root of the elasticity of the medium and inversely 
 proportional to the square root of its density. The velocity of 
 sound in water has been found by experiment to be 1435 "^* P^^ 
 sec. at 8.1°, — a velocity more than four times as great as in air. 
 Thus, in comparison with air, the retarding effect of the greater 
 density of water is more than offset by the accelerating effect of 
 its still greater relative elasticity. The same is true in a still greater 
 degree of soHds, the velocity in glass and steel being about fifteen 
 times as great as in air. 
 
 The velocity of sound increases with the temperature because, 
 with a rise of temperature, the air expands and its density dimin- 
 ishes, while its elasticity remains unchanged. An increase of 
 pressure increases the elasticity and density proportionally, hence 
 a change of pressure does not affect the velocity. 
 
 293. Reflection of Sound : Echoes. — When sound waves strike 
 a large surface, as a cliff or the side of a building, they are 
 reflected. The reflected sound is called an echo when it reaches 
 the ear long enough after the original sound to be distinguished 
 from it. This requires an interval not less than \ sec, during 
 which time sound travels about 68 m. ; hence a distinct echo will 
 not be heard unless the reflecting surface is at a distance not less 
 than about 34 m. from the source of sound. At less distances the 
 direct and the reflected sounds blend together. They are sensibly 
 coincident when the reflecting surface is not more than a few 
 meters from the source of the sound, and the result is an increased 
 loudness. It is for this reason that reflecting surfaces are often 
 erected behind band stands. When the distance is nearly suffi- 
 cient for an echo, the direct and the reflected sounds are mixed 
 confiisedly, causing indistinctness. This is often noticeable in 
 large halls. 
 
 The change in the shape and the direction of propagation of 
 sound waves caused by reflection from a plane surface is shown 
 in Fig. 139, which represents a section of a train of waves origi- 
 
226 
 
 Sound 
 
 nating at (9 and reflected by a plane surface, AB, The circular 
 arcs represent the wave fronts, and the lines OA, OB, and 
 /)N^ y^ / . ^^ directions of propagation 
 
 before reflection and AD, BR^ 
 and CF directions after reflec- 
 tion. The waves after reflection 
 have the same shape and direc- 
 tion of propagation as they 
 would have if they originated 
 at 0\ a point on the perpen- 
 dicular from the source of sound 
 to the reflecting surface and at 
 an equal distance behind it. 
 '*'■ *^ After reflection from a con- 
 
 cave surface, sound waves increase less rapidly in size, and are 
 consequently propagated with comparatively little loss of intensity, 
 almost as if they were confined in an inclosed space. Hence the 
 large reflecting walls at the rear of band stands are concave. 
 When a sounding body is at the proper distance from a concave 
 surface, the reflected waves decrease in size and increase in inten- 
 sity as they travel toward a point. Light is similarly reflected 
 from concave mirrors, such as are used behind wall lamps and the 
 head lights of locomotives and street cars. This efl'ect of concave 
 surfaces will be more fully discussed under the subject of light. 
 Laboratory Exercise 42, 
 
 PROBLEMS 
 
 1. {a) Give two reasons why sound travels farther through the rails of a 
 track than it does through the air. {b) Why does it travel faster through the 
 rails? 
 
 2. How would music be affected if sounds of different pitch or intensity 
 traveled with different velocities ? 
 
 3. A rifle is fired on one side of a canyon and 3.2 sec. later the echo is 
 heard from the opposite side. The temperature is 20°. What is the width 
 of the canyoQ ? 
 
Properties of Musical Sounds 227 
 
 4. A flash of lightning is seen 12.5 sec. before the thunder is heard. At 
 what distance did the lightning occur, the temperature being 20° ? 
 
 5. The mean distance of the sun from the earth is 93,(XX),C)00 miles. How 
 long after an explosion occurs upon the sun would we hear it if air at 0° 
 were provided as a medium for the transmission ? (Light reaches us from the 
 sun in 499 sec.) 
 
 II. Properties of Musical Sounds 
 
 294. Properties of Musical Sounds. — Musical sounds have 
 three characteristics or properties ; namely, (i) intensity or loud- 
 ness, (2) pitch, and (3) quality or timbre. 
 
 Loudness. — Intensity has already been considered. It is de- 
 termined by the energy of the sound waves. Loudness refers to 
 the sensation produced upon the ear. It increases with the 
 intensity ; but we cannot say that they are proportional, as loud- 
 ness cannot be measured. Loudness depends in part upon the 
 pitch, a shrill sound being more distinctly heard than one of 
 equal intensity but of low pitch. 
 
 Pitch. — Sounds of definite pitch are produced only by bodies 
 whose vibrations are regular and periodic. The waves of such a 
 sound are of equal length and are sent out from the sounding 
 body at equal intervals of time. An increase in the number of 
 vibrations produces what is called a higher pitch {Exp.). 
 
 Quality or timbre is that property by which we distinguish be- 
 tween sounds produced by different bodies, even when they have 
 the same pitch and intensity. We know at once from the quality of 
 the sounds whether a piece of music is being played on a violin, 
 a flute, or a cornet. We recognize familiar voices principally by 
 their quality, although pitch, loudness, and peculiarities of pronun- 
 ciation are also of assistance. The cause of quality will be con- 
 sidered later. 
 
 295. Differences between Musical Sounds and Noises. — The 
 pitch, intensity, and quality of musical sounds remain constant for 
 appreciable intervals of time and do not change irregularly. A 
 musical sound is often called a tone or note. All other sounds are 
 
228 Sound 
 
 called noises. A noise usually consists of a number of sounds pro- 
 duced by the vibration of the sounding body in parts or segments. 
 These vibrations not only differ among themselves, but are also 
 irregular in rate and amplitude ; they are discordant, unsteady, 
 and nonperiodic. A noise has therefore no definite pitch, and 
 its quality is nonmusical. 
 
 296. Measurement of Pitch. — The pitch of a note may be 
 expressed either relatively or absolutely. It is expressed rehi- 
 tively by stating its relation to some other note, generally the 
 keynote of the musical composition in which it occurs (Art. 302). 
 The relative pitch of a note is easily recognized by a trained ear. 
 
 ThQ abso/ufe pitch of a note is measured by the number of 
 vibrations per second of the sounding body. This is called the 
 vibration number ox frequency of the note. Thus the absolute 
 pitch of the C fork which corresponds to " middle C " of the 
 piano or organ is 256 ; />. the prongs of this fork make 256 
 vibrations per second. In physics the word pitch generally 
 signifies the absolute pitch or vibration number of the note. 
 
 'llie vibration number of a sound can be determined experi- 
 mentally either by causing the sounding body to make a perma- 
 nent record of its vibrations (called the graphic method), or by 
 comparing it with a sound of known pitch. The graphic method 
 
 is illustrated in part in 
 Fig. 140. A projecting 
 point attached to one 
 prong of the fork traces 
 f.,^, a wavy line on a piece 
 
 of smoked glass as the 
 vibrating tbrk is drawn over the glass, or the glass pulled from 
 under the fork. Each wave of the line is a record of one vibra- 
 tion of the fork. If by some additional device, not shown in the 
 figure, the time occupied by the fork in tracing the line is deter- 
 mined, the number of vibrations per second can be computed. 
 Comparison with sounds of known pitch, such as the notes of 
 tuning forks, will serve all the purposes of elementary physics. 
 
Properties of Musical Sounds 229 
 
 297. The Relation between Pitch, Wave Length, and Velocity. — 
 Let n denote the vibration number of a sounding. body, / the 
 length of the waves that it produces in a given medium, and v 
 the velocity of sound in that medium. Since a wave starts from 
 the body with each vibration, a train of n waves is sent out in one 
 second, the last of which will be on the point of leaving the body 
 at the end of the second. During the second the first wave of 
 the train travels the distance v ; hence, the n waves extend over 
 that distance. From which v = /n. 
 
 Assuming the velocity of sound in the medium to be known, we 
 can from this relation compute the wave length of a sound of 
 given pitch or the pitch of a sound of known wave length. 
 
 Laboratory Exercise 43. 
 
 298. Interference of Sound. — When a sounding fork is held 
 close to the ear and is slowly rotated about the stem as a vertical 
 axis, it will be observed that during one complete rotation there 
 are four positions of the fork in which it is not heard. The sound 
 is faintly heard when the fork is turned very slightly in either 
 direction from a position of silence, and swells to a maximum 
 midway between these positions. With the fork held in a posi- 
 tion of silence, sound is restored by covering either prong with a 
 small cylinder of paper or other material, care being taken not to 
 touch the prongs, as this would stop the vibration {Exp.). 
 
 These curious effects are explained by Figs. 141 and 142, which 
 represent the sound waves about a vibrating fork, as seen with the 
 ends of the fork 
 pointing toward 
 the observer. As 
 the prongs move 
 apart, a conden- 
 sation is set up 
 on the outside of 
 each, and a rare- 
 faction between 
 
 them ; as they 
 
 Fig. 141. 
 
 Fig. 142. 
 
230 Sound 
 
 move toward each other, opposite conditions are produced. If tlie 
 space about the fork were partitioned off into four compartments, 
 as indicated in the figures, there would be condensations and rare- 
 factions on opposite sides of the partitions at equal distances from 
 the fork, as shown in Fig. 141 ; but, without such partitions to 
 keep the condensations and rarefactions apart, their opposing ten- 
 dencies destroy each other where they meet, causing silence at 
 these places, as shown in Fig. 142. 
 
 The principle of the composition of motions as studied in 
 mechanics applies to the resultant vibration of any portion of a 
 medium when acted upon simultaneously by two or more trains 
 of waves. The waves from a tuning fork are an example of the 
 simplest case ; namely, that of two trains of waves of exactly 
 equal wave length and amplitude, traveling in the same direction, 
 the waves of one train being half a wave length in advance of 
 those of the other train. In the region where the two trains of 
 waves meet, the condensations of each unite with the rarefactions 
 of the other. The condensations would be transmitted by a for- 
 ward motion of the air, and the rarefactions by an equal backward 
 motion at the same time; hence the air in this region remains at 
 rest and there is neither condensation nor rarefaction. Silence is 
 thus the result of the inUrference of the waves with each other. 
 
 299. Beats. — The sounds of two forks of exactly the same 
 pitch unite perfectly into one sound. If the forks are sounded 
 together after slightly lowering the pitch of one of them (by stick- 
 ing a piece of soft wax near the end of one or both prongs), the 
 sound periodically swells and dies away in strongly marked pulsa- 
 tions, called beats {Exp.), 
 
 Let us suppose that two middle C forks are used (vibration 
 number = 256), and that, by loading with wax, one is reduced to 
 255. At intervals of one second the forks "keep step" in their 
 vibrations ; and the waves that they then set up approximately 
 coincide — condensations with condensations and rarefactions with 
 rarefactions, as at ^ and C (Fig. 143). These waves unite in 
 resultant waves of increased intensity, as represented at X and Z. 
 
Properties of Musical Sounds 231 
 
 Half a second after each coincidence, the forks vibrate oppositely, 
 the condensations produced by each approximately coinciding 
 
 with the rarefactions produced by the other, as at B; and the 
 resultant waves are of diminished intensity. A complete set of 
 intensified and weakened resultant waves is thus sent out from the 
 forks in one second. This constitutes one beat. 
 
 Beats may therefore be defined as regularly recurring pulsations 
 of sound caused by the successive reenforcement and interference 
 of two sets of sound waves differing slightly in wave length or pitch. 
 The number of beats per second is equal to the difference between 
 the vibration numbers of the two sounds. Hence the beats become 
 more frequent when the pitch of the loaded fork is further lowered 
 by using more wax (Exp.), 
 
 PROBLEMS 
 
 1. Show from the formula v = /n that the wave length of a sound of given 
 pitch is proportional to the velocity of sound in the medium. 
 
 2. Find the wave length of middle C in the air at 20*^; in water. 
 
 3. Why is the sound of a fork restored in the position of silence when one 
 prong is covered? 
 
 4. Why is the pitch of a fork lowered by loading its prongs? 
 
 5. What use can be made of beats in tuning two sounding bodies to the 
 same pitch? 
 
 300. Musical Intervals. — The in/erva/ between any two notes 
 is measured by the ratio of the vibration number of the higher to 
 that of the lower. Two notes are said to be in unison when they 
 have exactly the same pitch. The interval between two notes in 
 
232 Sound 
 
 unison is unity (the ratio of equal numbers). If the vibration 
 number of one note is twice that of another (interval = 2), the 
 first is said to be an octave above the second. Other intervals are 
 considered in Art. 302. 
 
 301. Harmony and Discord. — If the effect of two or more 
 tones when sounded together is pleasing to the ear, the tones are 
 said to be harmonious or consonant; if the effect is unpleasant, 
 they are said to be discordant or dissonant. 
 
 The two wires of a sonometer (Fig. 144) produce one contin- 
 uous sound when tuned to exact unison ; but, when the pitch 
 
 ^^Wy Fh; 144. 
 
 of one of the wires is gradually raised (either by shortening the 
 length of the vibrating portion with the movable bridge or by 
 increasing the tension), beats are heard. The beats increase in 
 frequency as the interval is increased ; and, as they become too 
 rapid to be recognized individually, they pass from an unsteady, 
 rattling sound into a discord. As the interval is still further 
 increased, the tones presently become less discordant, then har- 
 monious, then again discordant, etc. {Exp), 
 
 Four intervals can thus be found between the original tone and 
 its octave for which the tones are in harmony ; at all other inter- 
 vals between these limits they are more or less discordant. It 
 can be shown that these four intervals are {, J, f , and | respec- 
 tively. These ratios are frequently called simple ratios because 
 they are expressible in small numbers. The most perfect harmony 
 is that of a tone and its octave, and they are separated by the 
 simplest possible interval (|). 
 
 302. The Major Diatonic Scale. — Musical scales are deter- 
 mined by the comparatively few intervals that are pleasing to the 
 
Properties of Musical Sounds 233 
 
 ear. The major diatonic scale consists of a series of eight notes 
 whose syllable names are 
 
 do re mi fa sol la si do^ 
 The intervals between the first or keynote and the other notes 
 of the scale are as follows : — 
 
 I I * I f 4 ¥ ^ 
 
 These numbers are sometimes called the vibration ratios of the 
 notes. 
 The smallest whole numbers expressing the same ratios are 
 
 24 27 30 32 36 40 45 48 
 The intervals between successive notes of the scale are 
 do re mi fa sol la si do 2 
 
 I ¥ if I ¥ S H 
 
 The intervals f and y^ are called tones ; the interval |f is called 
 a semitone. 
 
 The eighth note, which is an octave above the first and is 
 called by the same name, is taken as the first of another series of 
 eight notes, each of which is an octave above the note of the same 
 name in the preceding series. The scale may thus be repeated 
 both upward and downwarcl over as many octaves as is desired. 
 
 The intervals remain the same whatever the absolute pitch of 
 the keynote may be. When middle C is taken as the keynote, 
 the letter names and the vibration numbers of the notes of the 
 scale are as given below : — 
 
 
 
 f , 
 
 
 Position on the stafi, 
 
 
 1 1 
 
 J -' ^ 
 
 
 SI _. /^ 
 
 €i 
 
 •■^ 1 
 
 
 7 \ \ ^\ ^ 1 
 
 
 S--<sU ^ 
 
 
 Letter names, c* //' ^' /' ^ a' 
 
 ^' r" 
 
 Syllable names, do re mi fa sol la 
 
 si do 
 
 Vibration numbers, 256 288 320 341J 384 426J 
 
 480 5 1 2 
 
 Vibration ratios, ^ \ \ \ \ % 
 
 ¥ 2 
 
 Intervals between 
 
 
 successive notes. 
 
 
 f ¥ il t 
 
 ¥ 1 
 
 U 
 
234 Sound 
 
 The notes do^ mi^ soly do^ are in harmony, and constitute the 
 common or major chord. Their vibration numbers are propor- 
 tional to the numbers 4, 5, 6, 8. 
 
 303. The Tempered Scale. — In order that any note of the 
 diatonic scale may be taken as the keynote, it is necessary to 
 introduce other notes within the octave. These notes are called 
 sharps or flats, and are played on the piano and the organ by 
 means of the black keys. Five notes are thus added in the 
 octave, forming the chromatic scale — 
 
 C Cj D Djl E F Fjf G G# A A# B C 
 
 It is further necessary to change the intervals slightly, making 
 them exactly equal, to avoid the necessity of introducing many 
 more notes. Thus the intervals C to D and D to E are equal, 
 and each is the square of the interval E to F. Only a trained 
 ear can recognize the slight imperfection of harmony that results 
 from this equalization of the intervals. The diatonic scale thus 
 modified is called the tempered scale to distinguish it from the 
 natural scale considered in the preceding article. Only the 
 natural scale is used in physics. 
 
 304. Limits of Audibility: Range of Pitch used in Music. — 
 We have seen that the vibrations of a body may be too slow to 
 produce an audible sound (Art. 278). The sound waves from 
 such bodies are too long to affect the ear. In all other respects 
 they are like the waves that produce the sensation of sound. The 
 slowest rate of vibration that will produce an audible sound differs 
 with different persons whose hearing for sounds of ordinary pitch 
 may be equally good. The limit of audibility generally lies be- 
 tween 16 and 30 vibrations per second. The vibration number 
 of the lowest note of a piano (A of the fourth octave below mid- 
 dle C) is 26.6. The lowest note of most pipe organs is the third 
 C below middle C, which gives 32 vibrations per second.^ 
 
 1 It is an interesting fact that the muscles vibrate when in action, sounding a note 
 near the lower limit of audibility. This note can be distinctly heard as a rapid pul- 
 sation by pressing the palms of the hands firmly over both ears. The sound comes 
 from the muscles of the arms, which are then contracted. 
 
Properties of Musical Sounds 235 
 
 The vibrations of a body may also be too rapid to produce an 
 audible sound. The upper limit of audibility differs much more 
 widely with different persons than the lower Hmit. From numerous 
 experiments it is concluded that the limit generally lies between 
 12,000 and 30,000 vibrations per second. The relative Hmits of 
 audibility of the members of the class can 
 
 be tested by gradually raising the pitch fTt&°tf[f^||) ' fr" ^ J 
 of a Galton's whistle (Fig. 145) {Exp.). 
 It is said that some persons whose hear- 
 ing is good cannot hear the shrill chirping of a cricket. The 
 highest note of a piano is the fourth C above middle C ; its 
 vibration number is 256 X 2^ or 4096. Pitch is not easily or 
 definitely appreciated beyond this limit. 
 
 Laboratory Exercise 46. 
 
 305. Conditions affecting the Vibration Rate of Strings. — Al- 
 though the sound of a stringed instrument comes almost wholly 
 from the body of the instrument, or from some part of the body, 
 the pitch of the sounds is determined by the rate of vibration of 
 the strings. (The word string is here used to include wires as well 
 as catgut strings.) It is well known, in a general way, that the pitch 
 of a string is raised by shortening it {i.e. the vibrating portion of 
 it), or by increasing its tension ; and that the pitch of a lighter 
 string is higher than that of a heavier one of the same length 
 when under the same tension. The effect of length is illustrated 
 by the different notes obtained from the same string of a violin, 
 mandolin, or guitar, by varying its length with the finger; the 
 effect of tension, by the use of the tightening pegs in tuning the 
 strings ; and the effect of mass, by the different strings of the in- 
 strument, the heaviest giving the lowest note. 
 
 The effects of the length, tension, and mass of a string can be 
 illustrated experimentally by means of a sonometer {Exp.), They 
 are explained in a general way {i.e. qualitatively, not quantita- 
 tively) as follows : — 
 
 306. Discussion of the Conditions. — Length. — It will be re- 
 membered that when a pendulum is shortened, it vibrates more 
 
236 
 
 Sound 
 
 rapidly because the accelerating force upon the bob for a given 
 length of arc is increased (Art. 135). Similarly, when a string is 
 shortened without change of tension, the accelerating force for a 
 given displacement (amplitude) is increased. This is illustrated 
 by the familiar fact that, with a constant tension, the shorter a 
 cord is, the greater is the resistance that it offers when its center is 
 pulled a given distance to one side. But the shorter cord vibrates 
 more rapidly for a second reason : the mass to be carried to and 
 fro diminishes in the same ratio as the length. 
 
 Tension. — The accelerating force for a given amplitude is pro- 
 portional to the tension ; hence an increase of tension increases 
 the rate of vibration. 
 
 Mass. — 'ITie acceleration produced by a given force is inversely 
 proportional to the mass upon which the force acts (Art. 113); 
 hence, other conditions being equal, the greater the mass of a 
 string, the slower will be its rate. The mass of a string per unit 
 length is proportional to its density and also to the square of its 
 diameter. 
 
 307. The Laws of Vibration of Strings. — The following laws 
 of vibration of strings have been established by experiment ; they 
 are derived in advanced physics by mathematical reasoning based 
 on the second law of motion. 
 
 I. Other conditions remaining constant, the vibration number 
 of a string is inversely proportional to its length. 
 
 II. Other conditions remaining constant, the vibration number 
 of a string is directly proportional to the square root of its tension. 
 
 III. Other conditions remaining constant, the vibration number 
 of a string is inversely proportional to the square root of its mass. 
 
 The first of these laws is the only one of importance in elemen- 
 tary physics. We find use for it in the following article. 
 
 PROBLEMS 
 
 I. Compute the wave length of the following notes in air at 20° : 
 Ci ( = 32), C ( = 64), ^ ( = 4096), and the highest audible sound, assum- 
 ing it to be 30,000. 
 
Properties of Musical Sounds 237 
 
 2. The pitch of a string 50 cm. long is r ( = 128). Compute its length 
 for each note of the octave, the tension remaining constant. 
 
 3. A string i m. long sounds C ( = 64) under a certain tension. The 
 tension remaining constant, compute the vibration number of the note sounded 
 by 2> h i» ^» 6> h *"<! i o^ '^s total length. All these notes but one are notes 
 of the diatonic scale in the first three octaves above the note of the whole 
 wire. Identify them. 
 
 308. Fundamental Tone and Overtones. — By properly timing 
 the motion of the hand in which one end of a long rubber tube or 
 coil of wire is held, the tube (or wire) can be made to vibrate 
 regularly as a whole (Fig. 146). When the frequency of the 
 
 Fig. 147. 
 
 impulses is doubled, the tube vibrates in two equal segments (Fig. 
 147) ; when trebled, it vibrates in thirds. The segments are 
 separated by points called nodes, which remain approximately 
 at rest {Exp.). 
 
 The strings of a musical instrument can also be made to vibrate 
 in two or more equal segments. To illustrate : When the string of 
 a sonometer wire is bowed near one end and, at the same time, is 
 lightly touched at its mid point with the finger nail or the tip of the 
 finger, it vibrates in halves, sounding the octave above its usual 
 note, even after the bow and the finger are removed (the finger 
 being removed an instant after the bow). When the string is 
 touched at one third its length from one end and bowed or plucked 
 near that end, it vibrates in thirds (Fig. 148). In the same man- 
 ner the string can easily be made to vibrate in any number of 
 
238 Sound 
 
 equal segments up to eighths or tenths. The division of the string 
 into segments is shown at a distance by placing upon it small 
 
 Fig. Z48. 
 
 folded bits of paper. These " riders " are thrown off by the vibra- 
 tion except at or very near the nodes (Ex^.). 
 
 The pitch of a string when vibrating in segments is the same as 
 it would be if the string consisted of but one of the segments. 
 The note that is sounded by a string when it is vibrating as a 
 whole is called its fundamental tone. A note that is produced by 
 vibration in segments is called an ot^ertone or a harmonic. The 
 overtones are numbered in order, beginning with the lowest : thus 
 the first overtone is produced by vibration in halves, the second 
 by vibration in thirds, etc. If we call the fundamental tone doi^ 
 the series of overtones is as follows : — 
 
 No. of segments, I 
 
 Names of the overtones. 
 Relative pitch, do^ 
 
 309. Quality of Sound. — The quality (Art. 294) of the tone of 
 a sonometer wire depends largely upon whether the wire is plucked 
 with the finger or the finger nail, bowed, or struck with a rubber 
 mallet. It also varies with the place where the wire is struck, 
 bowed, or plucked {Exp.). 
 
 These differences in quality depend upon the character of the 
 vibration of the string. Probably the vibration does not in any 
 case consist of a simple to-and-fro movement of the string as a 
 
 2 
 
 3 
 
 4 
 
 5 6 
 
 7 
 
 8 
 
 1st 
 
 2d 
 
 3ci 
 
 4th 5th 
 
 6th 
 
 7th 
 
 do^ 
 
 soli 
 
 do^ 
 
 w/3 W3 
 
 — 
 
 do^ 
 
Properties of Musical Sounds 239 
 
 whole. On the contrary, it is generally very complex, — vibration 
 in halves, thirds, fourths, etc., occurring simultaneously with the 
 vibration as a whole. Thus the note consists of the fundamental 
 tone together with one or more (generally several) overtones. 
 Figure 149 shows different forms assumed by a string while sound- 
 
 FIG. 149. 
 
 ing its fundamental and first overtone. The dotted lines indicate 
 the forms which the string would assume if it were sounding only 
 its fundamental tone. In some cases the first overtone can be 
 separately distinguished when an effort is made to do so ; but 
 generally the overtones are not recognized as separate sounds, 
 their effect being to impart a certain quality to the note. When 
 the wire is struck with a rubber mallet or plucked with the finger 
 near its center, the fundamental is loud and all the overtones 
 weak ; when it is bowed near one end, the overtones are much 
 stronger, adding what is commonly called richness to the sound. 
 When the wire is plucked near one end with the finger nail, the 
 fundamental and the lower overtones are wanting, or at most very 
 weak, and the higher overtones, some of which are discordant, 
 produce a shrill jangle. 
 
240 
 
 Sound 
 
 When a tuning fork is struck, it sounds one or more shrill over- 
 tones, which are perfectly distinct from the fundamental. These 
 overtones quickly die away, leaving the fundamental as a simple 
 tone {Ex/>.), 
 
 It was proved by Helmholtz, a noted German physicist, that 
 the quality of any sound is determined by the overtones which 
 accompany its fumlamental tone. Differences in the quality of 
 sounds are due to the presence of different overtones or to differ- 
 ences in their relative intensity. The tuning fork is the only in- 
 strument that gives a simple tone ; but the notes of the diapason 
 pipes of an organ are quite similar, being almost free from over- 
 tones. A musical sound in which a number of. the lower over- 
 tones are present is full and rich. The tones of the piano and 
 the violin are examples. The penetrating character of the tones 
 of brass instruments is due to their particularly strong overtones. 
 'ITie pitch of a sound is the pitch of its fundamental ; it is not 
 altered by the accompanying overtones. 
 
 III. Sympathetic and Forced Vibrations: Resonance 
 
 Laboratory Exercise 43. 
 310. Sympathetic and Forced Vibrations. 
 actions 
 
 1 
 
 Certain mechanical 
 which occur with 
 sounding bodies may be illus- 
 trated on a visible scale by 
 means of the apparatus repre- 
 sented in Fig. 15b. Two 
 pairs of pendulums are sus- 
 pended from a light rod, CZ>, 
 the pendulums of each pair 
 being of equal length. The 
 rod is supported by short 
 cords from a fixed support. 
 ^*°' '^- When any one of the pen- 
 
 dulums is set in vibration, it pulls the rod from which it is sus- 
 pended back and forth. 
 
Sympathetic and Forced Vibrations 241 
 
 This vibratory motion of the rod imparts a series of impulses 
 to the other pendulums. AUhough the effect of a single impulse 
 is almost inappreciable, a succession of such impulses causes the 
 other pendulum of the same length to vibrate with increasing 
 amplitude ; while the amplitude of the first steadily decreases as 
 it imparts its energy to the other. Since the two pendulums have 
 the same rate of vibration, the impulses imparted to the one that 
 was at rest are rightly timed to produce a cumulative effect. The 
 vibrations of this penduhmi are called sympathetic^ signifying that 
 its natural rate is in agreement with that of the impulses to which 
 it responds. 
 
 The other pair of pendulums make a few vibrations with in- 
 creasing amplitude, followed by an equal number during which 
 the amplitude decreases till the pendulums are brought to rest. 
 This series of increasing and decreasing vibrations is repeated in- 
 definitely. The explanation of this behavior is that the impulses 
 are not timed in agreement with the rate of these pendulums ; a 
 few successive impulses produce a cumulative effect, but these are 
 followed by an equal number which are opposed to the motion 
 already produced and hence destroy it. Since these pendulums 
 cannot be forced to vibrate in unison with the impulses, they 
 cannot accumulate any considerable store of energy. 
 
 When the rod is struck or drawn to one side and released (all 
 of the pendulums being at rest), it vibrates much more rapidly 
 than it does when under the control of a vibrating pendulum. 
 Thus, although the rod has a natural rate of vibration, it does not 
 persist in it as the pendulums do, but yields to the impulses of 
 either a long or a short pendulum. The motion of the rod when 
 thus controlled is called a forced vibration. 
 
 311. Forced Vibrations of Sounding Bodies. — The sound that 
 comes from a table, an empty box, or a large board when touched 
 by a vibrating fork is due to the forced vibration of the wood. 
 Since the sound thus produced always has the same pitch as the 
 fork, whatever this may be, it is evident that wood, especially in 
 the form of a thin board, is easily forced to vibrate in unison with 
 
242 Sound 
 
 periodic impulses of any frequency — a behavior like that of the 
 rod from which the pendulums were suspended (Fig. 150). 
 
 When a violin is played, the sound is produced by the forced 
 vibration of the body of the instrument. The pitch is . deter- 
 mined by the vibration rate of the strings ; the quality, partly by 
 the manner in which the string is bowed and partly by the kind 
 of wood of which the instrument is made and the workmanship. 
 The music of a piano comes from the large sounding board upon 
 which the wires are strung. 
 
 312. Sympathetic Vibration of Tuning Forks and Strings. — A 
 tuning fork may be made to vibrate sympathetically as follows : 
 The stems of a sounding and a silent fork of exactly the same 
 pitch are touched to the top of a table a short distance apart. 
 After one or two seconds, the fork that was sounded is stopped 
 with the fingers or removed from the table, and a sound is then 
 heard from the other fork. If the sound is too faint to be heard 
 at a distance, the vibration of the fork can be proved by touching 
 a suspended pith ball to one of its prongs. This vibration is 
 caused principally by impulses imparted through the stem by the 
 vibrating table. Probably from 500 to 1000 such impulses are 
 required to produce the observed effect {Exp.). 
 
 The silent fork can also be made to vibrate sympathetically by 
 holding it in the fingers close to the vibrating fork, held in the 
 other hand, the forks facing each other but not touching. In 
 this case the impulses are imparted by the sound waves in the 
 air {Exp.). 
 
 These experiments fail if the forks are not in perfect unison, 
 as will be found by repeating them with one of the forks loaded 
 with wax {Exp.). 
 
 If the two wires of a sonometer are tuned to exact unison, 
 either will vibrate sympathetically when the other is sounded, the 
 impulses being transmitted principally through the body of the 
 instrument. As the pitch of one of the wires is slowly changed, 
 the response of the silent wire immediately becomes very faint 
 and quickly ceases {Exp,). 
 
Sympathetic and Forced Vibrations 243 
 
 These experiments show that tuning forks and strings, like the 
 pendulums, persist in their natural rate of vibration, — a behavior 
 very different from that of sounding boards and the bodies of 
 stringed instruments. 
 
 Laboratory Exercise 44, 
 
 313. Resonance of Air Columns. — The reenforcement of the 
 sound of one body by the sympathetic or forced vibration of 
 another is called resonance; but the word most frequently refers 
 to the reenforcement of sound by the sympathetic vibration of 
 partly inclosed bodies of air. We shall use the word only in this 
 narrower sense. 
 
 Resonance may be secured by means of a tube provided with a 
 close-fitting piston (Fig 151). When the piston is in a certain 
 
 £ 
 
 Fig. 151. 
 
 position, the sound of a fork held at the end of the tube is strongly 
 reenforced by the sympathetic vibration of the column of air 
 extending to the piston. The sound from the tube is very much 
 fainter when the piston is moved even a very short distance in 
 either direction, from the position of maximum reenforcement 
 {Exp^. The column of air will vibrate strongly only at its natural 
 rate, which varies with the length. This is further shown by repeat- 
 ing the experiment with a fork of different pitch. It will be found 
 that the length of the column for maximum resonance is inversely 
 proportional to the vibration number of the note (^Exp.). 
 
 The vibration of the air column is longitudinal. As the nearer 
 prong of the fork moves toward the end of the tube, the air par- 
 ticles in the tube are driven a short distance toward the farther 
 end, producing a condensation which is greatest at the piston. 
 When the prong moves away from the tube, the air expands by 
 a motion of the particles toward the mouth of the tube. The 
 
244 
 
 Sound 
 
 expansion, like the compression, is greatest at the piston. Thus 
 the change of density of the air is greatest at the piston and di- 
 minishes toward the open end, where the density remains nearly 
 constant. The amplitude of vibration of the air particles is 
 greatest at the open end and diminishes to zero at the piston. 
 The tube controls the vibration of the air for a short distance 
 beyond its end, and it has been found by experiment that the true 
 length of the air column is equal to the length of the tube (to the 
 piston) plus half the diameter of the tube. It can be shown that 
 this length is one fourth of a wave length, i.e. is one fourth the 
 length of the sound wave set up by the vibration of the fork and 
 the air column. 
 
 Second resonance occurs when the length of the air column is 
 increased to three fourths of a wave length. The mode of vibra- 
 tion of the air is shown in Fig. 152, which represents the condi- 
 
 Y 
 
 | |!|lll ll li l lllll l li lll l!llli !| 
 
 
 ii:: uimwimii 
 
 liiiiil 
 
 m 
 ■ 
 
 I 
 
 \ I / 
 
 ami 
 
 Fi<; 
 
 tion of the air at intervals of one fourth of a vibration. AB is 
 the length for first resonance, AD for second resonance. AB, BC, 
 CD, are each one fourth of a wave length. At B and D there is 
 no motion, and the change of density is the greatest. These 
 
Sympathetic and Forced Vibrations 245 
 
 positions are called nodes. (Compare with the nodes of a string 
 vibrating in segments.) At A and C there is little or no change 
 of density and the amplitude of vibration is greatest. These 
 positions are called atiiinodes. The arrows in the second and 
 fourth parts of the figure indicate the direction of motion of the 
 air particles. At the instant represented in the first and third 
 parts of the figure the air is at rest throughout the tube. BD 
 is a vibrating segment of the air column; its length is half a 
 wave length. AB is half a segment. 
 
 If the tube were further lengthened by half a wave length, 
 another whole segment would be added to the vibrating air 
 column and third resonance would occur. 
 
 314. Resonance caused by Noise. — Any partly inclosed body 
 of air has a natural rate of vibration as a whole, and, when sub- 
 jected to a series of impulses rightly timed, it will vibrate sympa- 
 thetically. A noise always affords such a series of impulses, since 
 it consists of an indefinite number of different rates of vibration. 
 Hence a hollow body, as a tumbler, a glass or metal tube, or a 
 sea shell, continually sounds a faint note, which is distinctly heard 
 when the body is held to the ear. The pitch of this sound re- 
 mains constant when different noises are made in the vicinity, 
 but its loudness varies, in many cases swelling to a loud roar when 
 the foot is scraped on the floor {Exp.). 
 
 315. The Whistle. — In wind instruments the sounding body 
 is a column of air confined in a tube. The whistle is a familiar 
 illustration. Its action is explained as follows : A current of air 
 passing through a narrow slit, a (Fig. 153), is directed against 
 the farther edge of a lateral opening, b. Con- 
 tact with this edge causes the current of air to 3 ' r-—^ 
 flutter irregularly, producing a faint rustling 
 
 noise. The column of air, c, confined in the 
 body of the whistle is thrown into strong sympathetic vibration by 
 those impulses from the current of air which are in unison with its 
 natural rate. The note thus produced is generally so much louder 
 than the noise of the air current that the latter is not heard. By 
 
246 
 
 Sound 
 
 means of a whistle fitted with a piston it can be shown that the 
 pitch of the sound rises as the size of the air chamber is de- 
 creased {Exp. with Ga/ton's whistle or organ pipe with piston). 
 Within wide Hmits of pitch, the inclosed air always finds its own 
 note in the noise of the air current, and reenforces it. 
 
 316. Organ Pipes. — An organ pipe is merely a whistle of spe- 
 cial construction. The pitch of its note is determined by the 
 
 length of the air column (cor- 
 rected for the diameter of the 
 pipe). The quality of the 
 sound is modified by the shape 
 and material of the pipe. 
 Figure 154 represents a rec- 
 tangular wooden pipe and 
 Fig. 155 a cylindrical pipe 
 of metal. Some pipes are 
 provided with a tongue or 
 reed, against which the cur- 
 rent of air from the bellows 
 is directed, causing it to vi- 
 brate. The reed is tuned to 
 the natural rate of the air 
 column in the pipe, which 
 therefore vibrates sympatheti- 
 cally. The note of a reed 
 Organ pipes are made both 
 
 open and closed at the top ; the latter are called stopped pipes. 
 
 The following laws may be illustrated by means of a pipe provided 
 
 with a piston : — 
 
 I. The vibration number of an air column is inversely propor- 
 tional to its length. 
 
 II. The pitch of an open pipe is an octave above that of a closed 
 pipe of the same length. 
 
 317. Fundamental Tone and Overtones of Organ Pipes. — In 
 sounding its fundamental note the air in a closed pipe vibrates 
 
 Fig. 
 
 Fig. 
 
 155- 
 
 pipe has a characteristic quality. 
 
or THF 
 
 UNIVERSI7 
 Sympathetic and Forced Vibrati\jsc;,, /^^^v^ 
 
 in a half segment, with an antinode at the mouth and a node at 
 the closed end, as a resonance tube does for first resonance (Art. 
 313). Thus the length of a stopped pipe is one fourth the wave 
 length of its fundamental tone. The first overtone is produced 
 by vibration in one and one half segments, as for second reso- 
 nance (Art. 313). Its vibration number is three times that of 
 the fundamental, and hence corresponds to the second overtone 
 of a string. 
 
 The second overtone is produced by vibration in two and one 
 half segments, and corresponds to the fourth overtone of a string. 
 When a pipe is sounded by a gentle current of air, only its funda- 
 mental is heard ; with a greater pressure of air, this gives place 
 to the first overtone ; and, with still greater pressure, the second 
 overtone is sounded {Exp.). 
 
 The air in an open pipe necessarily vibrates with an antinode at 
 each end. When sounding its fundamental, there is but one node, 
 and this is at the middle ; i.e. the air vibrates in two half segments. 
 The length of an open pipe is therefore half the wave length of 
 its fundamental tone. The first overtone is produced by vibra- 
 tion with one segment between the half segments at the ends, the 
 second overtone with two segments between, the third with three, 
 etc. From the relative length of the segments thus produced it 
 will be seen that the complete series of overtones may be present 
 as with strings. The difference in the overtones present causes 
 a difference in the quality of open and closed pipes. 
 
 318. Wind Instruments. — Each pipe of an organ sounds only one 
 note ; hence in a complete set of pipes there is one for each note 
 
 of the instrument. An organ is provided with several sets of 
 pipes, the notes of each set differing in quality from those of the 
 
248 
 
 Sound 
 
 other sets. Most wind instruments are provided with but one 
 tube, the different notes being produced either by varying the 
 
 length of the tube or by 
 sounding overtones. 
 
 In the trombone (Fig. 
 156), one part of the tube, 
 SL^ sHdes within the other, 
 and the tube is shortened 
 by pushing the shding part 
 farther in. The tube of the 
 cornet (Fig. 157) forms sev- 
 
 FiG. 157. 
 
 eral turns or convolutions, which may either be included or cut 
 off from the remainder of the tube by means of pistons, <z, by r, 
 thus varying the length of the vibrating column of air. 
 
 The bugle y the trumpet (Fig. 158), and the coaching horn are 
 without any device for altering the 
 length of the air column ; hence their 
 notes are limited to the fundamental 
 and the complete series of overtones. 
 The different notes are produced by ^'^'- ^58. 
 
 varying the force of the breath and the tension of the lips. 
 
 The flute and clarinet are provided with holes at different dis- 
 tances, which are closed by the fingers or by keys. An antinode 
 is produced at an open hole ; this modifies the division of the air 
 column into vibrating segments, and hence determines the note. 
 
 PROBLEMS 
 
 1. Gjmpute the length of a stopped pipe that sounds Ci (= 32) at 20° C, 
 making no allowance for the diameter. 
 
 2. Compute the length of an open pipe that sounds ^' ( = 256) ; also the 
 length of one that sounds ^ (= 4096). 
 
 3. What are the letter and syllable names of the first five overtones of an 
 open pipe whose fundamental is c' or do\ ? 
 
 4. What are the letter and the syllable names of the first two overtones 
 of a closed pipe of the same pitch as in the preceding problem ? 
 
The Ear and the Voice 
 
 249 
 
 IV. The Ear and the Voice 
 
 319. The Parts of the Ear. — The ear consists of three parts ; 
 namely, the external ear^ the middle ear or tympanum, and the 
 ititernal ear or labyrinth. The sensation of sound originates in 
 the internal ear ; the other parts are accessory. 
 
 320. The External Ear. — The external ear consists of the 
 expanded part on the exterior of the head and the passage lead- 
 ing inward from it. This passage is a little over an inch in length, 
 
 and is closed at its inner end by a thin membranous partition, 
 called the tympanic or drum membrane. It is popularly called the 
 eardrum ; but this term is more properly applied to the middle 
 ear. The drum membrane is kept under tension by a small 
 muscle, which pulls it in at the center, giving it a conical shape. 
 Figure 159 represents the front view of a section of the right ear. 
 The expanded part of the external ear concentrates the sound 
 waves and directs them into the passage, at the inner end of 
 which they beat upon the drum membrane, setting it into forced 
 vibration. 
 
! 
 
 250 Sound 
 
 321. The Middle Ear. — The middle ear, or tympanum, P 
 (Fig. 159), is an irregular cavity in the temporal bone, separated 
 from the external ear by the dniin membrane. An enlarged view 
 
 of this cavity is shown in Fig. 160. 
 The tympanum is connected with the 
 back part of the mouth (pharynx) by 
 the Eustachian tube, R (Fig. 159). 
 This tube allows air to pass between 
 the throat and the middle ear, thus 
 equalizing the pressure on the two 
 sides of the drum membrane. 
 
 A chain of three small bones ex- 
 tends across the middle ear from the 
 *°* ' drum membrane to the internal ear. 
 
 They are called from their shape, the hammer, ;//, the anvil, i, and 
 the stirrup, s. A long, slender portion of the hammer (the handle) 
 is attached to the inner surface of the drum membrane ; its enlarged 
 end or head is fastened to an end of the anvil by an almost im- 
 movable joint The other end of the anvil is jointed to the 
 stirrup ; and the foot plate of the latter is fastened to a mem- 
 brane only slightly larger than itself, which closes an oval opening 
 (the avai window) in the bony partition between the middle and 
 the internal ear. 
 
 In vibrating, the drum membrane moves inward and outward 
 through a distance that is probably not more than .1 mm. at the 
 most, and in this motion carries the handle of the mallet with it. 
 The mallet and the anvil are supported by attachments not shown 
 in the figures, which permit their rotation as one body about an 
 axis passing through the head of the mallet and perpendicular to 
 the plane of the paper in the figures. Hence, as the two bones 
 rotate together, each inward and outward movement of the handle 
 of the mallet causes a like movement of the stirrup. Thus the 
 vibrations of the drum membrane are transmitted by means of the 
 chain of bones to the membrane covering the oval window, and, 
 by the latter, to the liquid in the internal ear. 
 
The Ear and the Voice 
 
 251 
 
 322. The Internal Ear. — The internal ear, or labyrinth^ occu- 
 pies several chambers and tubes hollowed out in the temporal 
 bone. The middle chamber, called the vestibule^ F(Fig. 159), is 
 adjacent to the inner side of the middle ear. Behind, the vesti- 
 bule opens into three semicircular canals, one of which is shown 
 at b in the figure ; in front, it opens into a spirally coiled tube, 5, 
 called the cochlea, from its resemblance to the shell of a snail. In 
 these bony chambers and tubes lie membranous chambers and 
 tubes, in certain parts of which the fibers of the auditory nerve. A, 
 end. The space both within and without the membranous parts 
 is filled with a watery liquid. The function of the semicircular 
 canals seems to be distinct from that of hearing. 
 
 323. The Cochlea. — The bony tube of the cochlea is coiled 
 into a spiral of two and a half turns. It contains a spiral ledge of 
 bone, which projects into it from the axis of the spiral and follows 
 the winding of the tube throughout its length (Fig. 159). A cross- 
 section of the bony tube of the cochlea is represented diagram- 
 matically in Fig. 161 ; in which Sp is the spiral ledge, and C a 
 triangular, membranous tube, called the canal of the cochlea (not 
 shown in Fig. 159), which is attached at its inner edge to the 
 spiral ledge and at its outer surface to the wall of the tube. The 
 space outside the canal is thus divided into an upper channel, A^ 
 and a lower channel, B. These channels unite at the tip of the 
 cochlea around the end of the canal, which is closed. The lower 
 end of the upper channel is connected with 
 the vestibule ; the lower channel terminates 
 at a round opening, r (Fig. 159), in the 
 bony partition between the internal and 
 the middle ear. This opening is called 
 the round window. It lies just below the 
 oval window, and, like the latter, is covered 
 with a membrane. The channels and the 
 canal are filled with a watery liquid. '°' '^^* 
 
 The lower side of the canal, vib (Fig. 161), is called the basilar 
 (basal) membrane. It consists of thousands of delicate fibers 
 
252 Sound 
 
 extending across the canal from side to side. This membrane 
 possesses upon its inner face, along the whole length of the tube, 
 from the top to the bottom of the spiral, a very remarkable cellular 
 structure known as the or-gan of Corti, The filaments of the audi- 
 tory nerve terminate among the cells of this structure. 
 
 The vibration of the membrane to which the stirrup is attached 
 causes a vibration of the liquid in the vestibule. This vibration is 
 transmitted along the upper channel of the tube of the cochlea to 
 its upper end; thence back along the lower channel to the round 
 window. The yielding of the membrane that covers this opening 
 permits vibrations of much larger amplitude in the cochlea than 
 would otherwise be possible. These vibrations are imparted to 
 the membranous canal of the cochlea and the liquid within it. 
 Now the lower side of the canal (the basilar membrane) gradually 
 widens from its lower to its upper end as 
 shown diagram matically in Fig. 162 ; and 
 is supposed to vibrate sympathetically in 
 transverse segments (as <x, b, and c in the 
 figure), the rate of each segment depending in part upon its length. 
 Thus when a number of vibrations of different frequencies, such as 
 constitute an ordinary musical sound, are transmitted to the 
 cochlea, they (it is supposed) throw into sympathetic vibration 
 those parts of the basilar membrane which have the same natural 
 rate, and the filaments of the auditory nerve ending in the over- 
 lying parts of the organ of Corti are stimulated. According to 
 this theory, the recognition of pitch is due to the stimulation of 
 different nerve fibers by different rates of vibration. 
 
 324. The Larynx and the Vocal Bands. — At the upper end 
 of the windpipe there is a short tubular box, called the larynx, 
 which opens above into the back part of the mouth, just below the 
 base of the tongue. Figure 1 63 represents a vertical section through 
 the larynx with the hinder half removed. The larynx is made of 
 plates of cartilage, movable on one another, and connected 
 together by joints, muscles, and membranes. 
 
 At the middle of the larynx there is a projecting ridge on each 
 
The Ear and the Voice 
 
 253 
 
 Fig. 163. 
 
 side, whose edges extend from front to back. These ridges are 
 composed of elastic tissue, and are cov- 
 ered with a mucous membrane : their sharp 
 free edges are the so-called vocal cords^ V 
 (Fig. 163), whose vibration produces the 
 voice. They are more appropriately called 
 vocal bands. The large angular plate of 
 cartilage, Thy to which the vocal bands 
 are attached in front, causes the promi- 
 nence in the neck known as " Adam's 
 apple." 
 
 The slitlike opening between the vocal 
 bands is called the glottis. In quiet 
 breathing this opening is narrow in front 
 and wider behind, leaving a free passage 
 for the air to and from the lungs as shown 
 in Fig. 164, By which represents a top view of the larynx and the 
 vocal bands. In using the voice, the bands are brought close 
 together at the rear by muscular action, leaving only a narrow slit 
 
 Epiglottis 
 False Vocal Cords 
 
 True Vocal Cords" 
 
 Fig. 164. 
 
 between their parallel edges (Fig. 164, A). When the bands are 
 in this position, they are set in vibration by the current of air from 
 the lungs, and the sound thus produced constitutes the voice. 
 
 325. The Voice. — The vocal bands alone would produce only a 
 very feeble sound ; but their vibration is reenforced and modified 
 in quality by the sympathetic vibration of the air in the pharynx 
 and in the mouth and nasal cavities, which together constitute a 
 resonance chamber. The loudness of the voice, aside from the 
 
254 Sound 
 
 reenforcement by resonance, depends upon the force with which 
 the air is driven past the bands, together with the size and condi- 
 tion of the bands themselves. 
 
 The pitch of the voice depends primarily on the size of the 
 lar)'nx, — the longer the vocal bands are, the lower is the pitch. 
 It is for this reason that the voices of men are lower in pitch than 
 those of women and children. Any note within certain limits can 
 by produced at will by the action of the muscles in the larynx, 
 which alter the tension of the vocal bands. 
 
 The quality of the voice depends primarily upon the natural 
 size and form of the larynx and the different resonance cavities, 
 and the condition of the mucous membrane lining these cavities 
 and covering the vocal bands. But the quality is also largely 
 affected by the manner in which the vibration of the vocal bands 
 and the form of the resonance cavities are controlled by muscular 
 action ; and it may therefore be modified and improved by culti- 
 vation. 
 
 326. Speech. — The modulation of the voice into speech is 
 effected by changing the size and form of the pharynx and the 
 mouth and nasal cavities by movements of the soft palate, tongue, 
 cheeks, and lips. " By movements of tongue, lips, and palate, the 
 air current, and therefore the sound, is interrupted from time to 
 time ; on other occasions the air is forced through a narrow pas- 
 sage in the mouth, giving rise to new sounds added on to those 
 originated by the vocal cords. In such manner the primitive, 
 feeble, monotonous tone due to the vocal cords is reenforced and 
 altered in various ways in throat and mouth, and voice is devel- 
 oped into articulate speech^ — Martin's The Human Body, 
 
CHAPTER X 
 
 LIGHT 
 
 I. Nature and Transmission of Light 
 
 327. Ways in which Energy can be Transmitted. — There are 
 only two conceivable ways in which energy can be transmitted. 
 The one consists in the transmission of matter possessing energy, 
 the other in the transmission of a disturbance of some sort through 
 matter. A flying bullet and a moving express train are exam- 
 ples of the first; sound waves and water waves, of the second. 
 In one case the energy remains associated with the same body 
 of matter during transmission ; in the other it is handed on to suc- 
 cessive portions of the medium. 
 
 328. Theories of Light : Historical. — The two ways in which 
 energy can be transmitted served as a basis for two rival theories 
 of light, which were supported by different physicists during the 
 latter part of the seventeenth century. They are known as the 
 emission or corpuscular theory, and the undulaiory or wave theory. 
 
 According to the emission theory, a luminous body emits streams 
 of minute particles (corpuscles), which travel in straight lines, and 
 on entering the eye, cause vision by their impact on the retina. 
 According to the wave theory, all space that is otherwise unoccu- 
 pied is filled with a substance called the luminiferous eiher^ or 
 simply the ether; and light consists in a periodic disturbance 
 transmitted through this medium from luminous bodies, in much 
 the same way as sound is transmitted through the air from sound- 
 ing bodies. 
 
 Neither theory, as at first advanced, was fully satisfactory ; but 
 Newton's support of the emission theory carried the decision 
 in its favor, and it met with almost universal acceptance during 
 
 255 
 
256 Light 
 
 the eighteenth centur>'. At the beginning of the nineteenth century 
 the subject was again taken up by a number of able physicists. 
 By a series of remarkable experiments adtiitional facts were 
 brought to light, which amounted to conclusive evidence against 
 the emission theory and in favor of the wave theory. The latter, 
 with some comparatively recent modifications, is now regarded as 
 fully established. The facts and experiments that led to the final 
 adoption of the wave theory lie almost wholly beyond the scope 
 of elementary physics ; we must, however, make use of the sim- 
 pler elements of the theory in explaining the phenomena with 
 which we have to deal. 
 
 329. The Ether. — The ether fills all space throughout the 
 known universe, for it is only by means of light that reaches us 
 from the heavenly bodies that we have any knowledge of them. 
 It cannot be excluded from a vacuum ; in fact, there is no evi- 
 dence that the quantity of it in a receiver is diminished in the 
 slightest degree by whatever means the receiver may be exhausted. 
 The velocity of light through transparent solids, liquids, and gases 
 is enormously greater than it would be if these bodies served as 
 media by means of which, as well as through which, the light is 
 transmitted ; hence it is concluded that light is transmitted 
 through bodies by means of the ether which fills the spaces 
 between their molecules. 
 
 While the ether must be regarded as a form of matter, it is in 
 all probability millions of times less dense than air under atmos- 
 pheric pressure, and there is no evidence either that it is subject 
 to gravitation or that it is composed of molecules. On the con- 
 trary, from the manner in which it transmits disturbances, it is 
 thought to be perfectly continuous and incompressible. It is 
 supposed to fill the intermolecular spaces in all bodies fi-om the 
 most highly rarefied to the densest. 
 
 330. Nature of Radiation. — Some helpful ideas in regard to 
 the origin and transmission of radiant energy (including light) 
 can be gathered from a comparison with the phenomena of 
 sound. A sounding body, as we have learned, is the center of a 
 
Nature and Transmission of Light 257 
 
 periodic disturbance which is radiated (transmitted radially) 
 from it in all directions through the surrounding medium, 
 in the form of concentric spherical shells, called sound waves. 
 Now the molecules of all bodies, as we have also learned, are sup- 
 posed to be in constant and inconceivably rapid vibration (Art. 
 186). According to the wave theory, the vibrations of each mole- 
 cule disturb the surrounding ether; and, as in the case of sound, 
 this disturbance is radiated with equal velocity in all directions 
 as a train of concentric spherical waves. Thus each molecule of 
 a body may be compared to a sounding fork, and the entire body 
 to a large group of forks, each of which is sending out a train of 
 waves. As the energy of a sounding body is gradually imparted 
 to the air or other medium, so too the heat of bodies is gradually 
 imparted to the ether as energy of wave motion, and in this form 
 it is called radiant energy. 
 
 It must not be supposed, however, that sound waves and ether 
 waves are at all alike in character ; we should rather expect the 
 contrary, since the properties of the ether are very different from 
 those of ordinary matter. Ether waves consist of a periodic dis- 
 turbance 0/ some sort; they have a definite length (measured 
 radially, as in the case of sound waves) and travel with a definite, 
 though very great, velocity ; but they do not consist of conden- 
 sations and rarefactions and the vibrations are not longitudinal. 
 
 331. Luminous and Nonluminous Radiation. — All known bodies 
 are a constant source of radiant energy, since all possess some 
 degree of heat ; but bodies colder than their surroundings lose less 
 heat than they receive by the absorption of radiation falling upon 
 them, and hence become warmer. With few exceptions, bodies 
 emit only nonluminous. radiation (the so-called "radiant heat") 
 unless they are very hot. A piece of iron, for example, becomes 
 luminous at about 525°, at which temperature it emits a dull red 
 light. As the temperature increases, the light grows stronger and 
 changes in color, at last becoming white. The nonluminous radi- 
 ation also increases in intensity as the temperature rises, as is shown 
 by the greater heating of the hand when held near the body. 
 
258 
 
 Light 
 
 When a body is heated, its molecules vibrate more rapidly and 
 hence set up shorter waves in the ether, — an effect similar to that 
 produced by raising the pitch of a sounding body. As there are 
 sound waves too long and others too short to cause the sensation 
 of sound, so too there are ether waves both too long and too short 
 to cause sight. Ether waves of the proper length to stimulate 
 the optic nerve and cause the sensation of vision are called light. 
 The distinction between light and nonluminous radiation is there- 
 fore primarily a physiological one (Arts. 221 and 222). 
 
 332. The Propagation of Light in a Homogeneous Medium. — 
 Any space or substance through which light can travel is called a 
 medium. A medium is said to be homogeneous when its chemical 
 composition and density are the same in all parts of it. Although 
 light is transmitted or, as we commonly say, propagated in any 
 medium by means of the ether in its intermolecular spaces, the 
 substance itself affects the process in different ways, as will be seen 
 later. 
 
 When a beam of sunlight is admitted into a darkened room, 
 its path, which is rendered visible by the dust particles in the air, 
 is seen to be perfectly straight {Exp.). This is perhaps the best 
 illustration of a very general law ; namely, In aiety homogeneous 
 medium light travels in straight lines. The most familiar conse- 
 quence of this fact is the 
 formation of shadows. The 
 light that passes on either 
 side of an object continues 
 in a straight line ; if it bent 
 round into the space behind 
 ttie object, there would be 
 no shadow. 
 
 Figure 165 is a section 
 diagram in which each circle 
 represents a wave of light 
 from a small luminous source 
 at/. The light that passes through an opening, (9, in a screen. 
 
Nature and Transmission of Light 259 
 
 > 
 > 
 > 
 
 m 
 
 Fig. 166. 
 
 AB, is represented by C/D. It has a conical shape and is called 
 
 a cone or pencil of lights regardless of the shape of the opening 
 
 through which it passes. When 
 
 the source of the light is at a 
 
 relatively great distance, any 
 
 small area of a wave front is 
 
 sensibly plane (Fig. 166), and 
 
 the light that passes through a 
 
 small opening is cylindrical in 
 
 shape and is called a beam of 
 
 light (represented by CEFD in 
 
 the figure) . A pencil or a beam 
 
 of light so slender that its cross-section has no appreciable area 
 
 is commonly called a ray of light. A ray is often regarded merely 
 
 as a mathematical line indicating a direction in which light travels. 
 
 In this sense any radius of a train of spherical waves is a ray. Rays, 
 
 in either sense of the word, are perpendicular to the wave fronts. 
 
 333. Why Light travels in Straight Lines. — The thoughtful 
 pupil will perhaps wonder why light does not spread out after 
 going through an opening or past the edge of an object. We 
 know that sound travels round buildings and other obstacles. If 
 light is a wave motion, why does it not do the same? It was the 
 want of an answer to this question that caused Newton and his 
 followers to reject the wave theory. 
 
 After more than a century, experiments were devised which 
 proved that this difference between the behavior of light and 
 sound is due to the very great difference in their wave lengths. 
 The waves of most sounds are from one to ten feet in length ; 
 light waves (which are also capable of accurate measurement) vary 
 from 33,000 to 64,000 to the inch. It was found that light waves 
 and sound waves behave in a similar manner when they pass 
 through openings or encounter obstacles of the same relative size in 
 comparison with their wave lengths. The propagation of light in 
 straight lines, leaving the space behind opaque objects in shadow, 
 is due to the fact that ordinary bodies and openings are enor- 
 
26o Light 
 
 mously large comj)ared with the wave length of light. Sound 
 l)ehaves in a similar manner under corresponding conditions, 
 forming what may be termed sound shadows. " Some few years 
 ago a jx)wd^r hulk exploded on the river Mersey. Just opposite 
 the spot there is an opening of some size in the high ground which 
 forms the watershed between the Mersey and the Dee. The 
 noise of the explosion was heard through this opening for many 
 miles, and great damage was done. Places quite close to the 
 hulk, but behind the low hills through which the opening passes, 
 were completely protected, the noise was hardly heard, and no 
 damage to glass and such like happened. The opening was large 
 compared with the wave length of the sound " — Glazebrook's Phys- 
 ical Optics, 
 
 When light passes through an opening that is not large com- 
 pared wth the wave length (as is generally the case with sound), 
 it spreads out into the region that is ordinarily occupied by the 
 shadow. A simple example of these effects is obtained by look- 
 ing through a handkerchief, held close to the face, at a brightly 
 illuminated pinhole or narrow slit about the width of a pin. (Try 
 it.) The pinhole or slit may be made in a piece of cardboard, 
 and illuminated by holding it in front of a lamp or gas jet. The 
 light spreads out in different directions in passing through the 
 narrow spaces between the threads of cloth, causing the slit to 
 look like a number of parallel slits, and the hole like a square 
 pattern of many holes. Phenomena of this class are studied in 
 advanced physics. It is by means of experiments involving such 
 phenomena that the wave lengths of light are determined. 
 Laboratory Exercise -//. 
 
 334. Shadows. — A cone of light is intercepted by an opaque 
 body, as ab (Fig. 167), when the source 
 of the light, Z, is so small that it may be 
 regarded as a point; and the light is 
 wholly excluded from the portion of this 
 6"'^^^^^B conical space that lies beyond the body. 
 Fig. 167. d When the source of light is of appreciable 
 
Nature and Transmission of Light 261 
 
 size, the light is wholly excluded from a portion of the space 
 beyond the body, as acdb (Fig. 168) ; and this space is surrounded 
 or enveloped by a space from which the 
 light is partially excluded. The latter space 
 receives light from a part of the source, 
 the light from the remainder being inter- 
 cepted by the object. 
 
 In physics the word shadow means the 
 space from which light is wholly or partly 
 excluded by an opaque body. The dark 
 area upon any surface where it intercepts a 
 shadow is a cross-section of the shadow, and should be so named. 
 The part of the shadow from which the hght is wholly excluded is 
 called the umbra; the partly illuminated space surrounding the 
 umbra is called the penumbra. The penumbra merges impercep- 
 tibly into fully illuminated space at its outer surface ; the boundary 
 between the penumbra and umbra is more sharply defined. 
 
 335. Solar Eclipses. — In Fig. 169, 6" represents the sun, ^ 
 the earth, M^ the moon at new moon, and M^ at full moon. The 
 
 Fig. 168. 
 
 ■'V 
 
 Fig. 169. 
 
 shadows of the earth and the moon are diminishing cones, termi- 
 nating in a point. (Why?) 
 
 On account of the varying distances of the sun and the moon 
 from the earth, the moon's shadow {i.e. the umbra) sometimes 
 reaches the earth and is sometimes too short to do so. The cross- 
 section of the moon's shadow is never more than 167 miles wide 
 at the earth's surface. Within the shadow the sun is totally 
 eclipsed ; within the penumbra, which covers a much larger area, 
 the eclipse is partial. 
 
 It is evident that an eclipse of the sun can occur only at new 
 moon ; but there is not an eclipse at every new moon, for the moon 
 
262 Light 
 
 generally passes to one side or the other (above or below the plane 
 of the paper in the figure) of the straight line between the sun and 
 the earth. The least number of solar eclipses that can occur in a 
 year is two, and the greatest number is five. 
 
 336. Lunar Eclipses. — The diameter of the earth's shadow at 
 the distance of the moon is about two and two thirds times the 
 diameter of the moon. When the moon passes entirely into 
 the shadow, it is totally eclipsed ; when only one side of it passes 
 through the shadow, the eclipse is partial. There is no perceptible 
 dimming of the moon within the penumbra until it almost reaches 
 the shadow. An eclipse of the moon, either total or partial, is of 
 course visible to half the earth simultaneously. Since the moon is 
 a nonluminous body, shining only by reflected sunlight, it would 
 be invisible when totally eclipsed if it were not for the fact that 
 some light is bent out of its course (refracted) into the shadow 
 in passing through the earth's atmosphere. The moon is thus 
 illuminated with a dull, copper-colored light. 
 
 An eclipse of the moon can occur only at -ftrfHnobn ; but the 
 moon generally escapes the shadow by passing to one side or the 
 other of it. The number of lunar eclipses in a year varies from 
 none to three. 
 
 337. Images produced by Small Openings. — When light from 
 any luminous or brightly illuminated object falls upon a screen after 
 passing through a minute opening, such as a pinhole, it forms 
 upon the screen an inverted image of the object. Figure 170 
 
 shows how such images are 
 produced. Every point on 
 the surface of the object 
 is a source of light which 
 travels outward in all direc- 
 tions from the surface. A 
 sle Jer cone of this light 
 from each point passes through the pinhol and illuminates a small 
 spot on the screen of the same shape as the opening. Since these 
 spots have the same relative positions as the corresponding points of 
 
Nature and Transmission of Light 263 
 
 liie object, and are illuminated by light of the same color as those 
 points, they unite into an image which reproduces the form and 
 color of the object. The inversion of the image is due to the 
 crossing of the cones of light at the opening. If the opening is 
 very small, the image is quite sharply defined, but faint. With a 
 larger opening, the image is brighter, but poorly defined (blurred) ; 
 for the light from each point of the object now covers a larger spot 
 pn the screen, and these spots overlap more and more as their size 
 increases.^ 
 
 338. Intensity of Illumination. — The intensity of illumination 
 on any surface is the quantity of light received per unit area of the 
 surface. 
 
 Effect of Distance, — The intensity of the light from any source, 
 or the intensity of the illumination that it produces on any surface, 
 is inversely proportional to the square of the distance from the 
 source. This " law of inverse squares " is the same as that for 
 sound (Art. 288), and for the same reasons. The reasoning, as 
 applied to light, may be briefly stated as follows: (i) Assuming 
 that there is no loss of light by absorption or other cause, the 
 same total quantity of light (radiant energy) passes all cross- 
 sections of a cone of light ; (2) hence the quantity of light per 
 unit area at any distance is inversely proportional to the area of 
 the cross-section of the cone at that distance ; (3) but the area of 
 the cross-section of a cone of any shape is proportional to the 
 square of the distance from its vertex (Fig. 171); (4) hence the law. 
 
 1 If the class room can be made perfectly dark, an interesting study of these //'«- 
 hole images, as they are called, can be made by admitting light into the darkened 
 room through a small hole in a window screen or shutter, and catching it up>on a 
 screen of oiled tissue paper or upon the opposite wall of the room. In the latter 
 case the opening must be a centimeter or more in diameter to give suflficent illumi- 
 nation. Under these conditions an image of the landscape in its natural colors will 
 be formed upon the wall or the screen. A pinhole camera can be made from a 
 pastelxjard box a foot or more in length. A large pinhole is made in the center of 
 one end to admit the light, which is caught upon a screen of oiled tissue paper 
 pasted in position across the center of the box. A hole half a centimeter or more 
 in diameter is made in the other end of the box. The image upon the screen is 
 viewed by placing the eye close to this hole. 
 
264 Light 
 
 Effect of Illuminating Power. — The total quantity of light con- 
 tinuously given out by any source is called 
 its illuminating power. The light given 
 out by a standard candle is taken as the 
 unit and is called a candle power. Thus an 
 incandescent lamp of sixteen candle power 
 gives out sixteen times as much light as 
 a standard candle. The intensities of iHumination produced by 
 two sources of light at equal distances are proportional to the 
 illuminating powers of the sources. 
 
 Effect of Distance and Illuminating Poiver Combined. — Let P 
 denote the illuminating power of a source of light, and / the in- 
 tensity of illumination which it produces at the distance D; then, 
 since / is proportional to P and inversely proportional to the 
 square of Z>, we may write p 
 
 ^^ D"' 
 
 339. Photometry. — Photometry deals with the comparison and 
 measurement of the illuminating powers of different sources of light. 
 Any apparatus by means of which such measurements are made 
 \%Z2^^^2i photometer. We can form no reliable estimate of the 
 relative brightness of unequally illuminated surfaces, but are able 
 to judge with considerable accuracy whether two adjacent parts of 
 the same surface are equally illuminated ; hence, with all forms of 
 photometers, the distances of the lights compared are adjusted to 
 give equal illumination. 
 
 Let Px and P^ denote the illuminating powers of two sources of 
 light, and let /, and L denote the intensities of illumination which 
 they produce at the distances D^ and Z>2, respectively ; then, 
 
 If the distances are such that the intensities of illumination are 
 equal {i.e. if I^ = /a), then 
 
 ' — ^^2 f.j.— p.p..r)'i.r)2 
 
Nature and Transmission of Light 265 
 
 This is the relation used in photometric measurements. Stated 
 in words : The illuminating powers of two sources of light are pro- 
 portional to the squares of the distances at which they produce 
 equal illumination. 
 
 Laboratory Exercise 48. 
 
 340. The Shadow Photometer (Rumf ord's Photometer) . — There 
 are several forms of photometers ; but the shadow photometer, 
 devised by Count Rumford, will serve as a sufficient illustration. 
 Figure 172 shows the adjustment of the apparatus for a comparison 
 
 172. 
 
 of the illuminating powers of a lamp and a candle. The rod, Ry 
 casts two shadows ; c is due to the candle, and / to the lamp. 
 Each source of light illuminates the shadow due to the other ; 
 hence when they are placed so that the shadows are equally dark 
 {i.e. equally illuminated), they give equal illumination at their 
 respective distances from the screen. The room must be dark- 
 ened, or other sources of light will make the shadows too foint for 
 accurate comparison. 
 
 341. The Velocity of Light. — The velocity of light in a vacuum 
 and in air is 186,000 miles per second in round numbers, — a 
 velocity sufficient to encircle the earth seven and one half times 
 in one second. Notwithstanding the difficulties involved in the 
 measurement of such rapid motion, this velocity has been re- 
 peatedly determined with consistent results by two experimental 
 methods ; it has also been computed by two independent methods 
 
266 Light 
 
 from astronomical data. We shall consider only the astronomical 
 method by which the first determination of the velocity of light 
 was made. 
 
 Jupiter, the largest of the planets, revolves about the sun in an 
 orbit whose diameter is about five times that of the earth, the 
 time occupied in one revolution being nearly twelve years. In Fig. 
 1 73, S represents the sun, £* and £ two positions of the earth in 
 
 ©\ its orbit, y* and J two positions of 
 
 \-v^ Jupiter. Jupiter is accompanied 
 \T^ by a number of satellites (moons), 
 s ]i which revolve round it as the moon 
 
 I does round the earth. The nearest 
 / of these satellites is shown in the 
 
 pftfc _ figure. It passes through the 
 
 _ / shadow of Jupiter in each revolu- 
 
 Fic. 173. / •' * 
 
 / tion, and is thus eclipsed at regular 
 
 intervals of 42 hr. 28 min. and 36 sec. An observer does not see 
 these eclipses when they occur ; for, after the satellite has entered 
 the shadow, it continues visible unAV the light that last Uft it 
 reaches the earth. If the eclipses were observed from any constant 
 distance, they would be seen at regular intervals ; for they would 
 all be seen the same length of time after their actual occurrence. 
 For example, if the earth remained at its least distance from 
 Jupiter, each eclipse would be seen 35 min. after it occurred, that 
 being the time required for light to travel the intervening distance, 
 EJ\ if the earth remained at its greatest distance, EJ, each 
 eclipse would be seen 51 min. and 40 sec. after it occurred, and 
 the observed intervals between the eclipses would be the same as 
 before. But, while the earth, in its annual revolution, is receding 
 from Jupiter, the observed interval between successive eclipses 
 exceeds the true interval by the time required for light to travel 
 the' added distance due to the earth's motion during the interval. 
 While the earth is advancing* toward Jupiter, the conditions are 
 reversed and the observed intervals are less than the true intervals. 
 Hence if the eclipses are computed ahead for one year, begin- 
 
Nature and Transmission of Light 267 
 
 ning when the earth is at E and assuming a constant interval, the 
 echpses as observed fall constantly more and more behind until 
 the earth is at E\ when they are 16 min. and 40 sec. late. Dur- 
 ing the second half of the year, the intervals are less than the 
 average, and the loss is gradually made up; so that when the 
 earth has arrived at E the eclipses are again on time. 
 
 These irregularities in the observed intervals were discovered 
 and explained by Roeme r. a Danish astronomer, in 1675. He 
 announced that the whole apparent retardation of the eclipses 
 from E to E' was the time required for light to travel across the 
 earth's orbit, which is known to be a distance of 186,000,000 
 mi.^ (Compute the velocity of light from the given data.) 
 
 PROBLEMS 
 
 1. {a) What is the necessary condition for a shadow without a penumbra? 
 (^) for an umbra of finite length? (r) for an umbra of indefinite length? 
 (</) for a cylindrical umbra? Make a section drawing, illustrating each case. 
 
 2. WTiy are shadows as we see them in nature not i)crfectly dark ? 
 
 3. What is the apparent shape of the moon two or three days after new 
 moon? at the first quarter? between the first quarter and full moon? Ac- 
 count for these apparent shapes of the moon. 
 
 4. State and account for the change in the size, brilliance, and sharpness 
 of outline of a pinhole image (<i) when the screen upon which it is caught is 
 moved farther from the opening; (/i) when the size of the opening is increased. 
 
 5. What determines the ratio of the size of a pinhole image to the size 
 of the object? 
 
 6. Two sources of light give equal illumination at loo cm. and 30 cm., 
 respectively. Hqw do they compare in illuminating power? 
 
 7. At what distance will a i6-candle-power lamp give the same intensity 
 of illumination as a standard candle at a distance of 70 cm. ? 
 
 1 Light reaches us from the sun in 8 min. and 20 sec. An express train travel- 
 ing at the rate 0(43 mi. per hour would require 254 years for the journey. Yet in- 
 conceivably great as this distance is, it is insignificant in comparison with that of 
 the fixed stars, the nearest of which, so far as is known, is so far away that light 
 from it requires 3.5 years to reach us. Light from the north star requires about 
 50 years for the journey. Indeed, so vast are stdlar distances that astronomers 
 regularly use the light year (the distance that light travels in one year) as the unit 
 in which to express them. 
 
268 Light 
 
 8. (tf) The cross-section of a cone of light is S scm. at a distance of 20 cm. 
 from its vertex. What is its cross.section at a distance of icx} cm. from the 
 vertex? (6) How does the intensity of the light at the second distance com- 
 pare with that at the first ? 
 
 II. Reflection of Light 
 
 342. Regular and Irregular Reflection. — The reflection of light 
 by different surfaces may be studied by means of a beam of sun- 
 hght admitted into a darkened room. A quantity of chalk dust 
 in the air makes the path of the light plainly visible. A plane 
 mirror reflects the beam in a definite direction, which depends 
 upon the angle at which the mirror is held. The light is called 
 incident light before reflection, and reflected light after reflection. 
 The angles made with the surface of the mirror by the incident 
 and reflected beams are always equal (as would be shown by 
 exact measurement) in whatever position the mirror is held. 
 When the incident beam is perpendicular to the mirror, the re- 
 flected beam coincides with it {Exp.). 
 
 The beam reflected by a mirror forms a small bright spot on 
 the wall ; but, when the light is reflected by a piece of tin, the 
 illuminated spot is larger, fainter, and less sharply defined, showing 
 that the reflection from the tin is less regular. When the beam 
 falls upon a sheet of writing paper or a piece of cardboard, the 
 reflected light is said to be irregularly reflected or diffused by the 
 paper. It will be observed that the paper is brilliantly illuminated 
 by the sunbeam and can be distinctly seen from all parts of the 
 room; while- the surface of the mirror is nearly" invisible, even 
 when the eye is in the path of the reflected beam {Exp.). 
 
 Although a sheet of writing paper may appear to be perfectly 
 smooth, a hand lens is sufficient to show minute irregularities 
 covering the entire surface. These irregularities are the cause of 
 the diffusion of light by the paper (Art. 345). 
 
 343. Definitions. — In Fig. 174, AB represents a section 
 through a mirror whose surface is perpendicular to the plane of the 
 paper; C^ represents a slender beam (ray), falling upon the 
 
Reflection of Light 
 
 269 
 
 mirror at iV (called \kiQ point of incidence)^ and ND, the reflected 
 beam ; MN is the perpendicular or 
 normal to the reflecting surfiice at the 
 point of incidence. 
 
 The angle between the incident beam 
 (or ray) and the perpendicular to the 
 reflecting surface at the point of inci- 
 dence is called the angle of incidence (angle CNM, or / ) ; and the 
 angle between the reflected beam and this perpendicular is called 
 the angle of reflection (angle MND, or r). 
 
 344. Laws of Reflection. — When light is reflected from polished 
 surfaces, the angles of incidence and reflection are equal, and they 
 are in the same plane. Since this plane contains the normal, it 
 is evidently perpendicular to the reflecting surface. 
 
 These laws can be proved by direct measurement, and can also 
 be deduced from the experimental study of images (Art. 347). 
 The reason for this behavior of light is illustrated in part by Fig. 
 175, which represents the reflection of a beam of light, CDFE, 
 
 by a mirror, AB. Consider the wave 
 that would be represented by LMN 
 if it did not meet the mirror. The 
 edge N met the mirror first, was 
 reflected, and is now at N\ having 
 traveled since reflection a distance 
 DN' equal to DN. The parts of 
 the wave from N Ko M were succes- 
 sively reflected in like manner, and now occupy the position N^M. 
 The laws of reflection are the result of the fact that each portion 
 of the wave travels as far after reflection as it would have traveled 
 in the same time if it had not met the reflecting surface. Hence 
 in the figure triangles DN'M and DNM are congruent ; from 
 which it is easy to prove that the angles of incidence and reflec- 
 tion are equal. (The proof is left to the pupil.) 
 
 345. Diffusion of Light. — The irregular reflection or diffiision 
 of light, as already stated, is due to microscopic irregularities of 
 
 Fig. 175. 
 
270 
 
 Light 
 
 Fu;. 176. 
 
 the reflecting surface. This is illustrated by Fig. 1 76, which repre- 
 sents a highly magnified section of an unpolished surface. Such 
 a surface is made up of minute parts in- 
 clined at all angles to one another. The 
 parts of a beam of light meet these minute 
 areas at different angles, and are reflected 
 in all directions. The irregularities on an 
 unjx)lished surface are so numerous that 
 any spot large enough to be seen includes 
 a sufficient number of them to reflect the light in all directions ; 
 so that, in 'effect, rvery point of the surface is a source of li^ht 
 which travels outivard in all directions as from a luminous 
 body. 
 
 The light received from the sky during the day is sunlight that 
 has been diffused by minute particles of dust in the air.^ 'J'his 
 diffusion takes place principally in the lower regions of the atmos- 
 phere, where the dust particles are the largest and the most 
 numerous ; hence the sky is much darker upon high mountains 
 than at lower altitudes. Shadows cast by objects in the sunlight 
 are only relatively dark, as they are generally quite strongly 
 illuminated by diffused light from the sky and from surrounding 
 objects. 
 
 346. The Light by which Objects are Seen. — Any object, either 
 luminous or illuminated, is seen by means of the cones of light 
 that enter the pupil of the eye from all points 
 of the visible portion of its surface. Figure 
 177 represents three such cones of light from 
 three points of a surface. When an object is 
 viewed directly {i.e. without the aid of mirrors 
 or lenses), the light follows a straight path to 
 the eye, and each point of the object is seen in its tnie position 
 at the vertex of the cone of light that enters the eye from the 
 point. 
 
 1 The process is illustrated by the effect of chalk dust ia the path of a beam of 
 light in a darkened room. 
 
 Fk;. 177. 
 
Reflection of Light 
 
 271 
 
 Laboratory Exercise 4g. 
 
 347. The Image of a Point in a Plane Mirror. — It is found by 
 experiment that the image of a point in a plane mirror (from 
 whatever point it may be viewed) is on the perpendicular to the 
 mirror from the point, and is as far behind the reflecting surface 
 as the point is in front of it (Lab. Ex. 49). 
 
 The image is at the vertex of the cone of 
 light by which it is seen ; it is, in fact, the 
 apparent source of this light. This is shown 
 in Fig. 178, in which AB is the mirror, O 
 the point object, and / its image. The image 
 is the center of the reflected waves ; hence 
 the light after reflection travels just as it 
 would if the image were its real source, the 
 mirror being removed. 
 
 This explains why light regularly reflected 
 by a mirror does not render the mirror itself 
 visible. The mirror is seen by the very small amount of light 
 that is diffused by its surface. A good mirror is therefore nearly 
 invisible when its surface is perfectly clean. 
 
 Having determined by experiment the relative position of a 
 point and its image in a plane mirror, the laws of reflection may 
 be established as follows : The experiment proves that OI is 
 perpehdicular to AB and that (9C= CI. Hence triangles NCO 
 and NCI are right triangles and are congruent ; from which it is 
 easily proved that the angle of incidence ONM is equal to the 
 angle of reflection MNE. (The completion of the proof is left 
 to the pupil, also the proof that the angles lie in the same plane.) 
 
 Conversely, if we a'ssume the law of reflection to be known, the 
 position of the image can be demonstrated geometrically. (An 
 exercise for the pupil.) 
 
 348. The Image of an Object. — Any image formed by a plane 
 mirror is so situated that the line joining any point of the object 
 and the image of that point is perpendicular to the mirror and 
 is bisected by it. Hence, to locate such an image in a drawing, 
 
272 
 
 Light 
 
 perpendiculars are drawn from a sufficient number of points of the 
 object to the mirror and extended equal distances behind it. The 
 extremities of the perpendiculars are the corresponding points of 
 the image. Thus in Fig. 179 the ends of the 
 image C and D' are found by dropping per- 
 pendiculars from C and D according to the 
 rule. 
 
 To construct the path of light to the eye 
 from any ix)int of the image, as C, draw a line 
 from that point to the eye, and from the point 
 of intersection of this line with the mirror 
 draw a line to the corresix)nding point of the object. Behind 
 the mirror the lines are dotted to indicate that they are only 
 apparent paths of the light. Such diagrams are commonly simpli- 
 fied by drawing only a single line to represent the cone of light 
 that enters the eye from any point. 
 
 Images formed by plane mirrors are evidently of the same size 
 as the objects. They are erect (unless the mirror is horizontal, 
 as the surface of still water) ; but object and image differ as the 
 right hand differs from the left. 
 Laboratory Exercise ^o. 
 
 349. Images by Multiple Reflection. — Images formed by suc- 
 cessive reflections from two mirrors are called multiple images or 
 images of images. The formation of multiple images and the rule 
 for locating them in diagrams are 
 illustrated in Fig. 180. After reflec- 
 tion from the mirror AB, the path 
 of light from O is the same as it 
 would be if the image /j were its real 
 source (Art. 347). Hence the por- 
 tion of this reflected light that falls 
 upon the mirror CD forms a second 
 image h, whose position is the same 
 as it would be if /j were the object, 
 the mirror AB being removed. In this sense I^ is the image of 
 
 U 
 
 Fig. 180, 
 
Reflection of Light 273 
 
 /i, and is located by first locating /j, then treating it as the object 
 whose image is L, 
 
 Light after reflection from AB and CD apparently comes from 
 A If some of this light again falls upon AB, a third image /, 
 will be formed, whose position is found by treating I-i as the object. 
 With the angle between the mirrors as shown in the figure, /, is 
 the last image of the series ; for, since it lies behind the plane of 
 the mirror CD, none of the light coming from it or from its 
 direction can fall upon this mirror. The smaller the angle between 
 the mirrors, the greater is the number of images in the series ; 
 when the mirrors are parallel, the series is indefinite, being limited 
 only by the gradually failing intensity of the light. There is, of 
 course, a second series of images, formed by reflection first from 
 CD, then from AB, etc. This series is not represented in the 
 figure. 
 
 Since the light by which each image is formed comes from the 
 direction of the preceding image of the series, we have the follow- 
 ing rule (illustrated in the figure for /,) for constructing the path 
 of the light to the eye for any image : Draw a line from the image 
 to the eye {L,E in the figure) ; from the point where this line 
 intersects the mirror draw a line to the preceding image of the 
 series (/i in this case) ; from the point where this line intersects 
 the other mirror draw a line to the next preceding image or, if 
 this is the end of the series, to the object. The path of the light 
 that reaches the eye is thus constructed backward without the 
 necessity of measuring angles. (Show that this method of con- 
 struction makes the angles of incidence and reflection equal.) 
 
 PROBLEMS 
 
 1. Sound is reflected regularly from a brick or a stone wall, and even from 
 cliffs whose faces are quite irregular. Show by a comparison of wave lengths 
 that these facts are not inconsistent with the irregular reflection of light from 
 unpolished surfaces. 
 
 2. (rt) In what respects does a pinhole image differ from an image in a 
 plane mirror ? {F) What purpose is served by the screen upon which a pin- 
 hole image is caught ? {c) Can an image formed by a plane mirror be caught 
 upon a screen ? 
 
274 
 
 Light 
 
 3. Why are stars invisible by day ? 
 
 4. Copy Fig. 180 ami locate the other series of images. Construct the 
 path of light to the eye for the third image of either series. 
 
 5. Draw a figure of mirrors at an angle of 90", locate the images of a 
 point object, and construct the path of light to the eye for each image. 
 
 6. Make a similar diagram with the mirrors at an angle of 6o'\ and the 
 object at uneciual distances from the mirrors. Prove that the object and 
 images lie on the circumference of a circle. 
 
 7. Make a similar diagram with the mirrors parallel ; locate the images 
 of each series; and construct the path of light to the eye for the first three 
 images of one series. Prove that the images lie in a straight line. 
 
 350. Spherical Mirrors. — A spherical mirror is a mirror whose 
 reflecting surface is a portion (usually a very small portion) of a 
 spherical surface. It is concave or convex according as the reflec- 
 tion takes place from the inside or the outside of the spherical 
 surface. Concave mirrors are not always spherical ; but in speak- 
 ing of them it is generally assumed that the curvature is spherical 
 unless it is otherwise stated. A spherical mirror is represented in 
 a section diagram by an arc of a circle, as MN (Fig. 181). The 
 
 Fig. 181. 
 
 center 0/ curvature of a spherical mirror is the center of the sphere 
 of which the mirror forms a part (C in the figure). The radius 
 of the sphere is called the radius of curvature of the mirror. The 
 angle MCN, formed by joining the center of curvature and extremi- 
 ties of the mirror, is its angular aperture^ or, simply, the aper- 
 ture. The straight line ACy which passes through the center of 
 the reflecting surface and the center of curvature, is called the 
 
Reflection of Light 
 
 275 
 
 \ 
 
 principal axis of the mirror ; any other line through the center of 
 curvature is a secondary axis. 
 
 351. Reflection of a Beam by a Concave Mirror. — The same 
 laws of reflection hold for curved surfaces as for plane ; but the 
 effect of the reflection is different. ^ When a beam of sunlight is 
 reflected by a concave mirror in a darkened room, the reflected 
 light is seen to have the form of a double cone {MFN and EFG^ 
 Fig. 181). The light is reflected as a converging cone; beyond 
 the point of convergence (the focus) it continues as a diverging 
 cone {Exp.). 
 
 In the study of reflection from plane surfaces we met with only 
 diverging cones of light. The waves of a diverging cone are 
 convex (viewed from the side toward which they are traveling) ; 
 the waves of a converging cone are concave (see figure). If the 
 direction of propagation of a diverging cone were reversed, the 
 light would return to its source as a converging cone with the con- 
 cave side of the waves in advance. ' A converging cone of light 
 increases in intensity toward the focus for the same reasons that 
 a diverging cone increases in intensity toward its source. When 
 the sunlight reflected by a concave mirror is caught upon white 
 paper at the focus, it is intensely bright, and the temperature may 
 be high enough to set fire to the paper. A match at this point is 
 quickly ignited (Exp.). 
 ^ The convergence of the light reflected from a concave surfiice 
 is explained by the laws of reflection. Consider the light incident 
 at il/when the incident beam 
 is parallel to the principal 
 axis. The perpendicular to 
 the mirror at M is MC, a 
 radius of the spherical sur- 
 face. Since the angles of 
 incidence and reflection are 
 equal, the reflected light evi- 
 dently crosses the princi- 
 pal axis at some point, Fj Fig. 182. 
 
276 Light 
 
 between the mirror and the center of curvature. An accurately 
 constnicted figure will show that light incident at any other point, 
 as at Hf also crosses the principal axis at or very near F ; hence 
 all the reflected light converges approximately to F, 
 
 It will be found by trial that the position of the focus changes 
 when the mirror is inclined at different angles to the beam. When 
 the incident beam is parallel to the principal axis, the focus lies 
 on this axis ; when the incident beam is inclined to the principal 
 axis, the beam and the focus, F\ are on opposite sides of it 
 (Fig. 182). 
 352. The Principal Focus and the Focal Length. — Let BAT 
 
 (Fig. 183) be any ray parallel to 
 the principal axis of a concave 
 mirror, and M the point of in- 
 cidence. C is the center of 
 curvature, and MC the perpen- 
 dicular at the point of incidence. 
 F is the point where the reflected ray cuts the principal axis. 
 
 Angles / and e are equal (alternate-interior angles of parallel 
 lines), and angles / and r are equal (law of reflection). Hence 
 angles r and e are equal, the triangle MFC is isosceles, and 
 MF=FC. If MA is not more than a tenth of the radius of 
 curvature (which must be the case if the angular aperture of the 
 mirror is not above 8° or 10°), AF is very nearly equal to MF^ 
 and hence also to FC \ i.e. AF=FC, approximately. The aper- 
 tures of concave mirrors are always small in order that this relation 
 may hold for light incident at any point of their surface. Hence 
 all rays parallel to the principal axis of a concave mirror are 
 reflected approximately to a point on the principal axis midway 
 between the mirror and its center of curvature. This point is 
 called \.\\t principal focus (F, Fig. 183), and is defined as the point 
 to which a beam of light parallel to the principal axis converges 
 after reflection. The distance from the mirror to the principal 
 focus is called the focal length of the mirror ; it is one half the 
 radius of curvature. 
 
 Fig. 183. 
 
Reflection of Light 277 
 
 Laboratory Exercise §1. 
 
 353. Conjugate Foci. — Figure 184 shows the effect of a concave 
 mirror on light radiated from a luminous point O on its principal 
 
 Fig. 184. 
 
 axis beyond the center of curvature. Since the angle of incidence 
 of the ray OM is less than that of a ray parallel to the principal 
 axis and incident at the same point, the reflected ray will evidently 
 cross the principal axis at some point / between the principal 
 focus and the center of curvature. It can be shown that (with a 
 mirror of small aperture) all the light of the incident cone MON 
 converges after reflection very approximately to the same point /, 
 which is therefore the focus of the reflected light and the image of 
 the source O. 
 
 This image can be seen and its position accurately determined 
 in two ways : (i) It can be viewed directly from any point within 
 the diverging cone of light, E/Gf and will then be seen in its 
 true position, if the eyes are directed toward it (not toward the 
 more distant mirror). (2) It can be caught upon a screen, which 
 should be held so as to intercept as little as possible of the inci- 
 dent light MON. When the screen is placed at /, the image 
 appears upon it as a bright point ; when it is placed at either a 
 greater or a less distance from the mirror, it is illuminated over an 
 area which is the cross-section of the cone of reflected light 2X 
 that distance (see the laboratory exercise). 
 
 If the source of light is removed from O to /, the former angles 
 of reflection at each point, as at M, become the angles of incidence, 
 and vice versa; and the reflected light will converge to O. 
 Hence the positions of image and object are interchangeable. 
 Any two points which, like O and /, are so situated that light 
 
2/8 Light 
 
 radiating from either converges after reflection to the other, are 
 called conjugate foci. Conjugate foci afford an ilhistration of the 
 fact that light can always traverse the same path in both directions. 
 
 354. Relative Positions of Conjugate Foci. — The position of 
 the focus conjugate to any given focus may be found by experi- 
 ment or determined from the laws of reflection. There are several 
 cases, as follows : — 
 
 {a) Conjugate Foci on the Principal Axis, — When the source 
 of light is a luminous point on the principal axis, the image (con- 
 jugate focus) is also a point on this axis. There are six cases. 
 
 (i) When the object is indefinitely far away, the image is at the 
 principal focus. This case is approximately illustrated by a sun- 
 beam parallel to the principal axis ; but only approximately, since 
 the sun, although very distant, is not equivalent to a point source. 
 The light at the focus covers a small round spot which is an image 
 of the sun. 
 
 (2) As the object approaches the center of curvature along 
 the principal axis from a great distance, the image moves from 
 the principal focus toward the center of curvature (Fig. 184). 
 
 (3) When the object is at the center of curvature, the image 
 coincides with it ; for each incident ray is perpendicular to the 
 mirror and retraces its path to the center. 
 
 (4) As the object moves from the center of curvature toward 
 the principal focus, the image recedes from the center of curva- 
 ture to an indefinite distance. This is the equivalent of case (2) 
 with the object and image interchanged. 
 
 (5) When the object is at the principal focus, the image is 
 indefinitely far away, or we may say that no image is formed; 
 This is case (i) with object and image interchanged. 
 
 (6) As the object is brought up from an indefinite distance 
 to the principal focus, the divergence of the incident cone of light 
 (measured by the angle at the vertex of the cone) steadily in- 
 creases, and the convergence of the reflected cone simultaneously 
 decreases. When the object is at the principal focus, the reflected 
 light is no longer convergent, but parallel. 
 
Reflection of Light 
 
 279 
 
 As the object approaches the mirror from the principal focus, 
 the divergence of the incident hght increases still further, and is 
 now only partly overcome by the reflection. The reflected light 
 is therefore divergent; and 
 its apparent source (or im- 
 age) is a point, / ( Fig. 185), ,^,'" 
 situated l^ehind the mirror ^"^^'^^ 
 on the principal axis (pro- """ 
 duced). The image is 
 
 located by producing back- * ' ^' 
 
 ward any two lines representing rays of reflected light. Such an 
 image (or focus) is called virtual^ while those in front of the mirror, 
 to which light converges after reflection, are called real. As the 
 object is moved up to the mirror from the principal focus, its 
 image (the conjugate focus) approaches the mirror, but is always 
 at a greater distance from it than the object is. (Why?) 
 
 (^) Conjugate Fofi on Secondary Axes. — Let O (Fig. 186) 
 be a luminous point not on the principal axis AC. The ray (9C, 
 
 passing through the cen- 
 ter of curvature, falls 
 upon the mirror perpen- 
 dicularly, and is reflected 
 back along the same 
 path. The image of O must therefore be at some point on this 
 secondary axis. The ray OMj parallel to the principal axis, 
 passes through the principal focus F after reflection. The point 
 of intersection of the reflected rays NC and MF is /, which is 
 therefore the point of convergence of ail the reflected light from 
 O ; i.e. the diverging incident cone of light MON is changed by 
 reflection into the converging cone MIN. Conversely, if the 
 object were removed to /, J//A^ would be the incident and MON 
 the reflected light, and the image would be at O. 
 
 In general : When the source of light is a luminous point 
 lying on one side of the principal axis, its image (the conjugate 
 focus) lies on the opposite side of this axis, and is on the same 
 
28o Light 
 
 secondary axis as the object. The relative positions of object 
 
 and image on a secondary 
 
 axis are the same as when 
 
 they are situated on the 
 
 principal axis. Figure 187 
 
 illustrates the case where 
 
 ,<.l-:r-:''^^"!r^ I! ' the image is virtual, the 
 
 7— ** Fig. 187. ,. r , , . ! . 
 
 distance of the object bemg 
 
 less than the focal length of the mirror. 
 
 355. Images formed by Concave Mirrors. — When the source 
 of light that falls upon a concave mirror is an object of appreci- 
 able size, the diverging cone of light from each point of the object 
 converges after reflection to the corresponding point of the image if 
 the image is real, or diverges as if it came from the correspond- 
 ing point of the image, if the image is virtual. A virtual image 
 formed by a concave mirror is in this respect like the virtual 
 images formed by plane mirrors. 
 
 In drawing figures to illustrate the formation of images by con- 
 cave mirrors, the size and position of the image are determined by 
 finding the conjugate focus of a point at each end of the object. 
 Since the conjugate focus of any point is the point of intersection 
 (real or apparent) of all rays from the point after reflection, it can 
 be determined in a diagram by constructing any two of these rays. 
 This, in general, requires the measurement of angles of incidence 
 and reflection ; but, unless the point is on the principal axis, there 
 are three rays from it whose directions after reflection are known, 
 and, by choosing any two of them, the measurement of angles is 
 avoided. These rays are as follows : (i) The ray (from the chosen 
 point of the object) through (or from the direction of) the center 
 of curvature. This is reflected back along the same path. 
 (2) The ray parallel to the principal axis. This passes through 
 the principal focus after reflection. (3) The ray through the 
 principal focus. This is reflected parallel to the principal axis. 
 
 This method of construction is employed in Figs. 188, 189, and 
 190 in locating the ends of the imige of an arrow. These figures 
 
Reflection of Light 
 
 281 
 
 . illustrate and explain the characteristics of the image for difierent 
 positions of the object. When the distance of the object from 
 the mirror is greater than the focal length, the image is real and 
 inverted^ as in the first four cases following. 
 
 (i) When the distance of the object is so great that the rays 
 from any point of it are sensibly parallel (a distance not less than 
 one hundred times the 
 
 focal length of the 
 
 mirror), the image is 
 
 at the principal focus 
 
 and is very small in 
 
 comparison with the 
 
 object. In Fig. 188 the rays A and A^ are from the same point 
 
 at the top of the object, and B and B^ from the same point at 
 
 the bottom : ab is the image. (As the object is at a great distance, 
 
 it cannot be- shown in the figure.) 
 / (2) When the object is beyond the center of curvature but not 
 
 at a great distance, the 
 image is between the center 
 of curvature and the princi- 
 pal focus, and is smaller 
 than the object (Fig. 189). 
 (3) When the object is 
 
 at the center of curvature, the image is also at the center of 
 
 curvature, and is of the same size as the object. (Draw figure.) 
 
 (4) When the object is between the center of curvature and 
 the principal focus, the image is 
 beyond the center of curvature, 
 and is larger than the object 
 (Fig. 189, taking ab as the ob- 
 ject, and AB as the image). 
 
 (5) When the object is at the 
 principal focus, the rays from 
 any point of it are parallel after reflection, and there is no per- 
 fect image (Fig. 188, taking ab as the object). 
 
 Fig. 189 
 
282 
 
 Light 
 
 Fig. 19X. 
 
 (6) When the object is between the principal focus and the 
 mirror, the image is virtita/ and /•/rr/, and is larger than the object 
 (Fig. 190). As the object approaches the mirror, the image also 
 approaches it and grows smaller. 
 
 356. Spherical Aberration : Parabolic Mirrors. — If the angular 
 aperture of a spherical mirror is large, the light that falls upon 
 
 different portions 
 of its surface from 
 the same point of 
 the object is not 
 nil brought to the 
 same focus after 
 reflection (Fig. 
 191). T\\\s aber- 
 ration or devia- 
 tion r)f the light 
 from the true focus increases as the distance of the point of inci- 
 dence from the axis increases ; hence the images formed by mir- 
 rors of large aperture are very imperfect. If the aperture (angle 
 MCN) does not exceed 10°, the focusing is approximately perfect. 
 
 When reflectors of large aperture are re- 
 quired, as in searchlights and the headlights of 
 locomotives, parabolic mirrors are used (Fig. 
 192). The curvature of a parabolic mirror 
 gradually decreases toward the edge, and light 
 radiating from the principal focus is reflected parallel to the prin- 
 cipal axis from all parts of the surface. The mirror of a reflecting 
 
 telescope is parabolic. 
 
 357. Convex Mirrors. 
 — A beam of light paral- 
 lel to the principal axis 
 of a convex mirror is 
 reflected as a diverging 
 cone whose vertex is the 
 point on the principal axis midway between the mirror and the 
 
 F.G. 192. 
 
 Fig. 193. 
 
Refraction of Light 283 
 
 center of curvature (Fig. 193). This point is called the prin- 
 cipal focus {Exp.), 
 
 The divergence of a cone of light is always increased by reflec- 
 tion from a convex mir- 
 ror ; hence the reflected 
 light is always divergent 
 and only virtual images 
 are formed (Fig. 194). 
 
 As an object approaches * ^^^ 
 
 a convex mirror from a great distance, its image, which at first 
 is at the principal focus and is very small, approaches the mirror 
 and increases in size. It is always erect and smaller than the object. 
 (The pupil may draw figures illustrating the above statements.) 
 
 PROBLEMS 
 
 1. (rt) Account for the difTerence in the brilliance ami distinctness of a 
 pinhole image and a real inmge formed by a concave mirror. {h) A pinhole 
 image is an imperfect real image. Why real ? Why imperfect ? 
 
 2. A pinhole image can be seen only when caught upon a screen. Why 
 can it not be seen in the air like a real image formed by a concave mirror ? 
 
 3. Prove that the divergence of a cone of light is not changed by reflection 
 from a plane mirror. 
 
 4. Why do plane and convex mirrors form only virtual images ? 
 
 5. What are the essential characteristics of a virtual image ? of a real 
 image ? (The answers involve a consideration of the convergence or diver- 
 gence of the reflected light. ) 
 
 6. Prove that the lengths of object and image (whether real or virtual) 
 are proportional to their distances from the center of curvature of a concave 
 mirror. 
 
 III. Refraction of Light 
 
 358. Refraction. — When a ray of light passes obliquely from 
 one transparent medium into another, it is bent, or refracted, at 
 the surface of separation of the two media. The refraction of a 
 beam of light on entering water from air can be shown in a dark- 
 ened room, by causing the beam to fall upon the surface of water 
 
284 
 
 Light 
 
 contained in a rectangular glass vessel (Fig. 195). The path of 
 the beam in the water is more distinct if the water is clouded with 
 a small- quantity of an alcoholic solution 
 of mastic or a little milk. It will be ob- 
 served that the refracted beam is less 
 oblique to the surface than the incident 
 beam ; on passing from air into waicr^ 
 Fig. 195. f^^j^f j^ refracted to^vard the perpendicular 
 
 to the surface at the point of incidence. The change of direction, 
 or deviation^ of the beam on entering the water is less when the 
 beam falls less obliquely upon the surface ; and when the inci- 
 dence is perpendicular, the beam enters the water without change 
 of direction {Exp.). 
 
 In Fig. 196, EOH represents a ray of light passing from air 
 into water at O. EO is the incident and OH the refracted ray. 
 MN is the perpendicular to the surface at 
 the point of incidence. The angle / is 
 called the angle of incidence ^ r the ajigle of 
 refraction^ and d the angle of dexnation. 
 (Give a definition of each without using 
 letters to designate the lines and angles.) 
 
 A ray of light passing from one medium 
 into another will retrace its course if started 
 back along the same path (see Art. 353, end). 
 Thus if HO (with the arrowhead reversed) be taken to represent 
 an incident ray passing from water into air, OE will be the 
 refracted ray, and the angles of incidence and refraction, as shown 
 in the figure, will be interchanged. On passing from water into 
 air J tight is refracted away from the perpendicular. 
 La bora tot y Exercise J2, Part I. 
 
 359. Some Effects of Refraction. — An object under water 
 always appears to be above its true position when viewed through 
 the surface. This is explained by Fig. 197. The cone of light 
 that enters the eye from any point of the object, as A, is refracted 
 from the perpendicular on entering the air ; and its apparent source 
 
 Fig. 196. 
 
Refraction of Light 
 
 285 
 
 is the vertex of the refracted cone A', — a point vertically above 
 
 A. All the light that enters the eye from 
 
 the object is similarly refracted, and the 
 
 object appears to be vertically above its 
 
 true position. Strictly speaking, what is 
 
 seen is not the object, but its image 
 
 formed by refraction. The apparent 
 
 elevation of the object increases as it is 
 
 viewed more and more obliquely, since 
 
 the deviation increases with the angle of 
 
 incidence (Fig. 198). 
 
 The broken appearance of a straight object, as a pencil, when 
 partly immersed in water in an oblique position, is similarly 
 explained (Fig. 199). There is also an apparent shortening of 
 the immersed part of the object. When the object is vertical, it 
 
 Vu,. 197. 
 
 Fig. 198. 
 
 Fig. 199. 
 
 appears to be shortened, but not bent. This is the simplest proof 
 that the direction of the apparent displacement is not obliquely 
 upward from the observer, as inaccurate observation often seems 
 to indicate, but vertical. 
 
 The apparent irregular motion and changing distortion of peb- 
 bles and other objects in the bed of a brook or along the shore of 
 a lake are due to the constantly varying angle of incidence at 
 which the light meets the surface of the water, as ripples and 
 larger waves pass over it. The varying angle of incidence causes 
 a varying deviation, and this determines the apparent position of 
 the point from which the light comes. The distortion is caused 
 by the unequal refraction of the light from different parts of the 
 
286 Light 
 
 object. A similar distortion is observed when objects are viewed 
 through common window glass, the surfaces of which, are more or 
 less wavy. Moving the head from side to side causes the distor- 
 tion to change. (Why?) 
 
 360. The Sine of an Angle. — In a right triangle the ratio of a 
 leg to the hypothenuse is called the sine of the angle opposite to 
 that leg. Thus, in the right angle ABC (Fig. 200), BC : BA \s 
 the sine of the angle A, and CA : BA is the sine of angle B. The 
 usual form of expression is — 
 
 sine^ = ^, sine^ = ^. 
 BA BA 
 
 The three right triangles ABC, AB'C, and AB"C' are similar ; 
 
 hence BC:BA=B'C : BA = B"C" : i9"^=sine A. From which 
 it will be seen that the sine of an angle 
 is independent of the size of the triangle. 
 The sine increases as the angle increases 
 (up to 90°), ///// no/ proportionally. 
 Angles and their sines are very nearly 
 proportional for small angles ; but, as an 
 angle approaches 90°, it increases much 
 
 more rapidly than its sine. The sine of 0° is o, the sine of 30° is 
 
 .5, the sine of 45° is — r or .707, and the sine of 90° is i. The 
 
 V2 
 sine of any angle can be found from a table of sines. 
 
 361. The Laws of Refraction. — The following laws of refrac- 
 tion have been established by experiment : — 
 
 I. Whatever the angle of incidence^ the ratio of the sine of the 
 angle of incidence to the sine of the angle of refraction is constant 
 for tlu same tivo media, hut imries with different media. 
 
 II. The angles of incidence and refraction lie in the same plane, 
 and are on opposite sides of the perpendicular to the surface at the 
 point of incidence. 
 
 To illustrate: I^t AOC (Fig. 201) represent a ray of light 
 passing from air into water at O, and BD the perpendicular to 
 
FlO. 20I. 
 
 Refraction of Light 287 
 
 the surface at this point. AB and CD are drawn perpendicular 
 to BD ; hence triangles ABO and CDO are 
 right triangles, and 
 
 • . AB . . CD 
 
 sine / = — and sine r = • 
 
 AO CO 
 
 Then, according to the first law, the ratio 
 
 AB CD . , , , 
 
 sine t ; sine r, or : , is constant for 
 
 AO CO 
 
 all angles of incidence ; and, according to 
 
 the second law, angles / and r lie in the 
 
 same plane. This plane is, of course, perpendicular to the 
 
 refracting surface, since it contains the perpendicular BD. 
 
 If the distances AO and OCf along the incident and refracted 
 
 rays respectively^ are taken equal (as they are in the figure), the 
 
 expression for the ratio of the sines reduces to 
 
 sine /■ : sine r=AB : CD. 
 
 362. Indices of Refraction. — The ratio of the sine of the angle 
 of incidence to the sine of the angle of refraction icdien light enters 
 a substance from a vacuum is called the index of refraction of the 
 substance. The methods employed in elementary physics are not 
 accurate enough to detect any difference between the index of 
 refraction from a vacuum into a substance and from air into the 
 same substance. Hence such differences may be disregarded. 
 
 According to the first law the index of refraction of a substance 
 is constant for all angles of incidence. It differs very slightly for 
 light of different colors. 
 
 If the index of refraction of one substance is greater than that 
 of another, the first is said to be more refractive or to have greater 
 refractive power than the second. It will be seen from the table 
 (p. 288) that the more refractive of two media is in some cases 
 the denser and in others the rarer of the two. The refractive 
 power of a substance cannot be determined from its density or 
 any other property ; it can be found only by experiment. 
 
288 
 
 Light 
 
 Indices of Refraction 
 
 Substance 
 
 Index of Refraction 
 
 Density (g. per ccm.) 
 
 Diamond 
 
 Carbon bisulphide 
 
 Glass, flint 
 
 (ilass, crown 
 
 Glycerine 
 
 Turpentine 
 
 Chloroform 
 
 Alcohol 
 
 Ether 
 
 Water 
 
 2.47102.75 
 
 1.68 
 
 1.58 to 1. 61 
 
 1.52 to 1.56 
 
 1.47 
 
 1.47 
 
 1-45 
 
 1.36 
 
 136 
 
 1.34 
 
 1.00029 
 
 1. 00 
 
 3-5 
 
 1.29 
 
 3.00 to 3.32 
 
 2.5 
 
 1.27 
 
 0.88 
 
 1-52 
 
 0.80 
 0.72 
 1. 00 
 
 Air 
 
 Vacuum 
 
 0.00129 
 
 When light passes from a less refractive to a more refractive 
 medium (as from water into glass), it is bent toward the perpen- 
 dicular ; when it passes from a more refractive to a less refractive 
 medium, it is bent from the perpendicular. The less the difference 
 between the indices of refraction of the two media, the less will be 
 the deviation for a given angle of incidence ; and if their indices 
 are equal, light will pass from one to the other at any angle with- 
 out deviation. The ratio of the sine of the angle of incidence to 
 the sine of the angle of refraction when Ifght passes from any 
 medium into another is called the relative index of refraction from 
 the former medium to the latter. 
 
 Laboratory Exercise S3- 
 
 363. Cause of Refraction. — The emission theory of light ex- 
 plained refraction upon the assumption that the velocity of light is 
 greater in the more refractive medium ; according to the wave 
 theory it is less. The velocities of light in air and in water were 
 determined experimentally by Foucault, a French physicist, in 
 1850. "He found the velocity in water to he less than in 
 air; from that moment Newton's emission theory was dead." 
 
Refraction of Light 
 
 289 
 
 It had, indeed, been little more than alive for a quarter of a 
 century. 
 
 Refraction may be explained in a general way as follows : Let 
 AH (Fig. 202) represent a beam of light 
 passing from air into some more refrac- 
 tive substance, as glass. When the inci- 
 dent beam is oblique, one side of a wave, 
 as B of the wave BD, enters the second 
 medium before the other and is retarded, 
 while the opposite side continues with 
 undiminished velocity in the air. Thus, 
 while one side of a wave travels from B to 
 F, the other side travels from D to E, and 
 the wave swings round so as to become more nearly parallel to the 
 refracting surface. 
 
 If Vx denotes the velocity of light in air and e'-j its velocity 
 in the second medium, then DE : BE= 7/, : fg. Angle DBE is 
 equal to the angle of incidence, and angle BEF is equal to the 
 angle of refraction. (Why ?) Hence, since triangles DBE and 
 
 DF Fi f^ 
 
 BEF are right triangles, sine/= -— and siner = ; from 
 
 BE BE 
 
 DE 
 
 which sine / : sine r = - — - = 
 
 /' 
 
 Hence the index of refraction 
 
 BF v^ 
 
 of a substance may be defined as the ratio of the velocity of light 
 in a vacuum (or, very approximately, in 
 air) to its velocity in the substance. 
 
 364. Construction for the Refracted 
 Ray. — Let EO (Fig. 203) be a ray of 
 light passing from air into water at O, 
 The construction for the refracted ray, 
 taking \ as the index of refraction of 
 water, is as follows : Draw MN perpen- 
 dicular to the surface AB at the point 
 
 Fig. 203. r • J J r 
 
 of mcidence ; and from any convenient 
 point, C, on the incident ray draw CD perpendicular to MN. 
 
 
 M 
 
 D 
 
 1 . 1 , 
 
 y 
 
 7 
 
 ^ 
 
290 Light 
 
 Liy off on AB a distance OF equal to } of CD, aod construct 
 FL perpendicular to AB and parallel to MN. With (? as a 
 center and a radius equal to (9C, describe an arc cutting FL 
 z\. G. OG is the refracted ray. (Prove it.) 
 
 To construct the refracted ray for light passing from air into 
 any substance, take OF of such length that CD : OF is equal to 
 the index of refraction of the substance. To construct the re- 
 fracted ray for light passing from any substance into air, take OF 
 of such length that OF : CD is equal to the index of refraction of 
 the substance. 
 
 PROBLEMS 
 
 For the following constructions take -J for the index of refraction of 
 crown glass, J for that of water, and ]} (which is f -r- J) for the relative index 
 of refraction from water into cniwn |{lass. 
 
 1. Construct the path of a ray of light from air into water for an angle of 
 incidence (a) less than 30", (*) between 40° and 50**, (c) between 80*^ and 
 
 2. Construct the path of a ray of light (r?) from crown glass into air; 
 (d) from air into crown glass. 
 
 3. Construct the path of a ray of light («) from water into crown glass; 
 (*) from crown glass into water. 
 
 365. Refraction through a Plate having Parallel Surfaces. — In 
 Fig. 204, FOO'F represents a ray of light passing from air through 
 a glass plate into air again, the surfaces 
 AB and CD being parallel. The angle 
 of incidence at O' is equal to the angle 
 of refraction at O ; hence the angle of 
 refraction into the air is equal to the first 
 angle of incidence. The emergent ray 
 O'F is therefore parallel to the incident 
 ray £0, but the two are not in the same 
 straight line. 
 
 iG. 204. rj.^^ point from which the ray comes 
 
 will appear to be on the line ^^; hence objects viewed obliquely 
 through a glass plate are apparently displaced to one side of their 
 
Refraction of Light 291 
 
 true position. This apparent lateral displacement increases with 
 the thickness of the plate, with its index of refraction, and with the 
 angle of incidence (Lab. Ex. 52). 
 
 366. Refraction through a Prism. — When the surfaces of a 
 medium through which light passes are planes incHned at an angle 
 with each other, the medium is called a prism, and the angle 
 between the refracting surfaces is called the refracting angle of the 
 prism. 
 
 In Fig. 205 the triangle represents a section of a glass prism. 
 A is the refracting angle ^ 
 
 for light passing through /\ 
 
 the surfaces ^^ and y4 C ^~>-^^ / \ ^,j^ 
 
 The ray EFGH is re- """^-^^ / \ ,-'-"'" 
 
 fracted toward the per- ^^ ^Z--i7fr^'~-\ ^ ^" 
 
 pendicular MNoxi enter- _^^,-^^^/a^\ '^'^"^^x'^Ell 
 ing the prism and from ' ^^ , ^ ^^^ 
 
 the perpendicular il/'iV' '" ^^* 
 
 on leaving it. The deviations d and (f are in the same direction 
 (away from the refracting angle) ; hence the total deviation is 
 their sum and is measured by the angle KPH. This angle is 
 called the afig/e of ikviation. 
 
 The deviation increases with the refracting angle of the prism 
 and its index of refraction ; it also varies with the angle of inci- 
 dence, being least when the angle of incidence is such that the 
 angle of emergence is equal to it. The deviation varies slightly 
 for light of different colors, producing effects which are considered 
 later. 
 
 The apparent source of the ray GH is some point on the line 
 HL ; hence an object viewed through a prism is apparently dis- 
 placed in the direction of the refracting edge of the prism. 
 
 Laboratory Exercise 52, Parts II and III. 
 
 367. Partial Reflection and Refraction. — In general, when 
 light meets a smooth surface separating two transparent media, 
 part of it is reflected and part refracted. A number of illustrations 
 are familiar where air is one of the media. For example, the 
 
292 
 
 Light 
 
 greater part of the light falling upon window glass passes through 
 it by refraction, as described in Art. 365 ; but a considerable por- 
 tion of it is regularly reflected from the front surface, forming an 
 image of its source as in a plane mirror. Such images are dis- 
 tinctly visible when the glass is backed with black cloth to pre- 
 vent the transmission of light from the opposite side. Similarly, 
 light falling upon the surface of still water is partly reflected, form- 
 ing images ; but the greater part is refracted into the water. 
 
 It is by partial reflection and refraction that a number of images 
 of a small, bright object, as a candle flame or a lighted match, are 
 formed by a single mirror when the images are viewed obliquely. 
 (Try it.) Tiiese multiple images are especially prominent in 
 mirrors of thick glass. Let ABDC (Fig. 
 206) represent a section of a mirror taken 
 at right angles to the reflecting surface 
 AH^ and O a luminous point in front of 
 the mirror. Part of the incident ray OE 
 is reflected from the front surface at E^ 
 forming (or helping to form) the image /i. 
 The greater part of the incident light 
 enters the glass, and is reflected at the rear 
 surface. The greater part of this light is refracted into the air at 
 /% forming the brightest of the images /, ; the remainder is inter- 
 nally reflected at /% is again reflected at the rear surface, and 
 the greater part of it passes into the air at G, forming the 
 image /j. This process is repeated, forming still other images ; 
 but their number is limited, as the light rapidly diminishes in 
 intensity with the successive reflections and refractions. 
 
 368. Total Reflection. — When 
 light is incident in the more re- 
 fractive of two media (as when it 
 passes from water into air), the 
 angle of refraction is always greater 
 than the angle of incidence. For a 
 certain angle of incidence, MOE Fig. 207. 
 
 
 
 /^ 
 
 u 
 
 
 / 
 
 A "*' 
 
 c/ 
 
 ^ /^ B 
 
 
 
 "^£:=i^s^ssyi=-^=-£:s- 
 
 '^KUtTrT^ 
 
 M 
 
 
 i=^"2^' - 1 
 
 
Fig. 208. 
 
 Refraction of Light 293 
 
 (Fig. 207), the angle of refraction is 90°, and the refracted ray 
 
 is parallel to the surface of the water. 
 
 For a greater angle of incidence, as angle 
 
 MOF, refraction cannot take place, and 
 
 all of the light is reflected internally 
 
 according to the laws of reflection. The 
 
 ray OFG is therefore said to be totally 
 
 reflected at F. Total reflection in water can be exhibited by 
 
 reflecting a beam of light upward through water in a rectangular 
 
 glass vessel (Fig. 208) {Exp.). 
 
 The angle of incidence in the more refractive medium for which 
 the angle of refraction is 90° is called the critical angle. When 
 the angle of incidence is less than the critical angle, refraction and 
 partial reflection take place ; when it is greater, total reflection 
 occurs. When light is incident in the less refractive medium, 
 refraction and partial reflection take place at all angles of 
 incidence. 
 
 The critical angle for water and air is 48.5° ; for crown glass 
 and air, about 41° ; for flint glass and air, about 38° ; for diamond, 
 about 24°. 
 
 Laboratory Exercise ^4. 
 
 369. Illustrations and Applications of Total Reflection. — When 
 a glass of water is held above the level of the eye, its surface, 
 viewed from below through the side of the glass, looks like a 
 mirror. When a spoon or a pencil is placed in the glass, the 
 part above the water is invisible, but a very distinct image of the 
 immersed part of it is seen in the surface. This image is formed 
 by total reflection. (Try it.) 
 
 Glass prisms afford excellent illustrations of total reflection. 
 
 Light that falls upon the inside of any face of the 
 
 prism at an angle greater than the critical angle is 
 
 totally reflected ; and, when the eye is in position 
 
 ^^''" to receive this light after refraction into the air, the 
 
 Fig. 209. reflecting surface has the appearance of a mirror 
 
 and forms a brilliant image of the source of the light (Fig. 209). 
 
294 Light- 
 
 Prisms having an angle of 90° and two angles of 45° are used in 
 astronomical telescopes and in other optical instruments to change 
 the direction of the light by 90° (Fig. 210). Incident light per- 
 pendicular to the face AB enters the prism without 
 deviation and meets the face BC 2X an angle of 45°. 
 \^^ | / Since this is greater than the critical angle, the light is 
 \J*"- totally reflected at the same angle, and passes out of the 
 ^ face AC without deviation. Such prisms are the most 
 •10. 210. pgj.fgj,^ mirrors known. They give only a single reflec- 
 tion, thus avoiding the faint, overlapping images due to multiple 
 reflection in ordinary mirrors. A totil-reflecting prism at the eye 
 end of a telescoi)e adds to the comfort of the observer, as it 
 enables him to look obliquely tlownward in viewing the heavenly 
 bodies, instead of in the direction in which the telescope points. 
 
 IV. Atmospheric Refraction 
 
 370. Atmospheric Refraction. — Although the refractive power 
 of the air is small, it gives rise to a number of interesting and 
 familiar phenomena. When we look over a bonfire or a hot stove 
 at any object situated beyond it, the object appears to undergo 
 a rapidly changing distortion, similar to that of a pebble in the 
 bottom of a brook. This appearance is due to the colistantly 
 changing refraction of the light as it passes through the currents 
 of air rising from the fire ; for these currents consist of bodies of 
 air of varying temperatures and hence of varying densities, and the 
 refractive power of air increases with the density. A similar effect 
 may often be observed when the line of sight passes near the 
 surface of some object that has become hot in the sunshine. 
 
 371., Twinkling of the Stars. — The twinkling of the stars con- 
 sists in a rapid, irregular variation in brightness. With the aid of 
 a telescope, this is seen to be accompanied by a dancing motion. 
 The phenomenon is wholly atmospheric ; the stars themselves are 
 fixed and shine with a steady light. The dancing is a rapid change 
 of apparent position, caused by changing refraction as currents of 
 
Atmospheric Refraction 
 
 295 
 
 
 
 air of varying density cross the line of sight. As a beam of light 
 passes through successive layers of air, the refraction at the irregu- 
 lar boundaries separating thorn may cause either a slight con- 
 vergence or divergence of the beam. The first increases the 
 intensity of the light, the second diminishes it ; and the twinkling 
 is largely due to the rapid alternation of these effects. Stars near 
 the horizon, the light from which traverses a greater stretch of 
 atmosphere, twinkle more than those overhead. The twinkling 
 also differs greatly on different nights according to the steadiness 
 of the air. 
 
 372. Regular Atmospheric Refraction. — The inconstant or 
 irregular refraction to which the twinkling of the stars is due is 
 small in comparison with the regu- 
 lar refraction, due to the increas- 
 ing density of the atmosphere 
 from its upper limit to the earth's 
 surface. Light traveling obliquely 
 downward through the atmos- 
 phere is bent continuously toward 
 the perpendicular (Fig. 211). The total deviation thus produced 
 varies from zero for heavenly bodies directly overhead to a little 
 more than half a degrfte at the horizon (it is greatly exaggerated 
 in the figure). 
 
 Since the angular diameter of the sun at the earth is about half 
 a degree, the sun is really just below the horizon when it appears 
 to be just above it. Thus, on account of atmospheric refraction, 
 sunrise occurs from two to four minutes earlier than it otherwise 
 would (depending upon the angle that the sun's path makes with 
 the horizon), and sunset is retarded by the same amount. 
 
 373. Mirage. — In sandy deserts the reflection of the sky and 
 of the scattered trees and other objects in the landscape is often 
 seen in the distance, on hot sunny days, as in the surface of a 
 calm lake. This optical illusion is called a mirage. It is due to 
 the heating and expansion of the air in contact with the hot sand ; 
 as a result of which the density of the air increases upward for 
 
 Fig. 211. 
 
296 Light 
 
 some distance from the ground. Light traveling obliquely down- 
 ward through this layer of air is gradually bent from the perpen- 
 dicular; and the angle of incidence, if nearly 90° at first, may 
 thus become greater than the critical angle. The light is then 
 totally reflected by the layer of rarer air into which it cannot pass, 
 
 and is refracted toivard the perpendicular as it returns through 
 the denser air alx)ve. By this total reflection images are formed 
 like those seen in the surface of still water (Fig. 212). The sky 
 and other objects are also seen, at the same time, erect and in 
 their true positions, by light that comes straight to the eye. 
 
 V. Lenses 
 
 374. Lenses. — A lens is a portion of a transparent medium 
 bounded by two curved surfaces or by a plane and a curved 
 surface. Lenses are usually made of glass, and their curved 
 surfaces are usually spherical. There are six forms of spherical 
 lenses, sections of which are represented in Fig. 213. From 
 their optical effects they are classed in two groups of three each, 
 as follows : — 
 
 Convex or Convergins^ Lenses. — The first three lenses repre- 
 sented in the figure belong to this class. The first of these is 
 
Lenses 
 
 297 
 
 called double convex, the second plano-convex^ and the third con- 
 cavo-convex. All are thicker in the middle than at the edges. 
 They are called converging lenses, because light is more converg- 
 ent or less divergent after passing through them than before 
 
 Fio. 213. 
 
 {Exp^. The three forms are equivalent in their effects so far as 
 the purposes of elementary physics are concerned ; and the double- 
 convex lens having surfaces of equal curvature is the only one that 
 will be considered. 
 
 Concave or Diverging Lenses. — To this class belong the last 
 three lenses represented in the figure. The first of these is 
 double-concave^ the second piano- concave , and the third convexo- 
 concave. They are all thinner in the middle than at the edges ; 
 and are called diverging lenses because they increase the diverg- 
 ence of light passing through them {Exp.). The double-concave 
 lens will be taken as the type of its class. 
 
 375. The Double-convex Lens. — The line joining the centers 
 
 of curvature of the spherical surfaces of a convex lens (C and C, 
 Fig. 214) is called iht principal axis of the lens. 
 
298 
 
 Light 
 
 Let OA be a ray of light from a luminous point on the prin- 
 cipal axis. On entering the lens, the ray is bent toward the 
 perpendicular AC ; on ^merging at B, it is bent from the per- 
 pendicular BC. The deviation of the ray is the same as it would 
 be if the lens were replaced by a prism ADB of the same mate- 
 rial and having faces tangent to the lens at A and B. The angle 
 between the tangent planes at the jwints of entrance and emerg- 
 ence of a ray increases toward the edge of the lens {g^. angle £ 
 is greater than angle D) ; hence the deviation also increases. A 
 ray of light traveling along the principal axis falls perpendicularly 
 upon both surfaces of the lens, and hence passes through it with- 
 out deviation. 
 
 The increased deviation toward the edge of the lens is almost 
 exactly what is required to bring a// the refracted light to the same 
 point, /. We shall for the present regard the focusing as perfect. 
 
 Thus the diverging 
 cone of incident light 
 A'OG undergoes re- 
 fraction at the surfaces 
 of the lens and emerges 
 as a converging cone, 
 L///, forming a real 
 '^' ^'^* image of its source at /. 
 
 The effect of the lens is also shown in Fig. 215, in which the 
 curved lines represent light waves. The points O and / are 
 called conjitgaU foci. Light radiating from either converges to the 
 other. 
 
 376. Conjugate Foci on the Principal Axis. — Real Foci. — As 
 the luminous point 6>(Fig. 214) is moved from the lens along the 
 principal axis, the incident cone of light becomes less divergent 
 and the refracted cone more convergent ; the image therefore 
 moves toward the lens along the axis. When the distance of the 
 object is relatively great (not less than one hundred times the 
 radius of curvature of the surfaces of the lens), the incident light 
 is sensibly parallel, and the point to which the refracted light con- 
 
Lenses 299 
 
 verges is called the principal focus of the lens {F, Fig. 216). 
 There is another principal focus at the same distance on the other 
 side of the lens, correspond- 
 ing to an incident beam 
 coming from the opposite 
 
 direction. Since light can 
 
 *u *u • F'G. 216. 
 
 traverse the same path m 
 
 both directions, it follows that light radiating from a luminous 
 
 point at either principal focus is refracted as a beam parallel to 
 
 the principal axis. 
 
 The distance of the principal focus from the lens is called the 
 focal length of the lens. The focal length depends upon the 
 refractive power of the glass of which the lens is made, as well 
 as upon the curvature of its faces. It can be shown that, when 
 the faces have equal curvature and the index of refraction of the 
 glass is 1.5, the focal length is equal to the radius of curvature. 
 When the index of refraction exceeds 1.5, the focal length is less 
 than the radius of curvature. 
 
 As the luminous point is moved toward the principal focus from 
 a greater distance, the incident light becomes more and more diver- 
 gent and the refracted light less convergent : the image, therefore, 
 recedes along the axis. When the object is at the principal focus, 
 the light is refracted as a parallel beam, as stated above, and the 
 image is indefinitely far away. 
 
 Virtual Foci, — When the object is nearer than the principal 
 focus, the divergence of the incident light is greater than the 
 
 lens can overcome, and the 
 refracted light is still divergent, 
 
 ,^^ ^arx>--------- | ^-p-| «^ though less so than the inci- 
 
 ^ '""'"'-^^^^^^^^^ dent light (Fig. 217). The 
 
 image (or focus) is therefore 
 * ^^^' virtual and at a greater dis- 
 
 tance than the object. As the object is moved up to the lens 
 from the principal focus, the image approaches the lens from an 
 indefinite distance on the same side. 
 
300 Light 
 
 377. Conjugate Foci on Secondary Axes. — When a ray of light 
 passes obliquely through the center of a double-convex lens, the 
 tangent planes at the points of entrance and emergence of the 
 
 ray (A and B, Fig. 218) 
 ^ are parallel. The emer- 
 gent ray lU is therefore 
 parallel to the incident 
 rays OA (Art. 365). 
 When the lens is thin and the angle of incidence small, as is gen- 
 erally the case, the lateral displacement of the refracted ray is 
 very slight and may be disregarded. Any ray through the center 
 of the lens, as OABI, is therefore regarded as a straight line, and 
 is so drawn in diagrams. 
 
 Any straight line through the center of a lens is called a second- 
 ary axis. It follows from the above that the conjugate focus of 
 any point not on the principal axis lies on the secondary axis 
 through that point. Conjugate foci on secondary axes are real or 
 virtual under the same conditions as for foci on the principal 
 axis ; i.e. if the distance of a point is greater than the focal length 
 of the lens, the conjugate focus is real ; if less, it is virtual. 
 Laboratory Exercise 55. 
 
 378. Real Images. — When light falls upon a convex lens from 
 an object situated beyond the principal focus, the diverging cone 
 of light from each point of the object converges to the conjugate 
 focus, and there forms the corresponding point of the image, 
 which in this case is real. The image can be caught upon a 
 screen, and can also be viewed directly from any point in the path 
 of the light diverging from it. Thus, in Fig. 214, the image can 
 be seen in mid-air from any point within the cone PIQ. 
 
 There are three rays from any point not on the principal axis 
 whose directions after passing through a lens are known : (i) The 
 ray through the center of the lens continues in the same straight 
 line. (2) The ray parallel to the principal axis is refracted so as 
 to pass through the principal focus. (3) The ray through the 
 principal focus (on the same side as the object) is refracted parallel 
 
Lenses 
 
 301 
 
 Fig. 219. 
 
 to the principal axis. These three rays are shown in Fig. 218. 
 In drawing figures, the conjugate focus of any point can be deter- 
 mined by means of any two of these rays, when the focal length is 
 known, without constructing angles of incidence and refraction. 
 Figures 219, 220, and 221 illustrate this method of construction. 
 In Fig. 219, AB may be regarded as the object and ab its 
 image, or vice versa. Since 
 
 any point of the object and a - ^ — ^ A-<^ b^ 
 
 its image are on the same 
 
 straight line through the 
 
 center of the lens, the 
 
 image, if real, is always inverted. Triangles AOB and aOl> are 
 
 similar; hence the lengths (and other similar dimensions) of 
 
 object and image are proportional to their distances from the 
 
 lens. 
 
 The following consequences of these geometrical relations are 
 of great importance in optical instruments : — 
 
 (i) When the distance of the object is so great that the rays 
 from any point of it are sensibly parallel, the real and inverted 
 
 image which is obtained of 
 it is at the principal focus 
 and is relatively very small 
 (Fig. 220). 
 
 (2) When the object is 
 at a distance only slightly 
 greater than the focal length, its image is relatively distant and 
 greatly enlarged or magnified. (Draw figure.) 
 
 (3) For a given object at a given distance, the size of the real 
 image increases with the focal length of the lens ; since, under 
 these conditions, the greater the focal length, the greater is the 
 distance of the image from the lens {Exp.). (Illustrate with two 
 drawings, taking lenses of unequal focal length.) 
 
 (4) When the object is at a relatively great distance (Fig. 220), 
 the length of the image is proportional to the focal length of the 
 lens. (Draw figures to illustrate.) 
 
 Fig. 220. 
 
-^^ 
 
 302 v,o- 
 
 X 
 
 Light 
 
 '--IM" 
 
 379. Virtual Images. — When the distance of the object from 
 the lens is less than the focal length, the light from each point of 
 the object is still divergent after refraction ; and its apparent 
 source is the corresponding point of the image, which in this case 
 is virtual. A virtual image can be seen only by looking through 
 the lens toward the object. What we really see through the 
 lens is not the " magnified object," but its magnified virtual 
 image. 
 
 Figure 221 illustrates the formation of a virtual image, following 
 „ the usual construction. It will 
 
 be seen from the figure that 
 
 ^"^irr ^-^^.A ^^ the virtual image is ahvays 
 
 *p!^ erect and magnified, and is at 
 
 a greater distance than the ob- 
 
 :: ^.s--- ject. A similar figure in which 
 
 *" ^^' the object is taken nearer the 
 
 lens will show that, as the object approaches the lens from the 
 
 principal focus, the image also approaches the lens and grows 
 
 smaller. (Draw the figure.) 
 
 From the similar triangles A OB and aOb it is evident that 
 the lengths of object and image are proportional to their distances 
 from the lens, as in the case of real images. The less the focal 
 length of the lens, the larger is the virtual image when formed at 
 a given distance from the lens. (Draw figure to illustrate.) 
 
 A convex lens, when used for observing the enlarged virtual 
 images of minute objects, is called a magnifying glass or simple 
 microscope. 
 
 380. Formulas Relating to Convex Lenses. — Formula for 
 Real Images. — In Fig. 222 
 
 the rays AM and BN are ' " ^- — /IT""^^ ^^^^ 
 
 drawn as if they were re- 
 fracted once midway between 
 the surfaces of the lens in- 
 stead of at each surface. This simplifies the present problem, 
 and involves no appreciable error. 
 
Lenses 303 
 
 From the similar triangles A OB and a Ob, 
 
 AB'.ahw CO: Oc. 
 From the similar triangles MFN and aFb, 
 
 MN'.ab-.'.OF'.Fc, 
 Since AB = MN^ we have from these proportions, 
 
 CO'.Oc'.'.OF'.Fc, 
 Let CO = D (the distance of the object), Oc = d (the distance 
 of the image), and OF=f (the focal length of the lens). Sub- 
 stituting these values in the last proportion, we have 
 D:d::f:{d-f). 
 From which df^D{d-f), 
 
 Transposing and combining, /(// + Z>)= Dd, 
 
 Dividing by /?./5^, . ^^ = 7- 
 
 JJa J 
 
 Separating the terms of the fraction and reducing, 
 
 -1 + 1 = 1. 
 D^ d f 
 ■f 
 By means of this formula any one of the three quantities Z>, d^ 
 
 and / can be found when the other two are given. 
 
 Formula for Virtual Images. — From the similar triangles fZil/^ 
 and aFO (Fig. 223), 
 
 aA\aO:\MA\FO. 
 From the similar triangles 
 OAC ".nd Oac, 
 
 aA'.aOwcC'.cO. 
 Hence, cC.cOw MA : FO. 
 
 I^t OC=D{=MA), Oc = d, d.n^.FO=f (the focal length 
 of the lens). Substituting these values in the last proportion, we 
 have (d-D):d:'.D:/, 
 
 d-D D 
 
 ->:^-~^. 
 
 Fig. 223. 
 
304 
 
 Light 
 
 Dividing by D, 
 
 tf-D I 
 
 or 
 
 I I _ I 
 
 Laboratory Exercise j6 
 
 
 
 PROBLEMS 
 
 The following problems will help to familiarize the pupil with a number 
 of facts which will be of assistance in the study of optical instruments. The 
 problems are to be solved by means of the formulas. 
 
 1. The distance of the image is how many times the focal length when 
 the distance of the object is (</) twice the focal length? (^) ten times the 
 focal length? (<-) one hundred times? (</) one thousand times? 
 
 2. The distance of the image is how many times the focal length when 
 the distance of the object is {a) .9 the focal length? (//) .8? {c) .5? (</) .1? 
 
 Do the distances of the object and its virtual image become more or less 
 nearly equal as the object approaches the lens from the principal focus? 
 
 3. An image seen in a simple microscope is at a distance of 10 in. What 
 is the distance of the object if the focal length of the lens is {a) 2 in.? {h) i 
 in.? (0 .5 in.? 
 
 For an image at a fixed distance, is the distance of the ol)ject more or less 
 nearly e(]ual to the focal length as the focal length is diminished? 
 
 4. What is the per cent of error in assuming that the distance of the ol>- 
 ject is equal to the focal length when the distance of the image is 10 in. and 
 the focal length of the lens (<i) i in.? {b) .5 in.? 
 
 5. An object. 2 cm. long is at a distance of 50 cm. from a lens whose focal 
 length is 15 cm. Find the distance of the image and its length. 
 
 6. An object i cm. long is^at a distance of 1.7 cm. from a lens whose focal 
 length is 2 cm. find the distance of the image and its length. 
 
 381. Spherical Aberration. — W'hen light from any object is 
 allowed to pass through only the central portion of a lens (an 
 opaque screen with a small circular ope#ing being used for this 
 purpose), the image is formed at a slightly greater distance than 
 it is when formed by light that passes through the lens near its 
 edge (the light through the central portion being cut off by an 
 opaque disk) {Exp.). 
 
 These experiments show that the deviation becomes too great 
 
Lenses 
 
 305 
 
 hlG. 224. 
 
 toward the edge of the* lens, causing convergence of this light to 
 a nearer focus than that of the light passing through the central 
 portion of the lens (Fig. 224). Hence, when the whole lens is 
 used, all the light cannot be 
 sharply focused on a screen 
 at the same time ; and the ^ 
 image is more or less blurred 
 by the light that is out of 
 focus. This unequal focusing of the light is called spherical 
 aberration, since it is the result of the spherical shape of the 
 surfaces. It increases toward the edge of the lens; and, for 
 lenses of equal size, it is greater, the less the focal length. Spher- 
 ical aberration is corrected in the large lenses of astronomical 
 telescopes by diminishing the curvature of the surfaces toward 
 the edge. The grinding and polishing of such surfaces is very 
 expensive, as it must be done by hand, hence spherical lenses 
 are used in the smaller telescopes and other optical instruments. 
 Excellent focusing is obtained with a camera by using a dia- 
 phragm with a small opening, which admits light only through the 
 central portion of the lens. This method of correction greatly 
 diminishes the intensity of the light ; but the intensity is still suffi- 
 cient for the purposes of photography. 
 
 382. The Concave Lens. — A beam of light parallel to the 
 principal axis of a concave lens emerges as a diverging cone 
 
 (Fig. 225) {Exp.), The 
 vertex F of this cone is 
 the apparent source of the 
 refracted light, and is called 
 the pnncipal focus. It is a 
 virtual focus. • 
 
 Since concave lenses al- 
 
 Fio. 225. 
 
 ways increase the divergence of the light passing through them, 
 the images formed by them are always virtual and are on the 
 same side of the lens as the object. These images, like those 
 formed by convex mirrors, are always erect, and smaller than the 
 
3o6 Light 
 
 object, and are nearer the lens than the' object is. When the 
 object is moved from a position at the lens to an indefinitely 
 great distance, the image recedes from the lens to the principal 
 
 focus. These characteristics 
 
 ^__ ^ :^f-^7"^^ of the image are explained 
 
 by figures constructed like 
 Fig. 226. In locating the 
 virtual conjugate focus of any 
 point, we make use of two 
 rays, namely : ( i ) The ray 
 through the center of the lens, which continues in the same direc- 
 tion after passing through the lens; and (2) the ray parallel to 
 the principal axis, the direction of which after emergence is from 
 the principal focus. 
 
 VI. The Eye 
 
 383. The Structure of the Eye. — The human eye is a nearly 
 spherical ball somewhat less than an inch in diameter. Figure 227 
 represents the top view of a horizontal section of the right eye. 
 The outer coat or wall of the eye is thick and horny ; it gives firm- 
 ness to the eye and protects the delicate parts within. Its front por- 
 tion, the cornea^ is transparent ; the remainder is opaque and white. 
 
 Behind the cornea is an opaque circular diaphragm called the 
 iris. This is the colored part of the eye, and the circular opening 
 at its center is called the pupil. The iris regulates the amount of 
 light that enters the eye by involuntary muscular action, which 
 causes the pupil to contract when the intensity of the light in- 
 creases and to expand when the intensity decreases. The iris 
 also corrects spherical aberration by permitting light to pass 
 through only the central portion of the crystalline lens, which is 
 just behind it. This lens is a double-convex, transparent solid 
 made up of concentric layers or shells, which increase in density 
 and refractive power toward the center. 
 
 The cavity between the cornea and the crystalline lens is filled 
 with a watery liquid called the aqueous humor. The large cavity 
 
The Eye 
 
 307 
 
 back of the lens is filled with a jelly-like substance called the 
 vitreous humor. 
 
 The rear half of the eyeball is lined with a semitransparent 
 membrane, called the retitia, which contains a network of nerve 
 
 Ciliary Muscle 
 
 Optic Nerve 
 
 Fig. 227. 
 
 Choroid 
 
 fibers branching from the optic nerve. Light incident on the 
 retina gives rise to the sensation of vision. A thin, black mem- 
 brane (the choroid coat) lies between the retina and the outer 
 coat of the eye, and extends forward to the iris, with which it is 
 continuous. This membrane absorbs all the light that falls upon 
 it, making the eye a dark chamber. 
 
 384. The Eye as an Optical Instrument. — The light by which 
 we see passes through the three refractive media of the eye, — the 
 aqueous humor, the crystalline lens, and the vitreous humor. The 
 index of refraction of the aqueous and the vitreous humors is 
 1.337, ^^^ ^^^^ average for the lens is 1.437. Consequently the 
 successive refractions at the cornea and the surfaces of the lens 
 
308 Light 
 
 together make the light strongly convergent. If the eye is perfect, 
 the convergence is just sufficient to bring the light to a focus on 
 the retina, which thus serves as a screen to receive the real and 
 inverted image of the source of light. How the formation of a real 
 image upon the retina results in sight is not to be considered 
 here ; it is a question to which no complete answer can be given. 
 
 " The inversion of images in the eye has greatly occupied both 
 physicists and physiologists, and many theories have been proposed 
 to explain how it is that we do not see inverted images of objects. 
 The chief difficulty seems to have arisen from the conception of 
 the mind or brain as something behind the eye looking into it 
 and seeing the image upon the retina ; whereas really this image 
 simply causes a stimulation of the optic nerve, which produces 
 some molecular change in some part of the brain, and it is only 
 of this change, and not of the image, as such, that we have any 
 consciousness. The mind has thus no direct cognizance of the 
 image upon the retina, nor of the relative positions of its parts, 
 and sight being supplemented by touch in innumerable cases, it 
 learns from the first to associate the sensations brought about by 
 the stimulation of the retina (although due to an inverted image) 
 with the correct position of the object as taught by touch." — 
 Ganot's Physics. 
 
 385. The Field of Distinct Vision. — Although you can see the 
 whole of both of these pages at the same time, you will find that 
 while you are looking steadily at one word you cannot read more 
 than one or two short words on either side of it. While the field 
 of vision (measured by the angle at the eye within which objects 
 can be seen at the same time) is very large, the field of most 
 distinct vision is surprisingly small, as is shown by this simple test. 
 We are seldom conscious of the fact, however, for we are accus- 
 tomed to fix the attention wholly on the spot at which we are 
 directly looking. In looking attentively at a large object, a rapid 
 shifting of the eyes brings successive portions of it into distinct 
 view. 
 
 When the eyes are directed toward any small area, its image 
 
The Eye 
 
 309 
 
 falls upon the middle of the back part of the retina. This is the 
 region in which vision is most distinct ; it is called the yellow spot 
 (see figure). 
 
 386. Adaptation to Different Distances. — If the parts of the eye 
 were all fixed in shape and relative position and the image of a 
 very distant object were exactly focused on the retina, the focusing 
 would remain perfect for shorter distances down to about twenty 
 feet ; but, for distances less than this, the focus would be beyond 
 the retina J causing the images upon the retina to be blurred, and 
 objects would appear indistinct. This imperfection of focusing 
 would rapidly increase for distances under three feet. 
 
 Since we are able to see both near and distant objects distinctly, 
 it is evident that the eye is capable of some form of adjustment 
 for distance. This adjustment is known as accointnodation. It 
 has been found by observations upon the eye, such as oculists 
 are able to make, that accommodation is effected by means of 
 the crystalline lens, the front surface of which moves forward 
 and becomes more convex when near objects are viewed. This 
 diminishes the focal length of the lens and increases the distance 
 
 CI).IARY MUSCLE 
 
 FAR NEAR 
 
 Fig. 228. 
 
 CIUARY PROCESS 
 
 of its center from the retina, both of which changes assist in 
 bringing the image forward to the retina. The left half of Fig. 
 228 represents the lens adjusted for distant vision and the right 
 half for near vision. 
 
 The mechanism of accommodation is, in its main features, as 
 follows : The lens is. inclosed in a membranous sac, round the 
 front surface of which, near the margin, is attached a membrane 
 
3IO Light 
 
 called the suspensory ligament. The outer edge of this ligament 
 is attached to the projecting edge of a circular muscle (the ciliary 
 muscle) extending round the eye just back of the iris. When this 
 muscle is relaxed, the suspensory ligament is under tension, causing 
 the membrane that incloses the lens to press upon its front sur- 
 face. The lens is slightly flattened by this pressure, and thus 
 brought into focus for distant vision. The contraction of the 
 muscle draws its projecting edge forward. This relaxes the 
 suspensory ligament, relieving the pressure from the front surface 
 of the lens, which moves forward and becomes more convex. 
 
 387. The Least Distance of Distinct Vision. — The power 
 of accommodation of the eye is limited. When the object is at 
 less than a certain distance, the effort to focus the eye upon 
 it becomes tiresome ; and within a somewhat shorter distance 
 focusing is impossible. These distances vary considerably with 
 different eyes ; but with perfect eyes the least distance of distinct 
 vision is about 25 cm. (10 in.). This is commonly taken as the 
 distance of most distinct vision in computing the magnifying power 
 of optical instruments. 
 
 388. Optical Defects of the Eye. — Short Sight. — In some eyes 
 the image of a distant object is formed in front of the retina, the 
 eyeball being too long or the curvature of the cornea or the lens 
 too great. Such eyes are said to be near-sighted, for the image 
 falls upon the retina only when the object is very near. This 
 defect is corrected by wearing concave glasses, which offset the 
 excessive convergence within the eyes by increasing the divergence 
 of the incident light. 
 
 Long Sight. — In some cases the eye is too short or the crystal- 
 line lens not sufficiently converging, and the focus, even for distant 
 objects, would fall behind the retina if the power of accommoda- 
 tion were not exercised. Such eyes are far-sighted, and cannot 
 be focused on near objects without a fatiguing effort, if at all. 
 Convex glasses correct the defect by supplementing the deficient 
 convergence within the eyes. 
 
 In old age the crystalline lens loses its elasticity and becomes 
 
The Eye 
 
 311 
 
 incapable of accommodation for near vision. Hence old people 
 whose eyes were perfect in earlier years see distant objects dis- 
 tinctly, but require convex glasses for reading. 
 
 Astigmatism. — This defect consists in unequal curvature of 
 the cornea or of the crystalline lens in different directions ; as a 
 result of which the light from any 
 point of an object does not con- 
 verge to a point on the retina, 
 but forms a line instead. An 
 eye that is astigmatic cannot be 
 exactly focused for vertical and 
 horizontal lines at the same time ; 
 hence Fig. 229 presents a simple 
 test for this defect. The radi.it- 
 ing lines are all alike ; but to 
 most persons they will appear 
 unequally distinct, for there are 
 very few eyes that are not astig- 
 matic in some degree. If the 
 eyes are strongly astigmatic (in- 
 dicated by decidedly unequal 
 
 Fig. 229. 
 
 distinctness of the radiating lines in the figure), glasses should be 
 worn, especially for reading. To correct astigmatism, one surface 
 of the lens is given a cylindrical curvature — in some cases concave, 
 in others convex. 
 
 389. Binocular Vision. — " The question as to how it is that we 
 see objects single with two eyes is not to be altogether explained 
 by habit and association. To each point in the retina of one eye 
 there is a corresponding point, similarly situated, in the other. An 
 impression produced on one of these points is, in ordinary circum- 
 stances, undistinguishable from a similar impression produced on 
 the other, and when both at once are similarly impressed, the 
 effect is simply more intense than if one were impressed alone ; 
 or, in other words, we have only one field of view for our two 
 eyes, and in any part of this field of view we see either one image, 
 
312 Light 
 
 brighter than we should see it by one eye alone, or else we see 
 two overlapping images. Tliis latter phenomenon can be readily 
 illustrated by holding up a finger between one's eyes and a wall, 
 and looking at the wall. We shall see, as it were, two transparent 
 fingers projected on the wall. One of these transparent fingers is 
 in fact seen by the right eye, and the other by the left, but our 
 visual sensations do not directly inform us which of them is seen 
 by the right eye, and which by the left." — Deschanel's Natural 
 Philosophy. 
 
 It is principally in the estimation of distances and in the per- 
 ception of relief that there is an advantage in having two eyes. 
 
 390. The Estimation of Distance. — The line through the cen- 
 ters of the cornea and the crystalline lens is called the optic axis 
 
 jiaC\ of the eye. When we look at 
 1 ■ ^s^ any point, the optic axes of both 
 
 ■ ^ i^I V ^y^^ ^^^ directed toward it (Fig. 
 
 Fig. 230. C^^y 230). This requires a greater 
 
 or less convergence of the axes according as the object is nearer 
 or farther away ; and, in the case of near objects, the amount 
 of the convergence is the principal means by which we estimate 
 the distance. 
 
 That we are in fact less able to judge of distances with one eye 
 than with both is easily proved by trying to snap a cork or other 
 small object from the back of a chair with a finger, while walking 
 rapidly past the chair with one eye shut. The object will almost 
 invariably be missed, because the eye misjudges the distance. 
 
 391. The Perception of Relief. — When we look with both eyes 
 at any object that is not very distant, the point of it to which the 
 eyes are at any instant directed appears single, for its image then 
 falls upon corresponding points of the retinas. At the same time 
 we receive a somewhat indistinct impression of the whole object ; 
 and this impression, when the attention is directed toward it 
 without shifting the position of the eyes^ is found to involve a large 
 amount of doubleness. For example, if we look at a long pencil, 
 held in the hand with the sharp end pointing toward the chin, 
 
The Eye 
 
 •313 
 
 a b c 
 
 Fig. 231. 
 
 it will appear as shown in «, b, or c of Fig. 231, according as 
 
 the eyes are directed toward the nearer end, 
 
 the middle, or the farther end of it. The 
 
 impression of distinct vision for the point 
 
 of direct observation and the impressions of 
 
 indistinct or double vision for the other points 
 
 combine to give us the perception of relief, 
 
 or of form in three dimensions. With a single eye, relief is 
 
 only imperfectly perceived, as in a picture. 
 
 392." The Principle of the Stereoscope. — Since the two eyes see 
 an object from different points of view, its image on the retina in 
 one eye differs slightly from that in the other, especially when the- 
 object is near ; and it is by the apparent union of these two dis- 
 similar views that we see the object in rehef. Hence if we pre- 
 sent to each eye a picture of an object as it appears from its point 
 of view and direct the eyes so that the pictures appear to be super- 
 posed, the appearance of relief will be perfectly reproduced. This 
 is illustrated by Fig. 232. The two pictures, A and B, represent 
 
 Fig. 232. 
 
 the tunnel as it appears to the left eye and the right eye respec- 
 tively. Hold the book so that one picture is immediately in front 
 of each eye, and place a card between the pictures and perpen- 
 dicular to the page, so that each eye can see only the picture on 
 its own side. Now direct the eyes as if you were looking through 
 the book at a point some distance behind it. When this is done, 
 the pictures will appear to move together and unite into a siijgle 
 
3H 
 
 Light 
 
 Fig. 233. 
 
 view, which has the appearance of real depth extending a foot or 
 more behind the book. With a httle practice, the card between 
 the pictures can be dispensed with ; but two 
 additional tunnels will then be indistinctly seen, 
 one on either side, the one on the left side being 
 the left picture as seen by the right eye and the 
 other the right picture as seen by the left eye. 
 
 The stereoscope is an instrument designed to 
 aid in the combination of slightly dissimilar 
 photographs taken with a double camera, or 
 with a single camera from two positions a short 
 distance apart. The two pictures, A and B 
 (Fig. 233), are mounted on the same card, and 
 are viewed through the half lenses M and N. 
 The magnified, virtual images of the two pictures coincide at C. 
 The partition P prevents each eye from seeing the picture 
 intended for the other. 
 
 393. Visual Angle and Angular Size. — The angle between the 
 rays that enter the eye from the extremities of an object is called 
 the visual angUy or the angle under which the object is seen. This 
 angle measures the angular size of the object. Thus in Fig. 234 
 the angular size of the ob- 
 ject AB is A OB. When a 
 small coin is held before the 
 eye at such a distance that it 
 is just large enough to con- 
 ceal the sun from that eye, 
 the coin at that distance has the same angular size as the sun. 
 For small angles, the angular size of an object is proportional to 
 its actual size and inversely proportional to its distance from the 
 observer. 
 
 The size of the image on the retina is proportional to the angu- 
 lar size of the object, and hence increases as the distance of the 
 object decreases (see figure). This explains why an object is 
 seen more and more distinctly as it is brought toward the eye, 
 
 Fig. 234. 
 
Optical Instruments 
 
 315 
 
 until the least distance of distinct vision is reached. No more 
 detail of the form and features of a man could be distinguished at 
 a distance of 100 yards than could be seen in a photograph of 
 him half an inch long when held at arm's length ; for the man and 
 the photograph, at these distances, would have the same angular size. 
 When the size of an object is approximately known, as the 
 height of a tr6e or a house, its angular size serves as a basis for 
 estimating its distance. The various forms of telescopes and 
 microscopes serve the purpose of increasing the visual angle. 
 
 VII. Optical Instruments 
 
 394. The Simple Microscope. — A converging lens, when used 
 as a simple microscope, forms a magnified, virtual image of the 
 object, as explained in Art. 379. Lenses for such use are mounted 
 in frames and supports of various forms; but these are merely 
 matters of convenience and need not be considered. 
 
 It can be shown that the magnification produced by a simple 
 microscope is greatest when the eye is placed close to the lens 
 and the distance of the object is such that the image is at the 
 least distance of distinct vision (Art. 387). This maximum mag- 
 
 ax- 
 
 Fig. 235. 
 
 nification is called the magnifying power of the miscroscope, and 
 may be defined as the ratio of the angular size of the image to the 
 angular size of the object, both being at the least distance of dis- 
 tinct vision (angle aOb : angle AOB\ Fig. 235), or as the ratig 
 
3i6 
 
 Light 
 
 of the length of the image at this distance to the length of the 
 object {ab : A'B' or ab : AB). 
 
 From similar triangles, ab : AB : : EO : DO \ hence the magni- 
 fying power is equal to the ratio of the least distance of distinct 
 vision, EOy to the distance of the object, DO. 
 
 If the focal length of the lens does not exceed 5 cm. (and it 
 is generally less), it is only slightly greater than DO and may be 
 substituted for it. Hence if L denote the least distance of dis- 
 tinct vision and / the focal length of the lens, we obtain the con- 
 venient ratio Z : / as the approximate measure of the magnifying 
 power of a simple microscope. L is taken as 25 cm. or 10 inches. 
 
 Laboratory Exercisf 57. 
 
 395. The Compound Microscope. — The images formed by a 
 single lens are imperfect if the magnification is very great ; hence 
 compound microscopes are used for observing very minute objects. 
 A compound microscope in its simplest form consists of two lenses 
 both of short focal length, called respectively the objective, O, and 
 the eyepUce^ E (Fig. 236). The objective is placed at a distance 
 
 Fig. 236. 
 
 only slighdy greater than its focal length from the object, AB ; 
 and a magnified, real image, ab, is formed at a much greater dis- 
 tance on the other side of the lens. The eyepiece is placed at a 
 distance a little less than its focal length from the real image, 
 and forms a virtual image of it, a'b'. The eyepiece serves as a 
 
Optical Instruments 317 
 
 simple microscope for viewing the real image formed by the 
 objective. 
 
 The magnification produced by the objective is measured by 
 the ratio of the distance of the real image from the objective to 
 the distance of the object from it (Art. 378). Let ^denote the 
 distance of the image and F the focal length of the objective. 
 The latter may be taken as the distance of the object with only 
 a slight error; hence the magnifying power of the objective is 
 approximately d -^ F, and is inversely proportional to its focal 
 length. The magnifying power of the eyepiece is L -5-/, / being 
 its focal length, and Z the least distance of distinct vision. The 
 magnifying power of the microscope is the product of the magni- 
 fications produced separately by the objective and the eyepiece, 
 or approximately {d -i- F)x(L -f-/). 
 
 If the objective and eyepiece consisted of only a single lens 
 each, as we have assumed, the image would be distorted, indis- 
 tinct, and bordered with the colors of the 
 rainbow. The coloring is explained in Art. 
 414; it is avoided by using one or more 
 achromatic lenses (Art. 415) for the object- 
 ive and two convex lenses, placed some dis- 
 tance apart, for the eyepiece. The distances 
 between the lenses of both objective and 
 eyepiece and the curvatures of their surfaces 
 are so chosen as to correct all defects of the 
 image. 
 
 Figure 237 represents a compound 
 microscope mounted on a stand. The 
 objective and eyepiece are at the ends 
 of the tube. The object is placed on _ 
 
 the stage and is brought into focus by 
 raising or lowering the tube by means of a thumb screw at the side. 
 
 Laboratory Exercise 62. 
 
 396. The Astronomical Telescope. — The astronomical tele- 
 scope, like the compound microscope, consists essentially of two 
 
3i8 
 
 Light 
 
 lenses, an objective, O^ and an eyepiece, E (Fig. 238). An 
 inverted image, ab^ of a distant object is formed by the objective 
 at its principal focus. The parallel rays JK and NO come from 
 a point on the lower side of the object, and converge to the cor- 
 
 FlG. 238. 
 
 responding point b in the image ; similarly the parallel rays MO 
 and HI from a point at the top of the object converge to a. The 
 eyepiece forms a magnified, virtual image, a^d\ of the real image, 
 as in the compound microscope. Both the real and the virtual 
 images are inverted with respect to the object. 
 
 The size of the image formed by the objective is proportional to 
 its focal length (4, Art. 378); and, since any increase in the size 
 of the real image would result in a proportional increase in the 
 size of the virtual image formed by the eyepiece, it follows that 
 the magnifying power of the telescope is proportional to the focal 
 length of the objective. It is for this reason that powerful tele- 
 scopes are long. The eyepiece is always of short focal length, and 
 increases the magnification. The magnifying power of a telescope 
 is defined as the ratio of the visual angle when an object is viewed 
 through it (angle a'Eb' or aEb) to the visual angle with the 
 naked eye (angle MON ox its equal aOb^. Since angles aEb 
 and aOb are subtended by the same line ab, they are (for small 
 angles) inversely proportional to the distances of their vertices 
 from this line ; i.e. angle aEb : angle aOb = OC: CE. Let F 
 denote the focal length of the objective, OC-, and / the focal 
 length of the eyepiece, which is (very nearly) equal to CE. Sub- 
 stituting in the proportion, we have angle aEb : angle aOb = F:/ 
 
Optical Instruments 
 
 319 
 
 (approximately) ; that is, the magnifying power is measured by 
 the ratio of the focal length of the objective to the focal length of 
 the eyepiece. 
 
 The objective is of large diameter in order to receive as much 
 light as possible from the heavenly body under observation. This 
 increases the brightness of the image, and renders visible count- 
 less stars that are never seen with the unaided eye. Even an 
 opera glass greatly increases the number of visible stars. The 
 objective consists of an achromatic pair of lenses, which, in large 
 instruments, are placed several inches apart. To secure the 
 greatest possible perfection, large objectives are ground and 
 polished by hand (Art. 381). The eyepiece is like that of the 
 compound microscope. 
 
 Laboratory Exercise 60. 
 
 397. The Terrestrial Telescope. — The fact that heavenly 
 bodies appear inverted through an astronomical telescope is 
 unimportant ; but for the observation of terrestrial objects the 
 image must be erect. An additional convex lens, Z (Fig. 239), is 
 
 
 b" 
 
 Fig. 239. 
 
 sufficient to reinvert the image. The inverted image ad is formed 
 by the objective (not shown in the figure). If the lens Z is at 
 twice its focal length from this image, it will form a real and erect 
 image a'b' at an equal distance on the other side. The eyepiece 
 forms an' erect image of a'b'. Telescopes are constructed with 
 
320 
 
 Light 
 
 two lenses to reinvert the image, since better resnlts are obtained 
 than with a single lens. A small terrestrial telescope is called a 
 spyg/ass. 
 
 398. The Galilean Telescope. — This telescope receives its name 
 from Galileo, who, if not the first, was among the first to constnict 
 such an instrument. It is the earliest form of telescope, and also 
 the simplest, since it gives erect images with only two lenses. The 
 
 Fir.. 240. 
 
 objective, O (Fig. 240), is a convex lens as in the astronomical 
 telescope, and would form a real and inverted image, ab, at its 
 principal focus if the light were not intercepted by the eyepiece, E, 
 This is a concave lens, and is placed at a distance equal to or only 
 very slightly exceeding its own focal length from the focus of the 
 objective. 
 
 The light incident upon the eyepiece from any point of the 
 object IS convergent ; after passing through the eyepiece it is very 
 slightly divergent Thus the parallel .rays JK and NO from a 
 point at the bottom of the object would converge to the point b of 
 the real image if it were not for the refraction produced by the 
 eyepiece ; but, as a result of this refraction, they appear to come 
 from b\ All the light transmitted by the eyepiece is similarly 
 refracted, forming the erect, virtual image a^b'. 
 
 The angular size of the image is a'Eb' or its equal aEbj and 
 the angular size of the object MON or its equal aOb. But for 
 small angles, angle aEb : angle aOb = OC: EC =F : /, i^ being 
 the focal length of the objective and / that of the eyepiece. 
 Hence the magnifying power is measured by the ratio of the focal 
 
Optical Instruments 
 
 32 
 
 length of the objective to the focal length of the eyepiece, as in the 
 astronomical telescope. For equal power the Galilean telescope 
 is the shorter of the two by twice the focal length of the eyepiece. 
 (Why?) It has a still greater advantage in this respect over the 
 
 Fig. 341. 
 
 terrestrial telescope, since the lens in the latter for erecting the 
 image increases the length without increasing the power. 
 
 An opera glass or a field glass is a pair of Galilean telescopes 
 mounted together, one for each eye. The magnityin^ power of the 
 opera glass is from two to three, that of the field glass is from four 
 to eight. 
 
 Laboratory Exercise 61. 
 
 399. The Optical Lantern. — This is an instrument by means 
 of which highly magnified images of transparent photographs and 
 drawings are projected upowa 
 white screen in a dark room. 
 Its essential parts are a strong 
 source of light, A (Fig. 241); 
 a condensing lens, L (gen- 
 erally double), for concen- 
 trating the light upon the 
 picture or "slide," F] and 
 an achromatic objective, L\ 
 placed at a little more than 
 its focal length from the slide. 
 The image is formed at a relatively great distance, and is corre- 
 
 FlG. 242. 
 
322 Light 
 
 spondingly magnified. In order that it may be erect, the slide is 
 inverted. 
 
 400. The Photographer's Camera. — The tube AD of this instru- 
 ment contains an achromatic convex lens, which forms an inverted 
 image upon a ground-glass screen at C. The image is focused by 
 moving either the part of the tube that carries the lens or the glass 
 screen. When the focus has been obtained, the screen is replaced 
 by a plate holder containing a sensitized glass plate (Fig. 242). 
 
 PROBLEMS 
 
 1. ^^^lat U the magnifying power of a simple microscope whose focal 
 length is (a) 2 in.? (//) .5 in.? 
 
 2. Show from the formula of Art. 394 that the magnifying power of a 
 simple microscope is inversely proportional to its focal length. Draw figures 
 to illustrate. 
 
 3. Assuming that the real image is formed 5 in. from the objective, find 
 the magnifying power of a compound microscope (a) with a .5 in. objective 
 and a 2 in. eyepiece ; {b) with a \ in. objective and a ^ in. eyepiece. 
 
 4. The great telescope of the Lick Observatory is 57 ft. long. What is its 
 magnifying power (a) when fitted with a 2 in. eyepiece? {b) when fitted with 
 a \ in. eyepiece? 
 
 5. The objective of a field glass has a focal length of 7.5 in., and the eye- 
 piece a focal length of 1.25 in. Find the length of the instrument and its 
 magnifying power. 
 
 6. How is the size of a photograph affected {a) by the distance of the 
 
 object from the camera? (^) by the focal length of the lens of the camera? 
 
 ». 
 
 YIU. Dispersion and Color 
 
 401. Decomposition and Recomposition of White Light ; Dis- 
 persion. — Sunlight is white ; but after refraction through uncolored, 
 
 transparent media, it is often highly 
 colored. This effect of refraction is 
 strikingly exhibited when a beam of 
 sunlight is admitted into a darkened 
 room through a long, narrow slit, and 
 Fig. 243. ggj^^ through a prism whose refracting 
 
 edge is parallel to the slit. If the slit and prism are vertical and 
 
Dispersion and Color 323 
 
 the path of the beam horizontal, Fig. 243 will represent a hori- 
 zontal section. When the refracted light is caught upon a white 
 screen at a distance of a few meters from the prism, it appears as 
 a horizontal band of light, VJ?, rounded at the ends and colored 
 in all the tints of the rainbow. This band of light is called the 
 so/ar spectrum. It consists of an indefinitely great number of 
 colors, which blend imperceptibly into each other. The seven 
 principail colors, named in order, are violet^ indigo, blue, greeriy 
 yellow, orange, and red. (The paths of only the red and the violet 
 light are shown in the figure.) The deviation is greatest for the 
 violet light and least for the red {Exp.). 
 
 When a convex lens is placed in the path of the refracted light 
 in such a position that the prism and 
 the screen are at conjugate focal dis- 
 tances (Fig. 244), the lens converges 
 the different colors to the same spot 
 upon the screen, and this spot is white .- 
 {Exp.). The prism, by unequal re- 
 fraction or dispersion, separates the 
 
 light of different colors, all of which are present in the incident 
 beam of white light. When these colors are reunited by the lens, 
 they again form white light. 
 
 When the prism and lens are removed and the screen alone is 
 placed in the path of the beam, an elongated and imperfect image 
 of the sun is formed upon it. (When nearly the whole length of 
 the slit is covered, the image of the sun is round.) This image is 
 white, since it is composed of all the colors of the incident light. 
 When the prism is interposed, the unequal deviation of the light 
 of different colors spreads the colored images of the sun out in 
 a line, forming the spectrum. Since the number of these images 
 is unlimited, there is much overlapping and mixing of the colors, 
 and the spectrum is said to be impure. 
 
 402. Production of a Pure Spectrum ; Fraunhofer's Lines. — 
 When a convex lens is placed in the path of the beam at conjugate 
 focal distances from the slit and the screen, the distance from the 
 
324 Light 
 
 slit being the shorter, an enlarged white image of the slit is formed 
 
 upon the screen.* When the prism 
 
 is placed near the lens (Fig. 245), it 
 
 produces dispersion of the colors, as 
 
 before ; but the Uns brings each color 
 
 to a focus upon the screen^ forming 
 
 Fig. 245. ^ spectrum which consists of a series 
 
 of narrow and only very slightly overlapping colored images of the 
 
 silt. This spectrum is rectangular, and is much wider than that 
 
 produced without the lens {Exp.). 
 
 Narrowing the slit diminishes the overlapping of the images, and 
 hence makes the spectrum more nearly pure ; but its brightness is 
 correspondingly diminished. When the slit is very narrow and the 
 lens exactly focused, the spectrum is crossed by many dark lines 
 at right angles to its length (/>. in a direction parallel to the slit). 
 These lines are called Fraunho/er's iines^ after the celebrated 
 optician of Munich, who first studied and gave a detailed descrip- 
 tion of them. They represent missing images of the s/it, and 
 indicate that light of certain colors is absent from sunlight. When 
 the width of the slit is only very slightly increased, the overlapping 
 of the images on each side of the dark lines obliterates them. 
 
 403. Virtual Spectrum. — If the preceding experiments cannot 
 be performed for want of a dark room or a porte lumiere for 
 directing a beam of sunlight into the room, the observation of a 
 virtual spectrum will serve as a substitute. The experiment consists 
 in looking through a prism at a slit about a millimeter wide in a 
 piece of black cardboard. The cardboard should be held up at 
 arm*s length before a window, with the sky for a background and 
 with the slit horizontal. The prism is held close to the eyes, with 
 its edges parallel to the slit. If the refracting edge of the prism 
 
 1 It b only the light from the same portion of the sun's disk that is parallel. The 
 rays in a sunbeam that come from opposite sides of the sun are at an angle of about 
 half a degree {AOB, Fig. 245). The lens, in the position described, brings all the 
 light that passes through any point of the slit to the same point of the image, and 
 hence forms an image of the slit. 
 
Dispersion and Color 
 
 325 
 
 ^ 
 
 ^;:- 
 
 Fig. 246. 
 
 is on the lower side, the cardboard will appear at an angle of about 
 40° below its true position (Fig. 246) ; and the observer, looking 
 obliquely downward at this angle, will see a 
 virtual spectrum consisting of a series of over- 
 lapping colored images of the slit. Since 
 the violet light is refracted the most, the vio- 
 let image of the slit will be the lowest (see 
 figure). 
 
 Laboratory Exercise 58, 
 404. The Nature of Color. — When any part of the light com- 
 posing a spectrum is allowed to pass through a second prism, it is 
 again refracted, but its color remains unchanged (Fig. 247). The 
 
 colors of the spec- 
 ^ trum are simple or 
 
 elementary; that is, 
 they cannot be fur- 
 ther decomposed 
 {Exp.). 
 
 It is found by ex- 
 periment ^ that the 
 wave lengths of the colors of the spectrum increase regularly from 
 the violet to the red ; and that the same elementary tolor always 
 has the same wave length (in the same medium) whether its 
 source is the sun or any other luminous body. It follows that the 
 physical cause of the sensation of color is nothing else than the 
 wave length of the light ; in other words, the sensation produced 
 by an elementary color is determined by its frequency {i.e. the 
 number of waves that pass any point in a second), just as the pitch 
 of a sound is determined by the vibration number of the sounding 
 body or the frequency of the sound waves. 
 
 As the temperature of a body rises, the vibration of its mole- 
 cules becomes more and more rapid, and shorter waves are set up 
 in the ether. At about 525° C. (Art. 331) some of the molecules 
 
 1 It is beyond the scope of elementary physics to discuss the methods by which 
 the wave lengths of light are measured. 
 
 Fig. 247. 
 
326 Light 
 
 vibrate with sufficient rapidity to give out red light. As the tem- 
 perature continues to rise, additional colors are given out in order 
 from red to violet, and the color of the body changes from red 
 through orange and yellow to white. 
 
 " That which we call white light is, in the state in which we 
 receive it from such a body as a white-hot bar of iron, or perhaps 
 in its purest form from the crater of the positive pole of the electric 
 arc, a mixture of long and short waves ; waves of all periods 
 within the range of visibility are either continuously present or, 
 if absent for a time, are absent in such feeble proportions or for 
 such short intervals that they are not appreciably missed by the 
 eye. White light of this kind is comparable to an utterly discord- 
 ant chaos of sound of every audible pitch ; such a noise would 
 produce no distinct impression of pitch ; and so white light is 
 uncolored." — Daniell's Principles of Physics. 
 
 405. The Invisible Spectrum. — The luminous ether waves vary 
 in length from .0000767 cm. for the extreme red to .0000397 cm. 
 for the extreme violet. The interval between these extremes, 
 expressed as in music, is somewhat less than one octave. Thus 
 we see that the range of sensibility of the eye is much less than 
 that of the ear (Art. 304). That there are many octaves of non- 
 luminous ether waves extending both above and below the visible 
 portion of the spectrum is proved by their chemical, heating, and 
 electrical effects. 
 
 406. Cause of Dispersion. — \Vhenever light is refracted, the 
 elementary colors of which it is composed are refracted unequally, 
 although in many cases the dispersion is not sufficient to be 
 noticeable. This unequal deviation indicates that the index of 
 refraction of a substance varies with the color of the incident 
 light; and, since the index of refraction is the ratio of the velocity 
 of light in a vacuum to its velocity in the substance (Art. 363), it 
 is evident that lights of different colors {or ether waves of different 
 lengths^ travel with unequal velocities in the same substance. 
 This, therefore, is the cause of dispersion. 
 
 With few exceptions, the deviation increases continuously from 
 
Dispersion and Color 327 
 
 the red to the violet, as in the preceding experiments, the velocity 
 of the shorter waves being less than that of the longer in most 
 substances. 
 
 407. Color of Opaque Bodies. — When a very narrow strip of 
 white paper, pasted on a piece of black cardboard, is viewed 
 through a prism as the slit was in the experiment of Art. 403, the 
 light from it is resolved into a complete spectrum, the colors of 
 which have the same relative intensity as in the spectrum of direct 
 sunlight. Any opaque body which, like the white paper, reflects 
 all the elementary colors of the incident light in equal proportions 
 appears white when white light falls upon it; but when the inci- 
 dent light is colored, the body appears of the same color. Thus 
 when a spectrum is thrown upon a white screen, the part of the 
 surface upon which the red light falls appears red, the part upon 
 which the blue light falls appears blue, etc. ; for each part reflects 
 the color that it receives. 
 
 When a narrow strip of colored paper is viewed through a prism, 
 the light from it is resolved into an incomplete spectrum ; gener- 
 ally half or more of the spectrum is either wanting or very faint. 
 The spectrum of a blue strip, for example, will probably be found 
 to consist of violet, indigo, blue, and green ; that of a yellow strip, 
 of green, yellow, orange, and some red. A similar analysis of the 
 light from different colored bodies shows that, with few exceptions, 
 the light reflected by them is composite^ i.e. it is composed of a 
 number of elementary colors. Any body that reflects some of 
 the elementary colors of white light in larger proportion than it 
 does others is colored, its color being determined by the combined 
 effect of all the colors that it reflects. A body reflecting no light 
 would be perfectly black. White, black, and the diff"erent shades 
 of gray differ only in brightness ; each reflects the different elemen- 
 tary colors in the same proportions in which it receives them. 
 
 Color, regarded as a property of opaque bodies, is therefore 
 merely the power of reflecting light of certain wave lengths either 
 exclusively or in larger proportions than others, the light that is 
 not reflected being absorbed. It is further evident that bodies 
 
328 Light 
 
 have no color of their own. A white body, as stated above, takes 
 the color of the incident light ; a colored body appears of its natu- 
 ral color only when the incident light cqntains all the elementary 
 colors that it is capable of reflecting. This is strikingly illustrated 
 by holding colored papers in different parts of the solar spectrum 
 thrown upon a screen in a darkened room. A piece of green 
 paper, for example, will appear black in the violet, indigo, orange, 
 or red, being incapablfe of reflecting these colors ; in the blue it 
 will probably appear a dark blue, and in the yellow a dirty yellow, 
 due to the reflection of a little of these colors ; in the green it will 
 appear at least very nearly of its natural color {Exp.). 
 
 This explains why some bodies do not appear of the same color 
 by artificial light as by daylight. Most artificial lights are deficient 
 in violet and blue, and hence are more or less yellowish. In 
 such a light, pale yellow is scarcely distinguishable from white, 
 and blue is often mistaken for green. The greenish appearance 
 of blue is due to the fact that blue pigments reflect violet and 
 green as well as blue light, and green predominates in the light 
 that they reflect when illuminated by light that contains little 
 violet and blue. 
 
 408. Color of Transparent Bodies. — A transparent body is 
 colored if it is more transparent to some of the colors of white 
 light than to others, its color being that which results from the 
 J mixture of all the transmitted colors. The remaining colors of 
 the incident light are absorbed on the way through the medium. 
 This action of a colored medium is called selective absorption. If 
 one or more of the elementary colors that a body can transmit are. 
 not present in the incident light, the body will not appear of its 
 natural color ; and if none are present, it will appear opaque, since 
 in this case no light will be transmitted. 
 
 The light transmitted by a transparent body, as a piece of glass, 
 may be analyzed by observing either its real or virtual spectrum. 
 To obtain the latter, a slit is observed through a prism, as de- 
 scribed in Art. 403, with one end of the slit covered by the trans- 
 parent body. This gives a complete spectrum from the uncovered 
 
Dispersion and Color 329 
 
 end of the slit, and, beside it, the spectrum of the light transmitted 
 through the body (Lab. Ex.). To obtain the real spectrum, a 
 solar spectrum is projected upon a screen in a darkened room by 
 means of a slit, lens, and prism, as described in Art. 402 ; and the 
 body is held so as to cover either the upper or lower end of the 
 slit. The solar spectrum and the spectrum of the transmitted 
 light will then be projected upon the screen, one above the other, 
 making a comparison of the two very easy. 
 
 It will be found by either of these methods of observation that 
 the light transmitted by colored bodies is composite and, with 
 few exceptions, gives a considerable portion of the complete 
 spectrum. Blue (cobalt) glass transmits violet, blue, green, and 
 some red ; yellow glass transmits red, orange, yellow, and green ; 
 red (ruby) glass transmits red and a little orange (Exp.). 
 
 When two transparent bodies of different color are placed before 
 the slit, one in front of the other, the light that passes through 
 both undergoes a double process of selective absorption ; and its 
 spectrum therefore consists only of the color or colors that are 
 common to the light transmitted by the two separately. Thus 
 green is the only one of the colors transmitted by either blue or 
 yellow glass that is also transmitted by the other ; hence the two 
 together appear green. Similarly, the combination of red and 
 blue, red an^ green, or orange and blue glass is very nearly 
 opaque, since no color that they separately transmit in considera* 
 ble quantity is common to both {Exp.). 
 
 409. Color of Bodies containing Suspended Particles. — A gas 
 or a liquid which, of itself, is colorless becomes colored when 
 it contains a multitude of minute particles in suspension. An 
 example of this is the sky-blue liquid obtained by adding to water 
 a very small proportion of milk or an alcoholic solution of mastic, 
 or by mixing a few drops of dilute nitrate of silver with a quantity 
 of water in which a little table salt has been dissolved.^ These 
 
 1 In this case chloride of silver is formed, which is insoluble in water, but 
 remains suspended in the form of extremely minute solid-^Ef&^e^r=-^K^^same is 
 true of the mastic. jf^^* ^^ Twr 
 
 'UNIVERSITY 
 
330 Light 
 
 liquids appear blue by reflected light ; but are yellow or orange 
 when viewed by transmitted light. This is due to the fact that 
 the suspended particles reflect a considerable part of the violet 
 and blue light, but reflect less and less of the other colors toward 
 the red end of the spectrum. Thus violet and blue predominate 
 in the reflected light, and red, orange, and yellow in the trans- 
 mitted light. 
 
 The blue color of the sky is similarly explained, " the air being 
 rendered visible against the dark background of black space by 
 sunlight reflected from its fine suspended dust or water particles ; 
 while the light transmitted is always more or less yellowish, and, 
 in the afternoon and evening, when sunlight comes to us through a 
 greater thickness of the more dusty layers, verges toward orange 
 or even red." — r3anieirs PrincipUs of Physics. 
 
 410. Mixture of Colors. — The unaided eye is wholly incapable 
 of distinguishing between composite and elementary colored light. 
 The light reflected from a piece of yellow paper or transmitted 
 through yellow glass may produce exactly the same color sensation 
 as the yellow of the spectrum, although it contains all the colors 
 of the spectrum from the red to the green inclusive. Moreover, 
 the sensation of yellow may be caused by light that contains no 
 elementary yellow at all ; in fact, the color sensation caused by any 
 elementary color except violet and red may also be caused by vari- 
 ous combinations of two or more of the other elementary colors 
 in certain proportions.^ In studying mixtures of colored lights, the 
 selected colors of the spectrum are focused by a lens, or reflected 
 by mirrors to the same spot upon a screen, or in some other 
 way are caused to enter the eye in a united beam. Among the 
 results established by such experiments are the following : — 
 
 I. The mixture of any two elementary colors named in alter- 
 nate spaces in Fig. 248 is like the intermediate one. For example, 
 
 1 If the ear were like the eye in this respect, it would be incapable of distinguish- 
 ing the constituents of a complex sound. The notes sounded simultaneously by 
 an orchestra would produce ^e sensation of a single note of average pitch, and 
 harmony and discord would alike be unknown. 
 
Dispersion and Color 
 
 331 
 
 U99Jl'J 
 
 Fig. 248. 
 
 Z»^^ 
 
 the mixture of red and yellow appears to the eye to be identical 
 
 with elementary orange. The mixture of red and violet light is 
 
 purple. There is no elementary 
 
 , . • , r , r Purple 
 
 purple, as is evident from the fact 
 
 that this color is not found in the 
 
 spectrum. 
 
 2. Complementary Colors. — The 
 mixture of any pair of opposite col- 
 ors in the figure appears white when 
 the colors are combined in the right 
 proportions. ** Ordinary white light 
 consists of all the colors of the spec- 
 trum combined ; but any one of the 
 elementary colors, from the extreme 
 red to a certain point in yellowish green, can be combined with 
 another elementary color on the other side of green in such 
 proportions as to yield a perfect imitation of ordinary white. The 
 prism would instantly reveal the differences, but to the naked eye 
 all these whites are completely undistinguishable from one another." 
 (Deschanel.) 
 
 Any two colors which yield white when mixed are called com- 
 plemeniary colors, 
 
 3. Primary Colors. — Any color of the spectrum can be pro- 
 duced (so far as the color sensation is concerned) by mixing 
 elementary red, green, and violet lights in proper proportions. 
 Red, green, and violet are therefore called the primary colors. 
 
 411. Newton's Disks. — One of the most convenient methods of 
 mixing colors is- by means of colored disks, each 
 of which is slit along a radius, thus permitting 
 any desired amount of overlapping when two 
 or more of the disks are placed together on 
 an axis through their common center (Fig. 
 249). When the disks are rapidly rotated 
 about the axis, only one color is seen, and this 
 Fig. 249. covers the entire circular area. This results 
 
332 Light 
 
 from the fact that the sensation of sight continues for a fraction 
 of a second after the light ceases to enter the eye or ceases to fall 
 upon the same part of the retina. The rapid rotation causes the 
 different colors to come from all parts of the surface in such rapid 
 succession that each of them produces a continuous impression ; 
 and the effect is the same as if they came simultaneously from all 
 parts of the surface. , 
 
 The colors reflected by the disks are, of course, composite; 
 but experiments have shown that a composite color produces the 
 same effect in a mixture as the elementary color that looks like 
 it. Hence the results described in the preceding article can be 
 obtained with the disks. In most cases, however, the total amount 
 of reflected light is so small that the resultant color is very defi- 
 cient in brightness. Complementary colors, for example, generally 
 yield a dark gray instead of white {Exp.). 
 
 412. After-images. — If one looks for half a minute or more at 
 a brightly illuminated piece of colored paper on a black back- 
 ground, then at a white surface, an image of the colored paper 
 will appear upon the surface in the complementary color. Thus if 
 the paper is green, the image will appear purple {Exp.). 
 
 The explanation is that the part of the retina upon which the 
 light from the colored paper falls becomes fatigued for that color, 
 and is less sensitive to it than to the other colors of white light ; 
 hence these other colors produce the stronger impression when 
 white light falls upon that part of the retina, and they together 
 give the complementary color. The image thus produced is called 
 a negative after-image. 
 
 When the object looked at is very bright, the after-image is 
 u'S\i2\\y positive, that is, of the same color as the object; and this 
 is frequently followed by a negative image. A positive after- 
 image niay be regarded as an extreme instance of the persistance 
 of impressions. After-images of bright objects can often be seen 
 with the eyes tightly closed. Newton is said to have suffered for 
 many years from an after-image of the sun, caused by incautiously 
 looking at it through a telescope. 
 
Dispersion and Color -- 333 
 
 413. Colors of Mixed Pigments. — We have seen that a mixture 
 of blue and yellow lights (either composite or elementary) is white 
 (Arts. 410 and 411) ; but that when pieces of blue and yellow 
 glass are placed together, they appear green (Art. 408, end). The 
 results are not inconsistent, for they are obtained by wholly differ- 
 ent methods of combination. In the first case the sensation of 
 white is due to the simultaneous action of the two colors upon the 
 retina ; the colors and the sensations that they produce are added. 
 If the colors are complex, as is the case with Newton's disks, they 
 together probably include all the elementary colors. In the sec- 
 ond case, the result is obtained by subtraction^ elementary green 
 being the only color that passes through both pieces of glass in 
 appreciable quantity. The other colors are absorbed, — some by 
 the blue glass, the remainder by the yellow. 
 
 The mixture of blue and yellow paints or powders is green 
 {Exp.). This is evidently a case of subtraction, like that of the 
 blue and yellow glass. The blue paint (or powder) absorbs the 
 red, orange, and yellow of the incident light, and the yellow paint 
 absorbs the violet, indigo, and blue. Thus green is the only color 
 not strongly absorbed by one or the other, and the mixture is 
 green. It is evident that the results obtained by mixing pigments 
 and by mixing lights of the same color are, in general, very differ- 
 ent. The light reflected by mixed pigments consists of the colors 
 which are not absorbed by either constituent. If the lights reflected 
 by two pigments have no elementary color in common, a mixture 
 of the two will be black or a dark gray. This is the case with 
 vermilion (a bright red) and ultramarine (a deep blue) {Exp.). 
 
 414. Chromatic Aberration. — The pupil has very probably 
 observed a fringe of color bordering objects when viewed through 
 a convex lens, and especially when viewed through a pair of 
 lenses, used either as a telescope or a 
 compound microscope. The coloring 
 is due to the unequal refraction of 
 the elementary colors by the lenses. ' ^^°* 
 
 When white light from any point, O (Fig. 250), falls upon a 
 
334 Light 
 
 lens, the greater refraction of the violet light brings it to a nearer 
 focus, Vy than that of the red light, r. The foci of the other colors 
 lie between these. When a screen is placed at the focus of the 
 red light, the image is surrounded by a border of violet and blue 
 light ; at the focus of the blue light it is surrounded by red {Exp.), 
 This unequal focusing of the different colors is called chromatic 
 aberration. 
 
 Since dispersion increases with the deviation, chromatic aberra- 
 tion is much greater for light passing through a lens near its edge 
 than for light passing through its central portion, and is greater the 
 less the focal length of the lens {Exp.). Thus the opaque 
 diaphragm with a small circular opening, which is placed in front 
 of the lens in cameras and other optical instruments, serves the 
 double purpose of diminishing both the spherical and the chro- 
 matic aberration. But the diaphragm only partially overcomes 
 the defect, and besides has the disadvantage of cutting off the 
 greater part of the light. During the seventeenth century the 
 remedy employed in the construction of telescopes was the use of 
 object glasses of great focal length — in some cases exceeding 
 lOo feet. The object glass in such cases was mounted at the top 
 of a high pole, and the eyepiece was on a separate stand below. 
 415. Achromatic Lenses. — About 150 years after the telescope 
 and the microscope were invented, it was discovered that chro- 
 matic aberration could be almost perfectly corrected by combining 
 a convex lens of crown glass with a concave lens of flint glass (Fig. 
 251). This depends upon the fact that, while the refracting power 
 of flint glass is only slightly greater than that of 
 crown glass, its dispersive power is nearly twice 
 as great, the spectrum formed by a prism of flint 
 Fig. 251. glass being nearly twice as long as that formed 
 
 by a prism of crown glass having an equal refracting angle {Exp.). 
 Hence a double lens, consisting of a convex lens of crown glass 
 and a concave lens of flint glass of nearly twice the focal length, 
 produces convergence of the light without dispersion ; for the dis- 
 persion due to the concave lens is equal and opposite to that 
 
Dispersion and Color 
 
 335 
 
 of the convex lens, and it neutralizes only half of the converg- 
 ence caused by the latter (Fig. 252). The objectives of tele- 
 scopes, opera glasses, and microscopes 
 (except the cheapest) are achromatic. 
 
 416. The Rainbow. — Rainbows are due 
 to the dispersion of sunHght by raindrops, 
 and by the drops of water in the spray of 
 fountains, waterfalls, etc. Sometimes one 
 
 fc 
 
 c 
 Fig. 25a. 
 
 bow is seen, sometimes two, each consist- 
 ing of the colors of the solar spectrum. 
 They are always arcs of circles, and, when 
 two are seen, they are concentric. The ~ 
 inner or lower one is always much the _ 
 brighter and is called the primary' bow. 
 In it the red is on the outside, the violet 
 on the inside. In the outer, or secondary bowy the colors occur in 
 the reverse order (Fig. 253). The rainbow is always seen in the 
 direction opposite to the sun, — the sun, the observer, and the 
 center of curvature of the bow being in the same straight line, EO. 
 This line is called the axis of the bow. 
 
 The formation of the rainbow can 
 be experimentally illustrated in a dark- 
 ened room by means of a globe (a 
 round-bottomed flask) filled with water. 
 It will be found that the results obtained 
 , .., when a slender beam of sunlight is 
 ^ «o-«HM ^^^"^ I ', \ caused to fall upon the globe depend 
 
 upon the angle at which the beam meets 
 its surface. At a certain angle a curved 
 spectrum is formed and may be caught upon a screen ; at another 
 angle the spectrum reappears with the colors in the reverse order. 
 At angles other than these the light is too widely scattered on 
 leaving the globe to produce visible effects. If a beam large 
 enough to cover the globe is used, one, and possibly both, of the 
 spectra will appear as complete circles {Exp.) . 
 
336 
 
 Light 
 
 42 
 
 Fig. 254. 
 
 The dispersion caused by a single drop of water is like that 
 obtained with the globe in the above experiment, except that 
 the total amount of reflected light is correspondingly less. Each 
 drop forms two complete spectra ; but the eye receives only a 
 slender ray of one color from any one drop, and the bow that is 
 seen is made up of light from a multitude of drops. 
 
 417. The Primary Bow. — Figure 254 represents a ray of light 
 entering a drop at the angle required for the primary bow. The 
 
 unequal refraction of the colors at A 
 'X causes dispersion, forming a spec- 
 trum, ^ y, at the back of the drop. 
 Here the greater part of the light is 
 refracted out (not shown in the 
 figure) ; the remainder is reflected 
 to /i'y, where the greater part of 
 it is refracted out with further dis- 
 persion. The red and the violet 
 rays make angles of about 42** and 40°, respectively, with the inci- 
 dent ray SA. The reason for the order of the colors in the 
 primary bow will be evident from a comparison of this figure with 
 the preceding one. The eye receives the same color fi-om all 
 drops at the same angular distance from the axis of the bow ; 
 hence the bow is circular. 
 
 At sunrise or sunset the rainbow, if complete, appears as a semi- 
 circle, its axis, £0(Fig. 253), being 
 horizontal. Since the center of the 
 bow is always at the same angle be- 
 low the horizon that the sun is above 
 it, the higher the sun is, the shorter 
 will be the arc of the bow. When 
 the sun is more than 42° above the 
 horizon, the primary bow is wholly 
 below it, and is therefore invisible. 
 
 418. The Secondary Bow. — The ^' ^''' '^* 
 secondary bow is formed by light that has undergone two reflec- 
 
Dispersion and Color 337 
 
 tions, as shown in Fig. 255. On leaving the drop, the red ray 
 makes an angle of about 51°, and the violet ray an angle of 54°, 
 with the incident ray. A comparison of Figs. 255 and 253 will 
 show why the secondary bow. is above the primary, and why the 
 order of the colors is reversed. The faintness of the secondary 
 bow is due to the additional loss of light at the second reflection. 
 
 419. Color by Interference. — When two pieces of plate glass are 
 pressed firmly together in the fingers or in a clamp, curved bands 
 of spectrum colors appear, surrounding the point where contact is 
 closest. The colors are brightest when the plates are looked at 
 from the more strongly illuminated side {Exp.). 
 
 In Fig. 256, MM' and NN' represent sections of the glass 
 plates, the distance between them being greatly exaggerated. 
 Light incident along the path AB is par- 
 tially reflected at C from the lower sur- 
 face of the upper plate, and also at E from 
 the upper surface of the lower plate. Some 
 of the light reflected at E is transmitted 
 through the upper plate parallel to and 
 nearly coincident with the light reflected 
 from C. But, in twice crossing the space ^^^' ^^^' 
 
 between the plates^ the waves reflected at E fall behind those 
 reflected at C; and the waves of some one of the elementary 
 colors in these two sets of reflected waves meet in opposite phase, 
 causing interference as in the case of sound waves (Art. 2C)8), and 
 that color is weakened or destroyed. The reflected light is com- 
 plementary to the light cut out by interference, and this differs at 
 different places, depending upon the distance between the plates. 
 
 Patches and bands of rainbow colors are similarly produced by 
 the interference of light reflected from the two surfaces of a very 
 thin film of any transparent substance, as a soap film or a film of 
 oil floating on water. In the above experiment the layer of air 
 between the glass plates acts as a thin film. Bodies whose colors 
 are due to the interference of the light reflected from them are 
 called iridescent. 
 
338 Light 
 
 The iridescence of some bodies is caused by interference of the 
 light reflected from minute parallel grooves and ridges (striations) 
 covering their surfaces, as in mother-of-pearl and the plumage of 
 many birds. The colors of an iridescent surface change with the 
 angle of incidence of the light, producing the beautiful effect 
 known as a ** play of colors." 
 
 The interference of light is the strongest evidence in support of 
 the wave theory. 
 
 PROBLEMS 
 
 1. \Miat is the function of the lens in producing a pure spectrum? 
 
 2. Why is it not possible to correct the chromatic aberration of a lens by 
 any change in the form of its surfaces? 
 
 3. (<i) State all the conditions necessary for a rainbow, (d) Do two ob- 
 ■ervers see exactly the same rainbow ? 
 
 4. Prove that the reflection within a raindrop takes place at less than the 
 critical angle, and is therefore not total. 
 
CHAPTER XI 
 
 MAGNETISM 
 
 I. Properties of Magnets 
 
 420. Natural and Artificial Magnets. — A magnet is a body 
 which has the property of attacting iron, and the term magnetism 
 is appHed to the unknown cause of this attraction. 
 
 Natural magnets, or lodestonesy were known to the ancients. 
 They are heavy, black stones consisting of a certain iron ore 
 called magnetic oxide of iron. The word magnet is derived from 
 Magnesia in Asia Minor, where these magnetic stones were first 
 found ; they were called lodestones (leading-stones) from their 
 property of pointing in a definite direction when suspended. 
 
 When a piece of highly tempered steel is rubbed with a magnet 
 or in any other way subjected to strong magnetic action, it 
 acquires permanent magnetic properties, and becomes a manu- 
 factured or artificial magnet. Magnets are made of various shapes, 
 each adapted to particular uses. 
 
 Laboratory Exercise 63, 
 
 421. Distribution of Attracting Power ; Polarity. — The differ- 
 ent parts of a magnet have very unequal power of attracting iron. 
 This is readily shown by placing 
 
 the magnet in a box of iron 
 
 filings or small tacks. On lifting 
 
 the magnet, it will be found that 
 
 Fig. 257. 
 the quantity of filings or tacks 
 
 clinging to the magnet is greatest at and near the ends, and 
 
 diminishes rapidly toward the center, where there are none (Fig. 
 
 257). The regions near the ends, where the force of attraction 
 
 is greatest, are called \\iQ poles {Exp). 
 
 339 
 
340 Magnetispi 
 
 Every magnet that is regularly or uniformly magnetized has 
 two poles, one at each end, whatever its shape may be. By 
 irregular magnetization additional poles may be developed between 
 the ends ; but such cases need not be considered. 
 
 Any straight magnet, when suspended or supported so that it is 
 free to turn in a horizontal plane, always comes to rest with the 
 same end pointing nearly north. This end of the magnet is called 
 the north pole (i>. north-seeking pole) and the other end the south 
 poh {Exp,). 
 
 422. The Bfagnetic Needle. — A slender magnet suspended at 
 
 its center by an untwisted fiber 
 or balanced on a pivot (Fig. 
 258) is called a magnetic needle. 
 After any displacement, a mag- 
 netic needle always returns to 
 
 *°* ^^ * one definite direction, which 
 
 is the direction of the magnetic meridian at that place. 
 
 423. The Mutual Action of Ifagnets. — Any force exerted be- 
 tween two magnets is most readily detected when one or both are 
 free to turn about an axis under the action of the force. Thus 
 when either pole of a bar magnet is brought up to the like pole of 
 a magnetic needle, the latter turns away from the magnet, show- 
 ing repulsion ; but the unlike pole turns toward the magnet, show- 
 ing attraction (Exp.). When the experiment is performed with 
 two needles, both will turn, showing that the attractions and repul- 
 sions are mutual. The law of equal action and reaction (third 
 law of motion) holds for magnetic forces as for all others. 
 
 The mutual action of magnets is expressed by the law : Like 
 poles repel and unlike poles attract each other. 
 
 424. The Effect of Distance on Bfagnetic Action. — The force 
 exerted by a pole of a magnet increases as the distance from the 
 pole decreases. This is shown by the fact that the nearer an end 
 of a bar magnet is brought to a magnetic needle, the more rapidly 
 will the unlike pole of the needle swing round and vibrate before 
 it ITie exact relation between the force and the distance varies 
 
Properties of Magnets 341 
 
 considerably under different conditions ; but the force is approxi- 
 mately inversely proportional to the square of the distance except 
 for very short distances. 
 
 This accounts for the fact that, when the nearer poles of two 
 magnets are unlike, the magnets tend to come together ; for the 
 attraction exerted upon the nearer pole of either exceeds the 
 repulsjon exerted upon its farther pole, giving a resultant force 
 which acts toward the other magnet. Similarly, when the nearer 
 poles are like, the resultant force upon each magnet tends to 
 move it from the other. 
 
 425. Magnetic and Nonmagnetic Substances. — Substances 
 that are attracted by a magnet are called magnetic substances ; 
 those that are not attracted are called nonmagnetic. The only 
 substances that are sufficiently magnetic to be attracted by mag- 
 nets of ordinary strength are iron, steel (a form of iron), some 
 compounds of iron, including magnetic iron ore, nickel, and cobalt. 
 All other substances are practically nonmagnetic ; although, under 
 the action of a powerful electro-magnet (Art. 457), all or nearly 
 all exhibit either weak attraction or repulsion. Nickel and cobalt 
 are less magnetic than iron. Iron in its different forms, such as 
 cast iron, wrought iron, and steel, is the only substance whose 
 magnetic properties are of any importance. 
 
 These properties are usefully applied in the telegraph, the tele- 
 phone, the dynamo, the motor, and many other electrical machines 
 and instruments. 
 
 426. Magmetic Action through Bodies. — The action of a mag- 
 net upon any magnetic body is not affected by placing a non- 
 magnetic body between them ; but is affected in a marked degree 
 by interposing another magnetic body. For example, a magnet 
 attracts or repels a magnetic needle through a board, a book, or a 
 plate of glass just as if nothing intervened ; but, when a sheet of 
 iron is placed between them, the needle is only slightly affected 
 by the presence of the magnet, if at all. The sheet of iron, espe- 
 cially if large, serves as a screen to cut off magnetic action from the 
 side opposite the magnet. This is explained in the next article. 
 
342 Magnetism 
 
 n. Kagnetization 
 
 Laboratory Exfrcise 64. 
 
 427. Bfagnetic Induction ; Penneability. — When either pole of 
 a magnet is held against or very near an end of a soft iron rod, 
 the other end of the rod attracts iron filings in considerable quan- 
 tity, and attracts or repels the poles of a magnetic needle as the 
 nearer pole of the magnet would do. Thus with the north pole 
 against the rod (Fig. 259), the farther end of the rod repels the 
 north pole of the needle. The rod, in fact, senies as a carrier 
 for the magnetic action of the magnet^ and is itself a magnet while 
 doing so, having poles as shown in the figure. Only magnetic 
 substances can thus modify and extend the action of a magnet, 
 and the property thus exhibited is called magnetic permeability. 
 We can now understand how a sheet of iron serves as a magnetic 
 screen. By itself becoming magnetized, it turns aside the mag- 
 netic action to its edges, which afe capable of exerting attractions 
 and repulsions (Exp.). 
 
 The iron rod in the above experiment is said to be magnetized 
 by induction. Magnetic induction always takes place when a 
 
 ^_ — ' magnet is brought near a magnetic 
 
 •*? -v body. The body is then attracted 
 
 Fig. 259. because its nearer pole is unlike the 
 
 nearer pole of the magnet (Fig. 259). Thus induction always 
 precedes attraction and is the cause of it. 
 
 428. Temporary and Permanent Kagnets. — When soft or 
 wrought iron, hard iron, untempered steel, and tempered steel 
 are subjected to equal inductive action, as by bringing them suc- 
 cessively in contact with the same magnet, a simple test with iron 
 filings will show that the soft iron becomes most strongly mag- 
 netized and the tempered steel the least. On the other hand, the 
 soft iron loses its magnetization, completely or nearly so, as soon 
 as it is removed from the influence of the magnet, while the tem- 
 pered steel retains all or nearly all of the magnetization induced 
 in it. Thus by magnetic induction a piece of soft iron may be 
 
Magnetization 
 
 343 
 
 made a temporary magnet and a piecfe of tempered steel a perma- 
 7ient one. Hard iron and untempered steel retain a considerable 
 part of induced magnetization ; they are subpermafient. 
 
 All manufactured magnets are pieces of highly tempered steel 
 that have been magnetized by induction. They may be demag- 
 netized or even magnetized with opposite polarity at any time by 
 sufficiently strong inductive action in the opposite direction. 
 
 429. Experimental Evidence on the Nature of Magnetization. — 
 There is much experimental evidence indicating that the magneti- 
 zation of a body is definitely related to its molecular condition. 
 The following are some of the facts that point most strongly to 
 this conclusion : — 
 
 Properties of a Broken Magnet. — When a magnet is broken 
 into any number of parts, each piece is a complete magnet having 
 the same polarity 
 and approximately 
 the same strength 
 as the original mag- 
 net (Fig. 260). This 
 can be readily shown by breaking a magnetized sewing or knitting 
 needle {Exp,). 
 
 Such experiments show that every part of a magnet is mag- 
 netized ; in fact, the neutral portion in the middle is generally 
 more strongly magnetized than the ends. The absence of attract- 
 ive power in the middle may be accounted for by regarding the 
 
 two halves of a magnet 
 
 ^ ^-^ as complete magnets 
 
 with their unlike poles 
 joined at the center. 
 These equal unlike poles, 
 situated at the same 
 point, exactly neutralize each other's action upon surrounding 
 bodies. A magnet may be regarded as composed of an in- 
 definite number of little magnets, with all their like poles pointing 
 in the same direction (Fig. 261). 
 
 
 !„i;i;i;li;:ii:;^J 
 
 
 Fig. 260. 
 
 ^1 
 
 M— 
 
 » 
 
 If- 
 
 —9 
 
 n 
 
 s 
 
 7f- 
 
 ■ 8 
 
 n 
 
 « 
 
 n 
 
 8 
 
 n 
 
 9 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 s 
 
 n 
 
 8 
 
 n 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 S 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 8 
 
 n 
 
 3 
 
 n 
 
 H 
 
 n 
 
 S 
 
 n 
 
 -2 
 
 n 8 
 
 2- 
 
 -2 
 
 n 
 
 8 
 
 n 
 
 8 
 
 7l 
 
 -J» 
 
 N 
 
 Fig. 261. 
 
344 Magnetism 
 
 Effect of Heat on Magnetization, — The strength of a magnet is 
 always diminished by heating it. This effect is only temporary for 
 moderate degrees of heat ; but a bright red heat causes permanent 
 demagnetization {Ex/>,), 
 
 This effect of heat suggests that magnetization depends upon or 
 consists in a certain molecular condition which is destroyed by 
 the violent motion of the molecules at a high temperature. 
 
 Effect of Mechanical Disturbance. — A magnet is weakened by 
 hitting it a number of sharp blows, and a magnetized knitting 
 needle by clamping it in a vise and causing it to vibrate vigorously 
 (Exp.). A magnetized piece of iron wire a foot or so in length 
 is almost completely demagnetized by twisting it once or twice 
 each way. (A small portion at each end may be bent at right 
 angles to the length for convenience in twisting.) {Exp.) 
 
 In such cases a loss of magnetization results from the disturbance 
 of the molecular condition of the magnet by mechanical forces. 
 On the other hand, such disturbances assist magnetic forces in 
 producing magnetization. Thus a number of blows upon a piece 
 of iron when it is near a magnet will cause it to become more 
 strongly magnetized. 
 
 430. Theory of Magnetization. — These facts and others of a 
 similar nature have led to the theory that every molecule of a 
 magnetic substance is a permanent magnet ; and that in an unmag- 
 netized body the poles of these molecular magnets point indis- 
 criminately in all directions (Fig. 262), while in a magnetized 
 
 body the greater 
 number of the mole- 
 
 r^^^^M^^M^i^ 
 
 
 !?,« ^.^ ...^ , cules lie with their 
 
 Fia a63. (IG. 263. 
 
 like poles pointing 
 in the same direction (Fig. 263). According to this theory, 
 the act of magnetizing consists in turning the molecules more 
 or less completely into one particular direction. If all the mole- 
 cules were turned in the same direction, the limit of possible 
 magnetization would be reached. Soft iron is more readily 
 magnetized than steel, because its molecules are more easily turned 
 
The Magnetic Field . 345 
 
 about, and it loses its magnetization more readily for the same 
 reason. This theory may be illustrated by means of a test tube 
 nearly full of steel filings, each particle of which plays the part of a 
 molecule on a greatly magnified scale. When the mass of filings 
 is magnetized, it exhibits polarity and acts as a magnet until 
 shaken up (Lab. Ex.). What constitutes the magnetism of the 
 molecule is not known. 
 
 III. The Magnetic Field 
 
 431. Magnetic Field and Lines of Force. — A magnetic field is 
 any space within which magnetic forces act. The intensity of 
 these forces is often referred to as the intensity of the tnagnetic 
 field. The field of a magnet is most intense near the poles, and 
 decreases rapidly with increasing distance. It really extends 
 indefinitely in every direction ; but at a comparatively short dis- 
 tance it becomes too weak to produce sensible effects. 
 
 When a magnetic needle is placed within the field of a magnet, 
 as at O (Fig. 264), its north pole is attracted by the south pole 
 of the magnet and repelled by the north pole. OB represents 
 the attraction and OA the re- 
 pulsion upon the north pole of 
 the needle when at 0\ and OR 
 their resultant (by the parallelo- 
 gram of forces). Hence the 
 needle behaves as if its north 
 pole were acted upon by a single ' pj^^ , ' 
 
 force in the direction of OR, 
 The component and resultant forces upon the south pole of the 
 needle, when at O^ are respectively equal in magnitude and 
 opposite in direction to those upon the north pole ; hence a 
 short needle placed with its center at O would come to rest 
 with its north pole pointing in the direction OR. 
 
 If the needle is moved constantly in the direction in which its 
 north pole points, it will trace the curved path OCS, the direction 
 
346 Magnetism 
 
 of which at any point is the direction of the resultant magnetic force 
 at that point. Such a line in a magnetic field is called a line of 
 force. Since the resultant forces upon a north and a south pole are 
 in opposite directions along a line of force, we avoid ambiguity by 
 defining the direction of a line of force as the direction of the 
 force acting upon a north pole. In accordance with this defini- 
 tion, the lines of force in the field of a magnet are said to extend 
 fix)m its north to its south pole (not from the south to the north 
 pole). In diagrams the direction of a line of force is often indi- 
 cated by an arrowhead placed on the line. 
 Laboratory Exercise dj, 
 
 432. Lines of Force in Certain Magnetic Fields. — The direction 
 of the lines of force in different parts of a magnetic field can be 
 
 determined by merely observing the 
 direction in which the north pole of a 
 magnetic needle points when moved 
 about in the field ; but fine iron filings 
 may be made to serve the same pur- 
 pose for the whole of a plane section 
 of a field at one time. The filings are 
 sprinkled from a pepper box or other 
 sifter upon a sheet of cardboard or 
 stiff paper placed over the magnet. 
 * A light tapping on the cardboard 
 
 assists the magnetic forces in bringing the filings into definite lines 
 coinciding with lines of force. 
 
 The lines thus obtained about . \ j //v'""^"^Vv • A'V 
 a bar magnet are shown in • \\\\' . l /v; ^rr:::- ^> '.'■'»!/'/// ^ 
 
 «=» — >^r- — > — 15>»» — -> f 
 
 Fig. 265. Beyond a distance ,y;, ' j ; \\.'^ :~~".-s^ y/i]\ v^\ 
 
 .of a few inches the field is 'V i \ 'v^srrirl'Cr' V i \ X'* 
 
 too weak to direct the filings ; ^ 
 
 hence the information that ^'^* ^^' 
 
 they afford is incomplete. All the lines of force extending from 
 the north pole are really continuous with lines of force coming to 
 the south pole (when no other magnet is in the vicinity). 
 

 
 N 
 
 TTT 
 /// 
 
 The Magnetic Field 347 
 
 When the north pole of one magnet is placed near the south pole 
 of another, the filings are 
 arranged in lines as shown in 
 Fig. 266. The lines of force 
 extend across between the i\^ 
 
 unlike poles of the two mag- \ \^ ^ . \ > / ' / , / i 
 
 nets. But when the like poles \ ^. \ \ "'i \ / >^ / / / ' 
 
 of two magnets are turned i i \ '. . .' . . • . 
 
 toward each other (Fig. 267), * 
 
 no lines are found to extend from one to the other ; on the con- 
 trary, the lines in one field turn away from those of the other. 
 
 433. Theory of Magnetic Action. — Magnetic action takes place 
 at a distance apparently without the aid of any medium by means 
 of which the force is exerted upon the distant body. In this 
 respect it is like gravitation ; and, like gravitation, it takes place 
 in a vacuum. But as action at a distance without an intervening 
 medium for its transmission is not considered possible (Art. 129), 
 it is assumed that a medium exists by means of which magnetic 
 forces are exerted, and this medium is thought to be no other than 
 the ether that pervades all space. 
 
 According to this view, the ether surrounding a magnet is under 
 certain stresses (Art. 206) due in some way to the presence of the 
 magnet; and the laws of magnetic action indicate that these 
 stresses consist of a tension along the lines of force and a pressure 
 across them. Thus we have a mental picture of an elastic sub- 
 stance which is in a state of tension between unlike poles, tending 
 to draw them together by contraction along the lines of force, and 
 in a state of compression between like poles, tending to push them 
 apart by expansion at right angles to the lines of force. 
 
 PROBLEMS 
 
 I, (a) When a pole of a strong magnet is brought toward the like pole 
 of a magnetic needle, repulsion may be followed by attraction as the magnet 
 is brought closer. Explain. {U) The same may happen when an end of a 
 weakly magnetized piece of iron is brought toward a needle. Explain. 
 
348 Magnetism 
 
 2. Why should decision as to the polarity of a magnetized body be based 
 on repulsion of the magnetic needle rather than on attraction ? 
 
 3. In what different ways may an unmagnetized magnetic substance be 
 dbtinguished from a magnet ? 
 
 4. Explain the effect of the soft iron bar, called the armature or keeper, 
 which is placed across the ends of a horseshoe magnet, when not in use, to 
 preserve the strength of the magnet. 
 
 5. In what respects does magnetic action resemble gravitation ? In what 
 respects docs it differ ? 
 
 IV. Terrestrial Magnetism 
 
 434. The Earth's Magnetic Field. — Everywhere upon the 
 earth's surface the magnetic needle, when removed from all mag- 
 netic substances, always comes to rest in a definite direction, 
 clearly indicating that it is controlled by a magnetic field. This is 
 the magnetic field of the earth. It varies largely in intensity and 
 in the direction of its lines of force over different portions of the 
 earth's surface ; but the changes are so gradual that the lines of 
 force are sensibly straight and parallel, and the intensity constant 
 over areas many miles in extent. In general, the lines of force 
 extend in a direction several degrees either to the east or west of 
 north, and are more or less inclined to the horizontal. 
 
 The magnetic field of the earth indicates that the earth is an 
 irregularly magnetized body. The cause of its magnetization is not 
 known. 
 
 435. Action of the Earth's Field on a Compass Needle. — How- 
 4Aaa*aaaa*4**a»*4 ever much or little the lines offeree of 
 
 I the earth's field may be inclined, it is 
 
 I only the horizontal component of the 
 
 I magnetic force that exerts a directive 
 
 j action upon magnets that are free to turn 
 
 only in a horizontal plane. Hence the 
 
 Fig. 268. directive action upon compass needles 
 
 is the same as it would be if the lines of force were horizontal, 
 
 as represented in Fig. 268. 
 
 The horizontal forces exerted upon the two poles of a compass 
 
Terrestrial Magnetism 
 
 349 
 
 needle by the earth's magnetic field are equal and opposite, since 
 the field is of uniform intensity and the lines of force are straight. 
 Hence when the needle is not parallel to these forces, they act 
 as a couple to swing the needle into Hne with them (A and B, 
 Fig. 268) ; and the needle is brought to rest in this position by 
 friction after a number of vibrations. 
 
 436. Magnetic Meridians and Declination. — A line extending 
 over the earth and having at every point the direction of the com- 
 pass needle is called a magnetic meridian. In Fig. 269 the 
 
 somewhat irregular heavy lines represent magnetic meridians. 
 The magnetic meridian at any place is sometimes called the mag- 
 netic north-and-south line ; and the angle that it makes with 
 the true north-and-south line is called the magnetic declina- 
 tion or simply the decHnation at the place considered. The curved 
 Hues in Fig. 270 are so drawn that each passes through all points 
 having the same declination. Such lines are called isogenic lines, 
 or lines of equal declination. The arrows in the figure show the 
 direction of the declination, whether east or west. 
 
 Magnetic declination is subject to a number of variations, only 
 one of which exceeds a small fraction of a degree. This variation 
 consists in a continuous change in declination in one direction for 
 about two hundred years, followed by a like change in the oppo- 
 
350 
 
 Magnetism 
 
 Fig, aTa 
 
 site direction. Recorded observations show that such changes 
 
 have amounted to over 35° in certain localities. 
 437. Inclination or Dip; Magnetic Poles of the Earth. — A 
 
 magnetic needle mounted on a horizontal axis through its center 
 of gravity is called a dipping needle (Fig. 
 271). If a dipping needle were unmag- 
 netized, it would be in neutral equilibrium 
 in any position in th6 vertical plane in which 
 it is free to turn ; hence its position in the 
 vertical plane is controlled only by magnetic 
 forces, as is the position of the compass 
 needle in a horizontal plane. Consequently, 
 I- 10. 271. when the axis of a dipping needle is placed 
 
 at right angles to the magnetic meridian, the needle comes to rest 
 
 in the direction of the lines of force of the earth's field. 
 
 The angle between the direction of the dipping needle and the 
 
 horizontal is called the inclination or dip. The irregular lines 
 
 extending across Fig. 272 are lines of equal dip. The line of no 
 
Terrestrial Magnetism 
 
 351 
 
 dip is called the magnetic equator. North of the magnetic equator 
 the north pole of the needle is depressed, and south of it the 
 south pole. 
 
 Arctic explorers have found a place where the dip is 90° ; this 
 is the north magnetic pole of the earth (so called from* its geo- 
 
 FlG. 272. 
 
 graphical position, not from its polarity). It is nearly 1400 miles 
 from the geographical north pole, and is shown in Fig. 269 as 
 the point in the northern hemisphere to which the magnetic 
 meridians converge. The south magnetic pole has never been 
 reached. Strictly speaking, the magnetic poles of the earth are 
 far below the surface. 
 
 438. Intensity of the Earth's Field; Inductive Action. — The 
 earth's magnetic field is much too weak to arrange iron filings in 
 lines or to appreciably modify magnetic action within a few cen- 
 timeters of a strong magnet ; but it is everywhere sufficiently 
 intense to control a magnetic needle and to cause considerable 
 magnetization in iron and steel. This inductive action is best 
 shown by means of a long rod of soft iron (Norway iron). When 
 
352 Magnetism 
 
 such a rod is held so as to point north and south or, better, in the 
 direction of the dipping needle, it will be found to be magnetized 
 with its north pole pointing north. When the rod is reversed, its 
 polarity is also instantly reversed, if the iron is very soft ; other- 
 wise it may be necessary to strike the rod on the end while it is 
 held in position {Exp.). 
 
 Any mass of iron or steel that remains in one position for a 
 long time becomes magnetized. This is especially true of the 
 rails of a track when extending in a northerly and southerly 
 direction. Natural magnets are very probably due to the earth's 
 induction. 
 
 439. Importance of the Earth^s Magnetism. — The importance 
 of the earth's magnetism is due to its directive action on the com- 
 pass needle. This is utilized on land in determining directions 
 in surveying, and on sea in directing the course of vessels. In 
 the use of the compass for either 6f these purposes the declination 
 at the place must be known. At sea this is given by the declina- 
 tion map or chart (Fig. 270) from the known latitude and longi- 
 tude of the vessel. 
 
 " Neither the inventor of the compass nor the exact time of its 
 invention is known. Guyot de Provins, a French poet of the 
 twelfth century, first mentions the use of the magnet in navigation, 
 though it is probable that the Chinese long before this had 
 used it. The ancient navigators, who were unacquainted with the 
 compass, had only the sun or pole star as a guide, and were 
 accordingly compelled to keep constantly in sight of land for fear 
 of steering in a wrong direction when the sky was clouded." — 
 Ganot's Physics. 
 
 The earth's magnetic field also plays an essential part in the 
 use of certain instnmients (galvanometers) in electrical measure- 
 ments, as will be seen later. 
 
CHAPTER XII 
 ELECTRICITY 
 
 440. " Electricity and magnetism are not in themselves forms 
 of energy ; neither are they forms of matter. They may perhaps 
 be provisionally defined as properties or conditions of matter ; but 
 whether this matter be the ordinary matter, or whether it be, on 
 the other hand, that all-pervading ether by which ordinary matter 
 is everywhere surrounded and permeated, is a question which has 
 been under discussion, and which is now held to be settled in 
 favor of the latter view." — Daniell's Principles of Physics, 
 
 I. The Voltaic CeU 
 
 441. Action of Dilute Sulphuric Acid on Zinc and Copper. — 
 When a strip of zinc is placed in dilute sulphuric acid (one part 
 by volume of acid to fifteen or twenty of water), it is acted upon 
 by the acid and is gradually dissolved or eaten away. This 
 action is accompanied by the rapid formation of small bubbles, 
 which adhere to the zinc until detached by the buoyant force of 
 the liquid. The escape of these bubbles at the surface gives the 
 liquid the appearance of boiling (Lab. Ex.). 
 
 » These effects are the result of chemical action. Sulphuric acid 
 is a compound substance, the constituents of which are hydrogen, 
 sulphur, and oxygen. The action of the acid upon the zinc con- 
 sists in the substitution of zinc for tlie hydrogen of the acid, by 
 which a compound (zinc sulphate) consisting of zinc, sulphur, 
 and oxygen, is formed and the hydrogen of the acid set free. The 
 bubbles observed in the experiment are bubbles of hydrogen ; the 
 zinc sulphate remains in solution in the liquid. ,The union of 
 the zinc and^he acid liberates chemical potential energy, which, 
 
 353 
 
354 
 
 Electricity 
 
 under the conditions of the experiment, is transformed into heat, 
 — much as the energy of coal is transformed into heat by union 
 with oxygen in the process of burning. 
 
 When copper is placed in dilute sulphuric acid, no bubbles are 
 formed and the copper does not waste away however long it may 
 remain in the liquid. There is no appreciable chemical action. 
 
 442. The Simple Voltaic Cell. — When a strip of zinc and a 
 strip of copper are in the same vessel of dilute sulphuric acid, but 
 
 are not in contact, the appear- 
 ance of each strip is the same 
 as if the other were not pres- 
 ent — bubbles, of hydrogen 
 form upon the zinc, but not 
 upon the copper. But on 
 connecting the strips by means 
 of a wire soldered to each 
 (Fig. 273), bubbles form upon 
 the copper as well as upon the 
 zinc. There is still, however, 
 no chemical action upon the copper, as is evident from the fact that 
 the copper does not waste away however long the strips may 
 remain connected. 
 
 If the wire connecting the strips be turned so as to extend 
 north and south, a compass needle held close to it, either above 
 or below, will be deflected, indicating the existence of a magnetic 
 field about the wire. This magnetic field indicates that something 
 is happening within and about the wire as the result of its connec- 
 tion with the zinc and copper strips in the acid ; and other effects 
 can be produced which prove conclusively that there is a transfer- 
 ence of energy along the wire (either through the wire or through the 
 ether surrounding it) from one of the metal strips to the other. 
 For example, an electric bell may be rung, a telegraph sounder 
 operated, or a piece of platinum wire heated by making proper 
 connections with the wires attached to the strips. 
 
 These effects and others are attributed to what is known as an 
 
 Fig. 273. 
 
The Voltaic Cell 
 
 355 
 
 electric current flowing through the wire from the copper strip to 
 the zinc, and through the liquid from the zinc to the copper, 
 making thus a complete circuit. We speak of electricity as if it 
 were a fluid, and think of it as flowing through wires and other 
 conductors in much the same way as water flows through pipes ; 
 but it is not known that anything is actually transferred round an 
 electric circuit except energy (of the kind known as electrical 
 energy), nor is it certain in which direction the transference of this 
 energy takes place. It is assumed, however, that the direction of 
 the current is as stated above. There is always a magnetic field 
 about a conductor carrying a current of electricity ; this is some- 
 times called an electro-magtietic field y because it is due to electricity 
 instead of magnetism. The magnetic fields of electric currents 
 serve as the readiest means of detecting and measuring them, and 
 also give rise to most of the industrial applications of electricity. 
 
 The vessel of acid, together with the zinc and copper strips, is 
 called a voltaic cell^ in honor of Alessandro Volta^ an Italian physi- 
 cist, who devised this method of producing an electric current in 
 1800.^ Two or more cells connected together constitute a voltaic 
 or electric battery^. In popular usage, a single cell is commonly 
 called a battery. The copper strip is called \.\\& positive plate, pole, 
 or electrode of the cell, and the zinc strip the negative. In diagrams 
 the signs -f and — are often used to indicate the positive and 
 negative plates respectively. 
 
 Laboratory Exercise 66. 
 
 443. Cause of the Current ; Potential and Electro-motive Force. 
 — In a voltaic cell such as we have described, the energy of the 
 current is derived from the chemical action of sulphuric acid on 
 zinc. How this chemical action causes an electric current is a 
 theoretical question that can hardly be discussed with profit in an 
 elementary course. It should be remembered, however, that while 
 the cell is in action, one of the constituents of sulphuric acid 
 (hydrogen) goes to 'the copper plate, where it accumulates as 
 
 1 The Xerm galvanic is sometimes used instead of voltaic; it is derived from the 
 name of Galvani, an Italian physician, who discovered current electricity in 1786. 
 
356 Electricity 
 
 bubbles of visible size, and that the other part of the acid, consist- 
 ing of sulphur and oxygen in chemical union, goes to the zinc 
 plate, with which it immediately unites, forming zinc sulphate. 
 More or less hydrogen is also liberated at the surface of the zinc, 
 depending upon its condition; but this is not essential to the 
 action of the cell and results only in wasted energy (Art. 447). 
 
 It can be shown experimentally that the zinc and the copper 
 plates differ in respect to a condition known as electrical potential ; 
 and the copper is said to be at a higher potential than the zinc* 
 The difference between the potentials of the plates is the direct 
 cause of the current when the plates are connected by a wire.^ A 
 difference of potential can be maintained between two parts of a 
 circuit by other means than chemical action (by a dynamo, for 
 example) ; but, however this may be brought about, the result is 
 always an electric current from the point at higher to the point at 
 lower potential when the points are connected by any conductor. 
 This behavior of electricity may be compared to the conduction 
 of heat between two points from higher to lower temperature, or 
 to the flow of water in a pipe from a higher to a lower level. 
 
 The agency that moves electflcity from any point to a point at 
 a lower potential is called electro-motive force (often denoted by 
 E. M. F.). " Just as in water pipes a difference of level produces 
 a pressure y and the pressure produces 2iflow as soon as the tap is 
 turned on, so difference of potential produces electro-motive force ^ 
 and electro-motive force sets up a current as soon as the circuit is 
 completed for the electricity to flow through." Electro-motive force 
 and difference of potential are commonly used as equivalent expres- 
 sions ; we shall have no occasion to distinguish between them. It 
 should be noted that electro- motive force is not a force at all, in 
 the proper sense of the word, since it does not act upon matter, 
 but upon electricity. 
 
 • 1 When the zinc and copper plates are connected by a wire, there is a continu- 
 ous fall of potential round tlie circuit in the direction of the current (through the 
 liquid from the zinc to the copper plate as well as through the wire), except in 
 passing from the zinc to the liquid and from the liquid to the copper, at each of 
 which places there is an abrupt rise of potentiaL The potential of the zinc is the 
 lowest in the entire circuit 
 
The Voltaic Cell 
 
 157 
 
 444. Conductors and Nonconductors ; Resistance. — All sub- 
 stances offer greater or less opposition to the passage of electricity 
 through them, and the property thus exhibited is called electrical 
 resistance. The resistance of a good conductor is small, that of a 
 poor conductor large. Substances through which electricity is 
 nearly or wholly unable to pass are called nonconductors or 
 insulators. 
 
 The metals are good conductors compared with other substances, 
 but differ largely among themselves, copper being much the best 
 with the exception of silver. Carbon and dilute acids are the 
 next best conductors, though not nearly so good as metals. The 
 resistance of the liquid in a voltaic cell is often much greater than 
 that of the remainder of the circuit. Silk, india rubber, vulcanite, 
 and glass are some of the best insulators. 
 
 445. The Electric Circuit. — A current of electricity requires a 
 complete circuit, i.e. a path that is continuous from any point 
 back to that point again without retracing any portion of it. The 
 circuit may be made up of any number of substances, and these 
 m'ay be of any size or shape. It is only necessary that all parts 
 of the circuit be of materials capable of conducting electricity and 
 that they be placed in close contact. The circuit is said to be 
 closed when it is complete, open or broken when there is a gap at 
 any point. 
 
 The wire of an electric circuit is either insulated by a non- 
 conducting cover or supported upon insulators (usually of glass) 
 to prevent the escape of the electricity from the path intended for 
 it. For currents of low potential, such as are used in ringing 
 electric bells, a cotton covering affords sufficiently good insula- 
 tion ; for higher potentials the covering is of silk or rubber. In 
 connecting wires for a circuit, the insulation must be removed for 
 a short distance at the ends, and the bare wires fastened together. 
 
 446. Materials used in Voltaic Cells. — An electric current may 
 be obtained with various dilute acids and solutions of different 
 silts, and the plates may be made of any two metals that are 
 unequally acted upon by the liquid. The greater the difference 
 
358 Electricity 
 
 in the chemical action upon the plates, the greater will be their 
 difference of potential, and hence also the greater will be the 
 current. The best results are therefore obtained when the nega- 
 tive plate is made of zinc and the positive plate of copper or 
 carbon ; for zinc is most readily acted upon by the acids and 
 solutions used in different cells, and copper and carbon are not 
 acted uf)on at all. 
 
 447. Local Action upon the Negative Plate. — We have seen 
 that some hydrogen is liberated at the zinc plate when a simple 
 voltaic cell is in action (Art. 443, first paragraph). This is due 
 to the presence of small particles of iron, carbon, or other impuri- 
 ties in commercial zinc. Any such particle on the surface of the 
 zinc and in contact with the liquid acts as a positive pole and 
 forms a minute voltaic cell with the adjacent zinc and liquid. 
 This causes a local or parasitic current at the spot and a continual 
 wasting of the zinc, whether the circuit is open or closed. 
 
 When the zinc used is chemically pure, this local action ^ as it 
 is called, does not occur. The zinc is consumed only when the 
 circuit is closed, and hydrogen bubbles apj)ear only upon the 
 copper. The same result is obtained with a plate of commercial 
 zinc when amalgamated^ i.e. when covered with a coating of 
 mercury. The mercury dissolves a portion of the zinc, forming 
 a pasty amalgam which covers the surface and keeps the acid 
 from contact with the impurities. As pure zinc is expensive, it is 
 more economical to use plates of commercial zinc and keep them 
 amalgamated. Amalgamation is necessary, however, only in the 
 case of cells in which acids are used, as in the chromic acid cell 
 (Art. 449). 
 
 448. Polarization of the Positive Plate. — The current supplied 
 by one or more cells of the type already described rapidly dimin- 
 ishes from the moment the circuit is closed. This may be shown 
 by measuring the current with a galvanometer (a process to be 
 considered later), or by arranging a battery of one or more cells 
 that is just sufficient to ring an electric bell, operate a telegraph 
 sounder, or run a small motor when the circuit is first closed. 
 
The Voltaic Cell 359 
 
 The current will very quickly become too weak to produce these 
 effects. The power of the battery may be restored by removing 
 the positive plates and wiping them thoroughly or by letting the 
 battery remain idle for some minutes {Exp.). 
 
 The weakening of the current is due to the accumulation of 
 hydrogen upon the positive plate. A cell that is in this condition 
 is said to be polarized. After the hydrogen has been wiped off or 
 has had time to escape, the current is as strong as at first. 
 Polarization causes a decrease in the current for two reasons. In 
 the first place, hydrogen is a nonconductor, and hence increases 
 the resistance of the cell by cutting off the current from the por- 
 tion of the surface that it covers. In the second place, the hydro- 
 gen tends to reunite with the other constituents of the acid, just as 
 the zinc does, though less strongly ; and hence it sets up an oppos- 
 ing electro-motive force, which tends to send a current in the 
 opposite direction. Polarization is avoided or diminished in vari- 
 ous ways in different forms of cells, some of the most common of 
 which are described in the following articles.^ 
 
 449. Chromic Acid or Bichromate Cell. — The zinc plate of this 
 cell (Fig. 274) is attached to a rod, by means of 
 which it can be raised from the liquid when the 
 cell is not in use. The positive pole consists of 
 two plates of carbon — one on each side of the 
 zinc — which are connected to the same binding 
 screw at the top. The liquid is dilute sulphuric 
 acid, containing in solution chromic acid or bi- 
 chromate of potassium or of sodium, which acts 
 as a depolarizer. These substances contain oxygen 
 which they give up readily to hydrogen, forming 
 water. The accumulation of hydrogen upon the 
 carbon plates is thus diminished, but not entirely ^^* ^^ 
 
 prevented. Polarization diminishes the current by one third or 
 more in a few minutes. 
 
 1 These descriptions may be most profitably studied in connection with the use 
 of the cells in the laboratory or the class room. Any or all of them may be passed 
 over for the present. 
 
360 Electricity 
 
 The electro-motive force of this cell is large compared with that 
 of most cells. Its resistance is small, for the current has only a 
 ver)' short path in the liquid, and the double carbon pole reduces 
 the resistance further by one half. As a result of the high E. M. F. 
 and low resistance, this cell is capable of supplying an exception- 
 ally strong current, and on this account is much used in experi- 
 mental work. The zinc should be kept thoroughly amalgamated : 
 it must be raised from the liquid when not in use^ for it is attacked 
 by the solution even when the circuit is open. 
 
 450. The Leclanch^ Cell. — The zinc of this cell is usually in the 
 form of a rod (Fig. 275) ; the positive plate is a block of carbon. 
 
 The latter is inclosed in a cylindrical cup 
 of porous earthenware, and is surrounded 
 by small fragments of carbon and man- 
 ganese dioxide, with which the cup is 
 filled. The liquid is a solution of am- 
 monium chloride (sal ammoniac) in water. 
 The zinc is not acted upon when the cir- 
 cuit is open, and does not require amal- 
 gamation. When the circuit is closed, 
 chlorine from the ammonium chloride 
 Fig. 275. unites with the zinc, forming zinc chlo- 
 
 ride and liberating the other constituents of the ammonium chloride 
 (ammonia and hydrogen). The zinc chloride and the ammonia 
 are held in solution ; the hydrogen passes through the porous 
 cup and unites slowly with oxygen from the manganese dioxide, 
 forming water. 
 
 This cell polarizes rapidly, and is suitable only for uses requiring 
 brief action with comparatively long intervals of rest, during which 
 it recovers from polarization. It is much used for ringing electric 
 bells, and has the merit of not requiring attention for months at a 
 time. 
 
 451. The Gravity Cell. — The positive pole of this cell (Fig. 
 276) consists of a number of strips of copper fastened together, 
 and is placed at the bottom of the jar. The zinc is near the top, 
 
The Voltaic Cell 
 
 361 
 
 and is made in various forms. In some cases it is supported by a 
 rod, as in the figure ; in others, it is hung from the edge of the jar. 
 The lower portion of the hquid is a strong 
 solution of copper sulphate (blue stone) ; 
 the upper portion is a weak solution of 
 zinc sulphate/ which, being of less spe- 
 cific gravity than the lower solution, rests 
 upon it without mixing except by the 
 slow process of diffusion. 
 
 While the cell is in use, copper from 
 the copper sulphate is continually de- 
 posited upon the copper plate. By 
 certain actions within the liquids which 
 need not be considered, the other con- 
 stituent of the copper sulphate is set free ' ^ 
 at the zinc, with which it unites, forming zinc sulphate. The solu- 
 tion of copper sulphate is continually renewed from a supply of 
 crystals of copper sulphate placed about the copper plate. When 
 not in use, this cell should be kept on a closed circuit through a 
 considerable resistance (about 20 ohms). 
 The small current then flowing tends to 
 prevent the mixing of the liquids by dif- 
 fusion ; otherwise the copper sulphate, 
 coming in contact with the zinc plate, will 
 deposit copper upon it, and in this con- 
 dition the cell will furnish little or no 
 current. The zinc is not amalgamated. 
 Polarization is entirely avoided in the 
 gravity cell, and the current that it sup- 
 plies through a given circuit is approxi- 
 mately constant. Its E. M. F. is about 
 half that of the chromic acid cell, and its 
 resistance is several times as great ; the 
 largest current obtainable from it is consequently comparatively 
 1 Water containing a very little sulphuric acid may be used in setting up the cell. 
 
 Fig, 277. 
 
362 Electricity 
 
 small. It is especially serviceable in experimental work requir- 
 ing a constant E. M. F., and is much used for purposes requiring 
 a current all or nearly all of the time, as in telegraphy. 
 
 452. The Daniell Cell. — This cell is essentially the same as the 
 gravity cell; but its parts are differently arranged and the two 
 solutions are kept separate by a partition of porous earthenware 
 in the form of a cylindrical cup (Fig. 277). The porous cup con- 
 tains the negative pole (a long bar of zinc) and a dilute solution 
 of zinc sulphate. This is placed in a glass jar containing a satu- 
 rated solution of copper sulphate. The positive plate is a sheet 
 of copper surrounding the porous cup. 
 
 II. The Electro-magnetic Field 
 
 453. Historical. — The first discovery of a definite relation 
 between electricity and magnetism was made by Oersted, a Danish 
 physicist, in 1819, — nineteen years after Volta's invention of the 
 electric battery. He found that, when a wire carrying a current 
 is placed in a horizontal position above a compass needle, the 
 needle is deflected from the magnetic meridian. When the 
 direction of the current is reversed or the wire held below 
 the needle, the deflection is in the opposite direction. 
 
 This is known as " Oersted's experiment." It marks the begin- 
 ning of the science of eUctro-magnetismy — that branch of electrical 
 science which treats of the relations between electricity and mag- 
 netism and which is usefully applied in such inventions as the 
 electric bell, the telegraph, the telephone, the dynamo, and the 
 motor. 
 
 Laboratory Exercise 6/. 
 
 454. Magnetic Field due to a Current in a Straight Conductor. — 
 Iron filings sprinkled on a horizontal piece of cardboard, through 
 which a vertical conductor passes, show circular lines of force 
 about the conductor when a sufficiently strong current is flowing ^ 
 
 1 In studying the magnetic field about a conductor carrying a current, a strong 
 current must be used ; otherwise the field would be too weak to bring iron filings 
 into line, and the behavior of a magnetic needle would be largely modified by the 
 
The Electro-magnetic Field 
 
 363 
 
 Fig. 278. 
 
 (Fig. 278). Since all cross-sections of the field taken at right 
 angles to the conductor are alike, it will be seen that a straight con- 
 ductor carrying a current is surrounded 
 by a cylindrical magnetic field, whose 
 lines of force are circles about the con- 
 ductor as a center and lie in planes at 
 right angles to it. The relation between 
 the direction of the lines of force round 
 the wire {i.e. the direction in which the 
 north pole of a needle points) and the 
 direction of the current is imix)rtant and 
 may easily be remembered from the following right-hand rule: 
 Grasp the wire with the right hand so that the extended thumb 
 points in the direction of the current ; then the fingers point round 
 the wire in the direction of the lines of force. 
 
 455. Field Due to a Current in a Circular Coil. — When a con- 
 ductor is curved, the lines of force are crowded together on the 
 concave side and spread apart on the convex side, the planes of 
 the lines of force being perpendicular to the con- 
 ductor at every point (Fig. 279) ; hence the field 
 is relatively strong at the center of a circular coil, 
 where lines of force round all parts of the coil 
 meet. The field at the center of a circular coil is 
 of special interest and importance, as it is utilized 
 in the simplest form o{ galvanometer m measuring 
 electric currents. It may be conveniently studied 
 by sending a strong current through all the turns 
 (usually fifteen) of a galvanometer coil. Iron filings sprinkled 
 on a piece of cardboard, placed within the coil, will show that the 
 lines of force at the center of the coil are straight and are perpen- 
 
 magnetic field of the earth. A battery of three or four chromic acid cells connected , 
 in parallel (Art. 481) gives a current sufficiently strong for this purpose; but even 
 better results are obtained from a single cell by sending the current round a rec- 
 tangle of insulated wire cohsisting of six or eight turns. The field is strengthened 
 in proportion to the number of parallel wires, but is otherwise the same as if there 
 were only one. 
 
 Fig. 279. 
 
364 Electricity 
 
 dicular to the plane of the coil. Their direction relative to the 
 direction of the current is given by the following right-hand rult 
 for coils : Close the right hand and place it within the coil with the 
 fingers pointing in the direction of the current: then the extended 
 thumb points in the direction of the lines of force through the coil, 
 
 456. The Helix or Solenoid. ^ A long, cylindrical coil of wire 
 is called a helix or solenoid. The successive turns of the coil may 
 be at Some distance apart or may be in contact ; in the latter case 
 insulated wire must be useci^ For some purposes coils are wound 
 upon woo<len spools and consist of several layers of turns. 
 
 The magnetic field about a helix through which a current is 
 passing is similar to that about a short, flat coil, as described in 
 the preceding article; but its shape is somewhat modified (Fig. 
 
 280). Within the coil, the lines 
 of force extend from end to end 
 in approximately straight lines. 
 Without the coil, the field is like 
 that of a bar magnet : the lines 
 of force spread out from one end 
 of the coil and return to the other ; 
 at one end the south pole of a magnetic needle is attracted, at the 
 other the north pole. In fact, a helix through which a current is 
 passing may be regarded as a magnet and as having a north and a 
 south pole. The right-hand rule for coils may be conveniently 
 stated for the helix as follows : Grasp the coil so that the fingers 
 point in the direction of the current; then the ex- 
 tended thumb points in the direction of the north pole 
 of the coil. 
 
 When a helix is suspended or supported horizon- 
 tally in such a way that it is free to turn in a hori- 
 zontal plane while a current is flowing through it, it 
 'comes to rest with its north pole pointing north, like 
 a compass needle {Exp). 
 
 457. The Electro-magnet. — When a soft iron rod is 
 placed within a helix through which a current is floW' fig. «8i.~ 
 
 Fig. a8o. 
 
The Electric Bell and the Telegraph 365 
 
 ing (Fig. 281), it becomes strongly magnetized, its polarity being 
 the same as that of the coil. The coil and iron rod together 
 constitute an electro-magnet. Electro-magnets are much more 
 powerful than steel magnets of equal size, and have the further 
 advantage that their magnetization is under perfect control, for they 
 become demagnetized the in- 
 stant the current is stopped. 
 Electro-magnets are made 
 of different shapes for differ- 
 ent purposes, various modi- 
 fications of the horseshoe 
 form (Fig. 282) being the 
 most generally useful. The 
 
 current flows in opposite di- 
 
 ' ^ Fig. 282. 
 
 rections round the two coils 
 
 of a horseshoe magnet, making one of the free ends of the iron 
 
 core north and the other south according to the rule given above. 
 
 PROBLEMS 
 
 1. Apply the right-hand rule to determine the direction in which the north 
 pole of a compass needle will be deflected when a current in a straight con- 
 ductor flows from north to south directly above it; directly below it j when 
 the current flows from south to north above it ; below it. 
 
 2. What would be the angle between the wire and the direction of the 
 needle in each of the above cases if the field of the earth were negligible in 
 comparison with that of the current ? Why would the angle be less if this 
 were not the case ? 
 
 3. How does the theory of magnetization account for the great strength 
 of electro-magnets ? 
 
 4. Does the current in a helix flow clockwise or anti-clockwise when its 
 north pole points toward the observer ? when the south pole points toward 
 the observer ? 
 
 III. The Electric Bell and the Telegraph 
 
 458. The Electric Bell. — An electric bell (Fig. 283) is rung 
 by the action of an electro-magnet. Different bells are somewhat 
 
366 
 
 Electricity 
 
 differently constructed. In most cases the metal frame of the 
 bell forms a part of the circuit ; but this is merely a matter of 
 »• convenience in construction. The connec- 
 
 tions and insulations must, however, be such 
 that the only path offered the current through 
 the bell is by way of the coils of the electro- 
 magnet and across between the free end of a 
 spring, a, and the end of a screw, <*. The 
 spring s, which carries the armature of the 
 magnet (a soft iron bar), draws it away from 
 the magnet, and, at the same time, presses 
 the spring a against the screw. (The springs 
 s and a are sometimes in one piece, as shown 
 in the figure.) 
 
 When the circuit is closed by pressing a 
 push button, placed at some convenient point 
 in the circuit, the electro-magnet attracts the 
 
 Fig. 383. 
 
 armature, and the clapper attached to it strikes the bell. At 
 the same time the spring is pulled away from the screw at ^, 
 breaking the circuit. With the stopping of the current, the mag- 
 net becomes demagnetized and ceases to attract the armature. 
 The spring s then pulls the armature back, making contact at c 
 again. This process is repeated as long as the button is pushed. 
 Laboratory Exercise j6. 
 
 459. The Electro-magnetic Telegraph. — The earliest and the 
 simplest system of transmitting messages by electricity is the elec- . 
 tro-magnetic telegraph. The credit for this invention belongs 
 principally to Samuel F. B. Morse of New York, who devised the 
 first practical instruments. The first telegraph line was established 
 by Morse in 1844 between Washington and Baltimore. The 
 instruments required for this system of telegraphy are the sounder^ 
 the key, and the relay. 
 
 460. The Sounder. — The sounder (Fig. 284) has an electro- 
 magnet, the poles of which point upward. An armature of soft 
 iron is fixed across a lever just above the poles. Whenever the 
 
The Electric Bell and the Telegraph 367 
 
 current passes through the coils of the electro-magnet, the arma- 
 ture is attracted down, carrying the lever with it, and a screw 
 near the end of the lever 
 makes a click as it strikes 
 against a stop. As soon as 
 the current ceases, the lever 
 is raised by a spring and 
 strikes against the end of a 
 screw. The two clicks sound 
 differently and are easily dis- 
 tinguished. They together 
 constitute a " dot " when one 1-24. 
 
 follows immediately after the other, and a "dash " when the inter- 
 val between them is an appreciable fraction of a second. The 
 letters of the alphabet, the punctuation marks, and the numbers 
 from zero to nine are represented by different numbers of dots, 
 dashes, and various combinations of the two. 
 
 461. The Key. — The key (Fig. 285) is a device by which 
 the operator makes and breaks the circuit in the act of sending a 
 message. It is fastened to a table by two screws, the one shown 
 
 at the left in the figure being 
 insulated from the metal 
 base. One wire of the line 
 is fastened to each screw. 
 There is a small platinum 
 point at the top of the insu- 
 lated screw, and another, /*, 
 just above it ontl)e under 
 side of the lever. When 
 the lever is depressed, these points come in contact, closing the 
 circuit. When the key is not in use, the circuit is kept closed by 
 the switch S, which connects the base of the instrument with the 
 insulated post, as shown in the figure. This switch is moved to 
 the right while a message is being sent, leaving the circuit open 
 and under the control of the operator by means of the lever. 
 
 Fig. 285. 
 
368 Electricity 
 
 462. The Relay. — The high resistance of a long telegraph line 
 makes the current too weak to operate a sounder. This difficulty 
 might be overcome by using a battery of a very great number of 
 
 cells ; but it is more conven- 
 ient and more economical to 
 
 D l^^g l^^^S^L. i9>^^^-* make use of an additional 
 
 instrument called the relay 
 (Fig. 286). The relay acts 
 on the same principle as the 
 Fig. 286. sounder, but the electro- 
 
 magnet is horizontal and the 
 armature vertical. The lever that carries the armature is light 
 and delicately balanced, and hence is easily moved to and fro by 
 much smaller forces than are required to operate the lever of a 
 sounder. 
 
 The coils of the electro- magnet are connected with the line by 
 means of the binding posts A and B. The wires of a local circuit 
 containing the sounder and a battery to operate it are connected 
 to the posts C and D. One of these posts is connected by a wire 
 under the base of the instrument with a metal column, the upper 
 end of which forms an arch above the lever, and the other post is 
 similarly connected with the lever. The local circuit is closed by 
 contact of the lever with the platinum tip of a screw, which it 
 strikes when drawn over by the electro-magnet, and is broken 
 when the magnet ceases to act and the lever is pulled back 
 by a spring against the hard rubber tip of the opposite screw. 
 Thus when the line circuit is closed or opened by means of 
 the key, the local circuit is at the same instant closed or opened 
 by the action of the relay. Since the resistance of the local 
 circuit is small, one or two cells are sufficient to operate the 
 sounder. The sounds made by the lever of the relay are nearly 
 inaudible. 
 
 463. The Telegraph System. — A diagram of a complete tele- 
 graph system connecting two cities is shown in Fig. 287. A key 
 and relay are included in the main line at each station (of which 
 
Electrical Measurements 
 
 369 
 
 there may be any number). There is a line battery^ at each of 
 the terminal stations, each consisting of many cells in series 
 (Art. 480), and the two are so connected to the line that they 
 exert an E. M. F. in the same direction through it. Some form of 
 cell that does not polarize must be used, as the gravity cell. The 
 
 New York 
 Key Sounder^ 
 
 Philadelphia 
 f2!""K Key, 
 
 \JX 
 
 Local Battery 
 
 Relay 
 
 Local Battery 
 
 Earth Y\G. 287. 
 
 line wire is connected with the earth at the terminal stations by 
 «ieans of metal plates sunk in moist ground. The earth com- 
 pletes the circuit, taking the place of a return wire. The sounder 
 at each station is in a local circuit connected with the relay. 
 
 When an operator at any station on the line wishes to send a 
 message, he opens the switch of his key. All other switches on 
 the line must be closed. (Why ?) The operator first calls the 
 station to which he wishes to send the message. The sounders at 
 all the stations deliver the message, but the operator at the station 
 called is the only one who pays attention to it. 
 
 It is possible, by means of different connections and instru- 
 ments of different construction from those here described, to send 
 two or more messages over the same wire at the same time. 
 
 IV. Electrical Measurements 
 
 464. Strength of Electric Currents. — The strength of an electric 
 current can be determined by observing how great an effect of one 
 kind or another it is capable of producing. Heating and chemical 
 
 1 In diagrams each cell of a battery is represented by two parallel lines ( 1 1) ; the 
 short heavy line represents the negative plate, and the long, thin line the positive. 
 
370 Electricity 
 
 effects may be made to serve this purpose ; but the simplest and 
 most convenient method is to use some form of instrument whose 
 action is due to the magnetic field of the current. Such an instru- 
 ment is called a galvanometer. There are various forms of galva- 
 nometers, but all of them depend in their action upon the fact 
 that the strength of an electric current is proportional to the inten- 
 sity of its magnetic field. 
 
 465. Ohm's Law. — Utilizing the magnetic action of the electric 
 current on a magnetic needle, (ieorg Ohm, a German physicist, 
 discovered that the strength of the current in an electric circuit 
 is proportional to the electro- motive force and inversely proportional 
 to the resistance of the circuit. This is known as Ohm's law. It 
 is stated in other terms in Art. 474. 
 
 The relations included in the law may be separately stated as 
 follows : — 
 
 (i) If the E. M. F. and the resistance of a circuit (including 
 the resistance of the battery) remain constant, the current remain^ 
 constant. 
 
 (2) If the resistance remains constant and the E. M. F. varies, 
 the current is proportional to the E. M. F. 
 
 (3) If the E. M. F. remains constant and the resistance varies, 
 the current is inversely proportional to the resistance. For exam- 
 ple, if the resistance is doubled, the current is decreased one half. 
 
 466. The Tangent Galvanometer. — The essential parts of a 
 tangent galvanometer (Fig. 288) are a vertical 
 coil of wire having one or more turns, and a 
 compass with a graduated scale, placed at the 
 center of the coil. The compass needle should 
 be very short in comparison with the diameter 
 of the coil, in order that it may be wholly 
 within the sensibly constant portion of the 
 magnetic field of the current at and near the 
 
 center of the coil, in whatever direction it may 
 Fig. 288. . , . . r y ' 
 
 turn. A long, nonmagnetic pomter of alumi- 
 num is commonly attached to the needle at right angles to it 
 
Electrical Measurements 
 
 371 
 
 Fig. 289. 
 
 (Fig. 289) ; and it is the position of the pointer on the scale 
 that is read. It is evident that the pointer and the needle turn 
 through equal angles. 
 
 At and near the center of the coil the lines 
 of force of the magnetic field of the current 
 are straight and are perpendicular to the plane 
 of the coil (Art. 455). Hence, if the current 
 in the coil were the only source of a magnetic 
 field where the compass needle is placed, the 
 needle would be brought to rest exactly at 
 right angles to the plane of the coil whenever a current of any 
 strength was flowing. The magnetic field of the earth, however, 
 plays a necessary part in the use of the instrument, the behavior 
 of the needle being determined by the relative intensity of the two 
 fields. 
 
 .In using a tangent galvanometer it must be set with the plane 
 of the coil in the magnetic meridian ; in which position the lines of 
 force of the coil are at right angles to those of the earth's field. 
 In Fig. 290, let O denote the position of the north pole of the 
 
 compass needle, ON the intensity and 
 direction of the earth's magnetic field, 
 and OD the intensity and direction 
 of the field of the current. The direc- 
 tion of the resultant of these forces is 
 OAj which is therefore the direction of the resultant force upon the 
 north pole of the needle. Since the resultant force upon the south 
 pole is equal and opposite to that upon the north pole, the needle 
 comes to rest in the line OA, and the deflection of the needle 
 caused by the current is the angle NO A, 
 
 Any increase in the strength of the current through the galva- 
 nometer increases the strength of its field i^OD) proportionally, and 
 hence increases the deflection, but not proportionally. This is 
 evident from the two parts of the figure. The part at the left 
 represents the field of the current equal to that of the earth, and 
 the deflection is consequently 45°. The part at the right shows 
 
 ^f 
 
 A % 
 
 Fig. 290. 
 
372 
 
 Electricity 
 
 V A B 
 
 O J} i' 
 
 Fig. 991. 
 
 the effect of doubling the current (which makes OD twice as 
 great) ; the deflection is increased, but is much less than 90°. In 
 fact, the deflection is never quite 90°, how- 
 ever great the current may be. 
 
 Let C and C denote two currents 
 through the same number of turns of the 
 same galvanometer, and let OD and 0£, 
 or JVA and NB (Fig. 291), denote the 
 intensities of their fields at the center of 
 the .galvanometer. OJV^, as before, is the intensity of the earth's 
 field. Then angles a and a' are the deflections caused by C and 
 C respectively. 
 
 Now the strengths of the currents are proportional to the inten- 
 sities of their magnetic fields (Art. 464), i>. 
 
 C: C::NA:NB, 
 
 Since the value of a ratio is not altered when its terms are 
 divided by the same quantity, we may write the above proportion 
 
 C: C: 
 
 NA , NB 
 ON ' ON 
 
 (I) 
 
 The ratio of one leg of a right triangle to the other is called the 
 
 NA 
 tangent^ of the angle opposite to the first leg. Thus is the 
 
 tangent of angle a (commonly abbreviated to tan tf), and is 
 
 1 The tangent of angle A (Fig. a^a) is DE : AD or PG : AG, DE and FG being 
 any line perpendicular to either side of the given angle. Since triangles ADE and 
 AFG are similar, the ratios DE : AD and EG : 
 AG are equal. It is evident, therefore, that 
 the tangent of an angle is a definite quantity 
 the value of which depends only upon the sire 
 of the angle. Angles are not prop>ortional to 
 their tangents, although small angles are very 
 nearly so. The tangent of any angle from o'' to 
 90^* may be found from a table of tangents 
 (Appendix, Table V). 
 
Electrical Measurements 373 
 
 the tangent of angle a! {tan a'). Hence proportion (i) may be 
 
 written /-> r^t 4. 4. t / \ 
 
 C : C : : tan a : tan a'. (2) 
 
 That is, currents sent through the same number of turns of the 
 coil of a tangent galvanometer are proportional to the tangents of 
 the angles of deflection that they produce. This is why the instru- 
 ment is called a tangent galvanometer. 
 
 Example. — A current C causes a deflection of 50°, and another current C, 
 a deflection of 25*^. It is found from a table of tangents that tan 50° =1.19 
 and tan 25° = .466. 
 
 Hence C: C: : 1.19: .466 ; 
 
 from which C — 2.55 C, 
 
 While a tangent galvanometer having a scale graduated in de- 
 grees may be thus used to determine the relative strengths of 
 currents, it does not give' their numerical values. The numerical 
 value (in amperes) may, however, be obtained by multiplying the 
 tangent of the angle of deflection by a constant factor, found by 
 experiment. 
 
 Laboratory Exercise 68. 
 
 467. Use of Different Numbers of Turns of the Coil. — When 
 equal currents are sent through different numbers of turns of a 
 tangent galvanometer, the tangents of the angles of deflection are 
 proportional to the number of turns used (Lab. Ex. -68). For 
 example, when a current is passed through fifteen turns, the tan- 
 gent of the angle of deflection is three times as great as when the 
 same current is sent through five turns. This is due to the fact 
 that the magnetic field of the coil is proportional to the number 
 of turns through which the current is sent (see note to Art. 454). 
 In measuring currents the most accurate results are obtained when 
 the number of turns used is the one that makes the deflection 
 nearest to 45°. 
 
 PROBLEMS 
 
 1. How would the current from a given battery be affected by making the 
 resistance of the circuit four times as great ? ten times as great ? 
 
 2. Make a diagram showing the deflection of the needle of a tangent 
 
374 Electricity 
 
 galvanometer due to a certain current, and also the deflection due ta 
 currents two, three, four, and five times as great. What change in the 
 increase of the deflection for successive equal increases of the current is 
 shown by the diagram ? 
 
 3. What value docs the deflection approach as the current increases 
 indefmitely ? 
 
 4. Why docs the stronger or weaker magnetization of a galvanometer 
 needle not atfect the deflection ? 
 
 Laboratory Exercise 6g. 
 
 468. Laws of Resistance. — The following laws of electrical 
 resistance have been established by experiment : — 
 
 I. Thf resistance of a conductor of uniform cross-section is pro- 
 portional to its length. 
 
 II. The resistance of a conductor is inversely proportional to the 
 area of its cross-section. The resistance of a wire is, therefore, 
 inversely proportional to the square of its diameter. For example, 
 the resistance of a wire 3 mm. in diameter is one ninth as great as 
 that of a wire of the same material and length and i mm. in 
 diameter. 
 
 IIL The resistance of a conductor depends upon the material of 
 which it is made. The following table gives the relative or specific 
 resistances of a number of substances in the form of wires of equal 
 length and cross- section, the resistance of copper being taken as 
 unity. 
 
 Specific Resistances (referred to copper) 
 
 Silver, annealed 0.94 Iron, telegraph wire . . . 9.4 
 
 Copper, annealed i.oo German silver 13.0 
 
 Aluminum 1. 81 Mercury 59.0 
 
 Iron, pure 6.03 Carbon (arc light) about . . 2500.0 
 
 IV. The resistance of metals and of mqst other substances in- 
 creases as the temperature rises. TJie resistance of carbon^ dilute 
 acidSf and solutions decreases with a rise of temperature. The 
 resistance of nearly all the pure metals increases about 40 per 
 cent with a rise of temperature of 100° C. The resistance of 
 German silver and other alloys is much less affected by change of 
 
Electrical Measurements 
 
 375 
 
 temperature ; hence they are used in making standard resistance 
 coils (Art. 470). The resistance of the carbon filament of an 
 incandescent lamp when hot is only about half what it is when 
 cold. 
 
 V. The resistance of a conductor is the same at the same tem- 
 perature whatever the strength of the current. 
 
 469. The Unit of Resistance. — The common unit of electrical 
 resistance is called the ohm^ in honor of the physicist of that 
 name. It is defined as the resistance of a uniform column of mer- 
 cury 106.3 cm. long and one square millimeter in cross-section, 
 at 0° C. It is very approximately the resistance of 157 ft. of 
 No. 18 copper wire (diameter = 1.024 mm-) or 349 ft. of No. 16 
 (diameter = 1.29 mm.). 
 
 470. Resistance Coils. — Standard coils of known resistance are 
 used in measuring resistances. They are called resistance coils, 
 and a box containing a set 
 of them is called a resist- 
 ance box (Fig. 293). The 
 coils are made of insulated 
 wire, generally of an alloy 
 having a high specific re- 
 sistance j and the ends of 
 each coil are connected to 
 brass blocks, A, By C 
 (Fig. 294), on the top of 
 the box. These blocks are separated a short distance ; but they 
 are connected electrically by means of the coils, and may also 
 
 be connected by inserting brass plugs, which 
 fit snugly into the spaces between them. The 
 resistance of the row of blocks and plugs is 
 practically zero ; but, wherever a plug is re- 
 moved, the resistance of the coil that bridges 
 the gap is introduced into the circuit in which 
 The amount of this resistance is marked 
 When two or more plugs are removed, 
 
 ^93- 
 
 Fig. 294. 
 
 the box is included, 
 on the top of the box. 
 
3/6 Electricity 
 
 the total resistance introduced is the sum of the resistances of 
 the coils that connect the gaps. A set of coils generally includes 
 .1, .2, .3, .4, I, 2, 3, 4, 10, 20, 30, and 40 ohms, and may extend 
 to much higher resistances. Any resistance from .1 ohm to 
 III ohms can be introduced into a circuit with such a box. In 
 finding a required resistance, the coils are tried in order from larger 
 to smaller, as weights are tried in weighing. 
 
 471. Measurement of Resistance. — The resistance of a con- 
 ductor can be measured in a number of ways ; but the method 
 of substitution is the only one that we shall consider. Suppose 
 the resistance of a coil of wire, R (Fig. 295), 
 j\->. is to be found. A constant cell (Art 451) is 
 
 connected with a galvanometer, <?, and the 
 coil is included in the circuit. The deflection 
 "* of the galvanometer is read as accurately as 
 
 * possible. The coil is then removed from the 
 
 circuit and a resistance box put in its place. Different resist- 
 ances are introduced into the circuit, by removing plugs from the 
 box, till the deflection of the galvanometer is exactly the same as 
 with the coil, R. 
 
 The sum of the resistances of the box then included in the 
 circuit is equal to the resistance of the coil, R. The proof of this 
 is as follows : ( i ) The equal deflections indicate equal currents 
 (formula (2), Art. 466). (2) The E. M. F. of the cell is constant ; 
 hence the current is inversely proportional to the resistance of the 
 circuit (Art 465, (3)), and, as the currents are known to be 
 equal, the resistances of the circuits must be equal. (3) Hence 
 the resistances introduced from the box must, be equal to the 
 resistance of R, for which they were substituted. 
 
 Laboratory Exercises yo and yi. 
 
 472. The Unit of Electro-motive Force. — The unit of electro- 
 motive force is called the volt, in honor of Volta (Art. 442). It is 
 defined as a certain fraction of the E. M. F. of a Clark standard 
 cell at 15°. This cell is chosen as the standard because of its 
 constancy. The E. M. F. of a gravity or a Daniell cell is approxi- 
 
Electrical Measurements 377 
 
 mately i volt ; of a chromic acid cell, 2 volts ; of a Leclanch^ 
 cell, 1.4 volts. The E. M. F. of a cell is independent of its size 
 and dimensions, but varies more or less with the condition of the 
 plates and the liquid. 
 
 473. The Unit of Current. — The unit of current is called the 
 ampere J in honor of Andre Ampere, a French physicist, noted for 
 his discoveries in electro-magnetism. It may be defined indepen- 
 dently of the ohm and the volt, in terms of either its magnetic or its 
 chemical effects ; but it will serve our purpose to define it as the 
 current produced by an E. M. F. of one volt in a conductor having 
 a resistance of one ohm. 
 
 474. Ohm's Law. — The units of current, E. M. F., and resist- 
 ance being thus chosen with reference to one another, Ohm's 
 law is expressed by the formula 
 
 c=|. (0 
 
 in which C denotes the current measured in amperes^ E the 
 E. M. F. measured in voltSy and R the resistance of the whole 
 circuit measured in ohms. From the formula, when E and R are 
 each one, C is one ; which agrees with the above definition of the 
 unit current. 
 
 475. Fall of Potential along a Conductor. — It can be shown 
 experimentally that there is a continuous fall 
 
 of potential round a circuit from the positive ^ iL 
 
 pole of the battery to the negative pole, and ^ ' 
 
 that the fall of potential is everywhere pro- 
 portional to the resistance to be overcome. 
 Thus if E^ denote the fall of potential or ^ 
 
 E.M.F. between A and B (Fig. 296) and ^'°* '^ 
 
 E^ the E. M. F. between B and C, and if Ry and R^ denote respec- 
 tively the resistance oi AB and BCy then 
 
 E^:E,:'.R,:R,, or |l = ^. (2) 
 
 Ri Aj 
 
3/8 Electricity 
 
 Let C denote the current (which is the same throughout the 
 circuit), then 
 
 C = | and C = |^ (3) 
 
 These equations express the fact that Ohm's law holds for parts 
 of circuits as well as for entire circuits. 
 
 476. Divided Circuits. — Electric currents often have two or 
 more branches between two points, as between A and B (Fig. 
 
 297). Either of two branches is called a 
 shunt to the other. The sum of the currents 
 in all the branches between two points is 
 equal to the current in the undivided part 
 of the circuit 
 
 In Fig. 297 let ^1 denote the resistance 
 of one branch of the circuit between A and 
 B, and ^, the resistance of the other ; and let Ci and Q denote 
 the currents in these branches respectively. The E. M. F. that 
 maintains the current in each branch is the difference of potential 
 between the points A and B, Let E denote this E. M. F. ; then, 
 by Ohm's law, 
 
 E= C,^i, and E= C^j. 
 Hence, C^R^ = C^R^ or Q: Q:: R^: Ri; (4) 
 
 that is, the currents in the two branches are inversely proportional 
 to the resistances of the branches. For example, if the resistance 
 of one branch is three times that of the other, the current in it will 
 be one third as great as in the other, or it will carry one fourth 
 of the whole current. 
 
 477. Resistance of Conductors in Series and in Parallel. — Two 
 or more conductors are said to be in series when they are con- 
 nected so that the entire current passes through each in succession. 
 Thus, in Fig. 296, AB and BC are in series. The combined 
 resistance of any number of conductors in series is the sum of 
 their separate resistances. 
 
 The conductors constituting the branches of a divided circuit 
 
Electrical Measurements 379 
 
 are said to be connected /// parallel. The resistance between any 
 two points of a circuit, as between A and B of Fig. 297, is always 
 diminislied by adding one or more conductors in parallel between 
 those points; for this is equivalent to replacing the given con- 
 ductor with one of larger cross-section. For example, in the 
 simple case where the branches are of the same material, length, 
 and cross-section, the resistance of three of them in parallel is 
 one third the resistance of one of them, — being the same as the 
 resistance of a single conductor of the same material and length 
 and three times the cross-section (Art. 468, Law II). 
 
 478. Measurement of E. M. F. ; The Voltmeter. — The coils of 
 some galvanometers consist of a great many turns of very fine wire, 
 and have a resistance of hundreds or even thousands of ohms. 
 On account of the large number of turns in the coil a very weak 
 current — often less than a thousandth of an ampere — causes a 
 considerable deflection of the needle. Such instruments are used 
 to determine the difference of potential, or E. M. F., between 
 different points of a circuit, and are called voltmeters. An 
 E. M. F., when measured in volts, is often called voltage. 
 
 To determine the E. M. F. between two points of a circuit, as 
 A and B (Fig. 298), the voltmeter, V^ is 
 connected as a shunt between these points. 
 The resistance of the voltmeter must be very 
 high in comparison with that of the circuit 
 between A and B^ in order that it may not 
 appreciably diminish the resistance of the 
 circuit between these points. This is neces- 
 sary ; for if the resistance of the voltmeter 
 were low, it would lower the resistance between the points to 
 which it was connected, and hence would diminish the E. M. F. 
 between these points in the act of measuring it (Art. 475). 
 
 By Ohm's law, the current through the voltmeter is proportional 
 to the E. M. F. between the points of the circuit to which it is 
 connected. Hence, if it is a tangent instrument, the E. M. F. 
 is proportional to the tangent of the angle of deflection. The 
 
380 Electricity 
 
 scale may be marked in degrees or graduated so as to give the 
 number of volts directly. To find the total E. M. F. of a battery, 
 its poles are connected directly with a voltmeter (Fig. 299). 
 Laboratory Exercise 72. 
 
 479. Arrangement of Cells in a Battery. — The cells of a 
 battery may be connected in three ways, namely: (i) in series ^ 
 (2) in parallel^ (3) in series and parallel combined. Which of 
 these arrangements will give the largest current through a given 
 circuit depends upon the relative resistance of the battery and the 
 remainder of the circuit, as shown in the following articles. In 
 this discussion E denotes the E. M. F. and r the resistance of 
 each cell, n the number of cells, and R the external resistance ; i.e, 
 the resistance of the circuit exclusive of the battery resistance. 
 Ohm's law, when applied to a circuit containing only one cell, 
 would then be represented by the formula 
 
 r-\- R being the total resistance of the circuit. 
 
 480. Cells in Series. — Cells are said to be connected in series 
 when the positive pole of the first is joined to the negative pole 
 of the second, the positive pole of the second to the negative 
 
 I I I I pole of the third, etc. (Fig. 299). The 
 
 r\ 'I ^1 ^K fall of potential in the short wires con- 
 
 V__,^____/T\ J necting the cells is negligible (their re- 
 
 ^-^►^ sistance being very small), hence the 
 
 Fig. 299. negative plate of each cell is at the same 
 
 potential as the positive plate of the preceding cell ; and, since 
 the rise of potential in each cell is E^ the E. M. F. of a battery 
 of n cells in series is nE, The resistance of the battery is the 
 sum of the resistances of the cells, or «r, according to the law 
 for conductors in series (Art. 477). Ohm's law, when applied to 
 a circuit containing a battery of n like cells in series, is therefore 
 represented by the formula 
 
 C=-^' (6) 
 
 nr + R ^ 
 
Electrical Measurements 
 
 381 
 
 When the external resistance, R^ is small in comparison with 
 the battery resistance, nr, the current is only very slightly increased 
 by adding more cells in series, for this increases the resistance of 
 the circuit in nearly the same ratio as it does the E. M. F. ; but, 
 when the external resistance is relatively large, an increase in the 
 number of cells increases the E. M. F. more rapidly than it does 
 the total resistance, and hence increases the current. 
 
 481. Cells in Parallel. — Cells are said to be connected in 
 parallel when the negative plates of all the cells are joined to form 
 the negative pole of the battery and __,^_ 
 the positive plates to form the positive -^ ' ^ ^:- ^-^ © 
 pole (Fig. 300). All the negative 
 plates are at one potential and all 
 
 the positive plates at another, the difference being the E. M. F. 
 of one cell. The resistance of a battery of n cells in parallel is 
 
 I T 
 
 - of the resistance of a single cell, or -, according to the law 
 
 n 
 
 Hence the formula for Ohm's law 
 C=^- (7) 
 
 for conductors in parallel, 
 in this case is 
 
 n 
 When the resistance of a single cell is small compared with the 
 external resistance, the current is only slightly increased by con- 
 necting any number of cells in parallel, for the total resistance 
 is only slightly diminished ; but, when the external resistance is 
 
 relatively small, any de- 
 crease in the battery 
 resistance decreases the 
 total resistance in nearly 
 the same ratio and pro- 
 duces a corresponding 
 increase in the current. 
 482. Cells in Series 
 and Parallel Combined. — In some cases the largest current is 
 obtained by arranging the cells of a battery in groups. The result 
 
 Fig. 301. 
 
 Fig. 302. 
 
382 Electricity 
 
 is the same whether the cells of each group are joined in series, 
 and the groups in parallel (Fig. 301), or the cells of each group 
 in parallel and the groups in series (Fig. 302). The formula for 
 the arrangement shown in either figure is 
 
 3-:+/. 
 
 2 
 
 It can be shown mathematically that, with a given number of 
 cells, the largest current through a given external resistance is 
 obtained when the cells are so connected that the resistance of the 
 battery is as nearly as possible equal to the external resistance. 
 
 Laboratory Exercise jj. 
 
 PROBLEMS 
 
 1. What is the combined resistance of three incandescent lamps in 
 parallel, the resbtance of each lamp being 2CX) ohms? 
 
 2. The fall of potential through a coil of wire is 1.5 volts when a current 
 of .2 ampere is flowing. What is the resistance of the coil ? 
 
 3. What E. M.F. will maintain a current of 1.5 amperes through a re- 
 sistance of 80 ohms? 
 
 4. If the E. M. F. of a chromic acid cell is 2 volts, and its resistance 
 .3 ohm, what current will it supply through an external resistance of .1 ohm? 
 
 5. What would be the current through the same external resistance from 
 a battery of 4 such cells (a) in parallel? {b) in series? 
 
 6. \\'hat current would be supplied by a battery of 12 Leclanche cells, 
 each having an E. M. F. of 14 volts and a resistance of i ohm, through an 
 external resistance of I.5 ohms {a) with the cells connected in series? {b) in 
 parallel? (<-) in three groups of four each, the cells of each group being in 
 series, and the groups connected in parallel? Draw a diagram of the last 
 arrangement 
 
 7. Show that, when the external resistance of a circuit is negligible in 
 comparison with the resistance of a cell, the current is proportional to the 
 number of cells connected in parallel, but a single cell furnishes as large a 
 current as any number of cells connected in series. 
 
 8. Show that, when the battery resistance is negligible in comparison 
 with the external resistance, the current is proportional to the number of 
 cells connected in series, but a single cell furnishes as large a current as any 
 number of cells in parallel. 
 
Electrical Measurements 
 
 383 
 
 483. The Astatic Galvanometer. — This instrument (Fig. 303) 
 is used for detecting and measuring extremely small currents. It 
 has an astatic needle (Fig. 304). This con- 
 sists of two magnetic needles placed parallel, 
 one above the other, with their like poles point- 
 ing in opposite directions. They are rigidly 
 attached to a short, vertical stem, and hence 
 turn together. If they were of exacdy the same 
 strength, they would come to rest indifferently 
 in any position, for the earth's magnetic field 
 would act upon them equally and in opposite 
 directions ; but a very slight directive action is ^'^' 3^* 
 secured by making one needle a little stronger than the other. 
 The needle carries a light, nonmagnetic pointer, and is suspended 
 
 by an untwisted silk fij^er, which en- 
 ables it to respond to the slightest 
 force. 
 
 The coil of the instrument is flat 
 and elongated horizontally. The lower 
 needle swings within the coil, the 
 upper one above it. The magnetic 
 field of the coil tends to turn both needles in tne same direction. 
 (Why?) Thus with the astatic needle the effect of the current is 
 increased while that of the earth's field is decreased, and a very 
 small current causes a considerable deflection. The astatic galva- 
 nometer is therefore called a sensitive instrument. 
 
 484. The d'Arsonval Galvanometer. — This galvanometer (Fig. 
 305) differs in principle from those previously considered. A coil 
 of fine wire, wound on an elongated frame, is suspended so as to 
 swing freely between the poles of a permanent horseslioe magnet. 
 The current is led to and from the coil through wires by means of 
 which it is suspended. When no current is flowing, the suspending 
 wires hold the coil in position with its axis at right angles to the 
 lines of force between the poles of the magnet ; but, with a current 
 flowing, the coil tends to set itself with its north side facing the 
 
 Fig. 304. 
 
384 
 
 Electricity 
 
 Fid. 
 
 305. 
 
 south pole of the magnet (Art. 456). This motion is opposed by 
 the torsion of the wire, and the coil turns through a greater or less 
 angle according to the strength of the current. In some instru- 
 ments the coil is provided with 
 a nonmagnetic pointer, which 
 moves over a scale, as in the 
 figure ; in others the coil carries 
 a small mirror, which indicates 
 the deflection by the angle at 
 which it reflects a beam of light. 
 One important advantage of 
 the d'Arsonval galvanometer is 
 that it is independent of the 
 earth's field, which is negligible 
 in comparison with the strong 
 field of the magnet ; hence the 
 instrument does not need to be turned in any particular direction. 
 The sensitiveness of this galvanometer is increased by decreasing 
 the size of the supporting wires, by increasing the strength of the 
 magnet, or by increasing the number of turns of the coil. A 
 sensitiveness sufficient to detect much less than a millionth of an 
 ampere can be secured. 
 
 485. The Weston Ammeter and Voltmeter. — A galvanometer 
 whose scale is graduated to read in amperes is called an ammeter. 
 Ammeters and voltmeters for indus- 
 trial use are usually of the d'Arsonval 
 type, and resemble each other in 
 appearance. Figure 306 represents 
 a Weston ammeter. The magnet is 
 horizontal, its poles being at the nar- 
 row side of the box, opposite the scale. 
 The magnetic force acting upon the 
 coil is opposed by a coiled spring. 
 The coil carries a long pointer 
 which moves over a scale graduated in amperes. In the ammeter 
 the resistance of the coil is low ; in the voltmeter it is very high. 
 
 Fig. 306. 
 
Heat, Light, and Power 385 
 
 V. Heat, Light, and Power from Electric Currents 
 
 486. Heating Effect of Electric Currents. — When the current 
 from a chromic acid battery is sent through a short piece of fine 
 German silver or platinum wire, the wire becomes white hot ; but 
 the remainder of the circuit, consisting of larger copper wire, is 
 not appreciably warmed (Exp.). 
 
 The greater heating of the German silver or platinum wire is 
 due to its greater resistance. A part of the energy of an electric 
 current is transformed into heat wherever there is resistance to be 
 overcome, whether in the battery or in the external circuit ; and 
 it is found by experiment that the amount of heat thus generated 
 in the different parts of the circuit is directly proportional to their 
 resistances. Heat is generated in a conductor as long as the cur- 
 rent flows ; but the temperature of the conductor ceases to rise as 
 soon as the rate at which it loses heat by conduction and radiation 
 becomes equal to the rate at which the heat is generated. 
 
 487. The Incandescent Lamp. — The heating of electric cur- 
 rents is utilized in electric lighting. In the incan- 
 descent lamp (Fig. 307) the current passes through 
 a slender carbon filament which, on account of its 
 high resistance, is heated to whiteness. The ends 
 of the filament are attached to short platinum 
 wires, which pass through the glass and make con- 
 nection with the circuit by means of the metal fittings 
 at the base of the lamp. The bulb is exhausted to 
 the highest possible degree to prevent combustion 
 of the filament when heated. The common sixteen- 
 candle lamp has a resistance of 160 to 200 ohms yig. 307. 
 when hot, and requires a potential difference of 100 
 
 or no volts between its terminals to maintain the necessary cur- 
 rent, which is about .5 to .6 ampere. The lamps are commonly 
 connected in parallel. 
 
 488. The Electric Arc. — If two pointed pieces of carbon are 
 placed in contact, then separated a short distance while a current 
 
386 
 
 Electricity 
 
 Fig. 308. 
 
 of sufficient strength and voltage is passing between them, the 
 heat generated at the points will vaporize some of the carbon, 
 and the current will continue to flow, being 
 conducted across the gap between the points 
 by the carbon vapor and heated air (Fig. 
 308). The conducting gases form a curved 
 luminous path, which is known as the electric 
 or voltaic arc. The greater part of the light 
 comes from the carbon points, especially from 
 the depression or crater which forms at the 
 end of the positive carbon. 
 
 The light of the electric arc is the most 
 powerful and its temperature the highest that 
 can be artificially produced. The tempera- 
 ture is estimated at from 3500" to 4800° C, 
 and is sufficient to melt the most infusible 
 substances, such as flint and diamond. An Y.. M. F. of 40 to 50 
 volts and a current of from 5 to 10 amperes are required to 
 produce a steady arc light ; and such a light will have from 1000 
 to 2000 candle power. The currents for both arc and incandes- 
 cent lights may be either direct or alternating and are supplied by 
 dynamos. 
 
 Arc lamps are provided with an auto- 
 matic device controlled by electro-mag- 
 nets, by means of which the carbons are 
 brought in contact when the current is 
 not flowing and separated as soon as it 
 is turned on, the action of the electro- 
 magnets being controlled by the current. 
 This device also "feeds" the upper 
 carbon toward the lower as fast as they 
 are consumed. 
 
 489. Joule's Law. — Dr. Joule (Art. 270) investigated the 
 heating effect of electric currents by passing a current through a 
 coil of wire of known resistance in a calorimeter containing water 
 
 Fig. 309, 
 
Heat, Light, and Power 387 
 
 or alcohol (Fig. 309). From the results thus obtained he found 
 that the heat developed in a conductor is proportional {i) fo the 
 resistance of the conductor^ (2) /'^ the time during which the cur- 
 rent flows ^ and (3) to the square of the strength of the current. 
 These relations are expressed in the equation 
 
 H=.2^C^Rt, (8) 
 
 in which H denotes the number of calories generated, C the 
 current in amperes, R the resistance of the conductor in ohms, 
 and / the time in seconds. This equation is known 2isfoule's law. 
 Since by Ohm's law E = CR^ Joule's law may be written 
 
 H=-.2^ECt, (9) 
 
 in which E denotes the difference of potential between the ends 
 of the conductor in which the heat is generated. 
 
 490. Electrical Energy. — 'Hie heat generated in a conductor 
 through which a current is flowing is the equivalent of the energy 
 lost by the current in overcoming the resistance of the conductor.. 
 This loss of energy is accompanied by a fall of potential, but the 
 current is not diminished — the reading of an ammeter is the same 
 at whatever point in the circuit it may be placed. It is evident 
 from this that electrical energy is not electricity ^ any more than the 
 energy of a stream of water is the water itself. Water falls to a 
 lower level, and hence loses potential energy, in turning a mill 
 wheel ; but there is no loss of water in the process. Similarly 
 there is a fall of electrical potential (as may be shown by a volt- 
 meter) whenever an electric current does work, whether it be in 
 overcoming the resistance of the conductor, in running a motor 
 (Art. 505), or in doing chemical work (Art. 511). 
 
 /// all cases the energy lost by the current, or the work done by it, 
 is proportional Jointly (i) to the fall of potential, (2) to the current, 
 and (3) to the time ; i.e. the work done is proportional to the prod- 
 uct ECt, 
 
 491. Power. — The, work done by an electric current in a 
 second is called \is power (Art. 153). Since the work done by 
 a current is proportional to ECt, the work done in a second, 
 
388 Electricity 
 
 or the power, is proportional to ^C; i>. stated more fully, the 
 power of a current utilized in any part of the circuit is pro- 
 portional to the strength of the current and to the fall of potential 
 in that part of the circuit. The electric unit of power is called 
 the watt; it is defined as the power of a current of one ampere 
 when driven by an E. M. F. of one volt. Hence, the power of a 
 current in watts is equal to the product of the volts and the 
 amperes; that is, 
 
 Power ^ EC watts. (10) 
 
 One thousand watts is called a kiloioatt; hence, 
 
 EC 
 
 Power = kilowatts. (11) 
 
 1000 
 
 It can be shown that one horse i)ower (Art. 153) is equal to 746 
 
 watts; hence, ^q 
 
 Power = — horse power. (12) 
 
 746 
 
 Examples. — i. If a iio-volt incandescent lamp takes a current of 
 .5 ampere, the power required to light it is i lo x .5, or 55 watts. A little 
 more than one horse power (770 watts) would be required to light fourteen 
 such lamps. 
 
 2. The power expended in lighting an arc lamp when the difference of 
 potential between the carbons is 45 volts, and the current 8 amperes, is 45 X 8, 
 or 360 watts (a trifle less than half a horse power). 
 
 PROBLEMS 
 
 1. What is the power of a battery that is able to maintain a current of 
 4 amperes through a resistance of 6 ohms? 
 
 2. A current of 5 amperes is passed for one minute through a coil of 
 4 ohms' resistance in a calorimeter containing 200 g. of water. Neglecting 
 the heat absorbed by the calorimeter, what is the rise of temperature of the 
 water? 
 
 3. A i6-candle-power lamp has a resistance of 200 ohms when hot, and is 
 used on iio-volt circuit. WTiat is the cost of running the lamp at the rate of 
 ten cents per kilowatt-hour? (A kilowatt-hour is a power of one kilowatt 
 supplied for one hour.) 
 
 4. An arc lamp takes a current of 7 amperes with an E. M. T. of 50 volts. 
 What is the cost of running the lamp at eight cents per kilowatt-hour? 
 
Induced Currents 
 
 389 
 
 VI. Induced Currents 
 
 310. 
 
 492. Induction of Currents by Magnets. — Electric currents can 
 be generated without chemical action by means of a process called 
 eleciro-7nagnetic induction, or simply induction. Induced currents 
 are generated on a large scale 
 by means of dynamos. The 
 laws of electro-magnetic induc- 
 tion can be studied by means 
 of the apparatus shown in Figs. 
 310 and 312. A coil of insu- 
 lated wire is connected with a 
 sensitive galvanometer (astatic 
 or d'Arsonval) . While a strong 
 magnet is being rapidly inserted 
 into the hollow of the coil or 
 rapidly withdrawn from it, the 
 galvanometer indicates a mo- 
 mentary current ; but there is no current while the magnet remains 
 at rest within the coil. Inserting the north pole of the mag- 
 net causes a deflection of the needle in one direction ; insert- 
 ing the south pole, a deflection in the opposite direction. When 
 either pole is withdrawn, the direction of the current is opposite 
 to that produced by inserting the same pole. 
 
 The induced current is called direct if its direction round the 
 coil is such that like poles of the coil and the magnet point in the 
 same direction ; inverse^ if their like poles point in opposite direc- 
 tions. The direction of the current round the coil can be deter- 
 mined from its connection with the galvanometer and the direction 
 in which the needle is deflected. It will be found that the current 
 is inverse when either pole of the magnet is inserted and direct when 
 it is luithdrawn ( Fig. 311). 
 
 Laboratory Exercise ^4. 
 
 493. Source of the Energy of the Induced Current. — It will be 
 seen from Fig. 311 that, whether the magnet is being inserted or 
 
390 
 
 Electricity 
 
 Fia 311. 
 
 removed, its motion is opposed by the magnetic field of the induced 
 current. Thus on account of the induced current more work must 
 be done in moving the magnet either 
 into or out of the coil than would other- 
 wise be required. This additional work 
 is the source of the energy of the induced 
 current. The transformation of mechan- 
 ical energy into the electrical energy of 
 the induced current is supposed to be 
 effected through the medium of the 
 ether. It takes place on a large scale in 
 the generation of currents by dynamos. 
 494. Induction of Currents by Currents. — The same effects are 
 produced when a second coil, in which a strong current is flow- 
 ing, is used instead of the magnet in the preceding experiments 
 (Fig. 312). The coil that takes the place of the magnet is called 
 the primaiy coil^ the 
 other the secondary 
 eoil. The primary 
 coil is connected 
 with a battery, and 
 its inductive action, 
 like that of a mag- 
 net, is due to its 
 magnetic field. The 
 inverse induced cur- 
 rent, which is caused 
 by inserting the pri- 
 mary coil into the 
 secondary, is oppo- 
 site in direction to 
 the primary current ; the direct induced current, which is caused 
 by withdrawing the primary coil, is in the same direction as the 
 primary current. 
 
 If the primary circuit is closed and broken while the primary 
 
 Fig. 312. 
 
Induced Currents 391 
 
 coil remains at rest within the secondary, the effects are respec- 
 tively the same as when the primary coil is inserted and with- 
 drawn with the circuit closed. In either case the induced current 
 is due to the change in the magnetic field within the secondary 
 coil. 
 
 The induced currents are in all cases much stronger when the 
 primary coil contains a soft iron core. This is because the iron 
 greatly increases the strength of the magnetic field. 
 
 495. Laws of Electro- magnetic Induction. — The following laws 
 of electro-magnetic induction have been established by experiment. 
 The pupil should consider to what extent the^ are illustrated by 
 the experiments previously discussed and by subsequent experi- 
 ments with induction coils. 
 
 I. An increase in the strength of the magnetic field within a 
 closed coil induces an inverse current^ and a decrease in the strength 
 of the field induces a direct current. 
 
 It will be helpful to remember that the direction of an inverse 
 induced current is anti-clockwise round the coil to an observer 
 looking in the direction of the lines of force of the magnetic 
 field that causes the induction, and that the direction of a direct 
 induced current is clockwise, when viewed in the same manner 
 
 (Fig. 311)- 
 
 II. The induced E.M.F. (i.e. the E.M.F. of the induced 
 current^ is proportional to the rate of increase or decrease of the 
 magnetic field within the coil, and also to the number of turns in 
 the coil. 
 
 The effect of the rate of change of the magnetic field is readily 
 shown by inserting and removing the primary coil at different 
 rates of speed with an iron core inserted. The deflection is con- 
 siderable when the motion is rapid, but may be indefinitely 
 decreased by moving the coil more and more slowly {Exp.). 
 
 III. The ratio of the induced E. M. F. to the E. M. F. of the 
 current in the primary coil is very nearly equal to the ratio of the 
 number of turns in the secondary coil to the number of turns in 
 the primary. 
 
392 Electricity 
 
 Thus if there are 150 turns in the primary coil and 30,000 in 
 the secondary, the E. M. F. of the induced current is approxi- 
 mately 200 times as great as that of the primary current. This 
 principle is applied in the generation of high potential currents 
 by means of induction coils (Art. 498). An induced current of 
 lower potential than the primary is obtained when the number 
 of turns in the secondary coil is less than that in the primary. 
 This principle is utilized in the transformer (Art. 507). 
 
 IV. There is no gain of energy in electro-magnetic induction. 
 The energy of the induced current is derived either from 
 mechanical work, 'as in the experiments with the coil and the 
 magnet and in the generation of currents by dynamos, or from 
 a current in another circuit (without transference of electricity 
 between the circuits), as in making and breaking the circuit 
 through the primary coil when at rest in the secondary and in 
 the action of induction coils and transformers. 
 
 496. Historical. — Induced currents were discovered in 1831 
 by Michael Faraday, one of the greatest of English physicists. 
 Knowing that electric currents act- upon magnets, he conducted 
 a series of experiments extending over seven years in the attempt 
 to discover any action of magnets upon currents, and was at last 
 rewarded by the discovery of electro-magnetic induction. Tyndall 
 pronounces this the greatest experimental result ever obtained. 
 Its importance can hardly be overestimated, since the action of 
 the dynamo depends upon induction, and the currents for all 
 industrial applications of electricity on a large scale are generated 
 by dynamos. 
 
 497. Self-induction. — Whenever a current commences or ceases 
 in a coil, the current in each turn exerts an inductive action upon 
 all the turns of the coil. This action of a current upon itself is 
 called self-induction^ and the current due to it is often called the 
 extra current. Self-induction is very considerable if the coil has 
 many turns, especially if it contains an iron core. When the cur- 
 rent is turned on in a coil, the growth of its own magnetic field 
 has the same effect as if a magnet were suddenly thrust into the 
 
Induced Currents 
 
 393 
 
 coil. The inverse E. M. F. thus induced opposes the current, 
 preventing an immediate rise to its full value. When the circuit 
 is broken, the effect of self-induction is the same as if a magnet 
 were suddenly withdrawn from the coil. The induced E. M. F. 
 in this case is direct. It is generally many times greater than 
 the original or pri- 
 mary E. M. F., and 
 continues the cur- 
 rent across the air 
 gap where the cir- 
 cuit is broken, caus- 
 ing a spark. 
 
 This effect of self- 
 induction can be 
 shown as follows : 
 A file is connected 
 to one pole of a 
 chromic acid bat- 
 tery and a piece of wire to the other. As the free end of this 
 wire is drawn over the file, the circuit is rapidly closed and broken, 
 and at each break a minute spark occurs. When the experiment 
 is repeated with the coils of a large electro-magnet included in the 
 circuit (Fig. 313), brilliant sparks are obtained, indicating a high 
 induced E. M. F. at each break {Exp.). 
 
 498. The Induction Coil. — The indue fion or Ruhmkorff coil 
 (Fig. 314) is an instrument for generating induced currents of 
 very high potential. A simplified diagram of the essential parts 
 is shown in Fig. 315. These are an iron core, AB \ a primary 
 coil consisting of one or two layers of turns of large insulated wire ; 
 a secondary coil of very fine wire, well insulated and often many 
 miles in length ; an automatic make-and-break device, or current 
 interrupter, CD, which is included in the primary circuit ; and a 
 condenser, E. There is generally also a device, called a confinu- 
 tator, for reversing the current through the primary coil without 
 changing the battery connections ; but this is not essential. 
 
 Fig. 313. 
 
394 
 
 Electricity 
 
 When a current from a battery is sent through the primary coil, 
 it magnetizes the iron core, and the core attracts the iron block 
 
 I'iG. 314. 
 
 C, which IS supported near the end of the core upon a spring. 
 This spring is the movable part of the interrupter, and the pri- 
 mary current passes between it and the point of a screw, D, against 
 which it rests. By the attraction of the magnetized core the 
 spring is drawn away from the 
 point, breaking the circuit. The 
 core instantly becomes demag- 
 netized, and the spring flies back 
 again, closing the circuit. The 
 primary circuit is thus closed and 
 broken many times every second, 
 causing alternately an inverse 
 and a direct induced E. M. F. in 
 the secondary coil. The ends ^'^- 3^5- 
 
 of the secondary coil are connected with the binding posts R and 
 G^ and may be extended by means of rods or wires attached to 
 the posts until the gap H is made as small as desired. When 
 this gap is not too great, an electric spark passes between the 
 terminals with every interruption of the primary current. The 
 
Induced Currents 395 
 
 length of the spark that can be obtained increases with the in- 
 duced E. M. F., and varies in different coils from a few milli- 
 meters to 30 cm. or more for very powerful coils. It is estimated 
 that an E. M. F. of 30,000 volts is required to cause a spark 
 across a distance of i cm. in air under atmospheric pressure and 
 at ordinary temperatures ; and that, under the same conditions, 
 an E. M. F. of at least 300 or 400 volts is required to start a spark, 
 however short. 
 
 The E. M. F. of the induced current increases with the number 
 of turns in the secondary coil and with the rate of change of the 
 magnetic field (Art. 495, Law II). The latter depends upon the 
 rate of increase or decrease of the current in the primary coil, both 
 of which are retarded by self-induction. The stopping of the 
 primary current is, however, much more abrupt than the starting 
 and induces a correspondingly greater E. M. F. In general, it is 
 only at the " break " of the primary circuit that a spark occurs 
 between the terminals of the secondary coil. The purpose of the 
 condenser is to prevent or at least diminish the spark at the inter- 
 rupter, and thus increase the abruptness of the " break." This it 
 does by serving as a temporary reservoir into which the extra cur- 
 rent flows instead of jumping across the gap. (The construction 
 and action of a condenser are discussed in Art. 524.) 
 
 499. Effects of the Induced Current. — While the E. M. F. of 
 the induced current is enormously higher than that of the bat- 
 tery current through the primary coil, its strength is exceedingly 
 small. The induced current may be compared to water flowing 
 through a very small pipe under very great pressure, and the pri- 
 mary current to water flowing through a large pipe under very little 
 pressure. 
 
 Although the energy of the induced current is necessarily less 
 than the total energy of the primary current, the induced current 
 is nevertheless capable of producing effects that are impossible 
 with the primary current. For example, a spark 3 or 4 cm. long 
 is capable of piercing a sheet of cardboard or thin pieces of other 
 nonconductors {Exp.). The induced current is also capable of 
 
396 
 
 Electricity 
 
 producing a shock and other physiological effects. These are the 
 well-known effects of the physician's battery, which is a small 
 induction coil operated by a voltaic cell. The handles which are 
 held by the patient are the terminals of the secondary coil. The 
 current from a powerful induction coil is very painful and may even 
 be dangerous. 
 
 The induction coil is also used for obtaining discharges in 
 rarefied gases inclosed in glass tubes and bulbs (Art. 526). 
 
 VII. The Dynamo and the Motor 
 
 500. The Principle of the Dynamo. — Suppose a single loop of 
 wire to be mounted on an axis at right angles to the lines of force 
 of a strong magnetic field as shown in Fig 316. When the 
 
 loop is vertical, the portion of the 
 magnetic field extending through it 
 in the direction of the lines of force 
 is as great as possible. As the loop 
 is turned from li.is position, the cross- 
 section of the field extending through 
 it (always taken at right angles to 
 Fig. 316. jj^g Ijj^gg Qf force) decreases, and be- 
 
 comes zero when the loop is horizontal. When the magnetic 
 field is thus Removed from within the loop, the inductive action 
 is the same as if it were removed in any other way. With 
 the direction of rotation indicated by the arrow, the induced 
 current is clockwise, looking in the direction of the lines of 
 force. As the rotation is continued in the same direction from 
 the horizontal to the vertical position again, an increasing cross- 
 section of the magnetic field extends through the coil. This 
 induces an anti-clockwise current, looking in the same direction as 
 before ; but, as the opposite face of the loop is now turned toward 
 the observer, the direction of the current round the loop is the 
 same as before. As the loop passes the vertical, the current is 
 reversed in it, and continues thus till it passes through the vertical 
 
The Dynamo and the Motor 397 
 
 again. (Why?) Thus a continuous rotation of the loop induces 
 an alternating current in it, the reversal of the current taking place 
 twice during each rotation, as the loop passes through the position 
 at right angles to the lines of force. 
 
 With a constant rate of rotation, the cross-section of the mag- 
 netic field extending through the loop changes most rapidly when 
 the loop is in .the neighborhood of the horizontal position, and 
 least rapidly in the neighborhood of the vertical position ; hence 
 the induced E. M. F. is greatest in the horizontal position and least 
 in the vertical. In fact, it diminishes to zero in the latter position 
 at the instant it reverses in direction. The induced current passes 
 through like changes.^ 
 
 The current thus generated in the loop can be sent through an 
 external circuit by means of either device shown in Figs. 317 
 and 318. In the first case each end of the loop is connected 
 
 Fig. 317. 
 
 Fig. 318. 
 
 with a copper ring surrounding the axis. The terminals of the 
 external circuit are connected with copper strips, called brushes^ 
 which press against the rings and make sliding contact with them 
 as they rotate. The current in the external circuit reverses with 
 every reversal in the loop. This is the principle of the alternating- 
 current dynamo. In the second case the ends of the loop are 
 connected with the two halves of a split copper tube. This device 
 is called a commutator. The brushes are adjusted so that the 
 contact of each changes from one segment of the commutator 
 
 1 If the lines of force of the field were vertical, it would be necessary to read hori- 
 zontal for vertical, and v/V<f versa, in the above discussion ; for the inductive action 
 depends upon the relative positions and motion of the loop and the lines of force. 
 
39^ Electricity 
 
 to the other when the current reverses in the loop; hence the 
 current in the external circuit is direct, i.e. it always flows in the 
 same direction. This is the principle of the direct-current dynamo. 
 501. Dynamo-electric Machines. — The E. M. F. induced in a 
 coil by rotating it in a magnetic Jield is proportional ( i ) to the 
 number of turns in the coil, (2) to the strength of the magnetic 
 field, and (3) to the rate of rotation. It is by the application of 
 these principles that electrical power is developed on a large scale 
 by means of dynamos. The current is generated in a series of 
 coils of wire, called the armature, which, in the more common 
 form of dynamo, is rotated at high speed between the poles of a 
 powerful electro- magnet, called the field magnet. In some large 
 dynamos the field magnet has several poles ; and powerful alter- 
 nating-current dynamos are sometimes made with stationary 
 armatures and revolving field magnets. In all cases the funda- 
 mental principle is the same, — an alternating current is gener- 
 ated in the coils of the armature by a rapid, periodic change of 
 the magnetic field within them. 
 
 Fig. 319. 
 
 502. The Direct-current Dynamo. — The Armature. — The coils 
 of an armature are wound upon a core of soft iron, which greatly 
 
The Dynamo and the Motor 399 
 
 intensifies the magnetic field within them. This core is generally 
 in the form of a ring or a long cylinder. An armature with a ring 
 core is called a ring armatiwe ; one with a cylindrical core is 
 called a drum armature. The dynamo shown in Fig. 319 has a 
 drum armature. The wire is wound in a number of coils, which 
 extend lengthwise and across the 
 ends of the core. The plan of a 
 four-coil armature is shown in Fig. 
 320. In practice the number of 
 coils is much greater than this. The 
 coils are connected in series and lie 
 in different positions about the axis 
 of the core ; as a result of which the 
 induced E. M. F. is practically con- 
 stant. The ends of the coils are also yw,. 320. 
 connected to the segments of the 
 
 commutator as shown in the figure. As the segments are insu- 
 lated from each other, the current generated in any coil can reach 
 the positive brush only by passing in series through all the coils 
 between it and the one whose commutator segment is at that 
 instant in contact with the brush. 
 
 The Field Magnet. — While a dynamo is in operation, the field 
 magnet must be strongly magnetized. This is accomplished in a 
 direct-current dynamo by sending a part or all of the current 
 generated by it through the coils of the field magnet. In either 
 case the energy consumed in the coils of the magnet is only a 
 very small fraction of the whole. In starting a dynamo, the 
 current is at first very weak, since the field magnet retains only 
 slight traces of magnetization; but this small current flowing 
 through the coils of the magnet strengthens it, causing a stronger 
 current. This mutual action continues until the magnet becomes 
 fully magnetized. 
 
 503. Transformation of Energy in a Dynamo. — We have seen 
 that, when a coil in which a current is flowing is placed in a 
 magnetic field, the magnetic forces tend to turn it so that its 
 
400 
 
 Electricity 
 
 north end (or side) will face in the direction of the lines of force 
 of the field, as a magnet would do (Arts. 456 and 484). It 
 follows from this, as the pupil may prove with the aid of Fig. 316, 
 that, as a coil is rotated in a magnetic field, the magnetic forces 
 acting upon the coil in consequence of the induced current flow- 
 ing in it tend to turn it in the opposite direction. It can be 
 shown that the magnetization induced in the core of the arma- 
 ture by the current also opix)ses the rotation by which the current 
 is induced. 
 
 As a result of these opposing magnetic forces, a much greater 
 expenditure of energy is required to rotate an armature while a 
 current is being generated than is necessary when the circuit is 
 open. An armature that requires 10 horse power to run it while 
 generating a current can be run at equal speed on open circuit by 
 a fraction of a horse power.' The additional energy delivered 
 to the dynamo when the circuit is closed is transformed by the 
 opposing electro- magnetic forces into the energy of the induced 
 current. 
 
 504. The Direct-current Motor. — When a current from some 
 
 other source is sent through the coils 
 of the field magnet and the arma- 
 ture of a dynamo in the same direc- 
 tion as the current that the dynamo 
 itself would generate, the magnetic 
 forces acting upon the armature 
 cause it to turn in the opposite 
 direction to that which would gen- 
 erate the current. The machine 
 then runs as a motor. The rotation 
 is in the same direction both as 
 a motor and as a dynamo if the 
 current is sent through the coils 
 Fig. 321. of the field-magnet in the same 
 
 1 The difference in the power required on ctosed and on open circuit is very 
 noticeable, even with a small hand-power dynamo (Exp.). 
 
The Dynamo and the Motor 401 
 
 direction in both cases and through the armature in opposite 
 directions. Any direct-current dynamo will run as a motor ; 
 but, for practical reasons, motors are made slightly different from 
 dynamos. 
 
 Figure 321 represents a simplified diagram of a small motor hav- 
 ing a three-coil armature. The current divides at the positive 
 brush, part going through one coil of the armature in one direc- 
 tion and part through the other two coils in the opposite direction. 
 This makes one outer pole of the armature north and the other 
 two south, or vice versa, according to the position of the armature. 
 By change of contact between the commutator segments and the 
 brushes, the current reverses in each coil as it passes through the 
 horizontal position on either side, the direction of the current in 
 the coils being always such that the attractions and repulsions 
 between the poles of the field-magnet and the poles of the arma- 
 ture all act in the same direction round the axis, causing rotation. 
 
 Laboratory Exercise 75. 
 
 505. Transformation of Energy in the Motor. — When the 
 current from a battery is sent through a small motor, a galva- 
 nometer (^, Fig. 322) placed in the circuit will show that the cur- 
 rent is much smaller when the motor is running than it is when 
 the armature is held stationary. By permit- 
 ting the motor to run at different speeds, it 
 will be found that the current decreases as 
 the speed increases. At the same time a 
 voltmeter (K in the figure) will show that 
 the fall of potential in the armature increases 
 as the speed increases {Exp.). 
 
 The fall of potential in the armature when '^" ^"' 
 
 at rest is no more than is necessary to send the given current 
 through the resistance of the coils (Art. 475). When the armature 
 is rotating, an additional fall of potential is required to overcome 
 an opposing E. M.F. induced in the armature in consequence of 
 its rotation ; for the machine, while running as a motor, tends as 
 a dynamo to generate a current in the opposite direction. This 
 
402 
 
 Electricity 
 
 counter E. M. F. increases with the speed, and is the cause of the 
 decrease of the current. 
 
 If the armature of a motor is not permitted to turn, the energy 
 lost by the current in the motor is all converted into heat in 
 overcoming the resistance of the coils, as in any other conductor 
 (Art. 490) ; but when a motor is running, part of the electrical 
 energy — generally 80 to 90 per cent of it — is expended in over- 
 coming the counter E. M. F. in the armature. It is this energy 
 that is transformed into mechanical energy by the motor in doing 
 work; and the rate at which the work is done is measured in 
 watts by the product of the current and the counter E. M. F. 
 
 506. Transmission of Electrical Energy. — In transmitting elec- 
 trical energy over a line for use at a distance, a portion of it is 
 lost as heat. This loss is due to the resistance of the conductor, 
 and limits the distance to which it is practicable to transmit power 
 by electricity. The conditions affecting the amount of power that 
 is lost in transmission are as follows : Let E^ denote the E. M. F. 
 of the current at the start and E^ its E. M. F. at the farther end of 
 the line, C the strength of the current, and R the resistance of the 
 line. The power generated is ^jC watts, and the power delivered 
 for use E2C watts (Art. 491). The fall of potential in the line is 
 Ex — Ei volts, and the power lost in transmission {E^ — E^) C or 
 C'R watts. Numerical results obtained from these formulas with 
 assumed values for E^t C, and R are given for comparison in the 
 following table : — 
 
 Given : — 
 
 z 
 
 3 
 
 3 
 
 4 
 
 E. M. F. delivered {E^, 
 
 500 volts 
 
 500 volts 
 
 1000 volts 
 
 5000 volts 
 
 Current (C), 
 
 10 amperes 
 
 20 amperes 
 
 10 amperes 
 
 2 amperes 
 
 Resistance of line {/d). 
 
 .25 ohms 
 
 25 ohms 
 
 25 ohms 
 
 25 ohms 
 
 Then : — 
 
 
 
 
 
 Power delivered i^E^C), 
 
 5000 watts 
 
 xo,ooo watts 
 
 10,000 watts 
 
 10,000 watts 
 
 Power lost {C^R), 
 
 2500 watts 
 
 10,000 watts 
 
 2500 watts 
 
 100 watts 
 
 Power generated {E^C), 
 
 7500 watts 
 
 20,000 watts 
 
 12,500 watts 
 
 10,100 watts 
 
 Fraction of power lost, 
 
 33-3% 
 
 50% 
 
 20% 
 
 1% (nearly) 
 
 E.M.F. at the start (£■!), 
 
 750 volts 
 
 1000 volts 
 
 1250 volts 
 
 5050 volts 
 
The Dynamo and the Motor 403 
 
 It will be seen from the formulas and from the numerical 
 examples that the power delivered over a given line can be 
 increased by increasing either the current or the voltage at which 
 it is delivered, but that the former greatly increases the relative 
 amount of power lost, while the latter decreases it. Hence the 
 economical transmission of electrical energy over long distances 
 requires currents of high potential. 
 
 In recent years electrical power stations have been established 
 on a large scale at Niagara Falls, at various points in the Sierra,, 
 Nevada Mountains, and elsewhere, where water power is utilized in 
 running dynamos of 3000 to 5000 horse power. The currents 
 generated at these stations are transmitted at an electrical pressure 
 of 40,000 to 60,000 volts to surrounding cities, in some cases to a 
 distance of 200 miles. Power thus obtained is rapidly taking the 
 place of the steam engine in manufacturing, mining, and trans- 
 portation. 
 
 507. The Transformer. — The high-potential currents that are 
 used for transmitting electrical power over long distances are not 
 adapted to any of the ordinary uses of electrical power, and, 
 besides, are extremely dangerous. The potential must therefore 
 be reduced without unnecessary loss of power at or near the place 
 
 Alternator 
 
 Loir Pressure ^fains 
 lAimps 
 
 Higli Pressure Mains 
 
 Fig. 323. 
 
 Lamps 
 
 where the power is to be used. With alternating currents this 
 can be accomplished by means of the transformer (Fig. 323). 
 This instrument is merely a reversed induction coil, consisting of 
 a primary coil of many turns and a secondary coil of fewer turns, 
 wound on the same core of soft iron. The high-potential current 
 is sent through the primary coil, and, by its rapid reversals (from 
 
404 Electricity 
 
 50 to 120 per second), induces an alternating current in the sec- 
 ondary coil. If there are - as many turns in the secondary coil as 
 there are in the primary, the E. M. F. of the induced current will 
 
 be approximately - as great as that of the primary, and its strength 
 n 
 
 n times as great (Art. 495, Law III). The loss of power in the 
 transformation is small. The motors or lamps to be supplied are 
 connected in the circuit of the secondary coil. 
 
 Alternating-current dynamos and motors are not discussed in 
 this book. They differ in some important respects from direct- 
 current machines. 
 
 PROBLEMS 
 
 1. Could the current generatefl l)y a dynamo be used to run a motor while 
 the motor was running the dynamo? 
 
 2. What relation to the law of the conservation of energy has the fact 
 that more power is requireil to run a dynamo on closed circuit than on open 
 circuit? 
 
 3. A direct-current dynamo delivers a current of 50 amperes at a pressure 
 of 500 volts to a line whose resistance is 2 ohms. Find (//) the power gener- 
 ated, (*) the power lost in the line, and (<•) the power delivered. (</) What 
 per cent of the power is lost in the line? 
 
 4. What per cent of the power would be lost in the line if the dynamo 
 generated 25 amperes at a pressure of looo volts? 
 
 VIII. The Telephone and the Microphone 
 
 508. The Telephone. — The instrument known as the receiver 
 of a telephone (Fig. 324) was invented by Graham Bell in 1876, 
 and was used by him both as a receiver and as a transmitter. It 
 contains a permanent bar magnet, A^ one end of which is sur- 
 rounded by a coil, B^ of fine copper wire. This coil is connected 
 with the binding posts at the end of the handle. A disk of very 
 thin sheet iron, EE, extends across the receiver at a distance of 
 about a millimeter from the end of the magnet. This disk is 
 forced into vibration by the sound waves that beat upon it when 
 the instrument is held before the mouth in speaking. The mag- 
 
The Telephone and the Microphone 405 
 
 Fio. 324. 
 
 net is strengthened as the disk moves toward it in vibrating, and 
 weakened as the disk recedes. This induces alternating currents 
 in the coil, as in the armature 
 of a dynamo. 
 
 The current thus gener- 
 ated flows through the coil 
 of tlie receiver at the other 
 end of the line, alternately 
 increasing and decreasing 
 the strength of the magnet as it flows first in one direction, then 
 in the other round the coil. When the magnet is strengthened, 
 it draws the disk more strongly; when it is weakened, the disk 
 springs back. The disk of the receiving telephone thus re- 
 l)eats the movements imparted by the sound waves to the disk 
 of the transmitting telephone ; and, by its vibration, reproduces 
 tlie sound in pitch, relative intensity, and quality with remarkable 
 accuracy. 
 
 A telephone line of this sort does not require a battery. It 
 works successfully over short distances ; but over a long line the 
 current is too weak to reproduce the sound with sufficient inten- 
 sity. The Bell telephone has continued in use as a receiver, 
 but has been superseded as a transmitter by a special form of 
 microphone. 
 
 509. The Microphone. — This instrument, as its name implies, 
 
 increases the intensity of faint 
 sounds. A common form of 
 microphone is shown in Fig. 325. 
 The pointed ends of a stick of 
 carbon rest loosely in cavities in 
 two horizontal supports, also of 
 carbon. These supports are 
 fixed to a small sounding board, 
 and are connected by means of 
 binding posts with a telephone 
 receiver and a battery. 
 
 Fig. 325. 
 
4o6 
 
 Electricity 
 
 Vibrations of the sounding board, due either to the presence of 
 a sounding body or to any slight disturbance of the instrument, 
 produce rapid changes of pressure at the points of contact of the 
 carbons with one another. The resistance at these points of 
 contact decreases when the pressure increases, and vice versa^ 
 producing an unsteady current in the circuit. This causes the 
 telephone receiver to emit sounds which are similar to the original 
 sounds, but generally very much louder. 
 
 510. The Transmitter. — The Bell telephone has been super- 
 seded as a transmitter by different forms of microphones. Figure 
 326 represents the Blake transmitter. Back of the 
 mouthpiece there is a thin disk, ^/, which is set in 
 vibration by the voice. The back of the disk touches 
 a platinum point attached to the end of a light 
 spring, /. A block of carbon, k, at the end of a 
 stiffer spring, g, also presses lightly against the plati- 
 num point, completing a local battery circuit through 
 the springs and the primary of a small induction 
 coil (Fig. 327). The receiver and the secondary of 
 the induction coil are included in the line circuit 
 which connects with the central station. 
 
 The vibration of the disk causes alternate increase and decrease 
 of pressure and corresponding changes of resistance at the point 
 of contact of the platinum and 
 the carbon. The changes thus 
 produced in the battery current 
 cause induced currents in the 
 secondary of the induction coil, 
 and these induced currents cause 
 the receiver at the other end of 
 the line to reproduce the message. 
 
 OtEarik 
 
 Fk;. 327. 
 
 The lever upon which the receiver of a telephone is hung when 
 not in use is pulled down by the weight of the receiver. When 
 the lever is in this position, the battery circuit is broken and the 
 call-bell is connected with the line. When the receiver is taken 
 
Chemical Effects of Currents 
 
 4»7 
 
 down, the lever is raised by a spring. This automatically discon- 
 nects the bell from the line, connects the line with the receiver 
 and the secondary of the induction coil, and closes the battery 
 circuit through the transmitter. The telephone is then ready for 
 use either in sending or receiving a message. 
 Laboratoiy Exercise 77. 
 
 IX. Chemical Effects of Currents* 
 
 511. Electrolysis. — By the chemical actions that take place 
 in a battery, chemical potential energy is liberated and converted 
 into electrical energy. The reverse transformation is also pos- 
 sible ; that is, an electric current can produce chemical changes 
 by which the energy of the current is stored up as chemical poten- 
 tial energy. For example, the energy of a battery current is 
 generally derived from the action of sulphuric acid on zinc, by 
 which zinc sulphate is formed (Art. 443). Conversely, zinc can 
 be separated from zinc sulphate by means 
 
 of an electric current, as in the following - — i,^^ r:— 
 
 experiment. A bent tube (Fig. 328) is 
 partly filled with a solution of zinc sul- 
 phate. A narrow strip of platinum, sol- 
 dered to a wire, is placed in the liquid in 
 one arm of the tube and connected with 
 the positive pole of a battery of two or 
 three chromic acid cells in series ; and 
 an iron or a copper wire is inserted in 
 the other arm of the tube and connected with the other pole of the 
 battery. While the current is flowing, bubbles continue to rise 
 from the platinum terminal ; and, after a few minutes, it will be 
 found that the iron or copper wire is covered with a layer of zinc. 
 The current in passing through the liquid decomposes the zinc 
 sulphate and carries the zinc with it to the negative terminal. 
 
 Fig. 328. 
 
 1 If the teacher wishes to discuss the theory of electrolysis, this section should 
 be preceded by the one on electrostatics. 
 
4o8 Electricity 
 
 The other part of the sulphate (consisting of sulphur and oxygen) 
 goes in the opposite direction and appears at the positive terminal. 
 Here it unites with hydrogen from the water, forming sulphuric 
 acid and setting oxygen free. This oxygen forms the bubbles that 
 rise from the strip of platinum {Exp.), 
 
 This experiment illustrates the decomposition of a chemical 
 compound by the passage of an electric current through it. The 
 process is called eUctrolysis, the substance decomposed is called 
 an eUctrolyU^ and the vessel in which the process is carried out 
 is called an electrolytic cell. The terminal by which the current 
 enters the liquid is called the positive electrode, or anode ; the one 
 by which it leaves the liquid is the negative electrode, or cathode. 
 
 If an anode of zinc be substituted for the platinum in the above 
 experiment, the compound of sulphur and oxygen that comes to 
 it wll unite with it, forming more zinc sulphate. The anode will 
 lose zinc and the cathode will gain it at the same rate, and the 
 strength of the solution will remain constant. Similar results are 
 obtained with copper electrodes and a solution of copper sulphate. 
 The anode wastes away and an equal weight of copper is deposited 
 upon the cathode. Water can be electrolyzed in a cell containing 
 dilute sulphuric acid and platinum electrodes. The acid takes part 
 in the chemical changes ; but it is only the water that is consumed, 
 hydrogen appearing at the cathode and oxygen at the anode. 
 
 512. Industrial Applications of Electrolysis. — "The earliest 
 industrial application of electrolysis was in electrotyping and elec- 
 troplating. These operations consist in depositing a thin film of 
 metal upon a surface. They are fundamentally the same, though 
 copper is the only metal used for producing electrotypes. 
 
 Electrotyping. — " Electrotypes are exact reproductions of the 
 original objects. The process of electrotyping is substantially 
 as follows : The page of type, or the woodcut, is first reproduced 
 in wax or plaster. This exact impression is next covered with 
 powdered graphite to make it conduct electricity. The coated 
 mold is then suspended as the cathode in an acid solution of 
 copper sulphate ; the anode is a plate or bar of copper. When 
 
Chemical Effects of Currents 
 
 409 
 
 the current is passed, electrolysis occurs ; copper is dissolved from 
 the anode and deposited upon the mold in a film of any desired 
 thickness. The exact copper copy is stripped from the mold, 
 backed with metal, and mounted on a wooden block, and used 
 instead of the type or woodcut itself. By this process exact copies 
 of expensive wood engravings can be cheaply reproduced, and 
 type can be saved from the wear and tear of printing. Most books, 
 magazines, and newspapers are now printed from electrotypes. 
 
 Electroplating. — "The process of electroplating differs from 
 electrotyping in only one essential ; viz., in electroplating the 
 deposited film is not removed from the object. The object to be 
 plated is carefully cleaned and made the cathode in a solution of 
 some salt of the metal to be deposited ; the anode is a bar or 
 
 Fig. 329. 
 
 plate of the same metal (Fig. 329). When the current passes 
 through the system, the metal is firmly deposited upon the object. 
 The electrolysis would take place, of course, if any anode were 
 present ; but anodes of the metal to be deposited are usually used 
 to prevent the solution or ' bath ' from weakening. They accom- 
 plish the purpose by replenishing the solution with metal as fast 
 as it is removed and deposited upon the cathode.. Silver, nickel, 
 and gold are the metals generally used in electroplating. 
 
 Electro-metallurgy. — " It is only within the last ten or fifteen 
 years that the electric current has been profitably applied to many 
 
4IO 
 
 Electricity 
 
 industries. But during this time the development of electro-chem- 
 istry has been very marked. The largest of these industries is the 
 refining of copper. The process is similar to that described 
 under electrotyping. Other metals, such as gold, silver, and lead, 
 are extracted from their ores and purified by electricity, though 
 the older processes are still used. All the aluminum, magnesium, 
 and sodium of commerce are now manufactured by passing 
 an electric current through their fused compounds." — Newell's 
 Descriptive Chemistry, 
 
 613. The Secondary or Storage Cell. — When a current is sent 
 through an electrolytic cell containing dilute sulphuric acid and 
 
 lead electrodes, the 
 electrolysisof the liquid 
 liberates oxygen at the 
 anode and hydrogen at 
 the cathode. The hy- 
 drogen gathers in small 
 bubbles and escapes. 
 The oxygen, or a part 
 of it, combines with 
 the anode, forming a 
 brownish layer of lead 
 peroxide (PbOg) upon 
 its surface. When the 
 anode is in this con- 
 dition, the electrolytic 
 cell is said to be 
 charged^ and is itself 
 capable of generating 
 an electric current. 
 This may be shown 
 by disconnecting the 
 electrodes from the battery and connecting them with a galva- 
 nometer {Exp.). It will be found that the direction of the 
 current generated by the cell is opposite to the current by which 
 
 Fig. 330. 
 
Electrostatics 411 
 
 it is charged ; hence the positive plate of the cell is the one 
 that receives the deposit of peroxide. 
 
 This experiment illustrates the principle of the secondary or 
 storage cell. If the electrodes are several inches square, the cur- 
 rent will probably be sufficient to ring an electric bell or run a 
 small motor ; but only for a moment. While the cell is generat- 
 ing a current, hydrogen goes to the positive plate and, uniting 
 with oxygen from the peroxide, reduces it to monoxide (PbO). 
 At the same time, oxygen is liberated at the negative plate and 
 forms a layer of monoxide upon it. When the plates have thus 
 been brought to the same condition, the cell is exhausted. It can 
 be charged again by sending a battery current through it, as before. 
 
 The plates of storage cells are commonly cast in the form of a 
 grating or grid, the holes of which are filled with a paste of an 
 oxide of lead. This greatly increases the capacity of the cell ; 
 and the capacity is further increased by the use of a number of 
 plates in each cell, positive plates alternating with negative plates, 
 as shown in Fig. 330. A charging current from a dynamo peroxi- 
 dizes the positive plates and reduces the oxide of the negative 
 plates to spongy lead. A charged secondary cell, like an ordinary 
 or primary cell, possesses a store of chemical potential energy. 
 The important advantage of secondary cells is that the materials 
 composing them can be used over and over again indefinitely. 
 Electric light and power stations are operated more economically 
 when equipped with large storage batteries, since they can be 
 charged when the dynamos are available for the purpose and used 
 when needed. 
 
 X. Electrostatics 
 
 614. Static and Current Electricity. — In the preceding sections 
 we have had to do only with electricity in motion, or ctirrent elec- 
 tricity. Under certain conditions electricity accumulates upon the 
 surface of bodies and remains at rest there for some time. In this 
 condition it is 0,2^^^ static electricity ; and the study of the phenom- 
 ena to which it gives rise is called electrostatics. 
 
412 Electricity 
 
 Electrostatics is the older branch of electrical science, and hence 
 possesses great historical interest. The experiments are also inter- 
 esting and curious to an exceptional degree. Static electricity is, 
 however, of but little importance compared with current electricity. 
 
 515. Electrification by Friction. — When a nonconductor or 
 an insulated conductor is mbbed with another substance, it acquires 
 the power of attracting bits of paper or pith, feathers, and other 
 light bodies (Exp.). The strongest effects are obtained by rub- 
 bing a vulcanite rod with fur or flannel, the attractive power in this 
 case being sufficient to rotate a slender wooden stick balanced on 
 a vertical pivot {Exp.). Sealing wax rubbed with furor flannel 
 and glass rubbed with silk generally give good results ; but in all 
 cases the substance must be dry, and the drier the atmosphere, the 
 better. 
 
 It was known to the ancient Greeks that amber possesses this 
 power of attraction when rubbed ; and it is from the Greek word 
 for amber, eUktron^ that the word eleciriciiy was derived as the 
 name of the agent to which these attractions are due. The process 
 of imparting a charge of electricity to a body, whether by friction 
 or other means, is called eUctrification^ and the body is said to be 
 in a state of eUctrificatioriy or to be eUctrified. 
 
 Static electricity differs from current electricity as a lake differs 
 from a running stream. When a charge of electricity is given the 
 opportunity of escaping through a conductor, it does so as an 
 electric current, and is capable of producing the usual effects of a 
 current in proportion to its strength and duration ; but the quantity 
 of electricity in even a large static charge is excessively small com- 
 pared with the quantity that a single voltaic cell can supply in a 
 fraction of a second. 
 
 516. Positive and Negative Electrification ; Attraction and 
 Repulsion. — An electrified vulcanite rod suspended in a wire 
 stirrup by a thread (Fig. 331) turns away when another electrified 
 vulcanite rod is brought near it, showing that it is repelled. Two 
 electrified glass rods also exhibit repulsion when tested in the same 
 way ; but a vulcanite rod rubbed with fur and a glass rod rubbed 
 
Electrostatics 4 1 3 
 
 with silk attract each other {Exp.). The action of electrified 
 bodies is, of course, mutual, — they attract each other equally or 
 repel each other equally (Newton's third 
 law) . 
 
 Any electrified body would repel either 
 the electrified glass or the electrified 
 vulcanite and would attract the other; 
 hence there are two states or conditions 
 of electrification. Electrification like 
 that of glass when rubbed with silk is 
 
 called positive electrification ; electrification like that of vulcanite 
 when rubbed with fur or flannel is called negative. Two bodies 
 repel each other if both are positively or both negatively electrified; 
 they attract each other if oppositely electrified. 
 
 Different theories have been advanced concerning the nature of 
 electricity and the difference between positive and negative elec- 
 trification ; but such questions are still undecided and are not con- 
 si lered in this book. It should be noted that the terms positive 
 and negative electrification refer to unlike electrical states or con- 
 ditions^ and do not imply that there are two kinds of electricity. 
 Experiment shows that, when oppositely electrified bodies are 
 connected by a conductor, the direction of the current, as the term 
 is used in current electricity (Art. 442, third paragraph), is from 
 the positively electrified to the negatively electrified body. 
 
 517. Electrification by Contact with an Electrified Body. — The 
 bits of paper or other light bodies that cling to an electrified rod 
 often dart away after contact (Exp.). The explanation of this 
 behavior is that, by contact with the rod, the pieces of paper 
 receive a portion of its charge and are then repelled, since they 
 are in the same state of electrification as the rod. Since the rod 
 is a nonconductor, it parts with its charge very slowly, even at 
 points of actual contact with the pieces of paper. This explains 
 why some of the pieces are not repelled. 
 
 518. Electrostaticlnduction. — When an electrified rod is brought 
 near an unelectrified pith ball suspended by a silk thread, the ball 
 
414 Electricity 
 
 is attracted and drawn far out from the vertical, although it is not 
 permitted to touch the rod {Exp.). The cause of the attraction is 
 illustrated in Fig. 332, assuming the charge of the rod to be 
 negative. The pith ball is perfectly insulated by 
 the silk thread, but is itself a conductor. Now 
 it can be shown by experiment that the presence 
 of an electrified body near an unelectrified, insu- 
 lated conductor causes a charge of the opposite 
 sign (unlike charge) on the nearer side of the 
 Fig. 33a. conductor, and an equal charge of the same sign 
 on its farther side. This action is called electro- 
 static induction^ and the resulting charges are called induced 
 charges. The nearer side of the pith ball is attracted by the rod, 
 their charges being unlike ; the farther side of it is repelled, but 
 with a less force, since the distance is greater. Hence the resultant 
 force is an attraction. 
 
 It is instructive to compare electrostatic induction with mag- 
 netic induction. Recall the fact that a piece of soft iron is 
 attracted by a magnet because it is first magnetized by the pres- 
 ence of the magnet. The electric attraction of an unelectrified 
 body is always due to induction. If the attracted body is not 
 insulated, the repelled charge escapes, as shown in later experi- 
 ments. 
 
 519. Permanent Electrification by Induction. — The equal and 
 opposite charges induced on the pith ball in the preceding experi- 
 ment are temporary; />. they continue only so long as the pres- 
 ence of the inducing charge keeps them separated. But if the 
 ball be touched by the finger while a negatively electrified rod is 
 held near it, the negative induced charge escapes, leaving only a 
 positive charge on the ball. This remains as a permanent induced 
 charge when the finger is removed before the rod is. The pres- 
 ence of a positive charge on the ball is proved if the ball is 
 repelled on bringing up an electrified glass rod. 
 
 It will be seen that a permanent charge obtained by induction 
 is unlike the inducing charge. 
 
Electrostatics 415 
 
 520. The Electroscope. — The suspenled pith ball can be used 
 as an electroscope to detect the presence of a charge upon any 
 body and to determine its sign. The pith ball is given a charge 
 of known sign, and the body to be tested is brought near it. If the 
 ball is repelled, the body is electrified, and its charge is like that 
 of the ball. The attraction of the ball is not a certain test of 
 electrification. (Why not?) 
 
 521. The Electric Field. — Electrostatic forces, like magnetic 
 forces, are supposed to act through the medium of the ether. The 
 two forces are of a different character, however, and are wholly 
 unrelated; for static electricity and magnetism have no effect 
 upon each other. The terms electric field and lines of electric force 
 correspond respectively to magnetic field and lines of magnetic 
 force. 
 
 522. Electrical Machines. — Machines for developing and col- 
 lecting charges of electricity are of two types ; namely, friction 
 machines and induction machines. 
 
 Friction Machines. — Figure 333 represents one form of friction 
 machine. A positive charge is developed on a large revolving 
 glass disk, A^ by the 
 friction of leather pads, 
 B. The charge is col- 
 lected on each side by 
 
 a number of points ^ ^^i^^n^jj^^ /\ Iw 
 
 which project from a 
 brass rod and nearly 
 touch the disk. The 
 rods carry the charge 
 to an insulated brass yig. 333. 
 
 cylinder, C, from which 
 
 it can be drawn off as a spark discharge by bringing the finger or 
 any other conductor near it. A spark a centimeter or more in 
 length can be obtained in this manner from a machine in good 
 condition. 
 
 Friction machines of various forms were invented during the 
 
4i6 
 
 Electricity 
 
 eighteenth century; but they are greatly inferior to the more 
 
 modern induction machines, by which they have been superseded. 
 
 Induction Machines, — The elecirophorus (Fig. 334) is the 
 
 simplest induction machine. It consists of a disk of vulcanite 
 
 or other resinous material, and 
 a metal disk of slightly smaller 
 diameter, provided with an insu- 
 lating handle. The vulcanite is 
 negatively electrified by striking 
 or rubbing it with catskin or 
 flannel, and the metal disk or 
 cover is then placed upon it. The 
 disks really touch at only a few 
 points ; and, as the vulcanite is a 
 nonconductor, it does not lose any appreciable part of its charge 
 to the cover. The entire charge, however, acts inductively on the 
 cover, producing a positive charge on its lower side and a nega- 
 tive charge on its upper side (Fig. 335, A). The negative charge 
 is repelled by the charge on the vulcanite, and is permitted to 
 escape by touching the cover with the finger. The positive 
 charge remains, being " bound " by the attraction of the charge 
 on the vulcanite (Fig. 335, B). This leaves the disk positively 
 
 Fig. 334, 
 
 cs 
 
 Fig. 335. 
 
 electrified when it is lifted from the vulcanite by means of llie 
 handle. It can then be discharged by bringing the finger or other 
 conductor near it, as its charge is no longer bound. The cover 
 can be repeatedly charged and discharged in this manner without 
 again rubbing the vulcanite. 
 
Electrostatics 
 
 417 
 
 Different forms of induction machines have been invented 
 which are wholly automatic in their action, and are much more 
 powerful than the electrophorus. One of these — the Toepkr-Holtz 
 machine — is shown in Fig. 336. The movable parts are carried 
 on a large glass disk. Positive and negative charges are induced 
 on different parts of the revolving disk as they pass certain points. 
 
 These charges are collected by projecting metallic points, as in 
 the friction machine, and accumulate on insulated conductors 
 until the difference between their potentials is sufficient to cause 
 a spark discharge across the gap between the knobs. Sparks from 
 5 cm. to 10 cm. long can be obtained from machines of moder- 
 ate size. Full descriptions of these machines are to be found in 
 larger works. 
 
41 8 Electricity 
 
 523. Potential of SUtic Electricity. — The same diffcience of 
 potential is required to produce a spark of given length whether it 
 is obtained from an electrified body or an induction coil. Hence 
 we know from the lengths of the sparks that the potentirls due to 
 electrification are often from tens of thousands to hundreds of 
 thousands of volts. A potential difference of 100,000 to 300,000 
 volts can be obtained with a Toepler-Holtz machine of medium 
 size. In fact, the potential to which it is possible to charge 
 the machine or any body, depends only upon the insulation and 
 the dryness of the atmosphere. Beyond this limit the charge 
 escapes as rapidly as it is developed or imparted. Charges at 
 even the highest potentials mentioned are not dangerous unless 
 the quantity of the charge is much larger than is generally the case. 
 
 An electrical machine is capable of furnishing small quantities 
 of electricity in the form of an intermittent current of very high 
 E. M. F., like the current produced by an induction coil (Art. 499). 
 Most experiments requiring a high-potentiaU current can there- 
 fore be performed either with af machine or a coil. 
 
 524. The Leyden Jar. — The Leyden jar (Fig. 337) is a device 
 for accumulating and storing a large charge, either positive or 
 
 negative. It received its name from the city of Leyden 
 in the Netherlands, where it was invented in 1746. 
 It consists of a glass jar coated inside and out for 
 about two thirds its height with tin foil. A brass rod, 
 terminating in a knob at the top, extends through the 
 stopper, and is connected with the inner coat of the jar 
 by means of a chain attached to its lower end. The 
 jar is charged by connecting the rod or the knob with 
 one of the terminals of an electrical machine. To dis- 
 ' charge it, one end of a short conductor is touched to 
 
 its outer coat and the other end brought near the knob. When 
 it is sufficiently near, a spark occurs, discharging the jar. The 
 discharging conductor must be provided with an insulating handle 
 to avoid a shock which, with a large jar, would be very painful and 
 possibly dangerous. 
 
Electrostatics 419 
 
 The action of the jar is illustrated by the following experiments : 
 (i) The jar is placed on the table and charged, by means of an 
 induction machine, for a certain length of time or until the handle 
 of the mafchine has made a certain number of revolutions. The 
 jar is then discharged, and the length and intensity of the spark 
 noted. (2) The jar is charged for the same length of time as 
 before, while standing on an insulator (a large sheet of vulcanite 
 or glass), and again discharged. The spark obtained with this 
 discharge fs much thinner and less brilliant than before, indicating 
 that the quantity of electricity in the charge is much less (^Exp.).. 
 The explanation of this difference is as follows : While the charge 
 is accumulating on the inner coat of the jar in the first experi- 
 ment, it attracts an opposite charge to the outer coat by induction 
 through the glass. In the accumulation of this charge the table 
 serves as a conductor. In the second experiment the insulation 
 of the jar prevents an induced charge on the outer coat. When 
 the outer coat of the jar is not insulated, the charge that is induced 
 on it attracts the charge on the inner coat. This attraction de- 
 creases the potential of the latter charge, and consequently increases 
 the rate at which the charging takes place. It follows that the 
 attraction of the induced charge increases the amount of the 
 charge on the inner coat for a given potential ; or, in other 
 words, the induced charge increases the capacity of the condenser. 
 
 The extent to which the capacity of a jar is thus increased is 
 perhaps best shown by the action of the jars of an induction 
 machine (Fig. ^2>^)' When the machine is operated, the charges 
 accumulate principally in the jars, the positive charge in one, the 
 negative in the other. The outer coats of the jars become oppo- 
 sitely charged by induction, each receiving its charge from the 
 other through a metal conductor by which they are connected 
 under the base of the instrument. Under these conditions, the 
 machine gives a thick, brilliant spark at intervals of several 
 seconds. When the outer coats of the jars are disconnected 
 by opening a switch (not shown in the figure), the sparks are 
 thin and much less brilliant than before, and, at the same time, 
 
420 Electricity 
 
 are much more frequent {Exp.), The quantity of electricity that 
 is discharged with each spark is evidently very much less than 
 before, indicating a corresponding decrease in the capacity of the 
 jars. This is due to the fact that the outer coats of the jars do 
 not become charged, the wooden base of the machine being 
 practically an insulator for such rapid action. 
 
 The Leyden jar is one form of condenseKf the essential parts 
 of a condenser being two conductors very near each other, but 
 separated by an insulator. A sheet of glass, mica, or paraffined 
 paper, with a smaller sheet of tinfoil attached to each side, leav- 
 ing a wide margin of the insulator round the foil, is a simple form 
 of condenser. 
 
 525. Distribution of an Electrical Charge ; Effect of Points. — 
 It is shown by experiments not to be described here that an elec- 
 trical charge resides wholly upon the surface of a solid body, and 
 only upon the outer surface of a hollow body unless there is an 
 opposite charge upon another body inside it. This is due to the 
 repulsion of all parts of a charge for every other part. 
 
 If the charged body is a conductor, this self-repulsion of the 
 charge causes a definite distribution of it, which depends only upon 
 the shape of the conductor (assuming that there are no other 
 charges in the vicinity to cause induction). A charge is distrib- 
 uted uniformly over the surface of a sphere ; upon other bodies 
 the quantity per unit of surface, or the electric density, is greater 
 where the curvature is greater, and is greatest at edges, corners, 
 and especially at points. 
 
 " At a point, indeed, the density of the collected electricity may 
 be so great as to electrify the neighboring particles of air, which 
 then are repelled, thus producing a continual loss of charge. For 
 this reason points and sharp edges are always avoided on electrical 
 apparatus, except where it is specially desired to set up a discharge. 
 The effect of points in discharging electricity from the surface of a 
 conductor may be readily proved by numerous experiments. If an 
 electrical machine be in good working order, and capable of giving, 
 say, sparks four inches long when the knuckle is presented to the 
 
Electrostatics 
 
 421 
 
 Fig. 338. 
 
 knob, it will be found that, on fastening a fine-pointed needle to the 
 
 conductor, it discharges the electricity so effectually at its point that 
 
 only the shortest sparks can 
 
 be drawn at the knob, while a 
 
 fine jet or brush of pale blue 
 
 light will appear at the point. 
 
 If a lighted taper be held in 
 
 front of the point, the flame 
 
 will be visibly blown aside 
 
 (Fig. 338) by the streams of 
 
 electrified air repelled from 
 
 the point. These air currents 
 
 can be felt with the hand. 
 
 They are due to a mutual repulsion between the electrified air 
 
 particles near the point and the electricity collected on the point 
 
 itself." — S. P. Thompson's Elementary Lessons in Electricity and 
 
 Magnetisfn. 
 
 526. The Electric Discharge in Rarefied Gases. — The electric 
 
 discharge is produced in rarefied gases by means of glass tubes or 
 
 bulbs, provided with electrodes of platinum wire fused into the 
 
 glass. Such tubes, when exhausted to a pressure of one or two 
 
 millimeters of mer- 
 cury, are known as 
 Geissler tubes ( Fig. 
 339). The differ- 
 ence of potential 
 required to pro- 
 duce a discharge 
 
 s^ 
 
 \^-^. 
 
 Fig. 339. 
 
 in any gas, between 
 
 electrodes at a given distance apart, decreases as the pressure of 
 the gas is diminished ; and an induction coil giving a spark i cm. 
 long in air will illuminate a Geissler tube 12 or 15 cm. long. An 
 induction machine can also be used for the purpose. 
 
 "Through such tubes, before exhaustion, the spark passes 
 without any unusual phenomena being produced. As the air is 
 
422 Electricity 
 
 exhausted, the sparks become less sharply defined, and widen out 
 to occupy the whole tube, becoming pale in tint and nebulous 
 in form. The cathode exhibits a beautiful bluish or violet glow, 
 separated from the conductor by a narrow dark space^ while at the 
 anode a single small bright star of light is all that remains. At a 
 certain degree of exhaustion the light in the tube breaks up into a 
 set of striip, or patches of light of a cup-like form, which vibrate to 
 and fro between darker spaces." (Thompson.) The color of the 
 discharge in Geissler tubes is different with different gases.^ 
 
 527. Atmospheric Electricity. — The sparks obtained from 
 electrical machines and Leyden jars suggested to a number of the 
 early experimenters in electricity that lightning was due to elec- 
 trical discharges in the atmosphere. Benjamin Franklin put this 
 theory to an experimental test in 1752. " He sent up a kite dur- 
 ing the passing of a storm, and found the wetted string to conduct 
 electricity to the earth, and to yield abundance of sparks. These 
 he drew from a key tied to the string, a silk ribbon being interposed 
 between his hand and the key for safety. Leyden jars could be 
 charged, and all other electrical effects produced, by the sparks 
 furnished from the clouds. The proof of the identity was com- 
 plete." (Thompson.) 
 
 It has been repeatedly shown by later experiments that the 
 atmosphere is generally electrified even in fair weather. In fair 
 weather the electrification is almost always positive ; in stormy 
 weather it is sometimes positive and sometimes negative. The 
 potential increases with the altitude ; but differs widely in different 
 localities and with different states of the weather. The rise of 
 potential has been found as great as 600 volts per meter of eleva- 
 tion above the ground. 
 
 1 As the exhaustion in a vacuum tube is continued beyond a pressure of one 
 millimeter, the dark space surrounding the cathode increases in width u:;;!). >Jicn 
 the pressure is reduced to about one millionth of an atmosphere, it completely fills 
 the tube. Tubes exhausted to this degree are called Crookes tubes. The eloctric 
 discharge in a Crookes tube produces new forms of radiation, called cathode rays 
 and X-rays. The latter are also known as Roentgen rays, from their discoverer. 
 The pupil is referred to other works for an account of these rays and their applications. 
 
Electrostatics 423 
 
 Various theories have been advanced to account for the electri- 
 fication of the atmosphere ; but very little is definitely known 
 about it. Evaporation is very probably one of the principal causes. 
 But, aside from this question, if we suppose the particles of water 
 vapor in the atmosphere to be even slightly electrified, the high 
 potential of clouds is easily explained ; for, as the particles unite 
 in the process of condensation, the charge increases much more 
 rapidly than the capacity of the drop. For example, if one million 
 equally charged particles unite, the potential becomes ten thousand 
 times as great ; and there are thousands of milHons of particles in 
 a drop. The great length of lightning sparks or flashes shows 
 that the potential of a thunder-cloud is enormously high. 
 
 528. Thunder. — Thunder corresponds to the snapping sound 
 produced by an electric spark. The sudden heating of the air 
 along the path of a lightning flash causes it to expand with explo- 
 sive violence, producing sound waves of great intensity. If the 
 flash is short and straight, the sound is a short clap ; if it is long 
 and zigzag, the sounds produced by its different parts have unequal 
 distances to travel to the observer and are heard in quick succes- 
 sion as a continuous rattle. The rolling sound of distant thunder 
 is due to various reflections of the sound from clouds, from the 
 ground, and often from neighboring hills. 
 
 529. Lightning Conductors. — The use of lightning conductors 
 to protect buildings was first suggested by Benjamin Franklin. 
 The usual device consists of one or more iron rods, extending 
 some distance above the highest points of a building and connected 
 by means of large iron or copper conductors with datnp earth or, 
 better, with water. If the conductor ends in dry earth, it is not 
 only useless but even dangerous. (Why?) Each rod is terminated 
 by a gilded copper point. 
 
 The action of a lightning conductor depends largely upon in- 
 duction. A charged cloud induces an opposite charge on the 
 ground under it and on houses, trees, and other objects within 
 this area. This inductive action is strongest upon the highest 
 objects, and causes lightning rods to become highly electrified. 
 
424 
 
 Electricity 
 
 Under these conditions a rapid and continuous discharge takes 
 place from the sharply pointed tips of the rods (Art. 525). This 
 quiet discharge of opposite electrification toward the cloud is 
 often sufficient to prevent lightning ; but, if a stroke does occur, 
 the rod receives the discharge and the building is preserved. 
 
 530. The Aurora Borealis. — "In the northern regions of the 
 earth the aurora horfalis, or northern lights^ is an occasional phe- 
 nomenon ; and within and near the Arctic Circle is of almost 
 nighdy occurrence. Similar lights are seen in the south polar 
 regions of the earth, and are denominated aurora ausiralis. As 
 
 
 Fig. 34a 
 
 seen in European latitudes, the usual form assumed by the aurora 
 is that of a number of ill-defined streaks or streamers of a pale 
 tint (sometimes tinged with red and other colors), either radiat- 
 ing in a fanlike form from the horizon in the direction of the 
 magnetic north, or forming a sort of arch across that region of 
 the sky, of the general form shown in Fig. 340. A certain flicker- 
 ing or streaming motion is often discernible in the streaks. Under 
 very favorable circumstances the aurora extends over the entire 
 
Electrostatics 425 
 
 sky. The appearance of an aurora is usually accompanied by a 
 magnetic storm, affecting the compass-needles over whole regions 
 of the globe. This fact, and the position of the auroral arches 
 and streamers with respect to the magnetic meridian, directly 
 suggest an electric origin for the light, — a conjecture which is 
 confirmed by the many analogies between auroral phenomena 
 and those of discharge in rarefied air. Yet the presence of an 
 aurora does not, at least in our latitudes, affect the electric condi- 
 tions of the lower regions of the atmosphere. 
 
 " The most probable theory of the aurora is that originally due 
 to Franklin; namely, that it is due to electric discharges in the 
 upper air, in consequence of the differing electrical conditions 
 between the cold air of the polar regions and the warmer streams 
 of air and vapor raised from the level of the ocean in tropical 
 regions by the heat of the sun." (Thompson.) 
 
APPENDIX 
 
 Table I 
 
 Metric Units of Lengthy Surface^ and Volume 
 
 lo millimeters (mm.) 
 lo centimeters (cm.) 
 lo decimeters (dm.) 
 loo sq. millimeters (smm.) 
 lOo sq. centimeters (scm.) 
 loo sq. decimeters (sdm.) 
 looo cu. millimeters (cmm.) 
 looo cu. centimeters (ccm.) 
 xooo cu. decimeters (cdm.) 
 
 : I centimeter 
 
 : I decimeter 
 
 : I meter (m.) 
 
 : I sq. centimeter 
 
 : I sq. decimeter 
 
 : I sq. meter 
 
 1 cu. centimeter 
 
 : I cu. decimeter 
 
 : I cu. meter (cu. m.) 
 
 Table n 
 
 Equivalents 
 
 Metric to English 
 
 cm. = .3937 in. 
 m- =39-37 in. 
 km. =.6214 mile 
 
 scm. =.1550 sq. in. 
 
 sq. m. = 1.196 sq. yd. 
 
 = 10.764 sq. ft. 
 
 I ccm. =.06103 cu. in. 
 I cdm. =1.0567 qt. (liquid) 
 I cu. m.= 1.308 cu. yd. 
 = 35-317 cu. ft 
 
 English to Metric * 
 
 I in. = 2.540 cm. 
 
 I ft. = 30.48 cm. 
 
 I yd. =.9144 m. 
 
 I mile = 1.6093 km. 
 
 I sq. in. =6.452 scm. 
 
 I sq.ft. =929.0 scm. 
 
 I sq. yd. = .8361 sq. m. 
 
 I cu. in. = 16.387 ccm. 
 
 I cu. ft. =28,315. ccm. 
 
 I cu. yd. = .7645 cu. m. 
 
 I qt. = .9463 cdm. (liters) 
 
 I gal. = 3.785 liters 
 
 I gram = .0353 oz. 
 I kg. = 2.2046 lb. 
 
 426 
 
 oz. 
 lb. 
 
 = 28.35 g. 
 = 453.6 g. 
 
Appendix 
 
 427 
 
 Table III 
 
 Mensuration Rules 
 
 ratio of the circumference of a circle to its diameter = 3.1416 
 
 Circumference of a circle (radius, r) = 2 Trr 
 
 Area of a circle = -nr^ 
 
 Surface of a sphere = 4 irr^ 
 
 Volume of a sphere = f Trr^ 
 Lateral surface of a right cylinder 
 
 (altitude h and radius of base r) = 2 irrh 
 
 Volume of a right cylinder = irr^h 
 
 Table IV 
 Densities (in grams per ccm.) 
 
 Aluminum . . 
 Antimony, cast . 
 Beeswax . . . 
 Bismuth, cast . 
 Brass .... 
 Copper . . . 
 Cork . . . . 
 Galena . . . 
 German silver . 
 Glass, crown . . 
 Glass, flint . . 
 Gold . . . . 
 
 Ice 
 
 Iron, bar . . . 
 Iron, cast . . . 
 Ivory . . . . 
 Lead . . . . 
 Marble . . . 
 Mercury, at o°C. 
 Platinum . . . 
 Quartz . . . 
 Silver . . . . 
 Steel . . . . 
 Sulphur, native . 
 Tin. ... . 
 Zinc,' cast . . . 
 
 2.67 
 6.72 
 
 .96 
 9.8 
 8.4 
 8.8 to 
 
 .14 to 
 7.58 
 8.5 
 
 2-5 
 
 3 to 
 
 19-3 
 .917 
 7.8 
 7.2 to 
 
 1.9 
 
 11.3 to 
 
 2.72 
 
 13-596 
 21.5 
 2.65 
 
 10.4 
 
 7.8 
 
 2.03 
 
 7-3 
 6.86 
 
 8.9 
 .24 
 
 3-5 
 
 7.3 
 
 11.4 
 
 to 10.5 
 to 7.9 
 
 Alcohol (95%) . . .82 
 
 Blood 1.06 
 
 Carbon disulphide . 1.29 
 Chloroform . . . 1.5 
 Copper sulphate solution i . 1 6 
 
 Ether 72 
 
 Glycerine . . . . 1.27 
 Hydrochloric acid . 1.22 
 Mercury, at 0° C. . 13.596 
 
 Milk 1.03 
 
 Nitric acid ... 1.5 
 Oil of turpentine . .87 
 
 Olive oil 915 
 
 Sulphuricacid(i5%) i.io 
 Sulphuric acid . . 1.8 
 Water (4° C.) . . i.ooo 
 Water, sea . . . 1.026 
 
 Gases at 0° C. and 76 cm. 
 Pressure 
 
 Air 001293 
 
 Carbon dioxide . . .001977 
 Hydrogen . . . .0000896 
 
 Nitrogen 001256 
 
 Oxygen 001430 
 
428 
 
 Appendix 
 
 Table V 
 
 Tangents of Angles 
 
 To find the tangent of an angle not measured by a whole num- 
 ber of degrees, find first the tangent of the integral part of the 
 number, and add to this the product obtained by multiplying the 
 difference between this tangent and the tangent of the next whole 
 number of degrees by the decimal part of the angle. For example, 
 to find the tangent of 38°.7, proceed thus : — 
 
 tan 38* =.781, tan 39* = .810. 
 
 .810— .781 = .029. 
 
 .7 X .029 = .020. 
 
 tan 38^.7 = .781 -f .020 = .801. 
 
 Anglk 
 
 Tangent 
 
 Angle 
 
 Tangent 
 
 Angle 
 
 Tangent 
 
 Angle 
 
 Tangent 
 
 0° 
 
 .0000 
 
 23° 
 
 .424 
 
 46^ 
 
 1.036 
 
 69° 
 
 2.61 
 
 I 
 
 .0175 
 
 24 
 
 .445 
 
 47 
 
 1.072 
 
 70 
 
 2.75 
 
 2 
 
 .0349 
 
 25 
 
 .466 
 
 48 
 
 I. Ill 
 
 71 
 
 2.90 
 
 3 
 
 .05 24 
 
 26 
 
 .488 
 
 49 
 
 1. 150 
 
 72 
 
 3.08 
 
 4 
 
 .0699 
 
 27 
 
 .510 
 
 50 
 
 1. 192 
 
 73 
 
 3-27 
 
 5 
 
 .0875 
 
 28 
 
 •532 
 
 51 
 
 1-235 
 
 74 
 
 3-49 
 
 6 
 
 .1051 
 
 29 
 
 •554 
 
 52 
 
 1.280 
 
 75 
 
 3-73 
 
 7 
 
 .1228 
 
 30 
 
 •577 
 
 53 
 
 1-327 
 
 76 
 
 4.01 
 
 8 
 
 .1405 
 
 31 
 
 .601 
 
 54 
 
 1.376 
 
 77 
 
 4-33 
 
 9 
 
 .1584 
 
 32 
 
 .625 
 
 55 
 
 1.428 
 
 78 
 
 4.70 
 
 ID 
 
 .1763 
 
 33 
 
 .649 
 
 56 
 
 1.483 
 
 79 
 
 5.14 
 
 II 
 
 .194 
 
 34 
 
 .675 
 
 57 
 
 1.540 
 
 80 
 
 5.67 
 
 12 
 
 •213 
 
 35 
 
 .700 
 
 58 
 
 1.600 
 
 81 
 
 6.31 
 
 13 
 
 .231 
 
 36 
 
 .727 
 
 59 
 
 1.664 
 
 •82 
 
 7.12 
 
 14 
 
 .249 
 
 37 
 
 .754 
 
 60 
 
 1-732 
 
 S3 
 
 8.14 
 
 '5 
 
 .268 
 
 38 
 
 .781 
 
 61 
 
 1.804 
 
 84 
 
 9-51 
 
 16 
 
 .287 
 
 39 
 
 .810 
 
 62 
 
 1.88 
 
 85 
 
 11-43 
 
 17 
 
 .306 
 
 40 
 
 •839 
 
 63 
 
 1.96 
 
 86 
 
 14.30 
 
 18 
 
 •325 
 
 41 
 
 .869 
 
 64 
 
 2.05 
 
 87 
 
 19.08 
 
 19 
 
 .344 
 
 42 
 
 .900 
 
 65 
 
 2.14 
 
 88 
 
 28.64 
 
 20 
 
 .364 
 
 43 
 
 .933 
 
 66 
 
 2.25 
 
 89 
 
 57-29 
 
 21 
 
 .384 
 
 44 
 
 .966 
 
 67 
 
 2.36 
 
 90 
 
 00 
 
 22 
 
 .404 
 
 45 
 
 1,000 
 
 68 
 
 2.48 
 
 
 
Appendix 
 
 429 
 
 VI — References to Chute's Physical Laboratory Manual 
 
 The following list of references to the revised edition of Chute's 
 Physical Laboratory Manual, published by D. C. Heath & Co., 
 is provided for the convenience of teachers who may wish to use 
 this manual in connection with the text. The numbers in the 
 first column refer to articles in the text ; those in the last column, 
 to articles in the manual. 
 
 Text 
 
 14,15 
 
 15 
 
 15 
 
 19 
 22-25 
 
 30-32 
 
 35 
 
 36 
 
 47 
 
 53 
 
 62, 63 
 
 67,68 
 
 96,97 
 
 119-123 
 
 130-138 
 
 155 
 
 159 
 
 162-164 
 
 166 
 
 190 
 
 199 
 
 231 
 
 233 
 235 
 237 
 241 
 241 
 
 245 
 249 
 257 
 263 
 
 Topic illustrated Manual 
 
 Extension 19-21 
 
 Capacity 25 
 
 Volume by displacement ... 28 
 
 Weighing. . 31 
 
 Liquid pressure 52 
 
 Buoyancy of liquids . . . . 55 
 
 Specific gravity of solids . . 56, 57 
 
 Specific gravity of liquids . . 58 
 
 Boyle's Law 53 
 
 The siphon 54 
 
 Concurrent forces 39 
 
 Parallel forces 40 
 
 Uniformly accelerated motion . 42 
 
 Curvilinear motion 41 
 
 The pendulum 43 
 
 The lever 44 
 
 The wheel and axle .... 46 
 
 Pulleys 45 
 
 The inclined plane .... 47 
 
 Tenacity 35 
 
 Capillarity 51 
 
 The fixed points of a thermometer 85 
 
 Coefficient of linear expansion . 86 
 
 Coefficient of cubical expansion 87 
 
 Law of Charles ..... 88 
 
 Specific heat of a solid ... 97 
 
 Specific heat of a liquid ... 98 
 
 Melting points 89 
 
 Heat of fusion of ice ... loi 
 
 Dew-point 91 
 
 Boiling points 90 
 
430 
 
 Appendix 
 
 Text 
 267 
 
 3»3. 
 
 313 
 
 337 
 
 339 
 
 344 
 
 347 
 
 351-355 
 
 357 
 
 362 
 
 375-380 
 382 
 
 394 
 
 396, 397 
 
 402 
 
 404 
 
 421 
 
 421 
 
 426 
 
 432 
 
 454-456 
 
 446 
 
 466 
 
 468 
 
 471 
 
 471 
 
 475 
 
 477 
 
 478 
 
 4781 485 
 492, 495 
 
 Topic illustrated Manual 
 
 Heat of vaporization of water . 102 
 
 Laws of vibrating strings ... 63 
 
 Resonance (velocity ol sound) . 59, 60 
 
 Vibration rate of a tuning fork . 62 
 
 Pinhole images 64, 65 
 
 Photometry 66 
 
 Reflection of light 67 
 
 Images in plane mirrors .... 69 
 
 Concave mirrors . . . . . . 70, 72, 73 
 
 Convex mirrors 7i> 73 
 
 Index of refraction 74. 75 
 
 Convex lens 76, 78, 79 
 
 Concave lens 77> 78, 79 
 
 The simple microscope .... 80 
 
 The telescope 80 
 
 Spectra 81 
 
 Wave length 82 • 
 
 The poles of a magnet .... 103 
 
 Distribution of magnetic action . 106 
 
 Magnetic transparency . . . . 104 
 
 Magnetic fields 107 
 
 Magnetic eflect of a current . . 109 
 
 Electro-motive series . . . . 110 
 
 The tangent galvanometer . . . 123, 124 
 Changeof resistance with temperature 1 20 
 
 Electrical resistance 118 
 
 The resistance of a cell ....122 
 
 Fall of potential along a conductor 125 
 
 Resistance of conductors in parallel 119 
 
 Electro-motive force of cells . . 126 
 
 Use of voltmeter and ammeter . 121 
 
 Induced currents 128,129 
 
INDEX 
 
 The Numbers refer to Pages 
 
 Aberration, chromatic, 333. 
 
 spherical, 282, 304. 
 Absolute temperature, 180-181. 
 Absorption, of radiation, 168, 171. 
 
 selective, 171, 327-329. 
 Acceleration, 71-74- 
 
 due to gravity, 76-78. 
 Accommodation, power of, 309. 
 Achromatic lens, 334. 
 Action and reaction, 8, 9, 89, 340, 413. 
 Adhesion, 149-150. 
 After-images, 332. 
 Air, buoyancy of, 46-48. 
 
 composition of, 3. 
 
 density of, 31. 
 
 water vapor in, 1 96-197. 
 Air pump, 41-42. 
 Amalgamating zinc, 358. 
 Ammeter, 384. 
 Ampere, 377. 
 
 Analysis, of light, 322, 327. 
 Angle, critical, 293. 
 
 of deviation, 284, 291. 
 
 of incidence and reflection, 269. 
 
 of refraction, 284. 
 
 refracting, of prisms, 291. 
 
 sine of, 286. 
 
 tangent of, 372. 
 
 visual, 314. 
 Anode, 408. 
 Antinode, 245. 
 
 Aperture, of mirror, 274, 282. 
 Arc, electric, 385. 
 Archimedes, principle of, 24, 25. 
 
 Armature, of dynamos and motors, 
 
 398. 
 Artesian wells, 20. 
 Astatic galvanometer, 383. 
 Atmosphere, heating of, 172-173. 
 
 height of, 23- 
 Atmospheric electricity, 422-425. 
 
 pressure, 31-36. 
 
 refraction, 294-296. 
 Attraction, electrostatic, 412. 
 
 magnetic, 340. 
 
 molecular, 147. 
 
 of gravitation, 98-103. 
 Audibility, limits of, 214, 234-235. 
 Aurora borealis, 425. 
 Axis, of lens, 297, 300. 
 
 of mirror, 274. 
 
 Balance, 12-13. 
 Balloon, 47. 
 Barometer, 34-36. 
 Battery, see Cells. 
 Beam, of light, 259. 
 Beats, 230-231. 
 Bell, electric, 365. 
 Bellows, 43. 
 Bichromate cell, 359. 
 Bicycle, 134. 
 Boiling, 200-202. 
 Boiling point, 175, 202. 
 Boyle's law, 39-40. 
 Buoyancy, center of, 65. 
 
 of air, 46-48. V 
 
 of liquids, 24-25. 
 
 431 
 
432 
 
 Index 
 
 Caloric theory, i6l. 
 Calorie, 182. 
 
 Calorimetry, 182-185, 190, 205. 
 Camera, photographer's, 322. 
 
 pinhole, 263. 
 Capillarity, 153-155- 
 Cathode, 408. 
 Cells, electric, 354, 357-362. 
 
 electro-motive force of, 376-377. 
 
 in battery, 380-382. 
 
 storage, 410. 
 Center, of buoyancy, 65. 
 
 of curvature, 274. 
 
 of gravity, 60-61. 
 Centrifugal force, ^6. 
 Centripetal force, 93-96. 
 Charge, electrostatic, 420. 
 Charles, law of, 179, 181. 
 Chemicar changes, i. 
 
 effects of electric current, 407-411. 
 
 energy, 120. 
 Chromatic aberration, 333. 
 
 scale, 234. 
 Circuit, electric, 357. 
 
 divided or shunt, 378-379. 
 Clouds, I9'i-I99. 
 Coal, energy of, iii, 120, 210. 
 Cohesion, 147-151. * 
 
 Coil, induction, 393. 
 Color. 325, 327-333. 
 
 by interference, 337. 
 Commutator, 397. 
 
 Compass, 340, 348, 352, ^ 
 
 Compressibility, of gases, 4, 37-40. 
 
 of liquids and solids, 4, 140. 
 Compression pump, 43, 
 Condensation, of gases, 205-206. 
 
 of water vapor, 198-200. 
 Condenser, electric, 420. 
 Conduction, electric, 357. 
 
 of heat, 164-165.. 
 Conservation, of energy, 132, 209. 
 Convection, of heat, 166. 
 Couple, 58. 
 Critical angle, 293. 
 
 Crookcs tubes, 422. 
 
 Current, electric, 354-355. 369-373- 
 
 chemical effects of, 407-411. 
 
 extra, 392. 
 
 first ideas of, 354-355- 
 
 heating effects of, 385-387. 
 
 induced, 389-396. 
 
 magnetic eflects of, 362-365. 
 
 measurement of, 369-373, 384. 
 
 unit of, 377. 
 Curvature, center of, 274. 
 Curvilinear motion, 93-^. 
 
 Dahon's laws, 195. 
 Daniell cell, 362. 
 D'Arsonval galvanometer, 383. 
 Declination, magnetic, 349. 
 Density, 13. 
 
 and pressure of gases, 40. 
 Deviation, angle of, 284, 291. 
 Dew, 199. 
 Dew-|X)int, 196. 
 Diffusion, of gases, 141-144. 
 
 of liquids, 145-146. 
 
 of light, 269-270. 
 Dip, magnetic, 350. 
 Dipping needle, 350. 
 Discord, 232. 
 
 Dispersion, of light, 322-326. 
 Distillation, 203-204. 
 Divisibility, of matter, 137-138. 
 Ductility, 159. 
 Dynamics, Chap. V. 
 
 definition of, 49. 
 Dynamo, 396-400. 
 
 Ear, 249-252. 
 
 Earth, effect of rotation on its shape, 
 
 lOI. 
 
 effect of rotation on weight, 102. 
 
 magnetic field of, 348-352. 
 
 revolution and rotation of, 100. 
 Echoes, 225. 
 Eclipses, 261-262. 
 Efficiency, of machines, 132. 
 
Index 
 
 433 
 
 Elasticity, 156-158. 
 Electric, arc, 385. 
 
 battery, 380-382. 
 
 bell, 365. 
 
 cells, 354-355» 357-362. 
 
 circuit, 357. 
 
 conduction, 357. 
 
 current, see Current. 
 
 discharge, in rarefied gases, 421. 
 
 energy, 354-355. 387. 389, 392. 399, 
 401, 402, 407. 
 
 light, 385-386. 
 
 measurements, 369-384. 
 
 motors, 400-402. 
 
 potential, see Potential. 
 
 power, 387. 
 
 resistance, see Resistance. 
 
 spark, 395. 
 
 telegraph, 366-369. 
 
 transmission of power, 402. 
 
 units, 375, 376, 377, 388. 
 Electricity, Chap. XII. 
 
 atmospheric, 422-425. 
 
 •current, 353-4". 
 
 nature of, 353, 413. 
 
 static, 411-425. 
 Electrification, by contact, 413. 
 
 by friction, 412. 
 
 by induction, 413-414. 
 
 two states of, 413. 
 Electrodes, 408. 
 Electrolysis, 407-410. 
 Electro-magnet, 364. 
 Electro-magnetic field, 362-364. 
 
 induction, 389-396. 
 Electro-metallurgy, 409. 
 Electro-motive force, 356. 
 
 measurement of, 379. 
 
 of cells, 376-377- 
 
 uftit of, 376. • 
 
 Electrophorus, 416. 
 Electroplating, 409. 
 Electroscope, 415. 
 
 Electrostatic attraction and repulsion, 
 412. 
 
 capacity, 419. 
 charge, 420. 
 condenser, 420. 
 field, 415. 
 induction, 413. 
 machines, 415-417. 
 potential, 418. 
 Electrotyping, 408. 
 Energy, Chap. VI. 
 
 conservation of, 132, 209. 
 dissipation of, 1 21, 223. 
 first ideas of, ii i . 
 forms of, chemical, 120. 
 electrical, 354-35 5» 387. 389. 392, 
 
 399,401,402,407. 
 kinetic, in, 116. 
 mechanical, 120. 
 molecular kinetic (heat), 118, 
 
 146-147. 
 molecular potential ("latent 
 
 heat"), 189, 204-205. 
 muscular, 119. 
 of light, 255-258. 
 of sound, 220-223. 
 potential, 118, 1 19, 1 20. 
 radiant, 167-172,255-258. 
 solar, 209-211. 
 sources of, 162-163, 210. 
 transference of, by electric current, 
 354-355» 402. 
 by machines, 123, 126, 128, 
 
 130. 
 conditions necessary for, 112. , 
 two modes of, 255. 
 transformation of, by dombustion, 
 163. 
 by compression and expansion of 
 
 gases, 162, 204. 
 by dynamos, 399. 
 by electrical resistance, 387. 
 by friction, 119, 161-162. 
 by fusion and solidification, 189. 
 by motors, 401. 
 
 by radiation and absorption, 168, 
 170. 
 
434 
 
 Index 
 
 by steam engine, 213. 
 by vaporization and condensation, 
 193, 204-207. 
 units of, electrical, 387-388. 
 mechanical, 116. 
 thermal, 182. 
 Engine, steam, 2 12-2 1 3. 
 Equilibrant, 54. 
 
 Equilibrium, of concurrent forces, 52. 
 of floating bodies, 65-66. 
 of parallel forces, 55-56. 
 of two forces, 49-50. 
 stable, unstable, and neutral, 60- 
 
 63. 
 Ether, luminiferous, 256-258, 347, 
 
 353. 
 Evaporation, 192. 
 
 cooling by, 206-207. 
 Expansion, by heat, 141, 176-180. 
 
 cooling by, 204. 
 
 force of, 177. 
 Extension, 10. 
 Eye, 306-315. 
 
 defects of, 310. 
 
 Falling bodies, 74-79. 
 
 Faraday, 392. 
 
 Far sight, 310. 
 
 Field, electro-magnetic, 362-364. 
 
 electrostatic, 415. 
 
 magnetic, 345-347- 
 Field magnet, 399. 
 Floating bodies, buoyancy upon, 24- 
 
 25. 
 
 equilibrium of, 65-66. 
 Fluids, characteristics of, 4-5. 
 Focal length, of lens, 299. 
 
 of mirrors, 276, 282. 
 Foci, of lenses, 297-300. 
 
 of mirrors, 276, 279. 
 Fog, 198. 
 Foot-pound, 116. 
 Force, 5-9. 
 
 buoyant, 24, 25, 46-48. 
 
 centrifugal, 96. 
 
 centripetal, 93-96. 
 
 electro-motive, 356. 
 
 elements of, 50. 
 
 graphic representation of, 50. 
 
 lines of, 345-347- 
 
 moments of, 57-59. 
 
 units of, 12. 
 Forces, balanced, 7, 8; Chap. IV. 
 
 composition of, 51-53,56. 
 
 concurrent, 51-55. 
 
 molecular, 147-155. 
 
 parallel, 55-56. 
 
 parallelogram of, 52. 
 
 resolution of, 54. 
 
 unbalanced, 7, 8; Chap. V. 
 Force pump, 45. 
 Franklin, 422, 423, 425. 
 Fraunhofer lines, 324. 
 Freezing, 186. 
 Freezing mixtures, 191. 
 Freezing point, 1 74. 
 Friction, 6. 
 
 heating effects of, 119, 161-162. 
 
 of the air, 74, 75, 78. 
 
 uses of, 90, 133, 134. 
 Frost, 199. 
 Fulcrum, 123. 
 
 Fundamental tone, 238, 246-247. 
 Fusion, 186-191. 
 
 change of volume during, 187. 
 
 heat of, 189-190. 
 
 Galilean telescope, 320. 
 Galileo, 76, 79, 92, 320. 
 Galvanometers, 370-373, 383-384. 
 Gases, characteristics of, 3. 
 
 compressibility of, 4, 39, 40. 
 
 condensation of, 205-206. 
 
 cooled by expansion, 204. 
 
 diffusion of, 141-143. 
 
 distinguished from vapors, 192. 
 
 effect of pressure on volume and 
 density of, 39, 40. • 
 
 effect of temperature on volume, of, 
 141, 179. 
 
Index 
 
 435 
 
 kinetic theory of 144-145. 
 
 mechanics of, Chap. III. 
 
 pressure of, 37-40. 
 Geissler tubes, 421. 
 Glaciers, flow of, 189. 
 Gram, mass and weight, 12. 
 Gram-centimeter, 116. 
 Gravitation, 98-103, 150-151. 
 
 law of, 98. 
 Gravity, 60. 
 
 acceleration due to, 76-78. 
 
 cell, 360. 
 
 center of, 60, 61. 
 
 specific, 26-30. 
 
 Hail, 200. 
 Hardness, 160. 
 Harmony, 232. 
 Hearing, 249-252. 
 Heat, Chap. VHI. 
 
 conduction of, 164-166. 
 
 convection of, 166. 
 
 expansion due to, 141, 176-180. 
 
 kinetic theory of, 146-147, 161-162. 
 
 mechanical equivalent of, 208. 
 
 of fusion, 189-190. 
 
 of vaporization, 193, 204-205. 
 
 sources of, 162. 
 
 specific, 183-184. 
 
 unit of, 182. 
 Helmholtz, 211, 240. 
 Horse power, 121. 
 Humidity, 197. 
 Hydrostatic press, 21. 
 
 Ice, 174, 186-190. 
 
 manufacture of, 207. 
 Illumination, intensity of, 263. 
 Images, by lens, 300-304. 
 
 by plane mirrors, 271-273. 
 
 by spherical mirrors, 280-283. 
 
 by small opening, 262-263. 
 
 real, 279-281, 300, 302. • 
 
 virtual, 279-283, 302, 303. 
 Incandescent lamp, 385. 
 
 Inclination or dip, 350. 
 
 Inclined plane, 54, 79, 1 14, 131-132. 
 
 Index of refraction, 287. 
 
 Induced currents, 389-396. 
 
 Induction, earth's, 351-352. 
 
 electro-magnetic, 389-396. 
 
 electrostatic, 413. 
 
 magnetic, 342-343. 
 
 self, 392. 
 Induction coil, 393. 
 Inertia, 5, 87, 89, 95, 96. 
 Insulators, electric, 357. 
 Interference, of light, 337. 
 
 of sound, 229-231. 
 Iridescence, 337, 338. 
 
 Joule, 162, 208, 386. 
 Joule's equivalent, 208. 
 law, 386. 
 
 Kilogram-meter, 116. 
 Kinetic energy, III, Il6. 
 Kinetic theory, of gases, 144, 146- 
 147. 
 of heat, 146-147, 161-162. 
 Kinetics, see Dynamics. 
 
 Lamp, arc, 385. 
 
 incandescent, 385. 
 Lantern, optical, 321. 
 Law, Boyle's, 39-40. 
 
 Dalton's, 195. 
 
 Joule's, 386. 
 
 »f Charles, 179, 181. 
 
 of gravitation, 98. 
 
 Ohm's, 370, 377. 
 
 Pascal's, 20. 
 Laws of motion, Newton's, 91. 
 Leclanche cell, 360. 
 Lenses, achromatic, 334. 
 
 concave, 305. 
 
 convex, 296-305. % 
 
 Lever, 123-126. 
 Leyden jar, 418. 
 Lifting pump, 44. 
 
436 
 
 Index 
 
 Light, Chap. X. 
 
 dispersion of, 322-326. 
 
 intensity of, 263-265. 
 
 interference of, 337.* 
 
 propagation of, 258-260. 
 
 reflection of, 268-270. 
 
 refraction of, 283-292. 
 
 theory of, 255-260. 
 
 velocity of, 265-267. 
 
 wave length of, 259, 325-326. 
 Lightning, 422-423. 
 
 rod, 423. 
 Lines of force, 345-347- 
 Liquids, characteristics of, 3, 5. 
 
 diffusion of, 145-146. 
 
 mechanics of, Chap. IL 
 Liter, 11. 
 
 Local action in voltaic cell, 358. 
 Lodestone, 339. 
 Loudness of sound, 227. 
 
 Machines, 123-135. 
 
 efficiency of, 132. 
 
 electrical, 415-417. 
 
 mechanical advantage of, 126. 
 Magdeburg hemispheres, 32. 
 Magnetic action, 340-341, 347. 
 
 declination, 349. 
 
 effects of a current, 362-365. 
 
 field, 345-347- 
 
 inclination or dip, 350. 
 
 induction, 342-345. 
 
 lines of force, 345-347. 
 
 meridian, 349. 
 
 needle, 340. 
 
 permeability, 342. 
 
 poles, 340. 
 
 substances, 341. 
 Magnetism, Chap. XL 
 
 terrestrial, 348-352. 
 Magnetization, permanent and 
 porary, 342-343- 
 
 theory of, 343-345- 
 Magnets, 338, 342. 
 Magnifying glass, 302, 315-316. 
 
 tem- 
 
 Major chord, 234. 
 Malleability, 159. 
 Manometers, 38-39. 
 Mass, detinition of, 11. 
 
 center of, 60-61. 
 
 measurement of, by weight, 12, 
 by inertia, 87. 
 
 units of, 12. 
 Matter, divisibility df, 137. 
 
 properties of. Chap. VI L 
 
 states of, 3, 147-148. 
 
 structure of, 137-147. 
 Measurement, 10-13. 
 Mechanical advantage, 126. 
 
 equivalent of heat, 208. 
 
 powers, 123. 
 Mechanics, definition of, 49. 
 
 of gases. Chap. IIL 
 
 of liquids. Chap. IL 
 
 of solids. Chaps. IV, V, VI. 
 Melting, 186. 
 [Celling points, 186. 
 ' effect of pressure on, 1 88. 
 Meter, lo. 
 
 Microphone, 405-406. 
 Microscope, compound, 316. 
 
 simple, 302, 315-316. 
 Mirage, 295. 
 Mirrors, parabolic, 282. 
 
 plane, 271-273. 
 
 spherical, 274-283. 
 Mobility of fluids, 159. • 
 Molecular forces, 14 7-1 51. 
 
 motion, 141-147. 
 
 structure of matter, 137-147. 
 Molecule, 138. 
 Moment of force, 57-59. 
 Momentum, 91. 
 Moon, revolution of, 100. 
 Motion, 68. 
 
 accelerated, 69, 71-74 
 
 curvilinear, 93-96. 
 
 laws of, '83-92. 
 
 of falling bodies, 74-78. 
 
 of pendulum, 104. 
 
Index 
 
 437 
 
 of projectiles, 80-83. 
 
 on an inclined plane, 79. 
 
 uniform, 68. 
 
 wiave, 217-220. 
 Motor, electric, 400-402. 
 Musical, instruments, 242, 246-248. 
 
 intervals, 231-233. 
 
 scales, 233-234. 
 
 sounds, 227. 
 
 Needle, dipping, 350. 
 
 magnetic, 340. 
 Newton, 92, 98, 255, 259. 
 Newton's disks, 331. 
 
 laws of motion, 91. 
 Nodes, 237, 244-245. 
 Noise, 227, 245. 
 
 Octave, 232. 
 
 Ohm, definition of, 375. 
 
 Ohm's law, 370, 377. 
 
 Opera glass, 321. 
 
 Optical instruments, 315-322. 
 
 Organ pipes, 246-247. 
 
 Overtones, 238, 246-247. 
 
 Parallelogram of forces, 52. 
 
 Pascal's law, 20. 
 
 Pendulum, 104-109. 
 
 Penumbra, 261. 
 
 Permeability, magnetic, 342. 
 
 Photometry, 264-265. 
 
 Physical changes, i. 
 
 Pinhole. camera, 263. 
 
 Pitch, of musical sounds, 214, 227- 
 
 229, 234. 
 Plasticity, 149, 158. 
 Polarization, in voltaic cell, 358. 
 Poles, magnetic, 340. 
 
 of voltaic cell, 355, 358-359- 
 Porosity, 138-139. 
 Potential, electric, 356, 377. 
 
 of induced currents, 395. 
 
 of static electricity, 418. 
 Potential energy, 118, 119, 120. 
 
 Pound, weight and mass, 12. 
 Power, 121. 
 
 electric transmission of, 402. 
 
 units of, 121, 387. 
 Pressure, atmospheric, 31-36 
 
 of gases, 37-40. 
 
 of liquids, 15-25. 
 
 of vapors, 193-195. 
 Pressure gauges, 33-35, 38-39- 
 Prism, 291. 
 Projectiles, 80-83. 
 Properties of matter. Chap. VII. 
 Pulleys, 129-130. 
 Pump, air, 41. 
 
 compression, 43. 
 
 force, 45. 
 
 suction, 44. 
 
 Quality of sound, 227, 238-240. 
 
 Radiant energy, 167-172, 255-258. 
 absorption of, 168-169, 171, 172. 
 emission of, 168. 
 luminous and non-luminous, 168, 
 
 257. 
 
 reflection of, 170, 268. 
 
 selective absorption of, 171,327-329. 
 
 transmission of, 169, 328-329. 
 Radiometer, 170. 
 Rain, 199. 
 Rainl>ow, 335-336- 
 Ray, of light, 259. 
 Reaction and action, 8, 9, 89, 90, 340, 
 
 413. 
 Reflection, of light, 268-283. 
 
 difi'used, 269-270. 
 
 of sound, 225. 
 
 regular, 268-269, 291. 
 
 total, 292-294. 
 Refraction, 283-292. 
 
 atmospheric, 294-296. 
 
 index of, 287. 
 
 laws of, 286. 
 
 of different colors, 322-326. 
 
 relation to velocity, 28S-289. 
 
438 
 
 Index 
 
 Relay, 368. 
 
 Resistance, electrical, 357. 
 
 laws of, 374. 
 
 measurement of, 376. 
 
 of conductors in parallel, 378. 
 
 specific, 374. 
 
 unit of, 375. 
 Resistance coils, 375. 
 Resolution, of a force, 54. 
 
 of a velocity, 70. 
 Resonance, 243-245. 
 Respiration, 46. 
 Resultant force, 54. 
 
 velocity, 70. 
 Retina, 307. 
 Rumford, Count, l6l, 265. 
 
 Scales, musical, 233-234. 
 
 Screw, 133. 
 
 Selective absorption, 171, 327-329. 
 
 Self-induction, 392. 
 
 Shadows, 260. 
 
 Short sight, 310. 
 
 Shunt circuit, 378-379. 
 
 Sine of an angle, 286. 
 
 Siphon, 45. 
 
 Size, angular, 314. 
 
 Sky, the color of, 330. 
 
 Snow, 200. 
 
 Soap bubbles, 152. 
 
 Solenoid, 364. 
 
 Solidification, 186-189. 
 
 Solids, characteristics of, 3, 5. 
 
 mechanics of. Chaps. IV, V, VI. 
 Solution, heat of, 191. 
 Sonometer, 232. 
 Sound, Chap. IX. 
 
 intensity of, 220-223. 
 
 interference of, 229-231. 
 
 loudness of, 227. 
 
 media, 216. 
 
 pitch of, 214, 227-229, 234. 
 
 properties of, 227. 
 
 quality of, 227, 238-240. 
 
 reflection of, 225. 
 
 sources of, 214. 
 
 velocity of, 224-225. 
 
 waves, 218-220, 229. 
 Sounder, telegraph, 366. 
 Speaking tubes, 223. 
 Specific gravity, 26-29. 
 
 heat, 183-184. 
 
 resistance, 374. 
 Spectrum, 322-326. 
 
 invisible, 326. 
 Speed, 68. 
 
 Spherical aberration, 282, 304. 
 Spyglass, 320. 
 Stability, 64. 
 
 of floating bodies, 66. 
 Stars, distance of, 267. 
 
 twinkling of, 294. 
 Static electricity, 411-425; see Elec- 
 trostatic. 
 Statics, of solids, Chap. IV. 
 Steam engine, 212-213. 
 Stereoscope, 313. 
 Stress and strain, 158. 
 Strings, vibration of, 235-240. 
 Sun, energy of, 209-211. 
 Surface tension, 151-153. 
 Sympathetic vibrations, 24^^248. 
 
 Tangent galvanometer, 370-373. 
 Tangent of an angle, 372. 
 Telegraph, 365-369- 
 Telephone, 404-407. 
 
 acoustic, 223. 
 Telescopes, 317-321. 
 Temperature, 163-164. 
 
 absolute, 180. 
 
 measurement of, 173-176. 
 Tenacity, 148. 
 
 Terrestrial magnetism, 348-352. 
 Theory, definition of, 145. 
 
 of electricity, 353,413- 
 
 of gases, 144-145, 146-147. 
 
 of heat, 146, 1 61-162, 
 
 of light, 255-260. 
 
 of magnetic action, 347. 
 
Index 
 
 439 
 
 of magnetization, 343-345. 
 
 of the structure of matter, 137-141. 
 Thermometers, 173-176. 
 Thunder, 423. 
 Tone, 233. 
 
 fundamental, 238, 246. 
 Torricelli, 35. 
 Total reflection, 292-294. 
 Transference of energy, see Energy. 
 Transformation of energy, see Energy. 
 Transformer, 403. 
 Tuning fork, pitch of, 228. 
 
 vibration of, 215. 
 
 Umbra, 261. 
 
 Units, of acceleration, 72. 
 
 of current strength, 377. 
 
 of electrical power, 387. 
 
 of electrical resistance, 375. 
 
 of E. M. F., 376. 
 
 of extension, lo-ll. 
 
 of fluid pressure, 39. 
 
 of force, 12. 
 
 of heat, 182. 
 
 of mass, 1 2. 
 
 of mechanical power, 1 21. 
 
 of velocity, 68, 69. 
 
 of work and energy, 116. 
 
 Vacuum, 35. 
 
 Vapor, atmospheric, 196-200. 
 
 pressure of, 193-195. 
 Vaporization, 192, 198. 
 
 heat of, 193, 204-205. 
 Vapors, 192. 
 Velocity, 68. 
 
 graphic representation of, 69. 
 
 of light, 265-267. 
 
 of sound, 224-225, 229. 
 
 resolution of, 70. 
 
 uniform, 68. 
 
 variable, 69-74. 
 Velocities, composition of, 69. 
 Vibration, forced and sympathetic, 
 240-242. 
 
 of air columns, 243-248. 
 
 of molecules, 146. 
 
 of pendulum, 104-105. 
 
 of strings, 235-240. 
 
 of tuning fork, 215. 
 Viscosity, 159. 
 
 surface, 153. 
 Vision, binocular, 311-314. 
 Visual angle, 314. 
 Vocal cords, 253. 
 Voice, 252-254. 
 Volt, definition of, 376. 
 Voltaic cell, 354. 
 Voltmeter, 379, 384, 
 
 Water, expansion of, 178, 179. 
 Watt, 388. 
 Wave motion, 217. 
 
 Waves, of light, 257, 259-260, 325- 
 326. 
 
 of sound, 218-220, 229. 
 
 of water, 218. 
 Weather, prediction of, 36. 
 Wedge, 133. 
 Weighing, 12. 
 Weight, 7, II, 102. 
 Welding, 149. 
 Wheel and axle, 127-128. 
 Whistle, 245. 
 
 Wind instruments, 246-248. 
 Windlass, 128. 
 Work and energy, 311-316. 
 
 units of, 116. 
 
 Zero, absolute, 1 80-1 81. 
 
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