A CALENDAR OF 
 LEADING EXPERIMENTS 
 
Bacon long ago listed in his quaint way the things 
 which seemed to him most needful for the advancement 
 of learning, and among other things he mentioned A 
 Calendar of Leading Experiments for the better Inter- 
 pretation of Nature. 
 
A CALENDAR OF 
 LEADING EXPERIMENTS 
 
 BY 
 
 WM. S. FRANKLIN AND BARRY MACNUTT 
 
 SOUTH BETHLEHEM, PA. 
 
 FRANKLIN, MACNUTT AND CHARLES 
 
 PUBLISHERS OF EDUCATIONAL BOOKS 
 
 1918 
 
 All rights reserved 
 
7 
 
 Copyright, 1918 
 By FRANKLIN AND MACNUTT 
 
 The authors -are -teachers, and thty consider 
 teaching to' be 'the 'greale'st of fun, but they never 
 yet have been helped in their work by anything 
 they have ever read concerning their profession. 
 
 PRESS'OF 
 
 THE NEW ERA PRINTING COMPANY 
 LANCASTER, HA. 
 
PREFACE. 
 
 (a) "A boatman sits on a seat, braces his feet against a cleat 
 and pulls on an oar. What forces act on the boatman's body?" 
 The earth pulls on the boatman, the seat pushes on the boatman, 
 the cleat pushes on the boatman and the oar pulls on. the boat- 
 man. This is all very simple to one who has acquired the 
 habit of analytical thinking, but a large group of sophomore 
 engineering students got an average of 45 per cent in their 
 answers to the question after two weeks of insistent coaching on 
 the fundamental notion of force action, and nearly every human 
 aspect of boating was represented in the answers, including 
 even the chance of a ducking, for several of the young men would 
 have it that the water pushes on the boatman's body. 
 
 (6) Ask a student of elementary mechanics how a body (a 
 particle) behaves when it is acted upon by an unbalanced force, 
 and the natural habit of thinking-in-terms-of-human-values 
 shows itself after the most exacting class-room drill. Instead 
 of giving his attention narrowly to what is taking place at an 
 instant, the student, in his groping for human values (which do 
 not exist in the bare elements of a subject), is pretty sure to 
 refer to past history and to future prospects; his idea as to 
 what is taking place is apt to remain widely inclusive, as if he 
 were thinking of a complex human experience like two hours at 
 a play, or a month's vacation, or four years of war. Very often 
 the homely wording of the question as stated above betrays the 
 student into a disclosure of his native contempt for precise ideas 
 and he answers naively that "the body moves in the direction 
 of the force." It is very difficult to develop the habit of analyt- 
 ical thinking. 
 
 Our teachers of mathematics seem to know that they are 
 responsible for a certain kind of training, and they are not 
 ashamed to apply themselves narrowly in its accomplishment. 
 
vi CALENDAR OF LEADING EXPERIMENTS. 
 
 Indeed, many of us who do not share their responsibility are inclined 
 to think that mathematics teachers apply themselves too nar- 
 rowly, because their students, as we get them, are deficient in 
 mathematical ideas although strong, it maybe, on forms. 
 
 Physics teachers also have a definite responsibility, and our 
 notion as to this responsibility is suggested by examples a 
 and b above. The physics teacher, however, is strongly imbued 
 with the idea that his function is to disclose to the student his 
 (the physicist's) "domain of nature," and he is inclined there- 
 fore to discuss results Helmholtz's theory of the origin of the 
 sun's heat, Lord Kelvin's calculations of the age of the earth, 
 and many other things of equal speculative interest but such 
 things prematurely considered divert the student's attention 
 away from elementary or irreducible ideas and conceptions with- 
 out which no analytical thinking is possible. 
 
 Results are fine inventions 
 
 For gentlemen who see; 
 But the micro-scope is needed 
 
 In this emergency. 
 
 Primarily this book has to do with class-room experiments in 
 physics. The best experiments are those that are homely and 
 simple, and suggestive rather than informing. The physics 
 lecturer should pull ideas out of things like a prestidigitateur, 
 and many of his demonstrations should be mere motion experi- 
 ments,* by which we do not mean experiments which have to do 
 with the theory of motion! We do not believe in the lecture 
 experiment which aims at a numerical result rather than at an 
 idea, decidedly we do not; and our pet aversion is the "study in 
 still life," if we may so describe the deadly lantern slide. Imag- 
 ine a sleight-of-hand performer making use of lantern slides! 
 It is unthinkable, and yet there never was a sleight-of-hand 
 performer who had one tenth of the resources of the physics 
 lecturer in college. In a two-years' course in elementary physics 
 
 * See pages 51 and 82 for examples. 
 
PREFACE. vii 
 
 we use, by actual count, 31 lantern slides; and out of a total of 
 48 lecture periods in the two years' course we have always used 8 
 or more for written tests on text-book and recitation work. This 
 may seem to some as a damning confession, but the constraint 
 which leads to analytical thinking cannot be made to operate 
 through any combination of talk and show. No Missourian was 
 ever regenerated by talk; and even when you have shown him he 
 remains what he was before a Missourian. 
 
 Secondarily this book is intended to set forth the possibilities 
 of an extended course in elementary dynamics, including the 
 dynamics of wave motion. 
 
 Some things in this book may seem to call for apology, for we 
 simply cannot refrain from an occasional diversion in the way of 
 poking fun, because so many things in teaching are funny, from 
 our point of view. 
 
 From our point of view. Let no one who reads this book lose 
 sight of the qualification. We might, of course, have dwelt 
 here and there on faults of our own, but such things are not funny 
 from our point of view. For example, we have many times 
 caught ourselves mistaking the fixity of an idea for its raison 
 detre, whereas the fixity of an idea is not the same thing as 
 reason, especially when it comes to making the idea clear to a 
 young student. Also we have many times been victims of the 
 "illusion of activity" by which we mean the sense of one's 
 effectiveness which comes from being wholly engaged in an 
 undertaking. "The lecture we gave this morning ; how complete 
 and perfect it was; what a masterpiece of edification!" The 
 only cure for this illusion is the reaction of the students, and no 
 teacher who seeks and uses this cure can be proud of his work; 
 although he may develop a humble but invincible self-respect 
 
 in that he never fails to do his best. 
 
 W. S. FRANKLIN, 
 
 BARRY MAcNurr, 
 
 October 27, 1917. 
 
TABLE OF CONTENTS. 
 
 PAGES 
 
 PART I. Mechanics 1-68 
 
 PART II. Heat 69-94 
 
 PART III. Electricity and Magnetism 95-138 
 
 PART IV. Light 139-156 
 
 PART V. Sound 157-162 
 
 PART VI. A simple treatise on wave motion 163-205 
 
 Appendix A. A visitor's laboratory of physics 207-210 
 
 DISCONNECTED ESSAYS. 
 
 On the Study of Science 2 
 
 Operative and inoperative definitions 70 
 
 The side-stepping of mathematics , 96 
 
 Bacon's New Engine 140 
 
 The philosophy of steam shovels and the philosophy of living 1 58 
 
 Science and technology versus the humanities in education. 164 
 The Traditive Lamp, or the proper method for handing down 
 
 the sciences to posterity 206 
 
PART I. 
 MECHANICS. 
 
 A very simple and complete treatment of the elementary 
 theory of elasticity is given on pages 182 to 218 of Franklin and 
 MacNutt's Mechanics and Heat, The Macmillan Co., New York 
 City, 1910; and some fundamental experimental demonstrations 
 in elasticity are described by Franklin and MacNutt on pages 
 90-100 of the Bulletin of the Society for the Promotion of Engineer- 
 ing Education, October, 1916. 
 
THE STUDY OF SCIENCE. 
 
 Everyone knows of the constraint which is placed upon men by the physical 
 necessities of the world in which we live, and although in one way this constraint is 
 more and more relieved with the advancement of the sciences, in another way it 
 becomes more and more exacting. It is indeed easier to cross the Atlantic ocean 
 now than it was in Lief Ericsson's time; but think of the discipline of the present 
 day shop and consider the rules of machine design, the rigors of the mathematical 
 sciences! Could even the hardy Norsemen have known anything as uncom- 
 promisingly exacting as these? 
 
 It is no wonder that easy-going believers in liberal education have always 
 looked with horror on the sciences, very much as softened men and women look 
 upon work. Liberalism means freedom, and "liberalism in education is the freedom 
 of development in each individual of that character and personality which is his 
 true nature." All this we accept in a spirit of optimism, believing men's true 
 natures to be good; but there is a phase of education which has but little to do, 
 directly, with character and personality, and we call to your attention this concep- 
 tion of liberalism in education in order that we may turn sharply away from it 
 as an incomplete conception which to a great extent excludes the mathematical 
 sciences. Indeed we wish to point out a condition in education which is the anti- 
 thesis of freedom; for the study of the mathematical sciences means a reorganiza- 
 tion of the work-a-day mind of a young man as complete in its sphere as the pupation 
 of an insect, and an exacting constraint is the essential condition of this reorganiza- 
 tion. 
 
 This statement is taken from The Study of Science, an introduction to Franklin 
 and MacNutt's Mechanics and Heat, The Macmillan Co., 1910. This essay is 
 reprinted in Bill's School and Mine, A Collection of Essays on Education, by W. S. 
 Franklin, published by Franklin, MacNutt and Charles, South Bethlehem, Pa. 
 In this reprint it is stated that the essay " is a sticker, and that any particular reader 
 who does not like it can leave it alone." But there is an increasing number of 
 young men in this world who must study science whether they like it or not; and 
 this essay is intended to explain this remarkable and in many respects distressing 
 fact. 
 
MASS AND WEIGHT. 
 
 " In defense of accuracy we must be zealous, as it 
 were, even to slaying." P. G. Tait. 
 
 Many of our teachers of physics and engineering are lacking 
 in complete precision of thought concerning elementary mech- 
 anics largely because they refuse to think of the fundamental 
 ideas of mechanics with a greater degree of mathematical or 
 logical precision than seems to them to be required by the degree 
 of measuring precision that satisfies the research physicist and 
 the engineer; but precision of thought is not dependent upon 
 precision of measurement. Indeed precision of measurement in 
 modern physics is mostly a tradition which we have inherited 
 from spherical astronomy; precision of thought is vastly more 
 important. 
 
 The term mass means quantity of material as measured by a 
 balance scale, and it is properly expressed in grams or pounds. 
 In commerce, however, we speak of the weight of a batch of 
 sugar or coal as weighed by a balance scale, and the word weight 
 so used means exactly the same thing as mass. 
 
 When we speak of the weight of a body in this text we mean 
 the force with which the earth pulls on the body, and it is properly 
 expressed in dynes or poundals. The pull of the earth on a 
 one-pound body in London is a definite force, it is extensively 
 used as a unit of force, and it is called the "pound." The 
 weight of a body, that is to say, the earth-pull on a body may be 
 properly expressed in "pounds" but the coal man's scales do not 
 determine the weight of a body in "pounds" 
 
 One "pound" of force gives an acceleration of 32.174 feet 
 per second per second to a mass of one pound, so that, according 
 to Newton's second law, one "pound" of force will give an 
 acceleration of one foot per second per second to 32.174 pounds 
 
 3 
 
4 CALENDAR OF LEADING EXPERIMENTS. 
 
 of material (one slug of material) . Therefore if force is expressed 
 in "pounds," mass in slugs and acceleration in feet per second 
 per second the simple form of the equation F = ma can be 
 used. 
 
 To reduce pounds of coal as measured by a balance scale to 
 slugs, divide by 32.174 (a pure number, not an acceleration). 
 The weight W of a body (the force with which the earth pulls 
 a body) is of course equal to mg, so that the mass of the body 
 m equals W/g. Thus the actual local weight of a body in 
 London "pounds 1 ' divided by the local value of g gives the 
 mass of the body in slugs*; but when your coal man sends you 
 a bill for what he calls a certain "weight" of coal, say 2,000 
 pounds, do not divide this kind of "weight" by g to get mass! 
 
 The system of mechanical units which involves the "pound" 
 as a unit of force is very widely and properly used, and every 
 teacher of elementary mechanics should use this system of units 
 unhesitatingly. We call the system the foot-slug-second system 
 for obvious reasons. There is no objection to the precisef use 
 of the foot-slug-second system except on the ridiculous assump- 
 tion that the primary business of the teacher is to promote the 
 metric system ; but it is improper for anyone to agree to use the 
 word weight for the force with which the earth pulls on a body 
 and then carelessly revert to the usage to the grocer and the 
 coal man. 
 
 The weight of a body in London in "pounds" is, of course, 
 equal to its mass in pounds, and it has, more than once, been 
 proposed to call the weight of a body in London its standard 
 
 * A perplexing mix-up of dimensions is involved in the double meaning of the 
 word pound. Out of respect for etymology we may postulate identical dimen- 
 sions to the pound of sugar and the "pound" of push or pull, or we may postulate 
 identical dimensions to pounds of sugar and slugs of sugar. We prefer the second 
 postulate, and, in accordance with this postulate, one must divide pounds of sugar 
 by 32.174 (a numeric) to get slugs of sugar, and one must divide the local weight 
 of a body in London "pounds" by the local acceleration of gravity (a denominate 
 number) to get mass in slugs. From this point of view the "pound" of force 
 has the same dimensions as the poundal of force. 
 
 t See page 50 of this volume for further discussion of this subject. 
 
MECHANICS. 5 
 
 weight and use the term standard weight instead of mass; but 
 there are two objections to this, namely, (a) The term mass is 
 almost universally recognized by physicists and chemists, and 
 (b) The mass of a given body is independent* of time and place, 
 it has to do only with an invariant* relation between the given 
 body and the standard kilogram (a piece of metal), and extra- 
 neous and confusing ideas would be introduced by the use of the 
 term standard weight because this term implies location and a 
 relationship between the given body and the earth. How awk- 
 ward it would be, for example, to be obliged always to speak and 
 think of the distance d between two points (x, y, z) and (#', y' ', 
 *') as [(x - x')* + (y - y')* + (z - z') 2 ] 1/2 . This is an- in- 
 variant and the most useful name or symbol for it is a name 
 or symbol which carries no redundant suggestions as to particular 
 axes of reference, and this would be true even if we had always to 
 make use of a particular set of reference axes in the measurement 
 of d. 
 
 Many engineering writers pretend never to use the poundal as 
 a unit of force, and we use the poundal chiefly as a name for a 
 step in an argument. It is better to name it than to refrain from 
 naming it, thereby pretending not to use it. Everyone who 
 expresses mass in pounds does use the poundal as a unit of force, 
 and every English-speaking person expresses mass in pounds 
 when he buys sugar or coal. If a man steals sugar no accounting 
 is necessary, at leas no book-keeping is necessary, and only 
 such men among English-speaking people can honestly claim not 
 to use the poundal as a unit of force! The question would 
 appear therefore to be a question of honesty versus dishonesty, 
 but it is not quite as bad as that, for the second horn of the 
 dilemma is unconscious dishonesty or sophistry, and this un- 
 conscious dishonesty has been carried to the limit by a recent 
 writer who speaks of the coal man's unit of mass as scholastic. f 
 
 * No consideration is here given to variations of mass as recognized in the 
 recent developments of the principle of relativity. 
 
 t Professor E. V. Huntington, of Harvard University. See American Mathe- 
 matical Monthly, January, 1917, page 16. In his recent articles on the fundamentals 
 
6 CALENDAR OF LEADING EXPERIMENTS. 
 
 of dynamics Professor Huntington has not contributed anything to the logical or 
 mathematical aspects of the subject which has not been known in all simplicity 
 and completeness since the days of Thomson and Tait, he is definitely astray 
 logically in his attempt to ignore the fundamental idea of mass (see page 10 
 of this volume), and he shows a peculiar lack of appreciation of the constraining 
 exactions which are placed upon the physicist by measuring operations. For 
 example, Professor Huntington says, with easy assurance that, "fundamental 
 units may be chosen at pleasure," whereas the choice of fundamental units is 
 governed by practical considerations; in the first place the fundamental units 
 must be preserved as material standards, and in the second place the fundamental 
 quantities must be susceptible of very accurate measurement because the definition of 
 a derived unit cannot be realized with greater accuracy than the fundamental quantities 
 can be measured. 
 
 Think of the years of confusion in electrical measurements when the theoretical 
 ohm could not be produced with greater accuracy than one or two per cent and 
 when anybody could make resistance measurements to one tenth of a per cent! 
 When we think of this old nightmare and read Professor Huntington 's argument 
 for the adoption of force as a fundamental quantity, what do you suppose we are 
 inclined to say? We refrain from saying it, at any rate. Professor Huntington 
 assures us that a spring can be easily preserved, whereas the best and most care- 
 fully "aged" measuring springs in existence grow perceptibly softer and softer in 
 use. Tempered steel is in a meta-stable condition and so also is hardened phosphor 
 bronze and fused quartz. Of course the "pound" of force, the pull of the earth 
 on the standard pound in London, is adequately preserved, but Professor Hunting- 
 ton seems nevertheless to think that a unit of force must be preserved in the form 
 of a stretched spring! 
 
 One of the very few redeeming features of the recent, long-drawn-out rehash of 
 elementary dynamics in Science and in the American Mathematical Monthly is the 
 article by Professor L. M. Hoskins of Stanford University, Science, April 23, 1915, 
 pages 608-611. The rehash is due primarily to Professor Huntington, and its 
 significance lies almost solely in its showing that nearly everybody seems to take 
 anybody seriously when anybody happens to be a Harvard professor. See page 
 34 of this volume for further comment. 
 
ON UNBALANCED FORCE. 
 
 When a car is being started by a locomotive the forward pull 
 of the locomotive on the car is greater than the backward pull 
 or drag of track and wind on the car; all the forces which act 
 on the car are together equivalent to a single force pulling for- 
 wards on the car, and this single force constitutes what is called 
 an unbalanced force, Whenever the velocity of a body is chang- 
 ing the body is being acted upon by an unbalanced force. As 
 an example let us consider a ball which is tied to a string and 
 twirled in a circle. All the forces acting on the ball, namely, the 
 gravity pull of the earth, the drag of the air and the pull of the 
 string, are together equivalent to a single force. 
 
 One of the most frequent perplexities of the student in regard 
 to unbalanced force comes from a wrong interpretation of the 
 principle of equality of action and reaction. In the above 
 example the string pulls inwards on the ball and the ball pulls 
 outwards on the string. We refer here to the two forces acting 
 at the point where the string is attached to the ball, and these 
 two forces act on different bodies as stated, namely, the string 
 pulls on the ball and the ball pulls on the string. Two such 
 forces which act on different bodies have nothing whatever to 
 do with the question as to whether the forces which act on a given 
 body are balanced or unbalanced. This matter is discussed more 
 at length on page 12 of General Physics. 
 
 The misunderstanding and false application of the principle 
 of equality of action and reaction is so common that every teacher 
 of physics should make a searching analysis of the matter; 
 he should not accept anyone as authority but he should think 
 for himself and he should by all means pay attention to simple 
 essentials. Do you agree as to what is said above concerning 
 unbalanced force? Do you agree that action and reaction are 
 equal and opposite forces acting on different bodies and there- 
 
 7 
 
8 CALENDAR OF LEADING EXPERIMENTS. 
 
 fore having nothing whatever, directly or indirectly, to do with 
 the question as to whether the forces which act on a given body 
 are balanced or unbalanced? If you are doubtful consider a 
 trade between two persons A and B. This trade is a purchase 
 from A's point of view and a sale from B's point of view, and 
 of course the sale aspect of this mutual operation is equal to its 
 purchase aspect; but everyone understands that a given person 
 may buy more than he sells or sell more than he buys. A trade 
 is a single thing, and yet it is very convenient to speak of a trade 
 as a sale or purchase in its relation to a given person. The force 
 action between two bodies A and B, involving the exertion 
 of equal and opposite forces on the two bodies, is a single thing, 
 a stress, let us say, and yet it is convenient to speak of a stress 
 as a force acting on A or as an equal and opposite force acting 
 on B. Indeed it is logically necessary to think of one aspect 
 of a stress when considering a given body. Iowa is north of 
 Missouri yes but Missouri is south of Iowa! He who would 
 argue that an unbalanced force action on a given body cannot 
 be because action and reaction are always exactly equal and 
 opposite is like the purist who would wish to invent a fancy way 
 of saying that Missouri is south of Iowa, fearing a verbal battle 
 with the man from Missouri who knows that Iowa is north of 
 Missouri! Imagine a man who, in considering that A is north 
 of B and that therefore B is the same distance south of A, 
 concludes that nothing can really be north or south of anything 
 else. The geometric sense of such a man would be on a par 
 with the mechanical sense of one who concludes that an un- 
 balanced force cannot exist because action and reaction are 
 always equal and opposite and on a par with the business sense 
 of a merchant who would conclude from the equal and opposite 
 aspects of a trade that it would be impossible to buy more than 
 he sells! 
 
 All the forces which act on a given body are often equivalent to a 
 single resultant unbalanced force, and it is such a force that changes 
 the velocity of the body. 
 
MECHANICS. 9 
 
 The recently developed electromagnetic theory (including the 
 theory of relativity) does not vitiate this point of view in the 
 least, although it does modify our notions of mass and point to 
 minute (ordinarily extremely minute) force actions between 
 accelerated bodies and the ether of space if we may use that term 
 merely to designate widespread associated energy. Indeed the 
 principle of relativity (for it should no longer be called a theory) 
 does not vitiate the principle of equality of action and reaction 
 if due attention be given to these etherial force actions. 
 
 A "body" includes everything inside of a certain bounding 
 surface, and at moderate velocities and accelerations only an 
 extremely minute fraction of the kinetic energy and momentum 
 of a body resides outside of this boundary, according ' to the 
 electromagnetic theory, and only an extremely minute fraction 
 of an accelerating force is balanced by force action across this 
 boundary;* and a recent attemptf to broaden the science of 
 dynamics by postulating an ether force which completely balances 
 every accelerating force is unjustifiable and indeed meaningless. 
 
 * The integral of the well-known Maxwellian stress across the bounding surface. 
 
 f Professor H. M. Dadourian's Analytical Mechanics. A very sane review of 
 this book by Professor E. W. Rettger is given in Science, January 23, 1914, pages 
 140-142. See page 34 of this volume for further comment. 
 
THE FUNDAMENTAL EQUATIONS OF DYNAMICS. 
 
 The recent attempt* to rule out mass as a fundamental con- 
 cept in dynamics makes it advisable to supplement what is 
 given in General .Physics on the fundamental equations of 
 dynamics (see pages 8 10). We are here concerned with the 
 acceleration of a body when acted upon by an unbalanced force. 
 The acceleration varies from body to body for a given identifiable 
 force, and it varies from force to force for a given body. These 
 are two equally fundamental modes of variation and both of 
 them must be formulated in the fundamental equation or 
 equations of dynamics, and of course this formulation must be 
 based on experiment. 
 
 Given three bodies A, B and C, and three identifiable 
 forces a, b and c. Let the acceleration of each body due to 
 each force be observed, the results being as shown in the accom- 
 panying table. 
 
 TABLE OF OBSERVED ACCELERATIONS. 
 
 Bodies 
 
 Forces 
 
 
 A 
 
 B 
 
 C 
 
 a 
 
 25 
 
 30 
 
 35 
 
 b 
 
 50 
 
 60 
 
 70 
 
 c 
 
 75 
 
 90 
 
 105 
 
 * By Professor E. V. Huntington who says that equation (2) below is a mathe- 
 matical consequence of equation (i) ! See Science March 3, 1916, page 315. 
 
 Equation (i) expresses the relation between the accelerations a and a' of a given 
 body due to two forces F and F', respectively, and pure logic would not know of 
 even the existence of any other body. See page 34 of this volume for further 
 comment. 
 
 10 
 
MECHANICS. II 
 
 Let us suppose that the table has been extended so as to include 
 a great many different forces and a great many different bodies, 
 then a careful inspection of the table would lead to the following 
 generalizations. 
 
 (a) If one force produces twice as much acceleration as another 
 force when acting on a given body, then the one force produces 
 twice as much acceleration as the other force when acting on 
 any body whatever. 
 
 (b) If one body is accelerated twice as much as another body 
 under the action of a given force, then the one body is accelerated 
 twice as much as the other body under the action of any force 
 whatever. 
 
 The experimental fact (a) makes it convenient to define the 
 ratio of two forces as the ratio of the accelerations they produce 
 when acting on a given body, because this ratio is the same for 
 all bodies. That is 
 
 Fa 
 
 where a is the acceleration of a given body produced by force 
 Fj and a! is the acceleration produced by force F f . 
 
 The experimental fact (b) makes it convenient to define the 
 ratio of the masses of two bodies as the inverse ratio of the 
 accelerations produced by a given force, because this ratio is 
 the same for all forces. That is 
 
 m a' 
 
 where a is the acceleration of body No. I and a' is the accelera- 
 tion of body No. 2 both produced by a given force, and m and 
 m' are the masses of the respective bodies. 
 
 Equations (i) and (2) are the fundamental equations of 
 dynamics. 
 
 Experiment shows that the ratio mfm' as defined by equation 
 (2) is exactly equal to the ratio of the masses of the two bodies 
 as measured on a balance scale, and since the balance scale 
 
12 CALENDAR OF LEADING EXPERIMENTS. 
 
 measures mass very conveniently and with extreme precision 
 it is best to define mass as measured by the balance scale and 
 accept equation (2) as an experimental discovery. 
 
 From equations (i) and (2) it is evident that the acceleration 
 of a body is proportional to the accelerating force and inversely 
 proportional to the mass of the body. 
 
 Experiment I. The authors have found it best to use only 
 extremely simple lecture experiments in the discussion of the 
 laws of motion. Taking a box in the hands, call attention to 
 the fact that you have to exert an upward force on the box to 
 balance the downward pull of the earth. Call on a young man 
 to pull on the box, and point out that in addition to supporting 
 the box against gravity you have to balance the force exerted 
 by the young man to keep the box stationary. 
 
 Point out that everything as stated applies to a box similarly 
 held in a steadily moving car or boat on a straight track or course, 
 so that the forces which act on a body are balanced when the 
 body is stationary or moving uniformly along a straight path 
 but surrounded on all sides by things moving along with it. 
 
 How about a body which moves steadily along a straight 
 path but not surrounded by bodies which move along with it? 
 Everyone knows that an active agent such as a horse or a steam 
 engine must pull steadily on such a body to keep it in motion. 
 If left to itself such a moving body quickly conies to rest. This 
 tendency for a moving body to come to rest is due to dragging 
 forces or friction exerted on the moving body by surrounding 
 bodies. Thus a moving boat is brought to rest by the drag of 
 the water when the propelling force ceases to act ; a train of cars 
 is brought to rest by a frictional drag when the pull of the 
 locomotive ceases; a box which is drawn steadily across the 
 table comes to rest when left to itself because of the dragging 
 force due to friction between the moving box and the table. 
 We must, therefore, always consider two forces when we think 
 of a body which is kept in motion like a car or boat, namely, the 
 propelling force due to some active agent and the dragging force 
 
MECHANICS. 13 
 
 of friction. Newton pointed out that when a body is moving 
 steadily along a straight path the propelling force is equal and 
 opposite to the dragging force of friction. See General Physics, 
 Art. 3, page 7. 
 
 Let us now consider the force* which must act on a body 
 which is changing its velocity, upon a body which is being started 
 or stopped, for example. Every one has noticed how a mule 
 strains at his rope when starting a canal boat, especially if the 
 boat is heavily loaded, and how the boat continues to move for 
 a long time after the mule ceases to pull. In the first case the 
 pull of the mule greatly exceeds the backward drag of the water, 
 and the velocity of the boat increases; in the second case the 
 backward drag of the water exceeds the pull of the mule, for the 
 mule is not pulling at all, and the velocity of the boat decreases. 
 When the velocity of a body is changing the forces which act 
 on the body are unbalanced. We may therefore conclude that 
 the effect of an unbalanced force on a body is to change the 
 velocity of the body, and it is evident that the longer the un- 
 balanced force continues to act the greater the change of velocity. 
 Thus if the mule ceases to pull on a canal boat for one second 
 the velocity of the boat will be but slightly reduced by the 
 unbalanced backward drag of the water, whereas if the mule 
 does not pull for ten seconds the velocity of the boat will be 
 reduced to a much greater .extent. In fact the change of velocity 
 of a body due to a given unbalanced force is proportional to the 
 time that the force continues to act. This is exemplified by a body 
 falling freely under the action of the unbalanced pull of the 
 earth on the body; during one second the body gains about 32 
 feet per second of velocity, during two seconds it gains twice as 
 much velocity (about 64 feet per second), and so on. 
 
 Everyone knows what it means to give an easy pull or a hard 
 pull on a body. That is to say, we all have an idea that a force 
 may be large or small. Everyone knows also that under the 
 
 * If several forces act we here refer to the single force which is equivalent to 
 them all. Translatory motion, only, is here considered. 
 
CALENDAR OF LEADING EXPERIMENTS. 
 
 action of a hard pull a canal boat will get under way more quickly 
 (gain velocity faster) than it will under the action of an easy 
 pull, and a precise statement of the effect of an unbalanced force 
 on a body must correlate the value of the force and the rate at 
 which it imparts velocity to the body. This seems a very difficult 
 thing, but its difficulty is in large part due to the fact that we 
 have not yet agreed as to what we mean when we say that one 
 force is exactly three or four or any number of times as large as 
 another force. Suppose therefore that we agree to call one force 
 twice as large as another when it will produce in a given body twice 
 as much velocity in a given time (remembering that we are talking 
 about unbalanced forces). As a result of this definition we may 
 state that the amount of velocity produced per second in a given 
 body by an unbalanced force is proportional to the force. 
 
 Note. This definition of the ratio of two forces is by no 
 means sufficient to establish the equation F/F' = a/a'. A very 
 wide range of experiment is necessary to show that this equation 
 is true for all bodies if it is true for one. See discussion on 
 pages 10 and n of this volume. 
 
 2. An experiment with two similar blocks. Drop two similar 
 blocks together and call attention to the fact that they reach 
 
 mass m 
 
 mass gm 
 
 yo\ 
 
 force F 
 
 'acceleration a 
 
 mass Z 
 
 force gf 
 
 acceleration 
 
 force F 
 
 Fig. 1. 
 
 acceleration a \ 
 I 
 
 Fig. 2. 
 
 Fig. 3. 
 
 the floor at the same time. Two blocks fall the same distance 
 in a given time, therefore they fall with the same increasing 
 velocity and with the same acceleration as one block.* 
 
 * The friction of the air is very small and it is ignored. 
 
MECHANICS. 15 
 
 The mass of the two blocks (as measured on a balance scale) 
 is 2w, where m is the mass of one block; and the pull of 
 gravity on the two blocks is 2F, where 'F is the pull gravity 
 on one block. Therefore when m and F are both doubled 
 the acceleration a remains unchanged. 
 
 This argument is not exactly in accordance with the fundamental equation (i) 
 on page ii of this volume. In all strictness it should be as follows: Let us adopt 
 F/F' = a/a' as the definition of the ratio of two forces, and let us assume that the 
 pull of the earth on the two blocks (mass 2m) is twice as great as the pull of the earth 
 on one block (mass m), as indicated in Fig. 2. Imagine the force in Fig. 2 to be 
 reduced to half-value; then according to the fundamental equation (FfF' = a/a') 
 the acceleration would be reduced to %a, as indicated in Fig. 3. Comparing 
 Figs, i and 3 we see that when the mass of a body is doubled a given force F pro- 
 duces half as much acceleration, or m/m' = a' /a, which is equation (2) on page 
 ii of this volume. Therefore the above assumption is justified, that is, the weight 
 of a body (pull of earth on the body) is proportional to the mass of the body as 
 measured by a balance scale, and the ratio of two masses as defined by the equation 
 m/m' = a' fa is the same as the ratio of two masses as measured by a balance scale. 
 
 3. Reaction experiment. An electrically driven toy loco- 
 motive runs on a circular track laid near the edge of a large 
 horizontal disk which is supported on ball bearings so as to turn 
 freely about a vertical axis. When the locomotive starts the disk 
 is set turning backwards, and when the locomotive stops the 
 disk is set turning forwards. An elevated railway structure has 
 to be braced near a station so as to withstand the backward 
 push of the locomotive as it starts a train and so as to withstand 
 the very great forward push of an entire train which is brought 
 to rest quickly by braking. 
 
 Remark. Is the kick of a gun the reaction which corresponds 
 to the push of the powder gases on the projectile? It is not. 
 The kick of the gun is due to the backward push of the powder 
 gases on the breech plug of the gun, and this is of course equal 
 to the forward push of the breech plug on the powder gases (ac- 
 tion is equal to reaction and opposite thereto) . Also the forward 
 push of the powder gases on the projectile is equal to the back- 
 ward push of the projectile on the powder gases (action is equal 
 to reaction and opposite thereto). But the forward push of the 
 
16 CALENDAR OF LEADING EXPERIMENTS. 
 
 breech plug on the powder gases is not equal to the backward 
 push of the projectile on the powder gases; the powder gases are 
 being rapidly accelerated and the forces which act on the powder 
 gases are not balanced. 
 
IMPULSE-VALUE OF A FORCE. 
 
 Consider an unbalanced force F = ma acting on a body, and 
 let us assume F to be constant. Multiply both members 
 of this equation by the time / during which the force continues 
 to act, and we have Ft = mat. But at (the rate of gain of veloc- 
 ity multiplied by elapsed time) is the total velocity v produced 
 by the force F during the time t. Therefore we have : 
 
 Ft = mv 
 
 The product mv is called the momentum of the body, and the 
 product Ft (the value of a force multiplied by the time it con- 
 tinues to act) is called the impulse-value of the force. 
 
 If the force F is not constant we have F'dt = m-dt = ni'dv, and 
 
 at 
 
 jF'dt = fm-dv = mv. The integral /F'dt is called the impulse value of the force 
 F and it is equal to the momentum mv which is produced. 
 
 The above equation applies not only to the starting of a body 
 but also to the stopping of a body. For example, a hammer of 
 mass m moving at velocity v strikes an obstacle and is brought 
 to rest. Let F be the average force with which the obstacle 
 pushes backwards against the hammer while stopping it (this is, 
 of course, equal to the average force exerted by the hammer on 
 the obstacle), and let / be the time during which this average 
 force acts. Then F = ma, where F is the average backward 
 force exerted on the hammer (or the average forward force 
 exerted by the hammer) and a is the average rate at which 
 the hammer is losing velocity. But during time / the ham- 
 mer loses all of its velocity so that at = v, or Ft = mat = mv, 
 where v is the initial velocity of the hammer (the total velocity 
 lost by the hammer). Therefore the impulse- value of the force 
 exerted by the hammer is equal to the momentum of the hammer. 
 The force exerted by a bullet is properly expressed in terms of its 
 impulse value (which is equal to the momentum mv of the bullet). 
 3 17 
 
l8 CALENDAR OF LEADING EXPERIMENTS. 
 
 4. The anvil experiment. A heavy anvil rests upon a yielding 
 support, and yet it gives a satisfactory base upon which to flatten a 
 piece of iron by a hammer blow. The enormous force exerted by 
 the hammer lasts for a very short time, say, one ten-thousandth 
 of a second, and, although the anvil is set in motion by this 
 force, the actual distance moved by the anvil in the ten-thous- 
 andth part of a second is very small and entirely negligible from 
 the point of view of the blacksmith. The anvil continues to 
 move, however, long after the hammer blow, and as it con- 
 tinues to move it compresses its elastic support for, say, a tenth 
 of a second. But to stop the anvil in a tenth of a second the 
 average force exerted on the anvil by the supporting structure 
 need be only one thousandth as great as the average value of 
 the force (exerted by the hammer) which set the anvil in motion 
 in one ten- thousandth of a second. 
 
 5. The coin and card experiment. Place a small card hori- 
 zontally on the end of the finger and on top of the card place a 
 small coin. A quick thump against the edge of the card drives it 
 out from under the coin, and the coin is left on the end of the 
 finger. During the very short time required for the card to 
 slide out from under the coin, the card exerts a forward drag 
 on the coin and imparts to the coin a small velocity; but the 
 coin has time to travel only a very short distance before the card 
 is gone, and the coin, then sliding along on the end of the finger, 
 is very soon brought to rest by friction. The whole distance 
 moved by the coin while being started by the forward drag of 
 the card and while being stopped by the backward drag of the 
 finger is, perhaps, one hundredth of an inch or less. 
 
 This coin-and-card experiment is worth while if it is carefully 
 analyzed as above, but, like most paradoxical experiments, it is 
 worse than useless if not accompanied by careful analysis. 
 
 6. The cord paradox. A heavy metal ball B is supported 
 by a small cord c as shown in Fig. 4, and a quick hammer blow, 
 as indicated, breaks the heavy cord C. The hammer causes a 
 
MECHANICS. 
 
 very large tension in cord C for a very short time thus setting 
 the ball B in motion. Then the ball continues to move for a 
 relatively long time, stretching cord c. 
 Thus cord c has a very long time in which 
 to stop the ball whereas cord C had an 
 extremely short time in which to start the 
 ball. Thus if cord C is under excessive 
 tension for a ten-thousandth of a second, 
 and if ball B thus quickly set in motion 
 continues to move downwards for a tenth 
 of a second before it is brought to rest by 
 the increasing tension of cord c, then the 
 average excess tension in cord c (in ad- 
 dition to tension due to the steady pull of the earth on B) 
 will be only one thousandth as large as the average tension in 
 cord C during the ten-thousandth of a second. 
 
UNIFORMLY ACCELERATED TRANSLATORY 
 MOTION. 
 
 7. Experiment with a rolling disk. The simplest method of 
 obtaining a close approximation to uniformly accelerated transla- 
 tory motion with small acceleration is to use a heavy disk with 
 a small axle rolling down an inclined track. According to equa- 
 tion (ii) on page 23, General Physics, this disk starting from 
 rest (PI = o) should travel a distance proportional to the square 
 of the elapsed time. Lay off from the starting point of the 
 rolling disk a series of distances proportional to I, 4, 9, 16, 25, 
 etc., and mark the points so located. Then as the disk rolls 
 down the inclined track it will mark time as it passes these 
 various points. In order that this experiment may be effective, 
 the distances should be chosen, or the inclination of the track 
 should be adjusted so as to make the rolling disk mark seconds 
 as it passes the various points, and the accuracy with which 
 the disk marks seconds may be appreciated by watching the 
 rolling disk and listening to a telegraph sounder operated by a 
 contact device on a clock pendulum. 
 
 8. Shooting at a falling ball. When the initial velocity v\ of 
 a body is zero, or when it is vertical, we have the ordinary case 
 of a falling body. In this case equation (ii) on page 23, General 
 Physics, can be solved by simple algebra, and all calculations 
 can be made by simple arithmetic, the only complication being 
 that vi is to be considered negative when it is upwards. When 
 the initial velocity v\ is not vertical, as in the case of a tossed 
 ball, the falling body is called a projectile. In this case the entire 
 argument as given on page 23, General Physics, holds good, but 
 geometric addition must be substituted for arithmetical addition 
 in equation (ii). This equation, being interpreted as shown in 
 Fig. 5, means that the position A' of the ball after / seconds is 
 
 20 
 
MECHANICS. 
 
 21 
 
 found by adding together the vectors v\t and \gfi. If the ball 
 
 at B is released at the instant the projectile leaves the muzzle 
 
 of the gun at A , then both 
 
 balls, A' and B', will always 
 
 be at the same distance \gfi 
 
 from the line AB. Therefore if 
 
 the gun is aimed accurately at 
 
 the point B the two balls will 
 
 always come together at a 
 
 point p. 
 
 The most satisfactory ar- 
 rangement is as follows : A wooden ball has a hole bored through 
 it so that it will slip over a half-inch brass tube, and a sling with 
 rubber bands is arranged to be released by a trigger and throw 
 the ball along the tube. This gun is sighted at ball B by look- 
 ing through the brass tube, and as ball A slips off the end of 
 the brass tube it comes against a very light wire bridge which 
 spans across between two brass clips thus breaking an electric 
 circuit. This circuit includes an electromagnet which supports 
 ball B. 
 
 Fig. 5. 
 
CENTER OF MASS. 
 
 The study of dynamics always begins with the study of 
 'translatory motion. Thus the discussion on pages 3-13, General 
 Physics, refers to translatory motion. A material particle is an 
 ideal body so small that the only sensible motion of which it is 
 capable is translatory motion, and the term material particle is 
 used in dynamics merely to direct one's attention narrowly to 
 translatory motion. Thus, if one is concerned only with trans- 
 latory motion, one may think of a body of any size and shape 
 as a particle located at the center of mass of the body. The 
 center of mass of a body is the point at which a single force must 
 be applied (that is the line of action of the single force must pass 
 through the center of mass) to produce translatory motion only. 
 
 9. Experiment with a slim stick. The experiment described 
 in Art. 9, page 14, General Physics, is quite effective if tried 
 before a class, but every student should try it for himself. 
 
 10. Experiment with hammer and stick as described on page 
 15, General Physics, is extremely important as elucidating, the 
 definition of center of mass. 
 
 11. Experiment showing the identity of center of mass and 
 center of gravity. The center of gravity of a body is the point 
 of application of the single force which is the resultant* of all 
 the forces with which the earth pulls on the various parts of the 
 body. When a body is supported by a single force the line of 
 action of the force must pass through the center of gravity of 
 the bcdy. Therefore the center of gravity of a slim stick can 
 be located by balancing the stick on a knife edge; and the point 
 so found coincides with the center of mass as located in experi- 
 ment 9. 
 
 * See Art. 23 of Franklin and MacNutt's Elementary Statics, published by 
 Franklin, MacNutt and Charles, South Bethlehem, Pa. 
 
 22 
 
TRANSLATORY MOTION IN A CIRCLE. 
 
 \ 
 
 Fig. 6. 
 
 It is important to understand that translatory motion does not 
 mean, necessarily, straight line motion. Thus Fig. 6 indicates 
 a stick performing translatory motion in a circle, and, as explained 
 on page 14, General Physics, the entire 
 stick may be thought of as concentrated 
 at its center of mass C. 
 
 I 
 
 When a ball is twirled on a string the 
 ball itself makes one revolution every time 
 it goes round the circle. This is evident 
 when one considers that one particular 
 part of the ball always faces inwards. The 
 motion of the ball is therefore a combina- 
 tion of translatory motion and rotatory 
 motion, but the rotatory motion once established is constant 
 (unchanging in value and about an axis which does not change 
 its direction) whereas the translatory motion is variable (chang- 
 ing in direction). 
 
 12. Experiment with ball and string. The authors have al- 
 ways found it worth while to go through with the discussion of 
 Art. 31, pages 3840, General Physics, in the class room with 
 ball and string in hand. It is well to point out that the pull of 
 the string is the only force acting on the ball, drag of air and pull 
 of gravity being assumed to be negligible. 
 
 A ball is tossed through the air; what forces act on the ball 
 after it leaves the hand? If any one in answer to this question 
 should seem to think that the continued motion of the ball 
 indicates a sort of continued driving force exerted by the hand, 
 let him remember that force means always an actual push or pull 
 BY something ON something; and very certainly the hand exerts 
 neither a push nor a pull on the ball after the ball has left the 
 
 23 
 
24 CALENDAR OF LEADING EXPERIMENTS. 
 
 hand! The gravity pull of the earth continues to act on the 
 ball and the air continues to exert a backward drag on the moving 
 ball; and these continued forces modify the motion of the ball 
 greatly. 
 
 But does not something pull outwards on a ball which is being 
 twirled on a string? If such a force exists it must be exerted by 
 something. The pull of gravity is exerted by the earth (although 
 the connecting mechanism between the earth and the ball is 
 invisible), the backward drag of the air on the moving ball is 
 due to actual contact of the air with the ball, and the pull of the 
 string on the ball is evident to anybody. There is no other force 
 acting on the ball. 
 
 Important example of motion in a circle. The belt wheel. 
 A wheel is driven by belt as shown in Fig. 7. When the belt is 
 
 driven at higher and higher speeds, 
 it presses more and more lightly 
 against the face of the wheel and 
 slips more and more easily. At a 
 certain critical speed the belt does 
 not press against the face of the 
 wheel at all. 
 
 Consider a short portion ab of 
 Fig. 7. 
 
 the belt. This portion or particle 
 
 travels in a circular path as it passes around the pulley. The 
 two forces TT act on the small portion due to the tension 
 T of the belt, the resultant of the two forces TT is a force F 
 acting towards the center of the wheel, and the force F is equal 
 to A/ T/r, where A/ is the length of the arc ab and r is the 
 radius of the wheel.* 
 
 At zero belt-velocity, the entire force F is effective in pushing 
 the portion ab of the belt against the pulley face. As the 
 velocity of the belt increases, a larger and larger portion of F 
 is used to constrain the portion ab to its circular path or orbit 
 
 * See Franklin and MacNutt's Mechanics and Heat, page 98- 
 
MECHANICS. 25 
 
 (that is, to produce the necessary acceleration of ab towards 
 the center of the pulley), and a smaller and smaller portion of 
 F is available for pushing the portion ab against the pulley face. 
 Let m-Al be the mass of the portion ab of the belt, and 
 let v be the velocity of the belt. Then the inward acceleration 
 of the portion ab is v z /r and the force required to produce this 
 inward acceleration is m A/ X v 2 /r. Therefore when 
 
 or when 
 
 . 
 
 m 
 
 then the portion ab of the belt is not pushed against the pulley 
 at all. It is interesting to note that this critical belt-velocity 
 (^T/m) is the velocity at which a wave or bend would travel, 
 with unchanged shape, along the belt. See pages 509-511, 
 General Physics. 
 
MOTION OF THE CENTER OF MASS OF A SYSTEM. 
 
 The idea of translatory motion can be made to serve as a basis 
 for the description of any complicated motion whatever; all 
 that is necessary is to look upon the moving body or bodies as 
 a collection of particles and consider the varying position, velocity 
 and acceleration of each particle. 
 
 A collection of particles treated in this way is called a system 
 of particles or simply a system. Thus a rotating wheel is a system 
 of particles, and a portion of flowing water is a system of particles. 
 
 When the vector sum of all the forces which act on the particles 
 of a system is zero, the center of mass of the system either remains 
 stationary or continues to move with uniform velocity in a straight 
 line. This may be illustrated by considering the rotation of a 
 balanced emery wheel. To say that a rotating wheel is balanced 
 means that the center of mass of the wheel lies in the axis of the 
 shaft so that the center of mass remains stationary as the wheel 
 rotates, and no force need be exerted on the shaft by the bearings 
 except the upward force required to support the wheel and axle 
 against the pull of gravity. 
 
 When the vector sum of all the forces which act on the particles 
 of the system is not equal to zero, then the center of mass of the system 
 is accelerated in the direction of the resultant of all the forces, the 
 acceleration is proportional to the resultant of all the forces and 
 inversely proportional to the total mass of the system. Indeed in 
 this case we have the equation F = MA, where A is the 
 acceleration of the center of mass of the system of particles, F 
 is the vector sum of all the forces which act on the system, and 
 M is the total mass of the system.* This may be illustrated by 
 considering the rotation of an unbalanced emery wheel, an 
 emery wheel of which the center of mass lies at a distance r 
 from the axis of rotation. Then, as the wheel rotates, the 
 
 * See Franklin and MacNutt's Mechanics and Heat, page 112 
 
 26 
 
MECHANICS. 27 
 
 center of mass describes a circular path of radius r, the accelera- 
 tion of the center of mass is equal to v 2 /r or to 4?r 2 w 2 r at each 
 instant, and a side force equal to ^.irVrM and parallel to r at 
 each instant must act on the axle to constrain the center of mass 
 to its circular path, precisely as if the entire mass of the wheel 
 were concentrated at its center of mass. The force here described 
 is the force which must be exerted on the emery wheel shaft by 
 the bearings, and, of course, the shaft must exert an equal and 
 opposite force on the bearings. This reacting force is usually 
 sufficiently large in the case of a rapidly rotating unbalanced 
 wheel to cause very violent vibrations of the supporting structure. 
 
 13. Emery-wheel experiment. Mount an iron disk on a 
 small spindle mounted in bearings and balance the disk with a 
 removable bolt or cap screw near one edge so that the disk can 
 be easily unbalanced by removing this cap screw. Remove the 
 spindle from its bearings and place it upon two parallel level 
 straight edges, and show that it is balanced whe"n the cap screw 
 is in place and unbalanced when the cap screw is removed. Then 
 replace the spindle in the bearings and drive in balanced and 
 unbalanced conditions. 
 
 14. Earth and moon experiment. The earth and moon rotate 
 about their common center of mass once every lunar month, 
 and this center of mass describes a smooth elliptical orbit about 
 the sun once a year. This motion of the earth and moon may 
 be illustrated as follows: A large ball and a small ball are tied 
 together with a string, and the center of mass of the two balls 
 (with the string stretched) is marked by a piece of red flannel 
 tied to the string. The balls are then tossed through the air in 
 such a manner as to cause them to rotate and keep the connecting 
 string stretched tight, and the piece of red flannel describes a 
 smooth curve. This experiment is not very striking because the 
 eye naturally, and in spite of every effort to the contrary, takes 
 the large ball as a basis of reference, so that the small ball and 
 the red flannel both seem to rotate around the large ball. 
 
SPIN-INERTIA.* 
 
 The spin-inertia K of a body about a given axis may be 
 defined as T/a, where T is the torque (unbalanced) which 
 acts on the body and a is the spin-acceleration produced by T 
 (the torque T is of course exerted about the given axis), or K 
 is numerically equal to the torque T required to produce unit 
 spin-acceleration (one radian per second per second). 
 
 15. Experiment with a slim stick. A round stick about two 
 feet long and half an inch in diameter is tapered to a point at 
 one end so that it may be set spinning by thumb and finger, 
 (a) Show that a moderately small torque acting for a very short 
 time sets the stick spinning rapidly about its longitudinal axis 
 (00, Fig. 15, General Physics) so that the spin-inertia of the 
 stick about the axis 00 is small, (b) Show that a much larger 
 torque acting for a longer time produces a much slower spin 
 about a transverse axis (00, Fig. 16, General Physics) so that 
 the spin-inertia of the stick about the transverse axis is large. 
 
 1 6. Direct determination of spin-inertia. A large wheel is 
 mounted on a shaft of which the radius is r feet, and the shaft 
 is supported in ball bearings. A string is wound around the 
 shaft, a spring scale is attached to the string, a steady pull of 
 F poundals is exerted on the string for, say, exactly 10 seconds 
 (= /), and the length / of string unrolled during this time is 
 measured. 
 
 The torque T exerted on the wheel is equal to Fr, where r 
 is the distance from axis of shaft to middle of string, and the spin- 
 acceleration of the wheel is equal to T/K radians per second 
 per second according to equation (7), page 22, General Physics. 
 Therefore the spin-velocity 5 which is gained by the wheel 
 during / seconds is s = (T/K)t. Now the average spin- velocity 
 
 * Usually called moment of inertia. 
 
 28 
 
MECHANICS. 29 
 
 during the t seconds is %s (see page 23, General Physics) and 
 therefore the total number of radians turned during the time t 
 is %st or %(T/K)t 2 , or the total number of revolutions turned 
 in the time t is %(T/K)P -T- 2?r so that the length, /, of string 
 unrolled during the time t is / = [%(T/K)P + 2ir} X 2irr so 
 
 TPr 
 that / = \~~JF~ in which K is the only unknown quantity 
 
 and its value may therefore be calculated. The result will be 
 expressed in pound (feet). 2 
 
 Note i. The force F may be expressed in "pounds," the 
 torque T in "pound "-feet and the moment of inertia in slug- 
 (feet) 2 . See pages 50 and 51 of this volume. 
 
 Note 2. This experiment may be arranged in accordance 
 with the discussion in Art. 34, General Physics. 
 
 Note 3. The wheel above mentioned may be a circular disk 
 of solid iron so that its spin-inertia may be calculated by the 
 formula given in the table on page 22, General Physics (the 
 spin-inertia of the shaft may be neglected because the above 
 measurement is not very precise) and thus the experimentally 
 determined value of K may be compared with its calculated 
 value. 
 
 Note 4. A very instructive modification of this experiment 
 is as follows: Instead of the wheel let four slim rods be fixed to 
 the shaft like the spokes of a wheel, and determine the value 
 of K of this structure as explained above. Then attach massive 
 steel balls to the spoke-like arms each at a distance p feet from 
 the axis of the shaft, and determine anew the spin-inertia K'. 
 Then K f K will be found equal to mp 2 where m is the 
 combined mass of the steel balls. It is best, however, according 
 to the authors' experience, relegate quantitative experiments 
 to the laboratory where the student makes all the measurements. 
 
 Derivation of the equation K = Sr 2 -Aw. The student of 
 mathematics is usually led to interpret the equation 
 
30 CALENDAR OF LEADING EXPERIMENTS. 
 
 fofi-dx = J* 3 + C 
 thus 
 
 f The function whose deriva- 1 
 
 i f == x I C~, 
 
 I tive with respect to x is x 2 \ 
 
 and from this point of view it is easy to determine the spin- 
 inertia K of a body of regular shape as on pages 594-597, 
 General Physics, where the derivative of K with respect to a 
 chosen variable is derived, etc. It is easy to alter this discussion 
 so as to use equation (7), page (22), General Physics, namely, 
 T = Ka, instead of the equation W = %Ks 2 . Consider the 
 added mass 2irAr'Ar\ its sidewise velocity (at right angles 
 
 to r) is rs, its sidewise acceleration is r -r or ra, and the 
 
 sidewise force AF which must act on it to accelerate it is 
 AF 2-jrAr-Ar X ra. Therefore a portion AT of the total 
 torque T (= Ka) which is acting on the body is used to 
 accelerate the added material, namely, 
 
 AT = r-AF = 2irAr-Ar X ra X r. 
 
 Therefore since AK = AT /a we have AK = 2wAr 3 -Ar so that 
 the derivative of K with respect to r is 2irAr 3 . 
 
 The more general equation K = Sr 2 -Aw may be derived as 
 follows: Let us consider a small particle of the body of mass 
 Am at a distance r from the axis of rotation. The sidewise 
 velocity of Am (at right angles to r) is rs and the sidewise 
 acceleration of Am is ra. Therefore the sidewise force which 
 must act on Am is AF = Am X ra and the torque value of this 
 force about the axis of rotation is AT = Am X ra X r, so that 
 the total torque T acting to produce spin-acceleration is 
 T = a2r*-Am so that K = 2> 2 -Aw. 
 
THE GYROSCOPE OR GYROSTAT. 
 
 It is scarcely possible to bring out the simple meaning of the 
 discussion of Figs. 390, 396 and 39^, General Physics without 
 showing the experiment as follows: 
 
 17. Experiment with a bicycle wheel. (a) Hold a bicycle 
 wheel (not spinning) as indicated in Fig. 8, supporting both ends 
 
 west O 
 
 north 
 
 north 
 
 west 
 
 Fig. 8. 
 
 of the spindle by the hands. Release the spindle at one end 
 and keep it supported at end 0, catch it again immediately 
 and call attention to the fact that eastward velocity has been 
 given to every particle in the upper half of the wheel and westward 
 velocity to every particle in the lower half of the wheel by the unbal- 
 anced torque due to the gravity pull of the earth on the wheel. 
 
 (b) Hold the spinning wheel and release it in the same way, 
 catch it again immediately and call attention to the fact that 
 eastward velocity has been given to every particle in the upper half 
 of the wheel and westward velocity to every particle in the lower 
 half of the wheel by the unbalanced torque due to the gravity pull of 
 the earth on the wheel. 
 
 (c) Hold the spinning wheel and release it as before and allow 
 
32 CALENDAR OF LEADING EXPERIMENTS. 
 
 it to precess steadily, and point out the meaning of Fig. 38, 
 General Physics. 
 
 The essential steps in the analysis of the sidewise accelerations 
 of the particles of a rotating disk in precession are given on pages 
 168-169 of Franklin and MacNutt's Mechanics and Heat. 
 
 There are two distinct aspects of the problem of the simple 
 gyroscope, namely, (a) To determine the torque required to 
 produce a specified precession, and (b) To determine the pre- 
 cession produced by a given torque. The first aspect of the 
 problem is very much simpler than the second. Indeed the 
 first aspect is a problem in differentiation, that is to say, it turns 
 out to be such when it is fully developed; whereas the second 
 aspect turns out to be a problem in integration. Now any prob- 
 lem in integration involves the evaluation of integration constants 
 (on the basis of initial conditions supposed to be given), and the 
 oscillatory movements of the gyroscope as very briefly described 
 on pages 42 and 43 of this volume are related to these integration 
 constants. 
 
 Why is it that one does not use the moment of inertia about 
 the axis T in Fig. 37, General Physics, in the equation T Ka? 
 One certainly would use the moment of inertia about T if the 
 wheel were not already spinning! Answer: One does use the 
 moment of inertia about T and several things not mentioned 
 in the bob-tailed theory (the simple theory as given in General 
 Physics'), unless one thinks of problem (b) as the simple inverse 
 of problem (a). 
 
WHAT IS LACKING IN THE DISCUSSION OF FIGS. 140, 
 AND 146 IN General Physics* 
 
 It is assumed in the discussion of Figs. 140, and 146 and again 
 in the discussion of Figs. 37 and 38 in General Physics that spin- 
 velocities can be added by the parallelogram law. The proof of 
 this is outlined in Franklin and MacNutt's Mechanics and Heat, 
 page 176. 
 
 It is the validity of the parallelogram law that justifies the 
 representation of spin-velocity by a line or arrow as specified 
 on page 19, General Physics. Similarly the parallelogram law 
 must be valid for the addition of torques to justify the repre- 
 sentation of a torque by a line or arrow as specified on page 19 
 of General Physics. The demonstration of the parallelogram 
 law for the addition of torques is given in Franklin and Mac- 
 Nutt's Mechanics and Heat, page 177. 
 
 It is utterly inadequate to say that spin-velocity and torque 
 are "directed quantities and therefore to be added by the 
 parallelogram law."f This is evident when we consider that a 
 displacement of a body through a certain angle <j> about a 
 given axis is not to be distinguished in its mode of specification 
 from such quantities as spin-velocity and torque, whereas angular 
 displacements cannot be added by the parallelogram law. 
 
 * One thing that is lacking grows out of the assumption that problem (6) as 
 described on page 32 is the simple inverse of problem (a). This deficiency most 
 beginners will accept uncomplainingly, we are sure of that because we have men- 
 tioned integration in connection therewith. We refer here, in particular, to 
 deficiencies that are comparatively easy to supply, but which simply must not be 
 lost sight of by accepting plausibility for rigor. 
 
 f Professor Rettger's review of the second edition of Dadourian's Analaytical 
 Mechanics (Science, August 26, 1916, page 279) calls attention to a fault which be- 
 comes more serious in Professor Dadourian's later attempt at defense. Plausi- 
 bility is in many cases necessary in the presentation of a mathematical subject to 
 young men, but to think of plausibility as the same thing as rigor is fatal. 
 
 4 33 
 
34 CALENDAR OF LEADING EXPERIMENTS. 
 
 One of the best discussions of vector quantity is to be found in 
 the first chapter of Abraham and Foppl's Theorie der Electrizitdt, 
 Vol. I, third edition, Leipzig, 1907. 
 
 Science, even in its elements, presents serious difficulties. 
 Anyone who thinks otherwise knows only pseudo-science as 
 P. G. Tait has said,* and pseudo-science is the bane of science 
 teaching.f Indeed the most serious fault with our physics teach- 
 ing is that our physicists who know all that has been thought 
 and all that has been done in physics (and there are many of our 
 physicists who do) hold themselves aloof while half-educated 
 men, as in the recent rehash of elementary mechanics in Science, 
 talking as oracles from out of their dignified institutional settings, 
 make pi of everything! 
 
 * See introduction to Tait's Heat. 
 
 t The most convincing statement of this fact has been made by a student of 
 the Classics. See remarkable article by Paul Shorey in the Atlantic Monthly 
 for June and July, 191?. 
 
THE USE OF THE GYROSCOPE AS A RUDDER 
 CONTROL. 
 
 Consider a freely precessing gyroscope, the precession being due 
 to the pull of gravity. If the precession is hindered by letting the 
 end of the gyroscope axle (see Fig. 39^, General Physics} come 
 against a stop, the gyroscope drops at once like any ordinary inert 
 body. In order that the gyroscope may stand out horizontally 
 in spite of the downward pull of gravity, the gyroscope must be 
 free to precess about a vertical axis. 
 
 An important use of the gyroscope is for automatically con- 
 trolling the rudder of a torpedo or of a flying machine. In 
 such a case the gyroscope constitutes a frame-work which keeps 
 an unchanged direction in space, and a sudden veering of the 
 torpedo or flying machine to right or left brings the gyroscope 
 frame against a lever which opens a valve and admits compressed 
 air to a piston which shifts a rudder. To exert a force in opening 
 the valve the gyroscope must be entirely free to precess in a 
 plane at right angles to the direction of the force, exactly as in 
 the case of a gyroscope precessing under 
 the pull of gravity as above explained. 
 
 Experiment 18. Support one end of the 
 axle of a spinning bicycle wheel in the hand, 
 allow the other end of the axle to come 
 against the side of a vertical post, and call 
 attention to the fact that the axle drops 
 exactly as if the wheel were not spinning, 
 or would so drop, if the powerful reaction 
 
 Fig. 11. 
 
 did not move the hand-support sidewise 
 
 and if it were not for the great friction due to the very large force 
 
 with which the moving end of the axle is pressed against the post. 
 
 19. Curious gyroscope toy. When the moving end of the 
 axle of a precessing gyroscope comes against an obstacle (even 
 
 35 
 
36 CALENDAR OF LEADING EXPERIMENTS. 
 
 when no outside torque is acting to push the axle against the 
 obstacle) the axle, if it is rotating with the wheel, starts to roll 
 along the obstacle. This motion (precession) of the axle causes 
 a reaction which pushes the axle against the obstacle so that 
 the rolling axle will follow an obstacle of almost any shape. 
 Fig. II shows a very interesting toy in which the axle of a 
 spinning top rolls round and round the entire boundary of a 
 double spiral made of a strip of metal. 
 
WHAT MAKES A HARD-BOILED EGG RISE ON END 
 WHEN SPUN ON ITS SIDE? 
 
 Experiment 20. Spin a sharp-pointed top with its axle in- 
 clined and note its motion of precession. This precession is 
 caused by the tipping action or torque action produced by 
 the pull of gravity, and the precession produces a reaction which 
 is just sufficient to keep the axis of the top at a constant incli- 
 nation. 
 
 Increase the speed of the precessional motion by holding a 
 thin rod above the top and helping the precession. This in- 
 creased precession produces more than enough reaction to support 
 the inclined top against gravity, and therefore the axis of the top 
 rises to the vertical. 
 
 The rolling of the blunt end of a spinning top (with its axis 
 inclined) on the table hastens the precessional motion of the 
 top by producing a side force against the lower end of the axle of 
 spin, and this hastening of the precession soon brings the axis 
 of the top into a vertical position. 
 
 The above effects can be easily shown by spinning two tops 
 which are alike except that one has a very blunt point and the 
 other a sharp point. A hard-boiled egg spun on its side quickly 
 stands up on one end, the action being the same as in case of the 
 blunt pointed top. 
 
 37 
 
THE SCOPE OF DYNAMICS. 
 
 It may seem that an excessive amount of space is devoted to 
 gyroscopic motion in General Physics and in this Calendar, but 
 rational mechanics is an important subject and it must be 
 adequately treated ! The plan which is commonly followed in our 
 technical schools is to give a course in analytical mechanics to 
 supplement an altogether inadequate course in elementary 
 mechanics; and this plan is worse than useless for two reasons, 
 namely, (a) Because analytical mechanics, the oldest branch of 
 theoretical physics, has less to do with engineering and less to 
 do with research than any other branch of mathematical physics, 
 and (b) Because the limitations of the engineering student and 
 the limitations of the engineering teacher have together led to 
 the development of a course in analytical mechanics which it is 
 a kindness to call mere formal nonsense. It sickens the brilliant 
 student and stupefies those who are not brilliant; and a close 
 competitor is the accepted course in thermodynamics which 
 has been developed under the same hopeless limitations; but, 
 according to our experience, there is no group of teachers so 
 stiffly proud of what they do as the teachers of theoretical 
 mechanics and thermodynamics in our technical schools. 
 
 What are authors to do who arrange an elementary treatise 
 on dynamics? Translatory motion must be taken up first, 
 then rotatory motion, then oscillatory and wave motion, and 
 then, were it not for the limitations of the student, a touch of 
 statistical mechanics which sticks its nose into Everything! 
 But we know the imperturbable satisfaction of Deans of Engineer- 
 ing; what are we to do? Having come to detest the technical 
 school no, not the school but the elaborate parts of its cur- 
 riculum, we cannot be satisfied merely to expound the elementary 
 theory of dynamics (we are willing to stop short of statistical 
 mechanics) and contemplate mortality tables! 
 
 38 
 
MECHANICS. 39 
 
 The old conflict in education was between the older classics 
 and science, but the conflict is now between science and pseudo- 
 science. We have the watery pseudo-science of the high school 
 and college which is intended to be enticing and plausible; and 
 we have the pseudo-science of the technical school which is 
 exacting but unintelligible, Dryer than Dust. It would desiccate 
 the Ocean if it were not far Up in the Air. 
 
THE BRENNAN MONO-RAIL CAR. 
 
 A gyrostat wheel is mounted on a horizontal spindle, this 
 spindle is carried in a frame FF, Fig. 12, this frame is free to 
 turn about a vertical axis, being pivoted in an outer frame GG 
 which is fixed to a platform PP, and this platform stands on 
 two pointed legs (one leg is behind the other in the figure). 
 
 The platform PP with attached 
 frame GG is shown as slightly 
 inclined to the left in Fig. 12 so that 
 the pull of the earth on the entire 
 arrangement exerts upon it a torque 
 Tj and the effect of this torque is to 
 make the spinning wheel precess, 
 bringing the end a of the spindle 
 towards the reader. The platform 
 does not fall over for a compara- 
 . tively long time (while the preces- 
 sion is taking place as stated) , and 
 the essential feature of the Brennan 
 
 balancer is a device for automatically hastening this precession. 
 The reaction of the unhastened precession is just sufficient 
 to balance the torque T, but the reaction of the hastened 
 precession is more than enough to balance T, and there- 
 fore the hastened precession lifts the end b of the spindle. 
 Indeed in the Brennan balancer the lifting of the end b of the 
 spindle is always enough to reverse the tilt of the platform and 
 reverse the torque T, whereupon the precession is reversed. 
 This reversed precession is allowed to continue until the end 
 a of the spindle has come back into the plane of the paper, and 
 then the reversed precession is hastened, the platform is tilted 
 
 40 
 
 O 
 
 Fig. 12. 
 
MECHANICS. 41 
 
 back to the left (as shown in the figure), the end a of the spindle 
 comes forwards to the plane of the paper, the precession is then 
 again hastened, and so on over and over again. A number of 
 devices have been used for automatically hastening the precession 
 as stated. 
 
 The only advantage of the mono-rail is that a car (if it is kept 
 erect) would run with less vibration on one crooked rail than on 
 two crooked rails! It certainly is better, however, to build two 
 straight rails and use ordinary cars than it would be to save a 
 little on a crooked mono-rail and spend a lot on complicated and 
 more or less unreliable balancing mechanisms! In fact the great 
 problem in heavy-traffic railroading is the problem of making 
 the track strong enough to carry the enormously heavy engines 
 and trains, and mono-rail construction would be an engineering 
 absurdity. 
 
 The gyro-platform. Two Brennan balancers can be arranged 
 to keep a platform very nearly level on board ship, and such a 
 gyro-platform may come to have important uses. 
 
 Experiment 21. A person sitting on the platform PP in 
 Fig. 12 can maintain his balance easily by properly hastening 
 the precession of the heavy lead-rimmed bicycle wheel which 
 has been set spinning by hand, especially if the two platform 
 legs are short. 
 
 22. The Schlick device for the steadying of a ship at sea. 
 Figure 13 shows a demonstration model of this device. The 
 ring swings freely about a horizontal precession axis pq, and the 
 axis of spin of the wheel is at right angles to pq as shown. 
 The boat is represented by the flat board AB which stands on 
 two legs 00. 
 
 Take the strings 55 in the hands and exert a repeatedly 
 reversed torque on the model, tending to rock the boat back 
 and forth about 00 as an axis. 
 
 Let us suppose that the axis of spin of the wheel is vertical 
 
42 CALENDAR OF LEADING EXPERIMENTS. 
 
 at the beginning, then pulling on either string does not* tip the 
 boat but causes precession about the axis pq. A steady pull 
 on either string soon brings the axis of spin into a horizontal 
 fore-and-aft position, and then the boat tips freely. The 
 gryostat is only effective in preventing the tipping or rocking of 
 
 T 
 
 Fig. 13. 
 
 the boat when repeatedly reversed pulls are exerted on the 
 strings, and the intensity and duration of a single pull must not 
 be sufficient to bring the axis of spin into the horizontal fore-and- 
 aft position. 
 
 23. Gyrostatic elasticity. A spinning gyroscope is held in the 
 position shown in Fig. 14, and we will speak of the end of the axle 
 which is opposite to as the outer end. The support of the 
 outer end is suddenly removed. To establish the precessional 
 motion ft the ring and wheel must of course be set in motion 
 about a vertical axis, and this means inertia reaction whose effect, 
 while it lasts, is exactly equivalent to an outside torque opposing 
 
 * Except for an elastic-like yielding which immediately recovers when the pull 
 See following discussion of gyrostatic elasticity. 
 
MECHANICS. 43 
 
 12. Therefore the outer end of axle and ring drops slightly while 
 precessional motion is being established. By the time 12 has 
 reached its normal mean value (the value due to gravity torque 
 on ring and wheel) the downward velocity of outer end is a 
 maximum and the downward momentum of ring and wheel carries 
 the outer end still farther downwards and doubles the value of 
 12. Then the unbalanced reaction of this doubled 12 (the reaction 
 due to mean value of 12 is just sufficient to balance the gravity 
 torque) lifts the outer end to its initial level and reduces 12 to 
 zero as at the beginning (the outer end having moved forwards, 
 however, in the direction of 12). The action as described is 
 repeated over and over again. 
 
 Fig. 14. 
 
 When one of the strings in Fig. 13 is pulled suddenly the boat 
 tips slightly while the precession is being established, and if the 
 pull then ceases the momentum which is associated with the 
 established precession causes the precession to persist for a short 
 time, and the unbalanced reaction of this continued precession 
 brings the boat again to a level position. If the strings are 
 removed in Fig. 13 a sudden blow causes the boat to oscillate 
 back and forth on its two legs exactly as if the boat were held 
 in a vertical position by an elastic spring, and this oscillatory 
 motion of the boat is accompanied by an oscillatory precession 
 of the gyrostat.* 
 
 * A very simple theoretical discussion of this kind of oscillatory motion is given 
 by W. S. Franklin, Physical Review, Vol. XXXIV, pages 48-52, January, 1912. 
 
CONSTANCY OF SPIN-MOMENTUM OF A CLOSED 
 
 SYSTEM. 
 
 When a body or system of bodies is not acted upon by any 
 forces from outside the body or system, the body or system of 
 bodies constitutes what is called a closed system. The only 
 force actions in a closed system are the mutual force actions 
 between the parts of the system. 
 
 The total momentum of a closed system cannot change. This 
 is evident when we consider two things, namely, (a) That an 
 unbalanced force acting on a body causes the momentum mv 
 of the body to change, and that the rate of change of the momen- 
 
 d(mv) . dv , 
 
 turn i is equal to the force m because m is a constant, 
 at at 
 
 and (b) That any mutual force action between two bodies in a 
 system consists of a pair of equal and opposite forces so that the 
 rates of change of the momenta of the two bodies must be equal 
 and opposite so that their combined momentum does not change. 
 The spin-momentum of a body is equal to Ks, where K is 
 the spin-inertia of the body and 5 is its spin-velocity. In a 
 closed system a mutual torque action between two parts of the 
 system means the exertion of equal and opposite torques on the 
 two parts, and equal and opposite rates of change of spin- 
 momentum by the two parts. Therefore the total spin-momen- 
 tum of a closed system cannot change. 
 
 24. Experiment with a pivoted stool. A platform or stool is 
 supported on a ball-bearing pivot. Stand on this platform with 
 weights in the hands and with arms outstretched, and have an 
 assistant set you rotating slowly about a vertical axis. Then 
 draw in your arms thus greatly decreasing the spin-inertia K 
 of the system. The spin-momentum Ks remains constant 
 and the spin-velocity s is greatly increased when K is reduced. 
 
 44 
 
MECHANICS. 45 
 
 25. Experiment with emptying bowl. A deep bowl, preferably 
 of glass, is filled with water and the water is set slowly rotating 
 by stirring the water with a stick. A hole at the center of the 
 bottom of the bowl is then opened and the water is allowed to 
 run out. The water in the bowl flows to some extent towards 
 the axis of the bowl and the spin-inertia of the system (water in 
 the bowl and the long " rod " of water which is to be thought of 
 as coming out of the hole) decreases greatly. Therefore the slow 
 rotatory motion of the water is greatly increased as shown by 
 the production of a deep whirl at the center of the bowl. See 
 description of cyclone and tornado, page 91, General Physics. 
 
 26. Experiment with pivoted stool and heavy rimmed bicycle 
 wheel. Stand on the pivoted stool, hold the bicycle with its 
 axis horizontal and set it spinning rapidly. Then when the axis 
 of the spinning wheel is brought into a vertical position your body 
 will be set rotating in a direction opposite to the spinning wheel. 
 Assuming the pivot of stool to be frictionless no torque about 
 a vertical axis can be exerted on your body from outside, and 
 therefore the spin-momentum of the pivoted system about a 
 vertical axis must remain what it was at first, namely, zero. 
 
STATIC AND DYNAMIC BALANCING. 
 
 A wheel and axle is said to be statically balanced when the 
 center of mass (or center of gravity) of the whole lies on the 
 axis of the axle. Such a wheel and axle will stand indifferently 
 in any position when the axle rests on two parallel, straight, 
 horizontal rails. 
 
 A wheel and axle is said to be dynamically balanced when the 
 axle has no tendency to quiver when the whole is spinning 
 rapidly. 
 
 Figure 15 shows a body B mounted on a shaft, and the 
 center of mass C of the whole is on the axis of the shaft; that 
 is to say, the body is statically balanced. But if the shaft 
 
 B 
 
 XC 
 
 shaft 
 
 \rope 
 
 Fig. 15. 
 
 Fig. 16. 
 
 is supported in bearings and the body set rotating, the bearings 
 must exert very considerable forces upon the shaft, and the arrows 
 FF show the directions of these forces for the given instantaneous 
 position of the body; or if hung by a rope as shown in Fig. 16 
 and set spinning, the shaft will take the position shown. 
 
 46 
 
MECHANICS. 
 
 47 
 
 Dynamic balancing of a dynamo armature. A 
 thin disk or narrow wheel and axle can be brought 
 into dynamic balance by placing the axle on hori- 
 zontal rails and adding material to or taking 
 material from one edge or the other until the 
 wheel and axle is in static balance. But a long 
 body like a dynamo armature, having been static- 
 ally balanced, must be hung as shown in Fig. 17 
 and set spinning rapidly about a vertical axis. 
 If the armature is dynamically out of balance op- 
 posite sides a and b of the shaft will be whit- 
 ened by chalk held lightly against the spinning 
 shaft. Equal weights must then be added at c 
 and d (or taken away from e and / ) ; and this 
 operation is repeated until the balancing is satisfactory. 
 
 rope 
 
 17. 
 
ONE, OILER, KNEW IT ALL! 
 
 The motion of a symmetrical spinning body like a wheel or a 
 top is extremely simple as compared with the motion of a non- 
 symmetrical body, and any one who cannot digest Euler's 
 (pronounced Oiler's) equations should go to Poinsot's wonderful 
 paper entitled Theorie Nouvelle de la Rotation des Corps, or make 
 for himself one of the non-symmetrical tops which Maxwell 
 devised many years ago. It is a common remark among thorough 
 students of rigid dynamics that Euler knew it all! and he did, in 
 fact, know it all. 
 
 Anyone can know what Euler knew by taking some pains, and 
 to one who knows what Euler knew it is extremely funny, and 
 more than funny, to read the following letter which was addressed 
 to the editor of an important scientific journal by a director of 
 the Aero Club of America asking an utterly meaningless question 
 about the simplest aspect of rotatory motion with a pretense of 
 forty thousand dollars' worth of seriousness; for, in fact, the 
 Cash Prize faded to nothing when investigation was made. 
 
 Dear Sir: I respectfully state that the following question is 
 before the members of the Aero Club of America for solution 
 and I would appreciate it very much if you would kindly put 
 
 the question to the readers of through its valuable 
 
 columns : 
 
 "What is the acceleration of precession when mass spins and 
 precesses with the same radius vector, and in the same plane, 
 tangential to the earth's surface?" 
 
 The above question is important and is put in consideration 
 of $40,000 Cash Prize offered. 
 
 I remain, Yours for the advancement of science, 
 
 1 Director. 
 
 48 
 
MECHANICS. 49 
 
 Ours for the advancement of science! Yes, now, ours. He 
 means it with all the good will he can muster without cost or 
 pains. But he was at one time yours, gentle teacher, whether of 
 the technical school or college, and he left you, untouched by 
 those kindly but rigorous exactions, which when shot through 
 and through with sympathetic intelligence do bring young men 
 to the pains of those who really learn and brace them for the 
 grief of those who are wise. 
 
WORK AND POWER. 
 
 The difficulty of the precise ideas of work and power as applying 
 to actual conditions and things is amusingly illustrated by the 
 following problem which we gave to a group of engineering 
 students at the end of a half-year's course in elementary 
 mechanics. "A cart moves due northwards at a velocity of 
 5^ feet per second. A man pushes vertically downwards on the 
 cart with a force of 200 pounds, and a mule pulls due northwards 
 on the cart with a force of 50 pounds. Find the rate at which 
 the man does work and the rate at which the mule does work." 
 We deliberately refrained from naming the part of the man's 
 body which exerted the force on the cart! In answer to the 
 question 44 per cent of the young men found that the man 
 developed 2 horse-power and the mule developed J horse-power. 
 Truly, mule driving would be strenuous labor for our pampered 
 college students! 
 
 Another example illustrates the dislike of many young men 
 for the hardest labor in the world, namely, thinking! "The 
 force required to tow a boat is, let us say, proportional to the 
 velocity of the boat. To tow the boat at a velocity of 2 miles 
 per hour requires 2 horse-power; how much power would be 
 required to tow the boat at a velocity of 4 miles per hour?" 
 More than 95 per cent of the class accepted unthinkingly the 
 suggestion of simple proportion and got 4 horse-power as the 
 answer ! 
 
 Kinetic energy and the engineer's system of units. We 
 usually speak of the system of units which is based on the foot 
 as the unit of length, the earth-pull-on-a-one-pound-body-in- 
 London as the unit of force and the second as the unit of time 
 as the foot-slug-second system (f.s.s. system) for the sake of the 
 sharp differentiation which is thereby made between this system 
 
 50 
 
MECHANICS. 51 
 
 and the foot-pound-second system (f.p.s. system). The units of 
 any system may be used in any equation in mechanics (including 
 hydrostatics and hydraulics) if the equation is in its simple gener- 
 alized form. Thus in the f.s.s system the kinetic energy of a 
 moving body in foot- "pounds" is equal to ^mv 2 where m is 
 the mass of the body in slugs and v is its velocity in feet per 
 second; the pressure p due to a column of fluid is p = hdg, 
 where, in the f.s.s system, p is expressed in "pounds" per 
 square foot, h is the height of the fluid column in feet, d is 
 the density of the fluid in slugs per cubic foot and g is the local 
 acceleration of gravity in feet per second per second. Thus if 
 the density of water is precisely 62^ pounds per cubic foot, 
 
 it is exactly slugs per cubic foot, and the equation 
 3 2 - r 74 
 
 p = hdg gives the pressure exactly in London "pounds" per 
 square foot if g is the local value of gravity. The numerical 
 precision which is required for most practical purposes (research 
 or engineering) does not justify us in carefully distinguishing 
 between the acceleration of gravity in London, 32.1740 feet per 
 second per second, and the acceleration of gravity at any other 
 place on the earth but precision of thought demands that we 
 make this distinction. See page 4 of this volume for further 
 discussion of this subject. 
 
 The principle of the conservation of energy. Standing before 
 the class take a stone and lower it from a high position A to a 
 low position B, thus getting work or energy out of the stone; 
 then bring the stone back from B to A , slipping it behind the 
 back in the hope or pretense of getting it back to A with a small 
 expenditure of work! This procedure will go far to give to the 
 student a clear idea of the principle of the conservation of energy 
 as discussed in Art. 48, pages 68-70, General Physics, and it 
 strongly suggests the point of view of every perpetual motion 
 promoter, namely, a vague and wholly unintelligent expectation 
 of success or downright cheating. 
 
HYDROSTATICS. 
 
 27. Experimental demonstration of Pascal's principle. Use 
 
 the arrangement described in Art. 51, pages 73-74, General 
 Physics. 
 
 Pascal's principle and the idea of hydrostatic pressure. The 
 force AF which is exerted by a fluid at rest on an exposed plane 
 area Aa is at right angles to Aa, the value of AF is the same 
 whatever the direction of Aa (Pascal's principle), and the ratio 
 
 AF 
 
 approaches a definite limiting value as Aa and AF approach 
 
 zero (AF is more and more nearly in exact proportion to Aa 
 when Aa and AF grow smaller and smaller) . 
 
 It is difficult and in fact unnecessary to distinguish sharply, 
 in this outline, between the elements which are based upon or 
 suggested by observation and experiment and the elements which 
 are of a purely postulate character, but the outline as it stands 
 must be considered and fully grasped by one who wishes to have 
 a clear idea of the hydrostatic pressure at a point in a fluid. 
 
 Having established the idea of hydrostatic pressure at a point 
 in a fluid, it is easy to show that the pressure must have the same 
 value everywhere in a body of fluid at rest when the fluid is not 
 acted upon by an outside force like gravity. Consider any 
 portion of the fluid in the form of an elongated rectangular 
 parallelepiped or prism. The forces exerted by the surrounding 
 fluid against the sides of the prism balance each other and these 
 forces have no components parallel to the axis of the prism. 
 Therefore the forces exerted by the surrounding fluid on the ends 
 of the prism are equal and opposite, and consequently the hydro- 
 static pressure has the same value at the ends of the prism. 
 
 Compare with the above the following line of development, 
 which is the best we know of in any elementary treatise on physics : 
 
 (a) It is pointed out that AF is at right angles to Aa. 
 
 52 
 
MECHANICS. 53 
 
 (b) Pressure is then defined as force per unit area. 
 
 (c) Then hydrostatic pressure is characterized as being the 
 same in every direction. (If the author had stated that the 
 value of A.F is the same whatever the direction of Aa the 
 statement would be intelligible, but it is non-sense* to say that 
 "the pressure is the same in every direction.") 
 
 (d) Then, after discussing the non-uniform distribution of 
 pressure in a fluid under the action of gravity, the author states 
 Pascal's principle as follows: "Pressure is transmitted equally 
 in all directions, or, if the pressure at any point is increased it is 
 increased everywhere throughout the fluid mass by the same 
 amount." The first clause is vague and unintelligible, and the 
 second clause is true only when properly qualified as to the time 
 required for the re-distribution of the pressure and as to incom- 
 pressibility of the fluid when an outside force like gravity acts 
 on the fluid. But a statement which has to be qualified cannot 
 be a statement of a principle. For example, a well known 
 writer on elementary mechanics illustrates equality of action 
 and reaction by pointing out that the pull of a mule on a rope 
 and the backward pull of a canal boat on the rope are equal 
 if the weight of the rope is negligible, from which it would be 
 proper to conclude that Newton's third law of motion is not true 
 because ropes always do have weight ! 
 
 The only comment needed on the above statement of Pascal's 
 principle (?) is to quote Pascal himself, and curiously enough 
 the passage which best serves our present purpose (which is to 
 show the absolute necessity of mathematical thinking in the 
 mathematical sciences) is used by Sir William Hamilton in his 
 "famous and terrific diatribe against mathematics." f Speaking 
 
 * Let no one who is not familiar with stress as a six-fold complex presume to 
 question this statement. 
 
 t This phrase is borrowed from a recent adoration of mathematics by an 
 "institutionally" well known mathematician. It is, of course, ridiculous. See 
 page 34 of this volume for further comment, and consider carefully what is 
 meant by an "institutional " reputation as applying to an individual. As applying 
 to the present case we would change the word pi on page 34 to mush. 
 
54 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 Fig. 18. 
 
 of the recognition of principles Pascal says that "Nothing is 
 wanted beyond a good sight; but good it must be, for principles 
 are so minute." 
 
 28. Pressure in a liquid depends only on depth. A valve V, 
 Fig. 1 8, is held up by a string which is attached to one end of 
 
 the beam of a balance scale, and the 
 weights on the scale pan are adjusted 
 so that the valve opens when the 
 wide vessel W is filled with water 
 to a certain level //. The valve is 
 then found to open when the narrow 
 vessel N is filled to the same level. 
 
 The use of engineer's units in 
 hydrostatics. The use of the two 
 equations (26) and (27) on page 
 75, General Physics, is very confus- 
 ing. It is much better to "go the limit" and use f.s.s units 
 for all of the quantities in equation (26), expressing d in slugs 
 per cubic foot. Then equation (27), which is strictly true in 
 London only, may be discarded. See discussion of equation 
 (26) on page 51 of this volume. 
 
 Boyle's law. Apparatus for the verification of Boyle's law is 
 familiarly known, but according to the authors' experience this 
 experiment should be done by the student in the laboratory. 
 
 29. Experiment on compressibility. Plug the outlet of an 
 ordinary bicycle pump. Stike the pump handle with a small 
 mallet (a) with the pump barrel filled with air and (b) with 
 the pump barrel filled with water. This experiment shows very 
 strikingly the enormous difference between the compressibilities 
 of air and water, indeed in this experiment the water seems to 
 be totally incompressible and the student can understand why, 
 for most practical purposes, water is thought of as incompressible. 
 
 Instead of a bicycle pump with its slim piston rod, a one- 
 
MECHANICS. 
 
 55 
 
 inch steel plunger fitting accurately in a one-inch barrel may be 
 used. In this case one may strike the plunger a heavy blow 
 without damaging the apparatus. 
 
 30. Torricelli's barometer. Fill a clean glass tube (closed at 
 one end) with clean mercury and invert in a tumbler of mercury. 
 Show that difference of level of mercury in tumbler and tube is 
 the same whether the tube be vertical or inclined if the space 
 above the mercury contains but little air. 
 
 The student should have an opportunity to take 
 hold of the barometer tube and lift it off the bot- 
 tom of the tumbler to feel the unbalanced down- 
 ward pressure of the air on the top of the tube. 
 
 31. Archimedes' principle. A solid cylinder C 
 fits accurately into the cylindrical pail P, the 
 two are hung from one pan of a balance, and the 
 weights are adjusted to give equilibrium. The 
 cylinder C is then submerged in water, and equi- 
 librium is again established by filling the pail P 
 
 with water. The buoyant force of the water on C Fig. 19. 
 is equal to the weight of the water in the pail. 
 
 32. Capillary elevation and depression. The capillary rise of 
 water and the capillary depression of mercury in small glass 
 
 - 
 
 VJ 
 
 Fig. 20. 
 Containing water. 
 
 Fig. 21. 
 Containing mercury. 
 
 tubes may be shown by placing in the lantern (horizontal pro- 
 jection arrangement) glass tubes arranged as shown in Figs. 
 20 and 21. 
 
56 CALENDAR OF LEADING EXPERIMENTS. 
 
 33. Surface tension experiments. (a) Mix alcohol and water 
 to get a liquid of the same density as olive oil, place the alcohol- 
 water in a lantern cell and drop in it several drops of olive oil. 
 Some prefer to use drops of carbon bisulphide floating in a care- 
 fully adjusted salt solution. 
 
 (b) Place a clean horizontal glass plate in the lantern (vertical 
 projection arrangement). On the plate place colored water to 
 a depth of about a millimeter. Allow a drop of alcohol to fall 
 on the middle of the plate. The water draws itself away from 
 the spot where the alcohol falls leaving this spot nearly dry 
 because the surface tension of the pure water is greater than the 
 surface tension of the mixture of alcohol and water. 
 
 (c) A fine German silver wire lies flat against a glass plate 
 which is placed horizontally in a lantern (vertical projection 
 arrangement). A thin layer of machine oil is poured on the 
 plate and the oil is seen to heap itself up near the wire because 
 of capillary action. Heat the wire by an electric current. The 
 oil draws itself away from the wire leaving the surface of the 
 glass nearly free from oil near the wire, because the surrounding 
 cool oil has a greater surface tension than the hot oil near the 
 wire. 
 
 (d) Place a shallow basin with glass bottom, thoroughly 
 cleaned and rinsed, in the lantern (vertical projection arrange- 
 ment). Fill with clean water. Skim the surface of the water 
 by blowing a portion of it out of the basin. Scrape a few very 
 small shavings of camphor gum and allow them to fall into the 
 basin. See General Physics, page 83. 
 
 Place in the basin above mentioned a very small cork boat with 
 a split stern into which split is inserted a bit of camphor gum. 
 See General Physics, page 83. 
 
 Note. The above mentioned glass basin should be made by 
 cementing two glass rings to a glass plate. Cut one of the rings 
 from a bottle 4 or 5 inches in diameter and the other from a 
 bottle 5 or 6 inches in diameter. Grind one edge of each ring 
 flat and cement to plate using a minute quantity of gutta percha 
 
MECHANICS. 57 
 
 or rubber cement or marine glue. Then in the skimming process 
 above mentioned the water that is blown off collects in the annular 
 space. Both experiments as above mentioned (and especially the 
 experiment with the cork boat) show more satisfactorily if the 
 basin is heaping full of water so that the water surface is convex 
 near the walls of the basin. 
 
 34. Experiments on oil flotation. Everyone is familiar with 
 the fact that water does not spread over an oily surface. Thus 
 drops of water stand without spreading on an oily surface 
 of glass whereas they spread out and cover a very clean glass 
 surface. Exactly these same effects are shown by a glass surface 
 under water. The water spreads under a bubble of air and 
 detaches it from a clean glass surface whereas the water does not 
 spread under a bubble of air on an oily glass surface, or, in other 
 words, the bubble of air clings to the oily surface. This may 
 be easily shown as follows: (a) Agitate the water in a clean 
 glass tumbler, using a clean egg beater, and the air bubbles all 
 rise to the surface and leave the walls of the tumbler clear, (b) 
 Do the same in a tumbler with greasy walls and numerous air 
 bubbles are left clinging to the tumbler walls. 
 
 There seems to be a marked difference in the tendency of oil 
 to cling to small particles of sand and to small particles of iron 
 pyrite or zinc blende. Place a mixture of fine sand and powdered 
 iron pyrite into a tumbler of water, add a drop or two of oil and 
 agitate with an egg beater as above. The particles of pyrite 
 become oily, bubbles of air cling to them and they float, whereas 
 bubbles of air do not cling to the particles of sand and they sink. 
 This experiment illustrates the oil flotation process which is now 
 extensively used for separating particles of ore from particles of 
 sand and rock. 
 
HYDRAULICS. 
 
 The limitations of mechanics. It is important to point out 
 to the student the fact that the science of hydraulics deals almost 
 entirely with ideal types of fluid motion, as explained on pages 
 85-87, General Physics, because, in the study of any branch of 
 science, nothing is more important than to understand to some 
 extent the limitations of the method used. 
 
 The actual phenomena of fluid motion are erratic in character, 
 visibly erratic, and it is quite certain that these phenomena can 
 never be correlated by the classical method of mechanics.* 
 This statement, it must be admitted, represents a very recent 
 point of view, one that is accepted only by a small group of 
 physicists, and we may be permitted therefore to set it forth 
 more strikingly by quoting from a well-known engineering 
 writer who is still deeply imbued with the ideals of the classical 
 method of mechanics. The laws of fluid motion, he says, have 
 not yet been discovered, although the movements of the heavenly 
 bodies have been completely formulated. This statement shows 
 that the writer assumes the existence and expects the discovery 
 of rigorous correlations in the minutice of fluid motion, only, 
 he would probably deny that laws of fluid motion would have 
 anything to do with such ephemeral and insignificant phenomena. 
 Every phenomenon that we contemplate in this world of ours, 
 even such extremely regular phenomena as the motion of the 
 planets, if examined with extreme care shows a substratum of 
 erratic behavior. In the case of fluid motion, however, this 
 erratic action rises to a level where it enters into human values 
 (if we may be permitted to use that word to designate the things 
 we consider and must make allowance for in our daily life) as 
 exemplified most strikingly in the phenomena of meteorology. 
 
 * Statistical mechanics is indeed not a branch of mechanics proper. It is a 
 theoretical structure with a purely postulate basis, and in its bearing on upon labora- 
 tory or research work it is a part, or, indeed the whole of the atomic theory. 
 
 58 
 
MECHANICS. 59 
 
 Read in this connection: 
 
 (a) Pages 119-123, General Physics, where the method of 
 thermodynamics is contrasted with the method of atomics (the 
 atomic theory), 
 
 (b) Pages 322-325, General Physics, where the method of 
 atomics is contrasted with the method of mechanics, and 
 
 (c) A brief and very simple paper on Statistical Physics, in 
 Science, Vol. XLIV, pages 158-162, August 4, 1916. 
 
 35. Turbulent fluid motion. The sensitive flame. Light 
 an ordinary fish- tail gas jet and increase the gas pressure (the 
 velocity of the gas in the jet) more and more until the flame 
 becomes turbulent and gives off a roaring sound. This sound 
 is produced by small eddies which develop in the border region 
 between the moving gas and the still air. In a high-pressure 
 steam jet these eddies are very violent and numerous and they 
 produce the familiar hissing sound of such a jet. 
 
 When a gas jet is on the point of changing from smooth to 
 turbulent type it is sometimes very sensitive as explained on 
 page 88, General Physics. To make a sensitive flame make an 
 assortment of nozzles by drawing short pieces of glass tube to 
 points and cutting off where the inner diameter is about a 
 millimeter. Use a file with its corner ground to a sharp serrated 
 edge, and break off the points clean and square. Trying several 
 nozzles one usually finds one that gives a very sensitive flame 
 with ordinary city gas. City gas pressure is, however, usually 
 not great enough to give the best results, and therefore it is 
 sometimes advisable to fill a small rubber gas bag with the gas 
 and increase the pressure. Natural gas and gas made from 
 gasolene do not burn satisfactorily at a small nozzle. If such 
 gas, only, is available it must be mixed with hydrogen in a gas 
 bag or gasometer. Acetylene gas is not satisfactory because 
 of the smoky character of a large acetylene flame. 
 
 36. Instability of the vortex sheet. The spit-ball. The border 
 region between moving and still fluid is called a vortex sheet. 
 
60 CALENDAR OF LEADING EXPERIMENTS. 
 
 The instability of the vortex sheet is illustrated by the behavior 
 of the sensitive flame. The behavior of the spit ball as explained 
 on page 89, General Physics , is another illustration. Drop a 
 marble in a tall jar of water, or exhibit the zig-zag motion of 
 small bubbles of air as they rise in water. 
 
DISCHARGE RATE OF A STREAM. 
 
 In ancient Rome the rate of delivery of water to a household 
 was legally gauged by the size of the delivery pipe. It is evident, 
 however, that a small pipe may deliver water at any rate what- 
 ever, depending on the velocity of flow, and commercial water 
 meters as now used in our cities measure the actual amount of 
 water delivered in a given time in cubic feet or in gallons. 
 
 The man of the street, or even the man of the farm has a 
 very vague notion indeed of rate of discharge of a stream; 
 his idea is about the same as that which was legalized in ancient 
 Rome. For example, the rate of discharge of a spring is fre- 
 quently specified as sufficient to fill a one-inch pipe or a two-inch 
 pipe, as the case may be. 
 
 Many people, however, seem not to have any idea at all as 
 to a rate of supply of water, as may be illustrated by a recent 
 discussion of the water question in Bethlehem, Pennsylvania, 
 where 2,000,000 gallons per day is needed. Many otherwise 
 intelligent citizens proposed a pipe line from Savior's Lake, the 
 outflow of which is less than 200,000 gallons per day; there is 
 evidently such a large quantity of pure clear water in the lake! 
 
 61 
 
PERMANENT AND VARYING FLOW. 
 
 A theoretical discussion of simple lamellar flow of a varying 
 type is given in Part VI of this volume in connection with wave 
 motion in air pipes. 
 
 37. The water hammer. A striking example of varying flow 
 is afforded by the well known water hammer. 
 
 When a water wheel is supplied through a long pipe a great 
 drop of pressure occurs at the wheel when the water is turned 
 on and while the water in the pipe is being started (accelerated) , 
 and a great rise of pressure, a dangerous rise of pressure, occurs 
 at the wheel when the water-wheel gates are closed. This 
 dangerous rise of pressure is prevented, however, by a relief 
 valve which opens with rise of pressure, or by a stand pipe out 
 of the top of which the water flows when the water-wheel gates 
 are closed. 
 
 62 
 
BERNOULLI'S PRINCIPLE. 
 
 The consistent use of engineering units. Equations (31), 
 (3 2 ) (33)f (34) and (35) on pages 93-96, General Physics, are 
 true as they stand if foot-slug-second units are used throughout. 
 That is, p must be expressed in "pounds" per square foot, 
 d must be expressed in slugs per cubic foot and so on. See 
 pages 51 and 54 of this volume for further discussion of this 
 point. 
 
 38. The disk paradox. See pages 96-97, General Physics. 
 A simple form of this experiment is to blow between the fingers 
 against a small piece of paper lying flat against the palm of the 
 hand. 
 
 The mechanics of the disk paradox may be made clear by 
 the following experiment. A jet of water falls centrally upon a 
 thin flat metal disk in a basin of water, and the disk is held up 
 (made to float) by the action of the jet. Everywhere on top 
 of the disk the water has high velocity and low level (low head 
 or pressure), and as the stream of water comes nearly to rest at 
 or near the edge of the disk it raises itself to a higher level (or 
 pressure); whereas the pressure everywhere in the still water 
 underneath the disk corresponds to the high-level water at or 
 near the edge of the disk. A 
 small needle should project up- 
 wards from the bottom of the 
 disk and pass through a small 
 hole in the disk to keep the disk 
 from moving sidewise. 
 
 39. Diminution of pressure 
 in a throat. A glass tube TT ', 
 Fig. 22, has a throat at a. 
 
 The tube should be about J inch bore and the throat should 
 
 63 
 
64 CALENDAR OF LEADING EXPERIMENTS. 
 
 be about f inch internal diameter. Two short glass nipples 
 are sealed to TT' at a and &, a glass U-tube is connected to 
 these nipples by short pieces of rubber tubing as shown, and a 
 small quantity of colored water is placed in the U-tube. 
 
 Blowing through TT 1 in the direction of the arrow draws the 
 colored water upwards in arm a as indicated in the figure. 
 The effect of friction, alone, would be to push the colored water 
 downwards in arm a, because friction alone would cause a lower 
 pressure at b than at a. 
 
 This diminution of pressure in a throat is utilized in the jet 
 pump. Water from city mains may, for example, flow through 
 TT f and the pressure at a may be low enough to draw water 
 out of a cellar, the discharge at T' being into the street. 
 
 The most striking form of jet pump is the steam boiler injector, 
 and the injector is sometimes operated in practice by exhaust 
 steam. The exhaust steam (because of its low density) attains a 
 very high velocity when it reaches the low-pressure region in the 
 injector throat. Let us assume this velocity to be 1,000 feet 
 per second for the sake of argument, and let us assume that 
 water would gain a velocity of 100 feet per second in issuing as 
 a jet from the boiler (boiler pressure about 75 "pounds" per 
 square inch). Then, neglecting friction, water moving towards 
 the boiler at a velocity of 100 feet per second would carry itself 
 into the boiler. But one unit mass of steam at 1,000 feet per 
 second in mixing with 9 units mass of cold still water gives 100 
 units mass of solid water moving at a velocity of 100 feet per 
 second, because the initial momentum of the moving steam must 
 be equal to the total momentum of the resultant water, friction 
 against walls of throat being neglected. 
 
 40. Ball riding on an air jet. A light ball like a ping-pong 
 ball is placed over a nozzle from which a blast of air is blowing, 
 and the ball rides in the jet without falling out. One can with 
 the lungs produce a blast sufficiently intense for this experiment. 
 Whenever the ball gets part way out of the moving stream of air 
 the higher pressure of the surrounding still air pushes it back. 
 
MECHANICS. 
 
 Fig. 23. 
 
 41. Ship suction. See page 98, General Physics. The effect 
 of ship suction can be shown by supporting two light balls as 
 shown in Fig. 23 and blowing between them. 
 
 The discussion of ship suction in Franklin and 
 MacNutt's Mechanics was questioned by a well- 
 known phsyicist, and to settle the matter a letter 
 was written to the Navy Department. The reply, 
 signed by the then Assistant Secretary of the Navy, 
 stated that the effect was unknown to the depart- 
 ment. It was added, however, that the attraction 
 of gravitation would pull two ships together! The 
 authors had at the time a dim recollection of the ex- 
 tensive experiments on ship suction that had been 
 carried out with great pains by Naval Constructor (now Admiral) 
 Taylor, and just before receiving the above mentioned reply from 
 the Navy Department one of the authors in a lecture to a group 
 of school teachers in Philadelphia, had found three out of a total 
 twenty-five who knew of ship suction by experience with small 
 boats; and, as a matter of fact, the attraction of gravitation 
 between two ships is really a repulsion ! ! How proud we School 
 Teachers might feel! if it were not largely our fault that an 
 officer of the Navy (for the above mentioned reply was no doubt 
 formulated by a navy officer acting in a clerical capacity) should 
 ever come to substitute the encyclopedia habit for critical 
 knowledge of science, and, also our fault, that silver-spoon youth 
 should never develop beyond hopeless dilettantism, and seldom 
 
 get even as far as that (for the 
 then Assistant Secretary of the 
 Navy had been a silver-spoon 
 youth with all the tremendous 
 possibilities and dangers that 
 go therewith). 
 
 42. Flat plate in a moving 
 fluid. Figure 24 shows the ap- 
 proximate trend of the stream lines of a fluid flowing round a flat 
 6 
 
 Fig. 24. 
 
66 CALENDAR OF LEADING EXPERIMENTS. 
 
 plate ab. The fluid velocity is greater at a than at b so that 
 the pressure of the fluid is greater at b than at a, and therefore 
 forces are exerted on the plate tending to turn the plate squarely 
 across the stream as indicated by the curved arrows. Drop a 
 small square of moderately stiff writing paper; it turns quickly 
 into a horizontal position and rocks back and forth, and this 
 rocking motion causes the paper to glide back and forth sidewise 
 as it falls. A flat disk like a coin dropped in clear water is easily 
 seen to behave in the same way. A kite with a plane surface 
 tends to rock back and forth, and therefore to dart to and fro 
 sidewise. 
 
 A bit of paper with its edges bent upwards falls steadily like a 
 clam shell (with its convex side downwards) in clear water, and a 
 kite with a convex front rides steadily in the air. 
 
 Note. The lines of flow in Fig. 24 represent the flow of an 
 ideal frictionless fluid. In an actual fluid a dead-water region 
 exists behind the plate. 
 
 43. Curved flight of a spinning ball. Cut corn-stalk pith 
 into square pieces and glue them together, thus forming a large 
 block from which make a ball about ij inches in diameter. 
 Such a ball thrown from a paste-board tube (a mailing tube) by a 
 quick sweeping motion rolls along the side of the tube and is 
 
 /-^ / 
 
 .S /\ x 
 
 Fig. 25. Fig. 26. Fig. 27. 
 
 set spinning rapidly. A sharply curved flight is the result. 
 Professor P. G. Tait, using a rubber balloon, has succeeded 
 in producing the three theoretically possible "up" curves as 
 shown in Figs. 25, 26 and 27. 
 
 44. The curved flight of a high foul-ball. A ball is set spinning 
 as indicated by the curled arrows c, Fig. 28. The catcher is apt 
 to estimate the drop of the ball at A as in the case of a thrown 
 
MECHANICS. 
 
 6 7 
 
 ball not spinning, but because of the rapid spin the high foul 
 turns in towards home base and drops at B. This error of a 
 catcher is sometimes very amusing to watch. 
 
 ?nd of bat 
 
 Fig. 28. 
 
 Fig. 29. 
 
 The curved flight of a high foul may be shown by shooting a 
 light ball marble fashion so as to give a nearly vertical trajectory 
 as indicated in Fig. 29. A ping pong or pith ball may be used, 
 but the best results can be obtained by using a small oak gall. 
 
 45. Viscous friction and eddy friction of fluids. It is perhaps 
 worth while to demonstrate the two simplest cases of -fluid 
 friction as follows: Measure the quantities of water v and v' t 
 that are discharged in a given time through a small-bore glass 
 tube, first, with head or pressure p and second with head or 
 pressure p f . Then v/v f = p/p'. 
 
 Make a pipe or duct consisting of a number of chambers 
 separated by brass disks with comparatively small holes through 
 the disks. Measure the quantities of water v and v' that are 
 discharged through the duct in a given time, first, with head or 
 pressure p and, second, with head or pressure p' . Then 
 v/v' 
 
 Remark. In fairly large pipes eddy friction only is appre- 
 ciable. At moderate velocities of flow the loss of pressure is 
 very nearly proportional to the square of the velocity. At very 
 high velocities the eddies become more and more fine grained 
 and the loss of pressure increases as v n , where n is greater 
 than 2. 
 
68 CALENDAR OF LEADING EXPERIMENTS. 
 
 In the chambered duct as above described the eddies are 
 fixed in location and the loss of pressure is quite accurately 
 proportional to z> 2 . The increasing fineness of grain of eddy 
 motion with increase of velocity in an ordinary pipe is, in a way, 
 analogous to a closer and closer spacing of the perforated parti- 
 tions in the above-described chambered duct. 
 
PART II. 
 HEAT. 
 
OPERATIVE VERSUS INOPERATIVE DEFINITIONS. 
 
 Nearly every physical definition, rightly understood, is an actual physical 
 operation. Thus you define a cow pasture by building a fence around it. See 
 page 73 of this volume for further comment. 
 
 We think of the flow of an electric current through a wire as being opposed by 
 a kind of frictional drag or resistance very much as the flow of water through a 
 pipe is opposed by a frictional drag. Thus we get the idea of electrical resistance, 
 or, rather, the beginnings of the idea. The rate of generation of heat in a wire is 
 proportional to the square of the current flowing through the wire, and the pro- 
 portionality factor has a definite value for the given wire. Therefore this propor- 
 tionality factor is adopted as the measure of the resistance of the wire. 
 
 In every case a preliminary idea of a physical quantity exists before the quanti- 
 tative definition itself is established, and the quantitative definition, rightly under- 
 stood, is a group of actual physical operations. 
 
 The preliminary idea of electrical resistance is expressed in terms of an analogy 
 and therefore it is easy to grasp and easy to talk about, but the recognition of an 
 idea, or the beginnings of an idea, in mechanics before the quantitative definition 
 is adopted is difficult. For example, John Smith knows that he has gotten some- 
 thing definite when he has secured possession of a given batch of sugar, although he 
 may have no idea as to how much sugar it is. Now the operation of weighing* 
 always gives the same numerical result when applied to a given batch of sugar and 
 therefore this result is a most satisfactory measure of quantity of sugar. 
 
 The recognition of heat as a measurable quantity depends upon the clear recogni- 
 tion of the fact that if body A is changed thermally by the dissipation of work 
 and brought back to its initial condition by being brought into contact with 
 another (cooler) body B, then the thermal change in B is exactly what would be 
 produced by the dissipation of the original amount of work on body B directly, 
 so that the body A by virtue of the thermal change which is produced in it by the 
 dissipated work "stores" something which is equivalent to the dissipated work. 
 The complex operation thus described is the very essence of the quantitative 
 definition of heat. See page 82 of this volume for further comment. 
 
 * An ideal operation of weighing is usually referred to in a brief statement of 
 this kind so as to avoid the necessity of referring to well known sources of error. 
 
 70 
 
HEAT. 
 
 Temperature as a condition. The recognition of temperature 
 as a condition or fact is explained on page 107, General Physics; 
 to find a temperature go to a closed cellar or to a stuffy, closed 
 room. A substance not in thermal equilibrium has, strictly 
 speaking, no temperature. Touch a high speed belt or stick a 
 finger into an electric arc and you will be burned, but the degree 
 of hotness cannot be expressed in either case as a temperature. 
 Certain portions of an electric arc, the hot tips of the carbons, 
 for example, seem to be approximately in thermal equilibrium 
 and they can be thought of as having a fairly definite temperature, 
 but it would be necessary to cage a portion of the arc gases, keep 
 them in a heat-insulating enclosure and wait awhile before they 
 could be said to have a definite temperature. 
 
 Definite quantitative notions in physics usually depend upon assumed uni- 
 formity of distribution in space and time, and to apply such notions to rapidly 
 varying conditions the idea of the limit, as used in the infinitesimal calculus must 
 be introduced. See pages 593-597. General Physics. For example the fundamental 
 idea of density refers to a homogeneous substance, and the density of a non-homo- 
 geneous substance at a point is defined as the limiting value of , where Am 
 
 is the mass and Av is the volume of a very small portion of the substance at or 
 near the point. 
 
 According to the atomic theory a portion of any substance (even of a substance 
 in thermal equilibrium) departs more and more widely from thermal equilibrium 
 as it is taken smaller and smaller. Therefore the idea of temperature cannot 
 be made to apply to a turbulent substance by using the method of limits. A 
 metal rod which is red hot at one end and cold at the other may, however, be thought 
 of as having a fairly definite temperature at each point because a portion of the 
 material containing many millions of molecules is pretty nearly in the steady 
 condition which we call thermal equilibrium. The idea of temperature is not 
 applicable to a few molecules of any substance unless the average behavior of the 
 few molecules during a long time is taken into consideration. 
 
 The atomic-theory conception of temperature. The atomic 
 theory is a logical structure which is built upon a postulate basis, 
 
72 CALENDAR OF LEADING EXPERIMENTS. 
 
 and it has a bearing on experimental work and research in that 
 by its means very definite and remarkable ideas can be developed 
 and put to experimental test. Read in this connection pages 
 119-122, General Physics. 
 
 The atomic-theory conception of temperature is entirely apart 
 from the things of experiment (entirely apart from thermo- 
 dynamics), and this conception has been fully developed only 
 for the ideal or perfect gas. See very brief discussion of the 
 atomic theory of gases on pages 325-328, General Physics, in 
 particular note the postulate basis as stated on page 325, see the 
 atomic prediction of Boyle's law as involved in equation (ii), 
 page 326, and note on page 328 an assumed conception of 
 temperature which makes our postulated atomic system con- 
 form to Gay Lussac's law and to temperature values as meas- 
 ured by the air thermometer. 
 
 The principle of the equal-partition of energy (which, by the 
 way, is at variance with many experimental results and cannot, 
 therefore, be true) among the various degrees of freedom of a 
 helter-skelter system seems to indicate that a certain tempera- 
 ture means always a certain average kinetic energy per molecule ; 
 but, considering the probable limitation of the principle of equal- 
 partition of energy to the simplest kind of an ideal gas, it is by 
 no means permissible to define temperature as the average 
 kinetic energy per molecule of a substance! In fact such a 
 definition is absurd from the laboratory man's point of view, 
 and it would be absurd even if it were entirely legitimate from 
 the point of view of the atomic theory. 
 
 Physicists long ago realized the limited usefulness in scientific 
 work of what may, perhaps, be called verbal philosophy. This 
 term unfortunately conveys a suggestion of contempt, but no 
 contempt is here intended, but quite the opposite of contempt, 
 for we refer broadly to that vital blend of intellect and morals 
 which has grown up with the use of words in our age-long 
 dealings with what the philosopher calls human values. Mathe- 
 matical philosophy is the thing, in the physical sciences. The 
 
HEAT. 73 
 
 idea of continuous quantity with the associated ideas of func- 
 tional relations and limits was the earliest phase of mathematical 
 physics; later comes the widened use of postulates and of the 
 theoretical structures based thereon; and already there is evi- 
 dence of a new method (we do not refer to statistical physics) 
 in the mathematics of discrete things, especially in the group 
 theory. 
 
 Any carefully ordered set of operations of the analytical 
 chemist is a group, the operations involved in the making of a 
 particular physical measurement constitute a group, and a long 
 step towards realism and adequacy of theoretical physics will 
 be taken when the operations of the physical sciences are con- 
 nected in a mathematical structure. It is now very difficult 
 to think of these things, but the group theory (or some other 
 kind of mathematics) when fully developed may be expected 
 to make this kind of thinking easy. 
 
 So difficult is it to think of the fundamental operations of the 
 physicist that every teacher of physics is more or less indulgenf 
 and charitable in his attitude towards the ever recurring re- 
 crudescence of verbal philosophy in his field even when he 
 knows that it is absurd. Mass? It is defined as quantity of 
 matter. Not by any means. It is defined by the operation of 
 weighing by a balance or by the group of operations involved in 
 showing that the unit of mass is accelerated m times as fast as 
 the given body, by a given force ; and these operations are simple 
 in comparison with many of the measuring operations in elec- 
 tricity and magnetism. Do you define a cow pasture as a 
 meadow sweet with clover and grass? You do not; you define 
 it, for the cow at least, by building a fence around it! 
 
 46. The Brownian motion. Every student of physics should 
 see the irregular and incessant to-and-fro motion of very fine 
 particles suspended in water, using a good microscope. This 
 motion was discovered by the English botanist, Brown, in 1827, 
 and it is called the Brownian motion. 
 
 Grind a small amount of insoluble carmine in a few drops of 
 
74 CALENDAR OF LEADING EXPERIMENTS. 
 
 water, rubbing with the finger in a shallow dish. Place a drop 
 of the mixture on a microscope slide and use a magnifying 
 power of about 400 diameters. The particles in India ink are 
 much finer than the particles of carmine, and a higher magnifying 
 power is required to see them satisfactorily. 
 
 47. The expansion of gases. To show the very great expan- 
 sion of air with ri'se of temperature place the long stem of a very 
 dry glass bulb in a- tall jar of water and heat the bulb. The 
 expansion drives a large amount of air out of the bulb as shown 
 by the bubbles which rise in the tall jar of water. When the 
 bulb cools the contraction of the air which is left in the bulb is 
 shown by the flow of water into the bulb. 
 
 The expansion of air, or, rather, its increase of pressure with 
 rise of temperature is utilized in the hot-air engine. A cheap form 
 of hot-air engine is sold by chemical supply houses. In the hot- 
 air engine the contained air is shifted from cold to hot part of 
 cylinder and back again repeatedly by the motion of a large 
 loosely fitting displacement plunger, and the consequent heating 
 and cooling of the air raises and lowers its pressure. The engine 
 piston moves outwards while the air is hot and its pressure is 
 high, and the energy of the fly wheel then pushes the piston 
 inwards and compresses the air while it is cold and its pressure 
 is low. A wooden model showing the large displacement plunger, 
 the long cylinder and the piston, and showing the red hot part 
 of the cylinder painted red is a great help in the explanation of 
 the action of the hot*air engine. 
 
 The expansion of a gas, or rather its rise of pressure with rise 
 of temperature is also utilized in the gas engine. In this case, 
 however, the gas which is used is a mixture of inflammable 
 gas or vapor and air, and the rise of temperature is produced 
 by the quick burning of the mixture. The toy gas cannon is the 
 same in principle as the gas engine and it makes a good piece of 
 demonstration apparatus. 
 
 Concerning the gas equation. Several years ago a physics 
 teaching friend sent us the following example of what he con- 
 
HEAT. 
 
 75 
 
 
 T 
 
 
 Fig. 30. 
 
 L is one half of a 
 
 tee-square. 
 
 sidered to be bad-medicine, and we agree with him. To fix 
 the formula L = \aP for the distance L traveled by a falling 
 body in t seconds the student is asked to remember that L 
 is in fact one half of a tee-square as indicated 
 in Fig. 30. 
 
 The unintelligent use of formulas is cer- 
 tainly a bad habit, and the following exam- 
 ple is taken from our own personal experi- 
 ence. It concerns the familiar gas equation 
 pv = RT. A physics teacher walked to the 
 University one morning in company with an 
 engineering instructor, and the conversation 
 turned to the gas turbine, concerning which the engineering 
 instructor had some new ideas which he hoped to develop 
 practically. After some discussion it appeared that the engineer- 
 ing instructor was perplexed concerning a theoretical phase 
 (as he would have expressed it) of gas action, and it soon became 
 clearly evident that the perplexity was this: "Since pv = RT, 
 how can a gas at high pressure be cold?" The young man did 
 not recognize the two variables p and v, and he did not have 
 the faintest appreciation of the fact that this equation applies 
 only to a given amount of a gas. He had allowed the formula 
 to take the place of the simplest kind of common sense, namely, 
 that anything may be as hot or as cold as one may care to make it. 
 A verse from one of the Bab Ballads comes, willy-nilly, into our 
 minds as we relate this actual occurrence : 
 
 "Oh, list to this incredible tale 
 Of Thomson Green and Harriet Hale 
 Its truth in one remark you'll sum; 
 
 Twaddle, twaddle, twaddle; twaddle twum." 
 
 Of course this engineering instructor had taken the regulation 
 dose of engineering thermodynamics and had passed it, for no 
 one can become an engineering instructor who has not graduated 
 from a technical school. 
 
 "Es erben sich Gesetz und Rechte wie eine ew'ge Krankheit fort." 
 
76 CALENDAR OF LEADING EXPERIMENTS. 
 
 and what a privilege (Recht) it is to become an engineering 
 instructor ! 
 
 Nernst has said that by far the most useful theoretical de- 
 velopments in thermodynamics are those which are expressed 
 explicitly in terms of physical operations, such as Carnot cycle 
 operations; and there are many other operations of equal im- 
 portance. It is doubtful if any experimental research in chem- 
 istry or physics or engineering has ever been suggested by the 
 elaborate algebraic developments to which thermodynamic theory 
 unfortunately lends itself. 
 
 A perfectly clear understanding of the physical meaning of 
 the second law of thermodynamics (which is possible because of 
 the postulate character of the law) and ability to interpret an 
 argument in thermodynamics in terms of definite physical opera- 
 tions, these are the all-important things; and no engineer, 
 however great his pretended predilection for practicality, can 
 ever arrive at these all-important things when he starts out with 
 a purely algebraic guess "that heat is expressible as the product 
 of two factors, one of which is entropy" and remains satisfied 
 with his algebraic guess as a definition of entropy ! 
 
 All there is to engineering thermodynamics, and a great deal 
 more, is included in pages 105-177, General Physics, supplemented 
 by Marks and Davis's Steam Tables. It may be that we should 
 include, as an essential supplement, the presumably unpublished 
 results of a recent research concerning the temperature-entropy 
 diagram for which an unnamed manufacturing concern is said to 
 have appropriated several hundred dollars.* Not knowing the 
 results of this investigation, however, we are inclined to point out 
 what would seem to us to be the only proper line of "research" 
 in this matter. Send a young man to, let us say, Berlin to take 
 a course under Nernst and Planck. This line of research is the 
 
 * See a reply by a Dean of Engineering to our statement as to the limited 
 usefulness of the temperature-entropy diagram. Bulletin of the Society for the 
 Promotion of Engineering Education, January 1917, pages 270-274. Our statement 
 is to be found in the same Bulletin, November 1915 pages 168-190. 
 
HEAT. 77 
 
 cheapest possible in this particular case, it would lead in three 
 years to perfectly definite results, and on the Cecil Rhodes scale 
 it would cost precisely 4,500 dollars. It is not often that one is 
 in a position to make these three definite specifications regarding 
 research; and also, alas, it is not very often that a manufacturing 
 concern is willing to appropriate even a small fraction of the 
 necessary cost of a piece of research! 
 
 48. Northrup's model of a gas. A glass jar containing many 
 thousands of small steel balls has a rapidly moving element or 
 agitator at the bottom which keeps the balls flying about, and 
 the mouth of the jar is covered by a disk which is freely sus- 
 pended from a balance arm. The flying balls in the jar exhibit 
 the properties of a gas and conform to Boyle's law and to Gay 
 Lussac's law* as follows : 
 
 A given speed of the agitator corresponds to a definite average 
 kinetic energy per ball, and doubling the speed of the agitator 
 quadruples the average kinetic energy per ball. Therefore 
 square of speed of agitator corresponds to or is proportional to 
 Kelvin temperature T. 
 
 (a) Boyle's law. Speed of agitator being kept constant, the 
 force exerted on the glass disk is found by test to be inversely 
 proportional to the volume of the space in which the balls fly 
 about (pressure inversely proportional to volume at constant 
 temperature) . 
 
 (b) Gay Lussac's law as relating to change of pressure at constant 
 volume. The force exerted on the glass disk is found by test to 
 be proportional to the square of the speed of the agitator, 
 volume of space in which balls fly about being constant (pressure 
 proportional to T at constant volume). 
 
 (c) Gay Lussac's law as relating to change of volume at constant 
 pressure. The force exerted on the glass disk is found by test 
 to be unchanged in value when the square of the speed of the 
 agitator and the volume of the space in which the balls fly 
 
 * We refer here particularly to Gay Lussac's Law as stated in Art. 75, page 113, 
 General Physics. 
 
78 CALENDAR OF LEADING EXPERIMENTS. 
 
 about are both doubled (volume is proportional to T at constant 
 pressure). 
 
 (d) The Brownian motion. A steel or wooden ball much larger 
 than the small steel balls or "molecules" is suspended in the 
 vessel and it moves erratically to and fro and up and down as 
 the small balls strike against it. A difficulty in this experiment 
 is that the larger ball must be suspended by a string, it tends to 
 vibrate to and fro regularly as a pendulum, and this regular 
 pendulous motion is mixed up with the erratic motion (the 
 Brownian motion). 
 
 49. Expansion of liquids and solids. The relative expansion 
 of water and glass may be shown by heating a glass bulb filled 
 with colored water which rises in a narrow glass tube. It is 
 most satisfactory to place this stem in the field of the projection 
 lantern. When the heating flame is brought against the bulb 
 the mean temperature of the glass walls is suddenly raised and 
 the sudden expansion of the glass is shown by a quick drop of the 
 water level in the stem. Then continued heating warms both 
 the water and the glass, and the greater expansion of the water 
 shows itself by a slow rise of water level in the stem. Very few 
 students seem to appreciate the fact that water contracts as it 
 is warmed from o C. to 4 C., and it is worth while, therefore, 
 to repeat this experiment starting with water bulb at o C. 
 
 The irregularity of expansion of steel in the neighborhood of 
 700 C. may very properly be shown at this point and the 
 phenomenon of recalescence referred to later. See Experiment 
 66. 
 
 Several samples of good chemical thermometers should be 
 exhibited, and it will do no harm to state that a thermometer 
 bulb and stem should be completely immersed in a fluid or region 
 in order that it may indicate the temperature of the region 
 accurately. In many cases, however, the stem must be allowed 
 to project into the outside air to make it possible to take a reading. 
 The authors know of a junior student in engineering who removed 
 
HEAT. 79 
 
 a thermometer from a calorimeter vessel and held it up to a light 
 so as to be able to read it accurately! 
 
 50. The Trevelyan rocker. A moderately hot soldering-copper 
 is laid on block of lead resting on the table, and, once started, 
 the copper continues to rock and produce a buzzing sound. At 
 each momentary contact of the rocking copper with the lead, the 
 lead near the point of contact is heated and the expansion of the 
 lead (a local swelling too small to see) gives a push to the receding 
 copper and thus keeps up the rocking motion. The lead should 
 present a cleanly cut surface where the copper comes into 
 contact with it, and the copper itself should be scraped clean. 
 With a little patience this curious 
 experiment can always be made to 
 work. 
 
 51. Shrinking bar and cast iron Fig 3L 
 pin. A steel bar bb, Fig. 31, has 
 
 a large pin P fixed to it permanently, and a removable cast 
 iron pin p. The middle part of the bar bb is heated nearly to 
 a red heat, the pin p is put in place, and the bar bb is then 
 lowered into position between two lugs / and two lugs /' as 
 indicated. The bar bb contracts as it cools, and breaks the pin 
 P. 
 
 52. Shrinking of collar on a shaft. The important shop opera- 
 tion of shrinking a collar on a shaft makes a good lecture demon- 
 stration. A steel collar turned to 1.99 inch inside diameter and 
 heated can be slipped over a 2-inch shaft if one does it quickly 
 so that the hot collar has no time to cool. 
 
 53. Thermo-stresses. The thin parts of an iron casting cool 
 quickly and contract, and as the thicker parts of the casting 
 then cool and contract excessive stresses are produced which 
 often result in the fracture of the casting. 
 
 When a pane of glass is heated quickly over a flame the under 
 surface which is in contact with the flame heats and expands, 
 the upper portions of the pane are subjected to excessive tension, 
 
80 CALENDAR OF LEADING EXPERIMENTS. 
 
 and the pane breaks. Holding the pane in the hand one expe- 
 riences a fairly severe shock due to the quick outward movement 
 which occurs at the instant of rupture. It is not entirely safe to 
 try this experiment. 
 
 The most interesting example of thermo-stresses is furnished 
 by the well-known Prince Rupert's drops. Hold the end of a 
 small glass rod in a blowpipe flame and allow the end of the rod 
 as it drops to fall into water. Most of the drops thus formed 
 explode before one can take them out of the water ; but by trying 
 patiently and especially by trying several varieties of glass one 
 can get drops which stand indefinitely, but which explode 
 suddenly when the surface is scratched or when the long tail is 
 broken off. One can feel a very decided shock when a drop 
 explodes between the fingers. This is a perfectly safe experiment 
 if one closes one's eyes. A drop may be held in the field of the 
 lantern and it suddenly disappears on explosion. The most 
 satisfactory proceeding is to place a drop on a horizontal flat 
 glass plate in the lantern (vertical projection arrangement), and 
 break off the tail of the drop by means of a small pair of pliers. 
 
 Fused quartz is remarkable in that its contraction when cooled 
 from a high red heat is not enough to produce in it a stress suffi- 
 ciently great to produce rupture. A clear quartz tube may be 
 heated in a blast lamp and quenched in water without breaking. 
 
 Heating by compression. Cooling by expansion. Any sub- 
 stance which expands with rise of temperature has its tempera- 
 ture raised by compression or lowered by expansion. This effect 
 is especially great in the case of a gas. 
 
 54. The fire syringe and Diesel-engine ignition. A small 
 plunger fits in a small cylinder and to the plunger is attached a 
 bit of dry cotton cloth. The air in the cylinder is compressed 
 by pushing the plunger in very quickly, and the cotton is found 
 to be burning if the plunger is quickly withdrawn. The air in 
 the cylinder is heated by the compression and the cotton is 
 ignited. This method of ignition is used in the Diesel engine. 
 
HEAT. 
 
 81 
 
 The moving piston draws in a cylinder full of fresh air which is 
 then compressed by the energy of the flywheel and heated so 
 that an oil spray injected into the compressed air is ignited and 
 the burning of the oil keeps the temperature (and pressure) 
 high as the piston recedes. 
 
 55. The formation of cumulous clouds. On a quiet, warm 
 summer's day the sun-heated air near the ground frequently 
 rises in chimney-like columns like aa, Fig. 32. As the air rises 
 its pressure decreases and it ex- 
 pands. As it expands its tem- 
 perature falls; and at a certain 
 level // the falling temperature 
 reaches the dew point and the 
 water vapor in the air begins to 
 condense and form cloud or fog. 
 The beautiful dome-like clouds so 
 commonly seen on a quiet sum- 
 mer's day are the fog-filled tops 
 of such rising columns of warm 
 moist air, and a striking characteristic of these clouds is their 
 level bottoms. 
 
 This cooling of air by expansion can be shown as follows: A 
 large glass flask containing a little water is warmed slightly over 
 a flame so that the contained air is warm and moist. Applying 
 the mouth to the flask a portion of the air can be sucked out 
 (causing a decrease of pressure) and allowed to flow back in 
 again (causing a rise of pressure). At each drop of pressure the 
 air in the flask is cooled and the flask is filled with cloud. At 
 each rise of pressure the air in the flask is warmed slightly and 
 the fog or cloud disappears. 
 
 Fig. 32. 
 
THE CONSERVATION OF ENERGY AND THE FIRST 
 LAW OF THERMODYNAMICS. 
 
 Any definition or principle in physics should be stated so as 
 to call to mind the operation upon which the definition or 
 principle is based, or, in other words, to suggest trial and verifica- 
 tion. Tell a woman she cannot drive a nail and she will, if she 
 takes you seriously, get hammer and nail and try. When Tom 
 Sawyer tells his playmates that they cannot whitewash a fence 
 they are all eagerness to submit the proposition to an experi- 
 mental test. But what young man ever dreamed of trial and 
 verification when told that energy can neither be created nor 
 destroyed? Indeed the statement does not suggest any physical 
 operation whatever, it is actually meaningless unless one knows 
 beforehand just what its meaning is intended to be. Read in 
 this connection Art. 48, pages 68-70, General Physics. 
 
 The extension of the principle of the conservation of energy 
 so as to cover heat effects (including all chemical effects) is 
 called the first law of thermodynamics, and stated in the lan- 
 guage of experiment, it is as follows : A certain change of state is 
 produced in a substance A by the dissipation of work, and the 
 substance A is brought back to its initial condition by contact 
 with another cooler in substance B; then the change produced 
 B is exactly the same as if the original work had been expended 
 (dissipated) on B directly. How much more intelligible this 
 statement than any abstract statement concerning the equival- 
 ence of work and heat ! 
 
 Take a coin and rub it on a board, pretending to heat only the 
 coin, then place the coin in contact with a colder object as if to 
 bring the coin back to its initial condition, and point out that the 
 change which would thus be produced in the colder object would 
 be exactly the same as if the original work had been expended 
 on the colder object directly. 
 
 82 
 
HEAT. 83 
 
 The molecular conception of heat. When heat is imparted to 
 a substance the temperature of the substance usually rises, but 
 not in every case. Thus when heat is imparted to ice at o C. 
 the ice is converted into water at the same temperature. It 
 may be that such a change as the melting of ice increases the 
 number of free molecules so that in spite of a constant average 
 kinetic energy per molecule (constant temperature) the whole of 
 the imparted heat goes to increase the kinetic energy of molecular 
 motion. If this were true in general it would be permissible 
 to speak of heat as the kinetic energy of molecular motion ; but 
 it is not true in general and therefore when heat is imparted to 
 a substance it is stored in the substance partly as an increase of 
 molecular kinetic energy and partly, no doubt, as an increase 
 of molecular potential energy due to a change of molecular 
 configuration of the substance. 
 
 The only case where heat imparted to a substance is known 
 to be stored solely as increased molecular kinetic energy is in 
 the "ideal" or " perfect" gas which does not change its tem- 
 perature during free expansion. This matter is discussed very 
 briefly on page 170, General Physics, and the important concep- 
 tion of the ideal or perfect gas as a gas which conforms to Boyle's 
 law and does not change its temperature during free expansion is 
 explained on pages 171-175, General Physics. Free expansion is 
 discussed more at length on pages 344-347 of Franklin and 
 MacNutt's Mechanics and Heat. 
 
 Heat of combustion. When we ask a young man in class to 
 give a definition we always insist or try to insist, which, alas 
 is not the same thing on a reply which clearly suggests or 
 definitely postulates the physical operation to which the definition 
 relates and in which it finds its meaning. Thus we would wish 
 a young man to say in defining the heat of combustion of coal 
 that it is the amount of heat you get out of a pound of coal when 
 you burn it. A freshman engineer was asked to give this defini- 
 tion; he had learned before to avoid the use of the word per in 
 the intimate and more or less informal talk of the classroom 
 
84 CALENDAR OF LEADING EXPERIMENTS. 
 
 and he managed to get all of the definition clearly stated except 
 to say that the coal was to be burned. This he could not be 
 led to say, and finally, being asked bluntly how to get heat out 
 of coal he replied "Why, professor, I don't know!" 
 
 This young man evidently did not believe that the study of a 
 mathematical science in a college (for even our technical students 
 "go to college" if you please) could possibly relate to so familiar 
 a thing as the burning of coal, and we could not blame him. 
 For, months before the question as to how to get heat out of 
 coal was put to him he had been studying calculus under a 
 class-room regime which had given from 40 to 60 percent of 
 failures year after year, and the unintelligibility of the text 
 book which was used may be fairly judged by the following which 
 is given as an example in a two-page discussion of discontinuity 
 (pages 22 and 23 in fact) : 
 
 "Another form of discontinuity is seen in the function 
 
 2 lfx -{- 2 
 
 y = l/x when x = o. Here y approaches two limits, 
 
 according as x approaches zero through positive or negative 
 values. 
 
 2^^ x -\- 2 2^^ x I 2 
 
 Lim -r- - = i. Lim 1/x - = 2. 
 
 ar=+0 2 s=-0 2 
 
 We see that when x = o the curve jumps from y = 2 to 
 y = I, that is from B to A [referring to a figure]." 
 
 When will our mathematics teachers learn that to be exacting 
 and unintelligible is fatal? When will they learn to appreciate 
 the tremendous importance of appealing to the intuition or 
 sense-complex of a student, especially in a brief and necessarily 
 incomplete introductory discussion of an idea? Any young man 
 can "see" a poor trapped sparrow butt his head against a window 
 pane, and a notion of discontinuity based upon such is incom- 
 parably more useful even in the study of calculus than any 
 purely formal notion, because the young man does not "see" 
 such things. 
 
HEAT. 85 
 
 Transfer of heat. The three processes by which heat is trans- 
 ferred, namely, conduction, radiation and convection, are very 
 briefly referred to on page 131, General Physics, and the elemen- 
 tary theory of heat conduction is there outlined. 
 
 An elementary presentation of the subject of radiation from 
 the point of view of the theory of heat (involving both the 
 atomic theory and thermodynamics) is given on pages 301-316 
 of Franklin and MacNutt's Light and Sound, The Macmillan 
 Co., 1909. 
 
 The transfer of heat by convection is exemplified in the heating 
 of buildings, and the purely empirical rules and equations which 
 are used by heating engineers are to be found on pages 653-687 
 of Kent's Handbook. 
 
 56. Safety lamp experiment. A horizontal piece of wire gauze 
 is lowered quickly into a flame, then for a few moments the gauze 
 conducts heat away from the flame and cools the gases to such 
 an extent that combustion ceases above the gauze. The flame 
 stops at the gauze. Turn on a Bunsen burner, place a piece of 
 wire gauze an inch or two above the burner, light the gas above 
 the gauze, and the flame burns steadily above the gauze without 
 striking through the gauze. This action of wire gauze is utilized 
 in the well-known miner's safety lamp, which was invented by 
 Sir Humphrey Davy. 
 
 57. Cloth singeing experiment. Draw a piece of fuzzy cloth 
 tightly around a metal cylinder (a short piece of steel shafting) 
 and run the flame of a Bunsen burner quickly over the cloth. 
 The body of the cloth is kept cool by contact with the metal, 
 but the fine fuzz is singed off. In the finishing of fine glossy 
 silk the cloth is run over a flame in close contact with a cool 
 metal cylinder. 
 
 Wrap a thin sheet of paper around a cylinder of which part is 
 solid metal and part is wood. Hold the paper-covered cylinder 
 in the flame of a Bunsen burner for a few moments. The paper 
 which is backed by wood will be badly scorched whereas the 
 paper which is backed by metal will not be perceptibly scorched. 
 
86 CALENDAR OF LEADING EXPERIMENTS. 
 
 Heat insulation. The most nearly complete heat insulation is 
 that of the Dewar bulb or thermos bottle, the action of which is 
 explained on pages 310-311 of Franklin and MacNutt's Light 
 and Sound. 
 
 Porous heat-insulating materials, such as saw-dust, wool, cork- 
 board, hair felt and magnesia pipe covering owe their heat- 
 insulating properties to the low heat conductivity of the air which 
 lies stagnant in the pores of the material. 
 
 The phase rule. The discussion of the phase rule as given on 
 pages 107-108, General Physics, should perhaps be given in 
 
 Chapter VIII. 
 
 
 
 58. Crystallization experiments. (a) Moisten a very clean 
 glass plate with a solution of ammonium chloride and place it 
 in the lantern. 
 
 (b) Make a hot concentrated solution of potassium chlorate, 
 and allow it to cool with occasional brisk stirring. After each 
 stirring great numbers of minute crystals form, and as these 
 crystals grow they settle towards the bottom as beautiful thin 
 plates, all alike. These thin plates show beautiful colors if the 
 vessel is placed in sunlight. The cooling solution can be placed 
 in a lantern cell and the floating crystals projected on the 
 screen. 
 
 Segregation. If a vessel of dilute salt solution is frozen, nearly 
 pure ice forms at first, nearly the whole of the salt accumulates in 
 the residual mother liquor at the center, and this residual liquor 
 then freezes if the temperature is sufficiently lowered. Something 
 similar to this generally takes place when any liquid mixture is 
 frozen. Thus certain of the impurities in cast iron or steel collect 
 near the axis of an ingot. It is advisable to exhibit a cut and 
 polished section of a small ingot of steel or cast iron or brass or 
 lead-antimony alloy. A small crucible full of the alloy is allowed 
 to cool slowly and the resulting ingot cut and polished. The 
 visible difference between core and rim of ingot is increased by 
 etching. 
 
HEAT. 87 
 
 59. The boiling paradox. A round-bottomed flask with 
 tightly fitting stopper and a 3O-inch stem contains water, it 
 is boiled until all of the air is driven off, and then quickly inverted 
 in a cup of mercury which stands in a deep tray or tub. See 
 Fig. 33- 
 
 Pouring cold water on the upturned bottom of the flask, 
 reduces the pressure momentarily, as indicated by the mercury 
 column, and causes the water to boil. 
 
 Fig. 33. 
 
 Fig. 34. 
 
 60. The cryophorus. The two connected bulbs A and B, 
 Fig. 34, contain nothing but water (in A) and water vapor. 
 Place bulb B in a mixture of ice and salt, and the water in A 
 freezes. The water vapor pressure is kept at a very low value 
 by the cooling of B, and the vaporization of the water in A 
 reduces its temperature sufficiently to freeze it. Bulb B should 
 be stirred round and round in the ice and salt, and bulb A 
 should be occasionally given a quick blow with the hand, other- 
 wise the water in A may be cooled far below the freezing point 
 without freezing. 
 
 61. The bursting of a cast-iron ball by freezing water. A 
 
 cast-iron ball 4 or 5 inches in diameter with walls about J inch 
 thick is filled full of water , plugged with a paraffine- or wax- 
 covered screw, and placed in a mixture of ice and salt. 
 
88 CALENDAR OF LEADING EXPERIMENTS. 
 
 A curious fact is that a pipe filled with boiled (air-free) water 
 is much more likely to burst on freezing than a pipe filled with 
 fresh water (containing air in solution). As the fresh water 
 freezes the air is segregated, a line of air bubbles is left along the 
 axis of the pipe, and this line of air bubbles may act as a duct 
 to relieve the pressure in a farther portion of the pipe. The 
 bubbles in ordinary "artificial ice" and most of the bubbles in 
 pond ice are the air which is segregated during the freezing. 
 
 62. Regelation. A loop of fine steel wire with two heavy 
 weights is hung over a block of ice, and in the course of an hour 
 the wire will cut through the ice, but leave the block as one piece. 
 In front of the wire where the pressure is g*eat the freezing point 
 (or melting point) of the ice is low, say, i C. Therefore 
 some of the ice (originally at o C.) in front of the wire melts and 
 the temperature of the region in front of the wire falls to 1 C. 
 and stands at that temperature. The water formed by the 
 melting flows to the back of the wire where it is free from excess 
 pressure and where it -stands at o C. Therefore heat flows 
 steadily across the wire from the warm region back of the wire 
 to the cooler region in front of the wire, the water back of the 
 wire is continuously frozen by this loss of heat, and the ice 
 in front of the wire is continuously melted by this supply of 
 heat. 
 
 63. Wood's metal. A striking example of low-melting-point 
 alloy is Wood's metal which consists of bismuth 50.1 %, cad- 
 mium 10.8 %, lead 24.9% and tin 14.2 %. Its melting point is 
 65.5 C., whereas the most fusible of its constituents is tin with 
 a melting point of 232 C. 
 
 Select a heavy-pattern teaspoon, imbed it in wax, cut away the 
 wax even with one face of the spoon, and cover spoon and wax 
 with a block of plaster of Paris. When the plaster block is 
 hard lift it off, varnish or oil its face, lay the spoon upon it and 
 cover with a second block of plaster. When this second block 
 of plaster is hard lift it off, loosen and remove the spoon and cut 
 
HEAT. 89 
 
 a gate through which Wood's metal may be poured into the 
 mould. 
 
 64. Bottle washer's experiment. A large dry bottle with a 
 tightly fitting stopper is warmed carefully over a flame, a small 
 sealed glass bulb of water is placed in the bottle, the bottle is 
 stopped, and the bulb is broken by shaking the bottle. If the 
 bottle is at 50 C. the pressure in the bottle rises to 92 milli- 
 meters above the atmosphere, and a decided burst of spray is 
 produced when the stopper is removed. 
 
 Transformation points or temperatures. Many solid sub- 
 stances can exist in several states or modifications as in the case 
 of ice as briefly described on page 147, General Physics, the change 
 from one state or modification to another state or modification 
 takes place at a definite transformation temperature (at a given 
 pressure), and the change is generally accompanied by the 
 evolution of heat (as in the freezing of water) or the absorption of 
 heat (as in the melting of ice). 
 
 This matter may be best exemplified by a detailed study in the 
 laboratory. An ounce or two of ammonium nitrate crystals is 
 melted in a small glass beaker, a thermometer reading up to 
 200 C. is placed in the melt, the heating flame is removed 
 and the thermometer readings are taken at intervals of 30 
 seconds while the beaker cools. The melt freezes at 168 C., 
 and changes its crystalline structure at 125 C., again at 85 C., 
 and again at 35 C. 
 
 Retarded transformations. In many cases a pure substance 
 in one state or modification can be heated above or cooled below 
 the transformation temperature at which it normally changes to 
 another modification before the actual change takes place. See 
 pages 151-152, General Physics. 
 
 65. Superheating and undercooling of water. Heat a small 
 quantity of very pure air-free water in a clean test tube. The 
 water will rise very considerably above its normal boiling point, 
 
90 CALENDAR OF LEADING EXPERIMENTS. 
 
 and when boiling does begin it takes place with almost explosive 
 violence, throwing most of the water out of the tube. This 
 experiment succeeds best if a small amount of sulphuric acid is 
 added to the water. In this case the normal boiling point is 
 raised, but the dilute acid does not boil perceptibly until it is 
 heated above its normal boiling point, as stated. The water is 
 freed from air by previous boiling in a larger vessel like a 
 flask. 
 
 Cool a small quantity of very pure air-free water in a clean 
 glass flask by dipping the flask repeatedly into a mixture of ice 
 and salt. The water cools 5 or 6 degrees below o C. (as may 
 be indicated by a thermometer with a very clean bulb) without 
 freezing, and then a sudden shock will cause almost instant 
 formation of fine ice crystals throughout the water. 
 
 To cool water 5 or 6 below o C. without freezing, the water 
 must be air free and extremely clean. It is difficult to keep the 
 water clean enough in an open vessel. Therefore this experiment 
 succeeds best by using an ordinary water hammer which con- 
 tains clean boiled water. Dip the containing bulb into slightly 
 salt water containing pieces of ice (temperature of mixture 
 about 5 C.), remove carefully and shake. Instead of a 
 sealed water hammer one may boil distilled water in a clean 
 flask and close the flask with a clean rubber stopper while the 
 boiling is taking place, and a clean-bulb thermometer may pass 
 through the stopper to indicate the temperature of the under- 
 cooled water in the flask. 
 
 Superheated and undercooled water are examples of retarded 
 transformations. Also a super-saturated solution of a salt is an 
 example of a retarded transformation. 
 
 Make a hot saturated solution of sodium acetate in a small 
 clean flask, stop the flask with a clean cork and set it aside 
 to cool. The cool supersaturated solution crystallizes suddenly 
 when a small crystal of sodium acetate is dropped into the flask 
 or when the solution is touched by a stick which has been wet 
 with sodium acetate solution and allowed to dry. 
 
HEAT. 
 
 This experiment may be projected by the lantern by using a 
 small flask and placing it in a water filled lantern cell. 
 
 An example of an almost permanently retarded transformation 
 is as follows: Make a strong syrup by dissolving granulated 
 sugar (or, better, white rock candy) in a small quantity of hot 
 water, boil the syrup over a slow fire until nearly all the water is 
 driven off, and set aside to cool. The solution does not freeze 
 (crystallize) at once but cools to a glass-like substance. Place 
 a small crystal of granulated sugar on the surface of the glass-like 
 substance, press it into close contact therewith, and allow the 
 whole to stand in a cool place for several days. A crystalline 
 growth will slowly spread out from the crystal of granulated 
 sugar, showing the slow transformation of the glass-like substance 
 (which is the stable form at higher temperature) to a crystalline 
 form (which is the stable form at lower temperature). 
 
 Pure sugar cannot be melted, because it is decomposed chem- 
 ically before its true melting point is reached, but nearly all of 
 the water in syrup can be driven off, leaving the pure or nearly 
 pure sugar in the melted condition (as a retarded state or con- 
 dition). 
 
 66. The recalescence of steel. A piano steel wire about 0.04 
 inch in diameter and 5 or 6 feet long is arranged as indicated in 
 Fig- 35* the terminals a and b 
 being connected to no- volt sup- 
 ply mains through a suitable 
 rheostat and a controlling switch. 
 Two small plates are clamped to 
 the wire at the lower end, and a 
 light wooden pointer catches un- 
 der a cleat at c, straddles the 
 wire and rests on the small 
 clamped plates as indicated. Fig 35 
 
 The movement of the pointer 
 while the wire is being heated or cooled shows a marked irregu- 
 
92 CALENDAR OF LEADING EXPERIMENTS. 
 
 larity of expansion at the transition temperature, which is about 
 700 C. 
 
 As the wire is being heated its temperature rises a little above 
 700 C. before the change of state or condition takes place, and 
 then when the change does take place a very noticeable momen- 
 tary drop in temperature (decrease of brightness of the hot wire) 
 occurs. 
 
 As the wire is being cooled its temperature falls a little below 
 700 C. before the change of state takes place, and then when 
 the change does take place a very noticeable momentary rise 
 of temperature occurs. 
 
 67. Hardening and tempering of steel. Heat a small rod of 
 tool steel to bright redness, quench it in water, and show that 
 it will scratch glass and that it is extremely brittle. Explain 
 the tempering process as a more or less complete conversion of 
 the hardened steel to the condition which is stable at low tem- 
 peratures (soft variety of steel). This conversion takes place at 
 ordinary room temperature but it is greatly hastened by rise 
 of temperature. A given degree of tempering is produced by 
 keeping a glass hard piece of steel at a certain temperature for a 
 certain time, and the higher the temperature the shorter the 
 time required for a given degree of tempering. 
 
 Make the points of three steel tools glass hard, then make 
 small polished areas at tips of tools, hold tools one at a time with 
 portions just back of tips in a Bunsen flame, watch the polished 
 areas carefully, quench the first tool when the polished area 
 shows a light straw color, quench the second tool when the 
 polished area shows a brownish color, and quench the third tool 
 when the polished area shows a brilliant blue color. These 
 colors depend upon a combination of time and temperature and 
 they are extensively used as indicators in the tempering of steel 
 tools. 
 
 Diffusion and osmosis. When a solution of salt is in thermal 
 equilibrium it is homogeneous, that is to say, it has the same 
 
HEAT. 93 
 
 degree of concentration everywhere. When a solution of salt 
 is more concentrated at the bottom of a vessel, for example, 
 than at the top, the inequalities of concentration are slowly 
 eliminated by the movement, the very slow movement, of the 
 dissolved salt through the solution from the region of high 
 concentration to the region of low concentration. This move- 
 ment of the dissolved salt is called diffusion. 
 
 When solutions of different concentration are separated by a 
 thin membrane, the diffusion of the salt through the membrane 
 is hindered more or less, whereas the water passes through more 
 freely. A thin bladder which is filled with water, tied and placed 
 in a vessel of brine will become flabby because of the loss of 
 water; and if the bladder is partly filled with brine, tied and 
 placed in a vessel of water it will swell until it bursts because of 
 the entrance of water. If the brine-filled bladder does not 
 burst the pressure in it rises to a fairly definite value, and this 
 limiting pressure, reckoned above the pressure of the surrounding 
 water, is called the osmotic pressure of the brine, and the action 
 of the membrane in permitting selective diffusion is called 
 osmosis. This phenomenon of osmosis which is a very widely 
 occurring phenomenon, has given rise to the precise conception 
 of the semi-permeable membrane, a conception which is of great 
 utility in the theory of solutions. 
 
 Sprinkle salt on fresh meat or fish, and the meat or fish appears 
 to become wet. Each individual cell of the meat or fish behaves 
 like a bladder filled with water (which indeed it is) and placed 
 in strong brine. Water flows out of the cells through the cell walls. 
 
 68. Solution and diffusion of CuSO 4 . Drop a few crystals 
 into a tall jar of water and place the jar where the class can see 
 from day to day the slow diffusion of the copper sulphate 
 upwards. 
 
 69. Osmosis. Fill a bladder partly full of strong brine, tie 
 the bladder, place it in a jar of water, and place the jar where the 
 slow distension of the bladder can be seen. 
 
94 CALENDAR OF LEADING EXPERIMENTS. 
 
 The second law of thermodynamics. Next to the principle 
 of the conservation of energy the most important generalization 
 of physics is the second law of thermodynamics or the principle 
 of entropy, as it is sometimes called, and no one can have any 
 sort of grasp of the philosophy of physics who does not have 
 some degree of understanding of both of these generalizations. 
 The student should, therefore, .be required to study very carefully 
 pages 152-169, General Physics. 
 
 70. Thermo-elastic properties of rubber. Cooling by expan- 
 sion and heating by compression is usually very small in liquids 
 and solids and difficult to detect. Indeed this is the case with 
 ordinary rubber insofar as change of volume due to change of 
 pressure is concerned. But when a rubber band is stretched it 
 is very perceptibly cooled, and when the stretch is relieved the 
 band is warmed. This may be shown by stretching and shorten- 
 ing a rubber band repeatedly and holding the band in contact 
 with the lips. 
 
 An interesting example of a widely applicable argument based on the second 
 law of thermodynamics is given on pages 393-396 of Franklin and MacNutt's 
 Mechanics and Heat. Knowing that a rubber band cools when stretched, this 
 argument shows that a rubber band stretched by a hanging weight must shorten 
 when heated; and this conclusion is easily verified by experiment. 
 
PART III 
 ELECTRICITY AND MAGNETISM 
 
 95 
 
THE SIDE-STEPPING OF MATHEMATI 
 
 'ICS. 
 
 The discussion of Pascal's principle on pages 52 and 53 of this volume illustrates 
 a wide-spread tendency among teachers of physics, a contempt for precise thinking, 
 which is deplorable. What are we to do? And the mathematics teacher also shows 
 his contempt for mathematics when he side-steps mathematical ideas and pins his 
 faith and his students to formalism and machinery. What are we to do ? 
 Our own inclination is that of P. G. Tait who in 1876 wrote that "In defense of 
 accuracy we must be zealous, as it were, even to slaying." But we thank Professor 
 Tait for the humorous turn in his qualifying clause; for indeed no human being, 
 and least of all a timid teacher, would wish in these terrible times to do no more 
 than run amuck, as it were, among his friends. See pages 53 and 106 of this 
 volume for further comment. 
 
MAGNETISM AND THE ELECTRIC CURRENT. 
 
 The traditional plan in the elementary treatment of electricity 
 and magnetism is to begin with electrostatics, but the authors 
 believe that it is much better to begin with magnetism and the 
 electric current because in this plan a wide range of simply 
 related experimental facts comes quickly into view; the many- 
 sided magnetic effect of the electric current (see pages 181, 196, 
 198, 199, 209, 254 and 276, General Physics), the chemical effect 
 of the electric current, and the heating effect of the electric 
 current. 
 
 ONE SET OF ELECTRICAL UNITS IS SUFFICIENT. 
 
 It is a fact, however reluctant some of our physicists may be 
 to admit it, that one set of units of measurement is sufficient. We 
 make exclusive use, therefore, of the units of the so-called 
 ' ' electromagnetic ' ' system. 
 
 Some familiarity with the "electrostatic" system of units is 
 demanded, however, by the wide use of this system in the 
 literature of electricity and magnetism, but essential demands 
 and conventional demands are not on the same plane, and the 
 authors emphasize this fact by ignoring the "electrostatic" 
 system of units entirely. It is sufficient to present the main 
 features of the electrostatic system of units orally in class. 
 
 97 
 
THE MYSTERIOUS ERUPTION OF "V" IN THE DIS- 
 CUSSION OF THE TWO SYSTEMS OF 
 ELECTRICAL UNITS. 
 
 Any discussion which places emphasis on the fact that the 
 "electromagnetic" unit of charge divided by the "electrostatic" 
 unit of charge is equal to the square of the velocity of light, but 
 which stops short of a complete elementary discussion of electro- 
 magnetic wave motion, is, in our opinion, misleading and fan- 
 tastic. Suppose we had two rival systems of mechanical units, 
 one of which was based on the arbitrary assignment of unity 
 as the density of the air and the other on the arbitrary assign- 
 ment of unity as the compressibility of air, and suppose that 
 the only reference to sound in our whole treatment of physics 
 were to mention the mysterious eruption of v (the velocity of 
 sound) in the ratios of the units in the two systems! would such 
 a thing make for clear understanding? It certainly would not. 
 We have, for this reason, chosen to treat the factor B in equa- 
 tion (i), page 293, General Physics, as a simple proportionality 
 factor to be determined by experiment, and it is exactly that. 
 Indeed it is nothing more than that to one who does not go into 
 the theory of electromagnetic wave motion. 
 
 98 
 
ELECTROMAGNETIC WAVE MOTION.* 
 
 Electrical effect of a magnetic field which moves in a direction 
 at right angles to itself; sidewise motion of magnetic lines of 
 force. The electromotive force along a path which encircles 
 an amount of magnetic flux <$> is 
 
 where E is expressed in ab volts and -rr is the rate of change 
 
 of the magnetic flux. See equation (81), page 256, General 
 Physics. The meaning of the negative sign is that the induced 
 electromotive force is in the direction around the path (or cir- 
 cuit) in which a left-handed screw would have to be turned to 
 travel in the direction in which the increasing flux <f> passes 
 through the opening of the circuit (in the direction of the force 
 with which the magnetic field which produces < would act on 
 a north pointing magnet pole). 
 Another form of equation (i) is 
 
 E = IHv (2) 
 
 where E is the electromotive force in abvolts induced in a 
 straight rod / centimeters long which travels sidewise at a 
 velocity of v centimeters per second across a magnetic field of 
 which the intensity is H gausses as indicated in Fig. 184, page 
 255, General Physics. See equation (80), page 255, General 
 Physics. 
 
 Let us think of the rod be, Fig. 184, page 255, General Physics, 
 as stationary and imagine the magnetic field as sweeping past 
 
 * This discussion is not intended to be entirely rigorous, or complete. Indeed 
 the idea of sidewise motion of identifiable lines of force is highly artificial. A 
 better point of view is that in which the differential equations of wave motion are 
 derived from the fundamental circuit-equations (i) and (8) below. This point of 
 view is developed in Part VI of this volume. 
 
 99 
 
IOO 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 the rod at velocity equal and opposite to v. Then equation (2) 
 still applies, E is the electromotive force in abvolts along the 
 stationary rod, and this electromotive force exists whether the 
 rod is there or not so that E/l is the electric field intensity in 
 abvolts per centimeter produced by the sidewise motion of the 
 magnetic field. Therefore, representing this electric field inten- 
 sity by / and remembering that v has been reversed in sign, 
 equation (2) becomes 
 
 f=-vH (3) 
 
 Magnetic effect of an electric field which moves in a direction 
 at right angles to itself; sidewise motion of electric lines of 
 force. A charged condenser A A BB, Fig. 36, is allowed to 
 
 B 
 
 edgewise view 
 
 side view 
 
 Fig. 36. 
 
 discharge through a wire. The current / is equal to the rate 
 of decrease dQ/dt of the charge on either plate. But Q = CE, 
 where E is the electromotive force between the condenser plates 
 in abvolts and C is the capacity of the condenser in abfarads, 
 and 
 
 ;;;;;; . .--. C = B (4) 
 
 where a is the area of one face of a condenser plate, x is the 
 distance between the condenser plates and B is a proportionality 
 factor whose value is to be determined. See Art. 206, page 293, 
 
ELECTRICITY AND MAGNEtlSM. ' 
 
 General Physics. Therefore, using the value of C from equation 
 (4) in the expression Q = CE, we get 
 
 - (5) 
 
 36 
 
 But E/x is the intensity of the electric field in the air space 
 between the condenser plates, and a X E/x is the electric 
 flux ^ from plate to plate. Therefore 
 
 Q = B* (6) 
 
 Differentiating we get 
 
 Now the magnetomotive force M along any path which 
 encircles / abamperes of current is equal to 4^7*. Therefore, 
 using M/4TT for I, equation (7) becomes 
 
 M = ^B- d -j t (8) 
 
 The magnetomotive force may be thought of as being taken 
 along the dotted line in the side view in Fig. 36. A changing 
 electric flux through the opening of any loop always means the 
 existence of a magnetomotive force around the loop in accordance 
 with equation (8) which is identical in form to equation (i) 
 except for the presence of the proportionality factor 4irB and 
 the absence of the negative sign [the positive sign is proper in 
 equation (8) because the magnetic field surrounding an electric 
 current is in the direction in which a right-handed screw would 
 have to be turned to travel in the direction of the current]. 
 Considerations almost identical in form to those involved in the 
 discussion of Fig. 184, page 255, General Physics, would lead to 
 an equation formally identical in form to equation (3), namely 
 
 H = 4*Bvf (9) 
 
 where H is the intensity in gausses of the magnetic field pro- 
 
 * See pages 79-81, Advanced Electricity and Magnetism. Franklin and MacNutt, 
 The Macmillan Co., 1915. 
 
ioi2 v CALENDAR OF LEADING EXPERIMENTS. 
 
 duced by the sidewise motion at velocity v of an electric field 
 of intensity / abvolts per centimeter. 
 
 Meaning of opposite signs in equations (3) and (9). Con- 
 sidering that we have chosen to think of the magnetic field 
 moving to left in Fig. 184, page 255, General Physics, as being 
 exactly equivalent to motion of rod be to the right, and knowing 
 that induced electromotive force is from c towards b when the 
 magnetic field is directed away from the reader, it is evident that 
 the relation between /, v and H as expressed by equation (3) 
 must be as shown in Fig. 37. 
 
 Multiplying H by v in equation (3) turns H in a clock- 
 wise direction as seen from the left in Fig. 37. 
 
 Multiplying /by -f v in equation (9) turns / in a counter- 
 clock-wise direction as seen from the left in Fig. 38. 
 
 The electromagnetic plane wave. It is convenient to turn 
 Fig. 38 so that directions of v, H and / are the same as in 
 - 37 1 thus giving two identical figures, 39 and 40. 
 
 Fig. 37. 
 
 H is perpendicular to the plane of 
 the paper and directed away from 
 reader. This figure shows the relation 
 between /, v and H as expressed by 
 equation (3). 
 
 O 
 
 Fig. 38. 
 
 / is perpendicular to the plane of the 
 paper and directed towards reader. 
 This figure shows the relation between 
 H, v and / as expressed by equation 
 (9). 
 
 Now it is evident that the / which is produced by sidewise 
 motion of H moves along with H because the induced / is 
 always where H is; and, similarly, the H which is produced 
 by sidewise motion of / moves along with /. Therefore: 
 
 (a) The H in Fig. 40 which is produced by the sidewise mo- 
 tion of / may be the H in Fig. 39, and 
 
ELECTRICITY AND MAGNETISM. 
 
 103 
 
 (b) The / in Fig. 39 which is produced by the sidewise motion of 
 H may be the / in Fig. 40; v being the same in both figures. 
 
 Fig. 39. 
 
 / is the electrical field produced by 
 sidewise motion of H at velocity v. 
 
 Fig. 40. 
 
 H is the magnetic field produced by 
 the sidewise motion of / at velocity v. 
 
 Two such mutually sustaining magnetic and electric fields 
 traveling along together constitute a plane electromagnetic wave. 
 
 A front view (as seen from v) of Figs. 
 39 and 40 is shown in Fig. 41, in which 
 the full lines represent magnetic lines 
 of force and the dotted lines represent 
 electric lines of force. 
 
 It is evident from what is stated 
 above that equations (3) and (9) are 
 to be thought of as simultaneous equa- 
 tions inasmuch as /, H and v have 
 identically the same meanings in both 
 equations. Therefore, multiplying equa- 
 tions (3) and (9) member by member 
 we get 
 
 
 
 
 i 
 
 ; t! 
 
 
 
 
 
 
 { 
 
 
 
 A 
 
 
 
 
 ; 
 
 "! "I i 
 
 
 i (_ . 
 
 
 i 
 
 
 
 
 / 
 
 ^ ! 
 
 
 
 
 K i 
 
 j j | ' 
 
 
 I 
 
 
 
 
 k 
 
 r, J i 
 
 
 i 
 
 
 
 
 
 )! ' .1 
 
 
 i 
 
 
 [ 
 
 
 i 
 
 i 
 
 
 
 
 
 
 r 
 
 u 
 
 
 
 
 
 
 
 ! I 
 
 
 
 Fig. 41. 
 
 Front view of plane elec 
 tromagnetic wave (coming 
 towards reader). Full lines 
 represent magnetic lines of 
 force, dotted lines represent 
 electric lines of force. 
 
 or, ignoring the negative sign which here relates to the quaternion 
 interpretation of the square of a vector quantity, we get 
 
 V = 
 
 (II) 
 
 Also, dividing equation (3) by equation (9) member by member 
 and (ignoring negative sign as above) we get 
 
 / 
 
 H 
 
 (12) 
 
104 CALENDAR OF LEADING EXPERIMENTS. 
 
 Equation (n) gives the velocity at which two mutually sustain- 
 ing electric and magnetic fields must travel, and equation (12) 
 gives the necessary ratio of the two mutually sustaining fields. 
 The value of v is 2.99778 X io 10 centimeters per second in air 
 (the velocity of light), and therefore the value of ' B is 
 8.84 X io~ 23 so that the capacity of an air condenser in abfarads 
 as expressed by equation (4) is 
 
 abfarads 
 
 (l 3 ) 
 
 THE ELECTROSTATIC UNIT OF CHARGE. 
 
 According to equation (6) in the above discussion the electric 
 flux which emanates from Q abcoulombs of charge (comes in 
 to the charge if it is negative) is equal to Q/B. This relation is 
 general, and we may apply it to a concentrated positive charge 
 Q r . The electric flux passes out from Q f symmetrically in all 
 directions. Let / be the electric field intensity at a point distant 
 r from Q'. Then 4?rr 2 /* is the electric flux from Q f so that 
 
 or 
 
 Let another charge Q" be placed at a point distant r from 
 Q'. Then, according to equation (95), page 308, General Physics t 
 
 i O'O" 
 fQ" or n'^T~ > is the force F in dynes exerted on Q" by 
 
 47T-O T 
 
 Q'. That is 
 
 r* 
 
 * Argument similar to Art. 135 page 135 General Physics. 
 
ELECTRICITY AND MAGNETISM 105 
 
 Consider two equal concentrated charges g (expressed in ab- 
 coulombs) which exert a force of one dyne on each other when they 
 are at a distance of one centimeter apart. Putting F = I and 
 r = I inequation (15), and writing qq or q z for Q'Q"j we have 
 
 or 
 
 g = A/4^B (16) 
 
 But A/47TJ3 = i/v according to equation (n) so that equation 
 (16) becomes 
 
 g = - abcoulombs (17) 
 
 The charge g as here specified is the electrostatic unit of charge, 
 and therefore there are i/v abcoulombs in one electrostatic 
 unit of charge. 
 
POTENTIAL. 
 
 The velocity potential of a fluid at a point (when it exists) 
 is the height at that point of an imagined hill whose slope is 
 everywhere equal to the fluid velocity. For example, consider 
 a layer of fluid lying on a plane and moving at constant uniform 
 velocity v in the direction of the #-axis of reference. Then if 
 \f/ is the height of the potential hill at a point whose abscissa 
 
 d\l/ 
 is x we have = v, whence ^ = vx + a constant. 
 
 There are distributions of fluid velocity which have no velocity 
 potential ; for example, a layer of water on a rotating disk. 
 
 The potential of an electric field at a point (when it exists) 
 is the height at that point of an imagined hill whose slope is 
 everywhere equal to the electric field intensity. For example, 
 the electric potential is ^ = fx + a constant in a uniform electric 
 field of intensity /, the x-axis being chosen in the direction of 
 the field. If / is expressed in volts per centimeter and x in 
 centimeters, the height of the potential hill is expressed in volts. 
 
 The idea of potential is useful because the precise mode of 
 distribution of an electric field (or of a magnetic field or fluid 
 velocity) in space may be easily thought of and easily formulated 
 in terms of the potential distribution, and especially because 
 certain mathematical transformations, such, for example, as are 
 involved in a change of axes of reference,* are easily made 
 when the field distribution is expressed in terms of potential. 
 
 We carefully avoid the use of the word potential in our General 
 Physics for three reasons, namely, (a) Because electric field 
 (or magnetic field or fluid velocity) is the physically real thing, 
 and potential is merely a mathematical idea, (b) Because the 
 
 * A good example of such a transformation is given on page 246 of Franklin, 
 MacNutt and Charles's Calculus. 
 
 106 
 
ELECTRICITY AND MAGNETISM. 107 
 
 idea of potential is useful only when one is considering the space 
 distribution of a field, and such matters are largely beyond the 
 scope of our General Physics, and (c) Because the common use 
 of the term potential is hopelessly vague and shot through and 
 through with absurdity from the rigorous mathematical point 
 of view. We believe in physical reality, and we believe in 
 mathematical thinking; therefore we cannot tolerate the common 
 use of the word potential. 
 
 Let no one imagine that by common usage we refer to the 
 habit of the man in the street. Recently, in discussing the 
 question of potential with a teacher of physics in a technical 
 school, he said "Potential, potential; you mean the potential 
 function." Exactly so; but his implication was that he always 
 meant something very much simpler than the ' potential function ' 
 when he used the term! At a given time temperature has a 
 definite value at every point on the map, and even a complex 
 minded mathematician would be shocked if his inquiry as to the 
 temperature at Cape Cod were to be met by the response 
 " Temperature, temperature; you mean the temperature func- 
 tion," and he might reply, somewhat impatiently, "Yes, in the 
 name of all that is precise and definite, that is what I do mean." 
 
 It is very greatly to be desired that our teachers of physics 
 and electrical engineering should understand the essentials of 
 what is nowadays called vector analysis, the mathematics of the 
 distribution of scalar and vector values in space. A very simple 
 discussion of this general subject is given on pages 210253 of 
 Franklin, MacNutt and Charles' Calculus.* The idea of electric 
 potential and its use in the study of the electric field is discussed 
 
 * This book is recommended by a large western university for use by its students 
 in correspondence courses for private study, it has been used, and to some extent 
 successfully, in the night school and in the high school and in the college, and it 
 has been highly praised as a good book for the graduate student. To such a state 
 of topsy-turvydom has our vague and unintelligible collegiate instruction in the 
 mathematical sciences brought us! Any mathematical treatise that is intelligibly 
 straightforward and definitely clear and precise is, it would seem, good medicine 
 in our time for anybody. 
 
108 CALENDAR OF LEADING EXPERIMENTS. 
 
 at length on pages 165-192 of Franklin and MacNutt's Advanced 
 Electricity and Magnetism, The Macmillan Co., 1915. 
 
 71. The electromagnet. An experiment which is very striking 
 is to exhibit a large and powerful electromagnet. To eliminate 
 the spectacular features, however, let the magnet be of moderate 
 size, consisting of a straight bar of iron or mild steel with a coil 
 of wire upon it. Show that the bar loses nearly all of its magne- 
 tism when the exciting current is reduced to zero by opening the 
 supply switch. Magnetize a bar of hardened steel by placing 
 it in the coil and show that it retains a large part of its magnetism 
 when removed from the coil. 
 
 72. The heating effect of the electric current, although al- 
 ready familiar to every student, may be strikingly shown by 
 stretching an iron or German silver wire across the lecture table 
 and connecting it through a supply switch to no- volt supply 
 mains. 
 
 73. The chemical effect of the electric current may be shown 
 by placing two lead electrodes in a jar of dilute sulphuric acid 
 and connecting them through a heavy-current rheostat to direct- 
 current supply mains, or a thin coating of copper may be de- 
 posited on a bright piece of platinum or silver foil using a solution 
 of copper sulphate as the electrolyte and using a current of not 
 more than, say, half an ampere; an ordinary glow lamp in 
 circuit will provide for a suitable current from no- volt direct- 
 current mains. See Experiment 84. 
 
 74. The magnetic compass. The most convenient form of 
 demonstration compass is a magnetized steel needle six or eight 
 inches long mounted on a pivot stand. 
 
 75. Poles of a magnet. Dip a bar magnet into a box of iron 
 filings and point out that filings cling only to end parts of bar 
 as shown in Fig. 114, page 187, General Physics. 
 
 76. Attraction of unlike magnet poles. Repulsion of like 
 magnet poles. Most young men are familiar with magnetic 
 
N 
 
 V 
 
 handle^ 
 
 S 
 
 ELECTRICITY AND MAGNETISM. 109 
 
 attraction, whereas but few are familiar with magnetic repulsion. 
 Mark north-pointing ends of two compass needles by painting 
 them red. Remove one of the needles from its pivot and show 
 that its red end repels the red end of the other 
 needle and attracts the unpainted end of the 
 other needle. 
 
 Place a short straight cylindrical steel mag 
 net NS, Fig. 42, on a horizontal glass plate 
 in the lantern (vertical projection arrange- Fig 42 
 ment) and bring another magnet N'S' near 
 
 to it, as shown. In one position NS is repelled by N'S', 
 and in the reverse position NS is attracted. 
 
 77. Distributed poles and concentrated poles. Magnetize a 
 short thick bar and a long slim bar of hardened steel, as explained 
 in experiment 71. Dip both magnets into a box of iron filings 
 and point out that filings cling to extended portions near the 
 ends of the short -thick bar, but only to short portions near the 
 ends of the long slim bar. 
 
 Magnetize a long slim steel bar, remove it from the magnetizing 
 coil, turn it end for end, and bring one end of the bar thus 
 reversed into the end of the coil. Dip the bar into filings and a 
 widely distributed pole will be shown to be spread over the middle 
 portions of the bar, with two moderately concentrated poles at its 
 ends. Show by using a compass needle that the end poles are 
 alike in kind and that the middle pole is unlike the end poles. 
 
 Remark. It may be explained to the student that the idea 
 of the concentrated magnet pole is a differential. Thus the 
 element of force action between any two magnet poles is 
 
 dm' -dm" C Cdrn'-dm" 
 AF = so that ^=11 , and this equation 
 
 is a sextuple integral although set in the form of a double integral. 
 Of course it does no good to explain the concentrated pole to the 
 student as a differential, and our easy reference to double and 
 sextuple integrals is made to deter the too eager teacher from 
 making the attempt. 
 
no 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 78. The unit pole. It is a great help to the understanding of 
 Art. 127, page 188, General Physics, to have a number of pairs 
 of long slim magnets; taking one pair in the hands at a time, hold 
 them so that two of their poles are about one centimeter apart 
 and the other two poles very far apart (assumed to be indefinitely 
 far apart), pretend to estimate the force of attraction or repulsion. 
 The student will thus see that it is possible to find a pair of 
 magnet poles which attract or repel each other with a force of 
 one dyne when they are one centimeter apart. 
 
 79. Magnetic figures. Place a short bar magnet on a hori- 
 zontal glass plate in the lantern (vertical projection arrangement), 
 place a thin glass plate over the magnet with steadying wedges 
 or blocks under its edges, dust fine iron filings very sparingly on 
 the thin glass plate and tap the thin glass plate lightly with a 
 hard object like a short glass tube. It is desirable to show also 
 the filaments of filings between two opposite poles of two broad 
 ended magnets as indicated in Fig. 119, page 191, General 
 Physics. 
 
 80. Behavior of a magnet in a non-uniform magnetic field. 
 
 The two poles of a magnet which is placed in a uniform magnetic 
 
 field (a field which has everywhere 
 the same intensity and of which 
 the lines of force are parallel) are 
 acted upon by equal and opposite 
 forces, as indicated by the arrows 
 in Fig. 120, page 191, General 
 Physics. Therefore a uniform 
 magnetic field tends only to turn 
 a magnet into a certain direction, 
 the direction of the field. The 
 two poles of a magnet which is 
 placed in a non-uniform magnetic 
 field (a field which does not have everywhere the same intensity 
 and of which the lines of force are not parallel) are not equal and 
 
 Fig. 43. 
 
ELECTRICITY AND MAGNETISM. 
 
 Ill 
 
 opposite, and they tend not only to turn the magnet into the 
 direction of the field but also to impart to the magnet a motion 
 of translation. Thus the two arrows F and F' show the two 
 forces exerted on the small magnet ns in Fig. 43. 
 
 The attraction of a particle of iron by a magnet depends in the 
 first place on the conversion of the particle into a magnet, and in 
 the second place on the non-uniformity of the magnetic field in 
 which the particle finds itself. 
 
 The magnetic field near the flat end of a large magnet pole is 
 nearly uniform as indicated by the lines of force in Fig. 44; 
 
 S^^fSSS*S85 yj i! i^'W'^w^i^f^' 
 
 Fig. 44. 
 
 Fig. 45. 
 
 near the sharp corners, however, the field is distinctly non- 
 uniform. Therefore small particles of iron are not perceptibly 
 attracted by the flat face of the pole, but only by the sharp 
 corners. Pass the square end of a magnet over a table on 
 which fine iron filings are dusted very sparingly, and the filings 
 are gathered only on the sharp corners of the pole. The lines 
 of force in the neighborhood of a sharply pointed pole diverge 
 strongly as indicated in Fig. 45, that is to say, the magnetic 
 field in the neighborhood of the point is non-uniform to a high 
 degree, and a pointed pole exerts a strong attraction for a small 
 particle of magnetic material. The eye-magnet which is used 
 for removing particles of steel from the eye has a sharp point, 
 and the essential feature of the magnetic ore separator as used 
 to separate the particles of magnetic material from finely crushed 
 ore is a sharply pointed magnet. 
 
112 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 81. Oersted's experiment. Connect a flexible wire (one strand 
 of a lamp cord) to no- volt direct-current mains through a 
 rheostat so as to have a current of about 2 or 3 amperes in the 
 wire. Stretching a portion of the wire in a north-south direction 
 between the hands, bring it down over a compass needle and 
 note the deflection of the needle. Interpret this experiment as 
 explained in Art. 137, page 197, General Physics. Connect a 
 copper plating arrangement in the circuit so as to deposit copper 
 on a bright platinum or silver surface, and point out the fact that, 
 according to the indication of the magnetic needle, the current 
 flows through the copper sulphate solution towards the platinum 
 or silver electrode on which copper is deposited. 
 
 Wrap the flexible wire several times around a small soft iron 
 bar, test the magnetic polarity of one end of the bar, and, having 
 previously established the direction of flow of current through 
 wire as above explained, point out the meaning of Fig. 130, page 
 198, General Physics. 
 
 82. Side push of the magnetic field on an electric wire. (a) 
 Stretch a 6 or 7-foot span fine copper wire loosely in front of a 
 strong electromagnet as indicated in Fig. 46. The wire should 
 
 wire 
 
 current 
 
 wire 
 
 side view 
 (wire seen endwise) 
 
 top view 
 
 Fig. 46. 
 
 be covered with white cotton or silk so as to be easily visible. 
 Connect the wire through a suitable rheostat and through a 
 reversing switch to no-volt, direct-current supply mains, and 
 
ELECTRICITY AND MAGNETISM. 113 
 
 show that the wire is pushed sidewise as indicated by the dotted 
 arrow in the side view. Reverse the current and show that 
 side force is reversed. 
 
 It is remarkable how widely prevalent is the idea of attraction; 
 point out explicitly that the wire is not attracted by the pole of 
 the magnet but pushed sidewise as indicated. 
 
 (b) Project the image of a direct-current arc (using ordinary 
 arc lamp carbons) on the screen. Bring the north pole of a bar 
 magnet up behind the arc and call attention to sidewise movement 
 of image of arc on screen. Reverse the magnet and call attention 
 to reversed sidewise movement of image. Bring magnet nearer 
 and nearer to arc until arc is blown out. 
 
 (c) Arrange a single-pole single-throw switch in a circuit in 
 which 10 or 15 amperes of current flows from no-volt, direct- 
 current supply mains. Open the switch slowly and call attention 
 to the destructive arc which is produced. Show how greatly 
 this arc is reduced by opening the switch quickly. Place a 
 magnet as indicated in Fig. 140, page 203, General Physics, and 
 show that arc is almost instantly blown out if the switch 
 is opened slowly or rapidly. 
 
 83. The electric motor. Exhibit a commercial type of direct- 
 current motor, and take out the armature so that the armature 
 wires may be seen lying on the armature surface parallel to the 
 armature shaft. 
 
 The small, non-enclosed, two-pole, ring-wound, Crocker- 
 Wheeler motor is one of the best motors for this experiment. 
 Excite the field magnet of the motor by connecting the field 
 winding through a controlling switch to no- volt, direct-current 
 supply mains; connect the brushes through a suitable rheostat 
 and a controlling switch to no- volt direct-current mains; and 
 operate the motor. Show that direction of running is reversed 
 by reversing field connections, or armature connections; but 
 that reversal of both leaves direction of running unchanged. 
 
 If a motor having a drum- wound armature is used, the student 
 9 
 
114 CALENDAR OF LEADING EXPERIMENTS. 
 
 should be required to read the fine print on page 209, General 
 Physics. 
 
 84. The chemical effect of the electric current may be strik- 
 ingly shown by the decomposition of lead nitrate as explained in 
 Art. 147, page 214, General Physics. The best arrangement is to 
 use a narrow cell with plate glass sides and place it in the lantern 
 (horizontal projection arrangement) . After the deposition of lead 
 crystals on one electrode, reverse the battery connections when 
 these crystals will be seen to redissolve, and crystals will grow 
 on the other electrode. 
 
 Where lead crystals are being deposited the density of the 
 solution is reduced as may be seen by the upward streaming of 
 the solution near the cathode, and where the previously deposited 
 crystals are being redissolved the density of the solution is 
 increased as may be seen by the downward streaming of the 
 solution near the anode. Point out that the chemical action 
 in an electrolytic cell takes place only at the electrodes. 
 
 The voltaic cell. It is very important to direct the students' 
 attention very particularly to Art. 149, page 216, General Physics. 
 A voltaic cell is any electrolytic cell in which the chemical action 
 produced by the current is a source of energy. 
 
 85. The simple voltaic cell. Voltaic action and local action. 
 
 Place the above described glass cell in the lantern (horizontal pro- 
 jection arrangement), fill the cell with dilute sulphuric acid, and 
 introduce a strip of clean pure zinc. No perceptible chemical 
 action takes place, that is to say, no bubbles of hydrogen are 
 seen to be produced. 
 
 Connect the strip of zinc to a strip of copper by a soldered 
 wire connection and place the two strips into the cell. An 
 electric current now flows through the wire from copper to zinc 
 and through the acid from zinc to copper, this current decom- 
 poses the H2SO4, the SC>4 radical is liberated at the zinc electrode 
 where it combines with the zinc to form ZnSO 4 , and the hydrogen 
 is liberated at the copper strip where it appears in the form of 
 
ELECTRICITY AND MAGNETISM. 115 
 
 bubbles. Cut the wire and this action ceases. This is voltaic 
 action. 
 
 Dip the clean strip of zinc momentarily into a dilute solution 
 of copper sulphate, and minute particles of copper will be left 
 on the zinc. Replace the contaminated strip of zinc in the 
 dilute sulphuric acid, and chemical action takes place as indicated 
 by the formation of hydrogen bubbles. This is local action. 
 
 Connect the strip of contaminated zinc to a strip of copper 
 by a soldered wire connection and place both strips in the 
 sulphuric acid. Hydrogen bubbles will be seen to form at both 
 electrodes, that is to say, local action and voltaic action both 
 take place. 
 
 Amalgamate the contaminated strip of zinc by rubbing a 
 little mercury over it, replace the two strips in the sulphuric 
 acid cell, and voltaic action, only, will be in evidence by the 
 liberation of hydrogen bubbles at the copper strip. Cut the 
 wire and this voltaic action will cease. 
 
 86. Experiment showing the use of oxidizing agent around 
 the cathode (copper or carbon) of a voltaic cell. This matter 
 is explained in Art. 153, page 220, General Physics, and it may 
 be shown experimentally as follows: Connect an ammeter to 
 a simple voltaic cell, and allow the current to flow until the 
 ammeter reading falls considerably (until the dissolved atmo- 
 spheric oxygen near the copper electrode is used up and hydrogen 
 bubbles are given off copiously). Then pour into the cell a 
 solution of potassium bichromate. The ammeter reading is 
 thereby increased and hydrogen bubbles are no longer liberated. 
 
 87. Change of resistance with temperature. (a) A battery 
 jar is provided with two lead electrodes. Fill the jar nearly 
 full of water, connect through an ammeter (reading up to 20 or 
 30 amperes) to no- volt direct current supply mains, and connect 
 a voltmeter between the electrodes (no- volt meter). Place a 
 thermometer in the jar, and pour dilute sulphuric acid into the 
 jar very slowly with vigorous stirring until the ammeter reading 
 
n6 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 is about 5 amperes. Then read thermometer, voltmeter and 
 ammeter. 
 
 Allow the current to flow for a while until the temperature 
 rises considerably. Then again take the readings of thermom- 
 eter, voltmeter and ammeter. 
 
 Ignoring the small polarization voltage (2 or 3 volts) the 
 resistance of the acid solution at each temperature can be found 
 from the corresponding readings of voltmeter and ammeter. 
 
 (b) File off the corners of three 4" X 8" pieces of window 
 glass. Wind about 100 feet of No. 28 B. & S. cotton-covered 
 copper wire in one layer on each strip, leaving 3 inches of one 
 end of each piece of glass uncovered. Clamp the bare ends 
 of the glass strips to a wooden block, connect all of the windings 
 of wire in series and bring the terminals to two binding posts 
 on the wooden block. Place the wooden block across the top 
 of a battery jar with the wire-covered strips of glass projecting 
 into the jar. Fill the jar with kerosene so as to cover the wire 
 windings, bore a hole through the wooden cover for a ther- 
 mometer, and arrange a good stirrer. 
 
 Connect as under a above to no- volt direct-current supply 
 mains, and determine the resistance of the copper wire at, say, 
 25 C. and at 75 C. 
 
 88. Conductivity of glass at high temperatures. A thin- 
 walled glass tube about one inch in diameter and 8 or 10 inches 
 long is connected as shown in Fig. 47, and one side of the tube is 
 
 supply 
 mains 
 
 -T~" > -'^- % /W\/VvJ 3 
 
 rheostat g 
 
 o t 
 
 1 
 
 glass 
 
 gjloo volts 
 
 tube 
 
 1 
 
 
 Fig. 47. 
 
 heated by sweeping the flame of a blast lamp up and down 
 between the wire terminals. Keep the supply switch open until 
 all connections ate made, and use great caution to avoid contact 
 
ELECTRICITY AND MAGNETISM. 117 
 
 with high-voltage circuit after the supply switch is closed. The 
 current starts along a very narrow path, quickly heats the narrow 
 strip to a very high temperature, and the resistance of this path 
 of very hot glass drops to 200 ohms or less. The lo-ohm, 10- 
 ampere rheostat is necessary to prevent an excessive flow of 
 current after the glass begins to conduct appreciably. 
 
 Electricity or energy; which? Article 170, page 238, General 
 Physics, is intended to turn the student away from any premature 
 incursion into the atomic theory of electricity, and hold his 
 attention fast to the most important practical phase of electrical 
 science, namely, the purely mechanical phase. It is conceivable 
 that the atomic conceptions of electrical phenomena may some 
 time come to be important in everyday life and in everyday 
 engineering, but that time is certainly not yet; although the 
 atomic theory is nearly as important in engineering research as 
 in any other branch of physical research. 
 
 89. Experiment with fan blower and dynamo generator. 
 Drive a fan blower by a Drive a small shunt dynamo 
 small, direct-current, shunt as a generator by means of a 
 motor with an ammeter in the small, direct-current, shunt 
 armature circuit of the motor motor with an ammeter in the 
 to indicate roughly the power armature circuit of the motor 
 delivered to the blower. to indicate roughly the power 
 
 required to drive the generator. 
 
 Arrange a gate so that the Arrange a rheostat and switch 
 outlet of the blower can be so that the generator can be 
 closed or opened to any desired operated on open circuit (arma- 
 extent. ture delivering no current) or 
 
 so that the current delivery can 
 be varied at will. 
 
 Arrange an open-tube manom- Connect a voltmeter across 
 eter to measure the air pres- the armature terminals of the 
 sure behind the gate. generator. 
 
Il8 CALENDAR OF LEADING EXPERIMENTS. 
 
 With gate closed note power With armature circuit of 
 required to drive the fan and generator open note power 
 note value of air pressure E. required to drive the generator 
 
 and note voltage E across 
 terminals of generator arma- 
 ture. 
 
 Opening gate wider and Closing switch and adjusting 
 wider, note increase of power rheostat so as to take more and 
 required to drive the fan and more current from generator 
 note slight decrease of pressure armature, note increase of 
 E behind gate. power required to drive gen- 
 
 erator and note slight decrease 
 of voltage E between armature 
 terminals. 
 
 An interesting calculation is that which is suggested in problem 
 198, page 568, General Physics, and approximate data for such a 
 calculation can be obtained from the motor-generator arrange- 
 ment here described if an ammeter is placed in the armature circuit 
 of the generator. Thus if 5 amperes of current is taken by the 
 motor armature from no-volt direct-current supply mains when 
 generator is delivering no current, and if 15 amperes is taken by 
 motor armature when the generator is delivering 9 amperes. 
 Then, neglecting RI 2 losses in motor armature and in dynamo 
 armature, we may take (15 amperes 5 amperes) X no volts as 
 the part of the power which is supplied to the generator by the 
 motor and delivered as output by the generator so that the 
 voltage E of the generator must be approximately such as to 
 give 
 
 9 amperes X E = (15 amperes 5 amperes) X no volts. 
 
 90. Experiment on back electromotive force of a motor. 
 
 Connect the field winding F of a direct-current shunt motor 
 directly to no- volt direct-current supply mains through a con- 
 trolling switch 5; connect the armature M of the motor to the 
 
ELECTRICITY AND MAGNETISM. 119 
 
 same mains through a rheostat, an ammeter A and a con- 
 trolling switch S'; and connect a voltmeter V to terminals 
 of M as shown in Fig. 48. 
 
 Fig. 48. 
 
 To determine resistance of M lock the armature so that it 
 cannot rotate, and reduce resistance in the rheostat until the rated 
 full-load current flows through A and M ; then take readings of 
 A and V. The required resistance of M is equal to V/A accord- 
 ing to Ohm's law. 
 
 Then unlock the motor armature and allow it to run, without 
 a belt load, let us say, and note the great decrease of current 
 through A and M and the great increase of voltage indicated 
 by V. Take readings A' and V when motor is running at 
 full speed. 
 
 We thus get data for a calculation like that of problem 215, 
 page 571, General Physics. 
 
 Note. The important thing in this experiment is to show the 
 great decrease of current through M when the motor is allowed 
 to start and rise to full speed. 
 
 91. The transformer experiment. A laminated iron core about 
 i" X i" X 8" long has about 150 turns of No. 16, B. & S. 
 copper wire wound upon it, the wound core is mounted in an 
 upright position on a wooden base, and the terminals of the 
 winding are brought out to binding posts as indicated in Fig. 
 198, page 266, General Physics. This winding can be safely 
 connected directly to no- volt, 6o-cycle, alternating-current 
 supply mains. 
 
120 CALENDAR OF LEADING EXPERIMENTS. 
 
 (a) A copper ring 2 or 3 inches in diameter, J inch wide and 
 J inch thick is held in a pair of strong pliers and brought down 
 over the above wire-wound core. The repeated reversals of 
 magnetism of the core induce an alternating current in the ring 
 and the ring becomes red hot. 
 
 (6) Hold the cold ring loosely by the fingers so as to encircle 
 the core about 2 inches from its upper end, and suddenly close 
 the supply switch. The ring is thrown violently upwards. 
 
 (c) A small coil of insulated copper wire three or four inches in 
 diameter and containing 15 or 20 turns of wire has a short piece 
 of German silver wire (No. 22, B. & S., let us say) connected 
 across its terminals, and the coil is held in the hand and brought 
 slowly down over the above wire-wound core. The small coil 
 acts as the secondary coil of a transformer (a step-down trans- 
 former), and an alternating current is induced in it as indicated 
 by the heating of the German silver wire. 
 
 92. Eddy currents in a solid iron rod. A coil of insulated, 
 No. 16, B. & S., copper wire containing about 1,000 turns and 
 having an opening about 3 inches in diameter is connected to 
 no- volt, 6o-cycle, alternating current-supply mains through a 
 suitable rheostat (seven or eight ohms and ten-ampere capacity) 
 and a controlling rheostat. A short iron rod 2 or 2\ inches in 
 diameter with a strong handle like a soldering iron is thrust into 
 the opening of the coil, and the rod soon becomes very hot 
 because of eddy currents. 
 
 Eddy currents due to motion of a good electrical conductor 
 near an unvarying magnet may be shown by dropping a thick 
 sheet of copper into the space between the two facing poles of a 
 strong electromagnet. 
 
 93. Demagnetization by a weakening series of reversals. 
 A steel magnet placed in the coil which is described in experiment 
 92 and slowly withdrawn is left almost wholly demagnetized. 
 A magnet (or a magnetized watch) may be demagnetized by 
 holding it near the pole of a strong magnet and turning it slowly 
 
ELECTRICITY AND MAGNETISM. 121 
 
 while moving it farther and farther away from the magnet pole. 
 The axis of turning must be at each place at right angles to the 
 lines of force of the magnetic field at that place. 
 
 94. Spark at break. Connect the coil and core which are 
 described in Experiment 96 to two cells of storage battery in 
 series, and arrange a strip of spring brass BB as shown in Fig. 
 49 for quickly breaking the cir- 
 cuit at p (see also Fig. 215, 
 
 page 276, General Physics) . Also. ^ block 
 
 arrange a piece of German sil- 
 ver wire of the same resistance 
 
 as the coil to be put in circuit in place of the coil and permit 
 the same steady current to flow. A small glow lamp may be 
 included in the circuit to show by its brightness that the cur- 
 rent has approximately the same value in each part a, b and 
 c of the experiment, as follows: 
 
 (a) With German silver wire in place of the coil, break the 
 circuit suddenly at p and note that there is no perceptible 
 spark at break. 
 
 (b) Place the coil in circuit without the laminated iron core, 
 and show that there is a moderate spark at break 
 
 (c) Place the laminated core in the coil and show that there 
 is a very intense spark at break. 
 
 An interesting experiment for the individual student is to 
 connect a telegraph relay as a buzzer (see problem General 
 Physics), operate it by three or four dry cells, and connect hand- 
 electrodes to the terminals of the magnet winding. At each 
 break of the circuit a moderately high voltage exists across 
 the magnet winding which may be felt by taking hold of the 
 hand electrodes. 
 
 95. Choking effect of an inductance in an alternating-current 
 circuit. Connect a 6owatt, no- volt tungsten lamp to no-volt, 
 6o-cycle, alternating-current supply mains in series with the 
 German silver resistance referred to in the previous experiment. 
 
122 CALENDAR OF LEADING EXPERIMENTS. 
 
 Connect a similar lamp to the same supply mains in series with 
 the coil referred to in the previous experiment. The resistances 
 of the two circuits are the same, but the lamp which is in series 
 with the coil is dimmer than the other lamp (showing that less 
 alternating current flows through it), and if the laminated iron 
 core is placed in the coil the current will be very greatly reduced. 
 See page 279, General Physics. 
 
 Note. Change-over switches may be arranged to shift the 
 above circuits from no-volt, alternating-current supply mains 
 to no-volt, direct-current supply mains. 
 
 96. Experiment showing slow growth of current in an inductive 
 circuit. A small glow lamp is connected to a battery through a 
 controlling switch and in series with a coil of wire in which is a 
 removable iron core. When the switch is closed the lamp comes 
 almost instantly to full brightness when the iron core is removed, 
 but it takes a very perceptible time for the lamp to come to full 
 brightness after closing the switch when the iron core is in place. 
 
 A delay of about 0.6 second with iron core in place will be 
 obtained under following specifications: Use a half-ampere, 2- 
 volt, tungsten filament lamp, and two storage battery cells 
 connected in series. Make the coil about 6 inches long with i 
 inch depth of winding and containing about 1,350 turns of No. 
 14, B. &S., double cotton-covered copper wire. Make the core 
 1 1 inch X 1 1 inch X 13 inches long, built up of soft sheet-iron 
 stampings bound together by one layer of strong twine. The 
 corners of the core should be rounded off to avoid cutting of 
 twine. Soak the whole core in shellac varnish and bake. Each 
 end of the core for about 3^ inches should be bare so that the 
 edges of the laminations of the core can come as close as possible 
 to the edges of laminations of a yoke. Make area of contact 
 of each end of core with yoke about ij X 3 inches. The coil 
 must have a 2|-inch hole to admit the laminated core. The 
 yoke should be mounted on a base board with lugs projecting 
 upwards to connect with and support the ends of the core. 
 
ELECTRICITY AND MAGNETISM. 123 
 
 A very interesting experiment with the above apparatus is 
 the following: Connect lamp and coil in series to the 2 cells of 
 storage battery, placing the coil on its side on the table (yoke 
 not used). Allow the magnetic forces to draw the core quickly 
 into the coil and the lamp will be momentarily dimmed. Draw 
 the core quickly out of the coil and the lamp will become momen- 
 tarily very brilliant. (Be careful not to burn out the lamp by 
 withdrawing core too quickly.) 
 
 While the core is being drawn into the coil the power, El, 
 delivered to the coil by the battery is being only in part used to 
 heat the wire of the coil (RI Z ) because some power is evidently 
 used to pull the core into the coil. Therefore RP is less than 
 El, or RI is less than E, or J is less than E/R, which is 
 the value of the steady current according to Ohm's law. While 
 the core is being drawn into the coil the magnetic flux through 
 the coil is increasing, this increasing flux induces a back electro- 
 motive force in the coil (opposing the current), and the work 
 done in forcing the current to flow in opposition to this back 
 electromotive force is spent in two ways, namely, in pulling the 
 core into the coil and in magnetizing the core. 
 
 Inductance of a coil or circuit. The discussion of inductance 
 as given in General Physics is seriously incomplete. Indeed we 
 have never written a book without being unduly influenced by 
 a feeling of the hopelessness of the undertaking of bringing young 
 men to a fairly complete understanding of the fundamentals of 
 physics, and we therefore cut out things that should be included. 
 We might excuse such omissions by saying that we do not wish 
 to make a perfect book and thereby deprive all physics teachers 
 of their work! 
 
 A coil, having Z turns of wire, has a current i flowing in it, 
 as indicated in Fig. 50. The current in the coil produces a 
 magnetic field in the surrounding region as indicated by the lines 
 of force in Fig. 50, and it can be shown* from the magnetic 
 definition of current strength (see Art. 144, page 209, General 
 
 * It is not worth while to introduce this proof here. 
 
124 CALENDAR OF LEADING EXPERIMENTS. 
 
 Physics) that the intensity of the magnetic field is everywhere 
 doubled in value if i is doubled in value, the trend of the lines 
 
 of force remaining unchanged. There- 
 of wire fore the amount of magnetic flux 
 through each turn of wire or the av- 
 erage flux per turn, <, is doubled if 
 i is doubled. Therefore $ is pro- 
 portional to i so that we may write 
 
 Fig. SO. $> = bi (i) 
 
 where b is a constant for a given coil or circuit. If i changes 
 it is evident that $ must change b times as fast, or 
 
 d* di 
 
 But the changing $ induces an electromotive force 
 
 d$ 
 
 in each turn of wire or a total electromotive force Z -3- in 
 
 at 
 
 the whole coil, according to equation (82), page 257, General 
 
 d^& fi^i 
 
 Physics, or using the value of -j- from (ii) we get bZ 
 
 at at 
 
 as an expression for the electromotive force induced in the coil 
 by the changing current, and, if we represent the quantity bZ 
 by the single letter L, we have 
 
 where E f is the electromotive force induced in a coil or circuit 
 by the changing of the current. The negative sign indicates 
 
 di 
 that E f is opposed to i when is positive, that is, when i 
 
 is increasing. Therefore, to make the current increase at the 
 
 di 
 
 rate , an outside electromotive force (the electromotive force 
 at 
 
 of a battery, for example) equal to L but helping the current 
 
ELECTRICITY AND MAGNETISM. 125 
 
 must act or push on the circuit. That is 
 
 ; '. B-L$ .. V: . (iv) 
 
 which is equation (84), page 278, General Physics, and in which 
 E is the outside electromotive force required to make the 
 
 current in a coil or circuit increase at the rate , and L is a 
 
 CLt 
 
 constant for the given circuit. The quantity L is called the 
 inductance of the circuit. 
 
 Let R be the resistance of the circuit, and let E be the total 
 electromotive force acting on the circuit. Then a portion, Ri, 
 of the electromotive force is used to overcome resistance, and the 
 remainder, E Ri, causes the current to increase in accordance 
 with equation (iv). 
 
 Note. If there is an iron core in the coil in Fig. 50 the flux 4? 
 will be nearly proportional to i for small values of i, but as the 
 iron approaches magnetic saturation the flux $ is no longer 
 even approximately proportional to i. 
 
 Flux-turns. The quantity 3> in the above discussion is the 
 average flux per turn of wire, and the product $Z might properly 
 be called the total flux passing through the coil (each part of the 
 actual flux in Fig. 50 being counted n times, where n is the 
 number of turns of wire surrounding that part). This product 
 $Z is usually called the number of linkages of lines of force and 
 turns of wire, and it is expressed as flux-turns. Multiply both 
 members of equation (i) by Z and we get 3>Z = bZi. But bZ 
 is the inductance L of the coil, so that 
 
 $Z = Li. (v) 
 
 Consider a canal boat on Consider a circuit on which 
 which a propelling force E is an electromotive force E is act- 
 exerted by the tow rope. As- ing. The portion of E which 
 sume the backward drag of the is used to overcome friction is 
 water to be proportional to the Ri, when R is a constant and 
 
126 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 velocity i of the boat. Then i is the current, and the net 
 
 the net accelerating force is accelerating electromotive force 
 
 E Ri, where R is a constant, is E Ri, so that 
 and we have 
 
 E - Ri = L 
 
 di 
 dt 
 
 E - Ri = L 
 
 (vi) 
 
 di 
 
 dt 
 
 where L is the mass of the 
 
 di 
 
 boat and -T. is its acceleration. 
 at 
 
 When the boat reaches full 
 
 where L is the inductance of 
 
 di 
 
 the circuit and -j-. is the rate of 
 dt 
 
 increase of the current. 
 
 When the current reaches full 
 
 speed (corresponding to given value (corresponding to given 
 
 force E) then -r = 
 aquation (vi) becomes 
 
 o and 
 
 di 
 
 value of E) then -r = 
 at 
 
 equation (vii) becomes 
 
 o and 
 
 p 
 
 E = Ri or i= (viii) 
 J\. 
 
 E = Ri or i = (ix) 
 K. 
 
 At the instant that the mule At the instant that the elec- 
 begins to pull, i= o andequa- tromotive force E begins to act 
 
 tion (vi) becomes 
 
 (x) 
 
 on the circuit, i = o and equa- 
 tion (vii) becomes 
 
 E = L- (xi) 
 
 When the mule stops pulling When the electromotive force 
 
 after having set the boat in E ceases to act after having 
 
 motion, E = o and equation established a current, E = O 
 
 (vi) becomes and equation (vii) becomes 
 
 di r di 
 
 L T; = la (xn) L = - Ri (xm) 
 
 Note. Equation (vi) is usually written and thought of in the 
 
 di 
 
 form E = Ri + L , that is to say, the total electromotive force 
 dt 
 
 acting on a simple circuit of resistance R and inductance L is 
 partly used to " overcome " resistance and partly used to " over- 
 come " inductance; the former part is Ri and the latter part 
 
 s 
 
ELECTRICITY AND MAGNETISM. 127 
 
 97. Elimination of spark at break by a condenser. Figure 51 
 shows the coil L and the quick-break arrangement p (as 
 described in experiment 94) connected to 2 storage battery cells 
 
 Fig. 51. 
 
 B. As stated in experiment 94 an intense spark at break is 
 produced when the iron core is in the coil, but when a condenser 
 C is connected as shown the spark at break is eliminated. The 
 condenser should have about one microfarad capacity. 
 
 The reversed surge of current which flows through the coil 
 L and the battery B after breaking the circuit at p may be 
 shown by its demagnetizing action on the iron core as explained 
 on page 283, General Physics. 
 
 The so-called discharge resistance. Connect a glow lamp 
 in place of the condenser in Fig. 51 and show that spark at break 
 is eliminated. It is common practice to use a resistance thus 
 connected across a switch which is 
 used to open a highly inductive cir- 
 cuit, such as the field winding of 
 > a large dynamo, and the switch is 
 always arranged as a two-step Fig 52> 
 
 switch. The first step inserts the 
 
 resistance in circuit and the second step opens the circuit after 
 the current has been greatly reduced by the insertion of the re- 
 sistance. A common arrangement of switch with ''discharge 
 resistance" is shown in Fig. 52. 
 
 98. Flow of alternating current through a circuit containing a 
 condenser. Connect a 25-watt, no- volt, tungsten filament, 
 glow lamp in series with a 10- or 2O-microfarad condenser, and 
 arrange a change-over switch so that the condenser and lamp 
 
128 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 can be connected to no-volt, direct-current, supply mains or to 
 6ocycle, no- volt, alternating-current, supply mains as indicated 
 in Figs. 222 and 223, page 284, General Physics. 
 
 99. The lightning arrester. A very interesting experiment is 
 to show a simple type of lightning arrester in action. Thus 
 Fig. 53 shows a single-pole direct-current arrester of the magnetic 
 
 supply mains 
 
 ground 
 
 ifr 
 
 spark 
 
 Fig. 53. 
 
 blow-out type. The ground connections in the figure should be 
 good metallic connections to the same piece of gas or water pipe 
 because the action of the magnetic blow-out is most interesting 
 when the spark across gap g produces a dead short circuit. 
 If 22O-volt direct-current supply is available it should be used 
 in preference to no- volt supply. 
 
 Connect one terminal of the electric machine or induction coil 
 to ground and the other terminal to the metal ball B. When the 
 spark jumps from B to W it is choked by the winding L, and 
 
 s E therefore the sudden spark discharge 
 
 M 03 , 
 
 }' I passes across gap g to ground. But 
 
 the spark across g starts an electric 
 arc and introduces a dead short cir- 
 cuit between the supply mains. The 
 blow-out magnet L is to eliminate 
 this short circuit. 
 
 Fig. 54. 
 
 brass strip B (faced 
 short-circuit when G 
 
 100. Electrostatic attraction. A 
 
 with varnished paper so as to avoid 
 touches it), and a suspended strip of 
 
ELECTRICITY AND MAGNETISM. 
 
 129 
 
 gold leaf G as indicated in Fig. 54 are placed in the lantern 
 (horizontal projection arrangement), and the binding posts are 
 connected to no- volt, direct-current supply mains. 
 
 Show that the attraction is sensibly the same if the binding 
 posts are connected to I lO-volt, alternating-current supply mains. 
 
 Exhibit an electrostatic voltmeter. 
 
 10 1. Capacity of a condenser. The experiment which is de- 
 scribed in Art. 204, page 289, General Physics, is very helpful in 
 giving to the student a clear understanding of equation (89), 
 page 290, General Physics. 
 
 The term capacity is misleading in that it suggests that a con- 
 denser can hold only a certain amount of charge just as a pail 
 can hold only a certain amount of water. 
 
 102. Residual charge on a condenser. So-called electric 
 absorption. 
 
 A rubber tube is stretched by A Leyden jar condenser is 
 a very considerable steady force charged by a very high steady 
 E and certain elongation Q is voltage R and certain charge 
 produced. Q is drawn out of one coating 
 
 and pushed into the other coat- 
 ing thus producing what may 
 be thought of as an electrical 
 stress in the glass wall. 
 
 The two coatings of the 
 Leyden jar are connected by 
 a conductor so that current can 
 flow, and the charge on the 
 coatings together with the elec- 
 trical stress in the glass wall 
 disappears. 
 
 The wire connection is then 
 taken away so that current 
 cannot flow. 
 
 The end of the rubber tube 
 is momentarily released so as 
 to be free to move and the 
 stretch of the tube is relieved 
 by the movement. 
 
 The free end of the momen- 
 tarily relieved rubber tube is 
 then made fast so that it cannot 
 
 move. 
 10 
 
130 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 A minute later the end of the 
 tube is released and the move- 
 ment of the end shows that a 
 small amount of stretched condi- 
 tion has developed in the tube. 
 
 When a stretched rubber 
 band is momentarily released 
 most of the stretch disappears 
 at once, but a small residue of 
 stretch disappears very slowly; 
 and this residue of stretch ac- 
 cumulates as sensible stretch 
 if the momentarily released end 
 of the rubber band is quickly 
 made fast. 
 
 A minute later the two coat- 
 ings are connected by wire and 
 a minute spark shows that a 
 small amount of charged condi- 
 tion has developed. 
 
 When the plates of a charged 
 condenser are momentarily 
 connected by a wire, most of 
 the electrical stress in the di- 
 electric disappears at once, but 
 a small residue of the stress 
 disappears very slowly; and 
 this residue of stress accumu- 
 lates as sensible charge on the 
 condenser plates if the momen- 
 tary wire connection is taken 
 away so that no current can 
 flow. 
 
 103. The use of the spark gauge. The simple experiment 
 which is described in Art. 210, pages 298-299, General Physics, 
 is best carried out by the student in the laboratory. 
 
 104. The ringing spark. The oscillatory character of the dis- 
 charge of a condenser through an inductance can be shown by 
 
 
 O 
 
 to 
 high voltage 
 supply 
 
 ~~ o 
 
 
 
 o 
 o 
 
 b 
 
 spark gap 
 * J 
 
 Fig. 55. 
 
 the arrangement shown in Fig. 55. The condenser C is charged 
 to about 20,000 or 30,000 volts by a large influence machine, 
 and when it discharges across the spark gap the back and forth 
 
ELECTRICITY AND MAGNETISM. 131 
 
 surging of the discharge produces 15 or 20 sparks in rapid 
 succession and if the frequency is low enough this multiple 
 spark gives a very distinct ringing sound like the stroke of a 
 hammer on an anvil. Also the spark when reviewed in a rotating 
 mirror is seen to consist of a series of sparks in rapid succession. 
 
 The pitch of the tone emitted by the spark can be easily 
 determined by adjusting a calibrated Gal ton whistle to the 
 same pitch. 
 
 It is interesting to vary the pitch of the spark by using fewer 
 Leyden jars (smaller capacity) or by using a portion only of the 
 sections of the coil (smaller inductance). It is advisable there- 
 fore to bring out terminals from intermediate pancake coils of 
 the coil L. 
 
 A condenser C, Fig. 55, consisting of 6 fairly large Leyden 
 jars grouped in parallel and a coil L having about 2,500 
 turns of double cotton covered No. 18, B. & S., copper 
 wire wound on a spool with winding space 8 inches long and I 
 inch deep, inside diameter of winding 3 inches, outside diameter 
 of winding 5 inches, will give a frequency of oscillation of about 
 2,000 cycles per second, and at this frequency the spark will 
 give a tone having a pitch of 4,000 complete vibrations per 
 second which is suitable for this experiment. 
 
 Note. Very high voltages come into existence across the 
 terminals of L in this experiment and therefore the coil should 
 be wound as a set of, say, 20 pancake coils each 3" inside 
 diameter X 5 inches outside diameter X J inch thick, and these 
 pancake coils should be assembled on a hard rubber tube 2 
 inches inside diameter and about 2f inches outside diameter 
 with thin disks of wood between; and the whole coil should be 
 impregnated with beeswax and rosin compound. 
 
 105. The use of the electric arc for the maintenance of electric 
 oscillations. An electric arc is maintained between a carbon 
 rod A and a water cooled copper vessel B, and the arc is 
 shunted by a circuit containing an inductance L and a con- 
 
132 CALENDAR OF LEADING EXPERIMENTS. 
 
 denser C as shown in Fig. 56. Under these conditions a 
 steady back and forth surging of current through the circuit LC 
 is maintained. 
 
 Use as an inductance a winding of about 75 turns of No. 16, 
 B. & S., cotton-covered copper wire wound in five or six sections 
 around a short wooden cylinder 6 inches in diameter, and bring 
 out terminals from intermediate sections so as to be able to use 
 15 or 30 or 45 or 60 or 75 turns at will. Use one one-microfarad 
 condenser or two such condensers connected in parallel. 
 
 f 000000 
 
 I L * 
 
 110- volt 
 
 d. c. 
 
 supply 
 
 carbon 
 
 arc 
 copper vessel 
 
 000000 
 
 Fig. 56. 
 
 With the two condensers and with all of the 75 turns in cir- 
 cuit the frequency of oscillation will be about 4,000 or 5,000 
 cycles per second and the arc will emit a shrill whistling sound 
 of which the pitch is 8 or 10 thousand complete vibrations per 
 second. 
 
 Make an open coil of 10 or 12 turns of No. 18 insulated copper 
 wire, the coil being large enough to slip over the coil L as above 
 described, connect a small glow lamp to the terminals of this 
 coil, and bring the coil slowly over the coil L while the arc is 
 whistling. Be careful not to burn out the lamp. Under these 
 conditions the auxiliary coil acts as the secondary of a transformer 
 as indicated by the lighting of the lamp. 
 
 Reduce the value of L by using fewer sections as above 
 described, or reduce C, or both, and the tone emitted by the 
 arc will rise higher and higher in pitch and eventually pass 
 beyond the pitch limit of audibility; but the oscillatory char- 
 
ELECTRICITY AND MAGNETISM. 133 
 
 acter of the current through L and C will still be shown by the 
 lamp.* 
 
 This experiment operates more satisfactorily if the arc is in 
 an atmosphere of hydrogen, but it usually operates quite satis- 
 factorily with the arc in air. A few trials is sufficient to make it 
 go, especially if the current through the arc is reversed once or 
 twice to get the more favorable direction. 
 
 The inductances L^ L% are helpful in that they tend to keep 
 the supply current at a constant value. 
 
 1 06. The electric field. The region around a charged body 
 is an electric field, and a match stick suspended by a fine thread 
 may be used for indicating the direction of electric field in a 
 manner very similar to the use of the compass needle for indicat- 
 ing the direction of a magnetic field. See page 305, General 
 Physics. 
 
 107. Experiment giving a basis for the definition of intensity 
 of an electric field. See pages 306-308, General Physics. 
 
 A very amusing experiment is to place small pith figures 
 between two metal plates A A and BB as shown in Fig. 57, 
 the plates being connected to the termi- 
 nals of a small influence electric machine. ^ 
 
 The pith figures should be made slightly j^ 
 
 conducting by rubbing a trace of salt on # B 
 
 them, and the plates AA and BB should Fi s- 57 - 
 
 be slightly dished as shown. 
 
 1 08. The rosin experiment is described on page 309, General 
 Physics. To get the best results the metal ladle should be con- 
 nected to one terminal of a large influence machine, and the 
 other terminal of the machine should be grounded. 
 
 The gold-leaf electroscope. A very convenient arrangement 
 of the gold-leaf electroscope is shown in Fig. 58. The leaves / 
 
 * The explanation of the production of oscillations by the arc in Fig. 56 is given 
 on pages 141-142 of W. S. Franklin's Elements of Electrical Engineering, Vol. I, 
 The Macmillan Co., 1917. 
 
134 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 of the electroscope are in the field of the lantern, and the 
 metal plate P of the electroscope is connected by a fine wire to an 
 insulated metal plate P' which stands on the lecture table. The 
 fine wire should be held loosely by small weights as shown, and 
 the insulating supports S and S' should be blocks of cast 
 
 
 
 S 
 
 S 
 
 
 
 Y 
 
 
 
 
 
 1 
 
 
 \ 
 
 
 
 
 K 
 
 
 1 
 
 \ 
 
 
 M 
 
 7 
 
 \' 
 
 1 
 
 
 
 Fig. 58. 
 
 sulphur and their surfaces should be freshly scraped. The 
 support 5 should be made of four brick-like slabs (as shown in 
 section) thus shielding the gold leaves from currents of air that 
 might enter the case around the rod R. MM are metal strips 
 connected to earth. Of course the case has glass front and back. 
 In using a gold leaf electroscope it is usually most convenient 
 to charge the electroscope by influence, using a glass rod which is 
 positively charged by rubbing it with silk, or a hard-rubber rod 
 which is negatively charged by rubbing it with fur. 
 
 109. Charging by influence. Bring a positively charged glass 
 rod near plate P', Fig. 58, touch P' with the finger and with- 
 draw the charged glass rod, and the entire system PP' will be 
 left negatively charged. 
 
 no. Use of the gold-leaf electroscope. (a) When the system 
 PP', Fig. 58, is neutral a positive or a negative charge brought 
 near to P' will cause the gold leaves to spread apart. 
 
 (b) When the system PP' is charged, a like charge brought 
 near to P' increases the spread of the leaves. 
 
 (c) When the system PP' is charged, an unlike charge, if 
 large enough, brought nearer and nearer to P' lessens the spread 
 of the leaves to zero and then causes the leaves to spread apart 
 again as it comes still nearer to P'. 
 
ELECTRICITY AND MAGNETISM. 135 
 
 Note. These effects should be explained, let us say, in terms 
 of the two fluid theory so as to enable the student to hold them 
 in mind for a while at least. The behavior of the gold-leaf 
 electroscope is complicated by changes of capacity as may be 
 understood from experiment 114, and therefore the statements 
 under (b) and (c) should in all strictness be qualified. 
 
 in. Sharing of charge by two insulated metal balls brought 
 into contact. Take two similar metal balls 4 or 5 inches in 
 diameter with insulating handles. Charge one ball positively, 
 let us say, note its effect on gold leaves when it is brought near 
 to P f in Fig. 58 (electroscope being already slightly charged), 
 touch it to the other ball, and then show that the two balls have 
 nearly equal positive charges, and that the charge on each ball 
 is considerably less than what was initially on the first ball. 
 
 Note. The greatest amount of charge that can be held on a 
 5-inch metal ball in air would produce about a thirty-thousandth 
 of an ampere for a thousandth of a second and therefore one 
 cannot easily study the transfer of charge from one ball to the 
 other by measuring the electric current which is associated with 
 (which indeed constitutes) the transfer of charge. 
 
 112. Giving up of entire charge by internal contact. Charge 
 a metal ball and touch it to the inside of a metal can which is 
 itself supported by an insulating handle, and show that the ball 
 is left almost entirely without charge whereas the charge on 
 the can increases more and more with each internal contact of 
 the freshly charged ball. 
 
 113. The electric doubler. The experiment which is described 
 in Art. 221, page 315, General Physics, is very interesting and 
 instructive. Strands of loose cotton threads serve very well in 
 place of the slabs of pith. The threads should be drawn re- 
 peatedly between the fingers to make them slightly conducting. 
 
 It is instructive to place the metal balls in the position shown 
 in Fig. 254, page 315, General Physics, and show that they have 
 approximately equal but opposite charges when they are sepa- 
 
136 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 rated and withdrawn. This experiment shows charging by 
 influence in its most complete and simplest form. 
 
 114. Change of capacity of plate P' in Fig. 58 and its effect 
 on the gold leaves. Demonstration of inductivity. 
 
 Two elastic bags A and A' Two insulated metal bodies 
 are connected by a pipe, and air P and P' are connected by a 
 
 P fl 
 
 wire 
 
 ft P' 
 
 Fig. 59. 
 
 is pumped into the bags so as 
 to inflate them. 
 
 If the walls of bag A' could 
 be weakened by making them 
 thinner some of the air would 
 flow out of A into A', thus 
 reducing the amount of air in 
 A. 
 
 If the material of bag A' 
 could be made of more yielding 
 stuff (without changing its 
 thickness) some of the air would 
 flow from A into A'. 
 
 Fig. 60. 
 
 wire and charged with elec- 
 tricity. 
 
 If the dielectric which sur- 
 rounds P' is made more yield- 
 ing by making it thinner some 
 of the charge will flow out of P 
 into P', thus reducing the 
 amount of charge on P and 
 decreasing the spread of the 
 gold leaves. By bringing a 
 grounded metal body near to 
 P' as shown the specified effect 
 is produced. 
 
 If a slab of paraffine is placed 
 between P' and B the air 
 dielectric is replaced to some 
 extent by a more yielding 
 dielectric (a dielectric with 
 greater inductivity) and some 
 of the charge will flow out of 
 P into P', thus reducing the 
 spread of the gold leaves. 
 
ELECTRICITY AND MAGNETISM. 137 
 
 115. The corona discharge may be shown in a dark room by 
 stretching two very fine wires two to six inches apart and con- 
 necting them to an influence electric machine or to the secondary 
 coil of a step-up transformer giving a secondary voltage of 
 twenty or thirty thousand volts. 
 
 The familiar old-time name of the corona discharge is brush 
 discharge, and it is seen on the projecting points on an influ- 
 ence machine which is operated in a dark room. 
 
 116. Electrical smoke deposition. This experiment is de- 
 scribed and explained in Art. 225, page 319, General Physics. 
 
 117. The ozonizer is described and its action is explained in 
 Art. 226, page 320, General Physics. 
 
 Discharge of electricity through gases and radio-activity. 
 
 One of the best teachers of elementary physics in the United 
 States, who is well known for the splendid research he has done 
 in recent years, once remarked to the authors that he never 
 could find the time to devote to the discharge of electricity 
 through gases and to the atomic theory of electricity in his 
 rather severe course in elementary physics (the authors had 
 admitted their own difficulties in this respect) ; but, he added, 
 "I often have a spare lecture hour or two near the end of the 
 term which enables me to show and discuss a few experiments, 
 and," he added, in a highly significant semi-humorous vein, 
 ' I generally prepare myself for these lectures by looking up the 
 subject in 's High School Physics." 
 
 Following is a list of experiments which we have found satis- 
 factory : 
 
 (a) Show the decreasing dielectric strength of a gas (air) with 
 decreasing pressure by connecting a vacuum tube in parallel 
 with an adjustable spark gap, and exhausting the vacuum tube 
 slowly. 
 
 (&) Show the characteristic Geissler tube discharge. 
 
 (c) Show the characteristic Crookes tube discharge. 
 
 (d) Exhibit cathode ray shadow and show magnetic deflection 
 
138 CALENDAR OF LEADING EXPERIMENTS. 
 
 of cathode rays. Note: Cathode rays and canal rays are both 
 shown by the vacuum tube devised by Professor C. T. Knipp, 
 of the University of Illinois. See Science, Vol. XLII, page 942, 
 December 31, 1915. 
 
 (e) Show luminescence produced by cathode rays. Common 
 forms of Crookes tubes for this purpose are (i) A Crookes tube 
 in which several varieties of minerals are mounted and (2) A 
 Crookes tube in which the back of a metal butterfly is painted 
 with several varieties of finely crushed minerals. 
 
 (/) Give a demonstration of X-rays using a fluoroscope. 
 It is advisable for the operator to shield himself from X-rays by 
 using screens of sheet lead. 
 
 (g) Show the ionizing action of X-rays by holding a Crookes 
 tube so that the X-rays may pass near to the plate P', Fig. 58. 
 
 (h) Show the ionizing action of the a-rays, /3-rays and 7-rays 
 from radium by holding near P', Fig. 58, a metal plate upon 
 which a minute quantity of radium has been deposited. This 
 plate is to be kept in a small box when it is not in use. 
 
 (i) Exhibit the spinthariscope. 
 
 (j) Exhibit the paths in a fog chamber of individual a-particles 
 from radium. A fog chamber for this purpose is for sale by 
 The Cambridge Scientific Instrument Company, Cambridge, 
 England. 
 
PART IV. 
 LIGHT. 
 
 139 
 
A NEW ENGINE, OR A HELP TO THE MIND CORRESPONDING TO 
 TOOLS FOR THE HAND. 
 
 Bacon listed long ago, in his quaint way, the things which seemed to him most 
 needful for the advancement of learning, and among other things he mentioned 
 A New Engine, or a Help for the Mind corresponding to tools for the Hand; and 
 the most remarkable aspect of present-day physical science is the aspect in which it 
 constitutes a realization of Bacon's New Engine. We now force upon extremely 
 meager data (obtained directly through our senses) interpretations which would 
 seem to be entirely incommensurate* with the data themselves, and we exercise 
 over physical things a kind of rational control which greatly transcends the native 
 cunning of the hand. The possibility of this forced interpretation and of this 
 rational control depends upon the use of two complexes. 
 
 (a) A logical structure, that is to say, a body of mathematical and conceptual 
 theory which is brought to bear upon the immediate materials of sense, and 
 
 (&) A mechanical structure, that is to say, either a carefully planned arrangement 
 of devices or a carefully planned order of operations such as the successive operations 
 of solution, reaction, filtration, and weighing in chemistry. 
 
 These two complexes do indeed constitute a New Engine for the Mind, and the 
 study of elementary physical science is intended to lead to the realization of this 
 Engine (i) By the building up of the logical structure in the mind of the student, 
 (2) By training in the use of devices, as in the making of measurements, and in the 
 performance of ordered operations, and (3) By exercises in the application of (i) 
 and (2) to the actual phenomena of physics and chemistry at every step and all of 
 the time with every possible variation. 
 
 That surely is an exacting program; but the only alternative is to place the 
 student under the instruction of Jules Verne where he need not trouble himself 
 about foundations, but may follow his teacher pleasantly on a care-free trip to the 
 moon or with easy improvidence embark on a voyage of twenty thousand leagues 
 under the sea. 
 
 From The Study of Science, an introduction to our Mechanics and Heat, The 
 Macmillan Co., 1910; reprinted in Bill's School and Mine by W. S. Franklin, 
 published by Franklin, MacNutt and Charles, South Bethlehem, Pa. 
 
 * An astronomer, for example, looks at a spec of light as it crosses the field of his 
 telescope, and he listens to the ticks of a clock to note the time of day when the spec 
 crosses the center line of the field. He then examines a set of fine lines on a divided 
 circle to find the angular altitude of the spec above the horizon. This he does three 
 times in succession. Then he proceeds to calculate when the spec (a comet) will 
 be nearest the sun, how far it will then be from the sun, how fast it will be moving, 
 and when it will return, maybe a hundred years later! 
 
 I 4 
 
LIGHT. 
 
 118. Unit sensory areas on the skin. A very interesting 
 experiment is to blind-fold an assistant and apply the points 
 of a compass to finger tips, to back of hand, to arm and to the 
 back of the neck, and bring the points of the compass closer 
 and closer together until in each case the two points can no 
 longer be distinguished as two. 
 
 The approximate diameter of the unit areas in the central 
 part of the retina can be determined by placing two minute 
 globules of mercury very close together (distance apart being 
 measured) in bright sunlight, and finding greatest distance from 
 the eye at which the two points of light are distinguishable 
 as two points. 
 
 119. The astigmatic pencil of rays. A clear understanding of 
 the difference between the homocentric pencil of rays and the 
 astigmatic pencil of rays is necessary if one 
 
 wishes to understand the important subject 
 of lens imperfections. 
 
 A large wooden torus or ring is very help- 
 ful in the discussion of Fig. 276, page 357, 
 General Physics. . Also the wire model 
 which is shown in Fig. 61 is very helpful. 
 The wire AA forms an arc of a circle of Fig 61 
 
 which the center is at C. By rotating the Heavy lines are stiff 
 model through a small angle about DD as wires, light lines are fine 
 
 strings. 
 
 an axis, the arc AA describes a portion of 
 
 a ring surface; thinking of this surface as a wave front, the 
 successive positions of the strings represent rays, and these rays 
 intersect along the line C (perpendicular to the plane of the 
 paper) and along the line DD. 
 
 141 
 
142 CALENDAR OF LEADING EXPERIMENTS. 
 
 Note. A narrow astigmatic pencil of rays may be obtained 
 by passing a beam of light from an electric arc obliquely through 
 the central portions of a long-focus converging lens, as indicated 
 in Fig. 277, page 358, General Physics, and the linear foci at 
 and at DD may be thrown upon a small piece of cardboard 
 or on a translucent screen. The central portion, only, of the 
 lens must be used, that is, the lens must be covered with an 
 opaque disk with a hole at its center, otherwise the astigmatic 
 effect is hidden by the complicated effects of the coma. See 
 page 400, General Physics. A much broader astigmatic pencil, 
 not perceptibly confused by coma, can be obtained by reflecting 
 the light from an electric arc (or the light from the sun) obliquely 
 from a concave mirror. 
 
 1 20. Total reflection. The best demonstration of total reflec- 
 tion is to pass around the class several pieces of plate glass 
 asking the students to note the silvery appearance of the squarely 
 cut edge as seen by looking obliquely through the glass (showing 
 how to look while making the statement) ; also several small 
 45-45-9O glass prisms may be used. 
 
 A brilliant beam of light may be passed through a tank of clear 
 water and focused in a small aperture through which the water 
 issues as a jet. The beam of light is kept in the smooth jet by 
 total reflection until the surface of the jet becomes deeply rippled 
 or breaks into drops. This experiment, which, it must be 
 admitted, is of little value, may be modified by using a bent 
 glass rod instead of the water jet. 
 
 121. Refraction of a spherical wave at a plane surface. A 
 very interesting and instructive experiment, which must be 
 carried out by each student for himself, is as follows: The point 
 0, Fig. 62, is a small bright object (a bit of chalk or a globule 
 of mercury) on the blackened bottom of a basin of clear still 
 water. Looking vertically downwards at O it appears to be 
 at F (the center of curvature of the portion of the refracted 
 wave WW at or near V). Looking along the line aa DD, 
 
LIGHT. 143 
 
 the point appears to be at DD if the line joining the two 
 eyes is horizontal, but the point appears to be at C if the 
 line joining the two eyes is vertical. The pencil of rays at aa 
 is astigmatic. 
 
 AIR 
 
 AIR 
 
 GLASS 
 
 Fig. 62. 
 
 122. The actual lens and the ideal lens. Pass a beam of light 
 from an electric arc through a large, short-focus lens, such a 
 lens, for example, as is used in the condenser of a projection 
 lantern, and render the convergent beam of light beyond the 
 lens visible by a cloud of chalk dust (using black board erasers) 
 or by a cloud of ammonium chloride smoke. The appearance of 
 the beam is shown in Fig. 342, page 397, General Physics, and 
 described in Art. 287, page 397. This beam becomes very much 
 more complicated if it passes obliquely through the lens. 
 
 Point out the fact that the whole theory of the simple lens as 
 developed in Chapter XX, General Physics, applies to the ideal 
 simple lens as explained in Art. 267, pages 373-374, General 
 Physics. 
 
 123. Model of a compound microscope. Mount as an object 
 a small flat piece of fine wire gauze A , as object glass a one-inch 
 
144 CALENDAR OF LEADING EXPERIMENTS. 
 
 focal length lens 0, and as an eye piece a longer focal-length 
 lens E on a supporting bar as indicated in Fig. 63. Have this 
 model in hand while explaining Art. 277, page 386, General 
 Physics, and then pass it around the class. 
 
 axis 
 
 Fig. 63. Fig. 64. 
 
 124. Model of a simple telescope. Mount as object glass a 
 long focal length lens O and as eye piece a shorter focal length 
 lens on a supporting bar as indicated in Fig. 64. Have this 
 model in hand and use it to look at a distant object through an 
 open window while explaining Art. 278, pages 387-388, General 
 Physics; and give to each student an opportunity to use it. 
 
 125. Experiment on vision. We base our sense of distance of 
 an object (a) On a knowledge of the size of the object as com- 
 pared with its apparent size, thus the finger at arm's length 
 looks larger than, and can be made to cover, a man at a hundred 
 yards, (b) By the muscular sense associated with the turning 
 of both eyes directly towards the object, and (c) By the muscular 
 sense associated with accommodation (see page 382, General 
 Physics). If we look at an unfamiliar object with one eye 
 through a pin hole held near the eye, bases (a) and (b) evidently 
 fail, and base (c) also fails because accommodation is almost 
 unnecessary when we look through a pin hole. 
 
 Stretch a cord across the room, using a kind of cord which 
 comes in a wide range of sizes with same kind of twist. Looking 
 at the cord with one eye through a pin hole walk towards it, 
 then stop, and bring the fingers straight up from below as if to 
 pinch the cord! The result is very amusing. The experiment 
 works quite satisfactorily even when one has handled the cord 
 and is familiar with its size. Sidewise or up-and-down move- 
 
LIGHT. 145 
 
 ments of the head tend to supply the missing sense of distance 
 which is dependent on two-eye vision, and therefore the cord 
 should be on a level with the experimenter's eyes and as the 
 experimenter moves towards the cord he should not raise or 
 lower his head nor move it sidewise. 
 
 The absence of any need for accommodation when one looks 
 through a fine pin hole can be shown by looking through such 
 a hole (held very close to the eye) at a printed page or preferably 
 at a piece of wire gauze and bringing the gauze closer and closer 
 to the eye. 
 
 126. Experiment on vision. The coin box. Nothing, perhaps, 
 is more familiar than a silver coin, and in a certain United States 
 coinage one face is exactly similar in the dollar, the half-dollar, 
 the quarter-dollar and the dime, and a very amusing device is 
 a box with a fine peep hole at one end with these similar coins 
 arranged at distances from the peep hole proportional to their 
 respective diameters. The coins should be supported on pedes- 
 tals the widths of which are in proportion to the diameters of 
 the coins, and the pedestals should be high enough (box deep 
 enough) so that the bottom of the box can be made invisible by 
 a small screen at a short distance from the peep hole. 
 
 We, the authors, find that the coin box is almost sure to get 
 out of order from one year to the next, because the coins get 
 loose and are lost ; but such things have never greatly disturbed 
 us. We manage to give our best efforts to our students in spite 
 of the ever insistent call of pessimism, and, as Professor Wm. 
 Lyon Phelps says, teaching is great fun. But we know teachers, 
 college teachers, who for such reasons, and often, alas, for 
 reasons much less respectable and adequate, never take any 
 pains at all! Professor Phelps has the advantage of us in that 
 he teaches English. What unlimited opportunity for humor an 
 English teacher has, and how shameful it is that English teaching 
 should ever have the fatal quality of dullness! shameful even in 
 the teacher of "crude Pennsylvania Dutch boys," who, according 
 to one English teacher's comfortable belief, are unworthy of any 
 ii 
 
146 CALENDAR OF LEADING EXPERIMENTS. 
 
 effort on his part! We can say no more, lest "the gals hold their 
 fans before their faces" as they did at Miss Meadow's party 
 when Ole Brer Tarrypin led them to think that Brer Fox had 
 cussed when he met him on the big road. Brer Fox hadn't 
 cussed, he had merely said, Hello Stinkin Jim. Think of calling 
 a nice man, like Brer Tarrypin, Stinkin Jim! His only fault is 
 that he is constitutionally slow, but his patience is illustrated 
 by what he tells Brer Fox about his being caught in a fire- 
 swept field. Well, but what did you do Brer Tarrypin? Sot 
 and tuck 'er, Brer Fox, sot and tuck 'er. 
 
 One of the most remarkable things in this world is the settled 
 character of a teacher's position. Where a teacher begins he 
 usually stays for life; and a teacher should make of his im- 
 mobility an opportunity for virtue, like Ole Brer Tarrypin, even 
 to the extent of teaching English to "crude Pennsylvania Dutch 
 boys " ; for all boys are crude; yes, even the far away groups that 
 that never bother us, let us say. 
 
 127. Experiment on vision. Why is an object seen erect when 
 its image on the retina is inverted? In answer to this question 
 the equally sensible question is sometimes asked: When one 
 hears a baby cry with two ears, why does one not take it for 
 twins? Our sensations are symbols which we interpret in ac- 
 cordance with experience. 
 
 Make a very small kink or curl at the end of a bit of very fine 
 wire, hold the curl near the eye, beyond the curl hold a fine pin 
 hole and look towards a broad bright surface. An erect shadow 
 (an erect image) of the curl is thrown on the retina, and the curl 
 is seen inverted. If we always looked at curls in this way 
 we would develop the habit of seeing them right side up. 
 
 The compound microscope and the astronomical telescope 
 always invert an object which is being examined, and this 
 inversion is confusing to a novice, but to the experienced astron- 
 omer or microscopist it is not in the least confusing or even 
 troublesome. 
 
LIGHT. 147 
 
 128. Experiment on vision. Curious effect of two-eye vision. 
 
 Stretch a thread or string on 
 
 thread 
 a frame as indicated in Fig. 65, 
 
 and look at the string with the 
 two eyes as indicated by the Fig 65 
 
 arrow. Bring the eyes to focus 
 
 on any point p of the string and one will see two strings inter- 
 secting at the point p. 
 
 129. Shadows of blood veins on the retina. Any persistent 
 phase of retinal excitation which is independent of external 
 objects and conditions is by long habit ignored and therefore 
 imperceptible. For example, a complex system of arteries and 
 veins lies in front of the retina and casts shadows on the retina, 
 but ordinarily these shadows are not perceptible. However, if 
 the image of. a kerosene lamp flame is thrown by a moderately 
 short focus lens on the eye ball (eye being turned away from 
 direction of lamp so that image may fall on the eye ball at some 
 distance to one side of the cornea) in a darkened room, the small 
 quantity of light which penetrates the thick walls of the eye 
 casts shadows of arteries and veins' in an unusual manner and the 
 entire net-work of veins and arteries becomes visible. 
 
 130. Persistence of vision and the stroboscope. The light 
 sensation produced by a flash of light lasts from one half of a 
 second to one second or more, according to the intensity of the 
 light, and an intermittent light is sensibly continuous when the 
 flashes recur at a frequency of 15 or 20 flashes per second or more. 
 
 Move the hand rapidly to and fro, with outspread fingers, 
 in the light of an alternating-current electric arc, and many 
 duplicate images of the hand are seen because of the inter- 
 mittent character of the light from such an arc. 
 
 Mount a slotted metal disk on the shaft of a small electric 
 motor and arrange an electric arc (direct-current arc) near the 
 edge of the disk so that flashes of light may pass through the 
 slot or slots as the disk rotates. A cardboard disk with figures 
 
148 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 on it marked in broad black lines like Fig. 66 is mounted on a 
 second motor, the cardboard disk is illuminated by the flashes 
 of light, and the speed of one or both 
 motors is adjusted until the ratio of the 
 motor speeds is an integer. With the 
 
 OS~*\ \ particular marking shown in Fig. 66 
 \^t j the widest range of effects can be ob- 
 tained if the slotted metal disk has four 
 equidistant slots, and with a 10 or 12- 
 inch disk the slots should be about J an 
 inch wide. 
 
 A most amusing experiment is to use a cardboard disk marked 
 as indicated in Fig. 67. 
 
 Fig. 66. 
 
 Fig. 67. 
 
 131. Reversal of a sense of fatigue. In many cases it seems 
 that an even blend of excitation of many nerve elements gives, 
 let us say, a neutral or colorless sensation, whereas the predomi- 
 nance of excitation of certain elements of the blend gives a sharp 
 and distinctive character to the corresponding sensation. This 
 is exemplified by the sensations of color according to the Young- 
 Helmholtz theory (see General Physics, pages 479-481), and it 
 
LIGHT. 149 
 
 is also shown by the reversal by fatigue of the sense of motion. 
 When one looks at the floor after looking for a long time out of 
 the window of a moving railway car, one sees the floor in motion 
 in a reversed direction. Mark a broad black spiral on a card- 
 board disk, rotate the disk steadily at a speed of three or four 
 revolutions per second, look intently at the rotating disk for 
 half-a-minute or more, and then turn and look at the face of a 
 nearby companion. Your companion's head will be seen to swell 
 or shrink according to the direction of motion of the disk and 
 spiral. 
 
 This reversal of the sense of direction by fatigue seems to show 
 that all the nerve excitation which is necessary for a sense of 
 motion in any direction is present in a complex blend when we 
 look at a stationary field; but that steady gazing at a field 
 moving in one direction produces fatigue which results in the 
 weakening of a certain element of excitation in what would 
 ordinarily be a neutral or colorless blend when we look again 
 at a stationary field.* 
 
 132. Imperfection of focal spot of a perfect lens. By passing 
 the light from a distant electric arc through a very fine pinhole 
 p, Fig. 68, and looking through , 
 
 the magnifying glass M, the ^ 
 
 distribution of the light on the v 
 
 plane FF can be seen. If > 
 
 the pin hole is very fine, the ^ 
 
 distribution of light gives an Fig 6g 
 
 exaggerated example of the 
 
 imperfections of the focal spot of a perfect lens as explained in 
 Art. 283, pages 392-394, General Physics. 
 
 133. Axial spherical aberration. See experiment 122. 
 
 134. Astigmatism of a lens of narrow aperture. (a) See note 
 under experiment 119. 
 
 * This matter is discussed in a brief note in Science, Vol. IX, page 70, Jan. 13 
 1899. 
 
150 CALENDAR OF LEADING EXPERIMENTS. 
 
 (6) The astigmatism of a lens may be shown by looking 
 obliquely through a low power magnifying glass at the cross- 
 rulings on cross-section paper, the lens being held so that its 
 central plane intersects the plane of the paper in a line parallel 
 to one set of rulings. In this case the pupil of the eye limits the 
 effective beam of light to a very narrow pencil, and one or the 
 other sets of cross rulings may be sharply focused by adjusting 
 the distance of the lens from the paper. 
 
 (c) Substitute a simple lens for the object lens of a projection 
 lantern, cover the lens with an opaque disk with a hole at its 
 center, turn the lens so that the light passes obliquely through it, 
 and use a piece of wire gauze as a lantern slide. By adjusting 
 the distance of the oblique simple lens from the gauze the vertical 
 or horizontal wires may be focused on the lantern screen. Of 
 course the central plane of the oblique lens must intersect the 
 plane of the gauze along a line parallel to one set of wires. 
 
 135. Image distortion. Using a piece of wire gauze as a 
 lantern slide, and a simple lens as the object lens of the lantern, 
 the two kinds of image distortion can be shown as explained in 
 Art. 291, pages 402-403, General Physics. It is worth while 
 also to use the regular orthoscopic lantern objective and to pro- 
 ject an undistorted image of the gauze on the lantern screen. 
 
 136. Chromatic aberration. The image of the wire gauze as 
 projected by the simple lens in experiment 135 shows traces of 
 color due to the chromatic aberration of the lens, but the beam 
 of light passes chiefly through the central part of the lens as 
 indicated in Figs. 352 and 353, page 403, General Physics, so 
 that the chromatic aberration is small as explained in connection 
 with Fig. 362, page 411, General Physics. 
 
 The regular orthoscopic lantern objective is corrected for 
 achromatic aberration and it projects a nearly color-free image 
 of the wire gauze on the screen. 
 
 The lenses of the eye are by no means achromatic, but we 
 have come by habit to neglect, and therefore not to see, the color 
 
LIGHT. 
 
 fringes even of brilliant objects. Any unusual condition, how- 
 ever, makes this habit ineffective. Thus by looking at a distant 
 brilliant lamp a very decided show of color is seen as the finger 
 is held near the eye and moved slowly sidewise until it nearly 
 covers the pupil. 
 
 An experimental study of chromatic aberration suitable for 
 the laboratory is described in Franklin, Crawford and MacNutt's 
 Practical Physics, Vol. Ill, The Macmillan Co. 
 
 137. The telephoto lens. Figure 69 represents the object lens 
 O and the eye piece of a simple telescope, the eyepiece being 
 drawn out so that its distance from the image i is greater than 
 
 V 
 
 Fig. 69. 
 
 its focal length EF. Under these conditions an enlarged real 
 image of i is formed at I. Figure 70 represents an opera-glass 
 type of telescope with the eyepiece E pushed in so that its 
 distance from the image i is greater than its focal length EF. 
 Under these conditions the eyepiece forms an enlarged real image 
 of i at /. Either arrangement, Fig. 69 or Fig. 70, can be used 
 as a telephoto lens but the arrangement in Fig. 70, being shorter, 
 is more convenient. The individual lenses and E are always 
 compound lenses. Thus the left-hand group of lenses in Fig. 
 373, page 417, General Physics, takes the place of in Fig. 70, 
 and the right-hand group of lenses in Fig. 373 takes the place 
 of E in Fig. 70. 
 
 An experiment of great interest is to use a simple telescope 
 as indicated in Fig. 69 for projecting an image of the sun on a 
 sheet of cardboard. If a large astronomical telescope is not 
 
152 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 available a small reading telescope from the laboratory will 
 answer. The telescope should be fixed in a hole through a 
 
 Fig. 70. 
 
 large board (to darken the region back of the telescope), and 
 the image should be thrown on a translucent screen of ground 
 glass so as to be easily seen by a group of persons. 
 
 138. Spectroscope demonstrations. To project a large and 
 sharply defined spectrum upon a screen requires very expensive 
 apparatus and therefore the authors always arrange several 
 laboratory spectroscopes so that the members of a class may one 
 by one see the following : 
 
 (a) The continuous spectrum of a gas flame or glow lamp. 
 
 (b) The bright-line spectrum of an alkali metal in a Bunsen 
 
 flame. 
 
 (c) The reversal of the sodium lines, and 
 
 (d) The dark-line spectrum of the sun. 
 
 The reversal of the sodium lines may be seen by looking through 
 a Bunsen flame at the crater of a direct-current carbon arc and 
 placing in the Bunsen flame a small roll 
 of asbestos cloth which has been previ- 
 
 ously soaked in strong brine and dried. 
 
 139. Projection of a soap film. The 
 
 interference colors of a thin film may be 
 p . 71 beautifully shown by projection. A thin 
 
 metal disk dd, Fig. 71, with a circular 
 
 hole two or three inches in diameter is dipped into a soap solu- 
 tion and placed as shown in front of the lantern condenser CC 
 
LIGHT. 153 
 
 and an image of the soap film S is projected on the screen by 
 the lantern objective 0. 
 
 140. Spectroscopic analysis of the light reflected from a thin 
 film. Arrange a thin film of mica so as to reflect the light from 
 a gas flame or glow lamp into the slit of a spectroscope. It is 
 desirable in this experiment to use a film too thick to show inter- 
 ference color to the unaided eye. The spectrum will show a 
 large number of bright and dark bands, and no color is seen 
 directly by the unaided eye because the intensified wave lengths 
 (and the weakened wave lengths) are distributed throughout the 
 spectrum. When light from a soap film is reflected into the slit 
 of a spectroscope one or two parts of the spectrum are intensified 
 (and one or two parts weakened) at a time. 
 
 141. The diffraction grating. A most striking demonstration 
 is to look through a glass grating (200 or 300 lines to the inch) 
 at an electric arc between impregnated carbons. This gives the 
 arrangement shown in Fig. 400, page 440, General Physics, LL 
 being the lens of the eye and PP being the retina. 
 
 142. Polarized waves on a rubber tube. Tie one end of a 
 rubber tube fast and hold the other end in the hand with the tube 
 stretched, and give the discussion of Art. 311, page 442, General 
 Physics. Pass the rubber tube through two narrow slits, produce 
 irregular waves on the tube and point out the polarizing effect 
 of the slit near the hand. Turn the second slit so as to be 
 parallel to the first and then turn it so as to be at right angles 
 to the first and show the effects as described. 
 
 143. Group of experiments on polarized light. Polarizing 
 apparatus for projection is expensive and the authors have there- 
 fore always arranged what our students have called a "seven- 
 ring circus" for demonstrations in polarized light. The demon- 
 strations we use in spectroscopy also constitute a "seven-ring 
 circus." Each set of apparatus is usually duplicated, and the 
 entire teaching staff of the department usually serve as "ring- 
 masters." 
 
154 CALENDAR OF LEADING EXPERIMENTS. 
 
 (a) Show tourmaline tongs and direct the observer to look 
 through the tongs and turn one of the crystals slowly. 
 
 (b) Look through a Nicol prism at a bright varnished surface 
 and turn the prism slowly. 
 
 (c) Place a small rhomb of Iceland spar on a sheet of paper on 
 which is a small black dot and turn the rhomb slowly. Look at 
 the rhomb and dot through a Nicol prism and turn the prism 
 slowly. 
 
 (d) Look through two Nicol prisms and turn one of the prisms 
 slowly. 
 
 (e) Using a simple polariscope (parallel beam) examine a plate 
 of mica or other doubly refracting crystal, turning polarizer and 
 crystal plate slowly, one and then the other. 
 
 (/) Using a polariscope (arranged for convergent beam) 
 examine a plate of Iceland spar cut perpendicularly to optic 
 axis; also examine a plate of a biaxial crystal, aragonite for 
 example. 
 
 (g) Using a simple polariscope (parallel beam) examine a 
 cube of glass under stress. 
 
 Note. The inexpensive Norrenberg polariscope can be used 
 for e and g; and for / a pair of tourmaline tongs held close to 
 the eye is perhaps most satisfactory. 
 
 Note. Experiment c can be easily projected. Use a metal 
 lantern slide with a pin hole in it, and place a rhomb of Iceland 
 spar against the slide covering the pin hole, and turn the rhomb 
 slowly. 
 
 Note. A large cardboard model of a rhomb of Iceland spar 
 with equal length edges and with a rod set in the position of the 
 optic axis (the axis of symmetry )is useful. 
 
 Note. A large block of wood cut to the form of a Nicol prism 
 and sawed in two like a Nicol prism is useful. 
 
 Note. The following experiments are very interesting but 
 inconvenient and unsatisfactory for general purposes. 
 
 (h) Fill a tall glass jar with a milky solution made by dropping 
 
LIGHT. 155 
 
 a few drops of extremely dilute rosin-alcohol into water and 
 stirring vigorously. Throw a beam of light downwards into the 
 jar in a dark room and look sidewise at the jar through a Nicol 
 prism, turning the prism slowly. Throw a beam of plane 
 polarized light downwards into the jar, rotate the plane of 
 polarization, and look at the jar from one side. In this second 
 experiment a test tube may be used instead of a jar so that a 
 small sized Nicol prism can be used as a polarizer. 
 
 (i) Fill a test tube with a strong solution of syrup made 
 milky by the addition of a small quantity of extremely dilute 
 rosin-alcohol. Throw a beam of plane polarized light downwards 
 into the tube in a dark room and look at the tube from one side. 
 The tube will show faint helices of colored bands. 
 
 Note. The authors have always arranged for students to 
 use the saccharimeter in the laboratory. See Franklin, Crawford 
 and MacNutt's Practical Physics, Vol. Ill, pages 68-73. 
 
 144. Group of experiments on color. 
 
 (a) Light from a gas flame or glow lamp passes into the slit 
 of a spectroscope and the observer places before the slit in 
 succession a series of colored glasses or a series of colored solutions 
 in narrow glass cells. 
 
 (b) Light from a gas flame or glow lamp is reflected into the 
 slit of a spectroscope from a series of colored pieces of paper or 
 cloth. It is advisable to have a piece of white paper placed 
 permanently in position to reflect light into the spectroscope 
 when the colored paper or cloth is removed. 
 
 (c) Illuminate a batch of brightly colored objects (a selection 
 of small skeins of brilliantly colored worsted such as is used for 
 color-blind tests)' by the light from a sodium flame in a dark room 
 and arrange an electric glow lamp so that it can be easily turned 
 off and on so as to give white light illumination. 
 
 (d) Demonstrate the familiar rotating sector-colored disk, 
 and adjust the sectors so as to give a neutral gray resultant or 
 blend. 
 
156 CALENDAR OF LEADING EXPERIMENTS. 
 
 (e) Contrast effect. Place side by side on an inclined board 
 a large sheet of green tinted paper and a large sheet of red tinted 
 paper with good illumination. Tear off two bits of grayish 
 paper (from the same piece) placing one of the bits on each 
 sheet of tinted paper, and tear off a bit of each sheet of tinted 
 paper and place the bit upon the other sheet. The colors of the 
 small bits of paper are so modified by the effects of contrast that 
 it is almost impossible to believe that the two bits of grayish 
 paper are the same, and to believe that each bit of colored paper 
 is the same as the nearby large sheet. This experiment is most 
 effective if the bits of paper are torn off and placed after calling 
 the students' attention to the experiment and while they are 
 looking attentively. 
 
 (/) The authors usually arrange for color-blind tests of the 
 entire class. We ordinarily use the Holmgren test,* and usually 
 we can persuade a color-blind person to undergo the test before 
 the entire class. 
 
 * Colored worsteds for this test may be obtained from E. B. Meyrowitz, 
 N. Y. City. 
 
PART V. 
 SOUND. 
 
 157 
 
THE PHILOSOPHY OF STEAM SHOVELS AND THE PHILOSOPHY OF 
 
 LIVING. 
 
 Imagine a never-to-be-escaped human need of a twenty-foot arm! What age- 
 long development and what unthinkable pains! It is easier to build a steam shovel. 
 All of which means that homo sapiens is now bent towards Social Inheritance; but 
 social inheritance has own pains as many know who burn the midnight oil. 
 
 How shocking to reduce the tender-minded philosopher's love of perfect pre- 
 cision* to a materialistic preference for steam shovels as opposed to immeasurable 
 pains of birth; and to make mathematical philosophy appear as a dire necessity! 
 rather than as a thing to be chosen for its own sake. And then to urgej with that 
 lover of paradox, Gilbert Chesterton, that the serious spiritual and philosophic 
 objection to steam shovels is not that men work at them and pay for them and 
 make them very ugly, nor even that men are killed by them, but merely that men 
 do not play at them ! Think of a group of sportsmen cavorting over a ten-thousand 
 acre field tossing and catching a Brobdignagian ball in steam shovels! It is 
 conceivable that the one objection to the steam shovel might have been overcome 
 if the Great War had not come upon us. 
 
 The great danger of our time has been the confusion of boundaries between 
 thing-philosophy and human philosophy, between the philosophy of material 
 conquest and power and that intimate philosophy of comfort which makes life 
 not easy but worth while. When these boundaries are rectified there will be one 
 philosophy of steam shovels, recognized and used as such, and another philosophy 
 of living. Science will then stand as the essence of man's inescapable responsibilities 
 in practical affairs, and we shall seek God, a finite God, in that which is intimately 
 and even narrowly human, if narrowness there be in that supreme and illimitable 
 mystery. 
 
 From Education After the War by Franklin and MacNutt; reprinted in Bill's 
 School and Mine. 
 
 * This is a reference to the point of view of Professor Keyser of Columbia 
 University as hazily set forth in his recent articles On the Human Worth of Precise 
 Thinking. See page 34 of this volume for further comment. 
 
 t See our Introduction to Mechanics (The Study of Science), as referred to on 
 page 140 of this volume. 
 
 J See Preface to our Elements of Electricity and Magnetism, The Macmillan 
 Co., 1908. 
 
 158 
 
SOUND. 
 
 The senses of touch and sight are preeminently space senses, 
 as everyone knows, whereas the sense of sound might almost 
 be called the time sense, because the order in sequence of the 
 elements of a given sound sensation is all-important. This is true 
 of fleeting, irregular sounds as in speech, and it is equally true of 
 musical sounds as is shown by the use of rythm and sequences 
 of tones in music. 
 
 145. An extremely amusing experiment, and one which illus- 
 trates the importance of order or sequence in the elements of a 
 complex sound sensation, is to arrange a phonograph so that it 
 can be driven forwards or backwards at will and produce, 
 forwards and backwards, a familiar melody like "Yankee 
 Doodle" or a familiar speech like "Mary Had a Little Lamb." 
 
 A melody or a speech when reversed produces, of course, 
 precisely the same sense elements but nothing is more utterly 
 unlike than a melody and its reverse or a speech and its reverse. 
 
 146. Musical sticks. To show that musical tones commonly 
 are component parts of many noises thump at different points on 
 a table or chair and call attention to the fact that the sound 
 differences are mostly differences of pitch in the very brief tones 
 that are produced. 
 
 Drop a stick on the floor and the sound is mostly noise, but 
 drop a carefully adjusted series of sticks and the tone differences 
 stand out by contrast as a clearly recognized melody or musical 
 scale. 
 
 147. The Galton whistle. An interesting experiment for a 
 very small group of listeners is to determine the pitch limit of 
 audibility of musical tones by means of the Galton whistle. 
 
 A series of steel bars giving tones in an ascending scale up to 
 
 159 
 
160 CALENDAR OF LEADING EXPERIMENTS. 
 
 and beyond the limit of audibility can be used. These bars can 
 be cut from cold rolled steel shafting and if d is the diameter 
 and / the length of a bar in inches the number of complete 
 vibrations per second (transverse vibrations) corresponding to 
 its fundamental mode is 
 
 d 
 
 n = 15010^- 
 
 148. Simple modes of vibration of a string. Produce the 
 successive harmonics on the string of a violin or guitar. Using 
 a sonometer place paper riders on the string and show the 
 experiment which is described at the top of page 509, General 
 Physics. 
 
 149. Simple modes of vibration of an air column. Produce 
 the successive notes on a bugle. Using a long narrow organ 
 pipe blow it with increasing air pressure, and a series of tones 
 corresponding to successive simple modes will be clearly heard. 
 
 A long glass tube an inch or an inch and a half in diameter is 
 arranged as a whistle or organ pipe, closed at one end and blown 
 steadily by bellows (the moist breath will not do) , and the vibrat- 
 ing segments are made visible by means of lycopodium powder. 
 Increasing the air pressure will cause the air column to break up 
 into shorter and shorter vibrating segments. 
 
 150. Simple modes of vibration of a plate. Chladni's figures. 
 The simple modes of vibration of a plate may be shown as 
 explained on pages 517-518, General Physics. 
 
 151. Resonance. (a) Free the strings of a piano by pressing 
 the damper pedal and sing a series of clear notes loudly with the 
 mouth held near to the sounding board. Each note will set in 
 vibration that particular piano string which vibrates in unison 
 with it, and each note will be heard (as produced by the vibrating, 
 string) when the sung note ceases. A piano is an important 
 part of the equipment of a physics lecture room. 
 
 (b) Hold a vibrating tuning fork over the mouth of a tall 
 
SOUND. 161 
 
 slender glass jar, and pour water slowly into the jar. The 
 air in the jar will be set vibrating by the fork when the frequency 
 of vibration of the air column coincides or nearly coincides 
 with the frequency of the vibrating fork and the sound produced 
 will be greatly intensified. This experiment is most satisfactory 
 when the water flows into (or out of) the tall jar quietly through 
 a rubber tube connection at the bottom of the jar. 
 
 (c) An ordinary telephone connected to alternating-current 
 supply mains (through a high resistance, of course) produces a 
 musical tone which is very rich in over- tones, and if such a 
 singing telephone is held near the mouth of the tall jar above 
 described the successive over-tones come out with wonderful 
 distinctness as the jar is slowly filled with water. The over- 
 tones may be brought out also by holding the singing telephone 
 near one's open mouth and changing the mouth cavity as if to 
 speak the following vowels in succession u, 5, a, a, e 1 and e (see 
 General Physics, page 522). Ordinary 6o-cycle alternating cur- 
 rent is not very satisfactory for use in this experiment, but 
 the experiment is wonderful if one has 133-cycle or i5O-cycle 
 current. One can produce sufficient alternating current by 
 an electrically driven tuning fork, the telephone (very low re- 
 sistance telephone) being connected between the terminals of 
 the windings of the driven .magnet, and, if necessary, a large 
 capacity condenser being connected across the break. A tuning 
 fork of 250 or 300 complete vibrations per second should be 
 used. An ordinary pitch pipe can be used instead of a singing 
 telephone but the overtones do not come out very sharply. 
 
 152. Experiments on vowel sounds. Shape the mouth as if to 
 produce the vowels u, 5 and a in succession and thump the cheek, 
 and the characterizing tones (173, 517 and 775 vibrations per 
 second, respectively, see page 522, General Physics) will be 
 distinctly heard. 
 
 An extremely amusing experiment is to fill the lungs with 
 carefully purified hydrogen and repeat deliberately a familiar 
 speech like "Mary Had a Little Lamb," or speak deliberately 
 
 12 
 
162 CALENDAR OF LEADING EXPERIMENTS. 
 
 the following words in order: rude, no, paw, part, pay, pet and 
 see, as explained on page 523, General Physics. The use of 
 familiar words which are recognizable by the accompanying 
 consonants tends to correct one's perception of the false vowel 
 sounds in spite of any effort of the will on the listener's part. 
 Therefore it is better to pronounce the successive vowels u, o, 
 a, a, a, e and e, and point each time to a word (written on the 
 blackboard) which contains the particular vowel. 
 
 Note. Hydrogen made by dissolving zinc in sulphuric acid 
 is apt to contain arsenic hydride. Purify by passing the hydro- 
 gen through a strong solution of potassium permanganate. 
 
 153. Illustration of the action of the ear in the perception of 
 tone quality or timbre. Free the strings of a piano and sing a 
 vowel sound loudly against the sounding board as described and 
 explained in Art. 373, page 524, General Physics. 
 
 154. Beats. Blow two similar organ pipes together and alter 
 the pitch of one of the pipes slightly by means of a paper or 
 cardboard extension of the open end of one of the pipes. 
 
 155. Combination tones. A combination tone (difference 
 tone) which can be heard throughout a large room can be pro- 
 duced as explained on page 528, General Physics. 
 
PART VI. 
 THE DYNAMICS OF WAVE MOTION. 
 
 The substance of this entire part VI was given in a lecture by Wm. S. Franklin 
 before a joint meeting of the Western Society of Engineers and the Chicago Section 
 of the American Institute of Electrical Engineers on May 28, 1917. Incredible 
 as it may seem, nearly all of the mathematics here given and all of the experiments 
 here described were included in the lecture. 
 
 163 
 
SCIENCE AND TECHNOLOGY VERSUS THE HUMANITIES IN 
 EDUCATION. 
 
 The worst cant of our time, touching the Idolatry of Science*, which is our sin- 
 cerest religion, and handled to perfection by our easier college product, is the semi- 
 serious wail of regret that a silver-spoon smartness was not transmuted by a pleasant 
 college course into Knowledge and Appreciation of Science. Knowledge and 
 Appreciation alas! It were better to say easy acquisition instead of appreciation, 
 and fabulous riches instead of knowledge. Riches in which nearly every man 
 has had a share, unearned. 
 
 Unearned, indeed, and yet much more than earned! Every loafer knows some- 
 thing of the unpleasant exactions of effective labor, but there never yet was a 
 dilettante who could even dream of the pains of those who really learn nor any 
 self-satisfied Philistine who could sense the grief of those who are wise! 
 
 What do you think we are driving at with all this apparent rhetoric? Merely 
 to express the mood we fall into when we read such things as Paul Shorey's "The 
 Assault on Humanism" (see page 34 of this volume for further comment) and 
 think of the cockey attitude of many of our friends who have to do with technical 
 education. What our friends need is to be obliged to defend their ground before a 
 jury of Paul Shoreys, and what the Paul Shoreys need is to hear us confess how 
 little we are able to accomplish in our teaching of the mathematical sciences. But 
 much as we approve of Paul Shorey's point of view we are very far from capitulating 
 to the easy talk of educationalists concerning the real aim of education as the 
 development of personality and character. 
 
 Our greatest comfort as teachers comes from the exaggerated idea among those 
 who do not teach (and among others who only pretend to teach) as to what can 
 be accomplished by teaching. That teacher of philosophy, we know that he ac- 
 complishes but little however bigity his talk. The manufacturer who prates about 
 character, he knows what he is talking about all right, from his point of view, 
 but his talk gives us no distress at least no distress concerning our work. And 
 we know that personality and character cannot be developed by sentimentality, 
 however articulate. Science and technology versus the humanities; the antithesis 
 does not disturb us, we are at war with the Philistines! 
 
 * Contempt for science has been much in evidence in England and America in 
 the past, and of recent years science has been more and more idolized. With no 
 abatement of Contempt, an Idolatry of science has developed, and these two ex- 
 tremes are blended together in the same men I 
 
 Science is Finding Out and Learning How, but most men think of science in 
 terms of results. These results have fascinated the crowd, and the great majority 
 of men "have adopted a scale of physical values for everything in life with a 
 consequent neglect of quality and a denial of human value in everything. We 
 have a philosophy of rectangular beatitudes and spherical benevolences, a theology 
 of universal indulgence, a jurisprudence which will hang no rogues; all of which 
 means, in the root, incapacity of discerning worth and unworth in anything and 
 least of all in man. Whereas, Nature and Heaven command us at our peril, to 
 discern worth from unworth in everything and most of all in man. 
 
 164 
 
THE DYNAMICS OF WAVE MOTION. 
 
 One of the most important items of needed improvement in 
 the curriculum of the engineering school is the rejuvenation 
 of the usual, dried-up and unfruitful course in theoretical 
 mechanics, and this discussion of the simplest aspects of the 
 dynamics of wave motion is intended to show what can be 
 done to make this important branch of mechanics intelligible 
 to the engineering student. As Heaviside says, the physics of 
 wave motion is extremely simple and the mathematics extremely 
 difficult for the wave pulse, whereas the physics is extremely 
 complicated and the mathematics extremely simple for the 
 sine- wave train. Therefore the consideration of periodic waves* 
 is almost entirely excluded from this discussion. 
 
 The important subject of wave-diffusion or wave-distortion 
 is discussed in an extremely simple manner in Arts. 121-123 
 (pages 222-228) of Franklin and MacNutt's Advanced Electricity 
 and Magnetism, The Macmillan Co., 1915. 
 
 The exhaustive mathematical treatment of wave-distortion 
 (due to resistance and leakage of a telephone line, for example) 
 is based almost entirely on considerations such as are involved 
 
 * A well-known elementary treatise on physics, before making use of the wave 
 theory of light "proves" that light is wave motion, as follows: Any effect which is 
 periodic and which travels at finite velocity is wave motion; light is periodic and 
 it travels at finite velocity; therefore light is wave motion! The idea of the 
 Catling gun popped into our minds when we first came across this argument 
 and the effect of the Catling gun is periodic and is propagated at finite velocity I 
 In fact periodicity of waves in an extended medium always depends upon and is 
 determined by actions which take place in the system which produces or absorbs 
 the waves (water waves produced by wind of a given velocity constitute an excep- 
 tion), and waves have, in general, no definite velocity, as everyone knows who 
 understands what Heaviside calls wave-diffusion or who understands the difference 
 between group velocity and wave velocity in Helmholtz's theory of dispersion. The 
 fact is that the theory of periodic waves lends itself easily to unlimited algebraic 
 formulation and few men know anything beyond these formulas and the pictures of 
 sine curves which underlie theml 
 
 165 
 
166 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 in the discussion of the alternating-current transmission line on 
 pages 141-153 of W. S. Franklin's Electric Waves, The Macmillan 
 Co., 1909. This discussion is unique in the simplicity with 
 which the troublesome boundary conditions are set forth in 
 Figs. 132, 140-147. 
 
 The equation of a traveling curve. The curve cc, Fig. 72, is 
 stationary with respect to the origin of coordinates 0', and the 
 curve and the origin 0' are assumed to be traveling together 
 towards the right at velocity v so that the abscissa of the 
 
 -stationary 
 
 y-axis 
 
 
 Amoving 
 r y-axis 
 
 X"v 
 
 
 
 
 > 
 
 / > 
 
 ^ 
 
 
 
 -j^/ 
 
 V 
 
 
 
 ct 
 
 
 
 x^axis 
 
 t et- 
 
 
 . JC 
 
 ' >i 
 ,1 
 
 
 
 I 
 
 T ig. 72. 
 
 
 
 moving origin 0' as referred to the fixed origin O is at each 
 instant equal to vt as indicated in the figure. Therefore the 
 abscissa x of the point p on the moving curve is x = x' -f- vt, 
 so that x' = x vt. Let the equation to the moving curve cc 
 as referred to the moving origin 0' be 
 
 y = 
 
 Then, substituting x vt for x' t we have 
 
 y = F(x- vt) 
 
 (i) 
 
 as the equation of a curve traveling to the right at velocity v; 
 and in a similar manner it may be shown that 
 
 y =f(x + vt) 
 
 (2) 
 
 is the equation of a curve 'traveling to the left at velocity v. 
 It is highly instructive to derive a differential equation which 
 
THE DYNAMICS OF WAVE MOTION. 167 
 
 is satisfied by both equations (i) and (2); and, inasmuch as the 
 rules for differentiating are not usually understood by the student, 
 especially when applied to such general equations as (i) and (2), 
 the following discussion is arranged to appeal to one's funda- 
 mental arithmetical sense. Awkwardness of notation is the 
 chief difficulty, as usual, and one must remember the following 
 points : 
 
 If y increases always u times as fast as x, then u is called 
 the derivative of y with respect to x. If y depends on x as 
 the only variable, the derivative is represented by the symbol 
 dy/dx. If y depends on more than one variable, the derivative 
 with respect to x is called a partial derivative and it is usually 
 represented by the symbol dy/dx. The two symbols dy/dx 
 and dy/dx have, however, precisely the same arithmetical mean- 
 ing; thus in the first case y actually increases dy/dx times 
 as fast as x, and in the second case y would increase dy/dx 
 times as fast as x if x only, among all the variables upon 
 which y depends, were allowed to change. 
 
 If u (= dy/dx or dy/dx) increases w times as fast as x, 
 then, of course, w is the derivative of u with respect to x and 
 of course it is represented by du/dx or du/dx. In thinking of the 
 derivative of u, however, it is often desirable to keep clearly 
 in mind the fact that u itself is the derivative of y with respect 
 to x. This is shown by the following notation: 
 
 du d*y dy 
 
 = when u = 3- 
 dx dx 2 dx 
 
 du d 2 y dy 
 
 T~ = Tl when u = 
 dx dx 2 dx 
 
 Partial differential equation of travel. The two equations (i) 
 and (2), above, are particular solutions of a partial differential 
 equation which may be called the partial differential equation of 
 travel, namely 
 
 d z y d 2 y 
 
 * + (3) 
 
168 CALENDAR OF LEADING EXPERIMENTS. 
 
 and this differential equation is of fundamental importance in 
 the mathematical theory of wave motion. 
 
 For the sake of simplicity let the quantity (x vt) of equa- 
 tion (i) be represented by z so that 
 
 z = x vt (i) 
 
 from which we have dz/dx = i (which means that if x, only, 
 increases z must increase at the same rate), and dz/dt = v 
 (which means that if /, only, increases z must increase v 
 times as fast or decrease v times as fast). 
 
 Equation (i) is to be thought of for the moment as y F(z), 
 and the derivative dy/dz may be represented by F'(z), meaning 
 that y increases F'(z) times as fast as z. 
 
 Proposition: y increases F'(z) times as fast as z, and when 
 x, only, changes z increases dz/dx times as fast as x. Therefore 
 when x, only, changes y must increase F'(z) times dz/dx 
 times as fast as x. Therefore dy/dx = F'(z)> (dz/dx). But 
 dz/dx = i as stated above. Therefore* 
 
 Proposition: y increases F'(z) times as fast as z, and when 
 /, only, changes z increases dz/dt times as fast as /. Therefore 
 when /, only, changes y must increase F'(z) times dz/dt times 
 as fast as /. Therefore dy/dt = F'(z)- (dz/dt). But dz/dt = - v 
 as stated above. Therefore 
 
 g= -v-F'(z) (iii) 
 
 From equations (ii) and (iii) it is evident that dy/dt = 
 v (dy/dx), and this may be called the differential equation 
 of travel to the right. If we had started with y = f(x + vt) 
 
 * To the student who is familiar with the rules for partial differentiation this 
 argument may seem ridiculous, but it is not ridiculous, by any means. No one 
 can understand partial differentiation who has not at some time followed this 
 kind of an argument in pure arithmetic. 
 
THE DYNAMICS OF WAVE MOTION. 169 
 
 and z = x -f- ^ the above discussion would have led to 
 dy/dt = + v- (dy/dx), which may be called the differential equa- 
 tion of travel to the left. Neither of these differential equations 
 is of importance in the theory of wave motion. 
 
 Let us think of the function F'(z) as changing F"(z) times 
 as fast as z. 
 
 When x alone, changes, z changes dz/dx times as fast as x, 
 and therefore F'(z) changes F"(z) times dz/dx times as fast 
 as x. But F'(z) is equal to dy/dx according to equation (ii). 
 Therefore dy/dx increases F"(z) times dz/dx times as fast as x. 
 Therefore, since dz/dx = I, we have 
 
 When / alone, changes, z changes dz/dt times as fast as t\ 
 therefore F'(z) increases F"(z) times dz/dt times as fast as t, and 
 therefore v-F'(z) increases v times F"(z) times dz/dt times 
 as fast as t. But v-F'(z) is equal to dy/dt according to 
 equation (iii). Therefore dy/dt increases - v X F"(z) X dz/dt 
 times as fast as /, and therefore, since dz/dt = v, we have 
 
 fjr = *F'(?) (v) 
 
 From equations (iv) and (v) we have 
 
 and this same result would be obtained from equation (2) by 
 taking z = x + vt. That is to say, equation (3) is a differential 
 equation which is satisfied by both of the equations (i) and (2), 
 and it is of very great importance in the mathematical theory 
 of wave motion. 
 
 The principle of superposition. A property of linear differ- 
 ential equations. A principle of extremely wide application in 
 physics is the so-called principle of superposition. From the 
 
170 CALENDAR OF LEADING EXPERIMENTS. 
 
 physical point of view a general statement of the principle is 
 scarcely possible, and therefore the following examples must 
 suffice: (i) A person at A in Fig. 73 can see window No. i 
 
 and another person at B 
 can see window No. 2 at 
 the same time. This means 
 that the two beams of light 
 a and b can travel through 
 the same region at the same 
 time without getting tangled 
 
 Fig 73 together, as it were; each 
 
 beam behaves as if it were 
 
 traveling through the region alone. (2) Two sounds can travel 
 through the same body of air simultaneously, and each sound 
 travels as though it occupied the space by itself. (3) Two 
 systems of water waves can travel over the same part of a pond 
 simultaneously, each system behaving as if the other were not 
 present. (4) Two messages* can travel over a telegraph wire 
 simultaneously and not get mixed up together. (5) Two forces 
 F and G exerted simultaneously upon an elastic structure 
 produce an effect which is the sum of the effects which would be 
 produced by the forces separately, provided the sum of the forces 
 does not exceed the elastic limit of the structure, therefore each 
 force may be thought of as producing the same effect that it 
 would produce if acting alone. 
 
 All of the effects in physics which are superposable and this 
 includes by far the greater portion of the effects in mechanics, 
 heat, electricity and magnetism, light and sound, and chemistry 
 are expressible in terms of linear differential equations, and 
 the principle of superposition is a clearly defined property of 
 such equations as follows: // y is a function of x which satisfies 
 
 * Indeed any number of distinct messages can travel over a telegraph wire 
 in either direction or in both directions simultaneously. The only limiting feature 
 in multiplex telegraphy is the design of the sending and receiving apparatus; 
 and the same is true in wireless telegraphy. In each of the above examples the 
 word two means two or more. 
 
THE DYNAMICS OF WAVE MOTION. lyi 
 
 a linear differential equation, and if z is another function of x 
 which satisfies the same differential equation , then (y + z) is a 
 function which satisfies the differential equation. 
 
 This proposition is true for both ordinary and partial linear 
 differential equations, and indeed nearly all of the superposable 
 effects in physics are expressible in terms of partial linear dif- 
 ferential equations. The proof of the proposition is, however, 
 nearly the same for ordinary and for partial differential equations, 
 and therefore it is sufficient to give the proof for ordinary 
 differential equations only. Let the given linear differential 
 equation be: 
 
 Let y be a function of x which satisfies this differential equation, 
 then: 
 
 Let z be another function of x which satisfies (i), then: 
 
 Now 
 
 2) _ dy dz d z (y 
 
 ~ + 
 
 dx ~ dx dx dx 2 ~ dx z dx* 
 
 Therefore, adding equations (ii) and (iii) , we get : 
 
 4- z) 
 
 But equation (iv) is exactly the same form as equation (i), 
 and therefore (y + z) is a function of x which satisfies equation 
 
 (o.* 
 
 * The above proposition is true for any linear differential equation whatever; 
 that is when the coefficients A, B, etc., are constants, and when the coefficients 
 A, B, etc., are functions of the independent variable x. The latter type of linear 
 differential equation does not, however, concern us here. 
 
172 CALENDAR OF LEADING EXPERIMENTS. 
 
 The above proposition is the basis of Fourier's method of 
 analysis as applied to the flow of heat and as applied to the 
 motion of strings, and it is the basis of the use of spherical, 
 zonal, and cylindrical harmonics. The importance of the propo- 
 sition can scarcely be overestimated. 
 
 Undetermined constants in the solution of an ordinary differ- 
 ential equation. It is sufficient, perhaps, to illustrate this matter 
 by two very simple examples. 
 
 Example I. Consider the simple ordinary differential equation 
 
 
 
 dy 
 
 where a is a given constant. The increase of y during / 
 seconds is at, and the value of y at the end of the / seconds is 
 
 y = at + c 
 
 where C is the unknown value of y at the beginning (/ = o). 
 Example 2. Consider the simple ordinary differential equation 
 
 where a is a given constant. Then 
 
 and 
 
 + Bt + C 
 
 where C is the unknown value of y at the beginning, and B 
 is the unknown value of dyjdt at the beginning. 
 
 Note. From the point of view of the physicist the unknown 
 constants which appear in the general solution of an ordinary 
 differential equation are thought of as disposable constants because 
 the solution may be made to fit any special case by assigning 
 proper values to these constants. The number of disposable 
 
THE DYNAMICS OF WAVE MOTION. 
 
 173 
 
 constants in the general solution of an ordinary differential 
 equation is always equal to the order of the differential equation. 
 
 Undetermined functions in the solution of a partial differential 
 equation. It is sufficient, perhaps, to illustrate this matter by 
 a few simple examples. 
 
 Example I. Concerning a hill it is known only that its slope, 
 dy/dx, in the direction of the #-axis is constant and equal to a 
 so that 
 
 (0 
 
 Before the complete hill or surface can be constructed from 
 this differential equation an arbitrary starting curve cc, Fig. 74, 
 must be chosen. Let y = F(z) be the equation to the curve cc, 
 then F(z) is the height of the 
 
 hill at any point z on cc, and 
 ax + F(z) is the height of the hill 
 at any point x, z. That is to 
 say, the integration of (i) gives 
 
 w-axis 
 
 y = ax + F(z) 
 
 (ii) 
 
 Fig. 74. 
 
 Example 2. If all boys were of 
 the same ability we might say 
 that any boy saves money at 
 the average rate of $5 per year 
 beginning at fourteen years of age. Integrating with respect 
 to b, the boy, from fourteen years to twenty-one years we get 
 $35; but the amount of money a boy has when he comes of 
 age is not $35 plus a constant, but $35 + F(m), where F(m) 
 is what the boy's "old man" has saved for him; b and m are 
 independent variables, let us say, and a "constant" of integra- 
 tion with respect to b turns out to be an unknown function of m. 
 
 Example j. Concerning a hill it is known only that its slope, 
 dy/dx, in the direction of the # -axis increases a times as fast 
 
174 CALENDAR OF LEADING EXPERIMENTS. 
 
 as x t or, expressed in symbols, we have 
 
 Before the complete hill or surface can be constructed from this 
 differential equation two things must be given, namely, (i) An 
 arbitrary starting curve like cc, Fig. 74, and (2) An arbitrary 
 value of the starting slope, dy/dx, at each point of cc. 
 
 The integration of (i) or (iii) is called partial integration, but 
 example 2 should make it clear that partial integration is identical 
 to ordinary integration, the only difference being that in the 
 former the "constant" of integration turns out to be a function 
 of the other independent variable or variables. Therefore, 
 integrating (iii) twice we gee 
 
 y = J* + */() + F(z) (iv) 
 
 where f(z) and F(z) are unknown functions of z. Indeed 
 y = F(z) is the equation to the starting curve cc in Fig. 74, 
 and the value of the starting slope (dy/dx) x=0 , at each point of 
 cc is (dy/dx) x=0 , =/(*). 
 
 Note. From the point of view of the physicist the unknown 
 functions which appear in the general solution of a partial 
 differential equation are thought of as disposable functions 
 because the solution may be made to fit any special case by 
 properly choosing the forms of these functions. The number of 
 these disposable functions in the general solution of a partial 
 differential equation is always equal to the order of the differential 
 equation. 
 
 General solution of equation (3). Equations (i) and (2) are 
 particular solutions of (3), and therefore 
 
 y = F(x-vt) + /(* + /) (4) 
 
 is also a solution, according to the principle of superposition. 
 But equation (4) involves two independent and disposable func- 
 tions and it is therefore the general solution of (3). 
 
THE DYNAMICS OF WAVE MOTION. 
 
 175 
 
 Equation (4) represents the piling on top of each other of two 
 curves of any shape, one of the curves traveling to the right at 
 velocity v, and the other traveling to the left at velocity v. 
 
 Differential equation of motion of a stretched string. When 
 a stretched string is in equilibrium it is of course straight. Let 
 us choose this equilibrium position of the string as the #-axis 
 of reference. We will assume that each particle of the string 
 moves only in a direction at right angles to the string (parallel 
 to the j-axis of reference), and we will assume that the string 
 is perfectly flexible which means that the only forces to be con- 
 sidered are the forces due to the tension of the string. An 
 important consequence of the first assumption is that the x- 
 component of the tension of the string has always and every- 
 where a certain value T which is equal to the tension of the 
 string when it is in equilibrium. 
 
 Fig. 75. 
 
 Let the curve ccc, Fig. 75, be the configuration of the string 
 at a certain instant, that is, ccc is what the photographer would 
 call a snapshot of the moving string. The shape of the curve 
 ccc defines y as a function of x and the steepness of the curve 
 at any point is the value of dy/dx* at that point. 
 
 Consider the very short portion ab of the string. The length 
 of this portion when the string lies along the #-axis (in equi- 
 librium) is dxj and the mass of the portion is m-dx, where m 
 is the mass per unit length of string. An enlarged view of the 
 very short portion ab of the string is shown in Fig. 76. The 
 
 * Any reader who fails to appreciate the fact that we are here talking about the 
 state of affairs at a given instant, or that time is supposed to stop, as it were, 
 may wonder why we use the notation dy/dx instead of dy/dx. 
 
176 CALENDAR OF LEADING EXPERIMENTS. 
 
 adjacent portions of the string pull on the portion ab, the pull 
 at a is represented by R and it is parallel to the string at a, 
 and the pull at b is represented by R' and it is parallel to the 
 string at b. The ^-component of R is the force T to the left, 
 and the ^-component of R' is an equal force T towards the 
 right. Therefore the downward force D (see Fig. 76) is equal 
 to T tan 6, the upward force U is equal to T tan B' ', and the 
 net upward force acting on the portion ab of the string is : 
 
 dF = U - D = T tan tf - T tan (i) 
 
 But tan B is equal to the value of dy/dx at a, and tan 6' is 
 equal to the value of dy/dx at b. Therefore the value of 
 tan B' tan B is the increase of dy/dx from a to 6, and this 
 increase is equal to (d 2 y/dx 2 ) -dx. This is evident when we con- 
 sider that d 2 y/dx 2 means the rate of increase of dy/dx with 
 respect to x. Therefore, substituting (d^/dx^'dx for 
 tan 0' tan B in equation (i) we get: 
 
 Now, according to Newton's laws of motion, the net upward 
 force dF acting on the portion ab of the string is equal to the 
 mass m-dx of the portion multiplied by the upward acceleration, 
 B z y/dP, of the portion. Therefore substituting m(d 2 y/dP)-dx 
 for dF in equation (ii) we get : 
 
 d*y d*y 
 
 m ~W ~ T M 
 or 
 
 d z T d* 
 
 The general solution of this differential equation, as explained 
 above, is: 
 
 y = F(x - vf) + f(x + vt) (4) bis 
 
THE DYNAMICS OF WAVE MOTION. 177 
 
 where v is given by the equation : 
 
 .- 
 
 Of course particular solutions of equation (3) are 
 
 y = F(x - vt) (6) 
 
 and 
 
 y=f(x + vt) (7) 
 
 Equation (6) represents a bend or curve on the string traveling 
 to the right at velocity v and retaining its shape unchanged, 
 and equation (7) represents a bend or curve on the string traveling 
 to the left at velocity v and retaining its shape unchanged. 
 These traveling bends of unchanging shape are called pure waves. 
 
 Pure wa've traveling to the right. Equation (6) expresses 
 what is called a pure wave traveling to the right and it is important 
 to consider the necessary relation between V and s where V 
 is the sidewise velocity and 5 is the slope of the string at a point ; 
 of course, V = dy/dt and s = dy/dx. 
 
 From equation (6) we have s = (dy/dx) = F'(x vt), and 
 V = dy/dt = v-F'(x vt), and therefore for a pure wave 
 traveling to the right we have 
 
 . - ; r 7 = - ,; ( 8 ) 
 
 Pure wave traveling to the left. From equation (7) we have 
 
 ^ = dy/dx = F'(x + vt) and V = dy/dt = + v-F'(x + v), and 
 therefore for a pure wave traveling to the left we have 
 
 j=+v (9) 
 
 Remark. Equations (8) and (9) are useful in that they enable 
 one to pick out particular solutions of equation (3). Any dis- 
 tribution of sidewise velocity V and slope 5 which satisfies (8) 
 is a pure wave traveling to the right, and any distribution of 
 13 
 
178 CALENDAR OF LEADING EXPERIMENTS. 
 
 sidewise velocity V and slope 5 which satisfies (9) is a pure wave 
 traveling to the left. 
 
 Reflection and change of phase thereby. Let us consider a 
 simple form of pure wave traveling to the right along a string, 
 a wave throughout which the sidewise velocity of the string has 
 everywhere the same value V, and throughout which the slope 
 of the string has everywhere the same value s. Such a simple 
 wave we will call a ribbon wave, and it may be symbolized by the 
 arrow in Fig. 77. The head of the arrow shows the direction of 
 travel, everywhere between the points a and b the string is 
 moving sidewise at velocity V, and everywhere between a 
 and b the slope of the string is s. This slope is not actually 
 shown however. 
 
 What happens when the wave V s reaches the end of the 
 string? Whatever happens on the string it must be of the nature 
 of a pure wave traveling to the right and a pure wave traveling 
 to the left, one heaped on top of the other, because this is the 
 physical interpretation of the general equation (4). Therefore if 
 we assume a pure wave (sidewise velocity V and slope s f ) to 
 
 ; o string fc E string <Y* *' 
 
 Fig. 77. Fig. 78. 
 
 Reflection from rigid end of string. 
 
 shoot to the left from E when ab reaches E we are sure to 
 cover every possible outcome, and then if this assumption is 
 justified by showing that it satisfies all of the necessary conditions 
 we may be sure that the correct solution has been found. In 
 Fig. 78, therefore, is shown the tail end of the original wave V s, 
 and the beginning of an assumed reflected wave V s'. Now, 
 according to equations (8) and (9) we must have 
 
 and 
 
 V 
 
THE DYNAMICS OF WAVE MOTION. 179 
 
 Furthermore the actual end of the string cannot move and 
 therefore we must have 
 
 V + V = o (iii) 
 
 and from equations (i), (ii) and (iii) it follows that V = V 
 and that s' = s. That is to say, a wave is wholly reflected from 
 the rigid end of a string (numerical values of V and V the 
 same, and numerical values of s and s f the same), and the side- 
 wise velocity of the string in the reflected wave is the reverse of 
 the sidewise velocity of the string in the original wave 
 (V = - V). 
 
 Figure 79 represents an ideal condition in which the end of a 
 string is held by an indefinitely long weightless thread. In this 
 case the actual end of the string cannot slope, and therefore we 
 
 must have 
 
 5 + s f = o (iv) 
 
 which in conjunction with equations (i) and (ii) gives s' = s 
 
 string <-* weightless 
 ^ i thread 
 
 Fig. 79. Fig. 80. 
 
 Reflection from "free" end of string. 
 
 and V = V. That is, a wave is wholly reflected (numerical 
 values of 5 and s' the same, and numerical values of V and 
 V 1 the same) at the "free" end of a stretched string, and the 
 slope in the reflected wave is opposite to the slope in the original 
 wave (s f = s). The "free" end of a stretched string as 
 shown in Fig. 79 is merely an ideal, but the analogous condition 
 in electric waves and in air waves is very common. 
 
 Examples of wave motion on a string. i . A complete picture 
 of a ribbon wave traveling to the right on a stretched string is 
 shown in Fig. 80. The region ab occupied by the wave has a 
 uniform slope 5, and every part of the string ab is moving down- 
 wards at velocity V as indicated by the small parallel arrows. 
 
ISO CALENDAR OF LEADING EXPERIMENTS. 
 
 2. An indefinitely long stretched string is struck sharply by a 
 square-faced hammer so as to set every particle of the portion ab 
 of the string moving sidewise at a given velocity V as indicated 
 by the small parallel arrows in Fig. 81. To determine the motion 
 of the string it is only necessary to resolve the uniform sidewise 
 
 V - 
 string B ^ V + s 
 
 llUUlr 
 
 Fig. 82. 
 
 velocity and zero slope of portion ab into two oppositely traveling 
 pure waves as indicated in Fig. 82, in which the arrow A repre- 
 sents a ribbon wave traveling to the right and the arrow B 
 represents a ribbon wave traveling to the left. Half of the given 
 sidewise velocity is associated with each ribbon wave as indicated, 
 and equal and opposite slopes [according to equations (8) and (9)] 
 are associated with the respective ribbon waves. The state of 
 affairs a moment later is shown in Fig. 83, and Fig. 84 shows the 
 
 Fig. 83. Fig. 84. 
 
 state of affairs after the two ribbon waves A and B have 
 entirely separated from each other. 
 
 3. Motion of a plucked string. A stretched string is pulled 
 sidewise into the position APB, Fig. 85, and released; and it 
 is required to determine the motion of the string. For the sake 
 of simplicity the point P (or C) is taken at the middle of the 
 string. 
 
 To determine the motion of the string the initial condition 
 of the string, namely, slope + a between A and C, and 
 
THE DYNAMICS OF WAVE MOTION. 
 
 slope a between C and B, with no sidewise velocity any- 
 where, must be resolved into pure waves traveling to right and 
 left. Thus the slope a is resolved into the two waves s' V 
 and s n V", and the slope + Q> is resolved into the two waves 
 s'" V" and s iy F IV . 
 
 Now V must be equal to V" because there is zero side- 
 wise velocity between C and B, therefore, according to equa- 
 
 Fig. 85. 
 
 tions (8) and (9), s' must be equal to s" and the same in sign. 
 But s' + s" = -a so that s' = s" = - a/2. 
 
 Similarly, V" must be equal to F IV , so that s'" must be 
 equal to 5 IV and the same in sign. But S"' -f 5 IV = + a 
 so that S'" = 5 IV = + a/2. 
 
 Therefore, using a/2 for the common numerical value of 
 s', s", s'" and $ IV , and using V for the common numerical 
 value of F', F", V" and F IV , and indicating the proper sign 
 in each case we may simplify Fig. 85 as indicated in Fig. 86. 
 
 To find the state of affairs after lapse of time t it is only neces- 
 sary to consider that the ribbon waves will all have traveled 
 forwards a distance vt, and that reflection at A and B always 
 takes place with reversal of F. Thus after time sufficient for 
 waves to travel over ij times the length of the string (arrow- 
 head p, for example, will have traveled to A and back to q, 
 
 _ 
 
 1 1 1 1 t* F t t 1 1 1 r 
 
 +V 
 
 moving 
 
 C 
 
 
 -40 - 
 
 TV 
 
 Fig. 87 
 
182 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 and V will have been converted into -f- V by reflection at 
 A) the state of affairs will be as shown in Fig. 87. A portion 
 of the string near A is still and its slope is a; the middle 
 portion of the string is moving upwards at velocity 2V and 
 its slope is zero; and the portion near B is still and its slope is 
 + O" Slope and velocity of each part of string in Fig. 87 is found 
 by adding slopes and velocities associated with the overlapping 
 portions of the respective ribbon waves.* 
 
 The Kelvin ladder. A number of equidistant bars are fixed 
 to a fine steel wire or ribbon and suspended from a small crank 
 as indicated in Fig. 88. The rotatory motion of each portion of 
 this arrangement satisfies the equation 
 
 E 
 
 dt? ^ k dx 2 
 
 where <f> is the angular displacement at the instant / of the 
 bar which is at a distance x below the upper end of the ladder 
 (jc-axis directed downwards), k is the moment; 
 inertia of unit length of the ladder, and E is the 
 torque required to produce one radian of twist per 
 unit length of the ladder. This equation may be 
 established by an argument which is nearly identical 
 in form to the argument leading to equation (3) 
 on page 176, and what we may call a ribbon wave 
 may be symbolized by the arrow in Fig. 88. Every 
 portion of the ladder between the points a and 
 b is rotating at uniform spin-velocity, and the 
 whole of the portion ab of "the ladder is uniformly 
 twisted. Let o> be the spin-velocity and h the 
 degree of twist (radians per unit length of ladder). 
 Then the ratio u/h has a certain value, a, which 
 is positive or negative according to whether the 
 wave is traveling upwards or downwards. 
 
 * A more elaborate example of the motion of a plucked string as analyzed by 
 this same method is given in Journal of the Franklin Institute, Vol. CLXXIX, 
 May, 1915. 
 
 Fig. 88. 
 
THE DYNAMICS OF WAVE MOTION. 
 
 183 
 
 When a ribbon wave is reflected at the free end B of the ladder, 
 the twist h is reversed without change of sign of co. That is 
 to say, if the ladder is twisted like a right-handed screw or helix 
 in the original wave it will be twisted as a left-handed screw or 
 helix in the reflected wave, but the rotation or spin in the 
 reflected wave, will be in the same direction as the rotation or 
 spin in the original wave. 
 
 When the end bar at B is rigidly fixed, a ribbon wave is 
 reflected at B with reversal of rotation or spin but without 
 reversal of twist. 
 
 What takes place when an elastic rod strikes endwise against 
 a rigid wall and rebounds. A longitudinal wave on a rod is a 
 state of endwise compression C (negative value of C means 
 tension) associated with a certain velocity V of the material of 
 the rod, and the ratio V/C has a certain value, a, which is 
 positive or negative according to the direction of travel of the 
 wave. The initial condition of the elastic rod just before it 
 strikes the wall is a uniform velocity of the material of the rod 
 towards the wall, this initial condition is to be thought of as 
 always being in existence except in so far as it is modified by 
 the superposition of new conditions, and this initial condition 
 of uniform unchanging motion is symbolized by the heavy dotted 
 line in Figs. 89 and 90. 
 
 V +C 
 
 Fig. 89. 
 
 B moving still 
 
 
 .' .'/'.'.*::'': '* 
 
 not compressed 
 
 compressed 
 
 [ -V +C 
 
 
 -V -C 
 
 Fig. 90. 
 
 After the rod strikes the wall it continues for a definite time 
 to push steadily against the wall, a long-drawn-out ribbon wave 
 continues to shoot out from the wall as indicated by the heavy- 
 
184 CALENDAR OF LEADING EXPERIMENTS. 
 
 line arrow in Figs. 89 and 90. The first lap of this ribbon wave 
 (velocity V of the material of the rod and an associated 
 compression C) annuls or literally wipes out the initial velocity 
 V of the rod, and lays down a condition of uniform compression 
 as indicated in Fig. 89. The second lap of the ribbon wave 
 (velocity V and compression C, because reflection at 
 the free end B of the rod takes place with reversal of C so 
 that the compression in the first lap becomes tension in the second 
 lap) then wipes out the uniform compression and lays down a 
 uniform condition of reversed motion as indicated in Fig. 90. 
 Therefore when the second lap of the ribbon wave reaches the 
 wall the entire rod is relieved of compression and it is moving 
 away from the wall at velocity V. 
 
 Sudden stopping of flow of water in a rigid pipe. When a valve 
 is suddenly closed at the end of a pipe the stoppage and re- 
 bounding of the column of moving water takes place precisely 
 in the manner of the stoppage and rebounding of the rod as 
 represented in Figs. 89 and 90, on the assumption that the pipe 
 is rigid. 
 
 Experiment with a cast iron rod. A short slug moving at 
 velocity 2 V comes squarely against the end of a rod as indicated 
 by diagram A in Fig. 91. Let us assume that slug and rod are 
 of the same diameter and made of the same material. Then 
 as long as the slug pushes on the rod It may be thought of as a 
 part of the rod, and the state of affairs as shown in diagram A 
 is resolvable into two oppositely moving waves (ribbon waves) 
 V C and V + C as shown. The wave V C merges 
 into V ~\- C as it is reflected from the free end of the slug, and 
 soon a ribbon wave of length 2L (where L is the length of the 
 slug) develops and travels to the right along the rod as indicated 
 in diagrams B and C. This ribbon wave is reflected from the 
 right-hand end of the rod as indicated in diagram D, a region 
 of doubled velocity 2V and zero compression develops until 
 the ribbon wave is half reflected as shown in diagram E, and 
 then a region R of tension begins to develop as indicated in 
 
THE DYNAMICS OF WAVE MOTION. 
 
 185 
 
 diagram F. This tension is equal to the original compression, 
 and, if the rod is made of cast iron which withstands very great 
 compression but which does not withstand great tension it may 
 easily be that the rod separates in the narrow region R in dia- 
 
 D 
 
 2V +C 
 
 
 1 
 
 c 
 
 
 Fig. 91. 
 
 j.-.y *rt7 
 
 -C 
 
 gram .F. The imperfect elasticity and lack of homogeneity of 
 a cast-iron rod modifies the ideal action as represented in Fig. 91, 
 and some appreciable time is required for the cast-iron rod to 
 separate in the region R. Therefore the slug which flies off 
 the end of the cast-iron rod is sure to be somewhat longer than 
 the original slug in diagram A, and it will be moving as a whole 
 at a lower velocity than 2 V. Nevertheless it is a very interesting 
 and instructive experiment to shooc a steel slug against the end 
 of a cast-iron rod. 
 
 Use a No. 10 single-barrel shot gun; use a snugly fitting 
 steel slug about 2 inches long; make the cast-iron rod of same 
 diameter as steel slug with a squarely ground end; and use 
 about half-a-gram of black powder with an empty space of two 
 inches or more between powder and slug. This experiment is 
 suggestively frightful but in fact quite safe. 
 
186 CALENDAR OF LEADING EXPERIMENTS. 
 
 An interesting example of the action which is represented in 
 Fig. 91 is afforded by some experiments which 
 were made at Woolwich, England, to determine 
 the effect of exploding a charge of dynamite, or 
 other high explosive, against a thick plate of steel 
 armor. In some of the experiments a high bulge 
 was formed on the back of the plate, and when 
 the plate was cut open its section was as shown 
 in Fig. 92. A deep dent was formed where 
 
 p v . 92 the explosion took place, a layer of steel / was 
 
 thrown off the other side of the plate, and a hol- 
 low space h was left. 
 
 Experiment with billiard balls. Place a number of billiard balls 
 together in a straight row. Let one ball, two balls, three balls, 
 etc., come against the end of the row, and one ball, two balls, 
 three balls, etc., will fly off from the other end of the row. The 
 action is essentially the same as that shown in Fig. 91. 
 
 Whip action. An interesting modification of the above de- 
 scribed experiment of shooting a steel slug against the end of a 
 cast-iron rod is to use a tapering cast-iron rod and shoot the steel 
 slug against the larger end of the rod. As the ribbon wave 
 travels towards the smaller end of the rod the values of V and 
 C increase, and a much lower initial velocity of slug is sufficient 
 to throw off the tip of the tapered rod. 
 
 This wave intensification along a tapered rod is sometimes 
 troublesome in modern high-power guns which taper from breech 
 to muzzle. A quick-acting backward force on the breech block 
 produces a ribbon wave of tension-and-backward-motion, and 
 as this ribbon wave travels towards the muzzle it is intensified, 
 and the tension may rise to a value exceeding the strength of the 
 material, thus breaking off the muzzle of the gun, or, what is 
 quite as bad for the gun, leaving a permanent stretch of the 
 muzzle portions of the gun. 
 
 The intensification of a wave which travels along a tapered 
 
THE DYNAMICS OF WAVE MOTION. 187 
 
 rod is an example of what Professor P. G. Tait called whip 
 action, and Professor Tait's explanation of the cracking of a 
 whip is as follows: First, any object moving through the air 
 at a velocity greater than the velocity of sound produces a 
 sharp snap or crack. Second, a wave of moderate sidewise 
 velocity may be started in the thick and heavy butt portions of 
 a whip, and the sidewise velocity increases more and more 
 as the wave travels towards the small end of the whip, thus caus- 
 ing the whip cracker to reach velocities exceeding 1,000 feet per 
 second. 
 
 An interesting experiment is to stand on a ladder, hold the 
 butt end of a long flexible whip in the hand, let the whip hang 
 vertically downwards, move the butt quickly to one side, and 
 watch the increasing visible amplitude of motion of the resultant 
 wave as it travels down the whip.* 
 
 Gun recoil. Simple ideal case. A portion of the recoil of a 
 heavy gun (backward momentum of gun, the gun being assumed 
 to be free) is equal to the forward momentum of the projectile, 
 and the remainder is equal to the forward momentum of the 
 powder gases. The first part of the recoil momentum is quite 
 easily calculated whereas no rational method has hitherto been 
 developed for calculating the second part of the recoil momen- 
 tum, f A very simple and more or less ideal example of gas- 
 recoil would be afforded by suddenly opening one end of a tube 
 which contains very slightly compressed air. The initial condi- 
 tion of uniform compression is symbolized by the heavy dotted 
 line in Figs. 93 and 94, and the heavy arrow symbolizes the long- 
 drawn-out ribbon wave (of outward-motion-and-raref action, 
 + V and C) which continues to shoot inwards from the 
 open end of the tube. The first lap of this ribbon wave wipes 
 out the existing compression and lays down a condition of out- 
 
 * Change of wave velocity due to changing mass and tension have an important 
 effect here, but it is not worth while to discuss the matter further. 
 
 f This deficiency has recently been met in a paper by Wm. S. Franklin, Journal 
 of the Franklin Institute, Vol. CLXXIX, pages 559-577, May, 1915. 
 
188 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 ward motion. The ribbon wave is reflected from the closed 
 end of the tube with reversal of V, and the second lap of the 
 ribbon wave wipes out the outward motion and lays down a 
 condition of rarefaction. The ribbon wave is then reflected 
 
 initial compression C 
 moving 
 
 still 
 
 initial Compression^ (7 
 still 
 
 compressed [not compressed ;' 
 
 -C 
 
 Fig. 93. 
 
 Fig. 94. 
 
 from the open end with reversal of C, and the third lap of the 
 ribbon wave wipes out the rarefaction and lays down a condition 
 of inward motion. The ribbon wave is then reflected from the 
 closed end of the tube with reversal of V and the fourth lap of 
 the ribbon wave wipes out the inward motion and lays down a 
 condition of uniform compression as at the beginning. 
 
 If such a simple wave motion existed in the powder gases of 
 a gun, the gas- recoil momentum would be due to the fact that 
 the gases would continue to push backwards on the breech block 
 after the opening of the muzzle for the length of time required 
 for the first lap of the ribbon wave to reach the breech block. 
 As a matter of fact, however, the wave motion in the powder 
 gases is greatly complicated by the velocity already existent in 
 the gases when the muzzle is opened and by the adiabatic cooling 
 of the powder gases as they expand. 
 
 Experiment with Kelvin ladder showing the effect described 
 in connection with Figs. 93 and 94. Clamp the crank at the top 
 of the ladder, take hold of the bar at the bottom and twist the 
 ladder slightly, and wait for the ladder to become still. Then 
 release the bottom bar. The initial state of twist is analogous to 
 the initial compression of the air in a tube, and the initial twist 
 may be thought of as symbolized by the heavy dotted line in 
 
THE DYNAMICS OF WAVE MOTION. 189 
 
 Figs. 93 and 94. Releasing the bottom bar is analogous to the 
 opening of the end of the tube. A ribbon wave (of rotation-and- 
 reversed-twist) then wipes out the existing twist and lays down 
 a condition of uniform rotation. At the instant this ribbon 
 wave reaches the top of the ladder the ladder will be seen to be 
 flat (entirely freed from twist) and the entire ladder will be in 
 uniform rotation. The further details of behavior of ladder 
 need not be described but they should be fully described if the 
 experiment is shown to a class, otherwise the student will not 
 know what to look for. 
 
 Differential equations of electrical wave motion on a trans- 
 mission line. The theory of electrical wave motion is usually 
 developed in terms of electric and magnetic field intensities 
 in space,* and the equations are not easy to understand, especially 
 by the electrical engineer who is accustomed to express every- 
 thing in terms of voltage and current. Therefore the following 
 discussion of electric wave motion on a transmission line is 
 expressed in terms of voltage and current. Throughout the 
 discussion the resistance of the line wires is assumed to be 
 negligible and the line wires are assumed to be perfectly insulated. 
 
 The horizontal lines in Fig. 95 represent the wires of a trans- 
 
 wire 
 
 > AX >i+Ai wire 
 
 
 \a 
 
 b\ 
 
 
 
 i 
 
 i 
 
 
 
 L 
 
 |e+4e 
 
 
 wire 
 
 [d 
 
 c; 
 
 wire 
 
 Fig. 95. 
 
 mission line, and abed is an element of the line. Let e be the 
 voltage across the transmission line at the point ad and let i 
 
 * A very simple development of this general theory is given on pages 186-195 
 of Franklin's Electric Waves, The Macmillan Co., 1909. The student must be 
 familiar with the elements of vector analysis to be able to understand this electro- 
 magnetic theory. See Electric Waves, pages 158-185, or see Franklin, MacNutt 
 and Charles's Calculus, pages 210-253. 
 
190 CALENDAR OF LEADING EXPERIMENTS. 
 
 be ''the current in the line at the same point" (meaning outflow- 
 ing current in one wire and returning current in the other wire) 
 as shown in Fig. 95. The voltage across the line at the point be 
 is e + Ae, and the current in the line at the same point is 
 i + At. 
 
 The capacity of the element abed is C-Ax, where C is the 
 capacity of unit length of the line. Therefore the charge q 
 "on the element" (positive charge on ab and negative charge on 
 cd) is q = C-Ax X e,* and the rate of decrease of g, namely, 
 
 "C'Ax, is equal to Ai. Therefore we have: 
 
 c de - - in 
 
 c di~ ~ e x 
 
 The net electromotive force around the elementary circuit 
 abed is (e + Ae) e, and this electromotive force causes the 
 current in the circuit to decreasef at a definite rate such that 
 
 di 
 
 Ae = L'&xX, according to Art. 196, pages 276-277, 
 ot 
 
 General Physics, where L is the inductance per unit length of the 
 transmission line and L-kx is the inductance of the elementary 
 circuit abed. Therefore we have 
 
 di de 
 
 Equations (i) and (ii) contain the two unknown dependent 
 variables e and i, and it is necessary to eliminate one to get 
 an equation involving the other alone. By differentiating equa- 
 tion (i) with respect to / and equation (ii) with respect to x 
 
 * The charge is greater than C-Ax X e and less than C-Ax X (e + Ae), and 
 when Ax approaches zero, the expression for q approaches C-Ax X e as a limit. 
 
 t The electromotive force e + Ae is associated with an electric field from wire 
 to wire and the arrow shows the direction of this field or the direction of e + A 
 as it would be indicated by a voltmeter. Evidently, however, an excessive charge 
 on the wires at be and a large electromotive force from b to c would tend to 
 create a current opposite to i. 
 
THE DYNAMICS OF WAVE MOTION. 191 
 
 we get 
 
 r - d 2 * dH _d% 
 
 C W = ~ dx-dt ' dt-dx ~ " dx* 
 
 but d 2 i/(dx-dt) = d 2 i/(dt-dx)* and therefore we get 
 
 d*e i d*e 
 dfi "LC'dx 2 
 
 By differentiating equation (it) with respect to / and equation 
 (i) with respect to x, and eliminating as before, we get 
 
 ^ _ 1?_ i 
 dt* ~ LC'dx* 
 
 The general solution of equation (10) is of course 
 
 e = F(x - vt) + f(x + vt) 
 where 
 
 but it is most convenient for present purposes to consider the 
 two particular solutions e = F(x vt) and e = f(x + vt) 
 as follows: 
 
 A pure wave traveling to the right in Fig. 95 is represented by 
 the particular solution 
 
 e = F(x - vt) (13) 
 
 and to adopt this solution of (10) is to fix the corresponding 
 solution of (u) as follows: From equations (i) and (13) we have 
 
 - Cv-F'(x - vt) = - j- (iii) 
 
 and from equations (ii) and (13) we have 
 
 L -^ = - F'(x - vt) (iv) 
 
 * This is by no means self-evident as an arithmetical proposition; but a full 
 discussion of it would be out of place here. 
 
192 CALENDAR OF LEADING EXPERIMENTS. 
 
 Therefore, by integrating (iii) and (iv), we get 
 
 i = Cv F(x vt) + any function of t (v) 
 
 and 
 
 i = j- F(x vt) + any function of x (vi) 
 
 Lit) 
 
 i fC 
 
 But, according to equation (12), Cv = -=- = + yj- , and there- 
 fore the unknown function of / in (v) must be identical to the 
 unknown function of x in (vi), which means, in the most 
 general case, that both functions are constants; and for present 
 purposes these constants may be taken as zero. Therefore 
 equations (v) and (vi) become 
 
 i = a-F(x-vt) (14) 
 
 where 
 
 'L 
 whence dividing (14) by (13) we get 
 
 ; - + a (16) 
 
 e 
 
 From equations (13), (14) and (15) it is evident that a pure 
 wave traveling to the right in Fig. 95 is any distribution of voltage 
 e traveling to the right at velocity v, the current at each point in the 
 line being equal to ae; or any distribution of current i traveling 
 to the right at velocity v, the voltage across the line at each 
 point being equal to i/a. 
 
 A pure wave traveling to the left in Fig. 95. Starting with 
 e = f(x + vt) we may find, by an argument similar to the above, 
 that i ae. That is 
 
 \ = - a (17) 
 
 and it follows that a pure wave traveling to the left in Fig. 95 is 
 any distribution of voltage e traveling to the left at velocity v, the 
 current at each point in the line being equal to ae. 
 
THE DYNAMICS OF WAVE MOTION. 193 
 
 Remark. Equations (16) and (17) are useful in that they 
 enable one to pick out particular solutions of equations (io)and 
 
 en). 
 
 Above discussion from another point of view. Professor 
 P. G. Tait's classical discussion of wave motion on a stretched 
 string which does away with all necessity for the solution of 
 differential equations* has its analog in the theory of electrical 
 waves, and it is interesting to note that the avoidance of integra- 
 tion comes from the postulate of travel which is introduced at 
 the start. This fact makes it all the more clear that the classical 
 differential equation of wave motion [see equation (10) or (n) 
 above] is merely a differential equation of travel. 
 
 Effect of traveling distribution of current. Imagine current to 
 be distributed over a transmission line in any arbitrary manner, 
 the current i at any given point ad of the line (meaning out- 
 flowing current i in one wire and back flowing current i in 
 the other wire) being represented by the prdinate i of any 
 given curve CC, Fig. 96, and suppose this current distribution 
 to travel as a whole to the right in Fig. 96 at velocity v, that is 
 to say, let us imagine the curve CC to travel to the right at 
 velocity v and the current distribution to change so as to be 
 represented at each instant by this traveling curve. It is strictly 
 meaningless to think of the current itself as moving along, but 
 it is convenient to think of the current and the magnetic field 
 (also the charges on the wires and the electric field between the 
 wires) as moving along with CC. Such a traveling current dis- 
 tribution would produce a voltage distribution (traveling along 
 with it) such that 
 
 e = Liv (18) 
 
 where e is the voltage across the line at the place where the 
 current in the line is i. This equation may be established as 
 follows : The inductance of the element abed is L Ax, and the 
 magnetic flux between the wires ad and be is equal to the 
 
 * See General Physics, Art. 359, pages 499-501. 
 14 
 
194 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 inductance of the element multiplied by the current, everything 
 being expressed in c.g.s. units. Now, the current distribution 
 and the associated flux are assumed to travel to the right at 
 velocity v so that all of the flux between ad and be will sweep 
 across the line be in Ax/v of a second. Therefore the voltage 
 e induced along be by the traveling flux is Li- Ax divided by 
 Ax/v which gives Liv abvolts. 
 
 Effect of traveling distribution of voltage. Imagine electric 
 charge to be distributed over the transmission line in Fig. 96 
 
 wire 
 
 wire 
 
 Fig. 96. 
 
 (positive charge on one wire and an equal negative charge on 
 the other). This charge means a definite voltage between wires 
 at each point along the line, and we may imagine the voltage e 
 at each point to be represented by the ordinate of a curve CC. 
 Imagine the electric charge and the associated voltage distribu- 
 tion to travel along the line at velocity v. Such a traveling 
 voltage distribution would produce a definite current distribution 
 over the line such that 
 
 i = Cev (19) 
 
 where i is the current in the line at the place where the voltage 
 across the line is e, and C is the capacity of the line per unit 
 of length. This equation may be established as follows: The 
 capacity of the element ab cd is C-Ax so that C-Ax X e is 
 the positive charge on ab (or negative charge on cd). But 
 all of the charge on ab must flow past the point b during 
 
THE DYNAMICS OF WAVE MOTION. 
 
 195 
 
 Ax/v of a second, so that the current in the line is found by 
 dividing C-kx X e by Ax/v which gives equation (19). Of 
 course the current distribution travels along with voltage dis- 
 tribution which produces it. 
 
 Mutually sustaining current and voltage distributions. Let e, i 
 and v refer to identically the same quantities in equations (18) 
 and (19) respectively, that is to say, e is produced by the motion 
 of i, and in turn i is produced by the motion of e (see pages 
 100-105 of this volume). Then equations (18) and (19) are 
 simultaneous equations, and by elimination we get 
 
 i 
 LC 
 
 and 
 
 (20) 
 (21) 
 
 A clear understanding of an electric wave traveling along a 
 transmission line may be obtained from Fig. 97, resistance of 
 
 wire * *. * 
 
 current 
 
 wire 
 
 
 x 
 
 ^ 
 
 " 
 
 . 
 
 
 
 - 
 
 : HKffl 
 
 :[{:: ::::;: - 
 
 ; 
 
 
 
 1 
 
 ' 
 
 I 
 
 
 
 + charge' 
 
 X 
 
 
 
 
 
 
 
 
 
 
 '!!;;' ill 
 
 ;::::;; i . 
 
 * 
 
 
 
 
 , 
 
 .' 
 
 
 V ^ 
 
 charge^ 
 
 
 ^ ' 
 
 . 
 
 . 
 
 / 
 
 
 
 
 ;'. : :: : :': 
 
 iiji-i-:' ' 
 
 ; 
 
 
 ^ 
 
 
 
 ' 
 
 
 
 wire 
 
 S 
 
 v 
 
 + 
 
 ^ 
 
 1 
 
 ^ 
 
 
 
 
 :ji : .;;j - 
 
 
 h- 
 
 
 / 
 
 
 
 * 
 
 wire 
 
 current 
 
 Fig. 97. 
 Showing an electromagnetic wave on a transmission line. 
 
 line wires being negligible and wires being perfectly insulated. 
 The curve WW corresponds to CC in Fig. 96, and the ordi- 
 nate y represents the voltage e across the line or the current i 
 
196 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 in the line at the point p. The upper wire is positively charged 
 and the fine vertical lines represent the lines of force of the electric 
 field which emanates from the positively charged wire and con- 
 verges upon the lower wire which is negatively charged. Cur- 
 rent flows to the right in the upper wire and to the left in the 
 lower wire as represented by the short horizontal arrows, and 
 the fine dots represent the lines of force of the magnetic field 
 between the wires, this magnetic field being perpendicular to the 
 plane of the paper in Fig. 97. The heavy arrow shows the 
 direction of travel of the wave at velocity v. 
 
 The direction of travel and the directions of e and i may be 
 correlated as follows: That particular wire is positively charged 
 out of which the electric lines of force emanate; the voltage e is 
 from positively charged wire to negatively charged wire; and 
 the current in the positively charged wire may be thought of as 
 carrying the positive charge forwards from the back of the wave 
 and laying it down on the wire in front of the wave. 
 
 What determines wave form? A device which produces a 
 varying voltage e and which can deliver current ad libitum is 
 connected across the end of a transmission line, and the curve in 
 Fig. 99 (which is of course traveling to the right) which represents 
 the shape of the wave which shoots out along the transmission 
 line is exactly like the curve in Fig. 98 which shows e as a 
 
 axis of e 
 
 axis of time 
 
 axis of e 
 
 axis of x 
 
 wire 
 
 transmission line 
 
 Fig. 98. 
 
 Showing voltage e as a function 
 of the time. 
 
 wire 
 
 Fig. 99. 
 
 Showing shape of wave which shoots out 
 from end of a transmission line when volt- 
 age e is connected across end of line. The 
 wave involves a certain current / = ae at 
 each part of line according to equation (16). 
 
THE DYNAMICS OF WAVE MOTION. 
 
 197 
 
 function of the time. This is evident when we consider, first, 
 that the assumed wave satisfies the differential equations (10) 
 and (n), second, that the assumed wave is consistent with 
 the fact that the transmission line is entirely quiescent up to 
 the instant of connecting e, and third, that the assumed wave 
 in passing out on the line in Fig. 99 involves at each instant a 
 voltage e across the end of the line which is in fact equal to 
 the voltage acting at that instant according to Fig. 98. That 
 is to say, first, the differential equations are satisfied, second, the 
 initial conditions are satisfied, and third, the boundary conditions 
 are satisfied. 
 
 The wave train. When a periodic electromotive force acts 
 across the end of the line, for example when an alternator is 
 connected to the line, then what is called a wave-train passes 
 out along the line, and the state of affairs (before matters are com- 
 plicated by the reflection of the waves from the distant end of 
 the line) is shown in Fig. 100. This figure shows a wave- train 
 
 direction of 
 
 progression 
 
 Fig. 100. 
 
 which is produced by a harmonic alternating voltage, and the 
 curve WW is a curve of sines. The short horizontal arrows 
 represent the current at various places in the wires, the fine 
 vertical lines represent the lines of force of the electric field, 
 and the dots represent the lines of force of the magnetic field 
 as in Fig. 97. 
 
 The ribbon wave. When a battery of constant voltage and 
 negligible resistance is connected across the end of a transmission 
 
198 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 line, a wave shoots out on the line, the voltage e in the wave is 
 everywhere of the same value and equal to battery voltage, and 
 the current is everywhere the same in value and equal to 
 e X ilC/L, according to equation (16). Such a wave we will 
 call a ribbon wave. Thus Fig. 101 represents a ribbon wave d/v 
 
 wire 
 
 ribbon wave 
 
 Fig. 101. 
 
 Showing the ribbon wave which shoots out from a battery which is suddenly 
 connected to the end of a line. 
 
 seconds after the battery is connected. The short horizontal 
 arrows represent current, the fine vertical lines represent the 
 lines of force of the electric field, and dots represent magnetic 
 lines of force as in Figs. 97 and 100. 
 
 Reflection. For the sake of brevity and clearness we will 
 discuss only the reflection of the ribbon wave, and such a wave 
 will be represented by a single heavy horizontal arrow as in Fig. 
 101. When such a wave is turned back or reflected at the end 
 of a line, the heavy arrow will be shown as turned back as in 
 Figs. 102, 103 and 104. The voltage and current in the original 
 wave are represented by E and / and the voltage and current 
 in the reflected wave are represented by E r and I r as shown. 
 
 Reflection from open end of line. The doubled arrow in Fig. 
 1 02 represents a wave which has been turned back or reflected 
 at the open end of a transmission line. The necessary condition 
 which must be satisfied at the open end of a line is that the actual 
 current there be zero. Therefore we have: 
 
 / + /, = o 
 
THE DYNAMICS OF WAVE MOTION. 199 
 
 In addition to this we must have : 
 
 E JL 
 
 7 = + \ r M 
 
 J. * Lx 
 
 and 
 
 f=->i (iii) 
 
 Now the voltage and current in the original wave are supposed 
 to be given and the voltage and current in the original wave do, 
 as a matter of course, satisfy equation (ii). From equation (i) 
 we have: 
 
 Ir = - / (iv) 
 
 and from equations (ii) and (iii) we have: 
 
 E r = + E (v) 
 
 Reflection at the open end of a line is therefore complete and it 
 takes place with reversal of current. 
 
 wire 
 
 open end 
 wire of H 
 
 wire 
 
 short-circuited 
 end of line 
 
 --m E / 
 
 . v V X E 1 
 
 Fig. 103. 
 
 Showing a ribbon wave partly reflected Showing a ribbon wave partly reflected 
 from the open end of a line. from the short-circuited end of a line. 
 
 Reflection from short-circuited end of line. The doubled arrow 
 in Fig. 103 represents a wave which has been turned back or 
 reflected from the short-circuited end of a line. The necessary 
 condition which must be satisfied at the short-circuited end of a 
 line is that the actual voltage across the end be zero. Therefore 
 
 we have: 
 
 E + E r = o (vi) 
 
 whence 
 
 E r = - E (vii) 
 
2OO 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 and, since equations (ii) and (iii) always apply, we get: 
 
 Ir = + / (Viii) 
 
 Reflection at a short-circuited end of a transmission line is 
 therefore complete and it takes place with reversal of voltage. 
 
 Transmission line oscillations which follow the switching on 
 of a generator. When a generator of negligible resistance and 
 
 wire 
 
 wire 
 
 open end of line 
 
 +E 
 
 -E +1 
 
 +E +1 
 
 axis of I 
 
 axis of time 
 
 curve of current at B 
 
 Fig. 104. 
 
 The upper part of the figure shows a ribbon wave which shoots out from a 
 suddenly connected generator and is repeatedly reflected at both ends of the line. 
 The lower part of the figure shows the value of the current at B as a function of 
 elapsed time. 
 
 inductance is suddenly switched on to a line, a ribbon wave of 
 generator voltage and corresponding current shoots out over the 
 line. . Assuming line resistance to be negligible and line insulation 
 to be perfect, this ribbon wave is reflected back and forth as 
 represented in Figs. 104 and 105, and by adding voltages and 
 
THE DYNAMICS OF WAVE MOTION. 
 
 201 
 
 currents in the successive laps of the ribbon wave a precise 
 knowledge of the distribution of voltage and current over the 
 line at any instant may be obtained. Thus, after a thousandth 
 of a second, the total length of the ribbon wave would be 186 
 
 short-circuited end of line 
 
 wire 
 
 wire 
 
 x 
 
 =1 wire 
 
 wire 
 
 s\ \\vvx ribbon wave i-E +1 
 
 
 -E +1 
 
 ) 
 
 L i-E +1 
 
 
 -E +1 
 
 ) 
 
 axis of I 
 
 curve of growth of 
 current at B 
 
 axis of time 
 
 Fig. 105. 
 Same as Fig. 104 except that distant end of line is short-circuited. 
 
 miles which would give a definite number of complete laps and 
 a fraction of the next lap. By adding a number of laps in this 
 way, curves can be plotted showing the actual distribution of 
 voltage and current over the line at any given instant; or values 
 of current and voltage at a given point of the line can be found for 
 successive instants and from this data curves can be plotted 
 showing voltage or current at any point as functions of elapsed 
 
2O2 
 
 CALENDAR OF LEADING EXPERIMENTS. 
 
 times. The ampere-time curves in the lower parts of Figs. 104 
 and 105 were obtained in this wav 
 
 wire 
 
 wire 
 
 wire 
 
 wire 
 
 R 
 
 M 
 
 -kE 
 
 -il 
 
 axis of I 
 
 axis of I 
 
 curve 'of current growth at B curve tf current growth at R 
 
 Fig. 106. 
 
 Same as Fig. 104 except that distant end of line is connected to a non-inductive 
 circuit of which the resistance is R = 3 v . In this case the ribbon wave is only 
 
 partially reflected from the end R and the successive laps of the ribbon wave are 
 thereby greatly reduced in intensity as represented by the shading. 
 
 Note. The character of the reflection at B in Figs. 104, 105 
 and 106 is determined by the condition that the sum of the volt- 
 ages in the successive laps must always be equal to battery 
 
THE DYNAMICS OF WAVE MOTION. 203 
 
 voltage E, and of course there is always an odd number of 
 laps at B, one, or three, or five, etc. 
 
 Note. Figures 104, 105 and 106 show what takes place when 
 a direct-current generator or battery is switched on to the line, 
 but on a line of moderate length a number of laps are formed in an 
 excessively short interval of time, and therefore Figs. 104, 105 and 
 1 06 show quite accurately what takes place immediately after an 
 alternator is switched on to the line, E being the value of the voltage 
 of the alternator at the instant of closing the switch. 
 
 Transmission line surges which are produced when a circuit 
 breaker opens. When a circuit breaker opens, the arc which 
 is formed persists for a very long time, relative speaking, and 
 the open gap in the circuit is filled with a fairly good conducting 
 material which slowly loses its conductivity. It is about as 
 nearly impossible to produce characteristic line surges by opening 
 a circuit breaker as it would be to set up an abrupt water wave 
 in a canal by allowing a cubic mile of soft mud to flow into the 
 canal prism to stop a troublesome flow of water in the canal ; and 
 yet the moon, as a 6o-cycle generator (60 cycles per month!), 
 might produce a troublesome tidal wash in a large estuary while 
 the attempt was being made to "open-circuit" the estuary in 
 this Brobdignagian fashion! Let the reader consider this 
 hydraulic analog carefully. It reproduces nearly all of the es- 
 sentials of the electrical case. The conducting vapor in the arc 
 of a circuit breaker is somewhat analogous to mud as a dam- 
 building material. 
 
 Very little need be said of characteristic line surges in connec- 
 tion with opening of switches, except in the case of very long lines. 
 When a line is short, or only moderately long, what takes place 
 may be described quite accurately in terms of the simple ideas 
 of the elementary alternating-current theory, where current 
 values are supposed to rise and fall simultaneously throughout 
 an entire circuit. The formation of a long arc between line wires 
 in air and the quick snapping out of such an arc does, however, 
 
204 CALENDAR OF LEADING EXPERIMENTS. 
 
 produce an electric wave disturbance on a transmission line of 
 moderate length, and the essential features of this case are shown 
 in Fig. 107. Imagine the system to be short-circuited by an arc 
 
 /*~large inductance 
 S wire wire 
 
 V 000000 
 
 D 
 
 T OQOOOQ wire _____ wire 
 
 fjg -I 
 
 +E +1 
 
 +1 
 
 Fig. 107. 
 
 The dotted line AB symbolizes the initial current in the transmission line. 
 The long heavy arrow with laps represents the ribbon wave which comes from D 
 after the line at D is opened. 
 
 at the distant end D of the transmission line, voltage being 
 reduced to a negligibly small value over the whole line, and a 
 large current I being established in the line. The generator is 
 to be thought of as having a large inductance so that the generator 
 current cannot change perceptibly during the very short time 
 required for the characteristic line surges. When the distant 
 end of the line is opened, a ribbon wave shoots towards the gen- 
 erator, is completely reflected at the generator with reversal of 
 current, again completely reflected at the opened end D of the 
 line with reversal of current, and so on. The first lap of this 
 ribbon wave wipes out the current in the line and lays down a 
 certain voltage E ( = A/L/C times the initial value of the 
 current) . The second lap of the ribbon wave lays down a double 
 voltage, and the original current. The third lap wipes out the 
 current again, and lays down a voltage equal to $E, and so on. 
 
 Note. The character of the reflection at the generator in Fig. 
 
THE DYNAMICS OF WAVE MOTION. 205 
 
 107 is determined by the condition that che sum of the currents 
 in the successive laps at A must always be equal to the initial 
 current / because the current in the highly inductive generator 
 does not have time to change perceptibly. 
 
 Note. It must not be imagined that the sending out of a 
 ribbon wave depends upon a continued supply of energy at the 
 point where the ribbon wave originates. Thus in Fig. 107 the 
 ribbon wave is superposed upon the initial current /, and the 
 first lap of the ribbon wave wipes out this current. Therefore 
 since the current is zero, there is no energy flow at all from the 
 opened end. The second lap of the ribbon wave lays down the 
 original current / and a doubled voltage 2E, and this combina- 
 tion of voltage and current represents a flow of energy from the 
 generator into the line. 
 
THE TRADITIVE LAMP. 
 
 Long ago Bacon gave a list of the things needed for the Ad- 
 vancement of Learning, and, among other things, he mentioned 
 A Traditive Lamp, or the Proper Method of Handing Down the 
 Sciences to Posterity. This Lamp has not yet been discovered ! 
 
 206 
 
APPENDIX. 
 A VISITORS' LABORATORY OF PHYSICS. 
 
 The authors' experience shows that the average visitor who 
 comes into the physical laboratory of a college or technical school 
 is greatly interested in the simplest kind of a non-dazzling experi- 
 ment if the experiment has a rational appeal and if this rational 
 appeal is set forth in a simple explanation. Indeed the interest 
 which the average visitor takes in genuine natural philosophy 
 has led the authors to devote a great deal of time to visitors. 
 A number of simple experiments, varied from time to time, are 
 always set up, and every member of the department, including 
 Clarence and Pete, has taken a share of the fun of edifying 
 visitors. 
 
 This phase of the museum idea is certainly important, and the 
 authors believe that our generously supported public institutions, 
 especially, should have fully equipped Visitors 1 Laboratories of 
 Physics and provide for the work of demonstration by members 
 of the Physics Department staff. 
 
 The authors have tried all of the following experiments on 
 visitors, some of them many times, and the greater portion of 
 these experiments could be set up permanently in a space 
 25 feet X 80 feet with two or three small dark rooms adjoining, 
 and the entire equipment need cost no more than four or five 
 thousand dollars. 
 
 VISITORS' EXPERIMENTS. 
 
 5. Coin and card experiment. 
 
 17 and 1 8. The bicycle wheel as a gyroscope. 
 
 19. Curious gyroscope toy. 
 
 20. Sharp and blunt pointed tops and hard-boiled egg. 
 22. Model of Schlick balancer. Gyroscope oscillations. 
 
 207 
 
208 CALENDAR OF LEADING EXPERIMENTS. 
 
 24. Experiment with pivoted stool. 
 
 33. Gum camphor on water. 
 
 34. Experiments relating to oil flotation. 
 
 35. The sensitive flame. 
 
 36. The spit-ball experiment. 
 
 37. The water hammer. 
 
 38. The disk paradox and its water analog. 
 
 39. The jet pump and the atomizer. 
 
 40. Ball riding on air or steam jet. 
 
 41. Experiment on ship suction. 
 
 43. Curved flight of spinning ball. 
 
 44. High foul. 
 
 46. The Brownian motion. 
 50. The Trevelyan rocker. 
 
 53. Prince Rupert's drops. 
 
 54. The fire syringe. 
 
 55. Cloud formation experiment with large flask. 
 
 58. Crystallization of NH 4 C1 and of KC1O 3 . 
 
 59. The boiling paradox. 
 63. Wood's metal spoon. 
 
 66. The recalescence of steel. 
 
 67. Retarded transformations. Hardening and tempering of 
 
 steel. Exhibit sample of partly crystallized candy. 
 
 70. Thermo-elastic properties of rubber. 
 
 71. Large electro-magnet; the magnet should have pole pieces 
 
 coming near together, (a) Show powerful attraction for 
 a piece of iron ; (b) Show eddy-current damping of motion 
 of thick sheet of copper ; (c) Show eddy-current damping 
 of motion of closed copper ring; and (d) Show absence 
 of eddy-current damping of motion of open-circuited 
 copper ring. 
 
 84. Lead tree experiment. 
 
 88. Conductivity of glass at high temperatures. 
 
 91. Transformer experiment. 
 100. Electric dancers. 
 
APPENDIX. 209 
 
 105. Electric oscillations maintained by an electric arc. 
 108. Rosin experiment. 
 113. The electric doubler. 
 
 115. The corona discharge. 
 
 116. Smoke deposition by Cottrell's method. 
 
 117. The ozonizer. 
 
 Discharge of electricity through gases, (a) Geissler tube 
 discharge, (b) Crookes tube and cathode-ray shadow, 
 (c) Magnetic deflection of cathode rays, (d) Heating 
 effect of cathode rays, (e) Luminescence by cathode rays, 
 (/) X-rays and fluoroscope, (g) Ionizing action of X- 
 rays, (h) Ionizing action of alpha, beta and gamma rays, 
 (i) The spinthariscope, and (j) Fog chamber for showing 
 paths of alpha particles. See pages 139 and 140 of this 
 volume. 
 
 1 1 8. Unit areas of sense of touch. 
 
 121. Looking at bright point under water. 
 
 Phantom boquet. A large concave mirror projects on top 
 of an empty base a real image of a hidden boquet. 
 
 122. Visible beam beyond a large short-focus single lens. 
 
 123. Model of compound microscope. 
 
 124. Model of simple telescope. 
 
 125. Vision. Looking through a pin hole at a cord. 
 
 126. Vision. The coin box. 
 
 127. Vision. Seeing an object inverted when its image is erect. 
 
 128. Vision. Curious effect of two-eye vision. 
 
 129. Vision. Shadows of blood vessels on retina. 
 
 130. Vision. The stroboscope. 
 
 131. Vision. Reversal of sense of motion. 
 
 134. Astigmatism of simple lens of narrow aperture. 
 
 Looking at distorted drawings in cylindrical and conical 
 mirrors. See catalogue of Max Kohl. 
 
 136. Chromatic aberration of the eye. 
 
 137. Projection of large image of sun by telescope used as a 
 
 telephoto-lens. 
 
210 CALENDAR OF LEADING EXPERIMENTS. 
 
 138. Spectroscope demonstrations, (a) Continuous spectrum, 
 (b) Bright-line spectrum, (c) Reversal of sodium lines, 
 and (d) Dark-line spectrum (solar spectrum). 
 
 140. Spectroscopic analysis of light reflected from thin film. 
 
 (a) Very thin soap film, (6) Moderately thick mica plate. 
 
 141. Looking at flaming arc through coarse diffraction grating. 
 
 143. Experiments on polarized light, a, b, c, d, e, f and g. 
 
 144. Experiments on color, a, 6, c, d and e. 
 
 145. Reversal of order of sound sensations. 
 
 146. Musical sticks. 
 
 147. The Galton whistle. Determination of upper pitch limit 
 
 of audibility. 
 
 148. Simple modes of vibration of string. 
 
 149. Simple modes of vibration of air column. Long glass tube 
 
 whistle with lycopodium powder. 
 
 150. Chladin's figures. 
 
 151. Resonance, (a) With piano and voice, (b) With tuning 
 
 fork and adjustable air column. A most striking experi- 
 ment is to connect a telephone to an alternating current 
 supply (the frequency must be about 150 cycles per 
 second to give best results) and hold the telephone over 
 the mouth of a tube which is slowly lowered in a jar of 
 water. 
 
 152. Experiments on vowel sounds. 
 
 153. The action of the ear in the perception of tone color or 
 
 timbre. 
 
 154. Beats. 
 
 155. Combination tone. 
 
 Note. Details of experiments are described in main body of this volume (under 
 given number) and in Franklin and MacNutt's General Physics. 
 
FRANKLIN, MAcNurr AND CHARLES 
 
 Publishers of Educational Books 
 
 That our interest in selling books is not narrowly commercial 
 is evident from the wide departures from traditional points of 
 view in every one of the twenty-six volumes. This number in- 
 cludes Nichols and Franklin's Elements of Physics, three volumes, 
 and Franklin and Williamson's Elements of Alternating Currents. 
 We are propagandists rather than commercialists, and as prop- 
 agandists we address this prefatory statement to particular 
 groups of men as follows : 
 
 (a) To those having a general interest in education, we recommend Bill's 
 School and Mine as well worth reading, and re-reading. 
 
 (6) To those specially concerned with scientific and technical education, 
 we recommend the philosophy and the biting humor that is scattered 
 throughout the Calendar of Leading Experiments. 
 
 (c) To teachers of physics and to teachers of electrical engeneering. The 
 
 earlier books on these subjects are widely known, but General Physics 
 (1916) and Elements of Electrical Engineering Vol. I. (1917) represent 
 wide departures from the earlier books, and they are, we believe, great 
 improvements thereon from a teaching point of view. 
 
 (d) To teachers of mathematics we earnestly recommend a careful examina- 
 
 tion of our Elements of Calculus. This book does not side-step the rigors 
 of mathematics. It represents a point of view which is meeting with 
 increase of approval, and more than approval, among those who are 
 broadly and vitally concerned with the mathematical sciences. 
 
 (e) To teachers in trade schools and night schools we recommend Elementary 
 
 Statics (No. 8) and Elementary Electricity and Magnetism (No. 9). 
 
 (/) To students of the non-mathematical sciences. Franklin, MacNutt 
 and Charles take pride in the fact that what they stand for is quoted in 
 William James' book on Pragmatism as an extreme example of pragmatic 
 philosophy. The references which are brought together on page 8 of 
 this circular constitute a very simple and highly pragmatic treatise on 
 the philosophy of the mathematical sciences. 
 
 South Bethlehem, Pennsylvania 
 
 8^" Any book in this list will be sent post paid on approval, to be 
 returned post paid if not satisfactory. 
 
No. 1 
 
 GENERAL PHYSICS 
 
 (1916} 
 
 BY 
 
 WM. S. FRANKLIN AND BARRY MAcNUTT. 
 
 A TREATISE ON NATURAL PHILOSOPHY 
 
 Published and for sale by McGraw-Hill Book Company 
 
 Price $2.75 post paid 
 A Key to Problems is supplied with teacher's desk copies of this book 
 
 This textbook, with its brevity and clearness, its directness 
 of treatment and its freedom from tedious repetition of the 
 commonplaces of High School Physics, was intended to do away 
 with the usual excessive amount of classroom coaching which 
 is the despair of every physics teacher who aims to develop the 
 power of analytical thinking and who tries therefore to hold 
 his students accountable for mathematical ideas, mathematical 
 formulations (not mathematical formulas, please note) and 
 numerical calculations. And the use of the book in the class- 
 room seems to show that it does in some measure accomplish 
 what was intended. The students do read it, it tends to focus 
 their attention on fundamentals, it helps to an unprecedented 
 extent (in the authors' experience) to form precise ideas, and it 
 enables the student to devour numerical problems. 
 
 For sale by 
 FRANKLIN, MacNUTT AND CHARLES 
 
No. 2 
 
 A CALENDAR OF LEADING 
 EXPERIMENTS 
 
 (1918) 
 
 BY 
 
 WM. S. FRANKLIN AND BARRY MAcNUIT 
 
 Published and for sale by 
 
 FRANKLIN, MacNUTT AND CHARLES 
 South Bethlehem, Pa. 
 
 Price $2.50 post paid 
 
 This book has to do primarily with lecture demonstrations in 
 physics. The authors believe that High School Physics would 
 be greatly helped by the use of more class-room demonstrations, 
 they believe that such demonstrations can be provided for at a 
 very small expense, and they consider that many of the demon- 
 strations described in this book would be usable in the High 
 School. The book has an Appendix on A Visitors' Laboratory 
 of Physics, and certainly our State Colleges and High Schools 
 could do a great work by developing this phase of the museum 
 idea. 
 
 Secondarily this book sets forth the possibilities of an extended 
 course in elementary dynamics, including the dynamics of wave 
 motion ; and from beginning to end the book is rilled with semi- 
 humorous reflections on the problems of the teacher of physics. 
 
No. 3 
 
 AN ELEMENTARY TREATISE 
 ON CALCULUS 
 
 BY 
 
 FRANKLIN, MAcNUTT AND CHARLES 
 Published by the authors. Price $2.00 post paid 
 
 Nearly everyone who has examined this book has been favor- 
 ably impressed by its mathematical quality and by its simplicity 
 and clearness. It does not side-step mathematical precision and 
 rigor, and yet the textual discussion is vividly intelligible so that 
 the student can reasonably be expected to understand the subject 
 without the usual excessive amount of class-room coaching. It 
 contains more than enough carefully graded problem and exercise 
 work to keep the average student busy for a year. 
 
 "It will teach many a mathematician many a useful thing if 
 only the mathematicians will take heed unto it." Professor 
 E. B. Wilson, Massachusetts Institute of Technology. 
 
 "It seems to me that you have succeeded in making a book 
 that will cause enthusiasm rather than discouragement in the 
 student." A professor of mathematics. 
 
No. 4 
 
 BILL'S SCHOOL AND MINE 
 
 (mi) 
 
 A COLLECTION 
 OF ESSAYS ON EDUCATION 
 
 BY WM. S. FRANKLIN 
 
 Published and for sale by 
 
 FRANKLIN, MacNUTT AND CHARLES 
 
 South Bethlehem, Pa. 
 
 Printed on India paper and beautifully bound. 
 Price $1.00 post paid 
 
 These essays are so compact and so forcible that the Inde- 
 pendent called the book "A Package of Dynamite," and the 
 reviewer in The Elementary School Journal (University of Chi- 
 cago) says "Impossible to read these essays lying down." 
 
 From Nature (London), May 10, 1917. "This new edition of 
 Professor Franklin's brightly written essays, with their advocacy 
 of education in the ' Land of Out-of-Doors ' and of the claims of 
 sensible science to a prominent place in school curricula, is en- 
 riched by a new essay on Education after the War." 
 
 "These brief papers are written in an entertaining style that 
 is gripping and interesting. . . . There is much that is good and 
 much that is of value in a careful study of this purposeful little 
 volume." School and Science Review. 
 
 "Yesterday I read every word of your book how splendid it is! Two 
 things I wish, yea three, I desire 
 First, that every American should read your book. 
 Second, that I had had your experiences. 
 
 Third, that I could have gone to school to you in Mathematics and Science, 
 You are right in your Science-teaching ideas absolutely right! But we 
 both agree that teaching is great fun, what? 
 
 Yours 
 
 WM. LYON PHELPS of Yale University 
 
COMPLETE LIST 
 OF FRANKLIN AND MacNUTT BOOKS 
 
 1. General Physics. McGraw-Hill Book Co., 1916. Price $2.75 
 
 2. Calendar of Leading Experiments. Franklin, 
 
 MacNutt and Charles, 1918 " 2.50 
 
 5. Mechanic sand Heat. TheMacmillanCo., 1910, " 1.75 
 
 6. Elements of Electricity and Magnetism. The 
 
 Macmillan Co., 1909 " 1.60 
 
 7. Light and Sound. The Macmillan Co., 1909. . . " 1.60 
 
 NOTE Books 5, 6 and 7 constitute a fairly complete treatise on 
 the elements of physics. 
 
 8. Elementary Statics (Mechanics). Franklin, Mac- 
 
 Nutt and Charles, 1915 " 0.50 
 
 NOTE This book is in pamphlet form. 
 
 9. Elementary Electricity and Magnetism. The 
 
 Macmillan Co., 1914 " 1.25 
 
 NOTE This book is suitable for trade schools and night schools. 
 
 10. Advanced Electricity and Magnetism. The Mac- 
 
 millan Co., 1915 " 2.00 
 
 NOTE Every student of electrical engineering needs something be- 
 yond the bare elements of electricity and magnetism, and this book 
 is intended to supply this need. Contents areas fol'ows: Pages 
 1-74 summary of elements, pages 75-103 magnetism of iron; pages 
 104-120 ship's magnetism and the compensation of the compass; 
 pages 121-164 electrostatics; pages 165-192 the theory of poten- 
 tial; pages 193-273 electric waves; and pages 274-297 the electron 
 theory. 
 
 Practical Physics. By Franklin, Crawford and 
 MacNutt. TheMacmillanCo., 1903. A Labora- 
 tory Manual of Physics in three volumes. 
 
 11. Volume I. Precise Measurements. Me- 
 
 chanics and Heat " 1.25 
 
 12. Volume II. Elementary and Advanced 
 
 Measurements in Electricity and Magne- 
 tism " 1.25 
 
 13. Volume III. Photometry. Measurements 
 
 in Light and Sound .90 
 
 14. Simple Tables for students of physics and chemistry. 
 
 Price per 100 2.50 
 
 For Sale by the Publishers and by 
 
 FRANKLIN, MacNUTT AND CHARLES 
 
 South Bethlehem, Pa. 
 
Books on Electrical Engineering 
 
 For sale by 
 FRANKLIN, MacNUTT AND CHARLES 
 
 and also by the publishers, The Macmillan Co., N. Y. City 
 
 ELEMENTS OF ELECTRICAL ENGINEERING. 
 
 Franklin and Esty. 
 
 15. Vol. I. Direct Currents 1906 Price $4.50 
 
 16. Vol. II. Alternating Currents 1908 " 3.50 
 
 ELEMENTS OF ELECTRICAL ENGINEERING. 
 
 W. S. Franklin. 
 
 17. Vol. I. D. C. and A. C. Machines and 
 Systems 1917 " 4.50 
 
 18. Vol. II. Electric Lighting and Miscellaneous 
 Applications 1912 " 2.50 
 
 Note. A Key to Problems is supplied with teachers' desk 
 copies of these two books. 
 
 DYNAMO LABORATORY MANUAL. Franklin 
 and Esty. 
 
 19. Vol. I. Direct Currents 1906 " 1.75 
 
 20. Vol. II. In preparation. 
 
 21. DYNAMOS AND MOTORS. Franklin and 
 
 Esty, 1909 " 4.00 
 
 This volume contains the portions of Franklin and Esty's 
 Elements which relate to D. C. and A. C. machines. 
 
 22. ELECTRIC WAVES. W. S. Franklin, 1909.. " 3.00 
 
 The more elementary portions of this volume are given 
 in a carefully revised form on pages 193-273 of Frank- 
 lin and MacNutt's Advanced Electricity and Magnetism, 
 book No. 10 in the above list. 
 
Many biologists and many purely experimental chemists would like to 
 have a clear insight into the methods of the mathematical sciences ; but every 
 science is now presented as for the narrow specialist, or for the dilettante, and 
 the student must choose everything in a particular field, or nothing. An exclu- 
 sive interest in a particular science seems to be the only interest that is recog- 
 nized as serious. 
 
 The following references bring together a simple and highly 
 pragmatic treatise on 
 
 THE PHILOSOPHY OF THE 
 
 MATHEMATICAL SCIENCES 
 
 and with the help of this simple treatise, which it is hoped some 
 time to publish separately, any mature student can get a clear 
 understanding of the methods of the mathematical sciences with 
 a very moderate expenditure of time and effort. 
 
 The infinitesimal calculus. See pages 593-597 of book No. i for a 
 discussion of the meaning of the infinitesimal calculus, and see chapter one of 
 book No. 3 (40 pages) for a precise and clearly intelligible survey of differential 
 and integral calculus. This chapter is presented in carefully revised form in 
 the Supplement which accompanies the book. 
 
 Methods in physics. The four fundamental methods in physics are (a) 
 The method of mechanics, (b) The method of thermodynamics, (c) The method 
 of atomics (the atomic theory), and (d) The method of statistics. 
 
 The philosophy of elementary mechanics is discussed at some length in 
 book No. 2. See pages 119-123 of book No. i for a discussion of the atomic 
 theory in contrast with thermodynamics; see pages 322-325 of book No. I for 
 a discussion of the method of mechanics as contrasted with the method of 
 atomics and see a very simple and suggestive discussion of the use of the 
 statistical method in physics in Science, pages 158-162, August 4, 1916. 
 
 The conservation of energy and the law of entropy. Any scientific 
 work, to be fruitful, must be to some extent conditioned by the great generali- 
 zations of physics; but these generalizations need to be tempered, in their wide 
 acceptance, by a full appreciation of the fact that they are very largely of 
 postulate content. The great value of precise ideas (mathematical ideas) is 
 that they open the mind for the perception of the simplest evidences of a sub- 
 ject, and mathematical thinking is a necessity in the mathematical sciences; 
 but any kind of an idea unconditionally held closes the mind completely to 
 contrary evidences. This matter is discussed somewhat naively on pages 75- 
 90 and 99-101 of book No. 4. 
 
 The law of entropy is, perhaps, rigorously true for a world in which no 
 organizing agencies are at work coordinating things which are apart in time and 
 apart in space, but living forms seem to be subject to the law of entropy only in 
 that the easiest line of delevopment of an organism which must use free energy 
 is the line of development in which an excessive burden of coordination is not 
 necessary at the beginning. 
 
 The conservation of energy is discussed on pages 60-70 and 119-125 of 
 book No. i. 
 
 The law of entropy (the second law of thermadynamics) is discussed on 
 pages 153-168 of book No. i. 
 
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