UNIVERSITY OF CALIFORNIA c c A LABORATORY MANUAL PHYSICS AND APPLIED ELECTRICITY A LABORATORY MANUAL OF PHYSICS AND APPLIED ELECTRICITY ARRANGED AND EDITED BY EDWARD L.. NICHOLS . * 7SV PROFESSOR OF PHYSiqg ^N CORMELL UNIVERSITY IN" TWO VOLUMES VOL. I JUNIOR COURSE IN GENERAL PHYSICS BY ERNEST MERRITT AND FREDERICK J. ROGERS MACMILLAN AND CO. AND LONDON 1894 All rights reserved IGHT, 1 BY MACMILLAN AND CO. Set up and electrotyped June, 1894. Reprinted October, 1894. Norfooolf J. S. Gushing & Co. Berwick & Smith. Boston, Mass., U.S.A. PREFACE. THIS work has been written to supply in some measure the needs of a modern laboratory, in which the existing manuals of physics have been found inadequate. In its present form the book is the work, chiefly, of Assistant Professors George S. Moler, Ernest Merritt, and Frederick Bedell, of Instructors Frederick J. Rogers, Homer J. Hotchkiss, Charles P. Matthews, and of the editor. Certain parts, however, have been taken from written directions to students which had been prepared by instructors who are no longer members of the department from which the book emanates, and who have taken no immediate hand in its final preparation. No attempt has been made to provide a complete and sufficient source of information for laboratory students. On the contrary, it has been thought wise to encourage continual reference to other works and to original sources. It is assumed that in all laboratories in which a work of this kind will be found useful, there is accessible to the student a small collection of reference volumes, including the Laboratory Manuals of Kohlrausch, Glazebrook and Shaw, Stewart and Gee, Witz, and of Wiedemann and Ebert ; also that the larger treatises on experimental physics of Jamin, Winkelmann, Violle, Wiedemann, Preston, etc., together with the best known of the lesser works in English, are available. The Manual has been divided into two volumes ; and it is designed for three classes of students, differing from each other in experience, maturity, and purpose. The method of treatment has been varied in accordance with the principle, that with increasing experience the student should be divorced more and more from the use of the Manual v i PREFACE. and also from the close supervision of the instructor, and that he should be thrown gradually upon his own resources, and be led to make a wider and wider use of the literature of the science. It will be found that the first volume, which is intended for beginners, affords explicit directions, together with demonstrations and occasional elementary statements of principles. This volume is the outgrowth of a system of junior instruction which has been gradually developed during a quarter of a century. No attempt has been made to include the whole of physics. On the other hand, the principle has been followed here, as indeed throughout the book, of incorporating only such experi- ments as have been in actual use. It is assumed that the student possesses some knowledge of analyti- cal geometry and of the calculus ; also that he has completed a text- book and lecture course in the principles of physics. It is not expected that the experiments will be taken consecutively, nor that a student, in the time usually given to the work, will complete more than a third of them. The experiments have been divided into groups, an arrange- ment of the work for which there were two reasons. On the one hand, it serves to guide the practicant and the instructor in the selection of experiments ; on the other hand, it furthers the development of the system by making it easy to add or to exclude material. It is expected, indeed, that the book will be used thus by those into whose hands it may come, each one adding such experiments to the various groups as he may desire to include in his course, and dropping out those which he may deem useless. In the second volume more is left to the individual effort and to the maturer intelligence of the practicant. This volume differs from the first also in another respect. In the junior course no attempt is made to leave the beaten track. The very nature of the subjects with which we have to deal in Volume II, however, has compelled the introduction of new materials. The writers trust that where the ripeness and maturity of treatment which comes from long-continued experience in the teach- ing of a subject is missing, some recompense may be found in the freshness and novelty of the themes. PREFACE. vii A large proportion of the students, for whom primarily this Manual is intended, are preparing to become engineers, and especial attention has been devoted to the needs of that class of readers. In Parts I, II, and III of Volume II, especially, a considerable amount of work in applied electricity, in photometry, and in heat has been introduced, with particular reference to the training of students of engineering. It is believed, nevertheless, that selections from these parts may be made which will be of value to students of pure physics also. The final chapters (Part IV), which are intended for those who have already had two years or more of laboratory instruction, consist simply of certain hints for advanced work. These are accompanied by typical results, the object of which is to show in brief form what has already been accomplished by the methods proposed, and to lead the student to a suitable starting-point for further investigation. Through- out this portion of the book theory and experimental detail alike have been omitted. The outlines which have been given are designed to afford suggestions only, and by virtue of their very meagreness to com- pel the student to read original memoirs in preparation for his work. EDWARD L. NICHOLS. CORNELL UNIVERSITY, ITHACA, NEW YORK, May, 1894. TJNIVEI, :IT7 *OI THB = -f TABLE OF CONTENTS. VOLUME I. PAGE INTRODUCTION i Record of Observations. Units. Graphical Representation of Results. Errors of Observation and Method of Least Squares. CHAPTER I 26 Curvature of a Lens by the Spherometer. Adjustment of a Cathetometer. Calibration of a Thermometer Tube. Volume and Density by Measure- ment. Periodic Motion by Method of Middle Elongations. Parallelo- gram of Forces. Parallel Forces. Principle of Moments. Coefficient of Friction. Wheel and Axle. Efficiency of a System of Pulleys. Atwood's Machine. Gravity from Motion of a Freely Falling Body. General Statements concerning Moment of Inertia and Simple Har- monic Motion. Gravity by the Physical Pendulum. Gravity by Rater's Pendulum. Variation of Period of Bar Pendulum with Position of Knife Edges. Moment of Inertia. Young's Modulus by Stretching. Moment of Torsion. Moment of Inertia by Torsion. CHAPTER II 79 General Statements concerning Density. Approximate Determination of Density by weighing in Water. Specific Gravity Bottle. Density with Corrections for Temperature and Air Displacement. Jolly Balance. Nicholson's Hydrometer. Fahrenheit's Hydrometer. Graduation of a Hydrometer. Density of a Solid by Variable Immersion Hydrometers. Hare's Method of determining Density of a Liquid. Verification of Boyle's Law. Comparison of Barometers. Expansion of Air. CHAPTER III ioj General Statements concerning Calorimetry. Heat of Vaporization of Water. Heat of Fusion of Ice. Specific Heat of a Solid. Radiating and Absorbing Powers of Surfaces. CHAPTER IV 122 General Statements concerning Static Electricity. Electrostatic Induc- tion. The Principle of the Condenser. The Holtz Machine. The Holtz Machine (continued). CHAPTER V 138 General Statements concerning Magnetism. Lines of Force and Study of Magnetic Fields. Magnetic Moment by Method of Oscillations. Mag- netic Moment by the Magnetometer. Measurement of the Intensity of a Magnetic Field. Distribution of " Free " Magnetism in a Permanent Ma.gnet. x TABLE OF CONTENTS. PAGE CHAPTER Vl 153 General Statements concerning the Electric Current. Law of the Tangent Galvanometer. Measurement of Current by Electrolysis. Measurement of the Constant of a Sensitive Galvanometer. Theory of Shunts. Appli- cations of the Galvanometer to the Measurement of Current. CHAPTER VII 181 General Statements concerning Difference of Potential and Electromotive Force. Comparison of two Electromotive Forces. Ohm's Method for the Measurement of E. M. F. Potential Difference Study of Flame Spectra of Various Metals. Determination of the Distance between the Lines of a Grating. Measurement of Candle Power by the Bunsen Photometer. . ^"^^^5^. TABLE OF CONTENTS. tfls* x i frar/fe* ^ &*+ y VOLUME II. PART I. EXPERIMENTS WITH DIRECT CURRENT APPARATUS. GENERAL INTRODUCTION Study of a Dynamo. Introductory to Characteristics of Dynamos. Char- acteristics of a Series Dynamo. Characteristics of a Shunt Dynamo. Armature Characteristic. Characteristics of a Compound Dynamo. Comparison of Magnetization Curves of Dynamos. Characteristics of the Waterhouse Dynamo and Study of Third Brush Regulation. Charac- teristics of Edison Arc Dynamo. Characteristics of Thomson-Houston Arc Dynamo. Characteristics of the Ball Dynamo. Study of Brackett- Cradle Dynamometer. Efficiency of Double Transformation of a Small Motor and Dynamo. Efficiency of a Small Dynamo. Efficiency of a Small Motor by Use of a Rafford Dynamometer. Efficiency of a Motor without a Dynamometer. Efficiency of a Dynamo without a Dynamom- eter. Efficiency of a Motor with a Cradle Dynamometer. Efficiency of a Dynamo with a Cradle Dynamometer. Reversing Motor. Study of Arc-lamps. Determination of the Constants of a Tangent Galva- nometer from its Dimensions. Use of Great Tangent Galvanometer and Verification of Constants by Experiment. Conditions of Maximum Sensi- tiveness for Great Tangent Galvanometer. Determination of H by the Tangent Galvanometer Method. Computation of the Resistance neces- sary to render a Potential Galvanometer Direct Reading. Determina- tion of Magnetic Dip. Introductory to the Calibration of Instruments. Calibrating an Ammeter; Voltmeter; Electrodynamometer. Constants of Graded Ammeter; Voltmeter. Reliability Test of a Voltmeter; Am- meter. Exploration of the Field of a Dynamo. Exploration Curves by Means of a Dynamo Indicator. Determination of the Coefficient of Magnetic Leakage in a Dynamo. Distribution of Waste Magnetic Flux in a Dynamo. Measurement of Resistance by the Ammeter and Volt- meter Method. Study of a Resistance. Construction and Test of Fuse Wires. Adjustment and Test of Accumulators. Loss of Electromotive Force Due to Self-induction in a Dynamo. Compounding a Dynamo from its Armature Characteristic. Compounding a Dynamo by Added Turns. Economic Coefficient of a Series Dynamo. Economic Coeffi- cient of a Shunt Dynamo. Determination of Air Gap; of Armature Turns; of Back Turns; of Field Turns. Power lost in an Armature Due to Hysteresis and Foucault Currents. Separation of Losses in a Dynamo. Speed Curves of a Series Dynamo. Speed Curves of a Shunt Dynamo. Curve of Torque of a Motor by running it as a Dynamo. Mechanical Characteristic of a Motor. xii TABLE OF CONTENTS. PART II. EXPERIMENTS IN ALTERNATING CURRENTS. INTRODUCTION TO ALTERNATING CURRENT EXPERIMENTS Curve of Magnetization of Alternating Current Generator. Study of Alter- nating-Current Generator. External Characteristic of Alternating Current Generator. Alternating Current Potentiometer; Adjustment and Test for Sensitiveness. Curve of Magnetization of Alternating Current Generator : Ballistic Method. Exploring Field of Alternator. Measurement of the Coefficient of Self-induction : Impedance Method. Measurement of the Coefficient of Self-induction : Three Voltmeter Method. Variation in the Coefficient of Self-induction with the Current. Variation in the Co- efficient of Self-induction with the Saturation of the Iron Core. Effects of the Variation of the Resistance in a Series Circuit. Effects of the Variation of the Self-induction in a Series Circuit. Electromotive Forces in a Series Circuit. Measurement of Power : Three Voltmeter Method. Measurement of Power : Three Ammeter Method. Equivalent Resist- ance and Self-induction of Parallel Circuits. Effect of Frequency upon Impedance of a Circuit containing Resistance and Self-induction. Effect of Frequency upon Angle of Lag in a Circuit containing Resistance and Self-induction. Measurement of Mutual Induction : Ballistic Method. Measurement of Mutual Induction : Alternating Current Method. Study of a Transformer. Transformer Test : Three Voltmeter Method at No Load. Transformer Test : Three Voltmeter Method at All Loads. Trans- former Test : Three Ammeter Method at No Load. At All Loads. Trans- former Test: Variations in Transformer Diagrams. Operation of a Synchronous Motor. Introductory to Experiments with Condensers. Study of Standard Condenser. Curves of Condenser Discharge : Ballistic Method. Curves of Condenser Discharge : Potential Method. Curves of Condenser Discharge: Deflection Method. Measurement of Capacity by Curves of Condenser Discharge. Measurement of Resistance by Curves W Condenser Discharge. Comparison of Capacities : Direct Deflection Method. Comparison of Capacities : Method of Mixtures. Comparison of Capacities : Gott's Method. Comparison of Capacities : Bridge Method. Comparison of Capacities: Divided Charge Method. Comparison of Capacities : Diminished Charge Method. Capacities in Parallel and Series. Comparison of Electromotive Forces by a Condenser. Study of Residual Discharges. Measurement of Capacity by Alternating Current Method. Effects of the Variation of the Resistance in a Series Circuit containing a Condenser. Effects of the Variation of the Capacity in a Series Circuit. Effects of the Variation of Frequency in a Circuit contain- ing Capacity but No Self-induction. Neutralization of Self-induction and Capacity in Series. Self-induction and Capacity in Parallel. Instan- taneous Measurement with a Revolving Contact Maker and Electrostatic Voltmeter. Instantaneous Measurement with a Revolving Contact Maker : Telephone Method. Instantaneous Measurement with a Revolv- TABLE OF CONTENTS. x iii ing Contact Maker : Ballistic Method. Irregularities in Alternating Current Curves. Measurement of Power by the Method of Instantaneous Contact. Transformer Test by the Method of Instantaneous Contact. Study of the Effects of Capacity by Method of Instantaneous Contact. Determination of Dielectric Hysteresis. Test of a Non-inductive Resist- ance by the Method of Instantaneous Contact. Investigation of Liquid Resistance. Calibration of a Hot-wire Voltmeter. Calibration of a Hot- wire Ammeter. Determination of the Constant of a Ballistic Galvanom- eter. Calibration of D'Arsonval or Ballistic Galvanometer for Potential. Magnetic Qualities of Iron : Ring Method. Magnetic Qualities of Iron : Instantaneous Contact Method. Illustrative Experiments with Alternat- ing Current Magnet. PART III. SENIOR COURSES IN PHOTOMETRY AND HEAT. CHAPTER I Standardization of Instruments. Distribution of Candle-power about an Incandescent Lamp. Characteristic Curves of an Incandescent Lamp. Photometry of the Arc-light. CHAPTER II Specific Heat by the Method of Mixtures. Specific Heat by the Method of Mixtures, Liquids. Specific Heat by the Method of Cooling. Specific Heat by the Bunsen Ice Calorimeter. Use of Favre and Silbermann's Calorimeter. Pressure of Saturated Vapors at Low Temperatures. Pressure of Saturated Vapors at High Temperatures. Density of Vapors (Dumas' Method). Heat of Vaporization. Heat of Combustion of Metals. Cubical Expansion of Solids by the Method of Balance Transits. Mechanical Equivalent of Heat by Current Calorimeter. Measurement of Temperatures by the Thermal Element. PART IV. OUTLINES OF ADVANCED WORK. INTRODUCTION CHAPTER I Critical Study of Thermometers. Influence of Temperature upon Young's Modulus. Thermal Conductivity of a Copper Bar. Volume of Liquid Mixtures. Volume of Substances near the Melting-point. Influence of Temperature upon the Color of Pigments. Transparency of Solutions. CHAPTER II Efficiency of Artificial Light Sources. The Glow-lamp. The Arc-lamp. The Magnesium Light. The Drummond Light. Other Sources.- CHAPTER III Life Studies of Light Sources. Incandescent Oxides. The Glow-lamp. Candles, Oil, and Gas. xiv TABLE OF CONTENTS. CHAPTER IV Spectrophotometry. The Construction of Spectrophotometers. Compari- son of Artificial Light Sources. Daylight. Radiation as a Function of Temperature. The Study of Pigments and Solutions. CHAPTER V Studies of the Invisible Spectrum. The Infra-red. The Ultra-violet. CHAPTER VI Physiological Optics. Duration of Impressions. Luminosity and Dichroic Vision. The Neutral Zone in the Spectrum of the Color-blind. Per- sonal Errors in Photometry. The Light passing through Sectored Disks in Rapid Rotation. CHAPTER VII Explorations of the Earth's Magnetic Field. The Method of the Kew Mag- netometer. The Method of the Tangent Galvanometer. Appendix. A LABORATORY MANUAL OF PHYSICS AND APPLIED ELECTRICITY. VOLUME I. JUNIOR COURSE IN GENERAL PHYSICS. BY ERNEST MERRITT AND FREDERICK J. ROGERS. INTRODUCTION. THE object of all of the experiments described in the fol- lowing pages is twofold : (i) to illustrate, and therefore impress more thoroughly on the mind, the principles and laws which have previously been taught by text-books or lectures ; (2) to familiarize the student with proper methods of observation and physical experimentation. These two aims should be kept in view throughout the work which follows. GENERAL DIRECTIONS. Before beginning any experimental work, the student is advised to read carefully the directions for the experiment that is to be performed, making sure that the object of the experi- ment and the means to be employed in accomplishing this object are fully understood. If the experiment involves prin- ciples which are unfamiliar, the matter should be looked up in 2 JUNIOR COURSE IN GENERAL PHYSICS. some reference book before the observations are begun. If this is done, the significance of each step in the experimental work will be appreciated, and the experiment will therefore be more instructive. The likelihood of essential observations being omitted is also less when the bearing of each observation upon the result is fully understood. Record of Observations. All original observations should be recorded in a note-book at the time when they are taken, and should be preserved. It is a saving in the end to devote enough time to the original records to make them neat and clear, and so complete as to enable any person who is familiar with the experiment to understand the meaning of each figure recorded. In all cases, it is the original observations that are to be recorded. A derived result should in no case be recorded as an observation, no matter how simple may be the process of derivation. For example, it may be required to find the duration of a certain phenomenon ; let us say that it begins at half-past three o'clock and lasts until twenty-two minutes of four ; the time is eight minutes, but this is a derived result obtained by subtracting 3.30 from 3.38. The actual time of beginning and end should be recorded, and the sub- traction performed afterward. The uniform observance of this rule will save annoyance from simple mistakes due to carelessness or haste, which are frequently made even by the best observers, and which, without the original observations, it would be impossible to correct. The time at which each observation is taken should always be recorded, including the day, hour, and minute. An example of the possible usefulness of this rule might occur in the case of observations taken with a sensitive galvanometer. Let us suppose that it is found on working up the data that the results of certain observations disagree with those of the remainder. The very annoying question then arises whether this disagreement represents an actual change in the phenomena observed, or whether it is due to the INTRODUCTION. 3 effect on the galvanometer of some outside disturbance. Knowing the time at which the observations were taken, it will be possible to investi- gate the matter, and if it is found that there was a change in the mag- netic conditions (due to varying currents, moving iron, or other causes) at the time when the irregularities were observed, then these observa- tions may be legitimately discarded. Observations. It is to be remembered that the object of scientific observations is not to confirm preconceived theories, or to obtain a series of results which shall arouse admiration on account of their uniformity, but to discover the truth in regard to the phenomenon investigated, no matter what the truth may be. It is of the greatest importance, therefore, that the observer should be entirely unprejudiced, either by a knowledge of the results of other experimenters, or by any preconceived notion as to what the results should be. It is not meant by this that the observer must be ignorant of the probable results : but that his observations should be taken with as much care as though he were ignorant; and that great. precautions must be taken to avoid the almost unconscious tendency, to which all observers are more or less subject, of making the observations correspond with what is thought to be the truth. In many cases artificial devices can be used to insure unprejudiced observations. For example, the scale of a micrometer screw may be covered, so that it is kept out of sight until the setting is made. Or, in an experiment like that on the Coefficient of Friction (No. d), one experimenter may adjust the weights while the other observes whether the motion obtained is uniform. Since the latter does not see the weights, his judgment is uninfluenced by any assumption as to the law by which they vary. In the measurement of almost all physical quantities the results will be better if the observation is repeated several times. The individual observations will doubtless differ from one another on account of slight unavoidable errors ; but the mean of the results will in all probability be nearer the truth than any single observation. To gain the advantages of taking 4 JUNIOR COURSE IN GENERAL PHYSICS. an average, however, it is necessary that each observation should be independent of all the rest. Knowing that all the measurements should be alike except for accidental errors, there is an unconscious tendency to make them agree. This tendency must be carefully guarded against, as in the cases cited above. Each observation should be taken as carefully as though the final result depended upon it alone. Estimation of Tenths. In measurements in which a grad- uated scale of any kind is used it often happens that the result sought cannot be expressed by any exact number of scale divis- ions. For example, in using a thermometer graduated to single degrees, the top of the mercury column will probably come between two divisions on the scale. In such cases always estimate the fractional part of a division by the eye, expressing the fraction in tenths. Even if the estimation is poor, it gives results nearer to the truth than if the fraction were disregarded ; while after a little practice it will be found possible to estimate tenths with great accuracy. Choice of Conditions. It often happens that the accuracy of the results of an experiment can be improved by a proper choice of the conditions under which the observations are made. An example of this fact occurs in the experiment where the internal resistance of a cell is determined by measurements of the current sent by the cell through two different external resistances. If I v R v and 7 2 , R^ represent the corresponding values of current and resistance, the internal resistance of the cell is It is evident that if 7 1 and 7 3 are nearly alike, a slight error in the measurement of either may cause a very large error in x. To make the results reliable it is therefore necessary to choose R and R so that the two values of the current shall differ INTRODUCTION. 5 widely. There are many cases similar to this, where an inspec- tion of the formula by which the results are to be computed will suggest what conditions will make the influence of acci- dental errors as small as possible.* Computations. In computing results every precaution should be used to avoid simple numerical mistakes. Mistakes due to careless adding or subtracting, to incorrect copying from one sheet to another, to the misplacing of a decimal point, etc., are a source of great annoyance, and unless care is used to avoid them they will appear with a frequency that is startling to one unaccustomed to computing. The best safeguard against mis- takes is neatness and an orderly arrangement of the work. In many cases four or five place logarithms are a help, not so much on account of any saving of time, as because of the dimin- ished liability of mistakes. Tables of squares, reciprocals, etc., can often be used to advantage, and the slide rule, when one is accustomed to its use, affords a considerable saving in time and worry. When a number of similar computations are to be made, the work should be done systematically and the results arranged in tabular form. In working up the results of an experiment time is often wasted by carrying the results to a degree of refinement that is not warranted by the observations upon which the computa- tions are based. Very few of the experiments that are described here will give results that are accurate to within less than one- tenth of one per cent. In most cases, therefore, it is useless to express the result by more than three, or at most four, sig- nificant figures. If it is decided from an inspection of the observations that the result should be carried to three places, then the computations should be made with four places in order to insure the accuracy of the last significant figure of the result. * In this connection see the paragraph dealing with the " Influence of errors of observation upon derived results," p. 18. 6 JUNIOR COURSE IN GENERAL PHYSICS. In the progress of the work numbers may be obtained in which five or six significant figures appear ; in such cases all beyond the fourth may be discarded. In many cases approximate methods may be used which will effect a considerable saving in time without diminishing the accuracy of the results. For example, it often happens that a factor of the form - appears as a multiplier, k being a very \+k small quantity. In most cases it is sufficiently accurate to say that 1 and in general (i +/)"= i -\-nk when k is small. * Units. In almost all physical measurements, the units employed are based upon the centimeter-gram-second system. Since this system differs in several important particulars from that generally used in engineering work, it is essential that these differences should be clearly understood. In physics, all derived units are defined in terms of the fun- damental units of length, mass, and time. In the foot-pound- second system, commonly employed in engineering work, the fundamental units are length, weight, and time. Now the terms " weight " and " mass," although technically quite different in meaning, are frequently confused in ordinary conversation, and it is probably from this cause that the relation between the two systems is so often misunderstood. It must be remembered that the weight of a body is defined as the force with which the body is pulled downward by gravity. By the word pound is meant, not the block of metal which weighs a pound, but the force by which that block is drawn toward the center of the earth. Since a force is numerically equal to the product of the mass moved into the acceleration, we have * Other examples of the use of approximations will be found in Kohlrausch, Glazebrook and Shaw, and in Stewart and Gee, Appendix to vol. i. INTRODUCTION. W = Mg, and in order to find the mass of a body whose weight in pounds is known, we must divide the weight by g\ i.e. The mass of a pound weight is therefore y^, an< ^ tne mass in the foot-pound-second system is the mass of a body which weighs 32.2 Ibs. In the C. G. S. system the gram is the unit of mass. By the word gram, therefore, is meant the amount of matter contained in a certain standard piece of metal. The weight of this piece of metal is found by multiplying its mass by the acceleration of gravity, and for the latitude of Ithaca (about 40) is a little more than 980 dynes. The process of weighing a body by means of a balance con- sists of choosing the weights so that both scale pans are pulled downward by gravity with the same force. When the adjust- ment is correct, the weight is therefore the same on each pan. But since, so long as g remains unaltered, the mass of a body is proportional to its weight, the two masses must also be equal. The balance may therefore be used either for comparing weights, or for comparing masses. In physical experiments the weight is seldom required, so that the balance is used almost entirely for the measurement of mass. The standards used, being grams, or multiples of a gram, are standards of mass, and the term " weights," which is so commonly applied to them, is really a misnomer. If it is found, therefore, in making a weighing by the balance that 100 grams are required to produce equilibrium, the mass of the body weighed is shown to be 100 grams.- The weight 'of the body is 100 x g = 98,000 dynes. If care is used in distinguishing between the terms " weight " and "mass," no difficulty shouk} be experienced in passing from one system of units to the other. The two systems are per- fectly consistent with each other when properly used, and each 8 JUNIOR COURSE IN GENERAL PHYSICS. has special advantages for the kind of work in which it is com- monly employed. Graphical Representation of Results. When a series of observations has been taken to show the manner in which one quantity depends upon another, it is often of advantage to present a summary of the results to the eye by means of a curve. Points upon such a curve are located on cross-section paper by using the values of one quantity as abscissas, and the corresponding values of the other quantity as ordinates, the scales used in measuring the various co-ordinates being any that are convenient. It is customary to use the values of the inde- pendent variable as abscissas. As an example of the use of the graphical method, we may consider the experiment on the coefficient of Friction (Ci). In this experi- 5 K 1O K 15 K 2O K 25 K PRESSURE Fig. 1. ment the force necessary to overcome the friction between iron and wood is measured for a number of different values of the pressure between the two. It is natural to suppose that the amount of friction depends in some way upon the pressure. To determine the law of this dependence, a curve is platted, in which pressures are used as abscissas, and the INTRODUCTION. 9 corresponding values of the friction as ordinates. If the observations have been carefully taken, the points located in this way will be found to lie very nearly upon a straight line passing through the origin. If the divergence from a straight line is not great, it is proper to assume that such divergence in the case of individual points is due to the accidental errors of observation, and that a straight line, passing as nearly as possible through all the points, really represents the relation sought. Now the equation of a straight line passing through the origin is y = mx, in which m is a constant. But the x's of our line represent pressures, while the ys represent the corresponding values of the friction. The law established by the experiment is therefore that F= mP; i.e. friction is proportional to pressure. It is to be observed that when a curve is platted in order to show the relation between two variables, it is by no means necessary that the horizontal and the vertical scale should be the same. Either scale may be assumed at pleasure, and with- out reference to the other. In the case just cited, for example, the horizontal scale may be taken as 5 kilograms to the inch, while the vertical scale may be i kilogram, -fa kilogram, or any other quantity that proves convenient. In taking readings from the curve, however, regard must be paid to the scale employed. If, for example, the horizontal scale adopted is 5 kilograms to the inch, 5 inches would be read 25 kilograms. If the vertical scale at the same time is -fa kilogram to the inch, 5 inches on the vertical scale would be read i kilogram. The example referred to above, where the curve obtained is a straight line passing through the origin, illustrates the sim- plest case that could arise. In other cases where the graphical method is used the curve obtained may prove to be a straight line which does not pass through the origin ; or it may be any form of curved line, such as a parabola or a hyperbola. In any case the law sought is determined as before, and can be expressed, either in words or by a formula, as soon as the curve is recognized. Since the straight line is the curve which is most readily tested, it is often convenient to transform the results of an 10 JUNIOR COURSE IN GENERAL PHYSICS. experiment in such a way that they will give a straight line when platted. Suppose, for example, that the volume of a gas has been measured when subjected to a number of different pressures. We know from Boyle's Law that PV= a constant = k. If the results were platted, therefore, with pressures and corresponding volumes for co-ordinates, the resulting curve would be a hyperbola whose equation is xy = k. If, however, we plat instead of volumes the products PV, the curve will be a straight line with the equation y = k. By observing whether this line is accurately straight, the law can be tested more readily than if the first curve had been used, while if the line is not straight it affords a simple means of exhibiting the deviation from Boyle's Law to the eye. If the method described in Exp. H x for verifying Boyle's Law is employed, the data may be platted in still a different way to advantage. In this method the total volume V is not measured, but merely a portion v, while a part V Q of the volume remains unknown, but constant. Then F=(z; + z; ), (i) P(v + v*) = k. (2) If now P and V are taken as co-ordinates, a hyperbola should be obtained. But if v and - are used, the resulting line should be straight, P its equation being ky = x + v<>. (3) If the data are platted in this way, a means is therefore afforded of determining both v and k. Since the line obtained is straight, we know that the form of its equation must be y = ax + b, (4) and the numerical values of a and b can be at once computed. From Boyle's Law, however, , = ^ + f- (5) Since these two equations represent the same line, we must have i = *,and=. (6) INTRODUCTION. TI Graphical methods are of such great value in all branches of physical investigation that their use is recommended in a large number of the experiments which follow. The student is strongly advised .to make himself familiar with graphical methods and their interpretation, as early as possible. Reports. As soon as the observations required in an experi- ment have been completed, and the results computed, a report is to be written, describing in detail the work that has been done. This report should be sufficiently clear and complete to enable it to be understood by any person having a good general knowl- edge of physics, even though the particular experiment described is entirely unfamiliar to him. Each report should therefore contain the following : (1) A statement of the object of the experiment and an explanation of the means employed to accomplish this object. (2) A description of the apparatus used.* (3) All formulas used, which express relations between phys- ical quantities, should be proven.f The object of putting such demonstrations in the report is to make it clear to the instructor that the principles involved are fully understood. The student will find, also, that there is no better way of making a subject perfectly clear to himself than by presenting it in such a form as to be readily intelligible to some one else. Those steps or details of a demonstration which are merely referred to in the text-books should therefore be very clearly explained. Origi- nality in the methods of proof is desirable, but of course cannot be expected in every case. (4) The report should contain all the original data, and an indication of the numerical work by which the results are ob- tained. It is not necessary to include all the computations in * In case the same apparatus has been employed in previous experiments, how- ever, it is not necessary to describe it a second time. t The proof of purely mathematical formulas, such as the trigonometrical rela- tions used in solving triangles, is not required. 12 JUNIOR COURSE IN GENERAL PHYSICS. the report, although where this can be done systematically and neatly, it is an advantage. In case the results are obtained by substitution in a formula, the numerical work should be given in detail in at least one case. (5) When possible the results obtained should be compared with the results of previous experiments as found in various reference books. When graphical methods have been used in connection with an experiment, the curves obtained are to be included in the report. In such cases the scale by which the co-ordinates have been measured should be clearly indicated on the drawing itself. In writing reports, it is always to be borne in mind that one important benefit which practice in this work may accomplish is the acquirement of clearness and facility of expression in the description of scientific investigations. The arrangement and wording of each report should therefore be carefully considered with this object in view. INTRODUCTION. ERRORS OF OBSERVATION AND METHOD OF LEAST SQUARES. Sources of Error in Physical Measurements. All physical measurements are subject to error from a variety of sources. Although the choice of proper methods, the employment of carefully constructed instruments, and great care in the obser- vations themselves may enable results to be reached which are quite close to the truth, yet absolute accuracy can in no case be expected. The effort of the experimenter should always be to reduce these errors to a minimum ; yet he may feel per- fectly sure that to completely eliminate them is quite impossible. As an example of the different ways in which inaccuracies can occur, we may consider a case which represents probably the simplest measure- ment imaginable ; namely, the measurement of a length by means of a graduated scale. The chief sources of error in this measurement may be summarized as follows : 1. The scale may be incorrect either in total length or in graduation. 2. Even if it were possible that the scale were constructed with perfect accuracy, it can only be correct at one definite temperature. The coefficient of expansion of the scale must therefore be known, while its temperature must be determined at the instant of making the measurement. Two sources of error are here introduced. 3. The end of the length to be measured will in all probability lie between two divisions of the scale. The fractional part of a scale divis- ion must therefore be estimated, and on account of a variable illumina- tion of the scale, an improper location of the observer's eye, or lack of experience on the part of the experimenter, this estimation is always subject to error. 4. Lastly, the observer may make a mistake ; i.e. may read 10 for 20, Jg- for ^, etc. A little consideration will show that all possible errors may be made to fall under four classes : 1. Errors of method. 2. Inaccuracies in instruments. 3. Accidental errors of observation. 4. Mistakes. I 4 JUNIOR COURSE IN GENERAL PHYSICS. The avoidance of errors due to the employment of faulty methods is largely a matter of judgment and experience on the part of the experimenter. No general rule can be given. Probably the best means of testing for the presence of errors in the method of measurement employed is to repeat the deter- mination by several radically different methods. If the results agree, it is to be presumed that the methods contain no funda- mental errors. The presence of inaccuracies in the instruments used may similarly be tested by making the same measurement with several different instruments. Special methods may also in most cases be devised by which the errors of any given instru- ment may be determined. These methods are different for each particular case, so that it is useless to give illustrations here. After the errors of method and of apparatus are as far as possible eliminated, there still remain the " accidental errors of observation." Two measurements of the same quantity made by the same observer, with the same instrument, and to all appearances under the same conditions, will, in the great majority of cases, differ from each other by an appreciable amount. Such discrepancies are entirely accidental, and a cause for the disagreement in the two results can in no case be assigned. The discussion of these errors can therefore only be undertaken with the aid of the theory of probabilities, and numerous treatises have in fact been written which deal with the "theory of errors" and the " method of least squares."* The principal results of such discussion, in so far as they have an application to physical measurements, will be briefly stated here. * Merriman, Method of Least Squares; Violle, Cours de Physique (see Introduc- tion to vol. i) ; Weinstein, Handbuch der Physikalischen Maassbestimmungen, vol. I ; Holman's Precision of Physical Measurements; also, for brief discussions, Kohlrausch, Physical Measurements, and the Appendix to Stewart and Gee, vol. I. INTRODUCTION. 15 Probable Error, etc. If a large number of independent* measurements of the same quantity are made, it is evident that one result is as likely to be correct as any other. As a matter of fact, all of the results are doubtless in error. It is also evident that the most probable value of the result sought will be found by taking the average of all the values found. This average will probably be more correct than any one of the single determinations. For this reason it is always advisable to repeat a determination a number of times when the condi- tions are such as to make this possible. If a series of independent observations has been taken under favorable conditions and by a skillful observer, so that the indi- vidual results do not differ greatly from one another, it is obvious that the average has greater probable accuracy than if the con- ditions had been unfavorable so that the individual results showed a wide divergence among themselves. From an inspec- tion of a series of determinations we may therefore form an esti- mate of the probable accuracy of the average. In order to express this estimate numerically the term " Probable Error " has been introduced, which is defined as follows : The probable error of a result is a quantity e such that the probability that the actual error is greater than e is the same as the probability that the actual error is less than e. f A result whose probable error is small is thus in all proba- bility more accurate than one whose probable error is large. The probable reliability of a result is often indicated by writing the probable error with the sign after the result itself: e.g. 7=27.36 0.21. * Too much stress cannot be laid on the condition that the observations must be independent ; i.e. the observer must be entirely uninfluenced by results previously obtained, or by his own opinion as to what the result " ought " to be. The avoidance of this bias in making a series of readings of the same quantity is one of the most difficult things which an observer has to learn. f The name " probable error " is an unfortunate one and is apt to lead to confu- sion. That the probable error of a result is e does not mean that the result is proba- bly in error by this amount. i6 JUNIOR COURSE IN GENERAL PHYSICS. If another series of measurements of the same quantity gave the result /= 27.51 0.38, it is clear that the first result is more reliable. If a series of observations a v a 2 , ..., a n has been taken, the average being a, then the probable error of the average may be shown to be * e = (-!) It may happen that it is desired to determine the probable accuracy of the result obtained from a single reading. The probable error of a single observation is given by the formula = o.6 7 449\/ (a w n I (8) That is to say, if a single observation of the quantity in question is made, the error is as likely to be greater than e 1 as it is to be less. As an example of the computation of the probable error we may consider the following case where ten independent settings are made with a spherometer on the same surface. (See Exp. Aj.) Readings of micrometer. Deviation (d) from the mean. * 3.445 mm. 0.001 0.00000 1 3.448 + 0.002 0.000004 3-442 O.OO4 0.000016 3-45 + 0.004 0.000016 3-45 * + 0.005 0.000025 3-444 0.002 0.000004 3.446 o.ooo o.oooooo 3-442 0.004 0.000016 3-445 0.00 1 O.OOOOOI 3-447 + o.ob i 0.00000 1 Mean 3.446 2d 2 = 0.000094 Probable error of the mean e = .0007 Probable error of single observation e' .0022 * For the derivation of this formula, see any text-book of least squares. It is to be observed that the computation of the probable error has no signifi- cance unless n is large. Unless at least ten observations have been taken, it is useless to compute e. INTRODUCTION. ! 7 On account of the annoyance in computing the probable error, the "average deviation" is often used instead; i.e. the average (disregarding signs) of the deviations of the individual observations from the mean. It is to be observed that the probable error affords no means of estimating the so-called " constant errors " that are caused by improper methods of measurement or by imperfections in the instruments used. These may be very large even when the "probable error" is quite small. The use of the "probable error" may be looked upon as merely an arbitrary means of showing at a glance how closely the individual observations have agreed among themselves, and it indicates, therefore, to what extent the accidental errors of observation have been eliminated. Assignment of Weights in taking an Average. When the same quantity has been measured by several different methods, the results will in general differ, and it is often desirable to combine all the results by taking an average. In such cases "weights" should be assigned to the different determinations in accordance with their probable accuracy. The theory of probabilities shows that in taking an average, each quantity should be given a weight equal to the reciprocal of the square of its probable error ; i.e. if the various values determined by the different methods are A v A 2 , A z , etc., the probable errors being, respectively, e lt e 2 , e B , etc., the most probable value of the quantity in question, as determined from all of the observa- tions, is (9) In Exp. Aj, for example, the length / of one side of the triangle formed by the three legs of the spherometer may be determined in several different ways. Let the result obtained by one method be /= 6.1 2 cm 0.03, while that determined by another and less accurate IS JUNIOR COURSE IN GENERAL PHYSICS. method is /= 6.2O cm o.i i. It is certainly not right to use the average .12 + 20 ( == 6 tI 6), for much more reliance can be placed on the 2 first result than on the second. According to the rule above stated the most probable value of / is 1 i (o.03) 2 (o.n) 2 Influence of the Errors of Observation upon Derived Results. It often happens that the final result sought must be computed from the observations themselves by substitution in some formula. In such cases it is of importance to know how the final result will be influenced by possible errors in the indi- vidual observations. If an error in one of the quantities involved will produce a large error in the result, then this quantity must be observed with especial care. On the other hand, if an error in another of the observed quantities has only a slight influence on the result, it is needless to occupy one's time in measuring this quantity with a high degree of refine- ment. By considering this question before the actual meas- urements are begun, it is thus possible not only to obtain better final results, but also to save time in the observations themselves. The general case may be discussed as follows : Let the result R, which is sought, be some function < of the quantities to be observed ; i.e. R = (x,y, 2, ). Now if x, y, 2, etc., are measured with absolute accuracy, R will be correct. But if one of the quantities x is in error by the amount e, then an error E x will be introduced into the result, and E x = (x+e lt y, z, ) $ (x,y, z, ). (11) INTRODUCTION. ! 9 >ince e will in general be quite small in comparison with x, no great inaccuracy will be introduced by treating it as an infinitesimal ; i.e. neglecting powers higher than the first : -T-t 7- d , / \ Then E x = e 1 -xy is therefore not equal to 2.r2y. 24 JUNIOR COURSE IN GENERAL PHYSICS. In the general case, where the relation between x and y is expressed by an equation of any form, the method of pro- cedure is the same as that illustrated by the example above. Let y = $(x, a, b, c, ), where a, b, c, etc., are the constants to be determined. The observed values of y are then y lt j/ 2 , y s , etc., while the computed values are y, j 2 ', etc. The principle of least squares requires that the sum / i~J/) 2 +(j2~~-7 / 2 / ) 2 H shall be a minimum ; i.e. [.7i-0(*i, a, b, c, )] 2 +|>2-(x, a, ft c, )] 2 = a minimum. *( )] 2 =o, (27) ~2Lr- should be the result of an independent attempt to obtain an exact setting. EXPERIMENT A 2 . Adjustment of a cathetometer, and deter- mination of the sensitiveness of the level. A general description of the cathetometer will be found in any text-book of physics. The various adjustments should be made in the following order. I. To make the level parallel to the axis of the telescope. Adjust the vertical column and bring the bubble to the middle of the level tube ; then reverse the telescope in its Y's. If the bubble settles away from the middle, it must be brought back, partly by the screws that attach the level to the telescope, and partly by changing the direction of the telescope itself. Repeat until the bubble remains in the middle of the tube when the telescope is reversed. II. To adjust the telescope to a right angle with the colttmn, and to make the column vertical. Unclamp the column and turn it till the telescope is parallel to the line joining two of the leveling screws at the base ; bring the bubble to the middle of the tube, and then turn the column through 180. If the bub- ble moves from the center, it must be brought back, partly by the screw that adjusts the angle between telescope and column, and partly by the leveling screws at the base, using only the two to which the telescope is parallel. Turn now through 90, and adjust the third leveling screw. 'Turn back to the first position, and repeat the adjustments till the bubble will remain in the middle of the tube for the entire revolution. III. To adjust the line of collimation. Bring the point ; crossing of the spider lines exactly upon some well-define 30 JUNIOR COURSE IN GENERAL PHYSICS. . point and turn the telescope upon its axis. If the spider lines move away from the point, they must be brought back, partly by the small screws in the ring near the eye-piece, and partly by moving the telescope. Repeat until the point of intersection of the spider lines remains fixed while the telescope is rotated. IV. To determine the angular value of one division of the level. Incline the telescope as far as possible, at the same time mak- ing sure that the position of both ends of the bubble can be read on the scale ; raise or lower the telescope till the inter- section of the spider lines coincides with some well-defined point, and take the reading on the vertical scale ; then incline the telescope in the other direction, and raise or lower it till the spider lines again coincide with the fixed point ; read the ver- tical scale, and measure the distance from it to the point. This may be taken as the radius of a circle, of which the difference in readings upon the vertical column may be considered as an arc. The angle subtended by it may then be computed. This, divided by the number of divisions through which the bubble has moved, is the angle sought. EXPERIMENT A 3 . Calibration of a thermometer tube. The object of this experiment is to determine how completely the variations in the bore of a thermometer tube have been corrected by the graduation of its scale. The experiment is also of value in affording practice in the use of the dividing engine. As usually employed in the Physical Laboratory, the divid- ing engine is merely an instrument for the accurate measure- ment of lengths. The more or less complicated modifications which make it possible to use the dividing engine in construct- ing scales and in ruling diffraction gratings need not be here considered. The essential parts of one of the common forms of dividing LENGTH, TIME, AND MASS. 3! engine are shown in Fig. 5. The most important part of the instrument is the carefully constructed screw, which extends the whole length of the " engine." The figure shows only a portion of the screw and bed. The reading microscope m is attached to a carriage which rests upon a nut fitting this screw. By the rotation of the latter, the carriage can be moved through any distance within the range of the instrument. If the pitch of the screw is known, the distance through which the micro- scope has been moved can be computed from the number of rotations of the screw. A large divided circle at one end enables fractional parts of a complete revolution to be measured. Fig. 5. In using the engine, the object whose length is to be measured is placed upon the massive support underneath the reading microscope, and in such a position that the line to be measured is parallel with the screw. The microscope is then moved until the intersection of the cross-hairs is directly above one end of this line. After the reading of the divided circle has been recorded, the microscope is again moved until the other end of the line to be measured lies directly below the cross-hairs. From the number of turns of the screw necessary to accomplish this the length is computed. The chief source of inaccuracy in the use of the dividing engine is the "lost motion " between nut and screw. To avoid errors arising from this source, the screw should be turned during each measure- ment always in the same direction. If the microscope has by accident been carried too far, do not attempt to correct this by 32 JUNIOR COURSE IN GENERAL PHYSICS. moving the carriage backwards, but begin the measurement again. No matter how carefully the screw of a dividing engine has been cut, it is impossible to obtain one that is perfect. For the most accurate determinations the screw must therefore be calibrated ; i.e. the pitch must be determined at different points along the screw by comparison with a standard scale. In most of the experiments which follow, the results will, however, be sufficiently accurate if the errors of the screw are neglected.* The method of the experiment is as follows : (1) Invert the thermometer, and allow a portion of the mercury to run into the bulb at the upper end of the tube. (2) Separate from the rest a thread of mercury whose length is about one-tenth as great as that of the whole tube. Assuming that a 40 thermometer is used, this thread will be about 4 long. (3) Let the end of this thread be at the 40 mark, and measure its length on the dividing engine ; then by gently jarring the thermometer while in a slanting position, move the thread until the end that was at 40 arrives at 36. Having measured its length in this position, move the thread through another 4, and continue in this way until it has reached the bottom of the tube. A curve is to be platted from the results of these measure- ments in which the positions of the middle point of the thread are used as abscissas, and the reciprocals of its length as ordi- nates. The reciprocal of the length of the thread is obviously proportional to the average cross-section of the tube at the place where the thread is measured. This curve, therefore, shows the relative values of the cross-section at different points along the tube. Fig. 6 (I) shows such a curve. * For more detailed description of the dividing engine, see Anthony and Brackett; also Stewart and Gee, vol. I. LENGTH, TIME, AND MASS. 33 (4) Measure the length of each four-degree space on the dividing engine. The product of the length of a four-degree space by the reciprocal of the length of the thread of mercury in this space will be a quantity proportional to the volume of the space. If the thermometer is accurately graduated, this product should be a constant in all parts of the tube. Fig. 6 (II) shows the actual result attained in the graduation of a fine thermometer. From the above measurements, the error in graduation at any point may be determined in the following manner : Suppose .370 .360 .350 .340 ~ I Cross-section of bore .970 .960 -.01 - II Relative volumes of 4 Spaces: -j- 5 - l n 12 16 20 24 28 CALIBRATION CURVES OF A THERMOMETER Fig. 6. 32 36 that the range of the thermometer is from o to 40. Let v be the volume of a thread of mercury which is very nearly equal in length to a four-degree space. Let / x , / 2 , , / 10 be the measured length of this thread when its mid-point is at the two-degree mark, six-degree mark, and so on. Let L lf L 2 , , Z 10 be the measured length of ist, 2d, , loth four-degree space. len we shall have for the mean cross-section of the th four- Jgree space Sn = j> (30) VOL. I I ) 34 JUNIOR COURSE IN GENERAL PHYSICS, and for the volume of the nth four-degree space r. = v -j* (3D Now the error of any graduation is a cumulative one; i.e. it depends on all the errors preceding it. The volume up to the end of the nth four-degree space is = ^2^, (32) ^n and the total volume is The volume up to the end of the nth four-degree space should be of the whole volume. Therefore the volume error at the 10 end of the nth four-degree space is E ^\L L (33) 10 1 4 i The error in degrees will be L * ~r~ 10 1 4 Finally a curve should be platted (See Fig. 6, III), with values of e n as ordinates, and with corresponding values of n as abscissas. EXPERIMENT A 4 . Determinations of volumes and densities of solids by measurement of their dimensions. I. Determination of the volume of a regular solid by measure- ment of its dimensions. If the solid is a parallelepiped, measure each of its twelve edges on the dividing engine. If it is a cylinder, measure its altitude in four places, and measure the diameter of each base in four different places. In each case great care should LENGTH, TIME, AND MASS. 35 be taken that the microscope moves parallel to the line measured. From the data obtained compute the volume. If the solid proves to be pyramidal or conical, treat it as a frustum. As a check upon the result, weigh the solid in air and in water. The difference of these weights, in grams, is numerically equal to the mass of the displaced water, and this quantity divided by the density of water at the observed temperature will give the volume of the solid. In weighing in water, free the solid from air bubbles, and correct for the weight of the suspending wire. More accurate results may be obtained by correcting for the buoyancy of the air. (See Exp. G 3 .) II. Determination of the volume and density of a wire, from measurements of length, diameter, etc. If the wire is insulated, it should first be carefully stripped in such a way as not to scratch the surface or change the shape of the cross-section. Then measure the diameter, at ten or twelve different points throughout the length, with a micrometer wire gauge. Before using the micrometer, its zero point should be tested ; if it is found to be incorrect, a suitable correction must be made to each reading. Measure the length of the wire as accurately as possible, and compute its volume, treating it as a cylinder whose diameter is the mean of the diameters measured. (If, however, the diameter is found to decrease progressively from one end to the other, the wire should be treated as the frustum of a cone.) Finally, weigh the wire and compute its density. Check the last result by determining the specific gravity by weighing in water. (See Exp. G r ) III. Measurement of the diameter of a zvire by the microscope, and determination of density from diameter, length, and mass. First determine the value in millimeters of one division of the micrometer eye-piece. To do this, focus the microscope on JUNIOR COURSE IN GENERAL PHYSICS. an accurate scale, and observe how many divisions of the scale are covered by any convenient number of micrometer divisions. Measure the diameter of the wire at ten or twelve different points, by means of the micrometer eye-piece, and then compute the volume and density of the wire as in II, above. Check the result by finding the specific gravity as directed in Exp. Gj. EXPERIMENT A 5 . Determination of the time of a periodic motion by the method of middle elongations. The method illustrated by this experiment affords a means of determining the vibration period of any vibrating body with great accuracy. It is used, for example, in determining the time of vibration of the sus- pended magnet of a magnetometer, in deter- mining the period of a pendulum, etc. With the object of affording practice in the use of the method, the apparatus is arranged as de- scribed below : A heavy disk (Fig. 7), having a black spot or a pencil line on the edge, is suspended by a long wire, and is kept in vibration, when once started, by the torsion of the wire. Place a telescope in some convenient position near a clock, and adjust it so that the vertical cross- hair is in the prolongation of the wire. The black spot will then move back and forth in the field, passing the cross-hair twice in each vibration. Note the time of day (hour, minute, second, and tenth of a second) of each passage of the spot across the hair, for ten successive transits. To obtain the time accurately, ol> Fig. 7. -Disk for Tor- serve the second hand of the clock and count sional Vibrations. seconds as indicated by it. Continue the count while observing the transit, looking occasionally at the clock to LENGTH, TIME, AND MASS. 37 see that no mistake is made. In most cases the time of transit will not correspond exactly to the beginning of a second. Observe the position of the spot at the second just before, and again at the second just after the transit : from the relative distances of these two positions from the cross-hair the tenths of a second can be estimated. This will doubtless at first be somewhat dif- ficult, but after a little practice the estimation can be made with considerable accuracy. An experienced observer should be able to estimate twentieths of a second with certainty. Repeat the ten readings mentioned above at intervals of about fifteen min- utes until three sets have been taken of ten observations each. To utilize these data in computing the period in question, add together the fifth and sixth time of transit in each set and divide by two. The result will be the time of the " Middle Elongation," or the time at which the spot was at its greatest distance from the cross-hair between the fifth and sixth transits. If all the observations were correct, the same time of middle elongation would be found by adding together the fourth and seventh, the third and eighth, etc., and in each case dividing by two. In general, however, the five values obtained for the time of middle elongation will differ slightly on account of errors in the observations, and their average should be used. Subtract- ing the time of one middle elongation from that of the next, and dividing by the number of vibrations in the interval, gives the time of vibration with great accuracy. It is not necessary to count the vibrations ; the number may be deduced from the observations themselves. Between the first and ninth, or second and tenth observations of each set, there were four vibrations. Dividing the interval between the first and ninth observations by four, gives an approximation to the periodic time. If the interval between two middle elongations is divided by this quan- tity, the quotient would, if the observations were all exact, be a whole number ; * i.e. the number of vibrations in the interval. * It is to be observed that this quotient might also be a whole number plus a half. This will be the case if the disk moved in opposite directions at the beginning the two sets of observations. 38 JUNIOR COURSE IN GENERAL PHYSICS. It should, with reasonably accurate observations, be near enough to a whole number to leave no doubt as to the true number of vibrations. Dividing the interval by this number gives the periodic time desired. As a check, the time of vibration should also be computed from the interval between the second and third middle elongations. It is to be observed that the interval which it is safe to allow between two sets of observations depends upon the accuracy of the observations, and upon the length of the period to be deter- mined. If the period is short, the interval between two sets of observations must also be short. Determine, from a comparison of your observations, how long an interval would have been safe. . / /^ A "\ \ A &* TAN \ ^ ^ b \tf a \ \ Fig. 11. The object of the experiment is to verify by means of this apparatus the solutions of problems involving the principle of moments. 1. Four forces are given in direction, magnitude, and point of application. It is required to find the magnitude of a fifth force which will produce equilibrium when applied at a given point and acting in a given direction. 2. Four forces are given as above, together with the magni- tude and point of application of a fifth force. It is required to find the direction in which this last force must act in order to produce equilibrium. To verify the results obtained by calculation, fasten the frame by means of a pin in one other point besides the axis, so that it is no longer free to revolve. Then attach the cords to 44 JUNIOR COURSE IN GENERAL PHYSICS. the frame, and adjust the position of the pulleys, so that the forces to which the frame is subjected are as given in the problem. Adjust in like manner the direction and intensity of the force which is to produce equilibrium. Now remove the pin, so that the frame is again free to rotate. If the computa- tions are correct, the position of the frame should remain unal- tered. To prove that the frame is not held in position by friction, it may be displaced through a small angle by the hand. The action of the various forces will then cause it to return, after a few vibrations, to its original position. In the report on this experiment the method of working the problems should be clearly explained. Diagrams of the appa- ratus should also be given, in which are shown the directions and magnitudes of the forces actually used in each test. GROUP C: FRICTION AND SIMPLE MACHINES. Coefficient of friction; (C 2 ) Law of wheel and axle ; (C 3 ) Law of systems of pulleys. EXPERIMENT C r To determine the coefficient of friction between two surfaces. w J 1 \ \\J. \ Fig. 12. Coefficient of Friction. The apparatus, which is shown in Fig. 12, consists (i) of a smooth plate made of one of the materials to be tested and capable of being adjusted so that its upper surface is accu- FRICTION AND SIMPLE MACHINES. 45 rately horizontal ; (2) a small block of the second material in question which can be made to slide across the plate by means of a cord passing over a pulley and loaded with suitable weights. Observations should be taken as follows : First adjust the plate so that its surface is horizontal. Place the block upon it, and add enough weights to make the total pressure five kilograms. Then hang weights on the cord until the force is just sufficient to keerj the block moving uniformly when once started. Repeat the observations with pressures of 10, 15, 20, etc., kilos on the block until a pressure of 50 kilos is reached. It is to be observed that the weights upon the cord do not represent exactly the force required to overcome the friction between plate and block. A correction must be applied in each case on account of the friction of the pulley itself. To determine this correction, a cord may be passed over the pul- ley, carrying equal weights at its two ends. A definite press- ure is thus exerted on the bearings of the pulley, and to overcome the resulting friction, a slight additional weight, whose amount is determined by experiment, must be placed on one side. In this way the relation between the friction of the pulley and the pressure on its bearings can be determined, after which the corrections to be applied to the former observa- tions can be readily computed.* The results may now be best shown by platting a curve on cross-section paper, using pressures, W (Fig. 12) as abscissas and forces necessary to overcome friction, w (Fig. 12) as ordi- nates. If friction is proportional to pressure, this curve should be a straight line passing through the origin. Find its equation by the method of least squares, and so deduce the coefficient. A typical curve of the kind described is shown in Fig. 13. Each of the observations at different pressures should be * It is to be observed that the friction of the pulley is determined by the pressiire on its bearings, and is independent of the direction of this pressure. The weight of the pulley itself is usually so small that it can be neglected. 4 6 JUNIOR COURSE IN GENERAL PHYSICS. independent, and uninfluenced by any assumption as to the probable result. Friction, under the best of conditions, is irregular, so that it need not be at all surprising if the observa- tions are somewhat discordant. The best final results will be obtained by making a number of entirely independent observa- 8.00 i4.00 2.00 FRICTION 10 PRESSURE (W) Fig. 13. tions, each one being as carefully made as though it alone were to determine the coefficient. The same apparatus may be employed to determine the influence of the area of contact upon the coefficient of friction, and also to study the "friction of rest," or "starting friction." EXPERIMENT C 2 . Law of the wheel and axle and deter- mination of efficiency. In this experiment a small weight suspended by a cord from a large wheel is made to lift a larger weight which hangs from the axle of the wheel.* The object of the observations is to * The experiment will perhaps be more instructive if a compound wheel and axle is used, or a compound system consisting of an endless screw and gear wheel. In these cases the influence of friction on the results will be much more marked. FRICTION AND SIMPLE MACHINES. 47 determine experimentally the relation between the two weights when the smaller is just sufficient to keep the system moving. It is to be observed that the conditions differ from those con- sidered in the simple theory of the wheel and axle, in the fact that the friction of the various parts is not negligible. The system forms, in fact, a simple type of machine, whose object we may consider to be the raising of weights. The effect of friction in reducing the efficiency of this simple machine is exactly the same in kind as it is in larger and more complicated machines, and the experiment thus affords an opportunity of studying the influence of friction in a simple case where the various disturbing factors may be readily isolated. Observations are to be taken as follows : Find by experiment the weights necessary to raise loads of 5, 10, 15, up to 50 kilos, the small weight being adjusted in 20 30 LOAD Fig. 14. each case until it is just sufficient to keep the system moving with a slow, uniform motion, when started by the hand. Make several trials with each load and use the mean of the results. It is essential that each observation should be entirely inde- mdent of all the rest, and uninfluenced by any assumption as what the relation should be between "power" and "load." From the data thus obtained, plat curves showing the rela- ion between the power and the load in each case. Fig. 14 lows such a curve. To locate points on these curves (which 4 8 JUNIOR COURSE .IN GENERAL PHYSICS. should be accurately drawn on cross-section paper), the loads are to be used as abscissas and the corresponding powers as ordinates.* From the appearance of the curves decide upon the form of their equations, and find the constants by the method of least squares. The lines represented by the equa- tions that are obtained by least squares should be drawn on the .90 .80 .70 WHEEL AND AXLE CURVE 10 20 30 LOAD Fig. 15. 40 50 same sheet as the original ones, in order to see how closely they represent the observations. Having determined the velocity ratio in each case, show what the behavior of the apparatus would be if there were no friction, and compute the efficiency of the apparatus, considered as a machine for lifting weights, for loads of 5, 10, 25, and 50 kilos. A curve showing the relation between efficiency and load may then be drawn (see Fig. 15). The velocity ratio may be roughly computed from the diameters of the wheel and axle ; but on account of the appre- ciable thickness of the rope used, it is better to obtain the velocity ratio by actually measuring the distance passed over by the load when the wheel is turned a known number of times. * Note that the horizontal and vertical scales need not be the same. See Introduction. FRICTION AND SIMPLE MACHINES. 49 Addenda to the report : (1) Interpret the curves obtained in detail. For example, the friction of the machine consists of two parts : (i) a constant frictional resistance, which is independent of the load; (2) a variable resistance becoming greater as the load increases. Each of these is readily determined from the curve. (2) Indicate the greatest possible efficiency that can be attained by the machine, and the load to which this corresponds. EXPERIMENT C 3 . To determine the efficiency of a system of pulleys. In this experiment a system of pulleys is used by which a small weight moving through a considerable distance is enabled to lift a much larger weight through a comparatively small distance. The objects of the experiment are : (i) To determine experimentally the relation between "power" and "load "for uniform motion ; (2) to determine the efficiency of the system considered as a machine for raising weights. The procedure is as follows : (1) Find by experiment the weights necessary to raise loads of 0.5, i.o, 1.5 , up to 6 kilos, the small weight being adjusted in each case until it is just sufficient to maintain uniform motion when the system is started by the hand. Make several trials with each load, and use the mean of the results. (2) With the data obtained, plat curves showing the relation between power and load for both cases, and from the appear- ance of the curves decide upon the form of their equations. The constants are to be determined by the method of least squares. The lines represented by the equations obtained by least squares should be drawn on the same sheet as the original curves. (3) Having determined the ratio of the distances passed over by the two weights, show what powers would be necessary to raise the same loads if there were no friction, and compute the VOL. I E 50 JUNIOR COURSE IN GENERAL PHYSICS. efficiencies of the two systems for loads of 0.5, i, 3, and 6 kilos. The results are to be discussed in the manner explained in the previous experiment. GROUP D: UNIFORMLY ACCELERATED MOTION. (Dj) Atwood's machine. (D 2 ) Determination of gravity from the motion of a freely falling body. EXPERIMENT D r Atwood's machine. In Atwood's machine a vertical standard, from two to three meters high, carries at the top a light pulley, P (Fig. 16), which is mounted in such a way as to make the friction of its bearings as small as possible. To the standard is attached a scale grad- uated in centimeters or inches for conven- ience in measurement. Over the pulley hangs a light silken cord, to which weights, w lt w v may be hung. If equal weights are hung on the two sides of the pulley, it is evident that the system will remain at rest. But if a small additional weight be placed on one side, the condition of equi- librium will be destroyed, and the heavier side will begin to fall with a uniformly accelerated motion. The force of gravity acting on the small added mass, or "rider," *- (Fig. 17), is thus utilized to set in motion a much larger mass, and the acceleration is, in consequence, smaller than if the rider alone were moved. By suitably choosing 16 the various weights, the motion may be made so slow that the velocity can be readily measured. The apparatus thus affords a means of illustrating the laws of uni- UNIFORMLY ACCELERATED MOTION. formly accelerated motion, and can also be used, as explained below, to determine the acceleration of gravity, g. For convenience in measuring time, most forms of Atwood's machine are provided with an electric bell or sounder, which can be connected with a seconds pendulum. By ; means of an electromagnet, m (Fig. 16), placed at the top of the vertical standard, and con- nected with the same circuit as the sounder, the weights may be released exactly at the beginning of a second, so that the necessity of estimating fractions of a second is avoided. A bracket, s s (Fig. 16), movable along the upright standard, may be adjusted so as to stop the fall at any point desired, while a ring, s 2 , also adjustable in position, serves to remove the rider at any desired time without disturbing the motion of the weights them- selves. Fig. 17. I. To test the laws of uniformly accelerated motion. Hang equal weights on the two sides of the pulley, and then put enough additional weight on the side which is to fall during the experiment to overcome the friction of the apparatus. This can be done by adding small pieces of paper or tin-foil until the weight will continue to move uniformly downward when once started. When this adjustment is completed, place the rider in position, and adjust the ring by trial to such a position on the vertical bar that it will remove the added weight after a fall of exactly two seconds. Measure the distance traversed by the rider and record it, together with the time of fall. To deter- mine the velocity acquired, adjust the bracket to such a position that the space between it and the ring shall be traversed in some exact number of seconds. This distance between ring and table being measured, the velocity can be computed. To insure JUNIOR COURSE IN GENERAL PHYSICS. accuracy, each of these observations should be repeated several times and the average of the results used. Now shift the posi- tion of the ring until the time of fall is three seconds ; then four L. ATWOOD'S MACHINE 20cm. 40cm. 60cm. 80cm. 100cm. SPACE TRAVERSED Fig. 18. seconds ; and so on, repeating the observations described above in each case. The results can be best shown by platting two curves, one / / / / / / / / / z 10cm.20 30 40 50 60 70 80 90 10 VELOCITY ACQUIRED Fig. 19. showing the relation between time of fall and the space trav- ersed (Fig. 1 8), the other showing the relation between time of UNIFORMLY ACCELERATED MOTION. 53 fall and the velocity acquired (Fig. 19). Discuss the results and show whether or not they are in agreement with the laws of uniformly accelerated motion. II. To use Atwood's machine for the determination of g. If the mass of the rider is m, the resultant force acting on the system is mg. This force is equal to the product of the total mass moved into the acceleration imparted. If, therefore, the total mass except the rider be denoted by M, and the measured acceleration by a, we have mg=(m+M)a; (34) g can therefore be computed as soon as m, M, and a are known. The mass m can be at once determined by weighing, while a can be computed from the observations. But the value of M cannot be so simply obtained, since the pulley itself forms a part of the mass set in motion. The " equivalent mass" of the pulley must therefore be first determined. To accomplish this, proceed as follows : (1) Remove one of the small weights from each side of the cord ; adjust again with tin-foil to overcome friction, and deter- mine by experiment the spaces corresponding to falls of two, three, four, etc., seconds, respectively. (2) Determine also the mass upon the cord. (3) Repeat the observations after removing another weight from each side, and continue until only one weight remains. From these observations, compute the acceleration imparted by the rider in each case. Since the equivalent mass of the pulley is known to be a constant, it may now be readily com- puted, either algebraically or graphically. The graphical method which follows is, however, recommended. Plat a curve (see Fig. 20) upon cross-section paper in which the masses hung upon the pulley are used as abscissas, and the 54 JUNIOR COURSE IN GENERAL PHYSICS. reciprocals of the corresponding accelerations as ordinates. This curve should, if the observations are good, be nearly a straight line. The equation of the line is, in fact, (35) where M denotes the constant equivalent mass of the pulley, and m + M the sum of the masses hung from the cord. The co-ordinates of the curve are therefore x= m + M and jj> = - ; i.e. a (36) RECIPROCAL OF ACCELERATION 8 S 8 fe 8 8 3 S 8 ^ ATWi )OD'J MAC MINE / / / / X / / / s / / / / / / x^ aoo 300 MASSES Fig. 20. 300 400 500 This is an equation of the first degree, and therefore repre- sents a straight line. Owing to errors of observation, the curve obtained will not be exactly straight. A straight line should, however, be drawn which passes as nearly as possible through all the points platted. A little consideration will show that the intercept of this line on the axis of abscissas is equal to the equivalent mass of the pulley. The computation of g can now be easily performed. UNIFORMLY ACCELERATED MOTION. 55 It may be readily proved that what has been called the equiva- lent mass of the pulley is really its moment of inertia divided by the square of the distance from its center to the cord. The work done by gravity when the rider has moved a distance, /, is mgl, but this work must be equal to the kinetic energy gained. .'. mlg=(m+M)v* + K, this equation reduces to (38) in which r is the radius of the pulley. EXPERIMENT D 2 . Determination of g from the motion of a freely falling body. The apparatus for this experiment, Fig. 21, is so arranged that a piece of smoked glass may be allowed to fall freely in Fig. 21. front of a vibrating tuning-fork of known pitch. A stylus attached to one prong of the fork is adjusted to trace a sinuous line on the glass as it falls. By measuring the length of the successive waves of this curve, it is possible to compute the acceleration of gravity. As a means of measuring g t the method is not at all accurate, since any friction in the apparatus will introduce a considerable error. The experiment is valuable, however, in illustrating the laws of falling bodies, and in familiar- 56 JUNIOR COURSE IN GENERAL PHYSICS. izing the student with the use of the dividing engine as an in- strument for measuring length. Having covered the glass with a thin layer of smoke (pref- erably from burning camphor), adjust the stylus until it traces a smooth and distinct curve when the glass is allowed to fall. Several trials may be necessary before this adjust- ment is satisfactory. When a good curve has been obtained, stop the vibration of the fork, and allow the glass to fall a* ) second time without changing the position of the glass. The stylus will then be made to trace a straight line ) nearly through the center of the sinuous curve. (See Fig. 22.) Now adjust the glass under the microscope of the ) dividing engine, assume some sharply defined intersection of the straight line and curve as a starting-point, and measure the distance from this to the third, fifth, seventh, ) etc., intersection. These distances evidently represent the spaces passed over during one, two, three, etc., com- ( plete vibrations of the fork. It is best not to start with the beginning of the curve, since the line may be more or less blurred and irregular in this region. From these measurements, the acceleration of gravity can be determined in the following manner : Let V Q be the velocity with which the falling body passed the point < > of the sinuous curve taken as origin. Let L be the dis- (46) Fig. 25. Moment of inertia of a cylinder about any axis perpendicular to its geometrical axis. Let L be the length of the cylinder, a the radius, 8 the density, and h (Fig. 26) the distance of the axis from one end of the cylinder. Taking as element of mass a lamina of thick- ness dx, we have MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 6l From equations 42 and 46 we have for the moment of inertia of this lamina, with respect to the axis a, dK*=x*dM+ - dM. (47) 4 *& K /*&-* ,T^x>4 /*&-* = TT^B I x *dx + I dx, J-h J-h and K a =M-hL+k*+ (48) L3 4j Two particular cases are of especial importance ; namely, when /& = o and when h = L. o dx h \ L-7i Fig. 26. Simple harmonic motion. An oscillating body is said to have simple harmonic motion when its distance, either linear or angular, from a fixed position is a simple harmonic function of the time of either of the forms, (49) in which A, p, and t Q are constants. The distance of the body, either linear or angular, from the fixed position is called its displacement. The maximum displacement occurs when the cosine becomes unity. This maximum displacement is called the amplitude of the simple harmonic motion. In the above equation t is a constant in- terval of time. This constant is obviously zero if time be reckoned from the instant when the displacement is a maximum in the positive direction, using the first of equations 49. When the time t has increased to the value , the displacement x is 62 JUNIOR COURSE IN GENERAL PHYSICS. exactly equal to what it was at the instant, to. Moreover, at any time, t^ the displacement has the same value that it had at The constant interval of time P the time 2 ' Irn during which the displacement takes all possible values, and the motion begins to repeat itself, is called the period of the simple harmonic motion. It is usually represented by T or 7! Simple harmonic motion of translation. The rectangular projection of uniform circular motion upon any diameter is simple harmonic motion ; i.e. if the point N (Fig. 27) revolves about a circle with a uniform velocity, the point P will move along the diameter BC with simple harmonic motion. Let A be the radius of the circle, let p be the constant angular velocity of the radius ON, and suppose time to be reckoned from the instant that the point P leaves the right- hand end of the diameter BC; then at any time, t, we shall have * x=A cos 13= A cos//. (50) The distance, x, is the displace- B ment, and A the amplitude, of the simple harmonic motion. If the point P moves through the distance dx in the time dt, we have, from the definition of velocity, ^=-/^sin// dt Fig. 27. (so If the velocity of the point P changes by the amount dv in the time dt, we have, from the definition of acceleration, tf= = dt (52). * It is to be understood that these equations hold for any simple harmonic motion, that the circle is an auxiliary circle, and that the motion of N is only to aid in understanding the real motion, which is along BC. MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 63 Substituting for A cos// its value from equation 50, we have = -/** (53) Equation 53 shows that the acceleration of a point whose motion is simple harmonic is at any instant proportional to its displacement from the ' mid-point. The negative sign shows that the acceleration is always directed oppositely to the dis- placement ; i.e. when the point is at the right of the mid-point, its acceleration is directed towards the left, while the reverse is true when the point is on the left. Multiplying both sides of equation 53 by the mass of the moving point, and remembering the dynamical equation FMa, we have F=Ma=-Mp*x. (54) This is a dynamical equation, and shows that the force which produces the acceleration of the mass M in simple harmonic motion is directed towards the mid-point and is proportional to the displacement. Conversely, it may be proved that whenever the resultant force acting on a body is proportional to its displacement from a fixed position, the motion of the body will be simple harmonic. From the equations 50, 51, and 52, it follows that the dis- placements, velocities, and accelerations of the point P begin to repeat themselves when t has increased from o to - This ? TT constant value of the time is the period of the simple har- monic motion ; and is obviously the same as the time required for the point TV 7 " to revolve about the auxiliary circle. Substituting Tior , equations 50, 51, and 52 now become P x=Acos^J, (55) "=-sin^/, (56) cost. (57) 6 4 JUNIOR COURSE IN GENERAL PHYSICS. It is often especially desirable to know the velocity with which the moving body passes the mid-point. It will first pass the mid-point after one quarter of a period has elapsed, and it will pass the same point for every odd number of quarter periods. Substituting for t in the equation for velocity any of the values \T,\T,\T> , we have Simple harmonic motion of rotation. Let M be a body oscillating about an axis O (Fig. 28) per- pendicular to the plane of the paper. The line OA, fixed in Fig. 28. the body, oscillates between the extreme positions OA' and OA". The motion of the body will be simple harmonic motion, according to the definition above given, if we have at any time /, < = Scos/A (59) S and/ being constants. If d$ be the angle turned through during the time dt> we shall have, from the definition of angular velocity, <> .*. w = -f- = po sin pt. at (60) MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 65 If da) is the change of angular velocity in the time dt t we shall have, from the definition of angular acceleration, (6 1) at Substituting the value of 8 cos// from 59, we have = -pt, (62) from which it follows that in simple harmonic motion of rotation the angular acceleration at any instant is proportional to the angular displacement. Multiplying both sides of equation 62 by the moment of inertia of the rotating body with respect to the axis of rotation, we have G=Ka=-Kp*. (63) Since, however, Ka is equal to the resultant moment, with respect to the axis of rotation, of the forces acting on the body, it follows that the moment of the force producing the angular acceleration in simple harmonic motion is directly proportional to the angular displacement. Conversely, it may be proved that if the resultant moment of the forces acting on a body with respect to the axis of rota- tion is proportional to the angular displacement from a fixed position, the resulting motion of the body will be a simple harmonic motion. Here, as in simple harmonic motion of translation, the motion begins to repeat itself in all respects after a time P has elapsed. This constant time is the period of the simple harmonic motion, and, calling it T, equations 59, 60, and 61 become = 5cos/ f (64) (65) (66) VOL. I F ^^, Of THE 66 JUNIOR COURSE IN GENERAL PHYSICS. If &) be the angular velocity with which the body passes its mid-position, we have, in a manner similar to equation 58, "o"*^*- (67) Examples of simple harmonic motion of translation : (1) If a mass be suspended by a spiral spring, it will oscillate along a vertical line with simple harmonic motion, if it is first displaced upwards or downwards from its position of equilibrium, and then set free. (2) Any molecule in a sounding body or a sound-wave, when the sound is absolutely simple, i.e. without harmonics or overtones. (3) The bob of a simple pendulum, or any point in a compound pendulum, when the arc of vibration is very small. (4) Any point in a magnet vibrating in a uniform magnetic field when the arc of vibration is very small. Examples of simple harmonic motion of rotation : (1) A mass suspended by a wire or cord, and rotating about a vertical axis, the only force acting being the force of torsion. (2) A compound pendulum when the arc of vibration is very small. (3) A magnet vibrating in a uniform magnetic field when the arc of vibration is very small. In these examples, as well as in all other cases, there are certain retarding forces, as friction, imperfect elasticity, or induced currents of electricity, which prevent the motion from being absolutely simple harmonic. This " damping," as it is called, has an extremely small effect upon the period of the simple harmonic motion, and may be safely neglected when the period is the quantity desired. When the amplitude of the simple harmonic motion is the quantity to be used, a correction for " damping" must generally be introduced.* * See Nichols, The Galvanometer, Lecture 3; also Stewart and Gee, Vol. 2, p. 364 et seq. MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 67 EXPERIMENT E r Determination of g by the physical pen- dulum. If a physical pendulum be displaced from its position of equilibrium through an angle so small that the angle may be substituted for its sine without appreciable error, the moment of the force acting on the pendulum will be proportional to the angular displacement. The pendulum must therefore have simple harmonic motion. From the principle of the conservation of energy, in any transformation, the two forms of energy must be equal to each other. As the energy dissipated in a single swing of the pendulum is small enough to be negligible, we are justified in equating the kinetic energy of the pendulum when at its lowest point to the gain in potential energy when it reaches its highest point. The kinetic energy of a rotating body is ^ K a (o 2 . Since the pendulum has simple harmonic motion, the angular velocity at the mid-position will be -~r- (See equation 67.) The potential energy at the highest point is equal to the work required to turn the pendulum through the angle from the lowest point. This work is equal to the average moment multiplied by the angular distance moved, or, Since E K =E P , we have in which T is the period of a complete oscillation, K a the moment of inertia of the pendulum with respect to the axis of suspension, M its mass, R the distance of the center of 68 JUNIOR COURSE IN GENERAL PHYSICS. gravity from the axis of suspension, and g the acceleration of gravity. If T and R be observed, and K a computed, the acceleration of gravity may be determined. In this experiment a uniform bar of metal, provided with an adjustable pair of knife-edges (Fig. 29), is to be used as a pendulum. The method of procedure is as follows : (1) Fasten the knife-edges firmly at some point not at the end of the bar, and set the pendulum to vibrating through a small arc. (2) Determine the time required for the pendulum to make some large number of oscillations. (3) From the result compute the period of the pendulum. An ordinary watch or clock may be used for deter- mining this time, although a stop- watch is better. In any case, several determinations of the period should be made, in each of which the time is at least five or six minutes. (4) After the period has been determined, measure the dimensions of the bar and the distance of the knife-edges from one end. From these data, the moment of inertia can be computed. As the bar is homogeneous, the center of gravity will be at the center of the figure, and thus R is known. M will- be found to cancel out ; consequently g may be computed. The knife-edges and clamp slightly affect the moment of inertia and the center of gravity of the pendulum, thus slightly changing the period. If greater accuracy is desired, the effect of the knife-edges and clamp on the period may be made zero by fastening to the knife-edges an auxiliary mass, a portion of which extends above the axis of suspension, and varying the center of gravity of this mass until the period of vibration of Fig. 29. MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 69 the knife-edges without the bar is approximately the same as the period of the bar pendulum. The observations and results of the experiment are to be tabulated in the manner indicated below : GRAVITY BY THE PHYSICAL PENDULUM. No. of Transit to Right. Time. Duration of 100 Oscillations. Dimensions of Pendulum and Results. hr. min. sec. Distance of axis from upper I 27 oo end of bar, 3.5 cm. IOI 30 24 204S. Length of bar, L= 159.6 cm. 201 33 49 20 5 Diameter of bar, zr 1.6 cm. 3 OI 37 13 204 Distance of axis from center 4OI 40 38 205 1 of gravity of bar, R= 76.3 cm. hr. min. sec. Moment of inertia, K a 7945 M. Periodic time, 7^=2.047 I 42 50 Computed value of gravity, IOI 46 14 204 g 981+ cm. per sec. per sec. 201 49 40 2O6 Most careful determination 3 OI 53 3 203 for Cornell laboratory, 980.28 4OI 56 29 206 Addenda to the report: (1) Explain how the mass of the pendulum cancels so that it does not need to be known. Compute the acceleration of gravity in centimeters per second per second and in feet per second per second. (2) Explain what is meant by the statement, The force of gravity is g dynes. (3) Explain why a small error in the period will make an error relatively twice as great in g. EXPERIMENT E 2 . Determination of g by Kater's pendulum. The equation for the physical pendulum may be put into the form (69) jO JUNIOR COURSE IN GENERAL PHYSICS. in which JT is the moment of inertia of the pendulum with respect to an axis through the center of gravity parallel to the axis of suspension. (See equations 42 and 68.) If the pendu- lum be inverted, and the time of vibration determined for an axis on the opposite side of the center of gravity, a new equation will be obtained similar to the above, except that there will be new values of R and T. Between these two equations K Q may be eliminated and g determined. To this end we substitute / for R l + R 2 in the equation derived by the elimina- tion of KQ from two equations like the above. It will then reduce to the equation for the simple pendulum, provided that the two times of oscillation are the same, and the two values of R are not the same. Kater's pendulum is an apparatus which makes use of this fact. The use of Kater's pendulum depends upon the principle that the center of oscillation and the center of suspension of any pendulum are interchangeable ; i.e. if a pendulum is reversed, so that the point which was the center of oscillation is made the center of suspension, the time of vibration will remain unchanged. The distance between these two points being equal to the length of the corresponding simple pendulum, the measurement of this length, together with the observation of the time of vibration, is sufficient to determine the force of gravity. The experiment consists, therefore, in adjusting the positions of the two knife- edges by trial until the time of vibration about one pair as an axis is the same as that about the other. The pendulum used consists of a hollow cylindrical bar, one end of which is loaded by a filling of lead (Fig. 30).* There are two pairs of knife-edges, one being placed near each end of the bar ; both are capable of adjustment along the bar, so that the distance between them can be altered. * The pendulum here referred to is a very simple one, but with careful observa- tions is capable of giving quite accurate results. A homogeneous cylindrical bar is sometimes used, but with such a pendulum one pair of knife-edges will be at such a point that considerable variation of its position will produce but little change in the time of vibration. (See Exp. E 3 , and Fig. 31.) MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. The method of the experiment is as follows : (1) Fasten one pair of knife-edges to the bar at some point near the end which is not weighted. (2) Determine the rate of vibration roughly by observing with a watch or clock the time occupied by some large number of oscillations (20-50). (3) Locate approximately the posi- tion of the center of oscillation by hang- ing a simple pendulum (a small weight suspended by a cord) near by, and adjusting its length until it vibrates nearly in unison with the bar. The length of this simple pendulum is then a rough approximation to the distance from the center of suspension of the bar to its center of oscillation. To obtain this distance more accurately, set the second pair of knife-edges at a distance from the first equal to the length just determined ; then reverse the bar, and determine its time of vibration as before. The period should now be nearly the same as at first. If the two periods differ, one or both of the knife-edges should be shifted until the time of vibration is very closely the same with either suspension. (4) The final determination of the time of vibration must be made very carefully, and should be repeated several times. (5) Finally, the distance between the knife-edges is to be measured as accurately as possible. From this distance and the two times of vibration, the value of g is to be computed. Determine the value of g in the C. G. S. system, and also in the foot-pound-second system. Fig. 30. JUNIOR COURSE IN GENERAL PHYSICS. EXPERIMENT E 3 . Relation between the time of vibration and the position of the knife-edges in a uniform cylindrical pendulum. In the equation for the physical pendulum given in the preceding experiment, everything is determined for any given pendulum except R and T. The object of this experiment is to show the relation which exists between these two variables. To this end, fasten the knife-edges at one end of the bar, and determine the period of vibration by observing the time occupied by a large number of oscillations. Then shift the 3.6 3.2 O 2.8 a 2 T 2.0 1.6 1O 2O 3O 4O 50 6O 7O SO 90 DISTANCE OF KNIFE-EDGE FROM CENTER OF PENDULUM Fig. 31. knife-edges down the bar three or four centimeters, and deter- mine the new period. Continue shifting the point of suspension and observing the period until the center of the bar is reached. From eight to ten different positions of the knife-edges should be used, and the distance of the knife-edges from one end of the bar should be carefully measured in each case. In deter- mining the time of vibration, a stop-watch will be found of considerable assistance. From the data obtained, plat a curve, similar to that given in Fig. 31, using for abscissas the distances MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 73 of the point of suspension from the center of the bar, and for ordinates the corresponding times of vibration. Discuss and explain the shape of this curve, and determine the form of its equation from a knowledge of the law of the physical pendulum. The following is a tabulated statement of a set of observations taken as indicated above. The results are shown graphically in Fig. 31. RELATION OF PERIODIC TIME TO POSITION OF KNIFE-EDGES. Distance from Center of Gravity. No. of Transit to Right. Time. Duration of IOO Oscillations. Periodic Time. Other Data and Results. hr. min. sec. I 2 16 50 Length of bar =183 cm. 9 IOI 2O 30 2 2O 2.21 Diam. " " =2.5 " 201 I 24 12 2Q OO 222 Equation of curve from 80 IOI J7 32 34 214 2.15 theory of pendulum : 201 36 10 216 2 4 W 22 I 42 oo T^R = - R^ 1 + 70 IOI 45 30 210 2.10 g S 201 I 49 oo 3 22 OO 2IO From pairs of points on 60 IOI 25 27 2O7 2.07 curve having equal ordi- 2O I 28 54 20 7 nates. 55 I IOI 38 25 205 2.O5 T=2.20, -=970; 20 1 4i 5 205 T 2.16, -=989; 5 IOI 5 1 4 55 6 206 2.O6 T=2.i2, Young's modulus may be computed if the quantities on the right of this equation are determined in the proper units. Fasten a wire two or three meters in length to some firm support. A small vice rigidly attached to the brick or stone wall of the room makes a support which is satisfactory in most cases.* Suspend from the end of the wire a weight which is just sufficient to take out the kinks ; for a wire whose diameter is i mm. a weight of from two to four kilos will be required. A horizontal microscope containing, an eye-piece microm- eter is now to be adjusted so that a slight scratch on the wire is sharply focused in the lower part of the field. As the tension of the wire is increased by the addition of weights, this mark will move across the field, and by means of the micrometer the elongation corresponding to each increment in weight can be measured. Measure in this way the elonga- tions produced by successive increments in weight until the mark has passed out of the field. Each increment in weight should be sufficient to cause an elongation of three or four scale divisions. * If there is any reason to suspect that the support is not rigid, two microscopes must be used, one at the upper and the other at the lower end of the wire. 7 6 JUNIOR COURSE IN GENERAL PHYSICS. After the wire has been fully loaded the weight is to be gradually reduced and the measurements repeated until the original load is reached. Note whether equal increments of tension produce equal increments of length, and whether the elastic limit has been passed. The results can be best shown by a curve in which abscissas represent force applied, and ordinates the increments in length produced. Determine the value of one division of the micrometer as described in Exp. A 4 (III), and measure the length and diameter of the wire. Since the square of the radius enters in the above equation, a small error in determining it will be relatively doubled in the computed value of the modulus. For this reason the diameter of the wire must be measured with unusual care. The cross-section may be most accurately determined by com- putation from the density and the mass of a given length. Since the density of various specimens is liable to differ, the density should be carefully determined by weighing in water. EXPERIMENT F 2 . Determination of the moment of torsion of a wire. When a wire of elastic material, such as steel, bronze, or hard drawn copper, is twisted by a moderate amount, the moment of the couple by which it tends to regain its original condition is proportional to the angle of torsion ; i.e. if Q is the angle, and G the moment of the elastic return force, G=G Q 6. The constant G Q is called the moment of torsion, and depends upon the length, diameter, and material of the wire. To determine the value of G Q , a heavy weight, of such shape that its moment of inertia can be readily computed, is hung upon the end of the wire, and set to vibrating through an angle of twenty or thirty degrees. Since the moment of the return force is proportional to the angular displacement, the weight ELASTICITY. jj will have simple harmonic motion, and the vibration will be isochronous. From equation 63 we will have (72) U,l,~ in which G 6 is the resultant moment due to torsional displace- /2/3 ment through an angle 0, and -is the angular acceleration of the suspended weight. An integration of this equation gives (73) in which T is the period of the harmonic motion. The same equation may be derived more easily from the energy relations. If 8 is the maximum angular displacement, the kinetic energy of the rotating weight as it passes the mid- position will be The potential energy of the twisted wire, when the suspended weight is at its greatest displacement, is equal to the work that must be done on the wire to twist the lower end through the angle 8. The moment of the force at any instant to be over- come is G Q ; as this varies between o and G Q 8, the average moment is J G 8, and hence the work done and the potential energy gained is P =G Q 8.8. (75) As the dissipation of energy during a single vibration may be neglected, the potential energy at the extreme, when the weight has no motion, must be equal to the kinetic energy when the weight is at its mid-position and there is no twist in the wire. Hence we have r=27r VJ- < 76) To determine the period of vibration, the method of Exp. A 5 should be used. It is to be observed that since it is the square ;8 JUNIOR COURSE IN GENERAL PHYSICS. of T that appears in the formula, an error in the determination of the period will introduce a considerable error in the result. The moment of inertia is to be computed from the mass and linear dimensions of the vibrating weight. From the result obtained for G compute the force, both in dynes and in pounds, which would twist the wire through a complete revolution when acting at a distance of one centimeter from the center. EXPERIMENT F 3 . To determine the moment of inertia of an irregular body by torsion. The preceding experiment offers one of the best means of determining the moment of inertia of an irregular body. G Q is a constant for any given wire, independent of the mass sus- pended by it and of the period of oscillation. Therefore, if it is already known, the moment of inertia of the suspended weight may be computed after the period has been determined. If G Q is not known, it may be eliminated between two equations of the form (76), in one of which the moment of inertia of the suspended weight is known. To perform the experiment, suspend the body by a wire of phosphor bronze or some other elastic material, the upper end of the wire being rigidly fastened. The axis about which the moment of inertia is required should lie in the prolongation of the wire. Set the system to vibrating, and determine the period as in the previous experiment. Then hang upon the wire a body whose moment of inertia is known, and determine the vibration period as before. If the two periods are T v T 2 , then *2 = *i (77) CHAPTER II. GROUP G: DENSITY. (G) General statements ; (G 1 ) Rough determination of specific gravity by weighing in water ; (G 2 ) Specific gravity of solids and liquids by the specific gravity bottle ; (G 3 ) Determination of density with corrections for air displacement and tem- perature ; (G 4 ) Specific gravity by the Jolly balance; (G 5 ) Nicholson s hydrometer ; (G 6 ) Fahrenheit 's hydrometer ; (G 7 ) Graduation of a hydrometer of variable immersion; (G 8 ) Specific gravity of a solid by means of a variable immer- sion hydrometer ; (G 9 ) Density of a liquid by Hare's method. (G.) General statements concerning specific gravity and density. The specific gravity of a substance is the ratio of the weight of a given volume of the substance to the weight of an equal volume of water at its maximum density. Specific gravities being ratios of like quantities are abstract numbers, and hence the same for all systems of units. The unit used in comparing the weights may indeed be entirely arbitrary, such as the unit extension of a spring made use of in the Jolly balance. The density of a substance is the ratio of the mass of a given volume of the substance to the volume which it occupies ; or in symbols />= (78) Since density is not an abstract number, its numerical value in any particular case must depend upon the units used. For 79 8o JUNIOR COURSE IN GENERAL PHYSICS. example, the density of water in the foot-pound-second system is 62|. The density of water in the C. G. S. system is unity, for the reason that the unit of mass is equal to the mass of a cubic centimeter of water. Hence it follows that the densities of all substances in the C. G. S system are numerically equal to their specific gravities. This is not absolutely true, however, for the mass of a cubic centimeter of water at its maximum density is not exactly a gram. The term " relative density " is sometimes used. It has the same meaning as "specific gravity." EXPERIMENT G r Rough determination of specific gravity by weighing in water. Specific gravity of a body more dense than water. Weigh the body in air; then suspend it from a hook under one of the scale-pans of a balance, immerse it in water, and weigh again. The specific gravity is to be computed from these two weights, no correction being made for the temperature of the water or the buoyancy of the air. The wire used for suspending the body must be quite fine. It should be immersed in water to the same extent that it will be when the body is attached, and balanced with shot or sand, before the second weighing above is made. II. Specific gravity of a body lighter than water. First weigh the body in air. Then suspend a heavy sinker from one scale- pan, and find its weight when immersed in water. Finally attach the body to the sinker, and find the weight of the two when entirely submerged. From these three weights compute the specific gravity. The results should be tested by placing the vessel of water on the scale-pan and suspending the substance in the water from some outside support. The gain in weight of the vessel should be the same as the loss of the substance. DENSITY. 8l Errors in this method of determining specific gravity are apt to arise from small bubbles of air adhering to the substance when immersed ; such bubbles must, therefore, be carefully shaken off before the weighings are made. EXPERIMENT G 2 . Specific gravity of solids and liquids by the specific gravity bottle. The specific gravity bottle is simply a small bottle which is provided with an accurately fitting ground-glass stopper. A very^mall hole through the center of this stopper leads to the interior of the bottle, its object being to allow the bottle to be completely filled with any liquid. To use the specific gravity bottle, proceed as follows : I. Specific gravity of a liquid. First weigh the bottle alone, when perfectly clean and dry. Next fill with distilled water and weigh again. Finally fill the bottle with the liquid whose density is required, and weigh a third time. These three weights are sufficient for the computation of the specific gravity. II. Specific gravity of a solidl Place the substance in the specific gravity bottle and determine the combined weight. Then add sufficient distilled water to entirely fill the bottle, insert the glass stopper, and after wiping off any drops which may adhere to the outside, weigh again. Finally determine the weight of the bottle when filled with water alone. These three weights, together with the weight of the bottle, are sufficient to determine the specific gravity of the substance. This method is of course only available when the substance is insoluble in water. In the case of soluble substances some liquid of known density must be used in which the substance does not dissolve. 1 The specific gravity bottle is especially useful when the solid is in the form of small fragments or powder. VOL. I G 82 JUNIOR COURSE IN GENERAL PHYSICS. It sometimes happens that difficulty is met with in shaking off the small bubbles of air which tend to adhere to the sub- stance, and which will introduce a considerable error. In such cases the bottle containing the substance, and about half full of water, should be placed under the receiver of an air-pump, and the air exhausted until bubbles are no longer formed. If greater accuracy is required, corrections for temperature and air displacement must be made, similar to those described in Exp. G 3 . EXPERIMENT G 3 . Determination of density, with corrections for air displacement and temperature. The balance is nearly always used for comparing masses, but it should be remembered that it is merely a lever with equal arms, by which two forces may be proved to be equal. Each of the two equal forces is the resultant of the weight of the body on the scale-pan acting downwards, and the buoyant effect of the weight of the fluid displaced by the body acting upwards. This gives W^-w^ *F 2 -w/ 2 . Since weights are directly proportional to masses, we have M 1 -m 1 = M 2 -w zt (79) in which M 1 and M 2 are the masses of the bodies on the two scale-pans, and m^ and m^ are the masses of the displaced fluid in the two cases. Nearly always in using the balance, m 1 and m 2 are supposed to be equal, or at least it is assumed that their difference is negligible. ^ If M, is the mass of the substance of density S s , then from the definition of density the volume of the displaced fluid will be . If B a is the density of the displaced air, then its mass S, <> is M,. If M is the mass of the counterpoise, and S c its density, S equation 79 becomes (So) DENSITY. 83 In order to determine M s from the known mass of the counter- poise, Sat &> and S c must be known. Approximate values for these quantities will serve quite as well as more accurate values, because the term in which they appear is always a very small quantity. The object of this experiment is to determine the density of the substance s with all possible accuracy. If the substance is suspended from the scale-beam, so as to be immersed in water of density p, and is then counterpoised with the mass M f , equa- tion 79 becomes M, M a =M' M'^. (81) If equation 80 be divided by 81, and the resulting equation solved for S s , we shall have S s = ^^(p-S a ). (82) In deriving equation 82 it has been assumed (1) That the density of the counterpoise M is the same as that of M'. (2) That the density of the air has not changed between the two weighings. (3) That the density of the substance or of the weights has not been changed during the experiment on account of expan- sion. The value of p depends on the temperature of the water. The value of S a depends on the temperature, pressure, and humidity of the atmosphere at the time of performing the experiment. The effect of humidity in altering the density of the air may be neglected except when the substance weighed is a gas or a vapor. Use the most accurate balances that are available, counter- poise the substance whose specific gravity is required, first in air, and then when suspended by a fine wire, in distilled water. Observe also the temperature of the water and the temperature and barometric pressure of the atmosphere. The distilled water 8 4 JUNIOR COURSE IN GENERAL PHYSICS. used should first be thoroughly boiled in order to expel the dissolved air. The values of p for different temperatures can be found in most reference books, while S a can be computed from the tem- perature and pressure of the air.* In this experiment all observations must be taken with great care. A suitable correction should be made for the weight of the wire used in suspending the substance in water, and all air bubbles that may adhere to the wire or specimen must be care- fully removed. When the value of 8 S is finally obtained, it must be remem- bered that this is the density of the substance at the tempera- ture of the water in which it was weighed. For comparison, this must be reduced to o by using the coefficient of cubic expansion. EXPERIMENT G 4 . Specific gravity by the Jolly balance. The Jolly balance consists of a spiral spring hanging in front of a vertical graduated scale, and carrying at its lower end two small scale-pans. The lower of these should always be kept immersed in water,, as shown in Fig. 32, and in order to render this possible, the bracket which supports the vessel of water is made adjustable in position. The instrument may be used in determining specific gravity in two different ways : I. Fig 32 Place the body whose specific gravity is required upon the upper scale-pan and observe the elonga- tion of the spring. The weight of the body is now determined by finding what known weight is required to produce the same * Tabulated values of p and 5 a will be found in Landoldt and Bornstein, in Stewart and Gee, Vol. I, etc. DENSITY. elongation. Then place the body on the lower scale-pan (under water), and observe what weight must be placed on the upper pan to make the elongation the same as before. These two observations are evidently sufficient to determine the specific gravity. II. The specific gravity may also be determined without the use of weights, upon the assumption that the elongation of the spring is proportional to the force tending to stretch it. The specific gravity of some solid should be determined by each of the above methods, and the assumption made in method II (i.e. that the elongation is proportional to the weight) should be tested by observing the elongations pro- duced by five or six different weights. Compute the value in grams of an elongation of one scale division. Find also the number of dynes required to produce an elongation of one division. EXPERIMENT G 5 . Specific gravity by Nicholson's hydrom- eter. This hydrometer (Fig. 33) consists of a hollow cylinder which is made to float with its rifk axis vertical by means of a heavy weight at the bottom. At the top a wire projects two or three inches above the end of the cylinder and supports a small scale-pan. At the bottom an- . other pan is provided, upon which can be placed the object whose density is required. To determine the specific _~E gravity of a solid, place the Fi s- 33. hydrometer in water, and find by trial the weight which must be placed on the scale-pan in order to bring some well-defined 86 JUNIOR COURSE IN GENERAL PHYSICS. mark to the surface of the water. In the instrument shown in Fig. 33> which is a slight modification of the hydrometer of Nicholson, this mark consists of the point of a wire which projects downward from the center of the scale-pan. Then place upon the scale-pan the body whose density is required, and add weights until the instrument has sunk again to the same level. Finally place the body upon the lower pan or basket, and again determine the weight necessary to sink the hydrometer. From these three weights the specific gravity can be computed. In case the specimen is lighter than water it must be fastened in some way to the bottom of the instrument to prevent it from floating away. The instrument may also be used in determining the specific gravity of a liquid. This form of hydrometer is not very sensitive, and there- fore cannot be expected to give results of great accuracy. In this experiment, however, as in all specific gravity determi- nations, the most common source of error is the presence of air bubbles, which will adhere both to the specimen and to the instrument unless carefully shaken off. The report should contain a full explanation of the principles involved, including Archimedes' Law. EXPERIMENT G 6 . Fahrenheit's hydrometer. This hydrometer consists of an elongated glass bulb (Fig. 34), weighted at the bottom, and carrying at the top a small scale-pan supported by a wire sealed into the bulb. To determine the density of a liquid, first float the instrument in distilled water and place weights on the scale-pan until some well-defined mark on the stem is brought to the surface of the water. It will be found preferable to use bits of tin-foil for weights. The tin-foil corresponding to each separate obser- vation should then be wrapped in a piece of paper, labeled, and afterwards weighed on a pair of balances. Then place the hydrometer in the liquid whose specific gravity is required and determine the weight necessary to sink it to the same point. DENSITY. From these two weights, together with the weight of the hydrometer, the specific gravity of the liquid can be computed. A correction should be made for the temperature of the water. I Use the instrument in the manner just described, to deter- mine the variation in the density of a salt solution as its degree of concentration is altered. To accomplish this, first dissolve in water sufficient salt to make a nearly saturated solution, weigh- ing both the salt and the water. Having determined the density of this solution, dilute it by the addition of a known weight of water, and again determine its density. Continue in this way until the solution is so dilute as to have nearly the same specific gravity as water. At least eight or ten different observations Fi8f> 34- should be taken. With the results obtained, plat a curve in which the strengths of the solution are used as abscissas and the corresponding densities as ordinates. II. Fahrenheit's hydrometer may be used for determining the density of water at different temperatures. In this experi- ment it will not be allowable to assume that the volume of the submerged portion of the hydrometer is constant, for as the temperature is changed, the hydrometer expands or contracts. The volume of the hydrometer at temperature / may always be expressed in the form F (i +//), in which F is its volume at o, and k is the coefficient for cubical expansion of glass. If M is the mass of the hydrometer, m the additional mass 88 JUNIOR COURSE IN GENERAL PHYSICS. necessary to sink the hydrometer to the point of reference, and S t the density of the water at the temperature t, we shall have (83) If another observation be now taken in which t, and conse- quently m, are different, we shall have another equation similar to the above. Between these two equations F may be elim- inated, and the ratio of B t to S t - determined. If one of these values is already known, the other may be computed in absolute measure. For this experiment fill a vessel nearly full of distilled water, and cool it by means of ice and salt down to 2 or 3 C. Make eight or ten determinations of corresponding values of m and t. The values of / should differ by approximately equal increments, with two or three additional observations at temperatures as near as possible to 4. From the observations near 4, determine by interpolation the value of m that would correspond to a temperature of 4. Assume the density at 4 to be unity, and compute the densities at the other observed temperatures. From the results thus obtained, plat a curve with temperatures as abscissas and densities as ordinates. The ordinates should be on a greatly enlarged scale, the axis of abscissas not being shown at all. Upon the same paper, plat the results of some standard deter- mination. EXPERIMENT G 7 . Graduation of a hydrometer of variable immersion. The density of liquids is very frequently determined by means of variable immersion hydrometers. These hydrometers consist of an elongated glass bulb weighted at the bottom with mercury, and supporting a graduated stem of uniform cross- section. The graduations on the stem are not equidistant, how- ever, for equal increments of submersion in different liquids do not correspond to equal decrements in density. DENSITY. 8 9 If J/is the mass of the hydrometer, V the volume of that part below the lowest division of the scale, a the cross-section of the stem, and / the added length of stem submerged in a liquid of density 8, we have, from Archimedes' principle and the definition of density, M=(VQ + ta)8 = (V + / f a )S' ' = (F 4-/"tf)S" ' = . (84) The product of density by volume submerged is a constant. Therefore the volumes submerged in different liquids vary inversely as their densities. It also follows that as the densities increase in arithmetical progression, the vol- a, umes submerged must decrease in a corresponding har- a z monic progression. For example, if the series of densi- ties is i, i.i, 1.2, 1.3, , the series of volumes must be 5 numerically ^, ^, ^, ^~, If M, F , and a were i i.i 1.2 1.3 known, the values of / could be computed corresponding to any particular arithmetical series of densities. How- ever, if the values of two points on the scale be experi- mentally determined as described below, it will be A possible to determine the values of other points by the harmonic law. Suppose ^ and # 5 , Fig. 35, to be the two experi- Figr ' 3J mentally determined points of the scale corresponding to den- sities of i and 1.4, respectively. Let it be required to find a point a 2 which shall correspond to a density of i.i. Let A be a point such that distances measured from A to a v a 2 , , will be proportional to the volume of the hydrometer up to these points. Then we shall have If Aa & be eliminated between these equations, the position of # 2 may be determined, that of a v a 5 being already known. Other points may be determined in the same way. The graduation may be performed much more easily and quite as accurately by the following graphic method. Let it be 90 JUNIOR COURSE IN GENERAL PHYSICS. remembered that the end will be attained if the scale is divided harmonically so that the fixed points a and a 5 shall correspond to densities I and 1.4, respectively. Take any line PP l (Fig. 36), and divide it so that PP lt PP Z , , PP*> are m the harmonic progression i, , , , . From any point, O, at a con- i.i 1.2 1.3 1.4 venient distance from PP^ draw lines through the points P lt P z , . Any line drawn across this series of diverging lines Fig. 36. parallel to PP^ will be divided by them in the same harmonic progression as PP\. Now take the paper scale and lay it across these lines (keeping it parallel to / > / > 1 ) so that a^ will fall on the line drawn through P lt and a 5 will fall on the line drawn through P 6 . If the distances between the graduations so deter- mined are not great, these distances may be subdivided into equal parts without introducing an appreciable error. To determine experimentally the two fixed points, proceed as follows : Place the instrument in distilled water, and adjust the paper scale in the tube until its zero is at the level of the water. Next determine (by one of the methods of Exp. G 2 or G 9 ) the density of some liquid considerably heavier than water (e.g. a strong DENSITY. 91 salt solution). Place the hydrometer in this liquid and observe the reading. Having now two points on the scale, the inter- mediate divisions can be determined by either of the methods described above. Finally test the accuracy of the calibration by using the hydrometer to measure the specific gravities of one or two liquids of intermediate density. EXPERIMENT G 8 . Density of a solid by means of a variable immersion hydrometer.* If a hydrometer of variable immersion is provided with two pans, one above and the other below the surface of the liquid, it may be used for the determination of the density of a solid. Let a be the cross-section of the stem, / the added length of the stem submerged when a substance of mass m is placed on the upper scale-pan. From Archimedes' principle, the increased mass of liquid displaced must equal the mass of the substance. Therefore, we have, from the definition of density, m = a, (86) in which 8 is the density of the liquid in which the hydrometer is placed. If the substance is placed in the lower pan, the same volume of liquid will be displaced as before, but since the sub- stance itself is below the surface, a shorter length of the stem will be submerged. And we have ^-8, (86 a) ~ being the volume of the mass m. From (86) and (86 a) we have 6 8 (87) 8 /-/' From this equation it follows that either 8 e or 8 may be deter- mined if the other is known. * This form of hydrometer is due to G. H. Failyer of the Kansas Agricultural College. 9 2 JUNIOR COURSE IN GENERAL PHYSICS. To perform the experiment, first observe the division of the scale which coincides with the surface of the liquid when no substance is placed on the hydrometer. Next observe the scale reading when the substance is placed successively on the upper and lower scale-pans. From these observations the density of the subtance may be computed if the density of the liquid is known. Determine in this way the densities of several solids. The determination will be most accurate when the sample tested is as large as possible. To verify the statement that the volume of liquid displaced by a floating body is proportional to its mass, proceed as follows : Place a known mass on the upper pan, and observe the corre- sponding scale reading. Repeat these observations with a series of masses, varying from zero to the maxi- mum that can be used. Plat a curve with masses as abscissas and scale readings as ordi- nates. Show that this curve verifies the above statement. From its constants determine the cross-section of the stem, assuming the density of the liquid to be unity. EXPERIMENT G 9 . Density of a liquid by Hare's method. The apparatus used in this experiment con- sists of two vertical tubes open below and con- nected above to a common tube ; the latter tube is provided with stop-cock (see Fig. 37). The two tubes dip into separate vessels, one containing distilled water, and the other the liquid whose density is to be determined. The p . 37 tubes are fastened to an upright board on which there is a scale. If the pressure of the air in the common tube is reduced by suction, the liquid will rise in each tube, the heights of PROPERTIES OF GASES. 93 the two columns being inversely proportional to the densities of the liquids used. This may be demonstrated as follows : Let a be the atmos- pheric pressure, b the pressure of the air in the common tube above the two columns of liquid, both measured in dynes per square centimeter. Let k, h' , and S, &' be the heights and densities of the two columns of liquid. From Pascal's law we have for any point within the first tube on a level with the surface of the liquid in the open vessel, a = b + hfa (88) and for the corresponding point within the second tube, Put distilled water into one of the vessels, and the liquid whose density is to be determined into the other. By suction cause the liquids to rise in the tubes until the top of the highest column is near the end of the scale. Adjust the level of the liquid in each vessel until it is at the zero of the scale, and read the heights of the two columns. Then open the stop- cock until the columns have fallen through 6 or 8 cm. Adjust as before, and again read the height of each column. Repeat these readings for several different heights. Compare in this way the densities of three different liquids with that of distilled water. The tubes should be rinsed with distilled water before and after using each different liquid. GROUP H: PROPERTIES OF GASES. (Hj) Verification of Boyle s law ; (H 2 ) Comparison of the cistern barometer and the siphon barometer; (H 3 ) Coefficient of expansion of air. EXPERIMENT H r Verification of Boyle's law. The apparatus consists of two glass tubes mounted vertically upon some suitable support and connected at the bottom. One 94 JUNIOR COURSE IN GENERAL PHYSICS. tube is left open at the top, while the other can be closed so as to be air tight. Both are provided with scales to enable the height of the mercury contained in them to be measured. (See F'g. 38.) To test the law for pressures greater than one atmosphere. For this purpose one tube should be considerably shorter than the other. (i) In case the two tubes are not provided with a common scale, determine two points, one on each scale, which are at the same level. This can be done by observing the height to which mercury rises in the two tubes when both are open to the air ; or the same thing may be accomplished by means of a spirit level. (2) The end of the shorter tube being tightly closed, observe the height of the mercury in each tube. Then increase the pressure by pour- ing more mercury into the longer tube, and again observe the two levels. Continue in this way until the longer tube is filled nearly to the top, taking in all about ten observations. To check these observations the pressure should now be gradually diminished by allowing mer- cury to escape from the stop-cock at the bottom of the apparatus. Ten more readings should be taken as the pressure falls to its original value. In each of the observations above, the total pressure to which the air in the short tube is subjected is measured by the difference in level between the two columns of mercury phis the pressure of the atmosphere. In tabulating the results each difference in level should therefore be increased by the height of the barometer at the time of the experiment. If the tube containing the air is of uniform cross-section, the volume of the confined air is proportional to the length of the Fig. 38. PROPERTIES OF GASES. 95 tube. In this experiment it is sufficiently accurate to assume the tube to be uniform, except at the closed end, where the cross- section is apt to be irregular. The zero point of the scale used with the shorter tube is therefore placed, not at the top of the tube, but a little below the top. If / is the reading on this scale, and V Q the unknown volume of that portion of the tube above the zero point, then the total volume is V= V^ + IA, in which A is the cross-section. If Boyle's law is true, we should havePF=^; or P(V^ + IA) = K. With the exception of P and /, all the quantities in this equation are constant. If a curve is platted with the observed values of / as abscissas and the cor- responding values of i-t-P for ordinates, this curve should therefore be a straight line. Determine the equation of this line by the method of least squares, and from this equation compute the values of V and K. Reduce both quantities to C. G. S. units. The cross-section A may be determined by weighing a small amount of mercury which is allowed to escape from the appa- ratus, and at the same time observing the alteration in the readings of the two columns. Since the density of mercury is known, these observations are sufficient for the computation of A. If more accurate results are desired, the short tube must be calibrated by means of mercury,* and the air used must be care- fully dried. In all cases great care must be taken to keep the temperature of the air constant. The student will find it interesting to compute the constant PV k in the equation =, which is true for a perfect gas at all temperatures. If this is done, the results should be put in such a form as to refer to the volume and pressure of one gram of air. It will then be possible to compare the value computed for k with those given in various reference books, and a check on the results of the whole experiment is obtained. * See Stewart and Gee, vol. I. 9 6 JUNIOR COURSE IN GENERAL PHYSICS. II. To test the law for pressures less than one atmosphere. In this case the two tubes should be of the same length. Both tubes are first filled with mercury to within 10-20 cm. of the top. One tube is then tightly closed, and observations are taken as the pressure is reduced by drawing off mercury from the bottom. The pressure should then be gradually increased again until the confined air has returned to its original condition. In other respects this experiment is exactly the same as the preceding. EXPERIMENT H 2 . Comparison of a cistern barometer and a siphon barometer. The apparatus for this experiment consists of two barom- eters of the types indicated above, an accurate vertical scale A a divided to millimeters, and a read- ing telescope mounted upon a ver- tical rod. The length of this rod should be at least 80 cm., and the vertical scale should be of the same length. It is essential that the telescope turn freely upon its support with an accurately hori- zontal motion. The arrangement of the appa- ratus which is shown in Fig. 39 is as follows : The two barometers are mount- ed side by side upon a substantial block. At points a and b, situated at distances equally distant to the right of barometer B+ and to the B, Fig. 39. left of barometer B^ are pins from which the scale 5 may be suspended. The latter must be adjusted beforehand so that PROPERTIES OF GASES. 97 when the A -shaped opening is placed on either pin the scale will swing freely into a vertical position. The reading telescope should be set up at as small a distance from the barometers as the length of the draw-tube will permit, and should be in such a position that the meniscus of either mercury column can be seen, and also the scale, in good defini- tion, without change of focus. These adjustments having been completed, the following observations are to be made : (1) Scale hanging at the right. (a) The telescope is focused upon the upper meniscus of barometer B l (siphon), and the distance from the cap of the meniscus to the fixed cross-hair in the eye-piece is measured by means of a micrometer.* (b) The telescope is then swung to the right until the vertical scale comes into the field. (In case the scale is not in proper focus, further adjustment must be made by moving it towards or away from the telescope, and not by refocusing the latter.) (c} The scale divisions nearest the fixed cross-hair are identi- fied and noted, and their distances from the latter are measured by means of the micrometer screw. (d) These operations are repeated in the case of barometer B (89) where t is the temperature of the steam bath. The tempera- ture t is to be determined from the pressure of the vapor in 80. 76. 74. 72. 70. 98 99' Fig. 41. Boiling Point and Pressure. the steam bath, for which purpose the results of Regnault may be used. That portion of his curve which applies to pressures in the neighborhood of one atmosphere is given in Fig. 41.* By means of it the temperature of a steam bath for any ordinary barometric pressure can be obtained without the use of a thermometer. To compute Vol and Vol,, the density of mercury must be known at o and at t. This may be conveniently obtained from the curve of densities, given in Fig. 42.* * For the data from which the curves are obtained, see Landolt and Bornstein, Tabellen, pp. 41 and 58. PROPERTIES OF GASES. 103 The most important corrections are those arising from the depression of the mercury in the neck of the manometer at a, and from the temperature of the mercury in the manometric and barometric columns. The former is to be ascertained by isolating the bulb from the manometer, by turning V to 13.58 13.56 13.54 13.52 13.50 13.48 13.46 13.44 13.42 13.4O 13.38 13.36 \ N \ J NSIT YOI ME ICUF Y \ \ \ \ \ f\ s S \ \ \ \ > 20' 4O 6O Fig. 42. 100 the proper position, and bringing the mercury to the mark by means of the plunger. The difference of level at a and in the open tube when the two surfaces are subjected to the same (the barometric) pressure is then noted. This reading is most accurately performed by means of the vertically suspended scale already described, and the reading telescope. The latter correction is that which it is necessary to apply to all barometric and manometric readings on account of the change of density of the mercury with temperature, viz. *=*. Un (90) where h n is the corrected barometric or manometric height, 104 JUNIOR COURSE IN GENERAL PHYSICS. //, the observed height, and is the ratio of the densities of "o mercury at the temperature of observation and at o C. A less important correction than the foregoing is that due to tha fact that a certain volume of the air contained in the apparatus, that, namely, in the tube connecting the bulb with the neck of the manometer does not change temperature. If the bulb is of considerable size (300 cu. cm. or more), and the tube has the contracted bore described above, the volume under consideration will be found to be of little influence. The student should, however, make the measurements and compu- tation necessary to assure himself of the fact. For this purpose a piece of the tubing which is used in the apparatus is provided. The diameter of this is to be gauged and the contents of the connecting tube estimated therefrom, and from the length of the latter. The neck of the manometer from a to the joint b is frequently of greater volume than the connecting tube. Its contents may be directly determined as follows : (1) By means of the plunger drive the mercury into the neck of the manometer until it reaches the stopcock V, which must have been previously turned so as to connect the manom- eter neck with the open air. (2) Turn the stopcock vS at the base of the manometer tube so as to isolate the tube leading to the neck. (3) Turn the stopcock 6* so as to drain the above tube into a clean beaker, taking care to close 5 at the moment the level of the mercury reaches the mark a. The mercury which has escaped measures the contents of the neck. The computation of the coefficient of expansion of the air within the bulb follows readily from these measurements by the application of the laws of Boyle and Gay Lussac. Thus : A gas possessing a coefficient of expansion a, when heated from o to T, expands from F to V T , according to the law F r = (i+7)F . (91) PROPERTIES OF GASES. ^05 If reduced, when at the higher temperature, to a volume V by the pressure P T , we have V P rfe- = v *p; (92) where P is the pressure at which the gas is measured while still at o. Under the conditions of the present experiment, we have two volumes of gas to consider : that within the bulb, which has a volume V at o and of V$(i +kT) at T, and that within the neck, the volume of which is z> (i +kt). The full expression for the relations between volumes, pressures, and temperatures therefore is T _i_ T T J_ bt\ / r -I- bt\ o. (93) in which equation k is the cubical coefficient of the expansion of the bulb, and z/ is the volume of the neck and connecting tube at the temperature o. In equation 93 we may ignore the influence of temperature upon the volume of the neck, and write for V Q (I +kt) the simpler form v (volume of the neck at temperature of the room t). Equation 93 then becomes which may be written v (i + kT}+vl - P . (95) P T and PQ are quantities obtained by adding the atmospheric pressure H^ and H^ observed respectively at the times of the first and last adjustment of the manometer and the corresponding manometric pressures h^ and h^ (which may be positive or neg- ative). 106 JUNIOR COURSE IN GENERAL PHYSICS. To solve equation 95 we assume the value a = 0.003665 for the right-hand member, an approximation to our final actual value which will in no appreciable manner influence the result, since it enters only into the correction of the very small quan- tity v. The accuracy of the determination of a depends upon our knowledge of the quantities P T , P Q , T, V^ and k* * For a description of Regnault's classical research upon this subject, see Memoires de 1'Academie Royale des Sciences, XXI (1847). CHAPTER III. GROUP I: CALORIMETRY. (I) General statements ; (1^ Heat, of vaporization; (I 2 ) Heat of fusion; (I 3 ) Specific heat ; (I 4 ) Radiating and absorbing power. (I). General statements concerning calorimetry. It may be said in general that calorimetric determinations are subject to a great variety of annoying errors, which can be avoided only by the exercise of especial care and patience on the part of the experimenter. The student is therefore advised to plan his work very carefully before beginning the experiment itself, so that he shall run no risk of omitting essential observations and precautions. It will generally be found that the greatest source of error in calorimeter experi- ments is the inaccurate determination of temperatures. This may be due to several causes : (1) The thermometer may indicate the temperature of a portion of the liquid ; the rest of the liquid being at a different temperature. (2) The thermometer may not have had time to acquire the temperature of the surrounding liquid. (3) The thermometer itself may be inaccurate. (4) The reading of the thermometer may be at fault. These sources of error should be guarded against with especial care. The equations required for the computation of results in calorimetry may all be derived from one general principle. 107 108 JUNIOR COURSE IN GENERAL PHYSICS. This principle may be stated as follows : The amount of heat lost by one system of bodies is equal to the amount gained by another system. This, of course, treats potential energy due to change of state as latent heat. The heat lost or gained by a body may be due to two causes : (1) Change in temperature ; the amount in this case is equal to the continued product of the mass, specific heat, and change in temperature of the body. (2) Change of state ; this amount is equal to the product of the mass so changed by a constant quantity of heat necessary to produce such a change in unit mass. The amount of heat lost by radiation to the air cannot be expressed in either of these ways ; but it may be expressed as equal to the product of the time during which radiation takes place, the average difference of temperature between the radiating body and the air, and the radiation constant of the body. I. Comparison of thermometers. When two or more thermometers are used in an experiment, their indications should always be compared, to determine whether their indications agree. Even the best thermometers are apt to differ in "zero point," so that they may give different readings for the same temperature, and yet measure differences in temperature accurately. To compare thermometers, they should be placed together in a vessel of water (at any convenient temperature), and alternate readings taken for several minutes, the water being kept thoroughly stirred. If they are found to differ, a suitable correction must be made to all subsequent readings. The numbers, or other distinguishing marks, of the ther- mometers used should in all cases be recorded. CALORIMETRY. II. Determination of the water equivalent of a calorimeter. When a calorimeter containing water, etc., is heated or cooled, heat is absorbed or given out by the vessel itself in addition to that absorbed or liberated by its contents. The water equivalent of a calorimeter is a quantity of water which would absorb the same amount of heat, when warmed through a certain number of degrees, as is absorbed by the calorimeter when heated through the same range of temperature. To determine the water equivalent, proceed as follows : (i) Fill the calorimeter nearly three-fourths full of water three or four degrees colder than the air, the weight of the water being known. This water should be kept thoroughly stirred, and its temperature should be observed by means of a thermom- eter hanging in it. Add enough hot water, of known temperature, to fill the calorimeter to within one or two centimeters of the top. Stir thoroughly, and record the reading of the thermometer in the mixture at intervals of half a minute, until the temperature becomes practically constant. The hot water should be stirred immediately before it is poured in, and the temperature of both hot and cold water should be observed just the instant before mixing. It is best to choose the tem- perature of the hot water so that the mixture will come to about the temperature of the air, corrections for radiation being un- necessary if this is done. The mass of the hot water used may be determined by weighing the mixture after the obser- vations are completed. From the data obtained, the water equivalent is to be computed.* The student should make at least three determinations. * The amount of heat that the calorimeter absorbs is very small compared with the amount absorbed by the water which it contains. For this reason slight errors of observation will generally cause a very great error in the computed result. A common source of error is the following : while the hot water is being poured into the cold 110 JUNIOR COURSE IN GENERAL PHYSICS. If the material from which the calorimeter is made is known, the water equivalent may also be computed, as a check on the above results, from the mass and specific heat. In the determination of the water equivalent, great care must be used in all temperature readings, or the results of successive determinations will be discordant. This is especially true in the case of small calorimeters. To obtain the best re- sults, a number of separate determinations should be made, and the average of all the results used. No single result should be discarded merely because it differs widely from the rest. A result can be legitimately discarded only when something has occurred during the experiment which tends to throw discredit on some of the observations, or when there is an obvious mistake in one of the readings. In the most accurate calorimetric experiments it is necessary to determine not only the water equivalent of the calorimeter, but also the water equivalents of the thermometers, stirring- rods, etc. In the experiments which follow, however, this is unnecessary. In all calorimetric experiments, the temperatiire of the room should be recorded, as it will be found necessary in making corrections for radiation. III. Determination of the radiation constant of a calorimeter. The loss of heat from a body which is a few degrees warmer than its surroundings is proportional : (i) to the time during which radiation takes place ; (2) to the difference in temperature between the body and the room ; (3) to a constant called the constant of radiation, depending upon the nature and extent of the radiating surface. water, it will lose some heat to the air. In the computations this small quantity of heat is necessarily treated as if it were absorbed by the calorimeter, thus giving too large a value to the water equivalent. CALORIMETRY. 1 1 1 Note that this constant depends only on the surface, and not upon the nature of the interior of the body. The radiation constant of a calorimeter is, for example, the same when it contains mercury as when it is filled with water. But the rate of cooling will be different in the two cases on account of the difference in the two specific heats. Radiation is essentially a phenomenon which occurs at the surface of a body, and depends wholly upon the nature and temperature of this surface. The gain of heat by absorption when the body is colder than its surroundings obeys the same laws. The law above stated is known as Newton's law of cooling, and is really only an approximation to the truth. In the case of bodies differing in temperature from their surroundings by not more than 10, the approximation is, however, good. The radiation constant may be defined as the amount of heat which is lost by radiation in one minute when the radiating body is one degree hotter than the air. For a difference in temperature of 0, the radiation is 6 times as great; and for / minutes instead of one minute the loss is t times as great. It will thus be seen that if the radiation constant is known, the loss of heat from a body such as a calorimeter can be readily computed. In most calorimetric work, corrections must be made for the loss of heat by radiation, or the gain by absorption, during the time of the experiment. The first step in any calorimetric experiment should therefore be the determination of the radia- tion constant. The method is as follows : (i) Fill the calorimeter to within I or 2 cm. of the top with water considerably warmer than the air (say io-2O warmer). The mass of the water should be known. Suspend a thermom- eter in the center of the calorimeter, and observe the tempera- ture at intervals of one minute as the water cools. These observations should be continued for at least an hour, the water being thoroughly stirred before each reading. The temperature 112 JUNIOR COURSE IN GENERAL PHYSICS. of the room, as indicated by a thermometer hanging near, should also be occasionally recorded. (2) With the data obtained plat two curves, using times as abscissas in each case, and temperatures of air and water as ordinates. A smooth curve should now be drawn in each case, passing as nearly as possible through all the points platted. Any slight deviations from such smooth curves are probably due to accidental errors in the observations. From the data given by these new curves, and knowing the mass of water, the statements made above may be verified, and the radiation constant computed. It should be observed that an approximation must here be made, viz., that the temperature of the surface of the calorim- eter is the same as that of the liquid contained in it. If the liquid is kept thoroughly stirred, and if the material from which the calorimeter is made is a good conductor, no great error is, however, introduced. For example, let the mass of water plus the water equivalent of the calorimeter be 500 grams. Suppose that the temperature fell from 30 to 28 in five minutes, the temperature of the room being 20. The temperature of the water having changed 2, the loss of heat is equal to 2 x 500, or 1000 calories. Since this loss took place in five minutes, the loss in one minute was iooo-f- 5, or 200 calories. The average difference in temperature between water and air was 9. The loss for one minute, and for i difference in temperature, would there- fore be 200 -7-9 = 22 + minor calories, which is the radiation constant. Similar computations made with different portions of the data should give nearly the same result. Make eight or ten such computations and use the mean. In using the constant thus obtained to correct for radiation losses, it usually happens that the temperature of the calorimeter does not remain constant throughout the experiment, so that the rate at which heat is lost by radiation is continually changing. The method to be used in such cases is illustrated by the following example : Suppose that the temperature of the calorimeter is observed at intervals of one minute and is found to vary as follows : 29, 26. 5, 24, 22.6, 2i.4, 20.8, 20.6, 20.5, the temperature of the air being CALORIMETRY. 22. The average temperature of the calorimeter during the seven minutes is therefore 23. 1 8, (found by adding all the readings and dividing by 8) . Radiation has therefore taken place for seven minutes at a rate whose average value is that corresponding to a difference in temperature of i.i8 from the air. If the radiation constant is 20, the loss of heat is 20 x 1.18 x 7 = 165.2 calories. EXPERIMENT l v Determination of the heat of vaporization of water. The apparatus for this experiment may be arranged in a great variety of ways. The essential parts are : (1) Some vessel in which steam may be generated. (2) A calorimeter, which may be any metallic vessel of suitable size. (3) Tubes of metal or glass by which the steam may be conveyed to the calorimeter. The latter should be sheltered 11 il KMU Fig. 43. from the heat radiated from the boiler, and some device should be supplied to prevent the water which condenses in the tubes from entering the calorimeter. Fig. 43 shows a convenient form of apparatus for this determination. The water equivalent and radiation constant of the calorim- 114 JUNIOR COURSE IN GENERAL PHYSICS. eter used should first be determined as previously described. Observations may then be made as follows to determine the heat of vaporization. (1) Fill the calorimeter to within 2 or 3 cm. of the top with a known mass of water considerably colder than the air (from 8 to 12 colder). (2) Pass steam into the calorimeter from a vessel of boiling water by means of the tubes provided for the purpose, keeping the water in the calorimeter thoroughly stirred, and observe its rise in temperature at intervals of one minute, until it has been heated as far above the temperature of the room as it was previously below it. (3) Determine the mass of steam condensed by weighing the calorimeter and contents at the end of the experiment, the weight of the vessel and of the cold water having been previ- ously determined. These weighings should be made with con- siderable care, as the mass of the condensed steam may be quite small. To make sure that the steam is dry, it should be slightly superheated by a flame placed under the tube which leads to the calorimeter. The temperature of the steam just before entering the water may be observed by means of a thermometer inserted in the tube. The steam should be allowed to pass through the tubes for a considerable time before beginning the experiment, in order to make sure that they are thoroughly warmed (to avoid condensation). (4) From the data obtained compute the heat of vaporization of water, or the latent heat of steam. Corrections should be made for the loss or gain of heat due to radiation and absorp- tion, and for the heat capacity of the calorimeter itself. This correction, due to radiation, may be reduced to a minimum by allowing the flow of steam to continue until the water in the calorimeter reaches a temperature as much above that of the air as it was initially below that temperature. But the correction should always be computed. At least three determinations should be made. CALORIMETRY. The following tables show the character of the data per- taining to this experiment and the method of arranging them. COMPARISON OF THERMOMETERS. No. 12975. No. 12319. No. 3. 7-53 7-55 8-3 7.60 7.62 8-3 26.80 26.82 27.4 25-83 25-87 27.2 38.03 38.08 39. 36-89 36.91 37.6 No. 12327 registers 99 in boiling water. RADIATION CONSTANT. Time. Tern, of Vessel. Tern, of Room. Radiation Constant. Time. Tern, of Vessel. Tern, of Room. Radiation Constant. 3-34 30.8 II. 2 3-47 28.0 II. I 35 30.56 48 2 7 .8 36 30.36 49 2 7 .6 4.15 37 30.12 50 27.4 38 29.90 "3 5i 27.23 39 29.70 3.98 52 27.03 40 29.46 53 26.88 II. 4i 29.28 54 26.70 3-93 42 29.03 55 26.50 43 29-83 II. 2 56 26.33 44 28.60 4.22 57 26.15 45 28.40 58 25-95 10.8 46 28.20 59 25.80 4 .l6 Mass of Calorimeter + Water = 492.5 grams. Mass of Calorimeter =154.7 Water Equivalent Radiation Constant 337-8 = 14.8 352.6 grams. = 4.09 calories. i6 JUNIOR COURSE IN GENERAL PHYSICS. WATER EQUIVALENT OF CALORIMETER. .. II. III. Mass of Calorimeter, 154.7 !54-7 154.7 Cal. + Cold Water, 334-0 344- 340.2 " Cold Water, 179-3 189.3 185.5 " Cal. + Mixture, 478.5 480.5 486.5 Warm Water, H4.5 136-5 146.3 Tern, of Room, No. 3, 24.0 21. 21.0 " Cold Water, No. 12975, 9.8 10.2 8.25 " Warm Water, No. 12319, 35-6 36.6 37-4 " Mixture, No. 12975, 20.88 20.85 20.4 Water Equivalent, 12.6 12.6 19.2 Water equivalent = 14.8. HEAT OF VAPORIZATION OF WATER. I. II. III. Mass of Calorimeter, 154-7 !54-7 J 54-7 Cal. + Cold Water, 438.0 427.7 434-o " Cold Water, 283-3 273.0 279.3 " Cal. + Mixture, 450.0 440.6 447.2 " Condensed Steam, 12.0 12.9 13.2 Temperature of Room, 21.0 21.0 21.0 Cold Water, 10.4 8.2 7 10.82 No. 12319, { TIME. IO S 105 103 " Steam, im. 107 105 104 No. 12327, 3 108 105 [ 4 107 o 13-4 14.0 12.8 5 16.2 19.6 13.6 I.O 1 8.0 2 7 .2 . 17.8 " " Mixture, 1.5 20.7 344 22.0 2.O 23-7 35-0 24.7 No. 12319, 2.5 26.5 35-45 28.5 3-0 31-0 32.9 3-5 34-4 37-0 4.0 34-9 37-7 Heat of Vaporization, 545 544 540 CALORIMETRY. H 7 EXPERIMENT I 2 . Determination of the heat of fusion of ice. The radiation constant and the water equivalent of the calorimeter used are first to be determined, as previously described. Observations may then be taken to determine the heat of fusion as follows : (1) Fill the calorimeter to within 2 or 3 cm. of the top with a known mass of water, 3 or 4 warmer than the air. (2) Stir thoroughly and observe the temperature. Then drop in a piece of ice ; hold it under water by means of a stirrer arranged for the purpose, and observe the temperature of the water at intervals of half a minute until the ice is melted, and a fairly constant temperature is reached. In case the melting of the ice cools the calorimeter below the temperature of the room, it is well to continue observations of temperature, stirring thoroughly before each reading, until the calorimeter begins to warm again by absorption of heat from the air. The ice used should be at its melting-point. This is assured by keeping it for some time inside the warm room. It should be carefully dried by means of filter paper just before dropping in the calorimeter. The mass of ice used may be obtained by weighing the calorimeter and contents after the observations are completed, the weight of the vessel and of the warm water being already known. From the data obtained compute the heat of fusion. Corrections are to be made for the loss of heat by radiation, and for the water equivalent of the calorimeter. Make at least three complete determinations. EXPERIMENT I 3 . Determination of the specific heat of a solid. (i) Place the metal whose specific heat is to be determined in the calorimeter, and support it in such a way that it does not touch the sides or bottom. Enough water of known weight Il8 JUNIOR COURSE IN GENERAL PHYSICS. should now be placed in the calorimeter to just cover the metal, the temperature of the water being from 8 to 15 above that of the air. (2) Allow the calorimeter and contents to stand for at least ten minutes in order to make sure that the metal has acquired the temperature of the water. Then add cool water, stir thoroughly, and observe the temperature at half-minute intervals until it reaches a practically constant value. The temperature and amount of the cold water should be such as to bring the final temperatures of the mixture very close to that of the air. A few preliminary trials will show about what the temperature should be. The temperature of hot and cold water, each thoroughly stirred, should be observed immediately before mixing. The weight of the cold water added is to be found by weighing the calorimeter and contents after the other observations are completed. As the specific heat of any metal is much less than that of water, it will be advisable to take a rather large mass of the metal. For good results, its heat capacity should be comparable with that of the mass of water used. If the metal is not a good conductor of heat, it should be in small pieces. The method here described is merely one of many which may be used in the determination of specific heat. The student will find it instructive, if time is available, to check his results by one of the numerous other methods which will be found described in various text-books. The water equivalent and the radiation constant of the calorimeter used are to be determined as described in the general directions at the beginning of this group. The weight of the metal being known, its specific heat may now be computed. Corrections are to be made for radiation and for the absorption of heat by the calorimeter itself. At least three determinations should be made. CALORIMETRY. II9 EXPERIMENT I 4 . Radiating and absorbing powers of dif- ferent surfaces. The objects of this experiment are to investigate the radia- tion and absorption of heat from different surfaces, and to determine the relation between the radiating and absorbing powers of the same surface. The radiating constant of a surface may be defined as the number of calories that will be radiated from one square centi- meter of the surface in one minute, for a difference in tempera- ture of one degree between the surface and its surroundings. In like manner the constant for absorption may be defined as the number of calories that will be absorbed by one square centimeter of the surface under similar conditions. The radiation constant of a surface may be determined by dividing the heat lost by a vessel in a given time, by the time, the average difference in temperature between the surface and the air, and the area of the vessel. The absorption constant may be computed in a similar manner from the heat gained in a given time. It is to be observed that radiation and absorption depend upon the temperature of the radiating or absorbing surface, and not upon the temperature of the contents of the vessel. If the walls of the vessel are thin, however, and of some highly conducting material, no great error is introduced by assuming that the contents of the vessel are at the same temperature as the surface. The method of the experiment is as follows : (1) Fill the vessel for whose surface the radiation constant is to be determined with water 1 5 or 20 warmer than the air, and place it upon a poorly conducting support, such that the vessel will be free to radiate its heat in all directions. (2) Observe the temperature by means of a thermometer hang- ing in the center of the vessel, at intervals of two minutes, stirring the water thoroughly before each reading. The temperature of the air should also be observed at intervals of about five 120 JUNIOR COURSE IN GENERAL PHYSICS. minutes, and for good results must remain nearly constant throughout the experiment. Continue these observations for at least half an hour. A curve should now be platted with times as abscissas and temperatures as ordinates. From this curve, or from the data themselves, make four or five independent computations of the radiation constant. If the constant is computed from the curve, it will be necessary to find the "pitch," dt-^dT (^tempera- ture; T = time), at different points on the curve, by drawing tangents. From Newton's law of cooling, the radiation constant R is given by the equation where c is the heat capacity of the vessel, A its superficial area, and 4 the temperature of the air. The value of c is deter- mined by adding the water equivalent of the vessel to the weight of the water contained in it. The following method of computing the results will be found instructive as an example of the employment of graphical methods, and may be used instead of the above if desired. ). (97) ( 9 8) By integration : log (/- 4) = T + K, (99) where K is the constant of integration. If, therefore, a curve is platted whose co-ordinates are T and log(/ f a ) respectively, the result should be a straight line. In the RA equation of this line, - - enters as one of the constants. Note that the logarithm which occurs in the above equation is the CALORIMETRY. 12I Napierian logarithm. Ordinary logarithms may, however, be used until the final result is reached. The constant for absorption can be determined in a similar manner by filling the vessel with water 15 or 20 colder than the air, and observing the gradual rise in temperature due to absorption. These observations should be repeated with three or four vessels which are of the same size and shape, but differ widely in the character of the radiating surface. Polished metal and lampblack surfaces will probably be found to differ most widely in their radiating powers. No difficulty should be experienced in carrying on the observations with four vessels at the same time. It is to be observed that a slight error is introduced in this experiment by assuming that all the heat is lost by radiation, for part of the loss is really due to convection. For small differences of temperature, however, the loss by convection is small, and may be treated as though it obeyed Newton's law. The radia- tion constants obtained represent, therefore, the sum of the losses due to the two causes. (In connection with this experiment, see the general direc- tions for calorimetric work.) % CHAPTER IV. GROUP P : STATIC ELECTRICITY. (P) General statements ; (P t ) Electrostatic induction; (P 2 ) The principle of the condenser; (P 3 ) The Holtz machine; (P 4 ) Further experiments with the Holtz machine. (P). General statements concerning static electricity. Whenever a body or system of bodies becomes electrified, equal quantities of positive and negative electricity are pro- duced. Many experimental facts lead to the conclusion that the energy of electrification exists in the insulating medium between the bodies containing these two equal quantities of positive and negative electricity. These experimental facts prove that the insulating medium is in a state of strain. Therefore the energy of electrification is the potential energy of an electrical field, in an insulating medium, bounded by bodies containing what are called "charges of electricity." If an electrified body or system of bodies be placed within a closed conducting surface, the charge of electricity on this sur- face is equal, and of opposite sign, to the charge of the body or system of bodies. This law has been deduced directly from experiment. However, it may be shown to be directly deducible from the following theorem. Let F denote the resultant electrical force at a point on a small element of the surface of a charged body : the integral of the quantity FdA, taken over the entire surface of the charged body, is numerically equal to 4 TrQ, in which Q is the number of 122 STATIC ELECTRICITY. 123 units of electricity in the body.* This is known as Green's theorem. Another way of stating this fact is as follows : The number of lines of force, or of unit tubes of force, issuing from the surface of a body charged with Q units of electricity, is 47rQ. These lines of force, or tubes of induction, must end on some other body or bodies. On the surfaces of the conductors where these 47r<2 lines of force end, there must be Q units of induced electricity of the opposite sign to the electricity on the first conductor.! To completely discharge a conductor, and cause to vanish the field surrounding it, it will be necessary for these two equal quantities of electricity of opposite signs to unite. The conception of free and bound elec- tricity helps to the understanding of this and other phenomena of static electricity. The term "free electricity," or "free charge," is applied to that portion of a charge which will escape to the earth, when the conductor containing it is connected to earth, while a bound charge is that portion which is held by the induction of some other near-by insulated charge. Suppose A (Fig. 44) to be an insulated conductor charged with Q units of positive electricity. Suppose B to be a conductor which has been grounded and afterwards insulated. The charge Q induces on B, q' units and on the walls of the room, q" units of negative electricity, such that <2= -(?'+*")- * Gray, Absolute Measurements in Electricity and Magnetism, vol. I, p. 10. t This is more general than the second law above given, but it is based on the assumption that an electrical field does not extend indefinitely in a direction in which there are no charged bodies. 124 JUNIOR COURSE IN GENERAL PHYSICS. That part q' of the electricity induced on B is bound by the charge Q. None of it will escape to the earth, for its potential has been reduced to zero by grounding it. The charge q' on B binds, by induction, a portion of the charge on A ; so that if A were grounded, only a portion of the Q units of electricity would escape. That which escapes is free electricity ; the remainder is bound by the negative charge on B. It is very important to keep clearly in mind the distinction between the character and the potential of a charge of electricity. In the above example, before A was grounded, B was at zero potential, but it had a negative charge ; after A was grounded, the potential of B became negative, although its charge was unchanged. A, however, was reduced to zero potential, but it still retained a positive charge. Positive and negative electricity always exist at the positive and negative ends respectively of electrical lines of force ; or, as some may prefer to put it, at the positive and negative boundaries of an electrical field of force. The potential of the body containing the positive charge must always be positive with respect to the body containing a negative charge at the other boundary of the field ; but the potential of either or both of these bodies may be anything with respect to the earth, whose potential is usually taken as zero. The potential of a conductor is positive, when, upon being grounded, positive electricity is discharged to the earth ; when negative electricity is thus discharged, the potential is negative, and when no discharge occurs, the conductor is at zero potential. It is a very instructive exercise to map out a field of force with equipotential surfaces and lines of force.* It is not difficult to do this in an approximate manner, if the student keeps clearly in mind the definitions, the fact that lines of force and equipotential surfaces are mutually perpendicular, * In this connection the beautiful maps of the electrostatic field at the end of the first volume of Maxwell's Electricity and Magnetism should be inspected. STATIC ELECTRICITY. 125 and the fact that the surface of every conductor is an equi- potential surface. Let it be required to map a section of the field within a hollow conductor at zero potential, containing two insulated conductors. One of these conductors is positively charged, and the other has only an induced charge. It will be found easier to draw the lines of force first. (1) They must always be drawn between conductors of different potentials. (2) They must issue from a conductor at right angles to its surface. (3) Lines of force must always issue from a body containing a positive charge, and end on a body containing a negative charge. If lines of force are drawn fulfilling these conditions, they will be as indicated in Fig 45.* It may be assumed, approximately, that along the shortest distance between the two conductors the potential falls uniformly. Assume that the difference of potential between them is nine. Divide the distance into nine equal parts, and through each point of division draw a line, and continue it so that it is everywhere perpendicular to lines of force. Each of these lines must be a closed curve. And from definition,' each of them must lie in an equipotential surface. By definition, the same amount of work is done in carrying a charge from one point in an equipotential surface to any * The equipotential lines and lines of force in Fig. 45 were computed by C. D. Child. This computation was made as follows : Known charges were supposed to be concentrated at points. A series of points were then found which had the same potential, according to the formula V = 2 -^ . All the equipotential surfaces, from 3 K to 15 inclusive, were determined in this way. A conductor connected to the ground was then supposed to coincide with the equipotential surface 3. This reduced the potential of every point within by three units. A conductor was then supposed to surround a charge of + 40 units, and coincide with the equipotential surface 12; while another conductor was supposed to surround charges of + io> 5> and 5 units, and coincide with the equipotential surface 3. These two conductors in no wise changed the potential of any point in the field of force, while it was perfectly allowable to suppose the charges within them to be transferred to their outside surfaces. 126 JUNIOR COURSE IN GENERAL PHYSICS. point in another equipotential surface. Therefore, the field must be strongest where these surfaces are closest together. Strength of field is sometimes represented by the number of lines of force per square centimeter. Therefore, where the A is a conductor [Potential 12 ; charge 40?]. B is a conductor [Potential 3 ; induced charge 10 q and + I0q~\ Fig. 45. equipotential surfaces are closest together, the lines of force should be most numerous. It should be noticed that one of the conductors has lines of force issuing from it and also ending on it. It follows that there is a positive and a negative charge on this conductor, although the whole of it is at the same potential. STATIC ELECTRICITY. I2 / Nearly all experiments on static electricity are more suc- cessful in cold weather than in warm. This difference is prob- ably due to a difference in humidity. In moist air, bodies are rapidly discharged, so that it is relatively much more difficult to accumulate charges when the surrounding atmosphere is moist. In cold weather the absolute humidity is usually much less than in warm weather. In an artificially heated room, the absolute humidity remaining the same as the outside air, the relative humidity is very much lessened on account of the higher temperature. From the definition of a line of force, a positively charged body (an insulated pith-ball,, for example) tends to move along the lines of force from positively charged bodies towards nega- tively charged bodies. If the body were negatively electrified, it would tend to move in the opposite direction. This offers a means of testing the direction of lines of force, and con- sequently the character of the charge on a charged body. If an insulated pith-ball be positively electrified (by being brought in contact with a glass rod which has been previously rubbed with silk), and then be suspended in a region supposed to be a field of electrical force, surrounding a charged con- ductor, one of three things will occur : (1) The pith-ball will tend to move from the body supposed to be charged. This proves that the region is an electrical field with lines of force issuing from the body, which is there- fore positively charged. (2) The pith-ball will not tend to move at all. In this case we infer that the electrical field is too weak, or that the charge on the pith-ball is too weak to produce a perceptible effect. (3) The pith-ball will tend to move towards the charged body. This indicates that the region is a field of force with lines of force entering the body, which is therefore negatively electri- fied. We say " indicates," for it only proves that there is now a field between the pith-ball and the body, and that one of them was originally electrified before they were brought near together. 128 JUNIOR COURSE IN GENERAL PHYSICS. We know that if the pith-ball were originally neutral, it would move toward a strongly charged body when brought near it. If the conductor were strongly charged, and the pith-ball weakly charged, both with positive electricity, the motion would be the same. Moreover, if the conductor were neutral, a charged pith-ball brought near it would tend to move towards it. These facts may be readily explained on the theory of induc- tion. What is to be learned from this is that an electrical field of force, and the character of the charge on a body, cannot be certainly determined from the motion of a charged pith-ball towards the body. Under these circumstances, the pith-ball should be charged negatively (by being brought in contact with vulcanite previously rubbed with fur), and again experimented upon. Sometimes it is better to first bring the pith-ball into contact with the body supposed to be charged, and then to test the nature of the charge on the pith-ball by bringing it near electri- fied glass and vulcanite rods in turn. The electrification of a body may often be tested more reli- ably by the use of a proof -plane and a gold-leaf electroscope. In using the electroscope, it must be remembered that it is the first motion of the leaves, as the charged proof-plane is brought near it, that is to be noted. If the proof-plane has a considera- ble charge, whose sign is opposite to that of the leaves, it will cause them to collapse, and afterwards to diverge, as it is brought quite near to the electroscope. EXPERIMENT P x . Electrostatic induction. Every insulated conductor in the neighborhood of a positively charged body has induced upon its surface equal quantities of positive and negative electricity. The positive electricity is on the side farthest from the charged body, and the negative on the side nearest. The object of this experiment is to investigate this and other phenomena of electrostatic induction. STATIC ELECTRICITY. I2 9 For this experiment, a rather large insulated conductor is required. A Leyclen jar, in which the small knob is replaced by a sphere 10 or 12 cm. in diameter, and connected with the inner coating, is excellent for this purpose on account of its great capacity. Such a conductor will usually retain its charge for the whole time of the experiment. A second insulated conductor, preferably an elongated cylinder with hemispherical ends, is also required. Charge the sphere connected to the Leyden jar by means of an electrical machine. Place the second conductor in the elec- trical field produced by the charged conductor. The two con- ductors should be not more than 2 or 3 cm. apart. The nature of the charge on different parts of the second conductor should now be investigated. This may be done, either by means of a pith-ball suspended by a silk fiber, or by means of a proof-plane and gold-leaf electroscope. Some idea may be formed of the direction of the lines of force in the electrical field surrounding the conductors by the direction in which a positively charged pith-ball tends to move. After testing as above, remove the conductor, still insulated, to a distance from the charged body, and test again. Place the conductor again in proximity to the charged body, ground the conductor, and test with the pith-ball as before. Then move the conductor closer to the charged body, and note any change in its condition. Finally, remove the conductor to a distance from the inducing body (the connection with the ground having been first broken), and test again. Throughout the experiment care must be taken not to allow any discharge from the charged body to the second conductor. To secure uniform results it will generally be necessary to repeat these tests several times. This experiment will give satisfactory results only when the air is rather dry. It succeeds best in cold weather, when the room is artificially heated. VOL. I K 130 JUNIOR COURSE IX GENERAL PHYSICS. Addenda to the report: (1) Define the following : unit electrical charge ; electrical field of force ; field of unit intensity ; electrical difference of potential ; unit difference of potential ; electrical potential at a point ; equipotential surfaces ; electrical line of force ; unit line of force, or unit tube of force ; electrical capacity ; unit of capacity. (2) Give a demonstration of the fact that lines of force and equipotential surfaces are mutually perpendicular. (3) State the general relation which the quantity of induced electricity on the conductor bears to the quantity on the charged body and the distance between them. (4) Assuming that the charge of the charged body is posi- tive, what is the potential of the conductor in each of the five cases investigated ? What would be its potential if the charged body were negative ? (5) Draw a vertical section of the two conductors showing equipotential surfaces and lines of force. (6) Draw two such figures, one when the conductor having the induced charge is insulated, and one when it is grounded. EXPERIMENT P^ The principle of the condenser. When a conductor connected to the earth is brought near a charged body, the potential of the charged body is reduced. If the conductor almost surrounds the charged body, and is very close to it, its potential will be very greatly reduced, although the amount of the charge remains absolutely unchanged. Another way of viewing this fact is to consider that the conductor lessens the quantity of free electricity on the charged body. The remainder of the charge is bound by the electricity induced on the near-by conductor. If, instead of maintaining the charge constant, the potential of the charged body is main- tained constant, it will be found that the charge must be rapidly increased as the conductor connected with the earth is brought very near. STATIC ELECTRICITY. ! 3I A combination of two conductors, very close together, one of which is connected to the earth, is called a condenser. The capacity of such a condenser is enormously greater than the capacity of either of the conductors of which it is composed when measured in the absence of the other. In order to become familiar with the phenomena of the condenser, two forms are to be experimented with : I. The first form is an apparatus consisting of two vertical, parallel metal plates. These plates are both insulated, and are capable of motion along a line joining their centers. (1) Fasten a pith-ball by means of a conducting thread to one plate, so that the ball rests against the plate. (2) Connect the second plate to the earth, and charge the first one by means of an electrical machine. (3) Move the plates to and from each other, and note the effect on the pith-ball. (4) When the plates are quite near together, insulate the plate that was formerly grounded, and afterwards discharge the other plate by grounding it. Then separate the plates, and note the effect on the pith-ball. (5) Fasten a pith-ball, as above, to the second plate also. Charge the plates while I or 2 cm. apart by connecting them to the opposite terminals of an electrical machine. Insulate the two plates without grounding either of them, and determine the character of the charge on each plate, by bringing a charged body whose condition is known near each pith-ball in turn. (6) Connect one of the plates to the ground for an instant, and observe the effect on the sign and magnitude of the charges. Do the same with the second plate. Continue grounding alter- nately the two plates until they are both very nearly discharged. (7) Charge the plates again, and observe the effect of con- necting them by means of a good conductor. Repeat these 132 JUNIOR COURSE IN GENERAL PHYSICS. observations with a glass plate between the metal conductors, the latter being very close, or in contact with the glass. II The other form of condenser to be experimented with is a Ley den jar. (1) Place the jar on an insulating support, and charge it by connecting the two coatings to the opposite terminals of an electrical machine. (2) Disconnect from the electrical machine without ground- ing either coating, and experiment as with the plate condenser. (3) Determine the number of alternate groundings of the two coatings necessary to reduce the charge to a definite fraction of its original value. For the purpose of this determination, the assumption may be made that the charge is proportional to the length of spark, when either coating is grounded, the other coating having been previously grounded and then insulated. (4) When the jar is fully charged, make metallic connection between the two coatings. After a few minutes connect the coatings again, and note the existence of the "residual charge." (5) Try to charge the jar by connecting only one coating to the electrical machine, the other coating being insulated. Investi- gate the nature of charges on the two coatings, and afterwards discharge the jar, observing whether the spark is comparable with that obtained when the jar was charged by the method first given. The above-described experiments should be repeated several times in order to be certain of the results and to become familiar with the phenomena. Addenda to the report: (i) Indicate whether in the case of a condenser with a gas or a liquid as dielectric, there would be anything com- parable to the residual charge of a Leyden jar. STATIC ELECTRICITY. 133 (2) Indicate why it requires a very large number of alter- nate groundings of the two coatings of a condenser to per- ceptibly reduce its charge. (3) Assume that the alternate groundings of the two coat- ings are at equal intervals of time, and draw two curves with times as abscissas and potentials of the two coatings as ordi- nates. (4) Draw a vertical section of the jar, with coatings quite wide apart and showing lines of force and the vertical sections of equipotential surfaces. (5) Draw two such figures, one in which the potential of one coating is zero, and one in which the surface of zero potential lies between the coatings. (6) Determine approximately the electrostatic capacity of the jar from its dimensions. (7) Assume the difference of potential between the coatings to be 100 electrostatic units, and compute the electrostatic force in the glass between the coatings. (8) Compute the total charge in the jar under the above conditions. (9) Compute the energy of the charge. EXPERIMENT P 3 . The Holtz machine. In all influence machines, mechanical energy is directly transformed into the energy of electrification. The object of this experiment is to familiarize the student with the use of such machines and the principles involved in their action. Any type of influence machine may be used. The following is the procedure : (i) Run the machine a few seconds until it is fully charged. The poles should be a few centimeters apart. Then stop the machine, and determine, by means of a pith-ball, the character of the charge on every part of the machine. Repeat these observations several times, and observe whether the polarity of the machine becomes reversed. 134 JUNIOR COURSE IN GENERAL PHYSICS. (2) While the machine is charged and at rest, gradually bring the terminals together until a discharge takes place, and observe the effect upon the pith-ball. Determine the character of the charges on different parts of the machine when it is running steadily with the terminals too far apart to allow a discharge. (3) Observe the difference in the discharge when the Leyden jars are removed, also when they are replaced by larger ones. (4) Determine the maximum distance between the terminals/ at which a discharge will pass when the machine is running stead- ily, but not very rapidly. Remove the crossbar, and determine the maximum length of spark between the terminals when the machine is running at the same rate as before. (5) Reverse the direction of rotation, and determine under what conditions the machine will work. (6) Take the machine into a dark room, and run it steadily (a) with the crossbar in position, and the terminals in contact ; (b) with the crossbar in position, and the terminals very wide apart ; (c) without the crossbar, and with the terminals first in contact, and afterwards widely separated. Observe carefully the brush discharge between the revolving plate and the combs in all these cases. Addenda to the report: (1) Indicate the results, by positive and negative signs, upon carefully drawn diagrams of the machine. (2) Explain how the machine becomes highly charged when one armature is given a small initial charge, and the plate is steadily revolved. (3) Indicate the function of the crossbar, and the most advantageous position for it. (4) Indicate the function of the Leyden jars. EXPERIMENT P 4 . The Holtz machine (continued}. After performing Exp. P 3 , the following further experiments with an electrical machine will be found very instructive : STATIC ELECTRICITY. I. 135 Remove the Leyden jars, and connect to each terminal of the machine one coating of a condenser whose capacity may be varied in a known manner. Connect together the remaining coatings of the two condensers. Condensers formed by coating the whole of one side of a glass plate with tin-foil, while on the other side are several pieces of tin-foil insulated from each other, and of equal area, serve very well for this purpose. Place the terminals at a fixed distance apart of 2 or 3 cm., and run the machine uniformly, counting the number of dis- charges per minute. Vary the capacity of the condensers connected to the terminals, and repeat these observations. If the machine works uniformly, it will be found that the number of sparks per minute varies inversely as the capacity of the condensers. This fact may be readily shown to follow from the as'sumption that the amount of electrical work done when the machine is running uniformly is directly proportional to the time, and independent of the capacity of the condenser used. On account of the uncertainty of the conditions, it will be necessary to take a large number of observations, and to use their mean in testing the truth of the above statement. II. An electrical machine is a generator of electricity, and under conditions that are not variable it has a constant electromotive force, and a constant internal resistance. If the machine is in good working condition, and is rotated uniformly in an atmosphere of constant humidity, its electro- motive force and internal resistance will not be greatly variable, provided that the external resistance is not greater than a few million ohms. To verify this statement, connect the terminals of the machine to the terminals of a high-resistance sensitive galva- nometer. Turn the machine uniformly, and observe the galva- 136 JUNIOR COURSE IN GENERAL PHYSICS. nometer readings for current, both direct and reversed. Place in series with the machine and galvanometer a resistance of 100,000 ohms. The change in the galvanometer deflection will probably be imperceptible. This shows that the internal resist- ance of the machine is enormous, and that the electromotive force is probably very great. In order to determine these constants, the method of Exp. S 2 may be used. Place in series with the galvanometer and machine a variable resistance of several megohms.* Observe the galvanometer readings for several different resistances pro- ducing quite different deflections. For each deflection, the current in amperes may be calculated if the constant of the galvanometer is known. As in Exp. S 2 , we have E If E and R Q are constant, they may be determined from any pair of observations giving two values of R and two of /. If several different values of R are used, plat a curve with resistances as abscissas, and reciprocals of currents as ordinates. This will be a straight line from whose constants E and R may be determined. Owing to the difficulty of maintaining a uniform rate of rotation, and other unavoidable causes affecting the current, the values of the current used in computations should be the mean of several observations. It must be remembered that the potential difference of the terminals of the machine, and consequently the electromotive force, is enormously greater when the terminals are separated by several centimeters, and the resistance between them is thou- sands of megohms. * Heavy black-lead lines on wood will serve for high resistances. If divided into sections of about a million ohms, these resistances may be measured by means of a Wheatstone bridge. STATIC ELECTRICITY. 137 III. Replace the Ley den jars by large ones. Separate the terminals to a distance of 8 or 10 cm., and run the machine until the jars are charged. Then slip off the belt, and stop the revolving plate with the finger. Under favorable circum- stances, the plate will start to rotate backwards, and continue to do so for quite a number of turns. After a successful trial, it will be found that the jars are very nearly discharged when the plate ceases to rotate. Addenda to the report: (1) If the electrostatic capacity of the condenser used is known, as well as the electrostatic difference of potential producing the sparks of known length, calculate the electrical work done in ergs per second, in watts. (2) Prove upon theoretical grounds that the number of sparks per minute is inversely proportional to the capacity of the condenser used. (3) Indicate the kind of battery that would produce an effect similar to that of the machine in the second experiment of this section. (4) Indicate the cause of the backward rotation in the third experiment above. CHAPTER V. GROUP Q : MAGNETISM. (Q) General statements ; (QJ Lines of force and the study of the magnetic field ; (Q 2 ) Determination of the magnetic moment of a bar magnet by the method of oscillations ; (Q 3 ) Determination of magnetic moment by the magnetom- eter ; (Q 4 ) Measurement of the intensity of a magnetic field ; (Q 5 ) Distribution of free magnetism in a permanent magnet. (Q). General statements concerning magnetism. The phenomena of current electricity and of magnetism are almost, if not quite, inseparably connected. In the medium surrounding a conductor conveying a current of electricity, magnets are acted upon by a force. Such a region is naturally called a magnetic field of force. Imaginary lines showing at all points the direction in which the force acts are called lines of force. Greater intensity of a field of force is usually represented by a greater number of these lines intersecting a given area. If masses of iron are brought into such a magnetic field of force, the intensity of the field is greatly increased, in the neighborhood of those parts of the iron where the lines enter and emerge. The same is also true of some other substances. This fact may be explained by saying that these substances are much better conductors of lines of force than the air or ether, or that their "permeability" for lines of force is greater than the permeability of the air. Such good conductors of magnetic lines of force are called magnetic substances. Those portions of the magnetic lines which lie within a mag- netic substance are called lines of magnetization. A magnetic 138 MAGNETISM. 139 substance containing these lines is said to be magnetized, and is called a magnet. Some magnetic substances, steel for example, may be removed from the magnetic field where they have been magnetized, without losing their magnetic properties. The magnetic field surrounding the magnet moves with the magnet, and seems to have a fixed connection with it, indepen- dent of any other magnetic field. Such a body is called a permanent magnet. In all localities where the experiment has been performed, such a magnet is acted on by a force. It follows that there is a magnetic field surrounding the earth. A magnet suspended in a magnetic field so as to turn freely about its center of gravity always comes to rest with its longer axis in a particular direction. The direction of this axis is always tangent to lines of force, the positive direction of the line being the direction in which the end of the magnet points, which points north in the earth's field. The end of a magnet which points north in the earth's field is called the positive end, the other end being called negative. If such a suspended magnet be brought into a field about the negative end of another magnet, it will set itself with its positive end pointing towards the negative end of the other magnet. The reverse is true in the field about the positive end of the second magnet. It follows from this that the lines of force of the field due to a magnet diverge from its positive end, and converge towards its negative end. Such a region within a magnet, towards which the lines of force converge, or from which they diverge, is called a pole. A convenient statement of the fact that a magnet always tends to point in a particular direction in a magnetic field may be based upon the principle just laid down ; viz. : The positive pole of a magnet always tends to move along magnetic lines of force in the positive direction, and the nega- tive pole in the negative direction. The mutual action two magnets when brought near together may also be 140 JUNIOR COURSE IN GENERAL PHYSICS. in the following form : Like poles repel cadi oilier, and unlike poles attract each other. In a real magnet, lines of force diverge from a region in the positive half, curve around through space, and converge to a region in the negative half, and then pass on through the mag- net as lines of magnetization. The idea of a pole, as a point towards which lines of force converge, is a highly idealized conception. It is a very useful conception, however, and by most authors is made the basis of the whole system of electro- magnetic units. This ideal conception of a magnet pole is not likely to lead to error except in one case ; namely, when the intensity of the force at a point in a magnetic field is expressed as a function of the strength of its poles, and the distance between them, as in Exp. O 3 . An ideal magnet with ideal poles of a given strength and a given distance between them may be conceived, such that the magnetic field would be at four symmetrical points, exactly like the field produced by a real magnet. But the field of the real magnet would be different at all other points from the field of the ideal magnet. For example, the field of a magnet, quite close to its middle point, is such as would be produced by an ideal magnet with poles comparatively close together, while the reverse is true for a point near either end of the magnet. The error introduced into Exp. Q 3 by the assumption made is quite small whenever L > 3 /. Notwithstanding what has been said about magnet poles, the term "magnetic moment " has a perfectly definite physical meaning. If a magnet be placed in a magnetic field with its axis at right angles to the lines of force, it will be acted upon by a turning force. If the moment of this turning force be represented by G, and the intensity of the field by H, the magnetic moment of the magnet may be defined by the re,la- tion MH=G, (101) in which M is the magnetic moment. MAGNETISM. 141 EXPERIMENT Q r Lines of force and the study of magnetic fields. Surrounding every magnet and every current of electricity there is a magnetic field. The earth also has a magnetic field surrounding it. The object of this experiment is to investigate the direction in which the force acts in such fields; that is to say, the direction of the lines of force. For this purpose place a sheet of glass immediately above the magnet whose field is to be investigated, and scatter over it iron filings, allowing them to drop from a height of 8 or 10 inches. If the magnet is sufficiently strong, the filings will arrange them- selves in " lines offeree." A slight tapping or jarring of the glass will probably make the magnetic curves more perfect. Sheet metal (not iron) or paper may be used instead of glass if desired, but the glass plate has the advantage of allowing the position of the magnet to be clearly seen. Permanent records of the curves may be obtained by allowing the filings to arrange themselves upon a sheet of blue print paper, and exposing the latter to the sun while the filings are still in position. Among cases which may be studied to advantage in this manner are the following : (1) The field of a single " horseshoe " magnet. (2) Two magnets with like poles near each other. (3) Two magnets with unlike poles near each other. (4) A bar magnet placed in the neighborhood of a horse- shoe magnet. (5) The field of two horseshoe magnets placed vertically, their four poles forming a square. Many other more complicated arrangements will suggest themselves. Observe also the effect of pieces of soft iron, placed in different positions in the field, upon the form of the curves obtained. If the piece of soft iron seems to produce little effect, bring it in contact with one pole. The direction of the magnetic force at any point will be indicated by the direction in which a small compass needle 142 JUNIOR COURSE IN GENERAL PHYSICS. will set itself when placed at that point. By shifting the compass from place to place, the direction of the force can thus be found at any number of points. To study the field by this method, place one or more magnets in the middle of a large board ruled in squares, which has been previously set with two opposite edges parallel to the magnetic meridian. The board should be so large that at the edges the field due to the magnets is decidedly weaker than the earth's "field. By means of a small compass determine the direction of the lines of force for a large number of points. There should be enough of these observations, so that the direction in which the compass needle would point if placed anywhere on the board may be known within rather narrow limits. Make a diagram (to scale) of the board and magnets, and at each point where the compass was placed draw a little arrow to show the direction of the force. An arrow should also be drawn somewhere on the board to show the direction of the earth's field. Map the whole field on the board by means of lines representing the lines of force. These lines do not need to pass through the arrows, but shquld be so drawn as to represent the direction in which the compass needle would point if placed upon corresponding points on the board. The field so mapped is the resultant field of the magnets and of the horizontal component of the earth's field. Therefore, it must not be expected that all the lines of force will enter a magnet. In the neighborhood of every magnet or system of magnets there are in general two or more points where the magnetic field due to them exactly neutralizes the earth's field. At these points there will be no directive force acting on the compass needle, and on opposite sides of these points the needle will point in opposite directions. Locate these points on your diagram. MAGNETISM. 143 Addenda to the report: (1) State the law of magnetic force. (2) Define : unit magnet pole ; magnetic field of force ; field of unit intensity ; magnetic difference of potential ; magnetic potential at a point ; equipotential surfaces ; mag- netic lines of force ; unit line of force, or unit tube of force. (3) Show that lines of force and equipotential surfaces- are mutually perpendicular. (4) Indicate the reason why, in triis experiment, the filings move away from points directly above the magnet, especially in the neighborhood of the poles. (5) Draw the horizontal sections of several equipotential surfaces whose potentials differ by equal amounts. (6) Assume that the potential of any point a centimeter from the north pole is 100, and that the surface of zero potential bisects the distance between the poles, and determine approxi- mately from the map and from the assumptions already made, the magnetic force at- several points 10 or 20 cm. distant from the magnet. EXPERIMENT Q 2 . Determination of the magnetic moment of a bar magnet by the method of oscillations. A magnet suspended by a torsionless fiber with its axis horizontal will come to rest with its magnetic axis in the mag- netic meridian. If the magnet is turned so as to make a small angle with the magnetic meridian, the moment of the force tending to restore the magnet to its position of equilibrium is directly proportional to the angular displacement. The result- ing motion of the magnet, when left free to vibrate, is therefore a simple harmonic motion. The periodic time is dependent upon the magnetic moment of the magnet, its moment of inertia, and the horizontal intensity of the magnetic field in which it is suspended. The following equation may be derived by equating the kinetic energy of the 144 JUNIOR COURSE IN GENERAL PHYSICS. magnet at its mid-position to the potential energy when at the end of its swing. The method of derivation is the same as that pursued in Exp. E r (102) To perform the experiment : (1) Place the magnet in a small wire stirrup, and suspend it by a few untwisted silk fibers. It should be suspended in a box with glass ends, to avoid the effect of air currents, and a position should be chosen at a distance from movable masses of iron. If the bar is rather strongly magnetized, the torsion of the silk fiber may be neglected, or, if desired, it may be elimi- nated by determining the ratio of the moment of torsion to the moment of the magnetic forces.* (2) Set the magnet to vibrating through an arc of not more than five degrees. Determine the period of oscillation by the method of Exp. A 6 , or by counting the number of passages in the same direction during five or six minutes. The period should be determined several times, and as carefully as possible. (3) Measure the length, diameter, and mass of the bar, and from these data compute its moment of inertia. If the value of the horizontal component of magnetism for the place where the magnet was suspended is known, the value of M may be com- puted from the above equation. The method of oscillations may be used, if desired, to deter- mine the value of H in different parts of the laboratory ; in which case the period of oscillation must first be determined from observations taken in a locality where H is known. If this experiment and the following one be performed with the same magnet and at the same place, both H and M may be determined in absolute measure. * Kohlrausch, Physical Measurements, p. 128. MAGNETISM. ! 4 5 Addenda to the report: (1) State the effect upon the period of oscillation, if in the above experiment the magnet were not quite horizontal. (2) State the effect upon the period, if the magnet were bent into the form of a horseshoe, without changing its intensity of magnetization. (3) Determine the average intensity of magnetization in the magnet experimented with, and indicate in what part of the magnet it is the greatest, and in what part the least. (4) Compute the ergs of work that would be required to rotate the magnet 180 about a vertical axis in the earth's field from the position of rest. (5) Assuming the dip 75, compute the work required to rotate the magnet from a vertical position through 180 about a horizontal axis. EXPERIMENT Q 3 . Determination of magnetic moment by the magnetometer. When two forces act at right angles to each other, their resultant makes an angle with each of the forces such that the tangent is the ratio of the two forces. See Fig. 46, in which a and b are the forces and and /3 the angles which their resultant r makes with them respectively. Obviously, tana=- and tan/3 = -- a b This fact may be used to determine the ratio of the intensity of two magnetic fields at any given point. When a magnet is so placed that the field due to it at a given point is at right angles to the field due to the earth, a short magnetic needle placed at that point will be deflected from the magnetic meridian through an angle whose tangent is equal to the ratio between the intensities of the two components of the field at that point. VOL. I L 146 JUNIOR COURSE IN GENERAL PHYSICS. The needle must be short with respect to the distance to the magnet, for otherwise its ends would extend too far beyond the point at which the field of the magnet has the intensity given below. The strength of the field at any point due to a magnet is a known function of its magnetic moment, the distance between its poles, and the distance of the point from the magnet. H B I Fig. 47. For example, the strength of the field at A, due to a magnet whose poles are at n and s (Fig. 47), is 2 ML ( . At B, the strength of the field due to the same magnet is (104) in which M is the magnetic moment of the magnet, 2. 1 the dis- tance between its poles, and L the distance of A or B from the center of the magnet. If the magnet is at right angles to the magnetic meridian, a magnetic needle placed at A will be deflected from the meridian through an angle B according to the relation = (i 1 3). in which 7 is the constant per scale division. The error in using this approximate value for the current is as follows : When = , the error is about 0.0025. L 10 When = -, the error is about o.oioo. L 4 When = -, the error is about 0.0600. L 2 From an inspection of Fig. 5 2 a, it is obvious that a scale might be so curved that deflection of the ray of light, as measured along the scale, would be directly proportional to THE ELECTRIC CURRENT. 161 tan S. This curve would not be one easy to construct, but it can easily be proved that if a scale S' be made upon the arc of a circle whose radius is f of the distance from the mirror to the scale immediately under the telescope, the error in assuming that tan S is proportional to the deflection along the scale does not exceed o.ooi for any deflection of the needle not exceeding 24. When the current is reversed in a galvanometer, it often takes several minutes for the needle to come to rest. Time Fig. 52 a. may be saved by closing the circuit as the galvanometer needle gets about to the end of its free swing in the direction in which the current would deflect it. There are two principal methods of "damping" the oscil- lations of galvanometer needles, and making the instrument nearly or quite "dead-beat." (1) By the attachment of a mica or aluminum vane to the needle. This vane, by friction against the air in an inclosed place, brings the needle to rest much more quickly than would otherwise be the case. The action may be increased by suspending the vane in a vessel of oil. (2) By suspending the magnetic needle within a cavity in a small mass of copper. As the magnet moves in this cavity, causing the lines of force to sweep through the copper, 162 JUNIOR COURSE IN GENERAL PHYSICS. currents of electricity are induced. These currents, as stated in Lenz's law, are always in such a direction as to oppose the motion which produced them. The best example of this is found in the type of instrument first designed by Siemens. In this form of galvanometer the magnetic needle is of the horseshoe type, ordinarily called a bell magnet. This magnet is suspended in a hole but little larger than itself in a copper sphere. Fig. 53 shows a vertical section of the cop- per sphere, with the inclosed magnet ; Fig. 54 represents a cross-section of the same magnet. In the use of sensitive galvanometers, the question will often arise as to what type of galvanometer will be most sensitive for a particular purpose. This question cannot be settled in general terms, on account of the great difference between the different types. The more restricted question as to what number of turns will be most sen- sitive for a particular purpose may be easily determined. If the type of galvanometer, the size of the coils, the mass of copper in them, and the closeness of the turns to the needle remain the same, it may readily be proved that that instrument will be most sensitive whose internal resistance is equal to the external resist- *ance in series with it. This relation may be demonstrated as follows : Let Fbe the volume, L the total length, s the cross-section, R g the resistance, and p the specific resistance of the wire to be used in the galvanometer coils. Then we have Flg . 54 . (H4) THE ELECTRIC CURRENT. 163 If 7 is the constant per scale division of the galvanometer, we have 7 = , in which C is a constant depending on the type -L> t of galvanometer. If E is the electromotive force, R the external resistance, and S the galvanometer deflection, we have If this equation is differentiated with respect to L, we have * dL Consequently S will be a maximum when R=L*=R f (116) a result that coincides with the statement which we desire to verify. EXPERIMENT R 1 . Law of the tangent galvanometer. If a galvanometer coil is placed with the plane of its wind- ings in the magnetic meridian, the magnetic field due to a current circulating in the coils will be (in the axis of the coil) at right angles to the earth's magnetic field. The resultant of these two fields will therefore make an angle with the magnetic meridian whose tangent is the ratio of the intensity of the earth's field to the field due to the current in the galvanometer coils. A short magnetic needle suspended anywhere in the axis of the coil will set itself along this resultant direction. If the needle is not a short one, it will, when considerably deflected JUNIOR COURSE IN GENERAL PHYSICS. from the meridian, extend beyond the axis to points where the two components of the field are not at right angles to each other. Therefore the tangent law will no longer hold. This experiment is intended to give a method of testing experimentally whether a given galvanometer obeys the law of tangents (i.e. whether the tangent of the angle of deflection is proportional to the current). Connect the galvanometer in series with a resistance box and cell, and place a reversing key somewhere in the circuit so that the direction of the current in the galvanometer can be readily reversed. Then observe the reading of the galvanom- eter, with current both direct and reversed, for ten or twelve different values of the resistance in the box, choosing the resist- ances so that the deflection of the galvanometer is varied from the smallest that can be accurately observed up to the largest that can be used. From the data thus obtained, plat a curve, using resistances as abscissas and cotangents of deflections as ordinates. If a reflecting galvanometer is used, it will be necessary to compute the angle of deflection from the linear deflection and the dis- tance between mirror and scale. If the galvanometer obeys the tangent law, the curve obtained by platting as above should be a straight line. Draw a straight line, therefore, that passes as nearly as possible through all the points, and produce this line backward until it intersects the horizontal axis. The dis- tance between the origin and this point of intersection is a measure of the resistance in the circuit outside the resistance box ; i.e. if we call this resistance R^ then ^ = galvanometer resistance + battery resistance + resistance of connecting wires. Now, if E is the E. M. F. of the cell, and R the resistance in the box, then (117) If E is known, and R Q determined from the curve, as stated above, the constant of the galvanometer can be computed. In THE ELECTRIC CURRENT. I6 5 making this computation, take the E. M. F. of a gravity cell as I volt.* In some reflecting galvanometers in which the deflection is small, the deflection itself is nearly proportional to the current ; i.e. if the deflection on the scale is 8, then 7=/ S. To test whether this is true in the case of the galvanometer used, plat a curve with box resistances as abscissas, and reciprocals of deflec- tions as ordinates. If / = / S, the line obtained will be straight. By some students, the following method of using the data may be preferred. From the data, determine the value of ^ , as in Exp. T 5 . Compute the value of the current in each case \ .,[_ LAWC >F TAN iENTO M.VANC METER ^^ ^* +*~ \ 4 I 1 1 ^^t^ l^f* ^* ^ lil 3 | oj ^^- ^^ ^ 8 1 ^f^ ^ ^ " ^^ ,4- "**'^ B0> C RES STAN CES 10 12 14 Fig. 55. 16 IB 20 22 24 26 28 from Ohm's law. If the galvanometer obeys the tangent* law, the ratio of current to tangent of deflection will be a constant. The following data are typical of the results to be obtained, and will serve to indicate the method of arranging and tabulat- ing readings. The curve given in Fig. 55 shows the graphical method of testing the deviation of the instrument from the law of tangents. * The E. M. F. of the battery used in taking the following data was 4 volts. 1 66 JUNIOR COURSE IN GENERAL PHYSICS. LAW OF TANGENT GALVANOMETER. RESISTANCE IN Box. GALVANOMETER READINGS. MEAN DEFLECTION. tan 8 cot 8 /o Direct. Reversed. r N. 62.5 t S. 62.0 N. 6 3 .o 1 S. 6 3 .5 I 62. 70 1-937 0.516 0.6 1 3 2 r N. 5o.6 1 S. so.o N. 50 .0 \ S. so.2 J 50. 20 I.2OO 0-833 0.625 4 / N. 4 i.5 I S. 4i.o N. 4 i.o | S. 4i.2 / 4I.20 0.875 I.I42 0.62 3 6 / N. 34 .8 \ S. 34 .2 N. 3 4.5 \ S. 35 .i / 34-8o 0.695 1-439 0.616 10 r N. 250.5 I S. 25.o N. 250.5 I S. 26 .0 / 25-5Q 0.477 2.096 0.628 14 f N. 2o. 4 1 S. i9.8 N. 20. I ) S. 20.6 / 20. 20 o. 3 68 2.718 0.627 18 f N. i7.o 1 S. i6-5 N. i7.o 1 S. i7.6 J i7.oo o. 3 o6 3-271 0.610 24 f N. i 3 . 4 1 S. I2.9 N. 130.2 ) S. i 3 .8 / I3-30 0.2 3 6 4-230 0.620 30 f N. io.9 1 S. io. 3 N. io.9 ) S. ii. 4 J io.90 0-193 5-193 0.621 50 f N. 7 .o I S. 6.s N. 70.1 1 S. 7.6 / 7-05 0.124 8.086 o.6i 3 From curve JR Q = 3 . 33 ohms. " / = 0.620 amp. Last column computed assuming value of ^ obtained from plat. EXPERIMENT R 2 . Measurement of current by electrolysis. One of the most accurate methods of measuring current is by means of the amount of copper or silver deposited in a vol- tameter through which the current flows. The voltameter deposit represents the integrated value of the current extending over considerable time ; that is, it is a measure of the total quantity of the current which has flowed through the voltameter. This instrument, therefore, can only give an average value of the current. On account of this and other disadvantages, the voltameter is chiefly used to calibrate THE ELECTRIC CURRENT. l6 7 or determine the constants of instruments which depend for their indications on the magnetic field produced by the current. In this experiment the spiral coil voltameter devised by Professor H. J. Ryan is to be used.* Two coils are to be prepared for each cell by wrapping copper wire on cylindrical forms. The size of the coils depends somewhat on the strength of the current used. With a current of from one to three amperes, a coil made of one and a half meters of wire of 1.5 mm. diameter will give satisfactory results. The coils should be of about the same length, but should differ in diameter by 3 or 4 cm., so that the smaller may be placed inside the other without danger of touching. (See Fig. 56.) At one end of each coil the wire is to be brought out parallel with the axis for several inches for convenience in making connections. These two coils are to be used as the electrodes of a voltameter cell, current passing in through the outer coil and leaving the cell by the inner coil. The amount of copper deposited in a known time is then sufficient to determine the average current flow- ing. (One coulomb deposits 0.000328 gram of copper.) The amount of copper dissolved is always slightly in excess of the amount deposited, and for various rea- sons is not so reliable a measure of the current. In preparing the gain coils, great care must be used to have them thoroughly clean. A wire of suitable length for the pur- pose should be fastened by one end and then cleaned with sand- paper. When thoroughly cleaned, the wire is coiled upon a Fig. 56. * See Ryan, Transactions of the American Institute of Electrical Engineers, vol. 6, p. 322. i68 JUNIOR COURSE IN GENERAL PHYSICS. suitable form, the latter being first covered with clean filter- paper. After cleaning with the sandpaper, the coil should not be touched by the hand anywhere except at its terminal. If this work has been well done, the coils will be ready for use without any further cleansing. If not, pass the coil through a non-luminous Bunsen flame to remove oil, plunge it in a very dilute solution of sulphuric acid, and then into distilled water. To dry the coil rapidly and without danger of oxidation, it is first rolled on filter or blotting paper until only a thin film of water remains. This is rinsed off by dipping into strong alco- hol. After again rolling on filter-paper, what little alcohol is left will quickly evaporate, leaving the coil dry and ready for Fig. 57. weighing. The loss coil should also be cleansed with sand- paper, but it is unnecessary to use the precautions that are required in the case of the gain coil. The density of the copper sulphate solution should lie between i.io and 1.18. A few drops of sulphuric acid will improve the action of the solution. The direction of the current should be determined by a compass needle before the voltameter is placed in the circuit. The connections can then be made in such a way as to make the deposit occur on the inner coil. In this experiment the voltameter is to be used in determin- ing the constant of a tangent galvanometer. Two voltameter cells (Fig. 57) are used as a check on the weighings, the two cells being connected in series with the galvanometer and with each other. Figure 58 gives a diagram of the connections. A THE ELECTRIC CURRENT. I6 9 steady current is sent through the circuit for some measured length of time, and the strength of the current is computed from the amount of copper deposited. The deflection of the galvanometer having been also observed, the constant is readily computed. A reversing key should be used in connection with the galvanometer, the construction of the key being such that the current through the galvanometer can be reversed without breaking the current in the main circuit. Deflections, both direct and reversed, are to be observed at intervals of two or three minutes throughout the experiment. The gain coils must be weighed with great care, and placed in the solution only a few minutes before the current is started. At the end of the experiment they should be immediately removed, dipped in distilled water, and dried, as described above. The second weighing should be made as soon as possible after the coils are dry. The constant of any galvanometer may be measured by this method. Fi s- 58. In the case of instruments, the sensitiveness of which is so great that currents of the magnitude adapted to the voltameter cannot be measured directly, a shunt of suitable resistance, ^ (Fig. 58), should be placed across the galvanometer terminals. The ratio of the current in the voltameter to that which flows through the coils of the galvanometer can readily be computed. The following table gives the results obtained in the cali- bration of a tangent galvanometer, and shows the method of arranging them : 170 JUNIOR COURSE IN GENERAL PHYSICS. GALVANOMETER CONSTANT BY COPPER VOLTAMETER. GALVANOMETER READINGS. Other Data and Results TIME. Current Direct. Current Reversed. Distance of mirror from scale 50, scale Average double deflection = 32.27, ' div. tan 28 o ^227 25 17 C'*' hr. mm. 5 = 8 56^' tan 5 = 0.1573 9 49 Circuit completed. 51 42.40 Two voltameter cells in series : 54 74.62 Before. After. Gain. 57 10 I 5 74.62 42.43 42-43 Cathode A ... 27.434 28.686 " B . . . 27.5715 28.624 1.2520 1.2525 9 74.68 Duration of run = 3600 sec. 13 42.39 17 74.70 Intensity of current / = 1.059 amp. 21 42.38 For tangent galvanometer, /= 7 tan5; 25 74.67 /. 7 = 6.73 amp. 2 9 42.32 33 74.70 Galvanometer, one turn, needle at center : 37 42.31 Diameter of ring =77.7 cm. 45 74.57 42.30 True constant G 0.1617. 49 74.56 /- I0 ^ ; .;ff= 0.109. 49 Circuit broken. G * EXPERIMENT R 3 . Measurement of the constant of a sensitive galvanometer. It is frequently impracticable to calculate the constant of c. sensitive galvanometer from its dimensions and from the value of the horizontal intensity of magnetism at the point where the needle hangs. The constant of such a galvanometer can best be determined by measuring the deflection of the needle which a known current produces. The constant can then be deter- mined from one of the equations : /=/ tanS, 7=/ sinS, (nS) 7=/ S, according to the law of the galvanometer. THE ELECTRIC CURRENT. l j l There are three principal methods of determining the con- stant of such a galvanometer, depending upon the method of determining the current flowing through the galvanometer coils. I. The current may be measured by means of a tangent galvanometer whose constant is already known. For this pur- pose it will be necessary to put a shunt across the terminals of the sensitive galvanometer, since the latter will usually be very much more sensitive than the instrument whose constant is already known. The method of procedure is as follows : (1) Connect the tangent galvanometer in series with a battery of constant E. M. F., a reversing key, and a variable resistance. (2) Connect the sensitive galvanometer so that it shall be in multiple with a portion of the variable resistance, as in Fig. 59. The variable resistance should be so adjusted that the deflection of the tangent galvanometer is the most suitable (about 45, if the deflection of the needle is read directly, or nearly as large a deflection as the scale will per- mit, if the reading is made by Fig. 59. means of a mirror). (3) Adjust the resistance in multiple with the sensitive galvanometer, until its deflection is nearly across the scale. Observe the readings of both galvanometers for direct and reverse current, and repeat these observations several times to get a good average. As a check, take another series of observations with a different resistance in multiple with the sensitive galvanometer, producing a deflection varying con- siderably from the first. From the deflection of the tangent galvanometer, the cur- JUNIOR COURSE IN GENERAL PHYSICS. rent flowing in the main circuit is known ; and from the law of divided circuits, the fraction of the current flowing through the sensitive galvanometer can be computed, provided the ratio of the galvanometer resistance to the shunt resistance is known. II. If the tangent galvanometer be replaced by a voltameter, the current in the main circuit can be measured as in Exp. R THE MEASUREMENT OF RESISTANCE. 213 and the temperature coefficient for resistance is defined by the equation (133) in which R t and R are the resistances at temperatures t and o, respectively, and a is the coefficient. For metals a is positive. Method of the Wheatstone bridge. The wire to be tested should be insulated, and coiled in the form of a solenoid sufficiently small to slip into a long test tube, or ordinary glass tube sealed at one end. Heavy insulated copper wires are to be soldered to the two ends of the coil and brought to the terminals of the Wheatstone's bridge. Make the test wire of such size and length that its resistance will be from two to ten ohms (the higher, the better). Place a thermometer in the tube with the wire, the bulb being at the center of the coil, and fill the upper part of the tube with waste cotton, asbestos, or similar material to prevent the circulation of air. Then immerse the tube in a water bath and measure its resistance at different temperatures, ranging from zero to the boiling-point. Readings of resist- ance should be taken both for increasing and decreasing tem- peratures, and the thermometer should be read before and after each measurement, the mean of the two readings being used. Let the changes of temperature take place very gradually, and keep the water thoroughly stirred. For practice determinations of the temperature coefficient of carbon, an incandescent lamp may be used instead of a wire. From the results obtained, plat a curve on cross-section paper, using temperatures as abscissas, and resistances as ordi- nates. This curve, in the case of most metals, will be very nearly a straight line. Draw a straight line as nearly as possi- ble through all the points, and determine its equation. From this equation determine the temperature coefficient a and the resistance at o. 214 JUNIOR COURSE IN GENERAL PHYSICS. II. Fall of potential method. (See Exp. T 2 .) In this case, a wire of low resistance can be used to advan- tage. Two wires, not necessarily large, are to be soldered to each end of the test wire, one pair serving to carry the current, and the other pair leading to the galvanometer. When the coil is at the temperature of the room, adjust the resistance in series with it, so that the galvanometer deflection is about two-thirds the distance across the scale. It is very important that the temperature remain very nearly constant during an observation of the galvanometer reading. The tem- perature should be taken as nearly as may be at the same instant the galvanometer reading is observed, both direct and reversed. The mean of these two temperature observations is to be used in the computations. For the determination of the temperature coefficient it is not necessary to have any absolute standard of resistance. Since galvanometer deflections are proportional to resistance, we may substitute for R t and R Q the deflections 8 t and S (equation 133), or their tangents, if a tangent galvanometer is used. After making the necessary readings, a curve should be platted, with temperatures as abscissas and galvanometer deflections as ordinates. The equation of this line is then to be determined, and from its constants the temperature coefficient and deflection for o are to be calculated. Addenda to the report: (1) Justify the substitution of galvanometer deflection for resistances in the above equation. (2) Using the coefficient determined, calculate the resistance at absolute zero of wire whose resistance is 100 ohms at o C. EXPERIMENT T 6 . Measurement of the internal resistance of a battery by Ohm's method. This experiment requires the same observations as Exp. S 2 , and the battery resistance may be calculated from the observa- THE MEASUREMENT OF RESISTANCE. 215 tions taken in that experiment, provided the resistance of the galvanometer and of the connecting wires is known. It is not necessary, however, to know the constant of the galvanometer. From Ohm's law we have 7 = in which R is a known resistance, R* and R g the battery and galvanometer resistances respectively. The last named includes the resistance of the connecting wires. For / may be substi- tuted 7 S (or 7 tan 8, in case the current is proportional to the tangent of the deflection of the galvanometer needle). If two different values of R be taken, and the corresponding galvanometer deflections observed, we shall have two equations similar to 134. If one of these equations be divided by the other, both E and 7 will be eliminated, and R b will be a function of known quantities. This experiment furnishes an excellent example of the general principles discussed on page 4. The precautions there suggested should be followed here ; that is to say, the difference between the two currents in the observations by means of which E and 7 are eliminated, and R b is determined, should not be far from the value of the smaller one. Further- more, the resistances used should be comparable in magnitude with the battery resistance. In order to meet these conditions it will be necessary to use a non-sensitive galvanometer of low resistance, or to adjust a sensitive galvanometer with a shunt of proper resistance placed across its terminals. The procedure is as follows : (1) Connect the battery in series with a resistance box, the galvanometer, and a reversing key. (2) Observe the galvanometer readings for eight or ten different resistances. These readings should be taken several times for each resistance used, and the mean deflection derived from them should be utilized in the computations. 2i6 JUNIOR COURSE IN GENERAL PHYSICS. (3) From each suitable pair of observations compute the resistance of the battery. It will be found instructive to determine the resistance from the observations graphically, and if a considerable number of observations are taken, that will usually be the least laborious method and sufficiently accurate. To do this, plat a curve with known resistances as abscissas, and reciprocals of currents, or of galvanometer deflections as ordinates. The intercept on the axis of abscissas will be the resistance of the circuit outside of the resistance box. The resistance of a cell is sometimes determined by con- necting two cells, first in series, and then in multiple, and observing the galvanometer deflections in each case. Between two equations representing these observations, E and 7 may then be eliminated, and R may be determined. This method assumes that the E. M. F.'s and internal resistances of the two cells are identical. To test this assumption, connect the two cells in series, but so that their E. M. F.'s are opposed. If no current flows, their E. M. F.'s are equal. Next connect each cell in turn in series with the galvanometer and the same low resistance. If the currents are equal, their internal resist- ances are equal (provided their E. M. F.'s are equal). The following modification of the method, which is especially useful when the law of the galvanometer is unknown, may be used to check the results : (1) Connect the two cells in multiple, and observe the deflection produced when some known resistance is used in the box. (2) Join the cells in series and adjust the box resistance until the deflection is the same as before. (3) From the values of the two box resistances and the galvanometer resistance, compute the resistance of a single cell. The method of this experiment is not applicable to batteries that suffer marked variation from polarization. THE MEASUREMENT OF RESISTANCE. EXPERIMENT T 6 . Resistance of a battery by Mance's method. It is a rather difficult matter to secure a satisfactory measurement of the internal resistance of a battery. Mance's method is perhaps the most accurate for a battery that is not subject to rapid polarization. The battery whose resistance is required is made one arm of a Wheatstone bridge, the other three arms being adjust- able resistances of known value. The battery usually employed with the bridge is removed and replaced by a wire, the battery key being retained in its old place. In this use of the bridge, a current flows through the galvanometer at all times, and it will be found advantageous to keep the galvanometer key closed. The measurement now consists in so adjusting the resist- ances in the bridge that the opening or closing of the battery key has no effect upon the deflection of the galvanometer needle. When this adjustment has been obtained, the resistance of the cell can be computed from the ordinary law of the bridge. If the deflection of the galvanometer is too great to be read on the scale, a permanent magnet may be used to bring the needle back. This magnet should be kept as far away as is possible, however, in order not to diminish the sensitive- ness of the galvanometer. Judgment must be used in the choice of the resistances placed in the various arms, so as to secure the greatest sensitiveness and at the same time as little inconvenience as possible from large and variable deflections. If the resistance of the battery is not very great (thousands of ohms), it will be best to adjust the resistance of the three arms of the bridge, so that the greatest resistance is in series with the battery and galvanometer. If the battery polarizes, even very slowly, there will be a drift of galvanometer reading. This change of the current through the galvanometer must, of course, be disregarded. Sometimes the observations are still further complicated by the existence of some small self-induction in the bridge coils. 2i8 JUNIOR COURSE IN GENERAL PHYSICS. The effect of this is to give the galvanometer needle a slight inductive throw, even though the conjugate condition of the four arms of the bridge has been reached. Addenda to the report : (1) Prove that the ordinary law of the bridge holds for Mance's method. (2) A dynamo is like a battery in the fact that it is the seat of an E. M. F., and has internal resistance. What difficulty would be experienced in measuring, by this method, the internal resistance of a dynamo while running ? EXPERIMENT T 7 . Resistance of electrolytes. When a current is passed through an electrolyte, the electro- lyte is decomposed, and a counter E. M. F. is always set up. Often there is also an evolution of gas at one or both ends of the electrodes. These effects complicate the experimental determination of electrolytic resistance, but the difficulties which they introduce may be, in great part, avoided by the use of an alternating current of short period. The Wheatstone's bridge method of measuring resistance may be adapted to the determination of electrolytic resistance as follows : (1) An alternating current is supplied by replacing the battery by the secondary circuit of an induction coil. (2) The galvanometer is replaced by some means of detect- ing alternating currents. A telephone will serve this purpose very well. The method of working is analogous to that described in Exp. T r The resistance of the bridge arms is varied until no sound is heard in the telephone, and the un- known resistance is determined by the ordinary law of the bridge. Since the current flowing is a rapidly fluctuating one, it is of the utmost importance that the bridge arms have no self-induction. For this reason, a special form of bridge is generally used. THE MEASUREMENT OF RESISTANCE. 219 If the vessel containing the electrolyte is a tube or a pris- matic trough with electrodes filling the ends, the specific resist- ance may be computed as in Exp. T 3 . In this way we may determine the specific resistance of different solutions, or of the same solution at different temperatures and densities. The temperature of the solution should always be noted at the time of the experiment. If the vessel used does not admit of accurate measurement, it should be standardized a follows : (1) Make a 10 per cent solution of zinc or copper sulphate. (2) Fill the vessel and determine its resistance. (3) From this resistance and the specific resistance of the electrolyte taken from tables, compute what must be the length of the electrolyte if its cross-section is one square centimeter. The apparatus having been thus standardized, the specific resistance of any other solution may be determined. Before putting a solution into the vessel, care should always be taken to scrupulously clean the vessel, and to rinse it with distilled water. The resistance of a solution is sometimes greatly changed by even slight traces of other substances. CHAPTER IX. GROUP U: ELECTRICAL QUANTITY. {U) General statements ; (Uj) Constant of a ballistic galvanom- eter ; (U 2 ) Logarithmic decrement ; (U 3 ) Comparison of capacities ; (U 4 ) Capacity in absolute measure. (U). General statements concerning electrical quantity. The electromagnetic unit of quantity is that quantity of electricity which is transferred by unit current in unit time. The practical unit of quantity, or the coulomb, is the amount transferred by a current of one ampere in one second. The total quantity of electricity transferred by any current is the product of the current by the time during which it con- tinues. If the current is variable, this becomes taken between the proper limits. Quantities of electricity are considered when we deal with, (1) The total amount of an electrolyte decomposed. (2) The charge and discharge of condensers. (3) Momentary induced currents. In cases 2 and 3 the duration of the current is usually very brief, and since the magnetic field produced is equally transient, it is obvious that the quantity of electricity transferred cannot be measured by means of a galvanometer used in the ordinary manner. The quantity of electricity transferred through the coils of a galvanometer by a momentary current can be meas- 220 ELECTRICAL QUANTITY. 221 ured, however, by the " throw " or " swing " of the needle due to the magnetic impulse of the momentary current. A galvanometer used for measuring such impulses is called a ballistic galvanometer from its analogy to a ballistic pendulum. Any galvanometer can be used as a ballistic galvanometer, simply by observing "throws" instead of permanent deflections, provided that the motion of the needle be so slow that the end of the swing can be determined accurately. It is also desirable, in the case of galvanometers used ballistically, that the damping should be as small as possible. These two requi- sites are secured by making the needle heavy, thus securing slow motion and small factor of decrement. In using a ballistic galvanometer, it must be remembered that the magnetic moment of the needle enters the constant of the instrument. Therefore the needle should be a magnet whose moment is not subject to rapid change. EXPERIMENT Uj. Measurement of the constant of a ballistic galvanometer. There are three methods for determining this constant : (1) By measuring the throw of the galvanometer needle due to the discharge of a condenser. (2) By measuring the throw of the gal- vanometer needle produced by the induced current due to the rotation of a coil in a magnetic field. (3) By computation from the periodic time of the galvanometer needle, and the constant of the instrument used as a tangent galvanometer. The last is the most instruc- tive, and is the one here given. The amount of work done against mag- ' netic forces in turning a magnet through an angle 8 (Fig. 72) in a magnetic field of horizontal intensity H> is (135) 222 JUNIOR COURSE IN GENERAL PHYSICS. The kinetic energy of the magnet when it has its greatest angular velocity is The kinetic energy of the moving magnet at the mid-point is equal to the work necessary to turn the magnet through the angle 8. (137) A current of / amperes flowing in the galvanometer coils exerts a force on the galvanometer needle whose moment is -^MGf, G being the true constant of the galvanometer. If the moment of this force be integrated over the time during which the cur- rent lasts, it must equal the moment of momentum produced. whence, -^MGQ = K<*^ (138) Force multiplied by time is equal to the momentum produced. In the same way moment of a force multiplied by time is equal to the moment of momentum produced : but the moment of momentum of a body is also equal to its moment of inertia multiplied by its angular velocity. If T is the periodic time of the magnetic needle for small oscillations, we have (see equation 102) ^im- If in the equations (137), (138), and (139), K and Mbe expressed in the form of their ratio, this ratio and o> may be eliminated between the three equations. This will give Q= io^ sin J5. (140) 7T Lr TT Now 10 is the working constant of the galvanometer. Gr .-. Q= r / sini-8. (141) 7T ELECTRICAL QUANTITY. 223 The constant factor multiplied into the term sin \ B is the constant of the instrument used as a ballistic galvanometer. Calling this quantity Q Q , we have (2 = Go sin} 8. (142) In order to determine <2 , first determine 7 as in Exp. R lt and then the periodic time of the galvanometer needle as in Exp. Q 3 . From these values compute Q Q . If 8 is quite small, the quantity of electricity Q is propor- tional to 8, but 8 is proportional to the deflection on the scale. Therefore we have in which S is the throw in scale divisions, and <2 is the constant per scale division. The above demonstration assumes that the whole of the kinetic energy of the needle after the current has ceased to flow is used in overcoming magnetic forces. This is not quite true. The friction of the needle against the air and the current induced in the galvanometer coil by the moving magnetic needle, both require the expendi- ture of energy, and therefore make S less than it otherwise would be. The theory of damping leads to the conclusion that (i + }X) sin} 8 should be substituted for sin }8 in the above equation, in which X is the logarithmic decrement of the gal- vanometer needle. (See Exp. U 2 .) EXPERIMENT U 2 . Determination of the logarithmic decre- ment of a ballistic galvanometer needle. It has already been shown that the quantity of electricity that passes through the coils of the ballistic galvanometer is proportional to the impulse imparted to the needle, which, in its turn, is proportional to the sine of half the angle of throw, or to the angle itself, if the latter be small. This is true, how- ever, only when there is no lost energy due to air friction and induced currents, which damp the oscillation of the needle, and finally bring it to rest. 224 JUNIOR COURSE IN GENERAL PHYSICS. Since it is by means of the throw that the quantity is to be measured, we must know the correction that is to be applied to the actual throw of the needle to give the throw that would have resulted had there been no damping. When a magnetic needle oscillates under the influence of damping, the ratio of any amplitude to the succeeding one in the opposite direction is very nearly constant, or i So % n f=f = ^ = r ' 044) 2 63 On+l This constant is the " ratio of damping," and its Napierian logarithm is called the logarithmic decrement, and is generally designated by X. We have, therefore, * = log e -^ (145) o n +i The equation of motion of a body oscillating under the action of a force whose moment is proportional to the angular displacement, as has been shown under the head of simple har- monic motion, is K^+G^ = o. (146) If the motion is not simply harmonic, but is damped by friction or otherwise, a third term must be introduced. In the case of an oscillating magnet, damping is produced : (1) By air friction. (2) By induced currents due to the motion of the magnet near conductors. Both of these retarding forces are very nearly proportional to the angular velocity; consequently the term that must be added to the above equation is k &-, in which dt k is a constant. The complete equation of motion of the damped magnetic needle is therefore ELECTRICAL QUANTITY. If we integrate this equation, we have sn 225 (148) in which S is a constant, and T is the period of oscillation of the needle under the influence of damping. Let time be reckoned from the instant the needle passes the position of equilibrium; and let 8 V S 2 , be the values of 6 at the times = , - These values of will be the 4 4 successive actual amplitudes of the oscillatory motion ; and o n e SkT From (149) we have (149) (150) and by substituting for this quantity X, as in (145), equation 149 gives Transposing and expanding the exponential in terms of X, and neglecting powers of X higher than the first, we obtain When there is no damping, i.e. when k = o, we have, from (149), S 1 = S . Therefore, it follows that S is the quantity that should be substituted for the first actual throw in using a ballistic galvanometer, and that equation 143 becomes The above demonstration is based upon the assumption that both S and X are small. If S is 4 and the ratio of damping is 1.05, equation 153 will be in error by about one * See Gray's Absolute Measurements in Electricity and Magnetism, vol. 2, p. 393. VOL. i Q . 226 JUNIOR COURSE IN GENERAL PHYSICS. part in a thousand. If 8 is 10 and the ratio of damping is 1.2, the error will be about one in a hundred. The object of this experiment is to determine the logarith- mic decrement of a galvanometer needle, and to show the relation of the decrement to the resistance in circuit with the galvanometer. It is obvious that the decrement must depend on the resistance, since the damping is, in large part, due to the currents induced in the galvanometer coils by the moving needle, and because these currents are inversely pro- portional to the resistance of the circuit. In the performance of the experiment, a galvanometer should be used in which the needle is not strongly damped. From equation 144, we have 7^- = ^> (154) n+m whence x = l log a- (155) n+m. Errors of observation have the least influence when the ratio of S n to S n+m is about 3. The method of procedure is as follows : (1) Set the needle to vibrating, and observe the limits of the successive swings to the right and left by means of a telescope and scale. (2) From these observations determine the successive amplitudes. The position of equilibrium of the needle will generally be obtained by noting the scale reading when the needle is at rest. Sometimes this position changes during the progress of an experiment. It may then be obtained as follows : Let S v S 2 , and 5 3 be three scale readings corresponding to the extremes of successive throws. We shall then have in which 5 is the zero position at the instant when the scale ELECTRICAL QUANTITY. 22/ reading is 5 2 . The deflection required, then, is in scale divisions, If the angles are not small, these amplitudes should be reduced to circular measure by means of the known distance of the scale from the mirror. Several values of the ratio of damping should be obtained in the following manner: Suppose the (+i)st amplitude to be about one-third of the first ; X should then be determined from the ratios Wl n+2 Determine in this way the logarithmic decrement when the galvanometer coils are short-circuited, and are in open circuit, and also for several different resistances, comparable with the galvanometer resistance. Finally, from these deter- minations plat a curve, with resistances as abscissas and corre- sponding values of the decrement as ordinates. This curve will have an asymptote parallel to the axis of abscissas, at a distance from that axis equal to the decrement on open circuit. If the axis of abscissas be made to coincide with this asymptote, the ordinates to the curve will be the decrements due solely to induced currents. These decrements are inversely proportional to the resistance of the circuit. From this relation and from the curve, compute the resist- ance of the galvanometer. EXPERIMENT U 3 . Comparison of the capacities of two condensers. When the coatings of a condenser are charged to a potential difference, pd, the charge or quantity of electricity stored in the condenser is Q = Cpd, (156) in which C is the capacity of the condenser. It has already been shown in preceding experiments that if the quantity of 228 JUNIOR COURSE IN GENERAL PHYSICS. electricity Q is discharged through a ballistic galvanometer producing the deflection 8, we have If a condenser of capacity C v charged to a potential differ- ence, pd^ be discharged through the ballistic galvanometer, we have (158) If another condenser of capacity C 2 , charged to a potential difference, pd^ be discharged through the same ballistic galva- nometer, we shall have a similar relation. And if the first equation be divided by the second, we shall have 7^ = 4^sT ( J 59> A still simpler relation follows if the condensers have been charged to the same potential difference. In experimenting with condensers it is generally necessary to use rather large potential differences (from 50 to several hundred volts). Such potential differences may be produced by a water battery of a sufficient num- ber of very small cells. Each cell Fi - 73< consists of a short test-tube filled with very slightly acidulated water. The plates are made by soldering together, short strips of copper and zinc. Each "couple" is bent into a U-shape, and the copper dipped into one cell, and the zinc into the next cell, as illustrated in Fig- 73- In condenser work it is also necessary to use great care in securing good insulation, not solely on account of the use of high potentials, but because the condenser must sometimes remain charged for a few minutes while unconnected with a battery. The procedure in this experiment is as follows : ELECTRICAL QUANTITY. 22 9 Fig. 74. (1) Connect the condenser in series with the battery and ballistic galvanometer, and place in the circuit a double con- tact key, as shown in Fig. 74. (2) Make contact at A, and thus charge the condenser through the galvanometer. The corresponding galvanometer throw should be determined as in Exp. U 2 . (3) Break contact at A, and immediately make contact at B, thus discharging the condenser through the galvanometer. The galvanometer needle will receive an impulse in the opposite direction, which should be very nearly equal to the former throw. These observations should be repeated several times in order to get a good average. A similar series of observations should now be taken with the condenser replaced by the one with which it is to be compared. If the capacities of the two condensers do not differ greatly, that is, if one is not more than two or three times as great as the other, the same number of cells should be used. If the difference of capacity is very large, the E. M. F.'s of the batteries in the two cases should be adjusted to suit the two condensers, and they should be compared as in Exp. S r When condensers are connected as shown in Fig. 75, they are said to be connected in multiple. If C is the capacity in multiple, we have the relation I I I Fig. 75. (160) This relation follows directly from the fact that the capacity of a condenser is proportional to the area of either of its coatings. Fig. 76. When condensers are connected as shown in Fig. 76, they 230 JUNIOR COURSE IN GENERAL PHYSICS. are said to be connected in series. If C. is the capacity of the system in series, we have -++-. (I6D L a L l C 2 This relation may be readily derived by making use of the following facts : (i) The potential difference at the terminals is equal to the sum of the potential differences between the coatings of each condenser, or pd a =/ /, 7 1 , etc., compute the pitch of the string from 252 JUNIOR COURSE IN GENERAL PHYSICS. the formula given above, and note how closely the result agrees with the known pitch of the fork. The law should be tested in this way for at least three strings of different diameter and density, and several forks should be used with each string. On account of the great difference in quality between the note of the string and that of the fork, great care must be used in adjusting the former. If the ear is untrained, a mistake of an octave is not unusual. EXPERIMENT W 6 . The study of the transverse vibration of cords and wires, by Holer's method. The apparatus* for this experiment is a modification of Melde's well-known device. It consists of an instrument de- O Fig- 91. signed for the purpose of maintaining in continued circular vibration a cord or wire, the tension of which is adjusted by weights until well-defined loops and nodes are produced. * For a fuller description, see G. S. Moler's article in the American Journal of Science, Vol. 36, p. 337. SOUND. The cord in question is attached at one end to a crank of small throw which is driven at a high speed by means of an electric motor or water wheel. The velocity is maintained con- stant by the action of an electric brake. The arrangement of these parts is shown in Fig. 91. In that figure, A is the main shaft, and C the crank pin, upon which a hook is placed carrying the cord or cords D to be put into vibration. To counteract the tension of these cords, which would other- wise produce too great friction, the crank is attached to the bearing E by a stout cord, the strain upon which is adjustable by means of a tightening pin at F. In order to drive simultaneously two parallel cords at differ- ent speeds, there is a second shaft and crank which can be put into motion at will, by means of the pinion wheels at G and H. Fig. 92. The cords, the vibrations of which are to be studied, one end of each of which is fastened to the crank hooks, are carried over pulleys, attached at any desired distance upon the base of the apparatus (see Fig. 92), and are subjected to tension by the application of weights. When these weights are in proper rela- tion to the velocity of the crank, the cord breaks into nodes and falls into a stable condition of vibration which is maintained as long as the conditions upon which it depends continue. One of the conditions of equilibrium is the speed, and it is for the purpose of regulating this factor that the electric brake is used. This part of the apparatus is shown in Figs. 93 and 94- In the former figure, J is a lever pivoted at K (see also /, Fig. 91), and this is forced outward with increasing speed until 254 JUNIOR COURSE IN GENERAL PHYSICS. it bends the spring L and makes a contact with M. Thus an electric circuit through the electromagnet P (Fig. 94) is closed and the brake Q is thrown against the periphery of a wheel upon the main shaft. In this way the speed is checked when- ever it exceeds a certain desired rate. It should be noted that in Fig. 93. practice this electric regulator has but little to do, since a heavy cord 2 m. in length, or even less, is in itself a very effective regulator of speed as soon as it has once been brought into definite vibration. It is indeed entirely practicable to perform the experiment without putting the brake into function, provided that two cords be used, one of which is maintained under constant tension and serves as a governor while variations of length and load are made in the case of the other. The governing cord should be of considerable mass. The experiment consists in varying the factors upon which the transverse vibration of strings depends, and verifying the rela- tions which exist between them and the rate. These factors are : (1) The cross-section (S f ), which may be conveniently varied by using several strands of a light cord in common. (2) The length (L), which is to be changed by shifting pul- leys along the base of the instrument. (3) The tension. Fig. 94. SOUND. 255 The rate itself may be subject to one change by fastening the cords first to the main shaft and then to the shaft H. The results are to be arranged as shown in the following table. TABLE. Observa- tion. Cross- Section. 5. Length. Rate of Vibra- tion. N. Number of Segments. . Tension in Grams. />. Square Root of Tension. Sf. Constant H ^^ NL id No. i 2 I 627. 25.04 12.52 " 2 T3 2 2 162. 12.70 12.70 " 3 2 3 74- 8.60 12.90 " 4 "to 2 4 41. 6.40 1 2. 80 " 5 2 5 26. 5.10 12-75 No. 6 T3 C I 2 i 627. 25.04 12.52 " 7 M & I 2 i 266. 16.30 12.22 " 8 In i 2 i 1 66. 12.90 I2.9O No. 9 2 i 2500. 50.00 12.50 " 10 t *1 2 2 625. 2O.OO 12.50 " ii a 2 3 280. 16.70 12.52 " 12 1/5 2 4 158. 12. 60 12. 60 No. 13 . I i 1 66. I2.9O 12.90 " 14 *"O c M rt I 2 41. 6.40 1 2. 80 " J 5 5 I 3 1 8. 4.20 12. 6O " 16 I 4 9-5 3 .10 I2.4O If N be the number of vibrations per unit of time, L the length of the cord, n the number of segments, and V the velocity of transmission of an impulse transmitted to the cord, we have the familiar formula expressing the transverse vibra- tions of flexible cords : = JL T V - 2 L (179) If P is the tension of the cord, s its cross-section, and d its density, we have also sd 256 JUNIOR COURSE IN GENERAL PHYSICS. Finally, if X is the wave length, we may write X = , (181) (182) - (183) CHAPTER XII. GROUP X: LENSES AND MIRRORS. (X 1 ) Radius of curvature of a lens (by reflection) ; (X 2 ) Focal length of a concave mirror ; (X 3 ) Focal length of a convex lens ; (X 4 ) Magnifying power of a telescope ; (X 6 ) Mag- nifying power of a microscope and focal lengths of same. EXPERIMENT X r Determination of the radius of curvature of a lens by reflection. The apparatus consists of a telescope placed midway be- tween two small gas jets (g, g \ Fig. 95), the distance between 30 Fig. 95. the jets being capable of adjustment. The lens (L, Figs. 95 and 96) whose curvature is desired is placed at a distance of from i to 2 m., and in such a position that the reflected images of the two flames can be seen in the telescope. The apparent distance between the images is measured by means 257 VOL. I S 2 5 8 JUNIOR COURSE IN GENERAL PHYSICS. of a scale (Fig. 97) fastened to the surface of the lens, and from this measurement, together with the distance from the Fig. 96- lens to the flames, and the actual distance between the flames, the radius of curvature can be computed. The problem with which this experiment deals consists in finding the radius of curvature r(=co, Fig. 98) in terms of L, the distance between the gas jets (gg 1 ) ; of D t the dis- Fig. 97. Fig. 98. tance from telescope to lens (cT), and of s, the apparent distance of the images (g n , g nl ) as measured upon the scale on the face of the lens (aa 1 ). From the relation of conjugate foci we have 47--L- i (184) gc l g"c' oc and in case the telescope is at a distance from the lens much greater than L, we may write as an approximation I _ I _'2 L Tc tc oc Dr or where (185) (186) = ct. LENSES AND MIRRORS. 259 From the geometry of similar triangles we have, also, where / = g", g n ', which is the distance between the images. The quantities / and d are to be eliminated, and r is to be expressed as stated above. Combining equations 187 and 188, we have s(r+D) L(r-d] Lr D (189) '-TIT; (I90) To obtain accurate results, the conditions of the experi- ment should be varied by changing the position of the lens, and by altering the distance between the flames. Make five or six determinations for each side of the lens and use the average of each set. It may happen that two pairs of images are seen by reflec- tion. This is due to the fact that a part of the light from the flames passes through the first surface and suffers reflection at the second. One pair of images will probably be erect and the other inverted, so that no difficulty need be experi- enced in distinguishing between the two. Addendum to the report : Rays from the gas jet g are reflected from the face of the lens, and enter the telescope T. The angles which the incident and reflected rays make with the normal to the surface are equal. From this consideration deduce formula 190, without using the relation of conjugate focal lengths, or the position of the 260 JUNIOR COURSE IN GENERAL PHYSICS. EXPERIMENT X 2 . Focal length of a concave mirror. The object of this experiment is to verify the formula which shows the relation between the conjugate foci and the principal focus of a concave mirror ; viz. _ , _ = = Pi Pi r f (191) The apparatus required consists simply of the mirror m y a glass scale S, a screen S f , and a gas flame. The mirror should first be mounted (see Fig. 99) in such a way that its principal axis is nearly horizontal. The glass S'- ; Gas Flame Fig. 99- scale * may then be placed at some point in this axis, with the gas flame a short distance behind it. It will be found more convenient to work in a room which is partially darkened. The position of the image of the scale may now be found by trial, a screen S f (preferably of ground glass) being placed * A glass scale is recommended merely because it constitutes an " object '* whose image will, in general, be especially sharp. LENSES AND MIRRORS. 2 6l in such a position that the image thrown upon it is as sharp as possible. This adjustment may be made more accurately if the mirror is partly covered, so that only a comparatively small portion near the center is used. The distances of object and image from the mirror are now to be measured, together with the distance between the lines in the image of the scale. Repeat these measurements for three or four different positions of the scale, the position in each case being such that the image lies between the scale and the lens. The focal length and the radius of curvature are to be computed from each of the observations. As a cheek upon the results, the center of curvature may be located by placing a needle, or other pointed object, in such a position that the image of its point shall coincide in position with the point itself. This may be done quite accurately by moving the eye about and noting whether the relative positions of image and object vary. Addenda to the report: (1) From the data obtained, verify the formula which shows the relation between the size of the image and its distance from the mirror; i.e. if the lengths of object and image are respectively / x and / 2 , l \ : 4=/i : A- (2) Give a demonstration of the formula above referred to ; also the formula for conjugate foci. (3) Indicate the advantage of using only a small central portion of the surface of the mirror. EXPERIMENT X 3 . Determination of the focal length of a convex lens. ' The focal length of the lens used is to be determined by each of the four methods described below, a number of obser- vations being made in each case, and the average used. 262 JUNIOR COURSE IN GENERAL PHYSICS. (i) The lens is made to form an image F, of some object whose distance is so great that light proceeds from it to the lens in rays that are very nearly parallel. The focal length is then equal to the distance between lens and image. The sun is usually the most convenient source of light for use in this method, the rays being rendered horizontal by reflection from a mirror M (Fig. 100). A screen of ground glass or paper Fig. 100. is adjusted until the image thrown upon it is as distinct and sharp as can be obtained. The focal length is then equal to the distance from the screen to the center of the lens. This method is not capable of great accuracy, but is more direct than those which follow. (2) A telescope which has been focused for parallel rays is used to observe some sharply defined object as seen through the lens. The position of the lens having been adjusted until the object is seen to be properly focused in the telescope, the distance between lens and object is equal to the focal length required. In principle this method is practically the same as that first described, and the degree of accuracy that can be attained is about the same in each. (3) An object is placed at any convenient distance in front of the lens, and a screen is adjusted until the image received upon it is sharply defined. The focal length can then be computed from measurements of the distances of object and LENSES AND MIRRORS. 263 image from the lens. If / x and / 2 are these two distances, we have 7^7 (I92) The luminous object used may be the flame of a candle or gas jet. There are some objections, however, to the use of a flame, on account of the flickering caused by air currents. Better results can usually be obtained by using a fine thread or wire which is stretched across an opening in an opaque screen (Fig. 101). When the aperture is illuminated by means Fig. 101. of a lamp, the shadow of the wire forms an image which is unaffected by the flickering of the flame, and which can be very sharply focused. (4) Placing the object at any convenient distance from the lens, adjust the position of the screen until the image is sharply focused. Then, without changing the position of the screen, move the lens until a second position is found, such that a 264 JUNIOR COURSE IN GENERAL PHYSICS. sharp image is formed. From the distance between object and screen, and the distance through which the lens is moved, the focal length can be computed. If / and a are the two distances, This method of determining focal length has the advantage of being uninfluenced by any uncertainty as to the thickness of the lens and the position of the principal points. Since it is merely the distance through which the lens is mcved that is required, measurements can be made to any convenient point on the support of the lens, and no correction need be made for the thickness of the glass. For this reason the method will probably give better results than can be obtained by any of the three methods first described. Addenda to the report : (1) Sharper images, and therefore more accurate results, will be obtained if the lens is covered, so that only a small region near the center is used. (Explain.) (2) From the curvature of each face of the lens and your determination of the focal length, compute the index of refrac- tion of the glass from which it is made. 'EXPERIMENT X 4 . Magnifying power of a telescope. Focus the telescope upon some large object, such as a scale, which contains sharply defined portions of equal length. The bricks in the wall of a building, or the pickets of a fence, will serve for this purpose. Looking through the telescope with both eyes open, the magnified image of the scale will be seen by one eye, while with the other the scale is observed directly. By a comparison of the two images the magnifying power is determined. For example, if one division of the image seen in the telescope covers ten divisions of the unmagnified image, LENSES AND MIRRORS. 26 5 the magnifying power is ten. To guard against errors due to a difference in the two eyes, it is best to use the left eye in observing the telescopic image as often as the right. The magnifying power should be determined in this way when the object observed is at several different distances, ranging from a distance that is so great as to be practically infinite, to the least distance for which the telescope can be focused. If any difference is found in the magnifying power, the variation should be shown by a curve in which distances and magnifying powers are used as co-ordinates. For some one distance of the object observed the distances between the various lenses should be accurately measured when the telescope is focused. Addenda to the report : (1) Determine the focal length of each of the lenses, and compute the magnifying power, drawing a diagram to scale to show the position and size of the various images. (2) Explain the cause of the variation in magnifying power with the distance of the object. EXPERIMENT X 5 . Magnifying power of a microscope and determination of the focal length of its lenses. I. The "open-eye" method. This method is similar to that described for the determin- ation of the magnifying power of a telescope. (i) Focus the microscope upon a finely divided scale and place another scale at the side of the instrument at a distance from the eye equal to the distance of distinct vision (about 25 cm.). By observing the scale with one eye and the image formed in the microscope with the other, the apparent size of the magnified image is determined. The ratio of this to the actual size is the magnifying power. 266 JUNIOR COURSE IN GENERAL PHYSICS. (2) Measure the distance between the object glass and eye piece and determine the focal length of the latter by one of the methods of Exp. X 3 . From a knowledge of the magni- fying power it will now be possible to compute the focal length of the object glass. (3) Construct a diagram to scale to explain the action of the instrument, showing the position and size of each image. (See Gage, Microscopical Methods, p. 65.) II. Franklin s method. The object of this method is to find the focal lengths of microscope lenses from the magnifying power with short and long draw tubes. The apparatus required is a compound microscope, a microm- eter caliper, a stage micrometer, an unsilvered glass microscope slip, and a scale divided to millimeters. The experiment consists in the following determinations : (1) The magnifying power (m) of the microscope with short tube. (2) The magnifying power (m 1 ) with long tube. (3) The change (/) in the length of the microscope tube. (4) The movement of the objective in refocusing for deter- mination (2). This measurement is to be made by means of the caliper. The magnifying power, as in the previous method, is the ratio of the apparent size of an object as seen with the micro- scope to its apparent size as seen with the naked eye at a distance of D centimeters. Instead of measuring magnifying power by the " open-eye " method, the glass slip is mounted obliquely before the eye piece (Fig. 102), so as to bring an image of the scale (S) into the field. The theory of the method of computing focal lengths is as follows. Let / be the equivalent focal length of the eye piece ; /', the LENSES AND MIRRORS. 267 focal length of the objective; a, the distance from the object to the center of the objective ; and b, the distance of the image from the center of the objective. I s' Fig. 102 We have the following equations : m =(!+')-> \p Ja '-(-+*)> I- \p Ja-k V p a -k b+f from which these may be obtained. 2km' m' m SV __ (m' = 0. (194) (195) (196) (197) (198) (199) From equation 199, a may be computed, then b from (198), then finally / and p' by combining (194) or (195) with (196) or (197). 268 JUNIOR COURSE IN GENERAL PHYSICS. The distance D is the distance from the focal plane in the eye piece (which in the case of a negative [Huyghenian] eye piece lies midway between the two lenses, and in the positive eye piece is distant from the eye lens by an amount equal to three-fourths of the distance between the lenses) to the scale (s), Fig. 102. GROUP Y : THE SPECTROSCOPE AND PHOTOMETER. (Yj) Index of refraction of a prism ; (Y 2 ) Flame spectra of the metals; (Y 3 ) Distance between the lines of a grating; (Y 4 ) The Bunsen photometer. EXPERIMENT Y r Measurement of the index of refraction of a prism by means of a spectrometer. In the spectrometer used (see Fig. 103), the rays of light from the source are made to pass through a narrow vertical slit, Fig. 103. and are then rendered parallel by means of a lens. This lens and slit, mounted in a tube to exclude stray light, constitute what is known as the collimator. After leaving the collimator, the light is made to pass through a prism, and is finally observed by means of a telescope. Both collimator and telescope are adjusted so as to point towards the center of a horizontal grad- uated circle, and the telescope is free to rotate about a vertical THE SPECTROSCOPE AND PHOTOMETER. 269 axis passing through this center. By means of a vernier attached to its support, the angular position of the telescope can be read. Before beginning observations with the prism, the collimator must be so adjusted that the light leaves it in parallel rays. To accomplish this, take the instrument to an open window and focus the telescope on some object which is so far away that the rays from it are practically parallel. Then turn the tele- scope so that it points directly at the collimator, and, without changing the focus of the telescope, alter the length of the collimator tube until the image of the slit, as seen in the tele- Fig. 104. scope, is sharply denned. Both telescope and collimator are now properly focused, and should not be altered during the experiment. In order to obtain the index of reflection, it is necessary to know the angle of the prism, and the angle through which it bends the %'ays from the collimator when in the position of " minimum deviation." (i) To determine the angle of the prism. Place the prism near the center of the graduated circle, with its refracting edge turned toward the collimator (Fig. 104). Turn the telescope 2/0 JUNIOR COURSE IN GENERAL PHYSICS. until the slit is seen by reflection from one face of the prism, and adjust the position of the telescope until its cross-hair coincides with the image of the slit. Record the position of the telescope as read by the vernier. Then set the cross-hairs in the same way upon the image of the slit, as reflected from the other face of the prism, position T f , and again read the vernier. The angle between the two positions of the telescope is then equal to twice the angle of the prism. A number of readings should be taken. (2) To determine the angle of minimum deviation. Up to this point in the experiment any source of light will serve equally well ; but since different colors are bent by refraction in different degrees, it will now be necessary to use some mono- chromatic light. The most convenient light of a single color is that obtained by burning some salt of sodium in the Bunsen flame.* Place the prism near the center of the horizontal circle, and in such a position that light from the collimator will be refracted through it and pass into the telescope. If, while observing the image of the slit in the telescope, the table which carries the prism is slowly turned, the image will be seen to move across the field, and it may be necessary to shift the position of the telescope in order to keep it in sight. In this way the prism can be set by trial to the position which causes the light to deviate least from its original direction on leaving the collimator ; when this position is reached, a slight motion of the table in either direction will cause the image to move toward a position of greater deviation. When this setting is made as accurately as possible, bring the cross-hair into coinci- dence with the image of the slit, and take the reading of the vernier. Several settings should be made in this way, with * If light of a different wave length were used, the index of refraction obtained would, of course, be different. Since this experiment is, however, merely intended to illustrate the use of the spectrometer, it will be found best to use the most con- venient monochromatic light; viz. sodium. THE SPECTROSCOPE AND PHOTOMETER. 271 deviations both to the right and left. The index of refraction can be then computed from the formula > sm |- a in which a is the angle of the prism, and 8 the angle of mini- mum deviation. Indices of refraction are to be obtained in this way for several prisms of different materials. EXPERIMENT Y 2 . Study of the flame spectra of various metals. This experiment should follow that on the determination of indices of refraction (No. Yj), and can be more conveniently performed if there are two observers. The ^apparatus required is a spectrometer, or spectroscope, similar to the one used in that experiment. The substances whose flame spectra are most readily studied are Na, Li, K, Sr, Ba, Ca, Rb, and Cs. A salt of one of these metals, usually either the chloride or the carbonate, is placed in the colorless flame of a Bunsen burner, immediately in front of the slit of the spectrometer. The spectrum of the flame, when colored by the incandescent vapor of the metal, will be seen to consist of several bright lines, whose color and arrangement are characteristic of the metal studied. To map the position of the different lines, the prism should first be adjusted to the position of minimum deviation for sodium light. Then, without altering the adjustment of the prism, set the telescope successively on each of the lines of the spectrum, reading its angular position by means of the vernier. The spectrum can be mapped for comparative purposes from these results. The approximate limits of the visible spectrum, as determined by substituting white light for the metallic spectrum, should be marked on each diagram, and the colors and relative intensities of the lines should be indicated. If it is desired to determine the wave 272 JUNIOR COURSE IN GENERAL PHYSICS. length of each of the lines, a grating may be used instead of a prism; or, if this is not convenient, the prism may be "cali- brated " by reference to the Fraunhofer lines. To accomplish this, the slit should be illuminated by bright daylight, or direct sunlight, and adjusted until the dark lines in the spectrum are clearly seen. The prism is then set to the position of least deviation for the D line, and the angular position of the tele- scope is observed for several of the more prominent lines. The wave lengths corresponding to these lines being known, a curve can now be constructed, in which angles of deviation are taken as abscissas, and wave lengths as ordinates. By reference to this curve, the wave length corresponding to any observed deviation is readily determined. The most instructive method of mapping bright line spectra is that employed by Lecoq de Boisbaudran in his work on the Fig. 105. spectra of the metals.* An example is given in Fig. 105. As will be seen from the diagram, each spectrum is mapped twice, once above and once below the median line. The former x gives the normal, the latter the prismatic spectrum of the substance in question. This method should be employed in reporting upon the results of this experiment. Considerable difficulty is sometimes met with in working with flame spectra in obtaining sufficient permanence and brilliancy for accurate observation. To obtain the best results the methods of heating must be suited to the salt used. In some cases, a small amount of the salt, when placed on a wire * Spectres Lumineux; Lecoq de Boisbaudran, Paris, 1874. THE SPECTROSCOPE AND PHOTOMETER. 273 and heated in the flame, will form a bead which lasts for a considerable time and gives a good spectrum. In other cases the supply of salt will need to be constantly renewed. A piece of asbestos, or wire, which has been moistened by a strong solution of the salt, will sometimes give good results. In general, the results will be more satisfactory when the flame is quite hot. For this reason, the substitution of a blast lamp for the ordinary Bunsen burner is sometimes advisable. The observations can usually be made more rapidly if one observer devotes his attention to keeping the flame in proper condition, while the other observes the spectrum. EXPERIMENT Y 3 . Determination of the distance between the lines of a grating by the diffraction of sodium light. The object of this experiment is to illustrate the principles involved in the formation of spectra by a diffraction grating. It is expected, therefore, that the report should contain a clear explanation of the phenomena observed. (See Kohlrausch, Glazebrook's Physical Optics, p. 183, and other books on light.) The apparatus consists of a horizontal arm, which may for convenience be provided with a scale, mounted upon a suitable support, and having at its center a narrow vertical slit which may be illuminated by sodium light. To obtain the pure yellow light of sodium it is only necessary to place a wire carrying a bead of some sodium salt in the flame of a Bunsen burner. The grating to be studied is placed in front of the slit with its rulings vertical, and should be mounted on some support so that its distance from the slit can be varied. On looking through the grating, several images of the slit will be seen on either side, the distance between these images depending upon the distance apart of the lines of the grating. If white light were used instead of the sodium flame, these images would become spectra. By measuring the distance between the grating and the slit (D, Fig. 106), and the displace- ment of one of the images d, the angle through which the ligh VOL. I T ~* A 274 JUNIOR COURSE IN GENERAL PHYSICS. is bent by diffraction can be determined. From this angle, and the wave length of sodium light, the distance between the lines of the grating is to be computed. In measuring the displace- ment of the images the fol- lowing method will be found convenient : Adjust a small rider, which can be clamped to the horizontal arm, until it coincides with the correspond- ing image on the opposite side of the slit. The distance be- tween the two riders is then equal to twice the displacement of the image. Measurements should be made in this way for four or five different positions of the grating and with different pairs of images. If the meas- urements are carefully made, the individual results should agree fairly well, and the final average will not be far from the truth. The wave length of sodium light may be taken as 0.000059 cm. G Fig. 106. EXPERIMENT Y 4 . Measurement of candle power by the Bunsen photometer. In the Bunsen photometer a screen of white paper (D, Fig. 107), a portion of which has been made translucent by the LX- *-- Fig. 107. application of oil, is placed in a blackened box (technically called the carriage), and mirrors, M,M', are adjusted so that THE SPECTROSCOPE AND PHOTOMETER. 2/5 both sides of the paper may be observed at the same time. By means of openings in the carriage, light is admitted from the two sources whose intensities are to be compared. The carriage being placed between the two lights, each face of the screen is illuminated only by light from the source toward which it is turned, while the translucent portion of the paper receives light from both sources. In using the instrument, the carriage is shifted in position until both sides of the screen are seen to be equally illuminated. The distances of the two lights from the screen are then measured, and the relative intensities of the two sources are computed by the law of inverse squares. Transmitted Light Reflected; Light Fig. 108. The translucent spot on the screen merely serves to locate the position of equal illumination with greater accuracy than could otherwise be obtained. If the adjustment is not quite correct, this spot will appear dark on one side and bright on the other (see Fig. 108) ; but when the proper position has been found, it will almost entirely disappear. The standard source of light in the apparatus used consists of an Argand burner placed just back of an opaque screen, in which a small oblong slit has been cut. The slit is made so small that it appears entirely covered with light when viewed from the front, and should be so adjusted as to receive only that light which comes from the central portion of the flame. In this way the irregularities due to the flickering of the edges of the flame are avoided. It is to be observed that under the conditions mentioned, the slit itself is the source of light. 276 JUNIOR COURSE IN GENERAL PHYSICS. Distances should be measured to the plane of the slit, and not to the center of the flame. To perform the experiment, place the standard and the light to be measured at opposite ends of the photometer bar, and before beginning any actual observations, practice setting the carriage to the position of equal illumination until nearly the same reading is obtained several times in succession. After each reading, the carriage should be shifted two or three feet, in some cases to the right and in others to the left of the proper setting, and then brought back again, without reference to the previous reading, until the two sides of the screen appear to be equally illuminated. The uncertainties of the observation, together with slight variations in the intensities of the two lights, will make it impossible to obtain coincident settings, but after a little practice the succes- sive readings should agree to within three or four per cent. Constant differences are often observed between the settings of different persons. These are due to differences in the eye, and cannot be avoided. After sufficient practice has been gained in reading, the photometer may be used to measure candle power in some one of the cases which follow. It is to be observed that the scale on the bar is located without regard to the positions of the two lights, so that suitable corrections will have to be made at each end. (1) By comparison with a standard candle the absolute intensity of the standard light may be determined. At least ten or twelve readings should be taken to get a good result. (2) The light from a fish-tail burner or an oil lamp may be measured as the flame is seen from different directions. It will probably be found that the flame differs in illuminating power according as the broad surface or the edge is turned toward the photometer. To investigate this variation first set the flame so that its plane is parallel with the photometer bar, and measure its intensity. Then turn it about a vertical axis THE SPECTROSCOPE AND PHOTOMETER. 277 and measure the candle power at intervals of 30 until the flame has been turned through a complete revolution. The results should be shown graphically by a curve, in which the angular position of the flame, and the observed intensity of the light, are used as polar co-ordinates. Such a curve is very commonly used to show the distribution of the light from any source, and has the advantage of showing at a glance the intensity of the light in all horizontal directions. (3) The absorbing power of substances which are nearly transparent may be determined. To accomplish this, measure the intensity of any source as seen direct ; then interpose the substance to be investigated, and see how much the light is diminished. From the two measurements the percentage absorption can be computed. Investigate in this way the absorption of sheets of glass of different thickness, and of cells containing various liquids. It must be remembered, however, that some of the light which is apparently absorbed is really lost by reflection. If it is desired to separate the effects of reflection and absorption, more elaborate methods will be necessary. Numerous other interesting problems will suggest them- selves in the solution of which the photometer may be used. For further details concerning photometry, see Vol. II of this Manual ; also Palaz, Photometric Industrielle. TABLES. [In these tables the admirable arrangement made use of in Bottomley's Four-Figure Mathematical Tables has been followed.] 280 LOGARITHMS. 1 2 3 4 5 6 7 8 9 123 456 789 10 oooo 0043 0086 0128 0170 0212 0253 0294 0334 0374 4 8 12 17 21 25 29 33 37 11 12 13 0414 0792 "39 0453 0828 "73 0492 0864 1206 Q53 1 0899 1239 0569 0934 1271 0607 0969 I33 0645 1004 1335 0682 1038 1367 0719 1072 1399 0755 1 106 1430 4811 3 7 I0 3 6 10 15 19 23 14 17 21 13 16 19 26 30 34 24 28 31 23 26 29 14 15 16 1461 1761 2041 1492 1790 2068 1523 2095 1553 1847 2122 1584 2148 1614 1903 2175 1644 i93i 2201 1673 1959 2227 i73 1987 2253 1732 2014 2279 369 368 3 5 8 12 15 18 II 14 17 ii 13 16 21 24 27 20 22 25 18 21 24 17 18 19 2304 2553 2788 2330 2577 2810 2355 2601 2833 2380 2625 2856 2405 2648 2878 2430 2672 2900 2455 2695 2923 2480 2718 2945 2504 2742 2967 2529 2765 2989 2 5 7 257 2 4 7 IO 12 15 9 12 14 9 " 13 17 20 22 16 19 21 16 18 20 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 246 8 ii 13 15 17 19 21 22 23 3222 3424 3617 3243 3444 3636 3 26 3 3464 3655 3284 3483 3674 334 3502 3692 3324 3522 37 11 3345 354i 3729 3365 356o 3747 3385 3579 3766 3404 3598 3784 246 246 246 8 10 12 8 10 12 7 9 ii 14 16 18 H 15 17 13 15 17 24 25 26 3802 3979 415 3820 3997 4166 3838 4014 4183 3856 4031 42OO 3874 4048 4216 3892 4065 4232 3909 4082 4249 3927 4099 4265 3945 4116 4281 3962 4133 4298 2 4 5 2 3 5 235 7 9 " 7 9 10 7 8 10 12 14 16 12 14 15 II 13 15 27 28 29 4314 4472 4624 4330 4487 4639 4346 4502 4654 4362 4518 4669 4378 4533 4683 4393 4548 4698 4409 45 6 4 4713 4425 4579 4728 4440 4594 4742 445 6 4609 4757 2 3 5 2 3 5 i 3 4 689 689 679 II 13 14 II 12 14 IO 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 i 3 4 679 10 ii 13 31 32 33 4914 5Q5 1 5i85 4928 5 65 5198 4942 5079 5211 4955 5092 5224 4969 5105 5237 4983 5 ir 9 5250 4997 5132 5263 5011 5H5 5276 5024 5159 5289 5 38 5172 5302 i 3 4 i 3 4 1 3 4 678 5 7 8 568 IO II 12 9 II 12 9 10 12 34 35 36 5315 544i 5563 5328 5453 5575 5340 5465 5587 5353 5478 5599 5366 5490 5611 5378 55 02 5623 539i 55 J 4 5635 5403 5527 5 6 47 5416 5428 5539 555i 5658] 5670 i 3 4 I 2 4 I 2 4 568 5 6 7 5 6 7 9 10 ii 9 10 ii 8 10 ii 37 38 39 5682 5798 59" 5 6 94 5809 5922 5705 5821 5933 5717 5832 5944 5729 5843 5955 5740 5855 5966 5752 5866 5977 5763 5877 5988 5999 5786 5899 6010 I 2 3 I 2 3 I 2 3 5 6 7 5 6 7 457 8 9 10 8 9 10 8 9 10 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 I 2 3 4 5 6 8 9 10 41 42 43 6128 6232 6335 6138 6243 6345 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 63^ 6415 6222 6325 6425 I 2 3 I 2 3 I 2 3 4 5 6 4 5 6 4 5 6 7 8 9 789 7 8 9 44 45 46 6435 6532 6628 6444 6542 6637 6 454 6& 6464 6561 6656 6474 6 57i 6665 6484 6580 6675 6493 6590 6684 6503 6599 6693 65 : 3 6609 6702 6522 6618 6712 I 2 3 I 2 3 I 2 3 4 5 6 4 5 6 4 5 6 789 7 8 9 7 7 8 47 48 49 6721 6812 6902 6730 6821 6911 6739 6830 6920 6749 6839 6928 6758 6848 6937 6767 6857 6946 6776 6866 6955 6785 6875 6964 6972 6803 6893 6981 I 2 3 I 2 3 I 2 3 4 5 5 4 4 5 445 678 678 678 50 6990 6998 7007 7016 7024 733 7042 7050 7059 7067 I 2 3 3 4 5 678 51 52 53 7076 7160 7243 7084 7168 725 1 7093 7177 7259 7101 7185 7267 7110 7193 7275 7118 7202 7284 7126 7210 7292 7135 7218 7300 7 J 43 7226 73o8 7152 7235 73^6 I 2 3 I 2 2 122 3 4 5 3 4 5 3 4 5 678 6 7 7 667 54 7324 7332 7340 7348 735 6 7364 7372 738o 7388 7396 1223 4 5 667 LOGARITHMS. 281 1 2 3 4 5 6 7 7459 1 8 9 123 456 789 55 7404 7412 7419 7427 7435 7443 745 i 7466 7543 7619 7694 7474 122 3 4 5 5 6 7 5 6 7 5 6 7 5 6 7 56 57 58 7482 7559 7 6 34 749 7566 7642 7497 7574 7649 755 7582 7 6 57 75'3 7589 7664 7520 7597 7672 7528 7604 7679 7752 7825 7896 7536 7612 7686 755i 7627 7701 2 2 2 2 I 2 345 345 344 59 60 61 7709 7782 7853 7716 7789 7860 7723 7796 7868 773i 7803 7875 7738 7810 7882 7745 7818 7889 7760 7767 7832 7839 7903! 7910 7774 7846 7917 2 2 2 344 344 344 5 6 7 5 6 6 5 6 6 62 63 64 7924 7993 8062 7931 8000 8069 7938 8007 8075 7945 8014 8082 7952 8021 8089 7959 8028 8096 7966 8035 8102 7973 798o 8041! 8048 8109 8116 7987 8055 8122 2 2 2 334 334 334 5 6 6 5 5 6 5 5 6 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 I 2 334 5 5 6 5 5 6 5 5 6 4 5 6 66 67 68 8195 8261 8325 8202 8267 8331 8209 8274 8338 8215 8280 8344 8222 8287 8351 8228 8293 8357 8235 8299 8363 8241 8306 8370 8248 8312 8376 8254 8319 8382 I 2 I 2 I 2 334 334 334 69 70 71 8388 845 i 8513 8395 8457 8519 8401 8463 8525 8407 8470 8531 8414 8476 8537 8420 8482 8543 8426 8488 8549 8432 8494 8555 8439 8500 8561 8445 8506 8567 2 2 2 234 234 234 4 5 6 4 5 6 4 5 5 72 73 74 8573 8633 8692 8579 8639 8698 8585 8645 8704 8591 8651 8710 8597 8657 8716 8603 8663 8722 8609 8669 8727 8615 8621 8675 8681 8733 8739 8627 8686 8745 2 2 2 234 234 234 4 5 5 4 5 5 4 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 2 233 455 4 5 5 4 4 5 445 4 4 5 4 4 5 445 4 4 5 4 4 5 445 76 77 78 8808 8865 8921 8814 8871 8927 8820 8876 8932 8825 8882 8938 8831 8887 8943 8837 8893 8949 8842 8899 8954 8848 8854 8904 8910 8960 8965 8859 8915 8971 2 2 2 233 233 233 79 80 81 8976 9031 9085 8982 9036 9090 8987 9042 9096 8993 9047 9101 8998 9053 9106 9004 9058 9112 9009 9063 9117 9015 9069 9122 9020 9074 9128 9025 9079 9133 9186 9238 9289 2 2 2 233 233 233 82 83 84 9138 9191 9243 9H3 9196 9248 9149 9201 9253 9154 9206 9258 9159 9212 9263 9165 9217 9269 9170 9222 9274 9175 9227 9279 9180 9232 9284 2 2 2 233 233 233 85 9294 9299 9304 9309 9315 9320 93 2 5 933 9335 9340 9390 9440 9489 I 2233 4 4 5 86 87 88 9345 9395 9445 9350 9400 945 9355 9405 9455 9360 9410 9460 9365 9415 9465 937 9420 9469 9375 9425 9474 938o 943 9479 9385 9435 9484 I 2 I O I 233 223 223 4 4 5 344 344 89 90 91 92~ 93 94 9494 9542 9590 9499 9547 9595 954 9552 9600 959 9557 9605 95*3 9562 9609 95^8 9566 9614 9523 957i 9619 9528 957 6 9624 9533 9581 9628 9538 9586 9633 O I O I O I 223 223 223 344 344 344 9638 9685 973 1 9643 9689 9736 9647 9694 974i 9652 9699 9745 9657 973 975 9661 9708 9754 9666 9713 9759 9671 9717 9763 9675 9680 9722J 9727 9768 9773 O I O I I 223 223 223 344 344 344 95 9777 9782 9*86 9791 9795 9800 9805 9809 9814' 9818 l| 2 2 3 344 96 97 98 9823 9868 9912 9827! 9832 9872 9877 9917 9921 9836 9881 9926 9841 9845 9886 9890 9930 9934 9850 9894 9939 9854 9899 9943 9859 9863 9903 9908 9948 9952 Ij 2 2 3 o 1223 o 1223 344 344 344 99 995 6 9961 9965 9969 9974 9978J 9983 9987 9991 9996 o i i 2 2 3J 3 3 4 282 NATURAL SINES. 0' 6' 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 oooo 0017 0035 0052 0070 0087 0105 OI22 0140 OI 57 369 12 15 1 2 3 OI 75 0349 0523 0192 0366 0541 0209 0384 0558 0227 0401 0576 0244 0419 0593 0262 0436 0610 0279 0454 0628 0297 0471 0645 0314 0488 o663 0332 0506 ob8o 369 369 3 6 9 12 I 5 12 I 5 12 I 5 4 5 6 0698 0872 1045 0715 0889 1063 0732 0906 1080 0750 0924 1097 0767 0941 "15 0785 0958 1132 0802 0976 1149 0819 0993 1167 0837 IOII 1184 0854 1028 I 20 1 369 369 369 12 I 5 12 I 4 12 14 7 8 9 1219 1392 i5 6 4 1236 1409 1582 1253 1426 1599 1271 1444 1616 1288 1461 1633 1305 1478 1650 1323 1495 1668 1340 1513 1685 1357 1530 1702 1374 X 547 1719 369 369 369 12 I 4 12 14 12 14 10 i73 6 1754 1771 1788 1805 1822 1840 1857 1874 1891 369 12 14 11 12 13 1908 2079 2250 1925 2096 2267 1942 2113 2284 1959 2130 2300 1977 2147 2317 1994 2164 2334 2OII 2181 2351 2028 2198 2368 2045 2215 2385 2062 2232 2402 369 369 3 6 8 II 14 II 14 II 14 14 15 16 2419 2588 2756 2436 2605 2773 2453 2622 2790 2470 2639 2807 2487 2656 2823 2504 2672 2840 2521 2689 2857 2538 2706 2874 2554 2723 2890 257i 2740 2907 368 3 6 8 368 II 14 II 14 II 14 17 18 19 2924 3090 3256 2940 3107 3272 2957 3 I2 3 3289 2974 3MO 3305 2990 3i5 6 3322 3007 3i73 3338 3024 3190 3355 3040 3206 3371 3057 3223 3387 374 3239 3404 368 3 6 8 3 5 8 II 14 II 14 II 14 20 3420 3437 3453 3469 3486 3502 35i8 3535 355 1 35 6 7 3 5 8 II 14 21 22 23 3584 3746 3907 3600 3762 3923 3616 3778 3939 3633 3795 3955 3649 3811 397i 3665 3827 3987 3681 3843 4003 3697 3859 4019 37H 3875 4035 3730 3891 405 l 3 5 8 3 5 8 3 5 8 II 14 II 14 II 14 24 25 26 4067 4226 4384 4083 4242 4399 4099 4258 4415 4"5 4274 443 i 4I3 1 4289 4446 4H7 435 4462 4163 4321 4478 4179 4337 4493 4195 4352 459 4210 4368 45 2 4 3 5 8 3 5 8 3 5 8 II I 3 II 13 10 13 27 28 29 4540 4695 4848 4555 4710 4863 457 1 4726 4879 4586 4741 4894 4602 475 6 4909 4617 4772 4924 4633 4787 4939 4648 4802 4955 4664 4818 497 4679 4833 4985 3 5 8 3 5 8 3 5 8 10 13 10 13 10 13 30 5000 5 OI 5 53Q 545 5060 575 5090 5 I0 5 5120 5135 3 5 8 10 13 31 32 33 5150 5 2 99 5446 5165 53H 546i 5180 5329 547 6 5195 5344 5490 5210 5358 555 5225 5373 5519 5240 5388 5534 5255 5402 5548 5270 5417 55 6 3 5284 5432 5577 2 57 2 5 7 2 5 7 10 12 IO 12 IO 12 34 35 36 5592 5736 5878 5606 575 5892 5621 57 6 4 5906 5635 5779 5920 5650 5793 5934 5664 5807 5948 5678 5821 5962 5 6 93 5835 597 6 577 5850 5990 572i 5864 6004 2 5 7 2 5 7 257 IO 12 IO 12 9 12 37 38 39 6018 6l 57 6293 6032 6170 6307 6046 6184 6320 6060 6198 6334 6074 6211 6347 6088 6225 6361 6101 6239 6374 6115 6252 6388 6129 6266 6401 6i43 6280 6414 2 5 7 257 247 9 12 9 I' 9 ii 40 6428 6441 6 455 6468 6481 6494 6508 6521 6534 6 547 247 9 ii 41 42 43 6561 6691 6820 6 574 6704 6833 6587 6717 6845 6600 6730 6858 6613 6743 6871 6626 6756 6884 6639 6769 6896 66^2 6782 6909 6665 6794 6921 6678 6807 6934 247 2 4 6 246 9 ii 9 ii 8 ii 44 |6947 6959 6972 6984 6997 7009 7022 734 7046 759 246 8 10 NATURAL SINES. 283 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 123 4 5 45 7071 7083 7096 7108 7120 7i33 7H5 7i57 7169 7181 246 8 10 46 47 48 7193 73H 743i 7206 7325 7443 7218 7337 7455 7230 7349 7466 7242 7361 7478 7254 7373 7490 7266 7385 75 01 7278 7396 75 J 3 7290 7408 7524 7302 7420 7536 2 4 6 2 4 6 246 8 10 8 10 8 10 49 50 51 7547 7660 7771 7558 7672 7782 757 7683 7793 758i 7694 7804 7593 7705 7815 7604 7716 7826 7615 7727 7837 7627 7738 7848 7638 7749 7859 7649 7760 7869 246 2 4 6 2 4 5 8 9 7 9 7 9 52 53 54 7880 7986 8090 7891 7997 8100 7902 8007 8111 7912 8018 8121 7923 8028 8131 7934 8039 8141 7944 8049 8151 7955 8059 8161 7965 8070 8171 7976 8080 8181 245 235 2 3 5 7 9 7 9 7 8 55 8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 2 3 5 7 8 56 57 58 8290 8387 8480 8300 8396 8490 8310 8406 8499 8320 8415 8508 8329 8425 8517 8339 8434 8526 8348 8443 8536 8358 8453 8545 8368 8462 8554 8377 8471 8563 2 3 5 2 3 5 235 6 8 6 8 6 8 59 60 61 8572 8660 8746 8581 8669 8755 8590 8678 8763 8599 8686 8771 8607 8695 8780 8616 8704 8788 8625 8712 8796 8634 8721 8805 8643 8729 8813 8652 8738 8821 3 4 3 4 3 4 6 7 6 7 6 7 62 63 64 8829 8910 8988 8838 8918 8996 8846 8926 9003 8854 8934 9011 8862 8942 9018 8870 8949 9026 8878 8957 9033 8886 8965 9041 8894 8973 9048 8902 8980 9056 3 4 3 4 3 4 1 1 5 6 65 9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 2 4 5 6 66 67 68 9135 9205 9272 9M3 9212 9278 915 9219 9285 9157 9225 9291 9164 9232 9298 9171 9239 9304 9178 9245 93" 9184 9252 9317 9191 9259 9323 9198 9265 9330 2 3 2 3 2 3 5 6 4 6 4 5 69 70 71 9336 9397 9455 9342 9403 9461 9348 9409 9466 9354 94i5 9472 936i 9421 9478 9367 9426 9483 9373 9432 9489 9379 9438 9494 9385 9444 9500 9391 9449 9505 2 3 2 3 2 3 4 5 4 5 4 5 72 73 74 95 11 95 6 3 9613 95 l6 9568 9617 952i 9573 9622 95 2 7 9578 9627 9532 9583 9632 9537 9588 9636 9542 9593 9641 9548 9598 9646 9553 9603 9650 9558 9608 9655 2 3 2 2 2 2 4 4 3 4 3 4 75 9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 I 2 3 4 76 77 78 973 9744 978i 9707 9748 9785 9711 9751 9789 97'5 9755 9792 9720 9759 9796 9724 9763 9799 9728 9767 9803 9732 9770 9806 9736 9774 9810 9740 9778 98i3 2 2 I 2 3 3 3 3 2 3 79 80 81 9816 9848 9877 9820 9851 9880 9823 9854 9882 9826 9857 9885 9829 9860 9888 9833 9863 9890 9836 9866 9893 9839 9869 9895 9842 9871 9898 9845 9874 9900 I 2 O 2 3 2 2 2 2 82 83 84 9903 9925 9945 9905 9928 9947 9907 993 9949 9910 9932 995 i 9912 9934 9952 9914 9936 9954 9917 9938 995 6 9919 9940 9957 9921 9942 9959 9923 9943 9960 O O 2 2 I 2 I I 85 9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 001 I I 86 87 88 9976 9986 9994 9977 9987 9995 9978 9988 9995 9979 9989 9996 9980 9990 9996 9981 9990 9997 9982 9991 9997 9983 9992 9997 9984 9993 9998 9935 9993 9998 O O I 000 O O O I I O O 89 9998 9999 9999 9999 9999 rooo nearly. rooo nearly. rooo nearly. rooo nearly. rooo nearly. O O O O O 284 NATURAL COSINES. 0' 6' 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 I '000 I '000 nearly. rooo nearly. rooo nearly. I'OOO nearly. 9999 9999 9999 9999 9999 000 o o 1 2 3 9998 9994 9986 9998 9993 9985 9998 9993 9984 9997 9992 9983 9997 9991 9982 9997 9990 9981 9996 9990 9980 9996 9989 9979 9995 9988 9978 9995 9987 9977 o o o o o o O O I o o I I I I 4 5 6 9976 9962 9945 9974 9960 9943 9973 9959 9942 9972 9957 9940 9971 995 6 9938 9969 9954 9936 9968 9952 9934 9966 995 i 9932 9965 9949 9930 9963 9947 9928 o o O I I I I I 2 I 2 7 8 9 9925 9903 9877 9923 9900 9874 9921 9898 9871 9919 9895 9869 9917 9893 9866 9914 9890 9863 9912 9888 9860 9910 9885 9857 -9907 9882 9854 9905 9880 9851 O I O I I 2 2 2 2 2 2 10 9848 9845 9842 9839 9836 9833 9829 9826 9823 9820 112 2 3 11 12 13 9816 9781 9744 9813 9778 9740 9810 9774 9736 9806 9770 9732 9803 9767 9728 9799 9763 9724 9796 9759 9720 9792 9755 97 J 5 9789 975 1 9711 9785 9748 9707 I I 2 I I 2 I I 2 2 3 3 3 3 3 14 15 16 973 9659 9613 9699 9655 9608 9694 9650 9603 9690 9646 9598 9686 9641 9593 9681 9636 9588 9677 9632 9583 9673 9627 9578 9668 9622 9573 9664 9617 9568 I 2 2 2 2 2 3 4 3 4 3 4 17 18 19 9563 95 11 9455 9558 955 9449 9553 9500 9444 9548 9494 9438 9542 9489 9432 9537 9483 9426 9532 9478 942i 9527 9472 9415 952i 9466 9409 95 l6 9461 9403 2 3 2 3 2 3 4 4 4 5 4 5 20 9397 939i 9385 9379 9373 9367 936i 9354 9348 9342 2 3 4 5 21 22 23 933 6 9272 9205 9330 9265 9198 9323 9259 9191 9317 9252 9184 93U 9245 9178 9304 9239 9171 9298 9232 9164 9291 9225 9157 9285 9219 915 9278 9212 9H3 2 3 2 3 2 3 4 I 4 6 5 6 24 25 26 9135 9063 8988 9128 9056 8980 9121 9048 8973 9114 9041 8965 9107 9033 8957 9100 9026 8949 9092 9018 8942 9085 9011 8934 9078 9003 8926 9070 8996 8918 I 2 4 i 3 4 i 3 4 5 6 5 6 5 6 27 28 29 8910 8829 8746 8902 8821 8738 8894 8813 8729 8886 8805 8721 8878 8796 8712 8870 8788 8704 8862 8780 8695 8854 8771 8686 8846 8763 8678 8838 8755 8669 i 3 4 i 3 4 i 3 4 5 7 6 ? 30 8660 8652 8643 8634 8625 8616 8607 8599 8590 8581 i 3 4 6 7 31 32 33 8572 8480 8387 8563 8471 8377 8554 8462 8368 8545 8453 8358 8536 8443 8348 8526 8434 8339 8517 8425 8329 8508 8415 8320 8499 8406 8310 8490 8396 8300 2 3 5 235 2 3 5 6 8 6 8 6 8 34 35 36 8290 8192 8090 8281 8181 8080 8271 8171 8070 8261 8161 8059 8251 8151 8049 8241 8141 8039 8231 8131 8028 8221 8121 8018 8211 8111 8007 8202 8100 7997 2 3 5 235 2 3 5 7 8 7 8 7 9 37 38 39 7986 7880 7771 7976 7869 7760 7965 7859 7749 7955 7848 7738 7944 7837 7727 7934 7826 7716 7923 7815 775 7912 7804 7694 7902 7793 7683 7891 7782 7672 245 245 246 7 9 7 9 7 9 40 7660 7649 7638 7627 76i5 7604 7593 758i 757 7559 2 4 6 8 9 41 42 43 7547 743i 73H 7536 7420 7302 7524 7408 7290 75'3 7396 7278 75 01 ll 7266 7490 7373 7254 7478 736i 7242 7466 7349 7230 7455 7337 7218 7443 7325 7206 2 4 6 2 4 6 246 8 10 8 10 8 10 44 7 J 93 7181 7169 7i57 7H5 7133 7120 7108 7096 7083 246 8 10 N.B. Numbers in difference-columns to be subtracted, not added. NATURAL COSINES. 28 S 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 123 4 5 45 7071 7059 7046 7034 7022 7009 6997 6984 6972 6959 2 4 6 8 10 46 47 48 IsT 50 51 6947 6820 6691 6934 6807 6678 6921 6794 6665 6909 6782 6652 6896 6769 6639 6884 6756 6626 6871 6743 6613 6858 6730 6600 6845 6717 6587 6833 6704 6574 2 4 6 2 4 6 247 8 ii 9 ii 9 ii 6561 6428 6293 6 547 6414 6280 6534 ^401 6266 6521 6388 6252 6508 6374 6239 6494 6361 6225 6481 6347 6211 6468 6334 6198 6455 6320 6184 6441 6307 6170 2 4 7 247 2 5 7 9 ii 9 ii 9 ii 52 53 54 6l 57 6018 5878 6i43 6004 5864 6129 5990 5850 6115 597 6 5835 6101 5962 5821 6088 5948 5807 6074 5934 5793 6060 5920 5779 6046 5906 5764 6032 5892 575 257 257 2 5 7 9 12 9 12 9 12 55 le" 57 58 5736 572i 577 5 6 93 5678 5664 5650 5635 5621 5606 257 10 12 5592 5446 5299 5577 5432 5284 5563 54i7 5270 5548 5402 5255 5534 5388 5240 55i9 5373 5225 5505 5358 5210 5490 5344 5195 5476 5329 5180 546i 53H 5165 257 257 257 IO 12 10 12 10 12 59 60 61 5150 5000 4848 5135 4985 4833 5120 497 4818 5io5 4955 4802 5090 4939 4787 5075 4924 4772 5060 4909 475 6 5045 4894 4741 5030 4879 4726 5015 4863 4710 3 5 8 3 5 8 3 5 8 10 13 10 13 10 13 62 63 64 4695 4540 4384 4679 4524 4368 4664 459 4352 4648 4493 4337 4633 4478 4321 4617 4462 4305 4602 4446 4289 4586 443i 4274 457i 4415 4258 4555 4399 4242 3 5 8 3 5 8 3 5 8 10 13 10 13 II 13 65 4226 4210 4195 4179 4163 4H7 4131 4H5 4099 4083 3 5 8 II 13 66 67 68 4067 3907 3746 4051 3891 3730 4035 3875 37 H 4019 3859 3697 4003 3843 3681 3987 3 ?? 7 3665 397 1 3811 3649 3955 3795 3633 3939 3778 3616 3923 3762 3600 3 5 8 3 5 8 3 5 8 II 14 II 14 II 14 69 70 71 ~72~ 73 74 3584 3420 3256 3567 3404 3239 355i 3387 3223 3535 337 1 3206 35i8 3355 3190 35 2 3338 3173 3486 3322 3*5 6 3469 3305 3MO 3453 3289 3123 3437 3272 3107 3 5 8 3 5 8 368 II 14 II 14 II 14 3090 2924 2756 3074 2907 2740 3057 2890 2723 3040 2874 2706 3024 2857 2689 300,7 2840 2672 2990 2823 2656 2974 2807 2639 2957 2790 2622 2940 2773 2605 368 368 3 6 8 II 14 II 14 II .14 75 2588 2571 2554 2538 2521 2504 2487 2470 2453 2436 368 II 14 76 77 78 2419 2250 2079 2402 2233 2062 2385 2215 2045 2368 2198 2028 235 l 2181 2OII 2334 2164 1994 2317 2147 1977 2300 2130 1959 2284 2113 1942 2267 2096 1925 368 369 369 II 14 II 14 II 14 79 80 81 ~S2 83 84 1908 1736 i5 6 4 1891 1719 1547 1874 1702 1530 1857 1685 I5U 1840 1668 H95 1822 1650 1478 1805 1633 1461 1788 1616 1444 1771 1599 1426 1754 1582 1409 369 369 369 12 I 4 12 14 12 14 1392 1219 1045 1374 I2OI 1028 1357 1184 IOII 1340 1167 0993 1323 1149 0976 1305 1132 0958 1288 "i5 0941 1271 1097 0924 1253 1080 0906 1236 1063 0889 369 369 369 12 I 4 12 14 12 I 4 85 0872 0854 0837 0819 0802 0785 0767 0750 0732 0715 369 12 I 5 86 87 88 0698 0523 0349 0680 0506 0332 o663 0488 03H 0645 0471 0297 0628 454 0279 0610 0436 0262 593 0419 0244 0576 0401 0227 0558 0384 0209 0541 0366 0192 369 369 369 12 15 12 I 5 12 I 5 89 oi75 0157 0140 OI22 0105 0087 0070 0052 0035 0017 369 12 I 5 N.B. Numbers in difference-columns to be subtracted, not added. 286 NATURAL TANGENTS. 0' & 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 oooo 0017 0035 0052 0070 0087 0105 OI22 0140 OI 57 369 12 14 1 2 3 oi75 0349 0524 0192 0367 0542 0209 0384 0559 0227 0402 577 0244 0419 0594 0262 437 0612 0279 0454 0629 0297 0472 0647 03H 0489 0664 0332 0507 0682 369 369 369 12 15 12 15 12 I 5 4 5 B 0699 0875 1051 0717 0892 1069 0734 0910 1086 0752 0928 1104 0769 0945 1122 0787 0963 H39 0805 0981 "57 0822 0998 "75 0840 1016 1192 0857 103? I2IO 369 369 369 12 I 5 12 I 5 12 I 5 7 8 9 1228 1405 1584 1246 1423 1602 1263 1441 1620 1281 H59 1638 1299 1477 1655 1317 H95 1673 1334 1512 1691 1352 1530 1709 1370 1548 1727 1388 1566 1745 369 369 369 12 I 5 12 15 12 I 5 10 1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 369 12 I 5 11 12 13 1944 2126 2309 1962 2144 2327 1980 2162 2345 1998 2180 2364 2016 2199 2382 2035 2217 2401 2053 2235 2419 2071 2254 2438 2089 2272 2456 2107 2290 2475 369 369 369 12 I 5 12 I 5 12 I 5 14 15 16 2493 2679 2867 2512 2698 2886 2530 2717 2905 2549 2736 2924 2568 2754 2943 2586 2773 2962 2605 2792 2981 2623 2811 3000 2642 2830 3019 2661 2849 3038 369 369 369 12 l6 13 J 6 13 16 17 18 19 3057 3249 '3443 3076 3269 3463 3096 3288 3482 3"5 3307 3502 3134 3327 3522 3153 3346 3541 3172 3365 356i 3i9i 3385 358i 3211 3404 3600 3230 3424 3620 3 6 10 3 6 10 3 6 10 13 16 13 16 13 17 20 3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3 7 10 13 17 21 22 23 3839 4040 4245 3859 4061 4265 3879 4081 4286 3899 4101 437 3919 4122 4327 3939 4142 4348 3959 4163 4369 3979 4183 4390 4000 4204 4411 4020 4224 443 i 3 7 I0 3 7 I0 3 7 I0 13 17 H 17 14 17 24 25 26 445 2 4663 4877 4684 4899 4494 4706 4921 45i5 4727 4942 4536 4748 4964 4557 4770 4986 4578 4791 5008 4599 4813 5029 4621 4834 5051 4642 4856 573 4 7 10 4 7 ii 4 7 ii 14 18 14 18 15 18 27 28 29 5095 5317 '5543 5"7 5340 5566 5139 5362 5589 5161 5384 5612 5184 5407 5635 5206 5430 5658 5228 5452 5681 525 5475 5704 5272 5498 5727 5295 5520 5750 4 7 ii 4 8 ii 4 8 12 15 18 15 19 15 19 30 '5774 5797 5820 5844 5867 5890 59H 5938 596i 5985 4 8 12 16 20 31 32 33 6009 6249 6494 6032 6273 6519 6056 6297 6544 6080 6322 6569 6104 6346 6594 6128 637 1 6619 6152 6395 6644 6176 6420 6669 6200 6445 6694 6224 6469 6720 4 8 12 4 8 12 4 8 13 16 20 16 20 I 7 21 34 35 36 6745 7002 7265 6771 7028 7292 6796 754 7319 6822 7080 7346 6847 7107 7373 6873 7!33 7400 6899 7 J 59 7427 6924 7186 7454 6950 7212 7481 6976 7239 7508 4 9 13 4 9 13 5 9 H 17 21 18 22 18 23 37 38 39 7536 7813 8098 75 6 3 7841 8127 7590 8156 7618 7898 8185 7646 7926 8214 7 6 73 7954 8243 7701 7983 8273 7729 8012 8302 7757 8040 8332 7785 8069 8361 5 9 H 5 I0 M 5 10 15 18 23 19 24 20 24 40 8391 8421 8451 8481 8511 8541 857i 8601 8632 8662 5 I0 J 5 20 25 41 42 43 8693 9004 9325 8724 9036 9358 8754 9067 939i 8785 9099 9424 8816 9131 9457 8847 9163 9490 8878 9195 9523 8910 9228 9556 8941 9260 9590 8972 9293 9623 5 10. 16 5 ii 16 6 ii 17 21 26 21 27 22 28 44 ,9657 9691 97 2 5 9759 9793 9827 9861 9896 993 9965 6 ii 17 23 29 NATURAL TANGENTS. 28 7 45 47 48 0' 6' 12' 18' 24' ' 30' 36' 42' 48' 54' 123 4 5 roooo 0035 0070; 0105 0141 0176 O2 1 2 0247 0283 0319 6 12 18 24 30 1-0724 1-1106 0392 0761 "45 0428 0464 0799 0837 1184 1224 0501 0875 1263 0538 0913 1303 0575 095 ! 1343 0612 0990 1383 0649 1028 1423 0686 1067 1463 6 12 18 6 13 19 7 13 20 2 5 31 25 32 26 33 49 50 51 1-1504 1-1918 1-2349 1544 1960 2393 1585 1626 2OO2 ! 2045 2437 2482 1667 2088 2527 1708 2131 2572 1750 2174 2617 1792 2218 2662 1833 2261 2708 1875 2305 2753 7 14 21 7 14 22 8 15 23 28 34 29 36 30 38 52 53 , 54 1 1-2799 1-3270 'I-3764 2846 33*9 3814 2892 3367 3865 2938 3416 3916 2985 3465 3968 3032 35'4 4019 3079 3564 4071 3127 3613 4124 4176 3222 4229 8 16 23 8 16 25 9 17 26 3i 39 33 4i 34 43 55 1-4281 4335 4388 4442 4496 4550 4605 4659 47i5 4770 9 18 27 36 45 56 57 58 1-4826 1-5399 1-6003 4882 4938 4994 5458 5517! 5577 6066 6128 6191 5051 5637 6255 5108 5697 6319 5 l66 5757 6383 5224 5818 6447 5282 5880 6512 5340 6577 10 19 29 10 20 30 II 21 32 38 48 40 50 43 53 59 60 61 1-6643 1-7321 1-8040 6709; 6775 7391 7461 8115 8190 6842 7532 8265 6909 6977 7603! 7675 8341! 8418 745 7747 8495 7"3 7820 8572 7182 7893 8650 725 i 7966 8728 " 23 34 12 24 36 13 26 38 45 56 48 60 5 1 64 62 63 64 1-8807 1-9626 2-0503 8887 8967 97" 9797 0594 0686 9047 9883 0778 9128 9970 0872 9210 0057 0965 9292 0145 1060 9375 0233 "55 945 s 0323 1251 9542 0413 1348 14 27 41 15 29 44 16 31 47 55 68 58 73 63 78 65 2-1445 1543! 1642 1742 1842 1943 2045 2148 2251 2355 1 7 34 5 1 68 85 66 67 68 2-2460 2-3559 2-475 i 2566 2673 3673 3789 4876 5002 2781 3906 5129 2889 4023 5257 2998 4142 5386 3109 4262 5517 3220 4383 5649 3332 454 5782 3445 4627 18 37 55 20 40 60 22 43 65 74 92 79 99 87 108 69 70 71 2-605 l 2-7475 2-9042 6187 6325 7 62 5 | 7776 92o8j 9375 6464 7929 9544 6605 8083 97 J 4 6746 8239 9887 6889 8397 0061 734 8556 0237 7179 8716 0415 7326 8878 595 24 47 7i 26 52 78 29 58 87 95 "8 104 130 "5 H4 72 73 74 3-0777 3-2709 J4874 0961 2914 5105 1146 3122 5339 3332 5576 3544 5816 1716 3759 6059 1910 3977 6305 2106 6554 2305 4420 6806 2506 4646 7062 32 64 96 36 72 108 41 82 122 129 161 144 i 80 162 203 75 3-732I 7583 7848 8118 8391 8667 8947 9232 9520 9812 46 94 139 186 232 76 77 78 4-0108 4-33I5 4-7046 0408 3662 7453 0713 4015 7867 IO22 4374 8288 1335 4737 8716 1653 5107 9152 1976 5483 9594 2303 5864 0045 2635 6252 0504 2972 6646 0970 53 107 160 62 124 186 73 146 219 214 267 248 310 292 365 79 80 81 5^446 5-67I3 6-3138 1929 7297 3859 2422 7894 4596 2924 8502 5350 3435 9124 6122 3955 9758 6912 4486 0405 7920 5026 1066 8548 5578 1742 9395 6140 2432 0264 87 175 262 35 437 Difference-columns cease to be useful, owing to the rapidity with which the value of the tangent changes. 82 83 84 7'"54 8-1443 2066 2636 9-677 3002 3863 9-845 3962 5126 IO-O2 4947 6427 IO'2O 5958 7769 10-39 6996 9152 10-58 8062 0579 10-78 9158 2052 10-99 0285 3572 1 1 -20 85 ii-43 i'-66 11-91 12-16 12-43 12-71 13-00 13-30 13-62 I3-95 86 87 88 14-30 19-08 28-64 14-67 I9-74 30-14 15-06 20-45 31-82 15-46 2I'2O 3J69 I5-89 22-O2 35-80 16-35 J 6'83 22-90 23-86 38-19140-92 17-34 24-90 44-07 17-89 26-03 47-74 18-46 27-27 52-08 89 57-29 63-66 7I -62 81-85 95'49 114-6 143-2 191-0 286-5 573-o ! 288 NATURAL COTANGENTS. 0' 6' 12' 18' 24' 3O' 36' 42' 48' 54' Difference-columns not useful here, owing to the rapidity with which the value of the cotangent changes. Inf. 573-o 286-5 191-0 143-2 114-695-49 81-85 71-6263-66 1 2 3 57-29 28-64 19-08 52-08 27-27 18-46 47-74 26-03 17-89 44-07 24-90 17-34 40-92 23-86 16-83 38-I9 22-90 16-35 35-80 22-02 15-89 33-69 2 1 -2O I5-46 31-82 20-45 15-06 30-I4 '19-74 14-67 4 5 6 14-30 "'43 9"5 J 44 I3-95 11-20 3572 13-62 10-99 2052 iJ30 10-78 579 13-00 10-58 .9152 I2-7I 10-39 7769 12-43 IO'2O 6427 12-16 IO'O2 5126 11-91 n-66 9-845 9-677 3863 2636 7 8 9 8-1443 7-ii54 6-3138 0285 0264 2432 9158 9395 1742 8062 8548 1066 6996 7920 0405 5958 6912 9758 4947 6122 9124 3962 535 8502 3002 4596 7894 2066 3859 7297 10 5'67i3 6140 5578 5026 4486 3955 3435 2924 2422 1929 123 4 5 11 12 13 5-I446 4-7046 4-33I5 0970 6646 2972 0504 6252 2635 0045 5864 2303 9594 5483 1976 9152 5107 1653 8716 4737 1335 8288 4374 IO22 7867 4015 0713 7453 3662 0408 74 148 222 63 125 188 53 107 160 296 370 252 314 214 267 14 15 16 4-0108 37321 3'4874 9812 7062 4646 9520 6806 4420 9232 6554 4197 8947 8667 6305 6059 3977 3759 839i 5816 3544 8118 5576 3332 7848 5339 3122 7583 5105 2914 46 93 J 39 41 82 122 36 72 I 08 i 86 232 163 204 144 180 17 18 19 372709 3-0777 . 2-9042 2506 595 8878 2305 0415 8716 2106 0237 8556 1910 0061 8397 1716 9887 8239 5 2 4 97H 8083 ^334 9544 7929 1146 0961 9375! 9208 777 6 ! 7625 32 64 96 2 9 5 8 87 26 5 2 7 8 129 161 115 J 44 104 130 20 2 7475 7326 7179 734 6889 6746 6605 6464 63251 6187 24 47 71 95 n8 21 22 23 2-6051 2'475 I 2-3559 5916 4627 3445 5782 454 3332 5649 4383 3220 55!7 4262 3109 5386 4142 2998 5257 4023 2889 5 I2 9 3906 2781 5002 4876 3789 3673 2673 2566 22 43 65 20 40 60 18 37 55 87 108 79 99 74 92 24 25 26 '2-2460 2-1445 2-0503 2355 1348 0413 2251 1251 0323 2148 "55 0233 2045 1060 0145 1943 0965 0057 1842 0872 9970 1742 0778 9883 1642 0686 9797 1543 0594 9711 17 34 5 1 16 31 47 15 29 44 68 85 63 78 58 73 27 28 29 1-9626 1-8807 1-8040 9542 8728 7966 9458 8650 7893 9375 8572 7820 9292 8495 7747 9210 8418 7675 9128 8341 7603 9047 8265 7532 8967 8190 746i 8887 8115 739i 14 27 41 1 3 26 38 12 24 36 55 68 5 1 64 48 60 30 17321 7251 7182 7"3 745 6977 6909 6842 6775 6709 ii 23 34 45 56 31 32 33 ~3T 35 36 1-6643 1-6003 1-5399 6577 594i 5340 6512 5880 5282 6447 5818 5224 6383 5757 5166 6319 5697 5108 6255 5637 5051 6191 5577 4994 6128 5517 4938 6066 5458 4882 II 21 32 10 20 30 10 19 29 43 53 40 5 38 48 1-4826 1-4281 1-3764 477 4229 3713 4715 4176 3663 4659 4124 3613 4605 4071 3564 4550 4019 35H 4496 3968 3465 4442 3916 34i6 4388 3865 3367 4335 3814 3319 9 18 27 9 17 26 8 16 25 36 45 34 43 33 4i 37 38 39 1-3270 1-2799 1-2349 3222 2753 2305 3175 2708 2261 3127 2662 2218 3079 2617 2174 3032 2572 2131 2985 2527 2088 2938 2482 2045 2892 2437 2OO2 2846 2393 1960 8 16 23 8 15 23 7 14 22 3i 39 30 38 29 36 40 1-1918 1875 1833 1792 '75 1708 1667 1626 1585 1544 7 '4 21 28 34 41 42 43 1-1504 1-1106 1-0724 1463 1067 0686 H23 1028 0649 1383 0990 0612 1343 095 i 0575 1303 0913 0538 1263 0875 0501 1224 0837 0464 1184 0799 0428 "45 0761 0392 7 13 20 6 13 19 6 12 18 26 33 25 32 25 3i 44 I !-355 0319 0283 0247 O2 1 2 0176 0141 0105 OO7O 0035 6 12 18 24 30 N. B. Number difference-columns to be subtracted, not added. NATURAL COTANGENTS. 289 45 0' 6' 12 18' 24' 3O' 36' 42' 48' 54' 123 4 5 ro 0-9965 0-9930 0-9896 0-9861 0-9827 0-9793 o'9759 0-9725 0-9691 6 ii 17 23 29 46 47 48 ~49~ 50 51 9657 9325 9004 9623 9293 8972 9590 9260 8941 955 6 9228 8910 9523 9195 8878 9490 9163 8847 9457 9131 8816 9424 9099 8785 939i 9067 8754 9358 9036 8724 6 ii 17 5 ii 16 5 i 16 22 28 21 27 21 26 8693 8391 8098 8662 8361 8069 8632 8332 8040 8601 8302 8012 8571 8273 7983 8541 8243 7954 8511 8214 7926 8481 8185 7898 8451 8156 7869 8421 8127 7841 5 I0 '5 5 I0 '5 5 I0 J 4 2O 25 20 24 19 24 52 53 54 7813 7536 7265 7785 7508 7239 7757 7481 7212 7729 7454 7186 7701 7427 7i59 7673 7400 7133 7646 7373 7107 7618 7346 7080 7590 73i9 7054 7563 7292 7028 5 9 H 5 9 H 4 9 13 18 23 18 23 18 22 55 7002 6976 6950 6924 6899 6873 6847 6822 6796 6771 4 9 13 17 21 56 57 58 6 745 6494 6249 6720 6469 6224 6694 6445 6200 6669 6420 6176 6644 6395 6152 6619 6371 6128 6594 6346 6104 6569 6322 6080 6544 6297 6056 6519 6273 6032 4 8 13 4 8 12 4 8 12 17 21 16 20 16 20 59 60 61 6009 '5774 '5543 5985 5750 55 20 596i 5727 5498 5938 5704 5475 59H 5681 5452 5890 5658 5430 5867 5 6 35 5407 5844 5612 5384 5820 5589 5362 5797 5566 5340 4 8 12 4 8 12 4 8 ii 16 20 15 19 15 19 62 63 64 5317 '595 4877 5295 573 4856 5272 55i 4834 5250 5029 4813 5228 5008 479i 5206 4986 477 5184 4964 4748 5161 4942 4727 5139 4921 4706 5"7 4899 4684 4 7 ii 4 7 ii 4 7 ii 15 18 15 18 14 18 65 ~66~ 67 68 4663 4642 4621 4599 4578 4557 4536 45'5 4494 4473 4 7 10 14 18 4452 4245 4040 443i 4224 4020 4411 4204 4000 4390 4183 3979 4369 4163 3959 4348 4142 3939 4327 4122 3919 4307 4101 3899 4286 4081 3879 4265 4061 3859 3 7 10 3 7 10 3 7 10 14 17 14 17 13 17 69 70 71 3839 3640 '3443 3249 3057 2867 3819 3620 3424 3799 3600 3404 3779 358i 3385 3759 356i 3365 3739 354i 3346 3719 3522 3327 3699 35 2 3307 3679 3482 3288 3659 3463 3269 3 7 10 3 6 10 3 6 10 13 17 13 17 13 16 72 73 74 3 2 3o 3038 2849 3211 3019 2830 3 l 9 l 3000 2811 3172 2981 2792 3153 2962 2773 3134 2943 2754 3H5 2924 2736 3096 2905 2717 3076 2886 2698 3 6 10 369 369 13 16 13 16 13 16 75 2679 2661 2642 2623 2605 2586 2568 2549 2530 2512 3 6 9 12 16 76 77 78 2493 2309 2126 2475 2290 2107 2456 2272 2089 2438 2254 2071 2419 2235 2053 2401 2217 2035 2382 2199 2016 2364 2180 1998 2345 2162 1980 2327 2144 1962 369 369 369 12 I 5 12 I 5 12 15 79 80 81 1944 i7 6 3 1584 1926 38 1908 1727 1548 1890 1709 1530 1871 1691 1512 1853 1673 H95 1835 l6 55 H77 1817 1638 H59 1799 1620 1441 1781 1602 1423 369 369 3 6 9 12 I 5 12 I 5 12 I 5 82 83 84 1405 1228 1051 1388 I2IO 1033 1370 1192 1016 1352 "75 0998 1334 0981 1317 "39 0963 1299 1122 0945 1281 1104 0928 1263 1086 0910 1246 1069 0892 369 369 369 12 I 5 12 I 5 12 I 5 85 0875 0857 0840 0822 0805 fZ!Z. 0612 0437 0262 0769 0752 0734 0717 369 12 I 5 86 87 88 0699 0524 0349 0682 0507 33 2 0664 0489 0314 0647 0472 0297 0629 454 0279 0594 0419 0244 577 0402 0227 559 0384 0209 0542 0367 0192 369 3 6 9 369 12 I 5 12 I 5 12 I 5 89 0175 0157 j 0140 0122 0105 0087 0070 0052 0035 0017 3 6 9 12 14 N,B. Numbers in difference-columns to be subtracted, not added. VOL. I U INDEX TO VOLUME I. Absorbing power for heat radiation, 119; for light radiation, 277. Acceleration of gravity by Atwood's ma- chine, 53; by free fall, 55; by physical pendulum, 67 ; by Kater's pendulum, 69 ; value of, at Cornell Laboratory, 69. Air, expansion of, 99. Air displacement, correction for, 82. Ampere, definition of, 155. Atwood's machine, laws of uniformly ac- celerated motion, 50 ; gravity by, 53. Ballistic galvanometer, 221 ; constant of, 221. Barometer, cistern, 96; siphon, 96; com- parison of, 96. Barometric height corresponding to boiling points of water, 102. Battery resistance, measurement by Ohm's method, 214; Mance's method, 217; half deflection method, 192; Beetz's method, 197. Beetz's method for measuring E. M. F. and resistance of a battery, 197. Boiling point under different pressures near one atmosphere, 102. " Bound " electricity, 123. Boyle's law, example, 10 ; verification of, 93. Bunsen photometer, 274. Calibration, of a thermometer tube, 30; of a hydrometer, 88 ; of a galvanometer, 174 ; of a prism for wave-lengths, 272. Calorimetry, 107. Candle-power, by 3unsen photometer, 274. Capacity, definition of, 227 ; comparison of, 227 ; measurement in absolute measure, 230. Cathetometer, adjustment of, 29. Cell, standard Daniell, 173 ; E. M. F. of, by Ohm's method, 189; by comparison, 187 ; resistance (see Battery Resistance) . Collimator, 268. Commutator, 158. Computations, 5. Concave mirror, focal length of, 260. Condenser, principle of, 130 ; variable, 135 ; capacity of, 228 ; arranged in series and in multiple, 229. Conditions, choice of, 4, 20. Conductivity, electric, 204. Conjugate foci, for convex mirror, 258 ; for concave mirror, 260; for convex lens, 261. Constant of galvanometer, true, 156 ; work- ing, 156; by copper voltameter, 167; of sensitive galvanometer, 170; of ballistic galvanometer, 221. Convex lens, radius of curvature, 257 ; focal length, 261. Copper voltameter, 167. Current of electricity, definition, 153 ; pro- portional to magnetic field, 153 ; absolute unit, 155 ; practical unit, 155 ; measured by electrolysis, 155, 166; measurement of, 177. Curvature, radius of, by spherometer, 26; by reflection, 257; of concave mirror, 260. Curves, plotting of, 8, 21. Damping, 66; of galvanometer needle, 161, 223 ; theory of, 223 ; ratio of, 224. Daniell cell, standard, 173. Decrement, logarithmic, 223. Density, from mass and dimensions, 34; definition, 79; with corrections for tem- perature and air displacement, 82; by specific gravity bottle, 81 ; of liquid, 81, 291 2Q2 INDEX. 86, 88 ; of salt solution, 87 ; of water, 87 ; by Hare's method, 92. Deviation, minimum, of light through prism, 269. Difference of potential, 181 ; definition, 181 ; electromagnetic unit, 186; at terminals of battery, 191. Diffraction, 273. Dip of earth's magnetic field, 236. Distribution of "free" magnetism, 151. Dividing engine, 31. Earth inductor, 236. Efficiency, of wheel and axle, 46 ; of system of pulleys, 49 ; curve, 48. Elasticity, 74. Electrical Congress, definition of ampere, 155; definition of ohm, 205. Electrical machine, 133. Electrical quantity, 220. Electricity " bound " and " free," 123. Electrification, energy of, 122. Electrolysis, measurement of current by, 155, 166. Electrolytes, resistance of, 218. Electromagnetic induction, 232. Electromagnetic, unit of current, 155 ; unit of E. M. F. and potential difference, 186. Electromotive force, 181 ; of an electrical machine, 135 ; definition, 182 ; compari- son of, 187 ; Ohm's method of measur- ing, 189; Beetz's method of measuring, 197 ; of a thermo-element, 201 ; of in- duced currents, 223. Electroscope, 128. Equipotential lines in liquid conductor, 199 ; how to construct, 125. Errors, sources of, 13; accidental, 14; probable, 15 ; constant, 17 ; influence of, 18 ; relative, 20. Expansion, coefficient for air, 99. Fahrenheit's hydrometer, 86. Fall of potential, in wire-carrying current, 194; method of measuring resistance, 208. Field of force, electrical, how to map, 125 ; magnetic, definition, 138 ; due to a mag- net computation of, 146 ; due to a cur- rent, 154. Focal length, of a concave mirror, 260 ; of a convex lens, 261. Forces, parallelogram of, 42 ; parallel, 42. Franklin's method for magnifying power of microscope, 266. " Free," electricity, 123 ; magnetism, 151. Friction, coefficient of, 8, 44. Fusion of ice, heat of, 117. Galvanometer, tangent for what angle most sensitive, 20; definition of, 153-4; true constant of, 156 ; reduction factor, 157 ; working constant, 157; to set, 157; de- flections, how measured, 159; most suitable number of turns, 162; best re- sistance of, 162; law of tangent, 163; determination of constant, 163, 169 ; sen- sitive, 162; constant of sensitive, 170; potential galvanometer, 192; ballistic galvanometer, 221. Gases, properties of, 93. Graphical representation of results, 8. Grating, diffraction, 273. Gravity, by Atwood's machine, 53 ; by free fall, 55 ; by physical pendulum, 67 ; by Kater's pendulum, 69 ; for Cornell labo- ratory, 69. Hare's method for determining density, 92. Harmonic scale for hydrometer, 89. Henry, unit of induction, 243. Heat, 107; of fusion of ice, 117; of vaporiza- tion of water, 113; specific, 117. Holtz machine, 133; experiments with, 134. Hydrometer, Nicholson's, 85 ; Fahrenheit's 86; of variable immersion, 88, 91. Index of refraction, 268. Induced currents, direction of, 233. Induction, electrostatic, 128 ; electromag- netic, 232 ; self, 209-236 ; mutual, 240. Internal resistance of batteries, 214-217. Interference of sound waves, 245. Jolly balance, 84. Konig's apparatus, 245. Kundt's method for velocity of sound in brass, 249. Laplace's law, 154. Latent heat of steam, 113; of water, 117. Least squares, method of, 21. Lens, curvature of, by spherometer, 26; by reflection, 257; focal length of, 261. INDEX. 293 Leyden jar, 132. Lines of equal potential, 199. Lines of force electrical, 123, 125, 127; how to determine direction, 127 ; magnetic lines, definition, 138 ; positive direction of, 139 ; study of, 141 ; around wire carrying current, 154; of a permanent magnet, 238. Logarithmic decrement of magnetometer needle, 223 ; definition of, 224. Magnet pole, definition, 139. Magnet field, definition, 138; lines of force in, 141; measurement of, 149, 238. Magnetic moment, 140 ; by oscillations, 143 ; by magnetometer, 145. Magnetism, 138. Magnetization, lines of, 138. Magnetometer, 145. Magnifying power, of a telescope, 264 ; of a microscope, 265. Map, of an electrostatic field, 126 ; of mag- netic field, 142; of field around a cur- rent, 154. Mance's method of measuring resistance of a battery, 217. Manometric capsule, 245. Middle elongation, 36. Modulus of elasticity, 74. Moler's method of studying vibrations, 252. Moment of inertia, 57; for parallel axes, 58 ; of a thin rod, 59 ; of a cylinder, 60 ; of a circular lamina, 60; measurement of, 74. Moment of momentum, 57, 222. Moment of torsion, 76, 144. Moments, principle of, 42. Mutual induction, coefficient of, 243. Newton's law of cooling, in. Nicholson's hydrometer, 85. Observations, 3 ; record of, 2. Ohm, definition of, 205. Ohm's law, 182; method of measuring E. M. F., 189 ; of measuring resistance of battery, 214. Open-eye method of determining magnify- ing power, 265. Parallel forces, 42. Parallelogram of forces, 40. Pendulum, physical, 67; Kater's, 69; uni- form bar, 72. Periodic motion, time of, 36. Permeability, magnetic, 138. Photometer, Bunsen, 274. Pitch measured by syren, 245. Polarization, effect upon current, 179. Potential, electrostatic, 124. Potential difference, 181 ; definition of, 181 ; electromagnetic unit of, 186; practical unit of, 186 ; at terminals of cell, 191. Potential galvanometer, 192, 195. Principle of moments, 42. Proof plane, 128. Pulleys, system of, 49. Quantity of electricity, 220 ; produced by induction, 235. Radiating and absorbing power for heat, 119. Radiation constant of a calorimeter, no. Ratio of damping, 224. Refraction, index of, 268. Regulating magnet, 156, 174. Reports, u. Residual charge, 132. Resistance, definition, 204; absolute unit, 204 ; practical unit, 205 ; coils, 205 ; measurement of, by Wheatstone's bridge, 206 ; by fall of potential method, 208 ; specific, 211 ; temperature coefficient of, 212 ; of battery, 214, 217 ; of electrolytes, 218. Resonance of air columns, 247. Reversing key, 158. Rheostat, 205. Self-induction, 209; definition, 236. Sensitive galvanometer, 162; constant of, 170. Shunt for galvanometer, 169 ; theory of, 175. Simple harmonic motion, 61 ; of transla- tion, 62 ; of rotation, 64 ; examples, 66. Sonometer, 250. Sound, 244. Specific gravity, 79 ; by weighing in water, 80; by specific gravity bottle, 81 ; of liquid, 81 ; by Jolly balance, 84 ; by Nich- olson's hydrometer, 85 (see Density). Specific heat, 117. Specific resistance, 211 ; measurement of, 211 ; of liquid, 219. 294 INDEX. Spectra, of metals, 271. Spectrometer, 268, 271. Spectroscope, 271. Spherometer, 26. Standard cell, Daniell, 173, 178. Static, electricity, 122; induction, 128. Strings, laws of vibrating, 250. Syren, 245. Tangent galvanometer, most sensitive de- flection, 20 ; law of, 163. Telescope and scale, 159, 161 ; magnifying power of, 264. Temperature, errors in determining, 107; coefficient for resistance, 212. Tenths, estimation of, 4. Thermo-element, E. M.F. of, 201. Thermometer, calibration of, 30 ; compari- son of, 108. Torsion, moment of, 76. Transverse vibration, study of, 252. True constant of a galvanometer, 156. Uniformly accelerated motion, 50. Units, 6. Vaporization, heat of, 113. Velocity of sound, in air, 247 ; in brass, 249. Vibrating strings, laws of, 250, 252. Vienna method of measuring current, 178. Volt, definition, 187. Voltameter, silver, specifications for, 155; copper, 167 ; spiral coil, 167. Wave-length, measurement of,' 245; of sodium light, 274. Water, battery, 228; equivalent of calor- imeter, 109. Weight, 6-7 ; in taking an average, 17. Wheatstone bridge, 206. Wheel and axle, 46. Young's modulus, 74, THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE.