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 BOUGHT WITH THE INCOME OF THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF 
 
 HENRY W. SAGE 
 
 1891 
 
Cornell University Library 
 QC 37.N61 1912 
 A laboratory manual of physics and jj|PP« 
 
Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924005401926 
 
A LABORATORY MANUAL 
 
 OF 
 
 PHYSICS AND APPLIED ELECTRICITY 
 
THE MACMILLAN COMPANY 
 
 NEW YORK • BOSTON • CHICAGO 
 SAN FRANCISCO 
 
 MACMILLAN & CO., Limited 
 
 LONDON ■ BOMBAY • CALCUTTA 
 MELBOURNE 
 
 THE MACMILLAN CO. OF CANADA, Ltd. 
 
 TORONTO 
 
A LABORATORY MANUAL 
 
 OF 
 
 Physics and Applied Electricity 
 
 ARRANGED AND EDITED 
 
 BY 
 
 EDWARD L. NICHOLS 
 
 PROFESSOR OF PHYSICS IN CORNELL UNIVERSITY 
 
 IN TWO VOLUMES 
 
 Vol. I 
 
 JUNIOR COURSE IN GENERAL PHYSICS 
 
 REVISED AND REWRITTEN BY 
 
 ERNEST BLAKER 
 
 ASSISTANT PROFESSOR OF PHYSICS IN CORNELL UNIVERSITY 
 
 Neto got* 
 
 THE MACMILLAN COMPANY 
 
 1912 
 
 All rights reserved 
 
 A 
 
/Vri^oio 
 
 Copyright, 1894, 1912, 
 By THE MACMILLAN COMPANY. 
 
 Set up and electrotyped. Published February, 1912. 
 
 NoTinooti ftress 
 
 J. 8. Gushing Co. —Berwick & Smith Co. 
 
 Norwood, Mass., U.S.A. 
 
PREFACE TO FIRST EDITION. 
 
 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 
 
vi PREFACE TO FIRST EDITION. 
 
 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 ana- 
 lytical geometry and of the calculus ; also that he has completed a 
 textbook 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 arrangement 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 experi- 
 ments 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 teaching of a subject is missing, some recompense 
 may be found in the freshness and novelty of the themes. 
 
PREFACE TO FIRST EDITION. ' vii 
 
 A large proportion of the students, for whom primarily this Manual 
 is intended, are preparing to become engineers, and especial atten- 
 tion 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. 
 Throughout 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 compel the student to read original memoirs in prepa- 
 ration for his work. 
 
 EDWARD L. NICHOLS. 
 Cornell University, Ithaca, New York, 
 May, 1894. 
 
PREFACE TO SECOND EDITION. 
 
 Since the first edition of this manual was published, changes have 
 
 been made in the treatment of many of the experiments. A few are 
 
 no longer given and about forty others have been added to the list. 
 
 The publication of a new edition has made possible the incorporation 
 
 of these changes and new experiments. As a result, the book has 
 
 been almost entirely rewritten. The changes and additions have 
 
 extended over a period of years, and many of them have been due 
 
 to members of the teaching staff no longer in the work. It is a great 
 
 pleasure to make acknowledgment to all such, and especially to 
 
 Professor John S. Shearer of Cornell University, Professor Oscar M. 
 
 Stewart of the University of Missouri, Professor O. A. Gage of the 
 
 University of Wisconsin, and Professor F. K. Richtmyer of Cornell 
 
 University. 
 
 ERNEST BLAKER. 
 Cornell University, 
 November, 1911. 
 
TABLE OF CONTENTS. 
 
 VOLUME I. 
 
 PAGE 
 
 Introduction i 
 
 Record of Observations. Units. Graphical Representation of Results. 
 Errors of Observations and Method of Least Squares. 
 
 Chapter I 29 
 
 Curvature of a Lens by the Spectrometer. Calibration of a Thermometer 
 Tube. Volumes and Densities by Measurement. Time of Periodic Mo- 
 tion by the Methods of Middle Elongation and Transits. The Balance, 
 its Adjustments and Use, and the Calibration of a Set of Weights. The 
 Planimeter, its Calibration and Use. Parallelogram of Forces. Parallel 
 Forces. The Principle of Moments. Coefficient of Friction. Wheel 
 and Axle. Efficiency of a System of Pulleys. The Differential Pulley. 
 Atwood's Machine. Determination of Gravity from Motion of a Freely 
 Falling Body. Angular Acceleration, Angular Velocity, and Rotational 
 Inertia. General Statement concerning Moment of Inertia and Simple 
 Harmonic Motion. Gravity by the Physical Pendulum. Gravity by 
 Kater's Pendulum. Variation of the Period of a Bar Pendulum with 
 Position of Knife Edges. Young's Modulus by Stretching. Moment 
 of Torsion and Slide Modulus. Young's Modulus by Flexure. Elastic 
 and Inelastic Impact. 
 
 Chapter II 115 
 
 General Statements concerning Density. Specific Gravity Bottle. Den- 
 sity with Correction for Temperature and Air Displacement. Nichol- 
 son's Hydrometer. Fahrenheit's Hydrometer. Hare's Method of 
 determining Density of a Liquid. Verification of Boyle's Law. Com- 
 parison of Barometers. Variation of Pressure of Saturated Vapor with 
 Temperature. 
 
 Chapter III 133 
 
 General Statements concerning Calorimetry. Heat of Vaporization of 
 Water. Heat of Fusion of Ice. Specific Heats of Solids and Liquids. 
 Radiating and Absorbing Power of Surfaces. Electrical Method of 
 determining Joule's Equivalent. Coefficient of Linear Expansion. Co- 
 efficient of Volume Expansion of Liquids. 
 
 Chapter IV 158 
 
 Radius of Curvature by Reflection. Focal Length of a Concave Mirror. 
 Focal Length of a Convex Lens. Focal Length of a Concave Lens. 
 Magnifying Power of a Telescope. Magnifying Power of a Microscope. 
 
xii TABLE OF CONTENTS. 
 
 The Adjustments of a Spectrometer. Index of Refraction of a Prism. 
 Calibration of a Prism. A Study of Flame Spectra. The Diffraction 
 Grating and Measurement of Wave Lengths. 
 
 Chapter V . 182 
 
 General Statements concerning Photometry. Horizontal Distribution of 
 Light by the Bunsen or Lummer-Brodhun Photometer. Variation of 
 Candle Power with Voltage. Calibration and Use of the Weber Pho- 
 tometer. 
 
 Chapter VI 192 
 
 Measurement of Pitch by the Syren. Interference and Measurement of 
 Wave Length by Koenig's Apparatus. Resonance of Columns of Air 
 and the Velocity of Sound. Velocity of Sound in Brass. The Sonom- 
 eter. Study of Transverse Vibrations of Cords by Melde's Method. 
 
 Chapter VII 202 
 
 General Statements concerning State Electricity. Notes on Electric and 
 Magnetic Potential. Electrostatic Induction. The Principle of the 
 Condenser. The Holtz Machine. The Holtz Machine (continued*). 
 
 Chapter VIII . . 219 
 
 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 
 Magnet. 
 
 Chapter IX 234 
 
 General Statements concerning the Electric Current. Law of the Tangent 
 Galvanometer. Law of the d'Arsonval Galvanometer. Measurement 
 of Current by Electrolysis. Theory of Shunts. (Measurement of the 
 Constant of a Sensitive Galvanometer. Applications of the Gal- 
 vanometer to the Measurement of Current. Calibration of an Am- 
 meter. 
 
 Chapter X 268 
 
 General Statements concerning Difference of Potential and Electromotive 
 Force. Ohm's and Kirchhoff's Laws. Ohm's Method for the Measure- 
 ment of E. M. F. Fall of Potential in a Series Circuit. Potential Differ- 
 ence at the Terminals of a Battery. Fall of Potential in a Wire carrying 
 a Current. Beetz's Method of measuring E. M. F. To trace the Lines 
 of Equal Potential in a Liquid Conductor. (Variation of the E. M. F. of 
 a Thermo-element. Calibration of a Voltmeter. Comparison of Elec- 
 tromotive Forces by PoggendorPs Method and by Lord Rayleigh's 
 Method. The Potentiometer. 
 
 Chapter XI 303 
 
 General Statements concerning Resistance. Measurement of Resistance 
 by the Wheatstone Bridge. Calibration of a Slide Wire. Carey- Foster's 
 Method of measuring Low Resistances. Measurement of Resistance by 
 
TABLE OF CONTENTS. xiii 
 
 PAGB 
 
 the Fall of Potential Method. Measurement of Specific Resistance. 
 'Temperature Coefficient for Resistance. Kelvin Double Bridge Method 
 of measuring Low Resistances. Resistance of Electrolytes. Internal 
 Resistance of a Battery by Ohm's, Thomson's, Mance's, Modified Mance's, 
 Fall of Potential, and Benton's Methods. Resistance of a Galvanometer 
 by Half Deflection, Equal Deflection, and Thomson's (Lord Kelvin's) 
 Methods. 
 
 Chapter XII 337 
 
 General Statements concerning Electrical Quantity. Constant of a Ballistic 
 Galvanometer. Logarithmic Decrement of a Galvanometer Needle. 
 Comparison of the Capacities of Two Condensers. Measurement of 
 Capacity in Absolute Measure. 
 
 Chapter XIII 352 
 
 General Statements concerning Induction. Dip and Intensity of the 
 Earth's Magnetic Field. (Method of the Earth's Inductor.) Measure- 
 ment of the Lines of Force of a Permanent Magnet. Mutual Induction. 
 General Statements concerning Self-induction. Measurement of Self- 
 induction by Comparison with a Variable Standard Self-induction, by 
 Rimington's Modification of Maxwell's Method, and by Anderson's 
 Method. 
 
 Chapter XIV 373 
 
 General Statements concerning the Magnetic Properties of Iron. Test of 
 the Magnetic Properties of Iron by the Magnetometer Method, by the 
 Ring Method using a Ballistic Galvanometer. 
 
 Tables 392 
 
 Some Useful Numbers. Work and Power. Densities of Some Substances. 
 Coefficients of Friction. Elastic Constants. Data on Change of State. 
 Variation in Boiling Point of Water with Change in Barometric Pressure. 
 Specific Heats and Coefficients of Expansion. Vapor Pressure of Satu- 
 rated Vapor at Various Temperatures. Vapor Pressure and Relative 
 Humidity. Index of Refraction for Sodium Light. Bright Line Spectra. 
 Velocity of Sound. Specific Resistances and Temperature Coefficients. 
 Electrochemical Equivalents. Specific Resistances of Electrolytes. 
 Electromotive Force of Cells. Wire Gauge and Resistance of Copper 
 Wire. Logarithms of Numbers. Natural Trigonometric Functions. 
 
A LABORATORY MANUAL OF PHYSICS 
 AND APPLIED ELECTRICITY. 
 
 Volume I. 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 Revised and Rewritten by Ernest Blaker. 
 
 INTRODUCTION. 
 
 The object of all of the experiments described in the follow- 
 ing pages is twofold: (i) to illustrate, and therefore impress 
 more thoroughly on the mind, the principles and laws which 
 have previously been taught by textbooks 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 
 some reference book before the observations are begun. On 
 account of the large number of textbooks and other books of 
 
 VOL. 1 — B I 
 
2 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 reference and the ever increasing number, few references ex- 
 cept to original sources are given in the manual. A very com- 
 plete list of references to original articles on the topics treated 
 in Electricity and Magnetism will be found at the ends of the 
 appropriate chapters in Henderson's Pi~actical Electricity and 
 Magnetism. 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 are to 
 be entered in the regulation notebook at the time they are made. 
 
 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. 
 
 It is a saving in the end to devote enough time to the origi- 
 nal records to make them neat and clear, and so complete and 
 logically arranged 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 subtrac- 
 tion performed afterward. 
 
 It is advised that formulas, proofs, the solution of problems, 
 notes, and questions also be entered in the notebook. It will 
 often be found convenient to have numerical work carried out 
 
INTRODUCTION. 3 
 
 in the notebook. Always allow at least two pages for each 
 experiment, the first page being a left-hand page. Put in 
 sketches and diagrams of apparatus, and always put in working 
 diagrams of all connections in electrical experiments. 
 
 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 r 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. Q) one ex- 
 perimenter 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 an average, 
 however, it is necessary that each observation should be inde- 
 pendent of all the rest. Knowing that all the measurements 
 
4 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 should be alike except for accidental errors, there is an un- 
 conscious 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 divi- 
 sions. 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 resist- 
 ances. If I v R x , and / 2 , R 2 , represent the corresponding values 
 of current and resistance, the internal resistance of the cell is 
 
 „ _ f l R \ ~ ^2 
 
 / — / 
 
 It is evident that if I x and 7 2 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 x and R 2 so that the two values of the current shall differ 
 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 accidental 
 errors as small as possible. 
 
INTRODUCTION. 5 
 
 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 unac- 
 customed to computing. The best safeguard against mistakes 
 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 diminished 
 liability of mistakes. Tables of squares, reciprocals, etc., can 
 often be used to advantage. When a number of similar compu- 
 tations are to be made, the work should be done systematically 
 and the results arranged in tabular form. 
 
 The slide rule is perhaps the most valuable aid in computa- 
 tion and is a great time saver. The division of its scales is 
 based on logarithms. Since multiplication and division of num- 
 bers, using logarithms, are performed by adding and subtracting 
 logarithms, the same operations are performed by means of a 
 slide rule, by adding and subtracting lengths of scale which are 
 proportional to the logarithms of the numbers treated. Great 
 care should be used in the use of the slide rule. With a little 
 practice great proficiency in its use is attained, and an accuracy 
 of from one to five tenths of one per cent can be assumed. 
 This means that computations are correct to three significant 
 figures, and sometimes the fourth may be estimated in that part 
 of the scale where the graduation is coarsest. The slide rule is 
 especially valuable where one number is to be multiplied by a 
 series of numbers ; as, for example, the multiplication of a gal- 
 vanometer constant by the tangents of angles of deflections, 
 in finding currents as measured by a tangent galvanometer. 
 In this case one setting of the slide, or at most two, is 
 necessary, and results may be readily read off the settings of 
 the indicator. Do not use the slide rule when the method of 
 least squares is involved. 
 
6 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 computations 
 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 ex- 
 press the result by more than three, or at most four, significant 
 figures. If it is decided from an inspection of the observations 
 that the result should be carried to three places, then the com- 
 putations should be made with four places in order to insure the 
 accuracy of the last significant figure of the result. 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. The slide rule will automatically do 
 away with the undesirable figures. Zeros must often be used 
 as significant figures. For example, suppose a certain mass 
 is twenty-five grams to within a tenth of a milligram. The 
 following number indicates such an accuracy: 25.000. If the 
 mass be known only to tenths of a gram, it would be written 
 25.0. If a certain computation indicates a magnitude of thirteen 
 places to the left of the decimal point, as in Young's Modulus 
 for steel, it is neither to be written as 2,173,890,604,514 nor 
 2,170,000,000,000, but as 21.7 x io 11 . The result written as 
 indicated in the last number shows that the significant figures 
 are three, and that the third one, to the right of the decimal 
 point, is in doubt. It requires the complete expression to indi- 
 cate the magnitude of the result. This method of writing the 
 result of computation has the merit of brevity, but its greatest 
 usefulness is in indicating the accuracy of the work. It is often 
 convenient to represent such quantities as that representing 
 the electro-chemical equivalent of copper, not as 0.000328 but as 
 328 x io~ 6 . 
 
 In many cases approximate methods may be used which will 
 effect a considerable saving in time without diminishing the ac- 
 curacy of the results. For example, it often happens that a factor 
 
INTRODUCTION. 7 
 
 of the form appears as a multiplier, k being a very small 
 
 quantity. In most cases it is sufficiently accurate to say that 
 
 1 +k 
 and in general 
 
 (1 + k) n = 1 + nk, when k is small.* 
 
 When the angle 6 is small, it is often convenient to make the 
 following approximations : 
 
 For sin 6, the angle 6 in radians, 
 for tan 6, the angle 6 in radians, 
 and for cos 6, 1. 
 
 Units. — In almost all physical measurements the units em- 
 ployed 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 denned 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 to- 
 ward 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, in 
 Stewart and Gee, Appendix to vol. 1, and in Watson's Practical Physics. 
 
8 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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. 
 
 s 
 
 The mass of a pound weight is therefore jj^, and the unit of 
 mass in the foot-pound-second system is the mass of a body 
 which weighs 32.2 lbs. 
 
 In the C. G. S. system the gram is the unit of mass. By 
 the word gram, therefore, is meant the amount of matter con- 
 tained, in a certain standard piece of metal. The weight of this 
 piece of metal is found by multiplying its mass by the accelera- 
 tion 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 in 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 oi the body is 100 x g = 
 98,000 dynes. 
 
 If care is used in distinguishing between the terms "weight" 
 and " mass," no difficulty should 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 
 
INTRODUCTION. 9 
 
 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 pre- 
 sent 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 corre- 
 sponding 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 independ- 
 ent 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 experiment 
 
 4 
 
 3 s' 
 
 2 ,S 
 
 IS 20 25 
 
 PRESSURES IN KGS.WT. 
 
 Fig. 1. 
 
 30 
 
 the force necessary to overcome the moving friction between two surfaces 
 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 depend- 
 ence, a curve is plotted, in which pressures are used as abscissas, and 
 the corresponding values of the tangential forces to overcome friction as 
 
io JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 indi- 
 vidual 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. Since the line is straight, it shows that 
 there is a constant relation between the various normal pressures and 
 their corresponding moving forces. The slope of the line with respect 
 to the axis of abscissas, i.e. the tangent of the angle it makes with the 
 x axis, is constant and is equal to the moving force divided by the normal 
 pressure. But the moving force divided by the normal pressure gives 
 the value of the coefficient of friction. Thus the slope of the line, in 
 terms of the scale used, gives the coefficient of friction sought. Such a 
 curve not only shows at once the relation between the quantities plotted, 
 but also makes it possible to get the mean value of the coefficient of 
 friction from the curve. 
 
 It is to be observed that when a curve is plotted in order to 
 show the relation between two variables, it is by no means nec- 
 essary that the horizontal and the vertical scale should be the 
 same. Either scale may be assumed at pleasure, and without 
 reference to the other. 
 
 In the case just cited, for example, the horizontal scale may be taken 
 as s kilograms to the inch, while the vertical scale may be i kilogram, 
 j 1 ^ kilogram, or any other quantity that proves convenient. In taking 
 readings from the curve, however, regard must be paid to the scale em- 
 ployed. 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 T \ kilogram to the inch, 5 inches on the vertical scale 
 would be read 1 kilogram. 
 
 The equation of a straight line passing through the origin is y = mx, 
 in which m is a constant. But the *'s of the line represent pressures, 
 while the jy's represent the corresponding values of the moving force. 
 The law established by the experiment is therefore that F— mP ; i.e. 
 friction is proportional to pressure. 
 
 But —=m, and m gives the value of the coefficient of friction //,. 
 Therefore, substituting /a for m gives 
 
 F=yJ>, (1) 
 
INTRODUCTION. n 
 
 which is the physical equation of the curve, i.e. the equation of the curve 
 in terras of the physical quantities used. 
 
 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. Suppose that the normal pressure 
 be made up of two parts, a known variable part and an un- 
 known but constant part. If in this case the known variable 
 parts of the normal pressure be plotted as abscissas and the 
 corresponding moving forces as ordinates, the resultant curve 
 will be a straight line, but will not pass through the origin. The 
 slope of the line will give the coefficient of friction as before, 
 as will be shown in the following discussion. Let the constant 
 unknown pressure be denoted by P , and P, F, and /* represent 
 the known part of the pressure, the moving force, and the 
 coefficient of friction, respectively. 
 
 Then u = 
 
 or F=i*P + fiP , (2) 
 
 which is the physical equation of the straight line, in which /u. is 
 the slope of the line, fiP the intercept on the y axis, and P the 
 negative intercept on the x axis. Thus it is possible to find the 
 constant unknown pressure from the curve. It should be 
 noted that in finding the slope of such a curve it is necessary 
 to divide the value of any ordinate by the corresponding 
 abscissa plus the negative intercept, or to use the expression 
 
 2=$-- « 
 
 ■*2 1 
 
 The cases above outlined should be understood thoroughly, 
 for the principles there given will be found to have wide 
 application. 
 
 Many times it is found from the physical theory that the 
 form the curve will take will be hyperbolic, parabolic, or ex- 
 ponential. It is well to plot such curve, develop its physical 
 equation, and get such physical constants from it as are possible. 
 
12 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The intercepts, the slope, asymptotes, area, maxima, and 
 minima should be fully interpreted. 
 
 Since the straight line is the curve which is most readily 
 tested, it is often convenient to transform the results of an 
 experiment in such a way that they will give a straight line 
 when plotted. 
 
 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 plotted, 
 therefore, with pressures and corresponding volumes for co-ordinates, 
 the resulting curve would be a hyperbola whose equation is xy = k. If, 
 however, we plot 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. Hj for verifying Boyle's law is 
 employed, the data may be plotted in still a different way to advantage. 
 In this method the total volume V\% not measured, but merely a 
 portion v, while a part v of the volume remains unknown, but constant. 
 
 Then V={v + v <i ), (4) 
 
 PV=P{v + v»)=k. (5) 
 
 If now /'and Fare taken as co-ordinates, a hyperbola should.be 
 
 obtained. But if v and - are used, the resulting line should be straight, 
 
 its equation being 
 
 }~k + k- (6) 
 
 If the data are plotted in this way, a means is therefore afforded of 
 determining both » and k. Since the line obtained is straight, we 
 know that the form of its equation must be 
 
 y = mx + b, (7) 
 
 and the numerical values of m and b can be at once computed. 
 Since these two equations represent the same line, we must have 
 
 | = W ,and|» = 3. (8) 
 
INTRODUCTION. 13 
 
 It is sometimes very useful to put an equation into the loga- 
 rithmic form and plot logarithms as in the following example. 
 
 Suppose that the bending of a beam in terms of some con- 
 stant k, the length, breadth, depth, and deflecting force f be 
 indicated as follows (see Exp. F 4 ) : 
 
 e = kfHWd*. (9) 
 
 Let it be required to find how the deflection varies with the 
 length, all other factors remaining constant. The equation may 
 then be written in the form 
 
 e = k'l? 
 
 in which k' includes all of the constant factors. The equation 
 may be put into the form 
 
 log e = j3 log / + log k'. (10) 
 
 If observations be made of the deflections e for various lengths 
 and values of log e and log / be plotted as ordinates and ab- 
 scissas and a straight line result, then the slope of that line will 
 give the value of /3 in the above expression, for it is in the form 
 
 y = mx + b. 
 
 Curves may also be plotted which have no regular "form, such 
 as thermometer calibration curves in which the forms of the 
 curves depend on the variation in bore of the tubes of which 
 the thermometers are made. In such cases smooth curves are 
 to be drawn through the points which are taken to represent 
 the progressive variations of the tubes. Such curves will show 
 more nearly the conditions the closer the observed points are to 
 each other. The allowable interval between points will depend 
 upon the irregularity of the changes and the accuracy desired. 
 
 Curve Plotting. — The following rules should be followed in 
 plotting curves : 
 
 1. The arbitrary variable should generally be plotted along 
 the horizontal or x axis. 
 
 2. Choose such scales for plotting the variables that the 
 largest values of them will be near the right-hand and upper 
 
14 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 edges of the plot, using a scale that may easily be read, such 
 as one division for one, two, five, or ten units or like sub-mul- 
 tiple units. Do not use three as a factor in plotting, as such a 
 plot is hard to read. 
 
 3. In case two or more curves are plotted for comparison the 
 scales should be the same, and they should be drawn on a single 
 sheet. 
 
 4. Points indicating the experimental values to be plotted 
 should be marked plainly by means of crosses, dots, or dots sur- 
 rounded by circles. If two or more curves are to be plotted on 
 the same sheet, use different symbols to indicate the points on 
 each curve. 
 
 5. A smooth curve is then to be drawn so that it may best 
 fit the observed points, but not necessarily passing through them. 
 This curve is to represent the continuous changes of the vari- 
 ables plotted ; the deviation of points from the curve usually 
 indicates errors of observation. 
 
 6. Indicate clearly on the plot what the co-ordinates mean 
 and the scale chosen. Number on margin close to outer ruling 
 and only the heavy lines of the paper. Always draw curves in 
 ink and bind next to data sheet. 
 
 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.* 
 
 * In case the same apparatus has been employed in previous experiments, it is 
 not necessary to describe it a second time. 
 
INTRODUCTION. 15 
 
 (3) All formulas used, which express relations between 
 physical quantities, should be proven.* The object of putting 
 such demonstrations in the report is to make it clear to the in- 
 structor 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 textbooks should therefore be very clearly explained. 
 Originality 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 
 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. 
 
 (6) Data and results are to be tabulated and kept separate 
 from the text of the report and should follow it. 
 
 (7) In all reports, whenever possible, curves are to be plotted, 
 the physical equations given and interpreted, and graphical meth- 
 ods of obtaining physical quantities used. Care should be taken 
 to use readable scales. 
 
 When two students work together, observations and compu- 
 tations should be made in common, but the two reports are to 
 be written independently. 
 
 In writing reports it is always to be borne in mind that one 
 important benefit which practice in this work may accomplish 
 
 *The proof of purely mathematical formulas, such as the trigonometrical rela- 
 tions used in solving triangles, is not required. Formulas once proved need not be 
 proved a second time, but the reports in which they were proved should be cited. 
 
16 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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. 
 
 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 perfectly 
 sure that to completely eliminate them is quite impossible. 
 
 As an example of the different ways in which inaccuracies 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 : 
 
 i. 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 measure- 
 ment. 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 divi- 
 sion 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, Jj for fa, etc. 
 
 A little consideration will show that all possible errors may 
 be made to fall under four classes : 
 
INTRODUCTION. 17 
 
 1. Errors of method. 
 
 2. Inaccuracies in instruments. 
 
 3. Accidental errors of observation. 
 
 4. Mistakes. 
 
 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."* 
 
 * For brief discussions see Kohlrausch, Physical Measurements, and the appen- 
 dix to Stewart and Gee, vol. 1. For more complete discussions see Merriman, 
 Method of Least Squares ; Violle, Cours de Physique (see Introduction to vol. I ) ; 
 Weinstein, Handbuch der Physikalischen Maassbestimmungen, vol. I ; and Hol- 
 man's Precision of Physical Measurements. 
 VOL. I — C 
 
1 8 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The principal results of such discussion, in so far as they have an 
 application to physical measurements, will be briefly stated here. 
 
 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 
 estimate 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.\ 
 
 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 errorwith the sign ± after the result itself : e.g. /= 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. 
 
 t 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. 
 
INTRODUCTION. 
 
 10 
 
 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, 
 of which the average is a, then a — a v a — a 2 , •■■ a — a n are 
 called the errors. If the sum of the squares of the errors 
 be divided by the number of observations less one, the result 
 is the square of the mean error. The probable error of a single 
 observation * is found by multiplying the square root of the 
 square of the mean error by 0.67449 (f° r ordinary work use 
 0.674) and prefixing the sign ± as follows : 
 
 e'=±o.6 74 J ( -^ a ^ + ^- a ?f + - +("-*») a . (II) 
 * n— 1 v ' 
 
 The probable error of the mean is found by the use of the follow- 
 ing equation : 
 
 *= ±0.674 V ( ^^ + (a -/s )2 + - +^^~> 2 . ( I2 ) 
 
 n{n — 1 ) / 
 
 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. k x .) 
 
 Readings on Disk 
 
 Deviations from Mean 
 a— a lt etc. 
 
 Square of Deviations 
 
 44-5 
 
 — O.I 
 
 O.O I 
 
 
 44.8 
 
 + 0.2 
 
 O.04 
 
 
 44.2 
 
 -0.4 
 
 O.16 
 
 
 45.0 
 
 + O.4 
 
 O.16 
 
 
 45- 1 ' 
 
 + 0.5 
 
 O.25 
 
 
 44.4 
 
 — 0.2 
 
 0.04 
 
 
 44.6 
 
 ± O.O 
 
 O.OO 
 
 
 44.2 
 
 -0.4 
 
 O.16 
 
 
 44-5 ■ 
 
 — O.I 
 
 O.OI 
 
 
 44-7 
 
 + O.I 
 
 0.0 1 
 
 
 Mean 44.6 
 
 Mean 0.24 
 
 2rf 2 = 0.84 
 
 
 Probable error of a single observation e' = ± 0.204. 
 Probable error of the mean e = ± 0.065. 
 
 * For the derivation of this formula, see any textbook 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. 
 
20 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 observations 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 3 , etc., the probable errors 
 being, respectively, e v e 2 , e & , etc., the most probable value of the 
 quantity in question, as determined from all of the observa- 
 tions, is 
 
 «» + *» + - 
 
 In Exp. Aj, for example, the length / of one side of the tri- 
 angle formed by the three legs of the spherometer may be 
 determined in several different ways. Let the result obtained 
 by one method be / = 6.12™ ± 0.03, while that determined by 
 
INTRODUCTION. 21 
 
 another and less accurate method is /= 6.2O 0m ± o. n. It is 
 
 certainly not right to use the average — — — — : — (=6.16), for 
 
 2 
 
 much more reliance can be placed on the first result than on 
 the second. According to the rule above stated the most prob- 
 able value of / is 
 
 - l „ 6.I2 + , ' „ 6.20 
 
 , (0.03 f (0.11) 2 . , . 
 
 1= — ~ L — - — = 6.126. (14) 
 
 , + : 
 
 (0.03 ) 2 (0.11) 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 for- 
 mula. In such cases it is of importance to know how the final 
 result will be influenced by possible errors in the individual 
 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 refinement. By considering 
 this question before the actual measurements 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 <f> of the quantities 
 to be observed ; 
 i.e. R = <j>(x,y, z,--). 
 
 Now if x, y, z, 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 
 
 R' = R + E X = <j>(x + e v y, z, •••) 
 from which E x = <f>(x + e v y, z, •••)— ${x, y, z, •■•). (15) 
 
 Since e will in general be quite small in comparison with x, 
 no great inaccuracy will be introduced by treating it as an 
 
22 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 infinitesimal ; i.e. neglecting powers higher than the first. Let it 
 be supposed that 
 
 R = ax 2 + by — z. (16) 
 
 But x is in error by an amount e ; 
 then R' = R + E x = a(x + ef+by-z 
 
 = ax 2 + 2 axe + ae 2 + by — z. 
 Substituting for R its value in equation 16 and remembering 
 that terms in e 2 may be neglected as infinitesimals of a higher 
 order, it follows that 
 
 E x = 2 axe. (17) 
 
 Now this same result might have been attained by finding 
 
 the rate of change of R with respect to x ; that is, differentiating 
 
 equation 16 with respect to x and multiplying by the error in x. 
 
 Returning to the general case we may, therefore, write (\ 
 
 Similarly, E v = ej*-${x, y, z, 
 
 E z = e 3 ^4>{x,y, z, • 
 
 etc. 
 If the probable errors e v e 2 , etc., are known, the correspond- 
 ing errors E x , E y , etc., in the result may thus be readily com- 
 puted. The probable error in the result due to the combined 
 effect of the errors in all of the observed quantities may then 
 be shown to be * 
 
 E=^E * + £*+-... (19) 
 
 Take, for example, the case which occurs in Exp. A x . The radius 
 of curvature is given by the formula 
 
 b a 2 
 Let us suppose that /= 7.14 ± 0.05, and a = 0.423 ± 0.004. Sub- 
 stituting in the formula /= 7.14 and a = 0.423, we obtain 
 
 r = 20.09. 
 * See Merriman's Method of Least Squares, etc. 
 
INTRODUCTION. 23 
 
 To compute E a and E lt we have : 
 
 T£+*\ («) 
 
 da da 
 
 6 a 1 2 
 
 r 2 
 
 : O.OO4 ' ^ 2 + - 
 
 L 6 x 0.423 2 . 
 
 = —0.188. 
 
 -ff,=<f,-*(*^)=* I -r— + ?> \ (22) 
 
 1 Vr v ' y l dl\6a 2) v 
 
 / 7.14 o 
 
 = *, = 0.05 X — - — - — =0.28. 
 
 3« 3X0-423 
 
 E = V .i88 2 -f- 0.28* = 0.34. (23) 
 
 .•. r = 20.09 ± 0.34. 
 When the formula is known by means of which the result 
 of an experiment is to be computed, it is often possible to deter- 
 mine the most favorable conditions before beginning the obser- 
 vations. The method of procedure in such cases can best be 
 explained by means of the following example : * 
 
 A tangent galvanometer is to be used in measuring a current. What 
 are the most favorable conditions for making the measurement ? 
 
 The formula for computing the result is : 
 
 7= /„ tan e. (24) 
 
 The only observed quantity is $ ; and if this angle is read from a 
 circular scale, it is subject to the same error, e, no matter from what part 
 of the scale it may be read. Let the resulting error in I be E. 
 
 Then ^ = ^./ o tan0 = -^-. (25) 
 
 do cos" 8 
 
 The relative error is 
 
 jE , = j E = _ £ 4_ H _ /()tan = (; _i _!£_ ( 26) 
 
 I cos 11 sin cos sin 2 6 
 
 It is, however, evident that £' reaches its smallest value when sin 2 6 
 reaches its greatest value, namely, unity. In other words, the relative 
 error in the result will be least when the deflection of the galvanometer 
 is 45 . If the galvanometer used has several coils, these should there- 
 fore always be so connected as to make the deflection as near 45 as 
 possible. 
 
 * Numerous instructive examples will be found discussed in detail in Holman's 
 Discussions of the Precision of Physical Measurements. 
 
24 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Determination of Constants by the Method of Least Squares. 
 — It often happens that a series of observations is made, not of 
 the same quantity, but of quantities which are known to be 
 related to one another. If the form of the equation expressing 
 this relation is known (as is usually the case), the question then 
 arises as to what values should be given to the constants of the 
 equation in order that it should represent the results of experi- 
 ment as accurately as possible. 
 
 A case of this kind is illustrated by Exp. C 2 , where a series of obser^ 
 vations is made to determine the relation between "working force" and 
 "load" in the case of a wheel and axle. If the results are plotted, the 
 points corresponding to the different observations will probably be found 
 to lie nearly in a straight line, as shown in Fig. 2. Although it is impos- 
 
 Fig. 2. 
 
 sible to draw a straight line which shall pass through all the observed 
 points, yet it seems probable that these points would have formed a 
 straight line, had it not been for accidental errors of observation. The 
 problem, therefore, is to draw a straight line which shall pass as nearly 
 as possible through all the points. This can often be done by the eye ; 
 but when the highest degree of accuracy is required, the Method of 
 Least Squares should be used as explained below. 
 
 The method of procedure in all such cases rests upon the 
 principle * that the results will best be represented by the equa- 
 
 * For the proof of this principle see any textbook of Least Squares. 
 
INTRODUCTION. 25 
 
 tion in question when the constants are so chosen that the sum 
 of the squares of the deviations of the individual ohsexYations 
 
 from the values Computed from the pqnatinn is a minimum 
 
 In the example just cited, the formula which expresses the relation 
 between " working force " (y) and " load " (x) is evidently 
 
 y = ax + b; (27) 
 
 for the observations, when plotted, have been found to give roughly a 
 straight line, and the general equation of a straight line is of the form 
 stated. As the result of experiment a number of values y lt y 2 , y s , etc., 
 of the force have been observed, corresponding respectively to loads of 
 x lt x 2 , x 3 , etc. Now, if the constants a and b were known, it 
 would be possible to compute y from x : e.g. 
 
 yl = ax x + b, 
 
 y 2 ' = ax 2 + b, (28) 
 
 etc. 
 
 [The y's have been primed in order to distinguish them from the 
 observed values y lt y 2 , etc.] 
 
 The principle of least squares, which has been stated above, 
 now says that a and b must have such values that the sum 
 
 shall be a minimum. 
 
 Interpreted graphically, this means that the line must be so 
 drawn that the sum of the squares of the distances 1 P v 2 P 2 , 
 etc., shall be as small as possible. (See Fig. 2.) 
 
 To determine a and b, it is therefore merely necessary to 
 apply the ordinary methods for maxima and minima : 
 
 Oi -y\f + (y-t ~y*? + - = s O ~y'f = a minimum. 
 But y = ax + b. 
 
 •'• {y\ — ax i — b f + {y% — ax i — tf + ■•• = %(y — ax—bf 
 
 = a minimum. 
 
 It is to be observed that x x y v x 2 y 2 , etc., are not variables, 
 but constants, being the quantities determined by observation. 
 
26 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 It is a and b that must be varied until such values are found 
 that the above expression is a minimum.* 
 The conditions are therefore that 
 
 — Z(y-ax-bf = o and ~ ■ ~L(y- ax- Vf = o. (29) 
 
 da do 
 
 On performing the differentiation the following equations 
 result : 
 
 — 2 ( j/i — ax x — b )x x — 2( y 2 — ax 2 — b )x 2 — ■■■ 
 
 = — 2 2 (j/ — ax — b )x = o, 
 
 (30) 
 
 — 2(jj/j — ax x — b)— 2(jj/ 2 — ax % — b)— ••• 
 
 = -22(j/-«jr-^) = o. 
 
 These equations may be more readily utilized if written in 
 the following form : f 
 
 ~S,xy — a^x 2 — bH,x — o, 
 
 (30 
 
 2y — a ~Lx — bn = o. 
 
 In the last equation n represents the number of observations. 
 
 Since the quantities "Zxy, l.x\ etc., are readily computed 
 from the observations, these two equations make possible the 
 determination of both a and b. In fact, 
 
 _ 2 x Sjj/ — ttLxy 
 
 U ~ CZxf-nix* , . 
 
 (32) 
 and b= ^xl.xy-^y^x\ 
 
 (txf-n-Zx* 
 
 * Note that this variation of a and b in the algebraic work corresponds to shifting 
 the line AB in the graphical consideration of the problem. In the one case a and b 
 are varied until certain mathematical considerations indicate that X(y — y') 2 has 
 reached a minimum; in the other case the line is shifted until it looks to the eye as 
 though a good intermediate position had been reached. 
 
 t The student is cautioned in regard to the use of the sign of summation. 'Zxy 
 means x^yi + x 2 y 2 + •••, while 2/ = y x + y 2 + y a -f .... 2*/ is therefore not 
 equal to 2*2/. 
 
INTRODUCTION. 27 
 
 In the general case, where the relation between x and y is 
 expressed by an equation of any form, the method of procedure 
 is the same as that illustrated by the example above. 
 
 Let y = (f>(x, a, b, c, ••■), 
 
 where a, b, c, etc., are the constants to be determined. 
 
 The observed values of y are then y lt y 2 , y s , etc., while the 
 computed values are y-[, y 2 ', etc. The principle of least squares 
 requires that the sum 
 
 (j'l ~Ji') 2 + (J2 ~J / i) 2 + "•■ shall be a minimum ; 
 i.e. [/j-c^Oi. a > b, c , •*")] 2 + l>2-0(-*2> a > b > c < •■■)] 2 + ■■■ 
 
 = 2 \_y— <f>{x, a, b, c, ■■•)] 2 = a minimum. 
 
 j- b ^[y-4>{)f = o, (33) 
 
 ^2[jy-0()]* = o, 
 etc. 
 
 The number of equations obtained in this way is always 
 equal to the number of constants sought, so that the problem is 
 in all cases determinate. 
 
 In applying the method of least squares, the numerical work 
 is always somewhat tedious, especially when the number of ob- 
 servations is large. For this reason the computations should 
 be made with especial care ; for if a mistake occurs, considerable 
 difficulty will be met with in discovering it. The following 
 example illustrates a systematic way of arranging the computa- 
 tions, which will be found of advantage : 
 
 Equation is known to be of the form y = ax+ b. 
 
 _'2x'Zy — n1,xy 
 " a ~ CZxY-nZx*' 
 
 , = 2_£S S - L 2x2£?. , > 
 
28 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 X 
 
 y 
 
 -rr 
 
 jr* 
 
 5 
 
 0.20 
 
 1. 00 
 
 2 5 
 
 IO 
 
 0.34 
 
 340 
 
 100 
 
 '5 
 
 0.48 
 
 7.20 
 
 225 
 
 20 
 
 0.64 
 
 12.80 
 
 400 
 
 25 
 
 0.80 
 
 20.00 
 
 625 
 
 3° 
 
 o-93 
 
 27.90 
 
 900 
 
 35 
 
 1. 10 
 
 38.50 
 
 1225 
 
 40 
 
 1.24 
 
 49.60 
 
 1600 
 
 45 
 
 1.38 
 
 62.10 
 
 2025 
 
 5° 
 
 1.54 
 
 77.00 
 
 1 2500 
 
 275 
 
 8.65 
 
 299.5 
 
 9625 
 
 Sj; = 275 
 2> = 8.65 
 
 2*j» = 299.5 
 S* 2 = 9625 
 
 a = 
 
 275 x 8.65 — 10 x 299.5 
 ^5~ 2 - 10 x 9625 
 
 = 00.299, 
 
 A _ 275 x 299.5 -8.65 X9625 
 
 o = ^ = O.433. 
 
 2752-10x9625 * 33 
 
 The equation which most accurately represents the relation 
 between the quantities measured is therefore 
 
 . y = 0.299^+0.433. 
 
 A simple means of detecting large mistakes in computation 
 is always afforded by plotting the curve represented by the 
 equation found by least squares upon the same diagram as 
 the original data. This curve should then pass close to all of 
 the observed points, although it may not actually pass through 
 any one of them. 
 
CHAPTER I. 
 
 GROUP A: LENGTH, TIME, AND MASS. 
 
 (A x ) Curvature of a lens ; (A 3 ) Calibration of a thermometer tube ; 
 (A 4 ) Volumes and densities by measurement ; (A 6 ) Time of 
 periodic motion; (A 6 ) The balance and its use ; (A 7 ) The 
 planimeter. 
 
 Experiment A v Measurement of the curvature of a lens 
 by means of the spherometer. 
 
 The spherometer, as indicated by its name, is intended 
 primarily for the determination of the radius of a spherical sur- 
 face. It can also be employed, however, 
 for other measurements : for example, the 
 thickness of plates of glass or other 
 materials can be determined by means of 
 the spherometer, although unless the 
 plates are almost perfectly plane, the 
 results will not possess a high degree of 
 accuracy. 
 
 J Fig. 3. —The Spherometer. 
 
 As will be seen by reference to Fig. 3, 
 which represents a simple type of spherometer, the instrument 
 consists essentially of four metallic rods connected in the manner 
 shown, each rod being sharply pointed at the lower end. Three 
 of these rods are fixed in position, and constitute a tripod upon 
 which the instrument rests. The three supporting points are 
 made to form an equilateral triangle. The fourth rod may be 
 moved in a direction at right angles to the plane of this triangle 
 by means of a micrometer screw. It is situated equidistant 
 from the other legs of the instrument. 
 
 In using the spherometer to determine the curvature of a 
 
 29 
 
3° 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 surface (see Fig. 4) which is known to he spherical, such for 
 
 example as that of a lens, the reading of the micrometer is taken, 
 
 first when the instrument 
 rests upon a plane sur- 
 face, and then when it is 
 placed upon the lens in 
 question. All four points 
 must in each case be in 
 contact with the surface. 
 The difference between 
 the two micrometer read- 
 ings then gives the height 
 of the fourth point above 
 the plane of the other 
 three. If this height be 
 represented by a, and the 
 length of one side of the 
 triangle formed by the 
 
 three fixed points by / (l=pp' ' ; Fig. 4), then the radius of 
 
 curvature is readily shown to be * 
 
 / 2 
 
 Fig. 4. 
 
 oa 2 
 
 (35) 
 
 Since a, which enters the formula for r, is in general a 
 very small quantity, its value must be determined with especial 
 care. It is therefore advisable to make a number of independ- 
 ent settings of the micrometer, and to use the mean of the 
 readings obtained. Make at least ten disk readings for the 
 plane surface, and the same number for the spherical surface. 
 
 In order to avoid errors due to the fact that the disk is not 
 plane or is not perpendicular to the axis of the screw, it is well 
 to count the number of complete turns, and use disk readings 
 for the fraction of a revolution ; then, knowing the pitch of the 
 screw, compute a. Be careful to tabulate actual readings. 
 
 * For a more detailed description of the spherometer and derivation of this for- 
 mula, see Stewart and Gee, vol. 1. 
 
LENGTH, TIME, AND MASS. 31 
 
 In making a series of readings with the spherometer upon 
 the same surface, it is best not to look at the scale until the 
 setting has been made. Otherwise it is difficult to avoid being 
 influenced by a knowledge of previous readings. Each reading 
 should be the result of an independent attempt to obtain an 
 exact setting. 
 
 In determining /, two methods may be employed ; viz. 
 ( 1 ) the three sides of the triangle may be measured directly by a 
 millimeter scale ; (2) the instrument may be placed upon a piece 
 of paper, and the distances between the impressions left by the 
 feet may be measured. After making a number of measure- 
 ments by both methods, obtain the mean and compute r. 
 
 After the value of r has been computed, determine the in- 
 fluence upon the result of the probable error in measuring the 
 vertical height ; also the influence of the error which is likely to 
 occur in measuring the distance between the legs ; and finally 
 the probable error in the result, arising from both these causes. 
 
 Computation of Probable Errors. 
 
 From the disk readings taken on the plane surface compute 
 the "probable error of the mean" and the "probable error of a 
 single observation" in disk divisions. (See pp. 18 and 19.) 
 
 Treat the readings on the lens surface in a similar manner. 
 
 From the " probable error of the mean " of the readings on 
 the plane surface and the " probable error of the mean " of the 
 readings on the spherical surface, compute the probable error of 
 a in scale divisions. Knowing the pitch of the screw, reduce 
 the probable error to centimeters. In this case, a is the derived 
 result sought and the general equation 
 
 R = <j>(x,j/, z) (See pp. 21 and 22.) 
 becomes R = (x — y). 
 
 Estimate the error which is likely to occur in measuring /. 
 In many cases of direct measurement probable error has little or 
 no meaning. It is not at all difficult to estimate the degree of 
 accuracy obtained in a direct measure of length with a scale or 
 
32 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 tape. For example, if a carpenter measures a piece of lumber, 
 he usually knows whether he is certain of his result within an 
 inch or one eighth of an inch. By close attention to his work, 
 he could tell more accurately. 
 
 From the probable error in a and the estimated error in /, 
 find the probable error in r. (See pp. 22 and 23.) 
 
 Queries : Is there any limit to the radius which can be measured 
 by a given spherometer? Explain. 
 
 Would it be necessary in using an instrument of this kind to observe 
 the direction of rotation of the disk ? 
 
 Experiment A 3 . Calibration of a Thermometer Tube. 
 
 The ideal thermometer tube is one in which the bore has 
 a uniform cross section throughout its length. This very desir- 
 able feature is not obtainable in practice ; hence, for accurate 
 determinations of temperature, the thermometer must either be 
 calibrated or compared with a standard instrument whose errors 
 are known. The calibration will include errors due to gradu- 
 ation as well as those due to varying cross section of the bore 
 of the tube. In the experiment which follows, either an ordi- 
 nary traveling microscope or the unaided eye may be used to 
 make readings on the ends of a mercury thread of suitable 
 length at its several positions as it is moved from place to place 
 in the tube to be calibrated. In order that the volume of the 
 mercury thread remain constant, the experiment should be per- 
 formed where no temperature changes occur. 
 
 In the method of calibrating a thermometer tube here 
 described,* if the unaided eye is used, it is well to place the 
 tube on a mirror, making readings when the ends of the thread 
 are in line with the image seen in the mirror, estimating tenths 
 of the smallest divisions into which the scale is divided. 
 
 Obtain a thread equal to about one tenth of the graduated 
 stem, but not over three centimeters long. If the thread is too 
 long, it gives the average volume of too long a section of stem, 
 
 * See Heat for Students, Edser, Chap. II. 
 
LENGTH, TIME, AND MASS. 33 
 
 and if too short, the percentage errors in reading the lengths of 
 the thread are liable to be too great for the desired accuracy. 
 If the thermometer has an expansion chamber at its upper end, 
 it may be inverted and a portion of the mercury run into the 
 chamber. Then gently heat the bulb over a Bunsen flame until 
 the desired length of thread is forced into the capillary, when 
 a quick but gentle jerk of the thermometer, parallel with its 
 length, will give the desired thread. If the thermometer has 
 no expansion chamber, it often happens that a sudden move- 
 ment of the thermometer, parallel with its axis in a direction 
 away from the bulb, will separate a thread due to a contraction 
 where the bulb is joined to the stem. The desired length may 
 be obtained by trial, heating or cooling the bulb to vary the 
 length of the column in the stem beyond-the contraction. Some- 
 times it is necessary to heat the stem with a small pointed gas 
 flame at the point where separation is desired. In order to 
 study the tube at the lower end, it may be necessary to produce 
 a contraction of the remaining mercury by covering the bulb 
 with filter paper, or two or three layers of cheesecloth, and satu- 
 rating it with alcohol or ether. 
 
 It is sometimes necessary to obtain two threads, using one 
 over the lower range and the other over the upper range, caus- 
 ing them both to traverse an intermediate portion of the tube. 
 This permits the determination of the ratio of the two lengths 
 and the reduction of the readings to those of a single thread. 
 In general, the length of the thread to be used depends on the 
 length of the stem to be calibrated and the accuracy to be 
 attained. 
 
 The thread being detached, it is now necessary to measure 
 its length in scale divisions at different parts of the stem, work- 
 ing from one end to the other. If the graduations are equally 
 spaced and the tube be of uniform bore, the length of the thread 
 will be the same in all parts. As, in general, the tube is not of 
 uniform cross section, the lengths will vary, since the volume of 
 the thread is supposed to remain constant. The variations in 
 
 VOL. I — D 
 
34 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 length of thread are to be used in calibrating the tube as out- 
 lined in the following example. 
 
 Assume a thermometer whose scale is graduated in degrees 
 to be calibrated from o° to 100°. The two ends of the thread 
 give the readings in the following table in moving it stepwise 
 along the tube. 
 
 Readings 
 
 in Thread. 
 
 Length of Thread. 
 
 Variation of 
 Length from Mean 
 
 
 End nearest o° 
 
 End nearest 100° 
 
 Corrections. 
 
 Mark. 
 
 Mark. 
 
 
 L-l. 
 
 
 — .2 
 
 94 
 
 h =9-6 
 
 + .09 
 
 + .09 = -+- .1 nearly 
 
 9.6 
 
 19.1 
 
 k =9-5 
 
 + .19 
 
 + .28 = + .3 nearly 
 
 19.0 
 
 28.5 
 
 h =9-5 
 
 + •19 
 
 + .47 = + .5 nearly 
 
 28.6 
 
 38.3 
 
 U =9.7 
 
 — .01 
 
 + .46 = + .5 nearly 
 
 38.6 
 
 48.2 
 
 k =9-6 
 
 + .09 
 
 + -55 = + .5+ nearly 
 
 48.1 
 
 59-7 
 
 k =9.6 
 
 + .09 
 
 + .64 = + .6 nearly 
 
 59-5 
 
 (>9-3, 
 
 h =9.8 
 
 — .11 
 
 + .53 = + .5 nearly 
 
 69.4 
 
 79-3 
 
 k =9-9 
 
 — .21 
 
 + .32 = + .3 nearly 
 
 79.6 
 
 89.4 
 
 h =9-8 
 
 — .11 
 
 + .21 = + .2 nearly 
 
 89.3 
 
 99.2 
 
 ho = 9-9 
 
 — .21 
 
 — .00 = nearly 
 
 Mean length Z = 9.69. 
 
 The third column in the table gives the thread length, the 
 fourth column gives the variation from the mean length, and 
 the fifth column the correction to be applied. Since the errors 
 
 Ul 
 
 Ul 
 O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 z 
 
 m 
 O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 EC 
 O 
 O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 20 30 40 50 60 70 80 
 
 TUEBMONETES SCALE READINGS IN DEGREE8 
 
 Fig. 5. 
 
 90 100 
 
 are accumulative, the fifth column is obtained by taking the 
 algebraic sum of the errors from the zero up. 
 
 The corrections at the right of the last column are to be 
 applied at the upper readings of the corresponding sections of 
 the tube. A curve, Fig. 5, is to be plotted with corrections as 
 ordinates and corresponding scale readings as abscissas. Draw 
 a smooth curve through the points plotted. 
 
LENGTH, TIME, AND MASS. 35 
 
 If the zero and boiling points are correct, the calibration 
 curve will give the true temperature corrections to be applied. 
 But if either or both of these fixed points are in error, this is not 
 the case. If the errors of the fixed points be plotted on the 
 same sheet as the calibration curve and a straight line be drawn 
 connecting them, the ordinates intercepted between it and the 
 calibration curve will give the true temperature correction curve. 
 By taking a sufficient number of these temperature corrections 
 and using them as ordinates, plot another curve, using tempera- 
 tures as abscissas. This new curve will give temperature cor- 
 rections djjfeetly. 
 
 Follow the method outlined above in the calibration of a 
 thermometex.tube. Determine the freezing and boiling point 
 corrections for the thermometer supplied, and finally plot a 
 calibration^ curve for the thermometer. 
 
 To find the boiling-point correction, place the thermometer in 
 a steam bath in such a manner that just enough of the stem is 
 exposed to give a reading, being careful to wait at least five 
 minutes before making a reading, in order that the thermome- 
 ter may attain a steady reading. Next expose about 20° of the 
 stem and make another reading of the boiling point. Repeat 
 this process until nothing but the thermometer bulb is immersed 
 in the steam. From computations based on these readings a 
 stem-correction curve is to be plotted. 
 
 To find the zero-point correction, the thermometer bulb is to 
 be surrounded with finely broken ice placed in a funnel, so that 
 the water runs off. 
 
 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 parallelopiped, measure each of its twelve 
 edges on the dividing engine. If it is a cylinder, measure its 
 
36 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 altitude in four places, and measure the diameter of each base 
 in four different places. In each case great care should 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. However, the original readings 
 are to be entered in the notebook and corrections for zero error 
 of the instrument are to be made later. 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 diame- 
 ters 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 x .) 
 
LENGTH, TIME, AND MASS. 37 
 
 III. 
 
 Measurement of the diameter of a wire by the microscope, and 
 determination of density from diameter, length, and mass. 
 
 First determine the value in millimeters of one division of 
 the micrometer eyepiece. To do this, focus the microscope on 
 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 eyepiece, and then compute 
 the volume and density of the wire as in II, above. 
 
 Experiment A 6 . Determination of the time of a periodic 
 
 motion. 
 
 I. 
 
 By the method of middle elongations. 
 
 The method illustrated by this experiment affords a means 
 of determining the vibration period of an oscillating body with 
 great accuracy. It is used, for example, in determining the 
 time of vibration of the suspended magnet of a magnetometer, 
 in determining the period of a pendulum, etc. With the object 
 of affording practice in the use of the method, the apparatus 
 is arranged as described below : 
 
 A heavy disk (Fig. 6), having a black spot or a pencil line 
 on the edge, is suspended by a long wire, and is kept in vibra- 
 tion, 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, observe the second 
 hand of the clock and count seconds as indicated by it. Con- 
 tinue the count while observing the transit, looking occasion- 
 ally at the clock to see that no mistake is made. In most 
 
38 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 doubt- 
 less at first be somewhat difficult, 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 minutes until three sets have been 
 taken of ten observations each, noting care- 
 fully whether for the first vibration of each 
 set the disk was moving to the right or to the 
 left. 
 
 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. Sub- 
 tracting 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 
 
 Fig. 6. — Disk for Tor- 
 sional Vibrations. 
 
LENGTH, TIME, AND MASS. 39 
 
 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 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, and also between the first 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. 
 
 Repeat the experiment with the cross-hair to one side of the 
 center, and show that the method pursued eliminates any such 
 want of symmetry. 
 
 The principle of this method may perhaps be more clearly 
 understood if the motion of the disk is represented graphically, 
 as in Fig. 7. Horizontal distances here represent times, while 
 vertical distances correspond to the displacement of the disk 
 from its middle position. 
 
 The sinuous line in Fig. 7 thus represents graphically the 
 displacement of the disk as a function of the time. The pas- 
 sage of the spot across the cross-hairs of the telescope corre- 
 sponds in the figure with the intersection of the curve with the 
 line A'B'. When the cross-hairs are placed in the prolongation 
 
 * It is to be observed that this quotient might also be a whole number plus a 
 half. This would be the case if the disk moved in opposite directions at the beginning 
 of the two sets of observations. 
 
4 o 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 of the suspending wire, this line coincides with the middle line 
 AB. In general it is displaced as shown. The time of middle 
 elongation corresponds to the point M on the curve, and lies 
 midway between the times 5 and 6, 4 and 7, etc. It is evident 
 also that M lies midway between 5' and 6', 4' and 7', etc. In 
 other words, the method is independent of the position of the 
 
 A,-J- 
 
 ,M 
 
 -/-B, 
 
 7T 
 
 \4 
 
 \ 
 
 Fig. 7. 
 
 \<3 
 \ 
 V 
 
 cross-hairs. Since M corresponds to the time at which the 
 vibrating body was at rest, it is clear that the time of middle 
 elongation is independent of the position of the telescope. A 
 movement of the latter between two sets of observations is 
 therefore without effect on the result. 
 
 When the time of vibration is less than four or five seconds, 
 the observations become difficult, and in such cases an electrical 
 contact is provided by means of which the successive transits 
 are automatically recorded upon a chronograph. The principle 
 of the method remains, however, unaltered. 
 
 As an example of the employment of the method, the fol- 
 lowing set of observations is appended : 
 
 Date: Jan. 4, 1894. 
 
 
 First Set. 
 
 
 Second Set. 
 
 No 
 
 h. m. sec. 
 
 
 No 
 
 h. m. sec. 
 
 
 I 
 
 ...3:14:10.2 
 
 
 I 
 
 • • -3 = 35 : 9-3 
 
 
 2 
 
 ...3:14:23.7 
 
 Middle Elongations. 
 
 2 
 
 ...3:35:22.8 
 
 Middle Elongations. 
 
 3 
 
 ...3:14:35.9 
 
 5-6.... 3: 15: 8.25 
 
 3 
 
 ■••3:35 : 35-° 
 
 5-6. ...3:36:7.45 
 
 4 
 
 ...3:14:49.3 
 
 4-7.... 3: 15:8.25 
 
 4 
 
 ...3:35:48.4 
 
 4-7. ...3:36:7.4 
 
 5 
 
 ...3:15: 1.5 
 
 3-8. ...3:15:8.30 
 
 5 
 
 ...3:36: 0.7 
 
 3-8.... 3: 36: 7.4 
 
 6 
 
 ...3:15:15.0 
 
 2-9.... 3: 15:8.35 
 
 6 
 
 ...3:36:14.2 
 
 2-9. ...3:36:7.45 
 
 7 
 
 ...3:15:27.2 
 
 1-10 3: 15:8.30 
 
 7 
 
 ...3:36:26.4 
 
 i-io....3:36:745 
 
 8 
 
 ...3:15:40.7 
 
 Average, 3 : 15 : 8.29 
 
 8 
 
 ...3:36:39.8 
 
 Average, 3 : 36 : 7.43 
 
 9 
 
 ■••3= l5:53-0 
 
 
 9 
 
 ...3:36:52.1 
 
 
 10 
 
 ...3:16: 6.t 
 
 
 10 
 
 ...3:37: 5.6 
 
 
LENGTH, TIME, AND MASS. 41 
 
 Interval between first and second middle elongations = 20 m. 
 59.14 second = 1259.14 second. 
 
 Approximate time of one vibration computed from interval 
 between 1st and 9th observation of first set = 25.7 second. 
 
 —48.9 + ; nearest whole number = 49. 
 
 25.7 
 
 „ . , 1259.14 , 
 
 •. Period = — £^_2 = 25.697. 
 49 
 
 It is probable that the result is correct to within a unit in 
 the third place of decimals. 
 
 II. 
 
 Method of transits. 
 
 In several of the experiments which follow, the method of 
 finding the periodic time which is described below will be found 
 useful. 
 
 If the disk described in part A 5 I be vibrating rapidly, so that 
 it is difficult to estimate accurately the time of each transit, it 
 may be possible to estimate the time of every fifth transit to the 
 right ; say, as for example, transits 1, 6, 11, 16 ••• 56. 
 
 If the time of the 1st transit be subtracted from the time of 
 the 56th, the interval will be the time of fifty-five transits. In a 
 like manner the time of the 6th transit subtracted from that of 
 the 51st gives the time of forty -five transits and, finally, the time 
 of the 26th transit subtracted from the time of the 31st transit 
 gives the time for five transits. If these time intervals be added, 
 the sum will be equivalent to the time of 1 80 vibrations. This 
 method uses each observation once and only once, while a 
 method used frequently to compute the periodic time from 
 such data, by subtracting the time of the 1st from the 6th 
 transit, the 6th from the 1 ith, and so on, and computing the 
 mean period from these results, is equivalent to using the first 
 and last observations alone and discarding all intermediate 
 observations. The latter method is of use to check the accu- 
 
4 2 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 racy of the count. The following example will show how com- 
 putations of the periodic time are to be made by this method. 
 
 Number of Transit to 
 
 Time of Transit. 
 
 Time of 5 Oscillations 
 
 Time Intervals between 
 
 the Right. 
 
 in Seconds. 
 
 Transits as Indicated. 
 
 
 h. m. sec. 
 
 
 min. sec. 
 
 I 
 
 9 40 10 
 
 21 
 
 36-I 2 23 
 
 6 
 
 3i 
 
 2O.5 
 
 31-6 I 41 
 
 II 
 
 51-5 
 
 20 
 
 26-I I I OO 
 
 16 
 
 41 11.5 
 
 20.5 
 
 2I-l6 20.5 
 
 21 
 
 32 
 
 19-5 
 
 80 5 24.5 = 
 
 26 
 
 51-5 
 
 20.5 
 
 324.5 sec. for the 
 
 3i 
 
 42 12 
 
 21 
 
 equivalent of 80 vi- 
 
 3° 
 
 33 
 
 
 brations. 
 
 Periodic time = •* ^'- 
 80 
 
 : 4.06. 
 
 701, being equivalent to 1600 
 
 ■ 351, being equivalent to 800 
 
 176, being equivalent to 400 
 
 91, being equivalent to 250. 
 
 Note. — The above method of finding periodic time is to be used in 
 experiments that follow as outlined below, it being kept in mind that 
 observations are to be made always when the transits are in the same 
 direction. 
 
 E x . Observe transits 1, 101, 201, 
 vibrations. 
 
 E 2 . Observe transits 1, 51, 101, 
 vibrations. 
 
 £ 3 . Observe transits 1, 26, 51, • 
 vibrations. 
 
 Q 2 . Observe transits 1, 11, 21, 
 vibrations, if the period is not greater than 10 seconds. Observe 
 transits 1, 6, 11, 16, ••• 61 if the period is between 10 and 15 seconds. 
 For periods greater than 10 seconds the method of A s I may be used, 
 and must be used if period is greater than 15 seconds. 
 
 Experiment A 6 . The Balance and Its Use.* 
 The equal arm balance for comparing masses is one of the 
 most useful of instruments in the physical and chemical labora- 
 
 * The following books should be consulted on the general theory of the balance : 
 Stewart and Gee, vol. I, pp. 63-73; Physical Measurements, Kohlrausch, 3d 
 English Edition, pp. 30-43; Watson's Physics, § 95. 
 
 For a more complete account see Walker on the Balance. 
 
LENGTH, TIME, AND MASS. 43 
 
 tory. If properly used, measurements made by it are among 
 those of greatest accuracy obtainable, often being to one part 
 in a million in determining a mass of a hundred grams. 
 
 The balance consists of a light rigid beam supported at the 
 center on a horizontal knife-edge. Near the two ends of the 
 beam and in the same horizontal plane with the central knife- 
 edge are two other knife-edges which support two scale pans on 
 which the substance whose mass is to be determined and the 
 known determining masses, " weights," are placed. A long 
 pointer is attached to the middle of the beam. This pointer 
 sweeps in front of a scale on which readings are made to deter- 
 mine a balance, as is described hereafter. The parts of the beam 
 between the central knife-edge and the knife-edges near the 
 ends of the beam are called the arms of the balance. The bal- 
 ance is supplied with counterweights, the ones moving vertically 
 to raise or lower the center of gravity, and the ones moving on a 
 horizontal axis to bring the knife-edges into a horizontal plane. 
 
 The knife-edges are protected from being made dull by jar- 
 ring and slipping by a mechanism, called an arrest, which sepa- 
 rates them from the planes against which they press when the 
 balance is in use. The mechanism is operated by means of a 
 jnilled head or lever at the front of the protecting balance case 
 near the base. 
 
 Some of the chief mechanical conditions required of a 
 balance by theory are : 
 
 ( 1 ) The knife-edges should be in the same plane and parallel. 
 
 (2) The knife-edges at the ends of the beam should be at 
 the same distance from the middle knife-edge. 
 
 (3) The pans should have equal mass 1 — it being an advan- 
 tage to make observations with the beam horizontal. 
 
 (4) The center of gravity of the beam should be in the same 
 vertical plane as the central knife-edge. 
 
 From theory we find for a balance whose knife-edges are in 
 the same plane that the sensibility is greater, — 
 (a) The longer the beam. 
 
44 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 (b) The lighter the beam. 
 
 (c) The smaller the distance between the central knife-edge 
 and the center of gravity of the beam. 
 
 (d) The smaller the friction of the knife-edges. 
 
 Since an increase in length of the beam makes an increase 
 in its weight, it is necessary to make a compromise between 
 conditions (a) and (b). The nature of this compromise is not 
 entirely settled. Among other things the solution depends on 
 the purpose of the balance ; that is, whether it is to be used for 
 weighing large or small masses. 
 
 If, according to the third condition the distance between the 
 center of gravity and the central knife-edge be very small, the 
 vibrations of the beam will become very slow and much time 
 will be consumed in weighing. Usually, therefore, a compro- 
 mise on this is made and the knife-edge adjusted as near the 
 center of gravity as is possible without making the period of 
 vibration too great. 
 
 If the balance is " knocked down " when assigned for use, it 
 is to be put in adjustment by the student, great care being taken 
 to place the beam on its arrest supports, and the pan supporting 
 planes and pans on their proper sides, protecting the knife-edges 
 from touching. Lower the beam by means of the milled head 
 at the base of balance which shifts the arrests and raises the 
 beam, and note if the pointer moves off the center. If it does, 
 adjust the horizontal counterweights until it remains in the same 
 position whether the beam is raised or lowered. This assumes 
 that the arrest has been properly adjusted by the maker. If 
 the maker has not properly taken care of this adjustment, it may 
 be made by raising or lowering the supports used to free the 
 end knife-edges, by carefully turning the capstan-headed screws 
 of this part of the arrestment. See that the instrument is prop- 
 erly leveled, by means of the plumb line or level, put on by the 
 maker, or if these are lacking, use a small hand level on the base, 
 leveling by means of the three leveling screws of the base. 
 
 Now adjust the vertically moving counterweight to such a 
 
LENGTH, TIME, AND MASS. 45 
 
 position that the time of a complete swing is about that sug- 
 gested by the instructor assigning the experiment. 
 
 The following precautions are always to be taken in using 
 the balance : 
 
 Dust the pans with a camel' s-hair brush before using. 
 
 See that the rider is in its place and that nothing interferes 
 with the swing of the balance. 
 
 Always determine the zero of the scale before and after any 
 weighing. 
 
 Never put anything on the pans which is likely to injure 
 them. 
 
 In releasing and arresting the beam, be careful to avoid all 
 jerks and always arrest the beam as it passes its position of 
 equilibrium. 
 
 As the sensitiveness of a balance depends, among other things, 
 on the sharpness of its knife-edges, it is therefore best that they 
 should not be in contact with their bearings any longer than 
 necessary. Under no circumstances should weights be added or 
 taken from the pan if the balance is resting on its knife- 
 edges. 
 
 If, when the balance is in equilibrium, there is difficulty in 
 getting up a vibration, gently waft the air over one of the pans : 
 Or the arrest may be raised and lowered again. 
 
 It is best to try the marked masses methodically in their 
 proper order and to arrange them in order of magnitude on the 
 pans. 
 
 Never touch the pans or marked masses with the fingers or 
 with anything likely to injure them. A measurable change in a 
 mass may be caused by a single touch. 
 
 Final weighings must be made with the balance case closed, 
 and care must be taken that the pans do not swing. 
 
 Do not weigh a body when hot ; the air currents will affect 
 the weighings. 
 
 All liquids must be weighed in stoppered bottles. Water 
 vapor will ruin knife-edges. 
 
46 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 See that the heam is arrested and everything put away when 
 the weighings are finished. 
 
 This experiment is divided into several parts as follows : 
 
 I. Weighing by equal swings. 
 
 II. Weighing by vibrations. 
 
 III. Double weighing. 
 
 IV. Weighing with a tare. 
 V. Reduction to vacuo. 
 
 VI. Sensibility of balance for various loads and periodic 
 times. 
 
 VII. Ratio of balance arms. 
 
 VIII. Calibration of a set of weights. 
 
 The work to be done depends on the parts assigned. If A 6 
 + G 2 is the assignment, the specific gravity bottle is to be 
 weighed by all of the methods, but only Paragraphs II, V, VI, 
 and VII need be used to find the density of the three substances 
 which are to be assigned by an instructor. 
 
 I. 
 
 Weighing by equal swings. 
 
 The method of weighing here outlined is very commonly 
 used, but for accurate work it is never to be adopted. It is 
 assumed that the balance is in perfect adjustment and that the 
 pointer swings equal distances both sides of the middle scale 
 division, which may be assumed the zero, on no load, allowance 
 being made, of course, for ordinary damping. 
 
 The mass to be determined is placed in one pan and 
 "weights "put on the other pan until the second and third 
 swings on the two sides of the zero are alike. This necessitates 
 the use of a rider. The first swing is discarded and the excur- 
 sions to the right and left of the central division should not be 
 over five or six divisions. 
 
 Weigh the given mass several times by this method, putting 
 down each determination as made. 
 
LENGTH, TIME, AND MASS. 47 
 
 II. 
 
 Weighing by vibrations. 
 
 The most convenient as well as the most accurate method of 
 weighing is "weighing by vibrations." Owing to the time 
 required for the balance to come to rest and the fact that due 
 to friction and other causes the balance does not always come 
 to rest at its true equilibrium position, it is best to determine 
 the zero by the vibrations themselves. 
 
 What seems at first sight to be the most simple method of 
 weighing, that of adjusting the known masses on the pan until 
 the balance is at rest at the zero position, is in reality much 
 slower and less accurate than weighing by vibrations. In addi- 
 tion to the time required in waiting for the balance to come to 
 rest, it usually takes a large number of trials to bring the bal- 
 ance to exactly the same position of equilibrium as that corre- 
 sponding to zero load. Even after many trials there is often 
 uncertainty about it. Furthermore, by the method of vibra- 
 tions it is not necessary to adjust the masses until the pointer 
 indicates exactly the zero point, i.e. the position of equilibrium 
 corresponding to empty pans. 
 
 The method of getting the position of equilibrium consists 
 in reading the " turning points " or the scale readings corre- 
 sponding to the extreme positions of the 
 pointer in its vibration and from these com- 
 puting the mean. In reading the scale, it 
 should be numbered as shown in Fig. 8, and 
 tenths should be estimated by the eye, care being taken to avoid 
 parallax. Since the amplitude of vibration decreases, it is not 
 correct to take a reading at each end of the swing and then 
 take the mean of these two readings as the position of equi- 
 librium. One more reading should be taken on one side than 
 on the other. 
 
 The following example illustrates the method of taking the 
 readings. In the first column are given the readings of the 
 " turning points " on the left-hand side and in the second column 
 
 5 10 15 
 
 Fig. 8. 
 
48 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 those on the right-hand side of the scale. 
 
 The readings were 
 
 of course taken in the order 3.2, 
 
 18.3, 4.0, 
 
 17.8, 4.4, 17.2, etc. 
 
 Left 
 
 Right 
 
 
 3- 2 
 
 18.3 
 
 
 4.0 
 
 17.8 
 
 
 4-4 
 
 17.2 
 
 
 5-° 
 
 16.8 
 
 
 5-4 
 
 4)70.1 
 
 
 5)22.0 
 
 I7-S 2 
 
 
 4.40 
 
 4.40 
 
 
 
 2)21.92 
 
 
 10.96 
 The method of finding the mean is shown in the table. Or 
 if a v a 2 , a 3 , etc., represent, the readings of the turning points on 
 the left-hand side and & v 6 2 , etc., those on the right-hand side, 
 
 then 2* 23 
 2 x= h 
 
 n n— 1 
 
 where x is the reading of the position of equilibrium. On rough 
 preliminary weighings two readings on one side and one on 
 the other will usually be found sufficient. 
 
 Before placing the mass to be weighed on the balance, the 
 beam should be lowered and set into vibration over about ten 
 or more scale divisions, and after the pointer has made one or 
 two swings, the exact turning points should be read. The read- 
 ing corresponding to the position of equilibrium does not need 
 to be exactly at the center of the scale, but should be some- 
 where near it. If not to be found so, the attention of an in- 
 structor should be called to it, unless the student has been 
 directed to adjust the balance. 
 
 The readings for the position of equilibrium for the empty 
 pans having been taken, the mass to be determined is placed in 
 the left-hand pan and marked masses placed in the right-hand 
 one until an approximate equilibrium is obtained.* 
 
 * Stewart and Gee, vol. I, p. 74. 
 
LENGTH, TIME, AND MASS. 49 
 
 Equilibrium having been approximately obtained, the turn- 
 ing points are carefully read, and from these the reading corre- 
 sponding to the position of rest obtained. This may be several 
 scale divisions from the position of equilibrium for empty 
 pans. 
 
 The next step is to determine the sensibility of the balance, 
 or the number of divisions one milligram will shift the position 
 of rest. The rider is shifted a distance on the arm correspond- 
 ing to one milligram and the position of rest is again obtained 
 from the turning points. The difference between this position 
 and the other gives the sensibility or the value of a milligram 
 in scale divisions. Knowing the zero taken with empty pans 
 and the position of equilibrium with the masses on, the differ- 
 ence in scale divisions may be found. Say that it is 2.25 scale 
 divisions. Suppose that it had been found that one milligram 
 deflected the pointer 1.10 scale divisions. It follows that 
 2.25/1. 10 milligrams is, as the case may be, to be added to or 
 subtracted from the marked masses in the pan to obtain the 
 correct result. After the masses have been removed the equi- 
 librium position should again be observed, and the mean of this 
 and the one taken at the beginning of the set used instead of 
 one alone. 
 
 Determine the given mass twice by the method outlined 
 above. 
 
 III. 
 
 Mass by double weighings. 
 
 In accurate weighing it is necessary among other things to 
 take account of or to eliminate the effect of the inequality of 
 the lengths of the balance arms. 
 
 Gauss' method consists in making weighing with the mass to 
 be determined first in one pan and then in the other. If a 
 body whose true mass is W when placed in the left-hand pan, 
 whose lever arm is /, balances weights R in the right-hand pan, 
 
50 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 whose lever arm is r; and when placed in the right-hand pan, 
 balances a mass L in the left-hand pan, 
 
 Wl=Rr 
 and Wr= LI, 
 
 from which W 2 = RL, 
 
 or IV = VRL. (36) 
 
 Thus, it is seen that the true weight, neglecting bouyant force 
 of the air, is the geometric mean of the two weighings. 
 
 Determine the mass of the given substance by the above 
 method, using the method of weighing by vibrations. 
 
 IV. 
 
 Weighing with a tare. 
 
 Another method of eliminating errors due to inequalities 
 in the lengths of the balance arms is that suggested by Borda, 
 which consists of placing the mass to be determined in one pan 
 and in the other to put any mass just sufficient to balance it. 
 Then remove the body whose mass is desired and replace it 
 with standard masses sufficient to produce another balance. 
 It is convenient to use two riders in this method, one being con- 
 sidered a part of the tare whose mass may not be known. 
 
 Make two determinations of the mass by this method. 
 
 V. 
 
 Reduction of weighings to vacuo* 
 
 Another source of error which must be taken into account 
 in accurate weighings, if the mass weighed has a density dif- 
 ferent from the weights, is the buoyant force of the air. If 
 8„ is the density of the air at a particular temperature, 8„ the 
 density of the weights, and 8, that of the mass to be deter- 
 mined, M, the true mass and M the apparent mass, then 
 
 M - M (' + j.-f) 
 
 * See Exp. G 3 . 
 
LENGTH, TIME, AND MASS. 51 
 
 Assume the weights used in part III to be brass and reduce 
 results there obtained to vacuo. 
 
 VI. 
 
 The sensibility of the balance. 
 
 The sensibility of a balance may be defined as the number 
 of scale divisions shift of the pointer for a change of 1 milligram 
 load on one arm. As has already been noted, the sensibility 
 depends on the same factors as the periodic time, and also on 
 the load. In parts II and III the sensibility was determined 
 for each load. In this part of the experiment the sensibility 
 is to be determined for a given load, of 5 grams say, at three 
 different periodic times, and also for a given periodic time for 
 loads varying from zero to 50 grams at 5 gram intervals. 
 From the data obtained for varying loads a curve is to be 
 plotted, using loads as abscissas and corresponding sensibilities 
 as ordinates. The curve may then be used to compute final 
 weighings, the rest position of the pointer being known for 
 "no load" and for the approximate weight of the body under 
 observation, instead of finding the sensibility for every weighing. 
 
 The method to be used in both parts of VI is given in the 
 following outline : 
 
 Determine the sensibility for zero load by placing a rider 
 on one arm of the balance at such a place as to displace the 
 pointer three or four divisions to one side of the middle of the 
 scale. Then using the second rider, place it at such a position 
 on the opposite arm of the balance as to shift the position of 
 the indicator to a point three or four divisions on the opposite 
 side of the middle division. Note the position of the second 
 rider, from which the added mass is determined. Knowing the 
 shift in the position of the pointer and the added mass, the 
 sensibility may be obtained. Use the method of vibrations for 
 determining the rest positions in every case. 
 
 In like manner determine the sensibility for the loads 
 previously indicated. 
 
52 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 VII. 
 
 Ratio of the balance arms. 
 
 If the balance arms are not equal in length, a single deter- 
 mination of a mass, putting it on one pan only, will be in error. 
 If the ratio of the arms be known, the ma^s may be computed. 
 However, the method is not good for fine weighing unless the 
 ratio be known for the mass used, since the ratio is likely to vary 
 under different loads. 
 
 It will be seen from the equations in part III on double 
 
 weighings that 
 
 Wl = Rr 
 
 and Wr = Ll, 
 
 from which -=-v/— • (37) 
 
 From the double weighings made in part III compute the 
 ratio of the arms of the balance used. 
 
 VIII. 
 
 Calibration of a set of weights.* 
 
 In general for fine work the weights used must be calibrated. 
 For a great deal of work done it is sufficient to know the varia- 
 tion within a given set of weights from their marked values. 
 The following method of calibration is taken from an article by 
 Richards, and is to be applied to the set of weights assigned, 
 beginning with the 0.1 gram and running up to the 50-gram 
 weight. 
 
 All comparisons are to be, made by the method of weighing 
 by vibrations (part II), the sensibility being determined for 
 each load. 
 
 " According to this method the weights to be standardized 
 are weighed wholly on one side of the balance, the comparison 
 being made by substitution. This procedure, of course, elimi- 
 nates a possible inequality in the length of the arms of the bal- 
 ance, which must otherwise be computed. A more important 
 * Richards, The Jour, of Am. Chem. Soc, vol. 22, 1900. 
 
LENGTH, TIME, AND MASS. S3 
 
 advantage is the fact that it also obviates the mental confusion 
 resulting from the continual interchanging of weights between 
 the opposite pans. Thus is avoided one of the common sources 
 of error in the beginner's work." 
 
 " It is, of course, necessary that all of the fractional weights 
 should, taken together, constitute a gram; and because the 
 milligram weights are never used, it is convenient to add an 
 extra centigram weight from another set to supplement the 
 other small weights. The different weights of the same de- 
 nomination should be marked in a recognizable way, and should 
 always be arranged in the same order in the box. The com- 
 parison usually begins with centigrams and proceeds upwards. 
 One of the centigram weights is placed upon the left-hand scale- 
 pan, and is balanced by any suitable tare, care being taken that 
 the rider is not too near either end of its path.* The zero point 
 of the balance need not have been taken in the first place." 
 
 The equilibrium position of the pointer of the balance with its 
 centigram load is now carefully determined, and then another 
 centigram weight is substituted for the first. By noting the shift 
 in the equilibrium position and determining the sensibility for 
 the load, the variation of the second weight from the first is 
 determined. 
 
 " The first weight is then replaced upon the left-hand pan, and 
 if the swings correspond to the first observation, it is reasonably 
 certain that the balance has remained in a constant condition 
 throughout the trial, and hence that the difference between the 
 two weights has been accurately determined. In this way every 
 weight is compared with every other weight of the same denomi- 
 nation, as well as with the combination of all the smaller weights. 
 Thus are obtained a number of independent equations one less 
 
 * "A crude set of weights is, of course, the most convenient tare, and a 5 -milli- 
 gram weight may be kept on the left-hand pan so that the rider may assume a 
 convenient position. The use of the left-hand scalepan for the weights to be stand- 
 ardized renders a confusion of the sign of the correction less likely, because the rider 
 is on the right. In this case, the weights are the objects to be weighed, and hence 
 naturally take the left-hand position." 
 
54 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 than the number of weights ; and by assuming the value of any 
 one of the weights the others may all be calculated." 
 
 " We have found it most convenient for the purposes of cal- 
 culation to make the temporary assumption that the first cen- 
 tigram weight is correct. From it by the simplest possible 
 process of addition and subtraction may be built up quickly the 
 values of all the other weights. While the values thus com- 
 puted are wholly consistent among themselves, they are usually 
 far too different from the face values of the weights for conven- 
 ient use. The reason of this is because the assumed standard 
 is so small a quantity. It is necessary then to translate these 
 consistent values into other terms by dividing every value by 
 the value of one of the larger weights, to be taken as the new 
 and permanent standard." 
 
 " The table below presents all the data and results of a sam- 
 ple standardization. ... In the first column the weights are 
 named by their face values, which are inclosed in parentheses 
 in order to show that they do not signify true grams. In the 
 second column are given the results of the mutual comparison 
 of these weights copied from a notebook in which every detail 
 of each weighing was recorded. The third column gives the 
 actual values of the weights based upon the first centigram 
 weight ; these values are obtained by simply adding together 
 the appropriate preceding values in the third and the last minute 
 fractional weight enumerated in the second column. The ali- 
 quot parts of the value for the io-gram piece, which is now to 
 be taken as the permanent standard, are recorded in the fourth 
 column, while the corrections sought, obtained by simply sub- 
 tracting the numbers in the fourth column from those in the 
 third, are given in the last vertical row." 
 
 " Owing to neglected fractions, the figures in the last column, 
 when added together, are sometimes slightly discordant with 
 those given in the second column. This is inevitable ; of course 
 such corrections should always be calculated to one decimal 
 place beyond the figure which one wishes to have exact." 
 
LENGTH, TIME, AND MASS. 
 
 55 
 
 Nominal 
 
 Data obtained by Substitution Method. 
 
 Preliminary- 
 Values (actual) . 
 
 Aliquot Parts 
 of 10.01768 
 
 Corrections 
 in Milligrams 
 
 Values. 
 
 Grams. 
 
 
 
 (Ideal). 
 
 (Actual 
 
 
 
 
 Grams. 
 
 Grams. 
 
 Minus Ideal). 
 
 (O.OI) 
 
 = Standard of comparison 
 
 Standard 
 
 O.OI002 
 
 — 0.02 
 
 (o.oi') 
 
 = (0.01) 
 
 + 0.00006 
 
 0.01006 
 
 0.0I002 
 
 + O.04 
 
 (o.oi") 
 
 = (0.01') 
 
 — 0.0000 1 
 
 0.01005 
 
 O.OIO02 
 
 + O.03 
 
 (0.02) 
 
 = (0.01) + (0.01' 
 
 ) — 0.0000 1 
 
 0.02005 
 
 O.02004 
 
 + O.OI 
 
 (0.05) 
 
 = (0.02) + etc. 
 
 — 0.00007 
 
 0.05009 
 
 O.05009 
 
 ± 0.00 
 
 (O.I) 
 
 = (0.05) + etc. 
 
 — 0.00006 
 
 0.10019 
 
 O.IOO18 
 
 + O.OI 
 
 (O.I') 
 
 = (o-i) 
 
 + 0.0000 1 
 
 0.10020 
 
 O.IOO18 
 
 + 0.02 
 
 (0.2) 
 
 = (0.1) + (0.1') 
 
 — 0.00004 
 
 0.20035 
 
 O.20035 
 
 ± 0.00 
 
 (0.5) 
 
 = (0.2) + etc. 
 
 — O.OOOII 
 
 0.50088 
 
 0.50088 
 
 ± 0.00 
 
 (I) 
 
 = (°-5) + etc. 
 
 — 0.00004 
 
 1.00183 
 
 I.OOI77 
 
 + 0.06 
 
 (I') 
 
 = (0 
 
 — 0.00002 
 
 1.00181 
 
 I.OOI77 
 
 + 0.04 
 
 (I") 
 
 = (1) 
 
 — 0.00006 
 
 1.00177 
 
 I. OOI 77 
 
 ±0.00 
 
 (2) 
 
 = (1') + (1") 
 
 + 0.00025 
 
 2.00383 
 
 2.00354 
 
 + 0.29 
 
 (5) 
 
 = (2) + etc. 
 
 — 0.00040 
 
 5.00884 
 
 5.00884 
 
 ± 0.00 
 
 (10) 
 
 = (S) + etc. 
 
 — 0.00040 
 
 10.01768 
 
 IO.OI768 
 
 Standard 
 
 
 etc. 
 
 
 etc. 
 
 etc. 
 
 etc. 
 
 In performing the experiment the 20-gram weight of the 
 set to be calibrated is to be compared with a standard 20-gram 
 weight which is to be taken as the standard in place of a 10-gram 
 weight as indicated in the example given above. 
 
 Experiment A 7 . The planimeter. 
 
 It is often desirable to know the area of a closed curve, as an 
 engine indicator diagram or a hysteresis loop. There are vari- 
 ous methods used for determining areas, such as drawing the 
 curves to scale on cross-section paper and counting the squares, 
 or cutting out the inclosed area from the paper on which it is 
 traced, determining its mass and comparing it with the mass of a 
 known area of the same kind of paper. A third method is to 
 determine the area by a mechanical integrator. The type of 
 integrator considered in the experiment given below is called a 
 polar planimeter. 
 
 The planimeter consists of two arms hinged to permit rela- 
 tive motion in a plane. One arm has a needle point at its end 
 
56 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 to fix its position and the other arm has a tracing point at its 
 free end. In the arm carrying the stylus there is a wheel free 
 to rotate about an axis in the line connecting the hinge and the 
 tracing point. In connection with the wheel there is a counting 
 device to indicate the number of turns, and a vernier to read 
 fractions of a turn. Rotation of the wheel can take place only 
 when the arm moves so that there is a component of the motion 
 at right angles with the arm. 
 
 In moving the stylus about any closed area not embracing 
 the needle point, the wheel revolves through an angle propor- 
 tional to the inclosed area, the direction of motion of the stylus 
 being in the same sense in all parts of its path. The principle 
 involved is outlined briefly as follows : * 
 
 The area swept out by a line of length / having any motion 
 whatever in a plane may be considered as made up of transla- 
 tions and rotations. For the translational part the area swept 
 out will be equal to the product of the length of the line and the 
 distance dx which the line moves in a direction measured per- 
 pendicular to its length 
 
 dA t = Idx. (38) 
 
 The part of the area swept out by the line due to its rotation 
 about one end as an axis is equal to the length of the arc trav- 
 ersed by the other end multiplied by one half the length of the 
 line, 
 
 dA r = ld$- = Pd<f>/2 ; (39) 
 
 the total area swept out is therefore 
 
 dA = Idx + Pd<j>/2. (40) 
 
 If a wheel of radius r whose axis is in the line /be used to meas- 
 ure dx, then dx = rdd and 
 
 dA = IrdO + Pd<j>/2. (41) 
 
 * More extended proofs may be found in Williamson's Integral Calculus ; Ferry 
 and Jones, Practical Physics, and Miller's Laboratory Physics. 
 
LENGTH, TIME, AND MASS. 57 
 
 If one end of the line / is free and the other end is attached 
 by means of a hinged joint, h, to another line of length R, whose 
 opposite end is fixed at O, as in Fig. 9, the system may be con- 
 sidered to represent the arms of a polar planimeter. There are 
 two cases of importance to be considered : I, when the tracing 
 point p in tracing the outline of the figure 
 does not comprise the fixed end O of the arm 
 R, and II, when it does comprise the point 
 O in going completely around the figure 
 whose area is to be determined. 
 
 Case I. The line R in rotating about O 
 to the final position, which coincided with the 
 initial position, sweeps out just as much area 
 in rotating clockwise as in rotating in the 
 opposite direction ; consequently the net area 
 is zero, considering clockwise rotation as positive and counter- 
 clockwise rotation as negative. The line / will sweep over a 
 part of the area covered in its motion just as many times in 
 a counterclockwise as in a clockwise direction, and since the 
 final position of the line coincides with the initial position, the 
 rotational part of the area covered will be zero, as is indicated 
 in the following expression, 
 
 m /rdO +]/-*!>. (42) 
 
 From which A = /rO = ks, (43) 
 
 in which k is a constant and s is the net number of divisions the 
 wheel has turned and is proportional to the angle 6. The factor 
 k is determined by the manufacturer of the planimeter, or may 
 be easily determined by the user thereof. 
 
 Case II. It is seen that if the tracing point p moves com- 
 pletely around and returns to its original position in such a 
 manner that the plane of the wheel lies along a radius from the 
 point O, no rotation of the wheel will take place, although a 
 circular area will be inclosed within the line followed by the 
 tracing point. The circle including this area is called the zero 
 
58 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 or datum circle. If the tracing point is at a distance from O 
 greater than the radius a of the datum circle and moves around 
 O in a clockwise direction, say, the wheel w will roll in one direc- 
 tion and the area indicated is to he added to that of the datum 
 
 circle. If the distance to * is less than 
 h r 
 
 /V that of the radius of the datum circle, as 
 
 o / .*W- <*p" and the rotation about O again be 
 
 / /' \\ X clockwise, the wheel will roll in the oppo- 
 
 £1 a \^>p site direction. The indicated area in this 
 
 _. ,„ case is to be subtracted from the area of 
 
 Fig. 10. 
 
 the datum circle. In the general case, 
 when the tracing point moves completely around O, the area 
 indicated on the wheel is to be added to or deducted from 
 the area of the datum circle, depending on whether the net 
 rotation of the wheel has been in one direction or the other. 
 
 The experiment consists of the calibration of a planimeter 
 and its use in finding the areas of several figures. In all of the 
 experimental work, if a student is working alone, he should 
 make at least five readings on each part of the experiment. If 
 two students are working together, each student should make 
 at least three readings on each part. 
 
 In making the calibration of the planimeter, you are fur- 
 nished with a metal plate on which there is a triangle, a circle, 
 and a square. The dimensions of these figures are to be meas- 
 ured with an ordinary centimeter scale, and readings made to 
 j 1 ^ of a millimeter in order to determine the areas of these fig- 
 ures. The planimeter is then to be used on these figures, the 
 movable point moving completely around the figures, and read- 
 ings made on the vernier and counter to determine the number 
 of divisions of the wheel necessary to indicate i square centi- 
 meter. Then readings are to be made of the areas of the other 
 figures furnished. There is also to be drawn by the student, on 
 cross-section paper or on a page of his notebook, which may be 
 cut out and added to the report, a blocked figure 8 like sample 
 herewith, in which the upper part of the 8 is smaller than the 
 
STATICS. 59 
 
 lower part. The area of each part of this figure 8 is to be meas- 
 ured, and then the planimeter is to be used on the whole figure, 
 first going around the 8 by traversing the whole 
 of the left side first, and then the right side back 
 to the starting point. The figure 8 is then to be 
 traversed by following the upper left side, the 
 lower right, and around clockwise, then the lower 
 left and upper right to the starting point. Com- 
 pare these results of traversing the figure 8 in this 
 
 Fig. 1 1. 
 
 manner with the sum and difference of the in- 
 dependent determinations of the two parts of the figure. In 
 at least one case the area of a figure is to be determined, in 
 which the fixed point O is within the figure. 
 
 GROUP B: STATICS. 
 
 (Bj) Parallelogram, of forces ; (B 2 ) Parallel forces, and principle 
 
 of moments. 
 
 Experiment B v The parallelogram of forces. 
 
 When a single force acts on a body, a change of motion takes 
 place in the direction and sense in which the force acts. When 
 more than one force acts on a body, each force produces its 
 own effect, whether acting alone or with the other forces. Two 
 equal forces acting in the same direction along the same line but 
 in opposite sense produce no change in motion. A single force 
 of proper magnitude, direction, and sense may be made to re- 
 place two or more forces acting at a point, since it will produce 
 the same effect. Such a force is called the resultant of the 
 several forces. A force of the magnitude and direction of the 
 resultant force but in the opposite sense acting along the line of 
 the resultant is called the anti-resultant, and if introduced into 
 the force system of the several forces will produce equilibrium. 
 Such a balanced force system is called a closed system of forces. 
 In any closed system of forces any force may be considered the 
 anti-resultant of all the other forces of the system. The resultant 
 
6o 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Fig. 12. 
 
 of any force system may be found graphically by drawing the 
 force polygon ; that is, by plotting to scale the several forces in 
 their proper directions and senses, in any order, care being taken 
 to join the arrows representing the forces so that in going from 
 one arrow to the next the progression is always 
 in the direction of the points of the arrows. 
 The resultant will be represented in direction, 
 sense, and amount by the arrow necessary to 
 close the figure with its point at the point of 
 the last force drawn. The anti-resultant will, 
 of course, have the opposite sense, its point being at the 
 beginning of the polygon. Such a force system is shown in 
 the figure, in which the force R is the resultant of forces 
 a, b, c, and d. 
 
 In the case of two forces only, a parallelogram may be drawn 
 in which the opposite sides represent, respectively, the two forces 
 and the proper diagonal represents the resultant. This parallelo- 
 gram shows that the same resultant follows no matter in what 
 order the forces are taken. In the experiment which follows, 
 closed or balanced systems of forces are used. Pass two cords, 
 having hooks at their ends, over two pulleys fixed to a horizontal 
 bar. At two ends of the cords attach weights, the opposite ends 
 of the cords being attached to a ring 
 as in the figure. Attach the third cord 
 to the ring and suspend weights by it. 
 Let the weights be suspended freely, 
 and the system will come to rest with 
 the angles, a, /3, 7, of such values as to 
 produce equilibrium of the system. 
 Measure the angles a, /3, and 7 by 
 means of a protractor and make a note of the weights used, 
 taking care to see that the system may move into its equi- 
 librium position freely. For purposes of the experiment it 
 is well to have the weights of such values as to have none 
 of the angles very small. Make another set of readings, 
 
STATICS. 61 
 
 using different weights. Make two other sets of readings, 
 using systems in which there are four forces acting at a 
 point. 
 
 For each set of readings plot the forces to scale, using a large 
 plot. In each of the cases of the force systems made up of 
 three forces, find the resultant of any two of the forces graphi- 
 cally, and compare the magnitude, direction, and sense of the 
 resultant with the magnitude, direction, and sense of the third 
 force. For the sets of readings where four forces were used 
 find the resultant of any two forces ; then combine this resultant 
 with a third force, and compare the resultant of the three forces 
 so obtained with the fourth force as noted in the cases of the 
 three forces above. Compute and compare in every case the 
 vertical components of all the forces and also the horizontal 
 components. 
 
 A force table may be used in the above experiment. On 
 this table the degrees may be marked at the edges. The posi- 
 tion of the pulleys must be adjustable so that when the proper 
 adjustment is attained, the center of the ring is in the center of 
 the board. The student is to explain why this adjustment is 
 necessary. 
 
 Experiment B 2 . Parallel forces and principle of moments. 
 
 For equilibrium of concurrent forces it is not enough that the 
 components of the forces in any direction must be zero, but also 
 that the resultant moments must also 
 
 be equal to zero, since the forces ' ""T 
 
 B 
 
 might produce rotation even when 
 
 balanced for translation. Two cases ' ' I ' F? 
 
 will be studied : I, when the forces 
 
 Fig. 14. 
 
 lie in the same plane and are parallel, 
 
 and II, when the forces lie in the same plane but have various 
 
 directions. 
 
 I. Suspend a uniform bar B, Fig. 14, horizontally by means 
 of a pair of spring balances S, S, and suspend weights W, W 
 
62 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 from the bar by means of hooks and pins. Adjust one or both 
 supports of the spring balances until the bar is approximately 
 horizontal. Observe the weights suspended, the mass of the 
 bar, the readings of the spring balances, the horizontal distances 
 from the points of application of all of the forces involved (con- 
 sidering the mass of the bar as concentrated at its center of 
 gravity) to any point whatever, not a point of application of a 
 force. 
 
 Find the difference between the upward and downward 
 forces and the percentage variation from the mean. Find 
 also the difference between the sums of the clockwise and 
 contraclockwise moments and the percentage variation from 
 the mean. It may be necessary to calibrate the spring balances, 
 and also to determine their zero errors. If so, the corrected 
 readings should be used in making the computations noted 
 above. 
 
 Change the positions of applications of all of the forces, their 
 
 values, by changing the values of the suspended weights, and 
 
 also increase or decrease the number of suspended weights; 
 
 'make observations and computations as indicated in the first set 
 
 of observations. 
 
 II. Build up a derrick model as shown in the figure, with 
 uniform wood bars A and B, whose masses may be considered 
 
 as concentrated at their geometrical 
 centers, the spring balances at C and 
 D, and the suspended weights at E. 
 There are pin bearings at O and P 
 which may be adjusted so that the bars 
 A and B are vertical and horizontal, 
 respectively. By means of force dia- 
 grams and the principle of moments, 
 find the direction, sense, and amounts 
 of the reactions at and P. After these reactions have been 
 determined take some point outside the derrick as a center 
 of moments, so chosen that none of the moments about it 
 
FRICTION AND SIMPLE MACHINES. 
 
 63 
 
 will be zero, and determine the sum of the moments with 
 regard to the point.* 
 
 Find the variation of the sum of the moments thus found 
 from the mean of the positive and negative moments. 
 
 GROUP C: FRICTION AND SIMPLE MACHINES. 
 (C x ) Coefficient of friction; (C 2 ) Law of wheel and axle ; 
 (C 3 ) Law of systems of pulleys ; (C 4 ) Law of differential 
 pulley. 
 
 Experiment C v To determine the coefficient of friction 
 between two surfaces. 
 
 The apparatus, which is shown in Fig. 16, consists (1) of a 
 smooth plate made of one of the materials to be tested and 
 
 I w 
 
 Fig. 1 6. — Coefficient of Friction. 
 
 capable of being adjusted so that its upper surface is accu- 
 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 
 
 * It will be found advantageous to draw to scale a figure of the derrick, inserting 
 the forces at their proper places with their proper directions and senses; assume a 
 point on the drawing as the center of moments and base computations on distances 
 measured on the diagram. 
 
64 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 the force is just sufficient to keep the block moving uniformly 
 when once started. Repeat the observations with pressures of 
 10, 15, 20, etc., kgs. on the block until a pressure of 50 kgs. is 
 reached. 
 
 Each of the observations at different pressures should be 
 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- 
 tions, each one being as carefully made as though it alone were 
 to determine the coefficient. 
 
 Care should be taken in every case to start the load, because 
 starting friction is greater than moving friction. The coeffi- 
 cient of moving friction is the quantity desired. 
 
 The surfaces should be carefully rubbed with filter paper 
 before beginning the experiment, and should not be touched 
 during the experiment, since the condition of the surfaces must 
 not be altered. 
 
 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 it is necessary to find the coefficient 
 of frictional torque of the pulley. This coefficient is to be de- 
 termined by passing a cord over the pulley and suspending 
 1 kg. from each end, then adding a known mass, first to one 
 side, then to the other, sufficient to produce uniform motion, 
 and taking the mean from the proper mass. Repeat the process 
 for 2, 3, 4, and 5 kgs. on each side. Then 
 
 ^^m+M+M+P' (44) 
 
 where /* p is the coefficient of frictional torque, m the mass 
 necessary to produce uniform motion, M the mass suspended 
 from each side, and P the mass of the pulley. 
 
FRICTION AND SIMPLE MACHINES. 
 
 65 
 
 Remembering that the force necessary to overcome pulley 
 friction is equal to the normal pressure on the bearing times /&„, 
 a factor may be found, which, if multiplied into the mass sus- 
 pended from the cord to produce sliding, will give the required 
 tension in the horizontal part of the cord which is necessary in 
 finding the coefficient of friction desired. In finding the nor- 
 mal pressure it is to be noted that the pull w' on the horizontal 
 
 
 
 
 
 
 FRIC 
 
 noN 
 
 
 
 
 A 
 
 ^8.00 
 
 
 
 
 
 
 
 
 
 
 jf 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 z 
 < 
 .-. 6.00 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 
 
 
 
 
 (2L, 
 
 
 
 
 
 o 
 
 a. 
 o 
 
 
 
 
 
 
 /u 
 
 ) 
 
 
 
 
 a 
 z 
 > 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 s 
 
 
 
 // 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 20 30 40 
 
 PRESSURE (W) 
 
 Fig. 17. 
 
 part of the cord is not very different from the suspended mass 
 w. Therefore the normal pressure N on the bearing may be 
 shown to be approximately equal to wV2. 
 
 Compute the value of the coefficient of friction for the sur- 
 faces for each load used. 
 
 Plot two curves to the same scale from the same origin : (i) 
 using pressures W {¥\g. 17) as abscissas and moving forces w' 
 (w corrected for pulley friction) as ordinates, and (2) using 
 pressures and total moving forces as co-ordinates. Find the 
 equations of the first of the above-mentioned curves by the 
 method of least squares. 
 
 VOL. I — F 
 
66 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 If the mass of the block be considered, the curve should pass 
 through the origin ; in which case b = o, and the value of a found 
 by least squares is to be obtained from the expression 
 
 ~2,xy — a1,x 2 = o. (45 ) 
 
 (See Manual, p. 26, Eq. 31.) 
 
 Give the physical equation of the curve and interpret its slope 
 and intercept, and get all possible physical constants from it. 
 From the two curves get the coefficient of frictional torque of 
 the pulley and compare it with that computed from the data 
 taken. 
 
 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 determina- 
 tions^ 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 
 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 : 
 
 * 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. 
 
 67 
 
 Find by experiment the weights necessary to raise loads of 
 1. 2, 3. S. 7. r °, IS. 25, 35, and 50 kgs., the small weight being 
 adjusted in 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 
 independent of all the rest, and uninfluenced by any assumption 
 
 Fig. 18. 
 
 as to what the relation should be between "moving force" and 
 "load." 
 
 From the data thus obtained, plot curves showing the relation 
 between the moving force and the load in each case. Figure 18 
 shows such a curve. To locate points on these curves (which 
 should be accurately drawn on cross-section paper), the loads 
 are to be used as abscissas and the corresponding moving forces 
 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 equations that 
 are obtained by least squares should be drawn on the 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 
 
 * Note that the horizontal and vertical scales need not be the same. 
 Introduction. 
 
 See 
 
68 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 friction, and compute the efficiency of the apparatus, considered 
 as a machine for lifting weights, for loads used as indicated 
 above. A curve showing the relation between efficiency and 
 load may then be drawn. (See Fig. 19.) 
 
 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 
 
 .90 
 
 > 
 
 z 
 u 
 
 
 
 
 WHEEL 
 
 AND AXLE 
 
 
 
 
 
 ^2^ 
 
 yi CURVE 
 
 
 
 it .80 
 
 UJ 
 
 h 
 z 
 ui 
 
 
 DC M 
 
 uj .70 
 
 EL 
 
 
 
 
 
 m^^^^ 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 20 
 
 30 
 LOAD 
 
 Fig. 19. 
 
 40 
 
 50 
 
 velocity ratio by actually measuring the distance passed over 
 by the load when the wheel is turned a known number of times. 
 
 Addenda to the report: 
 
 (1) Interpret in detail the curves obtained. For example, 
 the friction of the machine consists of two parts : (1) 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 
 
FRICTION AND SIMPLE MACHINES. 69 
 
 to lift a much larger weight through a comparatively small 
 distance. The objects of the experiment are : (1) To determine 
 experimentally the relation between " moving force " 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 1, 2, 3, 5, 10, 15, etc., .up to 50 kgs., the moving force 
 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, plot a curve showing the relation 
 between moving force and load, and from the appearance of the 
 curve decide upon the form of its equation. The constants are to 
 be determined by the method of least squares. 
 
 (3) Having determined the ratio of the distances passed over 
 by the two weights, show what moving forces would be necessary 
 to raise the same loads if there were no friction, and compute 
 the efficiencies of the two systems for the various loads used. 
 
 Plot a curve based on the least squares on the same sheet 
 as the experimental curve. Also plot on the same sheet an 
 " ideal curve " of no friction, no pulley mass, based on the theory 
 of the apparatus. This curve will pass through the origin and 
 have a less slope than the experimental curve. Explain these 
 points in detail and discuss the curves fully. An efficiency- 
 load curve is to be plotted, its equation derived and discussed, 
 and its asymptote located. 
 
 Queries : What change in the apparatus would cause the je-intercept 
 of the line to increase ? The slope of the line ? 
 
 Is it true that the tension of the string on two sides of a pulley is the 
 same when motion is taking place ? 
 
 Experiment C 4 . The differential pulley. 
 
 Study the pulley supplied, and briefly explain in your report 
 the principle upon which it works. 
 
70 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Find the moving force just necessary to raise loads of 5, 10, 
 15 • • • 50 kgs. at a uniform rate. Find also the relative dis- 
 tances traversed by the working force and load. Compute the 
 efficiency for each load. 
 
 Plot a curve, using loads as abscissas and working forces as 
 corresponding ordinates. Plot another curve, using loads and 
 efficiencies as co-ordinates. 
 
 Derive the constants of the first of the above curves by the 
 method of least squares (Manual, pp. 24-28) and plot a curve 
 based on these constants. 
 
 Discuss the curves fully. 
 
 Addendum : 
 
 Explain fully why the machine will not run backward if 
 sufficient load is applied. 
 
 GROUP D: UNIFORMLY ACCELERATED MOTION. 
 
 (D x ) Atwood's machine ; (D 2 ) Determination of gravity from 
 the motion of a freely falling body; (D 3 ) Angular accelera- 
 tion, angular velocity, and rotational inertia. 
 
 Experiment D x . 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. 20), which 
 is mounted in such a way as to make the friction of its bearings 
 as smalLas possible. To the standard is attached a scale grad- 
 uatedun'rcentimeters or inches for convenience in measurement. 
 Over the pulley hangs a light silken cord, to which weights, 
 w v w 2 , 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 equilibrium 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," 
 r{¥ig. 21), is thus utilized to set in motion a much larger mass, 
 
UNIFORMLY ACCELERATED MOTION. 
 
 71 
 
 and the acceleration is, in consequence, smaller than if the rider 
 alone were moved. By suitably choosing the various weights, 
 the motion may be made so slow that the 
 velocity can be readily measured. The 
 apparatus thus affords a means of illus- 
 trating the laws of uniformly 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 pro- 
 vided with an electric bell or sounder, 
 which can be connected with a seconds 
 ; pendulum. By means 
 
 of an electromagnet, m 
 (Fig. 20), placed at the 
 top of the vertical stand- 
 0,11 1 ard, and connected with 
 
 I U r tne 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. 20), 
 movable along the upright standard, may be 
 adjusted so as to stop the fall at any point 
 desired, while a ring, j 2 , also adjustable in 
 Fig 21, position, serves to remove the rider at any 
 
 desired time without disturbing the motion of the weights 
 
 themselves. 
 
 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 
 
 Fig 20. 
 
 fP S ° 
 
72 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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. Move the 
 ring a few centimeters, in order to get independent readings, 
 and make two more settings as before. Set the ring for the 
 mean of the three readings ; then place the bracket so that the 
 mass will strike it after a uniform motion of 3, 4, and 5 seconds, 
 making one careful setting for each. This constitutes one com- 
 plete set of readings. Make three additional complete sets of 
 readings for accelerations for 3, 4, and 5 seconds. 
 
 If the timing apparatus is arranged to beat half seconds, 
 use 1 \, 2, 2|-, and 3 seconds for the acceleration periods in 
 
 place of 2, 3, 4, and 5 seconds, 
 and 2, 3, and 4 seconds for the 
 periods of uniform velocity in- 
 dicated above. If it is not 
 possible to get uniform motion 
 for the periods noted above, 
 try some other series of three 
 periods, as 1, 2, and 3. 
 
 From the data obtained 
 find a value of the acceleration 
 based on the mean distance trav- 
 ersed for uniformly accelerated 
 motion, and another value of the 
 acceleration from the computed velocity attained based on the 
 determination of the velocity after the removal of the rider. 
 Do this for each set and tabulate results. 
 
 The results are also to be shown by plotting two curves on 
 the same sheet of cross-section paper, to the same scale, from 
 the same origin (Fig. 22), using time in seconds for ordinates 
 
 
 
 
 
 
 
 
 
 
 
 V u 
 
 ,fs" 
 
 
 
 
 
 & 
 
 ?/<&' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 40 60 80 100 120 140 
 
 DISTANCE TRAVERSED AND VELOCITY ATTAINED 
 Fig. 22. 
 
UNIFORMLY ACCELERATED MOTION. 73 
 
 in both cases, and using for abscissas velocities attained in one 
 case and distances traversed in the other. Discuss the results 
 and show whether or not they are in agreement with the laws 
 of uniformly accelerated motion. Discuss the curves fully, 
 giving the meaning of the slopes, the intersection of the two 
 curves, and the relation between them. 
 
 II. 
 
 To use AtwoocFs 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) it ; (46) 
 
 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 : 
 
 Hang half of the weights supplied with the machine on 
 each side of the pulley. Add enough tin foil, as in part I, to 
 overcome moving friction and produce uniform motion in the 
 proper direction when the machine is started. 
 
 Make two settings of the ring that takes off the rider, so 
 that it is removed after 2, 3, 4, and 5 seconds acceleration, 
 making note of the distances passed through. 
 
 Remove one weight from each side, readjust for uniform 
 motion with the rider off, and proceed as before. Continue 
 to take off weights and make observations until only one of 
 the equal weights remains on each side. If a reading for 
 5 seconds cannot be made, try the series, 1, 2, 3, and 4, 
 dropping the 4 and 3 if necessary for the smaller masses. 
 
74 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Find the masses of one of the equal weights, the rider, and 
 the pins and supports, to three significant figures. 
 
 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- 
 
 ..1 
 
 .09 
 
 p 
 
 fe.07 
 
 0= 
 
 u 
 
 u.06 
 o 
 
 o 
 
 < .04 
 
 o 
 
 o 
 
 £.03 
 
 o 
 u 
 
 K m 
 
 .01 
 
 
 
 
 
 ATW 
 
 JOD'i 
 
 MAC 
 
 HINE 
 
 
 
 
 17^ — ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 100 
 
 200 
 MASSES 
 
 Fig. 23. 
 
 200 
 
 400 
 
 500 
 
 puted, either algebraically or graphically. The graphical 
 method which follows is, however, recommended. 
 
 Plot upon cross-section paper a curve (see Fig. 23) in which 
 the masses hung upon the pulley are used as abscissas, and the 
 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, 
 
 a mg mg 
 
 (47) 
 
 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^ = m + Ma.ndy = - ; i.e. 
 
 a 
 
 y = — x + ■ 
 mg mg 
 
 (48) 
 
UNIFORMLY ACCELERATED MOTION. 75 
 
 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 
 plotted. 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. 
 
 From the data obtained compute the values of g for each set, 
 using the value of M obtained from the curve. Get a value of 
 g from the curve. 
 
 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. 
 
 .-. mgl= \{m + M)v i + l KaP, (49) 
 
 in which v is the final velocity of the suspended masses, to the 
 final angular velocity of the pulley, and K its moment of inertia. 
 Remembering that v % = 2 al and v = rco, this equation reduces to 
 
 mg=(m + M+ ~\a, (50) 
 
 in which r is the radius of the pulley. 
 
 Addendum : 
 
 Find the tension in the cord on both sides of the pulley for 
 both uniform velocity and uniformly accelerated motion. Ac- 
 count for the difference in the tension on the two sides in the 
 second case. 
 
 Experiment D 2 . Determination of g from the motion of a 
 freely falling body. 
 
 Two forms of apparatus are used in this experiment, one of 
 which, Fig. 24, is so arranged that a vibrating tuning fork of 
 known pitch falls in front of a glass plate on which is a thin 
 
7 6 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 „ : 1 
 
 a 
 
 Fig. 24. 
 
 coating of " Bon Ami" soap. In the second form of apparatus 
 a small glass plate, treated in a similar manner, falls freely in 
 front of an electrically driven tuning fork. 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 a successive 
 equal number of waves, 5 or 10 for the first form of apparatus 
 and 1 or 2 for the second form, it is possible to compute the 
 acceleration of gravity. As a means of measuring £-, the method 
 is not at all accurate, since any friction in the apparatus will 
 
 introduce a consider- 
 able error. The ex- 
 periment is valuable, 
 however, in illustrat- 
 ing the laws of fall- 
 ing bodies. Having 
 covered the glass 
 with a thin coating 
 of " Bon Ami " with a wet cloth and allowing it to dry, 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 adjustment 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. 25.) 
 
 It may be found better, however, to use a straight edge and 
 the point of a knife blade to get the center line. Mark an 
 intersection of the straight line with the curve near one end, 
 and calling that intersection number one, mark off intersections 
 number eleven, twenty-one, thirty-one, and so on until ten spaces 
 have been marked off. Then lay a scale down on the glass plate 
 and note the positions of the marked intersections on the scale. 
 For curves traced by the second form of apparatus, the dividing 
 engine may be used, as in this form of apparatus it is necessary 
 to measure intersections much closer together to get ten readings 
 
UNIFORMLY ACCELERATED MOTION. 77 
 
 on the plate. In this case, adjust the glass under the microscope 
 of the dividing engine, assume some sharply denned 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, in either case, evidently /' 
 represent the spaces passed over during the appropriate 
 intervals of time as measured by the tuning fork. It is 
 well 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 distance from this point to an intersection passed t 
 vibrations later. Then we shall have 
 
 L = v Q t+%gt\ (51) 
 
 in which g represents the acceleration of gravity. If 
 the series of observations taken be plotted, with values 
 of L as ordinates and values of t as abscissas, the 
 resulting curve will be a parabola. The constants v 
 and g may be determined from this curve by the method 
 of least squares. As this is a quadratic equation, the 
 numerical computations will be very laborious. It will 
 therefore be desirable to use a linear equation if pos- 
 sible. This may be done as follows : Let / be the 
 distance traversed during the tth. interval as measured 
 by the fork, counting from the assumed origin ; then 
 we shall have v 
 
 l=v*-\g+gt. (52) 
 
 If a series of corresponding values of / and t be plotted, this 
 will give a straight line, from which the constants v and g may 
 be determined either directly by measurement or indirectly by 
 the method of least squares. 
 
 Fig. 25. 
 
78 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The constants may also be derived from two independent 
 equations like the above. The two values of / taken should 
 differ considerably, one being about twice the other. 
 
 In the above discussion, the unit of time is some multiple of 
 the period of the fork. Therefore the values of v andg obtained 
 will be referred to it as the unit of time. Since v varies 
 inversely as the time, it is necessary, in order to express that 
 constant in centimeters per second, that the values obtained 
 be divided by the multiple of the fork period. For a similar 
 reason, the value of g obtained must be divided by the square 
 of the proper period if the acceleration of gravity is to be 
 expressed in centimeters per second per second. 
 
 Experiment D 3 . Angular acceleration and velocity ; torque, 
 and rotational energy. 
 
 The experiments in the D group preceding this one deal with 
 the accelerations, velocities, and distances traversed by bodies 
 having constant forces acting on them. This experiment, which 
 is divided into two parts, deals first (part I) with the analogous 
 relations of angular accelerations, angular velocities, and angular 
 distances traversed by bodies acted on by constant torques, and 
 second (part II) with the energy relations involved. 
 
 I. 
 
 Angular acceleration and velocity, and angular distance 
 traversed. 
 
 If a body is acted on by a constant torque, it is found that 
 the angular acceleration is constant. Let <j> and a> , respec- 
 tively, be the angular displacement and the angular velocity 
 at the instant from which time is counted, and a be the constant 
 angular acceleration. 
 
 a = doa/dt, (53) 
 
 from which a>= ( da>= I adt = at + » , (54) 
 
UNIFORMLY ACCELERATED MOTION. 79 
 
 or if the body starts from rest at the instant from which time is 
 counted, 
 
 » = **. (55) 
 
 Remembering that the angular velocity is the rate of change 
 of angular position, equation 54 may be written 
 
 d<f>/dt =ai + &> , (56) 
 
 which when integrated gives 
 
 <j> = at 2 /2 + co t + <j> , (57) 
 
 in which &> and <£ are, respectively, the angular velocity and the 
 angular displacement when t is zero. 
 
 If the initial angular displacement is zero, </> is zero and 
 
 equation 57 becomes 
 
 <f> = afi/2 + <o t. (58) 
 
 As in D 2 this equation is an inconvenient one to use in rela- 
 tion to the apparatus at hand, but from it an equation may be 
 obtained giving the angle turned through during any interval 
 of time. The angle turned through during the ^th second 
 
 = o) o -a/2+ctt. (59) 
 
 The above equation is of the first degree, so that if successive 
 equal time intervals be plotted as abscissas and corresponding 
 values of 6 be plotted as ordinates, 
 a straight line will result. From 
 the curve the angular acceleration 
 may be found and also the com- 
 plete time during which the an- 
 gular acceleration has taken place. 
 The apparatus consists of a 
 heavy disk, Fig. 26, which is 
 caused to revolve with a uni- 
 formly accelerated angular motion 
 about a fixed axis. The torque to 
 
 produce the motion is obtained by a weight attached to a cord 
 passing around the drum. A stylus actuated by an electrically 
 driven tuning fork traces a sinuous line on the broad edge of 
 
80 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 / 
 the disk, which has been previously coated with a mixture of 
 
 whiting and alcohol. The fork is attached to a support which 
 may be moved so that the sinuous line does not overlap as 
 the wheel turns. The disk is graduated in degrees on one 
 side. An adjusting screw permits variation in the pressure of 
 the stylus on the edge of the disk. 
 
 Place the apparatus on a table so that the cord to which the 
 weight w is attached may have a free motion as it unwinds from 
 the drum. Connect the terminals of the electromagnet of the 
 electrically driven fork in series with a suitable variable resist- 
 ance and a storage battery. Coat the edge of the disk with the 
 mixture of whiting and alcohol, being careful to get a thin, 
 smooth, uniform coating. Adjust the stylus to bear on just 
 hard enough to give a good clear tracing on the edge of the 
 disk. Suspend such a mass w by means of the cord that, when 
 it is released by hand from a height of 80 to 100 cm. from the 
 floor, it will reach the floor in from 5 to 8 seconds. When 
 the above conditions have been met, wind the cord on the drum, 
 bringing the mass w up to a known distance above the floor, 
 start the electrically driven fork, and allow the weight to drop. 
 Move the support carrying the fork in such a way as to get a 
 good sinuous curve not overlapping, continuing the curve for at 
 least one complete turn of the wheel after the mass w reaches 
 the floor, in order to determine the negative acceleration due to 
 frictional torque alone. Assuming a unit of time of 50 com- 
 plete vibrations of the fork and starting near the beginning of 
 sinuous curve mark off every 50th wave throughout the length 
 of the curve, so as to get both positive and negative angular 
 accelerations. Read the angular position of each 50th wave, 
 note the frequency of the fork, and measure the diameters of 
 the disk and drum. The difference of the angular positions 
 noted above will give the angles passed through in successive 
 time intervals. Find these intervals and plot the successive 
 differences in radians as ordinates, using as abscissas the corre- 
 sponding successive time intervals 1, 2, 3, etc. It will be found 
 
UNIFORMLY ACCELERATED MOTION. 8l 
 
 that the points lie along two straight lines, making such an 
 angle that one has a positive slope and the other a negative 
 one. (Why ?) Draw the two curves until they intersect. (Why 
 not draw a smooth curve to connect the two tangents ?) From 
 these curves, find the positive and negative angular accelera- 
 tions in terms of the unit of time used and also in radians per 
 sec 2 . (See D 2 for method of reduction.) Find also the angular 
 velocity at the instant of beginning to count time ; also at the 
 instant the mass struck the floor, in radians, and also find the 
 time that elapsed from the instant the wheel started to revolve 
 until the mass w struck the floor. 
 
 Compute the values of w and « by elimination, from two sets 
 of suitably chosen pairs of observations, and compare them with 
 the results obtained graphically. 
 
 Make a total of two sets of observations, using different 
 masses as widely different as possible, yet keeping within the 
 time limits noted above. 
 
 II. 
 
 Torque, rotational inertia, and rotational energy * 
 
 When a force acting on a body produces an acceleration, the 
 acceleration is found to be directly proportional to the force 
 acting and inversely proportional to the inertia of the body, 
 which is measured by the mass. The acceleration is in the 
 direction in which the force acts, and is uniform for a given 
 mass so long as the force remains constant. It is necessary to 
 bear in mind that if the force varies, the acceleration varies 
 directly with it and that the usual types of motion studied in a 
 laboratory course are those in which the force remains constant 
 or varies in some simple way, as in the cases of uniform circular 
 motion or simple harmonic motion. In the cases just cited the 
 forces are constant in amount but changing in direction for the 
 
 * The general statements in the E group regarding moments of inertia should be 
 studied in connection with this part of the experiment. 
 VOL. I — G 
 
82 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 first case, and in the second case the return force is proportional 
 to the displacement, whether it be linear or angular. 
 
 In part I of this experiment a constant force acting at the 
 end of a constant lever arm comprises a constant moment' or 
 torque L which produces a constant angular acceleration a. It 
 is found that the angular acceleration produced is directly pro- 
 portional to the torque applied and inversely proportional to 
 the rotational inertia of the body. The rotational inertia of a 
 body depends upon the distribution of the material of the body 
 with respect to the axis about which the body rotates. The 
 rotational inertia of any body with respect to an axis is called 
 the Moment of Inertia, K, with respect to that axis, and it may 
 be shown that the rotational inertia K is equal to the summation 
 of products of the linear inertia or mass of each small element 
 of the body and the square of the distance of the element from 
 the given axis; i.e. 
 
 K = l.^mt 2 = I dmr*. (60) 
 
 From the relations- indicated above, 
 
 a = L/K or L = aK. (61) 
 
 The work done when a constant torque twists through an angle 
 6 is equal to the product of force times distance or 
 
 W= LB. (62) 
 
 The kinetic energy of a rotating body may be shown to be 
 
 E K =\K«? (63) 
 
 in a manner analogous to finding the kinetic energy of trans- 
 lation, and is equal to the work done on the body to give it the 
 given angular velocity. 
 
 It is the object of part II of this experiment to show (1) that 
 the loss of potential energy of the mass dropping to the floor 
 from its maximum height is equal to the total kinetic energy 
 gained by the system, taking account of the work done against 
 friction; and (2) that the resultant torque (taking account of 
 the frictional torque) is equal to the product of the rotational 
 inertia of the system and the angular acceleration. 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 83 
 
 The data taken in part I of the experiment together with the 
 mass of the disk and the vertical distance the mass drops are 
 sufficient to show the above relations. 
 
 GROUP E: MOMENT OF INERTIA AND SIMPLE HARMONIC 
 
 MOTION. 
 
 (E) General statements ; (Ej) The physical pendulum ; (E 2 ) 
 Kater' s pendulum ; (E 3 ) Relations between the time of vibra- 
 tion and the position of the knife-edges in a uniform cylin- 
 drical pendulum. 
 
 E. General statements concerning the moment of inertia 
 and simple harmonic motion. 
 
 The moment of inertia of a body, with respect to a right line 
 taken as an axis, is the sum of the products of each element of 
 ■ mass by the square of its distance from the axis. If K a is the 
 moment of inertia of any body with respect to the axis a, dM 
 any element of mass, and r its perpendicular distance from the 
 axis a, we have 
 
 K a =fr*dM, (64) 
 
 in which the integral is extended to every element of mass of 
 the body. 
 
 Moment of inertia bears the same relation to motion of rota- 
 tion that mass bears to motion of translation. The following 
 dynamical relations may readily be derived from definitions and 
 Newton's second law of motion. 
 
 J Momentum = mass x velocity. 
 
 [Moment of momentum = moment of inertia x ang. vel. 
 
 J Resultant force = mass x acceleration. 
 
 I Resultant moment = moment of inertia x ang. ace. 
 
 J Kinetic energy = £ mass X (velocity) 2 . 
 
 I Kinetic energy = J moment of inertia x (ang. vel.) 2 . 
 The moment of inertia of a body with respect to any axis is 
 equal to the moment of inertia of the same body with respect to 
 
8 4 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 a parallel axis through the center 
 of gravity, plus the mass of the 
 body multiplied by the square of 
 the distance between the two axes. 
 Let dM be an element of mass 
 whose co-ordinates with respect 
 to an axis through the center of 
 gravity are x, y. Let c be an axis 
 parallel to the axis through the 
 center of gravity (Fig. 27) at dis- 
 Let K and K^ 
 
 tance a from it. 
 
 be the moments of inertia with 
 Then we shall have 
 
 Fig. 27. 
 
 respect to the two axes. 
 
 K c = f[(a+xf+y*]dM=a*fdM+f(x*+y*)dM+2 afxdM, or 
 K e = M&+K . (65) 
 
 The* term I xdM is zero, for the origin is at the center of 
 
 gravity of the body ; this means that the sum of the positive 
 products xdM is just equal to the sum of the negative products 
 -xdM. 
 
 Moment of inertia of an infinitely thin rod about an axis per- 
 pendicular to its length. 
 
 Let L be the length of the rod, 5 its cross section, 8 its den- 
 sity. If dx is the length of the element of mass, we shall have 
 
 zzr 
 
 dx 
 
 L-h 
 
 Fig. 28. 
 
 dM= BSdx. If x (Fig. 28) is the distance of the element of 
 mass from the axis a, and h is the distance of the axis from 
 either end of the rod, we shall have 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 85 
 
 K. 
 
 fL-h 
 
 a = SS) x 2 dx. 
 
 If M is the mass of the whole rod, this reduces to 
 K a = M\ — - kL + h*\ 
 
 (66) 
 
 (67) 
 
 Two particular cases are of especial, interest ; namely, when 
 
 h = o and when h = — . 
 2 
 
 Moment of inertia of a cylinder about its axis of figure. 
 
 Let L be the length, a the radius, and 8 the density of the 
 cylinder. Take as element of mass an indefinitely thin, hollow 
 cylinder, concentric with the axis, of radius r and thickness dr. 
 
 We shall then have dM = 2 vrLBdr. Therefore we have 
 
 K = 2 ttBL Cr 3 dr = M-, (68 ) 
 
 •^o 2 
 
 in which M is the mass of the whole cylinder. 
 
 Moment of inertia of a circular lamina about any diameter. 
 
 Let a be the radius of the lamina, e its thickness, and 8 its 
 density. Taking the element of mass in polar co-ordinates, we 
 
 have dM=eSrd0dr. 
 
 As the distance of the element of mass from the diameter is 
 
 r sin 6 (Fig. 29), we have 
 
 K =2 eSf" [sinW^yWr] = M-. 
 
 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. 
 30) the distance of the axis from one 
 end of the cylinder. Taking as ele- 
 ment of mass,a lamina of thickness dx, 
 
 we have 
 
 dM=ira*hdx. 
 
 (69) 
 
 Fig. 29. 
 
86 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From equations 65 and 69 we have for the moment of inertia of 
 this lamina, with respect to the axis a, 
 
 dK a = x*dM + -dM.- (70) 
 
 x 2 dx-\ I dx, 
 
 ■h 4 J -h 
 
 = M[—-AL+k 2 + -\ (71) 
 
 4 
 and K. 
 
 Two particular cases are of especial importance ; namely, when 
 
 h = o and when h = | Z, 
 
 n n?r 
 
 da; 
 
 Fig. 30. 
 
 If h be given the value \ L, then 
 
 !].* (72) 
 
 ^0 = ^ 
 
 .12 4- 
 
 The value of K from equation 72 may be substituted in 65 to 
 find K c . 
 
 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, 
 
 x^Acostpit+to)], . 
 
 x=Asm\_p(t + t )-], K/5) 
 
 in which A, p, and t 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 
 
 * A more elegant proof of this equation may be found in Shearer's Problems In 
 Physics, problem 356. 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 87 
 
 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 73. When 
 
 2 7T 
 
 the time t has increased to the value — , the displacement x is 
 
 P 
 exactly equal to what it was at the instant t = o. Moreover, at 
 any time, t 2 , the displacement has the same value that it had at 
 
 • • 2 7T7L 2 77" 
 
 the time t lt if t 2 — t 1 = The constant interval of time — » 
 
 P P 
 
 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 r or T. 
 
 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. 31) 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/ 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 
 
 Fig. 31. 
 
 any time, t, we shall have * 
 
 x = A cos /8 = A cospt. (74) 
 
 The distance, x, is the displacement, and A the amplitude, of 
 the " '^ple harmonic motion. 
 
 * 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. 
 
88 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 If the point P moves through the distance dx in the time dt, 
 we have, from the definition of velocity, 
 
 dx 
 
 v = ^=-pAsinpt. (75) 
 
 If the velocity of the point P changes by the amount dv in 
 the time dt, we have, from the definition of acceleration, 
 
 a = ^-= - p 2 A cos pt. (76) 
 
 dt 
 
 Substituting for A cos pt its value from equation 74, we have 
 
 a=-fx. (77) 
 
 Equation 77 shows that the acceleration of a point having 
 simple harmonic motion 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 77 by the mass of the 
 
 moving point, and remembering the dynamical equation F. = Ma, 
 
 we have 
 
 F= Ma=- Mfx. .(78) 
 
 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 body will have simple harmonic motion. 
 
 From the equations 74, 75, and 76, it follows that the dis- 
 placements, velocities, and accelerations of the point P Vgin to 
 
 2 7T 
 
 repeat themselves when t has increased from o to Thib 
 
 constant value of the time — is the period of the simple har- 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 89 
 
 monic motion ; and is obviously the same as the time required 
 for the point iVto revolve about the auxiliary circle. 
 
 Substituting T for^^c? equations 74, 75, and 76 now become 
 
 (79) 
 
 (80) 
 
 (81) 
 
 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, 4 T, ■■-, we have 
 
 (82) 
 
 X 
 
 = 
 
 A 
 
 cos 
 
 T ' 
 
 
 V 
 
 = 
 
 — 
 
 2 7T 
 
 T 
 
 A sin 
 
 
 a 
 
 = 
 
 - 
 
 4 7T 
 
 J~1 
 
 A cos — ► 
 T 
 
 v,= T 2 -fA=T P A. 
 
 Simple harmonic motion of rotation. 
 
 Let M be a body oscillating about an axis O (Fig. 32) per- 
 pendicular to the plane of the paper. The line OA, fixed in 
 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 t, 
 
 <p = 8 cos pt, (83) 
 
 8 and / being constants. 
 
 If dcj> be the angle turned 
 through during the time dt, we 
 shall have, from the definition of 
 angular velocity, 
 
 (84) 
 
 , = W=-j,8sinpt. 
 dt 
 
 Fig. 32. 
 
90 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 If dm is the change of angular velocity in the time dt, we 
 shall have, from the definition of angular acceleration, 
 
 a = ~-=-fhcospt. (85) 
 
 Substituting the value of S cos ft from 83, we have 
 
 «=-/»*, (86) 
 
 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 86 by the moment of 
 inertia of the rotating body with respect to the axis of rotation, 
 we have the torque 
 
 L = K*=-Kf§. (87) 
 
 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 rotation 
 is proportional to the angular displacement from a fixed posi- 
 tion, 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 — has elapsed. 
 
 P 
 This constant time is the period of the simple harmonic motion, 
 and, calling it T, equations 83, 84, and 85 become 
 
 <f> = 8cos^t, (88). 
 
 2 7T s . 2 7T . /ON 
 
 o> = -— Ssm— -t, (89) 
 
 4^5 27T. , . 
 
 «=-^ ¥ 8cos — t. (90) 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 91 
 
 If w be the angular velocity with which the body passes its 
 mid-position, we have, in a manner similar to equation 82, 
 
 *>0 = ^S. (91) 
 
 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 over- 
 tones. 
 
 (3) The bob of a simple pendulum, or any point in a com- 
 pound 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 cer- 
 tain retarding forces due to friction, imperfect elasticity, or in- 
 duced 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 har- 
 monic motion is the quantity to be used, a correction for " damp- 
 ing" must generally be introduced.* 
 
 * See Nichols, The Galvanometer, Lecture 3 ; also Stewart and Gee, vol. 2, p. 364 
 et seq. ; and Exp. U2. 
 
92 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Experiment E x . 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 pen- 
 dulum is small enough to be negligible, we are justified in equat- 
 ing 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 \ Kja 1 . Since the 
 pendulum has simple harmonic motion, the angular velocity at 
 
 the mid-position will be ——• (See equation 51.) 
 
 . „ _27r2S2 
 
 The potential energy at the highest point is equal to the work 
 required to turn the pendulum through the angle 8 from the 
 lowest point. This work is equal to the average moment multi- 
 plied by the angular distance moved, or, 
 
 2 
 
 Since E K = E P , we have 
 
 £*.«*&*. (92) 
 
 * Proof of the equation for the periodic time of the physical pendulum from the 
 equation of motion. 
 
 Let Z = torque. 
 
 Then L = K* = Kjf = K«<?±, (93) 
 
 in which K is the moment of inertia and u the angular acceleration. (Franklin and 
 MacNutt, Elements of Mechanics, p. 131.) 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 93 
 
 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 gravity 
 from the axis of suspension, and g the acceleration of gravity. 
 
 In the case of the physical pendulum the equation for the return torque may be 
 
 written 
 
 L = — MgR sin 0, (94) 
 
 or, when 4> is so small that for sin <p may be written the angle in radians, 
 
 L=-MgR<t>; (95) 
 
 combining (93) and (95) and putting —S— = C, a constant, 
 
 By multiplying the last equation by 2 -^ and integrating, it becomes 
 
 at 
 
 (*)' = - C*+C (97) 
 
 in which C is the constant of integration. \ 
 
 When — = o, then <p is a maximum, <j> m . \ \\\>- 
 
 dt Vr p^ 
 
 By putting -2- = o, and solving for C, C/tf^X 
 
 C = C4> V . ' 
 
 ,.(*) 2 = C( ^-«, ^' 
 
 >y 
 
 A 
 
 (98) 
 
 from which ^f = C*(0 2 m - 2 )^ 
 
 at 
 
 ',V 
 
 (0 2 ro - * 2 )* 
 
 C^rfZ. (99) 
 
 Integrating equation 99, 
 
 sin-i -1 = cW C". (100) 
 
 x 0m 
 
 Taking * = o when <p = o, the new constant of integration C" is shown to be equal to 
 
 zero. 
 
 T 
 When t = — , <)> = <p m , 
 
 4 
 
 then sln- 1 -^ = C* — = sin" 1 1. (101) 
 
 0™ 4 
 
 But the angle whose sign is 1 is — • 
 ' 2 
 
 ... r=ci^. (102) 
 
 \ 
 
94 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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. 33), 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. (See A 6 II.) 
 
 (3) From the result compute the period of the pendulum to 
 four significant figures in the manner explained in A 6 II. An 
 
 ordinary watch or clock may be used 
 for determining 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 de- 
 termined, 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. 
 
 (5) Change the position of the knife-edges a few centimeters 
 and make another complete set of observations for periodic 
 time. Determine the position of the knife-edges as before. 
 
 ? 
 
 4 
 
 TO 
 
 Fig. 33. 
 
 From, equation 102, 
 
 
 
 7-2 _4* 2 . 
 C 
 
 But 
 
 c _MgR 
 
 
 MgR 
 
 (103) 
 
 The same method may be applied to find the expression for the periodic time of 
 the torsional pendulum, F<l, or of the vibrating magnet, Q2, using the appropriate 
 initial equations. 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 95 
 
 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 
 the knife-edges without the bar is approximately the same as 
 the period of the bar pendulum. 
 
 The observations and computations for determining the peri- 
 odic time are to be tabulated as shown in the example in A 5 II, 
 and the dimensions of the pendulum and results as follows : 
 
 Gravity by the Physical Pendulum 
 
 Dimensions of Pendulum and Results 
 
 Distance of axis from upper end of bar, 3.5 cm. 
 
 Length of bar, Z = 159.6 cm. 
 
 Diameter of bar, 2 r = 1.6 cm. 
 
 Distance of axis from center of gravity of bar, ft = 76.3 cm. 
 
 Moment of inertia, K a = 7945 M. 
 
 Periodic time, T = 2.047 
 Computed value of gravity, g = 981 + cm. per sec. per sec. 
 
 Most careful determination for Cornell laboratory, 980.28 
 
 Addenda to the report : 
 
 (1) Derive all formulas used and show that there is justifica- 
 tion for assuming S. H. M. in this case. 
 
 (2) Derive the formula for the moment of inertia of the 
 pendulum about its axis of suspension. Show that K a = K 
 + MR*. 
 
 (3) 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. 
 
96 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 (4) Explain what is meant by the statement, The force of 
 gravity is g dynes. 
 
 (5) 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 
 
 4-^Vo + MR*)= MgR v (104) 
 
 J i 
 
 in which K 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 65 and 92.) 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 equa- 
 tion will be obtained similar to the above, except that there 
 will be new values of R and T. Between these two equations 
 R~ may be eliminated and g determined. To this end we substi- 
 tute / for R 1 + R 2 in the equation derived by the elimination of 
 K 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 re- 
 versed, 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 value of 
 
 * See Franklin and MacNutt, Elements of Mechanics, Theory of the Reversion 
 Pendulum, pp. 142-144. 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 97 
 
 Ej2? 
 
 gravity. The experiment consists, therefore, in adjusting the 
 positions^dL the two knife-edges by trial until the time of vibra- 
 tion abojBPne 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. 34).* 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 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 (40-50). 
 
 (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 dis- 
 tance 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 
 
 * 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. Es, and Fig. 35. 
 
 vol. 1 — H 
 
98 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 before. The period should now be nearly the same as at 
 first. If the two periods differ, one or both of thadjwfe-edges 
 should be shifted until the time of vibration is ver^^Bisely the 
 same with either suspension. 
 
 (4) The final determination of the time of vibration must be 
 made very carefully, by the method outlined in A 5 II, making 
 computations to the fourth significant figure. 
 
 (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. 
 
 If the values of the periodic time obtained for the two posi- 
 tions of the pendulum vary by more than .2 per cent, the follow- 
 ing equation, which may be obtained from two equations of the 
 form of 104, is to be used : * 
 
 4^1'-^,'). (I05) 
 
 R x and R 2 may be obtained with sufficient accuracy by bal- 
 ancing the Kater's pendulum on a horizontal bar to determine 
 the position of the center of gravity, and measuring the dis- 
 tances from it to the knife-edges. 
 
 Tabulate data and results as in A 5 II and E x . 
 
 Addenda : 
 
 Answer addenda 1, 2, 3, and 4 of Experiment E x . 
 
 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 the method of A 5 II, making 
 readings as directed in the note under that heading. Then shift 
 
 * Franklin and MacNutt, Ibid. 
 
MOMENT OF INERTIA AND SIMPLE HARMONIC MOTION. 99 
 
 the knife-edges down the bar five or six 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. 
 Seven or eight 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 determining the 
 time of vibration, a stop-watch will be found of considerable 
 
 3.2 
 
 E 
 
 O 2-8 
 
 5 
 
 O 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '" 10 20 30 40 50 60 70 SO 90 
 
 DISTANCE OF KNIFE-EDGE FROM CENTER OF PENDULUM 
 
 Fig. 35. 
 
 assistance, but a good watch whose " rate " is known will give 
 satisfactory results. From the data obtained, plot a curve, simi- 
 lar to that given in Fig. 35, using for abscissas the distances 
 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- 35- 
 
I0O 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Relation of Periodic Time to Position of Knife-Edges. 
 
 Distance 
 from Center 
 
 No. of 
 Transit to 
 
 Time. 
 
 Duration of 
 100 
 
 Periodic 
 Time. 
 
 Other Data and Results. 
 
 of Gravity, 
 
 Right. 
 
 
 Oscillations. 
 
 
 
 
 hr. min. sec. 
 
 
 
 
 
 I 
 
 2 16 50 
 
 
 
 Length of bat = 183 cm. 
 
 90 
 
 101 
 
 20 30 
 
 220 
 
 2.21 
 
 Diam. of bar = 2.5 cm. 
 
 
 20I 
 
 24 12 
 
 222 
 
 
 Equation of curve from 
 
 
 I 
 
 29 OO 
 
 
 
 theory of pendulum : 
 
 80 
 70 
 
 IOI 
 20I 
 I 
 IOI 
 20I 
 
 32 34 
 36 10 
 42 00 
 
 45 3o 
 49 00 
 
 214 
 2l6 
 
 2IO 
 2IO 
 
 2'5 
 2.10 
 
 S S 
 From pairs of points on 
 curve having equal ordi- 
 
 
 I 
 
 3 22 00 
 
 
 
 nates. 
 
 60 
 
 IOI 
 
 25 27 
 
 207 
 
 2.07 
 
 T = 2.20, g — 970 ; 
 
 55 
 
 201 
 I 
 
 IOI 
 
 28 54 
 35 0° 
 38 25 
 
 207 
 205 
 
 2.05 
 
 T = 2.16, ^=989; 
 7'=2.I2, g=911- 
 
 
 201 
 
 41 50 
 
 205 
 
 
 
 
 1 
 
 51 40 
 
 
 
 
 5° 
 
 IOI 
 
 55 ° 
 
 206 
 
 2.06 
 
 
 
 201 
 
 58 32 
 
 206 
 
 
 
 
 1 
 
 4 6 30 
 
 
 
 
 40 
 
 IOI 
 
 10 1 
 
 211 
 
 2.IO 
 
 
 
 201 
 
 13 3° 
 
 209 
 
 
 
 
 I 
 
 17 10 
 
 
 
 
 3° 
 
 IOI 
 
 20 52^ 
 
 222J 
 
 2.225 
 
 
 
 201 
 
 24 35 
 
 222j 
 
 
 
 
 I 
 
 28 40 
 
 
 
 
 20 
 
 IOI 
 
 32 54 
 
 254 
 
 2-535 
 
 
 
 20I 
 
 37 7 
 
 253 
 
 
 
 
 I 
 
 42 40 
 
 
 
 
 10 
 
 IOI 
 
 48 24 
 
 344 
 
 3-435 
 
 
 
 20I 
 
 54 7 
 
 343 
 
 
 
 Addenda to the report: 
 
 Answer addenda 1 and 2 in Exp. E x and 
 
 (3) From any pair of points on the curve compute the 
 values of the constants of its equation, and hence determine the 
 acceleration of gravity. 
 
ELASTICITY. ioi 
 
 (4) Prove that the abscissa corresponding to the minimum 
 ordinate is equal to the radius of gyration of the pendulum with 
 respect to its center of gravity. 
 
 GROUP F: ELASTICITY. 
 
 (F^) Young's modulus by stretching; (F 2 ) Moment of torsion; 
 (F^) Young 's modulus by flexure ; (F b ) Impact. 
 
 Experiment F v Young's modulus by stretching. 
 
 Elasticity of tension is denned as the ratio of force applied to 
 extension produced. If the elastic limit has not been reached, 
 the extension of a weighted rod or wire is proportional to the 
 force applied, and to the length ; and is inversely proportional 
 to the cross-section, or 
 
 l=E^l, (106) 
 
 in which E is the coefficient of elasticity. From this equation 
 it is seen that E is the increase in length produced by unit force 
 applied to a rod of unit length and unit cross-section. 
 
 Young's modulus is defined as the reciprocal of the coefficient 
 of elasticity. Calling this modulus M, we have 
 
 "'%! (io7 > 
 
 Young's modulus may be computed if the quantities on the right 
 of this equation are determined in the proper units. 
 
 The experiment consists of finding Young's modulus for two 
 different wires, the elongations being measured by means of either 
 a micrometer microscope or an optical lever. In either case the 
 wires are suspended from a rigid support.* 
 
 Suspend from the end of the wire a weight which is just 
 sufficient to take out the kinks; for a wire whose diameter is 
 1 mm. a weight of from two to four kilos will be required. 
 
 * 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. 
 
102 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Microscope method. A microscope containing an eyepiece 
 micrometer 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 elongations produced by 
 successive increments in weight until ten elongations have been ■ 
 measured or the safe load limit of the wire has been reached. 
 The limit of load on the wire is to be obtained from an instructor. 
 Each increment in weight should be sufficient to cause an elon- 
 gation of three or four scale divisions. Usually i kg. applied 
 at a time will be sufficient change in load. Make all settings 
 in such a way as to eliminate back lash. 
 
 After the wire has been fully loaded the weight is to be 
 gradually reduced to the initial load and a new set of measure- 
 ments made. Note whether equal increments of tension produce 
 equal increments of length, and whether the elastic limit has 
 been passed. 
 
 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. Use a micrometer caliper to determine the diameter, 
 noting the zero error of the instrument, making measurements 
 at eight or ten places. 
 
 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. 
 
 If the first and last readings, the second and next to last, and 
 so on, be used in a manner like that explained in A 5 II, for 
 finding periodic times, an average value of the elongation per 
 unit change of load may be obtained, giving equal weight to all 
 the observations made. From these data compute Young's 
 modulus. 
 
 The results are also to be shown by plotting curves in which 
 
ELASTICITY. 103 
 
 abscissas represent forces applied, and ordinates the increments 
 in length produced. 
 
 These curves are to be discussed in the usual manner. 
 
 Optical lever method. This method of finding Young's 
 modulus differs from the microscope method only in the method 
 of determining the elongation. Computations and curves are 
 to be obtained as explained above. 
 
 The optical lever consists of a light frame supported on three 
 sharp pins and carrying a mirror which may be rotated, in a 
 plane containing the points of two of the pins, about these points 
 as an axis. In this experiment these two pins are supported on 
 a fixed plane in a groove. The third pin is supported on the 
 upper surface of a cylinder whose axis is the axis of the wire 
 under investigation, to which the cylinder is clamped. Any 
 change in tension in the wire moves the cylinder parallel to its 
 axis, rotating the optical lever about its fixed axis. If the per- 
 pendicular distance R between the movable pin and the axis of 
 rotation of the optical lever be known and the angle through 
 which the plane of the points is rotated be also known, the 
 elongation may be computed. When the angle is small, the 
 elongation / will be equal to the product of the angle 6 in radians 
 and the distance R. The distance R is to be determined by 
 pressing the points on paper and making the proper measure- 
 ments. 
 
 The angle 6 is to be determined by setting up a telescope with 
 a horizontal cross wire and a vertical scale at a convenient dis- 
 tance from the mirror (not less than a meter) in such a position 
 that the image of the scale may be seen in the telescope. The 
 axis of the telescope should be approximately horizontal. The 
 height of the telescope should be such that when half the maxi- 
 mum load is suspended from the wire, the scale number seen in 
 the telescope is about on a level with the axis of the telescope. 
 Make scale readings by means of the telescope for increasing 
 and decreasing loads as indicated under the first method. 
 Measure the distance from the mirror of the optical lever to the 
 
104 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 scale, and compute values of 6. Remember that the beam of 
 light reflected into the telescope moves through twice the angle 
 that the mirror, and consequently the plane of the pins of the 
 optical lever, moves through. 
 
 Example of Method of Finding /. 
 
 Value of S =5 cm. 
 
 Dist mirror to scale, d = 100 cm. 
 
 Mean radius of wire, r = 0.043 cm - 
 
 Length of wire under observation, L = 148 cm. 
 
 Force producing 
 
 Elongation in 
 
 Kg Wt. 
 
 Scale 
 Readings. 
 
 Differences in 
 
 Readings for 
 
 Indicated Loads. 
 
 „ rjiir. 
 
 tan 2 8 = =20 
 
 100 
 
 (approx ). 
 
 9 per 
 Kg Wt. 
 
 1= Re. 
 
 O 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 14 
 
 30.2 
 28.7 
 27-3 
 257 
 24.2 
 22.7 
 21.3 
 19.8 
 
 I4-0 = 10.4 
 12-2 = 7.4 
 IO-4 = 4.6 
 
 8-6= 1.5 
 
 O.IO4 
 
 O.074 
 
 0.046 
 
 O.O15 
 * 
 
 0.00743 
 0.00740 
 0.00767 
 0.00750 
 0.00750 
 
 °-°375 
 
 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 6 is the 
 angle, and L the moment of the elastic return force, L=L 6. 
 The constant L is called the moment of torsion, and depends 
 upon the length, diameter, and material "of the wire. 
 
 To determine the value of L , 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. 105 
 
 will have simple harmonic motion, and the vibration will be 
 isochronous. From equation 87 we will have 
 
 l 6=-k^, (108) 
 
 in which L Q is the resultant moment due to torsional displace- 
 ment through an angle 0, and — - is the angular acceleration 
 of the suspended weight. An integration of this equation gives 
 
 T 
 
 = 2iryj—, (109) 
 
 'A> 
 
 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 
 
 E K =\K** = K 2 -^t- ' (no) 
 
 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 L o ; as this varies between o and L 8, the average 
 moment is | L 8, and hence the work done and the potential 
 
 energy gained is 
 
 E P = I L 8-8. (in) 
 
 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 
 
 
 (112) 
 
 For another proof of this equation, see Exp. Ei . 
 
106 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 It is to be observed that since it is the square of T that 
 appears in the formula, an error in the determination of the 
 period will introduce a considerable error in the result. 
 
 To get the period accurately, proceed by method bf A 5 I ; but 
 the following is given as the regular method of procedure (Exp. 
 A 5 II). 
 
 Permitting the bob to rotate with an amplitude of about 15°, 
 observe the transit to the right of some point on the bob past a 
 marker of some sort on a stationary support. If the period is 
 longer than 10 seconds, observe the time of the 1st, 6th, nth, 
 16th, 21st, 26th, 31st, 36th, 41st, 46th, 51st, 56th, and 61st 
 transits, otherwise take the 1st, nth, 21st, 31st, ••• 91st transits. 
 
 If the period is the same for a large and a small amplitude, 
 then it must be true that the return torque due to the elasticity 
 of the wire is proportional to the angular displacement. To test 
 this point determine the period for an amplitude of about 90 . 
 
 Measure the diameter of the wire in eight or ten places, with 
 a micrometer caliper ; also determine its length. Find the diam- 
 eter of the cylindrical weight, making several measurements. 
 If the weight is easily detached (i.e. without unscrewing any- 
 thing), take it off and determine its mass; otherwise the mass 
 may be obtained from an instructor. 
 
 From the data taken, find Z and n from the equations 
 
 L = L Q e=:^fe = ^e, ( 1I3) 
 
 {Mm 
 
 in which n is the slide modulus of the wire, r its diameter , and 
 / its length. 
 
 From the result obtained for L 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. 
 
 Addenda. (1) Define moment of torsion, slide modulus. 
 (2) Prove that the torsional pendulum has true simple har- 
 monic motion. (The formulas for S. H. M. may be assumed.) 
 
ELASTICITY. 107 
 
 Experiment F 4 . Young's modulus by flexure.* 
 
 A bar supported by two parallel knife-edges is sub- 
 jected to a vertical force applied for convenience at a point 
 midway between the supports. Along its upper surface it 
 suffers compression and along its lower surface there is a 
 tension. The compression and tension decrease from the 
 surfaces inward to a neutral interior surface where neither 
 exists. 
 
 The amount of deflection e depends upon the distance / be- 
 tween the supports, the depth d of the bar, its breadth b, 
 the applied force/, and also on a constant k which depends on 
 the material of the bar. The relation may be expressed by the 
 following equation : 
 
 e = kfHWd*. (114) 
 
 It is the object of this experiment to determine the constant 
 k and the exponents «, /3, 7, and 8. 
 
 The apparatus consists of a heavy bed plate with sliding sup- 
 ports for the bar, an insulated carriage carrying a micrometer 
 screw with which to measure deflections, a battery, and an elec- 
 tric bell or other current detector. The deflecting force is ap- 
 plied midway between the supports and the screw adjusted to 
 a point just over the point of application of the force in order 
 to measure the deflections. 
 
 Place the bar on the supports which have been adjusted to 
 the desired distance apart. Suspend a kilogram weight from the 
 center of the bar and adjust the position of the screw. Connect 
 the battery to the current detector, the screw carriage, and the 
 bed plate. Adjust the screw until it is in contact with the sup- 
 port carrying the weight as indicated by the detector. Use this 
 reading as the reference point. Apply enough load to deflect 
 the center of the bar at least one complete turn of the screw 
 and take another reading. Then proceed as before until six 
 readings have been made. In this case the only variables have 
 
 * See Ferry and Jones, Practical Physics, vol. 1 , Exp. 27. 
 
108 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 been the applied force and the total deflections. The above 
 equation may therefore be written 
 
 < = C,f*, (US) 
 
 in which C f is a new constant including all the other constants 
 for this case. The expression may be written 
 
 log ^= log C f + « log/ (116) 
 
 Such an expression may be written for each observation as 
 
 log e z = log C f + a log / 3 . 
 
 log e s = log C f + a log/ 6 . 
 
 Taking the difference between two equations of the above form, 
 
 we have 
 
 log e 6 - log e 3 = a (log/ 6 - log / 3 ), ( 1 1 ;) 
 
 from which a may be readily computed. Use the observations 
 in pairs to compute three values of a, taking observations I and 
 4, 2 and 5, 3 and 6. 
 
 Now determine the mean deflection per kg. for five other 
 lengths of bar in the manner indicated above. In this case the 
 only variables are the deflections and length, and therefore we 
 may write expressions like the following : 
 
 e=C{* (118) 
 
 or 
 
 log* = logC,+ £log/, ( II9 ) 
 
 from which /8 may be determined. 
 
 Arrange the bars in a series of variable widths and constant 
 depths, and then in a second series of variable depths and con- 
 stant widths and find the deflections per kg.,* keeping the value 
 of / constant in either series. Compute the values of 7 and 8 
 from these series in a manner similar to that already outlined. 
 
 * Do not overload the bars. Do not use values of / less than 70 cm. In 
 finding y8 vary / by about equal amounts and not less than 10 cm. Tabulate 
 data carefully so that it may be easily understood. 
 
ELASTICITY. 109 
 
 After a, /8, 7, and 8 have been determined, make four inde- 
 pendent determinations of the constant k, one in each series. 
 Express k rationally and compare with values given in tables. 
 
 Plot a curve for each series, using values of log e as ordinates 
 and corresponding values of the logs of other variables as 
 ' abscissas. Discuss the forms of these curves and their con- 
 stants and get the values of a, /3, 7, and 8 from them. 
 
 Experiment F 5 . Elastic and inelastic impact. 
 Impulse, momentum, and coefficient of restitution. 
 
 When a body is moving, it is said to have a certain quantity 
 
 of motion or momentum. The momentum may be expressed 
 
 quantitatively as equal to the mass of the body multiplied by its 
 
 velocity, 
 
 M=mv. (120) 
 
 If the velocity of the body be changed, its momentum is 
 changed and the rate of change of momentum is .equal to the 
 force applied to produce it. 
 
 dM dv £. /.„.\ 
 
 ■*—*-* (I2I) 
 
 Equation 121 may be written 
 
 Fdt = mdv. ( I22 ) 
 
 Integrating equation 122 gives 
 
 j Fdt = J mdv — m(u — v), (123) 
 
 in which u and v are the initial and final velocities, respectively. 
 
 The first member of the equation, I Fdt, is called an impulse. 
 
 If F is constant, then the impulse is Ft. Usually the force is 
 a variable, and the time is not accurately and often not even 
 approximately measurable. The impulse is therefore measured 
 in terms of the change of momentum as is shown in equation 
 123. The case studied will be that of two bodies having direct 
 central impact ;■ that is, along the line joining their centers of 
 
no JUNIOR COURSE IN GENERAL PHYSICS. 
 
 mass. When they collide, in general they will have the same 
 velocity when the period of compression is just past and that 
 of restitution is about to begin. Call this common velocity w. 
 Let the masses be m x and m 2 , their velocities before impact 
 u x and u 2 , and those after impact v x and v 2 , respectively. 
 
 At any instant the elastic forces acting on the two bodies 
 during impact are equal and opposite (Newton's third law of 
 motion); then 
 
 A" -A- ( I2 4) 
 
 _r dv dv' , /,„,A 
 
 fl=mi d~t = ~ m% ~di = ~^ v ^ 4) 
 
 or m x dv = — m i dv'. (i 2 S) 
 Integrating over the whole time of impact, 
 
 m x (u x -v x )=-m 2 (u 2 -v 2 ), (126) 
 
 from which m x u x + tn 2 u 2 — m x v x + m 2 v 2 , ( I2 7) 
 
 which shows that the sums of the momenta before and after 
 impact are equal, that is, that 
 
 "Zmv = constant. ( I2 8) 
 
 Let the impact be considered in two parts, namely, impulse 
 R x from contact to maximum compression when w is the velocity 
 of both bodies, and impulse R 2 from maximum compression to 
 separation.' If the bodies were perfectly elastic, it would be 
 expected that R 2 would equal R v but as a matter of fact R- 2 is 
 less than R v The ratio of R 2 /Ri is constant so long as the 
 impact produces no permanent deformation. This constant is 
 called the coefficient of restitution. 
 
 RJR x = e. (129) 
 
 In the first impulse (^j), remembering that "2mv = C, that 
 the bodies have equal velocities w at the end of the impulse, 
 and that the impulse is measured by the change in momentum, 
 
 R x = m x {u y — w)= — m 2 (u 2 — w). ( I 3°) 
 
 In a similar manner 
 
 R 2 = m x {w — v x )= — m 2 (w — v % ), (131) 
 
from which 
 
 ELASTICITY. in 
 
 A = e _ mjjw-v x ) _ - w 2 (w - z> 3 ) 
 
 ^j w l( a l — W) — 7« 2 (^2 — W)' 
 
 w — v, w — v 
 
 e = *• == * 
 
 ■ W 
 
 and - eu x — ew=w — v x , ( I 3 2 ) 
 
 eu 2 — ew = w — v % . 
 Subtracting the second of these last two equations from the first, 
 we have e (u 1 -u 2 )=v 2 -v 1 (133) 
 
 or e = v -*=^- ( I33 ') 
 
 1 2 
 
 For perfectly inelastic bodies e = o, from which it is seen that 
 v 2 =v x ; that is, both bodies have the same velocity after impact. 
 
 Although the momentum after impact is equal to momentum 
 before impact there is a loss of kinetic energy during impact, 
 due to internal work on the bodies, the energy of motion chang- 
 ing into heat energy. 
 
 The loss in kinetic energy E K ' is given by the following 
 equation : 
 
 Ek = I ™ x ii? + \ m 2 u 2 2 - \ m x v? - \ m 2 v 2 2 ( 1 34) 
 
 = h m \( n \ - v i) + k ™iW ~ V) ( J 35) 
 
 = \ m x {n x — v 1 )(u 1 + vj)+ I m 2 (u 2 — v 2 )(u 2 + v 2 ). (136) 
 
 Now R t + R 2 = R = total impact, and adding ( 130) and( 131), 
 
 R = m x (u 1 -v l )=-m 2 {u 2 -v 2 ). (137) 
 
 Substituting in (136), we have 
 
 E^ = l : R(u l +v l )-\R(u % + v 2 ) (138; 
 
 = \R[_{u 1 -u 2 )-(v 2 -v 1 )]. (138') 
 
 Substituting in (138') the value of v 2 — v t from (133) gives 
 
 E K ' = J £(«!-*,)( !•-*). (139) 
 
 If the values of v x and v 2 obtained from (137) be substituted 
 
 in (133). then 
 
 * = -M-(«i-0(i+4 (140) 
 
 m \ + m % 
 
ii2 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Substituting (140) in (139), 
 
 ^' = ^(«i-» 2 ) 2 ('-4 040 
 
 2 7«j -+■ m<i 
 
 The percentage loss / of kinetic energy is (141) divided by 
 
 the initial kinetic energy 
 
 FJ 
 /= ^ (142) 
 
 '2 
 
 \ m x u^ + \ mtfif 
 
 In the case of inelastic impact when m 2 is initially at rest 
 (the case in the first part of the experiment), « 2 an< i e are zero 
 and (142) reduces to 
 
 /=—?—; (143) 
 
 when #z 2 is stationary for elastic impact but free to move, « 2 = o 
 and L = - m i m i uM 1 - e 2 ), ( 144) 
 
 /=. Z 
 
 J »*i*i a 
 
 #Z, 
 
 -(l-^ 2 )- (145) 
 
 »«! + #z 2 
 
 If »* 2 is not only at rest, but is very large compared to m v 
 
 then m 
 
 2 — = 1, approximately, 
 
 m 1 + m % 
 
 and /= 1 — e 2 . (146) 
 
 The following experiment is divided into two parts, I dealing 
 with inelastic impact and testing equations 128 and 143; and 
 II dealing with elastic impact and testing equations 128, 145, 
 and 146. 
 
 I. 
 
 Inelastic impact. 
 
 A suspended lead cylinder is struck by a suspended lead ball, 
 the contact surface being covered by a small amount of soft 
 wax. After impact they travel along together as one body, e 
 being zero. The cylinder pushes a light index along a metal 
 arc graduated in degrees. 
 
ELASTICITY. 
 
 "3 
 
 Adjust the suspensions until after impact the two masses 
 move along the arc without wobbling. Make some preliminary 
 trials to find the position of the index at the end of the path. 
 Place the index nearly out to this point, draw the ball back to 
 some known position, and let it fall along the arc. The ball 
 may be held in place by a thread. When 
 ready to make a reading, burn the thread. 
 This method reduces wobbling. 
 
 Make the following readings : the 
 position a of the center of the ball before 
 falling ; its position b when vertically 
 beneath the support, the cylinder having 
 been drawn out of the way, and its position 
 g when just in contact with the cylinder 
 in its position of rest; the position c of 
 the index when just in contact with the 
 forward side of the cylinder when free, its position d when 
 the ball and cylinder are in equilibrium and in contact, and the 
 reading/ of the index at the end of the swing. 
 
 The arc cd represents the difference between the position of 
 the center of gravity of the cylinder alone and the new center 
 of gravity of the system composed of the ball and cylinder. 
 Twice this distance, ce, is that which is not effective in carrying 
 the index to/ - . 
 
 The effective fall of the ball is that measured along J:he arc 
 ab — bg, and the resultant rise is measured by the arc df— de. Let 
 the arcs ab, df, de, or its equal dc, and bg subtend the angles <J3, 
 6, a, and /3, respectively. Then the velocity of the ball im- 
 mediately before impact and that of the system at the beginning 
 of the resultant rise may be expressed in terms of c£, 0, a, /3, and 
 the radius r. Based on the equation 
 
 v = V2gJ, (i47) 
 
 we have 
 and 
 
 « x = V2 gh x = V2£r(cos /3 — cos <£), 
 v % = -^2gk i =V2gr(cos a — cos 6). 
 
 VOL. I — I 
 
ii4 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Using the values of u x and v 2 thus determined, find the 
 momenta before and after impact and compare them. De- 
 termine the per cent loss of kinetic energy and compare it with 
 that computed, using equation 143. 
 
 Make three different sets of readings, using different initial 
 positions of the ball B. 
 
 II. 
 
 Elastic impact. 
 
 In the case of elastic impact two steel balls are used, and 
 two indices. One index is operated by the small ball, having a 
 longer projection with which to engage 
 the wire frame on the lower side of the 
 ball. The shorter index is operated by 
 the impinging ball. 
 
 In this case the values of u v v v and 
 w 2 are determined as in the previous 
 case. The value of « is determined by 
 holding B 1 just in contact with i? 2 in its 
 equilibrium position. Note that the 
 balls do not travel together after strik- 
 ing. With this fact in mind derive 
 formulas for ie v v v and v 2 , and from these find the values of 
 the momenta before and after impact. Find also the per 
 cent loss in kinetic energy and the value of e. 
 
 Make three determinations, two with the large sphere as B x 
 and one with the smaller sphere as B v In the latter case but 
 one index can be used since the small ball will rebound. Much 
 care is needed to make the reading of rebound in this case, so 
 make several trials. 
 
 Find the masses of all the balls and cylinders used. Measure 
 the values of r. Be very careful to take all necessary data 
 and tabulate them carefully. 
 
CHAPTER II. 
 
 GROUP G: DENSITY. 
 
 (G) General statements ; (G t ) Specific gravity of solids and 
 liquids by the specific gravity bottle ; (G 2 ) Determination 
 of density with corrections for air displacement and tem- 
 perature ; (G 3 ) Nicholson s hydrometer ; (G 4 ) Fahrenheit's 
 hydrometer ; (G 5 ) 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 
 
 D = ~- (148) 
 
 Since density is not an abstract number, its numerical value 
 in any particular case must depend upon the units used. For 
 example, the density of water in the foot-pound-second system 
 is 62J. 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 
 
 "5 
 
Ii6 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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." 
 
 A few methods for finding specific gravity are outlined in 
 experiments which follow. 
 
 Experiment G v 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 small hole through the center of this stopper leads to the 
 interior of the bottle, its object being to permit 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 computa- 
 tion of the specific gravity. 
 
 II. 
 
 Specific gravity of a solid* 
 
 Place the substance in the specific gravity bottle and deter- 
 mine 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 
 
 * The specific gravity bottle is especially useful when the solid is in the form of 
 small fragments or powder. 
 
DENSITY. 117 
 
 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 sub- 
 stances some liquid of known density must be used in which 
 the substance does not dissolve. 
 
 It sometimes happens that difficulty is met with in shaking off 
 the small bubbles of air which tend to adhere to the substance, 
 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 . 
 
 Find the specific gravity of two liquids and one solid. 
 Weighings are to be made by the method of vibrations as de- 
 scribed in Exp. A 6 II. 1 
 
 Experiment G 2 . Precise determination of density, by weigh- 
 ing in air and in water. 
 
 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 scalepan acting downwards, and the buoyant effect of 
 
 the weight of the fluid displaced by the body acting upwards. 
 
 This gives 
 
 W 1 — w x = W^ — w % . 
 
 Since weights are directly proportional to masses, we have 
 
 M 1 -m 1 =M % - m 2 , (149) 
 
 in which M 1 and M % are the masses of the bodies on the two 
 scalepans, and «z x and m 2 are the masses of the displaced fluid 
 in the two cases. Nearly always in using the balance, tn x and 
 
n8 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 7# 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 &„ then from 
 the definition of density the volume of the displaced fluid will be 
 
 -£-*• If & a is the density of the displaced air, then its mass 
 
 is M,~- If M is the mass of the counterpoise, arid B c its density, 
 
 equation 149 becomes 
 
 M.-M&=M-M^. (150) 
 
 In order to determine M, from the known mass of the counter- 
 poise, S a , S„ 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 j with all possible accuracy. If the substance is 
 suspended from the scalebeam, so as to be immersed in water 
 of density p, and is then counterpoised with the mass M', equa- 
 tion 149 becomes 
 
 M,-MP=M'—M'^- (151) 
 
 If equation 150 be divided by 151, and the resulting equation 
 solved for 8„ we shall have 
 
 In deriving equation 152 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 B a depends on the temperature, pressure, and 
 
DENSITY. 119 
 
 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, counterpoise 
 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 baro- 
 metric pressure of the atmosphere. The distilled water 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 3 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 S s is finally obtained, it must be remem- 
 bered that this is the density of the substance at the temperature 
 of the water in which it was weighed. For comparison, this 
 must be reduced to 0° by using the coefficient of cubic expansion. 
 
 Experiment G 3 . Specific gravity by Nicholson's hydrometer. 
 
 This hydrometer consists of a hollow cylinder which is made 
 to float with its 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 scalepan. 
 At the bottom another pan is provided, upon which can be 
 placed the object whose density is required. In the instrument 
 shown in Fig. 38, which is a slight modification of the hydrome- 
 ter of Nicholson, this mark consists of the point of a wire 
 which projects downward from the center of the scalepan. 
 
 * Tabulated values of p and S a will be found in Landoldt and Bornstein, in Stew- 
 art and Gee, vol. i, etc. 
 
120 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Fig. 38. 
 
 To determine the specific gravity of a solid, place the hy- 
 drometer in water, and find by trial the weight which must 
 be placed on the scalepan in order to bring some well- 
 defined mark to the surface of the water. Then place upon 
 the scalepan 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 hydrome- 
 ter. From these three weights 
 the specific gravity can be 
 computed. In case the speci- 
 men is lighter than water it 
 must be fastened in some way 
 to the bottom of the instru- 
 ment to prevent it from float- 
 ing away. The instrument may also be used in determining 
 the specific gravity of a liquid. 
 
 This form of hydrometer is not very sensitive, and therefore 
 cannot be expected to give results of great accuracy. In this 
 experiment, however, as in all specific gravity determinations, 
 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 prin- 
 ciples involved, including Archimedes' law. 
 
 Experiment G 4 . Fahrenheit's hydrometer. 
 
 This hydrometer consists of an elongated glass bulb 
 (Fig. 39), weighted at the bottom, and carrying at the top 
 a small scalepan supported by a wire sealed into the 
 bulb. 
 
 To determine the density of a liquid, first float the instrument 
 
DENSITY. 
 
 121 
 
 in distilled water and place weights on the scalepan 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 observa- 
 tion 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. 
 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 tempera- 
 ture of the water. 
 
 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 20 per cent solution, weighing 
 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 should be 
 taken. With the results obtained, plot a curve in which the 
 strengths of the solution are used as abscissas and the corre- 
 sponding densities as ordinates. 
 
 In making the various solutions use water at room tempera- 
 ture and let the solution when changed come to room tempera- 
 ture, before making readings. A Nicholson's hydrometer may 
 be used in this experiment in the same manner as the Fahren- 
 heit instrument. See Exp. G 3 for description. 
 
 Fig. 39 
 
122 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 dk 
 
 Experiment G 5 . Density of a liquid by Hare's method. 
 
 The apparatus used in this experiment consists of two verti- 
 cal tubes open below and connected above to a common tube ; 
 the latter tube is provided with stop cock (see 
 Fig. 40). The two tubes dip into separate 
 vessels, one containing distilled water, and the 
 other the liquid whose density is to be deter- 
 mined. The 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 the two columns 
 being inversely proportional to the densities 
 of the liquids used. 
 
 This may be demonstrated as follows : Let 
 a be the atmospheric 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 h, h', and 8, 8' 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 + hlg, (153) 
 
 and for the corresponding point within the second tube 
 
 a = b + h!l>g. 
 :. h:ti = l':S>. 
 
 The student is to perform either part (a) or part (b) as 
 instructed. 
 
 (a) 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 upper end of the scale. Adjust the level 
 
 Fig. 40. 
 
DENSITY. 123 
 
 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 8 or 10 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 four different liquids 
 with that of distilled water, and also two of the liquids with a 
 third. Compare the ratio of the densities thus determined with 
 those determined by comparison with distilled water. The tubes 
 should be rinsed with distilled water before and after using each 
 different liquid. 
 
 Plot curves for each set of data taken. When distilled water 
 is used as a standard of comparison, use values of the heights of 
 the water column as abscissas, and corresponding differences in 
 heights of the liquid tested and the water column as ordinates. 
 In the cases where two of the liquids are compared with a third 
 use heights of the third liquid as abscissas and corresponding 
 differences in heights of the liquids compared as ordinates. 
 Interpret the curves and obtain all the possible physical con- 
 stants from them. 
 
 (0) Make up a 20 per cent solution of the material supplied, 
 using distilled water as a solvent, and use as directed in part (a) 
 above. Repeat the experiment with five other known concen- 
 trations of the solution, diluting with the proper amounts of dis- 
 tilled water to give solutions varying by approximately 3 or 4 per 
 cent. Note the temperature in all cases. 
 
 Plot six curves, one for each concentration, using heights of 
 the distilled water column as abscissas and corresponding differ- 
 ences in solution and water heights as ordinates. 
 
 Plot one curve, using concentrations of the solutions as ab- 
 scissas and corresponding densities as ordinates. 
 
 Interpret the curves and find the possible physical constants 
 from them. 
 
124 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 GROUP H: PROPERTIES OF GASES. 
 
 (H x ) Verification of Boyle s law ; (H 2 ) Comparison of the cistern 
 barometer and the siphon barometer ; (H 3 ) Pressure of satu- 
 rated vapors. 
 
 Experiment H x . Verification of Boyle's law. 
 
 The apparatus consists of two glass tubes mounted vertically 
 upon some suitable support and connected at the bottom. One 
 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, or a 
 common scale may be provided. 
 
 I. 
 
 To test the law for pressures greater than one atmosphere. 
 
 For this purpose the closed tube should be considerably 
 shorter than the other, if the tubes are fixed in position 
 and the level of the mercury is adjusted by means of 
 a cistern whose height may be varied. If both tubes 
 are adjustable in height and connected by a rubber 
 tubing, as shown in Fig. 41, they may be of the same 
 length. In this experiment the tubes are not sup- 
 posed to be graduated in cubic centimeters, but the 
 cross section of the closed tube is supposed to be known 
 
 Iso that volumes may be computed. Near the top of the 
 closed tube, just below where the contraction begins, 
 there should be a reference mark. Above this mark 
 the volume may be unknown, but it is constant. The 
 volume below this mark above the mercury surface may 
 be computed from the readings and known cross 
 section. 
 
 Adjust the positions of the two tubes well down on 
 their supporting rods with both tubes open to the air, 
 so that the mercury is near the bottom of the tube that 
 is to be closed and near the top of the open tube. 
 
PROPERTIES OF GASES. 125 
 
 Close the stop cock on the closed tube and read the positions 
 of the mercury surfaces and the mark near the top of the 
 closed tube on the common vertical scale. Raise the open 
 tube through about one tenth of its range and make another 
 set of readings on the mercury surfaces and the reference 
 mark on the closed tube, waiting sufficient time for the 
 temperature to become constant. One or two minutes will 
 usually be sufficient. If the closed tube is not to be moved, 
 the first reading made on the reference mark near its top will 
 be sufficient for this part of the experiment. Continue, to 
 make reading in the manner just described until ten readings 
 have been made. Then gradually reduce the pressure on the 
 air within the closed tube to atmospheric pressure by lowering 
 the open tube, going down in five steps. 
 
 In each of the observations above, the total pressure to which 
 the air in the closed tube is subjected is measured by the dif- 
 ference in level between the two columns of mercury plus the 
 pressure of the atmosphere. In tabulating the results each dif- 
 ference 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 
 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. If / is the difference in 
 height of the mercury in the closed tube and the reference 
 mark near its closed end, and V the unknown volume of that 
 portion of the tube above that mark, then the total volume 
 is V= V Q + IA, in which A is the cross section. If Boyle's law 
 is true, we should have PV=K; or P(V Q + IA)=K. With 
 the exception of P and /, all the quantities in this equation are 
 constant. ' If a curve is plotted with the observed values of / as 
 abscissas and the corresponding values of 1 -s- 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 
 
126 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 this equation compute the values of V and K. Reduce both 
 quantities to C. G. S. units. Apply the method of least squares 
 to the ten observations made for increasing pressures. It is 
 necessary to use great care in the computations, carrying them 
 to four significant figures. The cross section A is to be obtained 
 from an instructor. The atmospheric pressure is to be read 
 by the student on a barometer in the laboratory, and the 
 temperature of the air noted. 
 
 II. 
 
 To test the law for pressures less than one atmosphere. 
 
 Adjust the positions of the two tubes near the tops of their 
 rods, the stop cock of the closed tube being open, so that the 
 mercury about half fills the closed tube and is as low as possible 
 in the open tube. Then close the stop cock and make readings 
 on the mark near the top of the closed tube and the mercury sur- 
 faces. Then lower the closed tube through about one fifth of 
 its range and proceed to make readings as in part I. Make 
 five readings down and five more on the return. Make compu- 
 tations as in case one, omitting the work using least squares. 
 In general the values of K obtained in the two cases will not 
 agree. Explain this difference. 
 
 If more accurate results are desired, the short tube must be 
 calibrated by means of mercury,* and the air used must be 
 carefully 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 k 
 
 PV 
 in the equation = k, 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. 
 
PROPERTIES OF GASES. 
 
 127 
 
 Experiment H 2 . Comparison of a cistern barometer and a 
 siphon barometer. 
 
 The apparatus for this experiment consists of two barometers 
 of the types indicated above, an accurate vertical scale divided 
 to millimeters, and a reading telescope mounted upon a vertical 
 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 ap- 
 paratus which is shown in Fig. 42 
 is as follows : 
 
 The two barometers are 
 mounted side by side upon a sub- 
 stantial block. At points a and b, 
 situated in positions equally dis- 
 tant to the right of barometer B x 
 and to the left of barometer Z? 2 > 
 are pins from which the scale 5 
 may be suspended. The latter 
 must be adjusted beforehand so 
 that 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 drawtube 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 
 
 Fig. 42. 
 
128 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 barometer B x (siphon), and the distance from the cap of the 
 meniscus to the fixed cross-hair in the eyepiece 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 focusing 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 
 i? 2 (cistern). 
 
 (2) Scale hanging at the left. 
 
 (e) The various operations described as a, b, c, and d are 
 carefully repeated. 
 
 (/) The telescope is shifted to a position opposite the cistern 
 of barometer i? 2 , and the level of the mercury in the same is 
 obtained by readings similar to those described under a, b> 
 and c. 
 
 (g) The level of the lower meniscus or barometer B 1 is de- 
 termined as above. 
 
 (3) Scale hanging at the right. 
 
 (h) The levels of cistern and lower meniscus are redeter- 
 mined as above. 
 
 (4) The reading of a thermometer placed midway between the 
 two mercury columns is noted. 
 
 If the conditions indicated in the description of this experi- 
 ment are fulfilled, that is to say, if the scale hangs vertically 
 both at the right and left, and the telescope moves smoothly in 
 a nearly horizontal plane, the height of mercury column {B x and 
 B 2 respectively) will be found nearly the same, whether com- 
 
 * In case the reading telescope is not provided with a micrometer eyepiece the 
 common eyepiece should be furnished with a suitable ruling on glass, which, placed 
 in the focus, makes a very good substitute. 
 
PROPERTIES OF GASES. 129 
 
 puted from readings with scale left or scale right. Any dis- 
 crepancy approaching 0.01 cm. should indicate the advisability 
 of repeating the measurements. The height of the two mer- 
 cury columns in B x and i? 2 will, however, differ very appreci- 
 ably, even when the vacuum is good in both instruments. The 
 difference is due to depression by capillary action, which influ- 
 ences the cistern barometer only. The next step is to determine 
 whether the correction for capillarity will account for the differ- 
 ence of barometric height. 
 
 (5) To calibrate the cistern barometer for capillarity, note the 
 reading of the meniscus when the screw by means of which the 
 height of the mercury in the cistern * is adjusted, is at almost 
 its lowest position ; then add a weighed quantity of pure mer- 
 cury to the cistern sufficient to produce a rise of about one cen- 
 timeter in the surface of the contents. The meniscus will rise 
 through a distance precisely corresponding to the change of 
 level in the cistern, and in case the ratio in the cross section's be 
 not very large indeed, the change of level as compared with 
 that which would have occurred had there been no loss of mer- 
 cury from the cistern to supply the increase in the column 
 within the barometric tube, will afford a fair approximation to 
 the diameter of the latter. This determination involves the 
 measurement of the dimensions of the cistern and the compu- 
 tation of its contents per centimeter of vertical height. 
 
 In case the difference in the observed height for B x and 2? 2 is 
 not entirely accounted for by means of the correction for cap- 
 illarity (concerning which see any one of the larger treatises in 
 physics), it is probable that the vacuum in one or both barom- 
 eters is imperfect. Gross errors of filling may be detected by 
 driving the column of B 2 to the top of the tube, by means, of the 
 screw, and watching for a bubble which cannot be made to dis- 
 appear by pressure, and, in the case of the siphon barometer, 
 
 * The cistern barometer to be used in this experiment should be provided with a 
 cistern which has a flexible leather bottom, upon which a screw impinges as in the 
 Fortin barometer, giving considerable range of level. 
 vol. 1 — K 
 
130 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 reaching the same end by the direct application of pressure to 
 the open end of the tube. 
 
 To reduce the readings obtained in this experiment to absolute 
 measure,* the scale should be placed upon the dividing engine, 
 and compared with some good standard of length, or with the 
 screw itself, if the constant of the instrument is known. 
 
 Experiment H 3 . Vapor pressure of saturated vapor, f 
 
 When a vapor is in contact with its liquid in an inclosed 
 space, there will be a pressure exerted by that vapor which de- 
 pends only upon the temperature and consequently not at all 
 upon the volume which the vapor occupies. If the temperature 
 be raised, liquid will evaporate until the pressure of the vapor 
 has risen to a new value corresponding to that for the given 
 new temperature. If the temperature is lowered, then the 
 vapor pressure will be too high for the new temperature, and 
 enough vapor will be condensed to lower the vapor pressure to 
 that corresponding to the lower temperature. If at a given 
 temperature the volume be increased, the first effect will be to 
 lower the pressure, but immediately evaporation begins to take 
 place and will continue to do so until enough liquid has been 
 vaporized to bring the vapor pressure back to its original value. 
 If under like circumstances the volume be decreased, the con- 
 verse of the above will take place. 
 
 This experiment is one to determine the vapor pressure of a 
 saturated vapor. • The vapor is trapped over mercury in a sealed 
 tube. By the side of this sealed tube is another tube containing 
 mercury at the barometric height. Surrounding the whole is a 
 water jacket in which there is a spiral coil of wire by means of 
 
 * The apparent height of the barometer depends upon the temperature. To find 
 the true 'height it is therefore necessary to make correction for the density of the 
 mercury and the length of the scale used, as both of these factors depend on tempera- 
 ture. It is frequently desirable to reduce the readings to sea level at a given 
 latitude since the value of g varies from point to point on the earth's surface. For 
 methods of making corrections for the above see Kohlrausch, Physical Measure- 
 ments, pp. 76-78, and Edser's Heat for Advanced Students, pp. 23-27. 
 
 t Reference : Edser's Heat for Advanced Students, pp. 94-95, 102-104, 220-224, 
 
PROPERTIES OF GASES. 
 
 131 
 
 which the temperature of the water may be raised when a 
 current of electricity is sent through the wire. Connect the 
 spiral coil in series with a resistance, a key for making and 
 breaking circuit, and a 55-volt alternating E. M. F., leaving the 
 circuit open. Then fill the water jacket with ice water by 
 means of the circulation pump attached to the 
 apparatus. After having caused the water to 
 circulate for a few minutes to bring the vapor 
 to the temperature of the water, make a reading 
 on the top of the mercury column in each tube, 
 by means of the sliding index and scale, taking 
 care to eliminate parallax. Read the tempera- 
 ture by means of the thermometer suspended 
 within the water bath. Close the circuit and 
 allow current to flow for sufficient time to raise 
 the temperature of the bath about io°, stirring 
 continuously by means of the pump. Open 
 the circuit, but continue the pumping so that 
 the water and the vapor may take approxi- 
 mately the same temperature. Make another 
 set of readings, as noted above. Continue this 
 operation at intervals of io°, approximately, 
 until the limits of the apparatus have been 
 reached or the water is about 8o° C. Make a 
 scale reading on the upper end of the inside of 
 the vapor tube. Fi e- 42a - 
 
 Plot a curve, using as abscissas the temperature in centigrade 
 degrees and as ordinates the differences of reading on the two 
 mercury columns. These differences give approximately the 
 vapor pressures of the liquid. This method neglects the change 
 in density of the mercury due to change in temperature and 
 also the depression due to the small quantity of liquid. 
 
 The pressure readings may be reduced to 0° C. by using the 
 expression 
 
 A = /i (i+{3t), (154) 
 
132 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 in which h is the observed height at a temperature t, /3 is the 
 volume coefficient of expansion of mercury, and A is' the height 
 of the column at o° C. This correction will usually be less than 
 1 per cent, on account of the comparatively small range of tem- 
 perature used or within the errors of observation, and need not 
 be made. 
 
 In order to compare the action of a saturated vapor at dif- 
 ferent temperatures with a perfect gas, two more curves are to 
 be drawn, one for the saturated vapor and the other for an ideal 
 gas, using the products pv as ordinates and corresponding values 
 of the absolute temperature 8 as abscissas. The curves are to be 
 plotted on the same sheet to the same scale from the same origin. 
 The true origin = for the ^r-co-ordinates may not appear on 
 the sheet, so that the curves may be shown to a better scale. 
 
 Remembering that the volume v of the vapor is proportional 
 to the length of the tube above the surface of the liquid, the 
 cross section being constant, the products pi and 6 may be used 
 as co-ordinates instead of pv and 9. Find the products pi, and 
 plot such a curve for the saturated vapor. 
 
 To get data for the curve for a perfect gas assume the perfect 
 gas to occupy the same volume and to be at the same tempera- 
 ture and pressure as the saturated vapor at the highest tem- 
 perature used in the experiment. The following relation holds 
 for the perfect gas, 
 
 pi=ce, (155) 
 
 in which / is proportional to volume, as noted above. Find the 
 value of C for the perfect gas. C being a constant, it can be 
 used for any other temperature. Taking the lowest tempera- 
 ture used in the experiment and C as obtained above, compute 
 the final product // at the corresponding temperature. Since 
 the line is a straight line, connect these two points. This will 
 give the line representing the change for a perfect gas starting 
 with the same final conditions as with the saturated vapor. 
 
 Compare the two curves and discuss some causes for the dif- 
 ference between them. 
 
CHAPTER III. 
 
 GROUP I : CALORIMETRY. 
 
 (I) General statements ; (Ij) Heat of vaporization ; (I 2 ) Heat 
 of fusion ; (I 3 ) Specific heat ; (I 4 ) Radiating and absorbing 
 power; (I 6 ) Joule 's equivalent. 
 
 (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 will run no risk of omitting essential ob- 
 servations and precautions. It will generally be found that 
 the greatest source of error in calorimetric experiments is the 
 inaccurate determination of temperatures. This may be due to 
 several causes : 
 
 (1) The thermometer may indicate the temperature of a por- 
 tion 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. 
 This principle may be stated as follows : The amount of heat 
 lost by one system of bodies is equal to the amount gained by 
 
 133 
 
134 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 another system. This, of course, treats as potential energy the 
 amount of heat necessary to produce changes of state. The 
 heat lost or gained by a body may be due to two causes : 
 
 ( i ) 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 radi- 
 ating 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 they 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 temperature differ- 
 ences accurately. It is often necessary, particularly in prob- 
 lems involving the use of the method of mixtures, to measure 
 various temperature differences with different thermometers, 
 and to know them in terms of a single thermometer. 
 
 To compare thermometers, they should be placed in a vessel 
 of water and readings made at frequent temperature intervals, 
 over the whole range covered in the experiment. The water 
 should be well stirred, and readings made on the thermometers 
 compared as rapidly as may be consistent with accurate read- 
 ings, the readings being estimated to tenths of the smallest 
 graduated temperature intervals. If the temperature of the 
 water is more than 20 different from room temperature, it is 
 well to read the thermometers in direct and reverse order for each 
 
CALORIMETRY. 
 
 J 35 
 
 temperature of comparison, the mean readings being compared, 
 thus eliminating errors due to temperature changes. The num- 
 bers, or other distinguishing marks, of the thermometers used 
 should in all cases be recorded. 
 
 An example of a comparison of three thermometers to be 
 used in I 1 (the heat of vaporization) is given in the following 
 table, the first reading being made in melting ice, and the others 
 in water, hot water being added to that in which the thermome- 
 ters are immersed for successive readings, the water being 
 thoroughly stirred before readings, so that the whole may come 
 to a uniform temperature. 
 
 Comparison of Thermometers. 
 
 No. 
 
 725. 
 
 No. 736. 
 
 No. 
 
 108. 
 
 Readings. 
 
 Mean. 
 
 Readings. 
 
 Mean. 
 
 Differences. 
 
 Readings. 
 
 Differences. 
 
 — O.I2 
 
 — 0.12 
 
 — 0.09 
 
 — O.O9 
 
 + O.O3 
 
 — 0.2 
 
 - O.08 
 
 + 4.36 
 4.40 
 
 + 4-38 
 
 + 4-38 
 4.40 
 
 + 4-39 
 
 + 0.01 
 
 + 4-3 
 
 -O.08 
 
 9.27 
 
 9-3° 
 
 9.29 
 
 9.28 
 9-30 
 
 9.29 
 
 O.00 
 
 9.2 
 
 — O.09 
 
 12.47 
 
 
 12.46 
 
 
 — O.OI 
 
 12.4 
 
 — 0.07 
 
 16.85 
 
 
 16.83 
 
 
 — 0.02 
 
 16.8 
 
 — O.05 
 
 21.10 
 
 
 20.97 
 
 
 -O.03 
 
 21. 1 
 
 O.OO 
 
 25-73 
 
 
 25.71 
 
 
 — 0.02 
 
 25.7 
 
 -0.03 
 
 30.06 
 
 
 30.05 
 
 
 — O.OI 
 
 30.0 
 
 — 0.06 
 
 34-23 
 
 
 34-25 
 
 
 + 0.02 
 
 34-2 
 
 -0.03 
 
 38.76 
 38-73 
 
 38-75 
 
 38-79 
 38-77 
 
 38.78 
 
 + 0.03 
 
 38-7 
 
 — O.05 
 
 43.16 
 43.12 
 
 43 H 
 
 43.16 
 
 43-H 
 
 43-IS 
 
 -|- O.OI 
 
 43-1 
 
 — 0.04 
 
 Thermometer No. 725 taken as the standard of comparison. 
 
 From the data obtained, assuming thermometer No. 725 as 
 the standard, compute and tabulate the temperature differences 
 and plot a curve, using temperature variations as ordinates and 
 temperatures of the respective thermometers as abscissas. 
 
136 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 In making computations corrected thermometer readings 
 should be used, the corrections being read directly from the 
 curves. 
 
 
 
 
 
 
 
 THERMOMETER COMPARISON 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 STANDARD THERMOMETER No. 726 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 « REFER TO NO. 109 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 30 
 
 THERMOMETER READINGS IN DEGREES 
 
 40 
 
 SO 
 
 Fig. 43. 
 
 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 equiva- 
 lent of a calorimeter is a quantity of water which would absorb 
 the same amount of heat, when warmed through a certain num- 
 ber of degrees, as is absorbed by the calorimeter when heated 
 through the same range of temperature. 
 
 To determine the water equivalent, proceed as follows : 
 
 ( 1) Fill the calorimeter nearly two thirds full of water four or 
 five 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 thermometer 
 hanging in it. 
 
 Add enough hot water, of known temperature, eight or ten 
 degrees warmer than the air, 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 fifteen 
 seconds, 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 
 
CALORIMETRY. 
 
 137 
 
 the temperature of the hot water so that the mixture will come 
 to about the temperature of the air, corrections for radiation 
 being unnecessary if this is done. The mass of the hot water 
 used may be determined by weighing the mixture after the 
 observations are completed. From the data obtained, the water 
 equivalent is to be computed.* The student should make at 
 least three determinations. 
 
 Water Equivalent of Calorimeter. 
 
 
 1. 
 
 11. 
 
 in. 
 
 Mass of Calorimeter, No. 12, 
 
 1 547 
 
 1547 
 
 1547 
 
 Mass of Cal. + Cold Water, 
 
 334-o 
 
 344-o 
 
 340.2 
 
 Mass of Cold Water, 
 
 179-3 
 
 189.3 
 
 185.5 
 
 Mass of Cal. 4- Mixture, 
 
 478-5 
 
 480.5 
 
 486.5 
 
 Mass of Warm Water, 
 
 144.5 
 
 I36-5 
 
 146.3 
 
 /Tern, of Rqoiju No. 108, 
 
 24.0 
 
 21.0 
 
 21.0 
 
 Tern, of Cold Water, No. 725, 
 
 9.8 
 
 10.2 
 
 8.25 
 
 Tem. of Warm Water, No. 736, 
 
 35-6 
 
 36.6 
 
 37-4 
 
 Tem. of Mixture, No. 725, 
 
 20.88 
 
 20.85 
 
 20.4 
 
 Water Equivalent, 
 
 12.6 
 
 12.6 
 
 19.2 
 
 Water equivalent = 14.8. 
 
 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 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 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. 
 
138 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 temperature 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: (1) 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. 
 
 Note that this constant depends only on the surface, and not 
 upon the nature of the interior of the body. The radiation con- 
 stant of a calorimeter is, for example, the same when it contains 
 mercury as when it is filled with water. But the rate of cool- 
 ing will be different in the two cases on account of the differ- 
 ence 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 
 
CALORIMETRY. 139 
 
 known as Newton's law of cooling, and is really only an ap- 
 proximation to the truth. In the Case of bodies differing in 
 temperature from their surroundings by not more than io°, 
 the approximation is, however, good. 
 
 The radiation constant may be denned 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 tempera- 
 ture of 6°, the radiation is 8 times as great ; and for t 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 calorimetic ex- 
 periment should therefore be the determination of the radiation 
 constant. The method is as follows : 
 
 (1) Fill the calorimeter to within 1 or 2 cm. of the top with 
 water considerably warmer than the air (say, io°-20° 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 
 of the room, as indicated by a thermometer hanging near, should 
 also be occasionally recorded. 
 
 (2) With the data obtained plot 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 plotted. 
 Any slight deviations from such smooth curves are probably 
 due to accidental errors in the observations. The curve for the 
 temperature of the radiating surface will be convex toward the 
 x-a.xis, since the rate of loss of heat decreases as the surface 
 temperature approaches air temperature. 
 
140 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From the data given by these curves, and knowing the mass 
 of water, the statements made above may be verified, and the 
 radiation constant computed. Since the slope of the curve of 
 surface temperature at any point gives the rate of drop of tem- 
 perature, and the ordinate between the two curves at the point 
 gives the difference of temperature between the radiating . sur- 
 face and the air, the mass of water plus the water equiva- 
 lent of the calorimeter multiplied by the slope of the curve at 
 the point chosen, and divided by the ordinate intercepted be- 
 tween the two curves at this point, will give a value of the radia- 
 tion constant. Note that the slope of the curve and the length 
 of the ordinate must be in terms of the scales used in plotting 
 the curves. Find three values of the radiation constant from 
 the curves plotted. 
 
 It should be observed that an approximation must here be 
 made, viz., that the temperature of the surface of the calorimeter 
 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 cal- 
 orimeter 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 352.6 grams. Suppose that the tempera- 
 ture fell from 30.°8 to 29.° jo in five minutes, the average tem- 
 perature of the room being n. "2 5. The temperature of the 
 water having changed i.°i, the loss of heat is equal to 1.1 x 352.6 
 or 387.9 calories. Since this loss took place in five minutes, the 
 loss in one minute was 387.9^5, or 77.6 calories. The average 
 difference in temperature between water and air was 19 . The 
 loss for one minute, and for 1° difference in temperature, would 
 therefore be 76.6 -f- 19 = 4.08 + minor calories, which is the radia- 
 tion constant. Similar computations made with different por- 
 tions 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 
 
CALORIMETRY. 
 
 Radiation Constant. 
 
 141 
 
 Time. 
 
 Tem. of 
 
 Tem. of 
 
 Radiation 
 
 Time. 
 
 Tem. of 
 
 Tem. of 
 
 Radiation 
 
 Vessel. 
 
 Room. 
 
 Constant. 
 
 Vessel. 
 
 Room. 
 
 Constant. 
 
 3-34 
 
 30.8 
 
 II.2 
 
 
 3-47 
 
 28.O 
 
 II. I 
 
 
 35 
 
 30.56 
 
 
 
 48 
 
 27.8 
 
 
 
 36 
 
 30-36 
 
 
 
 49 
 
 27.6 ' 
 
 
 4.16 
 
 37 
 
 30.12 
 
 
 
 50 
 
 27.4 
 
 
 
 38 
 
 29.90 
 
 "•3 
 
 
 51 
 
 27.23 
 
 
 
 39 
 
 29.70 
 
 
 4.08 
 
 52 
 
 27.03 
 
 
 
 40 
 
 29.46 
 
 
 
 53 
 
 26.88 
 
 II.O 
 
 
 4i 
 
 29.28 
 
 
 
 54 
 
 26.70 
 
 
 3-93 
 
 42 
 
 29.03 
 
 
 
 55 
 
 26.50 
 
 
 
 43 
 
 28.83 
 
 II. 2 
 
 
 56 
 
 26.33 
 
 
 
 44 
 
 28.60 
 
 
 4-33 
 
 57 
 
 26.15 
 
 
 
 45 
 
 28.40 
 
 
 
 58 
 
 25.95 
 
 IO.8 
 
 
 46 
 
 28.20 
 
 
 
 59 
 
 25.80 
 
 
 4.14 
 
 Mass of Calorimeter + Water = 492.5 grams. 
 Mass of Calorimeter = 154.7 
 
 337-8 
 Water Equivalent = 14.8 
 
 Radiation Constant 
 
 352.6 grams. 
 = 4.13 calories. 
 
 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 follow- 
 ing example : 
 
 The temperature of a mixture of ice and water in a calo- 
 rimeter is observed at intervals of one minute and is found to 
 vary as follows: 29°, 26.°$, 24°, 22.°6, 2i.°4, 20. '*8, 20.°6, 20.°5, 
 the temperature of the air being 22 . The average temperature 
 of the calorimeter during the seven minutes is therefore 23.°i8 
 (found by adding all the readings and dividing by 8). Radia- 
 tion has taken place for seven minutes at a rate whose average 
 value is that corresponding to a difference to temperature of 
 i.°i8 from the air. If the radiation constant is 4.13, the loss of 
 heat is 4.13 X 1. 18 x 7= 34.2 calories. 
 
142 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Experiment I r 
 water. 
 
 Determination of the heat of vaporization of 
 
 The apparatus for this experiment may be arranged in a 
 great variety of ways. The essential parts are : 
 
 ( i ) Some vessel in which steam may be generated. 
 
 (2) A calorimeter, which may be any metallic vessel of suit- 
 able size. 
 
 (3) Tubes of metal or glass by which the steam may be con- 
 veyed to the calorimeter. The latter should be sheltered from 
 
 the heat radiated from 
 the boiler, and some 
 device should be sup- 
 plied to prevent the 
 water which condenses 
 in the tubes from en- 
 tering the calorimeter. 
 Figure 44 shows a 
 convenient form of 
 apparatus for this 
 determination. 
 
 The thermometers 
 to be used should be 
 compared and the 
 water equivalent and 
 radiation constant of 
 the calorimeter 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 
 
 3T 
 
 Fig. 44. 
 
CALORIMETRY. 143 
 
 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 pre- 
 viously below it. Cut off the steam supply, continue to stir the 
 water in the calorimeter, and read the thermometer every 1 5 sec- 
 onds until it reaches its maximum temperature and begins to drop, 
 thus insuring the getting of the highest mixture temperature. 
 If the steam is superheated, its temperature should also be deter- 
 mined at minute intervals. The barometer should be read and 
 tables or curves consulted to find the true boiling point of water 
 at atmospheric pressure, which is approximately the pressure at 
 which the steam is condensed, so that a correction can be made 
 for the heat given up by the superheated steam. 
 
 (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 heat of condensation of steam. Corrections 
 should be made for the loss or gain of heat due to radiation 
 and absorption, and for the heat capacity of the calorimeter 
 itself. 
 
 This correction, due to radiation, may be reduced to a mini- 
 mum 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 
 
144 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 correction should always be computed. At least three deter- 
 minations should be made. 
 
 Heat of Vaporization of Water. 
 
 
 1. 
 
 n. 
 
 III. 
 
 Mass of Calorimeter, 
 
 1 547 
 
 1547 
 
 1547 
 
 Mass of Cal. + Cold Water, 
 
 438.0 
 
 427.7 
 
 434-0 
 
 Mass of Cold Water, 
 
 283-3 
 
 273.0 
 
 279-3 
 
 Mass of Cal. + Mixture, 
 
 450.0 
 
 440.6 
 
 447.2 
 
 Mass of Condensed Steam, 
 
 12.0 
 
 12.9 
 
 13.2 
 
 Temperature of Room, 
 
 21.0 
 
 21.0 
 
 21.0 
 
 Temperature of Cold Water, 
 
 10.4 
 
 8.27 
 
 10.82 
 
 No. 725 
 
 
 
 
 r Time. 
 
 i°S° 
 
 105° 
 
 1 03 
 
 Temperature of Steam, 1 ij m. 
 
 107 
 
 i°5 
 
 104 
 
 No. 108 | 3 
 
 108 
 
 
 io S ° 
 
 I 4 
 
 107 
 
 
 
 
 
 
 134 
 
 14.0 
 
 12.8 • 
 
 
 5 
 
 16.2 
 
 19.6 
 
 13-6 
 
 
 1.0 
 
 18.0 
 
 27.2 
 
 17.8 
 
 Temperature of Mixture, 
 
 i-5 
 
 20.7 
 
 344 
 
 22.0 
 
 No. 725 
 
 2.0 
 
 237 
 
 35-° 
 
 24.7 • 
 
 
 2.5 
 
 26.5 
 
 3545 
 
 28.5 
 
 
 3-o 
 
 31.0 
 
 
 32-9 
 
 , 
 
 3-5 
 
 344 
 
 
 37-° 
 
 
 4-0 
 
 34-9 
 
 
 37-7 
 
 Heat of Vaporization, 
 
 545 
 
 544 
 
 540 
 
 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 and the thermom- 
 eters compared, 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 
 
CALORIMETRY. 145 
 
 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 
 it into 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 I. Determination of the specific heat of a 
 solid. 
 
 (1) 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 
 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 
 
 VOL. 1 — L 
 
146 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 should be. The temperature of hot and cold water, each 
 thoroughly stirred, should be observed immediately before mix- 
 ing. The weight of the cold water added is to be found by 
 weighing the calorimeter and contents after the other observa- 
 tions 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 com- 
 parable 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 textbooks. 
 
 The water equivalent and the radiation constant of the calo- 
 rimeter used are to be determined, and the thermometers used 
 are to be compared 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. 
 
 Experiment I 8 II. Specific heat of a liquid by the method 
 of cooling. 
 
 From Newton's law of cooling given in the introduction to 
 this group it follows that the radiation constant of a vessel does 
 not depend on the contents of a vessel, but only on the nature 
 and extent of the surface. The radiation constant R is denned 
 as the amount of heat lost per minute per degree of difference 
 of temperature between the radiating surface and the surround- 
 ing medium. From this it follows that • 
 
 R = < e \- e i) t ( IS 6) 
 
 A* . 
 
CALORIMETRY. 147 
 
 in which c is the heat capacity of the vessel, 6 X and 2 the tem- 
 peratures of the radiating surface at the beginning and end of 
 the time interval t, and A the average difference in temperature 
 
 between the radiating surface and the air during the time t. 
 
 a . a 
 (In general A will not be * 2 — 6 air. Why ?) 
 
 2 
 
 The temperature of the surface may be taken as that of the 
 liquid within the calorimeter if the liquid is well stirred and the 
 calorimeter has thin walls and is a good conductor. 
 
 The heat capacity of the system is the amount of heat lost 
 per degree of drop in temperature and is therefore equal to the 
 water equivalent of the vessel and contents, or m 1 s l + mfy in 
 which m 1 and m 2 are the masses of the calorimeter and contents, 
 respectively, and jj and s 2 their specific heats. 
 
 Then equation 156 may be written 
 
 R _ («iJi + w^Xfl, - ft,) / I57 x 
 
 At v ' 
 
 Some other liquid of specific heat s 3 would, of course, give the 
 same radiation constant. This fact makes it possible to deter- 
 mine the specific heat of a liquid by the method of cooling, for 
 we may write 
 
 R = (m^ + m^(6 x - fl a ) = Q^ + ^iOCY ~ e 4) . ( ICi8 \ 
 
 Vi vv 
 
 The water equivalent of the calorimeter is given by m^s^, or 
 may be determined by experiment. The quantity m 2 is the 
 mass of water used in determining the value of R, and s 2 , the 
 specific heat of water which may be taken as unity. The 
 mass of the liquid whose specific heat s 3 it is desired to 
 find is m 3 . 
 
 It is necessary to perform the same preliminary experiments 
 to compare thermometers and find the water equivalent of the 
 calorimeter as in the other experiments in the group, following 
 the instructions given in the introduction to the group. 
 
 A run of an hour is to be made to determine the value of R 
 
148 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 for the calorimeter, using water in the calorimeter. Then a 
 second run of an hour is to be made, using the liquid whose 
 specific heat is to be determined in the calorimeter in place of 
 the water. 
 
 The initial temperature in each of these determinations should 
 be about 15 degrees above room temperature. Readings of 
 the temperature of the liquids are to be taken every minute, the 
 liquids being well stirred in order that the temperature of the 
 surface of the calorimeter be as nearly that of the contained 
 liquid as possible. In each case the room temperature should 
 be read every five minutes. 
 
 Twelve determinations of R are to be made from the first 
 set, taking 5-minute intervals. The mean value of R thus deter- 
 mined is to be taken from which to compute the specific heat of 
 the liquid. Make* six sets of computations for the determination 
 of the specific heat of the liquid investigated, using 10-minute 
 intervals of the run made with it in the calorimeter. 
 
 Plot two radiation curves. In each case use calorimeter 
 temperatures and room temperatures as ordinates and time as 
 abscissas. From the curve drawn for the water content, find 
 two values of R from the slope and then apply the mean of 
 these values to the second curve to find two values of the 
 specific heat. See equation 159, p. 150. ' 
 
 Experiment I 4 . Radiating and absorbing powers of different 
 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 
 
§ 
 
 ». 
 
 CALORIMETRY. 149 
 
 the number of calories that will be absorbed by one square cen- 
 timeter 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 areu of the vessel. The absorption constant 
 ay be computed in a similar manner from the heat gained in 
 given time. K %■#$& 
 
 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 con- 
 ducting 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 con- 
 stant is to be determined with water 15 or 20 degrees 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 
 hanging in the center of the vessel, at intervals of two minutes, 
 stirring the water thoroughly before each reading. The tem- 
 perature of the air should also be observed at intervals of about 
 five 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 plotted 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," d6 -*- dt (0= tempera- 
 ture ; t — time), at different points on the curve, by drawing 
 tangents. 
 
ISO JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From 
 
 Newton's 
 
 law of cooling, 
 
 the radiation constant R is 
 
 given by 
 
 the equation 
 
 
 
 
 
 
 
 R 
 
 _ c 
 
 ~ A 
 
 8-0, 
 
 dt' 
 
 (159) 
 
 where c is the heat capacity of the vessel, A its superficial area, 
 and 9 a the temperature of the air. The value of c is determined 
 by adding the water equivalent of the vessel to the weight 
 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. 
 
 , M^ZA.y-O.). (160) 
 
 at c 
 
 do RA j. ia\ 
 
 — ■ dt. ( r 6i) 
 
 e-e, 
 
 R A 
 By integration : log e (<? -0 a )= — t+K, ( 162) 
 
 where K is the constant of integration. 
 
 If, therefore, a curve is plotted whose co-ordinates are t and 
 
 log(# — #„), respectively, the result should be a straight line. In 
 
 RA 
 the equation of this line, enters as one of the constants. 
 
 Note that the logarithm which occurs in the above equation is 
 the Napierian logarithm. Ordinary logarithms may, however, 
 be user! until the final result is reached. 
 
 The constant for absorption can be determined in a similar 
 manner by filling the vessel with water 1 5 or 20 degrees 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 
 
CALORIMETRY 151 
 
 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 differ- 
 ences of temperature, however, the loss by convection is small, 
 and may be treated as though it obeyed Newton's law. The 
 radiation 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.) 
 
 Experiment I 5 . Mechanical equivalent of heat. Joule's 
 law. Electrical method. 
 
 When an electric current flows through a wire, heat is devel- 
 oped. Joule's law affirms that the amount of heat developed 
 varies with the square of the current, with the resistance of 
 the wire, and with the time, in accordance with the following 
 expression, 
 
 Heat developed = RIH —pdlt. (163) 
 
 The heat is expressed in work units. If R, I, pd, and t be 
 expressed in C. G. S. units, the heat will be expressed in ergs, or 
 if they be expressed in the practical units, the ohm, the ampere, 
 and the volt, / being expressed in seconds in either case, the 
 heat will be given in joules. 
 
 The first law of thermodynamics states that " whenever me- 
 chanical energy is converted into heat, or heat into mechanical 
 energy, the ratio of the mechanical energy to the heat is con- 
 stant." That is, 
 
 W=JH, (164) 
 
 where H'\% the amount of heat developed, Wthe work done, and J 
 the reduction factor, called the mechanical equivalent of heat. 
 
 Since the electrical energy used up in a circuit or a part of a 
 circuit may be expressed in terms of mechanical units, it is pos- 
 sible to determine the value of J electrically. 
 
152 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Fig. 45. 
 
 The electrical circuit is to be made up as follows : A storage 
 battery is to be used as a source of E. M. F. In series with the 
 storage battery B, Fig. 45, connect a variable resistance R, 
 capable of carrying two or three amperes, an ammeter A, 
 and the electric calorimeter C. Shunt a voltmeter V around 
 the calorimeter terminals. A single electrical measuring in- 
 strument, a millivoltmeter, may be used to measure both the 
 current and voltage if connected as in Fig. 45 b. When the 
 double pole, double throw, switch K is closed so as to connect 
 the instrument to the proper terminals of a suitable shunt S, put 
 
 in the main circuit in 
 place of the ammeter, 
 it will give the current 
 in amperes. When the 
 switch is closed in the 
 opposite way, the instru- 
 ment is connected to the 
 terminals of the heating coil, and with a suitable resistance R' 
 in series it will read in volts. 
 
 Fill the calorimeter about two thirds full of distilled water 
 at a temperature of io° C. or 12° C. below room temperature, the 
 mass of water being known. 
 
 Place the calorimeter in its support, lower the cover carrying 
 the resistance coil into position, suspend a thermometer so that 
 the bulb will be covered with water, and start the current flow- 
 ing. Adjust the resistance in the circuit to give about two am- 
 peres of current. 
 
 Begin making observations of the temperature of the 
 water about a minute after the current has been adjusted, 
 and every minute thereafter until the water has attained 
 a temperature about as much above room temperature as 
 it was originally below, then break the circuit. Make read- 
 ing of the room temperature every two minutes and read 
 the current and voltage carefully every minute throughout the 
 run. 
 
CALORIMETRY. 153 
 
 Caution. Be careful to stir the water thoroughly before each 
 temperature reading, but do not stir it so violently that water 
 will spill out of the calorimeter. 
 
 Consider the first run merely as a preliminary one to get 
 acquainted with the method of operation. No computations 
 of J are to be based on this run, unless upon inspection by an 
 instructor, the data is considered sufficiently accurate to be 
 worked up. 
 
 After breaking the circuit continue to read the temperature 
 of the water and the room temperature at intervals of one and 
 five minutes, respectively, for ten minutes in order to determine 
 the radiation constant of the calorimeter. One set of observa- 
 tions for the radiation constant will be sufficient. 
 
 Make two runs for the determination of J in the manner 
 above described, but with different current values. Use cur- 
 rents of about 1.8 and 2.0 amperes. Read the current and 
 voltage as nearly simultaneously as possible every minute. Be 
 sure to know the mass of water for each run. 
 
 Plot two curves on the same sheet to the same scale for 
 each run, using time as abscissas in both, water temperature as 
 ordinates in one and room temperatures as ordinates in the other. 
 The computations to be made are to be based on the curves 
 drawn. Two values of J are to be computed from each set of 
 data, by choosing two pairs of points such that one of the points 
 is just as much above room temperature as the other is below. 
 Good judgment should be used in selecting the points to be 
 used. Points too near the beginning of the curves may belong 
 to such a period of the run that steady conditions had not yet 
 been reached. 
 
 The water equivalent of the apparatus may be determined by 
 using masses and specific heats of those parts of the apparatus 
 which are subjected to change of temperature. Glass parts are 
 to be considered as nonconducting. If a Dewar tube be used 
 as a calorimeter, its mass should be determined and a careful 
 estimate made as to the part in contact with the water. The 
 
154 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 water equivalent of the immersed parts of the thermometer and. 
 :oil support should be determined, since in this experiment the 
 mass of water used may not be great in comparison to the water 
 equivalent of these parts. If a metal calorimeter be used," its 
 svhole mass is to be used in determining the water equivalent. 
 
 From the amount of heat developed in calories and the work 
 lone by the current, as expressed in equations 163 and 164, the 
 :our values of J are to be computed. 
 
 GROUP J: EXPANSION. 
 
 (J x ) Linear expansion ; (J 2 ) Volume expansion. 
 
 Experiment J v Coefficient of linear expansion. 
 
 In genera] when materials are subjected to changes in tem- 
 
 Derature there will be changes in dimensions. If the body is 
 
 sotropic, that is, if it has similar properties no matter the direc- 
 
 :ion taken within the body, the change in length per unit of 
 
 ength will be the same. In crystals the change of length per 
 
 mit of length may vary with the direction taken. In general 
 
 :he change in length per unit of length per degree change of 
 
 :emperature is called the coefficient of expansion. -The general 
 
 :xpression for the length of a body is represented by the follow- 
 
 ng equation: 
 
 l=l a {i+at + Pfi+-). (165) 
 
 [t is usually sufficient, where the changes of temperature are 
 
 lot great, to use only the first two terms within the brackets 
 
 ind write 
 
 l=/ (i+at), (166) 
 
 n which / is the length at the final temperature, / the length at 
 he initial temperature, t the temperature difference, and a. the 
 :oefficient of expansion. The coefficient « is often determined 
 )etween the temperatures of melting ice and boiling water 
 inder standard conditions of atmospheric pressure. If the 
 :oefficient obtained between intermediate temperatures is to be 
 eferred to the temperature of melting ice, as is the case in the 
 
EXPANSION. 
 
 155 
 
 following experiment, the length at both temperatures must be 
 referred to the melting ice temperature. 
 
 The experiment is to determine the average coefficient of ex- 
 pansion between two temperatures, one of which is the boiling 
 point of water, and also to find the average coefficient when 
 referred to the melting point of ice. 
 
 Focus a micrometer microscope on a scratch on the free end 
 of one of the tubes supplied, so fixing the microscope that it may 
 be moved in a direction parallel with the axis of the tube, and 
 the amount of motion be accurately measured. 
 
 Insert a thermometer within the free end of the tube and 
 take its reading after sufficient time has elapsed for it to have 
 attained the temperature of the tube. Make readings on the 
 thermometer and the micrometer. Pass steam through the tube 
 until the thermometer within the tube reaches a steady value, 
 then take its reading, and also the new micrometer reading, 
 having followed the scratch on the tube. Measure the distance 
 from the scratch to the block at the fixed end of tube. Cool 
 the tube by passing cold water through it from the tap. 
 
 Make three sets of readings as outlined above on each tube. 
 From the data obtained compute the values of the coefficients 
 for each tube as indicated above. 
 
 Does the coefficient of linear expansion depend on the mate- 
 rial studied, the unit of length used, and the scale of tempera- 
 ture ? For each tube supplied show the effect of changing the 
 unit of length used, and the temperature scale. 
 
 Why is it necessary to measure the elongation so carefully 
 while the total length is measured with a meter scale estimating 
 only to tenths of millimeters ? 
 
 Experiment J 2 . Cubical expansion of solids and liquids. 
 
 The coefficient of volume expansion of a solid or liquid is 
 defined as the change in volume per unit volume per degree 
 change of temperature. 
 
 V=V (i+/3t). (167) 
 
156 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 It is easily shown for isotropic bodies, in which the linear 
 coefficients of expansion along the three principal axes are the 
 same, that /3 = 3 a, neglecting terms of the second and third 
 order of magnitude. In the case of liquids the volume coeffi- 
 cient alone has a physical meaning. 
 
 This experiment may be used to find the value of /3 for either 
 solids or liquids. In the case of finding the value of /3 for 
 a glass bottle the value of the absolute volume coefficient of 
 mercury as determined by Regnault's method,* which is inde- 
 pendent of the expansion of the containing vessel, is used. 
 
 The bottle should be clean and thoroughly dried. It is then 
 weighed, and pure mercury is poured into it by means of a 
 small funnel. Small bubbles of air on the glass can be removed . 
 by tilting the bottle when not quite full. Mercury is added or 
 withdrawn, as the case may require, so that the level of the 
 mercury surface coincides with a scratch on the bottle's neck. 
 The temperature of the mercury having been obtained, the 
 bottle and its contents are weighed on a nonsensitive balance. 
 
 The bottle containing the mercury is then placed in a beaker 
 filled with sufficient water to bring its level above the mark on 
 the neck of the bottle. The whole is to be heated on a sand 
 bath up to about 90 C. When this temperature, observed by 
 means of a thermometer dipping in the water, has been main- 
 tained constant for ten or fifteen minutes, mercury is carefully 
 drawn out of the neck of the bottle by means of a small pipette, 
 until the surface of the remainder coincides with the scratch. 
 The mass of the mercury withdrawn is to be carefully deter- 
 mined, a chemical balance being used for this purpose. 
 
 Make two complete determinations of the desired coefficient. 
 
 Let W be the mass of mercury filling the bottle up to the 
 index at the original temperature, 
 co be the mass of the mercury removed at the final 
 temperature, 
 
 * Edser's Heat for Advanced Students; pp. 76-80. 
 
EXPANSION. 157 
 
 p and p' the densities of the mercury at the initial and 
 
 final temperature respectively, 
 Fand V be the volumes of the bottle at the initial and 
 
 final temperatures respectively, 
 t the temperature difference, and 
 ft, fig, and g be the absolute coefficient of expansion of 
 
 mercury, the apparent coefficient of expansion of 
 
 mercury in glass, and the coefficient of volume 
 
 expansion of glass, respectively. 
 The initial volume of the bottle filled with mercury is 
 
 i?= V, (168) 
 
 P 
 and the final volume is 
 
 Z=±-r. (169) 
 
 V'=V{i+gt), (170) 
 
 or g=— y t — (170 
 
 Substituting in equation 171 the values of V and V from 168 
 and 169. W -a> W 
 
 _P[ P_ 
 
 (172) 
 
 p 
 
 Since p'=p{i-^t), (173) 
 
 w-~-mi-et). (I74) 
 
 From which z= ft ( 1 75 ) 
 
 The quantity — — is approximately equal to the apparent coeffi- 
 cient of mercury with respect to glass, so that 
 
 g = ft-ftg- (176) 
 
 From the above equation it may be stated that the absolute 
 coefficient of mercury is equal to the coefficient of expansion of 
 the glass plus the apparent coefficient of mercury in glass. 
 
CHAPTER IV. 
 
 GROUP L: LENSES AND MIRRORS. 
 
 (Lj) Radius of curvature of a lens (by reflection); (L 2 ) Focal 
 length of a concave mirror ; (L 3 ) Focal length of a convex 
 lens ; (L 4 ) Focal length of a concave lens ; (L 5 ) Magnifying 
 power of a telescope ; (L 6 ) Magnifying power of a microscope 
 and focal lengths of same. 
 
 Experiment L r Determination of the radius of curvature of 
 a lens by reflection. 
 
 The apparatus consists of a telescope placed midway between 
 two small gas jets (g, g', Fig. 46), the distance between the 
 
 C 10 
 
 Fig. 46. 
 
 jets being capable of adjustment. The lens (L, Figs. 46 and 
 47) whose curvature is desired is placed at a distance of from 1 
 to 2 m., and in such a position that the reflected images of the 
 two flames can be seen in the telescope. The apparent dis- 
 tance between the images is measured by means of a scale 
 (Fig. 48) fastened to the surface of the lens, and from this 
 
 '58 
 
LENSES AND MIRRORS. 
 
 159 
 
 measurement, together with the distance from the 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. 49) in terms of L, 
 
 *=jfe 
 
 I 
 
 Fig. 47. 
 
 the distance between the gas jets (gg ! ) ; of D, the distance from 
 telescope to lens (cT), and of s, the apparent distance of the 
 
 Fig. 48. 
 
 Fig. 49. 
 
 images (g", g"') as measured upon the scale on the face of the 
 lens (««'). 
 
 From the relation of conjugate foci we have 
 
 (177) 
 
 I I _2_ 
 
 gc' 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 
 
 or 
 
 where 
 
 I 
 Tc~ 
 
 1 
 
 tc 
 
 2 
 1 
 
 OC 
 
 d — 
 
 
 Dr 
 
 
 2 
 
 D + r 
 
 d= 
 
 ct 
 
 
 (178) 
 
 (179) 
 
160 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From the geometry of similar triangles we have, also, 
 
 l-.®±4. (.80) 
 
 >-&=£. os.» 
 
 where l = g"g"', 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 180 and 181, we have 
 
 s(r+D) _ L(r-d) _ Lr ( 
 
 D "~ (D + d) ~2D' { } 
 
 2 sD , ■. 
 
 r=j—^- (183) 
 
 To obtain accurate results, the conditions of the experiment 
 should be varied by changing the position of the lens, and by 
 altering the distance between the flames. Make readings for 
 three positions of the flame for each of four positions of the 
 lens. 
 
 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 experienced in dis- 
 tinguishing 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 183 without 
 using the relation of conjugate focal lengths, or the position of 
 the image g". 
 
LENSES AND MIRRORS. 
 
 161 
 
 Experiment L 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. 
 
 i_ j__ 2 __ I 
 ' ~~r~~f 
 
 P\ A 
 
 (184) 
 
 RTOWRWWWV.WWW 
 
 F 
 
 The apparatus required consists simply of the mirror m, 
 a metal screen .S with holes in it to be used as a source or 
 "object," a screen S', and a gas flame. 
 
 The mirror should first be mounted (see Fig. 50) in such a 
 way that its principal axis is nearly horizontal. The metal 
 screen 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 par- 
 tially darkened. 
 
 The position of the image 
 may now be found by trial, 
 a screen S' (preferably of 
 ground glass) being placed 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 a dimension of the 
 object and the size of the corresponding part of the image. 
 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. Make four settings of the 
 
 VOL. I — M 
 
 ;; Gas Flame 
 Fig. 50. 
 
162 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 object, giving images farther from the mirror than the object. 
 These settings can be easily made if the object be placed just 
 to one side of the axis of the mirror and the ground-glass screen 
 just to the other, and shading the screen from the source of light. 
 The focal length and the radius of curvature are to be computed 
 from each of the observations. 
 
 As a check 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. Make five independent settings. 
 Find the radius of curvature by means of a spherometer. 
 (See Exp. A v ) 
 
 Addenda to the report ; 
 
 (i) 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, l x and / 2 , 
 
 4 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 L 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, six observations 
 being made in each case, three with one face of the lens 
 toward the object and the other three with the lens reversed. 
 In methods 3 and 4 the distance from the object to the lens 
 is to be changed for each set of readings. Find the average 
 value of the focal length for each method. 
 
LENSES AND MIRRORS. 163 
 
 (1) 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. A screen of ground 
 glass or paper 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 51 
 
 to the focal length re- 
 quired. 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 
 image from the lens. If pi and p 2 are these two distances, 
 we have 
 
 A A / 
 
 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. 52). When the aperture is illuminated by means of a 
 
164 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 lamp, the shadow of the wire forms an image which is un- 
 affected by the flickering of the flame, and which can be very 
 sharply focused. 
 
 (186) 
 
 Fig. 52. 
 
 For four of the settings make measurements of some dimen- 
 sion of the aperture and the corresponding dimensions of the im- 
 ages, and test the relation 
 
 p 2 d 2 ' 
 in which d x and d % refer to the dimensions measured. 
 
 (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 
 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, 
 
 / = 
 
 / 2 -q 2 
 
 (i87) 
 
 This method of determining focal length has the advantage 
 of being uninfluenced by any uncertainty as to the thickness of 
 
LENSES AND MIRRORS. 165 
 
 the lens and the position of the principal points. Since it is 
 merely the distance through which the lens is moved 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, by method 4, compute the in- 
 dex of refraction of the glass from which it is made. The 
 radius of curvature of each face is to be determined by means 
 of a spherometer. (See Exp. A v ) 
 
 Experiment L 4 . Focal length of a concave lens. 
 
 Since a concave lens is a diverging one, its focal length 
 cannot be measured directly. The auxiliary lens method is to 
 be used in this experiment. A concave lens put in the path of 
 the convergent light from a convex lens may cause it to become 
 divergent, parallel, or less convergent, depending on conditions. 
 In the third case the position of the image produced of a given 
 object will be farther away from the convex lens than when 
 it is used alone. In effect the image produced by the con- 
 verging lens may be considered a virtual object for the diverg- 
 ing lens, and the image produced by the combination as the 
 real image of that virtual object. If the distances of the 
 virtual object and its image from the diverging lens be known, 
 its focal length may be determined, using the usual expression 
 for lenses 
 
 due regard being paid to signs. 
 
1 66 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Set up a luminous object, such as a metal screen with holes 
 drilled in it, back of which is placed a gas flame, a converging 
 lens, and a ground-glass screen on which to get an image. 
 After obtaining a sharp image place the diverging lens between 
 the converging lens and the ground-glass receiving screen and 
 find a new focus. From the two distances of the screen to the 
 concave lens the focal length may be determined. Make eight 
 different settings for the determination of the focal length 
 sought, four with one face of the concave lens toward the screen 
 and four with the lens reversed. 
 
 Find the radii of curvature of the lens surfaces by the 
 method of Exp. A 1 and compute the index of refraction for the 
 material of which the lens is composed. 
 
 Addendum to the report: 
 
 Show by diagram how to find the position and size of the 
 image produced by the two lenses, explaining fully. 
 
 Experiment L s . 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, 
 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, 
 
LENSES AND MIRRORS. 167 
 
 the variation should be shown by a curve in which distances 
 and magnifying powers are used as co-ordinates. 
 
 For each 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 powers. Draw a diagram to scale to 
 show the position and size of the various images in one case. 
 
 (2) Explain the cause of the variation in magnifying power 
 with the distance of the object. 
 
 Experiment L 6 . Magnifying power of a microscope and 
 determination of the focal length of its lenses. 
 
 The "open-eye" method. 
 
 This method is similar to that described for the determina- 
 tion of the magnifying power of a telescope. 
 
 (1) 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 tp 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. 
 
 (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. L 3 . From a knowledge of the magnifying 
 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. 
 
1 68 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 GROUP M: THE SPECTROSCOPE, DIFFRACTION GRATING, AND 
 
 SPECTRUM. 
 
 (Mj) Index of refraction of a prism ; (M 2 ) Flame spectra of the 
 metals ; (M 3 ) Distance between the lines of a grating. 
 
 Experiment M v Measurement of the angles of a prism and 
 its index of refraction by means of a spectrometer. 
 
 The spectrometer consists of a collimator C, a telescope T 
 (Fig. 53), a table to carry a prism P, a divided circle, and two 
 verniers 180 apart for reading angles. These parts are all sup- 
 ported on a heavy base and 
 in such a way that all except 
 the collimator may be re- 
 volved about a common axis. 
 The collimator carries a 
 lens at its end nearest the 
 prism table, and an adjust- 
 able slit in a draw tube (not 
 shown in the figure) at the 
 other end. The telescope 
 is fitted with cross-hairs in 
 the eyepiece. The instrument is fitted with clamps to fix the 
 general positions of telescope and circle, and slow-motion 
 screws for fine adjustment in making settings. 
 
 When the spectrometer is in proper adjustment, the plane 
 in which the telescope revolves -about the axis of the instru- 
 ment is perpendicular to that axis and comprises the axis of the 
 collimator ; the axes of the telescope and collimator intersect in 
 the axis of the instrument, for all positions of the telescope; 
 the refracting edge of the prism is parallel with the axis of the 
 instrument; and the telescope and collimator are focused for 
 parallel rays. Before making measurements with the instru- 
 ment, it must be put into adjustment to conform to the re- 
 quirements given above, as outlined in the following numbered 
 
 Fig. 53. 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 169 
 
 sections. The method of making these adjustments varies 
 somewhat with the type of eyepiece used. 
 
 The' Gauss eyepiece in its simplest form is composed of 
 two lenses; two cross-hairs at right angles, their intersection 
 being in the axis of the telescope; an unsilvered glass mirror 
 set with its plane making an angle of 45 to the telescope axis 
 in order to illuminate the cross hairs by reflecting light, which 
 is permitted to enter the eyepiece through a hole in its side, 
 toward the object glass. The illumination of the cross-hairs is 
 best effected by placing a ground-glass screen between the light 
 source and the opening in the side of the Gauss eyepiece. 
 
 Adjustments using a Gauss eyepiece. 
 
 (1) Focus the eyepiece so that the cross-hairs are distinctly 
 seen. All other focusing is to be done without disturbing this 
 adjustment by moving the eyepiece as a whole. 
 
 (2) To adjust the telescope for parallel rays, proceed as fol- 
 lows : Mount a piece of plane parallel glass (a piece of good 
 
 plate glass will do) on edge on the prism ^ . 
 
 table, with a little wax, so that the plane of / / 
 either face is parallel to the line joining two / j/_ 
 of the leveling screws supporting the table, \<y"~ 
 (Fig. 54). Turn the telescope until its axis \ "^\ 
 is approximately perpendicular to a face of ^~_ 
 the glass plate. Illuminate the cross-hairs. Flg ' 54, 
 
 If light cannot be seen reflected from the plate, revolve the tele- 
 scope about the axis of the instrument or change its elevation 
 by means of its vertical adjusting screw, or both until reflected 
 light is found. Change the position of the eyepiece until an 
 image of the cross-hairs is distinctly seen and no parallax is 
 noticeable, i.e. no relative motion of the cross-hairs and their 
 images is observed when the eye is shifted about. The tele- 
 scope is now focused for parallel rays. 
 
 (3) Turn the table carrying the glass plate until the vertical 
 cross-hair and its image coincide. Vary the inclination of the 
 
170 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 telescope by means of its vertical supporting screw until the 
 horizontal cross-hair and its image coincide. Turn the prism 
 table through 180 , bringing the vertical hair and its image into 
 coincidence. If the horizontal hair and its image do not coincide, 
 take out one half the variation by means of the third prism table 
 leveling screw, and the remainder by means of the vertical tele- 
 scope leveling screw. Turn the prism table through 180 and 
 repeat the process. When the intersection of the cross-hair and 
 its image coincide for either position of the prism table, the axis 
 of the telescope is perpendicular to the axis of the instrument. 
 This adjustment may be made, using a prism set as indicated 
 in (5), the adjustment of one face being made in such manner 
 as not to disturb the second, as explained in that section. 
 
 (4) The collimator may now be adjusted by illuminating the 
 slit, turning the telescope to observe the slit through the collima- 
 tor. Change the position of the slit by means of the draw tube 
 until it is sharply focused with no parallax between its image 
 and the cross-hairs. Bisect the slit with the horizontal cross- 
 hair, using the vertical adjusting screw beneath the collimator. 
 The collimator is now focused for parallel rays, and its axis is in 
 the plane of revolution of the axis of the -telescope and perpen- 
 dicular to the axis of the instrument. 
 
 (5) The prism may now be set so that its refracting edge is 
 parallel with the axis of the instrument. Mount the 60° prism, 
 as in Fig. 54, with the base lines at right angles with the lines 
 connecting the table leveling screws, the refracting edge being 
 nearer the center of the table than the other edges. With the 
 cross-hairs illuminated as noted, rotate the prism until one face 
 containing the refracting edge is approximately perpendicular 
 to the axis of the telescope. Make the face perpendicular to 
 the telescope axis by bringing the intersection of the cross-hairs 
 and its image into coincidence by means of the prism table 
 leveling screws. Rotate the prism table until the other face 
 containing the refracting edge may be viewed in the telescope. 
 Bring the cross-hairs intersection and its image into coincidence 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 171 
 
 again by means of the table adjusting screw that will rotate the 
 first face in its own plane. Test the first face again and make 
 further adjustment if necessary in such manner as not to dis- 
 turb the adjustment of the second face. Repeat the process 
 until the intersection of the cross-hairs and its image coin- 
 cide as viewed with either face toward the telescope. The 
 instrument is now ready for use, it being assumed that the verti- 
 cal cross-hairs and slit are parallel to the axis of the instrument. 
 This may be tested by viewing the image of the slit in the tele- 
 scope as reflected from either face of the prism, the refracting 
 edge being turned toward the collimator. 
 
 Adjustments using the ordinary eyepiece. 
 
 The ordinary eyepiece is made up of two lenses, one carried 
 in a drawtube, the other in a stationary tube which also con- 
 tains a diaphram supporting two cross-hairs at right angles, 
 intersecting in the axis of the instrument. 
 
 (6) Adjust the eyepiece by means of the drawtube until 
 the cross-hairs are sharply focused. This adjustment should not 
 be disturbed in any subsequent adjustment of the instrument. 
 
 (7) To obtain parallel rays from the collimator to the tele- 
 scope proceed as follows : Focus the telescope on some object 
 which is so far away that the rays from it may be con- 
 sidered practically parallel. In focusing, the eyepiece is to be 
 moved as a whole, and the parallax reduced to zero. Then 
 turn the telescope 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- 
 scope, is sharply defined with no parallax. Both telescope and 
 collimator are now properly focused, and should not be altered 
 during the experiment. 
 
 (8) Rotate the telescope about the vertical axis of the in- 
 strument until the vertical cross-hair and the image of the 
 collimator slit coincide. By means of the vertical supporting 
 screws rotate the collimator and telescope in a vertical plane 
 
172 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 until their axes are coincident and the horizontal cross-hair 
 bisects the slit. 
 
 (9) Place the prism on the table as indicated in (5), Fig. 54. 
 Turn the refracting edge toward the collimator and view the 
 image of the slit reflected from one face of the prism. By 
 means of the leveling screws of the prism table cause the ver- 
 tical hair to bisect the slit. Turn the telescope and view the 
 image of the slit from the other side of the prism. Bisect the 
 image of the slit with the vertical hair, taking care to use 
 the leveling screw that will rotate the face, previously adjusted, 
 in its own plane. Turn to the first position and readjust if 
 necessary by means of the proper screw. If the slit image is 
 bisected by both cross-hairs in both positions of the telescope, 
 the adjustments are complete, and the instrument is ready for 
 use. If the horizontal cross-hair does not bisect the slit, it may 
 mean that the vertical hair and the slit are not parallel with the 
 axis of the instrument. To make further adjustment rotate 
 the slit about the axis of the collimator through a small angle, 
 contraclockwise if the horizontal cross wire is too high when 
 the slit is viewed from the right-hand face of the prism, or too 
 low if viewed from the left-hand face, and clockwise if the 
 opposite is true. View the slit direct, again bringing the verti- 
 cal hair parallel with the slit. Adjust the prism again as 
 before, and repeat until the proper adjustment is obtained. 
 
 I. 
 
 To determine the angles of a prism. 
 
 After the spectrometer has been adjusted as described 
 above, place the prism near the center of the graduated circle, 
 with its refracting edge turned toward the collimator (Fig. 55). 
 Turn the telescope 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 both verniers. Then set 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 173 
 
 Fig. 55. 
 
 the cross-hairs in the same way upon the image of the slit, as 
 reflected from the other face of the prism, position V , and again 
 read the verniers. The angle between the two positions of the 
 telescope is then equal to 
 twice the angle of the prism. 
 Loosen the clamp securing 
 the prism table to the spindle. 
 Turn the table through twenty 
 or thirty degrees and clamp 
 again. Turn the graduated 
 circle and table until the re- 
 fracting edge is again turned 
 toward the collimator. Clamp 
 it and proceed as before. 
 Make a third set of readings 
 after having again made a 
 change in the part of the graduated circle used to read the 
 angle of the prism. Mount a second prism, whose angles are 
 different, on the prism table and measure all the angles three 
 times as indicated above. 
 
 II. 
 
 To determine the angle of minimum deviation, and the index 
 of refraction. 
 
 The spectrometer having been adjusted and the angle of the 
 prism determined, the index of refraction may be found. 
 
 Up to this point in the experiment any source of light will 
 serve equally well; but since different colors are bent by refrac- 
 tion in different degrees, it will now be necessary to use some 
 monochromatic light. The most convenient light of a single 
 color is that obtained by burning some salt of sodium in the 
 Bunsen flame.* 
 
 * If light of a different wavelength 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. 
 
174 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 tele- 
 scope 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 direc- 
 ' tion will cause the image to move toward a position of greater 
 deviation. When this setting is made as accurately as possible, 
 bring the vertical cross-hair into coincidence with the image of 
 the slit, and take the reading of the verniers. 
 
 Without disturbing the position of the graduated circle 
 turn the telescope and set the vertical cross-hair on the slit 
 viewed directly, removing the prism if necessary, and read both 
 verniers. The differences of the readings of the verniers in the 
 two positions gives the angle of minimum deviation. Make 
 another setting for the angle of minimum deviation. Then turn 
 the prism so that the light enters the opposite face, using the 
 same refracting edge, and proceed as before, making two sets of 
 readings. In this way find the angle of minimum deviation for 
 one angle of each prism. 
 
 From the angles of the prisms and their angles of minimum 
 
 deviation the indices of refraction are to be computed from the 
 
 formula . , / * , \ 
 
 sini(fi+«) (l88) 
 
 in which a is the angle of the prism, 8 the angle of minimum 
 deviation, and p the index of refraction. 
 
 Addenda to the report ; 
 
 ( i ) Prove that the angle turned through by the telescope in 
 measuring the angle of the prism by the method in (I) is equal 
 to twice the angle to be determined. 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 175 
 
 (2) Show that the index of refraction is the ratio of the ve- 
 locities of light in the two media considered. 
 
 (3) Show that for minimum deviation the angle of incidence 
 is equal to the angle of immergence. 
 
 (4) Show that equation 188 is true. 
 
 Experiment M 2 . Study of various bright line spectra. 
 
 When a ray of light passes through a prism it will be bent 
 from its path and, if made up of different wave lengths, each 
 wave length will have its own emergent ray. If the incident 
 ray be made up of parallel light, the separate emergent rays will 
 be made up of parallel light. If a lens be placed in the path of 
 the emergent light, it will focus the various wave lengths on the 
 screen and a spectrum will be produced. The type of spectrum 
 will depend upon the source of light. Glowing solids and 
 liquids give continuous spectra, glowing vapors and gases 
 usually have bright line spectra. Dark line spectra are produced 
 by the absorption of particular wave length of the incident light 
 depending on the nature of the vapor or gas. The bright line 
 and dark line spectra are characteristic of the various gases 
 producing them. 
 
 It is the object of this experiment to study the spectra of dif- 
 ferent substances when giving their bright line spectra. It is to 
 be noted that a given gas will give out different bright line 
 spectra depending on conditions of excitation as in the flame 
 spectrum, the spark spectrum, the vacuum tube spectrum, and 
 the arc spectrum. The first three types may be used in the ex- 
 periment given below. 
 
 The study will be made with the spectrometer, using a simple 
 prism. Since the spectrum produced by a given prism depends 
 on its angle and material, it is necessary to know the dispersion 
 and deviation produced by it ; that is, to know its calibration. If 
 the calibration be not known, the first part of the experiment will 
 consist in such a determination, after which the spectra of vari- 
 ous unknown substances may be studied. 
 
176 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The calibration of a prism. 
 
 The spectrometer should be adjusted, as in Experiment M v 
 The slit should be narrow in order to get as pure a spectrum as 
 possible, the spectrum seen in the telescope being made up of 
 images of the slit, one for each wave length ; consequently the 
 narrower the slit the less the overlapping. The calibration may 
 be made 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 are ordinates. By reference to 
 this curve, the wave length corresponding to any observed devia- 
 tion is readily determined. 
 
 Another method which may be more convenient is by 
 means of the spark, vacuum tube, or flame spectrum, or any 
 combination of them. To produce the spark or vacuum tube 
 spectrum, the vacuum tube containing a known gas or the 
 spark gap between the ends of two pieces of a pure metal, 
 between which an electric spark is to be produced, is to be 
 placed directly in front of the collimator slit. The terminals 
 of the vacuum tube or the two pieces of metal are connected 
 to the secondary of an induction coil, which, when properly 
 excited, will produce the desired discharge. Better results 
 may sometimes be obtained if a small Leyden jar be put in 
 multiple with the spark gap. The prism having been set at 
 minimum deviation for sodium, the positions of several bright 
 lines are to be located and read on the verniers, taking care not 
 to disturb the setting of the prism. If flame spectra are to be 
 used in calibration, a Bunsen burner is placed immediately in 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 177 
 
 front of the slit and the salts introduced into the colorless 
 flame, using a wire of platinum supported on a glass rod. The 
 positions of the bright lines are to be determined as in case of 
 the spark or vacuum tube discharge. A good combination to 
 use is the flame spectra of sodium and potassium, the vacuum 
 tube spectrum of hydrogen and the spark spectrum of lead- 
 The chlorides or carbonates of the salts are generally used in 
 producing the flame spectra. 
 
 A suitable calibration may be based on the following table : 
 
 Element 
 
 Spectrum 
 
 Color 
 
 Wave length 
 in 10—8 cm. 
 
 K 
 
 Flame 
 
 Red 
 
 7669 
 
 Pb 
 
 Spark 
 
 a 
 
 6657 
 
 H 
 
 Vacuum tube 
 
 a 
 
 6563 
 
 Na 
 
 Flame 
 
 Yellow 
 
 5893 
 
 Pb 
 
 Spark 
 
 Yellow Green 
 
 5608 
 
 U 
 
 a 
 
 Green 
 
 5373 
 
 ti 
 
 it 
 
 a 
 
 5046 
 
 (i 
 
 it 
 
 a 
 
 5006 
 
 H 
 
 Vacuum tube 
 
 Blue Green 
 
 4862 
 
 Pb 
 
 Spark 
 
 Violet 
 
 4387 
 
 H 
 
 Vacuum tube 
 
 a 
 
 4341 
 
 Pb 
 
 Spark 
 
 a 
 
 4245 
 
 it 
 
 a 
 
 it 
 
 4058 
 
 Plot a curve using wave lengths as abscissas and deviations 
 from the sodium line as ordinates. 
 
 II. 
 
 Study the spectra of the substances furnished, determining 
 the wave lengths of their bright lines and mapping their spectra 
 on the same sheet, arranging the substances one below the 
 other with their bright lines located to scale from left to 
 right. 
 
 A most instructive method of mapping bright line spectra 
 is that employed by Lecoq de Boisbaudran in his work on the 
 
 VOL. I — N 
 
178 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 spectra of the metals.* An example is given in Fig. 56. As 
 will be seen from the diagram, each spectrum is mapped twice, 
 once above and once below the median line. The former 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 for one element having sev- 
 eral lines. 
 
 Considerable difficulty is sometimes met with in working 
 with flame spectra in obtaining sufficient permanence and bril- 
 liancy for accurate observation. To obtain the best results the 
 
 Fig. 56. 
 
 methods of heating must be suited to the salt used. In some 
 cases, a small amount of the salt, when placed on a wire and 
 heated in the flame, will form a bead which lasts for a consid- 
 erable 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 solu- 
 tion 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 atten- 
 tion to keeping the flame in proper condition, while another 
 observes the spectrum. 
 
 Experiment M 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. 
 
 * Lecoq de Boisbaudran, Spectres Lumineux, Paris, 1874. 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 179 
 
 There are two types of gratings, the reflection and the trans- 
 mission grating. The transmission type is used in this experi- 
 ment. It consists of aj, plane glass plate on which are ruled 
 parallel lines quite near together. The ruling is done with a 
 dividing engine. The spaces between the rulings act as parallel 
 line sources of light. If parallel monochromatic light from a 
 slit be incident upon the grating, each grating space being 
 parallel to the slit may be taken a§_a source of a new disturbance 
 sending out like cylindrical waves in all directions. jPlanes 
 may be passed tangent to these new wave fronts such that the 
 difference of path from any two adjacent wave fronts is equal to 
 zero, one, two, three or more wave lengths. These planes may 
 be considered as wave fronts of parallel light which may be 
 focused on a screen by a converging lens, giving a series of 
 linear images of the source. The central image corresponds to 
 zero difference of path. The first image to the right or left of 
 the central image corresponds to a difference of path of one wave 
 length and is called a spectrum of the first order. The second 
 image to the right or left corresponds to a path difference of 
 two wave lengths from consecutive grating spaces and is said to 
 be a spectrum of the second order. The order of the spectrum 
 indicates the path difference between consecutive grating spaces, 
 and the image on the screen. It is readily seen from Fig. 57 
 that 
 
 ^=sin0„ (189) 
 
 a 
 
 in which n is the order of the spectrum, \ the wave length, d 
 
 the grating space, or distance between corresponding 
 
 parts of the lines and 0„ the deviation of the spectrum 
 
 of the «th order from the normal to the grating. If "j &. 
 
 white light be incident on the grating, a series of 
 
 colored spectra will be found on either side of a 
 
 r Fig. 57. 
 
 white central image, with violet nearest the central 
 
 image and progressing through all the colors of the spectrum to 
 
 red at the outside. 
 
 Zx 
 
d 
 
 180 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The grating may be used with a spectrometer to measure the 
 angular deviation, but in the following experiment very simple 
 apparatus is to be used. 
 
 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 ; or to 
 wrap some asbestos paper around the burner tube, permitting it 
 to extend above the tube and saturating it with some sodium 
 
 salt solution. The grating to 
 be studied is placed in front 
 of the slit with its rulings ver- 
 tical, 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. By measuring the distance between the grating and 
 the slit (D, Fig. 58), and the displacement of one of the 
 images d, the angle through which the light is bent by diffrac- 
 tion can be determined. From this angle, and the distance 
 between the lines of the grating, the wave length of sodium 
 light is to be computed. 
 
 In measuring the displacement of the images the following 
 method will be found convenient: Adjust a small rider, which 
 can be placed on the horizontal arm, until it coincides with the 
 corresponding image on the opposite side of the slit. The 
 distance between the two riders is then equal to twice the dis- 
 placement of the image. Sometimes in a dimly lighted room, 
 
 Fig. 58. 
 
SPECTROSCOPE, DIFFRACTION GRATING, SPECTRUM. 181 
 
 if the images are bright enough, the positions of the various 
 images may be read directly when looking through the grating. 
 Read the positions of five images on each side of the central slit, 
 if possible; for each of three positions of the grating, making D 
 vary by about equal amounts for values not less than 80 nor 
 more than 140 centimeters. Knowing these distances, compute 
 the angular deviation and the value of the wave length of the 
 monochromatic source used. Make readings in a similar manner 
 for a grating for which the distance between the lines is to be 
 computed, using the wave length determined by the use of the 
 grating whose grating space is known. 
 
 Addenda to the report : 
 
 (1) Explain what is meant by deviation and dispersion. 
 
 (2) Prove the equation for the grating. 
 
CHAPTER V. 
 
 GROUP N: PHOTOMETRY. 
 
 (N) General statements ; (N x ) Horizontal distribution of light by 
 the Buns en or the Lummer-Brodhun photometer ; (N 2 ) Varia- 
 tion of candle power with voltage. Use of the Weber pho- 
 tometer. 
 
 (N) If the properties of a medium transmitting energy are 
 the same in all directions, the amount of energy transmitted 
 across equal areas from a small source is inversely proportional 
 to the squares of the distances of the areas from the source. 
 This principle, applied to light, is stated as follows : The 
 intensity of illumination produced on a surface by a light source 
 is proportional to the brightness L of the light source, and 
 inversely proportional to d 2 , d being the distance from the 
 source to the surface, 
 
 If two sources L x and L 2 produce equal intensities of illumina- 
 tion on a given surface, then the relative brightness of the two 
 sources may be compared, or the brightness of one source 
 determined in terms of the other. 
 
 I=L crrw (I9I) 
 
 d 2 
 L\ = L %fv 092) 
 
 The comparison of light sources and the study of the distribu- 
 tion of light is called photometry. Instruments used to com- 
 pare intensities of illumination are called photometers. 
 
 182 
 
PHOTOMETRY. 
 
 183 
 
 Photometers. — Three different kinds of photometers are 
 used in performing the experiments which follow ; namely, the 
 Bunsen, the Lummer-Brodhun, and the Weber. 
 
 In the Bunsen photometer a screen of white paper (Z>, 
 Fig. 59), a portion of which has been made translucent by the 
 application of oil, is placed in a blackened box (technically 
 
 LA- 
 
 IH^M 7 
 
 -XL' 
 
 Fig. 59. 
 
 called the carriage), and mirrors, M, M' , are adjusted so that 
 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 posi- 
 tion 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. 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. 60) ; but when the proper position 
 
 Transmitted Light 
 
 Reflected Light 
 
 Fig. 60. 
 
184 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 has been found, it will almost entirely disappear. Since the 
 screen may not be alike on its two sides the photometer should 
 be reversed in any set of readings, so that both faces be exposed 
 to each source and the mean reading for the set used in making 
 computations. 
 
 The Lummer-Brodhun photometer (Fig. 61) consists of a 
 blackened box or carriage, through two opposite sides of which 
 are cut holes which permit light from the sources to be com- 
 pared to fall on the two sides of a white opaque screen 5, 
 which has matte surfaces. The two sides of the screen are 
 viewed at the same time through the observing telescope T, by 
 means of properly placed mirrors M and M', 
 and a Lummer-Brodhun cube P The cube 
 is made up of two 90 prisms. On one of the 
 prisms a part of the face opposite the right 
 angle is cut away. The remainder of this face 
 is then brought into optical contact with the 
 >1 T\ corresponding face of the other prism, and 
 6 N,. X P^ced in the carriage as indicated in the 
 Fig- 6i. figure. Light from the mirror M may pass 
 
 through the area of contact of the prisms into 
 the telescope, but the light incident outside of this area is 
 absorbed by the blackened ground surface. Light from 
 the mirror M' passes through the area of contact of the two 
 prisms and does not enter the telescope. But that light strik- 
 ing the polished surface of the right-hand prism is totally 
 reflected, and enters the telescope. Thus the two fields are 
 observed side by side, at the same time, in the telescope. If 
 the two sources are emitting the same quality of light and the 
 two paths from the like surfaces of the screen S are similar, 
 the brightness of the sources may be compared, as in the case 
 of the Bunsen photometer, by moving the carriage until the 
 observed field is uniform. To guard against errors due to dis- 
 similarity of path, settings may be made first with one face of 
 the screen exposed to one source of light, then the other face 
 
PHOTOMETRY. 
 
 185 
 
 S* 
 
 TT 
 
 exposed to the same source by rotating the carriage through 
 180° about the horizontal axis AB, perpendicular to the pho- 
 tometer bar. If the sources are not of the same quality, the 
 estimation of equal intensities becomes a much more difficult 
 matter, but the process is the same as with sources of like 
 quality. 
 
 The Weber photometer is a portable instrument which makes 
 use of the Lummer-Brodhun cube described above. It consists 
 of two tubes (Fig. 62), blackened on the inside to absorb light 
 incident on these surfaces, joined together with their axes at 
 right angles. The center of the 
 Lummer-Brodhun body B is placed 
 at the intersection of the two axes. 
 At one end of the principal tube 
 there is an observing telescope T, 
 and at the other end an opening to 
 receive light from the source S. At 
 this end there is also an opening into 
 which absorption screens or test 
 plates P may be inserted. At the 
 outer end of the side tube there is 
 placed the "comparison lamp" L. 
 Within the side tube there is an 
 adjustable "comparison screen" C, whose position is indicated 
 by an index moving over a scale on the outside of the tube. 
 Equality of fields is obtained by moving the "comparison 
 screen " C along the axis of the side tube. 
 
 By using different absorption screens at P, light sources of 
 greatly varying intensities may be compared, or 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 inves- 
 tigated, 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 dif- 
 
 Fig. 62. 
 
186 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 ferent 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 elabo- 
 rate methods will be necessary. 
 
 The Weber photometer is suitable for many purposes. 
 It is often used as an illuminometer. 
 
 Light Standards. — Although the brightness of light 
 sources is expressed in candle power, the candle is seldom 
 used as a standard of comparison. Many suggestions have 
 been made regarding what sources to use as standards of com- 
 parison, and the question is even yet a debatable one. Among 
 the requirements for a standard light source that of reproduci- 
 bility is of prime importance. The two most important flame 
 standards in use at the present time are the pentane lamp and 
 the Hefner lamp. The Hefner lamp burns pure amyl acetate. 
 The wick tube has a certain prescribed length and diameter. 
 There is a flame gauge mounted on the lamp in order to get a 
 flame of a certain height. The following precautions should be 
 taken in using the Hefner lamp. 
 
 The lamp should be well cleaned, filled, and the wick should 
 be trimmed square off. 
 
 The height of the flame should be carefully adjusted until 
 the image of its tip exactly meets the line on the optical sight. 
 The height of the flame is then 40 mm. and the intensity 
 of the light is normal. Measurements should not be begun 
 until the lamp has been burning at least ten minutes. Draughts 
 in the room should be avoided and the lamp used without box- 
 ing it up. Two observers are necessary for the best use of the 
 lamp, one to make the photometric observations and the other 
 to indicate the moments when the height of the flame is exactly 
 right. 
 
 After using, and while the lamp is still hot, the upper part 
 of the wick tube should be wiped clean with a soft cloth. The 
 brightness of the Hefner lamp is influenced by the barometric 
 
PHOTOMETRY. 187 
 
 pressure, the amount of C0 2 , and of water vapor in the air. 
 Usually the effects of the variation of barometric pressure from 
 the normal, and of the amount of C0 2 present in a well- 
 ventilated room are so small as to be negligible. In accurate 
 work it is necessary to take account of the effect of water vapor 
 present. The normal state is taken as 8.8 liters of water vapor 
 per cubic meter of air, the lamp then having 0.9 candle power 
 in terms of the new International Standard. The following 
 relation has been found to hold regarding intensity and water 
 
 vapor : 
 
 y = 1.049 -0.0055 x, (193) 
 
 in which y is the intensity in terms of the Hefner unit, and x 
 is the number of liters of moisture per cubic meter of dry air. 
 The factor x depends on the temperature and the relative 
 humidity (the percentage of saturation of the air). It is equal 
 to the mass of saturated water vapor per cubic meter times the 
 relative humidity divided by the mass of saturated water vapor 
 per liter. All of these factors depend on temperature. Ex- 
 perimental determinations have been made of them and in terms 
 of these determinations equation 193 may be written 
 
 y= 1.049 — 0.0055 — h, (194) 
 
 in which m, m„ and h are the masses of saturated vapor per 
 meter and liter, and the relative humidity respectively. The 
 
 factor 0.0055 — ma y be computed for various temperatures and 
 m e 
 
 a curve plotted using temperatures and it as co-ordinates, as 
 
 shown in Fig. 63. Having found the relative humidity (see 
 
 table 10), obtain the value of the factor C[ =0.0055 — ) from the 
 
 \ mj 
 
 curve for the proper dry bulb reading, and compute the value 
 of y. 
 
 The standards now in most general use are carefully cali- 
 brated glow lamps. These standards are used to calibrate 
 
188 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 secondary standards, which in turn are used to determine the 
 candle power of the working standards of comparison. The 
 above process is necessary, owing to the variation of candle 
 power with voltage, age, or service. A standard lamp should 
 be carefully brought up to its proper voltage and should never 
 be carried above it, even for an instant. The lamp should be 
 
 ./* 
 
 .22 
 
 .20 
 
 .18 
 o 
 fe.16 
 
 111 
 
 3 
 |,4 
 
 1,2 
 
 ho 
 
 .08 
 .06 
 ,04 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 12 14 16 18 20 22 24 26 
 
 DRY BULB TEMPERATURES IN DEGREES CENTIGRADE 
 
 2b 30 
 
 Fig. 63. 
 
 kept burning only when in actual use in making comparisons, 
 and should not be used longer than a specified time in the 
 aggregate. 
 
 In general it is best to use a lamp at constant wattage 
 rather than constant voltage. 
 
 Experiment N r Horizontal distribution of light by the 
 Bunsen or Lummer-Brodhun photometer. 
 
 Set up a Hefner lamp, a Bunsen or Lummer-Brodhun pho- 
 tometer, and a bat-wing gas burner, capable of being rotated 
 about a vertical axis through definite angles, on a photometer 
 bar. The gas burner should be connected to a line in which 
 
PHOTOMETRY. 189 
 
 there is a water manometer in order to determine the gas pres- 
 sure. 
 
 Adjust the height of the flame of the Hefner lamp to the 
 proper value. Set the gas burner so that the plane of the 
 flame is at right angles to the axis of the photometer bar. 
 Adjust the height of the flame to that which gives steadiness 
 without spluttering. Note the manometer pressure. Change 
 the position of the photometer carriage to give equality of 
 illumination on both sides of the screen and make the proper 
 photometer bar readings. 
 
 After each reading, the carriage should be shifted 20 or 
 30 cm., 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 intensi- 
 ties of the two lights, will make it impossible to obtain coinci- 
 dent settings, but after a little practice the successive readings 
 should agree to within three or four per cent. Constant differ- 
 ences are often observed between the settings of different per- 
 sons. These are due to differences in the eye, and cannot be 
 avoided. Make at least two readings with each face of the 
 screen toward the standard lamp. It is found to be advan- 
 tageous to use black screens, which may be mounted on the 
 photometer bar, to keep direct light out of the eyes. Measure 
 the candle power of the gas flame for different angular 
 positions, at intervals of 15 from o° to 180 . In making 
 computations, assume the candle power of the Hefner lamp 
 as 0.9. 
 
 The candle power of the Hefner lamp should really be cor- 
 rected for moisture of the air. This correction, however, may 
 be omitted for purposes of this experiment, unless otherwise 
 directed. Be careful to keep the Hefner flame at exactly the 
 right height. 
 
 Also measure the candle power of the flame for ten pressures 
 
190 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 partly above and partly below normal for one angular position 
 only. 
 
 Plot a curve, using polar co-ordinates, showing the distri- 
 bution of intensity around the gas flame. Also compute the 
 percentage difference between maximum and minimum candle 
 power. Plot another curve, showing the relation between 
 candle power and gas pressure. 
 
 Experiment N 2 . Variation of candle power with voltage. 
 Use of the Weber photometer. 
 
 There are numerous ways of calibrating and using the Weber 
 photometer, but when it is not desired to measure the absorp- 
 tion coefficients of the several screens used, the following simple 
 method may be used : 
 
 Set up an incandescent lamp of known candle power at a 
 measured distance (i or 1.5 m.) in front of the "test plate " of 
 the photometer. (See Fig. 62.) 
 
 The plane of the filament should be perpendicular to the 
 photometer axis. Adjust the voltage around the lamp, and 
 also adjust the height of the "comparison flame" to the pre- 
 scribed value and see that these are kept constant during the 
 experiment. Then adjust the " comparison screen " until the 
 field of the photometer appears uniform. Note the position 
 of the index on the photometer scale. The intensity of illumi- 
 nation produced by the standard lamp on the test plate is L/D L 
 meter candles, where L is the candle power of the standard 
 lamp and D is distance in meters from the test plate. Obtain 
 in this way three different illuminations on the test plate and the 
 corresponding positions of the comparison screen for a balance, 
 the latter positions being well distributed over the photometer 
 scale by proper choice of illuminations. It will readily be seen 
 how this process may be worked backward and the candle power 
 of any unknown lamp measured. 
 
 Now replace the standard lamp by a 50-volt carbon filament 
 incandescent lamp and, by use of the calibration just described, 
 
PHOTOMETRY. 191 
 
 measure the candle power and current of the 50-voIt lamp at 
 the following voltages: 40, 42, 44, 46, 48, 50, 51, 52, 53, 54, 55. 
 Plot four curves with co-ordinates as follows : 
 
 1. Candle power — Volts. 
 
 2. Candle power — Amperes. 
 
 3. Candle power — Watts. 
 
 4. Candle power — Watts per candle power. 
 
CHAPTER VI. 
 
 GROUP 0: SOUND. 
 
 (Oj) Measurement of pitch by the syren ; (0 2 ) Wave length by 
 Koenig's apparatus; (0 3 ) Resonance of columns of air with 
 determinaton of the velocity of sound ; (0 4 ) Velocity of sound 
 in brass ; (0 5 ) The sonometer ; (0 6 ) Study of the transverse 
 vibration of cords, Melde's method. 
 
 Experiment O v Measurement of pitch by the syren. 
 
 This experiment consists in the determination, by means 
 of a syren, of the pitch of an organ pipe, first when closed at 
 one end and then when open. Each determination should be 
 made several times. Two observers are needed to make these 
 measurements successfully, one devoting his attention to keep- 
 ing the syren in unison with the pipe, while the other operates 
 the counter and observes the time. To form an estimate of 
 the degree of accuracy that is attainable, several measurements 
 should be made with a tuning fork of known pitch before begin- 
 ning observations with the pipe. 
 
 Experiment 2 . Interference and measurement of wave 
 length by Koenig's apparatus. 
 
 In the form of apparatus used a manometric capsule (m v m 2 ) 
 is attached to one end of each of two tubes. The opposite 
 ends of the tubes are brought together at a common opening 
 (Fig. 64), where some sounding body, such as a tuning fork or 
 organ pipe, is to be placed. The two tubes are initially of the 
 same length, but one of them (£ 2 ) is capable of adjustment 
 so that its length can be increased by about 50 cm. From each 
 of the two capsules a tube (g v g 2 ) leads to a small gas jet, 
 
 192 
 
SOUND. 
 
 193 
 
 The latter will be set in vibration when the membrane of the 
 capsule is disturbed, and can be observed in a revolving mirror. 
 There is also a third jet attached to g z which is connected, by 
 tubes of equal length, to both capsules ; so that if a pressure or 
 
 condensation is sent to it by one of the capsules at the same time 
 that an equal rarefaction is sent by the other, the two acting 
 on the flame at the same instant will not affect it. Each of the 
 single jets will, however, still show the disturbance. In order 
 that a condensation may exist at 
 one capsule at the same time that 
 a rarefaction exists at the other, it 
 is obvious that the two tubes must 
 differ in length by one half the 
 wave length of the sound that is 
 producing the disturbance, or by 
 some odd multiple of a half wave 
 length. This can be brought about 
 by sliding the movable tube in or 
 out. When the proper adjustment is obtained, the jet that is 
 connected with both capsules should show a minimum disturb- 
 ance. It will not be found possible to produce complete 
 quiescence in the image of g z , such as is indicated in Fig. 65. 
 
 Fig. 65. 
 
 VOL. I — O 
 
194 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Care must be taken to have the tubes which supply this jet 
 of the same length, and also to have the pressure of gas the 
 same in each. To adjust the pressure, pinch shut one of the 
 tubes, and note the height of flame due to the other; then 
 pinch the second tube and release the first, and adjust the supply 
 of gas until the height of flame is the same in both cases. 
 
 The wave length being determined as described above, and 
 the pitch of the fork used being known, the velocity of sound 
 can be computed. The result obtained should be compared 
 with the velocity given by the formula 
 
 v- 332 Vi + 2J3/, (195) 
 
 where t is the temperature of the room. If the sounding body 
 used is of unknown pitch, the wave length can be determined as 
 before, the velocity of sound obtained from the above formula, 
 and from these two quantities the pitch can be computed. 
 
 Experiment O a . Resonance of columns of air and deter- 
 mination of the velocity of sound. " 
 
 The length of a closed pipe giving its fundamental in reso- 
 nance with a source of sound such as a tuning fork held near its 
 open end, is not strictly proportional to the wave length of the 
 sound produced. The length of the pipe must have a correc- 
 tion factor added which is proportional to the radius of the pipe, 
 due to the disturbance of the open end. Consequently, to find 
 the velocity of sound by the resonant air column method, it is 
 necessary to find the correction factor or to find the pipe of 
 zero radius which would give resonance. 
 
 A satisfactory arrangement for performing this experiment 
 is to use a vertical cylinder 10 or 12 cm. in diameter and 70 or 
 80 cm. long, closed at its lower end, a set of five tubes varying 
 from 1 to 5 cm. in diameter and about the length of the cylinder, 
 and two tuning forks of suitable frequencies. The vertical 
 cylinder is to be filled with water to within 3 or 4 cm. of the 
 top. Place one of the tubes within the cylinder, leaving a short 
 
SOUND. 195 
 
 length above the water surface. Cause one of the forks to 
 vibrate, and holding it a constant short distance above the 
 tube, raise or lower the tube until a point of resonance is 
 found. Clamp the tube in position and measure the distance 
 from the open end of the tube to the water surface. Repeat 
 the above operation three or four times in order to get a good 
 average. With care settings may be made with variations of 
 only a few millimeters. If the tube be long enough, find the 
 other possible points of resonance in the same manner. This 
 process is to be followed with each tube for each fork. Note 
 the temperature of the air. 
 
 The correction to be added to the mean resonance lengths 
 obtained for any tube may be taken as 0.6 of the radius for that 
 tube. The averages of the appropriate means may be used to 
 compute the velocity of sound. As a check upon the results 
 the velocity of sound may be computed for the temperature of 
 the air at the time of the experiment, upon the assumption that 
 the velocity at o° is 332 m. per second, and that the velocity of 
 sound increases 60 cm. per second for each degree rise of tem- 
 perature on the centigrade scale. 
 
 Plot points on cross-section paper for each fork, using tube 
 radii as abscissas and resonance lengths as ordinates. Locate 
 straight lines best suiting the points plotted, continuing them 
 backwards to the Faxis. Interpret the meaning of the slope 
 and the Y intercept and find a value of the velocity of sound 
 based on each curve. 
 
 The report should contain a full explanation of the resonance 
 phenomena observed, with derivation of formulas. 
 
 Experiment 4 . Velocity of sound in brass. Kundt's 
 method. 
 
 A brass rod about a meter long, Fig. 66, is placed in a 
 horizontal position, and firmly supported at its center. To 
 one end of the rod is fastened a disk of cardboard or cork, 
 whose diameter is almost equal to that of a glass tube in 
 
196 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 which it is inserted. The opposite end of this tube is closed 
 by means of an adjustable piston, so that the length of the air 
 column in the tube can be altered. On setting the rod into 
 vibration (by rubbing its free end with leather covered with 
 rosin), the air in the tube will also vibrate, and by placing some 
 
 sfc 
 
 Fig. 66. 
 
 light powder in the tube (such as lycopodium or cork dust), 
 these vibrations are made evident to the eye. If the length of 
 the tube is properly adjusted, the dust will be seen to distribute 
 itself regularly in little heaps, these heaps corresponding to 
 nodes in the stationary waves set up in the air. Frequently 
 the dust figures are similar to those shown in Fig. 67. 
 
 The experiment consists in so adjusting the length of the 
 air column as to make this regular distribution of the dust as 
 marked as possible. 
 
 Determine the distances between the first and last nodes, as 
 indicated by the dust heap, the second and next to the last and 
 
 Js-^lMllli-'.e'iJDfliraitt'&'alMl'iLte'' 
 
 Fig. 67. 
 
 so on, noting in each case the number of vibrating segments 
 included between the nodes considered. From the sum of 
 these distances and of the number of vibrating segments find 
 the average length of a vibrating segment. From it and the 
 velocity of sound in the air within the tube the frequency may 
 be determined ; and also the frequency of the longitudinal vibra- 
 
SOUND. 197 
 
 tions in the rod. Since the rod is clamped at its center, the 
 wave length of the vibrations produced in it is equal to twice 
 the length of the rod. Shake up the dust in the tube, change 
 the position of the plunger on the end of the rod by moving 
 the tube, and proceed as before, making in all three sets of 
 readings. Having the wave lengths of the same note in air 
 and in brass, the ratio of the two gives the ratio of the two 
 velocities of sound. 
 
 To compute the velocity of sound in air at the temperature 
 of the experiment, make use of the formula v = 332 Vi +-%\-gt. 
 Find the mass and dimensions of the rod and compute Young's 
 Modulus for material of the rod, remembering that 
 
 the velocity ^elasticity . 
 * density 
 
 The apparatus may be used to find the velocity of sound in any 
 other gas, the above being considered a calibration. 
 
 Experiment 6 . The sonometer. 
 
 All the laws of vibrating strings or wires may be expressed 
 by the formula 
 
 where N is the number of complete oscillations per second, / the 
 length of the vibrating segment of the string, r its radius, d its 
 density, and T the tension to which the string is subjected. 
 This formula may also be put in the form 
 
 N 
 
 2 / y m 
 
 where m is the mass of unit length. This form of the equation 
 is often the more convenient. 
 
 The object of the experiment is to verify this formula experi- 
 mentally. The apparatus used is a sonometer (Fig. 68), which 
 consists of a long wooden box, upon which may be stretched 
 two or more wires. One of these wires is stretched by turning 
 
198 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 a key. The tension of the other one must be known. To 
 secure this, one end of the string is fastened to the box, and 
 the other end to a lever which moves about a knife-edge as an 
 axis. From the other end of this lever weights are suspended. 
 If the two lever arms are made equal, the tension of the string 
 is equal to the weight suspended. The length of the vibrating 
 segment of either of these strings may be varied by changing 
 the position of a movable bridge. 
 
 In performing the experiment suspend enough weights to 
 give the string to be tested the desired tension. By varying the 
 
 Fig. 68. 
 
 position of the movable bridge make three independent determi- 
 nations of the length of a segment of the string under investigation 
 which will vibrate in unison with a tuning fork of known pitch. 
 In making computations use the mean length. For two other 
 tensions make sets of readings as indicated above. Having 
 measured the values of r, I, T, etc., compute the pitch of the string 
 from 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. It may be found advantageous to 
 
SOUND. 199 
 
 put light paper riders on the string and bring the stem of the 
 vibrating fork in contact with the top of the sonometer. As 
 the bridge is moved to that position giving resonance the rider 
 will be agitated violently and may jump off. In such cases it is 
 well to approach the resonance length from both too long 
 and too short a length of string, taking the mean length as that 
 giving resonance. 
 
 Experiment 6 . The study of the transverse vibration of 
 cords by Melde's method. 
 
 If one end of a horizontal cord be fastened to a prong of a 
 heavy tuning fork, which is caused to vibrate, while the other 
 end passes over a bridge and a pulley to a scalepan or other 
 arrangement for suspending weights, it will break up into vibrat- 
 ing segments when the tension is of the proper value. If the plane 
 of the prongs be horizontal and the stem of the fork be parallel to 
 the cord, when the prong moves in one direction the cord will move 
 in the same direction, and when the direction of motion of the 
 prong is reversed the direction of motion of the cord will be 
 reversed ; thus the frequency of the cord will be the same as of 
 the fork. If the prongs and cord be in a horizontal plane but 
 with the axis of the cord perpendicular to the fork stem, then 
 the string will have one half the frequency of the fork. This 
 may be explained as follows : suppose the prong to which the 
 cord is attached move toward the bridge, the tension in the cord 
 is reduced and the cord sags. A moment later the prong is 
 moving in the opposite direction, the tension in the cord is in- 
 creasing, the cord is moving up. The prong reaches the end of 
 its excursion, the cord reaches its neutral position, but having 
 inertia passes on through it, and as the prong moves again 
 toward the bridge, the string moves up, becoming concave down- 
 ward ; thus*the fork makes a complete vibration while the string 
 makes a half vibration. 
 
 For a definite frequency, a certain cord will vibrate as a 
 whole, at a definite tension. Let the tension be reduced. A 
 
200 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 value will finally be reached at which the string will vibrate 
 in two segments, since the frequency does not change. By 
 still further reduction of the tension the string will break up 
 into three segments, and so on. 
 
 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 vibrations of 
 flexible cords : 
 
 N=f- L V. ( I98 ) 
 
 If P is the tension of the cord, s its cross section, and d its 
 density, we have also 
 
 ^=Vg- (199) 
 
 Finally, if X is the wave length, we may write 
 
 \ = — , (200) 
 
 n 
 
 V=N\, (201) 
 
 The experiment consists in setting up a heavy electrically 
 driven fork and a braided silk fishline with a bridge, pulley, 
 and appliances for varying the tension in the cord in accordance 
 with the above discussion and making the following observa- 
 tions. For the first position of the fork noted above vary the 
 tension in the cord until it is vibrating in one segment in unison 
 with the fork. Note the applied tension and the distance be- 
 tween nodes. Change the tension until the cord breaks up into 
 two vibrating segments. Note the tension, number of vibrating 
 segments, and distance between the extreme nodes. Repeat the 
 
SOUND. 201 
 
 process, obtaining three and then four vibrating segments of the 
 cord. 
 
 Change the relation of fork and cord, making them corre- 
 spond to the second case discussed above and again cause the 
 cord to vibrate in one, two, three, and four segments, noting 
 the corresponding tensions and lengths. Obtain the mass of 
 the cord per unit length and solve for the frequency of the 
 cord. Compare the frequency of the cord with that of the 
 fork. 
 
 Discuss the derivation of the formulas in your report. 
 
CHAPTER VII. 
 
 GROUP P: STATIC ELECTRICITY. 
 
 (P) General statements ; (P x ) Electrostatic induction; (P 2 ) The 
 principle of the condenser ; (P 3 ) The Holts 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 produced. 
 
 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 irQ, in which Q is the number of 
 units of electricity in the body.* This is known as Green's 
 theorem. 
 
 * Gray, Absolute Measurements in Electricity and Magnetism, vol. I, p. 10. 
 
 202 
 
STATIC ELECTRICITY. 203 
 
 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 4ttQ. 
 These lines of force, or tubes of induction, must end on some 
 other body or bodies. On the surfaces of the conductors where 
 these 4 ttQ 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 electricity 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. 69) to be an insulated 
 conductor charged with Q units of positive Fi 69 
 
 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 = -(?' + ?")• 
 
 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 
 
 * 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. 
 
204 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 elec- 
 tricity. 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 diffi- 
 cult 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, and the 
 fact that the surface of every conductor is an equipotential 
 surface. 
 
 Let it be required to map a section of the field within a hol- 
 low conductor at zero potential, containing two insulated con- 
 
 * 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. 205 
 
 ductors. One of these conductors (A) 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 dif- 
 ferent 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. 70.* It may be assumed, approxi- 
 mately, 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 dis- 
 tance into nine equal parts, and through each point of division 
 draw a line, and continue it so that it is everywhere perpen- 
 dicular to lines of force. Each of these lines must be a closed 
 curve. And from definition, each of them must lie in an equi- 
 potential surface. 
 
 By definition, the same amount of work is done in carrying 
 a charge from one point in an equipotential surface to any 
 point in another equipotential surface. Therefore, the field 
 
 * The equipotential lines and lines of force in Fig. 70 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 
 
 q 
 potential, according to the formula V ■=■ 2 — All the equipotential surfaces, from 
 
 3 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 re- 
 duced 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 con- 
 ductors 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. 
 
206 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 
 
 Fig. 70. 
 
 equipotential surfaces are closest together, the lines of force 
 should be most numerous. 
 
 It should be noticed that one of the conductors (B) has lines of 
 force issuing from it and also others 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. 
 
 Potential as used in electricity and magnetism is analogous to gravita- 
 tional potential, in so much as it means work done to bring a unit pole to the 
 
STATIC ELECTRICITY. 207 
 
 designated point in a magnetic field, or a unit charge to a point in an electric 
 field, from a place where the field is zero; that is, from a point without the 
 field, which is assumed to be of zero potential. This is analogous to the con- 
 ception of potential energy ; for. if sea level be assumed as a level of zero 
 potential energy, a kilogram weight raised 10 meters above that level is said 
 to have 10 kilogram-meters of potential energy and the gravitational potential 
 of the point is called 10, since a unit mass was used. If for example a mass 
 of 20 kilograms be moved from the sea level to a place where the gravitational 
 potential is 50, the amount of work done would be 50 x 20 kilogram-meters, 
 since the gravitational potential gives the amount of work that must be done on 
 one kilogram to carry it from sea level to the point in question. Just as this 
 potential energy is independent of path, so also are magnetic and electric po- 
 tentials. Assuming always a positive unit pole or charge to be used, then 
 as work is done against or by the magnetic or electrical forces, so is the sign 
 of the potential arbitrarily taken as plus or minus. 
 
 Considering two points in a field of force as A and B, let V„ and V b be 
 the potentials at a and b ; V a — V h , the work per unit charge or pole required 
 to move it from a to b. Therefore V a — Vi, — average force per unit quantity 
 times the length ab or (V a — V h )/ab = average force per unit quantity or 
 pole. When b is very close to a and the force does not change abruptly, 
 we have 
 
 - r F. (204) 
 
 In some cases we may proceed from a known relation between force and 
 distance and find the potential at points required. As an example we may 
 consider the potential at a point outside an isolated and insulated conducting 
 sphere. For external points we may consider the entire charge as concen- 
 trated at the center of the sphere. 
 
 Then if r is the distance from the center of a charged sphere, with + Q 
 units, to a point P, the force is given by 
 
 P=9- (205) 
 
 r' 2 
 
 The element of work done in moving a unit charge a distance dr toward Q is 
 
 dlV=Fdr=Q—- (206) 
 
 r" 
 
 The total work done in moving the unit charge from without the field to the 
 point Pis then 
 
 W = (pdr = Q C— = 9l = V (207) 
 
 J J x r 1 r 
 
 the potential at P. 
 
 When other charges are in the field, the potential at a point may be found 
 
 by adding potentials at the point due to each charge, giving due regard to the 
 
 signs. 
 
208 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The method indicated above is easily applied where the attracting or 
 repelling region may be considered as concentrated in certain points, as in the 
 case of charges in static electricity uniformly distributed on spheres and 
 of poles in magnetism, which may be looked upon as points in certain extreme 
 cases. When, as is usually the case, the charge or pole is distributed, the 
 determination of the potential becomes a problem in integration and often of 
 considerable difficulty. 
 
 Nearly all experiments on static electricity are more success- 
 ful in cold weather than in warm. This difference is probably 
 due to a difference in humidity. In moist air, bodies are rapidly 
 discharged, and it is relatively much more difficult to accumulate 
 charges upon their surfaces. In cold weather the absolute hu- 
 midity is usually much less than in warm weather. In an arti- 
 ficially heated room, the absolute humidity remaining the same 
 as the outside air, the relative humidity is very much les- 
 sened 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 ball were negatively electri- 
 fied, it would tend to move in the opposite direction. This 
 offers a means of testing the direction of lines of force, 
 and consequently 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 conduc- 
 tor, one of three things will occur : 
 
 (i) 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 
 
STATIC ELECTRICITY. 209 
 
 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 
 electrified. 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. 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 consider- 
 able 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. 
 
210 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Experiment P r 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. 
 
 For this experiment, a rather large insulated conductor is 
 required. A Leyden 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 con- 
 ductor, 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 elec- 
 trical 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, the connection with 
 the ground having been first broken, and note any change in its 
 
STATIC ELECTRICITY. 211 
 
 condition. Finally, remove the conductor to a distance from 
 the inducing body 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 suc- 
 ceeds best in cold weather, when the room is artificially heated. 
 
 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 two vertical sections of the two conductors show- 
 ing equipotential surfaces and lines of force, one when the con- 
 ductor having the induced charge is insulated, and one when it 
 is grounded. 
 
 II. 
 Faraday's ice pail. 
 
 The object of this experiment is to show that when one kind 
 of electricity is induced, an equal amount of the opposite kind is 
 also induced. 
 
 Support a metal cylinder on an insulating stand, and connect 
 
212 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 it by means of a fine wire to a gold-leaf electroscope. The top 
 of the cylinder should be closed except for a circular opening 
 large enough to admit a proof-plane or a small metal sphere 
 with an insulating handle, without danger of touching. 
 
 I. Obtain a charge on the proof -plane or carrier sphere from 
 an electrical machine and note carefully the action of the elec- 
 troscope during the following operations : 
 
 (i) Bring the charged body near the cylinder. 
 
 (2) Hold it well within the cylinder but not making contact. 
 Move the charged sphere about within the cylinder, taking care 
 not to make contact. 
 
 (3) Finally touch the cylinder with the charged sphere. 
 
 II. Ground the cylinder and electroscope temporarily so that 
 they are in a neutral state. (In cold, very dry weather touch- 
 ing with the hand may not be sufficient to ground the apparatus. 
 Always make connection with a wire to a gas pipe.) Also dis- 
 charge the case of the electroscope in the same way. Obtain 
 another charge on the insulated carrier sphere and proceed as 
 follows, noting the indications of the electroscope : 
 
 (1) Introduce the charged sphere within the cylinder, taking 
 care not to make contact, as before. 
 
 (2) Ground the cylinder or electroscope. 
 
 (3) Break the grounding circuit and make contact between 
 the carrier sphere and the insulated cylinder. 
 
 Make careful notes of each step in both sets of observations 
 outlined above, repeating several times, and illustrate the dis- 
 tribution of charges and lines of force by diagram. 
 
 Experiment P 2 . 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 
 
STATIC ELECTRICITY. 213 
 
 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. 
 
 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 con- 
 denser, 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 1 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. 
 
2i 4 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 (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 
 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 
 Leyden 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 frac- 
 tion 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. In- 
 vestigate 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. 
 
STATIC ELECTRICITY. 215 
 
 Addenda to the report: 
 
 (1) Indicate whether in the case of a condenser with a gas 
 or a liquid as dieleetric there would be anything comparable 
 to the residual charge of a Ley den jar. 
 
 (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 two vertical sections of the jar, with coatings 
 quite wide apart and showing lines of force and the vertical 
 sections of equipotential surfaces, one in which the potential 
 of one coating is zero, and one in which the surface of zero 
 potential lies between the coatings. 
 
 (5) Determine approximately the electrostatic capacity of 
 the jar from it's dimensions. 
 
 (6) Assume the difference of potential between the coatings 
 to be 100 electrostatic units, and compute the electrostatic force 
 in the glass between the coatings. 
 
 (7) Compute the total charge in the jar Under the above 
 conditions. 
 
 (8) 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 : 
 
 (1) 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 
 
216 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 of the charge on every part of the machine. Repeat these 
 observations several times, and observe whether the polarity of 
 the machine becomes reversed. 
 
 (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 
 steadily but not very rapidly. Remove the crossbar, and deter- 
 mine 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 ; 
 {&) with the crossbar in position, and the terminals very wide 
 apart ; (c) without the crpssbar, 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. 
 
STATIC ELECTRICITY. 217 
 
 Experiment P 4 . The Holtz machine {continued). 
 
 After performing Exp. P 8 , the following further experiments 
 with an electrical machine will be found very instructive : 
 
 I. 
 
 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 con- 
 nected 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 assumption 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. 
 
 Replace the Leyden jars by large ones. Separate the termi- 
 nals 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 circumstances, 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 
 
218 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 the jars are very nearly discharged when the plate ceases to 
 rotate. 
 
 Addenda to the report : 
 
 (i) If the electrostatic capacity of the condenser used is 
 known, as well as the electrostatic difference of potential pro- 
 ducing the sparks of known length, calculate the electrical work 
 done in ergs per second and in watts. 
 
 (2) Prove upon theoretical grounds that the number of sparks 
 per minute is inversely proportional to the capacity of the con- 
 denser used. 
 
 (3) Indicate the cause of the backward rotation in the second 
 experiment above. 
 
CHAPTER VIII. 
 
 GROUP Q: MAGNETISM. 
 
 (Q) General statements ; (Q x ) 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 ) Deter- 
 mination of magnetic moment by the magnetometer ; (Q 4 ) 
 Measurement of the intensity of a magnetic field ; (Q 6 ) Dis- 
 tribution 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 
 magnetic substance are called lines of magnetization. A mag- 
 
 219 
 
220 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 netic 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. 
 
 That there is a magnetic field surrounding the earth is shown by 
 the fact that in all localities where the experiment has been per- 
 formed, a magnet is acted on by a torque tending to bring a cer- 
 tain line of the magnet called the magnetic axis into a particular 
 position. 
 
 A magnet suspended freely in a magnetic field always comes to 
 rest with its magnetic axis tangent to the lines of force. The 
 positive direction of a line is 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 negative 
 pole in the negative direction. The mutual action of two mag- 
 nets when brought near together may also be stated in the fol- 
 
MAGNETISM. 221 
 
 lowing form : Like poles repel each other, 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 con- 
 ception. It is a very useful conception, however, and by most 
 authors is made the basis of the whole system of electromag- 
 netic 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. Q 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 D > 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 L, and the intensity of the field by H, the mag- 
 netic moment of the magnet may be defined by the relation 
 
 MH= L, (208) 
 
 in which M is the magnetic moment. 
 
222 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 of force." 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 blueprint 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 : 
 
 ( i ) 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 
 
MAGNETISM. 223 
 
 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 should 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. 
 
224 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Addenda to the report: 
 
 (i) 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 ; magnetic 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 this 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 
 magnet at its mid-position to the potential energy when at the 
 
MAGNETISM. 225 
 
 end of its swing. The method of derivation is the same as that 
 pursued in Exp. E r 
 
 MH=^f. (209) 
 
 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 eliminated 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^Ag II, making two independent sets of observa- 
 tions at each of two assigned stations. 
 
 (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 once be determined 
 from observations taken in a locality where His known. 
 
 The data obtained by performing both Q 2 and Q 3 is sufficient 
 for the computation of M and H in absolute measure. Two 
 conditions, however, are assumed and must be complied with. 
 
 I. 
 
 The magnetic moment of the magnet used in both experi- 
 ments must be the same. To comply with this condition the 
 same magnet must be used for both experiments, and the mag- 
 
 *Kohlrausch, Physical Measurements, p. 128. 
 vol. 1 — Q 
 
226 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 netic moment must not have suffered any change between the 
 times of performing the two experiments. 
 
 II. 
 The horizontal intensity of the field where the magnet oscil- 
 lated in Exp. Q 2 is the same as the field where the magnetic 
 needle was placed in Exp. Q 3 . This condition may be complied 
 with by performing the two experiments at the same place, and 
 at a distance from all movable magnets or magnetic substances. 
 
 Addenda to the report : 
 
 (i) 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. 71, in which a 
 and b are the forces and a and /3 the angles which their result- 
 ant r makes with them, respectively. 
 
 Obviously, tan a = - and tan /3 = -. 
 a b 
 
MAGNETISM. 
 
 227 
 
 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. 
 
 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. 
 
 Fig. 71. 
 
 -H, 
 
 A Fig. 72. 
 
 For example, the strength of the field at A, due to a magnet 
 whose poles are at n and s (Fig. 72), is 
 
 2 MB 
 
 H A = - 
 
 A (£>2 _ pf 
 
 At B, the strength of the field due to the same magnet is 
 
 (210) 
 
 H B = 
 
 M 
 
 (211) 
 
 (B 2 + Pf 
 
 in which M is the magnetic moment of the magnet, 2 / the dis- 
 tance between its poles, and D the distance of A or B from 
 the center of the magnet. 
 
228 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 8 according to the relation 
 
 _ = L___^tanS, ( 2I2 ) 
 
 and there will be a corresponding relation for the position B. 
 
 These equations may also be derived by an application of 
 the principle of moments to the magnetic forces acting on the 
 suspended needle. 
 
 The magnetometer consists, essentially, of a suspended 
 magnetic needle, a bar, i m. or more long, upon which to place 
 the magnet with which the experiment is to be performed, and 
 some means of measuring the angle through which the needle 
 is turned. 
 
 In preparation for this experiment, adjust the magnetometer 
 bar at right angles to the magnetic meridian. This may be 
 done with sufficient accuracy with the aid of a small compass 
 needle. If greater accuracy is desired, adjust the magnetom- 
 eter bar by trial so that the same deflection is produced when 
 the magnet is placed on opposite sides of the magnetometer 
 needle, at the same distance from it, and with the same pole 
 pointing towards it. 
 
 Having completed the adjustment, proceed as follows: 
 
 (i) Place the magnet on the bar with its poles pointing east 
 and west, and at a distance of not less than 20 or 30 cm. east 
 of the magnetometer needle. 
 
 (2) Observe the magnetometer reading by means of a tele- 
 scope and scale, and the mirror on the magnetic needle. Then 
 turn the magnet end for end, keeping its distance from the needle 
 the same, and again observe the reading. ' Half the difference 
 of the two readings is a measure of the deflection of the needle 
 from its position of rest on account of the presence of the magnet. 
 
 (3) Reverse the magnet in this way several times so as to 
 get the average of a number of observations. 
 
MAGNETISM. 
 
 229 
 
 (4) Finally place the magnet at tfie same distance to the 
 west of the magnetometer needle and proceed as before. From 
 the average of all the deflections observed, and the distance 
 between the mirror and scale, compute the angle through which 
 the needle is deflected. 
 
 As a check the observations should be repeated with the 
 magnet at such a distance from the needle as to produce a 
 deflection which is considerably greater or less than that first 
 used. 
 
 The following table gives typical data and shows the method 
 of presenting them : 
 
 Magnetic Moment by the Magnetometer. 
 
 North Pole 
 
 Distance of 
 
 Scale 
 
 
 Pointing 
 
 Center of Mag. 
 from Needle. 
 
 Reading, 
 
 Scale Div. 
 
 
 
 48.37 
 
 
 West 
 
 54 W. 
 
 3-°4 
 
 45-33 
 
 East 
 
 54 W. 
 
 94.62 
 
 46.25 
 
 East 
 
 54 E. 
 
 95-63 
 
 47-37 
 
 West 
 
 54 E. 
 
 3-57 
 48.26 
 
 44.69 
 
 West 
 
 80 W. 
 
 35.86 
 
 12.40 
 
 East 
 
 80 W. 
 
 60.68 
 
 12.42 
 
 East 
 
 80 E. 
 
 60.85 
 
 12.59 
 
 West 
 
 80 E. 
 
 35-95 
 48.26 
 
 12.31 
 
 Distance between poles 
 
 of magnet, 2 / = 22 cm. 
 
 Horizontal intensity = 0.145 
 Average deflection = 45.91 
 
 Distance from mirror to 
 
 scale =91.1 scale division 
 tan 2 S = 0.5037 
 
 28 =26° 44' 
 
 tan S = 0.2376 
 
 Magnetic moment = 2436 
 
 Average deflection 
 
 tan 2 8' 
 
 28' 
 
 tan 8' 
 
 Magnetic moment 
 
 = 12.43 
 = 0.1364 
 
 = 7° 46' 
 = 0.0679 
 = 2426 
 
 In the above formula, 2 / is the distance between the poles 
 of the magnet, and is therefore less than the length of the bar 
 itself. The position of the poles, and therefore the length 2 /, 
 may be approximately determined by the aid of a small compass. 
 
 If H is known, M may be computed ; or, if the product MH 
 is known, both M and H may be computed in absolute measure. 
 This product may be obtained by the method described in 
 Exp. Q 2 . 
 
230 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 If the magnetometer admits of it, a similar series of obser- 
 vations should be taken with the magnet placed at points north 
 and south of the magnetometer needle, its poles pointing east 
 and west as before. 
 
 Addenda to the report : 
 
 (1) Determine the strength of each pole of the magnet ex- 
 perimented with. 
 
 (2) Calculate the magnetic force and potential due to the 
 magnet for two or three points in its neighborhood. 
 
 (3) Calculate the work required to carry a magnet pole of 
 strength equal to either pole of the magnet, from one pole face 
 to the other pole face, along any path. 
 
 (4) Compare the pull on either magnetic pole with the pull 
 of gravity on one gram for a case in which the inclination of 
 the earth's lines of force is 75°. 
 
 Experiment Q 4 . Measurement of the intensity of a mag- 
 netic field. 
 
 The intensity of a magnetic field at different points may be 
 compared with the intensity of the earth's field by either of the 
 methods made use of in Exps. Q 2 and Q 3 . In the following 
 experiment, these methods are to be used in measuring the 
 magnetic field at a series of points in the neighborhood of a 
 permanent magnet. 
 
 I. 
 
 Place the magnet with its axis in the magnetic meridian, its 
 negative pole pointing north. For all points to the east or west 
 of the middle of the magnet, the intensity of the field will be 
 the arithmetical sum of the earth's horizontal intensity and the 
 intensity of the field at that point, due to the magnet. For all 
 points to the north or south of the magnet, the intensity of the 
 field will be the difference of these two quantities. 
 
 Determine the period of oscillation of a small magnet of any 
 shape, for a series of points on a line at right angles to it, and 
 bisecting the distance between its poles. Do the same for a 
 
.MAGNETISM. 231 
 
 series of points north or south of the magnet. For each point, 
 the number of oscillations produced in three or four minutes 
 should be determined. 
 
 As the intensity of the field due to the magnet varies most 
 rapidly near it, the points of observation should be closer 
 together the nearer they are to the magnet. A good series 
 of distances is the geometric series ^L, \L, \L, ■■■ 2 L, in 
 which L is the length of the magnet. 
 
 As in Exp. Q 2 , we have 
 
 H ^ H = A M = h (2I3) 
 
 in which H P is the intensity of the field due to the magnet at 
 the point where the time of vibration is T P , H is the horizontal 
 intensity of the earth's magnetism, and C is a constant depend- 
 ing upon the magnet. C may be eliminated by taking the time 
 of vibration at a distance from the magnet where H P is zero. 
 H P can then be computed in terms of H, or if H is known it 
 may be computed in absolute measure. 
 
 Plot a curve with distances from the magnet as abscissas, 
 and corresponding values of H P as ordinates. 
 
 II. 
 
 Place a large bar magnet at right angles to the magnetic 
 meridian, as in Exp. Q 3 . For points " A " and " B " the ratio 
 of the intensity of the field, due to the magnet and the earth's 
 horizontal intensity, will be equal to the tangent of the angle 
 through which a magnetic needle will be deflected from, the 
 magnetic meridian, if placed at that point. 
 
 Place a compass with a circle graduated to degrees at a 
 series of points " A," at distances from the magnet as in I 
 above. For each point determine the deflection from the 
 meridian as follows : Read both ends of the needle, then 
 reverse the magnet and read again. The angle through which 
 the needle has been turned is double the angle of deflection 
 from the meridian. In the same manner make a series of 
 
232 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 observations for points " B." Plot a curve with distances from 
 the magnet as abscissas, and the intensity of the field due to the 
 magnet as ordinates. 
 
 Addenda to the report: 
 
 (i) From the equations in Exp. Q 3 , and several points on 
 one of the above curves, compute values of M. The points 
 taken should not be observed points unless these points happen 
 to fall exactly upon the curve. 
 
 (2) Note whether these values of M show a progressive 
 increase or decrease, and if so, indicate cause of the variation. 
 
 Experiment Q 5 . Distribution of " free " magnetism in a per- 
 manent magnet. 
 
 For purposes of calculation, the distribution of imaginary 
 magnetic matter in a magnet may be considered in two ways : 
 as a distribution throughout its volume, or over its surface. 
 The volume distribution or intensity of magnetization is greatest 
 midway between the poles. The surface distribution is greatest 
 near the ends, and is vanishingly small midway between the 
 poles. This imaginary magnetic matter is supposed to be so 
 distributed as to produce by its attraction or repulsion the same 
 field of force that the magnet produces. From this we see that 
 the quantity of surface magnetism, or "free" magnetism, as it 
 is called, is everywhere proportional to the number of unit lines 
 of force which enter or emerge from the magnet. 
 
 The distribution of magnetism may be determined by measur- 
 ing the force necessary to detach a small, soft iron armature from 
 the magnet. For measuring this force use a pair of balances 
 or a spiral spring, whose extension can be readily determined. 
 
 Determine in this way the force necessary to detach the 
 armature for ten or twenty points along the magnet from one 
 end to the other. The magnet may not be symmetrically 
 magnetized ; if not, the forces at symmetrical points will not be 
 
MAGNETISM. 233 
 
 equal. Plot a curve with distances from the center of the 
 magnet as abscissas, and the forces necessary to detach the 
 armature as ordinates. 
 
 In considering that this curve represents the distribution 
 of magnetism along the magnet, two things should be re- 
 membered : 
 
 (1) By induction, the distribution of magnetism is slightly 
 changed on account of the presence of the armature. If the 
 armature is quite small with respect to the magnet, this may be 
 neglected. 
 
 (2) The force with which the soft iron armature is attracted 
 to the magnet is proportional to the square of the magnetism 
 at that point. This is true, since the force is proportional to 
 the product of the magnetism of the magnet and the magnetism 
 of the armature, in the immediate neighborhood of the point 
 of contact, but the induced magnetism of the armature is itself 
 proportional to the magnetism of the permanent magnet. 
 
 II. 
 
 The distribution of magnetism may also be determined by 
 the method of oscillations. 
 
 Place a bar magnet in a vertical position, and determine the 
 period of oscillation of a small magnet for a series of positions 
 along the magnet, and quite close to it. These points should be 
 north or south of the magnet. 
 
 As in Exp. Q 4 , we have 
 
 H P ±H=-^, (214) 
 
 in which H P is the intensity for the point P of the field due to 
 the magnet, resolved in a direction perpendicular to its length. 
 
 If the point P is quite close to the magnet, H P will be pro- 
 portional to the free magnetism at the corresponding point of 
 the magnet. As in (1), plot a curve with distances from the 
 center of the magnet as abscissas and corresponding values of 
 H P as ordinates. 
 
CHAPTER IX. 
 
 GROUP R: THE ELECTRIC CURRENT. 
 
 (R) General statements ; (R x ) The law of the galvanometer ; 
 (R 2 ) Measurement of current by electrolysis ; (R 3 ) Theory 
 of shunts ; (R 4 ) Measurement of the constant of a sensitive 
 galvanometer ; (R 6 ) Measurement of current by means of 
 the galvanometer ; (R 6 ) Calibration of an ammeter. 
 
 (R) General statements concerning the electric current. 
 
 The electric current may be Refined as the rate at which 
 electricity is transferred, or the amount of electricity which 
 passes through a given plane cutting the circuit at right angles 
 to the lines of flow, in a unit time. When two bodies which 
 differ in potential are connected by means of a conductor, the 
 fleeting phenomena which accompany the electric discharge 
 occur, and we have a transient current; if the difference of 
 potential be maintained constant by the expenditure of work, 
 there will be a permanent current. 
 
 If current were always measured by electrolysis, the idea of 
 current would be a derived conception involving time. Since, 
 however, when a current flows in a conductor there is a mag- 
 netic field surrounding the conductor, the intensity of which at 
 any given point is always directly proportional to the current, 
 it is more convenient to measure the latter by means of the 
 field which it produces. In this way we reach a conception of 
 current which does not directly involve time. 
 
 Those instruments which measure currents by comparing 
 the field produced with the earth's magnetic field, with the 
 earth's field as modified by controlling or regulating magnets, 
 
 234 
 
THE ELECTRIC CURRENT. 
 
 235 
 
 or with the field of a permanent magnet, are called galva- 
 nometers. 
 
 The lines of force surrounding a wire carrying a current 
 may easily be mapped by the aid of iron filings. Figs. 73 and 
 74 are such maps showing the field around a straight wire. 
 The former was obtained by cutting a sensitive plate and 
 passing the conductor, the field of which was to be mapped, 
 through the hole. The plate was then fastened in a position at 
 right angles to the conductor. Current was sent through the 
 
 C^JE^S), 
 
 Fig. 73. — Map of the Field around a Wire carrying 
 Current (from a Photograph). 
 
 Fig. 74. 
 
 latter, and the surface of the film was strewn with iron filings. 
 These operations having been completed by the red light of the 
 developing room, the plate was then exposed for three seconds 
 to gaslight, after which the photograph was developed, giving 
 the map. 
 
 The direction of the lines of force, at any point in the field, 
 produced by a current, is always at right angles to the plane 
 containing the point and the current. The intensity of this 
 field may be deduced from Laplace's law, of which the following 
 equation is a statement : 
 
 /sin 6ds 
 
 dHp = : 
 
 (21S) 
 
236 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Iji this expression, dH P is the intensity of the magnetic field 
 at the point P, due to the short element ds of the current of 
 intensity /; r is the distance from the point P to the element 
 ds, and 8 is the angle which the direction of the element ds 
 makes with the line drawn from it to the point P. 
 
 The absolute unit of current in the electromagnetic system 
 is that current, a unit length of which will produce unit magnetic 
 field at unit distance from the current. It follows that the 
 C. G. S. unit current in the electromagnetic system, flowing 
 around a circle of 1 cm. radius, will produce at the center of 
 the circle a field whose intensity is 2 ir units. 
 
 The Chamber of Delegates of the Electrical Congress at 
 Chicago adopted " as a [practical] unit of current the interna- 
 tional ampere, which is one tenth of the unit of current of the 
 C. G. S. system of electromagnetic units, and which is represented 
 sufficiently well for practical use by the unvarying current which, 
 when passed through a solution of nitrate of silver, in water, and 
 in accordance with the accompanying specifications, deposits silver 
 at the rate of 0.001 1 18 gram per second." 
 
 The Tangent galvanometer. 
 
 The intensity of the field at the center of the galvanometer 
 coils produced by unit current flowing in the coils is called the 
 true constant of the galvanometer, and is generally denoted by 
 G. For many galvanometers, this constant may be computed 
 from Laplace's law, and the dimensions, position, and number 
 of turns of the coils. 
 
 If a galvanometer coil is placed with the plane of its 
 windings in the magnetic meridian, the field produced by it 
 at the center of the coil (or at any point on its axis) will be' 
 at right angles to the earth's field. Let CO (Fig. 75) repre- 
 sent the horizontal section of the galvanometer coil. The 
 intensity of the field at the point O, due to the current / flowing 
 in the coils, is GI. If H represents the horizontal intensity of 
 the field at theg^int O due to the earth (and regulating mag- 
 
THE ELECTRIC CURRENT. 
 
 237 
 
 nets), the resultant of these two fields will make an angle 8 
 -with the plane of the coils such that 
 
 7=^tanS. 
 
 (216) 
 
 If a short magnetic needle be suspended at the point O, it will 
 come to rest with its magnetic axis in the plane of the resultant 
 magnetic field through the point O. That is, it 
 will turn through the angle 8 from the position of 
 equilibrium when no current flows. If the mag- 
 netic needle is not short, its ends are liable to 
 extend too far beyond the point O at which the 
 field due to the current I has the intensity Gl. 
 If the current is required in amperes instead of 
 in absolute units of current, the above equation 
 becomes 
 
 (217) 
 
 K 
 
 1= io^tanS. 
 
 The constant quantity 10 — is called the "reduc- 
 
 G 
 
 tion factor," the "working constant," or, for 
 
 brevity, simply the constant of the galvanometer. 
 
 If this be represented by 7 , we have 
 
 f—f ta.n8. 
 
 Fig. 75. 
 
 (218) 
 
 From (218) it is obvious that the galvanometer constant 7 is 
 that current which will produce a deflection of 45 . 
 
 In this discussion it is assumed that the friction of the 
 needle on the pivot or the torsion of the suspending fiber is 
 negligible. This is generally a safe assumption except in sensi- 
 tive galvanometers, where a very small needle or an astatic 
 system is used. In such cases, the moment of the force of. 
 torsion tending to bring the needle back to its position of 
 equilibrium may be very considerable compared with the 
 moment of the magnetic forces tending to return the needle 
 to the magnetic meridian. Moreover, there may be a twist in 
 
238 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 the fiber, such that the needle does not return to the magnetic 
 meridian when the current ceases to flow in the galvanometer 
 coils. 
 
 In Fig. 75, above, it is obvious that if the galvanometer 
 current is reversed, the direction of the field GI will be 
 reversed. In this case the magnetic needle will be turned 
 through the same angle 8, with its north end pointing on the 
 opposite side of the meridian. This offers a means of setting 
 the galvanometer coils in the plane of the magnetic meridian, if 
 they are not already so adjusted. 
 
 For this purpose, we send a current through the galvanome- 
 ter, and observe the angle through which the needle has turned 
 when it comes to rest. We then reverse the current through 
 the galvanometer, and observe the corresponding angle of 
 deflection on the opposite side of the position of equilibrium. 
 If these angles are not equal, we turn the galvanometer coils in 
 such a direction as to increase the smaller angle of deflection, 
 and repeat until the difference of \ the two angles is a small 
 fraction of either one of them. In doing this it must be remem- 
 bered that if the scale is turned with the galvanometer coils, the 
 needle will come to rest at a new position with respect to the 
 scale ; i.e. the galvanometer will have a new " zero point." 
 
 In measuring current by means of a galvanometer, angles 
 of deflection should be determined for the current, both direct 
 and reversed. There are two reasons for this : 
 
 (1) If the galvanometer coil makes a small angle with the 
 magnetic meridian, it may be proved * that the mean of the 
 deflections for direct and reversed current will be in error by a 
 small quantity of the second order. 
 
 (2) The equilibrium position or zero point of a galvanometer 
 needle is constantly varying, the fluctuations being due to 
 variations in the earth's magnetic field. Now it is quite as 
 
 * See Mascart and Joubert, Lecons sur ' '""'-icite et le magnetism, vol. 2, 
 p. 235 ; also Nichols, The Galvanometer, Lecture 2. 
 
THE ELECTRIC CURRENT. 
 
 2 39 
 
 Fig. 76. 
 
 convenient to observe the reading for reversed current as it is 
 'to observe the zero reading for every measurement. 
 
 In measurements by this "method of direct and reversed 
 deflections a commutator, or reversing key, is used. A double- 
 pole double-throw switch may be used for reversing the 
 direction of the current in any part of the 
 system desired by connecting the points af 
 and dc. When the blades of the switch 
 are closed, as shown in Fig. 76, the current 
 flows from the battery along the path afeghbcd. When the 
 blades are thrown into the reverse position, the current flows 
 along the path abhged, being in a reverse direction between 
 the points g and h. 
 
 The angle of deflection of the galvanometer needle may be 
 determined directly from the reading of a long pointer moving 
 O _ over a circular graduated 
 
 scale. In this case both 
 ends of the pointer should 
 be read in order to eliminate 
 eccentricity, as well as to 
 get a more accurate value 
 of the deflection. 
 
 In many cases the angle 
 of deflection is determined 
 by means of a small mirror 
 permanently attached to the 
 magnetic needle. With 
 mirror galvanometers -either 
 a telescope and scale, or a 
 lamp and scale, may be 
 used. In either case, the 
 angle of deflection is computed in the same way. 
 
 Let OM (Fig. 77) be the horizontal section of the mirror 
 attached to the galvanometer needle, .S the scale, and T the 
 telescope. The telescope and scale should be adjusted at right 
 
 
 t*WM»^;*A»>SA»»,. 
 
 ^x^^aWfeasfl 
 
 B 
 
 Fig. 77. 
 
240- JUNIOR COURSE IN GENERAL PHYSICS. 
 
 angles to each other, and so placed that the portion of the 
 scale immediately below (or above) the telescope is seen 
 reflected from the mirror through the telescope when no cur- 
 rent flows through the galvanometer. If the galvanometer 
 needle be now deflected through the angle 8, a new portion of 
 the scale will be seen reflected from the mirror. From the law 
 of reflection, the angle which the ray reflected from the mir- 
 ror into the telescope makes with the normal must be equal 
 to the angle which the incident ra.y AO makes with the same 
 normal. It follows that the angle A T between the reflected 
 and incident rays is equal to 2 8. If s is the deflection along 
 the scale from the scale reading when no current is flowing, and 
 L the distance of the scale from the mirror, we have 
 
 tan 2 8= y. (219) 
 
 From this equation and a table of tangents, 8, and hence tan 8, 
 may be deduced. For simplicity of computations, the distance 
 L from the scale to the mirror is often made equal to 50 scale 
 divisions. In this case, the difference between the scale read- 
 ings for the same current for the two positions of the reversing 
 switch, direct and reversed, divided by 100 gives directly 
 
 tan 2 8 = H- = double deflection . ( 2 19') 
 
 2-50 100 
 
 It may be readily shown that when 8 is quite small, tan 8 is 
 very nearly proportional to s. Under this condition it follows 
 that current is very nearly proportional to deflection, and we 
 may use the equation f = ^ (22o) 
 
 in which /„' is the constant per scale division. The error in 
 using this approximate value for the current is as follows : 
 
 s 1 
 When — = — , the error is about 0.0025. 
 L 10 
 
 When — = -, the error is about 0.0100. 
 L 4 
 
THE ELECTRIC CURRENT. 
 
 241 
 
 When the current is reversed in a galvanometer, it often 
 takes several minutes for the needle to come to rest. Time 
 may be saved after a reading has been made in one direction, 
 by opening the switch and leaving it open until the needle in 
 its free swing has nearly come to rest, then closing and open- 
 ing the switch quickly several times in the reversed position, 
 finally leaving it closed in that position. A little practice will 
 make possible quite rapid readings. 
 
 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 suspend- 
 ing 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, caus- 
 ing the lines of force to sweep through the 
 copper, 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. 78 shows a vertical 
 section of the copper sphere, with the inclosed magnet ; Fig. 
 79 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 sensi- 
 
 Fig. 78. 
 
 VOL. I — R 
 
242 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 tive 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 sensitive 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. 
 
 The d Arsonval galvanometer. 
 
 The d'Arsonval galvanometer is an instrument 
 
 based on the fact that a wire carrying a current which 
 
 is in a magnetic field has a force acting on it which is 
 
 proportional to the length, /, of the wire in the field, 
 
 the strength of the current, /, flowing in the wire, and to the 
 
 strength of the field, /. 
 
 The following equation gives an expression for the force : 
 
 F= Ilf sinfl, 
 
 (221) 
 
 in which 6 is the angle between the direction of the magnetic 
 field and the wire carrying the current. 
 
 If a coil of wire, free to rotate about its axis, be suspended 
 in a magnetic field, there will be a torque action tending to turn 
 the coil about its axis when current flows in the coil. 
 
 The d'Arsonval galvanometer is based on the above outline. 
 It consists of a coil of wire suspended by a phosphor bronze 
 or steel ribbon above and a small wire coil below. The suspen- 
 sions serve as conductors for the current into and out of the coil. 
 The coil is suspended to rotate freely between the poles of a 
 strong horseshoe magnet. 
 
 When a current flows in the coil, the magnetic torque causes 
 the coil to turn about its" axis. The coil will come to rest in 
 
THE ELECTRIC CURRENT. 
 
 243 
 
 such a position that the return torque due to the suspensions is 
 equal to the magnetic torque. 
 
 The expression for the magnetic torque is obtained as fol- 
 lows : The force acting on each vertical length of wire / is 
 
 Ilf sin 6 and the corresponding torque is Ilf sin 6 - cos 8. In 
 
 the case here taken the field is supposed to be uniform and at 
 right angles to the length of the elements / of the wire as 
 shown in Fig. 8o, which gives a vertical sec- 
 tion (a) and a plan (b) in which the coil is 
 displaced through an angle 8. Since = 90 , 
 the total torque will be given by the expression 
 
 L — 2 nllf- cos 8 = IAf cos 8, (222) 
 
 in which n is the total number of turns and A 
 is the " equivalent area " of the coil, A = nla. 
 If the pole pieces be properly shaped and a 
 soft iron core be placed within the open space 
 of the coil, the magnetic field will be approxi- 
 mately radial and the plane of the coil will be 
 in the plane of the field. Under these conditions the magnetic 
 torque will be constant, 8 will be zero, and therefore 
 
 L = IAf. (223) 
 
 The mechanical torque tending to return the coil to its rest 
 
 position for no current is- approximately proportional to the 
 
 angular deflection 8. An expression for the value of the 
 
 mechanical torque may be written 
 
 L m = L 8, (224) 
 
 in which Z is the mechanical constant. (See F 2 .) When cur- 
 rent is flowing in the coil a deflection will be produced and 
 the coil will come to rest as noted above when 
 
 IAf=L 8. (225) 
 
 Solving this equation for /, 
 
 
 (226) 
 
244 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 that is, the current is proportional to the angular deflection. 
 This principle is used in the manufacture not only of the sta- 
 tionary form of d'Arsonval galvanometers, but also of portable 
 types such as ammeters and voltmeters for direct current 
 measurement. 
 
 If the field be uniform and of constant direction in the 
 neighborhood of the suspended coil from equations 222 and 224 
 it may be shown that 
 
 / = ^a -A- = k' tan B. (227) 
 
 ,4/ cos B v " 
 
 If I is expressed in amperes, the right-hand member must be 
 multiplied by 10. If a telescope and scale be used to measure 
 8, the same reasoning may be applied as that given above in the 
 case of the tangent galvanometer. For small deflections it may 
 be found that currents flowing through the coil of the d'Arsonval 
 galvanometer are very nearly proportional to the scale deflec- 
 tions. Indeed, the galvanometer may be so designed as to give, 
 very nearly a constant relation between the current to be meas- 
 ured and the deflection. In such a case the law for the galva- 
 nometer may be expressed as 
 
 I = ks, (228) 
 
 in which k is the current constant in amperes per scale division 
 of single deflection and s is the number of scale divisions of 
 single deflection. In all cases the d'Arsonval galvanometer 
 should be carefully calibrated over the range it is to be used. 
 
 For a given current the deflections from the "zero," or n® 
 current scale reading should be alike. Readings should be 
 made both direct and reversed for each current value. Rapid 
 reading is facilitated by the operations noted above in the use 
 of the tangent galvanometer. 
 
 Experiment R x . Law of the galvanometer. 
 
 The introduction to the R group should be carefully read 
 before performing either of the following experiments. 
 
THE ELECTRIC CURRENT. 245 
 
 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 
 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). 
 
 Use a galvanometer in which the angles are to be observed 
 directly. 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. Set the plane of the coil to be used in the 
 magnetic meridian by the method indicated in the introduction 
 to this group. It is usually not sufficient to bring the pointer to 
 the zero of the circular scale. In general the scales will be found 
 to be so mounted as to make this impossible. The " zero " read- 
 ings will consequently be the readings of the ends of the pointer 
 when no current is flowing. In making readings always read 
 both ends of the pointer for both directions of current through 
 the coil. Designate the two ends of the pointer so that there 
 may be no confusion in interpreting the readings of its two ends 
 for direct and reverse readings. They are designated as TV and 
 
246 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 5 in the sample of data given below, but that may not be satis- 
 factory with the galvanometer assigned. 
 
 Two sets of readings are to be made, using coils of different 
 number of turns as directed. Enough gravity cells should be 
 used to give a deflection of about 70° when using the coil of the 
 larger number of turns, with no resistance in the resistance 
 box. Make readings for current in both directions, " direct and 
 reverse," for each of at least ten different box resistances from 
 
 to 40 ohms at approximately equal steps. 
 
 Using the data thus obtained, plot two curves, one for each 
 coil used, on the same sheet to the same scale with cotangents 
 as ordinates, and box resistances as abscissas. If the galvanom- 
 eter obeys the tangent law, each curve obtained by plotting as 
 above should be a straight line. Draw a straight line, there- 
 fore, that passes as nearly as possible through all the points, 
 and produce this line backward until it intersects the horizontal 
 axis. The distance between the origin and this point of inter- 
 section is a measure of the resistance in the circuit outside the 
 resistance box ; i.e. if we call this resistance R , then R = gal- 
 vanometer resistance + battery resistance 4- resistance of con- 
 necting wires. The slope of line is -9 • 
 
 E 
 
 Now, if E is the E. M. F. of the cell, and R the resistance 
 
 in the box, then 
 
 /=/ tanS = -A-. ( 229 ) 
 
 R + R 
 
 If E is known, and R determined from the curve, as stated 
 above, the constant of the galvanometer can be computed. In 
 making this computation, take the E. M. F. of a gravity cell as 
 
 1 volt. In making computations it will be found advantageous 
 to use the tables in the back of the Manual. The values of 
 both these unknown quantities are to be obtained from the 
 curves as well as from the data. 
 
 In your report, derive the law of the tangent galvanometer. 
 Discuss the effect on the galvanometer constant of the torsion 
 
THE ELECTRIC CURRENT. 
 
 247 
 
 of the fiber, the length and magnetic moment of the needle. 
 From Ohm's law, derive the equation of the curve and show 
 that the intercept and the slope have the meanings indicated 
 above. (See introduction on " Curves.") 
 
 
 
 
 LAWI 
 
 )F TAN! 
 
 ENT 
 
 ILVANC 
 
 METER 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 z 
 < 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 BO) 
 
 : RES 
 
 STAN 
 
 3ES 
 
 
 
 
 
 
 
 10 12 14 
 
 Fig. 81. 
 
 18 20 22 24 26 28 
 
 Addenda to the report: 
 
 (1) Explain, using diagram showing the relation of the mag- 
 netic field due to the current following in the galvanometer coils 
 and the earth's field, why the plane of the coils must be in the 
 magnetic meridian. 
 
 (2) What additional data, if any, would be necessary in 
 order to find the value of HI Show how H would be 
 computed. 
 
 (3) Show that for the two coils used the ratio of the 7 s is in- 
 versely as the number of turns. The mean radii of the two 
 coils are considered as equal. 
 
 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. 8 1 shows the graphical 
 method of testing the deviation of the instrument from the law 
 of tangents. 
 
248 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Law of Tangent Galvanometer. 
 
 Resist- 
 
 Galvanometer Readings. 
 
 Mean 
 De- 
 flection. 
 
 tan 5. 
 
 cot 6. 
 
 
 Box. 
 
 Direct. 
 
 Reversed. 
 
 /<>• 
 
 O 
 
 jN. 62°-5 
 ( S. 62°.o 
 
 N. 63°.o ) 
 S. 6 3 °. 5 J 
 
 62 -70 
 
 1-937 
 
 0.516 
 
 0.613 
 
 2 
 
 ( N. 5o°.6 
 I S. so°.o 
 
 N. so°.o > 
 S. 5o°.2 J 
 
 50°.20 
 
 1.200 
 
 -833 
 
 0.625 
 
 4 
 
 jN. 4 i°-S 
 < S. 4i°.o 
 
 N. 4i°.o > 
 S. 4i°.2 J 
 
 4I°.20 
 
 0.875 
 
 1. 142 
 
 0.623 
 
 6 
 
 J N. 34 °.8 
 1 S. 34 °.2 
 
 N- 34°-5 I 
 S. 35M) 
 
 34°-8o 
 
 0.695 
 
 '•439 
 
 0.616 
 
 10 
 
 1 N. 2 S °. S 
 1 S. 25°.o 
 
 N. 2 S °. 5 ) 
 
 S. 26°.o) 
 
 2S°-S° 
 
 0.477 
 
 2.096 
 
 0.628 
 
 H 
 
 | N. 2o°.4 
 (S. i9°.8 
 
 N. 20°.I > 
 
 S. 2o°.6 f 
 
 2O°.20 
 
 0.368 
 
 2.718 
 
 0.627 
 
 18 
 
 jN. i7°.o 
 iS. i6°. 5 
 
 N. i7°.o) 
 S. i7°.6i 
 
 i7°.oo 
 
 0.306 
 
 3-271 
 
 0.610 
 
 24 
 
 J N. i 3 °. 4 
 
 1 S. I2°. 9 
 
 N. I3°.2[ 
 S. I3°.8 \ 
 
 i3°-3° 
 
 0.236 
 
 4.230 
 
 0.620 
 
 3° 
 
 j N. io°.9 
 Is. io°. 3 
 
 N. io°. 9 > 
 S. n°.4i 
 
 io°.9o 
 
 0.193 
 
 5-193 
 
 0.621 
 
 5° 
 
 } N. 7°.o 
 \ S. 6°-5 
 
 n. y.i \ 
 
 S. 7°.6) 
 
 7°.o 5 
 
 0.124 
 
 8.086 
 
 0.613 
 
 From curve J? = 3.33 ohms. 
 " " /„ = 0.620 amp. 
 Last column computed assuming value of R obtained from plot. 
 The E.M.F. of the battery used in taking the data in the table \vas4v0Its. 
 
 II. 
 
 The d' Arsonval galvanometer. 
 
 The d' Arsonval galvanometer has several advantages over 
 instruments of the tangent type, since its governing field is 
 very strong and practically constant in direction and amount. 
 It is, therefore, possible to use it in the neighborhood of elec- 
 trical machinery, even where the external fields are varying 
 and have considerable strength. Since the moving coil must 
 
THE ELECTRIC CURRENT. 249 
 
 have metallic leads for carrying the current into and out from 
 the coil, the coil may be supported in such manner as to make 
 the instrument portable. Portable d'Arsonval galvanometers are 
 in general use in portable Wheatstone bridges (Exp. T x ) and 
 commercial electrical testing sets. The commercial direct 
 current ammeters and voltmeters, station or portable', are gen- 
 erally nothing more than d'Arsonval galvanometers. 
 
 Connect the galvanometer in series with a battery, a known 
 variable resistance, and a reversing switch, in such a manner 
 as to make it possible to reverse the direction of the current 
 through the galvanometer. 
 
 Make scale readings for direct and reverse current for each 
 of at least ten different values of box resistances. It will be 
 found that a very satisfactory set of readings may be obtained 
 by taking resistances from 4 to 40 ohms at intervals of 4 ohms. 
 
 From Ohm's law 
 
 1= „ " „ = ks, (230) 
 
 R + R ' K * J 
 
 in which E is the E. M. F. of the battery ; R the known box 
 resistance ; Rq the unknown resistance of the circuit, which 
 includes the galvanometer, battery, and connecting wires ; k is 
 the constant of the galvanometer expressed in amperes per 
 scale division of single deflection ; and s is the number of scale 
 divisions of single deflection ; that is, the mean deflection for a 
 given value of the known resistance, from the no current or 
 " zero " reading. 
 
 Using values of R as abscissas and corresponding values of 
 the reciprocals of s as ordinates plot a curve. The negative 
 intercept on the axis of abscissas will give the value of R . 
 Taking the value oij? obtained from the curve, the known 
 values of E, R, and s, compute values of k for each value of R. 
 
 Make a second set of readings as indicated above, using 
 another battery, and treat the readings in the same manner, 
 plotting a curve and computing values of k. 
 
 Derive the physical equation of the curves, interpret it, and 
 
250 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 obtain all the physical constants possible from the curves. (See 
 notes on curves in the Introduction, and the interpretation of 
 curves in R x I.) 
 
 Discuss the factors on which the sensibility of the d'Arson- 
 val galvanometer depends. If the galvanometer assigned is 
 very sensitive, i.e. a very small current produces a large deflec- 
 tion, it will be found necessary to shunt across its terminals a 
 suitable resistance to reduce the sensibility of the instrument 
 so far as the current flowing in the main circuit is concerned. 
 The constant k thus obtained will refer to the current in,'the main 
 circuit necessary to produce one large scale division of deflection. 
 The theory of divided circuits is discussed in Experiment R 3 . 
 
 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 electricity 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 
 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 pre- 
 pared for each cell by wrapping copper wire on cylindrical 
 forms. The size of the coils depends somewhat on the strength 
 of the current used. There should be not less than 50 sq. cm. 
 of surface of the coil per ampere. 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. 
 
 * See Ryan, Transactions of the American Institute of Electrical Engineers, 
 vol. 6, p. 322. 
 
THE ELECTRIC CURRENT. 
 
 251 
 
 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. 
 82). 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 flowing. (One cou- 
 lomb deposits 0.000328 gram of 
 copper.) The amount of copper dis- 
 solved is always slightly in excess of 
 the amount deposited, and for various 
 reasons 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 purpose should be 
 fastened by one end and then cleaned 
 with sandpaper. The wires should be carefully wiped with 
 filter paper to remove all copper dust, sand, and grit, taking 
 care to keep the fingers off that portion of the wire which is 
 to receive the deposit. A failure to observe these precautions 
 may make the deposit flake off and necessitate a new set of 
 observations. 
 
 When thoroughly cleaned, the wire is coiled upon a 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 
 
 Fig. 82. 
 
252 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 alcohol. After again rolling 
 on filter paper, what little alcohol is left will quickly evaporate, 
 leaving the coil dry and ready for weighing. The loss coil 
 should also be cleansed with sandpaper, 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 connec- 
 tions can then be made 
 in such a way as to make 
 the deposit occur 071 the 
 inner coil. 
 
 In this experiment the 
 voltameter is to be used 
 in determining the con- 
 stant of a tangent galvanometer. Two voltameter cells . 
 (Fig. 83) are used as a check on the weighings, the two cells 
 being connected in series with the galvanometer and with 
 each other. The connections are outlined in Fig. 84. A re- 
 versing switch must be placed in the circuit so that the galva- 
 nometer may have the current reversed through it without 
 reversing the current in the other parts of .the circuit. There 
 must also be a short circuiting switch in such a position that 
 the current through the galvanometer may be reversed without 
 breaking the remainder of the circuit. The short circuiting key 
 is to be kept open excepting when reversing the direction of the 
 current through the galvanometer. 
 
 You will be assigned a galvanometer, a station at which to 
 set up the voltameter cells, and the numbers on the switchboard 
 
 Fig. 83. 
 
THE ELECTRIC CURRENT. 253 
 
 of the battery (storage cells) and of the resistance R. Con- 
 nect these four stations in series on the switchboard, having in all 
 of the adjustable resistance R. Then by means of a compass 
 find the direction of current. Before breaking the circuit adjust 
 R until a single deflection (deflection one way) equal to about 
 15 large scale divisions is obtained. Open the key at the galva- 
 nometer. Adjust the voltameter coils in the circuit so that cur- 
 rent flows from the larger to the smaller coils, pour the 
 CuS0 4 into the jars, and close the circuit at the galvanometer, 
 observing the hour, minute, and second. Be sure you know 
 how to manipulate the keys so as to reverse the galvanometer 
 current without breaking the circuit through the voltameters. 
 
 A steady current is sent through the circuit for some meas- 
 ured 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. Deflections, both 
 direct and reversed, are to be observed 
 at intervals of two or three minutes 
 throughout the experiment. I f y\ 
 
 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, 
 thoroughly washed with tap water, rolled on filter paper to take 
 off "the surplus water, and dried over a Bunsen burner, holding 
 them high enough not to oxidize. The second weighing should 
 be made as soon as possible after the coils are dry and cool. 
 
 The constant of any galvanometer may be measured by this 
 method. 
 
 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, 
 R, (Fig. 84), should be placed across the galvanometer ter- 
 
2 54 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 minals. 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 : 
 
 Galvanometer Constant by Copper Voltameter. 
 
 
 Galvanomeu 
 
 r Readings. 
 
 Other Data and Results. 
 
 Time. 
 
 
 
 
 
 Current 
 
 Current 
 
 Distance of mirror from scale = 50 scale div. 
 
 
 Direct. 
 
 Reversed. 
 
 Average double deflection = 32.27 " 
 
 hr. min. 
 
 
 
 
 9 49 
 
 Circuit 
 
 completed. 
 
 tan 28 = 0.3227 28 =17° 53' 
 
 51 
 
 — 
 
 42.40 
 
 8 = 8°56|' tan 8 =0.1573 
 
 54 
 
 74.62 
 
 ■ — 
 
 Two voltameter cells in series : 
 
 57 
 
 — 
 
 42-43 
 
 Before After Gain 
 
 10 1 
 
 "74.62 
 
 
 
 Cathode A . . 27.434 28.686 1.2520 
 
 5 
 
 — 
 
 42-43 
 
 " B . . 27.5715 28.624 1.2525 
 
 9 
 
 74.68 
 
 — 
 
 
 '3 
 
 — 
 
 42-39 
 
 Duration of run = 3600 sec. 
 
 17 
 
 74.70 
 
 — 
 
 Intensity of current / = 1 .059 amp. 
 
 21 
 
 — 
 
 42.38 
 
 For tangent galvanometer, I = I tan 8 ; 
 
 25 
 
 74.67 
 
 — 
 
 ■• ^0= 6-73 amp. 
 
 29 
 
 — 
 
 42.32 
 
 
 33 
 
 74.70 
 
 — 
 
 Galvanometer, one turn, needle at center : 
 
 37 
 
 — 
 
 42-31 
 
 Diameter of ring = 77.7 cm. 
 
 4i 
 
 74-57 
 
 
 True constant G = 0.1617 
 
 45 
 
 
 
 42.30 
 
 
 49 
 
 74.56 
 
 — 
 
 /„ = 10 — : .-.// = 0.109 
 
 49 
 
 Circuit 
 
 broken. 
 
 G 
 
 In your report state Faraday's laws and their bearing on 
 your experiment. Define the C. G. S. unit of current and the 
 practical unit, stating the relation between them. From experi- 
 mental laws and definitions prove that G, the true or coil con- 
 stant of a tangent galvanometer, the needle of which is in the 
 
 plane of the coils, is 
 
 2 irn 
 
 What physical meaning has G? I J 
 
THE ELECTRIC CURRENT. 255 
 
 Compute G and find H. How will the sensibility be affected 
 by changing the magnetic field where the needle hangs ? Ex- 
 plain the method used of testing the direction of the current. 
 Explain the telescope and scale method of reading angles and 
 their tangents. 
 
 Experiment R 3 . Theory of shunts. 
 
 When a current flows in a divided circuit in which there is 
 no E. M. F. the currents in the branches are inversely as the 
 resistances in those branches. For a branch circuit of two parts 
 the relation may be written as follows : 
 
 r=f- too 
 
 This simple case has many practical applications in electrical 
 measuring instruments ; for example, in the ammeter, the volt- 
 meter, and the galvanometer when too sensitive to measure the 
 current directly. In using a sensitive galvanometer to measure 
 comparatively large currents it is shunted by means of a suitable 
 resistance through which a definite portion of the current flows. 
 The current flowing through the galvanometer multiplied by a 
 certain constant called the " multiplying power of the shunt " 
 gives the value of the current in the main circuit at the point of 
 division. 
 
 If /, be used to denote the current through the shunt R, and 
 /„ the current through the galvanometer branch R g analogous, 
 to the equation above, the relation showing the division of the 
 current is 
 
 Remembering that /, + /„ is equal to the current / in the 
 main circuit the following equation may be written 
 
 /, + /„_/ _ R„ + R. 
 
 In In R. 
 
 (232) 
 
 a 
 
 - r l\. n -h .A. „ , 
 
 or 1=1 g °x ', (233) 
 
256 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 from which the multiplier of /„ is the "multiplying power of 
 the shunt." Shunting an instrument does not change the sensi- 
 bility of the instrument itself, but it does alter its constant as a 
 measuring instrument for finding current in the 
 main circuit. 
 
 The following experiment is intended to verify 
 the relation existing between currents in a simple 
 divided circuit, as given in equations' 231 and 233. 
 
 Connect a battery, a galvanometer with a revers- 
 ing key, and a resistance R, at least 100 times the 
 galvanometer resistance, in series. Shunt across the 
 galvanometer terminals a known variable resist- 
 ance R t . Under these circumstances the current 
 in the main circuit may be assumed to be constant, 
 no matter how much the resistance R, in the resistance box is 
 varied. If / stands for current in the main circuit, the above 
 equation to be verified becomes 
 
 Fig. 85. 
 
 I a R, + 1 '„R. =a IR, 
 
 (234) 
 
 in which the variables are R, and I g . 
 
 Observe the reading of the galvanometer with current, 
 both direct and reversed, for a number of different resistances 
 in the box R„ including the reading when the circuit is broken 
 through the box. This last reading obviously represents the 
 constant current in the main circuit. The resistances taken 
 should be such as to make the galvanometer readings vary by 
 approximately equal steps. 
 
 If the resistance of the galvanometer (including connecting 
 wires in multiple with the resistance box) be now measured, 
 sufficient data will be obtained to make a number of verifica- 
 
 tions of the theory. Since /„ 
 written 
 
 I—.Ig, equation 231' maybe 
 
 I- 
 
 JjL-Eji 
 
 L 
 
 R. 
 
 (23s) 
 
 This equation can be verified, since every quantity except R„ 
 
THE ELECTRIC CURRENT. 
 
 257 
 
 is known. This is more apparent when the equation is written 
 in the form 
 
 R^R. 1 -/*- 
 
 (236) 
 
 By substituting in the right hand number of this equation 
 the R„ / — /„, and I t of each observation, a series of values of 
 the resistance of the galvanometer R g is obtained. If these 
 values of R g thus computed agree, the equation is verified. 
 
 As current appears in both numerator and denominator of 
 equation 236 in the first degree, any quantity which is propor- 
 tional to current may be substituted for it; as the galvanometer 
 deflection, or its tangent. 
 
 If the observations be plotted with box resistances as ab- 
 scissas, and tangents of galvanometer deflections as ordinates, 
 
 Fig. 86. 
 
 the curve obtained will be a hyperbola, with an asymptote 
 parallel to the axis of abscissas. (See Fig. 86.) 
 
 If the observations be plotted with reciprocals of box resis- 
 tances as abscissas, and cotangents of deflections as ordinates, 
 the resulting curve should be a straight line, the intercept on 
 
 the axis of abscissas being equal to — — . 
 
 R g 
 
 The following table gives a typical set of data from readings, 
 
 taken with a tangent galvanometer, and shows the method of 
 
 arranging them. If these results be plotted as indicated above, 
 
 they will be found to give a curve the form of which is that of 
 
 Fig. 86. 
 
 VOL. I — S 
 
2 5 8 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Currents in a Divided Circuit. 
 
 Shunt 
 Resist- 
 
 Galvanometer 
 Readings. 
 
 Double 
 
 De- 
 flections. 
 
 Tan 8. 
 
 h- 
 
 h = T-I a . 
 
 Kg. 
 
 ance 
 
 Current 
 
 Current 
 
 
 
 Direct. 
 
 Reversed. 
 
 
 
 
 
 
 8 
 
 37-55 
 
 6.40 
 
 31-15 
 
 .1522 
 
 .01578 
 
 
 
 
 
 40 
 
 35-30 
 
 8.82 - 
 
 26.48 
 
 .1302 
 
 .01350 
 
 .00228 
 
 6.72 
 
 3° 
 
 34.60 
 
 9-38 
 
 25.22 
 
 .1241 
 
 .01287 
 
 .OO29I 
 
 6.78 
 
 20 
 
 33.60 
 
 IO.52 
 
 23.08 
 
 .II40 
 
 .OIl82 
 
 .OO396 
 
 6.70 
 
 15 
 
 32.70 
 
 II.42 
 
 21.28 
 
 .1052 
 
 .OIO9I 
 
 .00487 
 
 6.69 
 
 12 
 
 31.98 
 
 I2.I8 
 
 19.80 
 
 .0980 
 
 .OIOI6 
 
 .OO562 
 
 6.63 
 
 8 
 
 30.50 
 
 13-77 
 
 16.73 
 
 .0830 
 
 .00862 
 
 .OO716 
 
 6.64 
 
 6 
 
 29-33 
 
 I4.9O 
 
 14-43 
 
 .0718 
 
 .00745 
 
 .00836 
 
 6.71 
 
 5 
 
 28.64 
 
 15-54 
 
 13.10 
 
 .0652 
 
 .00675 
 
 .OO903 
 
 6.71 
 
 4 
 
 27.87 
 
 16.50 
 
 "•37 
 
 .0566 
 
 .00587 
 
 .OO994 
 
 6.76 
 
 3 
 
 26.76 
 
 17-33 
 
 9-43 
 
 .0471 
 
 .00488 
 
 .OIO9O 
 
 6.72 
 
 2 
 
 25.67 
 
 18.63 
 
 7.04 
 
 .0352 
 
 .OOJ65 
 
 .OI2I3 
 
 6.66 
 
 1 
 
 24.13 
 
 20.17 
 
 396 
 
 .OI98 
 
 .00205 
 
 ■OI373 
 
 6.68- 
 
 
 
 22.06 
 
 21.99 
 
 0.07 
 
 .0004 
 
 .00004 
 
 
 
 
 
 103.7 X I0 ~ 
 
 Derive the physical equations of both curves, interpret 
 them, and get all possible physical constants from them. If a 
 d'Arsonval galvanometer be used, be careful to use the proper 
 terms in expressing the current. 
 
 Experiment R 4 . Measurement of the constant of a sensi- 
 tive galvanometer. 
 
 It is frequently impracticable to calculate the constant of a 
 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: 
 
THE ELECTRIC CURRENT. 259 
 
 / = 7 tan S, 
 
 1= I sin 8, (237) 
 
 I=I'>s, 
 ■ I=ks, 
 
 according to the law of the galvanometer. 
 
 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 galvanometer 
 of small sensibility whose constant is already known. For this 
 purpose 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 galvanometer whose constant is known in 
 series with a battery of constant E. M. F., a reversing key, and 
 two variable resistances. One 
 variable resistance is to be placed 
 at the station with the _ known 
 galvanometer and the other re- 
 sistance at the station of the 
 sensitive galvanometer where 
 there is to be also a reversing 
 key. 
 
 (2) Connect the sensitive galvanometer in multiple with the 
 variable resistance box at its station. The resistance in multi- 
 ple with the sensitive galvanometer should not be less than 10 
 ohms unless very well known and the variable resistance in the 
 main circuit should be so adjusted as to give a deflection of 
 about 45 if the galvanometer whose constant is known is of 
 the direct reading tangent type, or nearly as large a deflection 
 
260 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 as the scale will permit, if the reading is made by means of a 
 mirror. 
 
 (3) Adjust the resistance in multiple with the sensitive gal- 
 vanometer, until its deflection is nearly across the scale, heed- 
 ing the precaution noted above. It may be found necessary to 
 insert in series with the sensitive galvanometer a high resist- 
 ance in order to bring the readings within the required range. 
 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 two other series of 
 observations with different resistances in the circuit, produc- 
 ing deflections varying considerably from the first. 
 
 From the deflection of the galvanometer of known constant, 
 the current 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. 
 
 Answer the addenda at the end of the R± group. 
 
 II. 
 
 If the galvanometer whose constant is known be replaced 
 by a voltameter the current in the main circuit can be meas- 
 ured as in Exp. R v 
 The rest of the experi- 
 ment is the same as 
 above. 
 
 It may happen with 
 a very sensitive galvan- 
 ometer that a current 
 strong enough to pro- 
 duce a suitable deposit 
 in the voltameter will 
 require a shunt of ex- 
 cessively low resistance in multiple with the galvanometer. 
 Under these circumstances, it will be advisable to connect the 
 
 Fig. 88. 
 
THE ELECTRIC CURRENT. 261 
 
 galvanometer in multiple with a branch which itself is in mul- 
 tiple with a portion of the main circuit, as in Fig. 88. 
 
 If the ratio of the resistance in each pair of branches is 
 known, the current flowing in the coils of the galvanometer 
 may be calculated from the current flowing in the main circuit. 
 
 It is to be observed that both of the above methods deter- 
 mine the constant independently of the value of any resistance. 
 They simply depend upon a knowledge of the ratio of resistances. 
 
 III. 
 
 The current flowing may be determined from Ohm's law, 
 provided the E. M. F. of the battery is known in volts and the 
 resistance of each portion of the circuit is known in ohms. 
 
 For the purpose of this determination, a standard Daniell 
 cell may be very easily constructed as follows : Place an amal- 
 gamated zinc rod in a porous cell containing a saturated solu- 
 tion of zinc sulphate. Coil around the porous cell eight or ten 
 turns of rather large copper wire, which has been previously 
 cleaned with sandpaper. Place the porous cell in a larger ves- 
 sel containing a semi-saturated solution of copper sulphate. 
 The two vessels should be thoroughly cleaned before using. 
 
 Such a cell at 15° has an E. M.F. of 1.074 volts. It should 
 be used immediately, although its E. M.F. will change very 
 little for several hours. The internal resistance of such a cell 
 is usually negligible compared with 10,000 ohms. But if it is 
 thought desirable to do so, its resistance may be afterwards de- 
 termined by the method described in Exp. T 7 . Either the 
 Clark or cadmium cell affords a more accurate standard, but 
 either is more difficult to construct, and possess the disadvan- 
 tage of high internal resistance. 
 
 The following is the procedure : 
 
 (1) Connect the galvanometer in series with a resistance of 
 at least 10,000 ohms, and the standard cell. 
 
 (2) Observe the galvanometer readings, and repeat them 
 several times to get a good average. As a check, repeat these 
 
262 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 observations with two or three different resistances in series 
 with the cell and galvanometer. The galvanometer deflection 
 may be deduced from the readings, and the current flowing 
 may be calculated from Ohm's law. An application of one of 
 the above equations will then give the galvanometer constant. 
 
 It may happen that the galvanometer used is so sensitive 
 that the deflection is too great to be read even when all the 
 available resistance is in the circuit. In this case the deflection 
 may be diminished as described in part I of this experiment. 
 
 In the above it has been assumed that the law of the gal- 
 vanometer is known. In nearly all galvanometers, some one of 
 the equations given above hold pretty accurately up to io° or 
 20°, which is the maximum deflection that should be used with 
 reflecting galvanometers. If the galvanometer deflection is read 
 directly by means of a pointer moving over a graduated scale, 
 the maximum deflection may be much greater. 
 
 In all such cases the galvanometer should be calibrated. 
 For this purpose proceed as in any of the above experiments, 
 and observe the resistances that correspond to deflections, vary- 
 ing by approximately equal increments from zero to the maxi- 
 mum reading that the scale admits. At least ten or twelve such 
 observations should be taken. 
 
 Plot a curve with currents flowing through the galvanom- 
 eter as abscissas, and galvanometer deflections, as ordinates. 
 This curve is called the calibration curve of the instrument. 
 
 If the galvanometer is furnished with a regulating magnet, 
 its exact position should be noted at the time of performing the 
 experiment. 
 
 Addenda to the report : 
 
 (i) Indicate the difficulties which make it impracticable to 
 calculate the constant of a sensitive galvanometer from its 
 dimensions. 
 
 (2) If the instrument considered is a tangent galvanometer, 
 from its constant and the horizontal intensity of the field where 
 
THE ELECTRIC CURRENT. 263 
 
 the magnet hangs, calculate the true constant of the galvanom- 
 eter. 
 
 (3) If the instrument is a reflecting galvanometer, calculate 
 the constant per scale division. 
 
 (4) Discuss the influence of the position of a regulating 
 magnet upon the sensitiveness of the galvanometer. Where 
 should the magnet be placed in order to change the zero point 
 by a few scale divisions and yet have the least effect in chang- 
 ing the galvanometer constant ? 
 
 (5) How could a magnet be placed quite near a tangent 
 galvanometer and yet have an inappreciable effect, either to 
 turn the needle, or to change the galvanometer constant. 
 
 Experiment R 5 . Applications of the galvanometer to the 
 measurement of current. 
 
 I. 
 
 Measurement of the current from a battery with different 
 arrangements of the cells. 
 
 This experiment serves as a study of the conditions under 
 which certain groupings of cells are most advantageous. 
 
 Theory indicates that the only case where a multiple arrange- 
 ment of cells gives a larger current than any other is when the 
 external resistance is small. Since the resistance of the galvanom- 
 eter and connecting wires are included in the external resist- 
 ance as well as that inserted by means of the resistance box, 
 it follows that if a galvanometer coil is used, the resistance of 
 which approaches that of a single cell, no point will be found 
 where the current for series arrangement is less than that for 
 multiple, and one of the objects of the experiment cannot be 
 accomplished. 
 
 For this experiment a galvanometer of small sensitiveness 
 and of very low resistance is required. The galvanometer 
 should have two or three different degrees of sensitiveness, 
 either by having separate coils if a tangent galvanometer or by 
 means of shunts. 
 
264 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 (1) Connect a closed-circuit battery* of four or six cells in 
 series with the tangent galvanometer. The circuit should con- 
 tain also a variable known resistance and a reversing key. 
 
 (2) Arrange the cells in series and measure the current for 8 
 or 10 different box resistances, ranging from o to 20 ohms; as, 
 for example, o, 1, 2, 3, 5, 7, 10, 15, and 20. 
 
 (3) Measure the current for two other groupings of cells, (a) 
 all in multiple, and (b) a multiple-series arrangement. The latter 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "> 
 
 -^ 
 
 L v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■, ALL 
 
 SERIES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 AUtoU 
 
 LTJPle 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *V K 
 
 5 6 7 8 9 10 
 
 TOTAL RESISTANCE OUTSIDE OF BATTERY 
 
 12 13 14 15 
 
 Fig. 89. 
 
 grouping may be arranged by dividing the battery into two 
 equal parts, connecting the cells in each part in multiple and 
 then connecting the two parts in series. In both of these 
 arrangements make galvanometer readings for resistances as 
 follows: o, .5, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, and 20 ohms. In 
 all cases use the galvanometer coil or shunt that will give the 
 greatest deflection. Changes may be made within a set of 
 readings, noting the resistance at which the change is made. 
 
 (4) Obtain the resistance of the galvanometer coils, or 
 shunted galvanometer, and connecting wires including lines 
 leading to the battery. Plot, on the same sheet to the same 
 
 * Any battery which does not suffer marked polarization will serve for this pur- 
 
 pose. 
 
THE ELECTRIC CURRENT. 265 
 
 scale, curves with resistances outside of the battery as abscissas, 
 and currents in amperes as ordinates. 
 
 From the curves determine under what conditions each sepa- 
 rate grouping of the cells would produce a greater current than 
 any other grouping. Each curve is an hyperbola. The point 
 of intersection of the all-series all-multiple curves gives the 
 resistance of a single cell, provided they are alike. Prove this, 
 and find what the other intersections indicate. 
 
 Prove that a group of cells gives a maximum current 
 through a given external resistance when the cells are so 
 arranged that the internal resistance equals that of the external 
 resistance as nearly as possible. 
 
 It is well to test separately the cells used in order to find 
 whether each gives the same current under like conditions. 
 
 II. 
 
 Measurement of current by the Vienna method. 
 
 This method is usually employed with currents so large that 
 they cannot be measured directly by ordinary galvanometers or 
 ammeters. The main current is sent through a heavy wire of 
 German silver or similar material, whose resistance changes very 
 little with temperature, and a galvanometer is connected in 
 multiple with this resistance. A constant small proportion of 
 the current will always pass through the galvanometer, arid can 
 be measured. From the resistance of the German silver coil, 
 together with that of the galvanometer, the ratio of this meas- 
 ured current to the total current can be computed, and the latter 
 is therefore determined. 
 
 The practice of the method may be illustrated by the meas- 
 urement of the current from a nonpolarizing cell of high elec- 
 tromotive force and low internal resistance. In place of a cell 
 a commercial thermo-battery may be used. 
 
 Before beginning the experiment, compute the resistance of 
 the shunt that must be put across the terminals of the galva- 
 
266 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 nometer, in order that the maximum readable galvanometer de- 
 flection will be produced when the maximum current to be 
 measured flows in the main circuit. This can be done if the 
 galvanometer constant is known ; but it will be necessary to 
 assume approximate values for the electromotive force and 
 internal resistance of the battery. 
 
 Having determined the proper resistance of the shunt, pro- 
 ceed as follows : 
 
 ( i ) Connect the cell in series with a variable resistance, and 
 with the shunt which is in multiple with the galvanometer. 
 
 (2) Observe the galvanometer readings when several differ- 
 ent resistances are used in series with the cell. These resist- 
 ances should vary from one to ten ohms. 
 
 (3) Plot a curve with resistances as abscissas, and reciprocals 
 of currents flowing in the main circuit as ordinates. This curve 
 should be a straight line, and from its constants the electromo- 
 tive force and internal resistance of the cell may be computed. 
 
 III. 
 
 To investigate the effect of polarization upon current. 
 
 (1) Connect a Le Clanche cell, or some other cell that polar- 
 izes rapidly, in series with five or ten ohms resistance. 
 
 (2) Connect a sensitive galvanometer whose constant is 
 known in multiple with a portion of this resistance, such that 
 the galvanometer deflection is quite large. 
 
 (3) Observe the galvanometer readings both direct and re- 
 versed every three or four minutes for half an hour or longer. 
 Then break the circuit, stir the solution in the cell, and in the 
 course of ten or fifteen minutes close the circuit, measure the 
 current flowing, and repeat three or four times. 
 
 (4) Take another cell as nearly as possible like the first one, 
 and make a similar series of observations, but with a resistance 
 of 50 or 100 ohms in series with it. 
 
 (5) Compute the current flowing in the main circuit, and 
 
THE ELECTRIC CURRENT. 267 
 
 plot a curve for each cell, with times as abscissas and currents 
 as ordinates. 
 
 Experiment R 6 . Calibration of an ammeter. 
 
 Connect a storage battery, a variable iron resistance, an am- 
 meter, and a suitable galvanometer in series. Do not put the 
 ammeter so near the galvanometer as to change the controlling 
 field if it be a tangent instrument. Take readings on the gal- 
 vanometer, direct and reversed, and corresponding readings on 
 the ammeter, using such values of current as to give ten or 
 twelve nearly equally spaced readings on the ammeter scale. 
 
 Compute the galvanometer currents and put ammeter read- 
 ings in a column parallel with said computed currents. Find 
 the error of the ammeter reading. 
 
 Find the zero correction for the ammeter and correct read- 
 ings taken for the zero error. Find the errors of these cor- 
 rected readings and their percentage errors. 
 
 Plot a curve, using calculated galvanometer currents as ab- 
 scissas and corresponding ammeter readings as ordinates. The 
 same scale should be used on both axes. 
 
 Plot another curve, taking ammeter readings as abscissas and 
 corrections, not considering the zero error, as ordinates. It is 
 advisable to put the jr-axis in the middle of the sheet on account 
 of having plus and minus errors to plot. 
 
 Describe the construction of the ammeter used, and if it is 
 "dead beat," explain why. 
 
CHAPTER X. 
 
 GROUP S: DIFFERENCE OF POTENTIAL AND ELECTRO- 
 MOTIVE FORCE. 
 
 (S) General statements; (Sj) Ohm's method for the measure- 
 ment of the E. M. F. of a battery; (S 2 ) Fall of potential in a 
 series circuit; (S 3 ) Potential difference at the terminals of a 
 battery as a function of the external resistance ; (S 4 ) Fall of 
 potential in a wire carrying current ; (S 5 ) Beets' method of 
 measuring electromotive forces ; (S 6 ) Lines of equal poten- 
 tial in a liquid conductor; (S T ). Variation in the E.M.F. of 
 a thermo-element with change of temperature ; (S 8 ) Calibra- 
 tion of a voltmeter ; (S 9 ) Comparison of electromotive forces ; 
 (S 10 ) The potentiometer. 
 
 (S) General statements concerning difference of potential and 
 electromotive force. 
 
 The indiscriminate use of the terms " electromotive force " 
 and "difference of potential" has given rise to much confusion. 
 The following treatment of the subject, though different from 
 that of many writers, is believed to be entirely consistent with 
 the facts. Moreover, it is hoped that it will make clear to the 
 mind of the student the relation between two ideas which, 
 though intimately related, are nevertheless entirely distinct. 
 
 The difference of potential between two points is that differ- 
 ence in condition which tends to produce a transfer of electricity 
 from one point to the other point. The measure of this differ- 
 ence of potential is the amount of work that would be done by 
 or against electrical forces in carrying unit quantity of electricity 
 from the one point to the other point. From definitions it follows 
 
 268 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 269 
 
 that whenever electricity is transferred along a circuit, between 
 
 two points a and b, work is done according to the relation 
 
 z x 
 
 W=k(V a - V h )It. 
 
 The difference of potential between a and b is one electro- 
 magnetic unit, if work is done at the rate of 1 erg per second, 
 when the current flowing is one electromagnetic unit. This 
 choice of unit potential difference makes k unity in the above 
 equation. The electromagnetic unit of electromotive force is 
 that electromotive force which is capable of producing unit 
 difference of potential. It may be proved that the electromag- 
 netic unit of electromotive force is produced whenever unit 
 magnetic lines of force are cut at the rate of one per second. 
 The practical unit of difference of potential is called a "volt." 
 It is equal to io 8 electromagnetic units. 
 
 Any generator of electricity (whether it be a battery, dynamo, 
 or electrical machine) is capable, when energy is supplied to it, 
 of maintaining a. difference of potential between its terminals, 
 even though they are connected by a conductor. It is to this 
 capability of maintaining a difference of potential that we apply 
 the name of electromotive force. The electromotive force of a 
 generator is measured by the maximum difference of potential 
 which it is capable of producing when no current flows. Or, 
 when a current is allowed to flow, it is measured by the differ- 
 ence of potential at the terminals, plus the fall of potential due 
 to the resistance of the generator. 
 
 From these definitions it follows : 
 
 (1) That there is a difference of potential between any 
 two points of a circuit conveying a current. 
 
 (2) That the electromotive force of a circuit is always 
 located in the generator. The source of a counter electromotive 
 force may always be looked upon as a negative generator. So 
 far as our present knowledge extends, there is never any electro- 
 motive force in a perfectly homogeneous conductor which is not 
 moving relatively to a magnetic field. 
 
370 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The above meaning of the term "electromotive force" is 
 always in mind when it is said that a given conductor is the seat * 
 of an electromotive force, as in the case of a wire moving in a 
 magnetic field ; also when it is said that there is no electromotive 
 force in a given branch of a multiple circuit. Counter electromo- 
 tive force is the negative of electromotive force, as above defined. 
 
 Ohm's law as originally stated, using modern terms, is : The 
 current flowing in a {perfectly homogeneous) conductor {not moving 
 relatively to a magnetic field) is directly proportional to the dif- 
 ference of potential between the terminals of the conductor. If the 
 conductor between two points is in any way varied subject to 
 the above conditions, the current will be equal to the difference 
 of potential between the points divided by a quantity known as 
 the resistance of the conductor between the points. From which 
 we have „ „ ,„ 
 
 r=^=^_\ oxi= d -V. (238) 
 
 R* dR v ' 
 
 The statement that the current flowing in a circuit is equal 
 
 to the total electromotive force in the circuit, divided by the 
 
 total resistance of the circuit, is a deduc- 
 
 r. * A 
 
 tion from Ohm's law.* 
 
 The above discussion may be fixed 
 in the mind of the student by the fol- 
 lowing graphic treatment of two par- 
 Fig- 90. 
 
 ticular cases. 
 
 Let BC, Fig. 90, be the two poles of a cell whose electro- 
 motive force is two volts, and internal resistance four ohms. 
 The negative pole of the cell is maintained at zero potential 
 by being grounded, and the two poles are connected by a 
 homogeneous conductor of twelve ohms' resistance : CA four 
 ohms, and AB eight ohms. 
 
 If a curve be plotted showing the relation between potential 
 and resistance, with resistances counting from A in the direc- 
 
 * See Gray's Absolute Measurements in Electricity and Magnetism, pp. 142-146. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 
 
 271 
 
 tion of the current as abscissas and potentials for ordinates, the 
 result will be as given in Fig. 90 a. From A to B the potential 
 falls uniformly ; between the negative pole and the liquid there 
 is a finite difference of potential represented by BB' ; in the 
 liquid, supposed homogeneous, there is a fall of potential at 
 the same rate as in the outside conductor; between the liquid 
 and the positive pole there is a finite difference of potential 
 
 Ftg. 90 a. 
 
 represented by C'C, and from C to A the potential falls at the 
 same rate as before, reaching, of course, the original value. 
 
 By the application of Ohm's law the current, I =— , may 
 
 dR 
 
 be derived from any part of the circuit that is homogeneous. 
 The result is obviously the same whether increments of poten- 
 tial and resistance are infinitesimal or of any magnitude. 
 
 If the conductor is cut at A, Fig. 90, then the potential of 
 AB will immediately fall to zero. As no current flows, there 
 will be no fall of potential in the liquid ; the potential of C will 
 therefore immediately rise by the amount of the former fall 
 through the liquid. The broken line represents the potential in 
 the cell and conductor after the circuit is broken. The electro- 
 motive force of the cell E is measured by the maximum differ- 
 
!272 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 ence of potential between its terminals when no current flows. 
 This is obviously equal to pd+pd', in which pd is the difference 
 of potential between the terminals before the circuit is broken, 
 and pd' is the fall of potential in the cell due to its resistance. 
 
 It is obvious from the geometry of the figure that — , in which 
 
 R 
 
 R is the total resistance of the circuit, gives the same value for 
 
 the current as —— taken in any homogeneous part of the circuit. 
 dR 
 
 Figure 91 represents the potential as a function of the 
 
 resistance in a circuit, in which the generator is a dynamo. 
 
 Fig. 91. 
 
 In this case the potential is not a discontinuous function of the 
 resistance. The electromotive force is not located at a point 
 (or at a surface) in the circuit as in the case of a cell, but it 
 exists in all those parts of the armature which cut lines of force. 
 With this exception, the above discussion applies to the present 
 case, word for word. If the circuit is broken as before, the 
 broken line shows the condition of affairs, provided that the 
 resultant* magnetic field remains unchanged after the circuit is 
 
 * By resultant field is meant the field that is the resultant of the field due to the 
 field magnets and to the current in the armature coils. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 273 
 
 broken. This assumption does not hold true, of course, in the 
 case of the series-wound dynamo. 
 
 The above graphic representations of the potentials in a 
 conductor carrying a current bring prominently forward the 
 fact that in a conductor not containing an electromotive force 
 the current always flows from points of higher potential to 
 points of lower potential; but that in a conductor or in that 
 part of it containing an electromotive force producing a current, 
 the current always flows from points of lower to points of higher 
 potential. 
 
 From the discussion given above, it is seen that Ohm's law 
 may be stated in two forms, 
 
 '-%> (239) 
 
 for a part of a simple circuit not containing a generator or 
 source of electromotive force, in which a circuit is flowing,/^ 
 being the potential difference between two points separated by 
 a resistance R ; and 
 
 /=-, (240) 
 
 for a complete simple circuit in which E is the sum of all the 
 electromotive forces and R the total resistance. 
 
 The two equations are not identical, the distinction arising 
 from the difference between " potential difference," pd, and 
 electromotive force, E. M. F., made above.* 
 
 A more general form of Ohm's law, which may be applied 
 to a complete simple circuit or to a part of a circuit in which 
 there is or is not an E. M. F., is expressed in the form : 
 
 ,JV^±E). (24I) 
 
 Let Fig. 92 represent a part of a complete circuit in which 
 there is included between A and B a resistance R and an 
 E. M. F. acting to send current in the direction BA, although 
 
 * See also Ayrton, Practical Electricity, vol. 1, pp. 359-364. 
 VOL. I — T 
 
274 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 current is flowing from A to B, as indicated by the arrow, due 
 to some generator not shown. The difference of potential be- 
 tween A and B may be considered as 
 — *- / w\aa<w-||-a<vwvww. made up of two parts, one being that 
 Fig. 92. due to tne current f flowing through 
 
 the resistance R between A and B, the 
 other due to the electromotive force, which in the case shown 
 would make the potential of A higher than B by an amount E. 
 
 V A - V B =pd= IR + E. (242) 
 
 If the conditions of the circuit were as indicated above, except 
 that the E. M. F. be acting in the opposite direction, then 
 
 V A -V B =pd=IR-E. (243) 
 
 The general expression for the current in a simple circuit may 
 be written, therefore, as in equation (241), or 
 
 , (E-pd) . , 
 
 I= R ■ (244) 
 
 Equations 239' and 240 are readily seen to be particular 
 cases of equation 244. If the points A and B are taken far- 
 ther and farther from the generator, they may be made to coin- 
 cide; then V A — V B =pd = o, and equation 244 becomes that 
 for a complete circuit, equation 240. If by the path chosen 
 from A to B there be no E. M. F., then equation 244 reduces 
 to the form for a part of a simple circuit carrying current, in 
 which there is no E. M. F., equation 239. 
 
 The potential difference at the terminals of a generator, be 
 it battery or dynamo, is often called the external E. M. F., and 
 the E. M. F. proper is called the total E. M. F. The difference 
 of potential at the terminals of a motor, the primary of a trans- 
 former, etc., is often called in practice an impressed E. M. F. It 
 must be remembered that an external E. M. F. and an impressed 
 E. M. F. are properly " potential differences." 
 
 For many circuits that appear complex there may be found 
 equivalent simple circuits, as in Exps. R 4 , S 3 , or S 4 and in S 5 , 
 S 8 , S 9 , and S 10 , when the galvanometers in the parts of the 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 275 
 
 circuits containing the standard cells show no current flowing 
 through them. 
 
 In the more complex circuits, those containing electro- 
 motive forces in more than one branch, there are two simple 
 methods of solution : first, the superposition method, and, second, 
 a method based on Kirchhoff's laws. In the superposition 
 method the E. M. F's in each branch are taken as effective 
 separately in sending current through all parts of the circuit, 
 the other E. M. F.'s not acting. The final values of the cur- 
 rents are found by taking the algebraic sums of the currents 
 in the various branches produced by the separately considered 
 E. M. F.'s. In many cases the superposition method is very 
 cumbersome and another method based on Kirchhoff's laws is 
 found to be simple and more direct. 
 
 Kirchhoff's laws are derived from Ohm's law and may be 
 expressed as follows : 
 
 1. In any closed circuit, whether simple or made up of shunts 
 or mesh, the sum of the electromotive forces around any closed 
 path is equal to the fall of potential around that closed path, 
 due regard of course being paid to the signs. Expressed mathe- 
 matically, the law takes the following form : 
 
 Z(IR) = ?,E. (245) 
 
 2. The sum of the currents flowing to a point must be equal 
 to the sum of the currents flowing away from it ; or, at a point 
 
 2* = o, (246) 
 
 the currents flowing away from the point taking the minus sign. 
 
 In any but constant currents the laws must be used with 
 caution (and the second breaks down entirely with variable cur- 
 rents if there be appreciable capacity). 
 
 For an example of the application of Kirchhoff's laws, see 
 Exp. T 7 III, Mance's Method of Measuring the Internal Resist- 
 ance of a Battery. 
 
 In applying Kirchhoff's laws, the following suggestions may 
 be found useful : 
 
276 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 1st. Assume a direction of current through the network, and 
 then build up the equations consistent with this assumption. 
 
 2d. Use both laws ; for if only the first is used, the equa- 
 tions will not be independent. 
 
 3d. If there are n unknown currents to be found, there 
 must be n simultaneous equations. 
 
 The accompanying example will illustrate the method. 
 I, E, and R refer to the current E. M. F. and resistance in 
 each branch, including the resistance of the battery. The 
 direction of current is assumed as indicated, 
 From law 2, 7j + 7 2 + 7 3 = o. 
 
 From law 1, I\R\ — ^2-^2 — ^i~ ^v 
 From law 1, 7 2 i? 2 — f a R s = E 2 — E 3 . 
 If the E. M. F. and resistance of each branch are given, then 
 from these equations the currents may be found. The absolute 
 values of the currents will be correct, since 
 the' resistances are known and also the 
 E. M. F.'s are known in direction as well 
 as amount. Since all of the currents 
 cannot flow up to the junction point of 
 the branches, the sign of one or more will 
 be found to be negative, thus indicating 
 the current to be flowing in the opposite direction to that 
 assumed. 
 
 Experiment Sj. Ohm's method for the measurement of the 
 E. M. F. of a battery. 
 
 The object of this experiment is the determination of an 
 E. M. F. in absolute measure, without reference to any standard 
 cell. From Ohm's law we have 
 
 in which R is the constant unknown resistance of the battery, 
 connecting wires, and galvanometer; and R is the known 
 resistance which may be varied at pleasure. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 
 
 277 
 
 The procedure is as follows : 
 
 (1) Place in a simple series circuit the battery whose E. M. F. 
 is to be determined, a known variable resistance, and a tangent 
 or d'Arsonval galvanometer whose constant is known. A re- 
 versing key should be placed in the circuit to enable readings of 
 the galvanometer to be made for both directions of current 
 through it. 
 
 (2) Make a series of direct and reverse readings of the galva- 
 nometer for at least ten known box resistances, within the range 
 
 E. M. F. 
 
 BY OHMS N 
 
 ETHOD 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 30 40 50 60 70 
 
 RESISTANCE 
 Fig. 9*. 
 
 o to 40 ohms, beginning with the smallest resistance that will 
 keep the galvanometer readings on the scale, and proceeding 
 by approximately equal steps. 
 
 Make another set of readings within the same range for 
 some other battery, as directed. 
 
 From any pair of observations in either set two equations 
 may be obtained between which J? may be eliminated and E 
 determined in volts. The best results will be obtained if the 
 two resistances are so chosen as to make the two values of the 
 current quite different. It is to be observed that the method 
 depends upon the assumption that the E. M. F. is unaffected 
 
278 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 by changes in the current. With some cells this is only 
 approximately true. 
 
 The results can be readily computed by graphical methods. 
 To accomplish this, a curve should be plotted with resistances as 
 abscissas, and reciprocals of currents as ordinates. If the E. M. F. 
 remains constant, the curve obtained should be a straight line, 
 and from the " pitch " of this line E can be computed. 
 
 Plot such curves for each set of observations taken. From 
 the values of R w obtained from the curves, the observed 
 data, and the galvanometer constants compute values of E for 
 all resistances used. Derive the physical equation of the curves, 
 and interpret it. Obtain as many physical constants as pos- 
 sible from the curves. From two pairs of suitably chosen 
 observations * in each set find the values of J? by elimination. 
 
 The following table presents typical data, using a tangent 
 galvanometer, the results of which are shown graphically in 
 Fig. 94. 
 
 E. M. F. by Ohm's Method. — Two Gravity Cells in Series. 
 
 
 Galvanometer 
 
 
 
 
 
 
 
 
 Box 
 
 Readings. 
 
 Double 
 
 
 
 
 
 
 
 Resist- 
 
 
 Deflec- 
 tion. 
 
 tan 2 5. 
 
 28. 
 
 tan 5. 
 
 Current, 
 
 1 
 
 E.M.F. 
 
 ance. 
 
 
 
 Current 
 
 
 
 Right. 
 
 Left. 
 
 
 
 
 
 
 
 
 7 
 
 95-70 
 
 49.04 
 
 46.66 
 
 O.4666 
 
 25 I' 
 
 0.2218 
 
 O.1686 
 
 5-93 
 
 2.I4 
 
 10 
 
 90.66 
 
 53.88 
 
 36.78 
 
 O.3678 
 
 20° II' 
 
 O.I780 
 
 0-1353 
 
 7-39 
 
 2.12 
 
 20 
 
 83-IS 
 
 61.28 
 
 21.87 
 
 O.2187 
 
 12° 20' 
 
 O.I080 
 
 O.082I 
 
 12.18 
 
 2. II 
 
 30 
 
 79.92 
 
 64.47 
 
 '5-45 
 
 O.I545 
 
 8° 47' 
 
 O.O767 
 
 O.O583 
 
 17.15 
 
 2.08 
 
 40 
 
 78.I9 
 
 66.09 
 
 12.10 
 
 O.I2IO 
 
 6° 54' 
 
 O.0603 
 
 O.0458 
 
 21.84 
 
 2.O9 
 
 5° 
 
 77.II 
 
 67-I3 
 
 9.98 
 
 O.O998 
 
 5° 42' 
 
 O.O498 
 
 O.0384 
 
 26.05 
 
 2.14 
 
 60 
 
 76-35 
 
 67.85 
 
 8.50 
 
 O.085O 
 
 4° 52' 
 
 O.O425 
 
 O.O323 
 
 30.96 
 
 2 12 
 
 70 
 
 75.90 
 
 68.42 
 
 7.48 
 
 O.O748 
 
 4° 17' 
 
 O.0374 
 
 O.0284 
 
 35.22 
 
 2.15 
 
 Distance of mirror from scale = 50 scale divisions. 
 
 Galvanometer constant / u = 0.76 amperes. 
 
 From curve R^ = 5.7 ohms, E = 2.14 volts. 
 
 Last column computed assuming value of R obtained from curve. 
 
 * See Introduction, p. 4. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 279 
 
 Experiment S 2 . Fall of potential in a series circuit.* 
 
 The potential difference between two points in a circuit is 
 denned as the work done in transferring a unit quantity of 
 plus electricity from one point to the other. From Ohm's law 
 it follows that the potential difference between two points be- 
 tween which there is no E. M. F. along the path traveled may 
 be expressed as follows : pd = Ir, in which pd is the potential 
 difference, I the current flowing, and r the resistance between 
 the points considered. Whenever a current flows through a 
 resistance, there is always a disappearance of electrical energy. 
 The energy reappears in some other forms, as in heat. 
 
 The E. M. F. of a battery or a dynamo is equal to the 
 greatest possible potential difference between its terminals; 
 that is, when the generator is allowed to give no current, being 
 on "open circuit." As soon as the battery or dynamo is 
 allowed to give current, the circuit being closed, the difference 
 of potential between its terminals no longer equals its E. M. F., 
 but is less, owing to a loss of potential due to the work done 
 in sending current through the generator itself, the generator 
 having resistance. The loss of potential inside the dynamo or 
 battery will be equal to the internal resistance multiplied by 
 the current flowing. From the above we get the following 
 expressions for a simple circuit. 
 
 When no current is flowing, pd=E between the generator 
 terminals. When a current is flowing, then pd = E — Ir b , where 
 pd is the potential difference between the generator terminals, 
 E. the E. M. F. of the generator, r b its resistance, and / the 
 current flowing through it. 
 
 If there is more than one source of E. M. F. in a circuit, it 
 may be that the current will flow through one generator in a 
 direction opposite to that in which it would send current if free 
 to act alone. If this is the case, the pd between its terminals 
 
 * This experiment is taken from Blaker and Fisher, Experiments in Physics, 
 second edition, 1910. 
 
280 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 will be expressed by the following relation : pd = E + Ir b , be- 
 cause tYiz pd must not only be sufficient to send current through 
 the cell owing to its having resistance, but also to overcome an 
 opposing E. M. F. (See p. 273.) 
 
 Instruments used to measure E. M. F. and pd are called 
 potential galvanometers or voltmeters. Such instruments must 
 have comparatively high resistances and must be sensitive, since 
 very little current is supposed to flow through them. If such 
 an instrument be placed in series with a generator, the current 
 flowing is very small, consequently the loss of potential within 
 the generator is also very small and the pd is very nearly equal 
 to the E. M. F. of the generator. 
 
 If a potential galvanometer or a voltmeter be shunted around 
 a resistance, very low in comparison, the current flowing through 
 the instrument will be negligible in comparison to that flowing 
 through the resistance, and the pd between the terminals of the 
 resistance will not be appreciably disturbed by the presence of 
 the potential measurer. 
 
 Suppose a high resistance galvanometer of the tangent or 
 d'Arsonval type be in shunt with a resistance, around which the 
 pd is wanted. Then , pd=Ir. (248) 
 
 The current through the galvanometer is 
 
 7 e = |^=/ o tan0or^, (249) 
 
 according to the type of galvanometer used, R g being the total 
 resistance in the galvanometer branch of the circuit. 
 
 From equation 249, pd = RJ tan 6 or R g ks. (250) 
 
 Equation 250 shows that the pd is proportional to the 
 tan 9 or the number of scale divisions of single deflection, the 
 products R g l q or R„k being the potential constant of the in- 
 strument. 
 
 (1) To calibrate the potential galvanometer, arrange a circuit 
 to send current from one or two cells through a tangent gal- 
 vanometer whose constant is known or an ammeter and a resist- 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 
 
 281 
 
 ance box in series. Connect the potential galvanometer to the 
 terminals of the resistance box, or a part of known value. Ob- 
 serve the deflections of both galvanometers for five or six differ- 
 ent parts of the scale. 
 
 Tabulate observed data and the corresponding computed 
 currents and potential differences. Give a working diagram 
 of the connections. 
 
 Plot a curve with volts as ordinates and scale readings of 
 the potential galvanometer as abscissas. From this calibration 
 curve the number of volts corresponding to any given deflection 
 may be easily found. If the line is straight, find the fraction 
 of a volt represented by each large scale division. 
 
 (2) Connect four cells, three resistances, and an ammeter or 
 non-sensitive galvanometer in series as shown in Fig. 95. Note 
 
 T 
 
 -*l|^V\AAAA- c l|- d -VVV^AA*||^vVV^A^|l- 
 
 Fig. 95. 
 
 that one of the cells is so connected as to oppose the other 
 three. Connect the point a to a gas pipe. Its potential will 
 then be zero. Connect the potential galvanometer terminals to 
 points a and b and observe the deflection. Then connect the 
 terminals to the points b and c and observe the deflection as 
 before, being careful to note whether the potential rises ox falls. 
 Remember that the potential can only fall in a resistance travel- 
 ing with the current. In like manner make potential readings 
 when connected successively to c and d, d and e, and so on to 
 h and a. The deflection for current in the main circuit should 
 be observed occasionally to determine whether the current 
 remains constant or not. When ihejtd's and current have been 
 
282 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 measured for a closed circuit, break the circuit and measure the 
 E. M. F. of each cell. 
 
 Tabulate the observed data and computations as indicated 
 below. Plot a potential-resistance diagram with resistances as 
 abscissas, taking values from the last two columns of the table. 
 This curve must show the potential differences in the cells as 
 well as in the external resistances. 
 
 Draw conclusions from the diagram in regard to the fol- 
 lowing : 
 
 ( i ) The fall of potential in the external resistances. 
 
 (2) The fall or rise of potential through the voltaic cells. 
 
 (3) The meaning of the slope of the lines. 
 
 (4) The tests for accuracy. 
 
 Variation of Potential in a Series Circuit. 
 
 Current 
 Galv. 
 
 Parts of 
 Circuit. 
 
 Potential Galv. 
 
 Resist- 
 ance 
 
 in 
 Ohms. 
 
 E. M. F. 
 of Cells. 
 
 Total from a 
 
 Read- 
 ing. 
 
 * 
 
 Amps. 
 
 Read- 
 ing. 
 
 * 
 
 Volts 
 Rise in 
 Poten- 
 tial. 
 
 Volts 
 Fall in 
 Poten- 
 tial. 
 
 Read- 
 ing. 
 
 * t 
 
 Volts. 
 
 To 
 
 Volts. 
 
 Ohms- 
 
 
 0.2l8 
 O.218 
 0.2I8 
 0.218 
 O.218 
 O.218 
 0.218 
 O.218 
 
 a 
 
 Cell No. i* 
 b 
 
 /?! C 
 
 c 
 
 Cell No. id 
 d 
 
 R 2 e 
 e 
 
 Cell No. 3/ 
 
 / 
 
 R s g 
 g 
 
 Cell No. 1J1 
 h 
 R t a 
 Totals 
 
 
 I.582 
 
 .128 
 
 1. 29I 
 
 •436 
 
 .654 
 
 .872 
 
 I.0I8 
 
 .022 
 
 1.0 
 2.0 
 4.0 
 3° 
 
 •5 
 
 4.0 
 
 I.O 
 
 .1 
 
 
 I.80 
 1. 00 
 
 1.40 
 
 .80 
 
 b 
 
 C 
 
 d 
 e 
 f 
 g 
 h 
 a 
 
 I.582 
 1. 146 
 I.274 
 
 .620 
 1. 91 1 
 I.039 
 
 .012 
 -.001 
 
 1.0 
 
 3° 
 
 7.0 
 
 10.0 
 
 10.5 
 14.5 
 
 15-5 
 15.6 
 
 One scale division of potential galvanometer = . 
 One scale division of ammeter = 
 
 volts, 
 amperes. 
 
 * Observed. 
 
 t Circuit broken. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 
 
 283 
 
 In plotting the diagram it is well to arbitrarily assume that 
 the E. M. F. of the cell consists of two parts ; half being between 
 the zinc and liquid and half between the liquid and copper (or 
 carbon), and the liquid itself offering a resistance to the current 
 which causes the pd to be less when the current is flowing than 
 on open circuit. Thus pd = E — Ir h where r h is almost entirely 
 the resistance of the liquid in the cell. 
 
 Following this statement, plot one half the E. M. F. of the 
 first cell on zero resistance, 1.8/2 = 0.9 volt in the diagram. 
 Then the fall in potential through the cell resistance of 1 ohm is 
 
 2.0 
 
 a 1.5 
 
 0.5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /< 
 
 
 
 
 
 
 
 ( 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 s k 
 
 
 
 
 
 \ d 
 
 
 9' 
 
 
 
 
 
 
 
 
 
 \ 
 
 >c 
 
 
 
 
 
 
 \ 
 
 > 
 
 
 
 Ni» 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 
 
 
 
 
 
 
 
 Y« 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 h 
 
 , a 
 
 
 8 10 
 
 RESISTANCE IN OHMS 
 
 Fig. 96. 
 
 1.8 — 1.582 = 0.218 volt. 0.9 — 0.218 = .682 volt, which is the 
 potential to be plotted at 1 ohm. Then we have the other half of 
 the E. M. F. of the cell (0.9) to plot vertically above the second 
 point, bringing the pd up to 1.582 for 1 ohm as shown in the 
 last tw*o columns of the table and to the point b of the diagram. 
 From point b to c there is no E. M. F. but there is a fall of 
 potential of 0.436 volt through a resistance of .2 ohms, hence 
 1.582 — 0.436= 1. 146 volts, which is plotted at the total resist- 
 ance of 3 ohms, the point c of the diagram. Here the potential 
 rises again by one half the E.M.F. of the second cell, then falls 
 0.872 volt through the cell liquid resistance of 4 ohms, and then 
 rises the other half of the E. M. F. to the point d. 
 

 284 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The same process is followed for the remainder of the circuit 
 excepting cell g-h, which being in opposition to the other cells 
 causes the potential to drop instead of to rise. 
 
 Notice that the lines of the diagram have the same slope. 
 The slope =pd/r = current, and since the current remains nearly 
 constant the lines should have nearly the same slope. 
 
 Experiment S s . The potential difference at the terminals of 
 a battery considered as a function of the external resistance. 
 
 The difference of potential between the terminals of a cell 
 has its greatest value when the external resistance is infinite 
 (when the circuit is broken), and is then equal to the electro- 
 motive force. As the external resistance is diminished, the 
 E. M. F. remains constant ; but the differ- 
 ence of potential between the poles steadily 
 grows less, until the external resistance is 
 zero, when the two poles are at the same 
 potential. 
 
 Fig. 97. The relation between these two quan- 
 
 tities may be investigated as follows : 
 
 (1) Put a battery and a variable known resistance in 
 series. 
 
 (2) Connect a high resistance galvanometer through a re- 
 versing key to the terminals of the resistance box. (See 
 Fig. 97.) The resistance of the galvanometer used should be 
 so great (1000 ohms or more) that the current passing through 
 it is too small to modify appreciably the current in the main 
 circuit. Under these circumstances, the galvanometer merely 
 serves to measure the difference of potential between the 
 terminals of the cell. If the galvanometer is very sensitive, it 
 may be necessary to put a resistance in series with it in order 
 to decrease the current in that branch of the circuit. The 
 added resistance is to be treated as a part of the galvanometer 
 resistance. 
 
 Let /„ be the current flowing in the galvanometer, R g its 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 285 
 
 resistance, and pd the potential difference between its terminals ; 
 then we have 
 
 ^=/ = / tanS. 
 
 The product R g I is a constant for which may be substituted 
 the symbol pd . This is the constant of the instrument used as 
 a potential galvanometer, 
 
 ,\pd = R g / tan h=pd ta.n 8. (25 1 ) 
 
 When R g is very great, this will be very nearly the potential 
 difference that would exist if no galvanometer were used. It 
 may be here noted that the potential difference between the 
 terminals of the galvanometer, the battery, and the resistance 
 box are practically identical, since the connecting wires are 
 supposed to have negligible resistance. 
 
 (3) Observe the reading of the galvanometer for the follow- 
 ing box resistances : 00, 40, 30, 25, 20, 15, 12, 10, 8, 6, 4, 3, 2, 
 1, 0.5 ohms. These resistances will be such that the galva- 
 nometer deflections vary by approximately equal steps. 
 
 If / is the current in the resistance box, R its resistance, 
 and R b the resistance of the cell, including the connecting wires 
 to the box, we shall have 
 
 I= E ^pd 
 R h + R R' 
 
 or pdR b +pdR = ER. (252) 
 
 If . the observations taken be plotted with box resistances as 
 abscissas, and potential differences as ordinates, the resulting 
 curve should be a hyperbola, with an asymptote parallel to the 
 axis of abscissas. 
 
 If the observations be plotted with reciprocals of box resist- 
 ances as abscissas, and cotangents of galvanometer deflections 
 as ordinates, the curve should be very nearly a straight line, 
 
 whose intercept on the axis of abscissas is equal to — — ■ R b 
 
 Rb 
 
 may also be obtained from the first curve. It is the abscissa 
 
286 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 corresponding to the ordinate which is half the maximum 
 ordinate. 
 
 If the constant pd is not known, any quantity that is pro- 
 portional to the potential difference may be substituted for it 
 in plotting the curve, as tangents if a tangent galvanometer 
 
 20 25 
 
 RESISTANCE 
 
 Fig. 98. 
 
 be used, or single deflections if the galvanometer be of the 
 d'Arsonval type. 
 
 Two sets of observations are to be made, one set with a 
 single cell, and another set with two similar cells in series, one 
 of which is the cell used in the first set. 
 
 Compute the potential differences for all observations, and 
 also two values of the battery resistances for each set. 
 
 Plot two curves for each set of data obtained. Derive the 
 physical equations of the curves, interpret them, and get the 
 possible physical constants. 
 
 Results obtained by the method just described are given 
 in the following table. The relation between resistance and 
 potential difference is shown graphically in Fig. 98. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 287 
 
 Potential Difference between Terminals of Gravity Cell. 
 
 
 Galvanometer Readings. 
 
 Galvanometer 
 
 Potential 
 Difference in 
 
 Resistance in 
 
 
 
 Deflection 
 Proportional to 
 
 Box. 
 
 
 
 
 Direct. 
 
 Reversed. 
 
 pd. 
 
 
 00 
 
 65.26 
 
 9.80 
 
 55.46 
 
 I.065 
 
 40 
 
 58.06 
 
 17.07 
 
 40.99 
 
 O.787 
 
 3° 
 
 56.38 
 
 18.76 
 
 37.62 
 
 0.722 
 
 2 5 
 
 55-3° 
 
 19.87 
 
 35-43 
 
 O.680 
 
 20 
 
 53.86 
 
 21.32 
 
 32.56 
 
 O.624 
 
 ! S 
 
 51.90 
 
 23.30 
 
 28.60 
 
 O.549 
 
 12 
 
 50.40 
 
 24.87 
 
 25-53 
 
 O.49O 
 
 IO 
 
 49.25 
 
 26.10 
 
 2315 
 
 O.444 
 
 8 
 
 4774 
 
 27.62 
 
 20.12 
 
 O.386 
 
 6 
 
 45-95 
 
 29.23 
 
 16.60 
 
 O.3I9 
 
 4 
 
 43-9° 
 
 31-5° 
 
 12.40 
 
 0.227 
 
 3 
 
 42.67 
 
 32.76 
 
 9.91 
 
 O.I9O 
 
 2 
 
 41.20 
 
 34.28 
 
 6.92 
 
 0-'33 
 
 1 
 
 39 73 
 
 35-67 
 
 4.06 
 
 O.O78 
 
 °-5 
 
 38-75 
 
 36.61 
 
 2.14 
 
 O.O4I 
 
 d'Arsonval galvanometer No. 48. 
 
 Resistance of galvanometer = 340 ohms. 
 
 Resistance in series with galvanometer = 5000 ohms. 
 
 Resistance of galvanometer branch = 5340 ohms. 
 
 Current constant of galvanometer per scale division = 360 x io -8 . 
 
 Potential constant = 360 x io -8 x 5340 = 192 x io -4 . 
 
 From curve R b = 13.6 ohms. 
 
 Experiment S 4 . Principle of fall of potential in a wire car- 
 Tying a current. 
 
 This experiment is intended to illustrate the fact that the 
 difference in potential between any two points on a simple cir- 
 cuit in which a current is flowing is proportional to the resist- 
 ance between these points. This proportionality of fall of 
 potential and resistance holds true in the case of any simple 
 circuit, provided that there is no electromotive force between 
 the two points considered. It is a direct consequence of Ohm's 
 law, and may be stated as follows : 
 
 pd=Ir, (253) 
 
288 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 in which pd is the difference of potential, r the resistance be- 
 tween any two points of a simple circuit, and / the current 
 flowing. 
 
 The most direct method of testing this proportionality would 
 undoubtedly be to measure the difference of potential between 
 selected points of a circuit by means of an electrometer. In 
 this case the measurement would depend upon electrostatic 
 forces, and the current flowing in the circuit would not be 
 modified. The following method will, however, give results 
 
 that are quite closely cor- 
 rect if the galvanometer 
 resistance is sufficiently 
 large. 
 
 The procedure is as 
 follows : 
 
 (i) Connect a resist- 
 ance box in series with a, 
 gravity battery of one or 
 more cells, and take out 
 all the plugs corresponding to the low resistances. 
 
 (2) Connect a high resistance galvanometer through a re- 
 versing key s to side plugs, and by this means put the galva- 
 nometer in multiple with a.- portion of the resistance in the box 
 (Fig. 99). In order that the galvanometer shall not perceptibly 
 alter the fall of potential in the main circuit, the resistance be- 
 tween the points to which it is connected should not be greater 
 than 0.005 that of the galvanometer. If the galvanometer de- 
 flection is not a suitable one, it should be made so by varying 
 the resistance or E. M. F. in the main circuit. 
 
 The side plugs should now be shifted from place to place on 
 the box so as to include different resistances between them; 
 and for each value of the included resistance the deflection of 
 the galvanometer (both direct and reversed) should be observed. 
 Ten or twelve different values of the resistance included between 
 the plugs should be used. A suitable series is the following 
 
 Fig. 99. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 289 
 
 one: 30, 20, 17, 14, 10, 9, 7, 6, 5, 4, 3, 2, and 1 ohms. If the 
 galvanometer reading is off the scale when 30 ohms is included 
 between the side plugs, a resistance may be put in series with it. 
 
 Since the resistance of the galvanometer remains constant 
 throughout the experiment, the current passing through it is in 
 each case proportional to the difference of potential between 
 the side plugs. 
 
 The results may be best used to test the principle of fall of 
 potential by plotting a curve in which resistances between side 
 plugs are used as abscissas and the corresponding galvanometer 
 deflection as ordinates. (If the galvanometer is a tangent gal- 
 vanometer, tangents of deflections must be used ; if a sine gal- 
 vanometer, sines of deflections, etc.) The curve should be very 
 nearly a straight line, very slightly concave towards the axis of 
 abscissas. 
 
 The principles involved in the use of a galvanometer as a 
 voltmeter will be brought out quite clearly if a new series of 
 observations is taken in which the resistances between the side 
 plugs range up to \ or \ of the resistance of the galvanometer. 
 This may be done by very greatly increasing the resistance of 
 the main circuit, or by using a low resistance coil, if the galva- 
 nometer be a tangent galvanometer of coils of various resistances, 
 or by shunting the galvanometer with a suitable resistance. 
 The same range of resistances may be used as before, but if 
 the deflections are off the scale for the highest resistance, suffi- 
 cient resistance is to be inserted in the main circuit to bring the 
 readings on the scale. If in this case the observations be 
 plotted as before, there will be a very decided curvature toward 
 the axis of abscissas ; but if the true multiple resistance be- 
 tween the side plugs be used as abscissas, the curve rigorously 
 becomes a straight line. 
 
 Three curves are to be plotted as indicated above, one for 
 the first case and two for the second. The physical equations 
 of the curves are to be derived and discussed, and physical con- 
 stants obtained. 
 vol. 1 — u 
 
290 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Addenda to the report: 
 
 ( 1 ) Indicate the circumstances under which there would be 
 no curvature in a series of observations plotted as above. 
 
 (2) Why does the curve become straight when plotted as 
 described in the next to the last paragraph of the directions ? 
 
 (3) Discuss the characteristics of instruments to be used for 
 measuring current, and for measuring voltage. 
 
 Experiment S 5 . Beetz's method of measuring electromotive 
 forces. 
 
 This experiment depends upon finding two points, A and B 
 (Fig. 100), in the circuit of the battery whose E. M. F. is re- 
 
 3 vVW\AM/VV J 
 
 --MSM'- 1 
 
 Fig. 100. 
 
 quired, such that their potential difference shall be equal to the 
 E. M. F. of a standard cell. If r is the resistance between 
 these points, R the total resistance of the circuit exclusive of 
 the battery, whose resistance is R h , pd the fall of the potential 
 between A and B (which is equal to the E. M. F. of the stand- 
 ard cell), the current in the principal circuit will be equal to 
 
 E = pd 
 
 R + R h r' 
 
 (254) 
 
 If a new value of R is taken, r must also be changed. This 
 will give another equation similar to the above, and between 
 them R b may be eliminated and the ratio of E to pd computed. 
 
 The method may also be employed to determine the battery 
 resistance R b , but good results cannot be expected unless R is 
 always comparable with R b . 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 291 
 
 To perform the experiment : 
 
 (1) Connect the unknown E. M. F. in series with two known 
 resistances which may be varied at pleasure. If the current in 
 the main circuit flows from A to B, connect the negative pole 
 of the standard cell to B and the positive pole to a galvanom- 
 eter. The other terminal of the galvanometer is connected to 
 A through a key K. 
 
 (2) Make the value of the resistance r 200 ohms, say; then 
 adjust the resistance / until no current flows through the gal- 
 vanometer on closing the key K. 
 
 Under these circumstances the fall of potential from A to B 
 due to the current in the main circuit is equal to the E. M. F. 
 of the standard cell. 
 
 That is, 
 
 r + S + R-T" ^ 5S) 
 
 in which r and r' are known box resistances and/*/= E e , the 
 E. M. F. of the standard cell. Make a series of ten determina- 
 tions of values of r 1 necessary to produce no flow of current 
 through the galvanometer, increasing r by 200 ohms at each 
 determination. From any pair of values of r and / both E 
 and R b may be determined. Use three pairs of suitably chosen 
 values to determine E and R b . 
 
 Plot a curve using as co-ordinates values of R{= r+r 1 ) 
 and r 1 . This will be a straight line from whose constants 
 E and R h may be determined. Derive the physical equation 
 of the curve, interpret it, and determine the physical constants 
 from it. 
 
 Taking R b from the curve, compute values of E for each 
 observation made. 
 
 If the battery whose E. M. F. is to be measured is subject to 
 rapid polarization, the current should be allowed to flow only for 
 an instant before closing the circuit of the galvanometer. 
 
 By this method it is obvious that the E. M. F. to be measured 
 must be greater than that of the standard cell. Unless the 
 
292 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 galvanometer is very sensitive, it should be three or four times 
 as large. 
 
 Addendum to the report: 
 
 Show that when no current flows in the galvanometer the 
 potential difference between A and B is equal to the E. M. F. of 
 the standard cell. 
 
 Experiment S 6 . To trace the lines of equal potential in a 
 liquid conductor. 
 
 The apparatus for this experiment consists of a shallow 
 vessel provided with a glass bottom and filled with some poorly 
 conducting liquid, such as ordinary water. A telephone is also 
 required, and some means of obtaining an alternating or inter- 
 rupted current. A small induction coil is suitable for this 
 purpose. 
 
 If two electrodes are placed in the liquid and a current 
 passes between them, the current will flow from one electrode 
 to the other by every possible path. The potential varies along 
 each of these paths, having its greatest value at the positive 
 pole and its least value at the negative pole. For each value 
 of the potential between these limits, there is therefore a point 
 on each of these " lines of flow." Since all these points are at 
 the same potential, they lie upon one of the equipotential lines 
 of the liquid. The object of this experiment is to determine 
 the shape of these equipotential lines. 
 
 Connect two wires to the terminals of the telephone, and 
 fasten one of them so that its end dips into the liquid. If the 
 end of the second wire is also placed in the liquid, a sound will 
 in general be heard in the telephone, due to the rapid make and 
 break of the current. By shifting the position of the second 
 wire, however, a position can be found such that this sound is 
 no longer heard. When this position is reached, the ends of 
 the two wires must be at the same potential, and are therefore 
 points on the same equipotential line. Keeping the position of 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 293 
 
 the first wire unaltered and varying that of the second, enough 
 points can be found in this way to locate the equipotential line 
 with considerable accuracy. These lines should be traced quite 
 carefully near the edge of the conductor, and in the neighbor- 
 hood of a line separating a good conductor from a bad con- 
 ductor. It will be found convenient to place a board ruled with 
 equidistant lines beneath the glass bottom of the vessel, and to 
 record the position of the points by reference to these lines. 
 A diagram can afterwards be drawn on which the equipotential 
 curves are accurately represented. To avoid annoyance from 
 the noise of the interrupter on the induction coil, it is advisable 
 to place the latter in a separate room. 
 
 In the manner described above, the form of the equipoten- 
 tial lines may be investigated when electrodes of different 
 shapes are used, or when the relative position of the electrodes 
 is altered. In each case, at least five or six lines should be 
 located, the intervals between them being so chosen that the 
 field in all parts of the liquid is clearly shown. Diagrams should 
 be drawn to scale, representing the position of the electrodes 
 and the limits of the vessel, as well as the equipotential curves. 
 Since the lines of flow must be at all points perpendicular to 
 the equipotential lines, the former can also be drawn. 
 
 Very instructive results may be obtained by placing between 
 the electrodes a piece of metal of high conductivity. Since 
 the resistance of the metal is less than that of the liquid, the 
 field will be distorted, and the modified form of the equipoten- 
 tial lines can be determined by the telephone. A piece of some 
 poorly conducting substance, such as glass or paraffin, will also 
 give instructive results. 
 
 It may sometimes be desirable to use a galvanometer in- 
 stead of a telephone in tracing the equipotential lines. In 
 this case, a continuous instead of an alternating current must 
 be used, and the liquid conductor may be replaced by a sheet 
 of tinfoil. The equipotential lines are determined by finding 
 a series of points, such that if the galvanometer terminals be 
 
294 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 connected to any two of them, no current will flow through 
 the galvanometer. 
 
 On account of the analogy that has been found to exist 
 between lines of flow and magnetic lines of force, the results 
 of this experiment have important bearings on magnetic prob- 
 lems, such as occur in dynamo work. 
 
 Addenda to the report : 
 
 (i) Indicate the reason why the equipotential lines are 
 always normal to the edge of the conductor. 
 
 (2) Indicate the part of the conductor in which the current 
 density is the greatest. 
 
 (3) Indicate the part of the conductor in which the fall 
 of potential is most rapid. 
 
 Experiment S 7 . Variation in the E. M. F. of a thermo- 
 element with change in temperature. 
 
 There is always a difference of potential between points on 
 opposite sides of the junction between two different metals. If 
 two metals be joined so as to make a complete circuit, there will 
 be a fall of potential at each junction. Since these two changes 
 of potential are equal and are opposed to each other, no current 
 will be produced. In a word, the whole of one metal will be at 
 one potential, while the whole of the other metal will be at a 
 different potential. 
 
 This contact difference of potential depends upon tempera- 
 ture. Therefore, if the two junctions are at different tempera- 
 tures, these two differences of potential will not, in general, 
 annul each other, and a constant current will flow through the 
 circuit. Such a combination of two metals with the two junc- 
 tions at different temperatures constitutes a thermo-element. 
 It is the seat of a true E. M. F., as that term has already been 
 defined. 
 
 It is the object of this experiment to determine the relation 
 between this E. M. F. and the temperatures of the two junctions. 
 
 The procedure is as follows : 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 295 
 
 ( 1 ) Construct a simple form of element by soldering together 
 the ends of two wires made of different metals : for example, 
 German silver and copper, or copper and iron. Then cut one 
 of the wires in the center, so that the free ends will form the 
 terminals of the element. 
 
 (2) Connect the terminals of the element through a revers- 
 ing key to a sensitive galvanometer. It will be advisable to 
 place a resistance box somewhere in the circuit, so that the re- 
 sistance of circuit may be under control. The resistance of the 
 whole circuit should be great enough so that it will not be 
 appreciably altered by changes in the temperature of the 
 element. 
 
 (3) One junction of the element is now to be kept at a con- 
 stant temperature, while the other is placed in a bath of oil or 
 water whose temperature can be readily varied. The E. M. F. 
 corresponding to any observed difference in temperature be- 
 tween the junctions is then proportional to the galvanometer 
 deflections, or its tangent, as the case may be. It is important 
 that the terminals of the element be kept at the same tempera- 
 ture. This can usually be accomplished by placing them side 
 by side and wrapping them with paper. 
 
 The junction whose temperature is to be varied should be 
 inserted in a test tube for protection against the chemical action * 
 of the bath. Its temperature may be measured by a thermome- 
 ter placed in the same tube, the bulb of the thermometer being 
 on a level with the junction. To prevent air currents it is best 
 to fill the upper part of the tube with cotton waste or asbestos. 
 For rough work, the other junction may be left in the air, pro- 
 vided it is protected from draughts. It is better, however, to 
 place the junction in some constant temperature bath, such as 
 boiling water, melting ice, or water that is nearly at the tempera- 
 ture of the room. In this case the junction should be inserted 
 in a test tube, as described above. 
 
 If two elements mounted in the same frame be supplied, ob- 
 servations are to be made on each element for ten different 
 
296 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 temperature differences approximately equally spaced between 
 room temperature and 95° C, the cold junctions being in an ice 
 bath. The bath whose temperature is variable is to be con- 
 stantly stirred. The connections of the circuit should be so 
 made that by throwing a switch either element may be put 
 in circuit with the resistance box and galvanometer. Such an 
 arrangement will obviate the necessity of heating the variable 
 temperature bath twice. When the bath is at a temperature 
 at which observations are desired, read the temperature of the 
 bath, following it with a direct galvanometer reading on each 
 element, then another temperature reading, then a reverse gal- 
 vanometer reading on each element taken in the reverse order, 
 and finally a third temperature reading. By taking the mean 
 of the three temperature readings errors due to uncertainty as 
 to bath temperatures will be greatly reduced. 
 
 (4) From the galvanometer constant, the resistance of the 
 circuit, and the galvanometer deflections, compute the E. M. F.'s of 
 the thermo-elements for each observed difference of temperature. 
 
 (5) Plot a curve with temperature differences as abscissas 
 and E. M. F.'s in microvolts as ordinates for each element used. 
 
 Addenda to the report: 
 
 (1) Define thermo-electric power and find it for three points 
 on each curve. 
 
 (2) Explain how by the method of making observations out- 
 lined in part (3) errors, of bath temperature are reduced. 
 
 Experiment S 8 . Calibration of a voltmeter. 
 
 The principle of this experiment is the same as that of Exp. 
 S 5 , and it should be preceded by that experiment. 
 
 Connect a storage battery B of suitable E. M. F., a protecting 
 resistance such as a starting rheostat, one 100,000-ohm and two 
 10,000-ohm resistance boxes R, R v and R 2 in series." Connect 
 the voltmeter to be tested in multiple with the two io.ooo-ohm 
 boxes. Connect a standard cell S, a galvanometer, a resistance 
 for the protection of the standard cell, and a contact key in mul- 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 297 
 
 tiple with one of the 10,000-ohm boxes. Figure 101 shows the 
 connections in outline. 
 
 Put 5000-ohms in each box R t and i? 2 and vary the resist- 
 ance in R until the voltmeter reading is about one tenth of the 
 maximum of the scale. 
 
 Adjust the resistances in R 1 and R 2 , keeping the sum equal 
 to 10,000 ohms, until on closing the galvanometer circuit no cur- 
 rent flows through the 
 
 f-AA/VSAil|l||----Hl|l|l|lV\AAAAAA,/V v - 1 
 
 s 
 
 VWWWW-MAAMA 
 
 r^AA-ll®- 
 J— ^/va/va/v 
 
 Ri 
 
 <^ 
 
 R 2 
 
 Fig. 101. 
 
 galvanometer. Note 
 the resistances R v R 2 , 
 and the voltmeter 
 reading. 
 
 The potential dif- 
 ference pd 2 at the 
 terminals of the resist- 
 ance R 2 is then equal to the E. M. F. of the standard cell. 
 Knowing pd % and R 2 , the current flowing through R-y and R% 
 may be found. The current multiplied by Ry + R 2 will give the 
 pd around R x and R 2 , which will be the correct voltage. 
 
 For the next reading change the resistance of R until the 
 voltmeter reading is about two tenths of the maximum readable 
 voltage on the instrument and proceed as before. 
 
 Make ten different determinations of the pd at the voltmeter 
 terminals. 
 
 Plot a curve, using computed values of the potential differ- 
 ences around R x + R 2 as ordinates and corresponding voltmeter 
 readings as abscissas. 
 
 Compute the percentage error of each voltmeter reading 
 when corrected for the zero error of the instrument. 
 
 Plot another curve, using the true voltages as abscissas and 
 instrumental errors, plus or minus, as ordinates. 
 
 Addenda to the report : 
 
 (1) Howmaya voltmeter be used to measure voltages ten times 
 as great as the range for which it is designed ? Explain in detail. 
 
298 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 (2) Explain fully how a voltmeter may be used to measure 
 current. 
 
 Experiment S 9 . Comparison of electromotive forces.* 
 
 I. 
 
 Poggendorf's method. 
 
 A steady current is sent through a wire resistance AB, Fig. 
 102, and a regulating resistance R from a storage battery. One 
 of the cells to be tested is placed in series with a sensitive gal- 
 
 -llllll^VWWNA 
 
 Fig. 102. 
 
 vanometer and a high resist- 
 ance, which are then connected 
 to one end of the wire AB at 
 A and to a sliding contact a. 
 The cell is connected so that 
 its E. M. F. opposes that of 
 the storage battery. The posi- 
 tion of the sliding contact a is 
 varied until a point is found where no current passes through 
 the galvanometer. The E. M. F. of this cell is then equal to the 
 fall of potential through the resistance Aa. By means of the 
 three-point switch K, the second cell is inserted in the circuit 
 in place of the first. The position of the connection to AB 
 is varied to some point b until no current flows through the 
 galvanometer. If the current in AB has been constant, the 
 ratio of the resistances Aa and Ab equals the ratio of the 
 E. M. F.'s of the two cells. 
 
 Great care should be taken in handling a standard cell. 
 The E. M. F. of a standard cell changes due to polarization 
 when it is permitted to give more than a very small current. 
 To prevent this the high resistance is inserted in series with the 
 cell. However, greater sensitiveness is afforded if this resist- 
 ance be decreased as a, or b, as the case may be, approaches 
 
 * The principles upon which these methods depend are essentially those of 
 Exp. S5. 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 299 
 
 the position of balance. If, under these circumstances, the 
 resistance has been decreased, the student must remember to 
 increase it before changing to the other cell. 
 
 In connecting up the cells to be tested, if any difficulty is 
 found in getting them in the right direction or the current in 
 the main circuit sufficiently large, the trouble may be ascer- 
 tained by observing the galvanometer deflection as follows : If 
 while moving a from A to B, the direction of the deflection does 
 not reverse, either the fall of potential along AB is not great 
 enough or the cell is not connected to oppose this potential 
 difference. If the current in the main circuit is reversed and 
 again no reversal of the direction of the deflection is found, the 
 fall of potential in AB is too small. 
 
 In this method a wire is used for the resistance AB, along 
 which a sliding contact may be moved (Fig. 102). The lengths 
 of wire Aa and Ab are observed and the ratio of these lengths 
 is equal to the ratio of the E. M. F.'s of the cells. 
 
 The current in this wire should be such that the fall of po- 
 tential wire is but little greater than the largest E. M. F. to be 
 tested. This may be accomplished by varying a resistance in 
 series with the storage cells and the wire AB. 
 
 Take three readings for each cell without changing the cur- 
 rent in AB. Increase the current so that about 0.9 of the wire 
 is needed to balance largest E. M. F., and take three more read- 
 ings for each cell. Make a similar set by increasing the cur- 
 rent so that about 0.75 of the wire is used for the larger E. M. F. 
 
 II. 
 
 Lord RayleigKs potentiometer method. 
 
 , If a high resistance galvanometer that is quite sensitive is 
 available, this method is more accurate than the slide-wire 
 method, but the principles are the same. 
 
 Two accurately adjusted resistance boxes of 10,000 ohms 
 each replace the wire used in I. The resistances R and R' are 
 varied, care being taken to keep the sum of the two constant, 
 
■|l|l|l— ^vww^ 
 
 300 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 say 8000 ohms, until the galvanometer gives no deflection on 
 closing the galvanometer circuit. The value for R when a 
 balance is obtained may be denoted as R v The other cell is 
 now placed in the circuit and the resistances are changed, 
 
 keeping their sum the same 
 as before, until the circuit is 
 again balanced. If R 2 be the 
 new value of R, 
 
 7S/U& R X _E 1 
 
 where E x and E 2 are the 
 E. M. F.'s of the cells that are to be compared. The readings 
 should be repeated to be certain that the current in the main 
 circuit has not varied. Repeat, using a constant sum of 10,000, 
 then of 12,000, making for each case the reading described 
 above. 
 
 Experiment S 10 . The potentiometer. 
 
 The principles discussed in Exps. S 5 and S 9 are the basis 
 of a very useful measuring instrument, the potentiometer. The 
 potentiometer may be used to compare E. M. F.'s, resistances, 
 and currents. In any case it is potential differences that are 
 used in making comparisons. If resistances are to be com- 
 pared, the comparison may be arrived at by comparing the 
 J>d's between their terminals when the same current is flow- 
 ing in each. Currents may be compared or measured by com- 
 paring or determining the fld's produced by a known current 
 through a known resistance, and of the unknown current through 
 a known resistance, or by comparing the pd of a standard 
 cell with that produced by the unknown current through a 
 known resistance, generally a standard. The following descrip- 
 tion applies to a particular type of instrument, but the principles 
 involved may be generally used. 
 
 The potentiometer consists of a circuit made up of a low 
 resistance battery B (Fig. 104) of small polarization, as of two 
 
POTENTIAL AND ELECTROMOTIVE FORCE. 301 
 
 or three storage cells, a variable resistance R, 29 coils of equal 
 resistances between b and d, and a wire ab a meter long, divided 
 into millimeters, whose resistance is equal to that of each coil, 
 all in series. A branch circuit, including a standard cell S, 
 a key K, and a sensitive galvanometer G, is connected to the 
 main circuit by the sliding contacts in and n. The contact n 
 
 l|l|l[l|l— ) AAAA/ysAAAA- 
 
 10 8 8 7 6 6 4 8 2 1 1 2 8 4 12 13 H 15 16 17 26 27 28 28 
 6 7l\ • d 
 
 may be made to include any whole number of the equal poten- 
 tiometer coils between b and «T, and any equivalent fraction of a 
 coil may be added by means of the contact m sliding over the 
 wire, reading being made on a scale running parallel with the 
 wire. 
 
 If the potential difference between a and d due to the current 
 in the main circuit is greater than the E. M. F. of the standard 
 cell S, positions of the contacts m and n can be found such that 
 on closing the key K no current will flow through the galvanom- 
 eter branch. Under these conditions the pd between m and n 
 is equal to the E. M. F of the standard cell. If E 1 represents 
 this E. M. F. and JV t represents the number of coils included 
 between m and n, then thcpd per coil will be 
 
 — l = k (volts per coil). ( 2 S6) 
 
 If E x is replaced by a cell of unknown E. M. F. E v and a new 
 
 balance obtained for which N % coils are included between m and 
 
 n, then 
 
 E 2 = kN 2 , (257) 
 
 provided the current in the main circuit has not changed. 
 
 Computations may be much simplified if k be made some 
 
302 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 such factor as o.i which will make the instrument direct reading. 
 This may be readily done by making the number of coils in- 
 cluded between m and n equal to a multiple of the E. M. F. of 
 the standard cell and changing the current in the main circuit 
 by varying the resistance R until a balance is obtained. As an 
 example suppose a Clark cell, whose E. M. F. is 1.4240 at the 
 particular temperature at which it is used, be put in circuit. If 14 
 potentiometer coils be included between b and d, the sliding 
 contact m set at 0.240 on the wire, and the resistance R be varied 
 until a balance is produced, the coil factor is then 0.1. 
 
 Great care must be used to keep the coil factor constant in 
 making determinations with the instrument. After each ob- 
 servation on the unknown, as well as before, the instrument 
 should be checked by putting the standard cell in circuit. If 
 necessary, the resistance R should be adjusted before each new 
 determination of the unknown. 
 
 Before beginning experimental work, carefully trace out the 
 connections of the instrument, making a rough diagram in the 
 notebook with the parts properly designated. 
 
 Make connections carefully, find the coil constant, and make 
 a series of observations for determining the unknown potential 
 difference. Make another series of determinations, using a sec- 
 ond coil constant. One of the constants may be the factor o.i, 
 but the other factor should be an odd one for which approxi- 
 mately the maximum range of the coils should be used. Com- 
 pare results obtained by the two methods. 
 
 The problem to be solved will be assigned by an instructor. 
 
CHAPTER XI. 
 GROUP T: THE MEASUREMENT OF RESISTANCE. 
 
 (T) General statements ; (Tj) Measurement of resistance by the 
 Wheatstone bridge ; (T 2 ) Measurement of resistance by the 
 method of fall of potential ; (T 3 ) Specific resistance ; (T 4 ) De- 
 termination of the temperature coefficient for resistance of 
 carbon and of various metals ; (T 5 ) Kelvin double bridge 
 method for measuring low resistances ; (T 6 ) Resistance of 
 electrolytes ; (T T ) Measurement of the internal resistance of a 
 battery ; (T 8 ) Measurement of the resistance of a galvanometer. 
 
 (T). General statements concerning resistance. 
 
 When two points of a homogeneous conductor are main- 
 tained at different potentials, a current will flow in the conductor. 
 The magnitude of this current depends upon the substance and 
 dimensions of the conductor. The conductivity of a conductor 
 is that quantity which must be multiplied into the potential 
 difference at its terminals to give the current which flows. The 
 resistance of a conductor is the constant ratio between the 
 difference of potential at its terminals and the current which 
 this potential difference produces. 
 
 The absolute unit of resistance is the resistance of a conductor 
 such that unit electromagnetic difference of potential at its ends 
 will cause unit electromagnetic current to flow. It may be 
 shown experimentally that the resistance of a conductor varies 
 directly as its length, and inversely as its cross-section. On 
 account of the relative ease with which a conductor of some 
 standard substance of given length and section may be con- 
 structed, it is more usual to define the practical unit of 
 resistance in these terms. The Chamber of Delegates at the 
 Chicago Electrical Congress adopted, " As a unit of resistance 
 
 3°3 
 
304 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 the international ohm, which is based upon the ohm equal to io 9 
 units of resistance of the C. G. S. system of electromagnetic units, 
 and is represented sufficiently well by the resistance offered to an 
 unvarying electric current by a column of mercury at the tempera- 
 ture of melting ice, 14.4521 grains in mass, of a constant cross- 
 sectional area, and of a length of 106. 3 centimeters." 
 
 In current electricity it is necessary to have variable resist- 
 ances, such that any known value may be inserted in a circuit 
 at pleasure. This demand is met by constructing a series of 
 coils of wire of different resistances and enclosing them in a 
 box, the whole being called a rheostat or resistance box. These 
 coils are constructed of insulated wire, usually of manganin. 
 This alloy of copper, manganese, and nickel has a high specific 
 resistance, thereby giving considerable resistance with com- 
 paratively small lengths of wire. It has a low temperature 
 coefficient and a very small thermo-electric force against 
 copper. These coils are non-inductively or doubly wound, so 
 that their self-induction shall be as small as possible. The 
 ends of each coil are connected to separate brass blocks 
 which are electrically connected by removable brass plugs. 
 When all of these plugs are in place, the resistance between 
 the binding screws of the box is inappreciable. Any desired 
 resistance may be introduced into the circuit by removing the 
 plugs corresponding to the proper resistance coils. 
 
 In the use of resistance boxes it should always be remem- 
 bered that the resistance apparently in circuit is not the true 
 resistance unless each plug in place makes good connection 
 between the adjacent brass blocks. If the plug be simply 
 dropped into place, or if it be not thoroughly cleaned, the resist- 
 ance between it and either brass plug, instead of being infini- 
 tesimal, may have a large value. Unless care is taken, the 
 unknown resistance thus introduced into the circuit is likely to 
 be a considerable fraction of an ohm. If the resistances used 
 are small, this becomes of great relative importance. To avoid 
 this difficulty, each plug when it is inserted should be twisted 
 
THE MEASUREMENT OF RESISTANCE. 
 
 3°5 
 
 in its seat, thus securing good contact. Sometimes it is nec- 
 essary to clean the plugs and brass blocks with sandpaper. 
 Emery paper should never be used. 
 
 The coils of resistance boxes are generally wound with 
 small wire, hence they should only be used for weak currents. 
 
 Experiment T v Measurement of resistance by the bridge 
 
 method. 
 
 I. 
 
 The Wheatstone bridge. 
 
 One of the most useful and accurate methods of measuring 
 resistance is by means of the Wheatstone bridge. 
 
 Let ABC and AB' C (Fig. 
 105) be the two parts of a 
 divided circuit containing no 
 E. M. F. If by means of a 
 battery a current is made to 
 flow from A to C, the poten- 
 tial will fall from A to C 
 along both branches. Let 
 B and B' be points in the 
 two branches having the 
 same potential. Let pd and 
 pd' be the differences of potential between A and B and 
 between B and C respectively. Let the resistances be r x and x. 
 As no current can flow through the branch BB', we have, from 
 Ohm's law, 
 
 M = PA. ( 2S 8) 
 
 r x x 
 
 As B and B' are at the same potential, the difference of 
 potential between A and B must equal that between A and B'. 
 Therefore, in the lower branch we have 
 
 pd pd' 
 
 Fig. 105. 
 
 whence 
 
 x = r -±R. 
 r 
 
 (259) 
 (260) 
 
306 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 A Wheatstone bridge is an apparatus consisting of three 
 sets of wire coils whose resistances are known. R is a rheostat 
 or variable resistance in which any resistance may be obtained 
 from o.i to 10,000 ohms. r x and r 2 are called "ratio arms"; 
 each consists of a series of resistances, which may be made 1, 
 10, 100, or 1000 ohms at pleasure. 
 
 To measure a resistance with this apparatus, connect the 
 three sets of resistance coils, r v f 2 ,.and R, the unknown resist- 
 ance, a sensitive galvanometer, and a battery, as in the diagram. 
 By removing plugs, make r t /r 2 any convenient ratio, say 
 10/100. Vary the resistance in the rheostat until no current 
 flows through the galvanometer connected between B and B'. 
 The unknown resistance may then be computed from the 
 known resistances of three of the four branches. 
 
 In measuring resistances with the Wheatstone bridge, two 
 contact keys should be used — one in the battery branch and 
 one in the galvanometer branch. In order to eliminate the 
 effect of thermo-currents, a reversing key should be included in 
 the battery branch. 
 
 It is essential for accurate results that the battery key 
 should be closed first, and held closed long enough for the cur- 
 rent to become steady, before the galvanometer circuit is com- 
 pleted. Otherwise a deflection may be produced on closing the 
 battery circuit even when the bridge is properly balanced. 
 This is due to the fact that the distribution of a current when 
 first started is determined largely by the relative values of the 
 self-induction in different branches of the circuit, and does not 
 depend solely on the resistances, as is the case when the cur- 
 rent has become steady. The effect of disregarding this pre- 
 caution when measuring inductive resistances, such as electro- 
 magnets or the field coils of a dynamo, is always to make the 
 resistance appear larger than it really is. 
 
 This fact may be illustrated by selecting as one of the resist- 
 ances to be measured an electromagnet of rather high self- 
 induction. After the resistance in the rheostat has been so 
 
THE MEASUREMENT OF, RESISTANCE. 307 
 
 adjusted that no current passes through the galvanometer when 
 the keys are closed in the proper order, observe the effect of 
 closing the keys in the reverse order. After the galvanometer 
 needle has come to rest, observe the effect of opening the gal- 
 vanometer key while the battery key remains closed. 
 
 Wheatstone's bridge is often made in the form known as the 
 slide-wire bridge. In this pattern r is a rheostat, and the 
 branch AB'C is a straight wire, a meter long, of uniform cross 
 section. At B' there is a key which makes contact with the 
 bare wire. This key is moved along the wire until a point is 
 found having the same potential as B. Since resistance is pro- 
 portional to length (assuming the wire to be cylindrical and 
 homogeneous), we have 
 
 i=v (26l) 
 
 in which a and b are the lengths of the two segments of AB'C. 
 The requirements of the experiment are as follows : 
 Trace out the connections of the bridge. To do this make 
 a rough diagram of the bridge to be used, as well as an ideal 
 bridge diagram, in the notebook. Then starting from a 
 battery terminal, follow the branch until a point of division is 
 found. Mark this point, and follow out each of these branches, 
 one at a time, until other points of division are found, and so 
 on until the four division points have been located on the actual 
 bridge, and the resistance to be measured has been inserted in 
 the proper place. Note that three, and only three, wires meet 
 at any division point. 
 
 Measure at least three resistances, one of which is to be an 
 inductive resistance. Measure the resistances all in series, and 
 also at least two of them in multiple. Use not less than three 
 different ratios in the bridge arms in determining each resist- 
 ance. Note how to change the ratio arms of the bridge. Com- 
 pare the measured series and multiple resistances with the 
 computed values, using the single measured values, and explain 
 the principle involved in the computation. 
 
3 o8 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Addenda to the report: 
 
 (i) Explain the effect observed in measuring an inductive 
 resistance if the galvanometer key is closed first. 
 
 (2) Prove that the battery and galvanometer may be inter- 
 changed without affecting the balance of the bridge. 
 
 (3) Define the ohm, not using Ohm's law. 
 
 Fig. 106. 
 
 II- 
 
 The Carey Foster method of measuring resistance* 
 
 The ordinary form of Wheatstone bridge is not sensitive 
 enough for extremely accurate measurement of resistance, and 
 
 so other methods have been 
 devised. One of these is the 
 Carey Foster method. Figure 
 106 shows the usual arrange- 
 ment for the measurement of 
 resistance by a slide-wire 
 bridge. The point c is moved 
 until the galvanometer shows no deflection, and then from the 
 law of the bridge 
 
 R-^ _r ■ • ac 
 R 2 r ■ be 
 
 in which r is the resistance per unit length of slide wire. If R x 
 is unknown and R 2 
 known, then R t may 
 be determined. 
 
 The Carey Foster a 
 method differs from 
 this in that the known 
 and unknown resist- 
 ances k and x are 
 placed in series with the bridge wire, as shown in Fig. 107. 
 
 * Carhart and Patterson, Electrical Measurements, pp. 64-78; Henderson, Practi- 
 cal Electricity and Magnetism, pp. 58-62. 
 
 Fig. 107. 
 
THE MEASUREMENT OF RESISTANCE. 309 
 
 If the point c is moved until a balance is obtained, then the 
 following equation must hold, 
 
 R x _k+ r-ae + m 
 
 R, 
 
 2 x + r ■ be + n 
 
 (262) 
 
 m being the resistance of the connections used to join a, k, and 
 point d; and n the connecting resistance for the other arm. 
 
 If k and x are interchanged, then the point c must be moved 
 to obtain a new balance, unless k = x. Let c x be this new point, 
 
 R, x+r-ac-t+m , .- n 
 
 then 1^ = , ¥^~- ( 26 3) 
 
 -^2 k + r • oc l + n 
 
 Equations 262 and 263 may be written 
 
 *1 =— k+r-^+m (2g4) 
 
 R\ + R% k + x+ r{bc + ac)+m + n 
 
 R t x + r ■ ac-f + m 
 
 R\ + R% k +x + r^bc-t + ac t ) + m + n 
 
 in which be + ac = be t + ac x = length of slide wire. 
 
 Therefore the numerators of second members of these 
 equations must be equal. 
 
 k + r-ae + m = x+r- ac x + m, 
 
 x— k = r(ae — ac^), (265) 
 
 in which ac — ac-^ equals the distance the point c was moved to 
 obtain the new balance. This method really measures the dif- 
 ference between k and x. R x and R 2 do not enter into the final 
 expression, so that any resistance may be used, but it is advisable 
 to have them approximately equal and about the same values as 
 k and x. The resistance of the slide wire per unit length r 
 must be known. The simplest way to determine it is to obtain 
 first the two readings with k and x, then to obtain two more 
 readings with an additional resistance R, about 100 times as 
 large as k, shunted around k. This additional resistance should 
 be fairly accurate, but need not have the same accuracy as k. 
 
310 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The following equations will then hold : 
 x = k + r (ac — acj), 
 x = k' + r {ad — ac'-f), 
 
 Eliminating x, 
 
 k 2 
 
 (267) 
 
 (k + R)(ac' — ac x + ac x — ac) 
 
 In addition to the apparatus commonly used in measuring 
 resistances with the ordinary slide-wire bridge, the resistances 
 x, k, R, noted above, together with a suitable standard resistance 
 and a commutator switch, whereby the positions of k and x may 
 be interchanged without otherwise disturbing the circuit, are 
 necessary. 
 
 The experimental work to be performed is as follows : 
 
 The resistances k and x are each to be measured in terms of 
 the standard resistance. Then k and x are to be compared, 
 one in terms of the other. Three independent determinations 
 are to be made for each case. It is necessary to note the por- 
 tion of the wire in series with the standard resistance in the first 
 case in order to find whether k or x is larger. Determine r by 
 the method already described. 
 
 The above determinations assume that the value of r re- 
 mains constant throughout the length of the wire and that the 
 scale on which the lengths are read is correctly graduated. In 
 practice this is not true, hence it is necessary to determine r for 
 various portions of the wire. This determination constitutes 
 the calibration of the wire. To perform this calibration it is 
 necessary to have two coils k^ and k 2 whose difference in resist- 
 ance is of such a value as to cause a shift of the point cof a 
 desired length of the portion of the wire to be calibrated. This 
 length will depend upon the accuracy desired. In this calibra- 
 tion the middle two thirds of the wire is to be used, and not less 
 than ten segments compared. The resistances k x and k 2 are to be 
 
THE MEASUREMENT OF RESISTANCE. 311 
 
 placed in the positions of k and x in the first part of this experi- 
 ment. The resistances R x and R % must be adjustable, so that the 
 point of balance may be located at any desired part of the wire. 
 
 Place the contact point c near one end of the part of the 
 wire to be calibrated and adjust R t and R 2 until a balance is 
 obtained, or as near a balance as possible, completing it by 
 moving the point c. Then by means of the commutator inter- 
 change k x and k 2 , and move c until a new balance is obtained. 
 Leaving c in its new position, the commutator is to be put in its 
 original position, interchanging k x and k 2 again. Now readjust 
 ^1 and R 2 until there is a balance for the new position of c, as nearly 
 as possible, completing the balance by a movement of the point c, 
 if necessary. The operations just described are to be used to 
 shift the point c from one end of the section studied to the other. 
 
 By these operations the difference in the resistance of k x 
 and £ 2 is determined at different portions of the wire in' terms 
 of segment lengths, a comparison of which therefore shows the 
 variation in the resistance per unit length of the bridge wire. 
 
 The methods of computation are the same as those used in 
 the calibration of a thermometer in Exp. A 3 . (See p. 34.) The 
 readings and computations are to be tabulated as there shown, 
 and a curve is to be plotted, using readings on the bridge scale 
 and corrections as co-ordinates. 
 
 Experiment T 2 . Measurement of resistance by the method 
 of fall of potential. 
 
 In any part of a simple circuit not containing an E. M. F., 
 we have, from Ohm's law, 
 
 pd=IR, (268) 
 
 in which pd and R are the difference of potential and resistance 
 between the points, and / is the current flowing. In any other 
 part of the same circuit, we have 
 
 pd' = IR', (269) 
 
 the current being the same in all parts of the circuit. 
 
 If one of these resistances is known (r, Fig. 108), and the 
 
312 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 ratio of the two differences of potential is determined, the un- 
 known resistance may be readily calculated. This ratio may 
 most easily be determined by means of a potential galvanometer. 
 (See Exps. S 3 and S 4 .) In order that the fall of potential to be 
 measured shall not be perceptibly lessened, when the galvanom- 
 eter is connected to the two points, its resistance should be 
 iooo or more times as great as the unknown resistance. If the 
 galvanometer resistance is not large, the unknown resistance 
 may still be determined if we know the galvanometer resistance. 
 The relation is proved as follows : 
 
 Let pd and pd' be the potential differences between the 
 terminals of the galvanometer, when connected in multiple with 
 the standard and unknown resistance, respectively; then we 
 have 
 
 This assumes that the total current in the main circuit re- 
 mains constant throughout the experiment. 
 
 This method is especially useful in measuring very small re- 
 sistances. It is commonly used, for example, in determining 
 the specific conductivity of a wire of which only a small sample' 
 
 is available. (See Exp. T 3 .) 
 It is also a convenient method 
 of measuring resistance in 
 determining the temperature 
 coefficient of a wire (see Exp. 
 T 4 ), or when the variation of 
 resistance is used to measure 
 temperature changes. 
 
 The unknown resistance x 
 (Fig. 108) is connected in series with the standard r, a variable 
 resistance R, and a battery of constant E. M. F. (The variable 
 resistance may be a standard resistance box.) Since the 
 unknown resistance and the standard are in series, the same 
 current is flowing in each. 
 
 Fig. 108. 
 
THE MEASUREMENT OF RESISTANCE. 313 
 
 The fall of potential is measured by connecting the galva- 
 nometer, through a reversing key, first in multiple with the un- 
 known resistance, and then in multiple with the standard, and 
 observing the deflection in each case. The ratio of the two re- 
 sistances is then equal to the ratio of the tangents of the two 
 deflections, if a tangent galvanometer be used, or to the deflec- 
 tions if a d'Arsonval galvanometer is employed. 
 
 In preparing for this experiment, the galvanometer should 
 first be put in multiple with the unknown resistance, and the 
 resistance or E. M. F. in the main circuit varied, until the de- 
 flection is a suitable one. Then put the galvanometer in mul- 
 tiple with r by means of the key k^ ard change the value of r 
 until about the same deflection is obtained as when in multiple 
 with x. 
 
 The resistance in the main circuit should not be very small, 
 for under these circumstances the battery is more apt to polarize, 
 and the current to change during the experiment. 
 
 To eliminate various errors, it is best to have two reversing 
 keys, one k z in the battery circuit and the other k x in the galva- 
 nometer circuit. For each position of the key in the main cir- 
 cuit, the direct and reversed reading of the galvanometer should 
 be observed. The reversal of the main current eliminates errors 
 due to the thermo-currents caused by differences in temperature 
 between different portions of the circuit ; while the reversal of 
 the galvanometer circuit eliminates any error that might be 
 caused, when using a tangent galvanometer, by a direct magnetic 
 action of the current in the unknown resistance upon the gal- 
 vanometer needle. 
 
 In taking observations, it is well to alternate between the 
 unknown and standard resistances, so as to eliminate the error 
 which might be introduced by a progressive change in the 
 conditions. 
 
 Several independent determinations should be made. This 
 may be done as follows : 
 
 (1) Take direct and reversed readings with the galvanome- 
 
314 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 ter in multiple with x and then with r for both positions of the 
 switch k z . Repeat these readings as a check. 
 
 (2) Change the value of R cutting down the deflection 
 about one third and repeat (1). 
 
 (3) Change the value of r by a few ohms and repeat (1) and 
 (2). 
 
 Addenda to the report: 
 
 (1) Explain by diagram the necessity of having two revers- 
 ing keys. 
 
 (2) Compute the error introduced in your case by using a 
 galvanometer whose resistance was not infinite compared with 
 the unknown resistance. 
 
 Experiment T 3 . Measurement of specific resistance. 
 
 The specific resistance of a substance is usually defined as 
 the resistance in absolute units of a conductor, 1 cm. long and 
 1 sq. cm. in cross-section. Specific resistance is sometimes de- 
 fined in terms of mass instead of volume ; i.e. it is the resistance 
 of a conductor 1 cm. long whose mass is 1 gram. 
 
 If the resistance, length, and cross-section of a wire be meas- 
 ured, it is obvious, since resistance varies directly as length, and 
 inversely as cross-section, that its specific resistance may be 
 readily calculated. The temperature at which the resistance 
 has been determined should be noted and stated. 
 
 I. 
 
 If the sample furnished has a resistance of several ohms, 
 the resistance may be measured by the method of the Wheat- 
 stone bridge. The measurement should be made with great 
 care, using several different ratios, reversing the ratio arms, 
 reversing the battery current, and taking every precaution to 
 make the determination accurate. The temperature of the 
 bridge coils, as well as that of the wire whose resistance is 
 being determined, should be observed. From these data, know- 
 
THE MEASUREMENT OF RESISTANCE. 315 
 
 ing the temperature coefficients of the wire and the bridge coils, 
 and the temperature at which the bridge is correct, the resist- 
 ance of the wire at 0° can be computed. 
 
 II. 
 
 If the resistance of the sample to be experimented on is one 
 ohm or less, it should be measured by the fall of potential 
 method and the same series of readings be made as in Exp. T 2 . 
 
 In either case, the length and diameter should be measured 
 with the greatest care. The diameter may be directly measured 
 in a number of places by means of a micrometer wire gauge; 
 or, better, the mean cross-section may be indirectly determined 
 from the mass, length, and density of the specimen. The 
 density should be determined by weighing in water. 
 
 Addenda to the report: 
 
 (1) Calculate the specific resistance in terms of volume and 
 in terms of mass. 
 
 (2) Compute the relative conductivity, assuming that of 
 copper to be 100. 
 
 Experiment T 4 . Determination of the temperature coeffi- 
 cient for resistance of carbon and of various metals. 
 
 The resistance of all conductors varies with the temperature, 
 
 and the temperature coefficient for resistance is defined by the 
 
 equation 
 
 R t = R Q {i+at°), (271) 
 
 in which R t and R are the resistances at temperatures t° and 
 0°, respectively, and a is the coefficient. For metals a is 
 positive. 
 
 If the material whose temperature coefficient is to be deter- 
 mined is a metal it should be in the form of wire. In using 
 the methods here described, wire should be used which has a 
 resistance of several ohms, in order to have comparatively 
 large absolute changes of resistance, thus decreasing percentage 
 errors in determinations of the changes. The wire should be 
 
316 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 wound in the form of a coil, the various turns being insulated 
 from each other. 
 
 If the wire is large and stiff, it need not be wrapped with insu- 
 lating material since it will hold its form, but small wire should be 
 very well insulated and wound on a thin metal cylinder. There 
 should be heavy low resistance current leads soldered to the ends 
 of the coil. For use in the " fall of potential " method there should 
 be a second pair of leads soldered to the terminals of the coil, 
 but it is not necessary that these leads have low resistance. 
 
 The coil should be placed in an oil bath whose temperature 
 may be varied at will. The bath should be constantly stirred. 
 Before making readings sufficient time should elapse to permit 
 the coil to attain the desired temperature. This time will be 
 short if the coil be free or wound on thin copper. The tempera- 
 tures are to be read with a thermometer inserted in the bath. 
 
 I. 
 
 Method of the Wheatstone bridge. 
 
 The heavy insulated copper wires soldered to the two ends 
 of the coil are to be connected to the terminals of the Wheat- 
 stone bridge. Make at least ten determinations of the resist- 
 ance of the coil at approximately equal temperature differences 
 between room temperature and 95 C. 
 
 Readings of resistance should be taken both for increasing 
 and decreasing temperatures, 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 oil thoroughly stirred. 
 
 From the results obtained, plot 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°. 
 
THE MEASUREMENT OF RESISTANCE. 317 
 
 II. 
 Fall of potential method. (See Exp. T 2 .) 
 
 In this case, a wire of low resistance may be used. The 
 connections to be made are as indicated in Exp. T 2 , the heavy 
 current leads being in the battery circuit. Make the preliminary 
 adjustment of r as in Exp. T 2 . 
 
 Resistance determinations are to be made at temperature 
 intervals of seven or eight degrees between room temperature 
 and 95° C. Make readings for each determination as follows : 
 
 (1) Direct and reverse reading with the galvanometer in 
 multiple with r. 
 
 (2) Thermometer reading. 
 
 (3) Direct and reverse reading with the galvanometer in 
 multiple with the coil to be tested. 
 
 (4) Thermometer reading. 
 
 (5) Take reading (3), (2), and (1) in the order named with 
 the battery switch k z reversed. The temperature is to be taken 
 as the mean of the three thermometer readings. Compute the 
 unknown resistance, using mean deflections. 
 
 For the determination of the temperature coefficient it 
 is not necessary to have any absolute standard of resistance. 
 Since galvanometer deflections are proportional to resist- 
 ance, we may substitute for R t and R the deflections h t and 
 B (equation 271), or their tangents, if a tangent galvanometer 
 is used. 
 
 After making the necessary readings, a curve should be 
 plotted, with temperatures as abscissas and galvanometer de- 
 flections as ordinates. The equation of this line is then to be 
 determined, and from its constants the temperature coefficient 
 and the deflection for o° C. are to be calculated. < 
 
 Addenda to the report: 
 
 (1) Justify the substitution of galvanometer deflections for 
 resistances in the above equation. 
 
3i8 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 (2) Using the coefficient determined, calculate the resist- 
 ance at absolute zero of a wire whose resistance is 100 ohms 
 at o° C. 
 
 Experiment T 5 . Kelvin double bridge method for low 
 resistance measurements. 
 
 The method devised by Lord Kelvin for measuring low 
 resistances gives determinations of very great accuracy. It 
 may be used to advantage in determining specific resistances 
 
 and temperature coeffi- 
 cients of small samples. 
 The specimens to be 
 tested may be made up 
 in the form of bars and 
 may be compared with 
 Fi & 109 - standard bars. 
 
 An outline of the theory of the method is given in connec- 
 tion with Fig. 109. 
 
 The resistance X to be measured is connected in series with 
 the standard N and a battery. The connection between X and 
 N should be of low resistance, so that the quantity d, represent- 
 ing the resistance between the points q and r, may be small. 
 The auxiliary resistances a, b, c, and e are connected to the un- 
 known and standard at p, q, r, and s by sliding contacts if X and 
 N are bars, or otherwise by mercury cups or other definite low 
 resistance contact devices. If convenient, the resistances com- 
 prised between the contact points/^ and rs should be made 
 approximately equal and then adjusted to give zero reading of 
 the galvanometer connecting the points m and n, the ratio a/b 
 and c/e being each equal to unity. When this condition is 
 obtained, then 
 
 (272) 
 
 X 
 
 a 
 b' 
 
 The above equation will be true even if the ratios a/b and 
 c/e are not unity, so long as they are equal. If they are not 
 equal, the following relation holds : 
 
THE MEASUREMENT OF RESISTANCE. 319 
 
 X_a d^ ( e \(a 
 
 N~ b N\c + e + , 
 
 This equation is general and is derived in the usual manner 
 from the principle of equal fid's between fi and n, and q and m 
 on the one side of the zero reading galvanometer and equal 
 fid's between n and s, and m and r on the other side. 
 
 It is readily seen that the so-called correction factor 
 
 -Tj[ — — T~^)( I ) * s zero when a/b is equal to c/e. It is also 
 
 seen that in the event that such is not quite true, that if d is 
 small compared with N, the correction factor may be neglected. 
 When measuring very small resistances, d may be comparable 
 with iVand the ratio a/b not equal to the ratio c/e. It is then 
 necessary to compute the value of the correction factor. In the 
 following experiment, which is mainly one of method, the cor- 
 rection factor may be neglected. 
 
 Build up a double bridge, using well-adjusted resistance 
 boxes of the range .1 to 1000 ohms for the auxiliary arms a, b, c, 
 and e, a carefully calibrated box or standard resistance, and the 
 coil or wire whose resistance is to be determined. Use a short 
 length of heavy copper wire to connect the standard and the 
 unknown, and make all connections with the auxiliary resist- 
 ances of as low resistance as convenient. This does not apply 
 to the galvanometer connections. 
 
 Knowing the approximate resistance of X, choose a value 
 of N of the same range. Obtain a "balance" by changing 
 the values of the auxiliary resistances, being careful to keep 
 the ratios a/b and c/e as nearly equal as possible, keeping the 
 smaller resistances in the neighborhood of 100 ohms. 
 
 Make two other determinations of the unknown, using dif- 
 ferent initial values of N. It may be found convenient to use 
 standard resistances of .1, 1, and 10 ohms. If the length and 
 cross-section of the unknown be obtainable, compute its specific 
 resistance. 
 
320 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Experiment T 6 . 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 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 bridge method of measuring resistance 
 may be adapted to the determination of electrolytic resistance 
 as follows: 
 
 (i) An alternating. current is supplied by replacing the bat- 
 tery by the secondary circuit of an induction coil, or by con- 
 necting with an alternating current generator of high frequency 
 and low E. M. F., the circuit being protected by resistance if 
 necessary. 
 
 (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 v The resistances of the bridge arms are 
 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, such as 
 the Kohlrausch bridge, is one commonly used. It is a slide- 
 wire bridge with a non-inductive variable resistance, the slide 
 wire being wound on a drum. 
 
 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. 
 
THE MEASUREMENT OF RESISTANCE. 321 
 
 If the vessel used does not admit of accurate measurement, 
 it should be standardized as follows : 
 
 (1) Fill the vessel with a solution of known concentration of 
 zinc or copper sulphate at a known temperature and determine 
 its resistance. 
 
 (2) 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 and a little of the solution to be tested next. 
 The resistance of a solution is sometimes greatly changed by 
 even slight traces of other substances. 
 
 Determine the resistances of three solutions and try also 
 ordinary water and distilled water. 
 
 Make at least three independent determinations of each 
 liquid used, including the standardizing liquid, using the middle 
 third of the slide wire only. Determine the temperatures of 
 the solutions, and note any changes in temperature during the 
 tests. 
 
 Experiment T 7 . Measurement of the internal resistance of 
 a battery. 1 
 
 Open circuit cells or those that do not give constant current 
 when used continuously require different treatment in deter- 
 mining their resistances than those whose E. M. F.'s remain 
 practically constant. Several methods are given below ; I- 
 VI being for cells of constant E. M. F.'s, and VII and VIII for 
 cells that polarize. 
 
 I. 
 
 Ohm's method. 
 
 This experiment requires the same observations as Exp. S lt 
 and the battery resistance may be calculated from the obser- 
 
322 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 vations taken in that experiment, provided the resistance of 
 the galvanometer and connecting wires is known. If R is 
 a known resistance, R b the resistance of the battery, and R g 
 that of the galvanometer including the connecting wires, we 
 have 
 
 '- X. + % + R (274) 
 
 It is not necessary, however, to know the constant of the gal- 
 vanometer. In the above equation ks or 7 tan 8 may be sub- 
 stituted for I, the value of the current, depending on the kind 
 of galvanometer used in the experiment. 
 
 If two different values of R be taken, and the corresponding 
 galvanometer deflections observed, we shall have two equations 
 similar to (274). If one of these equations be divided by the 
 other, both E and I will be eliminated, and R b will be a 
 function of known quantities. 
 
 This experiment furnishes an excellent example of the gen- 
 eral principles discussed on page 4. The precautions there 
 suggested should be followed here ; that is to say, the differ- 
 ence 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. Furthermore, 
 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 ten different 
 resistances as in Exp. S r These readings should be taken 
 several times for each resistance used, and the mean deflection 
 derived from them should be utilized in the computations. 
 
THE MEASUREMENT OF RESISTANCE. 
 
 323 
 
 (3) From each suitable pair of observations compute the 
 resistance of the battery. 
 
 The resistance of the battery is also to be found graphically, 
 using the method of " least squares " (see p. 24) for locating the 
 line and determining its slope and intercept. The co-ordinates 
 of the curve are to be the known box resistances as abscissas 
 and the reciprocals of currents, or quantities proportional to 
 them, as ordinates. 
 
 Derive the physical equation of the curve and interpret it, 
 obtaining physical constants from it. 
 
 II. 
 
 Thomson's method. 
 
 Connections are made as shown in 
 Fig. no. The resistances R and R t 
 are adjusted until a suitable deflection 
 is given by the galvanometer. R is 
 then removed and R x adjusted until 
 the galvanometer has the same de- 
 flection as before. Call this new 
 value of the resistance R 2 - 
 
 In the first case, 
 
 R 
 
 A/VWVW- 
 
 z 
 
 R, 
 AAAAA- 
 
 Fig. 110. 
 
 /„ = 
 
 R g + R + R x 
 
 R„ + 
 
 (Rj + R^R' 
 
 (275) 
 
 R x +R g + R 
 
 where ./J, is the current through the galvanometer, R b and R g are 
 the resistance of the battery and galvanometer respectively. 
 In the second case, 
 
 R b + R g + R 2 
 
 Equating the right-hand members of these equations and 
 solving for R b gives 
 
 RiR.-R,) 
 
 R„ = 
 
 R g + R x 
 
 (277) 
 
3 2 4 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From this equation the resistance of the battery may be 
 computed. As the equation shows, it is necessary to know the 
 resistance of the galvanometer. If, however, the galvanometer 
 resistance is small as compared with J? v it may be neglected. 
 
 Determine the resistance of two different cells, making five 
 independent sets of readings for each, using different initial 
 conditions for each set. 
 
 III. 
 
 Mance's method* 
 
 This method is one of the most accurate ones for a battery 
 that is not subject to rapid polarization. It was originally ap- 
 plied to the measurement of the resistances of cable and tele- 
 graph lines as well as of battery resistances. 
 
 Although it is not a Wheatstone bridge method, and should 
 not be confused with it in any sense, the equation for finding 
 the unknown resistance has the same form, but cannot be de- 
 rived in the same manner. The form of circuits for Mance's 
 method and the Wheatstone bridge may be made to look alike. 
 In the circuit for Mance's method the battery to be tested takes 
 the place of one of the four arms, as x in Fig. 105, the galva- 
 nometer is in the usual posi- 
 tion, and in place of the bat- 
 tery B' usually used there is 
 a make and break key K, 
 Fig. in. 
 
 The measurement consists 
 in so adjusting the resistances 
 A, B, and R that the opening 
 or closing of the key K has no 
 effect on the deflection of the galvanometer through which cur- 
 rent is flowing all the time. When this condition is reached, the 
 difference of potential between the galvanometer terminals re- 
 mains constant, but it is not zero. When the key K is open the 
 current divides at a, part flowing to b through the resistances A 
 
 * Mance, Proc. Royal Soc, London, 1870, Vol. 19, p. 248. 
 
 Fig. 111. 
 
THE MEASUREMENT OF RESISTANCE. 
 
 3 2 5 
 
 and B in series, and the remainder through the galvanometer. 
 The circuit is completed through R and the battery. It may be 
 shown that the current flowing in the galvanometer branch is 
 
 E 
 
 /„ = 
 
 (*+&&+* 
 
 A+B+G 
 
 (278) 
 
 A+B 
 
 If the key K is closed, the battery current will divide at a, 
 part flowing to the other battery terminal at c through A, con- 
 sidering the key circuit as of negligible resistance ; the other 
 part flowing through the galvanometer to b, where it divides and 
 flows through the multiple resistances B and R to the negative 
 terminal of the battery c and d. In this case the current flow- 
 ing through the galvanometer may be shown to be 
 
 E 
 
 la 
 
 R b + 
 
 A G + 
 
 RB 
 R+B. 
 
 A + G + 
 
 RB 
 
 G + 
 
 RB 
 R + B 
 
 + A 
 
 A 
 
 (279) 
 
 R + B_ 
 
 For the condition of equal galvanometer deflections for open 
 or closed key K the values of /„ in equations 278 and 279 
 will be equal, and therefore the denominators of the right-hand 
 members may be equated and solved for R b> giving 
 
 *> = p. 
 
 (280) 
 
 If a tangent galvanometer is used in performing the exper- 
 iment, and 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. If a d'Arsonval galvanometer is used, 
 its sensibility may be changed by a suitable shunt. Judgment 
 must be used in the choice of the resistances placed in the va- 
 rious branches of the circuit, so as to secure the greatest sen- 
 sitiveness and at the same time as little inconvenience as possible 
 
326 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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 branches A, B, and R, so that 
 the greatest resistance is in series with the battery and galva- 
 nometer. 
 
 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. 
 The effect of this is to give the galvanometer needle a slight 
 inductive throw, even though the proper relation of the resist- 
 ances A, B, R, and R b has been reached. 
 
 Make ten independent determinations of a battery resistance. 
 
 Addenda to the report: 
 
 (i) Prove equation 280 for Mance's method by Ohm's or 
 Kirchhoff's laws. 
 
 (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 ? 
 
 IV. 
 
 Mance's method as modified by Lodge and by Guthe* 
 
 Mance's method has been further extended by Lodge and 
 later by Guthe by using a condenser in series with a ballistic 
 galvanometer which makes a " zero " method of it. 
 
 The circuit of the modified method is made up as fol- 
 lows: 
 
 The cell whose resistance is desired is put in series with a 
 key K and the resistances Q, P, and R. The point D is con- 
 nected to a blade of the switch 5" in such a way that when the 
 
 * O. J. Lodge, Phil. Mag., 1877, vol. 3, p. 515; Arthur W. Smith, Science, 
 vol. N. S. 22, p. 434. 
 
THE MEASUREMENT OF RESISTANCE. 
 
 3 2 7 
 
 switch is moved the blade passes over the point a connected to 
 A. B is connected to a condenser and C is connected to the 
 galvanometer. The second blade of the switch is connected to 
 the galvanometer and to the condenser from b. Its movable 
 end slides from the point c to the point e, in both of which 
 positions it short-cir- 
 
 ~d>- 
 
 D <2 
 =-AAAAAA 
 
 _4?ir 
 
 X3- 
 
 l> 
 
 B 
 
 AAAAA/*- 
 
 R C 
 
 AAAAAA-^ 
 
 cuits the galvanom- 
 eter, c and e being 
 permanently con- 
 nected to the galva- 
 nometer. The switch 
 is so made that the 
 contact at c is made 
 before that of the 
 other blade is broken 
 at a. This is to eliminate the kick of the galvanometer when 
 the condenser is being charged to its original potential when 
 the switch is moved from e to c. 
 
 With the switch on e it is seen that contact at a is broken 
 and the condenser is charged to a pd, e v equal to the fall of 
 potential through P and R from B to C. This drop is 
 
 E 
 
 Fig. 112. 
 
 pd=e 1 = f(R + P) = 
 
 R+P+Q+X 
 
 (R + P), (281) 
 
 where E is the E. M. F. of the cell under test and X its 
 resistance. 
 
 Now the switch is moved so that the contact at e is broken, 
 the galvanometer is put in the circuit between B and C, the con- 
 tact at a is made, thus short-circuiting Q and P, reducing all 
 points between A and D to the same potential. Contact at a 
 is then broken and that at c made, thus restoring conditions in 
 the lower circuit and short-circuiting the galvanometer. 
 
 When P and Q are short-circuited, thus bringing them to 
 the same potential, the condenser is charged to some new value 
 e' equal to the pd between B and C or A and C, since B and A 
 
328 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 are at the same potential. Call this new -potential difference 
 pd', then 
 
 pd' = e'=I'R = ^-^R. (282) 
 
 If on throwing the switch from e to c there is no throw 
 of the galvanometer, it indicates that no electricity has passed 
 through the galvanometer. Therefore the potential difference 
 between the terminals of the condenser has not changed and 
 d = e v 
 
 Equating the right-hand members of (281) and (282), 
 
 R + P R 
 
 R + P+Q + X R + X' 
 from which X = % R - 8 3) 
 
 The experiment consists in so adjusting the resistances Q, 
 P, R, that no galvanometer throw takes place when the switch 
 is thrown from e to c. If the cell used is one that polarizes 
 rapidly, make Q large (1000 ohms or more). Give R different 
 values, and for each value change P and Q until a balance is 
 obtained. 
 
 Make at least a half-dozen determinations of X. If the 
 resistance is very low, put a known resistance in series with it 
 and proceed as before, finding the resistance of the combination. 
 
 The method may be used to find the temperature coefficient 
 of the cell resistance. 
 
 V. 
 
 Fall of potential method for cells of constant E. M. F. 
 
 A sensitive galvanometer of high resistance, used to measure 
 potential differences, is connected to the terminals of the bat- 
 tery to be tested. A variable resistance R is also connected to 
 the battery terminals in multiple with the galvanometer. The 
 circuit is connected up as shown in Fig. no, excepting R v which 
 may be omitted. When the circuit through R is broken, i.e. 
 
THE MEASUREMENT OF RESISTANCE. 329 
 
 when R is infinite, the deflection of the galvanometer is propor : 
 tional to the E. M. F. of the battery, 
 
 d x = kE, (284) 
 
 in which d x is the deflection in scale divisions. 
 
 When the circuit is closed through the resistance R, the 
 deflection is proportional to the fall of potential in R, 
 
 d-i = Hva — Vj>) = kIR. 
 
 Substituting for / the value obtained from Ohm's law, 
 
 d * = kR RTR b - (285) 
 
 Dividing (285) by (284), 
 
 d 2 /d x = R/(R + R b ), (286) 
 
 or R b = R d ^~ d * . (287) 
 
 When R = R b , d 2 is one half of d v A direct method is 
 therefore given by adjusting R until the initial deflection is 
 halved. R is then equal to R b . 
 
 In performing this experiment use two different cells, taking 
 five different values of R for each. The deflection with R 
 infinite should be taken alternately with the readings for the 
 different values of R. Take first a reading with the circuit 
 through R broken, then with R closed, next with R broken, and 
 then with a new value for R, and so on. Remember that the 
 best results are obtained when d 2 is about one half as large as d v 
 
 If the circuit is connected as in the figure, the resistance of 
 R b as found by this method includes the connecting wires a and 
 b for which, if their resistance is appreciable, correction must 
 be made. 
 
 VI. 
 
 Fall of potential method for open ciraiit cells. 
 
 When the cell is one that polarizes, a condenser and a bal- 
 listic galvanometer are necessary. The connections are shown 
 
33© JUNIOR COURSE IN GENERAL PHYSICS. 
 
 in Fig. 113. The switch K x is for the purpose of charging the 
 condenser, and then discharging it through the galvanometer. 
 
 The method pursued is similar to that in V. When the 
 external circuit through R is open, the condenser C is charged 
 and immediately discharged through the gal- 
 vanometer and the throw noted. As this throw 
 is proportional to the E. M. F., it gives d 1 in 
 equation 284, part V. The resistance R is 
 now inserted and the discharge throw again 
 noted. As this throw is proportional to the 
 fall of potential in R, it gives d 2 in equation 
 285, part V, above. A number of different 
 resistances should be used. Alternately with 
 
 Fig. 113. . 
 
 these, the discharge for R infinite should be 
 observed. The best results are obtained when the value of d 2 
 is about half as large as that of d x . 
 
 The circuit through R should be closed for a very short 
 time, as the cell may quickly polarize. The condenser must 
 however be disconnected from the battery by opening K x before 
 this circuit is opened. 
 
 Make at least six independent determinations of the inter- 
 nal resistance of a battery. 
 
 VII. 
 
 Benton's method.* 
 
 This method may be used for cells that do or do not polarize 
 rapidly. 
 
 In Fig. 1 14, Cisa sensitive galvanometer of low resistance, 
 E 1 the cell whose internal resistance is to be found, E 2 an aux- 
 iliary cell, k a key which closes the circuits ckE x R"ac and 
 ckE 2 dc at the same time. The auxiliary cell E 2 may have 
 either a greater or less voltage than E v 
 
 If the internal resistance of E 2 is too small, then another 
 resistance R 2 ' must be used in series with it. 
 
 * Physical Review, vol. 1 6, April, 1903, p. 253. 
 
THE MEASUREMENT OF RESISTANCE. 331 
 
 (1) Make R" equal to zero and R 1 any suitable small resist- 
 ance. Then change R% until such a value is found that on 
 closing the key k no galvanometer deflection is produced. 
 
 (2) Give R 1 some much larger value R x ' , and leaving R 2 
 unchanged, make R" such a value 
 that no deflection is produced. 
 
 Then if R b be the resistance of 
 
 cell E v 
 
 R R" 
 
 **~Rf=Ri (288) 
 
 Proof. Let the upper and 
 lower branches be known as 1 and 
 
 Fig. 1 14. 
 
 2. Since no current flows through 
 
 the galvanometer for a "balance," the current in all parts of 
 branch 1 is the same. The current in all parts of branch 2 
 also has a single value. 
 
 For the first adjustment the current in the upper circuit is 
 
 ^-R. + R,- R,' (289 > 
 
 and in the lower circuit 
 
 E A *pd. 
 
 2' + RJ R* 
 
 Since the galvanometer has no deflection, 
 
 yd c = d pd c . (291) 
 
 For the second adjustment the current in the upper circuit is 
 
 E x _'pd a < 
 
 h ~ RJ + R" + R b ~ RJ ^ 292) 
 
 Since no change has been made in the lower circuit the current 
 has the same value as in the first case, therefore, 
 •pd. = 'pd.'. 
 
 From equations 289 and 292, 
 
 E.R, _ E& 
 
 T >= R* + R< + Rr R- (290) 
 
 R 1 + R h R t ' + R" + R b 
 
 (293) 
 
332 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From equation 293, 
 R» 
 
 RJi" 
 
 (294) 
 
 R{-R x - 
 Make six determinations of R b , varying R v but keeping it 
 
 comparatively low, not over 20 ohms. 
 
 Experiment T 8 . Measurement of the resistance of a galva- 
 nometer. 
 
 The following methods of measuring the resistances of gal- 
 vanometers use their own deflections and require no auxiliary 
 galvanometers. 
 
 I. 
 
 The following method is best suited for high resistance gal- 
 vanometers. 
 
 Connections are made as in Fig. 115. R s is made so small 
 that its resistance may be neglected in comparison with the re- 
 sistance of the galvanometer. The re- 
 sistances are adjusted until a large deflec- 
 tion is obtained, then R 2 is changed until 
 the deflection of the galvanometer is 
 halved. Call this new value of the resist- 
 ance R 2 '. Since the resistance in the 
 galvanometer branch of the circuit has 
 been doubled, we have 
 
 RJ + R g = 2 (R 2 + R g ), 
 
 zR % . (295) 
 
 R, 
 
 R 3 
 
 -WWW^ 
 
 Fig. 115. 
 
 or 
 
 R, = RJ- 
 
 If i? 2 is zero, the method is further simplified. The resist- 
 ance R x is inserted in the battery circuit as an aid in controlling 
 the value of the initial deflection of the galvanometer. 
 
 If a gravity cell is used or any other cell of fairly high 
 resistance that does not polarize, the cell may simply be short- 
 circuited by a wire having a resistance of not over a few tenths 
 of an ohm. 
 
THE MEASUREMENT OF RESISTANCE. 333 
 
 Make six independent determinations, varying the initial 
 conditions for each. 
 
 II. 
 
 Method of equal deflections. 
 
 The circuit is connected as in Fig. 115, but no resistance 
 is placed in series with the galvanometer. The lettering and 
 description below apply to that figure. 
 
 A known resistance R s is inserted in multiple with the gal- 
 vanometer, and the resistance R 1 is varied until a good-sized 
 deflection of the galvanometer is obtained. The shunt R s is 
 then removed, and the resistance R 1 is changed until the de- 
 flection is the same as before. Let this new value be R 2 . 
 From the data obtained in these two adjustments the resistance 
 of the galvanometer may be computed. 
 
 In the first case, when the galvanometer is shunted, 
 
 T _Rj E 
 
 ' R, + R. R , _ , R,R. ' ( 2 96) 
 
 K ^ + r + R^R s 
 
 where I g is the current in the galvanometer, R q and R s are the 
 resistances of the galvanometer and shunt respectively, and r is 
 the resistance of the battery and connecting wires. 
 In the second case, 
 
 h= (297) 
 
 R 2 + r+R g yW; 
 
 Since the values of /,, are equal, the right-hand members of 
 (296) and (297) may be equated and E eliminated. Solving for 
 R g gives 
 
 R ' = Rl Rl+ R r' X (298) 
 
 In case R 1 is so large that r may be neglected, 
 
 R g = ^{R*-R x \ (299) 
 
 where all the quantities are known. 
 
334 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 The best results are obtained when R, is nearly equal to R g , 
 that is, when R x is one half of R%. 
 
 In performing this experiment several determinations should 
 be ipade. In the first one a value for R, is selected at random. 
 Noting the values obtained for R^ and R 2 and remembering 
 that the most accurate results are obtained when R x is about 
 one half of R 2 , more suitable values of R a can now be chosen. 
 For a complete set of observations, at least six different values 
 of R 1 are to be used. If the resistance of the battery and con- 
 necting wires is not accurately known, R 1 must be so large 
 that their resistance may be neglected. On account of their 
 low internal resistance, Edison-Lalande cells are particularly 
 well adapted for this experiment. 
 
 Two restrictions on this method may be noted : 
 
 (i) When the galvanometer is of low resistance, the resist- 
 ance R, will be small and the resistance of the connecting 
 wires must be taken into account. 
 
 (2) When the galvanometer is not very sensitive, the resist- 
 ance R 1 is not usually large enough, so that r, the resistance of 
 the battery and connecting wires, may be neglected. In this 
 case the method is tedious, as a determination of r must be 
 made. 
 
 III. 
 
 Thomson's {Lord Kelvin's) method. 
 
 In this method the galvanometer to be tested is inserted as 
 the unknown resistance in one of the arms of a Wheatstone 
 bridge. In the usual position of the galvanometer (see Fig. 105) 
 a key is inserted. When the bridge is properly " balanced," the 
 key is connected to points having the same potential. Clos- 
 ing the key will therefore cause no change in the current 
 from the battery. But if the bridge is not properly balanced, 
 closing the key will change the current through the galva- 
 nometer. Therefore instead of adjusting the resistance until 
 the galvanometer gives no deflection, the bridge is adjusted 
 
THE MEASUREMENT OF RESISTANCE. 335 
 
 until no change in the deflection of the galvanometer is observed 
 on closing the key. 
 
 For low resistance galvanometers, the slide-wire bridge is 
 best. In using the slide-wire bridge it is always well, after one 
 adjustment has been made, to interchange the positions of the 
 standard and the unknown, the galvanometer in this case. 
 Usually it will be found convenient to build up a bridge, using 
 resistance boxes for that purpose. 
 
 In many cases the deflection of the galvanometer will be so 
 great that it will be necessary to diminish it. If the galva- 
 nometer is not of the d'Arsonval type, the deflection may be 
 decreased by a magnet. The magnet should be placed so that 
 its field opposes the field due to the current in the galvanometer 
 coils, but does not materially change the earth's field ; that is, 
 the magnet should be placed so that the north and south com- 
 ponent of its field at the center of the galvanometer is weak. 
 Otherwise, the sensibility of the galvanometer may be so 
 decreased that small changes in the current cannot be detected. 
 In case the galvanometer has two coils that can be connected to 
 different circuits, it may be used differentially, that is, one coil 
 may be measured at a time, and through the other may be sent 
 a current from another circuit in a direction such as to oppose 
 the effect from the coil in the bridge circuit. When these two 
 currents are thus balanced, the resultant deflection is small, yet a 
 slight change in either current is indicated by the needle. This 
 neutralizing current may be obtained by a multiple circuit from 
 the battery that is used in the bridge circuit, and can be 
 regulated by placing a resistance box in series with the galva- 
 nometer coil. 
 
 When the field due to the coil that is being measured is 
 neutralized by a field either from a magnet or from another 
 current, the observer may be bothered in balancing the bridge by 
 having to readjust the magnet or auxiliary current each time the 
 resistance in the bridge is changed. This trouble is entirely 
 avoided if the adjustment of the bridge is made by changing 
 
336 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 both r % and R in such a way that their sum remains constant. 
 If a slide-wire bridge is used, the difficulty is avoided by con- 
 necting the battery wires to the terminals of the slide wire. 
 
 In the case of a d'Arsonval galvanometer, where neither 
 method of decreasing the deflection is available, it will be neces- 
 sary to weaken the current by placing resistance in the battery 
 circuit or by using two cells of slightly different electromotive 
 force connected so as to oppose each other. 
 
 Make six independent determinations of the resistance of the 
 assigned galvanometer, varying conditions of the circuit. 
 
CHAPTER XII. 
 
 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 unity of quantity is that quantity of 
 electricity which is transferred by unit current in unit time. 
 The practical unit of quantity, the coulomb, is the quantity 
 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 
 
 a-f. 
 
 Idt 
 
 taken between the proper limits. 
 
 Quantities of electricity are considered when we deal with 
 (i) 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 
 coil of a galvanometer by a momentary current can be meas- 
 ured, however, by the "throw " or " swing " of the moving parts 
 due to the magnetic impulse of the momentary current. 
 
 A galvanometer used for measuring such impulses is called 
 vol. 1 — z 337 
 
338 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 a ballistic galvanometer from its analogy to a ballistic pen- 
 dulum. It may be of either the tangent or d'Arsonval type. 
 
 Any galvanometer can be used as a ballistic galvanometer, 
 simply by observing " trirows " instead of permanent deflections, 
 provided that the motion of the moving parts be slow enough 
 to determine the end of the swing with accuracy. It is also 
 desirable, in the case of galvanometers used ballistically, that 
 the damping should not be very great. These two requisites 
 are secured by making the moving parts heavy, thus securing 
 slow motion and small factor of decrement. In using a tangent 
 galvanometer ballistically, it must be remembered that the mag- 
 netic moment of the needle enters the constant of the instru- 
 ment. Therefore the needle should be a magnet whose moment 
 is not subject to rapid change. 
 
 Experiment U v Measurement of the constant of a ballistic 
 galvanometer. 
 
 The following are three methods for determining this 
 constant : 
 
 (i) By measuring the throw due to the discharge of a con- 
 denser. 
 
 (2) By measuring the throw produced by the induced cur- 
 rent due to the rotation of a coil in a magnetic field, or a known 
 change of magnetic flux in a solenoid. 
 
 (3) By computation from the periodic time of the moving 
 parts of the galvanometer, and the constant of the instrument 
 used as a current measurer. 
 
 Methods (1) and (2) may be applied to all galvanometers 
 having little damping. They are also applicable to d'Arsonval 
 galvanometers, which are heavily magnetically damped.* 
 
 The third method, the one used in this experiment, may be 
 applied in determining the " quantity constant " of galvanome- 
 ters in which the damping is small. 
 
 * O. M. Stewart, The Damped Ballistic Galvanometer, Physical Review, Vol. 
 XVI, 1903, p. 158. 
 
ELECTRICAL QUANTITY. 339 
 
 The theory on which the third method is based is the sa^ne 
 in principle whether applied to the d'Arsonval or tangent gal- 
 vanometer, as will be seen from the following brief discussion. 
 
 The d'Arsonval ballistic. 
 
 It is assumed that the total quantity of electricity to be 
 measured passes through the galvanometer before the moving 
 coil has moved appreciably from rest ; and that the initial veloc- 
 ity a> given to the coil due to the magnetic forces produced 
 by the transient current is the maximum velocity. 
 
 The instantaneous value of the magnetic torque (see p. 243) is 
 
 L q = ^. (300) 
 
 But the torque is equal to the moment of inertia multiplied 
 by the angular acceleration, therefore 
 
 iAf= K ^. (301) 
 
 \ 10 dt 
 
 Integrating over the time the current flowed; 
 
 41 C idt = 4fQ=Kwv (302) 
 
 10J 10 
 
 that is, quantity of electricity is proportional to the angular 
 momentum or moment of momentum Ka> . 
 
 The equation for the quantity of electricity just given is not 
 of practical application. It is possible to get it into a usable 
 form by applying the principles given in Exp. F 2 . 
 The initial kinetic energy is 
 
 E K = \K^, (303) 
 
 which is used up in twisting the suspension. At the end of 
 the swing the energy is all potential, 
 
 E P = \ L a &, (304) 
 
 in which L is the moment of torsion of the suspension and 8 
 the angle of twist. 
 Since E P = E K , 
 
 \L^=\K<o*. (305) 
 
34CX' JUNIOR COURSE IN GENERAL PHYSICS. 
 
 ;The return torque being proportional to the angular dis- 
 placement, the motion is S. H. M. and the periodic time is 
 
 T 
 
 = 2 *->/£. (306) 
 
 'A, 
 
 " . By combining equations 302, 305, and 306 and noting that for 
 small angles S may be written in terms of the scale divisions j and 
 the distance r from the scale to the minor as — , K and a> may 
 
 2 Y 
 
 be eliminated, giving 
 
 -' 2 7T Af 2 7T Af2r u n 
 
 From the theory of the d'Arsonval galvanometer used as a 
 current measuring instrument (see pp. 242-244) the constant 
 per scale division is 
 
 £ = I°£a. ( 3o8) 
 
 Af2 r y:> ' 
 
 .:Q = —ks=Q 's, (309) 
 
 2 7T 
 
 that is, the quantity constant Q ' per scale division is equal to 
 the current constant k per scale division multiplied by 7"/2 it. 
 
 The tangent ballistic. 
 
 In the case of the tangent galvanometer the following con- 
 ditions are to be noted : 
 
 (1) The instantaneous value of the magnetic torque acting 
 to displace the needle is 
 
 T MGi t»r n \ 
 
 £ 9 = -^-> (3IO) 
 
 in which Mis the magnetic moment of the needle, G the true 
 constant of the galvanometer, and * the instantaneous value of 
 the current. 
 
 (2) The potential energy of the needle at the end of its 
 swing is equal to the amount of work done against magnetic 
 forces in turning it through an angle 8 (Fig. 116) in a magnetic 
 field of strength H, 
 
 E P = MH(\ - cos S)= 2 Mil sin 2 ±8/ (311) 
 
ELECTRICAL QUANTITY. 
 
 341 
 
 (3) The periodic time of the needle is 
 T=27r\[J^. 
 
 (See Q 2 .) 
 
 The theory outlined for the d'Arsonval galvanometer under 
 the conditions named above gives the follow- 
 ing expression for the quantity of electricity 
 passing through the galvanometer, 
 
 £?= 10- — sin \h. 
 
 IT Cr 
 
 (312) 
 
 If. 
 
 Now 10— is the working constant of the 
 G 
 
 galvanometer, 
 
 Q 
 
 T 
 
 7 sin \ 8. 
 
 (313) 
 
 The constant multiplying factor of sin ^ S is Fig- 1 ' &■ 
 
 the ballistic constant of the instrument. Calling this quantity 
 
 <2o- 
 
 S=0o sin 2 8 - (314) 
 
 If the deflections are small, a constant per scale division may 
 be computed as indicated for the d'Arsonval galvanometer. 
 
 The above demonstrations assume that the whole of the 
 kinetic energy of the moving parts after the current has ceased 
 to flow is used in overcoming magnetic forces. This is not 
 quite true. The friction of the moving parts against the air 
 and the current induced in the galvanometer coil by the moving 
 magnetic needle both require the expenditure of energy, and 
 therefore make 8 or s less than they otherwise would be. The 
 theory of damping leads to the conclusion that (1 4- \ X) should 
 be used as a multiplying factor in the above equations, in which 
 X is the logarithmic decrement of the galvanometer needle. 
 (Sge Exp. U 2 .) 
 
 The experiment is to be performed as follows : 
 
342 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 I. 
 
 Find the current constant k, "figure of merit," per scale 
 division of the d'Arsonval galvanometer by one of the methods 
 of Exp. R 4 or by a potentiometer method of measuring current. 
 
 Determine the periodic time by the method of transits, 
 Exp. A 5 II, taking the time at the first, sixth, eleventh, . . . 
 thirty-sixth transits, and repeating the process three times, if the 
 period is not more than 15 seconds. In case of periods greater 
 than 15 seconds use the method of Exp. A 5 I. 
 
 Compute the quantity constant in the manner indicated in 
 the theory given above. 
 
 II. 
 
 If a tangent galvanometer is used, perform the experiment 
 in the manner indicated in I, finding the 7 of the instrument. 
 
 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 ballistic galvanometer is proportional 
 to the impulse imparted to the needle or moving coil, 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 or 
 coil and finally bring it to rest. 
 
 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 vibrating system oscillates under the influence of 
 damping, the ratio of any amplitude to the succeeding one in 
 the opposite direction is very nearly constant, or 
 
 | = ^ = A =n ( 3I5 ) 
 
 \ <>3 °n+l 
 
ELECTRICAL QUANTITY. 343 
 
 This constant is the "ratio of damping," and its Napierian 
 logarithm is called the logarithmic decrement, and is generally 
 designated by \. We have, therefore, 
 
 \ = log,A_. (316) 
 
 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<t> = o- (317) 
 
 If the motion is not simple harmonic, but is damped by friction 
 or otherwise, a third term must be introduced. In the case 
 of an oscillating magnet or moving coil damping is produced : 
 
 (1) By air friction. 
 
 (2) By induced currents. Both of these retarding forces 
 are very nearly proportional to the angular velocity; conse- 
 quently the term that must be added to" the above equation 
 
 is k-y, in which k is a constant. The complete equation of mo- 
 at 
 
 tion of the damped magnetic needle is therefore 
 
 K^ + kf t+ MH*- a . (3,8) 
 
 If we integrate this equation, we have 
 
 *=V 2A 'sm^/,* (319) 
 
 in which S is a constant, and T is the period of oscillation of 
 
 the needle or moving coil under the influence of damping. 
 
 Let time be reckoned from the instant the needle or coil 
 
 passes the position of equilibrium; and let 8 V & 2 , , be the 
 
 T % T 
 values of d> at the times = — , - — These values of <f> will 
 
 4 4 
 
 * See Gray's Absolute Measurements in Electricity and Magnetism, vol. 2, p. 393. 
 
344 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 be the successive actual amplitudes of the oscillatory motion ; 
 and 
 
 _kT 
 
 S 1 = S € SK , (320) 
 
 SkT 
 °2 — °0 e 
 
 From these equations we have 
 
 l0g | = 4^' (32I) 
 
 and by substituting for this quantity A., as in (316), equation 
 320 gives 
 
 fix = v* (322) 
 
 Transposing and expanding the exponential in terms of X, and 
 neglecting powers of \ higher than the first, we obtain 
 
 S = S 1 (i + |\). (323) 
 
 When there is no damping, i.e. when k = o, we "have, from 
 (320), \ = 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 310 becomes 
 
 (2=<2 (i + jx)S 1 . (324) 
 
 The above demonstration is based upon the assumption 
 that both 8 and \ are small. If 8 is 4° and the ratio of damp- 
 ing is 1.52, equation 324 will be in error by about one part in 
 a thousand. If 8 is io° 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, 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 circuit and because these currents are 
 inversely proportional to the resistance of the circuit. 
 
 In the performance of the experiment a galvanometer 
 
ELECTRICAL QUANTITY. 345 
 
 should be used in which the damping is not very great. From 
 equation 315, we have 
 
 ^-=^> (325) 
 
 whence \ = — log e — — ■ (326) 
 
 Errors of observation have the least influence when the ratio of 
 K to S n+m is about 3. 
 
 The method of procedure is as follows : 
 
 (1) Set the needle or coil 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 or coil will gen- 
 erally 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 lt S 2 , and S s be three scale readings corresponding to the 
 
 extremes of successive throws. We shall then have 
 
 * 
 
 S — f\_S 1 + S 3 + 2 Sq], 
 
 in which S is the zero position at the instant when the scale 
 reading is 6* 2 . The deflection required, then, is, in scale 
 divisions, 
 
 "2 ™ ^2 — "-V 
 
 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 (n + 1 )st amplitude 
 to be about one third of the first ; \ should then be determined 
 from the ratios 
 
 8, Ja_... 
 
346 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 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, in series 
 with the galvanometer as follows : 20,000, 15,000, 10,000, 7,000, 
 5,000, 3,000, 1,000, 500, 200, and 100 ohms. Finally, from 
 these determinations plot a curve, with box resistances as ab- 
 scissas and corresponding 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 coik 
 densers. 
 
 I. 
 
 Ballistic galvanometer method. 
 
 When the coatings of a condenser are charged to a potential 
 
 difference^, the charge or quantity of electricity stored in the 
 
 condenser is 
 
 Q=Cpd,-^0 (327) 
 
 in which C is the capacity of the condenser. It has already 
 been shown in preceding experiments that if the quantity of 
 electricity Q is discharged through a ballistic galvanometer pro- 
 ducing the deflection 8, we have 
 
 <2=<2 (i+H)S. (328). 
 
 If a condenser of capacity C lt charged to a potential differ- 
 ence pd-i be discharged through the ballistic galvanometer, we 
 have 
 
 Ci-^t+lX)^. (329) 
 
ELECTRICAL QUANTITY. 347 
 
 If another condenser of capacity C 2 , charged to a potential 
 difference pd 2 , 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 
 
 C 2 pd x h % < 33 °> 
 
 A still simpler relation follows if the condensers have been 
 charged to the same potential difference. 
 
 The procedure in this experiment is as follows : 
 
 (1) Connect the condenser* in series with the battery and 
 ballistic galvanometer, and place in the cir- 
 cuit a double contact key, as shown in Fig. HI I 1 1 I 
 
 117. 
 
 (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 Fi ^ 1 ' 7 - 
 galvanometer needle will receive an impulse in the opposite di- 
 rection, 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 differ- 
 ence 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. 
 
 * In condenser work it is necessary to use great care in securing good insulation, 
 because the condenser must sometimes remain charged for a few minutes while un- 
 connected with a battery. 
 
348 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 When condensers are connected as shown in Fig. 118, they 
 are said to be connected in multiple. If C is the capacity in 
 multiple, we have the relation 
 
 HI 
 
 y 
 
 C m = C 1 + C i + -. (331) 
 
 When condens- 
 ers are connected 
 as shown in Fig. 
 119, they are said 
 
 Fig. us. t0 be connec ted in Fig. 119. 
 
 series. If C s is the capacity of the system in series, we have 
 
 111 , 
 
 r = r+7- + -- (332) 
 
 W °1 '-'2 
 
 This relation may be readily derived by making use of the 
 following facts : 
 
 (1) The potential difference at the terminals is equal to the 
 sum of the potential differences between the coatings of each 
 condenser, or 
 
 pd s =pd 1 +pd i +-. (333) 
 
 (2) When several condensers are connected in series, the 
 quantity of electricity on each coating of every condenser is the 
 same, or 
 
 0i=0. = -. (334) 
 
 Equation 332 then follows directly from the definition of 
 capacity. It is also to be remembered that the capacity of a 
 system of condensers connected in series is the ratio of the 
 charge on either extreme coating divided by the potential differ- 
 ence between the extreme coatings. 
 
 The relations expressed in equations 331 and 332 should be 
 verified experimentally. This may be done as above by com- 
 paring the series or multiple system with a condenser whose 
 capacity is known. 
 
 The following observations are to be made : 
 
 Three readings of the throw of the ballistic galvanometer are 
 
ELECTRICAL QUANTITY. 
 
 349 
 
 to be made for both charging and discharging a condenser of 
 known capacity. To eliminate errors due to lack of symmetry 
 of galvanometer throws on the two sides of the rest position, re- 
 verse the switch leading to the galvanometer and repeat the 
 readings as noted above. 
 
 In a similar manner make sets of readings for each of the 
 unknown condensers and also when the unknowns are in 
 multiple and in series. Compare the capacities of the measured 
 multiple and series groupings with their computed values, based 
 on the individual determinations. 
 
 Addenda to the report: 
 
 (i) Give a physical definition of capacity, of unit capacity, 
 and explain upon what the capacity of a condenser depends. 
 
 (2) Derive expressions for the capacity of a number of con- 
 densers in series and in multiple. 
 
 (3) Show that the expression for the capacity of a simple 
 plate condenser is 
 
 C = 
 
 kA 
 
 4-yrd 
 
 II. 
 
 Bridge method. 
 
 The method outlined below may be applied to comparison of 
 capacities of condensers whose times of charge and discharge 
 are very small. It will not apply 
 where there is absorption, or much 
 electromagnetic induction, such as 
 in cables. 
 
 Build a Wheatstone bridge, using 
 two resistance boxes R 1 and R 2 , 
 two condensers C x and C 2 to be 
 compared, a ballistic galvanometer 
 G, a battery B, and a key K. The 
 use of the key is to charge and cischarge the condensers. 
 
 The resistances R x and R 2 must have no self-induction. 
 
3 SO JUNIOR COURSE IN GENERAL PHYSICS. 
 
 When the proper values of the resistances have been chosen, 
 the bridge will be balanced, and there will be no throw of the 
 galvanometer either on charging or discharging, there being no 
 difference of potential existing between the points E and F < 
 
 In order that there be no pd between E and F the following 
 relation must hold : 
 
 r=4 2 - < 33S > 
 
 If no charge flows through the galvanometer, all the quantity 
 passing to one set of plates for condenser C x must pass through 
 the resistance R v and the electricity charging one set of plates of 
 C 2 must pass through the resistance R 2 . The transfer of the two 
 quantities of electricity must take place in the same time interval. 
 
 The experiment may also be performed by substituting for 
 the ballistic galvanometer a telephone receiver, and for the bat- 
 tery the secondary of an induction coil or an alternating E. M. F. 
 If the lighting circuit is used as a source of alternating E. M. F., 
 a protecting resistance should be placed in series with it. 
 
 Put a standard condenser of known capacity in one arm of 
 the bridge and the condenser whose capacity is to be determined 
 in another arm as shown in the diagram. Adjust the resistances 
 until the detector indicates no charge passing between E and F. 
 Read the resistances. Do not use a resistance of less than 500 
 ohms in either box. Vary one resistance by a few hundred 
 ohms and proceed as before. Make five such determinations 
 for each of the unknowns and also for the unknowns when they 
 are in series, and in multiple. 
 
 Compare the capacities of the measured multiple and series 
 groupings with their computed values, based on the separate 
 determinations. 
 
 Addenda to the report: 
 
 Answer the addenda to part i and prove that for method II 
 
 £i_fe 
 
 v 
 
ELECTRICAL QUANTITY. 351 
 
 Experiment U 4 . Measurement of the capacity of a con- 
 denser in absolute measure. 
 
 If the condenser is charged or discharged through a ballistic 
 galvanometer, we shall have, as in the preceding experiment, 
 
 C =|f( I+ ^) 8 - (336) 
 
 If the quantities on the right of this equation are all deter- 
 mined in absolute measure, the capacity will be determined 
 independently of the capacity of any standard condenser. The 
 constants Q ,pd, and X should be determined as described in 
 previous experiments. It should be remembered that the value 
 of \ to be used in this experiment is that obtained when the 
 galvanometer circuit is open. 
 The procedure is as follows : 
 
 (1) The throw of the needle 8 is to be determined, as in 
 the preceding experiment, by charging and discharging the 
 condenser through the galvanometer. 
 
 (2) The values of 8', which always differ in the case of the 
 charge and of the discharge, respectively, should be averaged 
 separately. The former value will correspond to the instan- 
 taneous capacity, while the latter corresponds to the capacity 
 of the condenser after a greater or less absorption has taken 
 place. 
 
 Determine the capacities of three condensers separately, in 
 series and in multiple, making observations in the manner indi- 
 cated in Exp. U 3 I. 
 
 Answer the addenda of U 3 I. 
 
CHAPTER XIII. 
 
 GROUP V: ELECTROMAGNETIC INDUCTION. 
 
 (V) General statements ; (Vj) Dip and intensity of the earth's 
 magnetic field {method of the earth inductor); (V 2 ) Lines 
 of force of a permanent magnet ; (V 3 ) Mutual induction ; 
 (V 4 ) Self-induction. 
 
 (V) General statements concerning induction. 
 Faraday discovered that when any portion of a complete 
 circuit is moved through a magnetic field, an electric current 
 circulates in all parts of it. 
 
 This fact may be viewed as follows : 
 
 Let there be a conductor, shown in 
 cross-section (Fig.' 121), which forms 
 part of a complete circuit. Suppose 
 it to be moving in the direction of 
 the arrow in a magnetic field origi- 
 nally uniform. The arrangement of 
 the lines of force of this field is indi- 
 cated by the dotted lines. During 
 the motion of this conductor the other- 
 wise uniform field will be distorted. 
 The field of force on the side towards 
 which the conductor is moving will 
 be stronger than before, and the lines of force will be crowded 
 together, and concave towards the conductor. On the oppo- 
 site side, the lines of force will be more widely separated, and 
 convex towards the conductor. Immediately around the con- 
 ductor, and extending to a greater or less distance, according 
 to the intensity of the induced current, the lines of force will be 
 closed curves surrounding it. The positive direction of the lines 
 
 35 2 
 
 Fig. 121. 
 
ELECTROMAGNETIC INDUCTION. 
 
 353 
 
 z 
 
 Fig. 122. 
 
 of force in these closed curves are as indicated in the figure. It 
 follows that if the direction of motion is to the right, and the 
 positive direction of the lines of force ver- 
 tically upward, the current will be directed 
 towards the observer. Or if the motion is 
 along the jr-axis, and the lines of force . 
 along the .e-axis, the current will be directed 
 along the j/-axis, each in the positive di- 
 rection. (See Fig. 122.) 
 
 Since a current may be produced in this way, it must be that 
 there is an E. M. F. generated in the moving conductor. This 
 E. M. F. exists whether the circuit is closed or not. In the 
 latter case, if the motion is uniform and in a uniform field, there 
 will simply be a static rise of potential along the conductor in the 
 direction in which current would flow if the circuit were completed. 
 
 It has been experimentally demonstrated that the E. M. F. 
 generated in this way is directly proportional : 
 
 (1) To the velocity of motion. 
 
 (2) To the intensity of the magnetic field. 
 
 (3) To the length of the moving conductor ; the three direc- 
 tions being mutually perpendicular. 
 
 Let ABCD (Fig. 123) be 
 a rectangular circuit with one 
 open side, and let it be placed 
 in a magnetic field of inten- 
 sity H, the lines of force being 
 Fig- 123. supposed perpendicular to the 
 
 plane of the paper. Let mn be a conductor resting on the two 
 parallel conductors and completing the circuit. If the length of 
 BC is I, and mn moves in the direction of the arrow with a ve- 
 
 dx 
 locity V= — the E. M. F. generated in the circuit will be, in volts, 
 
 A- 
 
 :D 
 
 B: : 
 
 io* H dt 
 
 (337) 
 
 * For the significance of the numerical factor io B see introduction to group S, p. 269. 
 VOL. I — 2 A 
 
354 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 When different parts of a circuit cut lines of force at different 
 rates, the total E. M. F. generated in the whole circuit is 
 
 io 8 J dt 
 
 (338) 
 
 ■*■ 
 
 
 J] 
 
 
 
 w 
 
 d 
 
 ( 
 
 a 
 
 
 I 
 d 
 
 ' f T T T 
 
 1 J 
 
 \ 
 
 ■ 
 
 . 
 
 It is obvious that E may be zero both when no lines of force 
 are cut, also when the E. M. F.'s in different parts of the circuit 
 
 due to cutting lines of force are 
 oppositely directed, and exactly 
 balance each other. For ex- 
 ample, let AS, Fig. 124, be a 
 rectangular circuit whose plane 
 is parallel to the lines of force, 
 and capable of rotation about 
 an axis parallel to the lines of 
 force. If this circuit be rotated 
 clockwise, E. M. F.'s will be gen- 
 Fig. 124. erated in the different parts, as 
 indicated by the arrows. These obviously annul each other 
 when added around the complete circuit. There is, however, 
 an E. M. F. between a and b producing a rise of potential from 
 a to b, from a to b' , from c to d, and from c to d'. 
 
 The equation for the E. M. F. generated in a complete cir- 
 cuit may often be simplified in the following manner : Let TV be 
 the total number of lines of force that at any instant pass through 
 the circuit. Now if the position of the circuit is changed in the 
 time dt in such a manner that the change in the number of 
 lines of force that pass through the circuit is dN, we have, for 
 
 the complete circuit, 
 
 1 dN 
 
 E = 
 
 io 8 dt 
 
 (339) 
 
 If the circuit is composed of n turns, through each of which 
 the N lines pass, we have 
 
 n dN 
 
 E = 
 
 io 8 dt 
 
 (340) 
 
ELECTROMAGNETIC INDUCTION. 355 
 
 Furthermore, since Q = ( Idt, we have, for the total quantity of 
 electricity produced, 
 
 1 TV. * 
 G-^. (340 
 
 in which N x is the number of lines of force cut, and R is the 
 resistance of the circuit in ohms. Q depends not at all on the 
 rate of cutting lines of force, but only on the total number of 
 lines cut. 
 
 In making application of the law of induced E. M. F., the 
 following fundamental principles are of service : 
 
 (1) The source of the magnetic field is immaterial. It may 
 be due to a permanent magnet, to the earth, or to an electric 
 current. 
 
 (2) It is immaterial whether the conductor is moved in a 
 magnetic field, or a magnetic field is moved past the conductor. 
 
 (3) Movements of the lines of a magnetic field may be pro- 
 duced in two ways : 
 
 (a) By moving bodily a magnet, or a circuit conveying a 
 current. 
 
 (b) By causing a current to change, in which case its lines, 
 of force will move outwards when the current increases, or move 
 inwards and disappear when the current decreases. 
 
 (4) An E. M. F. may be induced in a conductor already con- 
 veying a current, and this may either increase or decrease the 
 current flowing. 
 
 (5) If the current flowing in a circuit is decreased, the mag- 
 netic field due to the current will decrease, the lines of force 
 collapsing on the conductor. This motion of the field will pro- 
 duce an E. M. F. in the conductor tending to produce a current 
 in the same direction as the original current. If the current is 
 increased, the induced E. M. F. changes sign. From this it 
 follows that when current is changed in a circuit, an induced 
 
 * The factor fi?\ takes into consideration the number of turns » as well as the 
 change in the magnetic field. 
 
356 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 E. M. F. is set up, which opposes that change, 
 induction is called self-induction. 
 
 This kind of 
 
 Experiment V r Dip and intensity of the earth's magnetic 
 field. (Method of the earth inductor.) 
 
 The earth inductor consists essentially of a coil of wire C, 
 Fig. 125, capable of revolution about ah axis A in its own plane. 
 
 Usually this coil is mounted in a 
 frame F, which is itself capable of 
 rotation about an axis A 1 in its plane, 
 perpendicular to the axis A. To this 
 frame is attached a graduated circle 
 .S ; by means of this circle the angle 
 which the axis A makes with a hori- 
 zontal plane can be measured. The 
 instrument is also furnished with stops, 
 which enable the coil to be turned 
 Fig- 125. through exactly 180 ; and the base 
 
 is furnished with leveling screws, by means of which the plane 
 containing the two axes of revolution may be made truly 
 horizontal. 
 
 To determine the angle of dip. 
 
 The angle of dip is defined as the angle which the direction 
 of the lines of force makes with the horizon. If H and fare 
 the horizontal and vertical components of the intensity of the 
 earth's field at any point, we have 
 
 H 
 
 tan/3 = 
 
 V 
 
 (342) 
 
 To determine /3, which is the object of this part of the 
 experiment, proceed as follows : 
 
 (1) Turn the whole apparatus until the axis about which 
 the square frame revolves is perpendicular to the magnetic 
 meridian. This may be done with the aid of a small compass. 
 
ELECTROMAGNETIC INDUCTION. 357 
 
 (2) Adjust the leveling screws until the square frame con- 
 taining the two axes of revolution is truly horizontal. 
 
 (3) Adjust the stops so that the plane of the movable coil is 
 horizontal in both of the extreme positions. 
 
 When thus adjusted, the vertical component of the magnetic 
 field passes through the coil. In other words, V lines of force 
 per square centimeter pass through the coil. If n is the 
 number of turns of the coil, and A is the mean area of the 
 coil, the number of lines of force passing through the coil will 
 be nA V. 
 
 (4) If the coil be now turned through 180°, all the lines of 
 force will be cut twice, and we have from equation 341 
 
 n 2nAV , . 
 
 Qr-rtJT' (343) 
 
 in which R is the resistance of the circuit. If a ballistic galva- 
 nometer forms part of the circuit, and if the coil be turned 
 quickly, we shall have 
 
 2 nAV 
 itfR 
 
 <2r~J2 (i + *M*r=?^r. (344) 
 
 in which B r is the throw of the galvanometer needle. If a 
 tangent galvanometer be employed and the angular motion of 
 the needle is not small, sin \ 8 must be used instead of 8. 
 
 (5) If the square frame be now rotated through exactly 90 , 
 as measured by the divided circle, the number of lines of force 
 passing through the coil in its new position will be nAH\ and 
 if S B is the corresponding galvanometer throw, we have 
 
 k-y (345) 
 
 The constants n, A, R, Q , and X, being the same for both 
 positions, are eliminated, and it is not necessary to know their 
 values. 
 
 The values S r and B B used in this computation should each 
 be the mean of ten or twelve determinations. 
 
 When the square frame makes an angle with the horizontal 
 
358 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 equal to the dip, no lines of force thread through the coil in any 
 position ; consequently, no current will be produced when it is 
 rotated about its own axis. The position in which no current is 
 produced by the rotation of the coil should be found by trial. 
 The angle through which the frame was turned from the hori- 
 zontal position furnishes a second determination of the dip. 
 
 II. 
 
 To determine intensity. 
 
 From equation 343 it is obvious that both the vertical and 
 horizontal intensity may be determined in absolute measure, 
 provided Q Q and X have previously been determined for the 
 ballistic galvanometer, and n, A, and R are known. 
 
 The lines of force make but a small angle with the vertical, 
 and on this account a small error in leveling the coil will pro- 
 duce a relatively great error in the determination of H. This 
 should be remembered in determining the dip as well as in de- 
 termining the horizontal intensity. 
 
 Addenda to the report : 
 
 ( 1 ) Explain fully the direction of the induced current when 
 the coil is rotated about a vertical and a horizontal axis. 
 
 (2) Explain from, first principles why there is no current 
 when the coil is rotated about an axis parallel to the lines of 
 force. 
 
 Experiment V 2 . Measurement of the lines of force of a 
 permanent magnet. 
 
 The object of this experiment is the determination of the 
 number of lines of force that emerge from the positive half of a 
 permanent magnet. Before beginning these measurements the 
 constant and the logarithmic decrement of a ballistic galvanom- 
 eter must have been determined (Exps. Uj and U 2 ). These 
 values having been ascertained, the procedure is as follows : 
 
 Connect a test coil of a known number of turns in series 
 with a variable known resistance and a ballistic galvanometer. 
 
ELECTROMAGNETIC INDUCTION. 359 
 
 Place the coil at the center of the magnet, and when the 
 galvanometer needle has come to rest, observe the throw 
 of the needle produced by quickly slipping the test coil off 
 the end of the magnet. This test coil should consist of a 
 considerable number of turns of small copper wire, Nos. 24-36, 
 
 according to the resistance of 
 
 the galvanometer and the sen- 
 
 sitiveness of the latter. It 
 
 should be of such a form as to 
 
 fit easily over the bar magnet 
 
 to be tested. (See Fig. 126.) Fi e- I26 - 
 
 Place the coil again at the center of the magnet and move 
 it stepwise a centimeter at a time to the end of the magnet; 
 then remove it entirely, observing throws of the ballistic gal- 
 vanometer for each change of position. 
 
 Make three sets of readings in this manner from the center 
 of the magnet to each end. 
 
 Plot a curve, using distances from the center of the magnet 
 to the mid-point of each space as abscissas and mean throws as 
 ordinates, paying due regard to signs. 
 
 Plot another curve on the same sheet to the same scale, using 
 distances as abscissas as before, but using the corresponding 
 sum of the throws as ordinates. 
 
 Discuss the results and curves. 
 
 If /Vis the number of lines of force that emerge from the 
 magnet, and n the number of turns in the coil, we have, from 
 equations 324 and 341, 
 
 e= J% =eo(l+W (346) 
 
 in which R is the resistance of the circuit. 
 
 The most suitable number of turns for the test coil will 
 depend upon the strength of the magnet, the sensitiveness of 
 the galvanometer, and the resistance of the circuit. These 
 quantities should be so adjusted that the galvanometer throw 
 is rather large. Within certain limits this result can be most 
 
360 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 easily obtained by varying the resistance in circuit with the 
 galvanometer. 
 
 If the bar is not symmetrically magnetized, the magnetic 
 center must be found experimentally. To do this, move the test 
 coil along the bar stepwise. When the magnetic center is 
 reached, a slight motion of the coil in either direction may be 
 made without producing a reversal of current in the galva- 
 nometer circuit. 
 
 By this method the flow of induction from several magnets 
 should be determined, selecting for the purpose both bar 
 magnets and those of the horseshoe type. 
 
 Addenda to the report : 
 
 (1) From the readings obtained compute the induction per 
 square centimeter through the center of each magnet. 
 
 (2) If the magnetic moment is known, compute the distance 
 between the poles, or, more properly, the distance between the 
 " centers of gravity " of the two distributions of magnetism. 
 
 Experiment V 3 . Mutual induction. 
 
 The objects of this experiment are : I, to observe certain of 
 the phenomena of mutual induction ; II, to measure the quantity 
 of electricity which circulates in a secondary circuit when the 
 magnetic field in its vicinity produced by a current in a primary 
 circuit is varied. 
 
 I. 
 
 The primary and secondary circuits consist of two coils of 
 the same size. The primary coil, however, is wound with con- 
 siderably coarser wire than the secondary coil. 
 
 The method is as follows : 
 
 (1) Connect the primary coil P (Fig. 127) in circuit with a 
 battery of constant E. M. F., a variable resistance R and an 
 ammeter or galvanometer suitable for measuring currents of 
 about an ampere, and insert a make and break key K. 
 
 (2) Connect the secondary coil 5 in series with a ballistic 
 galvanometer and a resistance box. The latter is placed in the 
 
ELECTROMAGNETIC INDUCTION. 361 
 
 circuit to enable the observer to adjust the throws of the 
 galvanometer needle. 
 
 (3) Place the primary and secondary coils close together, 
 with their axes coincident. 
 
 (4) Observe the galvanometer throws when a current of 
 about one ampere is made and broken in the primary circuit. 
 
 (5) The circuit being closed, the current flowing steadily in 
 the primary circuit, observe the galvanometer throws produced 
 by quickly moving the secondary to a distance of a meter ; also 
 when the coil is quickly replaced. 
 
 (6) Repeat these observations, this time moving the primary 
 instead of the secondary coil. 
 
 (7) Observe the galvanometer throw when one of the coils 
 is quickly turned and placed with its opposite face next to the 
 other coil. 
 
 (8) Observe the effect upon the galvanometer when a per- 
 manent magnet is moved in the vicinity of the secondary coil. 
 
 Repeat all the above as a check. 
 
 II. 
 
 (A) The quantity of electricity which is produced in the 
 secondary circuit is directly proportional to the intensity of the 
 current that is made and 
 broken in the primary circuit. 
 To prove this relationship, use 
 the following method : 
 
 (1) Make connections as 
 in I, Fig. .127, the secondary 
 coil being close to the primary 
 with the axes coincident. 
 
 (2) Observe the throws of Fig ' 127 ' 
 
 the ballistic galvanometer needle when the primary circuit is 
 made and broken, the current being about 0.1 ampere. 
 
 (3) Note the current flowing in the primary circuit. 
 Repeat these observations with different currents in the 
 
362 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 primary circuit ranging from 0.1 to 1.0 ampere, varying the 
 current each time about o. 1 ampere. 
 
 The resistances of the two circuits should be so adjusted that 
 for the maximum current used the throws on the ballastic gal- 
 vanometer may be the largest that the scale will allow. The 
 resistance of the secondary circuit must not be changed during 
 the experiment. 
 
 If currents in the primary be plotted as abscissas, and 
 throws of the ballistic galvanometer as ordinates, the resulting 
 curve will be found to be a straight line passing through the 
 origin. This verifies the relation 
 
 Q s ^Ip, (347) 
 
 in which I p is the current in the primary, and Q a is the quantity 
 of electricity which circulates in the secondary. 
 
 The apparatus described above should be further utilized to 
 establish the following relations : 
 
 (B) The quantity of electricity which is induced in the 
 secondary circuit is inversely proportional to the resistance of 
 that circuit. 
 
 To determine this fact, the same connections as in (A) 
 should be made. Now observe the throws of the ballistic 
 galvanometer when the primary circuit, in which about 1.0 
 ampere is flowing, is made and broken, for several different re- 
 sistances in the secondary circuit. The resistances should vary 
 by ten equal steps from the smallest that will keep the throw 
 just on the scale to a resistance which will cut down the throw 
 to about one tenth of the maximum. Judgment should be used 
 in selecting convenient values of resistances. If a curve be 
 plotted with resistances in the secondary circuit as abscissas, 
 and the reciprocals of throws as ordinates, it will be found to be 
 a straight line ; thus verifying the relation 
 
 fiocf (348) 
 
ELECTROMAGNETIC INDUCTION. 363 
 
 If the results of (A) and (B) be combined, we have 
 
 Qa = Mj£> (349) 
 
 in which the constant M is defined as the coefficient of mutual 
 induction of the two coils. The value of M depends solely on 
 the construction of the two coils and their relative position. If 
 Q, I, and R be measured in coulombs, amperes, and ohms, 
 respectively, M will be expressed in henrys. 
 
 (C) If the distance between the primary and secondary coils 
 be varied, the mutual induction will also vary. The relation 
 between these two quantities may be experimentally determined 
 as follows : Make connections as above, place the two coils on 
 a common axis, and observe the throws of the ballistic galva- 
 nometer needle corresponding to several different distances 
 between the two coils. From (349) we have 
 
 M=&£*. (350) 
 
 1 F 
 
 If the current which flows in the primary while that circuit 
 is closed has a constant value throughout the experiment, the 
 mutual induction will be proportional to the product of resist- 
 ance in the secondary circuit and the throw of the galvanometer 
 needle, and we may write 
 
 Mk&K* (350 
 
 These observations should be repeated with a soft iron core in 
 the primary coil. In both cases use the same current in the 
 primary, about 1.0 ampere. Take readings for ten different 
 distances between the frames on which the coils are wound as 
 follows: O, 1, 2, 3, 5, 7, 10, 15, 20, 30, and 50 centimeters. In 
 each case use such a resistance in the secondary as will give 
 approximately the same maximum readable throw. 
 
 Plot two curves, one for each case, on the same sheet, to 
 the same scale, using the same origin, with distances in centi- 
 meters as abscissas and corresponding throws as ordinates. 
 
364 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Compute five values of the product &R S for similarly located 
 points on each curve. 
 
 If the coefficient of mutual induction is known for any one 
 position, it can now be computed for any other position by a 
 simple proportion between the known and unknown coefficients, 
 and the corresponding ordinates to the curves. 
 
 Experiment V 4 . Self-induction. 
 
 In a circuit containing a generator, a key, and coil of wire, 
 when the key is closed, the E. M. F. causes a current to flow, and 
 about the wire a magnetic field is set up, the lines of force 
 being closed curves. These lines may be thought of as spread- 
 ing out from the wire as the current grows, taking up their final 
 positions with respect to the wire when the current reaches its 
 full value. As they are spreading out, especially in the coil, 
 they will cut across other parts of the circuit, and therefore in 
 those parts an E. M. F. will be generated which opposes the 
 setting up of the current in the circuit of which the coil is a 
 part. If, after the current has been established, the generator 
 be cut out of the circuit, the current will drop to zero ; but on 
 account of the lines of force of the collapsing field cutting 
 across the circuit itself, an E. M. F. is set up which opposes 
 the stopping of the current. This phenomenon is called self- 
 induction. 
 
 The E. M. F. of self-induction of any coil depends upon the 
 rate of cutting of lines of force. (See equation 337.) The rate 
 of cutting depends upon the number of turns, the area of the 
 coil, and the rate of change of the current. The E. M.F. may 
 be expressed as 
 
 E = -dN/dt=-L d f, (352) 
 
 at 
 
 in which dl/dt is the rate of change of the current and L a 
 constant for the given coil called the coefficient of self-induction 
 of that coil. 
 
 The physical meaning of L may be obtained in two ways : 
 
ELECTROMAGNETIC INDUCTION. 
 
 36S 
 
 First, from the equation above, L is equal to the E. M. F. pro- 
 duced when the rate of change of the current is unity. Second, 
 if the equation 
 
 he integrated, 
 
 dNldt=L — 
 1 dt 
 
 N=LI. 
 
 (353) 
 
 If this integration be made between the limits of o and 1 for the 
 current, then /= 1 and N=L, that is, the coefficient of self- 
 induction of a coil is equal to the number of lines of force cut 
 when the current is changed by unit amount. 
 
 The introduction to the V group should be carefully studied, 
 for to successfully perform the following experiments it is neces- 
 sary to understand the theory of induced electromotive forces. 
 It is necessary to use much care, patience, and judgment in 
 order to choose the ppper conditions, to eliminate or neutralize 
 thermo-currents, and to get good steady current balances, since 
 the methods are based on the Wheatstone bridge. 
 
 I. 
 
 Measurement of self-induction by comparison, using a variable 
 standard self-induction. 
 
 The connections to be made are those of a Wheatstone 
 bridge, Fig. 128, in which R x andi? 2 are non-inductive resistances 
 used as ratio arms, L x 
 the unknown induc- 
 tive resistance whose 
 resistance is R s , L, 
 the standard variable 
 self-induction of resist- 
 ance R it B the bat- 
 tery, and G the galva- 
 nometer. S, S are 
 commutators for re- 
 versing the direction R g. 128. 
 
366 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 of the current from the battery through the system, excepting the 
 galvanometer. The galvanometer is so connected that the im- 
 pulses due to the extra current are in the same direction and 
 additive. The system is first balanced in the ordinary manner 
 of the bridge for steady currents. After this balance is ob- 
 tained, the commutators are revolved, in general producing a 
 deflection of the galvanometer due to the currents of self- 
 induction. The variable self-induction is now changed until no 
 deflection is noted either for steady or variable currents, if such 
 a double balance be possible. In such an event the E. M. F.'s 
 of self-induction oppose each other in such ratio that no current 
 flows through the galvanometer. In this case 
 
 LJL, = R 1 /R 2 , 
 
 L X = L,(RJR 2 ). (354) 
 
 In the actual operation of the experiment it may happen 
 that the range of the standard self-induction is not such as to 
 make it possible to get the ratio of the self-inductions to be 
 equal to that of the resistances. In such a case it is necessary 
 to put in series with the standard self-induction or with the in- 
 ductance to be determined a non-inductive resistance R, get a 
 new steady current balance, and try again. If the balance for 
 variable currents is a very small reading on the standard so that 
 the error in setting is large compared with the actual reading, 
 it is best to change the steady current balance by means of 
 auxiliary resistances in one or the other of the inductance arms 
 in order to get a greater reading on the standard. 
 
 The connections having been made as indicated in Fig. 128, 
 get a steady current balance. Then change the value of the 
 standard until for varying currents no deflection of the galvanom- 
 eter is obtained. If a balance cannot be obtained for variable 
 currents, then insert a non-inductive resistance in series with 
 the standard, putting in such a resistance to change the ratio of 
 the bridge arm by 50 per cent, and try again. This trial will 
 indicate what changes must be made to get both steady and 
 variable current balances. 
 
ELECTROMAGNETIC INDUCTION. 367 
 
 Make three determinations of the unknown inductance, using 
 different ratios. Vary the ratio of the arms within allowable 
 limits. 
 
 Note. The proof of equation 354 is based on the fact that the 
 galvanometer and battery branches of the circuit are conjugate, and 
 consequently an auxiliary E. M. F. in an arm of the bridge will produce 
 the same current through the galvanometer branch whether the battery 
 circuit is closed or open. Therefore we may write the equations for 
 the currents through the galvanometer due to induced E. M. F.'s in 
 arms R s and R t as if the battery branch were open. 
 
 / o = / 4 (y? 1 + J ff 2 )/0ff 1 + j? 2 + j? G ) (355) 
 
 and 
 
 /_ L,dijdt - . 
 
 Substituting for 7 4 its value, 
 
 / dt= Z,di i (R 1 + R 2 + R )(R 1 +R 2 ) , . 
 
 ° [(R 1 +R 2 +R )(R i +R s )+(R 1 +R 2 )R a ](R 1 +R 2 +R o y W,J 
 
 Integrating (357), the quantity through the galvanometer due to the 
 induced E. M. F. in branch 4 is shown to be 
 
 Q. = L e i i (R 1 +R 2 )/[(R 1 +R 2 +R a ) (R t +R s ) + (R.+R^Ra], (358) 
 
 and in like manner that due to L x is 
 
 Q x = L x i s {R x + R 2 ) I [(R, + R 2 + RoXR.+ R^ + iR.+R^Ro]. (359) 
 
 When a balance is obtained for variable currents as well as steady 
 currents, Q.= Q X and L,i t =LJ s , or LJR i = Z x /R s , since i i =pd/R i 
 and i s =pd/ R s for steady currents. 
 
 Z x /L, = R i /R i , 
 but RJR^RJR* 
 
 .-. L X = L B R X IR 2 . (360) 
 
 II. 
 
 Measurement of self-induction by comparison with a capacity 
 (Rimington's modification of Maxwell's method). 
 
 A Wheatstone bridge is arranged with the inductance to 
 be determined in one arm of the bridge (Fig. 129). In the 
 
368 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 same arm is also placed a variable non-inductive resistance, the 
 use of which is noted below. The other arms of the bridge are 
 made up of non-inductive resistances. A condenser of known 
 
 capacity is arranged 
 so that it may be 
 shunted about various 
 resistances in branch 
 R v while keeping the 
 resistance constant. 
 This is done in the 
 experiment as here 
 performed by using 
 two resistance boxes r 1 
 
 Fig. 1 29. 
 
 and r", so shifting their values as to keep their sum R x constant. 
 The bridge is balanced in the usual manner for steady cur- 
 rents, being careful to close the battery branch key before that 
 in the galvanometer branch. Then the resistance around which 
 the condenser is shunted is varied until no kick of the galva- 
 nometer is noted when the galvanometer key is closed before 
 the battery key. When this condition is obtained, 
 
 L a =Cr'*RJR v 
 
 (36i) 
 
 m which C is the capacity of the condenser in farads, r" the 
 resistance about which the capacity is shunted, R i the total 
 resistance of the arm in which the self-induction is placed. 
 The galvanometer used should be quite sensitive. 
 
 In one arm of the bridge, R a say, put a wire in series with 
 the box by means of which, by varying the length of the portion 
 of the wire used, an accurate steady current balance may be 
 obtained. Two boxes of 1000 ohms each may be used in the 
 arm R v Pull out all of the plugs of one of these boxes and 
 connect the condenser (0.5 or 1 m. f.) across its terminals. A 
 box having adjustable side plugs may be substituted for r 1 and 
 r" . Balance the bridge for steady currents. When this is ob- 
 tained a kick of the galvanometer will, in general, be produced 
 
ELECTROMAGNETIC INDUCTION. 369 
 
 by closing the galvanometer key and then the battery key. 
 Change the value of r" around which C is connected, keeping 
 r 1 + r" constant, until no kick is observed, if this be possible. 
 If such a balance cannot be obtained, add non-inductive resist- 
 ance to branch R i (50 ohms or more) and try again. Continue 
 the preliminary adjustments until changes of r" and r 1 will pro- 
 duce deflections in opposite directions. Then get a steady cur- 
 rent balance, after which change r" and r 1 , keeping their sum 
 constant, until no deflection is obtained, no matter in what order 
 the keys are closed. From the ordinary law of the bridge R t 
 may be computed, after which L x may be determined from the 
 equation given above. 
 
 Make three determinations of L x , using two different capaci- 
 ties, and then a new value of the non-inductive resistance in 
 series with the unknown inductance. 
 
 Since the bridge is balanced for steady currents, the gal- 
 vanometer and battery branches are conjugate, and the current 
 through the galvanometer due to an E. M. F. in any branch will 
 not depend on the resistance or E. M. F. of the battery branch. 
 Equations may be written for the current through the gal- 
 vanometer branch as if the battery branch were open, as indeed 
 it is, on breaking the circuit. 
 
 Instead of the instantaneous value of the current the total 
 quantity of electricity passing through the galvanometer may 
 be written. That quantity discharged through the galvanom- 
 eter due to the inductance is 
 
 O - L x ijR x + R 2 ) . , 6 v 
 
 Ul ~ {R 1 + R* + R e ) {R a +R t )+(R 1 + R 2 )R G ' U ° ; 
 
 and that quantity due to the discharge of the condenser in 
 which the quantity Cr"i x is stored is 
 
 Q <>"»/'(*, + * 4 ) . (363) 
 
 If no throw of the galvanometer takes place, Q L = Q c . Putting 
 the right-hand members of the equations equal, remembering 
 
37° 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 that RJR^ = RJR iy and that i 1 /i i = RJR Z , and simplifying, 
 we have 
 
 L x =Cr"*RJR v 
 
 III. 
 
 Measurement of self-induction by comparison with a capacity 
 {Anderson' s method).* 
 
 This modification of Maxwell's method of measuring self- 
 induction is very useful and accurate for finding small induc- 
 tances. The circuit 
 is made up in the 
 Wheatstone bridge 
 form (Fig. 130), with 
 the unknown induc- 
 tance L in ofte arm 
 of the bridge; the 
 other arms, R v R 2 , 
 and R 3 , being non-in- 
 ductive resistances. 
 In series with the 
 galvanometer, which is placed in its usual position, there is 
 inserted a variable non-inductive resistance r connecting it to 
 the point b of the bridge. A capacity C is inserted between the 
 point a and the point /(between the resistance ^and the galva- 
 nometer G). A buzzer is put in the battery branch, in multiple 
 with the key k 2 , to produce a variable current when the key 
 is opened and the buzzer armature given a start. To detect 
 the presence of the variable current in the branch fd, a tele- 
 phone T is placed in multiple with the galvanometer, the key 
 k x making it possible to insert either the galvanometer or the 
 telephone. 
 
 The bridge is balanced in the usual manner for steady cur- 
 
 * Anderson, Phil. Mag., S. 5, vol. 31, 1891, p. 333 ; Fleming, Phil. Mag., 
 S. 6, vol. 7, 1904, p. 586. 
 
ELECTROMAGNETIC INDUCTION. 371 
 
 rents, using the galvanometer. The buzzer is then set in oper- 
 ation, and the value of r changed until the sound heard in the 
 telephone reaches a minimum. The resistance r in no way 
 affects the steady current balance, no matter what changes are 
 made in its value. The two balances being obtained, the value 
 of L may be computed from the equation 
 
 L=C\r{R 2 + R i ) + R 2 R z \. (364) 
 
 It may be shown from this equation that in order to get a 
 balance for variable currents, the product CR 2 R S must be less 
 than L, otherwise r would have to be a negative resistance, 
 manifestly an absurdity. If, a steady current balance having 
 been obtained, it is impossible to get a balance for variable 
 currents, the product CR 2 R 3 must be increased. It is to be 
 noted that this may be attained by increasing the capacity 
 alone, which will not destroy the steady current balance. If 
 R% or -^3 is increased, a new steady current balance must be 
 obtained before trying again for a variable current balance. It 
 has been shown also that the best results are obtained when 
 the resistances R 1 and R 2 are comparatively large, and R 3 and 
 r are small. 
 
 In performing the experiment, make three independent de- 
 terminations of the value of the inductance of the unknown, 
 varying any or all the factors. The following is an outline of 
 the derivation of equation 364. 
 
 When the bridge is balanced for both constant and variable 
 currents, the pd between f and d is zero. 
 
 Let g v q 2 , q & , and q r be the quantities of electricity that have 
 flowed through the resistances R v R 2 , R 3 , and r, respectively, in 
 the time /. 
 
 Then ?i + ?r = 0s> (365) 
 
372 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Substitution in (368) of the values of -^, -|r, -|r, and -J^ 
 obtained from (367), (365), and (366), respectively, gives 
 
 But for a steady current balance the theory of the Wheat- 
 stone bridge shows that 
 
 |-t-°' (370) 
 
 Consequently the right-hand member of (369) must be zero, and 
 since the variable current is not in general zero, its coefficient 
 must be equal to zero. 
 
 ■'■ r + R > + -R?-CR 2 = °- (37I) 
 
 Solving (371) for L and remembering the relation given in 
 (37o), 
 
CHAPTER XIV. 
 GROUP W: MAGNETIC PROPERTIES OF IRON. 
 
 (W) General statements ;* (W x ) Magnetometer method; (W 2 ) 
 Ring method, with ballistic galvanometer. 
 
 (W) General statements concerning the magnetization of iron. 
 
 When a piece of iron in a neutral magnetic state is placed 
 in a magnetic field, no matter how produced, it becomes mag- 
 netized and in general exhibits free poles. Within the iron 
 there will be more magnetic lines than exist in the same space 
 when the iron is not present, the additional lines being due to 
 the iron being a magnet. Each unit positive pole will supply 
 4 7r lines, called lines of magnetisation to distinguish them from 
 the lines of the magnetizing field. The total number of lines, 
 including lines of force and lines of magnetization, is called 
 the magnetic induction. The number of unit poles per unit 
 of cross-section is called the intensity of magnetisation. The 
 intensity of magnetization is also defined as the magnetic mo- 
 ment per unit of volume. 
 
 Let H be the strength of the magnetizing field, /„ the 
 intensity of magnetization, B the magnetic induction per unit 
 of area, <j> the total induction, and A the area considered. 
 Assuming the induction uniform, then 
 
 <f) = 4 irm + HA. (372) 
 
 The induction per unit area is 
 
 i=^2 +i1 ' (373) 
 
 or B = 4 tt/„ + H. (374) 
 
 * See Introduction to Group Q. 
 373 
 
374 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 If a piece of iron initially unmagnetized be subjected to a 
 magnetic field which is increased with no breaks or reversals 
 whatever, the ratio of the induction produced to the magnetiz- 
 ing field is called the permeability, 
 
 B 
 
 (375) 
 
 and the ratio of the intensity of magnetization to the magnetiz- 
 ing field is called the susceptibility, 
 
 
 (376) 
 
 Let the curve oabc, Fig. 131, represent the increase of B 
 with H. Such a curve is called a magnetization curve. The 
 
 permeability (J. applies 
 only to the ratio of cor- 
 responding values of 
 B and H used in ob- 
 taining this curve. If 
 instead of B and H as 
 co-ordinates, I n and H 
 be used, a curve of the 
 same general shape may 
 be drawn. ^ However, 
 when the iron becomes 
 fully magnetized, /„ be- 
 comes constant, and the 
 curve will then be par- 
 allel with the jr-axis; 
 while when B and H 
 are plotted, B will con- 
 Fig. 131. tinue to increase with 
 H even after the iron is fully magnetized. 
 
 If, after the iron has been magnetized, the magnetizing field 
 be decreased to zero,. it is found that the iron retains some of 
 its magnetism. The curve cd may be taken to represent the 
 
 
 
 
 
 
 B 
 
 14000 
 
 w 
 
 
 
 
 — -Sj 
 
 
 
 
 
 
 
 12000 
 
 
 
 >* 
 
 >* 
 
 »•* 
 
 
 
 
 
 
 
 10000 
 
 4 
 
 y 
 
 r 
 
 
 
 
 
 
 
 
 
 800D 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 flofo 
 
 
 ! 
 
 
 
 
 
 
 
 
 
 
 4QO0 
 
 al 
 
 
 
 
 
 
 
 
 
 
 
 2tt>0 
 
 1 
 
 
 
 
 
 
 H_, 
 
 3 -a 
 
 -3 
 
 -2 
 
 -1 
 
 \eo 
 
 /■ 
 
 Q 2 
 
 ) 3 
 
 a « 
 
 5 
 
 . 
 
 
 
 
 
 
 2000 
 
 r 
 
 
 
 
 
 & 
 
 
 
 
 
 
 £000 
 
 
 
 
 
 
 
 
 
 
 
 
 ■POOCI 
 
 
 
 
 
 
 
 
 
 
 
 
 ■8000 
 
 
 
 
 
 
 
 
 
 
 
 
 10000, 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 1^600 
 
 
 
 
 
 
 
 
 
 
 
 
 14000 
 
 
 
 
 
 
 
 P~ 
 
 
 
 
 
 B 
 
 
 
 
 
 
 
MAGNETIC PROPERTIES OF IRON. 375 
 
 decrease of B as H is reduced to zero. The ordinate od repre- 
 sents the residual magnetism. To demagnetize the iron com- 
 pletely requires the application of a magnetizing field in the 
 opposite direction, as oe. The strength of this field is called the 
 coercive force. If the field be further increased in the negative 
 direction, the iron will become magnetized in the opposite sense, 
 as indicated by the curve ep. Now let the negative field be 
 decreased to zero and then increased in the original direction 
 to its original maximum os, the iron will be demagnetized and 
 remagnetized in the original sense as indicated by the curve 
 pqrc. The initial and final values of B will not be the same, 
 however, unless the cycle of magnetization has been repeated sev- 
 eral times. In this case the shape and size of each succeeding 
 loop will be the same. This means that the maximum values of 
 B for the magnetization curve and the loop are different for the 
 same maximum value of H, and that the magnetization curve 
 and loop are to be plotted independently. The lagging of the 
 magnetization behind the magnetizing force is called magnetic 
 hysteresis, and the loop obtained by plotting B and H is called a 
 hysteresis loop. Its area is proportional to the work done in 
 carrying the iron through one complete cycle of magnetization. 
 
 The susceptibility and permeability of a given sample of 
 iron vary with an increasing magnetizing field, being small for 
 a very weak field, increasing very rapidly to a maximum as the 
 field increases, and then decreasing to values smaller than the 
 initial one. The maximum values are reached at compara- 
 tively weak fields. The relations between k, fi, and H are best 
 shown graphically in susceptibility and permeability curves, 
 using values of H for abscissas and corresponding values of k 
 and jtt for ordinates. 
 
 The intensity of magnetization is found to depend upon (1) 
 the intensity of the magnetizing field, (2) the quality of the 
 iron, and (3) the previous state of magnetization. When dif- 
 ferent samples of iron are to be compared, they should have no 
 residual magnetism and should be subjected to magnetizing 
 
376 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 fields that can be varied at will, progressively increasing or 
 decreasing them. 
 
 Two methods of studying the magnetization of iron follow, 
 one being a magnetometer method in which the field, at a mag- 
 netometer needle, produced by a rod, of iron, is compared with a 
 known field ; the other being a method of measuring the changes 
 of induction in an iron ring by the use of a ballistic galvanom- 
 eter. 
 
 Experiment W v Magnetic properties of iron — magnetom- 
 eter method. 
 
 The rod or wire of iron or steel whose magnetic properties 
 are to be studied should have a length which is great compared 
 with its diameter. This is desirable on account of the demag- 
 netizing effect of its own poles. The residual magnetism is 
 much greater for a long than a short bar of the same material 
 of equal diameter, even after having been subjected to such 
 magnetizing fields as to produce the same intensity of mag- 
 netization in each. The value of the magnetizing field with the 
 iron in place must be corrected for the field due to the poles. 
 According to Ewing, the value of the magnetizing field should 
 be H' = H— NI n , in which N is a factor depending on the ratio 
 of the length to diameter as given in the following table : 
 
 Length 
 
 Diameter 
 
 N 
 
 S° 
 
 O.OI817 
 
 IOO 
 
 O.OO540 
 
 200 
 
 O.OOI57 
 
 300 
 
 O.OO075 
 
 400 
 
 O.OO045 
 
 5O0 
 
 O.OOO30 
 
 However, if the rod has a length at least 400 times its diameter, 
 the correction may be assumed to be negligible. 
 
 In the experiment as here discussed, there are two positions 
 in which the rod may be placed, both of which produce fields at 
 the magnetometer at right angles to the earth's or other govern- 
 ing field, whose value is known. 
 
MAGNETIC PROPERTIES OF IRON. 377 
 
 I. 
 
 The rod is placed in a solenoid somewhat longer than the 
 specimen to be tested, with its axis in a horizontal plane and in 
 a magnetically east and west line. The field produced at the 
 magnetometer needle will be at right angles to the known hori- 
 zontal component of the governing field, and the tangent of the 
 angle the needle is deflected from its position of rest in the 
 governing field will give the ratio between the two fields. 
 Under these conditions the equation of Exp. Q 3 may be applied, 
 
 — = i — '- tan 8, 
 
 f -2-L 
 
 in which f is the known governing field, a is half the distance 
 between the poles, and the other terms have the significance 
 given in the experiment referred to above. 
 
 The strength of the magnetizing field within the solenoid 
 may be computed from the expression 
 
 H.iSf. ( 377 ) 
 
 in which n/l is the number of turns per unit length of the sole- 
 noid and / is the current measured in amperes. 
 
 Knowing the dimensions of the bar, the distance of its mid- 
 point from the magnetometer, the number of turns and length 
 of the solenoid, the strength of the comparison field ; and 
 
 R B 
 
 J-. A/VWy wVW — |l|l|l|l|l 
 
 mmmm^\_K^ 
 
 s cm 
 
 ■jtflflflaaflgjmm. m 
 
 c d efghijk 
 
 Fig. 132. 
 
 observing the current flowing and the deflections of the mag- 
 netometer needle, the values of H, I n , k, B, and p may be found. 
 Arrange the circuit as in Fig. 132. A solenoid 5 is con- 
 nected in series with a neutralizing coil C, a storage battery B, 
 
378 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 a regulating resistance R, an ammeter or current measuring 
 galvanometer A, a variable resistance r, and a reversing key K. 
 The axis of the solenoid produced passes through the magnet- 
 ometer needle at M, at right angles to the governing field. 
 
 Since the field due to the current in the solenoid affects the 
 magnetometer, the neutralizing coil must be so placed as to 
 reduce this effect to zero. The value of the magnetizing field 
 is controlled by the resistance r. The steps be, cd, de, etc., are 
 adjusted to give the desired changes in magnetizing field as they 
 are successively short-circuited or added. When increasing the 
 field, the largest resistance be is to be cut out first, then cd, and 
 so on. When decreasing the field, the resistance units are to be 
 added, beginning at the opposite end and progressing to the 
 largest resistance. An ordinary resistance box may not be used, 
 since it will not carry the required currents safely. 
 
 The regulating resistance is used to obtain the desired maxi- 
 mum current. Before making readings the following adjust- 
 ments should be made : 
 
 With r equal to zero vary the value of the regulating resist- 
 ance R to give a maximum current of 1.2 to 1.5 amperes. This 
 adjustment is not to be disturbed during the experiment. 
 
 Place the neutralizing coil in such a position that no de- 
 flection of the magnetometer needle is produced when the 
 direction of the current in the coils is reversed, no iron being 
 present. 
 
 Center the iron rod to be tested within the solenoid and 
 place the solenoid in such a position that the deflection of the 
 needle for direct or reversed maximum current will be the 
 desired maximum. If the deflections are the same, then the 
 field produced by the rod is at right angles to the governing 
 field. 
 
 The rod must be demagnetized before beginning the obser- 
 vations. Either of the following methods may be used : (1) Re- 
 move the rod from the solenoid and hold it in the earth's 
 magnetic field in such a direction that this field tends to 
 
MAGNETIC PROPERTIES OF IRON. 379 
 
 demagnetize it. Tap the rod lightly with a block of wood until 
 it no longer deflects the magnetometer needle when one end is 
 held within 50 or 60 centimeters. (2) Place the bar within the 
 solenoid and, starting with the current at a maximum, gradually 
 reduce its value, at the same time rapidly changing its direction ; 
 or, note the direction of deflection of the magnetometer and send 
 a current through the solenoid so as to produce a field to demag- 
 netize the bar, starting with a small current and gradually in- 
 creasing it until the deflection is reduced to zero. Observations 
 are now to be made for finding /„ and /i. Read the ammeter 
 and magnetometer zeros, the switch being open. Close the 
 switch, the resistance r being all in, and make ammeter and 
 magnetometer readings. Cut out the largest step in the resist- 
 ance r by short-circuiting be, and read the instruments as 
 before. Continue this process until all of r has been short-cir- 
 cuited and the current is a maximum. Corresponding values of 
 H, I n , k, B, and /t are to be computed from the data obtained ; 
 and three curves are to be drawn, a magnetization curve show- 
 ing the relation between B and H, a susceptibility curve show- 
 ing the relation between k and H, and a permeability curve 
 showing the relation between p and H. 
 
 To obtain data for the magnetic hysteresis, reverse the 
 direction of the current at its maximum value ten or fifteen 
 times, then close the switch in either direction and take ammeter 
 and magnetometer readings. Add resistance to the circuit by 
 beginning at the low resistance end of r and make readings as 
 before. Continue the process of adding resistance and making 
 readings until the resistance of r is a maximum, then open the 
 reversing switch and make readings. The magnetometer read- 
 ing for zero current will enable the computation of the residual 
 magnetism. Close the reversing switch in the opposite direction 
 and make readings as in obtaining data for the magnetization 
 curve. The rod will now be magnetized in the opposite sense. 
 To return to the original magnetic condition completing the 
 cycle, decrease the current to zero, then increase it to a max- 
 
3 8o 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 imum in its original direction, making appropriate readings for 
 each step in varying r as before. From the data obtained, 
 compute values of H and corresponding values of /„ and B, and 
 plot the hysteresis loop. 
 
 II. 
 
 The difficulty of finding accurately the positions of the poles 
 is obviated by Ewing's one-pole method. The magnetizing 
 solenoid and the rod to be tested are 
 placed with their coincident axes in a 
 vertical position, the upper pole, say, of 
 the rod being magnetically east or west 
 of the magnetometer needle. The dis- 
 tance from the needle to the rod is 
 made considerably less than the length 
 of the rod. The arrangement of coils, 
 solenoid, rod, and magnetometer is shown 
 in Fig. 133. 
 
 The field at the needle due to the 
 magnetized rod is 
 
 Fig. 133. 
 
 H m = 
 
 
 m 
 
 cos</>. 
 
 (378) 
 
 d 2 +4a 2 
 
 If for the pole strength m its equal I n q, intensity of magnetiza- 
 tion times area of cross-section, be written in the above equa- 
 tion, then 
 
 H m = lj\, - _ d . . 1 (379) 
 
 -/*[; 
 
 (aT 2 + 4« 2 ) S J 
 
 The second term within the brackets is a correction factor intro- 
 duced on account of the effect of the pole at S. This factor 
 becomes smaller the closer the rod is placed to the magnetom- 
 eter needle and the longer the rod used. If / is the strength 
 of the governing field of the magnetometer and 8 is the angular 
 deflection, then 
 
 ^=tanS. 
 
 (380) 
 
MAGNETIC PROPERTIES OF IRON. 381 
 
 Solving equation 379 for /„ and substituting the value of H„ 
 from equation 378 gives 
 
 /.= r . /tang , f (381) 
 
 q \d* (^ + 4^)* J 
 
 The last equation involves the distance between the poles of the 
 rod, but since the correction is comparatively small, an approxi- 
 mation of the distance 2 a will not greatly affect the results. 
 The values of /„ having been obtained, the values of k, H, B, 
 and p may be computed in the same manner as in part I. 
 
 The circuit is made up in the same way as in part I. An 
 auxiliary circuit, shown in Fig. 133, composed of a coil to 
 neutralize the magnetizing effect of the vertical component of 
 the earth's field on the bar, a battery, and a resistance are 
 connected in series. The coil is wound on the form carrying 
 the solenoid. The resistance is adjusted to give a current in 
 the proper direction and sufficient to produce a field within the 
 neutralizing coil equal to the vertical component of the earth's 
 field, and in the opposite sense. The auxiliary circuit is neces- 
 sary only when testing soft iron. 
 
 A second neutralizing coil to neutralize the magnetic effect 
 of the solenoid is necessary as in part I. 
 
 To get the upper pole in the same horizontal plane as the 
 magnetometer needle, the iron rod is centered within the sole- 
 noid, the current given, its maximum value, and the frame carry- 
 ing the solenoid and rod is raised or lowered until the deflection 
 of the needle is a maximum. The frame is then clamped in 
 place, the rod removed, and the small neutralizing coil C adjusted 
 to counteract the magnetic effect of the solenoid on the needle. 
 
 The iron rod is then replaced in the solenoid and demagnet- 
 ized as in the second method of part I. 
 
 The connections, excepting as noted above, are the same as 
 indicated in Fig. 132. The same method of making observa- 
 tions and computations is to be followed as that given in part I. 
 
3 82 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Magnetic Properties of Iron, Magnetometer Method. 
 Hysteresis Loop. 
 
 
 
 Magnetom- 
 
 
 
 
 
 / 
 
 H 
 
 eter 
 Reading. 
 
 Deflection. 
 
 tan S 
 
 U 
 
 B 
 
 1.263 
 
 + 54-0' 
 
 10.60 
 
 + 14-4° 
 
 141 1 
 
 +959 
 
 + 12,090 
 
 .670 
 
 28.6 
 
 12.00 
 
 13.00 
 
 1278 
 
 869 
 
 10,940 
 
 •363 
 
 I5-S 
 
 13-45 
 
 11-55 
 
 "39 
 
 775 
 
 9.745 
 
 .158 
 
 6.75 
 
 I5.60 
 
 9.40 
 
 0934 
 
 635 
 
 7,985 
 
 .063 
 
 2.69 
 
 I7.4O 
 
 7.60 
 
 0757 
 
 5'5 
 
 6,465 
 
 .000 
 
 0.00 
 
 19.05 
 
 5-95 
 
 °595 
 
 404 
 
 5,080 
 
 ■033 
 
 -1. 41 
 
 20.50 
 
 4.50 
 
 0450 
 
 358 
 
 4,495 
 
 .061 
 
 2.61 
 
 2I.85 
 
 3-i5 
 
 0315 
 
 214 
 
 2,685 
 
 .112 
 
 4.78 
 
 25.9O 
 
 -.90 
 
 0090 
 
 -61 
 
 -775 
 
 .220 
 
 9.40 
 
 30.70 
 
 5.70 
 
 0570 
 
 387 
 
 4,870 
 
 •363 
 
 '5-5 
 
 34.IO 
 
 9.10 
 
 0908 
 
 613 
 
 7,715 
 
 .680 
 
 29.1 
 
 37.IO 
 
 12.10 
 
 1 192 
 
 810 
 
 10,200 
 
 .920 
 
 39-4 
 
 38-25 
 
 13-25 
 
 1304 
 
 887 
 
 11,170 
 
 1.268 
 
 54-2 
 
 39-35 
 
 14-35 
 
 1407 
 
 957 
 
 12,065 
 
 .670 
 
 28.6 
 
 38.05 
 
 13-05 
 
 1284 
 
 873 
 
 10,990 
 
 •37° 
 
 15.8 
 
 36.65 
 
 11.65 
 
 1 1 50 
 
 782 
 
 9,835 
 
 .160 
 
 6.84 
 
 34-65 
 
 9.65 
 
 0956 
 
 650 
 
 8,175 
 
 .063 
 
 2.69 
 
 32-55 
 
 7-55 
 
 0751 
 
 5" 
 
 6,415 
 
 .OOO 
 
 0.00 
 
 30.80 
 
 5.80 
 
 0578 
 
 393 
 
 4,93o 
 
 •°33 
 
 + 1.41 
 
 29.30 
 
 4-30 
 
 0430 
 
 292 
 
 3,660 
 
 .06l 
 
 2.61 
 
 27.90 
 
 2.90 
 
 0290 
 
 197 
 
 2,470 
 
 .115 
 
 4.91 
 
 24.80 
 
 + .20 
 
 0020 
 
 + 14 
 
 + 175 
 
 .220 
 
 9.40 
 
 19.25 
 
 5-75 
 
 0574 
 
 390 
 
 4,905 
 
 .361 
 
 15.4 
 
 16.15 
 
 8.85 
 
 0879 
 
 598 
 
 7,530 
 
 .680 
 
 29.1 
 
 12.90 
 
 12.10 
 
 1 192 
 
 810 
 
 10,200 
 
 .920 
 
 39-4 
 
 11.65 
 
 13-35 
 
 1312 
 
 892 
 
 11,240 
 
 1.268 
 
 54.2 
 
 10.65 
 
 14-35 
 
 1407 
 
 957 
 
 12,065 
 
 Station 148. 
 
 Length of rod . . . 78.3 cm. 
 Diameter of rod . .630 cm. 
 
 Cross-sectional area . .3iisq. cm. 
 Volume of rod, v . . 24.3 cu. cm. 
 Dist. between poles, 20 75.0 cm. 
 Distance, center of rod 
 
 to needle, L . . . 126.7 cm - 
 Distance of scale from 
 
 mirror 50.0 cm. 
 
 Length of solenoid . . 
 Turns of wire in sol- 
 enoid 3230 
 
 95.0 cm. 
 
 Turns per cm. . . 
 
 " = ^=42.73/ 
 10/ 
 
 34 
 
 Governing field, f . 
 (L 2 - a 2 ) 2 / 
 
 /.= - 
 
 iLv 
 
 .195 C.G.S. 
 tan 8 = 68 x io 2 tan 8. 
 
MAGNETIC PROPERTIES OF IRON. 
 
 Magnetization and Permeability. 
 
 383 
 
 
 
 Magnet- 
 
 
 
 
 
 
 
 Current. 
 
 H* 
 
 ometer 
 Reading. 
 
 flection. 
 
 tan 5 
 
 In 
 
 * 
 
 B 
 
 ^ 
 
 .000 
 
 
 25.OO 
 
 
 
 
 
 
 
 .032 
 
 1-37 
 
 24.70 
 
 ■3° 
 
 .0030 
 
 20.4 
 
 I4.9 
 
 257 
 
 188 • 
 
 .063 
 
 2.69 
 
 24-34 
 
 .66 
 
 .0066 
 
 44.9 
 
 16.7 
 
 566 
 
 2IO 
 
 .112 
 
 4.78 
 
 23.25 
 
 1 -75 
 
 .0175 
 
 II 9 . 
 
 24.9 
 
 1,498 
 
 3H 
 
 .162 
 
 6.92 
 
 21.67 
 
 3-33 
 
 ■0333 
 
 226. 
 
 32-7 
 
 2,845 
 
 412 
 
 .219 
 
 936 
 
 19-33 
 
 5.67 
 
 .0565 
 
 384- 
 
 41.0 
 
 4,830 
 
 516 
 
 •370 
 
 15.8 
 
 16.08 
 
 8.92 
 
 .0886 
 
 606. 
 
 38.3 
 
 7,630 
 
 483 
 
 •542 
 
 23.2 
 
 14.15 
 
 10.85 
 
 .1069 
 
 727. 
 
 31.2 
 
 9,265 
 
 39S 
 
 .683 
 
 29.2 
 
 13.12 
 
 11.88 
 
 .1175 
 
 798. 
 
 27.4 
 
 10,050 
 
 345 
 
 •932 
 
 39-8 
 
 II.90 
 
 13.10 
 
 .1290 
 
 877- 
 
 22.0 
 
 11,050 
 
 278 
 
 1.268 
 
 54-2 
 
 IO.75 
 
 14.25 
 
 .1400 
 
 952. 
 
 17.6 
 
 I2,OIO 
 
 222 
 
 Experiment W 2 . Magnetic properties of iron — ballistic 
 method. 
 
 In the ballistic method of studying the magnetic properties 
 of iron, the specimen is best used in the form of a closed ring. 
 No corrections are then necessary in the value of the magnet- 
 izing field as computed in terms of the number of turns per 
 centimeter length of the coil used to produce it, since there are 
 no free magnetic poles. Values of the magnetizing field and 
 changes of induction can be computed directly from the obser- 
 vations. 
 
 It is the object of the experiment to study the relation be- 
 tween B and H for iron initially unmagnetized, to get values of 
 the permeability, and also after the specimen has been fully 
 magnetized to find the hysteresis. 
 
 The arrangement of the apparatus is shown in Fig. 134. A 
 ring C of the iron or steel to be tested is wound with two coils, 
 a primary and a secondary. The primary coil is connected in 
 series with a secondary battery B, an ammeter or current 
 
 * No correction has been made for the demagnetizing effect of the poles. 
 
3«4 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 measuring galvanometer A, a constant governing resistance R, 
 and a variable resistance r. A reversing key K is inserted in 
 the primary circuit to change the direction of the current in the 
 primary coil. Since the value of the magnetizing field is 
 
 Fig. 134. 
 
 governed by the variable resistance r, and this field must be 
 capable of continuous stepwise changes, either increasing or 
 decreasing, the resistance must be so arranged as to permit of 
 such changes without breaking circuit or making steps in the 
 wrong direction. 
 
 A satisfactory resistance may be constructed of resistance 
 lamps. The lamps should have such resistances and be so ar- 
 ranged as to permit the desired changes in current strength 
 when short-circuited or inserted into the circuit. 
 
 The secondary coil is connected in series with a ballistic gal- 
 vanometer BG and a resistance box R'. A short-circuiting key 
 K' is connected across the galvanometer terminals. When the 
 current in the primary circuit is changed, the number of lines 
 cutting through the secondary is changed, and an induced cur- 
 rent lasting but a very short time is set up in the secondary 
 circuit. This momentary discharge through the. galvanometer 
 produces a throw of the galvanometer which, as will be proved 
 later, is proportional to the change in the number of lines of 
 force in the ring. If the ballistic galvanometer is not well 
 damped, to facilitate work some method of damping may be used 
 after the throw has been read. The relative motion of the gal- 
 vanometer coil and permanent magnet, whether the instrument 
 be of the tangent or d'Arsonval type, produces an E. M. F. which 
 
MAGNETIC PROPERTIES OF IRON. .385 
 
 will cause an induced current to flow at the expense of the 
 motion necessary to produce it. The current will be large, and 
 consequently the damping will be large if the resistance in the 
 circuit is small. Hence, to bring the galvanometer needle to 
 rest at zero quickly, it should be short-circuited by means of the 
 key K' in the secondary circuit. In using this key, care must 
 be taken to open it before any change is made in the primary 
 current, otherwise no throw will be obtained. 
 
 The rings that are wound for this experiment are rectangu- 
 lar in cross-section and have a primary coil of 1500 turns. For 
 this number of turns the maximum current should be about one 
 ampere. The resistance R in series with the storage battery 
 must be adjusted once for all, so that the maximum value of the 
 current is about one ampere when all the resistance ris cut out. 
 
 Before beginning observations for the magnetization curve, 
 the ring must be thoroughly demagnetized. To do this the 
 current in the primary circuit is given its maximum value, and 
 while being rapidly reversed by the key K it is gradually de- 
 creased to zero, the key being finally left open. If the current 
 is decreased by steps, it should be reversed several times for 
 each change. 
 
 The ring having been demagnetized, the following observa- 
 tions are to be made. The key K being open, # read the rest 
 position of the ballistic galvanometer, and also of the ammeter 
 to find its zero error. 
 
 All of the resistance in r being in circuit, close the key K, 
 read the throw of the ballistic galvanometer and note the corre- 
 sponding ammeter reading. When the galvanometer needle 
 is at rest, the key K' in the secondary circuit being open, short 
 circuit the first lamp at the high resistance end of r, noting the 
 corresponding galvanometer throw and ammeter reading. 
 
 Continue the above process, cutting out the sections of the 
 resistance r progressively, and reading corresponding galvanom- 
 eter throws and ammeter readings until the maximum current 
 is reached. 
 
 VOL. 1 — 2 c 
 
386 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 From the data obtained, the values of the magnetizing field 
 may be obtained from the expression 
 
 in which n is the number of turns of the primary coil, / its 
 mean length, and / the true value of the current in amperes. 
 This formula assumes that H is uniform over the whole cross- 
 section of the coil, which is sufficiently accurate if the radius 
 of the coil is small compared to the radius of the ring on which 
 it is wound. 
 
 The method of obtaining B is not so simple. The theory 
 of the method is as follows : As shown in Exp. Uj, in a ballis- 
 tic galvanometer the throw is proportional to the quantity of 
 electricity that passes through the galvanometer, 
 
 <2=<2o(i+^)sinlS, 
 or Q = Q<f(i + | tys. (See equation 324.) 
 
 The quantity Q that flows in the secondary circuit due to a 
 change in the number of lines of force passing through it is 
 given by equation 341 (see p. 354), 
 
 where N is the change in the number of lines of force, n 2 the 
 number of turns, and R s the total resistance of the secondary 
 circuit. If after the ring has been demagnetized, a current is 
 sent through the primary, there will be, say, B 1 lines of mag- 
 netic induction set up. The N in equation 382 will equal AB V 
 where A is the area of cross-section of the iron. Therefore 
 the quantity of electricity that goes through the secondary cir- 
 cuit will be given by 
 
 «-$& ««> 
 
 Combining this with equation 324 and solving for B v we 
 have 
 
MAGNETIC PROPERTIES OF IRON. 387 
 
 B ^x-Mt + W ,,, (384) 
 
 or 
 
 ^1=% (385) 
 
 If now the current be again increased, say, to 7 2 , and the 
 throw s 2 obtained, we have 
 
 ■"a = "&%> 
 where B 2 is the change in the induction. 
 
 The total induction in the ring due to the current 7 2 is equal 
 'to the sum of the changes, or 
 
 B = B (s 1 + sJ. (386) 
 
 If Oe, in Fig. 135, represents the first magnetizing force 
 applied, em is the resulting in- 
 duction. If the current be 
 increased and the magnetiz- 
 ing force thus increased by 
 the amount ef, the change 
 in the induction is an. If 
 another increment, say, fg, 
 be added to H by another 
 increase in the current, the 
 change in B is bp. Thus the 
 total induction for the point 
 p is given by the sum of all 
 the changes up to that point, 
 
 or 
 
 B, = B 1 + B t + B a + 
 
 = B (s 1 +s i + s 3 + 
 In general 
 
 B = B ts, 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 9 „ 
 
 
 
 Y\ 
 
 
 
 v s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 7- 
 
 1 
 
 
 
 
 
 
 
 f\ 
 ■ 
 
 
 
 
 
 
 
 0/ 
 
 'e 
 
 f \ 
 
 1 
 
 JA 
 
 
 H 
 
 \i 
 
 Fig. 135- 
 
 •)• 
 
 ' (387) 
 
 where the summation includes all the preceding throws. In 
 this way a series of points is obtained and the magnetization 
 curve drawn. 
 
 As is evident from the figure, the magnitude of the change 
 in induction and the corresponding galvanometer throw depends 
 
388 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 on the amount of the change in H and the place on the curve 
 where the change is made. Where the curve is very steep, 
 small changes in H should be made, or the galvanometer deflec- 
 tion may be too large to be read. 
 
 A preliminary set of observations should be made to deter- 
 mine whether the proper value of resistance R' has been used 
 in series with the ballistic galvanometer. If the largest throw 
 of the galvanometer is not great enough, the resistance should 
 be decreased in approximately the inverse ratio of the maxi- 
 mum throw obtained to the maximum that is desired. 
 
 Magnetization of Iron by the Ring Method. 
 
 
 Ballistic 
 
 
 
 
 
 
 
 Galvanometer. 
 
 
 
 
 
 
 / 
 
 
 
 Throw. 
 
 S.S 
 
 B 
 
 H 
 
 
 
 Zero. 
 
 Reading. 
 
 
 0.081 
 
 45.OO 
 
 45.82 
 
 0.82 
 
 0.82 
 
 62 
 
 0.68 
 
 91.2 
 
 0.043 
 
 45.OO 
 
 46.82 
 
 1.82 
 
 2.64 
 
 200 
 
 1.95 
 
 102.5 
 
 0-055 
 
 45.OO 
 
 46.10 
 
 I. IO 
 
 3-74 
 
 283 
 
 2.50 
 
 113.0 
 
 0.081 
 
 45.OO 
 
 47-83 
 
 2.83 
 
 6-57 
 
 498 
 
 3-66 
 
 '35-0 
 
 0-135 
 
 45.OO 
 
 54-97 
 
 9-97 
 
 16.54 
 
 1253 
 
 6.14 
 
 204.0 
 
 0.183 
 
 45.OO 
 
 57-65 
 
 12.65 
 
 29.19 
 
 2210 
 
 8.32 
 
 268.0 
 
 0.266 
 
 45.OO 
 
 61.10 
 
 16.10 
 
 45.29 
 
 3430 
 
 12. 1 
 
 284.0 
 
 0.321 
 
 45.IO 
 
 52.29 . 
 
 7.19 
 
 52.48 
 
 3980 
 
 14.6 
 
 272.0 
 
 0.438 
 
 45.05 
 
 56-31 
 
 11.26 
 
 63-74 
 
 4830 
 
 20.0 
 
 242.0 
 
 0.605 
 
 45.24 
 
 57-23 
 
 11.99 
 
 75-73 
 
 5740 
 
 27.5 
 
 209.0 
 
 1.065 
 
 45.05 
 
 63-93 
 
 18.88 
 
 94.61 
 
 7160 
 
 48.4 
 
 148.0 
 
 Cast-iron Ring, Rectangular Cross-section. 
 
 External diameter of 
 
 
 H-^ n T- 45-45/- 
 
 nng 
 
 15.24 cm. 
 
 10 / 
 
 Internal diameter of 
 
 
 Turns in secondary coil . . . 300 
 
 nng 
 
 11. 17 cm. 
 
 Resistance of secondary coil . 2.8 
 
 Thickness of ring . . 
 
 2.06 cm. 
 
 Resistance of galvanometer . . 12.1 
 
 Area of cross-section of 
 
 
 
 ring 
 
 4.19 sq. cm. 
 
 Resistance of secondary circuit 1 14.9 
 
 Mean radius of ring . 
 
 6.60 cm. 
 
 A = 0.12. go' = 781 X IO -8 . 
 
 Turns in primary coil . 
 
 1500 
 
 B = 75.8. 
 
MAGNETIC PROPERTIES OF IRON. 389 
 
 To obtain data for the study of the hysteresis of the ring, 
 observations must be begun with both B and H at their maxi- 
 mum value, that is, at the point c in Fig. 131. When the current, 
 and thus H, is decreased to zero the history of the change in the 
 iron is given by the line cd. The ordinate od is the residual 
 magnetization of the iron. The sum of the throws in changing 
 H from the maximum value to zero when multiplied by B gives 
 the total change in induction, wd. For any point on the line 
 cd the change in B is proportional to the sum of the throws 
 from c to that point. 
 
 At d the current is reversed and increased to a maximum in 
 the opposite direction. The resultant curve is dep. From p the 
 current is decreased to zero at q. The current is reversed at q 
 and increased to a maximum, thus bringing the iron back to c. 
 The sum of the throws in going from c to p should equal the 
 sum from p to c. In general, when starting at the point c, the 
 initial induction is unknown. If the magnetization curve has 
 just been taken, and the loop is taken as a continuation of it, 
 then it is known. But if this is done, the curve obtained may 
 not be re-entrant. It is only after the iron or steel has been 
 carried through the cycle several times that a symmetrical loop 
 will always be obtained. For this reason before commencing 
 observations, the current while at its maximum value should be 
 reversed fifteen or twenty times. 
 
 The initial value of the induction, at the point c, may be ob- 
 tained in the following way : The sum of the throws from c to 
 p by the path cdep is found. This sum when multiplied by B 
 gives the total change in induction from c top. As the curve 
 must be symmetrical, half of this change will give the initial 
 value of the induction, i.e. the ordinate for the point c. 
 
 It is not worth while to begin observations until the process 
 is clearly understood. If, while taking the observations for the 
 loop, it be kept in mind that abscissas are proportional to cur- 
 rent, and the change in the ordinate in going from one point to 
 the next is proportional to the galvanometer throw, there will 
 
39° 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 be little trouble in following the various steps in the process. 
 Observations are to be made in the same manner as in obtaining 
 data for the magnetization curve and the permeability. It 
 should be noted, however, that sections of the resistance r must 
 be added in the reverse order to obtain a decreasing magnetiz- 
 ing field of approximately equal steps. 
 
 Hysteresis Loop. Ring Method. 
 
 I 
 
 Galv. Zero. 
 
 Reading . 
 
 Throw. 
 
 2i 
 
 B 
 
 H 
 
 .752 
 
 20.IO 
 
 
 
 
 
 37-40 
 
 + 14,700 
 
 + 58.08 
 
 
 181 
 
 20.IO 
 
 I4.85 
 
 -5.25 
 
 32-15 
 
 + 12,700 
 
 + I3-98 
 
 
 000 
 
 20.IO 
 
 12.90 
 
 — 7.20 
 
 24.85 
 
 + 9,800 
 
 0.00 
 
 - 
 
 °53 
 
 20.IO 
 
 8.50 
 
 — II.60 
 
 13-85 
 
 + 5,300 
 
 —4.10 
 
 - 
 
 075 
 
 20.20 
 
 6.65 
 
 -13-55 
 
 — .20 
 
 -80 
 
 -5-79 
 
 - 
 
 103 
 
 20.15 
 
 IO.55 
 
 —9.60 
 
 -9.80 
 
 -3,860 
 
 -7-95 
 
 - 
 
 138 
 
 20.15 
 
 I2.5O 
 
 -7.65 
 
 -17-45 
 
 — 6,900 
 
 — 10.66 
 
 - 
 
 184 
 
 20.13 
 
 H-53 
 
 — 5.60 
 
 -23.05 
 
 — 9,IOO 
 
 — 14.21 
 
 - 
 
 324 
 
 20.14 
 
 12.84 
 
 -7-3° 
 
 -30-35 
 
 — 1 1 ,900 
 
 — 25.02 
 
 - 
 
 752 
 
 20.14 
 
 13.09 
 
 -7.05 
 
 -37-4° 
 
 - 14,700 
 
 -58.08 
 
 - 
 
 183 
 
 20.10 
 
 25.40 
 
 5-3° 
 
 -32.10 
 
 -12,630 
 
 -14-13 
 
 
 000 
 
 20.I0 
 
 27.25 
 
 7-15 
 
 -24.95 
 
 — 9,800 
 
 0.00 
 
 
 °53 
 
 20.I0 
 
 31 65 
 
 11.55 
 
 -13.40 
 
 -5,272 
 
 +4-io 
 
 
 075 
 
 20.15 
 
 33-65 
 
 13-5° 
 
 .10. 
 
 + 40 
 
 + 5-79 
 
 
 i°3 
 
 2O.I5 
 
 29.70 
 
 9-55 
 
 9.65 
 
 + 3,800 
 
 + 7-95 
 
 
 138 
 
 20.I5 
 
 27.85 
 
 7.70 
 
 17-35 
 
 + 6,800 
 
 + 10.66 
 
 
 183 
 
 20.15 
 
 25.80 
 
 5.65 
 
 23.00 
 
 + 9^50 
 
 + 14-13 
 
 
 321 
 
 2O.I5 
 
 27.50 
 
 7-35 
 
 30.35 
 
 + 11,900 
 
 +24.79 
 
 
 756 
 
 20.15 
 
 27.10 
 
 6.95 
 
 37-3° 
 
 + 14,700 
 
 + 58.38 
 
 Annealed Tool Steel Ring, 
 
 External diameter of ring . 14.96 cm. 
 Internal diameter of ring . 10.92 cm. 
 Thickness of ring . . . 1.70 cm. 
 Area of cross-section of 
 
 ring 3-434 
 
 Mean radius of ring . . 6.47 
 Turns in primary coil . 2500 
 
 10/ 
 
 Rectangular Cross-section. 
 
 Turns in secondary coil 500 
 
 Resistance of secondary 
 
 coil 5 ohms 
 
 Resistance in galv. . . no ohms 
 
 Resistance in box . . 10,000 ohms 
 
 Total resistance of sec- 
 ondary to, 1 ^ ohms 
 
 X = .10. 
 
 Qa = 6 39 x IO_9 - 
 
 #o = 393-5- 
 
MAGNETIC PROPERTIES OF IRON. 391 
 
 A trial run should be made to obtain a desirable resistance 
 in the secondary circuit. In general the same secondary resist- 
 ance will not be suitable for the two sets of data. 
 
 Three curves are to be plotted and presented in the report : 
 the magnetization curve, with H and B ; the permeability curve, 
 with H and fi ; and the hysteresis loop, with H and B as co- 
 ordinates. 
 
 The observations and results given on the preceding page 
 are for a ring of annealed tool steel. In this table the initial 
 value for 2j of 37.40 was obtained as follows : The sum of the 
 throws in going from a maximum value of the current in one 
 direction to the maximum value in the reversed direction was 
 found to be 74.80. One half of this was taken as the initial 
 value. No values for (i are given, since permeability applies 
 only to iron originally unmagnetized. 
 
TABLES. 
 
 [In the tables of logarithms and natural trigonometric functions the admirable 
 arrangement made use of in Bottomley's Four-Figure Mathematical Tables has been 
 followed.] 
 
 i. Some Useful Numbers. 
 
 7r = 3.1416 27r = 6.283 4?r = 12.57 
 
 71^=9.87 47r 2 = 39.48 i/tt = 0.318 
 
 1/2 7r = 0.159 7r /4 = 0.785 log ir = 0.4971 
 
 Napierian base ^ = 2.7183. 
 l 0g8 AT = log 1Q ^ = 2 3Q26 , iv; 
 lo gio* 
 togio* = -43429- 
 
 1 radian = 5 7° 3. 
 
 1 degree = 0.01745 radian. 
 
 1 inch = 25.4 mm. 
 
 1 meter = 39.37 inches. 
 
 1 kilogram =2.2 pounds. 
 
 1 pound = 453-6 grams. 
 
 Approximate value of g= 980. V9^o = 14 V 5 = 3 I -3 + 
 1 / y 980 = 0.032 
 
 2. Work and Power. 
 
 1 joule = io 7 ergs. 
 1 watt = 1 joule per second. 
 1 horse power = 746 watts = 550 ft.-lb./sec. 
 1 French horse power = 75 kg. m./sec. 
 Mechanical equivalent of heat = 4.18 x io 7 ergs. 
 392 
 
TABLES. 
 
 3. Densities of Some Substances. 
 
 393 
 
 Solids. 
 
 Aluminum 2.6 
 
 Lead 1 1.3 
 
 Beeswax . 
 
 
 
 0.96 
 
 Nickel 8.9 
 
 Brass .... 
 
 
 
 8.1-8.7 
 
 Paraffin 0.87-0.93 
 
 Copper . . . 
 
 
 
 8.9 
 
 Platinum 21.5 
 
 Cork .... 
 
 
 
 •14- -3 
 
 Silver 10.5 
 
 German silver . 
 
 
 
 8. S 
 
 Tin 7.3 
 
 Glass, common 
 
 
 
 2.4-2.7 
 
 
 Glass, flint . . 
 
 
 
 3-o-S-9 
 
 Woods seasoned : 
 
 Hard rubber 
 
 
 
 1. 15 
 
 Oak 0.7-1.0 
 
 Iron, cast . . 
 
 
 
 7.1-7.7 
 
 Pine ... . 0.5 
 
 Iron, wrought 
 
 
 7-7 
 
 
 Iron, steel . . 
 
 . 7-8 
 
 Zinc . ... 7.1 
 
 Liquids at 20 C. 
 
 Alcohol, ethyl .... 0.789 
 
 
 
 Mercury . . . . 13.596 
 
 Ammonium chloride, 10%. 1.030 
 
 Sodium chloride, 10%. 1.071 
 
 Carbon bisulphide . . . 1 .264 
 
 Zinc sulphate, 10% . 1.107 
 
 Copper sulphate, 10% . .1.107 
 
 
 Gases under Standard Conditions, i.e. at 0° C. and 76 cm. of 
 
 Hg. Pressure. • 
 
 Air . . 
 
 0.001293 g. per cu. cm. 
 
 Carbon dioxide 
 
 
 
 0.001974 g. per cu. cm. 
 
 Coal gas . . . 
 
 
 
 0.00046 g. per cu. cm. 
 
 
 
 
 0.0000900 g. per cu. cm. 
 
 Nitrogen 
 
 
 
 0.001257 g. per cu. cm. 
 
 Oxygen . . 
 
 
 0.001430 g. per cu. cm. 
 
 4. Coefficients of Friction. 
 
 Materials. 
 
 fi (Approximate). 
 
 
 O.I9 
 O.48 
 0.22 
 0.20 
 0.2O 
 O.I4 
 O.04 
 
 
 Smooth and well-lubricated surfaces .... 
 
394 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 5. Elastic Constants. 
 
 Substances. 
 
 Young's Modulus. 
 Dynes per sq. cm. 
 
 Simple Rigidity. 
 Dynes per sq. cm. 
 
 
 6.5 X IO 11 
 
 8-3- 97 
 9.8-12.0 
 6.0- 7.0 
 
 19.3-20.9 
 
 19.0-22.0 
 1.0 
 I.I 
 
 2.4-3.3 X IO 11 
 
 3.I-3.6 
 
 4-S 
 
 2.4 
 7.7-8.0 
 8.0-8.8 
 
 0.07 
 
 0.10 
 
 6. Data on Change of State. 
 
 Substances. 
 
 Melting 
 Point. 
 
 Heat of 
 Fusion.* 
 
 Boiling 
 Point. 
 
 Heat of 
 Vaporiza- 
 tion.* 
 
 Heat of 
 Combustion.* 
 
 Charcoal .... 
 Coal, soft . . . 
 Coal, anthracite 
 Petroleum . . . 
 Wood .... 
 
 
 
 
 
 7070 to 8080 
 7400 to 8800 
 
 7850 
 1 1050 
 
 4100 to 4500 
 
 Acetic acid . . . 
 Alcohol, ethyl . . 
 Alcohol, methyl . 
 Mercury .... 
 Water .... 
 
 -39 
 
 
 
 2.8 
 80.00 
 
 188 
 78 
 64.5 
 
 357 
 100 
 
 84.9 
 205 
 267.5 
 
 62 
 
 537 
 
 7180 
 
 53'° 
 
 Acetylene f . . . 
 Coal gas f . . . 
 Hydrogen f . . . 
 Water gas f . . . 
 
 
 
 
 
 14500 
 1 coo to 1400 
 
 3090 
 
 2000 tO 35OO 
 
 * Expressed in calories per gram. 
 
 t Expressed in calories per liter at '0° C. and 76 cm. Hg. pressure. 
 
TABLES. 
 
 395 
 
 7. Variation in Boiling Point of Water with Change 
 in Barometric Pressure. 
 
 Pressure in cm. 
 of Hg. 
 
 Boiling Point in °C. 
 
 Pressure in cm. 
 of Hg. 
 
 Boiling Point in °C. 
 
 70 
 
 71 
 
 72 
 
 73 
 74 
 
 97-71 
 98.II 
 
 98.49 
 98.88 
 99.26 
 
 75 
 76 
 
 77 
 78 
 79 
 
 99- 6 3 
 100.00 
 100.37 
 100.73 
 101.09 
 
 8. Specific Heats and Coefficients of Expansion. 
 
 Substance. 
 
 Aluminum . . 
 
 Brass . . . 
 
 Copper . . . 
 
 German silver 
 
 Glass 
 
 Ice . . . . 
 
 Acetic acid . . 
 Alcohol, ethyl , 
 
 Specific 
 Heat. 
 
 0.206 
 
 O.093 
 
 O.0927 
 
 O.095 
 
 0.188 
 
 O.474 
 
 0.50 
 O.58 
 
 Linear Co- 
 efficient of 
 Expansion. 
 
 O.OOOO23 
 O.OOOOI9 
 O.OOOOI7 
 O.OOOOI8 
 0.000008 
 O.OOOO38 
 
 Volume 
 Coefficient. 
 
 O.OOI07 
 O.OOIIO 
 
 Substance. 
 
 Iron . . 
 Lead . . 
 Platinum 
 Silver 
 Zinc . . 
 
 Mercury 
 Water . 
 
 Specific 
 Heat. 
 
 0.II7 
 
 0.0305 
 
 O.O324 
 
 O.0561 
 
 O.0939 
 
 0-334 
 
 1.0 
 
 Linear Co- 
 efficient of 
 Expansion. 
 
 O.OOOOII 
 O.OOOO29 
 O.OOOOO9 
 O.OOOOI9 
 O.OOOO29 
 
 Volume 
 Coefficient. 
 
 O.OOOl8l 
 O.OOO065 
 
 Substance. 
 
 Air .... 
 Carbon dioxide 
 Hydrogen . . 
 Nitrogen . . 
 Oxygen . . 
 Steam . . . 
 
 Specific Heat 
 
 at Constant 
 
 Pressure, 
 
 O.238 
 O.206 
 3-238 
 O.244 
 O.218 
 O.48 
 
 I.403 
 I.264 
 I-4'4 
 I-4I4 
 I.408 
 
 Pressure Coefficient 
 
 at Constant 
 
 Volume. 
 
 O.OO3663 
 O.OO3668 
 O.O03667 
 O.OO3668 
 O.OO3668 
 
396 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 Vapor Pressure of Saturated Vapors in Centimeters 
 of Mercury at Various Temperatures. 
 
 Tempera- 
 
 Ethyl 
 
 Water. 
 
 Tempera- 
 
 Ethyl 
 
 
 ture °C. 
 
 Alcohol. 
 
 ture °C. 
 
 Alcohol. 
 
 Water. 
 
 O 
 
 1.22 
 
 .46 
 
 55 
 
 
 n.75 
 
 5 
 
 
 •65 
 
 60 
 
 3S-o 
 
 14.89 
 
 IO 
 
 2.38 
 
 .91 
 
 65 
 
 
 18.71 
 
 IS 
 
 
 , 1.27 
 
 70 
 
 54. 1 
 
 23-33 
 
 20 
 
 4.40 
 
 I.74 
 
 75 
 
 
 28.88 
 
 25 
 
 
 2-35 
 
 80 
 
 81.2 
 
 35-49 
 
 3° 
 
 7.81 
 
 3-15 
 
 85 
 
 
 43-32 
 
 35 
 
 
 4.18 
 
 90 
 
 118.7 
 
 52-55 
 
 40 
 
 13-37 
 
 549 
 
 95 
 
 
 63-37 
 
 45 
 
 
 7.14 
 
 100 
 
 169.2 
 
 76.00 
 
 5° 
 
 22.0 
 
 9.20 
 
 150 
 
 736.9 
 
 358.1 
 
 10. Vapor Pressure and Relative Humidity. 
 
 The ratio of the mass of aqueous vapor in a given volume of air at a given 
 temperature, to the mass of aqueous vapor in the same volume at the same tempera- 
 ture when saturated, is called the relative humidity. Since this ratio is very nearly 
 equal to the ratio of the vapor pressures under like conditions for ordinary atmos- 
 pheric temperatures, the vapor pressures are commonly used in finding the relative 
 humidity. 
 
 An instrument used to determine relative humidity is called an hygrometer. 
 The wet and dry bulb hygrometer consists, as its name implies, of two thermometers, 
 the bulb of one being kept dry and the other wet by means of « wick which con- 
 ducts water up to a muslin wrapping about the bulb. The evaporation of moisture 
 from the wet bulb requires heat, some of which is abstracted from the wet bulb, thus 
 reducing its temperature. The depression varies with the temperature and the 
 amount of water vapor present in the air. 
 
 From the following table, based on Table 170 of the Smithsonian Physical 
 Tables, showing the relation between the temperatures of the two thermometers and 
 the vapor pressure, the relative humidity may be computed. The left-hand column 
 gives dry bulb thermometer readings, and the succeeding columns give vapor pres- 
 sures corresponding to the wet bulb thermometer depressions indicated in the top 
 row. Saturation pressures are those corresponding to no wet bulb thermometer de- 
 pressions. 
 
TABLES. 
 
 397 
 
 Dry Bulb Temperatures and Wet Bulb Depressions. 
 
 T°C. 
 
 0. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 s. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 O 
 
 4.6 
 
 3-7 
 
 2.9 
 
 2.1 
 
 i-3 
 
 
 
 
 
 
 
 I 
 
 4.9 
 
 4.1 
 
 3-2 
 
 2.4 
 
 1.6 
 
 
 
 
 
 
 
 2 
 
 5-3 
 
 4.4 
 
 3-6 
 
 2.7 
 
 1.9 
 
 1.1 
 
 o-3 
 
 
 
 
 
 3 
 
 5-7 
 
 4.8 
 
 3-9 
 
 3-i 
 
 2.2 
 
 1.4 
 
 0.6 
 
 
 
 
 
 4 
 
 6.1 
 
 5-2 
 
 4-3 
 
 3-4 
 
 2.6 
 
 1.8 
 
 0.9 
 
 
 
 
 
 S 
 
 6-5 
 
 5-6 
 
 4-7 
 
 3-8 
 
 2.9 
 
 2.1 
 
 1.2 
 
 
 
 
 
 6 
 
 7.0 
 
 6.0 
 
 5-1 
 
 4.2 
 
 3-3 
 
 2.4 
 
 1.6 
 
 
 
 
 
 7 
 
 7-5 
 
 6. S 
 
 5-5 
 
 4.6 
 
 3-6 
 
 2.8 
 
 1.9 
 
 I.I 
 
 0.2 
 
 
 
 8 
 
 8.0 
 
 7.0 
 
 6.0 
 
 5-° 
 
 4.1 
 
 3-2 
 
 2-3 
 
 1.4 
 
 0.6 
 
 
 
 9 
 
 8.6 
 
 7-5 
 
 6-5 
 
 5-5 
 
 4-5 
 
 3-6 
 
 2.7 
 
 1.8 
 
 0.9 
 
 
 
 IO 
 
 9.2 
 
 8.1 
 
 7.0 
 
 6.0 
 
 5.0 
 
 4.0 
 
 3-i 
 
 2.2 
 
 1-3 
 
 
 
 ii 
 
 9.8 
 
 8.7 
 
 7.6. 
 
 6.5 
 
 5-5 
 
 4-5 
 
 3-5 
 
 2.6 
 
 i-7 
 
 
 
 12 
 
 10.5 
 
 9-3 
 
 8.2 
 
 7-i 
 
 6.0 
 
 5.0 
 
 4.0 
 
 3-o 
 
 2.1 
 
 1.2 
 
 o-3 
 
 '3 
 
 11. 2 
 
 10.0 
 
 8.9 
 
 7.6 
 
 6.5 
 
 5-5 
 
 4-5 
 
 3-5 
 
 2.5 
 
 1.6 
 
 0.7 
 
 H 
 
 11.9 
 
 10.7 
 
 9.4 
 
 8-3 
 
 7-i 
 
 6.1 
 
 5.0 
 
 4.0 
 
 3-o 
 
 2.0 
 
 1.1 
 
 IS 
 
 12.7 
 
 11.4 
 
 10. 1 
 
 9.0 
 
 7.8 
 
 6.6 
 
 5-5 
 
 4-5 
 
 3-4 
 
 2.5 
 
 i-5 
 
 16 
 
 13-5 
 
 12.2 
 
 10.9 
 
 9-7 
 
 8.4 
 
 7-3 
 
 6.0 
 
 5.0 
 
 4.0 
 
 3-o 
 
 1.9 
 
 17 
 
 14.4 
 
 13.0 
 
 11. 7 
 
 10.4 
 
 9.1 
 
 8.0 
 
 6.7 
 
 5.6 
 
 4-5 
 
 35 
 
 2.4 
 
 18 
 
 15.4 
 
 13-9 
 
 12.5 
 
 11. 2 
 
 9.9 
 
 8.6 
 
 7-4 
 
 6-3 
 
 5-i 
 
 4.0 
 
 3° 
 
 r 9 
 
 16.3 
 
 14.9 
 
 13-4 
 
 12.0 
 
 10.7 
 
 9.4 
 
 8.1 
 
 6,9 
 
 5-7 
 
 4.6 
 
 3-5 
 
 20 
 
 17.4 
 
 15.9 
 
 H-3 
 
 12.9 
 
 11.5 
 
 10.2 
 
 " 8.8 
 
 7.6 
 
 6.4 
 
 5.2 
 
 4.1 
 
 21 
 
 18.5 
 
 16.9 
 
 IS -3 
 
 13.8 
 
 12.4 
 
 11.0 
 
 9.6 
 
 8.4 
 
 7-i 
 
 5-9 
 
 4-7 
 
 22 
 
 19.7 
 
 18.0 
 
 164 
 
 14.8 
 
 •3-3 
 
 11.9 
 
 10.5 
 
 9.1 
 
 7.8 
 
 6.6 
 
 5-4 
 
 23 
 
 20.9 
 
 19.2 
 
 17-5 
 
 15.9 
 
 H-3 
 
 12.8 
 
 "•3 
 
 10. 
 
 8.6 
 
 7-3 
 
 6.1 
 
 24 
 
 22.2 
 
 20.4 
 
 18.6 
 
 17.0 
 
 '5-3 
 
 13.8 
 
 12.3 
 
 10.9 
 
 9.4 
 
 8.1 
 
 6.8 
 
 2 5 
 
 23-5 
 
 21.7 
 
 19.9 
 
 18. 1 
 
 16.4 
 
 14.8 
 
 13-3 
 
 11. 8 
 
 10.3 
 
 9.0 
 
 7.6 
 
 26 
 
 25.0 
 
 23.1 
 
 21. 1 
 
 19.4 
 
 17.6 
 
 15.9 
 
 14-3 
 
 12.8 
 
 "•3 
 
 9.8 
 
 8.4 
 
 27 
 
 26.5 
 
 24.5 
 
 22.5 
 
 20.7 
 
 18.8 
 
 17. 1 
 
 15.4 
 
 138 
 
 12.3 
 
 10.8 
 
 9-3 
 
 28 
 
 28.1 
 
 26.0 
 
 24,0 
 
 22.0 
 
 20.1 
 
 18.3 
 
 16.6 
 
 14.9 
 
 13-3 
 
 11. 8 
 
 10.2 
 
 29 
 
 29.8 
 
 27.6 
 
 25.5 
 
 23-5 
 
 21.5 
 
 19.6 
 
 17.8 
 
 16.1 
 
 14.4 
 
 12.8 
 
 11. 2 
 
 30 
 
 31-5 
 
 29-3 
 
 27.1 
 
 25.0 
 
 22.9 
 
 21.0 
 
 19.1 
 
 17-3 
 
 15-5 
 
 13-9 
 
 12.3 
 
 31 
 
 33-4 
 
 3 1 -! 
 
 28.8 
 
 26.6 
 
 24.5 
 
 22.4 
 
 20.4 
 
 18.5 
 
 16.7 
 
 15.0 
 
 13-4 
 
398 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 ii. Index of Refraction for Sodium Light. 
 
 Alcohol 
 
 Canada balsam .... 
 Carbon bisulphide . . . 
 
 Glass, crown 
 
 Glass, flint 
 
 Water 
 
 Calcite, ordinary ray . . 
 Calcite, extraordinary ray 
 Quartz, ordinary ray . 
 Quartz, extraordinary ray 
 
 1.362 
 
 1.54 
 1.628 
 
 1.515-1.615 
 1. 609-1 .754 
 
 1-333 
 1.659 
 1.486 
 
 1.544 
 1-553 
 
 12. Bright Line Spectra. 
 
 Some Wave Lengths in Air at Ordinary Temperatures and 
 
 Pressures. 
 
 Element. 
 
 
 How'Produced. 
 
 Wave Length 
 in io~* Cm. 
 
 Color. 
 
 Calcium .... 
 
 Ca 
 
 Flame 
 
 4227 
 
 Blue 
 
 Cadmium .... 
 
 Cd 
 
 Flame and spark 
 
 6439 
 5086 
 4800 
 4679 
 
 Red 
 Green 
 Blue 
 Blue 
 
 Helium .... 
 
 He 
 
 Vacuum tube 
 
 7065 
 6678 
 5876 
 5016 
 4922 
 
 4713 
 4472 
 
 Red 
 
 Red 
 
 Yellow 
 
 Green 
 
 Blue 
 
 Blue 
 
 Violet 
 
 Hydrogen .... 
 
 H 
 
 Vacuum tube 
 
 6563 
 4862 
 
 4341 
 4102 
 
 Red 
 Blue 
 Violet 
 Violet 
 
TABLES. 
 
 399 
 
 Element. 
 
 
 How Produced. 
 
 Wave Length 
 in io- 8 Cm. 
 
 Color. 
 
 
 Pb 
 
 Spark 
 
 6657 
 5608 
 
 4387 
 4245 
 4058 
 
 Red 
 
 Green 
 
 Violet 
 
 Violet 
 
 Violet 
 
 Lithium 
 
 Li 
 
 Flame 
 
 6708 
 6104 
 
 Red 
 Red 
 
 Mercury .... 
 
 Hg 
 
 Vacuum tube 
 
 6IS3 
 5791 
 
 5770 
 5461 
 
 4359 
 4047 
 
 Orange 
 
 Yellow 
 
 Yellow 
 
 Green 
 
 Violet 
 
 Violet 
 
 
 Ni 
 
 Spark 
 
 S893 
 4874 
 4866 
 4856 
 
 4715 
 4402 
 
 Yellow 
 
 Blue 
 
 Blue 
 
 Blue 
 
 Blue 
 
 Violet 
 
 Potassium . . . 
 
 K 
 
 Flame 
 
 7702 
 7669 
 4047 
 4044 
 
 Red 
 Red 
 Violet 
 Violet 
 
 Sodium .... 
 
 Na 
 
 Flame 
 
 5896 
 5890 
 
 Yellow 
 Yellow 
 
 Strontium . . . 
 
 Sr 
 
 Flame 
 
 4608 
 
 Blue 
 
 Thallium .... 
 
 Tl 
 
 Flame 
 
 53Si 
 
 Green 
 
 
 Zn 
 
 Spark 
 
 6364 
 6103 
 4925 
 481 1 
 
 4723 
 4680 
 
 Red 
 Red 
 Blue 
 Blue 
 Blue 
 Blue 
 
400 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 13. Velocity of Sound at o° C. and 760 mm. Hg. 
 Pressure. 
 
 Substance. 
 
 Velocity in Meters per Second. 
 
 Air 
 
 332-5 
 
 261.6 
 
 490.0 
 1286.4 
 
 317-2 
 3500 
 356o 
 
 5000-6000 
 5000 
 
 
 Hydrogen . 
 
 
 14. Specific Resistance and Temperature Coefficients. 
 
 Substance. 
 
 Specific Resist- 
 ance in 
 Ohms X io-«. 
 
 Temperature 
 
 Coefficient. 
 
 Aluminum 
 
 Copper (annealed) . ... 
 Copper (hard-drawn) .... 
 German silver (4 Cu + 2 Ni + 1 Zn) 
 Iron (annealed) .... 
 Manganin (Cu 84%, Mn 12%, Ni 4%) . 
 
 2.9I 
 
 I.59 
 
 1.62 
 
 20.24 
 
 9.69 
 
 4I.OO 
 
 94.07 
 
 8.96 
 
 1.52 
 
 O.OO435 
 
 O.OO428 
 
 O.OO388 
 
 O.OOO273 
 
 O.O0625 
 
 0.000025 
 
 O.OO0883 
 
 O.OO345 
 
 O.OO377 
 
 
 15. Electro-chemical Equivalents. 
 
 Element. 
 
 Atomic Weight. 
 
 Valence. ' 
 
 Equivalent in Grams 
 per Coulomb. 
 
 Copper 
 
 Hydrogen 
 
 Lead 
 
 Oxygen 
 
 Silver 
 
 Zinc ... .... 
 
 63.6 
 I.008 
 206.9 
 
 16.O 
 107.9 
 
 65.4 
 
 2 
 I 
 
 2 
 2 
 I 
 
 2 
 
 O.OOO329 
 
 O.OOOOI05 
 
 O.OOI072 
 
 O.OOO0829 , 
 
 O.OOIII8 
 
 O.OOO339 
 
TABLES. 
 
 401 
 
 16. Specific Resistances of Electrolytes. 
 
 
 Per Cent of 
 
 Density at 
 
 Gram Molecules 
 
 Specific Resist- 
 
 
 Solution. 
 
 18° C 
 
 per Liter. 
 
 Ohms per Co 
 
 CuS0 4 . . 
 
 5 
 
 I.0 53 I 
 
 O.658 
 
 50.9 
 
 
 10 
 
 1. 1073 
 
 I-387 
 
 30.0 
 
 
 IS 
 
 1.1675 
 
 2.I94 
 
 22.8 
 
 H 2 S0 4 . . . 
 
 5 
 
 I-033I 
 
 i-°S3 
 
 4.61 
 
 
 10 
 
 1 .0673 
 
 2.176 
 
 2.46 
 
 
 'S 
 
 I. IO36 
 
 3-376 
 
 1-7.7 
 
 
 20 
 
 I.I414 
 
 4.655 
 
 1.47 
 
 NaCl .... 
 
 5 
 
 I -0345 
 
 0.884 
 
 14-3 
 
 
 10 
 
 I.0707 
 
 1.830 
 
 7-94 
 
 - 
 
 IS , 
 
 I. I087 
 
 2.843 
 
 5.86 
 
 
 20 
 
 1. 1477 
 
 3-9 2 4 
 
 4.91 
 
 ZnS0 4 . . . 
 
 5 
 
 I.0509 
 
 0.653 
 
 50-3 
 
 
 10 
 
 1. 1069 
 
 1-365 
 
 29.9 
 
 
 IS 
 
 I-ID7S 
 
 2.169 
 
 23.2 
 
 
 20 
 
 1-2323 
 
 3-°53 
 
 20.5- 
 
 17. Electromotive Force of Cells. 
 
 Edison Lalande .... 0.90 
 
 Daniell 1.08 
 
 Gravity I . I 
 
 Dry Cell 1.4 
 
 LeClanche" 1 .4-1.7 
 
 Bichromate 2.0 
 
 Storage 2.0 
 
 E. M. F. Standard Cells at 20 C. 
 
 Clark 1.427 
 
 Weston (Cadmium) . . . 1.019 
 
 vol. 1 — 2D 
 
402 
 
 JUNIOR COURSE IN GENERAL PHYSICS. 
 
 1 8. Wire Gauge and Copper Resistance Table. 
 
 Size. 
 B. & S. Gauge. 
 
 Diameter in Inches. 
 
 Ohms per 1000 Feet. 
 
 Feet per Ohm. 
 
 
 
 0.32495 
 
 O.0983 
 
 IOI66 
 
 4 
 
 0.20431 
 
 O.2487 
 
 4021 
 
 8 
 
 0.12849 
 
 O.6288 
 
 1590 
 
 10 
 
 0.10189 
 
 I .OOOO 
 
 1000 
 
 12 
 
 0.08081 
 
 I.590 
 
 629 
 
 14 
 
 0.06408 
 
 2.528 
 
 396 
 
 16 
 
 0.05082 
 
 4.OI9 
 
 248.8 
 
 18 
 
 0.04030 
 
 6.391 
 
 • 1565 
 
 20 
 
 0.03196 
 
 IO.163 
 
 98.4 
 
 24 
 
 0.02010 
 
 25.695 
 
 38.92 
 
 28 
 
 0.01264 
 
 64.97 
 
 '5-39 
 
 32 
 
 0.00795 
 
 164.3 
 
 6.09 
 
 36 
 
 0.00500 
 
 415.2 
 
 2.41 
 
 40 
 
 0.00314 
 
 IO49.7 
 
 Q-953 
 
LOGARITHMS. 
 
 403 
 
 10 
 
 11 
 
 12 
 13 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 ~W 
 
 21 
 22 
 23 
 
 24 
 25 
 26 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 45 
 46 
 
 47 
 48 
 49 
 
 50 
 
 51 
 52 
 53 
 
 54 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 4 5 6 
 
 7 8 9 
 
 0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 4 8 12 
 
 17 21 25 
 
 29 33 37 
 
 0414 
 0792 
 "39 
 
 0453 
 0828 
 
 "7"3 
 
 0492 
 0864 
 1 206 
 
 °53i 
 0899 
 1239 
 
 0569 
 
 0934 
 1271 
 
 0607 
 0969 
 'i3°3 
 
 0645 
 1004 
 '335 
 
 0682 
 1038 
 1367 
 
 0719 
 1072 
 1399 
 
 °755 
 1 106 
 
 1430 
 
 4 8 11 
 3 7 10 
 3 6 10 
 
 15 '9 23 
 14 17 21 
 13 16 19 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 1461 
 1761 
 2041 
 
 1492 
 1790 
 2068 
 
 1523 
 1818 
 2095 
 
 •553 
 l8 47j 
 
 2122 
 
 1584 
 
 JL8.75 
 2148 
 
 1614 
 1903 
 
 2175 
 
 1644 
 
 '93 1 
 2201 
 
 1673 
 1959 
 2227 
 
 1703 
 1987 
 
 2253 
 
 1732 
 2014 
 2279 
 
 369 
 368 
 3 5 8 
 
 12 15 18 
 11 14 17 
 11 13 16 
 
 21 24 27 
 20 22 25 
 18 21 24 
 
 2304 
 
 2 553 
 2788 
 
 2330 
 
 2577 
 2810 
 
 2355 
 2601 
 
 2833 
 
 2380 
 2625 
 2856 
 
 2405 
 2648 
 2878 
 
 2430 
 2672 
 
 ,29©0 
 
 2455 
 2695 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 2765 
 2989 
 
 2 5 7 
 2 5 7 
 247 
 
 10 12 15 
 9 12 14 
 9 " 13 
 
 17 20 22 
 16 19 21 
 16 18 20 
 
 3010 
 
 3°3 2 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 246 
 
 8 11 13 
 
 15 17 19 
 
 3222 
 3424 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 3655 
 
 3284 
 3483 
 3674 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 
 37" 
 
 3345 
 3541 
 3729 
 
 3365 
 3560 
 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 3598 
 3784 
 
 246 
 246 
 246 
 
 8 10 12 
 8 10 12 
 7 9 11 
 
 14 16 18 
 14 15 17 
 '3 15 17 
 
 3802 
 
 3979 
 4150 
 
 3820 
 
 3997 
 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 4031 
 4200 
 
 3874 
 404S 
 4216 
 
 3892 
 4065 
 4232 
 
 3909 
 4082 
 4249 
 
 3927 
 4099 
 4265 
 
 3945 
 4116 
 4281 
 
 3962 
 
 4133 
 4298 
 
 245 
 235 
 
 235 
 
 7 9 11 
 7 9 10 
 7 8 10 
 
 12 14 16 
 
 12 14 is 
 " 13 15 
 
 4314 
 4472 
 
 4624 
 
 4330 
 4487 
 
 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 4518 
 4669 
 
 4378 
 4533 
 4683 
 
 4393 
 4548 
 4698 
 
 4409 
 4564 
 47'3 
 
 4425 
 4579 
 4728 
 
 4440 
 4-5*94 
 
 4742 
 
 4456 
 4609 
 4757 
 
 235 
 235 
 1 3 4 
 
 689 
 689 
 679 
 
 11 13 14 
 11 12 14 
 10 12 13 
 
 4771 
 
 4786 
 
 .4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 I 3 4 
 
 679 
 
 10 11 13 
 
 4914 
 5051 
 5i85 
 
 4928 
 5°6S 
 5198 
 
 4942 
 5079 
 5211 
 
 4955 
 5092 
 
 5224 
 
 4969 
 5105 
 5237 
 
 4983 
 5"9 
 5250 
 
 4997 
 5'32 
 5263 
 
 501 1 
 
 5»45 
 5276 
 
 5024 
 5159 
 5289 
 
 5038 
 5172 
 5302 
 
 I 3 4 
 1 3 4 
 1 3 4 
 
 678 
 5 7 8 
 5 6 8 
 
 10 11 12 
 9 11 12 
 9 10 12 
 
 5315 
 S44i 
 5563 
 
 5328 
 5453 
 5575 
 
 534o 
 5465 
 5587 
 
 5353 
 5478 
 5599 
 
 5366 
 
 549° 
 5611 
 
 5378 
 55°2 
 5623 
 
 5391 
 55M 
 5635 
 
 5403 
 5527 
 5647 
 
 5416 
 
 5539 
 5658 
 
 5428 
 
 5551 
 5670 
 
 « 3 4 
 I 2 4 
 
 I 2 4 
 
 568 
 567 
 
 5 6 7 
 
 9 10 11 
 9 10 11 
 8 10 11 
 
 5682 
 
 5798 
 591 1 
 
 5694 
 5809 
 5922 
 
 5705 
 5821 
 
 5933 
 
 5717 
 5832 
 5944 
 
 5729 
 5843 
 5955 
 
 5740 
 
 5855 
 5966 
 
 5752 
 5866 
 
 5977 
 
 5763 
 5877 
 5988 
 
 5775 
 5888 
 
 5999 
 
 5786 
 
 5899 
 6010 
 
 1 2 3 
 1 2 3 
 1 2 3 
 
 567 
 5 6 7 
 4 5 7 
 
 8 9 10 
 8 9 10 
 8 9 10 
 
 6021 
 
 6031 
 
 6042 
 
 6°53 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 61 17 
 
 I 2 3 
 
 456 
 
 8 9 10 
 
 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 
 
 6314 
 6415 
 
 6222 
 
 6325 
 6425 
 
 1 2 3 
 1 2 3 
 1 2 3 
 
 4 5 6 
 
 4 5 6 
 456 
 
 7 8 9 
 7 8 9 
 7 8 9 
 
 6435 
 6532 
 6628 
 
 6444 
 6542 
 6637 
 
 6454 
 6551 
 6646 
 
 6464 
 6561 
 6656 
 
 6474 
 6571 
 6665 
 
 6484 
 6580 
 6675 
 
 6493 
 6590 
 6684 
 
 65°3 
 6599 
 6693 
 
 6513 
 6609 
 6702 
 
 6522 
 6618 
 6712 
 
 1 2 3 
 1 2 3 
 1 2 3 
 
 456 
 4 5 6 
 456 
 
 7 8 9 
 7 8 9 
 
 7 7 8 
 
 6721 
 6812 
 6902 
 
 6990 
 
 6730 
 6821 
 691 1 
 
 6739 
 6830 
 6920 
 
 6749 
 6839 
 6928 
 
 6758 
 6848 
 6937 
 
 6767 
 
 6857 
 6946 
 
 6776 
 6866 
 6955 
 
 6785 
 
 6875 
 6964 
 
 6794 
 6884 
 6972 
 
 6803 
 6893 
 6981 
 
 1 2 3 
 1 2 3 
 1 2 3 
 
 4 5 5 
 4 4 5 
 4 4 5 
 
 678 
 678 
 678 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7°33 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 1 2 3 
 
 3 4 5 
 
 678 
 
 7076 
 7160 
 7243 
 
 7084 
 7168 
 7251 
 
 7°93 
 7177 
 
 7259 
 
 7101 
 7185 
 7267 
 
 7110 
 7'93 
 7275 
 
 7118 
 7202 
 7284 
 
 7126 
 7210 
 7292 
 
 7'35 
 7218 
 7300 
 
 7'43 
 7226 
 73o8 
 
 7'52 
 7235 
 73i6 
 
 1 2 3 
 122 
 1 2 2 
 
 3 4 5 
 3 4 5 
 3 4 5 
 
 678 
 6 7 7 
 667 
 
 7324 
 
 7332 
 
 734° 
 
 7348 
 
 735^ 
 
 7364 
 
 7372 
 
 738o 
 
 7388 
 
 7396 
 
 1 2 2 
 
 3 4 5 
 
 667 
 
4°4 
 
 LOGARITHMS. 
 
 55 
 
 56 
 57 
 58 
 
 59 
 60 
 61 
 
 62 
 63 
 64 
 
 65 
 
 66 
 67 
 68 
 
 69 
 70 
 
 71 
 
 72 
 
 73 
 74 
 
 75 
 
 76 
 77 
 78 
 
 79 
 80 
 81 
 
 82 
 83 
 84 
 
 85 
 
 86 
 87 
 88 
 
 89 
 90 
 91 
 
 92~ 
 
 93 
 
 94 
 
 95 
 
 96 
 97 
 98 
 
 99 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 4 5 6 
 
 7 8 9 
 
 7404 
 
 7412 
 
 74i9 
 
 7427 
 
 7435 
 
 7443 
 
 745 1 
 
 7459 
 
 7466 
 
 7474 
 
 122 
 
 3 4 5 
 
 5 6 7 
 
 7482 
 
 7559 
 7634 
 
 7490 
 7566 
 7642 
 
 7497 
 7574 
 7649 
 
 75°5 
 7582 
 7657 
 
 75'3 
 7589 
 7664 
 
 7520 
 
 7597 
 7672 
 
 7528 
 7604 
 7679 
 
 7536 
 7612 
 7686 
 
 7543 
 7619 
 7694 
 
 755 1 
 7627 
 7701 
 
 122 
 122 
 112 
 
 3 4 5 
 3 4 5 
 3 4 4 
 
 5 6 7 
 567 
 5 6 7 
 
 7709 
 7782 
 7853 
 
 7716 
 7789 
 7860 
 
 7723 
 7796 
 7868 
 
 773i 
 7803 
 7875 
 
 7738 
 7810 
 7882 
 
 7745 
 7818 
 
 7889 
 
 7752 
 ■7825 
 7896 
 
 7760 
 7832 
 7903 
 
 7767 
 
 7839 
 7910 
 
 7774 
 7846 
 7917 
 
 112 
 112 
 1 1 2 
 
 3 4 4 
 3 4 4 
 3 4 4 
 
 5 6 7 
 566 
 566 
 
 7924 
 
 7993 
 8062 
 
 7931 
 8000 
 8069 
 
 7938 
 8007 
 8075 
 
 7945 
 8014 
 8082 
 
 7952 
 8021 
 8089 
 
 7959 
 8028 
 8096 
 
 7966 
 
 8035 
 8102 
 
 7973 
 8041 
 8109 
 
 7980 
 8048 
 8^16 
 
 7987 
 
 8055 
 8122 
 
 1 1 2 
 1 i- 2 
 1 1 2 
 
 3 3 4 
 3 3 4 
 3 3 4 
 
 5 6 6 
 5 5 6 
 5 5 6 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 112 
 
 3 3 4 
 
 5 5 6 
 
 8195 
 8261 
 
 8325 
 
 8202 
 8267 
 833' 
 
 8209 
 8274 
 8338 
 
 8215 
 8280 
 8344 
 
 8222 
 8287 
 8351 
 
 8228 
 8293 
 8357 
 
 8235 
 8299 
 8363 
 
 8241 
 8306 
 8370 
 
 8248 
 8312 
 8376 
 
 8254 
 
 8319 
 8382 
 
 1 1 2 
 1 1 2 
 112 
 
 3 3 4 
 3 3 4 
 3 3 4 
 
 5 5 6 
 5 5 6 
 
 456 
 
 8388 
 8451 
 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 
 
 112 
 112 
 1 1 2 
 
 2 3 4 
 2 3 4 
 2 3 4 
 
 4 5 6 
 4 5 6 
 
 4 5 5 
 
 8573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
 8585 
 8645 
 8704 
 
 8591 
 8651 
 8710 
 
 8597 
 8657 
 8716 
 
 8603 
 8663 
 8722 
 
 8609 
 8669 
 8727 
 
 8615 
 8675 
 8733 
 
 8621 
 8681 
 8739 
 
 8627 
 8686 
 8745 
 
 112 
 112 
 112 
 
 2 3 4 
 2 3 4 
 2 3 4 
 
 4 5 5 
 4 5 5 
 4 5 5 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 112 
 
 2 3 3 
 
 4 5 5 
 
 8808 
 8865 
 8921 
 
 8814 
 8871 
 8927 
 
 8820 
 8876 
 8932 
 
 8825 
 8882 
 8938 
 
 8831 
 8887 
 8943 
 
 8837 
 8893 
 8949 
 
 8842 
 8899 
 8954 
 
 8848 
 8904 
 8960 
 
 8854 
 8910 
 
 8965 
 
 8859 
 8915 
 8971 
 
 1 1 2 
 112 
 112 
 
 2 3 3 
 2 3 3 
 2 3 3 
 
 4 5 5 
 4 4 5 
 4 4 5 
 
 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 
 
 112 
 1 1 2 
 1 1 2 
 
 2 3 3 
 2 3 3 
 2 3 3 
 
 4 4 5 
 4 4 5 
 4 4 5 
 
 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 
 
 9186 
 9238 
 9289 
 
 112 
 112 
 1 1 2 
 
 2 3 3 
 2 3 3 
 2 3 3 
 
 4 4 5 
 4 4 5 
 4 4 5 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 933° 
 
 9335 
 
 9340 
 
 112 
 
 2 3 3 
 
 4 4 5 
 
 9345 
 9395 
 
 9445 
 
 935° 
 9400 
 
 945° 
 
 9355 
 9405 
 
 9455 
 
 9360 
 9410 
 9460 
 
 9365 
 9415 
 9465 
 
 937° 
 9420 
 9469 
 
 9375 
 9425 
 9474 
 
 9380 
 9430 
 9479 
 
 9385 
 9435 
 9484 
 
 939° 
 9440 
 9489 
 
 112 
 
 Oil 
 Oil 
 
 2 3 3 
 223 
 223 
 
 4 4 5 
 3 4 4 
 3 4 4 
 
 9494 
 9542 
 9590 
 
 9499 
 9547 
 9595 
 
 95°4 
 9552 
 9600 
 
 9509 
 
 9557 
 9605 
 
 95'3 
 9562 
 9609 
 
 9518 
 9566 
 9614 
 
 9523 
 9571 
 9619 
 
 9528 
 9576. 
 9624 
 
 9533 
 958i 
 9628 
 
 9538 
 9586 
 
 9633 
 
 Oil 
 Oil 
 Oil 
 
 223 
 223 
 223 
 
 3 4 4 
 3 4 4 
 3 4 4 
 
 9638 
 9685 
 9731 
 
 9643 
 9689 
 
 9736 
 
 9647 
 9694 
 974' 
 
 9652 
 9699 
 9745 
 
 9657 
 97°3 
 975° 
 
 9661 
 9708 
 9754 
 
 9666 
 97'3 
 9759 
 
 9671 
 9717 
 9763 
 
 9675 
 9722 
 9768 
 
 9680 
 9727 
 9773 
 
 Oil 
 
 I I 
 
 Oil 
 
 223 
 223 
 223 
 
 3 4 4 
 3 4 4 
 3 4 4 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 Oil 
 
 223 
 
 3 4 4 
 
 9823 
 9868 
 9912 
 
 9827 
 9872 
 9917 
 
 9832 
 
 9877 
 9921 
 
 9836 
 9881 
 9926 
 
 9841 
 9886 
 993° 
 
 9845 
 9890 
 
 9934 
 
 9850 
 9894 
 9939 
 
 9854 
 9899 
 
 9943 
 
 9859 
 
 9903 
 9948 
 
 9863 
 9908 
 9952 
 
 Oil 
 Oil 
 Oil 
 
 223 
 223 
 223 
 
 3 4 4 
 3 4 4 
 3 4 4 
 
 9?56 
 
 9961 
 
 9965 J 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 Oil 
 
 223 
 
 3 3 4 
 
NATURAL SINES. 
 
 4°5 
 
 0° 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 oooo 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 3 6 9 
 
 12 15 
 
 0175 
 
 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 
 0680 
 
 3 6 9 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 0698 
 0872 
 i°45 
 
 071S 
 0889 
 1063 
 
 0732 
 0906 
 1080 
 
 0750 
 0924 
 1097 
 
 0767 
 0941 
 1115 
 
 0785 
 0958 
 1132 
 
 0802 
 0976 
 1 149 
 
 0819 
 
 0993 
 1167 
 
 0837 
 
 IOII 
 
 1 184 
 
 0854 
 1028 
 1 201 
 
 369 
 369 
 369 
 
 12 15 
 12 14 
 12 14 
 
 7 
 8 
 9 
 
 10 
 
 11 
 12 
 13 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 20 
 
 21 
 22 
 23 
 
 24 
 25 
 26 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 
 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 
 
 *357 
 1530 
 1702 
 
 1374 
 1547 
 1719 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 1736 
 
 1754 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 369 
 
 12 14 
 
 1908 
 2079 
 2250 
 
 1925 
 2096 
 2267 
 
 1942 
 2113 
 2284 
 
 1959 
 2130 
 2300 
 
 1977 
 2147 
 
 23!7 
 
 1994 
 2164 
 2334 
 
 201 1 
 2181 
 
 235 ' 
 
 2028 
 2198 
 2368 
 
 2045 
 2215 
 2385 
 
 2062 
 2232 
 2402 
 
 3 6 9 
 369 
 
 3 6 8 
 
 11 14 
 11 14 
 11 14 
 
 2419 
 2588 
 2756 
 
 2436 
 2605 
 2773 
 
 2 453 
 2622 
 2790 
 
 2470 
 2639 
 2807 
 
 2487 
 
 2656 
 
 2823 
 
 2504 
 2672 
 2840 
 
 2521 
 2689 
 2857 
 
 2538 
 2706 
 2874 
 
 2554 
 2723 
 2890 
 
 2571 
 2740 
 2907 
 
 368 
 368 
 3 6 8 
 
 11 14 
 11 14 
 11 14 
 
 2924 
 3090 
 325 6 
 
 2940 
 
 3107 
 3272 
 
 2957 
 3 I2 3 
 3289 
 
 2974 
 3140 
 3305 
 
 2990 
 
 3156- 
 3322 
 
 3007 
 3173 
 3338 
 
 3024 
 3190 
 3355 
 
 3040 
 3206 
 337 1 
 
 3057 
 3223 
 3387 
 
 3074 
 3239 
 3404 
 
 368 
 368 
 
 3 5 8 
 
 11 14 
 11 14 
 11 14 
 
 3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 35'8 
 
 3535 
 
 355i 
 
 3567 
 
 3 5 8 
 
 11 14 
 
 3584 
 3746 
 3907 
 
 3600 
 3762 
 3923 
 
 3616 
 3778 
 3939 
 
 3633 
 3795 
 3955 
 
 3649 
 381 1 
 
 3971 
 
 3665 
 3827 
 3987 
 
 3681 
 
 3843 
 4003 
 
 3697 
 3859 
 4019 
 
 37 J 4 
 3875 
 4035 
 
 373o 
 3891 
 4051 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 11 14 
 11 14 
 11 14 
 
 4067 
 4226 
 4384 
 
 4083 
 4242 
 4399 
 
 4099 
 4258 
 4415 
 
 4"5 
 4274 
 
 443 1 
 
 4131 
 4289 
 4446 
 
 4H7 
 4305 
 4462 
 
 4163 
 4321 
 4478 
 
 4179 
 4337 
 4493 
 
 4195 
 4352 
 45°9 
 
 4210 
 4368 
 4524 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 11 13 
 11 13 
 10 13 
 
 4540 
 4695 
 4848 
 
 4555 
 4710 
 4863 
 
 457 l 
 4726 
 4879 
 
 4586 
 
 4741 
 4894 
 
 4602 
 
 4756 
 4909 
 
 4617 
 4772 
 4924 
 
 4633 
 4787 
 4939 
 
 4648 
 4802 
 4955 
 
 4664 
 4818 
 4970 
 
 4679 
 4833 
 4985 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 5000 
 
 5 01 5 
 
 5030 
 
 5°45 
 
 5060 
 
 5°75 
 
 5090 
 
 5105 
 
 5120 
 
 5i35 
 
 3 5 8 
 
 10 13 
 
 515° 
 5299 
 
 5446 
 
 5165 
 
 53H 
 5461 
 
 5180 
 5329 
 5476 
 
 5'95 
 5344 
 5490 
 
 5210 
 
 5358 
 
 55°5 
 
 5225 
 5373 
 55"9 
 
 5240 
 5388 
 5534 
 
 5255 
 5402 
 5548 
 
 5270 
 5417 
 5563 
 
 5284 
 5432 
 5577 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 10 12 
 10 12 
 10 12 
 
 5592 
 5736 
 5878 
 
 5606 
 
 575° 
 5892 
 
 5621 
 
 5764 
 5906 
 
 5 6 35 
 5779 
 592o 
 
 5650 
 5793 
 5934 
 
 5664 
 5807 
 5948 
 
 5678 
 5821 
 5962 
 
 5 6 93 
 5835 
 5976 
 
 57°7 
 5850 
 5990 
 
 5721 
 5864 
 6004 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 10 12 
 
 10 12 
 
 9 12 
 
 6018 
 6157 
 6293 
 
 6032 
 6170 
 6307 
 
 6046 
 6184 
 6320 
 
 6060 
 6198 
 6334 
 
 6074 
 621 1 
 6347 
 
 6088 
 6225 
 6361 
 
 6101 
 6239 
 6374 
 
 6115 
 6252 
 6388 
 
 6129 
 6266 
 6401 
 
 6143 
 6280 
 6414 
 
 2 5 7 
 2 5 7 
 247 
 
 9 12 
 9 " 
 9 11 
 
 6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 247 
 
 9 11 
 
 6561 
 6691 
 6820 
 
 6574 
 6704 
 
 6833 
 
 6587 
 6717 
 6845 
 
 6600 
 6730 
 6858 
 
 6613 
 
 6743 
 6871 
 
 6626 
 6756 
 6884 
 
 6639 
 6769 
 6896 
 
 6652 
 6782 
 6909 
 
 6665 
 6794 
 6921 
 
 6678 
 6807 
 6934 
 
 247 
 246 
 246 
 
 9 11 
 9 11 
 8 11 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7°34 
 
 7046 
 
 7059 
 
 246 
 
 8 10 
 
406 
 
 NATURAL SINES. 
 
 45° 
 
 46 
 47 
 48 
 
 49 
 50 
 51 
 
 52 
 53 
 54 
 
 55 
 
 56 
 57 
 58 
 
 59 
 60 
 61 
 
 63 
 63 
 64 
 
 65 
 
 66 
 67 
 68 
 
 69 
 70 
 71 
 
 72 
 73 
 74 
 
 75 
 
 76 
 
 77 
 78 
 
 79 
 80 
 81 
 
 82 
 83 
 84 
 
 85 
 
 86 
 87 
 88 
 
 89 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7'33 
 
 7i45 
 
 7i57 
 
 7169 
 
 7181 
 
 246 
 
 8 10 
 
 7*93 
 73H 
 743 « 
 
 7206 
 7325 
 7443 
 
 7218 
 7337 
 7455 
 
 7230 
 
 7349 
 7466 
 
 7242 
 7361 
 7478 
 
 7254 
 7373 
 7490 
 
 7266 
 
 7385 
 7501 
 
 72781 
 7396 
 75i3 
 
 7290 
 7408 
 7524 
 
 7302 
 7420 
 7536 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 7547 
 7660 
 7771 
 
 7558 
 7672 
 7782 
 
 757° 
 7683 
 7793 
 
 758i 
 7694 
 7804 
 
 7593 
 7705 
 
 78i5 
 
 7604 
 7716 
 7826 
 
 7615 
 7727 
 7837 
 
 7627 
 7738 
 7848 
 
 7638 
 7749 
 7859 
 
 7649 
 7760 
 7869 
 
 246 
 246 
 245 
 
 8 9 
 7 9 
 7 9 
 
 7880 
 7986 
 8090 
 
 7891 
 
 7997 
 8100 
 
 7902 
 8007 
 8m 
 
 7912 
 8018 
 8121 
 
 7923 
 8028 
 8131 
 
 7934 
 8039 
 8141 
 
 7944 
 8049 
 8151 
 
 7955 
 8059 
 8161 
 
 7965 
 8070 
 8171 
 
 7976 
 8080 
 8181 
 
 245 
 235 
 235 
 
 7 9 
 7 9 
 7 8 
 
 8192 
 
 8202 
 
 821 1 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 2 3 5 
 
 7 8 
 
 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 
 2 3 5 
 
 6 8 
 6 8 
 6 8 
 
 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 
 
 1 3 4 
 1 3 4 
 1 3 4 
 
 6 7 
 6 7 
 6 7 
 
 8829 
 8910 
 8988 
 
 8838 
 8918 
 8996 
 
 8846 
 8926 
 9003 
 
 8854 
 8934 
 901 1 
 
 8862 
 8942 
 9018 
 
 8870 
 
 8949 
 9026 
 
 8878 
 8957 
 9033 
 
 8886 
 8965 
 9041 
 
 8894 
 
 8973 
 9048 
 
 8902 
 8980 
 9056 
 
 1 3 4 
 1 3 4 
 1 3 4 
 
 5 7 
 5 6 
 
 9063 
 
 9070 
 
 9078 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 1 2 4 
 
 5 6 
 
 9135 
 9205 
 9272 
 
 9H3 
 9212 
 9278 
 
 9150 
 9219 
 9285 
 
 9157 
 9225 
 9291 
 
 9164 
 9232 
 9298 
 
 9171 
 9239 
 93°4 
 
 9178 
 9245 
 93" 
 
 9184 
 9252 
 9317 
 
 9191 
 9259 
 9323 
 
 9198 
 9265 
 933° 
 
 1 2 3 
 1 2 3 
 1 z 3 
 
 5 6 
 4 6 
 4 5 
 
 9336 
 9397 
 9455 
 
 9342 
 9403 
 9461 
 
 9348 
 9409 
 9466 
 
 9354 
 9415 
 9472 
 
 936i 
 9421 
 
 9478 
 
 9367 
 9426 
 
 9483 
 
 9373 
 9432 
 9489 
 
 9379 
 9438 
 9494 
 
 9385 
 9444 
 9500 
 
 939i 
 9449 
 9505 
 
 1 2 3 
 1 2 3 
 1 2 3 
 
 4 5 
 4 5 
 4 5 
 
 9511 
 95 6 3 
 9613 
 
 9516 
 9568 
 9617 
 
 9521 
 9573 
 9622 
 
 9527 
 9578 
 9627 
 
 9532 
 9583 
 9632 
 
 9537 
 9588 
 9636 
 
 9542 
 
 9593 
 9641 
 
 9548 
 9598 
 9646 
 
 9553 
 9603 
 9650 
 
 9558 
 9608 
 
 9655 
 
 1 2 3 
 122 
 122 
 
 4 4 
 3 4 
 3 4 
 
 9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 1 1 2 
 
 3 4 
 
 9703 
 9744 
 9781 
 
 9707 
 9748 
 9785 
 
 97" 
 9751 
 9789 
 
 9715 
 9755 
 9792 
 
 9720 
 
 9759 
 9796 
 
 9724 
 97 6 3 
 9799 
 
 9728 
 9767 
 9803 
 
 9732 
 9770 
 9806 
 
 9736 
 9774 
 9810 
 
 9740 
 9778 
 9813 
 
 112 
 112 
 112 
 
 3 3 
 3 3 
 2 3 
 
 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 
 
 112 
 
 Oil 
 Oil 
 
 2 3 
 2 2 
 2 2 
 
 9903 
 9925 
 
 9945 
 
 9905 
 9928 
 9947 
 
 9907 
 993° 
 9949 
 
 9910 
 9932 
 9951 
 
 9912 
 9934 
 9952 
 
 9914 
 9936 
 9954 
 
 9917 
 
 9938 
 9956 
 
 9919 
 9940 
 9957 
 
 9921 
 9942 
 9959 
 
 9923 
 9943 
 9960 
 
 Oil 
 Oil 
 Oil 
 
 2 2 
 1 2 
 1 1 
 
 9962 
 
 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 001 
 
 1 1 
 
 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 
 
 9985 
 9993 
 9998 
 
 001 
 000 
 000 
 
 1 1 
 1 1 
 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 rooo 
 
 nearly. 
 
 rooo 
 
 nearly. 
 
 rooo 
 
 nearly. 
 
 rooo 
 
 nearly. 
 
 rooo 
 
 nearly. 
 
 000 
 
 
 
NATURAL COSINES. 
 
 407 
 
 0° 
 
 1 
 
 2 
 3 
 
 i 
 5 
 
 
 
 7 
 8 
 9 
 
 10 
 
 11 
 12 
 13 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 20 
 
 21 
 22 
 23 
 
 24 
 25 
 26 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 
 O' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 rooo 
 
 9998 
 
 9994 
 9986 
 
 i-ooo 
 
 nearly. 
 
 1 000 
 
 nearly. 
 
 1 000 
 
 nearly. 
 
 I'OOO 
 
 nearly. 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 000 
 
 
 
 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 
 
 000 
 000 
 001 
 
 
 
 1 1 
 1 1 
 
 9976 
 9962 
 9945 
 
 9974 
 9960 
 
 9943 
 
 9973 
 9959 
 9942 
 
 9972 
 
 9957 
 9940 
 
 9971 
 
 995 6 
 9938 
 
 9969 
 9954 
 9936 
 
 9968 
 9952 
 9934 
 
 9966 
 
 995 " 
 9932 
 
 9965 
 9949 
 993° 
 
 9963 
 9947 
 9928 
 
 001 
 
 Oil 
 
 1 1 
 
 1 1 
 
 1 2 
 1 2 
 
 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 
 
 1 1 
 
 Oil 
 
 I I 
 
 2 2 
 2 2 
 2 2 
 
 9848 
 
 9845 
 
 9842 
 
 9839 
 
 9836 
 
 9833 
 
 9829 
 
 9826 
 
 9823 
 
 9820 
 
 I I 2 
 
 2 3 
 
 9816 
 9781 
 9744 
 
 9813 
 9778 
 9740 
 
 9810 
 9774 
 9736 
 
 9806 
 9770 
 9732 
 
 9803 
 9767 
 9728 
 
 9799 
 97 6 3 
 9724 
 
 9796 
 
 9759 
 9720 
 
 9792 
 9755 
 97'5 
 
 9789 
 975 « 
 97" 
 
 9785 
 9748 
 9707 
 
 I I 2 
 I I 2 
 I I 2 
 
 2 3 
 
 3 3 
 3 3 
 
 97°3 
 9659 
 96i3 
 
 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 I 2 
 I 2 2 
 12 2 
 
 3 4 
 3 4 
 3 4 
 
 95 6 3 
 95" 
 9455 
 
 9558 
 95°5 
 9449 
 
 9553 
 9500 
 
 9444 
 
 9548 
 9494 
 9438 
 
 9542 
 9489 
 9432 
 
 9537 
 9483 
 9426 
 
 9532 
 9478 
 9421 
 
 9527 
 9472 
 9415 
 
 9521 
 9466 
 9409 
 
 95 l6 
 9461 
 
 9403 
 
 I 2 3 
 
 1 2 3 
 
 I 2 3 
 
 4 4 
 4 5 
 4 5 
 
 9397 
 
 9391 
 
 9385 
 
 9379 
 
 9373 
 
 9367 
 
 9361 
 
 9354 
 
 9348 
 
 9342 
 
 I 2 3 
 
 4 5 
 
 9336 
 9272 
 9205 
 
 933° 
 9265 
 9198 
 
 93 2 3 
 9259 
 9191 
 
 93'7 
 9252 
 9184 
 
 93" 
 9245 
 9178 
 
 93°4 
 9239 
 9171 
 
 9298 
 9232 
 9164 
 
 9291 
 9225 
 9157 
 
 9285 
 9219 
 9150 
 
 9278 
 9212 
 
 9H3 
 
 I 2 3 
 I 2 3 
 I 2 3 
 
 4 5 
 
 4 6 
 
 5 6 
 
 9135 
 9063 
 8988 
 
 9128 
 9056 
 8980 
 
 9121 
 9048 
 8973 
 
 9114 
 9041 
 8965 
 
 9107 
 9033 
 8957 
 
 9100 
 9026 
 8949 
 
 9092 
 9018 
 8942 
 
 9085 
 901 1 
 8934 
 
 9078 
 9003 
 8926 
 
 9070 
 8996 
 8918 
 
 I 2 4 
 I 3 4 
 
 1 3 4 
 
 5 6 
 5 6 
 5 6 
 
 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 
 
 1 3 4 
 
 1 3 4 
 ' 3 4 
 
 5 7 
 
 6 7 
 6 7 
 
 8660 
 
 8652 
 
 8643 
 
 8634 
 
 8625 
 
 8616 
 
 8607 
 
 8599 
 
 8590 
 
 8581 
 
 ■ 3 4 
 
 6 7 
 
 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 
 2 3 5 
 2 3 5 
 
 6 8 
 6 8 
 6 8 
 
 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 
 Si 1 1 
 8007 
 
 8202 
 8100 
 7997 
 
 2 3 5 
 2 3 5 
 2 3 5 
 
 7 8 
 7 8 
 7 9 
 
 7986 
 7880 
 7771 
 
 7976 
 7869 
 7760 
 
 7965 
 7859 
 7749 
 
 7955 
 7848 
 7738 
 
 7944 
 7837 
 7727 
 
 7934 
 7826 
 7716 
 
 7923 
 7815 
 7705 
 
 7912 
 7804 
 7694 
 
 7902 
 7793 
 7683 
 
 7891 
 7782 
 7672 
 
 2 4 5 
 2 4 5 
 246 
 
 7 9 
 7 9 
 7 9 
 
 7660 
 
 7649 
 
 7638 
 
 7627 
 
 7615 
 
 7604 
 
 7593 
 
 7581 
 
 7570 
 
 7559 
 
 246 
 
 8 9 
 
 7547 
 7431 
 73H 
 
 7536 
 7420 
 7302 
 
 7524 
 7408 
 7290 
 
 75'3 
 7396 
 7278 
 
 7501 
 
 7385 
 7266 
 
 7490 
 7373 
 7254 
 
 7478 
 736i 
 7242 
 
 7466 
 
 7349 
 7230 
 
 7455 
 7337 
 7218 
 
 7443 
 7325 
 7206 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 7'93 
 
 7181 
 
 7169 
 
 7157 
 
 7'45 
 
 7'33 
 
 7120 
 
 7108 
 
 7096 
 
 7083 
 
 246 
 
 8 10 
 
 N.B. — Numbers in difference-columns to be subtracted, not added. 
 
408 
 
 NATURAL COSINES. 
 
 45° 
 
 46 
 47 
 48 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 4 5 
 
 7071 
 
 7059 
 
 7046 
 
 7°34 
 
 7022 
 
 7009 
 
 6997 
 
 6984 
 
 6972 
 
 6959 
 
 246 
 
 8 10 
 
 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 
 
 246 
 246 
 
 247 
 
 8 11 
 
 9 11 
 9 11 
 
 49 
 SO 
 51 
 
 52 
 53 
 
 54 
 
 55 
 
 56 
 57 
 58 
 
 59 
 60 
 61 
 
 62 
 63 
 
 64 
 
 65 
 
 66 
 67 
 68 
 
 69 
 70 
 71 
 
 72 
 73 
 74 
 
 75 
 
 76 
 
 77 
 78 
 
 79 
 80 
 81 
 
 82 
 83 
 84 
 
 85 
 
 86 
 87 
 88 
 
 89 
 
 6561 
 6428 
 6293 
 
 6 S47 
 6414 
 6280 
 
 6534 
 6401 
 6266 
 
 6521 
 6388 
 6252 
 
 6508 
 
 6374 
 6239 
 
 6494 
 6361 
 6225 
 
 6481 
 
 6347 
 621 1 
 
 6468 
 
 6334 
 6198 
 
 6455 
 6320 
 6184 
 
 6441 
 6307 
 6170 
 
 247 
 
 247 
 
 2 5 7 
 
 9 11 
 9 ii 
 9 11 
 
 6157 
 6018 
 5878 
 
 6143 
 6004 
 5864 
 
 6129 
 599o 
 5850 
 
 61 15 
 
 5976 
 5835 
 
 6101 
 5962 
 5821 
 
 6088 
 5948 
 5807 
 
 6074 
 5934 
 5793 
 
 6060 
 5920 
 5779 
 
 6046 
 5906 
 5764 
 
 6032 
 5892 
 575° 
 
 257 
 2 5 7 
 2 5 7 
 
 9 12 
 9 12 
 9 12 
 
 5736 
 
 5721 
 
 5707 
 
 5693 
 
 5678 
 
 5664 
 
 5650 
 
 5635 
 
 5621 
 
 5606 
 
 2 5 7 
 
 10 12 
 
 5592 
 5446 
 
 5 2 99 
 
 5577 
 5432 
 5284 
 
 5563 
 54'7 
 5270 
 
 5548 
 5402 
 
 5255. 
 
 5534 
 5388 
 5240 
 
 55 J 9 
 
 5373 
 5225 
 
 55°5 
 5358 
 5210 
 
 549° 
 5344 
 5195 
 
 5476 
 5329 
 5180 
 
 5461 
 53i4 
 5165 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 10 12 
 10 12 
 10 12 
 
 5150 
 5000 
 4848 
 
 5135 
 4985 
 4833 
 
 5120 
 4970 
 4818 
 
 5105 
 
 4955 
 4802 
 
 5090 
 4939 
 4787 
 
 5°75 
 4924 
 4772 
 
 5060 
 4909 
 4756 
 
 5045 
 4894 
 4741 
 
 5030 
 
 4879 
 4726 
 
 5015 
 4863 
 4710 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 4695 
 454° 
 4384 
 
 4679 
 4524 
 4368 
 
 4664 
 45°9 
 4352 
 
 4648 
 4493 
 
 4337 
 
 4633 
 4478 
 4321 
 
 4617 
 4462 
 
 43°5 
 
 4602 
 4446 
 4289 
 
 4586 
 4431 
 4274 
 
 457i 
 44?5 
 4258 
 
 4555 
 4399 
 4242 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 
 10 13 
 
 11 13 
 
 4226 
 
 4210 
 
 4'95 
 
 4179 
 
 4163 
 
 4147 
 
 4131 
 
 4"5 
 
 4099 
 
 4083 
 
 3 5 8 
 
 11 13 
 
 4067 
 
 3907 
 3746 
 
 4051 
 3891 
 373° 
 
 4°35 
 3875 
 37!4 
 
 4019 
 3859 
 3697 
 
 4003 
 
 3843 
 3681 
 
 3987 
 3827 
 3665 
 
 397 ' 
 38" 
 3649 
 
 3955 
 3795 
 3633 
 
 3939 
 3778 
 3616 
 
 3923 
 3762 
 3600 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 11 14 
 11 14 
 11 14 
 
 3584 
 3420 
 
 3256 
 
 3567 
 3404 
 3239 
 
 3551 
 3387 
 3223 
 
 3535 
 3371 
 3206 
 
 3518 
 
 3355 
 3190 
 
 3502 
 3338 
 3173 
 
 3486 
 3322 
 3i5 6 
 
 3469 
 33°5 
 3'4° 
 
 3453 
 3289 
 3123 
 
 3437 
 3272 
 3107 
 
 3 5 8 
 3 5 8 
 368 
 
 11 14 
 11 14 
 II 14 
 
 3090 
 2924 
 2756 
 
 3074 
 2907 
 2740 
 
 3°57 
 2890 
 
 2723 
 
 3040 
 2874 
 2706 
 
 3024 
 
 2857 
 2689 
 
 3007 
 2840 
 2672 
 
 2990 
 2823 
 2656 
 
 2974 
 2807 
 2639 
 
 2957 
 2790 
 2622 
 
 2940 
 
 2773 
 2605 
 
 368 
 368 
 3 6 8 
 
 11 14 
 11 14 
 11 14 
 
 2588 
 
 2571 
 
 2554 
 
 2538 
 
 2521 
 
 2504 
 
 2487 
 
 2470 
 
 2453 
 
 2436 
 
 3 6 8 
 
 11 14 
 
 2419 
 2250 
 2079 
 
 2402 
 2233 
 2062 
 
 2385 
 2215 
 
 2045 
 
 2368 
 2198 
 2028* 
 
 235 1 
 2181 
 201 1 
 
 2 334 
 2164 
 1994 
 
 2317 
 2147 
 1977 
 
 2300 
 2130 
 1959 
 
 2284 
 2113 
 1942 
 
 2267 
 2096 
 1925 
 
 3 6 8 
 369 
 369 
 
 11 14 
 II 14 
 11 14 
 
 1908 
 1736 
 1564 
 
 1891 
 1719 
 '547 
 
 1874 
 1702 
 1530 
 
 1857 
 1685 
 
 i5'3 
 
 1840 
 1668 
 1495 
 
 1822 
 1650 
 1478 
 
 1805 
 
 1633 
 1461 
 
 1788 
 1616 
 
 1444 
 
 1771 
 
 1599 
 1426 
 
 r 754 
 1582 
 1409 
 
 3 6 9 
 369 
 
 369 
 
 12 14 
 12 14 
 12 14 
 
 1392 
 1219 
 1045 
 
 •374 
 1 201 
 1028 
 
 1357 
 1 184 
 
 IOII 
 
 1340 
 1 167 
 
 0993 
 
 1323 
 1 149 
 0976 
 
 1305 
 1132 
 0958 
 
 1288 
 iiiS 
 0941 
 
 1271 
 1097 
 0924 
 
 1253 
 1080 
 0906 
 
 1236 
 1063 
 0889 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 0872 
 
 0854 
 
 0837 
 
 0819 
 
 0802 
 
 0785 
 
 0767 
 
 0750 
 
 0732 
 
 0715 
 
 3 6 9 
 
 12 15 
 
 0698 
 0523 
 0349 
 
 0680 
 0506 
 0332 
 
 o663 
 0488 
 °3 '4 
 
 0645 
 0471 
 0297 
 
 0628 
 0454 
 0279 
 
 0610 
 0436 
 0262 
 
 °593 
 0419 
 0244 
 
 0576 
 0401 
 0227 
 
 0558 
 0384 
 0209 
 
 0541 
 0366 
 0192 
 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 0175 
 
 OI57 
 
 0140 
 
 0122 
 
 0105 
 
 0087 
 
 0070 
 
 0052 
 
 0035 
 
 0017 
 
 3 6 9 
 
 12 IS 
 
 N.B. — Numbers in difference-columns to be subtracted, not added. 
 
NATURAL' TANGENTS. 
 
 409 
 
 0° 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 10 
 
 11 
 12 
 13 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 20 
 
 21 
 22 
 23 
 
 24 
 25 
 26 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 •0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 3 6 9 
 
 12 14 
 
 ■0175 
 ■0349 
 •0524 
 
 0192 
 0367 
 0542 
 
 0209 
 0384 
 o559 
 
 0227 
 0402 
 
 0577 
 
 0244 
 0419 
 0594 
 
 0262 
 
 0437 
 0612 
 
 0279 
 0454 
 0629 
 
 0297 
 0472 
 0647 
 
 °3'4 
 0489 
 0664 
 
 0332 
 0507 
 0682 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 •0699 
 
 •087s 
 
 •1051 
 
 0717 
 0892 
 1069 
 
 o734 
 0910 
 1086 
 
 0752 
 0928 
 1 104 
 
 0769 
 
 0945 
 1 122 
 
 0787 
 0963 
 "39 
 
 0805 
 0981 
 1157 
 
 0822 
 0998 
 1175 
 
 0840 
 1016 
 1 192 
 
 0857 
 
 i°33 
 1210 
 
 369 
 369 
 369 
 
 12 15 
 
 12 15 
 12 IS 
 
 •1228 
 ■1405 
 •1584 
 
 1246 
 
 1423 
 1602 
 
 1263 
 
 1441 
 1620 
 
 1281 
 
 '459 
 1638 
 
 1299 
 <477 
 1655 
 
 •317 
 H95 
 i673 
 
 '334 
 1512 
 1691 
 
 '352 
 1530 
 1709 
 
 '37° 
 1548 
 1727 
 
 1388 
 1566 
 
 1745 
 
 369 
 369 
 369 
 
 12 IS 
 12 15 
 12 15 
 
 ■1763 
 
 1 781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 3 6 9 
 
 12 15 
 
 •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 
 
 3 6 9 
 369 
 
 12 I 5 
 12 15 
 12 15 
 
 •2493 
 •2679 
 •2867 
 
 2512 
 2698 
 2886 
 
 2530 
 2717 
 2905 
 
 2549 
 2736 
 
 2924 
 
 2568 
 2754 
 2943 
 
 2586 
 
 2773 
 2962 
 
 2605 
 2792 
 2981 
 
 2623 
 281 1 
 3000 
 
 2642 
 2830 
 3019 
 
 2661 
 2849 
 3038 
 
 369 
 369 
 369 
 
 12 16 
 
 13 16 
 13 16 
 
 ■3057 
 •3249 
 ■3443 
 
 3076 
 3269 
 3463 
 
 3096 
 3288 
 3482 
 
 3"5 
 3307 
 3502 
 
 3134 
 3327 
 3522 
 
 3153 
 3346 
 3541 
 
 3172 
 3365 
 35 61 
 
 3191 
 
 3385 
 358i 
 
 321 1 
 
 3404 
 3600 
 
 323° 
 3424 
 3620 
 
 3 6 10 
 3 6 10 
 3 6 10 
 
 13 16 
 13 16 
 13 17 
 
 ■3640 
 
 3659 
 
 3679 
 
 3699 
 
 37'9 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3 7 10 
 
 13 17 
 
 ■3839 
 
 •4040 
 
 •4245 
 
 3859 
 4061 
 4265 
 
 3879 
 4081 
 4286 
 
 3899 
 4101 
 
 4307 
 
 3919 
 4122 
 
 4327 
 
 3939 
 4142 
 
 4348 
 
 3959 
 4163 
 
 4369 
 
 3979 
 4183 
 439° 
 
 4000 
 4204 
 4411 
 
 4020 
 4224 
 4431 
 
 3 7 IO 
 3 7 10 
 3 7 IO 
 
 >3 <7 
 
 14 17 
 14 17 
 
 ■4452 
 ■4663 
 
 •4877 
 
 4473 
 4684 
 4899 
 
 4494 
 4706 
 4921 
 
 45»5 
 4727 
 
 4942 
 
 4536 
 4748 
 4964 
 
 4557 
 4770 
 4986 
 
 4578 
 
 479' 
 5008 
 
 4599 
 4813 
 5029 
 
 4621 
 4834 
 5°5" 
 
 4642 
 4856 
 5°73 
 
 4 7 10 
 4 7 11 
 4 7 11 
 
 14 18 
 
 14 18 
 
 15 18 
 
 •5095 
 •5317 
 
 ■5543 
 
 5117 
 5340 
 5566 
 
 5139 
 5362 
 5589 
 
 5161 
 
 5384 
 5612 
 
 5184 
 5407 
 5635 
 
 5206 
 5430 
 5658 
 
 5228 
 5452 
 5681 
 
 5250 
 5475 
 57°4 
 
 5272 
 5498 
 5727 
 
 5295 
 552o 
 575° 
 
 4 7 n 
 4 8 11 
 4 8 12 
 
 15 18 
 15 19 
 15 19 
 
 ■5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 59'4 
 
 5938 
 
 596i 
 
 5985 
 
 4 8 12 
 
 16 20 
 
 •6009 
 ■6249 
 ■6494 
 
 6032 
 6273 
 6519 
 
 6056 
 6297 
 6544 
 
 6080 
 6322 
 6569 
 
 6104 
 6346 
 6594 
 
 6128 
 
 6371 
 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 
 
 17 21 
 
 ■6745 
 •7002 
 •7265 
 
 6771 
 7028 
 7292 
 
 6796 
 7°54 
 7319 
 
 6822 
 7080 
 7346 
 
 6847 
 7107 
 
 7373 
 
 6873 
 7'33 
 7400 
 
 6899 
 7159 
 7427 
 
 6924 
 7186 
 7454 
 
 6950 
 7212 
 748i 
 
 6976 
 7239 
 75°8 
 
 4 9 '3 
 
 4 9 13 
 
 5 9 14 
 
 17 21 
 
 18 22 
 18 23 
 
 7536 
 7813 
 •8098 
 
 7563 
 7841 
 8127 
 
 759° 
 7869 
 8156 
 
 7618 
 7898 
 8185 
 
 7646 
 7926 
 8214 
 
 7673 
 7954 
 8243 
 
 7701 
 7983 
 8273 
 
 77?9 
 8oi2 
 8302 
 
 7757 
 8040 
 
 8332 
 
 7785 
 8069 
 8361 
 
 5 9 14 
 5 IO '4 
 5 IO '5 
 
 18 23 
 
 19 24 
 
 20 24 
 
 •8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 5 10 '5 
 
 20 25 
 
 ■8693 
 •9004 
 
 •9325 
 
 8724 
 9036 
 9358 
 
 8754 
 9067 
 
 9391 
 
 8785 
 9099 
 9424 
 
 8816 
 9131 
 
 9457 
 
 8847 
 9163 
 9490 
 
 8878 
 9195 
 9523 
 
 8910 
 9228 
 9556 
 
 8941 
 9260 
 959o 
 
 8972 
 9293 
 9623 
 
 5 10 16 
 
 5 11 16 
 
 6 11 17 
 
 21 26 
 
 21 27 
 
 22 28 
 
 •9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 993o 
 
 9965 
 
 6 11 17 
 
 23 29 
 
4io 
 
 NATURAL TANGENTS. 
 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 45° 
 
 46 
 
 47 
 48 
 
 49 
 50 
 51 
 
 52 
 53 
 54 
 
 I'OOOO 
 
 0035 
 
 0070 
 
 0105 
 
 0141 
 
 0176 
 
 0212 
 
 0247 
 
 0283 
 
 0319 
 
 6 12 18 
 
 24 30 
 
 i-°35S 
 10724 
 i- 1 106 
 
 0392 
 0761 
 1 145 
 
 0428 
 0799 
 1 184 
 
 0464 
 
 0837 
 1224 
 
 0501 
 
 0875 
 1263 
 
 0538 
 0913 
 
 1303 
 
 °575 
 0951 
 
 1343 
 
 0612 
 0990 
 1383 
 
 0649 
 1028 
 1423 
 
 0686 
 1067 
 1463 
 
 6 12 18 
 
 6 13 19 
 
 7 '3 20 
 
 25 3' 
 
 25 32 
 
 26 33 
 
 1-1504 
 1-1918 
 1-2349 
 
 1544 
 i960 
 
 2393 
 
 1585 
 2002 
 2437 
 
 1626 
 
 2045 
 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 
 
 1-2799 
 1-3270 
 13764 
 
 2846 
 3319 
 3814 
 
 2892 
 3367 
 3865 
 
 2938 
 34i6 
 39i6 
 
 2985 
 
 3465 
 3968 
 
 3032 
 35 '4 
 4019 
 
 3079 
 3564 
 4071 
 
 3127 
 3613 
 4124 
 
 3i75 
 3663 
 4176 
 
 3222 
 
 37i3 
 4229 
 
 8 16 23 
 
 8 16 25 
 
 9 17 26 
 
 3« 39 
 
 33 4i 
 
 34 43 
 
 55 
 
 56 
 57 
 58 
 
 59 
 60 
 61 
 
 62 
 63 
 
 64 
 
 65 
 
 66 
 67 
 68 
 
 69 
 70 
 71 
 
 72 
 73 
 
 74 
 
 75 
 
 76 
 77 
 78 
 79 
 80 
 81 
 
 82 
 83 
 84 
 
 85 
 
 86 
 87 
 88 
 
 89 
 
 1 -428 1 
 
 4335 
 
 4388 
 
 4442 
 
 4496 
 
 455° 
 
 4605 
 
 4659 
 
 47i5 
 
 4770 
 
 9 18 27 
 
 36 45 
 
 1-4826 
 
 1-5399 
 1-6003 
 
 4882 
 5458 
 6066 
 
 4938 
 5517 
 6128 
 
 4994 
 5577 
 6191 
 
 5051 
 5637 
 6255 
 
 5108 
 5697 
 6319 
 
 5166 
 5757 
 6383 
 
 5224 
 5818 
 6447 
 
 5282 
 5880 
 6512 
 
 534° 
 594i 
 6577 
 
 10 19 29 
 
 10 20 30 
 
 11 21 32 
 
 38 48 
 40 50 
 
 43 53 
 
 1-6643 
 17321 
 1 8040 
 
 6709 
 
 7391 
 8115 
 
 6775 
 7461 
 8190 
 
 6842 
 
 7532 
 8265 
 
 6909 
 7603 
 8341 
 
 6977 
 7675 
 8418 
 
 7045 
 7747 
 8495 
 
 7"3 
 7820 
 
 8572 
 
 7182 
 
 7893 
 8650 
 
 7251 
 7966 
 8728 
 
 " 23 34 
 
 12 24 36 
 
 13 26 38 
 
 45 56 
 48 60 
 51 64 
 
 1-8807 
 1-9626 
 20503 
 
 8887 
 971 1 
 0594 
 
 8967 
 
 9797 
 0686 
 
 9047 
 
 9883 
 0778 
 
 9128 
 
 9970 
 0872 
 
 9210 
 
 o°57 
 0965 
 
 9292 
 0145 
 1060 
 
 9375 
 0233 
 "55 
 
 9458 
 0323 
 125 1 
 
 9542 
 
 0413 
 1348 
 
 14 27 41 
 
 15 29 44 
 
 16 31 47 
 
 55 68 
 58 73 
 63 78 
 
 21445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2148 
 
 2251 
 
 2355 
 
 17 34 5 1 
 
 68 85 
 
 22460 
 
 2-3559 
 2-4751 
 
 2566 
 3673 
 4876 
 
 2673 
 
 3789 
 5002 
 
 2781 
 3906 
 5129 
 
 2889 
 4023 
 5257 
 
 2998 
 4142 
 5386 
 
 3109 
 4262 
 5517 
 
 3220 
 
 4383 
 5649 
 
 3332 
 45°4 
 5782 
 
 3445 
 4627 
 5916 
 
 18 37 55 
 20 40 60 
 22 43 65 
 
 '74 92 
 79 99 
 87 108 
 
 2-6051 
 2-7475 
 2-9042 
 
 6187 
 7625 
 9208 
 
 6325 
 7776 
 
 9375 
 
 6464 
 7929 
 
 9544 
 
 6605 
 8083 
 
 97'4 
 
 6746 
 8239 
 9887 
 
 6889 
 8397 
 0061 
 
 7°34 
 8556 
 
 0237 
 
 7179 
 8716 
 0415 
 
 7326 
 8878 
 
 0595 
 
 24 47 7 1 
 26 52 78 
 
 29 58 87 
 
 95 "8 
 104 130 
 
 115 144 
 
 30777 
 3-2709 
 
 J4874 
 
 0961 
 2914 
 5105 
 
 1 146 
 3122 
 
 5339 
 
 1334 
 3332 
 
 5576 
 
 1524 
 
 3544 
 5816 
 
 1716 
 
 3759 
 6059 
 
 1910 
 
 3977 
 6305 
 
 2106 
 
 4197 
 6554 
 
 2305 
 4420 
 6806 
 
 2506 
 4646 
 7062 
 
 32 64 96 
 36 72 108 
 41 82 122 
 
 129 161 
 144 180 
 162 203 
 
 37321 
 
 7583 
 
 7848 
 
 8118 
 
 8391 
 
 8667 
 
 8947 
 
 9232 
 
 ,9520 
 
 9812 
 
 46 94 139 
 
 186 232 
 
 4-0108 
 4-33I5 
 47046 
 
 0408 
 3662 
 
 7453 
 
 0713 
 4015 
 
 7867 
 
 1022 
 4374 
 8288 
 
 1335 
 4737 
 8716 
 
 1653 
 5 io 7 
 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 
 
 5- 1446 
 5 -6 7'3 
 63138 
 
 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. 
 
 7 - "54 
 8-1443 
 95 '44 
 
 2066 
 2636 
 9-677 
 
 3002 
 
 3863 
 9-845 
 
 3962 
 5126 
 
 10-02 
 
 4947 
 6427 
 
 IO - 20 
 
 5958 
 7769 
 1039 
 
 6996 
 
 9152 
 
 10-58 
 
 8062 
 
 °579 
 10-78 
 
 9158 
 2052 
 1099 
 
 0285 
 3572 
 
 II20 
 
 II-43 
 
 n-66 
 
 11-91 
 
 I2 - l6 
 
 12-43 
 
 12-71 
 
 1300 
 
 1330 
 
 13-62 
 
 I3-95 
 
 14-30 
 19-08 
 28-64 
 
 1467 
 I9-74 
 3014 
 
 15-06 
 20-45 
 31-82 
 
 15-46 
 21-20 
 33-69 
 
 15-89 
 22-02 
 
 35-80 
 
 i6-35 
 22-90 
 
 38-19 
 
 16-83 
 23-86 
 40-92 
 
 I7-34 
 24-90 
 44-07 
 
 17-89 
 2603 
 47-74 
 
 18-46 
 27-27 
 52-08 
 
 5729 |63-66 
 
 71-62 
 
 8r85 
 
 95'49 
 
 1 146 
 
 143-2 
 
 1910 
 
 286-5 
 
 573-Q 
 
 
 
NATURAL COTANGENTS. 
 
 411 
 
 0° 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 12 
 13 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 
 
 Inf. 
 
 573-o 
 
 286-5 
 
 191-0 
 
 143-2 
 
 114-6 
 
 95-49 
 
 81-85 
 
 71-62 
 
 63-66 
 
 Difference-columns 
 not useful here, owing 
 to the rapidity with 
 which the value of the 
 cotangent changes. 
 
 57-29 
 28-64 
 19-08 
 
 52-08 
 27-27 
 1846 
 
 47-74 
 2603 
 
 17-89 
 
 44-07 
 24-90 
 I7-34 
 
 40-92 
 23-86 
 16-83 
 
 38-19 
 22-90 
 
 16-35 
 
 35-80 
 
 22-02 
 15-89 
 
 3369 
 
 21-20 
 15-46 
 
 31-82 
 
 20-45 
 15-06 
 
 3014 
 
 I9-74 
 1467 
 
 1430 
 "■43 
 9-5J44 
 
 I3-95 
 
 II-20 
 
 3572 
 
 13-62 
 1099 
 2052 
 
 !3-3° 
 10-78 
 
 0579 
 
 1300 
 10-58 
 9152 
 
 1271 
 1039 
 7769 
 
 12-43 
 
 IO-20 
 6427 
 
 I2l6 
 I002 
 5126 
 
 11-91 
 
 9-845 
 3863 
 
 u-66 
 9-677 
 2636 
 
 8'i443 
 7'"54 
 63138 
 
 O285 
 O264 
 2432 
 
 9158 
 9395 
 1742 
 
 8062 
 8548 
 1066 
 
 6996 
 7920 
 0405 
 
 5958 
 6912 
 9758 
 
 4947 
 
 6l22 
 9124 
 
 3962 
 
 535° 
 8502 
 
 3002 
 4596 
 7894 
 
 2066 
 
 3859 
 7297 
 
 5-6713 
 
 614O 
 
 5578 
 
 5026 
 
 4486 
 
 3955 
 
 3435 
 
 2924 
 
 2422 
 
 1929 
 
 12 3 
 
 4 5 
 
 5-1446 
 47046 
 4-33I5 
 
 O97O 
 6646 
 2972 
 
 ,0504 
 6252 
 2635 
 
 0045 
 5864 
 2303 
 
 9594 
 5483 
 1976 
 
 9152 
 5107 
 1653 
 
 8716 
 4737 
 1335 
 
 8288 
 
 4374 
 1022 
 
 7867 
 4015 
 °7!3 
 
 7453 
 3662 
 0408 
 
 74 148 222 
 63 125 188 
 53 107 160 
 
 296 37° 
 252 3'4 
 214 267 
 
 4-0108 
 37321 
 
 3-4874 
 
 9812 
 7062 
 4646 
 
 9520 
 6806 
 4420 
 
 9232 
 
 6554 
 4197 
 
 8947 
 6305 
 3977 
 
 8667 
 6059 
 3759 
 
 8391 
 5816 
 
 3544 
 
 8118 
 
 5576 
 3332 
 
 7848 
 5339 
 3122 
 
 7583 
 5105 
 2914 
 
 46 93 139 
 41 82 122 
 36 72 108 
 
 186 232 
 163 204 
 144 180 
 
 32709 
 
 3-0777 
 2-9042 
 
 2506 
 
 °595 
 8878 
 
 2305 
 0415 
 8716 
 
 2106 
 0237 
 8556 
 
 1910 
 0061 
 8397 
 
 1716 
 9887 
 8239 
 
 1524 
 
 97 H 
 8083 
 
 £334 
 9544 
 7929 
 
 1 146 
 
 9375 
 7776 
 
 0961 
 9208 
 7625 
 
 32 64 96 
 
 29 58 87 
 26 52 78 
 
 129 161 
 115 144 
 104 130 
 
 20 
 
 21 
 22 
 23 
 
 ~24~ 
 25 
 26 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 
 2-7475 
 
 7326 
 
 7179 
 
 7034 
 
 6889 
 
 6746 
 
 6605 
 
 6464 
 
 6325 
 
 6187 
 
 24 47 71 
 
 95 118 
 
 2-605 1 
 
 2-4751 
 2-3559 
 
 5916 
 4627 
 
 3445 
 
 5782 
 4504 
 3332 
 
 5649 
 4383 
 3220 
 
 55"7 
 4262 
 3109 
 
 5386 
 4142 
 2998 
 
 5 2 57 
 4023 
 2889 
 
 5129 
 3906 
 2781 
 
 5002 
 3789 
 2673 
 
 4876 
 
 3673 
 2566 
 
 22 43 65 
 20 40 60 
 '8 37 55 
 
 87 108 
 79 99 
 74 92 
 
 2-2460 
 2-1445 
 20503 
 
 2355 
 1348 
 0413 
 
 2251 
 1251 
 
 0323 
 
 2148 
 "55 
 0233 
 
 2045 
 1060 
 
 oi45 
 
 '943 
 0965 
 
 0057 
 
 1842 
 0872 
 9970 
 
 1742 
 0778 
 
 9883 
 
 1642 
 0686 
 
 9797 
 
 1543 
 0594 
 
 9711 
 
 17 34 51 
 '6 31 47 
 15 29 44 
 
 68 85 
 63 78 
 58 73 
 
 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 
 7461 
 
 8887 
 8115 
 7391 
 
 14 27 41 
 i3 26 38 
 12 24 36 
 
 55 68 
 5 1 64 
 48 60 
 
 1 7321 
 
 7251 
 
 7182 
 
 7"3 
 
 7°45 
 
 6977 
 
 6909 
 
 6842 
 
 6775 
 
 6709 
 
 'i 23 34 
 
 45 56 
 
 1-6643 
 1-6003 
 1-5399 
 
 6577 
 5941 
 534° 
 
 6512 
 5880 
 5282 
 
 6447 
 5818 
 
 5224 
 
 6383 
 5757 
 5166 
 
 6319 
 
 5697 
 5108 
 
 6255 
 5637 
 5051 
 
 6191 
 
 5577 
 4994 
 
 6128 
 5517 
 4938 
 
 6066 
 5458 
 4882 
 
 11 21 32 
 10 20 30 
 10 19 29 
 
 43 53 
 40 50 
 
 38 48 
 
 1-4826 
 1-4281 
 I-3764 
 
 4770 
 4229 
 3713 
 
 4715 
 4176 
 
 3663 
 
 4659 
 4124 
 
 3613 
 
 4605 
 4071 
 3564 
 
 455° 
 4019 
 
 35M 
 
 4496 
 3968 
 3465 
 
 4442 
 39i6 
 34i6 
 
 4388 
 3865 
 3367 
 
 4335 
 3814 
 3319 
 
 9 18 27 
 9 17 26 
 8 16 25 
 
 36 45 
 34 43 
 33 4' 
 
 1-3270 
 1-2799 
 1-2349 
 
 3222 
 
 2753 
 2305 
 
 3175 
 2708 
 2261 
 
 3127 
 2662 
 2218 
 
 3079 
 2617 
 2174 
 
 3°3 2 
 2572 
 2131 
 
 2985 
 2527 
 2088 
 
 2938 
 2482 
 2045 
 
 2892 
 
 2437 
 2002 
 
 2846 
 
 2393 
 i960 
 
 8 16 23 
 
 8 15 23 
 7 14 22 
 
 3i 39 
 30 38 
 29 36 
 
 1-1918 
 
 1875 
 
 1833 
 
 1792 
 
 1750 
 
 1708 
 
 1667 
 
 1626 
 
 1585 
 
 1544 
 
 7 '4 21 
 
 28 34 
 
 1-1504 
 1-1106 
 1-0724 
 
 1463 
 1067 
 0686 
 
 1423 
 1028 
 0649 
 
 1383 
 0990 
 0612 
 
 '343 
 
 °95 1 
 °575 
 
 1303 
 0913 
 
 0538 
 
 1263 
 0875 
 0501 
 
 1224 
 
 0837 
 0464 
 
 1 184 
 0799 
 0428 
 
 1 145 
 0761 
 0392 
 
 7 13 20 
 6 13 19 
 6 12 18 
 
 26 33 
 25 32 
 25 31 
 
 i-°355 
 
 0319 
 
 0283 
 
 0247 
 
 0212 
 
 0176 
 
 0141 
 
 0105 
 
 0070 
 
 0035 
 
 6 12 18 
 
 24 30 
 
 JV.B. — Numbers in difference-columns to be subtracted, not added. 
 
412 
 
 NATURAL COTANGENTS. 
 
 45° 
 
 46 
 47 
 48 
 
 49 
 SO 
 SI 
 
 52 
 53 
 54 
 
 55 
 
 56 
 57 
 58 
 
 59 
 60 
 61 
 
 62 
 63 
 64 
 
 65 
 
 66 
 67 
 68 
 
 69 
 70 
 71 
 
 72 
 73 
 74 
 
 75 
 
 76 
 
 77 
 78 
 
 79 
 80 
 81 
 
 82 
 
 83 
 84 
 
 85 
 
 86 
 87 
 88 
 
 89 
 
 0' 
 
 6' 
 
 12 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 ro 
 
 0-9965 
 
 0-9930 
 
 0-9896 
 
 0-9861 
 
 0-9827 
 
 0-9793 
 
 °"97S9 
 
 0-9725 
 
 C969 1 
 
 6 11 17 
 
 23 29 
 
 9657 
 •9325 
 ■9004 
 
 9623 
 
 9293 
 8972 
 
 9590 
 9260 
 8941 
 
 955° 
 9228 
 8910 
 
 9523 
 9195 
 8878 
 
 9490 
 9163 
 8847 
 
 9457 
 9131 
 8816 
 
 9424 
 9099 
 8785 
 
 939i 
 9067 
 
 8754 
 
 9358 
 9036 
 8724 
 
 6 11 17 
 5 11 16 
 5 10 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 
 
 85" 
 
 8214 
 7926 
 
 8481 
 8185 
 7898 
 
 8451 
 8156 
 7869 
 
 8421 
 8127 
 7841 
 
 5 >° "5 
 5 1° 15 
 5 IO M 
 
 20 25 
 20 24 
 19 24 
 
 7813 
 7536 
 7265 
 
 7785 
 7508 
 7239 
 
 7757 
 7481 
 7212 
 
 7729 
 7454 
 7186 
 
 7701 
 7427 
 7i59 
 
 7673 
 7400 
 
 7'33 
 
 7646 
 
 7373 
 7107 
 
 7618 
 
 7346 
 7080 
 
 759° 
 7319 
 7°54 
 
 75°3 
 7292 
 7028 
 
 5 9 14 
 5 9 H 
 4 9 '3 
 
 18 23 
 18 23 
 18 22 
 
 •7002 
 
 6976 
 
 6950 
 
 6924 
 
 6899 
 
 6873 
 
 6847 
 
 6822 
 
 6796 
 
 6771 
 
 4 9 13 
 
 17 21 
 
 •6745 
 •6494 
 ■6249 
 
 6720 
 6469 
 6224 
 
 6694 
 
 6445 
 6200 
 
 6669 
 6420 
 6176 
 
 6644 
 
 6395 
 6152 
 
 6619 
 
 637 1 
 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 
 
 ■6009 
 •5774 
 •5543 
 
 598s 
 575° 
 5520 
 
 596i 
 5727 
 5498 
 
 5938 
 5704 
 5475 
 
 59H 
 5681 
 
 5452 
 
 5890 
 5658 
 543° 
 
 5867 
 5 6 35 
 54°7 
 
 5844 
 5612 
 5384 
 
 5820 
 5589 
 5302 
 
 5797 
 5566 
 534° 
 
 4 8 12 
 4 8 12 
 4 8 11 
 
 16 20 
 15 19 
 15 19 
 
 •53'7 
 •509S 
 •4877 
 
 5295 
 5073 
 4856 
 
 5272 
 
 5°5i 
 4834 
 
 5250 
 5029 
 4813 
 
 5228 
 5008 
 479i 
 
 5206 
 4986 
 477° 
 
 5184 
 4964 
 4748 
 
 5161 
 
 4942 
 4727 
 
 5*39 
 4921 
 4706 
 
 5"7 
 4899 
 4684 
 
 4 7 " 
 4 7 11 
 4 7 11 
 
 iS 18 
 15 18 
 14 18 
 
 •4663 
 
 4642 
 
 4621 
 
 4599 
 
 4578 
 
 4557 
 
 453° 
 
 4515 
 
 4494 
 
 4473 
 
 4 7 10 
 
 14 18 
 
 •445 2 
 ■4245 
 ■4040 
 
 443 1 
 4224 
 4020 
 
 441 1 
 4204 
 4000 
 
 439° 
 4183 
 3979 
 
 4369 
 4163 
 3959 
 
 4348 
 4142 
 
 3939 
 
 4327 
 4122 
 
 39'9 
 
 43°7 
 4101 
 
 3899 
 
 4286 
 4081 
 3879 
 
 4265 
 4061 
 3859 
 
 3 7 IO 
 3 7 IO 
 3 7 1° 
 
 14 17 
 14 17 
 13 17 
 
 3839 
 •3640 
 
 3443 
 
 3819 
 3620 
 
 3424 
 
 3799 
 3600 
 
 3404 
 
 3779 
 358i 
 3385 
 
 3759 
 35 61 
 3365 
 
 3739 
 3541 
 334° 
 
 3719 
 3522 
 3327 
 
 3699 
 35° 2 
 33°7 
 
 3679 
 3482 
 3288 
 
 3659 
 3463 
 3269 
 
 3 7i° 
 3 6 10 
 3 6 10 
 
 13 17 
 13 17 
 13 16 
 
 •3249 
 ■3057 
 •2867 
 
 3230 
 3038 
 2849 
 
 321 1 
 3019 
 2830 
 
 3i9» 
 3000 
 281 1 
 
 3172 
 2981 
 2792 
 
 3153 
 2962 
 
 2773 
 
 3134 
 2943 
 2754 
 
 3"5 
 2924 
 2736 
 
 3096 
 2905 
 2717 
 
 3076 
 2886 
 2698 
 
 3 6 10 
 3 6 9 
 3 6 9 
 
 13 16 
 13 16 
 13 16 
 
 •2679 
 
 2661 
 
 2642 
 
 2623 
 
 2605 
 
 2586 
 
 2568 
 
 2549 
 
 253° 
 
 2512 
 
 3 6 9 
 
 12 16 
 
 •2493 
 ■2309 
 •2126 
 
 2475 
 2290 
 2107 
 
 2456 
 2272 
 2089 
 
 2438 
 2254 
 2071 
 
 2419 
 
 2235 
 2053 
 
 2401 
 2217 
 2°3J 
 
 2382 
 2199 
 2016 
 
 2364 
 2180 
 1998 
 
 2345 
 2162 
 1980 
 
 2327 
 2144 
 1962 
 
 3 6 9 
 3i 6 9 
 3 6 9 
 
 12 15 
 
 12 15 
 12 15 
 
 •1944 
 
 •763 
 ■1584 
 
 1926 
 
 1745 
 1566 
 
 1908 
 1727 
 1548 
 
 1890 
 1709 
 1530 
 
 1871 
 1691 
 1512 
 
 1853 
 •°73 
 1495 
 
 1835 
 '655 
 >477 
 
 1817 
 1638 
 H59 
 
 1799 
 1620 
 1441 
 
 1781 
 1602 
 1423 
 
 3 6 9 
 3 ° 9 
 3^9 
 
 12 15 
 12 15 
 12 I 5 
 
 •1405 
 1228 
 •105 1 
 
 1388 
 1210 
 1033 
 
 1370 
 1192 
 1016 
 
 "35 2 
 "75 
 0998 
 
 1334 
 "57 
 0981 
 
 1317 
 "39 
 0963 
 
 1299 
 1122 
 
 0945 
 
 1281 
 1 104 
 0928 
 
 1263 
 1086 
 0910 
 
 1246 
 1069 
 0892 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 ■0875 
 
 0857 
 
 0840 
 
 0822 
 
 0805 
 
 0787 
 
 0769 
 
 0752 
 
 °734 
 
 0717 
 
 3 6 9 
 
 12 15 
 
 •0699 
 •0524 
 0349 
 
 0682 
 0507 
 0332 
 
 0664 
 0489 
 0314 
 
 0647 
 0472 
 0297 
 
 0629 
 
 0454 
 0279 
 
 0612 
 
 °437 
 0262 
 
 0594 
 0419 
 0244 
 
 0577 
 0402 
 0227 
 
 °559 
 0384 
 0209 
 
 0542 
 0367 
 0192 
 
 3 6 9 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 •0175 
 
 0157 
 
 0140 
 
 0122 
 
 0105 
 
 0087 
 
 0070 
 
 0052 
 
 0035 
 
 0017 
 
 3 6 9 
 
 12 14 
 
 N.B. — Numbers in difference-columns to be subtracted, not added. 
 
INDEX 
 
 Absorbing power for heat radiation, 148. 
 
 Acceleration, angular, 78 ; linear, 78. 
 
 Acceleration of gravity, by Atwood's ma- 
 chine, 73 ; by freely falling body, 75 ; by 
 physical pendulum, 92; by Kater's pen- 
 dulum, 96; value of, at Cornell Labora- 
 tory, 95- 
 
 Air displacement, correction for, 117. 
 
 Ammeter calibration, 267. 
 
 Ampere, definition of, 236. 
 
 Amplitude of S. H. M., 87. 
 
 Approximations in computing, 7. 
 
 Assignments of weights, 20. 
 
 Atwood's machine, 70 ; gravity by, 73. 
 
 Balance, 42. 
 
 Ballistic galvanometer, 338 ; constant of 
 d Arsonval, 339 ; constant of tangent, 340. 
 
 Barometer, cistern, 127; siphon, 127; com- 
 parison of, 127. 
 
 Battery resistance, measurement of, by 
 Beetz's method, 290; Benton's method, 
 330; fall of potential method for constant 
 E. M. F. cells, 328 ; for open circuit cells, 
 329; Mance's method, 324; Mance's 
 method as modified by Lodge and by 
 Guthe, 326; Ohm's method, 321; Thom- 
 son's method, 323. 
 
 Beetz's method for measuring E. M. F. and 
 resistance of a battery, 290. 
 
 Benton's method of measuring battery re- 
 sistance, 330. 
 
 *' Bound " electricity, 203. 
 
 Boyle's law, example of, 12; verification of, 
 124. 
 
 Bunsen photometer, 183. 
 
 Calibration, of an ammeter, 267; of a slide 
 wire, 310; of a thermometer tube, 32; 
 of a voltmeter, 296 ; of weights, 52. 
 
 Calorimetry, general statements regarding, 
 
 133- 
 Candle power, distribution of, by Bunsen 
 
 and Lummer-Brodhun photometers, 188 ; 
 
 standards, 186 ; variation of, with voltage, 
 
 190. 
 Capacity, 346 ; measurement of, in absolute 
 
 terms, 351; by comparison, ballistic 
 
 method, 346 ; bridge method, 349. 
 
 Cells, E. M. F. of, by Beetz's method, 290; 
 by Ohm's method, 276; comparison of 
 E. M. F.'s by Lord Rayleigh's method, 
 299; by Poggendorf's method, 298; by 
 potentiometer method, 300 ; arrangement 
 of, giving maximum current, 263; resist- 
 ance (see Battery resistance) ; Standard 
 Daniell, 261. 
 
 Coefficients, cubical expansion, 155 ; friction, 
 63; frictional torque, 63; linear expan- 
 sion, 154; mutual induction, 363; self- 
 inductjon, 365. 
 
 Coercivity, 375. 
 
 Collimator, 168. 
 
 Commutator or reversing "switch, 239. 
 
 Comparison, of barometers, 127; of ther- 
 mometers, 134; of E. M. F.'s (see Cells, 
 E.M. F. of). 
 
 Computations, 5. 
 
 Concave mirror, focal length of, 161 ; radius 
 of curvature of, 162; center of curvature, 
 162. 
 
 Condenser, capacity of, 346, 351; capacity 
 in multiple, 348 ; in series, 348 ; principle 
 of, 212; variable, 213. 
 
 Conditions, choice of experimental, 4. 
 
 Conductivity, 303. 
 
 Conjugate foci, for convex mirror, 159 ; for 
 concave mirror, 161 ; for convex lens, 163. 
 
 Constant of galvanometer, ballistic, 340; for 
 d'Arsonval, working, 244 ; d'Arsonval 
 ballistic, 339 ; for tangent, true, 236 ; tan- 
 gent, working, 237. 
 
 Convex lens, focal length, 162; radius of 
 curvature by reflection, 158 ; by sphe- 
 rometer, 29. 
 
 Copper voltameter, 250. 
 
 Counter electromotive force, 270. 
 
 Current electricity, general statements, 234; 
 lines of flow, 292; measurement by elec- 
 trolysis, 250 ; by magnetic effect, 245, 248 ; 
 by Vienna method, 265 ; unit, 236. 
 
 Curvature, radius of, by reflection, 158 ; by 
 spherometer, 29; of concave mirror, 161. 
 
 Curve plotting, 9, 24. 
 
 Damping, of balance, 47 ; of vibrations, 91 ; 
 of galvanometer needle, 341 ; theory of, 
 342; ratio of, 343. 
 
 413 
 
414 
 
 INDEX 
 
 Daniell cell, standard, 261. 
 
 d'Arsonval galvanometer, constant of, 248; 
 
 theory of, 242 ; used ballistically, 339. 
 Decrement, logarithmic, 342. 
 Demagnetizing effect of poles of a magnet, 
 
 376- 
 Density, general statements, 115; from 
 
 mass and dimensions, 35; definition, 
 
 115 ; with correction for temperature and 
 
 air displacement, 117 ; by specific gravity 
 
 bottle, 116; of liquids, by Hare's method, 
 
 122. 
 Deviation, minimum, of light through a 
 
 prism, 173. 
 Difference of potential (see Potential) . 
 Differential pulley, 69. 
 Diffraction, grating, 179; theory of, 179. 
 Dip of earth's magnetic field, 356. 
 Distance traversed with linear velocity, 72, 
 
 77- 
 Distribution, of light, horizontal, 188 ; of free 
 
 magnetism, 232, 259. 
 Double weighing, 49. 
 
 Earth inductor, 356. 
 
 Efficiency, curve, 68; of pulley system, 68; 
 of wheel and axle, &/. 
 
 Elasticity, 101, 104, 107, 109. 
 
 Electric, charge, bound, 203; free, 203; po- 
 tential, 204, 205, 206; lines of force, 205, 
 
 Electrical machine, 215, 217. 
 
 Electrical quantity, general statements, 210, 
 
 337- 
 
 Electrification, energy of, 202. 
 
 Electrolysis, measurement of current by, 236, 
 250. 
 
 Electrolytes, resistance of, 320. 
 
 Electromagnetic induction, 352; mutual, 
 360; relf, 356, 364. 
 
 Electromagnetic, unit of current, 236; unit 
 of E. M. F. and potential difference, 269. 
 
 Electromotive force, general statements, 268 ; 
 counter, 270; external, 274; impressed, 
 274; of a thermo-element, 294; total, 
 274; comparison of, by Lord Rayleigh's 
 method, 299; by Poggendorfs method, 
 298 ; by potentiometer method, 300 ; 
 measurement of, by Beetz's method, 290 ; 
 by Ohm's method, 276; unit of, 269. 
 
 Electroscope, 209. 
 
 Electrostatic, lines of force, 205 ; potential, 
 206; induction, 210; 
 
 Emissivity, 148. 
 
 Equipotential, lines, «o5; surfaces, 204, 205. 
 
 Errors, accidental, 17; constant, 20; in- 
 fluence of, 21 ; probable, 18 ; sources of, 
 16. 
 
 Estimation of tenths, 4. 
 
 Expansion, coefficient of cubical, 15s ; linear, 
 IS4- 
 
 Fahrenheit's hydrometer, 120. 
 
 Fall of potential, method of resistance 
 measurement, 311 ; in a circuit, 271, 275, 
 279 ; in wire carrying current, 287. 
 
 Faraday's ice pail, 211. 
 
 Field of force, electrical, 202; mapping, 205 ; 
 unit, 207 ; magnetic, 219 ; computation of, 
 due to current, 235 ; due to magnet, 227. 
 
 Focal length, of concave lens, 165 ; of con- 
 cave mirror, 161 ; of convex lens, 162. 
 
 Forces, parallelogram of, 59 ; parallel, 61. 
 
 " Free" charge, 203 ; electricity, 203 ; mag- 
 netism, 232. 
 
 Friction, coefficient of moving, 9, 63 ; of 
 frictional torque, 64. 
 
 Fusion of ice, heat of, 144. 
 
 Galvanometer, d'Arsonval, theory, 242 ; re- 
 duction factor, 244 ; tangent, theory, 245 ; 
 reduction factor, 237 ; coil or true con- 
 stant, 236 and 254 ; working constant, 
 237 ; to set in meridian, 238 ; to deter- 
 mine deflection with scale and mirror, 
 239 ; determination of constant of, 244, 
 248, 254, 259, 260, 261 ; damping, 241, 
 341, 342 ; ballistic, 338 ; potential, 280, 
 284 ; potential constant, 287 ; resistance 
 (see Resistance) . 
 
 Gases, properties of, 124. 
 
 Graphical representation of results, 9. 
 
 Gratings, diffraction, 179. 
 
 Gravity, by Atwood's machine, 73 ; by free 
 fall, 75 ; by physical pendulum, 92 ; by 
 Kater's pendulum, 96. 
 
 Guthe, modification of Mance's method, 326.. 
 
 Hare's method of determining density, 122. 
 
 Heat of fusion, of ice, 144 ; of vaporization 
 of water, 142 ; specific heat by cooling, 
 146 ; by method of mixtures, 145 ; me- 
 chanical equivalent, 151. 
 
 Hefner lamp, candle power and its varia- 
 tion, 186, 187. 
 
 Henry, unit of induction, electromagnetic, 
 
 363- 
 Holtz machine, 215, 217. 
 Hydrometer, Fahrenheit's, 120 ; Nicholson's, 
 
 119. 
 
 Impact, 109; elastic, 114; inelastic, 112. 
 Impressed E. M. F., 274. 
 Impulse, 109. 
 Index of refraction, 173. 
 Induced, currents, direction of, 353 ; E. M. F.„ 
 353- 
 
INDEX 
 
 415 
 
 Induction, electrostatic, 203; electromag- 
 netic, 352 ; mutual, 360 ; self, 364. 
 Interference of sound waves, 192. 
 Internal resistance of batteries (see Battery 
 
 resistance) . 
 Interpretation of curves, 12. 
 
 Joule's equivalent, 151. 
 
 Kater's pendulum, 96. 
 
 Kelvin, measurement of resistance of bat- 
 tery, 373 ; of galvanometer, 334 ; double 
 bridge, 318. 
 
 Kirchhoff 's laws, 275. 
 
 Koenig's apparatus, 192. 
 
 Kundt's method for velocity of sound in 
 solid rods, 195. 
 
 Laplace's law, 235. 
 
 Latent heat, of steam, 142 ; of water, 144. 
 
 Least squares, method of, 24. 
 
 Lens, curvature 'of, by reflection, 158 ; by 
 spherometer, 29 ; focal length of con- 
 cave, 165 ; of convex, 162. 
 
 Leyden jar, 214. 
 
 Light standards, Hefner lamp, 186 ; glow 
 lamp, 187. 
 
 Lines of equal potential in a conductor, 292. 
 
 Lines of force, electrical, 203, 205, 208 ; de- 
 termination of direction of, 208 ; mag- 
 netic, 219, 220 ; positive direction of, 220 ; 
 study of, 222 ; around a wire carrying 
 current, 235 ; of a permanent magnet, 358. 
 
 Lodge, modification of Mance's method, 
 326. 
 
 Logarithmic decrement, of galvanometer 
 needle, 341 ; theory of, 342 ; determina- 
 tion of, 345. 
 
 Magnet, axis of, 220 ; magnetic moment of, 
 221 ; earth a, 220 ; permanent, 220 ; 
 poles, 220 ; force action between poles, 
 221. 
 
 Magnetic field, study of, 222 ; measurement 
 of intensity of, 230; lines of force, 219, 
 220 ; direction of, 220 ; of earth, study of, 
 by inductor, 356. 
 
 Magnetic hysteresis, 375. 
 
 Magnetic induction, 373. 
 
 Magnetic moment, 221 ; determination of, by 
 oscillations, 224; by magnetometer, 226. 
 
 Magnetic potential, 206. 
 
 Magnetic properties of iron, general state- 
 ment, 373 ; study of, by ballistic method, 
 383 ; by magnetometer method, 376. 
 
 Magnetism, general statement, 219. 
 
 Magnetization, lines of, 219, 373; intensity 
 of. 373- 
 
 Magnetometer, 226. 
 
 Magnifying power of microscope, 167; 01 
 telescope, 166. 
 
 Map, of an electrostatic field, 206; of a 
 magnetic field of a magnet, 223; of a 
 wire carrying a current, 235. 
 
 Mance's method of measuring battery resist- 
 ance, 324. 
 
 Manometric capsule, 192. 
 
 Mechanical equivalent of heat, 151. 
 
 Melde's experiment, 199. 
 
 Middle elongation, 38. 
 
 Mirror, focal length of concave, ifei. 
 
 Modulus, slide, 106; Young's, 101. 
 
 Moment of inertia, 82, 83-; about parallel 
 axis, 83; of a thin rod, 84; of a cylinder, 
 85 ; of a circular lamina, 85. 
 
 Moment of torsion, 104. 
 
 Moments, principle of, 61. 
 
 Momentum, 109 ; moment of, 83, 339. 
 
 Mutual induction, coefficient of, 360. 
 
 Newton's law of cooling, 139, 130. 
 Nicholson's hydrometer, 119. 
 
 Observations, 3 ; record of, 2. 
 
 Ohm, definition of, 304. 
 
 Ohm's law, 270, 273 ; method of measuring 
 
 E. M. F., 276 ; of measuring resistance of 
 
 a battery, 321. 
 Open-eye method of determining magnifying 
 
 power, 167. 
 Optical lever, 103. 
 
 Parallel forces, 61. 
 
 Parallelogram of forces, 59. 
 
 Pendulum, Kater's, 96; physical, 92; sim- 
 ple, 96; uniform bar, 98. 
 
 Periodic motion, 37; time of, by middle 
 elongations, 37; by transits, 41; of uni- 
 form bar pendulum for varying position 
 of knife-edges, 98. 
 
 Permanent magnet, 220; measurement of 
 lines of force of, 358. 
 
 Permeability, 218, 374. 
 
 Photometer, Bunsen, 183; Lummer-Brod- 
 hun, 184; Weber, 185. 
 
 Photometry, 182; of gas burner, 188; of 
 glow-lamp, 190. 
 
 Physical equations of curves, 11. 
 
 Physical pendulum. 92. 
 
 Pitch of sound by syren, 192. 
 
 Planimeter, 53. 
 
 Polarization, effect up Mi current, 266. 
 
 Potential, electrostatic, 207 ; magnetic, 207; 
 fall in a series circuit, 279 ; in a wire, 287. 
 
 Potential difference, 268 ; definition of, 268 ; 
 electromagnetic unit of, 269; practical 
 
4i6 
 
 INDEX 
 
 unit of, 269; variation at generator ter- 
 minals, 284. 
 
 Potential galvanometers, and measurers, 280, 
 285. 
 
 Potential-resistance diagrams, 271, 272. 
 
 Potentiometer, 300 ; Lord Rayleigh's method, 
 299 ; Poggendorf s method, 298. 
 
 Principle of moments, 61. 
 
 Prism, determination of angles of, 172; of 
 angle of minimum deviation, 173 ; cali- 
 bration of, 176. 
 
 Probable error, 18. 
 
 Proof plane, 209. 
 
 Pulleys, system of, 68 ; differential, 69. 
 
 Quantity of electricity, 337 ; produced by in- 
 duction, 355 ; measurement of, by ballistic 
 galvanometer, 337. 
 
 Radiating and absorbing power of surfaces, 
 148. 
 
 Radiation constant, 138. 
 
 Radius of curvature, by reflection, 158 ; by 
 spherometer, 29. 
 
 Ratio, of balance arms, 52 ; of damping, 343. 
 
 Refraction, index of, 173. 
 
 Regulating magnet, 236, 262. 
 
 Relative error, 23. 
 
 Reports, 14. 
 
 Residual charge, 214. 
 
 Resistance, of battery {see Battery resist- 
 ance) ; box, 304; coils, 304; general 
 statements regarding, 303 ; methods of 
 measurement, Carey Foster, 308; fall of 
 potential, 311; Kelvin double bridge, 
 318 ; potentiometer, 300 ; Wheatstone 
 bridge, 305 ; slide-wire bridge, 307 ; of 
 electrolytes, 320; specific, 314; temper- 
 ature coefficient, 315 ; unit, absolute, 303 ; 
 practical, 304. 
 
 Resonance of air columns, 194. 
 
 Retentivity, 375. 
 
 Reversing key, 239. 
 
 Rheostat, 304. 
 
 Rotational, energy, 82 ; inertia, 82. 
 
 Self-induction, 364 ; coefficient of, 364 ; 
 measurement of, by Anderson's method, 
 370; by comparison, 365; by Rimington's, 
 method, 367. 
 
 Sensibility of a balance, 51. 
 
 Sensitive galvanometer, 241 ; constant of, 258. 
 
 Shunts, theory of, 255. 
 
 Significant figures, 6. 
 
 Simple harmonic motion, 86; amplitude of, 
 86; period of, 87; of rotation, 89; of 
 translation, 87; examples of, 91. 
 
 Slide rule, 5. 
 
 Sonometer, 197. 
 
 Sources of error, 16. 
 
 Specific gravity, 115; by specific gravity 
 
 bottle, 116; by Nicholson's hydrometer, 
 
 119 {see Density). 
 Specific heat, method of cooling, 146; of 
 
 mixtures, 145. 
 Specific resistance, 314; measurement of, 
 
 314 ; of electrolytes, 320. 
 Spectra, 175; of various substances, 177. 
 Spectrometer, 168 ; adjustments, using 
 
 Gauss' eyepiece, 169 ; using ordinary 
 
 eyepiece, 171. 
 Spectroscope, 168. 
 Spherometer, 29. 
 
 Standard cell, 290, 298, 301; Daniel], 261. 
 Static, electricity, 202; induction, 210. 
 Strings, law of vibrating, the sonometer, 
 
 197 ; Melde's method, 199. 
 Susceptibility, 374. 
 Syren, 192. 
 
 Tables, 392-412. 
 
 Tangent galvanometer, theory, 236 ; working 
 constant, 237 ; constant per scale division, 
 240 ; as a quantity measurer, 340, 
 
 Telescope, and scale, 239; magnifying power 
 of, 166. 
 
 Temperatures, errors in determining, 133; 
 coefficient of expansion, cubical, 155 ; 
 linear, 154 ; coefficient of resistance, 315. 
 
 Tenths, estimation of, 4. 
 
 Thermo-electromotive force, 294; variation 
 with temperature, 295. 
 
 Thermo-element, 294. 
 
 Thermometers, calibration of, 32 ; compari- 
 son of, 134. 
 
 Torque, 82. 
 
 Torsion, moment of, 104. 
 
 Total E. M. F., 274. 
 
 Transverse vibrations, study of, 199. 
 
 True constant of a galvanometer, 236, 254. 
 
 Uniformly accelerated motion, circular, 78 ; 
 
 linear, 71. 
 Units, 7. 
 
 Vapor, pressure of saturated, 130. 
 
 Vaporization, heat of, 142. 
 
 Variation of periodic time of a uniform 
 
 cylindrical pendulum with variation of 
 
 position of knife-edges, 98. 
 Velocity attained with angular acceleration, 
 
 '79; with linear acceleration, 72. 
 Velocity of sound, in air, 194; in brass, 195. 
 Vibrating strings, laws of, 197, 199. 
 Vienna method of measuring current, 265. 
 Volt, definition of, 269. 
 
INDEX 
 
 417 
 
 Voltameter, copper, 250; spiral coil, 250. 
 Voltmeter calibration, 296. 
 Volume determinations by measurement of 
 dimensions, 35. 
 
 Water equivalent, of calorimeter, 136. 
 Wave-length, measurement of, of sound, 
 
 192 ; of light, 178. 
 Weighing, method of equal swings, 46; with 
 
 a tare, 50; by vibrations, 47; precau- 
 tions in, 45 ; reduction to vacuo, 50. 
 
 Weight, 7 ; in taking an average, 20. 
 
 Wheatstone bridge, 305. 
 
 Wheel and axle, 66. 
 
 Young's modulus, by flexure, 107; by 
 stretching, 101; microscope method, 
 102; optical lever method, 103. 
 
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 Physical Optics 
 
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 Modern Theory of Physical Phenomena, Radio-Activity, 
 
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 Notes and Questions in Physics 
 
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