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PRACTICAL PHYSICS

BY

R. T. GLAZEBROOK, M.A., F.RSS.

FELLOW OF TRINITY COLLEGE, AND

W. N. SHAW, M.A.

FELLOW OF EMMANUEL COLLEGE

Demonstrators at the Cavendish Lcei'oratory, Cambridge

THIRD EDITION

LONDON

LONGMANS, GREEN, AND CO

AND NEW YORK : 15 EAST 16"' STREET 1889

Been *

, < • • • ii-1

PREFACE.

THIS book is intended for the assistance of Students and Teachers in Physical Laboratories. The absence of any book covering the same ground made it necessary for us, in conducting the large elementary classes in Practical Physics at the Cavendish Laboratory, to write out in MS. books the practical details of the different experiments. The increase in the number of well- equipped Physical Laboratories has doubtless placed many teachers in the same position as we ourselves were in before these books were compiled ; we have therefore collected together the manuscript notes in the present volume, and have added such general explana- tions as seemed necessary.

In offering these descriptions of experiments for publica- tion we are met at the outset by a difficulty which may prove serious. The descriptions, in order to be precise, must refer to particular forms of instruments, and may there- fore be to a certain extent inapplicable to other instruments of the same kind but with some difference, perhaps in the arrangement for adjustment, perhaps in the method of graduation. Spherometers, spectrometers, and katheto- meters are instruments with which this difficulty is particu- larly likely to occur. With considerable diffidence we have thought it best to adhere to the precise descriptions referring

viii Preface.

to instruments in use in our own Laboratory, trusting that the necessity for adaptation to corresponding instruments used elsewhere will not seriously impair the usefulness of the book. Many of the experiments, however, which we have selected for description require only very simple apparatus, a good deal of which has in our case been constructed in the Laboratory itself. We owe much to Mr. G. Gordon, the Mechanical Assistant at the Cavendish Laboratory, for his ingenuity and skill in this respect.

Our general aim in the book has been to place before the reader a description of a course of experiments which shall not only enable him to obtain a practical acquaintance with methods of measurement, but also as far as possible illustrate the more important principles of the various sub- jects. We have not as a rule attempted verbal explanations of the principles, but have trusted to the ordinary physical text-books to supply the theoretical parts necessary for understanding the subject ; but whenever we have not been able to call to mind passages in the text-books sufficiently explicit to serve as introductions to the actual measurements, we have either given references to standard works or have endeavoured to supply the necessary information, so that a student might not be asked to attempt an experiment without at least being in a position to find a satisfactory explanation of its method and principles. In following out this plan we have found it necessary to interpolate a considerable amount of more theoretical information. The theory of the balance has been given in a more complete form than is usual in mechanical text-books ; the introductions1 to the measure- ment of fluid pressure, thermometry, and calorimetry have been inserted in order to accentuate certain important prac- tical points which, as a rule, are only briefly touched upon ;

Preface. ix

while the chapter on hygrometry is intended as a complete elementary account of the subject. We have, moreover, found it necessary to adopt an entirely different style in those chapters which treat of magnetism and electricity. These subjects, regarded from the point of view of the practical measurement of magnetic and electric quantities, present a somewhat different aspect from that generally taken. We have accordingly given an outline of the general theory of these subjects as developed on the lines indicated by the electro-magnetic system of measurement, and the arrangement of the experiments is intended, as far as possi- ble, to illustrate the successive steps in the development. The limits of the space at our disposal have compelled us to be as concise as possible ; we have, therefore, been unable to illustrate the theory as amply as we could have wished. We hope, however, that we have been suc- cessful in the endeavour to avoid sacrificing clearness to brevity.

We have made no attempt to give anything like a com- plete list of the experiments that may be performed with the apparatus that is at the present day regarded as the ordinary equipment of a Physical Laboratory. We have selected a «few — in our judgment the most typical — experi- ments in each subject, and our aim has been to enable the student to make use of his practical work to obtain a clearer and more real insight into the principles of the subjects. \Vith but few exceptions, the experiments selected are of an elementary character ; they include those which have formed for the past three years our course of practical physics for the students preparing for the first part of the Natural Sciences Tripos ; to these we have now added some ex- periments on acoustics, on the measurement of wave-lengths,

x Preface.

and on polarisation and colours. Most of the students have found it possible to acquire familiarity with the contents of such a course during a period of instruction lasting over two academical terms.

The manner in which the subjects are divided requires perhaps a word of explanation. In conducting a class in- cluding a large number of 'students, it is essential that a teacher should know how many different students he can accommodate at once. This is evidently determined by the number of independent groups of apparatus which the Laboratory can furnish. It is, of course, not unusual for an instrument, such as a spectrometer, an optical bench, or Wheatstone bridge, to be capable of arrangement for working a considerable number of different experiments ; but this is evidently of no assistance when the simultaneous accommo- dation of a number of students is aimed at. For practical teaching purposes, therefore, it is an obvious advantage to divide the subject with direct reference to the apparatus required for performing the different experiments. We have endeavoured to carry out this idea by dividing the chapters into what, for want of a more suitable name, we have called ' sections,' which are numbered continuously throughout the book, and are indicated by black type headings. Each section requires a certain group of apparatus, and the teacher knows that that apparatus is not further available when he has assigned the section to a particular student. The different experiments for which the same apparatus can be employed are grouped together in the same section, and indicated by italic headings.

The proof-sheets of the book have been in use during the past year, in the place of the original MS. books, in the following manner:— The sheets, divided into the section.*

Preface. xi

above mentioned, have been pasted into MS. books, the re- maining pages being available for entering the results obtained by the students. The apparatus referred to in each book is grouped together on one of the several tables in one large room. The students are generally arranged in pairs, and be- fore each day's work the demonstrator in charge assigns to each pair of students one experiment — that is, one section of the book. A list shewing the names of the students and the experiment assigned to each is hung up in the Laboratory, so that each member of the class can know the section at which he is to work. He is then set before the necessary apparatus with the MS. book to assist him ; if he meets with any difficulty it is explained by the demonstrator in charge. The results are entered in the books in the form indicated for the several experiments. After the class is over the books are collected and the entries examined by the demonstrators. If the results and working are correct a new section is assigned to the student for the next time ; if they are not so, a note of the fact is made in the class list, and the student's attention called to it, and, if necessary, he repeats the experiment. The list of sections assigned to the different students is now completed early in the day before that on which the class meets, and it is hoped that the publication of the description of the experiment will enable the student to make himself acquainted beforehand with the details of his day's work.

Adopting this plan, we have found that two demon- strators can efficiently manage two classes on the same day, one in the morning, the other in the afternoon, each con- taining from twenty-five to thirty students. The students have hitherto been usually grouped in pairs, in consequence of the want of space and apparatus. Although this plan

xii Preface.

has some advantages, it is, we think, on the whole, undesir- able.

We have given a form for entering results at the end of each section, as we have found it an extremely convenient, if not indispensable, arrangement in our own case. The numerical results appended as examples are taken, with very few exceptions, from the MS. books referred to above. They may be found useful, as indicating the degree of accuracy that is to be expected from the various experi- mental methods by which they are obtained.

In compiling a book which is mainly the result of Labora- tory experience, the authors are indebted to friends and fellow- workers even to an extent beyond their own knowledge. We would gladly acknowledge a large number of valuable hints and suggestions. Many of the useful contrivances that facilitate the general success of a Laboratory in which a large class works, we owe to the Physical Laboratory of Berlin ; some of them we have described in the pages that follow.

. For a number of valuable suggestions and ideas we are especially indebted to the kindness of Lord Rayleigh, who has also in many other ways afforded us facilities for the development of the plans and methods of teaching explained above. Mr. J. H. Rand ell, of Pembroke College, and Mr. H. M. Elder, of Trinity College, have placed us under an obligation, which we are glad to acknowledge, by reading the proof-sheets while the work was passing through the press. Mr. Elder has also kindly assisted us by photograph- ing the verniers which are represented in the frontispiece.

R. T. GLAZEBROOK.

W. N. SHAW.

CAVENDISH LABORATORY : December I, 1884.

CONTENTS.

CHAPTER I.

PHYSICAL MEASUREMENTS.

PAGE

Direct and indirect Method of Measurement I Indirect Measurements reducible to Determinations of Length

and Mass .......... 4

Origin of the Similarity of Observations of Different Quantities , 7

CHAPTER I!.

UNITS OF MEASUREMENT.

Method of expressing a Physical Quantity . . . . . 9

Arbitrary and Absolute Units . . . . . . .10

Absolute Units . . . . . . . . 13

Fundamental Units and Derived Units . . . . 17

Absolute Systems of Units . . . . . . ..17

The C. G. S. System . . ..... 21

Arbitrary Units at present employed . . . . 22

Changes from one Absolute System of Units to another. Dimen- sional Equations ........ 24

Conversion of Quantities expressed in Arbitrary Units . . . 28

CHAPTER III.

PHYSICAL ARITHMETIC.

Approximate Measurements ....... 30

Errors and Corrections . . . . . . . . 31

Mean of Observations ........ 32

xiv Contents

PAGE

Possible Accuracy of Measurement of different Quantities . . . 35 Arithmetical Manipulation of Approximate Values . . 36

Facilitation of Arithmetical Calculation by means of Tables.

Interpolation .......... 40

Algebraical Approximation— Approximate Formulae — Introduc- tion of small Corrections . . . . . . .41

Application of Approximate Formulae to the Calculation of the

Effect of Errors of Observation . . . . . . 44

CHAPTER IV.

MEASUREMENT OF THE MORE SIMPLE QUANTITIES.

SECTION

LENGTH MEASUREMENTS 50

1. The Calipers . . . . . . . 50

2. The Beam-Compass ........ 54

3. The Screw-Gauge . . . . . . • • 57

4. The Spherometer . . . . . . . -59

5. The Reading Microscope — Measurement of a Base-Line . 64

6. The Kathetometer . . . . . . * . .66

Adjustments .... ... 67

Method of Observation . . . . . 7 1

MEASUREMENT OF AREAS 73

7. Simpler Methods of measuring Areas of Plane Figures . 73

8. Determination of the Area of the Cross-section of a Cylin-

drical Tube — Calibration of a Tube . . . -75 MEASUREMENT OF VOLUMES .78

9. Determination of Volumes by Weighing . . . -78

10. Testing the Accuracy of the Graduation of a Burette . . 79 MEASUREMENT OF ANGLES ...... 80

MEASUREMENTS OF TIME . . . . ' . . . 80

11. Rating a Watch by means of a Seconds-Clock . . .81

CHAPTER V.

MEASUREMENT OF MASS AND DETERMINATION OF SPECIFIC GRAVITIES.

12. The Balance 83

General Considerations . , . i 8 j

The Sensitiveness of a Balance . . . 84

The Adjustment of a Balance . . . . -87

Contents. xv

ECTION PAGE

Pra:tical Details of Manipulation — Method of Oscillations . . . . . . ..91

13. Testing the Adjustments of a Balance . . . .98

'Determination of the Ratio of the Arms of a Balance and of the true Mass of a Body "when the Arms

of the Balance are. unequal . . . . . 100

Comparison of the Masses of the Scale Pans . . 101

14. Correction of Weighings for the Buoyancy of the Air . . 103 DENSITIES AND SPECIFIC GRAVITIES — Definitions . 105

15. The Hydrostatic Balance ....... 107

Determination of the Specific Gravity of a SoliJ heavier than Water . . . . . .107

Determination of the Specific Gravity of a Solid lighter than Water . . . . . . 109

Determination of the Spcdjic Gravity of a Liquid . 1 1 1

16. The Specific Gravity Bottle 112

Determination of the Specific Gravity of small Frag- ments of a Solid . . . . . .112

Determination of the Specific Gravity of a Powder . 1 16 Determination of the Specific Gravity of a Liquid . 1 16

17. Nicholson's Hydrometer . . . . . . . 117

Determination of the Specific Gravity of a Solid . 117 Determination of the Specific Gravity of a Liquid . 119

18. Jolly's Balance ........ 120

Determination of the Mass and Specific Gravity of a small Solid Body . . . . . ..121

Determination of the Specific Gravity of a Liquid . 122

19. The Common Hydrometer ...... 123

Method of comparing the Densities of two Liquids by the Aid of the Kathetometer . . . . . 125

CHAPTER VI.

MECHANICS OF SOLIDS.

20. The Pendulum . . . . . . . .128

Determination of the Acceleration of Gravity by Pendulum Observations . . . . . 128

Comparison of the Times of Vibration of two Pen- dulums— Methoi of Coincidences . . .132

a

xvi Contents.

SECTION

21. Atwood's Machine .... . i-»-» SUMMARY OF THE GENERAL THEORY OF ELASTICITY . 139

22. Young's Modulus I4I

Modulus of Torsion . . . . . . . .144

Moment of Inertia . , . . . . . 144

Maxwell's Vibration Needle I46

Observation of the Time of Vibration . . . . 148

Calculation of the A iteration of Moment of Inertia « 150

CHAPTER VII.

MECHANICS OF LIQUIDS AND GASES.

Measurement of Fluid Pressure 152

24. The Mercury Barometer . . . . . . . 1^3

Setting and reading the Barometer . . . -154 Correction of the Observed Height for Tempera- ture, drv.' 155

25. The Aneroid Barometer . . . . , . -157

Measurement of Heights . . . . . . 158

26. The Volumenometer . . . . . . .160

Verification of Boyle's Law . . . . . 160 Determination of the Specific Gravity of a Solid . 163

CHAPTER VIII.

ACOUSTICS.

Definitions, £c. . . . . . . . .164

27. Comparison of the Pitch of Tuning-forks — Adjustment of

two Forks to Unison 165

28. The Siren 168

29. Determination of the Velocity of Sound in Air by Measure-

ment of the Length of a Resonance Tube corresponding

to a given Fork . . . . . . . 172

30. Verification of the Laws of Vibration of Strings — Determina-

tion of the Absolute Pitch of a Note by the Monochord 175

31. Determination of the Wave-Length of a high Note in Air

by means of a Sensitive Flame . . . . .180

Contents. xvii

CHAPTER IX.

THERMOMETRY AND EXPANSION.

ECTION PAGE

Measurement of Temperature . . . . . . 183

32. Construction of a Water Thermometer .... 190

33. Thermometer Testing 193

34. Determination of the Boiling Point of a Liquid . . . 196

35. Determination of the Fusing Point of a Solid . . . 197 COEFFICIENTS OF EXPANSION . . . .198

36. Determination of the Coefficient of Linear Expansion of a

Rod -. . 200

37. The Weight Thermometer 202

38. The Air Thermometer ....... 208

CHAPTER X. C A L O R I M E T R Y.

39. The Method of Mixture 212

Determination of the Specific Heat of a Solid . .212 Determination of the Specific Heat of a Liquid . . 218 Determination of thj Latent Heat of Water . .219 Determination of the Latent Heat of Steam . .221

40. The Method of Cooling 225

CHAPTER XI.

TENSION OF VAPOUR AND HYGROMETRY.

41. Dalton's Experiment on the Pressure of Mixed Gases and

Vapours ......... 228

HYGROMETRY 231

42. The Chemical Method of determining the Density of

Aqueous Vapour in the Air . . . . . . 233

43. Dines's Hygrometer — The Wet and Dry Bulb Thermometers 238

44. Regnault's Hygrometer ....... 241

CHAPTER XII. PHOTOMETRY.

45. Bumen's Photometer ..... t .. 244.

46. Rumford's Photometer ..... . 748

xviii Contents.

CHAPTER XIII.

MIRRORS AND LENSES.

SECTION PAGE

47. Verification of the Law of Reflexion of Light . . . 250

48. The Sextant 253

OPTICAL MEASUREMENTS 259

49. Measurement of the Focal Length of a Concave Mirror . 261

50. Measurement of the Radius of Curvature of a Reflecting

Surface by Reflexion . . . . . . 263

Measurement of Focal Lengths of Lenses . . . . 267

51. Measurement of the Focal Length of a Convex Lens (First

Method) . . 267

52. Measurement of the Focal Length of a Convex Lens

(Second Method) 268

53. Measurement of the Focal Length of a Convex Len (Third

Method) 269

54. Measurement of the Focal Length of a Concave Lens . . 274

55. Focal Lines 276

Magnifying Powers of Optical Instruments . . . . 278

56. Measurement of the Magnifying Power of a Te'escope

(First Method) 279

57. Measurement of the Magnifying Power of a Telescope

(Second Method) . . 281

58. Measurement of the Magnifying Power of a Lens or of a

Microscope . . . . . . . 283

59. The Testing of Plane Surfaces 287

CHAPTER XIV.

SPECTRA, REFRACTIVE INDICES AND WAVE-LENGTHS.

Pure Spectra ......... 295

60. The Spectroscope 297

Mapping a Spectrum . . . . . -297

Comparison of Spectra . . . . . 301

Refractive Indices ........ 302

61. Measurement of the Index of Refraction of a Plate by

means of a Microscope ....... 303

62. The Spectromeler 305

The Adjustment of a Spectrometer . . . . 306

Contents. xix

SECTION PAGE

Measurements with the Spectrometer ..... 308

(1) Verification of the Law of Reflexion . . . 308

(2) Measurement of the Angle of a Prism . . 308

(3) Measiuemcnt of the Refractive Index of a Prism

(First Method] 309

Measurement of the Refractive Index of a Prism (Second Method] 313

(4) Measurement of the Wave-Length of Light by

means of a Diffraction Grating . . .315

Optical Bench 318

Measurement of the Wave-Length of Light l>y means of FresneVs Bi-prism . . . . . . 319

Diffraction Experiments ..... 324

CHAPTER XV.

POLARISED LIGHT.

On the Determination of the Position of the Plane of Polarisation ......... 325

64. The Bi-quartz ......... 327

65. Shadow Polarimeters ........ 332

CHAPTER XVI.

COLOUR VISION.

66. The Colour Top 337

67. The Spectro-Photometer ....... 341

68. The Colour Box ........ 345

CHAPTER XVII.

MAGNETISM.

Properties of Magnets 347

Definitions 348

Magnetic Potential ........ 353

Forces on a Magnet in a Uniform Field .... 355

Magnetic Moment of a Magnet 356

Potential due to a Solenoidal Magnet .... 358

Force due to a Solenoidal Magnet 359

Action of one Solenoidal Magnet on another . . .361

xx Contents.

SECTION PAGE

Measurement of Magnetic Force . . .'.•». 364 Magnetic Induction. ....... 366

69. Experiments with Magnets ....... 367

(a) Magnetisation of a Steel Bar .... 367 (p] Comparison of the Magnetic Moment of the same Magnet after different Methods of Treatment, or of two different Magnets , . . 370

(c) Comparison of the Strengths of different Magnetic

Fields of approximately Uniform Intensity . 373

(d) Measurement of the Magnetic Moment of a

Magnet and of the Strength of the Field in which it hangs . . . . • • 373

(e) Determination of the Magnetic Moment of a

Magnet of 'any shape . . . . -375

(f) Determination of the Direction of the Earth 's

Horizontal Force . . . . . 375

70. Exploration of the Magnetic Field due to a given Magnetic

Distribution 0 379

CHAPTER XVIII.

ELECTRICITY — DEFINITIONS AND EXPLANATIONS OF ELECTRICAL TERMS.

Conductors and Non-conductors . . . . . . 382

Resultant Electrical Force 382

Electromotive Force ........ 383

Electrical Potential 383

Current of Electricity 386

C.G.S. Absolute Unit of Current 388

Sine and Tangent Galvanometers . ... 390

CHAPTER XIX.

EXPERIMENTS IN THE FUNDAMENTAL PROPERTIES OF ELECTRIC CURRENTS— MEASUREMENT OF ELECTRIC CURRENT AND ELECTROMOTIVE FORCE.

71. Absolute Measure of the Current in a Wire . . .391 GALVANOMETERS ........ 395

Galvanometer Constant . . - . .... 397

Contents. xxi

ECTION PAGE

Reduction Factor of a Galvanometer . . . , . 401

Sensitiveness of a Galvanometer ..... 402

TV .Adjustment of a Reflecting Galvanometer . . . 404

72. Determination of the Reduction Factor of a Galvano-

meter .......... 405

Electrolysis ......... 406

Definition of Electro-chemical Equivalent . . . . 406

73. Farnday's Law— Comparison of Electro-chemical Equiva-

lents . . . . . . . . . . 411

74. Joule's Law— Measurement of Electromotive Force . .416

CHAPTER XX.

OHM'S LAW— COMPARISON OF ELECTRICAL RESISTANCES AND ELECTROMOTIVE FORCES.

\

Definition of Electrical Resistance . , . . 421

Series and Multiple Arc ....... 422

Shunts .......... 424

Absolute Unit of Resistance ...... 425

Standards of Resistance ....... 426

Resistance BDXCS ........ 427

Relation between the Resistance and Dimensions of a Wire

of given Material ........ 428

Specific Resistance ........ 429

75. Comparison of Electrical Resistances . . ... 430

76. Comparison of Electromotive Forces .... 435

77. Wheatstone's Bridge ........ 437

Measurement of Resistance . . . . .443

Measurement of a Galvanometer Resistance — Thom- son's Method ....... 445

Measurement of a Battery Resistance— Mance's Mdhod ........ 447

78. The British Association Wire Bridge . . . . 451

Measurement of Electrical Resistance . . .451

79. Carey Foster's Method of Comparing Resistances . . . 455

Calibration of a Bridge- Wire . . . .460

80. Po^gendorff's Method for the Comparison of Electromotive

Forces -Latimer Clark's Potentiometer . . . . 461

xxii Contents

CHAPTER XXI.

GALVANOMETRIC MEASUREMENT OF A QUANTITY OF ELECTRICITY.

SECTION I-AGE

. Theory of the Method 466

Relation between the Quantity of Electricity which passes through a Galvanometer, and the initial Angular Velocity produced in the Needle . . 466 Work done in turning the Magnetic Needle through

a given Angle 467

Electrical Accumulators or Condensers . ... 470 Definition of the Capacity of a Condenser . . . . 47 1

The Unit of Capacity 471

On the Form of Galvanometer suitable for the Comparison of Capacities 472

81. Comparison of the Capacities of two Condensers . . . 473

(1) Approximate Method . . . . -473

(2) Null Method 476

82. Measurement in Absolute Measure of the Capacity of a

Condenser ......... 479

INDEX ......... .483

PRACTICAL PHYSICS.

CHAPTER I.

PHYSICAL MEASUREMENTS.

THE greater number of the physical experiments of the present day and the whole of those described in this book consist in, or involve, measurement in some form or other. Now a physical measurement — a measurement, that is to say, of a physical quantity— consists essentially in the comparison of the quantity to be measured with a unit quantity of the same kind. By comparison we mean here the determination of the number of times that the unit is contained in the quantity measured, and the number in question may be an integer or a fraction, or be composed of an integral part and a fractional part. In one sense the unit quantity must remain from the nature of the case perfectly arbitrary, although by general agreement of scientific men the choice of the unit quantities may be determined in accordance with certain general prin- ciples which, once accepted for a series of units, establish cer- tain relations between the units thus chosen, so that they form members of a system known as an absolute system of units. For example, to measure energy we must take as our unit the energy of some body under certain conditions, but when we agree that it shall always be the energy of a body on which a unit force has acted through unit space, our choice has been exercised, and the unit of energy is no longer arbitrary, but

B

2 Practical Physics. [CHAP. I.

defined, as soon as the units of force and space are agreed upon ; we have thus substituted the right of selection of the general principle for the right of selection of the particular unit.

We see, then, that the number of physical units is at least as great as the number of physical quantities to be measured, and indeed under different circumstances several different units may be used for the measurement of the same quantity. The physical quantities may be suggested by or related to phenomena grouped under the different headings of Mechanics, Hydro-mechanics, Heat, Acoustics, Light, Electricity or Magnetism, some being related to phenomena on the common ground of two or more such subjects. We must expect, therefore, to have to deal with a very large number of physical quantities and a correspond- ingly large number of units.

The process of comparing a quantity with its unit — the measurement of the quantity— may be either direct or in- direct, although the direct method is available perhaps in one class of measurements only, namely, in that of length measurements. This, however, occurs so frequently in the different physical experiments, as scale readings for lengths and heights, circle readings for angles, scale readings for galvanometer deflections, and so on, that it will be well to consider it carefully.

The process consists in laying off standards against the length to be measured. The unit, or standard length, in this case is the distance under certain conditions of temperature between two marks on a bar kept in the Standards Office of the Board of Trade. This, of course, cannot be moved from place to place, but a portable bar may be obtained and com- pared with the standard, the difference between the two being expressed as a fraction of the standard. Then we may apply the portable bar to the length to be measured, deter- mining the number of times the length of the bar is contained in the given length, with due allowance for temperature, and

CHAP. I.] Physical Measurements. 3

thus express the given length in terms of the standard by means of successive direct applications of the fundamental method of measurement. Such a bar is known as a scale or rule. In case the given length does not contain the length of the bar an exact number of times, we must be able to determine the excess as a fraction of the length of the bar ; for this purpose the length of the bar is divided by transverse marks into a number of equal parts — say 10 — each of these again into 10 equal parts, and perhaps each of these still further into 10 equal parts. Each of these smallest parts will then be -^-^ of the bar, and we can thus determine the number of tenths, hundredths, and thousandths of the bar contained in the excess. But the end of the length to be measured may still lie between two consecutive thou- sandths, and we may wish to carry the comparison to a still greater accuracy, although the divisions may be now so small that we cannot further subdivide by marks. We must adopt some different plan of estimating the fraction of the thousandth. The one most usually employed is that of the 'vernier.' An account of this method of increasing the accuracy of length measurements is given in § i.

This is, as already stated, the only instance usually oc- curring in practice of a direct comparison of a quantity with its unit. The method of determining the mass of a body by double weighing (see § 13), in which we determine the number of units and fractions of a unit of mass, which to- gether produce the same effect as was previously produced by the mass to be measured, approaches very nearly to a direct comparison. And the strictly analogous method oi substitution of units and fractions of a unit of electrical re- sistance, until their effect is equal to that previously produced by the resistance to be measured, may also be mentioned, as well as the measurement of time by the method of coinci- dences (§ 20).

But in the great majority of cases the comparison is far from direct. The usual method of proceeding is as follows :—

B 2

4 Practical Physics. [CHAP. I.

An experiment is made the result of which depends upon the relative magnitude of the quantity and its unit, and the nume- rical relation is then deduced by a train of reasoning which may, indeed, be strictly or only approximately accurate. In the measurement, for instance, of a resistance by Wheatstone's Bridge, the method consists in arranging the unknown resist- ance with three standard resistances so chosen that under cer- tain conditions no disturbance of a galvanometer is produced. We can then determine the resistance by reasoning based on Ohm's law and certain properties of electric currents. These indirect methods of comparison do not always afford perfectly satisfactory methods of measurement, though they are sometimes the only ones available. It is with these in- direct methods of comparing quantities with their units that we shall be mostly concerned in the experiments detailed in the present work.

We may mention in passing that the consideration of the experimental basis of the reasoning on which the various methods depend forms a very valuable exercise for the student. As an example, let us consider the determination of a quantity of heat by the method of mixture (§ 39). It is usual in the rougher experiments to assume (i) that the heat absorbed by water is proportional to the rise of temperature ; (2) that no heat is lost from the vessel or calorimeter ; (3) that in case two thermometers are used, their indications are identical for the same temperature. All these three points may be con- sidered with advantage by those who wish to get clear ideas about the measurement of heat.

Let us now turn our attention to the actual process in which the measurement of the various physical quantities consists. A little consideration will show that, whether the quantity be mechanical, optical, acoustical, magnetic or electric, the process really and truly resolves itself into measuring certain lengths, or masses.1 Some examples will

1 See articles by Clifford and Maxwell : Scientific Apparatus. Hand- book to the Special Loan Collection, 1876, p. 55.

CHAP. I.] Physical Measurements. 5

make this sufficiently clear. Angles are measured by read- ings of length along certain arcs ; the ordinary measure- ment of time is the reading of an angle on a clock face or the space described by a revolving drum ; force is measured by longitudinal extension of an elastic body or by weighing ; pressure by reading the height of a column of fluid sup- ported by it ; differences of temperature by the lengths of a thermometer scale passed over by a mercury thread ; heat by measuring a mass and a difference of temperature ; lu- minous intensity by the distances of certain screens and sources of light ; electric currents by the angular deflection of a galvanometer needle ; coefficients of electro-magnetic induction also by the angular throw of a galvanometer needle. Again, a consideration of the definitions of the various physical quantities leads in the same direction. Each physical quantity has been denned in some way for the purpose of its measurement, and the definition is insuffi- cient and practically useless unless it indicates the basis upon which the measurement of the quantity depends. A definition of force, for instance, is for the physicist a mere arrangement of words unless it states that a force 'is mea- sured by the quantity of momentum it generates in the unit of time ; and in the same way, while it may be interest- ing to know that * electrical resistance of a body is the oppo- sition it offers to the passage of an electric -current,' yet we have not made much progress towards understanding the precise meaning intended to be conveyed by the words ' a resistance of 10 ohms,' until we have acknowledged that the ratio of the electromotive force between two points of a con- ductor to the current passing between those points is a quan- tity which is constant for the same conductor in the same physical state, and is called and is the ' resistance ' of the conductor ; and, further, this only conveys a definite mean- ing to our minds when we understand the bases of measure ment suggested by the definitions of electromotive force and electric current.

Practical Physics.

[CHAP. I.

When the quantity is once defined, we may possibly be able to choose a unit and make a direct comparison ; but such a method is very seldom, if ever, adopted, and the measurements really made in any experiment are often sug- gested by the definitions of the quantities measured.

The following table gives some instances of indirect methods of measurement suggested by the definitions of the quantities to be measured. The student may consult the descriptions of the actual processes of measurement detailed in subsequent chapters : —

Name of quantity measured

MECHANICS. Area Volume . Velocity Acceleration . Force

Work . Energy . Fluid pressure (in abso- lute units) . Coefficients of elasticity

SOUND.

Velocity . .

Pitch .

HEAT.

Temperature . Quantity of heat Conductivity .

LIGHT.

Index of refraction . Intensity

MAGNETISM.

Quantity of magnetism Intensity of field .

Magnetic moment .

Measurement actually made

Length (§ 1-6).

Length.

Length and time.

Velocity and time.

Mass and acceleration, or extension

of spring. Force and length. Work, or mass and velocity.

Force and area (§ 24-26). Stress and strain, i.e. force, and length or angle (§§ 22, 23).

Length and time (§ 29). Time (§ 28).

Length (§ 32).

Temperature and mass (§ 39). Temperature, heat, length, and time.

Angles (§ 62). Length (§ 45).

Force and length (§ 69).

Force and quantity of magnetism

(§ 69). Quantity of magnetism and length

(§ 69).

CHAP. I.] Physical Measurements. J

Name of quantity measured Measurements actually made

ELECTRICITY.

Electric current . . Quantity of magnetism, force, and

length (§71)-

Quantity of Electricity . Current and time (§ 72). Electromotive force . Quantity of electricity and work

(§ 74).

Resistance . . . Electric current and E. M. F. (§ 75). Electro-chemical equivalent. Mass and quantity of electricity

(§ 72).

The quantities given in the second column of the table are often such as are not measured directly, but the basis of measurement has, in each case, already been given higher up in the table. If the measurement of any quantity be reduced to its ultimate form it will be found to consist always in measurements of length or mass.1 The measurement of time by counting ' ticks ' may seem at first sight an exception to this statement, but further consideration will shew that it, also, depends ultimately upon length measurement.

As far as the apparatus for making the actual observations is concerned, many experiments, belonging to different subjects, often bear a striking similarity. The observing apparatus used in a determination of a coefficient of tor- sion, the earth's horizontal magnetic intensity, and a coefficient of electro-magnetic induction, are practically identical in each case, namely, a heavy swinging needle and a telescope and scale ; the difference between the experi- ments consists in the difference in the origin of the forces which set the moving needle in motion. Many similar in- stances might -be quoted. Maxwell, in the work already referred to ('Scientific Apparatus,' p. 15), has laid down the grounds on which this analogy between the experiments in different branches of the subject is based. * All the physical sciences relate to the passage of energy under its various forms from one body to another,' and, accordingly,

1 The measurement of mass may frequently be resolved into that of length. The method of double weighing, however, is a fundamental measurement sui generis.

8 Practical Physics. [CHAP. I.

all instruments, or arrangements of apparatus, possess the following functions : —

' i. The Source of energy. The energy involved in the phenomenon we are studying is not, of course, produced from nothing, but enters the apparatus at a particular place which we may call the Source.

' 2. The channels or distributors of energy, which carry it to the places where it is required to do work.

'3. The restraints which prevent it from doing work when it is not required.

'4. The reservoirs in which energy is stored up when it is not required.

1 5. Apparatus for allowing superfluous energy to escape.

' 6. Regulators for equalising the rate at which work is done.

* 7. Indicators or movable pieces which are acted upon by the forces under investigation.

' 8. Fixed scales on which the position of the indicator is read off.'

The various experiments differ in respect of the functions included under the first six headings, while those under the headings numbered 7 and 8 will be much the same for all instruments, and these are the parts with which the actual observations for measurement are made. In some experi- ments, as in optical measurements, the observations are simply those of length and angles, and we do not compare forces at all, the whole of the measurements being ultimately length measurements. In others we are concerned with forces either mechanical, hydrostatic, electric or magnetic, and an experiment consists in observations of the magni- tude of these forces under certain conditions ; while, again, the ultimate measurements will be measurements of length and of mass. In all these experiments, then, we find a foundation in the fundamental principles of the measure- ment of length and of the measurements of force and mass. The knowledge of the first involves an acquaintance with

CHAP. I.] Physical Measurements. 9

some of the elementary properties of space, and to under- stand the latter we must have some acquaintance with the properties of matter, the medium by which we are able to realise the existence of force and energy, and with the pro- perties of motion, since all energy is more or less connected with the motion of matter. We cannot, then, do better than urge those who intend making physical experiments to begin by obtaining a sound knowledge of those principles of dynamics, which are included in an elementary account of the science of matter and motion. The opportunity has been laid before them by one — to whom, indeed, many other debts of gratitude are owed by the authors of this work — who was well known as being foremost in scientific book-writing, as well as a great master of the subject. For us it will be sufficient to refer to Maxwell's work on ' Mattel and Motion ' as the model of what an introduction to the study of physics should be.

CHAPTER II.

UNITS OF MEASUREMENT.

Method of Expressing a Physical Quantity.

IN considering how to express the result of a physical experi- ment undertaken with a view to measurement, two cases essentially different in character present themselves. In the first the result which we wish to express is a concrete physical quantity^ and in the second it is merely the ratio of two physical quantities of the same kind, and is accordingly a number. It will be easier to fix our ideas on this point if we consider a particular example of each of these cases, instead of discussing the question in general terms. Con- sider, therefore, the difference in the expression of the result of two experiments, one to measure a quantity of heat and the second to measure a specific heat — the measurements

IO Practical Physics. [CHAP. II.

of a mass and a specific gravity might be contrasted in a perfectly similar manner — in the former the numerical value will be different for every different method employed to express quantities of heat ; while in the latter the result, being a pure number, will be the same whatever plan of measuring quantities of heat may have been adopted in the course of the experiment, provided only that we have adhered through- out to the same plan, when once adopted. In the latter case, therefore, the number obtained is a complete expression of the result, while in the former the numerical value alone conveys no definite information. We can form no estimate of the magnitude of the quantity unless we know also the unit which has been employed. The complete expression, therefore, of a physical quantity as distinguished from a mere ratio consists of two parts : (i) the unit quantity employed, and (2) the numerical part expressing the number of times, whole or fractional, which the unit quantity is contained in the quantity measured. The unit is a concrete quantity of the same kind as that in the expression of which it is used.

If we represent a quantity by a symbol, that must likewise consist of two parts, one representing the numerical part and the other representing the concrete unit. A general form for the complete expression of a quantity may therefore be taken to be q [Q], where q represents the numerical part and [Q] the concrete unit. For instance, in representing a certain length we may say it is 5 [feet], when the numerical part of the expression is 5 and the unit i [foot]. The number q is called the numerical measure of the quantity for the unit [Q].

Arbitrary and Absolute Units.

The method of measuring a quantity, q [Q], is thus resolved into two parts : (i) the selection of a suitable unit [Q], and (2) the determination of q, the number of times which this unit is contained in the quantity to be measured. The second part is a matter for experimental determination, and

CHAP. II.] Units of Measurement. 1 1

has been considered in the preceding chapter. We proceed to consider the first part more closely.

The selection of [Q] is, and must be, entirely arbitrary — that is, at the discretion of the particular observer who is making the measurement. It is, however, generally wished by an observer that his numerical results should be under- stood and capable of verification by others who have not the advantage of using his apparatus, and to secure this he must be able so to define the unit he selects that it can be .repro- duced in other places and at other times, or compared with the units used by other observers. This tends to the general adoption on the part of scientific men of common standards of length, mass, and time, although agreement on this point is not quite so general as could be wished. There are, however, two well-recognised standards of length1 : viz. (i) the British standard yard, which is the length at 62° F. between two marks on the gold plugs of a bronze bar in the Standards Office ; and (2) the standard metre as kept in the French Archives, which is equivalent to 39*37079 British inches. Any observer in measuring a length adopts the one or the other as he pleases. All graduated instru- ments for measuring lengths have been compared either directly or indirectly with one of these standards. If great accuracy in length measurement is required a direct com- parison must be obtained between the scale used and the standard. This can be done by sending the instrument to be used to the Standards Office of the Board of Trade.

There are likewise two well-recognised standards of mass , viz. (i) the British standard pound, a certain mass of platinum kept in the Standards Office ; and (2) the kilogramme des Archives, a mass of platinum kept in the French Archives, originally selected as the mass of one thou- sandth part of a cubic metre of pure water at 4° C. One

1 See Maxwell's Heat, chap. iv. The British Standards are now kept at the Standards Office at the Board of Trade, Westminster, in accordance with the * Weights and Measures Act,' 1878.

12 Practical Physics. [CHAP. II.

or other of these standards, or a simple fraction or multiple of one of them, is generally selected as a unit in which to measure masses by any observer making mass measure- ments. The kilogramme and the pound were carefully com- pared by the late Professor W. H. Miller ; one pound is equivalent to '453593 kilogramme.

With respect to the unit of time there is no such divergence, as the second is generally adopted as the unit of time for scientific measurement. The second is -g-^V^i. of the mean solar day, and is therefore easily reproducible- as long as the mean solar day remains of its present length.

These units of length, mass, and time are perfectly arbi- trary. We might in the same way, in order to measure any other physical quantity whatever, select arbitrarily a unit quantity of the same kind, and make use of it just as we select the standard pound as a unit of mass and use it. Thus to measure a force we might select a unit of force, say the force of gravity upon a particular body at a particular place, and express forces in terms of it. This is the gravitation method of measuring forces which is often adopted in practice. It is not quite so arbitrary as it might have been, for the body generally selected as being the body upon which, at Lat. 45°, gravity exerts the unit force is either the standard pound or the standard gramme, whereas some other body quite unrelated to the mass standards might have been chosen. In this respect the gallon, as a unit of measurement of volume, is a better example of arbitrariness. It contains ten pounds of water at a certain temperature^

We may mention here, as additional examples of arbitrary units, the degree as a unit of angular measurement, the thermometric degree as the unit of measurement of tem- perature, the calorie as a unit of quantity of heat, the standard atmosphere, or atmo, as a unit of measurement of fluid pressure, Snow Harris's unit jar for quantities of electricity, and the B.A. unit of electrical resistance.

CHAP. II.] Units of Measurement. 13

Absolute Units.

The difficulty, however, of obtaining an arbitrary standard which is sufficiently permanent to be reproducible makes this arbitrary method not always applicable. A fair example of this is in the case of measurement of electro-motive force,1 for which no generally accepted arbitrary standard has yet been found, although ic has been sought for very diligently. There are also other reasons which tend to make physicists select the units for a large number of quantities with a view to simplifying many of the numerical calculations in which the quantities occur, and thus the arbitrary choice of a unit for a particular quantity is directed by a principle of selection which makes it depend upon the units already selected for the measurement of other quantities. We thus get systems of units, such that when a certain number of fundamental units are selected, the choice of the rest follows from fixed principles. Such a system is called an ' absolute ' system of units, and the units themselves are often called 'absolute,' although the term does not strictly apply to the individual units. We have still to explain the principles upon which absolute systems are founded

Nearly all the quantitative physical laws express relations between the numerical measures of quantities, and the general form of relation is that the numerical measure of some quantity, Q, is proportional (either directly or inversely) to certain powers of the numerical measures of the quan- tities x, Y, z . . . If q^ x, y, z, . . . be the numerical measures of these quantities, then we may generalise the physical law, and express it algebraically thus : q is propor- tional to xa, y*3, zr, . . ., or by the variation equation

q oc xa. ft . £y. . . .

where a, /3, y may be either positive or negative, whole or frac- tional. The following instances will make our meaning clear :

1 Since this was wittcn, Lord Ka) leigh has shewn that theE.M.F. of a Latimer-Ouk's cell is very nearly constant, and equal to 1-435 volt at 15° G

14 Practical Physics. [CHAP. II.

(i.) The volumes of bodies of similar shape are propor- tional to the third power of their linear dimensions, or

(2.) The rate of change of momentum is proportional to the impressed force, and takes place in the direction in which the force is impressed (Second Law of Motion), or

m a.

(3.) The pressure at any point of a heavy fluid is propor- tional to the depth of the point, the density of the fluid, and the intensity of gravity, or

(4.) When work produces heat, the quantity of heat produced is directly proportional to the quantity of work expended (First Law of Thermo-dynamics), or

(5.) The force acting upon a magnetic pole at the centre of a circular arc of wire in which a current is flowing, is directly proportional to the strength of the pole, the length of the wire, and the strength of the current, and inversely proportional to the square of the radius of the circle, or

and so on for all the experimental physical laws.

We may thus take the relation between the numerical measures —

q oc xay* zy . . .

to be the general form of the expression ot an experimental law relating to physical quantities. This may be written in the form

q = kxaylszf ...... (i)

when k is a 'constant.'

This equation, as we have already stated, expresses a

CHAP. Il.j Units of Measurement. 15

relation between the numerical measures of the quantities involved, and hence if one of the units of measurement is changed, the numerical measure of the same actual quan- tity will be changed in the inverse ratio, and the value of k will be thereby changed.

We may always determine the numerical value of k if we can substitute actual numbers for q, x, y, z, ... in the equation (i).

For example, the gaseous laws may be expressed in words thus: —

* The pressure of a given mass of gas is directly pro- portional to the temperature measured from —273° C., and inversely proportional to the volume,' or as a variation equation —

or

We may determine k for i gramme of a given gas, say hydrogen, from the consideration that i gramme of hydro- gen, at a pressure of 760 mm. of mercury and at o° C., occu- pies IT 200 cc.

Substituting / = 760, 6= 273, v — 11200, we get

and hence

/=3ii8o- . , . (2).

Here/ has been expressed in terms of the length of an equivalent column of mercury ; and thus, if for v and 0 we substitute in equation (2) the numerical measures of any volume and temperature respectively, we shall obtain the corresponding pressure of i gramme of hydrogen expressed in millimetres of mercury.

This, however, is not the standard method of expressing

1 6 Practical Physics. [CHAP. II.

a pressure ; its standard expression is the force per unit of area. If we adopt the standard method we must substitute for/ not 760, but 76 x 13*6 x 981, this being the number of units of force l in the weight of the above column of mercury of one square-centimetre section. We should then get for k a different value, viz. : —

, I,OI4,OOOX II200 K = — - — --- =41500000,

so that

A

p = 41500000- . . . (3),

and now substituting any values for the temperature and volume, we have the corresponding pressure of i gramme of hydrogen expressed in units of force per square centimetre.

Thus, in the general equation (i), the numerical value of k depends upon the units in which the related quantities are measured ; or, in other words, we may assign any value we please to k by properly selecting the units in which the related quantities are measured.

It should be noticed that in the equation

we only require to be able to select one of the units in order to make k what we please ; thus x, y, z, . . . may be beyond our control, yet if we may give q any numerical value we wish, by selecting its unit, then k may be made to assume any value required. It need hardly be mentioned that it would be a very great convenience if k were made equal to unity. This can be done if we choose the proper unit in which to measure Q. Now, it very frequently happens that there is no other countervailing reason for selecting a different unit in which to measure Q, and our power of arbitrary selection of a unit for Q is thus exercised, not by selecting a particular quantity of the same kind as Q as unit, 1 The units offeree here used are dynes or C.G.s. units offeree.

CHAP. II.] Units of Measurement. 17

and holding to it however other quantities may be mea- sured, but by agreeing that the choice of a unit for Q shall be determined by the previous selections of units for x, y, z, . . . together with the consideration that the quancity k shall be equal to unity.

Fundamental Units and Derived Units. It is found that this principle, when fully carried out, leaves us free to choose arbitrarily three units, which are therefore called fundamental units, and that most of the other units employed in physical measurement can be defined with reference to the fundamental units by the consider- ation that the factor k in the equations connecting them shall be equal to unity. Units obtained in this way are called derived units, and all the derived units belong to an absolute system based on the three fundamental units.

Absolute Systems of Units.

Any three units (of which no one is derivable from the other two) may be selected as fundamental units. In those systems, however, at present in use, the units of length, mass, and time have been set aside as arbitrary fundamental units, and the various systems of absolute units differ only in regard to the particular units selected for the measure- ment of length, mass, and time. In the absolute system adopted by the British Association, the fundamental units selected are the centimetre, the gramme, and the second re- spectively, and the system is, for this reason, known as the C.G.S. system.

For magnetic surveying the British Government uses an absolute system based on. the foot, grain, and second ; and scientific men on the Continent frequently use a system based on the millimetre, milligramme, and second, as fun- damental units. An attempt was also made, with partial success, to introduce into England a system of absolute units, based upon the foot, pound, and second as funda- mental units.

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PH

LECTRO-MAGNE TIC UNITS.1

'he force acting upon a magnetic pole at the centre of a circular arc of wire carrying a current, is proportional to the strength of the current, the length of the wire, and the strength of the

pole, ana inversely pro- portional to the square of the radius of the arc. he quantity of electricity which passes across any

sii«l|§ g.ita.g01

?^ O^Q >,'3 JJ #« « „•« 0.-S rt 0.3 g'C ? o

1^I||F|^ IHlUlls Ilil^lili

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CHAP. II.] Units of Measurement. 21

The C.G.S. System.

The table, p. 18, shows the method of derivation of such absolute units on the C.G.S. system as we shall have occasion to make use of in this book. The first column contains the denominations of the quantities measured ; the second contains the verbal expression of the physical law on which the derivation is based, while the third gives the expression of the law as a variation equation j the fourth and fifth columns give the definition of the C.G.S. unit obtained and the name assigned to it respectively, while the last gives the dimensional equation. This will be explained later (p. 24).

The equations given in the third column are reduced to ordinary equalities by the adoption of the unit defined in the next column, or of another unit belonging to an absolute system based on the same principles.

Some physical laws express relations between quantities whose units have already been provided for on the absolute system, and hence we cannot reduce the variation equations to ordinary equalities. This is the case with the formula for the gaseous laws already mentioned (p. 15).

A complete system of units has thus been formed on the C.G.S. absolute system, many of which are now in practical use. Some of the electrical units are, however, proved to be not of a suitable magnitude for the electrical measurements most frequently occurring. For this reason practical units have been adopted which are not identical with the C.G.S. units given in the table (p. 20), but are immediately derived from them by multiplication by some power of 10. The names of the units in use, and the factors of derivation from the corresponding C.G.S. units are given in the following table : —

22

Practical Physics.

[CHAP. II.

TABLE OF PRACTICAL UNITS FOR ELECTRICAL MEASUREMENT RELATED TO THE C.G.S. ELECTRO-MAGNETIC SYSTEM.

Quantity

Unit

Equivalent in C.G.S. units

Electric current Electromotive force

Ampere Volt

IO ~*

I08

Resistance

Ohm

I09

Capacity . Rate of working Quantity of Electricity

Farad Watt Coulomb

io-9

IO7

io-1

To shorten the notation when a very small fraction or a very large multiple of a unit occurs, the prefixes micro- and mega- have been introduced to represent respectively divi- .sion and multiplication by io6. Thus:— -

A mega-dyne = i o6 dynes.

A micro -farad = i— , farad. 10°

Arbitrary Units at present employed.

For many of the quantities referred to in the table (p. 18) no arbitrary unit has ever been used. Velocity, for instance, has always been measured by the space passed over in a unit of time. And for many of them the physical law given in the second column is practically the definition of the quantity ; for instance, in the case of resistance, Ohm's law is the only definition that can be given of resistance as a measurable quantity.

For the measurement of some of these quantities, how- ever, arbitrary units have been used, especially for quan- tities which have long been measured in an ordinary way as volumes, forces, &c.

Arbitrary units are still in use for the measurement of temperature and quantities of heat; also for light intensity, and some other magnitudes.

We have collected in the following table some of the arbitrary units employed, and given the results of experi- mental determinations of their equivalents in the absolute

CHAP. II.]

Units of Measurement.

23

units for the measurement of the same quantity when such exist : —

TABLE OF ARBITRARY UNITS.

Quantity

Arbitrary unit employed

Equivalent in absolute units

Angle

Degree (\-§ part of two

right angles)

Radian (unit of circular

measure)

Force

Pound weight

32-2 poundals (British

absolute units)

Gramme weight

981 dynes

Work

Foot-pound

32-2 foot-poundals

Kilogramme-metre

981 x io7 ergs

Temperature

Degree Centigrade, corre-

sponding to T^ of the

expansion of mercury

in glass between the

freezing and boiling

points ; degree Fahren-

heit, corresponding to

•~ of the same quantity

Quantity of

Amount of heat required

The gramme - centi-

heat

to raise the temperature

grade unit is equi-

of unit mass of water

valent to 4*214 x io7

one degree

ergs

Intensity of

Standard candle. Sperm

light

candles of six to the

pound, each burning 120

grains an hour

The Paris Conference stan:

dard. The light emitted

by I sq. cm. of platinum

at its melting point

Electrical re-

The B.A. unit (originally

•9867 true ohm '

sistance

intended to represent the

r- ohm)

The 'legal ohm' adopted

•9976 true ohm '

by the Paris Conference.

The resistance at o° C.

of a column of mercury

1 06 cm. long, and of I

sq. mm. cross-section

Cavendish Laboratory determinations.

24 Practical Physics. [CHAP. II.

Changes from one Absolute System of Units to another. Dimensional equations.

We have already pointed out that there are more than one absolute system of units in use by physicists. They are deduced in accordance with the same principles, but are based on different values assigned to the fundamental units. It becomes, therefore, of importance to determine the factor by which a quantity measured in terms of a unit be- longing to one system must be multiplied, in order to express it in terms of the unit belonging to another system. Since the systems are absolute systems, certain variation equations become actual equalities ; and since the two systems adopt the same principles, the corresponding equations will have the constant k equal to unity for each system. Thus, if we take the equation (i) (p, 14) as a type of one of these equa- tions, we have the relation between the numerical measures

holding simultaneously for both systems.

Or, if q, x, y, z, be the numerical measures of any quan- tities on the one absolute system ; q' , x', y, zf, the numerical measures of the same actual quantities on the other system, then q = x»fz, ..,.(,)

and ?' = *'•/ *'" • • • • (2)-

Now, following the usual notation, let [Q], [x], [Y], [z] be the concrete units for the measurement of the quantities on the former, which we will call the old, system, [Q'], [x'], [Y'], [z'] the concrete units Tor their measurement on the new system.

Then, since we are measuring the same actual quantities,

y[v]=f [V]

« W - * [z'J 1 The symbol = is used to denote ^bspiuie identity, as distinguished from numerical equality.

CHAP. II.] Units of Measurement. 25

In these we may see clearly the expression of the well-known law, that if the unit in which a quantity is measured be changed, the ratio of the numerical measures of the same quantity for the two units is the inverse ratio of the units.

From equations (i) and (2) we get

and substituting from (3).

Thus, if £, 17, £ be the ratio of the new units [x'], [Y;], [z'] to the old units [x], [Y], [z] respectively, then the ratio p of the new unit [Q'] to the old unit [Q] is equal to £*vft?, and the ratio of the new numerical measure to the old is the reciprocal of this.

Thus

P = *Vfr . . . (4).

The equation (4), which expresses the relation between the ratios in which the units are changed, is of the same form as (i), the original expression of the physical law. So that whenever we have a physical law thus expressed, we get at once a relation between the ratios in which the units are changed. We may, to avoid multiplying notations, write it, if we please, in the following form : —

[Q] = [X]-[Y]«[Z]' (5),

where now [Q], [x], [Y], [z] no longer stand for concrete units, but for the ratios in which the concrete units are changed. It should be unnecessary to call attention to this, as it is, of course, impossible even to imagine the multiplication of one concrete quantity by another, but the constant use of the identical form may sometimes lead the student to infer that the actual multiplication or division of concrete quantities

26 Practical Physics. [CHAP. II.

takes place. If we quite clearly understand that the sen- tence has no meaning except as an abbreviation, we may express equation (5) in words by saying that the unit of Q is the product of the a power of the unit of x, the ft power of the unit of Y, and the y power of the unit of z ; but if there is the least danger of our being taken at our word in express- ing ourselves thus, it would be better to say that the ratio in which the unit of Q is changed when the units of x, Y, z are changed in the ratios of [x] : i [Y] : i and [z] : i re-1 spectively is equal to the product of the a power of [x], the /? power of [Y], and the y power of [z].

We thus see that if [x], [Y], [z] be the ratios of the new units to the old, then equation (5) gives the ratio of the new unit of Q to the old, and the reciprocal is the ratio. of the new numerical measure to the old numerical measure.

We may express this concisely, thus : — If in the equa- tion (5) we substitute for [x], [Y], [z] the new units in terms of the old, the result is the factor by which the old unit of Q must be multiplied to give the new unit ; if, on the other hand, we substitute for [x], [Y], [z] the old units in terms of the new, then the result is the factor by which the old numerical measure must be multiplied to give the new numerical measure.

If the units [x], [Y], [z] be derived units, analogous equations may be obtained, connecting the ratios in which they are changed with those in which the fundamental units are changed, and thus the ratio in which [Q] is changed can be ultimately expressed in terms of the ratios in which the fundamental units are changed.

We thus obtain for every derived unit

[L], [M], [T] representing the ratios in which the funda- mental units of length, mass, and time, respectively, are changed.

The equation (6) is called the dimensional equation for

CHAP. IT.] Units of Measurement. 27

[Q], and the indices a, (3, y are called the dimensions of Q with respect to length, mass, and time respectively.

The dimensional equation for any derived unit may thus be deduced from the physical laws by which the unit is denned, namely, those whose expressions are converted from variation equations to equalities by the selection of the unit.

We may thus obtain the dimensional equations which are given in the last column of the table (p. 18). We give here one or two examples.

(i) To find the Dimensional Equation for Velocity.

Physical law

s-=vtt or

Hence

(2) To find the Dimensional Equation for Force. Physical law

f = m a. Hecce

W-MWs

but

W-WW-*

••• M = [M]M[T]-'.

(3) To find the Dimensional Equation for Strength of Magnetic Pole.

Physical law

Hence

23 Practical Physics. [CHAP. II.

But

or

When the dimensional equations for the different units have been obtained, the calculation of the factor for con- version is a very simple matter, following the law given on p. 26. We may recapitulate the law here.

To find the Factor by which to multiply the Numerical Measure of a Quantity to convert it from the old System of Units to the new, substitute for [L] [M] and [T] in the Dimen- sional Equation the old Units of Length, Mass, and Time respectively, expressed in terms of the new.

We may shew this by an example.

To find the Factor for converting the Strength of a Mag- netic Pole from C.G.S. to Foot-gram-second Units —

i cm. = 0-0328 ft. i gm.= 15-4 grs.

Writing in the dimensional equation

M=[M]i[L]l[T]^

[M]=i5'4 [L] = 0-0328 [T] = I, we get

M = (15-4)* (-0328)!,

or the factor required

= -0233.

That is, a pole whose strength is 5 in C.G.S. units has a strength of '1165 foot-grain-second units.

Conversion of Quantities expressed in Arbitrary Units.

This method of converting from one system to another is only available when both systems are absolute and based on the same laws. If a quantity is expressed in arbitrary

CHAP. II.] Units of Measurement. 29

%

units, it must first be expressed in a unit belonging to some absolute system, and then the conversion factor can be cal- culated as above. For example : —

To express 15 foot-pounds in Ergs.

The foot-pound is not an absolute unit. We must first obtain the amount of work expressed in absolute units. Now, since g= 32-2 in British absolute units, i foot-pound = 32-2 foot-poundals (British absolute units).

.*. 15 foot-pounds = 15 X32'2 foot-poundals.

We can now convert from foot-poundals to ergs. The dimensional equation is

M-MWM-*.

Since

i foot = 30-5 cm. i Ib. = 454 gm. Substituting

[M]=454, [L] = 3o-s we get

[w] = 454 x (30-5)2.

Hence

15 foot-pounds = 15 x 32-2 x 454 x (30-5)2 ergs. = 2'04X i o8 ergs.

Sometimes neither of the units belongs strictly to an absolute system, although a change of the fundamental units alters the unit in question. For example : —

To find the Mechanical Equivalent of Heat in C. G. S. Centigrade Units, knowing that its Value for a Pound Fahrenheit Unit of Heat is 772 Foot-pounds.

The mechanical equivalent of heat is the amount of work equivalent to one unit of heat. For the C.G.S. Centi- grade unit of heat, it is, therefore,

2x -— X772 foot-pounds. 5 454

3° Practical Physics. [CHAP, III

This amount of heat is equivalent to

2x—- X772 x i'36x io7 ergs, 5 454

or the mechanical equivalent of heat in C.G.S. Centigrade units

= 4*14 x io7.

If the agreement between scientific men as to the selection of fundamental units had been universal, a great deal of arithmetical calculation which is now necessary would have been avoided. There is some hope that in future one uniform system may be adopted, but even then it will be necessary for the student to be familiar with the methods of changing from one system to another in order to be able to avail himself of the results already published. To form a basis of calculation, tables showing the equiva- lents of the different fundamental units for the measure- ment of the same quantity are necessary. Want of space prevents our giving them here ; we refer instead toNos. 9-12 of the tables by Mr. S. Lupton, recently published. We take this opportunity of mentioning that we shall refer to the same work * whenever we have occasion to notice the necessity for a table of constants for use in the experiments described.

CHAPTER III.

PHYSICAL ARITHMETIC

Approximate Measurements.

ONE of the first lessons which is learned by an experimenter making measurements on scientific methods is that the number obtained as a result is not a perfectly exact expres- sion of the quantity measured, but represents it only within

1 Numerical Tables and Constants in Elementary Science^ by S. Lupton.

CHAP. III.] Physical Arithmetic. 31

certain limits of error. If the distance between two towns be given as fifteen miles, we do not understand that the distance has been measured and found to be exactly fifteen miles, without any yards, feet, inches, or fractions of an inch, but that the distance is nearer to fifteen miles than it is to sixteen or fourteen. If we wished to state the distance more accurately we should have to begin by defining two points, one in each town — marks, for instance, on the door- steps of the respective parish churches— between which the distance had been taken, and we should also have to sped .y the route taken, and so on. To determine the distance with the greatest possible accuracy would be to go through the laborious process of measuring a base line, a rough idea of which is given in § 5. We might then, perhaps, obtain the distance to the nearest inch and still be uncertain whether there should not be a fraction of an inch more or less, and if so, what fraction it should be. If the number is expressed in the decimal notation, the increase in the accuracy of measurement is shewn by filling up more decimal places. Thus, if we set down the mechanical equivalent of heat at 4*2 x io7 ergs, it is not because the figures in the decimal places beyond the 2 are all zero, but because we do not know what their values really are, or it may be, for the purpose for which we are using the value, it is immaterial what they are. It is known, as a matter of fact, that a more accurate value is 4*214 x io7, but at present no one has been able to determine what figure should be put in the decimal place after the 4.

Errors and Corrections.

The determination of an additional figure in a number representing the magnitude of a physical quantity generally involves a very great increase in the care and labour which must be bestowed on the determination. To obtain some idea of the reason for this, let us take, as an example, the case of determining the mass of a body of about 100

32 Practical Physics. [CHAP. III.

grammes. By an ordinary commercial balance the mass of a body can be easily and rapidly determined to i gramme, say 103 grammes. With a better arranged balance we may shew that 103-25 is a more accurate representation of the mass. We may then use a very sensitive chemical balance which shews a difference of mass of o'i mgm., but which requires a good deal of time and care in its use, and get a value 103*2537 grammes .as the mass. But, if now we make another similar determination with another balance, or even with the same balance, at a different time, we may find the result is not the same, but, say, 103 2546 grammes. We have thus, by the sensitive balance, carried the measurement two decimal places further, but have got from two observations two different results, and have, there- fore, to decide whether either of these represents the mass of the body, and, if so, which. Experience has shewn that some, at any rate, of the difference may be due to the balance not being in adjustment, and another part to the fact that the body is weighed in air and not in vacuo. The observed weighings may contain errors due to these causes. The effects of these causes on the weighings can be cal- culated when the ratio of the lengths of the arms and other facts about the balance have been determined, and when the state of the air as to pressure, temperature, and moisture is known (see §§ 13, 14).

We may thus, by a series of auxiliary observations, determine a correction to the observed weighing correspond- ing to each known possible error. When the observations are thus corrected they will probably be very much closer. Suppose them to be 103 2543 and 103 '2542.

Mean of Observations.

When all precautions have been taken, and all known errors corrected, there may still be some difference between different observations which can^j&nly arise from causes beyond the knowledge and control of the observer. We

CHAP. III.] Physical Arithmetic. 33

must, therefore, distinguish between errors due to known causes, which can be allowed for as corrections, or elimi- nated by repeating the observations under different con- ditions, and errors due to unknown causes, which are called 'accidental ' errors. Thus, in the instance quoted, we know of no reason for taking 103-2543 as the mass of the body in preference to 103 '2542. It is usual in such cases to take the arithmetic mean of the two observations, i.e. the number obtained by adding the two values together, and dividing by 2, as the nearest approximation to the true value.

Similarly if any number, n, of observations be taken, each one of which has been corrected for constant errors, and is, therefore, so far as the observer can tell, as worthy of confidence as any of the others, the arithmetic mean of the values is taken as that most nearly representing the true value of the quantity. Thus, if q\, q^ q$ • • • • qn be the results of the n observations, the value of q is taken to be

It is fair to suppose that, if we take a sufficient number of observations, some of them give results that are too large, others again results that are too small ; and thus, by taking the mean of the observations as the true value, we approach more nearly than we can be sure of doing by adopting any single one of the observations.

We have already mentioned that allowance must be made by means of a suitable correction for each constant error, that is for each known error whose effect upon the result may be calculated or eliminated by some suitable arrangement. It is, of course, possible that the observer may have overlooked some source of constant error which will affect the final result. This must be very carefully guarded against, for taking the mean of a number of obser-

D

34 Practical Physics. [CHAP. III.

vations affords, in general, no assistance m the elimination of an error of that kind.

The difference between the mean value and one of the observations is generally known technically as the ' error ' of that observation. The theory of probabilities has been applied to the discussion of errors of observations !, and it has been shewn that by taking the mean of n observations instead of a single observation, the so-called 'probable error ' is reduced in the ratio of i / >J~nI

On this account alone it would be advisable to take several observations of each quantity measured in a physical experiment. By doing so, moreover, we not only get a result which is probably more accurate, but we find out to what extent the observations differ from each other, and thus obtain valuable information as to the degree of accuracy of which the method of observation is capable. Thus we have, on p. 54, four observations of a length, viz. —

3'333 in. 3'332 » 3*334 „ 3 "334 „ Mean =: 3-3332 „

Taking the mean we are justified in assuming that the true length is accurately represented by 3*333 to the third decimal place, and we see that the different observations differ only by two units at most in that place.

In performing the arithmetic for finding the mean of a number of observations, it is only necessary to add those columns in which differences occur— the last column of the example given above. Performing the addition on the other columns would be simply multiplying by 4, by which number we should have subsequently to divide.

An example will make this clear.

1 See Airy's tract on the Theory of Errors of Observations.

CHAP, in.] Physical Arithmetic. 35

Find the mean of the following eight observations : — 56-231 56-275 56-243 56-255 56-256

56-267

56-273 56-266

Adding (8 x 56-2 +) -466 Mean"! . 56-2582

The figures introduced in the bracket would not appear in ordinary working.

The separate observations of a measurement should be made quite independently, as actual mistakes in reading are always to be regarded as being within the bounds of pos- sibility. Thus, for example, mistakes of a whole degree are sometimes made in reading a thermometer, and again in weighing, a beginner is not unlikely to mis-count the weights. Mistakes of this kind, which are to be very care- fully distinguished from the * errors of observation,' would probably be detected by an independent repetition of the observation. If there be good reason for thinking that an observation has been affected by an unknown error of this kind, the observation must be rejected altogether.

Possible Accuracy of Measurement of different Quantities.

The degree of accuracy to which measurements can be carried varies very much with different experiments. It is usual to estimate the limit of accuracy as a fractional part or percentage of the quantity measured.

Thus by a good balance a weighing can be carried out to a tenth of a milligramme ; this, for a body weighing about 100 grammes, is as far as one part in a million, or -oooi per cent. — an accuracy of very high order. The measurement

D 2

36 Practical Physics. [CHAP. III.

of a large angle by the spectrometer (§ 62) is likewise very accurate ; thus with a vernier reading to 20", an angle of 45° can be read to one part in four thousand, or 0*025 per cent. On the other hand, measurements of temperature cannot, without great care, be carried to a greater degree of accuracy than one part in a hundred, or i per cent., and sometimes do not reach that. A length measurement often reaches about one part in ten thousand. For most of the experiments which are described in this work an accuracy of one part in a thousand is ample, indeed generally more than sufficient.

It is further to be remarked that, if several quantities have to be observed for one experiment, some of them may be capable of much more accurate determination than others. It is, as a general rule, useless to carry the accuracy of the former beyond the possible degree of accuracy of the latter. Thus, in determining specific heats, we make some weighings and measure some temperatures. It is useless to determine the weights to a greater degree of accuracy than one part in a thousand, as the accuracy of the result will not reach that limit in consequence of the inaccuracy of the temperature measurements. In some cases it is necessary that one measurement should be carried out more accurately than others in order that the errors in the result may be all of the same order. The reason for this will be seen on p. 48.

Arithmetical Manipulation of Approximate Values.

In order to represent a quantity to the degree of accuracy

of one part in a thousand, we require a number with four

digits at most, exclusive of the zeros which serve to mark the

position of the number in the decimal scale. ! It frequently

1 It is now usual, when a very large number has to be expressed, to write down the digits with a decimal point after the first, and indicate its position in the scale by the power of 10, by which it must be mul- tiplied : thus, instead of 42140000 we write 4*214 * io7. A corre- sponding notation is used for a very small decimal fraction : thus, instead of -00000588 we write 5-88 x io~6.

CHAP, ill.] Physical Arithmetic. 37

happens that some arithmetical process, employed to deduce the required result from the observations, gives a number containing more than the four necessary digits. Thus, if we take seven observations of a quantity, each to three figures, and take the mean, we shall usually get any number of digits we please when we divide by the 7. But we know that the observations are only accurate to three figures; hence, in the mean'obtained, all the figures after the fourth, at any rate, have no meaning. They are introduced simply by the arithmetical manipulation, and it is, therefore, better to discard them. It is, indeed, not only useless to retain them, but it may be misleading to do so, for it may give the reader of the account of the experiment an impression that the measurements have been carried to a greater degree of accuracy than is really the case. Only those figures, there- fore, which really represent results obtained by the measure- ments should be included in the final number. In dis- carding the superfluous digits we must increase the last digit retained by unity, if the first digit discarded is 5 or greater than 5. Thus, if the result of a division gives 3 2 '3 1 6, we adopt as the value 32*32 instead of 32*31. For it is evident that the four digits 32*32 more nearly re- present the result of the division than the four 32*31.

Superfluous figures very frequently occur in the multi- plication and division of approximate values of quantities. These have also to be discarded from the result ; for if we multiply two numbers, each of which is accurate only to one part in a thousand, the result is evidently only accurate to the same degree, and hence all figures after the fourth must be discarded.

The arithmetical manipulation may be performed by using logarithms, but it is sometimes practically shorter to work out the arithmetic than to use logarithms ; and in this case the arithmetical process may be much abbreviated by discarding unnecessary figures in the course of the work.

38 Practical Physics. [CHAP. III.

The following examples will show how this is managed:— Example (i). — Multiply 656-3 by 4-321 to four figures.

Ordinary form Abbreviated form

656-3 656-3

4-32I 4-32I

6563 (656-3x4) =2625-2

I3I26 (656x3) = 196-8

19689 (65x2) = 13-0

26252 (6 x i) = 6

2835-8723 2835-6

Result 2836 Result 2836

The multiplication in the abbreviated form is conducted in the reverse order of the digits of the multiplier. Each successive digit of the multiplier begins at one figure further to the left of the multiplicand. The decimal point should be fixed when the multiplication by the first digit (the 4) is completed. To make sure of the result being accurate to the requisite number of places, the arithmetical calculation should be carried to one figure beyond the degree of accuracy ultimately required.

Example (2). — Divide 65-63 by 4-391 to four figures,

Ordinary form Abbreviated form

4-391) 65-63000 (14946 4'390 65-630 (14948

4391 4391

21720 17564

(439) -4156 3951

•20410 (43) -205

17564 172

•2846 (4) '33

Result 14-95 Result 14-95

In the abbreviated form, instead of performing the successive steps of the division by bringing down o's, sue-

CHAP. III.] Physical Arithmetic. 39

cessive figures are cut off from the divisor, beginning at the right hand ; thus, the divisors are for the first two figures of the quotient 4391 ; for the next figure, 439 • for the next, 43. It can then be seen by inspection that the next figure is 8. The division is thus accomplished.

It will be seen that one o is added to the dividend ; the arithmetic is thus carried, as before, to one figure beyond the accuracy ultimately required. This may be avoided if we always multiply the divisor mentally for one • figure beyond that which we actually use, in order to determine what number to ' carry ' • the number carried appears in the work as an addition to the first digit in the multipli- cation.

The method of abbreviation, which we have here sketched, is especially convenient for the application of small corrections (see below, p. 42). We have then, gene- rally, to multiply a number by a factor differing but little from unity ; let us take, for instance, the following : —

Example (3). — Multiply 563*6 by 1*002 to four places of decimals.

Adopting the abbreviated method we get— 563*6

1*002

I'l

5647

Result 5647 or

Example (4). — Multiply 563-6 by '9998. In this case '9998 =» I - -0002.

I — *OOO2

— 1*1

Result 562*5

4O Practical Physics. [CHAP. III.

It will be shewn later (p. 44) that dividing by '9998 is the same, as far as the fourth place of decimals is concerned, as multiplying by 1-002, and vice versa-, this suggests the possibility of considerable abbreviation of arithmetical cal- culation in this and similar cases.

Facilitation of Arithmetical Calculation by means of Tables. — Interpolation.

The arithmetical operations of multiplication, division, the determination of any power of a number, and the ex- traction of roots, may be performed, to the required degree of approximation, by the use of tables of logarithms. The method of using these for the purposes mentioned is so well known that it is not necessary to enter into details here. A table of logarithms to four places of decimals is given in Lupton's book, and is sufficient for most of the calculations that we require. If greater accuracy is necessary, Cham- bers's tables may be used. Instead of tables of logarithms, a * slide-rule ' is sometimes employed. The most effective is probably 'Fuller's spiral slide rule,' which is made and sold by Stanley of Holborn. By this two numbers of four figures can be multiplied or divided.

Besides tables of logarithms, tables of squares, cubes, square roots, cube roots, and reciprocals may be used. Short tables will be found in Lupton's book (pp. 1-4); for more accurate work Barlow's tables should be used. Besides these the student will require tables of the trigono- metrical functions, which will also be found among Lupton's tables.

An arithmetical calculation can frequently be simplified on account of some special peculiarity. Thus, dividing by 5 is equivalent to multiplying by 2, and moving the decimal point one place to the left. Again, 7r2 = 9-87 = 10 — -13, and many other instances might be given ; but the student can only make use of such advantages by a familiar acquaint- ance with cases in which they prove of service.

CHAP. III.] Physical Arithmetic. 41

In some cases the variations of physical quantities are also tabulated, and the necessity of performing the arith- metic is thereby saved. Thus, No. 31 of Lupton's tables gives the logarithms of (i + 'oc^y/) for successive degrees of temperature, and saves calculation when the volume or pressure of a mass of gas at a given temperature is required. A table of the variation of the specific resistance of copper with variation of temperature, is given on p. 47 of the same work.

It should be noticed that all tables proceed by certain definite intervals of the varying element ; for instance, for successive degrees of temperature, or successive units in the last digit in the case of logarithms ; and it may happen that the observed value of the element lies between the values given in the table. In such cases the required value can generally be obtained by a process known as 'interpolation.' If the successive intervals, for which the table is formed, are small enough, the tabulated quantity may be assumed to vary uniforntly between two successive steps of the varying element, and the increase in the tabulated quantity may be calculated as being proportional to the increase of the vary- ing element. We have not space here to go more into detail on this question, and must content ourselves with say- ing that the process is strictly analogous to the use of ' pro- portional parts' in logarithms. We may refer to §§ 12, 19, 77 for examples of the application of a somewhat analogous method of physical interpolation.

Algebraical Approximation. Approximate Formula, Introduction of small Corrections.

If we only require to use a formula to give a result accurate within certain limits, it is, in many cases, possible to save a large amount of arithmetical labour by altering the form of the formula to be employed. This is most frequently the case when any small correction to the value of one of the observed elements has to be introduced, as in the case,

42 Practical Physics. [CHAP. III.

for instance, of an observed barometric height which has to be corrected for temperature. We substitute for the strictly accurate formula an approximate one, which renders the calculation easier, but in the end gives the same result to the required degree of accuracy.

We have already said that an accuracy of one part in a thousand is, as a rule, ample for our purpose ; and we may, therefore, for the sake of definiteness, consider the simplifi- cation of algebraical formulae with the specification of one part in a thousand, or o'i per cent., as the limit of accuracy desired. Whatever we have to say may be easily adapted for a higher degree of accuracy, if such be found to be necessary.

It is shewn in works on algebra that

(i + x)n = i + n x + n±-^^'x2 + terms involving higher

2

powers of x ........ (i).

This is known as the * binomial theorem/ and is true for all values of n positive or negative, integral or frac- tional. Some special cases will probably be familiar to every student, as : —

If we change the sign of x we get the general formula in the form

We may include both in one form, thus : —

where the sign ± means that either the + or the — is to be taken throughout.

CHAP. III.] Physical Arithmetic. 43

Now, if x be a small fraction, say, i/iooo or o'ooi, xz is evidently a much smaller fraction, namely, 1/1000,000, or o-oooooi, and .v3 is still smaller. Thus, unless n is very large indeed, the term

will be too small to be taken account of, and the terms which follow will be of still less importance. We shall probably not meet with formulae in which n is greater than 3. Let us then determine the value of x so that

— > 1 3*

may be equal to !ooi, that is to say, may just make itself felt in the calculations that we are now discussing. Putting n = 3 we get

3A;2 = 'ooi x — ^ -00033 = '02 roughly.

So that we shall be well within the truth if we say that (when n = 3), if x be not greater than o'oi, the third term of equation (i) is less than *ooi, and the fourth term less than -oooo i. Neither of these, nor anyone beyond them, will, therefore, affect the result, as far as an accuracy of one part in a thousand is concerned ; and we may, therefore, say that, if x is not greater than o'oi,

To use this approximate formula when x = o'oi would be inadmissible, as it produces a considerable effect upon the next decimal place ; and, if in the same formula, we make other approximations of a similar nature, the accumulation of approximation may impair the accuracy of the result.

In any special case, therefore, it is well to consider

44 Practical Physics. [CHAP. III.

whether x is small enough to allow of the use of the approxi- mate formula by roughly calculating the value of the third term ; it is nearly always so if it is less than -005. This in- cludes the important case in which x is the coefficient of expansion of a gas for which x = '00366.

If n be smaller than 3, what we have said is true within still closer limits ; and as n is usually smaller than 3, we may say generally that, for our purposes,

(i+.v)"= i •!••«#, and

(i—x)n — i — nx,

provided x be less than 0*005.

Some special cases of the application of this method of approximation are here given, as they are of frequent occur- rence : —

(l±#)2= I±2X

(i±x)3 = i ±3*

</I±x = (i ±*)t = i ±?

i±x

The formulae for +x and — x are here included in one expression ; the upper or lower sign must be taken through- out the formula.

We thus see that whenever a factor of the form (i±^)' occurs in a formula where x is a small fraction, we ma) replace it by the simpler but approximate factor i±_nx\ and we have already shown how the multiplication by such a factor may be very simply performed (p. 39). Cases o> the application of this method occur in §§ 13, 24 etc.

Another instance of the change of formula foi fhe pur

CHAP. III.] Physical Arithmetic. 45

poses of arithmetical simplicity is made use of in § 13. In that case we obtain a result as the geometric mean of two nearly equal quantities. It is an easy matter to prove algebraically, although we have not space to give the proof here, that the geometric mean of two quantities which differ only by one part in a thousand differs from the arithmetic mean of the two quantities by less than the millionth of either. It is a much easier arithmetical operation to find the arithmetic mean than the geometric, so that we substi- tute in the formula (x+x')/2 for *J x x'.

The calculation of the effect upon the trigonometrical ratios of an angle, due to a small fractional increase in the angle, may be included in this section. We know that

sin (6 + d) = sin 6 cos d 4- cos 6 sin d.

Now, reference to a table of sines and cosines will shew that cos d differs from unity by less than one part in a thousand if d'be less than 2° 33', and, if expressed in circular measure, the same value of d differs from sin d by one part in three thousand; so we may say that, provided dis less than 2|°, cos d is equal to unity, and sin d is equal to d expressed in circular measure.

The formula is, therefore, for our purposes, equivalent to

sin 0 + d = sin Q + d cos 6.

We may reason about the other trigonometrical ratios in a similar manner, and we thus get the following approximate formulae : —

sin (0±d) = sin 6±Jcos 0.

cos (Q±d) = cos <9zp</sin (9.

tan (0±d) = tan <9±</sec 2 0.

The upper or lower sign is to be t*aken throughout the formula.

If d be expressed in degrees, then, since the circular

46 Practical Physics. [CHAP. III.

measure of i° is 7r/i8o, that of d° is */7r/i8o, and the formulae become

sin (6±d) = sin ^

180 &c.

It has been already stated that approximate formulae are frequently available when it is required to introduce correc- tions for variations of temperature, and other elements which may be taken from tables of constants. There is besides another use for them which should not be overlooked, namely, to calculate the effect upon the result of an error of given magnitude in one of the observed elements. This is practically the same as calculating the effect of a hypothe- tical correction to one of the observed elements. In cases where the formula of reduction is simply the product or quotient of a number of factors each of which is observed directly, a fractional error of any magnitude in one of the factors produces in the result an error of the same frac- tional magnitude, but in other cases the effect is not so simply calculated. If we take one example it will serve to illustrate our meaning, and the general method of employ- ing the approximate formulae we have given in this chapter.

In § 75 electric currents are measured by the tangent galvanometer. Suppose that in reading the galvanometer we cannot be sure of the position of the needle to a greater accuracy than a quarter of a degree. Let us, there- fore, c onsider the following question : — * To find the effect upon the value of a current, as deduced from observations with the tangent galvanometer, of an error of a quarter of a degree in the reading?

The formula of reduction is

c = k tan 0.

Suppose an error^S has been made in the reading of 0, so that the observed value is

(p. 45)

CHAP, in.] Physical Arithmetic. 47

The fractional error q in the result is c'--c/&8sec2(9 8

_ _ c k tan 0 sin 0 cos 0

= 28 sin 2 0'

The error 8 must be expressed in circular measure ; if it be equivalent to a quarter of a degree, we have

_ -0087^2 ' * sufafl."

The actual magnitude of this fraction depends upon the value of 0, that is upon the deflection. It is evidently very great when 0 is very small, and least when 0 = 45°, when it is 0-9 per cent. From which we see not only that when 0 is known the effect of the error can be calculated, but also that the effect of an error of reading, of given magnitude, is least when the deflection is 45°. It is clear from this that a tangent galvanometer reading is most accurate when the deflection produced by the current is 45°. This furnishes an instance, therefore, of the manner in which the approxi- mate formulae we have given in this chapter can be used to determine what is the best experimental arrangement of the magnitudes of the quantities employed, for securing the greatest accuracy in . an experiment with given apparatus. The same plan may be adopted to calculate the best arrangement of the apparatus for any of the experiments described below.

In concluding this part of the subject, we wish to draw special attention to one or two cases, already hinted at, in which either the method of making the experiments, or the formula for reduction, makes it necessary to pay special attention to the accuracy of some of the elements observed. In illustration of the former case we may mention the weighing of a small mass contained in a large vessel. To

43 Practical Physics. [CHAP. III.

fix ideas on the subject, consider the determination of the mass of a given volume of gas contained in a glass globe, by weighing the globe full and empty. During the interval between the two weighings the temperature and pressure of the air, and in consequence the apparent weight of the glass vessel, may have altered. This change, unless allowed for, will appear, when the subtraction has been performed, as an error of the same actual magnitude in the mass of the gas, and may be a very large fraction of the observed mass of the gas, so that we must here take account of the variation in the correction for weighing in air, although such a precaution might be quite unnecessary if we simply wished to determine the actual mass of the glass vessel and its contents to the degree of accuracy that we have hitherto assumed. A case of the same kind occurs in the determination of the quantity of moisture in the air by means of drying tubes (§ 42).

Cases of the second kind referred to above often arise from the fact that the formulas contain differences of nearly equal quantities ; we may refer to the formulae employed in the correction of the first observations with Atwood's machine (§ 21), the determination of the latent heat of steam (§ 39), and the determination of the focal length of a concave lens (§ 54) as instances. In illustration of this point we may give the following question, in which the hypothetical errors introduced are not really very exaggerated.

' An observer, in making experiments to determine the focal length of a concave lens, measures the focal length of the auxiliary lens as 10-5 cm., when it is really 10 cm., and the focal length of the combination as 14-5 cm., when it is really 15 cm. ; find the error in the result introduced by the inaccuracies in the measurements.'

We have the formula

1 - T J^

F~/i 7,

CHAP. III.] Physical Arithmetic. 49

whence

putting in the true values of F and/i.

and putting the observed values

7 14-5x10-5 =_i5£^5 14-5-10-5 4

The fractional error thus introduced is 8-06

or more than 25 per cent., whereas the error in either observation was not greater than 5 per cent.

It will be seen that the large increase in the percentage error is due to the fact that the difference in the errors in F and/! has to be estimated as a fraction of F— /", ; this should lead us to select such a value of /i as will make F— /i as great as possible, in order that errors of given actual magnitude in the observations may produce in the result a fractional error as small as possible.

We have not space for more detail on this subject. The student will, we hope, be able to understand from the in- stances given that a large amount of valuable information as to the suitability of particular methods, and the selectior of proper apparatus for making certain measurements, can be obtained from a consideration of the formulae of reduc- tion in the manner we have here briefly indicated.

$0 Practical Physics. [Cn. IV. § i.

CHAPTER IV.

MEASUREMENT OF THE MORE SIMPLE QUANTITIES. LENGTH MEASUREMENTS.

THE general principle which is made use of in measuring lengths is that of direct comparison (see p. 2); in other words, of laying a standard, divided into fractional parts, against the length to be measured, and reading off from the standard the number of such fractional parts as lie between the extremities of the length in question. Some of the more important methods of referring lengths to a standard, and of increasing the accuracy of readings, may be exemplified by an explanation of the mode of using the following instruments.

i. The Calipers.

This instrument consists of a straight rectangular bar of brass, D E (fig. i), on which is engraved a finely-divided scale.

From this bar two steel jaws project. These jaws are at right angles to the bar ; the one, D F, is fixed, the other, c G, can slide along the bar, moving accurately parallel to itself. The faces of these jawrs, which are opposite to each other, are planed flat and parallel, and can be brought into contact. On the sliding piece c will be observed two short scales called verniers, and when the two jaws are in contact, one ^nd of each vernier, marked by an arrowhead in the figure, coincides with the end of the scale on the bar.1 If then, in any other case, we determine the position of this end of the vernier with reference to the scale, we find the distance between these two flat faces, and hence the length of any object which fits exactly between the jaws.-

It will be observed that the two verniers are marked ' out- sides and ' insides J respectively. The distance between the

1 If with the instrument employed this is found not to be the case, a correction must be made to the observed length, as described in § 3. A similar remark applies to § 2.

2 See frontispiece, fig. 3.

Cn. IV. § i.] Measurement of the Simple Quantities. 5 1

FIG. i.

jaws will be given by the outsides vernier. The other pair of faces of these two jaws, opposite tc the two plane parallel ones, are not plane, but cylindrical, the axes of the cylinders being also perpendicular to the length of the brass bar, so that the cross section through any point of the two jaws, when pushed up close together, will be of the shape of two U's placed opposite to each other, the total width of the two being exactly one inch. When they are in contact, it will be found that the arrowhead of the vernier attached to the scale marked insides reads exactly one inch, and if the jaws of the calipers be fitted inside an object to be mea- sured— e.g., the internal dimensions of a box— the reading of the vernier marked insides gives the distance required.

Suppose it is required to measure the length of a cylinder with flat ends. The cylinder is placed with its axis parallel to the length of the calipers. The screw A (fig. i) is then turned so that the piece attached to it can slide freely along the scale, and the jaws of the calipers are adjusted so as nearly to fit the cy- linder (which is shown by dotted lines in the diagram). The screw A

is then made to bite, so that the attached piece is ' clamped ' to the scale. Another screw, B, on the under side of the scale, will, if now turned, cause a slow motion of the jaw c G, and by means of this the fit is made as accurate as possible. This is considered to be attained when the cylinder is just held firm. This screw B is called the ' tangent screw,' and the adjustment is known as the 'fine adjustment.'

It now remains to read upon the scale the length of the cylinder. On the piece c will be seen two short scales — the ' outsides ' and ' insides ' already spoken of. These short scales are called ' verniers.' Their use is to increase the

£ 2

-1

" : ,

D

in

'>ji ' i >i i

11

.,,

C

1!

-

C G

-Tig

52 Practical Physics. [Cn. IV. § i.

accuracy of the reading, and may be explained as follows : suppose that they did not exist, but that the only mark on the piece c was the arrowhead, this arrowhead would in all probability lie between two divisions on the large scale. The length of the cylinder would then be less than that corresponding to one division, but greater than that corre- sponding to the other. For example, let the scale be actually divided into inches, these again into tenths of an inch, and the tenths into five parts each ; the small divisions will then be ^ inch or -02 inch in length. Suppose that the arrowhead lies between 3 and 4 inches, between the third and fourth tenth beyond the 3, and between the first and second of the five small divisions, then the length of the cylinder is greater than S + T^+^OJ i-e- >3'32 inches, but less than 3 + yV + ^V> i-e- <3'34 inches. The vernier enables us to judge very accurately what fraction of one small division the distance between the arrowhead and the next lower division on the scale is. Observe that there are twenty divisions on the vernier,1 and that on careful ex- amination one of these divisions coincides more nearly than any other with a division on the large scale. Count which division of the- vernier this is — say the thirteenth. Then, as we shall show, the distance between the arrowhead and the next lower division is -JJ of a small division, that is T-o"<hy='OI3 inch, and the length of the cylinder is therefore 3+A+A+Tiiw=3>32 + -oi3=3<333 inch.

We have now only to see why the number representing the division of the vernier coincident with the division of the scale gives in thousandths of an inch the distance between the arrowhead and the next lower division.

Turn the screw-head B till the arrowhead is as nearly coincident with a division on the large scale as you can make it. Now observe that the twentieth division on the vernier is coincident with another division on the large scale, and that the distance between this division and the first is nineteen small divisions. Observe also that no other 1 Various forms of vernier are figured in the frontispiece.

CH. IV. § i.] Measurement of the Simple Quantities. 5 3

divisions on the two scales are coincident. Both are evenly divided ; hence it follows that twenty divisions of the vernier are equal to nineteen of the scale— that is, one division on the vernier is -J-|ths of a scale division, or that one division on the vernier is less than one on the scale by Jo-th of a scale division, and this is -nnjffti1 of an inch.1

Now in measuring the cylinder we found that the thirteenth division of the vernier coincided with a scale divi- sion. Suppose the unknown distance between the arrowhead and next lower division is x. The arrowhead is marked o on the vernier. The division marked i will be nearer the next lower scale-division by irnroth of an inch, for a vernier division is less than a scale division by this amount. Hence the distance in inches between these two divisions, the one on the vernier and the other on the scale, will be

•^ ~~ TTJTFO"'

The distance between the thirteenth division of the vernier and the next lower scale division will similarly be

x ~~TOO &•

But these divisions are coincident, and the distance between them is therefore zero ; that is ^=Ti§-0-. Hence the rule which we have already used.

The measurement of the cylinder should be repeated four times, and the arithmetic mean taken as the final value. The closeness of agreement of the results is of course a test of the accuracy of the measurements.

The calipers may also be used to find the diameter of the cylinder. Although we cannot here measure surfaces which are strictly speaking flat and parallel, still the portions of the surface which are touched by the jaws of the calipers are very nearly so, being small and at opposite ends of a diameter.

Put the calipers on two low supports, such as a pair of glass rods of the same diameter, and place the cylinder on end upon the table. Then slide it between the jaws of the

1 Generally, if n divisions of the vernier are equal to n — I of the scale, then the vernier reads to i/«th of a division of the scale.

54 Practical Physics. [CH. IV. § 2.

calipers, adjusting the instrument as before by means of the tangent screw, until the cylinder is just clamped. Repeat this twice, reading the vernier on each occasion, and taking care each time to make the measurement across the same diameter of the cylinder.

Now take a similar set of readings across a diameter at right angles to the former.

Take the arithmetic mean of the different readings, as the result.

Having now found the diameter, you can calculate the area of the cross section of the cylinder. For this area is

— , d being the diameter. 4 •

The volume of the cylinder can also be found by

multiplying the area just calculated by the length of the cylinder.

Experiments.

Determine the dimensions (i) of the given cylinder, (2) of the given sphere. Enter results thus : —

1. Readings of length of cylinder, of diameter.

3-333 in. D}am r J 1-301 in.

3332 „ (1303 „

3-334 „ Diam. 2 J 1-303 „

3334 „ 11302 „

Mean 3-3332,, Mean 1-3022,,

Area = i'33 1 8 sq. in. Volume — 4*4392 cu. in.

2. Readings of diameter of sphere.

Diam. i 5-234 in.

2 5-233 „

» 3 5-232 „

„ 4 5^33 „

Mean 5-233 „

2. The Beam-Compass.

The beam-compass, like the calipers, is an instrument for measuring lengths, and is very similar to them in con- struction, consisting essentially of a long graduated beam

Cn. IV. § 2.] Measurement of the Simple Quantities. 55

with one steel compass-point fixed at one end of it, and ;another attached to a sliding piece provided with a fiducial mark and vernier. These compass-points take the place of the jaws of the calipers. It differs from them however in this, that while the calipers are adapter! for end-measures such as the distance between the two flat ends of a cylinder, the beam-compass is intended to find the distance between two marks on a flat surface. For example, in certain experiments a paper scale pasted on a board has been taken to represent truly the centimetres, millimetres, &c. marked upon it. We now want to know what error, if any, there is in the divisions. For this purpose the beam-compass is placed with its scale parallel to the paper scale, and with the two compass points lying in a convenient manner upon the divisions. It will be found that the beam-compass must be raised by blocks of wood a little above the level of the paper scale, and slightly tilted over till the points rest either just in contact with, or just above, the paper divisions.

One of the two points is fixed to the beam of the com- pass ; we will call this A. The other, B, is attached to a sliding piece, which can be clamped by a small screw on a second sliding piece. First unclamp this screw, and slide the point B along, till the distance A B is roughly equal to the dis- tance to be measured. Then clamp B, and place the point A (fig. 2) exactly on one of the marks. FIG 2

This is best effected by gentle taps at the end of the beam with a small mallet. It is the inside edge of the compass- point which has to be brought into co- incidence with the mark. Now observe that, although B is clamped it is capable of a slow motion by means of a second screw called a * tangent screw,' whose axis is parallel to the beam. Move this screw, with so light a touch as not to disturb the position of the beam-compass, until the point B is on the other mark, i.e. the inside edge of B coincides with

56 Practical Physics. [CH. IV. § 2.

the division in question. Suppose that the point A is on the right-hand edge of the paper scale division, then B should also be on the right-hand edge of the corresponding division. To ensure accuracy in the coincidence of the edges you must use a magnifying-glass.

You have now only to read the distance on the beam- scale. To do this observe what are the divisions between which the arrowhead of the vernier1 falls. Then the reading required is the reading of the lower of these divisions + the reading of the vernier. The divisions are each i milli- metre. Hence, if the arrowhead falls between the i25th and 1 2 6th, the reading is 125 mm. -f the reading of the vernier.

Observe which division of the vernier is in the same straight line with a division of the scale. Suppose the 7th to be so situated. Then the reading of the vernier is T7g mm. and the distance between the points is 125-7 mm.

Repeat the observation twice, and suppose that 125*6 and 125-7 are the readings obtained, the mean of the three will be 125-66, which may be taken as the true distance between the marks in question.

Suppose that on the paper scale this is indicated by 126 mm., then to make the scale true we must reduce the reading by -34 mm. This is the scale correction for this division.

Experiment. — Check by means of the beam-compass the accuracy of the divisions of the given centimetre scale. Enter results thus : —

Division of scale at Division of scale at Vernier readings

which A is placed which B is placed (mean of 3 obs.) o i cm. 1-005 cm.

„ 2 „ 2-010 „

» 3 » 3"0io „

» 4 ,, 4-015 »

5 » 5*oi5 -

etc.

* 55ee frontispiece, fie. z.

CH. I V. § 3. ] Measurement of the Simple Quantities. 5 7

3. The Screw-Gauge.

This instrument (fig. 3) consists of a piece of solid metal s, with two arms extending perpendicularly from its two ends. To the one arm a FlG>

steel plug, p, with a care- fully planed face, is fixed, [~ and through the other L arm, opposite to the plug, a screw c passes, having a plane face parallel and opposite to that of the plug. The pitch of the screw is half a millimetre, and consequently if we can count the number of turns and fractions of a turn of the screw from its position when the two plane faces (viz. that of the plug and that of the screw) are in contact, we can determine the distance in millimetres between these two parallel surfaces when the screw is in any position.

In order to do this the more conveniently, there is at- tached to the end of the screw farther from the plug a cap x, which slides over the cylindrical bar through which the screw passes ; this cap has a bevelled edge, the circumference of which is divided into fifty equal parts. The circle on the cylindrical bar, which is immediately under the bevelled edge, when the two opposing plane surfaces are in contact, is marked L, and a line drawn parallel to the length of the cylinder is coincident (if the apparatus is in perfect adjust- ment) with one of the graduations on the bevelled edge; this we will call the zero line of that edge. Along this line a scale is graduated to half-millimetres, and hence one division of the scale corresponds to one complete turn of the cap and screw. Hence the distance between the parallel planes can be measured to half a millimetre by reading on this scale.

We require still to determine the fraction of a turn. We know that a complete revolution corresponds to half a millimetre ; the rotating edge is divided into fifty parts, and

58 Practical Physics. [Cn. IV. § 3.

therefore a rotation through a single part corresponds to a separation of the parallel planes by T J-^ mm. Suppose, then, that the scale or line along which the graduations on the cylinder are marked, cuts the graduations on the edge of the cap at 1 2 '2 divisions from the zero mark ; then since, when a revolution is complete, the zero mark is coincident with the line along which the graduations are carried on the cylinder, the distance between the parallel planes exceeds the number of complete revolutions read on that scale by -^2 ths of a turn, i.e. by -122 mm.

If then we number every tenth division on the bevelled edge successively i, 2, 3, 4, 5, these numbers will indicate tenths of a millimetre; 5 of them will be a complete turn, and we must go into the next turn for 6, 7, 8, 9 tenths of a millimetre. It will be noticed that on the scale gradu- ated on the fixed cylinder the smaller scratches correspond to the odd half-millimetres and the longer ones to the com- plete millimetres. And on the revolving edge there are two series of numbers, i, 2, 3, 4, 5 inside, and 6, 7, 8, 9, 10 out- side. A little consideration will shew that the number to be taken is the inside or the outside one according as the last visible division on the fixed scale is a complete millimetre division or an odd half-millimetre division.

We can therefore read by this instrument the distance between the parallel planes to y-J-^th of a millimetre, or by estimating the tenth of a division on the rotating edge to the TuVotn °f a millimetre.

We may use the instrument to measure the length of a short cylinder thus. Turn the screw-cap, holding it quite lightly, so that, as soon as the two parallel planes touch, the fingers shall slip on the milled head, and accordingly shall not strain the screw by screwing too hard.1 Take a reading when the two planes are in contact; this gives the zero read-

1 Special provision is made for this in an improved form of this apparatus. The milled head is arranged so that it slips past a rntchet wheel whenever the pressure on the screw-face exceeds a certain limit,

CH. IV. §4.] Measurement of the Simple Quantities. 59

ing, which must be added to any observation reading if the zero of the scale has been passed, subtracted if it has not been reached. Then separate the planes and introduce the cylinder with its ends parallel to those of the gauge, and screw up again, holding the screwhead as nearly as possible with the same grip as before, so that the ringers shall slip when the pressure is as before. Then read off on the scales. Add or subtract the zero correction as the case may be ; a reading of the length of the cylinder is thus obtained. Read the zero again, and then the length of the cylinder at a different part of the area of the ends, and so on for ten readings, always correcting for the zero reading.

Take the mean of the readings for the length of the cylinder, and then determine the mean diameter in the same way.

The diameter of a wire may also conveniently be found by this instrument.

The success of the method depends on the touch of the screwhead, to make sure that the two planes are pressed together for the zero reading with the same pressure as when the cylinder is between them.

Be careful not to strain the screw by screwing too hard.

Experiment.— Measure the length and diameter of the given small cylinder.

Enter result thus : —

Correction for zero + "0003 cm.

Length (mean of ten) '9957 „

True length -9960 „

4. The Spherometer.

The instrument consists of a platform with three feet, whose extremities form an equilateral triangle, and in the middle of the triangle is a fourth foot, which can be raised or lowered by means of a micrometer screw passing perpendi- cularly through the centre of the platform, The readings

6o

Practical Physics.

[CH. IV. § 4.

of the spherometer give the perpendicular distance between the extremity of this fourth foot and the plane of the other three.

It is used to measure the radius of curvature of a spherical surface, or to test if a given surface is truly spherical.

The instrument is first placed on a perfectly plane sur- face— a piece of worked glass — and the middle foot screwed down until it touches the surface. As soon as this is the case, the instrument begins to turn round on the middle foot as a centre. The pressure of the hand on the screw should be very light, in order that the exact position of contact may be observed. The spherometer is then care- fully removed from the glass, and the reading of the micro- meter screw is taken.

The figure (fig. 4) will help us to understand how this is done. ABC are the ends of the three fixed feet ; D is the

movable foot, which can be raised by turning the milled head at E. This carries round with it the graduated disc F G, and as the screw is turned the disc travels up the scale H K. The graduations of this scale are such that one complete revolution of the screw carries the disc from one graduation to the next. Thus in the figure the point F on the screw-head is opposite to a division of the scale, and one complete turn would bring this point opposite the next division. In the instrument in the figure the divisions of the scale are half-millimetres, and the millimetres are marked o, i, 2. Thus only every second division is numbered.

But the rim of the disc F o is divided into fifty parts,

FIG. 4.

CH. IV. § 4.] Measurement of the Simple Quantities. 6 1

and each of these subdivided into ten. Let us suppose that division 12 of the disc is opposite to the scale at F, and that the milled head is turned until division 36 comes oppo- site. Then the head has been turned through 24 (i.e. 36 — 1 2) larger divisions ; but one whole turn or fifty divisions carry the point D through \ mm. Thus a rotation through twenty-four divisions will carry it through |-J of \ mm. or •24 mm.

Hence the larger divisions on the disc F G correspond to tenths of a millimetre, and these are subdivided to hundredths by the small divisions.

Thus we might have had opposite to the scale in the first instance 12 6 large divisions, and in the second 36*9. Then the point D would have moved through "243 mm.

It will be noticed that in the figure division o is in the centre of the scale H K, which is numbered i, 2, 3, &c., from that point in both directions up and down. The divisions numbered on the disc F G are the even ones * — 2, 4, 6, &c. — and there are two numbers to each division. One of these numbers will give the parts of a turn of the screw when it is turned so as to lower the point D, the other when it is turned so as to raise D. Thus in the figure 1 2 and 38 are both opposite the scale, and in the second position, 36 and 14. We have supposed the head to be turned in such a way that the point D has been lowered through -24 mm. If the rotation had been'in the opposite direction, D would have been raised through 0*26 mm.

Let us for the present suppose that all our readings are above the zero of the scale.

To take a reading we note the division of the scale next above which the disc stands, and then the division of the disc which comes opposite to the scale, taking care that we take the series of divisions of the disc which corresponds to a motion of the point D in the upward direction — the

1 These numbers are not shewn in the figure.

62 Practical Physics. [CH. IV. § 4.

inner ring of numbers in the figure. Thus the figured reading is 1*380.

If the instrument were in perfect order, the reading when it rested on a plane surface would be o€o. This is not generally the case, so we must observe the reading on the plane. This observation should be made four times, and the mean taken. Let the result be -460. Now take the instrument off the plane and draw the middle foot back some way. We will suppose we are going to measure the radius of a sphere from the convex side.

Place the instrument on the sphere and turn the screw E until D touches the sphere. The position of contact will be given as before, by noticing when the instrument begins to turn round D as a centre.

Read the scale and screw-head as before ; let the scale reading be : —

2*5 ; and the disc -235.

Then the reading is 2*735

Take as before four readings.

We require the distance through which the point D has been moved. This is clearly the difference between the two results, or 2735 — -460 ; if .we call this distance a we have

a — 2*275 mm-

It may of course happen that the reading of the instru- ment when on the plane is below the zero ; in this case to find the distance a we must add the two readings.

We must now find the distance in millimetres between the feet AB or AC. We can do this directly by means of a finely divided scale ; or if greater accuracy is required, lay the instrument on a flat sheet of card or paper, and press it so as to mark three dots on the paper, then measure the distance between these dots by the aid of the beam- compass (§ 2).

CH. IV. § 4.] Measurement of the Simple Quantities. 63

Let us call this length /. Then we can shew ' that, ii" r be the radius required,

The observation of / should be repeated four times.

If we wish merely to test if a given surface is spherical, we must measure a for different positions of the apparatus on the surface, and compare the results ; if the surface be spherical, the value of a will be the same for all positions.

Experiments.

(1) Test the sphericity of the given lens by observing the value of a for four different positions.

(2) Determine the radius of the given sphere for two posi- tions, and compare the results with that given by the calipers.

Enter results thus : —

Readings on plane Readings on sphere 0-460 2735

0-463 2733

0-458 2734

Q'459 _27_39

Mean 0-460 Mean 2-735

a = 2-275 mm' Obs. for / 43*56 43-52

43-57 43-59 Mean 43^56

r = 140-146 mm. By calipers r — 5-517 in. = 140-12 mm.

1 Since the triangle formed by the three feet is equilateral, the

radius of the circumscribing circle is . -- .i.e. — . But a beinir

2 sin 60° ^3-

the portion of the diameter of the sphere, radius r, cut off by the plane of the triangle, we have (Euc. iii. 35)

whence r= -^- + -•

6a 2

64 Practical Physics. [CH. IV. § 5.

5. Measurement of a Base-Line.

The object of this experiment, which is a working model of the measurement of a geodetic base-line, is~to determine with accuracy the distance between the scratches on two plugs so far apart that the methods of accurate measurement described above are impracticable.

i ne general plan of the method is to lay ivory scales end to end, fixing them by placing heavy weights on them, and to read by means of a travelling reading microscope the distance between the extreme graduations of the two ivory scales, or between the mark on the plug and the extreme graduation of the ivory scale placed near it. We have then to determine the real length of the ivory scales, and by add- ing we get the total length between the plugs.

The experiment may therefore be divided into three parts.

(i). To determine the Distance between the End Gradu- ations of the Ivory Scales placed end to end.

This is done by means of the travelling microscope. Place the scales with their edges along a straight line drawn between the two marks and perpendicular to them, and fix them so that the extreme graduations are within \ inch. Next place the microscope (which is mounted on a slide similar to the slide-rest of a lathe, and moved by a micrometer screw the thread of which we will suppose is -sVn °f an mcn) s° tna^ the line along which it travels on its stand is parallel to the base line, and focus it so that one of its cross-wires is parallel and coincident with one edge of the image of the end graduation of the one ivory scale. (It is of no conse- quence which edge is chosen, provided it be always the same in each case.)

Read the position of the microscope by its scale and micrometer screw, remembering that the fixed scale along which the divided screw-head moves is graduated to 5oths of an inch, and the circumference of the screw-head into

CH. IV. § 5.] Measurement of the Simple Quantities. 65

200 parts j each part corresponds, therefore, to T^Urr incn- So that if the reading on the scale be 7, and on the screw- head 152, we get for the position—

7 divisions of the scale=/^-in. =0-14 in. 152 divisions of the screw-head =0-0152 in. Reading=o-i552in.

Or if the scale reading be 5 and the screw-head read- ing 15, the reading similarly is 0-1015 in.

Next turn the micrometer screw-head until the last division on the other ivory scale comes into the field of view, and the corresponding edge of its image is coincident with the cross- wire as before. Read again ; the difference of the two readings gives the required distance between the two graduations.

In the same way the distance between the scratch on the plug and the end division of the scale maybe determined.

Place one -ivory scale so that one extremity is near to or coincident with the scratch on the plug ; read the dis- tance between them ; then place the other scale along the line and end-on with the first, and measure the distance between the end divisions of the two scales. Then transfer the first scale to the other end of the second ; measure the distance between them again ; and so on.

(2). To Estimate the Fraction of a Scale over.

This may be done by reading through the microscope the division and fraction of a division of the scale corre- sponding to the scratch on the second plug. This gives the length of a portion of the scale as a fraction of the true length which is found in (3).

(3). To Determine the true Length of the Ivory Scales.

This operation requires two reading microscopes. Focus these two, one on each extreme division of the scales to be measured, taking care that the same edge of the scratch is used as before. Then remove the scale, introduce a standard whose graduation can be assumed to be accurate,

F

66 Practical Physics. [CH. IV. § 5.

or whose true length is known, and read by means of the micrometer the exact length, through which the microscopes have to be moved in order that their cross-wires may co- incide with two graduations on the standard the distance between which is known accurately.1

The lengths of all the separate parts of the line between the marks, which together make up the whole distance to be measured have thus been expressed in terms of the standard or of the graduations of the micrometer screw. These latter may be assumed to be accurate, for they are only used to measure distances which are themselves small fractions of the whole length measured (see p. 41). All the data necessary to express the whole length in terms of the standard have thus been obtained.

Experiment. — Measure by means of the two given scales and the microscope the distance between the two given points.

Enter the results thus : — Distance from the mark on first plug to the end

graduation of Scale A 0-1552 in.

Distance between end graduations of Scales A and B(i) 0-1015 „

» » (2) 0-0683 „

» » » (3) Q'0572 „

» i} » (4) 0-1263 „

(5) 0-1184,, Total of intervals .... -6269 in.

Reading of Scale B at the mark on the second plug . 10-631 „ True length of Scale A 12-012,,

» B n'993,,

Total distance between the marks

= 3 x 12-012 + 2 x 1 1*993 + 10-631 + 0-6269 = 71-280 in.

6. The Kathetometer.

This instrument consists ot a vertical beam carrying a scale. Along the scale there slides a brass piece, support- ing a telescope, the axis of which can be adjusted so as to be horizontal. The brass slide is fitted with a vernier

1 For less accurate measurements the lengths of the scales may also be determined by the use of the beam -compass § 2.

CH. IV. § 6.] Measurement of the Simple Quantities. 67

FIG. 5.

which reads fractions of the divisions of the scale, thus determining the position of the telescope.

The kathetometer is used to measure the difference in height between two points.

To accomplish this, a level fitted so as to be at right angles to the scale is permanently attached to the instru- ment, and the scale is placed vertical by means of levelling screws on which the instrument rests.

Let us suppose the instrument to be in adjustment, and let p, Q be the two points, the vertical distance between which is required.

The telescope of the instrument has, as usual, cross-wires in the eye-piece. Focus the telescope on the mark p, and adjust it until the image of P coincides with the horizontal cross-wire. Then read the scale and vernier.

Let the reading be 72*125 cm.

Raise the telescope until Q comes into the field, and ad- just again till the image of Q coincides with the cross- wire; let the reading be 33*275 cm.

The difference in level be- tween p and Q is

72'125 — 33'275> or 38>85° cm-

The adjustments are :— (i) To level the instrument so that the scale is vertical in all positions.

(2) To adjust the telescope so that its axis is horizontal.

(3) To bring the cross-wire in the focal plane of the telescope into coincidence with the image of the mark which is being ob- served.

(i) The scale must be vertical, because we use the instru- ment to measure the vertical height between two points. The scale and level attached to it (fig. 5) can be turned

F 2

68 Practical Physics. [CH. IV. § 6.

round an axis which is vertical when properly adjusted, carrying the telescope with them, and can be clamped in any position by meatis cf a screw.

(a) To test the Accuracy of the Setting of the Scale-level and to set the Axis of Rotation vertical.

If the scale-level is properly set it is perpendicular to t.he axis of rotation ; to ascertain whether or not this is so, turn the scale until its level is parallel to the line joining two of the foot screws and clamp it; adjust these screws until the bubble of the level is in the middle. Unclamp, and turn the scale round through 180°. If the bubble is still in the middle of the level, it follows that this is at right angles to the axis of rotation ; if the bubble has moved, then the level and the axis of rotation are not at right angles. We may make them so by adjusting the screws which fix the level to the instrument until the rotation through 180° produces no change, or, without adjusting the level, we may proceed to set the axis of rotation vertical if, instead of adjusting the levelling screws of the instrument until the bubble stands in the centre of the tube, we adjust them until the bubble does not move relatively to the tube when the instrument is turned through 180°.

This having been secured by the action of two of the screws, turn the scale until the level is at right angles to its former position and clamp. Adjust now in the same manner as before, using only the third screw.

It follows then that the bubble will remain unaltered in position for all positions of the instrument, and that the axis about which it turns is vertical.

If the scale of the instrument were parallel to the axis, it, too, would be vertical, and the instrument would be in adjustment.

(b) To set the Scale vertical.

To do this there is provided a metallic bracket-piece. One arm of this carries a level, while the other is a flat surface at right angles to the axis of the level, so that when

CH. iv. §6.] Measurement of the Simple Quantities. 69

the level is horizontal this surface is truly vertical. The adjustment can be tested in the following manner. The level can rotate about its axis, and is weighted so that the same part of the tube remains uppermost as the bracket is rotated about the axis of the level. Place then the flat face of the bracket with the level uppermost against a nearly vertical plane surface ; notice the position of the bubble. Then reverse it so that the level is lowest, and read the posi- tion of the bubble again. If it has not changed the level is truly set, if any displacement has taken place it is not so.

The scale of the instrument can be adjusted relatively to the axis of rotation and fixed by screws.

Press the flat surface of the bracket- piece against the face of the scale. If the scale be vertical, the bubble of the level on the bracket-piece will occupy the middle of its tube. Should it not do so, the scale must be adjusted until the bubble comes to the central position. We are thus sure that the scale is vertical.

For ordinary use, with a good instrument, this last ad- justment may generally be taken as made.

Now turn the telescope and, if necessary, raise or lower it until the object to be observed is nearly in the middle of the field of view.

(2) It is necessary that the axis of the telescope should be always inclined to the scale at the same angle, for if, when viewing a second point Q, the angle between the axis and the scale has changed from what it was in viewing p, it is clear that the distance through which the telescope has been displaced will not be the vertical distance between p and Q.

If, however, the two positions of the axis be parallel, the difference of the scale readings will give us the distance we require.

Now the scale itself is vertical. The safest method, therefore, of securing that the axis of the telescope shall be always inclined at the same angle to the scale is to adjust

Practical Physics.

[Cn. IV. § 6.

the telescope so that its axis shall be horizontal. The method of doing this will be different for different instru- ments. We shall describe that for the one at the Cavendish Laboratory in full detail ; the plan to be adopted for other instruments will be some modification of this.

FIG.

In this instrument (fig. 6) a level L M is attached to the telescope T T'. The telescope rests in a frame Y Y'. The lower side of this frame is bevelled slightly at N ; the two surfaces Y N, Y' N being flat, but inclined to each other at an angle not far from 180°.

CM. IV. § 6.] Measurement of the Simple Quantities. 71

This under side rests at N on a flat surface c D, which is part of the sliding-piece c D, to which the vernier v v' is fixed.

A screw passes through the piece Y Y' at N, being fixed into c D. The hole in the piece Y Y' is large and somewhat conical, so that the telescope and its support can be turned about N, sometimes to bring N Y into contact with c N, sometimes to bring N Y' into contact with N D.

Fitted into c D and passing freely through a hole in N Y' is a screw Q ; p is another screw fitted into c D, which bears against N Y'. Hidden by P and therefore not shown in the figure is a third screw just like p, also fitted into c D, and bearing against N Y'. The screws N, p, and Q can all be turned by means of a tommy passed through the holes in their heads. When p and Q are both screwed home, the level and telescope are rigidly attached to the sliding- piece c D.

Release somewhat the screw Q. If now we raise the two screws p, we raise the eye-piece end of the telescope, and the level-bubble moves towards that end. If we lower the screws p, we lower the eye-piece end, and the bubble moves in the opposite direction.

Thus the telescope can be levelled by adjusting the screws P. Suppose the bubble is in the centre of the level. Screw down the screw Q. This will hold the telescope fixed in the horizontal position.

If we screw Q too firmly down, we shall force the piece N Y' into closer contact with the screws P, and lower the eye-piece end. It will be better then to adjust the screw p so that the bubble is rather too near that end of the tube. Then screw down Q until it just comes to the middle of the tube, and the telescope is level.

(3) To bring the image of the object viewed to coincide with the cross-wires.

The piece c D slides freely up and down the scale. EF F'E' is another piece of brass which also slides up and down.

72 Practical Physics. [CH. IV. § 6.

H is a screw by means of which E F' can be clamped fast to the scale. A screw R R' passes vertically upwards through E F' and rests against the under side of a steel pin G fixed in c D. Fixed to EF' and pressing downwards on the pin G so as to keep it in contact with the screw R R' is a steel spring s s'. By turning the screw R R', after clamping H, a small motion up or down can be given to the sliding piece c D and telescope.

Now loosen the screw H and raise or lower the two pieces c D, E F' together by hand, until the object viewed is brought nearly into the middle of the field of view. Then clamp E F' by the screw H.

Notice carefully if this operation has altered the level of the telescope ; if it has, the levelling must be done again.

By means of the screw R R' raise or lower the telescope as may be needed until the image is brought into coincidence with the cross-wire. Note again if the bubble of the level is in its right position, and if so read the scale and vernier.

It may happen that turning the screw R R' is sufficient to change the level of the telescope. In order that the slide c D may move easily along the scale, a certain amount of play must be left, and the friction between R' and the pin is sometimes sufficient to cause this play to upset the level adjustment. The instrument is on this account a trouble- some one to use.

The only course we can adopt is to level ; and then adjust R R' till the telescope is in the right position, levelling again if the last operation has rendered it necessary.

This alteration of level will produce a small change in the position of the line of collimation of the telescope rela- tively to the vernier, and thus introduce an error, unless the axis round which the telescope turns is perpendicular both to the line of collimation and to the scale. If, however, the axis is only slightly below the line of collimation and the change of level small, the error will be very small indeed and may safely be neglected.

It is clear that the error produced by an error in levelling

CH IV. § 7.] Measurement of the Simple Quantities. 73

will be pioportional to the distance between the instrument and the object whose height is being measured. We should therefore bring the instrument as close to the object as is possible.

Experiment. — Adjust the kathetometer, and compare by means of it a length of 20 cm. of the given rule with the scale of the instrument.

Hang the rule up at a suitable distance from the kathe- tometer, and measure the distance between division 5 cm. and 25 cm.

The reading of the kathetometer scale in each position must be taken three times at least, the telescope being displaced by means of the screw R R' between each observation.

Enter results as below : —

Kath. reading, upper mark Kath. reading, lower mark 253I5 25305 25^20

Mean 25-3133 Difference 20-0167

Mean error of scale between divisions 5 and 25, -0167 cm.

MEASUREMENT OF AREAS.

7. Simpler Methods of measuring Areas of Plane Figures.

There are four general methods of measuring a plane area: —

(a) If the geometrical figure of the boundary be known, the area can be calculated from its linear dimensions — e.g. if the boundary be a circle radius r.

Area = TT r* where TT — 3*142.

The areas of composite figures consisting of triangles and circles, or parts of circles, may be determined by addition of the calculated areas of all the separate parts.

A table of areas which can be found by this method is given in Lupton's Tables, p. 7.

74 Practical Physics. [Cn. IV. § 7.

In case two lengths have to be measured whose product determines an area, they must both be expressed in the same unit, and their product gives the area expressed in terms of the square of that unit.

(b) If the curve bounding the area can be transferred to paper divided into known small sections, e.g. square milli- metres, the area can be approximately determined by count- ing up the number of such small areas included in the bounding curve. This somewhat tedious operation is facili- tated by the usual grouping of the millimetre lines in tens, every tenth line being thicker. In case the curve cuts a square millimetre in two, the amount must be estimated ; but it will be generally sufficient if portions greater than a half be reckoned a whole square millimetre and less than a half zero.

(c) By transferring the curve of the boundary to a sheet of paper or metal of uniform thickness and cutting it out, and cutting out a square of the same metal of known length of side, say 2 inches, and weighing these two pieces of metal. The ratio of their weights is the ratio of the areas of the two pieces of metal. The one area is known and the other may therefore be determined.

(d) By the planimeter. A pointer is made to travel round the boundary, and the area is read off directly on the graduated rim of a wheel.

For the theory of this instrument see Williamson's In- tegral Calculus (§ 149). Practical instructions are issued by the makers.

Experiment. — Draw by means of a compass a circle of 2 in. radius. Calculate or determine its area in all four ways, and compare the results.

Enter results thus : —

Method a b c. d

12-566 sq. in, 12-555 scl- in« 12-582 sq. in. 12-573 sq. in,

CH. IV. §8.] Measurement of the Simple Quantities. 75

8, Determination of the Area of the Cross-Section of a Cylindrical Tube, — Calibration of a Tube.

The area of the cross-section of a narrow tube is best determined indirectly from a measurement of the volume of mercury contained in a known length of the tube. The principle of the method is given in Section 9. The tube should first be ground smooth at each end by rubbing on a stone with emery-powder and water, and then very care- fully cleaned, first with nitric acid, then with distilled water, then with caustic potash, and finally rinsed with dis- tilled water, and very carefully dried by passing air through it, which has been dried by chloride of calcium tubes.1 The different liquids may be drawn up the tube by means of an air-syringe. If any trace of moisture remain in the tube, it is very difficult to get all the mercury to run out of it after it has been filled.

The tube is then to be filled \\ithfltre* mercury ; this is best done by immersing it in a trough of mercury of the necessary length. [A deep groove about half an inch broad cut in a wooden beam makes a very serviceable trough for the purpose.] When the tube is quite full, close the ends with the forefinger of each hand, and after the small globules of mercury adhering to the tube have been brushed off, allow the mercury to run into a small beaker, or other con- venient vessel, and weigh it. Let the weight of the mercury be w. Measure the length of the tube by the calipers or beam-compass, and let its length be /. Look out in the table (33) the density of mercury for the temperature (which may be taken to be that of the mercury in the trough), and

1 For this and a great variety of similar purposes an aspirating pump attached to the water-supply of the laboratory is very convenient.

- A supply of pure mercury may be maintained very conveniently by distillation under very low pressure in an apparatus designed by Weinhold (see Carl's Rep. vol. 15, and Phil. Mag., Jan., 1884).

76 Practical Physics. [CH. IV. § 8.

let this be p. Then the volume v of the mercury is given by the equation

and this volume is equal to the product of the area A of the cross-section and the length of the tube. Hence

If the length be measured in centimetres and the weight in grammes, the density being expressed in terms of grammes per c.c., the area will be given in sq. cm.

The length of the mercury column is not exactly the length of the tube, in consequence of the fingers closing the tube pressing slightly into it, but the error due to this cause is very small indeed.

This gives the mean area of the cross-section, and we may often wish to determine whether or not the area of the section is uniform throughout the length. To do this, care- fully clean and dry the tube as before, and, by partly im- mersing in the trough, introduce a thread of mercury of any convenient length, say about 5 centimetres long. Place the tube along a millimetre scale, and fix it horizontally so that the tube can be seen in a telescope placed about six or eight feet off.

By slightly inclining the tube and scale, adjust the thread so that one end of it is as close as possible to the end of the tube, and read its length in the telescope. Displace the thread through 5 cm. and read its length again ; and so on, until the thread has travelled the whole length of the tube, taking care that no globules of mercury are left behind. Let /1} 72, /3 . . . . be the successive lengths of the thread. Then run out the mercury into a beaker, and weigh as before. Let the weight be w, and the density of the mercury be p.

CH. IV. § 8.] Measurement of the Simple Quantities. 77

Then the mean sectional areas of the different portions of the tube are

w w w

.. etc.

— — , — -, — -,

P i\ P/2 P *3

The mean of all these values of the area should give the mean value of the area as determined above. The accu- racy of the measurements may thus be tested.

On a piece of millimetre sectional paper of the same length as the tube mark along one line the different points which correspond to the middle points of the thread in its different positions, and along the perpendicular lines through these points mark off lengths representing the correspond- ing areas of the section, using a scale large enough to shew clearly the variations of area at different parts of the length. Join these points by straight lines. Then, the ordinates of the curve to which these straight lines approximate give the cross-section of the tube at any point of its length.

Experiment. — Calibrate, and determine the mean area of the given tube.

Enter the result thus : —

[The results of the calibration are completely expressed by the diagram.]

Length of tube . . . .25*31 cm. Weight of beaker . . . 10-361 gm. Weight of beaker and mercury . 1 1 786 gm.

Weight of mercury . . 1*425 gm. Temperature of mercury . '14° C. Density of mercury (table 33) 13*56

Mean area of section = * ^25 — Sq. cm.

25*31 x 13-56

= 0*415 sq. mm. Mean of the five determinations for calibration 0*409 sq. mm.

78 Practical Physics. [CH. IV. § 9.

MEASUREMENT OF VOLUMES.

The volumes of some bodies of known shape may be de- termined further by calculation from their linear dimensions ; one instance of this has been given in the experiment with the calipers.

A Table giving the relations between the volume and linear dimensions in those cases which are likely to occur most frequently will be found in Lupton's Tables, p. 7.

9. Determination of Volumes by Weighing.

Volumes are, however, generally determined from a knowledge of the mass of the body and the density of the material of which it is composed. Defining c density ' as the mass of the unit of volume of a substance, the relation between the mass, volume and density of a body is ex- pressed by the equation M=vp, where M is its mass, v its volume, and p its density. The mass is determined by means of the balance (see p. 91), and the density, which is different at different temperatures, by one or other of the methods described below (see pp. 107-1 1 2). The densities of certain substances of definitely known composition, such as distilled water and mercury, have been very accurately de- termined, and are given in the tables (Nos. 32, 33), and need not therefore be determined afresh on every special occa- sion. Thus, if we wish, for instance, to measure the volume of the interior of a vessel, it is sufficient to determine the amount and the temperature of the water or mercury which exactly fills it. This amount may be determined by weigh- ing the vessel full and empty, or if the vessel be so large that this is not practicable, fill it with water, and run the water off in successive portions into a previously counterpoised flask, holding about a litre, and weigh the flask thus filled. Care must be taken to dry the flask between the successive fillings ; this may be rapidly and easily done by using a hot clean cloth. The capacity of vessels of very considerable

CH. IV. § 10.] Measurement of the Simple Quantities. 79

size may be determined in this way with very great accuracy.

All the specific gravity experiments detailed below involve the measurement of a volume by this method.

Experiment. — Determine the volume of the given vessel. Enter results thus : —

Weight of water

Filling i . . i ooi -2 gms.

2 . . 9987 „

3 . . 1002-3 »

4 . . 999-2 „

5 . . 798-1 „

Total weight . 4799-5 gms. Tenlperature of water , Volume. . 4803-5 c.c. in vessel, 1 5°.

10. Testing the Accuracy of the Graduation of a Burette,

Suppose the burette to contain 100 c.c. ; we will suppose also that it is required to test the capacity of each fifth of the whole.

The most accurate method of reading the burette is by means of afloat, which consists of a short tube of glass loaded at one end so as just to float vertically in the liquid in the burette ; round the middle of the float a line is drawn, and the change of the level of the liquid is determined by reading the position of this line on the graduations of the burette. The method of testing is then as follows : —

Fill the burette with water, and read the position of the line on the float. Carefully dry and weigh a beaker, and then run into it from the burette about £th of the whole contents ; read the position of the float again, and weigh the amount of water run out into the beaker. Let the number of scale divisions of the burette be 20-2 and the weight in grammes 20-119. Read the temperature of the water ; then, knowing the density of water at that temperature (from table 32), and that i gramme of water at 4° C. occupies i c,c.,

So Practical Physics. [CH. IV. § 10.

we can determine the actual volume of the water correspond- ing to the 20-2 c.c. as indicated by the burette, and hence determine the error of the burette. Proceeding in this way for each -Jth of the whole volume, form a table of cor- rections.

Experiment.— Form a table of corrections for the given burette.

Enter results thus : —

Burette readings Error

0-5 c.c --007 c.c.

S-io „. . . . --020 „

10-15 ,, • -'on „

15-20 „ . -ooo „

20-25 » ' -'036 „

MEASUREMENT OF ANGLES.

The angle between two straight lines drawn on a sheet of paper may be roughly measured by means of a protractor, a circle or semi-circle with its rim divided into degrees. Its centre is marked, and can therefore be placed so as to coin- cide with the point of intersection of the two straight lines ; the angle between them can then be read off on the gradua- tions along the rim of the protractor. An analogous method of measuring angles is employed in the case of a compass- needle such as that required for § 69.

The more accurate methods of measuring angles depend on optical principles, and their consideration is accordingly deferred until the use of the optical instruments is explained (see §§ 62, 71).

MEASUREMENTS OF TIME.

The time-measurements most frequently required in practice are determinations of the period of vibration of a needle. To obtain an accurate result some practice in the use of the * eye and ear method ' is required. The experi-

CH. IV. § ii.] Measurement of the Simple Quantities. 8 1

ment which follows (§ n) will serve to illustrate the method and also to call attention to the fact that for accurate work any clock or watch requires careful * rating,' /.*. comparison of its rate of going with some timekeeper, by which the times can be referred to the ultimate standard — the mean solar day. The final reference requires astronomical obser- vations.

Different methods of time measurement will be found in §§ 21 and 28. The 'method of coincidences ' is briefly discussed in § 20.

ii. Rating a Watch by means of a Seconds-Clock.

The problem consists in determining, within a fraction of a second, the time indicated by the watch at the two instants denoted by two beats of the clock with a known interval between them. It will be noticed that the seconds- finger of the clock remains stationary during the greater part of each second, and then rather suddenly moves on to the next point of its dial. Our object is to determine to a fraction of a second the time at which it just completes one of its journeys.

To do .this we must employ both the eye and ear, as it is impossible to read both the clock and watch at the same instant of time. As the watch beats more rapidly than the clock, the plan to be adopted is to watch the latter, and listening to the beating of the former, count along with it until it can be read. Thus, listening to the ticking of the watch and looking only at the clock, note the exact instant at which the clock seconds-finger makes a particular beat, say at the completion of one minute, and count along with the watch-ticks from that instant, beginning o, i, 2, 3, 4, . . and so on, until you have time to look down and identify the position of the second-hand of the watch, say at the instant when you are counting 21. Then we know that this time is 2 1 ticks of the watch after the event (the clock-beat) whose

G

82 Practical Physics. [CH. IV. § i*.

time we wished to register ; hence if the watch ticks 4 times a second, that event occurred at ^ seconds before we took the time on the watch.

We can thus compare to within ^ sec. the time as indicated by the clock and the watch, and if this process be repeated after the lapse of half an hour, the time indicated by the watch can be again compared, and the amount gained or lost during the half-hour determined. It will require a little practice to be able to count along with the watch.

During the interval we may find the number of ticks per second of the watch. To do this we must count the number of beats during a minute as indicated on the clock. There being 4 or 5 ticks per second, this will be a difficult operation if we simply count along the whole way; it is there- fore better to count along in groups of either two or four, which can generally be recognised, and mark down a stroke on a sheet of paper for every group completed ; then at the end of the minute count up the number of strokes ; we can thus by multiplying, by 2 or 4 as the case may be, obtain the number of watch-ticks in the minute, and hence arrive at the number per second.

Experiment. — Determine the number of beats per second made by the watch, and the rate at which it is losing or gaining.

Enter results thus : —

No. of watch-ticks per minute, 100 groups of 3 each. No. of ticks per second, 5.

hr. m. s.

Clock-reading. . . . . . . . n 38 3

Estimated watch-reading, n hr. 34 m. and 10 ticks = u 34 2

Difference . 4 I

Clock-reading. . . . . . 12 8 3

Estimated watch-reading, I2hr. 4m. and 6 ticks -12 4 1-2

Difference . . . 41-8

Losing rate of watch, I -6 sec. per hour.

CHAPTER V.

MEASUREMENT OF MASS AND DETERMINATION OF SPECIFIC GRAVITIES.

12. The Balance,

General Considerations.

THE balance, as is well known, consists of a metal beam, supported so as to be free to turn in a vertical plane about an axis perpendicular to its length and vertically above its centre of gravity. At the extremities of this beam, pans arj sus- pended in such a manner that they turn freely about axes, passing through the extremities of the beam, and parallel to its axis of rotation. The axes of rotation are formed by agate knife-edges bearing on agate plates. The beam is provided with three agate edges; the middle one, edge down- wards, supporting the beam when it is placed upon the plates which are fixed to the pillar of the balance, and those at the extremities, edge upwards ; on these are supported the agate plates to which the pans are attached.

The effect of hanging the pans from these edges is that wherever in the scale pan the weights be placed, the vertical force which keeps them in equilibrium must pass through the knife-edge above, and so the effect upon the balance is independent of the position of the weights and the same as if the whole weight of the scale pan and included masses were collected at some point in the knife-edge from which the pan is suspended.

In order to define the position of the beam of the balance, a long metal pointer is fixed to it, its length being perpen- dicular to the line joining the extreme knife-edges. A small scale is fixed to the pillar of the balance, and the motion of the beam is observed by noting the motion of the pointer along this scale. When the balance is in good adjustment, the scale should be in such a position that the pointer is

G 2

84 Practical Physics. fCn. V. § 12.

opposite the middle division when the scale-beam is hori- zontal. The only method at our disposal for altering the relative position of the scale and pointer is by means of the levelling screws attached to the case. Levels should be placed in the case by the instrument-maker, which should shew level when the scale is in its proper position.

In the investigation below we shall suppose the zero position of the balance to be that which is defined by the pointer being opposite the middle point of its scale, whether the scale is in its proper position, and the pointer properly placed or not.

The other conditions which must be satisfied if the balance is in perfect adjustment are : —

(1) The arms must be of equal length.

(2) The scale pans must be of equal weight.

(3) The centre of gravity of the beam must be vertically under the axis of rotation when the beam is in its zero position. This can always be ensured by removing the scale pans altogether, and by turning the small flag of metal attached to the top of the beam until the latter comes to rest with the pointer opposite the middle of its scale. Then it is obvious from the equilibrium that the centre of gravity is vertically under the axis of support.

On the Sensitiveness of a Balance.

Let us suppose that this third condition is satisfied, and that the points A, c, B (fig. 7) represent the points in which FlG- 7' the three knife-edges

cut a vertical plane at right angles to their edges, and let c A, c B make angles a, a' with a horizontal line ^- through c. [If the balance is in perfect adjustment a==o/.]

We may call the lengths c A, c B the lengths of the arms

CH. V. § 12.] Measurement of Mass. 85

of the balance, and represent them by R, L respectively. Let the masses of the scale pans, the weights of which act ver- tically downward through A and B respectively, be P and Q. Let G, the centre of gravity of the beam, be at a distance //, vertically under c, and let the mass of the beam be K. It the balance be in adjustment, R is equal to L, and P to Q. Now let us suppose that a mass w is placed in the scale pan P, and a mass w + x in Q, and that in consequence the beam takes up a new position of equilibrium, arrived at by turning about c through an angle 6, and denoted by B' c A', and let the new position of the centre of gravity of the beam be G'. Then if we draw the vertical lines B' M, A' N to meet the horizontal through c in M and N, a horizontal line through G' to meet c G in x, and consider the equilibrium of the beam, we have by taking moments about the point c

(Q + W+X) CM = (P + W) CN -f K . G'X. Now

c M = c B' cos (a' — 6) = L (cos a! cos 0 + sin a! sin 6). c N = c A' cos (a + 6) = R (cos a cos B — sin a sin 6).

G' x = c G' sin 6 = h sin 0. Hence we get

L (Q + W + X) (cos a' cos 0+sin a' sin 6) =R(P + W) (cos a cos 6— sin a sin 0) + K/i sin 0.

Since 0 is very small, we may write tan 0=0,

sin a'— RP-J-«/ sin

This gives us the position in which the balance will rest when the lengths of the arms and masses of the scale pans are known, but not necessarily equal or equally inclined to the horizon; and when a difference x exists between the masses in the scale pans.

It is evident that 0 may be expressed in pointer scale divisions when the angle subtended at the axis of rotation Dy one of these divisions is known.

86 Practical Physics. [CH. V. § 12.

DEFINITION. — The number of scale divisions between the position of equilibrium of the pointer when the masses are equal and its position of equilibrium when there is a given small difference between the masses is called the sensitiveness of the balance for that small difference. Thus, if the pointer stand at 100 when the masses are equal and at 67 when there is a difference of *ooi gramme between the masses, the sensitiveness is 33 per milligramme.

We have just obtained a formula by which the sensi- tiveness can be expressed in terms of the lengths of the arms, &c.

Let us now suppose that the balance is in adjustment, i.e.

L=R, Q=P, a=a'

L* COS a

_ xv

Hence the angle turned through for a given excess weight x increases proportionally with x, and increases with the length of the arm.

Let us consider the denominator of the fraction a little more closely. We see that it is positive or negative ac- cording as

K/Z> or < Now it can easily be shewn that the equation

is the condition that c should be the centre of gravity of the beam and the weights of the scale pans, &c. supposed col- lected at the extremities of the arms. If this condition were satisfied, the balance would be in equilibrium in any position. If K h be less than L(2P + 2w + x) sin a, tan 6 is negative, which shews that there is a position of equilibrium with the centre of gravity of the whole, above the axis ; but it is reached by moving the beam in the opposite direction to that

CH. V. § 12.] Measurement of Mass. 87

in which the excess weight tends to move it : it is therefore a position of unstable equilibrium. We need only then discuss the case in which K h is > L(2P + 2ze/+#)sino, i.e. when the centre of gravity of the whole is below the axis of rotation.

With the extreme knife-edges above the middle one, a is positive and the denominator is evidently diminished, and thus the sensitiveness increased, as the load w increases; but if the balance be so arranged that a=o, which will be the case when the three knife-edges are in the same plane, we have

or the sensitiveness is independent of the load ; if the extreme knife-edges be below the mean, so that a is nega- tive, then the denominator increases with the load w, and consequently the sensitiveness diminishes. Now the load tends to bend the beam a little ; hence in practice, the knife-edges are so placed that when half the maximum load is in the scale pans, the beam is bent so that all the knife- edges lie in a plane, and the angle a will be positive for loads less than this and negative for greater loads. Hence, m properly made balances, the sensitiveness is very nearly independent of the load, but it increases slightly up to the mean load, and diminishes slightly from the mean to the maximum load.

The Adjustment of a Balance.

I. Suppose the balance is not known to be in adjust- ment.

Any defect may be due to one of the following causes:—

(i) The relative position of the beam and pointer and its scale may be wrong. This may arise in three ways : (a) the pointer may be wrongly fixed, (ft) the balance may not be level, (y) the pointer when in equilibrium with the pans unloaded may not point to its zero position. We

88 Practical Physics. [CH. V. § 12.

always weigh by observing the position of the pointer when at rest with the scale pans empty, and then bring its position of equilibrium with the pans loaded back to the same point. It is clear that this comes to the same thing as using a pointer not properly adjusted. In all these cases a will not be equal to a! in equation (i).

(2) The arms may not be of equal length, i.e. L not equal to R.

(3) The scale pans may not be of equal weight.

We may dispose of the third fault of adjustment first. If the scale pans be of equal weight, there can be no change in the position of equilibrium when they are interchanged ; hence the method of testing and correcting suggests itself at once (see p. 101).

The first two faults are intimately connected with each other, and may be considered together. Let the pointer be at its mean position when there is a weight w in P and w'+x in Q, w and w1 being weights which are nominally the same, but in which there may be errors of small but un- known amount,

Then 6=0 .'. tan 6=0 .', from (i) (assuming P=Qy

cos a'=R(p + ze/)cos a . . . (3)

Interchange the weights and suppose now that w in Q balances w' +y, in P, then

L (P + W) cos a'=R(p-fze/4-jy) cos a , (4)

And if the pointer stands at zero when the pans are un- loaded, we have

L.PCOS a' = R. P COS a .... (5)

Hence equations (3) and (4) become

L (w1 +X) COS a' = R W COS a.

L w COS a! =R (w' +y) rus a.

Cn. V. § 12.] Measurement of Mass. 89

Multiplying

L2 cos V (w' + x)=R2 (w1 +7) cos 2a . . . (6)

. L COS a! /W1

R COS a ~~ \/ W'

= i +^-~ approximately (p. 44).

It will be seen on reference to the figure that L cos a' and R cos a are the projections of the lengths of the arms on a horizontal plane — i.e. the practical lengths of the arms considered with reference to the effect of the forces to turn the beam.

If the balance be properly levelled and the pointer straight a=a', and we obtain the ratio of the lengths of the actual arms. We thus see that, if the pointer is at zero when the balance is unloaded, but the balance not properly levelled, the error of the weighing is the same as if the arms were unequal, provided that the weights are adjusted so as to place the pointer in its zero position. The case in which a = — a' and therefore cos a = cos a' will be an im- portant exception to this; for this happens when the three knife-edges are in one plane, a condition which is very nearly satisfied in all delicate balances. Hence with such balances we may get the true weight, although the middle point of the scale may not be the equilibrium position of the pointer, provided we always make this equilibrium position the same with the balance loaded and unloaded. If we wish to find the excess weight of one pan from a knowledge of the position of the pointer and the sen- sitiveness of the balance previously determined, it will be

pO Practical Physics. [CH. V. § 12.

a more complicated matter to calculate the effect of not levelling.

We may proceed thus : Referring to equation (i), putting p = Q we get

tan 0— L(p + w+*) cos a' — R (P + ?e/) cos a ~ sin a/-

And since 0=o when no weights are in the pans, we get L P cos O/=R p cos a.

L X COS a'

K/J — L W + P + .X Sin a' — R w + p sin a

/. tan 0 =

Since a and a! are always very small, we may put cos a = i and sin a'=a', and so on, the angles being measured in circular measure (p. 45).

/. tan 0= —f—

_^

Neglecting x and the difference between L and R, in the bracket, since these quantities are multiplied by a or a', we have

The error thus introduced is small, unless

is a very large quantity, compared with a, and it well may be so, since h is small and W + P may be many times K; but a in a well-made balance is generally so small that the effect is practically imperceptible, and if the knife-edges be in a plane, so that a = — a', the correction vanishes.

CH. v. § 12.] Measurement of Mass. 91

Practical Details of Manipulation. Method of Oscillations.

All delicate balances are fitted with a long pointer fixed to the beam, the end of which moves over a scale as the beam turns.

The middle point of this scale should be vertically be- low the fulcrum of the beam, and if the balance be in perfect adjustment, when the scale pans are empty and the beam free, the end of the pointer will coincide with the middle division of the scale. This coincidence, however, as we have seen, is not rigorously necessary.

To weigh a body we require to determine first at what point of the scale the pointer rests when the pans are empty. We then have to put the body to be weighed in one pan and weights in the other, until the pointer will again come to rest opposite to the same division of the scale. The weight of the body is found by adding up the weights in the scale pan.

We shall suppose that the weights used are grammes, decigrammes, &c.

The weights in the boxes usually supplied are some of them brass and the others either platinum or aluminium.

The brass weights run from i gramme to 50, 100 or icoo grammes in different boxes.

We may divide the platinum and aluminium weights into three series : —

The first includes, -5, -2, -i, -i gramme

The second -05, -02, -or, -01 „

The third '005? '002, 'ooi, *ooi „

that is, the first series are decigrammes, the second centi- grammes, and the third milligrammes.

The weights should never be touched with the fingers ; they should be moved by means of the small metal pliers provided for the purpose. In the larger boxes a brass bar is provided for lifting the heavier weights.

When the balance is not being used, the beam and the scale pans do not rest on the knife-edges but on independent

92 Practical Physics. [CH. V. § 12.

supports provided for them. The balance is thrown into action by means of a key in the front of the balance case. This must always be turned slowly and carefully, so as to avoid any jarring of the knife-edges from which the beam and scale pans hang.

When it is necessary to stop the beam from swinging, wait until the pointer is passing over the middle of the scale, and then turn the key and raise the frame till it supports the beam. The key must not be turned, except when the pointer is at the middle of the scale ; for if it be, the sup- porting frame catches one end of the beam before the other, and thus jars the knife-edges.

. The weights or object to be weighed when in the scale pans must never be touched in any way while the beam is swinging ; thus, when it is required to change the weights, wait until the pointer is passing across the middle point of the scale, turn the key, and fix the beam, then move the weights from the scale pan.

In the more delicate balances, which are generally en- closed in glass cases, it will be seen that the length of each arm of the beam is divided into ten parts.

Above the beam, and slightly to one side of it, there is a brass rod which can be moved from outside the balance case. This rod carries a small piece of bent wire, which can, by moving the rod, be placed astride the beam. This piece of wire is called a 'rider.' The weight of the rider is usually one centigramme.

Let A c B, fig. 8, be the beam, c being the fulcrum; the divisions on the arm are reckoned from c.

Suppose now we place the centigramme rider at division i, that is one-tenth of the length of the arm away from the FIG. s. fulcrum, it will clearly

A require one-tenth of its own weight to be placed in the scale pan sus- pended from B, to balance it. The effect on the balance-

CH. V. § 12.] Measurement of Mass. 93

beam of the centigramme rider placed at division i, is the same as that of a weight of T^ centigramme or i milligramme in the pan at A. By placing the rider at division i, we practically increase the weight in the pan at A by i milli- gramme. Similarly, if we place the rider at some other division, say 7, we practically increase the weight in A by 7 milligrammes.

The rider should not be moved without first fixing the balance beam.

Thus without opening the balance- case we can make our final adjustments to the weights in the scale pan by moving the rider from outside.

The object of the case is to protect the balance from draughts and air currents. Some may even be set up in- side the case by opening it and inserting the warm hand to change the weights ; it is therefore important in delicate work to be able to alter the weight without opening the case.

We proceed now to explain how to determine at what point of the graduated scale the pointer rests when the pans are empty. If the adjustments were quite correct, this would be the middle point of the scale. In general we shall find that the resting-point is somewhere near the middle.

We shall suppose for the present that the stand on which the balance rests is level. This should be tested by the spirit-level before beginning a series of weighings, and if an error be found, it should be corrected by moving the screw- feet on which the balance-case rests.

We shall find that the balance when once set swinging will continue in motion for a long period. The pointer will oscillate across the scale, and we should have to wait for a very long time for it to come to rest

We require some method of determining the resting- point from observations of the oscillations.

Let the figuie represent the scale, and suppose, reckoning from the left, we call the divisions o, 10, 20, 30. . « .

94 Practical Physics. [CH. V. § 12.

A little practice enables us to estimate tenths of these divisions.

Watch the pointer as it moves ; it will come for a moment

to rest at P1 suppose, and then move back again. Note the

FIG. 9. division of the scale,

1

I M I n I I

63, at which this hap- MM! I I M 1.1 P6115'1 The P°inter

|

O 1020 30 3050 6070 80 90aOOJ10120130M01501G017<U80J90200 SWingS On

resting-point, and comes to instantaneous rest again in some position beyond it, as P2, at 125 suppose.

Now if the swings on either side of the resting-point were equal, this would be just half-way between these two divi- sions, that is at 94 ; but the swings gradually decrease, each being less than the preceding. Observe then a third turning point on the same side as the first, P3 suppose, and let its scale reading be 69.

Take the mean 66, between 69 and 63. We may assume that this would have been the turning-point on that side at the moment at which it was 125 on the other, had the pointer been swinging in the opposite direction. Take the mean of the 125 and 66, and we have 95 -5 as the value of the resting-point.

Thus, to determine the resting point : —

Observe three consecutive turning points, two to the left and one to the right, or vice versd. Take the mean of the two to the left and the mean of this and the one to the right ; this gives the resting-point required.

The observations should be put down as below.

Turning-points Resting-point

Left Right

Mean66Jj?3 125 95-5

We may, if we wish, observe another turning-point to the right, 120 suppose; then we have another such series.

1 A small mirror is usually fixed above the scale, the planes of the two being parallel. When making an observation the observer's eye is placed so that the pointer exactly covers its own image formed in the mirror ; any error due to parallax is thus avoided.

CH. v. § 12.] Measurement of Mass. 95

Proceeding thus we get a set of determinations of the resting-point, the mean of which will give us the true position with great accuracy.

Having thus found the resting point with the pans empty, turn the key or lever, and fix the beam ; then put the object to be weighed in one scale pan. Suppose it to be the left-hand, for clearness in the description. Then put on some weight, 50 grammes say, and just begin to turn the key to throw the balance into action. Suppose the pointer moves sharply to the left, 50 gms. is too much. Turn the key back, re- move the 50 and put on 20 ; just begin to turn the key ; the pointer moves to the right, 20 is too little. Turn the key back, and add 10 ; the pointer still moves to the right ; add 10 more, it moves to the left ; 40 is too much. Turn the key back, remove the 10 and add 5. Proceed in this way, putting on the weights in the order in which they come, re- moving each weight again if the pointer move sharply to the left, that is, if it be obviously too much, or putting on an additional weight if the pointer move to the right

There is no necessity to turn the key to its full extent to decide if a weight be too much or too little until we get very nearly the right weight ; the first motion of the pointer is sufficient to give the required indication.

It saves time in the long run to put on the weights in the order in which they come in the box.

Caution.— The beam must always be fixed before a weight is changed.

Suppose now we find that with 37*68 grammes the pointer moves to the right, shewing the weight too little, and that with 37-69 the motion is to the left, shewing that it is too much. Close the balance-case, leaving on the lighter weight, 37*68 grammes. Turn the key, and notice if the pointer will swing off the scale or not. Suppose it is quite clear that it will, or that the resting-point will be quite at one end near the division 200. Fix the beam, and put on the rider say

g6 Practical Physics. [Cn. V. § 12.

at division 2. This is equivalent to adding '002 gm. to the weights in the scale pan, so that the weight there may now be reckoned as 37-682 gms. Release the beam, and let it oscillate, and suppose that this time the pointer remains on the scale.

Read three turning-points as before.

Turning-points Resting-point

Left Right

Mean 1 70 1 '72 gS I34

Thus we find that with no weights in the scale pans, the resting-point is 95 -5 — we may call this 96 with sufficient ac- curacy— while, with the object to be weighed in the left pan, and 37*682 grammes in the right, the resting-point is 134.

Hence 37*682 gms. is too small, and we require to find what is the exact weight we must add to bring the resting- point from 134 to 96, that is, through 38 divisions of the scale.

To effect this, move the rider through a ew divisions on the beam, say through 5 ; that is, place it at division 7. The effective weight in the scale pan is now 37*687 gms.; observe