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LABORATORY PHYSICS 
 
 A STUDENT'S MANUAL FOB COLLEGES 
 AND SCIENTIFIC SCHOOLS 
 
 BY 
 
 DAYTON CLARENCE MILLER, D.Sc. 
 
 Pkofessor of Physics in Case School of Applied Science 
 
 BOSTON, U.S.A. 
 
 GINN & COMPANY, PUBLISHERS 
 
 Cl)e Stbensenm preec! 
 
 1903 
 
COPYKIGHT, 1903, BY 
 
 DATTON CLAKENCE MILLER 
 
 ALL BIGHTS BESEBVED 
 
PREFACE 
 
 This Manual is designed to be a student's handbook for the 
 performance of experimental problems in physics. The selection 
 of problems and their treatment is the result of twelve years of 
 teaching experience, and the descriptions of most of the exercises 
 have been used in typewritten form for the past six years. The 
 grade of work is that of the course in general physics in colleges 
 and technical schools. It is presumed that the student has had 
 a course in preparatory laboratory physics, and that these exer- 
 cises wUl be accompanied, or preceded, by a full course of lec- 
 tures and recitations in general physics, including instruction in 
 the principles and manipulations of the various experimental 
 operations. 
 
 One hundred and twenty-eight exercises are described. Some 
 of these, however, such as reading the barometer, determining 
 the heat equivalent of a calorimeter, etc., are usually performed 
 in connection with other problems. There is a break in the 
 serial numbers of the exercises at the end of each part, to 
 permit the assignment of numbers to additional exercises in 
 their proper groups. 
 
 The definite problems given under each heading have been 
 formulated with much care, to cover as wide a range of the 
 subject as is possible. They are based upon the work of the 
 sophomore class in the Case School of Applied Science, and 
 each one represents what may be expected of one or two 
 students in a laboratory exercise of three hours' duration. 
 Shorter exercises, or exercises covered by particular articles, 
 may be conveniently assigned in the manner explained in 
 the Introduction. 
 
IV PREFACE 
 
 The book is not merely a compilation, though oi course most 
 of the exercises have been used and described before. The 
 writer has made free use of all sources of information, sufficient 
 acknowledgment for which will be found in the many refer- 
 ences given. It is thought that several important exercises 
 appear for the first time in a laboratory manual, and that others 
 have been made more efficient by the method of treatment. 
 Among these may be mentioned those concerning the compari- 
 son and calibration of graduated scales, the balance, density by 
 hydrostatic weighing. Reed's method for i?ating a fork, ther- 
 mometry and calibration of thermometers, high-temperature 
 measurements, calorimetry by heating, mechanical equivalent 
 of heat, hygrometry, photometry, spectrometry, concave-grating 
 spectroscope, interferometer, silvering glass, refractometer, 
 capUlary electrometer, and magnetic variometer. 
 
 In making references the endeavor has been not to refer to 
 original sources but rather to those books which contain infor- 
 mation in the most useful form for the experiments described, 
 and which are most likely to be available to the student. It is 
 believed that the descriptions are ample for the performance 
 of the exercises ; the references indicate where other methods 
 are explained, or where information for advanced work may be 
 found. 
 
 The writer is greatly indebted to Professor Edward W. Morley 
 for many valuable suggestions in general, and in particular in 
 connection with the balance, the thermometer, and the inter- 
 ferometer. Messrs. H. W. Springsteen, H. S. Hower, S. R. 
 Cook, and J. W. Easton, present or former Instructors in Physics 
 in Case School of Applied Science, have given efficient aid in 
 reading the manuscript and proof sheets of the book. 
 
 DAYTON C. MILLER. 
 Cleveland, Ohio, 
 
 July, 1903. 
 
CONTENTS 
 
 PAET I — INTEODUCTION 
 CHAPTER I. GENERAL INSTRUCTIONS 
 
 PAGB 
 
 Objects of Laboratory Practice — Available Problems — Equipment — 
 Descriptions of Exercises — Assignment of Exercises — Notes and 
 Records — Reports — Observations, Errors, and Corrections — Prob- 
 able Error — Least Squares — Weighted Observations — Significant 
 Figures — Graphical Methods — Units — Reading Divided Scales . 1 
 
 PAET II — MECHANICS 
 
 CHAPTER n. GENERAL MEASUREMENTS, LENGTH, AND MASS 
 
 I. Length with the Calipers. Length — The Vernier — The Vernier 
 
 Caliper — The Micrometer Caliper . . 15 
 
 II. Length and Radius of Curvature with the Spherometer. The 
 
 Spherometer 19 
 
 III. Constant of a Micrometer Microscope. The Micrometer Micro- 
 
 scope .... 20 
 
 IV. Length with the Comparator. The Comparator 23 
 
 V. Length with the Dividing Engine. The Dividing Engine — 
 
 Dividing Engine Comparator 25 
 
 VI. Circular Arc with the Dividing Engine. Dividing Circles . 27 
 VII. Errors of a Graduated Scale by Gay-Lussac's Method. 
 Calibration of Graduated Scales — Calibration with a Dividing 
 Engine and One Microscope — Calibration with a Dividing 
 Engine and Two Microscopes — Calibration with a Micrometer- 
 Microscope Comparator 28 
 
 VIII. Erboks op a Graduated Scale by Precision Methods. Cali- 
 bration of Scales by Precision Methods 32 
 
 IX. Length with the Optical Micrometer. The Optical Micrometer 
 
 — Three Legged Optical Micrometer — The Reading Telescope 86 
 X. Constants of a Level with the Level Trier. The Spirit Level 39 
 
 v 
 
VI CONTENTS 
 
 , PAGE 
 
 XI. Areas with the Planimeter. The Planimeter — Mensuration 
 
 of Areas 42 
 
 XII. Angular Distance with the Sextant. The Sextant ... 46 
 
 XIII. Laws of the Equilibrium of Forces by the Triangle of 
 
 Forces. Equilibrium of Forces 49 
 
 XIV. Conservation of Momentum with the Ballistic Pendulum. 
 
 Momentum and Impact 51 
 
 XV. Mass by the Equilibrium of Moments. Equilibrium of 
 
 Moments 54 
 
 XVI. Mass with the Balance by the Method of Vibrations and 
 Double Weighing. The Balance — Weighing by Vibrations 
 — Double Weighing and Eatio of Balance Arms — Weighing 
 by the Rider Method — Sensibility of Balance — Practical 
 
 Hints on the Use of the Balance 55 
 
 XVII. Absolute Mass. Weighing by Keversal — Weight in Vacuo . 64 
 XVIII. Errors of a Set of Weights by Comparison. Calibration of 
 
 Weights — Adjustment of Weights 66 
 
 XIX. Volume by Weighing. Volume by Weighing — Volume and 
 
 Eadius of a Tube 68 
 
 XX. Atmospheric Pressure and Altitudes with the Barometer. 
 The Barometer — Beduction of Barometer Readings — Baro- 
 metric Pressure — The Barye and Standard Pressure — Alti- 
 tudes by the Barometer 69 
 
 CHAPTER III. TIME, ACCELERATION, AND GRAVITY 
 
 XXI. Laws of Accelerated Motion with a Falling Tuning Fork. 
 Time — Accelerated Motion — Acceleration with Variable 
 
 Time Unit 74 
 
 XXII. Laws of Accelerated Motion with Atwood's Machine. 
 Atwood's Machine — Gravity and Inertia of Pulley with 
 Atwood's Machine 77 
 
 XXIII. Properties of the Path of a Projectile. The Trajectory . 79 
 
 XXIV. Gravity by the Simple Pendulum. Simple Pendulum — Peri- 
 
 odic Time by the Method of Coincidences 81 
 
 XXV. Gravity with the Reversible Pendulum. Kater's Pendu- 
 lum — Formula for the Compound Pendulum — The Chrono- 
 graph — Mendenhall's Optical Method for comparing Two 
 
 Pendulums 84 
 
 XXVI. Gravity by Free Fall. Freely Falling Body — The Tuning- 
 Fork Chronograph — Smoked Paper and Glass 90 
 
CONTENTS 
 
 Vll 
 
 CHAPTER IV. ELASTICITY, AND PROPERTIES OF MATTER 
 
 PAGE 
 
 XXA'II. Young's Modulus by Stretching. CoeflScients of Elasticity 
 — Young's Modulus and Elastic Limit — Young's Modulus 
 with a Suspended Wire and Scale — Young's Modulus with 
 the Cathetometer — Adjustment of the Cathetometer . . 93 
 XXVIII. Modulus of Elasticity bt: Flexure. Flexure of a Rectan- 
 gular Bar supported at Both Ends — Bar supported at One 
 End ; Cylindrical Bar 98 
 
 XXIX. Coefficient of Rigidity with the Torsion Lathe. The 
 
 Modulus of Rigidity — Simple Torsion Apparatus ... 99 
 XXX. Modulus of Torsion by the Torsion Pendulum. Modulus 
 
 of Torsion — Periodic Time by the Method of Transits . 101 
 
 XXXI. Moment of Inertia by the Torsion Pendulum. Moment of 
 Inertia by Torsion Pendulum — Formulse for Moments of 
 Inertia 104 
 
 X X XTT . Compressibility of a Liquid with the Piezometer. The 
 
 Piezometer 106 
 
 XXXIII. Boyle's Law with the U-Tube. Boyle's Law .... 108 
 
 XXXIV. Errors of an Aneroid with an Air Pump. Testing the 
 
 Aneroid Barometer 110 
 
 XXXV. Coefficient and Laws of Friction by Several Methods. 
 
 Coefficient of Friction 112 
 
 XXXVI. Surface Tension with Capillary Tubes. Capillarity and 
 
 Surface Tension — Cleaning Glassware . ....... 11.3 
 
 XXXVII. Surface Tension by Direct Measure. Surface Tension . 116 
 
 XXXVni. Specific Viscosity with Torsion Pendulum. Viscosity . 117 
 
 CHAPTER V. DENSITY AND SPECIFIC GRAVITY 
 
 XXXIX. Density by Hydrostatic Weighing. Density and Specific 
 Gravity — Density by Hydrostatic Weighing — Other 
 Methods for Hydrostatic Weighing. . . ... 119 
 
 XL. Density with the Pyknometer. Density with the Pyknom- 
 
 eter . 124 
 
 XLI. Density of Air by Weighing and Exhaustion. Density of 
 
 Air 126 
 
 XLII. Density of a Gas by Weighing and Comparison with Air. 
 
 Density of a Permanent Gas . 127 
 
 XLIII. Density op Liquids with a U-Tube. Density with a U-Tube 
 
 — Relative Density by Hare's Method 128 
 
 XLIV. Specific Gravity with the Jolly Balance. The Jolly 
 
 Balance 129 
 
 XLV. Specific Gravity with Hydrometers. The Constant Immer- 
 sion Hydrometer — The Variable Immersion Hydrometer 131 
 
VUl CONTENTS 
 
 PAET III — SOUND 
 CHAPTER VI. FREQUENCY OF VIBRATION 
 
 PAGB 
 
 L. Frequency of a Sound by Beats. The Tuning Fork — Fre- 
 quency of Sound with a Tuning Fork 133 
 
 LI. Frequency of a Sound with the Siren. Frequency of Sound 
 
 with the Siren 134 
 
 LII. Frequency of Vibration by Graphic Methods. Frequency of 
 
 Vibration by Graphic Methods 136 
 
 LIII. Ratio of Vibration FrbqiJencies by Lissajous's Figures. Fre- 
 quency of Vibration by Optical Method — Copying a Standard 
 
 Fork — Projection of Lissajous's Figures 137 
 
 LIV. Frequency of a Fork by Optical Comparison. Reed's Method 
 
 for Rating- a Tuning Fork — The Stroboscopic Method . . . 139 
 
 CHAPTER VII. VELOCITY AND WAVE LENGTH OF SOUND 
 
 LV. Velocity OP Sound by Resonance. Velocity of Sound in Air — 
 
 Wave Length of Sound by Resonance 144 
 
 LVI. Velocity of Sound and Young's Modulus by Dust Figures. 
 Relative Velocity of Sound by Stationary Waves — Young's 
 
 Modulus by the Velocity of Sound 145 
 
 LVII. Velocity and Wave Length of Sound by Interference. Inter- 
 ference of Sound 147 
 
 CHAPTER VIII. VIBRATING STRINGS AND AIR COLUMNS, AND 
 SOUND ANALYSIS 
 
 LVIII. Laws op Vibrating Strings with the Sonometer. Laws of 
 Vibrating Strings — Melde's Experiments with Vibrating 
 
 Strings ■ 149 
 
 LIX. Forms of Vibration in Organ Pipes by Manometric Flames. 
 Nodes in Organ Pipes — Combined Sounds — Organ Pipe with 
 
 Water Trough 151 
 
 LX. Composition op a Sound with Resonators. Sound Analysis . 152 
 
CONTENTS 
 
 IX 
 
 PART IV — HEAT 
 CHAPTER IX. EXPANSION 
 
 PAGE 
 
 LXX. Linear Expansion of a Solid with the Comparator. 
 
 Coefficient of Linear Expansion . . 153 
 
 LXXI. Cubical Expansion of Glass with the Weight Thermom- 
 eter. The Weight Thermometer . 155 
 
 LXXII. Expansion of Liquids with the Dilatometer. Coefficient 
 
 of Expansion of Liquids ... 157 
 
 CHAPTER X. THERMOMETRY 
 
 LXXTTT. Constants and Errors of a Thermometer by Calibration. 
 Thermometry — Calibration of a Thermometer — Funda- 
 mental Interval of a Thermometer — Table of Corrections 
 for a Thermometer — Exposed Column of Mercury — Com- 
 parison of Mercury-iu-Glass and Hydrogen Thermometers 159 
 
 LXXIV. Temperature and Expansion of Gases with the Air Ther- 
 mometer. The Air Thermometer 166 
 
 LXXV. High Temperatures with Resistance and Thermo-Electric 
 Pyrometers. Measurement of High Temperatures — The 
 Platinum Thermometer — The Thermo-Electric Pyrometer 
 — Fixed Points for High Temperatures . . .... 168 
 
 CHAPTER XL CALORIMETRY 
 
 LXXVI. Calorimeter Constants by Calculation and Experiment. 
 Calorimetry — Heat Capacity of a Calorimeter — Radiation 
 
 Corrections .... 173 
 
 LXXVII. Specific Heat by Black's Method. Black's Ice Calorimeter 178 
 LXXVIII. Specific Heat with Bunsen's Ice Calorimeter. Bunsen's 
 
 Ice Calorimeter 180 
 
 LXXIX. Specific Heat of a Solid by the Method of Mixtures — 
 
 Specific Heat by Method of Mixtures 182 
 
 LXXX. Specific Heat of a Liquid by the Method of Mixtures. 
 
 Specific Heat of Liquids by Method of Mixtures .... 184 
 LXXXI. Specific Heat of a Liquid by Cooling. Specific Heat by 
 
 Method of Cooling 186 
 
 LXXXII. Specific Heat of a Liquid by the Method or Heating. 
 
 Specific Heat of Liquids by Method of Heating .... 188 
 
X CONTENTS 
 
 PAGE 
 
 LXXXIII. Heat Equivalent of Fusion of Ice. Heat Equivalent of 
 
 Fusion 191 
 
 LXXXIV. Heat Equivalent of Vaporization of Water. Heat 
 
 Equivalent of Vaporization 192 
 
 LXXXV. Mechanical Equivalent of Heat by Puluj's Method. 
 
 Mechanical Equivalent of Heat 194 
 
 CHAPTER XII. HYGEOMETRY 
 LXXXVI. Htgrometkic State by Various Methods. Hygrometry . 197 
 
 PAET V — LIGHT 
 
 CHAPTER XIII. PHOTOMETRY 
 
 C. Luminous Intensity with Simple Photometers. Photometry 
 — Candle Power — Horizontal Candle Power — Simple 
 
 Photometers 203 
 
 CI. Intensity of Light by Precise Methods. Standards of 
 Light — Mean Spherical Candle Power — Gas Meter — 
 Photometry of Intense Lights — The Lummer-Brodhun 
 Photometer — The Rood- Whitman Flicker Photometer . 207 
 
 CHAPTER XIV. MIRRORS AND LENSES, MAGNIFYING POWER 
 
 CII. Radii and Foci of Mirrors by Various Methods. Con- 
 jugate Foci — Concave Mirrors — Convex Mirrors . . . 213 
 
 cm. Radii and Foci of Lenses by Various Methods. Lenses — 
 Convex Lenses — Concave Lenses — Radius of Curvature of 
 a Lens — Focal Length of a System of Lenses . . . 216 
 
 CIV. Magnifying Power. Magnifying Power .... . . 221 
 
 CHAPTER XV. GONIOMETRY 
 
 CV. Angle of a Crystal with a Goniometer. The Reflecting 
 
 Goniometer 228 
 
 CVI. Adjustment of the Spectrometer. The Spectrometer — 
 
 Adjustment of the Spectrometer — Reading Divided Circles 225 
 CVII. Angle of a Prism with the Spectrometer. Adjustment of 
 Prism — Angle of Prism ; First Method — Angle of Prism ; 
 Second Method — Angle of Prism; Third Method . . . 230 
 
CONTENTS XI 
 
 CHAPTER XVI. INDEX OF REFRACTION 
 
 PAGE 
 
 CVIII. Index of Refraction by Minimum Deviation. Index of 
 
 Refraction — Angle of Minimum Deviation 234 
 
 CIX. Index of Refraction with a Microscope. Index of Refrac- 
 tion with a Microscope . . .... .... 236 
 
 ex. Index of Refraction by Displacement. Index of Refraction 
 
 by Displacement . 238 
 
 CXI. Index of Refraction by Total Reflection. Total Re- 
 flection — Total-Refleotometer — Refractometer — Crystal- 
 Refractometer ... 239 
 
 CHAPTER XVII. WAVE LENGTH OF LIGHT 
 
 CXn. Wave Length of Light by Fresnel's Interference Method. 
 Interference of Light — Wave Length of Light with the 
 Bi-Prism — Fresnel's Mirrors ... 246 
 
 CXIII. Wave Length of Light with the Plane Diffraction Grat- 
 ing. Wave Length of Light by Diffraction 249 
 
 CXIV. Wave Length with the Concave-Grating Spectroscope. 
 The Concave-Grating Spectroscope — Normal Diffraction 
 Spectrum . . . . . 251 
 
 CHAPTER XVIII. THE INTERFEROMETER 
 
 CXV. Small Lengths with the Interferometer. The Interferom- 
 eter — Finding the Interference Bands — Circular Bands — 
 Counting Fringes — Monochromatic Light . . 255 
 CXVI. Plane Surfaces by Optical Methods. Plane and Plane- 
 Parallel Surfaces 264 
 
 CXVII. Silvering Glass by Chemical Methods. Brashear's Process 
 for Silvering Glass — Roohelle-Salts Process for Silvering 
 Glass — Cleaning Optical Surfaces for Silvering 266 
 
 CHAPTER XIX. THE SPECTROSCOPE AND POLARIMETER 
 
 CXVIII. Chemical Composition with the Spectroscope. Spectrum 
 
 Analysis — The Heliostat 271 
 
 CXIX. Optical Activity with the Polarimeter. Rotatory Polari- 
 zation — The Polarimeter — Saccharimetry 275 
 
xu 
 
 CONTENTS 
 
 PAET VI — ELECTRICITY AND MAGNETISM 
 
 CHAPTER XX. RESISTANCE 
 
 PAGE 
 
 CXXV. Resistance by the Wheatstone's Bridge. Electrical 
 Measurements and Ohm's Law — Wheatstone's Bridge 
 — The Slide- Wire-Meter Bridge — The Galvanometer — 
 Adjusting a Galvanometer — Telescope, or Lamp, and 
 Scale — Standard International Ohm — Resistances and 
 Rheostats — Keys 282 
 
 GXXVI. Resistance WITH THE Box Bridge. The Box Bridge — Max- 
 well's Rule for Wheatstone's Bridge 293 
 
 CXXVII. Resistance with a Tangent Galvanometer. The Tangent 
 Galvanometer — Resistance by Three Observations with a 
 Tangent Galvanometer, and Battery Resistance by Ohm's 
 Method — Commutators, Reversing Keys 295 
 
 CXXVIII. Resistance by the Differential Method. The Differential 
 Galvanometer — To measure a Resistance greater than that 
 of One of the Galvanometer Coils — To measure a Resistance 
 less than that of One of the Galvanometer Coils .... 299 
 
 CXXIX. 
 
 CXXX. 
 
 Resistance by the Fall of Potential. Ammeters and 
 Voltmeters — Resistance with Voltmeter and Ammeter — 
 Battery Resistance with Voltmeter and Ammeter — Battery 
 Resistance with Voltmeter and Standard Resistance — Bat- 
 tery Resistance with Condenser and Resistance .... 302 
 
 Galvanometer and Battery Resistance by the Wheat- 
 stone's Bridge. Thompson's Method for Galvanometer 
 Resistance — Mance's Method for Battery Resistance . . 
 
 CXXXI. Battery Resistance by Compensation. Beetz's Method for 
 Battery Resistance — Benton's Method for Battery Resist- 
 ance 
 
 CXXXII. Electrolyte Resistance by Kohleausch's Method. Elec- 
 trolyte Resistance with Alternating Currents — Electrolyte 
 Resistance ... 
 
 305 
 
 307 
 
 309 
 
 CXXXIII. Insulation Resistance by Direct Deflection. Measure- 
 ment of High Resistance by Direct Deflection — High Resist- 
 ance by Direct Deflection and Double Readings .... 311 
 
 CXXXIV. Specific Resistance by the Comparison of Potentials. 
 Specific Resistance — Resistance by Method of Comparison 
 of Potentials — The Low-Resistance Bridge 313 
 
CONTENTS 
 
 xiu 
 
 CXXXV. Temperature Coefficient of Resistance bv Foster's 
 Method. Temperature Coefficient of Resistance — Carey 
 Foster's Metliod for Comparison of Resistances — Resist- 
 ance of the Slide Wire 316 
 
 CXXXVI. Errors of a Bridge Wire bv Barus's Method. Calibra- 
 tion of Bridge Wire . 319 
 
 CHAPTER XXI. CURRENT STRENGTH 
 
 CXXXVn. Galtajtometer Cosstant WITH THE Voltameter. Constant 
 of Tangent Galvanometer — The Copper Voltameter — 
 Standard International Ampere ; Silver Voltameter . . 322 
 CXXXVin. Jodle's Equivalent with the Electrocalorimetek. 
 
 Joule's Equivalent ; the Joule ; the Watt .... 324 
 
 CXXXIX. Figure of Merit of a Galvanometer by Direct Deflec- 
 tion. Figure of Merit of a Galvanometer — Sensitiveness 
 
 of Galvanometers ; Shunts ... 325 
 
 CXL. Current Strength with the Electkodtnamometer — The 
 
 Electrodynamometer 330 
 
 CHAPTER XXII. ELECTROMOTIVE FORCE 
 
 CXLL Electromotive Force by Compensation. - Standard Inter- 
 national Volt ; Standard Cells — Electromotive Force by 
 
 Compensation — ^The Potentiometer 332 
 
 CXLII. Electromotive Force with a Condenser. Comparison of 
 Electromotive Forces with the Condenser — The Ballistic 
 Galvanometer 336 
 
 CXLni. Electromotive Force with the Quadrant Electrometer. 
 The Quadrant Electrometer — Electrometer for measuring 
 Large Potential Differences 339 
 
 CXLIV Electromotive Force with the Capillary Electrom- 
 eter. The Capillary Electrometer — Contact Difference 
 
 of Potential . , 342 
 
 CXLV. Electromotive Force with an Ammeter. Electromotive 
 Force and Resistance of Cells with a Milliammeter and 
 a Resistance 345 
 
 CXLVI. Thekmo-Electromotite Force by Direct Measure. 
 Thermo-Electromotive Force — Thermo-Electric Power 
 
 and Thermo-Electrio Diagram 347 
 
 CXLVn. Errors of a Voltmeter by Compensation. Calibration of 
 
 Voltmeter 348 
 
XIV 
 
 CONTENTS 
 
 CHAPTER XXIII. CAPACITY 
 
 PAOE 
 
 CXLVIII. Relative Capacity ■with a Ballistic Galtanometer. 
 Standard International Coulomb and Farad; Standard 
 Condensers — Condenser Keys — Comparison of Conden- 
 ser Capacities 350 
 
 CXLIX. Relative Capacity by Bridge Methods. Bridge Methods 
 
 for Comparison of Capacities .... 353 
 
 CL. Relative Capacity by the Method op Mixtures. Compari- 
 son of Capacities by the Method of Mixtures 355 
 
 CLI. Absolute Capacity of a Condenser with a Ballistic Gal- 
 vanometer. Absolute Capacity with a Ballistic Galva- 
 nometer — The Ballistic Constant 356 
 
 CHAPTER XXIV. INDUCTION 
 
 CLII. Inductance by Comparison. Standard Henry ; Coefficient 
 
 of Self-induction 360 
 
 CLIII. Coefficient of Mutual Induction by Comparison. Mutual 
 
 Induction — Comparison of Two Mutual Inductances . . 361 
 
 CHAPTER XXV. MAGNETIC QUANTITIES 
 
 CLIV. Earth's Horizontal Magnetic Intensity with the Mag- 
 netometer. The Earth's Magnetic Elements — The Mag- 
 netometer — The Kew Magnetometer 363 
 
 CLV. Variation op the Earth's Horizontal Magnetic Intensity 
 with the Variometer. The Magnetic Variometer — Mag- 
 netic Declination — Comparison of Horizontal Intensities 
 — Oscillation and Deflection Methods for Comparison of 
 Horizontal Intensities . 366 
 
 CLVI. Earth's Magnetic Inclination and Intensity with the 
 Earth Inductor. The Earth Inductor — Magnetic Incli- 
 nation — Magnetic Intensity 370 
 
 CLVII. Distribution of Magnetism by Rowland's Method. Flow 
 
 of Induction from a Magnet — Distribution of Magnetism . 374 
 CLVIII. Intensity of a Magnetic Field by Rowland's Method. 
 
 Comparison of Magnetic Fields with the Earth's Field . . 375 
 
 CLIX. Temperature Coefficient op a Magnet by Deflection 
 
 Methods. Temperature Coefficient of a Magnet . . . 376 
 
CONTENTS XV 
 
 PAET VII — APPENDIX 
 
 CHAPTER XXVI. TABLES, CONSTANTS, AND KEFEEENCES 
 
 PAGE 
 
 Tables. Explanation of Tables 378 
 
 1. Reduction to Vacuum of Weighings made with Brass Weights in Air 379 
 
 2. Density of Various Substances 379 
 
 3. Density of Water and of Mercury 380 
 
 4. Volume of a Glass Vessel at 20° 380 
 
 5. Coefficient of Static Friction 380 
 
 6. Relative Viscosity at 19° ... 380 
 
 7. Elastic Constants of Solids ... 381 
 
 8. Compressibility of Water, Mercury, and Glass 381 
 
 9. Surface Tension at 20° ... 381 
 
 10. Absolute Value of Gravity 381 
 
 11. Local Geographical Data for Cleveland 381 
 
 12. Specific Gravity 'of Air 382 
 
 13. Capillary Depression of Mercury in Glass Tubes 382 
 
 14. Reduction of Barometer Readings to 0° 383 
 
 15. Reduction of Mercurial Thermometer Reading to the Normal Scale 383 
 
 16. Boiling Temperature of Water at Various Pressures 384 
 
 17. Fixed Points for High Temperatures 384 
 
 18. Heat Constants of Liquids ... ... 384 
 
 19. Heat Constants of Solids ... . . . 385 
 
 20. Reduction of Psychrometric Observations 385 
 
 21. Relative Humidity 386 
 
 22. Declination of the Sun, and Equation of Time 386 
 
 23. Tension and Mass of Aqueous Vapor in Saturated Air ... . 387 
 
 24. Index of Refraction of Various Substances . 387 
 
 25. Wave Length of Lines of Solar Spectrum 387 
 
 26. Specific Resistance of Various Substances 388 
 
 27. Electromotive Force and Internal Resistance of Cells 388 
 
 28. Vibration Frequency of Tones of the Musical Scale 388 
 
 29. Reference Books 389 
 
 30. Miscellaneous Constants and Numbers 390 
 
 31. Metric and English Equivalents and Abbreviations 390 
 
 32. Natural Trigonometrical Functions 391 
 
 33. Four-Place Logarithms 392, 393 
 
 Index 394 
 
LABORATORY PHYSICS 
 
 PaET I IlSTTRODTJCTIOF 
 
 chapter i 
 
 gej^eeal instructions 
 
 1. Objects of Laboratory Practice A course in general 
 
 physics has for its primary object the inculcation of the funda- 
 mental principles of the science. This is best accomplished 
 for college classes by the proper coordination of lectures and 
 demonstrations by the instructor, and of recitations and labora- 
 tory practice by the student. The lectures, demonstrations, and 
 recitations should consume half or more of the time allotted 
 to physics, and should be founded upon a complete systematic 
 text-book. This part of the course should be in advance of the 
 laboratory work, so that the student shall have studied the theory 
 of an experiment before attempting to perform it. 
 
 A course of physical experimentation by the student has for 
 its objects the fixing in mind of the essential principles studied 
 in the class room, the training of the student's thinking and 
 reasoning powers, and the furnishing of direct proof of some 
 of the fundamental laws of the science ; and it has the further 
 objects, of almost equal importance, of giving the student an 
 acquaintance with the methods and instruments of physical and 
 technical operations, and of developing skill in the manipulation 
 of delicate apparatus and in the making and reducing of meas- 
 urements of precision. 
 
 1 
 
2 INTEODUCTION 
 
 This book is designed as a student's handbook for use in 
 the laboratory; it contains such practical explanations of the 
 principles involved in the experiments, and such details of 
 manipulation and reduction, as experience has shown to be 
 necessary and helpful to the student. 
 
 With these objects in view, a laboratory course is not con- 
 sidered as consisting of a certain number of exercises to be 
 worked out by each student, and to be complete when these 
 are finished, but rather as consisting of a definite amount of 
 time spent in judicious experimenting. Several days of work 
 put upon some troublesome process which fails at first to give 
 the desired results is as valuable and as much to the student's 
 credit, if honestly done, as the completion of several probleins 
 in which no difficulties were encountered. 
 
 2. Available Problems. — Certain problems in each branch of 
 the subject are fundamental, and should be required of each 
 student; but many may be given to some students and not to 
 others, as circumstances determine. Only those problems which 
 have a very clear object in view, and which demand some skill 
 in manipulation and the use of apparatus neither too simple nor 
 too delicate, should be used for a general course ; and usually an 
 experiment should require not less than three nor more than 
 six hours for its completion. These conditions greatly limit 
 the number of available problems. 
 
 Mechanics furnishes a larger number of valuable laboratory 
 exercises than any other branch of physics, and some of these 
 are best introduced at the end of the course after the student 
 has acquired considerable skill. If each problem contains a 
 distinct lesson, either in principle or in manipulation, it matters 
 not whether one student receives this exercise at the same time 
 that it is assigned to another. It is often advantageous to have 
 students at work in several branches of the subject at the same 
 time. With the work properly systematized, this method is not 
 extravagant of teaching force. 
 
 3. Equipment — The duplication of sets of apparatus of 
 cheap and simple construction, in order that all students may 
 be at work upon the same experiment at the same time, is not 
 
GENERAL INSTRUCTIONS 3 
 
 advocated for college classes. The laboratory equipment should 
 be of as great a variety as is possible, and the student should be 
 trained in the use of the actual instruments employed in scien- 
 tific work. If several instruments for one purpose are required, 
 let them be of all varieties as regards size, adjustments, and 
 attachments, and though a student may work with but one form, 
 he cannot fail to obtain information from merely seeing other 
 forms in use. 
 
 Each piece of important apparatus should have a definite 
 place in the laboratory ; it should be in position and in perfect 
 order at the beginning of an exercise, and the student should 
 leave it in the same condition at the completion of his work. 
 Subsidiary apparatus, such as calipers, weights, thermometers, 
 and tools, may be obtained from the instructor in charge, and 
 these are to be returned by the one who borrowed them. 
 
 Much of the apparatus of the physical laboratory is delicate 
 and costly, and is liable to disarrangement by improper handling. 
 If any instrument is out of order, the fact should be at once 
 reported ; the student will be held accountable for all damages 
 which result from neglect or carelessness. When a piece of 
 apparatus is to be used with which the student is unfamiliar, 
 the peculiarities of construction and adjustment should be care- 
 fully studied before manipulation is begun, both to save time 
 and to prevent injury to the instrument. 
 
 "With large classes the making of apparatus by the individual 
 student is not practicable. An apparatus made from the odds 
 and ends of the laboratory will not command the beginner's 
 respect and careful attention, even though it would do good 
 work in the hands of an experienced manipulator. 
 
 An essential part of the equipment of every laboratory is a 
 reference library, which need not be large, but should contain 
 the principal standard works on practical and theoretical physics. 
 The student should have free access to the reference books, and 
 should use the privilege constantly to obtain information in regard 
 to the various forms of apparatus and the different methods of 
 observation and reduction. A short list of reference books is 
 given in the Appendix. 
 
4 INTRODUCTION 
 
 4. Descriptions of Exercises. — The titles to the various 
 experiments might all be preceded by the phrase, " Determina- 
 tion of," "Measurement of," or "Proof of"; the titles them- 
 selves mention what is to be determined, and specify the method 
 or the principal piece of apparatus which is to be employed. 
 
 There follows a concise and specific statement of just what is 
 required of the student. In some cases more than one problem 
 is given under one heading; these are then designated as (a), 
 (b), etc. A problem is referred to in the laboratory work, both 
 in assignment and in records, by its number ; as 8, or 117 (b). 
 
 The articles grouped under each exercise describe the partic- 
 ular methods of work to be adopted, and, together with the 
 articles to which reference is made, they constitute sufficient 
 information to enable the student who has already studied the 
 principles involved to perform the experiment intelligently. 
 These descriptions are largely explanations of methods of work 
 rather than of particular pieces of apparatus. If more than one 
 important form of instrument is used with a given method, brief 
 descriptfions of these are given. This makes the manual useful 
 in laboratories of various equipment, and it is also advantageous 
 for the student using one form to know that other forms are 
 often employed. 
 
 5. Assignment of Exercises. — For assigning the problems, 
 small cards bearing the numbers of the exercises to be used 
 are prepared ; the assignment may be announced in two ways. 
 On each card is written the name of the student to whom the 
 corresponding experiment is assigned and the card is given to 
 him at the beginning of the exercise, to be returned at the end. 
 These names are then crossed off, and for the next day new 
 names are written on the same cards. Or there may be in the 
 laboratory a bulletin board containing a list of the names of the 
 students, and the cards with the numbers may be placed opposite 
 the names selected ; then the students have only to refer to the 
 book for the statements of the problems. 
 
 The card system has the advantage that any variation desired 
 in the specific statement of a problem may be written on the 
 card, while the description of the method would be the same as 
 
GENERAL INSTKUCTIONS 5 
 
 given under the corresponding exercise in the book. Shorter 
 exercises may thus be announced, or entirely different ones, 
 according to the requirements of various laboratories. 
 
 6. Notes and Records. — Blank forms arranged for the records 
 of each experiment are not recommended ; instead the student 
 is required to arrange for himself the data and results of his 
 work in a systematic way. The efficiency of experimental work 
 will be greatly increased by intelligent and skillful records. 
 Before beginning an experiment the student should carefully 
 study the various operations and measurements involved, and 
 plan a method of procedure and of recording the observations. 
 Tabular and symmetrical forms are to be used whenever possible. 
 Numerical results of several exercises are given in forms suit- 
 able for the reports; the original notes should be similarly 
 arranged, though they may of necessity contain more of detail 
 and computation. 
 
 Make the first right-hand page of the notebook a general 
 title-page; several following leaves are to be used for a table 
 of contents, which will be filled in as the experiments are per- 
 formed throughout the course. When the book is filled with 
 notes an index should be prepared and placed at the end. 
 Begin the record of each experiment on a new right-hand page, 
 giving a liberal amount of space to the title. There should be 
 recorded all the original data of the experiment: date, fellow- 
 observer, object of experiment, description, with the identifying 
 numbers or names of the particular pieces of apparatus used, 
 method, and all data of the observations, calculations, and results. 
 All observations, whether used in calculating the final result or 
 not, should be preserved, and any reason for suspecting certain 
 results as being inaccurate should be noted. Never erase the 
 result of an observation once recorded, even if it is deemed 
 worthless. 
 
 Important items should be conspicuous enough to be readily 
 found, and they should be so arranged as to make comparisons 
 and computations convenient. The record of every measurement 
 should specify the unit employed. The scales used in graphical 
 representations are to be explained. There should be diagrams 
 
b INTEODUOTION 
 
 showing the arrangement of apparatus ; special formulae used 
 should be analyzed and proved. 
 
 Numerical calculations in multiplication, division, involution, 
 and evolution must not be made by arithmetical processes. For 
 the greater part of the work in this course, four-place logarithms 
 (Appendix) or the slide rule are sufficient, though for a few 
 exercises, five-place, or even six- and seven-place logarithms 
 are required. The figures of calculation, logarithms, etc., must 
 be arranged in an orderly manner. The final result and' its 
 exact meaning should be made conspicuous ; if it is the mean 
 of several determinations, compute and specify the probable 
 error (Art. 9). When a result has been obtained it should be 
 compared with what is considered the correct result, obtained 
 by some independent method of proving, or from tables of 
 constants, or from published results of research work. 
 
 A notebook is valuable only in so far as it is a complete 
 record of all that has been done in making the original experi- 
 ment, and it should contain just so much of detail as will make 
 it intelligible to the writer or to others at any time after the 
 completion of the work. The necessity for full, unaltered, origi- 
 nal notes cannot be too strongly impressed upon the beginner. 
 
 Upon the completion of each exercise, before the student leaves 
 the laboratory, he should present his notebook to the instructor for 
 examination, criticism, and grading. If the work has been care- 
 lessly or improperly done, a part or all of it may have to be repeated. 
 When the notes and report mentioned in the next article are satis- 
 factory, both will be approved and signed by the instructor. 
 
 The notebooks for the entire class should be uniform in style 
 and of a permanent kind, having a cloth or leather cover, and 
 paper of good quality. The leaves are preferably of quadrille 
 paper, while an unruled paper is better than that with lines in 
 one direction only. The quadrille ruling facilitates tabulation 
 and is convenient for graphic solutions. A suitable size for the 
 page is fifteen by twenty-three centimeters. 
 
 The original notebook is to remain the permanent property of 
 the student, and if the records are properly made it will have 
 become a valuable possession at the end of the course. 
 
GENEEAL INSTRUCTIONS 7 
 
 7. Reports. — Upon the completion of each experiment the 
 student must make out a fuU and systematic report to be given 
 to the instructor, for grading and permanent filing in the records 
 of the laboratory. To secure uniformity, this report should be 
 made upon a blank form which, whUe containing no form for 
 the experiment proper, bears the title of the laboratory, and has 
 a place for the student's name, title of experiment, and sufficient 
 space for the data mentioned as necessary in the original notes. 
 The report may be somewhat condensed in the data of observa- 
 tions, giving only the essential results, and in the calculation, 
 which may be indicated. The numerical results of several of 
 the exercises which are given exhibit the proper forms for the 
 reports. Credits and grades for the exercises will be given only 
 after both notes and reports have been approved. 
 
 8. Observations, Errors, and Corrections. — When the highest 
 precision is attempted it is found that repeated measures of 
 the same quantity differ by appreciable amounts. The meas- 
 ures are affected by errors, some of which are accidental and 
 variable in magnitude, while there are others which are con- 
 stant. The constant errors may be due to imperfections in 
 the measuring instrument, to a faulty method, or to bias in the 
 observer. Study will often discover the errors of instruments 
 and methods, and they may be eliminated or made inappreciable 
 by determining their values and applying corrections for them. 
 
 Errors due to bias or prejudice are only to be reduced by the 
 exercise of special care and skill, or by a change of observers. 
 Whenever practicable the observations of one person should 
 be checked by another. While it is not necessary that the 
 student be in ignorance of the nature or value of a result to 
 be obtained, yet it may require an effort on his part to prevent 
 this knowledge from prejudicing the result, and to check the 
 tendency to make a second measure agree with the first. 
 
 Besides the constant errors there are always present accidental 
 errors of observation, which in a large series of independent, 
 equally trustworthy measures, are likely to be half positive and 
 half negative; most of these errors will be of small magnitude, 
 and only a very few will be large. The arithmetical mean is 
 
8 INTRODUCTION 
 
 the best result that can be obtained from such a series, and it 
 is much more trustworthy than is a single measure. Theory- 
 indicates that the probability of the mean increases as the 
 square root of the number of observations; hence the mean 
 of nine measures has a probable error one third as* large as the 
 probable error of one measure, and the mean of twenty-five 
 measures has a probable error one fifth as large. The proba- 
 bility increases so slowly in comparison with the labor of obser- 
 vation that a practical limit is soon reached. Usually ten 
 measures of the same quantity are sufficient. 
 
 The accidental errors are often small in comparison with the 
 constant errors, and time which might be spent in making a 
 longer series of measures is more profitably used in obtaining 
 a second result after making a complete readjustment of the 
 apparatus, or with new apparatus, or with a new method. Since 
 in laboratory practice mere numerical results are not primary 
 objects, the obtaining of a measure which by accident is correct 
 will not excuse the student from making the repeated adjust- 
 ments required in original work. Usually much more time is 
 consumed in arranging and adjusting a set of apparatus than 
 is required to make the final observations ; for the student this 
 is the most instructive part of laboratory work, and it should 
 be performed thoroughly and understandingly. 
 
 Another class of errors which may be called mistakes are 
 sometimes present. These generally result from inattention 
 and lack of skill, which are inexcusable even in the beginner. 
 
 The methods of procedure given in connection with the labo- 
 ratory problems are largely concerned with the elimination and 
 correction of errors. 
 
 Reference. — Lupton, Notes on Observations. 
 
 9. Probable Error — From a number of observations, depend- 
 ing upon their concordance, there may be computed a criterion 
 of the precision with which the measures can be made, which 
 is called the probable error. The probable error of a single 
 observation is the quantity which when added to and sub- 
 tracted from the mean gives limiting values, such that if a 
 
GENERAL INSTRUCTIONS 9 
 
 single measure of the same kind is made, it is as likely to lie 
 outside of these limits, on either the positive or negative side, 
 as it is to lie between them. The. probable error of the mean is 
 a quantity which when added to and subtracted from the mean 
 gives limiting values such that if another mean value is deter- 
 mined as was the first, its value is as likely to lie outside of 
 these limits, in either the positive or negative direction, as it is 
 to lie between them. 
 
 The probable error is not a limiting value such that there is 
 no probability of a greater error ; it is such a value that in a 
 large series there are certain to be as many errors greater than 
 it as there are smaller ones. In other words, a result is as 
 likely to have an error (compared with the mean) larger than 
 the probable error as it is to have a smaller one. Probability 
 shows that an error twice as large as the probable error is 
 likely to occur in one out of 5 measures, that an error three 
 times the probable error may be expected once in 23 cases, an 
 error four times as large once in 142, while an error five times 
 as large occurs only once in 1314 observations. 
 
 To give reasonable security, in the ratio of 142 to 1, that 
 the mean may not have an error exceeding four times the proba- 
 ble error of a single observation, the mean must be the result 
 of 16 measures. 
 
 If m is the arithmetical mean of a series of n measures, aj, a^, 
 fflg . . . Oj,, the probable error of a single observation is 
 
 ± Q 6745 J(w - a^f -F (wT- a^f +---(m- 
 * n — 1 
 
 ^ a„)2 
 
 n 
 and the probable error of the mean is 
 
 E=± 0.6745 J (w - ai)^ + (m - a^f + ■ ■ ■ (m - a,,)^ ^ _^_ 
 * w(w— 1) Vn 
 
 The numerical coefficient usually may be taken as |. 
 
 The differences between the mean and the several measures 
 are called the residuals or the deviations. The average of the 
 deviations, without regard to sign, is often used as an indication 
 of pre6ision. The percentage deviation, the value of which is 
 
10 INTRODUCTION 
 
 the deviation divided by the quantity, is useful when only a 
 few measures are made. 
 
 These precision measures, the probable error, the percentage 
 deviation, etc., only indicate the agreement of the measures 
 among themselves as regards the variable accidental errors ; they 
 indicate nothing with respect to constant errors, which may 
 be very large, and hence they do not show the relation of the 
 result to the absolute truth, a relation which usually is wholly 
 indeterminate. 
 
 10. Least Squares. — When a number of independent direct 
 measurements of the same quantity have been made, the arith- 
 metical mean is the most probable value. But if the quantity 
 is measured indirectly, being involved with others, the obser- 
 vations furnish equations expressing the relations of these 
 quantities. If the number of the equations equals that of 
 the unknown quantities, only one solution is possible. If there 
 are more equations than unknown quantities, no one set of 
 values for the unknown quantities is likely to satisfy all the 
 equations ; that is, an exact solution is impossible. Various 
 approximate solutions may be made, and theory shows that 
 the most probable one is that which makes the sum of the 
 squares of the residual errors the least possible. This method 
 of solution is known as the Method of Least Squares. The cal- 
 culation is apt to be laborious, and is required only in impor- 
 tant reductions. The student may consult the references for 
 descriptions of the operations, and mathematical text-books for 
 the full theory. 
 
 Repekences. — Concerning the reduction of observations, the student is 
 referred to Stewart and Gee, Practical Physics ; Kohlrausch, Physical Measure- 
 ments ; Nichols, A Laboratory Manual of Physics ; JBolman, Precision of 
 Measurements ; Merriman, Method of Least Squares. 
 
 11. Weighted Observations. — When a final result is to be 
 calculated from a number of different determinations of the 
 quantity, made perhaps by different observers, by different 
 methods, with different apparatus, or resulting from different 
 numbers of separate observations, these determinations will not 
 all be equally trustworthy. In the computation each one must 
 
GENERAL INSTRUCTIONS 11 
 
 be allowed to affect the result only in proportion to its trust- 
 worthiness, and weights must be assigned to them according 
 to their probable accuracy. When the separate values are each 
 the result of a number of observations, the weight is set equal 
 to this number. If the probable error, e, is known (Art. 9), 
 
 the weight is proportional to — • In other cases the weights 
 
 must be assigned according to the judgment of the computer. 
 In the calculation each value is multiplied by its weight, and 
 the sum of these products is divided by the sum of the weights. 
 The final mean, r, of several results, rj, r^, etc., having weights, 
 »„ »,, etc., is 
 
 12. Significant Figures In order that numerical operations 
 
 should have all the accuracy possible and yet not appear to 
 claim an unwarranted precision, and to save time, the data of 
 observation, computation, and results should contain only the 
 proper number of significant figures. A significant figure is 
 any one of the ten digits other than a zero which is used merely 
 to locate the decimal point. 
 
 In a series of measurements the significant places should be 
 carried one place farther than that in which differences first 
 occur. In casting off figures, increase by 1 the last figure 
 retained when the following was 5 or more. 
 
 When several numbers are to be added or subtracted, whose 
 significant figures by the above rule reach to different decimal 
 places, no significant places are to be retained beyond those of 
 the number whose last significant figure is in the highest place. 
 
 When numbers are to be multiplied or divided, there should 
 be the same number of significant figures in all the data, con- 
 stants, products, quotients, and result, as there are in that factor 
 which has the fewest significant figures. 
 
 In logarithmic computation there should be as many signifi- 
 cant figures in the mantissse as there are significant places in the 
 numbers operated upon. The characteristic is not considered 
 in counting the significant places. 
 
12 INTRODUCTION 
 
 All records of measurements should show clearly the full 
 degree of accuracy attained, and to this end final zeros in a decimal 
 fraction should be considered as significant as any other figure 
 in the same place. For example, if in weighing a mass with 
 a balance and by a method capable of determining single milli- 
 grams, the result is exactly twenty-one and six-tenths grams, 
 record the result thus: 21.600 g. If the last figure in a num- 
 ber representing a quantity is 9, this figure would of course be 
 written; if the figure is 10, in the decimal system, the 1 of 
 the figure is "carried" and added to the preceding figures, 
 but the is as significant now as was the 9 in the first case. 
 If the in the last place is dropped, it would indicate that no 
 value had been found for this place, which is contrary to the 
 supposition. 
 
 13. Graphical Methods. — When a series of observations has 
 been made of the dependence of one quantity upon another, 
 often the most perspicuous presentation of the results is by a 
 curve representing this dependence. Using the values of the 
 independent variable as abscissae, and the corresponding values 
 of the dependent variable as ordinates, a series of points may 
 be plotted. A curve drawn through these points as nearly as 
 possible, so that very few points are far from the curve, and 
 that with respect to any small portion of the curve there are as 
 many points on one side of it as on the other, wiU graphically 
 represent the observations (Art. 74). The exact form of the 
 curve will be a matter of judgment, and hence the solution 
 is somewhat indeterminate. An error in an observation may 
 be indicated by the erratic location of a point. The values of 
 unknown quantities may often be measured from such a plot ; 
 that is, a graphic solution may be made. Whenever it is pos- 
 sible, even though the solution has not been a graphic one, the 
 results of an exercise should be shown graphically. 
 
 14. Units All measurements in mechanics are to be recorded 
 
 in centimeter-gram-second units unless distinctly required other- 
 wise. Do not record some measurements in meters and others 
 in millimeters, but use the centimeter as the universal unit of 
 length, and the gram as the only unit of mass. In final results 
 
GENERAL INSTRUCTIONS 13 
 
 all forces are to be expressed in dynes. Attention to these points 
 may save much confusion in calculation. 
 
 The centimeter is ^^-^ of the distance, at the temperature of 0°, 
 between two certain marks on a certain bar of platinum-iridium, 
 known as the International Prototype Meter, which is preserved 
 at the International Bureau of Weights and Measures near Paris. 
 
 The gram is y^^-jj of the mass of a certain piece of platinum- 
 iridium, known as the International Prototype Kilogram, which 
 is preserved at the International Bureau of Weights and Meas- 
 ures near Paris. 
 
 The second is ^g f-g-j of a mean solar day. 
 
 Upon these three fundamental units are founded many derived 
 imits, such as the dyne, erg, and joule ; there are also related 
 units, such as the calorie, and many independent units, such as 
 the degree centigrade, candle power, etc. When deemed neces- 
 sary these units are described in connection with their first use 
 in the experimental work. 
 
 The employment of one system of units for all quantities 
 sometimes results in numbers which are very large or very 
 small. Such a number is best expressed as the product of two 
 factors, one of which is the number with one integral place and 
 as many decimal places as are significant, the other factor being 
 a power of 10. Since logarithmic computation is almost uni- 
 versal, this method is simple; the first factor is the number 
 corresponding to the mantissa alone, while the second is 10 
 with an exponent equal to the characteristic of the logarithm. 
 An illustration is the answer to the example in Art. 74. 
 
 Reference. — For a complete treatment of the units of physios together 
 with tables of constants, reference is made to Everett, Illustrations of the 
 C. G. S. System of Units. 
 
 15. Reading Divided Scales. — In quantitative work the read- 
 ing of some form of divided scale is almost universal, and when 
 the highest precision is desired fractions of the smallest divisions 
 must be determined. Often this is accomplished by means of 
 either the vernier or the micrometer screw, described in Arts. 17 
 and 19. When this is not the case it is a general rule that the 
 
14 mTKODUCTION 
 
 fraction of a division is to be estimated in tenths. Never esti- 
 mate in fourths or eighths, but always in tenths for recording 
 decimally. This is to apply to straight and circular measuring 
 scales; to the divided heads of measuring screws; to balance, 
 galvanometer, and thermometer scales ; to the time intervals 
 represented by pendulum beats ; and to all other readings wh^re 
 such an estimation is possible. 
 
Paet II MECHAlSriCS 
 
 CHAPTER II 
 GENERAL MEASUREMENTS, LENGTH, AND MASS 
 
 I. LENGTH WITH THE CALIPERS 
 
 Measure the length of a steel rod with the vernier caliper, in both inches 
 and centimeters, and find the ratio of the inch to the centimeter. 
 Determine the dimensions of a steel cylinder with English and metric 
 micrometer calipers, and find the ratio of the inch to the centimeter. 
 Calculate the probable error of a single observatipn and of the mean 
 for each set of measurements. 
 
 16. Length. — Measurements of length are always referred 
 to real or virtual divided scales, and when precision is desired 
 fractions of divisions must be determined. One of three 
 methods is employed for this purpose. The simplest is that 
 of estimation, mentioned in Art. 15, which depends upon the 
 judgment of the observer. A second method, more precise and 
 largely independent of judgment, is by means of the vernier. 
 The greatest precision is attained through the application of the 
 micrometer screw. These latter methods are directly applied 
 in the calipers described below, and their use in more elaborate 
 instruments for length measures and comparisons is described 
 in connection with other experiments. 
 
 17. The Vernier. — To facilitate the reading of fractions of a 
 division of a divided scale, the vernier is often employed. It 
 consists of a short subsidiary scale, movable along the main 
 
 15 
 
16 MECHANICS 
 
 scale, the zero line of which is the index to the main scale. If 
 
 the smallest indication desired is — of a division, the vernier is 
 
 n 
 
 usually n — 1 scale divisions in length, and is divided into n parts. 
 If a vernier is applied to determine tenths of a scale division, 
 it may be constructed so that its total length is nine scale divi- 
 sions, while it is divided into ten equal parts ; each vernier part 
 will therefore be equal to nine tenths of a scale division. Sup- 
 pose, as is represented in Fig. 1, that the index or vernier line 
 lies between the main scale graduations corresponding to 1.4 
 and 1.5, the fraction of a division from 1.4 to the index Une can 
 be determined as follows. The vernier line No. 1 is nearer to 
 its corresponding scale line by one tenth of a division (a vernier 
 
 11 
 
 10 
 
 Fig. 1. Scale with Versier 
 
 division being nine tenths of a scale division), and the vernier 
 line No. 2 is two tenths of a division nearer the next scale 
 line, and so on. Each succeeding vernier line being one tenth 
 of a division nearer its corresponding scale line, the number of 
 the vernier line which exactly coincides with a scale line will 
 represent the number of tenths of a division by which the first, 
 or 0, line is separated from the preceding scale line. If, as shown, 
 line No. 7 of the vernier thus coincides with a scale line, since 
 it is seven tenths of a division nearer to this line than the 
 vernier line is to its corresponding scale line, the reading to 
 the nearest tenth of a division is 1.47. 
 
 The following rule for reading vernier scales is usually appli- 
 cable. From an examination of the main scale determine the 
 value of its smallest division. This smallest division will be 
 further subdivided by the vernier into as many parts as there 
 are divisions in the entire vernier scale: determine what this 
 
THE CALIPEE 17 
 
 value is ; it may be called the value of a vernier division. A 
 reading then consists of the sum of two parts. The first part is 
 the exact reading of the main scale up to the line which imme- 
 diately precedes the index or line of the vernier. The second 
 part is the vernier reading for the fraction of a division. Look 
 along the vernier until a line is discovered which exactly coin- 
 cides with some line (no matter which) on the main scale. The 
 vernier reading is then the product of the number, on the vernier, 
 of this coinciding line, and the value of one vernier division. 
 
 Fig. 2 represents a circular vernier in which the smallest division of the 
 circle is ^°, or 10' ; the vernier subdivides this into sixty parts, therefore 
 indicating 10" The first part of the reading (at a) is 158° 40', the vernier 
 
 a 
 
 b 
 Fig. 2. Vernier for Divided Circle Eeading to 10" 
 
 reading (at 6) is 7' 10", and the whole reading is 158° 47' 10"- Other forms 
 of verniers are sometimes used, but it is not necessary to describe them, as 
 their indications may be analyzed in a manner similar to that given above. 
 
 Reference. —SteZey, Gillespie's Surveying, Vol. I, pp. 201-213. 
 
 18. The Vernier Caliper. — For quickly measuring various 
 small lengths with moderate precision, the vernier caliper is 
 useful. As compared with the screw caliper described in the 
 next article, it is not so precise, but it has the advantage of 
 greater range and of facility in setting for measures of different 
 lengths. The vernier caliper (Fig. 3) consists of a straight scale 
 having two jaws, whose plane and parallel surfaces are at right 
 angles to the length of the scale. One jaw is fixed, and the 
 other, movable along the length of the scale, carries an index 
 and vernier, which indicates the distance between the jaws. 
 The object whose length is to be measured is placed between 
 the jaws, which are adjusted by the slow-motion screw until it 
 
18 
 
 MECHANICS 
 
 is very lightly held between them. The movable jaw is then 
 clamped in position, and the reading of the vernier taken. The 
 scale is sometimes graduated on one side in inches and on the 
 
 other in centimeters, 
 setting of the caliper. 
 
 l<M.l.li|jl,lil.l.LliljliLl.lil 
 
 UjImiI.IiLi.I 
 
 Fig. 3. Vernier Caliper 
 
 In this problem read both sides at each 
 Each reading must be corrected for the 
 index error by adding, 
 with the sign changed, 
 the reading of the ver- 
 nier when the jaws are 
 closed. 
 
 19. The Micrometer 
 Caliper. — One of the 
 
 \J 
 
 iiJiU.mJ 
 
 UT 
 
 most useful devices for the measurement of ordinary small lengths 
 is the micrometer screw, which depends upon the principle that 
 if an accurate screw is turned in a fixed nut, the end of the screw 
 will move through spaces proportional to the rotation. The 
 distance in terms of the pitch of the screw is thus obtained, which 
 may be expressed in any desired unit if the pitch is known. 
 
 The micrometer caliper consists of a screw moving in a fixed 
 nut toward or from a stop carried by the frame that holds the nut. 
 The distance between the end of the screw and the stop can be 
 determined in terms of the pitch of the screw, there being a 
 longitudinal scale (Fig. 4) to indicate whole turns of the screw, 
 and divisions upon the head to indicate fractions of a turn. The 
 screws of such calipers 
 often have forty 
 threads per inch, or 
 twenty per centimeter. 
 The heads of the for- 
 mer being divided into 
 twenty-five parts, and 
 those of the latter into fifty parts, the instruments read, by 
 estimating tenths of the divisions, to a ten-thousandth of an 
 inch, or a ten-thousandth of a centimeter. 
 
 Place the object to be measured between the screw and the 
 stop, and turn the screw upon it to hold it lightly. The read- 
 ing is then taken in a manner which will be obvious from an 
 
 4. Micrometer 
 Caliper 
 
THE SPHEROMETEE 
 
 19 
 
 examination of the caliper. The error of the zero, with its sign 
 changed, must be applied as a correction to all readings, and it 
 should be determined both before and after making a series of 
 measurements by taking the reading when the caliper is closed. 
 Often the zero error can be eliminated by learning, from several 
 trials, just how much pressure milst be put upon the screw head 
 to bring the index exactly to zero. If the screw is provided 
 with a ratchet head for securing uniform pressure, it should be 
 turned gently until the ratchet slips one notch. 
 
 n. LENGTH AND RADroS OF CURVATURE WITH THE 
 SPHEROMETER 
 
 Measure the thickness of a metal disk, and the depth of a hole in it. 
 Determine the radius of curvature of one surface of a lens. Calcu- 
 late the probable error of a single observation and of the mean for 
 each set of measurements. 
 
 20. The Spherometer. — For certain kinds of measurements 
 the micrometer screw is conveniently made in the form of a 
 spherometer, which consists of a tripod (Fig. 6) carrying an 
 accurate screw, the axis of which passes through the center of 
 the circle in which lie the three feet. The head 
 of the screw is a large divided disk, and attached 
 to the tripod is an upright scale which serves 
 as an index to indicate the number of whole 
 turns given to the screw. 
 
 The screw may have a pitch of one twentieth 
 of a centimeter, and the disk be divided into 
 five hundred parts. Then one whole turn of 
 the screw represents a longitudinal motion of 
 the point of five hundredths of a centimeter, 
 and a turn through one division of the large 
 disk represents a motion of the point through 
 one ten-thousandth of a centimeter. By estimating tenths of a 
 division the reading can be taken to a hundred-thousandth of 
 a centimeter. 
 
 Fig. 5 
 Spherometer 
 
20 MECHANICS 
 
 An essential part of the instrument is a plane surface for it 
 to stand upon ; usually a piece of plate glass may be employed. 
 A series of readings should first be taken to determine the zero 
 point, by placing the spherometer upon the plane and so adjust- 
 ing the screw that its point shall be in the plane of the three 
 fixed points. This can be done with great accuracy by attempt- 
 ing to rotate the instrument about the central point ; if it rests 
 more heavily upon this point than upon the others, it will easily 
 rotate about it, and the screw should be so turned that this 
 tendency just disappears. 
 
 The object whose thickness is required is placed under the 
 central point, and the screw again adjusted as described. A 
 series of readings equal in number to those made for the zero 
 point should be taken, and the difference between the means 
 of the two series wiU be the thickness. 
 
 The spherometer is especially useful in determining the radius 
 of curvature of spherical surfaces. Make a series of settings 
 with the instrument standing upon the spherical surface. Let 
 d be the difference between this reading and the zero, determined 
 as before, I the mean of the lengths of the sides of the triangle 
 formed by the points of the three legs, and R the radius of 
 curvature of the surface; then 
 
 ^ V^ d 
 ^ = 6^+2' 
 
 The length I may be determined by holding the edge of a 
 steel scale to the different points, or by measuring between the 
 impressions of the points upon a piece of smooth paper. 
 
 III. CONSTANT OF A MICROMETER MICROSCOPE 
 
 Calibrate a microscope in both inches and centimeters, and find the 
 ratio of the inch to the centimeter. Calculate the probable error 
 of a single observation and of the mean for each set of measure- 
 ments. 
 
 21. The Micrometer Microscope. — A most useful instrument 
 for measuring very small lengths with precision is the micrometer 
 
THE MICROMETER MICROSCOPE 
 
 21 
 
 microscope, which is a combination of the micrometer screw 
 
 and the ordinary microscope. The objective of the microscope 
 
 produces an enlarged image of the object, which is adjusted, by 
 
 focusing, to lie in the plane MN (Fig. 6) of 
 
 a system of fixed and movable spider lines. 
 
 These spider lines and the image are observed 
 
 mth a magnifying eyepiece. The position 
 
 of the movable cross wires is controlled by a 
 
 fine screw, M. The measurement of a length 
 
 consists in determining exactly the number 
 
 of turns of the screw required to move the 
 
 cross wires over the magnified image. This 
 
 value win vary with the focal length of the 
 
 objective and also with the length of the 
 
 microscope body. 
 
 The head of the screw is usually a disk 
 divided into one hundred parts to facilitate 
 the reading of fractions of a turn. Estimate 
 tenths of these divisions. To facilitate the 
 counting of whole turns there is in the field 
 of view a saw-toothed iudex (Fig. 7), the distance from one tooth 
 to another corresponding to the motion of the wire in one com- 
 plete turn of the screw. Every fifth notch is cut deeper than 
 
 the others to aid 
 in counting. 
 
 The deep notch 
 farthest from the 
 screw head (a, 
 Fig. 7) is taken 
 as the zero posi- 
 tion for the cross 
 wires. When the 
 index has been 
 set upon any mark in the field of view, the position is to be 
 referred to this zero. The reading consists of the number 
 of notches between the initial deep notch and the position of the 
 index (which equals the number of whole turns of the screw) 
 
 Fig. 6. Miceometee 
 
 MiCEOSCOPE 
 
 Fig. 7. 
 
 Field of View op Miceometee 
 
 MiCEOSCOPE 
 
22 MECHANICS 
 
 plus the reading on the disk for the fractional turn. If the 
 appearance of the field of view and of the head of the screw is 
 as shown in the figure, the reading is 28.397. 
 
 In measuring a length, set the movable wire first on one end 
 and then on the other, reading each position as described; the 
 difference between the two readings is the required length. Do 
 not try to set one end of the space at the zero position of the 
 cross wires ; the space to be measured should as nearly as pos- 
 sible be central in the field of view. The fixed cross wires will 
 assist in adjusting the object to be measured so that it is parallel 
 to the line of motion. 
 
 The disk on the screw head is often held in position by a 
 friction washer, to permit setting it to position. When the 
 index wire is exactly over the central fixed wire the disk should 
 read zero. 
 
 When a micrometer screw is used for measurement, the final 
 motions of the screw in making settings should always be in the 
 same direction, to eliminate errors due to lost motion; usually 
 that direction is best which gives increasing numbers on the 
 divided head of the screw'. 
 
 Where the readings to be made are the distances between 
 lines of a divided scale instead of a single wire or cross wires, 
 it is desirable to have two parallel wires for the movable index, 
 the distance between them being a little more than the width 
 of the line. To set upon a line, adjust the parallel wires so 
 that the line bisects the space between them. 
 
 Whenever using a microscope or telescope with cross wires in 
 the field of view, always first focus the eyepiece, by a sliding 
 motion, upon the cross wires ; then focus upon the object by the 
 rack and pinion, or by a second draw tube. The perfection of 
 focus is indicated by the absence of parallax ; that is, when the 
 eye is moved sideways there should be no apparent motion 
 between the cross wires and the image. If there is such motion 
 the adjustments must be repeated. To obtain a measurement 
 in any given system of units the microscope must be calibrated 
 by determining how many turns of the screw would be necessary 
 to move the cross wires over the enlarged image of a standard 
 
THE COMPARATOR 
 
 23 
 
 unit of this system. This number is a constant for a given 
 microscope under fixed conditions. For metric measures a stand- 
 ard length of one tenth of a centimeter will be convenient, since 
 this is about the length easily seen in the field of view. Any 
 measurement made in terms of the pitch of the screw will be 
 reduced to centimeters by dividing the number of turns by the 
 calibration constant. 
 
 Thus, if it is found that to move the cross wires over one tenth of a 
 centimeter requires 16.43 turns of the screw, the calibration constant is 
 164.3 turns = 1 cm; and if to move the wires over the bore of a small 
 tube seen endwise requires 12.46 turns, the diameter of the bore is 12.46 
 -=- 164.3 = 0.0758 cm. 
 
 IV. LENGTH WITH THE COMPARATOR 
 
 Compare a steel meter with a standard meter at each decimeter of its length, 
 and plot the results. Determine the correction to the steel meter at 0°. 
 
 22. The Comparator An apparatus for facilitating the com- 
 parison of two nearly equal lengths, and for determining the 
 difference between them, is a comparator. It is essentially a 
 rigid bar (Fig. 8) to which two micrometer microscopes can be 
 
 Fig. 8. Comparator 
 
 firmly attached at any desired distance apart. This bar is fas- 
 tened to a base on which rolls a carriage. The carriage is pro- 
 vided with two adjustable tables on which the scales to be 
 compared are placed, and its movement permits either scale to 
 be quickly brought into view under the microscopes. 
 
24 MECHANICS 
 
 / 
 The microscopes are set at a distance apart approximately 
 equal to the lengths to be compared, and are adjusted so that 
 both shall focus upon a line parallel to the supported bar. The 
 scales are placed upon their supports, and are so adjusted that 
 when brought under the microscopes the end graduations will be 
 visible near the center of the field of view, and in perfect focus. 
 The microscopes are first set upon the standard scale, then 
 upon the other, and the difference between their lengths read in 
 turns of the micrometer screws ; make a second determination of 
 this difference by resetting upon the standard scale. Often one 
 microscope serves as an index to which one end of each scale 
 can be set by means of suitable screws, and the difference is 
 measured with the other microscope. The value of the microm- 
 eter screw may be determined from a known distance graduated 
 on the standard bar. 
 
 If the two bars are of different materials, their coefficients of 
 expansion will be different, and it will be necessary to reduce 
 the observations to 0° in finding the correction to the unknown 
 bar. If t is the temperature of the two bars when compared, 
 d the difference in their lengths, positive when the standard is 
 the longer, Sj the correction to the standard at 0°, ej its coefficient 
 of expansion, and Sj and e^ the corresponding constants for the 
 other bar, then the correction for the unknown bar at 0° is 
 
 i^ = d + h^ + t{e^-e^). 
 
 The constants of the standard bar are supposed to be known, and 
 the coefficient of expansion of the material of the unknown bar 
 may be taken from tables, or it may be experimentally determined. 
 
 The relation between these various quantities is apparent 
 from the following diagram (Fig. 9), in which M represents the 
 true length of the meter at 0°, S the actual length of the standard 
 at t°, and L the length of the bar to be compared. The signs in 
 the diagram indicate the algebraic signs of the quantities for the 
 case supposed. 
 
 In work of precision great care is necessary to ascertain exactly 
 the temperatures of the bars. The comparator should be mounted 
 in a constant-temperature room, and the bars adjusted the day 
 
THE STRAIGHT-LINE DIVIDING ENGINE 25 
 
 before the comparisons are made. Several thermometers whose 
 bulbs lie upon the bars at various points may be used to show 
 their temperatures. 
 
 In the comparison of long bare flexure may introduce serious 
 error. For some work it is sufficient to assume that the surfaces 
 of the supports are planei If the bars are of the H form, the 
 
 r 
 
 s 
 
 
 'te,' ; 
 
 • 
 
 -6, r 
 
 1 
 
 1 
 
 
 1 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 .& 
 
 
 
 
 ' 
 
 
 
 . .j; 
 
 
 
 ♦ 
 
 1 
 
 
 ; _.. 
 
 *yj. 
 
 -.J 
 
 Fig. 9. Comparison of Lengths 
 
 flexure effects are negligible. In other cases the bars should 
 be supported upon two rollers placed at the neutral points, 
 which for meter bars are 54 cm apart. 
 
 V. LENGTH WITH THE DIVIDING ENGINE 
 
 (a) Divide 10 cm in half-millimeter spaces, and an inch in twentieths and 
 
 thirty-seconds. 
 
 (b) Make a 30-cm brass scale divided to single millimeters. 
 
 (c) Determine the pitch of the screw, and measure an unknown length. 
 
 23. The Dividing Engine. — A machine for ruling lines which 
 shall divide a given length into any required number of sub- 
 divisions is a dividing engine. It usually consists of a screw 
 of known pitch, S (Fig. 10), which moves a table, T, designed to 
 carry the object to be graduated. A steel or diamond graver is 
 supported by an adjustable mechanism, G, which permits the 
 cutting of lines at right angles to the axis of the screw. An, 
 arrangement of disks and screws regulates the length of the 
 stroke so that long and short marks may be made in such groups 
 as are desired. 
 
 Suitable stops and ratchets on the divided head of the screw 
 permit the latter to be turned repeatedly through any deter- 
 mined angle. 
 
26 
 
 MECHANICS 
 
 Having determined the style of graduation, the stops on the 
 divided ratchet head are set to allow the screw to move the 
 table forward the length of the smallest division by each motion 
 of the crank. The disks and screws in the tool holder must 
 
 Fig. 10. Dividing Engine for Straight Scales 
 
 be properly set, and the blank scale secured to the bed with its 
 edge parallel to the screw. A mark is made with the graver, 
 and the scale is moved forward one division by a motion of the 
 crank ; this operation is repeated to the end of the scale. 
 
 Care must be taken to avoid all lost motion in the ratchet, 
 and to turn it against the stops gently. An index is provided 
 by which any error in turning the screw may 
 be detected. The steel graver should be 
 ground and set as shown in Fig. 11, the 
 front surface of the cutting point being per- 
 pendicular to the scale. Often the piece to 
 be graduated is coated with wax, in which 
 the graduations are made, and afterwards 
 etched with acid. 
 
 24. Dividing-Engine Comparator The 
 
 dividing engine is often provided with micro- 
 scopes, Jlfj and ilfj (Fig. 10), so that it may 
 be used as a comparator, or for measuring 
 lengths if the exact pitch of ,the screw is 
 known. The pitch is determined by placing 
 a standard length upon the table and observing the number 
 
 Fig. 11. Gkaver 
 
 FOR Dividing 
 
 Engine 
 
THE CIRCLE DIVIDING ENGINE 
 
 27 
 
 of turns of the screw necessary to move the table this length. 
 Such measurements should be made for different parts of the 
 screw, applying corrections for the temperature of the scale and 
 the screw. The dividing engine is adapted to the calibration of 
 thermometer and other tubes, and to the measurement of all small 
 lengths which are too large for the micrometer microscope. 
 
 VI. CIRCULAR ARC WITH THE DIVIDING ENGINE 
 
 Graduate an arc of a circle in half degrees, and make a vernier reading to 
 
 single minutes. 
 
 25. Dividing Circles. — In experimental work divided circles, 
 arcs, or verniers are often required. The circle dividing engine 
 is a machine for facilitating their production. It consists of 
 a master circle, C (Fig. 12), arranged to be rotated by a worm 
 gear, G. The worm is provided with a ratchet head having 
 adjustable stops, to permit 
 the turning of the circle 
 any required amount. An 
 adjustable tool carriage, T, 
 holds the graver, and a 
 microscope, M, attached to 
 the bedplate serves as an 
 index to the master circle. 
 
 The blank circle is cen- 
 tered and clamped upon the 
 master circle, the ratchet 
 is set to give the required 
 divisions, and the disks 
 and screws regulating the motion of the graver are adjusted. 
 The graver is set as explained in Art. 23. Make a mark with 
 the graver, and then turn the worm ; repeat the operation until 
 the circle is graduated, watching the master circle to avoid 
 mistakes. 
 
 For the vernier the ratchet stops are adjusted to make the 
 divisions of the proper size, according to the principles of Art. 17. 
 
 Fig. 12. Dividing Engine for Circles 
 
28 MECHANICS 
 
 VII. ERRORS OF A GRADUATED SCALE BY GAY-LUSSAC'S 
 
 METHOD 
 
 (a) Determine the corrections to the ten-degree divisions of a thermometer 
 scale from 0° to 100°. 
 
 (b) Determine the corrections to the decimeter divisions of a meter scale. 
 
 26. Calibration of Graduated Scales. — Before any graduated 
 scale, such as a scale of lengths, a thermometer scale, or a 
 divided circle, can be used as an instrument of precision, the 
 errors of its subdivisions must be investigated and corresppnd- 
 ing corrections must be applied to the observations made with 
 the scale. The fundamental error in the total length of the 
 scale, or of its principal length, must be determined by com- 
 parison with known standards ; as by the method of Art. 22 for 
 scales of length. The following method, due to Gay-Lussac, is 
 the simplest which can be employed to determine the errors of 
 the subdivisions of a scale, considered as equal parts of a known 
 length. (See Art. 30.) Though the method is described as applied 
 to the calibration of a scale in ten subdivisions, it may be readily 
 adapted to any number of parts ; however, when this number is 
 large, more than ten for instance, the cumulative errors in the 
 results may be serious. The probable errors in the positions of the 
 lines increase from the ends toward the center of the scale. 
 
 27. Calibration with a Dividing Engine and One Microscope 
 
 Let it be required to find the corrections to the scale of a 
 thermometer reading from 0° to 100°, at each of the ten-degree 
 points. (For the calibration of a thermometer, see Art. 129.) 
 A dividing engine used as a comparator is a convenient form 
 of measuring apparatus (Art. 24). The thermometer may be 
 conveniently supported, parallel to the line of motions in grooves 
 cut in two large, flat corks placed upon the dividing-engine 
 table. By means of the engine screw bring the zero -line of 
 the scale under the stationary index microscope. The stops 
 being removed, the divided screw head, to which the crank is 
 attached, may be turned backward to the zero of its index 
 without turning the screw. Now turn the screw forward till 
 the 10° mark is under the microscope, noting the number of 
 
CALIBEATION OF GRADUATED SCALES 
 
 29 
 
 whole turns given to the screw. The direct reading of the 
 screw head will give the fraction of a turn in excess of this 
 number as it started from zero. With a screw of the usual 
 pitch, the threads being 1 mm apart, this reading is the length 
 of the space in millimeters. Without disturbing the thermom- 
 eter, turn the screw head backward again to zero, and then 
 forward till the 20° mark is under the microscope, noting the 
 full turns and fractions of a turn, as before. Repeat this process 
 for each of the ten spaces on the scale, from 0° to 100°. If the 
 spaces are all of equal length within the limits of the errors of 
 observation, the graduation errors are negligible. Otherwise 
 the correction to each space is the quantity which must be alge- 
 braically added to it to make it equal to the average length of 
 all the spaces. The corrections usually required are those to 
 the spaces included between each of the several lines and the 
 zero line of the scale. These are equal to the sums, taken 
 successively, of the corrections to each of the ten-degree spaces. 
 Thus the correction to the space between 0° and 30° is equal to 
 the sum of the corrections to the spaces between 0° and 10°, 
 between 10° and 20°, and between 20° and 80°. 
 The following is an example of this method. 
 
 Calibration op the Scale of Thermometer, Gerhardt 4248, 
 AT THE Ten-Degree Points from 100° to 200° 
 
 May 7, 1898 
 
 Spaces 
 
 Obsekved Lengths 
 
 COBEECTIONS 
 
 Sums of Cobkectioks 
 
 100°-110° 
 
 25.600 mm 
 
 -0.116 mm 
 
 -0.116 mm 
 
 110 -120 
 
 .615 
 
 - 0.131 
 
 - 0.247 
 
 120-130 • 
 
 .420 
 
 + 0.064 
 
 -0.183 
 
 130 -140 
 
 .475 
 
 + 0.009 
 
 - 0.174 
 
 140 -150 
 
 .405 
 
 + 0.079 
 
 - 0.095 
 
 150 -160 
 
 .395 
 
 + 0.089 
 
 -0.006 
 
 160 -170 
 
 .470 
 
 + 0.014 
 
 + 0.008 
 
 170 -180 
 
 .480 
 
 + 0.004 
 
 + 0.012 
 
 180 -190 
 
 .505 
 
 -0.021 
 
 - 0.009 
 
 190 -200 
 
 .475 
 
 + 0.009 
 
 0.000 
 
 Average 25.484 
 
 
 
30 MECHANICS 
 
 28. Calibration with a Dividing Engine and Two Microscopes. 
 
 — The method just described is useful only for short scales 
 and those in which great precision in the measurement of length 
 is not required. The following method, when it can be applied, 
 is more expeditious, and is more accurate, as it does not involve 
 the cumulative errors of many turns of the dividing engine 
 screw. 
 
 Two fixed index microscopes are placed a distance apart 
 slightly less than the length to be calibrated. Designate the 
 marks defining these various spaces by (0), (1), (2), etc. Adjust 
 the scale on the dividing-engine table, parallel to the line of 
 motion, so that mark (1) is exactly under the microscope farthest 
 from the screw head, and mark (0) is nearly under the other 
 microscope. Turn the screw head backward to the zero of its 
 index, and then forward till the mark (0) is under the corre- 
 sponding microscope. The fraction of a turn given to the screw 
 measures the excess of the length of the space on the scale over 
 the distance between the microscopes. Now move the scale on 
 the table about the length of one subdivision till line (3) is 
 exactly under one microscope and line (2) is nearly under the 
 other. Turn the screw head backward to zero reading, and 
 then forward till line (2) is in position ; the amount the screw 
 is turned measures the excess of the second space over the 
 fixed length between the microscopes. Measure each subdivi- 
 sion of the scale in this manner. Each space is thus found to 
 consist of an unknown constant length plus a known length 
 which diifers for the several spaces. The corrections to the 
 scale parts are such quantities as, when algebraically added to 
 the observed excess lengths, will make them, and therefore the 
 separate spaces on the scale, all equal. The solution will be 
 the same as that given above in the numerical example, except 
 that instead of the total lengths of the subdivisions, the second 
 column will contain the small excess lengths. Average these 
 excess lengths, and the difference between the average and each 
 observed length is the correction to the space. The corrections 
 are to be summed successively as before. 
 
CALIBEATION OF GRADUATED SCALES 
 
 31 
 
 29. Calibration with a Micrometer-Microscope Comparator. — 
 When the highest precision of calibration is required a microm- 
 eter microscope is substituted for one of the index microscopes 
 mentioned in the previous article, and the small excess lengths 
 are measured with the micrometer screw instead of with the 
 dividing-engine screw. The observations are thus given in 
 terms of turns of the micrometer screw, which must be cali- 
 brated (Art. 21) and the readings, or the results obtained from 
 them, reduced to terms of microns or centimeters. Instead of 
 a dividing engine any other convenient form of comparator 
 (Art. 22) for conveniently carrying the scale and microscopes 
 may be used. 
 
 The following is an example of this method. 
 
 Calibration op SociiT:6 Genevoise Nickel Meter No. 11 
 
 fok the decimeter spaces 
 
 With Comparator and Micrometer Microscope No. 3 
 
 May 13, 1898 
 
 SPACES 
 
 em 
 
 Obseetations 
 Turns of Screw 
 
 Cokbections 
 Turns of Screw 
 
 Coebections u 
 20.791 turns = 1 mm 
 
 S Coekections 
 
 0-10 
 
 23.256 
 
 + 0.091 
 
 + 4.4 
 
 + 4.4 
 
 10-20 
 
 .391 
 
 - 0.044 
 
 -2.1 
 
 + 2.3 
 
 20-30 
 
 .347 
 
 0.000 
 
 0.0 
 
 + 2.3 
 
 30-40 
 
 .348 
 
 + 0.001 
 
 0.0 
 
 + 2.3 
 
 40-50 
 
 .291 
 
 + 0.056 
 
 + 2.6 
 
 + 4.9 
 
 50-60 
 
 .412 
 
 - 0.065 
 
 -3.2 
 
 + 1.7 
 
 60-70 
 
 .465 
 
 -0.118 
 
 -5.7 
 
 -4.0 
 
 70-80 
 
 .145 
 
 + 0.202 
 
 + 9.7 
 
 + 5.7 
 
 80-90 
 
 .416 
 
 - 0.069 
 
 - 3.3 
 
 + 2.4 
 
 90-100 
 
 .396 
 
 - 0.049 
 
 -2.4 
 
 0.0 
 
 Average 23.347 
 
 
 
 
32 MECHANICS 
 
 VIII. ERRORS OF A GRADUATED SCALE BY PRECISION 
 METHODS 
 
 Determine the corrections to the decimeter divisions of a meter scale. 
 
 30. Calibration of Scales by Precision Method. — A complete 
 method for the calibration of a divided scale which determines the 
 positions of all the lines of the scale with equal precision, while 
 requiring a much larger number of observations than Gay-Lus- 
 sac's method (Art. 27), is, however, usually to be preferred. The 
 method, including a least^squares solution, is due to Hansen; 
 while a much simpler solution of the same observations, which 
 can in every case be substituted for the least-squares solution 
 without lessening the precision of the result, has been devised 
 by Neumann and Thiesen. This latter method is so useful and 
 simple that, while no attempt will be made to explain the theory, 
 a numerical example is given of the calibration of the Societe 
 Genevoise Nickel Meter No. 11, by means of which one can 
 adapt the method to any problem in calibration that is likely to 
 occur. 
 
 To determine the corrections to the decimeter subdivisions of 
 a meter scale, the following observations are required. With 
 the microscopes of the comparator set approximately 1 dm apart, 
 each of the ten decimeter spaces is measured as described in 
 Art. 29, the example there given being a part of the present 
 example. With the microscopes approximately 2 dm apart, the 
 spaces from (0) to (2), from (1) to (3), and so on to (8) to (10) 
 are measured. With the microscopes approximately 3 dm apart, 
 the spaces (0) to (3) . . . (7) to (10) are measured. The process 
 is continued until, 'with the microscopes 9 dm apart, the obser- 
 vations are completed by measuring the spaces (0) to (9) and 
 (1) to (10). 
 
 Table I contains these observations, which are fifty-four in 
 number for a ten-space calibration. The first column in each 
 group designates the space measured; the second column con- 
 tains the readings expressed in thousandths of a turn of the 
 micrometer-microscope screw, which represent the excesses in 
 
CALIBRATION OF GRADUATED SCALES 
 
 33 
 
 the lengths of the several spaces over the fixed distance between 
 the microscopes; the third column in each group contains the 
 numbers that must be added to each number of column 2 to 
 give the number next below it in the same column. 
 
 To obtain the corrections to the graduations from these obser- 
 vations, arrange a square form of ten rows and ten columns, 
 Table II; place a row of ciphers in the diagonal from the upper 
 left to the lower right comers ; write the numbers in the third 
 column of group 1 of Table I, in the diagonal next below the 
 cipher diagonal, and write the same numbers, with their signs 
 changed, in the diagonal just above it; write the numbers of 
 the third column of group 2 of Table I in the next lower 
 diagonal, and the same numbers, with their signs changed, in the 
 
 
 
 
 
 
 
 
 
 ; 
 
 
 
 - 
 
 
 
 
 
 
 > 
 
 
 
 ) 
 
 \ 
 
 
 
 
 y 
 
 "\ 
 
 
 
 j^ 
 
 \ 
 
 
 / 
 
 ■\ 
 
 
 
 
 y 
 
 ^\ 
 
 
 
 ^ 
 
 ^ 
 
 
 / 
 
 ^ 
 
 
 
 
 y 
 
 
 
 
 
 
 1 
 
 / 
 
 
 "^ 
 
 
 /J 
 
 y 1 
 
 2 
 
 S 
 
 ) A 
 
 5 
 
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 3 / 8 
 
 J £ 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 Length in Centimetcra 
 
 Fig. 13. Coerections to Geneva Nickel Meter No. 11 
 
 next upper diagonal ; and so continue until the square form is 
 fiUed. The algebraic sum of all the numbers in each column of 
 the square, divided by ten, is the correction to the correspond- 
 ing decimeter space, expressed in thousandths of a turn of the 
 micrometer screw. These corrections are reduced to microns 
 by dividing by 20.791, the number of turns of the micrometer 
 screw per millimeter. The successive sums of these numbers 
 are the corrections to the spaces 1, 2, 3, . . . decimeters %,s meas- 
 ured from the zero line ; these are given in the bottom row of 
 Table II. Fig. 13 is a graphic representation of these results. 
 
 The actual temperature of the scale during these observations 
 is of no importance, but it is necessary that, during any one set 
 of measurements, the temperature should not change. This is 
 particularly-important when the microscopes are far apart. The 
 comparator should therefore be in a constant-temperature room. 
 
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 O CO t^ CO lO It- (N 
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 o 
 to 
 
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 (M 
 IN 
 
 05 
 
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 + 
 
 s 
 
 C0rHC»Q0OThOOi-IC0 
 ,-HU3OlO5C0?O(M t-'^t* 
 r-((Ni-HrHTH(MCO (N<N 
 + + + + + + + + + 
 
 « 
 
 05 
 Oi 
 l-H 
 
 + 
 
 CO 
 
 d 
 
 + 
 
 CO 
 
 + 
 
 
 OS^iO^TiCOOOtMOS 
 Olr-i-HrHCOiO (NWSCO 
 
 IM I-H irH f-H CO 
 
 1 1 1 1 i 1 111 
 
 7 
 
 1-t 
 00 
 
 1 
 
 ■*■ 
 
 1 
 
 1 
 
 r-OS'^iOi-HOCo-^Tjir- 
 Tt< O CD ci »o <© (N 
 
 1 1 1 1 1 +1 + 1 
 
 o 
 
 1 
 
 CO 
 
 o 
 
 CO 
 
 1 
 
 + 
 
 1 
 
 1— 'COiOt-Oi— ii— lOCOO 
 COOCDO (NOOCOCOi-H 
 
 f— 1 i-H i-H »— 1 »-t T— 1 
 1 + + + + + 1 + + 
 
 to 
 
 + 
 
 + 
 
 + 
 
 1 
 
 1 + 1 1 + 7- 7 + + 
 
 d 
 
 1 
 
 d 
 
 1 
 
 l-H 
 
 + 
 
 S 
 
 COtJ^Oi-iOtHiCCOOO 
 
 1 + + 1 + + 7 + + 
 
 1 
 
 CO 
 
 l-H 
 
 d 
 
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 l-H 
 
 + 
 
 iH 
 
 lOO-^TPCOOSTpT-Ht-CO 
 CO '**<'* o *- S '-' 
 
 1 1 1 1 + + 1 + 1 
 
 d 
 
 1 
 
 CO 
 
 1 
 
 CD 
 
 l-H 
 
 + 
 
 o 
 
 oocoosi-it-oaco '■* i-" 
 
 COOOOOCO-^Oi-H^OCO 
 r-l r- 1 (N T-1 i-H 1— 1 
 
 + + + + + + 1 + + 
 
 I— 
 
 CO 
 
 + 
 
 05 
 
 + 
 
 03 
 l-H 
 
 + 
 
 
 ■s ie 
 
 Hi 
 
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 o 
 o 
 
 4. 
 
 o 
 
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 a. 
 
 § 
 
 1 
 
 1 
 
 35 
 
36 MECHANICS 
 
 To eliminate, as well as may be, the unavoidable small progres- 
 sive changes produced while observations are being made, the 
 measurements of the several parts of the scale with one spacing 
 of the microscopes should be made successively from one end to 
 the other of the scale, and repeated in reverse order to the start- 
 ing point. The average of these two sets will be free from 
 progressive errors. 
 
 The correction to the total length of the scale must be deter- 
 mined by an independent comparison with a standard, as 
 explained in Art. 22. See also Art. 129 for thermometer cali- 
 bration. An error in the total length of a scale affects sub- 
 divisions in proportion to their lengths. 
 
 Kefekence. — For a full explanation of this method of calibration, also for 
 the calculation of the probable errors, and for references to the original memoirs, 
 see Guillaume, Thermom^trie, pp. 46-73. 
 
 IX. LENGTH WITH THE OPTICAL MICROMETER 
 
 Measure the thickness of two pieces of thin glass. 
 
 31. The Optical Micrometer. — A ray of light may be made 
 to serve as a very long pointer without mass, for indicating 
 small angular displacements. A mirror is attached to the body 
 whose motion is to be measured, by which light is reflected from 
 a fixed source to a suitable scale. The optical micrometer 
 employs this device for measuring small lengths. It consists 
 of a rod supported by four short legs, one at each end, and two, 
 close together, so placed that the line joining them bisects at 
 right angles the line joining the end legs. The instrument must 
 stand upon a plane surface, such as a piece of thick plate glass. 
 One of the end legs is adjustable in length, so that by the sense 
 of touch the four supporting points may be brought into the same 
 plane. This adjustment should be made with great care. 
 
 A reading telescope with a vertical scale is used for deter- 
 mining the deflection of the ray of light. Place the telescope at 
 a distance from the micrometer of several meters, more or less, 
 according to the degree of accuracy required, with its line of 
 
THE OPTICAL MICROMETER 
 
 37 
 
 sight in the direction of the length of the micrometer. Adjust 
 the telescope, scale, and mirror, any one or all, so that the image 
 of the scale, reflected by the mirror, is seen in the telescope 
 (Art. 33). The scale should be made perpendicular to the line 
 joining the mirror and that part of the scale seen in the tele- 
 scope. Place the object to be measured under the two central 
 legs of the micrometer; the instrument not being balanced 
 about the liue of the central legs, one end will be raised above 
 the plane and the micrometer will stand upon three legs only. 
 In this position read in the telescope the division of the scale 
 which coincides with the cross wires. Next place a small weight 
 upon the raised end of the micrometer, causing it to rest upon 
 the plane, and again read the scale. 
 
 Let d (Fig. 14) represent the thickness of the object under 
 the central legs, I the half length of the base, L the distance of 
 the scale from the mirror, and n the difference between the scale 
 readings. MC and MD represent the normal to the mirror in 
 
 Fig. 14. Principle of Optical Miorometeii 
 
 its two positions, and A and B the two divisions of the scale 
 seen in the telescope. In the similar triangles, d:m=CE:L. 
 But in any case in which this method is useful, m may be 
 taken as equal to I without sensible error, and CE as equal to 
 AB 
 
 = - : therefore 
 4 
 
 d = 
 
 nl 
 4l' 
 
38 
 
 MECHANICS 
 
 The length of the micrometer may be determined by measuring 
 between impressions of its four points on a firm piece of paper. 
 
 Ebperbncb. — When the thickness of the object is considerable as compared 
 with the half length of the base, the complete trigonometric formula should be 
 applied. See Stewart and Gee, Practical Physics, Vol. I, p. 54. 
 
 32. Three-Legged Optical Micrometer. — A modification of 
 this method may be used in case the object to be measured is 
 too small to permit both central legs to rest upon it. The 
 instrument is allowed to stand upon its three fixed legs only, 
 and the telescope reading taken while it rests upon the plane. 
 The object is then placed under the end leg, and the reading 
 again taken. Using the same notation as before, 
 
 nl 
 
 d = 
 
 2L 
 
 33. The Reading Telescope. — In experimental work the read- 
 ing telescope is extensively used to give definite direction to a 
 line of sight, or for magnifying a divided scale or a small dis- 
 placement. A small mirror may be attached to 
 the body to be observed, by which the image of 
 a stationary scale is reflected to the telescope. 
 Any rotation of the mirror results in an apparent 
 displacement of the scale. 
 
 Usually the telescope and scale are separately 
 
 attached to the same support so that they may 
 
 be set at various heights and adjusted as to 
 
 direction. A convenient support is a tall, upright 
 
 rod on a tripod base, as shown in Fig. 15. The 
 
 telescope is often mounted on a short stand, 
 
 which is then supported on a small tripod table 
 
 of variable height. If possible, the mirror should 
 
 be so turned that the scale when in position shall 
 
 be well illuminated. To find quickly the posi- 
 
 FiG. 15. Reading ^^^ foj, ^t^q telescope and Scale, the observer, 
 
 holding a piece of white paper in the hand, 
 
 moves about in front of the mirror till the reflection is seen. 
 
 Then, following the reflection with the eye, the motion is 
 
THE LEVEL TRIER 39 
 
 continued till the paper and eye are in positions convenient for 
 the scale and telescope respectively. 
 
 After locating the telescope and scale, first focus the sliding 
 eyepiece so that the cross wires are distinct and then alter the 
 focus of the telescope by the rack and pinion or the large draw 
 tube till the scale is clearly seen. In searching for the scale, 
 remember that it is optically twice as far away as the mirror, 
 and the telescope must be focused accordingly. 
 
 Telescopes usually will focus only upon objects distant a 
 meter or more. A subsidiary convex lens may be placed over 
 the object glass to facilitate the focusing on nearer objects and 
 to increase the magnifying power; the telescope then becomes 
 in effect a low-power microscope. 
 
 X. CONSTANTS OF A LEVEL WITH THE LEVEL TRIER 
 
 (a) Determine the value of one division of a level, in seconds of arc, for 
 various parts of the scale. Find the radius of curvature of the level. 
 
 (b) Calculate the value of one turn of the screw of the level trier. 
 
 34. The Spirit LeveL — A very sensitive instrument for meas- 
 uring minute angles, particularly the deviation of lines from the 
 true horizontal, is the spirit level. Its essential part is a closed 
 glass tube nearly filled with a mixture of ether and alcohol. 
 
 This tube has its inner sur- 
 
 face, on the upper side, f^^^^s^-z :^^^=^'- 
 
 accurately ground so that a v ^~" 
 
 longitudinal section (Fig. ^^^ ^g Chambeked Level Vial 
 
 16) would show an arc of a 
 
 circle of large radius. The longer this radius the more sensitive 
 the level. If the tube is inclined through a small angle, the bubble 
 will rise to the highest part of the curve in the vial, and by the 
 change in position will measure the angle. The tube is held in 
 a suitable frame, which carries a graduated scale for indicating 
 the position of the bubble. Changes in temperature will vary 
 the length of the bubble, and a fine level may have a chamber 
 at one end which permits the length of the bubble to be altered. 
 
40 MECHANICS 
 
 To eliminate errors due to improper adjustments, a level 
 should be read in direct and reversed positions. Half the 
 difference between the readings is the inclination of the line. 
 If the surface is altered till the readings in the two positions 
 are the same, the surface is level, and if the level is then made 
 to read zero, it is in adjustment. With sensitive levels this is 
 difficult to accomplish, and the two readings should always be 
 taken. The level should be at a uniform temperature, as the 
 bubble tends to move toward the warmest portion of the vial. 
 
 For any particular measurement, a level of suitable sensitive- 
 ness should be chosen. It is therefore necessary to determine 
 the inclination corresponding to a displacement of the bubble by 
 one division, and to determine that this relation is constant 
 
 throughout the scale. 
 This measurement is 
 made by means of the 
 level trier, which con- 
 sists of a horizontal 
 plate, arranged to 
 carry all forms of levels, pivoted at one end, and movable 
 through very small known arcs by means of a screw placed at 
 the other end. The level having been adjusted in position 
 (Fig. 17), the greater part of the weight is taken from the point 
 of the screw by the counterpoise, and a series of observations is 
 made to determine how much the screw must be turned to move 
 the bubble through a certain number of level divisions. Do 
 not attempt to set the bubble on a particular mark, but read 
 the positions of both its ends, and consider the mean the level 
 reading. Turn the screw to move the bubble approximately ten 
 scale divisions, and record the positions of both screw and bubble ; 
 move the bubble another ten divisions, and continue until the 
 limit of the scale is reached. Reverse the level in the Y's, and 
 repeat the observations. The final motion of the screw in mak- 
 ing settings should always be in a direction to lift the arm carrying 
 the level, as this will diminish the errors due to lost motion. 
 
 Compute the value of one division of the level, in seconds 
 of arc, from each observation. If there is no indication of 
 
 Fig. 17. Levkl Teiek 
 
THE LEVEL TRIER 41 
 
 irregularity, the mean of all may be taken as the value of one 
 division. Usually the head of the screw is so graduated that 
 it indicates seconds of arc directly ; if this is not so, the value of 
 the screw may be determined as described below. 
 
 If the scale is I centimeters long and has n divisions, and if a 
 is the value in seconds of one division, then the length of one 
 
 second of arc of the level vial is — , and the radius of the level is 
 
 „ 206265 Z 
 an 
 
 206265 being the number of seconds in a radian. 
 
 If s is the distance between the threads of the screw, and L 
 the length of the arm, the value of one turn of the screw, in 
 seconds of arc, is 
 
 ^_ 206265 s 
 
 The length of the arm may be obtained from direct measure- 
 ments of the triangle formed by the point of the screw, and the 
 two pivots, or by measuring the impressions of these points on 
 paper. 
 
 The nominal value of the pitch of the screw, known from 
 its construction, is usually sufBciently accurate for use in the 
 above formula. If not, it may be determined in the following 
 manner. Having noted the reading of the level, a small, thick 
 piece of glass, the thickness of which has been measured with 
 the micrometer caliper, is placed under the screw, and the level 
 reading is restored by turning the screw. The thickness divided 
 by the number of turns is the required distance between the 
 threads. The pitch may be determined by direct measurement, 
 as described in Art. 24, the screw and nut being taken from the 
 level trier and arranged to move either the microscope or the 
 standard scale of the dividing-engine comparator. 
 
 The angular motion produced by the screw may also be 
 measured by a ray of light reflected by a mirror attached to the 
 
42 MECHANICS 
 
 level tri«r. Calculate the angle from the measured deflection 
 caused, at a known distance, by a certain number of turns of 
 the screw. If the incident and reflected rays and the arm of 
 the level trier are in the same plane, the ray will be deflected 
 through twice the required angle. 
 
 XI. AREAS WITH THE PLANIMETER 
 
 Determine the area of the datum circle in square centimeters, and measure 
 two large and two small areas. Measure the small areas in square 
 inches. Measure the area, in square feet, of a figure drawn to scale. 
 
 35. The Planimeter. — A mechanical integrating machine 
 termed the planimeter may be used to measure directly the area 
 of a plane surface. It is employed in engineering work for 
 measuring the areas of cross sections, the contour lines of rivers 
 and ponds, drainage areas, displacement areas of vessels, indi- 
 cator diagrams, etc. The polar planimeter (Fig. 18) consists of 
 two arms hinged together, the outer end of one being pivoted at 
 P in the plane of the figure, the end of the other arm carrying 
 
 Fig. 18. Planimeter 
 
 a style, /S, is caused to trace the periphery. A graduated wheel 
 and disk, If, partly roUs and partly slides over the surface, and 
 by means of a counting mechanism, indicates the area, the read- 
 ings usually being given by means of the vernier to one tenth 
 of a square centimeter. The instrument may be set to read in 
 various systems of units and to different scales by varying the 
 length of the arm WS. 
 
THE PLANIMETER 
 
 43 
 
 When the angle between the two arms is such that the line 
 from the pole to the wheel contact, FW(Fig. 19), is perpendicu- 
 lar to the axis of the wheel, motion of the whole instrument 
 about the pole will produce no rotation of the wheel and hence 
 no record. The style will trace a 
 circle of radius PS, called the datum 
 circle, the area of which may be found 
 as explained later, and recorded as a 
 constant of the instrument. If the 
 distance from the pole P to the joint 
 J is a, and from the joint to the style 
 S is b, and from the joint to the wheel 
 
 contact is o, then the radius PS is 
 
 _^_^^ Fig. 19. Planimeter Tra- 
 
 '• 4- 2 he. ciNG Datum Circle 
 
 /•o=Va2 + , 
 
 When the style is outside the datum circle and is moved 
 around the pole in a clockwise direction, the wheel gives posi- 
 tive readings, while motion in the same direction inside the circle 
 
 gives reversed rotation to the wheel. 
 To explain the theory of the pla- 
 nimeter an element of area will be 
 assumed which is bounded by an 
 arc of the datum circle, S-^S^ = B 
 (Fig. 20), by the arc of a concentric 
 circle, N^N-^, of radius r, and by two 
 radial lines, N^S^ and N^S^. 
 
 As already noted, tracing S-^S^ 
 will produce no indication. Tracing 
 S^N^ will cause some positive rota- 
 tion, while tracing N-^S-^ will cause 
 an equal contrary rotation. There- 
 fore when the style traces the entire 
 periphery, S-^S^N^N^Sy, the final reading will be due to rotation 
 produced while tracing the arc NJ^x ^"^1- I* i^ to ^^ shown 
 that this is proportional to the inclosed area. 
 
 When the arc N^N-^^ = is traced, W revolves around P in the 
 direction WX through an equal arc of length P Wd. If a is the 
 
 ""Or 
 
 Fig. 20. Measlrement 
 OF Areas 
 
44 MECHANICS 
 
 angle between PW and the axle, the axle makes the angle 
 90° — a with the direction of motion. This motion may be 
 resolved into the component PWd cos (90° — a) in the direction 
 of the axle, producing only sliding, and 
 
 F we sin! (90° - a ) = P WO cos a, 
 
 perpendicular to the axle, which turns the wheel. 
 In the triangles FWN^ and FWJ, 
 
 r^ = (h + cf + FW^ - 2 {h + o)FWcQS a, 
 
 and «2 ^c2 +pw^ - 2 cFW eos a. 
 
 .-. FW cos a = ^ (a2 + 52 + 2 5c - r2) = ^ (r^^ - r^), 
 
 and the turning of the wheel equals — (r^^ — r^) 6, 
 
 A 
 
 Area S-^S^N^N-^ = sector FS-^S^ — sector FN^N^ 
 
 This is h times the distance traveled by the roller ; h is con- 
 stant, therefore the area is proportional to the travel of the roller. 
 
 If the element of area lies outside of the datum circle, as 
 B^E^S^S^, similar reasoning will show that when the style has 
 traced the lines -^2*^2' '^2*^1' ^^^ i^i^v *^® wheel will indicate no 
 change, but that when the line ^1^2 ^^ traced, the wheel will 
 travel a distance proportional to the inclosed area. 
 
 Let the element of area be S^N^N^B^, partly within and partly 
 without the datum circle. Tracings on the radial lines will 
 neutralize each other; the motion (counter-clockwise around 
 the pole) while tracing iVjiVj will indicate positively the area 
 within the circle, and the tracing of B^B^ will add to this the 
 area of the part without the circle ; the final reading wiU there- 
 fore indicate the total area. 
 
 If the element of area is B^B^O^O^, entirely without the 
 datum circle, tracing^jJJg will give a positive reading for the 
 area between this line and the datum circle, while the tracing 
 of 0^0^ will give a negative reading for the area O^O^S^S-i, and 
 
THE PLANIMETER 45 
 
 the final result is the area of B^E^O^O^. Similarly tracing the 
 element N^^^^Q^Qi will give the difference between (>2^i'S'i'^2 
 and N^X^S-^S^, or the area Q^Qi^i^^i- 
 
 Since any area which does not inclose the pole may be repre- 
 sented by the sum of elements such as have been considered, it 
 foUows that the tracing of the periphery by the style moving 
 around it in a clockwise direction will cause the vs^heel to indicate 
 directly the area inclosed. 
 
 If the area boundary incloses the entire datum circle and the 
 pole, it will be made up of elementary lines such as i^j-Bg ' ^^^ 
 tracing of this line entirely around the circumference will cause 
 the wheel to indicate, in the manner explained, only the area 
 between the boundary and the datum circle. In this case, there- 
 fore, the wheel indication must be added to the known area of 
 the datum circle. 
 
 When the boundary incloses the pole and lies within the 
 datum circle, the wheel gives a negative indication (for clock- 
 wise motion of the style) which is the area between the bound- 
 ary and the circle ; hence the desired area, being the difference 
 between that of the circle and the ring, is equal to the algebraic 
 sum of the circle area and the wheel indication. The same rule 
 naturally applies if the boundary inclosing the pole is partly 
 inside and partly outside the circle. 
 
 To find the Area of the Datum Circle. — In a strip of cardboard 
 pierce two needle holes at a measured distance apart. Fasten 
 the pole through one hole, and by resting the style in the other 
 a circle of knovm area may be readily traced. The wheel read- 
 ing is the area to be subtracted algebraically from this known 
 circle to give the area of the datura circle. It is best to trace 
 two known circles, one a little smaller than the datura circle, 
 and one larger. 
 
 General Rule for measuring Areas. — If possible place the pole 
 outside the area ; trace the periphery with a clockwise motion of 
 the style ; the wheel reading is the area. When the pole is inside 
 the area trace the boundary with a clockwise motion of the style ; 
 the wheel indication, attention being given to the sign, algebra- 
 ically added to the area of the datum circle is the required area. 
 
46 MECHANICS 
 
 If in the last case a trial shows a negative turning of the 
 wheel, it may be convenient to trace the boundary in a counter- 
 clockwise direction when the direct reading of the wheel is the 
 area to be subtracted from the datum circle. 
 
 The setting of the wheel is so easily accomplished that it is 
 usually best to start with the index at 0. Use a straightedge 
 to facilitate the tracing of straight lines. Large areas may be 
 subdivided and measured in parts. 
 
 The details of setting the instrument to various scales will 
 be obvious from the marks and figures engraved on the arm, or 
 from the maker's instructions. The setting for a particular 
 scale may be found or tested by trial measurements of a known 
 area, such as a rectangle, or a circular groove in a metal plate. 
 
 36. Mensuration of Areas. — For figures of regular shape 
 the following formulae may be used, the symbols requiring 
 no explanation. 
 
 Triangle = base x i altitude. 
 
 Circle = trr^. 
 
 Parabola = base x f height. 
 
 Ellipse = irah. 
 
 Cylinder = 2 -n-rZ + 2 nrr". 
 
 Cone =-7rr^ + 2 Trr x -J- slant height. 
 
 Sphere = 4 Trr\ 
 
 For irregular plane figures the curved boundary may be drawn 
 upon cross-section paper, and the number of squares inclosed 
 counted, estimating fractions of squares ; or the figure may be 
 cut out of a sheet of paper or metal of uniform thickness, and 
 its weight compared with the weight of a piece of the same 
 material whose area is known. 
 
 XII. ANGULAR DISTANCE WITH THE SEXTANT 
 
 Determine the angle between two distant points from two slightly different 
 stations. Determine the angular elevation of a distant object. 
 
 37. The Sextant. — The angle in any plane between two points 
 may be very conveniently measured, with considerable precision. 
 
THE SEXTANT 47 
 
 by means of the sextant. It is a small, light-weight instrument, 
 with which the angle between the two objects is measured not 
 by pointing first at one and then at the other but by observing 
 both at the same time, the objects being brought into apparent 
 coincidence by mirrors. The adjustment for coincidence can be 
 made while the instrument is held in the hand, and even when 
 the observer is in motion, as on board ship. This portability 
 makes the instrument very useful in the laboratory and field 
 as well as at sea. 
 
 The sextant consists of an arc of a graduated circle of about 
 60°, at the center of which is pivoted an index arm with a 
 vernier, V (Fig. 21), carrying the index mirror M. Attached to 
 the frame of the sextant is a small 
 telescope, T, and the horizon glass, H, 
 both fixed in position. The horizon 
 glass is half silvered and half trans- 
 parent, so that upon looking through 
 the telescope one sees objects in the 
 direction JffB, and also, by reflection 
 from the mirrors M and II, objects in 
 the direction Jf^. j,^^ 2^ g^^^^^^ 
 
 If the two mirrors are parallel the 
 two lines of sight are parallel, while if one mirror is turned through 
 any angle the lines of sight are inclined at twice this angle ; 
 hence every half-degree division on the arc is numbered as a 
 whole degree, aiid the readings give directly the required angle. 
 
 Colored shade glasses are provided to reduce, the intensity of 
 one or the other of the beams of light entering the telescope, in 
 order that both objects may be seen equally well. 
 
 The parts of a sextant, properly constructed, should fulfill the 
 following conditions : the index glass and horizon glass should 
 be perpendicular to the plane of the circle, the axis of the tele- 
 scope should be parallel to this plane, and when the mirrors are 
 parallel the vernier should read 0°. Sextants are provided with 
 means for making these adjustments, and the following methods 
 may be used to verify them ; but having been once properly made 
 they shoidd not need altering. 
 
48 MECHANICS 
 
 Set the index at about 100°, and place the eye close to the 
 index glass so that part of the divided circle is seen reflected by 
 the glass and part by direct vision. These two portions of the 
 arc should appear to be in the same plane. If they do not so 
 appear, alter the inclination of the mirror till the condition 
 is satisfied ; then the index mirror is perpendicular to the plane 
 of the arc. 
 
 Point the telescope at a small, distant object, and move the 
 index arm till the reflected image of the same object comes into 
 the field o^vievsr. If the horizon glass is perpendicular to the 
 plane of the arc, the two images can be made to coincide by 
 moving the index arm ; if not, alter the inclination of the horizon 
 glass till this is possible. 
 
 That the axis of the telescope is parallel to the plane of the 
 sextant is sufficiently tested as follows. Support the sextant 
 in a horizontal position, and place two sights on the ends of the 
 divided arc. The sights should be of the same height and such 
 that their tops are in a plane with the axis of the telescope, 
 parallel to the plane of the arc. Observe a distant point in line 
 with the sights, which should also be visible in the center of the 
 field of the telescope. The method of correcting, if necessary, 
 will be obvious from the construction of the instrument. 
 
 To test the fourth condition, point the telescope to a distant 
 object and turn the index arm till the reflected image coincides 
 with the direct one, as for the second correction. The two 
 mirrors are now parallel and the vernier should read 0°. If it 
 does not, the construction may permit the correction of the error. 
 Usually it is better to determine the amount of this index error, 
 and to apply to all angle readings the index correction, which is 
 the vernier reading for the above setting with its sign changed. 
 
 When altitudes above the horizon are to be measured, the 
 observer, if at sea, points the telescope to the visible horizon and 
 brings the image of the object tangent to this line by moving the 
 index arm ; but if the observer is on land the true horizon can 
 rarely be seen, and an artificial horizon consisting of a shallow 
 basin of mercury is used. Such a position is taken that the image 
 of the sun, for instance, may be seen reflected from the mercury, 
 
EQUILIBRIUM OF FORCES 
 
 49 
 
 and the angle between the apparent direction of the image and 
 of the sun is measured, which is twice the altitude of the sun. 
 
 Measure the angle between two selected points distant half 
 a mile or more, as seen from a given point. Make three settings 
 to determine the index correction ; the sum of the means of the 
 two determinations is the required angle. 
 
 Move ten feet towards or from the objects (a half mile distant) 
 and repeat the measurements. 
 
 Using an artificial horizon, measure the altitude of an assigned 
 point. 
 
 References. — Chauvenet, Practical Astronomy, Vol. II, pp. 92-118 ; John- 
 son, Surveying, pp. 108-112. 
 
 Xm. LAWS OF THE EQUILIBRIUM OF FORCES BY THE 
 TRIANGLE OF FORCES 
 
 Prove the laws for the equilibrium of three forces about a point, using 
 various combinations of forces. 
 
 38. Equilibrium of Forces. — The relation between three forces 
 which, acting simultaneously at a point, result in equilibrium is 
 of great importance in engineering work. By the polygon law, 
 such forces are in equilib- 
 rium if they are propor- 
 tional to the sides of a 
 triangle, the sides taken in 
 order having directions 
 parallel to the forces. Or 
 the forces are in equilib- 
 rium if each one is equal 
 and opposite to the results 
 ant of the other two. 
 
 These relations are illus- 
 trated by Fig. 22, in which 
 
 A, B, and C represent the three forces, and B the resultant of 
 A and B, which is equal to — C. In a triangle the sides are pro- 
 portional to the sines of the opposite angles. It is evident that 
 
 Fig. 22. Triangle and Paral- 
 lelogram OF Forces 
 
50 
 
 MECHANICS 
 
 the angles a, h, and e are the supplements respectively of a, /3, and 
 7, the angles between the forces as originally applied to the point. 
 Hence if these forces applied to a point are in equilibrium, they 
 are proportional to the sines of the opposite angles ; that is 
 
 ^ ^ B ^ C 
 sin a sin /3 sin 7 
 
 This relation may be proved as follows. 
 
 Arrange apparatus consisting of ball-bearing pulleys, cords, 
 
 and weights in the manner shown in 'Fig. 23, preferably in 
 
 front of a blackboard. The friction 
 of the pulleys and the stiffness of 
 the cords may cause the position 
 of equilibrium to be a little uncer- 
 tain, but by moving the system out 
 of equilibrium and allowing it to 
 return in various directions, a mean 
 position can be found which -will 
 be sufficiently exact. On the black- 
 board draw lines parallel to the 
 three cords, and with a large pro- 
 tractor measure the three angles, 
 a, /3, 7. 
 Use different combinations of weights, varying their relations 
 
 as much as possible, and record the results in the following 
 
 tabular form. 
 
 Equilibrium of Forces 
 
 Fig. 23. Equilibeium of Forces 
 
 
 
 
 December 20, 1901 
 
 
 
 
 Sekies 
 
 FOKCBS, grams 
 
 A^fGrLES, degrees 
 
 RATIOS 
 
 No. 
 
 A 
 
 JB 
 
 C 
 
 a 
 
 /3 
 
 y 
 
 A 
 Sin a 
 
 B 
 
 siu)3 
 
 C 
 
 sin y 
 
 1 
 
 59.8 
 
 69.8 
 
 102.8 
 
 146 
 
 139 
 
 75 
 
 106.5 
 
 106.9 
 
 106.4 
 
 2 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
THE BALLISTIC PENDULUM 51 
 
 XIV. CONSERVATION OF MOMENTUM WITH THE BALLISTIC 
 
 PENDULUM 
 
 Prove the conservation of momentum by the impact of elastic bodies, a 
 larger body striking a smaller one, and also the smaller striking the 
 larger ; find the coefficient of restitution in each case. Prove the prin- 
 ciple for the impact of inelastic bodies. Compare the kinetic energies 
 before and after impact. 
 
 39. Momentum and Impact. — The changes in motion im- 
 pressed upon bodies by their impact is in general a complicated 
 problem, but the study of several simple cases gives valuable 
 illustrations of Newton's Second and Third Laws of Motion. 
 The cases of impact to be considered are those of bodies whose 
 centers of mass may move only in the same line, and for which 
 the point of contact is also in this line. 
 
 By the Second Law of Motion any change in momentum is 
 the result of the action of a force ; if the only forces acting are 
 those due to impact, then, since by the Third Law action and 
 reaction are equal and opposite, regard being had to algebraic 
 sign, the quantity of motion is unchanged by impact. In other 
 words, the sum of the momenta before impact is equal to the 
 sum of the momenta after impact ; this is the principle of the 
 Conservation of Momentum. 
 
 Perfectly elastic bodies after collision would separate with the 
 same velocity as that with which they approach ; inelastic bodies 
 would not separate at all. Actual bodies lie between these two 
 extremes, and the ratio of the relative velocity after collision to 
 the relative velocity before is the coefficient of restitution. 
 
 The fact of imperfect restitution is not contrary to the Con- 
 servation of Momentum nor to the Third Law of Motion ; it 
 may diminish the action, but the reaction is diminished by the 
 same amount, always remaining equal to the action ; thus the 
 change in momentum of the system is zero. 
 
 The usual result of imperfect restitution is a change in the 
 kinetic energy of the system at the time of impact, this energy 
 causing vibration, change of shape, etc., and being largely 
 dissipated in the form of heat. 
 
52 MECHANICS 
 
 The conditions of collision above mentioned are very approx- 
 imately secured by using bodies of spherical shape supported 
 from parallel axes by bifilar suspensions, so that when they are 
 at rest the. spheres are in contact without pressure, and their 
 centers are in the same horizontal line in the plane of vibration. 
 One body may be cylindrical, when four 
 cords from the same axis are required for 
 \ its proper support. These cords may be 
 
 \ about a meter and a half long, and a 
 
 \ graduated arc of 40° is suitable for measur- 
 
 \ ing the swings of the pendulums. The 
 
 \ separation of the two axes will slightly 
 
 -■-■—--^4 displace them from the center of the arc, 
 4i^^r:Tr...J but the errors resulting are inappreciable. 
 
 Pointers on the under sides of the balls 
 Fig. 24. Relation of ,^^ move paper riders placed on the edge 
 Fall and Amplitude -,, i.T,,i , ., 
 
 OP Pendulum ^^° mdicate the extreme pomts 
 
 of motion. 
 If a sphere under the influence of gravity swings through 
 an arc AB (Fig. 24), it will have acquired at 5 a velocity the 
 same as if it had fallen vertically from A to -D, that is, through 
 the height A, and its velocity will be ^2gh, or it is propor- 
 
 tional to VA. By geometry, h = — — , C being the chord of the 
 
 arc- and R its radius; hence, since VX is proportional to C, 
 the velocity is proportional to the chord of the arc through 
 which the ball swings. But further, for arcs not exceeding 
 20°, since the length of the chord of 20° is 0.348 and of the 
 arc 0.349, the velocities may in this exercise be taken as pro- 
 portional to the arcs for the purposes of comparing momenta 
 and energies. 
 
 Mastic Bodies. — Suspend two steel balls of about 3.5 cm 
 and 5 cm diameter, so that the line joining the centers, the 
 balls being at rest and in contact without pressure, is horizontal 
 and in the plane of vibration. Withdraw the smaller ball to 
 the end of the arc and allow it to fall through a measured arc, 
 «! (Fig. 25), against the larger ball at rest. After collision the 
 
THE BALLISTIC PENDULUM 
 
 53 
 
 large ball will move forward through an arc, a^, while the small 
 baU rebounds through the arc a^. The momenta are propor- 
 tional to the products of the masses M and m into the respective 
 arcs ; and it is to be shown, by allowing m to fall from different 
 heights, that in each case 
 
 waj = Ma^ — ma^. 
 
 The coefficient of restitution is 
 
 «2 + «3 
 
 Calculate the relations of the kinetic energies before and 
 after collision. In this particular case the energy before colli- 
 sion is proportional to ma^, and after collision to Ma^ + ma^. 
 
 Make similar observations for 
 the momenta, kinetic energies, 
 and the restitution when the 
 large ball is allowed to strike 
 the small one. 
 
 Inelastic Bodies. — A brass 
 sphere and a lead cylinder serve 
 for these experiments, being 
 made inelastic in effect by plac- 
 ing a piece of soft wax at the 
 point of contact. They are to 
 be supported as before, so that 
 when the two hang at rest they 
 are in contact without pressure, 
 and the line joining the centers 
 of mass is horizontal and in the plane of vibration. Allow the 
 sphere to strike the cylinder, falling from various heights, and 
 calculate the relations between the momenta before and after 
 impact, and also between the kinetic energies. 
 
 Collision Pendulums 
 
54 
 
 MECHANICS 
 
 XV. MASS BY THE EQUILIBRIUM OF MOMENTS 
 
 Determine an unknown mass with a balanced bar ; determine the weight 
 of the bar, using known masses ; determine an unknown mass, the 
 bar not balanced. 
 
 40. Equilibrium of Moments. — A rigid body free to rotate 
 about a fixed point will be in equilibrium only when the sum 
 of the moments of the forces acting upon it is zero ; that is, 
 when the sum of the moments tending to produce rotation in 
 one direction is equal to the sum of the moments tending 
 to produce rotation in the opposite direction. 
 
 A sliding knife-edge is provided to support a meter stick 
 from a stirrup attached to a suitable stand, as shown in Fig. 26. 
 Slide the bar in its holder till it rests in a horizontal position. 
 
 Fig. 26. Equilibbium of Moments 
 
 By means of knife-edge hooks suspend a known mass near one 
 end and an unknown mass near the other. Alter the position 
 of the hooks to restore equilibrium, keeping them as near the 
 ends as possible. Calculate the value of the unknown mass. 
 The knife-edge support being balanced, its mass need not be 
 considered; but the hangers must be considered parts of the 
 suspended masses. 
 
 Suspend masses of 200 g and 500 g 98 cm apart (Fig. 26), 
 and shift the bar in its support until the moments are again in 
 
THE BALANCE 55 
 
 equilibrium. The three moments, due to the suspended masses 
 and the mass of the bar, are now in equilibrium. With the 
 aid of the principle that the weight of a body may be con- 
 sidered as concentrated at its center of gravity, the mass of 
 the bar is to be calculated. 
 
 Substitute an unknown mass for one of the known masses, 
 and shift the point of support to secure equilibrium ; deter- 
 mine the unknown mass, taking into account the weight of the 
 unbalanced bar. 
 
 Verify the results by weighing the unknown masses on a 
 balance. 
 
 XVI. MASS WITH THE BALANCE BY THE METHOD OF 
 VIBRATIONS AND DOUBLE WEIGHING 
 
 Determine an unknown mass by vibrations. Determine a mass by double 
 weighing, and find the rartio of the arms of the balance. 
 
 41. The Balance Perhaps the most generally useful and 
 
 the most precise of scientific measuring instruments is the beam 
 balance, with which masses are compared by the principle of 
 the equilibrium of moments. 
 
 The balance consists of a double lever or beam, B (Fig. 27), 
 supported in a horizontal position upon a knife-edge through 
 the center ; near the ends of the beam are knife-edges from 
 which are suspended pans, P, in which can be placed the 
 masses to be compared. The distances from the center knife- 
 edge to the end knife-edges are the arms of the balance. A 
 pointer attached to the beam and moving over a scale, S, indi- 
 cates rotational displacements of the beam. An arrest is pro- 
 vided which removes the beam and pan supports from the 
 knife-edges, and in addition it often firmly supports the pans 
 from beneath. It is usually operated by a milled head or lever 
 at A. A system of lever rods, R, is provided for placing small 
 weights called riders upon the beam at any point of its length, 
 and for lifting them when they are not needed. A case to 
 protect the balance from air disturbances is necessary. There 
 
56 
 
 MECHANICS 
 
 are numerous adjusting devices and conveniences, which need 
 not be described. 
 
 The theoretical conditions which a balance should fulfill to 
 secure sensitiveness and precision are the following: the three 
 knife-edges should be in the same plane and parallel to each 
 other; the beam should be inflexible; the arms should be of 
 equal length ; the center of gravity should be below the central 
 knife-edge, and as near to it as possible. That the balance 
 
 may be operated rapidly 
 the beam should be short, 
 and its center of gravity 
 far from its support. Some 
 of these requirements are 
 practically inconsistent 
 with each other, and one or 
 another may be sacrificed 
 according to the use for 
 which the balance is 
 desired. 
 
 The highest sensibility 
 and precision are attained 
 in balances with long and 
 heavy beams. Such bal- 
 ances indicate variations 
 of 1 part in 500 000000 
 in the maximum load of 500 g or 1000 g. The period of 
 vibration is very long, being as much as 60 seconds in some 
 instances. A balance which will indicate a variation of 1 part 
 in 1 000000 for the maximum load of 200 g is usually amply 
 sensitive. Such balances may be made with short and light 
 beams and with a period of vibration of 10 seconds. 
 
 If a perfect balance were used, a theoretical weighing would 
 consist simply in placing the unknown mass in one pan and 
 known masses or weights in the other pan, and in adjusting 
 the latter to bring the balance to a position of rest exactly the 
 same as its position of rest when no loads are in the pans ; 
 but in practice it would be very difficult or impossible to 
 
 T3 ^*V~ 
 
 Fig. 27. The Balance 
 
10 
 
 '1' 
 
 20 
 
 THE BALANCE 57 
 
 secure these conditions. In general it is easier to make a 
 precise determination of the difference between two quantities 
 than to secure their equality. The practical method of weigh- 
 ing is, then, to select two groups of weights differing by a very 
 small quantity, as for instance by 1 mg, such that one weight 
 is smaller than the unknown mass and the other larger; and 
 after comparing each of these with the unknown mass, to find 
 the true value of the latter by interpolation. The practice of 
 this method will be developed in the following articles. 
 
 42. Weighing by Vibrations. — A concrete example will aid 
 in explaining the method of weighing by vibrations. Let it be 
 required to determine the apparent weight of a brass cylinder, 
 to the nearest tenth of a milligram, 
 the smallest weight used being one 
 milligram. 
 
 The scale of the balance is usually ' I ' '( 1 1 
 graduated as shown in Fig. 28. The 
 figures are often omitted, which fact, ^ „„ „ 
 
 , ° .,, . J. . , , I^iG- 28. Scale of Balance 
 
 however, will not interfere with mak- 
 ing the readings described. Allow the pointer to swing two or 
 three divisions each side of the center. Determine the zero 
 point by reading one swing to the left, one to the right, and 
 a second one to the left, estimating tenths of a division; or 
 the readings may begin and end with a swing to the right. 
 Suppose the reading, indicated in the figure, as the pointer 
 swings to the left, is 6.0 ; then that it swings to the right to 
 13.1, and back to the left to 6.6. The average of the two 
 left readings is 6.3, and the sum of the average left swing 
 and the right swing, that is the sum of the excursions, is 19.4. 
 The resting point is half the sum of the excursions ; but it is 
 shorter and somewhat more precise to use the sums of the excur- 
 sions instead of the equilibrium points in calculating the weights. 
 Record the observations on one line as follows : 
 
 Zero point 6.0 1-3.1 G.6 19.4 
 
 When beginning work with the balance it will be well to 
 determine the zero point three times, arresting the beam after 
 
58 MECHANICS 
 
 each determination. The proper agreement of the results will 
 indicate that the balance is in good condition. 
 
 Now place the brass cylinder in the left pan, and weights to 
 balance it as nearly as can be estimated, 93.217 g, in the right 
 pan. Allow the balance to swing, and read three excursions 
 as before, — to the right 10.4, to the left 8.6, and again to the 
 right 10.2. Record thus: 
 
 Cylinder left 93.217 g right 10.4 8.6 10.2 18.8 
 
 The weight in the right pan is evidently too great, as indi- 
 cated by the sum of the excursions being smaller than for the 
 zero point. Remove one milligram and observe another set 
 
 of vibrations : 
 
 93.216 g right 13.0 7.9 12.7 20.7 
 
 Now the weight in the pan is too small, and the apparent 
 weight of the cylinder lies between the two amounts tried. 
 From these observations the exact weight can be calculated. 
 The observations show that the removal of 1 mg caused a 
 change in the sum of the excursions of (20.7 — 18.8) 1.9 divi- 
 sions,, while the change desired was (19.4 — 18.8) 0.6 divisions. 
 Therefore, if the weight removed had been ^g mg, the sum of 
 the excursions would have been the same as for the zero point. 
 This shows that a weight of (93.217 g - ^g mg) 93.2166 g 
 would exactly balance the cylinder. 
 
 Collecting the observations, the whole record for a complete 
 single weighing by vibrations is as follows : 
 
 Mass of Cylinder No. 5 
 
 METHOD OF SINGLE ' WEIGHING BY VIBRATIONS 
 
 Balance No. 7, Weights No. 1 
 
 May SI, 189S 
 Zero point 6.0 13.1 6.6 19.4 
 
 Cylinder left j 93.217 g right 10.4 8.6 10.2 18.8 
 
 ■^ ( 93.216 g right 13.0 7.9 12.7 20.7 
 
 Cylinder in left pan = 93.217 g - t\ mg = 93.2166 g. 
 
 The assumptions made in this article, that the arms of the 
 balance are of equal length, that the density of the weights and 
 
THE BALANCE 59 
 
 of the body weighed are equal, and that the weights are per- 
 fectly adjusted, are never realized. The method must be 
 extended to suit these circumstances, as explained in the next 
 article and in Arts. 47, 48, and 49. 
 
 43. Double Weighing and Ratio of Balance Arms The 
 
 method of double weighing described below theoretically gives 
 the true apparent mass of a body by the elimination of the 
 effects of inequality of the balance arms. It assumes that the 
 position of equilibrium of the loaded balance is the same as 
 that of the unloaded balance. Since this assumption is not 
 justified, the method is not trustworthy for the determination 
 of the true mass. The 
 
 method of weighing ^ I ^ r ^ 
 
 by reversal, described 
 in Art. 47, is always 
 to be preferred for 
 
 this purpose, as it is # \ j.^,,. 29. Balance with 
 
 precise in practice / \ Unequal Akms //J^ 
 
 and requires fewer 
 observations. 
 
 The method of this article is the only one possible for finding 
 the ratio of the balance arms, and it must therefore be used in 
 investigating the constants of a balance. 
 
 Let a body be weighed by vibrations, as described in the pre- 
 ceding article, with a balance whose arms are unequal ; let the 
 length of the left arm be represented by I, and of the right by 
 r (Fig. 29) ; let X be the unknown mass in the left pan, and W^ 
 the weight that balances it. The body is transferred to the right 
 pan and a second complete weighing is made by vibrations ; let 
 W^ be the weight balancing it in this position. Then from the 
 principle of the equilibrium of moments the following equations 
 are true. 
 
 XI = W^r, Xr = W^l. 
 
 The product of these equations is XHr = W^ W^lr ; therefore 
 
 x=Vw^^, 
 
60 MECHANICS 
 
 which is the weight of the body independent of the lengths of the 
 arms. Instead of the geometrical mean, the arithmetical mean, 
 
 X 2 
 
 is sufficiently precise for a balance in which the arms are made 
 as nearly equal as possible. 
 
 Dividing one of the equations for the moments by the other, 
 
 I 
 
 r 
 
 
 which gives the ratio of the arms. When the arms are of 
 nearly equal length their ratio is more easily computed, and 
 with sufficient accuracy, by the approximate formula, 
 
 I _^ , w,-w^ 
 
 A numerical illustration follows. 
 
 r 2W^ 
 
 Mass of Cylinder No. 5 and Ratio of Balance Arms 
 
 method op double weighing by vibrations 
 
 Balance No. 7, Weights No. 1 
 
 May SI, 1895 
 Zero point 
 
 Cylinder left [f'T.^'^^''' 
 
 ^ \ 93.216 g right 
 
 Zero point 
 
 Cylinder right {93;2jJgLt 
 
 Cylinder in right pan = 93.210 g + t\ mg = 93.2102 g. 
 
 6.0 
 
 13.1 6.6 
 
 19.4 
 
 10.4 
 
 8.6 10.2 
 
 18.8 
 
 13.0 
 
 7.9 12.7 
 
 20.7 
 
 mg = 
 
 93.2166 g. 
 
 
 6.8 
 
 12.3 7.1 
 
 19.3 
 
 11.4 
 
 7.7 11.2 
 
 19.0 
 
 15.0 
 
 6.1 14.4 
 
 20.8 
 
 Apparent mass = V93.2102 x 93.2166 g = 93.2134 g. 
 
 „,.,,, I /93.2166 
 
 Katio of balance arms, - = V — = 1.000034. 
 
THE BALANCE 61 
 
 44. Weighing by the Rider Method. — A more rapid method 
 of weighing than that described in Art. 42 is often required. 
 Foi>the determination of specific gravity, and for chemical analy- 
 sis, relative mass only is necessary. If bodies are always weighed 
 on the same side of the balance, their masses are affected in the 
 same proportion by inequality in the arms, and weighing by 
 reversal is not necessary. The method of single weighing by 
 vibrations may be simplified for these purposes. 
 
 Small weights are troublesome to handle, and those of less 
 than 10 mg value may be replaced by a rider, which is a piece 
 of wire usually weighing exactly 10 mg, of such a shape that it 
 may ride astride the beam of the balance at different distances 
 from the central knife-edge. It may be manipulated by the 
 handle shown at R in Fig. 27. If it rides over the end knife- 
 edge it produces the same effect as though it were in the pan. 
 But if it rides at a tenth of the distance from the center to the 
 end its effect is one tenth as great; that is, it may be used in 
 this position as a substitute for 1 mg in the pan. The arm may 
 be divided into ten parts with subdivisions, and then when the 
 rider is moved to a position for equilibrium, its effect in milligrams 
 and fractions is read from the figures engraved on the beam. 
 
 Often two riders are \ised, one on each arm ; or one rider may 
 move over both arms. The effect of the rider, whether to be 
 added to or subtracted from the weights in the pan, will be 
 evident from inspection. 
 
 The complete method of vibrations with interpolation may be 
 employed as described, or it may be shortened with only a slight 
 loss of precision, as follows. The sensibility of the balance for 
 various loads (Art. 45) is carefully determined once for all, and 
 a table is constructed showing the sensibility for all loads within 
 the capacity of the instrument. When weighing, the zero point 
 of the unloaded balance is determined as before. The body is 
 placed in the pan and balanced as nearly as may be with weights 
 and the rider. The difference between the equilibrium point and 
 zero point, taken in connection with the table of sensibilities, will 
 determine the amount by which the weight in the pan differs 
 from the true weight. 
 
62 MECHANICS 
 
 Instead of finding the zero point in this manner, the unloaded 
 balance may be made to swing equally on each side of the center 
 of the scale by means of one rider; the center of the scale is 
 then the zero point, which simplifies the readings. A second 
 rider is used in balancing the body. Instead of reading three 
 turning points on the scale, the equilibrium position may be 
 taken as the mean of two readings. 
 
 Each simplification slightly diminishes the precision, and the 
 nature of the work must determine in each case the method to 
 be adgpted. 
 
 45. Sensibility of Balance. — The scale displacement of the 
 equilibrium position of the pointer of a balance, caused by a 
 change of one milligram in the load in either pan, is the sensi- 
 bility. The sensibility varies greatly with the loads in the pans ,- 
 it may be determined for various loads, and a table or curve pre- 
 pared for reference. It is somewhat more precise to determine 
 the sensibility at each weighing, as explained in Arts. 42 and 47. 
 
 The sensibility of a balance may be adjusted by altering the 
 position of the center of gravity of the beam. A small screw 
 is often attached above the central knife-edge for this purpose. 
 The center of gravity of the beam must be below the point of 
 support, otherwise the beam wiU be in unstable equilibrium. 
 The nearer the center of gravity is brought to the point of sup- 
 port, the more sensitive the balance; but increase of sensitive- 
 ness results in a longer period of vibration and a less rapid use. 
 The limits of precision of a balance vary greatly according to the 
 design and quality of construction, and it is useless to increase 
 the sensitiveness beyond the point of trustworthy indications. 
 Often it is advantageous to make the sensibihty less than its 
 maximum value, altering it so that it is not greater than is 
 required for the particular work in hand. 
 
 46. Practical Hints on the Use of the Balance In addition 
 
 to the general methods already described, the student should 
 carefully observe the following hints. 
 
 Remove dust from the pans and weights with a camel's-hair 
 brush. 
 
 Level the base. 
 
THE BALANCE 63 
 
 The unloaded balance should be in equilibrium when the 
 pointer is near the center of the scale; this condition may be 
 secured by adjusting the small screw attached to the beam. The 
 student should not alter this, or any other adjustment of the 
 balance, except upon deiinite orders. 
 
 The beam and pans should be arrested when the balance is 
 not in use, and also whenever weights are added or removed. 
 
 Arrest the beam only when it is passing through its position 
 of equilibrium. 
 
 Avoid parallax in reading by having the line of sight perpen- 
 dicular to the scale. 
 
 The release of the arrest will usually cause the beam to vibrate 
 as desired. If not, a breath of air under one of the pans will be 
 sufficient. A small stream of air from a rubber bulb held in the 
 hand may be directed against the pans to regulate the swing. 
 The pans should have no pendular motion. 
 
 Weighings should be made with the balance case closed, and 
 precautions are required to avoid the effects of air currents in 
 the closed case caused by uneven temperature. 
 
 The larger weights should be placed near the center of the 
 pan, and the others so distributed that the pan will not swing 
 to one side when released. 
 
 The weights should be handled only with the pincers and 
 forks provided. Small weights may be removed from the pan 
 with a camel's-hair pencil. 
 
 To avoid mistakes, the weights should be counted several 
 times. Count them while they are in the pan ; remove them 
 from the pan, arranging them in groups in the order of magni- 
 tude, and count again ; and finally they may be counted by the 
 unoccupied places in the box, or as they are returned to their 
 places in the box. 
 
 When using gilded weights care should be taken that they 
 do not come into contact with small particles of mercury. 
 
 Volatile liquids and fuming acids should be weighed only 
 in closed vessels. It is sometimes necessary, as when 
 weighing hygroscopic bodies, to keep drying material in the 
 balance case. 
 
64 MECHANICS 
 
 Arrest the balance, remove weights from the pan, arranging 
 them in the proper order in the box, and close the balance case, 
 upon the completion of the weighing. 
 
 XVII. ABSOLUTE MASS WITH THE BALANCE BY THE 
 METHOD OF VIBRATIONS AND REVERSAL 
 
 Determine the true mass of a body. 
 
 47. Weighing by Reversal. — Any weighing which depends 
 upon the determination of the zero point must be untrustworthy, 
 as the position of equilibrium of the unloaded balance may not 
 coincide with its position of equilibrium when loaded. The 
 method of reversal here described does not require the position 
 of the zero point to be known ; it eliminates the effect of the 
 inequality of the arms, and is both short and accurate. 
 
 For example, let it be required to compare the kilogram 
 weight (lOOO)p with the normal kilogram Kj,. With Kj^ in the 
 right pan and (1000)^ in the left, determine the sum of the 
 excursions as described in Art. 42. Interchange the loads in 
 the pans and again find the sum of the excursions. If the appar- 
 ent weights of the two masses are equal, the sums of the excur- 
 sions will be identical, regardless of the lengths of the beam 
 arms ; while if the weights are unequal, the difference between 
 the sums of the excursions will correspond to twice the difference 
 between these weights. 
 
 This difference will be expressed in milligrams by making the 
 observed change in the sum of the excursions, caused by reversal, 
 the numerator of a fraction whose denominator is the change in 
 the sum of the excursions produced by adding a two-milligram 
 weight to either side. The two-milligram weight should be 
 dropped into the pan without arresting the balance. 
 
 The complete record of such a comparison is given in the 
 next article. 
 
 48. Weight in Vacuo. — When a weighing is performed in 
 the air, as is usually necessary, all bodies in the pans are buoyed 
 up by forces proportional to their volumes. If the volumes in 
 
THE BALANCE 65 
 
 the two pans are unequal, the apparent weight is in error because 
 of this buoyancy. A correction for this is not always applied, 
 though it is of sufficient importance to be regarded even in ordi- 
 nary weighing. If m represents the weight which balances a 
 body in air, I the density of the air, s the density of the body, 
 and d the density of the weights, the true weight in vacuo is 
 (Table 1, Appendix), 
 
 ""•{'^'-ri)- 
 
 Frequently this correction can be made with greater ease and 
 precision from the known volumes of the bodies and the weights. 
 Its amount is equal to the difference of volumes in the two pans 
 multiplied by the density of the air at the temperature and 
 pressure of the experiment, the apparent weight being too small 
 by this amount, if the body has the larger volume. 
 
 When the masses to be compared are of different materials, it 
 is also necessary to consider their volumetric expansions. The 
 volumes are to be reduced to 0°, using the coefficients of cubical 
 expansion, which are three times the coefficients of linear 
 expansion. 
 
 A numerical illustration follows. 
 
 Absolute Mass of Kilogram (1000) q 
 
 by cojfparison with noemal kilogram kj, method of keveksal 
 
 Staudinger Balance, Rueprecht Weights 
 
 June 15, 1895 
 
 The true mass of the standard kilogram, Kj,, is known to be 999.9998 g. 
 Barometer 74.6 cm; attached thermometer 2'2°.4; thermometer, air, 20°.7. 
 
 (1000)q left 
 
 Kif right 
 
 7.7 13.3 
 
 8.0 
 
 21.1 
 
 Ky left 
 
 (1000)q right 
 
 8.1 11.6 
 
 8.3 
 
 19.8 
 
 Bd 2 mg left 
 
 
 8.7 14.0 
 
 8.9 
 
 22.8 
 
 (1000)g = ii:^+Hmg. 
 
 Volume of iT^r at 0° referred to water at 4° = 118.876 ccm. 
 Volume of (1000)<j at 18°.6 referred to water at 4° = 120.75 ccm. 
 Both masses made of brass. 
 
66 MECHANICS 
 
 To compare the volumes of the two bodies, that of (1000)q observed at 
 18°.6 must be reduced to 0°. The cubical expansion of brass is 0.000057 
 for 1°, from -which the volume at 0° is found to be 120.62 ccm. The 
 difference in volume of the bodies, 1.74 ccm, gives for the reduction to 
 vacuum, the density of air being 0.00118, + 0.00208 g. Hence 
 
 (1000) Q = ifar + 0.00043 g + 0.00208 g 
 
 Ks= 999.9998 g 
 (1000) Q= 1000 .0023 g 
 
 This represents the absolute mass of the weight, since it refers to the 
 body when it is in a vacuum. 
 
 XVni. ERRORS OF A SET OF WEIGHTS BY COMPARISON 
 
 Determine the correction to each weight of a set by comparisons among 
 themselves and by comparison with a normal weight. 
 
 49. Calibration of Weights. — The errors of adjustment in a 
 set of weights are frequently of considerable magnitude, and it 
 cannot be assumed that they are so small as to be negligible, 
 even in weights presumed to be of the finest quality. Weights 
 for use in work of precision must always have their apparent 
 relative masses determined, and often their absolute masses. 
 The latter requires that the volume of each weight be known ; 
 this may be ascertained by hydrostatic weighing (Art. 96), though 
 it may be sufficient to assume a density for the material of which 
 the weights are constructed and to compute the volumes from 
 this and their nominal values. 
 
 All the weights are to be compared among themselves accord- 
 ing to a systematic scheme, in which there are as many weighings 
 as separate weights, each weight being compared with another 
 single weight or combination of weights. These weighings 
 furtiish equations from which may be calculated the errors of 
 each weight. When the absolute mass is required, one weight 
 must be compared with a standard weight. A numerical example 
 will illustrate the method. 
 
CALIBRATION OF WEIGHTS 
 
 67 
 
 Calibration op Rukprecht Platinized Weights 
 
 Spoerhase Balance No. 1 
 
 December 6, 1895 
 
 Besignate each weight by its nominal value in parenthesis ; where there 
 are several weights of the same nominal value distinguishing marks must 
 be placed upon them. Let the weights in the set be 
 
 (100) = a, (50) = J, (20) = c, (10) = d, (10') = e, 
 
 and (10") = (5) + (2) + (1) + (1') + (1") =/, 
 
 the letters representing the true values. 
 
 The weight a is compared with the standard 100-gram weight, and the 
 weights are compared among themselves, as indicated in the following equa- 
 tions, by the method of double weighing described in Art. 47. 
 
 100 g = a - 0.07 riig 
 
 a = 
 
 b+c+d+e 
 
 : +/- 0.91 
 
 b = 
 
 c -ird + e. 
 
 14-/- 0.95 
 
 c = 
 
 d + ( 
 
 ! - 0.76 
 
 d = 
 
 t 
 
 '. - 0.03 
 
 e = 
 
 
 /- 0.04 
 
 Solving these equations, 
 
 
 / = 
 
 f 
 
 
 e = 
 
 f- 0.04 mg 
 
 
 d = 
 
 /- 0.07 
 
 
 c = 
 
 2/- 0.87 
 
 
 6 = 
 
 5/- 1.93 
 
 
 a = 
 
 10/- 3.82 
 
 
 a — 
 
 100.00007 g 
 
 
 From the last two equations. 
 
 
 10/= 
 
 100.00389 g 
 
 
 / = 
 
 10.00039 g 
 
 
 and therefore 
 
 
 
 a = 
 
 100.00007 g. 
 
 correction -1- 0.00007 g 
 
 6 = 
 
 50.00002 
 
 + 0.00002 
 
 c = 
 
 19.99991 
 
 - 0.00009 
 
 d = 
 
 10.00032 
 
 + 0.00032 
 
 e = 
 
 10.00035 
 
 + 0.00035 
 
 /= 
 
 10.00039 
 
 + 0.00039 
 
 A repetition of the process with the smaller weights contained in the 
 set /will determine their values ; and the values of the subdivisions of the 
 gram are to be found in the same manner. 
 
68 MECHANICS 
 
 50. Adjustment of Weights. — Usually it will be more con- 
 venient to make a table of corrections for a set of weights than 
 to adjust them. When adjustment is necessary the following 
 method may be used ; if other methods are required the circum- 
 stances will determine the procedure. 
 
 Analytical weights are usually constructed with knobs screwed 
 into the bodies ; the knobs may be removed, exposing cavities 
 in which small bodies are placed in adjusting. Make each 
 weight a trifle lighter than its normal value, removing small 
 particles from the interior if necessary. The entire set of weights 
 is then calibrated by comparison with a normal weight and among 
 themselves,, as described in the preceding article. The error of 
 each weight is thus determined. Weigh a measured length of 
 fine wire which may have mass of about one milligram per centi- 
 meter of length ; by measurement it will then be possible to cut 
 pieces from this wire which will very nearly correct the several 
 errors. Place these pieces of wire inside the respective weights. 
 
 After adjustment the whole set of weights should again be 
 calibrated. The errors will be very small if the work has been 
 carefully done. 
 
 XIX. VOLUME BY WEIGHING 
 
 (a) Determine the volume of a glass bulb, and the volume per centimeter 
 
 of length of its tubular stem. 
 
 (b) Determine the radius of bore of a capillary tube. 
 
 51. Volume by Weighing, — This method is useful for verify- 
 ing or correcting the volumes of measuring vessels and for 
 determining the volumes of bulbs and tubes. 
 
 If the fluid used to fill the vessel weighs m grams in air and 
 has the density s, its volume is 
 
 _ m /^ I l\ 
 s\ a dj^ 
 I being the density of the air and d that of the weights. 
 
 For water and brass weights in air it wiU be sufficient to 
 take the volume as 
 
 v = m(2.00106-s). 
 
THE BAEOMETER 69 
 
 For mercury weighed with brass weights in air the volume is 
 
 V = 0.999945 -, 
 
 s 
 
 the density of the mercury at t° being (Appendix, Table 3) 
 s = 13.595(1- 0.000182 «). 
 
 The mass of water which balances 1 g of brass in air has at 
 20° the volume of 1.0029 ccm. 
 
 The mass of mercury which balances 1 g of brass in air has 
 at 20° the volume 0.07382 ccm. 
 
 Tables 3 and 4 in the Appendix facilitate the calculation of 
 volumes for the conditions which occur most frequently. 
 
 52. Volume and Radius of a Tube. — A mass of mercury of 
 m grams at the temperature t° has the volume 
 
 m(\ + 0.000182 «) 
 
 V = — ^ • . 
 
 13.595 
 
 To find the volume, c, per centimeter of length of a cylindrical 
 tube, determine the weight, m grams, of the column of mercury 
 at the temperature t° which occupies I centimeters of length of 
 the tube ; then 
 
 _ 7ra(l + 0.00018^) 
 ''" 13.595 Z 
 
 If m is the weight of the mercury which fills I centimeters of 
 the cylindrical tube at 20°, the radius is 
 
 r = 0.1534 \/^. 
 
 XX. ATMOSPHERIC PRESSURE AND ALTITUDES WITH THE 
 
 BAROMETER 
 
 (a) Read the barometer and reduce the reading to standard conditions. 
 
 (b) Determine altitudes from the barometric reading. 
 
 53. The Barometer In its usual form the barometer con- 
 sists essentially of a glass tube closed at one end, filled with 
 mercury, and then inverted in a cistern of mercury. The tube 
 
70 
 
 MECHANICS 
 
 is of such a length that in its upper part there is a vacuum. 
 Instead of the cistern, the open lower end of the tube is some- 
 times extended and bent upward, forming the siphon barometer. 
 The mercury column being maintained by atmospheric pressure, 
 variations in its height will measure changes in this pressure. 
 
 By the height of the barometer is meant the height of the mer- 
 cury column at 0°, which at the specified point would exert the 
 same pressure as is exerted by the atmosphere 
 at that point. If the barometric pressures at 
 different localities are to be compared, it is not 
 sufficient to compare the heights as defined ; 
 but the actual pressures in dynes per square 
 centimeter must be compared, since the weight 
 of a given column of mercury varies with the 
 intensity of gravity. 
 
 The Fortin cistern barometer is the form 
 usually employed for laboratory and meteor- 
 ological observations. In reading it the fol- 
 lowing operations should be performed in the 
 order given. Read the attached thermometer. 
 Gently tap the tube near the top of the column 
 to avoid adhesion of the mercury. Adjust 
 the surface of the mercury in the cistern 
 (Fig. 30) by the screw S at the bottom of the 
 instrument until it just touches the/tip of the 
 ivory pointer P. Again gently tap the tube 
 and set the vernier index so that its lower edge 
 is tangent to the meniscus. Read the vernier. 
 In the siphon barometer the scale is often etched on the glass 
 tube, with its zero near the middle; readings are then to be 
 made upwards and downwards to the mercury surfaces, and the 
 apparent height is the sum of the two, readings. 
 
 54. Reduction of Barometer Readings. — A reading, taken as 
 described, requires several corrections ; viz. for the temperature 
 of the mercury, the temperature of the scale which is supposed 
 to be correct at 0°, the capillary depression of the mercury 
 column, and for the depression due to the presence of air or 
 
 Fig. 30. Cistern 
 OF Bakometer 
 
THE BAEOMETEE 71 
 
 water vapor in the tube above tlie mercury. ' Sometimes it may 
 also be necessary to include a correction for the index error of 
 the scale. The temperature correction is usually large and should 
 never be omitted, while for many laboratory purposes the other 
 corrections are so small as to be negligible. 
 
 Temperature Corrections. — If Z is the height of the column 
 as read at- 1°, and e the coefficient of expansion of the material of 
 the scale, the height corrected for the two temperature errors is 
 
 h = l- (0.000182 -e)Zi; 
 from which, 
 
 for a brass scale, h = l — 0.000163 It, 
 for a glass scale, h = l- 0.000174 It. 
 
 The values of the temperature correction for a barometer with 
 a brass scale are given in the Appendix, Table 14. 
 
 Index and Other Errors. — The tubes of barometers are often 
 so small that the capillary depression is considerable ; and often 
 the barometer contains air or water vapor above the mercury, 
 causing an appreciable depression. The makers usually elimi- 
 nate these errors from the readings by setting the scale so that 
 it reads correctly, as compared with a true standard barometer, 
 though it does not give the actual height of the mercury above 
 the index. If such an apparent index error is discovered, it does 
 not necessarily introduce an error into the readings ; it probably 
 corrects several errors. Whether it does so correct the errors can 
 be conveniently determined only by recomparison with a standard 
 barometer. In the siphon barometer it is not difficult to determine 
 such an error, by altering the quantity of mercury so that the 
 space in the tube above the mercury is changed. If this space 
 be diminished to one half, the pressure due to inclosed air will be 
 doubled, and it may be detected by comparison of the readings. 
 
 If a true index error is present it may be corrected in a manner 
 which will be evident from the construction of the instrument. 
 
 Correction for Capillary Depression. — If this has been cor- 
 rected in setting the scale as described above, it will need no 
 further consideration. But mercury columns are used for many 
 measurements where independent correction is desired. 
 
72 MECHANICS 
 
 A table is given in the Appendix (Table 13) from which the 
 correction for capillary depression may be found. Its amount 
 is somewhat uncertain, and for tubes of more than one centimeter 
 internal diameter it may usually be neglected. 
 
 55. Barometric Pressure. — The observed height of a mercury 
 column, corrected as described in the previous article, will be 
 reduced to the true pressure in dynes per square centimeter 
 when it is multiplied by the density of mercury and by the 
 intensity of gravity at the place of observation. 
 
 The density of mercury at 0° is 13.5950, while the values 
 of g for various localities may be found in tables (Table 10, 
 Appendix). 
 
 56. The Barye and Standard Pressure. — The unit of pressure 
 is a pressure of one dyne per square centimeter ; it is called the 
 harye. For measurements of the elasticity of gases, and for all 
 other cases where pressures have been heretofore expressed in 
 atmospheres, it has been decided (International Congress of 
 Physics, Paris, 1900) to adopt the megaharye, equal to a pressure 
 of 10^ djTies per square centimeter, as the standard pressure. 
 For Cleveland, where g has the value 980.240, this corresponds 
 to a mercury column having at 0° the height of 76.04 cm. 
 
 57. Altitudes by the Barometer. — If the height of the barom- 
 eter be observed at the same time at two different stations, the dif- 
 ference in altitude may be calculated by the following foimula. 
 Let 6j and h^ denote the two corrected barometer readings, t the 
 mean of the temperatures t.^ and t^ of the air at the two stations, 
 gj and ^2 the tensions of the vapor of water at the two stations, 
 h the mean height in centimeters above the sea level of the two 
 places, 4> the latitude, and II the required difference in altitude 
 in centimeters ; then, 
 
 H=l 843000 (log \ - log h^ (1 + 0.00367 t) 
 
 (1 -f- 0.0026 cos 2 ^ -I- 0.00002 h + lh). 
 
 The following approximate formula, which assumes the 
 value of gravity as that of latitude 45°, that gravity does not 
 
THE BAROMETER 73 
 
 vary with altitudes, and that the air is half saturated with 
 moisture, is sufficient for differences of altitude not exceeding 
 100000 cm. 
 
 ^=1 600000 ^L^l? (1 + 0.004 1). 
 
 "1 "r "2 
 
 References. — Kohlrausch, Physical Measurements, pp. 76-80; Staley, 
 Gillespie's Surveying, Vol. II, pp. 273-296 ; Everett, C. G. S. System of Units, 
 pp. 43-47. 
 
CHAPTER III 
 TIME, ACCELEEATION, AND GRAVITY 
 
 XXI. LAWS OF ACCELERATED MOTION WITH A FALLING 
 TUNING FORK 
 
 Verify the laws of accelerated motion and find the acceleration of gravity. 
 
 58. Time. — Of the three fundamental units of mechanics, 
 determinations of length and mass are considered as belonging 
 to the physicist, while the determination of time is usually dele- 
 gated to the astronomer. In the physical laboratory it is required 
 either to find the ratio of relatively short time intervals or to 
 compare a periodic interval with the infiications of a standard 
 timepiece. The methods for such comparisons are described in 
 connection with the several exercises of this chapter. 
 
 59. Accelerated Motion. — A tuning fork is attached to a frame 
 arranged to fall vertically about a meter, between guides which 
 offer the least possible resistance. The fork may be held at the 
 top of the guides by a catch having a trigger, which will release 
 the fork and at the same time set it in vibration. One prong 
 of the fork carries a style for recording the vibrations in a sinu- 
 ous line upon a long plate of smoked glass. The glass may be 
 lightly smoked with burning camphor gum. Make the guides 
 vertical, so that the slide may fall freely, tracing a sinuous line. 
 If necessary adjust the style and repeat the tracing after having 
 moved the glass plate sideways about a centimeter. In this way 
 obtain three good tracings to be measured as directed below. 
 
 Remove the plate from the frame ; select a point, a (Fig. 31), 
 near the beginning of a tracing where the waves are perfectly 
 foi-med. From this point mark off spaces each containing exactly 
 
 74 
 
THE FALLING TUNING FOEK 
 
 75 
 
 ten waves, as b, c, d, etc. When the fork was at a it had already 
 acquired some velocity in falling, which is represented by v^y. 
 In the interval of time t, corresponding to ten vibrations, the 
 fork, because of this initial velocity alone, would pass over a 
 space, v^t; but gravity also acts, and adds to 
 this the space igt^. Hence in the first interval, 
 from a to b, the fork passes over a space. 
 
 In two intervals of time it would pass over a 
 space, from a to c, represented by the same 
 formula if for t we substitute 2 t : 
 
 Similarly for the successive intervals, ad, ae, 
 etc., we must substitute 3 i, 4 i, etc., obtaining 
 
 S3 = 3 f o« + f 9t^ 
 
 84 = 4 v + ¥-^^'. 
 
 The differences between these distances are 
 
 the lengths of the spaces passed over in the 
 
 successive ten-vibration intervals. These are 
 
 s^ — Bq = ab = v^t + J gt\ 
 
 g^-s^ = bo = v^t + lgt'^. 
 
 s^ = cd ■ 
 
 v + 
 
 9^^': 
 
 These equations show that 
 
 be — ab = cd — be = = gfi. 
 
 Fig. 31. Trace 
 made by a tun- 
 ING FOKK 
 
 Measure as many of the ten-vibration spaces of the trace as is 
 possible, and find the differences between successive distances. 
 The differences should all be equal, proving that the force act- 
 ing on the fork, gravity, is constant. Compute the mean of 
 these measured accelerations, which is equal to / = gfi. If the 
 frequency of the fork, n, is known, the time interval is 
 
 n 
 
76 MECHANICS, 
 
 Compute g, the acceleration due to gravity ; the result will be 
 a little too small, because of the friction of the falling slide. 
 
 Having found t and g, calculate the initial velocity, t)g, from 
 the value of any of the measured internals ; as 
 
 ah = v^t + \gt\ 
 
 60. Acceleration with Variable Time Unit. — Using the second 
 and third wave traces, made ' as above described, mark off on 
 one spaces of twenty and on the other spaces of thirty wave 
 lengths. Find the average increase in lengths of the successive 
 equal time intervals. Compare the accelerations thus obtained 
 for the double and triple time intervals with that for the single 
 interval obtained before. The relations should be 
 
 /i • /a '/s ~ *i • ^2 • ^3 =1:4:9. 
 
 In other words, the acceleration is proportional to the squares 
 of the lengths of the time unit. A further >analy sis of the con- 
 crete case of the experiment will make the meaning clear. The 
 acceleration is obtained for two different units of time which 
 have the ratio of 1:3. In the first instance the acceleration is 
 measured by the increase of velocity per single interval. If, 
 now, the force acts during the triple interval, it will produce 
 three times as much increase in velocity as before, the velocity 
 being measured, as before, per single interval. But if the velocity 
 is now measured by the space passed over in the triple interval, 
 its value will be again multiplied by three, or in aU by nine ; 
 that is, the acceleration increases as the square of the unit of 
 time taken for measuring. But liaving a fixed unit of time, as 
 the second, the effect of a constant acceleration is proportional 
 to the first power of the time during which it acts. Gravity in 
 three seconds will increase the velocity of a falling body only 
 three times as much, expressed in centimeters per second, as it 
 would do in one second. 
 
ATWOOD'S MACHINE 77 
 
 XXII. LAWS OF ACCELERATED MOTION WITH ATWOOD'S 
 
 MACHINE 
 
 Verify the laws of uniformly accelerated motion. Determine gravity and 
 the moment of inertia of the pulley. 
 
 61. Atwood's Machine. — The essential features of Atwood's 
 machine are two equal weights, of mass w, suspended by a fine 
 cord passing over a light pulley mounted on antifriction bear- 
 ings. A third mass, r, is allowed to ride upon one of the 
 masses, m, causing it to descend. After falling for a known 
 time interval, the rider is removed by a ring through which m 
 passes. A stop is placed to receive m a known time after the 
 removal of the rider. The distance through which the rider 
 falls is measured, and also that through which the weight falls 
 after the lifting of the rider. 
 
 The time intervals may be determined with a metronome, 
 with a sounder in connection with a break-circuit clock, or 
 with a chronograph. Many machines have pendulums or clock 
 attachments for giving the time signals. 
 
 Neglecting the inertia of the pulley, the moving mass may be 
 represented by ^m + r; and, not considering friction, the force 
 moving this mass is the weight, / = rg, of the rider. If a is the 
 acceleration imparted to the moving mass by the rider, measured 
 by the distance moved over in one succeeding second after the 
 rider is lifted, divided by the time during which the rider acted, 
 
 f=rg = mass X acceleration = {2m + r)a. 
 
 Prove from measurements of the spaces passed over in one, 
 two, and three seconds, that after the rider is lifted the masses 
 move with constant velocity. 
 
 Allow the rider to act for one, two, and three seconds, and 
 show that the spaces through which it falls are as the squares 
 of the times. 
 
 Increase the size of the masses and that of the rider in the 
 same proportion, and show that the fall in three seconds is the 
 same as with the small masses, proving that heavy and light 
 bodies fall equally fast. 
 
78 MECHANICS 
 
 Show that the acquired velocities are proportional directly to 
 the times of the fall, by allowing the rider to act for one, two, 
 and three seconds and measuring the velocities produced in each 
 instance. 
 
 Prove that a force is proportional to the product of the mass 
 and the acceleration, as expressed by the above equation, by 
 using one rider on two different masses. 
 
 Prove the preceding proposition by measuring the accelera- 
 tions produced by two different riders on the same mass. In 
 this experiment the sum 2m + r should be kept constant by 
 means of small auxiliary weights. 
 
 62. Gravity and Inertia of Pulley with Atwood's Machine 
 
 The friction of the wheelwork and of the air is very difficult to 
 determine, and it is therefore reduced to a minimum by suitable 
 construction. The remaining friction may be compensated for 
 by adding to the descending mass small pieces of tin foil of such 
 a weight that after the motion is started by hand the mass will 
 descend uniformly. The amount of foil required should be 
 determined for each different load carried by the pulley. This 
 adjustment is best made when the two movable masses are at 
 the same level, so that the weights of the cord on the two sides 
 of the pulley are equal. 
 
 The rider in falling not only accelerates the masses m but it 
 also must turn the wheel with increasing speed. The pulley is 
 equivalent to a small additional balanced mass, P, for which 
 allowance should be made. 
 
 The equation of motion may now be written 
 
 rg = (2 m + r + P) a.^^. 
 
 Alter the masses m (for instance, double them) and represent 
 the new masses by M. With the same rider as before, the equa- 
 tion of motion is /n nr , , t,\ 
 
 rg =(2M+ r + P) a^. 
 
 By eliminating P from these two equations the following result 
 is obtained for the acceleration due to gravity. 
 
 «,«, 2M— 2 m 
 
PATH OF A PROJECTILE 79 
 
 Determine a^ and a2 as explained in the preceding article and 
 calculate g. When the rider under the influence of gravity falls 
 through a space s, it does work represented by rgs. This work 
 imparts velocity to the masses and the wheel, and it is measured 
 by their kinetic energies. The masses, 2m + r, acquire a velocity 
 V, and the wheel, whose radius is H and moment of inertia is I, 
 acquires an angular velocity ta. Therefore 
 
 rgs = ^(2m + r)v^+ ^ la?'. 
 But t)2 = 2as = i22(B2. 
 
 hence '''9 = \ 2m + r -\ — ;)«» 
 
 and -P = -4. 
 
 That is, the equivalent mass of the pulley is its moment of 
 inertia divided by the square of its radius. 
 
 By eliminating g from the first two equations, 
 
 _ (a^ — aj)r + a^2 M — a-j^2 m 
 
 *i ~ '^2 
 Measure the radius of the wheel and compute /. 
 /= PE^. 
 The moment of inertia might be experimentally determined 
 with the torsion pendulum, as described in Art. 84. 
 
 XXm. PROPERTIES OF THE PATH OF A PROJECTILE 
 
 Prove that the acceleration of gravity is constant and show that the path 
 of a projectile is a parabola. 
 
 63. The Trajectory. — Arrange apparatus for projecting a ball 
 horizontally in front of a blackboard or a sheet of paper, by 
 allowing it to roll down a semicycloidal curve, as indicated in 
 Fig. 32. The height of this curve may be 30 cm. A mechan- 
 ical trigger should be arranged to release the ball uniformly. 
 
 Determine the path of the ball by holding 3 straightedge in 
 different parts of the path, so that the ball just grazes it in falling. 
 
80 
 
 MECHANICS 
 
 Fifteen or more points should be determined, and a smooth 
 curve drawn through them. A parallel curve separated from 
 this by the radius of the ball will be the path of the center. 
 From the level of the center of the ball at the lowest point 
 of its cycloidal path draw a horizontal line, and measure 
 a series of distances, i, increasing 
 by two centimeters each. Since 
 the ball will move with a uniform 
 velocity in a horizontal direction, 
 these distances represent both the 
 times of flight and the horizontal 
 motion in these times. Measure the 
 vertical distances, I, from each divi- 
 sion to the plotted path. These rep- 
 resent the spaces fallen in the several 
 intervals of time, due to gravity. 
 A level and thumb line will aid in plotting the coordinates. 
 From the laws of falling bodies each pair of values of I and t 
 should satisfy the equation Z = J at^ ; prove this by showing that 
 the values of o / 
 
 « = ^' 
 
 computed for each point measured, are all equal. This value 
 of a represents the acceleration produced by gravity in the time 
 taken for the ball to move over 1 cm in a horizontal direction ; 
 this would be equal to g if the unit of horizontal measure, 
 instead of being 1 cm, was of such a length that it represented 
 the horizontal distance traveled in one second. 
 
 Since the acceleration produced is proportional to the square 
 of the time, the unit of time represented by t is 
 
 Fig. 32. Path of a 
 Pkojbctile 
 
 -^- 
 
 and the velocity of the horizontal projection of the ball is 
 
 1 
 
 t 
 
 Calculate both t and v. 
 
THE SIMPLE PENDULUM 81 
 
 The constancy of a, as determined above, proves the patli 
 
 2 
 to be a parabola, whose equation in the usual form is t^ — -l, 
 
 a 
 
 from which it appears that the distance of the focus from the 
 origin is 
 
 2 a 
 The length of the semi-latus rectum is 
 
 Plot the focus and measure the latus rectum. 
 
 XXIV. GRAVITY BY THE SIMPLE PENDULUM 
 
 Determine the intensity of gravity. 
 
 64. Simple Pendulum. — An excellent pendulum consists of 
 a metal sphere of from 3 to 5 cm diameter, suspended from a 
 rigid support by a fine steel wire .about 0.01 ^.^ 
 
 cm in diameter. A simple and sufficient (^)) 
 
 support is made by fastening the wire to a ^^ 
 
 light steel ring, such as a screw eye, and JU 
 
 hanging this on a knife-edge, which may be 
 a three-cornered file, ground smooth, as indi- 
 cated in Fig. 33. ; 
 
 The length of the pendulum for the pur- '; 
 
 poses of this experiment may be about 99 cm 
 or 396 cm. This length must be determined 
 as accurately as possible : if it is about a /^'^^^ 
 meter, a meter bar and a beam compass may / \ 
 
 be used for measuring ; if longer, a steel tape ( 1 
 
 is convenient. The distance, d, from the V J 
 
 point of support to the center of the sphere ^ ^ 
 
 is required. The distance to the bottom of ^"o- 33. Simple 
 the ball may be measured, from which is to ^ ^ 
 
 be subtracted the radius of the ball, r, determined with the 
 
82 MECHAIflCS 
 
 calipers. The length of the equivalent simple pendulum is, 
 neglecting the effect of the suspending wire, 
 
 The pendulum is made to vibrate through a small arc, a, not 
 exceeding 4°, and its period, j?, is determined, preferably by the 
 method of coincidences as described in the next article. The 
 time of vibration in an infinitely small arc is approximately 
 
 t = I 
 
 l + ^sin^l 
 From the theory of the pendulum the acceleration of gravity is 
 
 Since a dyne is the force which imparts unit acceleration to unit 
 mass, and gravity imparts g units of acceleration, its intensity in 
 dynes is represented by the numerical value of g. 
 
 This result might be further corrected for the buoyancy of 
 the air and the mass of the suspension wire ; but these are both 
 very small, and moreover are of opposite sign, and they may be 
 neglected. The error due to motion of the support is difficult 
 of determination. 
 
 If the formula without corrections, 
 
 ^ = -2"' 
 
 is used, the errors may be one part in two thousand. 
 
 A pendulum about four meters long will swing in about two 
 seconds, and will be convenient when the clock signals are given 
 once in two seconds, as is frequently the case. 
 
 65. Periodic Time by the Method of Coincidences Arrange 
 
 a telescope with its line of sight at right angles to the line of 
 vibration, so that the vertical wire in the field of view coincides 
 with the pendulum wire when it is at rest. The vibrations are 
 
METHOD OF COINCIDENCES 83 
 
 then compared with the audible beats of the seconds pendulum 
 of a standard clock, or, better^ with the ticking of a sounder in 
 connection with a break-circuit clock or chronometer. For many 
 purposes this latter arrangement is very useful, as sounders may 
 be placed in aU parts of a laboratory to distribute accurate time 
 signals. In the absence of these devices an ordinary watch 
 having a second-hand can be made to serve satisfactorily. An 
 assistant looking at the second-hand may count the seconds to 
 himself, and in any convenient manner beat them audibly for 
 the observer. 
 
 The comparisons between the pendulum and the clock can 
 best be made by the valuable method of coincidences, which is 
 useful for comparing two periodic phenomena whose periods are 
 nearly equal. Let the pendulum be of such length that its 
 period is very nearly one second. If the signals occur every 
 second, a time will come when the pendulum wiU pass the 
 center of its swing, as seen in the telescope, exactly as one of 
 the signals is given; the transit of the pendulum and the signal 
 coincide. But at the next swing they will not coincide, as the 
 period of the pendulum differs from the interval between 
 the two signals. This lack of coincidence will increase until 
 the transit occurs midway between the two signals ; then it will 
 apparently decrease until coincidence again occurs. The inter- 
 val between these two coincidences should be determined as 
 accurately as possible ; it will be an integral number of seconds. 
 In this interval the number of vibrations of the pendulimi is 
 one more or less than the number of time signals. Hence the 
 period is obtained by dividing the number of seconds by this 
 number plus or minus one, according as the pendulum is shorter 
 or longer than the seconds pendulum. Whether the pendulum 
 gains or loses on the time signals may be determined by a simple 
 auxiliary observation. If there appears to be coincidence at 
 several transits, note the first and last, and consider the mean 
 as the true time of coincidence. Observe three or more coin- 
 cidence intervals, and compute the period from each independ- 
 ently. If the rate of the clock is known, and it exceeds one 
 second per day, a further correction should be made because of 
 

 
 March 5, 1902 
 
 Time of Coincidenc3 
 
 Interval 
 
 11" SSm 
 
 15« 
 
 10™ 108 
 
 12 8 
 12 18 
 12 28 
 
 25 
 20 
 19 
 
 9 55 
 9 59 
 
 84 MECHANICS 
 
 it ; if the clock is gaining, the computed period is too large. A 
 numerical example of a gravity determination follows. 
 
 Gravity with the Simple Pendulum 
 Riefler Mean-Time Clock 
 
 Period 
 
 f|f = 0.99836 s 
 III = 0.99832 
 IJf = 0.99833 
 
 Mean = 0.99834 
 Rate of clock known to be less than 0.5 s per day. 
 
 Average arc = 8°. Corrected period, t = 0.99830 s 
 From the point of support to bottom of sphere, 101.41 cm 
 
 Radius of sphere, 2.45 
 
 Length of equivalent simple pendulum, I — 98.985 
 
 '^ = ^ = 980.30^. 
 ^ fi s2 
 
 If the pendulum has a period of neariy two seconds, and the 
 coincidences of its transits with a time signal occurring once in 
 two seconds are observed, the number of vibrations made in the 
 coincidence interval is half the number of seconds, plus or minus 
 one. Otherwise the method is as above described. 
 
 XXV. GRAVITY WITH THE REVERSIBLE PENDULUM 
 
 Determine the intensity of gravity. 
 
 66. Kater'5 Pendulum. — The reversible pendulum has the 
 advantage over other forms, that it is easier to obtain an accurate 
 value of the length of the equivalent simple pendulum. It con- 
 sists of a rigid bar (Fig. 34) with a fixed knife-edge near each 
 end. A heavy mass is fastened at one end, and lighter masses 
 are adjustable along its length. The pendulum is to be swung 
 first upon one knife-edge and then upon the other, and the 
 adjustable masses moved nearer that knife-edge about which 
 
THE REVERSIBLE PENDULUM 
 
 85 
 
 A 
 
 IS 
 
 the time of swing is greater, until the times of vibration about 
 the two knife-edges become equal. When this condition is 
 secured the distance between the knife-edges is equal to the 
 length of an ideal pendulum of equal period. 
 
 To determine this distance the pendulum may be placed upon 
 a comparator (Art. 22) and the microscopes set upon the knife- 
 edges ; a standard scale is substituted for the 
 pendulum and the length read directly. If 
 proper apparatus is available, the length may 
 be determined with a cathetometer while the 
 pendulum is suspended. But probably the 
 best method is to provide a steel rod with a 
 screw end, whose length can be adjusted to 
 reach from one knife-edge to the other while 
 the pendulum is suspended. The length of 
 this rod is then measured with an end com- 
 parator, or in any other convenient manner. 
 
 The period may be determined by the 
 method of coincidences described in Art. 65. 
 The comparisons with the standard clock 
 will be facilitated by the use of the optical 
 method of Art. 69, or by making the com- 
 parison with the chronograph (Art. 68). For 
 the latter method a platinum wire is attached 
 to each end of the pendulum, so that when 
 it is stationary the wire at the lower end 
 just touches a small globule of mercury 
 held in a small tube, completing an electric 
 circuit through the chronograph. The 
 swinging of the pendulum may thus be recorded, 
 contact does not occur exactly at the middle of the swing, 
 the resulting error will be eliminated by considering only 
 intervals containing an even number of vibrations. It is advan- 
 tageous to have an adjustment by which the mercury globule 
 can be withdrawn from the contact wire after one coincidence, 
 to be replaced when another approaches. The pendulum may 
 be allowed to swing during several coincidence intervals, and 
 
 Fig. 34. Risversible 
 Pendulum 
 
 If the 
 
86 MECHAjnCS 
 
 by taking the mean, the uncertainty due to the difficulty of 
 determining the exact coincidence will be somewhat reduced. 
 The period thus determined should be corrected for the ampli- 
 tude of swing by the formula given in Art. 64. 
 
 If the period of the pendulum about one knife-edge is exactly 
 equal to that about the other, the time of a single oscillation, 
 reduced to an infinitely small arc, being t, and the distance 
 between the knife-edges being I, the acceleration of gravity is 
 
 67. Formula for the Compound Pendulum. — Instead of con- 
 tinuing the adjustments until exact reversal is secured, when 
 the times are approximately equal the following formula may 
 be applied. Let \ and h^ be the distances from the center of 
 gravity to the knife-edges, about which the times of vibration 
 are respectively t-y and t^. Then 
 
 g 21 2(Ai-/g' 
 
 The center of gravity can be determined with sufficient exact- 
 ness by balancing the pendulum upon a metal rod and measuring 
 the distances hy and h^ with a meter stick. 
 
 Further corrections" for the amplitude of swing, motion of the 
 support, the drag of the air, etc., may be made when extreme 
 precision is required, by methods described in the references. 
 
 Refebences. — Stewart and Gee, Practical Physics, Vol. I, pp. 247-256 ; 
 Bouth, Eigid Dynamics, pp. 72-85 ; Price, Integral Calculus, Vol. Ill, p. 582 ; 
 Vol. IV, p. 290. 
 
 68. The Chronograph. — When it is desired to make a per- 
 manent record of the time of the occurrence of a phenomenon, 
 or of its duration, some form of chronograph may be used. This 
 may print the time in plain figures, or the interval of time may 
 be determined by comparison with the vibrations of a tuning 
 fork (Art. 71), or by measuring a length. ■ The latter form is 
 the more common. It consists of a cylinder (Fig. 35) covered 
 
THE CHRONOGEAPH 
 
 87 
 
 with paper, which is made to revolve upon its axis at a uniform 
 rate, usually one turn per minute. A pen carried on the arma- 
 ture of an electromagnet is moved slowly along the length of 
 the cylinder as it revolves, tracing a spiral mark upon the 
 
 Fig. 35. Ctlindek Chkonogbaph 
 
 paper. Making or breaking an electric circuit through this 
 magnet causes a little indentation in the trace, as represented 
 in Fig. 36, which shows part of a chronograph record. The 
 reading at a is 2" 26'° 4'.7. 
 
 The chronograph is used in connection with a standard time- 
 piece, having an attachment for either momentarily making or 
 breaking the circuit at intervals of one or two seconds. To 
 
 ^ 
 
 2h 30m Os 
 
 ^ 
 
 
 
 ^ 
 
 
 
 „ 
 
 
 ^ 
 
 
 lOs 
 
 „ 
 
 
 ^ 
 
 
 
 
 
 
 „ 
 
 
 „ 
 
 
 „ 
 
 
 
 
 
 
 ,f^ 
 
 
 ^ 
 
 
 
 „ 
 
 
 ^ 
 
 
 
 
 
 
 
 
 „ 
 
 
 
 „ 
 
 
 
 
 
 „ 
 
 finnr 
 
 innn 
 
 — ^ 
 
 3, 
 
 » n 
 
 „ 
 
 
 ^ 
 
 ^ 
 
 
 
 
 „ 
 
 
 „ 
 
 
 n 
 
 „ 
 
 
 
 2h2Sin0s 
 
 Fig. 36. Ebcobd of Cylinder Chronograph 
 
 indicate the beginning of each minute there may be some pecu- 
 liarity in the circuit breaking. If the signals occur at intervals 
 of one second, the fifty-ninth in each minute may be omitted ; 
 while for an interval of two seconds, the odd second's signals, 
 except the fifty-ninth in each minute, may be omitted. 
 
88 
 
 MECHANICS 
 
 A signal key, either to be held in the hand or attached to 
 any apparatus as a pendulum, is included in the pen circuit, 
 and permits the recording of the time of any phenomenon along 
 with the clock signals. From this record the actual time of the 
 occurrence, or the interval of time between two phenomena, can 
 be determined by comparing the positions of the special signal 
 marks with the nearest second's mark. A scale which divides 
 the interval between two successive clock signals into parts cor- 
 responding to tenths of a second is often convenient for measur- 
 ing a chronograph record. 
 
 Two pens may be used, one for the clock signals only, and 
 the other for the observation signals. Sometimes provision 
 is made for rotating the drum at higher rates, as one turn in 
 
 Fig. 37. Flashing Appabatus 
 
 ten seconds, or even one turn in one second. It is difficult to 
 maintain uniform rotation at these high speeds. Observations 
 which would require a rapid rotation, if the ordinary chroiio- 
 graph is employed, are usually better made with the tuning- 
 fork chronograph. 
 
 69. Mendenhall's Optical Method for comparing Two Pendu- 
 lums. — This method employs the principle of coincidences 
 (Art. 65), the coincidences being easily and accurately observed 
 by optical means. 
 
 The standard pendulum should have a break-circuit attach- 
 ment, which produces a flashing line signal of the pendulum 
 beats in the following manner. The clock breaks the primary 
 circuit of an induction coil, the secondary circuit of which is 
 connected to a Pliioker tube (Fig. 37). As the a,mount of 
 current required in the primary of the coil would be likely 
 
MENDENHALL'S METHOD 89 
 
 to ruin the break-circuit mechanism, or to interrupt the clock, 
 a relay is introduced. In the clock circuit a single cell of 
 gravity battery is sufficient, while the coil circuit may contain 
 one or more storage cells, or a few dry cells, for this experiment. 
 When properly adjusted each beat of the pendulum will cause 
 the tube to flash. 
 
 If the pendulum breaks the circuit by swinging through a 
 globule of mercury, a large current will do no injury, and the 
 relay may be dispensed with. Instead of the coil and tube, an 
 electrically controlled ^..^^ 
 
 shutter may illuminate a ^^lisj' J-'''fl^- 
 
 slit with flashes of light. ^\~\ - - ' ' ' iH-^ — 
 
 A small mirror, perhaps ,,--'' 
 
 2cm wide and ,--''' 
 
 5cm long, is 3-^'"" I^">- 38. Pendulum for 
 
 , T , Jcl,.^ — Mendenhall's Method 
 
 attached to t»--^ 
 
 the pendulum to be compared near its point of sup- 
 port, the normal to the mirror lying in the plane of 
 vibration. A second nlirror, m (Fig. 38), is held by 
 the stationary support of the pendulum at one side 
 of the flrst mirror, close to it, and in the same plane 
 with it when the pendulum is at rest. The capillary 
 part of the Pliicker tube, T, is placed horizontally so 
 that it may be seen reflected in the two mirrors by 
 means of a telescope, R. The image should appear 
 straight and unbroken, one mirror being adjusted to 
 secure this if necessary. If, now, the tube flashing with the 
 clock beats, the pendulum is set in vibration, the two parts of 
 the image of the flash made by the mirrors will appear discontinu- 
 ous except when the two pendulums coincide ; this coincidence 
 is thus determined by noting that flash which gives a perfectly 
 continuous image. 
 
 When this method is used with the reversible pendulum in 
 addition to the mirror on the support, two are required on the 
 pendulum, one near each knife-edge. 
 
 A numerical example is given of the determination of the 
 period of a pendulum to be used in Exercise LIV. 
 
90 MECHANICS 
 
 Period of a Pendulum; Mendenhall's Method 
 
 Time from Standard Self-winding Clock 
 
 Jii^y 26, 1902 
 
 An auxiliary pendulum was compared with the standard clock. Pen- 
 dulum gaining: as nearly as can be judged, it makes 14 vibrations in 13. 
 seconds. 
 
 COINCinENCES 
 
 5h 26" 108 
 
 
 Number of Inteb- 
 vals between 
 Obsekvations 
 
 27 
 
 15 
 
 
 5 
 
 
 29 
 
 14 
 
 
 9 
 
 Period 
 
 30 
 
 30 
 40 
 09 
 
 
 6 
 10 ' = 
 16 
 
 858 
 
 — \ 
 
 32 
 36 
 
 858 + 66 
 
 38 
 
 33 
 
 
 11 
 
 
 40 
 
 28 
 
 
 _9 
 
 
 Interval 14™ 
 
 188 = 
 
 858= 
 
 Total 66 
 
 
 0.9286 s 
 
 XXVI. GRAVITY BY FREE FALL 
 
 Determine the acceleration of gravity by observing a freely falling body. 
 
 70. Freely Falling Body. — An iron ball is supported by an 
 electromagnet and released by a key which also records a 
 signal upon an electric high-speed chronograph (Art. 68) or 
 upon a tuning-fork chronograph (Art. 71). A contact piece is 
 placed below by which a second signal is recorded when the 
 ball strikes, thus permitting the time of fall to be determined. 
 The contact may be placed at several known distances below 
 the starting point of the ball, as 100, 200, and 300 cm ; and 
 with the results the equations of accelerated motion given in 
 Art. 59 may be verified. The method is more useful for a ball 
 dropped from the top of a tall shaft, the acceleration of gravity 
 being directly obtained. The distance fallen is measured with 
 a steel tape, the temperature correction being applied. 
 
 If I is the distance fallen in the time t, the acceleration of 
 gravity is 2 ; 
 
 .9 = l2-- 
 
THE TUNING-FORK CHEONOGRAPH 91 
 
 71. The Tuning-Fork Chronograph For accurately deter- 
 mining an interval of time not exceeding a few seconds in 
 duration, the tuning-fork chronograph is suitable. It is a 
 machine by means of which a piece of smoked paper is moved 
 under a style attached to one prong of a tuning fork of known 
 vibration number, the vibrations of the fork causing the style to 
 trace a sinuous line upon the paper. The paper may be attached 
 to the surface of a rotating cylinder or may be in the form of a 
 ribbon upon reels. There are two methods by which an obser- 
 vation may be recorded. 
 The cylinder and the fork 
 may be connected to the 
 terminals of the secondary 
 
 circuit of an induction ^'''- ^^- Record of Tuning-Fork 
 
 -1 .!_ . 1 • Chronograph 
 
 coil, so that makmg or 
 
 breaking the primary circuit causes a spark to pass from the 
 style through the paper to the cylinder, producing a record. Or 
 a second style attached to the armature of an electromagnet may 
 trace a line beside the sinuous mark of the fork, and making or 
 breaking the circuit through this magnet will produce an inden- 
 tation in the line. Fig. 39 shows such a record made by an 
 instrument having two recording styles, one on either side of 
 the fork. Lines are drawn from the notches on one side of the 
 wave to those on the other side; by counting the number of 
 waves between two cross lines, the time interval is determined 
 with great precision. 
 
 A fork commonly used giyes one hundred complete vibra- 
 tions per second, and is maintained in motion automatically 
 by an electromagnetic attachment. The paper may be moved 
 by clockwork, by hand, or in any other convenient manner. 
 Uniform motion is desirable but not essential. 
 
 In the absence of a tuning-fork chronograph, a pen chrono- 
 graph, arranged for a high-speed rotation of the cylinder, may 
 be used (Art. 68). 
 
 72. Smoked Pap'er and Glass. — Glass or paper may be smoked, 
 as is required in many experiments, by passing it through the 
 smoking flame of a candle or, better, by holding it in the, smoke 
 
92 MECHANICS 
 
 produced by burning camphor gum. Usually a very thin 
 coating is preferable to a thick one. 
 
 A paper ribbon may be conveniently smoked by passing it 
 over the surface of a brass cylinder iilled with cold water, while 
 a smoking kerosene flame plays against it. The water absorbs 
 the heat, preventing burning and condensing the smoke. 
 
CHAPTER IV 
 ELASTICITY, AISTD PROPEETIES OF MATTER 
 
 XXVn. YOUNG'S MODULUS BY STRETCHING 
 
 (a) Determine the modulus of elasticity and the elastic limit of a short 
 iron wire. 
 
 (b) Determine the modulus of elasticity of a long steel wire. 
 
 73. Coefficients of Elasticity. — The elasticity of an isotropic 
 body under any form of stress may be described with the aid of 
 two coeificients or moduli of elasticity, the modulus of bulk 
 elasticity, the reciprocal of which is the compressibility, and the 
 modulus of elasticity of shape or the modulus of rigidity. Only 
 solids possess the latter form of elasticity, hence rigidity is 
 chiefly used in describing them ; the modulus of bulk elasticity 
 is used for fluids, though it applies also to solids. 
 
 Besides the above a somewhat simpler modulus is frequently 
 employed, called Young's modulus; it is the modulus for simple 
 longitudinal strain, in which change of lateral dimensions is not 
 considered. 
 
 Each modulus is measured by the ratio of the applied stress 
 to the resulting strain. The bulk modulus is denoted by K, 
 the modulus of rigidity by -B,,and Young's modulus by M. 
 
 When a varying stress not exceeding a certain limit is 
 applied to a solid, the resulting strain is proportional to the 
 stress. That limiting value of the stress beyond which the 
 strain is no longer proportional to the stress is called the elastic 
 limit of the substance. A body subjected to a stress greater 
 than the elastic limit is permanently deformed. 
 
 74. Young's Modulus and Elastic Limit. — Attach the wire to 
 be tested to a comparator, in the manner shown in Fig 40. The 
 
 93 
 
94 
 
 MECHANICS 
 
 wire is held at one end by a firm screw, S, which permits of 
 longitudinal adjustment; while the other end is attached to a 
 right-angled arm in order that the weight of a load, F, placed 
 in the pan shall be transmitted as a horizontal stress on the 
 wire. Sufficient load should be placed in the pan to take the 
 kinks out of the wire. An index microscope, X, is adjusted upon 
 a fine mark placed near the end of the wire, and a micrometer 
 microscope, M, is set upon a mark at the other end, about a meter 
 distant. Note the reading of the micrometer microscope. Place 
 a weight, for instance a kilogram, in the pan ; readjust the mark 
 under the index microscope by the screw S, set the microm- 
 eter microscope again on the mark, and note reading. The 
 
 5 -^iT l|nni 
 
 i"i&. 40. Apparatus fob MEASnEiuG Elongation 
 
 J^ difference between the first and second readings will 
 give the elongation produced in the length of wire 
 between the two marks by a weight of one kilogram. Remove 
 the kilogram, readjust the screw, and repeat the zero reading. 
 Make the load two kilograms, then one kilogram; three kilo- 
 grams, and then two ; adjusting the screw S, and observing the 
 elongation produced in each case. In this manner increase the 
 load a kilogram at a time, removing the last added kilogram to 
 determine whether the wire has been permanently stretched. 
 Continue until the elastic limit is reached. 
 
 Plot the results, the abscissae representing the stresses, and 
 the ordinates the strains. 
 
 From the observations within the elastic limit find the elon- 
 gation produced by each kilogram of change in the load. 
 
YOUNG'S MODULUS 
 
 95 
 
 expressing the mean value in centimeters (Art. 21). This is 
 the quantity s of the formula, and F is the load producing the 
 stretch, in this case 1000 g. Measure a number (ten) of diam- 
 eters of the wire in different directions and in different parts, 
 and let r be the mean radius. With a meter bar placed upon 
 the table of the comparator determine the original length, I, of 
 the wire between the marks. 
 
 Fa 
 Then the stress in dynes per square centimeter is -—, and 
 
 g 
 
 the strain per unit length is -• Young's modulus is 
 
 Trr" 
 
 M 
 
 _ stress _ Fgl 
 
 strain 
 
 The report may be in the following form. 
 
 Young's MoDtri.us or Annealed Iron Wike 
 
 WITH A MICEOMETER-MICKOSCOPE COMPARATOE 
 
 August 15, 1902 
 
 Loads, grams 
 
 Eeadings 
 
 Averages 
 
 Elongatioks 
 
 Diameters 
 
 2100 + 
 
 1st 2(1 
 
 Turns 
 
 per 1000 g 
 
 cm 
 
 
 
 0.00 0.08 
 
 0.04 
 
 
 
 1000 
 
 2.67 2.69 
 
 2.68 
 
 2.64 
 
 0.0568 
 
 2000 
 
 5.63 5.50 
 
 5.56 
 
 2.88 
 
 9 
 
 3000 
 
 8.62 8.50 
 
 8.56 
 
 3.00 
 
 5 
 
 4000 
 
 11.72 11.84 
 
 11.78 
 
 3.22 
 
 9 
 
 5000 
 
 14.66 14.60 
 
 14.63 
 
 2.85 
 
 5 
 
 6000 
 
 17.46 17.44 
 
 17.45 
 
 2.82 
 
 8 
 
 7000 
 
 20.36 20.41 
 
 .20.39 
 
 2.94 
 
 6 
 
 8000 
 
 23.69 61.4 
 
 elastic limit 
 
 
 7 
 
 9000 
 
 61.4 
 
 
 
 9 
 
 10000 
 
 wire broke 
 
 
 
 6 
 
 Means 
 
 2.907 
 
 0.0567 
 
96 
 
 MECHANICS 
 
 Micrometer microscope, No. 2, 169.7 turns per cm. 
 
 2 907 
 Elongation, s = 7^777-;^ = 0.01712 cm. Load, F= 1000 g. 
 
 Radius, 
 
 169.7 
 r = 0.0284 cm. 
 
 M= — = 2.02 X 1012. 
 irr's 
 
 Length, Z = 89.3 cm. 
 
 10.000 
 Load 
 grams 
 
 — 
 
 ^ 
 
 
 n 
 
 
 ■^ 
 
 ^~ 
 
 
 '~~ 
 
 
 
 
 
 ~ 
 
 r— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V~ 
 
 
 
 , 
 
 ~ 
 
 "~ 
 
 
 
 
 
 ~ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■" 
 
 
 
 
 
 
 
 
 SiOOO 
 
 
 
 
 
 
 "~ 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~~ 
 
 ~^ 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 ) 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 El 
 
 on 
 
 9" 
 
 ion 
 
 
 20 Turns 0/ Screw 
 
 Plot of the Observations 
 
 75. Young's Modulus with a Suspended Wire and Scale. — 
 
 Suspend from the same support two pieces of wire, each from 
 3 m to 10 m long, and to one piece attach at its lower end 
 a millimeter scale and a small weight to keep it stretched. 
 To the other wire fasten a vernier arranged to move over the 
 scale, and below all a pan for weights. Measure with a steel 
 tape the length of the wire to be stretched when there is suf- 
 ficient weight in the pan to keep it straight. Then proceed to 
 determine the elongation produced by the successive addition of 
 weights in the manner described in the preceding article. The 
 modulus of elasticity is calculated by the same formula. 
 Refebence. — Glazehrook and, Shaw, Practical Physics, p. 141. 
 
 76. Young's Modulus with the Cathetometer. — Suspend ver- 
 tically from a firm support a wire about a meter long, to the 
 lower end of which is attached a pan for weights. The elon- 
 gation produced by the addition of weights is then measured 
 with the cathetometer (Art. 77), having one or two reading 
 telescopes. Needles may be fastened to the wire with wax, to 
 serve as indices. The method of procedure and calculation is 
 as described in Art. 74. 
 
 Reference. — Nichols, A Laboratory Manual of Physics, Vol. I, p. 74. 
 
THE CATHETOMETER 
 
 97 
 
 ^ 
 
 77. Adjustment of the Cathetometer A cathetometer (Fig. 
 
 41) is essentially a vertical scale carrying a horizontal telescope 
 capable of motion up and down, the amount of this motion being 
 determined by a vernier attached to the telescope. The follow- 
 ing are the adjustments usually required. The second and third, 
 having once been carefully made, should not thereafter be dis- 
 turbed, so that ordinarily it is necessary only, 
 to adjust for parallax, to set the scale vertical 
 and the telescope horizontal. 
 
 To adjust for Parallax. — Adjust the sliding 
 eyepiece until the cross wires are seen with 
 perfect distinctness ; then focus by the rack 
 and pinion, or long draw tube, upon the ob- 
 ject or mark. If, now, a motion of the eye 
 produces a relative motion of the cross wires 
 and the image of the object, the preceding 
 adjustments must be altered and the process 
 repeated until no such motion is produced by 
 moving the eye, and both the cross wires and 
 object are seen distinctly. 
 
 To adjust the Line of Collimation. — Set the 
 cross wires exactly upon some well-defined 
 point, and rotate the telescope upon its own 
 axis. If the cross wires move away from the 
 point, they mtist be brought back partly by 
 adjusting the reticule held by small screws in 
 the eyepiece and partly by altering the direc- 
 tion of the telescope. Repeat this process until the cross wires 
 remain exactly upon the point during a complete rotation of 
 the telescope. 
 
 To set the Level Parallel to the Optical Axis of the Telescope. 
 — Move the telescope until the level bubble is central. Reverse 
 the telescope with the attached level in its Y's. If the bubble 
 is not central it must be made so, partly by means of the screws 
 that attach it to the telescope and partly by moving the whole 
 telescope. Repeat this operation until the bubble remains cen- 
 tral, when the telescope is reversed. 
 
 Fig. 41 
 Cathetometer 
 
98 
 
 MECHANICS 
 
 To set the Scale Vertical and the Telescope Horizontal. — Turn 
 the column till the level is parallel to the line joining two of 
 the leveling screws in the base, and make the bubble central. 
 Turn the column through 180°, and if the telescope is not level, 
 adjust it, partly by means of the leveling screws and partly by 
 altering the angle between telescope- and scale. Turn the col- 
 umn through 90° and level by the third leveling screw. Repeat 
 the whole process until the bubble remains central during an 
 entire revolution of the column. 
 
 Reference. — Stewart and Gee, Practical Physics, Vol. I, pp. 27-35. 
 
 XXVIII. MODULUS OF ELASTICITY BY FLEXURE 
 
 Determine the modulus of elasticity of a rectangular steel bar when it is 
 resting on its broader side; also when resting on its edge. Make the 
 same measurements for a wood bar. Find the modulus of elasticity 
 of a round brass rod. 
 
 78. Flexure of a Rectangular Bar supported at Both Ends. — 
 The ends of the bar are supported upon knife-edges resting on 
 
MODULUS OF RIGIDITY 99 
 
 a rigid bedplate, as shown in Fig. 42. Midway between the 
 knife-edges there hangs upon the bar a hook or pan for carrying 
 weights. Above the bar, supported by this hook, is a divided 
 scale or a fine wire, which serves as an index. Attached to the 
 bedplate is an adjustable microscope with which the flexure 
 may be measured. 
 
 Adjust the microscope and observe the reading. Add a weight 
 (500 g, more or less, according to the strength of the beam) and 
 determine the flexure produced; add weights, one at a time 
 untU. five have been used, observing the increase of flexure 
 produced by each one. 
 
 Let P be the value of each of the weights expressed in grams, 
 F the average deflection produced by this load in its successive 
 additions expressed in centimeters, L the. distance between the 
 supports of the bar, a the vertical dimension of the cross section 
 of the bar, and h the horizontal dimension ; then the modulus of 
 elasticity of the bar is p rs 
 
 M^—^ — 
 
 4:Fa^ 
 
 79. Bar supported at One End ; Cylindrical Bar. — If the bar 
 
 is fixed at one end and the weight is applied at the other end, 
 
 the modulus is a p^jz 
 
 ^= ^• 
 
 Fa% 
 
 If the section of the bar in either case is a circle of radius r, 
 instead of a% substitute 3 Trr*. 
 
 Keperences. — Stewart and Gee, Practical Physics, Vol. I, pp. 179-186; 
 KoMrausch, Physical Measurements, pp. 128-130. 
 
 XXIX. COEFFICIENT OF RIGIDITY WITH THE TORSION LATHE 
 
 Determine the coeflBcients of rigidity of one- steel rod and of two brass rods 
 of different diameters. 
 
 80. The Modulus of Rigidity. — When a substance is in the 
 form of a solid rod its modulus of rigidity may be determined 
 from measures of its torsion. If a rod of length I and radius r 
 
100 MECHANICS 
 
 is twisted through an angle 6 (radians) by a couple C (gram- 
 centimeters), the modulus of rigidity is 
 
 2gCl 
 
 E- 
 
 Ti^e 
 
 Clamp the ends of a metal rod, from 3 mm to 6 mm in diam- 
 eter, in the chucks of the torsion lathe, so that the length of the 
 rod to be twisted shall be about a meter. One chuck is rigidly 
 attached to the base, while the other is capable of rotation about 
 the axis of the rod by means of a large pulley (Fig. 43), the 
 amount of rotation being determined by a divided circle. This 
 circle may be divided into two hundred parts, when, since the 
 circumference equals 2 ir radians, each division is 0.01 tt radians. 
 
 Apparatus fob Measuring Torsion 
 
 Place a 500 g weight in each of the two pans, 
 which are attached by cords to the pulley so that they act as a 
 couple to twist the rod. Observe the amount of twist produced. 
 To eliminate the effects of friction disturb the cii-cle from its posi- 
 tion of rest several (five) times, reading its indications each time, 
 and take the average position as a single measure. Add a second 
 500 g weight to each pan, then a third, and then remove the 
 weights one at a time ; determine the twist resulting from each 
 change of couple. Find the average effect. The. couple is 
 measured by the mass, 1000 g as described, multiplied by the 
 radius of the pulley. 
 
 Rods of the same material, of equal lengths but of different 
 radii, of the same radius but of different lengths, and rods of 
 various materials, should be experimented with. In this manner 
 all the laws of torsion and rigidity may be demonstrated. 
 
 81. Simple Torsion Apparatus The laws of torsion may be 
 
 illustrated by simple apparatus, the rod or wire being suspended 
 
TORSION AND RIGIDITY 
 
 101 
 
 m> 
 
 MP 
 
 CZ) 
 
 vertically in a firm clamp, which is attached to the wall or sup- 
 ported on a tripod. To the bottom is fastened a heavy weight 
 carrying a pointer moving over a 
 divided circle of paper (Fig. 44). 
 Two spring balances or weights 
 are attached to cords passing 
 around the weight, acting as a 
 couple, of any desired moment, 
 tending to twist the wire. 
 Pointers attached to the wire by 
 wax and moving over divided 
 circles, placed at various posi- 
 tions along its length, may be 
 used to show that the angle of 
 torsion varies as the length of 
 the wire. To prove the relation 
 of torsion to the radius of the 
 wire, fasten together two pieces 
 of unequal diameter and of 
 equal length, and suspend as 
 before with a pointer attached to the bottom of each piece. 
 The respective angles of torsion should satisfy the formula, 
 
 2gCl 
 
 Tig. 44. Simple Torsion 
 Appakatus 
 
 = 
 
 ■Er^ 
 
 being the angle of torsion of a wire whose length is I, radius r, 
 and rigidity B, which is twisted by a couple C. 
 
 If is expressed in radians, the modulus of rigidity, S, may 
 be computed as described in the preceding article. 
 
 XXX. MODULUS OF TORSION BY THE TORSION PENDULUM 
 
 Determine the modulus of torsion of several wires with the torsion pendulum. 
 
 82. Modulus of Torsion. — Suspend a heavy weight of some 
 simple, regular, geometrical shape by a fine wire (Fig. 45), the 
 upper end of the wire being rigidly clamped. Turn the weight 
 
102 
 
 MECHANICS 
 
 through an angle of 30°, twisting the wire, and release it ; it wUl 
 oscillate with a rotary motion, and constitutes a torsion pendu- 
 lum. The formula expressing the condition of vibration is 
 
 T 
 
 = 2 7rV-' 
 
 in which T is the time of a complete vibration, / the moment of 
 inertia of the weight, and r the modulus of torsion of the wire. 
 The time of vibration is to be determined by the method of the 
 following article, and the moment of inertia 
 may be calculated by the formula given in 
 Art. 85. 
 
 83, Periodic Time by the Method of Transits. 
 — This method of determining the time of a 
 slow periodic movement is so valuable that 
 it will be described in detail. Adjust a 
 telescope so that the vertical cross wire is 
 seen in the prolongation of the torsion wire 
 and covers an index mark placed on the edge 
 of the disk when it is stationary. Cause 
 the disk to vibrate, stopping any pendular 
 vibration with the hand, and the index will 
 move back and forth in the field of view, 
 crossing the wire twice in each complete 
 vibration. Let one observer, watching the 
 vibration, give the signal "tick" each time 
 the index crosses the wire, for ten successive transits ; while 
 another observer records the time of each signal from a clock, 
 chronometer, or watch, noting the day, hour, minute, second, 
 and tenths of a second. With a clock beating seconds or a 
 chronometer beating half seconds, a single observer can note 
 the time by the eye-and-ear method, observing the index at the 
 second just before and just after transit to assist in estimating 
 the tenths of seconds. Repeat the series of ten readings 
 twice, at intervals of fifteen or twenty minutes, being careful 
 that the disk swings in the same direction at the beginning of 
 each series. , 
 
 Fig. 45 
 
 TOKSION PENDULtrM 
 
PERIODIC TIME BY TRANSITS 103 
 
 Add together the fifth and sixth times of transit of each set ; 
 also the fourth and seventh, the third and eighth, the second 
 and ninth, and the first and tenth. Divide each sum by two, 
 and average the five quotients of each series. These results will 
 give the times of the middle elongations, or the times when the 
 index was at its greatest distance from the cross wires between 
 the fifth and sixth transits of each series. Dividing the interval 
 between the two middle elongatipns by the number of vibrations 
 win give the periodic time with great accuracy. The number 
 of vibrations may be determined from the observations without 
 the trouble of counting. Between the first and ninth, or second 
 and tenth transits of each set, there were four complete vibra- 
 tions, and this interval divided by four is approximately the 
 periodic time. If all the observations were exact, the period 
 thus found would be exact. To reduce the unavoidable errors 
 of observation it is desirable to find the period from a longer 
 interval. The interval between any two elongations would 
 contain the exact period an integral number of times, provided 
 the disk moved in the same direction at the beginning of each 
 set. The quotient will be a whole number plus a half, if the 
 disk moved in opposite directions. Divide the interval between 
 two elongations by the approximate period; in general the 
 result wiU not be an integer, but will be so near a whole 
 number that there will be no doubt as to the true number of 
 vibrations. Having thus found the number of vibrations in 
 the long interval, the interval divided by this number wiU give 
 a more precise value of the period. 
 
 The length of time to be allowed between the sets of observa- 
 tions will depend upon the accuracy of the work and the peri- 
 odic time ; other things being equal, the longer the interval the 
 more exact the final result. Two sets of observations are sufii- 
 cient, but it is better to make three, as directed, to guard against 
 errors. If the interval between the first and third sets is too 
 long to give a sure result, one of the shorter intervals can be 
 used. The middle set may also aid in determining the number 
 of vibrations in the long interval. A numerical example of 
 such a determination will illustrate the method. 
 
104 
 
 MECHANICS 
 
 Period of Torsion Pendulum ; Method of Transits 
 
 Time from Chronometer 
 
 May 2^, 1895 
 
 Transit 
 
 Time 
 
 Blongation 
 
 Transit 
 
 Time 
 
 Elosgatiok 
 
 1 
 
 h m B 
 2:10: 8.2 
 
 
 1 
 
 h m s 
 2:30:51.3 
 
 
 2 
 
 2 : 10 : 14.6 
 
 
 2 
 
 2 : 30 : 58.0 
 
 
 3 
 
 2 : 10 : 20.8 
 
 
 3 
 
 2:31. 4.1 
 
 
 4 
 
 2 : 10 : 27.0 
 
 
 4 
 
 2 : 31 : 10.5 
 
 
 5 
 6 
 
 2 : 10 : 33.7 
 2 : 10 : 40.0 
 
 h in s 
 5-6 2:10:36.85 
 
 5 
 6 
 
 2:31:16.9 
 2:31 :23.5 
 
 h m s 
 5-6 2:31:20.20 
 
 7 
 
 2 : 10 : 46.9 
 
 4-7 2:10:36.95 
 
 7 
 
 2 : 31 : 30.0 
 
 4-7 2 : 31 : 20.25 
 
 8 
 
 2:10:53.0 
 
 3-8 2:10:36.90 
 
 8 
 
 2 : 31 : 36.3 
 
 3-8 2:31:20.20 
 
 9 
 
 2 : 10 : 59.3 
 
 2-9 2:10:36.95 
 
 9 
 
 2 : 31 : 42.4 
 
 2-9 2 : 31 : 20.20 
 
 10 
 
 2 : 11 : 5.6 
 
 1-10 2 : 10 : 36.90 
 
 10 
 
 2:31:48.9 
 
 1-10 2:31:20.10 
 
 Average 2 : 10: 36.91 
 
 Average 2:31 :20.19 
 
 The interval between the two sets is 20 m 43.28 s, equail to 1243.28 s. 
 The approxiiliate time of vibration, determined from the first and ninth 
 observations of the first set (51.1 -^ 4), is 12.8 s. 
 
 1243.28 H- 12.8 = 97.1+. 
 
 From this result it is certain that there were 97 vibrations in the interval, 
 which gives the final value of the period, 
 
 P = 1243.28 -^ 97 = 12.817 s. 
 
 XXXI. MOMENT OF INERTIA BY THE TORSION PENDULUM 
 
 Determine the moment of inertia of an irregularly shaped body. 
 
 84. Moment of Inertia by Torsion Pendulum If a body is 
 
 suspended by a wire and caused to vibrate as a torsion pendulum, 
 
 47r2' 
 
 its moment of inertia is 
 
MOMENT OF INERTIA 105 
 
 T being the period, and t the modulus of torsion of the wire. 
 To determine the moment of inertia of a body, it may be sus- 
 pended by a wire whose modulus of torsion is known (Art. 82), 
 in such a manner that the axis about which the moment of inertia 
 is required shall be in the prolongation of the wire. The system 
 is then made to vibrate as a torsion pendulum, and its period is 
 determined by the method of Art. 83. 
 
 If the modulus of torsion of the wire is unknown, it will be 
 necessary to make two observations. The period of the pendu- 
 lum is determined when the body, the moment of inertia of 
 which is required, is suspended as described above ; the second 
 observation is for the period of the pendulum when the sus- 
 pended body is one whose moment of inertia is known, by cal- 
 culation from its dimensions (Art. 85) or otherwise. If Jj is the 
 unknown moment of inertia, and Zj the known, and T^ and T^ 
 the respective periods, by elimination between two equations of 
 the previous form, 
 
 Instead of substituting the body with known moment of inertia 
 for the one whose moment of inertia is to be determined, the 
 former may be added to the latter, and the period of the pendu- 
 lum, consisting of the combined bodies, is determined. Let /j 
 be the unknown moment of inertia, and Jj the known moment 
 of inertia, and Tj the period when the first body only is sus- 
 pended, and Tj^2 ^^ period with the combined bodies; then 
 
 7, = /, -^ 
 
 1 2 /•n2 _ ya' 
 
 -' 1 + 2 ^\ 
 
 85. Formulae for Moments of Inertia. — The moment of inertia 
 of a solid cylinder about its axis of figure is 
 
 r being the radius and m the mass. 
 
106 MECHANICS 
 
 For a hollow cylinder about its axis of figure, the moment of 
 inertia is 
 
 I=m ^ ^ ^' , 
 
 rj and r^ being the inner and outer radii of the ring, and m its 
 mass. 
 
 For a sphere of radius r and mass m, the moment of inertia 
 about a diameter is 
 
 2 mr^ 
 
 . ~b~' 
 
 The moment of inertia of a parallelopipedon of mass m, about 
 an axis passing through its center of mass and perpendicular to 
 the face whose edges are a and b, is 
 
 a2 + P 
 Eeference. — Stewart and Gee, Practical Physics, Vol. I, p. 243. 
 
 XXXII. COMPRESSIBILITY OF A LIQUID WITH THE 
 PIEZOMETER 
 
 Determine the compressibility of water. 
 
 86. The Piezometer. — The compressibility of a liquid may be 
 determined with a piezometer, which consists of a strong glass 
 cylinder (Fig. 46) the ends of which are closed by metal caps, 
 so that when the cylinder is filled with water, a great internal 
 pressure may be produced by means of a screw plug. A glass 
 bulb, B, with a narrow, open stem, is provided for containing 
 the liquid whose compressibility is to be determined. This bulb, 
 a thermometer, T, and a manometer, M, are attached to a suitable 
 support, and during the experiment are placed in the water in 
 the cylinder. 
 
 It is necessary to know the volume, V, of the liquid experi- 
 mented upon, and the change in volume, v, which is caused by 
 a change in pressure, P. Carefully weigh the bulb when it is 
 
THE PIEZOMETEE 
 
 107 
 
 empty and dry. Then draw into the stem a thread of mercury 
 which is nearly as long as the stem, and measure the length of 
 the thread. Allow the mercury to drop into the bulb, and again 
 weigh to determine the weight of the mercury. Complete the 
 calibration by the method of Art. 52, finding the volume, c, of one 
 centimeter of length of the tube. Now drive the mercury out 
 of the bulb, and fill it and about half of the stem with water. 
 Immerse the bulb in a water bath; notice 
 the temperature and the exact position of 
 the end of the water column in the stem. 
 Wipe the outside of the bulb and again 
 weigh ; by the formulae of Art. 51 find the 
 volume, F, of the bulb. 
 
 A drop of mercury is placed in the stem 
 for an index, to show the position of the 
 end of the liquid column. The bulb is 
 attached to its support and is placed in the 
 cylinder, which is fiUed with water. The 
 caps are firmly fastened, the vent screw, S, 
 being open. 
 
 The manometer may be a glass tube, Jf, 
 closed at the top and open at the bottom, 
 which is filled with air while it is placed in 
 the water. The pressure, H, of the air in 
 M is equal to the barometric pressure plus 
 the pressure due to the depth of the 
 water. The latter is expressed in terms of" a mercury column 
 by dividing the distance from S to the bottom of M in centi- 
 meters by 13.6. Observe the position of the mercury index, and 
 the length, I, of the air column in M, and the temperature, t. 
 Close the vent and increase the pressure as much as is con- 
 venient, to 5 or 10 atmospheres for instance. Again observe 
 the length, l,^, of the air column, and the position of the index. 
 The volumes of the air may be taken as proportional to the lengths 
 of the column, a proper allowance being made for the change in 
 shape of the tube where it is sealed. It would be more precise 
 if the tube were graduated and calibrated as to volume. 
 
 Fig. 46. Piezometer 
 
108 MECHANICS 
 
 Make ten determinations, and let n be the mean change of 
 the index. Then the apparent change in volume is w = w x c. 
 The change of pressure in dynes is 
 
 P = ^^.^x 13.6x980. 
 '2 
 
 The apparent compressibility is then 
 
 7 v 
 
 PV 
 
 The pressure in the piezometer will decrease the volume of 
 the bulb, notwithstanding its being open, by the same amount 
 as though it were of solid glass. Hence the observed compres- 
 sion is the difference between that of the liquid and the glass. 
 The real compressibility, at the temperature t°, is 
 
 k^ being the eompressibiUty of glass, which is 2.6 x 10 ~" per 
 dyne per square centimeter. 
 
 XXXTTT. BOYLE'S LAW WITH THE U-TUBE 
 
 Verify Boyle's Law for pressures both greater and less than one atmosphere. 
 
 87. Boyle's Law. Pressures greater than One Atmosphere. — 
 An apparatus for verifying Boyle's Law consists of a U-tube 
 about a meter in length, one branch ending in a funnel, the 
 other capable of being closed by a stopcock (Fig. 47). At the 
 bottom is an outlet closed by a stopcock, and between the tubes 
 is a scale. The upper stopcock being open, pour mercury into 
 the tube till it stands at the zero of the scale in both branches ; 
 if too much mercury is introduced, some may be drawn off at the 
 bottom. Close the stopcock at K tightly. Pour mercury in at 
 the funnel till it stands at the height of 10 cm in the open arm. 
 Denote this height by h, and read the height, A', of the column 
 in the left branch. Add mercury to make h 20 cm, and read h'. 
 Continue increasing A by 10 cm at a time till it becomes 100 cm, 
 finding the corresponding values of A'. Then make a set of 
 
BOYLE'S LAW 
 
 109 
 
 K 
 
 D 
 
 measures in reversed order, by drawing off mercury at the bottom 
 so that A is 90 cm, 80 cm, etc. The top of the mercury column 
 is represented in each case by the plane tangent to the meniscus. 
 
 The products of the volumes and the corresponding pressures 
 of the inclosed gas should all be equal provided 
 the temperature has remained constant. The 
 volumes may be taken proportional to the lengths 
 of the tube occupied by the air. Let K be the 
 division of the scale at the top of the inclosed 
 part of the tube (so chosen that it corrects for 
 the distortion of the tube where the stopcock is 
 sealed on) ; then the volumes are represented by 
 K— h'. A further correction may be made by 
 adding a small quantity, e, which represents the 
 volume of air contained between the tangent 
 plane to the mercury meniscus and its actual sur- 
 face. The pressures are equal to the barometer 
 reading plus the differences between heights of 
 the two columns of mercury in the U-tube, P = B 
 + (A — A'). Corrections for the temperature of the 
 mercury are not required if the barometer ajid 
 manometer columns are at nearly the same tem- 
 perature. Record the observations in tabular 
 form (see table on following page), and plot them, 
 using the pressures as abscissae and the volumes 
 as ordinates. 
 
 Pressures less than One Atmosphere. — With 
 the stopcock, K, open, fill the U-tube with mer- 
 cury till the top of the column is about 10 cm 
 below K. Close X, and read the heights, h and 
 h', of the right and left columns. Draw off mer- 
 cury at the bottom to lower the column h by 10 cm, and record 
 the height h'. Continue lowering A 10 cm at a time till it becomes 
 zero, finding the corresponding values of h'. Repeat the meas- 
 ures by increasing h 10cm at a time till it is again 100 cm. Record 
 and plot as described in the preceding article, the pressures in 
 this case being P = B — (h' — h). 
 
 Fig. 47 
 Boyle's Law 
 
110 
 
 MECHANICS 
 
 Other forms of apparatus are frequently used ; for instance, 
 the mercury may be forced from a cistern into the tubes by com- 
 pressed air, or the two tubes may be joined by a flexible tube, or 
 a tube closed at one end may be dipped into a jar of mercury. 
 But with any form the measurements outlined are to be made. 
 
 h 
 
 K 
 
 P=B + (h-h') 
 
 V=K-K+c 
 
 PV 
 
 10 
 20 
 30 
 
 100 
 
 
 
 
 
 XXXIV. ERRORS OF AN ANEROID WITH AN AIR PUMP 
 
 Find the corrections to an aneroid barometer for each centimeter of pres- 
 sure from 65 to 76 cm. 
 
 88. Testing the Aneroid Barometer. — The aneroid barometer 
 has the advantage of portability, and if frequently compared 
 with a mercurial barometer and adjusted, it will serve excel- 
 lently for approximate determinations of altitude. 
 
 The aneroid may be set, by turning the adjusting sprew, so 
 that its reading agrees with that of the mercurial barometer, 
 though this will not be necessary if the error is small. 
 Place it in a receiver (Fig. 48) having a plate-glass 
 cover with a ground joint. The receiver connects 
 through a stopcock with an 
 air pump and with a U-tube -5^ 
 
 I Aneroid \ 
 
 mercury manometer. 
 
 Partially exhaust the 
 receiver till the aneroid 
 indicates exactly an even 
 centimeter (or half-inch) mark on its scale. It may be easier 
 to exhaust a little too much, allowing air slowly to leak in 
 
 Fio. 48. Testing Aneroid 
 
THE ANEROID BAEOMETEE 
 
 111 
 
 till the pressure is that desired. Read the difference in the 
 level of the mercury columns in the manometer with a scale, or, 
 better, with a cathetometer. The actual pressure on the aneroid 
 is measured by the external barometric pressure less the manom- 
 eter pressure. Both of the quantities should be corrected for 
 the temperature of the mercury; but if the barometer reading 
 is corrected at the beginning, usually appreciable errors wUl 
 not be introduced by neglecting the temperature correction to 
 the manometer. 
 
 Reduce the pressure till the aneroid indicates the next point 
 on its scale for which the correction is desired, and repeat the 
 observations. Continue until 'Cach point has been tested. 
 
 If the cover of the receiver is firmly held in place, it will be 
 possible slightly to compress the air and then to test the aneroid 
 at pressures greater than the external atmospheric pressure. 
 
 The corrections are the quantities which must be added to 
 the aneroid indications to make them equal to the actual pressure. 
 Record the work in tabular form as indicated below, and plot a 
 curve of corrections. The table shows the comparison of an aneroid 
 having an inch scale with a barometer having a centimeter scale. 
 
 The temperature corrections to an aneroid must be determined 
 empirically ; therefore the table of corrections can only be relied 
 upon for the temperature of the trial, which should be recorded. 
 
 Calibration op Aneroid No. 1 R 
 February 27, 1902 
 
 
 Barometric reading, 74.30 cm 
 
 ; « = 2r.6; ^ = 
 
 74.04 cm 
 
 Aneroid 
 
 Manom, 
 
 B — Manom. 
 
 B — iViANOM. 
 
 C0RKEUT10N.S 
 
 inches 
 
 cm 
 
 cm 
 
 inches 
 
 inches 
 
 30.00 
 
 
 
 
 
 29.50 
 
 
 
 
 
 29.16 
 
 0.00 
 
 74.04 
 
 29.15 
 
 -0.01 
 
 29.00 
 
 0.43 
 
 73.61 
 
 28.98 
 
 -0.02 
 
 28.50 
 
 
 
 
 
112 
 
 MECHANICS 
 
 XXXV. COEFFICIENT AND LAWS OF FRICTION BY SEVERAL 
 
 METHODS 
 
 Determine the coefficients of friction, both static and kinetic, for pine wood 
 sliding on a horizontal cast-iron plane ; determine the kinetic friction 
 when the pressure is increased by one and by two kilograms. Deter- 
 mine the kinetic friction for the same body on an inclined cast-iron 
 plane ; determine the kinetic friction for a brass block sliding on each 
 of its unequal faces upon the inclined plane. Determine the kinetic 
 rolling friction for a wood and a brass cylinder. 
 
 89. Coefficient of Friction. — Adjust the friction table so that 
 its surface is horizontal, and place upon it the body to be tested, 
 adding weights to make the total pressure three or more kilo- 
 grams. By means of a cord and weights arranged to pull the 
 block in a horizontal direction (Fig. 49) determine what force 
 
 Zl 
 
 ^ 
 
 ] c 
 
 Fig. 49. Fkiction Apparatus 
 
 n. 
 
 D 
 
 ft 
 
 is necessary to start the body from rest. The ratio of this force 
 to the total pressure is the coefficient of static friction. Deter- 
 mine also the force necessary to keep the body in uniform 
 motion after it is started, and the ratio of this to the pressure 
 is the coefficient of kinetic friction. A correction to these obser- 
 vations may be made for the friction of the pulley over which 
 the cord passes, determined as described below. 
 
 Make a number of experiments with different substances, as 
 pine, oak, iron, and brass, for the sliding body. Also make a 
 series of observations to determine the manner in which the 
 friction increases with the increase in pressure between any two 
 given surfaces, by repeating the observations with the pressure 
 
CAPILLARITY 113 
 
 increased by one, two, and three kilograms. Represent the 
 results graphically. 
 
 The work may be simplified by using a spring balance 
 attached to the cord and drawn horizontally. When used in 
 this position a correction must be applied for the index error of 
 the spring balance. The body can be weighed with the spring 
 balance. 
 
 * 
 
 Friction on an Inclined Plane. — Place the blocks used in the 
 preceding experiment, loaded as before, upon an inclined plane 
 (Fig. 49) -whose surface is of the same material as that of the 
 horizontal plane. So adjust the angle of inclination that the 
 body starts to slide downward. Then the ratio of the height 
 of the inclined plane to its base, that is the tangent of the angle 
 of inclination, is the coefficient of static friction, which should 
 agree with the result already obtained. The tangent of the 
 angle of iaclination necessary to maintain the body in uniform 
 motion after it is started is the coefficient of kinetic friction. 
 
 Determine the coefficient of kinetic friction of a brass block 
 the dimensions of which may be 2.5 X 5 x 10 cm, sliding in 
 turn upon each of the three unequal faces. 
 
 Rolling Friction. — Arrange a cylinder to roll upon a track of 
 two horizontal and parallel bars (Fig. 49). Suspend two equal 
 weights by a cord passing around the cylinder, and by applying 
 additional weights to one side determine the coefficients of 
 static and kinetic rolling friction. Vary the experiment by 
 using different substances ; also use tracks of several materials, 
 and vary the loads in each case. 
 
 Reference. — Barker, Physics, pp. 81-87. 
 
 XXXVI. SURFACE TENSION WITH CAPILLARY TUBES 
 
 Determine the surface tension of water, using three different capillary tubes. 
 
 90. Capillarity and Surface Tension. — If a tube is held ver- 
 tically, dipping into a liquid which from the nature of the 
 materials wets the tube, the liquid is drawn upward along the 
 walls of the tube. Since liquids possess surface tension, this 
 
114 
 
 MECHANICS 
 
 effect alone would cause the liquid to rise and fiU the tube to 
 its top, regardless of length or diameter ; but when the liquid 
 rises above the normal surface, gravity begins to act to draw 
 it downward. These two forces are oppositely directed, and 
 tend to stretch the surface of the liquid. It will rise, therefore, 
 only so high that the maximum tension which the surface can 
 exert is counterbalanced by the weight of the liquid lifted. 
 The tension exerted by the surface on a line one centimeter 
 long is called the surface tension, and is designated by T. 
 
 For a tube of radius r the tension exerted along the circum- 
 ference is 2 irrT. The weight (in dynes) of the liquid column 
 is irr^Mg, h being its height and 8 its density. These two forces 
 are equal. 
 
 2 irrT = irr^Sg, 
 
 from which 
 
 2 ■ 
 
 To correct for the meniscus, h should be measured from the 
 level of the liquid in the outer vessel to a point one third of 
 the radius of the tube above the lowest surface of the meniscus, 
 
 as indicated in Fig. 50. 
 
 The tubes may have a diameter 
 varying from 0.1 cm to 0.01 cm. 
 These tubes may be readily drawn 
 from larger ones if those of the 
 desired size are not provided. The 
 radius of the tube required is that 
 at the top of the capillary column ; 
 the tube may be broken at this point 
 after the experiment, and its diam- 
 eter measured with the micrometer 
 microscope. If the tube is of uni- 
 form bore, its radius may be conveniently determined by mercury 
 calibration. The height of the liquid column may be measured 
 with a glass scale or, better, with a cathetometer. 
 
 The surface tension varies with the temperature; therefore 
 record the temperature of the liquid. 
 
 
 1 
 
 
 
 
 T 
 
 
 
 <t" — ^ 
 
 
 
 
 
 i 
 
 M 
 
 f^.'-.\^'l? 
 
 y 
 
 
 : - z v_ 
 
 ti-'-ririr-m -_ 
 
 'r 1 --.'. 
 
 
 ^s-^i^~~fy^.yy?^^n 
 
 Fig. 50. Capillakity 
 
CLEANING GLASSWARE 115 
 
 It is necessary that the tubes and vessel for the liquid should 
 be chemically clean (Art. 91), and that the liquid should be pure. 
 On account of the difficulty of cleaning tubes of small bore 
 required in this experiment, it is desirable to use only tubes 
 which have never become soiled. The tubing should be sealed 
 at the time of its manufacture, and such pieces as are required 
 are cut from the stock, and used for capillary measurements 
 but once. 
 
 Eefekencb. — Ostwald, Physico-Chemical Measurements, pp. 163-168. 
 
 91. Cleaning Glassware. — Glass vessels and tubes are com- 
 monly best cleaned with a solution of caustic potash. The use 
 of caustic soda should be avoided, because glass is dissolved 
 more by it than by caustic potash. The entire surface to be 
 cleaned must be scrubbed with a brush ; after thorough rinsing 
 with distilled water, the vessel may be allowed to drain and dry 
 spontaneously. The surface will be left in better condition if it 
 is dried by passing through the vessel a current of cold, dry air. 
 
 When the surface cannot be reached by a brush, small lead 
 shot may be introduced with the caustic, and, by shaking, the 
 entire surface should be scrubbed. After rinsing with water, 
 wash with nitric acid to remove all trace of the lead; then 
 rinse with distilled water and dry with a current of air. 
 
 Surfaces that cannot be treated as above may be washed with 
 warm aqua regia or with chromic acid. The latter should 
 remain in contact with the surface for several hburs. Rinse 
 and dry as before. 
 
 Warm air may be used for drying to save time, but it is not 
 so good as cold air, as it usually leaves a stain on the surface. 
 
 If much grease is on the surface this should first be removed 
 with carbon bisulphide, and then the cleaning may be continued 
 by one of the above methods. 
 
 Instructions for cleaning optical glass surfaces are given in 
 Art. 200. 
 
116 
 
 MECHANICS 
 
 XXXVn. SURFACE TENSION BY DIRECT MEASURE 
 
 Determine the surface tension of water and of soap solution. 
 
 92. Surface Tension. — A direct measurement of surface ten- 
 sion may be made with a beam balance in the manner described 
 below. If a Jolly balance (Art. 102) is used, the counterbal- 
 ancing of the surface tension is conveniently made by gradually 
 increasing the tension of the spring. A second observation is 
 then required to determine the weight in grams 
 necessary to produce the same tension in the 
 spring. 
 
 Make of fine platinum or glass wire a | |- 
 shaped fork with prongs about 1.5 cm long 
 and 3 cm apart. Suspend the fork by a fine 
 wire from the hook of a balance (Fig. 51), and 
 place a beaker containing the liquid on a bridge 
 over the pan at such a height that when the 
 beam is horizontal the prongs only dip into 
 the liquid. Exactly counterbalance the fork 
 and its suspension while it is in position. Now 
 cause a film of the liquid to fill the space 
 between the fork and the surface by tipping 
 the beam to immerse the whole fork and then carefully raising 
 it out. This film will tend to contract and puU the fork back 
 to the surface. Determine by repeated trials the additional 
 weight necessary to balance this tension ; that is, the weight 
 necessary to maintain the balance beam in horizontal position 
 while the film is intact. If w is this weight, and I the width 
 of the fork, the surface tension in dynes per centimeter of 
 breadth of surface is 
 
 7' = ^ 
 21' 
 
 &- 
 
 ^ 
 
 Fig. 51 
 Sdeface Tension 
 
 Record the temperature of the liquid. Exercise great care 
 to have the fork and beaker clean and the liquid pure. Care 
 in manipulation is required to secure the liquid film. Exces- 
 sive difficulty in obtaining it is probably caused by insufficient 
 
VISCOSITY 117 
 
 cleaning of the fork. If the fork is of platinum, it may be 
 cleaned by bringiag to a red heat in a Bunsen flame. Methods 
 for cleaning glass are described in Art. 91. 
 
 Reference. — Hastings and Beach, General Physios, pp. 138-159. 
 
 XXXVm. SPECIFIC VISCOSITY WITH TORSION PENDULUM 
 
 Determine the viscosity, at various temperatures, of lubricating and 
 cylinder oils, compared with that of water. 
 
 93. Viscosity, — The resistance which a fluid offers to change 
 of shape is its viscosity; it is ascribed to a kind of internal 
 friction. The coefficient of viscosity is the tangential force per 
 unit area of either of two horizontal planes, one of which is 
 relatively fixed while the other moves with unit velocity, the 
 space between being filled with the viscous material. 
 
 Viscosities are experimentally determined in two ways, from 
 the resistance to flow through small tubes, and from the resist- 
 ance to the motion of a solid body through the fluid. 
 
 If a torsion pendulum (Art. 82) is caused to vibrate while 
 immersed in a liquid, part of its energy is expended in impart- 
 ing motion to the surrounding liquid. The work done in 
 imparting equal velocities to different liquids is proportional to 
 their viscosities. The oscillations of the pendulum are isoch- 
 ronous, but their amplitudes continually diminish, and its 
 motion ls not simple harmonic. 
 
 When any other motion which would otherwise be simple 
 harmonic has impressed upon it a retarding force which varies 
 as the velocity, it is said to be damped. The arcs of successive 
 swings of the pendulum diminish in a constant geometrical 
 ratio ; that is, the logarithms of these arcs are in arithmetical 
 progression. The natural logarithm of the ratio of successive 
 arcs of such a damped vibration is the logarithmic decrement 
 (Art. 267). If R is the ratio of any amplitude of the vibration 
 to the next succeeding amplitude, the logarithmic decrement is 
 
 \ = loge E. 
 
118 MECHANICS 
 
 It can be shown that the viscosity of a liquid in which the 
 pendulum is vibrating is proportional to 
 
 T' 
 where T is the period of the pendulum. 
 
 A heavy disk has a short rod attached to it in the prolonga- 
 tion of its axis, and this is attached to a suspended wire, form- 
 ing a torsion pendulum. The dimensions of the parts are such 
 that the period is long, 30 seconds for instance. A pointer 
 attached to the rod swings over a divided circle the center of 
 which is in the axis of the wire, permitting the angular ampli- 
 tude of vibration to be observed. The liquid to be investigated 
 is contained in a suitable receptacle, preferably such that it may 
 be placed in an oil bath while observations are being made. A 
 gluepot is a suitable vessel. The vessel is then placed so that 
 the disk of the pendulum is immersed, and the liquid is heated 
 to the desired temperature. 
 
 The disk is displaced, 180° for instance, and caused to 
 vibrate without lateral displacement. The period, T, is deter- 
 mined as explained in Art. 83 ; when the liquid is very thick 
 it ma.y be possible to use only the first part of this method, as 
 the pendulum will soon come to rest. 
 
 Successive turning points, as indicated on the divided circle, 
 are observed. From these are found the successive arcs of vibra- 
 tion, independent of direction. Let a^ be any arc, and «!+„, 
 the wth following arc, which for accuracy may be about \a-^. 
 The logarithmic decrement is 
 
 ^ , ^ 2.3026,, , 
 
 \ = log,:B = — -— (logio aj - logio «! + »)• 
 
 With these values of \ and T calculate K, the viscosity 
 factor. Make three determinations with different mean ampli- 
 tudes of vibration. 
 
 In this manner find the viscosity factor for water at about 
 20° ; and for lubricating oil at the temperature at which it will 
 be used in machinery, about 40° ; and for cylinder oil at about 
 180°. The ratio of the factor for the oil to that for water is 
 the specific viscosity. 
 
CHAPTER V 
 DENSITY AND SPECIFIC GRAVITY 
 
 XXXIX. DENSITY BY HYDROSTATIC WEIGHING 
 
 Determine the density of aluminum. 
 
 94. Density and Specific Gravity. — The density of a homo- 
 geneous substance is the ratio of its mass to its volume at 0° ; 
 it is the mass per unit volume. The specific gravity of a sub- 
 stance is the ratio under specified conditions of its weight to the 
 weight of an equal volume of water. In the centimetfer-gram- 
 second system, if the water is at 4°, density and specific gravity 
 differ by 1 part in 20,000, specific gravity being the larger. 
 (The density of water at 4° is found to be 0.99995 : Inter- 
 national Congress of Physics, Paris, 1900.) If the specific 
 gravity is referred to water at 20°, it differs from the density 
 by 1 part in 500. The term " density " should always have the 
 meaning given, except that, in case it is not possible to reduce 
 the .volume of the body to 0°, the temperature of the determi- 
 nation must be specified. 
 
 In density determinations the mas§ of the body may be found 
 by methods already described. Th/ direct measurement of vol- 
 ume is usually difficult, and the method involved in the Prin- 
 ciple of Arc'himedes may be employed. The loss in the weight 
 of the body when immersed in water is the weight of the water 
 displaced. The density of water at various temperatures hav- 
 ing been determined with great precision, its volume per gram 
 is known, and thus the volume of the displaced water is found. 
 
 The most useful methods for density determinations are 
 described in connection with the hydrostatic balance and the 
 
 119 
 
120 MECHANICS 
 
 pyknometer. Other methods, the results of which are specific 
 gravities rather than densities, are often more convenient, and 
 give results which differ from true densities by amounts which 
 for many purposes are insignificant. 
 
 95, Density by Hydrostatic Weighing. — The hydrostatic bal- 
 ance is one arranged to weigh a body while it is immersed in 
 water, the operation best adapted for determining the volume 
 in density investigations. Often a beaker of water is placed on 
 a bridge over the pan, and the body whose volume is to be deter- 
 mined is suspended by a fine wire from a hook on the pan sup- 
 port; or the water may be placed below the pan. The latter 
 arrangement, since it leaves the pan at all times accessible for 
 the adjustment of weights, facilitates the operation. A pan with 
 a short hanger may be provided, so that the dish of water may 
 be placed in the balance case ; or there may be a hole in the 
 bottom of the case permitting the suspension of the body from 
 the bottom of the ordinary pan in a dish of water placed under- 
 neath the balance. 
 
 The density of a substance is the ratio of its mass to its 
 volume, the accurate determination of which requires correc- 
 tions for the buoyancy of the air, the density of the water, 
 weight of the harness, inequality of the balance arms, and vari- 
 ation in the sensibility of the balance. The method of weighing 
 by substitution with a constant load eliminates several of these 
 errors and makes it possible to correct for the others in the 
 simplest manner. 
 
 Arrange a harness of wire to support the body, and attach 
 this harness to the underside of the pan by a single wire about 
 one hundredth of a centimeter in diameter. Submerge the har- 
 ness in recently boiled distilled water in the position required 
 for the hydrostatic weighing of the body. The harness is to be 
 left in this position throughout the experiment. Place in the 
 second pan any convenient load (weights from a second set) 
 which is a trifle heavier than the body ; this also is to remain 
 unaltered during the weighings. Put precision weights in the 
 pan to which the harness is attached, and adjust with the great- 
 est care to bring the pointer to the middle of the scale. This 
 
THE HYDEOSTATIC BALANCE 121 
 
 should be done to the nearest milligram, and then by vibrations 
 and interpolation (Art. 42) fractions of a milligram may be deter- 
 mined. The hanger suspended in the water will dampen the 
 vibrations and quickly bring the beam to rest ; even though the 
 vibrations are of little assistance, interpolation for the fractions 
 of a mUligram may be employed for loads of less than a hundred 
 grams. The surface tension of the water around the wire will 
 have a slight influence, but with wire of the smallest suitable 
 diameter this may be neglected. 
 
 Remove the weights and put the body whose density is 
 required in their place. Then add such weights as will exactly 
 restore the balance to its equilibrium position, making the adjust- 
 ment with the same care as before. The difference between the 
 weights in the pan at the first and second adjustments is the 
 apparent mass of the body in air. 
 
 Place the body in the harness in the water, removing all air 
 bubbles from its surface. A camel's-hair brush or a piece of 
 wire may be used for this. Stir the water, if necessary, and 
 read its temperature. Add weights to the pan to restore equi- 
 librium, adjusting with the greatest care. The weights added in 
 this operation represent the apparent-loss of weight of the body 
 in water, and correspond approximately to its volume. 
 
 The observations are completed by reading the barometric 
 height and the temperature of the air in the balance case. 
 
 The ratio of the weight of the body in air to the loss of weight 
 in water is its approximate specific gravity. This may differ 
 from its true density by as much as one part in three hundred. 
 There must be applied corrections for the buoyancy of the air 
 and for the density of the water of the experiment. 
 
 Since on one side of the balance containing the ballast load, 
 the volume has not changed during the experiment, the buoy- 
 ancy has been constant, and it may be considered as a part of 
 the load, needing no further consideration. On the first side, 
 however, the volumes have differed in each of the three weigh- 
 ings, and the result of each observation must be corrected by 
 subtracting an amount equal to the weight of the air displaced. 
 The volumes in the pan in the first and third weighings are the 
 
122 MECHANICS 
 
 volumes of the weights used. These will be known if the weights 
 have been properly calibrated (Arts. 49 and 96); if not thus 
 known, the volumes are to be calculated from the assumed density 
 of the weights, — 8.4 for brass weights. In the second weigh- 
 ing the volume of air displaced is that of the weights in the pan 
 plus the volume of the body whose density is being determined. 
 The latter is approximately known from the uncorrected weigh- 
 ings, as already noted. The volumes in the pan multiplied by 
 the mass of one cubic centimeter of air at the pressure and 
 temperature of the experiment (Table 12) give the values of the 
 required corrections. 
 
 The difference between the first and second corrected weigh- 
 ings is the true mass of the body, and the difference between 
 the third and second is the mass of the water displaced. The 
 temperature of the water being known, the volume of one gram 
 of water is found (Table 3), and thus the true volume of the 
 body at the temperature, t°, of the experiment. The ratio of 
 mass to volume is the density of the body at t°. 
 
 If thjB coefficient of linear expansion, a, of the substance is 
 known, the volume expansion factor is 1 -|- 3 at, and the density 
 at t° multiplied by this factor will be the true density at 0° of 
 the substance. 
 
 A numerical example of the determination of the density of 
 a piece of aluminum alloy is given. 
 
 A piece of copper wire 0.01 cm in diameter is attached to the left pan, 
 and extends downward into a jar of distUled water beneath the balance. 
 Under the water the wire ends in a harness adapted ta support the piece 
 of aluminum. A ballast load of 300 g is placed in the right pan. Weights 
 from a set which had been calibrated and whose volumes had been deter- 
 mined are placed in the left pan to bring the pointer to the center of the 
 scale, the adjustment being possible under this load only to the nearest 
 mUligram. The weights required are found to be 292.560 g," having a 
 volume of 35.1 ccm. The piece of aluminum is placed in the left pan, and 
 weights of only 88.480 g, having a volume of 10.6 ccm, are needed to bring 
 the balance to the same position as before. The aluminum is now placed 
 in the harness under water, the water is stirred, and its temperature is 
 observed to be 22°.2. The weight required to restore equilibrium is 
 148.627 g of 17.8 ccm volume. The approximate volume of the alumi- 
 num is 148.6 — 88.5 = 60.1 ccm. The corrected barometric pressure is 
 73.8 cm, and the temperature of the air in the balance case is 22°.7. 
 
THE HYDROSTATIC BALANCE 
 
 123 
 
 From tables the density of the air is found to be 0.00115, the density of 
 the water 0.99772, and the coefficient of expansion of aluminum 0.000023. 
 The complete record may be made as follows. 
 
 Density or Aluminum Alloy 
 January IS, 1902 
 
 Weighings by method of constant load, with Staudinger balance and 
 Rueprecht weights. 
 
 Barometer, 74.07 cm at 22°.5 = 73.80 cm at 0°; air, 22°.7 ; water, 22°.2. 
 1 com air = 0.00115 g; density of water = 0.99772. 
 
 Obsebved Weighings 
 
 Vol. in 
 Pan 
 
 COKBEO- 
 TION 
 
 COEBECTEn 
 
 Weighings 
 
 Masses 
 
 To counterbalance 292.560 g 
 Al. in air + 88.480 
 Al. in water + 148.627 
 
 35.1 ccm 
 
 70.7 
 
 17.8 
 
 0.040 g 
 
 0.081 
 
 0.020 
 
 292.520 g 
 
 88.399 
 148.607 
 
 Al. 204.121 g 
 Water 60.208 
 
 204 121 X 99772 
 
 Density of alloy at 22°.2 = '- '- ■ = 3.3825. 
 
 •' •' 60.208 
 
 Expansion factor, 1.00153 ; density of alloy at 0° = 3.3877. 
 
 96. Other Methods for Hydrostatic Weighing. — The dish of 
 water may be placed on a bridge over the pan, and the body 
 whose volume is to be determined is suspended by a fine wire 
 from a hook on the pan support ; or the dish of water may be 
 placed on the pan, the body being suspended from a fixed support. 
 In the latter case the increase in the apparent weight of the water 
 is the weight of the water displaced. With these arrangements 
 the details of making the weighings and corrections for the 
 buoyancy of the air differ from those described, but the general 
 principles are the same. 
 
 For substances that float in water a sinker of lead may 
 be attached to the harness before beginning the observations, 
 remaining in this position throughout the experiment. 
 
 If the substance is soluble in water, some other suitable liquid 
 of known density may be substituted, the result obtained being 
 multiplied by this density. 
 
124 
 
 MECHANICS 
 
 The volumes and densities of a set of weights for calibration 
 purposes may be determined by a single hydrostatic weighing 
 of each piece. Arrange a fine wire to support the weight in 
 water, and counterbalance the wire by the rider or otherwise. 
 Determine the weight of each piece when it is immersed in 
 water. The difference between this weight and its nominal 
 value is its volume ; the ratio of its nominal mass to its volume 
 is its density. The results will be sufficiently accurate for the 
 purpose mentioned. 
 
 XL. DENSITY WITH THE PYKNOMETER 
 
 Determine the density of a liquid and of fragments of glass. 
 
 97. Density with the Pyknometer. — The specific-gravity 
 bottle, or pyknometer, is a small flask designed to hold a definite 
 volume of a liquid, and sometimes also to receive small solid 
 bodies. It is made in a variety of shapes, the forms shown in 
 Fig. 52 being the most useful. Form a may have either a solid 
 
 stopper or one with a capillary 
 tube, and it is to be completely 
 filled with the liquid. Form 
 b is provided with a thermom- 
 eter for indicating the tempera- 
 ture of the liquid; it is to be 
 fiUed to a mark on the slender 
 stem. These two forms may 
 be used for either solids or 
 liquids. Form g is for liquids 
 only. It has a thermometer 
 sealed in and two tubular arms. One arm is provided with 
 an extension tube to dip into the supply of liquid, which is 
 caused to flow into the pyknometer by suction applied to the 
 other arm. This latter tube is of such small diameter that 
 when the bottle is filled the liquid rises by capillary action 
 to the end of the tube. By applpng a piece of filter paper to 
 the end of this arm, a small amount of the liquid may be drawn 
 
 Fig. 52. Pyknometees 
 
THE PYKNOMETER 125 
 
 out, so that at the given temperature the liquid shall stand at a 
 mark placed on the larger arm. The unfilled portion of the 
 latter permits a slight increase of temperature during the weigh- 
 ing without the overflowing of the liquid. The two arms are 
 provided with caps to prevent loss by evaporation. 
 
 Density of a Liquid. — The pyknometer being clean and dry 
 (Art. 91), place it on one pan of the balance and accurately 
 weigh it or counterpoise it. Fill the bottle with distilled water, 
 and determine the weight of the water. Fill the bottle with 
 the liquid whose density is required and weigh again. These 
 apparent weights are to be corrected for the buoyancy of the air. 
 For making this coiTection, the volume of the bottle may be 
 taken as equal to the weight of the water filling it, and the 
 volumes of the weights are to be computed from the assumed 
 density, — 8.4 for brass weights. The excess of the volume of the 
 bottle over that of the weights used in each case, multiplied by 
 the density of the air at the temperature and pressure of the 
 experiment (0.0012 with sufficient approximation), is the correc- 
 tion which is to be added to the apparent weight to give the 
 weight in vacuo. The ratio of the weight of the liquid to the 
 weight of the water is the specific gravity as compared with water 
 at the temperature of the experiment, and this result multiplied 
 by the density of water at this temperature (Table 3) is the 
 density of the liquid at the temperature of the experiment. 
 
 The neglect of the correction for the buoyancy of the air 
 amounts, for liquids of density 0.5 or 2.0, to about one part in 
 a thousand, while it vanishes for liquids of density 1.0. The 
 correction for the density of the water is commonly about one 
 part in five hundred. 
 
 The temperature of the bottle when filled with the liquid 
 should be the same as when filled with water; otherwise the 
 corrections required are more complicated than above outlined. 
 The pyknometer may be filled while immersed in a water bath, 
 and it should be handled only by the neck. 
 
 Density of a Solid in Fragments. — Weigh the fragments of 
 the solid in air. Fill the pyknometer with distilled water and 
 determine the weight of bottle and water. Put the fragments 
 
126 MECHANICS 
 
 into the bottle, displacing part of the water ; replace the stopper, 
 being sure that the bottle is filled properly with water, wipe dry, 
 and weigh again. The difference between the sum of the first 
 two weighings and the last is the weight of the water displaced. 
 The ratio of the weight of the solid in air to the weight of the 
 water displaced is its specific gravity. 
 
 The bottle used in this method should be as small as is con- 
 venient. If the solid is soluble in water, weigh it in some liquid 
 of known density in which it is insoluble. Its specific gravity 
 as compared to this liquid, multiplied by the density of the liquid, 
 is its true specific gravity. 
 
 In work of precision corrections for the temperature of the 
 water and the buoyancy of the air miist be applied according to 
 the principles given above and in Art. 95. 
 
 Other Methods. — In some cases it may be sufficiently accurate 
 to assume that the capacity of the bottle has been adjusted by 
 the maker to an amount marked upon the bottle. A "100 
 Grams " specific-gravity bottle may be assumed to hold 100 g 
 of water, or to have a capacity of 100 ccm. Hence it wiU not 
 be necessary to weigh the bottle filled with water; and if the 
 bottle is accompanied by a counterpoise, the determination of 
 the specific gravity of a liquid is reduced to a single weighing. 
 
 RBrERENCE. — Stewart and Gee, Vol. I, p. 127. 
 
 XLI. DENSITY OF AIR BY WEIGHING AND EXHAUSTION 
 
 Determine the mass of one cubic centimeter of dry air at 0°, under a pres- 
 sure of 76 cm. 
 
 98. Density of Air. — A convenient form of receiver for this 
 experiment is a glass globe of the form shown in Fig. 53, hav- 
 ing a capacity of one or more liters. If the globe contains any 
 moisture, this must be removed by drawing dry air through 
 it; moderate and careful heating wiU assist in the operation. 
 Exhaust the receiver until the pressure is reduced to less than 
 one centimeter, and let p be the pressure indicated by the gauge 
 when the receiver is closed. Weigh the globe as accurately and 
 
DENSITY OF A GAS 
 
 127 
 
 quickly as possible. Fill the receiver with dry air, by allowing 
 the air slowly to flow into it through a drying bottle containing 
 concentrated sulphuric acid. Again weigh the globe carefully. 
 The difference between these two weighings 
 wUl be the weight of the air exhausted. 
 Repeat this operation twice, securing the 
 same degree of exhaustion in each case, and 
 let m be the mean of the three determina- 
 tions. Let t be the temperature of the air 
 with which the globe is filled, and P the cor- 
 rected barometer reading. Fill the globe 
 with water of known temperature, and weigh 
 to the nearest tenth of a gram. The differ- 
 ence between this weight and that of the 
 jexhausted globe is the weight in grams of 
 the water, whose volume is equal to that 
 of the globe. This quantity divided by the density of water at 
 the given temperature (Table 3) is the volume, v, of the receiver. 
 The mass of 1 ccm of air at 0° and under a pressure of 76 cm, 
 that is, its density, is 
 
 Fig. 53. Globe for 
 Wbighing Aie 
 
 D = !?^. 
 
 76 273 -f- 1 
 
 ■P 
 
 273 
 
 (Empty the receiver at the end of the experiment and place 
 it so that all water can drain out.) 
 
 XLU. DENSITY OF A GAS BY WEIGHING AND COMPARISON 
 
 WITH AIR 
 
 Determine the density of carbon dioxide. 
 
 99. Density of a Permanent Gas. — Fill a light glass globe 
 provided with a stopcock with the gas. This may be done by 
 displacing mercury over a mercury trough, or simply by displac- 
 ing the air if there is sufficient gas for this. The globe is closed, 
 and its weight, Wj, is determined. Displace the gas by air, and 
 find the weight, w^, of the globe open. Fill the globe with 
 
128 MECHANICS 
 
 mercury (or water), and let Jf be its weight. From the observed 
 temperature and the barometric height, find the density of the 
 air, I, with the aid of Table 12. The temperature and tables 
 will give the density, Q, of the mercury (or water). If the gas 
 and air with which the globe was filled were both at the same 
 temperature and pressure, the density of the gas is 
 
 ^_ Wi-w^ Q-l ^ ^ 
 M—w^ I 
 
 If the temperature and pressure of the gas were t-^ and py, and 
 of the air t^ and p^^, then the value obtained from the above 
 formula must be multiplied by 
 
 £2 1 + 0.00367 ^1 
 p^'l +0.00367^2' 
 
 References. — Kohlrausch, Physical Measurements, pp. 56-63; Ostwald, 
 Physico-Chemical Measurements, pp. 99-107. 
 
 XLm. DENSITY OF LIQUIDS WITH A U-TUBE 
 
 Determine the density of mercury -with the U-tube. Determine the density 
 of sulphuric acid by Hare's Method. 
 
 100. Density with a U-Tube. — Provide a U-tube with arms 
 about a meter long. Pour me'rcury into the tube until it stands 
 at a height of five or six centimeters in each arm ; into one arm 
 now pour distilled water nearly to fill it. The heights of the 
 free surfaces of the liquids above their common surface are 
 inversely proportional to the densities. Determine these heights 
 carefully with a cathetometer, setting the cross wires tangent to 
 the meniscus, and calculate the relative density of the mercury. 
 This result is the density of mercury at its temperature, t°, 
 compared with water at t°. Multiply this by the density of 
 water at t° (Table 3), which will give the density of merc\iry 
 at t°. 
 
 This method can be used for comparing the densities of any 
 two liquids which do not mix. For liquids which will mix, the 
 following method is often convenient. 
 
THE JOLLY BALANCE 129 
 
 101. Relative Density by Hare's Method. — Two upright tubes, 
 each about a meter long, dip into separate vessels containing the 
 liquids to be compared, — for example, distilled water and an acid. 
 The tops of these tubes are connected to a single tube provided 
 with a stopcock, which leads to a small air pump or other suc- 
 tion apparatus. The withdrawal of air will cause the liquids to 
 rise in the tubes to heights inversely proportional to their den- 
 sities, neglecting capillary action. Determine these heights 
 with a cathetometer, and calculate the relative densities. This 
 result when multiplied by the density of water at the tempera- 
 ture of the experiment, t°, will give the density of the liquid 
 at t°. 
 
 Instead of the cathetometer, any convenient scale may be 
 placed behind the tubes for measuring the heights of the liquid 
 columns. 
 
 XLIV. SPECIFIC GRAVITY WITH THE JOLLY BALANCE 
 
 Find the specific gravity of five solids and one liquid, making three deter- 
 minations of each. 
 
 102. The Jolly Balance. — A spring balance is often a con- 
 venient substitute for the beam balance ; it is especially suitable 
 .when relative weight only Jg' required, as in specific gravity 
 determinations. In one form the Jolly balance consists of a 
 long spiral spring suspended in front of a mirror millimeter 
 scale. Attached to the lower end of the spring is a small weight 
 pan, and hanging from this by a fine wire is a second pan which 
 is always to be immersed in water. In reading the elongations 
 any convenient point at the bottom of the spring may be taken 
 as the index, which point should always be made to coincide 
 with its image as seen in the mirror scale, to avoid parallax. 
 
 The necessity for continual adjustment of the position of the 
 dish of water is somewhat troublesome ; this is avoided by the 
 form of construction shown in Fig. 54. The pans are suspended 
 at a constant level by raising or lowering the upper end of the 
 spring, which is accomplished by turning the milled head S ; 
 
130 
 
 MECHANICS 
 
 the amount of extension is indicated by a scale graduated on the 
 telescoping tubes. The pans are prevented from slipping away 
 when the load is changed by the short glass tube and stops 
 shown at T. Marks on the wire and tube are to be brought to 
 the same level when making a setting. 
 
 Upon the assumption that the stretch of the 
 spring is proportional to the load it carries, 
 this apparatus permits a very convenient deter- 
 mination of specific gravity with moderate 
 precision. 
 
 Spedfio Gravity of a Solid. — Ascertain the 
 elongation of the spring produced when the 
 solid is supported by the upper pan, the lower 
 one being in water; and also the elongation 
 produced when the body is placed upon the 
 lower pan in water. The ratio of the first 
 elongation to the difference between the two 
 elongations is the specific gravity required. 
 
 A method perhaps more accurate, but involv- 
 ing more trouble, is to determine the elonga- 
 tions as above described, and to find by trial 
 in each case what weights alone placed in 
 the pan wiU produce the same extension of the 
 spring. The ratio of the fii-st weight to the 
 difference between the two is the specific 
 gravity. 
 
 The relation between the stiffness of the 
 spring employed and the weight of the body 
 experimented with should be such that a considerable elongation 
 of the spring is produced. 
 
 Specific Gravity of a Liquid. — Suspend from the spring, in 
 place of the pans, any solid which is not affected by the liquid. 
 Read the index when the solid is suspended in the air, when 
 immersed in water, and also when immersed in the given liquid. 
 The ratio of the difference between the first and third of these 
 readings to the difference between the first and second, is the 
 specific gravity of the liquid. 
 
 Fig. 54 
 Jolly Balance 
 
THE HYDROMETER 131 
 
 XLV. SPECIFIC GRAVITY WITH HYDROMETERS 
 
 Determine the specific gravity of sevei-al solids and liquids, using various 
 forms of hydrometers. 
 
 103. The Constant Immersion Hydrometer. — A very simple 
 instrument for measuring the specific gravity of a solid or liquid 
 is the constant immersion hydrometer, which consists of a glass 
 or metal elongated bulb so constructed as to float in water in 
 an upright position. At its lower end is a small basket, and a 
 slender stem on the upper end carries a small weight pan above 
 the water. 
 
 To determine the specific gravity of a solid, ascertain what 
 weight, TTj, on the pan will cause the instrument to sink to a 
 mark on the stem ; with the solid to be tested on the pan, find 
 what weight, W^, must be added to produce the same result; 
 place the solid in the basket in the water and determine again 
 the weight, W^, required to sink the bulb as before. The ratio of 
 the weight of the solid in air to its loss of weight in water, that 
 
 jf Jfr 
 
 is, — -, is the required specific gravity. Bodies that float 
 
 W^ — Jr 2 
 
 in water may be tied to the basket or held in place otherwise. 
 
 If the hydrometer is made of glass, it may be used to deter- 
 mine the specific gravity of a liquid. If m is the weight of the 
 hydrometer itself, W-y the weight required to sink it in water to 
 the index mark, and W^ the weight to sink it in the given 
 
 liquid, then is the specific gravity of the liquid. 
 
 m+ Wy 
 
 104. The Variable Immersion Hydrometer. — For indicating 
 directly the density of a liquid, variable immersion hydrometers 
 are used. These are bulbs of glass with elongated stems so 
 constructed that they will float in liquid, sinking to a depth 
 depending upon the density of the liquid. An empirical scale 
 is provided, the reading of which, at the surface of the liquid, 
 indicates directly the required density. The scale should be 
 read through the liquid, the eye being level with the surface. 
 By using properly constructed hydrometers, and applying 
 
132 MECHANICS 
 
 corrections for the temperature of the liquid, a fair degree of 
 accuracy can be obtained, while the method is the simplest 
 possible. It however requires an amount of liquid sufficient to 
 float the instrument. 
 
 These hydrometers are constructed with scales of great variety, 
 adapted to special uses, indicating the percentage of alcohol, 
 sugar, or of a salt in a solution, and for other similar purposes. 
 
 Repeeehcb. — Stewart and Gee, Practical Physics, Vol. I, pp. 146-154. 
 
PaET III SOTJIN^D 
 
 CHAPTER VI 
 FEEQUENCY OF VIBRATION 
 
 L. FREQUENCY OF A SOUND BY BEATS 
 
 Determine the frequency of a tuning fork by the method of beats. Make 
 a copy of a " Standard A " fork. 
 
 105. The Tuning Fork The most important of acoustical 
 
 instruments is the tuning fork, which, when properly constructed 
 and mounted, gives a sound of great purity and constancy of 
 pitch. Forks should be carefully protected from rust and from 
 rough usage. They should never be struck with metal. They 
 may be sounded with a violin bow, or with a hammer of wood 
 or ivory, or with one made of a rubber cork on a spring metal 
 rod for a handle. A felt piano hammer attached to a spring 
 handle is excellent for striking forks. When the fork is pro- 
 vided with an electromagnetic vibrator, the vibrations may be 
 started with the felt hammer. 
 
 The number of vibrations given by a steel tuning fork varies 
 slightly with the temperature. The standard temperature at 
 which forks are usually rated is 20°. The temperature coeflB- 
 cient has been found nearly constant for forks of any pitch, and 
 to have the value — 0.00011. A fork giving 256 vibrations 
 per second at 20° will have its frequency diminished by 0.0282 
 for one degree increase in its temperature. 
 
 133 
 
134 SOUND 
 
 A fork can be adjusted in pitch by filing. Filing the prongs 
 at the ends wUl raise the pitch, while filing them near the yoke 
 will lower it. The student should not alter any fork except upon 
 definite orders. 
 
 106. Frequency of Sound with a Tuning Fork. — Two sounds 
 nearly in unison produce beats the number of which per second 
 is equal to the difference of their vibration frequencies. Sets of 
 standard forks are often provided such that one may be selected 
 which, when sounded with an unknown fork, will give a num- 
 ber of beats that can be easily counted. The frequency of the 
 given sound is, therefore, equal to that of the standard fork 
 increased or diminished by the observed number of beats. To 
 determine whether the unknown fork is higher or lower than 
 the standard, load one prong of it with a piece of wax ; if the 
 beats are now fewer, the fork altered was originally the higher 
 in pitch, and vice versa. 
 
 A fork may be brought into unison with a standard, that is, 
 the standard may be copied, by adjusting the fork to make the 
 beats fewer, and finally to cease. But to count the beats when 
 they are fewer than one per second is difficult. A convenient 
 number for counting is four per second. One can count the 
 number, at about this rate, made in ten seconds with ease and 
 accuracy. Hence, to tune a fork to a standard, an auxiliary 
 fork which is exactly four beats per second sharper than the 
 standard is provided. The given fork can now be adjusted till 
 it is four beats per second flatter than the auxiliary fork, in 
 which case it must be exactly in unison with the standard. 
 
 LI. FREQUENCY OF A SOUND WITH THE SIREN 
 
 Determine the number of vibrations of two organ pipes and a tuning fork. 
 
 107. Frequency of Sound with the Siren. — The siren (Fig. 55) 
 consists of a metal disk, pierced with several series of holes in 
 concentric circles, which is caused to rotate and periodically 
 to interrupt jets of compressed air. This rotation may be pro- 
 duced by the impact of the jets upon the oblique sides of the 
 
THE SIREN 
 
 135 
 
 holes, the rapidity of rotation being controlled by varying the 
 air pressure. A better form has the holes bored through 
 the disk perpendicularly, the rotation being produced by an 
 electric motor controlled with a rheostat. A counting device 
 is arranged to record the number of rotations. Often the siren 
 has two disks, each with several series of holes, any one or any 
 combination of which may 
 be used as desired. 
 
 The interruption of the 
 air jets wiU, in general, give 
 rise to a musical tone, the 
 pitch of which depends upon 
 the rapidity of these inter- 
 ruptions. To determine the 
 number of vibrations of a 
 given sound, so adjust the 
 speed of the siren disk that 
 its tone is in unison with 
 the given one. If sufficient 
 variation in air pressure can- 
 not be obtained, a different 
 number of holes in the disk 
 must be used. It is neces- 
 sary to adjust with care, 
 that the sounds may remain in unison during the counting. 
 When the sounds differ slightly beats may be heard. If these 
 are caused by increased speed of the disk, the unison may be 
 restored by a very light touch of the finger on the disk. If 
 perchance this causes the beats to increase in frequency, the 
 disk was before rotating too slowly ; a slight pressure with the 
 hand on the wind chest bellows may restore it to the proper 
 speed. When the unison has become constant determine the 
 number of rotations of the disk in one minute. This number 
 multiplied by the number of holes in the series used, and divided 
 by sixty, is the number of puffs of air per second producing the 
 siren tone, and is the number of vibrations of the given sound 
 with which it is in unison. It may be more convenient, using a 
 
 Fig. 55. The Siren 
 
136 , SOUND 
 
 stop watch, to determine the time required for the disk to make 
 exactly 1000 or 2000 rotations. 
 
 If it is inconvenient to secure exact unison, the number of 
 beats per second may be counted, and their number subtracted 
 from the frequency determined as above, when the disk rotates 
 too rapidly, or added, if the rotation of the disk is too slow. 
 
 LII. FREQUENCY OF VIBRATION BY GRAPHIC METHODS 
 
 Determine the frequency of a tuning fork. 
 
 108. Frequency of Vibration by Graphic Methods. — A graphic 
 method will often be the most convenient for comparing a tun- 
 ing fork or other vibrating body with a standard fork or with 
 a standard clock. 
 
 The fork must be provided with a style for marking on 
 smoked paper; if the style is permanent, it may be of thin 
 sheet brass cut to a point and so attached as to swing edgewise ; 
 if it is temporary, it may be a stiff bristle, a piece of quill or of 
 sheet celluloid, attached with wax. 
 
 The smoked paper may be carried on the cylinder of a chrono- 
 graph if this has sufficient speed, or a special cylinder may be 
 arranged ; perhaps a ribbon of smoked paper on reels (Art. 72) 
 is the most efficient device. The fork is firmly supported so 
 that the style will mark lightly on the paper when the fork is 
 in vibration and the paper is moving underneath. If a drum 
 is used, either the fork or the drum should have a motion in the 
 direction of the axis making the trace in a spiral form. 
 
 A break-circuit clock is connected through a relay (Fig. 37) 
 to the primary circuit of an induction coil ; the secondary of the 
 coil is connected to the metallic drum carrying the paper and 
 to the fork, or to a metallic point placed beside the fork style. 
 As the clock interrupts the current through the coil, a spark 
 will pass from the point through the paper, producing a small 
 spot in the smoke, thus marking the clock beats. By counting 
 the number of fork waves between two such marks, or, better, 
 between the marks corresponding to an even number of seconds 
 
FEEQUENCY BY OPTICAL METHODS 137 
 
 to eliminate eccentricity in the break, the frequency of the fork 
 is directly determined. A varying rate of moving the paper 
 introduces no error, as the frequency depends upon the number 
 of waves between two points and not upon their size. 
 
 The addition of the style may appreciably change the period. 
 By determining the change in frequency caused by a known 
 added load, a correction for the style may be computed. 
 
 Two forks are compared by having their traces made side by 
 side simultaneously. 
 
 Lm. RATIO OF VIBRATION FREQUENCIES BY LISSAJOUS'S 
 
 FIGURES 
 
 Determine the number of vibrations of several vibrating bodies by means 
 of Lissajous's optical comparator. 
 
 109. Frequency of Vibration by Optical Method. — The optical 
 comparator of Lissajous depends upon the principles of the com- 
 position of simple harmonic motions. If the periods of two rec- 
 tangular simple harmonic motions are in certain simple ratios, the 
 results of their composition will be characteristic curves which 
 are easily recognized. If the ratio of the periods is exact, the 
 curve will have a fixed form, the shape of which will depend 
 upon the relative phases of the two vibrations. When the 
 periods are such that they differ slightly from the exact ratio, 
 their relative phases will change continuously and slowly pass 
 through all possible values, with the result that the curve of 
 their combination will successively exhibit all forms belonging 
 to the given ratio. The curve will pass through a complete 
 cycle of changes in the time required for one vibrating body to 
 gain or lose one whole vibration on the number which it would 
 have made in the same time if the ratio of the frequencies were 
 exactly that corresponding to the observed curve. 
 
 In the most convenient form of the optical comparator the 
 objective of a microscope is carried by one prong of a standard 
 tuning fork, the body tube being on a separate support and per- 
 pendicular to the plane of vibration of the fork. (See Fig. 56.) 
 
138 
 
 SOUND 
 
 The body whose vibration frequency is to be determined- is 
 adjusted so that its line of vibration is at right angles to that 
 of the fork, and so that some distinct point of it can be seen 
 through the microscope. Usually it will not be difficult to find 
 a suitable point as the object, — a speck of adhering chalk dust 
 will often suffice ; or the surface may be slightly smoked and 
 
 a point made in the smoke with a 
 needle. It is convenient to main- 
 tain the vibration of the standard 
 fork by an electromagnetic attach- 
 ment. For universal comparisons 
 a number of forks, or forks whose 
 frequencies can be altered by slid- 
 ing weights, will be necessary. 
 
 A standard fork is selected for 
 the comparator such that its motion 
 compounded with that of the given 
 body will produce a figure which 
 may be recognized as representing 
 a certain ratio of frequency. The 
 time required for the figure to pass 
 through a complete cycle of phase 
 changes is observed. Thence the 
 frequency of the given body may 
 be obtained with great accuracy. If the standard fork has a 
 frequency S, and the figure corresponds to the ratio 1 : n, and 
 if the cycle of changes occurs in c seconds, the unknown fork 
 has a frequency 
 
 N=nS±-. 
 
 
 
 Often it may be necessary to observe the effect of loading the 
 fork with a small piece of wax to determine whether it gains or 
 loses a vibration in the time of the phase cycle. 
 
 The method is applicable to vibrating strings as well as to 
 forks. 
 
 Fig. 56. Lissajous's Tuning 
 Fork Comparator 
 
 Reference. — Zahm, Sound and Music, p. 416. 
 
FEEQUENCY BY OPTICAL METHODS 139 
 
 110. Copying a Standard Fork. — Two independent forks may 
 be easily compared with such a comparator as has just been 
 described, or a standard fork may be copied. So load the com- 
 parator fork in any manner that it gives a known figure with 
 the standard fork, and adjust the second fork to give the same 
 figure. The pitch of a fork may be raised by filing the ends of 
 its prongs, and lowered by filing the prongs near the yoke. 
 
 111. Projection of Lissajous's Figures. — Instead of the micro- 
 scope method described, it is sometimes more convenient, espe- 
 cially for demonstration purposes, to use forks with mirrors 
 attached to their ends, a pencil of light being reflected from 
 one mirror to the other, and finally into a telescope or upon a 
 screen. The pencil of light will move with the compounded 
 motion of the two forks, and will describe Lissajous's figures. 
 
 UV. FREQUENCY OF A FORK BY OPTICAL COMPARISON 
 
 Determine the frequency of a tuning fork, including the measuring of the 
 period of the auxiliary pendulum. 
 
 112. Reed's Method for rating a Tuning Fork. — Often it is 
 desired to know the absolute frequency, or to determine slight 
 changes in the frequency, of a fork with great precision. Lis- 
 sajous's optical method (Art. 109) will give the ratio of two 
 frequencies with precision only when this ratio is very approx- 
 imately represented by a small integer. Of the various methods 
 for accurately comparing any fork with a standard clock, Reed's 
 method is probably the simplest that has been devised. The 
 feature of the method is a flashing apparatus which may be 
 adjusted to give flashes of any desired frequency, of very short 
 duration (0.00001 second) and of great regularity. The fre- 
 quency of the fork being investigated must be known to the 
 nearest integer (Art. 108); this frequency should not be more 
 than 300, and the satisfactory application of the method requires 
 that a mirror be attached to one prong of the fork. 
 
 The flashing apparatus consists of an auxiliary pendulum, 
 about a meter long, whose period is adjustable by movable 
 
140 
 
 SOUND 
 
 weights. Near the middle of the pendulum rod and parallel 
 to it is attached a fine needle, n (Fig. 57). A short vertical 
 axis, a, is pivoted in a stationary support; attached normally 
 to this axis is a second fine needle or glass tube, t, a fraction 
 of a millimeter in diameter, which rests against the needle n. 
 A light mirror, m (a galvanometer mirror 2 cm 
 in diameter), is fastened to the axis a. A small 
 drop of oil joins the two needles, so that as the 
 pendulum swings the mirror rotates through 
 an arc of considerable magnitude. This arc 
 may be made' larger or smaller by varying the 
 lengths of the several levers involved. 
 
 For observation the apparatus is arranged as 
 shown in Fig. 58. ^ is a bright source of 
 lQ__v] 4K-| light which is condensed by the lens i^ on a 
 I / rial vertical slit /Sj. The light from the slit is 
 reflected by the flashing mirror, when the 
 pendulum is at rest, to the observing instru- 
 ment which is placed three or more meters 
 from the pendulum. For a fork of high 
 frequency this distance and the intensity of 
 the light have both to be increased. The 
 I I ' observing instrument consists of a collimator 
 
 |T^ and a view telescope ; a spectroscope or spec- 
 
 ■"^ trometer serves for this purpose, though an 
 
 Abbe-Zeiss autocollimating telescope, as 
 described by Reed, is simpler. The light from 
 the flashing mirror is thrown upon the colli- 
 mator, the image of the slit S-^ being focused on 
 the collimator slit S^ by the lens L^ The light 
 passing through the collimator falls upon the mirror attached to 
 the tuning fork to be rated, and from this it is reflected to the 
 view telescope. The various parts are to be so adjusted that a 
 bright, well-defined narrow image of the slit is seen. 
 
 113. The Stroboscopic Method. — If the pendulum is caused 
 to vibrate, the slit will flash into view once in each swing. 
 When the fork is also caused to vibrate, the flash will still be 
 
 Fig. 57 
 
 Reed's Flashing 
 
 Pendulum 
 
FREQUENCY BY OPTICAL METHODS 141 
 
 "seen at every swing of the pendulum, but its position in the 
 field of view will vary unless the fork makes an exact integral 
 number of vibrations in one swing of the pendulum. Suppose^ 
 for illustration, that the fork makes 30^ vibrations in one swing 
 
 Ls 
 
 t.::: 
 
 
 ■ 
 
 ..-A- 
 
 
 
 s. f 
 
 -^ 
 
 
 
 a 1 
 
 Fig. 58. Stboboscopio Method 
 
 of the pendulum. If the first flash is observed in the center of 
 the field, at in Fig. 59, when the second flash occurs the fork 
 will have made 30|^ vibrations, and the flash will appear at the 
 point 1, I of a vibration past the center ; at the end of the next 
 swing the fork will have made 60| vibrations, the flash appear- 
 ing at point 2 ; the flash will thus appear successively at the 
 points 3, 4, 5, 6, 7, until, at the end of the eighth swing, when 
 the fork has vibrated 240| times, the flash again occurs at the 
 starting point. Since the fork vibrates in a straight line and 
 not in an ellipse, the various points all lie in one line ; whether 
 a given flash coiTesponds to point 4 or 8 is determined by the 
 direction in which the flashes approach this point. Such a 
 series of images from an intermittently illuminated vibrating 
 body constitutes a Btrolo- 
 scopic cycle. 
 
 Usually one cycle will 
 not take place in an inte- 
 gral number of swings of - g 
 the pendulum. If the flash ^^^^^ Stroboscopic Motion 
 occurs once at the center, 
 
 and then after a time occurs again at the center, by counting the 
 number of flashes between the two coincidences and dividing by 
 the number of cycles, the time of one cycle in pendulum swings 
 is determined. 
 
142 SOUND 
 
 The fraction of a cycle between two successive flashes is the 
 fraction of a vibration which added to some integer gives the 
 frequency of the fork referred to the period of the flash. This 
 integer must be determined from an independent source, this 
 method giving only the fraction. 
 
 If the number of vibrations per pendulum beat is a whole 
 number minus a small fraction, a similar stroboscopic cycle will 
 occur, taking place in the opposite direction. With a straight- 
 line motion this direction cannot be discovered, and some sub- 
 sidiary method is necessary to determine whether the fraction is 
 to be added or subtracted. 
 
 A pendulum swinging freely will slowly diminish in ampli- 
 tude, and the period will slightly decrease at the same time. A 
 study of the conditions of the stroboscopic cycle will show that, 
 as the period of the pendulum slowly decreases, the number of 
 flashes per cycle will increase if the number of vibrations is a 
 whole number plus a fraction, and that the number of flashes 
 per cycle will decrease if the number of vibrations is a whole 
 number minus a fraction. 
 
 For convenience of observation the period of the pendulum 
 should be so adjusted that with the given fork the number of 
 flashes per cycle is about ten. 
 
 If the flash image of the sht is too wide, it may be made nar- 
 rower by increasing the distance of the collimator from the flash- 
 ing mirror, by narrowing either one of the slits or by increasing 
 the arc through which the mirror swings. 
 
 In general, if c is the number of swings per cycle, and w, to the 
 nearest integer, is the number of complete vibrations of the fork 
 in one swing, then the exact number of vibrations per swing is 
 
 n±-; and if the exact period of the pendulum is i, the absolute 
 
 frequency of the fork at the temperature of the experiment is 
 
 ,1 
 
 w ±- 
 
 N= f. 
 
 t 
 
 This value should be reduced to 20° by means of the tempera- 
 ture coefficient for tuning forks explained in Art. 105. 
 
FEEQUENCY BY OPTICAL METHODS 143 
 
 A numerical example follows. 
 
 FilEQUENCY OF AN ADJUSTABLE KoENIG FORK ; ReED'8 METHOD 
 
 Time by an Auxiliary Pendulum from Standard Self-winding Clock 
 
 July 36, 1903 
 
 To find the frequency of a fork known to give 32+ vibrations per 
 second. Preliminary observations sliowed that there were 8+ flashes per 
 cycle. 
 
 Cycles 
 
 Flashes 
 
 SWINC 
 
 IS PER C 
 
 6 
 
 51 
 
 
 8.500 
 
 10 
 
 84 
 
 
 8.400 
 
 8 
 
 67 
 
 
 8.375 
 
 11 
 
 92 
 
 
 8.364 
 
 
 
 Mean, c = 
 
 = 8.410 
 
 Temperature of fork, 23°. 6. 
 
 Since the observed values of c decrease as the amplitude of the pendu- 
 lum diminishes, c has the negative sign. 
 
 The period of the flashes, t, was determined by a subsidiary experiment 
 (see example under Art. 69) to be 0.92858 seconds. In this time a fork 
 of frequency about 32 would make about 30 vibrations. Hence the fre- 
 quency of the fork is 
 
 1 
 
 "~c 30-0.1189 „„,„ 
 
 Nos'R = = = 32.179. 
 
 ^ ■" t 0.928.58 
 
 To reduce to 20°, the correction is 32.2 x 0.00011 x 3.6 = 0.013 ; there- 
 fore 
 
 iVao- = 32.179 -I- 0.013 = 32.194. 
 
 Reference. —Beed, Physical Review, Vol. 12, pp. 279-291, 1901. 
 
CHAPTER VII 
 VELOCITY AND WAVE LENGTH OF SOUND 
 
 LV. VELOCITY OF SOUND BY RESONANCE 
 
 Determine the velocity of sound in air, using a resonance tube and a fork 
 of known vibration number. 
 
 114. Velocity of Sound in Air. — Theory shows that the veloc- 
 ity of sound in a gas is unaffected by changes in pressure, but 
 that it varies with temperature and hygrometric state. If v^^ is 
 the velocity at 0°, then the velocity at a temperature t is 
 
 Vf = Vq Vl + at. 
 
 The velocity in dry air at 0° is found by experiment (Inter- 
 national Congress of Physics, Paris, 1900) to be 
 
 Vq = 33136 cm per sec. ; 
 
 whence Vf = 88186 Vl -f- at cm per sec. ; 
 
 or with sufi&cient approximation, 
 
 Vf = 83186 -f- 60.7 t cm per sec. 
 
 115. Wave Length of Sound by Resonance. — The resonance 
 tube may be of glass about 100 cm long and 6 cm in diameter, 
 arranged with a scale and water reservoir, as shown in Fig. 60. 
 A rubber tube connects the reservoir with the bottom of the 
 glass tube, so that the latter may be filled to any desired height 
 with water. A fork of known vibration number, for instance 
 256, is firmly fastened above the tube as shown. The fork is 
 made to vibrate by means of a felt piano hammer or by a violin 
 bow, and the height of the water is adjusted till the sound is 
 
 144 
 
VELOCITY OF SOUND 
 
 145 
 
 most strongly reenforced. Several positions of maximum reen- 
 forcement may be found. Determine each by at least ten 
 observations. The difference between successive positions of 
 the water surface is one half of the wave length in air of the 
 sound of the given vibration number. From 
 this determine the velocity at the temper- 
 ature of the air, and reduce it to 0° (Art. 114). 
 Repeat the experiment with other forks of 
 higher pitch. Find the wave length for a 
 fork of unknown frequency, and from the 
 velocity in air determine the frequency. 
 
 LVI. VELOCITY OF SOUND AND YOUNG'S 
 MODULUS BY DUST FIGURES 
 
 Determine the velocity of sound in a gas, and in 
 brass, steel, and ■wood. Compute Young's 
 modulus for the solids. 
 
 116. Relative Velocity of Sound by Station- 
 ary Waves. — Kundt's apparatus for deter- 
 mining the relative velocity of sound in 
 solids and fluids consists of a glass tube 
 about 150 cm long and 3 cm, or more, in 
 diameter, provided with a brass fitting at 
 each end. On each fitting is a stopcock. 
 Near one end of the tube is a closely fitting 
 piston which may be moved a few centimeters by a handle 
 passing though one end cap. The other end is provided with 
 a clamp to grasp a rod about a meter long at its center. The 
 end of the rod which projects into the tube carries a cork 
 diaphragm which fits the tube loosely. (See Fig. 61.) Inside 
 the tube between the piston and the diaphragm is placed cork 
 dust, amorphous silica, or lycopodium powder. 
 
 Rub the free end of the rod with a piece of rosined leather if 
 it is of wood or metal, or with a wet cloth if it is of glass. It 
 will vibrate longitudinally and give a clear, shrill tone. A slow 
 
 Fig. 
 
 60. Resonance 
 Apparatus 
 
146 SOUND 
 
 motion of the rubber with light pressure is most effective. If 
 the cork diaphragm on the rod rubs too heavily upon the glass 
 tube, or if the rod slips so that it is not held at its middle part, 
 it may be difficult to produce the tone. By properly adjusting 
 the movable piston, the air column in the tube can be tuned 
 
 = Mw ^^^^|<^^' W ^«gi»'"-itii/i'"'iigi< 'iiy' 'ajs./ y Q:: 
 
 Fig. 01. Ddst Figueb Appabatus 
 
 to resonance with this tone, when the positions of the nodes 
 and segments will be clearly shown by the cork dust. After 
 the adjustment appears to be perfect, lift one side of the appa- 
 ratus and jar it, so that the dust will lie in a line part way up 
 on the side of the tube; then give the rod a single stroke, 
 making a 'clear sound. If the adjiistment is perfect, the dust in 
 the segments of the waves will be lifted into the air in flutings 
 and will fall to the bottom, lying in loops like festoons himg 
 from the points corresponding to the nodes. These nodal cusps 
 where the dust has not been disturbed should be obtained as 
 sharp as possible. By measuring from a well-formed node near 
 one end to one near the other end, determine the average dis- 
 tance between the nodes. Shake the dust back against the 
 side of the tube, and repeat the observations. The distance 
 between the nodes is half the wave length of the sound in air 
 at the temperature of the experiment. The length of the rod 
 is half the wave length of the same sound in the material of 
 the rod. The velocities in the two substances are proportional 
 to the wave lengths. Knowing the velocity in air (Art. 114), 
 calculate the velocity in the material of the rod. 
 
 Fill the tube with a gas and determine the wave length of the 
 sound in the gas. Since the frequency of the sound is the same 
 whether the tube is filled with gas or air, the velocities in the 
 two media are proportional to the wave lengths. Calculate the 
 velocity in the gas. 
 
 Reference. — Kohlrauach, Physical Measurements, p. 140. 
 
INTEEFERENCE OF SOUND 
 
 147 
 
 117. Young's Modulus by the Velocity of Sound. — The velocity 
 of sound in a solid is 
 
 V- 
 
 Mheing Young's modulus (Art. 73) and I) the density. If the 
 material is in such form that the velocity of sound in it can 
 be determined by the method of the preceding article, and its 
 density is known, Young's modulus may be computed at once 
 from this relation. The method is a valuable one. 
 
 LVn. VELOCITY AND WAVE LENGTH OF SOUND BY 
 INTERFERENCE 
 
 Determine the velocity of sound in air and in a gas, and And the wave length 
 and frequency of the sound of an organ pipe. 
 
 118. Interference of Sound. — An apparatus for studying 
 sound interference consists essentially of a tube which between 
 its extremities is divided into two branches, one of which can 
 
 I. Ill 
 
 |l"l I' 
 
 Fig. 62. ApPAEATUS for SoDND iNTEEFERENdE 
 
 be lengthened a known amount by a slide. A sound produced 
 at one end will be divided into two parts, which, after traversing 
 the two branches, will again be united. If one branch is longer 
 than the other by an odd number of half wave lengths of the 
 
148 SOUND 
 
 sound, the two parts will meet in opposite phases, and inter- 
 ference will result. The intensity of the combined vibrations 
 may be clearly shown by placing a manometric capsule on the 
 united tube (Fig. 62) and observing the small flame of the gas 
 jet in a revolving mirror. 
 
 Sound a fork of known frequency before the open end of the 
 tube. To reenforce the sound a resonator may be placed in 
 the tube, or the open end of the resonant box of the fork may 
 be held to the tube. If the branches of the tube are of the 
 same length, the jet wiU. be strongly agitated, being affected by 
 the sum of the two parts of the original sound. Now gradually 
 lengthen one branch till the jet ceases to vibrate or until a 
 minimum is found. The quiescence of the jet signifies that the 
 two parts of the original sound, having traversed different paths, 
 reach the ends of the tubes in opposite phases and interfere; 
 hence one branch of the tube has been lengthened by a half 
 wave length of the sound. The velocity of sound is equal to 
 the product of the wave length and frequency of vibration. 
 Determine the velocity of sound in air at the temperature of 
 the experiment, and reduce the result to 0°. 
 
 Place a thin rubber diaphragm over the open end of the tube 
 and fill it with gas. Determine the velocity of the same sound 
 in the gas. 
 
 Remove the diaphragm and gas from the tube and, using the 
 velocity of sound in air just determined, find the wave length of 
 the sound of a giveji organ pipe, and thence its frequency. 
 
 Instead of unitipg the two branches, a manometric capsule is 
 sometimes attached to the end of each, the gas jets of which will 
 indicate the state of vibration in the tube. A third gas jet is 
 connected to both capsules and is under the influence of the 
 combined effect of the vibrations in both branches, indicating 
 the condition of interference. 
 
 Ebfbeence. — Koenig, Experiences d'Aoostique, p. 76. 
 
CHAPTER VIII 
 
 VIBEATING STEINGS AND AIE COLUMNS, AND 
 SOUND ANALYSIS 
 
 LVm. LAWS OF VIBRATING STRINGS WITH THE SONOMETER 
 
 Prove the laws of transverse vibrations of musical strings. 
 
 119. Laws of Vibrating Strings. — For studying the phenomena 
 of vibrating strings the sonometer is often used. It consists of 
 a resonant box upon which may be stretched several musical 
 strings, whose tensions and lengths can be varied at will. Com- 
 monly there are two strings : one permanently attached to the 
 sonometer, its tension being adjustable by a thumb screw; while 
 the other string may be readily attached to the sonometer or 
 removed from it, and at one end it passes over a pulley, so that 
 it may be stretched by weights. The permanent wire may be 
 called the comparison wire. The wires usually pass over fixed 
 bridges one meter apart; this is therefore their maximum 
 sounding length, while their effectual lengths may be made less 
 by movable bridges. The sound is produced with a violin bow 
 or by plucking with the finger. 
 
 The laws of the transverse vibration of strings are contained 
 in the formula . — 
 
 n being the number of vibrations per second given by a string of 
 length I, radius r, density d, and under a tension of t dynes. 
 
 (a) Calculate the number of vibrations per second given by a 
 string, using the above formula ; also determine its frequency by 
 comparison with a standard tuning fork. 
 
 149 
 
150 SOUND 
 
 (b) The number of vibrations of a string is inversely as its 
 length. Tune two strings in unison. When they are nearly in 
 unison beats may be heard, which diminish in frequency as unison 
 is approached. Shorten the effectual length of one string till 
 it gives the octave of its original tone as compared with the 
 unaltered string. The number of vibrations has been doubled 
 and the length halved. Show that the law is true for tones 
 two and three octaves higher than the original tone. 
 
 Prove the law by comparing the lengths of a string which 
 gives tones in unison with two tuning forks of known frequency. 
 
 (c) The number of vibrations is inversely as the radius of the 
 string. Select two wires of the same material but of different 
 diameters. Attach the heavier wire to the sonometer, stretch- 
 ing it with a weight sufficient to cause it to give a good tone. 
 TUne the comparison wire in unison with this. Substitute the 
 smaller wire for the first, stretching it with the same weight. 
 Determine the effectual length of the comparison wire, which 
 gives a tone of this pitch, using the movable bridge. The 
 effectual lengths of the comparison wire should be proportional 
 to the radii of the two given wires. 
 
 (d) The number of vibrations is inversely as the square root 
 of the density of the string. Select two strings of the same 
 diameter but of materials differing greatly in density, as, for in- 
 stance, wires of steel and platinum, or aluminum and copper, or 
 catgut and brass. Stretch the denser string upon the sonom- 
 eter, and tune the comparison wire in unison. Replace the 
 dense wire with the less dense, using the same load for stretch- 
 ing ; and with the movable bridge find the effectual length of 
 the comparison wire when in unison, and thus prove the law. 
 
 (e) The number of vibrations of a string is directly as the 
 square root of the tension. Place a string upon the sonometer 
 and stretch it with a load of one kilogram. Tune the compari- 
 son wire in unison with it. Make the load four kilograms, and 
 the pitch should be varied one octave. By finding the effectual 
 lengths of the comparison wire when it is in unison with the given 
 wire under tensions of two or three kilograms, prove the law. 
 
 Ebfbkence. — Daniell, Physics, pp. 434-440. 
 
VIBRATING AIR COLUMNS 151 
 
 120. Melde's Experiments with Vibrating Strings. — The vari- 
 ous phenomena of transversely vibrating strings may be beauti- 
 fully shown by attaching one end of a string to the prong of an 
 electrically vibrated tuning fork, and so adjusting the length, 
 tension, and density of the string that it will vibrate with one 
 or more visible loops and nodes. Often vibrations of one and 
 more segments can be seen superposed. The complex vibrations 
 of a string attached to two forks are very instructive. 
 
 LIX. FORMS OF VIBRATION IN ORGAN PIPES BY 
 MANOMETRIC FLAMES 
 
 Determine the relation of the nodes and ventral segments of the vibrating 
 air column in open and closed organ pipes. Study the combinations 
 of the sounds of two or more organ pipes. 
 
 121. Nodes in Organ Pipes. — Using an open organ pipe pro- 
 vided with manometric capsules, show, when the pipe is sounding 
 a fundamental tone, that there is but one node, this being at the 
 center. Show that there are two nodes when the first overtone is 
 sounded, and three nodes when the second overtone is produced. 
 
 Experiment similarly with a closed pipe, finding the locations 
 of the nodes and ventral segments. 
 
 Represent the results of the various experiments by sketches. 
 
 122. Combined Sounds. — Study the composition of the sounds 
 of various pairs of organ pipes by connecting their manometric 
 capsules to a single gas flame, which therefore vibrates under 
 the combined effects of the two pipes. Use pipes nearly in 
 unison, to show the effects of beats, and pipes in various ratios, 
 such as 3 : 4, 2 : 3, 8 : 15, etc. 
 
 123. Organ Pipe with Water Trough A stopped pipe is 
 
 sometimes used in which a manometric capsule connection can 
 be placed at any point of its length through a longitudinal slot, 
 the slot being closed air tight by laying the pipe in a trough of 
 water. With such a pipe study the condition of vibration of 
 the entire air column. 
 
 Keference. — Koenig, Experiences d'Acoustique, pp. 47-56 and 206-217. 
 
152 SOUND 
 
 LX. COMPOSITION OF A SOUND WITH RESONATORS 
 
 Determine the number and relative intensities of the overtones present in 
 the sound of an open and a closed organ pipe. 
 
 124. Sound Analysis. — The sound analyzer consists of a 
 series of adjustable resonators which may be connected to a 
 number of manometric capsules, and the vibrations of the gas 
 flames of the capsules can be observed in a rotating mirror. 
 
 Adjust the resonators connected to the capsules so that they 
 are in tune with the first eight possible partial tones of a sound 
 whose pitch is Ut^. Sound a pipe of this pitch in front of the 
 resonators ; careful observation of the series of flames will 
 determine what partial tones are present and what are their 
 relative intensities. 
 
 It is necessary that the resonators should be exactly tuned. 
 For a tone to which they have not been adjusted by the maker, 
 the following method may be used. Tune a sonometer string 
 in unison with the given fundamental. Then tune the reso- 
 nators to the fundamental and harmonics of this string, which 
 latter can be easily produced. The proper adjustment of a 
 resonator can be determined quickly by detaching the rubber 
 tube from the capsule and placing the end in the ear. 
 
 Refekence. — Koenig, Experiences d'Acoustique, p. 70. 
 
Part IY — Heat 
 chapter ix 
 
 EXPANSION 
 
 LXX. LINEAR EXPANSION OF A SOLID WITH THE 
 COMPARATOR 
 
 Determine the coefficient of linear expansion of a glass tube. 
 
 125. Coefficient of Linear Expansion. — The increase in unit 
 length of a substance resulting from one degree increase in its 
 temperature is its coefficient of linear expansion. As this value 
 varies with the temperature it is necessary to specify the tem- 
 perature corresponding to any given numerical value. If a rod 
 at two temperatures t^ and t^, has the lengths Zj and l^, the 
 mean coefficient of expansion between these temperatures is 
 
 hih-h) 
 
 To measure the coefficient of expansion with precision requires 
 elaborate apparatus, both for determining the increase in length, 
 which is relatively a small quantity, and for measuring the 
 temperature change. When the substance can be had in the 
 form of a tube or rod about a meter long, the following simple 
 method is usually sufficient. 
 
 A glass tube about 130 cm long and 4 or 5 cm in diameter 
 has two openings blown in one side near the ends, at a distance 
 apart somewhat less than the length of the inaterial to be 
 
 153 
 
154 
 
 HEAT 
 
 investigated. Corks are fitted to the large tube to support the 
 rod in the manner shown in Fig. 63, These corks are inserted 
 under the holes in the glass tube, and the cork is cut away so 
 that fine marks placed on the specimen may be observed with 
 microscopes. These corks are perforated with short tubes. 
 The ends of the large tube are closed by other corks also per- 
 forated with short tubes. A mineral wool jacket covers the 
 large tube to protect it from outside temperature changes. By 
 
 4p^m 
 
 X- 
 
 ¥ 
 
 a:)&mi^s:ssm^ssse^^!^s^S!S^^^i^m: 
 
 5v3iTistssj»3Hi;^ss«^2SB5^ses3a»^aia® 
 
 Fig. 63. Linear Expansion Apparatus 
 
 
 this arrangement the inner rod may be surrounded with a cur- 
 rent of water or steam while it is observed. A metal frame of 
 the form sketched in Fig. 40 conveniently supports the micro- 
 scopes and the expansion apparatus. 
 
 Pass a current of water through the tube till the temperature 
 becomes constant. Place the tube so that the mark at one end 
 is under the index microscope, X, and then set the micrometer 
 microscope, M, to the other mark. Empty the water from the 
 tube and connect a steam supply to the apparatus. Allow the 
 steam to pass freely till the temperature is again constant, that 
 is till the rod ceases to expand. Set one end to the index 
 microscope, and with the micrometer determine the increase in 
 length, Zj — Zj of the formula. Do not disturb the microscopes 
 in any way between the first and second observations, not even 
 for focusing ; adjust the tube by its supports for this purpose. 
 The large tube may be moved as convenience requires. 
 
 Remove the expansion apparatus from the bed of the com- 
 parator, place a meter bar under the microscopes, and focus by 
 adjusting the supports of the bar, not by moving the micro- 
 scopes. Determine the length of the rod when it was cold, to 
 
CUBICAL EXPANSION 
 
 155 
 
 the nearest half millimeter ; this is the quantity l^ of the formula. 
 The temperature of the water is directly measured, and that of 
 the steam is found with the aid of the barometric pressure. 
 
 @ 
 
 LXXI. CUBICAL EXPANSION OF GLASS WITH THE WEIGHT 
 THERMOMETER 
 
 Determine the mean coefficient of cubical expansion of glass. 
 
 126. The Weight Thermometer. — In experiments on the vol- 
 umetric expansion of liquids and solids, an instrument called the 
 weight thermometer is often used. It consists of a cylindrical 
 bulb of about 15 ccm capacity, with a stem of about 0.5 mm 
 bore, open at the end, and bent in the form shown in Fig. 64. 
 The bulb is held upright in a small 
 stand, which also supports a cup under 
 the end of the stem. If the weight 
 of mercury which fills the bulb at 0° 
 and 100° has been determined, any 
 other temperature may be determined 
 by finding the weight which fills it 
 at this temperature. Weighings can 
 be performed with great precision, and 
 the method obviates the necessity for 
 calibrating the stem and scale of a 
 thermometer; but the operations of 
 weighing are too slow for ordinary 
 temperature observations. The 
 instrument may be used to determine 
 the relative expansions of liquids, and 
 if the absolute expansion of mercury is assumed, the volumetric 
 expansion of the glass envelope may be measured. The present 
 exercise is that of the latter operation. 
 
 The thermometer, including the stand and cup, empty, clean 
 and dry, is carefully weighed. The bulb is to be filled with 
 mercury : gently warm it, dip the stem into mercury, and upon 
 cooling some mercury will be drawn in ; repeat the operations 
 
 wi 
 
 Fig. 64. Weight 
 Thekmombtek 
 
156 HEAT 
 
 till the bulb and stem are completely filled. A better method, 
 which, however, requires care and skill to prevent breaking the 
 glass, is to boil the mercury which partially fills the bulb, when 
 upon dipping the stem into mercury the vapor will be con- 
 densed and the bulb will be entirely filled. With the stem 
 still in mercury, place the bulb in a bath of crushed ice, and 
 cool it to 0°. Remove the mercury supply, being sure that the 
 stem is full to the end. Holding the empty cup under the 
 stem to catch all the mercury that may run out, take the bulb 
 from the ice, dry it and place it in its support. Weigh again 
 to determine the weight of mercury that filled it at 0°; the 
 mercury will run out as the bulb warms, but as this is caught 
 by the cup it does not interfere with the weighing. Place the 
 bulb in a thermometer boiler or other convenient steam bath, 
 leaving it until its temperature is that of the steam. Again 
 place the bulb in its support and weigh it, the cup being empty. 
 Let TOj be the mass of the mercury filling the bulb at 0°, and 
 m^ that which fills it at the temperature of the steam, t°. The 
 
 mi 
 
 volume at 0° of the mass of mercury m, is -jl, S being the den- 
 
 o 
 
 sity of mercury ; this is also the volume of the bulb at 0°. If 
 
 the coefficient of cubical expansion of the bulb is 3 /3, its volume 
 
 at ^° is -r;! (1 + 3 ^f). The mass of mercury which fills the bulb 
 
 ^ m 
 
 at t° is m, ; the volume of this mass of mercury at 0° is -^ , but 
 
 o 
 
 at t° it has the volume -^ (1 + 3 at), 3 a being the coefficient of 
 
 o . 
 
 cubical expansion of mercury. These two volumes are equal; 
 that is, 
 
 ^(l + 8/3«) = ^(l-(-3a«). 
 
 from which it follows that 
 
 3/3 = 3a^-- 
 
 wij _ 1 Wi — fn^ 
 
 m^ t Wj 
 
 The mean coefficient of cubical expansion of mercury, 3 a, 
 between 0° and 100° is 0.000182. 
 
THE DILATOMETER 
 
 157 
 
 A y 
 
 LXXn. EXPANSION OF LIQUIDS WITH THE DILATOMETER 
 
 Determine the coefficient of cubical expansion of ■water. 
 
 127. Coefficient of Expansion of Liquids. — The dilatometer is 
 a simple glass bulb with an open stem, used for measuring the 
 expansion of liquids after the manner of the ordinary ther- 
 mometer. Two convenient forms are shown in Fig. 65. The 
 capUlary tube attached to the bottom of the bulb and bent up- 
 ward facilitates filling by suction, after which 
 the end is closed with sealing wax or by a 
 stopcock. The other form is filled by gently 
 heating, placing some liquid in the enlarged 
 top, coohng the bulb to draw the liquid into 
 it, and repeating the operation till the bulb is 
 full. The stem may be graduated, or an 
 auxiliary scale may be used to measure the 
 apparent expansion. 
 
 The constants of the dilatometer are to be 
 determined by preliminary experiment. These 
 are the weight of the instrument when empty, 
 clean, and dry; the volume of the bulb at 0° 
 to the zero line of the graduated scale, and 
 the volume of the bore of the stem per unit 
 of the divided scale ; and the coefficient of 
 the voluminal expansion of the glass bulb. 
 If the observations for the coefficient of expan- 
 sion of the bulb are carried out as described 
 in the preceding exercise (Art. 126), one addi- 
 tional weighing, that of the weight of a measured length of a 
 thread of mercury in the stem (Art. 62), suffices for the 
 determination of all these constants. 
 
 The dilatometer is filled with the liquid to be investigated, so 
 that when the temperature is that of the ice bath, 0°, the column 
 of liquid in the stem rises only a small distance above the zero 
 line. Let V^ be the volume of the liquid when the temperature 
 has become constant, equal to the volume of the bulb to the 
 
 / 
 
 /P 
 
 v_y 
 
 ^J 
 
 Fig. 65 
 
 DiLATOMETERS 
 
158 HEAT 
 
 zero line plus the volume of that portion of the stem above 
 this line which is occupied by liquid. 
 
 Transfer the dilatometer to the warm bath (steam) and ascer- 
 tain the apparent volume of the liquid, V^, that is, the standard 
 volume of the bulb plus the volume of that portion of the stem 
 now occupied by liquid. 
 
 If 3 a is the coefficient of cubical expansion of the liquid, by 
 heating it from 0° to t°, its volume has been increased by an 
 amount represented by FjS at. It now occupies a volume of the 
 glass which at 0° is V^, but the coefficient of cubical expansion 
 of the glass bulb being 3 /8, the increase in this volume at t° is 
 FjS 0t. The observed apparent increase in volume, V^— Fj, is 
 equal to the difference between the actual increases in the 
 volumes of the liquid and the envelope; that is, 
 
 Fg - Fj = Fi 3 a« - Fg 3 /3e, 
 
 from which the mean coefficient of cubical expansion of the 
 liquid is found to be 
 
 V V — V 
 
 F, r,t 
 
 Repekence. — Ostwald, Physioo-Chemloal Measurements, p. 108. 
 
CHAPTER X 
 THERMOMETRY 
 
 T.XXTTT. CONSTANTS AND ERRORS OF A THERMOMETER 
 BY CALIBRATION 
 
 (a) Determine the fundamental interval of a thermometer. 
 
 (b) Calibrate the scale of a thermometer at ten equidistant points. 
 
 (c) Form a complete table of corrections to a thermometer. 
 
 128. Thermometry. — The mercury-in-glass thermometer is 
 the almost universal laboratory instrument for the measurement 
 of temperature. Such skill has been attained in the manufacture 
 of thermometers that the indications of those now obtainable 
 from reliable makers may be depended upon to a few hundredths 
 of a degree. When greater precision is needed the thermome- 
 ter may be submitted to one of the several national standardiz- 
 ing bureaus, who will furnish certificates of such constants and 
 corrections as are desired. Still it will often be necessary 
 for the user himself to make the many and somewhat difficult 
 observations for standardizing the thermometer. 
 
 The mercury thermometer depends upon observations of the 
 relative expansion of mercury and glass. The properties of glass 
 are such that the bulb of a thermometer, having been expanded 
 by exposure to a high temperature, upon being transferred to a 
 low one contracts quickly for the greater part of the possible 
 contraction ; while there remains a small residual contraction 
 which proceeds very slowly over a period of days or weeks. 
 The amount of this residual contraction depends mainly upon 
 the nature of the glass and the difference between the two 
 temperatures. Hence for the same temperature a thermometer 
 may give many different indications, depending upon its previous 
 
 159 
 
160 HEAT 
 
 treatment. For thermometers of Jena Normal Glass the error 
 of this kind, due to a change of temperature of 100°, is about 
 0°.l. To obviate this difficulty, in precise thermometry, a tem- 
 perature is always referred to that zero point which is found, or 
 would be found, immediately after the temperature observation. 
 
 The measurement of temperatures by the mercury-in-glass 
 thermometer is founded upon the following procedure. The 
 thermometer is subjected to the temperature of steam under 
 standard conditions until the indication is constant; the point 
 corresponding to what would be the end of the mercury column, 
 if the steam were under a pressure of 76 cm of mercury, is 100°. 
 The thermometer is quickly transferred to melting ice ; the lowest 
 point reached by the mercury is 0°. A one-hundredth part of 
 the volume between the two points is one degree. The ther- 
 mometer is subjected to the unknown temperature tiLl its indica- 
 tion is constant; it is then quickly transferred to melting ice, 
 and the lowest indication is observed, which may be different 
 from the fotmer ice point. The temperature 'in degrees is the 
 ratio of the volume between these last two points to the degree 
 volume as defined. 
 
 To standardize a thermometer its fundamental interval must 
 be found by locating the fixed points ; then the relation of the 
 internal volume of the tube to the scale which has been pro- 
 vided is to be determined, that is the scale is to be calibrated. 
 In addition to these corrections, which should never be neglected, 
 there are several others of less importance which are due to the 
 following causes : the internal pressure of the mercury column 
 which varies in height with the temperature, the external pres- 
 sure due to varying barometric pressure, and the varying depth 
 to which the thermometer may be immersed in a liquid. 
 
 129. Calibration of a Thermometer. — The unavoidable variar 
 tions in the internal bore of the tube, anxi the errors of gradua- 
 tion, make it necessary to calibrate a thermometer in order that 
 its readings may be reduced to the ideal scale. 
 
 The calibration of volume requires the separation of a thread 
 of mercury from the column the length of which shall be approxi- 
 mately an aliquot part of the total length of the scale, — for 
 
THEEMOMETRY 161 
 
 example 10°, on a thermometer scale from 0° to 100°. The ther- 
 mometer is inverted, and by gently tapping the end on the hand 
 the mercury is caused to run down into the tube. Sometimes 
 a portion is separated by this operation alone ; if not, a sudden 
 turning of the thermometer with the proper jerking or tapping 
 will often cause a separation at the neck of the bulb. This 
 separated column is moved in the tube till its lower end is at 
 some observed convenient mark, — for instance 20° ; the thermom- 
 eter is then placed horizontally, and the bulb is gently heated 
 with the hand or otherwise till the mercury in the bulb expands 
 and joins the column in the tube. The heat is withdrawn, and 
 as the temperature falls the entire column of mercury is slowly 
 withdrawn. When the upper end of the column is at that 
 point of the scale which is above the point of junction by the 
 length of thread desired, — at 30° in this instance, — the ther- 
 mometer is given a longitudinal shock, when the column will 
 again separate at the previous point of junction, leaving a thread 
 of the desired length. Patience and repeated trials are some- 
 times necessary to success in this operation. The separation 
 results from the presence of a small bubble of gas which adheres 
 to the glass at the place of junction. 
 
 If the mercury thread is very fine, the separation in this man- 
 ner is not practicable, and it may be necessary to heat the tube 
 very carefully in the tip of a minute flame, continually turning 
 the tube on its axis till the mercury is vaporized at the desired 
 point of separation. Further heating will distill mercury to 
 the separated thread, or from it, to make its length exactly that 
 required. This method is rather dangerous, and requires skill'; 
 even when it has been applied successfully the thermometer, if 
 of thick-waUed tubing, may at some future time break where 
 the heat was applied. If several threads are required after one 
 has been produced by heating, the others may usually be obtained 
 by the first method. 
 
 Place the thread of mercury between 0° and 10°, a cooling 
 mixture being placed on the bulb to prevent the loss of the 
 thread. To avoid the use of the cooling mixture, some mercury 
 might have been placed in the bulb at the top of the thermometer 
 
162 HEAT 
 
 before the ten-degree thread was separated. Read the length of 
 the thread, in degrees, as accurately as possible, using a low- 
 power microscope. Place the thread between 10° and 20°, and 
 read its length again; place it between 20° and 30°, and so on 
 to the end. Find the average of the ten lengths observed ; the 
 amount that must be added algebraically to each observed length 
 to make it equal to the average, is the correction to the corre- 
 sponding portion of the thermometer. The successive sums of 
 these corrections are the corrections to the volumes included 
 between the zero mark and the successive ten-degree marks. 
 
 The most complete calibration of a thermometer at ten points 
 requires the making of fifty-four observations with mercury 
 threads 10°, 20°, 30°, . . ., 90° in length. The making of the 
 observations and their reduction is fully described in Art. 30. 
 A calibration in other aliquot parts may be made in the manner 
 described for ten parts. 
 
 Somewhat greater precision may be secured by making a double 
 calibration with the dividing engine, as explained in Art. 27. 
 The scale is first calibrated as there illustrated. Then mercury 
 threads are measured with the dividing engine in the same 
 manner as was the scale ; this will give the corrections to the 
 'volumes referred to an ideal scale of lengths in millimeters. 
 The differences between the several volume corrections and the 
 corresponding scale corrections, each referred to the ideal scale, 
 are the corrections of the volumes referred to the actual ther- 
 mometer scale. These corrections, expressed in millimeters, 
 are reduced to degrees by the aid of the measured length of a 
 degree in millimeters. 
 
 130. Fundamental Interval of a Thermometer. — To deter- 
 mine the boiling point the thermometer is placed in a steam 
 bath obtained by means of the well-known form of boiliug-point 
 apparatus. The entire mercury column should be in the steam, 
 and the bulb must always be above the water. The steam 
 should be generated copiously, and it must have a free exit 
 from the boiler. If the steam does not escape freely, the excess 
 of pressure produced in the boiler may be measured and added 
 to the barometer pressure. The reading is best made with a 
 
THEEMOMETRY 163 
 
 telescope, the thermometer being withdrawn from the steam 
 just sufficiently to allow the mercury to be seen. If the cor- 
 rected barometric pressure, including the excess of pressure in 
 the boiler, is B, then the temperature of the steam is 
 
 < = 100°-0°.375(76.0-£). 
 
 It may be more convenient to take the temperature of the 
 boiling point from tables (Table 16). 
 
 The thermometer is to be transferred from the steam to the 
 ice as quickly as is consistent with safety. It may without 
 danger be plunged in ice from a temperature of 40°. To facili- 
 tate this transfer, two dishes of water may be prepared, — one 
 at a temperature of about 60°, and the other at about 30° ; 
 the thermometer is dipped into these successively, and finally 
 into the ice. 
 
 The ice bath is made of pure, clear ice, finely crushed, and 
 closely packed into a receptacle of several liters capacity, the 
 interstices being filled with distilled water. The thermometer 
 should be immersed so that only the end of the mercury column 
 is visible in a reading telescope whose line of sight is on a level 
 with the top of the dish. The thermometer should be gently 
 tapped on the end, and the lowest indication is the zero point 
 desired. This will occur in perhaps two minutes after removal 
 from the steam, and will be followed by a slow rising of the 
 mercury. 
 
 The correction to the boiling point is the quantity which 
 must be added to the interval in scale degrees between the 
 observed steam and ice points to make this equal to the degrees 
 of the true boiling point. This correction for the error in the 
 fundamental interval affects the 100° point by its full value. 
 There is no correction because of this error at 0°. All other 
 readings require corrections, which are in the same proportion 
 to the boiling-point correction as the given reading is to 100°. 
 
 131. Table of Corrections for a Thermometer. — The explana- 
 tion of the method of forming a table of corrections for a ther- 
 mometer can best be made by means of a complete numerical 
 example. 
 
164 
 
 HEAT 
 
 Calibration or Jena Normal Glass Thermometer, Gerhardt 
 4234, AT the Ten-Degree Points from 0° to 100° 
 
 Scale from -6° to + 106°, divided in ft" 
 
 July Zlf, 1902 
 
 A thread of mercury about ten degrees long was obtained by heating ; 
 by putting a little cotton wet with ether on the bulb, this was placed 
 between 0° and 10°, and then without the ether it was placed between 10° 
 and 20°, between 20° and 30°, etc., and finally between 90° and 100°. 
 Three sets of such observations were made. 
 
 Position 
 
 . 
 
 Length 
 
 
 Mean 
 
 COEBBCTION 
 
 S COBEECTION 
 
 0° 
 
 
 
 
 
 
 0°.000 
 
 0°-10 
 
 9°. 88 
 
 9°. 88 
 
 9°. 88 
 
 9°. 880 
 
 +0°.0648 
 
 + .065 
 
 10-20 
 
 .90 
 
 .90 
 
 .90 
 
 .900 
 
 + .0448 
 
 .+ .110 
 
 20-30 
 
 .93 
 
 .93 
 
 .92 
 
 .937 
 
 + .0078 
 
 + .117 
 
 30-40 
 
 .94 
 
 .95 
 
 .95 
 
 .947 
 
 - .0022 
 
 + .115 
 
 40-50 
 
 .96 
 
 .96 
 
 .96 
 
 .960 
 
 - .0152 
 
 + .100 
 
 50-60 
 
 .96 
 
 .96 
 
 .96 
 
 .960 
 
 - .0152 
 
 + .084 
 
 60-70 
 
 .96 
 
 .96 
 
 .96 
 
 .960 
 
 - .0152 
 
 + .069 
 
 70-80 
 
 .95 
 
 .96 
 
 .96 
 
 .957 
 
 - .0122 
 
 + .057 
 
 80-90 
 
 .98 
 
 .98 
 
 .98 
 
 .980 
 
 - .0352 
 
 + .022 
 
 90 -100 
 
 .97 
 
 .97 
 
 .96 
 
 .967 
 
 - .0222 
 
 .000 
 
 
 
 
 Averag 
 
 'e 9°. 9448 
 
 
 
 The thermometer was placed in a steam bath, the reduced barometer 
 pressure being 74.61 cm, corresponding to a boiling point of 99°.48. The 
 thermometer read 99°.44. It was immediately transferred to an ice bath, 
 in which the lowest indication was — 0°.01. The fundamental interval of 
 the scale is thus 99°.97, requiring a correction at the 100° point of +0°.03. 
 
 The sum of the calibration correction and that for the fundamental 
 interval gives the correction which when added to the reading at any 
 part of the scale reduces this to the true mercury-in-glass temperature. If, 
 further, we add the correction to reduce to the hydrogen thermometer 
 scale (Art. 133), the result is the correction which when added to the 
 scale reading of any temperature reduces this to the true temperature 
 according to the normal hydrogen scale. The following table shows all 
 these corrections. 
 
THE THERMOMETER 
 
 165 
 
 Readings 
 
 Calibration 
 
 Fundamental 
 intebval 
 
 Mhroukv- 
 
 in-Glass 
 
 Scale 
 
 H. Scale 
 
 Total Corkec- 
 TIONS TO Bead- 
 ING.S FOR H. Scale 
 
 0° 
 
 0°.000 
 
 0°.000 
 
 0°.000 
 
 0°.000 
 
 0°.000 
 
 10 
 
 + 
 
 .065 
 
 + .003 
 
 + .068 
 
 - .055 
 
 + .013 
 
 20 
 
 + 
 
 .110 
 
 + .006 
 
 + .116 
 
 - .090 
 
 + .026 
 
 30 
 
 + 
 
 .117 
 
 + .009 
 
 + .126 
 
 - .109 
 
 + .017 
 
 40 
 
 + 
 
 .115 
 
 + .012 
 
 + .127 
 
 - .115 
 
 + .012 
 
 50 
 
 + 
 
 .100 
 
 + .015 
 
 + .115 
 
 - .109 
 
 + .006 
 
 60 
 
 + 
 
 .084 
 
 + .018 
 
 + .102 
 
 - .096 
 
 + .006 
 
 70 
 
 + 
 
 .069 
 
 + .021 
 
 + .090 
 
 - .076 
 
 + .014 
 
 80 
 
 + 
 
 .057 
 
 + .024 
 
 + .081 
 
 - .053 
 
 + .028 
 
 90 
 
 + 
 
 .022 
 
 + .027 
 
 + .049 
 
 - .027 
 
 + .022 
 
 100 
 
 
 .000 
 
 + .030 
 
 + .030 
 
 .000 
 
 + .030 
 
 These corrections are founded upon the assumption that the zero mark of 
 the scale is the true ice point. According to the principles explained above, 
 the ice point is to be determined immediately after each temperature meas- 
 urement. Then the true temperature in every case is the scale reading 
 plus the corresponding tabular correction, plus the reading of the ice point 
 mUh its sign changed. The ice point will be found of such constancy that 
 it will need only occasional determination. 
 
 It appears from the calibration that the maker has pm-posely constructed 
 the scale of the thermometer to indicate temperatures on the hydrogen 
 normal scale instead of mercury-in-glass temperatures ; and that within 
 the ordinary range of laboratory work the indications of the thermometer 
 without any correction will never be in error more than 0°.02. 
 
 To test the calibration, the temperature of a large jar of water was 
 measured with this and another standard thermometer. The thermom- 
 eters were wholly immersed, the water was frequently stirred, and after 
 some time the results were : 
 
 Beading 
 
 Calibration correction 
 Zero correction . . . 
 Corrected reading . . 
 
 No. 1721 
 
 No. 4234 
 
 24°. 30 
 
 24°. 12 
 
 .003 
 
 + .12 
 
 - .06 
 
 - .01 
 
 24 .24 
 
 24 .23 
 
 132. Exposed Column of Mercury. — The method described for 
 measuring temperature requires that all the mercury be at one 
 temperature. It often happens that a part of the column is 
 necessarily exposed to a temperature different from the temper- 
 ature of the bulb. Let I be the length in degrees of the exposed 
 
166 HEAT 
 
 column, ^1 its mean temperature determined by an auxiliary- 
 thermometer, and <2 the temperature of tKe bulb; then for a 
 Jena Normal Glass thermometer the correction to the reading is 
 
 + 0.000157 I (t^ - tj). 
 
 133. Comparison of Mercury^in-Glass and Hydrogen Thermom- 
 eters. — The scale of the hydrogen thermometer is now accepted 
 as the scientific standard for all temperature measurements. 
 The indications of a mercury thermometer made of Jena Nor- 
 mal Glass (16 III) require the following corrections to corre- 
 spond to the hydrogen thermometer (Chappuis, International 
 Congress of Physics, Paris, 1900). 
 
 Eeadings 0" 10 20 30 40 60 60 70 8P 90 100 
 
 Corrections 0°.000 -0.055 —0.09a -0.109 -0.115 -0.109 -0.096 -0.076 —0.053 -0.027 0.000 
 
 LXXIV. TEMPERATURE AND EXPANSION OF GASES WITH 
 THE AIR THERMOMETER 
 
 Standardize an aix thermometer and measure an unknown temperature. 
 Calculate the coefficient of expansion of the gas. 
 
 134. The Air Thermometer. — The normal temperature scale 
 is that of the hydrogen gas thermometer, founded upon the 
 principles described below. Hydrogen is best suited for low- 
 temperature measurements, while for temperatures above 180° 
 it attacks some kinds of glass and permeates metals. For gen- 
 eral use the best practical standard thermometer uses nitrogen 
 as the thermometric material. 
 
 The gas, or air, thermometer may consist of a bulb of 50 ccm 
 or more capacity, filled with dry air or hydrogen, or other suit- 
 able gas, communicating with a mercury manometer by means 
 of a tube of small bore. Sometimes the bulb is rigidly attached 
 to the manometer as shown in Fig. 66, and sometimes it is con- 
 nected by a long, flexible metal tube. The manometer consists 
 of a long and a short glass tube (the diameter being a centimeter 
 or more) which are joined together and to a mercury cistern. 
 A screw plunger in the cistern regulates the height to which the 
 
THE AIR THERMOMETER 
 
 167 
 
 mercury rises in the manometer. Inside the short manometer 
 tube, near its junction with the capillary tube, is a small index, 
 to which the height of the mercury is always to be adjusted, 
 causing the gas in the bulb to occupy a constant volume. 
 
 The use of the apparatus as a thermometer depends upon 
 the fact that the pressure of a gas at constant volume varies 
 directly as its absolute temperature. 
 The pressure of the inclosed gas is 
 always equal to the barometric height 
 plus the difference between the levels 
 of the mercury in the open and closed 
 arms of the manometer. Corrections 
 must be applied to reduce the barom- 
 eter readings to 0°, and also for 
 the temperature of the manometer 
 (Art. 54). 
 
 To calibrate the thermometer, place 
 the bulb in a steam bath and adjust 
 the manometer till the surface of the 
 mercury touches the index. Deter- 
 mine the difference between the mer- 
 cury levels by reference to the glass 
 scale or, better, with a cathetometer. 
 Let Pj be the sum of the manometer 
 and barometer pressures (reduced to 
 0°) at the temperature, *,, of the steam 
 (Art. 130). Place the bulb in an ice 
 bath, and let P^ be the total pressure 
 of the gas at the same volume as 
 before, and at the temperature, 0°, of 
 the melting ice. 
 
 Now place the bulb in the region 
 the temperature of which, t^, is to be determined, and observe P^,, 
 the total pressure. (Read the barometer at each observation, to 
 note changes in the atmospheric pressure.) Then approximately 
 
 Fig. 66. Air Thermometek 
 
 
168 HEAT 
 
 In this formula it is assumed that the space occupied by the 
 air in the connecting and manometer tubes, which is not varied 
 in temperature, is comparatively insignificant; and the expan- 
 sion of the glass bulb is neglected. If v is the volume of the 
 air space which is heated, and v' the volume which is not 
 heated (determined by mercury calibration), 3 /8 the coefficient 
 of cubical expansion of the glass, and t,. the temperature of the 
 air in the room, then 
 
 " \P,-PoJl -Po \0.003Q7 V 1 + 0.00367 ej J' 
 
 in which 3 /3 may be taken as equal to 0.000025. 
 
 The Coefficient of Expansion of Gas. — From the above meas- 
 urements the coefficient of eipansion of the gas, referred to its 
 volume at 0°, is P _ P 
 
 a = —5 2, 
 
 the symbols having the meanings given above. 
 
 References. — Kohlrausch, Physical Measurements, pp. 93-97; Guillaume, 
 Thermom^trie, Chap. V. 
 
 LXXV. HIGH TEMPERATURES WITH RESISTANCE AND 
 THERMO-ELECTRIC PYROMETERS 
 
 Determine the melting point of zinc. 
 
 135. Measurement of High Temperatures. — An instrument 
 for measuring temperatures, especially those which are above the 
 range of the ordinary mercury thermometer, is called z. pyrometer. 
 A mercury thermometer with a vacuous space is useless for tem- 
 peratures above 300°, and even above 100° the mercury begins 
 to distill from the heated column to the cold part of the stem, 
 causing inaccuracies. Mercury thermometers made of special 
 hard Jena glass to "withstand the heat, and containing carbon 
 dioxide under a pressure of twelve or more atmospheres above 
 the mercury to prevent its vaporization, are capable of measuring 
 temperatures up to 550°. 
 
THE PLATINUM THERMOMETER 169 
 
 For high temperatures, as well as for ordinary and low tem- 
 peratures, the air thermometer is the standard ; but for general 
 use either one of two other instruments has been found more 
 practicable. For precise work the platinum resistance thermom- 
 eter of Callendar and Griffiths is best adapted. It will measure 
 temperatures as high as 1200°, though its precision diminishes 
 above 700°. The thermo-electric thermometer of Le Chatelier 
 is simpler in construction and manipulation, 
 and it can be used by workmen in technical 
 industrial operations. It measures temper- 
 atures up to 1500°, or even to the melting 
 point of platinum, 1775°, but with somewhat 
 less precision than is attained with the platinum 
 thermometer. 
 
 136. The Platinum Thermometer. — The 
 variations of the electrical resistance of a coil 
 of platinum wire, as its temperature changes, 
 furnishes a valuable means of measuring tem- 
 perature. The platinum thermometer (Fig. 67) 
 consists of a coil of pure platinum wire wound 
 on a mica frame, which is protected by a glass 
 or porcelain tube ; copper or platinum wires 
 pass from the coil to binding posts on the end 
 of the tube so far removed from the coil that 
 they are uninjured when the coil is heated. ^'thermomet17" 
 The coil may have "a resistance of about 
 2.6 ohms at 0°, which increases about one ohm for each hundred 
 degrees increase in temperature. The resistance is measured by 
 a Wheatstone's bridge, which, if especially designed for the pur- 
 pose, will indicate variations in the resistance of 0.00001 ohm, 
 corresponding to 0°.001 in the temperature. 
 
 To avoid breaking, the thermometer should be gradually 
 heated to the temperature to which it is to be subjected. This 
 may be done in a muffle, or by placing it in the furnace or cru- 
 cible before the heat is applied. The platinum thermometer is 
 so fragile and requires such care in making the measurements 
 that it is not adapted to indu^rial operations. 
 
170 HEAT 
 
 To measure a temperature, the fundamental interval is first 
 determined, much as for a mercury thermometer (Art. 130), by 
 finding the resistances, -Kg and B^^q, when the coil is at the tem- 
 perature of melting ice and of steam at 76 centimeters pressure. 
 
 The change in resistance per degree is, then, ^'^^qq — ~ I and if 
 
 the resistance at some unknown temperature is Bt the platinum 
 temperature is 
 
 tp-{Bt-B^)-- — ■ 
 
 ■""lOO "~ -'*'o 
 
 This platinum temperature differs from that of the air ther- 
 mometer ; for temperatures up to 600°, the relation between the 
 true temperature, t, and the platinum temperature, tp, is given 
 by the following equation. 
 
 137. The Thermo-Electric Pyrometer. — The electromotive 
 force of a thermo-electric couple varies with its temperature in 
 such a manner that it may be used as a trustworthy measure 
 of temperature. The thermo-electric pyrometer consists of a 
 thermo-electric couple made of pure, platinum and an alloy of 
 platinum with ten per cent of iridium, fused together. The 
 junction is protected by a porcelain tube and is to be placed 
 in the high temperature to be measured. Commonly the cold 
 junction does not exist, though for research work it may be 
 used, being placed in steam or ice. A thermo-electromotive 
 force is generated which is approximately proportional to the 
 temperature. If one junction is at 0° and the other at 300°, 
 the electromotive force is 2.26 millivolts ; at 600°, 5.14 milli- 
 volts; at 900°, 8.33 miUivolts; at 1200°, 11.83 millivolts; and 
 at 1500°, 15.64 millivolts. 
 
 The thermo-electromotive force is very conveniently meas- 
 ured with a portable D'Arsonval galvanometer (Art. 209), in 
 which the direct deflections are observed, the scale being an 
 arbitrary one, or corresponding approximately to the tempera- 
 tures. The method possesses the great advantage that the 
 
HIGH TEMPERATUEES 
 
 171 
 
 galvanometer may be permanently placed, while the thermo- 
 element is taken to any desired distance for use. The lead 
 wires should be protected from any variation in temperature. 
 The indications of the pyrometer are almost instantaneous, and 
 the tube with the couple may be small in size ; in some cases 
 the tube may be dispensed with, only the junction being heated. 
 
 No simple equation has been found connecting temperature 
 and thermo-electromotive force in general, and the best pro- 
 cedure is to calibrate a given thermo-electric couple by reference 
 to well-established fixed points (Art. 138), interpolating graphi- 
 cally for the unknown temperature. Two or more fixed points 
 should be chosen which are as near as possible to the tempera- 
 ture being measured. The error of such a measurement need 
 not be greater than one per cent. 
 
 For research work the electromotive force, e, should be 
 accurately measured by comparison with a standard cell by the 
 compensation method described in Arts. 250 and 251. Then 
 any temperature, t, measured with a given thermo-couple is 
 determined by the equation 
 
 log e = a log t + b, 
 
 the constants a and h having been determined by measuring the 
 electromotive force for two known temperatures. Le Chatelier 
 found the constants to give the equation 
 
 log e = 1.2196 log t + 0.302. 
 
 With this method the error of a determination may not be 
 greater than one part in five hundred. 
 
 138. Fixed Points for High Temperatures. — The following 
 well-determined high temperatures are suitable for standardizing 
 any form of pyrometer. 
 
 Ebullition of water . . 100° 
 
 Ebullition of naphthalene 218° 
 
 Fusion of tin .... 232° 
 
 Fusion of zinc .... 419° 
 
 Ebullition of sulphur 
 
 445°.2 
 
 Fusion of aluminum . 
 
 657° 
 
 Ebullition of zinc 
 
 930° 
 
 Fusion of gold . . . 
 
 1064° 
 
 Fusion of copper . . 
 
 1084° 
 
 Fusion of platinum 
 
 1775° 
 
172 HEAT 
 
 When measuring the fusion point of metals which are obtain- 
 able in any desired quantities the metal is melted in a crucible, 
 and then the pyrometer is inserted. The crucible is allowed to 
 cool, and the stationary temperature of solidification is easily 
 observed. With small quantities of metal, and for the ebulli- 
 tion points, special methods are required which are described in 
 the references. 
 
 References. — Le Cfiatelier, High Temperature Measurements, pp. 83-128 
 and 200-208 ; Chappuis, and Bams, Rapports au Congrfes International de 
 Physique, Paris, 1900, Tome I, pp. 144-177 ; Chappuis, Travaux et M^moires 
 du Bureau International des Poids et Mesures, Tome XII, Part 3. 
 
CHAPTER XI 
 GALORIMETEY 
 
 LXXVI. CALORIMETER CONSTANTS BY CALCULATION AND 
 EXPERIMENT 
 
 Determine the heat capacity, of a calorimeter and the correction for radia- 
 tion. (This problem is to be performed as part of some calorimetric 
 measvirement.) 
 
 139. Calorimetry. — The measurement of the quantity of heat 
 which appears or disappears during a physical or chemical change 
 is the object of calorimetry. The changes usually considered are 
 fusion, vaporization, combustion, solution, and change of tem- 
 perature. The unit of heat is ^-^^ of the quantity of heat 
 required to change the temperature of 1 gram of water from 0° 
 to 100° ; it is called the mean calorie. The calorie represents 
 the actual quantity of heat required to change the temperature 
 of 1 gram of water from 17° to 18°, or from 72° to 73°. At 
 all other temperatures the quantity required is either slightly 
 more or less than 1 calorie, having a maximum value of 1.008 
 calories at 0°, and a minimum of 0.997 at 40°. In terms of the 
 mean calorie, the calorie at 17°, the heat equivalent of fusion 
 of ice is 80.1 calj^, and the mechanical equivalent of the mean 
 calorie is 4.184 x 10^ ergs. 
 
 A calorimeter often consists of a cup of about 500 ccm capacity, 
 for containing water or other hquid, which may be provided with 
 a cover, a stirrer, and a thermometer. The cup is supported on 
 nonconducting points inside another cup, somewhat larger in 
 size, forming an air jacket around the inner cup. Both cups 
 should have bright, polished surfaces. The qualities desired in 
 the material for a calorimeter cup are that it should be thin, 
 
 173 
 
174 HEAT 
 
 bright, a'good conductor of heat, and unaffected by the liquids 
 experimented with. Platinum or silver are the best materials, 
 while for ordinary purposes nickel-plated copper or brass is 
 suitable. Glass is less suitable because of its low conductivity. 
 
 The stirrer may be disk shaped, to be moved up and down 
 through the liquid by means of a handle. A turbine-wheel 
 stirrer is more efficient, and it may be kept in continuous opera- 
 tion by connection with a small motor ; this is almost a necessity 
 if many measurements are to be made. 
 
 Other forms of calorimeters are used according to the require- 
 ments of the method, such as the ice and cooling calorimeters 
 described in Arts. 142, 143, and 146. 
 
 Thermometers for calorimetry measure only changes of tem- 
 perature, which rarely exceed 10°. Such need not have the 
 boiling point on their scales ; the scales may cover only a few 
 degrees, and can therefore be open, the graduations being in J^°, 
 •ji^°, or y^-g^". Such a thermometer needs calibration as to its 
 scale, and must be compared with a standard thermometer to 
 determine the value of the degree of its scale. To overcome 
 capillary resistance, the thermometer may be gently tapped on 
 the end before a reading is taken. Since the observed changes 
 are small, the readings should be made with the greatest pos- 
 sible care, the tenths of the smallest division of the scale being 
 estimated or read with a micrometer microscope. 
 
 140. Heat Capacity of a Calorimeter. — The number of calories 
 required to raise the temperature of the entire calorimeter and 
 contents one degree is its heat capacity. It is the sum of the 
 heat capacity of the water (or other liquid) and the water equivor 
 lent of the other parts that are subject to the temperature changes. 
 These parts are commonly the cup, the stirrer, and the thermom- 
 eter. A direct experiment may be made to determine the water 
 equivalent of these parts, but it is more often calculated from 
 known constants. 
 
 The cup and stirrer are usually of the same material; the 
 product of their combined mass and the specific heat of the 
 substance is the water equivalent. It is assumed that the entire 
 apparatus is equally heated. Since the mass is small, — and the 
 
CALORIMETRY 175 
 
 material is always a good conductor of small specific heat, — and 
 the mass of the water is relatively large, this assumption cannot 
 cause an appreciable error. 
 
 In calculating the heat equivalent of the thermometer, glass 
 being a poor conductor of heat, only that part is to be considered 
 which is immersed. This is of glass and mercury combined in 
 uncertain proportions. Since the heat capacities for equal vol- 
 umes of glass and mercury are nearly the same, 0.46 calories 
 per cubic centimeter, the water equivalent is most conveniently 
 determined from the volume of the immersed portion. This 
 volume may be determined by dipping the thermometer into 
 water contained in a tube with volumetric graduations ; or it 
 may be easily determined while the calorimeter cup and water 
 are on the balance for the determination of their jnass. When 
 the cup has been balanced the thermometer is dipped into the 
 water to the same depth as it will be in the heat experiment ; 
 the apparent increase in the weight of the water, in , grams, is 
 the volume in cubic centimeters of the immersed portion of the 
 thermometer. 
 
 If, then, mj is the mass of the liquid in the calorimeter whose 
 specific heat is Cj, m^ and m^ are the masses of the -cup and stirrer 
 of specific heats c^ and Cg, and v is the volume of the immersed 
 portion of the thermometer, the heat capacity of the calorim- 
 
 K = WjCj + mgffg + "'sCs + 0.46 v. 
 
 If water is the liquid used, Cj may be assumed to be 1, and if the 
 cup and stirrer are of the same material, the formula becomes 
 
 K=m■^+ (m^ + m^ c^ + 0.46 v. 
 
 If the body tested is contained in a receptacle which is intro- 
 duced into the calorimeter with it, the heat capacity of the 
 receptacle must be considered in a manner which will be obvious 
 from the conditions of the experiment. 
 
 141. Radiation Corrections. — In some methods of calori- 
 metric measures the errors due to radiation or absorption are 
 obviated by surrounding the calorimeter with a protecting jacket. 
 This is illustrated by the ice calorimeter and by the cooling 
 
176 HEAT 
 
 calorimeter. In the method of heating, the radiation effects are 
 negligible. In the other methods described, these errors may 
 be of considerable magnitude, and precaution must be taken to 
 diminish and to determine their effects. It is often assumed 
 that the radiation errors will be compensated by starting with 
 the calorimeter at a temperature as much below that of the room 
 as at the end of the experiment the temperature will be above 
 that of the room. The assumption that the calorimeter will 
 gain as much heat during the first half of the experiment as it 
 will lose during the second half is not justified unless the rise 
 in temperature of the calorimeter is proportional to the time. 
 It rarely happens that this is true ; hence it is necessary in precise 
 work to determine the exchange of heat for each observation. 
 The methods employed depend upon Newton's law of cooling, 
 — that the exchange of heat by a body with its surroundings 
 is proportional to the difference in temperature and to the time. 
 When the difference in temperature is small, not exceeding 10°, 
 this law may be applied for the purposes of correction ; but for 
 large temperature differences it is not sufficiently exact. 
 
 It is necessary to determine the radiation errors for two 
 methods of measurement. In one, the method of mixtures, 
 the change in temperature of the calorimeter is very rapid at 
 first, and very slow towards the end of the observation. In the 
 second case, as in determining the heat equivalent of vaporizar 
 tion, the heat is supplied to the calorimeter at a nearly constant 
 rate, causing a more uniform rise in temperature. 
 
 Radiation Correction for Method of Mixtures. — In calorimeter 
 observations by the method of mixtures (Art. 144) the tempera- 
 ture changes very rapidly at first, and as the two substances 
 come nearer to the final common temperature the exchange of 
 heat between them takes place very slowly. It should therefore 
 be arranged that this final temperature shall be about that of 
 the surroundings. The calorimeter should be cooled below the 
 room temperature by an amount equal to the expected tempera- 
 ture change. This change is found either by calculation from 
 the assumed specific heats or by a preliminary experiment. 
 Observations are made of the temperature of the calorimeter at 
 
CALORIMETKY 177 
 
 intervals of a minute, to determine the amount of warming, per 
 minute per degree difference in temperature. The heated body 
 is then placed in the calorimeter at a noted time, and the tem- 
 perature of the calorimeter is read every fifteen or twenty seconds 
 till it has become constant, or until the temperature change is uni- 
 form. The average temperature during the mixture operation is 
 calculated. The difference between the average and the room tem- 
 perature, multiplied by the duration of this part of the experiment 
 and the rate of warming, is the temperature gained by absorption. 
 The correction is this quantity with the negative sign. 
 
 A complete numerical example is given to illustrate this 
 method. It shows that the error due to absorption is small 
 when the operations are carried out as described, being only 
 one part in two hundred in this particular case. 
 
 Specific Heat or Iron 
 
 METHOD OF MIXTURES 
 
 April IS, 1903 
 
 Weight of copper cup and stirrer, 53. .3 g; specific heat of copper, 0.093 ; 
 volume of the immersed thermometer, 2.6 ocm ; weight of water, 452.3 g; 
 from which 
 
 K = 53.3 X 0.093 + 2.6 x 0.46 + 452.3 =-458.5. 
 
 Weight of iron which ia in the form of a hollow cylinder, 243.2 g, heated 
 to 99°.42; temperature of room, 18°.5; thermometer No. 1721, divided 
 
 I o 
 TiT • 
 
 Time 
 
 Tempebatuke 
 
 
 21' 35™ 
 
 00» 
 
 12°. 34 
 
 
 36 
 
 00 
 
 12 .37 
 
 
 37 
 
 00 
 
 12 .40 
 
 
 38 
 
 00 
 
 (12 '.43) 
 
 Iron introduced 
 
 
 38 
 
 20 
 
 16 .80 
 
 
 
 38 
 39 
 
 40 
 00 
 
 17 .09 
 17 .17 
 
 
 . Average 16°. 31 
 
 39 
 
 20 
 
 17 .18 
 
 
 
 39 
 
 40 
 
 17 .18 
 
 Highest temperature . 
 
 
 40 
 
 00 
 
 17 .18 
 
 
 41 
 
 00 
 
 17 .18 
 
 
 ' 
 
 The temperature of the calorimeter at the instant the iron was intro- 
 duced was calculated from the rate of warming. 
 
178 HEAT 
 
 Dviring the first observations the calorimeter gained heat at the rate of 
 0°.0S> per minute, with a mean temperature difference from that of the 
 room of 6°.2, a gain of 0°.005 per degree per minute. After the iron was 
 introduced the average temperature for the two-minute duration of the 
 experiment was 16°.31, differing from that of the room by 2°.2. Therefore 
 the gain in temperature during the experiment because of absorption is 
 0°.005 X 2.2 X 2 = 0°.022. 
 
 The observed temperature change of the calorimeter was 17°.18 — 12°.43 
 = 4°.75 ; while the iron changed 99°.42 - 17°.18 = 82°.24. 
 
 The specific heat of the iron is 
 
 ^ 458.5 X (4.75- 0.02) _p^p3^ 
 243.2 X 82.24 
 
 Radiation Correction when Temperature Change of Calorimeter 
 is Uniform. — In some calorimetric observations the calorimeter 
 changes temperature at a nearly uniform rate (Arts. 147, 148, 
 149, and 150). If the temperature of the calorimeter at the 
 beginning is as much below the room temperature as at the 
 end it will be above that of the room, the calorimeter will gain 
 about as much heat during the first half of the experiment as 
 it will lose during the second half. When the highest precision 
 is required it may be desirable to verify or correct this assump- 
 tion by determining the heat exchanges for each experiment. 
 
 Observations are made, as described in the preceding article, 
 to determine the increase in temperature because of absorption. 
 Similar observations made during the time the calorimeter is 
 above room temperature will permit the loss because of radiation 
 to be computed. The algebraic sum of the two effects is the 
 error ; the correction has the same magnitude but the opposite 
 algebraic sign. This method is illustrated in the numerical 
 example given in Art. 150. 
 
 LXXVn. SPECIFIC HEAT BY BLACK'S METHOD 
 
 Determine the specific heats of iron, zinc, lead, and brass. 
 
 142. Black's Ice Calorimeter. — If the heat given out by a 
 body is caused to melt ice, the quantity of heat may be deter- 
 mined from the mass melted. The calorimeter consists of a 
 
THE ICE CALORIMETEE 179 
 
 block of ice with a cavity to contain the warmed body and the 
 water produced in cooling, and a cover to prevent loss of heat. 
 The method is both simple and accurate when solid ice, such 
 as manufactured ice, is used. 
 
 Weigh the body to be tested, which may conveniently be in 
 the shape of a sphere of four centimeters diameter, with a small 
 hook attached by which to lift it. Heat the body to a known 
 temperature in a water or steam bath. 
 
 A piece of solid ice, selected to be free from air bubbles and 
 cracks, is shaped so that its upper surface is slightly convex. 
 In the middle of this surface make a cavity large enough to 
 contain any ode of the bodies whose specific heat is to be deter- 
 mined. The cavity may be made of the proper size and with 
 a smooth surface, by causing one of the warmed spheres to melt 
 its way into the ice. Provide a smooth slab of ice, slightly con- 
 cave on its under surface, to serve as a cover. The shapes of 
 surfaces suggested are to prevent the water produced by surface 
 melting from running into the cavity. 
 
 Select a sponge of a size corivenient to wipe out the cavity, 
 moisten it, squeeze it dry, place it in a small beaker, cover with 
 a watch glass, and weigh carefully. Wipe the ice cavity with 
 another sponge, transfer the warmed body to the cavity with the 
 least possible loss of heat, and quickly cover it with the ice 
 slab. As soon as the body has cooled to 0°, which may be in 
 ten minutes under the circumstances described, remove the cover 
 and body, wipe up all the water in the cavity with the weighed 
 sponge, and weigh again to determine the amount of ice melted. 
 
 If w is the mass of the body, t its temperature when placed 
 
 in the ice, and M the mass of the ice melted, the specific heat 
 
 of the substance is 
 
 80.1 Jf 
 
 c = 
 
 mt 
 
180 
 
 HEAT 
 
 LXXVIII. SPECIFIC HEAT WITH BUNSEN'S ICE CALORIMETER 
 
 Prepare a Bunsen ice calorimeter and use it to determine the 
 specific heat of copper. 
 
 143. Bunsen's Ice Calorimeter. — If the heat given out by a 
 body is caused to melt ice, the quantity of heat may be deter- 
 mined from the change of volume which results. One gram of 
 ice has the volume 1.0908 ccm, while one gram of water at 0° 
 has the volume 1.0002 ccm. The quantity of ice that must be 
 melted to cause a change of 1 ccm in volume is ^.-jy^ o 6 ~ 11'04 g ; 
 and the heat required to melt it is 11.04 x 80.1 = 884 calories. 
 
 When properly applied this method is not subject to the 
 errors of radiation, and it may be used to measure small and 
 slow thermal changes. It is suitable for determining the specific 
 heat of solids and liquids, and especially when only a small 
 quantity of the substance to be experimented upon is available. 
 The disadvantage of the ^ , , 
 
 method is the trouble of 
 preparation, and its accuracy is not greater 
 than that of other common methods. 
 
 The calorimeter is a peculiarly shaped 
 glass vessel (Fig. 68) into which is sealed 
 a tube closed at its inner end, to receive 
 the body to be tested ; this will be called 
 the test tube. Another tube, or stem, 
 leads into the calorimeter; through this 
 the calorimeter is filled with distilled water 
 which has been recently boiled to remove 
 air. Mercury is poured into the stem, dis- 
 placing some of the water, till the stem is 
 wholly filled with mercury, and the lower portion of the body 
 is partially filled, as indicated. 
 
 Place the calorimeter carefully in crushed ice, which may be 
 conveniently contained in a wooden pail. Pack the ice closely 
 and fill the interstices with ice-cold water. 
 
 A tube about 30 cm long, the bore of which is iiniform and has 
 a diameter of 0.5 mm, is fitted by a rubber cork or a ground joint 
 
 Fig. 
 
 68. Bdnsen's Ice 
 Calorimeter 
 
THE ICE CALOKIMETER 181 
 
 into tKe upper end of the stem, so as to steind in a horizontal 
 position. Calibrate this tube with mercury by the method of 
 Art. 52, determining the volume of the bore of the tube per cen- 
 timeter of length. Insert the tube in the stem and push it down 
 tUl it is filled with mercury, taking care that no air remains in 
 the stem between the cork and mercury. 
 
 When the calorimeter has been packed long enough for the 
 whole to have become cooled to 0°, pour a little ether or carbon 
 bisulphide into the test tube, and rapidly evaporate it by forcing 
 air through it with a foot blower. This will cause some of the 
 water in the body to freeze around the test tube. Since ice has 
 a greater volume than the water from which it is formed, the 
 progress of freezing can be inferred by the flow of mercury from 
 the calorimeter. Continue the freezing until about a cubic 
 centimeter or more of mercury has been forced out. When the 
 carbon bisulphide or ether has all been evaporated or removed, 
 pour into the test tube sufficient water at 0° to half fill it, and 
 place a cork in the mouth. 
 
 The body whose specific heat is to be determined may be in 
 the form of a wire spiral. Weigh it and heat it in a water bath 
 to the boiling point. 
 
 If the mercury in the fine tube is moving outward, the water 
 in the body of the calorimeter is freezing, which may be caused 
 by impurities in the packing ice producing a temperature below 
 0° ; if the mercury is moving inward, the packing is probably 
 imperfect and must be improved ; if the mercury is stationary 
 for fifteen minutes or more, the calorimeter is ready for use. It 
 is often impossible to secure a stationary condition, in which case 
 the rate of motion may be observed and a correction applied 
 for the error thus introduced into the observations. Taking 
 great care to avoid loss of heat, transfer the warmed body to 
 the inside of the test tube ; care is also necessary to prevent 
 breaking the tube. The heat given up by the body as it cools 
 to 0° will melt some of the ice around the test tube, causing a 
 contraction of volume, which is indicated by the inward move- 
 ment of the end of the mercury column in the fine tube. Meas- 
 ure the motion which has thus taken place when the mercury 
 
182 HEAT 
 
 has become stationary, which may require an hour's tiihe. A 
 scale is often etched upon the tube for making this measure- 
 ment, but a separate scale is equally convenient. 
 
 If the motion of the mercury column produced by the cool- 
 ing of the body to 0° is I, and v is the volume of unit length of 
 the bore of the tube, the heat given out by the body is 884 Iv 
 calories ; and if m is the mass of the body, and t its temperature 
 when put into the calorimeter, its specific heat is 
 
 884 Zv 
 
 = . 
 
 mt 
 
 Refeeences. — Ostwald, Physico-Chemical Measurements, p. 136 ; Eohl- 
 rausch, Physical Measurements, p. 121 ; Preston, Theory of Heat, p. 217; 
 Nicliols, A Laboratory Manual of Physics, Vol. II, p. 237. 
 
 LXXIX. SPECIFIC HEAT OF A SOLID BY THE METHOD OF 
 
 MIXTURES 
 
 Determine the specific heat of aluminum and of iron. 
 
 144. Specific Heat by Method of Mixtures. — One of the most 
 useful methods for determining the specific heat of solids and 
 liquids is that known as the method of mixtures. In this 
 method the body to be tested has its heat mixed with the heat 
 of the calorimeter ; usually there is no physical mixture of the 
 substances. 
 
 In Regnault's apparatus the heating of the body is accom- 
 plished without wetting it. A large vessel to hold steam or hot 
 water has a tubular hole through its middle, passing in either 
 a vertical or inclined direction. The body to be heated is held 
 in the middle of the tube by a string; there are contrivances 
 for stopping the ends of the tube, and a thermometer is passed 
 through the upper stopper, so that its bulb is near the body. 
 The heating is to be continued until the water has been boiling 
 for some time, and the indication of the thermometer is 
 constant. 
 
 The calorimeter, usually of the form described in Art. 139, 
 is mounted on a small carriage running on a track so that it 
 
THE METHOD OF MIXTUKES 183 
 
 may be quickly brought under the lower end of the tube through 
 the heater, to receive the warmed body with little loss of heat. 
 A protecting screen should be placed between the heater and 
 the calorimeter. 
 
 The measurement of a specific heat, with the determination 
 of the heat capacity of the calorimeter and the radiation correc- 
 tion, as described in Arts. 140 and 141, requires the following 
 observations. 
 
 Weigh the body to be tested and place it in the heater. 
 Weigh the dry calorimeter cup and stirrer, and measure the 
 volume of that part of the thermometer to be used. Pour into 
 the cup sufficient water to cover the body tested when the 
 latter lies on the bottom of the cup, and determine the mass of 
 the water. 
 
 In order that the radiation correction may be small, the tem- 
 perature of the calorimeter when the body is introduced should 
 be so much below that of the surroundings that the final tem- 
 perature shall be very nearly the same as that of the room. A 
 preliminary trial of the experiment may be made to determine 
 approximately the temperature change that will occur ; or the 
 change to be expected may be calculated from assumed values 
 of the quantities involved. 
 
 After the body has become thoroughly heated, cool the calorim- 
 eter a little below the temperature desired for the beginning 
 of the test, and make observations of the temperature at inter- 
 vals of a minute to determine the rate of warming. At a 
 noted time observe the temperature of the calorimeter, that of 
 the heated body, and transfer the latter to the calorimeter as 
 quickly as possible and without splashing the water. Remove 
 the calorimeter from the heater. Stir the water and note the 
 temperature at intervals of 20 seconds till it becomes constant, 
 or until it changes uniformly because of radiation or absorption. 
 Continue the temperature observations for several minutes at 
 intervals of a minute, to determine the rate of change. Calcu- 
 late the radiation correction as explained in Art. 141. 
 
 If K is the heat capacity of the calorimeter, t^ its temperature 
 when the body is introduced, t^ the highest temperature attained 
 
184 HEAT 
 
 by the calorimeter, B the radiation correction, M the mass of the 
 body being tested, and ^3 its temperature when introduced into 
 the calorimeter, 
 
 from which c, the specific heat of the substance, is to be calculated. 
 
 LXXX. SPECIFIC HEAT OF A LIQUID BY THE METHOD OF 
 
 MIXTURES 
 
 Make two determinations of the specific heat of glycerin. 
 
 145. Specific Heat of Liquids by Method of Mixtures. — The 
 
 method of mixtures described in the preceding article may be 
 used for determining the specific heat of a liquid, the variations 
 in the details of construction and operation of the calorimeter 
 being mentioned below. 
 
 The liquid to be tested is placed in a heater, H (Fig. 69), 
 which is supported in a larger vessel in such a manner that it 
 may be entirely surrounded by water. Heating arrangements 
 and a stirrer for the water bath are provided, while a ther- 
 mometer with its bulb in the heater will indicate the tempera- 
 ture of the liquid. If the liquid is not too volatile, the water 
 bath may be boiled ; for very volatile liquids an ice bath or a 
 freezing mixture may be substituted for the hot water, and the 
 variations in procedure required will not need description. In 
 either case the liquid must be left in the bath till the indication 
 of the thermometer is constant. 
 
 A tube having a suitable stopcock leads from the bottom of 
 the heater through the water bath, and through a double wooden 
 protecting screen, into the calorimeter. The calorimeter cup is 
 supported, inside the usual protecting case, on an adjustable 
 stand. In the interior of the cup, fastened so as to be entirely 
 surrounded by the water of the calorimeter, is a receptacle,- R, 
 for the liquid. The tube from the heater leads into the upper 
 part of this receiver ; while above it, and connected with it by 
 a short tube, is a smaller vessel, C, which serves to condense 
 
THE METHOD OF MIXTURES 
 
 185 
 
 any vapor that may rise from the warm liquid. The condenser 
 opens to the air through a tube. The calorimeter is provided 
 with a finely divided thermometer and a stirrer. 
 
 The calorimeter cup, with the receptaple, stirrer, etc., are 
 weighed when clean and dry. Sufficient water is then placed 
 in the cup to cover the receptacle and condenser, the mass of 
 the water being deter- 
 mined. To minimize 
 the errors of radiation 
 the calorimeter should 
 be cooled so much be- 
 low the temperature of 
 the room that the final 
 temperature shall be 
 nearly the same as that 
 of the room. If the 
 liquid to be tested is 
 cooled instead of 
 heated, the calorimeter 
 must be warmed at the 
 beginning of the ex- 
 periment. The change 
 in the temperature of 
 the calorimeter to be 
 expected is determined 
 
 by calculation from , 
 
 known constants or by I. 
 a preliminary trial of 
 the experiment. 
 
 When the liquid has 
 become thoroughly heated, cool the calorimeter a little below 
 the temperature desired for the beginning of the test, and make 
 observations of the temperature at intervals of a minute to deter- 
 mine the rate of warming. Place the calorimeter in position to 
 receive the liquid from the heater. At a noted time observe the 
 temperature of the calorimeter and of the liquid, open the stop- 
 cock, and, by air pressure applied to the tube T of the heater. 
 
 Fig. 69. 
 
 Apparatus for Specific Heat 
 OP Liquids 
 
186 HEAT 
 
 , force the liquid quickly into the calorimeter. Remove the calo- 
 rimeter from the heater. Stir the water and note the temper- 
 ature at intervals of 20 seconds tUl it becomes constant or until 
 it changes uniformly because of radiation or absorption. Con- 
 tinue the temperature observations for several minutes at inter- 
 vals of a minute to determine the rate of change. Calculate the 
 radiation correction as explained in Art. 141. 
 
 Weigh the calorimeter and contents to determine the mass 
 of the liquid which has been tested. 
 
 If K is the heat capacity of the calorimeter (Art. 140), <j its 
 temperature when the liquid is introduced, t^ the highest tem- 
 perature attained by the calorimeter, B the radiation correction, 
 M the mass of liquid tested, and t^ its temperature when 
 introduced into the calorimeter, 
 
 from which c, the specific heat of the liquid, is to be calculated. 
 
 LXXXI. SPECIFIC HEAT OF A LIQUID BY COOLING 
 
 Determine the specific heat of turpentine. 
 
 146. Specific Heat by Method of Cooling. — The determina- 
 tion of specific heats by the method of cooling is based upon this 
 principle, that when a body cools in a given inclosure the quan- 
 tity of heat radiated depends only upon the difference of tempera- 
 ture between the body and the inclosure, and upon the nature of 
 the surface of the body. In other words, the times required 
 for equal masses of different substances, under identical 
 conditions, to cool through a given temperature range, are 
 proportional to the quantities of heat they hold ; that is, to their 
 specific heats. 
 
 If a substance of mass m^ and specific heat Cj cools through a 
 given temperature range in the time t-^, and a second substance 
 with an identical radiating surface, in the same cooling cham- 
 ber, having a mass m^ and a specific heat Cg, requires a time t^ 
 
SPECIFIC HEAT BY COOLING 
 
 187 
 
 for the same temperature change ; and if K is the heat capacity 
 of the receptacle and thermometer (Art. 140), then 
 
 WjCj + -ff _ ^1 
 
 From this equation one of the specific heats may be calculated 
 if the other quantities are known. 
 
 This method is useful for liquids, and especially for those 
 which can be obtained in small quantities only, since they may 
 be given the identical radiating sur- 
 faces required, by placing equal 
 volumes successively in the same 
 containing vessel. 
 
 The calorimeter consists of a con- 
 taining vessel and a cooling chamber, 
 as represented in Fig. 70. The small 
 flask for containing the liquid is made 
 of silver, platinum, or glass, and the 
 thermometer serves as stopper and 
 handle. The cooling chamber may be 
 filled with water at the room temper- 
 ature, though crushed ice is preferable. 
 
 The flask is filled with a known 
 mass of the liquid investigated, which 
 is then heated to a temperature of 50° or more; the flask is 
 transferred to the cooling chamber, and times are noted at 
 which the thermometer indicates 40°, 35°, 30°, etc., down 
 to a temperature near that of the cooling chamber. The same 
 flask is then filled with water to the same depth as before, 
 and the mass of the water is determined. This is then heated 
 and cooled, and the time observations are made as before. The 
 rates of cooling through the same temperature range are thus 
 determined ; a graphic solution may be the most convenient. 
 Substitution of the observed quantities in the formula will 
 enable the determination of the specific heat required. No 
 corrections are needed for water equivalent of the calorimeter, 
 loss by radiation, etc, 
 
 Fig. 70. Cooling 
 Caloeimbtbr 
 
188 HEAT 
 
 LXXXII. SPECIFIC HEAT OF A LIQUID BY THE METHOD OF 
 
 HEATING 
 
 Determine the specific heat of a solution of 1 gram-molecule of cane sugar 
 (molecular weight 342) in 50 gram-molecules of water (molecular 
 weight 18) ; find the molecular heat of sugar in solution. 
 
 147. Specific Heat of Liquids by Method of Heating The 
 
 specific heat of a liquid contained in one calorimeter may be 
 compared with that of a standard liquid contained in a similar 
 calorimeter, as proposed by Pfaundler, by observing the increases 
 in temperature produced when equal quantities of heat are 
 imparted to the calorimeters. These equal quantities of heat 
 are generated by sending a current of electricity through two 
 equal resistance coils connected in series, one of the coils 
 being placed in each calorimeter. If the quantities of liquids 
 employed are so proportioned that the rise in temperature is 
 the same, or nearly the same, in each calorimeter, the correc- 
 tions for the water equivalent of the cups, etc., and for radia- 
 tion are negligible. . 
 
 Magic's form of this apparatus consists of two exactly similar 
 and equal calorimeters, the details of which are shown in Fig. 71. 
 Through insulating bushings in the cover pass two heavy copper 
 wires, between which beneath the surface of the liquid is the 
 resistance coil of about 4 ohms. This coil, is preferably made 
 of a German silver alloy which has a zero temperature coeffi- 
 cient of resistance. The spiral is stretched downward and the 
 middle is held in place by a glass rod attached to the cover. 
 The coil and lead wires are covered with an insulating varnish 
 which is not affected by the liquids. 
 
 A turbine stirrer and thermometer are arranged as shown. 
 The turbines of the two calorimeters are turned by the same 
 belt of rubber cord, as any inequality in the stirring produces 
 appreciable error. The thermometers are provided with auto- 
 matic electric tappers. 
 
 The two calorimeters must be similarly placed with respect 
 to the surroundings, and they may be inclosed in a box or in a 
 water tank, for further protection. 
 
SPECIFIC HEAT BY HEATING 
 
 189 
 
 One cup contains the standard liquid, for example 700 g of 
 water, and the other cup as much of the liquid to be tested as 
 will give an equal rise in temperature. This amount is deter- 
 mined either by trial or by computation from the assumed 
 specific heats. Cool the two 
 cups to a temperature about 5° 
 below that of the room, place 
 them in position, and stir the 
 liquids. If the temperatures are 
 imequal, warm the cold cup with 
 the hands. When the two are 
 approximately equal read the 
 thermometers, turn on the cur- 
 rent, about 5 amperes for the 
 conditions described, and start 
 the stirring. Continue the cur- 
 rent till the temperature is as 
 much above that of the room as 
 at the beginning it was below 
 this temperature. Continue the 
 stirring and observe the highest 
 uniform temperature attained. 
 
 From the observations deter- 
 mine the rise in temperature of 
 each liquid; let 0i and 6^ ^^ 
 these increases, Wj and wig the 
 masses of the two liquids, whose specific heats are Cj and Cj ; then 
 
 from which the unknown specific heat c^ may be calculated. 
 If the standard liquid is water, 
 
 e =^1. 
 
 For those investigations to which it is applicable, this method 
 is probably the most precise known. Details as to the com- 
 parison of thermometers and resistances, and of precautions 
 necessary for extreme precision are given in the references. 
 
 Fie. 71. Caloeimetek roi: 
 Electric Heating 
 
190 HEAT 
 
 For a solid which has such poor conductivity that the usual 
 methods are not suitable, the following may be used for deter- 
 mining its specific heat. The solid in a finely powdered state 
 is placed in one calorimeter with some liquid which does not 
 dissolve it. If the specific gravity of the solid is not too great, 
 when the liquid is stirred the powder travels through the whole 
 cup, and takes the temperature quickly. The second cup con- 
 tains some of the same liquid without the solid. The com- 
 parison of the specific heats as described will permit that of 
 the powder to be determined. 
 
 Professor Magie gives the following example of the use of 
 this calorimeter. 
 
 Specific Heat op Milk Sugar Solution 
 
 METHOD OF HEATING 
 
 April 1, 1902 
 
 A solution of 1 gram-molecule of milk sugar in 200 gram-molecules of 
 water was prepared. The molecular weights of these substances being 
 342 and 18, the masses were in the proportion of 342 to 3600. In cup A 
 was 700 g of water, and in cup B 736.57 g of solution. 
 
 Cup a Cup B 
 
 Temperature before heating 15°.130 15°.115 
 
 Temperature after heating 25 .324 25 .309 
 
 Rise of temperature 10 .294 10 .294 
 
 Specific heat of solution = '- = 0.9504. 
 
 ^ 736.57 X 10.294 
 
 Hence the heat capacity of 3942 g of solution would be 3746.4 calories ; of 
 this amount of solution 3600 g are water, the heat capacity of which is 
 3600 calories. Therefore the apparent heat capacity of the 342 g, or 
 1 gram-molecule, of the milk sugar in solution is 146.4 calories, which is 
 then the apparent molecular heat. 
 
 Reference. — Magie, Physical Review, Vol. 9, p. 72, 1899 ; Vol. 14, pp. 198- 
 203, 1902; Vol. 16, p. 881, 1908. 
 
HEAT EQUIVALENT OF FUSION 191 
 
 LXXXin. HEAT EQUIVALENT OF FUSION OF ICE 
 
 Determine, by at least three trials, the heat equivalent of fusion of ice. 
 
 148. Heat Equivalent of Fusion. — The quantity of energy 
 in the form of heat required to change a unit mass of the sub- 
 stance from the solid to the liquid state without changing its 
 temperature is its heat equivalent of fusion. The details of its 
 experimental determination in the case of ice may be arranged 
 as follows. 
 
 Besides the usual errors of calorimetry there may be, in this 
 experiment, the serious one of the carrying of water into the 
 calorimeter by the ice. 
 
 The calorimeter cup may have a capacity of about a liter; 
 determine its water equivalent, and that of the stirrer and ther- 
 mometer (Art. 140). Pour water into the cup till it is two 
 thirds fuU, and determine the mass of the water. The radia- 
 tion error is made inappreciable by the following procedure. 
 Warm the water till its temperature is abotit 10° above that of 
 the surroundings, note the temperature of the calorimeter, and 
 then introduce dry ice. The ice may be dried with a cloth or 
 filter paper just before being used. Stir the water, keeping the 
 ice beneath the surface by means of the stirrer, which may be a 
 wire ring covered with gauze. Continue the cooling till the 
 temperature is as much below that of the room as at the begin- 
 ning it was above this temperature. More ice may be added if 
 needed, or, if too much has been placed in the cup, the remaining 
 pieces may be removed. Note the temperature when the last 
 particle of ice is melted ; that is, the lowest temperature attained. 
 A final weighing will determine the mass of ice melted. 
 
 If K is the heat capacity of the calorimeter, t-^ the tempera- 
 ture of the water when the ice is introduced, t^ the temperature 
 when the ice is all melted, and M the mass of ice melted, then 
 
 from which x, the heat equivalent of fusion, is to be calculated. 
 
 When the observations are made as described above, the 
 
 combined errors of absorption and radiation are inappreciable. 
 
192 
 
 HEAT 
 
 Instead of following this method the calorimeter may be cooled 
 only to the room temperature, and the correction for radiation 
 may be determined as explained in Art. 141. 
 
 Reference. — Preston, Theory of Heat, pp. 283-287. 
 
 LXXXIV. HEAT EQUIVALENT OF VAPORIZATION OF WATER 
 
 Determine; by at least three trials, the heat equivalent of 
 vaporization of water. 
 
 149. Heat Equivalent of Vaporization The quantity of 
 
 energy in the form of heat required to change a unit mass of 
 the substance from the liquid to the gaseous state without 
 
 chaiiging its temperature is its 
 heat equivalent of vaporization. 
 For water it may be determined 
 experimentally as described below. 
 The apparatus is to be arranged 
 as indicated in Fig. 72, the boiler 
 being a three-liter flask, and the 
 calorimeter having a capacity of 
 one liter. Between the boiler and 
 the calorimeter is an asbestos or 
 double wooden protecting screen. 
 The steam delivery tube should, 
 as far as possible, incline towards 
 the boiler and be covered with a 
 nonconductor. A water trap is 
 placed in this tube just above the 
 calorimeter, and the tube is bent 
 in a horizontal direction about five 
 centimeters from the end. 
 
 Besides the errors common to 
 calorimetry, evaporation, radiation, and absorption, there is in 
 this experiment another of relatively large amount, — the enter- 
 ing into the calorimeter of water from steam already condensed. 
 The latter is reduced to a minimum by the construction described, 
 
 Fig. 72. Apparatus for Heat 
 OF Vaporization 
 
HEAT EQUIVALENT OF VAPORIZATION 193 
 
 while the following method of procedure will reduce the effects 
 of the other errors. 
 
 Weigh the dry calorimeter cup and stirrer, and determine 
 the water equivalents of these and also of the thermometer 
 (Art. 140). Nearly fill the cup with water and reduce its tem- 
 perature 10° or 12° below that of the surroundings. Weigh 
 to determine the mass of the water. When steam is coming 
 abundantly from the boiler, and the delivery tube is hot so that 
 little condensation takes place, thoroughly stir the water in the 
 calorimeter, observe its temperature, and quickly place it under 
 the delivery tube, so that the horizontal portion of the latter 
 shall be a few centimeters below the surface of the water. Stir 
 the water continually while the steam is condensing, and con- 
 tinue the condensation till the temperature of the calorimeter 
 is as much above the room temperature as it was below this 
 temperature at the beginning. Remove the calorimeter, stir 
 the water, and note the highest temperature indicated by the 
 thermometer. Again weigh the calorimeter cup, as quickly as 
 possible to avoid evaporation, determining the mass of the steam 
 condensed. 
 
 If ^ is the heat capacity of the calorimeter, t-^ the temperature 
 when the steam is first admitted, t^ the highest temperature of 
 the calorimeter, tg the temperature of the boiling point corre- 
 sponding to the barometric pressure at the time of the experi- 
 ment, and Mthe mass of steam condensed, then 
 
 from which x, the heat equivalent of vaporization, is to be 
 calculated. 
 
 When the observations are made as described above, the 
 combined errors of absorption and radiation are inappreciable. 
 Instead of following this method the calorimeter may be heated 
 only to the room temperature, and the correction for absorption 
 may be determined as explained in Art. 141. 
 
 Reference. —Preston, Theory of Heat, pp. 304-314. 
 
194 
 
 HEAT 
 
 LXXXV. MECHANICAL EQUIVALENT OF HEAT BY PULUJ'S 
 
 METHOD 
 
 Determine the mechanical equivalent of heat. 
 
 150. Mechanical Equivalent of Heat. — The number of ergs 
 which when transformed into heat will equal one heat unit, the 
 calorie at 17°, is the mechanical equivalent of heat. The exact 
 measurement of the work done and of the resulting heat, while 
 theoretically simple, is much complicated in practice by numer- 
 ous corrections of relatively large magnitudes. The method to 
 be described, in which friction between metals produces the heat, 
 has been found to be both convenient and of moderate precision. 
 
 Two metallic cones, 
 one fitting in the other, 
 are supported by non- 
 conducting rings in a 
 cup, -^(Fig. 73), so that 
 the outer cone may be 
 rotated, while the 
 inner one is prevented 
 from rotating by the 
 action of a mass, M, 
 attached to a wooden 
 disk, D. The inner 
 cone is hollow and is nearly filled with water. A thermometer 
 passes through the disk into the water, and a small stirrer may 
 be added. There is a counting device to register the number 
 of rotations of the outer cone which are caused by turning the 
 hand wheel, To secure the proper amount of friction between 
 the cones, weights are placed on the disk. 
 
 By properly regulating the speed of the rotating cone, the 
 work represented by the friction between the cones may be just 
 enough to keep the mass M suspended, so that its weight, the 
 force Mg, acts on the circumference of the disk. If to keep 
 this force acting during the experiment it is necessary to turn 
 the support n times, the result is the same as though the 
 
 Fig. 73. Appabatcs tor Mechanical 
 Equivalent op Heat 
 
MECHANICAL EQUIVALENT OF HEAT 195 
 
 support had remained stationary and the disk had turned n times 
 in the opposite direction. Thus, to turn the disk the force Mg 
 attached to its circumference would haTe had to move through 
 the distance nc, where c is the circumference of the disk, and the 
 work done would have been ncMg. This amount of work in 
 the experiment is transformed into heat, most of which is 
 absorbed by the cones and their contents and can be measured ; 
 some of the heat will be lost by passing to the supports and 
 disk and by radiation; the latter part may be estimated and 
 taken into account. 
 
 If K is the heat capacity of the cones, stirrer, and thermome- 
 ter (Art. 140), tj the initial temperature of the water, t^ the 
 temperature at the end of the experiment, and B radiation 
 correction (Art. 141), the heat produced is K\{t^ — tj) + -E], and 
 the work per unit heat, that is, the mechanical equivalent of 
 heat, in ergs, is ^^^^ 
 
 The weighings and temperature observations required are 
 the same as for specific heat determinations, already described. 
 While making the experiment, one observer should give atten- 
 tion to stirring the water in the calorimeter and to the tem- 
 perature readings. It will rfiquire some care on the part of the 
 second observer to maintain the proper rotation. 
 
 Before beginning an experiment, clean the rubbing surfaces 
 and put a very little oil on them to prevent injury. A tempera- 
 ture change of from 10° to 15° is convenient, the room tem- 
 perature being midway between the extremes. A uniform rate 
 of turning is not required, but it is necessary that the mass 
 shoidd be kept suspended by the friction alone. After the 
 cones are in place and before the temperature readings are made, 
 a few turns may be given to the cone to make sure that the 
 proper friction is secured to lift the mass. The pressure weights 
 on the disk may be changed during the experiment to accommo- 
 date changes in friction. 
 
 The details of such a determination are illustrated by the 
 following numerical example. 
 
196 HEAT 
 
 Mechanical Equivalent op Heat 
 
 METHOD OF METAL FRICTION 
 
 July S3, 1902 
 
 Weight of brass cones and stirrer 114.00 g 
 
 Weight of water in cone 24.02 g 
 
 Volume of thermometer immersed (No. 1721) . . . 1.5 ccm 
 
 Specific heat of brass 0.093 
 
 Heat capacity of calorimeter 35.25 
 
 Time Tempeeatcbe 
 8" 41"'.0 17°.78 
 
 42 .0 18 .08 
 
 43 .0 18 .38 Turning begun. Counter 2180 
 45 .8 24 .50 Room temperature 
 
 49 .5 Turning ceased. Clounter 4630 
 
 50 .0 31 .70 Highest temperature 
 
 51 .0 31 .55 
 
 52 .0 31 .40 
 
 From the first observations the rate of warming is 
 
 0°.30 -^ (24.50-18.08) = 0°.047; 
 
 and the gain in temperature during the first part of the experiment is 
 
 0°.047 X (45.8-43.0) x J (24.50- 18.38) = 0°.40. 
 
 From the final observations the rate of cooling is 
 
 0°.15 H- (31.55-24.50) = 0°.021; 
 
 and the loss in temperature during the second portion of the experiment is 
 
 0°.021 X (50.0-45.8) x ^31. 70- 24.50) = 0°.32. 
 
 The corrected temperature change is 
 
 31°.70-18°.38-0°.08 = 13°.24. 
 
 The circumference of the disk is 80.7 cm; the suspended mass is 100 g; 
 the number of turns is 2450. Therefore 
 
 ^ 80.7 X 2450 X 100 x 980 , 
 •^= 35.25 X 13.24 = ^'^^^ ^ ^0' ^'S^' 
 
CHAPTER XII 
 HYGEOMETEY 
 
 LXXXVI. HYGROMETRIC STATE BY VARIOUS METHODS 
 
 Determine the dew-point, and the absolute and relative humidity of the 
 air by four methods. 
 
 151. Hygrometry. — The determination of the absolute and 
 relative amount'of vapor of water present in the atmosphere 
 is the purpose of hygrometry. Such determinations are of 
 value in many scientific and technical operations, as well as in 
 weather prognostications. 
 
 The density of the water vapor in the air is the absolute 
 humidity ; it is the mass in grams of the water contained in one 
 cubic centimeter. For convenience, hygrometric tables usually 
 give the quantity of water in grams per cubic meter. The ten- 
 sion of the water vapor is the pressure which it alone would 
 produce if measured by a mercury gauge. The dew-point is the 
 temperature at which the anjiJunt of water actually present 
 in the air is sufficient to produce saturation ; it is the tempera- 
 ture to which the air must be reduced to cause the deposition 
 of water just to begin. The relative humidity is the ratio of the 
 amount of water actually present in the air to the amount that 
 would saturate it, the temperature remaining unchanged. Since 
 the vapor of water follows Boyle's Law, the relative humidity 
 is also equal to the ratio of the actual pressure, /, of the vapor 
 
 in the air, to the maximum possible pressure, F, at the same 
 
 / 
 temperature ; that is, the relative humidity is equal to -- . 
 
 The four most useful forms of hygrometers are described below. 
 For the convenient use of any one of these forms, hygrometric 
 tables are necessary (Appendix, Tables 20, 21, 23). 
 
 197 
 
198 
 
 HEAT 
 
 Absorption Hygrometer. — By means of an aspirator (Fig. 74) 
 a known volume of air is drawn through a drying bottle. If 
 the aspirator contains just five liters of water, and it is allowed 
 to operate twice, ten liters of air will have been drawn through 
 the dryer. The rate of flow of the water should be such that 
 
 this will require about an hour's 
 time. If the flow is too rapid, the 
 moisture in the air may not all be 
 absorbed by the dryer. 
 
 The dryer may consist of a flask 
 of about 250 ccm capacity, filled 
 with small fragments of pumice 
 stone saturated with sulphuric acid. 
 The air is allowed to enter at the 
 bottom and is drawn out at the 
 top. This flask must be carefully 
 weighed to the nearest milligram, 
 both before and after the aspirator 
 operation. The mass of water per 
 cubic centimeter of air, which is the 
 absolute humidity, is determined 
 from the increase in weight. From 
 the temperature of the air, by means 
 of the tables, the mass of the water 
 required to produce saturation is found. From these quantities 
 calculate the relative humidity. From the absolute humidity 
 and the tables find the dew-point. 
 
 This method may be made the most precise of all, in which 
 case the influence of temperature and water in the aspirator 
 must be considered. 
 
 Fig. 74. Aspiratob foe 
 Htgkometrt 
 
 Refebbnce. — Preston, Theory of Heat, p. 361. 
 
 Dew-Point Hygrometer (Daniell, Regnault). — In this form of 
 instrument a polished surface has its temperature reduced till 
 moisture from the air is condensed upon it. This temperature, 
 t^, is the dew-point. From the tables find the tension of 
 the vapor corresponding to this dew-point, and also to the 
 
HYGEOMETEY 
 
 199 
 
 temperature of the air; the ratio of the former to the latter 
 is the relative humidity. 
 
 The absolute humidity is found by taking from the tables the 
 density of vapor corresponding to the dew-point. This value of 
 the density needs correction, because the air close to the instru- 
 ment has been cooled. The tabular value multiplied by 
 
 273 + tg 
 
 273 -h i ' 
 
 where t^ is the dew-point, and t the air temperature, is the 
 absolute humidity. 
 
 The dew-point hygrometer usually consists of two thermom- 
 eters, one designed to measure the temperature of the air and 
 the other the temperature of a polished surface which is cooled 
 to the dew-point by the evaporation of ether. In one form one 
 thermometer has a hollow, cup- 
 shaped bulb, in which ether is 
 poured. The evaporation is facil- 
 itated by blowing air into the 
 ether. In another form the ther- 
 mometer is inclosed in a bulb 
 containing ether ; this bulb is con- 
 nected with a second on the out- 
 side of which is a muslin cover. 
 This cover is saturated with ether, 
 which is then evaporated by blow- 
 ing upon it ; this cools the second 
 bulb, causing condensation of 
 ether vapor inside, which in turn 
 causes the evaporation of the ether 
 in the first bulb, reducing its tem- 
 perature to the dew-point. 
 
 It is preferable to have the 
 two thermometers of the same size 
 and shape, and symmetrically placed with respect to the other 
 parts of the apparatus, as shown in Fig. 75. The thermometer 
 bulbs are inclosed in similar glass cylinders, whose ends are 
 
 Fig. 75. Dew-Poiht 
 Htgkometer 
 
200 
 
 HEAT 
 
 closed by polished silver cups. One cup contains ether, and is 
 connected to an aspirator. By allowing water to run out of 
 the aspirator, air is drawn through the ether, evaporating and 
 cooling it. 
 
 Observing the polished surface carefully, cool it tUl the dew 
 just appears, and note the temperature; allow it to warm till 
 the dew just clears away, and note the temperature. By repeated 
 trials it will be possible to locate these two temperatures within 
 a degree of each other; or even to adjust the rate of flow of 
 water to keep the temperature of the bulb exactly at the dew- 
 point. Make several determinations. Care should be taken 
 that air currents and the heat of the body do not affect the 
 instrument. A telescope may assist in reading the thermometer 
 from a distance. 
 
 Wet- and Dry-Bulb Hygrometer, — Psyehrometer (Auguste). — 
 This instrument consists of two similar thermometers, one hav- 
 ing its bulb covered with muslin arranged to be kept wet with 
 water. If the air is not saturated, the water will evaporate from 
 this muslin cover and cool the thermometer; the rapidity of 
 evaporation, and therefore the amount of cooling, depends upon 
 the humidity and the relative motion of the thermometer and 
 the surrounding air. By means of empirical formulae and 
 tables, the absolute and relative humidity and the dew-point 
 
 may be determined 
 from the readings of 
 the wet- and dry- 
 bulb thermometers. 
 The formulae here 
 given assume that 
 the relative motion 
 of the thermometers 
 and air is three 
 meters per second or 
 more, this velocity 
 producing the" great- 
 est depression of the wet-bulb temperature. This condition may 
 be secured by swinging the instrument as a pendulum or by 
 
 Fig. 76. Pstchrometer 
 
HYGEOMETKY 201 
 
 blowing the air against the thermometer with a fan. Such a 
 ventilating fan is often made a part of the apparatus, as shown 
 in Fig. 76. 
 
 If t is the temperature of the air as indicated by the dry bulb, 
 f„ the temperature of the wet bulb, £ the barometer reading, 
 and c„, the tension of water vapor for the temperature t^, as taken 
 from the tables, then the actual tension is 
 
 e = e^~ 0.0066 £(t- t„) [1 + 0.00115 {t - yj. 
 
 With tables such as the Smithsonian Meteorological Tables, 
 this formula is both convenient and accurate, the value of e^ be- 
 ing found in one table (Table 23, Appendix), while the value of 
 
 0.0066 £{t- tj) [1 + 0.00115 {t - tj] 
 
 is given in another table (Table 20). Having found e, the dew- 
 point is the temperature taken from the table, for which this 
 tension is sufficient to produce saturation. Still another table 
 gives the relative humidity (Table 21), the temperature and 
 dew-point being known. The absolute humidity is 
 
 /= 1.060 
 
 1-h 0.00367 « 
 
 When the only available table is one of the tension of aqueous 
 vapor, the following formulae are more convenient, though less 
 
 ®^^°*- e = e^- 0.0066 5 («-Q. 
 
 It will often be sufficient to assume the barometric pressure as 
 
 75 cm; then e = e^-Q.bO {t-t„). 
 
 Having found e, the temperature at which this tension is suffi- 
 cient to produce saturation is the dew-point. The relative 
 humidity is the ratio of e to the tension e^ found in the table, 
 which would produce saturation at the temperature t of the dry- 
 
 bulb thermometer ; it is — . 
 
 The absolute humidity is 
 /= 1.060 
 
 1 + 0.00367 i 
 
202 HEAT 
 
 If the temperature ^„ is below the freezing point, the cover 
 of the wet bulb should be freshly moistened with water before 
 reading the temperature, and the formula should have the con- 
 stant as given below. 
 
 e = e^-0M58£{t-t^). 
 Beferbnce. — Smithsonian Mete6rological Tables. 
 
 Sair Hygrometer. — A very useful hygrometer consists of a 
 prepared hair suspended over a wheel cariTing a pointer. The 
 hair changes length with change in the humidity, and by 
 empirical adjustment a scale may be made to show the relative 
 humidity. From specially prepared tables the absolute humidity 
 and the dew-point may be found. The setting of the index 
 should be tested occasionally by closing the case of the instru- 
 ment and introducing a sheet of muslin which is wet with 
 water. When the inclosed air has become saturated with water 
 vapor, the reading should be 100. 
 
Paet Y — Light 
 chapter xhi 
 
 PHOTOMETRY 
 
 C. LUMINOUS INTENSITY WITH SIMPLE PHOTOMETERS 
 
 Show that the Law of Inverse Squares applies to light intensity. Find the 
 mean horizontal candle power of various light sources (flat flame, 
 Argand, and Welsbach mantle gaslights, and kerosene lamp) by means 
 of several photometers. 
 
 152. Photometry. — The photometer is an apparatus fOr com- 
 paring the luminous intensities of two light sources, or for deter- 
 mining the distribution of light about a source. There is no 
 practicable direct method for measuring light intensity ; it then 
 becomes necessary to compare a light with a chosen standard 
 light. That feature of lights which is of most practical impor- 
 tance is their illuminating power. The light being investigated 
 and the standard light are placed to illuminate the same or 
 similar screens, and the relative distances of the sources from 
 the screen are altered to make the two illuminations equal. 
 The luminous intensities of the sources are then proportional 
 to the squares of their distances from the screen. 
 
 This relation follows from the law that the intensity of illu- 
 mination at different distances from, a source of light varies 
 inversely as the square of the distance. When the light is 
 incident upon a surface at an oblique angle, the intensity fur- 
 thermore varies as the cosine of the angle of incidence. 
 
 203 
 
204 LIGHT 
 
 That the Law of Inverse Squares is applicable may be approx- 
 imately shown by comparing the illuminations produced by 
 different sources of known relative intensities ; as, for instance, 
 three and five candles. 
 
 The degree of precision with which two light intensities may 
 be compared does not approach that attained in the comparison 
 of most other physical quantities. Some of the difficulties are 
 that no precise, constant standard has been devised, that the two 
 lights are usually of different colors, and that the determination 
 of the equality of illumination is a question of judgment, not of 
 measurement. 
 
 153. Candle Power; Horizontal Candle Power. — The unit of 
 intensity in which illuminating power is usually expressed is the 
 candle. This unit is neither very definite nor very precise, and 
 other units are sometimes used. Brief specifications for a few 
 light units are given in Art. 155. Horizontal candle power 
 expresses the intensity of the light emitted by the source in a 
 horizontal line as compared with that of a standard candle ; or 
 it expresses the number of candles which must be concentrated 
 to give an illumination equal to that under investigation. Since 
 most sources emit light of varying intensities in different direc- 
 tions, it is desirable to measure the candle power in a specified 
 number of lines in a horizontal plane ; and the average of these 
 js the mean horizontal candle power. Incandescent electric 
 lamps may be rotated about a vertical axis while being tested 
 so that the effective illumination in any one direction is the 
 average of that emitted in all directions ; a single measure then 
 gives the mean horizontal candle power. 
 
 For the purposes of this exercise it will be sufficient to 
 measure the candle power in four directions 90° apart, begin- 
 ning, for a flat flame, with one edge toward the photometer 
 screen. More complete methods are described in the next 
 exercise. 
 
 In order that the measures may be of value, the style of burner, 
 size of flame, rate of fuel consumption, etc., should be specified. 
 
 154. Simple Photometers. — Many devices for the comparison 
 of luminous intensities have been made, but only a few of the 
 
PHOTOMETRY 205 
 
 commonly useful forms will be mentioned. In all photometers 
 the essential part is the screen, together with the arrangement 
 for observing the illumination. 
 
 In some photometers the screen and the reference light are 
 fixed in position, the unknown light being moved to secure the 
 setting ; in other forms the two lights are fixed at the opposite 
 ends of the photometer bench, the screen being movable 
 between them. Two kinds of scales are in use, — one showing 
 the distances between the screen and the lights, the other indi- 
 cating, for a given position of the adjustable part, the ratio of 
 the intensity of the unknown light to the reference light. The 
 former is usually the more accurate, while the latter is very con- 
 venient. The scale of distances may be used for computation in 
 connection with tables giving the logarithms of the light ratios. 
 
 Rumford Photometer. — The two sources of light, a screen, 
 and an opaque object — such as a rod of wood — are arranged so 
 that two shadows of the rod are cast near each other upon the 
 screen. That portion of the screen which is shielded from one 
 light by the rod is illuminated by the other light. If, then, 
 the lights are placed at such distances that the two shadows 
 appear equally dark, the illuminations produced by the lights 
 upon the screen are equal, and the intensities of the lights are 
 proportional to the squares of their distances from the screen. 
 
 Foucault and Ritchie Photometers. — Instead of the opaque 
 screen of the preceding form, Foucault employed a translucent 
 screen arranged so that each light illuminated a different half 
 of the screen. A sight tube opposite the lights excluded from 
 view everything except these two portions of the screen. One 
 of the lights is moved until the greatest possible uniformity of 
 illuinination is secured. 
 
 In Ritchie's modification the two lights are fixed at the ends 
 of a photometer bench, while on the line between them are two 
 mirrors inclined to throw the lights at right angles on the 
 translucent screen. The screen and mirrors are moved towards 
 one light or the other till the illuminations are equal. 
 
 Paraffin-Diffusion Photometer. — Two thick plates of paraffin 
 are placed together, with an opaque layer — such as a sheet of tin 
 
206 
 
 LIGHT 
 
 foil — between them. These plates are placed in the line joining 
 the two lights, and perpendicular to this line. Each plate will 
 then receive light from only one source, which will diffuse into 
 the material of the plates. By looking at the edges of both 
 plates of wax, one may judge as to the equality of the illu- 
 mination which falls upon them. This simple device is very 
 satisfactory, even for lights differing in color. 
 
 Bunsen Photometer. — This form of photometer is probably 
 used more extensively than any other ; it is as convenient as 
 any, and is excelled in precision only by the Lummer-Brodhun 
 form (Art. 159). A piece of white paper is rendered translu- 
 cent over a part of its surface by saturation with paraffin or 
 stearin. If light falls normally upon such a screen, a consid- 
 erable part of it will be reflected from the opaque surface, while 
 
 the waxed surface will trans- 
 mit a large part of the light, 
 thus appearing darker. If a 
 second light be placed on the 
 opposite side of the screen at 
 such a distance as to produce 
 equal illumination, this light 
 will be transmitted and re- 
 flected from its side of the 
 screen in the same propor- 
 tions as that of the first light 
 from the first side of the 
 Under ideal condi- 
 
 
 Li. 
 
 \ 6 / 
 7\/v 
 
 E 
 
 Fig. 77. Bunsen Photometer Screen 
 
 screen. 
 
 tions the waxed spot should disappear when the illuminations are 
 equal ; in practice one must adjust the distances of the lights or 
 the position of the screen between the two lights so that the con- 
 trast between the waxed and unwaxed portions of the screen is 
 the same whether the screen is viewed from one side or the 
 other. Or the screen may be moved till the spot most nearly 
 disappears as seen from one side, and then be moved till the 
 same condition is fulfilled as seen from the other; the point 
 midway between these two positions is taken as the one from 
 which to measure. 
 
PHOTOMETRY 207 
 
 Often the Bunsen screen is provided with mirrors or prisms 
 which permit both sides to be seen at once ; this greatly facili- 
 tates its use. In Fig. 77 (S represents the Bunsen screen, which 
 is illuminated by light from L^ and L^. M-^ and M^ are mirrors 
 which permit both sides of the screen to be seen from E. 
 
 CI. INTENSITY OF LIGHT BY PRECISE METHODS 
 
 (a) Determine the mean spherical candle power of an incandescent electric 
 
 light. 
 
 (b) Determine the gas consumed per candle power per hour by a flat flame, 
 
 an Argand, and a Welabach mantle gaslight. 
 
 155. Standards of Light. — No altogether satisfactory stand- 
 ard of Uluminating power has been devised; the following are 
 the ones commonly in use in this country. The specifications 
 given are abridged ; for more detailed descriptions of these and 
 other standards, the treatises mentioned in the references may 
 be consulted. 
 
 The British Candle is made of spermaceti whose melting point 
 is 112° F. ; the candle is of 0.85 inch average diameter; its wick 
 is of 3 strands of cotton, each of 18 threads ; it should burn 120 
 grains of grease per hour with a flame 1.77 inch high. 
 
 In complete photometers the candle is supported on a balance 
 with which the amount of material consumed may be readily 
 determined. If this amount does not vary from that specified 
 by more than five per cent., the candle power of the light is taken 
 as proportional to the rate of consumption. 
 
 The Grerman Candle (Vereinskerze) is made of paraffin with 
 a melting point of 55°; it is 20mm in diameter; its wick is 
 made of 25 threads of cotton; the standard flame height is 
 50 mm. 
 
 The Mefner-Altenech Amyl Acetate Lamp burns pure amyl 
 acetate from a wick 8 mm in diameter and with a flame 40 mm 
 high. 
 
 This lamp, when made and used according to the specifica- 
 tions of the Physikalisch-Technische-Reichsanstalt, is considered 
 
208 LIGHT 
 
 the most reliable available standard. Its intensity, often called 
 the Hefner unit, is equal to 0.88 English Candles, and to 0.8-3 
 German Candles. 
 
 156. Mean Spherical Candle Power. — Sources of illumination 
 do not radiate light of equal intensity in all directions, and 
 hence the measure of the intensity in one or more directions may 
 give only a very incomplete idea of the actual quantity of light 
 radiated by the source. For purposes of general comparison 
 the mean spherical candle power is a more satisfactory measure 
 of the illuminating value of a light. The mean spherical candle 
 power of a light source expresses what would be the intensity 
 if the total quantity of light emitted were so redistributed that 
 the illumination over the entire surface of a sphere concentric 
 with the light were uniform. To find the mean spherical candle 
 power practically, it is necessary to measure the intensity in a 
 large number of directions, arranged systematically, and then 
 to average these measures so as to conform to the definition. 
 The following method is a convenient one. 
 
 Consider a sphere concentric with the light ; pass a horizontal 
 plane through the center, the light being in its normal vertical 
 position; designate the intersection of the plane and sphere the 
 equator. Let the surface of the sphere be divided into seven 
 zones each 30° wide, an equatorial zone extending 15° each side 
 of the equator, and zones above and below this, the centers of 
 which are "parallels of latitude" 30°, 60°, and 90° from the 
 equator. The areas of these zones, considered as portions of 
 the entire sphere, are 
 
 Upper pole ... . 0.0170 
 
 60° N 0.1294 
 
 30° N 0.2242 
 
 Equator 0.2588 
 
 30° S 0.2242 
 
 60° S 0.1294 
 
 Lower pole 0.0170 
 
 If the average intensity of the illumination on each zone is meas- 
 ured and is multiplied by the area of the zone, the sum of all the 
 products so obtained will be the mean spherical candle power. 
 
PHOTOMETRY 209 
 
 When investigating an incandescent electric lamp, it is con- 
 venient to have a lamp socket which can be continuously and 
 rapidly rotated by a motor. A photometric measurement is 
 taken when the axis is vertical and the lamp is rotatihg ; the 
 result is the average intensity in the equatorial direction, — that 
 is, the mean horizontal intensity. Incline the top of the axis 
 30° toward the photometer screen, and the measurement taken 
 with the lamp rotating in this position may be considered the 
 average intensity for the 30° zone. Incline the axis 60° and 
 then 90° toward the photometer, taking readings as before, and 
 proceed similarly for the lower hemisphere. The intensity for 
 the lower polar zone will be practically zero. 
 
 If it is not convenient to rotate the lamp continuously, the 
 mean of a number of readings for each zone, eight or more, 
 may be employed. Observe the intensity of the lamp, the axis 
 being vertical ; rotate the lamp on its axis 45°, and measure the 
 intensity; rotate it 45° again, and read; continue rotating it 
 45° at a time till it has been turned through 360°; the average 
 is the equatorial intensity. Incline the axis 30°, and repeat the 
 observations ; continue in this manner for the seven zones. Mul- 
 tiply the average for each zone by the zone area as given above, 
 and the sum of the products is the mean spherical intensity. 
 
 In the photometry of an electric lamp an ammeter should be 
 in series with the lamp, and a voltmeter should be connected to 
 the terminals. Keep the voltage at the value for which the lamp 
 is designed, using a rheostat to control this. From the readings 
 of these instruments compute the efficiency of the lamp; that 
 is, the watts required per candle power. 
 
 157. Gas Meter. — In using a standard wet gas meter the 
 instrument must be level, and the water in it must stand at 
 exactly the proper height, as indicated by the gauge. 
 
 In work of precision the temperature of the gas must be 
 noted; also its pressure and the barometer pressure. 
 
 158. Photometry of Intense Lights. — When the intensity of 
 a light exceeds 20 candle power it cannot be accurately com- 
 pared with the standard candle or the Hefner lamp. It is 
 then desirable to determine the true intensity of a nominal 16 
 
210 
 
 LIGHT 
 
 candle-power incandescent lamp, and to use this as an inter- 
 mediate or secondary standard. 
 
 Wlien lights of greater intensity, such as the electric arc, are 
 to be measured it may be necessary to reduce the effective inten- 
 sity by interposing in the path of the light a dispersion lens. 
 The methods to be employed for such investigations are described 
 in the references. 
 
 Refekences. . — Palaz-Fatterson, Industrial Photometry ; Stine, Photomet- 
 rical Measurements. 
 
 159. The Lummer-Brodhun Photometer. — In the Bunsen pho- 
 tometer (Art. 154) a field of view is presented, part of which is 
 illuminated from one source by reflected light and part is illumi- 
 nated .from the other source 
 by transmitted light. The two 
 portions having been differ- 
 ently treated their accurate 
 comparison is difficult. The 
 Lummer-Brodhun photometer 
 is a device for producing a 
 mixed field of view by opti- 
 cal means, in which the two 
 beams of light pass through 
 equal optical paths. 
 
 The two surfaces of an 
 opaque white screen, S (Fig. 
 78), are illuminated by the 
 two light sources i^ and L^. 
 By means of the mirrors Mj^ 
 and M^ ^^^ the two prisms 
 Pj and Pg these two surfaces are observed through a single 
 telescope in such a fashion as to permit great accuracy in the 
 estimation of the equality of the two illuminations. The two 
 right-angled prisms Pj and Pj have their hypothenuses ground 
 so that they are in close contact over a circular area. After this 
 the polish of the surface of Pg is destroyed by a sand blast over 
 portions of the shape shown black in Fig. 79. When the prisms 
 
 Fig. 78. Lwmmer- 
 
 Brodhun Photometer 
 
PHOTOMETRY 
 
 211 
 
 Fig. 79. Face of 
 Lummek-Brodhun Prism 
 
 are put together in the sight box, light will reach the telescope 
 from Xg and M^ only through the polished portion of the sur- 
 face of P^, where the contact is perfect. Light from M^ falling 
 upon these parts of the contact surface does not go to the tele- 
 scope, but is transmitted in the direction 
 of H; while light from M^ falling upon 
 those portions of the hypothenuse of Pj 
 which are not in contact with Pg is totally 
 reflected and enters the telescope. By 
 this device the field of view appears 
 elliptical in shape, with a vertical division 
 into halves (Fig. 80), in each of which is a 
 trapezoid. One source sends light to that 
 portion of the semiellipse f^, which sur- 
 rounds the trapezoid t^, and also to the 
 trapezoid t^ ; while the frame f^ and the trapezoid t^ receive light 
 only from the second source. Each trapezoid is made darker 
 in shade than the corresponding opposite semielliptical frame 
 by interposing in the paths of the light which goes to the trape- 
 zoids extra pieces of glass, G-^ and G^ (Fig. 78), the absorption 
 of which produces a slight reduction in intensity. This feature 
 is the contrast principle which aids in the 
 photometry of lights differing in color. 
 In Fig. 78 the paths of the light forming 
 the trapezoids are indicated by the dotted 
 lines. 
 
 In setting the photometer the screen is 
 to be moved to such a position between 
 the two lights that the two trapezoids 
 are both darker than the surrounding 
 backgrounds, and so that, neglecting dif- 
 ferences in color, the contrast between ^j 
 and /j is just the same as the contrast between t^ and /j. In 
 this position the two backgrounds /j and f^ are very nearly 
 the same shade, and the dividing line between them nearly 
 vanishes ; but it will entirely disappear only when the two 
 lights are of exactly the same color, which is very seldom. 
 
 Fig. 80. Field of View 
 
 OF Lcmmee-Brodhun 
 
 Photometer 
 
212 
 
 LIGHT 
 
 The determination, then, is one of equality of contrasts rather 
 than equality of illuminations. 
 
 160. The Rood- Whitman Flicker Photometer. — When lights 
 differ in color it is difficult, or impossible, to compare their lumi- 
 nous intensities by any of the methods previously described. 
 Even when the lights differ widely the flicker photometer per- 
 mits of convenient and accurate comparison. In this device a 
 screen, S (Fig. 81), is illuminated by one source, and is viewed 
 
 near its edge through a sight tube, 
 T. A second screen of the same 
 material as the first, circular in form 
 but with the alternate 90° sectors 
 cut out, is mounted so as to be 
 capable of rotation about an axis, 
 F, through the center of the circle, 
 and this is illuminated by the second 
 source. The rotating screen is of 
 such size that when the sectors 
 come in front of the sight tube they 
 eclipse the screen S and are them- 
 selves seen through the tube, while 
 during the alternate quarter revolutions the screen S only is 
 visible. The screen is rotated by hand or by a spring or 
 electric motor. If the illuminations of the two screens are 
 unequal in intensity, a flickering sensation is produced when 
 they are observed through the tube. If the apparatus is so 
 adjusted that the flicker disappears, the intensities of illumi- 
 nation are equal. The equality of luminous intensity is 
 determined independently of difference in color. 
 
 References. — Whitman, Physical Keview, Vol. 3, p. 241, 1896 ; Science, 
 Vol. 9, p. 734, 1899. 
 
 Fig. 81. Flickek Photometer 
 
CHAPTER XIV 
 MIREOES AND LENSES, MAGNIFYING POWEE 
 
 Cn. RADH AND FOCI OF MIRRORS BY VARIOUS METHODS 
 
 Determine the principal focus and radius of curvature of a concave mirror 
 by the method of conjugate foci, by the method of parallax, and by 
 the method of size of image. Determine the principal focus and radius 
 of curvature of a convex mirror by the method of conjugate foci, and 
 with sunlight. 
 
 161. Conjugate Foci. — The radius of curvature, R, the princi- 
 pal focus, F, and the conjugate foci, /^ and f^, of concave and 
 convex mirrors have the relation 
 
 1+1=1=1 
 /i A F R 
 
 A focal distance measured to a real focus (in front of the mirror) 
 is positive, while one measured to a virtual focus (behind the 
 mirror) is negative. If the resulting value of B is negative, it 
 indicates that the center of curvature is behind the mirror, — 
 that is, the mirror is convex ; F wiU also be negative, indicating 
 that the mirror renders convergent rays less convergent, or that 
 it is in effect a diverging mirror. 
 
 If /i is the distance from the mirror of an object whose length 
 is ly, and f^ is the distance from the mirror of the image the 
 length of which is 1.^,1 then 
 
 From these relations there arise various methods of determin- 
 ing the radii and focal lengths of mirrors. 
 
 In aU the following experiments with mirrors and lenses, 
 the foci should, as nearly as is practicable, lie on the line 
 
 213 
 
214 LIGHT 
 
 perpendicular to the mirror or lens at its center point; otherwise 
 the image will be more or less distorted and astigmatic. Also 
 the definition of the image may often be much improved by 
 covering the mirror or lens with a diaphragm permitting the use 
 of only the central portion of its surface. 
 
 162. Concave Mirrors. — Place a bright object, as a gas jet, 
 in front of a concave mirror, and adjust a screen to receive 
 the image in its best defined position. Using the measured 
 distances of the object and image from the mirror, /j and/^ of 
 the above formula, calculate the focus and radius of curvature. 
 A fine wire cross supported in front of the light will serve a 
 better purpose as an object, for the screen can be placed more 
 accurately at the conjugate focus. Measure pairs of conjugate 
 foci with the object at a considerable distance from the mirror, 
 when it is about twice as far from the mirror as is the image, 
 when the object and image are at equal distances from the 
 mirror, and when the object is nearer than the image. Calculate 
 the radius of curvature from each set of measurements. 
 
 Take note of the fact that when the conjugate foci are equal 
 to each other each is equal to the radius of curvature, and 
 that as one focal distance increases its conjugate approaches 
 half the length of the radius of curvature, which is also the 
 principal focal length. 
 
 Instead of receiving the image upon the screen, it is often 
 more accurate to locate its position in the air by the method 
 of parallax. Let the object be a brightly illuminated needle or 
 pointed strip, of paper. Place it in position in front of the 
 mirror, when by looking towards the mirror from a distance 
 the image can soon be found seemingly suspended in the air. 
 By trial a second index needle may be placed so that its point 
 is apparently just in contact with this image. If upon moving 
 the eye from side to side the image and index needle appear to 
 separate, or show parallax, it indicates that the image is nearer 
 the mirror or farther from it than the needle. Adjust the index 
 until the movement of the eye causes no relative displacement, 
 that is, till there is no parallax; then either needle is at the con- 
 jugate focus of the other. When looking at the image and 
 
CONVEX MIEEOES 215 
 
 index, keep the eyes distant about twenty-five centimeters ; or, a 
 magnifying glass may be used to aid in accurate setting. 
 
 The principal focus of a mirror may be approximately and 
 quickly determined by measuring the distance at which parallel 
 rays (sunlight) are brought to a focus. 
 
 Using for the object an illuminated scale and for the index 
 a duplicate scale, adjust, as explained for the needles, till the 
 index coincides with the image. Show, by measurements made 
 for several positions of the object, that the sizes of 'the object and 
 the image are in proportion to their distances from the mirror. 
 
 163. Convex Mirrors. — Virtual images only are produced 
 by convex mirrors ; these images cannot be projected upon a 
 
 Fig. 82. Conjugate Foci of Convex Mikrok 
 
 screen, hence it is necessary to use some indirect method for 
 obtaining conjugate focal points. 
 
 Let a convergent pencil of light produced from a source, S, 
 by a concave mirror, V (Fig. 82), or by a convex lens, be inter- 
 cepted by a convex mirror, X, so placed that the reflected beam 
 shall still be convergent, and let the real image thus produced 
 be received on a screen at p^. This is one real focal point, 
 while it» conjugate virtual focus is at the point p^ on the 
 opposite side of the mirror where the light would have been 
 brought to a focus by the concave mirror had it not been inter- 
 cepted. The distances of the points p^ and p^ from the mirror 
 are the conjugate focal distances /j and f^, which when substi- 
 tuted in the formula given above, regard being had to the 
 proper algebraic sign, will give the focus and radius of the 
 convex mirror. 
 
 The simplest method for obtaining the focal length of a con- 
 vex mirror is to allow a beam of sunlight to fall upon it through 
 an aperture cut in a screen, and to place the screen at such a 
 
216 LIGHT 
 
 distance from the mirror that the reflected pencil has, at the 
 screen, twice the dimensions of the aperture. Then the dis- 
 tance between the screen and mirror is equal to the focal length 
 of the mirror. 
 
 cm. RADU AND FOCI OF LENSES BY VARIOUS METHODS 
 
 Determine the focus of a double convex lens by the method of conjugate 
 foci, and by parallax. Determine the focus of a double concave lens, 
 by the method of conjugate foci, and with sunlight. Determine the 
 focus of a combination of lenses from its focal position. Determine 
 the focus, curvature, and index of refraction of a plano-convex and 
 of a plano-concave flint-glass lens. 
 
 164, Lenses A lens may be composed of one or more 
 
 pieces of glass of various optical densities and with surfaces of 
 different curvatures. The complete formulae connecting the 
 index of refraction, thickness, curvatures, and foci of a lens 
 are too complicated for our purpose. 
 
 For a single thin lens the principal focus, F, is related to the 
 conjugate foci, /j and /g, and to the index of refraction, n, and 
 to the radii of curvature of its surfaces, r^ and r,^, as follows : 
 
 A focal distance measured to a real focus is positive, and one 
 measured to a virtual focus is negative. The radius wf curva- 
 ture of a convex surface is positive, and the radius of curvature 
 of a concave surface Is^^egativeT- If the resultant value of the 
 principal focus, F, is positive, the lens is in effect converging ; 
 if it is negative, the lens is in effect diverging. 
 
 165. Convex Lenses. — Place a light on one side of a wire 
 cross and a convex lens on the other side. Arrange a screen 
 to receive the image of the cross in its best defined position. 
 Measure the distances of the cross and its image from the lens, 
 /j and/2, ^^^ calculate the focus from the above formula. Make 
 three sets of measurements, one with the object at a considerable 
 
CONCAVE LENSES 217 
 
 distance from the lens, one with object and image about equally 
 distant, and- one with the object nearer than the image. 
 
 Instead of receiving the image upon a screen, it is more accu- 
 rate to locate its position in space by the method of parallax. 
 Place a brightly illuminated needle on one side of the lens for 
 the object. Upon looking toward the lens from the other side, 
 at a considerable distance, the inverted image of the needle 
 may be found apparently suspended in the air. By trial a 
 second index needle may be placed so that its point is exactly 
 in contact with this image. This is determined by the absence 
 of parallax, when the eye is moved from side to side, as explained 
 more fully in connection with mirror measurements (Art. 162). 
 
 When a lens is midway between an object and its image, the 
 object and image are of equal size, and their distance apart 
 is four times the principal focal length of the lens. 
 
 Project an image of the sun upon a screen with the lens ; the 
 distance between the lens and screen when the image is best 
 defined is the focal length of the lens. 
 
 With the object and screen at a fixed distance apart greater 
 than four times the focal length of the lens, there may be found 
 two positions of the lens between them which give distinct 
 images. If I is the distance between object and screen, and d 
 is the distance between the two positions of the lens, the focal 
 length of the lens is 
 
 F = 
 
 il 
 
 This method is convenient when the lens is so mounted that 
 accurate measurements to its center cannot be made, its dis- 
 placement hemg measured instead. 
 
 166. Concave Lenses. — Virtual images only are produced by 
 concave lenses ; these images cannot be projected upon a screen, 
 hence it is necessary to use some indirect method for obtaining 
 conjugate focal points. 
 
 Let a convergent pencil of light produced from a source, S, 
 by a convex lens, X (Fig. 83), be intercepted by a concave lens, 
 V, so placed that the refracted beam shall still be convergent, 
 
218 LIGHT 
 
 and let the image thus produced be received on a screen at ip^. 
 This is one focal point for the concave lens, vfhile its conjugate 
 is the virtual focus 'p^, where the light would have converged 
 had it not been intercepted. Measure the distances /j and /g, 
 of the points f^ and ^j, from the concave lens, and apply, with 
 regard to the proper algebraic sign, in the formula given above 
 to compute the principal focus of the concave lens. 
 
 Allow sunlight to fall upon the lens over which has been 
 placed a diaphragm. Receive the emergent pencil of light 
 
 Fig. 83. Conjugate Foci of Concave Lens 
 
 upon a screen placed at such a distance that the pencil at the 
 screen has twice the dimensions of the aperture in the diaphragm. 
 Then the distance between the lens and screen is the required 
 focal length of the concave lens. 
 
 167. Radius of Curvature of a Lens. — If the focus and the 
 radii of curvature of the surfaces of a lens are known, the index 
 of refraction may be computed by the above formula. The 
 curvature may always be accurately found with the spherometer, 
 but for the purposes of this exercise the surfaces may be treated 
 as mirrors, and the curvature found as described below. A 
 plane surface has an infinite radius of curvature. Remember 
 that in the formulae for mirrors the radius of curvature of a 
 convex surface is negative, while the curvature of this surface, 
 considered as a lens, is positive ; and that the curvature of a 
 concave surface is positive as a mirror and negative as a lens. 
 
 For a concave surface, arrange cross wires in a small aper- 
 ture in a white screen, place a light on one side of the screen 
 and the lens on the other side so that its concave surface, acting 
 as a mirror, projects a well-defined image of the cross wires on 
 the screen near the aperture. The distance from the cross wires 
 to the surface of the lens is the radius of curvature. 
 
FOCAL LENGTH OF A SYSTEM OF LENSES 219 
 
 For a convex surface, arrange a convex lens, X (Fig. 84), to 
 converge light from a source, S^ towards the point pj ; the lens 
 whose curvature is to be measured is placed so that its convex 
 surface reflects the light to" a focus at f^. By the principles of 
 Art. 163 compute the radius of curvature. 
 
 If a lens is made of crown glass, its index of refraction is 
 approximately 1.5 ; and if one of its surfaces is plane, the corre- 
 sponding radius of curvature is infinite. For this special case 
 the formula given above reduces to 
 
 F /i /, in 
 
 Hence for a plano-convex or a plano-concave lens of crown glass 
 the radius of curvature is approximately half the focal length, 
 while for mirrors the radius of curvature is twice the focal length. 
 
 168. Focal Length of a System of Lenses. — If a lens has a 
 thickness which is considerable as compared with its focal length, 
 as is the case with microscope objectives and photographic com- 
 binations, the above methods are not sufficient. The following 
 may be applied. 
 
 Arrange the lens co'mbination in the manner commonly em- 
 ployed in lantern projections, to project an image of a strongly 
 illuminated object of known size (a graduated glass scale, or a 
 microscope stage micrometer) on a screen. Place the screen at 
 
 ~--h 
 
 Fig. 84. Badics of Ccrvatuke op Convex Surface 
 
 such a distance that the image is highly magnified. Then if s is 
 the length of a division of the scale whose image has the length 
 S, and d is the distance of the screen from the assumed principal 
 plane of the lens, the required focal length of the lens is 
 
 S+s 
 
220 
 
 LIGHT 
 
 A graphic method may be useful for finding the focal length 
 of a photographic combination. Allow sunlight to fall upon 
 the front lens in the direction of its axis. A piece of tissue 
 paper being placed against the back of the combination, meas- 
 ure carefully the diameter of the emergent beam. It often 
 happens that the interior mounting of the lens is such that it 
 cuts down the effective diameter of the incident light. In this 
 case a diaphragm must be placed over the front lens, before the 
 measures are made, the diameter of which is such that the full 
 incident beam is emergent. Place the lens in the camera and 
 accurately focus it for very distant objects, and measure the 
 
 Fig. 85. Focal Length op Combination of Lenses 
 
 distance of the ground glass from the back surface of the lens. 
 Then the graphic solution is as follows. Draw AB to repre- 
 sent the axis of the lens (Fig. 85). Plot CD, the diameter of 
 the incident beam, on the front lens ; and HF, the diameter of 
 the emergent beam, on the back lens. G is the position of the 
 focal plane, the ground glass. Draw lines through C and I) 
 parallel to the axis, and from G through H and F. At the inter- 
 sections I and If, draw the line IH, perpendicular to the axis ; 
 this represents the principal plane of the lens. The distance 
 from the focal plane to the principal plane, GJ, is the true 
 focal length of the lens combination. 
 
MAGNIFYING POWER 221 
 
 CIV. MAGNIFYING POWER 
 
 Find the magnifying power of a telescope when used for observing distant 
 objects, and for objects distant two meters ; find the power of an 
 opera glass; find the power of a microscope using its several eye- 
 pieces and objectives. 
 
 169. Magnifying Power. — The ratio of the angle subtended 
 by the image of an object seen through an optical instrument 
 to the angle subtended by the object when seen by unaided 
 vision is the magnifying power. 
 
 Telescope. — The power of a telescope increases with the near- 
 ness of the object observed. When the power is stated without 
 qualification, the amplification for an object at a very great dis- 
 tance, the astronomical magnifying power, is implied. This 
 may be measured as follows. 
 
 Focus the telescope for an object at a great distance, and 
 point it towards a bright surface, as the sky. Place the eye 
 about twenty-five centimeters from the eyepiece, in the direc- 
 tion of the axis of the telescope, when there may be seen, out- 
 side the telescope, before the eyepiece, a small, real image of the 
 objective and its mounting. The ratio of the diameter of the 
 aperture in the cell of the object glass (this is taken because it is 
 a well-defined object), to the diameter of its image, is the mag- 
 nifying power. The small image is best measured by means of 
 a fine scale ruled on glass, which is placed before the eyepiece. 
 A hand magnifying glass may be used in the measurement. 
 A specially constructed scale with magnifier attached, called a 
 dynameter, is often provided. A diaphragm in the body of the 
 telescope may reduce the cone of rays from the objective ; this 
 can be ascertained from an examination of the image with a 
 magnifjdng glass. If there is such a diaphragm there should 
 be placed over the object glass a screen having a reduced aper- 
 ture, the whole of which may be seen from the eyepiece ; this 
 aperture and its image are then to be measured. 
 
 For measuring the power of a telescope as used in the labora- 
 tory, or of a Galilean telescope (opera glass), the following 
 method is sufficient. Place a measuring rod, or other scale of 
 
222 LIGHT 
 
 equal parts, at the distance for which the telescope is to be used, 
 and focus the telescope upon the scale. Observe the scale 
 through the telescope with one eye, and with the other eye 
 unaided. Note that N divisions of the scale seen directly are 
 equaled in length by n magnified divisions as seen through the 
 telescope. Then the magnification is 
 
 M= — 
 n 
 
 Instead of a divided scale, a masonry wall, a window frame, 
 or other object with several well-marked, equal subdivisions 
 may be used. 
 
 Microscope. — Place an object of known length under the 
 microscope. A stage, or eyepiece, micrometer on glass with 
 divisions of 0.1 mm is convenient. When one eye is looking at 
 the object through the microscope, adjust a millimeter scale, held 
 near the stage, to be seen distinctly by the other eye unaided. 
 The scale is usually placed 25 cm from the eye, the normal dis- 
 tance of distinct vision. Seeing both scales simultaneously, 
 determine how many magnified divisions, n centimeters, are 
 equaled in length by N centimeters of the unmagnified scale. 
 The magnification is 
 
 M=—. 
 n 
 
 Sometimes it is difficult to see both scales clearly. It may 
 be more convenient, using a camera lucida, to draw the magni- 
 fied scale on a piece of paper placed 25 cm from the eyepiece. 
 This magnified image is then to be measured directly. 
 
 Reference. — Kohlrausck, Physical Measurements, pp. 184-188. 
 
CHAPTER XV 
 GONIOMETEY 
 
 CV. ANGLE OF A CRYSTAL WITH THE GONIOMETER 
 
 Measure the angles between the several faces of crystals of 
 calcite, galena, and garnet. 
 
 170. The Reflecting Goniometer. — Goniometry is the art of 
 measuring solid angles or the inclination of planes, such as the 
 angles between the faces of crystals and prisms. WoUaston's 
 goniometer consists of a vertical divided circle, 10 cm or more in 
 diameter, capable of rotation about a horizontal axis. Through 
 this axis is a second one supporting the crystal. The crystal is 
 attached to a small disk, d 
 (Fig. 86), capable of rotation 
 about three axes : the one, a, 
 passing through the axis of ^ 
 the circle ; a second axis, h, 
 perpendicular to a; and a 
 third one, c, which may have 
 any inclination to the plane 
 of the other two. 
 
 It is necessary that the 
 intersection of the two faces, 
 the angle between which is 
 to be measured, should be 
 parallel to the axis of the 
 circle, and it is desirable that it should coincide with this axis. 
 This adjustment can be made approximately by varying the 
 position of the crystal on the disk, to which it is usually 
 attached with wax. 
 
 223 
 
 EiG. : 
 
 Goniometer 
 
224 LIGHT 
 
 Place the instrument directly in front of a window at the 
 distance of a meter or more, with the axis of the circle parallel 
 to the horizontal bars of the window frame. Looking in the 
 direction of the bottom of the window, the eye being very 
 close to the crystal, rotate the crystal on the axis a, by means 
 of the milled head M, till the top edge of the window seen 
 reflected by the crystal face coincides with the bottom edge of 
 the window seen directly. 
 
 Other more convenient marks may be selected, as the top of 
 a chimney, a point on a roof, or any conspicuous feature in the 
 landscape for one mark, and a mark on the floor or wall for the 
 second. If necessary to make this coincidence exact, adjust 
 the crystal on any of its three axes. Keeping the eye in the 
 same position, turn the milled head Mto get reflection from the 
 second face, and secure coincidence between the two marks as 
 before. When the edge of the crystal is parallel to the axis a, 
 the crystal may be moved from the position of perfect coinci- 
 dence for one face to perfect coincidence for the other face, by 
 rotation on the axis a only ; continue the adjustments untU. this 
 condition is fulfilled. 
 
 By turning the milled head iV, both the circle and crystal 
 are rotated about the axis a; determine the angle between the 
 two coincidence positions; this is the supplement of the angle 
 between the faces of the crystal (Art. 176). 
 
 Make three or more determinations of each angle, repeating 
 the entire adjustment of the edge of the crystal to parallelism 
 with the axis of the circle each time. 
 
 If the circle is set to read 180°, and the crystal is turned by 
 the milled head M to give coincidence from one face, and then 
 the circle is turned by the head N to give the second coinci- 
 dence, the circle reading may be the prism angle directly. 
 
 Often a telescope is attached to give definite direction to the 
 line of sight, the cross wires taking the place of the lower win- 
 dow edge ; and again a collimator may be provided to give a 
 better object than the upper window edge. With these addi- 
 tions the instrument approaches the spectrometer in form, and 
 should be used in the manner described in Exercise CVII. 
 
THE SPECTEOMETER 
 
 225 
 
 CVI. ADJUSTMENT OF THE SPECTROMETER 
 
 Make the complete adjustments of a spectrometer. 
 
 171. The Spectrometer When a goniometer is constructed 
 
 of large size, being adapted particularly to the measurement of 
 prism angles and the deviation of light rays, it becomes a 
 spectrometer. It has for its essential parts (Fig. 87) a horizontal 
 table with a divided circle, a telescope, and a collimator. The 
 telescope and table are capable of independent, measurable 
 
 (a=a^ra 
 
 Fig. 87. Specteometer 
 
 rotations about the principal vertical axis of the instrument, 
 which passes through the center of the circle perpendicular to 
 its plane; the collimator is commonly attached to the base of 
 the instrument, and can be rotated about this axis only with the 
 instrument as a whole. For most purposes it is required that 
 the axes of the telescope and collimator should intersect the 
 axis through the center of the circle, and should always be in 
 the same plane perpendicular to it. In order that the spectrom- 
 eter may be made to satisfy these conditions, and that it may 
 be used for a variety of purposes, it is usually provided with 
 many movements, adjusting devices, and graduated scales, which 
 need not be described. 
 
226 LIGHT 
 
 Before the spectrometer is used for any work of precision 
 the accuracy of its adjustments should be verified. This may 
 be done in various ways, but the following method for all the 
 adjustments of a complete spectrometer is concise and accurate. 
 It is important that the various operations be performed in the 
 order described. 
 
 172. Adjustment of the Spectrometer. Preliminary. — Level 
 the table of the instrument approximately, by means of the 
 screws in the base. Make the axes of the telescope and 
 collimator approximately horizontal. 
 
 Set the telescope and collimator (or the table) at such a 
 height that they will be on the same level as the prism, or other 
 apparatus to be used, when the latter is in its mounting and 
 resting on the table. 
 
 Turn the telescope and collimator on their supports so that 
 their axes shall intersect the vertical axis through the center 
 of the circle. Often there are graduated scales to indicate 
 these positions ; if not, simple inspection will determine them 
 sufficiently. 
 
 For making the precise adjustments, a mirror of plane- 
 parallel glass, mounted on a support with leveling screws, is 
 required. Preferably this mirror should be silvered on both 
 sides, though it will serve silvered on one side or even if not 
 silvered at all. Place this mirror on the spectrometer table and 
 set it vertical by inspection. 
 
 To focus the Telescope for Infinity. — Place a collimating 
 (Gauss) eyepiece in the telescope. This eyepiece is one hav- 
 ing between the eye and the cross wires a clear, plane-parallel, 
 glass reflector inclined at about 45° (Fig. 88), so as to throw 
 light, received through an aperture in the side of the eyepiece, 
 along the optic axis of the telescope. Place a light to illumi- 
 nate this eyepiece. Looking through the telescope, slowly turn 
 the spectrometer table one way or the other till, when the 
 mirror is normal to the optic axis, a flash of light appears. 
 If the flash cannot be seen, alter the position of the light with 
 respect to the eyepiece, or repeat the preliminary adjustments 
 with greater care. Continue the search until the flash is found. 
 
THE SPECTEOMETEE 
 
 227 
 
 C 
 
 ci; 
 
 \ 
 
 when the light and mirror should be adjusted to cause the 
 entire field of view to be illuminated. 
 
 Focus the eyepiece upon the cross wires by sliding it in its 
 draw tube. By means of the rack and pinion slowly focus the 
 telescope, observing the field of view carefully for the appear- 
 ance of a second image of the cross wires. This may be faint, 
 and at first appear only in part. When it is found, readjust 
 the position of the light, rotate the eyepiece, 
 focus the eyepiece, focus the telescope or 
 alter the position of the eye, making any or 
 all of these adjustments until both images of 
 the cross wires appear distinct and motion 
 of the eye discloses no parallax. When this 
 is accomplished the telescope is focused for 
 infinity, and its focus should not again be 
 disturbed throughout the entire experiment. 
 
 To make the Optic Axis of the Telescope 
 Perpendicular to the Axis of the Table. — 
 Rotate the spectrometer table and alter the 
 leveling screws of the mirror, to bring the 
 reflected image of the cross wires into coin- 
 cidence with the direct image. 
 
 Rotate the spectrometer table (not the 
 mirror stand) till the second face of the mir- 
 ror is toward the telescope, and find the 
 image of the cross wires reflected from this face. Cause this 
 image to coincide with the direct image by the following 
 motions only. Rotate the table till the reflected image is ver- 
 tically above or below the other ; alter the inclination of the 
 telescope axis in the vertical plane to move the reflected image 
 half way towards the direct one ; alter the leveling screws in 
 the mirror stand till exact coincidence is obtained. 
 
 If the adjustments have been accurately made, coincidence 
 should be obtained from the first face when it is turned to the 
 telescope. If, upon trial, it is not exact, correct half the error 
 by altering the inclination of the telescope, and the other half 
 by tilting the mirror. 
 
 Lig ht 
 
 Fig. 88. Telescope 
 
 with collimat- 
 
 ING Eyepiece 
 
228 LIGHT 
 
 Continue in this manner till the reflected image from either 
 face coincides with the direct image. When this is true the 
 axis of the telescope is perpendicular to the axis of the table, 
 and the surfaces of the mirror are parallel to this axis. 
 
 To make the Optic Axis of the Telescope Perpendicular to its 
 Axis of Rotation. — The axis of rotation of the telescope should 
 be coincident with the axis of the table. As the spectrometer 
 is ordinarily constructed the relation of these two axes cannot 
 be altered. If the two axes are not parallel, the optic axis can 
 be made perpendicular to either one as desired, but of course not 
 to both. Evidently it should be adjusted with respect to that 
 axis about which rotation is to take place in the contemplated 
 measurements. 
 
 The preceding adjustment having been completed, the accu- 
 racy of construction may be quickly tested. For the mirror is 
 parallel to the axis of the table, and the telescope is perpendic- 
 ular to it. If, now, the telescope is turned 180° about the cen- 
 tral axis, it should be still perpendicular to the mirror if the 
 axis on which it turns is parallel to that of the table. 
 
 If it is desired to set the optic axis perpendicular to the axis 
 of rotation of the telescope, the method is similar to that of the 
 preceding adjustment, except that the telescope is rotated from 
 side to side of the mirror instead of the mirror being turned 
 from side to side by the rotation of the table. 
 
 Instead of turning the telescope, which requires also the 
 movement of the observer and light, the same result will be 
 obtained by holding the telescope and revolving the other parts 
 of the instrument on its base. 
 
 To test whether the Surface of the Table is Perpendicular to 
 its Axis. — The telescope and mirror being adjusted for coin- 
 cidence of the direct and reflected images of the cross wires, 
 carefully lift the mirror above the table, rotate the table 180°, 
 and replace the mirror. This operation should not have dis- 
 turbed the coincidence in a vertical direction. 
 
 To adjust the Collimator. — Remove the mirror and sight the 
 telescope, through the collimator, on the slit; cause the latter 
 to be seen distinctly, and without parallax in respect to the 
 
THE SFECTEOMETEE 229 
 
 cross wires, by sliding the draw tube of the slit. Do not alter 
 the focus of the telescope. 
 
 Turn the slit horizontal, and level the collimator by bring- 
 ing the slit to the cross wires. Replace the slit in the vertical 
 position, and do not again alter the adjustments. 
 
 If the spectrometer is not provided with a coUimating eye- 
 piece, the telescope may be focused on a distant object, the 
 telescope, collimator, and table may be leveled by means of a 
 spirit level, and the other adjustments can be made in ways 
 that will suggest themselves. There are also other methods 
 of adjustment which have not been mentioned, but in general 
 they are less precise than the one described above. 
 
 173. Reading Divided Circles. — Both the axis of the spec- 
 trometer table and the axis about which the telescope rotates 
 should pass exactly through the center of the divided circle. 
 In actual construction this condition is difficult to fulfill ; the 
 error introduced in the circle indications is called the eccentricity 
 of the circle. This error is wholly eliminated from the results 
 by reading two opposite verniers (Art. 17) and taking the mean. 
 Readings are sometimes made at three or four equidistant points 
 to eliminate eccentricity and graduation errors. 
 
 One vernier may be designated as vernier A, and the other as 
 vernier B. To avoid confusion the simple method may be 
 adopted of calculating the angle from the readings of vernier 
 A, and separately from vernier B, and taking the mean. If a 
 vernier between two readings is rotated through the zero point, 
 this must be taken into account in computing the angles. The 
 readings for an angle reduced by this method would appear as 
 follows : 
 
 Veknieb a Vernier B 
 
 On first face 335° 5' 20" 155° 2' 20" 
 
 On second face . . . . 95 7 10 275 6 40 
 
 Difference 120° 1' 50" 120° 4' 20" 
 
 Angle 120° 3' 5" 
 
 A somewhat more concise plan is to record the degrees of 
 vernier A only, but to use the mean of the minutes and seconds 
 
230 LIGHT 
 
 of the two rerniers. The above measurements, in this method, 
 would appear as follows : 
 
 On Fikst Face On Second Face 
 
 Vernier A 335° 5' 20" 95° 7' 10" 
 
 Vernier B 2 20 6 40 
 
 Mean 335° 3' 50" 95° 6' 55" 
 
 Angle 120° 3' 5" 
 
 The method of averaging the entire readings of the two 
 verniers for each setting should be avoided, as it is more 
 cumbersome. 
 
 CVn. ANGLE OF A PRISM WITH THE SPECTROMETER 
 
 Measure the angle of a prism -with the spectrometer by three methods. 
 
 174. Adjustment of Prism. Mounting the Prism. — Before 
 beginning the work with the prism, the spectrometer should have 
 been completely adjusted, as described in the preceding exercise. 
 These adjustments must not be disturbed while placing the prism 
 in its proper position. 
 
 Mount the prism on a support with three leveling screws, 
 
 place it upon the spectrometer table with one face towards the 
 
 telescope, and level the prism by inspection. For convenience 
 
 it is desirable that the angle between the two 
 
 lines joining one of the leveling screws in 
 
 the base of the prism holder to the two 
 
 other screws, should be the same as the 
 
 angle between the two faces of the prisms. 
 
 When this is true so place the prism on its 
 
 support that each of its two edges shall be 
 
 Pig. 89. Position of Perpendicular to one of these lines joining 
 
 Prism on Stand t^e leveling screws. Then (Fig. 89) the 
 
 face AB can be leveled by the screw D 
 
 without disturbing the level of the face AC ; and the face AC 
 
 may be leveled by the screw F without disturbing the face AB. 
 
 Before beginning the adjustment of the prism, it will be well 
 
 to turn the table and telescope into all the several positions 
 
ANGLE OF A PRISM 
 
 231 
 
 which they will later occupy in making the measurements, to 
 observe whether it will be possible to read the two verniers of 
 the divided circle for each of the settings. Sometimes one ver- 
 nier may come under the collimator or telescope and be inacces- 
 sible. This difficulty may usually be obviated by altering the 
 direction in which the prism stands on the table. 
 
 It should also be noticed whether in these positions the prism 
 faces are in the most advantageous relations to the objectives 
 of the collimator and telescope. The best position is not 
 always that in which the prism holder is centrally placed on 
 the spectrometer table. 
 
 To adjust the Prism. — The two prism faces must be parallel 
 to the vertical axis of the spectrometer table (or of the tele- 
 scope). Arrange the coUimating eyepiece and light to observe 
 the reflected image of the cross wires from the first face of the 
 prism, as described in Art. 
 172. By turning the spec- 
 trometer table and by alter- 
 ing the leveling screws in 
 the prism support only, 
 bring the reflected cross 
 into coincidence with the 
 direct image. The first face 
 is then in position. Turn 
 the spectrometer table till 
 the cross wires are seen 
 reflected from the second 
 face of the prism, and by 
 the principle illustrated in 
 Fig. 89 adjust the second 
 face. This should not have disturbed the first face ; but 
 reobserve the cross in the first face, correcting any displacement 
 which may have occurred. Continue thus until the reflected 
 image from either face coincides with the direct image, when the 
 prism will be in adjustment. 
 
 175. Angle of Prism ; First Method. Without CoUimating Eye- 
 piece; Table fixed. Telescope movable. — Place a light, L (Fig. 90), 
 
 Fig. 90. Angle of Pkisik 
 
 - Method I 
 
232 
 
 LIGHT 
 
 Fic. 91. Relation op Angles — Method I 
 
 to illuminate the slit. Turn the prism P with its refracting edge 
 toward the collimator C, dividing the beam of light, part falling 
 on one face of the prism and part on the other. The exact 
 
 position of the prism is 
 not important. Clamp 
 all the movable parts 
 of the spectrometer 
 except the telescope. 
 Set the telescope in 
 the position T-^, on the 
 image of the slit re- 
 flected from the first 
 prism face. Read the 
 two verniers of the divided circle, as explained in Art. 173. 
 Turn the telescope to the position T^, and set on the image of 
 the slit reflected from the second face. 
 
 The angle through which the telescope has been turned is 
 twice the prism angle. For (Fig. 91) it has moved through the 
 angle h + A + c. It is evident that the three angles b are all 
 equal, and the same is true of the three 
 angles c ; also J + c is equal to A ; therefore 
 b + A + c = 2A, twice the prism angle. 
 
 176. Angle of Prism; Second Method. 
 Without Collimating Eyepiece; Table mov- 
 able. Telescope fixed. — Turn the telescope 
 of the spectrometer as near to the colli- 
 mator as is convenient, and clamp all parts 
 except the table. Illuminate the slit. Rotate 
 the table until the image of the slit, reflected 
 from one face of the prism, is set on the 
 cross wires of the telescope (Fig. 92). Read 
 the two verniers, as described in Art. 173. 
 Rotate the table to bring the second prism 
 face into position to reflect the image of the slit to the cross wires. 
 The angle through which the table has been turned is the sup- 
 plement of the prism angle ; for in the first position the normal 
 to one face bisects the angle between the axes of the telescope 
 
 Fig. 92. Angle of 
 Prism — Method II 
 
AJvTGLE OF A PEISM 
 
 233 
 
 and collimator, and in the second position the normal to the 
 other face is turned to this same direction. The angle between 
 the normals is tlie same as the angle between the two faces 
 (Fig. 93), but the table has clearly been 
 turned through the supplement of this 
 angle. 
 
 177. Angle of Prism ; Third Method. 
 With Collimating Uyepiece ; either Tele- 
 scope or Table fixed, the other being 
 
 movable. — Clamp the telescope, and 
 
 ,. , , ,,. ,. Fig. 93. Relation of 
 
 arrange a light, the colhmatmg eye- angles-Methods Hand ill 
 piece, and the prism, to obtain the 
 
 direct and reflected images of the cross wires (Fig. 94), as 
 described in Art. 172. Secure coincidence between the two 
 images by rotating the table only, which is possible if the adjust- 
 ments have not been disturbed. Read 
 the verniers ; rotate the spectrometer 
 table till coincidence between the direct 
 and reflected images of the cross wires 
 is secured with the second prism face, 
 and again read the verniers. 
 
 From Fig. 93 it is clear that the table 
 has been turned through an angle which 
 is the supplement of the prism angle. 
 
 If the table is fixed, the telescope may 
 be turned from one coincidence to the 
 other, in which case the light must also 
 be moved ; or the spectrometer may be 
 turned on its base, the telescope being 
 held during the motion. This will avoid moving the light, and 
 may keep the telescope in a more convenient position. This 
 third method for measuring the angle between two reflecting 
 surfaces is the most precise one. 
 
 Fig. 94. Angle of Prism 
 — Method III 
 
CHAPTER XVI 
 INDEX OF REFRACTION 
 
 CVm. INDEX OF REFRACTION BY MINIMUM DEVIATION 
 
 Find the index of refraction of a prism for sodium light. 
 
 178. Index of Refraction. — Light moves with different veloci- 
 ties through different media. The ratio of its velocity in vacuum, 
 in which its velocity is greatest, to its velocity in any other 
 medium is the absolute index of refraction of this medium. It 
 is not practicable to compare these velocities directly. When 
 light passes from one medium into a second, its direction of 
 propagation in the^ second medium as compared with its direc- 
 tion in the first is a function of the angle of incidence upon the 
 bounding surface and of the indices of refraction of the two 
 media. If i is the angle of incidence, r the angle of refraction, 
 Wj and Wj ^^ indices of refraction of the two media, then 
 
 sin r = sin i — . 
 
 In laboratory measurements the first medium is usually air ; if 
 the index of refraction of air is taken as unity, the relative index 
 of refraction of the second medium is 
 
 sin i 
 n = —. — . 
 sinr 
 
 In the particular case of light passing through a prism in the 
 direction of minimum deviation, the relative index of refraction 
 of the prism is 
 
 \A 
 
 234 
 
INDEX OF REFRACTION 
 
 235 
 
 where D is the angle of minimum deviation (Art. 179) and A is 
 the refracting angle of the prism (Art. 175). 
 
 The index of refraction of a liquid may be determined by 
 this method, the liquid being contained in a hollow prism having 
 sides of plane-parallel plates which will have no deviating effect. 
 
 If the index of refraction of a second medium relative to the 
 first is multiplied by the absolute index of refraction of the first 
 medium, the result is the absolute index of refraction of the 
 second medium. The absolute index of 
 refraction of air at 20° under normal 
 pressure is 1.0002773. 
 
 179. Angle of Minimum Deviation 
 
 The spectrometer and prism are to be com- 
 pletely adjusted by the methods described 
 in Arts. 172-174. Illuminate the slit 
 with the kind of light for which the index 
 of refraction is desired, and so place the 
 prism and telescope (Fig. 95) that the 
 spectrum is visible in the telescope. For 
 finding the spectrum, a luminous flame ij 
 desirab].e, as it gives a bright, continuou i 
 spectrum. Even when the sodium spec- 
 trum is to be examined it is not necessary 
 to use the Bunsen flame ; a borax bead, 
 placed in the luminous gas flame, gives bright yellow sodium 
 lines in the continuous spectrum, which are very convenient 
 for observation. 
 
 Having found the spectrum in the telescope, rotate the prism, 
 following the motion of the spectrum with the telescope, in such 
 a manner as to cause the deviation D, of the light by the prism, 
 to have the least possible value. This position of the prism is 
 a definite one, and is readily found by trial; a rotation of the 
 prism in either direction from this point causes the deviation 
 to increase. Set the telescope on the sodium line when the 
 prism is at this turning point of minimum deviation. 
 
 Turn the prism (not the divided circle) on the table so that it 
 produces deviation on the opposite side, as indicated in Fig. 96, 
 
 Fig. 95. Minimum 
 Deviation 
 
236 
 
 LIGHT 
 
 and set the telescope on the sodium line when the prism is 
 adjusted for minimum deviation as before. 
 
 The angle between the two positions of the telescope is twice 
 the angle of deviation D. In making the circle readings, use 
 the two verniers as described in Art. 173. 
 
 Instead of finding the deviation on the second side, the prism 
 might have been removed after the first setting, and the tele- 
 scope sighted directly on the slit through the collimator. Then 
 
 the angle between the two posi- 
 tions of the telescope is the angle 
 of minimum deviation. The first 
 method is preferable. 
 C 
 
 CIX. INDEX OF REFRACTION 
 WITH A MICROSCOPE 
 
 .^' 
 
 ' Fig. 96. Second 
 Position for Minimum 
 Deviation 
 
 Determine the index of refraction of a 
 piece of plate glass, and of water, 
 alcohol, and carbon bisulphide, using 
 a microscope. 
 
 180. Index of Refraction with a 
 Microscope. — Any transparent sub- 
 stance which can be put in the form 
 of a plane-parallel plate of suitable 
 thickness may have its index of refraction determined with an 
 ordinary microscope, with an accuracy giving two or three deci- 
 mal places in the result. The objective of the microscope must 
 have such a focal length (working distance) as to permit an 
 object to be seen through the plate. The accuracy of the result 
 is increased by using a plate of the greatest possible thickness 
 together with a microscope objective of the shortest focal length 
 that will permit observation through the thick plate. 
 
 Let c (Fig. 97) be any object, as a fine scratcli on a piece of 
 glass, over which is placed the plate of thickness t whose index 
 of refraction is desired ; and let cam be the path of a ray of 
 light through the plate and air to the microscope when the 
 
INDEX OF KEFKACTION 
 
 237 
 
 latter is accurately focused. The light enters the microscope 
 as though it had come from the point b ; that is, the plate has 
 apparently elevated the object from o to b, represented by e. 
 The angles of incidence and refraction are represented by i and r 
 respectively. From the figure it is evident that, in the triangle 
 abc. 
 
 ac 
 ab 
 
 sin I 
 sinr 
 
 = 71, 
 
 the desired index. But in actual microscopic observation the 
 distance ao is very small as compared with ab and ac, and 
 
 ac , . , , . oc t 
 
 - = (approximately) -=^—; 
 
 to this degree of approximation, then, 
 
 _ t 
 ^^ t-e' 
 
 The thickness, t, may be measured with calipers, and the 
 apparent elevation, e, of the object by the plate is measured by 
 noting the difference between 
 the two positions of the mi- 
 croscope body when it is 
 focused on the object uncov- 
 ered and on the object cov- 
 ered by the plate. For 
 measuring this distance a 
 scale may be fastened with 
 wax to the stand of the mi- 
 croscope, and a -needle for a 
 pointer to the body tube ; or 
 the fine focusing screw, if of 
 known pitch, may be used. 
 If the pitch is unknown, 
 both the thickness of the 
 plate and the elevation may 
 
 be determined in terms of turns of the screw, giving the index 
 of refraction as before. The thickness may be measured, in 
 the first case, by focusing upon the uncovered object and then 
 
 Fig. 97. 
 
 Index ok Refkaction with 
 
 A MiCKOSCOPE 
 
238 LIGHT 
 
 upon the upper surface of the plate when it is in position, in 
 contact with the object. 
 
 If the plate has scratches or other marks on both surfaces, 
 and the thickness t has been measured with calipers or otherwise, 
 it is only necessary to focus upon the top surface and upon the 
 lower surface through the plate, noting the depression d of the 
 microscope. Then the index of refraction of the plate is 
 
 t 
 
 In determining the index of refraction of liquids, a crystallizing 
 dish is convenient for the receptacle. Make a fine mark on the 
 inside of the bottom of the dish. Focus upon this, and let the 
 scale reading be rj ; pour in the liquid to a depth as great as 
 will allow the mark to be seen through the liquid, focus, and 
 let r^ be the reading ; focus upon the upper surface of the liquid, 
 which may be indicated by some floating particle, and let r^ be 
 the scale reading. Then the index of refraction is, as before, 
 
 ri-ro 
 
 ex. INDEX OF REFRACTION BY DISPLACEMENT 
 
 Determine the index of refraction of a thick plane-parallel disk of glass. 
 
 181. Index of Refraction by Displacement, — A thick plate 
 with two plane-parallel surfaces, such as a glass disk, may have 
 its index of refraction determined by the following displace- 
 ment method. The measurements may be made with simple or 
 with more elaborate apparatus, according to circumstances and 
 the precision required. 
 
 Let a telescope be set to view a divided scale, *S', in the direc- 
 tion VS (Fig. 98). Set the plate whose index is desired between 
 the telescope and scale, with its plane faces perpendicular to the 
 line of sight. This condition is secured with sufficient accuracy 
 by observing with the telescope that there is no apparent dis- 
 placement of the scale seen through the plate. Let S be the 
 
TOTAL REFLECTION 
 
 239 
 
 point of the scale under the cross wires. Rotate the plate about 
 an axis perpendicular to the plane containing the line of sight 
 and the scale ; the scale will appear displaced, the point x now 
 being seen under the cross wires. Let d be the amount of this 
 displacement, t the thickness of the plate, and i the angle 
 through which the plate has been 
 turned. The angle of incidence of 
 the light on the plate is i, and the 
 angle of refraction, r, is to be found. 
 Without detailed explanation it is 
 evident that the following equations 
 are true. 
 
 fe 
 
 eh 
 tan r = — = 
 
 t 
 
 = tan i — 
 
 ce 
 
 cos ^ 
 
 t cos I 
 
 The index of refraction of the plate 
 is 
 
 sin i 
 
 n = 
 
 smr 
 
 . Displacement by 
 Refkaction 
 
 The rotation of the plate may be 
 measured with a protractor, or by 
 placing the plate on an engineer's transit, or on the table of a 
 goniometer or spectrometer. 
 
 CXI. INDEX OF REFRACTION BY TOTAL REFLECTION 
 
 Determine by Kohlrausch's method the index of refraction of a crown 
 glass plate immersed in carbon bisulphide, of turpentins in a glass 
 box, of a drop of Canada balsam, and of water by the immersion 
 of an air-filled box. 
 
 182. Total Reflection. — Light moving in one medium whose 
 index of refraction is n to the surface of another medium of 
 
240 
 
 LIGHT 
 
 greater refractive index, N, will be totally reflected when the 
 
 angle of incidence is such that its sine is greater than — . If 
 
 the limiting value, <^, of the angle of total reflection is observed, 
 and the index of refraction of one substance is known, the 
 other index may be found from the relation 
 
 sin (/) = 
 
 jsr 
 
 Various instruments using this principle have been devised, 
 which permit the determination to the fourth or fifth decimal 
 place of the index of refraction of any liquid, and of such 
 solids as have indices less than the index of a liquid used in 
 the apparatus. It is especially useful for liquids in small quan- 
 tities, for solids in thin plates or in powdered form, for crystals, 
 for bodies with only one small reflecting face, and for opaque 
 bodies. When the face of the body is of small reflecting power, 
 
 or is only approximately flat, the 
 method is still applicable. 
 
 183. Total-Reflectometer (Kohl- 
 rausch). Solids. — The body to be 
 investigated is supported in a 
 small flask filled with a liquid of 
 large and known refractive index. 
 Such liquids are carbon bisulphide 
 (1.63), a-bromonaphthalene (1.66), 
 and methyl iodide (1.74). The 
 index of the liquid may be deter- 
 mined from tables or with a hollow 
 prism and the spectrometer, as 
 described in Art. 178. The sup- 
 port of the body passes through 
 the center of a divided circle and 
 is provided with centering devices for placing the face of the 
 body in the axis of rotation, as shown in Fig. 99. 
 
 The front of the flask is a plane-parallel plate through which 
 a small telescope may receive light reflected from the body. 
 
 CEC 
 
 Fig. 99. Total-Reflectometer 
 
THE TOTAL-KEFLECTOMETER 
 
 241 
 
 Fig. 100. Kohlrausch 
 Total-Reflectometer 
 
 The telescope should be perpendicular to the axis of rotation, 
 and the face of the body and of the flask parallel to this axis. 
 A Gauss eyepiece or a collimator is provided for facilitating 
 these adjustments, which may be 
 made according to principles ex- 
 plained in Art. 172. 
 
 The flask is illuminated by sodium 
 light, L (Fig. 100), and by the inter- 
 position of a piece of oiled tissue 
 paper, diffused light from many 
 directions will fall upon the body. 
 All rays having a smaller angle of in- 
 cidence than the angle of total reflec- 
 tion (rays 1 and 2) will be partially 
 refracted and partially reflected, 
 while all rays having a greater angle 
 of incidence (3 and 4) will be totally 
 reflected. By turning the support of the body, the dividing 
 line corresponding to the angle of total reflection may be 
 brought to the center of the field of view of the telescope. 
 When this position has been found, a condensing lens may be 
 employed to increase the illumination. Read the position of the 
 body as indicated by the divided circle. Illuminate the opposite 
 side of the flask, and rotate the body until the limiting angle 
 for the opposite side is observed. The difference between the 
 two positions is twice the limiting angle, 0, of total reflection, 
 and the index of refraction of the body is 
 
 n = iVsin ^, 
 
 N being the index of the liquid. 
 
 The index of refraction of .carbon bisulphide is 1.6277 at 20°, 
 which changes 0.00080 per degree, being less for higher tem- 
 peratures. The light may be placed forty or more centimeters 
 from the flask, and a screen with a glass-covered aperture may 
 be used to protect the liquid from the heat of the flame. 
 
 Liquids. — If a solid plane plate having a known index of 
 refraction n is suspended in the liquid of unknown but greater 
 
242 LIGHT 
 
 index iV, and the angle of total reflection., is observed, the index 
 of refraction of the liquid is 
 
 n 
 
 N-- 
 
 siri ^ 
 
 If the liquid has an index less than that of the solid plane- 
 parallel plate, the plate is to be so immersed that there is 
 a film of air behind it. This is secured by making the plate 
 one side of a small box. The index of the plate should be 
 large, but it need not be known. Making the observations 
 as before, the index of refraction of the hquid in which the 
 box is immersed is 
 
 sm (^ 
 
 If the glass box is immersed in a liquid of known index, 
 less than that of its plane-parallel side but greater than that of 
 an unknown liquid, the index of the latter may be determined 
 by filling the box with it and proceeding exactly as for the 
 index of a solid. If only a small quantity of the unknown 
 liquid is available, instead of filling the box with it, a single 
 small drop placed on the inner surface of the plane-parallel 
 plate suffices for the experiment. 
 
 Crystals. — The refraction of a crystal can only be described 
 with the aid of two, or sometimes three, indices of refraction. 
 In order to determine them the direction of the axis of the 
 crystal must be known, and then the limiting angles of reflec- 
 tion corresponding to the various indices may be distinguished 
 with the aid of an analyzing Nicol prism. The method of pro- 
 cedure is described in special treatises and in the references. 
 
 Refekestces. — Kohlrausch, Physical Measurements, p. 161 ; Zeiss, Die 
 optischen Instrumente, pp. 52-61 ; Drude, Theory of Optics, pp. 339-344. 
 
 184. Refractometer (Pulfrich). — This instrument uses not 
 total reflection but refraction at grazing incidence. A glass 
 prism of known index of refraction, greater than that of any 
 substance to be investigated, has attached to its horizontal 
 face a glass cylinder to contain the liquid to be tested (Fig. 101). 
 
THE EEFEACTOMETEK 
 
 243 
 
 Behind the instrument in the plane of the prism is placed a 
 sodium light, which is condensed upon the surface of the prism 
 through the cylinder of liquid. The rays at grazing incidence 
 will be refracted at the angle of total reflection, while all other 
 rays will be partially refracted at less angles and partially 
 reflected. A telescope capable of motion in a vertical plane 
 about the axis of a divided circle measures the angle of emer- 
 gence a ; it is first set normal to the face of the prism (Art. 
 172), and then so that the field of view is half bright and half 
 dark. If N is the index of 
 the glass prism, the index of 
 refraction of the liquid in the 
 cylinder is 
 
 Vjv^ 
 
 sin-' a. 
 
 Fig. 101. PuLPKiCH 
 Eefkactometer 
 
 Tables are usually provided 
 
 which for a given prism show 
 
 the values of n for given 
 
 values of a. The index of refraction of water may be measured 
 
 to show that the circle and prism are properly set. For water 
 
 at 20°, n = 1.3332. 
 
 The index of refraction of liquids varies considerably with 
 the temperature, hence the temperature of the observations 
 must be noted ; usually this should be made 20°. If the room 
 temperature is higher than this, the liquid may be cooled a 
 small amount, and the observation made when its temperature 
 passes through 20°; and vice versa. 
 
 The cylinder may be surrounded by a hollow case through 
 which hot water or steam may be passed, or a coil of wire carry- 
 ing a current of electricity may be used, and the variations of 
 the index of refraction with temperature, or the index of a sub- 
 stance which is transparent only when melted, may be thus 
 investigated. In these cases a correction to the index of the 
 prism is required. 
 
 A partition of black glass may be cemented in the cylinder, 
 dividing it into two cells, permitting the convenient measure- 
 ment of the differential refraction of two liquids. 
 
244 LIGHT 
 
 A solid substance may be investigated if it has two surfaces 
 approximately at right angles, one of which is plane and well 
 polished, and the other polished sufficiently to transmit light. 
 The edge in which the surfaces intersect must be perfect. The 
 plane face is joined to the surface of the prism with a liquid of 
 high refractive index, the cyhnder being removed if necessary. 
 The index is then determined as described above. 
 
 If the solid is in the form of powder, its index of refraction 
 may be determined with moderate precision, provided its index 
 is intermediate in value between that of two available liquids 
 (alcohol, ether, or acetone, 1.36; benzene, 1.50; a-bromonaphtha- 
 lene, 1.66) which when mixed will not dissolve the powder. The 
 cylinder of the refractometer is filled with the powder and the 
 mixed liquids ; the proportions of the latter are so altered that 
 the index of the mixture is the same as that of the powder. 
 If the index of the liquid is too great, a bright band appears at 
 the boundary between the light and dark parts of the field of 
 view ; while if the index of the liquid is too small, the boundary 
 is indistinct. When the mixture of the liquids is correct the 
 dividing line indicating the limiting angle is quite sharp, and 
 then the index computed is that of both solid and liquid. 
 
 Reference. — Pulfrich, Astrophysioal Journal, Vol. 3, p. 259, 1896. 
 
 185, Crystal-Ref tactometer (Abbe). — In this method a hemi- 
 sphere of glass is substituted for the flask of liquid of the 
 
 Kohlrausch method (Fig. 102). It 
 requires but a small quantity of liquid 
 to join the crystal to the flat surface 
 of the hemisphere, and by rotating the 
 hemisphere on an axis perpendicular 
 to the surface of the crystal it per- 
 mits a convenient observation of the 
 relations of the several limiting curves 
 ^ BEiRAoxo^M™''" o^ *°*^1 reflection. It is not possible 
 
 to adjust the surface of the plate as 
 accurately as in Kohlrausch's method, and the construction 
 of the hemisphere and its adjustment introduce difficulties. 
 
THE EEFEACTOMETEE 245 
 
 Furthermore, it is not so well adapted to the investigation of small 
 and imperfectly reflecting surfaces. It is used according to the 
 principles explained for Kohlrausch's method, and is applicable 
 for a solid whose index is less than that of the hemisphere and 
 of the liquid with which it is joined to the hemisphere, and for 
 any liquid with an index less than that of the hemisphere. It 
 may be used with light at grazing incidence, or by total reflection. 
 The reference may be consulted for the method of use. 
 
 Kefeeence. — Leiss, Die optischen Instrumente, pp. 38-49. 
 
CHAPTER XVII 
 WAVE LENGTH OE LIGHT 
 
 CXII. WAVE LENGTH OF LIGHT BY FRESNEL'S 
 INTERFERENCE METHOD 
 
 Determine the wave length of sodium light. 
 
 186. Interference of Light. — If two series of exactly similar 
 waves of homogeneous light, originating in two near sources, 
 arrive at the same point in space, the disturbance at this point 
 is the resultant of the two waves. The loci of all points at 
 which the two waves simultanecjusly arrive in the same phase 
 will be brilliantly illuminated ; the loci of points at which the 
 waves arrive in opposite phases will be dark. These loci are 
 hyperboloids of revolution, having the two origins for foci. If 
 the two waves fall upon a screen, there will be seen the inter- 
 sections of the screen and the hyperboloid loci; these appear 
 as a series of alternately bright and dark interference bands or 
 fringes. The distance from one bright band to the next is a 
 function of the distance between the two sources, the distance 
 from the screen and the wave length of the light. Such inter- 
 ference phenomena may then be used in determining the wave 
 length of light. 
 
 187. Wave Length of Light with the Bi-Prism. — Since it is 
 impossible to make two exactly similar sources of light, it is 
 necessary for producing interference bands to divide one single 
 beam into two parts which shall have the same effect as the two 
 sources. Fresnel used for this purpose a bi-prism, which is equiv- 
 alent to two prisms, BP (Fig. 103), of very small angle (one- 
 half degree, for instance), placed base to base. If light from 
 a slit, S, set parallel to the refracting edge of the prism, falls 
 
 246 
 
THE BI-PRISM 
 
 247 
 
 upon the prism, it will be refracted into two beams which will 
 appear to come from two sources, S-^ and S^, very near to each 
 other. These beams are each divergent, and where they over- 
 lap interference bands may be produced. These bands may be 
 received upon a screen at J/; but for purposes of measurement 
 it is convenient to mount the slit and prism on an optical bench, 
 which also carries a micrometer eyepiece with which the bands 
 are to be observed. 
 
 Place the center of the slit, bi-prism, and eyepiece in the same 
 straight line, the plane face of the prism being perpendicular to 
 this line. Set the prism about twenty centimeters from the slit, 
 and the eyepiece fifty centimeters from the prism. It will be 
 well to place a bright light behind the slit, and first to find the 
 
 interference bands on a screen placed against the eyepiece. If 
 they are not at once visible, slightly alter the position of the 
 various parts, rotate the prism about the line of sight to make 
 its edge more nearly parallel to the slit, narrow the slit, and 
 correct the alignment, making any or all of these adjustments 
 till the bands are seen with satisfactory distinctness through 
 the eyepiece. 
 
 Illuminate the slit with the light whose wave length is 
 required, and carefully determine with the micrometer the 
 average distance between successive bright bands in the follow- 
 ing manner. Read the position of ten successive bright bands ; 
 subtract the first reading from the sixth, the second from 
 the seventh, etc. ; average these differences and divide by five ; 
 the result is the difference between consecutive bands. Make 
 three sets of readings, the mean of which is the quantity x of 
 
248 LIGHT 
 
 the formula. If d is the distance of the slit from the cross wires, 
 and c the distance between the two apparent sources of light, 
 then the wave length is 
 
 ex 
 
 The quantity e may be determined by either of the following 
 methods. 
 
 Select a convex lens of focal length less than one fourth the 
 distance d ; place it in such a position between the eyepiece 
 and bi-prism that it focuses the two images of the slit in the 
 field of view of the eyepiece. With the micrometer measure 
 the distance, Cj, between the centers of these two images. A 
 second- position of the lens may be found in which the images 
 are also focused, but of a different size; measure the distance, 
 t'o, between them. Then 
 
 If no position of the lens focusing the images can be found, the 
 eyepiece may be moved farther from the prism to permit the 
 convenient observation of the images as required. IJo not move 
 the prism or slit. 
 
 The second method for measuring a requires a spectrometer. 
 Completely adjust the spectrometer and point the telescope 
 toward the collimator. Having measured the distance a from 
 the slit of the interference apparatus to the bi-prism, remove 
 the bi-prism and place it on the spectrometer table with its plane 
 face towards the telescope. Adjust the prism until the two 
 images of the slit which are now seen in the telescope are as 
 distinct and as near together as possible. Measure the angle D, 
 in seconds of arc, between these two images. The distance 
 between the two apparent images of the slit in the interference 
 experiment is then a multiplied by D in radians, or 
 
 a,D 
 
 206265 
 Reference. — Glazebrook, Physical Optics, pp. 105-119. 
 
THE DIFFEACTION GRATING 249 
 
 188. Fresnel's Mirrors. — Instead of the bi-prism, two black- 
 glass mirrors lying nearly in the same plane may be used to pro- 
 duce the two virtual light sources. While the interference bands 
 so obtained may be sharper than those given by the bi-prism, 
 they are not so brilliant, and the adjustments required are more 
 complicated and troublesome. The quantities to be measured 
 and the formulae for determining a wave length, are the same 
 as given for a bi-prism. 
 
 References. — Glazdirook, Physical Optics, p. 119; Preston, Theory of 
 Light, p. 141 ; Drude, Theory of Optics, p. 130. 
 
 CXm. WAVE LENGTH OF LIGHT WITH THE PLANE 
 DIFFRACTION GRATING 
 
 Determine the wave lengths of the two lines, S^ and D^, of the sodium or 
 
 solar spectrum. 
 
 189. Wave Length of Light by Diffraction. — A diffraction 
 grating has for its essential feature a series of parallel, equidis- 
 tant apertures or surfaces from which light may come by 
 transmission or reflection. It is usually made by ruling on 
 glass or polished speculum metal a series of fine lines of from 
 three to ten centimeters in length, the series sometimes con- 
 sisting of one hundred thousand lines ruled at the rate of eight 
 thousand per centimeter. 
 
 When light from a slit with a coUimating lens falls upon a 
 grating, part of it is transmitted or reflected in regular manner, 
 and the image may be viewed with a telescope. If the spaces 
 of the grating are parallel to the slit, in addition to this direct 
 image, on either side of it will be found a series of diffracted 
 spectral images of great regularity and purity. The spectra 
 lying nearest to the direct image, on either side, are called the 
 first-order spectra ; while those successively more distant are of 
 the second, third, etc., orders. The relations between the angle 
 of incidence i, the angle of diffraction d, the order of the 
 
250 LIGHT 
 
 spectrum n, the wave length of the light X, and the distance s 
 between the centers of successive grating spaces, is 
 
 Q 
 
 \ = - (sin i + sin d). 
 n 
 
 The value of the grating space s is usually considered as 
 known from the constants of the ruling machine with which 
 the grating was made ; otherwise it might be determined by 
 comparison under a microscope, with a standard scale. The 
 angles i and d may be measured with a spectrometer. 
 
 The spectrometer must be in perfect adjustment (Art. 172). 
 Mount the grating in a suitable support with its lines vertical 
 and with its face approximately in the center of the circle of 
 the spectrometer table. If the grating support rests on three 
 leveling screws, the grating should be placed with its face per- 
 pendicular to the line joining two of the screws, that it may be 
 possible to adjust the inclination of the lines of the grating 
 without disturbing the perpendicularity of its face. Make the 
 grating perpendicular to the axis of the telescope as described 
 for a prism face in Art. 174. Before beginning the measure- 
 ments it is desirable to observe the spectrum, and to adjust the 
 width of the slit and the inclination of the lines of the grating 
 to secure the maximum distinctness of the spectrum lines. 
 
 For a reflection grating the necessary measures are most con- 
 veniently made in the following manner. Clamp the telescope 
 and collimator so that the angle between them shall remain con- 
 stant; it may be about 90°. Set the grating perpendicular to 
 the axis of the telescope and read the position of the divided 
 circle. Turn the spectrometer table carrying the grating till 
 the direct image of the slit is reflected into the telescope ; let 
 a be the angle through which it has been turned. Illuminate 
 the slit with the light to be investigated, and turn the grating 
 still farther till the nth order spectrum is visible in the tele- 
 scope, setting on the line whose wave length is desired ; let this 
 second angle be y8. The wave length of the light is 
 
 2 s 
 \ = — cos a sin ;S. 
 n 
 
THE CONCAVE GRATING 251 
 
 This formula is equivalent to the one previously given, for 
 (Fig. 104), 
 
 2 cos a sin /3 = sin (a + /3) + sin {a — fi) = sin i + sin d. 
 
 A transmission grating may be used as a reflection grating, 
 but it is equally convenient to pass the light through the grat- 
 ing and to use the first formula. If the grating is set perpen- 
 dicular to the incident light from the 
 collimator (by sighting the telescope 
 
 upon the slit and placing the grating y i /^\x „_ j 
 
 normal to the telescope), the angle ' /--<?/ /\'> 
 
 of incidence i becomes 0°, while the 
 angle of diffraction is at once meas- 
 ured by turning the telescope to the 
 desired spectrum line. The formula 
 
 then becomes 
 
 Fig. 104. Relations op 
 
 X ^ — sin d. Angles fob Gkating 
 
 '* Measures 
 
 A transmission grating may also be used in a manner analo- 
 gous to that for a prism (Art 178) by setting it in the position 
 of minimum deviation. The deviation D is directly measured, 
 and since D = i + d, and i = d, the wave length is 
 
 \ = — sin i- D. 
 
 n ^ 
 
 CXIV. WAVE LENGTH WITH THE CONCAVE-GRATING 
 SPECTROSCOPE 
 
 (a) Assuming the wave lengths of the solar spectrum lines Z>j and F, find 
 
 the wave lengths of the lines B, C, E, G, and H (second order). 
 
 (b) Construct a wave length scale for the spectroscope. 
 
 (c) Photograph the bands of the spectrum of carbon (second order). 
 
 (d) Find the wave lengths of the principal lines of any assigned spectrum 
 
 by photographic comparison with the solar spectrum. 
 
 190. The Concave-Grating Spectroscope. — If a diffraction grat- 
 ing the lines of which are ruled on a spherical concave reflecting 
 surface is used for spectroscopic observations, the coUimating 
 
25^ 
 
 LIGHT 
 
 and view telescopes of the plane-grating spectroscope are not 
 required. The absorption of the lenses is thus avoided, which 
 is an important consideration in some kinds of investigations, 
 and various other advantages are secured. 
 
 A concave grating is usually mounted at one end of a rigid 
 bar, at the other end of which, exactly at the center of curva- 
 ture of the concave surface, is an eyepiece or photographic 
 plate. An L-shaped track (Fig. 105) is provided with a slit, S, 
 
 mounted at the angle. The bar 
 is pivoted to wheel supports 
 resting on the tracks, so that 
 the grating G is always over 
 one arm of the L, and the eye- 
 piece S is always over the other 
 arm. As the bar is moved along 
 the tracks in the various possible 
 positions, the grating, eyepiece, 
 and slit will always lie on the 
 circumference of a circle whose 
 diameter is the radius of curva- 
 ture of the grating. As the bar 
 is moved to various positions 
 the several spectra of the light 
 illuminating the slit, which are 
 produced by the grating, are all brought to focus on the cir- 
 cumference of this circle. As the eyepiece is moved along the 
 track the spectra may be observed successively without any 
 other adjustment. 
 
 This arrangement is especially convenient for photographic 
 investigation, since a sensitized plate may be bent to coincide 
 with the circumference of the circle mentioned, and a consider- 
 able portion of the spectrum may be distinctly photographed at 
 one time. Glass plates about 5 by li inches are used for work 
 with a grating of 6 feet radius ; the camera is arranged to hold 
 the plate bent to the proper radius. 
 
 Adjustment. — It is to be determined that the two tracks are 
 at right angles, level, and in the same plane, and that the slit 
 
 Fig. 105. Concave-Grating 
 Spectroscope 
 
THE NORMAL SPECTRUM 253 
 
 is directly over their intersection and at the height of the grat^ 
 ing center. The instrument having been properly constructed, 
 and once mounted and adjusted, will presumably not require 
 alteration of these relations. 
 
 Place the grating, with the lines vertical, in its support at 
 one end of the bar, and, with a collimating eyepiece at the other 
 end, adjust the inclination of the grating and the length of the 
 bar until the cross wires of the eyepiece are exactly at the 
 center of curvature of the mirror. This is shown by the coin- 
 cidence of the direct and reflected images of the cross wires 
 and by the absence of parallax. 
 
 The slit being illuminated, as the diagonal bar is moved along 
 the tracks the several spectra should be seen in the center of 
 the field. If they rise or fall in the field of view, it indicates 
 imperfect mechanical construction as regards the surfaces of the 
 tracks, or the pivots which join the bar to the wheels. It may be 
 necessary to sacrifice the perfection of the previous adjustment 
 by altering the inclination of the grating until the spectra may 
 be seen throughout the required motion of the diagonal bar. 
 
 The slit being narrow, it is to be made accurately parallel to 
 the lines of the grating by rotating it until the definition of the 
 spectral lines is the best possible. 
 
 It should be ascertained, when the camera is substituted for 
 the eyepiece, that the plate coincides exactly with the previous 
 position of the cross wires, and that it is perpendicular to the 
 radius of curvature of the grating. 
 
 191. Normal Diffraction Spectrum Diffraction spectra pos- 
 sess the great advantage over prismatic spectra that the devia- 
 tion is proportional to the wave length and to the order of the 
 spectrum. For different gratings the deviation varies inversely 
 as the grating space. Thus all the spectra of any given sub- 
 stance produced by different gratings are exactly similar, and 
 may differ only in relative size. This is not true of spectra pro- 
 duced by other means. A disadvantage of diffraction spectra 
 is the small intensity of illumination. 
 
 The deviation being proportional to the wave length, the 
 comparison of wave lengths, with moderate precision, is reduced 
 
254 LIGHT 
 
 to a comparison of lengths. By measuring the distance between 
 two known lines, the value of the scale in wave lengths is deter- 
 mined ; the wave length of an unknown line is then found by 
 measuring its distance from one of the standard lines. The 
 value of the scale is inversely as the order of the spectrum ; for 
 instance, the distance between two given lines is twice as great 
 in the second-order spectrum as in the first. 
 
 The first-order spectrum is separate from the others, but the 
 red of the second order overlaps the violet of the third, and 
 in the higher orders the overlapping increases. Sometimes a 
 screen of colored glass or gelatine may be convenient for sepa- 
 rating the different spectra; for instance, if the overlapped 
 second- and third-order spectra be observed through a red glass, 
 the violet light is absorbed while the second-order red is seen 
 distinctly. 
 
 References. — Preston, Theory ol Light, pp. 226-242; Scheiner, Astro- 
 nomical Spectroscopy, p. 52. 
 
CHAPTER XVIII 
 THE INTEEFEEOMETEE, 
 
 CXV. SMALL LENGTHS WITH THE INTERFEROMETER 
 
 (a) Determine the distance between the two planes of an optical standard 
 
 of length. 
 
 (b) Determine the length of a division of a scale which is nominally one 
 
 tenth of a millimeter. 
 
 192. The Interferometer. — Perhaps the most precise method 
 for measuring small distances and displacements is by means of 
 the interferometer devised by Michelson. The measurements 
 are made in terms of the wave length of light as a unit, and 
 displacements as small as ^-^ of a wave length may be meas- 
 ured. The interferom- 
 eter consists essentially 
 of a half-silvered mir- 
 ror, ff (Fig. 106), which 
 divides incident light 
 from a source, S, into 
 two beams, one of 
 which is reflected to a 
 fixed mirror, Jfj, and 
 back, the other is trans- 
 mitted to a movable 
 
 „;,„^ ,»- J T ,1, Fig. 106. Plan of Interferometer 
 
 mirror, M^, and back. 
 
 The two beams which have returned to the half-silvered glass 
 
 are each partially transmitted or reflected to the observer in the 
 
 direction 0. The beam to M^ passes twice through the glass 
 
 plate supporting the half-silvered film, and to render the paths 
 
 of the two beams optically equal the beam to M-^ is caused to 
 
 255 
 
 Jtfp 
 
256 LIGHT 
 
 pass twice through a parallel clear glass plate, C, of the same 
 thickness as H; this is called the compensating glass, and it 
 may have any convenient position between H and M-^. 
 
 Let the light falling on H be parallel, and the mirrors 
 M^ and M^ perpendicular to the incident rays. If, then, the 
 paths of the two beams between separation and reunion differ 
 by any odd number of half wave lengths (including the phase 
 difference introduced by one internal and one external reflection), 
 upon reunion at H there will result interference, and no light 
 will be sent to 0. There is evidently an indefinite series of 
 positions of M^, with half wave-length intervals, which produce 
 interference; while whenever M^ is midway between any two 
 positions of this series, there will be reenforcement of the two 
 beams, with illumination in the direction 0. If M^ moves 
 slowly and continuously, one may count the number of times 
 it passes through an interference position, and thus determine 
 the distance moved. 
 
 Instead of adjusting the mirror M^ so that its entire surface 
 is in one interference plane of the series mentioned, it is usual 
 to give it a slight inclination, about 30" of arc, so that it cuts 
 across several planes, as shown in vertical projection in Fig. 107. 
 
 If the plate is observed in the direction 
 of the arrow, as is done in the inter- 
 ferometer, it is seen (as represented 
 in Fig. 108) crossed by a series of 
 parallel bands, which represent the 
 intersection of the plate with the series 
 of interference planes. Further, if the 
 plate is moved parallel to itself in 
 the direction of the arrow, the band 
 Fig. 107. Mikeor intek- representing intersection with any one 
 
 the edge a to the edge b. Let band 
 No. 4, for instance, be over an index mark (a cross scratched 
 in the silvering) on the surface of the plate ; as the plate is 
 moved this band will move toward h, and when band No. 5 
 has moved to the index, the plate has evidently been moved 
 
THE INTEEFEROMETER 
 
 257 
 
 one half a wave length ; if the motion has caused a displacement 
 of the bands of 0.7 of the distance between bands, the plate has 
 
 been moved 
 
 0.7 -, \ being the wave 
 
 7 6 5 4 3 2 1 
 
 length of the light employed (Art. 196). 
 Or, in general, if during the motion a 
 number of bands pass the index (whole 
 and fractions) equal to w, the distance 
 moved, in centimeters, is 
 
 
 Fig. 108. Field of View 
 OF Intekferometer 
 
 If the inclination of the plate is slight, it cuts few planes, and 
 the bands appear to be far apart. The distance between them 
 may be such that it is possible to estimate a hundredth part of 
 this distance, and thus to measure a displacement of M^ of -^-^ 
 of a wave length. The measurement of a fraction of a band 
 depends upon the fact that a slight change in the inclination of 
 the compensating glass displaces the system of bands. This 
 glass is mounted on a flexible support so that a micrometer 
 screw, F (Fig. 109), pressing against a spring, alters this inclina- 
 tion. By trial the amount by which the screw must be turned 
 to displace the system by the width of one band is determined, 
 thus facilitating the measurement of fractions. 
 
 The influence of the inclination of the mirror M^ upon the 
 simple theory of interference outlined need not be considered 
 here, since it does not alter the practical conclusions. 
 
 If, instead of the linear change in the path of either beam 
 described, there is a variation of any optical property of a medium 
 which the light traverses between the separation and reunion, 
 such as an alteration in density, the result is a displacement of 
 the bands. To measurements of this kind, and others which 
 cannot be described, the interferometer is especially adapted ; 
 its range of application to physical investigations is very wide. 
 
 References. — Michelsoyi,, Philosophical Magazine, Vol. 13, pp. 236-242, 
 1882 ; Morlev, Physical Review, Vol. 4, pp. 3-22, 1897 ; Wadsworth, Physical 
 Review, Vol. 4, pp. 480^91, 1897 ; Shedd, Physical Review, Vol. 11, pp. 304- 
 315, 1900 ; Michelson, Light Waves and Their Uses. 
 
258 
 
 LIGHT 
 
 193. Finding the Interference Bands. Monochromatic LigM. — 
 It is assumed that the interferometer has been properly con- 
 structed so that the diagonal glasses H and C(Fig. 109) are 
 held vertical, parallel, and in the direction of the bisector of 
 the angle between the two mirrors M-^ and M^, to the degree of 
 precision ordinarily attained in machine-shop practice. All the 
 glasses may be provided with adjusting screws, though it is 
 usual to have only one mirror, Jlfj, so arranged. 
 
 To protect the instrument against vibrations it 
 is best supported on a properly constructed pier ; 
 but it is usually suificient to place it on any table, 
 resting the feet of the instrument on soft rubber 
 
 corks. 
 
 Notice that the 
 half-silvered side 
 of H is toward the 
 compensator. 
 Turn the lead 
 screw by the han- 
 dle L to make the 
 distance of M^ from 
 the silver film H. 
 the same as that of 
 M^. A scale with 
 index, J, may indi- 
 cate this position ; if not, measure with a compass or scale from 
 any point on the edge of the silvered surface of H to My, and set 
 M^ at an equal distance from the same point of H. 
 
 Place a sodium or other monochromatic light near the focus 
 of a lens so that an approximately parallel beam of light shall 
 illuminate the whole of the half-silvered film, as shown in 
 Fig. 106. Since the interferometer is a symmetrical device, 
 the light may enter in the direction Oilfj, and the bands be 
 observed in the direction SM^^ without in any way altering the 
 phenomena ; either arrangement may be adopted according to 
 convenience or chance. Place fine wires or threads over the 
 surface of the lens to mark its horizontal and vertical diameters. 
 
 Fig. 109. Interfekometeu 
 
THE INTEEFEROMETEK 259 
 
 Look towards the interferometer in the direction OMp and alter 
 the position of the light or lens, and also of the diagonal mirror 
 IF if necessary, to cause the entire surface of the mirror to be 
 evenly illuminated. Usually there will also be visible three 
 images of the cross wires, the third image being due to the 
 reflection from the unsilvered side of the glass H. By slightly 
 altering the inclination of the adjustable mirror (Jfj or M^) in 
 its frame, either one cross will move and two will be stationary, 
 or two will move and one will be stationary. The odd image 
 must be brought into exact coincidence with one of the pair; 
 which one might be determined by simple theory, but usually 
 trial gives the quickest solution. Having produced coincidence 
 between two of the images, and after the vibration of the appa- 
 ratus caused by touching the screws has subsided, the interfer- 
 ence bands may be visible. Search for them by moving the eye 
 farther from or nearer to the mirrors, and by moving it cross- 
 ways. If they are not readily found, adjust the mirror to bring 
 the odd image of the cross into coincidence with the second one 
 of the pair. If the bands cannot be found now, the setting of 
 M^ ^t a distance from H equal to the distance of M^ was not 
 sufficiently exact. Turn the screw to move M^ a fifth of a 
 millimeter in one direction or the other, and repeat the search 
 for the bands as described. 
 
 Having found the bands, they may be given any direction 
 and any width by further careful adjustment of the inclination 
 of the mirror. Usually they should be made vertical and of 
 such a width that from five to ten bands are visible at one time. 
 Trial will best indicate what particular movements to give to 
 the mirror. ' 
 
 Bands in White Light. — Two beams of light of mixed wave 
 lengths (white light) will ordinarily produce interference phe- 
 nomena only when the relative retardation is a small number 
 of wave lengths, not over thirty. To obtain bands in the inter- 
 ferometer with white light, the distance of M^ from H may not 
 differ from that of M^ by more than fifteen wave lengths, a 
 quantity less than a hundredth of a millimeter. Having found 
 the bands with sodium light, as explained above, by means 
 
260 LIGHT 
 
 of the slow- motion screw S move the mirror M^ so slowly 
 that the bands may be distinctly seen as they pass across the 
 field. Examination will show that the distinctness of the bands 
 suffers periodic changes, there being about a thousand bands in 
 each group. These changes result from the fact that sodium 
 light consists of two wave lengths which differ from each other 
 about one part in a thousand. The combination produces fluc- 
 tuations analogous to beats in sound. Further, the bands in 
 the middle of one particular group will be found more distinct 
 than those in any other group on either side. Set the mirror 
 in this position of greatest maximum distinctness by quick 
 estimation ; it is then nearly in the position giving equality of 
 paths to the two beams. Place a luminous gas flame, or other 
 source of white light, just beyond the sodium flame without 
 disturbing the latter ; regulate the intensity of the white light 
 until the sodium bands are seen very faintly, perhaps in one 
 portion of the field only. Obsei-ving the mirror carefully, 
 slowly move M^ one way or the other till the group of brilliantly 
 colored bands due to white light is found. Care must be 
 taken that the motion is not so rapid that the bands pass 
 through the field unobserved ; the faintly visible sodium bands 
 permit the control of the speed. 
 
 If the colored bands are not readily found, it may be that the 
 wrong group has been selected ; therefore search in the middle 
 of the next group on either side of the one first tried. 
 
 With experience one may find the bands in white light cer- 
 tainly and more expeditiously by using the handle L, instead 
 of the slow-motion screw, for making these settings. 
 
 Another criterion for finding the colored bands is as follows. 
 Usually the bands with monochromatic light appear more or less 
 curved. A motion of the screw L in one direction increases 
 this curvature, while an opposite motion decreases it till the 
 bands appear straight, and a continued movement gives them 
 a curvature in the opposite direction. The position where the 
 curvature changes — that is, where the bands are straight — is 
 the one near which the colored bands will be found. Search 
 for them with white light as described. 
 
THE INTERFEROMETER 261 
 
 The middle band of the colored group is black. It corre- 
 sponds to light which has moved over paths of equal lengths, 
 but for which interference and not reenforcement takes place 
 because of the phase difference introduced by the reflection of one 
 part internally and the other part externally at the silver film H. 
 This center black band is the only one that can be identified ; it 
 is therefore taken as a starting point in making measurements. 
 
 194. Circular Bands. — If the movement of the mirror M^ 
 required in any measurement is small, — a millimeter or less, 
 for instance, — the straight bands described above are most con- 
 venient for observation. But when a much greater difference of 
 path of the two beams is necessary, circular fringes are more suit- 
 able. They are produced when the mirrors Jfj and M^ are both 
 exactly perpendicular to the incident light. The bands are con- 
 centric circles ; a motion of M^ causes them to enlarge and dis- 
 appear, while new ones continually form at the center, or else 
 they contract and vanish at the center. The center therefore 
 appears alternately light and dark, in accordance with the first 
 simple explanation. As the path to M^ becomes more nearly 
 equal to that to JIfj, the circles enlarge in size, and finally the 
 center of the system covers the entire plate. This is therefore 
 the position near which colored fringes may be iound. 
 
 In finding the circular bands it is convenient to begin by 
 
 making the path to M^ differ from that to Jfj by a thousand 
 
 or more wave lengths, as then the circles will be smaller and 
 
 several may be seen at once. The inclination of ^^-^^^ 
 
 the adjustable mirror is then to be changed so as f^^^\^p 
 
 to increase the curvature of the bands until they 
 
 become circles, with the center in the 
 
 center of the field of view. These circles ^^ ^ 
 
 may be conveniently observed with a | 
 
 telescope focused for distant obiects. „ ,,„ ^ 
 
 '- Fig. 110. Intekfekential 
 
 195. Counting Frmges. — For the pur- s^^^„^„„ ^^ ^^^^^^ 
 
 pose of laboratory exercises an optical 
 
 standard of length of about 0.1 mm is desirable. This consists 
 of a solid metal support carrying two plane-parallel plates of 
 equal thickness, Pj and P^ (Fig. 110). Each plate rests against 
 
 
262 LIGHT 
 
 three projecting points, with which it is kept in contact by suit- 
 able springs. The points against which Pj rests are shorter 
 than those for P^ by about a tenth of a millimeter, and they are 
 adjusted by careful grinding so that the front surfaces of the 
 plates are accurately parallel. A cross scratched through the 
 silvering in the center of each plate serves as an index. 
 
 The mirror M^ of the interferometer is removed, and the 
 standard is substituted for it. With sodium light the adjust- 
 ments are made so that the fringes are seen from both plates at 
 once. The fringes of the two sets should be of about the same 
 width, and in the same general direction, preferably vertical. 
 
 With white light find the colored system of fringes on one 
 plate, as Pj, and bring the black fringe exactly to the cross mark. 
 
 Illuminate with sodium light and give attention to the fringes 
 from plate P^. Having previously determined the direction in 
 which the bands will be displaced when the plate P^ is brought 
 toward the plane in which P^ is now set, turn the fraction-screw 
 F to displace the bands in this direction until the center of the 
 first dark band to come to the cross is exactly on the mark. 
 Thus the first fraction of a band is measured. Then from this 
 position as zero the slow-motion screw S is slowly turned, and 
 the bands are counted as they pass the mark. It is convenient 
 to count ten bands, then to take a new hold on the screw, to 
 count ten more, and thus to continue until a number has been 
 counted which is perhaps five less than the supposed number 
 in the length being measured. 
 
 Substitute white light. If the system of colored fringes is 
 not in sight, count five more bands with sodium light, and 
 again look for the bands in white light. Continue the count 
 until the colored system is found and the central black band is 
 on the cross mark. 
 
 If the colored system is not found, either to6 many bands 
 were counted before search was made with white light, or the 
 screw may have been turned in the wrong direction.' 
 
 The number of whole fringes plus the first fraction is the 
 length required expressed in half wave lengths of the mono- 
 chromatic light employed. 
 
MONOCHEOMATIC LIGHT 
 
 263 
 
 To give definite direction to the line of sight a telescope is 
 convenient; but the cross wires in the eyepiece must not be 
 used as the index for counting, — the marks on the surfaces 
 of the plates sefve better. The slow-motion screw may be 
 extended to the telescope by means of a glass tube, connected 
 to the screw by a short piece of rubber tube to prevent the 
 communication of vibrations to the instrument. The count 
 may be interrupted to rest the eye. Stopcocks for controlling 
 the gas may be placed within reach of the observer, so that 
 either monochromatic or white light may be used at will. 
 
 196. Monochromatic Light. Sodium Light. — The ordinary 
 laboratory requirements for monochromatic light are fulfilled by 
 placing some common salt held in a platinum wire loop or gauze 
 basket in the edge of a Bunsen flame, where it is vaporized and 
 rendered incandescent. The sodium light which 
 results is not of great intensity, but it may be 
 made of large surface. The light is made 
 steady, and somewhat more intense, by placing 
 an iron chimney over the flame, as indicated in 
 Fig. Ill, which shows a convenient form of 
 sodium lamp. Instead of salt, borax may be 
 fused into the platinum wire loop. 
 
 Sodium light consists of two principal wave 
 lengths, Rowland's values for which are 
 
 Z>i = 0.00005896154 cm, 
 and Da = 0.00005890182 cm. 
 
 It is not possible to separate these two com- 
 ponents of sodium light in large surface illu- 
 mination, and the wave length may be taken as 
 the mean, 0.00005893 cm. 
 
 Mercury Light. — A light which is more nearly 
 monochromatic and at the same time of greater 
 intensity is obtained from the mercury arc lamp. The essential 
 part of the lamp is a fl -shaped vacuum chamber (Fig. 112) con- 
 taining mercury. By means of platinum wires the mercury in 
 the interior is connected to two mercury cups, which receive the 
 
 
 Fig. Ill 
 Sodium Lamp 
 
.264 
 
 LIGHT 
 
 terminals of an electric circuit. The quantity of mercury in 
 the branches of the chamber may be so adjusted that the two 
 portions are separated by a small space. By gentle shaking 
 the two portions may be made to touch, and as they separate 
 an electric arc is formed which vaporizes 
 the mercury and emits a brilliant greenish- 
 white light. The current required is of 
 from 6 to 10 amperes, and from 24 to 32 
 volts. By immersing the lamp in a water 
 bath, it can be operated for hours without 
 interruption. 
 
 The mercury light consists of several 
 wave lengths which, however, may be 
 readily separated by a prism. The green 
 portion is the most useful, and its wave 
 length according to Fabry and Perot is 
 0.0000546074 cm. 
 
 -Michelson describes in Travaux et MSmoireB 
 du Bureau International des Poids et Mesures, Tome XI, a lamp 
 in which metallic cadmium is vaporized with a Bunsen burner, 
 and then rendered incandescent by the electric spark. By pris- 
 matic analysis a red light of great purity is obtained, the wave 
 length of which is perhaps determined with gi-eater precision 
 than that of any other light. The wave length of this red 
 cadmium light is 0.00006438472 cm. 
 
 Fig. 112. Mercury 
 Akc Lamp 
 
 Cadmium Light. ■ 
 
 CXVI. PLANE SURFACES BY OPTICAL METHODS 
 
 Test the surfaces of a prism for flatness, and the two surfaces of a glass 
 plate for plane parallelism. 
 
 197. Plane and Plane-Parallel Surfaces. — The surfaces of 
 prisms, of the plates used in interference experiments, some 
 surfaces of lenses and of other optical plates are required to be 
 optically plane. The skill of the optician is such that the sur- 
 face of a firs1>quality plate will not deviate from a true plane by 
 more than a tenth or a twentieth of a wave length of light. A 
 
PLANE SURFACES 265 
 
 plane-parallel plate is one the two surfaces of which are accu- 
 rately plane and parallel to each other. When two plane-parallel 
 plates of equal thickness are required they are constructed in 
 one piece, which is afterwards cut. 
 
 Plane Surfaces are conveniently tested only by optical com- 
 parison with a test true plane. This true plane is made by the 
 optician by grinding three planes together until they fit, two 
 and two, in all positions. 
 
 Such a true plane being available, it is placed upon the surface 
 to be tested; the surface of contact is illuminated with sodium 
 light by transmission through either plate as is convenient. 
 Before the plates are brought into contact the two surfaces 
 should be freed from dust and dirt, both to avoid injury to the 
 surfaces and to permit proper contact. To light the entire sur- 
 face a sodium flame is placed in the focus of a large lens, so 
 arranged that the bundle of parallel rays produced falls on the 
 surfaces and is then reflected 
 to the eye. Fig. 113 shows 
 the arrangement for testing a 
 prism. Interference bands 
 should appear over the entire 
 surface. If both surfaces are 
 plane, these bands are straight, 
 parallel, and equidistant. If 
 the plates are pressed closely 
 together, the bands will be far 
 apart and a very small depar- 
 ture from planeness may be 
 discovered. A width of bands 
 
 of from five to ten millimeters ^ ,,„ ,„ t, 
 
 Fio. 113. Testing a Plane 
 IS most useful. After the Surface 
 
 plates have been placed in 
 
 contact they must be left till they have come to a uniform 
 
 temperature throughout before the testing can be relied upon. 
 
 The form of the surface being tested may be inferred from 
 
 the interference pattern exhibited, since a band traces the locus 
 
 of all points in the unknown surface which are at a constant 
 
266 LIGHT 
 
 distance from the true plane. The bands are analogous to the 
 contour lines on a topographic map. 
 
 Plane-Parallel Surfaces. — The parallelism of the plane sur- 
 faces of a glass plate may be roughly tested by observing any 
 well-defined distant object as seen reflected from the plate 
 at an angle of about 45°. If the two surfaces are inclined, 
 two images will be seen, one of which revolves about the other 
 when the plate is turned in its own plane. If the surfaces are 
 parallel there will be but one image, which should be sharply 
 defined. 
 
 If the reflected image is observed with a telescope, the test 
 becomes much more delicate because of the magnification, and 
 it is then sufficiently sensitive for the usual requirements. 
 
 A spectrometer is most convenient for this test. Set the tele- 
 scope at right angles to the collimator, and place the plate to be 
 tested so that the illuminated slit is reflected in the telescope. 
 A more suitable object is obtained by opening the slit wide and 
 crossing fine wires over the opening. Then, as the plate is 
 rotated in its own plane, the image should remain single and 
 sharply defined. 
 
 CXVn. SILVERING GLASS BY CHEMICAL METHODS 
 
 (a) Make a thick-film mirror by Brashear's process. 
 
 (b) Make a half -silvered mirror by the Rochelle-Salts process. 
 
 198. Brashear's Process for Silvering Glass. — Mirrors for 
 optical experiments are often made by preparing a glass sur- 
 face of the desired form, upon which is then deposited a uni- 
 form film of silver. The following method, by which a hard, 
 •brilliant film is deposited from an ammoniacal solution of 
 nitrate of silver by an organic reducing agent, is usually the 
 most convenient and satisfactory. Two solutions are required : 
 one, the reducing solution, should be prepared at least a week 
 before it is used, and it may be made in large quantity and 
 kept in stock with advantage; the other solution is to be 
 prepared when used. 
 
SILVERING GLASS 267 
 
 Reducing Solution 
 
 Distilled water 700 ccm 
 
 Pure sugar (loaf, granulated, or rock candy) ... 80 g 
 When dissolved add 
 
 Alcohol . ... .... 175 ccm 
 
 Strong nitric acid (sp. gr. 1.42) ... ... 3 ccm 
 
 Add water to make . 1000 ccm 
 
 For silvering, the mirror may rest face up on the bottom of 
 a suitable dish; it may stand on edge, or be supported in any 
 manner, face downward, dipping into the upper part of the solu- 
 tion. In the latter case the mirror may be fastened with wax 
 to a stick laid across the dish, or it may be supported on glass 
 feet or on paraffined wood wedges. Dr. Brashear recommends 
 that the mirror, if round, form the bottom of the silvering dish, 
 which is completed by wrapping a strip of paraffined paper around 
 the edge of the mirror, this being held in place by rubber bands 
 or fastened with several wrappings of cord. 
 
 Having selected a dish and support for the mirror, measure 
 with water the quantity of solution that will be required to 
 make a layer a centimeter or two thick over the surface to be 
 silvered. For each 150 ccm of final solution, 1 g of silver nitrate 
 and 0.5 g of caustic potash (purified by alcohol) will be required. 
 Dissolve the silver and potash separately, using quantities of 
 water of the proportion of 100 ccm to 1 g of the solution. 
 Ordinary graduates or flasks are the most convenient form of 
 vessel in which to mix the solutions. Into the silver nitrate 
 solution pour a few drops of dilute aqua ammonia. The solu- 
 tion wiU turn to a dark brown color ; add ammonia little by little 
 till the precipitate is nearly but not quite redissolved. Now 
 add the potash solution, when a precipitate will again be formed. 
 This is to be nearly, but not entirely, redissolved by the addition 
 of more ammonia, a few drops being sufficient this time. After 
 the ammonia has been added, shake or stir the solution well and 
 wait a minute or two to be certain that it does not entirely clear. 
 If by chance too much ammonia has been used, a little silver 
 nitrate is to be dissolved and added, a few drops at a time, till 
 a permanent precipitate is formed. This excess of silver must 
 
268 LIGHT 
 
 be present, the solution showing a decided brown tint. The 
 solution may be filtered, though usually this is not necessary. 
 
 A quantity of reducing solution equal to about a twenty-fifth 
 part of the solution just prepared is measured out. The mirror, 
 having been properly cleaned and rinsed with distilled water, is 
 placed in position. The reducing solution is poured into the 
 silver and potash solution, and mixed by a quick shaking of 
 the graduate or stirring with a glass rod; the whole is then 
 poured into the dish. If the mirror is immersed face down, 
 care is necessary to remove air bubbles ; the min-or may well be 
 immersed after the solution is in, being dipped in at one side 
 first. If the mirror is at the bottom of the dish, after cleaning 
 it is covered with a thin layer of water, and the prepared 
 solutions are poured into the dish without further trouble. 
 In the latter case the dish must be rocked during the time 
 of deposition. 
 
 The solution soon turns to a black color, which in a few minutes 
 will turn to a brown ; and when it becomes a light gray and the 
 precipitate is flocculent, which may be in ten or fifteen minutes, 
 the operation is at an end. If the mirror is allowed to remain 
 in the solution too long, the surface will have a bleached appear- 
 ance, which polishing will hardly remove. Remove the mirror, 
 rinse with water, and carefully wipe off the sediment with a 
 tuft of absorbent cotton. It is then set on edge to dry ; a rins- 
 ing with alcohol wiU facilitate the drying, or all water may be 
 safely taken up by pressing clean blotting paper over the surface. 
 
 When dry the surface may be polished, if necessary, with a 
 small pad of chamois leather stuffed with cotton, on which is 
 spread a little rouge. Small, circular strokes of the pad, with 
 light pressure, will soon bring out the deep luster of the silver. 
 
 A uniform temperature of the bath and the glass, of about 
 20° is essential to success. 
 
 Since fulminating silver is liable to be produced by the action 
 of ammonia on silver oxide, especially in a warm room, all solu- 
 tions should be thrown away as soon as the silvering operation 
 is completed. The used solutions may be poured into a large 
 jar, in which is thrown some common salt ; this causes the silver 
 
SILVERING GLASS 269 
 
 to be precipitated as the chloride, and about ninety per cent, of 
 the original silver may be recovered. 
 
 199. Rochelle-Salts Process for Silvering Glass. — For deposit- 
 ing the uniform thin film of silver required on the half-silvered 
 glass of the interferometer, the following method is more suit> 
 able than the one described above, as the silver is deposited more 
 slowly. If a thick film is desired, two or more successive deposits 
 may be made, each of which may require an hour's time. 
 
 Dissolve 5 g of silver nitrate in 300 ccm of distilled water, 
 and add dilute aqua ammonia until the precipitate formed is 
 nearly, hut not entirely, redissolved in the manner explained in 
 the preceding method. Filter the solution and add water to 
 make 500 ccm. 
 
 Dissolve 1 g of silver nitrate in a small quantity of water, and 
 pour into about half a liter of boiling water; dissolve 0.83 g of 
 Rochelle salts in a small quantity of water, and add to the boil- 
 ing solution. Continue the boiling for half an hour, till the 
 gray precipitate collects as a powder in the bottom of the flask. 
 Filter hot, and add water to make 500 ccm. 
 
 These solutions may be kept in the dark for a month or two. 
 
 For silvering, equal volumes of the two solutions are mixed, 
 and the glass is supported in the mixture in whatever fashion 
 is convenient. Various methods are mentioned in the preceding 
 article. The thickest possible deposit may require an hour's 
 time. A second deposit may be made upon the first if necessary 
 to secure the desired thickness. The drying and polishing may 
 be carried out as described above. 
 
 A half-silvered film will be produced in about a minute ; only 
 experience can determine when the proper thickness has been 
 secured. The glass appears as though it were very lightly 
 smoked. A film that reflects a little more than half the light 
 incident at 45° is desirable for interferometer use. A simple 
 method of testing is to look at two similar gas flames, one seen 
 through the film and the other seen reflected by it. It is well 
 to silver at once all four surfaces of the two plane-parallel plates 
 of the interferometer and to select for use that film which is of 
 the proper and most uniform thickness. 
 
270 LIGHT 
 
 200. Cleaning Optical Surfaces for Silvering. — Probably the 
 most important part of the silvering process is the proper 
 cleaning of the surface to be silvered. 
 
 The surface is thoroughly cleaned of grease or other organic 
 matter by the usual methods (Art. 91), using alcohol or chromic 
 acid. Then it should be carefully cleaned with strong nitric 
 acid, the whole surface being firmly rubbed with pure cotton 
 tied to a rod of wood or glass. Care should be taken not to 
 injure the surface. Rinse with water, and then wash the surface 
 thoroughly with a strong solution of caustic potash, rubbing 
 with a cotton brush as before. Finally rinse with distilled 
 water, and keep the surface wet until it is placed in the silver- 
 ing solution. If the distilled water wets the whole surface 
 uniformly, the cleaning may be sufficient; if it does not wet 
 uniformly, the operations must be repeated. The fingers should 
 not touch the edges of the glass during the latter cleaning 
 operations, as a layer of organic matter is apt to spread over 
 the surface and render the silvering uneven. 
 
 Dr. Brashear recommends that the surface, after the washings 
 described above, be rubbed with prepared chalk on a cotton wad 
 until it is thoroughly dry and clean. .It may then be put into 
 the silvering solution at one's convenience. 
 
 The following method is sometimes convenient for holding 
 a mirror while cleaning and immersing it. A funnel having a 
 ground edge is placed against the back of the mirror with a ring 
 of sheet rubber between them. Connect the stem of the funnel 
 to an air pump with rubber tubing. A few strokes of the pump 
 will firmly attach the funnel, which will serve as a safe and 
 clean handle. 
 
CHAPTER XIX 
 THE SPECTROSCOPE AND POLARIMETEE, 
 
 CXVm. CHEMICAL COMPOSITION WITH THE SPECTROSCOPE 
 
 (a) Map the principal lines in the solar spectrum, and calibrate the scale 
 of the spectroscope. Find the wave lengths of the lines, and draw a 
 map of the spectrum of a solid and of a gas. 
 
 (b) Idsntify several unknown substances by comparing their spectra with 
 the spectra of known elements. 
 
 201. Spectrum Analysis. — A spectroscope usually differs from 
 the spectrometer (Art. 171) in that it has fewer adjustments, and 
 instead of a divided circle it has a linear scale made visible in 
 the field of the view telescope. A spectrometer may always 
 be used as a spectroscope. The dispersion part of a spectro- 
 scope may be one or several prisms, or a diffraction 
 •^ grating. Fig. 114 shows the relations of the essential 
 p.irts of a two-prism spectroscope. 
 
 The flame to be analyzed is placed at F, illumi- 
 nating the slit S. The slit must be in 
 the principal focus of the collimating 
 lens C, in order that the rays 
 v^ falling on the prism Pj shall be 
 parallel. It may have been 
 Fio.lM. SP.CTKOSCOPB - ^ permanently placed by the 
 
 maker, or a mark on the 
 draw tube may indicate its correct position; otherwise it must be 
 adjusted as follows. Remove the telescope T and focus it on a 
 distant object ; replace it so as to view the slit through the collima- 
 tor, the prisms being removed ; focus the slit by means of its draw 
 tube till it is seen distinctly in the telescope. (See Art. 172.) 
 
 271 
 
272 LIGHT 
 
 Adjust the prisms, P^, Pg, etc., to the position of minimum 
 deviation, as explained in Art. 179, placing first one and then 
 another till all are properly arranged. 
 
 Illuminate the scale JIf by a light at L, just bright enough to 
 render it visible. Cause the scale image to be reflected from 
 the prism face nearest the telescope so that it appears iu the 
 field of view crossing the whole spectrum. Focus it for dis- 
 tinctness, and so that there is no parallax displacement with 
 regard to the sht image, when the eye is moved sideways. 
 
 The relative positions of the lines of the spectrum of a given 
 substance produced by a prism vary with the composition of the 
 material of the prism. In order that one prismatic spectrum 
 may be compared with that from a different prism, an absolute 
 scale of reference must be adopted. The most useful of such 
 scales is one of actual wave lengths. To find the wave-length 
 values of the arbitrary spectroscope scale a standard spectrum 
 must be observed, and the relation between the positions 
 and known wave lengths of various lines may be plotted 
 on coordinate paper, from which the wave length of any 
 observed line may be found. The solar spectrum is a con- 
 venient standard. 
 
 If the spectrum is formed by a diffraction grating, it is a 
 normal spectrum; that is, the dispersion is proportional to the 
 wave length. The relative positions of the lines in the spectra 
 of any given substance formed by different gratings are always 
 identical (Art. 191). 
 
 If a spectrometer is employed, all the adjustments described 
 for determining index of refraction should be made. The spec- 
 tral lines may be located by reference to the divided circle, 
 the wave-length values being found by observing a standard 
 spectrum, as described above. 
 
 Arrange a heliostat (Art. 202) to throw sunlight upon the slit 
 of the spectroscope. If this is not convenient, the light from 
 a bright sky will serve. Alter the width of the slit until the 
 transverse (Fraunhofer) lines are distinct. If longitudinal lines 
 are visible, they are due to dust or irregularities in the slit. To 
 remove dust draw a fine sliver of soft wood throusrh the slit. 
 
SPECTEUM ANALYSIS 273 
 
 If this does not remove the lines, widening the slit may be 
 advantageous, though it may render the spectrum too bright or 
 the spectral lines too wide. 
 
 Identify and locate on the scale the principal Fraunhofer lines, 
 the B, C, D, A\ h, F, G, H, and A' lines, referring to a map of 
 the spectrum if necessary. Using the known wave lengths 
 of the lines (Table 26), plot a curve showing the wave-length 
 values of the scale. 
 
 The spectra of all the obtainable elements have been observed, 
 and the wave lengths of the principal lines have been tabulated. 
 Spectrum analysis in its simplest form consists in observations 
 of the lines given by the unknown substance, which by refer- 
 ence to. the tables enable the identification of the substance. 
 Besides position, features which aid in the identification are 
 brightness, width, color, and sharpness. Full descriptions of 
 the methods, which may become very complex, will be found 
 in the references. 
 
 To study the spectrum of a solid, such as the chlorides of 
 barium, calcium, lithium, sodium, strontium, thallium, etc., the 
 substance may be vaporized and rendered incandescent in the 
 Bunsen flame. Place a little of the material on a platinum 
 wire, and hold it just within the surface of the flame, near the 
 middle of its length. The temperature of the flame, and the 
 length of time the substance has been in it, influence the appear- 
 ance of certain lines. 
 
 Less volatile substances may be rendered incandescent by 
 being placed in the crater of an electric arc between carbons, 
 or an arc may be formed between electrodes of the substance 
 being studied. 
 
 The Bunsen flame and the carbon arc both give spectral lines, 
 which must be identified and differentiated from the lines of the 
 material being analyzed. The sodium line is almost universally 
 present in flame spectra. 
 
 Many lines are faint, and to observe them all outside light 
 must be excluded, by a screen behind the source, by covering 
 the prisms, and by covering the observer's head if necessary. 
 The slit must be widened when searching for faint lines. 
 
274 
 
 LIGHT 
 
 To observe the spectrum of a gas, the gas is usually placed 
 in a glass tube having platinum electrodes, between which the 
 discharge of an induction coil takes place, rendering the gas 
 incandescent. 
 
 Sun 
 
 Refekences. — Kohlrausch, Physical Measurements, pp. 169-173; Schuster 
 and Lees, Practical Physics, pp. 178-193 ; Scheiner, Astronomical Spectroscopy ; 
 Landauer, Spectrum Analysis. 
 
 202. The Hellostat. — In experimental work it is often desired 
 to project a beam of sunlight in a direction which shall remain 
 fixed throughout the day. This may be accomplished by means 
 of a heHostat, the essential part of which is a mirror provided 
 with various adjustments, moved by clockwork in such a manner 
 as to neutralize the effects of the earth's rotation on its axis. In 
 
 some forms of heliostat the 
 beam of light is thrown in 
 a direction parallel to the 
 earth's axis, the beam being 
 received on a second mir- 
 ror, which then reflects it 
 in any desired direction. 
 In other forms the beam 
 is sent in any given direc- 
 tion at the first reflection. 
 The first form is the simpler 
 and more certain in action, 
 its disadvantage being one 
 additional reflection. 
 
 A two-mirror heliostat is 
 shown in Fig. 115. The 
 polar axis P is to be placed 
 in the plane of the merid- 
 ian and elevated at an angle equal to the latitude of the sta- 
 tion. The graduated arc L indicates the proper latitude setting. 
 The axis is thus made parallel to the earth's axis. A clock, 
 C, is arranged to rotate this axis from east to west at a rate 
 corresponding to the diurnal motion of the sun. 
 
 Fig. 115. Heliostat 
 
ROTATOEY POLAEIZATION 275 
 
 At the upper end of the polar axis the mirror is held so as to 
 be movable about a line through a, which constitutes the decli- 
 nation axis. The normal to the surface of the mirror, iV, should 
 bisect the angle between the direction of the sun and the polar 
 axis. When the sun is on the equator the normal makes an 
 angle of 45° with the polar axis, and the declination circle D 
 reads 0°. As the sun moves north or south the mirror must be 
 inclined from this position by an amount equal to half the sun's 
 declination. The declination circle is usually ^radijated with 
 half-degree spaces numbered as whole degrees, and the setting 
 is therefore equal to the sun's declination. 
 
 Attached to the polar axis is an hour circle, which reads 
 when the dechnation axis is horizontal. In starting the 
 instrument this circle is to be set to the apparent solar time 
 (standard time reduced to local mean time, and then corrected 
 for the equation of time). 
 
 Approximate values of the sun's declination and the equation 
 of time, sufficient for heliostat settings, are given in Table 22, 
 in the Appendix. More complete tables will be found in the 
 Nautical Almanac. 
 
 Other forms of heliostats require similar settings. The man- 
 ner of making these will usually be evident from inspection. 
 
 Reference. — Le,isi, Die optisclien Instrumente, pp. 284-305. 
 
 CXrX. OPTICAL ACTIVITY WITH THE POLARIMETER 
 
 Determine the specific rotation of cane sugar, and find the percentage of 
 cane sugar in an unknown solution. 
 
 203. Rotatory Polarization. — Certain solids, liquids, vapors, 
 and solutions have the property, when plane-polarized light is 
 passed through them, of rotating the plane of polarization. 
 This phenomenon is called optical activity. Some substances 
 rotate the plane to the right and others to the left ; that rota- 
 tion which is related to the propagation of light, as the rotation 
 and translation in a right-handed screw, is said to be right- 
 handed. The amount of rotation for any active substance varies 
 
276 LIGHT 
 
 directly as the thickness of the layer through which the light 
 passes, and varies with the wave length of the light. For a 
 liquid it also varies with the density or the concentration of 
 the solution and with the temperature. The rotatory power of 
 fluids is much less than that of solids. About thirty crystals 
 are known to possess optical activity, while over seven hundred 
 liquids and solutions belonging to all groups of organic com- 
 pounds also have this property. The specific rotation of a 
 liquid or substance in solution is the rotation in degrees of arc, 
 per unit length of path of the light, per unit density of the 
 substance. If r is the rotation produced when the light passes 
 through I centimeters of length of the fluid, and d is its density, 
 the specific rotation is 
 
 If the active substance is mixed with an inactive liquid, d in 
 the above formula represents the number of grams of the sub- 
 stance in one cubic centimeter of the solution. A statement of 
 the specific rotation of a substance should include the tempera- 
 ture, the kind of light used, as the sodium D line, and the 
 direction of rotation, -I- for right-handed ; as 
 
 turpentine [af^ = - 3°.701. 
 
 The specific rotation multiplied by the molecular weight 
 gives the molecular rotation. 
 
 Founded upon these principles there are methods of analysis 
 useful for distinguishing one substance from another, or for 
 determining the nature or strength of a solution. Rotatory 
 polarization is employed as an aid in the analysis of various 
 carbon compounds, such as gum resins, camphor, essential oils, 
 alkaloids, dyestuffs, and many others ; and it is extensively used 
 in sugar analysis to determine the purity of various samples 
 and the proper concentration of syrups in manufacturing. 
 
 204. The Polarimeter. — An instrument designed to measure 
 the rotatory polarization of a substance is a polarimeter (Fig. 
 116). For general uses the rotation is measured in degrees 
 
THE POLAEIMETEE 
 
 277 
 
 of arc. Often the instrument is arranged for sugar analysis, 
 with a scale showing the percentage of sugar in solution as 
 compared with a standard solution; it is then a saccharimeter. 
 
 The field of view of a polariscope consisting of two Nicol 
 prisms is dark when the prisms are crossed. An optically 
 active substance placed in the path of the light between the 
 prisms rotates the plane of the polarized light and causes the 
 field to appear brighter. That rotation of either prism necessary 
 to restore darkness 
 measures the rotation 
 of the substance. To 
 determine the exact 
 position of the prism 
 producing this effect is 
 difficult, and various 
 devices have been 
 made to secure in- 
 creased sensitiveness. 
 Some of the more com- 
 mon forms are men- 
 tioned below. 
 
 Half Shade (Lau- 
 rent).— The field of 
 view embraces a plate 
 
 one half of which is glass while the other is quartz cut parallel to 
 the axis and of such -thickness as to introduce a retardation of 
 half a wave length of the particular color of light to be used 
 (commonly sodium) between the ordinary and extraordinary 
 rays. This causes the two halves of the field to appear 
 unequally illuminated, except when the principal plane of the 
 analyzing nicol is parallel to the axis of the quartz ; then the 
 two halves appear of uniform brightness. This brightness of 
 the field and therefore the sensitiveness may be regulated by 
 changing the position of the polarizer with respect to the 
 quartz plate. The sensitive position is that of the eyepiece 
 nicol producing uniformity of field illumination. Monochro- 
 matic (sodium) light is necessary. 
 
 Fig. 116. Polakimetek 
 
278 LIGHT 
 
 Interference Bands (Wild). — Two plates of calcite or quartz 
 are cut at an angle of 46° to the axis, and placed together with 
 their principal sections at right angles. This plate is inserted 
 in the path of the rays near the eyepiece nicol, and gives in 
 the field of view a series of bands alternately light and dark. 
 The polarizing nicol is rotated until these bands disappear in 
 the center of the field, a sensitive condition. The sensitiveness, 
 
 A NE X 1 S * P N 
 
 /S7D (DO] I I /S7 
 
 Fig. 117. Wild's Polabistrobombteb 
 
 depending upon the brightness of the light used, may be 
 adjusted by changing the position of the eyepiece nicol with 
 respect to the interference plate. Either monochromatic or 
 white light may be used, though the first is better, as the entire 
 field is then covered by the bands. 
 
 Fig. 117 represents the arrangement of the optical parts of 
 Wild's polaristrobometer. FN is the polarizing nicol, S the 
 active substance, / the interference plate, AN the analyzing 
 nicol, X the cross wires, and and E the observing lenses. 
 
 Sensitive Tint (Soleil). — Two quartz plates of equal thick- 
 ness but of opposite rotations are placed side by side in front of 
 the polarizing nicol. The sensitive condition is secured by 
 rotating the eyepiece nicol until it is parallel to the polarizing 
 nicol, in which case the two halves of the field appear equally 
 bright, or of the same color, a grayish violet, called the tint of 
 passage or sensitive tint. A slight displacement of the nicol 
 either way causes one half to appear red and the other blue, 
 with white light, or to appear unequally bright with monochro- 
 matic light. 
 
 With each of these devices the critical condition can be 
 secured by rotating one of the Nicol prisms, and the rotation 
 produced by an active substance put in the path of the light 
 between the nicols may be determined by the rotation of the 
 nicol required to compensate that of the substance. There are 
 always two positions of the nicol, 180° apart, satisfying this 
 
THE POLAEIMETEE 279 
 
 condition, and sometimes there are four positions 90° apart. 
 To determine whether the rotation of an active substance is 
 less or greater than 90° or 180°, or whether it is right- or left- 
 handed, it may be necessary to use two different lengths of the 
 substance, or two different strengths of solution. For example, 
 an observed rotation of 30° to the right may actually be one 
 of 60° to the left. A second observation with half the length 
 of substance would in the latter case show the sensitive condi- 
 tion at a point 30° to the left and at 60° to the right; while if 
 the first observation corresponded to an actual rotation to the 
 right of 80°, the second observation would show only 15° to 
 the right and 75° to the left. 
 
 Often, instead of measuring the angle of rotation produced 
 by the active substance by rotating one of the Nicol prisms an 
 equal amount in the opposite direction, as described, the rota^ 
 tion of the plane of polarization by the substance is neutralized 
 by the opposite rotation produced by a quartz compensator. 
 This consists of two quartz wedges having left-hand rotation, 
 one of which can slide over the other, varying their combined 
 thickness, together with a right-handed quartz plate. Any 
 desired rotation either right or left can be introduced, accord- 
 ing as the wedges combined are thicker or thinner than the 
 single plate. The thickness of wedge required to secure the 
 sensitive condition is measured by an arbitrary scale, which 
 usually indicates percentage of pure sugar in a solution made 
 
 W 
 
 E OAN 
 
 I D\Z\ H 
 
 Q S B PN CON 
 
 U I ~1 B\Z\ D\Z\ 
 
 Fig. 118. Quartz Wedge Saoohakimeter 
 
 according to rule. In instruments of German manufacture one 
 division of the scale equals 0°.346, and for sugar analysis 26.05 g 
 of the sample is dissolved in 10 ccm of solution. The scale reading 
 then gives at once the percentage of pure sugar. In instruments 
 of French manufacture one scale division equals 0°.217, and the 
 standard solution contains 16.35 g per 100 ccm of solution. 
 
280 LIGHT 
 
 Fig. 118 shows the optical parts of a saccharimeter. FN is 
 the polarizing nicol, B the quartz plate, S the active substance, 
 Q the right-handed quartz plate, W the left-handed quartz 
 wedges, AN the analyzing nicol, and and E are the observing 
 lenses, focused on B. To neutralize the effect of colored solu- 
 tions, or colored light, or to secure any desired tint of field, a 
 color regulator consisting of a nicol, CN, and a quartz plate, 
 C, are often provided. The rotation of these alters the color of 
 the field. 
 
 205. Saccharimetry. — Granulated sugar may be considered 
 pure cane sugar, and its specific rotation is determined in 
 accordance with the definition of Art. 203. For cane sugar 
 the specific rotation is nearly independent of temperature. 
 
 Dissolve 20 g of dry granulated sugar in about 80 ccm of 
 warm water, contained conveniently in a 100 ccm flask. When 
 the sugar is thoroughly dissolved, cool the mixture to 20° and 
 add water to make 100 ccm of solution. 
 
 Fill a 20 cm solution tube with distilled water, place it in the 
 saccharimeter, and make the critical adjustments as described 
 above, either for equal tint, equal shade, or for the disappear- 
 ance of the interference bands. Make a number of settings, 
 ten for example, and use the average as the zero reading of the 
 instrument. Fill the tube with the prepared solution of pure 
 sugar, and make the critical adjustments again, taking the aver- 
 age of the ten readings. Then if there are w g of sugar in v ccm 
 of solution, and the tube has a length of I centimeters, and the 
 observed rotation in degrees of arc is r, the specific rotation is 
 
 rv 
 W = ^- 
 
 Using a tube 10 cm long, repeat the observations and find the 
 specific rotation. 
 
 Dilute the solution still remaining to half strength, by adding 
 an equal volume of water, and with the 20 cm tube again 
 determine the specific rotation. 
 
 With the value of the specific rotation thus found, determine 
 the amount of cane sugar in an unknown solution provided for 
 testing. 
 
SACCHARIMETRY 281 
 
 (When the work is finished, wash all tubes and dishes used 
 for the sugar solutions.) 
 
 The method above described is of general application in the 
 measurement of the rotatory power of any active substance. 
 The treatment for clarifying colored solutions, and for the deter- 
 mination of sugar in the presence of invert sugar, or other active 
 substances, may be found in treatises on chemical analysis. 
 
 References. — Landolt, Optical Rotating Power; Kohlrausch, Physical 
 Measurements, pp. 190-198 ; Preston, Theory of Light, pp. 449-457. 
 
Paet yi — Electeicity ai^d 
 
 Magis^etism 
 
 CHAPTER XX 
 
 EESISTANCE 
 
 CXXV. RESISTANCE BY THE WHEATSTONE'S BRIDGE 
 
 Determine the resistance of a high- and of a low-voltage incandescent 
 lamp (cold) ; also the resistances of two wire coils, separately, in 
 series, and in parallel. Make three measures of each. 
 
 206. Electrical Measurements and Ohm's Law. — Electrical 
 measurements consist of the direct or indirect determinations of 
 resistances, current strengths, electromotive forces, quantities 
 of electricity, and magnetic quantities. Of these classes of 
 electrical measurements the first three are much the more com- 
 mon in technical work. The principles underlying many of the 
 various methods are contained in what is known as Ohm's Law, 
 and though this law is stated in very simple fashion, its wide 
 application renders it worthy of the student's careful study. 
 
 If a specified conductor is carrying a steady current, there 
 are three factors to which attention must be given in order that 
 the electrical conditions may be understood. These are the 
 difference of potential, E, between the ends of the conductor, 
 the strength of the current, /, and the resistance, B. Ohm's 
 Law expresses the relations between these quantities, that the 
 product of resistance and current strength is equal to the 
 
 282 
 
WHEATSTONE'S BEIDGE 
 
 283 
 
 difference of potential. It is also often stated by the following 
 equation : „ 
 
 The units in which these quantities are measured, the ohm, 
 the ampere, and the volt, are defined in Arts. 212, 244, and 
 249 respectively. Other electrical and magnetic units are 
 defined in Arts. 262, 269, and 272. 
 
 207. Wheatstone's Bridge. — Wheatstone's bridge is a device 
 by which the principles involved in Ohm's Law may be applied 
 to a variety of electrical measurements. 
 Before describing the methods specifically, 
 a general explanation will be given. 
 
 Let A and C (Fig. 119) be two points 
 connected by two conductors, ADC and 
 AUG, and B a source of electromotive force 
 producing a difference of potential between 
 A and C, that at A being the higher. There 
 will result a steady flow of current from A 
 to C and through the battery ; a part of the 
 current, Jq, will go from ^ to C through D ; 
 and a part, J^, from A to C through H. 
 Represent the resistances from ^ to Z>, D 
 to C, ^ to E, and U to C, by r-j, r^, r^, and r^ ; 
 and the potential difference between two points by V with proper 
 subscripts. The same current flowing from A to D must also 
 flow from D to C; therefore by Ohm's Law, 
 
 tB 
 
 Fig. 
 
 119. Branched 
 Circuit 
 
 It> = 
 
 _ yAD_ "DC 
 
 and similarly, 
 
 Ie = 
 
 Vae _ ^EC . 
 
 from which V^d : V^c = ''i = ''a' ^^^ ^ab ■ Vec = '"s = '*4- ^he 
 potential of any point on the branch D is less than the potential 
 of A and greater than the potential of C, and likewise for any 
 
284 ELECTRICITY 
 
 point on the branch E. Therefore, whatever may be the poten- 
 tial of a point on the D branch, there is always some point on 
 the E branch having the same potential. Let D and E be two 
 points having the same potential. Then 
 
 
 Vad= Vae, and F^c = 
 
 Vec; 
 
 and 
 
 r^: r^ = r^: r^, 
 
 
 or 
 
 r-^: r^ = r^: r^, 
 
 
 and 
 
 r^r^ = r^T^. 
 
 
 The last equation, stated in words, is that the products of 
 opposite, or alternate, resistances are equal. 
 
 From these equations, if the values of any three of the resist- 
 ances, Ty r^, j-g, and r^, are known, the fourth is determined; or, 
 if only the ratio of any two adjacent resistances, together with 
 the value of one of the other resistances, is known, the fourth 
 is determined. 
 
 There are two distinct methods of adjustment for locating the 
 point E, whose potential is the same as that of J), as required 
 in the above relations. The two forms of apparatus for this pur- 
 pose, the slide-wire-meter bridge and the post-office-box bridge, 
 are described in Arts. 208 and 214. 
 
 208. The Slide- Wire-Meter Bridge.— 
 The simplest form of the Wheatstone's 
 bridge is the slide-wire-meter bridge. The 
 conductor represented by AEC (Fig. 120) 
 is made of a uniform wire one meter long, 
 and the point E is determined by a slid- 
 ing contact which moves over a millimeter 
 scale. The lengths of the portions of the 
 wire, AE and EC, can be read directly. A 
 C resistance box, jB, of known value, corre- 
 
 FiG. 120. Wheatstone's sponds to r^ (Art. 207), while the Unknown 
 resistance, X, replaces r^. Any convenient 
 value being given to R, the potential of B becomes fixed. To find 
 the point E, having the same potential, a " bridge " conductor 
 containing a galvanometer is connected to D and to the slider. 
 
THE GALVANOMETER 285 
 
 If the bridge joins two points whose potentials are equal, evi- 
 dently no current will flow through the galvanometer. Usually, 
 however, when the key, K, of the bridge is closed, there will be 
 a deflection of the galvanometer needle caused by a flow of 
 current from Z> to E, or in the opposite direction. But it will 
 always be possible to find a point of contact by moving the 
 slider nearer to A or to C, such that there shall be no deflection 
 of the galvanometer ; then the bridge is said to be balanced, and 
 the potential at E is the same as at B. The resistances of the 
 two portions of the meter wire, AE and EC, are considered as 
 proportional to their lengths, l^ and l^. Therefore 
 
 With this form of apparatus it is desirable to have B and X 
 approximately equal, since in this case the contact will be near 
 the middle of the meter wire, and errors of reading and of the 
 wire will have a minimum effect. 
 
 Often the index on the slider is not over the contact, or the 
 resistances of the connections are unsymmetrical, or the wire is 
 of uneven size ; the result determined as above described may 
 then be slightly in error. A second determination should there- 
 fore be made after interchanging R and X; the mean of the two 
 values will be nearly free from these errors. 
 
 209. The Galvanometer. — The most common instrument for 
 measuring or detecting electrical currents is the galvanometer, 
 which is founded on Oersted's discovery of the mutual action 
 between a current and a magnet. In the galvanometer a 
 circuit is arranged with its plane parallel to the force of a 
 magnet ; when a current passes, there is a tendency for the 
 plane of the coil and the lines of magnetic force to set perpen- 
 dicular to each other. Either one of these parts, the circuit or 
 the magnet, may be fixed in position, the other being delicately 
 suspended so as to move under the force mentioned. Thus 
 there are two types of galvanometers, while there are many 
 varieties of each. 
 
 In one type a small magnet is freely suspended and allowed to 
 come to rest in the magnetic meridian ; coils of wire are placed 
 
286 
 
 ELECTRICITY 
 
 around the magnet, with the plane of the coils in the meridian. 
 Fig. 121 represents this type, some of the details of which are 
 shown in Fig. 123. Figs. 125, 130, and 134 show other forms 
 of this same type of galvanometer. When the current passes 
 
 through the wire the magnet 
 is deflected in one direction or 
 the other, depending upon the 
 direction of the current. 
 
 In the second type of gal- 
 vanometer a small coil of fine 
 wire is suspended by a delicate 
 metallic ribbon between the 
 poles of strong stationary per- 
 manent magnets. The torsion 
 of the suspension causes the 
 coil to seek its equilibrium 
 position, which is made to be 
 that in which the plane of the 
 coil is parallel' to the magnetic 
 force of the field magnets. 
 Fig. 122 represents a small 
 instrument of this type, while 
 a large one of the highest 
 sensibility is shown in Fig. 161. The essential parts are indi- 
 cated in Fig. 124. A current may enter the coil through the 
 suspension, and pass out through a delicate spiral of metal 
 ribbon attached to the bottom of the coil. When the current 
 passes, the coil tends to rotate so as to set itself perpendicular 
 to the field ; this rotation is counteracted by the torsion. This 
 type is often called the D'Arsonval galvanometer. It possesses 
 the distinct advantage of being practically independent of the 
 earth's magnetic field and of magnetic disturbances of all kinds. 
 Such' galvanometers are made of great sensitiveness, though the 
 extreme of sensitiveness has so far been obtained only with the 
 first type of instrument. 
 
 Galvanometers are used for three different kinds of measure- 
 ments. They may measure currents of definite strength, there 
 
 Fig. 121. Galvanometer 
 
THE GALVANOMETER 
 
 287 
 
 being produced a steady deflection. Such instruments are the 
 tangent galvanometer (Art. 216), the ammeter and voltmeter 
 (Art. 222), the electrodynamometer (Art. 
 248), etc. These are usually of small 
 or medium sensitiveness. 
 
 Instruments of another class are used 
 for detecting the presence of minute 
 currents, and are of the highest attain- 
 able sensitiveness. This class is perhaps 
 used more than any other in electrical 
 measurements. They are used with all 
 null methods, in which the adjustments 
 are made so that no current flows. 
 
 Ballistic galvanometers, constituting 
 the third class, measure, not the strength 
 of a steady current flowing as do the 
 first two classes, but the total quantity 
 of electricity which passes in an impulse 
 of short duration (Art. 253). 
 
 Galvanometers of any one of the three 
 classes may be of either of the two types 
 mentioned. 
 
 In instruments of the first and second 
 classes, which are used to measure steady 
 currents, it is very desirable that the 
 needle should quickly come to rest either 
 in its deflected or zero position. Gal- 
 vanometers in which the needle moves 
 to its equilibrium position without vibra- 
 tion are called aperiodic or dead-beat. 
 In a suspended magnet galvanometer 
 this condition is secured by causing the 
 magnet to move in a cavity in a solid 
 mass of copper. Induced currents are 
 generated in the copper by motion of the magnet ; the copper 
 has no effect when the magnet is stationary. The parts of such 
 a galvanometer may be arranged as indicated in Fig. 123. A 
 
 Fig. 122. D'Arsonval 
 Ga lv axom kte r 
 
 Fig. 123. Details of 
 Galvanometer 
 
288 
 
 ELECTRICITY 
 
 
 
 R, 
 
 bell or n -shaped magnet has its poles at iV" and S, inside the 
 copper mass A ; C-y and C^ are the coils of wire, and M is the 
 mirror attached to the hook by which the magnet is suspended. 
 A galvanometer with a suspended coil is made dead-beat by 
 winding the wire on a metal frame, or by surrounding it with a 
 metal tube in which the induced currents are 
 generated by the motion in the magnetic field. 
 Fig. 124 shows the arrangement of the parts 
 of this type, C being the coil suspended by 
 the ribbons of metal, -Bj and B^; iV and S 
 are the poles of the permanent magnets, and 
 M the mirror for scale reading. 
 
 The suspended part of a galvanometer 
 sometimes consists in effect of two magnets 
 rigidly fastened together with their poles 
 oppositely directed. Thus the earth's effect 
 upon the system is nearly neutralized. The 
 magnets are suspended inside the coils in such 
 a manner that the effect of the current is 
 twice as great as upon a single magnet. This 
 arrangement constitutes an astatio needle, and 
 makes the most sensitive galvanometer that 
 has been devised. This extreme sensitive- 
 ness renders the instrument somewhat trouble- 
 some to use. Fig. 125 shows the form of an 
 astatic galvanometer with two sets of coils. 
 
 210. Adjusting a Galvanometer Before 
 
 a galvanometer is moved it should be ascer- 
 tained that the weight of the needle is taken 
 from the suspension, to prevent breaking the 
 latter. The manner of doing this will be evident from the con- 
 struction. When the galvanometer is in position, on a pier or 
 other suitable support, if it is of the suspended magnet type, 
 the whole instrument should be turned on its base so that the 
 plane of the coils is approximately in the magnetic meridian. 
 Simple inspection will determine this sufficiently, unless the 
 deflections of the instrument are to be used for measurement. 
 
 Fig. 124. Parts 
 OF D'Arsonval 
 Galvanometer 
 
THE GALVANOMETER 
 
 289 
 
 When greater care is necessary in any of the exercises the 
 methods are mentioned (Arts. 216 and 253). 
 
 The instrument should be leveled. A- level on the base may 
 indicate this condition, but it often happens that the level is 
 itself out of adjustment. In any case the needle usually serves 
 as a better indicator of the proper position of the galvanometer. 
 When the needle is released it should swing freely and without 
 rubbing, and after a slight deflection it should 
 return to the starting point. 
 
 After the needle has taken its position in 
 the meridian it may be found that the mir- 
 ror faces in an inconvenient direction. In 
 some instruments it is possible, by carefully 
 holding the needle, to turn the mirror so 
 that it faces in any direction desired. It 
 should be so turned that when a scale is 
 placed in front of it the scale will be well 
 illuminated. 
 
 A control magnet may be advantageous 
 in adjusting a galvanometer. This may be 
 any permanent magnet, placed near the 
 instrument, or preferably attached to it in 
 some convenient manner, as shown at X, 
 Figs. 121 and 125. When a control magnet 
 is used it is not necessary to place the coils 
 in the magnetic meridian, as the magnet may be so placed as to 
 bring the axis of the needle into the plane of the coils, in what- 
 ever direction this may be. The control magnet may increase 
 the sensitiveness of the instrument by partially neutralizing 
 the earth's field. A consideration of the sensitiveness of a 
 galvanometer is given in Art. 247. 
 
 When the galvanometer is of the suspended coil type its direc- 
 tion with respect to the meridian is not important, and it may 
 be set facing in the most convenient direction, and leveled. 
 
 The replacing of a broken galvanometer suspension is such a 
 troublesome operation that especial care should be taken to 
 avoid breaking one. 
 
 Fig. 125. Astatic 
 Ballistic Galva- 
 nometer 
 
290 
 
 ELECTRICITY 
 
 211. Telescope, or Lamp, and Scale. — A galvanometer deflec- 
 tion may be observed by means of a pointer attached to the 
 needle ; but with a sensitive instrument it is usual to attach a 
 light mirror to the needle and to observe the deflections with a 
 telescope and scale, or with a lamp and scale. The adjustment 
 and use of a telescope and scale are sufficiently described in 
 Arts. 33, 253, and 264. 
 
 A light illuminating a slit or a wire may replace the tele- 
 scope. A lens is placed between the slit and mirror, and the 
 various parts are adjusted so that the image of the slit, reflected 
 by the mirror, is focused on the scale. The movements of this 
 image indicate the needle deflections. 
 
 212. Standard International Ohm ; Resistances and Rheostats. 
 — The international ohm is defined as the resistance of a column 
 of mercury, at 0°, 106.3 cm long, and 1 square mm area of cross 
 section. The mass of this quantity of mercury is 14.4521 g. 
 
 Standard resistances for electrical measurements are com- 
 monly made of wire. Various alloys have been devised such 
 that the specific resistance is high while the temperature coeffi- 
 cient and thermo-electric power are low, and such that the resist- 
 ance shall not change with time and use. Manganin is the 
 
 „„„^^^,,^ , most suitable alloy; it 
 
 ^ f 1 1 consists of 0.84 copper, 
 
 0.12 manganese, and 
 0.04 nickel. Other 
 alloys are constantan, 
 platinoid, and German 
 silver. 
 
 To avoid the self- 
 induction of a simple 
 helix, the wire for a 
 resistance coil is first 
 doubled at its middle, 
 and the two wires are 
 wound side by side in 
 such fashion that the current will flow through the two in 
 opposite directions. In coils of many turns this method gives 
 
 Fig. 126. 
 
 Standard Resistance in 
 Oil Bath 
 
RESISTANCES AND RHEOSTATS 
 
 291 
 
 considerable capacity. This, as well as the self-induction, may- 
 be avoided by winding a single wire, with freqxient changes in 
 the direction of the winding, so that in the end there are as 
 many turns in one direction as in the other. This method 
 is necessary only for coils of more than 400 ohms resistance. 
 
 Fig. 127. Resistance Box 
 
 Standards of single resistance are usually so constructed that 
 they may be placed in an oil bath to secure a definite and known 
 temperature. Connections are best made by means of mercury 
 cups, into which the terminals dip. Fig. 126 shows the con- 
 struction of such a standard. 
 
 Sets of resistances, mounted to permit various combinations, 
 are usually inclosed in wooden boxes, and are called resistance 
 boxes. On the top of the box are rows of metal blocks, and 
 between these are fitted taper plugs (Fig. 12T). The coils are 
 so joined to the blocks that when the plugs are all removed a 
 current connected to the terminals of the box must pass through 
 all the coils in succession, and the resistance is equal to their 
 sum. By the insertion of a plug any coil may be shunted by 
 a resistance so small that, for ordinary measurements, the resist- 
 ance of the plugged part may be considered 0. If all the plugs 
 are in place, the total resistance is 0. With this arrangement 
 any combination of resistances may be secured. 
 
 The actual resistance through a plug, when it is in good 
 order, with clean and well-fitting surfaces, is from 0.00005 to 
 0.0001 ohm. 
 
292 ELECTRICITY 
 
 The plugs are to be inserted with a firm pressure and a slight 
 twisting movement, but care must be taken that they are not 
 forced into the taper holes so tightly as to injure the apparatus. 
 Plugs belonging to different boxes should never be interchanged. 
 Sometimes the insertion of a plug warmed by contact with the 
 hand may produce thermo-electric currents. This is avoided by 
 the use of a nonconducting handle. 
 
 Resistance boxes are adjusted for a temperature of 20° ; if 
 the temperature differs much from this, a temperature correc- 
 tion may be necessary. The temperature coefficient of man- 
 ganin wire may vary from + 0.00001 to -|- 0.00004. 
 
 Resistance boxes should never be used to control large currents; 
 the heat developed may ruin the insulation or even melt the wire. 
 A variable resistance, not standardized, is often useful for 
 regulating a current. Such a rheostat may consist of coils of 
 wire with suitable switches, or wire arranged so that various 
 lengths of it may be used. A pile of short carbon rods, through 
 which the current passes, has a resistance which varies with the 
 contact pressure of the rods against each other. By a screw 
 press this pressure may be varied, and thus the current strength 
 may be continuously altered through a small range. If such 
 a rheostat is inserted in addition to a coil rheostat, it will be 
 possible to adjust a current exactly to a desired value. 
 
 213. Keys. — A great variety of keys and connectors are 
 required in arranging electrical apparatus. Single keys and 
 switches need not be described. For work with the Wheat- 
 stone's bridge, and other in- 
 struments, a double successive 
 contact hey is useful. There 
 are four contact points, 6j, h^, 
 ^1, and g^ (Fig. 128), which 
 are connected with the bind- 
 ing posts B,, B„, G,, and G„ 
 
 Fig. 128. Double Successive „ ,• i t) . i j 
 
 _ ^ respectively. Between o„ and 
 
 Contact Kbt . . ^ 
 
 ffi IS an insulating button. 
 
 The battery circuit is connected to ^j and B^, and the galvanom- 
 eter circuit to G^ and G^. When the knob K is pressed the 
 
THE BOX BRIDGE 293 
 
 battery circuit is first closed, and then the galvanometer circuit 
 is closed by a further depression of the knob. Thus the current 
 is started in the apparatus and assumes a steady condition before 
 the galvanometer is applied. The bridge contact often serves 
 as one key, in which case only a single key is required; or if 
 the battery is a constant cell, it may be put in a closed circuit, 
 when no key will be required. 
 
 Various forms of commutating keys are described in Art. 218, 
 a condenser key is shown in Art. 263, and other forms are men- 
 tioned in Arts. 229, 230, 239, 254, and 256. 
 
 CXXVI. RESISTANCE WITH THE BOX BRIDGE 
 
 Determine the resistances of the magnet coils of a sounder, of a relay, and 
 of the primary and secondary windings of an induction coil. Use 
 several ratios for each resistance. 
 
 214. The Box Bridge. — The slide-wire form of Wheatstone's 
 bridge (Art. 208) may be uncertain in its results because of 
 irregularities in the wire or in the contact. In it the sum of 
 the ratio resistances is constant, being equal to the resistance 
 of the meter wire, while it is often advantageous to change 
 this sum (Art. 215) ; the ratio is usually a fraction, precluding 
 mental computation, and the apparatus lacks portability. The 
 box bridge obviates these difficulties somewhat ; it is an appara- 
 tus in which the ratio resistances {r^ and r^ of the previous 
 form), as well as the standard resistance, are all wire coils con- 
 tained in one box. This form of bridge was introduced in 
 connection with the telegraph work of the postroffice depart- 
 ment in England ; hence it is often called the post-office 
 resistance box. 
 
 The sets of ratio coils, Bg and B^ (Fig. 129), take the place of 
 the meter wire and slider, and each usually contains resistances 
 of 1, 10, 100, and 1000 ohms. One resistance only from each 
 set is to be used at a time ; these may be selected so that the 
 ratio shall be either 1, 10, 100, 1000, 0.1, 0.01, or 0.001. If 
 the reference resistance has a total value of 10000 ohms, the 
 
294 
 
 ELECTEICITY 
 
 range of measurement is from 0.001 ohm to 10 000000 ohms, 
 there being from one to four significant figures in the result. 
 Often more significant figures may be obtained by interpolation 
 from the galvanometer readings. 
 
 Sometimes the three sets of resistances are permanently con- 
 nected, and sometimes the connections have to be made as 
 required. In the latter case arrange the apparatus in accord- 
 ance with the scheme of Fig. 129. Having selected a ratio, the 
 known resistance B is varied to secure a balance between 
 the points D and E, as indicated by zero 
 deflection of the galvanometer in the 
 bridge when the key K is closed. Then, 
 since (Art. 207) 
 
 -B ; JC = Ro '. Hi, 
 
 Fig. 129. Box Bridge 
 
 the unknown resistance is obtained by 
 the very simple calculation of multiply- 
 ing B by some power of ten. 
 
 When X is wholly unknown it is well 
 to select a ratio of equality, as 10:10; 
 and with this to find an approximate value 
 for X. Then change the ratio so that B shall have the largest 
 possible value, and make a final determination of X. For instance, 
 if with a ratio of 10 : 10, X is found to lie between 7 and 8 ohms, 
 make the ratio 1000 : 1, when the balance may be found with B 
 equal to 7341 ohms, giving for X the final value 7.341 ohms. 
 
 Sometimes, in addition to the ratio coils and the resistance 
 coUs, a key, galvanometer, and battery are included in the case 
 of the apparatus. The collection is then called a testing set, 
 and it may be exceedingly compact and convenient. It is also 
 evident that the method may be employed when the apparatus 
 is made up with three separate, simple, resistance boxes. 
 
 215. Maxwell's Rule for Wheatstone's Bridge An inspec- 
 tion of Fig. 129 indicates that the resistances B, X, Bg, and B^ 
 are represented by the four sides of a parallelogram, in one 
 diagonal of which the galvanometer is placed, and in the other 
 the battery. From the symmetry of the arrangement it does 
 
THE TANGENT GALVANOMETER 
 
 295 
 
 not matter, so far as the relations between the resistances are 
 concerned, which diagonal contains the battery. But to secure 
 the greatest sensitiveness Maxwell gives the following rule. 
 " Of the two resistances, that of the battery and that of the 
 galvanometer, connect the greater resistance so as to join the 
 two greatest to the two least of the four other resistances." 
 
 CXXVn. RESISTANCE WITH A TANGENT GALVANOMETER 
 
 Measure several unknown resistances, both large and small, by the method 
 of three observations with a tangent galvanometer; also find the 
 resistance of the battery employed and the electromotive force. 
 
 216. The Tangent Galvanometer. — A simple form of galva- 
 nometer consists of a number of turns of wire wound on a 
 circular frame of large diam- 
 eter, usually twenty centi- 
 meters or more, supported 
 with its plane in the plane 
 of the magnetic meridian, 
 and having a short perma- 
 nent-magnet needle sus- 
 pended in the center of the 
 coil (Fig. 130). The needle 
 carries a light pointer, and 
 a divided circle is provided 
 for measuring the angular 
 deflection of the needle. If 
 various steady currents be 
 sent through such a galva- 
 nometer, they will produce 
 deflections of the needle such 
 that the current strengths 
 are proportional to the 
 tangents of the angles of 
 deflection. Therefore these 
 current strengths may be expressed in any desired units by 
 multiplying the tangents of the angles of deflection by a 
 
 Fio. 130. Tangent Galvanometer 
 
296 ELECTEICITY 
 
 properly determined constant. If -K" is the constant found as 
 described in Art. 242, and 6 is the deflection observed, the 
 current strength in amperes is 
 
 I=Kta.n0. 
 Before a tangent galvanometer is used for measurement it 
 must be carefully adjusted to the magnetic meridian. Its own 
 needle wiU serve as a compass in making an approximate 
 adjustment. Connect a cell of battery, in series with a rheostat 
 if necessary, through a reversing key (Art. 218) to the galva- 
 nometer. Observe the deflection produced, by averaging the 
 readings of the two ends of the pointer ; reverse the current 
 and read again. If these opposite deflections are equal, the 
 instrument is properly placed ; if they are not equal, rotate 
 the plane of the coil and repeat the observations; continue 
 until the opposite deflections are equal. 
 
 References. — For an extended account of the tangent galvanometer, and 
 of work to be done with it, consult Stewart and Gee, Physical Measurements, 
 Vol. II, pp. 225-274 ; Gray, Absolute Measurements in Electricity and Mag- 
 netism, Vol. II, pp. 347 et seq. 
 
 217. Resistance by Three Observations with a Tangent Gal- 
 vanometer, and Battery Resistance by Ohm's Method. — Connect 
 the unknown resistance, X, a rheostat, B, and a constant battery, 
 B, in series, through a reversing key, K, 
 with the galvanometer, G (Fig. 131). By 
 changing the number of cells of battery 
 used, or by altering the resistance R, cause 
 the deflection of the galvanometer to be 
 about 45°. The battery should be so 
 arranged that B may be as small as pos- 
 sible, preferably 0. Observe the deflection 
 of the galvanometer, measured, as should 
 Fig. 131. Resistance always be the case, by the average of the 
 WITH Tangent Gal- ^^^ readings given by the two ends of 
 the needle when the current flows first in 
 one direction and then in the reversed direction. 
 
 If H represents the electromotive force of the battery, B its 
 resistance, G the resistance of the galvanometer, B^ the value 
 
THE TANGENT GALVANOMETEE 297 
 
 of E, X the unknown resistance, ^j the galvanometer deflection, 
 
 and ^the galvanometer constant, then, by Ohm's Law, from the 
 
 first observation, „ 
 
 = ^tan e.. 
 
 B+ G + R^-^X ^ 
 
 Leaving the battery unchanged, remove X, and alter B till a 
 deflection, 6^, of about 30° is obtained, and let R^ be the value 
 of R giving this result ; then, from the second observation, 
 
 E 
 
 = ^tan^. 
 
 B+ G + R^ 
 
 Further, alter R to give a deflection, ^3, of about 60°; if ^3 
 is the value of the resistance required, 
 
 = ^tan 0o. 
 
 B+ G + R^ ^ 
 
 Dividing the second equation by the third, 
 ^ + g + .^3 _ tan ^g 
 B+ G-\- R^~ t&zi e^ 
 and ^^^^Je^tan^.-iggtan^a 
 
 tan ^3 — tan 6^ 
 
 Calculate the value oi B -{■ G, which is required in finding X. 
 If the galvanometer resistance is known, the resistance of the 
 battery is determined by making the second and third observa- 
 tions only. 
 
 Dividing the third of the above equations by the first, 
 
 B-V g + -Si + X ^ tan^3 
 J? + G + i?3 ~ tan 6^ 
 
 tan 6. 
 
 -^iB+G + R^)-{B+G + R^). 
 ^1 
 
 from which X = 
 
 tan I 
 
 By eliminating B + G between the second and third equa- 
 tions given above, the electromotive force of the battery may 
 be obtained if the galvanometer constant K is known, as 
 expressed in the following relation. 
 
 ^ = JTtan e, tan 6, — ^ ~ ^^ 
 
 tan ^3 — tan 0^ 
 
298 
 
 ELECTEICITY 
 
 Fig. 132. Rotating Commctatok 
 
 The above methods for battery resistance and electromotive 
 force are known as Ohm's methods ; they may be used with 
 any nonsensitive galvanometer, for constant current cells. 
 
 218. Commutators, Reversing Keys. — When frequent changes 
 in the direction of a current through part of an apparatus are 
 
 required, some form of com- 
 mutator should be used. 
 An efficient and easily con- 
 structed one may consist of 
 a block of wood in which 
 are four holes, bored only 
 part way through, these 
 holes being filled with mer- 
 cury and connected as fol- 
 lows. One terminal of the 
 circuit is connected to hole 2 
 (Fig. 132), and the other to hole 3 ; while the apparatus 
 through which the current is to be reversed is connected to 
 holes 1 and 4. Two wire links are placed to join 1 and 2, 
 and 3 and 4, as shown by the heavy lines. If, now, the links 
 are lifted and rotated 90°, so that they connect 1 and 3, and 
 2 and 4, as indicated by the dotted lines, the current through 
 G will be reversed. 
 
 A plug commutator is sometimes made, in which the four 
 mercury cups are replaced by quadrants of brass, and the wire 
 links by plugs which join 
 the quadrants as desired. 
 
 The rocking commutator, 
 using six mercury cups, is 
 a most convenient form. 
 Gups 1 and 2, or 3 and 4 
 (Fig. 133), are connected to 
 the battery, and 5 and 6 to 
 the galvanometer or other 
 instrument ; 1 and 4, and 2 and 3 are permanently connected 
 by diagonal wires. A rocker of the form shown, consisting of 
 two three-pronged forks, joined by an insulating handle, H, 
 
 Fig. 133. Rocking Commctator 
 
THE DIFFERENTIAL GALVANOMETER 
 
 299 
 
 permits 5 to be joined to either 1 or 3, and at the same time 
 6 is joined to either 2 or 4. The removal of the rocker opens 
 all circuits. 
 
 If the diagonal wires of the commutator are removed, one 
 circuit whose terminals are joined to 5 and 6 may be connected 
 to either one of two other circuits whose terminals are 1 and 
 2, or 3 and 4. 
 
 There are many other forms rSP 
 
 of commutators, the operation 
 of which will be understood 
 from inspection. 
 
 CXXVm. RESISTANCE BY 
 THE DIFFERENTIAL METHOD 
 
 Compare several large and small 
 resistances of a resistance box 
 ■with approximately equal 
 standard resistances, by means 
 of a differential galvanometer, 
 using the methods of single 
 observation, substitution, and 
 reversal. 
 
 219. The Differential Galva- 
 nometer. — The coils of a gal- 
 vanometer are often wound in 
 two equal parts, Cj and C^ (Fig- 
 134), synunetrically placed with 
 respect to the needle, and so connected that the same current, 
 or equal currents, may flow through the coils in opposite direc- 
 tions. These currents in the two coils tend to turn the needle 
 in opposite directions, and it can then indicate only the differ- 
 ence in their effects. Often the terminals of each coil are 
 separately brought to binding posts in order that the coils may 
 be connected to give differential effects, or by joining them so 
 that the current flows through both in the same direction the 
 galvanometer becomes a sensitive one of the ordinary type. 
 
 Fig. 134. Differential 
 Galvanometer 
 
300 
 
 ELECTRICITY 
 
 Fig. 135. Magnetic 
 Adjustment 
 
 Magnetic Adjustment. — Connect the coils as indicated in 
 
 Fig. 135, so that the current going in at the inner end, i, of 
 
 the first coil enters the second by its outer end, o. If the termi- 
 nals of the coils are not marked, a simple 
 trial will indicate whether the coils are con- 
 nected for differential or summational effects. 
 By altering the position of one coil or a 
 part of one coil, the adjustment is to be made 
 so that the same current flowing through 
 the two coils in opposite directions shall 
 produce no effect upon th6 needle. 
 
 Resistance Adjust- 
 ment. — Connect the 
 
 coils in parallel to give differential effects ; 
 
 that is (Fig. 136), connect one terminal of 
 
 the battery to the outer end of one coil and 
 
 to the inner end of the other. Make the 
 
 other connections as shown; then, when 
 
 the resistances of the two branch circuits 
 
 are equal, equal currents will flow through 
 
 the two coils, and there will be no deflec- 
 tion. If necessary 
 insert external resistance in one branch 
 circuit until this condition is fulfilled. This 
 may be accomplished by using a rheostat, 
 or by varying the length of one of the lead 
 wires. 
 
 220. To measure a Resistance greater than 
 that of One of the Galvanometer Coils. By 
 Single Observation. — After completing the 
 adjustments insert a standard resistance, B, 
 and the unknown resistance, X, as indicated 
 in Fig. 137. Adjust R until no deflection 
 of the galvanometer is caused by closing the 
 battery circuit. Then Xis equal to R. If It 
 
 cannot be adjusted to produce an exact balance, calculate the true 
 
 value of X by interpolation between the nearest two values of B. 
 
 Fig. 
 
 136. Resistance 
 Adjdstment 
 
 Fig. 137. Method 
 OP Single Obsek- 
 vation 
 
THE DIFFERENTIAL GALVANOMETER 
 
 301 
 
 By Substitution. — In case the adjustment of the galvanom- 
 eter is uncertain, after securing a balance as nearly as possible 
 by the method described above, note the constant deflection 
 resulting ; substitute for X a standard resistance box, and adjust 
 it to produce the same deflection. The substituted resistance 
 is equal to A'. In this case the resistance B serves merely as a 
 rheostat, and need not be a standard. 
 
 By Reversal. — The galvanometer may be connected as 
 indicated in Fig. 138, without any previous adjustment. S, a 
 rheostat inserted in one branch circuit, is so adjusted that no 
 
 Fig. 138 
 
 Fig. 139 
 Method of Reversal 
 
 Fig. 140 
 
 deflection occurs when the battery circuit is. closed. Insert the 
 standard resistance B and the unknown resistance X (Fig. 139) ; 
 adjust B until a balance is again effected, and let B-^ be its value. 
 Interchange B and X (Fig. 140), and let B^ be the resistance 
 required to produce a balance. If g-^ and g^ are the entire 
 resistances (unknown) of the two branches of the circuit. 
 
 
 B^-.X^g^-.g^, 
 
 and 
 
 ^•■R'i=9\-9v 
 
 from which 
 
 X= ViJjiJ^ 
 
 (Compare the above methods of using a galvanometer with 
 the methods of using a balance described in Arts. 43 and 47.) 
 
302 
 
 ELECTRICITY 
 
 221. To measure a Resistance less than that of One of the 
 
 Galvanometer Coils When the unknown resistance is less 
 
 than that of one of the galvanometer coils, greater sensitiveness 
 will be obtained by a modification of the 
 previous methods. Connect the battery to 
 the galvanometer so that the same current 
 shall pass through the two coils in series in 
 opposite directions (Fig. 141) ; make the 
 standard resistance B a shunt to one coil, 
 and the unknown resistance X a shunt to 
 the other. Adjust B to secure a balance, 
 when, if the galvanometer is in perfect 
 adjustment both for magnetic effect and 
 resistance, X is equal to B, as in the method 
 of single observations previously described. 
 Without previous adjustment of the galvanometer the methods 
 of substitution and reversal may also be applied with this 
 system of connections. 
 
 Fig. 141. Method 
 OF Shunts 
 
 CXXIX. RESISTANCE BY THE FALL OF POTENTIAL 
 
 Measure the resistance of an incandescent lamp while it is lighted ; also 
 the internal resistance of a series of secondary cells, and of the several 
 cells separately. 
 
 222. Ammeters and Voltmeters. — The ammeter is a low- 
 resistance galvanometer in which a pointer, moving over an 
 empirical scale, indicates the strength of the current flowing 
 through the instrument. It is to be connected in series with 
 the circuit, its resistance being so low that for the currents to 
 which it is adapted its insertion does not appreciably alter the 
 current strength. 
 
 The voltmeter is a high-resistance galvanometer in which the 
 empirical scale indicates the potential difference which must be 
 applied to its terminals to cause a sufficient current to flow 
 through it to deflect the pointer by various amounts. It is to 
 be used on a shunt circuit, and preferably only for momentary 
 
RESISTANCE BY FALL OF POTENTIAL 303 
 
 observations. Its resistance is so high that when applied as a 
 shunt it does not appreciably alter the strength of the current. 
 The voltmeter may be applied to measure the difference of 
 potential of two points between which no current is flowing, 
 provided that the very small current which will flow through 
 the voltmeter does not disturb the potential difference. 
 
 Voltmeters and ammeters are commonly of the D'Arsonval 
 galvanometer type (Art. 209), though other forms are sometimes 
 used. 
 
 223. Resistance with Voltmeter and Ammeter. — The method 
 of fall of potential is based upon Ohm's Law. It may be very 
 
 Fig. 142. Resistance with Voltmetee and Ammetee 
 
 conveniently applied to the measurement of resistances of mod- 
 erate size, and also of small resistances, if suitable voltmeters 
 and ammeters are available. 
 
 Arrange to send a current of any convenient value through 
 the resistance to be measured, and a suitable ammeter in series. 
 Connect the voltmeter to the terminals of the unknown resist- 
 ance in a shunt circuit. (See Fig. 142.) If E is the difference 
 of potential indicated by the voltmeter while the current I is 
 flowing, the unknown resistance is 
 
 I 
 
 224. Battery Resistance with Voltmeter and Ammeter. — Con- 
 nect an ammeter, a rheostat, and a key in series with the bat- 
 tery, and a voltmeter as a shunt to the battery, as indicated in 
 Fig. 143. Adjust the rheostat so that when the key is closed 
 
304 
 
 ELECTRICITY 
 
 the current passing shall have any convenient value, /, and let 
 E^ be the difference of potential between the terminals of the 
 battery, as indicated by the voltmeter, while this current is flow- 
 ing ; let E^ be the voltmeter reading when the key is open and 
 
 no current is flowing ; then the 
 internal resistance of the battery 
 
 ^ 
 
 \^4\v^^ 
 
 IS 
 
 I 
 
 ^-\X\ ^ 
 
 ^^ 
 
 Internal Resistance 
 OF Battery 
 
 This method is particularly 
 applicable to the measurement 
 of the resistance of a secondary 
 battery. 
 
 225. Battery Resistance with 
 Voltmeter and Standard Resist- 
 In the place of the ammeter and rheostat, in the above 
 method, put a standard resistance. Let -fi^j be the difference of 
 potential while the current is flowing through the resistance B ; 
 and let E^ be the difference of potential of the battery on open 
 circuit ; then the battery resistance is 
 
 Fig. 143. 
 
 ance. 
 
 B = B 
 
 E, 
 
 226. Battery Resistance with Con- 
 denser and Resistance. — The method 
 last described will be better adapted to 
 the measurement of the resistances of 
 single cells if the voltmeter is replaced 
 by a condenser and ballistic galvanom- 
 eter. The charges received by the con- 
 denser are proportional to the potential 
 differences. Make the connections as 
 indicated in Fig. 144. Close the key 
 ^2, completing the battery circuit, and at once press the key K^ 
 firmly and for a very short time upon the lower contact ; release 
 the key quickly. The condenser will have been charged 'to a 
 potential difference equal to that of the battery terminals when 
 
 Fig. 144. Battery 
 Resistance with 
 Condenser 
 
THOMPSON'S METHOD 305 
 
 on the closed circuit, and the galvanometer will be deflected an 
 amount proportional to this charge (Art. 253). Let d-^ be the 
 deflection. Now, leaving the key K^ open, again charge and 
 discharge the condenser, and let d^ be the deflection produced. 
 Then, if B is the resistance in the battery circuit, the battery 
 resistance is 
 
 The observed deflections of the ballistic galvanometer, if 
 made with a telescope and scale, must be corrected, to make 
 them proportional to the quantities of electricity, as described 
 in Art. 252. 
 
 The value of B should not be much greater than that of B, 
 though if the latter is very small, too small a value of B will 
 cause too rapid a polarization of the cell. The method is there- 
 fore adapted to constant cells and those of high resistance. 
 
 A very full description of the method is given in the reference. 
 
 Refekence. — Carhart and Patterson, Electrical Measurements, pp. 100-109. 
 
 CXXX. GALVANOMETER AND BATTERY RESISTANCE BY THE 
 WHEATSTONE'S BRIDGE 
 
 Determine the resistance of a galvanometer by Thompson's method, and 
 the resistance of a gravity cell by Mance's m^ethod. 
 
 227. Thompson's Method for Galvanometer Resistance. — This 
 method is based upon the principle of Wheatstone's bridge. 
 The arrangement of apparatus differs from that previously 
 described (Fig. 120) in that the galvanometer is taken out of 
 the " bridge " and is placed as the unknown resistance, as shown 
 in Fig. 145. A current will now flow through the galvanom- 
 eter continually, producing a deflection. If this deflection is 
 too large, it may be modified by a control magnet, by a shunt, 
 by a resistance in the battery circuit, by using the galvanom- 
 eter differentially, or by using two cells of battery joined in 
 opposition, according to circumstances. 
 
306 
 
 ELECTEICITY 
 
 Making the bridge connection from D to E will usually alter 
 the current flowing from A to C, and thus cause a change in the 
 galvanometer reading. But by trial a point, E, may be found 
 which has the same potential as D, so that opening and closing 
 the bridge causes no change in the galvanometer deflection. 
 Then the resistances are related as follows (Art. 207) : 
 
 r^:r^ = R: G. 
 
 If the galvanometer is of high resistance, a box bridge is 
 preferable to the slide-wire form, as the resistance of the ratio 
 arms may then be made more nearly equal to that of the galva- 
 nometer, a condition securing greater sensitiveness. 
 
 228. Mance's Method for Battery Resistance. — The details 
 of this method are those of the preceding one, except that the 
 
 Fig. 145. THOiMPSON's Method 
 
 Fig. 146. Mange's Method 
 
 battery and galvanometer change places. The method is suit- 
 able only for a constant cell. The connections are shown in 
 Fig. 146. 
 
 Some current from the battery will always flow through the 
 galvanometer, causing a deflection. The adjustment consists in 
 finding such a point of contact, E, that opening and closing the 
 bridge does not alter the galvanometer reading. Then B and E 
 are at the same potential, and the resistances are in the following 
 proportion. 
 
 r^:r^ = R:B. 
 
COMPENSATION METHODS 
 
 307 
 
 CXXXI. BATTERY RESISTANCE BY COMPENSATION 
 
 Determine the resistances of various cells of bp,ttery. 
 
 229. Beetz's Method for Battery Resistance The following 
 
 method for measuring the resistance of a battery has the advan- 
 tage of requiring the circuit to be closed only momentarily, 
 thus avoiding to some extent the polarization of the cell. The 
 
 h 
 
 E' R 
 
 R, 
 
 B 
 
 D 
 
 Tvpjz 
 
 3 — r^v 
 
 Fig. 147. Beetz's Method for Battery Eesistance 
 
 principles of the method may be described in connection vrith 
 the apparatus as follows. Fig. 147 represents a slide-wire 
 bridge, the resistance of the slide wire of which is known, H^ 
 and B^ two small resistance boxes, X a three-point key, G a 
 galvanometer, U the cell whose resistance, JS, is to be deter- 
 mined, and e a cell of less electromotive force than U. The 
 connections having been made as indicated, the like poles of 
 both cells being connected to the key, it is evident that the 
 resistances Hi and Jt^ and the point of contact C can be so 
 adjusted that the difference of potential produced by e between 
 the points K and C will be counterbalanced by the opposite 
 potential difference between the same points produced by U. 
 When this condition is secured the galvanometer will show no 
 deflection. This must be determined by making only very 
 short contacts with the key. If the key is closed even for a 
 few seconds, the resistance of the cell changes, owing to polari- 
 zation, and thus becomes uncertain. Represent the total length 
 of the bridge wire, BD, by li, the length from 5 to C by l^, and 
 
308 
 
 ELECTRICITY 
 
 the resistance of unit length of the wire by p (Art. 240). Neg- 
 lecting the connections, the external resistance in the upper 
 circuit is E-^ + l^ p + JR^ ; represent this by a^, and let b^ be 
 
 equal to B^ + l^ 
 
 P\ 
 
 then by Ohm's Law the current in the 
 
 upper circuit is 
 
 
 j_ ^ _« 
 
 
 ^ + «! i>i' 
 
 from which 
 
 
 E JS + aj 
 e \ 
 
 Alter the resistance B^, or B^, so as to obtain a new point of 
 contact, differing considerably from the first, which will give 
 
 new values, a^ and h^, corresponding 
 to aj and 5j of the first observation. 
 Then, as before, 
 
 E _B + a^_B + a^ 
 
 and 
 
 "2 
 
 h 
 
 Fig. 148. Benton's Method 
 FOR Battery Eesistance 
 
 Instead of the two resistance boxes, 
 B^ and B^, and the slide-wire bridge, 
 a single thin stretched wire may be 
 used, simplifying the apparatus 
 required. 
 
 230. Benton's Method for Battery 
 Resistance. — Beetz's method can be 
 applied to the investigation of only 
 such cells as have an electromotive force greater than the auxili- 
 ary cell. The following method is not limited in this respect, 
 and it is better adapted for cells of low internal resistance. The 
 apparatus required consists of three resistance boxes, an auxiliary 
 cell, a low-resistance galvanometer, and a special key. These 
 are connected as indicated in Fig. 148, E' being the cell 
 investigated, and E" the auxiliary cell, the like poles of which 
 are connected to the key. The key has five connections, such 
 that the first pressure brings three points into contact, closing 
 
KOHLRAUSCH'S METHOD 309 
 
 the two battery circuits, and then a further pressure closes the 
 galvanometer circuit by means of an insulating contact piece. 
 
 With R'" = 0, and R' set at any small resistance, R" is 
 adjusted so that there is no deflection of the galvanometer when 
 the key is closed ; then A and C are at equal potentials. The key 
 is to be closed only for an instant, otherwise polarization may 
 result. If R^ and R^' are the values of the resistances in the 
 boxes, and B' and B" are the respective battery resistances, 
 the condition for equal potentials at A and C is 
 
 i?i' + B' R^' + B" 
 
 R' is now set at a large resistance, ^2'' while R" is left 
 unchanged. R'" is adjusted until no deflection occurs upon 
 closing the key ; let R^" be the resistance required. The 
 condition for equal potentials at A and C is 
 
 -B2 -^1 
 
 R^ + R^" + B' ~ R^' + B"' 
 
 Equating the first members of these equations and reducing, 
 the internal resistance of the cell is 
 
 R-i ~ -^1' 
 
 Reference. —J. R. Benton, Physical Review, Vol. 16, p. 253, 1903. 
 
 CXXXn. ELECTROLYTE RESISTANCE BY KOHLRAUSCH'S 
 
 METHOD 
 
 Find the specific resistance of a ten per cent, solution of copper sulphate 
 and the internal resistance of various cells singly, in series, and in 
 parallel. 
 
 231. Electrolyte Resistance with Alternating Currents.— 
 
 Kohlrausch's method for electrolyte resistance avoids the influ- 
 ence of polarization and allows the resistance to be measured 
 directly, just as that of a metallic conductor, by using in the 
 measuring device currents rapidly alternating in direction and 
 
310 
 
 ELECTEICITY 
 
 of exactly equal strength, between electrodes of large capacity. 
 This current is conveniently obtained from an induction coU, 
 and the detector is usually a telephone. The principle of 
 measurement is that of Wheatstone's bridge, though the bridge 
 is often of special construction to facilitate the use of the induc- 
 tion coil and telephone, which replace the battery and galva- 
 nometer respectively. The cell whose internal resistance is 
 required, or the vessel containing the electrolyte, is connected 
 as the unknown resistance. A separate cell is, of course, 
 required to operate the induction coil. A sketch of the con- 
 nections is shown in Fig. 149. ^3 and R^ represent the slide 
 
 Fig. 149. Telephone Bridge for Alternating Currents 
 
 wire, and A the sliding contact. The observation consists in 
 adjusting the contact until the sound of the induction current 
 in the telephone becomes a minimum. Then the ratios of the 
 various resistances are 
 
 The scale attached to the slide wire, instead of being a scale 
 of equal lengths, may be divided to show directly the ratio of 
 the lengths, ^3 to B^, for the various positions of the sliding 
 contact. -Bj may then be conveniently 1, 10, or 100 ohms, and 
 the unknown resistance is found at once by multiplying the 
 scale reading by Ry 
 
 232. Electrolyte Resistance. — The specific resistance of an 
 electrolyte may be calculated, as for a solid (Art. 235), from 
 
DIRECT DEFLECTION METHODS 311 
 
 measurements of the resistance of a column of the liquid of 
 known length and cross section. Geometrical measurement 
 of the space between two electrodes is seldom convenient, and 
 the more usual method is to compare the unknown electrolyte 
 with that of a standard solution in the same containing vessel. 
 A useful standard solution is a saturated solution (about twenty- 
 six per cent.) of sodium chloride. The liquid is to be shaken up 
 with an excess of salt just before being used. The conductiv- 
 ity of the solution is as follows (Kohlrausch) : 
 
 Temperature 15° 16° 17° 18° 19° 20° 21° 
 
 Conductivity, K 0.2015 0.2063 0.2112 0.2161 0.2210 0.2260 0.2310 
 
 Fill the electrolyte vessel with the salt solution and deter- 
 mine its resistance, rj, using the Kohlrausch method described 
 in the preceding article ; substitute the unknown electrolyte 
 and find its resistance, r^ ; then the conductivity desired is, 
 since the conductivities are inversely as the resistances, 
 
 '•2 
 
 Eefeeences. — Kohlrausch, Physical Measurements, pp. 317-321; Ostwald, 
 Physico-Chemical Measurements, pp. 222-229. 
 
 CXXXm. INSULATION RESISTANCE BY DIRECT DEFLECTION 
 
 Find the resistances of glass and porcelain insulators, and of the 
 insulation of a cable. 
 
 233. Measurement of High Resistance by Direct Deflection. — 
 
 A cup-shaped insulator may be arranged for testing by pouring 
 mercury into the interior and partially immersing the cup in a 
 mercury bath, the edge of the cup where not immersed hav- 
 ing been previously cleaned, dried, and coated with paraffin. 
 Connect one terminal of the circuit to the mercury in the cup, 
 and the other to the exterior bath. In the case of insulated 
 wire that is waterproof, make of it a coil of a known length, 
 and immerse it, except the ends, in a water bath. One circuit 
 terminal is to be connected to the wire and the other to the 
 
312 ELECTRICITY 
 
 water bath. When the material is in sheets, portions of the 
 surfaces may be silvered, and the electrodes connected to these 
 silvered portions. Care should be taken to have a wide margin 
 of clean and dry surface around the silvered portion. The con- 
 nections for tests of the insulations of permanently placed wires 
 and cables will suggest themselves. 
 
 Connect a sensitive galvanometer and the unknown resistance 
 in series with a battery or other source of electromotive force. 
 Usually as high an electromotive force as is available will be 
 required, such as that of a large storage battery, a commercial 
 lighting circuit, or a railway power circuit. Let E be the elec- 
 tromotive force used, measured with a voltmeter ; X, G, and B the 
 unknown, galvanometer, and battery resistances respectively ; d 
 the number of divisions of the scale at a meter distance, meas- 
 uring the constant deflection produced; and K the figure of 
 merit of the galvanometer. Then 
 
 dK ^ 
 
 X+B^ (?' 
 
 and X=^-{B+G). 
 
 dJi 
 
 The battery resistance may be neglected in comparison with 
 the other resistances. 
 
 234. High Resistance by Direct Deflection and Double Read- 
 ings. — ■ The measurement of high resistances by the method of 
 double readings is often more convenient than the one described 
 above. It is not necessary to know the electromotive force and 
 galvanometer constant, but a standard resistance of high value, 
 100000 ohms, for instance, is required. The electromotive 
 force is first connected to the galvanometer and unknown 
 resistance in series, and then to the 100000 ohms resistance and 
 galvanometer. The deflections produced, d^ and d^, may be 
 considered as inversely proportional to the resistances, from 
 which 
 
 X=4^ 100000. 
 
SPECIFIC RESISTANCE 313 
 
 It is desirable to have the two deflections nearly equal ; to 
 secure this it will usually be necessary to shunt the galva- 
 nometer when it is used with the 100000 ohms. If this shunt 
 
 is such that the galvanometer sensitiveness is reduced to - of 
 
 n 
 
 that which it had in the first trial, then 
 A-=w^ 100000. 
 
 In these formulte the resistance of the battery and galvar 
 nometer are considered as inappreciable in comparison with the 
 high resistance being measured. This is clearly permissible for 
 the comparison with the insulation resistance, and in comparison 
 with 100000 ohms the resistance of the shunted galvanometer is 
 also very small. If it is not permissible to neglect the galva- 
 nometer and battery resistance, the following complete formula 
 may be used. Represent the high comparison resistance by B, 
 the shunt resistance by S, and the battery and galvanometer 
 resistances by B and G ; then the unknown resistance is 
 
 dX 
 
 B+G + B + ^^±^\-{B+G). 
 
 CXXXIV. SPECIFIC RESISTANCE BY THE COMPARISON OF 
 POTENTIALS 
 
 Determine the resistances of short pieces of copper and German silver 
 wire, and And their specific resistances. 
 
 235. Specific Resistance. — The resistance in ohms of a centi- 
 meter of length of a substance when the area of cross section 
 is one square centimeter and its temperature is 0°, is its specific 
 resistance. 
 
 If Bf is the measured resistance at the temperature t of the 
 substance in the form of a wire of length I and sectional area s, 
 and a is the temperature coefficient of the substance (Appendix, 
 
314 
 
 ELECTEICITY 
 
 Table 26), then the resistance B^, of the piece of wire at 0°, is 
 found from the relation Bf = i?g (1 + at), and the specific resists 
 ance p is determined by the equation 
 
 236. Resistance by Method of Comparison of Potentials. — The 
 
 method of comparison of potentials is applicable to the meas- 
 urement of very small resistances, such as the resistances of 
 short lengths of conducting wires. A standard resistance is 
 required whose value is nearly equal to that of the unknown 
 
 resistance. If no single stand- 
 ard of a sufficiently small value 
 is available, several standards 
 of higher value may be used 
 in parallel; or, if the unknown 
 resistance is a wire, a piece of 
 it may be selected such that 
 its resistance is approximately 
 equal to that of the standard. 
 
 Connect the unknown resist- 
 ance X, a standard resistance, B, 
 a rheostat, r, and a constant bat- 
 tery, B, in series. By means of 
 a suitable key (Art. 218), or 
 switch, arrange to connect a high-sensibility galvanometer, G, 
 to the terminals of either B or X, as indicated in Fig. 150. 
 
 The readings of the galvanometer when so connected can be 
 given a convenient value by altering r. Since the fall of 
 potential is proportional to the resistance, the ratio oi B to X 
 will be the same as that of the currents which will flow through 
 the galvanometer when it is connected first to B and then to X. 
 When the apparatus is arranged as described, these currents 
 will be very small and nearly equal, and it may be assumed 
 that they are proportional to the observed galvanometer deflec- 
 tions, c?j and d^ respectively. Then 
 
 B-.X^d^-.d^. 
 
 Fig. 150. Compakison op 
 Potentials 
 
THE LOW-EESISTANCE BRIDGE 
 
 315 
 
 237. The Low-Resistance Bridge. — A form of apparatus 
 often employed for measuring the resistance of wires and rods, 
 or irregular low resistances such as armature windings, consists 
 of a Wheatstone's bridge specially constructed so that the 
 resistances of the connections 
 may be eliminated by a method 
 of double readings. 
 
 The known resistance, J?j 
 (Fig. 151), is a straight wire 
 attached to a graduated scale 
 which shows the actual resist- 
 ance in ohms of different- 
 lengths of wire. The portion 
 of this wire which is used in 
 balancing the bridge is deter- 
 mined by a sliding contact, A^. 
 The unknown resistance, X, if 
 a wire or rod, is stretched be- 
 tween two clamps, and the 
 length of this used in balancing 
 is limited by a movable contact, 
 Cj, and is measured by a centi- 
 meter scale. The ratio arms 
 are two resistances, S^ and JR^, 
 arranged to give various ratios. 
 The galvanometer and battery are connected as indicated. With 
 the contact on the unknown resistance at Cj, and with a suitable 
 ratio between Bg and B^, move the sliding contact ^j until a 
 balance is secured, indicated by no deflection of the galvanometer 
 when the bridge connection is made. Then 
 
 A^B : DC^ = ^3 : E^. 
 
 Move the contact on the unknown resistance to C^. Adjust 
 the slider to secure a balance, and if A^ is its new position, 
 
 from which it follows that 
 
 Fig. 151. Low-Eesistance Bridge 
 
 AA 
 
 c^c^ 
 
 B„: R. 
 
316 ELECTKICITY 
 
 The resistance A-^A^ is read directly from the graduated scale, 
 and this multiplied by the ratio of B^ to B^, usually a power of 
 10, gives the resistance of the known length of the substance 
 being investigated. 
 
 Any irregular resistance may be connected to the clamps, 
 and the contacts Cj and Cj may be made to these clamps, or 
 to other points between which the resistance is required. 
 
 A slide-meter bridge may be used for ^^ if the resistance of 
 its wire is known, and two resistance boxes will serve for ^3 and 
 B^, making the method one easily applied with simple apparatus. 
 
 CXXXV. TEMPERATURE COEFFICIENT OF RESISTANCE BY 
 FOSTER'S METHOD 
 
 Determine the constants of an ohm coil, that is, its resistance at 0° and its 
 temperature coefficient. 
 
 238. Temperature Coefficient of Resistance. — Standard resist- 
 ance coils are usually constructed, as described in Art. 212, 
 so that they may be surrounded with ice, water, or oil baths, to 
 give them definite known temperatures. Let the values of the 
 resistance whose coefficient is desired be determined as described 
 in Art. 239, when it is at two temperatures differing consider- 
 ably, at 10° and 40°, for example. Care must be taken that all 
 other parts of the apparatus remain at a constant temperature. 
 Let i?j and B^ be the observed resistances at temperatures t^ and 
 t^, and J?o its resistance at 0°, and a the temperature coefficient; 
 
 then Bj^ = i^o (1 + at^), 
 
 and -Eg = Bq(1 + at^) ; 
 
 , Be, — -K, 
 
 whence a = — — 
 
 R\h ~ R-ih 
 
 239. Carey Foster's Method for Comparison of Resistances. — 
 
 This method is adapted to the accurate comparison of two nearly 
 equal resistances, and it is very convenient for determining 
 the variations in the resistance of a coil due to changes in its 
 
CAREY FOSTER'S METHOD 
 
 317 
 
 temperature. The principle of the method is that of Wheat- 
 stone's bridge, modified so that, by two observations, the difference 
 between two resistances is determined in terms of the slide wire, 
 the resistance of all connections being eliminated. It is analogous 
 to the comparison of masses by the method of double weighing, 
 and, like this method, it is of the highest precision. 
 
 Designate the coil to be tested by X, a standard coil of the 
 same nominal value by S, and two auxiliary coils whose resist- 
 ances are each approximately equal to S, by iJj and B^. Con- 
 nect these four coils, a galvanometer, a battery, and a suitable 
 
 Fig. 152. The Carev Foster Method 
 
 bridge apparatus, as indicated in Fig. 152. Adjust the bridge 
 contact until a perfect balance is secured. Since the four resists 
 ances are approximately of the same value, the contact will be 
 near the middle of the wire (Art. 208). Interchange the resist- 
 ances S and X. If these are exactly equal, the balance will not 
 be disturbed ; if they are not equal, it will be necessary, in order 
 to restore the balance, to move the contact in the direction of 
 the greater resistance just so much that the resistance of the 
 short length of the slide wire between the two points of contact 
 is equal to the difference in the resistances of the standard, S, 
 and the unknown, X. Determine this length of wire by observa- 
 tion, and find its value in ohms by the aid of the methods of 
 the following article. 
 
318 ELECTRICITY 
 
 For conveniently interchanging the resistances, and for accu- 
 racy in the measurements by this method, elaborately constructed 
 bridges are often provided. 
 
 240. Resistance of the Slide Wire. — Usually the direct obser- 
 vations give the difference between the resistances compared in 
 terms of a length of the slide wire, and it is necessary to ascer- 
 tain the resistance, />, of the unit length of the wire, in order to 
 express this difference in ohms. 
 
 Method I. — If a single standard resistance is available, the 
 resistance of which is less than that of the slide wire, this 
 standard may be balanced against a heavy copper bar of inap- 
 preciable resistance put in place of X, as described in the pre- 
 vious article. By interchanging and again balancing, a length 
 of the slide wire whose resistance is equal to that of the standard 
 is determined. From this is deduced the resistance, p, of unit 
 length of the wire. 
 
 If no single standard of sufficiently small value is available, 
 two one-ohm coils and one ten-ohm coil may be used. Place 
 the ten- and one-ohm coils in parallel, on one side of the bridge, 
 and the other unit coil on the opposite side, and proceed as 
 before. The combined resistance of the pair of coils is 10 X 1 
 -i- 10 -|- 1 = 0.90909 ohms, giving for the difference between 
 the interchanged resistances, 0.09091 ohms. Other similar 
 combinations will suggest themselves if they are available 
 and more convenient. 
 
 Method U. — In case the only available standard has a resist- 
 ance greater than that of the slide wire, the following method, 
 requiring two complete observations, may be useful. 
 
 Let Q be any convenient but unknown resistance, a piece of 
 wire for example, whose resistance is less than that of the slide 
 wire. Balance this against the negligible copper bar resistance, 
 and by interchanging determine the length, Zj, of the slide wire 
 whose resistance is the same as that of Q\ then 
 
 Q = ply 
 
 Place the standard S in parallel with §, and balance the com- 
 bination, in two positions, against the negligible resistance, as 
 
CALIBRATION OF BRIDGE WIRE 319 
 
 before. Then if l^ is the length of the slide wire between the 
 two points of contact, 
 
 and p = s^' ~ ^" 
 
 tjig 
 
 Method m. — The resistance of the slide wire may be meas- 
 ured directly with an independent Wheatstone's bridge. (See 
 Arts. 208 and 237.) 
 
 Calibration of Slide Wire. — It has been assumed that the 
 shde wire is of uniform resistance throughout, and usually the 
 errors from such assumption are very small. But where extreme 
 accuracy is required, it becomes necessary to calibrate the wire 
 (Art. 241). 
 
 CXXXVI. ERRORS OF A BRIDGE WIRE BY BARUS'S 
 METHOD 
 
 Make a ten-part calibration of the wire of a Wheatstone's bridge. 
 
 241. Calibration of Bridge Wire. — Barus's method for cali- 
 brating a bridge wire is an adaptation of the Gay-Lussac method 
 of simple calibration, described in Art. 26. To use this method 
 there must be provided as many separate coils of approximately 
 equal resistance as there are parts in the proposed calibration ; 
 for instance, ten coils of one ohm each. Each coil should have 
 terminals arranged to dip into mercury cups for making connec- 
 tions. For the purposes of this problem provide a series of 
 eleven mercury cups so spaced that the ten coils may be con- 
 nected in series as indicated in Fig. 153. Designate the iirst 
 coil the calibrating coil, K, and the other coils by the numbers 
 II, III, IV, etc. Connect the series of coils to the bridge wire 
 by means of heavy copper wires or bars dipping into the two 
 end cups ; connect a cell of constant battery as indicated. A 
 sensitive galvanometer has one terminal joined to the first mer- 
 cury cup, and the other terminal to the slide contact. The 
 current flows from Ato C through the two branches of the circuit. 
 
320 
 
 ELECTRICITY 
 
 the series of ten coils, and the bridge wire. As explained in 
 Art. 207, corresponding to the potential of any point in one 
 branch, there is exactly the same potential at some point in 
 the other branch. Theoretically, then, it ought to be possible 
 to place the slide contact at a point, p^, having a potential 
 equal to that of cup 1, in which case no current will flow 
 through the galvanometer. The construction of the slider will 
 probably render it impossible to find this point, and it may be 
 assumed to be at the very beginning of the wire. Take the 
 galvanometer wire out of cup 1 and place it in cup 2, and find 
 a contact p^ such that there is no deflection of the galvanometer. 
 
 5 
 
 ii|iiM|iiii{iin|iiii|iiii|iiii|iiii|iiii|iiii|iiii|iiii|iiii|iiiiniii|iiii|iiii|iiii|iiii|iii 
 
 Fig. 153. Caweeation of Bkidge Wire 
 
 The potential of p^ is the same as that of cup 2. It then fol- 
 lows that the resistance of that portion of the bridge wire between 
 Pi and p^ is in the same ratio to the resistance of the whole wire 
 as the resistance of K is to the resistance of the series of ten 
 coils.' Now interchange coils K and II, and find a new contact 
 point, p^, giving no deflection while the galvanometer connection 
 is in cup 2, and find a similar point, p^, when the galvanometer 
 is connected to cup 3. The resistance of the portion of the 
 wire between pg and p^ must be equal to that between p-^ and p^. 
 Point pg will probably be near to p^, but will not coincide with 
 it. Interchange coils K and III, and find new contacts, p^ and 
 Pel giving no deflection when the galvanometer is connected to 
 cups 3 and 4, and continue the operation till the coil ^occupies 
 
CALIBEATION OF BRIDGE WIRE 
 
 321 
 
 the place of Z, and the last contact is the end of the wire. Thus 
 there will have been found ten successive portions of the wire, 
 
 h=P^-Pv h=Pi-Ps' h='P&-Pf.^ etc., 
 each having exactly the same resistance. 
 
 If the bridge wire were of perfectly uniform resistance through- 
 out, these observed lengths would aU. be equal. Otherwise, the 
 correction to each portion of the wire is the quantity which 
 must be added to it to make it equal to the average of the 
 observed lengths ; and the correction at any point of the wire 
 is the sum of the corrections to the several sections between 
 the point and the beginning of the wire. These corrections 
 being applied to any observed length of the wire, the length 
 becomes strictly proportional to the resistance of the wire. 
 
 The method can be employed for a calibration in any number 
 of equal parts other than ten. 
 
 Compare this description with that for the calibration of 
 scales and thermometers, Arts. 26-30 and 129. 
 
 An example of the calibration of a bridge wire in ten sec- 
 tions is given. 
 
 Calibration op Bridge Wire 
 
 by bakus's method 
 
 Hartmann & Bbaitn Wheatstone's Bridge No. 39 
 
 MarcTi 1, 1900 
 
 Position 
 OF Coil. K 
 
 Readings 
 
 Section 
 I 
 
 
 
 Section 
 
 COBREC- 
 TION 
 
 Point 
 
 COKKECTION 
 
 Pn 
 
 Pn+1 
 
 Kx 
 
 0.00 cm 
 
 9.89 cm 
 
 9.89 cm 
 
 0-10 
 
 + 0.073 cm 
 
 Oom 
 
 0.000 cm 
 
 K\ 
 
 9.90 
 
 19.89 
 
 9.99 
 
 10-20 
 
 -0.027 
 
 10 
 
 + 0.073 
 
 A-, 
 
 20.00 
 
 20.99 
 
 9.99 
 
 20-30 
 
 -0.027 
 
 20 
 
 + 0.046 
 
 X, 
 
 30.12 
 
 40.10 
 
 9.98 
 
 30-40 
 
 -0.017 
 
 30 
 
 + 0.019 
 
 Kr. 
 
 40.19 
 
 50.20 
 
 10.01 
 
 40-50 
 
 -0.047 
 
 40 
 
 + 0.002 
 
 Jr« 
 
 49.92 
 
 59.90 
 
 9.98 
 
 50-60 
 
 -0.017 
 
 50 
 
 -0.045 
 
 K, 
 
 59.91 
 
 69.90 
 
 9.99 
 
 60-70 
 
 -0.027 
 
 60 
 
 - 0.062 
 
 X, 
 
 70.00 
 
 79,93 
 
 9.93 
 
 70-80 
 
 + 0.033 
 
 70 
 
 -0.089 
 
 K„ 
 
 80.06 
 
 90.05 
 
 9.99 
 
 80-90 
 
 -0.027 
 
 80 
 
 -0.056 
 
 ^1. 
 
 90.12 
 
 100,00 
 
 9.88 
 9.963 
 
 90-100 
 
 + 0.083 
 
 90 
 100 
 
 -0.083 
 0,000 
 
 
CHAPTER XXI 
 
 CURRENT STRENGTH 
 
 CXXXVII. GALVANOMETER CONSTANT WITH THE 
 VOLTAMETER 
 
 Determine the constant of a tangent galvanometer by the electrolytic 
 • deposition of copper. 
 
 242. Constant of Tangent Galvanometer. — The constant of a 
 tangent galvanometer (Art. 216) is the factor K, which when 
 multiplied by the tangent of the angle of deflection, 6, produced 
 by a current, J, gives the measure of that current ; that is, 
 
 J=^tan^. 
 
 To determine the constant it is only necessary to observe the 
 deflection produced by any known current. The strength of 
 the current can be most conveniently measured by the work it 
 will do in a specially arranged electrolytic cell called a voltam- 
 eter. The most useful forms are those in which the amounts 
 of oxygen and hydrogen gas liberated in the water voltameter 
 are measured, or in which the amounts of silver or copper depos- 
 ited from solutions of these metals are ascertained. The silver 
 voltameter is the most accurate and is best suited for small cur- 
 rents ; for large currents the copper voltameter is sufficient. 
 
 243. The Copper Voltameter. — Prepare a nonsaturated solu- 
 tion of pure copper sulphate by dissolving 1 g of the crystals in 
 each 4 com of distilled water. The solution should have a 
 density of about 1.14. Provide an anode of pure copper and a 
 cathode of platinum or copper. The latter must be carefully 
 cleaned, dried, and weighed. 
 
 322 
 
STAJ^DARD AMPERE 323 
 
 Adjust the galvanometer to the magnetic meridian (Art. 216), 
 
 and arrange to pass a current through it and the voltameter in 
 
 series. Notice that the connections ai-e such as to make the 
 
 prepared plate the cathode. A rheostat may be included in the 
 
 circuit (Fig. 154) to regulate the current, which should not 
 
 exceed 1 ampere for each 25 square cm of cathode area. The 
 
 deflection of the galvanometer should not 
 
 be less than 30° nor greater than 60°. 
 
 Allow the current to pass for from a half 
 
 hour to two hours, and note the exact time 
 
 in seconds. The galvanometer should be 
 
 read frequently during this time, — every 
 
 five minutes perhaps, — to detect any 
 
 unsteadiness of the current. Adjust the 
 
 rheostat, if necessary, to keep the current 
 
 constant. The deflection of the galva^ 
 
 , , , , I- . , Fig. 154. Constant of 
 
 nometer is measured by the average of the Galvanometer 
 
 four readings given by the two ends of 
 
 the pointer when the current flows in one direction, and when it is 
 
 reversed (through the galvanometer only) by the commutator K. 
 
 At the end of the experiment the cathode is removed and 
 again carefully washed, dried, and weighed. 
 
 If w grams of copper are deposited in a copper voltameter in 
 t seconds, the current flowing is, in amperes, 
 
 0.0003294 t 
 
 244. Standard International Ampere; Silver Voltameter. — 
 
 The ampere is that current strength which, in a specified vol- 
 tameter, deposits 0.001118 g of silver in one second. The spe- 
 cifications for the voltameter are given in the references. The 
 silver voltameter is employed when the greatest accuracy is 
 desired, and especially for currents of less than one ampere. It 
 may be used to calibrate a standard ammeter. 
 
 Reference. — Carhart and Patterson, Electrical Measurements, pp. 158 and 
 328. 
 
324 
 
 ELECTRICITY 
 
 CXXXVm. JOULE'S EQUIVALENT WITH THE ELECTRO- 
 CALORIMETER 
 
 Determine the mechanical equivalent of heat with the electrooalorimeter. 
 
 245. Joule's Equivalent ; the Joule ; the Watt. — J-oule's 
 equivalent of heat (Arts. 139 and 150) must be distinguished 
 from the unit of energy, the joule. The joule is 10^ ergs, and 
 Joule's equivalent is 4.2 joules. The joule is defined electric- 
 ally as the quantity of work done in the transference of a 
 quantity of electricity of 1 coulomb under a pressure of 1 volt ; 
 it is also equal to the work done in 1 second by a current of 
 1 ampere passing through a resistance of 1 ohm. 
 
 The joule is the quantity of work equal to that defined, 
 regardless of the time in which it is performed. The power to 
 do this much work in one second — that is, to do a joule of work 
 per second — is the watt. From the relation of the various units 
 
 it follows that the power of a given 
 current in watts is measured by the 
 product of the current strength in 
 amperes and the pressure in volts. 
 Joule's equivalent of the unit of 
 heat may be determined with the 
 electrooalorimeter, which consists 
 of a coil of wire so placed in a water 
 calorimeter that the heat developed 
 by the passage of a current through 
 the coil can be measured. Fig. 155 
 shows a simple form of such a calo- 
 rimeter, while a more elaborate 
 form is shown in Fig. 71. 
 
 Weigh the calorimeter cup and 
 stirrer, and measure the thermometer; fill the cup with water, 
 cooled 10° below the air temperature, to such a depth that the 
 coil will be wholly immersed when it is in position, and find 
 the weight of the water. Let K be the heat capacity of the 
 calorimeter, determined as explained in Art. 140. 
 
 Fig. 155. Electrooalorimeter 
 
FIGURE OF MEEIT 325 
 
 Measure the resistance of the coil when it is in place. 
 
 Observe the temperature of the calorimeter. Connect the 
 current through a rheostat and an ammeter to the coil terminals, 
 and allow it to flow until the temperature is as much above that 
 of the room as at the beginning it was below this temperature. 
 The current strength, which may be about five amperes, should 
 be kept constant, and its average value, /, must be measured 
 as accurately as possible. Note carefully the temperature and 
 time when the current is started, and the temperature and time 
 at the end of the experiment. 
 
 Again measure the resistance of the coil, and let B be the 
 mean of its resistance before and after heating. 
 
 If t^— tj is the temperature change in the calorimeter while 
 the current is passing, the heat developed is 
 
 H=K{t^-t^). 
 
 The current, /, being expressed in amperes, the resistance, R, 
 in ohms, and the time, T, during which the current passes, in 
 seconds, the value of Joule's equivalent in C. G. S. units is 
 
 H 
 
 By assuming the value of J, the method may be used to 
 determine the current strength or the resistance. 
 
 CXXXIX. FIGURE OF MERIT OF A GALVANOMETER BY DIRECT 
 
 DEFLECTION 
 
 Determine the figure of merit of a galvanometer, including the measure- 
 ment of the battery and galvanometer resistances. 
 
 246. Figure of Merit of a Galvanometer. — The figure of 
 merit of a galvanometer is a statement of its sensitiveness, and 
 is usually expressed by the fraction of an ampere of current 
 that would cause an apparent deflection, in a reflecting galva- 
 nometer, of 1 mm on a scale placed 1 m from the mirror. It is 
 also sometimes expressed by stating the number of ohms resist- 
 ance through which an electromotive force of 1 volt would have 
 
326 
 
 ELECTRICITY 
 
 to be passed in order that the current shall be so reduced as to 
 produce 1 mm deflection on a scale 1 m distant from the mirror. 
 Connect the galvanometer in series with a cell of known 
 electromotive force and a high resistance. Adjust the resist- 
 ance to secure a readable deflection, preferably small. From 
 the known electromotive force and the resistance of the circuit, 
 including that of the galvanometer and battery, the current 
 strength can be calculated. This in connection with the 
 
 observed deflection enables the 
 calculation of the figure of merit 
 as defined. 
 
 If the galvanometer is a sensi- 
 tive one, it will not be convenient 
 to put a sufficiently high resistance 
 in series with it to keep the deflec- 
 tion within readable limits. It will 
 then be necessary to shunt the 
 galvanometer (Art. 247). Connect the galvanometer, shunt, 
 resistance, battery, and key, as indicated in Fig. 156. Let G, 
 S, B, and B represent the resistances of the several parts of the 
 apparatus, and B the electromotive force of the battery. Then 
 the current through the battery is 
 
 Fig. 156. Figuke of Mekit 
 
 /„ = . 
 
 B 
 
 B + B + 
 
 GS 
 G+ S 
 
 and that part of the current which flows through the galva- 
 nometer is 
 
 B S 
 
 Ia = - 
 
 B-\-B + 
 
 GS G+ S 
 
 If 
 
 -B, 
 
 -( 
 
 B + B + 
 
 G+ S 
 
 GS \G + S 
 
 G + SJ S 
 
 la 
 
 B 
 
FIGUEE OF MERIT 327 
 
 The quantity By may be called the virtual resistance ; it is the 
 resistance through which the given electromotive force would 
 produce a current equal to that which actually passes through 
 the galvanometer. 
 
 If the observed deflection of the needle is d centimeters on a 
 scale D centimeters from the mirror, the deflection expressed in 
 millimeters for a scale at a distance of one meter is 
 
 ^ 10 d 1000 i 
 
 A = = ; 
 
 _D_ D 
 
 100 
 and finally the figure of merit is 
 
 A ^,/l000^" 
 
 A Daniell cell is usually sufficient for this work, though a 
 standard ceU may be used. The galvanometer resistance may 
 be determined by Thompson's method (Art. 227) or otherwise, 
 and the battery resistance by Kohlrausch's method (Art. 231). 
 Usually, however, the resistance of the battery and of the 
 shunted galvanometer will be small, and may be neglected in 
 comparison with the resistance R. 
 
 A numerical example follows. 
 
 Figure of Merit 
 
 Queen D'ArsonTal Ballistic Galvanometer No. 92 
 
 May SO, 1897 
 
 Clark Standard Cell, H. & B. No. 100, at 19° . E= 1.43 volts 
 
 By the telephone bridge the resistance of cell . B = 20. ohms 
 
 Galvanometer resistance G= 1039.5 
 
 Series resistance ^ = 22000. 
 
 Shunt resistance •*>' = 2- 
 
 / „„ 2 X 1039.5 \ 1039.5 + 2 ,, -„-„«« i. 
 Rjr= (22000 + 20 + ^^^ggg^^ ) 2 " ^^ *^^°°*^ °^"''- 
 
328 
 
 ELECTRICITY 
 
 The following steady deflection readings were obtained with the scale 
 distance D = 100 cm. 
 
 Zero Beading 
 
 Second Keading 
 
 - Deflection 
 
 38.85 cm 
 
 56.35 cm 
 
 17.50 cm 
 
 39.30 
 
 56.60 
 
 17.30 
 
 39.33 
 
 56.60 
 
 17.27 
 
 39.33 
 
 56.68 
 
 17.85 
 
 89.20 
 
 56.40 
 
 17.20 
 
 39.30 
 
 56.40 
 
 17.10 
 
 39.30 
 
 56.43 
 
 17.13 
 
 89.30 
 
 56.40 
 
 17.10 
 
 39.30 
 
 56.40 
 
 17.10 
 
 39.35 
 
 56.45 
 
 17.10 
 
 
 Mean 
 
 d= 17.22 cm 
 
 F = 
 
 1.43 X 100 
 
 11 465000 X 1000 X 17.22 
 
 = 0.000000 00072 amperes. 
 
 247. Sensitiveness of Galvanometers ; Shunts. — A suspended 
 magnet galvanometer is made more sensitive by using a longer 
 suspension fiber, or one with less torsion. A fiber of quartz is 
 used for the most delicate work. The sensitiveness also varies 
 with the number of turns of wire in the coils and with their 
 nearness to the needle. The coils are often subdivided, are 
 made interchangeable, and adjustable as to distance, as shown 
 in Fig. 134. The sensitiveness will be increased by diminish- 
 ing the effect of the earth's field, through astatization or a con- 
 trol magnet (Art. 210), or by surrounding the galvanometer 
 with an iron shield. 
 
 The sensitiveness of a suspended coil galvanometer depends 
 upon its suspension, the number of turns of wire in the coil, and 
 the strength of the field magnets. It is nearly independent of 
 the earth's field. Practically the sensitiveness of this type of 
 instrument is altered by interchanging the entire suspended sys- 
 tem, the construction often making this a convenient operation. 
 
SHUNTS 329 
 
 Galvanometers such as are commonly used for laboratory work 
 have figures of merit ranging from 0.000000 3 to 0.000000 0002. 
 
 If a less sensitive arrangement of the galvanometer is desired, 
 it may be secured by using fewer turns of wire, either employ- 
 ing only a part of the windings already provided or substituting 
 new windings. The two coils may be connected differentially 
 (Art. 219) so that one portion only partially neutralizes the other. 
 Often the most convenient method is to use a shunt, — that is, 
 to connect the two galvanometer terminals by a small resistance. 
 
 A shunt does not reduce the sensitiveness proper of a gal- 
 vanometer, but it enables the instrument to be used for meas- 
 uring larger currents, since the shunt carries the greater part 
 of the current, and only the smaller part passes through the 
 galvanometer. If the ratio between the two parts is known, 
 and that part through the galvanometer is measured, the total 
 current may be determined. The current divides between the 
 galvanometer and shunt in the inverse ratio of their resistances. 
 If Is and Iq represent the currents through the shunt and gal- 
 vanometer, and S and G are their respective resistances, then 
 
 and 
 
 Is + Ia S+G, 
 
 but Is + Ig = I, the total current, and hence 
 
 S±G 
 I-Ig- ^ 
 
 li S=\ G, the current through G is J^ of the total current, 
 
 G 
 
 and similarly for other values of S. — — — is the multiplying 
 
 power of the shunt. 
 
 The actual resistance through the galvanometer and shunt 
 combined is found from the conductivity of the combination, 
 which is equal to the sum of the separate conductivities. This 
 conductivity is \ 1 8+ G 
 
 ^''''^7i^7s^~G^' 
 
330 ELECTRICITY 
 
 The resistance of the combination is equal to the reciprocal of 
 the conductivity ; that is, 
 
 _ 1 ^ GS 
 
 "' Kas S+G' 
 
 The resistance of any number of multiple circuits is found in 
 the same manner. 
 
 CXL. CURRENT STRENGTH WITH THE ELECTRO- 
 DYNAMOMETER 
 
 Calibrate an electrodynamometer and measure with it the strength of an 
 alternating current. 
 
 248. The Electrodynamometer. — An electrodynamometer is 
 an instrument for measuring the strength of a current by means 
 of the attractions and repulsions between a fixed and a movable 
 conductor carrying this current. The conductors usually con- 
 sist of flat rectangular coils of wire, the movable one being sus- 
 pended with its plane normally at right angles to that of the fixed 
 coil. The suspension is a silk thread and a spiral spring to give 
 increased torsion. The support of this spring can be turned 
 through a measurable angle to increase the torsidn at will. 
 
 In use the instrument is to be set up with the plane of the 
 fixed coil in the magnetic meridian, and with the movable coil 
 swinging freely. The torsion head is adjusted so that the 
 index of the movable coil is at 0, — that is, until the two coils 
 are at right angles. The current to be measured is then sent 
 through the coils, which will tend to rotate the movable coil. 
 This tendency is counteracted by turning the torsion head of 
 the spring to maintain the coil in its initial position. 
 
 If 6 is the twist of the spring and K the constant of the 
 instrument, the current strength is 
 
 I=KVe. 
 
 The instrumental constant may be determined by the copper 
 voltameter in a manner similar to that described for the tangent 
 galvanometer in Art. 242. 
 
THE ELECTEODYNAMOMETER 331 
 
 The electrodynamometer is particularly adapted to the meas- 
 urement of alternating currents, since a current in either 
 direction tends to turn the coil in one direction. 
 
 Reference. — Carhart and Patterson, Electrical Measurements, pp. 127 
 and 167. 
 
CHAPTER XXII 
 ELECTROMOTIVE FORCE 
 
 CXLI. ELECTROMOTIVE FORCE BY COMPENSATION 
 
 Compare the electromotive forces of several cells with that of a standard 
 cell, using the compensation method. 
 
 249. Standard International Volt; Standard Cells. — The inter- 
 national volt is that potential difference which will produce a 
 current strength of one ampere (Art. 2i4) through a resistance 
 of one ohm (Art. 212). It is impossible to produce directly an 
 electromotive force of exaxjtly one volt ; it is therefore necessary 
 to define it as a certain fraction of the electromotive force of a 
 standard cell. A standard cell must be capable of reproduction, 
 and must be constant. 
 
 The Clark Standard Cell is the almost universal standard of 
 electromotive force. It has mercury for its positive electrode, 
 and amalgamated zinc for the negative; the electrolyte is a 
 saturated solution of mercurous sulphate and zinc sulphate. 
 Minute specifications and notes for the preparation of standard 
 cells are given in the reference. The form of the Carhart- 
 Clark portable cell is shown in Fig. 157. The materials are 
 sealed in a small test tube, and the whole is mounted in a 
 protecting case which often contains a thermometer. 
 
 The electromotive force of the Clark cell at t° is 
 
 E^ = 1.4292 - 0.00123 {t - 18) - 0.000007 {t - 18)2 yolts. 
 
 At the usual laboratory temperatures this cell has the electro- 
 motive force, in volts, as given in this table. 
 
 I 
 
 15° 
 
 18° 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 25° 
 
 Ec 
 
 1.4328 
 
 1.4292 
 
 1.4267 
 
 1.4255 
 332 
 
 1.4242 
 
 1.4229 
 
 1.4216 
 
 1.4202 
 
COMPENSATION METHOD 
 
 333 
 
 The Weston Standard Cell is similar to the Clark cell except 
 that cadmium and cadmium sulphate are substituted for zinc 
 and zinc sulphate. It has the advantages aver the Clark cell 
 of a temperature coefficient which 
 is practically zero, and in having 
 an electromotive force which is 
 nearly one volt. If it is made with 
 a constant solution of cadmium 
 sulphate which is saturated at 4°, 
 its electromotive force at the usual 
 laboratory temperatures is 
 
 ^,,.= 1.0190 volts. 
 
 That the electromotive force of 
 a standard cell may remain constant 
 the cell must be used with great 
 care and only in open circuit or 
 compensation methods. To pre- 
 vent polarization and consequent 
 deterioration, the current taken 
 from such a cell should never 
 exceed 
 
 ^■jf-^-jj-g- of an ampere; that 
 is, it should not be used directly in 
 a circuit of less than 30000 ohms 
 resistance. If the cell is of small 
 size, a much higher safety resistance may be required. 
 
 Reference. — Carhart and Patterson, Electrical Measurements, pp. 176-186, 
 330-.335. 
 
 250. Electromotive Force by Compensation. — One of the most 
 useful methods for the comparison of electromotive forces, the 
 compensation method, may be described as follows. Connect 
 two similar resistance boxes, B and R' (Fig. 158), each of 10000 
 ohms or more, in series with a constant battery. The battery 
 may consist of one or more cells, such that its electromotive 
 force shall be somewhat in excess of that of any cell to be 
 measured. The total resistance in use in the two boxes between 
 A and B is always to be equal to the total resistance of one box. 
 
 Fig. 157. Carhaet-Clakk 
 Standard Cell 
 
334 ELECTEICITY 
 
 The battery B will maintain a constant potential difference, e, 
 between these points, while the fall of potential between A and 
 D will be proportional to that part of the total resistance which 
 is in E. Connect one of the cells to be tested, a sensitive 
 galvanometer, a safety high resistance (100000 ohms, for 
 instance), and a key, in series, as a shunt to the resistance B, as 
 indicated. The cell must be so placed that its electromotive 
 force will oppose that of the battery B. Keepirig the sum of the 
 resistances in use in R and B' always the same, so vary that 
 
 K— .^^ part of the total resistance 
 
 Gj Q I — >^ which is in B that upon 
 
 momentarily closing the 
 key the galvanometer shows 
 no deflection. If the two 
 resistance boxes are of the 
 same pattern, the adjust- 
 ment is facilitated by noting 
 that when a plug is removed 
 
 _ ,,Q „ ,, from any place in one box 
 
 Fig. 158. Compensation Method •' ^ 
 
 a plug is always put into 
 the corresponding place in the other box. If the arrangement 
 is not sufficiently sensitive, the resistance Q may be removed 
 after compensation has been approximately secured, and the 
 adjustment is then perfected. The resistance Q must be 
 replaced before beginning measurements for a second cell. 
 
 When the adjustment is completed, the electromotive force 
 tested, Ey is compensated by the potential difference between A 
 and D ; and if R^ is the resistance used in R, and B^i that in R\ 
 
 E^:e = R^:R-^ + R^. 
 
 Substitute for E^ a second cell of electromotive force E^, and 
 adjust as before; if R^ is the resistance in R necessary for 
 compensation, 
 
 E^:e = R^:R^ + R^; 
 
 and since R^ + R^ = R^ + R^, 
 
 E-^: E^ = R^: B^. 
 
THE POTENTIOMETEE 
 
 335 
 
 251. The Potentiometer With an especially arranged resist- 
 ance box, called a compensation apparatus or potentiometer, 
 the method described above for comparing electromotive forces 
 becomes very convenient, and is perhaps the most precise method 
 known. The resistance box is constructed so that the resist- 
 ance in one circuit remains constant while that in the branch 
 circuit is varied through all possible values. 
 
 One of the most convenient forms of compensation apparatus 
 has its parts arranged as shown in Fig. 159, its total resistance 
 being 14999.9 ohms. The following order of adjustment 
 may be followed. The standard cell, a Clark element for 
 instance, is connected at S; the unknown electromotive force, 
 
 Fig. 159. The Potentiometer 
 
 which may not exceed 1.5 volts, is connected at X; a galvanom- 
 eter at G; and a battery in series with a resistance of 5000 
 ohms or more is connected as shown at B and P. This battery 
 must have an electromotive force somewhat greater than 1.5 
 volts; it may be a single accumulator cell, or two gravity or 
 Leclanche cells. The several switches are to be set to indicate 
 the significant figures of the electromotive force of the standard 
 cell. If the Clark cell is at the temperature 19°, for example, 
 its electromotive force is 1.4279. Set the switch of the thou- 
 sands row at 14, the hundreds switch at 2, the tens at 7, the 
 units at 9, and the tenths switch at 0. Then the resistance of 
 
336 ELECTRICITY 
 
 that part of the main circuit to which the galvanometer is con- 
 nected is 14279.0 ohms. Turn the switch B to 100000, which 
 places a safety resistance of 100000 ohms in the circuit of the 
 standard cell corresponding to Q of Fig. 158. Now adjust the 
 resistance P so that upon closing the key K there will be no 
 deflection of the galvanometer. Turn Z) to 10000 and improve 
 the adjustment of P if required ; and finally turn B to 0, which 
 cuts out the safety resistance, and perfect the adjustment of P 
 till no deflection results from closing the key. 
 
 This makes the instrument direct reading, in that the number 
 of ohms required for compensation is exactly ten thousand times 
 the electromotive force of the cell attached, the standard in 
 this case. 
 
 Turn the switch B back to 100000, and turn the switch C 
 to X, which places the unknown cell in the compensation cir- 
 cuit. By adjusting the resistance levers of the potentiometer, 
 place them so that again there is no deflection when the key is 
 closed; turn B to 10000, and improve the adjustment, and 
 finally turn B to 0, and perfect it. The reading of the various 
 dials of the potentiometer is then ten thousand times E, the 
 electromotive force of the cell X being tested. 
 
 The potentiometer may be used without the extra resistance 
 P, but it is then not direct reading, and the proportion of the 
 previous article is to be used in computing the unknown elec- 
 tromotive force. In this case also the unknown electromotive 
 force may have any value not exceeding that of the battery B. 
 
 CXLII. ELECTROMOTIVE FORCE WITH A CONDENSER 
 
 Compare the electromotive forces of several cells with that of a standard 
 cell by means of a condenser. 
 
 252. Comparison of Electromotive Forces with the Condenser. 
 
 — If a condenser (Art. 262) is charged from several different 
 sources, the charges received will be proportional to the electro- 
 motive forces of the sources. By measuring these charges the 
 electromotive forces may be compared. This method has the 
 
THE BALLISTIC GALVANOMETER 337 
 
 advantage that the cells are used only on open circuit, thus 
 avoiding polarization. 
 
 Connect a ballistic galvanometer, G (Art. 253), a condenser, 
 C, a charge and discharge key, K (Art. 263), and one of the 
 cells to be tested, B, as indicated in Fig. 160. The key being 
 in charge position, the condenser will receive a charge propor- 
 tional to the electromotive force, E., i 
 of the cell. By pressing the key ( ' J I ^ 
 firmly, but for a very short time, c 
 
 upon the galvanometer contact, the "^ X 
 
 condenser vsrill discharge through ^-^ 
 
 the galvanometer, and wrill deflect ^ (p ) ■ — ^ 
 
 the needle through an angle, 9,, 
 
 I, u-u i. • . 1 a ■ ^- \ } ^«^- ISO- Condenser Method 
 
 such that sin iV^ is proportional to 
 
 the charge. By substituting for the cell first used a second one 
 of electromotive force E^, a deflection, 0^, will be obtained. If 
 the distance of the scale from the mirror is D and the two 
 observed scale readings are d^ and d^, 
 
 E^:E^ = sin \ 6^ : sin \0^ = sin \ tan"' - : sin ^ tan"' ^ . 
 
 If the angular deflections of the needle do not exceed 6°, the 
 sines and tangents may be taken as proportional to the corre- 
 sponding arcs, and then 
 
 I^\'- E^ = d^: c?2- 
 
 Any number of cells being thus compared, if one of them 
 is a standard, the electromotive forces of the others may be 
 determined. 
 
 (Caution Do not allow the cells to be short circuited nor to be directly- 
 connected to the galvanometer.) 
 
 253. The Ballistic Galvanometer A ballistic galvanometer 
 
 is one designed to measure the total quantity of electricity pass- 
 ing through it in a current of very short duration. The needle 
 must have a period so long that the current will have ceased 
 before it has moved appreciably from its position of rest. The 
 needle should have the least possible damping. The quantity of 
 
338 
 
 ELECTRICITY 
 
 3 
 
 electricity thus passing through the galvanometer is proportional 
 to the sine of half the angle of the first swing of the needle. 
 
 The deflections of the needle are usually measured by observ- 
 ing with a telescope a centimeter scale as seen reflected in a 
 mirror attached to the needle. The following adjustments are 
 essential. If the galvanometer is of the suspended magnet type 
 (Art. 209), the plane of the coils must 
 be in the plane of the magnetic meridian ; 
 and the suspension must be so adjusted 
 that when the needle is at rest the torsion 
 is zero, and so that the needle is magnet- 
 ically symmetrical with respect to the 
 plane of the magnetic meridian. If the 
 galvanometer is of the suspended coil 
 (D'Arsonval) type, the effect of the 
 earth's field may be neglected ; and it 
 is sufficient that the torsion of the sus- 
 pensions is zero when the needle is at . 
 rest, and that the plane of the coil is 
 parallel to the force of the stationary 
 field magnets. The direction of the 
 mirror will usually indicate whether 
 these conditions are approximately ful- 
 filled, while a sufficient test of the 
 accuracy of the adjustment is that 
 equal charges sent through the galva- 
 nometer in opposite directions produce 
 equal and opposite deflections. Fig. 161 
 represents a high sensibility D'Arsonval galvanometer, which is 
 exceedingly convenient for ballistic observations. 
 
 In addition to these adjustments of the galvanometer it is 
 necessary that the telescope and scale be so placed that the 
 mirror when at rest reflects to the cross wires of the telescope 
 that point of the scale which is in the same vertical plane as 
 the axis of the telescope. 
 
 When these adjustments are made the observed displacements 
 of the scale, when the needle is deflected, are proportional to the 
 
 D'Aksonvai, Galva- 
 
 NOMETEK 
 
THE QUADRANT ELECTROMETER 339 
 
 tangents of twice the angles of deflection. Tables are often pro- 
 vided, giving corrections, which, when added to the scale read- 
 ings, make these readings directly proportional to the arcs, or to 
 some function of the arcs, whose tangents they measure. 
 
 After a ballistic galvanometer has been deflected, some special 
 manipulation is required to bring the needle quickly to rest. If 
 it is of the suspended coil type, a short circuiting of the galva- 
 nometer is the most efficient method. The current induced by 
 the motion of the needle in the field of the magnet reacts upon 
 the needle, damping its vibrations. If the galvanometer is of the 
 suspended magnet type, the motions of the needle may be 
 checked by the manipulation of a permanent magnet held in the 
 hand ; or by temporarily connecting to the galvanometer a small 
 coil of wire which is arranged to slide over the end of a fixed 
 bar magnet. By moving this coil on the magnet an induced 
 current is produced whose direction and strength are easily con- 
 trolled, and which can be made to bring the needle to rest. When 
 this is accomplished the circuit of the auxiliary coil is broken. 
 If the needle cannot be brought entirely to rest, the condenser 
 may be discharged precisely when the needle is at a turning 
 point in its vibrations, and the deflection is to be calculated from 
 this turning point and not from the point of equilibrium. 
 
 For further consideration of the ballistic galvanometer, see 
 Arts. 267 and 268. 
 
 CXLIII. ELECTROMOTIVE FORCE WITH THE QUADRANT 
 ELECTROMETER 
 
 Compare the electromotive forces of various cells. 
 
 254. The Quadrant Electrometer. — The quadrant electrom- 
 eter has four metallic quadrants, which together form a short 
 cylindrical box, the quadrants being supported within a metal 
 case by insulating posts and separated from one another by 
 small spaces ; the alternate quadrants are connected by wires, 
 and each pair is connected to an insulated binding post outside 
 the case ; within the quadrants a flat aluminum needle is 
 
340 
 
 ELECTRICITY 
 
 A 
 
 lu 
 
 suspended either by a bifilar silk suspension or by an insulated 
 metallic wire suspension. In Fig. 162 the electrometer is shown 
 with the case open. The quadrants are at Q, one of them being 
 removed to show the needle N. In the bottom of the case is a 
 
 glass dish containing pure concen- 
 trated sulphuric acid. The needle 
 connects with the acid either by a 
 platinum wire or by a piece of mica. 
 The acid may serve three purposes : 
 it keeps the interior of the electrom- 
 eter dry, and it checks the vibra- 
 tion of the needle ; it may also serve 
 as the inner coating of a Leyden 
 jar, whose outer coating is the 
 metal case, giving to the needle 
 an increased capacity. This last 
 use of the acid is not necessary, 
 and if the connection between 
 needle and acid is mica, the con- 
 denser effect is not employed. 
 
 By means of sliding rings on 
 the quadrant binding posts, connect 
 the quadrants to the metal case, 
 and connect the case to the ground. 
 The needle being discharged, so 
 adjust its support that it hangs 
 symmetrically over one of the diam- 
 etral spaces separating the quad- 
 rants. The position of the mirror 
 will indicate this adjustment with 
 sufficient accuracy. Give the 
 needle a static charge from any convenient source, as an 
 electrophorus, through the acid or through the metallic sus- 
 pension, according to the construction of the instrument. If 
 the needle is permanently deflected by this charge, it must be 
 brought back to its zero position by adjusting one of the 
 quadrants provided with a movement for this purpose, or by 
 
 Fig. 162. Quadrant 
 Electbombtee 
 
THE QUADRANT ELECTROMETER 341 
 
 means of the leveling screws, or by a slight adjustment of the 
 torsion head. 
 
 Now insulate the quadrants and connect one pole of the cell 
 to be tested to one quadrant binding post, and the other pole 
 to the second post. The potential difference given to adjacent 
 quadrants will turn the needle until the electrostatic actions 
 are balanced by the couple generated in the twisted suspension. 
 The angle of deflection is proportional to the potential differ- 
 ence. It is desirable to deflect the needle first in one direction 
 and then in the other by reversing the cell connections ; this is 
 conveniently accomplished by means of a special reversing key 
 devised by Lord Kelvin. The mean of the two deflections is 
 free from errors of torsion. 
 
 If the average scale deflection produced by an electromotive 
 force, ^j, is d-^, and d^ is that due to an electromotive force, E^, 
 the scale being distant from the mirror D, then 
 
 ^i:i-„ = tan-i^:tan-'^. 
 
 To determine the potential of an insulated conductor, connect 
 one pair of quadrants to the ground, and the second pair to the 
 conductor in question. 
 
 The electrometer may also be used by connecting the quad- 
 rants to the poles of a large series battery, the middle of the 
 series being connected to the ground. This maintains the 
 adjacent quadrants at equal opposite potentials. The conduc- 
 tor whose potential is to be determined, as referred to the zero 
 potential of the earth, is connected to the needle. 
 
 255. Electrometer for measuring Large Potential Differences. 
 — If the electrometer is too sensitive when arranged as above 
 explained, it may be made less sensitive by connecting the needle 
 and one pair of quadrants to one pole, and the other pair of 
 quadrants to the second pole. In this case the deflection of the 
 needle is proportional to the square of the potential difference. 
 
 Arranged in this manner the quadrant electrometer is appli- 
 cable to the measurement of alternating potentials. 
 
 Refebence. — Stewart and Gee, Practical Physics, Vol. II, pp. 431-438. 
 
342 
 
 ELECTKICITY 
 
 CXLIV. ELECTROMOTIVE FORCE WITH THE CAPILLARY 
 ELECTROMETER 
 
 Determine the variation of contact difference of potential between copper and 
 copper sulphate solution, as the concentration of the solution varies. 
 
 256. The Capillary Electrometer. — Lippmann has devised a 
 simple electrometer based upon the fact that the surface tension 
 of mercury in contact with an electrolyte varies with the differ- 
 ence of potential at the surface of contact. When a current 
 passes from the acid to the mercury the area of the surface of 
 
 i separation tends to decrease, owing to an 
 increase of surface tension. A large mer- 
 cury surface and a small capillary surface 
 are connected by dilute sulphuric acid (one 
 part of acid to five of water) in a suitably 
 shaped glass vessel (Fig. 163). At C is a 
 capillary surface in a tube of about 0.5 mm 
 bore, and at S is the large surface. Plati- 
 num wires connect with the two portions 
 of mercury. 
 When a difference of potential is applied 
 to the two wires the capillary meniscus 
 changes its position by amounts approxi- 
 mately proportional to this difference when 
 it does not exceed a tenth of a volt. A small microscope with 
 an eyepiece micrometer is convenient for observing the menis- 
 cus. The deflection may be from five to ten scale divisions for 
 0.001 volt. 
 
 A more sensitive form of electrometer, giving certain indica- 
 tions to 0.0001 volt, employs a much finer capillary, and has its 
 parts arranged as shown in Fig. 164. The upper mercury tube 
 is 50 cm or more long ; connected to it by a flexible tube is a 
 reservoir, permitting the height of the mercury to be varied. 
 The capillary dips into dilute sulphuric acid which rests on 
 mercury in the lowest vessel. A microscope is provided to 
 
 Fig. 163. Capillaky 
 Electrometer 
 
 observe the capillary surface of separation. 
 
THE CAPILLARY ELECTEOMETER 
 
 343 
 
 The sensitiveness of the electrometer depends upon the fine- 
 ness of the capillary ; if the point is conical, the mercury may 
 be made to assume a desired position by varying the height of 
 the upper column. The displacement of the meniscus meas- 
 ured with the micrometer eyepiece may serve to determine the 
 potential difference ; or when the surface has been displaced it 
 may be restored to its former position by changing the pressure 
 above, the potential difference being 
 considered proportional to the change 
 in pressure. The electrometer is 
 standardized by comparison with a 
 standard cell ; but as the relation 
 between pressure and potential 
 changes and has therefore to be ire- 
 quently redetermined, the instrument 
 is satisfactory only for measuring very 
 small differences of potentials and for 
 indicating the absence of potential 
 difference in compensation methods 
 (Art. 250). 
 
 The mercury in the capillary should 
 always be connected to the negative 
 (zinc) pole, otherwise the mercury 
 becomes corroded. If the difference 
 of potential exceeds one volt, hydro- 
 gen is evolved, which interferes with 
 proper action. If either of these 
 effects has been produced, the capil- 
 lary meniscus is cleaned by forcing a drop of mercury entirely 
 through the capillary. 
 
 Occasionally the mercury should be drawn up out of the 
 capillary, allowing the acid to wet the small tube, as this is a 
 condition of sensitiveness. 
 
 The electrometer remains in condition for use only when it 
 is kept on closed circuit. Hence a three-point key (Fig. 170) 
 is desirable, arranged to short circuit the electrometer when the 
 lever is in its normal position, the short circuit being broken 
 
 Fig. 164. Capillary 
 Electrometer 
 
344 
 
 ELECTRICITY 
 
 i 
 
 :K] 
 
 and the external electromotive force being applied when the 
 key is pressed. 
 
 Instructions for making three forms of capillary electrometers 
 are given in the reference. 
 
 Eefekence. — Ostwald, Physico-Chemical Measurements, pp. 202-208. 
 
 257. Contact Difference of Potential. — Any two dissiniilar 
 substances in contact are at different potentials. In the cases 
 of two solids or two metals this difference is usually very small, 
 
 while for a metal and an electrolyte it 
 may be about a volt. The difference 
 depends, for a given metal and electro- 
 lyte, upon the concentration of the 
 ions • of this metal in the electrolyte. 
 The variation with concentration 
 may be observed in a concentration 
 cell, as suggested by Ostwald, arranged 
 as shown in Fig. 165. Two short glass 
 tubes each have one end conically 
 reduced; in the small end is placed a 
 plug of macerated filter paper. These 
 tubes are filled with the electrolytes, 
 either of diiferent salts or of different 
 concentrations of the same salt. The electrodes are supported 
 by corks, and the bottoms of the tubes are placed in a suitable 
 dish which contains a conducting liquid, — mercury, for instance. 
 The compensation method is the best for making the meas- 
 urements. The connections are to be made as shown in Fig. 166. 
 B and H' are two resistance boxes which for ordinary measure- 
 ments may be of 1000 ohms, or for more refined work may be 
 of 10000 ohms each, as described in Art. 250. B is a, cell 
 which furnishes a constant difference of potential greater than 
 that of the cell being investigated. It may be a small second- 
 ary cell or a gravity cell. C is the concentration cell to be 
 measured, JE is the capillary electrometer, and K is a three- 
 point short-circuiting key. The resistances in H and It' are to 
 be varied, their sum always being 1000 (or 10000) ohms, tiH 
 
 Fig. 165. Contact Cell 
 
ELECTEOMOTIVE FORCE WITH AMMETEE 345 
 
 compensation is secured. The value of the difference of potential 
 at C is determined by substituting for C a standard cell (Art. 249) . 
 The difference of potential measured is the difference of the 
 two contact differences of potential of the two electrodes of C, 
 since these are oppositely- 
 directed. For the laboratory 
 exercise iirst fill the two tubes 
 with equal solutions of 1 : 200 
 gram-molecular solution of 
 anhydrous copper sulphate. 
 The proportions for this are 
 copper sulphate crystals 
 (CuSO^ + S HgO), 250 g, and 
 water (3600 - 90), 3510 g. One 
 tube still containing the first 
 solution, the second is to be 
 filled with a 1 : 150 solution 
 (250 g of crystals to 2610 g of water) ; and again with a 
 1 : 100 solution (250 g of crystals to 1710 g of water) ; the 
 change of potential being measured in each case. 
 
 Reference. — Ostwald, Physico-Cliemical Measurements, pp. 209-216. 
 CXIV. ELECTROMOTIVE FORCE WITH AW AMMETER 
 
 Fig. 166. Compensation Method 
 WITH Electrometer 
 
 Determine the electromotive forces and resistances of several cells, using 
 a milliarameter or tangent galvanometer. 
 
 258. Electromotive Force and Resistance of Cells with a Milliam- 
 meter and a Resistance. — The following methods for investi- 
 gating constant cells are based upon Ohm's Law, and they 
 require very simple apparatus. A sensitive ammeter, a tan- 
 gent galvanometer, or pther galvanometer whose law is known, 
 and a resistance box, are all that is needed. For three of the 
 methods it is necessary to know the resistance of the galvanom- 
 eter, and for one its constant is required. The formulae will 
 be given for the ammeter, but if another instrument is used, it 
 is only necessary, except in the first formula, to substitute for 
 
346 ELECTEICITY 
 
 the current value /, the function which is proportional to the 
 current, as tan 6 for a tangent galvanometer, etc. 
 
 In the same manner as is developed in Art. 217, if /j and I^ 
 are the current strengths flowing through resistances i?^ and B^ 
 in series with the ammeter, produced by a cell whose electro- 
 motive force is required, the value of the electromotive force, 
 independent of the resistances of the battery and ammeter, is 
 
 and if G is the resistance of the ammeter, the battery resistance 
 is B = ^'^^ " ^'-^^ - G. 
 
 Electromotive forces may be compared, using a miUiammeter 
 and a resistance box, in four ways, as follows. 
 
 Let G be the ammeter resistance, and B the resistance of the 
 cell which produces a current Jj, through a resistance B^ in series 
 with the ammeter; and let B^, /g, and ^j ^^ ^^^ corresponding 
 quantities for a second cell. Representing the electromotive 
 forces of the cells by E^ and E^ respectively, 
 
 ^1 : ^2 = (5i + (? + i?i) Jj : (^2 + (? + B^ I^. 
 
 If B-^ + JSj be made equal to B^ + iJg, then 
 
 If B^ and B^ be so adjusted that ij equals Jj, then 
 
 E.^: E^= G + B-^ + B.^: G + B^ + B^. 
 
 Connect the two cells to be compared in series and let 7j be 
 the current flowing through any convenient resistance; inter- 
 change the poles of one cell so that the electromotive forces 
 are opposed, and let /j be the current .flowing, the resistance 
 remaining Unchanged ; then 
 
 J^i- ^i = k'^ k'- k~ k- 
 
THERMO-ELECTROMOTIVE FORCE 
 
 347 
 
 CXLVI. THERMO-ELECTROMOTIVE FORCE BY DIRECT 
 MEASURE 
 
 (a) Determine the thermo-electiomotive force of lead-copper, lead-iron, 
 and copper-iron thermo-elements at various temperatures between 0° 
 and 100°. Plot the results. Make a thermo-electric diagram for 
 copper and iron, and find their neutral temperature. 
 
 (b) Calibrate a thermo-electric pyrometer. 
 
 259. Thermo-Electromotive Force. — If one junction of a 
 thermo-couple is at a constant temperature, 0°, while the other 
 junction is raised to different temperatures, since the contact 
 difference of potential in general varies with the temperature, 
 there will result an electric current the electromotive force of 
 which is to be measured. 
 
 To the ends of a wire of one of the metals to be studied, 
 solder wires of the second material. Place the two junctions 
 in narrow test tubes in which are thermometers, and place the 
 tubes, one in an ice bath and the other in a water or oil bath. 
 
 Gu 
 
 -^l 
 
 Hi 
 
 1 
 9 
 
 >f 
 
 Fig. 167. Thekmo-Electeomotive roRCE 
 
 For better heat conduction the tubes may be partially filled 
 with petroleum. Connect the two ends of the thermo-couples, 
 through a resistance, to a galvanometer, as indicated.in Fig. 167. 
 By varying the temperature of one junction as much as is 
 convenient, the varying electromotive force is measured with 
 the galvanometer. This electromotive force may be taken as 
 
348 ELECTRICITY 
 
 proportional to- the deflection. The resistance is included in 
 the circuit, to regulate the current so that the deflection may 
 be readable. It is preferable that this should not be altered 
 during the measures, in which case, if the total resistance in 
 the circuit, including the galvanometer resistance, is over fifty 
 ohms, the value of the resistance need not be considered. If 
 the circuit resistance is low, the variation of resistance of the 
 thermo-couple with temperature must be taken into account. 
 
 In the several experiments the direction of the electromotive 
 forces must be noted. 
 
 Instead of measuring the deflection only, it is better to meas- 
 ure the actual electromotive force by the compensation method 
 of Art. 250. Or if the galvanometer constant is known (Art. 246) 
 the electromotive force may be computed from the resistance 
 of the circuit, and the deflection. 
 
 260. Thermo-Electric Power and Thermo-Electric Diagram. — 
 The ratio of the electromotive force of a thermo-element to the 
 difference in temperature of the junctions is the thermo-electric 
 power of the couple at the mean temperature. A thermo-electric 
 diagram is one showing these relations. Such diagrams are 
 usually constructed with reference to lead as the standard, since 
 hot lead and cold lead in contact show no thermo-electromotive 
 force, while with other metals there is a difference of potential 
 between the metal hot and the same metal cold. 
 
 The neutral point for two metals may be found from the 
 diagram; it is the temperature corresponding to the point of 
 intersection of their thermo-electric power lines. 
 
 CXLVII. ERRORS OF A VOLTMETER BY COMPENSATION 
 
 Calibrate a voltmeter at ten or more equidistant points of its scale. 
 
 261. Calibration of Voltmeter. — The compensation method 
 of Art. 250 may be adapted to the calibration of a voltmeter, as 
 follows. A storage or other battery of such size as to give the 
 various voltages corresponding to the points at which it is 
 desired to calibrate the voltmeter must be provided. Pass the 
 
CALIBRATION OP VOLTMETER 
 
 349 
 
 current from this battery through a standard resistance, R-^, of 
 large value, 100000 ohms for example, and through a standard 
 of variable resistance, B^, connected in series. Connect the 
 voltmeter V (Fig. 168), to indicate the total difference of poten- 
 tial; and to the terminals 
 
 ^ 
 
 R, 
 
 I 
 
 ^^^ 
 
 iB 
 
 \-J^\-\i-\[-\\- 
 
 of ^2 connect one or more 
 standard cells, S, a gal- 
 vanometer, G, and a key, 
 X, in series, so that the 
 electromotive force of the 
 cells shall oppose that of 
 the battery 5. It will now 
 be possible so to adjust 
 the resistance ^2 that the 
 difference of potential be- 
 tween its terminals shall 
 be exactly equal to that of the standard cells. That this condition 
 has been secured will be indicated by there being no deflection 
 of the galvanometer when the key K is closed. Since the fall 
 of potential is proportional to the resistance, the total potential, 
 a, measured by the voltmeter, is to that of the n standard cells, 
 ne, as Bj^ + B^is to B^ ; 
 
 Fig. 168. Voltmeter Calibration 
 
 E=ne —^ 
 
 B-i -f- -B, 
 
 i?2 
 
 The difference between this computed value of -E" and the 
 voltmeter reading is the correction to the scale at this point. 
 
 Alter the battery potential and repeat the process for each 
 point to be calibrated. 
 
CHAPTER XXIII 
 CAPACITY 
 
 CXLVm. RELATIVE CAPACITY WITH A BALLISTIC 
 GALVANOMETER 
 
 Compare the capacities of the various parts of a divided condenser. Deter- 
 mine the capacity of a condenser by comparison with a standard 
 condenser. 
 
 262. Standard International Coulomb and Farad; Standard 
 Condensers. — The coulomb is that quantity of electricity which 
 is transferred by one ampere (Art. 244) in one second. The 
 farad is the capacity of a conductor such that its potential is 
 increased by one volt (Art. 249) when it receives a charge of 
 one coulomb. The farad is too large a capacity for laboratory 
 uses, and the one millionth part of the farad, called the micro- 
 farad, is the working unit. 
 
 Standard condensers are usually constructed of tin-foil sheets 
 with mica insulation, and have capacities of ^ or 1 microfarad. 
 They are sometimes subdivided into five parts of 0.5, 0.2, 0.2, 
 
 Fig. 169. Plan of Divided Condenser 
 
 0.05, and 0.05 microfarads, which by means of the blocks and plugs 
 shown in Fig. 169, can be connected to give a great variety of 
 capacities. The first section of the condenser has one terminal 
 
 350 
 
CONDENSER KEYS 
 
 351 
 
 connected to the first block and the other to the second block ; 
 the second section of the condenser is connected between the 
 second and third blocks, etc. The charging current is to be 
 connected to the posts Pj and P^, which connect respectively 
 to the bars B^ and B^. If any two consecutive blocks are con- 
 nected to opposite bai's, it makes no difference which, that portion 
 of the condenser between the blocks will receive a charge. In 
 Fig. 169 the portion of the condenser connected is 0.45. 
 
 The method of connection described is that in which the 
 several parts are in parallel. They might be connected in 
 series, in which case the 
 capacity of the series is 
 the reciprocal of the sum 
 of the reciprocals of the 
 several parts. 
 
 Care must be taken not 
 to insert plugs at the two 
 ends of one block, for this 
 would short circuit the 
 charging battery. 
 
 263. Condenser Keys. — 
 For facilitating the charg- 
 ing and discharging of 
 condensers, keys are made 
 
 in many forms, the essential features of which may be explained 
 with the aid of Fig. 170. One terminal of the galvanometer is 
 to be connected to G, one pole of the battery to B, and one side 
 of the condenser to C, the other connections being as shown in 
 Fig. 171. A spiral spring draws the lever L downward in the 
 position shown. By pressing the knob K, the front of the lever 
 is brought downward till it catches under the two triggers, T-^ and 
 T^. This brings the end of L into contact with B, which is the 
 charge position. By pressing Tj the spring will pull the lever 
 downward, but the trigger T^ will catch it and hold it horizon- 
 tal so that L does not touch either B or G; this is the insulate 
 position. By pressing T^ the lever takes the position shown, 
 which is the discharge position. 
 
 Fig. 170. Condenser Key 
 
352 ELECTRICITY 
 
 In another form of condenser key, a flexible lever is normally 
 in'contact with the upper point (the charge position), while it- 
 may be pressed away from this point into contact with the 
 lower one (the discharge position). 
 
 264. Comparison of Condenser Capacities If different con- 
 densers, or different parts of one condenser, are charged to the 
 same potential, the quantities of electricity in the charges will 
 be proportional to the capacities of the condensers. These 
 quantities may be compared by discharging the condensers 
 through a ballistic galvanometer (Art. 253). 
 
 Connect a ballistic galvanometer, G, a condenser, C, a charge 
 and discharge key, K, and a cell, B, as indicated in Fig. 171. 
 
 I After charging the condenser, by 
 
 Jl ^ changing the key to the galvanom- 
 
 L 6ter contact, the condenser will be 
 
 K^ discharged and the galvanometer 
 
 needle will be deflected through an 
 
 pj angle, 6^, such that sin i ^j is pro- 
 portional to the charge and therefore 
 
 Fig. 171. Compakison of _ j.- i j. ^i, -j. /-» tj 
 
 proportional to the capacity C-. . By 
 Capacities . , , » 
 
 using a second condenser of capacity 
 
 Cg, a deflection 6^ will be obtained. If the distance of the scale 
 
 from the mirror is D, and the observed scale readings are c?j and d^, 
 
 Ci'. C^ = sin ^ ^1 : sin ^ ^2 = ^^^ i tan~' -J : sin ^ tan~^ -^ . 
 
 If the angular deflections of the needle do not exceed 6°, the 
 sines and tangents may be considered proportional to the 
 corresponding arcs; and then 
 
 (Caution Do not allow the cell to be short-circuited, nor to be 
 
 directly connected to the galvanometer.) 
 
 The following numerical example illustrates the comparison 
 of capacities. A part of these observations are used in the 
 determination of the absolute capacity of one of the condensers, 
 explained on page 358. 
 
COMPARISON OF CAPACITIES 
 
 353 
 
 Comparison of Capacities 
 Octoher 18, 1897 
 
 S. & H. Standard Condenser No. 6829; nominal capacity, 1.003 micro- 
 farad. Q Paper Condenser; nominal capacity, \ microfarad. Queen Bal- 
 listic Galvanometer No. 92. 
 
 The two condensers were charged successively from H. & B. Clark Cell 
 No. 100, and the following deflections were obtained, as read in centi- 
 meters on a scale distant 100 cm. 
 
 CONDEXSEK No. 
 
 S829 
 
 
 COKDENSEK 
 
 3 
 
 Zero 
 Beading 
 
 Second 
 Reading 
 
 Deflection 
 
 Zero 
 Reading 
 
 Second 
 Beading 
 
 Deflection 
 
 39.55 cm 
 
 76.10 cm 
 
 36.55 cm 
 
 39.8 cm 
 
 48.9 cm 
 
 9.1cm 
 
 .65 
 
 .25 
 
 .60 
 
 .7 
 
 
 8 
 
 ■ 
 
 .70 
 
 .20 
 
 .50 
 
 .7 
 
 
 9 
 
 
 .75 
 
 .25 
 
 .40 
 
 .7 
 
 
 8 
 
 
 .80 
 
 .20 
 
 .50 
 
 .7 
 
 
 8 
 
 
 .80 
 
 .30 
 
 .50 
 
 .8 
 
 
 9 
 
 
 .85 
 
 .40 
 
 .55 
 
 .7 
 
 
 8 
 
 
 .85 
 
 .40 
 
 .55 
 
 .7 
 
 
 9 
 
 
 .85 
 
 .40 
 
 .55 
 
 .7 
 
 
 8 
 
 
 .8-3 
 
 .30 
 
 .47 
 
 .7 
 
 
 8 
 
 
 
 Mean dx 
 
 = 36.52 cm 
 
 
 Mean < 
 
 is = 9.12 cm 
 
 . 36..52 
 
 .9.12 
 
 (No. 6829) : (Q) = sin J tan-i '-j^ : sin J tan-i y^ = 1 : 0.260. 
 
 Capacity of Condenser No. 6829 = 0.999 microfarad (p. 358). 
 Capacity of Condenser Q = 0.259 microfarad. 
 
 CXLK. RELATIVE CAPACITY BY BRIDGE METHODS 
 
 Determine the capacities of the parts of a subdivided condenser by com- 
 parison with a standard condenser. 
 
 265. Bridge Methods for Comparison of Capacities. — The fol- 
 lowing zero methods for condenser comparisons are similar to 
 the Wheatstone's bridge method for resistance comparisons. 
 
354 
 
 ELECTRICITY 
 
 De Sauty's Method. — Connect the two condensers, two resist- 
 ances, a dead-beat galvanometer, a battery of from 10 to 15 
 volts, and a charge and discharge key, as 
 shown in Fig. 172, the key being usually 
 on the discharge point. Adjust the resist- 
 ances until the galvanometer shows no deflec- 
 tion when the condensers are charged and dis- 
 charged. When this condition is fulfilled 
 
 The resistances, B^ and JEj, should have 
 values of from 1000 to 10000 ohms, and 
 must be noninductive. 
 
 Groifs Method. — Connect two condensers, 
 two resistances, a dead-beat galvanometer, a 
 battery of from 10 to 15 volts, and keys, as 
 indicated in Fig. 173. As compared with 
 the arrangement of the previous method, the battery and galva- 
 nometer have changed places. It will now be possible so to 
 adjust the resistances, E-^ and B^, that 
 upon first closing the key K-^, which 
 remains closed, and then closing the key 
 K^, no deflection of the galvanometer is 
 produced. When this condition is ful- 
 filled, the following relation is true. 
 
 After both keys have been closed as 
 described, and the resistances have been 
 altered, the condensers must be discharged 
 by closing K^ only, before again charging through Ky As before, 
 the resistances should be large, between 1000 and 10000 ohms. 
 
 Fig. 172. De Satjty's 
 Method 
 
 Fig. 173. Gott's 
 Method 
 
 Reference. 
 Vol. I, p. 440. 
 
 - Gray, Absolute Measurements In Electricity and Magnetism, 
 
COMPAEISON OF CAPACITIES 
 
 355 
 
 CL. RELATIVE CAPACITY BY THE METHOD OF MIXTURES 
 
 Determine the capacity of a condenser by comparison with a standard 
 
 condenser. 
 
 266. Comparison of Capacities by the Method of Mixtures. — 
 In this method two condensers are charged with opposite poten- 
 tials of such values that their charges when mixed neutralize 
 each other. Their capacities are inversely as these potentials. 
 
 Connect apparatus as indicated in 
 Fig. 174. A battery of from 10 to 
 15 volts is joined in series with two 
 large resistances, B^ and B^; the 
 condensers, Cj and Cg, are arranged 
 to be connected to ^j and E^ re- 
 spectively, by means of a special 
 double key, K. A rocking commu- 
 tator (Fig. 133) with the diagonal 
 connections removed is suitable for 
 this purpose. With the key in 
 proper position the condensers will 
 be charged to potentials which are 
 proportional to ^j^ and B^. When 
 the key is rocked to its second posi- 
 tion the connections as shown are 
 such that the two charges will be 
 mixed, and the residual charge is 
 only their difference. By closing 
 
 the key k, this residual will be discharged through the galva- 
 nometer G. The resistances are to be so adjusted that the 
 galvanometer shows no deflection, when the condensers are 
 charged and discharged as described; then 
 
 ^1 ■ ^2 ^^ 1 ■ 2' 
 
 The method is suitable for cable capacity measures, the con- 
 ductor of the cable being joined to K, and the sheath of the 
 cable or the water in which it is immersed being connected to 
 
 Earth | 
 
 Fig. 174. Capacity by Method 
 
 OP MlXTUKES 
 
356 ELECTEICITY 
 
 the earth. The ground connection is necessary for cable testing, 
 but may be omitted in other cases. Absorption by the condensers 
 may cause an error, and often the time required for mixing the 
 charges is appreciable ; hence high insulation is essential. 
 
 CLI. ABSOLUTE CAPACITY OF A CONDENSER WITH A 
 BALLISTIC GALVANOMETER 
 
 Determine the absolute capacity of a condenser. 
 
 267. Absolute Capacity with a Ballistic Galvanometer When 
 
 a ballistic galvanometer (Art. 253) is used for absolute measure- 
 ment, it becomes necessary to consider the period of vibration 
 of the needle as well as the deflection, and to make corrections 
 for the effect of the amplitude upon the period and for the 
 effect of damping on the deflection (Art. 93). 
 
 The Period of the needle may be found from observations 
 of transits, as explained in Art. 83, a set of ten transits being 
 observed, and, after a lapse of fifty or more vibrations, a second 
 set of ten. 
 
 This observed period, T, is reduced to the period in an infi- 
 nitely small arc, T^, by the approximate formula, 
 
 « 256 Z)2' 
 
 in which d is the observed scale reading (average) and D is the 
 distance of the scale from the mirror. Complete formulae and 
 tables to facilitate making this correction will be found in 
 the references. 
 
 The Logarithmic Decrement is the natural logarithm of the 
 ratio of decrease in the arcs of successive vibrations of the 
 needle. Let a^ be the arc of any swing of the needle, expressed 
 in scale divisions, and let a^^^ be the arc of the wth following 
 swing ; then the logarithmic decrement is 
 
 , 2.3026 ,, . 
 
 \ = -— — (log ai - log a^^ J. 
 
ABSOLUTE CAPACITY 357 
 
 Correction of the Tangent to the Arc. — If dis the observed 
 number of scale divisions from the center of the scale to the 
 turning point of the swing, expressed in centimeters, and the 
 scale distance is D, the number which is proportional to the arc 
 of swing is ^3 
 
 A = d 
 
 3i>2- 
 
 The Absolute Capacity of a Condenser may be found, after 
 the corrected period of the needle, Tq, and the logarithmic dec- 
 rement, \ of the galvanometer have been determined from the 
 results of the following deflection experiments. 
 
 Charge the condenser with a given cell and discharge it 
 through the galvanometer, as described in Art. 264. If c?j is 
 the observed scale reading when the distance of the scale from 
 the mirror is D, the corrected deflection is 
 
 A, =2sini tan-'^. 
 
 Then connect the same cell in series with the galvanometer 
 and a very high resistance, R, and let d^ be the deflection pro- 
 duced. Usually it will be necessary to shunt the galvanometer 
 in order to secure a readable deflection. This deflection properly 
 corrected will be t, 
 
 A, = tan 1- tan""'-^. 
 
 If the galvanometer resistance is G, the battery resistance 
 B, and if the galvanometer is shunted with a resistance S, then 
 the virtual value of the resistance through which the cell acted 
 in producing the observed deflection d^ is 
 
 Finally the absolute capacity of the condenser C is given by 
 the following formula. j, /i + i x\ a 
 
 References. — Stewart and Gee, Practical Physics, Vol. II, pp. 364-369, 
 407; KoMrausck, Physical Measurements, pp. 219-225, 347, 382; CarJiart 
 and Patterson, pp. 207-213, 227. 
 
358 
 
 ELECTRICITY 
 
 A numerical example of the determination of absolute capacity- 
 is given. 
 
 Absolute Capacity op Siemens & Halsee Condenser No. 6829 
 
 nominal capacity, 1.003 miohofabad 
 
 Queen D'Arsontal Ballistic Galvanometek No. 92 
 
 October 21, 1897 
 
 Two determinations of the period of the needle were made, the reduced 
 average of which gives 
 
 Tj, = 16.589 s. 
 
 Obsekvations and Computation for the Logarithmic Decrement 
 Scale distance, 100 cm. Center of scale, 40.00 
 
 TuBNiNG Point 
 P 
 
 d=p -40.00 
 
 3Z>2 
 
 Corrected 
 Turning Point 
 
 Arc 
 
 64.69 
 16.52 
 59.13 
 21.43 
 55.09 
 25.07 
 
 + 24.69 
 
 - 23.48 
 + 19.13 
 
 - 18.57 
 + 14.93 
 
 - 12.03 
 
 + 0.50 
 -0.43 
 + 0.23 
 -0.21 
 + 0.12 
 -0.11 
 
 64.19 
 16.95 
 58.90 
 21.64 
 54.97 
 25.18 
 
 47.24 = ax 
 41.95 = 02 
 37.26 = OS 
 33.33 = 04 
 29.79 = 05 
 
 J (log a, - log o ) = 0.05049 | ^^^^ ^ ^^^^^^ 
 T (log «6 - log S) = 0.04955 J 
 
 X = 2.3026 X 0.05002 = 0.11518. 
 
 The observations for the deflection produced by the discharge of the 
 condenser are given in the example on page 353. 
 
 di = 36.52. 
 
 Ai = 0.17485. 
 
 The observations for the steady deflection are given in the example on 
 page 328. 
 
 d^ = 17.22. \ = 0.08257. 
 
 iJ^= 11465000. 
 
 The absolute capacity of the condenser is 
 
 16.598 X 1.05759 x 0.17485 
 
 C = 
 
 •r- X 11 465000 X 0.08257 
 
 0.9991 microfarad. 
 
BALLISTIC CONSTANT 359 
 
 268. Ballistic Constant. — In the above method the electro- 
 motive force of the cell is eliminated by using the same cell in 
 both the condenser and deflection observations. If a standard 
 cell of known electromotive force is used in the deflection experi- 
 ment, the ballistic constant of the galvanometer may be deter- 
 mined and then the instrument may be used to measure capacities 
 or other quantities in absolute measure without the repetition of 
 this part of the above method. This constant is the factor by 
 which an observed ballistic deflection must be multiplied to give 
 the absolute measure of the quantity of electricity that passed 
 through the galvanometer. If E is the electromotive force of 
 the cell, and T^, R^, and Aj have the values given in the preced- 
 ing article, the ballistic constant of the galvanometer is 
 
 7rli^\ 
 
 The absolute capacity C, of a condenser charged to a potential 
 difference e, X and Aj having the values assigned in the preced- 
 ing article, is 
 
 C=K 
 
 a + ix)A^ 
 
CHAPTER XXIV 
 INDUCTION 
 
 CLII. INDUCTANCE BY COMPARISON 
 
 Determine the self-induction of several coils by comparison with a standard 
 
 of self-induction. 
 
 269. Standard Henry ; Coefficient of Self-induction The 
 
 unit of induction is the henry ; it is the induction due to a cur- 
 rent whose strength is changing at the rate of one ampere per , 
 second with a resulting induced electromotive force of one yolt. 
 The coefficient of self-induction of a coil is the total induction 
 through it per unit change of current 
 producing the induction. 
 
 The self-induction of two coils may 
 be compared with a Wheatstone's bridge 
 by Maxwell's method; if the self- 
 induction of one coil is known, that of 
 the other is determined. Let jBj and B^ 
 (Fig. 175)be the resistances of two arms 
 of a Wheatstone's bridge containing the 
 coils whose coefficients of self-induction, 
 L-^ and X^, are to be compared ; let ^3 
 and B^ be noninductive resistances. 
 Connect these with a galvanometer, a 
 battery, and a key, as indicated. If a steady current is flowing, 
 then, as in Art. 207, no current will flow across the bridge from 
 D to E when R^ and ^^ are so adjusted that 
 
 Fig. 175. Self- 
 Induotion 
 
 B-^-.B^ — Bg 
 
 B,. 
 
 Also it can be proved that if the current varies so that the 
 
 360 
 
MUTUAL INDUCTION 361 
 
 self-induction of the coils enters into the current relations, there- 
 wiU be no current through the galvanometer when 
 
 As it is not convenient to determine this latter condition alone, 
 both conditions must be satisfied at the same time. The adjusir 
 ments must be so made that no current flows through the galva- 
 nometer while the circuit is closed, and also there must be no 
 current when the battery circuit is made or broken. This means 
 that the resistances in the arms of the bridge apparatus which 
 contain the self-inductances must be made to have the same 
 ratio as the self-induction coefficients. If a standard of self- 
 induction is used for one coil, it is usually constructed so that 
 its self-induction may be varied. This alone may permit the 
 obtaining of a balance ; if not, noninductive resistance must be 
 inserted in series with one of the inductive coils so as to change 
 the value of ^j, for instance, without changing Ly By one or 
 both of these operations it will be possible to secure the condi- 
 tions for no deflection. Then 
 
 References. — Carhart and Patterson, Electrical Measurements, p. 255 ; 
 Stewart and Gee, Practical Physics, Vol. II, p. 394. 
 
 CLm. COEFFICIENT OF MUTUAL INDUCTION BY COMPARISON 
 
 "Verify the laws of mutual induction, and compare two mutual inductances. 
 
 270. Mutual Induction. — That the induced current in the 
 secondary coil varies directly as the current in the primary coil 
 and inversely as the resistance in the secondary circuit, and 
 
 Fig. 176. Mutual Induction 
 
362 
 
 ELECTRICITY 
 
 the dependence of the induction upon the relation of the pri- 
 mary and secondary, may be proved by comparing the currents 
 produced under various conditions, by means of a ballistic gal- 
 vanometer. If the constants of the current are known, the 
 coeificient of mutual induction may be calculated. If the 
 making of a current, /, in the primary (Fig. 176) induces a 
 quantity of electricity Q in the secondary, and the entire resist- 
 ance of the secondary circuit is B, the coefficient of mutual 
 induction, M, is obtained from the following relation. 
 
 Refebence. — Nichols, A Laboratory Manual of Physics, Vol. I, p. 240. 
 
 271. Comparison of Two Mutual Inductances. — Let it be 
 required to compare the mutual induction of two coils, Cj and 
 C„, with that of coils 
 
 Cg and C4. 
 
 Connect the two primary 
 coils, one from each pair, 
 Cj and C3. (Fig. 177), in 
 series with a battery, the 
 circuit containing a key, 
 K. Connect the second- 
 ary coil, C^, in series with 
 a resistance, -Bj' ^^^ ^ 
 sensitive galvanometer ; 
 connect the coil C^, and a 
 resistance, B^, to the same 
 galvanometer so that the 
 induced current in C^ shall 
 oppose that in C^. By adjusting B^ and B^ so that the making and 
 breaking of the primary circuit causes no deflection of the gal- 
 vanometer, the coefficients of mutual induction, lfi_2 and M^_^, 
 are in the same ratio as the resistances B^ and B^ ; that is, 
 
 Fig. 177. 
 
 Comparison of McTnAL 
 Inductances 
 
 7lf,_2 : M^. 
 
 .4 = -^2 ' 
 
 B, 
 
 Reference. — Carhart and Patterson, Electrical Measurements, p. 265. 
 
CHAPTER XXV 
 
 MAGNETIC QUANTITIES 
 
 CLIV. EARTH'S HORIZONTAL MAGNETIC INTENSITY WITH 
 THE MAGNETOMETER 
 
 Determine the horizontal intensity of the earth's magnetism, and 
 the magnetic moment of a magnet. 
 
 272. The Earth's Magnetic Elements. — The vertical plane 
 which contains the direction of the earth's magnetic force is 
 the magnetic meridian; the angle between this plane and that 
 of the geographic meridian is the magnetic declination. The 
 direction of the force is inclined to the horizontal in the mag- 
 netic meridian at an angle called the magnetic inclination or 
 dip. The total intensity of the earth's magnetism is the number 
 of dynes of force with which a unit magnet pole would be 
 urged along in the direction of the force. The unit of inten- 
 sity is called the gauss; the actual intensity varies from 0.28 
 to 0.78 of a gauss. This total intensity may be considered as 
 consisting of two components, — the vertical intensity, V, and 
 the horizontal intensity, H. The intensity and direction of this 
 force are subject to changes which may be analyzed into daily, 
 annual, secular, and irregular variations. It is important to 
 know the values of the' horizontal intensity and its variations ; 
 if the total intensity and its direction are required, they are 
 most readily determined from the values of the horizontal and 
 vertical components, or from the horizontal intensity and dip. 
 
 273. The Magnetometer. — If a magnet is suspended with 
 its axis horizontal, it will come to rest in the magnetic meridian. 
 After displacement from the meridian it will vibrate in the 
 manner of a torsion pendulum, with a period which varies 
 
 363 
 
364 ELECTEICITY 
 
 inversely as the square root of the restoring force. This force 
 is the product of the horizontal intensity of the earth's mag- 
 netism and the magnetic moment of the magnet. A determi- 
 nation of the period furnishes one equation of condition for 
 determining these two unknown quantities. 
 
 If the same magnet is placed at known distances from a com- 
 pass needle, in a line through the center of the needle perpen- 
 dicular to the plane of the magnetic meridian, the needle will 
 be deflected to a position such that the tendency of the earth's 
 field to restore the needle to the meridian is counterbalanced 
 by the moment due to the magnet tending to set the needle 
 east and west. The observed deflection furnishes a second 
 relation between the horizontal intensity and the magnetic 
 moment, permitting their values to be found. 
 
 The instrument for making these measurements, the magnet- 
 ometer, may be simple or elaborate, and while the formulae con- 
 cerning the general principles are simple, as given here, many 
 corrections may be made which greatly complicate both the 
 observations and their reduction. Even in simple measurements 
 care is required to avoid the effects of local magnetic disturbances 
 and variations in the strength of the earth's field. 
 
 Vibration Observations. — Suspend the magnet with its axis 
 horizontal, by a fiber having as little torsion as possible. The 
 torsion may be reduced to a minimum by first suspending in 
 place of thie magnet a bar of nonmagnetic material having the 
 same mass as the magnet, and adjusting the torsion head so 
 that the bar would come to rest with its axis in the magnetic 
 meridian. Displace the magnet from its position of equilib- 
 rium so that it shall vibrate with an amplitude not exceeding 
 5°, and carefully determine the period, T, of a complete vibra- 
 tion by the method of Art. 83. Then, if. 7 is the moment of 
 inertia of the magnet, M its magnetic moment, and H the hori- 
 zontal intensity of the earth's magnetism, 
 
 The moment of inertia, I, may be found by observing the 
 period of the needle when free and when a load of known 
 
THE MAGNETOMETER 365 
 
 moment of inertia is added to it, as explained in Art. 84 ; or it 
 may be calculated from the mass and dimensions of the magnet, 
 by the formula of Art. 85. 
 
 The value of MH thus determined may be corrected for 
 amplitude of swing, torsion of suspension, rate of clock, and 
 for the effects of temperature and induction upon the magnet, 
 as explained in the references. 
 
 Deflection Observations. — Place the magnet used in the pre- 
 vious experiment with its axis perpendicular to the plane of 
 the magnetic meridian and passing through the center of a 
 needle capable of horizontal deflections which can be measured. 
 The latter may be a form of compass, or a needle carrying a 
 mirror for telescope and scale observations. 
 
 With the magnet at a known distance from the center of the 
 needle, observe the deflection produced, reading both ends of 
 the pointer if it is of the compass form; turn the magnet end 
 for end, keeping it at the same distance, and observe the oppo- 
 site deflection produced ; make observations with the magnet in 
 two positions, at an equal distance on the opposite side of the 
 needle. Let 6.^ be the average deflection obtained from the 
 eight readings (or four readings if made with a telescope), and 
 let T-j be the distance between the centers of the magnet and 
 the needle, — that is, half the distance between the two positions 
 of the magnet. Place the magnet at a different distance, r^, 
 from the needle, and determine as before the average deflection, 
 ^2- Then ^^^ r^Han 6^ - r^Han 6^ 
 
 Corrections may be made for the effects of temperature and 
 induction upon the magnets. 
 
 Unless the deflections produced are too small for accurate 
 
 reading, the shortest distance of the deflecting magnet from the 
 
 needle should be not less than four times its length, and the 
 
 greatest distance should be 1.4 times the lesser. 
 
 M 
 Remits. — Using the values of MH and -- thus determined, 
 
 H 
 
 M and H are found separately, which are the magnetic moment 
 
366 ELECTKICITY 
 
 of the magnet and the horizontal intensity of the earth's mag- 
 netism respectively. 
 
 274. The Kew Magnetometer. — A form of magnetometer is 
 sometimes used, for which the method of observing the deflec- 
 tions differs slightly from that described above. The bar carry- 
 ing the deflecting magnet, instead of remaining perpendicular 
 to the plane of the magnetic meridian, is turned about the 
 center of the needle, so that the magnet is always perpendic- 
 ular to the axis of the needle. This requires the substitution 
 of the sine function of the angle of deflection for the tangent 
 in the formula given above; otberwise the method and formulae 
 are not changed. 
 
 ^ Repeeences. — Kohlrausch, Physical Measurements, pp. 240-260 ; Stewart 
 and Gee, Practical Physics, Vol. II, pp. 285-308 ; Gray, Absolute Measurements 
 in Electricity and Magnetism, Vol. II, p. 69 et seq. 
 
 CLV. VARIATION OF THE EARTH'S HORIZONTAL MAGNETIC 
 INTENSITY WITH THE VARIOMETER 
 
 Determine the relative horizontal intensity and direction of the earth's 
 magnetism at several assigned points about the laboratory, and plot 
 the results. 
 
 275. The Magnetic Variometer. — A magnetic needle which 
 is made to assume a position at right angles to the magnetic 
 meridian by means of a deflecting magnet acting in opposition 
 to the earth's magnetism is very sensitive to changes in the field 
 strength, and it may be used to determine the variations occur- 
 ring at one place ; or, by placing the instrument in several loca- 
 tions, the relative intensities at these points may be compared. 
 
 The deflecting magnet is supported below the needle so that 
 it may be rotated in a horizontal plane about the same axis as 
 that on which the needle swings. Its distance below the com- 
 pass may also be varied. An outline of such a variometer is 
 shown in Fig. 178. 
 
 The instrument may have a mirror needle and a telescope and 
 scale for observing the deflections, and the one deflecting magnet 
 
THE VARIOMETER 
 
 367 
 
 may be replaced by four small magnets placed around the needle 
 and in the same horizontal plane. 
 
 The adjustment and use of the variometer involve the following 
 operations. 
 
 (a) The Axis of Rotation of the instrument is to be made 
 vertical, as may be indicated by the level. 
 
 (b) Sensitiveness. — The magnet when in the meridian must 
 exert a force on the needle slightly in excess of that of the 
 earth. Turn the magnet to the zero of its divided circle and 
 
 rotate the whole instrument until the iV" 
 
 pole of the magnet is to the north and the 
 n end of the needle is to the south. It 
 should be possible to rotate the instrument 
 until the reversed needle is parallel to the 
 magnet ; that is, until its s end indicates 0° 
 on the compass circle. If such a position 
 cannot be found, the magnet must be set 
 nearer to the compass. The greater the 
 distance between magnet and needle at 
 which this condition is fulfilled, the greater 
 the sensitiveness. 
 
 (c) To place the Variometer in the Mag- 
 netic Meridian. — The operations described 
 in the first part of the preceding paragraph 
 place the instrument approximately in the 
 meridian ; but if the position of the magnet 
 is altered to secure proper sensitiveness, 
 it is necessary to make again the meridian adjustment. Set the 
 magnet to the zero of its divided circle. Turn the whole instru- 
 ment until the iVend of the magnet is to the north and the s end 
 of the needle indicates 0° on the compass circle. The needle is 
 then in the meridian. Clamp the instrument to the base. 
 
 (d) Angle of Rotation of the Magnet. — Turn the magnet to 
 one side till the needle points east and west, and fix a stop to 
 limit this position. Turn the magnet to the other side of the 
 meridian until the needle again points east and west, and place 
 the second stop to determine this position. 
 
 Fig. 178. The 
 Vakiometer 
 
368 , ELECTEICITY 
 
 It is not necessary for comparison that the deflection of the 
 needle should be exactly 90°, though it simplifies the computa- 
 tion if this exact deflection is produced when the instrument is 
 set in the place where the horizontal intensity is known. 
 
 Half of the angle of the rotation of the magnet between the 
 stops is the quantity 6 of the formulae. The stops must remain 
 unchanged in position during the entire set of comparisons. 
 
 (e) To perfect the Meridian Adjustment. — If the circle read- 
 ings for the positions of the two steps, determined as just 
 described, are different, it indicates that the 0° line of the mag- 
 net and compass circles are not in the same plane. To improve 
 the meridian adjustment, set the large magnet midway between 
 the two stops ; then, as in (c), turn the whole instrument until the 
 s end of the reversed needle indicates 0° on the compass circle. 
 The needle now indicates the magnetic meridian. The change 
 in the position of the instrument required for this is usually so 
 small that the adjustment is not disturbed. It may be repeated 
 if required. 
 
 In placing the instrument at a new station, the first setting is 
 made with the magnet in the position required in this adjust- 
 ment ; that is, in the position which corrects the zero error of 
 its circle. This correction is not constant because of possible 
 displacement of the circle when it is clamped in a new position 
 on the upright support. 
 
 276. Magnetic Declination The variometer being set in the 
 
 meridian as described, by means of sighting pieces on the com- 
 pass box the angle between the direction of the needle and any 
 given line, as the true north and south or the lines of a building, 
 may be found. 
 
 277. Comparison of Horizontal Intensities. — The variometer 
 having been adjusted as described above, in the first location, 
 preferably a place for which the absolute value of the horizontal 
 intensity is known, the magnet is turned against one stop, and 
 the amount by which the needle deviates from the magnetic 
 east and west (90° and 270°) is observed by reading both ends of 
 the needle. The magnet is turned to the second stop, and the 
 deviations of the two ends of the needle from east and west are 
 
THE VARIOMETER 
 
 369 
 
 again read. ■ The average of the four readings is 8 of the formulae, 
 and it is positive if, when the N end of the magnet is turned 
 toward the east, the n end of the needle is deflected more than 
 270° (that is, the n end is north of west, as shown in Fig. 179). 
 If the magnet and needle have the relative position shown in 
 Fig. 179, the forces acting on the s pole of the needle are the 
 field of the magnet in the direction sF, and the earth's field in 
 the direction sir. If the needle is at rest, the components of 
 
 Fig. 179. Field op Variometer 
 
 these two forces in the directions perpendicular to the needle 
 must be equal and opposite. If F is the strength of the mag- 
 net's field and -ff that of the earth, 6 and S having the meanings 
 already explained, jy pos S = J- cos (6-8). 
 
 The instrument is taken to the second station and adjustments 
 (a) and (e) only are performed. The observations just described 
 are then repeated. 
 
 By adopting suitable subscripts the results of the measures in 
 the two locations may be represented as follows : 
 _Hj = ^^(cos + sin tan Sj), 
 11^ = F{cos 6 + sin 6 tan 8^). 
 
370 ELECTRICITY 
 
 Dividing one equation by the other, the unknown field strength 
 of the magnet is eliminated, and by reducing, 
 
 ITj _ 1 + tan 6 tan Sj 
 H^ 1 + tan 6 tan Sj 
 
 If the deflection of the needle at the first station is exactly 
 90°, 8i is and ^^ = JTj (1 + tan 6 tan 8,). 
 
 In this manner any number of positions may be compared, and 
 if the intensity at one place is known, the intensity for all the 
 locations may be found. 
 
 Refekence. — Koklrausch, Physical Measurements, pp. 252-258. 
 
 278. Oscillation and Deflection Methods for Comparison of Hori- 
 zontal Intensities. — If a magnet is caused to vibrate as a torsion 
 pendulum in a location where the horizontal intensity has the 
 value JET^, and its period is Tj, determined as explained in Art. 83; 
 and if the same magnet has a period T^ when oscillating in a 
 field of horizontal intensity. JTj, then 
 
 If a magnet placed in the magnetic east and west line through 
 the center of a suspended needle which is in the location of hori- 
 zontal intensity -Hj, deflects the needle d-^°; and if the deflection 
 produced in the same apparatus when it is in a location of inten- 
 sity H^, is 6^°; then 
 
 H^: H^ = tan 6^ : tan 6^. 
 
 CLVI. EARTH'S MAGNETIC INCLINATION AND INTENSITY 
 WITH THE EARTH INDUCTOR 
 
 Determine the magnetic inclination with the earth inductor by two 
 methods; determine the component and total intensities. 
 
 279. The Earth Inductor. — A conductor which can be moved 
 in a determined manner in the earth's magnetic field may have 
 currents induced in it which are often useful for magnetic 
 measurements and comparisons. The earth inductor consists of 
 
THE EARTH INDUCTOR 
 
 371 
 
 a coil of wire 20 cm or more in diameter, suspended so as to 
 be turned about an axis in its own plane ; this axis is sup- 
 ported by a frame which rotates on a second axis at right angles 
 to the first, enabling the axis of the coil to be given any desired 
 inclination from horizontal to vertical. The coil may be rotated 
 continuously in this frame by means of a crank ; or stops may 
 be provided limiting its rotation to 180°, which rotation may 
 be produced by spiral springs. The terminals of the coil make 
 contact through rings and 
 brushes with an external 
 circuit. (See Fig. 180.) 
 
 If a circuit of wire be 
 moved in a magnetic field 
 so as to change the num- 
 ber of lines of force inclosed 
 by the coil, a current is in- 
 duced in the circuit which 
 is proportional to the 
 change in the number of 
 lines of force. If the coil 
 consists of n turns of wire, 
 each of area a, with its face 
 perpendicular to the lines 
 of a field whose strength 
 is F, the number of lines 
 
 inclosed is naF. If the plane of the coil is rotated 180°, the 
 same number of lines passes through it in the opposite direction, 
 and there has been a change of 2 naF lines. If the strength 
 of the earth's magnetic field is F, and the direction of the 
 lines of the force makes an angle of dip with the horizontal 
 represented by 6, the horizontal component, H, of this force is 
 F cos 0, and the vertical component, V, is F sin 6. If the 
 plane of the coil of the earth inductor is rotated about an axis 
 parallel to the horizontal component, from a position in which 
 one face of the coil is vertically upward, and therefore in the 
 position in which the maximum vertical component of the force 
 passes through it; to a position 180° from this, the coil will cut 
 
 Eakth Inductor 
 
372 ELECTRICITY 
 
 only the vertical component. If the entire resistance of the 
 circuit of which the coil is a part is R ohms, the quantity of 
 current induced by this rotation, in centimeter-gram-second 
 
 ^"^ R.IO^ ■ 
 
 If the plane of the coil rotates about an axis parallel to the 
 vertical component, from a position of its face perpendicular 
 to the horizontal component, to the position 180° from this, 
 the coil will cut only the horizontal component of the force, 
 and the induced current will be 
 
 _ 2 naF cos 6 
 ^" ^.108 ■ 
 
 The ratio of the first of these quantities to the second is the 
 ratio of the vertical to the horizontal components, and this is 
 the tangent of the angle of dip. The ratio of the two quantities 
 can be determined without knowing any of the factors by which 
 they are expressed above, by measuring the currents with a ballis- 
 tic galvanometer. For determining the absolute value of the 
 quantities these formulae or those of Art. 281 may be employed. 
 280. Magnetic Inclination. — Adjust the earth inductor so that 
 the two rectangular axes of rotation shall be level, and so that 
 the axis of the coil shall lie in the plane of the magnetic merid- 
 ian and the plane of the coil shall be horizontal. These con- 
 ditions may be secured by the aid of a compass and a level. 
 Connect the coil in series with a ballistic galvanometer (Art. 
 253) and a resistance for controlling the current. Then, by 
 means of the spiral springs, release and stop, cause the coil 
 to be suddenly rotated 180°, and observe the galvanometer 
 deflection produced. Make ten or more observations, and cor- 
 rect the average of the readings as described in Art. 252 ; let 
 di be the corrected average. Now place the axis of the coil 
 vertical, as determined by a divided circle, by a plumb line or 
 by other convenient method, and having the plane of the coil 
 perpendicular to the plane of the magnetic meridian, cause the 
 coil to be suddenly rotated 180°. Let d^ be the corrected 
 
MAGNETIC INTENSITY 373 
 
 average value of the observed galvanometer deflections, obtained 
 with the coil in this position. Then the magnetic inclination, 
 or the angle of dip, 6, is obtained from the following relation. 
 
 tan0 = 
 
 _^i 
 
 d^ 
 
 Second Method for Inclination. — If the coil of the earth 
 inductor is rotated about an axis parallel to the lines of force 
 of the field, there will be no variation in the number of lines of 
 force inclosed by the coil and no current will be induced. By 
 ti-ial such an inclination of the axis may be found, which gives 
 at once the magnetic dip. 
 
 281. Magnetic Intensity. — The absolute value of either com- 
 ponent of the earth's magnetic intensity may be determined 
 from the observations of the preceding article if the constants 
 of the earth inductor and the galvanometer are known. The 
 dip having been determined, the total intensity may also be 
 found. 
 
 In accordance with the explanation of Arts. 267, 268, and 
 279, there follow the relations: 
 
 2naFs\n6 ^ ,^ , , ^. ■, 
 
 and V — ^ oi" " — n 
 
 = i^sin = — =^ i~ — * , 
 
 Zna 
 
 ^ K(1 + \X) djt 108 
 
 and similarly n=F cos 6 = — !^ 2—^^ ; 
 
 •' Ana 
 
 V H 
 
 from which F = — — z. = ^ ■ 
 
 sin u cos o 
 
 The maker of an earth inductor usually supplies a certifi- 
 cate giving the number of turns, w, of wire in the coil, and the 
 diameters of the coils from which the average area, a, may be 
 computed ; or the wires may be exposed to view for counting 
 and measurement. The values of all the other quantities con- 
 tained in these formulae are sufficiently explained in the articles 
 to which reference is made. 
 
374 
 
 ELECTKICITY 
 
 CLVII. DISTRIBUTION OF MAGNETISM BY ROWLAND'S 
 METHOD 
 
 Determine the distribution and total flow of induction of a permanent 
 
 magnet. 
 
 282. Flow of Induction from a Magnet. — If a circuit be 
 moved from a- position in which it incloses all the lines of 
 force of a magnet, to a position in which it incloses none of 
 these lines, there will be induced in it a quantity of electricity 
 which measures the total flow of induction. 
 
 Provide a test coil of wire of a known number of turns, n, 
 which will fit around the bar magnet, but with sufficient clear- 
 ance to permit the coil to slide along the bar. Place the coil 
 over the center of the bar magnet, and connect its terminals, 
 through a resistance, B, to a ballistic galvanometer. Suddenly 
 draw the coil off the bar and observe the galvanometer deflec- 
 tion. Let this deflection, corrected as explained in Art. 252, 
 be d, and let the constant of the galvanometer (Art. 268) be K, 
 and its logarithmic decrement (Art. 267) be X; the number of 
 unit tubes of induction emanating from the magnet is 
 
 283. Distribution of Magnetism. — When the test coil is on 
 the bar magnet, stops may be clamped on each side of it to limit 
 
 Fig. 181. Distribution or Magnetism 
 
 its motion, and if the coil is suddenly moved from one stop to 
 the other a current will be induced which is proportional to 
 
COMPARISON OF MAGNETIC FIELDS 
 
 375 
 
 the average normal component of magnetization between the 
 two points. (See Fig. 181.) By placing the two stops so as to 
 include between them the successive portions of the length of 
 the magnet from one end to the other, and using the test 
 coil in each position, the galvanometer deflections will be pro- 
 portional to the flow of induction from the magnet at these 
 points. If these deflections are plotted as ordinates, with the 
 positions as abscissae, a curve representing the distribution of 
 magnetism is obtained. 
 
 References. — Stewart and Gee, Practical Physios, Vol. II, p. 388; Barker, 
 Physics, pp. 674 and 812. 
 
 CLVin. INTENSITY OF A MAGNETIC FIELD BY ROWLAND'S 
 
 METHOD 
 
 Find the number of unit tubes of induction per square centimeter in the 
 field between the two poles of an electromagnet. 
 
 284. Comparison of Magnetic Fields with the Earth's Field. — 
 
 Provide a test coil of wire of such size as to inclose that portion 
 of the field which it is desired to measure. Connect the coil C 
 
 Fio. 182. Intensity or Magnetic Field 
 
 (Fig. 182) in series with a resistance, B, an earth inductor, /, and 
 a ballistic galvanometer, G. Suddenly rotate the plane of the 
 coil 180° from a position in which the lines of force are perpen- 
 dicular to this plane ; let the galvanometer deflection produced 
 be d^. The earth inductor, being in adjustment for measuring 
 the horizontal component, H, of the earth's field (Art. 279), is 
 
376 ELECTRICITY 
 
 rotated 180° ; let the resulting galvanometer deflection be d^. 
 If the number of turns of wire in the test coil is N, each of area 
 A, and the number of turns in the earth inductor is n, each of 
 area a, then the strength of the field is 
 
 NAd^' 
 
 If the test coil, instead of being rotated 180° in the field, is 
 simply withdrawn from the field, the deflection d^ will be only- 
 half as large as in the above case, and 
 
 F-^2H 
 
 nad-i 
 NAd^' 
 
 Eefeeence. — Stmuart and Gee, Practical Physics, Vol. II, pp. 382-387. 
 
 CLIX. TEMPERATURE COEFFICIENT OF A MAGNET BY 
 DEFLECTION METHODS 
 
 Determine the temperature coefficient of a permanent magnet. 
 
 285. Temperature Coefficient of a Magnet. — The change in 
 magnetism, per unit magnetism, per degree change of tem- 
 perature, is the temperature coefficient of a magnet. It has 
 a negative value for increasing temperatures. It may be 
 determined by measuring the change in the effect of a magnet 
 upon a suspended needle, when the temperature of the magnet 
 changes. Two methods of manipulation by which the needle 
 will be more sensitive to the control of the magnet are given. 
 
 Zero Deflection Method. — Immerse the magnet to be experi- 
 mented upon in an oil bath, the temperature of which can be 
 conveniently changed. Place the bath magnetically east or 
 west and at a convenient distance from a suspended magnetic 
 needle, such for instance as the needle of a dead-beat galvanom- 
 eter. Let the angle of deflection produced (not exceeding 
 20° or 30°) be 6. Now, by means of a compensating magnet, 
 reduce the deflection of the needle nearly to zero. The magnet 
 remaining unchanged in position, let its temperature be altered 
 
TEMPERATURE COEFFICIENT OF A MAGNET 377 
 
 t°. Let the resulting change in the deflection be d, as read 
 with a telescope and scale distant D from the mirror on the 
 needle. Then the temperature coefficient is 
 
 M = 
 
 2 tan Dt 
 
 Ninety-Degree Deflection Method. — Place the magnet whose 
 coefficient is desired, in the oil bath, in the same horizontal plane 
 with a short magnetometer needle, and with its center in the 
 plane of the magnetic meridian through the center of the 
 needle. The magnet is to be adjusted accurately to the mag- 
 netic meridian by being turned so that the needle is unde- 
 flected. Now turn the magnet out of the meridian till the 
 needle is deflected about 90°, and let be the angle through 
 which the magnet has been turned. Change the temperature 
 of the magnet t° without altering its position, and, if the needle 
 deflection changes 8 degrees, the temperature coefficient is 
 
 1 8 
 ^ ~ 2 tun 6' t' 
 
 References. — Kohlrausch, Physical Measurements, p. 260 ; Stewart and Gee, 
 Practical Physics, Vol. II, p. 486. 
 
Paet YII — Appbitdix 
 chapter xxvi 
 
 TABLES, CONSTANTS, AND EEFEEENCES 
 
 286. Explanation of Tables. — The data contained in the 
 following tables have been selected and arranged from many 
 sources. The results of recent investigations have been incor- 
 porated as far as possible. The tables for the density and 
 volume of water and of mercury are based upon the determina- 
 tions of density made at the International Bureau of Weights 
 and Measures and reported to the International Congress of 
 Physics, held at Paris in 1900. The table of wave lengths of 
 light is founded upon Michelson's determination of the wave 
 length of cadmium light. 
 
 The tables have been limited both as to subject-matter and 
 range by the requirements of ordinary laboratory practice. 
 
 The values of some of the quantities- tabulated, such as the 
 elastic constants of solids, and electrical resistances, vary greatly 
 with circumstances. Sometimes limiting values are given for 
 these quantities, and at other times such single values as seem 
 best to represent the quantities as ordinarily observed. 
 
 It has not seemed convenient to indicate the sources of 
 information in general. The tables which have been most 
 frequently consulted are : Landolt und Bornstein, Physikalisch- 
 Chemische Tabellen ; ICohlrausch, Praktische Physik ; Everett, 
 The C. G. S. System of Units ; the Smithsonian Physical Tables 
 and the Smithsonian Meteorological Tables. 
 
 378 
 
TABLES 
 
 379 
 
 1. Reduction to Vacuum of Weighings made with Brass 
 Weights in Air 
 
 If a body of density s has in air the apparent weight of m grams, its weight 
 reduced to vacuum is m + »«fc grams, k is computed for air of density 
 0.0012 and for brass weights of density 8.4. (See Art. 48.) 
 
 ^ ^ 0.0012 0.0012 
 ~ s 8.4 ■ 
 
 s 
 
 k 
 
 s 
 
 k 
 
 
 k 
 
 s 
 
 k 
 
 0.7 
 
 + 0.00157 
 
 1.3 
 
 + 0.00078 
 
 3.0 
 
 + 0.00026 
 
 10 
 
 - 0.00002 
 
 0.8 
 
 136 
 
 1.4 
 
 71 
 
 3.5 
 
 20 
 
 12 
 
 4 
 
 0.9 
 
 119 
 
 1.6 
 
 61 
 
 4.0 
 
 16 
 
 14 
 
 6 
 
 1.0 
 
 106 
 
 1.8 
 
 52 
 
 5.0 
 
 10 
 
 16 
 
 7 
 
 1.1 
 
 095 
 
 2.0 
 
 46 
 
 6.0 
 
 06 
 
 18 
 
 8 
 
 1.2 
 
 086 
 
 2.5 
 
 34 
 
 7.0 
 
 03 
 
 20 
 
 8 
 
 1.3 
 
 078 
 
 3.0 
 
 26 
 
 8.0 
 
 01 
 
 22 
 
 9 
 
 2. Density of Various Substances 
 
 Solids 
 
 Alnminum 
 
 2.7 
 
 Beeswax 
 
 0.96 
 
 Brass 
 
 8.1-8.7 
 
 Calcspar 
 
 2.7 
 
 CaoatchOQC 
 
 0.95 
 
 Copper 
 
 8.5-8.9 
 
 Glass, Common 
 
 2.4-2.6 
 
 Flint 
 
 3.0-5.9 
 
 Gold 
 
 19.3 
 
 Hard Eubber 
 
 1.15 
 
 Ice 
 
 0.9167 
 
 Iridium 
 
 21.8-22.4 
 
 Iron, Cast 
 
 7.1-7.7 
 
 Wrought 
 
 7.8 
 
 Steel 
 
 7.8 
 
 Ivory 
 Lead 
 
 1.9 
 
 11.3 
 
 Nickel 
 
 8.8 
 
 Platinum 
 
 21.4 
 
 Quartz 
 
 2.65 
 
 Silver 
 
 Tin 
 
 Wood, Box 
 Cork 
 Ebony 
 
 Lignum 
 
 Vitse 
 Mahogany 
 Oak 
 Pine 
 Pitch Pine 
 
 10.5 
 
 7.3 
 
 0.95-1.16 
 
 0.2 
 
 1.2 
 
 1.2 
 0.56-0.85 
 0.60-0.90 
 0.35-O.50 
 
 0.84 
 
 LiQOIDS AT 20° 
 
 Alcohol 
 Amyl Acetate 
 Bromoform 
 Carbon 
 
 Bisulphide 
 Chloroform 
 Ether 
 
 0.789 
 
 0.88 
 
 2.86 
 
 1.264 
 1.489 
 0.715 
 
 Glycerin 
 Hydrochloric 
 
 Acid, 40% 
 Methyl Iodide 
 Nitric Acid 
 Olive Oil 
 Petroleum 
 
 1.23 
 
 1.20 
 
 3.34 
 
 1.522 
 
 0.92 
 
 0.88 
 
 Sulphuric Acid 
 Turpentine 
 
 * Water, Pure 
 
 Sea 
 
 * Mercury 
 
 *SeeTable3 
 
 1.832 
 0.87 
 0.998 
 1.024 
 13.546 
 
 Gases at 0° and 76 cm 
 
 Air 
 
 Carbon Dioxide 
 
 0.001 293 
 0.001 965 
 
 Chlorine 
 Hydrogen 
 
 0,003091 
 0.00008987 
 
 Nitrogen 
 Oxygen 
 
 0.001 251 
 0.001 429 
 
380 
 
 APPENDIX 
 
 3. Density of Water and 
 
 of Mercury 
 
 at the Temperature t of the 
 Hydrogen Thermometer 
 
 (Thiesen, Guillaume, International 
 Congress of Physics, Paris, 1900) 
 
 t 
 
 Water 
 
 t 
 
 Mercdrv 
 
 o 
 
 
 0.999 823 
 
 o 
 
 
 18.5930 
 
 1 
 
 .999 882 
 
 1 
 
 .5925 
 
 2 
 
 .999 923 
 
 2 
 
 .5901 
 
 3 
 
 .999 947 
 
 3 
 
 .5876 
 
 4 
 
 0.999 955 
 
 4 
 
 .5851 
 
 5 
 
 0.999 947 
 
 5 
 
 13.5827 
 
 6 
 
 .999 923 
 
 6 
 
 .5802 
 
 7 
 
 .999 884 
 
 7 
 
 .5777 
 
 8 
 
 .999 831 
 
 8 
 
 .5753 
 
 9 
 
 .999 763 
 
 9 
 
 .5728 
 
 10 
 
 0.999 682 
 
 10 
 
 13.5703 
 
 11 
 
 .999 587 
 
 11 
 
 .5679 
 
 12 
 
 .999 480 
 
 12 
 
 .5654 
 
 13 
 
 .999 359 
 
 13 
 
 .5629 
 
 14 
 
 .999 226 
 
 14 
 
 .5605 
 
 15 
 
 0.999 081 
 
 15 
 
 13.5581 
 
 16 
 
 .998 925 
 
 16 
 
 .5556 
 
 17 
 
 .998 756 
 
 17 
 
 .5531 
 
 18 
 
 .998 577 
 
 18 
 
 .5507 
 
 19 
 
 .998 387 
 
 19 
 
 .5482 
 
 20 
 
 0.998 185 
 
 20 
 
 13.5457 
 
 21 
 
 .997 974 
 
 21 
 
 .5433 
 
 22 
 
 .997 752 
 
 22 
 
 .5408 
 
 23 
 
 .997 520 
 
 23 
 
 .5384 
 
 24 
 
 .997 278 
 
 24 
 
 .5359 
 
 25 
 
 0.997 026 
 
 25 
 
 13.5335 
 
 26 
 
 .996 765 
 
 26 
 
 .5310 
 
 27 
 
 .996 496 
 
 27 
 
 .5286 
 
 28 
 
 .996 214 
 
 28 
 
 .5261 
 
 29 
 
 .995 926 
 
 29 
 
 .5237 
 
 30 
 
 0.995 628 
 
 30 
 
 13.5212 
 
 
 4. Volume of a Glass Vessel at 20° 
 
 If a glass vessel apparently contains 1 g 
 of water (or of mercury) at the temper- 
 ature t, being weighed with brass 
 weights in air of density 0.00120, the 
 volume of the vessel at 20° is as 
 given in the table. The coefficient of 
 cubical expansion of glass is assumed 
 to be 0.000025. 
 
 t 
 
 Volume-Water 
 
 t 
 
 VOLUME-MERCnBY 
 
 o 
 
 com 
 
 Q 
 
 com 
 
 5 
 
 1.001 49 
 
 5 
 
 0.073 647 
 
 6 
 
 .00149 
 
 6 
 
 .073 658 
 
 7 
 
 .00150 
 
 7 
 
 .073 669 
 
 8 
 
 .00153 
 
 8 
 
 .073 681 
 
 9 
 
 .00157 
 
 9 
 
 .073 693 
 
 10 
 
 1.001 63 
 
 10 
 
 0.073 704 
 
 11 
 
 .00170 
 
 11 
 
 .073 716 
 
 12 
 
 .001 78 
 
 12 
 
 .073 727 
 
 13 
 
 .001 88 
 
 13 
 
 .073 739 
 
 14 
 
 .00199 
 
 14 
 
 .073 750 
 
 15 
 
 1.002 10 
 
 15 
 
 0.073 762 
 
 16 
 
 .002 23 
 
 16 
 
 .073 773 
 
 17 
 
 .002 38 
 
 17 
 
 .073 785 
 
 18 
 
 .002 53 
 
 18 
 
 .073 797 
 
 19 
 
 .002 70 
 
 19 
 
 .073 808 
 
 20 
 
 1.002 87 
 
 20 
 
 0.073 820 
 
 21 ■ 
 
 .003 06 
 
 21 
 
 .073 831 
 
 22 
 
 .003 26 
 
 22 
 
 .073 843 
 
 23 
 
 .003 47 
 
 23 
 
 .073 854 
 
 24 
 
 .003 69 
 
 24 
 
 .073 866 
 
 25 
 
 1.003 92 
 
 25 
 
 0.073 877 
 
 26 
 
 .004 15 
 
 26 
 
 .073 889 
 
 27 
 
 .004 39 
 
 27 
 
 .073 900 
 
 28 
 
 .004 65 
 
 28 
 
 .073 912 
 
 29 
 
 .004 92 
 
 29 
 
 .073 923 
 
 30 
 
 1.005 19 
 
 30 
 
 0.073 935 
 
 5. Coefficient of Static Friction 
 
 6. Relative 
 Viscosity at 19° 
 
 Wood on Wood 
 Metal on Oak, Dry 
 Wet 
 Metal on Metal,Dry 
 Steel on Glass 
 
 0.20-0.50 
 0.50-0.60 
 0.24-0.26 
 0.15-0.20 
 0.10-0.20 
 
 Leather on Metal, Dry 
 Oily 
 Leather on Oak 
 Smooth Surfaces,Oiled 
 Rolling Friction 
 
 0.50 -O.60 
 0.15 -0.23 
 0.27 -0.38 
 0.03 -0.08 
 0.004-0.006 
 
 Cylinder Oil 
 Machine Oil 
 Wagon Oil 
 Olive Oil 
 Whale Oil 
 
TABLES 
 
 381 
 
 
 7. Elastic Constants of Solids 
 
 
 SCBSTASOK 
 
 Young's 
 Modulus 
 
 Rigidity 
 
 Elastic 
 Limit 
 
 Velocity op 
 
 SOUKD 
 
 Aluminum 
 Brass , 
 Copper 
 Glass 
 Iron, Cast 
 
 Iron, Wrought 
 
 Steel 
 Platinum 
 Silver 
 Wood 
 
 dynes per scm 
 6.5 X 10" 
 7.7-9.3 
 8.5-11.6 
 
 4-^3 
 6.9-10.8 
 
 19.3-20.9 X 1011 
 
 20.1-21.6 
 
 15. 
 
 7. 
 
 0.7-1.5 
 
 dynes per scm 
 
 2.4^3.3 X 10" 
 
 3.1-3.6 
 
 3.2-4.5 
 
 1.7-2.4 
 
 2.0-4.2 
 
 7.7-8.5 X 10" 
 
 8.0-8.8 
 
 6.1-6.5 
 
 2.6 
 
 0.07-0.12 
 
 dynes per scm 
 
 4.5-11 X 108 
 
 3-7 
 2.3 
 
 7. 
 
 20 X 108 
 
 33-40 
 
 14-23 
 
 3-11 
 
 1.5-2.4 
 
 cm per s 
 5.10 X 105 
 3.66 
 3.74 
 4.42 
 4.32 
 
 5.06 X 106 
 
 5.22 
 
 2.69 
 
 2.61 
 
 3. 
 
 Compressibility of Water, 
 Mercury, and Glass 
 
 9. Surface Tension at 20° 
 
 Substance 
 
 COMPBESSIOX 
 PEK MEGAEAKYE 
 
 Elasticity 
 OF Volume 
 
 Water 
 
 Mercury 
 
 Glass 
 
 5.0 X 10-5 
 3.9 X 10-6 
 2.6 X 10-6 
 
 2.03 X 10i» 
 2.60 X 10" 
 3.90 X 1011 
 
 Water, Pure 
 
 73 
 
 Water, Saturated with Olive 
 
 
 Oil 
 
 41 
 
 Water, Saturated with Oleate 
 
 
 of Soda 
 
 25 
 
 Olive Oil 
 
 35 
 
 Turpentine 
 
 28 
 
 Alcohol 
 
 22 
 
 10. Absolute Value of Gravity 
 
 Not reduced to sea level. By the United States Coast and Geodetic Survey, 
 founded upon the provisional value for Potsdam, 1900-1903 
 
 Atlanta 
 
 979.523 
 
 Denver 
 
 979.608 
 
 San Francisco 
 
 979.964 
 
 Austin 
 
 979.282 
 
 Ithaca 
 
 980.299 
 
 St. Louis 
 
 980.000 
 
 Boston 
 
 980.395 
 
 Kansas City 
 
 979.989 
 
 Terra Haute 
 
 980.071 
 
 Cambridge 
 
 980.397 
 
 Little Rock 
 
 979.720 
 
 Washington 
 
 980.111 
 
 Charleston 
 
 979.545 
 
 Montreal 
 
 980.742 
 
 Worcester 
 
 980.323 
 
 Charlottesville 
 
 979.937 
 
 New Orleans 
 
 979.323 
 
 London 
 
 981.201 
 
 Chicago 
 
 980.277 
 
 New York 
 
 980.266 
 
 Paris 
 
 980.941 
 
 Cleveland 
 
 980.240 
 
 Philadelphia 
 
 980.195 
 
 Potsdam 
 
 981.274 
 
 Cincinnati 
 
 980.000 
 
 Princeton 
 
 980.177 
 
 Para 
 
 978.03 
 
 Colorado Springs 
 
 979.489 
 
 Salt Lake City 
 
 979.802 
 
 Spitzbergen 
 
 983.08 
 
 11. Local Geographical Data for Cleveland 
 
 Case Observatory, Latitude, 41°30'14".53; Longitude, 6''26"25'.8; 
 Main Building, Altitude, 21210 cm; Bearing, West of North, 41° 15' 
 Reduction of Central Standard to Local Mean Time -I- 33" 34".l8 
 
382 
 
 APPENDIX 
 
 12. Specific Gravity of Air 
 
 At the temperature t, and under the pressure of Hem of mercury, the 
 specific gravity of air referred to water at 4° is 
 
 0.001293 H 
 
 
 
 
 1 + 0.00367 ( 
 
 76 
 
 
 
 
 t 
 
 Pbbssuke H in Centimeters 
 
 Pbopoetional 
 Pakts 
 
 72.0 
 
 73.0 
 
 74.0 
 
 75.0 
 
 76.0 
 
 77.0 
 
 o 
 10 
 
 0.001 182 
 
 0.001 198 
 
 0.001 215 
 
 0.001 231 
 
 0.001 247 
 
 0.001264 
 
 17 
 em 1 
 
 11 
 
 178 
 
 193 
 
 210 
 
 227 
 
 243 
 
 259 
 
 0.1 
 0.2 
 0.3 
 
 2 
 3 
 6 
 
 12 
 
 173 
 
 190 
 
 206 
 
 222 
 
 239 
 
 255 
 
 13 
 
 169 
 
 186 
 
 202 
 
 218 
 
 234 
 
 251 
 
 0.4 
 
 7 
 
 14 
 
 165 
 
 181 
 
 198 
 
 214 
 
 230 
 
 246 
 
 0.5 
 0.6 
 0.7 
 
 8 
 10 
 12 
 
 15 
 
 0.001 161 
 
 0.001 177 
 
 0.001 193 
 
 0.001 210 
 
 0.001 226 
 
 0.001 242 
 
 0.8 
 
 14 
 
 16 
 
 157 
 
 173 
 
 189 
 
 205 
 
 221 
 
 238 
 
 0.9 
 
 15 
 
 17 
 
 153 
 
 169 
 
 185 
 
 201 
 
 217 
 
 233 
 
 cm '' II 
 
 18 
 
 149 
 
 165 
 
 181 
 
 197 
 
 213 
 
 229 
 
 
 
 19 
 
 145 
 
 161 
 
 177 
 
 193 
 
 209 
 
 225 
 
 0.2 
 0.3 
 
 3 
 5 
 
 20 
 
 0.001 141 
 
 0.001 157 
 
 0.001 173 
 
 0.001 189 
 
 0.001 205 
 
 0.001 221 
 
 0.4 
 
 6 
 
 21 
 
 137 
 
 153 
 
 169 
 
 185 
 
 201 
 
 216 
 
 0.6 
 
 10 
 
 22 
 
 134 
 
 149 
 
 165 
 
 181 
 
 197 
 
 212 
 
 0.7 
 
 11 
 
 23 
 
 130 
 
 145 
 
 161 
 
 177 
 
 193 
 
 208 
 
 0.9 
 
 11 
 
 24 
 
 126 
 
 142 
 
 157 
 
 173 
 
 189 
 
 204 
 
 15 " 1 
 
 25 
 
 0.001 122 
 
 0.001 138 
 
 0.001 153 
 
 0.001 169 
 
 0.001 185 
 
 0.001200 
 
 cm 
 0.1 
 
 1 
 
 26 
 
 118 
 
 134 
 
 149 
 
 165 
 
 181 
 
 196 
 
 0.2 
 0.3 
 0.4 
 
 3 
 4 
 6 
 
 27 
 
 115 
 
 130 
 
 146 
 
 161 
 
 177 
 
 192 
 
 28 
 
 111 
 
 126 
 
 142 
 
 157 
 
 173 
 
 188 
 
 0.6 
 
 7 
 
 29 
 
 107 
 
 123 
 
 138 
 
 153 
 
 .169 
 
 184 
 
 0.6 
 0.7 
 0.8 
 
 9 
 10 
 12 
 
 30 
 
 0.001 104 
 
 0.001 119 
 
 0.001 134 
 
 0.001 150 
 
 0.001 165 
 
 0.001 180 
 
 0.9 
 
 13 
 
 13. 
 
 Capillary Depression of Mercury 
 
 in Glass Tubes 
 
 DIAMETBK 
 OF 
 
 Tdbb 
 
 Height of Meniscus in Centimeters 
 
 0.04 
 
 0.06 
 
 0.08 
 
 0.10 
 
 0.12 
 
 0.14 
 
 0.16 
 
 0.18 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 0.4 
 
 0.083 
 
 0.122 
 
 0.154 
 
 0.198 
 
 0.237 
 
 
 
 
 0.5 
 
 .047 
 
 .065 
 
 .086 
 
 .119 
 
 .145 
 
 0.180 
 
 
 
 0.6 
 
 .027 
 
 .041 
 
 .056 
 
 .078 
 
 .098 
 
 .121 
 
 0.143 
 
 
 0.7 
 
 .018 
 
 .028 
 
 .040 
 
 .053 
 
 .067 
 
 .082 
 
 .097 
 
 .113 
 
 0.8 
 
 
 .020 
 
 .029 
 
 .038 
 
 .046 
 
 .056 
 
 .065 
 
 0.077 
 
 0.9 
 
 
 0.015 
 
 0.021 
 
 0.028 
 
 0.033 
 
 0.040 
 
 0.046 
 
 .052 
 
 1.0 
 
 
 
 .015 
 
 .020 
 
 .025 
 
 .029 
 
 .033 
 
 0.037 
 
 1.1 
 
 
 
 .010 
 
 .014 
 
 .018 
 
 .021 
 
 .024 
 
 .027 
 
 1.2 
 
 
 
 .007 
 
 .010 
 
 .013 
 
 .015 
 
 .018 
 
 .019 
 
 1.3 
 
 V 
 
 
 .004 
 
 .007 
 
 .010 
 
 .012 
 
 .013 
 
 .014 
 
TABLES 
 
 383 
 
 14. Reduction of Barometer Readings to 0° 
 
 When the height of a mercury column has been measured with a brass 
 scale, the length of which is correct at 0°, the mercury and scale being 
 at the temperature t, the observed height will be reduced to 0° by 
 subtracting the quantity corresponding to the temperature and height, 
 taken from the following table. (See Art. 54.) 
 
 t 
 
 
 Observed Height in Centimeters 
 
 
 
 72.0 
 
 73.0 
 
 74.0 
 
 76.0 
 
 76.0 
 
 77.0 
 
 
 o 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 
 
 
 10 
 
 0.12 
 
 0.12 
 
 0.12 
 
 0.12 
 
 0.12 
 
 0.12 
 
 
 11 
 
 .13 
 
 .13 
 
 .13 
 
 .13 
 
 .14 
 
 .14 
 
 
 12 
 
 .14 
 
 .14 
 
 .14 
 
 .15 
 
 .15 ' 
 
 .15 
 
 
 13 
 
 .15 
 
 .15 
 
 .16 
 
 .16 
 
 .16 
 
 .16 
 
 
 14 
 
 .16 
 
 .17 
 
 .17 
 
 .17 
 
 .17 
 
 .17 
 
 
 15 
 
 0.17 
 
 0.18 
 
 0.18 
 
 0.18 
 
 0.18 
 
 0.19 
 
 
 16 
 
 .19 
 
 .19 
 
 .19 
 
 .19 
 
 .20 
 
 .20 
 
 
 17 
 
 .20 
 
 .20 
 
 .20 
 
 .21 
 
 .21 
 
 .21 
 
 
 18 
 
 .21 
 
 .21 
 
 .22 
 
 .22 
 
 .22 
 
 .22 
 
 
 19 
 
 .22 
 
 .22 
 
 .23 
 
 .23 
 
 .23 
 
 .24 
 
 
 20 
 
 0.23 
 
 0.24 
 
 0.24 
 
 0.24 
 
 0.25 
 
 0.25 
 
 
 21 
 
 .25 
 
 .25 
 
 .25 
 
 .25 
 
 .26 
 
 .26 
 
 
 22 
 
 .26 
 
 .26 
 
 .26 
 
 .27 
 
 .27 
 
 .27 
 
 
 23 
 
 .27 
 
 .27 
 
 .28 
 
 .28 
 
 .28 
 
 .29 
 
 
 24 
 
 .28 
 
 .28 
 
 .29 
 
 .29 
 
 .29 
 
 .30 
 
 
 25 
 
 0.29 
 
 0.30 
 
 0.30 
 
 0.30 
 
 0.31 
 
 0.31 
 
 
 26 
 
 .30 
 
 .31 
 
 .31 
 
 .32 
 
 .32 
 
 .32 
 
 
 27 
 
 .31 
 
 .32 
 
 .32 
 
 .33 
 
 .33 
 
 .34 
 
 
 28 
 
 .33 
 
 .33 
 
 .34 
 
 .34 
 
 .34 
 
 .35 
 
 
 29 
 
 .34 
 
 .34 
 
 .35 
 
 .35 
 
 .36 
 
 .36 
 
 
 30 
 
 0.35 
 
 .0.35 
 
 0.36 
 
 0.36 
 
 0.37 
 
 0.37 
 
 
 15. Reduction of Mercury-in-Glass Thermometer Reading to the 
 Normal Hydrogen Scale 
 
 for Jena Normal Glass, 16'^ 
 (See Art. 133) 
 
 Reading 
 
 0° 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 COEBECTION 
 
 0°.000 
 
 -0.055 
 
 -0,090 
 
 -0.109 
 
 -0.115 
 
 -0.109 
 
 Reading 
 
 50° 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 CORBBCTION 
 
 -0°.109 
 
 -0.096 
 
 -0.076 
 
 -0.053 
 
 -0.027 
 
 0.000 
 
384 
 
 APPENDIX 
 
 16. Boiling Temperature of Water / 
 
 at the Barometric Pressure B 
 
 B 
 
 t 
 
 B 
 
 t 
 
 B 
 
 t 
 
 B 
 
 t 
 
 B 
 
 t 
 
 B 
 
 t 
 
 cm 
 
 o 
 
 cm 
 
 o 
 
 cm 
 
 
 
 cm 
 
 o 
 
 cm 
 
 o 
 
 cm 
 
 o 
 
 72.0 
 
 98.49 
 
 73.0 
 
 98.88 
 
 74.0 
 
 99.26 
 
 75.0 
 
 99.63 
 
 76.0 
 
 100.00 
 
 77.0 
 
 100.37 
 
 .1 
 
 .53 
 
 .1 
 
 .92 
 
 .1 
 
 .29 
 
 .1 
 
 .67 
 
 .1 
 
 .04 
 
 .1 
 
 .40 
 
 .2 
 
 .57 
 
 .2 
 
 .95 
 
 .2 
 
 .33 
 
 .2 
 
 .70 
 
 .2 
 
 .07 
 
 .2 
 
 .44 
 
 .3 
 
 .61 
 
 .3 
 
 98.99 
 
 .3 
 
 .37 
 
 .3 
 
 .74 
 
 .3 
 
 .11 
 
 .3 
 
 .48 
 
 A 
 
 .65 
 
 .4 
 
 99.03 
 
 .4 
 
 .41 
 
 .4 
 
 .78 
 
 .4 
 
 .15 
 
 .4 
 
 .51 
 
 72.5 
 
 98.69 
 
 73.5 
 
 99.0T 
 
 74.5 
 
 99.44 
 
 75.5 
 
 99.82 
 
 76.5 
 
 100.18 
 
 77.5 
 
 100.55 
 
 .6 
 
 .72 
 
 .fi 
 
 .10 
 
 .6 
 
 .48 
 
 .6 
 
 .85 
 
 .6 
 
 .22 
 
 .6 
 
 .58 
 
 ,7 
 
 .76 
 
 .7 
 
 .14 
 
 .7 
 
 .52 
 
 .7 
 
 .89 
 
 .7 
 
 .26 
 
 .7 
 
 .62 
 
 .8 
 
 .80 
 
 .8 
 
 .18 
 
 .8 
 
 .55 
 
 .8 
 
 .93 
 
 .8 
 
 .29 
 
 .8 
 
 .66 
 
 .9 
 
 .84 
 
 .9 
 
 .22 
 
 .9 
 
 .59 
 
 .9 
 
 .96 
 
 .9 
 
 .33 
 
 .9 
 
 .69 
 
 17. Fixed Points for High Temperatures 
 
 (See Art. 138) 
 
 SCBSTANCE 
 
 BoiLiifG Point 
 
 Substance 
 
 MELTING POINT 
 
 Alcohol 
 
 o 
 78.26 
 
 Tin 
 
 o 
 232 
 
 Water 
 
 100.0 
 
 Zinc 
 
 419 
 
 Naphthalene 
 
 218.0 
 
 Aluminum 
 
 657 
 
 Mercury- 
 
 356.7 
 
 Gold 
 
 1064 
 
 Sulphur 
 
 445.2 
 
 Copper. 
 
 1084 
 
 Zinc 
 
 930. 
 
 Platinum 
 
 1775 
 
 18. Heat Constants of Liquids 
 
 (See also Table 17) 
 
 
 Cubical 
 
 Specific 
 
 Boiling 
 
 Heat of 
 
 
 Expansion 
 
 Hkat 
 
 Point 
 
 Vapobization 
 
 
 
 
 o 
 
 cal 
 
 Alcohol 
 
 0.001 10 
 
 0.58 
 
 78.3 
 
 210. 
 
 Benzene 
 
 0.00124 
 
 0.40 
 
 80.3 
 
 94.4 
 
 Glycerin 
 
 0.000 50 
 
 0.58 
 
 290. 
 
 
 Mercury 
 
 0.000 182 
 
 0.0332 
 
 356.7 
 
 62. 
 
 Sulphur 
 
 0.000 21 
 
 0.2 
 
 445.2 
 
 362. 
 
 Turpentine 
 
 0.000 94 
 
 0.42 
 
 159. 
 
 70. 
 
 Water 
 
 0.00018 
 
 1. 
 
 100.0 
 
 539. 
 
TABLES 
 
 385 
 
 19. Heat Constants of Solids 
 
 (See also Table 17) 
 
 SOBSTANCE 
 
 Linear 
 
 Specific 
 
 Melting 
 
 Heat of 
 
 Expansion 
 
 Heat 
 
 Point 
 
 Fusion 
 
 
 
 
 o 
 
 oal 
 
 Aluminum 
 
 0.0000 23 
 
 0.22 
 
 657 
 
 
 Brass 
 
 19 
 
 .093 
 
 900 
 
 
 Copper 
 
 17 
 
 .093 
 
 1084 
 
 
 Grerman Silver 
 
 18 
 
 .095 
 
 1000 
 
 
 Glass 
 
 0.0000 08 
 
 0.19 
 
 1100 
 
 
 Iron 
 
 12 
 
 .11 
 
 1300 
 
 30. 
 
 Nickel 
 
 13 
 
 .11 
 
 1470 
 
 4.6 
 
 Platinum 
 
 09 
 
 .032 
 
 1775 
 
 27. 
 
 Silver 
 
 0.000019 
 
 0.056 
 
 961 
 
 21. 
 
 Steel 
 
 11 
 
 .12 
 
 1350 
 
 
 Tin 
 
 23 
 
 .054 
 
 232 
 
 13. 
 
 Zinc 
 
 29 
 
 .094 
 
 419 
 
 28. 
 
 20. Reduction of Psychrometric Observations 
 
 Values of 0.00660 B (t - «„,) [1 + 0.00115 (t - «„)], t being the temperature 
 of the dry-bulb thermometer, („ that of the wet-bulb thermometer, and 
 B the barometric pressure. (See Art. 151.) 
 
 t-tw 
 
 Babometeic Pressure B in Centimeters 
 
 70.0 
 
 71.0 
 
 72.0 
 
 73.0 
 
 74.0 
 
 75.0 
 
 76.0 
 
 77.0 
 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 cm 
 
 1 
 
 0.047 
 
 0.048 
 
 0.048 
 
 0.049 
 
 0.050 
 
 0.050 
 
 0.051 
 
 0.052 
 
 2 
 
 .093 
 
 .094 
 
 .096 
 
 .097 
 
 .098 
 
 .100 
 
 .101 
 
 .103 
 
 3 
 
 .139 
 
 .141 
 
 .143 
 
 .145 
 
 .147 
 
 .149 
 
 .152 
 
 .154 
 
 4 
 
 .186 
 
 .189 
 
 .191 
 
 .194 
 
 .197 
 
 .199 
 
 .202 
 
 .204 
 
 5 
 
 0.232 
 
 0.2.36 
 
 0.239 
 
 0.243 
 
 0.246 
 
 0.249 
 
 0.252 
 
 0.256 
 
 6 
 
 .279 
 
 .283 
 
 .287 
 
 .291 
 
 .295 
 
 .299 
 
 .303 
 
 .307 
 
 7 
 
 ..?2(i 
 
 .331 
 
 .336 
 
 .340 
 
 .345 
 
 .350 
 
 .354 
 
 .359 
 
 8 
 
 .373 
 
 .379 
 
 .384 
 
 .389 
 
 .395 
 
 .400 
 
 .405 
 
 .411 
 
 9 
 
 .421 
 
 .427 
 
 .432 
 
 .438 
 
 .444 
 
 .450 
 
 .456 
 
 .462 
 
 10 
 
 0.468 
 
 0.474 
 
 0.481 
 
 0.488 
 
 0.494 
 
 0.601 
 
 0.508 
 
 0.515 
 
 11 
 
 .515 
 
 .522 
 
 .530 
 
 .537 
 
 .544 
 
 .551 
 
 .559 
 
 .566 
 
 12 
 
 .562 
 
 .570 
 
 .578 
 
 .586 
 
 .594 
 
 ,602 
 
 .611 
 
 .619 
 
 13 
 
 .610 
 
 .618 
 
 .627 
 
 .6.36 
 
 .645 
 
 .653 
 
 .662 
 
 .671 
 
 14 
 
 .658 
 
 .667 
 
 .676 
 
 .686 
 
 .695 
 
 .705 
 
 .714 
 
 .723 
 
 15 
 
 0.706 
 
 0.716 
 
 0.726 
 
 0.736 
 
 0.746 
 
 0.756 
 
 0.766 
 
 0.776 
 
 16 
 
 .754 
 
 .764 
 
 .775 
 
 .786 
 
 .796 
 
 .807 
 
 .818 
 
 .829 
 
 17 
 
 .802 
 
 .813 
 
 .824 
 
 .836 
 
 .847 
 
 .859 
 
 .870 
 
 .882 
 
 18 
 
 .850 
 
 .862 
 
 .874 
 
 .886 
 
 .898 
 
 .910 
 
 .922 
 
 .935 
 
 19 
 
 .898 
 
 .911 
 
 .923 
 
 .936 
 
 .949 
 
 .962 
 
 .975 
 
 ,987 
 
 20 
 
 0.946 
 
 0,960 
 
 0.973 
 
 0.987 
 
 1.000 
 
 1.014 
 
 1.027 
 
 1.041 
 
386 
 
 APPENDIX 
 
 21. Relative Humidity 
 
 The table gives the per cent, of satviration of air, t being the temperature 
 of the air, d the dew-point, and t — d the depression of the dew-point. 
 (See Art. 151.) 
 
 Depression 
 
 
 
 
 Dew-point, d 
 
 
 
 OF 
 
 Dew-Poikt 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t-d 
 
 -5° 
 
 0° 
 
 + 5° 
 
 10° 
 
 16° 
 
 20° 
 
 25° 
 
 o 
 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 1 
 
 93 
 
 93 
 
 93 
 
 94 
 
 94 
 
 94 
 
 94 
 
 2 
 
 86 
 
 87 
 
 87 
 
 88 
 
 88 
 
 88 
 
 89 
 
 3 
 
 80 
 
 81 
 
 81 
 
 82 
 
 83 
 
 83 
 
 84 
 
 i 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 78 
 
 79 
 
 ■5 
 
 69 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 6 
 
 64 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 70 
 
 7 
 
 60 
 
 61 
 
 62 
 
 63 
 
 65 
 
 66 
 
 67 
 
 8 
 
 56 
 
 57 
 
 58 
 
 60 
 
 61 
 
 62 
 
 63 
 
 9 
 
 52 
 
 53 
 
 55 
 
 56 
 
 57 
 
 58 
 
 60 
 
 10 
 
 49 
 
 50 
 
 51 
 
 53 
 
 54 
 
 55 
 
 56 
 
 12 
 
 42 
 
 44 
 
 45 
 
 47 
 
 48 
 
 49 
 
 50 
 
 14 
 
 37 
 
 38 
 
 40 
 
 41 
 
 43 
 
 44 
 
 45 
 
 16 
 
 32 
 
 34 
 
 35 
 
 37 
 
 38 
 
 39 
 
 
 18 
 
 28 
 
 30 
 
 31 
 
 33 
 
 34 
 
 35 
 
 
 20 
 
 25 
 
 26 
 
 28 
 
 29 
 
 30 
 
 .32 
 
 
 25 
 
 18 
 
 19 
 
 21 
 
 22 
 
 23 
 
 
 
 30 
 
 13 
 
 14 
 
 16 
 
 17 
 
 
 
 
 22. Declination of the Sun, and Equation of Time 
 
 
 Declina- 
 
 DiFF. 
 
 Equation 
 
 
 DECLDfA- 
 
 DlFF. 
 
 Equation 
 
 
 
 tion 
 
 IDAY 
 
 OF Time 
 
 
 
 TION 
 
 IDay 
 
 oe Time 
 
 
 
 o 
 
 o 
 
 m s 
 
 
 
 o 
 
 o 
 
 m s 
 
 Jan. 
 
 
 
 -23.1 
 
 0.11 
 
 + 3 15 
 
 July 
 
 9 
 
 + 22.4 
 
 0.15 
 0.21 
 0.26 
 0.30 
 0.34 
 
 + 4 49 
 
 
 10 
 
 -22.0 
 
 0.18 
 
 + 7 42 
 
 
 19 
 
 + 20.9 
 
 + 5 58 
 
 
 20 
 
 -20.2 
 
 0.25 
 
 + 11 13 
 
 
 29 
 
 + 18.8 
 
 + 6 13 
 
 
 30 
 
 -17.7 
 
 0.30 
 
 + 13 32 
 
 Aug. 
 
 8 
 
 + 16.2 
 
 + 5 27 
 
 Feb. 
 
 9 
 
 -14.7 
 
 0.34 
 
 + 14 27 
 
 
 18 
 
 + 13.2 
 
 + 3 44 
 
 
 19 
 
 -11.3 
 
 0.37 
 
 + 14 5 
 
 
 28 
 
 + 9.8 
 
 0.36 
 0.39 
 0.39 
 0.38 
 0.38 
 
 + 1 11 
 
 March 1 
 
 - 7.6 
 
 0.38 
 
 + 12 36 
 
 Sept 
 
 7 
 
 + 6.2 
 
 - 1 59 
 
 
 11 
 
 - 3.8 
 
 0.40 
 
 + 10 15 
 
 
 17 
 
 + 2.3 
 
 - 5 26 
 
 
 21 
 
 -f 0.2 
 
 0.39 
 
 + 7 23 
 
 
 27 
 
 - 1.5 
 
 - 8 55 
 
 
 31 
 
 + 4.1 
 
 0.38 
 
 + 4 19 
 
 Oct. 
 
 7 
 
 - 5.4 
 
 -12 4 
 
 April 
 
 10 
 
 + 7.9 
 
 0.35 
 
 + 1 23 
 
 
 17 
 
 - 9.2 
 
 0.35 
 0.32 
 0.26 
 0.22 
 0.16 
 
 -14 31 
 
 
 20 
 
 4-11.4 
 
 0.33 
 
 - 1 5 
 
 
 27 
 
 -12.7 
 
 -16 
 
 
 30 
 
 -M4.7 
 
 0.29 
 
 - 2 62 
 
 Nov. 
 
 6 
 
 -15.9 
 
 - 16 .16 
 
 May- 
 
 10 
 
 -M7.6 
 
 0.23 
 
 - 3 48 
 
 
 16 
 
 -18.7 
 
 -15 7 
 
 
 20 
 
 + 19.9 
 
 0.18 
 
 - 3 45 
 
 
 26 
 
 -20.9 
 
 -12 36 
 
 
 .SO 
 
 -1-21.7 
 
 0.12 
 
 - 2 49 
 
 Dec. 
 
 6 
 
 -22.5 
 
 0.08 
 0.01 
 0.08 
 
 - 8 54 
 
 June 
 
 9 
 
 + 22.9 
 
 0.05 
 
 - 1 11 
 
 
 16 
 
 -23.3 
 
 - 4 17 
 
 
 19 
 
 + 23.4 
 
 0.01 
 
 + 55 
 
 
 26 
 
 -23.4 
 
 + 41 
 
 
 29 
 
 + 23.3 
 
 0.09 
 
 + 32 
 
 Jan. 
 
 5 
 
 -22.6 
 
 + 5 34 
 
TABLES 
 
 387 
 
 23. Tension and Mass of Aqueous Vapor in Saturated Air 
 
 The table gives the tension e, and the mass / per cubic meter, of water 
 
 vapor in air saturated at the temperature t. (See 
 
 Art. 151.) 
 
 t 
 
 c 
 
 f 
 
 t 
 
 e 
 
 / 
 
 i 
 
 e 
 
 / 
 
 o 
 
 cm 
 
 g 
 
 o 
 
 cm 
 
 g 
 
 o 
 
 cm 
 
 g 
 
 
 
 0.46 
 
 4.9 
 
 10 
 
 0.92 
 
 9.4 
 
 20 
 
 1.75 
 
 17.3 
 
 1 
 
 0.49 
 
 5.2 
 
 11 
 
 0.98 
 
 10.0 
 
 21 
 
 1.86 
 
 18.3 
 
 2 
 
 0.33 
 
 5.6 
 
 12 
 
 1.05 
 
 10.7 
 
 22 
 
 1.98 
 
 19.4 
 
 3 
 
 0.57 
 
 6.0 
 
 13 
 
 1.12 
 
 11.4 
 
 23 
 
 2.11 
 
 20.6 
 
 4 
 
 0.61 
 
 6.4 
 
 14 
 
 1.20 
 
 12.1 
 
 24 
 
 2.24 
 
 21.8 
 
 5 
 
 0.65 
 
 6.8 
 
 15 
 
 1.28 
 
 12.8 
 
 25 
 
 2.38 
 
 23.1 
 
 6 
 
 0.70 
 
 7.3 
 
 16 
 
 1.36 
 
 13.7 
 
 26 
 
 2.53 
 
 24.5 
 
 7 
 
 0.75 
 
 7.8 
 
 17 
 
 1.45 
 
 14.5 
 
 27 
 
 2.68 
 
 26.8 
 
 8 
 
 0.80 
 
 8.2 
 
 18 
 
 1.55 
 
 15.4 
 
 28 
 
 2.84 
 
 27.3 
 
 9 
 
 0.86 
 
 8.8 
 
 19 
 
 1.65 
 
 16.3 
 
 29 
 
 3.01 
 
 28.8 
 
 10 
 
 0.92 
 
 9.4 
 
 20 
 
 1.75 
 
 17.3 
 
 30 
 
 3.18 
 
 30.4 
 
 24. Index of Refraction of Various Substances 
 
 for Sodium Light, D Line, X = 0.5893 /i 
 
 Air, Dry, 0° 
 
 1.0002945 
 
 20° 
 
 1.0002773 
 
 Ice 
 
 1.31 
 
 Water, 15° 
 
 1.33361 
 
 20° 
 
 1.33319 
 
 Alcohol 
 
 1.3616 
 
 Benzene 
 
 1.5005 
 
 Canada Balsam 
 
 1.54 
 
 Glass, 
 Light crown 
 Heavy crown 
 Light flint 
 Heavy flint 
 Heaviest flint 
 Carbon Bisul- 
 phide, 15° 
 20° 
 Phosphorus in 
 CS, 
 
 1.5153 
 1.6152 
 1.6085 
 1.7515 
 1.9 
 
 1.6317 
 1.6277 
 
 1.97 
 
 a-Bromonaphtha- 
 lene, 15° 
 20° 
 Methyl Iodide, 15° 
 20° 
 Calcite, 
 Ordinary Eay 
 Extraordinary 
 Quartz, 
 Ordinary Eay 
 Extraordinary 
 
 1.6518 
 1.6495 
 1.7429 
 1.7419 
 
 1.6585 
 1.4864 
 
 1.5442 
 1.5533 
 
 25. Wave Length of Lines of Solar Spectrum 
 
 in Air at 20°, Pressure 76 cm ; Unit, Micron = 0.001 mm 
 (See Art. 196) 
 
 Line 
 
 Element 
 
 Wave 
 Length 
 
 Line 
 
 Element 
 
 Wave 
 Length 
 
 Line 
 
 Element 
 
 Wave 
 Length 
 
 A 
 a 
 B 
 C 
 a 
 
 O 
 H 
 O 
 
 Na 
 
 0.7628 
 
 0.7185 
 
 0.68701 
 
 0.65629 
 
 0.62781 
 
 0.58960 
 
 6. 
 c 
 F 
 d 
 
 Na 
 Fe, Ca 
 Mg 
 Fe 
 H 
 Fe 
 
 0.58900 
 0.,52703 
 0.51837 
 0.49576 
 0.48614 
 0.46682 
 
 e 
 
 i 
 h 
 H 
 K 
 
 Fe 
 
 H 
 Fe, Ca 
 
 H 
 H, Ca 
 
 Ca 
 
 0.43836 
 0.43405 
 0.43079 
 0.41018 
 0.39685 
 0.39337 
 
388 
 
 APPENDIX 
 
 26. Specific Resistance of Various Substances 
 
 p is the resistance at 0° of a conductor 1 cm long and 1 scm in section, 
 a is the rate of increase in resistance per degree increase in tempera- 
 ture. The resistance of a conductor of length / and section s at the 
 temperature t is (see Art. 235) 
 
 Rt = p-(l +at). 
 
 SUBSTAKOE 
 
 p 
 
 a 
 
 SUBSTAHOB 
 
 P 
 
 a 
 
 
 ohms 
 
 
 
 ohms 
 
 
 Silver 
 
 0.16 X 10-6 
 
 0.0037 
 
 Iron 
 
 1.04 X 10-5 
 
 0.0060 
 
 Copper 
 
 0.17 
 
 43 
 
 Steel 
 
 5. 
 
 60 
 
 Gold 
 
 0.20 
 
 36 
 
 Lead 
 
 2.00 
 
 39 
 
 Aluminum 
 
 0.30 
 
 39 
 
 Manganin 
 
 4.20 
 
 003 
 
 Zinc 
 
 0.57 X 10-5 
 
 0.0036 
 
 German Silver 
 
 2.09 X 10-5 
 
 0.0003 
 
 Nickel 
 
 0.70 
 
 60 
 
 Brass 
 
 0.8 
 
 40 
 
 Platinum, Pure 
 
 1.08 
 
 32 
 
 Mercury 
 
 9.41 
 
 09 
 
 Commercial 
 
 1.40 
 
 32 
 
 Gas Carbon 
 
 500. 
 
 - .0005 
 
 
 27. Electromotive Force and Internal Resistance of Cells 
 
 Cell 
 
 E. M. F. 
 
 Besistakce 
 
 Cell 
 
 E. M. F. 
 
 Eesistanoe 
 
 
 volts 
 
 ohms 
 
 
 volts 
 
 ohms 
 
 Edison-Lalaude 
 
 0.7 
 
 0.03 
 
 Grove 
 
 1.9 
 
 0.1-0.2 
 
 Daniel 
 
 1.08 
 
 0.85 
 
 Bunsen 
 
 1.9 
 
 0.1-0.2 
 
 Gravity 
 
 1.1 
 
 1-5 
 
 Bicliromate 
 
 2.0 
 
 0.08-0.40 
 
 Silver Chloride 
 
 1.1 
 
 4. 
 
 Storage 
 
 2.0 
 
 0.004-0.02 
 
 Dry Cell 
 
 1.3 
 
 0.2-1.0 
 
 * Clark Standard 
 
 1.4267 
 
 20-50 
 
 Leclanchg 
 
 1.4-1.7 
 
 0.4-0.2 
 
 * Weston Standard 
 
 1.0190 
 
 20-50 
 
 28. Vibration Frequency of Tones of the Musical Scale 
 
 Higher or lower octaves are obtained by multiplying by some power of 2. 
 
 SoiENTiPic Diatonic Scale 
 
 Musical Equal-Tempered Chkomatio Scale 
 
 Ca=256 
 
 A3=435 
 
 0« 
 
 2S6. 
 
 Cs 
 
 258.65 
 
 Gs 
 
 387.54 
 
 Ds 
 
 288. 
 
 C»3 
 
 274.03 
 
 GSs 
 
 410.58 
 
 Es 
 
 320. 
 
 Da 
 
 290.33 
 
 As 
 
 43S. 
 
 Fs 
 Gs 
 As 
 
 341.33 
 
 384. 
 
 426.66 
 
 D^ 
 
 307.59 
 
 A#s 
 
 460.87 
 
 Es 
 
 325.88 
 
 Bs 
 
 488.27 
 
 Bs 
 
 480. 
 
 Fs 
 
 345.26 
 
 C4 
 
 517.30 
 
 Ci 
 
 512. 
 
 F«s 
 
 365.79 
 
 
 
REFERENCES 389 
 
 29. Reference Books 
 
 The following list specifies those books and papers to which reference has 
 been made in the text 
 
 Barker, George F. Physics. New York, 1892. 
 
 Barus, Cari.. Les Progres de la PyronxHrie. Rapports au Congrfes Interna- 
 tional de Physique. Paris, 1900. 
 
 Benton, J. R. Internal Resistance of Cells. Physical Review, 16, 1903. 
 
 Carhart, Henry S., and Patterson, George W. Electrical Measurements. 
 Boston, 1895. , 
 
 Chappuis, p. L'Echelle Themiom^trigue Normale, Rapports au Gongrfes Inter- 
 national de Physique ; Pyromitrie, Travaux et Memoires du Bureau Interna- 
 tional des Poids et Mesures, Tome XII, Part 3. Paris, 1900. 
 
 Chauvenet, William. Spherical and Practical Astronomy, 2 vols. Philadel- 
 phia, 1863. 
 
 Daniell, Alfred. Principles of Physics, 3d ed. New York, 1894. 
 
 Drude, Paul. TOeorj/ o/ Opiics, trans, by Mann and MiUikan. New York, 1902. 
 
 Everett, J. D. The C. G. S. System of Units, 5th ed. London, 1902. 
 
 Gdazebrook, R. T. Physical Optics. London, 1882. 
 
 Glazebrook, R. T., and Shaw, W. N. Practical Physics. London, 1884. 
 
 Gray, Andrew. Absolute Measureinents in Electricity and Magnetism, 3 vols. 
 London, 1888. 
 
 Gray, Thomas. Smithsonian Physical Tables. Washington, 1896. 
 
 GuiLLAUME, Ch. Ed. Thermometrie de Precision. Paris, 1889. 
 
 GuYOT, LiBBEY, aud others. Smithsonian Meteorological Tables. Washington, 
 1893. 
 
 Hastings, Charles S., and Beach, Frederick E. General Physics. Boston, 
 1899. 
 
 HoLM.iN, Silas W. Precision of Measurements. New York, 1892. 
 
 Johnson, 3. B. Theory and Practice of Surveying, 3d ed. New York, 1887. 
 
 Koenig, Rudolph. Quelques Experiences d' Acoustique. Paris, 1882. 
 
 Kohlrausch, F. Physical Measurements, 3d English from 7th German ed. 
 New York, 1894. 9th German ed., Leipzig, 1901. 
 
 Landauer, John. Spectrum Analysis, trans, by J. B. Tingle. New York, 1898. 
 
 Landolt, H. Optical Rotating Power, trans, by J. H. Long. Easton, 1902. 
 
 Landolt und Bornstein. Physikalisch^Chemische Tabellen, 2d ed. Berlin, 
 1893. 
 
 Le Chatelier, H. High-Temperature Measurevnents, trans, by G. K. Burgess. 
 New York, 1901. 
 
 Leiss, C. Die optischen Instrumente der Firma R. Fuess. Leipzig, 1899. 
 
 LuPTON, Sidney. Notes on Observations. London, 1898. 
 
 Magie, William Francis. The Specific Heat of Solutions. Physical Review, 9, 
 1899; 14,1902. 
 
 Merriman, Mansfield. Method of Least Squares, 7th ed. New York, 1897. 
 
 Michelson, Albert A. Interference Phenomena in a New Form, of Inter- 
 ferometer, Philosophical Magazine, 13, 1882; Valuer du Metre en Longueurs 
 d'Ondes Lumineuses, Travaux et Memoires du Bureau International du 
 Poids et Mesures, Tome XI. Paris, 1894. Light Waves and their Uses. 
 Chicago, 1903. 
 
 MORLEY, Edward W. Optical Principles of the Interferometer. Physical 
 Review, 4, 1897. 
 
 Nichols, Edward L. Laboratory Manual of Physics and Applied Electricity, 
 2 vols. New York, 1894. 
 
 OSTWALD, WiLHELM. Physico-Chcmical Measurements, trans, by James Walker. 
 London, 1893. 2d German ed. Leipzig, 1902. 
 
 Palaz, H. Indujitrial Photometry, trans, by G. W. Patterson. New York, 1896. 
 
 Preston, Thomas. Theory of Light, 2d ed. London, 1895 ; Theory of Heat, 
 London, 1894. 
 
 Price, B. Treatise on Infinitesimal Calculus, 2d ed., 4 vols. Oxford, 1868. 
 
 PULFRICH, C. ANewFormofRefractometer. Astrophysical Journal, S, 1896. 
 
390 
 
 APPENDIX 
 
 Rebd, J. O. Method of Eating Tuning Forks. Physical Review, 12, 1901. 
 EoUTH, E. J. Dynamics of a System of Rigid Bodies, 2 vols., 5tli ed. New York, 
 
 1890. 
 ScHEiNER, J. Astronomical Spectroscopy, trans, by E. B. Frost. Boston, 1894. 
 Schuster, Arthur, and Lees, C. H. Practical Pnysics. Cambridge, 1901. 
 Shedd, J. C. Forms of Curves presented by the Michelson Interferometer. 
 
 Physical Review, 11, 1900. 
 Staley, Cady. Gillespie's Surveying, 2 vols., revised ed. New York, 1897. 
 Stewart, Balfour, and Gee, W. W. Haldane. Practical Physics, 3 vols. 
 
 London, 1885. 
 Stine, Wilbur M. Photometrical Measurements. New York, 1900. 
 Wadsworth, F. L. O. Applications of the Interferometer. Physical Review, 4, 
 
 1897. 
 Whitman, F. P. On the Photometry of Different Colored Lights. Physical 
 
 Review, 3, 1896; Science, 9, 1899. 
 Zahm, J. H. Sound and Music. Chicago, 1892. 
 
 30. Miscellaneous Constants and Numbers 
 
 T = 3.14159265 ir^ = 9.869604 log ir = 0.49714987 
 
 4 5r = 12.566 Jtt =0.079577 log 4 tt = 1.09921 
 
 1 Radian = 57°.2958 = 3437'.75 = 206265" 
 Logarithms 1.758123 3.536274 5.314425 
 Base of natural logarithms, e = 2.7182818 ; log e = 0.434294. 
 Mean radius of the earth, 6.37106 X IQS cm. 
 Mechanical equivalent of heat, 4.184 X 10^ ergs. (See Art. 139.) 
 Candle power: 1 English candle = 0.95 German candle = 1.14 Hefner units. 
 
 (See Art. 155.) 
 Velocity of sound in dry air at 0°, 33136 cm per s ; in illuminating gas at 0°, 49000 
 
 cm per s. (See Table 7, and Art. 114.) 
 Velocity of light in vacuum, 2.9989 X 10i» cm per s. 
 Specific rotation of sugar, [a]2»° = 6°.65 per cm. (See Art. 203.) 
 Electro-chemical equivalents: silver, 0.001118; copper, 0.0003279; g per s per 
 
 ampere. (See Arts. 243, 244.) 
 Temperature coefficient of magnetism, from 0.0003 to 0.001. 
 
 31. Metric and English Equivalents and Abbreviations 
 
 1 inch = 2.539998 cm 
 
 1 micron = 0.001 mm 
 
 m meter 
 
 1 foot = 30.47997 cm 
 
 1 meter = 39.37011 inches 
 
 dm decimeter 
 
 lyard =91.43992 cm 
 
 1 kilometer = 0.621371 miles 
 
 cm centimeter 
 
 1 mile = 160934.3 cm 
 
 1 gram = 15.43236 grains 
 
 mm millimeter 
 
 Avoirdupois 
 
 1 milligram = 0.01543 grains 
 
 km kilometer 
 
 1 grain = 64.79892 mg 
 
 1 kilogram = 2.204622 pounds 
 
 fi, 0.001 millimeter 
 
 1 ounce = 28.34953 g 
 
 1 liter =1.75980 pints 
 
 HH 0.000001 millimeter 
 
 1 pound = 453.5924 g 
 
 Approximate Equivalents 
 
 tenth-meter = lO-" m 
 
 Troy 
 
 1 millimeter =A inch 
 
 g gram 
 
 1 grain = 64.79892 mg 
 
 1 meter = 3 ft, 3 in, J in 
 
 mg milligram 
 
 1 ounce = 31.10348 g 
 
 1 kilometer = 8 mile 
 
 kg kilogram 
 
 Apothecary 
 
 1 gram = 158 grains 
 
 scm square centimeter 
 
 1 ounce = 31.10348 g 
 1 quart = 1.13649 1 
 
 1 kilogram = 2J pounds 
 
 ccm cubic centimeter 
 
 1 liter = li pints 
 
 1 liter 
 
TABLES 
 32. Natural Trigonometrical Functions 
 
 391 
 
 Sine 
 
 
 1 
 
 2 
 
 3 
 4 
 
 5 
 6 
 7 
 8 
 9 
 
 10 
 11 
 12 
 13 
 14 
 
 15 
 16 
 17 
 18 
 19 
 
 20 
 21 
 22 
 23 
 
 24 
 
 25 
 26 
 27 
 28 
 29 
 
 30 
 31 
 32 
 33 
 34 
 
 35 
 36 
 37 
 38 
 39 
 
 40 
 41 
 42 
 43 
 44 
 
 0.0000 
 .0175 
 .0349 
 .0523 
 
 0.0872 
 .1045 
 .1219 
 .1392 
 .1564 
 
 0.1736 
 .1908 
 .2079 
 .2250 
 .2419 
 
 0.2588 
 .2756 
 .2924 
 .3090 
 .3256 
 
 0.3420 
 .3584 
 .3746 
 ..3907 
 .4067 
 
 0.4226 
 .4.384 
 .4540 
 .4695 
 .4848 
 
 0.5000 
 .5150 
 .5299 
 .5446 
 .5592 
 
 0.57.36 
 
 ..5878 
 .6018 
 .61.57 
 .6293 
 
 0.6428 
 .6.561 
 
 .6947 
 0.7071 
 
 175 
 174 
 174 
 175 
 174 
 
 173 
 174 
 173 
 172 
 172 
 
 172 
 171 
 171 
 169 
 169 
 
 168 
 168 
 166 
 166 
 164 
 
 im 
 
 162 
 161 
 160 
 159 
 
 168 
 156 
 155 
 153 
 152 
 
 150 
 149 
 147 
 146 
 144 
 
 142 
 140 
 139 
 136 
 135 
 
 133 
 130 
 129 
 
 127 
 124 
 
 Cosine 
 
 Tangent 
 
 0.0000 
 .0175 
 .0349 
 .0524 
 .0699 
 
 0.0875 
 .1051 
 .1228 
 .1405 
 .1584 
 
 0.1763 
 .1944 
 .2126 
 .2309 
 .2493 
 
 0.2679 
 .2867 
 .3057 
 .3249 
 .3443 
 
 0.3640 
 .3839 
 .4040 
 .4245 
 .4452 
 
 0.4663 
 .4877 
 .5095 
 .5317 
 .5543 
 
 0.5774 
 .6009 
 .6249 
 .6494 
 .6745 
 
 0.7002 
 .7265 
 .7536 
 .7813 
 
 0.8391 
 .8693 
 .9004 
 .9325 
 .9657 
 
 1.0000' 
 
 Cotangent 
 
 COTANOENT 
 
 oo 
 
 57.29 
 28.64 
 19.08 
 14.30 
 
 11.43 
 9.514 
 8.144 
 7.115 
 6.314 
 
 5.671 
 5.145 
 4.705 
 4.,331 
 4.011 
 
 3.732 
 3.487 
 3.271 
 3.078 
 2.904 
 
 2.747 
 2.605 
 2.475 
 2.356 
 2.246 
 
 2.145 
 2.0.50 
 1.963 
 1.881 
 1.804 
 
 1.732 
 1.664 
 1.600 
 1.S40 
 1.483 
 
 1.428 
 1.376 
 1.327 
 1.280 
 1.235 
 
 1.192 
 1.150 
 1.111 
 1.072 
 1.036 
 
 1.000 
 
 801 
 643 
 
 526 
 440 
 374 
 320 
 279 
 
 216 
 193 
 174 
 157 
 
 142 
 130 
 119 
 110 
 101 
 
 95 
 87 
 82 
 77 
 72 
 
 TANGENT 
 
 Cosine 
 
 Degree 
 
 1.0000 
 
 0.9998 
 
 .9994 
 
 .9986 
 
 .9976 
 
 0.9962 
 .9943 ■ 
 .9925 
 .9903 
 .9877 
 
 0.9848 
 .9816 
 .9781 
 .9744 
 .9703 
 
 0.9659 
 .9613 
 .9563 
 .9511 
 .9455 
 
 0.9397 
 .9336 
 .9272 
 .9205 
 .9135 
 
 0.9063 
 .8988 
 .8910 
 .8829 
 .8746 
 
 0.8660 
 .8572 
 .8480 
 .8387 
 .8290 
 
 0.8192 
 .8090 
 .7986 
 .7880 
 .7771 
 
 0.7660 
 .7547 
 .7431 
 .7314 
 .7193 
 
 0.7071 
 
 02 
 04 
 08 
 10 
 14 
 
 17 
 20 
 22 
 26 
 29 
 
 32 
 35 
 37 
 41 
 44 
 
 46 
 50 
 62 
 56 
 58 
 
 61 
 64 
 67 
 70 
 72 
 
 75 
 78 
 81 
 
 88 
 92 
 93 
 97 
 
 102 
 104 
 106 
 109 
 111 
 
 113 
 116 
 117 
 121 
 122 
 
 90 
 
 87 
 86 
 
 84 
 83 
 82 
 81 
 
 79 
 78 
 77 
 76 
 
 75 
 74 
 73 
 72 
 71 
 
 70 
 69 
 68 
 67 
 66 
 
 65 
 64 
 63 
 62 
 61 
 
 60 
 
 59 
 58 
 57 
 56 
 
 55 
 54 
 53 
 52 
 51 
 
 50 
 49 
 48 
 47 
 46 
 
 45 
 
 Degree 
 
392 
 
 APPENDIX 
 33. Logarithms 
 
 
 
 
 Propoktional Parts 
 
 , N 
 
 12 3 4 
 
 5 6 7 8 9 
 
 
 
 
 1 2 
 
 3 4 
 
 5 6 7 8 9 
 
 10 
 
 0000 0043 0086 0128 0170 
 
 0212 0253 0294 0334 0374 
 
 4 8 12 17 
 
 21 25 29 33 37 
 
 11 
 
 0414 0453 0492 0531 0569 
 
 0607 0645 0682 0719 0755 
 
 4 8 11 15 
 
 19 23 26 30 34 
 
 12 
 
 0792 0828 0864 0899 0934 
 
 0969 1004 1038 1072 1106 
 
 3 7 10 14 
 
 17 21 24 28 31 
 
 13 
 
 1139 1173 1206 1239 1271 
 
 1303 1335 1367 1399 1430 
 
 3 6 10 13 
 
 16 19 23 26 29 
 
 14 
 15 
 
 1461 1492 1523 1553 1584 
 
 1614 1644 1673 1703 1732 
 
 3 6 
 
 9 12 
 
 15 18 21 24 27 
 
 1761 1790 1818 1847 1875 
 
 1903 1931 1959 1987 2014 
 
 3 6 
 
 8 11 
 
 14 17 20 22 25 
 
 16 
 
 2041 2068 209S 2122 2148 
 
 2175 2201 2227 2253 2279 
 
 35 
 
 8 11 
 
 13 16 18 21 24 
 
 17 
 
 2304 2330 2353 2380 2405 
 
 243a 2455 2480 2504 2529 
 
 2 5 
 
 7 10 
 
 12 15 17 20 22 
 
 18 
 
 2553 2577 2601 2625 2648 
 
 2672 2695 2718 2742 2765 
 
 2 5 
 
 7 9 
 
 12 14 16 19 21 
 
 19 
 20 
 
 2788 2810 2833 2856 2878 
 
 2900 2923 2945 2967 2989 
 
 24 
 
 7 9 
 
 11 13 16 IS 20 
 
 3010 3032 3054 3075 3096 
 
 3118 3139 3160 3181 3201 
 
 2 4 
 
 6 8 
 
 11 13 15 17 19 
 
 21 
 
 3222 3243 3263 3284 3304 
 
 3324 3345 3365 3385 3404 
 
 2 4 
 
 6 8 
 
 10 12 14 16 18 
 
 22 
 
 3424 3444 3464 3483 3502 
 
 3522 3541 3560 3579 3598 
 
 24 
 
 6 8 
 
 10 12 14 15 17 
 
 23- 
 
 3617 3636 3655 3674 3692 
 
 3711 3729 3747 3766 3784 
 
 24 
 
 6 7 
 
 9 11 13 15 17 
 
 24 
 2R 
 
 3802 3820 3838 3856 3874 
 
 3892 3909 3927 3945 3962 
 
 24 
 
 5 7 
 
 9 11 12 14 16 
 
 3979 3997 4014 4031 4048 
 
 4065 4082 4099 4116 4133 
 
 2 3 
 
 5 7 
 
 9 10 12 14 15 
 
 26 
 
 4150 4166 4183 4200 4216 
 
 4232 4249 4265 4281 4298 
 
 2 3 
 
 5 7 
 
 8 10 11 13 15 
 
 27 
 
 4314 4330 4346 4362 4378 
 
 4393 4409 4425 4440 4456 
 
 2 3 
 
 S 6 
 
 8 9 11 13 14 
 
 28 
 
 4472 4487 4502 4518 4533 
 
 4548 4564 4579 4594 4609 
 
 23 
 
 5 6 
 
 8 9 11 12 14 
 
 29 
 30 
 
 4624 4639 4654 4669 4683 
 
 4698 4713 4728 4742 4757 
 
 1 3 
 
 4 6 
 
 7 9 10 12 13 
 
 4771 4786 4800 4814 4829 
 
 4843 4857 4871 4886 4900 
 
 1 3 
 
 4 6 
 
 7 9 10 11 13 
 
 31 
 
 4914 4928 4942 4955 4969 
 
 4983 4997 5011 5024 5038 
 
 1 3 
 
 4 6 
 
 7 8 10 11 12 
 
 32 
 
 5051 5065 5079 5092 5105 
 
 5119 6132 5145 6169 6172 
 
 1 3 
 
 4 5 
 
 7 8 9 11 12 
 
 33 
 
 5185 5198 5211 5224 5237 
 
 5250 5263 5276 5289 5302 
 
 1 3 
 
 4 6 
 
 6 8 9 10 12 
 
 34 
 
 5315 5328 5340 5353 5366 
 
 5378 5391 5403 5416 5428 
 
 1 3 
 
 4 5 
 
 6 8 9 10 11 
 
 5441 5453 6465 5478 5490 
 
 5502 5614 5527 5539 6661 
 
 1 2 
 
 4 5 
 
 6 7 9 10 11 
 
 36 
 
 5563 5575 5587 5599 5611 
 
 5623 5636 5647 5658 5670 
 
 1 2 
 
 4 6 
 
 6 7 8 10 11 
 
 37 
 
 5682 5694 5705 5717 5729 
 
 5740 6752 5763 5775 5786 
 
 1 2 
 
 3 5 
 
 6 7 8 9 10 
 
 38 
 
 5798 5809 5821 5832 5843 
 
 5855 5866 6877 5888 6899 
 
 1 2 
 
 3 5 
 
 6 7 8 9 10 
 
 39 
 40 
 
 5911 5922 5933 5944 5955 
 
 5966 5977 5988 5999 6010 
 
 1 2 
 
 3 4 
 
 5 7 8 9 10 
 
 6021 6031 6042 6053 6064 
 
 6076 6086 6096 6107 6117 
 
 1 2 
 
 3 4 
 
 5 6 8 9 10 
 
 41 
 
 6128 6138 6149 6160 6170 
 
 6180 6191 6201 6212 6222 
 
 1 2 
 
 3 4 
 
 5 6 7 8 9 
 
 42 
 
 6232 6243 6253 6263 6274 
 
 6284 6294 6304 6314 6326 
 
 1 2 
 
 3 4 
 
 5 6 7 8 9 
 
 43 
 
 6335 6345 6355 6365 6375 
 
 6385 6395 6405 6415 6425 
 
 1 2 
 
 3 4 
 
 5 G 7 8 9 
 
 44 
 4R 
 
 6435 6444 6454 6464 6474 
 
 6484 6493 6503 6513 6522 
 
 1 2 
 
 3 4 
 
 5 6 7 8 9 
 
 6532 6542 6551 6561 6571 
 
 6580 6590 6599 6609 6618 
 
 1 2 
 
 3 4 
 
 5 6 7 8 9 
 
 46 
 
 6628 6637 6646 6656 6665 
 
 6676 6684 6693 6702 6712 
 
 1 2 
 
 3 4 
 
 5 6 7 7 8 
 
 47 
 
 6721 6730 6739 6749 6758 
 
 6767 6776 6785 6794 6803 
 
 1 2 
 
 3 4 
 
 5 5 6 7 8 
 
 48 
 
 6812 6821 6830 6839 6848 
 
 6857 6866 6875 6884 6893 
 
 1 2 
 
 3 4 
 
 4 5 6 7 8 
 
 49 
 fiO 
 
 6902 6911 6920 6928 6937 
 
 6946 6955 6964 6972 6981 
 
 1 2 
 
 3 4 
 
 4 5 6 7 8 
 
 6990 6998 7007 7016 7024 
 
 7033 7042 7050 7059 7067 
 
 1 2 
 
 3 3 
 
 4 5 6 7 8 
 
 fil 
 
 7076 7084 7093 7101 7110 
 
 7118 7126 7136 7143 7152 
 
 1 2 
 
 3 3 
 
 4 5 6 7 8 
 
 fi2 
 
 7160 7168 7177 7185 7193 
 
 7202 7210 7218 7226 7235 
 
 1 2 
 
 2 3 
 
 4 5 6 7 7 
 
 m 
 
 7243 7251 7259 7267 7275 
 
 7284 7292 7300 7308 7316 
 
 1 2 
 
 2 3 
 
 4 5 6 6 7 
 
 54 
 
 7324 7332 7340 7348 7356 
 
 7364 7372 7380 7388 7396 
 
 1 2 
 
 2 3 
 
 4 5 6 6 7 
 
 N 
 
 12 3 4 
 
 5 6 7 8 9 
 
 1 2 
 
 3 4 
 
 5 6 7 8 9 
 
TABLES 
 33. Logarithms 
 
 393 
 
 N 
 
 12 3 4 
 
 5 6 7 8 9 
 
 Proportional Parts 
 
 
 12 3 4 
 
 5 
 
 6 
 
 7 8 9 
 
 
 55 
 
 56 
 57 
 58 
 59 
 
 60 
 
 61 
 62 
 63 
 64 
 
 65 
 
 66 
 67 
 68 
 69 
 
 70 
 
 71 
 72 
 73 
 74 
 
 75 
 
 76 
 77 
 78 
 79 
 
 80 
 
 81 
 82 
 83 
 84 
 
 85 
 
 86 
 87 
 88 
 89 
 
 90 
 
 91 
 92 
 93 
 94 
 
 95 
 
 96 
 97 
 98 
 99 
 
 N 
 
 7404 7412 7419 7427 7435 
 7482 7490 7497 7505 7513 
 7559 7566 7374 7582 7589 
 7634 7642 7649 7657 7664 
 7709 7716 7723 7731 7738 
 
 7443 7451 7459 7466 7474 
 7520 7528 7536 7543 7551 
 7597 7604 7612 7619 7627 
 7672 7679 7686 7694 7701 
 7745 7752 7760 7767 7774 
 
 12 2 3 
 12 2 3 
 12 2 3 
 112 3 
 112 3 
 
 4 
 4 
 4 
 4 
 4 
 
 5 
 5 
 6 
 
 4 
 4 
 
 5 6 7 
 5 6 7 
 5 6 7 
 5 6 7 
 5 6 7 
 
 
 7782 7789 7796 7803 7810 
 7853 7860 7868 7875 7882 
 7924 7931 7938 7945 7952 
 7993 8000 8007 8014 8021 
 8062 8069 8075 8082 8089 
 
 7818 7825 7832 7839 7846 
 7889 7896 7903 7910 7917 
 7959 7966 7973 7980 7987 
 8028 8035 8041 8048 8055 
 8096 8102 8109 8116 8122 
 
 112 3 
 112 3 
 112 3 
 112 3 
 112 3 
 
 4 
 4 
 3 
 3 
 3 
 
 4 
 
 4 
 4 
 4 
 4 
 
 5 6 6 
 
 6 6 6 
 
 5 6 6 
 
 6 5 6 
 5 5 6 
 
 
 8129 8136 8142 8149 8156 
 8195 8202 8209 8215 8222 
 8261 8267 8274 8280 8287 
 8325 8331 8338 8344 8351 
 8388 8393 8401 8407 8414 
 
 8162 8169 8176 8182 8189 
 8228 8235 8241 8248 8234 
 8293 8299 8306 8312 8319 
 8357 8363 8370 8376 8382 
 8420 8426 8432 8439 8445 
 
 112 3 
 112 3 
 112 3 
 112 3 
 112 3 
 
 3 
 3 
 3 
 3 
 3 
 
 4 
 4 
 4 
 4 
 4 
 
 5 5 6 
 
 5 5 6 
 
 6 6 6 
 4 5 6 
 4 5 6 
 
 
 8451 8457 8463 8470 8476 
 8513 8519 8525 8531 8537 
 8573 8579 8385 8591 8597 
 8633 8639 8645 8651 8657 
 8692 8698 8704 8710 8716 
 
 8482 8488 8494 8500 8606 
 8543 8549 8653 8561 8567 
 8603 8609 8615 8621 8627 
 8663 8669 8675 8681 8686 
 8722 8727 8733 8739 8745 
 
 112 2 
 112 2 
 112 2 
 112 2 
 112 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 
 
 4 
 4 
 4 
 4 
 
 4 5 6 
 4 5 5 
 4 5 5 
 4 5 5 
 4 6 5 
 
 
 8751 8756 8762 8768 8774 
 8808 8814 8820 8825 8831 
 8865 8871 8876 8882 8887 
 8921 8927 8932 8938 8943 
 8976 8982 8987 8993 8998 
 
 8779 8785 8791 8797 8802 
 8837 8842 8848 8854 8869 
 8893 8899 8904 8910 8915 
 8949 8954 8960 8965 8971 
 9004 9009 9015 9020 9025 
 
 112 2 
 112 2 
 112 2 
 112 2 
 112 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 4 6 5 
 4 6 6 
 4 4 5 
 4 4 5 
 4 4 5 
 
 
 9031 9036 9042 9047 9053 
 9085 9090 9096 9101 9106 
 9138 9143 9149 9154 9159 
 9191 9196 9201 9206 9212 
 9243 9348 9253 9258 9263 
 
 9058 9063 9069 9074 9079 
 9112 9117 9122 9128 9133 
 9165 9170 9175 9180 9186 
 9217 9222 9227 9232 9238 
 9269 9274 9279 9284 9289 
 
 112 2 
 112 2 
 112 2 
 112 2 
 112 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 4 4 5 
 4 4 5 
 4 4 6 
 4 4 6 
 4 4 6 
 
 
 9294 9299 9304 9309 9315 
 9345 9350 9355 9360 9365 
 9395 9400 9405 9410 9415 
 9445 9450 9455 9460 9465 
 9494 9499 9504 9509 9513 
 
 9320 9325 9330 9335 9340 
 9370 9375 9380 9385 9390 
 9420 9425 9430 9435 9440 
 9469 9474 9479 9484 9489 
 9518 9323 9528 9533 9538 
 
 112 2 
 112 2 
 112 
 112 
 112 
 
 3 
 3 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 4 6 
 4 4 5 
 3 4 4 
 3 4 4 
 3 4 4 
 
 
 9542 9547 9552 9557 9562 
 9590 9595 9600 9605 9609 
 9638 9643 9647 9652 9657 
 9685 9689 9694 9699 9703 
 9731 9736 9741 9745 9750 
 
 9566 9371 9576 9581 9586 
 9614 9619 9624 9628 9633 
 9661 9666 9671 9675 9680 
 9708 9713 9717 9722 9727 
 9754 9759 9763 9768 9773 
 
 112 
 112 
 112 
 112 
 112 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 
 
 9777 9782 9786 9791 9795 
 9823 9827 9832 9836 9841 
 9868 9872 9877 9881 9886 
 9912 9917 9921 9926 9930 
 9956 9961 9965 9969 9974 
 
 9800 9805 9809 9814 9818 
 9845 9850 9854 9859 9863 
 9890 9894 9899 9903 9908 
 9934 9939 9943 9948 9962 
 9978 9983 9987 9991 9996 
 
 112 
 112 
 112 
 112 
 112 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 3 3 4 
 
 
 12 3 4 
 
 5 6 7 8 9 
 
 12 3 4 
 
 6 
 
 6 
 
 7 8 9 
 
 
INDEX 
 
 [The numbers refer to pages] 
 
 Abbe, orystal-refractometer, 244 
 
 Abbreviations, Table 31. 
 
 Absolute, capacity, 356 ; humidity, 
 197 ; index of refraction, 234, 235 ; 
 mass, 64; units, 12. 
 
 Accelerated motion, with Atwood's 
 machine, 77 ; with falling tuning 
 fork, 74. 
 
 Acceleration with variable time unit, 
 76. 
 
 Accidental errors, 7. 
 
 Adjusting, ballistic galvanometer, 387 ; 
 cathetometer, 97 ; difierential gal- 
 vanometer, 300 ; galvanometers in 
 general, 288 ; heliostat, 274 ; inter- 
 ferometer, 258 ; reading microscope, 
 22 ; reading telescope and scale, 38, 
 290, 338 ; sextant, 47; spectrometer, 
 226 ; tangent galvanometer, 296 ; 
 variometer, 367 ; weights, 68. 
 
 Air, columns, vibrating, 151 ; density 
 of. Table 12 ; method for determin- 
 ing density of, 126 ; index of refrac- 
 tion of, 235 ; thermometer, 166. 
 
 Alternating currents, for resistance 
 measures, 309 ; method of measur- 
 ing, 330, 341. 
 
 Altitudes with the barometer, 72. 
 
 Ammeter, 302, 303, 345. 
 
 Ampere, the, 323. 
 
 Amyl acetate lamp, 207. 
 
 Analysis, sound, 152 ; spectrum, 271 ; 
 sugar, 280. 
 
 Aneroid, testing, 110. 
 
 Angle, of crystal, 223 ; of prism, 230 j 
 of minimum deviation, 235 ; from 
 scale readings, 337. 
 
 Arc, lamp, photometry of, 210; mer- 
 cury, 264 ; spectrum of, 273. 
 
 Archimedes, Principle of, 119. 
 
 Areas, measured with planimeter, 42 ; 
 
 formulae for, 46. 
 Assignment of exercises, 4. 
 Astatic galvanometer, 288. 
 Atmospheric pressure, barometer, 69. 
 Atwood, machine of, 77. 
 Auguste, psychrometer, 200. 
 
 B 
 
 Balance, the, 55, 64 ; hints on use of, 
 62 ; hydrostatic, 120 ; Jolly's spring, 
 129 ; ratio of arms of, 59 ; rider for, 
 55, 61 ; sensibility of, 62. 
 
 Ballistic, constant, 359 ; galvanometer, 
 337 ; pendulum, 51. 
 
 Bands, see fringes. 
 
 Barometer, the, 69; aneroid, 110; 
 Fortin, 70 ; pressure, 72 ; readings, 
 reduction of, 70, Table 14 ; siphon, 
 70. 
 
 Barus, method of calibrating bridge 
 wire, 319. 
 
 Barye, the, 72. 
 
 Battery, electromotive force of. Table 
 27; resistance of. Table 27; meas- 
 urement of resistance of : by Beetz's 
 method, 307; Benton's method, 308; 
 Kohlrausch's method, 309 ; by 
 method of fall of potential, with 
 voltmeter, ammeter, or condenser, 
 302 ; Mance's method, 306. 
 
 Beats, frequency of sound by, 134. 
 
 Beetz, method for battery resistance, 
 307. 
 
 Benton, method for battery resistance, 
 308. 
 
 Bi-prism for interference, 246. 
 
 Black, ice calorimeter, 178. 
 
 Boiling point, of various substances, 
 171, Tables 17, 18 ; of water, Table 
 16; of thermometer, 163. 
 
 394 
 
INDEX 
 
 395 
 
 Books, for notes, 5 ; reference. Table 
 29. 
 
 Boyle, Law of, 108, 197. 
 
 Brashear, method of silvering and 
 cleaning glass, 266, 270. 
 
 Bridge, the box, 293 ; calibrating of 
 wire of, 319 ; for measuring low re- 
 sistances, 315 ; method for capacities, 
 353 ; Wheatstone's, 283. 
 
 Bulk elasticity, modulus of, 93, 106, 
 Table 8. 
 
 Bunsen, ice calorimeter, 180 ; photom- 
 eter, 206, 210. 
 
 Buoyancy of air in weighing, 64, 121, 
 Table 1. 
 
 Cadmium, light, 264 ; cell, 333. 
 
 Calibration, of aneroid, 110 ; of bridge 
 wire, 319 ; with comparator, 30 ; of 
 graduated scales, 28, 32 ; of level, 39; 
 by precise methods, 32; of thermom- 
 eter, 160 ; of voltmeter, 348 ; of 
 weights, 66 ; with dividing engine, 28. 
 
 Calipers, micrometer, 18 ; vernier, 17. 
 
 CaMendar, platinum thermometer, 169. 
 
 Calorie, the, 13, 173. 
 
 Calorimeter, Black'sice, 178 ; Bunsen's 
 ice, 180 ; cooling, 186 ; described, 
 173; electro-, 188, 324; heat capacity 
 of, 174; Magie's, 188; Pfaundler's, 
 188 ; Eegnault's, 182, 184. 
 
 Calorimetry, 173. 
 
 Candle, British, 207; German, 207; 
 power, 13, 204, 208. 
 
 Capacity, absolute, 356 ; comparison 
 of, with ballistic galvanometer, 352 ; 
 with bridge, 353 ; by method of 
 mixtures, 355 ; thermal, of calorim- 
 eter, 174. 
 
 Capillarity, 113. 
 
 Capillary, correction for barometer, 
 71; depression. Table 13; electrom- 
 eter, 342. 
 
 Carbon rheostat, 292. 
 
 Card system of assignment of exer- 
 cises, 4. 
 
 Carey Foster, method of resistance 
 measurement, 316. 
 
 Carhart, standard cell, 332. 
 
 Cathetometer, the, 97; for determin- 
 ing Young's modulus, 96. 
 
 Cells, Carhart-Clark, 332; concentra- 
 tion, 344 ; electromotive force and 
 internal resistance of, Table 27; 
 
 measurement of resistance of, see 
 battery ; standard, 332 ; Weston, 
 333. 
 
 Centimeter, the, 13. 
 
 Charge, electrical, see capacity. 
 
 Chronograph, cylinder, 86 ; tuning 
 fork, 91, 136. 
 
 Circles, divided, to read, 17, 229 ; 
 dividing-engine for, 27. 
 
 Clark, standard cell, 332. 
 
 Cleaning glass for silvering, 270 ; 
 glassware, 115. 
 
 Cleveland, g and barometric pressure, 
 72 ; geographical data for, Table 11 . 
 
 Coefficient, of elasticity, 93 ; of expan- 
 sion, cubical, 155; of expansion, 
 linear, 153 ; of expansion, linear and 
 cubical. Tables 18, 19 ; of friction, 
 112, Table 5; of a gas, 168; of 
 mutual induction, 361 ; of restitu- 
 tion, 51 ; of rigidity, 99 ; of self-in- 
 duction, 360 ; temperature, of mag- 
 netism, 376, Table 30; temperature, 
 of resistance, 292, 316, Table 26 
 temperature, of tuning forks, 133 
 of viscosity, 117. See modulus. 
 
 Coincidences, method of, 82 ; optical 
 method of, 88. 
 
 Collimating eyepiece, 226. 
 
 Collimator, 225, 228. 
 
 Collision, 51. 
 
 Commutator, 298. 
 
 Comparator, 23 ; dividing-engine, 26 ; 
 Lissajous's optical, 137 ; micrometer 
 microscope, 30. 
 
 Compensation, apparatus, 335; method 
 for electromotive force, 333, 344. 
 
 Compressibility, with piezometer, 106, 
 Table 8. 
 
 Concave, grating spectroscope, 251 ; 
 lenses, 217 ; mirrors, 214. 
 
 Concentration cell, 344. 
 
 Condenser, capacity of, see capacity ; 
 for determining electromotive force, 
 336 ; key, 351 ; standard, 350. 
 
 Conductivity of electrolytes, 310. 
 
 Conjugate foci, 213. 
 
 Conservation of momentum, 51. 
 
 Constant, ballistic, 359 ; errors, 7 ; of 
 micrometer microscope, 20 ; of tan- 
 gent galvanometer, 322. 
 
 Constants, of a level, 39 ; elastic, of 
 solids. Table 7 ; heat, for liquids. 
 Tables 17, 18 ; heat, for solids. 
 Tables 17, 19; miscellaneous, Table 
 30. 
 
396 
 
 INDEX 
 
 Contact difference of potential, 344, 347. 
 
 Control magnet, 289, 328. 
 
 Convex, lenses, 216 ; mirrors, 215. 
 
 Cooling calorimeter, 186. 
 
 Copper voltameter, 322. 
 
 Corrections, in genei'al, 7 ; see calibra- 
 tion, index, radiation, scale. 
 
 Coulomb, the, 3oU; 
 
 Crystal, angle of, 223 ; refraction of, 
 242 ; refractometer, 244. 
 
 Cubical expansion, 155, 157, Table 18. 
 
 Current strength, with electrocalorim- 
 eter, 824 ; vfith electrodynamom- 
 eter, 330 ; with voltameter, 322. 
 
 Curvature, of a lens, 218 ; of a level, 
 41 ; radius of, with spherometer, 19. 
 
 Cycle, stroboscopic, 141. 
 
 Damped vibrations, 117, 356. 
 
 JDaniell, cell, 327 ; hygrometer, 198. 
 
 B^Arsonval, galvanometer, 170, 286, 
 303, 338. 
 
 Datum circle of planimeter, 43 ; area 
 of, 45. 
 
 Dead-beat galvanometer, 287. 
 
 Declination, magnetic, 363, 368 ; of 
 ■ the sun. Table 22. 
 
 Decrement, logarithmic, 117, 356. 
 
 Degree, centigrade, 13, 160. 
 
 Density, defined, 119 ; of air, Table 12; 
 of mercury. Table 3 ; of various 
 solids, liquids, and gases. Table 2 ; 
 of water, 119, Table 3. 
 
 Density, measurement of, of air, 126 ; 
 of a gas, 127 ; by Hare's method, 129 ; 
 with hydrometer, 131 ; by hydro- 
 static weighing, 120; with the Jolly 
 balance, 129 ; with the pyknometer, 
 124 ; with a U-tube, 128. 
 
 Depression, capillary, in barometer, 
 71, Table 13. 
 
 De Sauty, method for condenser capa- 
 city, 354. 
 
 Deviation, angle of minimum, 235. 
 
 Deviations of measures, 9. 
 
 Dew-point, 197 ; hygrometer, 198, 
 Table 23. 
 
 Difference of potential, contact, 344. 
 See electromotive force. 
 
 Differential galvanometer, 299. 
 
 Diffraction, concave grating, 251 ; 
 plane grating, 249 ; spectrum, 253. 
 
 Dilatometer, the, 157. 
 
 Dip, magnetic, 363, 372. 
 
 Divided, circles, to read, 17, 229; 
 
 scales, to read, 13. 
 Dividing engine, comparator, 26 ; for 
 
 circles, 27 ; for straight scales, 25. 
 Double weighing, method of, 59, 64. 
 Dust figures, velocity of sound by, 145. 
 Dynameterfor power of telescope, 221. 
 Dynamometer, electro-, 330. 
 
 Earth inductor, 370, 375. 
 
 Earth, magnetic elements of, 363; 
 radius of. Table 30. 
 
 Elastic, bodies, 52 ; constants of liq- 
 uids. Table 8 ; constants of solids, 
 Table 7. 
 
 Elasticity, 51, 52; limit of, 93, Table 
 7 ; modulus of, 93, 98, 147 ; of vol- 
 ume, 106, Table 8. 
 
 Electrical measurements, 282. 
 
 Electrocalorimeter, 188, 324. 
 
 Electrochemical equivalent, of copper, 
 323 ; of silver, 323. 
 
 Electrodynamometer, 330. 
 
 Electrolyte resistance, 309. SeebaUery. 
 
 Electrometer, capillary, 342; quad- 
 rant, 339. 
 
 Electromotive force, 332 et seq.; of 
 standard cells, 332; of thermo-couple, 
 170 ; of various cells, Table 27 ; for 
 methods of measurement of, see 
 Contents. 
 
 Elongation, middle, 103. 
 
 English, candle, 207 ; and metric 
 equivalents, Table 31. 
 
 Equation of time. Table 22. 
 
 Equilibrium, of forces, 49; of mo- 
 ments, 54. 
 
 Equivalent, electrochemical, 323; of 
 fusion, 173, 191, Table 19 ; Joule's, 
 324 ; mechanical, of heat, 173, 194, 
 324 ; of vaporization, 192, Table 18. 
 
 Equivalents, English and metric. Table 
 31. 
 
 Errors, 7 ; of barometer, 71 ; Index, of 
 calipers, 18 ; index, of sextant, 48 ; 
 probable, 8 ; of weights, 66. 
 
 Everett, Physical Constants, 378. 
 
 Excursions, sum of, 57. 
 
 Exercises, assignment of, 4; descrip- 
 tions of, 4. 
 
 Expansion, cubical, coefBcient of, 155, 
 157, Tables 8, 18; of a gas, 168; 
 linear, coefficient of, 153, Table 19. 
 
 Eyepiece, Gauss's coUimatlng, 226. 
 
INDEX 
 
 397 
 
 Fabry, wave length of light, 264. 
 
 Fall, free, gravity by, 90 ; of potential, 
 method of, 302. 
 
 Falling tuning fork, for gravity, 74. 
 
 Farad, the, 850. 
 
 Figure of merit, 826, 329. 
 
 Figures, significant, 11. 
 
 Flashing apparatus for time obser- 
 vations, Mendenhall's, 88; Reed's, 
 139. 
 
 Flexure, modulus of elasticity by, 
 98. 
 
 Flicker photometer, 212. 
 
 Foci, of lenses, 216, 219 ; of mirrors, 
 213. , 
 
 Forces, equilibrium of, 49; triangle 
 of, 49. 
 
 Fortin, barometer, 70. 
 
 Foucavlt, photometer, 205. 
 
 Fraunhofer, lines in spectrum, 272. 
 
 Freezing point, of thermometer, 163. 
 See fusion, melting point. 
 
 Frequency of sound, by beats, 133 ; 
 by graphic methods, 136; by Lis- 
 sajous's figures, 137; by optical 
 methods, 139; of musical scale. 
 Table 28 ; with the siren, 134. See 
 period. 
 
 Fresnel, bi-prism, 246 ; mirrors, 249. 
 
 Friction, coefiicients of, 112, Table 6 ; 
 mechanical equivalent by, 194 ; roll- 
 ing, 113 ; of viscosity, 117. 
 
 Fringes, interference, with bi-prism, 
 246; with interferometer, 258, 261; 
 with plane surfaces, 265 ; with polar- 
 imeter, 278. 
 
 Fundamental interval of thermometer, 
 162, 167, 170. 
 
 Fusion, of ice, heat equivalent of, 173, 
 191 ; of solids. Table 19. See melt- 
 ing point. 
 
 G 
 
 Galvanometer, the, general, 285 ; ad- 
 justing, 288; aperiodic, 287; astatic, 
 288: ballistic, 287, 337 ; constant of, 
 322; D'Arsonval, 286, -303, 338; 
 dead-beat, 287; differential, 299; 
 figure of merit of, 325 ; needle of, to 
 bring to rest, 339 ; resistance of, 
 305 ; sensitiveness of, 328 ; shunt 
 for, 328 ; tangent, 295. 
 
 Gas, density of, to measure, 127, 
 Table 2 ; coeflScient of expansion 
 
 of, 168 ; meter, 209 ; thermometer, 
 166 ; velocity of sound in, 144, 148, 
 Table 30. 
 
 Gauss, eyepiece, 226, 240. 
 
 Gauss, the, 363. 
 
 Gay-Lussac, method of calibration, 
 
 28, 32, 319. 
 
 Gerhardt, thermometer calibration, 
 
 29, 164. 
 
 German, Candle, 207; silver for resist- 
 ance, 290. 
 
 Glass, chemical, to clean, 115; opti- 
 cal, to clean, 270 ; to silver, 266 ; to 
 smoke, 91. 
 
 Goniometer, 223. 
 
 Gott, method for comparison of capa- 
 city, 354. 
 
 Gram, the, defined, 13. 
 
 Graphical methods, general, 12; for 
 frequency of sound, 136. 
 
 Gravity, with Atwood's machine, 78 ; 
 vrith falling fork, 76 ; by free 
 fall, 90 ; with simple pendulum, 81 ; 
 with reversible pendulum, 84 ; Table 
 10. 
 
 Griffiths, platinum thermometer, 169. 
 
 Hair hygrometer, 202. 
 
 Half-shade polarimeter, 277. 
 
 Half-silvered glass, 255, 269. 
 
 Hansen, method of calibration, 32. 
 
 Hare, method for density determina- 
 tion, 129. 
 
 Heat, capacity of calorimeter, 174 ; 
 constants for liquids, Tables 17, 18 ; 
 constants for solids. Tables 17, 19 ; 
 equivalent of fusion, 173, 191, Table 
 19; equivalent of vaporization, 192, 
 Table 18 ; mechanical equivalent of, 
 194 ; unit of, 173. 
 
 Heatingj method of, for specific heat, 
 188. 
 
 Hefner-Alteneck, standard lamp, 207.. 
 
 Heliostat, 274. 
 
 Henry, the, 360. 
 
 Hints on use of balance, 62. 
 
 Horizon, artificial, 48. 
 
 Horizontal, candle power, 204 ; inten- 
 sity, 363, 368, 373. 
 
 Humidity, absolute, 197; relative, 
 197, Table 21. 
 
 Hydrogen thermometer, 166 ; scale, 
 reduction to, 166, Table 16. 
 
 Hydrometers, 131. 
 
398 
 
 INDEX 
 
 Hydrostatic weighing, 120 ; other 
 methods of, 123. 
 
 Hygrometer, absorption, 198 ; Au- 
 guste's, 200; Daniell's, 198; dew- 
 point, 198 ; hair, 202 ; wet- and dry- 
 bulb, 200. 
 
 Hygrometry, 197, Tables 20, 21, 23. 
 
 Ice, calorimeter, Black's, 178 ; calorim- 
 eter, Bunsen's, 180; heat equiva- 
 lent of fusion of, 173, 191; volume 
 change in melting, 180. 
 
 Impact, laws of, 51. 
 
 Incandescent lamp, photometry of, 
 209. 
 
 Inclination, magnetic, 363, 372. 
 
 Inclined plane, friction on, 113. 
 
 Index error, of barometer, 71 ; of cali- 
 per, 18 ; of sextant, 48. 
 
 Index of refraction, of air, 235 ; of 
 carbon bisulphide, 241 ; by displace- 
 ment, 238 ; of a liquid, 235 ; with a 
 microscope, 236 ; of a prism, 234 ; 
 by total reflection, 239; of water, 
 243, Table 24. 
 
 Induction, coefBoient of mutual, 361; 
 coefficient of self-, 360; flow of, 
 374. 
 
 Inductor, earth, 370, 375. 
 
 Inertia, moment of, 104 ; formulse for, 
 105; of pendulum, correction for, 
 82 ; of pulley of Atwood's machine, 
 78. 
 
 Infinitely small arc, reduction to, 82, 
 356. 
 
 Insulation, resistance of, 311. 
 
 Intensity, of light, 203 ; magnetic, 363, 
 373. 
 
 Interference, of light, 246, 255, 265; 
 of sound, 147 ; polarimeter, 278. 
 
 Interferometer, 255; circular bands, 
 261; counting fringes, 261; finding 
 the bands, 258 ; lights for, 263 ; sil- 
 vering mirrors, 266; standard of 
 length, 261. 
 
 J 
 
 Jena Normal Glass thermometer, 160, 
 164, 166 ; to reduce to hydrogen 
 scale. Table 15. 
 
 Jolly, balance, 129. 
 
 Joule, mechanical equivalent of heat, 
 173, 194, 324. 
 
 Joule, the, 13, 324. 
 
 K 
 
 Kater, reversible pendulum, 84. 
 
 Kelvin, reversing key, 341. See 
 Thompson. 
 
 Kew, magnetometer, 366. 
 
 Key, condenser, 351 ; double succes- 
 sive, 292 ; reversing, 298. 
 
 Kilogram, the, 13. 
 
 Koenig, tuning fork, 143. 
 
 Kohlrausch, refractometer, 240, 245; 
 electrolyte conductivity, 309 ; vari- 
 ometer, 366. 
 
 Kundt, velocity of sound, 145. 
 
 Laboratory practice, objects of, 1. 
 
 Lamp, and scale, 290; for monochro- 
 matic light, 263. 
 
 Landolt, Tables of Constants, 378. 
 
 Latent heat, see heat equivalent. 
 
 Laurent, polarimeter, 277. 
 
 Laws, of friction, 112; of motion, 51 ; 
 of vibrating strings, 149; Ohm's, 
 282. 
 
 Least Squares, 10. 
 
 Le Chatelier, pyrometer, 169, 171. 
 
 Length, measurement of, 15. See 
 Contents. 
 
 Lenses, 20, 216, 219. 
 
 Level, 39 ; of cathetometer, 97; trier, 
 39. 
 
 Light, cadmium, 264; intensity of, 
 see photometry; interference of, 
 246,256,265; mercury, 263; sodium, 
 263 ; velocity of. Table 30 ; wave 
 length of, 246, 249, 251, Table 
 25. 
 
 Limit, elastic, 93, Table 7. 
 
 Linear expansion, coefficient of, 153, 
 Table 19. 
 
 Lippmann, capillary electrometer, 
 342. 
 
 Liquids, compressibility of, 106, Table 
 8 ; density of, see density. Table 2 ; 
 expansion coefficient of, 157 ; heat 
 constants of, Tables 17, 18 ; specific 
 heat of, 184, 186, 188. 
 
 Lissajous, figures, frequency by, 137 ; 
 projection of, 139. 
 
 Logarithmic, calculation, 6, 11; 13; 
 decrement, 117, 356. 
 
 Logarithms, Tables 30, 33. 
 
 Low-resistance bridge, 315. 
 
 Lummer, photometer, 206, 210. 
 
INDEX 
 
 399 
 
 M 
 
 Machine, Atwood's, 77. 
 Magie, calorimeter, 188, 190. 
 Magnet, control, 289, 328 ; tempera- 
 ture coefficient of, 376, Table 30. 
 Magnetic, fields, comparison of, 375 ; 
 shield for galvanometer, 328; dec- 
 lination, 363, 868 ; distribution, 
 374; dip, 363, 372; elements of 
 earth, 363 ; inclination, 363, 372 ; 
 intensity, 363, 373; meridian, 363; 
 moment, 365 ; variometer, 366. 
 Magnetometer, 363 ; Kew, 366. 
 
 Magnifying povrer, 221. 
 
 ilance, method for resistance, 306. 
 
 Manganin alloy for resistance, 290 
 
 Manometric flames, 148, 151, 152. 
 
 Mass, unit of, 13 ; determination of, 
 54, 55, 64, 120. 
 
 Maxwell, method of, 360 ; rule for 
 Wheatstone's bridge, 294. 
 
 Mechanical equivalent of heat, 173, 
 194, 324. 
 
 Megabarye, the, 72. 
 
 Melde, experiments with vibrating 
 strings, 151. 
 
 Melting point, 171, Tables 17, 19. 
 See fusion. 
 
 Mendenhall, optical method for period, 
 88 ; pendulum, 89. 
 
 Mercury, capillary depression of, 71, 
 Table 13 ; correction for exposed 
 column of, in thermometer, 165; 
 density of, Table 3 ; light, 263 ; ther- 
 mometer for high temperatures, 168 ; 
 thermometer, reduction to hydrogen 
 scale, 166, Table 15 ; volume of, 69, 
 Table 4. 
 
 Meridian, of Cleveland, Table 11 ; 
 magnetic, 363, 367. 
 
 Merit, figure of, 325, 329. 
 
 Meter, the, 13; calibration of, 31, 33; 
 gas, 209 ; slide-wire bridge, 284. 
 
 Method, of coincidences, 82, 85, 88 ; of 
 cooling, 186 ; graphical, 12 ; of heat- 
 ing, 188; Mendenhall's, 88; of mix- 
 tures, see mixtures ; of transits, 102. 
 
 Metric equivalents for English meas- 
 ures. Table 31 . 
 
 Michelson, interferometer, 255; wave 
 length of light, 264, 378. 
 
 Microfarad, the, 350. 
 
 Micrometer, caliper, 18; microscope, 
 20; microscope comparator, 30; 
 optical, 36, 38 ; screw, 18. 
 
 Microscope, index of refraction with, 
 
 236 ; micrometer, 20 ; power of, 221. 
 
 Milliammeter for resistance measures, 
 
 346. 
 Minimum deviation, 235. 
 
 Mirror, concave, 214 ; convex, 215 ; 
 foci of, 213 ; Fresnel's, 249 ; and 
 scale, see scale. 
 
 Mixtures, method of, for capacity, 
 355 ; radiation correction for, 176 ; 
 for specific heat of liquids, 184 ; for 
 specific heat of solids, 182. . 
 
 Modulus, of bulk elasticity, 93, 106, 
 Table 8 ; of rigidity, 93, 99 ; of tor- 
 sion, 101 ; Young's, 93, 98 ; Young's, 
 by velocity of sound, 147 ; Young's, 
 Table 7. See coefficient. 
 
 Molecular rotation, 276. 
 
 Moment, of inertia, formulse for, 105 ; 
 of inertia by torsion pendulum, 104 ; 
 magnetic, 365. 
 
 Moments, equilibrium of, 54. 
 
 Momentum, and impact, 51 ; conser- 
 vation of, 51. 
 
 Monochromatic light, 263. 
 
 Motion, accelerated, 74, 77 ; Laws of, 
 51. 
 
 Musical scales, frequency. Table 28. 
 
 Mutual induction, 361, 362. 
 
 N 
 
 Natural trigonometric functions. Table 
 32. 
 
 Neumann, method for calibration, 32. 
 
 Neutral point, of thermo-couple, 348. 
 
 Newton, Laws of Cooling, 176; Laws 
 of Motion, 51. 
 
 Nicol, prism, 277. 
 
 Nitrogen thermometer, 166. 
 
 Nodes in organ pipes, 151. 
 
 Noninductive winding, 290. 
 
 Normal, pressure, 72 ; scale of temper- 
 ature, 166, Table 16 ; spectrum, 253, 
 272. 
 
 Notes and records of experimental 
 work, 5. 
 
 Null methods, 287. 
 
 Numbers, miscellaneous. Table 30. 
 
 Numerical example of, Boyle's Law, 
 110 ; calibration of aneroid. 111 ; 
 calibration of bridge wire, 321 ; cal- 
 ibration of meter (simple) , 31 ; calibra- 
 tion of meter (complete), 34; cali- 
 bration of micrometer microscope, 
 23; calibration of thermometer 
 
400 
 
 INDEX 
 
 scale, 29; calibration of thermom- 
 eter, complete, 164; calibration of 
 weights, 67 ; capacity comparison, 
 353; capacity (absolute) determina- 
 tion, 358 ; density of aluminum, 123 ; 
 divided-circle reading, 229 ; equilib- 
 rium of forces, 50 ; figure of merit, 
 327 ; frequency of forlt. Reed's 
 method, 143; gravity with simple 
 pendulum, 84; mechanical equiva- 
 lent of heat, 196 ; period by coinci- 
 dences, 84 ; period by Mendenhall's 
 method, 90; period by method of 
 transits, 104; radiation correction, 
 177 ; significant figures, 12 ; speci- 
 fic heat of iron, 177; specific heat 
 by method of heating, 190 ; vernier 
 reading, 17; weighing, single, 58; 
 weighing, double, 60; weighing by 
 reversal, 65 ; Young's modulus of 
 iron, 95. 
 
 Observations, making, 7; weighted, 
 10. 
 
 Oersted, Law of, 285. 
 
 Ohm, Law of, 282; method of, 296, 
 345. 
 
 Ohm, the, 290. 
 
 Oil bath for resistances, 291. 
 
 Optical, activity, 275 ; axis of cathe- 
 tometer, 97 ; axis of spectrometer, 
 227 ; method for frequency, 137, 
 139 ; method for period, 88 ; microm- 
 eter, 36, 38; standard of length, 
 261 ; surfaces, to clean, 270. 
 
 Organ pipes, vibration in, 151. 
 
 Oscillation, see frequency, period. 
 
 Ostwald, capillary electrometer, 342, 
 344. 
 
 Overtones, 151, 152. 
 
 T, values and logarithms. Table 30. 
 
 Paper, to smoke, 91. 
 
 Paraffin-diffusion photometer, 205. 
 
 Parallax, of cathetometer, 97; method 
 of, for mirrors and lenses, 214, 217 ; 
 of reading microscope, 22 ; of spec- 
 trometer, 227. 
 
 Parallelogram of forces, 49. 
 
 Pendulum, compound, 86; flashing, 
 140 ; Mendenhall's, 89 ; reversible, 
 84 ; simple, 81 ; torsion, 101, 104, 
 117. 
 
 Percentage deviation, 9. 
 
 Period, by coincidences, 82, 89; of a 
 galvanometer, 356 ; by transits, 102. 
 See frequency. 
 
 Perot, wave length of light, 264. 
 
 PfauncUer, calorimeter, 188. 
 
 Photographic lens, focus of, 219. 
 
 Photometer, Bunsen, 206 ; flicker, 212 ; 
 Foucault, 205 ; Lummer-Brodhun, 
 210 ; Paraffiii-diffusion, 205 ; Ritchie, 
 205; Rood-Whitman, 212; Rum- 
 ford, 205. 
 
 Photometry, 203; of intense lights, 
 209; by precision methods, 210, 212 ; 
 of colored lights, 206, 212. 
 
 Piezometer, 106. 
 
 Pipes, vibration in organ, 151. 
 
 Pitch, of fork, to alter, 134. See fre- 
 quency, screw. 
 
 Plane, inclined, 113; optical surfaces, 
 264. 
 
 Plane-parallel surfaces, 264. 
 
 Flanimeter, 42. 
 
 Platinum thermometer, 169. 
 
 Plug, commutator, 298 ; for resistance 
 boxes, 291. 
 
 Polarimeter, 276. 
 
 Polaristrobometer, 278. 
 
 Polarization, rotatory, 275 ; of cell, 
 307, 309, 333. 
 
 Post-office bridge, 293. 
 
 Potential, contact difference of, 344 ; 
 method of comparison of, 314 ; 
 method of fall of, 302; large, to 
 measure with quadrant electrom- 
 eter, 341 ; small, to measure with 
 capillary electrometer, 342. See 
 electromotive force. 
 
 Potentiometer, 335. 
 
 Power, candle, 204 ; magnifying, 221 ; 
 thermo-electric, 348. 
 
 Precision, measures of, 9. 
 
 Pressure, barometric, 69; standard, 
 72; by Boyle's Law, 108. 
 
 Prism, angle of, 230 ; index of refrac- 
 tion of, 234. 
 
 Probable error, 8. 
 
 Projectile, path of, 79. 
 
 Psychrometer, 200, Table 20. 
 
 Pulfrich, refractometer, 242. 
 
 Pulley, inertia of, 78. 
 
 Puluj, mechanical equivalent of heat, 
 194. 
 
 Pyknometer, 124. 
 
 Pyrometer, platinum, 168; thermo- 
 electric, 170. 
 
INDEX 
 
 401 
 
 Quadrant electrometer, 339. 
 Quantity, electrical, see capacity, 
 
 current strength. 
 Quartz wedge, polarimeter, 279. 
 
 K 
 
 Radian, the, 41, 100, Table 30. 
 
 Radiation correction in calorimetry, 
 175, 176, 178. 
 
 Radius of curvature, of level, 41; of 
 lenses, 216, 218; of mirrors, 213; 
 with the spherometer, 20. 
 
 Reading, barometric, to reduce, 70, 
 Table 14 ; divided circles and scales, 
 13, 17, 229 ; telescope, 38. 
 
 Records of experimental work, 5, 12. 
 
 Reduction, of barometric readings, 70, 
 Table 14 ; of mercury thermometer 
 to hydrogen scale, 166, Table 15 ; 
 of psychrometer observations. Table 
 20 ; of weighings to vacuum, 64, 121, 
 Table 1 ; to small arc, 82, 356. 
 
 Beed, method for frequency of fork, 
 139 ; flashing pendulum, 140. 
 
 Reference books. Table 29. 
 
 Reflection, total, 239. 
 
 Reflectrometer, total, 240. 
 
 Refractive index, see index of refrac- 
 tion. 
 
 Refractometer, 242. 
 
 Regnault, calorimeter, 182, 184 ; hy- 
 grometer, 198. 
 
 Relative, humidity, 197, Table 21 ; in- 
 dex of refraction, 234 ; viscosity, 
 117, Table 6. 
 
 Reports of laboratory exercises, 7. 
 
 Residuals, 9. 
 
 Resistance, boxes, 291; of cells. Table 
 27; methods for measurement of, 
 see Contents ; oil bath for, 291 ; of 
 plug, 291 ; of slide wire, 318; spe- 
 cific, 313, Table 26 ; temperature 
 coefficient of, 316, Table 26 ; ther- 
 mometer, 169. 
 
 Resonance, wave length by, 144. 
 
 Resonators for analysis of sound, 152. 
 
 Restitution, coefficient of, 51. 
 
 Reversal, weighing by, 64. 
 
 Reversible pendulum, 84. 
 
 Reversing keys, 298, 341. 
 
 Rheostats, 290, 292. 
 
 Rider, for balance, 55, 61 ; weighing 
 with, 61. 
 
 Rigidity, 93, 99, Table 7. 
 Ritchie, photometer, 205. 
 Rochelle salts, method of silvering, 
 
 269. 
 Rood, photometer, 212. 
 Rotation, molecular, 276 ; specifio, 
 
 276. 
 Rotatory polarization, 275. 
 Rowland, wave length of light, 263 ; 
 
 induction, 374, 375. 
 Bueprecht, weights, 67. 
 Rumford, photometer, 205. 
 
 Saccharimeter, 277, 279, 280. 
 
 Saturated water vapor. Table 23. 
 
 Scale, calibration of, 28, 32 ; divided, 
 to read, 13; musical, Table 28; with 
 telescope or lamp, 37, 38, 41, 290, 
 338 ; with telescope, to correct read- 
 ings of, 837, 357. 
 
 Screw, micrometer, 18 ; pitch of, 26, 
 41. 
 
 Second, the, 13. 
 
 Self-induction, coefficient of, 360. 
 
 Sensibility of balance, 62. 
 
 Sensitive tint, polarimeter, 278. 
 
 Sensitiveness, of balance, 56, 62 ; of 
 galvanometer, 328 ; of electrometer, 
 343 ; of level, 40. 
 
 Sextant, 46. 
 
 Significant figures, 11. 
 
 Silver voltameter, 323. 
 
 Silvering glass, 266, 269. 
 
 Simple, pendulum, 81 ; photometers, 
 204 ; torsion apparatus, 100. 
 
 Siphon barometer, 70. 
 
 Siren, 134. 
 
 Slide-wire bridge, 284 ; resistance of, 
 318. 
 
 Smithsonian Tables, 201, 378. 
 
 Smoked paper and glass, 91. 
 
 SocOti Genevoise, meter, 31, 32, 34. 
 
 Sodium, chloride, conductivity of, 311 ; 
 lamp and wave length of light, 
 263. 
 
 Solar spectrum, wave lengths of. Table 
 25. 
 
 Soleil, polarimeter, 278. 
 
 Solids, density of. Table 2 ; elastic 
 constants of. Table 7 ; heat con- 
 stants of. Table 19. 
 
 Sonometer, 149. 
 
 Sound, analysis of, 152. See frequency, 
 velocity, wave length. 
 
402 
 
 INDEX 
 
 Specific, gravity, 119 ; gravity, see den- 
 sity ; gravity bottle, 124 ; gravity of 
 air, Table 12 ; heat of iron, example 
 of, 177; heat of liquids. Table 18; 
 heat of solids. Table 19 ; heat, see 
 calarimetry ; resistance, 313, Table 
 26 ; resistance of an electrolyte, 
 310 ; resistance. Table 26 ; rotation, 
 276 ; viscosity, 117. 
 
 Spectrometer, 225. 
 
 Spectroscope, 271 ; grating, 251. 
 
 Spectrum, analysis, 271 ; lines in solar, 
 Table 25; normal, 253, 272. 
 
 Spherical candle power, 208. 
 
 Spherometer, 19. 
 
 Spring balance. Jolly, 129. 
 
 Standard, cell, 332 ; interference, of 
 length, 261 ; lights, 207 ; pitch. 
 Table 28. 
 
 Steam, see boiling point, vaporization. 
 
 Strings, laws of vibrating, 149. 
 
 Stroboscopio method, 141. 
 
 Sugar analysis, 280. 
 
 Sun, declination of, Table 22. 
 
 Surface tension, 113, 110, Table 9; 
 with an electrolyte, 342. 
 
 Surfaces, plane and plane-parallel, 
 264. 
 
 System of lenses, foci of, 219. 
 
 .Tables, explanation of, 378. See Con- 
 tents or Appendix. 
 
 Tangent galvanometer, 295, 296 ; con- 
 stant of, 322. 
 
 Telephone bridge, 309. 
 
 Telescope, focusing for infinity, 226 ; 
 power of, 221; reading, 22, 38 ; with 
 scale, 290, 337, 338. 
 
 Temperature, coefficient of magnetism, 
 376, Table 30; coefficient of resist- 
 ance, 292, 316, Table 26; correc- 
 tion to barometer, 71, Table 14 ; 
 fixed points for high, 171, Table 17; 
 measurement of high, 168. See 
 thermometer, pyrometer. 
 
 Tension, surface, 113, 116, Table 9 ; 
 of water vapor, 197, Table 23. 
 
 Tenths, estimation of, 13. . 
 
 Testing, aneroid, 110 ; half-silvered 
 surfaces, 269 ; plane and plane-par- 
 allel surfaces, 264 ; set, 294. 
 
 Thermal capacity of calorimeter, 174. 
 
 Thermo-electric, diagram, 348; pyrom- 
 eter, 170. 
 
 Thermo-electromotive, force, 170, 347, 
 348 ; power, 348. 
 
 Thermometer, air, 166 ; calibration of, 
 29, 160, 162 ; for calorimetry, 174 ; 
 exposed column of mercury, 165; 
 fundamental interval of, 162 ; heat 
 capacity of, 175 ; for high tempera- 
 tures, 168 ; platinum, 169 ; reduc- 
 tion to hydrogen scale, 166 ; thermo- 
 electric, 170 ; weight, 155. 
 
 Thiesen, method of calibration, 32. 
 
 Thompson, method for resistance, 305. 
 
 Time, equation of, Table 22 ; meas- 
 urement of, 74 ; variable unit, 76. 
 See chronograph, frequency, period. 
 
 Torsion, lathe, 99 ; modulus of, 101 ; 
 pendulum, 101, 104, 117 ; simple 
 apparatus for, 100. 
 
 Total reflection, 239. 
 
 Total-reflectometer, 240. 
 
 Trajectory, 79. 
 
 Transits, periodic time by, 102. 
 
 Triangle of forces, 49. 
 
 Trigonometric functions. Table 32. 
 
 Tubes, capillary, 113 ; volume and 
 radius of, 69. 
 
 Tuning fork, 133, 134; falling, 74; 
 chronograph, 91. 
 
 U 
 
 Units, 12. See names of units. 
 U-tube, for Boyle's Law, 108; for den- 
 sity, 128. 
 
 V 
 
 Vacuum, reduction to, 64, 121, Table 
 1. 
 
 Vapor, mass of water. Table 23 ; ten- 
 sion of water. Table 23. 
 
 Vaporization, heat equivalent of, 192, 
 Table 18. 
 
 Variometer, magnetic, 366. 
 
 Velocity of sound, in air, 144 ; by dust 
 figures, 145; in gas, 145, 147, Table 
 30 ; by interference, 147 ; by reso- 
 nance, 144; in solids. Table 7. 
 
 Vernier, 15 ; caliper, 17. 
 
 Verniers, circle with two or more, 
 229. 
 
 Vibrating, air columns, 151 ; strings, 
 149. 
 
 Vibration, damped, 117, 356 ; fre- 
 quency of musical scale, Table 28 ; 
 in organ pipes, 151 ; weighing by, 
 57. See frequency, period. 
 
INDEX 
 
 403 
 
 Virtual, focus, 213, 216 ; resistance, 
 
 327, 357. 
 Viscosity, 117, Table 6. 
 Volt, the, 332. 
 
 Voltameter, copper, 322; silver, 323. 
 Voltmeter, 302 ; calibration of, 348. 
 Volume, elasticity, 93, 106, Table 8; 
 
 of a glass vessel, 69, Table 4; of 
 
 mercury, 69, Table 4 ■ of a tube, 69 ; 
 
 of water, 69, Table 4 ; by vfeighing, 
 
 68, Table 4. 
 Volumetric expansion of liquids. Table 
 
 18. 
 
 W 
 
 Water, boiling point of, 163, Table 
 16; density of, 119, Table 3; equiv- 
 alent of calorimeter, 174; mass of 
 vapor of. Table 23; trough for organ 
 pipe, 151 ; vapor tension of, Table 
 23; volume of, in glass vessel, 69, 
 Table 4. 
 
 Watt, the, 324. 
 
 Wave length, of light with bi-prism, 
 246 ; by diffraction, 249, 251 ; of 
 monochromatic lights, 363 ; of solar- 
 spectrum lines. Table 85 ; of sound 
 by interference, 147 ; of sound Tiy 
 resonance, 144. 
 
 Weighing, with constant load, 120 ; 
 double, 59 ; hydrostatic, 120 ; by 
 reversal, 64 ; with rider, 61 ; reduc- 
 tion to vacuum, 65, Table 1 ; by 
 vibration, 57 ; for volume, 68. 
 
 Weight in vacuo, 64. 
 
 Weight thermometer, 155. 
 
 Weighted observations, 10. 
 
 Weights, adjustment of, 68; calibra- 
 tion of, 66 ; counting, 63 ; of obser- 
 vations, 11. 
 
 Weston, standard cell, 333. 
 
 Wet- and dry-bulb hygrometer, 200. 
 
 Wheatstone, bridge, 169, 283. 
 
 White light, interference in, 259. 
 
 Whitman, flicker photometer, 212. 
 
 Wild, polarimeter, 278. 
 
 Wollaston, goniometer, 223. 
 
 Toung\i modulus, 93 ; by stretching, 
 93 ; by flexure, 98 ; by velocity of 
 sound, 147 ; values of, Table 7. 
 
 Zero point, of balance, 57, 64 ; of 
 thermometer, 159, 163. 
 
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